REFRIGERACIÓN DE COMPONENTES ELECTRÓNICOS USANDO ...

REFRIGERACIÓN DE COMPONENTES ELECTRÓNICOS USANDO CONVECCION INDUCIDA ELECTROHIDRODINÁMICAMENTE

JULIO 2019

JAVIER SALGADO GONZÁLEZ

DIRECTOR DEL TRABAJO FIN DE MASTER:

Jorge Muñoz Paniagua

TRABAJO FIN DE MASTER

PARA LA OBTENCIÓN DEL

TÍTULO DE MASTER EN

INGENIERÍA INDUSTRIAL

Refrigeración de componentes electrónicos usando convección inducida electrohidrodinámicamente

Javier Salgado González i

ABSTRACT

El rápido crecimiento y desarrollo de los componentes electrónicos hace que su tamaño sea cada vez más reducido y su potencia cada vez más alta. A pesar de que este tipo de componentes destacan por su fiabilidad al no tener partes móviles, el sobrecalentamiento puede causar diversos fallos reduciendo así su vida útil y fiabilidad. Por lo tanto, resulta fundamental el estudio de nuevas técnicas de refrigeración eficientes con el fin de garantizar el correcto funcionamiento y la vida útil de estos componentes.

Este Trabajo de Fin de Máster se centra en el estudio de un sistema electrohidrodinámico basado en el experimento de Moghanlou, F. S. et al. (2014): Experimental study on electrohydrodynamically induced heat transfer enhancement in a minichannel. Se realiza un análisis CFD con el fin de estudiar el efecto del campo eléctrico en la mejora de transferencia de calor y la caída de presión en un flujo laminar a través de un minicanal de sección cuadrada para diferentes números de Reynolds. El dispositivo se compone de tres partes diferentes: entrada, salida y sección de prueba. El fluido pasa a través de la sección de entrada para desarrollarse completamente hidrodinámicamente. La sección de prueba está compuesta por un cable de cobre situado en la parte superior que simula el electrodo de alto voltaje y una placa calentada ubicada en la parte inferior que constituye el componente electrónico a enfriar. Además, la parte inferior de la sección de prueba (la placa calentada) se considera como tierra. Las paredes del minicanal están aisladas térmica y eléctricamente. El electrodo de alto voltaje inyecta carga a través del fluido, produciendo un flujo secundario hacia la parte inferior del canal (placa calentada). Las moléculas neutras del fluido son empujadas por este flujo secundario, por lo que el perfil de velocidad del flujo se modifica.

Se realizan algunas simulaciones numéricas diferentes con el software ANSYS Fluent con el objeto de estudiar los fenómenos electrohidrodinámicos y determinar los efectos de la inyección de carga en el fluido. Con el fin de analizar la mejora de la transferencia de calor, la caída de presión y explicar el comportamiento del dispositivo electrohidrodinámico, se lleva a cabo un análisis de diferentes parámetros.

Tras analizar los resultados, se puede concluir que los efectos del campo eléctrico se acentúan cuando el número de Reynolds es menor. De hecho, para los escenarios estudiados con el número de Reynolds más bajo, la modificación del flujo es mayor. Se observa un fenómeno de recirculación de fluido en la salida de la sección de prueba. El campo eléctrico contribuye negativamente en la ecuación de conservación del momento en esta parte del minicanal, reintroduciendo el fluido hacia la sección de prueba, y por tanto, perjudicando la mejora de la transferencia de calor. Al aplicar un voltaje en el electrodo superior, se observa un aumento de la caída de presión, que se incrementa con el valor del voltaje aplicado y, por lo tanto, con la intensidad del campo eléctrico.

Analizando los parámetros calculados, los valores de PEC (eficiencia) obtenidos muestran que las configuraciones 2D con voltajes aplicados en el cable de 5 y 10 kV son más eficientes que la configuración sin voltaje. El PEC aumenta con el voltaje aplicado y, del mismo modo, la recirculación a la salida de la sección de prueba se vuelve más evidente con el valor del voltaje aplicado en el cable y por tanto con la intensidad del campo eléctrico. Por otro lado, ninguno de los escenarios con la configuración 3D analizados muestran valores de eficiencia mayores que la unidad. Sin embargo, dividiendo la placa en diferentes partes y analizándola exhaustivamente, se puede concluir que el aumento de temperatura debido a la recirculación está localizado en la parte final y que en el resto de la placa se observan temperaturas menores. Esta reducción de temperatura es más significativa en los escenarios con el menor número de Reynolds estudiado, y menos visible en los escenarios con el número de Reynolds mayor.

ii Escuela Técnica Superior de Ingenieros Industriales (UPM)


Javier Salgado González iii

AGRADECIMIENTOS

En primer lugar me gustaría agradecer a D. Jorge Muñoz Paniagua por la tutela de este Trabajo de Fin de Máster en mi universidad de origen. Fue mi profesor de “Máquinas Hidráulicas y Eólicas”, una de las asignaturas que más me han gustado y más he disfrutado del Máster de Ingeniería Industrial.

Del mismo modo, me gustaría agradecer a D. Miltiadis V. Papalexandris y a Dña. Valérie Gelbgras por sus consejos y tutela en la universidad de destino dónde se ha realizado este Trabajo de Fin de Máster (École Polytechnique de Louvain).

Me gustaría agradecer a mis padres y mi hermana por todo el apoyo que me han brindado durante los 7 años que he estado estudiando tanto el grado como el máster en la Escuela. Durante estos 7 años ha sucedido de todo. Hemos pasado muy buenos momentos juntos y hemos superado varios y muy duros sucesos que marcan para siempre, especialmente los dos últimos años. Algo he sacado en claro: estar unidos es fundamental. Gracias, “poco a poco, llegaremos lejos”.

Mención especial merece Lucía. Hace casi 3 años que nos conocemos y te has convertido en un pilar fundamental de mi vida. Pese a estar un año separados, hemos superado una oposición y un Erasmus y a día de hoy estamos mucho más unidos. Gracias por tus consejos, apoyo y visitas durante este último año, especialmente la última que fue fundamental para acabar esta tesis.

Me quiero acordar también de los “Chorvitos”. Sin duda, sois lo mejor que me llevo de la Escuela. Estoy seguro que seguiremos haciendo muchos años “el viaje de Coruña”, el partido de solteros contra casados, los doodles para cuadrar cenas de 25 personas y las barbacoas veraniegas.

También quiero agradecer a “mis amigos del cole”. Llevamos juntos desde los 3 años y pese a que cada uno elegimos diferentes carreras, universidades y países en los que estudiar, a día de hoy seguimos siendo esa piña que jugaba todos los días al fútbol en el recreo. Nuestros aperitivos seguirán siendo imprescindibles muchos años.

Por último, quiero agradecer también a mi nuevo grupo de amigos de Louvain la Neuve. Ha sido un año intenso, con altibajos, buenos y duros momentos y los hemos pasado como una gran familia. Fundamental juntar este buen grupo de gente tan lejos de casa.

Muchas gracias a todos.

Javier Salgado González

iv Escuela Técnica Superior de Ingenieros Industriales (UPM)


Javier Salgado González v

ÍNDICE

Abstract .................................................................................................................................................... i

Agradecimientos ..................................................................................................................................... iii

Índice ........................................................................................................................................................ v

Índice de figuras ..................................................................................................................................... vii

Índice de tablas ....................................................................................................................................... ix

1. Introducción .................................................................................................................................... 1

1.1 Contexto y motivación ............................................................................................................ 1

1.2 Objetivos ................................................................................................................................. 1

2. Estado del arte................................................................................................................................. 3

2.1 Conceptos generales de transferencia de calor ...................................................................... 3

2.2 Introducción y descripción de los componentes electrónicos: sobrecalentamiento ............. 4

2.3 Métodos de refrigeración de componentes electrónicos actuales......................................... 6

2.4 Líquidos refrigerantes dieléctricos: aceite mineral ................................................................. 6

3. Electrohidrodinámica: Principios Físicos ......................................................................................... 7

3.1 Hidrodinámica ......................................................................................................................... 7

3.2 Electrostática ........................................................................................................................... 7

3.3 Electrohidrodinámica .............................................................................................................. 8

4. Simulación CFD: ANSYS Fluent ........................................................................................................ 9

4.1 ANSYS Fluent ........................................................................................................................... 9

4.2 Modelos ................................................................................................................................... 9

4.3 UDS y UDF: Ecuaciones de transporte y funciones definidas por el usuario. ......................... 9

4.4 Discretización del dominio: mallado ..................................................................................... 10

4.5 Solver basado en presión. Algoritmo Acoplado .................................................................... 10

4.6 Criterios de convergencia ...................................................................................................... 11

4.7 Condiciones de frontera ........................................................................................................ 11

4.8 Recursos informáticos y limitaciones .................................................................................... 12

5. Metodología .................................................................................................................................. 13

5.1 Descripción general ............................................................................................................... 13

5.2 Parámetros a analizar ............................................................................................................ 13

5.3 Diseño CAD ............................................................................................................................ 13

5.3.1 Simulaciones 2D ............................................................................................................ 13

vi Escuela Técnica Superior de Ingenieros Industriales (UPM)

5.3.2 Simulaciones 3D ............................................................................................................ 14

5.4 Mallado .................................................................................................................................. 14

5.4.1 Mallado: parámetros para el modelo 2D ...................................................................... 15

5.4.2 Mallado: parámetros para el modelo 3D ...................................................................... 15

5.5 Configuración del solver ........................................................................................................ 15

5.6 Hipótesis y suposiciones ........................................................................................................ 16

5.7 Datos de entrada ................................................................................................................... 17

5.8 Implementación de las UDFs y UDS ...................................................................................... 18

6. Resultados ..................................................................................................................................... 19

6.1 Comentarios generales .......................................................................................................... 19

6.2 Simulaciones 2D .................................................................................................................... 19

6.2.1 Estudio paramétrico 2D: comentarios generales .......................................................... 19

6.2.2 Estudio paramétrico: escenarios 1.a - 1.d ..................................................................... 20

6.3 Simulaciones 3D .................................................................................................................... 24

6.3.1 Estudio paramétrico 3D: estudio del mallado ............................................................... 24

6.3.2 Estudio paramétrico 3D: escenarios 3.a – 3.h ............................................................... 25

6.4 Puntos débiles del modelo .................................................................................................... 30

7. Conclusiones .................................................................................................................................. 31

8. Impactos ........................................................................................................................................ 33

8.1 Impacto social ........................................................................................................................ 33

8.2 Impacto económico y medioambiental ................................................................................. 33

8.3 Impactos tecnológicos ........................................................................................................... 34

9. Planificación y presupuesto ........................................................................................................... 35

9.1 Planificación temporal ........................................................................................................... 35

9.2 Presupuesto ........................................................................................................................... 38


Javier Salgado González vii

ÍNDICE DE FIGURAS

Figura 1 — (a) Relación entre la temperatura ambiente y la vida de un condensador electrolítico. (b) Relación entre la temperatura ambiente y la tasa de fallo de semiconductor [11] ............................... 5

Figura 2 — Evolución de la temperatura de un componente electrónico [17]....................................... 5

Figura 3 — Flujos de calor que pueden eliminarse a una temperatura específica con los diferentes mecanismos de transferencia de calor [17] ............................................................................................ 6

Figura 4 — Visión general de los algoritmos del solver basado en presión [66] ................................... 11

Figura 5 — Modelo 2D: vista general, plano XY .................................................................................... 13

Figura 6 — Modelo 3D: vista general y vista inferior ............................................................................ 14

Figura 7 — Modelo 3D: (a) mallado, sección transversal. (b) mallado, sección de prueba .................. 15

Figura 8 — Escenario 1.d: Vectores de corriente eléctrica coloreados por la densidad de carga (C/kg) inyectada desde el electrode y absorbida por la placa (t = 10 s). Plano XY, x = 0.070 m. Borde derecho del electrodo y la placa. ........................................................................................................................ 20

Figura 9 — Componente y del campo eléctrico (Ey): (a) t = 0.005s. (b) t = 20 s. Plano XY, x = 0.070 m. Borde derecho del electrodo y la placa. ................................................................................................ 20

Figura 10 — Escenarios 1.a a 1.d: Temperatura media y máxima de la placa calentada ..................... 21

Figura 11 — Escenarios 1.a a 1.d: (a) Temperatura medida en el punto T2 (22, -2.5, 0) mm. (b) Temperatura medida en el punto T3 (45, -2.5, 0) mm. (c) Temperatura medida en el punto T4 (68, -2.5, 0) mm. ............................................................................................................................................ 22

Figura 12 — Escenario 1.d: Contornos en t = 16 s del minicanal: (a) Componente x del campo eléctrico (V/m). (b) Densidad de carga (C/kg). (c) Temperatura (K). (d) Componente u de la velocidad (m/s). Plano XY, t = 16 s. ........................................................................................................................ 22

Figura 13 — Escenario 1.d: Contornos en t = 26.5 s del minicanal: (a) Componente x del campo eléctrico (V/m). (b) Densidad de carga (C/kg). (c) Temperatura (K). (d) Componente u de la velocidad (m/s). Plano XY, t = 26.5 s. ..................................................................................................................... 23

Figura 14 — Ratio (a) Nu/Nu0 vs. voltaje aplicado (kV) (b) ∆𝑃/∆𝑃0 vs. voltaje aplicado (kV). (c) PEC vs. voltaje aplicado (kV) para los escenarios 1.a a 1.d................................................................................ 23

Figura 15 — Estudio del mallado: temperaturas medidas en los diferentes puntos definidos en la Tabla 7. (a) Caso a. (b) Caso b. (c) Caso c. ............................................................................................. 25

Figura 16 — Escenario 3.a: (a) Perfil de velocidad de la componente u a lo largo del minicanal. Planos XY en x = 0.004 m, x = 0.029 m, x = 0.054 m, t = 22 s. (b) Perfil de velocidad de la componente v a lo largo del minicanal. Planos XY en x = 0.0165 m, x = 0.029m, x = 0.0415 m, x = 0.054 m and x = 0.064 m, t = 22 s .............................................................................................................................................. 25

Figura 17 — Escenario 3.a: (a) Temperatura de la placa en funcionamiento de estado estable. Vista en planta, plano XZ, t = 22 s. (b) Temperatura a lo largo del minicanal. Planos YZ en x = 0.004m, x = 0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m y x = 0.064m, t = 22 s. ............................................ 26

Figura 18 — Escenario 3.a: Presión a lo largo del minicanal. Planos YZ en x = 0.004m, x = 0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m y x = 0.064m, t = 22 s. ................................................................... 27

viii Escuela Técnica Superior de Ingenieros Industriales (UPM)

Figura 19 — Escenarios 3.a a 3.d: (a) Temperatura media de la placa y (b) máxima temperatura de la placa calentada ...................................................................................................................................... 27

Figura 20 — Escenarios 3.a a 3.d: (a) Temperatura medida en el punto T2 (4, -2.5, 0) mm. (b) Temperatura medida en el punto T3 (29, -2.5, 0) mm. (c) Temperatura medida en el punto T4 (54, -2.5, 0) mm. ............................................................................................................................................ 27

Figura 21 — Escenario 3.d: Densidad de carga (C/kg).(a) t = 0.1 s (b) t = 26 s ...................................... 28

Figura 22 —(a) Escenario 3.b: Problemas de recirculación. Plano YZ a la salida de la sección de prueba x = 0.054 m. t = 26 s. (b) Escenario 3.d: modificación del perfil de velocidad en el plano medio del minicanal. Vectores de velocidad coloreados por temperatura. Plano XY, x = 0.070 m. t = 26 s ......... 28

Figura 23 — Escenario 3.b: Temperatura de la placa calentada. Vista en planta, plano XZ, t = 26 s. .. 28

Figura 24 — Ratio (a) Nu/Nu0 vs. voltaje aplicado (kV) (b) ∆𝑃/∆𝑃0 vs. voltaje aplicado (kV). (c) PEC vs. voltaje aplicado (kV) para los escenarios 3.a a 3.h................................................................................ 29

Figura 25 — Fuentes de energía [53] .................................................................................................... 33


Javier Salgado González ix

ÍNDICE DE TABLAS

Tabla 1 — Modelo 2D: geometría y puntos/planos de medida ............................................................ 14

Tabla 2 — Geometría y puntos/planos de medida ............................................................................... 14

Tabla 3 — Propiedades termofísicas del aceite mineral a 293.15K [23], del cobre y de la madera ..... 17

Tabla 4 — Condiciones de contorno ..................................................................................................... 17

Tabla 5 — 2D y 3D: Escenarios de simulación ....................................................................................... 18

Tabla 6 — Escenario 1.a: Números adimensionales ............................................................................. 21

Tabla 7 — Estudio del mallado: geometría y puntos/planos de medida .............................................. 24

Tabla 8 — Estudio del mallado: parámetros ......................................................................................... 24

Tabla 9 — Estudio del mallado: factores de fricción ............................................................................. 25

Tabla 10 — Escenario 3.a: Números adimensionales ........................................................................... 26

Tabla 11 — Escenarios 3.a to 3.d: Temperaturas de la placa calentada ............................................... 28

Tabla 12 — Distribución temporal de paquetes de trabajo .................................................................. 35

Tabla 13 — Presupuesto del proyecto .................................................................................................. 38

x Escuela Técnica Superior de Ingenieros Industriales (UPM)


Javier Salgado González 1

1. INTRODUCCIÓN

1.1 Contexto y motivación

El mundo aeroespacial, la industria del automóvil o el ámbito de la medicina son solo algunos de los ejemplos de la amplia gama de aplicaciones de los componentes electrónicos en nuestra sociedad. Estos componentes pueden producir una gran cantidad de calor durante su actividad normal y resulta fundamental enfriarlos para garantizar su correcto funcionamiento y alargar su vida útil. Es más, la tendencia actual es reducir el tamaño de estos dispositivos y aumentar su potencia, por lo que la generación de calor se acentúa.

El sobrecalentamiento es el principal problema de estos dispositivos. Reducir la dimensión de los dispositivos y ubicarlos en áreas cada vez más pequeñas contribuye negativamente a la extracción de calor [1]. Diseñar componentes con áreas grandes y lo suficientemente finos es una buena praxis que aumenta la extracción del calor generado [2]. Además, un mal diseño de los componentes puede crear contactos defectuosos del cableado que pueden generar chispas o incluso fuego [3].

Por otro lado, los factores ambientales y las condiciones de trabajo también juegan un papel importante. Una temperatura ambiente elevada contribuye directamente al sobrecalentamiento. Del mismo modo, diferencias de temperatura cíclicas o excesivas pueden inducir diferentes tensiones a los componentes del material y producir fallos. La humedad o la composición de la atmósfera de trabajo pueden erosionar los componentes metálicos y contribuir al deterioro de los dispositivos. Todos estos factores contribuyen a aumentar los problemas de sobrecalentamiento.

Es más, el sobrecalentamiento de un componente no afecta solo a dicho componente, sino que también puede afectar al resto del sistema y producir fallos importantes. La degradación del material, grietas, combustiones e incluso explosiones son las consecuencias de una mala extracción de calor [4].

Los métodos de refrigeración de componentes pueden clasificarse en dos grupos: activos o pasivos en función de si se aplica o no una energía externa. La aplicación de campo eléctrico, campo magnético o paredes vibratorias son algunos ejemplos de métodos activos, mientras que el uso de mini y microcanales, grandes superficies o la dispersión de nano partículas en un fluido son técnicas pasivas de refrigeración. Una importante variedad de microbombas se han desarrollado para la impulsión de fluidos a través de minicanales [23][24].

En resumen, nuevas tecnologías de refrigeración de componentes electrónicos están en desarrollo con el fin de cumplir las necesidades de extracción de calor requeridas. Este documento se centra en el estudio de un sistema electrohidrodinámico que impulsa un fluido dieléctrico a través de un minicanal.

1.2 Objetivos

El objetivo principal de este Trabajo de Fin de Máster es evaluar la mejora de transferencia de calor de un sistema electrohidrodinámico para refrigerar componentes electrónicos, mediante el bombeo de un líquido dieléctrico a través de un minicanal. El estudio se basa en el experimento realizado por Moghanlou, F. S. et al. [23]. La transferencia de calor se estudia gracias al análisis de varios parámetros. Los diferentes pasos para llegar al objetivo final son:

Realizar un estudio de los actuales métodos de refrigeración de componentes

Realizar un estudio del funcionamiento y los principios físicos de un sistema electrohidrodinámico

Diseñar y preparar un modelo óptimo en 2D y 3D en ANSYS Fluent para llevar a cabo el análisis CFD requerido para diferentes escenarios.

Introducción

2 Escuela Técnica Superior de Ingenieros Industriales (UPM)

Evaluar diferentes parámetros en la refrigeración de componentes electrónicos para determinar la eficiencia del dispositivo.



2. ESTADO DEL ARTE

2.1 Conceptos generales de transferencia de calor

El calor es la energía transferida entre sustancias o sistemas debido a una diferencia de temperatura entre ellos. De acuerdo con la primera ley de la termodinámica, si dos cuerpos a diferentes temperaturas se colocan juntos en ausencia de trabajo, la transferencia de calor ocurre de manera inmediata y espontánea desde el más caliente al más frío. Se mide en Julios (J) en el Sistema Internacional de Unidades. Hay tres tipos de modos de transferencia de calor: conducción, convección y radiación. Simplemente se explican brevemente la conducción y la convección, ya que la transferencia de calor por radiación no se estudia en este documento [5]. 1

Conducción: es una forma interna de transferencia de calor causada por vibraciones o movimientos rápidos de átomos y moléculas. Es el mecanismo de transferencia de calor más importante en sólidos, especialmente en metales porque los electrones libres pueden moverse y transferir fácilmente energía de una parte del metal a otra [6]. La conducción del calor sigue la Ley de Fourier. Indica que el flujo de calor transferido es proporcional a la magnitud del gradiente de temperatura con el signo opuesto [7].

�̇� = −𝑘 𝐴∇T (1)

La velocidad de transferencia de la conducción de calor depende de las propiedades del medio. Es relevante introducir el concepto de conductividad térmica. Es la capacidad de un material para conducir el calor. Para aplicaciones aislantes, se recomienda utilizar materiales con baja conductividad térmica, mientras que los materiales con valores más altos se utilizan para aplicaciones donde se necesita una buena conducción del calor [8].

Convección: se produce entre una superficie sólida y un fluido que se mueve sobre ella. Suele ser el principal método de transferencia en líquidos y gases. El calor se transfiere por la combinación de difusión (conducción) y por el movimiento de fluido (advección). La convección natural o libre se produce cuando el movimiento del fluido solo es causado por fuerzas de flotabilidad. La densidad del fluido varía debido a las diferencias de temperatura. Si la temperatura aumenta, la densidad normalmente disminuye y provoca el movimiento ascendente del fluido. Por otro lado, la convección forzada ocurre cuando el fluido se bombea sobre la superficie de un cuerpo. La convección con cambio de fase (ebullición) también puede ocurrir [6]. La expresión de transferencia de calor por convección es:

�̇� = ℎ𝐴(𝑇𝑠 − 𝑇𝑓) (2)

Hay diversos números adimensionales que caracterizan la transferencia de calor [12].

Número de Reynolds: es la relación de las fuerzas de inercia (propiedades del fluido y del flujo) y las fuerzas viscosas (solo las propiedades del fluido). Se utiliza para determinar el régimen de flujo (laminar o turbulento). La expresión es:

𝑅𝑒 = 𝜌𝑣𝑠𝐿

𝜇=

𝑣𝑠𝐿

𝜐 (3)

Para flujos internos en una tubería, la longitud característica es el diámetro hidráulico 𝐷ℎ (𝑚):

𝐷ℎ = 4𝐴

𝑃 (4)

1 Las variables de las expresiones están definidas en: List of Acronyms

Estado del arte


Número de Nusselt: es la relación entre el calor transferido por convección y la transferencia de calor conductiva. Es el parámetro adimensional que caracteriza la transferencia de calor por convección.

𝑁𝑢 = ℎ𝐿

𝑘 (5)

Número de Prandtl: es la relación entre la difusividad de la cantidad de momento y la difusividad térmica de un fluido. Depende de las propiedades del fluido.

𝑃𝑟 = 𝑐𝑝𝜇

𝑘 (6)

Número de Grashof: es la relación entre las fuerzas de flotabilidad y las fuerzas viscosas que actúan sobre un fluido. Cuantifica las fuerzas opuestas en la transferencia de calor por convección.

𝐺𝑟 = 𝑔𝛽(𝑇𝑠 − 𝑇𝑓)𝐷ℎ

3

𝜈2 (7)

𝛽 (𝐾−1) Es el coeficiente de expansión térmica. Para los gases, se puede calcular siguiendo la expresión 𝛽 = 1/𝑇; y para líquidos, se puede calcular si se conoce la variación de densidad con la temperatura a presión constante.

Número de Rayleigh: mide la relación entre los efectos de las fuerzas de flotabilidad y los efectos de las fuerzas de viscosidad y la conducción térmica.

𝑅𝑎 = 𝐺𝑟𝑃𝑟 (8)

El número crítico de Rayleigh para el caso de placas paralelas infinitas calentadas desde abajo es aproximadamente 1700. Cuando el número de Rayleigh está por debajo del valor crítico para un fluido dado, el mecanismo de transferencia de calor dominante es la conducción. Cuando se excede este valor crítico, la transferencia de calor por convección es dominante [44].

2.2 Introducción y descripción de los componentes electrónicos: sobrecalentamiento

Bajo un punto de vista mecánico, los componentes electrónicos son muy fiables al no tener partes móviles. Sin embargo, se convierten en dispositivos potenciales de sufrir problemas de sobrecalentamiento debido al calor que genera el flujo de corriente eléctrica a través de una resistencia. Por tanto, normalmente fallan después de un uso prolongado a altas temperaturas.

En primer lugar, los dispositivos electrónicos siguen la llamada "twice law": la vida aumenta al doble cuando la temperatura se reduce a 10 ° C. La tasa de fallo de los dispositivos electrónicos debido a la

temperatura se puede estimar con la Ley de Arrhenius ( 𝑘(𝑇) = 𝐴𝑒−𝐸𝑎

𝑅𝑇 ), dónde 𝑘(𝑇) es la constante

cinética, 𝐴 el factor de pre exponencial, 𝐸𝑎 (𝐽

𝑚𝑜𝑙) la energía de activación, 𝑅 (

𝐽

𝑚𝑜𝑙𝐾) la constante de

los gases ideales y 𝑇(𝐾) la temperatura absoluta [11]. Si la temperatura ambiente es inferior a 30 ° C, la tasa de fallos es inferior a la unidad, pero aumenta con el crecimiento de la temperatura. La Figura 1 muestra la relación entre la tasa de fallo de un semiconductor y la temperatura ambiente [11].

Es importante introducir algunos conceptos. Las uniones de un componente electrónico son los circuitos a través de los cuales fluye la corriente eléctrica. Estas uniones son los sitios potenciales de generación de calor y, normalmente, su temperatura está limitada para garantizar un funcionamiento seguro [17]. La resistencia térmica es una medida de la diferencia de temperatura entre dos superficies definidas de un material que resiste un flujo de calor. Para estos dispositivos, la resistencia térmica es la diferencia de temperatura entre el dispositivo y el ambiente cuando disipa un vatio de calor. Se mide en °C/W. Por tanto, se recomienda un valor bajo de este parámetro [18].



(a) (b)

Figura 1 — (a) Relación entre la temperatura ambiente y la vida de un condensador electrolítico. (b) Relación entre la temperatura ambiente y la tasa de fallo de semiconductor [11]

Hay varios tipos de componentes electrónicos, entre los que destacan los chip + cubierta y las placas de circuito impreso (PCB). Se deben utilizar buenos materiales aislantes para evitar averías eléctricas y buenos conductores térmicos para disipar el calor generado. Los esfuerzos térmicos deben evitarse usando materiales de alta resistencia y con buenos coeficientes de expansión térmica [17].

El primer paso para elegir o diseñar un método de refrigeración es determinar la cantidad de calor a eliminar: la carga térmica. El calor generado sigue la primera ley de Joule. El calor de efecto Joule o efecto óhmico es el proceso físico irreversible por el cual se genera calor debido al paso de la corriente eléctrica a través de un conductor eléctrico [19].

𝑊𝑒̇ = 𝜙 𝐼 = 𝐼2𝑅 (9)

Siguiendo la primera ley de la termodinámica, que establece la conservación de la energía, en ausencia de otra fuente de energía o interacción, el calor producido por un dispositivo electrónico en funcionamiento estable es igual a su consumo de energía. Sin embargo, esta condición ideal se ve alterada por la interacción con diferentes equipos que producen otras formas de energía. Por tanto, la carga de térmica se puede calcular como el consumo de energía menos todas estas interacciones con el resto de equipos. Otra forma de calcularlo es determinar y sumar todo el calor individual producido por todos los componentes. Normalmente, después de determinar la carga térmica a refrigerar, se añade un margen de seguridad para garantizar la fiabilidad y la seguridad de los componentes. Esta carga de enfriamiento adicional de seguridad normalmente aumenta el coste, el tamaño, el peso y el consumo del sistema. Por lo tanto, es importante ajustar bien este margen de seguridad para no sobredimensionar

considerablemente los inconvenientes. Del mismo modo, la temperatura ambiente también influye en los componentes. Ambientes extremos contribuyen al deterioro.

El estado térmico de un dispositivo electrónico puede dividirse en dos fases diferentes: una fase transitoria y otra estable. Cuando se enciende un dispositivo, los componentes comienzan a generar calor, por lo que la temperatura comienza a subir progresivamente. Cuando el calor generado es igual al calor eliminado por el método de refrigeración, la temperatura del dispositivo se estabiliza y comienza la operación estable [17].

Figura 2 — Evolución de la temperatura de un componente electrónico [17]

Estado del arte


2.3 Métodos de refrigeración de componentes electrónicos actuales

El calor generado varía según el dispositivo electrónico desde 5 W/cm2 de una placa de circuito integrado (PCB) hasta 20 kW cm2 para un láser semiconductor [16]. La temperatura de operación debe estar por debajo de la temperatura máxima especificada por los fabricantes.

Los principales métodos de refrigeración existentes son los métodos de conducción, la refrigeración por aire (convección natural y forzada), la refrigeración líquida, la refrigeración por inmersión y las nuevas técnicas avanzadas [17]. Los fabricantes generalmente proporcionan la tasa de disipación de calor y la temperatura máxima permitida para un funcionamiento fiable y seguro (Figura 3).

La refrigeración líquida es más efectiva que la refrigeración por aire. Los líquidos tienen una conductividad térmica mucho mayor que los gases, por lo que los coeficientes de transferencia de calor son más altos. La refrigeración líquida se recomienda para aplicaciones con altas cargas térmicas que los sistemas de refrigeración por aire no consiguen enfriar. Sin embargo, el uso de líquidos implica algunos inconvenientes, como los riesgos y problemas de fugas, corrosión y condensación.

Por lo general, los equipos electrónicos están sumergidos en sistemas de enfriamiento directo y la transferencia de calor puede ser natural o forzada por convección o ebullición. Se utilizan fluidos dieléctricos cuyas propiedades eléctricas son adecuadas para esta aplicación (ver sección 2.4).

2.4 Líquidos refrigerantes dieléctricos: aceite mineral

El uso de aceites minerales reduce el consumo de energía en la refrigeración de componentes electrónicos. La refrigeración líquida tiene múltiples ventajas sobre las técnicas tradicionales de aire, debido a las mayores capacidades de calor de los fluidos [32]. Además, el ahorro económico no solo es en términos de consumo de energía, sino que también se reducen los costes de diseño y fabricación [33]. Diferentes soluciones y casos de estudio con aceite mineral ([34] y [35]) han demostrado la efectividad y el ahorro económico de esta técnica de refrigeración.

Los beneficios de la tecnología de refrigeración con aceite mineral en vez de aire podrían resumirse en la reducción de los problemas operativos comunes y las principales causas de fallo. Típicamente, los sistemas de enfriamiento por aire tienen altas fluctuaciones en la temperatura y el perfil de humedad relativa. Sin embargo, las condiciones de operación de los sistemas de aceite se suavizan, se reducen los problemas de corrosión y la migración electroquímica. Además, no se usa ventiladores y se evita la exposición a descargas electrostáticas [33].

Los líquidos deben cumplir algunos requisitos especiales: alta conductividad térmica, alto calor específico, baja viscosidad, alta resistencia dieléctrica e inercia y estabilidad química [17]. Los aceites minerales pueden clasificarse en tres grupos diferentes, dependiendo de su composición: aceite parafínico, aceite nafténico y aceite aromático [38]. Para la función de refrigerante, los aceites deben tener un bajo punto de fluidez (medida del flujo de aceite a una temperatura relativamente baja) para garantizar el flujo correcto del fluido a cualquier temperatura. La temperatura del aceite en servicio debe ser controlada y debe ser más baja que su punto de inflamación. Además, el aceite mineral actúa como un aislante entre diferentes partes a diferentes potenciales eléctricos [37].

Figura 3 — Flujos de calor que pueden eliminarse a una temperatura específica con los diferentes mecanismos de transferencia de calor [17]



3. ELECTROHIDRODINÁMICA: PRINCIPIOS FÍSICOS

La electrohidrodinámica es una ciencia interdisciplinar que estudia la dinámica de los fluidos bajo la acción de campos eléctricos [41].

3.1 Hidrodinámica

Las ecuaciones de Navier-Stokes describen el movimiento de los fluidos viscosos. Este estudio se centra en el estudio de un flujo incompresible en el que las variaciones de densidad no están vinculadas a la presión, y la conservación de la masa es una restricción en el campo de la velocidad. Entonces, para un flujo incompresible, la divergencia de la velocidad del flujo es cero [42].

La aproximación de Boussinesq es una forma de resolver el flujo no isotérmico, como por ejemplo los problemas de convección natural, sin resolver la formulación completa de las ecuaciones de Navier-Stokes. Esta aproximación considera la densidad como un valor constante salvo en los términos dónde aparezca multiplicada por la aceleración de la gravedad. Es válida cuando las diferencias de temperatura y, por tanto, de densidad son pequeñas, con una formulación matemática y física sencilla [39]. La formulación es como sigue [40]. 2

La ecuación de continuidad para la conservación de la masa es:

𝜕𝜌

𝜕𝑡+ ∇ · (𝜌𝒖) = 0 (11)

Dónde 𝒖 (𝑚 ⁄ 𝑠) es la velocidad del fluido. Si la densidad es considerada constante:

∇ · 𝒖 = 0 (12)

La expresión de la densidad en función de la temperatura es:

𝜌 = 𝜌0 − 𝛽𝜌0∆𝑇 (13)

Dónde 𝛽 (𝐾−1) es el coeficiente de expansión térmica. Considerando 𝑭 = 𝜌𝑔 (𝑁) (14) la fuerza gravitacional, la ecuación de conservación del momento queda:

𝜕𝒖

𝜕𝑡+ (𝒖 · ∇)𝒖 = −

1

𝜌∇𝑝 + 𝜈∇2𝒖 − 𝒈𝛽∆𝑇 (15)

La expresión de conservación de la energía con la aproximación de Boussinesq es [41]:

𝜌𝐶𝑝 [𝜕𝑇

𝜕𝑡+ 𝒖 · ∇𝑇] = ∇ ∙ (𝜅∇𝑇) + 𝐽 (16)

3.2 Electrostática

La electrostática es la rama de la física que estudia las cargas en equilibrio en ausencia de un campo magnético significativo [72]. La fuerza eléctrica se aplicará en una determinada región del sistema y es equivalente a generar un circuito eléctrico. La fuerza electrostática debida a la carga espacial o la polarización del medio dieléctrico es la principal razón para la mejora de la transferencia de calor en los sistemas EHD [47]. La fuerza del campo eléctrico tiene la expresión [59]:

𝑭𝒆 = 𝜌𝑒𝑬 −1

2𝐸2 ∇𝜀 + ∇ (

1

2𝐸2𝜌 (

𝜕𝜀

𝜕𝜌) 𝑇) (17)

El primer término es el más relevante en el caso de los sistemas EHD y se llama fuerza de Coulomb. Es la fuerza por unidad de volumen en un medio que contiene carga eléctrica libre, responsable del

2 La numeración de las ecuaciones es coincidente con el documento completo

Electrohidrodinámica: principios físicos


movimiento del fluido (provoca un flujo secundario). El segundo término es la fuerza ejercida sobre un líquido dieléctrico no homogéneo por un campo eléctrico. Normalmente, es más débil que la fuerza de Coulomb y es relevante cuando aparece un gradiente de temperatura o cuando se considera un sistema monofásico AC-EHD. Finalmente, el tercer término muestra los cambios de permitividad debidos a la densidad para un campo eléctrico aplicado [23].

Para calcular la fuerza eléctrica, se consideran las ecuaciones de Maxwell. En los flujos electrohidrodinámicos, el efecto magnético puede ignorarse porque el tiempo característico para los fenómenos magnéticos (𝑡𝑚 ~ 𝜇𝑀𝐾𝑙2) es varios órdenes de magnitud menor que el tiempo característico para los fenómenos eléctricos (𝑡𝑒 ~ 𝜀 𝐾⁄ )[59] [60].

∇ · (𝜀𝑬) = 𝜌𝑒 (18)

∇ × 𝑬 = 0 (19)

Dónde 𝜌𝑒 (𝐶/𝑚^3 ) es la densidad volumétrica de carga. En términos del potencial eléctrico, 𝜙, el límite electrostático sigue la ecuación de Poisson:

𝐄 = −∇𝜙 (20)

∇ · (𝜀∇𝜙) = −𝜌𝑒 (21)

La densidad de carga puede expresarse como 𝜌𝑒 = 𝜌𝑧, siendo 𝑧 (𝐶/𝐾𝑔) (22) la carga por unidad de masa. La ecuación de conservación de la densidad de carga eléctrica es:

𝜕𝜌𝑒

𝜕𝑡+ ∇ · 𝑱 = 0 (23)

Dónde 𝑱 (𝐴/𝑚2) es la densidad de corriente:

𝑱 = 𝐾𝑬 + 𝜌𝑒𝒖 (24)

Considerando las relaciones electrostáticas (18) y (19), la ecuación de la conservación de la carga eléctrica se puede expresar:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 + 𝑬 ∙ (

𝐾

𝜀 ∇𝜀 − ∇𝐾) (25)

Considerando constantes las propiedades eléctricas 𝐾 and 𝜀, la ecuación (25) se reduce a:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 (26)

3.3 Electrohidrodinámica

La electrohidrodinámica describe los efectos de la electrostática en medios líquidos. La fuerza de Coulomb, 𝑭𝒆 = 𝜌𝑒𝑬, se añade a la ecuación de conservación de momento (15). En consecuencia, considerando un valor constante para 𝐾 y el calor generado por efecto Joule para la ecuación de conservación de la energía, las tres ecuaciones que rigen el sistema EHD sujeto a estudio son:

∇ · 𝒖 = 0 (12)

𝜕𝒖

𝜕𝑡+ (𝒖 · ∇)𝒖 = −

1

𝜌∇𝑝 + 𝜈∇2𝒖 − 𝒈𝛽∆𝑇 + 𝜌𝑒𝑬 (27)


𝜕𝑡+ 𝒖 · ∇𝑇] = κ∇2𝑇 + 𝜀|∇𝜙|2 (28)



4. SIMULACIÓN CFD: ANSYS FLUENT

4.1 ANSYS Fluent

La Dinámica de Fluidos Computacional (CFD) es una rama de la mecánica de fluidos que resuelve y analiza cuantitativa y cualitativamente de los flujos de fluidos gracias al modelado matemático y los métodos numéricos [62]. Resolver estos problemas de manera analítica es, en muchos casos, extremadamente difícil debido a los términos inerciales no lineales. Al discretizar el dominio en pequeños elementos (malla), es posible obtener una solución precisa. Las simulaciones descritas en el capítulo 5 se realizan con el software ANSYS Fluent.

4.2 Modelos

ANSYS Fluent proporciona una amplia lista de modelos para diferentes problemas de flujo de fluidos en estado estacionario o transitorios, incompresibles o compresibles, laminares o turbulentos.

Para el caso de estudio de este documento, se utilizan los modelos viscoso, energético y potencial. El modelo viscoso permite configurar y definir las características del flujo del fluido. El modelo energético resuelve la ecuación (28) y el modelo potencial resuelve la ecuación (18) y agrega el calor de efecto Joule a la ecuación de energía [83][84][86][87].

El modelo potencial de ANSYS Fluent resuelve la ecuación [83]:

∇ ∙ (𝜀∇𝜙) + 𝑆 = 0 (29)

Dónde:

𝜙 (𝑉) es el potencial eléctrico

𝜀 (𝐹/𝑚) es la permitividad eléctrica

𝑆 es el término fuente

Este modelo también añade la fuente del calor de efecto Joule (𝑊/𝑚3) generado cuando aparece un flujo de corriente, a la ecuación de la energía [83] [84]:

𝑆ℎ1 = 𝜀|∇𝜙|2 (30)

4.3 UDS y UDF: Ecuaciones de transporte y funciones definidas por el usuario.

Para simular el sistema electrohidrodinámico, es necesario añadir la fuerza de Coulomb a la ecuación de conservación del momento y resolver la ecuación de conservación de la densidad de carga. ANSYS Fluent permite resolver ecuaciones de transporte adicionales llamadas User-Defined Scalar (UDS) transport equations. Para un escalar arbitrario, ANSYS Fluent resuelve la ecuación [63]:

𝜕𝜌𝜙𝑘

𝜕𝑡+ ∇ ∙ (𝜌𝒖𝜙𝑘 − Γ𝑘∇𝜙𝑘) = 𝑆𝜙𝑘

(31)

Dónde:

(𝜕𝜌𝜙_𝑘)/𝜕𝑡 es el término transitorio

∇ ∙ (𝜌𝒖𝜙𝑘) es el término convectivo

Γ𝑘 es el coeficiente de difusión

∇ ∙ (Γ𝑘∇𝜙𝑘) es el término difusivo

𝑆𝜙𝑘 es el término fuente

La ecuación de conservación de la densidad de carga (26) se resuelve con una ecuación UDS.

Para introducir la fuerza de Coulomb en la ecuación de momento, se necesita una función definida por el usuario (UDF). Una UDF es una función de C que se puede ejecutar con ANSYS Fluent y permite, entre otros, personalizar las condiciones de contorno, agregar términos fuente en las

Simulación CFD: ANSYS Fluent


ecuaciones de transporte propias de ANSYS Fluent o en las ecuaciones de transporte de escalares adicionales definidas por el usuario (UDS) y mejorar los modelos de existentes. Los códigos UDF utilizan macros especiales proporcionadas por ANSYS Fluent para acceder a datos del solver y variables de dominio [73].

4.4 Discretización del dominio: mallado

El objetivo de la discretización de todo el dominio en celdas pequeñas de alta calidad es obtener la geometría del dominio y realizar el cálculo secuencial a través de todas las celdas para obtener una solución precisa. ANSYS Fluent permite generar mallados tetraédricos, hexaédricos o híbridos a partir de un mallado existente o un archivo CAD.

Un mallado bueno y fino ayuda al solucionador CFD a converger y obtener una solución precisa, minimizando los recursos empleados. Sin embargo, un mallado grueso puede ser una fuente importante de errores en una simulación. Por lo tanto, es importante encontrar un equilibrio correcto entre la definición de la malla y el coste computacional necesario para resolver las ecuaciones.

ANSYS Fluent proporciona indicadores para verificar la calidad de la malla. El primero se llama “calidad ortogonal” y sus valores van desde 0, que indica mala calidad, hasta 1, excelente calidad. La calidad ortogonal mínima debe ser mayor que 0.01, con un valor promedio significativamente más alto.

Otro indicador importante es la relación de aspecto que mide el estiramiento de una célula. Para un cubo unitario, la relación de aspecto es 1.732. Se recomienda evitar cambios bruscos y grandes en las relaciones de aspecto de las celdas en áreas donde los flujos sufren grandes cambios o gradientes significativos [64].

4.5 Solver basado en presión. Algoritmo Acoplado

ANSYS Fluent ofrece dos métodos numéricos diferentes: Pressure-Based Solver y Density-Based Solver. Normalmente, ANSYS Fluent recomienda usar el solver basado en presión para flujos incompresibles de baja velocidad y el solver basado en densidad para problemas de flujos compresibles de alta velocidad. Para las simulaciones de este proyecto, se selecciona el solver basado en presión.

Las ecuaciones integrales de los distintos principios físicos se resuelven siguiendo una técnica basada en control de volúmenes finitos. Primero, el dominio se divide en volúmenes de control discreto (malla). Las ecuaciones de control integradas se resuelven para cada uno de estos volúmenes de control individuales y se aproxima los valores de las variables en las caras y las derivadas con la información de las variables nodales. Finalmente, se resuelve el sistema de ecuación algebraico resultante [65] [70].

El solver basado en presión puede trabajar con dos algoritmos diferentes: segregado o acoplado. El algoritmo segregado resuelve las ecuaciones de control de forma secuencial: cada ecuación está desacoplada de las otras ecuaciones. Los requisitos de memoria necesarios son bajos porque las ecuaciones discretizadas solo necesitan almacenarse una vez. Por otro lado, el algoritmo acoplado resuelve simultáneamente el sistema de ecuaciones de continuidad y momento, aumentando la tasa de convergencia de la solución. Sin embargo, el coste de la memoria aumenta en 1.5-2 veces [66].



Figura 4 — Visión general de los algoritmos del solver basado en presión [66]

4.6 Criterios de convergencia

Generalmente, los problemas de flujo de fluidos no son lineales y deben resolverse mediante un cálculo iterativo con softwares de CFD. Los residuos miden los desequilibrios locales de una variable en cada volumen de control, y ANSYS Fluent los utiliza como criterio de convergencia. Para obtener una solución numéricamente precisa, deben ser lo más bajos posible. El criterio de convergencia predeterminado de ANSYS Fluent requiere que los residuos decrezcan 3 órdenes de magnitud para la ecuación de continuidad y de momento y 6 órdenes para la ecuación de energía [67].

Sin embargo, para problemas complicados, no siempre es un objetivo alcanzable. Monitorizar algunas variables como la fuerza, el coeficiente de resistencia o la temperatura media puede ayudar determinar cuándo una simulación converge. Si estas variables permanecen estables con más iteraciones, las simulaciones pueden considerarse como convergentes. La solución final debe garantizar la conservación de la masa, el momento y la energía [68].

4.7 Condiciones de frontera

En un análisis de CFD, es relevante definir cómo funciona el sistema. Las condiciones de frontera son el conjunto de restricciones y condiciones específicas para los valores límites del problema requeridos para resolver el modelo matemático. ANSYS Fluent ofrece una amplia lista de diferentes condiciones de frontera que permiten definir y configurar los valores de los límites y el comportamiento del flujo.

En las simulaciones realizadas, velocity inlet, pressure outlet y wall son las condiciones de contorno utilizadas [69].

Velocity inlet: define la velocidad del flujo y todas las propiedades escalares en la entrada del dominio. La presión total no se puede definir, pero adopta el valor necesario para proporcionar la velocidad definida.

Cuando se usa velocity inlet, ANSYS Fluent recomienda usar pressure outlet a la salida del dominio. Para flujos subsónicos, se define la presión estática a la salida. Si ocurren problemas

Simulación CFD: ANSYS Fluent


de reflujo en la salida del dominio, es posible definir ciertas propiedades de reflujo para evitar problemas de convergencia.

Wall: se utilizan para definir zonas sólidas y confinar el fluido. Para flujos viscosos, se aplica la condición de no deslizamiento: la velocidad del fluido tangencial es igual a la velocidad de la pared y la velocidad normal es nula. Para definir una pared adiabática, se debe establecer un flujo de calor nulo.

4.8 Recursos informáticos y limitaciones

Es importante remarcar que se usa la versión académica de ANSYS Fluent para estas simulaciones (v18.1 para las simulaciones 3D y v19.2 para las simulaciones 2D). Esta versión tiene una limitación importante en la generación del mallado: no está permitido crear una malla con más de 512k celdas.

Otra limitación importante son los recursos informáticos. Para este tipo de simulaciones, el coste computacional suele ser muy alto, por lo que normalmente se usan estaciones de trabajo especializadas. Para este Trabajo de Fin de Máster, los recursos informáticos son limitados y para trabajo adicional relacionado con este caso de estudio, se recomienda usar una estación de trabajo profesional especializada para simulaciones CFD y la versión profesional ANSYS Fluent para evitar la limitación de la generación del mallado.

Los recursos computacionales disponibles son:

Intel® Core™ i7-4500U, 2.4Ghz, 4 Gb RAM para las simulaciones 2D

2 x Intel® Core™ i5-6500, 3.20 GHz, 8Gb RAM para las simulaciones 3D

El coste computacional de las simulaciones es alto. A pesar de que se toman algunas consideraciones para mejorar el tiempo de simulación, los casos 3.a a 3.d tardaron unas 85-95 horas y los casos 3.e a 3.h tardaron alrededor de 60-70 horas. Los casos 2D tardaron aproximadamente 30-40 horas y los casos para el estudio del mallado son más rápidos y tardaron aproximadamente 10 minutos cada uno.



5. METODOLOGÍA

5.1 Descripción general

El caso de estudio de este documento consiste en simular un sistema electrohidrodinámico para un flujo laminar a través de un minicanal cuadrado. El objetivo es investigar cómo el campo eléctrico contribuye al proceso de refrigeración y la variación de la caída de presión para diferentes números de Reynolds. El acoplamiento de la electrostática y la hidrodinámica produce una modificación en las ecuaciones de conservación del momento, que modifican el perfil de velocidad. Se estudian dos métodos diferentes: un método activo basado en la aplicación de campo eléctrico a un fluido dieléctrico, y el uso de un minicanal como método pasivo. La mejora de la transferencia de calor se mide mediante el análisis de diferentes parámetros.

El dispositivo está formado por un cable de cobre (electrodo de alto voltaje) ubicado en la parte superior del minicanal, que inyecta carga eléctrica a través del líquido (aceite mineral). El componente electrónico a enfriar es una placa de cobre ubicada en la parte inferior del minicanal y el líquido dieléctrico bombeado es aceite mineral. La placa calentada también está puesta a tierra [23].

5.2 Parámetros a analizar

Se analizan los siguientes parámetros para determinar la mejora de transferencia de calor [23].

Coeficiente de transferencia de calor por convección ℎ =

�̇�

𝐴 (𝑇𝑤 − 𝑇𝑏) (32)

Número de Nusselt 𝑁𝑢 = ℎ𝐷ℎ

𝑘 (5)

Performance Evaluation Criterion (PEC)

𝜂 = 𝑗 𝑗𝑠⁄

(𝑓 𝑓𝑠⁄ )1 3⁄ (33)

Dónde:

𝑗 = 𝑆𝑡 ∗ 𝑃𝑟2 3⁄ (34)

𝑓 =Δ𝑃

(𝐿⁄𝐷)∗((𝜌𝑢^2)⁄2) es el coeficiente de fricción. (35)

𝑓𝑠 = 56.8/𝑅𝑒 es el coeficiente de fricción para conductos cuadrados. 𝑓𝑠 ∗ 𝑅𝑒 = 𝑃𝑜 dónde 𝑃𝑜 es el número de Poiseuille. [45] (36)

𝑆𝑡 = 𝑁𝑢 (𝑅𝑒 ∗ Pr)⁄ es el número de Staton. (37)

El sufijo “s” se refiere a la condición de superficie perfectamente lisa o método sin la mejora aplicada.

5.3 Diseño CAD

El diseño CAD de las simulaciones se realiza en SpaceClaim.

5.3.1 Simulaciones 2D

El diseño 2D es el siguiente (plano XY):

Figura 5 — Modelo 2D: vista general, plano XY

El origen de coordenadas está situado en el centro de la cara de la entrada del fluido. El cable está ubicado en la parte superior del canal y la placa calentada en la parte inferior de x = 0.020 m a x =

ENTRADA SALIDA

Metodología


0.070 m. El electrodo y la placa están alineados y tienen la misma longitud. Las partes están más detalladas en el siguiente apartado.

El conducto cuadrado tiene una altura de 5 mm y una longitud de 100 mm. Las dimensiones de las diferentes partes y algunos puntos/planos ubicados a lo largo del eje x para medir la temperatura y la presión se presentan en la siguiente tabla.

Zona Longitud (mm) Punto/plano de medida Posición de punto/plano

de medida (mm)

Entrada 20 P1 x = 18

Test 50

T2 (22, -2.5)

T3 (55,-2.5)

T4 (68,-2.5)

Salida 30 P2 x = 72

Tabla 1 — Modelo 2D: geometría y puntos/planos de medida

5.3.2 Simulaciones 3D

El diseño CAD de las simulaciones 3D es:

Figura 6 — Modelo 3D: vista general y vista inferior

El origen de las coordenadas se encuentra en el centro del área de la sección transversal de la entrada. El conducto cuadrado tiene un tamaño de 5x5 mm y una longitud de 74 mm. A lo largo del eje x, se ubican algunos planos/puntos para medir diferentes variables del fluido.


de medida (mm)

Entrada 4 P1 x = 2

Test 50

T2 (4, -2.5, 0)

T3 (29,-2.5,0)

T4 (52,-2.5,0)

Salida 20 P2 x = 56

Tabla 2 — Geometría y puntos/planos de medida

5.4 Mallado

Primero, para determinar y ver el impacto de la malla en los resultados finales, se realiza un estudio de malla para tres casos diferentes, para ver el impacto de la discretización del dominio en los

Entrada: entrada de flujo.

Salida: salida de flujo.

Paredes: las paredes confinan el fluido. Están térmica y eléctricamente aisladas.

Placa calentada: la placa a refrigerar. El flujo de calor considerado es 10 000 (𝑊/𝑚2).

Cable: es el electrodo de alto voltaje que inyecta la carga en el líquido. Está situado en el centro de la parte superior de la sección de prueba y mide 50 mm. En el modelo, el cable se simula como una placa de 0.3 mm de ancho.



resultados finales. La inyección de carga no se considera para este estudio. El mallado se realiza con Mesh Fluent. La versión académica tiene un límite total de celdas de 512k.

5.4.1 Mallado: parámetros para el modelo 2D Se toman algunas consideraciones en la generación del mallado:

Tipo de mallado: hexaédrico

Tamaño máximo de cada elemento = 0.1 mm para tener 50 celdas a lo largo del eje y.

Las características del mallado son:

Total de elementos: 50 000

Total de nodos: 51 051

Calidad ortogonal: 0.999237

Relación de aspecto máxima: 1.45265

5.4.2 Mallado: parámetros para el modelo 3D Las consideraciones para el mallado 3D son:

Tipo de mallado: hexaédrico

Tamaño máximo de cada elemento = 0.19 mm

Cable: tamaño máximo de elemento = 0.1 mm para tener 3 elementos a lo largo del eje z.

Las características del mallado son:

Total de elementos: 508 680

Total de nodos: 545 972

Calidad ortogonal: 0.99999

Relación de aspecto máxima: 2.95733

(a) (b)

Figura 7 — Modelo 3D: (a) mallado, sección transversal. (b) mallado, sección de prueba

La Figura 7 (b) muestra la transición de malla entre la sección de prueba y la sección de salida. Se afina la malla en la sección de prueba para garantizar mayor precisión en la solución de las ecuaciones en esta zona, ya que la inyección de la carga se realizará en esta parte del fluido. Además, se mejora también la malla en el cable.

5.5 Configuración del solver

La elección de la configuración del solver se basa en los tutoriales de aprendizaje realizados antes de la preparación de estas simulaciones y en las recomendaciones de la Guía del usuario de ANSYS.

Doble precisión, Pressure-Based Solver

Solutions Methods:

Metodología


- Pressure-velocity coupling: COUPLED

Spatial Discretization: - Gradiente: Least Squares Cell

Based - Presión: Body Force Weighted - Momento: Second Order Upwind

- Energía: Second Order Upwind - Potencial: Second Order Upwind - UDS-1 (densidad de carga): First

Order Upwind

Criterios de convergencia:

Continuidad: 10-3

Momento: 10-3

Energía: 10-9

UDS-1 (densidad de carga): 10-3

Potencial: 10-9

Under-Relaxation Factors:

Número de Courant: 5

Momento: 0.75

Presión: 0.75

Densidad: 0.8

Body forces: 1

Energía: 0.9

Potencial: 1

UDS-1 (densidad de carga): 0.7

El número de Courant de flujo (CFL) es una condición matemática de convergencia para la estabilidad al resolver problemas de convección o de fenómenos de ondas. Se utiliza para el esquema de presión-velocidad acoplado y relaciona la velocidad con el paso de tiempo y la longitud de los elementos de la malla (𝐶𝐹𝐿 = 𝑢∆𝑡/∆𝑥). En nuestro caso, para mejorar la estabilidad de la solución, el número CFL se establece en 5 [81].

5.6 Hipótesis y suposiciones

Se consideran los siguientes puntos para todas las simulaciones:

Simulación transitoria.

Flujo laminar completamente desarrollado.

Aproximación de Boussinesq para la densidad. La densidad varía solo con la temperatura en el término de flotabilidad en la ecuación de momento del eje y (capítulo 3).

Aceleración gravitacional que actúa en dirección negativa del eje y con un valor de 9.81 m/s2.

Temperatura de entrada del fluido igual a 293.15 K.

Presión de salida del fluido igual a 0 Pa.

Flujo de calor de la placa calentada igual a 10 000 W/m2. Este flujo corresponde a 2.5 W.

Muros adiabáticos. Condición de no deslizamiento: la velocidad tangencial del fluido es igual a la velocidad de la pared y la velocidad normal es nula.

Las diferencias de temperatura esperadas entre la entrada y la salida son bajas, por lo que, en caso de tener un flujo de retorno en la salida del minicanal, la temperatura del reflujo se establece en 296 K.

Se utilizan tres materiales diferentes: aceite mineral como fluido, cobre para el cable (electrodo) y la placa calentada y un aislante para las paredes de la tubería. En el capítulo 2.5 se presenta una descripción general de las propiedades de los aceites minerales. En resumen, el aceite mineral es adecuado como refrigerante debido a su baja viscosidad, buenas propiedades eléctricas y baja permitividad eléctrica. Garantiza una baja temperatura de operación, evita los problemas de oxidación y corrosión y reduce la contaminación ambiental como el polvo. Un material conductor se utiliza para el cable y para la placa calentada. El cobre es un material conductor con una alta conductividad y bajo valor de resistividad. Permite el flujo de carga libremente sobre su superficie. Una explicación más detallada de un conductor eléctrico se expone en el capítulo 6.1.



Finalmente, se utiliza un material aislante para las paredes de la tubería. Se selecciona madera debido a sus buenas propiedades aislantes. Un aislante no permite el flujo libre de carga y, por tanto, no permite la conducción eléctrica. Se caracterizan por un alto valor de resistividad eléctrica y un valor muy bajo de conductividad eléctrica [74][75].

Propiedad Nomenclatura Unidades Aceite mineral Cobre Madera

Densidad 𝜌 Kg/m3 856 8978 700

Viscosidad 𝜇 Pa s 0.03 — —

Calor específico 𝑐𝑝 J/kg K 1850 381 2300

Conductividad térmica 𝑘 W/m K 0.14 387.6 0.173

Permitividad eléctrica 𝜀 F/m 1.95 x 10-11

5.8 x 107 1 x 10

-30

Conductividad eléctrica 𝐾 S/m 3.3 x 10-12

1.7 x 10-8

1 x 1030

Tabla 3 — Propiedades termofísicas del aceite mineral a 293.15K [23], del cobre y de la madera

Como las diferencias de temperatura del fluido son bajas, las propiedades termofísicas del aceite mineral se consideran constantes. Las propiedades del cobre y de la madera se obtienen de la base de datos de Fluent. Se selecciona la madera como base pero es un material inventado.

Las condiciones de contorno son: Condición de contorno Ecuaciones

Entrada Velocity inlet 𝑢 = 𝑢𝑖𝑛 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑥= 0

𝜕𝜙

𝜕𝑥= 0

Salida Pressure outlet 𝜕𝑢

𝜕𝑥= 0

𝜕𝑣

𝜕𝑥= 0

𝜕𝑤

𝜕𝑥= 0

𝜕𝑧

𝜕𝑥= 0

𝜕𝜙

𝜕𝑥= 0

Placa calentada Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑛= 0 𝜙 = 0

Cable (electrodo)

Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝑧 = 𝑧0 𝜙 = 𝜙0

Paredes Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑛= 0

𝜕𝜙

𝜕𝑛= 0

Tabla 4 — Condiciones de contorno

Es necesario encontrar un equilibrio entre la duración de la simulación y cumplir con todos los criterios de convergencia en cada time step. Para las simulaciones 2D, el time step seleccionado es 0.001 s y para las simulaciones 3D, se elige un time step de 0.001 s para los primeros 0.1 s de simulación para asegurar una correcta inicialización del flujo y después de un se incrementa a 0.005 s. En algunos momentos de las simulaciones 2D, el time step se reduce a 5 x 10-4 s and 2 x 10-4 s para alcanzar los criterios de convergencia (casos 1.c y 1.d). En conclusión, todos los criterios de convergencia se cumplen para cada paso de tiempo, por lo que se garantiza la validez y la precisión de los resultados.

5.7 Datos de entrada

Se estudian escenarios diferentes para el modelo 2D y 3D: Caso 𝒖𝒊𝒏 (𝒎/𝒔) inicial Re 𝒛𝟎 (𝑪/𝒌𝒈) 𝝓𝟎 (𝒌𝑽)

1.a

0.01 1.43 6 x 10-3

0

1.b 5

1.c 10

1.d 15

3.a

0.01 1.43 6 x 10-3

0

3.b 5

3.c 10

3.d 15

3.e 0.05 7.13 6 x 10

-3

0

3.f 5

Metodología


3.g 10

3.h 15

Tabla 5 — 2D y 3D: Escenarios de simulación

5.8 Implementación de las UDFs y UDS

Este capítulo explica los pasos necesarios para calcular las ecuaciones eléctricas y cómo se combinan con las ecuaciones hidrodinámicas.

Respecto al modelo de potencial presentado en el capítulo 4.2, ANSYS Fluent resuelve la ecuación de potencial eléctrico [83]

∇ ∙ (𝜀∇𝜙) + 𝑆 = 0 (29)

Recordando las expresiones definidas en la sección 3.2

𝐄 = −∇𝜙; ∇ · (𝜀∇𝜙) = −𝜌𝑒 y 𝜌𝑒 = 𝜌𝑧 (20)(21)(22)

El modelo potencial resuelve la ecuación (21). Un UDF DEFINE_SOURCE se añade el término fuente (𝑆 = 𝜌𝑒 = 𝜌𝑧) (22)) a la ecuación de Poisson. Para introducir la ecuación de conservación de la carga, se utiliza una ecuación de transporte escalar adicional definida por el usuario (UDS).

𝜕𝜌𝜙𝑘


(31)

Recordando las expresiones de la sección 3.2:

𝜕𝜌𝑒

𝜕𝑡+ ∇ · 𝑱 = 0 (23)

𝑱 = 𝐾𝑬 + 𝜌𝑒𝒖 (24)

Considerando las relaciones electrostáticas (18) y (19), la ecuación de conservación de la carga es:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 + 𝑬 ∙ (

𝐾

𝜀 ∇𝜀 − ∇𝐾) (25)

Si las propiedades eléctricas del fluido 𝐾 y 𝜀 son constantes, la eq. (25) se puede escribir como:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 (26)

Para la ecuación UDS, el escalar 𝜙𝑘 es 𝑧 (𝐶/𝑘𝑔) y solo se consideran los términos transitorio y

convectivo. El término −𝐾

𝜀𝜌𝑧 es el término fuente 𝑆𝜙𝑘

de la UDS. Una UDF DEFINE_SOURCE

introduce el término fuente de la ecuación UDS:

𝑆𝜙𝑘= −

𝐾

𝜀𝜌𝑧 =

𝐾 ∇ · (𝜀∇𝜙)

𝜀=

∫ −∇ ∙ (𝐾𝑬)𝑑𝑉

∫ 𝑑𝑉=

∫ −∇ ∙ 𝐾𝑬𝒏𝜕𝑉

𝑉= −

Σ𝐾𝑬𝑨

𝑉 (39)

Tras resolver las ecuaciones (20), (21) y (26), es necesario introducir la fuerza de Coulomb en la ecuación de momento. Tres UDF DEFINE_SOURCE diferentes se diseñan para acoplar la electrostática y la hidrodinámica para las tres ecuaciones de conservación del momento.

El uso de una macro DEFINE_ADJUST Fluent es necesario para calcular el campo eléctrico y el término fuente para la ecuación de conservación de carga en cada iteración. Este tipo de macro se ejecuta al inicio de cada iteración antes de que se resuelvan las ecuaciones de transporte.

Las UDFs necesitan ser interpretadas o compiladas antes de hacer uso de ellas. Para el sistema operativo Windows, se utiliza Visual Studio para compilar los códigos.



6. RESULTADOS

6.1 Comentarios generales

Antes de analizar los diferentes escenarios propuestos, cabe destacar algunos comentarios sobre el método de refrigeración pasivo: el uso de un minicanal a través del cual fluye el líquido dieléctrico. Este método no se ha analizado numéricamente en las simulaciones, pero se pueden hacer algunos comentarios cualitativos.

El coeficiente de transferencia de calor es una función del número de Nusselt, la conductividad térmica del fluido y el diámetro hidráulico (eq. (5)). Para flujos laminares completamente desarrollados, el número de Nusselt se considera como un valor constante. Para un conducto cuadrado con un flujo de calor uniforme, el número de Nusselt es 3.61 [80]. Por tanto, para un flujo laminar completamente desarrollado, el coeficiente de transferencia de calor depende de la geometría del conducto y las propiedades térmicas del fluido. Para un fluido dado con una conductividad térmica constante, el coeficiente de transferencia de calor aumenta si la longitud característica disminuye (diámetro hidráulico para tuberías no circulares). A su vez, el diámetro hidráulico depende de la geometría del área de la sección transversal del minicanal (4).

Atendiendo ahora a la caída de presión, siguiendo la expresión del factor de fricción para el flujo laminar y conductos cuadrados (35), la caída de presión varía inversamente con el número de Reynolds, que es directamente proporcional al diámetro hidráulico. Por tanto, la caída de presión aumenta cuando se reduce el diámetro hidráulico. Ambas deducciones son:

ℎ ∝1

𝐷ℎ ∆P ∝

1

𝑅𝑒 → ∆P ∝

1

𝐷ℎ

Por lo tanto, es importante ajustar bien el diámetro hidráulico para asegurar una mejora del coeficiente de transferencia de calor sin penalizar en exceso la caída de presión [77].

Se han realizado 12 simulaciones diferentes para analizar el comportamiento del dispositivo electrohidrodinámico y 9 simulaciones para el estudio de la influencia del mallado en la precisión de los resultados. Este resumen recoge los puntos clave. Para un análisis en profundidad, por favor ver el documento completo.

6.2 Simulaciones 2D

6.2.1 Estudio paramétrico 2D: comentarios generales

Es importante recordar el mecanismo del dispositivo: un electrodo de alto voltaje ubicado en la parte superior del minicanal inyecta la carga a través del fluido dieléctrico. Esta carga junto con el voltaje aplicado en el electrodo, crea una fuerza de Coulomb que se añade a la ecuación del momento. En consecuencia, el perfil de velocidad se modifica: las partículas cargadas empujan las moléculas neutras del flujo hacia la placa calentada que está conectada a tierra.

En primer lugar, la carga eléctrica que genera la inyección de carga desde el electrodo, se mueve hacia la placa calentada, quedando cargada eléctricamente. La placa calentada está conectada a tierra y, como está hecha de cobre, un material conductor, la carga se distribuye por toda su superficie. Los conductores eléctricos permiten que las cargas se muevan libremente y se distribuye uniformemente en toda su superficie [74].

Es necesario mencionar algunos conceptos generales sobre los conductores eléctricos. El campo eléctrico dentro de un conductor es nulo. Los electrones se mueven fácilmente dentro de un material conductor y, si aparece un campo eléctrico dentro del conductor, se reorganizarán rápidamente para cancelar este campo eléctrico y alcanzar de nuevo el estado de equilibrio. Esta rápida reorganización está relacionada con la muy baja resistividad de los materiales conductores. La resistividad

Resultados


determina cómo un material resiste el flujo de corriente. Por lo tanto, un valor bajo de esta propiedad garantiza una buena conducción eléctrica. Se puede demostrar que la carga estará en la superficie del conductor gracias a la Ley de Gauss [74] [75]:

∫ 𝑬 ∙ 𝒅𝑺𝐴

= 1

𝜖0∫ 𝜌 𝑑𝑉

𝑉

(40)

Si no hay un campo eléctrico dentro de un conductor, la carga en el interior siempre será cero y, por lo tanto, residirá completamente en su superficie. En nuestro caso, parte de la carga inyectada desde el electrodo es recogida por la placa calentada y permanece en su superficie (Figura 8).

La Figura 9 muestra que el campo eléctrico es perpendicular al cable y la placa calentada. La Ley de Gauss también explica este fenómeno: si aparece un campo eléctrico tangencial cerca de la superficie de un material conductor cargado, las cargas se reorganizarán para alcanzar el estado de equilibrio y cancelar este campo eléctrico tangencial. Por lo tanto, para los materiales conductores, el campo eléctrico es siempre normal a su superficie. Además, en esta figura podemos observar cómo aumenta el campo eléctrico con el tiempo debido a la carga inyectada [74] [75].

(a) (b)

Figura 9 — Componente y del campo eléctrico (Ey): (a) t = 0.005s. (b) t = 20 s. Plano XY, x = 0.070 m. Borde derecho del electrodo y la placa.

La Figura 8 muestra la aparición de cargas inducidas cerca del electrodo. La carga se inyecta gradualmente y parte de ella es absorbida por la placa. Para cumplir con la ecuación de conservación de carga (26), aparecen cargas negativas inducidas cerca del electrodo. Bajo un punto de vista teórico, cuando un fluido dieléctrico está ubicado entre un electrodo de alto voltaje y una superficie conectada a tierra, las moléculas del fluido se polarizan. Estas moléculas polarizadas están alineadas con el campo eléctrico aplicado y todas las moléculas lo suficientemente alejadas del electrodo y la placa se neutralizan entre ellas. Sin embargo, las cargas inducidas cerca del electrodo permanecen, por la carga positiva absorbida por la placa [76].

6.2.2 Estudio paramétrico: escenarios 1.a - 1.d

Escenario 1.a

En primer lugar, es necesario explicar el escenario sin ningún voltaje aplicado. El estado de funcionamiento estable se alcanza después de 22 s y el perfil de velocidad corresponde a un flujo

Figura 8 — Escenario 1.d: Vectores de corriente eléctrica coloreados por la densidad de carga (C/kg) inyectada desde el electrode y absorbida por la placa (t = 10 s). Plano XY, x = 0.070 m. Borde derecho del electrodo y la placa.



laminar completamente desarrollado en 2D. La componente 𝑢 de la velocidad de un flujo laminar tiene un perfil parabólico que alcanza su valor máximo en el punto medio de la tubería. Considerando solo 2 dimensiones, la velocidad 𝑢 es una función de la coordenada 𝑦.

De acuerdo con el perfil de velocidad del flujo laminar, la componente 𝑢 de la velocidad es dominante sobre la componente 𝑣. En estas simulaciones, se utiliza la aproximación de Boussinesq, por lo que la densidad se considera constante para todas las ecuaciones, excepto para el término dónde va multiplicada por la gravedad en la ecuación de momento. Entonces, las variaciones de densidad del fluido debido a los cambios de temperatura influyen en la velocidad 𝑣. Se realiza una explicación en profundidad de este fenómeno para los casos 3D (sección 6.3.2). El fluido caliente con menor densidad tiende a subir y el fluido frío baja.

Número de Prandtl Número de Grashof Número de Rayleigh

396.43 38.16 1.51 x 104

Tabla 6 — Escenario 1.a: Números adimensionales

Los números adimensionales definidos en el capítulo 2.1 se calculan para este escenario. El número de Grashof representa la relación entre las fuerzas de flotabilidad y las fuerzas viscosas, y el número de Rayleigh cuantifica la importancia entre los efectos de las fuerzas de flotabilidad y los efectos de la viscosidad y la conducción térmica. Debido a los valores obtenidos, podemos concluir que hay transferencia de calor por convección. Con respecto al perfil de temperatura, el fluido extrae calor de la placa calentada y lo conduce hacia la salida del conducto.

Escenarios 1.b - 1.d

En primer lugar, la temperatura media de la placa disminuye para todos los escenarios y la temperatura máxima se reduce en el escenario 1.c (4 K). La reducción de la temperatura para todos los escenarios es significativa, siendo casi de 14 K en el escenario 1.c.

(a) (b)

Figura 10 — Escenarios 1.a a 1.d: Temperatura media y máxima de la placa calentada

Analizando la Figura 10, se puede concluir que el mejor comportamiento del dispositivo se obtiene con la configuración 1.c. (10 kV) debido a la disminución de la temperatura media y máxima de la placa calentada.

Se observan oscilaciones en las temperaturas medidas a lo largo de la placa en la Figura 11 (a), (b) y (c). Este hecho se debe al desarrollo del campo eléctrico y la contribución de la fuerza de Coulomb en la ecuación de conservación del momento (27). La interacción entre el momento de flujo y el momento que se aplica mediante la inyección de carga es mayor cuando el voltaje aumenta como muestran estas oscilaciones.

Resultados


(a) (b) (c) Figura 11 — Escenarios 1.a a 1.d: (a) Temperatura medida en el punto T2 (22, -2.5, 0) mm. (b) Temperatura medida en el punto T3 (45, -2.5, 0) mm. (c) Temperatura medida en el punto T4 (68, -2.5, 0) mm.

En primer lugar, se resuelve la ecuación de Poisson para el potencial eléctrico siguiendo la ecuación (21). El campo eléctrico se calcula como el gradiente del potencial (20), y finalmente se resuelve la ecuación de conservación de la densidad de carga (26). Por lo tanto, el campo eléctrico aumenta gradualmente con la inyección de carga y depende de la tensión aplicada en el cable.

Las oscilaciones son apreciables en los tres puntos de medida cuando el valor de la fuerza de Coulomb en la ecuación de momento comienza a ser relevante. El campo eléctrico presente en la salida de la sección de prueba se desarrolla hacia la entrada y el mismo fenómeno, pero con la dirección opuesta ocurre con el campo eléctrico de la entrada de la sección de prueba. Cargas inducidas aparecen cerca del electrodo debido a las cargas positivas presentes en la superficie de la placa. A lo largo del minicanal se observa: cuando el producto de carga y campo eléctrico es positivo, el fluido es empujado en el sentido positivo del eje x (por ejemplo, en la parte inferior de la entrada de la sección de prueba). Cuando este producto es negativo, el fluido es empujado en el sentido negativo del eje x (parte superior, límite derecho de electrodo al final de la sección de prueba).

La carga inyectada comienza a estar presente en el fluido y el flujo se modifica, siguiendo el razonamiento previo. Como resultado de esta interacción, se crean algunas "ondas" a lo largo de la sección de prueba. El fluido empujado por la fuerza de Coulomb hacia la salida del canal (sentido x positivo) se modifica cuando la contribución de la fuerza de Coulomb en la ecuación de momento es negativa. Este fenómeno produce estas oscilaciones en el perfil de velocidad.

(a) (b)

(c) (d)

Figura 12 — Escenario 1.d: Contornos en t = 16 s del minicanal: (a) Componente x del campo eléctrico (V/m). (b) Densidad de carga (C/kg). (c) Temperatura (K). (d) Componente u de la velocidad (m/s). Plano XY, t = 16 s.

Esta modificación del perfil de velocidad es la responsable de las oscilaciones de temperatura de la placa. El flujo con velocidad positiva logra extraer calor de la placa, mientras que el flujo recirculado con velocidad negativa reintroduce fluido caliente hacia la zona de prueba (Figura 12). Este hecho aumenta la temperatura máxima medida en la placa.



Cuando el campo eléctrico está casi totalmente desarrollado y es estable, la carga negativa se acumula en la parte superior del canal y la carga positiva cerca de la placa. El perfil de velocidad de la primera parte de la sección de prueba es más homogéneo y cerca del electrodo y la placa, el fluido es empujado hacia la sección de salida. Las "ondas" del flujo desaparecen.

(a) (b)

(c) (d)

Figura 13 — Escenario 1.d: Contornos en t = 26.5 s del minicanal: (a) Componente x del campo eléctrico (V/m). (b) Densidad de carga (C/kg). (c) Temperatura (K). (d) Componente u de la velocidad (m/s). Plano XY, t = 26.5 s.

Con respecto a la componente 𝑣 de la velocidad, se puede hacer el mismo razonamiento. El campo eléctrico en la dirección 𝑦 siempre es negativo en la sección de prueba, por lo que se observa un valor negativo de la velocidad 𝑣 cuando la carga es positiva, empujando hacia abajo el fluido.

Resumiendo, el campo eléctrico produce algunos problemas de recirculación a lo largo del minicanal que son más notables cuando el voltaje aplicado es mayor. La recirculación en el escenario 1.d es mayor que en el escenario 1.b. Del mismo modo, es importante mencionar el calentamiento generado por el efecto Joule. Un análisis más profundo se realizará en los escenarios 3D.

Análisis de los parámetros

Se calculan los parámetros definidos en el capítulo 5.2. En primer lugar, el número de Nusselt es la relación entre la transferencia de calor por convección y la transferencia de calor por conducción. Para calcularlo, se utiliza la expresión (5) definida en el capítulo 2.1. La relación Nu/Nu0 compara el número de Nusselt obtenido para los 3 escenarios con voltaje aplicado y el escenario sin voltaje. Un valor mayor que la unidad significa que la tasa de transferencia de calor aumenta.

(a) (b) (c)

Figura 14 — Ratio (a) Nu/Nu0 vs. voltaje aplicado (kV) (b) ∆𝑃/∆𝑃0 vs. voltaje aplicado (kV). (c) PEC vs. voltaje aplicado (kV) para los escenarios 1.a a 1.d

El ratio Nu/Nu0 es mayor que la unidad para los 3 escenarios con voltaje aplicado, lo que confirma la mejor refrigeración de la placa. La caída de presión aumenta con la tensión aplicada. Este crecimiento se acentúa cuando el voltaje aplicado es mayor ya que el campo eléctrico también es mayor y, por lo tanto, la fuerza de Coulomb en la conservación de las ecuaciones de momento también es mayor. La caída de presión observada es alta y se debe a la gran interacción entre el momento de flujo principal y el momento aplicado por la inyección de carga.

Resultados


Al analizar un método de refrigeración, es necesario estudiar la mejora de la transferencia de calor junto con el aumento de los requisitos de bombeo. Para ello se calcula el Performance Evaluation Criterion (PEC). El coeficiente de transferencia de calor puede ser mayor, pero si el aumento en los requisitos de bombeo es significativo, la eficiencia de la técnica de enfriamiento puede ser menor. Por lo tanto, los valores mayores que la unidad significan que la tasa de mejora de la transferencia de calor es mayor que la caída de presión necesaria para bombear el fluido.

Los escenarios 1.b y 1.c muestran una mejor eficiencia que el escenario 1.a. La mejora de la transferencia de calor es significativa y el aumento de la caída de presión no es demasiado alto. Sin embargo, en el escenario 1.d, pese a que la transferencia de calor mejora, el aumento de la caída de presión es demasiado alto. Los efectos del campo eléctrico son notables y los problemas de recirculación más altos por lo que la eficiencia es menor que el escenario sin voltaje.

6.3 Simulaciones 3D

6.3.1 Estudio paramétrico 3D: estudio del mallado

En primer lugar, se analizan nueve escenarios diferentes para determinar la importancia de la discretización del dominio en la precisión de una simulación de CFD. Se simulan tres escenarios diferentes (steady state) para tres mallas hexaédricas diferentes.

La geometría utilizada para este estudio previo es:


de medida (mm)

Entrada 20 Inlet_T1 (10, -2.5, 0)

P1 x = 18

Test 50

T2 (22, -2.5, 0)

T3 (50,-2.5,0)

T4 (68,-2.5,0)

Salida 25 Outlet_T2 (85, -2.5, 0)

P2 x = 72 Tabla 7 — Estudio del mallado: geometría y puntos/planos de medida

Los parámetros utilizados se resumen en:

Caso Mallado Tamaño máx. Del elemento (mm)

Elementos totales Calidad

ortogonal Relación de aspecto

máxima

2.1 10 x 10 0.5 19000 1 1.73205

2.2 20 x 20 0.25 152000 1 1.73205

2.3 29 x 29 0.17 470119 0.999942 1.75831 Tabla 8 — Estudio del mallado: parámetros

Para este estudio, las configuraciones utilizadas son las mencionadas en el capítulo 5.5, salvo que no se considera la aproximación de Boussinesq para la densidad.

Se selecciona una malla hexaédrica por diferentes razones. La geometría cuadrada del canal ayuda a discretizar el dominio con elementos hexaédricos frente a otros tipos de mallado. Un mallado hexaédrico garantiza la perfecta alineación de los elementos con el flujo de fluido para reducir y minimizar la difusión numérica. Un buen mallado está relacionado con la física que se desea resolver. Además, el uso de celdas hexaédricas debería reducir el coste computacional durante la simulación y, por lo general, necesita menos elementos para discretizar el dominio. Por lo tanto, debido a los recursos limitados disponibles para realizar estas simulaciones (celdas limitadas en la generación del mallado y tiempo de simulación grande), se elige una malla hexaédrica.

Se utilizan tres entradas de velocidad diferentes para analizar la precisión de la definición de la malla. La temperatura se mide en 5 puntos diferentes a lo largo del eje x y la presión se mide en la entrada y la salida de la sección de prueba. La Figura 15 y la Tabla 9 resumen los resultados obtenidos:



1.1 1.2 1.3

a b c a b c a b c

𝒖 𝒎/𝒔 0.01 0.05 0.1 0.01 0.05 0.1 0.01 0.05 0.1

Re — 1,43 7,13 14,27 1,43 7,13 14,27 1,43 7,13 14,27

𝒇 — 41,45 8,30 4,15 42,61 8,53 4,27 42,82 8,58 4,29

𝒇𝒔 — 39,81 7,96 3,98 39,81 7,96 3,98 39,81 7,96 3,98

Error % 7,03% 7,16% 7,16% 6,57% 6,69% 6,70% 3,94% 4,08% 4,08% Tabla 9 — Estudio del mallado: factores de fricción

(a) (b) (c) Figura 15 — Estudio del mallado: temperaturas medidas en los diferentes puntos definidos en la Tabla 7. (a) Caso a. (b) Caso b. (c) Caso c.

Tras analizar estos gráficos, el mallado correspondiente al caso 2.1 se puede descartar debido a la disparidad de los resultados con respecto a los otros dos casos, en los que las diferencias son bajas. Para determinar qué malla es más precisa, se calcula el factor de fricción (𝑓) y se compara con el factor de fricción calculado con la correlación clásica (𝑓𝑠). Se utilizan las expresiones (35) y (36) presentadas en el capítulo 5.2. Como se esperaba, los errores entre el factor de fricción calculado con la correlación clásica y los obtenidos en las simulaciones del caso 2.3 son menores que los de 2.1 y 2.2, debido a la malla más fina. Por lo tanto, se confirma que la mejor discretización del dominio garantiza la precisión de los resultados obtenidos.

6.3.2 Estudio paramétrico 3D: escenarios 3.a – 3.h

Se realizan ocho simulaciones para dos números de Reynolds diferentes. Se realiza un análisis general, con figuras y explicaciones relativas a los casos 3.a a 3.d. Para un análisis detallado, por favor ver documento completo.

Escenario 3.a y 3.e

Como se explicó antes para los casos 2D, el flujo estudiado en las simulaciones es un flujo laminar completamente desarrollado. La Figura 16 muestra cómo se desarrolla el flujo.

(a) (b)

Figura 16 — Escenario 3.a: (a) Perfil de velocidad de la componente u a lo largo del minicanal. Planos XY en x = 0.004 m, x = 0.029 m, x = 0.054 m, t = 22 s. (b) Perfil de velocidad de la componente v a lo largo del minicanal. Planos XY en x = 0.0165 m, x = 0.029m, x = 0.0415 m, x = 0.054 m and x = 0.064 m, t = 22 s

Debido a la condición de no deslizamiento en las paredes, la componente 𝑢 de la velocidad aumenta a medida que se aleja de las paredes hasta llegar al centro del minicanal, siguiendo un perfil parabólico. La velocidad alcanza su valor máximo en la mitad del conducto, por lo que, la velocidad 𝑢 disminuye para las coordenadas +/- 𝑦 y +/- 𝑧.

Resultados


La aproximación de Boussinesq se utiliza para la densidad. Esta aproximación considera una densidad constante para todas las ecuaciones, excepto para el término dónde va multiplicada por la gravedad en la ecuación de conservación del momento (eje y). A medida que el fluido pasa a través de la sección de prueba en la dirección x positiva (Figura 16 (b)), su temperatura aumenta y la temperatura de la placa disminuye. Debido al perfil de velocidad laminar, a medida que el fluido se aleja del eje vertical en la dirección 𝑧, la velocidad 𝑢 disminuye y los efectos de las fuerzas de flotabilidad son mayores, lo que aumenta la velocidad 𝑣.

Como se explica en el capítulo 2.1, el número de Grashof calcula la relación entre las fuerzas de flotabilidad y las fuerzas viscosas que actúan sobre un fluido. Los números de Grashof para estos escenarios no son especialmente altos, pero destaca el efecto de las fuerzas de flotabilidad sobre las fuerzas viscosas. Por lo tanto, la convección en la transferencia de calor es significativa debido a que el número de Rayleigh es mayor que el número crítico de Rayleigh mencionado en el capítulo 2.1 (1700).

Los números adimensionales presentados en el capítulo 2.1 se calculan para este escenario.

Escenario Re Número de Prandtl Número de Grashof Número de Rayleigh

3.a 1.43 396.43 35.38 1.40 x 104

3.e 7.13 396.43 22.22 8.88 x 103

Tabla 10 — Escenario 3.a: Números adimensionales

Como la velocidad media del fluido es mayor en el escenario 3.e, la velocidad en 𝑣 en todos los planos YZ considerados a lo largo del minicanal es más baja que la observada en el escenario 3.a. Este hecho puede explicarse analizando los números adimensionales calculados. El número de Grashof en este caso es menor que en el escenario 3.a (7), por lo que el efecto de las fuerzas de flotación es menor.

Respecto a la temperatura de la placa, la siguiente figura muestra el perfil de temperatura de la placa después de alcanzar el estado estable. En este experimento, la placa calentada tarda 22 segundos en el caso 3.a y 11 s en el caso 3.e en alcanzar este estado estable.

(a) (b)

Figura 17 — Escenario 3.a: (a) Temperatura de la placa en funcionamiento de estado estable. Vista en planta, plano XZ, t = 22 s. (b) Temperatura a lo largo del minicanal. Planos YZ en x = 0.004m, x = 0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m y x = 0.064m, t = 22 s.

La temperatura del fluido en el centro del conducto es más baja que en las paredes debido al perfil de velocidad laminar. El fluido es reemplazado por nuevo líquido más frío cerca del centro del canal y la temperatura aumenta gradualmente a lo largo del eje 𝑧 con el descenso de la velocidad 𝑢.

Finalmente, la presión a lo largo del canal es como muestra la siguiente imagen:



Figura 18 — Escenario 3.a: Presión a lo largo del minicanal. Planos YZ en x = 0.004m, x = 0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m y x = 0.064m, t = 22 s.

Escenarios 3.b – 3.d

Se analizan los escenarios 3.b a 3.d. Los escenarios 3.f a 3.h tienen un razonamiento análogo.

En primer lugar, como se puede observar en la Figura 20, solo la configuración del caso 3.b disminuye las temperaturas medidas, salvo en la última parte de la placa calentada (T4). La temperatura máxima alcanzada es similar y la temperatura media de la placa es más baja.

(a) (b) Figura 19 — Escenarios 3.a a 3.d: (a) Temperatura media de la placa y (b) máxima temperatura de la placa calentada

(a) (b) (c) Figura 20 — Escenarios 3.a a 3.d: (a) Temperatura medida en el punto T2 (4, -2.5, 0) mm. (b) Temperatura medida en el punto T3 (29, -2.5, 0) mm. (c) Temperatura medida en el punto T4 (54, -2.5, 0) mm.

El campo eléctrico modifica el perfil de velocidad debido a la fuerza de Coulomb introducida en la ecuación de conservación de momento (27). La primera conclusión evidente es que esta modificación del perfil de velocidad aumenta con el voltaje aplicado, ya que el campo eléctrico se calcula como el gradiente del voltaje (20). Por lo tanto, la contribución del campo eléctrico a las ecuaciones del momento en el caso 3.d y 3.h es mayor que la del caso 3.b y 3.f. Con respecto a la componente x del campo eléctrico, el comportamiento es análogo al caso 2D: un valor positivo en el borde derecho del cable (x = 0.054 m) y un valor negativo muy similar en la entrada de la sección de prueba (x = 0.004 m); y los respectivos valores opuestos de campo eléctrico en la placa (positivo en x = 0.004 m y negativo en x = 0.054 m). A medida que la carga se inyecta, el campo eléctrico crece.

Resultados


(a) (b)

Figura 21 — Escenario 3.d: Densidad de carga (C/kg).(a) t = 0.1 s (b) t = 26 s

El campo eléctrico negativo del borde de la placa ubicado en x = 0.054 m crea un efecto no deseado: reintroduce el fluido con alta temperatura hacia la sección de prueba con una velocidad constante a lo largo del área de la sección transversal. Esto produce un aumento localizado de la temperatura al final de la placa calentada. A medida que el fluido se aleja en la coordenada 𝑧, la velocidad 𝑢 disminuye y la recirculación es más perjudicial porque el fluido caliente aumenta la temperatura en las esquinas de la placa calentada. Además, la recirculación aumenta con el voltaje aplicado, y por tanto con el campo eléctrico. Las siguientes figuras ilustran esta explicación.

(a) (b)

Figura 22 —(a) Escenario 3.b: Problemas de recirculación. Plano YZ a la salida de la sección de prueba x = 0.054 m. t = 26 s. (b) Escenario 3.d: modificación del perfil de velocidad en el plano medio del minicanal. Vectores de velocidad coloreados por temperatura. Plano XY, x = 0.070 m. t = 26 s

Visto el perfil de temperatura para los escenarios con voltaje aplicado (Figura 17 (a)), se realiza un análisis por partes de la placa para analizar si el calentamiento provocado por la recirculación es general o simplemente está localizado en la última parte de la placa.

Figura 23 — Escenario 3.b: Temperatura de la placa calentada. Vista en planta, plano XZ, t = 26 s.

La placa se divide en 4 partes:

HP-1: de x = 0.004 m a x = 0.0165 m

HP-2: de x = 0.0165 m a x = 0.029 m

HP-3: de x = 0.029 m a x = 0.0415 m

HP-4: de x = 0.0415 m a x = 0.054 m

Para cada parte de la placa calentada, se calcula el valor de temperatura máximo y promedio.

Escenario Ave. Temp. (K) Max. Temp. (K)

HP-1 HP-2 HP-3 HP-4 HP-1 HP-2 HP-3 HP-4

3.a 323.48 343.34 353.46 359.24 347.88 364.36 377.09 392.11

3.b 322.84 341.26 348.80 358.98 354.09 358.98 363.9 391.76

3.c 323.90 340.57 345.85 365.25 341.66 361.55 363.37 396.5

3.d 325.67 344.33 359.81 376.84 340.38 358.06 381.21 406.67 Tabla 11 — Escenarios 3.a a 3.d: Temperaturas de la placa calentada



Como se puede observar en la Tabla 11, el aumento de la temperatura es localizado al final de la placa analizando las temperaturas obtenidas.

Otra causa del aumento de temperatura de la placa es el calentamiento por efecto Joule (eq (9)). Como se explica en el capítulo 6.2.1, parte de la carga inyectada desde el cable es absorbida por la placa debido al material conductor del que está hecha. Como la placa calentada no se considera como una pared adiabática (introduce el flujo de calor en el dominio), se genera un calor de efecto Joule debido al flujo de corriente. Sin embargo, esta contribución en el aumento de la temperatura es menor que la contribución del fenómeno de recirculación.

La mayor fuente de calentamiento por efecto Joule es el cable. No obstante, al ser considerado como una pared adiabática en el modelo, el fluido no aumenta su temperatura en la parte superior del canal a causa del efecto Joule.

Análisis de los parámetros

Los escenarios 3.b y 3.c tienen una mejora en la transferencia de calor debido al ratio mayor que la unidad Nu/Nu0. El coeficiente de transferencia de calor es más alto que el del escenario sin el voltaje aplicado y la temperatura media final de la placa es más baja que la obtenida en el escenario 3.a (0 kV). Sin embargo, en el escenario 3.d, el coeficiente de transferencia de calor disminuye debido al mayor valor de la temperatura media de la placa y, por lo tanto, el ratio Nu/Nu0 también es menor.

Por otro lado, se observan ratios inferiores a la unidad para los escenarios 3.e a 3.h. Esto es evidente ya que el coeficiente de transferencia de calor calculado para estos casos es más bajo que el que no tiene tensión aplicada (3.e), dada la mayor temperatura media de la placa.

(a) (b) (c)

Figura 24 — Ratio (a) Nu/Nu0 vs. voltaje aplicado (kV) (b) ∆𝑃/∆𝑃0 vs. voltaje aplicado (kV). (c) PEC vs. voltaje aplicado (kV) para los escenarios 3.a a 3.h

Se observa un aumento en la caída de presión. La caída de presión está relacionada con la modificación del perfil de velocidad. A medida que esta modificación es más significativa con el voltaje aplicado, la caída de presión también aumenta con él. La caída de presión es mayor en los escenarios con menor número de Reynolds. El campo eléctrico en estos casos logra tener más influencia en el flujo y su contribución en la conservación de la ecuación del momento es mayor. Además, como se explicó anteriormente, la modificación en el perfil de velocidad es mayor cuando se aplican voltajes más altos, por lo que es evidente que la caída de presión también aumentará con el aumento del voltaje. Al aumentar el número de Reynolds, la influencia que tiene el campo eléctrico en la conservación del momento es menor. Los efectos del campo eléctrico son más notables en los números de Reynolds bajos.

El Performance Evaluation Criterion (PEC) considera el aumento de la potencia de bombeo para evaluar la eficiencia de la técnica de refrigeración. Por lo tanto, los valores mayores que la unidad significan que la caída de presión generada con el uso del método de enfriamiento es menor que la tasa de mejora de la transferencia de calor.

Resultados


Analizando los valores de PEC, ninguno de los escenarios estudiados logra que la mejora de la transferencia de calor sea mayor que la caída de presión generada. Pese a que la tasa de transferencia de calor es mayor en los escenarios 3.b y 3.c, la caída de presión generada es mayor que la mejora de la transferencia de calor observada y no pueden considerarse técnicas de enfriamiento eficientes. Sin embargo, con respecto al análisis segmentado de la placa para los escenarios 3.b a 3.d, podemos concluir que las temperaturas máximas en el HP-1, HP-2 y HP-3 (de x = 0.004 m a x = 0.0415 m) de la placa calentada se reducen y la temperatura media de la placa también se reduce para estas partes en el escenario 3.b y 3.c.

Mediante el análisis de los escenarios del mayor número de Reynolds, podemos concluir que ninguna de las configuraciones logra mejorar el coeficiente de transferencia de calor global, por lo que los valores de PEC obtenidos son más bajos que la unidad. No obstante, respecto al análisis segmentado de la placa realizado, se observa cierta reducción en las temperaturas máximas alcanzadas.

6.4 Puntos débiles del modelo

En primer lugar, se puede hacer una mención especial de la malla del modelo 3D. El mallado 3D tiene una configuración multizona: la zona de prueba tiene elementos más pequeños que la sección de entrada y la sección de salida. Afinar el mallado en la zona donde se produce la mayor modificación del flujo es una buena práctica y una buena estrategia para una simulación CFD. Sin embargo, fijándose en la sección transversal (Figura 7 (a)), la transición entre los elementos más pequeños y los adyacentes más grandes no es progresiva. La relación de expansión es demasiado alta y para obtener mejores resultados y mejorar el mallado del modelo, se debe hacer un mejorar y afinar esta parte.

El tamaño máximo del elemento también se reduce en el mallado del cable, pero se observa el mismo problema en la parte superior del canal: la relación de expansión entre las caras del cable y las caras de la pared de la parte superior del canal es demasiado alta y no progresiva (Figura 7 (b)).

Para mejorar la malla cerca de la placa calentada, se pueden hacer los mismos comentarios y una malla más fina en las capas adyacentes proporcionaría resultados precisos en la zona donde el calor se introduce al dominio.

Las características de la versión académica de ANSYS Fluent son limitadas y no permite crear mallas con más de 512k celdas. Para trabajos adicionales relacionados con estas simulaciones, se debe usar una versión ANSYS Fluent profesional para evitar las limitaciones de la generación de la malla.



7. CONCLUSIONES

En primer lugar, se pueden mencionar algunas conclusiones sobre el análisis cualitativo del uso de un minicanal con un diámetro hidráulico reducido. Por un lado, el coeficiente de transferencia de calor es una función del número de Nusselt, la conductividad térmica del fluido y el diámetro hidráulico. Considerando el flujo laminar completamente desarrollado a través de una tubería, el número de Nusselt se puede considerar como un valor constante (3.61 para conductos cuadrados [80]). Por lo tanto, para un fluido dado, el coeficiente de transferencia de calor es inversamente proporcional al diámetro hidráulico del minicanal. Si se reduce el diámetro hidráulico, aumenta el coeficiente de transferencia de calor. Por otro lado, la caída de presión varía inversamente con el número de Reynolds, como muestra la expresión del factor de fricción (35). Por tanto, la caída de presión aumenta cuando el diámetro hidráulico disminuye.

Respecto a las simulaciones bidimensionales realizadas, la primera conclusión que puede extraerse es que el uso de un aceite mineral como refrigerante garantiza la refrigeración de la placa. Si se aplica un voltaje en el cable de la parte superior, aparece un flujo secundario desde el cable hacia la placa calentada. Este flujo secundario empuja las moléculas neutras del fluido hacia la placa, aumentando el coeficiente de transferencia de calor. La electrostática y la hidrodinámica se acoplan en la conservación de la ecuación del momento y la fuerza de Coulomb, producida por el campo eléctrico y la carga inyectada, modifica el perfil de velocidad. Esta modificación de la velocidad aumenta con la intensidad del campo eléctrico: cuando el voltaje aplicado es mayor, el campo eléctrico calculado también es mayor y, por lo tanto, la modificación del perfil de velocidad es más significativa. Se observa un fenómeno de recirculación al final de la sección de prueba debido a la interacción del campo eléctrico y la carga presente en el fluido. Esta interacción también es responsable de las diferentes modificaciones observadas en el fluido a lo largo de la sección de prueba.

Analizando el ratio del número de Nusselt de los diferentes escenarios, se puede concluir que la tasa de transferencia de calor aumenta con el voltaje aplicado y, por lo tanto, con la intensidad del campo eléctrico (escenarios 1.b a 1.d). Sin embargo, se observa un problema de recirculación debido a la velocidad negativa que crea el flujo secundario. Esta recirculación aumenta con la fuerza del campo eléctrico (su contribución a la ecuación del momento es mayor), por lo que es más significativa en el escenario 1.d. Pese a que se espera que mayores voltajes signifiquen un mayor flujo secundario y mayor coeficiente de transferencia de calor, la recirculación explica que el ratio Nu/Nu0 para el escenario 1.d (15 kV) sea menor que la del escenario 1.b (10 kV).

Por otro lado, la caída de presión aumenta con el voltaje aplicado, ya que la modificación del perfil de velocidad es más significativa. Mediante el análisis del PEC, se puede concluir que la eficiencia de los escenarios 1.b y 1.c (5 kV y 10 kV, respectivamente) es mayor que la unidad y, por lo tanto, la mejora de la transferencia de calor es mayor que la caída de presión. Además, el PEC es mayor para el escenario 1.c que para el 1.b, debido al mayor voltaje aplicado (mayor campo eléctrico). Por lo tanto, estas dos configuraciones garantizan una refrigeración más eficiente de la placa. No obstante, el valor de PEC para el escenario 1.d es más bajo que la unidad ya que, a pesar de la mejora de la transferencia de calor, la caída de presión es muy grande, por lo que no es un método eficiente.

Con el mismo propósito, se simulan algunos casos en 3D. Previamente, se realiza un estudio del mallado con el fin de determinar la influencia de la discretización del dominio en la precisión de los resultados obtenidos. Se estudian tres mallas hexaédricas diferentes con 10x10, 20x20 y 29x29 (límite de células alcanzadas) celdas en el área de la sección transversal. Al analizar los resultados obtenidos (Figura 15 y Tabla 9), se descarta el caso 2.1 debido a la disparidad de los resultados en comparación con los otros dos casos. Los valores del factor de fricción calculados de los escenarios 2.2 y 2.3 son similares y al compararlos con el factor de fricción teórico para flujos laminares en conductos cuadrados, se puede concluir que la malla más fina (caso 2.3) garantiza el error mínimo.

Conclusiones


Se calculan ocho simulaciones en 3D para dos números de Reynolds diferentes para analizar la técnica de refrigeración estudiada. Primero, las fuerzas de flotabilidad son más notables cerca de las paredes debido al perfil de velocidad laminar del flujo. A medida que el fluido se aleja del plano medio del canal hacia las paredes, la velocidad 𝑢 (dominante sobre las otras dos componentes) se reduce y las fuerzas de flotabilidad son más notables. Con respecto a los escenarios con el número de Reynolds más bajo (3.a a 3.d), la temperatura media de la placa calentada se reduce en los escenarios 3.b y 3.c y solo se reduce la temperatura máxima con la configuración 3.b. Se miden valores similares de temperatura en los tres puntos diferentes ubicados a lo largo de la primera parte de la placa calentada, y se observa un aumento de temperatura en la parte final de la placa.

Este hecho se explica por el problema de recirculación observado en la salida de la sección de prueba. La carga se inyecta desde el cable, creando un flujo secundario que empuja el flujo hacia la placa calentada. A medida que la carga se inyecta, la placa calentada se carga positivamente debido al material del que está hecha (cobre). La interacción del valor negativo de la componente x del campo eléctrico del final de la placa y la carga positiva presente en la placa produce una velocidad 𝑢 negativa que reintroduce el fluido caliente hacia la sección de prueba. Esta recirculación es más notable a medida que el fluido se aleja del plano medio en las coordenadas +/- 𝑧, debido al perfil de velocidad laminar. Este fenómeno explica que las temperaturas máximas observadas en la placa sean en sus esquinas, cerca del final de la sección de prueba. El análisis segmentado de la placa (Tabla 11) confirma que este aumento de temperatura es localizado. Esta recirculación aumenta cuando el voltaje aplicado es mayor y, por lo tanto, cuando la intensidad del campo eléctrico es mayor.

Los mismos comentarios sobre la modificación del perfil de velocidad y el flujo secundario se pueden aplicar para los escenarios 3.e a 3.h. Sin embargo, como el número de Reynolds es mayor, el momento aplicado por la inyección de carga tiene menos relevancia que en los escenarios 3.b a 3.d.

Respecto al análisis del ratio Nu/Nu0, ninguno de los escenarios 3.f a 3.h mejora la tasa de transferencia de calor y solo los escenarios 3.b y 3.c la mejoran. La caída de presión aumenta cuando el voltaje aplicado es mayor. De hecho, como el campo eléctrico se calcula como el gradiente de la tensión, mayor campo eléctrico significa mayor modificación del flujo y por tanto, mayor caída de presión. Además, los efectos del campo eléctrico son más notables en los números bajos de Reynolds. Ambos ratios tienen cambios más significativos en los escenarios con el número de Reynolds menor, y la pendiente de las líneas de los ratios es más acentuada en los casos con el número de Reynolds menor (3.b a 3.d).

Además, el análisis de los valores de PEC obtenidos muestra que ninguna de las configuraciones 3D es eficiente. Incluso si los escenarios 3.b y 3.c disminuyen la temperatura media de la placa, la caída de presión es mayor que la mejora de la transferencia de calor y no pueden considerarse técnicas de refrigeración más eficientes. Sin embargo, el análisis segmentado de la placa muestra algunas reducciones de la temperatura máxima para todos los escenarios, salvo en la parte final, donde se produce la recirculación y, por lo tanto, la temperatura aumenta.

En resumen, el efecto del campo eléctrico es más notable cuando el número de Reynolds disminuye. Además, el fenómeno de recirculación observado confirma que los efectos del campo eléctrico aumentan cuando la tensión aplicada es mayor y, por lo tanto, con la fuerza del campo eléctrico. En las zonas no influenciadas por la recirculación, la tasa de transferencia de calor aumenta con el voltaje aplicado debido a las temperaturas más bajas observadas. Además, la caída de presión aumenta con la tensión aplicada. Debido a los problemas de recirculación, ninguna de las configuraciones 3D alcanza un valor de eficiencia superior a la unidad. Por otro lado, la configuración 2D con 5 y 10 kV aplicados puede considerarse como técnicas de refrigeración más eficientes que la configuración 2D de voltaje cero.



8. IMPACTOS

8.1 Impacto social

Actualmente, los componentes electrónicos están presentes en casi todos los dispositivos y equipos del día a día, como televisores, ordenadores o electrodomésticos. El rápido desarrollo de los equipos electrónicos obliga a la sociedad a una permanente actualización y renovación para tener la última tecnología, descartando una gran cantidad de dispositivos. En 2016, la cantidad de basura electrónica aumentó a 44.7 millones de toneladas, y solo en la UE, se espera que los desechos de equipos eléctricos y electrónicos (WEEE) alcancen las 20 toneladas en 2020. Esta basura electrónica, en su mayoría mal gestionada, representa un peligro para la salud y el medio ambiente [43][46].

Además, la sociedad tiende a estar cada vez más conectada. La tendencia hacia las Smart City necesita la comunicación entre los habitantes, la ciudad y todos los dispositivos ubicados dentro de ella. Por tanto, los dispositivos electrónicos inteligentes (IED) desempeñarán un papel importante en nuestra sociedad futura. Para permitir esta interacción, se necesita la implementación de miles de dispositivos electrónicos inteligentes (IED), que deben operar sin interrupción. Es fundamental garantizar las condiciones de trabajo correctas para toda esta tecnología y se deben desarrollar nuevos métodos de refrigeración eficientes. Para garantizar una sociedad segura, saludable y fiable basada en el Internet de las cosas (IOT), la tasa de fallos de todos los componentes que crean esta red debe reducirse casi por completo.

8.2 Impacto económico y medioambiental

El uso de aceite mineral como fluido refrigerante para componentes electrónicos es una técnica cada vez más utilizada ya que supone considerables ahorros de energía. El bajo coste de producción, la amplia disponibilidad y sus buenas propiedades térmicas son factores que justifican su elección como fluido refrigerante por encima de otros líquidos, como por ejemplo el agua [37].

Las principales fuentes de obtención aceites minerales son los combustibles fósiles: petróleo, gas natural y carbón. Actualmente, la contribución a la producción mundial de energía de combustibles fósiles es realmente alta, pero tiende a reducirse en el futuro (del 85% actual al 74% en 2040) [53]. Sin embargo, pese a esta reducción en el uso de combustibles fósiles, que podría suponer un encarecimiento en el precio de obtención de aceites minerales, su uso va a seguir siendo importante, por lo que se puede suponer que el precio de los aceites permanecerá constante en los próximos años.

Además, la extracción de los combustibles fósiles tiene un impacto ambiental considerable e indirectamente, la producción de aceites minerales puede asociarse ligeramente con estos impactos (posibles fugas en la extracción y transporte, daño del subsuelo, alteración del hábitat y de las especies dónde se realizan las perforaciones, entre otros).

Figura 25 — Fuentes de energía [53]

Impactos


Generalmente, el coste de un sistema de refrigeración de alta fiabilidad para un centro de datos puede alcanzar el 50% del coste total. La inversión en la infraestructura necesaria para construir un sistema de refrigeración líquido, normalmente por inmersión, es considerablemente menor en comparación con los sistemas de aire, debido a la reducción del uso de maquinaria especializada como chillers o controladores de temperatura y humedad [54][55]. El aceite elimina aproximadamente 1200 veces más calor por unidad de volumen en comparación con el aire [37].

En la actualidad, alrededor de 500 millones de dólares se invierten en infraestructura de refrigeración en un centro de datos. El aceite tiene un mayor rendimiento y disponibilidad y puede reducir el coste de los sistemas de enfriamiento a la mitad ya que ahorra la energía utilizada para enfriar y bombear el aire. Aproximadamente, un ventilador necesita 0.13W de potencia para mover 1W de calor extraído hacia el circuito de agua refrigerada y la potencia para alimentar el ventilador al 100% es 10.5% más que la necesaria en un sistema de inmersión en aceite [54][55].

En consecuencia, los sistemas de refrigeración por inmersión con aceite mineral se implementan cada vez más.

8.3 Impactos tecnológicos

Una explicación detallada de los efectos negativos del sobrecalentamiento en componentes electrónicos se expone en el capítulo 1.1. Normalmente, los fabricantes recomiendan mantener la temperatura de trabajo por debajo de un valor concreto que puede superarse fácilmente si no se aplica una técnica de refrigeración correcta. Como se explicó anteriormente en el capítulo 2.2, los dispositivos electrónicos siguen la "twice law": la vida útil aumenta al doble cuando la temperatura se reduce a 10ºC [11].

Una de las aplicaciones más importantes en las que se pueden necesitar estas nuevas técnicas de refrigeración es en los centros de datos. Varios ejemplos de sistemas de refrigeración por inmersión se pueden encontrar implementados en diferentes centros de datos en todo el mundo, como la Agencia Nacional de Seguridad (NSA), que ha implementado un sistema de refrigeración por inmersión en su Laboratorio de Física. En Barcelona, las infraestructuras PIC (Port d’Informació Científia) tienen cuatro tanques que enfrían un clúster de cálculo científico.

Se han observado algunas ventajas al utilizar estas nuevas técnicas con aceites minerales en centros de datos. El Power Usage Efficiency (PUE) se reduce a valores de 1.05 en el PIC de Barcelona. El PUE es un estándar de referencia para determinar la eficiencia energética de los centros de datos. Se calcula como la relación entre el consumo eléctrico total de la instalación y el consumo exclusivo de los equipos IT. Valores inferiores a 1.2 significan una alta eficiencia del sistema [78].

Las propiedades térmicas del aceite mineral son mejores que las del aire y el agua. Además, evita las partículas de polvo, garantiza una limpieza adecuada del sistema y reduce el ruido al no usar ventiladores. Esto junto con una temperatura estable en todos los equipos también ayuda a reducir los fallos mecánicos [79].

En consecuencia, los métodos de refrigeración eficientes ayudan a aumentar la durabilidad y la vida útil de los componentes electrónicos. Por tanto, es necesario el desarrollo de estos métodos de refrigeración y tratar de reducir la tasa de desechos electrónicos producidos cada año.



9. PLANIFICACIÓN Y PRESUPUESTO

9.1 Planificación temporal

A continuación se muestra la Estructura de Descomposición del Proyecto (EDP) en la que se detallan los paquetes de trabajo en los que se ha dividido este Trabajo de Fin de Máster (Tabla 12 y Figura 26) Del mismo modo, el diagrama de Gantt asociado se muestra en la Figura 27.

Nombre de la tarea Fecha de inicio Fecha de fin

ACTIVIDADES INICIALES 1/10/18 26/10/18

Definición del proyecto 1/10/18 19/10/18

Elección del dispositivo 8/10/18 19/10/18

Planificación del proyecto 22/10/18 26/10/18

APRENDIZAJE 29/10/18 01/02/19

Documentación técnica 29/10/18 20/12/18

Software 19/11/18 01/02/19

DISEÑO DE LA CONFIGURACIÓN 04/02/19 29/03/19

Diseño del modelo 04/02/19 15/02/19

Diseño del setup 18/02/19 29/03/19

Datos de entrada y definición de parámetros de análisis 13/03/19 29/03/19

SIMULACIÓN 01/04/19 24/05/19

2D 01/04/19 26/04/19

3D 01/04/19 24/05/19

CONCLUSIONES 27/05/19 31/05/19

ENTREGA Y PRESENTACIÓN 15/04/19 23/07/19

Definición estructura del documento 15/04/19 17/04/19

Redacción del documento EPL 15/04/19 07/06/19

Entrega del documento EPL 10/06/19 10/06/19

Preparación de la defensa EPL 11/06/19 24/06/19

Defensa EPL 25/06/19 25/06/19

Redacción del documento UPM 12/06/19 28/06/19

Entrega del documento UPM 01/07/19 01/07/19

Preparación de la defensa UPM 02/07/19 19/07/19

Defensa UPM 23/07/19 23/07/19

Tabla 12 — Distribución temporal de paquetes de trabajo

Planificación y presupuesto


Figura 26 — Estructura de Descomposición del Proyecto (EDP)



Figura 27 — Diagrama de Gantt

Planificación y presupuesto


9.2 Presupuesto

El siguiente apartado recoge el presupuesto estimado para este Trabajo de Fin de Máster. Se analizan tanto los costes directos (costes materiales y coste de mano de obra) como los costes indirectos (material fungible y gastos generales).

Para el cálculo del coste de mano de obra, se ha utilizado el diagrama de Gantt para calcular los días trabajados. Del total de 212 días de duración de proyecto, se han supuesto 150 días de trabajo útil con 5 horas diarias de media para el alumno. Las horas de dirección se han estimado en función del número de reuniones y consultas realizadas con los distintos tutores.

Para llevar a cabo las simulaciones requeridas en este documento se han empleado 3 ordenadores:

Ordenador tipo 1: Intel® Core™ i7-4500U, 2.4Ghz, 4 Gb RAM para las simulaciones 2D

Ordenador tipo 2: 2 x Intel® Core™ i5-6500, 3.20 GHz, 8Gb RAM para las simulaciones 3D

También se han utilizado dos discos duros HDD de 1 y 2 TB, y material periférico como un ratón y un iPad a modo de pantalla adicional de apoyo. Se consideran 5 años de amortización para los equipos informáticos. Las licencias de ANSYS Fluent son gratuitas al haber utilizado la versión estudiante.

En cuanto a los gastos indirectos, se considera el material fungible (impresión, encuadernación) y los gastos generales. Estos gastos generales incluyen el consumo de luz, gas, etc. durante la realización del proyecto y se calcularán como el 10% de los costes directos.

Se considera también un beneficio industrial del 6% sobre los costes directos e indirectos y un IVA aplicable del 21%.

COSTE MATERIAL (CD)

Unidades Precio unitario

(€) Uso (meses)

Amortización (años)

Total (€)

Ordenador tipo 1 1 800,00 9 5 120,00

Ordenador tipo 2 2 1.000,00 3 5 100,00

iPad 1 700,00 9 5 105,00

Ratón 1 20,00 9 5 3,00

Discos duros HDD 2 80,00 4 5 10,68

Software 3 — 9 — —

TOTAL (€) 338,68

COSTE MANO DE OBRA (CD)

Horas Precio/hora (€) Total (€)

Tutor: Ingeniero con doctorado (EPL) 45 60,00 2.700,00

Tutor: Ingeniero con doctorado (UPM) 20 60,00 1.200,00

Co-tutor: Ingeniero con doctorado (EPL) 25 60,00 1.500,00

Alumno: Ingeniero Industrial 750 20,00 15.000,00

TOTAL (€) 20.400,00

TOTAL COSTE DIRECTO (€) 20.738,68

GASTOS GENERALES (CI) 10 % sobre CD 2.073,88

BENEFICIO INDUSTRIAL 6 % sobre CD + CI 1.368,75

MATERIAL FUNGIBLE Impresión + encuadernación 200,00

SUBTOTAL PRESUPUESTO (€) 24.381,31

IVA APLICABLE 21 % 5.120,08

TOTAL PRESUPUESTO (€) 29.501,39

Tabla 13 — Presupuesto del proyecto

École polytechnique de Louvain

Cooling electronic components using electrohydrodynamically induced convection

Authors : Javier SALGADO GONZÁLEZ Supervisors : Valérie GELBGRAS, Miltiadis V. PAPALEXANDRIS Readers : Philippe CHATELAIN Academic year 2018–2019 Master [120] in Mechanical Engineering


Javier Salgado González i

ABSTRACT

The rapid growth of the development of electronic components makes them more and more

powerful and smaller. Despite being very reliable components due to the absence of moving parts,

its reliability and lifespan can be reduced owing to failures caused by overheating. Thus, new efficient

cooling techniques should be studied to guarantee the correct operation and lifespan of this type of

components.

An electrohydrodynamic system is studied in the following Master Thesis based on the experiment of

Moghanlou, F. S. et al. (2014): Experimental study on electrohydrodynamically induced heat transfer

enhancement in a minichannel. A CFD analysis is performed in order to determine the effect of

electric field on heat transfer enhancement and pressure drop for different Reynolds numbers for

laminar flow through a square minichannel. The device is composed of three different parts: inlet,

outlet and test section. The fluid passes through the inlet section in order to become

hydrodynamically fully developed. The test section is composed of a copper wire located at the top

simulating the high voltage electrode, and a heated plate located at the bottom as the electronic

component to be cooled down. Besides, the bottom of the test section (the heated plate) is

considered as ground. The walls of the minichannel are thermally and electrically insulated. The high

voltage electrode injects charge through the fluid, producing a secondary flow towards the ground

(heated plate). The neutral molecules of the fluid are pushed by this secondary flow, thus the

velocity profile of the flow is modified.

Some different numerical simulations with the ANSYS Fluent software are performed in order to

study the electrohydrodynamic physics and determine the effects of the injection of charge in the

flow. Some different parameters are studied with the purpose of analysing the heat transfer

enhancement and the pressure drop and explain the behaviour of the electrodynamic device.

By analysing the results, it can be concluded that the effects of the electric field increase when the

Reynolds number decreases. In fact, for the scenarios with the lower Reynolds number, the

modification of the fluid is more significant. A recirculation phenomenon is observed at the outlet of

the test section. The electric field contributes negatively to the momentum equation at this part of

the minichannel, reintroducing fluid towards the test section and no contributing to the heat transfer

enhancement. An increase of pressure drop is observed with the appliance of the voltage and this

augmentation rises with the voltage applied, and thus with the strength of the electric field.

PEC values performed shows that the 2D configuration is efficient for scenarios with 5 kV and 10 kV

applied at the wire. PEC values increase with the voltage. The recirculation problem becomes more

noticeable when the voltage applied is greater and thus, with the strength of the electric field.

Regarding the 3D model, none of the scenarios shows a PEC value greater than unity so the overall

efficiency of the cooling technique studied is lower. However, by performing a segmented

temperature analysis of the heated plate, the temperature increase due to the recirculation

phenomenon is localised at the end part of the plate, and the maximum temperatures reached for

ii

the first part of the plate are lower. This temperature reduction is perfectly noticed in scenarios with

the lower Reynolds number and less visible in the scenarios with the higher Reynolds number.


Javier Salgado González iii

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor Mr. Papalexandris and Mrs. Gelbgras for the guidance in the

realisation of this work. Their tips and advice are been fundamental to accomplish the objectives

defined. I would also thanks Mr. Chatelain to be the third jury member of the committee and to read

this Master Thesis.

In addition, I would like to acknowledge my family and my girlfriend for all the support offered during

these months, and during all the degree and master studies.

Finally, I would like to thank my colleagues for the comfortable working environment in which I have

developed this Master Thesis.

Javier Salgado González


Javier Salgado González v

CONTENTS

Abstract .................................................................................................................................................... i

Acknowledgements ................................................................................................................................. iii

List of figures ........................................................................................................................................... ix

List of tables .......................................................................................................................................... xiii

List of acronyms ..................................................................................................................................... xv

1. Introduction ..................................................................................................................................... 1

1.1 Context and motivation ........................................................................................................... 1

1.2 Objectives ................................................................................................................................ 2

1.3 Thesis structure ....................................................................................................................... 2

2. State of the art ................................................................................................................................ 5

2.1 General heat transfer concepts ............................................................................................... 5

2.2 Introduction and description of electronics components: overheating problems ................. 8

2.3 Current electronic components cooling methods ................................................................ 12

2.4 Applications related to Electrohydrodynamics: Meso/micropumps: EHD pumps ............... 16

2.4.1 Mechanical micropumps ............................................................................................... 16

2.4.2 Non mechanical or dynamic micropumps ..................................................................... 16

2.4.3 EHD micropumps ........................................................................................................... 17

2.5 Dielectric fluids for cooling: mineral oils ............................................................................... 19

2.6 Social impacts ........................................................................................................................ 21

2.7 Economic impacts .................................................................................................................. 22

2.8 Technological impacts ........................................................................................................... 23

3. Governing equations ..................................................................................................................... 25

3.1 Hydrodynamics ...................................................................................................................... 25

3.2 Electrostatics ......................................................................................................................... 26

3.3 Electrohydrodynamics ........................................................................................................... 28

4. Simulation software: ANSYS Fluent ............................................................................................... 29

4.1 ANSYS Fluent ......................................................................................................................... 29

vi

4.2 Models ................................................................................................................................... 29

4.3 User-Defined Scalar transport equations and User-Defined Functions ................................ 30

4.4 Discretization of the domain: mesh generation .................................................................... 30

4.5 Pressure-Based Solver. Coupled Algorithm ........................................................................... 31

4.6 Convergence criteria ............................................................................................................. 32

4.7 Boundary conditions ............................................................................................................. 32

4.8 Computational resources ...................................................................................................... 33

5. Methodology ................................................................................................................................. 35

5.1 General overview .................................................................................................................. 35

5.2 Analysis parameters .............................................................................................................. 35

5.3 Geometry ............................................................................................................................... 36

5.3.1 2D simulations ............................................................................................................... 36

5.3.2 3D simulations ............................................................................................................... 37

5.4 Meshing ................................................................................................................................. 38

5.4.1 Mesh parameters for 2D simulations ............................................................................ 38

5.4.2 Mesh parameters for 3D simulations ............................................................................ 39

5.5 Solver Settings ....................................................................................................................... 40

5.6 Assumptions .......................................................................................................................... 41

5.7 Input data .............................................................................................................................. 43

5.7.1 2D simulations: scenarios .............................................................................................. 43

5.7.2 3D simulations: scenarios .............................................................................................. 43

5.8 UDF and UDS implementation .............................................................................................. 43

5.9 Limitations ............................................................................................................................. 45

6. Results and discussion ................................................................................................................... 47

6.1 Parametric study: general comments ................................................................................... 47

6.2 2D simulations ....................................................................................................................... 48

6.2.1 2D Parametric study: General comments ..................................................................... 48

6.2.2 2D Parametric study: scenarios 1.a to 1.d..................................................................... 50


Javier Salgado González vii

6.3 3D simulations ....................................................................................................................... 58

6.3.1 3D Parametric study: General comments ..................................................................... 58

6.3.2 3D Parametric study: Mesh study ................................................................................. 58

6.3.3 3D Parametric study: scenarios 3.a to 3.d..................................................................... 61

6.3.4 3D Parametric study: scenarios 3.e to 3.h .................................................................... 71

6.4 Weak points ........................................................................................................................... 79

7. Conclusions .................................................................................................................................... 81

8. Bibliography ................................................................................................................................... 85


Javier Salgado González ix

LIST OF FIGURES

Figure 1 —Conduction heat transfer [26] ............................................................................................... 5

Figure 2 — Natural and forced convection [9] ........................................................................................ 6

Figure 3 — Relation between ambient temperature of electrolytic condenser and life [11] ................ 9

Figure 4 — Relation between ambient temperature of semiconductor and failure rate [11] ............... 9

Figure 5 — Example of chip carrier [20] ................................................................................................ 10

Figure 6 — Common PCB [22] ............................................................................................................... 10

Figure 7 — Cooling load equal to power consumption [17] ................................................................. 11

Figure 8 — Temperature change of an electronic component with time [17] ..................................... 12

Figure 9 — Heat fluxes that can be removed at specified temperature with certain heat transfer

mechanisms [17] ................................................................................................................................... 13

Figure 10 — Heat extraction of a chip carrier [17] ................................................................................ 13

Figure 11 — Natural convection mechanism [17] ................................................................................. 14

Figure 12 — Indirect liquid cooling system [14] .................................................................................... 15

Figure 13 — Classification of meso/micro pumps [27] ......................................................................... 16

Figure 14 — Example: Debiotech’s Insulin Nanopump [57] ................................................................. 16

Figure 15 — Injection pump configuration [36] .................................................................................... 17

Figure 16 — Travelling wave induction pump configuration [24] ......................................................... 18

Figure 17 — Induction pump configuration [27] ................................................................................... 18

Figure 18 — Heterocharge layers and different types of electrode configuration [27] ....................... 19

Figure 19 — Hydrocarbons in the mineral oil [37] ................................................................................ 21

Figure 20 — Energy sources [53] ........................................................................................................... 22

Figure 21 — Secondary flow velocity profile [71] ................................................................................. 27

Figure 22 — Overview of Pressure-Based Solution Methods [66] ........................................................ 32

Figure 23 — Model 2D: general view: Plane XY .................................................................................... 36

Figure 24 — Model 3D: general view and bottom view ....................................................................... 37

Figure 25 — Model 3D: mesh-cross sectional view .............................................................................. 39

Figure 26 — Mesh: test zone ................................................................................................................ 40

x

Figure 27 — Scenario 1.d: Electric current vectors coloured by charge density (C/kg) injected from the

wire and collected by the heated plate (t = 10 s). Plane XY, x = 0.070m. Wire and heated plate right

edge. ...................................................................................................................................................... 48

Figure 28 — y-component of electric field (Ey): (a) t = 0.005s. (b) t = 10 s. (c) t = 20 s. (d) t = 26 s.

Plane XY, x = 0.070 m. Wire and heated plate right edge. .................................................................... 49

Figure 29 — Scenario 1.d: Induced charges near the electrode. Plane XY, x = 0.070 m, t = 26 s ......... 50

Figure 30 — Scenario 1.a: u-velocity profile at the middle plane. Plane XY, t = 22 s ............................ 50

Figure 31 — Scenario 1.a: v-velocity profile at the middle plane. Plane XY, t = 5 s .............................. 51

Figure 32 — Scenario 1.a: Temperature contour at the middle plane (XY plane). Plane XY, t = 22 s. .. 51

Figure 33 — Scenario 1.a: Pressure contour at the middle plane. Plane XY, t = 22 s. .......................... 51

Figure 34 — Scenarios 1.a to 1.d: Average temperature and maximum temperature of the heated

plate ....................................................................................................................................................... 52

Figure 35 — Scenarios 1.a to 1.d: (a) Temperature measured at point T2 (22, -2.5, 0) mm. (b)

Temperature measured at point T3 (45, -2.5, 0) mm. (c) Temperature measured at point T4 (68, -2.5,

0) mm. ................................................................................................................................................... 53

Figure 36 — Scenario 1.d: x- component of electric field (V/m) along the minichannel. Plane XY ...... 53

Figure 37 — Scenario 1.d: Different contours at t = 16 s along de minichannel: (a) x- component of

electric field (V/m). (b) Charge density (C/kg). (c) Temperature (K). (d) u-velocity component (m/s).

Plane XY, t = 16 s. .................................................................................................................................. 54

Figure 38 — Scenario 1.d: Different contours at t = 26.5 s along de minichannel: (a) x- component of

electric field (V/m). (b) Charge density (C/kg). (c) Temperature (K). (d) u-velocity component (m/s).

Plane XY, t = 26.5 s. ............................................................................................................................... 55

Figure 39 — Scenario 1.d: Different contours at t = 16 s along de minichannel: (a) y- component of

electric field (V/m). (b) v-velocity component (m/s). Plane XY, t = 16 s. ......................................... 55

Figure 40 — Ratio of Nu/Nu0 vs. applied voltage (kV) for scenarios 1.a to 1.d .................................... 56

Figure 41 — Ratio of ∆𝑃/∆𝑃0 vs. applied voltage (kV) for scenarios 1.a to 1.d .................................... 57

Figure 42 — PEC vs. applied voltage (kV) for scenarios 1.a to 1.d ........................................................ 57

Figure 43 — Mesh study: Details of different mesh studied at the outlet section: (a) Case 2.1. (b) Case

2.2. (c) Case 2.3 ..................................................................................................................................... 58

Figure 44 — Mesh study: temperatures measured at different point locations defined in Table 11. (a)

Case a. (b) Case b. (c) Case c. ................................................................................................................ 60

Figure 45 — Scenario 3.a: Velocity u profile along the minichannel. XY planes at x = 0.004 m, x = 0.029

m, x = 0.054 m, t = 22 s.......................................................................................................................... 61


Javier Salgado González xi

Figure 46 — Scenario 3.a: v-velocity profile along the minichannel. XY planes at x = 0.0165 m, x =

0.029m, x = 0.0415 m, x = 0.054 m and x = 0.064 m, t = 22 s. .............................................................. 61

Figure 47 — Scenario 3.a: v_velocity profile along the minichannel. (a) XY plane, x = 0.0165 m, (b) XY

plane, x = 0.029 m, (c) XY plane, x = 0.0415 m, (d) XY plane, x = 0.054 m, (e) XY plane, x = 0.064 m, (f)

XY plane, x = 0.074 m, t = 22 s. .............................................................................................................. 62

Figure 48 — Scenario 3.a: Temperature contour of the heated plate reached the steady state. Top

view, plane XZ, t = 21 s. ......................................................................................................................... 63

Figure 49 — Scenario 3.a: Temperature contour along the minichannel. YZ planes at x = 0.004m, x =

0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m and x = 0.064m, t = 22 s. ........................................ 64

Figure 50 — Scenario 3.a: Pressure contour along the minichannel. YZ planes at x = 0.004m, x =

0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m and x = 0.064m, t = 22 s. ........................................ 64

Figure 51 — Scenarios 3.a to 3.d: (a) Average temperature of the heated plate and (b) maximum

temperature of the heated plate .......................................................................................................... 65

Figure 52 — Scenarios 3.a to 3.d: (a) Temperature measured at point T2 (4, -2.5, 0) mm. (b)


0) mm. ................................................................................................................................................... 65

Figure 53 — Scenario 3.d: Electric field x - component. t = 0.1 s .......................................................... 66

Figure 54 — Scenario 3.d: Electric field y - component. t = 0.1 s .......................................................... 66

Figure 55 — Scenario 3.d: Electric field z - component. t = 0.1 s .......................................................... 67

Figure 56 — Scenario 3.d: Charge density. t = 0.1 s .............................................................................. 67

Figure 57 — Scenario 3.d: Charge density. t = 26 s ............................................................................... 68

Figure 58 — Scenario 3.b: Recirculation problems. YZ plane at the outlet of the test section x = 0.054

m. t = 26 s .............................................................................................................................................. 68

Figure 59 — Scenario 3.d: Recirculation problems. YZ plane at the outlet of the test section x = 0.054

m. t = 26 s .............................................................................................................................................. 68

Figure 60 — Scenario 3.d: velocity modification at the middle plane of the minichannel. Velocity

vectors coloured by temperature Plane XY, x = 0.070 m. t = 26 s ........................................................ 69

Figure 61 — Scenario 3.b: u_velocity modification at the middle plane of the minichannel. Plane XY, t

= 26 s ..................................................................................................................................................... 69

Figure 62 — Scenario 3.b: Temperature contour of the heated plate. Top view, plane XZ, t = 26 s. ... 70

Figure 63 — Scenario 3.b: Joule Heat source at the heated plate. Plane XY, t = 26 s ........................... 71

Figure 64 — Scenario 3.b: Joule heat source at the middle plane. XY planes at x = 0.004 m, x = 0.029

m, x = 0.054 m, t = 26 s.......................................................................................................................... 71

xii

Figure 65 — Scenario 3.e: v_velocity profile along the minichannel. XY planes at x = 0.0165 m, x =

0.029m, x = 0.0415 m, x = 0.054m and x = 0.064 m, t = 11 s. ............................................................... 72

Figure 66 — Scenario 3.e: Temperature contour along the minichannel. YZ planes at x = 0.004m, x =

0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m and x = 0.064m, t = 13 s. ........................................ 72

Figure 67 — Scenario 3.e: Pressure contour along the minichannel. YZ planes at x = 0.004m, x =

0.0165 m, x = 0.029 m, x = 0.0415 m, x = 0.054 m and x = 0.064m ...................................................... 73

Figure 68 — Scenarios 3.e to 3.h: Average temperature of the heated plate and maximum

temperature of the heated plate .......................................................................................................... 73

Figure 69 — Scenarios 3.e to 3.h: (a) Temperature measured at point T2 (4, -2.5, 0) mm. (b)


0) mm. ................................................................................................................................................... 74

Figure 70 — Scenario 3.h: u_velocity modification at the outlet of the test section. x = 0.054 m, t =

19.5s ...................................................................................................................................................... 74

Figure 71 — Scenario 3.h: u_velocity modification at the middle plane of the minichannel. Plane XY, t

= 19.5s ................................................................................................................................................... 75

Figure 72 — Scenario 3.h: v_velocity modification at the middle plane of the minichannel. Vectors

coloured by temperature Plane XY, t = 19.5s ........................................................................................ 75

Figure 73 — Scenario 3.f to 3.h: v_velocity and charge density at the middle plane of the

minichannel: (a) Scenario 3.f (5 kV). (b) Scenario 3.g (10 kV). (c) Scenario 3.h (15 kV). Plane YZ, t =

19.5s ...................................................................................................................................................... 76

Figure 74 — Scenario 3.h: Temperature and charge density at the heated plate. Plane XY, t = 19.5s 77

Figure 75 — Ratio of Nu/Nu0 vs. applied voltage (kV) for scenarios 3.a to 3.h .................................... 78

Figure 76 — Ratio of ∆𝑃/∆𝑃0 vs. applied voltage (kV) for scenarios 3.a to 3.h .................................... 78

Figure 77 — PEC vs. applied voltage (kV) for scenarios 3.a to 3.h ........................................................ 79


Javier Salgado González xiii

LIST OF TABLES

Table 1 — Typical values of the convective heat transfer [10] ............................................................... 6

Table 2 — Model 2D: geometry and measurement points................................................................... 37

Table 3 — Geometry and measurement planes/points ........................................................................ 38

Table 4 — Thermophysical properties of oil at 293.15K [23] ................................................................ 42

Table 5 — Thermophysical properties of copper .................................................................................. 42

Table 6 — Thermophysical properties of wood .................................................................................... 42

Table 7 — Boundary conditions ............................................................................................................ 42

Table 8 — 2D: Simulation scenarios ...................................................................................................... 43

Table 9 — 3D: Simulation scenarios ...................................................................................................... 43

Table 10 — Scenario 1.a: Dimensionless numbers ............................................................................... 51

Table 11 — Mesh study: geometry and measurement planes/points ................................................. 58

Table 12 — Mesh study: parameters .................................................................................................... 58

Table 13 — Mesh study: friction factors ............................................................................................... 59

Table 14 — Scenario 3.a: Dimensionless numbers ............................................................................... 63

Table 15 — Scenarios 3.a to 3.c: Temperatures of the heated plate .................................................... 70

Table 16 — Scenario 3.e: Dimensionless numbers ............................................................................... 71

Table 17 — Scenarios 3.f to 3.h: Temperatures of the heated plate .................................................... 76


Javier Salgado González xv

LIST OF ACRONYMS

∇ ∙ Divergence ∇ ∙ 𝐯 = 𝜕𝑣

𝜕𝑥+

𝜕𝑣

𝜕𝑦+

𝜕𝑣

𝜕𝑧

∇ Gradient ∇v = (𝜕𝑣

𝜕𝑥,

𝜕𝑣

𝜕𝑦,

𝜕𝑣

𝜕𝑧)

∇2 Laplacian ∇2v =𝜕2𝑣

𝜕𝑥2 +𝜕2𝑣

𝜕𝑦2 +𝜕2𝑣

𝜕𝑧2

𝐴 Heat transfer area (𝑚2)

𝑐𝑝 Heat capacity (𝐽/𝐾𝑔 𝐾)

𝐷ℎ Hydraulic diameter (𝑚)

𝑃 (𝑚) Wetted perimeter (𝑚)

𝐸 Electric field (𝑉/𝑚)

𝑓 Friction factor (−)

𝑓𝑠 Darcy friction factor (−)

𝐹𝑒 Electrical body force (𝑁/𝑚3)

ℎ Convection heat transfer coefficient (𝑊/𝑚2𝐾)

𝐿 Characteristic length (𝑚)

�̇� Mass flow rate (𝐾𝑔/𝑠)

𝑅𝑒 Reynolds number (−)

𝑇 Temperature (𝐾)

𝑇𝑠 Surface temperature (𝐾)

𝑇𝑓 Fluid temperature (𝐾)

𝑇𝑤 Surface average temperature (𝐾)

𝑇𝑏 Bulk temperature (𝐾)

�̇� Heat transfer rate (𝑊)

𝑞" Heat flux (𝑊/𝑚2)

𝜌 Density (𝐾𝑔/𝑚3)

𝑣𝑠 Characteristic velocity (𝑚/𝑠)

𝑢 Velocity in x-direction (𝑚/𝑠)

𝑣 Velocity in y-direction (𝑚/𝑠)

𝑤 Velocity in z-direction (𝑚/𝑠)

𝑔 Gravity acceleration (𝑚/𝑠2)

∆𝑃 Pressure drop (𝑃𝑎)

xvi

𝛼 Thermal diffusivity (𝑚2/𝑠)

𝑘 Thermal conductivity (𝑊/𝑚𝐾)

𝐾 Electrical conductivity (𝑆/𝑚)

𝜀 Electrical permittivity (𝐹/𝑚)

𝜂 Performance Evaluation Criterion (PEC) (−)

𝜇 Dynamic viscosity (𝑃𝑎 𝑠)

𝜌 Density (𝐾𝑔/𝑚3)

𝜌𝑒 Space charge density (𝐶/𝑚3)

𝑧 Charge density (𝐶/𝑘𝑔)

𝑃𝑟 Prandtl number (−)

𝐺𝑟 Grashof number (−)

𝑅𝑒 Reynolds number (−)

𝑅𝑎 Rayleigh number (−)

𝑃𝑜 Poiseuille number (−)

𝑆𝑡 Staton number (−)

𝑬 Electric field (𝑉/𝑚)

𝐸𝑥 Electric field in x-direction (𝑉/𝑚)

𝐸𝑦 Electric field in y-direction (𝑉/𝑚)

𝐸𝑧 Electric field in z-direction (𝑉/𝑚)

𝑱 Current density (𝐴/𝑚2)

𝜙 Electric potential (𝑉)

𝜙𝑘 User Defined Scalar

𝑆𝜙𝑘 Source term of a User Defined Scalar

Γ𝑘 Tensor diffusion coefficient (𝑚2/𝑠)

CFD Computational Fluid Dynamics

PCB Printed Circuit Board

EHD Electrohydrodynamic

𝑊�̇� Electric power consumption (𝑊)

𝑅 Electrical resistance (Ω)

𝐼 Electric current (𝐴)



1. INTRODUCTION

1.1 Context and motivation

Aviation, automotive or medical and healthcare industry are only some examples of the wide range

of electronic components applications in our society. Some devices can produce a relevant amount

of heat during their normal operation and must be cooled down to guarantee their working life. In

addition, the current trend is to develop smaller and more powerful electronics components, which

means an increase in heat generation within the system. This overheating can potentially produce

negative consequences and reduce the working life and the reliability of the electronic components.

Consumer devices such as laptops and smartphones tend to suffer overheating due to the reduction

of their physical dimensions. The electronics components are placed in small areas that do not

contribute to heat extraction [1]. A good practice is to design the electronic components with a large

area and thicker physical dimension to increase the heat extraction efficiency [2]. Other problems

related to the design are the risks of faulty contact or contact wiring. The possible sparks created

could generate fire [3].

Environmental factors or ambient working conditions have also an important role. Hot ambient

temperature contributes directly to overheating. In addition, cyclic or big temperature differences

can induce different stresses to the material components and produce a certain level of damage.

Humidity or different composition of the working atmosphere can erode metal components and

contribute to the deterioration of the devices. This material degradation results in overheating.

The overheating of a component affects not only the component, but it can also affect other

components of the system and produce important failures. Material degradation, cracks,

combustions and even explosions are the consequences of poor heat extraction. So, efficient cooling

methods should be developed to maintain a correct working temperature and guarantee electronic

devices safety and reliability [4].

The rapid development of electronic devices requires the research of new cooling methods and the

enhancement of the heat transfer techniques in order to remove a high heat flux in a limited space

and reduce the working temperature of these components. These new methods could be classified

into two main groups: active and passive methods, which main difference lies in the application or

not of external energy. Electric field, magnetic field or a vibrating wall are used in active methods,

while extended or rough surfaces, mini and microchannels or the dispersion of nanoparticles in the

fluid are the passive techniques. A great variety of micropumps have been developed to pump fluids

through minichannels [23][23].

In conclusion, new advanced cooling techniques are in development to meet the thermal

management demands. Spray, cryogenic or microchannels cooling are some examples. An

electrohydrodynamic system that pumps dielectric liquid through a microchannel is a new cooling

technique studied of this document.

INTRODUCTION

2

1.2 Objectives

The goal of this Master Thesis is to evaluate the heat transfer enhancement of an

electrohydrodynamic system to cooling electronics components, by pumping a dielectric fluid

through a microchannel for laminar flow. The simulations are based on the experiment performed by

Moghanlou, F. S. et al. [23]. The heat transfer enhancement is studied by the analysis of the effect of

different parameters. Consequently, the main steps which will be followed to reach the objectives

are:

To perform a study of current cooling methods of electronic components.

To conduct a depth study of the mechanism of an electrohydrodynamic system and their

theoretical principles.

To perform an optimal model in 2D and 3D in ANSYS Fluent software for the required

analysis and to simulate different scenarios.

To evaluate different parameters in electronics components cooling efficiency.

1.3 Thesis structure

This thesis is arranged in 7 chapters.

Chapter 2 presents the state of the art. It presents an introduction to the general heat transfer

concepts and a short description of different electronics components. After that, it includes a further

explanation of the different electronic cooling techniques used currently and an explanation of the

use of dielectric fluids for this application. In addition, an overall explanation of the EHD micropumps

is presented, which are based on the same theoretical principles that the device studied. It concludes

with an explanation of the different impacts that these new cooling techniques involve.

Chapter 3 is focused on the theoretical and physical concepts. Both, hydrodynamic and electrical

main concepts are explained to understand the EHD phenomena. The electrohydrodynamics

governing equations needed to explain the physics of the device are presented and detailed.

The following chapter 4 presents the software used for the Computational Fluid Dynamics: ANSYS

Fluent. It introduces a detailed explanation of the solver and different models used, the relevance of

the discretization of the domain and a review of the different boundary conditions and convergence

criteria. It is also mentioned the limitation that we have found in order to accomplish the work

proposed.

Chapter 5 presents the methodology followed in this thesis. First, a description of the device is done

followed by a description of the different analysis parameters studied, in order to analyse the heat

transfer enhancement. Two different models in 2D and 3D are performed, and a description of the

geometry and the mesh generation is exposed. This chapter furthermore presents all the

assumptions done and the parameters selected in the configuration of the simulations setup. Finally,

it concludes with a recap of different scenarios to be studied.



Chapter 6 covers the discussion of the results obtained for the 2D and 3D simulations performed.

Scenarios are compared and the results are highlighted in order to determine the heat transfer

enhancement. Chapter 7 summarises the most relevant findings encountered in the analysis and

presents the conclusions.

INTRODUCTION

4



2. STATE OF THE ART

2.1 General heat transfer concepts

Heat is the energy transferred between substances or systems due to a temperature difference

between them. According to the first law of thermodynamics, if two bodies at different temperatures

are placed together in absence of work, heat transfer occurs immediately and spontaneously from

the hotter one to the colder. Heat has the unit Joule (J) in the International System of Units [5].

There are three modes of heat transfer: conduction, convection and radiation. Only conduction and

convection modes are presented because radiation is not considered in the study performed in this

thesis.

Conduction heat transfer: it is an internally way of heat transfer caused by vibrations or

rapidly moves of atoms and molecules. It is the most important heat transfer mechanism in

solids, especially in metals because free electrons can move around and easily transfer

energy from one part of the metal to another [6].

Heat conduction is governed by Fourier’s Law. It states that the heat flux transferred is

proportional to the magnitude of the temperature gradient with the opposite sign [7].

�̇� = −𝑘 𝐴∇T (1)

Where:

�̇� (𝑊) is the heat transfer rate

𝑘 (𝑊/𝑚𝐾) is the thermal conductivity

𝐴 (𝑚2) is the cross-sectional area

∇T (𝐾/𝑚) is the temperature gradient

Figure 1 —Conduction heat transfer [26]

Heat conduction transfer rate depends on the properties of the medium. It is relevant to introduce

the concept of thermal conductivity. It is the ability of a material to conduct heat. For insulating

application, it is recommended to use materials with low thermal conductivities while materials with

higher values are used for applications where good heat conduction is needed [8].

Convection heat transfer: it occurs between a solid surface and a fluid that moves over it. It is

usually the main transfer method in liquids and gases. Heat is transferred by the combination

of diffusion (conduction) and by bulk fluid motion (advection). Natural or free convection

STATE OF THE ART

6

occurs when the flow is only caused by buoyancy forces. Density variations appear due to the

temperature differences within the fluid. If the temperature rises, the density normally

decreases and causes the upward movement of the fluid. On the other hand, forced

convection occurs when the fluid is pumped or pushed over the surface. Convection with

phase change (boiling) can occur too [6].

Figure 2 — Natural and forced convection [9]

The heat transfer expression for convection mechanism is:

�̇� = ℎ𝐴(𝑇𝑠 − 𝑇𝑓) (2)

Where:

�̇� (𝑊) is the heat transfer rate

ℎ (𝑊/𝑚2𝐾) is the convection heat transfer coefficient

𝐴 (𝑚2) is the heat transfer area

𝑇𝑠 (𝐾) is the surface temperature

𝑇𝑓 (𝐾) is the fluid temperature

The convection heat transfer is affected for the convection mode:

Process 𝒉 (𝑾/𝒎𝟐𝑲)

Free convection Gases 2 – 20

Liquids 50 - 100

Forced convection Gases 25 – 30

Liquids 100 – 40 000

Table 1 — Typical values of the convective heat transfer [10]

Some dimensionless numbers characterise the heat transfer [12] [13]:

Reynolds Number: it is the ratio of the inertial forces (fluid and flow properties) and the

viscous forces (only fluid properties). It is used to determine the flow regime (laminar or

turbulent). The expression is [42]:

𝑅𝑒 = 𝜌𝑣𝑠𝐿

𝜇=

𝑣𝑠𝐿

𝜐 (3)

Where:

𝑅𝑒 (−) is the Reynolds number

𝜌 (𝐾𝑔/𝑚^3 ) is the density of the fluid



𝑣𝑠 (𝑚/𝑠) is the characteristic velocity of the fluid

𝐿 (𝑚) is the characteristic length

𝜇 (𝑃𝑎 𝑥 𝑠) is the dynamic viscosity of the fluid

𝜐 (𝑚^2 ⁄ 𝑠) is the kinematic viscosity of the fluid

For internal flows through a pipeline, the term 𝐿 is expressed as the hydraulic diameter 𝐷ℎ as

follow:

𝐷ℎ = 4𝐴

𝑃 (4)

Where:

𝐷ℎ (𝑚) is the hydraulic diameter

𝐴 (𝑚2) is the cross-sectional area

𝑃 (𝑚) is the wetted perimeter

Nusselt Number: it is the ratio of convective to conductive heat transfer. It is the

dimensionless parameter that characterizes convective heat transfer.

𝑁𝑢 = ℎ𝐿

𝑘 (5)

Where:

𝑁𝑢 (−) is the Nusselt number


𝐿 (𝑚) is the characteristic length

𝑘 (𝑊/𝑚𝐾) is the thermal conductivity of the fluid

Prandtl Number: it is the ratio of momentum diffusivity to thermal diffusivity of a fluid. It

depends on the fluid properties.

𝑃𝑟 = 𝑐𝑝𝜇

𝑘 (6)

Where:

𝑃𝑟 (−) is the Prandtl Number

𝑐𝑝 (𝐽/𝐾𝑔 𝐾) is the specific heat

𝜇 (𝑃𝑎 𝑠) is the dynamic viscosity


Grashof Number: it is the ratio between the buoyancy forces and viscous forces acting on a

fluid. It is useful to quantify the opposing forces in convection heat transfer.

𝐺𝑟 = 𝑔𝛽(𝑇𝑠 − 𝑇𝑓)𝐷ℎ

3

𝜈2 (7)

STATE OF THE ART

8

Where:

𝐺𝑟 (−) is the Grashof Number

𝑔 (𝑚/𝑠2) is the gravity acceleration

𝛽 (𝐾−1) is the coefficient of thermal expansion. For gases, it can be calculated

following the expression 𝛽 = 1/𝑇; and for liquids, it can be calculated if it is known

the variation of density with the temperature at constant pressure.

𝑇𝑠 (𝐾) is the surface temperature

𝑇𝑓 (𝐾) is the bulk temperature

𝐷ℎ (𝑚) is the characteristic length

𝜈 (𝑚^2/𝑠) is the kinematic viscosity

Rayleigh Number: it measures the importance between the effects of the buoyancy forces

and the effects of the viscosity forces and thermal conduction.

𝑅𝑎 = 𝐺𝑟𝑃𝑟 (8)

Where:

𝑅𝑎 (−) is the Rayleigh Number

𝐺𝑟 (−) is the Grashof Number

𝑃𝑟 (−) is the Prandtl Number

The critical Rayleigh number for the case of infinite parallel plates heated from below is

approximately 1700. When the Rayleigh number is below the critical value for a given fluid,

the heat transfer mechanism dominant is conduction. When this critical value is exceeded,

convection heat transfer is dominant. [44]

2.2 Introduction and description of electronics components: overheating problems

Electronic equipment is widely applied in nearly every aspect of our lives. Under a mechanical point

of view, they are extremely reliable as results of no having moving parts. Concerning the thermal

environment, they can operate for many years without any problem in case of working at room

temperature. Nevertheless, they become potential devices to suffer overheating problems due to the

heat that generates the flow of electric current through a resistance. As a result, they normally fail

after prolonged use at these temperatures.

First of all, it is important to explain the effect of heat on electronics devices. Electronic devices

follow the “twice law”: life increases twice when the temperature reduces 10 °C. Figure 3 shows the

relationship between the temperature of electrolytic condenser and life [11].



Figure 3 — Relation between ambient temperature of electrolytic condenser and life [11]

With an ambient temperature increase of 10 °C (from 30°C to 40°C), the lifetime of the condenser is

reduced in half.

The failure rate of electronic devices due to temperature can be estimated with Arrhenius Law. If the

ambient temperature is less than 30°C, the failure rate is lower than unity, but highly increases with

the temperature growth. Figure 3 shows the relation between the failure rate of a semiconductor

and ambient temperature [11].

Figure 4 — Relation between ambient temperature of semiconductor and failure rate [11]

Furthermore, it is important to introduce some concepts. The junctions of an electronic component

are the circuits through which the electric current flows. These junctions are the potential sites of

heat generation and normally their temperature is limited to ensure safe operation [17].

Thermal resistance is a measurement of the temperature difference between two defined surfaces of

a material by which the material resists a heat flow. For devices, thermal resistance is the

temperature difference between the device and the surrounding ambient when it dissipates a Watt

of heat. It is measured in °C/W. Thus, a low value of this parameter is recommended [18].

There are different electronic components:

Chip + chip carrier: a ceramic, plastic or glass package that protects and contains integrated

circuits and provides the connexion to the circuit board. The silicon chip is normally placed in

a copper alloy plate at the bottom surface of the carrier. The copper thermal expansion is

similar to that of silicon, thus the possible thermal stresses problems of using plastic is

avoided. It is important to guarantee a good watertight to avoid problems of moisture.

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The chip carrier is the first thermal controller and some thermal considerations are taken in

its design. The heat generated is transferred to the package by conduction, convection and

radiation. Ceramic, plastic and glass are poor heat conductor and the interior cavity is filled

with a poor conductor gas. Consequently, the temperature difference between both parts is

large and the junction-to-case resistance vary from 10 °C/W to 100°C/W [17].

Figure 5 — Example of chip carrier [20]

Printed Circuit Boards (PCB): also called cards, a PCB is a wired plane board used as the

mechanical support of different electronic components such as diodes, transistors or

resistors. They are made by combining different non-conductive materials (fiberglass, plastic

or epoxy) and electrically connect the electronic components using tracks or thin strips of a

conducting material such as copper [21]. It is important to pay attention to the thermal

design of a PCB, because a simple failure of one of the electronic components could cause

the failure of all the electronic system.

Materials should be efficient insulators to prevent electrical breakdown and good heat

conductors to dissipate the heat generated. Thermal stresses should be avoided using high

strength and good thermal expansion coefficients materials.

Typically, an electronic system is formed with a few PCBs. The power dissipated varies from

5W to 30 W, and the cooling method most commonly used is the direct contact with a fluid,

normally air. Only if the electronic system is located in a tight enclosure, cooling is made with

a heat exchanger in contact with edge of the PCBs. The device-to-board to edge thermal

resistance could reach values from 20 to 60 °C/W, due to the low conductivity of the non-

conductive board materials [17].

Figure 6 — Common PCB [22]

All the electronic systems need an enclosure to place the circuit boards and the different required

connections and peripheral. The enclosure has two different important functions: guarantee the

protection of all the components and provide a cooling method. It is designed to easily access for



maintenance or replacement service and includes external indicators, such as lights or a screen to

give information and a keypad for user interface.

The materials are usually thin plates of aluminium or steel and have to face some mechanical issues

such as vibrations, shocks or moisture problems [17].

The first step to select or design a cooling method is determining the amount of heat to be removed:

the cooling load. The heat generated follows the Joule’s first law. Joule or Ohmic heating is the

physical process by which the pass of electric current through an electrical conductor produces

thermal energy [19].

𝑊𝑒̇ = 𝜙 𝐼 = 𝐼2𝑅 (9)

Where:

𝑊�̇� (𝑊) is the electric power consumption of the electronic device

𝑅 (Ω) is the electrical resistance

𝐼 (𝐴) is the electric current

𝜙 (𝑉) is the electric potential

The first law of thermodynamics states the conservation of energy. In consequence, in the absence of

other energy source or interaction, the heat produced by an electronic device in steady operation is

equal to its power consumption.

Figure 7 — Cooling load equal to power consumption [17]

Nevertheless, this ideal condition is perturbed by the interaction with different equipment that

outputs other forms of energy. The cooling load can be calculated as the power consumption minus

all these energy interactions. Another way to perform it is to determine and add up all the individual

heat produced by all the components.

Normally, after determining the cooling load of a system, it is usually to add a safety margin to

ensure the reliability and the safety of the components. This safety extra cooling load normally raises

the cost, the size, the weight and the consumption of the system. So, it is important to well adjust

the safety margin to not considerably oversize the drawbacks [17].

The thermal state of an electronic device could be divided into two different operations: transient

and steady operation. When a device is turned on, the device components start to absorb the heat

generated, so the temperature starts to rise progressively. When the heat generated is equal to the

heat removed for the cooling method, the device temperature stabilises at some point and starts the

steady operation. For the devices that operate for long periods, the cooling methods are designed for

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the steady operation. However, some devices do not reach the steady operation and the cooling

method can be designed only for a limited use during the transient state.

Figure 8 — Temperature change of an electronic component with time [17]

The thermal environment is also important to design the cooling method. The cooling requirements

needed for a device placed in a spacecraft differ extremely with those of a common TV. If the

ambient conditions are extremes or can deteriorate an electronic device, it is common to use a

conditioned fluid as heat sink intermediary between the device and the environment. Water,

dielectric fluid and especially air due to its availability, are the most common fluids used.

For example, electronic equipment designed for aircraft applications must meet several

requirements. They must be adequately located in odd-shaped and curved spaces and provide

efficient paths for the fluid and heat extraction. Normally, forced convection methods are used for

such applications, using the ambient air as cooling fluid. But, as the ambient temperature is high, the

fluid must be expanded and cooled before entering in the electronic cooling system. However, in

vehicles is typically used a liquid cooling method. The liquid passes through the components and

after is cooled down in a radiator.

2.3 Current electronic components cooling methods

Overheating is one of the most important issues that electronics industry has to face. Devices trend

to be more and more powerful and smaller, so they need innovative thermal management methods

in order to improve reliability and system performance.

The thermal management market is valued at USD 8.99 Billion in 2016 and is estimated to reach USD

14.24 Billion by 2022. The compound annual growth rate (CAGR), that describes a constant rate

growth over the years of the period considered, is equal to 7.91% during the forecast period (2017 –

2022) [15].

The heat generated varies depending on the electronic device from 5W/cm2 on a Printed Wiring

Board (PWB) to 20 kW/cm2 for a semiconductor laser [16]. The working temperature must be below

the maximum temperature allowed specified by manufacturers.

The main cooling methods existing are:



Conduction

Air cooling: natural and forced convection

Liquid cooling

Immersion cooling

Advanced cooling techniques

Manufacturers usually provide the rate of heat dissipation and the maximum allowable temperature

for reliable and safe operation. Figure 9 shows the different cooling mechanisms for a certain surface

heat flux and temperature difference.

Figure 9 — Heat fluxes that can be removed at specified temperature with certain heat transfer mechanisms [17]

First of all, conduction cooling is based on diffusion heat transfer. It is Important to establish effective

heat transfer paths to correctly extract the heat generated and transfer it to the heat sink.

Chip carriers are designed with several leads that remove the heat generated in the chip. The leads

are made with a highly conductive material. The heat is transferred to these leads with some bond

wires and the heat produced in the lead frame is transferred through the case material [17].

Figure 10 — Heat extraction of a chip carrier [17]

Air cooling is the simplest means of heat removal. It is based on natural or forced convection. For

low-power electronic systems, natural convection is widely used. This cooling method is highly

reliable because it does not use fans to pump the fluid, so mechanical problems are avoided.

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Normally, it is desirable when there are no big obstacles in the fluid path. The heat transfer rate is

directly related to the flow rate of the fluid: if the flow rate rises, the heat transfer rate increases.

Electronic components cooled with natural convection need a correct design of the enclosure. They

need sufficient vents to ensure the correct fluid flow through the components and guarantee the

easily fluid flow exit. They should be as large as possible and the inlet vents must be located at the

bottom of the case to allow the upward natural flow to the outflow upper vents. The flow regime

normally starts as laminar but it can turn to turbulent if the temperature difference between the air

and the heated surface is considerable. However, for temperature differences less than 100 °C and

body characteristic length less than 0.5 m, the flow can be considered as laminar. Radiation heat

occurs but it is only relevant if the temperatures reached are high [17].

Figure 11 — Natural convection mechanism [17]

When natural convection does not extract sufficient heat, it is common to use a fan to pump the air

(forced convection). Following the principle mentioned before (higher fluid flow rate, higher heat

transfer), the fan blows the air to the cooling system at a higher velocity and provide a higher air flow

rate. The mass air flow rate depends on the environmental condition. For harsh environment with

high temperatures, the air flow rate must be sufficient to avoid overheating and higher than the

mass flow needed for low temperatures environments. The contribution of radiation heat in this type

of cooling method is negligible because all the heat is removed throughout the air injected.

A good practice is to maintain de inlet-outlet temperature difference of 10 °C or K and a maximum

exit temperature of 70 °C. These conditions guarantee that the maximum surface temperature of the

components does not reach values higher than 100 °C.

The air flow can be internal or external, depending on the body geometry; and laminar or turbulent.

Turbulent flows guarantee a better heat transfer coefficient but it needs specific requirements for

the fan and the cooling system design. There are different types of fans. For the fan selection, there

are two important parameters to take into account: static pressure head and volume flow rate. Axial

fans are recommended for low static pressure head and they are cheap and smalls. For high static

pressure head, centrifugal fans are commonly used but they are more complex, bigger and expensive

[17].



Liquids are more effective than air. They have much higher thermal conductivity than gases, so

higher heat transfer coefficients. Liquid cooling is recommended for applications with high cooling

loads that air cooling systems cannot cope with. However, liquid cooling implies some drawbacks

such as leakage, corrosion and condensation risks and problems.

There are two different liquid cooling techniques: direct and indirect systems. Liquid removes the

heat from the heated surface in direct cooling systems, while there is no "liquid-component" contact

in indirect cooling systems. The heat is first transferred to another external medium such as a plate,

and the liquid extracts the heat from this plate. Another classification can be done regarding the

liquid cycle: if the liquid is recirculated, it is a closed-loop system and if the liquid is discharged after

the heat extraction, it is an open-loop system [17].

Typically, electronic equipment is immersed in direct cooling systems and the heat transfer can be

natural or forced convection or boiling. The fluids used are the dielectric fluids whose electrical

properties are extremely suitable for this application. In section 2.5, a detailed explanation of

dielectric liquids is done.

A closed-loop cycle can use water as cooling liquid. Typically, electronic components are mounted on

a plate made of very conductive material with some tubes and the fluid passes through the tubes.

After the heat extraction, the fluid is recirculated to a heat exchanger to reduce its temperature and

returns to the cooling system. An expansion tank absorbs the expansion and contractions of the fluid

in order to maintain a correct pressure and volume flow rate of the cooling fluid.

The liquids should meet some special requirements: high thermal conductivity, high specific heat,

low viscosity, high dielectric strength and chemical inertness and stability [17].

Figure 12 — Indirect liquid cooling system [14]

Boiling guarantee the highest heat transfer coefficients. Immersion cooling consists in submerge

high-power electronic component in a dielectric liquid. The temperature reaches very high values so

the liquid boils and provide a very high heat transfer rate. Regarding the thermodynamic properties,

a liquid boils at the saturation temperature for a given pressure, so the temperature of the bath is

constant [17].

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2.4 Applications related to Electrohydrodynamics: Meso/micropumps: EHD pumps

The limited space is one of the most difficult challenges to manufacture a reliable and efficient pump.

The length scales of the micropumps have the order of 100 nm to 0.1 mm. Micro/meso scale pumps

are classified into two different categories: mechanical and non-mechanical (dynamic) pumps,

considering if moving parts are used or not [24].

Figure 13 — Classification of meso/micro pumps [27]

2.4.1 Mechanical micropumps

The research in this new type of pumps starts in the early 1980 and it was the base to the

development of the microelectromechanical systems (MEMS) around 1990. The main applications of

MEMS focus on biomedical and biological fields. They allowed creating new devices to be applied in

insulin delivery or injection of glucose and electronic cooling systems, for example in micro

integrated circuits, currently still in development [25].

Figure 14 — Example: Debiotech’s Insulin Nanopump [57]

2.4.2 Non mechanical or dynamic micropumps

The main principle of dynamic micropumps is adding momentum to the fluid. Generally, these

micropumps provide higher flow rate and a much steadier discharge than mechanical pumps

nevertheless high-viscosity liquids cannot be pumped. Magnetohydrodynamic (MHD),

electroosmotic, bubble type or electrohydrodynamic (EHD) are different types of this technology

[25].



The EHD pumps consist in the interaction of electric fields and free charges in a dielectric fluid

medium. The fluid is pumped when electric fields push or pull charges through the fluid in a given

direction. Compared to other dynamic micropumps, the main advantages of EHD micropumps are

the low manufacturing costs and minimal power consumption, the absence of vibration (no moving

parts) and very low noise [27]. In addition, the control is very quickly by varying the electric field

applied and they are recommended for special environments. They could use different fluids,

depends on the application and in single and multiphase flow, with high reliability and high

efficiency. Usually, fluid flows will be laminar and with low Reynolds numbers. [24]

2.4.3 EHD micropumps

The EHD micropumps could be classified into three different types: ion-drag, induction and

conduction pumps [27].

Ion-drag pumps: ions are injected from a sharp–edged electrode (emitter) into a dielectric

fluid. The charged particles move towards the collector throughout the fluid due to a high

voltage electric field applied. This motion causes collisions with neutral molecules of the fluid

which exerts a drag force along the channel.

Figure 15 — Injection pump configuration [36]

Ion-drag pump could have two different configurations: positive or negative discharge. It

depends on which terminal of the power supply is connected to the high voltage electrode.

Positive discharge configuration (emitter connected to the positive terminal of the power

supply) produces the positively charge of the nearest molecules of the electrode, being

repelled them from the emitter towards the collector. In contrast, the negative terminal of

power supply is connected to the high voltage electrode in negative discharge configuration.

Negative ions appear in this case, and they are pulled by the collector. In both types, the

motion in the fluid is always from the high voltage electrode (emitter) to the ground

electrode (collector). [27]

Nevertheless, these pumps often could deteriorate the electrical properties of the working

fluid and they could be not easily operated.

Induction pumps: inductions pumps reduce the deterioration fluid problem of ion-drag [28].

Low conductive fluids contain particles charged positive and negatively in equilibrium that

could be disturbed if a gradient in electric conductivity of the fluid is applied. This contributes

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to the formation of positive and negative charges depending on the polarity of the applied

voltage. The free charges are pushed or pulled to the adjacent electrode by applying

electrical waves. This motion creates a drag force and it moves the fluid. [27]

Figure 16 — Travelling wave induction pump configuration [24]

A temperature gradient is used to generate this movement because mainly of two reasons:

the dissociation rate of neutral molecules depends on temperature and ionic mobility is

inversely proportional to viscosity so liquid conductivity depends on temperature too. The

way to induce charges is to apply a travelling wave perpendicular to the conductivity gradient

and the sign of this temperature gradient define the direction of the flow. Normally, this

temperature gradient is obtained by the Joule heating generated by the electric field but it

could be applied an external temperature gradient too [29].

Figure 17 shows the two different induction pump configuration. Figure 17 (a), the flow

direction and the travelling wave is the same while in Figure 17 (b) the direction is opposite.

Figure 17 — Induction pump configuration [27]

Conduction pumps: EHD conduction pumps are based on the dissociation and recombination

of neutral particles of the electric field [30]. In a neutral medium, the rate of dissociation and

recombination is in equilibrium. However, the intense electric field near the electrodes

creates a layer in which particles are not in equilibrium: the rate of dissociation is higher than

that of recombination. This layer is charged with the opposite sign from that of the adjacent



electrode and it is called heterocharge layer [31]. The thickness of the layer increases with

the electric field and the Coulomb force acting in this layer produce the drag force. [27]

Figure 18 — Heterocharge layers and different types of electrode configuration [27]

Figure 18 shows the heterocharge layers and different types of electrode configuration [27]:

a) Porous type electrodes

b) Flushed-type electrodes

2.5 Dielectric fluids for cooling: mineral oils

The use of minerals oils is a promising technique to reduce energy consumption for cooling electronic

components. Liquid refrigeration has multiples advantages over traditional air techniques, due to the

higher heat capacities of fluids [32]. In addition, the economic savings is not only in terms of energy

consumption, but also the design and manufacturing costs of the cooling systems are also reduced. A

correct choice of the heat transfer fluid can optimize the cooling procedure and the service life of the

electronic components to be cooled. [33] Different solutions and case studies using mineral oil ([34]

and [35]) proved the effectiveness and the economic savings of this immersion technique.

The benefits of oil immersion cooling technology versus air could be summarized in the reduction of

the common operational issues and the main causes of failures. Typically, air cooling systems have

high fluctuations in temperature and relative humidity profile. However, the oil systems operating

conditions are smoothed (low temperatures and no sensitivity to humidity or suspended particles),

corrosion problems and electrochemical migration are reduced. In addition, moving parts like fans

disappear and exposure to electrostatic discharge is avoided [33].

There are five different dielectric fluids used to cool electronic components [48]:

Mineral oil

White oil

Fluorinated oil

Vegetable oil

Synthetic isoparaffin

It is important to consider the following key parameters for each type of dielectric fluid [48] [49] [50]

[51].

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Heat Transfer Effectiveness: it is one of the most important characteristics of the cooling

fluid. Many physical characteristics of the fluid are temperature dependent as the viscosity

(dynamic viscosity has the largest effect) or density, so the operational temperature has an

important effect on cooling efficiency.

Electrical characteristics: all these dielectric fluid are good electrical insulators as result of

their non-polarity and their high dielectric strength.

Oxidation stability: this characteristic has a direct impact on the life cycle of oil because

hydrocarbons oils are prone to oxidize under heat and oxygen exposition. Sludge accelerates

the oxidation process and reduces the performance of the cooling system: it accumulates in

the corners and across the boundaries of pipes, reducing the speed of the cooling fluid

causing an overheat [37].

Material compatibility: the dielectric fluid does not influence, interference or modify the

physical characteristics of the equipment materials. The compatibility between all circuit

board and other materials of the system and the dielectric fluid must be guaranteed. One of

the ways that fluids are tested for potential incompatibilities is to determine the fluid's

solubility toward rubbers and similar compounds.

Worker health and safety: safety conditions to the workers and the environment must be

guaranteed. Workers exposure can be via inhalation, via ingestion or by skin exposure to

dielectric fluids. All these fluids are considered non-toxic and non-hazardous and whether

skin contact happens, washing the zone affected with soap and water should be enough.

The production of an electric arc as an ignition source is very small due to the low voltages

used in electronics systems. The fire point must be as high as possible.

Biodegradation and environmental fate: the toxicity to marine and soil organism and

biodegradation rate are important in the use of these cooling fluids. Biodegradation is the

process by which organic substances are broken down into smaller compounds by enzymes

produced by living microbial organisms [52]. The speed and the rate of the biodegradability

depend on the chemical composition of the fluid, temperature and type of soil or water.

Cost: it is important to consider spills, waste, leakage and waste to calculate the fluid cost.

In summary, these key parameters allow to choose the best dielectric fluid to use in cooling systems

for electronic components.

The mineral oils could be classified into three different groups, depending of its composition [38]:

Paraffinic oil

Naphtenic oil

Aromatic oil



Figure 19 — Hydrocarbons in the mineral oil [37]

Naphtenes do not contain paraffin waxes and are the most widely used in cooling application

because they form a smaller quantity of waxes and they have a very good dielectric behaviour [56].

The main properties of minerals oils are low viscosity, good electrical properties and low relative

permittivity. It can easily be refined from petroleum. For the coolant function, oils must have a low

pour point (measure of oil flow at a relatively low temperature) to ensure the correct flow of the

fluid at any temperature. Oil temperature in service must be controlled and must be lower than its

flash point (lowest temperature at which it can be flammable) [37].

The correct circulation at higher temperature is guaranteed because the viscosity decreases if the

temperature rises. Viscosity has a direct relationship with the heat rate dissipation and follows the

equation:

𝜇 = 𝐶1 ∗ 𝑒𝑥𝑝 (2797.3

𝑇 + 273.15) (10)

Where:

𝜇 (𝑐𝑃) is the dynamic viscosity

𝐶1 is a coefficient for scaling

𝑇 (º𝐶) is the temperature

Oil has also a safety function actuating as an insulator between different parts at different electrical

potential, so it should be not a hazardous material.

2.6 Social impacts

Nowadays, electronic components are present in almost all the common devices and equipment

such as TV’s, computers or household appliances. The rapid development of electronic equipment

forces the society to keep update to the last technologies, discarding a huge quantity of devices. By

2016, the amount of e-waste grew to 44.7 million metric tons, and only in EU, the waste of electrical

and electronic equipment (WEEE) is expected to reach 20 tons by 2020. This e-waste, mostly

inadequately treated, represents a healthy and environmental hazard [43] [46].

In addition, society trends to be more and more connected. Smart City initiatives need the

communication between the citizens, the city and all the elements located within the city. So,

intelligent electronic devices (IEDs) will play an important role in our future society. To allow this

interaction, it is needed the implementation of thousands of intelligent electronic devices (IEDs), that

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must operate without interruption. Thus, it will be critical to ensure the correct working conditions

for all this technology and new efficient cooling methods must be developed. To guarantee a safe,

healthy and reliable society based on the internet of things (IOT), failure rate of all the components

that create this network must be almost totally reduced.

2.7 Economic impacts

The use of mineral oil as a cooling fluid for electronic components is a very attractive technique to be

widely applied because offers an opportunity for important energy savings. The low production costs

and the wide availability are two factors that justify the choice of mineral oils as cooling fluid. In

addition, its good thermal properties place it above the water as cooling fluid [37].

The primary sources of mineral oils are fossil fuels: oil, natural gas and coal. Nowadays, the

contribution to the global energy production of fossil fuels is really high, but tends to be reduced in

the future (from the actual 85% to 74% in 2040) [53]. Therefore, the availability and the price of

mineral oil will should not highly vary next years.

Figure 20 — Energy sources [53]

Normally, the costs of a high-reliability cooling system for a data centre could reach the 50% of the

total costs. The investment in the infrastructure needed to build a liquid cooling system, typically an

immersion cooling system, is considerably lower compared to air systems due to the reduction of the

use of specialized machinery as chillers or temperature and humidity controls [54] [55]. Oil takes

away approximately 1200 times more heat by volume when compared to air [37].

At present, about 500 million dollars are invested in cooling infrastructure per data centre. Oil has

higher performance and availability and can reduce the cooling systems cost in half because the

energy used to cool and circulate the air is avoided. Approximately fan power needs 0.13W of power

to move 1W of waste heat into chilled water loop and the technical load to power fan at 100% for

fan-powered air is 10.5% more than oil immersion [54] [55].

Consequently, immersion cooling systems with mineral oil are increasingly implemented.



2.8 Technological impacts

An in-depth explanation of negative the effects of overheating in electronics components is

explained in chapter 1.1. Typically, manufacturers recommend maintaining the working temperature

below a defined value that easily can be overcome if a correct cooling technique is not applied. As it

explained before in chapter 2.2, electronic devices follow the “twice law”: life increases twice when

the temperature reduces 10ºC [11].

One of the most important applications in which it can be needed these new cooling techniques is in

data centres. Immersion cooling systems are implemented in different data centres over the world

such as the NSA (National Security Agency), which has implemented an immersion cooling system in

its Physics Lab. In Barcelona, the PIC (Port d’Informació Científia) infrastructures have four tanks that

cool down a scientific computing cluster.

Some positive aspects have been noticed by using these new techniques with mineral oils as cooling

methods in data centres. The Power Usage Effectiveness (PUE) is reduced to values of 1.05 in the PIC

of Barcelona. PUE is a benchmarking standard to determine how energy efficient data centres are. It

is computed as the ratio between the total facility power and the IT equipment power. Values less

than 1.2 mean a high efficiency of the system [78].

The mineral oil thermal properties are better than air and water ones. In addition, it avoids dust

particles, guarantees proper cleanliness of the system and reduces the noise due to the absence of

fans. This and stable temperature in all the equipment helps to reduce the mechanical failures too

[79].

Consequently, efficient cooling methods help to increase the durability and the lifetime of electronic

components. Thus, the development of efficient cooling methods is needed to increase the lifespan

of all the electronic devices and try to reduce the e-waste rate produced every year.

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3. GOVERNING EQUATIONS

Electrohydrodynamics is a science that relies on the study of the interaction of fluids with electrical

fields. It is an interdisciplinary science between the study of the movement of a fluid and the study

on how an electric field could produce this movement [41]. This chapter introduces the governing

equations and all the physics principles of EHD.

3.1 Hydrodynamics

First, in hydrodynamic, the Navier-Stokes equations describe the motion of viscous fluids. In this

study, it is only considered incompressible flows in which density variations are not linked to the

pressure and the mass conservation is a constraint on the velocity field. So, for an incompressible

flow the divergence of the flow velocity is zero that implies constant density value [42].

The Boussinesq approximation is a way to solve no isothermal flow, such as natural convection

problems, without solving the full compressible formulation of the Navier-Stokes equations. This

approximation considers density as constant value unless where it appears in terms multiplied by the

gravity acceleration. This approximation is accurate for many kinds of flows, when the temperatures

differences and thus, the density differences are smalls, with an easy physical and mathematical

formulation [39]. It is used to solve problems where the fluid temperature varies from one place to

another, driving a flow of fluid and heat transfer [40].

In this approximation, only the variation of the density when it appears multiplied by the gravity

acceleration is considered. The rest of fluid properties are considered as constants. The formulation

is as follow [40] [41]:

The continuity equation for conservation of mass is:

𝜕𝜌

𝜕𝑡+ ∇ · (𝜌𝒖) = 0 (11)

Where 𝒖 (𝑚 ⁄ 𝑠) is the local velocity of a parcel of fluid.

If density is considered as constant

∇ · 𝒖 = 0 (12)

The expression of the density is dependent on the temperature:

𝜌 = 𝜌0 − 𝛽𝜌0∆𝑇 (13)

Where 𝛽 is the thermal expansion coefficient.

If F (𝑁) is the gravitational force:

𝑭 = 𝜌𝑔 (14)

Then, the conservation of momentum equation:

GOVERNING EQUATIONS

26

𝜕𝒖

𝜕𝑡+ (𝒖 · ∇)𝒖 = −

1

𝜌∇𝑝 + 𝜈∇2𝒖 − 𝒈𝛽∆𝑇 (15)

Where:

𝜌 (𝐾𝑔/𝑚^3 ) is the density

𝑝 (𝑃𝑎) is the pressure

𝜈 (𝑚^2/𝑠) is the kinematic viscosity

𝛽 (𝐾−1) is the thermal expansion coefficient

First term: temporal variation of velocity

Second term: convective terms

Third term: pressure gradient

Fourth term: viscosity of the fluid

Fifth term: sum of all the forces involved

The expression for conservation of energy in hydrodynamics can be written as follows [41]:


𝜕𝑡+ 𝒖 · ∇𝑇] = ∇ ∙ (𝜅∇𝑇) + 𝐽 (16)

Where:

𝑇 (𝐾) is the temperature

𝐽 (𝐽/𝑚^3 ) is the rate per unit volume of internal heat production

𝜅 (𝑊/𝑚𝐾) is the thermal conductivity

𝐶𝑝 (𝐽/𝑘𝑔 𝐾) is the heat capacity

3.2 Electrostatics

Electrostatics describes the physics of charge in motion and at rest in absence of significant magnetic

field [72]. The electrical force will be applied in a certain region of the microfluidic system and it is

equivalent to the generator of an electrical circuit. The electrostatic force due to space charge of

polarization or dielectric medium is the main reason for heat transfer enhancement in EHD systems

[47]. The electric body force has the expression [59]:

𝑭𝒆 = 𝜌𝑒𝑬 −1

2𝐸2 ∇𝜀 + ∇ (

1

2𝐸2𝜌 (

𝜕𝜀

𝜕𝜌) 𝑇) (17)

Where:

First term: electrophoretic or Coulomb force

Second term: dielectrophoretic or dielectric force

Third term: electrostriction force

The first term is the most relevant in the case of EHD micropumps. It is the force per unit volume on

a medium containing free electric charge, responsible for the movement of the fluid. The second

term is the force due to the force exerted on a non-homogeneous dielectric liquid by an electric field.



Normally, it is weaker than Coulomb force and it is also relevant when a temperature gradient

appears or when an AC-EHD single phase system is considered. Finally, the third term shows the

permittivity changes to density to the applied electric field [23]

The electric force produces a secondary flow that pushes the neutral molecules of the dielectric fluid

towards the collector, and modifies the velocity profile.

Figure 21 — Secondary flow velocity profile [71]

To obtain the electric force, Maxwell’s equations should be considered. In electrohydrodynamic

flows, the magnetic effect could be ignored because the characteristic time for the magnetic

phenomena (𝑡𝑚 ~ 𝜇𝑀𝐾𝑙2) is several orders of magnitude smaller than the characteristic time for

electric phenomena (𝑡𝑒 ~ 𝜀 𝐾⁄ )[59] [60].

∇ · (𝜀𝑬) = 𝜌𝑒 (18)

∇ × 𝑬 = 0 (19)

Where:

𝜀 is the electric permittivity

𝑬 (𝑉/𝑚) is the electric field

𝜌𝑒 (𝐶/𝑚^3 ) is the volumetric charge density

In terms of the electrical potential, 𝜙, the electrostatic limit follows the Poisson equation:

𝐄 = −∇𝜙 (20)

∇ · (𝜀∇𝜙) = −𝜌𝑒 (21)

The volumetric charge density could be expressed as:

𝜌𝑒 = 𝜌𝑧 (22)

Where:

𝑧 (𝐶/𝐾𝑔) is the charge per unit mass

The charge conservation equation is:

𝜕𝜌𝑒

𝜕𝑡+ ∇ · 𝑱 = 0 (23)

GOVERNING EQUATIONS

28

Where:

𝑱 (𝐴/𝑚2) is the current density

𝑱 = 𝐾𝑬 + 𝜌𝑒𝒖 (24)

Where:

𝐾 (𝑆/𝑚) is the conductivity

First term: ohmic charge conduction

Second term: convection of charges

Considering the electrostatic relationship (18) and (19), the conservation equation of the charge (23)

can be written:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 + 𝑬 ∙ (

𝐾

𝜀 ∇𝜀 − ∇𝐾) (25)

If the electrical properties of the fluid 𝐾 and 𝜀 are constant, eq. (25) reduces to:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 (26)

3.3 Electrohydrodynamics

Electrohydrodynamics describes the effects of electrostatics in liquid media. The Coulomb force,

𝑭𝒆 = 𝜌𝑒𝑬, is considered and added to the conservation of momentum equation (15). Consequently,

considering constant value for K and the Joule Heating for the energy equation, the three governing

equations of our EHD system are:

∇ · 𝒖 = 0 (12)

𝜕𝒖

𝜕𝑡+ (𝒖 · ∇)𝒖 = −

1

𝜌∇𝑝 + 𝜈∇2𝒖 − 𝒈𝛽∆𝑇 + 𝜌𝑒𝑬 (27)


𝜕𝑡+ 𝒖 · ∇𝑇] = κ∇2𝑇 + 𝜀|∇𝜙|2 (28)



4. SIMULATION SOFTWARE: ANSYS FLUENT

4.1 ANSYS Fluent

Create a computer multiphysics model represents a truly complex and laborious task, consequently

there are some different computers tools in the market to perform such simulations. ANSYS is a

company focused on the development of engineering simulation for more than 45 years [61]. ANSYS

Fluent, one of its products, is a CFD software with the physical modelling capabilities needed to

accomplish the simulations defined in chapter 5.

Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that provides a qualitative and

quantitative prediction of fluids flows thanks to mathematical modelling and numerical methods

[62]. Solve these problems analytically are in many cases extremely difficult due to the non-linear

inertial terms. By discretizing the domain in small grids (mesh), it is possible to obtain an accurate

solution.

ANSYS Fluent solves the governing equations and generates the flow field data at each mesh

elements (nodes, faces, cells). The data generated are exported to a data processor called CFD-Post,

provided by ANSYS, to easily analyse the results. In addition, the data can be exported to Excel as a

.csv file, to allow the comparisons between simulations.

4.2 Models

ANSYS Fluent provides a wide list of models for different steady-state or transient, incompressible or

compressible, laminar or turbulent fluid flow problems.

For the case study of this document, viscous, energy and potential models are used. Viscous model

allows setting and defining the characteristics of the fluid flow. Energy model solves equation (28)

and potential model solves equation (18) and adds joule heating to the energy equation

[83][84][86][87].

ANSYS Fluent Potential model solves the equation [83]:

∇ ∙ (𝜀∇𝜙) + 𝑆 = 0 (29)

Where:

𝜙 (𝑉) is the electric potential

𝜀 (𝐹/𝑚) is the electric permittivity

𝑆 is the source term

When solving this equation, ANSYS Fluent adds the Joule heating (𝑊/𝑚3) generated to the energy

equation.

𝑆ℎ1 = 𝜀|∇𝜙|2 (30)

SIMULATION SOFTWARE: ANSYS FLUENT

30

Thus, the Joule heating is added for solid and fluid regions in which a flow of current appears [83]

[84].

4.3 User-Defined Scalar transport equations and User-Defined Functions

To simulate the electrohydrodynamic system, it is necessary to add the contribution of the Coulomb

force to the momentum equation. Fluent allows implementing extra scalar transport equations called

User-Defined Scalar (UDS).

For an arbitrary scalar 𝜙𝑘, ANSYS Fluent solves the equation [63]:

𝜕𝜌𝜙𝑘


(31)

Where:

(𝜕𝜌𝜙_𝑘)/𝜕𝑡 is the unsteady term

∇ ∙ (𝜌𝒖𝜙𝑘) is the convection term

Γ𝑘 is the tensor diffusion coefficient

∇ ∙ (Γ𝑘∇𝜙𝑘) is the diffusion term

𝑆𝜙𝑘 source term

Charge density conservation equation (26) is solved with an UDS equation.

To introduce the Coulomb force into the momentum equation, it is needed a User-Defined Function

(UDF). A UDF is a C function that can be loaded with the ANSYS Fluent solver and allow to, for

example, customize boundary conditions and material properties, add source terms in ANSYS Fluent

transport equations or in user-defined scalar transport equations and enhance ANSYS Fluent models.

It allows customizing the simulator to cover all the particular and specific requirements of different

simulations. UDF codes use special macros provided by ANSYS Fluent to access to solver data and

domain variables [73].

4.4 Discretization of the domain: mesh generation

The aim of discretization the entire domain into small high quality cells is to obtain the domain

geometry and make sequent calculation to obtain an accurate solution. ANSYS Fluent allows

generating tetrahedral, hex-core or hybrid volume mesh from an existing boundary mesh or a CAD

file.

While a good and fine mesh helps the CFD solver to converge to an accurate solution minimizing the

resources employed, a coarse mesh can be an important source of errors in a simulation.

Consequently, it is important to find a correct balance between the fineness of the mesh and the

computational cost needed to solve the equations.

ANSYS Fluent provides an indicator to check the quality of the mesh called orthogonal quality. For

each face, two different quantities are calculated:



The normalized dot product of the area vector of a face and a vector from the centroid of the

cell to the centroid of that face.

The normalized dot product of the area vector of a face and a vector from the centroid of the

cell to the centroid of the adjacent cell that shares that face.

The minimum value that results from calculating these two parameters is defined as the orthogonal

quality. The values go from 0, that indicates bad quality, to 1, excellent quality. The minimum

orthogonal quality should be more than 0.01, with an average value significantly higher.

Another important indicator is the aspect ratio that measures the stretching of a cell. It is computed

as the ratio of the maximum value to the minimum value of any of the following distances:

Normal distances between the cell centroid and face centroids.

Distances between the cell centroid and nodes.

For an unit cube, the aspect ratio is 1.732. It is recommended avoiding sudden and large changes in

cell aspect ratios in areas where the flows suffer large changes or strong gradients [64].

4.5 Pressure-Based Solver. Coupled Algorithm

ANSYS Fluent offers two different numerical methods: pressure-based and density-based solver.

Normally, pressure-based solver is recommended for low-speed incompressible flows and density-

based solver is more suitable for high-speed compressible flows problems. For the simulation

required for the case study of this project, pressure based solver is selected.

The governing integral equations are solved following a control-volume-based technique. First, the

domain is divided into discrete control volumes (mesh). The governing equations are integrated on

these individual control volumes and the discretized equations are linearized. Finally, the resultant

linear equation system is solved [65].

Pressure-based solver can work with two different algorithms: segregated or coupled. The

Segregated Algorithm solves the governing equations sequentially; each equation is decoupled from

the other equations. The memory requirements needed are low because the discretized equations

need only be stored one at time. On the other hand, Coupled Algorithm solves simultaneously the

system of momentum and pressure-based continuity equations, increasing the rate of solution

convergence. Nevertheless, the memory cost increases by 1.5-2 times [66].

The following graph shows the different steps for the calculations:


32

Figure 22 — Overview of Pressure-Based Solution Methods [66]

4.6 Convergence criteria

Typically, the fluid flow problems are not linear and must be solved by an iteratively calculation with

CFD solutions. Residuals measure as the local imbalances of a conserved variable in each control

volume, and ANSYS Fluent use them as convergence criteria. To obtain a numerically accurate

solution, they must be as lower as possible. The default convergence criterion of ANSYS Fluent

requires that the residuals drop 3 orders of magnitude for continuity and momentum equation and 6

orders for energy equation [67].

Nevertheless, for complicated problems, it is not always a target reachable. Monitoring some

variables like force, drag or average temperature can help the analysis to determine when a

simulation is converged. If these variables do not change with more iterations, the simulations can be

considered as converged. The final solution must guarantee the mass, momentum and energy

conservation [68].

4.7 Boundary conditions

In a CDF analysis, it is relevant to define how the system operates. Boundary conditions are the set of

constraints to boundary value required to solve the mathematical model. A wide list of different

boundary conditions is available in ANSYS Fluent, and allows defining and setting the boundary

values and flowing behaviour.

In the simulations performed, “velocity inlet”, “pressure outlet” and “walls” are the boundary

conditions used [69].



“Velocity inlet” boundary condition defines the flow velocity and all scalar properties of the

flow at the domain inlet. The total pressure is not fixed but will rise to the appropriate value

to provide the velocity defined.

When “velocity inlet” is used, ANSYS Fluent recommends using “pressure outlet” as

boundary condition outlet. For subsonic flows, it requires the static pressure at the domain

outlet. If backflow problems occur, it is possible to define “backflow conditions” to avoid

convergence issues.

“Wall” boundary conditions are used to define solid zones and confine the fluid. For viscous

flows, the no-slip boundary condition is applied: tangential fluid velocity equal to wall

velocity, and null normal velocity. To define an adiabatic wall, null heat flux must be set.

4.8 Computational resources

ANSYS Fluent v18.1 academic version is used for the 3D simulations of this document. The 2D

simulations are done in ANSYS Fluent v19.2 academic version. The computers used to perform the

simulations are two Intel® Core™ i5-6500, 3.20 GHz, 8Gb RAM for the 3D simulations and an Intel®

Core™ i7-4500U, 2.4Ghz, 4 Gb RAM for the 2D simulations.


34



5. METHODOLOGY

This chapter introduces the definition of the model to be simulated. It is based on the experiment

performed by Moghanlou, F. S. et al in 2014 [23]. The different considerations and assumptions

taken into account to accomplish with the required analysis are also presented.

5.1 General overview

The case study of this document consists in to simulate an electrohydrodynamic system for laminar

flow through a square minichannel. The aim is to investigate how the electric field contributes to the

cooling process and pressure drop for different Reynolds numbers. The coupled of electrostatics and

hydrodynamics produces a modification in the momentum equations, which implies the modification

of the velocity profile. The heat transfer enhancement is measured by the analysis of different

parameters.

The combination of two different methods of heat transfer enhancement is studied in this document.

An active method based on the appliance of electric field to a dielectric liquid flow, and a

minichannel as a passive method. The contact between the surface to be cooled and the fluid will be

increased due to the secondary flow produced by the electric field. Regarding the passive method,

the low hydraulic diameter of minichannels enhances the heat transfer coefficient and increases the

pressure drop.

A copper wire is the high voltage electrode placed at the top of the minichannel that injects electrical

charge through the liquid (mineral oil), producing an electrical force added to the momentum

equation. This induced secondary flow affects the primary flow: the velocity profile is modified near

the electrode. The electronic component to be cooled is a copper plate located at the bottom of the

minichannel and the dielectric liquid pumped is mineral oil. This heated plate is also considered as

ground [23].

The mechanism of the secondary flow is as follows: electrostatics and hydrodynamics are coupled in

the momentum equation, modifying the velocity profile. The charge injected from the high electrode

push the fluid neutral molecules towards the heated plate that is grounded. This vertical movement

of the fluid contributes to increasing the contact between the fluid and the plate to be cooled down.

In order to study the behaviour of the device, 2D and 3D simulations are performed.

5.2 Analysis parameters

This subsection will present the parameters to analyse the heat transfer enhancement and pressure

drop [23].

Convection Heat Transfer ℎ = �̇�

𝐴 (𝑇𝑤 − 𝑇𝑏) (32)

Where:

METHODOLOGY

36

�̇� 𝐴⁄ = 𝑞" (𝑊/𝑚2) is the heat flux of the heated plate

𝐴 (𝑚2) is the area of cross-section of the channel

𝑇𝑤 (𝐾) is the surface average temperature

𝑇𝑏 (𝐾) is the bulk temperature

Nusselt number 𝑁𝑢 = ℎ𝐷ℎ

𝑘 (5)

Where:

𝑁𝑢 (−) is the Nusselt number


𝐿 (𝑚) is the representative dimension


Performance Evaluation Criterion (PEC)

𝜂 = 𝑗 𝑗𝑠⁄

(𝑓 𝑓𝑠⁄ )1 3⁄ (33)

Where:

𝑗 = 𝑆𝑡 ∗ 𝑃𝑟2 3⁄ (34)

𝑓 =Δ𝑃

(𝐿⁄𝐷)∗((𝜌𝑢^2)⁄2) is the friction factor. (35)

𝑓𝑠 = 56.8/𝑅𝑒 is the friction factor for a square ducts. 𝑓𝑠 ∗ 𝑅𝑒 = 𝑃𝑜 where 𝑃𝑜 is the

Poiseuille number. [45] (36)

𝑆𝑡 = 𝑁𝑢 (𝑅𝑒 ∗ Pr)⁄ is the Staton number. (37)

𝑃𝑟 is the Prandtl number (6)

The suffix “s” refers to smooth surface or the condition without enhancement

5.3 Geometry

The geometry of the simulation is done in SpaceClaim. This chapter presents the geometrical

configuration for the 2D and 3D simulations.

5.3.1 2D simulations

The geometry for the 2D simulations (plane XY) is:

Figure 23 — Model 2D: general view: Plane XY

1. Inlet: flow inlet.

2. Outlet: flow outlet.

3. Walls: The walls confine the fluid. They are thermally and electrically insulated.

INLET OUTLET



4. Heated plate: the heated plate to be cooled. The heat flux considered is 10000

(𝑊/𝑚2)

5. Wire: high voltage electrode that produces the injection of the electric charge in the

oil.

The origin of the coordinates is located at the centre of the inlet boundary.

The wire is located at the top of the channel and the heated plate at the bottom from x = 0.020 m to

x = 0.070 m. Wire and heated plate are aligned and have the same length.

The square duct is 5 mm height and 100 mm length. The dimensions of the different parts and some

points/planes located along the x-axis to measure the temperature and the pressure are presented in

the next table.

Zone Length (mm) Measurement point Position of measurement

plane/point (mm)

Inlet 20 P1 x = 18

Test 50

T2 (22, -2.5)

T3 (55,-2.5)

T4 (68,-2.5)

Outlet 30 P2 x = 72

Table 2 — Model 2D: geometry and measurement points

5.3.2 3D simulations

The geometrical configuration of the simulation is:

Figure 24 — Model 3D: general view and bottom view

Where:

1. Inlet: flow inlet.

METHODOLOGY

38

2. Outlet: flow outlet.

3. Walls: The walls confine the fluid. They are thermally and electrically insulated.

4. Heated plate: the heated plate to be cooled. The heat flux considered is 10000

(𝑊/𝑚2)

5. Wire: high voltage electrode that produces the injection of the electric charge in the

oil. It is located on top of the test section and has the same length (50 mm). The

diameter of the wire is 0.3 mm. In the model, the wire is simulated as a plate of 50

mm long and 0.3 mm wide.

The origin of the coordinates is located at the centre of the cross-sectional area of the inlet.

The model is divided into three different parts: inlet, test and outlet section. The inlet section adapts

and hydrodynamically develops the fluid. The wire injects the charge through the fluid in the test

section, where the changes in the velocity profile are produced. In this part of the channel, the

charge injected and the electric field contributes to the momentum equation and modifies the

velocity profile. Finally, the fluid leaves the microchannel throughout the outlet section.

The square duct is 5x5 mm size and 74 mm length. Along the x-axis of the geometry, some

planes/points are located to measure different variables of the fluid.


plane/point (mm)

Inlet 4 P1 x = 2

Test 50

T2 (4, -2.5, 0)

T3 (29,-2.5,0)

T4 (52,-2.5,0)

Outlet 20 P2 x = 56

Table 3 — Geometry and measurement planes/points

5.4 Meshing

First, in order to determine and see the impact of the mesh in the final results, a mesh study is done

for three different cases. As it is explained before in chapter 4.4, the mesh plays a meaningful role in

a CFD simulation. Before defining the final mesh of our case study, an in depth analysis is performed

to see the impact of the discretization of the domain in the final results. The injection of charge is not

considered for this study. The results of this analysis are presented in chapter 6.3.2.

The geometry is modelled in SpaceClaim and meshed with Mesh Fluent. The academic version has a

total limit of cells of 512k for the mesh generation.

5.4.1 Mesh parameters for 2D simulations

Some considerations for the different parts of the domain are taken for the mesh generation:

Mesh type: hexahedral

Max element size = 0.1 mm in order to have 50 cells along the y-axis.



The statistics of the mesh are:

Mesh total elements: 50000

Mesh total nodes: 51051

Orthogonal quality: 0.999237

Maximum aspect ratio: 1.45265

5.4.2 Mesh parameters for 3D simulations

Some considerations for the different parts of the domain are taken for the mesh generation:

Mesh type: hexahedral

Max element size = 0.19 mm

Wire: maximum element size = 0.1 mm in order to have 3 elements along the z-axis in the

wire.

The statistics of the mesh are:

Mesh total elements: 508 680

Mesh total nodes: 545 972

Orthogonal quality: 0.99999

Maximum aspect ratio: 2.95733

Figure 25 — Model 3D: mesh-cross sectional view

Figure 26 shows the mesh transition between the test and the outlet sections. A refinement of the

mesh is needed in the test section to ensure the correct solution of the equations because the

injection of the charge will happen in this fluid part. In addition, a refinement at the wire is also

implemented.

METHODOLOGY

40

Figure 26 — Mesh: test zone

5.5 Solver Settings

The choice of solver settings is based on the previous training material done before the preparation

of these simulations and in the recommendations of the ANSYS User’s Guide.

The following settings were used for all the simulation:

Double Precision

Pressure-Based Solver

Solutions Methods:

- Pressure-velocity coupling: COUPLED

Spatial Discretization

- Gradient: Least Squares Cell Based

- Pressure: Body Force Weighted

- Momentum: Second Order Upwind

- Energy: Second Order Upwind

- Potential: Second Order Upwind

- UDS-1 (charge density): First Order Upwind

Convergence criteria:

Continuity: 10-3

Momentum: 10-3

Energy: 10-9

UDS-1 (charge density): 10-3

Potential: 10-9

Under-Relaxation Factors:

Flow Courant number: 5

Momentum: 0.75

Pressure: 0.75

Density: 0.8

Body forces: 1



Energy: 0.9

Potential: 1

UDS-1 (charge density): 0.7

The Flow Courant Number (CFL) is a mathematical convergence condition for the stability when

solving convection or wave phenomena problems. It is used for coupled pressure-velocity scheme

and relates the velocity with the time step and the length of the elements of the mesh (𝐶𝐹𝐿 =

𝑢∆𝑡/∆𝑥). In our case, to improve the stability of the solution, the CFL number is set to 5 [81].

5.6 Assumptions

The following points are considered for all the simulations:

Transient simulations.

Laminar fully developed flow.

Boussinesq approximation for density. The density varies only with temperature in the

buoyancy term in the y-momentum equation, as it explained in chapter 3.

Gravitational acceleration acting in negative y-direction at a rate of 9.81 m/s2.

The inlet temperature equal to 293.15 K.

Pressure gauge outlet equal to 0 Pa.

Heat flux of the heated plate equal to 10 000 W/m2. This heat flux corresponds to 2.5 W.

Adiabatic walls. No slip boundary condition: tangential fluid velocity equal to wall velocity,

and null normal velocity.

The temperature differences expected between inlet and outlet are low, so in case of having

backflow at the outlet of the minichannel, the “temperature backflow” is set equal to 296 K.

Three different materials are used: mineral oil as fluid, copper for the wire and the heated

plate and an insulator for the walls of the pipe.

An overall description of the properties of mineral oils is presented in chapter 2.5.

Summarizing, mineral oil is adequate as refrigerant due to its low viscosity, good electrical

properties and low electrical permittivity. It guarantees a low operation temperature, avoids

the problems of oxidation and corrosion and reduces environmental contamination like dust.

Conductor material is used for the wire and for the heated plate. Copper is a conductor

material with a high conductivity and low resistivity value. It allows the flow of charge freely

on its surface. An in depth explanation of electrical conductor is made in chapter 6.1, in

order to explain the phenomena observed in the simulations.

Finally, an insulator material is used for the walls of the pipe. Wood is selected due to its

good insulator properties. An insulator does not allow the free flow of charge and does not

METHODOLOGY

42

permit the electrical conduction. They are characterised by a high value of electrical

resistivity and a very low value of electrical conductivity [74][75].

The properties of the materials are:

OIL

Property Nomenclature Units Value

Density 𝜌 Kg/m3 856

Viscosity 𝜇 Pa s 0.03

Heat capacity 𝑐𝑝 J/kg K 1850

Thermal conductivity 𝑘 W/m K 0.14

Electric permittivity 𝜀 F/m 1.95 x 10-11

Electrical conductivity 𝐾 S/m 3.3 x 10-12

Table 4 — Thermophysical properties of oil at 293.15K [23]

As the fluid temperature differences are low, the thermos-physical properties of mineral oil are

considered as constant. The data of copper and wood are obtained from the Fluent database.

COPPER





Electrical conductivity 𝐾 S/m 5.8 x 107

Electrical resistivity 𝜌 Ω ∙ m 1.7 x 10-8

Table 5 — Thermophysical properties of copper

WOOD





Electrical conductivity 𝐾 S/m 1 x 10-30

Electrical resistivity 𝜌 Ω ∙ m 1 x 1030

Table 6 — Thermophysical properties of wood

The boundary conditions are:

Boundary location

Boundary condition Equations

Inlet Velocity inlet 𝑢 = 𝑢𝑖𝑛 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑥= 0

𝜕𝜙

𝜕𝑥= 0

Outlet Pressure outlet 𝜕𝑢

𝜕𝑥= 0

𝜕𝑣

𝜕𝑥= 0

𝜕𝑤

𝜕𝑥= 0

𝜕𝑧

𝜕𝑥= 0

𝜕𝜙

𝜕𝑥= 0

Heated plate Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑛= 0 𝜙 = 0

Wire Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝑧 = 𝑧0 𝜙 = 𝑉0

Walls Wall 𝑢 = 0 𝑣 = 0 𝑤 = 0 𝜕𝑧

𝜕𝑛= 0

𝜕𝜙

𝜕𝑛= 0

Table 7 — Boundary conditions



The selection of the time step can be a tricky process. It is necessary to find a balance

between the duration of the simulation and meet all the convergences criteria in each

time step. For the 2D simulations, the time step selected is 0.001 s and for the 3D

simulations, the time step selected is 0.001 s for the firsts 0.1 s of simulation to ensure a

correct initialization of the flow and after a time step equal to 0.005 s are used. In some

moments of the 2D simulations, the time step should be reduced to 5 x 10-4 s and 2 x 10-4

s to reach the convergences criteria (cases 1.c and 1.d). In conclusion, all the

convergences criteria are met for each time step so the validity and the accuracy of the

results are guaranteed.

5.7 Input data

5.7.1 2D simulations: scenarios

The next 4 different scenarios are simulated in order to carry out a parametric analysis:

Case 𝒖𝒊𝒏 (𝒎/𝒔) initial Re 𝒛𝟎 (𝑪/𝒌𝒈) 𝝓𝟎 (𝒌𝑽) 1.a

0.01 1.43 6 x 10-3

0

1.b 5

1.c 10

1.d 15

Table 8 — 2D: Simulation scenarios

5.7.2 3D simulations: scenarios

The following input data are considered for the simulations:

Case 𝒖𝒊𝒏 (𝒎/𝒔) initial Re 𝒛𝟎 (𝑪/𝒌𝒈) 𝝓𝟎 (𝒌𝑽) 3.a

0.01 1.43 6 x 10-3

0

3.b 5

3.c 10

3.d 15

3.e

0.05 7.13 6 x 10-3

0

3.f 5

3.g 10

3.h 15

Table 9 — 3D: Simulation scenarios

5.8 UDF and UDS implementation

This chapter explains the steps necessaries to calculate the electrical equations and how they are

coupled with hydrodynamics equations.

Regarding the potential model presented in chapter 4.2, ANSYS Fluent solves the electric potential

equation [83]:

∇ ∙ (𝜀∇𝜙) + 𝑆 = 0 (29)

METHODOLOGY

44

Remembering the expression defined in section 3.2:

𝐄 = −∇𝜙 (20)

∇ · (𝜀∇𝜙) = −𝜌𝑒 (21)

The volumetric charge density could be expressed as:

𝜌𝑒 = 𝜌𝑧 (22)

Where:

𝑧 (𝐶/𝐾𝑔) is the charge per unit mass

This potential model solves the equation (21). A DEFINE_SOURCE UDF is coded to add the source

term (𝑆 = 𝜌𝑒 = 𝜌𝑧) to the Poisson equation.

After solving the potential equation, it is needed to compute the electric field as the minus gradient

of electric potential (20).

In order to introduce the space charge equation, a User-Defined Scalar transport equation is used.

𝜕𝜌𝜙𝑘


(31)

Remembering the exposed expressions exposed in chapter 3.2:

The charge conservation equation is:

𝜕𝜌𝑒

𝜕𝑡+ ∇ · 𝑱 = 0 (23)

Where:

𝑱 (𝐴/𝑚2) is the current density

𝑱 = 𝐾𝑬 + 𝜌𝑒𝒖 (24)

Considering the electrostatic relationship (18) and (19), the conservation equation of the charge (23)

can be written:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 + 𝑬 ∙ (

𝐾

𝜀 ∇𝜀 − ∇𝐾) (25)

If the electrical properties of the fluid 𝐾 and 𝜀 are constant, eq. (25) reduces to:

𝜕𝜌𝑧

𝜕𝑡+ ∇ · (𝜌𝑧𝒖) = −

𝐾

𝜀𝜌𝑧 (26)

For the UDS equation, the scalar 𝜙𝑘 is 𝑧 (𝐶/𝑘𝑔) and only the unsteady and convection terms are

considered. The term −𝐾

𝜀𝜌𝑧 is the source term 𝑆𝜙𝑘

of the UDS.

A DEFINE_SOURCE UDF is coded to introduce the source term of the UDS equation:

𝑆𝜙𝑘= −

𝐾

𝜀𝜌𝑧 =

𝐾 ∇ · (𝜀∇𝜙)

𝜀=

∫ −∇ ∙ (𝐾𝑬)𝑑𝑉

∫ 𝑑𝑉=

∫ −∇ ∙ 𝐾𝑬𝒏𝜕𝑉

𝑉= −

Σ𝐾𝑬𝑨

𝑉 (39)



When equations (20), (21) and (26) are solved, it is necessary to introduce the Coulomb force in the

momentum equation. Three different DEFINE_SOURCE UDFs are coded to couple the electrostatics

and hydrodynamics for the three momentum conservation equations.

The use of a DEFINE_ADJUST Fluent macro is needed to calculate the electric field and the source

term for charge conservation equation at each iteration. This type of ANSYS Fluent macro is called at

the beginning of each iteration before transport equations are solved.

A UDF need to be interpreted or compiled before makes use of them. In our case, for Windows

operating systems, Visual Studio is used to compile the codes.

5.9 Limitations

First, it is important to remark that the student version of ANSYS Fluent is used for these simulations.

This version has an important limitation in the discretization of the domain: it is not allowed to create

a mesh with more than 512K cells.

Another important limitation is the computational resources. For these kinds of simulations, the

computational cost usually tends to be very high, so specialized workstations are typically used. For

this master thesis, the computational resources are limited and for further work related to this case

study, it is recommended to use a proper professional workstation for CFD simulations and the

professional ANSYS Fluent version to avoid the limitation of the mesh generation, if a refinement is

needed.

The computational resources available are:

Intel® Core™ i7-4500U, 2.4Ghz, 4 Gb RAM

2 x Intel® Core™ i5-6500, 3.20 GHz, 8Gb RAM

The computational cost of the simulations is high. Even if some considerations are taken in order to

improve the simulation time, the cases 3.a to 3.d took more than 85-95 hours, the cases 3.e to 3.h

took about 60-70 hours. The 2D cases took about 30-40 hours and the steady state cases for the

mesh study are quicker and took about 10 minutes.

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46



6. RESULTS AND DISCUSSION

6.1 Parametric study: general comments

First of all, before analysing the different simulations and scenarios proposed, it is important to

remark in this parametric study some comments about the passive heat transfer cooling method: the

use of a minichannel whereby the dielectric liquid flows. This parameter is not been numerically

analysed in the simulations but some qualitative comments are presented.

The expression of equation (5) shows that the heat transfer coefficient is a function of Nusselt

number, fluid thermal conductivity and hydraulic diameter. For fully developed laminar flows,

Nusselt number is considered as a constant value under a theoretical point of view. For a square duct

with a uniform heat flux, the Nusselt number is 3.61 [80].

Considering the fully developed region, the heat transfer coefficient depends on the duct geometry

and fluid thermal properties. For a given flow with a constant thermal conductivity, heat transfer

coefficient grows if the characteristic length decreases. For non-circular pipes, the hydraulic diameter

is used as characteristic length. Thus, the use of a channel with a reduced hydraulic diameter

guarantees the improvement of the heat transfer coefficient for a given fluid for a laminar fully

developed flow. It is also important to mention that the hydraulic diameter depends on the

geometry of the cross-sectional area of the minichannel (4).

ℎ ∝1

𝐷ℎ

Concerning the pressure drop, following the friction factor expression for laminar flow (35) and

square ducts, pressure drop varies inversely with Reynolds number. As the Reynolds number is

directly proportional to hydraulic diameter, the pressure drop increases when the hydraulic diameter

is reduced.

∆P ∝1

𝑅𝑒 → ∆P ∝

1

𝐷ℎ

So it is important to find a balance between these two phenomena to ensure an enhancement of the

heat transfer coefficient without penalizing in excess the pressure drop to drive the fluid through the

channel [77].

After this qualitative analysis of the heat transfer enhancement using a minichannel, this document is

focused in carry out an in-depth analysis of the active cooling method: the use of

electrohydrodynamically induced convection.

RESULTS AND DISCUSSION

48

6.2 2D simulations

6.2.1 2D Parametric study: General comments

In order to perform a parametric study of the device behaviour, some different scenarios are

proposed. Before start analysing some special facts and particularities for the different cases studies,

some general comments about the behaviour observed are explained in the following chapter.

First of all, it is important to remind the mechanism of the device: a high voltage electrode located on

the top of the minichannel injects charge through the dielectric fluid. This charge injected creates a

Coulomb force added to the momentum equation. Consequently, the velocity profile is modified: the

charged particles push the neutral molecules of the fluid towards the heated plate that is grounded.

This general analysis starts explaining the electrical part of the device. In order to show and explain

the concepts exposed, some contours of the scenario with 15 kV applied are provided.

The charge is injected from the electrode through the fluid. Figure 27 shows the electric current

coloured by the electric charge (the charge at the wire is not plotted). It can be observed how the

electric charge moves towards the heated plate that becomes electrically charged. The heated plate

is grounded and as it is made of copper, an electric conductor material, the charge is distributed

across its entire surface. Electric conductor materials allow the charges to move about freely, and the

charge is uniformly distributed across its entire surface [74].

Figure 27 — Scenario 1.d: Electric current vectors coloured by charge density (C/kg) injected from the wire and collected by

the heated plate (t = 10 s). Plane XY, x = 0.070m. Wire and heated plate right edge.

It is necessary to remark some general concepts about electrical conductors. There is no electric field

inside a conductor. As it is mentioned before, the electrons move easily within an electric conductor

material and if an electric field appears inside the conductor, they will rapidly move and rearrange

themselves owing to cancel this electric field and reach the equilibrium state. This quickly rearranged

is related to the very low resistivity of conductor materials. Resistivity determines how well the

conductor materials resist the flow of current. So, a low value of this parameter guarantees good



electricity conduction, demonstrating the rapid rearranged of the charges inside a conductor and,

thus, that there is no electric field inside a conductor [74] [75].

After explaining this principle, it can be demonstrated that the charge will remain in the surface of

the material conductor. This phenomenon is explained thanks to Gauss’s Law:

∫ 𝑬 ∙ 𝒅𝑺𝐴

= 1

𝜖0∫ 𝜌 𝑑𝑉

𝑉

(40)

If there is no electric field inside a conductor, the charge inside will be always zero and consequently

will reside entirely on its surface. In our case, part of the charge injected from the electrode is

collected by the heated plate and remain on its surface, as Figure 27 shows.

Figure 28 shows that the electric field is perpendicular to the wire and the heated plate. The Gauss

Law also explains this phenomenon: if a tangential electric field appears near the surface of a

charged conductor material, the charges will rearrange in order to reach the equilibrium state and

cancel this tangential electric field. So, for conductor materials the electric field is always normal to

the surface of the conductor. In addition, in this figure it can be observed how the electric field

increases with time due to the charge injected [74] [75].

(a) (b)

(c) (d)

Figure 28 — y-component of electric field (Ey): (a) t = 0.005s. (b) t = 10 s. (c) t = 20 s. (d) t = 26 s. Plane XY, x = 0.070 m. Wire

and heated plate right edge.


50

Figure 29 shows the appearance of induced charges near the electrode. The charge is injected

gradually and part of it is absorbed by the heated plate because it is made of conductor material

(Figure 27). In order to meet the charge conservation equation (26), induced negatives charges

appear near the electrode. Under a theoretical point of view, when a dielectric fluid is located

between a high voltage electrode and a surface grounded, the molecules of the fluid are polarized.

These polarized molecules are aligned with the electric field applied and all the molecules far enough

the electrode and the plate are neutralized between them. Nevertheless, charges induced near the

electrode remains. The dielectric fluid remains electrically neutral but suffers charge redistribution.

In this case, mineral oil is a liquid composed of carbon atoms bonded to hydrogen atoms. These

molecules create a very low polarity so in the absence of an electric field, they are considered as

neutral molecules. However, the appliance of a high electric field causes a sufficient separation of the

positive and negative charges of the molecules to make them slightly polar. Thus, induced charges

appear near the wire [76].

Figure 29 — Scenario 1.d: Induced charges near the electrode. Plane XY, x = 0.070 m, t = 26 s

6.2.2 2D Parametric study: scenarios 1.a to 1.d

Scenario 1.a

First of all, it is necessary to explain the scenario without the appliance of any voltage. The steady

state is reached after 22 s. The velocity profile corresponds to a fully developed laminar flow in 2D.

Next contour shows the velocity profile along the minichannel.

Figure 30 — Scenario 1.a: u-velocity profile at the middle plane. Plane XY, t = 22 s

Figure 30 shows how the flow is developed. The 𝑢-velocity of a laminar flow has a parabolic profile

reaching its maximum value at the middle of the duct. Considering only 2 dimensions, the 𝑢-velocity

is a function of the y-coordinate.



Figure 31 — Scenario 1.a: v-velocity profile at the middle plane. Plane XY, t = 5 s

According to the velocity profile of laminar flow, the 𝑢-velocity is dominant over the 𝑣-velocity. In

these simulations, the Boussinesq approximation is used so the density is considered as constant for

all the equations except for the gravity term in the momentum equation. So, the 𝑣-velocity is

influenced by the density variations where the temperature differences of the fluid occur. Figure 31

perfectly shows the variation of the 𝑣-velocity along the test section, where the heat transfer occurs.

A further explanation of this phenomenon is done for the 3D cases (section 6.3.2). The hot fluid with

lower density tends to go up and the cold fluid goes down.

Prandtl Number Grashof Number Rayleigh Number

396.43 38.16 1.51 x 104

Table 10 — Scenario 1.a: Dimensionless numbers

The dimensionless numbers presented in chapter 2.1 are computed for this scenario. The Grashof

number represents the ratio between the buoyancy forces and the viscous forces, and the Rayleigh

number quantifies the importance between the effects of the buoyancy forces and the effects of the

viscosity and thermal conduction. Due to the values obtained, it can be concluded that convection

occurs.

Regarding the temperature profile, Figure 32 shows how the temperature of the fluid increases since

it passes through the test section. The fluid extracts heat from the heated plate and drives it through

the outlet of the duct.

Figure 32 — Scenario 1.a: Temperature contour at the middle plane (XY plane). Plane XY, t = 22 s.

Finally, Figure 33 shows the pressure drop needed to drive the flow through the channel.

Figure 33 — Scenario 1.a: Pressure contour at the middle plane. Plane XY, t = 22 s.


52

Scenarios 1.b to 1.d

After presenting the overall case study, scenarios 1.b to 1.d, in which the active cooling method is

applied, are going to be studied. The voltage applied at the wire is 5, 10 and 15 kV (1.b, 1.c and 1.d

respectively).

Next graphs show the average and the maximum temperature of the heated plate after 26.5 s.

(a) (b)

Figure 34 — Scenarios 1.a to 1.d: Average temperature and maximum temperature of the heated plate

First of all, the average temperature of the heated plate for all the scenarios is lower than in case

without the appliance of the voltage and the maximum temperature reduction is observed in

scenario 1.c. The temperature reduction for all the scenarios is significant, and the best reduction

obtained is almost 14 K in scenario 1.c. Regarding the maximum temperatures of the heated plate,

only scenario 1.c reaches to decrease the maximum temperature (4 K).

Analysing Figure 34, it can be concluded that the best behaviour of the device is obtained with the

configuration 1.c. (10 kV) due to the decrease of the average and the maximum temperature of the

heated plate.

(a) (b)



(c)

Figure 35 — Scenarios 1.a to 1.d: (a) Temperature measured at point T2 (22, -2.5, 0) mm. (b) Temperature measured at

point T3 (45, -2.5, 0) mm. (c) Temperature measured at point T4 (68, -2.5, 0) mm.

Focused in the temperatures measured along the heated plate, oscillations are observed in Figure 35

(a), (b) and (c). This fact is due to the development of the electric field and its contribution to the

conservation of momentum equations (27). The interaction between the main flow momentum and

the momentum that is applied by the injection of charge is higher when the voltage increases owing

to the oscillations observed.

First, the Poisson equation is solved for the electric potential following the equation (21). Then, the

electric field is calculated as the gradient of the voltage (20), and finally the equation of conservation

of charge density (26) is solved. So, the electric field is gradually increasing with the injection of

charge and depends on the voltage applied at the wire.

With the purpose of understanding the evolution of the simulation with time, some different

contours at different time steps of scenario 1.d are presented to see the interaction between the

electric field and the flow in the test section. This scenario is selected because the modifications in

the velocity profile are more noticeable. The same reasoning can be done for scenarios 1.b and 1.c.

At t = 0.02 s, a positive value of electric field in x-direction is observed at the right edge of the wire (x

= 0.070 m) and a negative value at the left edge (x = 0.020 m). At the edges of the heated plate,

values of electric field in x-direction appear with opposite sign (positive value at x = 0.020 m and

negative value at x = 0.070 m). Since the charge injected is almost negligible, the velocity profile

observed is a typical laminar velocity profile without modifications, similar to Figure 30.

Figure 36 — Scenario 1.d: x- component of electric field (V/m) along the minichannel. Plane XY

At t = 6 s, the oscillations are appreciable in the three points of measure, so the value of Coulomb

force added to the momentum equation starts to be noticeable. The electric field present at the

outlet of the test section is developed towards the entry and the same phenomenon but with


54

opposite direction happens with the electric field present at the inlet of the test section, in order to

meet the conservation of charge density equation. Induced charges appear near the electrode due to

the positive charges located at the surface of the heated plate. At the outlet of the test section (x =

0.070 m), these negative charges near the electrode create a negative contribution to the x-

momentum equation due to its interaction with the positive electric field present in the upper part of

the minichannel. At the inlet, the electric field at the upper part is negative, so the fluid is pushed

towards the outlet. At the bottom of the minichannel, the same phenomenon is observed: the fluid is

pushed towards the inlet section at the outlet of the test zone (positive charge and negative x-

component of electric field) and is pushed in x-positive direction at the entry of the test section

(positive charge and positive x-component of electric field).

Charge injected starts to be present in the fluid and the flow is modified. The same phenomenon

explained before happens along the test section: where the product of the value electric field in x-

direction and the charge injected in the fluid is positive, the fluid is pushed towards the outlet (in x-

positive direction). But, when the value of this product is negative, the fluid is pushed towards the

inlet.

As a result of this interaction, some different “waves” are created along the test section. The fluid

pushed by the Coulomb force towards the outlet of the channel (positive x-direction) is modified

where the contribution of the Coulomb force to the momentum equation is negative: the fluid that

moves in the positive direction of the x-axis is altered by the fluid that is pushed in the negative

direction of the x-axis as a result of the negative product of the charge and the electric field. This

phenomenon produces these oscillations in the velocity profile. Some contours at t = 16 s are

presented in order to explain better this fact. They show perfectly the phenomenon explained. The

positive contribution of Coulomb force to the momentum equation push the fluid towards the outlet

and is interrupted and varied by the fluid with negative velocity value just below, creating a wave

(Figure 37). This negative velocity is produced due to the negative value of electric field at the

bottom of the minichannel and the positive value of the charge near the plate.

(a) (b)

(c) (d)

Figure 37 — Scenario 1.d: Different contours at t = 16 s along de minichannel: (a) x- component of electric field (V/m). (b)

Charge density (C/kg). (c) Temperature (K). (d) u-velocity component (m/s). Plane XY, t = 16 s.



This modification of the velocity profile is responsible for the temperature oscillations of the plate.

When the fluid is modified by the localized flow that moves in the negative direction of the x-axis,

the temperature profile is also altered. The flow with positive velocity manages to extract heat from

the plate while the recirculated flow with negative velocity re-introduces hot fluid towards the test

zone. This fact increases the maximum temperature measured in the heated plate. Figure 37 and

Figure 38 perfectly show this fact at the outlet of the test section (x = 0.070 m). The negative velocity

at the bottom and top of the minichannel (x = 0.070 m, y = +/- 0.0025 m) reintroduces hot fluid.

When the electric field is almost totally developed and stable, we can observe that the negative

charge is accumulated at the upper part of the channel and the positive charge is located near the

plate. The velocity profile of the first part of the test section is more homogeneous and near the

electrode and near the plate, the fluid is pushed towards the outlet section. The “waves” of the flow

disappear.

(a) (b)

(c) (d)

Figure 38 — Scenario 1.d: Different contours at t = 26.5 s along de minichannel: (a) x- component of electric field (V/m). (b)

Charge density (C/kg). (c) Temperature (K). (d) u-velocity component (m/s). Plane XY, t = 26.5 s.

Regarding the 𝑣-velocity, the same reasoning can be done for the interaction of the main flow

momentum and the momentum applied by the injection of charge. The electric field in y-direction is

always negative in the test section so a negative value of 𝑣-velocity is observed where the charge is

positive, pushing down the fluid. Nevertheless, as it is explained before, the 𝑢-velocity is dominant.

(a) (b)

Figure 39 — Scenario 1.d: Different contours at t = 16 s along de minichannel: (a) y- component of electric field (V/m).

(b) v-velocity component (m/s). Plane XY, t = 16 s.


56

Summarizing, the electric field produces some recirculation issues along the minichannel that are

more noteworthy when the voltage applied is greater. The recirculation in scenario 1.d is greater

than in scenario 1.b.

It is important to analyse the Joule heating generated. The Joule heating is equal to the electric

current that passes through a conductor multiplied by the voltage (9). In the model, the biggest

source of Joule heating is the wire. But, since it is treated as an adiabatic wall, it does not introduce

heat into the domain. Nevertheless, different behaviour is observed at the heated plate due to the

accumulation of charge. As it is explained before in section 6.2.1, the heated plate becomes

electrically charged due to the material that is made of (copper). As the heated plate is not

considered as an adiabatic wall because it introduces the heat flux into the domain, the Joule heating

generated contributes to the increase of the temperature of the plate. However, this contribution to

the augmentation of the temperature is lower than the temperature increase that produces the

recirculation problems. In the analysis of 3D cases, a deep explanation of this fact is presented.

Analysis of the parameters

The parameters defined in chapter 5.2 are computed. First, the Nusselt number is the ratio of

convective to conductive heat transfer. A high value of Nu means that convection is dominant in the

heat transfer process. To compute it, the expression (5) defined in chapter 2.1 is used. The ratio

Nu/Nu0 compares the Nusselt number obtained for the 3 scenarios with the appliance of the voltage

and the scenario with zero voltage applied. This ratio allows determining if there is a heat transfer

enhancement. A value greater than unity means that the heat transfer rate enhances.

Figure 40 — Ratio of Nu/Nu0 vs. applied voltage (kV) for scenarios 1.a to 1.d

The average temperature of the heated plate is lower for the 3 scenarios with applied voltage, and

thus the heat transfer coefficient is higher than the computed for scenario with zero voltage.

Regarding the pressure drop, Figure 41 shows that it increases with the voltage applied. This growth

is accentuated when the applied voltage is greater because the electric field is also greater, and

therefore the contribution of the Coulomb force in the conservation of momentum equations is also

higher.



The pressure drop observed is high and is due to the big interaction between the main flow

momentum and the momentum applied by the injection of charge. The modification in the velocity

profile is important and thus, the pressure needed to drive the fluid considerably increases.

Figure 41 — Ratio of ∆𝑃/∆𝑃0 vs. applied voltage (kV) for scenarios 1.a to 1.d

When analysing a cooling heat transfer method, it is not only important to evaluate the heat transfer

enhancement. It is necessary to study the increase in the pumping requirements. Performance

Evaluation Criterion (PEC) relates the heat transfer enhancement to the pumping power needed to

drive the fluid along the duct. The heat transfer coefficient can be higher but if the increase in the

pumping requirements is meaningful, the efficiency of the cooling technique can be lower. So, values

greater than unity mean that the rate of heat transfer enhancement is higher than pressure drop

needed to pump the fluid.

Figure 42 — PEC vs. applied voltage (kV) for scenarios 1.a to 1.d

Scenarios 1.b and 1.c show a better efficiency than the scenario 1.a without voltage applied. The

heat transfer enhancement is significant and the pressure drop increase is not too high. However, in

scenario 1.d, even if a heat transfer enhancement is noticed due to the value greater than unity of

ratio of Nu/Nu0, the pressure drop increase is high. The effects of the electric field are noteworthy

and the higher recirculation issues produces that the pressure needed to drive the fluid is high, and

the value of efficiency is lower than the one computed for the scenario with zero voltage condition.


58

6.3 3D simulations

6.3.1 3D Parametric study: General comments

After analysing the model in 2D, a three-dimensional model is prepared. First, a previous mesh study

is performed in order to determine the best discretization of the domain to guarantee accurate

results. It is extremely recommended in CFD calculations find a proper balance between the accuracy

of the results desired and the computational cost to achieve them.

6.3.2 3D Parametric study: Mesh study

First of all, nine different scenarios are analysed in order to determine the relevance of the

discretization of the domain in the accuracy of a CFD simulation. Three different steady-state

scenarios are simulated for three different hexahedral meshes.

The geometry used for this preliminary study is:


plane/point (mm)

Inlet 20 Inlet_T1 (10, -2.5, 0)

P1 x = 18

Test 50

T2 (22, -2.5, 0)

T3 (50,-2.5,0)

T4 (68,-2.5,0)

Outlet 25 Outlet_T2 (85, -2.5, 0)

P2 x = 72

Table 11 — Mesh study: geometry and measurement planes/points

Table 12 overviews the different parameters used.

Case Grid Max. element face size (mm)

Total elements Orthogonal

quality Maximum aspect

ratio

2.1 10 x 10 0.5 19000 1 1.73205

2.2 20 x 20 0.25 152000 1 1.73205

2.3 29 x 29 0.17 470119 0.999942 1.75831 Table 12 — Mesh study: parameters

(a) (b) (c)

Figure 43 — Mesh study: Details of different mesh studied at the outlet section: (a) Case 2.1. (b) Case 2.2. (c) Case 2.3



For this study, the solver settings used are the mentioned ones in chapter 5.5, with some

considerations: the Boussinesq approximation for density is not considered.

First, a hexahedral mesh is selected for these simulations for different reasons. The square geometry

of the channel helps to discretise the domain with hexahedral elements over other types of mesh

elements. Tetrahedral elements can fix better complex geometries, curves and shapes but can

introduce distortion in the elements, for example skewed triangles faces.

Another point to use a hexahedral mesh is the perfect alignment of the elements with the fluid flow

owing to reduce and minimize numerical diffusion. A good mesh is associated with the physics you

want to solve, and for the laminar flow of these simulations, hexahedral cells will obtain good

accurate results. In addition, the use of hexahedral cells should reduce the computational cost during

the simulation and usually need fewer elements to discretise the domain. Thus, due to the limited

resources available to accomplish these simulations (limited cells and huge simulation time), a

hexahedral mesh is preferred.

Three different velocity inlets are used to analyse the accuracy of the mesh definition. The

temperature is measured in 5 different points along the x-axis and the pressure is measured at the

inlet and the outlet of the test section. Table 13 and Figure 44 summarize the results obtained:

1.1 1.2 1.3

a b c a b c a b c

𝒖 𝒎/𝒔 0.01 0.05 0.1 0.01 0.05 0.1 0.01 0.05 0.1

Re — 1,43 7,13 14,27 1,43 7,13 14,27 1,43 7,13 14,27

𝒇 — 41,45 8,30 4,15 42,61 8,53 4,27 42,82 8,58 4,29

𝒇𝒔 — 39,81 7,96 3,98 39,81 7,96 3,98 39,81 7,96 3,98

Error % 7,03% 7,16% 7,16% 6,57% 6,69% 6,70% 3,94% 4,08% 4,08%

Table 13 — Mesh study: friction factors

Next graphs show the temperatures measured along the minichannel at the points presented in

Table 11:

(a) (b)


60

(c)

Figure 44 — Mesh study: temperatures measured at different point locations defined in Table 11. (a) Case a. (b) Case b. (c)

Case c.

Analysing these graphs, the mesh corresponding to the case 2.1 can be discarded due to the disparity

of the results regarding the two other cases, in which the differences are low. It can be observed as

well that temperature differences are larger in case c (Re = 14.27) than in case a (Re = 1.43). In order

to determine what mesh grid is more accurate, the friction factor (𝑓) is calculated and compared to

friction factor calculated with the classical correlation (𝑓𝑠). The expressions (35) and (36) presented in

chapter 5.2 are used.

As it is expected, errors between the friction factor performed with classical correlation and the

obtained in the simulations of case 2.3 are lower than the ones of 2.1 and 2.2, due to the refinement

of the mesh.

This mesh study does not pretend to study the velocity profile and the temperatures obtained, but

some comments can be done in order to discuss the accuracy of the values as a result of the mesh

quality. As a consequence of the velocity profile of fully developed laminar flow, the maximum

velocity is reached at the centre of the duct and the velocity near the walls trends to be null. This fact

explains why the maximum temperature of the heated plate is reached at the corners of the heated

plate. The different maximum temperatures obtained for the heated plate are 390.13 K, 392.06 K

and 392.08 K, for the 3 different meshes respectively (2.1, 2.2, 2.3) for the case with a velocity inlet

equal to 0.01 m/s. This disparity of the maximum temperature obtained in the mesh with the poorest

quality (2.1) confirms that the finest discretization of the domain guarantees the accuracy of the

results obtained.

A detailed explanation about the velocity profile and the temperatures obtained is going to be

presented in chapters 6.3.3 and 6.3.4.

These steady state simulations not only show the importance of the discretization of the domain in

order to have good accurate results, but also guarantee the validity of the model due to the errors

obtained for the frictional factor compared to the theoretical ones.

As a general conclusion, it is evident that the most accurate results are obtained with the use of the

mesh as finest as possible. However, it is important to find a balance between the discretization of

the domain and the computational cost of the simulations.



6.3.3 3D Parametric study: scenarios 3.a to 3.d

Scenario 3.a

At first, the temperatures and the velocity profile are going to be studied for the scenario 3.a without

the appliance of the voltage at the wire.

As it is explained before for the 2D cases, the flow studied in the simulations is a fully developed

laminar flow. Figure 45 shows how flow is developed.

Figure 45 — Scenario 3.a: Velocity u profile along the minichannel. XY planes at x = 0.004 m, x = 0.029 m, x = 0.054 m, t = 22

s.

In the previous Figure 45, it can be observed the typical laminar flow velocity profile mentioned

before. Due to the no-slip condition at the walls, the 𝑢-velocity increases as it moves away from the

walls until reach the centre of the minichannel, conforming a parabolic profile. The velocity reaches

its maximum value at the middle of the duct owing to meet the conservation of mass principle. The

origin of coordinates is considered at the middle point of cross-sectional area of the inlet boundary

section. So, the 𝑢-velocity decreases for +/- y and z coordinates.

According to the velocity along the y-axis, some contours are plotted in order to explain the effects

of the buoyance forces.

Figure 46 — Scenario 3.a: v-velocity profile along the minichannel. XY planes at x = 0.0165 m, x = 0.029m, x = 0.0415 m, x =

0.054 m and x = 0.064 m, t = 22 s.

The Boussinesq approximation is used for density. This approximation considers a constant density

for all the equations except for the gravity term in the conservation of momentum equation. The

gravity force is considered in the y-momentum equation, so the y-axis velocity (𝑣) is modified. As it is


62

explained in chapter 3.1, this approximation is useful when the temperature differences are low,

thus it is desirable for this case study.

Figure 47 shows the evolution of 𝑣-velocity along the minichannel. The velocity is increasing along

the channel at the same time that the heat transfer between the heated plate and the fluid occurs.

Next detailed contours show better this fact:

(a) (b) (c)

(d) (e) (f)

Figure 47 — Scenario 3.a: v_velocity profile along the minichannel. (a) XY plane, x = 0.0165 m, (b) XY plane, x = 0.029 m, (c)

XY plane, x = 0.0415 m, (d) XY plane, x = 0.054 m, (e) XY plane, x = 0.064 m, (f) XY plane, x = 0.074 m, t = 22 s.

As the fluid passes through the test section in x positive direction (Figure 47 a to f), its temperature

increases and the temperature of the heated plate decreases. Due to the laminar velocity profile, the

𝑢-velocity at the centre of the channel is considerably greater than the 𝑣-velocity. This explains that

the maximum velocity values on the y-axis are close to the walls. In fact, as it moves away from the

vertical axis in z-direction, the velocity in x decreases and the effects of the buoyancy forces are

greater, increasing 𝑣-velocity. Therefore, 𝑣-velocity of the fluid in the centre of the channel



decreases and near the walls increases. In addition, as the 𝑢-velocity is greater in the centre of the

duct, the fluid is replaced quicker than in the zones near the walls and the heat extracted with this

colder fluid is higher. When the fluid abandons the test section (figures e and f), the 𝑣-velocity

continues to rise near the walls.

As it is explained in chapter 2.1, the Grashof number computes the ratio between the buoyancy

forces and viscous forces acting on a fluid. In this case, the Grashof number is 35.38. It is not a high

value but remarks the effect of buoyancy forces over viscous forces. Both values of Grashof number

and Rayleigh number computed are very similar to the ones computed for the two dimensional case.

So, the same conclusion can be done: the role of convection in heat transfer is significant due to the

Rayleigh number is greater than the critical Rayleigh number mentioned in chapter 2.1 (1700).

The dimensionless numbers presented in chapter 2.1 are computed for this scenario.


396.43 35.38 1.40 x 104

Table 14 — Scenario 3.a: Dimensionless numbers

Concerning now the temperature analysis, next contour shows the temperature of the heated plate

after reached the steady state. Just to remind, in chapter 2.2, figure 8 shows how the temperature of

an electric component rises during a transient period, until its stabilization in the steady state. In this

experiment, the heated plate takes 22 seconds to reach this steady state and the temperature

contour is as follow:

Figure 48 — Scenario 3.a: Temperature contour of the heated plate reached the steady state. Top view, plane XZ, t = 21 s.

As it is expected owing to the velocity profile of laminar flow, the 𝑢-velocity at the centre of the

channel is higher than the 𝑢-velocity near the walls, and the 𝑣-velocity is greater near the walls as it

explained before. At the entrance of the heated plate, the fluid temperature is constant with a value

of 293.15 K. The fluid temperature increases along the test section, which explains that the heat

removed at the entrance of the test section (x = 0.004 m) is greater than at the outlet (x = 0.054m):

the temperature measured at the inlet of the test section is lower than the one measured at the

outlet.

In concordance with the 𝑣-velocity, the temperature of the fluid rises near the walls of the duct

where the buoyancy forces are more relevant. The fluid temperature along the middle plane of the

minichannel (plane XY, z = 0) gradually increases through its pass along the test section.


64

Figure 49 — Scenario 3.a: Temperature contour along the minichannel. YZ planes at x = 0.004m, x = 0.0165 m, x = 0.029 m,

x = 0.0415 m, x = 0.054 m and x = 0.064m, t = 22 s.

The previous contour confirms the explanation done before. The fluid temperature at the middle of

the duct is lower than in the walls due to the laminar velocity profile. The fluid is replaced by new

colder fluid quicker near the centre of the channel and the fluid temperature increases gradually

along the z-axis with the drop of the 𝑢-velocity. The 𝑣-velocity is lower at the centre of the

minichannel due to the higher 𝑢-velocity value and gradually increases with the z-coordinate until its

maximum near the walls, where the fluid reaches its maximum temperature and the buoyancy forces

are more meaningful.

Finally, the pressure contour shows a pressure drop along the minichannel owing to drive the fluid

within the duct.

Figure 50 — Scenario 3.a: Pressure contour along the minichannel. YZ planes at x = 0.004m, x = 0.0165 m, x = 0.029 m, x =

0.0415 m, x = 0.054 m and x = 0.064m, t = 22 s.

Scenarios 3.b to 3.d

After the analysis of the scenario 3.a, the scenarios with the appliance of the voltage at the wire are

analysed. Next graphs show the temperatures obtained measured at the different points defined in

chapter 5.3.2, and the maximum and average temperature of the heated plate:



(a) (b)

Figure 51 — Scenarios 3.a to 3.d: (a) Average temperature of the heated plate and (b) maximum temperature of the heated

plate

(a) (b)

(c)

Figure 52 — Scenarios 3.a to 3.d: (a) Temperature measured at point T2 (4, -2.5, 0) mm. (b) Temperature measured at point

T3 (29, -2.5, 0) mm. (c) Temperature measured at point T4 (54, -2.5, 0) mm.

First of all, as it can be observed in Figure 52, only the configuration of case 3.b reaches to decrease

the temperatures measured unless for the last part of the heated plate (T4). The maximum

temperature reached is similar and overall average temperature of the plate is lower.


66

Before performing an in-depth analysis of the temperatures obtained, some considerations of the

electric field created will be discussed. As it is explained before in chapter 6.2.1, the wire injects

charge through the fluid and has a defined voltage.

In 3D, a third component of the electric field appears in z-axis. Some contours are plotted in order to

see the electric field produced at the initial moments of the simulations. The electric field develops

with time as the voltage and the injection of charge do.

Regarding the electric field in y-axis, a similar behaviour as in 2 dimensions is observed.

Perpendicular negative electric fields lines are observed from the wire to the heated plate. This

negative electric field in y-axis creates a negative value of 𝑣-velocity where the charge injected is

positive. Near the electrode, where the charge introduced to the fluid is maximum, the 𝑣-velocity

reaches its maximum negative value and the fluid is pushed in the negative y-direction. Near the

heated plate, the buoyancy forces appears and the 𝑣-velocity is positive

Figure 53 — Scenario 3.d: Electric field x - component. t = 0.1 s

Figure 54 — Scenario 3.d: Electric field y - component. t = 0.1 s



Figure 55 — Scenario 3.d: Electric field z - component. t = 0.1 s

The electric field modifies the velocity profile owing to the Coulomb force introduced in the

conservation of momentum equation (27). The first evident conclusion is that this velocity profile

modification will increase with the voltage applied since the electric field is computed as the gradient

of the voltage (20). Thus, the contribution of the electric field to the momentum equations in case

3.d is higher than the one of case 3.b. Regarding the x-direction, which velocity component is

dominant in laminar flows, it can be observed in Figure 53 to Figure 55 how the electric field appears

in the device: a positive value at the right edge of the wire (x = 0.054 m) and a very similar negative

value at the entry of the test section (x = 0.004 m). As the heated plate becomes electrically charged

with the injection of charge due to the material that is made of, an electric field with opposite sign

appears at both edges of the plate (positive value at x = 0.004 m, and negative value at x = 0.054 m).

As the charge injected rapidly moves to the corners of the heated plate, these opposite electric fields

appear since the voltage is applied and grows gradually since the heated plate absorbs charge. The

charges progressively absorbed rearrange and redistribute themselves towards the corners of the

heated plate, increasing the electric field modulus at the lower part of the channel. This is explained

as a result of the electrical stability principles of conductor materials: the charges rearrange quickly in

conductors in order to reach the equilibrium state (an in-depth explanation of this fact is given at

chapter 6.2.1). Thus, the electric field grows with the injection of charge and the development of the

voltage. This electric field at the edge of the heated plate (x = 0.054 m) will be the main reason of the

fluid recirculation issues observed.

Figure 56 — Scenario 3.d: Charge density. t = 0.1 s


68

Figure 57 — Scenario 3.d: Charge density. t = 26 s

This evolution of electric field means that the modification of the velocity profile is gradual. The

negative electric field of the edge of the plate located at x = 0.054 m creates an undesired effect: it

reintroduces fluid with high temperature towards the test section with a constant velocity along the

cross-sectional area. This produces a localised augmentation of the temperature at the end of the

heated plate. Since z-coordinate grows in positive and negative values, the 𝑢-velocity decreases and

the recirculation is more undesirable because the hot fluid increases the temperature at the corners

of the heated plate. As it is explained for the case 3.a, the maximum temperature value is reached

just in the corners of the plate due to the laminar velocity profile, so this recirculation does not help

to cool down this localised zone. In addition, the recirculation increases with the voltage applied. As

examples, two contours are plotted for scenarios 3.b and 3.d.

Figure 58 — Scenario 3.b: Recirculation problems. YZ plane at the outlet of the test section x = 0.054 m. t = 26 s

Figure 59 — Scenario 3.d: Recirculation problems. YZ plane at the outlet of the test section x = 0.054 m. t = 26 s



The 𝑢-velocity is “narrowed” at the exit of the test section due to the electrical Coulomb force

present in this zone. The negative value of the charge and the positive electric field at the upper part

of the channel, near the electrode, create a negative velocity. At the bottom, the 𝑢-velocity is also

negative owing to the negative value of the electric field in x-direction and the positive value of the

charge near the plate. Thus, a negative 𝑢-velocity is as well created and is constant along the z-axis.

When the fluid moves away from the middle of the channel to the walls, in +/- z-coordinate, the 𝑢-

velocity due to the laminar profile reduces. So, this constant negative 𝑢-velocity that introduces the

hot fluid is more detrimental with increasing and decreasing z-coordinates.

Regarding the 𝑣-velocity, the electrical force added to the momentum equation modifies the velocity

profile. Near the wire, a negative 𝑣-velocity push the fluid towards the bottom of the minichannel. In

addition, Figure 60 shows the buoyancy forces effects at the last part of the heated plate, where the

hot fluid rises due to its density variation (considered in the y-momentum equation with the

Boussinesq approximation). The hot fluid tends to go up, extracting heat from the plate and flows

towards the outlet section and a part of it is recirculated when it reaches the end of the test section.

Figure 60 — Scenario 3.d: velocity modification at the middle plane of the minichannel. Velocity vectors coloured by

temperature Plane XY, x = 0.070 m. t = 26 s

The 𝑢-velocity profile modified by the Coulomb force for scenario 3.b, as an example, is as follow:

Figure 61 — Scenario 3.b: u_velocity modification at the middle plane of the minichannel. Plane XY, t = 26 s

Along the channel, the mean velocity of the fluid increases so the heat transfer rate is better than in

case 3.a without the appliance of the voltage. The temperature profile of the heated plate for case

3.a (Figure 48) shows that the temperature of the heated plate at the corners increases gradually

until its maximum value at the exit of the test section. With the appliance of the voltage, it can be

observed a decrease in the temperature of the corners and a smoother temperature profile, avoiding


70

the gap between the centre and the corners that it has noticed in case 3.a. A contour of scenario 3.b

is plotted as an example of this fact.

Figure 62 — Scenario 3.b: Temperature contour of the heated plate. Top view, plane XZ, t = 26 s.

After the phenomenon of recirculation exposed and the analysis of the velocity profile, an in-depth

analysis of the temperature of the heated plate is carried out. In order to determine if this

recirculation affects to the temperature of all the plate, a segmented analysis of the heated plate is

done. The heated plate goes from x = 0.004 m to x = 0.054 m and is divided in 4 different parts:

HP-1: from x = 0.004 m to x = 0.0165 m

HP-2: from x = 0.0165 m to x = 0.029 m

HP-3: from x = 0.029 m to x = 0.0415 m

HP-4: from x = 0.0415 m to x = 0.054 m

For each part of the heated plate, the maximum and the average temperature value is computed.

Case Ave. Temp. (K) Max. Temp. (K)


3.a 323.48 343.34 353.46 359.24 347.88 364.36 377.09 392.11

3.b 322.84 341.26 348.80 358.98 354.09 358.98 363.9 391.76

3.c 323.90 340.57 345.85 365.25 341.66 361.55 363.37 396.5

3.d 325.67 344.33 359.81 376.84 340.38 358.06 381.21 406.67 Table 15 — Scenarios 3.a to 3.c: Temperatures of the heated plate

As it can be observed in Table 15, the augmentation of the temperature is localised at the end of the

plate. The maximum temperatures obtained at HP-1, HP-2 and HP-3 (from x = 0.004 m to x = 0.0415

m) are lower for scenarios 3.b and 3.c (unless for HP-1 in scenario 3.b). In scenario 3.d, a reduction of

the maximum temperature measured is as well noticed but in HP-3 is higher. The maximum

temperature for the last part of the plate (HP-4) is higher for cases 3.c and 3.d and very similar for

case 3.b. This fact is a result of the recirculation at the end of the test section. Hot fluid re-enters to

this zone and locally increases the temperature value of the plate. Besides, due to the lower

Reynolds number of these cases, the Coulomb force contribution in the momentum equation is

significant and the recirculation increase with the voltage applied.

Another cause of the temperature increase of the plate is the Joule heating. As it is explained in

chapter 6.2.1, part of the charge injected from the wire is absorbed by the heated plate due to the

conductor material that it is made of. Charges can move easily through the surface of the heated

plate, so electric current flows through its surface. This current creates a source of Joule heating that



contributes to increase the temperature at the final part of the plate. As the heated plate is not

considered as an adiabatic wall (it introduces the heat flux into the domain), Joule heating is

generated. The charge is accumulated in the surface of the plate and tends to be higher in the last

part, where the temperatures are higher and where the Joule heating is higher too. Nevertheless,

this contribution in the increase of the temperature is lower than the contribution of the

recirculation phenomenon. Figure 63 shows the Joule heating at the plate.

Figure 63 — Scenario 3.b: Joule Heat source at the heated plate. Plane XY, t = 26 s

The joule heating is introduced due to the potential applied. Regarding the equation (9), the heat

generated is equal to the electric current that passes through an electrical conductor and the voltage

applied. The biggest source of joule heating is the wire, but it is considered as an adiabatic wall in the

model. So, the fluid does not increase its temperature in the upper part of the channel due to the

voltage applied at the wire.

Figure 64 — Scenario 3.b: Joule heat source at the middle plane. XY planes at x = 0.004 m, x = 0.029 m, x = 0.054 m, t = 26 s

6.3.4 3D Parametric study: scenarios 3.e to 3.h

Scenario 3.e

These next four scenarios consider another velocity inlet: 0.05 m/s (Re = 7.13). The device reaches

the steady state at 11 s. The same comments done for the velocity profile for scenario 3.a can be

applied in this case.

The dimensionless numbers presented in chapter 2.1 are computed for this scenario.


396.43 22.22 8.88 x 103

Table 16 — Scenario 3.e: Dimensionless numbers


72

However, some aspects should be mentioned about the 𝑣-velocity. As the mean velocity of the fluid

is higher in these scenarios, the 𝑣-velocity in all the YZ planes considered along the minichannel is

lower than the observed in scenario 3.a. This fact can be explained by analysing the dimensionless

numbers computed. Grashof number in scenario 3.a is 35.28 and in this case its value is 22.22 (7). As

it is explained in chapter 2.1, the Grashof number computes the ratio between the buoyancy forces

and viscous forces acting on a fluid. Even if both Grashof numbers are not too high, the value in

scenario 3.a is higher than in this case, so the buoyancy forces acting in the fluid in this scenario are

lower. Regarding the Rayleigh number, a lower value is as well obtained, so the convection heat

transfer is lower than in case 3.a, but we the convection occurs since this value is higher than the

critical value mentioned in chapter 2.1 (1700). Nevertheless, the same behaviour is observed when

analysing 𝑣-velocity contours. The 𝑣-velocity is increasing near the walls of the minichannel and a

decreasing value is observed in the centre of the duct. Therefore, the same detailed explanation of

the buoyancy forces at the different YZ planes done for scenario 3.a can be applied to the scenario

3.e. Figure 65 shows that the maximum 𝑣-velocity is lower than in scenario 3.a, as it is expected. In

addition, as the temperature gap between the plate and the fluid is lower, the contribution of the

density variation to the y-momentum equation is lower.

Figure 65 — Scenario 3.e: v_velocity profile along the minichannel. XY planes at x = 0.0165 m, x = 0.029m, x = 0.0415 m, x =

0.054m and x = 0.064 m, t = 11 s.

Next figures show the temperature of the heated plate after the steady state is reached. As it is

explained for scenario 3.a, the maximum temperature is reached at the corners but its value is lower.

Figure 66 — Scenario 3.e: Temperature contour along the minichannel. YZ planes at x = 0.004m, x = 0.0165 m, x = 0.029 m,

x = 0.0415 m, x = 0.054 m and x = 0.064m, t = 13 s.

The same pressure profile is observed than in scenario 3.a but with a bit higher values in order to

drive the fluid at this velocity.



Figure 67 — Scenario 3.e: Pressure contour along the minichannel. YZ planes at x = 0.004m, x = 0.0165 m, x = 0.029 m, x =

0.0415 m, x = 0.054 m and x = 0.064m

Scenario 3.f to 3.h

These three scenarios are simulated for 19.5 s, when the temperatures measured stabilize. Next

graphs (Figure 68) show the maximum temperature value and average temperature of the plate.

(a) (b)

Figure 68 — Scenarios 3.e to 3.h: Average temperature of the heated plate and maximum temperature of the heated plate

The maximum temperature and the average temperature of the heated plate are higher than in

scenario 3.e. This fact is due to three different factors: the modification of the velocity profile done

with the appliance of the voltage at the wire is not enough, the same recirculation problems at the

outlet of the test section mentioned above and the Joule heating generated at the plate.

(a) (b)


74

(c)

Figure 69 — Scenarios 3.e to 3.h: (a) Temperature measured at point T2 (4, -2.5, 0) mm. (b) Temperature measured at point

T3 (29, -2.5, 0) mm. (c) Temperature measured at point T4 (54, -2.5, 0) mm.

First, the problem of the modification of the velocity profile is treated. The same comments for the

generation of the electric field made for scenarios 3.b to 3.d can be applied. The mechanism of

charge injection, the evolution of the voltage and the electric field development are analogue.

Nevertheless, the velocity inlet used (0.05 m/s) is dominant over the modifications that introduce the

electrohydrodynamic system. The electrical Coulomb force contribution to the momentum equation

is not enough. In addition, the same recirculation problems are observed at x = 0.054 m (end of the

test section), so hot fluid re-enters to the test section increasing the temperature. Next contour

shows, as an example of these cases, the velocity modification and the recirculation of the fluid for

scenario 3.h.

Figure 70 — Scenario 3.h: u_velocity modification at the outlet of the test section. x = 0.054 m, t = 19.5s

By the analysis of the temperatures measured, the values obtained for T2 and T3 are similar. This is a

result of the no modification of the flow velocity in this region of the test section. However,

remarkable differences can be observed for T4. The biggest temperature increase at this point of the

heated plate is achieved in scenario 3.f.



Figure 71 — Scenario 3.h: u_velocity modification at the middle plane of the minichannel. Plane XY, t = 19.5s

The recirculation is highly undesirable. It is a result of the positive electric field at the right edge of

the heated plate (x = 0.054 m). The same explanation done for scenarios 3.b to 3.d is applied to these

cases: the negative electric field interacts with the positive charge injected in this zone of the fluid

and produces a negative Coulomb force added to the momentum equation, creating these negatives

values. As the velocity profile is laminar, this effect is more noticeable near the walls of the

minichannel.

Regarding the 𝑣-velocity, Figure 72 shows the contribution of the electrical force in the momentum

equation near the wire, modifying the fluid velocity with the purpose of pushing the fluid towards

the bottom of the minichannel. In addition, this figure shows the buoyancy effects at the last part of

the heated plate: the hot fluid rises up due to its density variations (considered in the y-momentum

equation).

Figure 72 — Scenario 3.h: v_velocity modification at the middle plane of the minichannel. Vectors coloured by temperature

Plane XY, t = 19.5s

In order to determine if the increase of the temperature is localised, the same procedure is

performed as in scenarios 3.b to 3.d to calculate the average temperature and the maximum

temperature of the four parts defined of the heated plate. Just to remind, the first part of the heated

plate (HP-1) goes from x = 0.004 to x = 0.0165, the second part (HP-2) from x = 0.0165 to x = 0.029,

third part (HP-3) from x = 0.029 m to x = 0.0415 m and the fourth part from x = 0.0415 m to x = 0.054

m. Next table shows the values obtained:

Case Ave. Temp. (K) Max. Temp. (K)


3.e 319.20 307.62 325.65 330.51 332.79 324.14 338.88 343.7

3.f 319.20 307.61 325.68 335.58 339.65 321.59 336.41 359.99


76

3.g 319.23 307.62 326.02 339.48 330.16 320.33 342.43 362.94

3.h 319.22 307.68 326.52 339.99 333.09 316.68 341.61 361.20

Table 17 — Scenarios 3.f to 3.h: Temperatures of the heated plate

The values of the average temperature of the first part, second and third part of the heated plate are

very similar. As it can be observed in the previous contours, the velocity profile is not highly modified

in these parts of the test section, so the same temperatures are expected. However, due to the

recirculation, the average temperature at the last part of the plate increases with the potential

applied. As it is explained before, a higher value of the potential applied means a higher value of the

electric field, so the modulus of Coulomb force added to the conservation of momentum equation is

higher when the voltage increases, becoming more significant the recirculation.

Regarding the maximum temperatures measured, it can be observed a decrease in the second part

of the heated plate. Even if the modification of the velocity profile do not reach to lower the average

temperature of the plate, its contribution near the corners help to decrease the temperature at

these points where the velocity due to the laminar flow is almost null. Next contour show, as an

example of this fact, the 𝑣-velocity at the end of the second part in the cross sectional area (just in

the middle of the heated plate, plane YZ, x = 0.029 m). It is higher when the voltage applied is

greater.

(a) (b)

(c)

Figure 73 — Scenario 3.f to 3.h: v_velocity and charge density at the middle plane of the minichannel: (a) Scenario 3.f (5

kV). (b) Scenario 3.g (10 kV). (c) Scenario 3.h (15 kV). Plane YZ, t = 19.5s



In the third part of the heated plate, the increase of the temperature starts to be noticeable and in

the final part, from x = 0.0415 m to 0.054 m the augmentation is more pronounced.

The augmentation of the temperature is produced not only for the recirculation phenomenon at the

end of the heated plate but also due to the Joule heating. When electric current flows through a

conductor produces Joule heating. As it is mentioned before, the material used for the walls is a

perfect electrical insulator, with a value of electrical conductivity very low. The mineral oil has as well

a low value of electrical conductivity that is one of the properties preferred to use it for cooling

applications. But, material conductors are characterised for having a high value of this property. The

wire and the heated plate are made of copper. As it is mentioned before, the biggest source of Joule

heating is the wire, due to its value of charge density and so its electrical current. However, the

thermal boundary condition applied for this part of the device is as an adiabatic wall, so it does not

introduce heat to the fluid.

On the other hand, the heated plate introduces a heat flux into the domain, so it is not treated as an

adiabatic wall. As it is made of conductor material, it absorbs charge that remains on its surface and

could move through it. This charge produces a Joule heat source at the different points of the heated

plate since it moves through it. As it can be noticed in the next contour, the maximum values of the

temperatures match with the maximum values of the charge absorbed by the heated plate. Thus, it

can be concluded that part of the temperature increase of the heated plate is due to the Joule

heating generated as a result of the flow of charge on its surface. Once again, a contour of scenario

3.h is provided to show this phenomenon.

Figure 74 — Scenario 3.h: Temperature and charge density at the heated plate. Plane XY, t = 19.5s

Analysis of parameters

The parameters presented in chapter 5.2 are computed for all the scenarios in 3D. As we can see in

Figure 75 shows that scenarios 3.b and 3.c have an enhancement in the heat transfer due to the

positive ratio Nu/Nu0. The heat transfer coefficient computed for those scenarios are higher than the

one of the scenario without the appliance of the voltage. The final average temperature of the

heated plate measured is lower than the one obtained in scenario 3.a (0 kV). Nevertheless, in

scenario 3.d, the heat transfer coefficient decreases due to the higher value of the average

temperature of the plate obtained, and thus, the Nusselt number is lower as well.


78

On the other hand, ratios lower than unity for the scenarios 3.e to 3.h are observed. This is evidence

since the heat transfer coefficient computed for these cases are lower than the one without the

appliance of the voltage (3.e). These configurations not only do not reach to lower the average

temperature of the plate, but also increase it due to the problems abovementioned.

Figure 75 — Ratio of Nu/Nu0 vs. applied voltage (kV) for scenarios 3.a to 3.h

Figure 76 shows the ratio of the pressure drop between different scenarios and the case without the

appliance of the voltage. As it is expected, an increase in the pressure drop is observed. The pressure

drop is related to the modification of the velocity profile. As this modification rises with the voltage

applied, the pressure drop increases as well with it.

Figure 76 — Ratio of ∆𝑃/∆𝑃0 vs. applied voltage (kV) for scenarios 3.a to 3.h

The pressure drop is higher in scenarios with lower Reynolds number. The electric field in these cases

achieve to have more influence in the flow and its contribution to the conservation of momentum

equation is higher. In addition, as it is explained before, the modification in the velocity profile is

greater when higher voltages are applied, so it is evident that the pressure drop will increase as well

with the increase of the voltage. By increasing the Reynolds number, the influence that the electric

field has in the conservation of momentum is lower. The effects of electric field are more noticeable

at low Reynolds numbers.



As it is explained before, the Performance Evaluation Criterion is a parameter that relates the heat

transfer enhancement to the pumping power needed. Even if the heat transfer coefficient shows a

better heat transfer behaviour of the devices, it is important to consider the pumping power increase

to assess the efficiency of the cooling technique. So, values greater than unity mean that the

pressure drop generated with the use of the cooling method is lower than the rate of heat transfer

enhancement.

Figure 77 — PEC vs. applied voltage (kV) for scenarios 3.a to 3.h

Regarding the PEC values, none of the scenarios studied achieves that the heat transfer improvement

is greater than the pressure drop generated. Even if the heat transfer rate is higher in scenarios 3.b

and 3.c, the pressure drop generated is greater than the enhancement of heat transfer observed and

they cannot be considered as efficient cooling techniques. However, regarding the segmented

analysis of the plate for scenarios 3.b to 3.d, we can conclude that the maximum temperatures in the

HP-1, HP-2 and HP-3 (from x = 0.004 m to x = 0.0415 m) of the heated plate are reduced and the

average temperature of the plate is also reduced for these parts is scenario 3.b and 3.c.

By the analysis of scenarios of the higher Reynolds number, we can conclude that none of the

configuration achieves to enhance the overall heat transfer coefficient, so the PEC values obtained

are lower than unity. Nevertheless, regarding the segmented analysis performed, some reduction in

the maximum temperatures reached is observed.

6.4 Weak points

Thanks to the analysis of the results obtained, some weak points have been noticed.

First, special mention can be done regarding the mesh of the model 3D. The 3D model mesh has a

multizone configuration: the test zone has smaller elements than the inlet and the outlet section.

This refinement of the mesh in the zone where the biggest modification of the fluids occurs is a good

practice and a good strategy to simulate this CFD study. However, concerning the cross-sectional

area (Figure 25), the transition between these smaller elements and the adjacent bigger ones is not

progressive. The expansion ratio is too high and to have better results and improve the simulation

setup, a refinement of this part should be done.


80

In addition, the maximum element size is also reduced at the wire but the same problem is observed

at the top of the channel, the expansion ratio between the faces of the wire and the faces of the wall

of the top of the channel is too high and not progressive (Figure 26).

In order to improve the mesh near the heated plate, the same comments can be done and a

refinement of the adjacent layers should provide accurate results in the zone where the heat flux

enters to the domain.

The use of ANSYS Fluent Academic version is limited and do not allow creating meshes with more

than 512k cells. For further work related to these simulations, a professional ANSYS Fluent version

should be use to avoid the limitations of the mesh generation.



7. CONCLUSIONS

This Master Thesis is focused in the study of the heat transfer enhancement using

electrohydrodynamically induced convection to cool down an electronic component.

First of all, some conclusions can be done by the qualitative analysis of use a minichannel with a

reduced hydraulic diameter. On one hand, the heat transfer coefficient is a function of Nusselt

number, fluid thermal conductivity and hydraulic diameter. Considering fully developed laminar flow

through a pipe, the Nusselt number is considered as constant value (3.61 for square ducts [80]).

Thus, for a given fluid, the heat transfer coefficient is inversely proportional to the hydraulic

diameter of the minichannel. If the hydraulic diameter is reduced, the heat transfer coefficient

increases. On the other hand, the pressure drop varies inversely with the Reynolds number following

the friction factor expression (35). So, pressure drop increases when the hydraulic diameter

decreases. Therefore, it is essential to find the balance between improving the heat transfer

coefficient and not excessively increasing the pressure drop necessary to pump the fluid.

Regarding now the simulations performed in 2D, the first conclusion is that the use of a mineral oil as

a refrigerant guarantees the cooling of a plate. By the appliance of a voltage at the wire of the top

section, a secondary flow appears from the wire to the heated plate. This secondary flow pushes the

neutral molecules of the fluid towards the plate, increasing the heat transfer coefficient.

Electrostatics and hydrodynamics are coupled in the conservation of momentum equations and the

Coulomb force, produced by the electric field and the charge injected, modifies the velocity profile of

the flow. This modification of the velocity profile is noteworthy 2D and increases with the strength of

the electric field: when the voltage applied is higher, the electric field computed is also higher and

therefore, the modification of the velocity profile is more noticeable. A recirculation phenomenon is

observed at the end of the test section due to the interaction of the electric field and the charge

present at the fluid. This interaction is also responsible for the different modifications observed in

the fluid along the test section.

Thanks to the analysis of the ratio between the Nusselt number of different scenarios, it can be

concluded than the heat transfer rate increases with the voltage applied, and thus with the strength

of the electric field. This fact is confirmed by analysing scenarios 1.a to 1.c. Nevertheless, a

recirculation issue is observed due to the negative velocity that creates the secondary flow. This

recirculation increases with the strength of the electric field (its contribution to the momentum

equation is higher), so it is more significant in scenario 1.d than in the other ones. Contrary as it is

expected (higher voltages mean higher secondary flow and higher heat transfer coefficient), this fact

explains that the ratio Nu/Nu0 for scenario 1.d (15 kV) is lower than the one for scenario 1.b (10 kV).

Regarding now the pressure drop, since the voltage applied rises, the secondary flow is greater. Thus,

the modification of the velocity profile is higher as well, increasing the pressure drop. Figure 41

shows this augmentation of the pressure drop ratio. By the analysis of Performance Evaluation

Criterion for these scenarios, it can be concluded that the efficiency of scenarios 1.b and 1.c (5 kV

and 10 kV, respectively) is greater than unity and thus the heat transfer enhancement is higher than

CONCLUSIONS

82

the pressure drop produced. Besides, as it is expected, the PEC is greater for the scenario 1.c with a

higher voltage applied (higher electric field computed). So, these two configurations are desirable to

cooling down the heated plate. Nonetheless, PEC value for scenario 1.d is lower than unity and

despite the greater heat transfer rate, the pressure drop needed is very large so it is not an efficient

cooling method.

In the same way, some 3D cases are simulated. Before analysing the same device in three

dimensions, a mesh study is done in order to determine the influence of the discretization of the

domain in the accuracy of the results obtained. Three different hexahedral meshes are studied with

10x10, 20x20 and 29x29 (limit of cells reached) cells at the cross-sectional area. By analysing the

results obtained (Figure 44 and Table 13), it is confirmed that the most accurate results are those of

the finest mesh generated. The mesh (2.1) with the poorest quality (10x10 elements at the cross

sectional area) is discarded due to the disparity of the results compared to the two other cases. The

values of the frictional factor computed for scenarios 2.2 and 2.3 are similar and when comparing

them with the classical correlation for laminar flows through square ducts, it can be concluded that

the finest mesh (case 2.3) guarantees the minimum error.

Furthermore, eight different simulations in 3D are computed for two different Reynolds numbers to

analyse the effect of the cooling technique studied. First, the buoyancy forces are more noteworthy

near the walls due to the laminar velocity profile of the flow: the u-velocity component is dominant

over the two other components and follows a parabolic profile. Since the fluid moves away from the

middle plane of the channel towards the walls, the 𝑢-velocity is reduced and the buoyancy forces are

more noticeable. Concerning now the scenarios with the lower Reynolds number (3.a to 3.d), the

average temperature of the heated plate is reduced in scenarios 3.b and 3.c and only the maximum

temperature is reduced with the 3.b configuration. Similar values of temperature are measured at

the three different points located along the first part of the heated plate, and a temperature increase

is observed at the final part of the heated plate.

This fact is explained because of the recirculation issue observed at the outlet of the test section. The

charge is injected from the wire, creating a secondary flow that pushes the flow towards the heated

plate. At the end of the wire (x = 0.054 m, y = 0.0025 m, z = 0 m) it can be observed a positive value

of electric field in x direction at the top of the channel. As the charge is injected through the fluid, the

heated plate becomes positively charged due to the material that it is made of, in our case copper. A

negative electric field in x-direction is observed at the bottom part of the channel at the end of the

test section, just in the edge of the heated plate (x = 0.054 m, y = -0.0025 m, z = 0 m). The interaction

of this negative value of electric field and the positive charge present at the plate produces a

negative u-velocity that reintroduces hot fluid towards the test section. This recirculation is more

noticeable since the fluid moves away from the middle plane in +/- z coordinates, due to the laminar

velocity profile. This phenomenon explains that the maximum temperatures observed at the heated

plate are at its corners near the end of the test section. The fluid recirculated in this zone is greater

than the main velocity of the fluid, so this hot fluid increases the temperature of the plate. The

segmented analysis of the heated (Table 15) confirms that this increase of temperature is localised at



the end of the heated plate. This recirculation increases when the voltage applied is greater and thus

when the strength of the electric field is higher.

The same comments about the velocity profile modification and the secondary flow can be applied

for the scenarios 3.e to 3.h. Nevertheless, as the Reynolds number is higher, the momentum applied

by the injection of charge has less relevance than in scenarios 3.b to 3.d.

Regarding now the analysis of the ratio Nu/Nu0, none of the scenario 3.e to 3.h reaches to improve

the heat transfer rate. Only scenarios 3.b and 3.c have values greater than unity of this ratio and

suppose an enhancement of heat transfer rate. The comparison of the pressure drop ratio shows

that it increases when the voltage applied is higher. In fact, as the electric field is computed as the

gradient of the voltage, a higher value of electric field means a higher modification of the velocity

profile and therefore a higher pressure drop. Another fact visible is that the effects of the electric

field are more noticeable at low Reynolds numbers. Both ratios have more significant changes in

scenarios with the lower Reynolds number, and the slope of the ratios lines is more accentuated in

the cases with low Reynold number (3.b to 3.d).

In addition, the analysis of the PEC values obtained shows that none of the 3D configurations is

efficient. Even if scenarios 3.b and 3.c reach to low the average temperature of the heated plate, the

pressure drop is greater than the heat transfer improvement and they cannot be considered as more

efficient cooling techniques. Nevertheless, regarding the segmented analysis performed, some

reductions at the maximum temperature are achieved for the plate for both Reynolds number

studied, unless for the final part, where the recirculation occurs and thus, its effects are greater.

Summarizing, the effect of electric field is noteworthy when the Reynolds number decrease due to

the values obtained for Nusselt number and pressure drop ratios. In addition, the recirculation

phenomenon observed confirms that the effects of the electric field increase when the voltage

applied is higher, and thus with the strength of the electric field. In fact, ratios of Nu/Nu0 greater

than unity are observed for scenarios with the lower Reynolds number with 5kV and 10 kV applied. In

the zones not influenced by the recirculation phenomenon, the heat transfer rate enhances with the

voltage applied owing to the lower temperatures observed. Besides, the pressure drop increases

with the voltage applied. Due to the recirculation issues, none of the 3D configurations reaches to

have a value of efficiency greater than unity. On the other hand, the 2D configuration with 5 and 10

kV applied can be considered as more efficient cooling techniques than the zero voltage 2D

configuration.

To conclude, the reliability and safety of the electronic components must be maximized and, thus,

the problem of overheating has to be minimized since it is the main cause of the reduction of the

lifespan of these components. So, the research and the development of these new cooling

techniques should continue in order to meet all these cooling requirements.

CONCLUSIONS

84



8. BIBLIOGRAPHY

[1] Liao, B. et al. (2016). Photo-excited charge carriers suppress sub-terahertz phonon mode in

silicon at room temperature, Nature Communications.

[2] Korotkov, A. N. et al. (1994). Effects of overheating in a single-electron transistor. Journal of

Applied Physics.

[3] Yang, W. B. et al. (2014). Numerical study of overheat fault in copper wire caused by bad

contact base on multi-physics coupling. Paper presented at the 7th International Conference

on Intelligent Computation Technology and Automation, Changsha, China.

[4] Almubarak, A. A., (2007). The Effects of Heat on Electronic Components. Int. Journal of

Engineering Research and Application ISSN : 2248-9622, Vol. 7, Issue 5, ( Part -5), pp.52-57.

[5] The Editors of Encyclopaedia Britannica, Erik Gregersen. (n.d.). Heat. Retrieved March 10,

2019 from https://www.britannica.com/science/heat

[6] (1999). Heat transfer. Retrieved March 10, 2019 from

http://physics.bu.edu/~duffy/py105/Heattransfer.html

[7] Davies, T. W. (2011). Fourier’s Law. Retrieved March 10, 2019 from

http://www.thermopedia.com/fr/content/781/

[8] Khan Academy. (n.d.) Retrieved March 10, 2019 from

https://www.khanacademy.org/science/physics/thermodynamics/specific-heat-and-heat-

transfer/a/what-is-thermal-conductivity

[9] Natural Convection. Forced Convection. Retrieved March 10, 2019 from https://www.cradle-

cfd.com/glossary/

[10] Newton’s Law of Cooling. Retrieved March 10, 2019 from https://www.nuclear-

power.net/nuclear-engineering/heat-transfer/convection-convective-heat-

transfer/newtons-law-of-cooling/

[11] Effect of heat on electronic devices. Retrieved 15 May, 2019 from http://www.apiste-

global.com/enc/technology_enc/detail/id=1262

[12] (2003). Dimensionless numbers. Retrieved March 12, 2019 from

https://ocw.mit.edu/courses/materials-science-and-engineering/3-185-transport-

phenomena-in-materials-engineering-fall-2003/study-materials/handout_numbers.pdf

[13] What is Grashof Number. Retrieved March 12, 2019 from https://www.nuclear-

power.net/nuclear-engineering/heat-transfer/introduction-to-heat-transfer/characteristic-

numbers/what-is-grashof-number/

[14] Hagedoorn H. (2008). Thermaltake PW850i Internal DIY Liquid cooling. Retrieved March 10,

2019 from https://www.guru3d.com/news-story/thermaltake-pw850i-internal-diy-liquid-

cooling.html

[15] Markets and Markets (2016). Thermal Management Market by Material Type (Adhesive,

Non-Adhesive), Devices (Conduction, Convection, Advanced, Hybrid), Service (Installation &

Calibration, Optimization & Post Sales), End-Use Application - Global Forecast to 2022.

Retrieved March 10, 2019 from https://www.marketsandmarkets.com/Market-

Reports/thermal-management-market-155049228.html

https://www.britannica.com/science/heat

http://physics.bu.edu/~duffy/py105/Heattransfer.html

http://www.thermopedia.com/fr/content/781/

https://www.khanacademy.org/science/physics/thermodynamics/specific-heat-and-heat-transfer/a/what-is-thermal-conductivity

https://www.khanacademy.org/science/physics/thermodynamics/specific-heat-and-heat-transfer/a/what-is-thermal-conductivity

https://www.cradle-cfd.com/glossary/

https://www.cradle-cfd.com/glossary/

https://www.nuclear-power.net/nuclear-engineering/heat-transfer/convection-convective-heat-transfer/newtons-law-of-cooling/



http://www.apiste-global.com/enc/technology_enc/detail/id=1262

http://www.apiste-global.com/enc/technology_enc/detail/id=1262

https://www.nuclear-power.net/nuclear-engineering/heat-transfer/introduction-to-heat-transfer/characteristic-numbers/what-is-grashof-number/



https://www.guru3d.com/news-story/thermaltake-pw850i-internal-diy-liquid-cooling.html

https://www.guru3d.com/news-story/thermaltake-pw850i-internal-diy-liquid-cooling.html

https://www.marketsandmarkets.com/Market-Reports/thermal-management-market-155049228.html

https://www.marketsandmarkets.com/Market-Reports/thermal-management-market-155049228.html

BIBLIOGRAPHY

86

[16] Chen, D. (n.d.) Need for Thermal Management in Electronic Systems. Retrieved March 10,

2019 from https://www.connector.com/need-thermal-management-electronic-systems/

[17] Cengel, Y. Ghajar, A. (2015). Cooling of electronic equipement. Heat and Mass Transfer:

Fundamentals and Applications. McGraw Hill. pp. 15.1 -15.70.

[18] C-Therm Technologies. Tech Library. Retrieved March 10, 2019 from

https://ctherm.com/products/tci_thermal_conductivity/helpful_links_tools/thermal_resista

nce_thermal_conductance/

[19] APS News (2009). December 1840: Joule’s abstract on converting mechanical power into

heat. Retrieved March 25, 2019 from

https://www.aps.org/publications/apsnews/200912/physicshistory.cfm

[20] TME. Retrieved March 25, 2019 from https://www.tme.eu/fr/katalog/memoires-eprom-

circuits-integres_100279/

[21] Techopedia. Retrieved March 25, 2019 from

https://www.techopedia.com/definition/2267/printed-circuit-board-pcb

[22] Thomas. Retrieved March 25, 2019 from https://www.thomasnet.com/articles/automation-

electronics/processing-printed-circuit-boards

[23] Moghanlou, F. S. et al. (2014). Experimental study on electrohydrodynamically induced heat

transfer enhancement in a minichannel. Experimental Thermal and Fluid Science 59, 24–31.

[24] Antonio Ramos, (2014). Microfluidic Technologies for Miniaturized Analysis Systems 2 (1, 2),

60-85

[25] Partha Kumar Das and A. B. M. Toufique Hasan. (2017). Mechanical Micropumps And Their

Applications: A Review. 7th BSME International Conference on Thermal Engineering. AIP

Conference Proceedings 1851

[26] Williams, M. (2014). What is heat conduction? Retrieved 20 March, 2019 from

https://phys.org/news/2014-12-what-is-heat-conduction.html

[27] Kiumars A. et al. (2016). A review on Electrohydrodynamic (EHD) pumps. 24th Annual

International Conference on Mechanical Engineering-ISME2016, Yazd University, Yazd, Iran

[28] Seyed-Yagoobi, J. (2005). Electrohydrodynamic pumping of dielectric liquids J. Electrostatics

63(6–10), pp. 861-869.

[29] Melcher, J. R. and M. S. Firebaugh, (1967). Traveling‐Wave Bulk Electroconvection Induced

across a Temperature Gradient, Physics of Fluids 10(6): 1178-1185.

[30] Siddiqui, M., Seyed-Yagoobi, J. (2009). Experimental study of liquid film pumping based on

conduction phenomenon. IEEE Trans. Ind. App 45, 3.

[31] Yazdani, M., Seyed-Yagoobi, J. (2009). Electrically induced dielectric liquid film flow based

on electric conduction phenomenon. IEEE Trans. Dielectr. Electr. Insul., 16.

[32] Jimil M. Shah, (2017). Recent Advancements in Oil Immersion Cooling. Electronic cooling

blog. Retrieved March 10, 2019 from https://www.electronics-cooling.com/2017/04/recent-

advancements-oil-immersion-cooling/

[33] Jimil M. Shah et al. (2016). Effects Of Mineral Oil Immersion Cooling On IT Equipment

Reliability And Reliability Enhancements To Data Center Operations. Conference paper.

[34] Liquid cool solutions. Retrieved March 28, 2019 from http://www.liquidcoolsolutions.com/

[35] Green revolution cooling. Retrieved March 28, 2019 from https://www.grcooling.com/

https://www.connector.com/need-thermal-management-electronic-systems/

http://highered.mheducation.com/sites/dl/free/0073398187/835451/Chapter15.pdf

http://highered.mheducation.com/sites/dl/free/0073398187/835451/Chapter15.pdf

https://ctherm.com/products/tci_thermal_conductivity/helpful_links_tools/thermal_resistance_thermal_conductance/

https://ctherm.com/products/tci_thermal_conductivity/helpful_links_tools/thermal_resistance_thermal_conductance/

https://www.tme.eu/fr/katalog/memoires-eprom-circuits-integres_100279/

https://www.tme.eu/fr/katalog/memoires-eprom-circuits-integres_100279/

https://www.techopedia.com/definition/2267/printed-circuit-board-pcb

https://www.thomasnet.com/articles/automation-electronics/processing-printed-circuit-boards

https://www.thomasnet.com/articles/automation-electronics/processing-printed-circuit-boards

https://phys.org/news/2014-12-what-is-heat-conduction.html

https://www.electronics-cooling.com/2017/04/recent-advancements-oil-immersion-cooling/

https://www.electronics-cooling.com/2017/04/recent-advancements-oil-immersion-cooling/

http://www.liquidcoolsolutions.com/

https://www.grcooling.com/



[36] Nishikawaraa, M. et al. (2016). Numerical investigation of the characteristics of an ion drag

pump. Journal of Electrostatics Vol 84, p 23-31

[37] Jimil M. Shah et al. (2016). Design Considerations Relating to Non-Thermal Aspects of Oil

Immersion Cooling. Proceedings of the ASME 2016 International Mechanical Engineering

Congress and Exposition

[38] Harvey, H. Types of mineral Oil: Retrieved 28 March, 2019 from

https://itstillruns.com/types-mineral-oil-7456026.html

[39] ANSYS Manual: Theory Guide. Retrieved April 8, 2019 from

https://www.sharcnet.ca/Software/Ansys/16.2.3/en-

us/help/flu_ug/flu_ug_sec_hxfer_buoy.html#flu_ug_uns_sec_boussinesq

[40] Tritton, D. J. (1977). Physical fluid dynamics. New York: Van Nostrand Reinhold Co.

[41] Antonio Castellanos (1998). The Boussinesq approximation in Electrohydrodynamics.

Electrohydrodanamic. pp 81.

[42] Incompressible and Compressible Flow. Chegg Study. Retrieved March 10, 2019 from

https://www.chegg.com/homework-help/definitions/incompressible-and-compressible-

flow-5

[43] European Comission: Waste Electrical & Electronic Equipment (WEEE): Retrieved 15 May,

2019 from http://ec.europa.eu/environment/waste/weee/index_en.htm

[44] Thermal Engineering Blog. How Rayleigh’s number links conduction and convection.

Retrieved May 10, 2019 from https://www.pirobloc.com/en/blog-en/how-the-rayleigh-

number-links-conduction-convection/

[45] Bejan, A. (2013): Convection Heat Transfer: John Wiley & Sons.

[46] Baldé, C.P. et al. (2017). The Global E-waste Monitor – 2017, United Nations University

(UNU), International Telecommunication Union (ITU) & International Solid Waste

Association (ISWA), Bonn/Geneva/Vienna. Retrieved 15 May, 2019 from

https://www.itu.int/en/ITU-D/Climate-Change/Documents/GEM%202017/Global-E-

waste%20Monitor%202017%20.pdf

[47] J.S. Paschkewitz, D.M. Pratt, (2000). The influence of fluid properties on

electrohydrodynamic heat transfer enhancement in liquids under viscous and electrically

dominated flow conditions, Exp. Thermal Fluid Sci. 21 (4) 187–197.

[48] Choosing the Right Fluid for Electronics Cooling: Seven Dimensions of a Decision. Retrieved

March 10, 2019 from https://dsiventures.com/library/

[49] ASTM Standard Specification D3487 “Standard Specification for Electrical Insulating Oil of

Mineral Origin”, American Society for Testing and Materials

[50] D. Sundin, Ph. D., Ambit (2013). Thermodynamic Analysis of Dielectric Heat Transfer Fluids,

Technical Consulting.

[51] Aluyor, E. Ori-Jesu, M, (2009). Biodegradation of mineral oils – A review, African Journal of

Biotechnology.

[52] Rajput, S., & Thakur, N. K. (2016). Geological Controls for Gas Hydrate Formations and

Unconventionals.

[53] (2018). IEA releases World Energy Outlook 2018. Retrieved 10 May, 2019 from

https://dieselnet.com/news/2018/11iea.php

https://itstillruns.com/types-mineral-oil-7456026.html

https://www.chegg.com/homework-help/definitions/incompressible-and-compressible-flow-5

https://www.chegg.com/homework-help/definitions/incompressible-and-compressible-flow-5

http://ec.europa.eu/environment/waste/weee/index_en.htm

https://www.pirobloc.com/en/blog-en/how-the-rayleigh-number-links-conduction-convection/

https://www.pirobloc.com/en/blog-en/how-the-rayleigh-number-links-conduction-convection/

https://www.itu.int/en/ITU-D/Climate-Change/Documents/GEM%202017/Global-E-waste%20Monitor%202017%20.pdf

https://www.itu.int/en/ITU-D/Climate-Change/Documents/GEM%202017/Global-E-waste%20Monitor%202017%20.pdf

https://dsiventures.com/library/

https://dieselnet.com/news/2018/11iea.php

BIBLIOGRAPHY

88

[54] D. Prucnal, (2013). Doing More With Less: Cooling Computers with Oil Pays Off, The Next

Wave, vol. 20, no. 2, pp. 20 - 29.

[55] Luke Werner, P. E. (2014). Submersion Cooling Evaluation, PG&E’s Emerging Technologies

Program

[56] Engineers Edge (n.d.). Naphthenic Oils Review. Retrieved 20 March, 2019 from

https://www.engineersedge.com/lubrication/naphthenic_oils.htm

[57] (2007). Debiotech’s Insulin Nanopump™. Retrieved 3 March, 2019 from

https://www.medgadget.com/2007/04/debiotechs_insulin_nanopump.html

[58] Melcher, J. R. (1981). Continium Electromechanics, 1st edn. MIT Press, Cambridge.

[59] J.M. López-Herrera et al. (2011). A charge-conservative approach for simulating

electrohydrodynamic two-phase flows using volume-of-fluid. Journal of Computational

Physics 230.

[60] D.A. Saville, (1997). Electrohydrodynamics: the Taylor–Melcher leaky dielectric model,

Annu. Rev. Fluid Mech. 29, 27–64.

[61] ANSYS website. Retrieved 5 May, 2019 from https://www.ansys.com/about-ansys

[62] Kuzmin, D. Introduction to Computational Fluid Dynamics. Institute of Applied Mathematics,

University of Dortmund. Retrieved 5 May, 2019 from http://www.mathematik.uni-

dortmund.de/~kuzmin/cfdintro/lecture1.pdf

[63] ANSYS Fluent User’s Guide: UDS Theory. Retrieved 6 May, 2019 from


us/help/flu_ug/flu_ug_sec_uds_theory.html

[64] ANSYS Fluent User’s Guide: Mesh Quality. Retrieved 6 May, 2019 from

https://www.sharcnet.ca/Software/Ansys/17.0/en-

us/help/flu_ug/flu_ug_mesh_quality.html

[65] ANSYS Fluent User’s Guide: Overview of Flow Solvers. Retrieved 6 May, 2019 from


us/help/flu_th/flu_th_SolveOverview.html

[66] ANSYS Fluent Theory Guide: Pressure-Based Solver. Retrieved 6 May, 2019 from

http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node361.htm

[67] ANSYS Fluent User’s Guide: Judging Convergence. Retrieved 6 May, 2019 from

https://www.sharcnet.ca/Software/Fluent6/html/ug/node1067.htm

[68] Kuron, M. (2015). 3 Criteria for Assessing CFD Convergence. Retrieved 6 May, 2019 from

https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/9296/3-

Criteria-for-Assessing-CFD-Convergence.aspx

[69] ANSYS Fluent User’s Guide. Boundary Conditions. Retrieved 6 May, 2019 from


[70] Osses, J (2016). El Método de los Volúmenes Finitos. Retrieved 6 May, 2019 from

https://www.esss.co/blog/es/el-metodo-de-volumenes-finitos/

[71] Vázquez González, P. Electrohidrodinámica con carga eléctrica en volumen: conceptos y

aplicaciones.

[72] Crowley, J.M. (1999). Fundamentals of Applied Electrostatics, 1st edn. Laplacian Press/

Morgan Hill, California

https://www.engineersedge.com/lubrication/naphthenic_oils.htm

https://www.medgadget.com/2007/04/debiotechs_insulin_nanopump.html

https://www.ansys.com/about-ansys

http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture1.pdf

http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture1.pdf

https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu_ug/flu_ug_sec_uds_theory.html

https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu_ug/flu_ug_sec_uds_theory.html

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_ug/flu_ug_mesh_quality.html

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_ug/flu_ug_mesh_quality.html

https://www.sharcnet.ca/Software/Ansys/17.2/en-us/help/flu_th/flu_th_SolveOverview.html

https://www.sharcnet.ca/Software/Ansys/17.2/en-us/help/flu_th/flu_th_SolveOverview.html



https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/9296/3-Criteria-for-Assessing-CFD-Convergence.aspx

https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/9296/3-Criteria-for-Assessing-CFD-Convergence.aspx


https://www.esss.co/blog/es/el-metodo-de-volumenes-finitos/



[73] ANSYS Fluent User’s Guide: What is a User-Defined Function? Retrieved 6 May, 2019 from


us/help/flu_udf/flu_udf_WhatIsAUDF.html

[74] Hughes, S. (2005). Lecture 5: Fields and Potentials around conductors. The electrostatic

uniqueness theorem. Massachusetts Institute of Technology, Department of Physics..

Retrieved 20 May, 2019 from http://web.mit.edu/sahughes/www/8.022/lec05.pdf

[75] Evans, M. (2013). Electromagnetism 3 Section 3: Gauss’s Law. University of Edinburg.

Retrieved 20 May, 2010 from https://www2.ph.ed.ac.uk/~mevans/em/lec3.pdf

[76] Capacitance and Dielectrics. University Central of Florida, Department of Physics. Retrieved

20 May, 2019 from https://physics.ucf.edu/~roldan/classes/Chap24_PHY2049.pdf

[77] Kandlikar, S. et al (2006). Single-phase liquid flow in minichannels and microchannels. Heat

Transfer and Fluid Flow in Minichannels and Microchannels. pp. 90-96.

[78] 42U Data Center Solutions. Retrieved 21, May 2019 from

https://www.42u.com/measurement/pue-dcie.htm

[79] Oil Cooling System for Data Centers. Gesab. Retrieved 21, May 2019 from

https://gesab.com/oil-cooling-system-for-data-centers/

[80] Zohuri, B. (2017). Laminar Incompressible Forced Convection. 10.1007/978-3-319-53829-

7_5.

[81] Caminha, G. (2019). The CFL Condition and How to Choose Your Timestep Size. Retrieved 10

March, 2019 from https://www.simscale.com/blog/2017/08/cfl-condition/

[82] Cimbala, Y. et al. (2006). Fluid mechanics: fundamentals and applications (1st ed.). Boston:

McGraw-Hill Higher Education. pp. 321, 322, 323, 324, 325, 326, 327, 328, 329.

[83] ANSYS Fluent User’s Guide: Electric Potential. Retrieved 6 May, 2019 from


us/help/flu_th/flu_th_models_elec_potential.html

[84] ANSYS Fluent User’s Guide: Heat Transfer Theory. Retrieved 6 May, 2019 from


us/help/flu_th/flu_th_sec_hxfer_theory.html#flu_th_eq_energy_E

[85] Overview of Physical Models in ANSYS Fluent. Retrieved 6 May, 2019 from https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu_th/flu_th_sec_models_overview.html

[86] ANSYS Fluent Continuity and Momentum Equations. Retrieved 6 May, 2019 from http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node11.htm

[87] ANSYS Fluent Viscous Model Dialog Box. Retrieved 6 May, 2019 from http://www.afs.enea.it/project/neptunius/docs/fluent/html/ug/node1022.htm

https://www.sharcnet.ca/Software/Ansys/15.0.7/en-us/help/flu_udf/flu_udf_WhatIsAUDF.html

https://www.sharcnet.ca/Software/Ansys/15.0.7/en-us/help/flu_udf/flu_udf_WhatIsAUDF.html

http://web.mit.edu/sahughes/www/8.022/lec05.pdf

https://www2.ph.ed.ac.uk/~mevans/em/lec3.pdf

https://www.42u.com/measurement/pue-dcie.htm

https://gesab.com/oil-cooling-system-for-data-centers/

https://www.simscale.com/blog/2017/08/cfl-condition/

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_th/flu_th_models_elec_potential.html

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_th/flu_th_models_elec_potential.html

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_th/flu_th_sec_hxfer_theory.html#flu_th_eq_energy_E

https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_th/flu_th_sec_hxfer_theory.html#flu_th_eq_energy_E

https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu_th/flu_th_sec_models_overview.html

https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu_th/flu_th_sec_models_overview.html


http://www.afs.enea.it/project/neptunius/docs/fluent/html/ug/node1022.htm

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