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MATHEMATHICAL MODEL

System Variables

SYSTEM VARIABLES

Controled Variable Manipulated Variable Disturbane VariablesAmmonia outlet

temperature !or s"ell

#ressure o! ammonia

inlet !or tube

Ammonia inlet

temperature !or pipe$

Energy balance of liquid ammonia (hot uid)

d (E )dt =E iEoQ

d (v Cv T(t))dt = fc hi fc hoUA T

Assuming that the change in density is constant and the volume of the

liquid does not change and because it is an incompressible uid, Cp=Cv

in the volume control

vCpd (T)dt =Cp ff(Tci(t)Tco(t))UAT

Considering that the average temperature is (TiTo ) , so

Tprom=TiTo

2

v Cp2

d (Tprom(t))dt

= Cp f(Ti(t)To(t))U A T

Ademas sabiendo que

T= T

1 T

2

2

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En un Sistema en contracorriente tenemos que

T1=Tce Tfs

T2=TcsTfe

v Cp2

d (Tprom(t))dt

= Cp f(Ti(t)To(t))UA

2 (Tci Tfs+TcoTfe)(1)

Steady state equation

v Cp

2

d (Tss prom )dt

=Cp f(TcissTcoss)UA

2 (Tciss Tfoss+Tco ssTfess )(2)

Ahora restamos la ecuacin 1 de la 2

v Cp

2

d (TpromTss prom )dt

=(TciTciss )( Cp fUA2)(TcoTco ss)(Cp f+UA2)(TfoTfoss)UA2 (Tfi

Sabiendo entonces que

f(t)=TfoTfoss

c(t)=TcoTcoss

Ahora, recordando que en el sistema la temperatura a la entrada es constante

f 0 ( t)=Tc 0Tc0ss

c(t)=Tf 0Tf0 ss

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Remplazando se nos reduce

v Cp

2

d (f(t))dt

=(f(t)) ( Cpf+UA2)(c ( t))UA2

Ahora dividimos todos los terminus entre Cp fUA

2

Y tenemos que

1=

v Cp

2Cp fUA

K1=

UA

2Cp fUA

Remplazando queda

1

d (f(t))dt

=(f(t))K1(c(t))

Aplicando la place nos queda

(f(s ))=1 s f( s )+f(0 )K1 (c ( s))

Tenemos entonces

f( s)=cK

1

1

+1 s