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UNIVERSIDAD NACIONALMAYOR DE SAN MARCOS
FACULTAD DE QUIMICA, ING.
QUIMICA E ING. AGROINDUSTRIAL
Curso: Control de ProcesosProfesor: Ing. Eder Vicua Galindo
Alumna: Joyssy Ticona Vilca
Lima, 2011
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Title:Temperature control at the bottom of an
aquarium by the variation of heat entering,ambient temperature and the partial pressure ofwater
One aquarium can be modeled bytwo perfectly mixed pools.
Be want derive an equations thatrepresent the response of the
temperature in bottom of anaquarium for the changes in heatinput, in the surroundingtemperature, and in thesurrounding water partial
pressure.
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The process is defined:There is a electric heater.
There are two perfectly mixed pools. The temperature at the bottom is T1(t),C . The temperature at the top is T2(t),C. The rate of vaporization of water from the tank: w = Ky*A*[p(T)-ps(t)] p[T(t)]= e A-B/T+C(Antoanie Equation )
OBJECTIVE:
Variable temperature control since this affects the stability of
operation of a tank which aims to maintain the life of the fishthat inhabit it.
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process scheme
f2 f1
f1
f2
VARIABLES:* Inputvariable: T1(t)* Disturbance
variable: ps(t)* Manipulationvariable: I(t)* State
variable: T2(t)
PARAMETROS:
f [m3/s]
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Assumptions:
Transfer of heat and mass to the surroundings isonly from the free surface of the water .
V1 = V2 = v
p[T2(t)]
The physical properties of water (Cp, Cv, and ) are constant.
Cp = Cv
1 = 2
The rate of vaporization is so small that the totalmass of water in the tank, M, kg, maybe assumedconstant.
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Mass balance1*f1 = 2*f2 1= 2 f1 = f2
Energy balance at the top:
1*f1*h1(t)- 2*f2*h2(t) - W(t)* = d[V**U(t)]dt
*f*Cp[T1(t)- T2(t)] - W(t)* = V**Cv*d[T2(t)] (1)dt
One equation with three unknows, T1, T2 and w
MATHEMATICAL MODELDEVELOPMENT
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Energy balance at the bottom2*f2*h2(t)- 1*f1*h1(t) + Fem = d[V**U(t)]
dt *f*Cp[T2(t)- T1(t)] + R*I(t)= V**Cv*d[T1(t)] (2)
dtTwo equations with four unknows, I
The rate of vaporization of waterw(t)= Ky*A*[p(T2(t))-p(t)]
Three equations with five unknows, p(T2)(3)
Antoanie equationP[T2(t)]= e A-B/T2+C
(4)
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4/17/12Establishing steady state equations
*f*Cp[ T1 T2] W* = 0 (6)
*f*Cp[ T2 - T1] + Fem*I = 0 (5)Linearizing and subtracting the aquations
corresponding to (1) and (2)*f*Cp[T1(t)- T2(t)] - W(t)* = V**Cv*d[T2(t)] (7)
dt*f*Cp[T2(t)- T1(t)] + R*I(t)= V**Cv*d[T1(t)] (8)
dt
Where:T2(t) = T2(t) T2T1(t) = T1(t) T1
I(t) = I(t) - IW(t)= w(t) w = Ky*A*[p(T2(t))-ps(t)]
= -
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4/17/12*Replacing (9) in (7)*f*Cp[T1(t)- T2(t)] - Ky*A*[a(T2(t))-ps(t)] * = v**cv*d[T2(t)](10)
dt
Using Laplace Transform and rearranging for equation (8) and(10)
T
1(s) = 1 T2(s) + K1 I(s)(11)
s 1+1 s 1+1
T2(s) = K2 T1s) + K3 Ps(s)(12)
s 2+1 s 2+1 Then:
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where
1= *V*Cv, = 20 segundos K1 = Fem= 0.98 K
*f*Cp *f*Cp
2= * V * Cv, = 60 segundos K2 = *f*Cp= 0.08 ,adimensional
Ky*A* *a+ fCp Ky*A* *a+ fCp
K3= Ky*A*, = 0.005 K/Pa
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Process transfer function:
Disturbance transfer function:
TRANSFER FUNCTION IN OPENLOOP
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block diagram for open loop system
K31s+1
K22s+1
K1
1 11s+1
I(s)
T1(s)
T2(s)
Ps(s)
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TT
TRC
T2
(t)
Tsp(t),K
m(t)
mA
outline of the process control system
Sensor
transmitter
Controller
Final
controlelement
T1(t)
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sensor transmitter:MBT 153
Static characteristics
Noise: 0,1mA p.p.
Sensor range : -10 50
Dynamic characteristics
Response Rate: 2 minPrecision: 0.1%
- Convert the millivolt output
current signal (typically 4-
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Transfer function of the sensor-transmitter
H(s) = Kt .ts+1
The transmitter can be represented by the
following linear relationship.
Gain:
Kt = 20 4 mA = 0.26750 (-10) C
Response time:t =2 min
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Is Tsp the set point:
e(t)= Tsp(t) Tm(t)
Mode of action: Reversible action
m(t)= m+ (Kc)e(t)
Mode of controller: Proportional
CONTROLLER
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4/17/12final control element:Electric actuator
Accion: fail closed (FC)
Linear valve characteristic.
Response time : 10 s
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Ky Kvp Kvqm(t)
mp(
t)
vp
(t)
K= 1 - 020 4
mA
Kvq=27.4
Kv= K*Kvq = 1* 27.416Kv= 1.712
Transfer function of thecontroller:
Gv (s)= 1.71210s + 1
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block diagram
0.98(20s+1)(s+0.016)(s+0.051)
0.005(s+0.016)(s+0.051)
1.712
10s+1
KcKsp
0.267
2s+1
Tsp(s)
Tm(s)
T1(s)I(t)
+
E(s)
M(s)
Ps(t)
+
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characteristic equation
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Stability analysis
1) ROUTH- HURWITZ TEST20 1.8196 0.0078+4.96Kc13.34 0.0764+29.7849Kc 0b1 b2c1 0
d1
For b1>0 :
b1= 1.896 44.65Kc >00.04>Kc Kcu= 0.04
For b2>0 :b2 = 0.00078 + 0.496Kc >0
Kc>-1.57*10^(-3)
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2) DIRECT SUBSTITUTION TEST
Grouping:
Is obtained:
Replacing: s = wcuiKc=Kcu
Then:
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Kc I D
PROPORTIONAL 0.175 - -
PROPORTIONAL-INTEGRAL
0.159 18.06 -
PROPOTIONALINTEGRAL
DERIVATIVO
0.206 10.83 2.71
controller tuning parameters
According to ZieglerNichols:
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RESPONSE TIMEAND
STABILITY
ANALYSIS
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PROPORTIONALCONTROLLER
Kc= 0.1749
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Response time
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Kc = 0.175rlocus(FTCA)
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Kc = 0.175Bode(FTCA)
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Kc = 0.175nyquist(FTCA)
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PROPORTIONALINTEGRAL
CONTROLLER
Kc = 0.159i= 18.06
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RESPONSETIME
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rlocus(FTCA)
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bode(FTCA)
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nyquist(FTCA)
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PROPORTIONAL
INTEGRALDERIVATIVE
CONTROLLER
Kc = 0.206i = 10.83D = 2.71
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RESPONSE TIME
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rlocus(FTCA)
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bode(FT
CA)
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nyquist(FTCA)
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ANALYSIS IN ACONTROLLER OFPROPORTIONAL
INTEGRALDERIVATIVE
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An increase of Kc in10%
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A decrease ofKc in 10%
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* Shows that the three response timegraphs for stability, but at the beginningintroduces instability. We conclude a
change in the controller (Kc) does notinfluence the search process stability
*The Kc Increase by 10% the amplitudedecreases and the same percentage
decrease in the amplitude increases,conclude that Kc varies the amplitude andincreasing its value the process will becloser to stability
ANALYSIS
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