CHEN3005 Process Instrumentation and Control
Review Exercise 3 Q.1 A jacketed vessel is used to cool a process stream as shown in the figure. The following information is available:
i. ii. iii. iv. v.
The volume of the liquid in the tank V and the volume of the coolant in the jacket V j remain constant. Volumetric flow rate F F is constant but F J varies with time. Heat loses from the jacketed vessel are negligible. Both the tank contents and the jacket contents are well mixed and have significant thermal capacitances. The thermal capacitances of the tank wall and the jacket wall are negligible. The overall heat transfer coefficient for the transfer between the tank liquid and the coolant varies with coolant flow rate:
U = KF J .
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Where U = [Btu/h ft F] 3 FJ = [ft /h] K = constant Derive a dynamic model for this system. (State any additional assumptions that you make) TF FF TJ FJ
Ti FJ
T F
Solution Q.1 Choose what type of balance equation to be used. Here, temperatures are of interest, so apply the energy balance equation.
Standard Statement: Acc = In – Out – Consumption + Generation + Qgain
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For the reactor system:
= − + −
Note: Assume exothermic reaction occurs, so heat is generated and the reactor need to be cooled down. Hence, Q loss represents heat generated that is transferred to the cooling jacket system. The terms in the equation above are given by
: ≅ = = , ℎ ℎ : = = = ∆; : = ,∆ = ℎ = ( −)
Substituting all terms into the equation and simplification leads to
= ( − ) +∆ − ( −) Likewise, energy balance in the cooling jacket system can be performed, but without reaction term and heat loss is replaced with heat gain. The equation is
= − +( −) Also, the constitutive equation is given for overall heat transfer coefficient
= . ,, , , , ,, , , , ,, ∆ = 8−3 = 5
Dependent variables:
Independent variables: Parameters:
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Solution Q.2
Control Objective: To maintain R = C A /CB at desired ratio or setpoint.
,, , ℎ
Disturbances:
Manipulated Variables:
Assumptions: 1. 2. The worst disturbance is The combined feedback-feedforward control strategy as shown below. The feedforward acts to remove the disturbance F B before it can seriously upset the process. This strategy works effectively provided the C B0 does not fluctuates that much, otherwise, the strategy will not be able to provide improved performance.
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Q.3 A stirred-tank reactor is operated with a feed mixture containing reactant A at a mass concentration C Ai. The feed flow rate is w i as shown below in the drawing. Under certain conditions the system operates according to the model: d ( ρ V )
= wi dt d ( ρ VC A )
=
dt
−
w
wi C Ai
−
wc A − ρ VkC A
wi CAi
w V
CAi
(a) For cases where the feed flow rate and feed concentration may vary and the volume is not fixed, simplify the model to one or more equations that do not contain product derivatives. The density may be assumed to be constant. Is the model in a satisfactory form for Laplace Transform operations? Why or why not? (b) For the case where the feed flow rate has been steady at w i for some time, determine how C A changes with time if a step change in C Ai is made from C A1 to CA2. List all assumptions necessary to solve the problem using Laplace transform techniques. Solution Q.3
If density is assumed constant, the total mass balance equation becomes
= − + = − − = ( − ) −
The species mass balance can be expanded using chain rule
Substituting the first equation into the second one:
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The equation above contains a nonlinear term and
, i.e., bilinear term consisting of
(two variables in combination becomes nonlinear term). Laplace Transform only
applies to linear equation, so the above equation cannot be directly transformed.
Let
=
, application of Taylor’s approximation
≅ ̅ + , ̅ ∆ +, ̅ ∆ ≅ ̅ + ̅ ∆ +∆, ℎ ∆ = − ∆ = − ̅ ∆ = (∆ − ∆)−( ̅ ∆ +∆) ∆() = ∆() −∆()− ̅ ∆() +∆() ∆ = 0 () = ∆∆()() = + +
Substituting into the equation above and convert to deviated form:
Apply Laplace Transform to the equation
Rearrangement and form transfer function, assuming
It’s first order transfer function, so the response will be monotonic shape (no oscillation whatsoever).
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Q.4 An expression has been found for the transient response of a second-order process:
d 2 y 2
dt
W 2
=−
dy dt
−
KV y + Qx 3 / 2 W
(a) What is the effect on the system characteristic time ( τ ) and damping coefficient of doubling V while keeping W constant? (b) What can you determine about the response time of the manometer? About its damping characteristic? (c) Calculate the response of the manometer to a pressure change, p (t ) = 2e
−
4 t
.
Solution Q.4
/ = ̅ 3 ≅ ̅ + 2 ∆, ℎ ∆ = −̅ ∆() = −∆() − ∆() + 32̅ / ∆() / ̅ 3 () = 2 () = ∆∆() + + () = +2 +1 / ̅ 3 = 2 = / = 2√ 1⁄√ 2.
Apply Taylor’s approximation to the nonlinear term
Substituting the approximation into the equation above and apply Laplace Transform
The transfer function obtained is
This is a second order process and can be written in the standard form of
The process gain
Time constant
Damping factor
Therefore, for constant W, by doubling the value of V, both time constant and dampin factor will be reduced by factor of
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Q.6 Consider the standard block diagram shown below, with the following transfer functions:
Gc
=
K c , Gv
=
Gm
=
1 , G p
1
=
(5s + 1)( s + 1)
,
G L
=
1 s+2
L
+
R +
C
+ -
(a) For a step change of 10 units in R, what is the offset? (b) Find the range of K c for which the closed-loop system is stable.
Solution Q.6 Need to find the overall closed-loop transfer functions.
For setpoint tracking or servo transfer function
= = 1 + = = 1 + = 10/ = →lim 1+10 = 101+
For disturbance rejection or regulatory transfer function
Step change of 10 units in R, so that
.
Apply final value theorem, the new steady-state value of C:
Substituting the transfer function components into the equation above
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The steady-state offset is
= ∆ − = 10 −101+ = 1+10
The characteristic equation (CE)
1 + = 0 (5 +1)( +1) + =0 5 + 6 +1+ = 0 1 + > 0 ⇒ > −1
Apply Routh stability, the necessary condition
In this case,
> −1
should give stable closed-loop system.
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