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Unformatted text preview: EIRETURN THIS SHEET WITH YOUR EXAM BOOKII E _;._. __ , ,,_.,. ,..._._ . .. . . . ._.. Chemical Process Dynamics and Control I Fall 2011
iir‘structor: Mattheos Koffas First Midterm ' o The exam is closed book and notes. Only a crib and the examination blue book are allowed.
0 Answer all questions. Write legibly, justify your answers, and state your assumptions. Problem 1 (20 points) Suppose that two liquid surge tanks are placed in series so that the outﬂow from the ﬁrst tank is the
inﬂow to the second tank as shown in the Figure above. The outﬂows from each tank are given by the
following equations:  Derive an unsteadystate model for the liquid levels hl and h; in response to changes in q. (10
points)   Bring the model in the form of deviation variables. (5 points) ' Derive a transfer function between Q; (S) and (215(3) where Q; (s) and 015(3) are the Laplace
transforms of the input variable q. and the output variable (12 in their deviation variable form. (5
points) You may make the following assumptions:
goon LUCK; IERETURN THIS SHEET WITH YOUR EXAM BOOKH '17. The two have different cross—sectional areas A; A2.
2. There are no temperature changes during the process and everything operates isothermally {11
(hint: that means no energy balances) Problem 2 110 points) _ Which of the following transfer functions do you expect to demonstrate an inverse response and which
not for a unit step change of the input variable? Which one will exhibit an overshoot? Explain your) answer in detail: on arts) = 5 . / 23+1 _ 5 , _ (b) GE (S) = (43+1)(s+1) 103+1'
(C) 01(3)  Problem 3 110 points! Consider the following transfer function:
K (—0.15 + 1) 0(3) = W Using Taylor approximation, derive:
(a) A ﬁrst—orderplustimedelay approximation for this transfer function. (5 points) (b) A secondorder—plus—time—delay approximation for this transfer function. {5 points) Problem 4 (20 points) A process is modeled by the following differential equations: d3’1 2371 = 3312 “i' 2111
dyz _
EF+ 6y2 — 4311 + 2u1 + 4n; In this process, 3/1 and y; are the controlled variables that also happen to be the measured output variables. The variables ul and u; are the only input variables.
(a) Derive the statespace model for this process. (10 points) 9000) LUCK; HRETURN THIS SHEET WITH YOUR EXAM BOOKI! *(b) Deterrnine the stability of the process using the statespace model. (10 points) ‘11 “2
a3 ‘14 — z 0 =15/12 — (a4, + a1)/l. "l' ((110.4. '"" (12613) = 0 Problem 5 [20 points! Note: The eigenvalues of a 2x2 matrix [ ] are the roots of the following equation: For unit step changes in the input variable, which of the following transfer functions correspond to
processes that oscillate? Which processes will converge and which will diverge? _ ___6<s:2_>___
3. C(57) — (32+9S+20)(s+4)
b 6(3) _ __32§i:3__
. — (32_6S+10)(S+2)
2
0. (HQ — m
2 +1 XErOblem 6 (20 points) For unit step changes in the input variablea which of the following transfer functions correspond to
processes that oscillate? Which processes will converge and which will diverge? __ _6_<£:33__
a. _ (52+93I20)(S+4r)
b 0(8) __ L
 _ (52—65+10)(3+2)
GC ) _ __2_
C’ S — (52+4S+3)
2 +1 Problem 7 {20 points)
90009 LUCK! EIRETURN THIS SHEET WITH YOUR EXAM BOOKI! Y ' i‘he dynamic behavior of a pressure sensor/transmitter can be expressed as a ﬁrst—order transfer function
(in deviation variables) that relates the measured value Pm to the actual pressure P: EMS) _ 1
P'(s) _ 603 + 1 Both P7; and P’ have units of psi and the time constant has units of minutes. Suppose that an alarm will
sound if Pm exceeds 45 psi. If the process is initially at steady state and then P suddenly changes from 35
to 50 psi at 12:40pm (a) At what time will the alarm sound? (10 points)
(b) Your colleague, Giorgos Papadomichalanikoladogeorgiopoulopoulos, suggests that you use a different sensor/transmitter that has the following transfer function:
P416) _ 1
‘ P’(s) _ s + 1
If it is important that the alarm will sound as soon as possible (Le. that Pm reaches 45psi as soon
as possible once P has changed), would you agree with his suggestion? Explain your answer (10
points) goon) LUCK; IlRETURN THIS SHEET WITH YOUR EXAM BOOKII ‘ rjhemical Process Dynamics and Control
' ram 201 1 Useful. equations Bernulli Equation 9:01:49:
p * I
where CV15 a constant AP is the pressure difference between the inlet and outlet of the valve
p is the liquid density Pin Pout Pressure P at the bottom of the tank P=Pa+£g~
gc Characteristic Equation for a First Order System KP
IS+1 Y (s) = X (s) Characteristic Equation for a Second Order System KP Y =______
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