CHE 322 Spring 2007 Final Exam

CHE 322 Spring 2007 Final Exam - Name Student Q1 Q2 Q3 Q4...

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Unformatted text preview: Name: Student #: Q1: Q2: Q3: Q4: Q5: Q6: Total: CHE3228 — Process Dynamics and Control Final Exam April 17, 2007 Closed Book All work to be marked must appear on front of page. Use back of page for rough work only. Use pen only. Tests in Pencils will got be graded. The last page of this exam includes an aid sheet which you may detach and keep aside for reference. Do not write more than what is necessary. When asked to provide a justification, please keep your answers to less than 2 sentences. lof16 1. (20 marks) Consider the following process flow diagram of a mixing tank that might be vaguely familiar to those who actively participated in the experimental laboratory. If we assume that the steam pipes were ruptured due to careless operation and the input CV2 is out of operation. However, fortunately, ABB Instruments were kind enough to donate a control valve that you could use for the control of the hot water flow rate. You assume that this configuration although not ideal will suffice for the control of this mixing process. Hot St W] [Net Water 32 CV1 Cold Water Outlet Water The outputs (y., y;) of this process include the temperatures T], and the tank level. The manipulated variables (11., uz) for this process are the valve positions of the inlet water flow CV1 and the hot water flow CV3. The transfer functions that relate the two inputs to the outputs are given below: ~03 1.00 _5 '5 + :.—__ + ‘ 1003+l“l(5) ll05+lu2(s) Ms) 90s+lul(s) ioss+i"2(‘) y I (S) = 3) Write down the steady state gain matrix and derive the relative gain array for the 2 x 2 system. Do you expect to see interactions? Justify your answer using the physical nature of the system. 20f16 b) Another student who has done this laboratory before has obtained the following relative gain array for the 2 x 2 system. Comment on which output-input pairing you would utilize based on the RGA. Justify your choice based on the RGA values. A_ 0.47 0.53 " 0.53 0.47 c) Supposing there exists a large time delay in the pairing you choose (say yl-ul), is it necessary to recalculate the RGA ? Will your choice change if the other pairing (say yl - u2) possibility does not have a time delay ? Explain in less than two sentences. 3of|6 2. (20 marks) Consider the heat exchanger process flow diagram shown below, where the objective is to heat a process stream to a desired temperature (Tpsouo using the hot stream flow rate as the manipulated variable (Fm). The process stream flow rate is Fp5 and the inlet temperature is 'l‘psin. a) Using the sensors and manipulated variables shown on the diagram, draw a feed- forward control strategy on the process flow diagram above to eliminate the effect of variations in Tpsin on the controlled variable Tm“. Draw also the block diagram representation of this feed—forward strategy below and mark the process variables in the diagram. 4ofl6 b) If the transfer functions relating TPs-m, F 115 to T950“. are given below, design a feed- forward controller to obtain perfect control. Is the controller designed realizable ? Provide a physical explanation for why or why not? Se—IJ e-I5.r F + T - s 63+! “(5) 305M ”“0 Tprmd (S) = c) If the delay in transfer function relating the FM to Tpsout is 20 min due to a different sensor placement, Is the controller designed realizable ? Provide a physical explanation for why or why not ? Answer in less than 2 sentences. Sofl6 3. (20 marks) After a particularly brutal selection process, you are recruited as a process engineer in your dream company, Boors Lite and are a part of the process operation team of their elite brewery situated right in the mountains next to the ski slope. Lately, there have been considerable variations in the ethanol composition of their final product and a decision to introduce process automation and control has been made and you have been assigned to develop a feedback control strategy for this process by your microbiologist boss. You still remember your ChE322 concepts and decide to develop a process model first for the control design. You start off by considering the femrenter, where microbial growth is taking place as a CSTR. Your boss has asked you to make the following assumptions and derive the model: 0 Microbial growth can be represented as a chemical reaction X + as ——> 2 X o The reaction rate per unit volume or the net production of cells/unit volume is equal to [1X S, where p is a constant and X, S are the concentration of cells, and the substrate respectively (reactants). o The substrate consumption rate per unit volume is given by -apX S. o The system has constant volume V and the feed flow rate F is fixed. The reactor is well mixed F1 Xin) Sin ——._l F, X,S a) Develop a dynamic mass balance for the system to describe the dynamics of X and S and then obtain the state space model. Is the underlying system linear ? Explain why or why not ? 60f16 b) If in addition to the variables cell concentration, X and substrate concentration, S, we are interested in the ethanol concentration dynamics. The ethanol concentration dynamics assuming that the substrate concentration is controlled at SSp using a secondary feedback control strategy in cascade are described below: 73 P+0.l V Incorporate the following deviation variables (y‘—=P(t)-PSS , u’=SsP(t)-SSP”) and obtain a linearized model of the product dynamics around the operating point (P”, 85““), where S‘p(t), the substrate concentration setpoint is the manipulated variable (u) and P(t), the ethanol concentration is the output, y. 80fl6 4. (20 marks) Consider the bottoms part of the distillation column control. Here, the goal of the control strategy is to maintain a constant temperature in the bottoms stream (T3). The heat supplied to the reboiler is controlled by manipulating the steam flow rate to the reboiler F31. The steam flow rate depends on the percent valve opening (SVP) and the steam supply pressure. Variations in the steam supply pressure P can impact the heat supplied to the reboiler even if the steam valve position is constant. Steam Header The transfer functions that relate the output T3 to the process variable PST and PST to the variables SVP and P are given below: 0.25 T = F 3(3) 10s+l 31(3) 2.2 1.5 F = SVP + P ”(8) 25+1 (s) 0.5s+1 (S) a) Explain why a cascade control strategy is possible and draw the cascade strategy on the process diagram above (add any sensors required). 9ofl6 b) Sketch the block diagram representation of the strategy below and indicate in the blocks all the transfer functions and the process variables. 0) Explain why a feedforward and feedback control strategy is possible and sketch the control strategy on the flow diagram below (add any sensors required). Steam Header l00fl6 d) Sketch the block diagram representation of the strategy below and indicate in the blocks all the transfer functions and the process variables. e) Under what conditions, will you choose the cascade control strategy over the combined feedforward/feedback strategy and or vice-versa ? llofl6 5. (20 marks) Consider the following process with a RHP zero that is controlled using a feedback strategy. (—s + 2.5) Gs =———————- U 32+IOS+25 a) Is a P-only controller with a gain Kc=2 stable ? Justify your answer. b) Show the closed loop poles on the complex plane and indicate the nature of the response of the P-only controller with a gain Kc=2 to a step change in the setpoint? l2 ofl6 c) If the process did not have the RHP zero and is described as below. I G(s) = —-———-—— s 2 + 10s + 25 Find the bound on the proportional gain to ensure closed stability for this process. 13 of16 6. (20 Marks) Consider the following first-order process that is controlled by a P—only controller with a gain Kc. G(S) = ___l__e-0.5: s +1 a) Calculate the transfer function representing the closed loop response. Using the gain of the closed loop transfer function, calculate the final value of the output when a unit step change in the set-point is given, in terms of Kc.‘ b) If the steady state off-set is 40%, calculate the value of the controller gain, Kc using the expression in a ? [4 of 16 c) Assume that the controller gain you calculate in the previous step is 2. If the Gain margin for this case is 2, what is the maximum value of the controller gain before the process becomes unstable ? What is the steady state off-set for this gain value ? Explain the reason for this performance limitation ? 15 ofl6 Dynamics: Poles: Roots ofdenominator polynomial, Zeros: Roots ofnumerator polynomial for 8150 It first Order Model: 6(5): 131’; First Order Time Domain Solution to Unit Step Response: y(!) = k(l - 9"") s dOd Mod I-G(S) : 2 2 econ rer e. ‘l' S +ZT€S+1 Second Order Time Domain Solution: Underdamped: {I 1 7. l~§2 _, 1—52 §<l,y(t)=k1——————e sm(at+¢) ,a=—————,¢=tan ,il — 52 T «f -tlr —tlr 1'18 ' — r e I Overdamped: 93> LY“) :57“ + 1' 12 ) 2 — l t _ Critically damped: 9: = l,y(t) = k(1_(1+ _;)2 Hr) Final Value Theorem: Lim y(t)= Lim(sy(s)) !-—)00 initial Value Theorem Lim y(t)= Lim(sy(s)) 5—)00 Feedback Control u(s) P- Control: =K eS( ) c 1. (s) 1 Pl Control: — K (1 +— —) 9( S) 1,3 u(s) l PlD-Conlrol -‘_ = Kc (I + — + MS) (3) 1,3 Stability Characteristic Equation: Polynomial defined by the equation l+L=0, L Is the Loop transfer function (product of all TFs In a loop) Frequency Response Phase Margin, PM: —,¢P.,,+l80 (me Is phase angle when AR IS 1 Gain Margin, GM=|IARW AR“, is the amplitude of Gpo at w”, where phase is -180 Loop Interaction: RGA element for a 2 x 2 system based on the process gain matrix: [in = —kl—k— I2 2I kllk22 l60fl6...
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