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Q1: Complete the following quiz using only materials distributed in this class. You mayuse any material in the class Google Drive and your textbook, but no othermaterials or people. You must do your work on your own. Enter your answers intothis document and convert it to a PDF prior to submission on eCampus. If your workinvolves long, handwritten solutions then you may attach those as an image eitherin this document or as a separate document. Final answers, however, MUST be inthis document. 1. (20 points) What is the input function and the Laplace transform of the following figure?See Answer
Q2: Consider a weight held in place by a spring. Initially suspended at rest, the height ofThe spring constant is kThis represents how "stiff" the spring is and is constant. The frictional losses areisthe weight is x=0The mass of the weight ismcharacterized by the constant b . The force applied to the weight after t=0 f(t) . This leads to the ODE m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=f(t) 2. (65 points) Imagine the spring is vertical. The weight is held by a platform atx=0 initially. If m=0.1kg , b=0.1 kg/ s , and k=1.0 kg/s . The platformis removed at t=0 O What are the initial conditions for this system? b. (15 points) What is the new steady-state position? That is, where willt - o ? Show this using the final-value theorem.the mass settle as c. (15 points) What is the maximum distance the mass will move from itsinitial position? (Hint: what is the force applied to the weight?)(15 peints) Whatvalue vwill recuultming d. (15 points) What spring constant value will result in the system comingto (and remaining within) 1% of its new steady-state position withoutovershooting that value as quickly as possible? e. (15 points) Imagine the spring system is now horizontal.represents how far the spring is from its resting position. If 1 N of forceis instantaneously transferred to the weight (in thedirection)described above att=0 ,what is the response? That is, what isx(t) when the weight is flicked with 1 N of force? Assume the weightis only free to move in one direction.See Answer
Q3: 3. (15 points) Prove that a second-order system that results from two first-ordersystems in series can never be underdamped.See Answer
Q4: In the process presented in Figure Q1, tomato pulp is heated as it passes through a steam heat exchanger and then enters an evaporator where the water boils off. The purpose of this process is to produce tomato paste, which has a lower water content than the pulp. As a chemical engineer, you are tasked to control the liquid level and temperature in the evaporator.Considering this information, answer the following: (a) Define the process variables and the manipulated variables, as well as possible disturbances. (b)Propose feedback control loops by sketching a schematic diagram. (c) Propose any additional features in order to assure the safe operation of the process. Illustrate these features using a schematic diagram.See Answer
Q5:Question 1 An engineer uses a temperature sensor mounted in a thermowell to measure the temperature in a CSTR. The temperature sensor behaves as a first-order process with a time constant of 3 seconds, and the thermowell behaves as a first order process with a time constant of 9 seconds. Using Matlab: (a) Plot the open-loop response of the system to a step-change in the CSTR temperature. Comment on whether the response in under or over damped (b) Plot the closed-loop response of the system for proportional only control with a controller gain of unity. (c) The engineer notes that the measured temperature has been varying sinusoidally between 180 and 186°C with a period of 20 seconds for at least 10 minutes. Determine the likely temperature fluctuation in the CSTR contents to the nearest degree.See Answer
Q6:I Preamble Noise Disturbance in Control Systems Control engineering, as one of the cornerstones of automation, has contributed enormously to the development of modern industrial society; and a control system is useful for regulating the behaviour of industrial systems. In this assignment, you are required to analyze the following control system in Figure 1 where X(s) is the signal input, N(s) the noise input and Y(s) the system output. G₁(s) = K G₂ (s) = G3(s) = G₁(s) = H₂(s) = 1 s+2 X(s) Figure 1 The gain values for the respective forward and feedback paths are given: 2 s+4 4 S H₁(s) = 1 10 s + 10 H3(s) = 2 G₁ G₂ H₂ H₂ N(s) G3 H3 G4 Y(s) Task 1: Analytically determine the signal and noise transfer function of the control system in Figure 1. If x(t) is a unit step input then how do the signal and noise vary at the output for any change in the value of K as t → ∞o. Task 2: Using Matlab determine the signal and noise transfer function of the control system in Figure 1 assuming K = 1. Indicate clearly the Matlab code applicable to appropriate block reduction. Plot the step response using Matlab to verify the final steady-state value obtained in Task 1 y(t) as t→∞o for signal and noise.See Answer
Q7:1 Preamble The aim of this coursework is to enhance your understanding of the dynamics of multi-degree of freedom systems to compliment what you have learnt in the EG5027 Dynamics and Control module. 2 Assignment Tasks (1) Derive the mass and stiffness matrices for the vibrating system shown in Figure 1. (2) Show that the natural frequencies and mode shapes of the coupled system are: f₁ = 1.24 Hz, f₂ = 2.59 Hz -0.89 = { 056 )}. {21 = {-000} {y} = 150 N m-1 www 80 Nm-1 2 kg (3) Write a Matlab program to compute the natural frequencies and mode shapes of the coupled system and compare the results obtained with the values given in Part (2). [20%] U/1 140 Nm wwww 1 kg Figure 1: A 2DOF vibrating system. [10%] U₂ [40%] [30%]/n3 Assignment Submission Format You are required to a PDF file of no more than four A4 pages showing: (a) The derivation of the mass and stiffness matrices of the system. (b) The derivation of the natural frequencies and mode shapes of the system. (c) The listing of your Matlab® script for calculating the natural frequencies and mode shapes of the system (Note: you should list the code within the PDF file as plain text, you are not required to submit your M-file.) (d) Comment(s) on the results obtained in (b) and (c). The page layout should be portrait, single column and margins should not be less than 20 mm. The file should be typeset with a minimum font size and linespacing of 12pt and 1.5, respectively. Equations prepared using technical typesetting software, such as LATEX or Math Type, are preferred, but if you are not able to do so, high-quality scanned clear and legible hand-prepared equations are acceptable. No title page is required for the PDF file. Pages of the file should be numbered consecutively and shown on the centre footer of each page. Your student number must be clearly shown on the right header of all pages. The PDF file should have the module code, your student number, assignment identifier as the filename; and 'pdf' as the extension, in the form of EG5027_u1234567_CW1R.pdf.¹See Answer
Q8:Use a trial-and-error approach in Matlab to determine the ultimate controller gain Kcu and the ultimate period Pu for control of a FOPDT process with a deadtime of 30 seconds and a process time constant of 50 s. Use the Ziegler-Nichols rules to tune a PI controller, providing Kc and reset time. Plot (a) the open loop response to a step-change in setpoint without control (b) the closed loop response to a step-change in setpoint without control, and (c) plot the closed-loop response to a step-change in setpoint with the ZN tuned controller.See Answer
Q9:The process described by the transfer function GP(S) Kp (T₁S+1) e-TDS (T₂S + 1)(T3S + 1) Gp(s) = is controlled by a P controller with an arbitrary gain of Kc. a) If Kc = 2; Kp = 5; T₁ = 2; Td = 1; T₂ = 2; T3 = 3 determine whether the closed-loop response will be stable using frequency response techniques. What is the limiting value of the controller gain to ensure stability? b) For the following scenarios, generate the Bode plots for the open- loop behavior using Excel. Comment on the effect of the parameter changes on plots 2, 3 and 4 using plot 1 as the reference. 1. 2. 3. 4. Kc = 2; Kp = 5; t₁ = 0; TD = 0; T₂ = 2; T3 = 3 Kc = 2; Kp = 5; t₁ = 0; TD = 5; T2 = 2; T3 = 3 Kc = 4; Kp = 5; T₁ = 0; TD = 0; T₂ = 2; T3 = 3 Kc = 2; Kp = 5; T₁ = 0; TD = 10; T₂ = 2; T3 = 3 c) Using the same general transfer function, plot the Nyquist plots using Excel for the following parameters. Comment on the behavior. 1. Kc = 2; Kp = 5; T₁ = 1; TD = 0; T₂ = 2; T3 = 3 2. Kc = 2; Kp = 5; T₁ = 3; TD = 0; T2 = 2; T3 = 3See Answer
Q10:Question 4 Two thermocouples are placed in an air stream whose temperature is varying sinusoidally. The temperature responses of each thermocouple are recorded for a range of frequencies, with the phase angle between the two measurements tabulated below. The standard thermocouple is known to have first-order dynamics with a time constant of 0.15 minutes when operating in air. Show the unknown thermocouple also displays first order behavior and determine its time constant. After approximately 30 seconds the standard themocouple shows a sinusoidal variation in air temperature between 96 and 104 degrees Celcius, at a rate of two cycles per minute. What error will the standard thermocouple report in this instance if left to measure the air temperature indefinitely? CHEM ENG 4050: Advanced Chemical Engineering Advanced Control Table 1. Thermocouple phase difference for Question 4. Frequency (cycles/min) 0.05 0.1 0.2 0.4 0.8 1.0 2.0 4.0 Phase Difference (deg) 4.5 8.7 16.0 24.5 26.5 25.0 16.7 9.2See Answer
Q11:Question 5 Determine the controller gain (Kc) for a proportional-only feedback controller for the system comprising the following elements that ensures a gain margin of at least 2 and a phase margin of at least 30°. Plot the controlled and uncontrolled closed-loop response of the system to a unit-step change in set- point. Gc= Kc; Gp = 50 30s +1' G₂ = 0.016 3s +1' Gm = 1 10s + 1See Answer
Q12:Question #1 The mass fraction of product A leaving a reactor (Y(s)) can be related to the reactor temperature (U(s)) by the transfer function G₁(s): Y(s) 12(s + 4) = G₁(s) = U(s) (s + x)(s + 3)(s+8) Where x = 4 a. If a step change is made to the reactor temperature will the mass fraction of A in the product reach a steady value? Clearly justify your answer. [3 marks] b. The relationship between the reactor temperature and the coolant flow rate (Z(s)) is given by the transfer function G₂(s). What is the transfer function that gives the relationship between the coolant flow and the mass fraction of product A leaving the reactor? [2 marks] c. G₂(s) can be represented by a first-order transfer function. Provide an example of such a function that will result in a stable system and an example that will result in an unstable system. Here stability is defined as the mass fraction of A reaching a steady value if a unit step change is made to the coolant flow rate. Include x in your functions. Clearly justify your choices. [5 marks]See Answer
Q13:Question #2 Dodecane is pumped into a well-insulated 2 m diameter cylindrical tank where it is heated to a temperature of 150 °C. The tank is 10 m tall, and the steady state liquid height is 6.5 m. The inlet temperature of the dodecane is 10 °C. In answering this question you can assume that the density and specific heat capacity of dodecane are constant, and that they have values of 750 kg m³ and 2200 J kg-¹ K¹, respectively. a. Draw a clear diagram of the system, and referring to this diagram derive suitable mass and energy balances for the system. Clearly state any assumptions that you have made. [5 marks] b. The outlet flow rate (Four) is proportional to the liquid height and the resistance of the outlet valve (R): h Fout R The inlet flow rate to the system is 120 L min¹¹. What is the value of R? [2 marks] c. If the inlet flow rate is 120 L min¹ how much energy needs to be supplied by the heater such that the outlet temperature is 150 °C? Here you can assume that the system is at steady state. [2 marks] d. The inlet flow-rate is increased stepwise from 120 to 180 L min-¹ and at the same time the energy supplied to the system is increased from the value calculated in part (c) to 550 kW. Plot the response of the system to these changes. Determine: • The steady-state value of the liquid height. • The steady-state value of the liquid temperature. • The values of the gain and time constant for the liquid height. 16 markslSee Answer
Q14:Question #3 A second system for heating dodecane is going to be built, this system is the same as the one from the previous question. In order to save money it is being proposed that the system is not insulated. Your manager would like to know how removing the insulation would affect the dynamics of the system. In performing this analysis you can assume that the air temperature is 10 °C and that the heat transfer coefficient between the tank and the air is 20 W m² K¹¹. In answering this question: • Clearly show how the changes to the design affect the mass and energy balances for the system. • Determine how much energy needs to be supplied by the heater to keep the uninsulated tank at a temperature of 150 °C for an inlet flow rate of 120 L min¹¹. • Compare how the insulated and uninsulated systems respond to a stepwise increase in the inlet flowrate from 120 to 180 L min¹. Use a value of 500 kW for the heater power, and an initial temperature of 150 °C for both cases. State whether or not you think building the system without insulation is a good idea. Make sure you justify your decisions. [10 marks] 3See Answer
Q15:Question #4 A system of three cylindrical tanks holding liquid is arranged as shown in Figure 1. Fint Fina Tank #1 R₁ F₁ h₁ h3 h₂ h₁ F₁ = R₁ Tank #2 R₂ F₂ Tank #3 Figure 1- Schematic showing layout of tanks. For all three tanks you can assume that the outlet flow rate is equal to the height in the tank divided by the valve resistance, i.e.: h3 hz F₂= F3: R₂ R3 a. Clearly derive the transfer function that relates the height in tank #1 (hi) to the inlet flow to the tank (Fin). [4 marks] b. Tank #1 is 0.5 m in diameter and the value of R₁ is 200 s m². At steady state the liquid height is 1 m. What is the flow rate of liquid entering the tank (i.c. what is Fin,1)? [1 mark] c. Calculate the gain and time constants for tank #1. [2 marks] d. Derive the transfer function that relates the height in tank # 3 (ha) to the flow entering the system (i.e. to Flis and F2.in). [5 marks] c. Tank #2 has the same diameter as Tank #1, but R₂ = 0.5R₁. Tank #3 has double the diameter of Tank #1 (i.e. it is 1 m in diameter), and R3= R₁. Fin1 = Fin2 = 0.01 m³ s¹. If the system is at steady state what is the liquid level in each of the tanks? [3 marks]See Answer
Q16:Q.1 Obtain the unit step response for the following in Simulink. Where possible, qualitatively justify the shape of the transient responseSee Answer
Q17:Q.2 The process is described by the following transfer functionSee Answer
Q18:Q.3 A process consists of two stirred tanks with input Q and outputs T1 and T2 (see Figure).See Answer
Q19:Q.4 An electronic PID temperature controller is at steady state with an output of 12 mA. The set point equals the nominal process temperature initially. At t=0, the set point is increased at the rate of 0.5 mA/min. If the current settings are K₂= 2, t=1.5 min, Tp=0.5 min. (a) Generate the controller response in Simulink. (b) Repeat part (a) for a PI controllerSee Answer
Q20:A control system has the following transfer functions: Gc = Kc, GvGp = 1/(s+1) (0.5s+1)2See Answer

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