Question 1 (a) Why is the final value theorem useful for process control purposes ? (b)...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Question 1 (a) Why is the final value theorem useful for process control purposes ? (b) Prove that lim f(1) = lim[$7(s)] 3e™* 10s +1 (i) What is the steady state gain and time constant ? (ii) If the forcing function is given as 4/s, what is the value of y(t) when →∞? (iii) For the same input, what is the value of the output when/=10 s? What is the output when expressed as a fraction of the new steady state value? (d) Using Laplace transforms find the complete time domain solution of the second order differential equation +5+6y()=7, y(0)=7, y(0)=1 [20] (c) Consider the following transfer function G(s)= Question 2 (a) Outline the steps one should take during the development of a mathematical model for chemical/metallurgical process control. (b) Consider the two systems shown below. System 1 differs from system 2 by the fact that the level of liquid in tank 2 does not affect the effluent flow rate from tank 1, which is the case for system 2. Tank I System I Tank 2 FLEL Tank I System 2 Tank 2 (i) Develop the mathematical model for each of the two systems. (ii) What are the state variables for each system, and what type of balance equations have you used? (iii) Which mathematical model is easier to solve, that for system 1 or that for system 2? Why? Assume that the flow rate of an effluent stream from tank 1 is proportional to the hydrostatic liquid pressure that causes the flow of liquid. The cross section are of tank 1 is 41 (f) and of tank 2 is A2 (fr) for both systems. The Flow rates F₁, F₂ and F3 are in ft/min. [20] Question 3 (a) What are the principal characteristics of the first order system? What causes the appearance of a purely capacitive process? (b) Show that a purely capacitive process is unstable with respect to a unit step change forcing function (c) A single-tank process has been operating for a long period of time with the inlet flow rate. q, equal to 30.1 ft/min. After the operator increases the flow rate suddenly at t=0 by 10%. the liquid level changes in the tank as shown below: 1.(min) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 5.50 5.75 5.93 6.07 6.18 6.26 6.32 6.37 6.40 6.43 6.45 6.50 6.51 6.52 Assuming that the process dynamics can be described by a first-order model, calculate the steady-state gain and the time constant (7) From the time required for the output to reach 63.2% of the total change. (ii) From the initial slope of the response curve. [20] Question 4 (a) With the aid of a well labeled diagram, describe the characteristics of an underdamped second order system. Your answer should be in reference to the sketch and it should include the appropriate equations. (b) The set point of the control system under proportional control (K-5) undergoes a step change of magnitude and assuming the transfer function of the measuring device is unity: 1 2. For G, (s)G, (s) = (s+1)(2s+1) (i) Determine the characteristic equation for the system (ii) Determine when the maximum value of y occurs (iii) Determine the offset (iv) Determine the period of oscillation (v) Sketch y(t) as a function of time showing key characteristics as determined in ii, iii and iv. [20] Question 5 (a) Compare and contrast the Cohen-Coon method with the Ziegler-Nichols method for tuning of feedback controllers (b) The response of a system to a unit step change in the input (occurring at time 0) is shown below. 1.6 1.4 1.2 1 y 0.8 0,6 0.4 0.2 10 20 25 15 Time (s) (i) Derive a second order plus dead time model approximation for the system. Provide values for the gain, time constants and time delay of the SOPTD model. (ii) Why would a FOPTD model be not accurate for fitting the data? 2 (iii) Select the controller settings for the Cohen-Coon method if a PI controller is used. Supposed a PID controller was used instead what significant differences in the tuning parameters would you expect to see? [20] Question 1 (a) Why is the final value theorem useful for process control purposes ? (b) Prove that lim f(1) = lim[$7(s)] 3e™* 10s +1 (i) What is the steady state gain and time constant ? (ii) If the forcing function is given as 4/s, what is the value of y(t) when →∞? (iii) For the same input, what is the value of the output when/=10 s? What is the output when expressed as a fraction of the new steady state value? (d) Using Laplace transforms find the complete time domain solution of the second order differential equation +5+6y()=7, y(0)=7, y(0)=1 [20] (c) Consider the following transfer function G(s)= Question 2 (a) Outline the steps one should take during the development of a mathematical model for chemical/metallurgical process control. (b) Consider the two systems shown below. System 1 differs from system 2 by the fact that the level of liquid in tank 2 does not affect the effluent flow rate from tank 1, which is the case for system 2. Tank I System I Tank 2 FLEL Tank I System 2 Tank 2 (i) Develop the mathematical model for each of the two systems. (ii) What are the state variables for each system, and what type of balance equations have you used? (iii) Which mathematical model is easier to solve, that for system 1 or that for system 2? Why? Assume that the flow rate of an effluent stream from tank 1 is proportional to the hydrostatic liquid pressure that causes the flow of liquid. The cross section are of tank 1 is 41 (f) and of tank 2 is A2 (fr) for both systems. The Flow rates F₁, F₂ and F3 are in ft/min. [20] Question 3 (a) What are the principal characteristics of the first order system? What causes the appearance of a purely capacitive process? (b) Show that a purely capacitive process is unstable with respect to a unit step change forcing function (c) A single-tank process has been operating for a long period of time with the inlet flow rate. q, equal to 30.1 ft/min. After the operator increases the flow rate suddenly at t=0 by 10%. the liquid level changes in the tank as shown below: 1.(min) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 5.50 5.75 5.93 6.07 6.18 6.26 6.32 6.37 6.40 6.43 6.45 6.50 6.51 6.52 Assuming that the process dynamics can be described by a first-order model, calculate the steady-state gain and the time constant (7) From the time required for the output to reach 63.2% of the total change. (ii) From the initial slope of the response curve. [20] Question 4 (a) With the aid of a well labeled diagram, describe the characteristics of an underdamped second order system. Your answer should be in reference to the sketch and it should include the appropriate equations. (b) The set point of the control system under proportional control (K-5) undergoes a step change of magnitude and assuming the transfer function of the measuring device is unity: 1 2. For G, (s)G, (s) = (s+1)(2s+1) (i) Determine the characteristic equation for the system (ii) Determine when the maximum value of y occurs (iii) Determine the offset (iv) Determine the period of oscillation (v) Sketch y(t) as a function of time showing key characteristics as determined in ii, iii and iv. [20] Question 5 (a) Compare and contrast the Cohen-Coon method with the Ziegler-Nichols method for tuning of feedback controllers (b) The response of a system to a unit step change in the input (occurring at time 0) is shown below. 1.6 1.4 1.2 1 y 0.8 0,6 0.4 0.2 10 20 25 15 Time (s) (i) Derive a second order plus dead time model approximation for the system. Provide values for the gain, time constants and time delay of the SOPTD model. (ii) Why would a FOPTD model be not accurate for fitting the data? 2 (iii) Select the controller settings for the Cohen-Coon method if a PI controller is used. Supposed a PID controller was used instead what significant differences in the tuning parameters would you expect to see? [20]
Expert Answer:
Answer rating: 100% (QA)
a final value theorem is value of response b value with this theorem ie at t4 t... View the full answer
Related Book For
Digital Signal Processing
ISBN: ?978-0133737622
3rd Edition
Authors: Jonh G. Proakis, Dimitris G.Manolakis
Posted Date:
Students also viewed these chemical engineering questions
-
Managing Scope Changes Case Study Scope changes on a project can occur regardless of how well the project is planned or executed. Scope changes can be the result of something that was omitted during...
-
The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-5. Ivan's grandfather died and left a portfolio of municipal bonds. In 2012, they pay Ivan...
-
Consider the following transfer function: Y(s) G(s) = 5 U(s) 10s +1 What is the steady-state gain? What is the time constant? If U(s) = 2/s, what is the final value of the output y(t)? For the same...
-
Question 5 [ 4 points ] In the table shown below is accounting equation information as it applies to Second Time Around Clothing. Calculate the missing amounts assuming that a . Assets decreased by $...
-
Red Devil Investors has a success rate of one project for every four funded. Red Devil has an average loan period of two years and requires a portfolio return of 25%. If you borrow from Red Devil,...
-
If the minimum wage were abolished, there would be a substantial increase in the employment of marginal workers only if the MRP for marginal labor was very __________ and the supply of marginal labor...
-
The 6-ft-long column has the cross section shown and is made of material which has a stress-strain diagram that can be approximated by the two line segments. If the column is fixed at both ends,...
-
On January 1, 2018, the general ledger of Freedom Fireworks includes the following account balances: During January 2018, the following transactions occur: Borrow $100,000 from Captive Credit...
-
Describe the complexities related to expatriate executive assignments and the complexity of the international compensation balance. Include in the analysis the perquisites management. Describe the...
-
1. Develop a master schedule using the information above. 2. A customer has just requested a major order of 45 pumps for delivery in week 5. What would you tell the customer about having such an...
-
What do you mean by vitamins, its type ,its function , and its importance in human life.
-
Assume that the recorded heights of 10 students are 120, 122, \(128,176,124,127,121,125,127\), and 129 centimeters. Which number do you think will be the outlier while calculating the average height...
-
Below is selected financial data extracted from the accounting records of Wilson Manufacturing Pty Ltd for the year ended 30 June 2019. Required (a) Prepare a cost of goods manufactured statement for...
-
During the year ended 30 June 2019, Beautiful Bottles Pty Ltd incurred the following costs in connection with its production activities. Required (a) Calculate the relationship between factory...
-
Bonnie and Clyde have a partnership to run their human resource management services firm. Account balances related to their equity for the year ended 30 June 2020 are as follows. Profit of $124 000...
-
Estimate the parameters in the linear equation Y = + X + using the data in Table 9.12. Now plot these data and draw in your fitted line. Explain why you either do or do not think that the fitted...
-
2. Draw the network and identify the critical path. Also calculate the earliest and latest starting and finishing times for each activity. Activity Preceding Activity Time (Weeks) A B C E F G H I A,...
-
A business had revenues of $280,000 and operating expenses of $315,000. Did the business (a) Incur a net loss (b) Realize net income?
-
Refer problem 6.29 for the DIF case. Develop an inverse redix-2 DIT FFT algorithm starting with the definition. Draw the flow graph for computation and compare with the corresponding flow graph for...
-
An AR(3) process is characterized by the prediction coefficients a3(1) = -1.25, a3(2) = 1.25, a3(3) = 1 (a) Determine the reflection coefficients. (b) Determine xx(m) for m 3. (c) Determine the...
-
Determine the input-output relationship, the system function, and plot the pole-zero pattern for the discrete-time system shown infigure. rcos e rsin 6 z- y(n) rsin 9 x(n) -rsin f -rcos e rcos 6
-
A system consisting of a gas confined in a cylinder undergoes a series of processes shown in Fig. 2.4. During the process A-1-B, \(60 \mathrm{~kJ}\) of heat is added while it does \(35 \mathrm{~kJ}\)...
-
What is internal energy? Prove that internal energy is a state function.
-
\mathrm{~kg}\) of water is vaporized in a container at the constant temperature of \(373 \mathrm{~K}\) and the constant pressure of \(1,01,325.0 \mathrm{~N} / \mathrm{m}^{2}\). The specific volume of...
Study smarter with the SolutionInn App