Homework

The purpose of the homework in this class is to reinforce concepts discussed in the reading material and re-emphasized in class. Unfortunately, due to time constraints, students often view homework as busywork necessary to get a grade. Getting the answer becomes all important rather than learning the concept.

Homework is due at the beginning of the class period. Late homework may be handed in for half credit within a week. When you go on an interview trip, please work the homework in advance to avoid the late penalty. The solution key may be consulted when doing late homework.

Special Problems

• Special Problem #1
• Special Problem #2
• Special Problem #3 - Excel and Simulink Files - SP3 Simulink Tutorial
• Special Problem #4 - Excel File
• Special Problem #5 - Excel and Simulink Files - SP5 Modeling Tutorial
• Special Problem #6 - Simulink Files
• Special Problem #7
• Special Problem #8
• Special Problem #9
• Special Problem #10 - Simulink Files
• Special Problem #11 - Excel Supplement - SP11 MPC Tutorial
• Special Problem #12 - See Hint Below
• Special Problem #13
• Special Problem #14 - T2 Laboratories and Deepwater Horizon Blowout

Homework Hints

SP8 - On 1A, remember that Kp = Delta y/Delta u, so you can calculate the value of y after the step change in u.

PDC 2.4 - The pressure drop across the valve can be written as a function of Pa, Pg, and rho*g*h.

PDC 2.10 - Although the flows are assumed constant, the concentrations are not necessarily constant. You will need to write 5 dynamic balance equations:

• 3 species equations and the energy equation for the reactor;
• 1 energy equation for the jacket.

In addition write the overall mass balance equations for the jacket and for the reactor reactor (even thought they do not contain time-dependent terms).

PDC 3.4 - Ramps are fun to use with Laplace transforms, but remember that the function goes on forever. There is no STOP function. If you start a ramp and want it to level off after a certain time, you have to put in a time delay function and then a ramp of opposite sign to make the function level off. Remember that for a time delay in the "time" space, you change all the t's to t-theta, and multiply my the unit step function S(t-theta), where theta is the time delay. In Laplace space, you just multiply the Laplace function by exp(-theta*s).

PDC 3.17 - You do not need to transform this equation into deviation variables. Just take the Laplace of both sides and manipulate it into a form compatible with something in the Laplace table.

PDC 4.7 - Generate four transfer functions as dynamic relationships between each input (2) and each output (2). See hints here and the solution here.

PDC 4.10 - Write the energy balance, including the substitution for U that involves the wind velocity. You will have to do the Taylor's expansion to get deviation variables.

PDC 5.15b - Use the equation for the time of the first peak and then using this in an equation to get the peak temperature. I would also like you to plot temperature versus time from 0 to 30 minutes, and see if your calculated peak temperature matches what your graph says.

PDC 6.7b

1. Problem 6.7 data file (Excel)
2. Get time-domain solution for P_m(t) in terms of K, zeta, and tau.
3. Use solver in Excel to perform least squares fit using K, zeta, and tau as variables or use Python SOPDT example code
4. Plot to see goodness of fit.

PDC 11.11 Homework help (pdf)

PDC11.7

• Start by solving for U. Follow algebra around the loop until you get U o both sides.
• Then manipulate the algebra to get an expression for U.
• Next solve for Y, and plug in the expression for U.

Please use the following video to check your work for 11.7 after you have completed the problem.

PDC 11.10 - Use stability criteria on the second line after Equation 11-93. In other words, all of the coefficients of the characteristic equation (the denominator of the transfer function) must be positive. Please use the following video to check your work for 11.10 after you have completed the problem.

PDC 11.14 - Use the shortcut method on the inner loop then on the outer loop to get Y/Ysp. You may get negative coefficients in the denominator, but the feedback controller is able to stabilize the system.

PDC 11.18 - On part a, you will get two equations and two unknowns (Kc and w). If you put the sin and cos terms on the left-hand side of the equation, you can divide the two equations, eliminating Kc and getting tan w. Then solve iteratively using Mathcad or Excel (guess w= 0.5 to start).

PPC 18.1 - I want you to run the jacketed reactor as follows:

1. without cascade control. You will need to do a doublet test, get tuning constants for a PI controller, and test response to a dissturbance change.
2. with cascade control. You will need to tune each controller (like the example in the book) and test the response to the same disturbance change as in part (a). Please comment on the improvment (if any).

SP 12 - For special problem #12, please assume a 4-20 mA signal from the measurement transducer. From Wikipedia - Current Loop: For industrial process control instruments, analog 4–20 mA and 10–50 mA current loops are commonly used for analog signaling, with 4 mA representing the lowest end of the range and 20 mA the highest. The key advantages of the current loop are that the accuracy of the signal is not affected by voltage drop in the interconnecting wiring, and that the loop can supply operating power to the device. Even if there is significant electrical resistance in the line, the current loop transmitter will maintain the proper current, up to its maximum voltage capability. The live-zero represented by 4 mA allows the receiving instrument to detect some failures of the loop, and also allows transmitter devices to be powered by the same current loop (called two-wire transmitters). Such instruments are used to measure pressure, temperature, flow, pH or other process variables. A current loop can also be used to control a valve positioner or other output actuator. An analog current loop can be converted to a voltage input with a precision resistor. Since input terminals of instruments may have one side of the current loop input tied to the chassis ground (earth), analog isolators may be required when connecting several instruments in series.

Generative AI Learning

Homework in this course is where concepts become skills, and Generative AI can now produce a full solution to most textbook problems in seconds. That makes the homework more valuable, not less: the scarce skill is knowing whether a solution is right. Use AI the way you would use a good TA - for hints, checks, and quizzes - and keep the problem-solving in your own hands. Three prompt patterns work well with the problems on this page:

Hint, not solution (works like the Homework Hints above, one step at a time):

Solution audit (after you have a complete answer):

Quiz me before the exam (name the topic of the current homework set):

Guidance by section:

• Modeling and control basics (SP1-SP8, PDC Ch. 1-2 problems): derive every balance equation by hand before asking AI to check it. For SP2 (speed control) run the Vehicle Speed Control Studio and for SP7 (tank blending) the Blending Tank Control Studio - both give the dynamic response and CSV export without writing code, so you can test your predicted gain and time constant against a simulation.
• Valves and equipment (SP6, PDC Ch. 9 problems): the Valve Design app shows inherent vs installed characteristics live - use it to sanity-check any Cv calculation before you trust it.
• Linearization and Laplace (SP9, PDC Ch. 3-4 problems): verify every analytic inversion numerically - simulate the ODE in MATLAB or Python and overlay your y(t). Agreement is the strongest evidence your transform algebra is right; disagreement localizes the error.
• Closed loop and stability (SP11-SP12, PDC Ch. 11 problems): after a Routh or direct-substitution result, confirm the ultimate gain in simulation: a closed-loop simulation at 0.9 Kc,max should be stable and at 1.1 Kc,max should oscillate with growing amplitude.
• Advanced control (SP10, SP13, PPC Ch. 18-19 problems): the Feedforward Cascade Control Studio runs feedback, feedforward, and cascade side by side on the same process, and the Distillation Column Control Studio is a digital twin for the SP10 column - use them to check whether your predicted improvement actually appears.
• Safety (SP14): watch the linked T2 Laboratories and Deepwater Horizon videos first, then reproduce a thermal runaway safely in the Stirred Reactor Control Studio - the 400 K alarm makes the heat-generation vs heat-removal argument concrete.

Quizzes on Learning Suite and exams are taken for a grade: no AI assistance there. The habits above are how you make sure the understanding is in your head when it counts.

What to Turn In

Homework submission requirements are unchanged (due at the beginning of class; solutions show your work). When Generative AI assisted with a problem, your submission must still demonstrate that the work is yours:

1. Show the setup - the balance equations, transform, or block diagram - in your own writing, with assumptions stated. This is the part AI is not allowed to do for you.
2. Include one verification for any AI-checked or AI-generated result: a unit check, a limiting-case check, or a simulation overlay against the analytic answer.
3. Note briefly where AI assisted (hint, audit, plotting, code) - one line is enough.
4. If an AI audit caught an error in your work, keep it in the write-up: show the wrong step, the catch, and the fix. Finding and repairing your own errors is graded understanding, not a penalty.