## Course Homework

#### 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
- 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

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

- 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.
- 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.