The following physical design of a reaction system is used for this entire problem:

A constant flowrate of an inert carrier liquid is mixed with a pure stream of component A in a mixing tank where no reaction occurs. This liquid is allowed to flow through a valve at the bottom of the tank into a reactor that holds 1000 gallons. The reactant, A, reacts according to the following rate law:

The products from the CSTR drain through the bottom of the tank into a holding tank where the reaction essentially stops. The contents of the holding tank are then pumped at a constant rate out of the plant. The outlet concentrations of A and B are measured with a spectroscopic technique that has first order dynamics with a rate constant of 2 min and a gain of 10.

Useful info: inert flowrate 1 gal/min, nominal flowrate of A into tank 1, 0.1 gal/min. Assume all liquids have the density of water and that the molecular weight of both the inert and A and B are 10 g/mol. The first tank has a capacity of 10000 gal and the flowrate leaving the tank varies linearly with height in the tank as 0.5 h, where h is in feet. All tanks are cylindrical with a diameter of 10 feet. The second tank has the same resistance in the valve leading to the third tank. The second tank also has a capacity of 10000 gal. The pump on the outlet from the last tank has a flowrate of 1.1 gal/min at steady state. You may assume that the rate constant for this reaction is 10¹³ cm⁶/mol minute.

Sketch this process and label everything.

Derive overall height balances on each tank. What is the overall steady state height in the first tank if it is initially empty?

Derive species balances on A and B for each tank, linearize if necessary.

Draw a block diagram for this process if you want to control the concentration of B in the outlet by varying the flowrate of A into the first tank. Use the balances from the previous part to find transfer functions for all of the important steps.

Assuming that the final control element has negligible dynamics but has a gain of 5, what are the upper and lower limits on stability for a proportional only controller added to the process?

Assuming that tank 3 all of a sudden sees a step increase in Ca of 1, but the flowrate is unchanged, find the general time dependent solution for the concentration of A leaving that tank.

Compare the controller setting you would need to implement a PI controller using a) Ziegler Nichols criteria and b) phase margin and gain margin general criteria.

If there is no controller on the process (Kc = 0) and the flowrate of A into the first tank doubles, what is the final steady state outlet value of B from tank 3?

A new pipe is added between the last tank and the sampling point and it takes 5 minutes for liquid to flow from the tank to the end of the pipe. Describe in your own words why the stability of the system might or might not change. What would the upper and lower bounds of Kc be?

Find the overall offset for a proportional only controller if a step change in inert flowrate is encountered.