$(45$ p$)$ In a well$-$mixed closed tank, a fermentation medium is heated with saturated steam that

flows through a coil immersed in the liquid. If the temperature of this system is to be

controlled by a feedback control structure;

a$)$ plot the system, indicate system information and list your assumptions $(5$ p$)$

b$)$ derive and solve the model equation for temperature change $(10p$

c$)$ develop the block diagram which represents the system $(5$p for basic structure $+$ for

transfer functions as $5p$ for each$)$

d$)$ Explain the properties of the first order systems and find if temperature change in this

system is a first order system or not, with your reason $(10$ p$)$

$(15$ p$)$ In a storage system where three non$-$interacting tanks are used in series, find the

transfer function that relates liquid height in the third tank to input flow rate to the first tank.

p$)$ Explain the meaning of mathematical model and write down the types of models.

$(15$ p$)$ For a continuously stirred tank reactor $($CSTR$)$ with first order reaction $($input material

A is partially converted to material B$)$ derive the model equations for the concentration

change of materials A and B with respect to time. Show the CSTR with all streams in a plot

and write down your assumptions.

$(10p$ Explain the meaning, importance and use of "Degrees of Freedom" for mathematical

modeling in process control studies.

$(10p)$ In a cooling system a pneumatic controller is used. If for the system error is evaluated

as a function of time given by the following relation, derive the transfer function for

proportional$-$derivative control. $(8$ p$)$

$(t)=t^2+e^{-2t}$

Attention please:

a$.$ Remembering our rule $"$Pc$"$ please consider that in order to control any system you should

know the properties of that system in detail, then you can control the systems. Therefore,

before answering the questions first try to imagine the systems.

b$.$ Please explain the properties of the systems you are considering in detail.

c$.$ Please indicate your assumptions in detail.