Please define the variables you named with letters.

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.

Explain the meaning of mathematical model and write down types of models. $(10p$

One of the famous milk processing companies noticed that in our country there exist lots of people

suffering from lactose intolerance. For this reason, reduction of lactose composition is obtained by the

enzymatic activity of lactase. Temperature is very important on enzyme activity and it should be above

$93\mathrm{.}5C$ and below $105C.$ For this process design a continuous process control system and indicate

control structure, sensor$,$ controller and final control element. Define all the variables. In addition,

discuss the advantages and disadvantages of process control structure you have chosen for this system.

$(15$ p$)$

For a continuously stirred tank reactor $($CSTR$)$ with second order reaction derive and solve the

model equation for concentration change with respect to time. Show the CSTR with all streams in a plot

and write down your assumptions. $(15$ p$)$

Explain the terms degrees of freedom, conservation equations, constitutive equations, and the

relationships between them. $(15$ p$)$

In a well$-$mixed closed tank, a liquid is heated with saturated steam, which 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 all the system information and list your assumptions $(10p$

b$.$ derive and solve the model equation for temperature change $(15$ p$)$

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

functions as $5$ p 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$)$

$($Bonus$)$ For the equation $(x(t)=(\frac{4}{3})e^{-t}-(\frac{1}{3})e^{-4t})$ show the use of Laplace transform for the solution.