Consider the blending process shown in figure $1,$ where a stirred tank is used to blend two streams.

Stream $1$ is a mixture of two chemical species A and B$,$ it's mass flow rate is ${}_1$ and it's mass fraction of A is $x_1,$ stream $2$ is a mixture of the same species A and B$,$ it's mass flow rate is ${}_2$ and it's mass fraction of A is $x_2,$ the mass fraction of $A$ in the exit stream is denoted by $x$ and the total mass flow rate is $.$

It's desired to control the mass fraction of A at the exit $"x".$ The volume of the liquid $"V"$ can change in the tank however, the liquid density $""$ is constant.

Let ${}_2$ be the manipulated variable and ${}_1$ be the disturbance, $x_1$ and $x_2$ have constant values $x_1=0\mathrm{.}2$ and $x_2=0\mathrm{.}1,$ the density $=1000K\frac{g}{m^3},$ the steady state values are: $V=10m^3,{}_1=1K\frac{g}{s}$ and ${}_2=3K\frac{g}{s}.$

Study the above system process, then do the following:

Build the dynamic model of the system.

$1$

Linearize the model around the steady state values.

Use Laplace transform to get the transfer function.

Draw the block diagram of the system.

Simulate the uncontrolled system using MATLAB.

Design a ratio controller to set the value of $x$ at $0\mathrm{.}5.$

Explain the advantages and disadvantages of ratio control.

Simulate the controlled system using MATLAB.

Write a report that includes all your work, MATLAB codes and results. Discuss your results.

Note: Report format is important $($font$,$ page number, figures' and tables' captions, table of contents, introduction, $...,$conclusion, references$).$