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.

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$).$