The blending tank shown in figure $(1)$ below, may be assumed to be perfectly mixed two streams.

Figure $1$: stirred$-$ tank blending system

Stream $1$ is a mixture of two chemical species A and B$,$ its mass flow rate is $w_1$ and its mass fraction of $A$ is $x_1.$ Stream $2$ is a mixture of the same species A and $B,$ its mass flow rate is $w_2$ and its 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 $w.$ It is 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 $w_2$ be the manipulated variable and $w_1$ 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 for the operating conditions are:

$V=10m^3$

$w_1=1k\frac{g}{s}$

$w_2=3k\frac{g}{s}$

Study the above system process, then do the following:

Identify state variables, input variables.

Derive the differential equation that describes the dynamics of state variable.

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 Simulink.

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

Describe the advantages and disadvantages of ratio control.

Simulate the controlled system using Simulink.