$A2B$

The inlet to the CSTR has a variable volumetric flow rate $($L$/$min$),q(t),$ with a constant

concentration of species $A(mo\frac{l}{L}),C_A.$ The stream exiting the reactor has a volumetric flow rate

equivalent to the inlet stream, due to a separate level control system. The concentration of

unreacted species A leaving the reactor is variable, denoted by $C_A(t).$ The reaction rate of species

A per unit volume in this process is defined as:

$r_A=k{C}_A(t{)}^2$

where $k$ is the reaction rate coefficient $(\frac{L}{m}ol-$min. The following assumptions can be made:

The volume in the tank is constant.

The densities and specific heat capacities of the inlet and outlet streams are equal and

constant.

The tank is perfectly mixed.

The reaction rate coefficient is constant.

a$.$ Derive a dynamic model for the exit concentration of species A from the CSTR$.$

b$.$ Linearize the dynamic model you derived in Part a$.$

c$.$ Using your linearized dynamic model from Part b$,$ find the transfer function $C_A^{\text{'}}\frac{s}{Q^{\text{'}}}(s)$

in standard form.

d$.$ Part c tells us the general dynamics of the process transfer function but does not

consider time delay. Experiments have shown that once the inlet volumetric flow rate is

changed, the exit concentration of species A begins changing after $0\mathrm{.}3$ minutes.

Based on your answer to Part c$,$ use the following process specifications to determine

numerical values of the process gain and time constant $($with units$),$ and state what

$G_p(s)$ is in standard form, including the given time delay.

Process Specifications:

$V=200L$

$\frac{?}{\text{b}\text{a}\text{r}}(q)=10\frac{L}{m}in$

$k=1\mathrm{.}5\frac{L}{m}ol*min$

$\frac{?}{\text{b}\text{a}\text{r}}(C{)}_A=0\mathrm{.}2mo\frac{l}{L}$

$C_{Ai}=3mo\frac{l}{L}$