Part $1$: Dynamic Modeling

A jacketed stirred tank is used to cool a process by causing cooling water to flow

trhough the jackes as shown in Figure $1.$ The process input variable is the flow of cooling water

$f_c(t),$ The inlet temperature of the process stream $T_i(t)$ is considered as a disturbance variable.

The output of interest is the outlet temperature of the process and the water $T(t),T_c(t),$

Figure $1$ Jacketed process

respectively. We assume a constant volume of the tank, both fluids densities and specific heat

coefficients are constant. Furthermore, both tank and jacket are assumed to be perfectly mixed

and heat losses are negligible

$1-$ From the unsteady$-$state energy balance

a$.$ derive the following differential equations that represent the dynamic response

of the process

$\frac{dT(t)}{dt}=\frac{f}{V}[T_i(t)-T(t)]-\frac{UA}{V{C}_v}[T(t)-T_c(t)]$

$\frac{d{T}_{c(t)}}{dt}=\frac{f_c(t)}{V}[T_{ci}(t)-T_c(t)]-\frac{UA}{V_c{}_c{C}_{vc}}[T(t)-T_c(t)]$

b$.$ Show that the transfer function model relating $T(s)$ to $T_i(s)$ and $T_c(s)$ can be

expressed as follows, and give the expression for ${}_1,K_1$ and $K_2$ as a function of

the process parameters

$T(s)=\frac{K_1}{{}_{1}s+1}{J}_i(s)+\frac{K_2}{{}_{1}s+1}{T}_c(s)$

where $T(s),T_i(s),$ and $T_c(s)$ are the deviation variables of $T(s),T_i(s),$ and $T_c(s)$

respectively

c$.$ Show that the transfer function model relating $T_c(s)$ to $F_c(s)$ and $T(s)$ can be

expressed as follows, and give the expression for ${}_2,K_3$ and $K_4$ as a function of

the process parameters

$T_c(s)=\frac{K_3}{{}_{2}s+1}{F}_c(s)+\frac{K_4}{{}_{2}s+1}T(s)$

where $F_c(s)$ is the deviation variable of $f_c(s)$

d$.$ Obtain the transfer function relating $T(s)$ to $F_c(s)$ and $T_i(s)$

$2-$ Draw the block diagram of the process

${}_1=1\mathrm{.}36,{}_2=2\mathrm{.}54,K_1=0\mathrm{.}6,K_2=0\mathrm{.}35,K_3=0\mathrm{.}2,$ and $K_4=1\mathrm{.}1$

$3-$ Provide the time$-$domine characteristics of the process, assuming $T_i(s)=0$

$4-$ Analyze the degrees of freedom of the process.