Project #$5$ Thermal tanks

Part $1$: Dynamic Modeling

Consider the set of tanks shown in Figure $1.$ The first tank provides some mixing and residence time to stream $A.$ Tank $2$ provides mixing of streams A and $B.$ We assume that the volumetric flows $f_A$ and $f_B$ are constant. The density $$ and heat capacity $C_p,$ of the streams are equal to each other and are constants. The heat losses to the surroundings and the paddle work are negligible. The volumes of tanks I and $2$ are $V_1$ and $V_2,$ respectively.

It is desired to know how the outlet temperature from the second tank $T_4(t)$ is affected by the inlet temperature of steam A $T_1(t),$ while that of steam $B,T_3(t),$ is considered as a disturbance.

$1-$ Based on the unsteady$-$state mass balance for each tank, show that the transfer function of the process can be obtained as the following equation and give the expression of $K_1,K_2,{}_1,$ and ${}_2$ as a function of process parameters

$T_4(s)=\frac{K_1}{({}_{1}s+1)({}_{2}s+1)}{T}_1(s)+\frac{K_2}{{}_{2}s+1}{T}_3(s)$

where $T_4,T_1,$ and $T_3$ are the deviation variables of $T_4,T_1$ and $T_3$ respectively.

Let ${}_1=8\mathrm{.}26,{}_2=12\mathrm{.}43,K_1=8\mathrm{.}31,$ and $K_2=9\mathrm{.}95$

$2-$ Draw the block diagram of the process

$3-$ Provide the time$-$domine characteristics of the process.

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

$5-$ If the pipe between tank $1$ and $2$ is long. How would this change affect the dynamies of the process?

Part $2$: Process Identification

Experimental data of the process is provided within the Project$\_5.$m file.

$1-$ Use data to derive a reasonable model between $T_4(s)$ and $T_1(s).$ Compare results obtained from dynamic model and explain the difference in results.

$2-$ Use Smith's method as well as nonlinear regression based on MATLAB to identify a model

$3-$ Provide a comparison of the model parameters and the sum of squared errors for each model

$4-$ Plot the step responses of fitted models with the original data.

$5-$ Conclude

Part $3$: Controller Design & Tuning

In what follows, use the identified SOPTD model from Part $2.$ For each question, compare the designed controllers for unit step changes in the set point and disturbances.

Consider the plot of results in a single graph, for a clear comparative analysis.

$1-$ Based on Direct Synthesis Method calculate PID controller settings for the process, for three different values of ${}_c$ of your choice.