Work a problem similar to P$3\mathrm{.}7\mathrm{.}6$ in the textbook. Differences:

a$.$ Find the degrees of freedom.

b$.$ What assumptions are inherent in the modeling?

c$.$ Calculate the final steady$-$state heights before you do any modeling. You can use these to check your Python code.

d$.$ Use Python instead of Matlab to solve the problem. Submit your graph for the heights as a function of time.

e$.$ Repeat part d with a $10\%$ decrease in F$1$spec. Submit your graph for the heights as a function of time.

f$.$ Submit your Python file.

$113$

$M\frac{dT}{dt}=F_1{T}_1+F_2{T}_2-(F_1+F_2)T$

To determine the initial product temperature, this equation is set equal to zero $($i$.$e$.,$ steady$-$state conditions$)$ and the initial process conditions are applied, i$.$e$.,$

$M\frac{dT}{dt}=0=(5k\frac{g}{s})(25C)+(5k\frac{g}{s})(75C)-(10k\frac{g}{s})T$

Therefore, the initial product temperature is $50C.$

The actuator is modeled using Equation $3\mathrm{.}7\mathrm{.}1$ and the sensor is modeled using Equation $3\mathrm{.}7\mathrm{.}4.$ Therefore, the model equations used to represent the CST thermal mixer are

Actuator

$\frac{d{F}_1}{dt}=\frac{1}{{}_v}(F_{1,\text{s}\text{p}\text{e}\text{c}}-F_1)$

Process

$M\frac{dT}{dt}=F_1{T}_1+F_2{T}_2-(F_1+F_2)T$

Sensor

$\frac{d{T}_s}{dt}=\frac{1}{{}_{Ts}}(T-T_s)$

Figure $3\mathrm{.}7\mathrm{.}5$ shows the resulting dynamic behavior of the measured temperature of the mixed liquid for this process. The process model is affected by changing $F_{1,\text{s}\text{p}\text{e}\text{c}},$ which changes $F_1,$ resulting in a change in $T,$ which is measured by the sensor as $T_s.$ After a change in the input, the process reaches a new steady$-$state condition; therefore, this process is referred to as a self$-$regulating

Figure $3\mathrm{.}7\mathrm{.}5$ Dynamic response of the CST thermal mixer to a step change in $(F_1{)}_{\text{s}\text{p}\text{e}\text{c}}.$

Figure $3\mathrm{.}7\mathrm{.}6$ Comparison between the dynamic response of the model of the process by itself and a model of the actuator$/$process$/$sensor system.