Process Dynamics and Control CHPE $501$

Assignment Three: Transfer Function and Linearization

Student's Name and ID$.$

Q$1)$ Where necessary linearise the following relationships about the normal condition $)=(1,0.$ Write down the linearised equations using:

i$)$ absolute variables

ii$)$ perturbation variables.

Sketch the linearised and non$-$linear relationships in each case:

a$)y=0\mathrm{.}5x+7$

b$)y=2e^{4x}$

Transform the linearised perturbation equations and draw the signal flow diagram with $x(t)$ as forcing variables.

Q$2)$ Linearize the relationship $z=x{y}^{0\mathrm{.}5}$ about the normal condition $(($:$(\stackrel{}{x})=4,(\frac{?}{\text{b}\text{a}\text{r}}(y))=9\}$ and construct the signal flow diagram for $x(t)$ and $y(t)$ forcing. Determine the error between the true and linearized models for perturbation of: $+-2$ in $x.$

With the aid of a sketch explain the reason for any deviations between the two.

Q$3)$ Transform the following equations making linear approximations where necessary. Draw the signal flow diagram and find the transfer functions $\frac{s}{{}_f}(s)$ :

$($i$),(t)=K\sqrt[\phantom{2}]{{}_f(t)}$

$($ii$),c\frac{d(t)}{dt}={}_f(t)$

$($iii$)T\frac{d(t)}{dt}+(t)={}_f(t)$

$($iv$)T^2\frac{d^2(t)}{d{t}^2}+(t)=\frac{d{}_f}{dt}(t)$

$($v$),(t)={}_2(t){}_3(t)$;$\frac{d{}_3(t)}{dt}={}_f(t)-{}_3(t)$

$\frac{d{}_2(t)}{dt}={}_3(t)-{}_2(t)$

Q$4)$ A dynamic model has been developed to describe the composition behaviour in the top part of a distillation column. The objective is to study the response in distillate composition, $x_d,$ to changes in the manipulated variable $($reflux flowrate, $L_R)$ and the disturbance $($ composition of vapour entering tray $1$ from below, $y_2).$ The model consists of the following equations:

$\frac{d{x}_1}{dt}=\frac{L_R}{M_t}(x_d-x_1)+\frac{V}{M_t}(y_2-\frac{*x_1}{1+(-1)x_1})$

$\frac{d{x}_d}{dt}=\frac{V}{M_d}(\frac{x_1}{1+(-1)x_1}-x_d)$

$M_t$ and $M_d$ are the molar holdups on the tray and in the reflux drum respectively $($both assumed constant in this model$),$ while V is the vapour flowrate up the column. $$ is the relative volatility of the material in the column, and is assumed to be constant. Determine the appropriate transfer functions of this model $($ie $).$