Chapter 6, Problem 6.13E

Interpretation Introduction

Interpretation:

The transfer functions $\frac{{P^{\prime }}_1\left(s\right)}{{P^{\prime }}_d\left(s\right)}$ and $\frac{{P^{\prime }}_2\left(s\right)}{{P^{\prime }}_d\left(s\right)}$ for the given system are to be determined. The results are to be shown in the standard gain/time constant form. Also, for the second-order model, it is to be estimated if the system is overdamped or underdamped.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances.

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The difference in the actual variable $\left(y\right)$ and the original variable $\left(\overline{y}\right)$ is known as deviation variable $\left(y^{\prime }\right)$ .It is generally used while modelling a process. Mathematically it is defined as:

$y^{\prime }=y-\overline{y}$

In steady-state process, the accumulation in the process is taken as zero.

For the second order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ...... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters, the system is overdamped if $\zeta >1$, critically damped if $\zeta =0$ and underdamped if $0<\zeta \le 1$.