Chapter 6, Problem 6.15E

Interpretation Introduction

Interpretation:

It is to be shown that overdamping is exhibited by the given process.

Concept introduction:

For chemical processes, dynamic models consisting of 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 the deviation variable $\left(y^{\prime }\right)$ .It is generally used while modeling a process. Mathematically it is defined as:

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

In the 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 $0<\zeta \le 1$.