Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of $t$.

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.

For a first-order system, the transfer function takes the form:

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

Here, $K\text{ and }\tau$ are the system parameters. Step response for the change in input of magnitude M is given by:

$y\left(t\right)=KM\left(1-e^{-t/\tau }\right)$

The transfer function for an integrating process is:

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{s}$

Here, $K$ is the system parameters. Step response for the change in input of magnitude M is given by:

$y\left(t\right)=KMt$

Interpretation Introduction

(b)

Interpretation:

For $t\ll \tau$, the relationship between $K_1\text{ and }{K}_0$ is to be derived if the two responses match.

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.

For a first-order system, the transfer function takes the form:

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

Here, $K\text{ and }\tau$ are the system parameters. Step response for the change in input of magnitude M is given by:

$y\left(t\right)=KM\left(1-e^{-t/\tau }\right)$

The transfer function for an integrating process is:

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{s}$

Here, $K$ is the system parameters. Step response for the change in input of magnitude M is given by:

$y\left(t\right)=KMt$

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find $K_0$ and time delay in an integrating process is to be explained.

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 transfer function for an integrating process with the time delay of $\theta$ is:

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{s}{e}^{-\theta s}$

Here, $K$ is the system parameters. Step response for the change in input of magnitude M is given by:

$y\left(t\right)=KM\left(t-\theta \right)$