Chapter 6, Problem 6.12E

Interpretation Introduction

(a)

Interpretation:

An approximate first-order-plus-time-delay transfer function for the given exact transfer function model is to be determined.

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 higher-order transfer function approximation, higher-order models are approximated using the time delays into lower-order models of approximate similar dynamics and steady-state characteristics. The formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$ ...................(1)

Provided the value of $\theta$ is very small.

The general form of a first-order-plus-time-delay transfer function is:

$G\left(s\right)=\frac{K{e}^{-\theta s}}{\tau s+1}$ ...................(2)

Here, $K$ is the gain of the system, $\theta$ is the time delay, and $\tau$ is the time constant.

Interpretation Introduction

(b)

Interpretation:

The response $y\left(t\right)$ for both exact and approximated transfer function are to be simulated plotted for a unit step change in the input on same graph.

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 higher-order transfer function approximation, higher-order models are approximated using the time delays into lower-order models of approximate similar dynamics and steady-state characteristics. The formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$ ...................(1)

Provided the value of $\theta$ is very small.

The general form of a first-order-plus-time-delay transfer function is:

$G\left(s\right)=\frac{K{e}^{-\theta s}}{\tau s+1}$ ...................(2)

Here, $K$ is the gain of the system, $\theta$ is the time delay, and $\tau$ is the time constant.

Interpretation Introduction

(c)

Interpretation:

The maximum error between the two responses for the exact and the approximated model is to be determined along with the time at which this error occurred.

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 higher-order transfer function approximation, higher-order models are approximated using the time delays into lower-order models of approximate similar dynamics and steady-state characteristics. The formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$ ...................(1)

Provided the value of $\theta$ is very small.

The general form of a first-order-plus-time-delay transfer function is:

$G\left(s\right)=\frac{K{e}^{-\theta s}}{\tau s+1}$ ...................(2)

Here, $K$ is the gain of the system, $\theta$ is the time delay, and $\tau$ is the time constant.