Chapter 6, Problem 6.6E

Interpretation Introduction

(a)

Interpretation:

The order of the overall transfer function is to be determined.

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 an additive process model, the output of the entire process is the sum of all the outputs of all the processes taking place internally of the system. Thus,

$Y\left(s\right)=Y_1\left(s\right)+Y_2\left(s\right)+\cdots +Y_n\left(s\right)$

Here, $n$ is the number of internal processes taking place in the system.

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

Interpretation Introduction

(b)

Interpretation:

The gain of $G\left(s\right)$ is to be determined.

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.

Interpretation Introduction

(c)

Interpretation:

The poles of $G\left(s\right)$ and their location in the complex $s-$ plane is to be determined.

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 any transfer function $G\left(s\right)$, the values of $s$ for which it approaches infinity are known as the poles of the transfer function. Poles are determined by equating the denominator polynomial of the transfer function to zero.

Interpretation Introduction

(d)

Interpretation:

The zeros of $G\left(s\right)$ and their location in the complex $s-$ plane is to be determined. Also, the condition for one or more zeros to be located on the Right Half Plane is to be stated.

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 any transfer function $G\left(s\right)$, the values of $s$ for which it approaches infinity are known as the poles of the transfer function. Poles are determined by equating the denominator polynomial of the transfer function to zero.

Interpretation Introduction

(e)

Interpretation:

The conditions for which the given process will exhibit both negative gain as well as a Right Half Plane zero is to be stated.

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 any transfer function $G\left(s\right)$, the values of $s$ for which it approaches infinity are known as the poles of the transfer function. Poles are determined by equating the denominator polynomial of the transfer function to zero.

Interpretation Introduction

(f)

Interpretation:

The functions of time present in the response of $y\left(t\right)$ for any input change are to be determined.

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 function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

Interpretation Introduction

(g)

Interpretation:

It is to be stated if the output for the given transfer function will be bounded for any bounded input change.

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 function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.