Chapter 6, Problem 6.7E

Interpretation Introduction

(a)

Interpretation:

An expression for $Q^{\prime }\left(t\right)$ is to be determined for 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.

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 that 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 values of $K,\tau ,\text{ and }\zeta$ are 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 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 that 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 $F\left(s\right)$.

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

Interpretation Introduction

(c)

Interpretation:

The overall transfer function $\frac{{P^{\prime }}_m\left(s\right)}{P^{\prime }\left(s\right)}$ 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.

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 that 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.

Interpretation Introduction

(d)

Interpretation:

An expression for the overall process gain 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.

Over process gain of a transfer function $G\left(s\right)$ is calculated as:

$K_{overall}={G\left(s\right)|}_{s=0}$