Chapter 5, Problem 5.24E

Interpretation Introduction

(a)

Interpretation:

The response $Y\left(s\right)$ and the quantitative value of $y\left(t\right)$ are to be derived.

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,

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

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 $ℒ$ 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 response in part (a) is to be simulated and its major characteristics are to be identified.

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 large value of time, the asymptotic value of $y\left(t\right)$ can be calculated using Final Value theorem (FVM) as shown below:

$\underset{t\to \infty }{\lim }y\left(t\right)=\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$   ..........(1)

This theorem is applicable only if $\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$ exists for all values of $\mathrm{Re}\left(s\right)\ge 0$.