Chapter 6, Problem 6.8E

Interpretation Introduction

(a)

Interpretation:

The most accurate transfer function $\frac{{C^{\prime }}_{out}\left(s\right)}{{C^{\prime }}_{in}\left(s\right)}$ for the given process is to be developed.

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 $F\left(s\right)$.

The difference in the actual variable $\left(y\right)$ and the original variable $\left(\overline{y}\right)$ is known as deviation variable $\left(y^{\prime }\right)$. It is generally used while modelling a process. Mathematically it is defined as:

$y^{\prime }=y-\overline{y}$

In steady-state process, the accumulation in the process is taken as zero.

Interpretation Introduction

(b)

Interpretation:

Approximated low order transfer function for the given system is to be determined.

Concept introduction:

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. Formula used for this approximation is:

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

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

Interpretation Introduction

(c)

Interpretation:

The conclusion regarding the need to model the mixing characteristics of the transfer pipe very accurately for this process is to be made.

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

(d)

Interpretation:

For a step change in $c_{in}$, the approximate and the full order model responses are to be simulated.

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.