Chapter 4, Problem 4.14E

Interpretation Introduction

(a)

Interpretation:

The transfer function which relates the exiting temperature $T$ to the inlet concentration $c_{Ai}$ is to be derived. Also, all the assumptions made during the derivation are 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.

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:

The sensitivity of the transfer function gain $K$ to the given operating conditions is to be determined. Also, expression for $K$ in terms of $\overline{q},\overline{T},\text{ and }{\overline{c}}_{Ai}$ are to be determined and their sensitivities are to be evaluated.

Concept introduction:

In any steady state, the ratio of the amplitude of input signal to the amplitude of the amplifier output is known as steady-state gain. This gain will be same for the entire input range if the amplifier output is linear. For a transfer function, steady-state gain occurs as times tends to infinity which means that in s-domain, this infinite time is represented by $s\to 0$.