Chapter 4, Problem 4.12E

Interpretation Introduction

(a)

Interpretation:

The transfer function assuming no change in the inlet temperature and no change in heater input needs to be derived.

Concept introduction:

The Laplace transform of a time-domain function $\text{f}\left(\text{t}\right)$ is represented by $\text{L}\left[\text{f}\left(\text{t}\right)\right]$ is defined as: $\text{L}\left[\text{f}\left(\text{t}\right)\right]\text{=f}\left(\text{s}\right)\text{=}{\int }_0^{\infty }\text{f}\left(\text{t}\right){\text{e}}^{\text{-st}}\text{dt}$.

An integral transform equation is named as Laplace transforms equation after the inventor Pierre-Simon Laplace. This transform equation has many applications in science and engineering field. This transformation is used to transfer a real variable $\left(t\right)$ into the function of a complex variable $\left(s\right)$.

Interpretation Introduction

(b)

Interpretation:

The transfer function when there is the negligible thermal capacitance of the heating element needs to be derived.

Concept introduction:

The theory of superposition, states that the net response induced for all linear systems by two or more stimuli is the sum of the responses that would have caused each stimulus individually. So, if input $\text{A}$ produces response $\text{X}$ and response $\text{Y}$ is produced by input $\text{B}$, response $\left(\text{A + B}\right)$ generates a response $\left(\text{X + Y}\right)$.