Chapter 5, Problem 5.23E

Interpretation Introduction

(a)

Interpretation:

For a change of unit rectangular pulse in the input of a heating process, the steady-state response is to be determined.

Concept introduction:

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 $\mathrm{Re}\left(s\right)\ge 0$.

Interpretation Introduction

(b)

Interpretation:

For a change of unit ramp in the input of a heating process, the steady-state response is to be determined.

Concept introduction:

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 $\mathrm{Re}\left(s\right)\ge 0$.

Interpretation Introduction

(c)

Interpretation:

The answers in part (a) and part (b) are to be explained physically along with any physical limitation that might be present on the ramping of the heating rate.

Concept introduction:

An input change when the process is subjected to a sudden step-change in the input which gradually then returns to its original value is known as a rectangular pulse change.

This change can be approximated as:

$u_{RP}\left(t\right)=\{\begin{array}{l}0t<0\\ h0\le t<t_w\\ 0t\ge t_w\end{array}$

Here, $h$ is the magnitude of the change and $t_w$ is the pulse width which can vary from very short to very long.

An input change when the process is subjected to a gradual upward or downward change in the input for some time period with a constant slope is known as ramp change.

This change can be approximated as:

$u_R\left(t\right)=\{\begin{array}{l}0t<0\\ att\ge 0\end{array}$

Here, $a$ is the magnitude or slope of the change.