Chapter 5, Problem 5.26E

Interpretation Introduction

(a)

Interpretation:

The difference between the measured value of temperature $\left(T_m\right)$ and the bath temperature $\left(T\right)$ at $t=0.1\text{ min and }t=1.0\text{ min}$ are to be calculated after the temperature change.

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 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 $F\left(s\right)$.

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

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

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

Interpretation Introduction

(b)

Interpretation:

The value of the maximum deviation between $T_m\left(t\right)$ and $T\left(t\right)$ is to be calculated. Also, the time when this maximum deviation is reached is to be determined.

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.

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

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

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

Interpretation Introduction

(c)

Interpretation:

The graph of $T\left(t\right)\text{ and }{T}_m\left(t\right)$ for 3 min is to be plotted. Also, the time lag between $T_m\text{ and }T$ for large values of $t$ is to be determined.

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.

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

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

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