Chapter 5, Problem 5.12E

Interpretation Introduction

(a)

Interpretation:

The transfer function for the given equation is to be determined and reported in standard gain/time constant form.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   ..... (1)

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 $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Laplace transform of higher order derivatives is given by:

$\mathcal{L}\left(\frac{d^{n}f}{d{t}^n}\right)=s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\left(1\right)}\left(0\right)-\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$   ..... (2)

Interpretation Introduction

(b)

Interpretation:

The qualitative form of the response for this system over the range of $-10\le K\le 10$ is to be discussed. The values of $K$ where the response will converge are to be specified along with the response forms without evaluating the coefficients.

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.