Chapter 6, Problem 6.23E

Interpretation Introduction

(a)

Interpretation:

It is to be estimated and explained if the given process will exhibit an overshoot for a step change in $u$.

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.

For $n$ processes in series, the output of the entire process is the product of all the outputs of all the processes taking place internally of the system. Thus,

$Y\left(s\right)=Y_1\left(s\right)Y_2\left(s\right)\dots Y_n\left(s\right)$   ...... (1)

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

From the definition of the overshoot, it can be written as:

$\text{OS}=\frac{y_{\max }\left(t\right)-KM}{KM}$   ...... (2)

Overshoot (OS) can also be defined as:

$\text{OS}=\exp \left(\frac{-\pi \zeta }{\sqrt{1-{\zeta }^2}}\right)$   ...... (3)

For higher order transfer function approximation, higher order models are approximated using the time delays into lower order models of approximate similar dynamics and steady-state characteristics. Formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$

Provided the value of $\theta$ is very small.

Interpretation Introduction

(b)

Interpretation:

The approximated maximum value of $y\left(t\right)$ for the given process at given conditions for a step change in $u$ of magnitude 3 is to be determined.

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.

For $n$ processes in series, the output of the entire process is the product of all the outputs of all the processes taking place internally of the system. Thus,

$Y\left(s\right)=Y_1\left(s\right)Y_2\left(s\right)\dots Y_n\left(s\right)$   ...... (1)

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

From the definition of the overshoot, it can be written as:

$\text{OS}=\frac{y_{\max }\left(t\right)-KM}{KM}$   ...... (2)

Overshoot (OS) can also be defined as:

$\text{OS}=\exp \left(\frac{-\pi \zeta }{\sqrt{1-{\zeta }^2}}\right)$   ...... (3)

For higher order transfer function approximation, higher order models are approximated using the time delays into lower order models of approximate similar dynamics and steady-state characteristics. Formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$

Provided the value of $\theta$ is very small.

Interpretation Introduction

(c)

Interpretation:

The time at which the maximum value occurs is to be calculated.

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.

For $n$ processes in series, the output of the entire process is the product of all the outputs of all the processes taking place internally of the system. Thus,

$Y\left(s\right)=Y_1\left(s\right)Y_2\left(s\right)\dots Y_n\left(s\right)$   ...... (1)

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

From the definition of the overshoot, it can be written as:

$\text{OS}=\frac{y_{\max }\left(t\right)-KM}{KM}$   ...... (2)

Overshoot (OS) can also be defined as:

$\text{OS}=\exp \left(\frac{-\pi \zeta }{\sqrt{1-{\zeta }^2}}\right)$   ...... (3)

For higher order transfer function approximation, higher order models are approximated using the time delays into lower order models of approximate similar dynamics and steady-state characteristics. Formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$

Provided the value of $\theta$ is very small.

Interpretation Introduction

(d)

Interpretation:

The response of the both the actual and the approximated model for the given process are to be simulated and plotted. Also, both the plots are to be generated for ${\tau }_1=1\text{ and }{\tau }_1=5$ and the results are to be compared graphically.

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.

For $n$ processes in series, the output of the entire process is the product of all the outputs of all the processes taking place internally of the system. Thus,

$Y\left(s\right)=Y_1\left(s\right)Y_2\left(s\right)\dots Y_n\left(s\right)$   ...... (1)

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

From the definition of the overshoot, it can be written as:

$\text{OS}=\frac{y_{\max }\left(t\right)-KM}{KM}$   ...... (2)

Overshoot (OS) can also be defined as:

$\text{OS}=\exp \left(\frac{-\pi \zeta }{\sqrt{1-{\zeta }^2}}\right)$   ...... (3)

For higher order transfer function approximation, higher order models are approximated using the time delays into lower order models of approximate similar dynamics and steady-state characteristics. Formula used for this approximation is:

$e^{-\theta s}=\frac{1}{e^{\theta s}}=\frac{1}{1+\theta s}$

Provided the value of $\theta$ is very small.