PROCESS DYNAMIC+CONTROL-EBOOK>I<
PROCESS DYNAMIC+CONTROL-EBOOK>I<
4th Edition
ISBN: 2819480255712
Author: Seborg
Publisher: INTER WILE
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Chapter 5, Problem 5.1E
Interpretation Introduction

Interpretation:

The time needed for output to reach the steady state depend on the magnitude of input needs to be determined.

Concept introduction:

Fourth order plus time delay system is used for describing the dynamics of chemical process.

Just by selecting proper choice of τDT and τ it represents the dynamics of several industrial processes.

Expert Solution & Answer
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Answer to Problem 5.1E

Magnitude of change increase with A increased with time and vice versa that is Y(s) is depend on the A.

Explanation of Solution

Given:

For given transformation,

G(S)=K(τ1s+1)(τ2s+1)(τ3s+1)(τ4s+1)eθs

Calculation:

  • To find the stead state value of Y(s):
  • Given transformation-

    G(S)=K(τ1s+1)(τ2s+1)(τ3s+1)(τ4s+1)e-θsG(S)=Y(s)U(s)

    Step input is applied in system input

    U(S)=1S

    As magnitude is 1

    So, Y(S)=1S×K(τ1s+1)(τ2s+1)(τ3s+1)(τ4s+1)e-θs

    At the steady state of output,

    limx¥

    f(t) to lims¥

    f(s),

  • For the steady state value of
  • Y(t)=lims0×S×1SK(τ1s+1)(τ2s+1)(τ3s+1)(τ4s+1)e-θsY(t)t0=K(1)(1)(1)(1)e(0)Y(t)t0=K

  • If step input magnitude A,
  • U(S)=AS

    When magnitude is A,

    Y(s)=ASK(τ1s+1)(τ2s+1)(τ3s+1)(τ4s+1)e-θsY(s)t0=lims¥×S×Y(s)Y(s)t0=A×K(1)(1)(1)(1)e(0)Y(s)t0=A×K 

Conclusion

Thus, steady state value of Y(s) is mainly depends on the magnitude of step input in A.

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