L19-21 Chapter 6

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CCB 3013 - Chemical Process Dynamics, Instrumentation and Control 1

Chapter 6

Dynamic Behavior of Higher-order

and Time delay Processes

6/30/2014

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Chapter Objectives

At the end of this chapter, you will be able to

Explain Interacting and non-interacting systems

Derive transfer functions for higher order systems

Explain delay times in processes

Develop transfer functions for delay times

Develop approximate expressions for delay times

6/30/2014 CCB 3013 - Chemical Process Dynamics, Instrumentation and Control 2

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Interacting and Noninteracting

Processes

Processes with variables that interact with each other, or

that contain internal feedback of material or energy

will exhibit interacting behavior

CCB 3013 - Chemical Process Dynamics, Instrumentation and Control 36/30/2014

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Non-interacting Processes

The two-storage tanks were connected in series in such a way that liquid level in the

second tank did not influence the level in the

first tank.


The following transfer functions were derived:

1)(

)(

1

11

s

K

sQ

sH

i (6.1)

11

1 1

)(

)(

KsH

sQ

(6.2)

1)(

)(

2

2

1

2

s

K

sQ

sH

(6.3)

22

2 1

)(

)(

KsH

sQ

(6.4)

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Non-interacting Processes

Tank 2 level h2 is related to qi by a second-ordertransfer function that can be obtained by simplemultiplication


11)()(

)(

)(

)(

)(

)(

)(

21

21

1

1

1

22

ss

K

sQ

sH

sH

sQ

sQ

sH

sQ

sH

ii (6.5)

A simple generalization of the dynamic

expression in (6.5) is applicable to n tanks in

series

n

i

i

n

i

n

s

K

sQ

sH

1

1)(

)(

n

i

ii

n

ssQ

sQ

1

1

1

)(

)(

(6.6)

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An interacting process


In the process shown above, h1 depends on h2as a result of interconnecting stream with flow

rate q1.

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The equation for flow from Tank 1 to Tank 2 must

be written to reflect that physical feature.


)(1

21

1

1 hhR

q (6.7)

For Tank 1, the level transfer function can be

derived as

1

1

)(

)(

121122

2

2121

21

22121

1

sARARARsAARR

sRR

ARRRR

sQ

sH

i

(6.8)

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Consequently, the overall transfer function

between h2 and qi is


12)(

)(22

22

ss

R

sQ

sH

i (6.10)

The transfer function relating h1 and h2 is

1)(

)(

21

221

21

2

1

2

sRR

ARR

RR

R

sH

sH (6.9)

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Processes with Time Delays

Time delays occur due to:

1.Fluid flow in a pipe

2.Transport of solid material (e.g., conveyor belt)

3.Chemical analysis

- Sampling line delay

- Time required to do the analysis (e.g., on-line

gas chromatograph)


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If a fluid is transported through a pipe in plug flow,


flowrate volumetric

pipe of volume

velocityfluid

pipe oflength (6.11)

Point 1 Point 2

Transportation time between points 1 and 2 is given by

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( )( )

( )

s Y sG s eU s


0 for (6-27)

for

ty t

u t t

(6.12)

Suppose that x is some fluid property at point 1, such as concentration, and y is the same property at point 2 and that both x and y are deviation variables. Then

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The output y(t) is simply the same input function shifted backward in time by the

amount of the delay translation in time.

The transfer function of a time delay of magnitude is given by


(6.13) sesG

sX

sY )()(

)(

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Polynomial approximations to (non-rational function)

Taylor series expansion:


...!5

!4

!3

!2

1

55443322

ssssse s

(6.14)

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Pad approximation (A ratio of two polynomials)

1/1 Pad approximation

Performing the long division in (6.15), we have

Comparison of (6.14) and (6.16) indicates they are correct through the first three terms.


21

21

1 s

s

(s)Ge s

(6.15)

...4

2

1

3322

ssse s (6.16)

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12

s

2

s1

12

s

2

s1

)(22

22

2

sGe s (6.17)

Comparison of actual

and approximate time

delay responses


2/2 Pad approximation


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An alternative first-order approximation consists of the transfer function,

where the time constant has a value of 0

These expressions can be used to approximate the pole or zero term in a transfer function.

0

0

0

1 1(6-58)

1

s

se

se



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Conclusions

Interacting and non-interacting systems were introduced

Transfer functions for higher order systems have been derived

Delay time concept has been introduced.

Transfer function and approximations are explained.


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L19-21 Chapter 6

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