Example Chap1

8/16/2019 Example Chap1

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CHAPTER I EXANPLE

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1.1 Illustrative Example: Blending system

Notation:

• w1, w2 and w are mass flow rates

• x1, x2 and x are mass fractions of component A


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Assumptions:

1. w1 is constant

2. x2 = constant = 1 (stream 2 is pure A)

3. Perfect mixing in the tank

Control Objective:

eep x at a desired !alue (or "set point#) xsp, despite !ariations in x1(t ). $low rate w2 can %e ad&usted for this purpose.

Terminology:

• 'ontrolled !aria%le (or "output !aria%le#) x

• anipulated !aria%le (or "input !aria%le#) w2

• *istur%ance !aria%le (or "load !aria%le#) x


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Design Question. +hat !alue of is reuired to ha!e2w

-SP x x=

Overall balance:

Component A balance:

1 2 (1/1)w w w= + −

1 1 2 2 (1/2)w x w x wx+ − =

(0he o!er%ars denote nominal stead/state design !alues.)

• At the design conditions, . u%stitute . 1/2,

and , then sol!e . 1/2 for

SP x x= SP x x=

2 1 x = 2w

12 1 (1/3)

1

SP

SP

x xw w

x

−=

−


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• uation 1/3 is the design euation for the %lending

sstem.

• 4f our assumptions are correct, then this !alue of will keep

at . 5ut what if conditions change-

x

SP x

Control Question. Suppose that the inlet concentration x1

changes with time. How can we ensure that x remains at or near

the set point ?

As a specific example, if and , then x 6 xSP.

SP x

1 1 x x> 2 2w w=

Some Possible Control Strategies:

Method 1 Measure x and adjust w2.

• 4ntuiti!el, if x is too high, we should reduce w27

2w


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• Proportional feed%ack control law,

( ) ( )2 2

(1/8)c SP

w t w K x x t = + −

1. where K c is called the controller gain.

2. w2(t ) and x(t ) denote !aria%les that change with time t .

3. 0he change in the flow rate, is proportional tothe de!iation from the set point, xSP 9 x(t ).

( )2 2 ,w t w−


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Method ! Measure x1 and adjust w2.

• 0hus, if x1 is greater than , we would decrease w2 so that

• One approach 'onsider . (1/3) and replace and with

x1(t ) and w2(t ) to get a control law

1 x

2 27w w<

1 x 2w

( )

( )12 1 (1/:)1

SP

SP

x x t

w t w x

−

= −


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• 5ecause . (1/3) applies onl at stead state, it is not clear

how effecti!e the control law in (1/:) will %e for transient

conditions.

Method " Measure x1 and x, adjust w2.

• 0his approach is a com%ination of ethods 1 and 2.

Method # Use a larger tan.

• 4f a larger tank is used, fluctuations in x1 will tend to %e damped

out due to the larger capacitance of the tank contents.

•;owe!er, a larger tank means an increased capital cost.


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1! Classi$ication o$ Control Strategies

Method Measured!aria"le

Manipulated!aria"le

#ategor$

1 x w2 $5a

2 x1 w2 $$

3 x1 and x w2 $$


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• 4t is important to make a distinction %etween negati%e &eed"ac

and positi%e &eed"ac .

ngineering sage !s. ocial ciences

• Advantages:

'orrecti!e action is taken regardless of the source of the

distur%ance.

>educes sensiti!it of the controlled !aria%le todistur%ances and changes in the process (shown later).

• 'isadvantages:

?o correcti!e action occurs until after the distur%ance hasupset the process, that is, until after x differs from x sp.

@er oscillator responses, or e!en insta%ilit


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%eed$or(ard Control:

*istinguishing feature measure a distur%ance

!aria%le

• Advantage:

'orrect for distur%ance %efore it upsets the process.

• 'isadvantage:

ust %e a%le to measure the distur%ance.

?o correcti!e action for unmeasured distur%ances.

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Example Chap1