files-2-Lectures_Lec18.ppt

8/17/2019 files-2-Lectures_Lec18.ppt

1/19


2/19

C h a p t e r 6

Example: Turbulent flow in a pipe

+et, fluid property (e.g., temperature or composition) at

point 1

fluid property at point 2

u

y

%ssume t!at t!e velocity profile is flat-, t!at is, t!e velocity

is uniform over t!e cross"sectional area. T!is situation is

analyed in /&ample .0 and Fig. ..

Fluid n Fluid ut

Figure .0

oint 1 oint 2


3/19

C h a p t e r 6


4/19

C h a p t e r 6

Example 6.5

(a) relating t!e mass flow rate of li$uid at 2, w2, to t!e mass flow

rate of li$uid at 1, wt,

(b) relating t!e concentration of a c!emical species at 2 to t!e

concentration at 1. %ssume t!at t!e li$uid is incompressible.

Solution

(a) First we ma5e an overall material balance on t!e pipe

segment in $uestion. #ince t!ere can be no accumulation(incompressible fluid),

material in 6 material out ( ) ( )1 2w t w t ⇒ =

For t!e pipe section illustrated in Fig. .0, find t!e transfer

functions:


5/19

C h a p t e r 6 (b) bserving a very small cell of material passing point 1 at time

t , we note t!at in contains Vc1(t ) units of t!e c!emical species of

interest w!ere V is t!e total volume of material in t!e cell. f, attime t 7 , t!e cell passes point 2, it contains units of

t!e species. f t!e material moves in plug flow, not mi&ing at all

wit! ad8acent material, t!en t!e amount of species in t!e cell is

constant:

utting ("3) in deviation form and ta5ing +aplace transforms

yields t!e transfer function,

( )

( )2

1 1

W s

W s

′

=′

' ( )2 'Vc t +

( ) ( )2 1' ("3)Vc t Vc t + =

or

( ) ( )2 1' ("31)c t c t + =


6/19

C h a p t e r 6

%n e$uivalent way of writing ("31) is

( ) ( )2 1 ' ("32)c t c t = −

if t!e flow rate is constant. utting ("32) in deviation form and

ta5ing +aplace transforms yields

( )

( )2 '

1("33)

sC s

eC s

−′

=′

Time Delays (continued)

Transfer Function 9epresentation:

( )

( )' ("2) s

Y se

U s

−=

;ote t!at !as units of time (e.g., minutes, !ours)'


7/19

C h a p t e r 6

Polynomial Approximations to

For purposes of analysis using analytical solutions to transfer

functions, polynomial appro&imations for are commonly

used. /&ample: simulation software suc! as


8/19

C h a p t e r 6

2. Padé pproxi!ations:


9/19

C h a p t e r 6


10/19

Approximation of Higher-rder Transfer

!unctions

'

1' ("0*) s

e s− ≈ −

n t!is section, we present a general approac! forappro&imating !ig!"order transfer function models wit!

lower"order models t!at !ave similar dynamic and steady"state

c!aracteristics.

n /$. "4 we s!owed t!at t!e transfer function for a time

delay can be e&pressed as a Taylor series e&pansion. For small

values of s,

C h a p t e r 6


11/19

B %n alternative first"order appro&imation consists of t!e transfer

function,

'

'

1 1("0)

1'

s

se

se

− = ≈+

w!ere t!e time constant !as a value of

B /$uations "0* and "0 were derived to appro&imate time"

delay terms.

B Cowever, t!ese e&pressions can also be used to appro&imate

t!e pole or ero term on t!e rig!t"!and side of t!e e$uation by

t!e time"delay term on t!e left side.

' .

C h a p t e r 6


12/19

S"ogestad#s $half rule%

B #5ogestad (22) !as proposed a related appro&imation met!od

for !ig!er"order models t!at contain multiple time constants.

B Ce appro&imates t!e largest neglected time constant in t!e

following manner.

B ne !alf of its value is added to t!e e&isting time delay (if any)and t!e ot!er !alf is added to t!e smallest retained time

constant.

B Time constants t!at are smaller t!an t!e largest neglected time

constant- are appro&imated as time delays using ("0).

C h a p t e r 6


13/19

Example 6.4

onsider a transfer function:

( ) ( )( ) ( ) ( )

.1 1("0A)

0 1 3 1 .0 1

$ s% s

s s s− +=

+ + +

Derive an appro&imate first"order"plus"time"delay model,

( )'

(")E 1

s $e% s

s

−

=+

%

using two met!ods:

(a) T!e Taylor series e&pansions of /$s. "0* and "0.

(b) #5ogestads !alf rule

C h a p t e r 6

ompare t!e normalied responses of %( s) and t!e appro&imate

models for a unit step input.


14/19

Solution

(a) T!e dominant time constant (0) is retained. %pplying

t!e appro&imations in ("0*) and ("0) gives:

.1.1 1 ("1) s s e−− + ≈

and

3 .01 1 ("2)3 1 .0 1

s se e s s

− −≈ ≈+ +

#ubstitution into ("0A) gives t!e Taylor series

appro&imation, ( ) :TS

% s%

( ).1 3 .0 3.

("3)0 1 0 1

s s s s

TS

$e e e $e% s

s s

− − − −

= =+ +

%

C h a p t e r 6


15/19

(b) To use #5ogestads met!od, we note t!at t!e largest neglected

time constant in ("0A) !as a value of t!ree.

' 1.0 .1 .0 2.1= + + =

C h a p t e r 6

B %ccording to !is !alf rule-, !alf of t!is value is added to t!e

ne&t largest time constant to generate a new time constant

B T!e ot!er !alf provides a new time delay of .0(3) 6 1.0.B T!e appro&imation of t!e 9C ero in ("1) provides an

additional time delay of .1.B %ppro&imating t!e smallest time constant of .0 in ("0A) by

("0) produces an additional time delay of .0.B T!us t!e total time delay in (") is,

E 0 .0(3) .0.= + =


16/19

and %&s' can be appro&imated as:

( )2.1

("4).0 1

s

S(

$e% s

s

−

=+

%

T!e normalied step responses for %&s' and t!e two appro&imate

models are s!own in Fig. .1. #5ogestads met!od provides

better agreement wit! t!e actual response.

C h a p t e r 6

Figure .1

omparison of t!e

actual and

appro&imate modelsfor /&ample .4.


17/19

Example 6.5

onsider t!e following transfer function:

( ) ( )( ) ( ) ( ) ( )

1 ("0)12 1 3 1 .2 1 .0 1

s

$ s e% s s s s s

−−= + + + +

Gse #5ogestads met!od to derive two appro&imate models:

(a) % first"order"plus"time"delay model in t!e form of (")

(b) % second"order"plus"time"delay model in t!e form:

( ) ( ) ( )

'

1 2 (")E 1 E 1

s $e

% s s s

−

=+ +

%

ompare t!e normalied output responses for %&s' and t!e

appro&imate models to a unit step input.

C h a p t e r 6


18/19

Solution

(a) For t!e first"order"plus"time"delay model, t!e dominant time

constant (12) is retained.

3.' 1 .2 .0 1 3.*02

3.E 12 13.0

2

= + + + + =

= + =

C h a p t e r 6

B ne"!alf of t!e largest neglected time constant (3) is allocated

to t!e retained time constant and one"!alf to t!e appro&imate

time delay.

B %lso, t!e small time constants (.2 and .0) and t!e ero (1)are added to t!e original time delay.

B T!us t!e model parameters in (") are:


19/19

(b) %n analogous derivation for t!e second"order"plus"time"delay

model gives:

1 2

.2' 1 .0 1 2.10

2

E 12, E 3 .1 3.1

= + + + =

= = + =

n t!is case, t!e !alf rule is applied to t!e t!ird largest time

constant (.2). T!e normalied step responses of t!e original and

appro&imate transfer functions are s!own in Fig. .11. C h a p t e r 6

Documents

files-2-Lectures_Lec18.ppt