##### Document Text Contents

Page 346

328 Chapter 13: Ideal Flow

Limit as C-b 0

t Figure 13.6 Motivation for Dirac delta function, s(t - b)

Consider an element of fluid (as “tracer”) entering the vessel at t = 0. Visualizing

what happens to the element of fluid is relatively simple, but describing it quantita-

tively as E(t) requires an unusual mathematical expression. The element of fluid moves

through the vessel without mixing with fluid ahead of or behind it, and leaves the vessel

all at once at a time equal to the mean residence time (f = V/q for constant density).

Thus, E(t) = 0 for 0 < t < t; but what is E(t) at t = t?

To answer this question, away from the context of PF, consider a characteristic func-

tion f(t) that, at t = b, is suddenly increased from 0 to l/C, where C is a relatively

small, but nonzero, interval of time, and is then suddenly reduced to 0 at t = b + C,

as illustrated in Figure 13.6. The shaded area of C(W) represents a unit amount of a

pulse disturbance of a constant value (l/C) for a short period of time (C). As C + 0

for unit pulse, the height of the pulse increases, and its width decreases. The limit of this

behavior is indicated by the vertical line with an arrow (meaning “goes to infinity”) and

defines a mathematical expression for an instantaneous (C + 0) unit pulse, called the

Dirac delta function (or unit impulse function):

8(t - b) = liio[j-(t)] = ~0; t = b

= 0;t # b (13.4-6)

such that

8(t - b)dt = 1 (13.4-7)

For E(t), b = t; and hence E(t) for PF is represented mathematically by

E(t) = LS(t - 4 =

Page 347

13.4 Age-Distribution Functions for Ideal Flow 329

Since 0 5 f < ~0, the area under the vertical line representation for PF is unity, as re-

quired by equation 13.3-1.

Another important property of the delta function is, for any function g(t):

I

tz

g(t)S(t - b)dt = g(b), if tl 5 b i- t2

t1

= 0, otherwise (13.4-10)

Wylie (1960, pp. 341-2) presents a proof of equation 13.4-10, based on approximating

8(t - t) by l/C (Figure 13.6), and using the law of the mean for integrals, but it is omitted

here.

In terms of dimensionless time, 8, for PF

E(B) = a(0 - 1) = ~0; 8 = 6 = 1

=o;ez1 (PF) (13.4-11)

For E(B) in PF, what are the mean value 6 and the variance ai?

SOLUTION

We use the property of the delta function contained in equation 13.4-10, and the definitions

of I!? and (T; in the 13 analogues of equations 13.3-14 and 13.3-16a, respectively, to obtain

I

m

fj= e&e - 1)de = 1

0

m

a ; =

I

e26(e - 1)de - fP = l2 - l2 = 0

0 I

The second result confirms our intuitive realization that there is no spread in residence

time for PF in a vessel.

13.4.2.2 F for PF

For PF, the F function requires another type of special mathematical representation.

For this, however, consider a sudden change in a property of the fluid flowing that is

maintained (and not pulsed) (e.g., a sudden change from pure water to a salt solu-

tion). If the change occurs at the inlet at t = 0, it is not observed at the outlet until

t = t. For the exit stream, F(t) = 0 from t = 0 to t = t; since the fraction of the

exit stream of age less than t is 0 for

Page 691

Directory of Examples 8olved Using E-Z !%lve

12-2 I 300 1 exl2-2.msp I performance of a amstant-volume batch reactor I

12-6

12-6

12-7

14-3

166

306

308

3 1 1

3 4 1

346

exl2S.msp adiabatic operaticn of a batch reach

exl2-6.msp optimization of batch-reactor operation

exl2-7.msp semibatch operation

exl4-3.msp translent operation of a CBTR

exl4-6.msc (yas&)e &(-jn in 9 r-!-l-R

14-7

14-8

14-g

14-11

~

153

154

156

157

348

3 6 1

367

369

372

373

376

377

..Y.s.. -m u, , . .

--.- dwity operation of a PFR

158 I 379 I exl6-8.msp IeffectofpreswedroponPFRperfcrmance

exl6-3.msp sizing of an LFR

ex17-2.msp comparison of CBTR and PFR performance

exl7-7.msp reactor combination for autocatalytic readion

exl8-l.msp effect of reach staging on conversion for a reversible readion

1 18-6 I 436 I exl8-6.msp I reactofdesignfor parallel readicn netwrk I

141 4 6 1 exlQ-l.msp RTD analysis for a pulse input

192 464 exl92msp RTD analysis for a step input

19-3 466 exlQ-3.msp RTD analysiswtth a reactive tracer

194 469 exlSl.msp RTD analysis - pulse input tith tailing

197 477 exl97.msp caiculation of gamma timction

198 I 478 1 exl9-3.msp I parameter estimation fortanks-irweries mcdel

IQ-9 1 481 I ex19-2.msp I parameter estimation fortanks-in-seties model

21-6 I 640

22-2 1 668

exZlb.msp I design ofafixed bed reactortith cokhhotccding

I ex22-2.msp I gassdid reaction: spherical particles in PF, particle size distribution

22-3 660 ex22-3.msp gas&d reaction: cylindrical particles in BMF

22-4 663 ex22Amsp gas-solid reaction: sensitivity of design to rate-limiting process

22-5 666 ex22S.m.y gas-solid reaction tith more than one rate limiting process

23-2 577 ex2%2.msp minimum fluidhtion velocity in a fluidized-bed reactor

23-3 I 687 I ex23-3.msp I design of a fluidized-bed reactor l~orderreaction- - - - - I

610

8 1 %

ex23-4.msp series reactions in a fluidized-bed reactor

ex23-6.msp operation of fluidized-bed reactor in intermediate particle regime

ex24-2.msp gas-liquid systems: perfcrmance of a bubbleGdumn reactor

ex24-3 msn aas-tinuid svstnms ripsinn nf R tank reactor

Page 692

udents read through this text, they’ll find a comprehensive, introduc

ent of reactors for single-phase and multiphase systems that expc 2733;

range of reactors and key design features. They’ll gain L ” DE G - CUCEI

sight on reaction kinetics in relation to chemical reactor design. They ~~~~~

CID

also utilize a special software package that helps them quickly solve systems of 1

Solving problems in chemical reaction engineering

and kinetics is now easier than ever!

q; $

V\*& Thorough coverage is provided on the relevant principles of kinetics in order8” \Yi \tt*a “* ; I3 1 $” ui* to develop better designs of chemical reactors.: 8 t.l 1“I *at E-Z Solve software, on CD-ROM, is included with the text. Bv utilizina this

; ;,i gq;“:

:‘,,I ,‘A!,. “), \

,,, \d$t*%f,,~.‘,:‘a,f~?;n$,, software, students’can have m&e time to focus on the de;elopm&t of.!>I ~>Q~*~/q~r~(“ti~,jr ’Y * : ; 1 i,y ,‘ , , * , !bts; ,\‘-i,,b:Y,$‘:,*, design models and on the interpretation of calculated results. The software“::*,,Fp\:h;‘I ;\‘,,lt?,,, ; , I * also facilitates exploration and discussion of realistic, industrial design prob-

*i\, 1 *v lems.

( :su

>I I ;“&

More than 500 worked examples and end-of-chapter problems are included

* ‘I

:, :Y$&:l

to help students learn how to apply the theory to solve design problems.

‘:*(t ~)\\\I”, I

A web site, www.wiley.com/college/missen, provides additional resources

\ \ \.’ :b’

including sample files, demonstrations, and a description of the E-Z Solve

‘- * ‘;*I

software.

““Z \I

About the Authors

1 :;$:; RONALD W. MISSEN is Professor Emeritus (Chemical Engineering) at the

’ University of Toronto. He received his BSc. and M.Sc. in chemical engineering

from Queen’s University, Kingston, Ontario, and his Ph.D. in physical chemistry

from the University of Cambridge, England. He is the co-author of CHEA4ICAL

I I REACTION EQUlLBRlUM ANALYSIS, and has authored or co-authored about

$\:;i;i;, 50 research articles. He is a fellow of the Chemical Institute of Canada and the

: ~~~~~~ Canadian Society for Chemical Engineering, and a member of the American

,‘;l>$$ Institute of Chemical Engineers and Professional Engineers Ontario.

* II0

yiI,tj:: CHARLES A. MIMS is a Professor of Chemical Engineering and Applied

,~i~~y;$ Chemistry at the University of Toronto. He earned his B.Sc. in chemistry at the

” “’ university of Texas, Austin, and his Ph.D. in physical chemistry at the

University of California, Berkeley. He has 15 years of industrial research experi-

ence at Exxon, is the author of over 65 research publications, and holds three

patents. His research interests focus on catalytic kinetics in various energy and

*i i hydrocarbon resource conversion reactions, and the fundamentals of surface.-I\ i

i- ,‘A” reactions.

\/

~ & BRADLEY A. SAVILLE is an Associate Professor of Chemical Engineering at

--I “>ii:;; the University of Toronto. He received his BSc. and Ph.D. in chemical engi-

* ;+$>;

*<Ye,

neering at the University of Alberta. He is the author or co-author of over 25

+;I ;2j research articles on enzyme kinetics, pharmacokinetics, heterogeneous reactions

%%% in biological systems, and reactors for immobilized enzymes. He is a member*r*+<

%&@ of the Chemical Institute of Canada, the Canadian Society of Chemical I5

‘%$%! Engineering, and Professional Engineers Ontario.g&-&&

ggi33

9 ;

328 Chapter 13: Ideal Flow

Limit as C-b 0

t Figure 13.6 Motivation for Dirac delta function, s(t - b)

Consider an element of fluid (as “tracer”) entering the vessel at t = 0. Visualizing

what happens to the element of fluid is relatively simple, but describing it quantita-

tively as E(t) requires an unusual mathematical expression. The element of fluid moves

through the vessel without mixing with fluid ahead of or behind it, and leaves the vessel

all at once at a time equal to the mean residence time (f = V/q for constant density).

Thus, E(t) = 0 for 0 < t < t; but what is E(t) at t = t?

To answer this question, away from the context of PF, consider a characteristic func-

tion f(t) that, at t = b, is suddenly increased from 0 to l/C, where C is a relatively

small, but nonzero, interval of time, and is then suddenly reduced to 0 at t = b + C,

as illustrated in Figure 13.6. The shaded area of C(W) represents a unit amount of a

pulse disturbance of a constant value (l/C) for a short period of time (C). As C + 0

for unit pulse, the height of the pulse increases, and its width decreases. The limit of this

behavior is indicated by the vertical line with an arrow (meaning “goes to infinity”) and

defines a mathematical expression for an instantaneous (C + 0) unit pulse, called the

Dirac delta function (or unit impulse function):

8(t - b) = liio[j-(t)] = ~0; t = b

= 0;t # b (13.4-6)

such that

8(t - b)dt = 1 (13.4-7)

For E(t), b = t; and hence E(t) for PF is represented mathematically by

E(t) = LS(t - 4 =

Page 347

13.4 Age-Distribution Functions for Ideal Flow 329

Since 0 5 f < ~0, the area under the vertical line representation for PF is unity, as re-

quired by equation 13.3-1.

Another important property of the delta function is, for any function g(t):

I

tz

g(t)S(t - b)dt = g(b), if tl 5 b i- t2

t1

= 0, otherwise (13.4-10)

Wylie (1960, pp. 341-2) presents a proof of equation 13.4-10, based on approximating

8(t - t) by l/C (Figure 13.6), and using the law of the mean for integrals, but it is omitted

here.

In terms of dimensionless time, 8, for PF

E(B) = a(0 - 1) = ~0; 8 = 6 = 1

=o;ez1 (PF) (13.4-11)

For E(B) in PF, what are the mean value 6 and the variance ai?

SOLUTION

We use the property of the delta function contained in equation 13.4-10, and the definitions

of I!? and (T; in the 13 analogues of equations 13.3-14 and 13.3-16a, respectively, to obtain

I

m

fj= e&e - 1)de = 1

0

m

a ; =

I

e26(e - 1)de - fP = l2 - l2 = 0

0 I

The second result confirms our intuitive realization that there is no spread in residence

time for PF in a vessel.

13.4.2.2 F for PF

For PF, the F function requires another type of special mathematical representation.

For this, however, consider a sudden change in a property of the fluid flowing that is

maintained (and not pulsed) (e.g., a sudden change from pure water to a salt solu-

tion). If the change occurs at the inlet at t = 0, it is not observed at the outlet until

t = t. For the exit stream, F(t) = 0 from t = 0 to t = t; since the fraction of the

exit stream of age less than t is 0 for

Page 691

Directory of Examples 8olved Using E-Z !%lve

12-2 I 300 1 exl2-2.msp I performance of a amstant-volume batch reactor I

12-6

12-6

12-7

14-3

166

306

308

3 1 1

3 4 1

346

exl2S.msp adiabatic operaticn of a batch reach

exl2-6.msp optimization of batch-reactor operation

exl2-7.msp semibatch operation

exl4-3.msp translent operation of a CBTR

exl4-6.msc (yas&)e &(-jn in 9 r-!-l-R

14-7

14-8

14-g

14-11

~

153

154

156

157

348

3 6 1

367

369

372

373

376

377

..Y.s.. -m u, , . .

--.- dwity operation of a PFR

158 I 379 I exl6-8.msp IeffectofpreswedroponPFRperfcrmance

exl6-3.msp sizing of an LFR

ex17-2.msp comparison of CBTR and PFR performance

exl7-7.msp reactor combination for autocatalytic readion

exl8-l.msp effect of reach staging on conversion for a reversible readion

1 18-6 I 436 I exl8-6.msp I reactofdesignfor parallel readicn netwrk I

141 4 6 1 exlQ-l.msp RTD analysis for a pulse input

192 464 exl92msp RTD analysis for a step input

19-3 466 exlQ-3.msp RTD analysiswtth a reactive tracer

194 469 exlSl.msp RTD analysis - pulse input tith tailing

197 477 exl97.msp caiculation of gamma timction

198 I 478 1 exl9-3.msp I parameter estimation fortanks-irweries mcdel

IQ-9 1 481 I ex19-2.msp I parameter estimation fortanks-in-seties model

21-6 I 640

22-2 1 668

exZlb.msp I design ofafixed bed reactortith cokhhotccding

I ex22-2.msp I gassdid reaction: spherical particles in PF, particle size distribution

22-3 660 ex22-3.msp gas&d reaction: cylindrical particles in BMF

22-4 663 ex22Amsp gas-solid reaction: sensitivity of design to rate-limiting process

22-5 666 ex22S.m.y gas-solid reaction tith more than one rate limiting process

23-2 577 ex2%2.msp minimum fluidhtion velocity in a fluidized-bed reactor

23-3 I 687 I ex23-3.msp I design of a fluidized-bed reactor l~orderreaction- - - - - I

610

8 1 %

ex23-4.msp series reactions in a fluidized-bed reactor

ex23-6.msp operation of fluidized-bed reactor in intermediate particle regime

ex24-2.msp gas-liquid systems: perfcrmance of a bubbleGdumn reactor

ex24-3 msn aas-tinuid svstnms ripsinn nf R tank reactor

Page 692

udents read through this text, they’ll find a comprehensive, introduc

ent of reactors for single-phase and multiphase systems that expc 2733;

range of reactors and key design features. They’ll gain L ” DE G - CUCEI

sight on reaction kinetics in relation to chemical reactor design. They ~~~~~

CID

also utilize a special software package that helps them quickly solve systems of 1

Solving problems in chemical reaction engineering

and kinetics is now easier than ever!

q; $

V\*& Thorough coverage is provided on the relevant principles of kinetics in order8” \Yi \tt*a “* ; I3 1 $” ui* to develop better designs of chemical reactors.: 8 t.l 1“I *at E-Z Solve software, on CD-ROM, is included with the text. Bv utilizina this

; ;,i gq;“:

:‘,,I ,‘A!,. “), \

,,, \d$t*%f,,~.‘,:‘a,f~?;n$,, software, students’can have m&e time to focus on the de;elopm&t of.!>I ~>Q~*~/q~r~(“ti~,jr ’Y * : ; 1 i,y ,‘ , , * , !bts; ,\‘-i,,b:Y,$‘:,*, design models and on the interpretation of calculated results. The software“::*,,Fp\:h;‘I ;\‘,,lt?,,, ; , I * also facilitates exploration and discussion of realistic, industrial design prob-

*i\, 1 *v lems.

( :su

>I I ;“&

More than 500 worked examples and end-of-chapter problems are included

* ‘I

:, :Y$&:l

to help students learn how to apply the theory to solve design problems.

‘:*(t ~)\\\I”, I

A web site, www.wiley.com/college/missen, provides additional resources

\ \ \.’ :b’

including sample files, demonstrations, and a description of the E-Z Solve

‘- * ‘;*I

software.

““Z \I

About the Authors

1 :;$:; RONALD W. MISSEN is Professor Emeritus (Chemical Engineering) at the

’ University of Toronto. He received his BSc. and M.Sc. in chemical engineering

from Queen’s University, Kingston, Ontario, and his Ph.D. in physical chemistry

from the University of Cambridge, England. He is the co-author of CHEA4ICAL

I I REACTION EQUlLBRlUM ANALYSIS, and has authored or co-authored about

$\:;i;i;, 50 research articles. He is a fellow of the Chemical Institute of Canada and the

: ~~~~~~ Canadian Society for Chemical Engineering, and a member of the American

,‘;l>$$ Institute of Chemical Engineers and Professional Engineers Ontario.

* II0

yiI,tj:: CHARLES A. MIMS is a Professor of Chemical Engineering and Applied

,~i~~y;$ Chemistry at the University of Toronto. He earned his B.Sc. in chemistry at the

” “’ university of Texas, Austin, and his Ph.D. in physical chemistry at the

University of California, Berkeley. He has 15 years of industrial research experi-

ence at Exxon, is the author of over 65 research publications, and holds three

patents. His research interests focus on catalytic kinetics in various energy and

*i i hydrocarbon resource conversion reactions, and the fundamentals of surface.-I\ i

i- ,‘A” reactions.

\/

~ & BRADLEY A. SAVILLE is an Associate Professor of Chemical Engineering at

--I “>ii:;; the University of Toronto. He received his BSc. and Ph.D. in chemical engi-

* ;+$>;

*<Ye,

neering at the University of Alberta. He is the author or co-author of over 25

+;I ;2j research articles on enzyme kinetics, pharmacokinetics, heterogeneous reactions

%%% in biological systems, and reactors for immobilized enzymes. He is a member*r*+<

%&@ of the Chemical Institute of Canada, the Canadian Society of Chemical I5

‘%$%! Engineering, and Professional Engineers Ontario.g&-&&

ggi33

9 ;