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ANALYSIS OF PERIODIC REACTOR OPERATION
A CASE STUDY

presented by

Eden Yee Tang T. Dionne

has been accepted towards fulfillment
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M . S - degree in QhemicaLEng .

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ANALYSIS OF PERIODIC REACTOR OPERATION
A CASE STUDY

By

Eden Yee Tang T. Dionne

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Chemical Engineering

1983

ABSTRACT

ANALYSIS OF PERIODIC REACTOR OPERATION
A CASE STUDY
By

Eden Yee Tang T. Dionne

The Van de Vusse reaction scheme, represented as
AtA-—E2a-D

is considered as a case study for investigating the effects
of periodically controlling the volumetric rate of through-
put to an isothermal CSTR and PFR. It is shown that a se-
lectivity shift to an enhanced production of the intermedia-
te product B is encountered when large fluctuations of the
cycling frequency is implemented in the CSTR. An adverse
effect to the yield of B is obtained for the periodic con-
trol of the volumetric flow rate in a PFR. The effects‘of
large cycling frequencies can not be surmised for a PFR
oPerating under a periodic control of the volumetric flow
rate because the physics describing the flow and mixing

Patterns are no longer similar to the plug flow behavior.

LIST OF FIGURESS . . . . . . .

TABLE OF CONTENTS

. . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vii
NOTATIONS--CHAPTER 1 . . . . . . . . . . .viii
NOTATIONS--CHAPTER 2 . . . . . . . . . . x
Chapter
1. SINE-WAVE CONTROL OF INPUT VOLUMETRIC FLOW
RATE IN AN ISOTHERMAL CSTR WITH A VAN DE
VUSSE KINETICS I I I I I I I I I I I I I I I I I I 1
IntrOduction I I I I I I I I I I I I I I I I I I 2
Basic Equations . . . . . . . . . . . . . . . . 6
Optimal Steady state . . . . . . . . . . . . . . 10
Perturbation SolutionzPeriodic Process . . . . 11
Di ScuSSi on I I I I I I I I I I I I I I I I I I I 15
Conclusion . . . . . . . . . . . . . . . . . #0
2. SINE-WAVE CONTROL OF INPUT VOLUMETRIC FLOW
RATE IN AN ISOTHERMAL PFR WITH A VAN DE I
VUSSE LINETICS I I I I I I I I I I I I I I I I I I [+4
Introduction . . . . . . . . . . : . . . . . . . 45
Basic Equations and The Perturbation Model . . . A9
Perturbation Solution and Optimal Periodic
Process I I I I I I I I I I I I I I I I I I I 52
Discussion . . . . . . . . . . . . . . . . . . . g6
Conclusion . . . . . . . . . . . . . . . . . . . 2
APPENDICES
Appendix
. DERIVATION OF THE CLOSED FORM SOLUTION FOR
REACTANT A (A PERTURBATION SOLUTION) . . . . . . . 97
B. THE INITIAL CONDITIONS FOR THE PERTURBATION
CAI'CULATION I I I I I I I I I I I I I I I I I I I 113
D- AVERAGE YIELD CALCULATION OF THE INTERMEDIATE

PRODUCT I I I I I I I I I I I I I I I I I I I I I 120

ii

Appendix
E. THE SECOND VARIATION IN THEFREQUENCY

DOMAIN I I I I I I I I I I I I I I I I I I I I I 121
F. CALCULATION OF THE VELOCITY PROFILE
IN THE PLUG FLOW TUBULAR REACTOR . . . . . . . . 126
G. GLOBAL SOLUTION TO x1 AND x2 IN A PLUG
FLOW TUBULAR REACTOR (PERTURBATION SOLUTION) . . 132
H. ASYMPTOTIC SOLUTION TO x1 AND x2 IN THE PLUG
FLOW TUBULAR REACTOR AT em . . . . . . . . . . . 149
I. EVALUATION OF BOUNDARY CONDITIONS FOR
1 2
P1 AND P1 I I I I I I I I I I I I I I I I I I I 155
J. EVALUATION OF BOUNDARY CONDITIONS FOR
1 2
P2 AND P2 I I I I I I I I I I I I I I I I I I I 158
K. PROOF OF THE DERIVATIVE OF THE BOUNDARY
CONDITIONS FOR P: AND Pg . . . . . . . . . 161
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 162

iii

LIST OF FIGURES

Chapter 1
1. Phase plane plot of a periodic
controlled isothermal CSTR with
a Van de VDsse kinetics. i2 and
1-2 are time-average yield and
contersion, respectively. . . . . . . . . . . . 4
2. Phase plane plot of a sine-wave

control of the.reactant feed concen-
tration in an isothermal CSTR wit
a Van de Vusse kinetics . . . . . . . . . . . . 7

3.(a-b) Perturbation solution versus nu-
merical solution with k1=k2=k3Aref

=t=1I ' A=2I €=OI5 I I I I I I I I I I I I I I I 16
4.(a-b)Perturbation solution versus nu;

merical solution with K1=1, K2=K2=5,

A:OI5’ €=OI25~ I I I I I I I I I I I I I I I I 19

5- Deviation of perturbation solution from
numerical simulation for

(a) K1=1, K2=R2=5, A;O.5, e=o.25
(b) K1=u.928, K2=24.642, fi2=o.493,.A=10., 6:0.1
(c) K1=K2=§2=1, Azio, 6:0.8
(d) K1=K2=K2=1, A=10, 6:0.5
(e) K1=20, K2=2, K2=30, A=1, e=O.5
(f) K1=20, K2=2, fi2=3o, Asl, 6:0.8
.
(g) K1=20, K2=2, K2=30, A:1, 6=O.1
(h) K1=1, K2=R2=5,.A=O.5, 6:0.025
(i) K1=1, K2=k2=5, A;0.05, 6:0.0495 . . . . . . . . 22

iv

6a.

6b.

Chapter 2

1.

(a)
(b)

(a)
(b)

AA
Frw-
l I
l—Ju.
vv

(m-n)

A
O

"U

V

x2-x1 phase plane for K1=K2=K2=1, A=1.

with varyingé. . . . . . . .

xz-x1 phase plane for K1=K2=K2=1. 5:0-5.

withvaryingA.

Comparison between perturbation solution

and Pi-criterion method with k =k =k A =1
94.=0.005 1 2 3 ref

e/A =0I01 I I I I I I I I I I I I I I I I I I
Velocity profile at £=1.0

E=A=0.05

e:OI01' A=OI]. I I I I I.I I I I I I I I I I I

Deviation of perturbation solution

from numerical simulation for the

velocity profile at 5:1.0

6=A=0.05

6:0.01, A?O.1 . . . .... . . . . . . . . . .

Behavior of Van de Vusse system for
=K2=K3=O.515, 6:0.01, A;0.05 at

=O.2

H

avior of Van de VUsse system for
=K3=O.515. 6:0.01, A;O.2 at

wwwm pqw wonmm p:
(D
Kn

II II II IIH
HOOO

OVU‘NN

Behavior of Van de Vusse system for

_ = = = . , = I t
K1- 2 K3 0.515. e 0 05 A.0 2 a
5:0.2
5=0-5
$=0-75
3:100 o o o o o o o o o o o o o o I 0 O 0 0 0

Deviation of perturbation solution from
numerical simulation at §=O.5, 6:0.01, A30.05
for the velocity profile .

for Van de VUsse system With K1=K2=K3=0.515 .

33

35

38

58

60

63

. 68

73

78

3. Deviation of perturbation solution from
numerical simulation at '5 =1.0, 6:0.01, A=0.05
(c) for the velocity profile.
(d) for Van de Vusse system with K1=K2=K3=O.515 . . 8O
3. Deviation of perturbation solution from
numerical simulation at 5:0.5, 6:0.01, AFO.2
(e) for the velocity profile.
(f) for Van de Vusse system with K1=K2=K3=O.515 . . 82
3. Deviation of perturbation solution from
numerical simulation at 3:1.0, €=O.01,.AFO.2
(g) for the velocity profile.
(h) for Van de Vusse system with K1=K2=K3=O.515 . . 84
1+. x2-x1 phase plane for K1=K2=K3=O.515. €=A=0ol
at
(a) §=O.2
(b) i=0-75
(c)'$=1.0 . . . . . . . . . . . . . . . . . . . . 86
4. xz-x1 phase plane for K1=K2=K3=O.515,
€=0.01, A=O.05 at
(d) 3:0.2
(e) i=0-75
(f) 3:1.0 . . . . . . . . . . . . . . . . . 88
4. x2-x1 phase plane for K1=K2=K3=O.515,
€=0.01, fF0.2, at
(g) S=O.2
(h)'$=0-75 O
(i)3=1.o .....9
Appendix F *
F-1. Characteristic transformation for the

independent variables . . . . . . . . . . . . 128

vi

LIST OF TABLES

Chapter 1
1. Yield enhancement . . . . . . . . . . .
2. Yield enhancement as predicted by the

pi-criterion and 8-D perturbation method .

Chapter 2
1. Percent yield and conversion enhancement
with varying reaction rate constants . .
2. Percent yield and conversion enhancement
with varyingJ\. . . . . . . . . . . .

vii

13

. 41

92

. 9h

Greek Symbols

 

6

NOTATIONS--CHAPTER 1

Reference concentration for species A
Hamiltonian
Objective function

Kinetic constants

0
(Ski

k2/}‘1

o
(sk3Aref
k3Aref/k1

kZ/kl or K2

Periodic functions in the 8-D perturbation me-
thod

Optimal cycle time =-%;
A/Aref, B/Aref - dimenSIOnless concentration
V/Vref - dimensionless reaction volume

Dimensionless optimal steady state concentra-
tion and reaction volume

Dimensionless residence time
Dimensionless reactant feed concentration

Dimensionless optimal steady state residence
time and reactant feed concentration

Perturbation amplitude

viii

1.2,):5, 1;, Multipliers evaluated at the optimal steady
s a e

A 001‘; , dimensionless cycling frequency
I?“ 72 Defined in table 2.
T; Optimal steady state resifence time

75.. , dimensionless lime (at)
3

Cycling frequency

ix

NOTATIONS--CHAPTER 2

 

AC Denotes the change in conversion of reactant A

E Enhancement of the yield of the intermediate pro--
duct B in the'Van de Vusse kinetic scheme.

J Objective function

K K3/K1

Ki Dimensionless reaction rate constants

Pl Periodic terms in the asymptotic sequences for the
perturbation solutions to Xj

r Independent variable in the application of the
method of characteristics

S Independent variable in the application of the
method of characteristics

T One cycle time

u Control variable

V Dimensionless fluid velocity

V Periodic terms in the asymptotic sequence for the
perturbation solution to v

X1 Dimensionless concentration for reactant (A) in the
Van de Vusse kinetic scheme

X2 Dimensionless concentration for the intermediate
product B in the Van de Vusse kinetic scheme

3 x as a function of r and s

Superscripts

0 Denotes the optimum state

X

Subscripts

 

S

m

Denotes the steady state

Approaching infinity

Greek Symbols

 

K/(K+1)

Defined in Appendix H
Perturbation amplitude
Dimensionless time
Dimensionless cycling freqency
Dimensionless reactor distance

Dimensionless reaction residence time

(K; 20(Ko_./_1)<2K °__ _)

8—K,
(KIWI w d: " I , 0
TA; E: (g; R, ) ”37: ‘K'

 

"a
«Doc/l ‘° ex. "a e . .
Ngz(€ +—Rg_’_) Sflgz-(cflflg

2 (mm. [12. ~12... +127.- +12.. -.a,.- +3.]

L=O

A’+O<,')’ g: H; _ 3,
2 (H ’28 :2)

 

Z<K;) R8 2:
“F
“A: e . U+—’—) — 4.
K5.K £8 £8 K8

 

-K
«5/:[8 54]
2(K.°)’ R.

xi

11,05.

 

 

 

ocA’[ e

k,°K 2,

43A“ [8494]

(K92 R9
429

«IN 9 "'

20491 K9

l

2

xii

I

29 = ((+2)KI°

CHAPTER 1

SINE-WAVE CONTROL OF INPUT VOLUMETRIC FLOW RATE IN
AN ISOTHERMAL CSTR WITH A VAN DE VUSSE KINETICS

INTRODUCTION

Numerous investigations[1-12], both theoretical and
eXperimental studies, have shown that forced periodic opera-
tion of chemical reactors in some cases lead to improved

conversion, enhanced selectivity, improved selectivity, and

reduced parametric sensitivity. Recently, Skerik and DeVera

[12] applied periodic control modes to an isothermal CSTR

with a selectivity reaction system that is described by a
Van de Vusse kinetic scheme [13] ,i.e.,

ALB—3C

AVA—k—a—u-D

This scheme exhibits interesting selectivity aspects. First,

from a reactor selection point of view, Gillespie and Carber-
ry [1A] and DeVera and Varma [15] have demostrated that a
level of macromixing, simulated by a PFR with recycle, in
some cases provides a maximum yield of B greater than can be
realized with the CSTR or the PFR modes of flow. Lee [16]
treated a similar partial mixing model Of the steady state
tubular reactor and have shown that when the recycle is fed

within the reactor length instead of the exit, the yield of

B is further enhanced. Second, from a periodic process view-

Point. Riddlehoover and Seagrave [17] and Lund and Seagrave

[18] have demonstrated that via simulated intermediate levels

of mixing, a superior yield of the desired product B compared

to the CSTR and PFR can be obtained.

In the dynamic but not periodic mode, the intermediate
B exhibits an absolute maximum concentration even though it

is already above the steady state it is approaching [19] .

Hence periodic operation provides an inviting method to capi-

talize on this phenomena. The objective is to determine a

control mode, certainly a periodic one, that will maintain

the reactor in a dynamic mode which is characterized by a

stable limit cycle. Such limit cycle should provide a time-

average yield of B greater than the corresponding Optimal

steady state yield. In the phase plane formalism, we re-

quire an (x1,x2) trajectory similar to trajectory 2 as depic-

ted schematically in figure 1. Trajectory 2 is characterized

by an unknown control vector g which is a function of a per-

turbation amplitudeE and a cycling frequency/L. Trajectory 1

is an unperturbed transient state which exhibits a transient
maximum yield. Now, Skeirik and DeVera [12] demonstrated
that, when the isothermal CSTR was subjected under the bang-
bang and sine-wave control modes of the reactant feed con-

centration, the time-average yield of B, regardless of pro-
cess parameters, is always less than the corresponding steady

state (optimal and non-optimal). However, the sine-wave con-

trol mode led to an enhancement of conversion. regardless
Of the magnitude of the perturbation amplitude and cycling

frequency. By applying the second variation in the frequency

domain method [20], a similar conclusion to the latter can

FIGURE 1. Phase plane plot of a periodic controlled

isothermal CSTR with a Van de Vusse kinetics.

x2 and l-il are time-average yield and
conversion, respectively.

X2, Yleld Of B

FIGURE 1.

 

 

locus of steady states
and transient yield

extrema
,/
/'
/
/
/
/ 1.D(€=O)
,/
x1,
/’
/’ s.s.
/
/’
amen

 

X1,

fraction of reactant remaining

 

6
be obtained, as in Sinéié and Bailey [10] calculations for

the second order kinetics. Figure 2 dipictes schematically

some of Skeirik and DeVera's [12] results. The shaded area
represents the region of the time-average x2 and x1. More-
over, the limit cycle which has a perturbation amplitude at
its maximum, (i.e.,€5=1) envelopes the transient maximum
yield. Although a negative enhancement of the yield was
observed for both bang-bang and sine-wave controls, a diffe-
rence in these two control modes was observed: a square-wave
control mode is more effective in shifting the product distri-
bution than a sine-wave control mode, and in almost every
case, the effect of the cycling mode increases as the osci-
llation increases.

In this work, we present an approximation to the glo-
bal dynamics of an isothermal CSTR which is subjected under
a sinusoidal control of the volumetric throughput. From the
approximate solution, we conduct a sensitivity analysis and
derive a predictive equation for the objective function.
Here, the objective function is the maximum yield of the in-
termediate product in a periodically controlled isothermal

CSTR. Finally, we compare the yield enhancement values ob—

tained from a perturbation method [11] and the enhancement

Prediction from the second variational in the frequency do-

main method [10.20.21].

BASIC EQUATIONS

The isothermal CSTR material balance for component A

FIGURE 2. Phase plane plot of a sine-wave control of
the reactant feed concentration in an iso-
thermal CSTR with a Van de Vusse kinetics.

 

 

 

FIGURE 2.
K2=K3=Kg=1.o
A°=1.0
/
/
2,e=1 / /
1
€=O.1
S. S.
1,t=o
X

 

9

(species 1) and B (species 2) in the Van de Vusse kinetic

scheme are eXpressed as

dXJ_lluuz x o 0 2 7C,

fi-‘£~“x—;_(lel+K5xI)-75(ul-')£f' (101)

41 o 9 X

29—2:- i-(‘I “17‘2”" 7")"21‘: (""0552 (1 - 2)
X

with initial conditions
9 = Ozx1=x1(0)=1; x2=0; x3=1 (1.4)

Equations (1.1-1.3) consider a constant output volumetric
flow rate and negligible composition variation of the reac-
tion solution density. The time derivative of x3 (3dimension-
less dynamic reaction volume) accounts for the changes in the
reaction volume due to the controlled volumetric throughput.
For the periodic process, the following integral constraints
are imposed as a basis of comparison with an optimal steady

state condition.

027T

_L9)u.uzde=u;u; ? 7"}? (2-1)
W'

 

“gémewg (2.2)
Eqn. (2.1) requires the values of u1 and u2 should be chosen
such that the amount of reactant input is equal to the opti-
mal steady state. Eqn. (2.2) requires the residence time for

both periodic and time-invariant processes should be the

same. When u2=ug =1, equations (2.1) and (2.2) collapse to
s

 

+fmwgzl (2.3)

Here, we chose

10

u.=l+esin/\°6 (2.4)
OPTIMAL STEADY STATE

The objective function for the time invariant process
is the yield maximization of the intermediate product with
respect to residence time and amount of reactant entering

the reactor; thus

objective function = J0 = x38 (3.1)

x3 is obtained from

S

FX (25:. 12:. 1:) = o (3.2)
Hu (kg. 1.1:: 2.2) - 0 (3-3)
.1: (22.112.13) = o (3.2+)
where H Hamiltonian = x2 + ATI (25. D. 2‘.) (3-5)
Hence, 1;: PJWfi/Ezr (3.6)
X20, = K? xi, (1+K7K,)-' (3'7)
x55=1 (3-8)
T;=[kz+(ku‘kz)xI;]-‘ (3.9)
At; xix; (I+K.'K.)'z(I-7<f,)-' (3.10)
2.2}: Cumin)" (3.11)
A}, = o (3.12)

A similar conclusion can be found in_[22. 23]-

11

PERTURBATION SOLUTION: PERIODIC PROCESS

The time-average yield of B at the limit cycle is

'22 = 34,552 (75.14MB... (4.1)

and thus, J, = Max}2 (4.2)
,xeX
ueU

When the perturbation method developed by Skeirik and DeVera
[12] is applied to equations (1.1) and (2.4), the folloWing
set of moderately nonlinear and non-autonomous differential

equations is obtained:

2—95: --I- K,’F- (HKDF (5'1)

0U). 605A 9 :F (H “DP” é(A°$t‘nA°9 ~ K7 Cos/('6) F

__ - h — -

 

AG .A°
+SI'nA°9—2:;FP.+ E-f-oosll'e-Fa (5.2)
’3‘ 11’9) - K.’co5A'9
%=——"$6-§efl—u “mm A "( A, -P.
-K;(2Fa+gz)+3f!-(cos/l'9)FR (5.3)

Where P1 and P2 are periodic functions while F satisfies

x,={-.(z<,u.;) , F(0)=7C.(0) (5-4)
The closed form solution to equations (5.1 - 3) is given in
APPendix A and has a structure described by
e He) + 6?, my e’?. (e) (5.5)
Eqn. (5.5) is valid only for values of j%-<1. Hence the sen-

sitivity of x1 strongly depends on this ratio. Moreover,

12

£— . .
for E:71, the reaction volume assumes negative values and

thus the model loses its physics.

The time-average conversion is
.. 0 ea
l-x.=l-XIS~ ‘Fégdem (506)

As in [12] , only the second order term contributes to;1

and F reduces to X? . Once the global dynamics for x1 is
known, the solutionsto x2 is readily Obtained by substituting
eqn. (5.5) into eqn. (1.2). Such results lead to a closed
form solution structure similar to eqn. (5.5). Thus, the
enhancement is calculated and is presented in table 1. A
sensitivity analysis of the forced periodic CSTR is also pre-
sented here using the second variation of the objective func-
tion in the frequancy domain. The latter technique was ini-
tially developed by Guardabassi and colleagues [20] and was
extended by SinCiC and Bailey [21] for variable time delay

process. The second variation of the objective function is

’ 2 6 (5.7)
Zgjezwwm
where V(A)Egr(-1A)fé(tfl)*§T§(1A)*ET('3M 3+ 8: ' 1': -’ (5. 8)
ew= £51111";

42 H, or, 1.5;)1
Q5 [in (52,119]
EEIH..(2.’.F;.A;)J
6,2 2 [qu (5.12:. 13)]
8 =-. [H... (3;, y;,3;>1 = o
and H is the Hamiltonian defined earlier.

The value of w(A) is presented in table 1.

13

TABLE 1. Yield enhancement

14

Q ...sr<...s .< +
m 3
CT is a ”a ”£5.28 . a

 

 

 

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~.s.s-.e.e..<§s~-sreset??? + mi. nine wee?are?s.$...s-e§§-seisximfii
C we {a ...s.< + Q. ..§.e..s.< Q. «.52 as? - 3.x... 2 E +

2.5 JS 4. .S r 4... ends + EL 3.4x 11.2 F
3.39.1. .3 We. ”a. Teena.-e._gi.s-e we: - Easiness a: M2 .8 3.1 M.x-_:.<..e...§- . a;

“Cowpmfifig wcooom was. sun :53. AS:

2.2+. mar . mxm.u~+..xi m .r

 

 

 

 

< ~\ r fid<+fl_~xv<. fil<$fl~§v< AN<V +N.r VAINW +~V ~N§ L: IN“ + — <Aflfl§ +N< V
. I. .l u n. . . u I
3.2.2 5 e.-. +1.1. E e: e... a z. .3 Essex...

ping Q38 «.52.? .553. .e

are are; . n
MR 1 .i 5 mi 1 _

_Ho~w.<o+.5fl<i.rfl_ oi . PK; l fi<fir+c+n<uég

 

Mel

 

+ T.<-.s+.s.e.i

when;
luau“ in

. H mam»;

15

DISCUSSION

The perturbation method developed by Skeirik and De-
Vera [12] for the sineflwave control of the reactant feed con—
centration was numerically verified. They have concluded,
that, regardless of the process parameters and frequency of
cycling, their perturbation method is a very reliable appro-
ximation to the global dynamics, even up to a perturbation
amplitude close to unity and any magnitude of the cycling
frequency. However, in applying the Skeirik-DeVera (or S—D)
perturbation method to our present work, we found that the
5% ratio characterizes the error sensitivity from the numeri-
cal simulation (vis IMSL's DGEAR implementation). Figures 3
and 4 illustrate the comparison of the dynamic profile using
the 5-D solution and the numerical simulation. Figures 5a-
5i exhibit the percent deviation of the 8-D perturbation so-
lution from the numerical simulation. The process parameters
chosen are not necessarily those that pertain to an optimal
steady state yield, since the 8-D perturbation solution gene-
rally holds for any process parameters. It is clear from
such plots that the deviation is strongly influenced by the
iclose to unity, regardless of the Evalue, and the devia-

A.

tion is very small when fi-is near zero. ‘Furthermore, the

deviation for x2 at the limit cycle is slightly less than the

deviation for x1 with almost any process parameters. For é?

value close to 1/2, the deviation is in the vicinity of 3 to

4 percent absolute when K2 is moderately large but the devia-

tion tend to approach a maximum of 13 percent absolute for

16

FIGURE 3(a—b). Perturbation solution versus numerical

Egéugion With k1=k2=k3Aref=f=1.,A;2.

17

:ON

mom

mmgowo mo mmmzbz
NON

ESQ

.MM mmDUHm

HON

Puma

CON
«.0

 

18

JON

mom

mesoso mo mmmspz
mom How

ESQ

puma

.Dm MMDme

OON
m.o

 

19

FIGURE 4(a-b). Perturbation solution yersus numerical
solution with K1=1,K2=K2=5,.A=O.5,
€=O.25

20

JON

mom

mmgowo mO mMmEDz
NON

HON

OON

 

L

Pawn

.m: mmDon

ESQ

no.0

 

21

:ON

mmquwo mo mmmzbz

MON NON HON OON

\/ ) )

puma

BBC

.9: mmDUHm

N.O

 

22

FIGURE 5. Deviation of perturbation solution from
numerical simulation for

(a) K1=1, K2=R2=5, A:0.5, 5:0.25
(b) K1=4.928, K2=2u.6u2, R2=o.493, A:10.,E=O.1
(c) K1=K2=K2=1, A=1o, 6:0.8

(d) K1=K2=K2=1, A=10, e=o.5

(e) K1=20, K2=2, fi2=3o, A=1, 6:0.5
(f) K1=20, K2=2, fi2=3o,.A=1, e=o.8
(g) K1=20, K2=2, fi2=3o,.A=1, 6:0.1
(h) K1=1, K2=k2=5, J\=O.5, e=o.025

(i) K1=1, K2=fi2=5, /\=o.05, 5=o.ou95

23

:ON

mmqowo mo mmmSDz

 

 

mom mom How oom
H a q - Cod-l
m
x .O.N|
.o.o
49m
A 0.:
Hum

.mm mmDUHm

LHHd SA WHN AHG 038d

24

mMADNO mo mmmzbz
AWN mom NON HON
4 q

‘ ‘ “ ‘3

:.O

K1 3 .2

 

.nm mmamHm

LHHd SA WflN A30 033d

25

d

.

mmqowo mo mmmzbz
NON

«2 « 4

Nun

.om mmDUHm

HON

l

 

LHHd SA WflN AHG 383d

26

:ON

memowo mo mmmzaz

 

 

. mmm mom Hom oom
. . N.O-
H
\(§ \ .‘ \
\ ( ( 4
\. .H.o
mx .m.o

.em mmauHm

LHHd SA WHN A30 083d

sow m

. «x,

mmqowo mo mmmzbz

HON OON
ON:

.»«/L»«/w»\vxo

 

ON
x

.mm mmeHm

‘LHEIcI SA IAIHN AHCI 083d

28

:ON

 

MON

 

mmgowo mo mmmzbz
NON

 

 

 

.Mm mmDUHm

HON

 

OON

 

 

lHHd SA WHN AHG 083d

29

memoso mo mmmsaz
mom

 

 

.mm mmDUHm

.LHHcI SA IAIle AHCI 083d

30

:ON

 

mom

mmqowo mo mmmzsz
mow Hom
Nun

r.

 

 

.Sm MMDUHM

 

 

nO.HI

.0.0

O.N

.O.m

 

LHHd SA WHN AHG 083d

31

JON

 

 

mom

 

 

mesowo mo mmmzaz
mom

 

 

 

"
~_ .

.Hm mmpuHm

HON

 

 

OON
own

2 21

To:

 

:00

lHHd SA WflN AHG 083d

32

very large values of K2.

Figures 6(a) and 6(b) show a limit cycle in the x2—x1
phase plane and a trajectory relative to a time-invariant
process at the optimal steady state. The trajectory in fi-
gures 6(a) and 6(b) which starts at x1 = 1.0 corresponds to
the dynamics of the system at zero perturbation amplitude.
Unlike in the reactant feed concentration cycling, the limit
cycle trajectory which oscillates about the optimal steady
state, has xl-xz boundaries less than those encountered for
the unperturbed transient maximum. Further, the limit cycle
patterns for various cycling frequencies are somewhat skewed.
(Figure 6(a)) compared to the more regular pattern found for
the system with amplitude variation (Figure 6(b)). An
increase of the cycling frequency produces a more profound
effect of shifting the trend of selectivity enhancement than
an increase in the magnitude of the perturbation amplitude.
In almost all kinetic and optimal steady state process para-
meters, increasing further the cycling frequency will change
a negative selectivity enhancement to positive. However, an
increase in the magnitude of the perturbation amplitude only
augments the selectivity enhancement, so that an already po-
sitive enhancement at some large cycling frequency is further
improved

Recently, Sinéié and Bailey [10] and Watanabe et al
[11] have extended Guardabassi's technique [20] for local
optimal periodic operation of an isothermal CSTR. (For bre-

Vity, such technique is referred to as the pi-criterion.)

33

FIGURE 6a. x2-x1 phase plane for K1=K =K =1,.A=1,

2 2
with varying e

34

'0
(x
.

 

.m
m mmOOHm

 

 

35

FIGURE 6b. xz-x1 phase plane for K1=K2=K2=1, 6 =0.5,

with varying A .

36

 

 

.90 mmDOHm

 

.o

J

37

The pi-criterion, which uses a variational approach to the
objective function is limited by very small magnitudes of
the perturbation amplitude. Such restriction is required in
order to linearize the system dynamics with reSpect to an
Optimal steady state, otherwise the second variation would
be difficult to evaluate. Characteristically, the pi-cri-
terion is generally useful in predicting only a locally im-
pr0per or proper optimal periodic process without excessive
calculations. Here, the pi-criterion is applied to the pre-
sent case and examine, as a function of 5%, the extent of
its deviation from the enhancement predicted by the 8-D per-
turbation solution. For comparison4-%-=0.005 and 0.01 were
chosen, since such values yield very small deviation of the
8-D perturbation solution from numerical simulation. These
results are presented in Figures 7(a) and 7(b). In the per-
turbation method, both-%—values yield a similar selectivity
enhancement trend, i.e., negative to positive with increas-
ing values of the cycling frequency and perturbation amplitu-
de. However, in contrast to the perturbation method, the
pi-criterion predicts a selectivity enhancement trend of pro-
ceeding from negative to positive and finally back to nega-
tive enhancement with increasing values of cycling frequency
and perturbation amplitude. It appears that the predictabi—
lity of the pi-criterion fails at.A somewhere between 0.01
and 0.03. Nevertheless, even at such small amplitudes, the

shift in selectivity enhancement is consistent with the per—

turbation method. However, for perturbation amplitude

38

FIGURE 7. Comparison between perturbation solution and

Pi-criterion method w1th k1=k2=k3Aref=1'

(a) €Ak=o.oo5

(b) e[A=0.01

(a)

39

FIGURE 7.

  
 

    
 
 

 

 

  
  
 
 
  

 

 

0»
. 0.3
'3 F perturbation
:gp 0.2 _ solution 1ncreasing c
1g increasinge
,U 0.1 ' P' 't .
1 cri erion
kg 00 l——— r; A 1 L 1 1 1 4L 4 1 ‘—‘
,9- 23456789101112131u15
Jo -o.1
UO
5%
-o.2 - A
(b)
1.u[
1,2. perturbation
solution
100.
*0
'1 0.8. . .
.: increa31ng E
-x
"(N 006’-
\. increasinge
'3 0.0‘
m
$2 0.2. . .
:5 P1 criterion
{3 0.0 J L 4 ‘ ‘ 4— l L ‘ #‘L A J J
- 123456789101112131a15
-002
-00“
-0.6
A.
-0.8.

 

40

greater than 0.03 (with.é%=0.005 to 0.01), the selectivity
enhancement results from the pi-criterion is less to be con-
fided. Furthermore, the E and A values at which the switch
from negative to positive enhancement occurs, lies ahead of
the 61A values obtained from the 8-D perturbation method.
Finally, when the pi-criterion is applied to the case
presented by Skeirik and DeVera [12] , the selectivity en-
hancement trend was correctly predicted for all values of
cycling frequencies and for small to moderately large values
of the perturbation amplitude. These results are presented

in table 2.
CONCLUSION

In general, for an isothermal CSTR subjected under a
sinusoidal control of the volumetric throughput, the yield
of the intermediate product in the van de Vusse kinetic
scheme is improved via implementation of large magnitude of
the cycling frequency. The magnitude of the yield enhance-
ment is further improved by imposing large magnitudes of the
perturbation amplitude. Regardless of the size of the per-
turbation amplitude, negative yield enhancement results from
small cycling frequency.

The application of the 8-D perturbation method yields
a good approximation to the global dynamics, and hence the
calculated trend of yield enhancement is reliable. The per-
turbation solution is sensitive to 5% such that large devia-
tions from the numerical simulation is encountered when fi-is

#1

TABLE 2. Yield enhancement as predicted by the pi-
criterion and 8-D perturbation method.

b2

 

 

TABLE 2.

B enhancement B enhancement

A. 6 pert. method Pi-criterion
0.2 0.025 -0.0019219 -0.00068700
0.075 —0.0172934 -0.00061838
1.0 0.100 -0.0001263 -0.00009820
0.25 -0.0007828 —0.00061380
0.75 —0.0070416 -0.00552420
5.0 0.25 —0.0001678 -0.000167uo
0.75 -0.0015057 -0.00150660

 

 

 

 

close to unity.

43

CHAPTER 2

SINE-WAVE CONTROL OF INPUT VOLUMETRIC FLOW RATE IN

AN ISOTHERMAL PFR WITH A VAN DE VUSSE KINETICS

45

INTRODUCTION

The study of periodic operation in tubular reactors
with plug flow model has not been as thoroughly investigated
as its counterpart, i.e., continuous stirred tank reactor.
Mainly, the reason is the lack of sufficient theory for
assessing whether or not a certain periodic control policy
is a locally proper optimal. Moreover, the specific case
of a persistent disturbance in the volumetric flow rate,
imposes a difficult description of the flow field, espe-
cially for cases where the disturbances are substantial.

The first chapter has set forth two techniques which in some
ways had advanced our understanding of periodic reactor
operation involving a lumped parameter system, e.g. an
isothermal CSTR with a mildly nonlinear kinetics. The tech-
niques discussed there had to utilize some approximation.
The Skeirik-DeVera (S-D) perturbation method assumes that
when a transient state is disturbed because of an advertent
control of the input stream, the response to such a con-
trolled disturbance should produce a state dynamics that can
be characterized by the nature of the disturbance. Such
assumption led them to assume the structure of the closed

form solution which entails a perturbation over an unper-

turbed transient state. The success of the 5-D perturbation

#6

method lies on the determination of a closed form solution
for the unperturbed dynamics of the yield of the inter-
mediate product in a Van de Vusse reaction scheme. The zero
order correction solution (or the unperturbed dynamic solu-
tion)along with the structure of the initial value are
important in obtaining the periodic terms in the perturba-
tion solution. Hence, the 8—D perturbation solution can be
extended to any control mode of an isothermal CSTR with a

Quadratic nonlinearity in the kinetic rate expression.

 

The second method due to Guardabassi [l] is a varia-
tional approach requiring limitations on the magnitude of
the perturbation amplitude. It was shown in Chapter 1 the
extent of applicability of Guardabassi's variational method.
Indeed the variational approach is restrained by very small
variations in the amplitude (i.e. only local fluctuations),
regardless of the cycling frequency. While the same
restrictions appeared to be innocuous for the 5-D perturba-
tion method, the variational method, unlike the 8-D pertur-
bation method, can be used for a host of different periodic
control of CSTR's, including periodic heating and cooling
rates, and CSTR's with kinetic rate expressions other than
a quadratic nonlinearity, e.g. Langmuir-Hinshelwood type.

The sets of research on distributed parameter systems,
for which the dynamics of a tubular reactor is a subset are
classified into two categories, viz (1) model systems that
involve a linear operator with a nonlinear nonhomogeneous

part and (ii) model systems described by a nonlinear

47

operator, e.g. a tubular reactor where the flow field is
characterized by a nonlinear hyperbolic partial differential
operator. Much of the mathematical analysis and actual con-
trol simulation have been done on the first category. Such
topics include state-space approximation of fixed—bed reactor
dynamics using collocation techniques [2]; approximate linear
dynamics of packed tubular reactors [3]; analysis of the
thermal and concentration traveling waves within a fixed bed
chemical reactor that yields optimal control policies [4];
experimental measurements of the temperature response of a
fixed bed reactor to sinusoidal disturbances in the feed
concentration, temperature and flow rate [5]; and, simula-
tion and control of a packed-bed tubular reactor which ex-
hibits the existence of hot spots [8].

In other areas typical of the first category, the
usual concern involves the establishment of criteria and
computational schemes of control and optimal control (non-
periodic control). For instance, Chang and Bankoff [61 has
extended Sirazetdinov's formulation [7] of determining the
necessary conditions for optimization. They have included
general objective functionals and boundary conditions such
as the recycle of an unconverted reactant with an appropri-
ate time delay for separation and a free choice of final
time. Chang and Bankoff's technique underlies a variational
approach which provides conditions (usually in terms of boun-
dary and final conditions of an adjoint partial differntial

equation) for the maximum of an objective functional. The

48

computational scheme that arose from their analysis involves
the simultaneous solution to an adjoint variable using the
method of characteristics, and the gradient method for estab—
lishing the minimal or maximal of the objective function.
This approach is different from a periodic operation, be-
cause the nature of the latter problem is concerned with
determining the optimal control such that when a new optimal
steady state is obtained, an objective functional is satis-
fied. Chang and Bankoff [6] has also provided a review of
optimal control in distributed parameter systems. Along the
same goal as the works earlier mentioned, Koppel and Shih [9]
has derived a strong minimum principle by converting a vec-
tor of linear hyperbolic partial differential equation to a
vector of ordinary differential equation and then defining a
performance index along a ground characteristic (i.e., t-z
plane). Unlike the rest, they accounted for the time fluc-
tuation of the fluid velocity over a certain steady state
velocity, i.e. plug flow. In an earlier paper, Koppel [10]
treated a similar differential operator but he aimed at com-
paring the exact solution and the solution from a linearized
dynamics of an isothermal tubular reactor with an nth order
irreversible reaction. Finally, for a completely periodic
operation, Fjeld and Kristiansen [11] applied a variational
method to establish conditions for local optimality of dis-
tributed parameter systems with a linear space-time differ-
ential Operator matrix e.g. transient plug flow and axial

dispersion tubular reactors. The search for the optimal

1+9

periodic control involves the solution to the process
equations and the associated adjoint differential equations.
The thrust of this work is twofold, viz, to find an
appropriate model description of an isothermal plug flow
tubular reactor under periodic perturbations of the volu-
metric flow rate and to compare the yield enhancement of
periodically forced plug flow tubular and continuous flow
stirred tank reactors. In this work, the local optimality
of an isothermal tubular reactor which is subjected under a
forced periodic volumetric throughput is determined by
employing a perturbation method developed earlier by Skerik-
DeVera [12]. It will be illustrated in the ensuing discus-
sions that the Skerik-DeVera perturbation method provides a
very reliable approximation of the state variable response
for small perturbation amplitudes and cycling frequencies.
Such results can be rationalized by comparing the equivalent
lumped parameter system (expressed in terms of a ground
characteristic variable, 5) and the process dynamics equation

of an isothermal CSTR.

BASIC EQUATIONS AND THE PERTURBATION MODEL

Consider a one-dimensional model of the plug flow tubu-
lar reactor. The material conservation equations for the
tubular reactor with a homogeneous Van de Vusse reaction

scheme is expressed as

50

for species A,

3X.+ o __o 2:
for species B,
3X¢+ ‘3 ° ° .5
3—5 25 ——(0x,— =t<.x.—sz. h, (1.2)

The mean fluid velocity which is associated in the material
conservation equations has space and time dependencies
caused by the continuous flow disturbance at the reactor
entrance. The flow disturbance is treated as small enough
so as to maintain the plug flow behavior and avoid the for-
mation of eddies which would transform the prescribed mass
conservation equations to one containing dispersive effects.
The velocity field is obtained by a momentum conser—
vation which assumes that pressure, viscous and gravity

effects are insignificant, hence

3V.+-2g_ :0 (2-1)

Perhaps, a rather general and yet simpler description of the
velocity field is due to Burger's equation [13, 14]. But the
simplified model given by equation (2.1) would temporarily
suffice since the present investigation is solely concerned
with local fluctuations to an already established velocity,

i.e. plug flow velocity at the steady state. These local

51
fluctuations arise because the input volumetric flow rate is
continuously perturbed by some periodic control. Here, the
local fluctuations, unlike in KOppel's work [9, 10], would
assume not only time but also spatial dependency. Such
formulation will be discussed later. Actually equation (2.1)
is a subclass of Burger's equation and thus the solution
contains a limiting function whose prOperties, in general,
contains shock-discontinuities. The limiting function is
derived from the solution to Burger's equation at the asymp-
totic limit of vanishing reaction mixture viscosity. There
is a number of both analytical and numerical investigations
of Burger's turbulence and a most recent review is found in
[15]. A very important contribution to the analysis of
Burger's turbulence is the description of formation and
decay of weak shock waves in a compressible flow [16, 17].
Here, the existence of these weak shock waves will be disre-
garded in spite of the continuous change in the mean velo-
city at the reactor entrance. Such assumption is tenacious
only at very small magnitudes of the perturbation amplitude
and the cycling frequency (or quasi-steady state). Hence,
the velocity field which is regular and smooth can be repre-

sented as

_, c- 2'

1]=I+€M+€\& (2.2)

and where v and V are fluctuation functions obtained from

1 2
the solution of equation (2.1), and where e ( <c<l) denotes

the perturbation amplitude.

52

Finally, the reactor is initially void of reactants and
time.

some inerts are already flowing inside the reactor at a mean

velocity corresponding to an Optimal steady state residence

PERTURBATION SOLUTION AND OPTIMAL PERIODIC PROCESS

The objective function which seeks for the optimal

yield of the intermediate product via a periodic control of
the input volumetric flow rate is expressed as

J=—+-,9§xz(te)49

(3)
where T° is a dimensionless cycle time relative to an opti-
mal steady state residence time.
subject to the following:

The optimal function x

2 is

X 3 _
%=f(§3§(v25),fi<9) (a)
iii--222
ae v33 (5)
l for x1
z§(o,e)={
o for x2

(6)
z(s,o)= 0

(7)
06.0)=l (8)
6(o,e)=u(6;e)=u;+3u

a control variable

(9)

53

+39§W°mde =' (10)

Here the yield enhancement is defined relative to an optimal
steady state (Jg) i.e.,
E=J-J° (11)
s
The optimal steady state can be found in [18, 19].

Now, in the region 9>>>5 , the velocity field accord-

ing to the perturbation model is expressed as

17: ”minim-9+e’a-sznzMe‘S) ' 97775 (12)

for Sus=69hne . Since the limit cycle exists in the

same region. 9>>75. , the asymptotic sequence

zc=x§+eE'(e,s>+e’£’(o.5) . a»); <13)

can be assumed. P1 and P2 are periodic functions obtained

from the solution to equation (4) and, in the region 6>>>5

. . . l _0
has initial (or boundary) conditions given by‘g (s- )
P2(s=0)=0. s is a ground characteristic variable. The

development of these initial conditions assumes that the

zero order term at 5:0 has the same value as 5(0,9). A dis-

cussion of the initial conditions is found in Appendices I

and J. Furthermore, here, the second order term 15 account-

ed because

-!-. ' ,5)Ae,=o . ems <14)
"PI-3‘9“

54

Hence, the first order terms do not contribute any informa—
tion about the yield enhancement. Such is typical of the
8-D perturbation method [12].

In the 6>zr§ region, 3; and}:2 satisfies the following
set of ordinary differential equations along a ground charac-

teristic s.

23' 2d. 2*" ' ace-'4‘ I<.°5.‘mir
° l+-—-s P,=-———:-=—'ACOSAY+—"‘V] (15)
M i K' [ 14.5“] k(I-a<.e"")[ we“

1 o no " 2-?!- " '91:]; -’ ' 4—11;
%+ K. ((+2sz,)Pf=’V'(N)[”5] Flag] 1,20,0[015 LY

 

,sr NY
“’ 1‘21 —k° '1 (16)
-x15 )3 5(Pl)
AW
4P, l o ' 406' "a
—_5 °g= P - 2‘ s.’ “Aw/1V1; (17)
M +K2 K4 1 [AS ]’;Ynl\. 5

a v ’
0 3 3 1x
% + K, P, - ~5l'ru‘LY [332—] .. 955‘. s.h2Ar[A-—;’] +AP3' eta/w
,6; ’3f

f

-[%S""W‘ M’cosw]i:, + «73’ (18)

55

where x0 = optimal steady state concentration profile.
‘5

N.

75, = 26:06“)
)6 = 3 and Y=9‘5

K

do = —_’+ K

The solution to equations (15) to (18) is found in Appendix

G. Thus, the average conversion and yield is expressed as

-K, 2
— o(. e .
(«7cI = I“ T—e—HW-[T 1- m[flrflgi-flgg + Fading”

(19)

, if K20: KIOU‘J.)
o ' 0 '
_. K. *2 “‘1
= —- o(
x: K 8 1:20 0 K:-I(:(‘fj) I
~ , otherwise

 

Kl.- K|.( "1.)

0
40

I3
+6 8 [5,3- + 2,? + K..[ 98 1,; (fl'fihfl (20)

where the 52's are defined in the notation section and the

9's are defined in Appendix H.

56

DISCUSSION

The periodic cycling of the input volumetric flow rate
in an isothermal tubular reactor would certainly create
fluctuations in the flow field within the tubular reactor.
In what manner can the fluid velocity inside the reactor be
mathematically described is a matter of how much refinement
is needed to adequately model the flow field. If only small
perturbations from an optimal steady state is desired perhaps
a simple perturbation model as described earlier would suf-
fice. Clearly, the enhancement is a function of the nature
of the flow field. Even in an already optimal steady state
regime, the differences in the steady state flow patterns or
the steady state flow models have created different maximum
yield [18, 19, 20]. Among different steady state flow pat-
terns that were investigated, the optimal yield corresponds
to an absolute maximum.

In this work, the velocity fluctuation is regarded as
smooth, regular and when random shock—discontinuities could
be present, is eliminated by assuming quasi—steady state.
The description of the quasi-steady state fluid velocity is
afforded by the limiting Burger's differential equation
(i.e.,/u-vO). A decomposition of the fluid velocity into
two fluctuating functions in terms of time and spatial coor-
dinate is necessary because the time-average of the periodic
functions 3} do not produce information regarding the yield

enhancement. Such result is typical of the 8-D perturbation

57

method. Hence, in retrospect the present work would assess
a properly local optimal yield in an isothermal tubular
reactor whereby the flow field response to a periodic
disturbance in the reactor is described by equation (12).

The sensitivity of the perturbation solution was tested
against the numerical simulation of the hyperbolic differen-
tial equations as suggested by Acrivos [21]. The numerical
solution to the material conservation equations employed the
perturbation equation for v. This was implemented here be—
cause of the good matching between the perturbation solution
for v and the numerical implementation of equation (2.1).
Hence, the numerical solution to the conservation equation
is partly simplified. For the parameters that were used in
the calculations (e.g. small A. and small 9 ), the percent
deviation between perturbation and numerical was generally
small (for example Figures la-ld). The absolute maximum
percent deviation is 0.02 at 6 =0.1 and A.=0.l. Usually the
sensitivity is largely influenced by the magnitude of 6 .
Hence, in subsequent calculations for the material balance
equations, particularly small values of € were chosen. The
requirement of small values of.A is imposed, not from a
standpoint of numerical matching of solution, but from a
standpoint of the required physics. However, the magnitude
Of.A somehow affects the matching between the numerical
simulation and the perturbation solution. Although the per-
cent deviation is yet small for the same value of E , an in-

crease in the cycling frequency leads to increase in the

58

( :.:5
a)
.O

(b)
6:0.01 A
. =0
.1

59

:8

:ON

mmqowo mo mmmzbz
MON NON

Es: Puma

mMHOWU mo mmmzbz
MON NON

85: Puma
.H MMDUHE

HON

HON

OON

 

,

l

3.0
mm.o
ooé

HO.H

OON

om.o
3.0

OO.H

 

mo.H

I'P'

3O

A3

60

FIGURE 1. Deviation of perturbation solution from
numerical simulation for the velocity
profile at §=4.0

(c) €=A= 0.05

(d) €=0.0l, A=O.1

61

 

mousse mo mmmssz
new mom How
I ‘ ‘

OON

 

JON

mmqoso mo mmmzsz
mom mom How

1

 

HOO.OI

OO0.0

HO0.0

NO0.0

OON

 

A

.H mmDOHm

j

 

HO.OI

O0.0

H0.0

N0.0

IHHd SA WON AHG 033d

IHHd SA WON AHO 333d

3:

3v

62

percent deviation (for example Figures 2a-2x and Figures

3a-3h). Furthermore, the large A. has percent deviation

propagating along 5 as compared to a more uniform percent
deviation along 3 for smaller values.

In the x1( 9 ; S) - x2( 6 :3 ) phase plane (Figures 4a-
4i), the size of the limit cycle increases from the reactor
entrance to the exit. Such is caused by the fluctuations in
the fluid velocity along the reactor. Finally, the impor-
tance of these effects render the yield enhancement to
achieve negative values but achieve positive values for con-
version enhancement (Tables 1 and 2). It appears that in-
creasing A creates a larger negative yield enhancement and
increasing conversion enhancement (Table 2). Hence increas-
ing A would tend to shift, in a way, the reaction to the

side reaction (A+A-*>D), and thus favoring a higher yield

for species D.

CONCLUSION

When an isothermal tubular reactor is allowed to
undergo a quasi-steady state periodic control of the input
volumetric flow rate, the time-average yield of B is less
than the corresponding optimal steady state yield with plug

flow behavior of the fluid. It appears that implementation

of larger frequencies would shift the direction of enhanced

selectivity to the side reaction.

Finally, the result presented here is not general for

all classes of forced input volumetric flow rate oscillation.

63

FIGURE 2. Behavior of Van de Vusse system for
K1=K2=K3=0.515, €=0.01,‘A=0.05 at

(a-b) 5:0.2
(c—d) 5:0.5
(e-f) S=O-75
(g-h) 3:1.0

new mom

64

mmgowo mo mmmzbz

55C

NON

Puma

mmflowo mo mmmEDZ

eom mom
A .

ESQ

NON
'

.N MMDUHm

HON

HON

Puma

oom
smo.o

mw0.0

owo.o

 

OO0.0

oom
OHm.o

ONw.O
.HNw.O

NNw.O

.mmm.o

 

lJNmO

ADO

Amv

65

mmnowo mo mmmspz
eom mom mom Hom oom
. . . oufi.o

muH.o

l

me.o Nx

. ONH.O

 

83g PHGQ

oma.o
Aev

mmqowo mo mmmzbz

JON MON NON HON OON

 

on
.m mmouHm

66

mmgowo mo mmmzaz

 

 

 

eom mom mom How oom
w . . . emm.o
.mmm.o
Nun
.omm.o
a it Lam . 0
SC
AHV
mmnowo mo mmmspz
eom mom mom aom
an

 

ESQ

 

. Amv
m mmome

67

mmgowo mo mmmzbz

 

 

 

 

eom mom mom How oom
. . . . nmm.o
Nun
A5
mmqowo mo mmmzoz
:ON MON NON HON CON
1» . . . mme.o
l dmjoo
tome—duo fix
.336
852 (\.\\ Puma . omJ.O

Amv
.m mmsuHm

68

FIGURE 2. Behavior of Van de Vusse system for
K =K =K =0.515, e=0.01,.A=0.2 at

1 2 3
(i-j) S: 0.2.
(k-l) 5= 0-5
(m-n) S: 0.75

(o-p) S: 1.0

69

JON

JON

mom

MON

”\ESC

ESQ

wMHOwO mo mmmzbz
NON

Puma

mmqowo mo mmmzbz
NON

Puma

.N mmeHm

HON

HON

OON
mmo.o

ww0.0

mmo.o

 

.omo.o

OON
“Hw.o
.Nw.o
Nw.o
\Nw.O
Nw.o

JNm.O

A:

A3

7O

JON

JON

mmqowo mo mmmzbz
MON NON HON

85C Puma

mmaowo mo mmmzbz
MON NON HON

as: when

.N mMDOHm

OON
tho

mmHo
02.0

 

OOH.O

 

A3

3:

71

MMHowo mo mmmEDz

O MO O O 00
J N .N NqN H.N NJNN.O

mmm.o
mx.

ONN.O

 

wmm.o
ESQ Puma

Adv

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MON NOW HON
4

‘3
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oom
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l

mam.o

320

wam.o

l

mam.o
omm.o

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85: Puma

 

l

Asv
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72

JON

mmHOMO mo mmmzbz
mom mom HON

 

ESQ (1|

mmqoso mo mmmssz
mom mom How

Puma

#909

.N mmDOHm

oommmm.o

l

J

emm.o

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L

00

l

A

t—H

OMN.O

N
NNJ.O

JNJ.O
ONJ.O
mNJ.O

 

one.o

A3

on

73

FIGURE 2. Behavior of Van de Vusse system for

K1=K2=K3=0.515, e=0.05, A=0.2 at
(q-r) 5==0.2
(s-t) 5= 0»5

(u-V) 5= 0.75
(w-x) 5:1.0

7H

mmqowo mo mmmzbz

sow . .mom mom How . com
. . .. mo.o

mo.o mx

 

OH.O

Anv

mmHONO mo mmmsbz

ewm mom mom _ How oom
. . om.o

Hm.o

 

85C“ N .. mm.0

Pawn
“so

.N mmDUHm

75

mmHOMO mo mmmSDz

 

eom mom) Now How oom
Q q u I “H00
mH.O
ESQ thm OH.O

mmgowo mo mmmEDZ

 

 

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. q . 3.0
.36
IMO-O
$0.0
.36
ESE thm

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G;

Am;

76

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mom mom . . Hm.o

NN.o

mm.o

 

 

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a» . . 4 03.0

.om.o

Hm.o H
mm.o
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ESQ Phwg

 

L

. Asv
m mmpme

77

mmqowo mo mmmzsz
:om mom mom mom oom

 

 

 

858 Puma AXV
mmqowo mo mmmzbz
:ON mom NON HON OON

. I q G 0:00
1H:.o
de.o

. fix
0 o o o Mdoo
.. . . . x .. . .. ..::.o
cnfifi\\ puma .md.o

“av
.m mmpmHm

78

FIGURE 3. Deviation of perturbation solution from

numerical simulation at 5:0.5, 6:0-01.
A:0.05.

(a) for the velocity profile.

(b) for Van de Vusse system with K1=K2=K3=O.515-

79

mmgowo mo mmmzbz

 

 

 

new mom Hom oom
[AJ-‘V. . . H0.0-
4" 4v. Av 4'. A-‘-v oo.o
Ho.o
H
N Nun
mmqowo mo mmmzpz
:om mom mom Hom oom
. . . q moo.o-
.fioo.ou
ooo.o
.Hoo.o

.m mmDUHm

lHHd SA WHN AHU 083d

$Hfld SA WHN AHG 083d

3v

A3

80

FIGURE 3. Deviation of perturbation solution from

numerical simulation at 3:1.0, 6:0.01,
AP0.05.

(c) for the velocity profile

(d) for Van de Vusse system with K1=K2=K3=0o515

81

mmgoso mo mmmzsz .
sow: Allmbm mom How com

a}??? imAWW

mmgowo mo mmmzpz
3mm mom mom How oom
. . moo.o-

1Hoo.oa

j000.0

L

Hoo.o

 

P

Noo.o

.m mmDUHm

lHHd SA WHN AHU 083d

lHHd SA WflN AHG 083d

2:

A3

82

FIGURE 3. Deviation of perturbation solution from

numerical simulation at 5:0.5, 6:0.01.
A=O.2.

(e) for the velocity profile.

(f) for Van de Vusse system with K1=K2=K3=Oo515-

83

dON

MON

mmqowo mo mmmzbz
NON

HON

OON

 

dON

mom

.
xiv

mmqowo mo mmmzbz
NON

.m mmDUHm

 

HON

3.0..
\woé-
.ao.o-
No.0-
.oo.o

 

 

20.0

00.0

OON
NOO.OI

HOO.OI

OO0.0

 

PHOO.O

IHH& SA WHN AHG 033d

Lflfla SA WflN AHG 083d

CV

8h

FIGURE 3. Deviation of perturbation solution from
numerical simulation at 5:1.0. €P=0.01.
A=O.2.

(g) for the velocity profile.

(h) for Van de Vusse system with K =K =K =0-515-

1 2 3

85

dON

dON

MON

O.)

MON

Hx

mmqowo mo mmmzbz
NOm

mmqomo mo mmmEDz
NON

.m mmDUHm

HON

Co-

H.OI

 

S
O
IHHd SA wnN AHG 388d

A5

OON
NO0.0|

HOO.OI
JOO0.0

.HO0.0

 

NO0.0

iHHd SA WfiN AHG 033d

Amv

86

FIG -
URE 4. x2 x1 phase plane for K =K =K =O.515
6=A=O.1 at 1 2 3 '
(a) 5:0.2
(b) i=0-75

(c) i=1.0

87

 

 

 

 

 

 

(a) FIGURE 4.
0.10,
0.09-
x2
0.08.
0.07 . . . is
0.80 0.81 0.82 0.83 0.84
X1
(b) 0.2h{
0.23 I
X2
0.22-
0021 n l l _J
0.4 0.50 0.52 0.50 0.56
X1
(e) 0.27 r
0.26 -
X2
0.25 -
0.24 . . - . _.
0.39 0.u1 0.u3 0.45 o.u7

88

FIGURE 4. x2-x1 phase plane for K =K =K =O.515.

1 2 3
6:0.01, A=0.05 at
(0) 5:0.2
(e) 5:0.75

(f) 8:1.0

(d)

(e)

(f)

89

 

 

 

 

 

 

FIGURE 4.
.090F
.089-
.088 ..
.087 l j l _;
0.820 0.821 0.822 0.823 0.824
X1
'227F
.226-
.225,
.224 , . . 4
0.510 0.512 0.514 0.516 0.518
x1
.256
F
.255-
.254-
0.2 - - - __.
g?423 0.425 0.427 0.429 0.431
X

1

9O

FIGUR
E 4
. x -
2 x ph
6:0 5 ase plane f
.1,A=02 orK=K
. at 1 2=K3=0.515

(8') 8:0 2
(h) S=O.75
('1) 5:1 0

(g)

(h)

(i)

0.090

0.089

0.088

91

FIGURE 4.

V

 

 

0.087
0

0.227

0.226

0.225

0.822 0.823 0.824

X1

fi

U

 

0.224

0.510 0.51

0.256

0.255
0.254

A

0.514 0.516 0.518

X1

‘-

 

0.253

‘ l ‘J

 

0.423 0.425

0.427 0.429 0.431

X1

92

TABL
E 1
. Per
. cent '
With ylel
varyingdrand conv
eaction iriion enh
a e an
const cement
ants

93

. AN PCMPmeh wSP

mo sonno>soo map CH mmswno may mmPoCop o¢t

 

 

 

 

 

 

 

 

mmm.m mos:.«- ooamo.o oodmm.o Ho.o Ho.o wmmm.o .H .NH .m
Hm8.o mmfio.m- mom:~.o mmmom.o Ho.o Ho.o om:H.o .m .m .NH
me.H mmmo.m- mammfi.o omomm.o H.o Ho.o .momm.o .H .m .m
cum.fi :an.N- mmmom.o womfis.o H.o 06.6 mmmo.o .H .H .m
w:m.~ m:mo.m- mmfimm.o nmfimm.o H.o Ho.o om:s.o .H .m .H
mma.H momm.m- mm:mm.o mmnmfi.o H.o Ho.o :mom.o .m H .H
mum.fi mmom.m- o:mmm.o w:msm.o H.o Ho.o oflmm.o .H .H .H
6H.o<e aoflmxme mwx me <_ m .0. ma “whamx Hg

 

.H mam<e

94

TAB
LE
2
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Wit t .
h varyield a
lng A nd con
. vers'
ion
enha
ncem
ent

95

.¢ PCMPoMmh mSP mo COHmhm>coo map CH mwcwno map mmPoGwU.o¢
*

 

 

 

 

 

 

m666.H 666m.~- mm666.6 wm66H.6 H66.6 H6.6 :mom.6 .m .H .H
m666.H momm.m- mmsmm.6 mmnmfi.6 H6.6 H6.6 666m.6 .m .H .H
m666.H 666m.m- mm6mm.6 mmmmfi.6 m6.6 «6.6 :mmm.6 .N .H .H
6Hm6.H 666m.m- mmsmm.6 6NR6H.6 H.6 «6.6 666m.6 .m .H .H
“H66.” 6m66.m- mm666.6 mmmmfi.6 m.6 H6.6 666m.6 .m .H .H
0 mx mH mm m mop H
2 as 63 me 0.. 6.. < 6 .6. x «E x

 

.N mgm¢a

96
Once the value of J\ is allowed to assume larger magnitudes,
another flow model which would allow the formation of shock-
discontinuity should be built into the flow model, even if
internal mixing can be neglected. The latter is obviously
more complex to handle because the material conservation
equation would have to include dispersion effects super-
imposed on convective material flow.

The 8-D perturbation method proved to be very reliable
even for problems involving a hyperbolic partial differen-
tial equation which does not, along with the boundary and
initial conditions, induce the presence of shock-
discontinuity. A similar conclusion was obtained for the
isothermal CSTR which is subjected under the same periodic

control of the input volumetric flow rate [22].

APPENDICES

APPENDIX A

DERIVATION OF THE CLOSED FORM SOLUTION FOR
REACTANT A (A PERTURBATION SOLUTION)

APPENDIX A

DERIVATION OF THElQLOSED FORM SOLUTION FOR
T NT PERT T 0 SO UTION

The material balance for reactant A (species 1) in

the Van de Vusse kinetic scheme

A—ah’ B—-r"’ c . IVA—5’ D
has a form of
_-— _ 2 3 1" 393 _
3M, $.41 (k,A+k5A)V A2t*V2t (A 1.1)

with the initial condition A(O)=Aref'
where

t=time

qf= the feed volumetric flow rate which is subjected

to perturbation.
§=the outlet volumetric flow rate which is controlled
at a fixed rate.

V=the reactor volume

A=concentration of species 1.

k1,k3=reaction rate constants.

Assuming the reaction solution density is independent
of composition, the following expression is true,

04"??? (IX-1'2)
97

98
integrating equation (A-1.2) from initial pointst=0 and V=V

yields

t
V=Vp+£(3f-§)dt (A-1.3)

For a sinusoidal perturbation of the inlet volumetric flow

rate, we specify
fif=§(l+esina)t) (A-1.4)

where
€=perturbation amplitude
50 =cycli c frequency .
Combining equations (A-1.3) and (A-1.4) gives the following

result:

=\7—-:)—e-casa)t (A‘1.5)

or,

g. «gesinwt mix-1.6)

Upon substituting equations(A-1.4), (A-1.5). (A-1.6) to equa-

tion (A-1.1) and introducing the following dimensionless va-

riables:

— A
; K'sfkl; K3: kgATEJfC '1 xg=m (A-107)

rmkr

f‘ 3A=f0;6=

=P¢<|

with some rearrangements, the dimensionless material balance

equation for species 1 is expressed as

e
dx 6 ' a-meAa
Z-él-(I’ KCQSA 9)+ XI[(‘+K')+(AGS'"A )A]
=/.esmAe-szf(I-x¢m9) (A-2.1)

99
with initial condition x1(0)=1.
An analytic solution to (A-2.1) is obviously not
possible. In order to develop an approximate one, the solu-
tion is assumed to be composed of a non-periodic term and a

sum of periodic terms:
70(9): F(9)+ 6P.(e>+ e’P.(e> (A-2.2)

This requires that 6be small enough so that terms of the
order eacan be neglected.

Substituting equation (A-2.2) into equation (A-2.1)
and collecting terms of same order in 6 yields the following

three differential equations:

d 1
EOE-.d'l‘gF—(HKOF (#34)
fl--'— 9”“: (I+K)P+J— $3011-
46 Am“ 3* ' . A011 a memo);
=smA9-2K.FP,+ £3 605/1an
(A-3.2)
LEE. - .L cos/19 aL’P. + (1+ map: A (AsinAO-K:¢0$A9)R
.19 A Z? M
= -k (2FP+P‘>+2K:-°-93—-Ffi _
’ ‘ ' ' A (A 3.3)

with the initial conditions

F(0)=1 and P1(O)=P2(0)=0.

Solution to F(9)

Equation (A—3.1) can be rewritten as

100

A HKI I
Eg=-K3(F2+—Ka-F'T<3) =-!<:<F—F.>(F-F-) (14-4-1)

where F: are readily obtained, thus

 

F1= mm): (I+K0‘+4k, (A42)

2k;

 

Integrate equation (A-4.1) by partial fraction from 0 to 0

 

yields
0)‘F .- A-u'o )
' ML;_MF(9) F— =-K39 ( 3
F.—F. F(o)- F+ F(0)- F.
define
o< , 5%.. (A-4.4)
F(o)- FL

and substituting equation (A-4.2) for F+ and F_ into equation

(A-4.3) gives

W -(X exp[-‘\/(I+Kc)a+4K5 9 ] (A-Ll' . 5)

F(6)=F+ I K3 (1- K-exP [" (Hsz‘KJ 6])

Notice that F+ is indeed the steady state solution that sa-
tisfies

z (A—4.6)
O: |— (1+Kn)f;,” kbfss

For convenience, we let

CD=2K
can 1’ (H'va‘f‘ug and 2 5

 

(A-4.7)

101

hence equation (A-4.5) simpifies to

2C3“ 9"? {-Cge)
Hen} + _ -
s: Cz[l~o(€xP(-C50)] (A 4.8)

Furthermore, using the binomial series and a little rearrange-

ment, equation (A-4.8) can be equivalently written as

2C3 a) . {304359
F<e)= tss’r '5— «we

2 j=0

(A-4.9)

Solution to P1(9)
Equation (A-3.2) is rearranged and rewritten as

“it—g: ‘|'[(‘+ K.) 1’ “(sf-1P:

-.- 71-(CaSAB+AanAQ)" A (005219 +ASn'nAo) F (A-S. 1)

the integrating factor needed to solve the above equation is

found to be
C36 , -C59 (A-5 2)

I: e —2a(+a<

substituting equation (A‘S-Z) into equation (A’5-1) and carry-
ing out the integration, we obtain along with the following

definition of terms.

20( K1 ..
41‘ ‘11—(“L21Ffl (A 5.3.1)
(A-5.3.2)

 

«2 (+KI4‘C; I
(I) :2 —' [If ] 1 2
2 A_ 2k; A*C3

102

= "is;
4)” A(A‘+c;) (A-5-3-3)

solution to P1(9) is found to be

‘C39 . 6
47,0 (fi—XA’ "' 603119) Lg[A(I-C5)SIHA9"(C,+A2)CO5A9]

 

 

 

F.(6)= ' -C 9 z + _
(I—exe ’) (,_ «emf
-c 9
+ 4’, [((3‘A2)C05A9+A(Hc,)s,‘,mg] e a
((14-112) 6-901 + 4:62 C(80))
’ 0’“ } (I-de J (A-5.4)

where C(P1(0)) is the constant of integration evaluated at

P1(0)=0, thus

C(fito))=-\g+(c,+A’)¢,—(c5—A’)w’ (A—5.5)

Solution to P2(9)

 

The process of solving equation (A-3.3) is similar to
the solution of equation (A-3.2). (A-3.3) is equivalently

written as

 

:16
1.7+ (/+K.+2K5F)Fz
A3; A9-K.CosA9 come 4?.
=-K.P.’+ 2193:543FR- " A P.+ ‘71—‘23 (A'6-1)

by comparing euqation (A-6.1) with (A—5.1), it is clear that

the integrating factor for selving equation (A-6.1) is simi—

lar to equation (A-5.1). The integration can be readily

performed. The procedure is straight forward but quite

103

Hence, the solution to P2(6) is expressed as

 

 

 

 

 

 

 

 

 

 

 

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(,‘(er-v>‘+4n‘

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let 2" 3,“ 2A9+A605A9 — ' CShAD+ACoSAD
( 5 n ) ACC’9A’)( 3

 

+ 2503(0))

 

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Ch 9 A

107

49 (33(93):?

3 ii! c'bém)‘ A [cngnrmemcosnv]
3 ~ 5

 

 

 

 

 

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A[c5(391+lf+ll’]
1 «#392909
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8-1659 ‘CQB
+ ( 0,605A6-Asm/19) - {’9 e Sin/19
(6:11“)
, 430'?”
'f' Zid1"581[ca(2fa) COS/l9 A5mfi9]
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“(30")9 .
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(if A
w w 1+1 {e 43([93909
+ 2' 24—15—— [6,(193'92) coma—Ashlie]
3:: 1:: c; (143' +2)’+ Az

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2 c;‘+4A c,'+4A

 

 

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4”Wk-4'

108

 

 

 

 

 

. ' 9
+ i c95'"ZA9-2AMSZA9 + “8-65 “IQ-26,9 '
2A can 4A? A GOSZAO F—IZ—(casnnzlwflllcosmo)
6 -C39 é2C59
a“ 6 0052A9_ _a(__
+7111 *[——_2A c5341” 11’ (c35mZA99 menu/19)]
2‘39
-— +§___'”ZA9 “*9 92(qu )e
lib“? T)e 63+ 4A1 (C5 COShO+ASm2A9-7A __3_
C
4““ 5A z)'(Catuft/w+AszAQ+--—z):l
+C;2+4A
2J3
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1 L144“ 4A )9
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18’265:
at
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“1+4 ( 5 O 1f)]

 

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&;[Q+M Lug) 9459+ ”(e (C coma “ma-4}]

 

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c '2 ’9 ‘
+W,(§_s____m2/19)e-C,9+ 1W“,- fle’ (c,&hf/19+As.th9-r if!)
+A 2 4A (”-9 4A "

3
C 7
_L ((+4 :IM -L-—(c,s.'n A9 A szA9+ 7%)
03+ 5

- o
-— C(P(0))(S—’————"M — 91511—12 96’ (A-6.2)

where C(P2(O)).is the constant of integration which has the

following form:

C(PW»: K _ 2___:___‘{,A 21A ‘D/A >511
2 3 C,“ 44111) 353 (759+ 4119+ C ,(‘§+4A’)+ K9199)

 

w 3 2 {A 2 t A: 2 15A
”1;“ [92691)94AJCXHJ) [<2 (1 cm M199?" 199*“ 14"?"

 

109

211’
Jig/l: _ ‘J [6371—02 4117] (a (2' .1)
(6; 31+ 4A’) 43,3

"I 9': I
4.

 

0 I 0”) 01‘)th

+§idé*‘[- . 212”. . ; - 3”!"

13! 1:! ('59-'11'08" (aflfl)+4A] 9(2+1‘5)[c;(j+1’3)’+mt]

 

 

27M + 214,"
C, (74* 01’ 597 74* 1)? 4A2] grid) [c,’g+z)’+ 411’]

 

 

 

__ 2 {5'11
c, (fix-03510414)? 4.47

76 2A 1 A‘)__13_—_ fro
______ + 2“ - _
-9c;+4/1’ (at, 5‘4) 4053 411’ ( 5 (5) 6 +411 (CHM 7;)

 

241———-——f ( )+ 1 J}...— C . + 2A3

+ 22"“ 5910*;"41‘ C, M 6(97’] C‘(2'u)’+4A‘ "7”) 43(ij
{a F , :Nlz _ {9 c .+ 2,1‘

- (flag)? 411’ (4"?) * c,(/+,")] 2).—33* M. ( 92 91)

 

 

2

I C (‘4)4 211 . 'H
C;(j-I)’+4A’ 57 (5 (3‘4) I 'f 3 J

 

 

o I Miami“

0 0 3‘ 16 [I ' + 2A: 1- 17 [C (' +z)+-2—A-:
+ 2):}? [- g‘qwafim‘ LCM“ 52+ [5 (2""5’J QzéwzfiM’Lfid 6W

 

|——J

-/ "I; 14’
99“" ")*4A [96“ I) 6492:") —:—-J 55‘} 4‘0“” aPM”) 6,742]

 

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(
C, (1144f +411 ”[6 3" H) (5(71‘1") :l

2AM _ 1(le _ 211(ilai'irc) _ {I4
+ 995-4342 4633+ 4-113 91+ 411’ 24

110

 

 

 

 

+2 a“; 21m _ 24,11 _ 24M _ 2ij
3:1 C52 (911' )‘1 4A2 Ca (3'12f4 411’ (3310413)? 4/13 can M’
211
fife-W 4A2 , if 3* f
-1"

.L

2’1 , o/Mrwise
”Liif, 2 {M 2 ”A film 211,41 21,541 ‘I
f! l= C56 (7212”) +441 C3(3+1+2)’+ 4A (5"(3'4u1y44 63"(711) ”+4.1? 5' (741—1), 4/! J

 

115A 4! 1
26 . - .. __L._ _ _'1
+ (PW) 46;”, 63,”, A

 

 

+2 if. _ __16_A_ _ ‘A _ full
f' 9’0*;')’+A’ o’ljw)’+A* agar
D i «11" _ '{ILA + ll), - '1" A
T I“ (fwflfi A. c;(;‘+1+,;’+A= 6724!)? A'

+ 31!: C, _ l/jC,
46324542 €3+A1 I
+zfdé[ {I my ) ’ 4W5“ .) - .1769?
c;(2*j): A: (52[/*z.)2+Aa (322-14;

 

”9 a 1‘4 dagmja) 4c, (14+!) 13‘3”“) ]
+1212: C,’(1+1+2)‘+ A ('3 (111“) +11 Ca"2l*""‘ N

[am] Li; +2.8

 

 

1' . . “+11
1652:”) +1421; (3(220‘0]

+117 [ng(/-2c,—A"’) 2991\(/+2C,—A)_ JAN!" ipgfl

 

 

a", 44’ 9’4 +‘4A
was A a ___:_]+cz c__;« (“23% _.-v_[$ira+A‘>(e‘+2A‘)0/z-so]
- 2A C;+4A’-X 63’4“ A (63544196,

+(1'fss)(C;+ZA1)(I'0(’) _ Maw/15+ ZA‘zfl-Cfi-M’cha) + “5“”) (A—6.2)

 

A’c,(c;+ 411’) E,A (g’uA’) c,(c,’+4A’) A

111

Other terms, such as 11,(i=1,19) appear in the P2(e) solu-

tion are defined as

1.=%+2%A’0«c;) (Ix-6.4.1)
73:43120’9" (xx-6.11.2)
79‘2““‘9’ (Ia—6.4.3)
4‘44““) (xx-6.11.4)
“WE/’7'“; (A-6.A.5>
is= ‘E’WA’Y (xx-6.4.6)
{7"MK’MZ) (A-6.u.7)
48:41:24,113 (Ci-1‘) H-648)
1,:2marf) (ls—6.4.9)
{mng’zgm’f (46.4.10)
1,: tg’Aa-gxcsm’) (AA-6.111)
163-4341 (MUELQA) (xx-6.4.12)

43:4(garz—c3119-A’1—gi-MA(”exam") (IX-6.4.13)

45-41; (fie-21-91) (Ix—6.11.14)
7% 1/3’1 (Mum-A“) (A-6.u.15)
tut/£11019) (A-6.’+.16)

1,7440%) (14611.17)

112

1,5: ¢,(c,+4‘) (A-6.4.18)

1w VJC‘J‘A') (xx-6.4.19)

Substituting the solution to F(9), P1(e) and P2(e)
into equation (A-2.2) gives the approximate global solution

to x1 as a function of e.

APPENDIX B

THE INITIAL CONDITIONS FOR THE
PERTURBATION CALCULATION

APPENDIX B

THE INITIAL CONDITIONS EOE THE
PEBTURBATION CALCULATION

In order to obtain the initial conditions for F, P1
and P2, we evaluate equations (A-2.1). (A-2.2), derivative
of equations (A-2.2), (A-3.1), (A-3.2) and (A-3.3) at 9=O.
However these six equations consist of a total of seven un-
knowns,viz., F(O), F'(O), P1(O), Pi(0), P2(o). Pé(0) and
Xi(0) which make the problem not well posed. So an "ad hoc"
assumption is then. attempted by imposing x1(0)=F(O) and
P1(O)=P2(O)=O, and thus achieve self-consistency of the pro-
blem. Of course, the validity of this assumption needs to
be verified by comparing the perturbation solution based on

these initial conditions against the full numerical solution.

As shown in figures 3, 4 and 5, the results are very satis-

factory and the error is large only at the initial transients

because if the initial conditions are important the error

could propagate in the limit cycle. Such were not observed

in the present work. But as clearly seen in the perturbation

solution at e, , the initial conditions vanish.

113

APPENDIX C

ASYMPTOTIC SOLUTION TO X AND X

APPENDIX C

ASYMPTOTIC SOLUTION TO x1 AND x2 AT eno

 

The objective function in the present work is the
optimization of the yield of the intermediate product B

(species 2) in the Van de Vusse kinetic scheme. The compo-

nent B material balance is

7;,6;«§5v(k25vk.A)V=5%+Vfi—f— (c— 1.1)

assuming there is no product B in the inlet stream, then,

B(O)=O.

Introducing the dimensionless variables

5 A ’ _
X‘SE‘} ’ thsz (C 1.2)

(where fi2=K1K2 and K2=k2/k1), and substituting (A—1.A),

(A-1.5). (A-1.6), (A—1.7). (C-1.2) into (c-1.1) yields the

dimensionless material balance for species 2 (or B):

(I- fees/ta) if; = -(1+es;nA9)xz —(12,x,-K.x,)(1-J§iwsne) (C— 2 . 1)

The global solution to (C-2.1) is not necessary for

calculation the average yield of B at the limit cycle.

Therefore, an asymptotic solution for B is sufficient.

11h

115

It is clear from (C-2.1) that the asymptotic solution

to A(species 1) is required. Referring to Appendix A, and

substituting F(9), P1(G), Pé(9) into (A-2.2) and then eva-

luating the resulting equation at 6=r&.yields the following

expression ( Sadenoting large reaction time).

where,

 

 

me...) =F(0..)+ eB(6..>+e"B(e..) (c— 3. 1)
F(9m)=f.ss (CZ-3'2)
P (e )= (I-fss) [(53 ’A’) “We *A(’*Ca)5"""9o] (c- 3. 3)
l O A (C3’+A‘)
. . 2A“
P2(6a.)= C;+4A’ 1,(c,3;nA90—As.n2AeO+-§)

+ 1°(c,ca5519b +1154) 2110., + 35’?) + 1,, (game, ’ZAcaSZABQ)
__ 12!; (”2:3 -AI) (13342118, -24 6.03211 0,)
4Léééiégsmzna,-2Acwqug
_ W) (c, as?! a. +A51h2/19a“ 3211;)

. 24’
+ -—-”Af;‘ (c,w’/16.+A5'"M9a +2.; )] (c—3.L))

Since equation (C—2.1) is structurally similar to

equation (A-2.1), an analytic solution is again not possible.

In order to obtain an approximate solution, the following

asymptotic sequence is assumed:

Xz(9a)=b.le.)+ eb.w.)+ flaw.) (c-A. 1)

where bo=non-periodic term

b and b2 = periodic terms

E<<<1.

116

Substituting equations (C-A.1), (C-3.1) into (C-2.1)
and collecting terms of the same orders in 6 yield the follow-

ing three differential equations:

 

b0 80) "
AOL—9(a— + (1+ Kz)bo(an)=’<'fs5 (C'5'1)
A @3119.» o ‘
dig?" +(1+1<.)b.(9.)=A (“(91) "17‘ss*’<zbo(9w)) ’5""A0obo<°a>*"'3‘5a) (c-5.2)

 

Q.15";44121416..)-“5"9‘1““‘d‘:"”--1<.m.)+1221:1619)“5""M"”'“""’*""”A(51») (c—5.3)

It is obvious that the integrating factor for equa-

- . , (”Kt) 8a:
tions (0-5.1), (C-5.2) and (C-5.3) are identical, 1.6.(1 ;
Notice that there is no need to specifiy the initial condi-
tions for above three equations since at the limit cycle,

the constants of integration vanish.

Solution to bQ

Equation (C-5.1) is readily integrated, at 6,. , bO

yields the steady state solution, or

mi. (046.1)

:5 = ,.
5° ’5 MK,

Solution to b1( 6..)

 

Equation (C-5.2) can be equivalently written as

dbl Cos/19’

EH! +k1)b,"-- “(fizhs'K'fis)’s'hA9°.b‘-$

+ KM,[(6,713)C0$A6¢+A(I+C4)Sin110m] (C' 7 - 1)

117

the integration is straight forward and the solution is

found to be

A.(9.)=[1<. w, (9.1:). as] ('*Kz>we.msm,
("22ft A:

 

+[ k: 43A (€34- ’) - bx] (H K031)”; 6,; 11945.4 9::

([+IE2)1+ A: ((3-702)

Solution to b2(e.)

 

A similar calculational process applies for the solu-
tion to equation (C-5.3). For convenience, we define the

following variables

120=[K1‘&(C.5'A1)' 92%,]; + [mg/10¢.) 'b‘SJ I

 

("#5415 (Ii-[21ft]?
b5: k‘:(l+£z) M 3
+I< (C'A’)---—-——"—- (C-A)
[ '4” 3 A ]fl[(/+IE,)’+/lz] A :3
+ the, __ K,¢,(c.-1’)c. + “LU-fat; (08.1.1)
(51* 411’ A ( (3'4 4A”) A (‘3 * 41")

 

A It ’2:
2" (6-3-5— A*C)'b __
11' [ HP; .3 A) A ](/+K‘2)‘+A‘ +[K¢§ (’ 3 55] (1+,22)2+A2
- K1170.) + m¢5(/+C;)(a/1 (C-8.1.2)

118

(I+£1)z+A2(/sz;t
K: (”'22) 2k'15'A + 2141.11 +2KF/I‘ica __ ("7(a)/(IQ)

 

 

éz%%'m1$(‘a-A’)JLLW[K1¢5 (1+:,)- 5“]—— AZ—Kflfow‘g)
1)
[

 

 

 

 

 

 

 

+ 141131<1+s)—b,,] fl. _
A[(/H<.)+Af] C,‘+4A’ ('5‘ +421 C,‘+ 411‘ A(C,‘+4A’)
- K196,(/+z(g-A’)C, _ ZK;5é(C;'/12) + 2K1(I' £15) + 2K1¢50+C§)A2
c;+ 4.42 c ,’+ 411’ A (1;. 4A“) c;+ 411‘
(0-80103)
K 41/ (I+2C.}—A2)-2AK,75-k1(1-1§ )
12,: ' 3 ’ ’ (C—8.1.L))
(‘3’+ 4112
23/4112 56(55'A’) "’ f“ —8.1.
z4= [1,+ I...“ 3‘77" — ACHmM A. (C 5)
(3(C;:+4A z)

the solution to b2( 9..) is then expressed as:

2
A
( A )1 1 10F”K1)C°SIA90+A3I"2A8"+—(,_f—_Kz)]
MK, 4411
F 31‘ 2A9,+ z 2
4 - 3 .. A
.12: (’*Kz)s’"A0° A n ”K1 ]

79(009) =

 

 

.1 121F011 [21)5131 2A9w-ZACOSZA940]

 

12:
J +, . (C—8.2)

+ 115[(1+ 12,)cosznew + 2.43 in 2A 9‘)

119

By simply substituting equations (C-6.1), (C-7.2) and
(C-8.2)into (C-u.1) will yield the solution to X2(9m).

APPENDIX D

AVERAGE YIELD CALCULATION OF

THE INTERMEDIATE PRODUCT

APPENDIX D

AVERAGE YIE D CA C TION F
THE INTERMEDIATE PRODUCT

The average yield of B at the limit cycle is obtained
by carrying out the cyclic integration of x2(9~). The cycle

time is defined as

T27? (D-1.1)
hence,
Qfiq‘
¢X2(eo) = ‘L' Xz(6¢,) (16¢, (D7' 1 . 2)
"r an

From equation (C-8.2) of Appendix C, the cyclic in-
tegration of sine and cosine functionSvanish and also the

terms that contain squares of sine and cosine will remain,

hence
I 1 -1)(IH32_+_2_4:) + ————1“ (D-2.1)
¢Xz(6m);(l*lzz)‘+4fi’ ( 2° 2’ 2 H K: "* ’21

where 120, 12,, I“ are defined in Appendix C.

120

APPENDIX E

THE SECOND VARIATIONAL IN THE FREQUENCY DOMAIN

APPENDIX E

THE SECOND VARIATIONAL IN THE EREQUENCY DOMAIN

Consider a system whose dynamics is given by

x=f(_>_<.u) (E-1.1)

2(0) =3 <E-1.2>

where

_x = state variable, u = a control vector.
Here, we wish to determine whether or not a control vector
.u in a periodic mode (x(6);x(9+T)) would yield a local opti-

mal of the objective function

J =-',':¢e(x.u)d6 (E-2.1)
In this work, g x u)=x2 or the yield of B. Let

J: = objective function at the optimal steady state, hence

for a small variation in u

_ o (E-3-1)
3 ~38 +82

. . _ _ 0
then 5;; will yield a local optimum if AJ—J JS>O. Hence we

wish to determine
Maxm13J
363

my
121

122

subject to the limit cycle dynamics

25 = £25.52) (E-4.1)
25(9...) =2<(6..+T) (E-5.1)
1w hLu)d6 = constant (E-6.1)

Now, by imposing $13, a priori we can relax equation (PS-6.1).

Now, define the Hamiltonian

Hum,» = sung) +AT(9>5<2<.2). <E-7.1>
H and f are twice differentiable and continuous in_xez, peg.

The Optimal staedy state is determined from

 

( H25: = 0 (E-8.1)
J0 , J H = o (Ii-8.2)
=g(Xg .Q :)<<F5 .3:
\ i: = o (E-8-3)
therefore,
J =-.i.'- g(25._u)d9 (E‘9-1)

At the cycle, we assume that A(6,) becomes time independent.

J =7r‘—.9§[H(x.u.2° )- (22>T5(_>5.2>] de (E-9.2)

J°= -+'¢H(2$O .28 °.2°)d9 (3‘10-1)

8

therefore,

. T
J-Jg =+¢[H(x.g.&g)-H(xg.yg22?] de.-(2.9 #:mnm

(E—11.1)

123

but,

=f(25_ u)

Qt Q:
ollx

hence, the last term in equation (E-9.2) becomes

d
fiqameffi-‘Egew 25(0)-25(T)=0. (342.1)

Therefore,

J-J° =1. [H(25 u._S)-H(25‘S’..gl_1§.>.x )] d6...

=§¢[H(X 113552 .l°)-H(258._L1:.6252)]d

29922—3. §x+zi—:L§ %[:Z§Z.,;

 

z__’H
*5st +ij_ 3“an 5X15“ J'

 

 

 

 

 

l T
— ~ h has,D
AJ‘TXVHX LOBEHBE 6.2)] de‘” +“I” “ =“
= BJJX-i-SZJ (E-13.1)

clearly'§J=O and SZJ is a quadratic form.

Here:

11 =[525 5.1.4]

LT [(525)T GMT]

 

 

124

E = [ Hxx(-)5:’B:’A(s))]

LO
II

T . . _ o o 0
Q (a symmetric matrix) - [ quLxS,uS,AS)]

um
ll

[ 151.152.2222]

Now, the condition that gives rise tOAAJ>O should depend on
au as well as E and ax. However, in order to derive a cri-
terion for the sgnQAJ), the relationship between 5; and.au
should be known. Such can be determined from linear con-
trol theory. So, if we perform the first variation of the

process dynamics,

dX _ d.— ___ ' = _ o 0
8dt “ (14°52 53 i(2§9.§) ELISSLBS)
= 2(2.2‘;+22>—2<2<;.2g>
= f(2_<s._u:+zu)+g52< + @512 +.....
o o _
£958,128) (E 14.1)
Hence,
25 = 5325 + 25.1.2 (3-15.1)
where
_ o o
g " ill-{(38938)
_ o o
g ' fB(l§SoBS)

Now, from linear control theory; if

Its-lo .
Where-gk=0 for k=0"£k = a constant, J =\/-1 ,GJ- cycling

125

frequency such that 62 forms a Fourier trigonometric series,
hence the solution to equation (E-16.1) should have a form

expressed by

53 = Z Xkejkwew (E-17.1)
and where xk=0 for k=0.

When equation (E-17.1) is substituted into equation

(E-15.1), the value of;k is determined. Hence

‘0

or,

25 = §(jkao)§k (E-18.2)

k

. . 2 .
with equations (E-18) the second variation 5 J is now eva-

luated, thus,

22J=%§5[<52)Tg(52> +(EB)TQT(6)_<) +<32<>Tg<52> +(811)T§(5_12)] de..

a) w m o
— T T T T +2 T c _
‘Zgfi-kgl‘fz ¥k9-kg 3k+2;#-kggk ng-kB—k (E 19.1)
replacing.xk by using equation (E—18.2)

AJ = $124311 kw)_9k .

APPENDIX F

CALCULATION OF THE VELOCITY PROFILE IN THE

PLUG FLOW TUBULAR REACTOR

APPENDIX F

CALCULATION OF THE VELOCITY PROFILE IN THE
PLUG FLOW TUBULAR REACTOR

Here, the momentum conservation equation leads to

W N _
——— Ifl——= (F 1.1)
at + 33 O

The disturbance in the inlet velocity, thus a disturbance in
the inlet volumetric flow rate, is assumed to be in the form

of a sine wave and is given as

Vfl’o) = <v>[1+c-5inwt] (F- 1 , 2)

where e is a small positive number indicating the degree of

disturbance away from its steady state and a>is the frequen-

Cy of the input disturbance.

Defining the residence time of the reactor as

(....L
-'<v>
Where L is the reactor length. Introducing the folowing

dimensionless variables:

-1: ’=—‘{—o S:
9—? 8 v (V),

A=w’c (F-1-3)

when equation (F-1.3) is substituted into equation (F-1.1),

We obtain the dimensionless equation of motion

126

127

20 75 " (F-2.1)

with the initial and boundary conditions

0(9=°,‘5)=l (F-3.1)

W6.3=0)= ”53MB (F'3'2)

It is obvious that an analytic solution to equation
(F-2.1) is not possible. In order to obtain an approximate
solution, we assume v to have a form composed of a steady

state or a non-periodic term and a sum of periodic terms

such that

0=l7(e=qs>+et'r,+e‘fi, (F-4.1)

where e is very small so that terms with 6 of order 3 or

higher are neglected.

Substituting (F-4.1) into equations (F-2.1) and (F-3)

and collecting terms of the same degree of 6 yield

Zi+flgo (F’Sol)
9D 25

with the following initial and boundary conditions
171(6=0,S)=o (F—5.1.1)
[Z(O,S=O)=5inA9 (F-5.1.2)
21-13 903 - 26,, (p-502)

and 33' 33 '23

128

with
initial condition 171(9=0,S)=0 (F-5.2.1)
boundary condition 1?;(6.‘S=o)=0 (F-5.2.2)

Notice that 0th order term of e in the equation yields the

steady state velocity.

The method of characteristic is applied here to solve
the partial differential equations (F-5), where the dependent
variable is treated along its characteristic path. As a
result, the partial differential equations for 91 and $2 in
9 and’S are transformed into ordinary differential equations
in terms of the variables r and s. The transformation of
the independent variables 6 and S into the characteristic

variables r and s is depicted in the figure below

Y

 

P

Y 3
FIGURE F-l. Characteristic transformation for the
independent variables

It is clear that for a fixed r, e and 3 can be related

to s.

129

Transforming;equation (F-5.1) into the new independent varia-

’. 1i} air ",
<::>=—'=‘J<‘§'>+ fl<fl =R (F-6.1)

where R is the right hand side of equation (F-5.1). Compar-

bles yields

ing equations (F—6.1) and (F-5.1) we have, at a fixed r,

25

3% (F-7.1)
fl-

2A " (F-7-2)

29 ,
hence a—S-sktanp org-45'

Integration yields

6-60==S-$O (F-8.1)
where'eO and 50 are some arbitrary initial condition for 9
and S , respectively.
Referring to figure F-l, two regions can be classified.
The variable r, in the lower region, represents an arbitrary

initial value for S , while in the region above OY, the para-

meter r represents an arbitrary initial value for 6. Mathe-

matically, the following relations for each region can be

written as

Region below OY:

9.2’6-81'np (F-9°1)

3=r+,5-ws‘9 (F-9.2)

130

Region above OY:

9=T+A-sin§ (F-10.1)

52AM? (F-10.2)

As far as the limit cycle is concerned, only the upp-
er region is used in the solution to 5. Furthermore, the
initial transient (i.e., 6(3) is washed out at 9=’C (or
reaction residence time) and does not provide information
for the solution at 6”. Hence, only the boundary condition
(5:0) is significant at this region. It follows that;O =0
and GO =r in equation (F-8.1).

As a conclusion, the foregoing calculations will be
only for this upper region (67!), therefore the integration

of equation (F—7.1) from 50:0 to 5 and sO=O to 8 yields

5:; (F‘llol)

while equation (F-8.1) gives

{gs-3 (F-11.2)

It is clearly seen that equations (F-ll) provide the trans-

formation between independent variables. Here, the trans-

formation of the partial differential equations (F—5) and

onditions can be easily implemented. The

After

their boundary c

solutions to the resulting ODE's are straight forward.

back transformation and using equations (F-11) yields the

solution in terms of the original independent variables.

131
Thus,
air.

36 =0

or, 71(S.r)=f(r). Applying the boundary condition (in terms
of r and 8) yields

17'. (f, 15:0) = Sin/1f

or 17', (if, ,5) = Sin/W

Upon back transformation, the above equation is

17. (6.§)=5£nA(e-$) (F-12.1)

Equation (F-5.1) is solved in a similar fashion.

Substituting v1(r,s) and [335912] , an ODE similar to (F-6.1)
fit

is obtained except, here R(s,r)= sinArcosAr. Integrating

the equation and using the boundary condition at 30:0 gives
V2(s,r). Back transformation yields
— g ' - (F-12 2)
v.(6,%)= —§-As.n2/l(6 S) .

Substituting equations (Fle) into (F-h.1), we obtain the

perturbation solution for V

0(ag)=l+e5inA(9—s)+ e‘%‘-s€n2/l(9'$). (F-13.1)

APPENDIX G

GLOBAL SOLUTION TO X1 AND x2

TUBULAR REACTOR (PERTURBATION SOLUTION)

IN A PLUG FLOW

APPENDIX G

GLOBAL SOLUTION TO x1 AND X2 IN A PLUG FLOW

TUBULAR REACTOjoPERTURBATION SOLUTION)

For the proposed Van de Vusse kinetic scheme,

ALBE-C , A+A—"—’—D

the material balance for A (species 1) and B (species 2) are,

respectively:
fiz-L . - - A: (G—l'l)
at ”(VA) k.A k,

and
715 9 _ _
fiz—é—j-(IFB)+IQ|A kza (G 102)

using the same T'as defined in Appendix F and introducing the

following dimensionless variables:
. -.Ei _ .
x'=_ 3 xiii—S; ; KFkaTi Kz'gkz'ci K5=k3AV4C' K‘ K, (G 1 3)

where A f is the inlet concentration of reactant A. Sub-
re

stituting equations(G-1.3) and (F-1.3) of Appendix E into

equations (G-1.1) and (G-1.2) yields the dimensionless ma-

terial balance equations for A and B as

(C—2.1)

132

133

and

31.2 9 '-
fl 4- 3;(VX.Z)=K1X.'K2X2 (CI-2'2)

respectively, where the solution to v can be found in Appen-
dix F.

It is clear the analytic solutionsto equations (G-2)
are not possible, and in order to obtain an approximate solu-

tion, we assume the following asymptotic sequences:

X.= X.,+eP.'+e’P.‘ (cs-3.1)
1.
X1=LS+GQI+EP3 (G-3.1)

E is restricted to small values in order to have the equa-

tions (G-3) meaningful. Furthermore, x1 and x2 satisfy

8 s
dxu 2 G-H'.1
Z75} ="K:7CIS-K;X:$ ( )
and 33“: KKK—K1705 (G-4.2)

Solution to x1(O.S)

 

Substituting equations (F+4.1) and (G-3.1) into equa-

tion (G-2.1), collecting terms of similar order in.6 gives

the following equations

' I
32-2- + %(Pu'*filx's):'K' P"-2I<3P,1C,5 (G'5'1)

13A

7.1 _ ' _ 1 :1
3&4. 50—5“); MP,+1&X.s)=-KIPI“K3[2P|2x15+(fi>] (G-5.2)

the boundary conditions (from Appendix I) are:

x1 (e,§=o)=1 (G-6.1)
S

Pi(e,s=o)=o (G-6.2)

Pi(9,$=0)=0 (G-6.3)

The solution to equation (G-4.1) is very simple,

 

thus,
doe—ms
7‘ = - (G—7 . 1)
5(5) KO" “0 8 K15)
where “oz—fiFE_' , where K=K3/K1

The method of characteristics is applied again to
solve for equations (G-5). Notice from the similarity of the
mathematical structure between equations (G-5) and equations
(F-5), we eXpect them to have the same ground characteristic

path, thus the transformations between independent variables

are

5-; (F—11.1)

Y‘G‘S (F—11.2)

The transformation in (s;r) converts the hyperbolic

O O O S
equations (i.e..(G-5)) into ordinary differential equation

vflfixfli have the independent variables given in terms of the

ground characteristic path, 8. The boundary conditions are

135

also transformed into (s;r). The solution to the ordinary
differential equations (in Pi and P?) are straight forward
but heavy handed, unfortunately.

The transformed equation in terms of ground charac-

teristic for equation (G-5.1) has the form

61?, 2d éKJ d 644.: K“ e-m
-—' I+—-—L—- 605A r———:——+s‘mr—'—3—— (G-8.1)
M +K’[ I- “01.9.42 A I<(I— we“) " I<(,—a."")‘

some clever manipulations are necessary on equation (G-8.1)
to Obtain the right integrating factor. Also using the bi-

nomial series expansion, the integrating factor, I, is
(G-9-1)

Completing the integration, and evaluating the constant of

integration using (G-6.2) yields,

 

'KM

! e “GA 4“ 8'“ do +Kld016 _____S‘. [W

Pl (Alf) (I‘doe K,’)3 (K K
back transformation yields
—KS

' 3 4° CosA(6-$)+KISIM(9‘S)]§

,(3,9 .. —[A
f’ .)= O—M ”m: K

-K,
- 93: _AwA(e-§)+ fig/1605MB S) e ( G" 10 . 1 )

A procedure similar to the above can be implemented,

2 . . . .
thus. in the transformed P1 differential equation.

2 am 212(05) 1%.
L8 4’"! “+2 K115”), r170.“ )HAT. -IW’ )[ 2; L; ”:2“; 451“
A»

_x .56.» .[Jééai’] - K,[P.'W)]z (G- 1 1 . 1 )

AX

136

comparing equations (G-11.1) and (G-8.1) yields an identi-
cal integrating factor for both equations. Therefore equa—
tion (G-11.1) can be readily integrated. Defining the

following groups for convenience:

(6': %€[AwSA(e—$)+K.anj\(9'$)] (G-12- 1)
a.” ,
(62: ——K;AcosA(9 S) (G-12-2)
o<.’ ,1
35: Elias/HOG) (cg-12.3)

with patience and care, the solution to PE was found. After

back transformation, the solution to PE is expressed as

«.5
P.2(fi5)*( e «.s. -sinA(9—s)[IP,(9,s)+4;(as)+@(35)“mama/#35)
I’doe
-HP65(9,$)] .
+A wsA(9-€)Ip.,<a,s)+ W,(as)+¢>,(e,5)
fig [42,- (6,5)+Ig,a-(6.S)+ Igzj(e,s)+ 1.932(96)
+5Q4j(915)*‘I°/;a‘(55)] (G‘13-1)
where ,_fis 2
2 d 1 01(6 ‘4) 03—92. _ .
13(33)=A$I'nfl(6"$)[-21%--2327§3—“ K3] (G 13.1 1)
Ip.(0,s>=Aw$M9'5> (G-13.1.2)
°‘°’<I - -.4. - . ‘ (G-13.1.3)
IP:(9.S)= T[55‘W\(9 ” 2 awe 5) 5]

«.1
W4(e.s>=-I<.[-glsz—%(€ —I)+3.S] (Ii-13.1.4)

137

 

«.5 ..
LP,(0.1)= midi-KR '(‘56 fig :14) $3 4193‘) gig“; )]

K .(ew-j)

 

- 4%ffi
«xx-I3)! e L, 3, 6190.3);
6 "KI g —-
'9‘“ 1): 3;,”[14668 + KAI-13') )*k.(1+y(e
I2,<a.s)= - Lia")

 

 

, KS
lPo(0£)= °(°K'A 35-32211 (649+ “K ”A 6052MB '§)[SeK +

NH; 13(4‘

 

Laws): - 53-A-SIn2A(9-§)[z+é (e 4] + -z§ooszA(9- s)

4“” )3
2 1 3 2; ' Z 2
mug-(6.5)= g, :('*3)°‘o[ KW) (5* km) Mup) K’O‘fl’]

 

 

 

e-Mflfj);
lpflj (9. $)'—' 7% 31 ;%(’*J)0‘°[“— K (112') (5+ W K! 2110—73.)

a . e—Wjfi’.
Lag-(95):: %; 3200”)“: I 100*?) ]

 

 

 

e'KI(2+a)S
I
I8:j<6.€)= =1$3aZ"*J""[ new (5 WI) WW)”
1 6.4401335,
‘8 10. =2 . ’0"[ - T
‘2 S) 8 3.51220 (”1 K. (2ft)

 

1 845.071)! 4
905/”): 3’ 1:01“ °[ K013) ]

J

(G-13.

)(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

(G-13.

Substituting equations (G-7.1), (G-10.1) and (G—13) into

...s

-5)

.6)

.7)

.8)

.9)

.10)

.11)

.12)

~13)

.1LI)

-15)

equation (G—3.1) yields the complete global solution to x1(9,5)

Solution to x2(9.5)

 

The solution to x2 is done in a similar fashion as

138
for solution to x1. Substituting the global solution to x1

together with equations (F-13.1), and (G-3.2) into equation

(G-2.2), collecting terms with similar orders in e yields:

dis
T; =I<x., “2X2: (94.2)
it? +d_P3
A? +‘9—1'SMA(9’g)i—:' 23‘K1P2+AW(9 $)XZS+K'PI,(G-1Llr,1)
AV}: Ly}; 4—P2' ‘x 25

43 4-” +Sm nA(9- S)- 6+ +-—5sin2A(9 S)-—-— 45

"'- 2. ' A ' - — 22 ' X *KIPI
- k,g+AcO$A(9-$)P,-[-Z-Smnle$) gamma] 1, .

(G-14.2)
with the following boundary conditions:
x2 (e,3=o)=o (c—15.1)
s
P:(e,$=o)=o (G-15.2)
P§(e,5=o)=o (G-15.3)

For convenience, we can rewrite equation (G-7.1) equi-

valently by applying the binomial series as

flow 1:<,.,e)"""a (G-16.1)

16.1 ) into equation (G-h. 2), with

‘25
the integrating factor of e .

15.1) yields the solution to

substituting equation (G-
The integration, with the
condition given in equation (G—

X2

53

139

s ' if KzzK1(1+j)
-K’gm '.
7C (9:195 «:3 G_ 6.
25 K jg, [ K2‘K:(‘*1)] S ( 1 1 )
e .—

4—— , otherwise

k:"‘: (”1)

 

The solutions to partial differential equations (G-14)
again apply the method of characteristics, since there is a
similarity in form of differential equations (G-1#) and (F-5),
the ground characteristic transformation, i.e.,(s,r) can
also be applied here, with the transformation, the following
ordinary differential equations for P: and P3 in r and s are

obtained, respectively, as:

 

l I
{at
fl; 7.. . 73h“) .. £1 ' 5.725] r. ’(AJ
M +19% --SMA’[_3filx 2A-Sln2AY[ 45 A: AW fl. )
_[ 13-5.": mr- A-A‘cosznr] 7615 (m) + “810:0 ( G’ 1 7 ° 2)

a;
The integrating factor for equations (G-17) 1s 8, .
Theoretically, the solutions to equations (G-17) are

Simple to solve. Upon applying the binomial series and

other manipulations, the actual number of terms in each equa-

tion increases as integration is carried out.
nd imposing the condi-

. 1 .
tion that P%(s=o)=o , and then, back transform1ng P2 in terms

Integrating equation (G-17.1) a

of 9 and 3 gives

1ho

 

I 4‘23
R.(9.s>-e (¢.,+¢23-*¢.J) (G-18.1)
where,
KS
n [We gszSm/MO 5)](——-§)—§.‘nfi(o “Me _') ,lf R lf-(O
gb,.(9.s)=—— {423a,
3 [WE-5)“?K35MA(9 3)];- - Sub/[(G'S) , otherwise
(G-18.1.1)
3.34313 ’ ' if R1:0
”(9' 3)'K'Z('*3>°<°3 (G-18.1.2)
3:0
&9
Z'IIe &(5-l)+RJ,J%(e q) r otherwise
8-K: ‘
' I if R1=O
. if R2=O (G-18.1.3)

fice. $)= M35; (ijxo

I otherwise

with R1=K2-K1(1+j) and hRZ=K2-K1(2+j)o

The first order term in the x2 solution dose not con-
tribute any information about the enhancement of x2 at the
limit cycle. This forces us to calculate the second order
term, presumably some insights can be obtained.

The compexity of the PS solution is directly related

to the number of terms in P: and Pi. It is found that the

2

single summation terms in the P1 solution generate double

summation terms, i.e., expansion involving binomial series.

In a more compact fashion, define the following terms:

lfil

I

do 8 ~
3, = 7(— [jl SnnA(9—g)-k.A comm-3)]

(G-19.1)
32’“ $112 sa'nMe—s)

: (G-19.2)
351g Sn'nMevS) (919.3)
R1=K2—K1(1+j) (G-20.1)
R2=K2-K1(2+j) (G-20.2)
R3=K2-K1(1+i+j) (G-20.3)
R4=K2-K1(2+i+j) (G—20.4)
R5=K2-K1(3+i+j) (G-20.5)
R6=K2-K1(u+i+j) (G-20.6)
R7=Kli (G-20.7)
R8=K1(i+1) (G-20.8)
R9=K1(i+2) (G'20°9)
R10=K1(i+3) (G-20.10)

Integrating with great care, and after back transforming, the

resulting equation for P3 , in terms of the original inde-

pendent variables, is

2 4‘25
P2 (6, S) = e {swim-QM; (6.5)+0(21' (9.9ng (9.5)} $15!?! 2A (9-?) “@189
+ A comm-9M; (Rindqlflfiydyj (615)14‘0‘0; (BISH H0§g'(9.5)} (G- 21 . 1)

142

 

 

 

 

 

where
5
( .5; [(A’—K{)sinA(a-;) -2K1Aco$A(9'§i +s‘[AcosA(9-€> +K25M(9'S)]
' ~gsrnme—s) . if R =0
“1‘95"? K “3:4”?4 1
£5 2
e -l S
\k-I-IKT-R—p -i)[(A——K1)Sm/1(9' $)~'2K2AC05A(9 3)]
+26”,z g '- . R" .
S)[AC¢SA(9'%)+Kzs.nA(e-$)]-(e «Mme-S)
, otherwise
(G-21.1.1) {
L
(%'-I< 53 I 5’ - -
' ’3')? +<3:+Zz-Kzg,)-; +325 ' 1‘5 Rfo
093mg)— «2:00.31...
5 65K. 2(efi's...) K05 ] '
I’;[($u- HEX-z“! +) -T]+$:[€ (5‘EV75J
+32(€R "')* (32” K232)eR[R “fl ’ otherwise
(G-21.1.2)
l , ePS-v -n$ .
f I. (as-K22.)(T+s)-$a<e ’0] '1f Bro.
I '5
«31(9129 kagoffldo 3! S[%3+ (SJ-«1%) I] a if R2=O
{15
I ‘25 ~
L583“ ")+(33I-k‘3’)[e}21 ' —g]] , otherwise

(G-21.1.3)

143

{($531}; .ifR =0

“43(915)’ —;/:0(+3 fl

 

Kl;
I R" | 2
k '12—»[8 (5.73)), i. -Kl[%l- (ng-L'y k1?- is :0 ,otherwise

(G—21.1.4)

u|m

“4-3 (9, = —— “20m

 

 

 

\Ammqynamw—s) en: "E— S -s'nn(e-S)(e fl")
Kn Kl: R1 .2— 2

otherwise

(G-21.1.5)

‘5’ 2 if R =0
r&7§*32%7 ’ - 1

'° é
d5j(915)=K3: (”jflo é

 

g R5 zflféO $19”_g’ ,otherwise
I, _—e( R+v)————-—-+ Ru

 

K, R.

(G—21.1.6)
49$
. =0

/ N(% W)‘ ,1f R1

°" . 1 2 ° R =0 G-21.1.7)
“75(9,S>=K';°("1)°‘°334 '2i ’1f 2 (
__ fie :—' ,otherwise

     

RZ(

mu

1 I . A
SA[Z-szA(9*$)-35WA(6-§)] , if R :0

dam) --—_Zo< ’ ‘

330 I A 636 em 52
E,[;&h2A(e-${‘_K—L-s]-AZ%COSZA(9J5)[ (3’ R”) El-E—H '

otherwise

(G-21.1.8)

49.3(95) =°<wj(015)+0<uj(6155* rosin/1(6- 9592;]- (6 S) + mag (96% 094.3- (6.5)

F424!—

+ «,5»,- (9,3) +00% (6, s) + 097,; (96)]

+ got [d M (9,;)+ 009,3 (9,§)+o(4,l (9,S)+0( 2,7(6/S)+dpd(9,i)+0‘37 (9,3)]

(G-21.1.9)
where
T.(9,$)+Tz(e,s)+1,(e,g)+ WM) .. if R1=O
a, .
- - 1
moi-(agrgmm
T%(elg)+T6j(elS)+ 771/6:5)‘733' (915)-ng (9,5) , otherwi s e
(G-21.1.10)
with
T(91$):~5do$lm(9‘5)[9hm0~$)[(‘E’1% 2;:me
K's).
+mm.s>I—,,—z:‘- - ' f’flfl] (Hi-1.10am)
J: E G-21.1.10b)
Meg): 5 [HSs'nMa—sHAcosvagfl (3,, ——+ /< +23) (
2
K' 5 £2. G-21.1.10c)
73(915)=OLA‘33'5""2A(9'3)( 77% -' 37‘- + ’9 (
5‘ do (G-21.1.10d)

mas): 54.21160: 2AIo-s>( '27?- 2,72,)

1145

1 M

e A’ e“; —-

(:k; ‘:T(5-i;>>(%- 3%)] x,

WM i"?— : i. K. .gt Mi)
(kRn (;-E)+KKII)(I- Z) - ZKRI - [(le ]

 

(G-21.1.10e)

 

 

 

 

 

RI;
‘@ 8 2; 3
753(Q$)=[K'Sa'nA(9'S)*A¢U$N9'3):| ‘2."[1—(71 1 R, ;,"3)"€?]+m 9 _TR(e 4)
eRIS
“(2,34 (5 12.)“? ,} (G-21.1.10f)
an! 2
773(64):“ 2“,? [—‘$a'nzA(9 --‘§)+A0052A(9 J$)][ ——($“:; -— +75)- %]

(G-21.1.10g)
2 6R6” p h)

723(9,§)=%° [é— co52A(e-S)- é srnzA(9~s)]—,;,,— (J’ 21- 1 ~ 10

$5

. _ «ammo-9r e _,_;__ L ,

793(9’S)- 2K L K! (6 R;)+Rf] (G‘21.1.101)
(45W 6

_ 8 " »o(9,$)_ dbl/‘2 _ ’Kl . _

( x. mg ”52"“ ”[9 (9%,)13 . 1f Rl-o
“113(94)::(prfiag toms) 3+“: coswe~5)5 . if R2=O

3:0
M 22 14s

8 4 d e .

\Tme) R, + :3 Comm-9b?“ -é-+-—] othermse

(G-21.1.11)

with

1&6

 

 

Twmsk Sf‘nnw W34 K K '55-"N9 9- ifcowa -.3) 3,]-Acom(e 5),; $3
o< N171
WOT —-'C«2A(9 $)--— K’sinzMe- i) (G-21.1.11a)
S oif R120
1; L» 2
, , ( ). d. , G-21.1.12
093(65):}:“1 3:: [R Fe '47] M ( )
6 -| .
—ET— ,otherw1se
5 .if R1=O
“M3168)=ZCHjfloZd:fl[—g—}+£3 3;] (G‘21-1-13)
1-0 L30 ‘3 R9 +R6 RI; .
___. ,otherw1se
RI
427;
C I 3' '2; 2 "R
(M. K1 (€+é'v)*/z,='xz] ”,7“ 19'? 3 (e ’9
if R1=O
(as): —Z( 5—5-23! Hi if R =0
d141,; ’ 33° "3)“ F4 2 5 R7 +3195 3

 

352
\%,[e—R—;(g -E';+) R'M] 3'1 (8" ____'_E) 1133‘” __,)

Ks
otherwise

(G-21.1.11+)

0(I‘i7(91$)= -X(H”d.1Z—R:J i}; 91f Ruzo (G’Zl - 1 ° 15)

?’° is!

 

\ 2(3’2‘59 ,otherwise

. m d C?!
“I61"(als): -:(Hj)dzz -.—'
a 3:0 in R8

. w 1
“17,2“ (61$) = 'Z("j)do
J

0997(9'5) _
(50

i=0

4

147

e. M- l

 

5 £3

“9

0 - a i
=Z(Hj)a(]Z(H'i) %<

Z! 4253 , .
(E[é,-e (5+fi-a)]-—R:-(%+gz) , 1f R1=O
;’ , .
Z,;+ §(%+gz) , 1f‘-RL,=O
R45
K43
\3, [£05749 1' é] +(,-§-;+zz)(ek;') , otherwi se

 

(G-21.1.16)

33 '55 . _
Hiya! -a)'1f Rl—O

- = (G-21.1.17)
,1f RL, 0

4 353

 

£
\ %(€ 75:) ,otherwise

if R1=O

2. ‘52 5 2 S J_ z . _
5%: (;+,2;+zg)+zz:525(z+g,)+zzs .lf Ru-O

5:3 K}!

e “3.5+; 3... £2. ,1.

K3 [2: 124 £491? +16%“ fly
as as

2 8 ‘0 8 _1_

*‘EZ’J*’3‘3‘[Z“'i>*/zz*

2
KKK:
, s
81%;” ]+32 (8&4).
K;

otherwise

(G—21.1.18)

1
I

 

 

148

“mam: ‘5): ‘Zdzofiéo-{U (14°— (‘3'71’2'

”it

(

o(10;j(9,6)= 42W. Zw— R—° { 23.39% 3+ it) + 231335

PO

 

K5

0917(915): 5: ('fjfloagofl) %2K

984K444;

 

Km
092.4,(919-442("3MozflW—35 s

 

 

em-I
Rs
4 5
w 4 1'0 d: z
095.} (ens)= {awa- ZW— ’33
3.30 4:444) RIO AS
8 -4
R:

%+3€) 44

6 S“; +—-.__._ 9'4”" _2*a -293
79)”; R9 2:, 8-9

,if R =0

1
(G-21.1.19)

,otherwise

if R1=O

, if R5=O

\ [% 8%?
2 (5 1+ ‘
335 72;) R 4 444

 

otherwise
(G_2101020)

if R1=O

(CT-2101.21)
otherwise
if R1=O

if R6=O (G-21.1.22)
otherwise

if R1=O
(G—21.1.23)

otherwise

Substituting equations (G-16.1 ), (G-18) and (G-21)

into equation (G-3.2) yields the global solution for x2(6.3).

APPENDIX H

ASYMPTOTIC SOLUTION TO x1 AND X2

PLUG FLOW TUBULAR REACTOR AT Ow

IN THE

APPENDIX H

ASYMPTOTIC SOLUTION TO X1 AND x2 IN THE

 

PLUG FLOW TUBULAR REACTOR AT4§Q

With the closed form solution to x1 and x2 available,
long time behavior of the concentration of species A and B
can be determined. This is done for comparison with res—
pect to a steady state (i.e., optimal or non-optimal). Here
it is assumed that at infinite time the process dynamics is

a limit cycle. So the comparison should be the time-average

at the limit cycle, i.e..

1
I"? 3 (19..

(The notations are described in chapter 2)
Recalling equations (G-7.1), (G-10.1), (G-13) and
equations (G-16.1), (G-18), (G—Zl) from Appendix G, the in-

dependent variable 6 appears only in the trigonometric func-

tions. Hence, these equations at 9. yields the exact equa-

tions with 6 replaced by Be . The length of each time pe-

riod is €%f for both x1 and x2 equations.
With the asymptotic form for equations (G-7.1),
13). (G-16.1). (G-18) and (G-21), the first

10.1) and (G—18)) are all composed

(CV-1001), (G—
order terms (equations (G-
of only sines and cosines. Their integrals over a period

1A9

150

vanish, and hence no information about enhancement can be

provided.

The second order terms,as shown in equations (G-13)

and (G-21) contain sines and cosines which drOp out after

time-averaging.

However, those same second order terms in

the perturbation solution also contain squares of sine and

cosine which do not vanish upon time-averaging.

In addition

to this, there are other terms which are 9 independent that

are non-zero after time-averaging.

Consequently, the con-

tribution of these terms in 62 makes the final solution to

the time-average x1 and x2 different from their respective

steady state solution.

The solutions to time-average x1 and x2 yield, after

“K.
simplifi cati on , (with ARG= l-doe

 

 

'KS
_ d.4
Xl($) Mq—[K+e Are—1(fll "alb- -9304’KJMD‘JI
where
12: lag—KM ___;(___ Kc; _,)
44K‘am do ’34”; 947;"
I 0 *g?’ e 7
< 7 )
.0 4 Zktiuollf «0&(8‘K8; + e. 'I
3' 2k 5.. K: S )
.044 =Z('+‘)°(° (125i —_a‘.-+_a-,, +123; qu-flm)
cm
with

 

 

for convenience),

(H—1.1)

(H-1.1.1)

(H-1.1.2)

(H-1.1.3)

(H-1.1.4)

(H-1.1.5)

Sfi_f'i-§‘

151

_du1
4i KaK

_o(_q_A 4(948 ____")
2 K: £8

83
4-094 4: do]? K” e
kak

x[ 9:4: (5f 72:) 2;]

 

     

 

4.4, ”7]

4a 44 «4.4,. 84271,
( ,2, )

 

9t .— K43

— 1
«.A‘ (e 29")

 

.0,“ 5;

and 4
S , if R =0

, otherwise

where

,0 [,[éf‘i—L 1.2%)- ("—2—- >

I I;
at.»

@223“ Kg20(__92:(,+4)d4
3:0
72-[ 2H6? “033:?

 

 

(H-1.1.6)

(H-1.1.7)

(H-1.1.8)
(H-1.1.9)

(H-1.1.10)

(H—2.1)

, if R1=O

412.5
T],otherwise

(Ii-2.1.1)

, if 31:0

30] _[e&S(S__4'_)+_4L]],otherwise
I I

(H-2.1.2)

 

 

152

 

 

 

 

5.? 5.5..
K“ 2 ( 9 D . if R1=O
$313: T‘K.zz(+3)d°1
I K K" 2 25 2 2 2.;
EH‘? <s-z+z,z>-z:2]—[e (5-22.23.
otherwise
(PI-2.1.3)
, S , if R1=O
I.
‘64.1=O(ok.z(Ha)doZd. RS (H’Zolou‘)
K7 I
3R" , otherwise
. g , if R1=O
M
5-4' -d°K——-'—-:(H3)doz 0“ (H-2.1.5)
{'1' L‘0 K: 6&5"
Ra , otherwise
‘7;
—2(6 "-J 55275 , if.R1=O
K7 37
_zdoKa an s _L , = _
593-2 —-Z( ng42— [uh—m) , 11" R3 0 (H 2.1.6)
’98 “>5
LL R(S_§’5)+— -—,’+§_’_'. ,otherwise
4,5 -
_2(e —:)_ 56 2’5 , if R1=o
( KI:z K3
2 0 D ‘2" 3
5-1%, and] 0.5;. 45+“) . = _
P7? 1K% 3 gfle 2 R3 , 1f Ru 0 (H 2.1.7)
1'45
L_%:g(g-i)+é+is " , otherwise

153

““3 nu, v2;
(5+2'")+.4.- 56 245 22-021

22:82 flake x,‘ g;
ifR=

 

 

 

 

 

. 2'; 1
Kai—AEKV 3” S3 51; 2; 2(62 _—|)
2222232222 “‘2 2212222222222§22222E 77:2? £51. .if 124:0
’36 em
i<§i£_£z)+2 _(§_J_ _) 2
22‘ R4 A To: R‘ 2212:0124
+ 2(ek‘g") 2C6 222-1) .
T“ , , otherw1se
Rah z”.
(H-2.1.8)
"-333
s g -
(_2%__)__(€Z 2+1) , if R1=O
a
5 2‘” 2 ‘2 RS
_doA +_9w‘0&_g g, L—I . :0
67,3.- K 320mg Ks 5(379) K37. , 1f Ru
5
‘8" £1.22: .
2%(57'2; +4272 5/24 Rafi, , otherW1se
(Ii-2.1.9)
42$
_(igl)-; if Rl- =0
/ 12,,
(€222; )
.. . - -
fioi—“j‘z’éZcmmZOMZ 222? , 11‘ 34.0 (H 2.1.10)
[ s 12.8
\8 -' _C -l

 

R4 R , otherwise
I

   

 

 

 

 

 

 

2(6 I) §
- 1 "_‘(e +IolfR—O
2 £9 g, ) 1
A” 2 i KS
do 5 1. e 4
. (”90“ (Ht)—— §(——+ - . = -
fi’dzd’i Kf. 2:; .1492 2 I.) M, .11" R5 0 (H 2.1.11)
‘65 m e,
Kr i2; 752,294-13}? . otherwise
"75
I .
-——- , fR O
( K9 1 1
0‘42!!! ‘
g2 =2 0222—222 (’ 2)0( L ——J 22-!
2225...:1ZQ2'3 3;:("2)fl; '3’?— , if R5: (H_2.1.12)
RS. 2,3
2 2 -e “2, otherwise
R7 RI
C'RIDS
.: . _
__Z;_n.3 , if R1-O
_L ° eflgS'
0“A 3 - ' = ..
€0,3- 2K, :(Hgfiozmfl-o-C RID 3-7 . 1f R6 0 (H 2.1.13)
&3
8 -4 6&i4
g‘ ‘T , otherwise

The equation for average conversion of A can be

readily obtained by subtracting equation (H—1.1) from unity.

r- “wk-11

APPENDIX I

2

EVALUATION OF BOUNDARY CONDITIONS FOR Pi AND P1

APPENDIX I

EVALUATION OF BOUNDARY CONDITIONS FOR Pi AND Pi

In the succeeding derivation of the boundary condi-

2

tions for Pi and?1 ..all perturbation terms are evaluated

at 3:0.

f““‘““' - -e-.[

From Appendix G,

X.(3.6)=X.5(6)+6P,2(5,9)+ e’F.’(S,9) (G-3.1)

At the reactor entrance, only pure A is present,

x.(3=o,e)=¢.o= x3). eP.'(S=0, 9) +e’P.’ (5:19)
[when there is no perturbation, i.e., €==O, we recover
795 (0) =1]
o= (4720.9) +6’P.’(0.9) (I-1.1)

evaluate equations (G-S) at 5:0:

2911+” 2E (-)<.'+r)5m/19 -AM$A9=-(K:+2K5)Pv2 (I—2.1)
3

z I ' z 2 -
3g+g§l+smnegg—p,onsA9+§-lsmm9=""afi""Ka[1Pv*€Pc2)2] (I 2'2)

evaluate equation (G-2.1) at S>=O yields:

(1+ wig-r1119) :21"(A3_2222 5‘"ZA9— 611609119)- - —;<, K, (1—3 . 1)

Differentiate equation (G-3.1) with respect to 3 and then

155

156
evaluate at 5 =0, thus,

UXI 7‘15 6262622182?
5‘5 3* ag“ 25 (141.1)

but

2
I}? = "K1X15- K3145

ax . .
substitute this into equation (1-4.1) for 3a? and multiplying

the resulting equation by (1+6sinA9), thus

P2

(l*€&nAe)—g"(K.+k,)(l+és:h/19)+e(He-5.21119)23§2222+€('*€'$5M9)%J2; (I-th-l)

Rearranging equation (I-3.1) and equating to (I-N.1.1). Upon

neglecting terms of 3rd or higher order in e , thus

1 . 3P2
"’(Ktth) ’6 gSmflai-GACOSAG =- (K1146)- €[(K'*"3)§mAD "" 5—5']

+9252" “2232222“? 7222) (14.1.2)

Collecting terms of similar order in.€, we obtain

2P2 2?.
—{2i312n2A6=SuhA05'$'2+ 3g (I-5.1)
aP.’ .
AMA9=-(K:+K3)5fn/le+ 5:5- (I-6.1)

:> 26% =AcagA9+(/<,+K9)Sa2nA9 (I—7.1)

Substitute equation (I-7.1) into euqation (1-6.1). thus

722‘:- A 9."..er - (KHL HEW/10 " A92" “-9 “5A9

— i
-

157

or, 222?,

——--(K.fK,)5f-nA9 AS'2’12A0 (I-8'1)

Multiply equation (I-2.2) by (-6) and then add to equation

(I-2.1). Yields
1

9' 2
,g- ""(K.+K.)5f—nA9 Ame-mesmeg-SPJ-egg ”5211051194125":er

=~ (k.+zk.)R'-*el<.n +6K.[2B‘+<P.')’J ”‘9'“
substitute equations (I-7.1) and (I-8.1) into equation (I-9.1),

éACosAO P.’= ~(A+zz,m'+ e K. R’+ e [9082+ (Pb’J

but from equation (I-1.1).

P =-€P

1 2
1 1

, 2
Amman“ = (mm P. + BK»? + 679??

0: 320211on + 1mm. + 6% F3)

thus, P§(O,6)=O and Pi(0,6)=0

or,

éAcasA9+ K. +619

226.6% - W
3

 

This root is disgarded because the periodic functions

do not explicitly and implicitly contain é dependency.

APPENDIX J

EVALUATION OF BOUNDARY CONDITIONS FOR P

1
2

AND

P

2
2

APPENDIX J

 

EVALUATION OF BOUNDARY CONDITIONS FOR P: AND P:
At the reactor entrance, x2(0,6)=0, or
12(0. 9) =X2,(o)+e g'(o,9) +e’P:(0, 9) :0.
from Appendix I, x.5(0)=0,
=> “X.5(0)=O.
1 2 _
=> P2(O,e)=-eP2(O,G) (J 1.1)
2?. e2?. (J-2.1)1
-—(o,9)=-€ ——-(o,9)
Evaluate equation (G-2.2) at 5 =0,
LX91+(‘+GS'°%EnA9) 23K| (J—Bol)
99 5
. 1 2 _ th
evaluate equations for x%;' P2 and P2 at 5 -0, us
dxn._K (J-4.1)
T
I
3?: ”2‘2 m, ‘=-K ._
5.5 game 5—9 .F.2 (J 5.1)
2 2 ' 2 l
{’32. . 2.". .s.-..A9}‘;’=— 19?, +AcosAeP. (J-6.1)

99 25

 

1see Appendix K

158

5‘6 "‘ 99 39 (J—7.1)
I 2
2X , ZEI 293* _ .
fifik|+e35+e as (J 8 1)
212

But equation (J-2.1) implies that 55 =0, thus,

(HGSmAG) 3 =)<.

substitute this into equation (J-8.1) yields

Bf: 93' 3228 2F.2 ..
l I e I +€3 5"S'IA
KGSHAQ-t 3‘) +€$nA9—- 23 35% 9“ 35

collecting terms in like ordersof e ,

g—E=—K.s;nzxe (J-9.1)
2

9.3“ =Ksin’Ae (J-10.1)

25 ‘

Multiplying equation (J-6.1) by 6 and then adding to equa-

tion (J-5.1),

9‘5

substituting equations (J-9.1) and (J-10.1) into the above

€L§+o+e§1nne>~ + K~SmA9=- K272 €K2F21*€A¢°‘A9' P2.

equation, yields:

1 2 I
6K.Sin"A9 - (H es.'nAe)(K.s.'nAe)-+ K.Su'nA 9 =—K. P, -ek.P. 4 e/LcosAoP,

or,

I
0= "(1%! “ é szza'tGA-USAQ' P1

1_ 2

160
I
o=eACISA9-P. (J-11.1)
T . . . 1_ _nW _
his implies PZ-O or .AQ—-§- ; n—O,1,2......

To sufficiently satisfy equation (J-11.1) for all 6,

P: must be zero. Hence,

P:(e, g=o>=p§(e, 3 =o)=o

APPENDIX K

PROOF OF THE DERIVATIVE OF THE BOUNDARY

CONDITIONS FOR P; AND Pg

APPENDI X K

PROOF OF THE DERIVATIVE OF THE BOUNDARY

CONDITIONS FOR P: AND P:

 

 

I
For a linear partial differential operator, 01H) L .

P,‘(s.e>=g.m3,<9) (K-1.1)
thus, at 5 =0,
P,'(o,e)= 2.00%.(6) (K-2.1)
and differentiating equation (K-2.1) with respect to 6
I
AP.(0,9)_ A «9) (K-3. 1)
7F"?‘” .9

For equation (J—2.1) to be valid, we follow the proof

given below.

Ame) 453:9»

 

75— : at“) 16
(rue
df’z'wfi) ._._ 3‘(0) .1452 LI,
“,9 5:0 (K’ o 1)

But the right hand side of equation (K-u.1) is identically

equal to that of equation (K-3.1), hence

9P7<°'°’... 2.8:
49 39

 

 

5‘0

161

LIST OF REFERENCES

REFERENCES--CHAPTER 1
1. Bailey, J.E., Chem. Eng. Commun. 1973.; 111.

2. Douglas, J.M. and Rippin, D.W.T., Chem. Eng. Sci., 1966
.21 305-

3. Doiowala, T.G. and Douglas, J.M., AICHE Jl., 1971 12
97 -

4. Ritter, A.B. and Douglas, J.M., Ind. Eng. Chem. Fundls.,
1970 ,2 21.

5- Bouglas, J.M., Ind. Eng. Chem. Proc. Des..Dev., 1967 6
3.
6. Renkan, A., Chem. Eng. Sci., 1972_2z 1925.

7. gandery, C., and Renken, A., Chem. Eng. Sci., 1977 32
#8.

8. Farhadpour, F.A., and Gibilaro, L.G., Chem. Eng. Sci.,
1981 36 183.

9. Beek, J., AICHE Jl., 1972.18 228.
10. Sinéié, D., and Bailey, J.E., Chem. Eng. Sci., 198o_15
1153

11. Watanabe, N., Onogi, K., and Matsubara, M., Chem. Eng.
Sci., 1981 36 809.

12. Skeirik, R.D., and DeVera, A.L., Chem. Eng. Sci., 1982
.32 1015.

13. Van de Vusse, J.G., Chem. Eng. Sci., 1964.19 99#.

1A. GilleSpie, B.M., and Carberry, J.J., Chem. Eng. Sci.,
1966,21 472.

15. DeVera,.A.L., and Varma, A., Chem. Eng. Jl.,1979,12 163.

16. Lee, H., Chem. Eng. 801.. 1977.32 332-

17. Riddlehoover, G.A., and Seagrave, R.C., Ind. Eng. Chem.

Fundls., 1973.12 “A“.

18. Lund, M., and Seagrave, R.C.,
#98.

Ind. Eng. Chem., 1971 19

162

19.

20.

21.

22.

23.

163

DeVera, A.L., and Varma, A., Chem. Eng. Sci., 1979 33
1377.

Bittanti, S., Foranza, G., and Guardabassi, G

., IEEE
Trans. Auto. Control, 1973 59-18 451.

J .
Sinéic, 1)., and Bailey, J.E., Int. J. Control, 1978 _22
547-

DeVera,A.L., and Varma, A., Chem. Eng. Jl., 1979 12 163.

DeVera, A.L., and Varma, A., Proc. Ninth Ann. Pitt. Conf.
on Math Mod. Simul., 1978 9 1425.

10.

11.

12.

13.

14.

15.

16.

REFERENCES--CHAPTER 2

Bittanti, S., Foranza, G..and Guardabassi, G., IEE Trans
Auto. Control.l973 AC-18.451.

Michelsen, M. L., Valeil, H. B.,and Foss, A. 8., Ind.
and Eng. Chem. Fundam.,13, 323 (1973).

Chen, C. C..and J. C. Friedley, Ind.

Eng. Chem. Fundam..
_13, 121 (1974).

Strangeland, B. B.,and Foss, A. S., Ind.

and Eng. Chem.
Fundam.,2, 38 (1970).

Sinai, J.,and Foss, A., AIChEng, 658 (1970).
Chang, K. S" and Bankoff, S. G., AIChEhlé, 410 (1969).

Sirazetdinov, T. K., Auto. Remote Control, 25, 431
(1964).

Kardos, P. W.,and Stevens, W., AIChELlZ, 1090 (1971).

Koppel, L. B.,and Shih, Y. P., Ind.

and Eng. Chem.
Fundam" Z, 414 (1968).

Koppel, L. B., Ind. and Eng. Chem. Fundam.hg, 269 (1965).

Fjeld, M..and Kristiansen, T., Int. J. Controlhlg, 601
(1969).

Skerik, R. D..and DeVera, A. L., Chem. Eng. Sci" 16,
1015 (1981).

Burgers, J. M., Proc. Acad. Sci.,(Amsterdam).g;, 2
(1940). .

Burgers, J. M., Proc. Acad. Sci.,(Amsterdam) 5;, 247
(1950).

Kida, 5., J.,Fluid Mech.,_9_}_, 337 (1979).

Lighthill, M. J., Surveys in Mechanics, (ed. G. K.

Batchelor and R. M. Davies) pp. 250-351, Cambridge
University Press, 1956.

161+

17.
18.

19.

20.
21.

22.

165

Tatsumi, T..and Tokunaga, J.,F1uid Mech.,65, 581 (1974).

gnu—_-

DeVera, A.,and Varma, A., Chem. Eng.,l7, 163 (1979).

_—

DeVera, A.,and Varma A., Proc. Ninth Ann. Pitt. Conf. on
Math. Mod. Simul. 9_, 1425 (1978).

Lee, H., Chem. Eng. Sci.,gg, 332 (1977).
Acrivos, A., Ind. Eng. Chem.,jfi, 703 (1956).

Dionne, E. T..and DeVera, A. L., submitted for
publication.

   

Elli: .

 

V-..