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This is to certify that the

thesis entitled

A HETEROGENEOUS CHEMICAL REACTOR WITH AN
INTERNALLY ISOTHERMAL BUT ESTERNALLY NON-ISOTHERMAL
CATALYST PELLET: SECOND-ORDER CATALYTIC REACTIONS

presented by
KYUNG JIN BAE

has been accepted towards fulfillment
of the requirements for

MS’. eeeeeeeeamsfi.
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Major professor

 

 

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A HETEROGENEOUS CHEMICAL REACTOR WITH AN
INTERNALLY ISOTHERMAL BUT EXTERNALLY NON-ISOTHERMAL
CATALYST PELLET: SECOND-ORDER CATALYTIC REACTIONS

By

Kyung Jin Bae

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

-‘r’5/C)

13¢

ABSTRACT
A HETEROGENEOUS CHEMICAL REACTOR WITH AN
INTERNALLY ISOTHERMAL BUT EXTERNALLY NON-ISOTHERMAL
CATALYST PELLET: SECOND-ORDER CATALYTIC REACTIONS
By

Kyung Jin Bae

A 1-3-1 multiplicity pattern was found for the
internally isothermal but externally non-isothermal catalyst
semi-infinite slab with second-order catalytic reactions.
The ranges of multiple steady states are generally rather
narrow but expands slightly for certain physico-chemical
parameters. For an isotropic and axisymmetric packed bed
reactor with plug flow in the bulk fluid phase, axial
convective heat transport and adiabatic conditions, a 3-1
multiple steady state pattern along the tube reactor was

obtained.

TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . v
NOTATIIONS . . . . . . . . . . . . . . . . . . . . . . vi
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1
THE KINETIC MODEL . . . . . . . . . . . . . . . . . . . 7

THE INTERNAL ISOTHERMAL, EXTERNAL NON-ISOTHERMAL
MODEL OF THE CATALYST PELLET . . . . . . . . . . . 9

NECESSARY AND SUFFICIENT CONDITIONS FOR UNIQUENESS
OF SOLUTION . . . . . . . . . . . . . . . . . . . 13

a. Dependency of Concentration on Temperature . 13

a-1. Variation of Concentration Profile with
Temperature . . . . . . . . . . . . . . 1A

b. Bounds for y . . . . . . . . . . . . . . . . 15
c. Necessary and Sufficient Conditions for
Uniqueness of the Steady State: A

Conservative Criteria . . . . . . . . . . . . 16

d. Van den Bosch and Luss Method for Uniqueness
and Multiplicity of Steady States . . . . . . 17

e. Exact Uniqueness and Multiplicity of Steady
State Criteria . . . . . . . . . 21

e-1. Bounds for u . . . . . . . . . . . . . . 21

e-2. Uniqueness of Internal Concentration
Profile . . . . . . . . . . . . . . . . 21

e-3. The Solution . . . . . . . . . . . . . . 22

THE HETEROGENEOUS CHEMICAL REACTOR MODEL . . . . . . . 26
NUMERICAL IMPLEMENTATION AND DISCUSSION OF RESULTS

a. Effectiveness Factor . . . . . . . . . . . . 29

ii

Page

b. Concentration Profile Inside the Catalyst
Pellet . . . . . . . . . . . . . . . . . . . 35

c. The Tubular Reactor . . . . . . . . . . . . . A3

APPENDICES

Appendix A. Computer Programs and Sample
Calculations . . . . . . . . . . . #8

LIST OF REFERENCES . . . . .

iii

LIST OF TABLES
Page
Table 1 Uniqueness and Multiplicity Criteria Comparison

Between Van den Bosch and Luss Analysis and the
Elliptic Integral Solution . . . . . . . . . . 36

iv

Figure 1

Figure 2
Figure 3
Figure A
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10

LIST OF FIGURES
Page

Effectiveness factor versus Thiele modulus for
various .
3': 10.0, Bim =00, Bih = 1.0 . . . . . . . . 31

Effectiveness factor versus Thiele modulus for
various 9.
I: 12.0, Bim :oo, Bih : 2.0 . . . . . . . . 32

Effectiveness factor versus Thiele modulus for
various Bih.

K: 10. 0, B: 0.0A, B1m = m) . . . . . . . . 33
Effectiveness factor versus Thiele modulus for
various Bim.

'2‘: 20.0, [3: 0.03, Bil,l : 0.5 . . . . . . . 3A

Concentration profile for a low .
15:10.0, (3: 0.03, d) = 0.3, Bim =00,
Bih : 1.0 . . . . . . . . . . . . . . . . . 37

Concentration profile for a case of multiple
solutions

i=10.0, p = 0.014, 4:: 0.22, 131m :00,

Bih : 100 o o o o o o o o o o o o o o o o o 38

Concentration profile for a high d>
‘6: 10.0, p: 0.03, 0: 4.0, Bim =00,
Bih = 100 o o o o o o o o o o o o o o o o o 39

Concentration profile for a finite biot mass
case

'6: 20.0, (s: 0.03, ct = 0.5, Bim = 50.0,

Bih : 0.5 . . . . . . . . . . . . . . . . . A0

Concentration profile along the reactor length
fvor unique solution case

.[.=10.0, (520.014, 42;:2.0,Bim=00,

Bih : ,gz3 0, Da 2.0 o o o o o o o 0 “5

Concentration profile along the reactor length
for multiple solutions case

25+ =10.0, (5+: 0.0a, ch: 1.11, Bim zoo,

Bih : 0.5, Q°= 3.0, Da : 2.0 . . . . . . . A6

A, B,C

F(y)
F(‘P,k)

NOTATIONS

Chemical Species

Ratio of external surface to volume of catalyst
particle

Defined by Equation (10.6a)

Shape factor (1 for semi-infininte slab, 2 for
cylinder 3 for sphere)

Defined by Equation (8.25a)
Biot heat number hS/Ae
Biot mass number kng/Dej
The upper bound of Equation (8.31)

The lower bound of Equation (8.31)

A normalization constant

Concentration of component j

Feed concentration of component 3

Heat capacity of component j

Damk‘cihler number (l-ZIZ) QVP 695 (NT I L1) kf't
Effective diffusivity of component 3

Diffusion coefficient of component j in radial
direction

Diffusion coefficient of component j in axial
direction

Constant in the Fourier-type expansion
Defined by Equation (8.18)
Elliptic integral of the first kind

vi

A!

F
‘3[u(0)]
fk
0,4910

8[u(0)]
H(u3)
(~AH)

h
h[u(o)]
ggk)
K3
Ke

k(T)

Defined by Equation (10.A)
Defined by Equation (8.53)
Defined by Equation (8.6a)

Defined by Equations (8.35), (8.36), (8.A0), and
(8.A1)

Defined by Equation (8.53b)

Defined by Equation (10.12)

Heat of reaction

Heat transfer coefficient

Defined by Equation (8.53a)

Jacobian matrix, defined by Equation (10.10)
Equilibrium constant Rj/kj

Reaction constant for the surface reaction, k3/E3
Arrhenius reaction rate constant

Mass transfer coefficient for component j
Rate constant for forward reaction

Rate constant for backward reaction

External surface area of catalyst particle

Total number of interior collocation points

Total effective flux of component j entering the
solid phase

Total number of catalyst pellet inside the reactor
Pressure
kth degree Jacobi polynomial

Rate of heat transfer to the surroundings/reactor
volume

Effective heat flux leaving (or entering) the
catalyst pellet

Rate of reaction
vii

 

Gas constant

 

R3 Observable rate of reaction

A

Rj Turnover frequency, number omeoles or molecules/cm2
active surfact area/time (or intrisic rate)

r Radial distance of the tubular reactor

8 Total active area/gm catalyst

S Fraction of unoccupied catalyst surface

T Temperature

T0 Bulk phase temperature

T? Feed temperature

t Time

uj Dimensionless concentration cj/cg

u? Dimensionless cgncentration defined relative to feed
conditions cj/cjf

ufi Defined by Equation (8.A7a)

U Specific molar volume of the fluid

Vp Catalyst pellet volume

<v> Fluid mean velocity

W(§2) Arbitrary weighting function

x Distance from the center of catalyst pellet

y Dimensionless temperature T/TO

yO Dimensionless temperature defined relative to feed
conditions TO/T%

Z Total length of the reactor

2 Distance along the reactor length

Superscripts

0

Bulk conditions

viii

Subscripts

 

f

Feed conditions

Greek Symbols

 

«a
fi?‘ 0“ ‘%3 c* o
x

23mmtm

Praeter temperature Dej(-AH)c3/WeT°
Defined by Equation (8.1Aa) @Bim/Bih

Dimensionle s adiabatic temperature rise
(-AH)03f/p° p1?

Dimensionless activation energy E/RgTo

Dimensionless activation energy defined relative to
feed conditions E/RgT?

Half-thickness of the catalyst particle
Krbnecker delta

cm3 solid/(cm3 solid + cm3 void)

cm3 catalyst/(catalyst + fluid)
Dimensionless distance along the reactor length z/Z
Effectiveness factor

Effective thermal conductivity

Thermal conductivity in radial direction
Thermal conductivity in axial direction
Dimensionless distance from center of pellet
Catalyst bulk density

Defined by Equation (8.55d)

Exp [5(1 - 1/y)]

Residence time Z/<v>

Thiele modulus. Pek(To)82Cg/De

Defined by Equation (8.25b)

Defined by Equation (8.3A)

Defined by Equation (8.39)

ix

INTRODUCTION

The strategy of operating heterogeneous chemical
reactors depends largely on the nature of the steady states
and on the stability of these steady states. The
understanding of the reactor dynamics is essential for
determining chemical reactor control policies. Normally,
the analysis of chemical reactor dynamics is preceded by
the analysis of these steady states.

For a heterogeneous chemical reactor, typified by a
packed bed reactor, the model that is usually employed
consists of material and energy balances, of the fluid phase
and the solid phase. Not unless the fluid velocity changes
drastically within the reactor, then the momentum
conservation equation can be discounted. Hence, for a truly
turbulent fluid having the mean velocity (V), the fluid

phase material balance for component j is

0

ac- _ 3 ac’ a 9’ __ ’ i—Z .
at "#0323? i tawfzsfit) <V’?§§ * 2 ”‘31 W
subject to CS (0. Y. b) = C50 (1.1)

0

ac _. '3 ._
"Dijsi’ = <v7icai~ CJCt»A-°U,A=n3 (1.2)

M _ ._ (1.3)
SI- --0 , .Q- R,Z.

2
Where Nej E Total effective flux of component j entering the

solid phase.

Other symbols are defined in the notation section. Notably,
this model presupposes axisymmetric concentration profile,

i.e” ru>6~dependency. The solid catalyst material balance
for component j is

80' A
__l + . . = e
at Y '38) 98 R1 (2)
where Rj E turnover frequency, no. of molecules or
moles/cm2 active surface area/time (or in- ‘
trinsic rate)
8 E

total active area/gm catalyst

’0
1

catalyst bulk density, (gm/cm3)

m

cm3 solid/(cm3 solid + cm3 void)
L191

9%

Assuming symmetric conditions,

m

" Dej‘ZCj

m

effective diffusivity of component j

E: C' = 0 0+ X = 0
2n 3 " (2.1)
where§% E derivative with respect to a normal
é : 0, represents the center of symmetry.
If there is an external mass transfer resistance,
__ a . .° . _
.3 :._ --—C :k-EC-C Q‘l‘ 23—352. (2.2)

wherelafl.represents the external surface of the catalyst

pellet.

3

The fluid phase energy balance is
N '3 9° N — ~91" 1 oaT
2.3.93 CP3(~§€ ‘l" (V795— ?) " “5:357“ “393) +‘F9r(7\r'—F)
91m 9?
.._.._.. __ +——— £01 Q
(NAT):ch at +07%— :1 “be. + (3)
where Q 2 rate of heat transfer to the surroundings/reactor

volume.

“beaeffective heat flux leaving (or entering) the

catalyst pellet
A
V ESpecific molar volume of the fluid

Other symbols are defined in the notation section.

The heat balance for the catalyst pellet is

393? P33: + 17:93.; = 698 (~0H1R3 “4)

where ifi E. ‘er i'r E. effective heat flux
7\e, :15 effective thermal conductivity

Assuming symmetric conditions,

and for an external heat transfer resistance,

$e(9£)=“7\e% = hCTO’T] 0+ 33:952- (11.2)

Now, the link between the fluid phase and the solid

phase is afforded by the concept of an effectiveness factor

of species 33%,. Thus

‘11- ‘2 EV? ARj dsr (5)
3 VP RSCC-j’, T°)

 

u
where V%'E.catalyst pellet volume

c131 adifferential catalyst pellet volume

Applying Gauss-Ostrograllskii's theorem,
Sm 11' Sej all]:
’71). — VP ePsfijcql-TO) (5.1)

 

Clearly, the numerator in equation (5.1) is interpreted as
the total moles of material j entering (or leaving) the
external surface of the catalyst pellet. Hence, the fluid

phase steady state balance can be rewritten as

ac° ° 96?
O : 5%(D55-a-gfl + J;%(Drg%%) -— <v>-a";

p

l'é , A. ° 0 5.1.—
._ :— 0fl1€€SR3CC1;T)\/PS£1E (5.2)

where NT‘E total number of catalyst pellet inside the
reactor. In the words of Carberry, the last term in
equation (5.2) is referred to as the observable rate.
Therefore, the solution to equation (5.2) requires the
calculation of the effectiveness factor.

In this work, a method based on the theory of elliptic
integrals and elliptic (Jacobian) functions is applied for
calculating the effectiveness factor of a second order
kinetics. The details of the diffusion-reaction model
assumptions is postponed to another section. A simulation
of the heterogeneous reactor model is also presented. Much
of the works involving diffusion-reaction systems have been

on first order kinetics and lately the focus of interest in

 

 

5
the literature have been on Langmurian form of the kinetic
rate expression. A comprehensive review of research
pertaining to diffusion-reaction systems up to 1975 is
provided by Aris [7]. Upon examining this reference [7] and
the recent literature, it is clear that studies on the
second order kinetics is not fully explored.

When chemical reactions take place within a porous
catalyst pellet, the system may have multiple steady state
solutions for certain combinations of physico-chemical
parameters. Many studies have treated internal and external
transport resistances separately or together. In cases of
gas-solid catalytic reactions, the external mass transfer
resistance (Bim) is relatively much greater than the
external heat transfer resistance (Bih), so that the major
temperature gradient exists in the external film and the
catalyst pellet may be considered to be isothermal. This
internal pellet isothermality has been carefully examined
for the first order reactions analytically [6,8] and
experimentally [A,5]. Pereira et. a1. analyzed the full
non-isothermal diffusion~reaction equation by regular
perturbation method for small Praeter temperature ((3) and
finite Bih/ ratios. The zeroth order solution of the full
problem justifies the internal isothermality [6]. Also
Cresswell studied the combined effects of internal and
external resistances on the uniqueness of the steady states
with internal isothermal model [8], while Hatfield and Aris
[9] studied the combined effects of the full non-isothermal

model with both linear and nonlinear kinetics.

6

A number of studies have been devoted to develop a
priori bounds on physico-chemical parameter values for
multiplicity of solutions [10-17]. But, because of the
difficulty of handling coupled nonlinear differential
equations, necessary and sufficient conditions could be
obtained only for lumped parameter system while only
sufficient conditions found for distributed system. Luss
[16] developed a priori prediction of the unique steady
state conditions with nonlinear heat generation term for
arbitrary kinetics. Later, Van den Bosch and Luss [12]
developed uniqueness and multiplicity criteria for lumped
parameter model and for the intraparticle concentration
gradient model. They found that for lumped model the higher
the order of the reaction, the smaller was the parameter
region where multiplicity could occur, and for the model
with intraparticle concentration gradients, the multiplicity
parameter region is smaller than that for a corresponding
lumped model.

For first order reaction model, Pereira et. al. found
analyticallgra 1-3-1 multiplicity pattern for relatively
small B and relatively large Bim/Bih model and their
analytic criteria were compared with exact numerical
integration of the heat and mass balance equations [10].
Lately, Morbidelli and Varma [1A] obtained explicit
approximate bounds for the multiple region.

For langmuir-Hinshelwood type of kinetic model, the

multiple steady state phenomenon depends on the adsorption

 

 

7
rate constant values. The assumption of fixed adsorption
rate constants with temperature was a good approximation for
the relatively small 9 cases. This assumption did not
change the multiplicity pattern for larger @ cases but
changed the numerical range [11].

Studies on multiplicity of the steady state solutions
is usually expressed in terms of the catalyst effectiveness
factor [13, 1A, 18-21]. HlavaCEk and KubICER [20] developed
a simple approximation for evaluating the dependence of the
effectiveness factor on the Thiele modulus. And Copelowitz
and Aris [18] calculated the effectiveness factor for small
and large Thiele modulus values asymptotically and applied
the reasonable approximation to the rising part of the
effectiveness factor versus Thiele modulus plot.

For bimolecular Langmuir-Hinshelwood Kinetics,
Morbidelli and Varma [1A] developed analytic solutions for
effectiveness factor for the full range of Thiele modulus,
in the limit of large adsorption inhibition constant, and
Wong and Szépe [21] found that for this kinetics, multiple
steady states might arise even when effectiveness factor is
less than 1, and that with higher reaction temperatures and
stronger surface chemisorption characteristics,

multiplicities in endothermic reactions are likely to occur.

THE KINETIC MODEL

The kinetics of interest here iS'typical of a

hydrogenation reaction involving an olefin, e.g” ethylene

8

on a transition metal supported on'U-alumina.

k1
——>
Chemisorption k
2.
B + * :3 9>~*
k1 (6.2)
£3. C
Surface A«* + B~* ‘R- / \ (6.3)
reaction 3 * *
In;
Product _ i ;K\ ;;3 C .+ 2.* (6.A)
desorption + *

Here, we suppose the surface-reaction is rate controlling.

Thus, invoking the pseudo-steady state assumption.

A K4 "2.
R3- : k,K.KzCCACB "W1 Cc) S (6.5)

where SzEE fraction of the surface that is unoccupied

-122,
Ké : ‘E‘ E reaction constant for the surface
'5
reaction
L153; _
Ki = [C E equilibrium constant
1

Assuming the forward reaction proceeds at a much faster rate
than the backward reaction, and the surface does not exhibit
poisoning (or S2 is constant for all purposes), hence

equation (6.5) is transformed into
A “/1
R3 = kBK, K13 CACB (6.6)

for an equimolar concentration of components A and B within

the catalyst pellet void,

1

R3 = kmca (6.7)

We will assume the Arrhenius dependency for k.

THE INTERNALLY ISOTHERMAL, EXTERNALLY NON-ISOTHERMAL
MODEL OF THE CATALYST PELLET
Hutchings and Carberry [1,2] have shown that the major

seat of temperature difference will most likely be in the
external fluid film, while the concentration gradient can be
expected to prevail primarily within the porous catalyst.
Lee and Luss have later refined this analysis [3]. Kehoe
and Butt's data [A] and Butt et. al. [5] presented some
results of an internally isothermal pellet but with a
limiting external heat transfer rate. Their experiments
yield various inter-intraphase temperature distribution data
for a single pellet in the Ni/Kieselguhr catalyzed
hydrogenation of benzyne. Hence with this consideration,

the steady state material balance is
Y~t>e- . Va, = seekmca‘ (7.1)

where k(T) is the Arrhenius reaction rate constant. Upon
assuming that the effective diffusivity is independent of
pore geometry and the other diffusion effects, e.g. Knudsen
and surface diffusion, thus for a semi-infinite catalyst
slab,

2.

935%??? = apSIdTm3 (7.2)

10

For the symmetric pellet with a surrounding mass transfer

 

 

film,
£10
= O
37% X=o (7.38)
66 -°tc — c1 1
37"§L1<=é>‘1 H 5 (7.3b)

where 3 E half the particle's thickness.

The energy balance is
Y-De‘ZT = ~ epSk(T)c;(-AH) (7.11)

Assuming an isotropic effective thermal conductivity,Ze

,thus for the semi-infinite catalyst slab,
1
38d 01:1 = — epgkmcj (-OH) (7.5)

Again, with the symmetric pellet and a heat transfer film

surrounding the pellet

QI' " 0
dx 7:0 ’ (7.6a)
= h o- _
MATH- 8 ET Thea] (7.6b)
Integrating equations (7. 5) and (7.6) yields
iii-Th:8 ~To] = 12936 (~0H)k(T)C; dx (7.7)

So, if the internal temperature gradient is zero, and the
heat of reaction is temperature independent, equation (7.7)

reduces to

3 1
RETlH ~T°] = eps (—oH) k(T1,:57§ c3 dx (7.8)

11
The material and energy balance equations can be

into the following dimensionless form

i.
I 1 U ‘- ) 9.

”5/ 7— ¢ 9 ( ‘8 U‘

05/1 = o

3:0

05",;1‘ = 87m [1 - Uj |¥=lj
131,, 2 1 b’(|*“g‘)
—— — = M- d ‘ e
(5 (l3 1) ‘9 So 3 ‘5

where (Pt-5 Pék(T°)31C;/D€j -=— Thiele modulus

% a dimensionless concentrations: Cj/Cg

3 E dimensionless temperature E T/TO
K Edimensionless activation energy E.E/(RgTO)
S zx/S

EEPraeter temperature E De; (~0H)C§l7\eT°
Blms Biot mass number 2 1(ng / Der]

BlhEBiot heat number E 118 / 7\e

casted

(7.9)

(7.9a)

(7.9b)

(7.10)

In the perturbation framework of Varma et. al. [6], the

full non-isothermal diffusion—reaction equations,

“3 = 4): 8111““? Hz . 5 e (0.1) (7.11)
‘1” = ‘94; (aw—9) n1 . S 6 (0‘0 (7°12)
5:1 ~. u'=B;m(1—U) 3 3’: 131,,(3—1) (7.13a, b)
3:0 I \A,=0 ; g’: o (7.1Aa, b)

has the zero order term in the regular perturbation

expansion yielding equations (7.9) and (7.10).

12

. = 2 _.L

Let R(u,g,[%) _ uexptb’u .33] (7.15)
E a perturbation parameter

thus assuming the following asymptotic sequences,

u 32°)-
‘9”)

(7.16)
3:0
N £2 5 -
‘3 3:09 ‘33 (7.17)
M” : ¢1R(u,g;(s)

~ (91 g R(u.,go) + @[Rucumgaw + Rg<“°"é°)‘3'j
1“ (52 [DRutumga u}. + 21?.acuo,g.1gz
+ 214.3. Rugcuoqg + um” (um-3.)
+ gfawwmgnj + —---}
N u: + (guy if (52 u; + (7.18)

Hence, the zero order terms consist of the solution to the

following equations:

'3! : o (7.19)
gzo : L3; :0 (7.19a)
5:1 : g", = o (7.19b)
thus yO : constant (7.20)
U: = 49 83(1- 1)") U: (7.21)

E = O . u, = o
g :1 : u; = 81m [141.] (7.22)
yO is obtained from the first order term, thus
lg” c)z R(u.,3.) (7,23,
3 = O : 3', = o (7.23a)
‘5 = l 1 $1,: Eéh'uéo“) (7.23b)
Upon integrating equation (7.23),
30-, = % S: Rcumg.) (15 (7.24)

The perturbation solution is not described here,

but
will be done in the future.

The perturbation solution will

provide stronger criteria (for small @ ) for the full non-
isothermal problem.

NECESSARY AND SUFFICIENT CONDITIONS FOR
UNIQUENESS OF SOLUTION

a.

Dependency of Concentration on Temperature

Consider again the catalyst pellet model equations
where both intraparticle diffusion and thermal conduction
occur,

_.L
”3” ___ 4): 87$“ 5) ué'n

m

(8.1)
ct" {(115 3g)

with

8-1.

111
J.
l- n
13” = — (3 11? e7“ '3) Mg (8.2)
115(0) :0 3 13(0) = 0 . (8.3)
1,15,“) =BTmCI-UJ'CD] 3 ‘j’(13=BThCl- 3(1):] (8.11)
Variation of Concentration Profile with Temperature

Assume there exists a maximal solution uj2 and

a minimal solution uj1 such that

HA?- __ ”5(13: g1) (8.5a)
U), E (1103: g.) (8.5b)
and g2 7 g, (8.50)

If the differential equations satisfying these

solutions are subtracted from each other, thus

H6: - Us”, = (big 1C1 - {:1} (8.6)
__L
where (k EU“ 3K) k:l,'2_ (8.6a)

Integrating from 5 :0 to g :1,
81,111,111- 83,1111f18g(1‘__.:_)1w 1.1118,, ,,
using the derivative definition

BTm{u3,m - 1132(1)} : q; 1:9“ 3‘3 3*

where y* €.(y1, y2). With the nth order kinetics

('12 $1M?) (8.8)

 

(n>0),equation(8.8)is written as

  
 

15
Triumor 11310)}- {15013 13 01%;)“ 37%” ~8(‘32-‘,11)6\S(8.9)

Since it was already assumed earlier that uj2
(a maximal solution) > uj1 (a minimal solution),
therefore the left-hand side is negative. This

indicates that,

. r *2
8M __ u
333-15111 < #3?— <0 (8.10)
01", 3.3.9; < o (8.11)

3‘3
Another contradictory proof is to assume that
if 31); /3% > 0, the integral is positive. Thus,
u32(1) > uj1(1) will make the left-hand side nega—

tive. This is a contradiction.

Bounds for y

By adding equations (8.1) and (8.2) the heat balance

and the mass balance are uncoupled to yield

+ u-m— (
3(1) (3’ 3 '3 5) (8.12)

For the internal isothermal-external non-isothermal

model,

Blh

(3-1) = uj’u) = BIWET— 1130)] (8.13)

and recognizing y(1) : y(g) ,thus,

015:

where %L

L
[1+p—13] (8.111)
(saw/131,, (8.1ua)

J:
(5

IN

16

3
Now, since —jL < 0 and u- éi(0,1), therefore
at” 3

fi 6. E ‘2 l+'@L J (8.

. Necessary and Sufficient Conditions for Uniqueness of
the Steady State: A Conservative Criteria

From the heat balance equation,

I

Ha (I)

H

Bi},
——-‘( *1)
(5 ‘3

the effectiveness factor defined relative to the bulk

conditions is

 

._ .L 1 1

Hence,

fl = l E () (8.
(57‘ ‘11" d F 13

9L}; = “17033” 7 fl (8
d3 (g-Ut)z

Furthermore,

FC‘3=|7 = + (>0

2 1 —l- 1
3' 8013’) = C? Sew 3'11; <13 (8.

18)

(8.20a)

F(g=1+@L) ”=’ o (8.20b)

Now, with equation (8.17),

__1_ I ~
'11 {lg-i (%—1) -1} + 2(g—1)e‘°’U WLUJCS‘)???‘ d

S

 

d3 — (3-1)1 (8.21)

 

11-1)

ARUBA; 2%?dE) (8.21a)

17
7f 7!
1—2( —1)—1}—2( —l)e
2.
.45. -1 1 3 1
-" 2.
3 (\3—1)
It is clear from equation (8.21) that the necessary
and sufficient condition for uniqueness of solution is

lug—1) —1 so

3
or, 32—- 33 + ‘0’ 20 (8.22)

Inequality (8.22) is satisfied iff
‘o’é4-
which is also the necessary and sufficient condition for

uniqueness of the steady state.

Van den Bosch and Luss Method for Uniqueness and
Multiplicity of Steady States

If we define the effectiveness factor relative to

the surface conditions,
I
"1 = UJ'UYZS U132 01% (8.23)
D

then the energy balance for the internal isothermal -

external non—isothermal model results into

B“ ‘ 7(1-L)
4.813-.) = 81506 8 115101);

(5

__L

_ 1 WU'é8 L z
'— ((51): e (”15“?) (8.211)
(7%? 131,

‘__—‘_—_—__ (8.25a)
M"

18

_1_
c1)1 :— 63“ g) (H- (BL—LA) (8.25b)
N _ _ (111(sL--13)11E§1
hence, B — 13(3) = T (8.26)

QmFUJ) = 9/111 C1320+pL-g)1+£n1—M(g~l)(8.26a)
thus,

11:11:

77:3)” = «818111.188;— !Wl —,_ (8.27)

Since F(y = 1) = + K), F(y = 1 + @L ) : O and F(y) Z 0,
the necessary and sufficient condition for uniqueness of

solution is

d 2. L 1 <
nghECHfi—gfl + 5138.61 — a: _ o
(%_D[gg{%$zm+ei_w Jrflinnflsl Ingé(1,HflLl8.28)

Using the identity

.11). .18..
”l_ lifinami

d
3 %%g_idflm[i~1jL;1+@— fl “11:75—31 (8.29)

substituting equation (8.29) to the inequality (8.28)

“‘1 L 3.11...
_ d 51/1113? (1113 11)] 811mg]
(3 01 d3 (1+—?L_ %@)+ 01% 1 El (8.30)

2.0-18513)

 

11

 

Tn (3:41.011?)

 

19

Suppose we can bound a. prior?

be 5. fl— 3 19, (8.31)

because §i5-< 0 for an nth (n > 0) order kinetics, it
follows a—pfiqfi < 0 and b1£ 0. Van den Bosch and Luss
[12] proved that for the slab, cylinder and sphere geo-
metrics,

b2 3 —1 (8.32)
It follows from the inequality (8.30) that a sufficient
uniqueness criterion can be obtained by substituting
dfinfi ldLn§> by its upper bound b1. Hence, for the

second order kinetics,
(1+ 12133 + c8L+711+1;—')~ 0%?)ng
— U(1+%3((3L+2)13 + 71(1+'321)(1+(3L) 20

to? 1 5‘3 i '1 (5L (8.33)

For a conservative criteria, b1 is set to zero, resulting
into a model where intraparticle diffusion is neglected.

This is tantamount to a lumped parameter model. Hence

for b1 : O,
L CT:(@L) 11°" (3L E1
ML ‘5 11’1“?) '5 L L (8.311)
”(5 511(8) 101‘ (5 7—1
where
2 L‘ V2 1‘18““ 20+?)

GM?) 5 1+21i(1*r——"’l+2m@1)]2 +3c1+211+w12 (8.35)

 

20

4(8L+2)((3L1+4@L+21 i2 8(216")

 

L __ . 8.36)
G111” = 3+2[‘ T ((122.1): ( (2.1-4.4)z (118x418?
For the case where Elm—30o ,
(s s 8,118.. 1°) = 4‘ (8.37)

It is clear that for (3" e (o, 00) , inequality (8.311) is
satisfied for X5.- 11. Hence, US 11 is the most conservative
uniqueness criterion.

Multiple steady states exist for some B values when-
ever inequality (8.30) is violated. Inspection of in-
equality (8.30) indicates that a sufficient condition for
its violation for some y e (1, 1 + (51‘) is to assume
equal to b2. Hence, multiplicity occurs for some 18

values if
(1+%)133 + [15L+(‘5—1)(1+-12§)] 13 — 3(|+—1’7—’_:)(21-(5)3

+ 1111+ 18111.1.) £0

(8.38)
for some yS in (1, 1 + (3L).
A more conservative bound is obtained when b2 : -1 (see
Van den Bosch and Luss [12]). Hence,
L
BL? C75((5L) for G El
—- >112 (8L) 5 (8.39)
L
”(3 ‘1 6111(8) fro. (1L 21
where
L2 Ji
L :3
(215(8) _ 6+2E11+C2+311+8L)]z) (8.110)

5114.1850— 6 (11119)]
[2'11 3c1+nL)j’-

 

+

 

21
L I L L? L L2
L __ 28 +3 2._ (28 +3)(2(3 +215 +3+413 )
In the case of Bim—> 0° or (3L -? 0° , a condi-

tion for multiplicity of steady states for some values of
B is given by 7528.
Exact Uniqueness and Multiplicity of Steady State
Criteria

Before proceeding to derive the exact criteria, a
proof of the uniqueness of the concentration distribution

for a fixed temperature is set forth at the outset.

e-1. Bounds for u

Upon integration of equation (8.1)

3 _.1_
u-’ = 4:25 127‘“ Wu,“ d);
O

3 (8.112)
Since the integrand is positive,
u; 7.0 1w 5 £1011) (8.113)
but, (A; (1) = 1— fig 113,11)
Hence, 0 s 113(1) 2 1 (8.111)
as a corollary,
O .4. (13(0) 5 115(1) f 1 (8.115)

e-2. Uniqueness of Internal Concentration Profile
Here again, the maximum principle is applied.
Consider the maximal solution Ué(§ ) and the minimal
solution G,(§ L Furthermore, let GE > G}. Hence,

given a y, and using a comparison theorem approach,

 

22

l "" ~ .
~ (201,313) - Rcuug)
11.01— 111(1) = (1)21 1,1,— u, (”fuddg (8.116)

 

upon using the limit theorem,

1
111,0) —- 011(1) : c125 ——-

3U (u.—u.)d8 (8.117)

 

11*
where 111* e (u. , 11.)

It is necessary and sufficient that if

9R >0

._.——

911 (8.48)

then equation (8.47) is readily violated. For
_.L
[z = 3.511 13)”?- , the uniqueness of u is quickly

established.

The Solution

The elliptic integral was used by DeVera and
Varma [22] to solve for the optimum bulk phase
composition in an isothermal catalyst slab with a
second order kinetics. This approach is extended
here. Applying the Clairut's transformation to the

material balance equation,

01 I z 2 ’
£1141): = legging-113- (8.119)
where 9(3) :3 8X? [6’ (1" ’lg‘)] (8.1198)

and integrating from g : 0 to g

I .L
”3' __. E 4) 92(3) 11013—1110)5 (8.50)

23
Equation (8.50) is an elliptic integral form [23],

whose solution is given by

2121(0)

F(1P(U; 11(0)),k) = T“? 92(3) 3 (8.51)
where F(W,kfl E elliptic integral of the first
kind
k E some modulus E— (\1—3—‘3127/213
cosxp =5. [(15 +1) uco)—u’_] / [(15—1) uco>+uj

For Bim-a M3, the centerline concentration is
evaluated by using the boundary condition at S :

thus the transcendental form for u(0) is

F(W(1;u(o)),k) 7’11”“? 0) 109213) (8.52)

Now, y is eliminated from equation (8.52) by using
the heat balance, and the result iscasted into the

form

= [3"(u1onfljflmfg11g k(wonj E 3021(0)) (8.53)

where

-o.2s
11111111») =5 3 F(IP(I;u10)). k) N117?) (8.53a)
301(0)) 2((5/Bih1111—u1013 1101(0)) (8.53b)

Furthermore, since 0 EIKO) 5-1, therefore for

11(0) = 1, (fi+1)_1 — O
1P(|;u(o):1): Amos—J —

(1155—011
F(01 k) = 0

M1110): 1) = O

H
1
3

and. 3'(1Mo)=l)
For MUD) =0

)

1PM, U(o)=o) = Arcos {-1) = Tr

F (TY, k)

[1

2K
hcuco)=o) = % = +I>°

and 3021mm) ” 1.113131111111117 = +00

It is clear that the necessary and sufficient
condition for uniqueness of the steady state for

all physico-chemical parameters is

{ghfigg h(u10>)] + 31%)%hdl—u(o)3h(U(o))
s 3.3. 1).. [1,313 1.1

101 0110) 6 (0,1)

(8.511)
where
__ 1—— {JUL _ 3u(o)zln(11(1>))
81(u10)) =%h[l-U(0)3du(o) 2W (8.5148)
3201(0)) E 1+ YET—“#111013 Hum) 70 (8.5%)
and ‘1 z _1 ~ ’%
0.25 51.11. = —211§{(11§—1)U110)+1} “Wampum/1101)}

 

dune) _ We) Sin 19(1;u10))

 

25
FWU; mo», k)
2U(O)\’U(o)

 

(8.54c)

Exactly the same approach is applied for finite
Bim. Here, the criteria for uniqueness is expressed
in terms of the external concentration, i.e., u(1),

instead of u(O). Thus, equation (8.53) assumes the

form,
mum) a [g"cuu»+l] Mfig‘ mum] =%_ (8.55)

-025
where Wm) '5 3 FttPcumMoLm/m (8658)

v
_ 3 3. '_ z 3
ac —{um — W)B‘mm W03} (8.55b)

3mm) Egh‘fuuf—ui k (mm 2 o (8.550

PLULD)Eexp[ZS(l—{I+EBT“U- Mm)”; )] (8. 55d)

The bounds for u(1) are established by
recognizing that since 0 £,u0 S.1 and for the
monotonic increasing kinetic rate function, u' > 0,
thus 0 ELR)< u S 1. The lowest bound for u(1)is

therefore the lowest root of

1.
3 s: 2
3 m
MU) ~ —— [1— them] 0
2¢‘Q(u*m) (8.56)
Moreover: 3(U*U)) :: + be and q(UU) :1) : _ DO ,

hence the necessary and sufficient condition for

uniqueness is

26

[I +3(u(-I))] dflmhmm) S égmhfég MIND]

dUU) due)
for (“(1) é um) 31 (8.57)

 

 

 

 

 

 

 

 

 

where
57m h elf/uh +~L‘I (8039381)“

30M») =¢J§e 87h ’é‘m l"“)]o\(zl(n+ 23, {1+ “ID-mu] ‘}(8.57a)
0.25,“, = CdLP [dam] __ F(7~P(M(I),Mo),k) duo (8 57b)

dun) “Mali-£39m 2mm d“) '

duo _

9.1L = ”2Cuc"du(1) 3°] (8.570)
dun) ‘ [(fi—nuo—uufl STnWCU(‘7)
due ’2/5 z B?1 tn—um‘} (I—UL\))U 5%

= o ( + m
dud) “ u {an 24>: WWWEE (HLBW‘U- muff) 57h }(8 57d)

THE HETEROGENEOUS CHEMICAL REACTOR MODEL

Here, an adiabatic tubular type reactor with a plug
flow velocity of the fluid phase model is employed. This
assumption is generally valid for a high aspect ratio (i.eq
reactor length/tube diameter). Hence, for the isotropic
bed, with negligible pressure drop and axisymmetric flow,

the steady state equation (5.2) reduces to

 

2

dc? l—E A
-0”??? _ onévfegsm—LI— RQCCZ T°>=0 (9.1)

The energy balance assumes that since the reactor is
adiabatic and for a large aspect ratio, the radial transport
of energy has a smaller contribution than the axial
transport of energy. In this work, it is further assumed

that the reactor bed is relatively dilute or low solids

 

 

27
density. This enables the model to neglect the axial thermal
conduction contribution and thus, the energy transport is
solely by a convective transport. For higher solids
density, the axial thermal conduction contribution along the
reactor length can no longer be neglected. This effect
would definitely create more exotic behavior of the steady
state multiplicity pattern; perhaps, more complex than the
number of steady states in a single catalyst particle.
Furthermore, the question of the steady state stability
would be difficult to address. These studies will be later
investigated in another research. With this model, the

energy balance collapses into

 

.—

A 0 ~
0 o (11” 1—6 ~ _
~QCF<V>T + MD: - 0 (9.2)
‘3 e

where %e is the effective thermal conductivity flux
entering or leaving the catalyst pellet.

Now, the heat balance in the solid phase, assuming an

effective thermal conduction flux, is expressed as

~ A
. = 6 —-A R. (9.3)
y 3;. cs ( H) a
and for a temperature independent heat of reaction,

SV-v— .$e dsi : S9511} . $641!: : ées (~OH)SVRJ‘ <13}:
(9.”)

jig—g9 dzr = Saga-99842:

(9.5)

and using the effectiveness factor definition and equations

 

28
(9.3) to (9.5), the energy balance in the fluid phase is
transformed into
" (1° - _I\_)__

:15 OH) €€SVFVIJ RjCijT°)-'— O

(9.6)
the initial boundary point conditions for equations (9A)
and (9.6) are provided by the feed conditions.
If the following dimensionlessvvariables are defined
Dd ':—: Damk'ohler number '5 [:26— OVP CQST; [(+12
T: E residence time 5 Z/(y7

where 35 2. total length of the reactor

§£ : feed concentration

60

E/RaT; = dimensionless activation energy
defined relative to feed conditions
flfio E feed temperature,
hence, the dimensionless material and energy balances are
expressed as

75°0-

Au° .‘fo BL") 81.
——-3* ‘+ ‘Da'ng (Lu, % ) 53 C(Jgj " O

dg (9.7)

o a __.L. 0
53—35— — (30D0‘m(u§’jo)exu g)CHj31=o (9.8)

with initial conditions of uj O(O) : 1 and yO(O) = 1.
° : — .° ° ° is the dimensionless
8 _ (oH)cH/E9 CF11]
adiabatic temperature rise. Equations (9.7) and (9.8) are
uncoupled to yield
3": l+(5°(|—Ul§) (9.9)

Qualitatively, u 0 £_1 for O s S £31 and although the

J
solution to equations (9.7) and (9.9) is unique at glance,

29

the multiplicity of solutions to equation (9.7) depends on
the continuity properties of the field, i.e” the second
term in equation (9.7). From theory of first order ordinary
differential equation, a unique solution is guaranteed if
and only if the field is Lifschitz continuous. This
requirement is satisfied sufficiently if the field is
piecewise continuous in the interval and domain of interest.
Since the field in equation (9A) is not necessarily
continuous because of the possibility of multiple solutions
for the effectiveness factor, therefore the concentration
profile of the fluid phase could likewise be multiple.
Their number‘is equal to the number of steady states
provided by the number of steady state effectiveness factor.

The steady state multiplicity pattern could change
along the reactor length and it is largely governed by how
the bulk fluid phase concentration change. It is possible

to have a 3 to 1 pattern, a 1 to 3 to 1, etc.

NUMERICAL IMPLEMENTATION AND DISCUSSION OF RESULTS

a. Effectiveness Factor
The complexity of the transcendental functions for
Bim—a #3 in inequality (8.54) prevented the development
of an a priori criteria for uniqueness. However since
u(O) is bounded, the numerical implementation of equation
(8.59) is not difficult but not practical if a quick
estimate of the uniqueness criteria is derived, and the

stronger criteria by Van den Bosch and Luss [12] would

 

 

3O
suffice. On the positive aspect, the violation of equa-
tion (8.54) can provide not an a priori criteria but
provides for the calculation of the multiple steady
states, their number, multiplicity patterns and the
bifurcation points. The bifurcation points are obtained

from the solution of

<15 <12?
du(o) = duco)‘ .= o . O 5 ”(”51 (10.1)

All the calculations involve a straight forward root
searching, e.g., the half-interval method, and they are
rather fast converging and not cumbersome. An example of
these calculations are found in Appendix A.

As in the case of Bim—9 “I, no a priori criteria can
be derived, however, the numerical calculation of the
criteria given by equation (8.57) is easily implemented
by a simple two-stage root searching technique. The
first involves the calculation of u*(1) and the second
involves the testing of the criteria for uniqueness,

i.e., equation (8.57).

Figures 1 to A exhibit a plot of effectiveness
factor versus Thiele modulus. The effectiveness factor
is written as
Rate of reaction in the presence of diffusion

a” “ Rate of reaction evaluated at bulk fluid phase
conditions

$EW EXP[%(1- 4%)] ‘For ELM—700 (10.2a)

 

L‘? [l- (1(1):) 4:015 4171116 BTm (10'2b)

(P

 

31

 

100.0

10.0

 

 

 

 

 

0.1 1.0 ¢ 10.0

Figure 1 Effectiveness factor versus Thiele modules

for various 0 .

1: 10.0, 131m =(><>, 81h = 1.0

 

32

 

 

 

 

 

 

100.0 ~
(3:0.1
$.08
“I u 06
0.011
10.0 —
1.0
1 1
0.1 1,0 ¢ 10.0

Figure 2 Effectiveness factor versus Thiele modules
for various 0 .

75: 12.0, Bi =00, Bi = 2.0
m h

 

33

 

100.0

10.0

 

 

 

l I
0.1 1-0 (p 10.0

 

Figure 3 Effectiveness factor versus Thiele modules

for various Bih.

25: 10.0, D: 0.09, 131m = (>0

3A

 

1000.0—

100.0—

10.0 —

 

 

 

 

0.1 1.0 ¢ 10.0

Figure A Effectiveness factor versus Thiele modules

for various Bim.

15: 20.0, (5: 0.03, Bih = 0.5

35
For all the physico-chemical parameters reported here,
the range of multiple steady states is relatively narrow.

However, fora given value ofBih, U andBi therange

m’
of multiple solutions expands as the Praeter temperature,
fi’ increases (Figures 1 to 3). A similar situation exists
for increasing Bimzflza given value of@,'6and Bih

(Figure A). The range of multiple solutions also expands
as the resistance to external heat transfer increases
(compare Figures 1 and 2). There was no multiplicity
pattern other than the 1-3-1 type that was found using

the physico—chemical parameters reported here and those
which are not reported here. Table 1 illustrates the
uniqueness and multiplicity criteria comparison between

the Van den Bosch and Luss analysis [12] and the elliptic
integral approach. Except for very small values of the
Praeter temperature, the Van den Bosch and Luss [12]

analysis indeed present a workable criteria.

Concentration Profile Inside the Catalyst Pellet

Figures 5 to 8 show the concentration profiles using
various physico-chemical parameters for the catalyst
pellet. Figure 5 in particular compares a Fourier—type
of orthogonal collocation solution (represented by A's)
with the closed form solution. The trial solution for

the orthogonal collocation is

. N“ 2:.
[AA N E‘ dug (10.3)

 

36
TABLE 1
Uniqueness and Multiplicity Criteria Comparison

between Van den Bosch and Luss Analysis
and the Elliptic Integral Solution

 

 

j Bim Bin @ Van den Bosch and Luss Elliptic
10 0° 1.0 0.02 yes no
0.03 yes yes
0.04 yes yes
0.06 yes yes
0.1 yes yes
0.5 yes yes
12 00 2.0 0.04 yes no
0.06 yes yes
0.08 yes yes
0.1 yes yes
10 00 0.1 0.04 yes yes
0.5 yes yes
1.0 yes yes
20 30 0.5 0.03 yes yes
50 yes yes
75 yes yes

 

where d; E a constant in the Fourier-type expansion
N

For the Bi

Ill

total number of interior collocation points
m —+ «acase, the collocated differential

equations (8. 1) and (7.10) or the weighted residuals are

n - 2 v
22:: (k—1)(21<— walks? 2'“ 2) 4) 9(g)[§:l‘dk33(k 0] =0 5 F1.

for i =1-) N (10.4)
N+l
2:.Ak — =

where €(g)

E EN‘H (10.5)
exp [210—15)]

‘3 = 131 4) ”(U

11;;(3 1> 13M?)

0

fl)
3
O.
,;3
a).
l
h

 

37

 

 

 

 

Low
.8 '
.6 —
.11 —
.2 -
1 1 I 1
oo 2 .4 6 8 1.

Figure 5 Concentration profile for a low ¢ .
0: 10.0, (3: 0.03, (t: 0.3,
Bim =83, Bih = 1.0
( _——_ : analytic calculation,

A I collocation calculation)

 

 

 

 

 

 

.8—
.6—
.4—
.2 _
I l 1 1
0.0 2 4 .6 8 l 0
Figure 6 'Concentration profile for a case of

multiple solutions.
25: 10.0, (I): 0.011, 4>= 0.22,
Blm =00, B1h = 0.1
( ——— 1 analytic calculation,

A :collocation calculation)

 

39

 

 

 

 

Figure 7 Concentration profile for a high (1) .
(1: 10.0, (s: 0.03,q>=11.0,

Bim =00, Bih =1.0

 

 

 

 

 

Figure 8 Concentration profile for a finite
Biot mass number case.
6: 20.0, (a: 0.03. d): 0.5,
Bim = 50.0, Bih = 0.5

 

111

for the Fourier-type trial function

u: __ N+1 2 + NH g?
3 — IE1 QL 521k“ Q°Qk (10°6)
kit
. 2(L-l7
where Q. E J; E (10.6a)
Hence,

0

1 1- N1 N ' ; -
r11. = '33 .435 SM nd‘s + 2+: fl 4.438 32: H‘ 2]
3 L=I o L=l E;

or,

N+l A: N+IM+| dadk
In) _ L=14(L—D+l + 1:1 '12:" 2(L+I<-2)+1 (10.7)
1

The exterior boundary was collocated because the trial
solution did not satisfy the exterior boundary condition.
The collocation points 3i are obtained from the

orthogonal property of the following Jacobi polynomial

l ~—
SW.) 11.191919)? ld; = 0.5-.1. .k=1»2»"‘5~"<1o.8>

and Si satisfies FL(§L) : C

where FkCE) is a kth degree Jacobi polynomial
01k E Kanecker delta

E a normalization constant

C.
0’.

m

shape factor (= 1 for semi-infinite slab, :2
for cylinder, : 3 for the sphere%
W(‘5‘)Earbitrary weighting function (usually 1-E?
or simply 1%
These orthogonal points are calculated efficiently from

the computer program developed by Michelsen and Villadsen

 

42
[2”].
The solution for g are secured by a Newton-Raphson

iteration, or

3“ .—. .1“ — g“(.1_‘<1’_13(<_1") (10.9)

The Jacobian matrix is
N

— 3F).
I |(4k) 3‘: 931 14k (10.10)

 

thus,
31:1; __ C2k 4)
BIT; — 2(k 1)(21<- 3)§
k-D
2 NZ‘HA g2(1c—1):I g3 >5.“
2(‘ 1) ‘3
+ 5L L fori:1—-)N (10.11)
3FN+1
.5}:— = (10.12)

Essentially, the calculation proceeds in the
following fashion:
i) assume g
ii) calculate y
iii) calculate the next guess via the Newton-Raphson
iteration
iv) calculate the convergence tolerance
A specimen of the computer program using this technique

is found in Appendix A.

 

 

43

The concentration profiles reveal that for a very
diffusion-limited reaction, the profile becomes rather
steep (Figure 7). An interesting case is when multiple
steady states do arise, the upper branch steady state (in
the context of the effectiveness factor plot) shows a
steep profile. This must be safeguarded whenever
instability indeed occur. The control poliCy should

always be to keep the lower branch steady state.

The Tubular Reactor
The numerical implementation employs a

discretization of the inverted fluid phase balance, i.e

’9

o .L
d‘; _ “3(1— ) o a 92 .4: _.
3713’ ‘8 ‘3' (Danjwyjfluljk —H(u3)(10.13)

It is assumed here that for an infinitesimal change in

u?, the effectiveness factor is approximately constant.

Hence,bye”)IMSLDGEARsubroutine,the valueof § atug

+ AU? is obtainedo from

U‘)
3k 3 kil . o
SHdS = . H(03)Auj (10.111)
5k ”Pk

o . ° 3' ° ‘3 .
and "(3 (L19 , 13) at some Us 6 (U3 , 1151-0113) is
evaluated from the previous discussions. In terms of the

bulk phase conditions the physico-chemical parameters are

redefined as 45‘ _=_ 4): 8111(1— 51’) US
75 751 / 3°
P ‘5 P1 ”3 M1"

[11

 

 

MU
where ¢; 2, Thiele modulus evaluated atfked condi-
tions
64 a dimensionless activation energy evaluated
at feed conditions
Q; g adiabatic temperature rise evaluated at
feed conditions

Figures 9 and 10 show the concentration profile of
the bulk fluid phase reactant. The bulk phase
temperature profile is not shown, but qualitatively, it
increases along the reactor length. Clearly, the
effectiveness factor decays along the reactor length, and
therefore the conversion is decreasing, especially
towards the reactor exit. Since the AME/d3 is steep at
the vicinity of the reactor entrance, the majority of the
catalytic activity is situated near the reactor entrance.
Furthermore, a 3-1 multiplicity pattern for the reactor
tube was found and noticeably, the lower steady state
profile has the highest catalytic activity. Hence, for
controllability and optimality, the choice should be the
lower steadystate butwith thereactorsize reduceto a
sizeable fraction equivalent to some tolerable outlet
conversion.

Finally, the case of high solids density should be
interesting, because the inclusion of the axial thermal
conductive heat transport can induce multiplicity
patterns other than those obtained from a single
particle. Furthermore, it should be realized that an

internally isothermal, but externally non-isothermal

L15

 

.98-

.96_

.92

.86-

.811 ~—

\\
Lt

 

 

 

Figure 9 Concentration profile along the reactor
length for unique solution case.
6+ = 10.0, {54: 0.01.1, c134: 2.0,
Bim =00, Bih = 0.1, e°= 3.0, Da = 2.0

 

 

 

 

1.0
.93
.96
_91; ..
92 “
90 r

fl:

1 1 L i
0.0 2 U .6 R 1.0

Figure 10 Concentration profile along the reactor
length for the multiple solution case.
U; = 10.0, {34: 0.015%2 1.1:,
O

Bim =00, Bih = 0.5, Q = ?.0, Da = 2.0

47
catalyst pellet was used as a model here such that the
consequences of these, as compared to the full non-
isothermal case, is only but the later's shadow. However
limited, it yet provides some insights into heterogeneous

chemical reactor design.

 

 

APPENDICES

 

Appendix A. Computer Program and Sample

Calculations

 

 

48

INPUT;*AP

(INPUT;CUTFUT9TAP
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50

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51

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CALCJLATE THE ROOT 0F V FOR A GIVEN SET OF D VECTOR

C
C
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T.

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T. c.

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+ AU

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A
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THE ROOT OF V(ROOT SEARCH)

ESTIMATE

8

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c

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—

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X CONCENTRATION
0.0050 .957059
.0500 .957165
.IOOO .957482
.1500 .958011
.2060 .958752
.2500 .959706
.3500 .960872
.3500 .962251
.4000 .963644
.4500 .965652
.5000 .967675
.5500 .959914
.6900 .972371
.6500 .975f46
.TFOO .977941
.7560 .981056
.8535 .984394
.8500 .9879F6
.9000 .991743
.9503 .995757
1.0000 1.030000
:03 05/19/83 1862628 357 Lines print.

ages print. Cost at R62 is s 3.36

 

55
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59

10E= .0014451905
TEMPERATURE: 5.9508423?
01F: 0.000000

EFFECTIVENESS: 44.20394969
YOE= .0181947795
TEMPERATURE: 2.28039119
DIF= .000000
EFFECTIVENESS: 11.43206424
YOE= .4171244597
TEMPERATURE: 1.13480076
DIP: 0.000000

EFFECTIVENESS: 1.20357822

UZERC= .990000

THI= .000075

EFFECTIVENESS FACTOR: 44.203950
UZERO= .990000

THI= .000292

EFFECTIVENESS FACTOR: 11.432064
UZERO: .990000

THI= .002770

EFFECTIVENESS FACTOR: 1.203578
YOE= .0015558120

TEMPERATURE: 5.62562623

DIF= 0.000000

EFFECTIVENESS: 35.46469569
YOE: .0181870902
TEMPERATURE: 2.24302814
DIF= .000000
EFFECTIVENESS: 9.5303019?
YOE= .3599677158
TEMPERATURE: 1.15474818
DIF= -.003000

EFFECTIVENESS: 1.18645495‘

 

 

 

6O

UZERO= .980000
THI= .000230
EFFECTIVENESS FACTOR:

quRo= .9écboo
THI= .000375
EFFECTIVENESS FACTOR:

UZERO= .980000
THI= .007895
EFFECTIVENESS FACTOR:

TOE: 00917283939

35.464696

9.530302

1.186455

TEMPERATURE: 5.2530188?

DIF= -.000000

EFFECTIVENESS: 28.40852006

YOE= .0175682262

TEMPERATURE: 2.23051404

DIF= .000000

EFFECTIVENESS= 8.21935759

YCE= .3110135554

TEMPERATURE= 1.17450726

DIP: “0030000

EFFECTIVENESS: 1.16564096

UZERO=' .970000

- THI= .000542

EFFECTIVENESS FACTOR:

UZERO= .970000
THI= .002040
EFFECTIVENESS FACTOR:

UZERO= .970000
THI= .017817
EFFECTIVENESS FACTOR:

28.408520

8.219358

10165641

 

61

YOE: .0019935515
TEMPERATURE: 4.83562908
_DIF= -.eooooa

EFFECTIVENESS: 22.61875262
YOE= .0162490481
TEMPERATURE: 2.24836615
DIF= .000000

EFFECTIVENESS: 7.36163081

YOE= .2700877915
TEMPERATURE: 1.19355393
«DIP: - 0.000050

EFFECTIVENESS: 1.14138995

UZERO= .960000

THI= .001174

EFFECTIVENESS FACTOR: 22.618753
UZERO= .960000

THI= .004354 '
EFFECTIVENESS FACTOR= 7.361631
UZERO= .960000

THI= .037397

EFFECTIVENESS FACTOR: 1.141390
YOE= .0024224507

TEMPERATURE: 4.3667104?

DIF= ‘ 0.000000

EFFECTIVENESS: 17.74185493
YOE: .0141333163
TEMPERATURE= 2.31049110
DIF= .030000
EFFECTIVENESS: 6.90601196
YOE= .2366469109
TEMPERATURE: 1.2112479?
DIE: 0.000000

EFFECTIVENESS: 1.11323228

 

62

UZERO=‘ .950000

THI= .002445

‘EFFECTIVENESS FACTOR= 17.741855
UZERO= .950000

THI= .008958

EFFECTIVENESS FACTOR: 6.906012
UZERO= .950000

THI: .076342

EFFECTIVENESS FACTOR: 1.113232

TOE: .0032401414

TEMPERATURE: 3.80965589
DIF= 0.000000

EFFECTIVENESS: 13.38010514
YOE= .0110275784
TEMPERATURE: 2.45970353
OIF: .OOOOOO
EFFECTIVENESS: 5.95138032
YOE: .2099453041
TEMPERATURE: 1.22687901
DIF= O.OOOOOO

EFFECTIVENESSz 1.08044015

UZERO= .940350

THIz .005001

EFFECTIVENESS FACTOR: 13.380105
UZERC: .940050

THI= 0018130

EFFECTIVENESS FACTOR: 6.951380
UZERO= 0900000

THI= 0154062

EFFECTIVENESS FACTOR: 1.080440

 

 

63

YCE= .1891399813
TEMPERATURE: 1.23978371
DIF: 0.000000

EFFECTIVENESS: 1.04256630

UZERO= .930300
THI= .309364
EFFECTIVENESS FACTOR: 1.042566

¥0E= .1733213659
TEMPERATURE: 1.24949420
DIF= 0.000000

EFFECTIVENESS: .99985193

UZERO: .920000
THI= .619856
EFFECTIVENESS FACTOR: .999852

YOEZ .1616009484
TEMPERATURE: 1.25584484

DIF= 0.000000

EFFECTIVENESS: .95331954

UZERO: .910000
THI= 1.240753
EFFECTIVENESS FACTOR: .953320

 

 

LIST OF REFERENCES

LIST OF REFERENCES

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2. Carberry, J. J., "On the Reactive Importance of
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3. Lee, J.CL M.,and Luss,‘D” "Maximum Temperature Rise
Inside Catalytic Pellets," Ind. Engng. Chem.
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H. Kehoe, J.Fh G” and Butt,.L B.,"Interactions of
Inter- and Intraphase Gradients in a Diffusion
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(1972).

5. Butt, J. 8., Downing, D. M., and Lee, J.VL, "Inter-
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6. Pereira, C..L, Wang, J.IB” and Varma, A.,"A
Justificatin of the Internal Isothermal Model for
Gas-Solid Catalytic Reactions," AIChE J. 25, 1036
(1979). ——

7. Aris, R., The Mathematical Theory of Diffusion and
Reaction in Permeable Catalysts, 2 Volih, Oxford
(1975).

 

8. Cresswell, IL I“, "On the Uniqueness of the Steady State
of a Catalyst Pellet Involving both Intraphase and
Interphase Transport," Chem. Engng. Sci. gg, 267
(1970).

9. Hatfield, B., and Aris, R., "Communications on the
Theory of Diffusion and Reaction-IV. Combined
Effects of Internal and External Diffusion in the
Non-Isothermal Case," Chem. Engng. Sci. 23, 1213
(1969).

69

 

65.

10. Pereira, C..L, Carberry, J..L, and Varma, A”
"Uniqueness Criteria for First Order Catalytic
Reactions with External Transport Limitationsfl'
Chem. Engng. Sci. 33, 239 (1979).

11. Pereira, C. J., and Varma, A., "Uniqueness Criteria of
the Steady State in Automotive Catalysis," Chem.
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12. Van den Bosch,IB” and Luss,lD” "Uniqueness and
Multiplicity Criteria for an nth Order Chemical
Reaction," Chem. Engng. Sci. 32, 203 (1977).

13. Gattifredi, J.(L, Gonzo, E.EL, and Quiroga, 0.1L,
"Isothermal Effectiveness Factor-I," Chem. Engng.
Sci. 39, 705 (1981).

13. Morbidelli, M” and Varma, A” "Explicit Multiplicity
Criteria for First-Order Catalytic Reactions with
External Transport Limitations," Chem. Engng. Sci.
36, 1211 (1981).

15. Jackson, R., "Some Uniqueness Conditions for the
Symmetric Steady State of a Catalyst Particle with
Surface Resistances," Chem. Engng. Sci. 21, 2205
(1972).

16. Luss, D” "Uniqueness Criteria for Lumped and Distributed
Parameter Chemically Reacting Systems," Chem.
Engng. Sci. _2_6_, 1713 (1971).

17. Drott,IL‘w” and Aris, R” "Communicationscnmthe
Theory of Diffusion and Reaction-I. A Complete
Parametric Study of the First—Order, Irreversible
Exothermic Reaction in a Flat Slab of Catalyst,"
Chem. Engng. Sci. 23, 531 (1969).

18. Copelowitz, I., and Aris, R., "Communications on the
Theory of Diffusion and Reaction-VI. The
Effectiveness of Spherical Catalyst Particles in
Steep External Gradients," Chem. Engng. Sci. 23,
885 (1970).

19. Maym6,.L A.,and Cunningham,fh E” "Effectiveness
Factors for Second Order Reactions," J. of Catal.
6, 186 (1966).

20. Hlavéc‘ék, V., and Kubicgk, M., "Modeling of Chemical
Reactors--XXII. Effect of Simultaneous Heat and
Mass Transfer Inside and Outside of a Pellet on
Reaction Rate--II," Chem. Engng. Sci. 23, 1761
(1970).

 

 

21.

22.

23.

23.

66

Wong, S. H., and Szepe, 8., "Effectiveness Factors for
the General Non-Isothermal Case of Bimolecular
Langmuir-Hinshelwood Kinetics," Chem. Engng. Sci.
31, 1629 (1982).

DeVera, A.I“, and Varma, A” "Optimal Bulk Phase
Composition for an Isothermal Second-Order Reaction
in a Catalyst Slab--Elliptic Integral Method," Ind.
Engng. Chem. Fundls. 13, 320 (1980).

Byrd,I%.F.,and Friedman,bL D” Handbookcfi‘Elliptic
Integrals for Engineers and Scientists, Spinger-
Verlag, New York, 1971.

 

Villadsen, J., and Michelsen, M., Solution of
Differential Equation Modelstanolynomial
Approximation, Prentice-Hall, 1978.

 

 

   

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