An Integral Equation Method for Evaluating the Effects
of
Film
and Pore Diffusion of Heat
and
Mass
on Reaction Rates
in
Porous
Catalyst
Particles
ARTHUR
S
KESTEN
The effective design of catalytic reactors requires ac curate procedures to predict the rates of chemical reaction on the catalyst surfaces. If the catalyst material is im pregnated on the interior and exterior surfaces of porous particles, the diffusion of reactants into the porous struc ture can greatly influence the overall reaction rate. In addi tion, the conduction of heat within the porous particles can affect the rate since the chemical reactions are ac companied by an evolution or absorption
of
heat. The rate of reaction can also be limited by the rate of heat and mass transfer from the bulk fluid, through a stagnant film surrounding the catalyst particles, to the outside surface of the particles. The effects of film and pore diffusion of heat and
mass
on reaction rates in porous infinitely long cylindrical cata lyst particles, have been considered for simple reaction kinetics by Miller and Deans
1)
and the effects of pore diffusion alone have been considered
by
Weisz and Hicks
2)
for spherical catalyst particles. In both of these cases, however, the pertinent differential equations were ex pressed in finite difference form and integrated numeri cally to determine concentration and temperature profiles within the catalyst particles. This procedure can be quite time consuming even with the use of a highspeed digital computer, particularly
if
the procedure is to be used re peatedly as in the analysis of a packedbed catalytic re actor where bulk fluid temperature and composition varies with position in the reactor. The appropriate differential equations can be converted to Fredholm integral equa tions which are much easier to solve numerically, particu larly for cases in which film diffusion of heat and mass significantly influence the reaction rate, An integral equa tion method is presented here for evaluating the effects
of
film and pore diffusion of heat and mass on reaction rates in porous catalyst particles. The method permits the com putation of temperature and concentration distributions within the particles for any reaction kinetics.
MATHEMATICAL MODEL
The system under consideration is a spherical porous catalyst pellet which is surrounded by a stagnant film
of
fluid. Reactant molecules must diffuse through this film and into the interstices of the porous particle before react ing on the catalytic surfaces. In describing the diffusion of mass within
a
porous pellet, it is assumed that Fick s law applies and that changes in the mass density of fluid within the particle are negligible in relation to changes in concentrations of the reacting species. Fourier s law is used to describe heat conduction within the catalyst pellet. Pressure changes within the particle resulting from non equimolar diffusion are neglected as is heat transported by pore diffusion of mass. Heat and mass transfer co efficients are used to describe film diffusion of heat and mdss. By assuming constant diffusion coefficients,
V
and effective thermal conductivities,
Kp,
within the porous structure, the equations describing heat and mass trans fer of a single reactant in a catalyst particle may be written as
Dp
V2
p~
h&
=
1)
United Aircraft Research Laboratories, East Hartford, Connecticut
where the concentration,
pp,
and rate of reaction,
rhet,
are expressed in mass units. The boundary conditions which consider diffusion of heat and mass through a film sur rounding
a
spherical particle are
3)
In a subsequent section of this paper Equations
1)
through
(4)
are further generalized to describe multi ple reactant systems and to include simultaneous catalytic reactions which are film diffusion controlled.
By
using Equations
1)
and
2),
Prater
3)
has pointed out that temperature and concentration are re lated quite simply by
(5)
P

Tp)s
=

DI
PP)
s

P
1
I p
The use of this relationship enables the reaction rate,
Thet,
to be written, for given surface conditions, as a function of concentration alone instead of concentration and tem perature. It is only necessary then to solve Equation
(1)
with
rhet
=
rhet
pp),
subject to boundary conditions
3)
and
4).
DEVELOPMENT
OF
INTEGRAL METHOD
If
the radius of the spherical catalyst particle is
01,
and if concentration
pp*
is defined such that
pp*
=
pp
i
Equation
1)
can
be
written as where
x
is the radial distance from the center of a sphere. The boundary conditions associated with Equation
(6)
are =Oatx=O
PP*
=pP*atx=a
PP
kc
(7)
dx
dx
DP
Equation
(6)
can be rearranged to get The solution to Equation
(8)
is most easily obtained by converting it into a Fredhdm integral equation of the form
4)
1
PB*(X)
=
x [u(x)u'(x)
u'(x)z)(x)]
where
u x)
is a solution of
du
dx
subject to the condition that
Page
128
AlChE
Journal
January,
1969
and
v
x)
is
a solution
of
d
dV
dX
x211
)
=
0
subject to the condition that The Green s function,
G(x,' )
s given by
u [)v x)
orOdELx
u(x)v('$)
orxL5da
(14)
(x,' )
=
{
The function
u x)
can be determined by first integrating Equation
(10)
to get If we apply Equation
(11)
together with the first of boundary conditions
(7)
to Equation
(15),
we find that
A1
=
0
and The function
v(x)
can be determined in a similar man ner by first integrating Equation
(12)
to get
u
=
B1
(16)
and then applying Equation
(13)
and the second of boundary conditions
(7)
to Equation
(
17)
to get Equations
(14), (16),
and
(18)
can now be combined to eet
(19)
In addition,
x'[u(x)
v'(x)

'(x)
u(x)]
=
AZBl
(20)
Equations
(19)
and
(20)
can now be substituted into Equation
(9)
to get or Equation
22)
is an implicit integral equation which can be solved numerically to determine the concentration at any point in a porous particle in terms of
pi,
the concen tration in the bulk fluid. Because of the dependence of the reaction rate, on particle surface temperature,
T,)
s,
and reactant concentration,
pP),,
it is necessary to solve Equation
(22)
simultaneously with Equations
3)
and
Vol.
15,
No.
AlChE
4)
o determine the concentration profile within the particle. Numerical methods have been developed to ac complish this and these have been programmed for digi tal computation. Typical computation time to generate concentration and temperature profiles by using a
UNI
VAC
1108
computer is
0.25
sec. This is at least an order of magnitude faster than the time required, with the same computer, to calculate these profiles by employing finite difference methods to solve the srcinal differential equa tions with the pertinent boundary conditions. In the special case of negligible film resistance to mass transfer (that is
k,
>>
Dp/cy)
Equation
22)
reduces to Since the reactant concentration at the particle surface is equal to the concentration in the bulk fluid in this case, the concentration profile within the particle can be deter mined from the simultaneous solution of Equations
(23)
and
(4)
only.
If
film resistance to heat transfer is negli gible also,
Tp)s
=
Ti
and Equation
(23)
can be solved alone for the concentration profile. For this special case the integral method has been used in analyses of the be havior of catalytic reactors for hydrazine decomposition
(5)
and for the dehydrogenation of methylcyclohexane
6).
SAMPLE CALCU LATlONS
This integral method has been used to compute con centration and temperature profiles in a porous catalyst particle used for the decomposition of ammonia in a gaseous mixture of ammonia, nitrogen, and hydrogen. The calculations pertain to a Shell
405
catalyst particle
7)
for which estimates have been made of the kinetics of the catalytic dissociation of ammonia
(8).
The reaction rate can be approximated by where
Tp
s in
OR
and the concentrations are expressed in lb/cu.ft. An illustrative example is considered for which the parameters are
Ti
=
2,024
OR
P
=
100
lb./sq.in.abs.
=
0.19
Mole Fractions
NH3
in Bulk Fluid
{
HZ
=
Oe5
N~
=
0.30
...
DpNH3
0.35
x
sq.ft./sec.
HNH3
=
1,405
B.t.u./lb.
kCNH3
=
5.0
ft./sec.
h
=
0.29
B.t.u./sq.ft.sec.OR.
Kp
=
0.40
x
low4
.t.u./ft.sec.OR. The temperature and ammonia concentration profiles within the catalyst particle are plotted for this case in Figure
1.
These profiles were computed by the simultane ous solution of Equations
(3), (4),
and
(22).
As pointed out previously, for cases like this one in which
kcNH3
>>
DpNH3/a
concentration and temperature distributions could be obtained by simultaneous solution of Equations
(4)
and
(23)
only. In catalytic reactor systems interest is centered pri marily on the flux
of
reactant material into the catalyst particles. This flux at the particle surface,
D,
dp,/dx)
or
k,[pi
P~)~],
s easily calculated once the concen tration profile is known. For the ammonia dissociation
a
=
103
ft.
Journal
Page
129
case discussed above, the mass flux
of
ammonia into the catalyst particle, normalized b dividing
by
kepi)
NH3,
is
2.
The reactant concentration profile and then the mass flux at the particle surface were obtained for temperatures between
1,700
and
2,700
OR ;
all other parameters were fixed at the same values used in computing the profiles shown in Figure
1.
The normalizing factor,
kcpi,
is the mass flux which would be obtained if the reaction were controlled by the film diffusion of heat and mass. For com parison purposes, normalized fluxes are also plotted in Figure
2
for the case in which film resistance to heat and mass transfer is negligible [that is
P~)~
pi
and
Tp)s
=
Ti]
and for the case where film and pore diffusion are sufficiently rapid
so
that the system is controlled by the rate of chemical reaction on the catalytic surfaces [that is
pp x)
=
pi
and
Tp x)
=
Ti].
or the case of negligi ble film resistance to heat and mass transfer, Equation
(23)
was used to determine the ammonia concentration distribution within the particle and then the flux at the particle surface. For the chemical reaction controlled sys tem, the rate of reaction was calculated from Equation
(24)
and the flux was then computed as
(a/3)
rhetNH3
(pi,Ti).
As shown in Figure
2,
these
two
cases repre sent lowtemperature asymptotes for the general case where the effects of film and pore diffusion on catalytic reaction rates are considered.
At
high temperature this flux asymptotically approaches the flux which would be ob tained if the reaction were controlled by the film diffusion of heat and mass. plotted as a function of bulk uid temperature in Figure
GENERALIZATION
OF
INTEGRAL
METHOD
The integral method described thus far can easily be
2
I
I
I
I
I
1700
1900
2100 2300 2500 2700
BULK FLUID TEMPERATURE.
T,
OEGR
Fig.
2.
Effect
of
bulk
fluid temperature on mass
flux
of
ammonia at surface of catalyst particle.
No
hydrazine present in
bulk
fluid.
adapted to include
a
treatment of simultaneous catalytic reactions, which are film diffusion controlled by simply modifying boundary condition
4)
o include the heat generated by these reactions. This can best be illustrated by considering the example of ammonia dissociation again, this time with hydrazine present in the bulk fluid surrounding the catalyst particle. At elevated tem peratures hydrazine decomposes catalytically at a rate
so
rapid that the decomposition is controlled by the rate of transport of hydrazine from the bulk fluid to the surface of a catalyst pellet [that is
pp)sN~H4
=
01
The integral expression for the ammonia concentration at any point in the porous catalyst is still Equation
(22).
The bound ary condition which considers diffusion of ammonia through a film surrounding the spherical particle remains as Equation
(3),
where the concentrations
pi
and
pp),
refer to ammonia. The boundary condition which con siders heat transfer through the film becomes
NH3
Hk,~i)~z~4
HNH3DpNH3
s
h [Ti
TP)J
0.016
0
0.2
0.4
0.6
0.8
1.0
Fig.
1
Temperature and ammonia concentration profiles within catalyst particle.
No
hydrazine present in
bulk
fluid. (see text for values of parometers required for numerical solution).
NORMALIZED RADIAL DISTANCE WITHIN CATALYST PARTICLE,
x
/a
4a)
where the first term on the left side of Equation
(4a)
represents minus the rate of heat generation
by
the film diffusioncontrolled catalytic decomposition of hydrazine. Equations
(3), (4a),
and
(22)
have been solved simul taneously for the temperature and ammonia concentra tion pronies in a porous catalyst particle for a case in which
Ti
=
2,000
OR
P
=
100
Ib./sq.in.abs.
NH~
=
0.43
{
~
=
0.23
HNH3
=
1,404
B.t.u./lb.
HNzH4
=
1,928
B.t.u./lb.
k,N
=
4.4
ft./sec.
k,N2H4
=
3.0
ft./sec.
h
=
0.29
B.t.u./sq.ft.sec.OR.
=
0.40
X
B.t.u./ft.sec.OR.
a
=
103
ft. Mole Fractions
N~H~
0.11
in Bulk Fluid
H~
=
0.23
DpNH3
=
0.34
X
sq.ft./sec. Both the temperature and the ammonia concentration distributions are plotted for this case in Figure
3.
Similar concentration profiles were calculated for bulk fluid tem peratures between
1,700
and
2,700
OR ;
all other param
AlChE
Journal January
1969
age
130
n
””
I
where
aA
+
bB
+
. .
.
tT
+
.
.
.
It
is apparent that the rate of the reaction can be expressed in terms of any of the reactants or products; however the rate may be a function of the concentration of any or all
of
those species. The same procedure used in
3)
can be employed to relate all reactant and product concentrations at any point within the porous catalyst particle to the con centration of reactant
A
at the same point. In addition, surface concentrations,
pp)
,
of all reactants and products can be written in terms of the surface concentration of reactant
A
and the known concentrations in the bulk fluid. Thus, as in the case of a single reactant, integral Equation
22)
(with concentrations referring to reactant
A)
may be solved simultaneously with the pertinent boundary con ditions to fully determine the concentration and tempera ture profiles within the porous catalyst particle.
0
0 2
04
0.6
08
10
NORMALIZED RADIAL DISTANCE WITHIN CATALYST PARTICLE,
x/a
Fig.
3.
Temperature and ammonia Concentration profiles within catalyst particle. Hydrazine present in
bulk
fluid. (see text for values of parameters required for numerical solution).
eters were fixed at the same values used in computing the profiles shown in Figure
3.
The flux of ammonia at the particle surface was then calculated, normalized by di viding by
(kcpi)NH3,
and plotted as a function of bulk fluid temperature in Figure
4
Here again, at high tem perature, the flux asymptotically approaches the values which would be obtained
if
the reaction were controlled
by
the film diffusion of heat and mass. At low tempera ture, in this case, film resistance to heat transfer remains important because of the heat generated at the particle surface
by
the decomposition of hydrazine.
A
second, perhaps more important generalization of the integral method presented here involves a description of multiple reactant systems. Consider a reacting system
riLM
OIFFUS ON
CONTROLLING EGLIGIBLE
FILM
/
lESISTLNCE
=

0
EFFECTS
CONSIDERED
I
I I
I
I
1900
2100
2300
2500
BULK
FLUID
TEMPERATURE.
T,
DEG
R
103
1700
Fig.
4.
Effect of
bulk
fluid temperature
on
mass flux of ammonia at surface of catalyst particle. Hydrazine present in
bulk
fluid.
NOTATION
a
b
D
h
H
k,
K,
P
=
pressure, Ib./sq.in.abs.
rhet
t
T
=
temperature,
OR
21
=
mathematical function
=
mathematical function
x
=
radial distance from the center of the spherical catalyst particle, ft.
=
coefficient defined
by
general chemical reaction
=
coefficient defined
by
general chemical reaction
=
diffusion coefficient of reactant gas in the porous
=
heat transfer coefficient, B.t.u./sq.ft.sec.OR.
=
heat of reaction per unit mass (negative for exo
=
mass transfer coefficient, ft./sec.
=
effective thermal conductivity
of
the porous cata lyst particle, B.t.u./ft.sec.OR. particle, sq.ft./sec. thermic reaction) B.t.u./lb.
=
mass rate of (heterogeneous) chemical reaction
=
coefficient defined
by
general chemical reaction on the catalyst surfaces, lb./cu.ft.sec.
Greek
Letters
01
[
pi
=
reactant mass concentration in bulk fluid,
lb./
pp
=
reactant mass concentration in gas phase within
ppQ
=
equals
p,

i,
lb./cu.ft.
=
mathematical function
Subscripts
p
s
=
radius of spherical particle, ft.
=
dummy variable in integral equations cu.ft. the porous particle, lb./cu.ft.
=
refers to bulk fluid phase
=
refers to gas within the porous catalyst particle
=
refers to surface of catalyst particle
LITERATURE CITED
1.
Miller, F.
W.,
and H.
A
Deans,
AIChE
J.,
13, 45 (1967).
2.
Weisz, P.
B.,
and J.
S.
Hicks,
Chem. Eng. Sci.,
17,
265
3. Prater,
D
C.,
ibid.,
8,
284
(
1958). 4. Irving,
J,,
and
N
Mullineux, “Mathematics in Physics and Engineering,” p. 731, Academic Press, New York (1959).
5.
Kesten,
A
S.
United
Aircraft
Res.
Lab.
Rept.
F91046112
Contract
NAS
7458 (May, 1967). 6. United Aircraft Corporation, “HydrocarbonFueled Scram jet (U),”
Vol
11
AFAPLTR67134
(Dec.,
1967). 7. Armstrong,
W.
E.,
D.
S.
LaFrance, and H. H. Voge,
Shell Deeelop.
Co
Rept.
S13864
(1962). 8. Kesten,
A
S.,
Paper presented at the Hydrazine Mono propellant Technology Symposium, Johns Hopkins Univ., Howard County, Maryland
Nov.,
1967). (1962).
Vol.
15,
No.
1
AlChE
Journal
Page
131