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A superposition integral equation for catalytic external surfaces
Chemical Engineering Science, 1970, Vol. 25, pp. 463-474.
Pergamon Press.
Printed in Great Britain.
A superposition integral equation for catalytic external surfaces R. MIHAIL Department of Chemical Engineering, Polytechnic Institute Bucarest, Polizu I, Bucarest 8, Romania (First received
20 May 1969; in revisedform
10 August
1969)
Abstract-A general superposition integral equation based on a suggestion by Eckert et al.[ 131 has been derived for the local rate of reaction on external catalytic surfaces of arbitrary shape. Proof of the validity of the equation is given for different kinetic systems and flow regimes. Results agree with previous work of Acrivos and Chambre[3] and Rosner[4]. Effectiveness due to the form of blunt catalytic surfaces can be inferred from wedge solutions. For porous catalysts both contributions of boundary layer and pore diffusion is taken into account by the introduction of an over-all effectiveness factor n70,a.This is a function of the local Thiele modulus and Damkohler number and characterizes the alteration of reaction rate as referred to the reactant concentration in an undisturbed stream. INTRODUCTION
IN RECENT years a great deal of work has been
carried out on the problem of catalytic reactions on external surfaces. The problem is both of theoretical importance, in determining the effect of the transport mechanism on the rate of surface reactions [l-6], and of practical importance: external flow reactions systems such as those encountered in the chemical industry[7-91 or in space techniques (e.g. frozen dissociated hypersonic flow) [lo- 121. In particular in kinetic analysis or design one is interested in knowing the influence of non uniform external reactant diffusion on the resulting reaction rate distribution. The known solutions to the problem are limited to isothermal catalytic flow reactions over flat plates with two dimensional boundarylayers or to axi-symmetrical flow in tubes, where the hydrodynamic conditions at the interface can be exactly described. In some of these solutions [ 1-33 the local surface concentration of the reactant depends formally on the local surface shear stress both variables having a similar form of variation along the surface of a flat plate, but not on the surface of a blunt body. Other solutions[4-6, lo] have been deduced by extending the von K&-man momentum integral equation, or the Falkner-Skan equation to concentration boundary-layers. In principle these solutions are valid for surfaces with pressure gradients, but were effectively applied only to the flat plate.
The primary purpose of this work is to present a simplified method for calculating the local concentration of a catalytic surface for either completely laminar or turbulent boundary-layer flows with and without pressure gradients over the surface. The solution is suggested by the previous work of Eckert [ 131 for calculating local heat flux to non isothermal surfaces. FORMULATION
OF THE
BASIC
EQUATIONS
An incompressible viscous fluid, containing a single reactant A flows past a solid catalyst of characteristic dimension L. In the carrier fluid the reactant A is present in dilute amounts. A boundary-layer of constant properties develops over the surface of the solid catalyst which consumes the reactant isothermaly at a rate R[c,(x)], determined by the local concentration of A next to surface. The two-dimensional continuum equations for continuity and momentum are:
(1)
~~+vau=-k!P+va2u ay
pbc
ay2'
(2)
The reactant concentration field in the vicinity of the solid catalyst satisfies the diffusion boundary-layer equation:
uac+u*~&c ax
463
ay
ay2*
(3)
R. MIHAIL
The boundary conditions of Eqs. (l-3) are: y = 0; u = 0,u = 0, Um,c=cm
y=w;u= x =
DE= R[c,(x)]
u,,
0; u =
(4)
c = cm.
N(x) = i
if for the differential linear equation, Eq. (3), a number of particular solutions Ci are available, which satisfy the boundary conditions (4), then the principle of superposition may be used and c=
If z, the distance from the which a step change occurs, parameter then the local mass is a function of coordinates x flux at x becomes:
2 K,ci
(5)
1
is also a solution of Eq. (3). The constants Kt can be used to adjust this new solution to the required boundary conditions. This method enables the substitution of a continuous variation of reactant concentration on the catalyst surface with a discontinuous one in steps, as is depicted in Fig. 1.
Each step behaves as a surface of constant concentration, for which the mass flux may be computed by known methods. For each step the flux of A to the catalyst surface is: o = k,,,,*Ac*= R(x)
(6)
where Aci is the concentration difference betwen surface and the fluid at the edge of the boundarylayer. With Eqs. (5) and (6) the flux of A to the surface is given by N(x)
= -D
i
1
Ki(2),
= i
1
Kfkm,dciv
k,(x,
t)Aci
(8)
the constants Ki being equal to unity[ 141. if the step changes are considered to be built from infinitesimal increments of concentration dc, which succeed each other at infinitesimal intervals dz then the sum of the right hand side of Eq. (8) may be replaced by: N(x) =
dcotz) Mx,z)ypk
(9)
which gives the continuous variation of the surface concentration and must be interpreted as a Stieltjes integral. At steady state the mass flux is identically equal to the local chemical rate of conversion at interface, thus for a surface of uniform temperature
R[cob)l=
Fig. 1. Step change of surface concentration.
1
stagnation point at is considered as a transfer coefficient and z and the mass
z k,(x, 4 !i!%&~ I0
(10)
Equation (10) may be used for bodies of different shapes and for laminar as well as for turbulent flow conditions provided that the required expressions for the local mass-transfer coefficient k,,, (x, E) following a step in surface concentration are available. Now, if k,(x, z) is referred to the mass transfer coefficient of a surface with locally constant concentration k,(x) by k,(x, z)/k,(x) = cp(x, z), Eq. ( 10) may be written as:
R[cotx)l = k,(x)~cotO)
+[
p(x,t)~dz] (11)
where the first term in the right hand side bracket indicates that the only concentration discontinuity occurs at the leading edge.
(7) 464
A superposition integral equation for catalytic external surfaces
Next the versatility of Eq. (15) will be illustrated by calculating some numerical examples. In order to compare the results with those known from other solutions the discussion will be limited to boundary layer flows past flat plates or wedges, more complicated geometries do not offer unusual difficulties.
Eckert [ 151 solved a similar integral for convective heat transfer from a nonisothermal surface; for the problem under consideration the solution is to approximate the actual wall concentration distribution by a series of sloping segments, thereby making dc,/dz a constant value over any interval Ax which allows its removal from the intergal sign. Eq. (11) then reads:
SQME NUMERICAL SOLUTIONS THE FLAT PLATE
R[c,b)l = k,(x+l~O) +
FOR
The irreversible reaction
2 ($jJ~,
cp(X~4q.
Consider the p-th order irreversible reaction:
(12)
i=l
pA + B The general form of the integrals in this equation may be written as follows:
with the rate law described by a power function of the local reactant concentration
cp(x,z) d ;
cp(x,z) dz=x
R(C‘J = kd,.
0
-I
x,-,/x
()I
cP(x,t)d; .
0
(13)
(x)
=
a(z)
+
b(:r.
(16)
For a laminar boundary layer flow past a flat plate the functioncp (x, z) is[14]:
The tabulated values of these integrals, which otherwise can be calculated without difficulties, can be approximated with good accuracy by the following simple expression: cp(x,t)d”
0)
cp(x,t) = [1-(;)3’4]-1’3
(17)
and the constants in Eq. (14) have the values [ 131:
(14)
a = O-895 (18)
b = 0.690. Assuming that the length of the catalytic surface is subdivided in II equal intervals, the substitution of the relation (14) into Eq. (12) gives:
On substituting Eq. (17) and the constants a and b into Eq. (15), we obtain after some simple rearrangements:
R[co(x)l = k,(x) AcoW)+a[Aco(n) -Aco(0)l +i
[
(2n-
a+:
[
1
1+2 ‘i c;Jo(llz)
l)Aco(n>
-AC,(O) -2 %1 c,(m)]}.
(15) where CL (0) = 1 and the Damkohler number
The values of the constants a and b in this equation depend only on the form of function (c (x, z) , specific for the geometry of the catalytic surface and for the flow conditions: laminar or turbulent. 465
CBSVdUNa3-H
Da
=- kd,’ km
is a measure for the relative contribution
of sur-
R. MIHAIL
face kinetics and mass transfer to the value of c,*,(x). If one choose a generalized coordinate along the catalyst surface, then a more detailed expression for Da is D Da = Ckcf&l -_S@Re L
-1
x
l/Z
L1/2
(21)
IO E
and by putting Da = n the reactant concentration distribution czO(Da; p) shown in Fig. 2 results. a=0+95 b=0.690
I
Fig. 2. pA
+ B;
2
3
4
5
6
7
6
9
IO
reactant concentration distribution ~,*,,@a;
p): -,
laminar; ----,
turbulent.
The generality of Eq. (19) with respect to the Schmidt number is included in the nature of the stretched coordinate Da, in particular in the numerical value of the constant C. By adapting the proper expression for k,, Eq. (19) applies for all Schmidt numbers so that in the range 0.5 < SC < lo4 C has the value (2/0*333 X O-215) l13. Forp = 1 the solution is easily compared with the corresponding results of Chambre and Acrivos [2] and Rosner[4], Fig. 3. While the numerical results obtained by Rosner using an approximate solution are 5 per cent less than the analytical solution of ChambrC and Acrivos, the present approach yields values 5 per cent higher. But in both cases the figures could be brought as near the analytical solution as wished either by improving the function of similarity in the Rosner Eq. (8)[4] or by evaluating exactly the integrals (13) instead of using the approximated formula (14) in the deduction of Eq. (15). First-order reversible reaction Let the surface reaction be A*B
01)
with the rate law of form R(cAor cs,) = k(c,,-2).
(22)
An Eq. (15) is written for each component: cAo
-.cBo= - 1 K D&A
I
Ac/io(O)
+a[ACA,(n) Chombri
-AC~o(o)l
and Acrivos [2] +;
(h-
I)fkA,(n)
[ -
-(CAo-2)
I
ACAO(O)
I
I
1
2
3
4
+;
Da
Fig. 3. A -+ B; reactant concentration distribution c& (Da); comparison with previous results [2] and [4].
2 7’
CAh)]}
(23)
= +-bh,(“)
+a[AC,,(n) I
-
b [
-b+,(O)1
(2n - l)A+,,(n) n-1
-Acs,(O)--
466
T
CUB,,
11
.
(24)
A superposition integral equation for catalytic external surfaces
reactions is illustrated with the most simple case
The boundary conditions are:
(25)
cil, = 0
in which both steps are first order. The reaction rates are RA = klcA
i.e. in the incident fluid there is no reaction product. Dividing both Eq. (23) and (24) by cA, and eliminating c& one obtains
RB = -klcA + kzcB R, = -kzce.
(28)
{K[a+(~)b]+Da,B}[a+~(l+2~cr.(m))]+Da,,%*2+Am)
c‘&(n) =
by substituting
(26)
[a+(~)b][K[Da,A+a+(~)b]+Da,J the calculated values of c+&(n) in Eq. (24) it results: c&(m) - (2n-
II
l)&(n)
1 .
Equation yielding
The results of numerical computations of the dimensionless surface concentrations CT” and c& are plotted in Fig. 4 against the generalized coordinate Da,A, with the equilibrium constant K as parameter. The influence of flow and diffusion on the concentration distribution is striking.
(15) is written
c&(n) = L Da,, +p
aLI
A*
0.9+-
07-
‘!,
0.5 0.6 -
‘\ yK .b : .4 ., ‘.\ ‘.>.
0.403-
.& .fl
0.2 -j’
-c*
1 = - Da,,
k2 ~c.?O(n) +Kct(n)
-ac&(n) I
.-.
+;[2c&(m)
00
-
‘.I?.--,_ +._.
-_ -.__
/Go
I
II
.-
‘.\ $ -o La
/4
I)cA*,(n)
(29)
,_‘~?~_=~=~=-------
DE-%
-G4n)l
1
The ability of the proposed method to predict reaction rate distribution for sequential surface I-O
for each component
1 + 2 ‘g c,&(m) - (2n-
First-order consecutive reactions
(27)
2
I 3
1 4
---_ ---._._I&--,
I 5
1 6
-+c&(n)
_ 1 7
I E
1 9
1
=+ ?C
@n-
l)cB(n)])
(30)
-ac&(n) 1
IO
Da sp.
Fig. 4. A 4 B; surface concentration distribution, with equilitkium constant K as parameter: -, K = 0.1;-----, = IO;-.-.-,=lOO.
467
(31)
R. MIHAIL
The present method can also be readily geneqlized to include turbulent and transition flows along a catalytic plate surface. One has only to consider the proper expression of the function cp(x, z). Of the different theoretical results known in the literature that of Seban[ 171 for turbulent flow with constant velocity over a flat plate, i.e.
in which it was assumed that the free stream concentration of B and C are nil. Also the components concentration satisfies the relation DaYA CZ~(~)+rc;O(n)
Da,* +D,c;O(n)
-B
= 1.
(32)
rc
The calculated surface concentrations are represented in Fig. 5 with Da,A as abscisa and k,lk, as parameter.
cp(x,z) = [t_(;~lro]-l+ was selected. On introducing this expression into Eq. (15) one obtains for constants a and b the values [ 131:
A-B-?-Ck k,
*y=
a = 0991
---.-._,
-0 3
b = 0.117.
I
2
3
4
5
6
7
6
9
-IO
DQqA Fig. 5. A 5 B 5 C; surface concentration distribution, with
/c,/k, as parameter: -.-.-,
k,/k,= 0.1; -,
=l.O; ----.
= 10. When the surface concentration mediary product is in the form
of the inter-
kZDa,, k,Da,8 cio
(33)
where
k2Da,A --= kl D&B
Sh (x) = 0.33 (Re,) WC~‘~, laminar; 0.6 < SC < 10. Sh (x) = O-0296(Re,) ‘YSC~/~,turbulent
’
it is possible to compare the results with those of AC&OS and ChambrC [3, Table 31;the maximum error is less than 8 per cent. Again this value could be reduced by calculating exactly the integrals (13) instead using the approximation (14). TURBULENT
For reaction (I) the surface concentration is shown in Fig. 2 which permits a comparison to be made with the results of Rosner[6]. In order to compare the effect of laminar and turbulent flow on the same surface reaction the use of Damkiihler number as coordinate could be misleading since the form of this coordinate is not identical for laminar and turbulent flow. A dimensionless coordinate x/L was selected. Consider a flat plate of length L whose surface catalyses a first order irreversible reaction (I). At Re8 = 160400 the boundary layer can be either laminar or turbulent. The respective equations for the local mass transfer coefficients
BOUNDARY
can be written as L l/2 k,(x)
= 0-33(ReL)1YW’3
k,(x)
= 0.0296(Re~)“Yc”3
y
0
;L O.* 0
(34)
it was accepted
that
LAYERS
Previous results concerning surface reactions with turbulent boundary layer includes those of Rosner[6] and Ruckenstein[l6].
where,
for
simplicity,
D/L = 1.
This expression 468
allows the local Damkohler
A superposition integral equation for catalytic external surfaces
number to be calculated so that for given values of Re,, SC and reaction velocity constant k, Eq. (19) can be used. Figure 6 shows the surface concentration of reactant A as a function of distance x/L for p = 1. As expected the surface concentration is higher IO
,
,
,
,
,
,
,
,
,
Re?eb”o&oo
0.9-
k
= 373m/h
utilize the local surface shear stress as independent variable is valid only for slender bodies, because for blunt bodies the surface shear stress and local mass transfer coefficients (consequently local surface concentration) have a dissimilar type of variation due to the contribution of form drag. Wedge type flows described by U, a xm are particularly suitable to simulate the behaviour of specific geometries which can be encountered as catalytic surfaces. The general form of the functioncp (x, z) for laminar flow is:
01 a
pOx,z)=
::[
,
,
,
,
0.1
0.2
03
O-4
, 05
,
,
06
0.7
, 06
1
, 09
IO
X/L
Fig. 6. A + B; turbulent boundary layer; c;.~ vs. x/L.
in turbulent than in laminar flow while the reactant supply is more intensive. Turbulent boundary layers are less sensitive than laminar layers to streamwise gradients of concentration exhibiting a more uniform depletion of the reactant and a higher effectiveness factor. WEDGE
FLOWS
As mentioned the above method of solution is general because it is applicable to arbitrary catalytic surfaces. In contrast, any solution which
k,(x)
b
C
P
m
Flate plate
0 215
3/4 0.895 0.690 0,332 0 114 15/16 0.837 0.635 0.412
f/2 213 617
-1 0.825 0.610 0.430 l/3 112 918 0.840 0.572 0.469 314 21/16 0.545 0.785 0.518
Sphere-forward stagnation point Cylinder-forward stagnation point
1
1
.
(35)
=
C(ll/x)Re,"S~"~
where the constants C and n are given in Table 1. The results are shown graphically in Fig. 7, with m as parameter. It is seen that for a given value of the reduced coordinate Da the surface concentration diminishes with increasing m, meaning that the rate at which reactant can be supplied to the catalyst surface is smaller. This becomes more
Surface
u
-113
When this expression is substituted in Eq. (13) and the integrals are calculated the results of Table 1 apply [ 131. For a first order irreversible reaction the surface concentration of reactant was computed using the values of a and b from Table 1. The coefficient for mass transfer of a surface with uniform concentration, needed for computing the Damkiihler number Da has the form:
Table 1. Parameters for wedge flow (CJ, = xm) and local mass transfer coefficient (k, = C (D/x)R~,~~~S~“~) a
[
1-f
312 0.192 0.538 0.560
R. MIHAIL
Fig. 7. A -+ B; wedge flows; reactant concentration distribution c&, (Da; ml.
NON-ISOTHERMAL
evident by calculating the external effectiveness factor, defined by &Ck&I.
REACTION
When a non-isothermal reaction occurs on a catalytic external surface there will generally be
(361 oAoor
Values of the integrated external effectiveness factor
have been computed and the results are shown as a function of m in Fig. 8, forDa = 7. From these results the behaviour of a blunt catalytic surface could be foreseen, at least for the front flow area. The external effectiveness factor depends on the surface geometry as does the internal effectiveness factor. For the front stagnation point the effectiveness is smaller in the order: fiat plate-sphere-cylinder. It is to be expected that in complex catalytic reactions variation in the effectiveness due to surface geometry will affect selectivity and yield.
,P
OS0-
O?‘GO’
0
‘ l/4
his, 8.A -, B; wedge
470
L I13
I 112
m
3pI
flows; ;? integrated external effective ness factor vs. m, for Da = 7.
A superposition
integral equation for catalytic external surfaces
continuous variation of concentration and temperature along surface. The method of solution of such a problem was first outlined by Chambt-6 [ l] and his analysis is readily applicable to the present method. The model of a non-isothermal reaction is described by the previous Eqs. (I-3) with an additional equation for conservation of energy. a
The boundary conditions are: c_46% m, = cz4(0, y) = T(x,
m)
-A,
T(0, y) = T,
=
o=
--DA
(
$
CA,
To(x)1
R[cAo(x)v
o = (-AHR) >
. R[cA,(x),
T&)1.
(39)
The differential equations describing the concentration and temperature field being similar we proceed as before. Finally one has to solve the two coupled equations:
(-A~idk(To)cAo(X)
=
+A,++
(40)
I”D(&$k] 0
and: e(x)
=k,[i\To+j-+t)~&]
(41)
where+(x,z)
= kh(x,z)lkt,(x). For a first-order irreversible reaction (I) and after introducing approximation (14), Eqs. (40) and (4 1) become ~(To)cA,(x)
=
km
AcA~(O) {
+~CACA~(~)
+i
[
-Ac~o(o)
-AcA~(O)I
l)AcAo(n)
(h-
-
2 2’
AC,.(d])
(42)
+o
+a[ATo(n) -AT,(O)1 +;[
(2n-
l)ATo(n)
The nonlinear equation system (42-43) can be solved by any common iterative procedure starting with known values of cAO(0) and T,(O) and specifying the amount of heat transfered to or from the catalytic surface. Finally CA0and To will result for each n. For some catalytic reactions of industrial interest such as NH, oxidation e.g. the high thermal conductivity of the metal catalyst produces a uniform temperature over the surface. The heat flux, which varies along the surface due to the variation of reactant concentration may then be computed by usual methods, provided that the local heat transfer coefficient is known.
OVER-ALL R[cA,(x)l
=
EFFECTIVENESS
FACTOR
The influence of a non-uniform environment on the rate of a catalytic surface was first considered by Petersen et al.[ 181 who discussed the effect of concentration and temperature gradients on surface reaction on a non-porous sphere. Recently Bischoff[ 191 solved a problem with varying surface conditions. The particular problem was that of a spherical, porous, catalyst particle immersed in a fluid with an axially linear temperature gradient at the particle surface. It would be of interest to consider the influence on the internal effectiveness factor, not of an arbitrary varying surface condition, but that due to a boundary-layer developed by fluid flow over a porous catalyst. The discussion will be limited to a flat plate of infinite thickness on which an isothermal catalytic reaction occurs; other geometries can be treated, but no new features are introduced. 471
R. MIHAIL
Let the surface reaction be (I); the diffusion equation inside the porous catalyst is:
It is obvious reaction (I)
that for a first-order
irreversible (49)
%.a = c* (x, 4.
(44) The integral over-all effectiveness where z is the depth of penetration and x a coordinate along the flat plate in the direction of flow. The boundary conditions are z=o,
%.a =
US Js RkA(.dl
cXh,Da)
The solution of Eq. (44) with the above boundary conditions is CA(XVt) = CA0(x) . e-+
(46)
where & is a local Thiele modulus and c,&,(x) is the surface concentration resulting from solution of the boundary layer equation. Depending on the desired accuracy and time allowed for computation one can substitute cAo(x) in Eq. (46) with one of the expressions derived by Chambre[l], ChambrC and Acrivos [2], Rosner[4] or with Eq. (19) from present work. Upon introducing reduced dimensionless coordinates for the axial position and depth of the porous plate we have
d-d
(50)
R [CAJ
or with reduced coordinates,
c‘4= c‘4(x)
factor will be
for reaction (I)
d(#,)
Wa).
(51)
The over-all effectiveness factor distribution inside a porous catalyst as calculated from Eq. (5 1) is shown in Fig. 9. P % t
Ii
I.0 09
+t\
(47) Thus a complete description of the concentration field inside a porous flat-plate catalyst is available as function of the two dimensionless coordinates, local Thiele modulus and Damkohler number. The internal (diffusion determined) effectiveness factor and the external (boundary-layer determined) effectiveness factor are coupled in an over-all effectiveness factor referred to reactant concentration in the undisturbed fluid %.a. =
R [CAk 2) 1
R[cA,I
*
(48) 472
Fig. 9. A --, B; over-all effectiveness factor distribu-
tion ?,.. (4.; Da). CONCLUSIONS
The computational technique of Eckert, Hartnett and Birkeback [ 131 for heat transfer to a non-isothermal surface has proven useful in the analysis of the significant variables involved in external catalytic reactions. The main feature of the superposition integral equation proposed in this work is its applicability to various surface
A superposition integral equation for catalytic external surfaces
Re
geometries, with or without pressure gradients and to both laminar or turbulent flows. The contribution of surface geometry to the effectiveness of an external catalyst was illustrated by examining wedge flows. The extension of the proposed method to nonisothermal reactions is easily possible along the lines shown by Chambre [ 11. An over-all effectiveness factor was introduced to account for all specific resistances’encountered in reactions catalyzed by porous solid bodies subject to boundary-layer flows.
number
for
flat plate,
Re = UJJv
Re,
local
Reynolds
number,
Re, =
U.&IV
Re8 Reynolds
number for boundary layer, Re = U,6/v S normalized surface area, S = 4I . Da SC Schmidt number, SC = v/D Sh Sherwood number, Sh = k,(x). x/D T temperature difference across therAT temperature mal boundary layer velocity in x or z direction in the I.4 boundary layer velocity at edge of boundary layer u velocity in y direction V distance measured parallel to catalyX tic surface, from leading edge Y distance measured perpendicular to catalytic surface 2 distance at which a step change occurs t depth of a porous catalyst
NOTATION
a, b constants defined by Eq. ( 14) reactant concentration dimensionless concentration c” = clc, difference across AC concentration mass transfer boundary layer AC = c, - c(x) c constant, in Eq. (21) D diffusion coefficient effective diffusion coefficient in a D” porous catalyst Damkijhler number heat of reaction (-*I$ k, k’, k,, kz reaction velocity constants local mass transfer coefficient k,(x) k,(x, z) local mass transfer coefficient for a step change at z, measured from leading edge local heat transfer coefficient k/i(x) local heat transfer coefficient for a k,, (x, z) step change at z, measured from leading edge K constant in Eq. (5) K equilibrium constant L characteristic dimension of catalyst m constant, velocity equation for wedge flows dimensionless coordinate mass flux to catalyst surface N(x; reaction order heat flux, Q(x) = -(AH,)k(T,)C,,(x) ecx? R rate of surface reaction
Reynolds
C
c*
Greeksymbols constant defined by Eq. (35) parameter for wedge angle, pm s boundary layer thickness 71 effectiveness factor, 7) = Cg (x) ?1 integrated effectiveness factor %.a. over-all effectiveness factor A thermal conductivity of fluid V kinematic viscosity of fluid P density of fluid (c (x7 z) ratio of mass transfer coefficients, cp(x, z) = k,(x, z)lkrt(x) Thiele modulus, & = #Jz local ;
zd(kC$;l/D*)
Iclk z) ratio of heat transfer coefficients, H-G z) = kh(x, z)lk,,(x) Subscripts 0 m i
A,B,C 473
surface conditions undisturbed flow conditions number of step change reactants and products
R. MIHAIL REFERENCES [l] [2] [3] [4]
CHAMBRE P. L., Appl. scient. Res. 1956 A6 97. CHAMBRE P. L. and ACRIVOS A., J. appl. Phys. 1956 27 1332. ACRIVOS A. and CHAMBRE L. P., Ind. Engng Chern. 1957 49 1025. ROSNER D. E., J. AerolSpace Sci. 1959 26 281. [5] ROSNERD.E.,A.I.Ch.EJl19639321. [6] ROSNER D. E., Chem. Engng Sci. 1964 19 1. [7] NOWAK J. E., Chem. Engng Sci. 1966 21 19. [8] MIHAIL R., Chem. Technik. 1957 9 344. [9] MIHAIL R. and HERSCOVICI J., Kinet. Katal. 1962 3 836. [lOI LI Y. T. and KIRKS. P., Int. J. Heat Mass Transfer 1967 10 257. I1 11 INGER G. R.. Int. J. Heat Mass Transfer 1963 6 815. [12] CHUNG M. P., LIU W. S. and MIRELS H., Int. J. Heat Mass Transfer 1963 6 193. 1131 ECKERT G. R. E.. HARTNETT P. J. and BIRKEBAK R.. J. Aero. Sci. 1957 24 549. il4j ECKERT G. R. E.‘and DRAKE M. R., Heat and Mass Transfer, p. 180,2nd Edn. McGraw-Hill 1959. [15] HARTNETT J. P., ECKERT E. R. G., BIRKEBAK R. and SAMPSOL R. L., SimpliJed Procedures for the Calculation of Heat Transfer to Surfaces with Non-Uniform Temperatures, WADC TR 5-373 July, 1956. 1161 RUCKENSTEIN E., Rev. Roumaine Phys. 1959 4 245. i17] SEBAN S., Appendix I of M.S.E. Thesis of S. SCESA, University of California 1951. [18] PETERSEN E. E., FRIEDLY C. J. and De VOGELAERE J. R., Chem. Engng Sci. 1964 19 683. [19] BISCHOFF B. K., Chem. Engng Sci. 1968 23451.
Resume- Une equation inttgrale a de superposition, generale, d’aprts une suggestion de Eckert et autres [13] a Ctt d&iv&e pour la vitesse locale de reaction sur des surfaces catalytiques extemes de formes diverses. La preuve de la validite de l’equation est indiquee pour differents systemes cinetiques et regimes d’ecoulement divers. Les resultats concordent avec ceux d’etudes anterieures par Acrivos et Chambre[3] et par Rosner[4]. L’efficacitt decoulant de la forme arrondie de surfaces catalytiques peut &tre deduite de solutions relatives aux catalyseurs de forme en pointe. En ce qui conceme 1es catalyseurs poreux, la contribution de la couche limite et celle de la diffusion par les pores sont prises en consideration par I’adoption d’un facteur d’efficacite general r)O.a.Celui-ci est fonction du module local de Thiele et de I’indice de Damkiihler et caracterise le changement de la vitesse de reaction relatif a la concentration du reactif dans un courant non perturbed. Zusammenfassung - Fur die ortliche Reaktionsgeschwindigkeit auf aussenseitigen katalytischen OberR&hen beliebiger Gestalt wurde auf Grund eines Vorschlags von Eckert et a1.[13] eine allgemeine Superpositionsintegralgleichung abgeleitet. Die Giiltigkeit der Gleichung fur verschiedene kinetische und Stromungssysteme wird bestatigt. Die Ergebnisse stimmen mit frtiheren Arbeiten von Acrivos und ChambrC [3] sowie Rosner[4] iiberein. Wirksamkeit infolge der Form abgestumpfter KatalysatorobertIachen ergibt sich aus Keillosungen. Bei porosen Katalysatoren wird durch Einftihrung eines allgemeinen Wirksamkeitsfaktors Q~, sowohl der Beitrag der Grenzschicht als such derjenige der Porendiffusion berijcksichtigt. Dieser Faktor ist eine Funktion des iirtlichen Thiele Moduls und der Damkohler Zah1 und kennzeichnet die Velidnderung der Reaktionsgeschwindigkeit in Bezug auf die Reaktandenkonzentration in einem ungestiirten Strom.
474
Pergamon Press.
Printed in Great Britain.
A superposition integral equation for catalytic external surfaces R. MIHAIL Department of Chemical Engineering, Polytechnic Institute Bucarest, Polizu I, Bucarest 8, Romania (First received
20 May 1969; in revisedform
10 August
1969)
Abstract-A general superposition integral equation based on a suggestion by Eckert et al.[ 131 has been derived for the local rate of reaction on external catalytic surfaces of arbitrary shape. Proof of the validity of the equation is given for different kinetic systems and flow regimes. Results agree with previous work of Acrivos and Chambre[3] and Rosner[4]. Effectiveness due to the form of blunt catalytic surfaces can be inferred from wedge solutions. For porous catalysts both contributions of boundary layer and pore diffusion is taken into account by the introduction of an over-all effectiveness factor n70,a.This is a function of the local Thiele modulus and Damkohler number and characterizes the alteration of reaction rate as referred to the reactant concentration in an undisturbed stream. INTRODUCTION
IN RECENT years a great deal of work has been
carried out on the problem of catalytic reactions on external surfaces. The problem is both of theoretical importance, in determining the effect of the transport mechanism on the rate of surface reactions [l-6], and of practical importance: external flow reactions systems such as those encountered in the chemical industry[7-91 or in space techniques (e.g. frozen dissociated hypersonic flow) [lo- 121. In particular in kinetic analysis or design one is interested in knowing the influence of non uniform external reactant diffusion on the resulting reaction rate distribution. The known solutions to the problem are limited to isothermal catalytic flow reactions over flat plates with two dimensional boundarylayers or to axi-symmetrical flow in tubes, where the hydrodynamic conditions at the interface can be exactly described. In some of these solutions [ 1-33 the local surface concentration of the reactant depends formally on the local surface shear stress both variables having a similar form of variation along the surface of a flat plate, but not on the surface of a blunt body. Other solutions[4-6, lo] have been deduced by extending the von K&-man momentum integral equation, or the Falkner-Skan equation to concentration boundary-layers. In principle these solutions are valid for surfaces with pressure gradients, but were effectively applied only to the flat plate.
The primary purpose of this work is to present a simplified method for calculating the local concentration of a catalytic surface for either completely laminar or turbulent boundary-layer flows with and without pressure gradients over the surface. The solution is suggested by the previous work of Eckert [ 131 for calculating local heat flux to non isothermal surfaces. FORMULATION
OF THE
BASIC
EQUATIONS
An incompressible viscous fluid, containing a single reactant A flows past a solid catalyst of characteristic dimension L. In the carrier fluid the reactant A is present in dilute amounts. A boundary-layer of constant properties develops over the surface of the solid catalyst which consumes the reactant isothermaly at a rate R[c,(x)], determined by the local concentration of A next to surface. The two-dimensional continuum equations for continuity and momentum are:
(1)
~~+vau=-k!P+va2u ay
pbc
ay2'
(2)
The reactant concentration field in the vicinity of the solid catalyst satisfies the diffusion boundary-layer equation:
uac+u*~&c ax
463
ay
ay2*
(3)
R. MIHAIL
The boundary conditions of Eqs. (l-3) are: y = 0; u = 0,u = 0, Um,c=cm
y=w;u= x =
DE= R[c,(x)]
u,,
0; u =
(4)
c = cm.
N(x) = i
if for the differential linear equation, Eq. (3), a number of particular solutions Ci are available, which satisfy the boundary conditions (4), then the principle of superposition may be used and c=
If z, the distance from the which a step change occurs, parameter then the local mass is a function of coordinates x flux at x becomes:
2 K,ci
(5)
1
is also a solution of Eq. (3). The constants Kt can be used to adjust this new solution to the required boundary conditions. This method enables the substitution of a continuous variation of reactant concentration on the catalyst surface with a discontinuous one in steps, as is depicted in Fig. 1.
Each step behaves as a surface of constant concentration, for which the mass flux may be computed by known methods. For each step the flux of A to the catalyst surface is: o = k,,,,*Ac*= R(x)
(6)
where Aci is the concentration difference betwen surface and the fluid at the edge of the boundarylayer. With Eqs. (5) and (6) the flux of A to the surface is given by N(x)
= -D
i
1
Ki(2),
= i
1
Kfkm,dciv
k,(x,
t)Aci
(8)
the constants Ki being equal to unity[ 141. if the step changes are considered to be built from infinitesimal increments of concentration dc, which succeed each other at infinitesimal intervals dz then the sum of the right hand side of Eq. (8) may be replaced by: N(x) =
dcotz) Mx,z)ypk
(9)
which gives the continuous variation of the surface concentration and must be interpreted as a Stieltjes integral. At steady state the mass flux is identically equal to the local chemical rate of conversion at interface, thus for a surface of uniform temperature
R[cob)l=
Fig. 1. Step change of surface concentration.
1
stagnation point at is considered as a transfer coefficient and z and the mass
z k,(x, 4 !i!%&~ I0
(10)
Equation (10) may be used for bodies of different shapes and for laminar as well as for turbulent flow conditions provided that the required expressions for the local mass-transfer coefficient k,,, (x, E) following a step in surface concentration are available. Now, if k,(x, z) is referred to the mass transfer coefficient of a surface with locally constant concentration k,(x) by k,(x, z)/k,(x) = cp(x, z), Eq. ( 10) may be written as:
R[cotx)l = k,(x)~cotO)
+[
p(x,t)~dz] (11)
where the first term in the right hand side bracket indicates that the only concentration discontinuity occurs at the leading edge.
(7) 464
A superposition integral equation for catalytic external surfaces
Next the versatility of Eq. (15) will be illustrated by calculating some numerical examples. In order to compare the results with those known from other solutions the discussion will be limited to boundary layer flows past flat plates or wedges, more complicated geometries do not offer unusual difficulties.
Eckert [ 151 solved a similar integral for convective heat transfer from a nonisothermal surface; for the problem under consideration the solution is to approximate the actual wall concentration distribution by a series of sloping segments, thereby making dc,/dz a constant value over any interval Ax which allows its removal from the intergal sign. Eq. (11) then reads:
SQME NUMERICAL SOLUTIONS THE FLAT PLATE
R[c,b)l = k,(x+l~O) +
FOR
The irreversible reaction
2 ($jJ~,
cp(X~4q.
Consider the p-th order irreversible reaction:
(12)
i=l
pA + B The general form of the integrals in this equation may be written as follows:
with the rate law described by a power function of the local reactant concentration
cp(x,z) d ;
cp(x,z) dz=x
R(C‘J = kd,.
0
-I
x,-,/x
()I
cP(x,t)d; .
0
(13)
(x)
=
a(z)
+
b(:r.
(16)
For a laminar boundary layer flow past a flat plate the functioncp (x, z) is[14]:
The tabulated values of these integrals, which otherwise can be calculated without difficulties, can be approximated with good accuracy by the following simple expression: cp(x,t)d”
0)
cp(x,t) = [1-(;)3’4]-1’3
(17)
and the constants in Eq. (14) have the values [ 131:
(14)
a = O-895 (18)
b = 0.690. Assuming that the length of the catalytic surface is subdivided in II equal intervals, the substitution of the relation (14) into Eq. (12) gives:
On substituting Eq. (17) and the constants a and b into Eq. (15), we obtain after some simple rearrangements:
R[co(x)l = k,(x) AcoW)+a[Aco(n) -Aco(0)l +i
[
(2n-
a+:
[
1
1+2 ‘i c;Jo(llz)
l)Aco(n>
-AC,(O) -2 %1 c,(m)]}.
(15) where CL (0) = 1 and the Damkohler number
The values of the constants a and b in this equation depend only on the form of function (c (x, z) , specific for the geometry of the catalytic surface and for the flow conditions: laminar or turbulent. 465
CBSVdUNa3-H
Da
=- kd,’ km
is a measure for the relative contribution
of sur-
R. MIHAIL
face kinetics and mass transfer to the value of c,*,(x). If one choose a generalized coordinate along the catalyst surface, then a more detailed expression for Da is D Da = Ckcf&l -_S@Re L
-1
x
l/Z
L1/2
(21)
IO E
and by putting Da = n the reactant concentration distribution czO(Da; p) shown in Fig. 2 results. a=0+95 b=0.690
I
Fig. 2. pA
+ B;
2
3
4
5
6
7
6
9
IO
reactant concentration distribution ~,*,,@a;
p): -,
laminar; ----,
turbulent.
The generality of Eq. (19) with respect to the Schmidt number is included in the nature of the stretched coordinate Da, in particular in the numerical value of the constant C. By adapting the proper expression for k,, Eq. (19) applies for all Schmidt numbers so that in the range 0.5 < SC < lo4 C has the value (2/0*333 X O-215) l13. Forp = 1 the solution is easily compared with the corresponding results of Chambre and Acrivos [2] and Rosner[4], Fig. 3. While the numerical results obtained by Rosner using an approximate solution are 5 per cent less than the analytical solution of ChambrC and Acrivos, the present approach yields values 5 per cent higher. But in both cases the figures could be brought as near the analytical solution as wished either by improving the function of similarity in the Rosner Eq. (8)[4] or by evaluating exactly the integrals (13) instead of using the approximated formula (14) in the deduction of Eq. (15). First-order reversible reaction Let the surface reaction be A*B
01)
with the rate law of form R(cAor cs,) = k(c,,-2).
(22)
An Eq. (15) is written for each component: cAo
-.cBo= - 1 K D&A
I
Ac/io(O)
+a[ACA,(n) Chombri
-AC~o(o)l
and Acrivos [2] +;
(h-
I)fkA,(n)
[ -
-(CAo-2)
I
ACAO(O)
I
I
1
2
3
4
+;
Da
Fig. 3. A -+ B; reactant concentration distribution c& (Da); comparison with previous results [2] and [4].
2 7’
CAh)]}
(23)
= +-bh,(“)
+a[AC,,(n) I
-
b [
-b+,(O)1
(2n - l)A+,,(n) n-1
-Acs,(O)--
466
T
CUB,,
11
.
(24)
A superposition integral equation for catalytic external surfaces
reactions is illustrated with the most simple case
The boundary conditions are:
(25)
cil, = 0
in which both steps are first order. The reaction rates are RA = klcA
i.e. in the incident fluid there is no reaction product. Dividing both Eq. (23) and (24) by cA, and eliminating c& one obtains
RB = -klcA + kzcB R, = -kzce.
(28)
{K[a+(~)b]+Da,B}[a+~(l+2~cr.(m))]+Da,,%*2+Am)
c‘&(n) =
by substituting
(26)
[a+(~)b][K[Da,A+a+(~)b]+Da,J the calculated values of c+&(n) in Eq. (24) it results: c&(m) - (2n-
II
l)&(n)
1 .
Equation yielding
The results of numerical computations of the dimensionless surface concentrations CT” and c& are plotted in Fig. 4 against the generalized coordinate Da,A, with the equilibrium constant K as parameter. The influence of flow and diffusion on the concentration distribution is striking.
(15) is written
c&(n) = L Da,, +p
aLI
A*
0.9+-
07-
‘!,
0.5 0.6 -
‘\ yK .b : .4 ., ‘.\ ‘.>.
0.403-
.& .fl
0.2 -j’
-c*
1 = - Da,,
k2 ~c.?O(n) +Kct(n)
-ac&(n) I
.-.
+;[2c&(m)
00
-
‘.I?.--,_ +._.
-_ -.__
/Go
I
II
.-
‘.\ $ -o La
/4
I)cA*,(n)
(29)
,_‘~?~_=~=~=-------
DE-%
-G4n)l
1
The ability of the proposed method to predict reaction rate distribution for sequential surface I-O
for each component
1 + 2 ‘g c,&(m) - (2n-
First-order consecutive reactions
(27)
2
I 3
1 4
---_ ---._._I&--,
I 5
1 6
-+c&(n)
_ 1 7
I E
1 9
1
=+ ?C
@n-
l)cB(n)])
(30)
-ac&(n) 1
IO
Da sp.
Fig. 4. A 4 B; surface concentration distribution, with equilitkium constant K as parameter: -, K = 0.1;-----, = IO;-.-.-,=lOO.
467
(31)
R. MIHAIL
The present method can also be readily geneqlized to include turbulent and transition flows along a catalytic plate surface. One has only to consider the proper expression of the function cp(x, z). Of the different theoretical results known in the literature that of Seban[ 171 for turbulent flow with constant velocity over a flat plate, i.e.
in which it was assumed that the free stream concentration of B and C are nil. Also the components concentration satisfies the relation DaYA CZ~(~)+rc;O(n)
Da,* +D,c;O(n)
-B
= 1.
(32)
rc
The calculated surface concentrations are represented in Fig. 5 with Da,A as abscisa and k,lk, as parameter.
cp(x,z) = [t_(;~lro]-l+ was selected. On introducing this expression into Eq. (15) one obtains for constants a and b the values [ 131:
A-B-?-Ck k,
*y=
a = 0991
---.-._,
-0 3
b = 0.117.
I
2
3
4
5
6
7
6
9
-IO
DQqA Fig. 5. A 5 B 5 C; surface concentration distribution, with
/c,/k, as parameter: -.-.-,
k,/k,= 0.1; -,
=l.O; ----.
= 10. When the surface concentration mediary product is in the form
of the inter-
kZDa,, k,Da,8 cio
(33)
where
k2Da,A --= kl D&B
Sh (x) = 0.33 (Re,) WC~‘~, laminar; 0.6 < SC < 10. Sh (x) = O-0296(Re,) ‘YSC~/~,turbulent
’
it is possible to compare the results with those of AC&OS and ChambrC [3, Table 31;the maximum error is less than 8 per cent. Again this value could be reduced by calculating exactly the integrals (13) instead using the approximation (14). TURBULENT
For reaction (I) the surface concentration is shown in Fig. 2 which permits a comparison to be made with the results of Rosner[6]. In order to compare the effect of laminar and turbulent flow on the same surface reaction the use of Damkiihler number as coordinate could be misleading since the form of this coordinate is not identical for laminar and turbulent flow. A dimensionless coordinate x/L was selected. Consider a flat plate of length L whose surface catalyses a first order irreversible reaction (I). At Re8 = 160400 the boundary layer can be either laminar or turbulent. The respective equations for the local mass transfer coefficients
BOUNDARY
can be written as L l/2 k,(x)
= 0-33(ReL)1YW’3
k,(x)
= 0.0296(Re~)“Yc”3
y
0
;L O.* 0
(34)
it was accepted
that
LAYERS
Previous results concerning surface reactions with turbulent boundary layer includes those of Rosner[6] and Ruckenstein[l6].
where,
for
simplicity,
D/L = 1.
This expression 468
allows the local Damkohler
A superposition integral equation for catalytic external surfaces
number to be calculated so that for given values of Re,, SC and reaction velocity constant k, Eq. (19) can be used. Figure 6 shows the surface concentration of reactant A as a function of distance x/L for p = 1. As expected the surface concentration is higher IO
,
,
,
,
,
,
,
,
,
Re?eb”o&oo
0.9-
k
= 373m/h
utilize the local surface shear stress as independent variable is valid only for slender bodies, because for blunt bodies the surface shear stress and local mass transfer coefficients (consequently local surface concentration) have a dissimilar type of variation due to the contribution of form drag. Wedge type flows described by U, a xm are particularly suitable to simulate the behaviour of specific geometries which can be encountered as catalytic surfaces. The general form of the functioncp (x, z) for laminar flow is:
01 a
pOx,z)=
::[
,
,
,
,
0.1
0.2
03
O-4
, 05
,
,
06
0.7
, 06
1
, 09
IO
X/L
Fig. 6. A + B; turbulent boundary layer; c;.~ vs. x/L.
in turbulent than in laminar flow while the reactant supply is more intensive. Turbulent boundary layers are less sensitive than laminar layers to streamwise gradients of concentration exhibiting a more uniform depletion of the reactant and a higher effectiveness factor. WEDGE
FLOWS
As mentioned the above method of solution is general because it is applicable to arbitrary catalytic surfaces. In contrast, any solution which
k,(x)
b
C
P
m
Flate plate
0 215
3/4 0.895 0.690 0,332 0 114 15/16 0.837 0.635 0.412
f/2 213 617
-1 0.825 0.610 0.430 l/3 112 918 0.840 0.572 0.469 314 21/16 0.545 0.785 0.518
Sphere-forward stagnation point Cylinder-forward stagnation point
1
1
.
(35)
=
C(ll/x)Re,"S~"~
where the constants C and n are given in Table 1. The results are shown graphically in Fig. 7, with m as parameter. It is seen that for a given value of the reduced coordinate Da the surface concentration diminishes with increasing m, meaning that the rate at which reactant can be supplied to the catalyst surface is smaller. This becomes more
Surface
u
-113
When this expression is substituted in Eq. (13) and the integrals are calculated the results of Table 1 apply [ 131. For a first order irreversible reaction the surface concentration of reactant was computed using the values of a and b from Table 1. The coefficient for mass transfer of a surface with uniform concentration, needed for computing the Damkiihler number Da has the form:
Table 1. Parameters for wedge flow (CJ, = xm) and local mass transfer coefficient (k, = C (D/x)R~,~~~S~“~) a
[
1-f
312 0.192 0.538 0.560
R. MIHAIL
Fig. 7. A -+ B; wedge flows; reactant concentration distribution c&, (Da; ml.
NON-ISOTHERMAL
evident by calculating the external effectiveness factor, defined by &Ck&I.
REACTION
When a non-isothermal reaction occurs on a catalytic external surface there will generally be
(361 oAoor
Values of the integrated external effectiveness factor
have been computed and the results are shown as a function of m in Fig. 8, forDa = 7. From these results the behaviour of a blunt catalytic surface could be foreseen, at least for the front flow area. The external effectiveness factor depends on the surface geometry as does the internal effectiveness factor. For the front stagnation point the effectiveness is smaller in the order: fiat plate-sphere-cylinder. It is to be expected that in complex catalytic reactions variation in the effectiveness due to surface geometry will affect selectivity and yield.
,P
OS0-
O?‘GO’
0
‘ l/4
his, 8.A -, B; wedge
470
L I13
I 112
m
3pI
flows; ;? integrated external effective ness factor vs. m, for Da = 7.
A superposition
integral equation for catalytic external surfaces
continuous variation of concentration and temperature along surface. The method of solution of such a problem was first outlined by Chambt-6 [ l] and his analysis is readily applicable to the present method. The model of a non-isothermal reaction is described by the previous Eqs. (I-3) with an additional equation for conservation of energy. a
The boundary conditions are: c_46% m, = cz4(0, y) = T(x,
m)
-A,
T(0, y) = T,
=
o=
--DA
(
$
CA,
To(x)1
R[cAo(x)v
o = (-AHR) >
. R[cA,(x),
T&)1.
(39)
The differential equations describing the concentration and temperature field being similar we proceed as before. Finally one has to solve the two coupled equations:
(-A~idk(To)cAo(X)
=
+A,++
(40)
I”D(&$k] 0
and: e(x)
=k,[i\To+j-+t)~&]
(41)
where+(x,z)
= kh(x,z)lkt,(x). For a first-order irreversible reaction (I) and after introducing approximation (14), Eqs. (40) and (4 1) become ~(To)cA,(x)
=
km
AcA~(O) {
+~CACA~(~)
+i
[
-Ac~o(o)
-AcA~(O)I
l)AcAo(n)
(h-
-
2 2’
AC,.(d])
(42)
+o
+a[ATo(n) -AT,(O)1 +;[
(2n-
l)ATo(n)
The nonlinear equation system (42-43) can be solved by any common iterative procedure starting with known values of cAO(0) and T,(O) and specifying the amount of heat transfered to or from the catalytic surface. Finally CA0and To will result for each n. For some catalytic reactions of industrial interest such as NH, oxidation e.g. the high thermal conductivity of the metal catalyst produces a uniform temperature over the surface. The heat flux, which varies along the surface due to the variation of reactant concentration may then be computed by usual methods, provided that the local heat transfer coefficient is known.
OVER-ALL R[cA,(x)l
=
EFFECTIVENESS
FACTOR
The influence of a non-uniform environment on the rate of a catalytic surface was first considered by Petersen et al.[ 181 who discussed the effect of concentration and temperature gradients on surface reaction on a non-porous sphere. Recently Bischoff[ 191 solved a problem with varying surface conditions. The particular problem was that of a spherical, porous, catalyst particle immersed in a fluid with an axially linear temperature gradient at the particle surface. It would be of interest to consider the influence on the internal effectiveness factor, not of an arbitrary varying surface condition, but that due to a boundary-layer developed by fluid flow over a porous catalyst. The discussion will be limited to a flat plate of infinite thickness on which an isothermal catalytic reaction occurs; other geometries can be treated, but no new features are introduced. 471
R. MIHAIL
Let the surface reaction be (I); the diffusion equation inside the porous catalyst is:
It is obvious reaction (I)
that for a first-order
irreversible (49)
%.a = c* (x, 4.
(44) The integral over-all effectiveness where z is the depth of penetration and x a coordinate along the flat plate in the direction of flow. The boundary conditions are z=o,
%.a =
US Js RkA(.dl
cXh,Da)
The solution of Eq. (44) with the above boundary conditions is CA(XVt) = CA0(x) . e-+
(46)
where & is a local Thiele modulus and c,&,(x) is the surface concentration resulting from solution of the boundary layer equation. Depending on the desired accuracy and time allowed for computation one can substitute cAo(x) in Eq. (46) with one of the expressions derived by Chambre[l], ChambrC and Acrivos [2], Rosner[4] or with Eq. (19) from present work. Upon introducing reduced dimensionless coordinates for the axial position and depth of the porous plate we have
d-d
(50)
R [CAJ
or with reduced coordinates,
c‘4= c‘4(x)
factor will be
for reaction (I)
d(#,)
Wa).
(51)
The over-all effectiveness factor distribution inside a porous catalyst as calculated from Eq. (5 1) is shown in Fig. 9. P % t
Ii
I.0 09
+t\
(47) Thus a complete description of the concentration field inside a porous flat-plate catalyst is available as function of the two dimensionless coordinates, local Thiele modulus and Damkohler number. The internal (diffusion determined) effectiveness factor and the external (boundary-layer determined) effectiveness factor are coupled in an over-all effectiveness factor referred to reactant concentration in the undisturbed fluid %.a. =
R [CAk 2) 1
R[cA,I
*
(48) 472
Fig. 9. A --, B; over-all effectiveness factor distribu-
tion ?,.. (4.; Da). CONCLUSIONS
The computational technique of Eckert, Hartnett and Birkeback [ 131 for heat transfer to a non-isothermal surface has proven useful in the analysis of the significant variables involved in external catalytic reactions. The main feature of the superposition integral equation proposed in this work is its applicability to various surface
A superposition integral equation for catalytic external surfaces
Re
geometries, with or without pressure gradients and to both laminar or turbulent flows. The contribution of surface geometry to the effectiveness of an external catalyst was illustrated by examining wedge flows. The extension of the proposed method to nonisothermal reactions is easily possible along the lines shown by Chambre [ 11. An over-all effectiveness factor was introduced to account for all specific resistances’encountered in reactions catalyzed by porous solid bodies subject to boundary-layer flows.
number
for
flat plate,
Re = UJJv
Re,
local
Reynolds
number,
Re, =
U.&IV
Re8 Reynolds
number for boundary layer, Re = U,6/v S normalized surface area, S = 4I . Da SC Schmidt number, SC = v/D Sh Sherwood number, Sh = k,(x). x/D T temperature difference across therAT temperature mal boundary layer velocity in x or z direction in the I.4 boundary layer velocity at edge of boundary layer u velocity in y direction V distance measured parallel to catalyX tic surface, from leading edge Y distance measured perpendicular to catalytic surface 2 distance at which a step change occurs t depth of a porous catalyst
NOTATION
a, b constants defined by Eq. ( 14) reactant concentration dimensionless concentration c” = clc, difference across AC concentration mass transfer boundary layer AC = c, - c(x) c constant, in Eq. (21) D diffusion coefficient effective diffusion coefficient in a D” porous catalyst Damkijhler number heat of reaction (-*I$ k, k’, k,, kz reaction velocity constants local mass transfer coefficient k,(x) k,(x, z) local mass transfer coefficient for a step change at z, measured from leading edge local heat transfer coefficient k/i(x) local heat transfer coefficient for a k,, (x, z) step change at z, measured from leading edge K constant in Eq. (5) K equilibrium constant L characteristic dimension of catalyst m constant, velocity equation for wedge flows dimensionless coordinate mass flux to catalyst surface N(x; reaction order heat flux, Q(x) = -(AH,)k(T,)C,,(x) ecx? R rate of surface reaction
Reynolds
C
c*
Greeksymbols constant defined by Eq. (35) parameter for wedge angle, pm s boundary layer thickness 71 effectiveness factor, 7) = Cg (x) ?1 integrated effectiveness factor %.a. over-all effectiveness factor A thermal conductivity of fluid V kinematic viscosity of fluid P density of fluid (c (x7 z) ratio of mass transfer coefficients, cp(x, z) = k,(x, z)lkrt(x) Thiele modulus, & = #Jz local ;
zd(kC$;l/D*)
Iclk z) ratio of heat transfer coefficients, H-G z) = kh(x, z)lk,,(x) Subscripts 0 m i
A,B,C 473
surface conditions undisturbed flow conditions number of step change reactants and products
R. MIHAIL REFERENCES [l] [2] [3] [4]
CHAMBRE P. L., Appl. scient. Res. 1956 A6 97. CHAMBRE P. L. and ACRIVOS A., J. appl. Phys. 1956 27 1332. ACRIVOS A. and CHAMBRE L. P., Ind. Engng Chern. 1957 49 1025. ROSNER D. E., J. AerolSpace Sci. 1959 26 281. [5] ROSNERD.E.,A.I.Ch.EJl19639321. [6] ROSNER D. E., Chem. Engng Sci. 1964 19 1. [7] NOWAK J. E., Chem. Engng Sci. 1966 21 19. [8] MIHAIL R., Chem. Technik. 1957 9 344. [9] MIHAIL R. and HERSCOVICI J., Kinet. Katal. 1962 3 836. [lOI LI Y. T. and KIRKS. P., Int. J. Heat Mass Transfer 1967 10 257. I1 11 INGER G. R.. Int. J. Heat Mass Transfer 1963 6 815. [12] CHUNG M. P., LIU W. S. and MIRELS H., Int. J. Heat Mass Transfer 1963 6 193. 1131 ECKERT G. R. E.. HARTNETT P. J. and BIRKEBAK R.. J. Aero. Sci. 1957 24 549. il4j ECKERT G. R. E.‘and DRAKE M. R., Heat and Mass Transfer, p. 180,2nd Edn. McGraw-Hill 1959. [15] HARTNETT J. P., ECKERT E. R. G., BIRKEBAK R. and SAMPSOL R. L., SimpliJed Procedures for the Calculation of Heat Transfer to Surfaces with Non-Uniform Temperatures, WADC TR 5-373 July, 1956. 1161 RUCKENSTEIN E., Rev. Roumaine Phys. 1959 4 245. i17] SEBAN S., Appendix I of M.S.E. Thesis of S. SCESA, University of California 1951. [18] PETERSEN E. E., FRIEDLY C. J. and De VOGELAERE J. R., Chem. Engng Sci. 1964 19 683. [19] BISCHOFF B. K., Chem. Engng Sci. 1968 23451.
Resume- Une equation inttgrale a de superposition, generale, d’aprts une suggestion de Eckert et autres [13] a Ctt d&iv&e pour la vitesse locale de reaction sur des surfaces catalytiques extemes de formes diverses. La preuve de la validite de l’equation est indiquee pour differents systemes cinetiques et regimes d’ecoulement divers. Les resultats concordent avec ceux d’etudes anterieures par Acrivos et Chambre[3] et par Rosner[4]. L’efficacitt decoulant de la forme arrondie de surfaces catalytiques peut &tre deduite de solutions relatives aux catalyseurs de forme en pointe. En ce qui conceme 1es catalyseurs poreux, la contribution de la couche limite et celle de la diffusion par les pores sont prises en consideration par I’adoption d’un facteur d’efficacite general r)O.a.Celui-ci est fonction du module local de Thiele et de I’indice de Damkiihler et caracterise le changement de la vitesse de reaction relatif a la concentration du reactif dans un courant non perturbed. Zusammenfassung - Fur die ortliche Reaktionsgeschwindigkeit auf aussenseitigen katalytischen OberR&hen beliebiger Gestalt wurde auf Grund eines Vorschlags von Eckert et a1.[13] eine allgemeine Superpositionsintegralgleichung abgeleitet. Die Giiltigkeit der Gleichung fur verschiedene kinetische und Stromungssysteme wird bestatigt. Die Ergebnisse stimmen mit frtiheren Arbeiten von Acrivos und ChambrC [3] sowie Rosner[4] iiberein. Wirksamkeit infolge der Form abgestumpfter KatalysatorobertIachen ergibt sich aus Keillosungen. Bei porosen Katalysatoren wird durch Einftihrung eines allgemeinen Wirksamkeitsfaktors Q~, sowohl der Beitrag der Grenzschicht als such derjenige der Porendiffusion berijcksichtigt. Dieser Faktor ist eine Funktion des iirtlichen Thiele Moduls und der Damkohler Zah1 und kennzeichnet die Velidnderung der Reaktionsgeschwindigkeit in Bezug auf die Reaktandenkonzentration in einem ungestiirten Strom.
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