high at, 3.3 X lo-*, but on the other hand the injection length
L1 \vas very small indeed a t 2 mm in a 30.3-cni diameter pipe. Despite a n est'eiisite literature on transpiration in boundary layers, there seems to be little reference to any possible influence of gas injection on eddy viscosity, although more recently ail equation of Wasan, et al. (1969), predict's a considerable increase in ef when y* > 10 for the conditions used in this study. In short, it is difficult to dismiss any suggestion that the act of tracer injection might interfere substantially \vit,h the eddy struct'ure near t'he wall. Severtheless, i t seems reasonable enough to conclude from curve I11 in Figure 13 that the gas eddy diffusivit,y near the wall is reduced very markedly solely 011 account of the presence of particles. Wall Shear Stress. For the flow conditions ill this work the wall shear stress acting 011 the fluid is 1.06 X lo-' pdl/ft' for air flowing alone and from the data in Figure 1 about 2.12 x 10-1 pdl/ft' when particles are present. However, from the velocity profile in Figure i the stress a t the wall which can be at,tribnted t,o shearing of the gas is much less than this. I t seems, therefore, that' most of the rvall shear stress is due to impact of t'he particles with the wall. Conclusions
I t has been shown esperimentally that small particles cause strong suppression of gas turbulence near the wall of a pipe. The particles also cause a much reduced gas velocity gradient near the wall. Tliw most of the wall shear stress can be attributed to particle impacts and not to viscous shearing by
the gas. The reduction in gas turbulence near the wall is sufficient to account for the low Nusselt number and friction factor often observed in this class of fluid. literature Cited
Arundel, P. A., Bibb, S. D., Boothroyd, R. G., Powder Techno/. 4 , 302 (1971). Boothroyd, R. G., Trans. Inst. Chem. Eng. 44,306 (1966). Boothrovd. R. G.. Trans. Inst. Chem. Ena. 45. 297 (1967). Boothroird, R. G.,'Haque, H., J . 3lech. Egg. Sit. 12, 191 (i970). Caseau, P., Deniau, R., Houzllc Blanche 24, 259 (1969). Chao, B. T., Min, K., -Vue/. Sci. Eng. 26, 534 (1966). Hall, W. B., Hashinii, J. A,, Proc. Inst. Jlech. Eng. Pt. 31, 178, 1 /10AAJ \ - V Y I , .
Hinze. J. 0..Aaol. Sci. Res. A l l . 33 11961). Hjelmfelt, A. T.: IIockros, L. F.; A p p l . Sei. Res. A16, 149 (1966). Jenkins, It., Proc. Heat Transfer and Fluid X w h . Inst. 147 (1951). Kays, W. M.,Noffat, R. J., Thielbahr, W. H., J . Heat Transfer 92C, 499 (1970). Laufer, J., aTat.Adv. Comm. Aeronaut. Tech. Rpt. 1174 (1954). Olson. R. 31.. Eckert. E. R. G.. J . A m . Jfech. 33. 7 11966). Quarmby, ii.; h a n d , R.K., J.'Fluid'-?fech. 38,433, 457 (1969). Saffman, P. G., J. Fluid Mech. 13, 120 (1962). Walton, P. J., Gammon, L. N., Boothrovd, R. G. Powder Technol. 4 , 293 (1971a). Walton, P. J., Ph.D. Thesis, University of Birmingham, England, 1971b. Wasan, D. T., Randhava, S. S.,Babu, P. S., Chem. Eng. Sei. 24, 595 (1969). RECEIVED for review January 10, 1972 ACCEPTEDOctober 4, 1972 This work was carried out at the Department of 11echanical Engineering, University of Birmingham, L-. K., and was supported by the Science Research Council.
A Theoretical Study of Pressure Drop and Transport in Packed Beds at Intermediate Reynolds Numbers Mohamed M. El-Kaissy and George M. Homsy* Department of Chemical Engineering, Stanford Cniversity, Stanford, Calif. 94305
Previously proposed theoretical cell models for transport in packed beds have been limited, with one exception, to the creeping flow regime. Two popular models ate reviewed and extended to finite particle Reynolds numbers b y regular perturbation techniques. The central goal i s to develop a theoretical framework and methodology b y which cell models may b e rationally extended using analytical representations. The predictions of pressure drop for spherical cells demonstrate the failure of these models to predict deviations from creeping flow conditions with any accuracy. One possible extension, that of distorted cells, i s briefly treated and i s shown to b e capable of representing experimental behavior. Transport in beds a t high Peclet number is then treated in some generality and it i s shown that predictions of the transport rates a t high Peclet number are quite insensitive to Reynolds number, thus offering theoretical confirmation for this well-known empirical fact.
D e h p i t e the widespread occurrence of fluid-particle ststems in n-liicli the ~.olurnefractioii of solids iz aljpreciable, the tenis by the use of models is still ability to describe these in a n early stage of dev pment. 'The main obstacle to be ovei come in the descriptioii of such tenis is that of satisfactorily treating pnrticle-liarticle iilteractio~is.One such \\-ell kiioivti model which reeks to surmouiit this Ijroblem is the cell 82
Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
model due to Happel (1958). In its formulation, the difficult many-body problem is replaced by a simple and coiicept~ua~lS more attractive coiitiiiuous one involvi~~g only one particle. K a l l effect,sand/or ent,ry and exit effects are neglected. The assembly of particles in t8hefluid is assumed to be uniform and each sljhere is fixed in space x i t h equal spacing separating: them. The inteiaction of a particular sphere viith its neigh-
bors is modeled by assuming each sphere to be surrounded by a hypothetical fluid sphere whose radius is related to the actual voidage in the assembly. The so-called free surface cell model proposed by Happel (1958) imposed a zero shear stress a t the outer hypothetical surface. It is, of course, fruitless to attach any significance to the point values of any physical variable (velocity, etc.) predicted when a given boundary value problem is solved within the cell. What is hopefully accomplished, however, is a prediction of bulk or continuum properties of the fluid-particle flow which may be obtained by integration over the physical domain in question This is of course the main purpose to which such cell models have been put. Some of the properties which have been predicted include bulk pressure drops, overall heat or mass transfer rates, and the effective viscosity of concentrated suspensions. It is safe to say, however, that these attempts have for the most part been failures in the sense that the interactions have been too over-simplified. For the case in point, Happel’s (1958) prediction of pressure drop and/or hindering in concentrated’ systems of spheres properly reduces to Stokes’ law a t infinite dilution, fails to predict hindered settling of sedimenting solutions in the dilute range, over-predicts the drag in fluidized beds, but for packed beds in the range of voidages 0.3 5 t 5 0.6, predicts with remarkable success the pressure drop a t low particle Reynolds numbers. Finally, we point out that the use of the cell model to predict t,he effective viscosity of suspensions fails to reduce properly to Einstein’s classical result a t low solids concentrations (Happel and Brenner, 1965). -4similar model for packed beds was developed by Kuwabara (1959), where the condition imposed a t the cell boundary was that of zero vorticity. H e calculated the drag in the creeping flow regime by integrating the dissipation energy function. This is conceptually in error due to the fact that the shear stress no longer vanishes a t the outer surface in this case and thus the cell does work against the surroundings. On the other hand, the corrected drag results, obtained by integrating surface forces as shown below, eshibit a dependence upon voidage which is somewhat stronger than experiments indicate. Happel and Brenner have also criticized the zero-vorticity model from the conceptual viewpoint that there should be no interaction between a particle with its surrounding cell and the remainder of the system. (Recall that the cell is thought to represent the total constraining effect of neighboring particles.) It thus seems that one area in which cell models appear promising is in the description of steady flow through fluidparticle assemblages with voidages in the packed bed range. The early work was limited to creeping flow. LeClair and Hamielec (1968) have estended Kuwabara’s zero-vorticity model to higher Reynolds numbers by generating numerical solutions to the Navier-Stokes equations. The results were in poor agreement with esperiments; reasons for this behavior are advanced below. Numerical solutions a t intermediate Reynolds numbers using the zero vorticity cell model were obtained for drag and transport in assemblages of cylinders and bubble swarms by LeClair and Hamielec (1970, 1971). Comparison with experimental data met with moderate to good success. I n addition, l’feffer (1964) has modeled transport in packed beds a t high Peclet numbers in the creeping flow regime with similar moderate success. Having remarked upon the apparent ability of cell models to predict continuum or overall properties of dense fluidparticle assemblages, we are now in a position to state the
objectives of this work. Foremost is the desire to explore to what extent the model may be rationally extended to higher Reynolds numbers. I n dealing with this question we have chosen to develop an analytical framework for the construction of higher order corrections to the creeping flow results for two reasons. The first is that in contrast to numerical techniques, the present treatment affords considerable latitude in the ability to easily vary such parameters as cell shape, outer conditions, etc. Secondly, we feel that some of the conclusions reached below have a generality which is not easily recognized if numerical techniques are employed. Momentum Transfer
The full Navier-Stokes equations for steady incompressible Newtonian axisynimetric flow in terms of the Stokes stream function are given in spherical polar coordinates by (Rosenhead, 1963)
- #,
b b6
-
+ 2 cot 6#,
-2 (1)
where 0 2E
b2 + __ sinZe b
br2
(1 2)
rz b6 sin 6 b6
The nondimensional velocities are related to # by UT= - -
w’
b*.ve = 1 r sin 6 dr rz sin 6 be’
Here the radial coordinate is scaled by the sphere radius, 160, and Re, = u0a/vis the particle Reynolds number. Using the cell model, the boundary conditions become: no slip
a , velocities by the uniform free-stream velocity,
0
(2)
*T(l,O) = 0
(3)
d4,O)
=
free stream conditions a t the outer surface #(A$)
1 X2 sinZ6 2
= -
(4)
and either free surface (5) or zero vorticity
where X = b / a , the ratio of the radius of the outer hypothetical sphere to that of the particle. X is related to the voidage by
We wish to develop a solution in terms of an expansion in the Reynolds number, Re,. While the correct treatment of the single isolated sphere (A + a ) is well known to require the use of singular perturbation techniques, the present expansion is regular. It is a fairly straightforward exercise to show that the creeping flow solution is uniformly valid for all finite h in the sense that given the voidage, and hence X, there exists a Reynolds number, Re,,k, below which fluid inertia is negligible in all regions of the flow field. The uniform validity of the creeping flow solution in this case allows it to be improved Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
83
by regular perturbation. Two important points must be stated, however. First, the rate of convergence of the resulting series becomes worse as A increases; i.e., Re,,x decreases with increasing voidage, and secondly as + 03 it is possible to recover the Stokes solution; the highest order corrections become singular as must be the case. Let $(r,O;Re,) be expanded as $(r,O;Re,)
=
+ K.e,h(T,O) + Rea2$dr,0) +
$&,e)
where
The solution of eq 9 can be seen to be proportional to Q z ( p ) alone, since the associated boundary conditions are homogeneous. A straightforward calculation yields
Azr6 + Bzra
(7) The aim of the calculation is then to determine the functions $0, $1, etc. Substituting eq 7 in eq I-5a and equating like powers of Re, generates the equations governing $0, iC1, etc. We have carried the calculations to 0(Rea2). The O(1) terms yield the well-known equations for the creeping flow case. These are
D44bO= 0 with the boundary conditions
The constants A z , Bz, CZ, and Dz, are evaluated from the imposed boundary conditions. Proceeding in exactly the same manner as above the O(Re,2) term was evaluated The solution for $2 was found to be of the form $2
1 $o(l,e) = 0; GOr(l,e) = 0; $o(x,6) = - A 2 sin2 6
= (A39
+ B3r4 + C3r-I + D ~ T - ~ ) Q ~+( P )
+ azr3+ a3r7+ a4r4In + a5r2+ + a7 + a8r + agr-2 + In r + a11~+ In r ) Q 3 ( p ) + (A3’r4 + B3’rZ+ C3’r + D3fr-1)Q1(p) + (al’+ + a2’r3+ + a4’r4In + a6‘r2 In r + + a?‘ + a8‘r In r + ( ~ g ’ r -+ ~ c ~ ~ o ‘ In r - ~r + all'^-^ + alz’r6)Q1(p) (11) (a@
2
T
sin2 6
(free surface)
T
ag’t-7
or =
sin2 6
(zero vorticity)
%~?.=k
The solution for the free surface case is (Happel and Brenner, 1965)
io=
- 1 Ar4 - Br + (5
where =
-5x- . B
w’
( ~ 6 ~ 5
a1Or-l
and either
A
B + C p + D r 2 - Ar4 20
=
-33x6
- 2x W
+
and 1%‘ = 3x5 2 - 2x6 p = cos 6! and in general
;c=
Again, the set of constants As, B3, CS, D3, and As’, Bs’, C3’, Ds’are evaluated by applying the boundary conditions. The CY and a’ terms are known functions of A. For compactness they are not reproduced here. They can be obtained from the authors upon request. We have shown detailed calculations for the free-surface model only. The results for the zero-vorticity model are obtained in an analogous manner: only the constants ( A , B , . . . Azl Bz,. . . A s ,Bs, . . . cy1, c y 2 , . . .) differ. Drag Calculation
- 3x.
QnG)=
ag‘r5
J:l PnG)dp
where Pnlp) is the Legendre polynomial of order n. Q n ( p ) are the well-known Sampson polynomials. The O(Re,) terms yield
The most important information to be obtained from the above solutions is the magnitude of the drag on a particle in the assemblage. We utilize here a modification of the method due to Chester and Breach (1969) which enables one to calculate the drag from a knowledge of the stream function alone. The drag is evaluated by integrating the surface forces around the sphere.
-
u,8
cos 6
-
( u r r cos 6
The corresponding boundary conditions are +l(1J6) $iT(l,O) = 0, $l(x,O) = 0, and either
or
=
0,
13 D,
1{( + 2) -p
2
(2-
t’* -
r
sin 6)r2 sin 6 d6
-)
+ lr b30v , -
=
sin 6 ) r* sin 6 de
Applying the boundary conditionb a t the sphere surface and noting that the integrand is evaluated at r = 1
IVith +bo substituted in the above equation one obtains In this expression D, is the Stokes drag on a single isolated particle and p is the usual dimensionless pressure. 84 Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
Table I. The Functions f(e), g(e) Free surface €
f(e)
0.3 0.4 0.5 0.6
2074 766.0 341.2 170.2
Zero vorticity
de) 5 . 9 1 X lo-* 8.49 x 8.25 X 8.09 X
e(€) - 3 . 8 7 X lo-' 8 . 7 9 x 10-2 8 . 9 4 X 10-2 8.73 X
f(e)
2385 916.4 421.6 215.6
Integrating the first part of eq 12 by parts, one obtains PS
r
( - p cos 0 sin e)de =
n
J
J O
- p sin e d sin 6
0
=
lS
sin2 e$
de
I ' I
L
2
10
10
ReJl-
Now from the 0 component of the equation of motion we have, after applying the boundary condition
9~
- 32
d61r=1
_-
(TuB)=
d2 1 d* br2 sin e br
---
----=
br2
(14) substituting (14) into (13) then yields
- sin e
3
dB
dr
+
LS
sin
dr
e de} r=l
3
10
e)
Figure 1 . f vs. Red/(l - E) at different fort he free surface cell model. The solid curve is the Ergun relation. The theoretical predictions are plotted as points. The dashed curve is the creeping flow result
Now, the familiar Carman friction factor is
Furthermore, by a force balance on the cell
We adopt the following representation of the solution
and hence
substituting for D from ey 16, so
Now a property of the Sampson polynomials is 1
J-lQn(g)
dp
=
so
substituting for
D
@I
=
f
2
Note that the Reynolds number in the parenthesis was converted to the more familiar form Red = u&/Y. Equation 17 was then compared with the classical Ergun relation
(n = -1) (n = 1)
0 (otherwise)
the solution obtained above, then
D 2 [f( 4 9
+ Reu2g(e)l+
(16)
where f ( e ) is the creeping flow contribution to the drag and is the O(Re2) correction. It should be noted that the correction is O(Re2)and not O(Re). f ( e ) and g(e) are complicated functions of t)he voidage. These functions are tabulated in Table I for the voidage mnge of interest. g(e)
Pressure Drop Calculations
T o test our final, improved drag relation it was first expressed in terms of a friction factor. f,and compared with the results obtained from the widely used Ergun relation.
Figures 1 and 2 show the coniparison between the results from eq 17 and 18 a t different voidages for the free surface and the zero vorticity cell model, respectively. It must be emphasized in these comparisons that the Ergun equation is a two-parameter best fit to data in the range of packed bed voidages 0.3 5 e 5 0.6. The data of several investigators scatter somewhat about relation 18, especially in the range Red > 10. However, the ability of a theory t o collapse its predictions when plotted in this form can be considered a test of its validity. The range of validity of the espansion in eq 17 is of course unknown; we have therefore only plotted theoretical points for values of Red below which the higherorder corrections to the creeping flow result amounts to
510%. It can be seen that the creeping flow solution is not affected by inertia for Reynolds numbers below 10. This is due to the Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
85
Confirming our remarks t o the free-surface model, the original formulation (eq 1-3) remains valid with the import a n t difference t h a t eq 4 and 5 must be modified to read #(~o,pC) = -QiG)ro2
(20)
Assuming the expansion (7) as before, and further assuming the analyticy of $0, $1, etc., considered as funct,ions of r O ( p , Rea), we may write (20-21) as conditions at the unperturbed spherical cell boundary as (Va,nDyke, 1964, p 13)
0 0 I
I
1 1
1 o2
10
+o(X,p)
-QiG)X2
(20a)
3 10
ReJl- E) Figure 2. f vs. Red/(l cell model
-
E)
at different eforthefreevorticity
fact that the correction to the drag is 0(Reu2),not O(Re,). Although for the two highest voidases the point of deviation from the Carman-Kozeny equation is predicted well by the models, it is seen that both models substantially underpredict the drag. This has previously been noted by LeClair and Hamielec (1968) for the case of the zero-vorticity model. We note that their numerical results are in agreement with the predictions developed herein. Happel's free surface model is also more successful in collapsing the predictions for different voidages. The stronger voidage dependence of the zero vorticity model is well known (Happel and Brenner, 1965) and is amply illustrated in Figure 2 . Thus in spite of its success in predicting pressure drop in the creeping flow regime, the cell model apparently does not afford a n attractive model for the rational extension of the theory to higher Reynolds numbers. The present analysis does, however, afford a n explanation for the surprisingly large range of Red over which the drag is linear in the superficial velocity. This has long been noted (e.g., LeClair and Hamielec, 1968, 1971) but never properly esplained.
+ 2&i(p)X-2(35i2(!J) - %(P))
Although i t is beyond the intent of this treatment to exhaust the possibilities implied by this modification, we will consider two limiting examples for the case of cells which continuously deform from spherical shells to prolate spheroids with their major axis oriented in the direction of flow. For this class of shapes, we have ro(p) =
Distorted Cell Models
Since i t is apparent that the simple cell model is incapable of accurately predicting the drag at, intermediate Reynolds numbers, we will turn our attention briefly to the possibility of altering the shape of the cell as a means of improving the model. This development has as its basis the suggest'ion of LeClair arid Hamielec (1968, p 548) that, one consider cells whose volume is constant b u t whose shape is a function of t.he Reynolds number. I n order to accommodate this modification within the framevork developed in the last sections, we will consider the class of cells whose shapes are axially symmetric bodies of revolution. We furthermore allow the dimensionless distance from the center of the particle to the cell boundary to be given by
r
=
TO^; Re,)
=
X(1
+ Re,El(p) + Rea2Ez(p) + . . . )
86 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
~ (1 e2))'/8/(1 -
e2p2)'/2
(22)
where e is the eccentricity. The shape is clearly an even function of p and hence of 8. Expanding (22) about the limiting case of a sphere (e = 0 ) , we have
Case 1. &(p)
z
0. F o r this case we parameterize t h e
problem by letting e2 = 6(1Re, where (1 is now the constant of proportionality between the Reynolds number and the eccentricity of the spheroid. Thus tl(p) = ( 3 p 2 - l)(t. It is possible to generate the first two terms in (7) by a straightforward calculation. $0 remains the same as before, while $1 must now satisfy (9) subject to the new conditions
(19)
S o t e that u-e ensure that the creeping-flow results remain valid as Re, + 0. Furthermore, in order to retain the relation b e h e e n X and voidage, we require the shape (19) t o be volume-preserving. This implies the constraints
(21c)
*l(X,P)
#1(1,P) = 0
(24)
$1,4,P) = 0
(25)
=
(% AX3 -
B
+ 4CX - 2DX-2 - 2X
-(6BX-4 - 8CX-a
+ 40Dh-6 + 4h-3)X(i(p)&i(p)
(27)
The complete solution to (9) is m
$1
+
(A,J,+~ B,rn+l
=
n=l
+ C,r2-n + D,r-,)Qn(p) +
Substitut,ion of (28) into (24-27) and making use of the orthogonality properties of the Sampson polynomials, viz.
J:l
E
Z
dp
=
0
(m# n)
we easily generate n 4 x 4 sets of linear algebraic equations for the sets of constants iL4,, B,, C,, D,} n = 1, 2, . . ., since as we have seen, the quantity of prime concern is the drag on the particle, which inlI>liesthat only the set foi n = 1 need be calculated. This set is
AI 4A1
A1X4
4Ai
+ B1X2 +
ClX
+ Bi + Ci + Di = 0 + 2B1 + Ci - Di = 0 + D1X-l =
- 2B1XP2 - 2C1X-3 + ~ D I X - ' = 6xj-I (6BXP4+ 40DX-6 + 4X?) 15
=
1 02
Re,/( 1 - E)
1 O3
Figure 3. f vs. Red/(l - E ) for distorted spheroidal cells. The theoretical predictions for different voidages collapse when plotted in these coordinates, and are shown as solid lines
(29%) (29b)
(29d)
This set may be solved in a straightforward manner, and the drag calculated to O(Re,) by eq (15). Case 2. &(p) = 0. I n this case, the eccentricity of the cell is of O(Re,), L e . e2
10
1
B~ZR~,~
The solution proceeds in an analogous fashion. $o, $ l J are identical n i t h those derived for spherical cells, while G2 differs by the terms in (21) which are proportional to &(p). The details of these calculations may be obtained from the authors upon request.
Transport in Packed Beds
I'feffer (1964) has applied the free surface cell model to high Peelet. number transport in packed beds with moderate success. Of particular int'erest is the fact t'liat predictions based upon creeping flo~vassumption remain accurat,e for Reynolds numbers as high as 50. We are nom in a position to assess the fiiiit'e Reynolds number effect upon the transport rates. (The treatment, here will be for spherical cells only.) The boundary layer form of the equation of coiivect'ive diffusion for high Peclet numbers is
where l'e
Red&
uod
= -
a,
Expanding the stream function about r Taylor series, n e have
Drag Results
Having obtaiiied the function %(r;Re,) for each of t.he tlvo cases t'reated above, \ve may determine a modified friction factor, j , whic,h reflect's the effect of cell distortion. At, this point, the model loses some of it.s original objectivity since we now must treat Cl,j-2 as free parameters and pick them so as to yield the best agreement with experimental data. hlthough we consider this object'ion a fundamental one, in that it essentially allows one t'o choose the distortion to fit the experiments, we report the results here for t'he important reason t h a t the class of spheroidal cells, with a suitable choice of Cl, {2> collapses t'he t'lieoretical predictions for different voidages onto the Ergun curve for Reynolds numbers with the region of validity of the expansion. Figure 3 illustrates this. It is possible to pick (SI, 12)such that the predictions off for a range of voidages collapse onto the esperiment'al data. !Case 2 accomplishes this slight'ly more efficient'ly, but considering the scatter in the esperinierit,al dat'a and the empirical nature of the Ergun relation, we do not consider this to be significaiit.)
=
$(?,e) = ic(1,@
+ $r(l,@(r- 1 ) +
=
+,TU
1 in the form of a
0)
2
( r - 1j2
+
but
&(l,O)
=
&,(l,O)
=
0
For high Peclet numberi a thin diffusional boundary layer is formed close to the surface. Thus
n-here y
=
r - 1j in addition, IW have
1 a+ 1 b\! 1 b$ ,ce=------r sin 8 br r sin e 30 - sin o
~ ' , = 2 - - N - - ~
-
1 a$ sill 6 by
substituting these espressioiis for 21, and Q into eq 19, t.heii
Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
87
10
E
i
0.L38
0
D a t a o f McCunr ?,Wilhalm llBL9).
0
D a t a of Williamson rt a1 11963).
X
Theorrtical results
3
10
102
10
1
- free surface model.
Re, Figure 4. j factor vs. Redat e = 0.436
Introducing the “stretched” y given by Y = (Pe/2)‘/“y we remove the explicit dependence of Pe and using eq 31 one obtains
This, of course, compares with hcrivos and Taylor’s result (1962, eq 42) with
Use of known velocity profiles enables us to evaluate the integral for nonzero Reynolds numbers. Of course, putting $ = $&-,e) recovers Pfeffer’s (1964) result. A s in the drag calculations, the first correction to the creeping flow prediction is O(Red*),and the final result may be written as Shw -Sh (Pfeffer) R’; (fzi’f(l)
{
1
Equation 33 then becomes
with the boundary conditions
c = l
Y=O
c = o
Y+
c = o
p
m
=1
which has the similarity solution given by Acrivos (1960). Of particular interest is the average Shern-ood number
where
88 Ind.
Eng. Chem. Fundam., Vol. 12, No. 1, 1973
+-
1
fzz”(1)
(-
fo”(1) -- 16 fo”(1)
1”(1)
+
=/3
rmy))}
(35)
Thus, we see that the effect of moderate Red upon Pfeffer’s creeping flow prediction is small. Figure 4 shows the results obtained from the analytical procedure when compared with the experimental results of previous investigators a t E = 0.436 plotted in the form of the Colburnj factor against Red. The theoretical curve is seen to overpredict the experimental data for all Red lees than about 60 above which agreement is excellent. Figure 5 plots the j factor against the modified Reynolds number as suggested by Pfeffer. This modified Reynolds number was suggested from the creeping flow solution in order to eliminate the voidage dependence. This is given by (Re)mod = 2.4495Red(-f 0‘ ’(1)) -’Iz
(36)
Plotting the results in this way is in fact successful in eliminating the voidage dependence for both the analytical and experimental results. The agreement between theory and experiments is good in general and becomes better as Red is increased. It should be noted that both types of cell models used give almost identical j factors for all the voidages and Reynolds numbers investigated. Although distorted cells increase the shear a t the surface, and hence increase the Sherwood number, i t is believed that t h e results would differ only slightly from those calculated using a spherical cell. This observation follows from the form of (34) which indicates that the effect of finite Reynolds number is small.
0.1
10
1
( Re’rnod
Figure 5. j factor vs. (Re),d
Conclusion
An extension to the cell-model results of Happel and Kuwabara has been shown to b e possible by a regular expansion in the Reynolds number. It is shown t h a t any spherical surface interaction model yields a n inertial correction to the creeping flow results t h a t is a t most O(RedZ). Results for the common free surface and zero-vorticity models are shown to underpredict the drag severely. I n addition, the zero-vorticity model yields too strong a voidage dependency. h rational framework is developed within which other surface interaction models may be easily tested, and one such extension, that of distorted cells, is briefly treated. T h e effect of inertia on mass or heat transport rates a t high Peclet number is shown to be very slight, in accord with experimental behavior. Thus, the ability of cell models t o account for t h e experimentally observed fact that predictions based upon creeping flow assumptions often remain valid for moderate Reynolds numbers is perhaps their most encouraging feature. Nomenclature
radius of particle radius of cell boundary = concentration = diameter of particle = 2a = spheroid eccentricity = Stokes drag on a n isolated particle = 67rpvaCo = corrected drag on a particle in a multiparticle system = mass diffusivity = Carman friction factor = known function of E f:”(l), f2:”(1), f22/1(1)= known function of E given by eq 8,10, and 11 = known function of E = Colburnj factor, Sh/RedSc’/3 = Legendre polynomials = pressure drop/unit length of bed = Peclet number = uod/D = Sampson polynomials = =
at different a
Re, Red (Re)mod r sc ShiPfeffer) Sh,, uo UT
us Y
Y
= = = = = = = = = = = =
Reynolds number = uau/v uod/v defined by eq 25 dimensionless radial distance Schmidt number = Y / B Sherwood number obtained by Pfeffer defined by eq 23 superficial velocity fluid velocity component in radial direction fluid velocity component in 0 direction (1 - T ) (Pe/2)’/ay
GREEKLETTERS known functions of E i = 1, 2, . . known functions of B i = 1 , 2 , . = 4 y ) U 7 P2)1’z = voidage in multiparticle systems = fluid kinematic viscosity = fluid density = radial stress component = tangential stress component = spherical angle = defined by eq 6 = b/a = cos0 = a function of r and Re, = Stokes stream function = components of expanded $ = expansion functions of the cell shape = parameters relating eccentricity to Re, = =
SUBSCRIPTS =
differentation with respect t o r , @
literature Cited
Acrivos, A., Phys. Fluids3,657 (1960). Acrivos, A., Taylor, T. D., Phys. Fluids5,387 (1962). Chester, W., Breach, D. R., J . Fluid Mech. 37,751 (1969). Happel, J., A.1.Ch.E. J . 4, 197 (1958). Happe!, !;, Brenner, H., ‘‘Low Re nolds Number Hydrodynamics. Prentice-Hall. Fnalewood Sliffs, N. J., 1965. Kuwabara, S., J . Phys. soc.J a p . 14,527 (1959). LeClair, B. P., Hamielec, A. E., IND. ENG.CHBM.,FUNDAX. 7, 542 (1968). Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
89
LeClair, B. P., Hamielec, A. E., IND.ENG.CHEM.,FUNDAM. 9, fin8 1 1 ~ -, 0). \ - -
LeCGir, B. P., Hamielec, A. E., Can. J . Chem. Eng. 49, 713 (1971). NcCune, L. K., Wilhelm, R. H., Ind. Eng. Chem. 41,1124 (1949). Pfeffer, R., IND. EKG.CHEY.,FUNDAM. 3,380 (1964). Ilosenhead, L., “Laminar Boundary Layers, I ’ Oxford University Press, London, 1963. Williamson, J. E., Bazaire, K. E., Geankoplis, C. J., IND.ENG. CHEM., FUNDAM. 2, 126 (1963).
Van Dyke, M., “Perturbation Methods in Fluid Mechanics,” Academic Press, New York, N. Y., 1964. RECEIVED for review March 20, 1972 ACCEPTED September 15, 1972 Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. The financial support of the National Science Foundation through Grant GK-27902 is also gratefully acknowledged.
A Model for the Kinetics of Oxygen Dissociation in a Microwave Discharge Alexis T. Bell” and Kam Kwong Department of Chemical Engineering, University of California, Berkeley, Calif. 9&Y?O
A theoretical model i s developed for describing the rate of oxygen dissociation in a microwave discharge. For this purpose the discharge i s viewed as a uniform cylindrical volume whose physical properties are determined b y the gas pressure, the discharge tube size, and the power density. Kinetic data taken from the literature are combined with the predicted electron temperature and density in order to determine the net rate of atomic oxygen formation. Based on this model a series of calculations were made of the conversion and yield for comparison with the experimental data of Mearns and Morris (1 971). From an examination of these results it i s concluded that the model gives reasonably good quantitative agreement with the data over a wide range of conditions.
I n a recent paper (Bell and Kwong, 1972), the authors proposed a model for the kinetics of oxygen dissociation in highfrequency electrical discharges. Calculatioiis of the extent of conversion and the yield of atomic oxygen were found to be in qualitative agreement wit,h the experimental result,s but a quantitative comparison \vas precluded, since the active volume of the experimental reactor could not be defined accurately. The present effort, n.as undert,aken to demonstrate that the proposed model could in fact predict quantitatively correct results for experimental situations which more suitably fitted the idealizations built’ into the model. For this 1)urpose it was decided t o make a comparison with the data of ;\learns and Morris (1971) on the dissociation of oxygen in a niicroivave discharge. Theory
Consider a cylindrical discharge tube containing a volume of discharge, 5’. The electron density and electrical field streiigth are assumed to be constant throughout the discharge and thus represent the volumetric averages of these quaiitities. Tlissociation of molecular oxygen can occur by one of two processes involving excitation from the ground state t o ritlier the 3 & + or the 3.&- excited state by electron collision. 1hese reactions can he expressed as r >
+ O2 +02*(A38,’) +O(3J’) + O ( T ) + e e + 0 2 + 02*(U3z,-)+O(3P) + O(lD) + e e
90 Ind.
Eng. Chern. Fundarn., Vol. 12, No. 1 ,
1973
(la) (Ib)
The production of atomic oxygen by dissociative attachment e+O%--tO-+O
(IC)
can be neglected since the reverse of reaction I Cis very rapid (12 = 1.4 x 10-lO cma/sec) and would serve t o remove atomic oxygen as quickly as it was formed by reaction IC. The experimentally determined cross sections for reactions Ia-c have been collected by Xyers (1969) and are illustrated in his paper as a function of electron energy. I n the present model the rate of oxygen dissociation is characterized by the sum of reactions I a and Ib. X rate constant can then be determined from the expression
where T , is the electron temperature, me is the mass of the electron, c is the electron energy, and ul is the total dissociation cross section. I n formulating eq 1 i t has been assumed that the electron energy distribution function is hIaswellian. Both Dreicer (1960) and Nyers (1969) have demonstrated that such an assumption is valid for excitations whose thresholds are not significantly larger than the average electron energy. Taking this point, into account k l was computed from eq 1 and is s h o r n in Figure 1 as a function of T,. The scale along the abscissa has also been represented in terms of E,/p, the ratio of the effective electric field strength to the gas pressure since the average electron energy ? can be represented by E,/p. In making this transformation llyers’
