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Creeping flow mass transfer to a single active sphere in a random spherical inactive particle cloud at high schmidt numbers
Chemical
Engineering Science, 1974, Vol. 29, pp. 863469.
CREEPING ACTIVE INACTIVE
Pergamon
Press.
Printed
in Great Britain
FLOW MASS TRANSFER TO A SINGLE SPHERE IN A RANDOM SPHERICAL PARTICLE CLOUD AT HIGH SCHMIDT NUMBERS KAMALESH
K. SIRKAR
Department of Chemical Engineering, Indian Institute ofTechnology, (Received
Kanpur 208016, India
30 July 1973)
Abstract - Using the solution by Tam of Navier-Stokes equations for creeping flow around an active sphere surrounded by a random cloud of inactive spheres, an asymptotic solution of the convective diffusion equation is obtained for high Schmidt numbers. The Sherwood number for the overall mass transfer coefficient to the active sphere has been analytically related to the Peclet number as Sh=0.992
2+1~5(1-e)+1~5{8(1-e)-3(1-e)*}“* l{2-3(1-e)}
1
1’3(Pe),,3,
It agrees very well with the experimental
mass transfer data on single active spheres for e = 0.476, l decreases to 0.33. Pfeffer’s model for the same problem has excellent agreement with the mass transfer data on single active spheres for E = 0.26, Re < 10 and SC = 1600. Pfeffer’s model seems to be quite satisfactory for the usual range of void volume fractions in packed beds. The present model seems to be more accurate at higher values of void volume fractions in packed and distended beds.
Re < 10 and large SC. This analytical result becomes invalid as
INTRODUCTION
Mass transfer particles
between a fluid and a large number of is utilized quite often in various chemical
engineering operations. The collection of particles may form a packed bed having varying geometry and pososity or a fluidized bed. The engineering objective in any study of mass transfer between a fluid and a collection of particles is to have an effective mean mass transfer coefficient for the multiparticle system. However, as proposed by Thoenes and Kramers [ lo] for a packed bed, the mass transfer coefficient to a single particle will give us a better understanding of the mass transport process between the fluid and the particles. Therefore, our objective in this paper is to derive an analytical result for the mass transfer coefficient to a single active sphere inside a random homogeneous isotropic cloud of inert spherical particles. Further we restrict ourselves to low Reynolds number flow and high Peclet numbers. Using the velocity profile derived from the “free surface model” of Happel[2] for a random assembly of spherical particles, Pfeffer[7] had analytically determined the dimensionless mass transfer coefficient as a function of the Peclet number and the fractional void volume of the multiparticle system. The above relationship is valid only for high Peclet
and low Reynolds numbers. In this work, we use the creeping flow solution of Navier-Stokes equations for a random spherical particle cloud as obtained recently by Tam[91 by means of a point force approximation and no particle-particle correlation. By adding a test sphere to the random homogeneous spherical particle cloud, Tam solved the equations for the mean fluid motion around the test particle and interpreted the solution as the most probable flow field around the test sphere. This flow field may, no doubt, be substituted into the convective diffusion equation for a sphere and a numerical solution would yield the Sherwood number for the whole range of Peclet numbers in the manner of Pfeffer and Happel[8]. However we are here interested in analytically obtaining an asymptotic relationship between Sherwood number and Peclet number when the latter takes very high values. The thinness of the concentration boundary layer at high Peclet numbers ensures this as exemplified by Levich[6].
FLOW FIELD AROUND
THE ACTIVE
SPHERE
The most probable flow field around the active spherical particle .of radius b in a homogeneous isotropic spherical particle cloud has been derived 863
864
KAMALESH
by Tam [9] to be
o =grad
K
K. SIRKAR
first three moments function n(b) as
u,,+A
e-ar(laT~r)-l-J$x]
m,, m,, m3 of the distribution
a = &rrmz+ [367r2m22+24~m~[1 - (3C/2)111’2
(5)
(2-3C)
+A?;
(1) where m, = J n(b) bn db, C = &rm, = volume concentration of particles and C = 1 -e, E being the void volume fraction. For a particle cloud containing only spheres of radius d/2, it can be shown from the above that
where A =-s
biJoeab
B=z[-azb2+3(eab-I-ab)].
l+a_d=2+1*5C+1~5[8C-3CZ]“2 2 (2-3C)
The (Yand u in Tam’s [9] nomenclature has been replaced here by a and b respectively. The unit vector in the positive x direction is Tand the superscript ‘-’ denotes a vector. The coordinate system for the flow around the active sphere is indicated in Fig. 1. The mean fluid velocity in the positive x-direction inside the particle cloud before the addition of the active particle is UO, which should be considered as the value of 0 far away from the active particle. The quantity a is defined by db
u
= r
u
case cl
I
2Bcose r3
n(b) = IV6 g-b (
1
where 6[ (d/2) -b] is the Dirac delta function. The quantity D,.(b) is such that the drag force experienced by the active particle of radius b is given as Drag force = D,(b) U,;.
1.
e+
I+ar a2r3
-1
(7)
Define a stream function I,!Jsuch that 1 a+ rz sin 6 a0
u,=---
(8) One then obtains from the velocity profile expression (7), the following expression for the stream function: $=-
Uor2 sin2 6 -~ B sin2 0+ A sin2 6 2 r azr X
[e-ar( 1 +ar) - 11.
(9)
In the limit of volume concentration of particles round the active sphere going to zero, the quantity a tends to zero and the above stream function reduces to that for the Stokes flow past a single sphere as it should.
CONVECTIVE
Fig.
u case
sin 0+ B sin 0 r3
_Asine
(4)
Tam[9] has evaluated the quantity a in terms of the
I
a2r3
UB = _ U0 sin e _Ae-ar
(2)
where n(b) is the number density distribution of particle sizes such that n(b) db is the number density of particles with radii between b and b+ db. For a particle cloud containing N particles/volume of uniform particle size d/2, n(b) is given as
(6)
This formula will be used often later. The velocity profile of Eq. (1) as derived by Tam [9] yields the following axisymmetric velocity profile around the sphere:
r /UP = j n(b)&(b)
’
DIFFUSION EQUATION FOR THE ACTIVE SPHERE AND ITS SOLUTION
Levich [6] has shown that for Stokes flow around a sphere the steady state convective diffusion equation for a species i at high Schmidt numbers
865
Creeping flow mass transfer
~A__’
can be simplified to
b sin B ay
3y sin 0 3 -26U,,(l+ab). At high Peclet numbers, the concentration change takes place in a region very close to the solid surface. This thinness of the concentration boundary layer permits the use of a velocity profile valid in this region only. It must be pointed out here that the creeping flow assumption need not restrict the sphere Reynolds number, Re, (based on superficial velocity) to less than 1. Happel and Brenner[3] has pointed out that in a packed bed, inertial effects are not important till Re - 5. Thus high Peclet numbers need not mean Pe G SC but Pe G 5Sc with the Schmidt number being much greater than 100 as is usually encountered in mass transfer between a liquid and a solid. Recent experimental measurements of mass transfer to a single active sphere in a close packed cubic array by Karabelas, Wegner and Hanratty[S], indicate that the mass transfer asymptote for very large Peclet numbers is valid for Re < 10.
If 9 is the distance from the surface of the solid sphere of any point in the thin concentration boundary layer, then r = b+y wherey <<< b, the radius of the active sphere. Changing the variables (r, 19) to (I/J.0) in the manner of Levich[6], the convective diffusion equation reduces to z=D,b’sin2H$
(1 +ab+a2b2)
z=-
Dib2sin2 6~(1+ob)]$[~f$$].
(15) The concentration boundary layer starts growing from 0 = r towards 0 = 0 in a direction opposite to that of the positive &coordinate. Define
ef=7r-e
(16)
so that 0’ = 0 when 0 = 7~and 0’ = 7~ when 0 = 0. The equation (15) may then be written in terms of independent variables I,IJand 8’ as
2 =Dib' sin' e'a(l
+ ab)$[
aas].
(17) The boundary conditions for this equation are: (a) AtJ,=O,Ci=O (at the particle surface) (b) AtfY=Oandr,!r=O,Ci=Ci, (front stagnation point) (c) As+-,--m,CI=Cio (far from the particle).
(18)
[ 1. (11) Uti$
Substituting r = b+y in the expression for I/J and expanding it in terms of powers ofy, we have
x ;
Substitution of this expression for lJB in the simplified mass balance equation (11) results in
1
+ terms with higher powers ofy.
Here Cio is the bulk concentration of species i in the solution that approaches the sphere. If in Eq. (17), the quantity U,,( 1 + ab) is replaced by Uo, the corresponding boundary value problem has been exactly solved by Levich [6] and the total diffusional flux to the particle surface I was derived by him to be 1 = 7.9gCiat’3U0”3b4’3.
(19)
(12)
For high Peclet numbers, terms containing y3 and higher powers ofy may be neglected, so that
In the present case, therefore, the total diffusional flux to the particle surface can be easily shown to be I = 7.98Cr,@~‘3U,1’3( 1 + ab)1’3b4’3.
The dimensionless diffusional flux to the particle surface, the Sherwood number, is then given as
and U= e
CES-Vol.
(20)
29, No. 3.-O
-- I alCr r sin 0 ar
Sh
=
lb
4?rb2DiCi,
= ?i.$?(FYI3 *
( 1 +
&,)
l/3.
(21)
866
KAMALESH
Since it is preferable to define the Sherwood number with respect to the diameter of the active particle d, the above result is changed to 1’3
(1 +c~b)“~.
(22)
In the expression (22), the term (U,b/D,) is some sort of a Peclet number based on the particle radius b and U,,, the mean fluid velocity in the particle cloud as the fluid approaches the test sphere from a distance. Normally, however, correlations for packed bed mass transfer are expressed as functions of the particle Peclet number based on the particle diameter d and the superficial velocity U,. If we make the reasonable assumption here that u, = U&
K. SIRKAR expression (6) for [ 1-t (ad/2)] diverges as E decreases to l/3 or as C 4 213. Consequently, both the y2 and the y3 term in the expression (12) for $, the stream function, diverge as e is reduced towards l/3, corresponding to the lowest void volume fraction in a randomly packed bed of identical spheres. The formula for the drag force as derived by Tam 191 as well as the expression (25) for the Sherwood number similarly diverge as E approaches l/3 from higher values. It is therefore necessary to know whether and how such divergence affects the high Peclet aumber assumption. Specifically, we will determine for which values of E and Pe, the neglect of the y3 term in comparison to the yz term is justified in the stream function expression (12). Dividing the y3 term in (12) by the y2 term we obtain,
(23) Y Y ----_=_
the Sherwood number based on particle diameter can be expressed as a function of the particle Peclet number (U,d/Di) in the following manner: 1’3
Sh = 0992 (’ :$)l”
(Pe)‘j3.
(24)
(25)
The above expression thus allows us to determine the Sherwood number for mass transfer to an active sphere immersed in a random homogeneous cloud of inactive spheres when the flow is in the Stokes regime and the Schmidt number is very high. It is worthwhile to mention here that both Pfeffer [7] and Pfeffer and Happel[8] had concluded earlier that for small Reynolds numbers and for Peclet number tending to infinity the solution of the mass balance equation for packed bed mass transfer to a sphere is of the form Sh = g(e)Pe1’3
3b
y3 term
a2b2
y2term’
3b(l+ab)
Following Levich [6], we can obtain the following expression for the concentration boundary layer thickness 6 around the active sphere from the solution of Eq. ( 17) subject to boundary conditions ( 18): 6 = 1.15 [8’-(sin28’/2]1’3 sin 8’
c
0.49 (l+ab)“3 (29) Defining as before, the Peclet number Pe as (U,26/ Di) the expression (29) is further reduced to 0.49 (2E) 1’3
0.49 (2~) 1’3a2b2
(Pe)1’3(1+ab)1~3-(Pe)1’3(1+ab)4’3* TAM’S VELOCITY PROFILE AND HIGH PECLET NUMBER ASSUMPTION FOR MASS TRANSFER
There analysis spherical of E, the
is an important limitation of Tam’s[9] of flow around a test sphere embedded in a particle cloud. It is not valid for all values fractional void volume. For example, the
(2g)
The function of angle 0’ in the above expression is such that it becomes infinite as 8’ -+ 7~ indicating a breakdown of our theory (but not of the results, since mass flux around the rear stagnation point is very small). At IV’= 0, this function is unity and at 8’ = 7r/2 it is (7r/2)l13 - l-162. So if y/b in (27) is written as 6/b, the ratio of the two terms in (27) is
(26)
where g(e) is a function of the void fraction E of the packed bed. The expression (25) as derived here is similar in form since [ ( 1+ ab)/e] l/3 is a function of E and the particle cloud geometry.
(27)
(30)
It is clear that since ab is much larger than 1 (it is 17.5 for E = O-436 and 8.05 for e = 0.53) the first term in (30) is less than 0.02 for all Peclet numbers greater than 1000. For the second term to have a value of O-1 so that the use of only y* term in the
Creeping
flow mass transfer
velocity profile isjustified, the Peclet number should be 24900 for E = O-436 and 5,000 for E = 0.53 etc. Thus for Tam’s velocity profile, as c is reduced, the Sherwood number expression (25) would be correct only if the Peclet number increases progressively. Somewhat similar behavior is observed if the velocity profile of Pfeffer[7] is used. The stream function that Pfeffer used may be written as 1cI=-
U, sin2 0 3(1 2W [
-y5)y2-5
x(1+4y5)+y4term+...
(31)
I so that the ratio of the y3 term to the y2 term is _y(l+49) b3(1-75)
where y is defined E= l-y3 and W=2-3y-t 3y5 - 2ys. Using the expression for the concentration boundary layer thickness of Levich[6], the ratio of (32) may be rewritten as -1.259 ($)“3(&J”3(3;;4$)).
(33)
Here also the functional dependence on c is such that in order for this ratio to be 0.1, Peclet number must continuously increase as E is progressively reduced. However, even at E = 0.3, the value of the Peclet number needed is only 720. As E --, 1 for a single sphere in an infinite fluid, the Sherwood number relationship with Peclet number in both cases (this paper’s and Pfeffer’s) reduce to the familiar Levich[6] solution: Sh = 0*992(Pe)1’3.
(34)
Thus it is concluded that the divergence of Tam’s velocity profile as E tends towards l/3 limits the applicability of relations derived here to values of E higher than l/3. At any of the acceptable values of E, the method of Acrivos and Goddard[ 11 may be adopted to calculate a first order correction term to the asymptotic mass transfer expression (25) derived here. For a first order correction term to Pfeffer’s[7] result, Sh = 1.26[ (1 -y”)/ W]1J3Pe1f3, no restrictions on E are necessary. COMPARISON
OF DERIVATIONS WITH AVAILABLE EXPERIMENTAL RESULTS
The expression (25) derived in this work should be compared with measurements of the overall rate of mass transfer to a single active sphere inside a random cloud of inactive spheres for high Schmidt numbers in low Reynolds number flow. Very few
867
measurements of the above type are available in literature. Jells and Hanratty[4] measured the overall rate of mass transfer to a single active sphere in a dumped bed of inert spheres (void fraction E = 0.41) for a Schmidt number of about 1700. The Reynolds number, Re, was varied from 5 to 1100. They could obtain satisfactory correlations for Re > 35 but for the low range laminar region, the scatter in their data was unfortunately large and no significant correlation was offered. The recent measurements by Karabelas et a1.[5] of the overall rate of mass transfer to single spheres in a close packed cubic array of inert spheres (E = 0.26) cannot be compared with our derivation because the expression (25) is not valid for the lower range of values of E. The experimental investigations of Thoenes and Kramers[lO] fortunately provide some data to compare with (25). These authors have determined the overall mass transfer coefficient from a single active sphere surrounded by inactive regularly packed spheres. The void fraction was varied from 0.26 to 0.48 by using different regular packing arrangements. For our purpose their data on a benzoic acid sphere dissolving in water with the packing arrangement corresponding to 4b (cubic, e = 0.476, see Thoenes and Kramers’ paper) is relevant since data for all other packing arrangements were taken at Reynolds number, Re, greater than 10. The benzoic acid-water system ensures high Schmidt number. Thoenes and Kramer’s[lO] data for 4b packing in the very low Reynolds number range, Re < 10, can be represented by the following equation: Sh = 3.05(Pe)1’3. For l = 0.476, the expression following:
(35)
(25) reduces to the
Sh = 3Nl(Pe)1’3.
(36)
Except for the fact that the express ion (25) is for a random assemblage of particles, the agreement between the theory and the meagre data that is available, is excellent. For E = 0.476, Pfeffer’s [7] derivation yields the following expression: Sh = 2.90(Pe) *‘3.
(37)
Thus Pfeffer’s derivation underestimates the data for e = O-476 by a very small amount of 5 per cent. It is, therefore, quite in order to compare Pfeffer’s result [7] with the measurements by Karabelas ef al. [5] of the overall mass transfer rate to an active
868
KAMALESH
sphere in a spheres. For numbers (Re e = 0.26, the measurements
close forced < 10) valid is
packed cubic array of inert convection at low Reynolds and high Peclet numbers and asymptotic relation for their
K. SIRKAR
Table 1. Comparison of the present model with Pfeffer’s model
volume
Sh = 4*58(Pe)1’3
Sh = 1.26 l-
(l-•)5’3
[ 2-3(1--~)“~+3(1-e)~‘~-22(1--e)~
X (Pe)‘j3.
fraction
0.26 0.30 0.37 0.436 0.476 0.53 0.60
(38)
Pfeffer’s result [7], given below,
’
g(e)
E, Void
1 113
This
work
Not valid Not valid 5.241 3.45 3.00 2.53 2.135
Pfeffer [71 4.54 4.25 3.775 3.13 2% 264 2.35
(39) CONCLUDING
reduces to the following expression for E = O-26: Sh = 4-54(Pe)1’3
(40)
indicating that Pfeffer’s derivation agrees very well with the experimental data available for E = 0.26. All the comparisons made so far deal with the overall mass transfer coefficient for the active sphere. Ideally one should compare the variation of the local mass transfer coefficient around the surface of the active sphere. No such data is available for Re < 10 and high Schmidt numbers. It must be mentioned, however, that at the location of contact points of the active sphere with the inactive spheres surrounding it, the predictions of the model of this work and Pfeffer’s [7] are not going to be in agreement with measurements. For example, in the dense cubic arrangement of spheres (E = O-26), there are contact points at 0 = 135,90 and 45” (see Wegner er a[.[1 11). Both the models predict finite mass transfer rate at f3= 135” and 45”. No comparison has also been made between the present model and the large amount of experimental data that is available for the overall mass transfer coefficient in packed or fluidized beds in which all the spheres are active. Jolls and Hanrattyl41 have pointed out that such data appear to be significantly lower than the results for a single active sphere since the effective concentration driving force is less when all the spheres are exchanging mass with the fluid. Pfeffer [7] compared his analytical results with only this type of data and as his Figs. 2 and 3 indicate, most of the data fall below his predicted correlation (39). It is appropriate at this stage to compare the analytical results of this work and Pfeffer’s[7]. Table 1 compares the two estimates of g(c) if the overall mass transfer coefficient correlation is represented as Sh = g ( e)Pe1’3.
REMARKS
The analytical results of the proposed model of mass transfer to a single active sphere in a random homogeneous inactive spherical particle cloud agree quite well with the experimental data for higher values of void fraction of the particle cloud. The nonapplicability of the present work for low values of void fraction in packed beds follows from the use of the point force approximation and no particle-particle correlation in Tam’s derivation of the velocity profile through the particle cloud. For packed beds of high l and distended beds, the analytical result of this work is likely to be quite accurate. Pfeffer’s model is however, likely to be very good for packed beds of low E. More experimental data on overall mass transfer coefficient to a single active sphere in packed and distended beds of inactive spheres are needed at various values of E, for high Schmidt numbers in low Reynolds number flow. Further, models such as the present one and Pfeffer’s do not predict local mass transfer coefficients correctly at the location of the contact points of the active sphere with the inactive spheres surrounding it.
NOTATION
a A b B
defined by Eq. (2) defined by Eq. (1) radius of a spherical particle defined by Eq. (1) c volume concentration of spherical particles in the particle cloud ci concentration of ith species in the fluid Ci0 bulk concentration of ith species d diameter of active sphere Di diffusivity of the ith species D, defined by Eq. (4) L? defined by Eq. (26) k mass transfer coefficient
Creeping flow mass transfer 4 4 m3
I
moments defined by Eq. (5)
number density distribution of particle sizes number of particles per unit volume of uniform particles of radius d/2 Pe Peclet number, (U,d/D,) r distance from origin of coordinates Re Reynolds number, (U&/v) Sh Sherwood number, (kd/Di) SC Schmidt number (v/DJ the fluid velocity vector in a particle cloud 0 UO mean fluid velocity in the positive x-direction in the particle cloud before the addition of the active particle Us superficial velocity in the particle cloud UT fluid velocity in the radial direction around the active particle Us fluid velocity in the direction of positive 0 coordinate around the active particle W defined by Eq. (32)
n(b)
0 $ F Y
869
spherical angle defined by Eq. (8), the stream function dynamic viscosity kinematic viscosity
N
Greek symbols y (1 -#‘3
E void volume fraction
REFERENCES
HI ACRIVOS A. and GODDARD J. D., J. Fluid Mech.
1965 23 273.
r21 HAPPEL J., A.Z.Ch.E. J 19.584 197. [31 HAPPEL J. and BRENNER H., Low Reynolds Number Hydrodynamics. Prentice-Hall, Englewood Cliffs, N.J. 1965. [41 JOLLS K. R. and HANRATI’Y T. J., A.1Ch.E. J 1969 15 199. [51 KARABELAS A. J., WEGNER T. H. and HANRATTY T. J..Chem.En~n~Sci. 197126 1.581. K51 LEVICH V. G., Physic&hYemical Hydrodynamics. Prentice-Hall, Englewood Cliffs, N.J. 1962. [71 PFEFFER R., Ind. Engng Chem. Fundls 1964 3 380.
[8] PFEFFER R. and HAPPEL J., A.Z.Ch.E. J 1964 10 605. [9] TAM C. K. W., .I. Fluid Mech. 1969 38 537. [lo] THOENES D. and KRAMERS H., Chem. Engng Sci. 19588291.
[l l] WEGNER T. H., KARABELAS A. J. and HANRATTY T. J., Chem. Engng Sci. 197126 59.
Engineering Science, 1974, Vol. 29, pp. 863469.
CREEPING ACTIVE INACTIVE
Pergamon
Press.
Printed
in Great Britain
FLOW MASS TRANSFER TO A SINGLE SPHERE IN A RANDOM SPHERICAL PARTICLE CLOUD AT HIGH SCHMIDT NUMBERS KAMALESH
K. SIRKAR
Department of Chemical Engineering, Indian Institute ofTechnology, (Received
Kanpur 208016, India
30 July 1973)
Abstract - Using the solution by Tam of Navier-Stokes equations for creeping flow around an active sphere surrounded by a random cloud of inactive spheres, an asymptotic solution of the convective diffusion equation is obtained for high Schmidt numbers. The Sherwood number for the overall mass transfer coefficient to the active sphere has been analytically related to the Peclet number as Sh=0.992
2+1~5(1-e)+1~5{8(1-e)-3(1-e)*}“* l{2-3(1-e)}
1
1’3(Pe),,3,
It agrees very well with the experimental
mass transfer data on single active spheres for e = 0.476, l decreases to 0.33. Pfeffer’s model for the same problem has excellent agreement with the mass transfer data on single active spheres for E = 0.26, Re < 10 and SC = 1600. Pfeffer’s model seems to be quite satisfactory for the usual range of void volume fractions in packed beds. The present model seems to be more accurate at higher values of void volume fractions in packed and distended beds.
Re < 10 and large SC. This analytical result becomes invalid as
INTRODUCTION
Mass transfer particles
between a fluid and a large number of is utilized quite often in various chemical
engineering operations. The collection of particles may form a packed bed having varying geometry and pososity or a fluidized bed. The engineering objective in any study of mass transfer between a fluid and a collection of particles is to have an effective mean mass transfer coefficient for the multiparticle system. However, as proposed by Thoenes and Kramers [ lo] for a packed bed, the mass transfer coefficient to a single particle will give us a better understanding of the mass transport process between the fluid and the particles. Therefore, our objective in this paper is to derive an analytical result for the mass transfer coefficient to a single active sphere inside a random homogeneous isotropic cloud of inert spherical particles. Further we restrict ourselves to low Reynolds number flow and high Peclet numbers. Using the velocity profile derived from the “free surface model” of Happel[2] for a random assembly of spherical particles, Pfeffer[7] had analytically determined the dimensionless mass transfer coefficient as a function of the Peclet number and the fractional void volume of the multiparticle system. The above relationship is valid only for high Peclet
and low Reynolds numbers. In this work, we use the creeping flow solution of Navier-Stokes equations for a random spherical particle cloud as obtained recently by Tam[91 by means of a point force approximation and no particle-particle correlation. By adding a test sphere to the random homogeneous spherical particle cloud, Tam solved the equations for the mean fluid motion around the test particle and interpreted the solution as the most probable flow field around the test sphere. This flow field may, no doubt, be substituted into the convective diffusion equation for a sphere and a numerical solution would yield the Sherwood number for the whole range of Peclet numbers in the manner of Pfeffer and Happel[8]. However we are here interested in analytically obtaining an asymptotic relationship between Sherwood number and Peclet number when the latter takes very high values. The thinness of the concentration boundary layer at high Peclet numbers ensures this as exemplified by Levich[6].
FLOW FIELD AROUND
THE ACTIVE
SPHERE
The most probable flow field around the active spherical particle .of radius b in a homogeneous isotropic spherical particle cloud has been derived 863
864
KAMALESH
by Tam [9] to be
o =grad
K
K. SIRKAR
first three moments function n(b) as
u,,+A
e-ar(laT~r)-l-J$x]
m,, m,, m3 of the distribution
a = &rrmz+ [367r2m22+24~m~[1 - (3C/2)111’2
(5)
(2-3C)
+A?;
(1) where m, = J n(b) bn db, C = &rm, = volume concentration of particles and C = 1 -e, E being the void volume fraction. For a particle cloud containing only spheres of radius d/2, it can be shown from the above that
where A =-s
biJoeab
B=z[-azb2+3(eab-I-ab)].
l+a_d=2+1*5C+1~5[8C-3CZ]“2 2 (2-3C)
The (Yand u in Tam’s [9] nomenclature has been replaced here by a and b respectively. The unit vector in the positive x direction is Tand the superscript ‘-’ denotes a vector. The coordinate system for the flow around the active sphere is indicated in Fig. 1. The mean fluid velocity in the positive x-direction inside the particle cloud before the addition of the active particle is UO, which should be considered as the value of 0 far away from the active particle. The quantity a is defined by db
u
= r
u
case cl
I
2Bcose r3
n(b) = IV6 g-b (
1
where 6[ (d/2) -b] is the Dirac delta function. The quantity D,.(b) is such that the drag force experienced by the active particle of radius b is given as Drag force = D,(b) U,;.
1.
e+
I+ar a2r3
-1
(7)
Define a stream function I,!Jsuch that 1 a+ rz sin 6 a0
u,=---
(8) One then obtains from the velocity profile expression (7), the following expression for the stream function: $=-
Uor2 sin2 6 -~ B sin2 0+ A sin2 6 2 r azr X
[e-ar( 1 +ar) - 11.
(9)
In the limit of volume concentration of particles round the active sphere going to zero, the quantity a tends to zero and the above stream function reduces to that for the Stokes flow past a single sphere as it should.
CONVECTIVE
Fig.
u case
sin 0+ B sin 0 r3
_Asine
(4)
Tam[9] has evaluated the quantity a in terms of the
I
a2r3
UB = _ U0 sin e _Ae-ar
(2)
where n(b) is the number density distribution of particle sizes such that n(b) db is the number density of particles with radii between b and b+ db. For a particle cloud containing N particles/volume of uniform particle size d/2, n(b) is given as
(6)
This formula will be used often later. The velocity profile of Eq. (1) as derived by Tam [9] yields the following axisymmetric velocity profile around the sphere:
r /UP = j n(b)&(b)
’
DIFFUSION EQUATION FOR THE ACTIVE SPHERE AND ITS SOLUTION
Levich [6] has shown that for Stokes flow around a sphere the steady state convective diffusion equation for a species i at high Schmidt numbers
865
Creeping flow mass transfer
~A__’
can be simplified to
b sin B ay
3y sin 0 3 -26U,,(l+ab). At high Peclet numbers, the concentration change takes place in a region very close to the solid surface. This thinness of the concentration boundary layer permits the use of a velocity profile valid in this region only. It must be pointed out here that the creeping flow assumption need not restrict the sphere Reynolds number, Re, (based on superficial velocity) to less than 1. Happel and Brenner[3] has pointed out that in a packed bed, inertial effects are not important till Re - 5. Thus high Peclet numbers need not mean Pe G SC but Pe G 5Sc with the Schmidt number being much greater than 100 as is usually encountered in mass transfer between a liquid and a solid. Recent experimental measurements of mass transfer to a single active sphere in a close packed cubic array by Karabelas, Wegner and Hanratty[S], indicate that the mass transfer asymptote for very large Peclet numbers is valid for Re < 10.
If 9 is the distance from the surface of the solid sphere of any point in the thin concentration boundary layer, then r = b+y wherey <<< b, the radius of the active sphere. Changing the variables (r, 19) to (I/J.0) in the manner of Levich[6], the convective diffusion equation reduces to z=D,b’sin2H$
(1 +ab+a2b2)
z=-
Dib2sin2 6~(1+ob)]$[~f$$].
(15) The concentration boundary layer starts growing from 0 = r towards 0 = 0 in a direction opposite to that of the positive &coordinate. Define
ef=7r-e
(16)
so that 0’ = 0 when 0 = 7~and 0’ = 7~ when 0 = 0. The equation (15) may then be written in terms of independent variables I,IJand 8’ as
2 =Dib' sin' e'a(l
+ ab)$[
aas].
(17) The boundary conditions for this equation are: (a) AtJ,=O,Ci=O (at the particle surface) (b) AtfY=Oandr,!r=O,Ci=Ci, (front stagnation point) (c) As+-,--m,CI=Cio (far from the particle).
(18)
[ 1. (11) Uti$
Substituting r = b+y in the expression for I/J and expanding it in terms of powers ofy, we have
x ;
Substitution of this expression for lJB in the simplified mass balance equation (11) results in
1
+ terms with higher powers ofy.
Here Cio is the bulk concentration of species i in the solution that approaches the sphere. If in Eq. (17), the quantity U,,( 1 + ab) is replaced by Uo, the corresponding boundary value problem has been exactly solved by Levich [6] and the total diffusional flux to the particle surface I was derived by him to be 1 = 7.9gCiat’3U0”3b4’3.
(19)
(12)
For high Peclet numbers, terms containing y3 and higher powers ofy may be neglected, so that
In the present case, therefore, the total diffusional flux to the particle surface can be easily shown to be I = 7.98Cr,@~‘3U,1’3( 1 + ab)1’3b4’3.
The dimensionless diffusional flux to the particle surface, the Sherwood number, is then given as
and U= e
CES-Vol.
(20)
29, No. 3.-O
-- I alCr r sin 0 ar
Sh
=
lb
4?rb2DiCi,
= ?i.$?(FYI3 *
( 1 +
&,)
l/3.
(21)
866
KAMALESH
Since it is preferable to define the Sherwood number with respect to the diameter of the active particle d, the above result is changed to 1’3
(1 +c~b)“~.
(22)
In the expression (22), the term (U,b/D,) is some sort of a Peclet number based on the particle radius b and U,,, the mean fluid velocity in the particle cloud as the fluid approaches the test sphere from a distance. Normally, however, correlations for packed bed mass transfer are expressed as functions of the particle Peclet number based on the particle diameter d and the superficial velocity U,. If we make the reasonable assumption here that u, = U&
K. SIRKAR expression (6) for [ 1-t (ad/2)] diverges as E decreases to l/3 or as C 4 213. Consequently, both the y2 and the y3 term in the expression (12) for $, the stream function, diverge as e is reduced towards l/3, corresponding to the lowest void volume fraction in a randomly packed bed of identical spheres. The formula for the drag force as derived by Tam 191 as well as the expression (25) for the Sherwood number similarly diverge as E approaches l/3 from higher values. It is therefore necessary to know whether and how such divergence affects the high Peclet aumber assumption. Specifically, we will determine for which values of E and Pe, the neglect of the y3 term in comparison to the yz term is justified in the stream function expression (12). Dividing the y3 term in (12) by the y2 term we obtain,
(23) Y Y ----_=_
the Sherwood number based on particle diameter can be expressed as a function of the particle Peclet number (U,d/Di) in the following manner: 1’3
Sh = 0992 (’ :$)l”
(Pe)‘j3.
(24)
(25)
The above expression thus allows us to determine the Sherwood number for mass transfer to an active sphere immersed in a random homogeneous cloud of inactive spheres when the flow is in the Stokes regime and the Schmidt number is very high. It is worthwhile to mention here that both Pfeffer [7] and Pfeffer and Happel[8] had concluded earlier that for small Reynolds numbers and for Peclet number tending to infinity the solution of the mass balance equation for packed bed mass transfer to a sphere is of the form Sh = g(e)Pe1’3
3b
y3 term
a2b2
y2term’
3b(l+ab)
Following Levich [6], we can obtain the following expression for the concentration boundary layer thickness 6 around the active sphere from the solution of Eq. ( 17) subject to boundary conditions ( 18): 6 = 1.15 [8’-(sin28’/2]1’3 sin 8’
c
0.49 (l+ab)“3 (29) Defining as before, the Peclet number Pe as (U,26/ Di) the expression (29) is further reduced to 0.49 (2E) 1’3
0.49 (2~) 1’3a2b2
(Pe)1’3(1+ab)1~3-(Pe)1’3(1+ab)4’3* TAM’S VELOCITY PROFILE AND HIGH PECLET NUMBER ASSUMPTION FOR MASS TRANSFER
There analysis spherical of E, the
is an important limitation of Tam’s[9] of flow around a test sphere embedded in a particle cloud. It is not valid for all values fractional void volume. For example, the
(2g)
The function of angle 0’ in the above expression is such that it becomes infinite as 8’ -+ 7~ indicating a breakdown of our theory (but not of the results, since mass flux around the rear stagnation point is very small). At IV’= 0, this function is unity and at 8’ = 7r/2 it is (7r/2)l13 - l-162. So if y/b in (27) is written as 6/b, the ratio of the two terms in (27) is
(26)
where g(e) is a function of the void fraction E of the packed bed. The expression (25) as derived here is similar in form since [ ( 1+ ab)/e] l/3 is a function of E and the particle cloud geometry.
(27)
(30)
It is clear that since ab is much larger than 1 (it is 17.5 for E = O-436 and 8.05 for e = 0.53) the first term in (30) is less than 0.02 for all Peclet numbers greater than 1000. For the second term to have a value of O-1 so that the use of only y* term in the
Creeping
flow mass transfer
velocity profile isjustified, the Peclet number should be 24900 for E = O-436 and 5,000 for E = 0.53 etc. Thus for Tam’s velocity profile, as c is reduced, the Sherwood number expression (25) would be correct only if the Peclet number increases progressively. Somewhat similar behavior is observed if the velocity profile of Pfeffer[7] is used. The stream function that Pfeffer used may be written as 1cI=-
U, sin2 0 3(1 2W [
-y5)y2-5
x(1+4y5)+y4term+...
(31)
I so that the ratio of the y3 term to the y2 term is _y(l+49) b3(1-75)
where y is defined E= l-y3 and W=2-3y-t 3y5 - 2ys. Using the expression for the concentration boundary layer thickness of Levich[6], the ratio of (32) may be rewritten as -1.259 ($)“3(&J”3(3;;4$)).
(33)
Here also the functional dependence on c is such that in order for this ratio to be 0.1, Peclet number must continuously increase as E is progressively reduced. However, even at E = 0.3, the value of the Peclet number needed is only 720. As E --, 1 for a single sphere in an infinite fluid, the Sherwood number relationship with Peclet number in both cases (this paper’s and Pfeffer’s) reduce to the familiar Levich[6] solution: Sh = 0*992(Pe)1’3.
(34)
Thus it is concluded that the divergence of Tam’s velocity profile as E tends towards l/3 limits the applicability of relations derived here to values of E higher than l/3. At any of the acceptable values of E, the method of Acrivos and Goddard[ 11 may be adopted to calculate a first order correction term to the asymptotic mass transfer expression (25) derived here. For a first order correction term to Pfeffer’s[7] result, Sh = 1.26[ (1 -y”)/ W]1J3Pe1f3, no restrictions on E are necessary. COMPARISON
OF DERIVATIONS WITH AVAILABLE EXPERIMENTAL RESULTS
The expression (25) derived in this work should be compared with measurements of the overall rate of mass transfer to a single active sphere inside a random cloud of inactive spheres for high Schmidt numbers in low Reynolds number flow. Very few
867
measurements of the above type are available in literature. Jells and Hanratty[4] measured the overall rate of mass transfer to a single active sphere in a dumped bed of inert spheres (void fraction E = 0.41) for a Schmidt number of about 1700. The Reynolds number, Re, was varied from 5 to 1100. They could obtain satisfactory correlations for Re > 35 but for the low range laminar region, the scatter in their data was unfortunately large and no significant correlation was offered. The recent measurements by Karabelas et a1.[5] of the overall rate of mass transfer to single spheres in a close packed cubic array of inert spheres (E = 0.26) cannot be compared with our derivation because the expression (25) is not valid for the lower range of values of E. The experimental investigations of Thoenes and Kramers[lO] fortunately provide some data to compare with (25). These authors have determined the overall mass transfer coefficient from a single active sphere surrounded by inactive regularly packed spheres. The void fraction was varied from 0.26 to 0.48 by using different regular packing arrangements. For our purpose their data on a benzoic acid sphere dissolving in water with the packing arrangement corresponding to 4b (cubic, e = 0.476, see Thoenes and Kramers’ paper) is relevant since data for all other packing arrangements were taken at Reynolds number, Re, greater than 10. The benzoic acid-water system ensures high Schmidt number. Thoenes and Kramer’s[lO] data for 4b packing in the very low Reynolds number range, Re < 10, can be represented by the following equation: Sh = 3.05(Pe)1’3. For l = 0.476, the expression following:
(35)
(25) reduces to the
Sh = 3Nl(Pe)1’3.
(36)
Except for the fact that the express ion (25) is for a random assemblage of particles, the agreement between the theory and the meagre data that is available, is excellent. For E = 0.476, Pfeffer’s [7] derivation yields the following expression: Sh = 2.90(Pe) *‘3.
(37)
Thus Pfeffer’s derivation underestimates the data for e = O-476 by a very small amount of 5 per cent. It is, therefore, quite in order to compare Pfeffer’s result [7] with the measurements by Karabelas ef al. [5] of the overall mass transfer rate to an active
868
KAMALESH
sphere in a spheres. For numbers (Re e = 0.26, the measurements
close forced < 10) valid is
packed cubic array of inert convection at low Reynolds and high Peclet numbers and asymptotic relation for their
K. SIRKAR
Table 1. Comparison of the present model with Pfeffer’s model
volume
Sh = 4*58(Pe)1’3
Sh = 1.26 l-
(l-•)5’3
[ 2-3(1--~)“~+3(1-e)~‘~-22(1--e)~
X (Pe)‘j3.
fraction
0.26 0.30 0.37 0.436 0.476 0.53 0.60
(38)
Pfeffer’s result [7], given below,
’
g(e)
E, Void
1 113
This
work
Not valid Not valid 5.241 3.45 3.00 2.53 2.135
Pfeffer [71 4.54 4.25 3.775 3.13 2% 264 2.35
(39) CONCLUDING
reduces to the following expression for E = O-26: Sh = 4-54(Pe)1’3
(40)
indicating that Pfeffer’s derivation agrees very well with the experimental data available for E = 0.26. All the comparisons made so far deal with the overall mass transfer coefficient for the active sphere. Ideally one should compare the variation of the local mass transfer coefficient around the surface of the active sphere. No such data is available for Re < 10 and high Schmidt numbers. It must be mentioned, however, that at the location of contact points of the active sphere with the inactive spheres surrounding it, the predictions of the model of this work and Pfeffer’s [7] are not going to be in agreement with measurements. For example, in the dense cubic arrangement of spheres (E = O-26), there are contact points at 0 = 135,90 and 45” (see Wegner er a[.[1 11). Both the models predict finite mass transfer rate at f3= 135” and 45”. No comparison has also been made between the present model and the large amount of experimental data that is available for the overall mass transfer coefficient in packed or fluidized beds in which all the spheres are active. Jolls and Hanrattyl41 have pointed out that such data appear to be significantly lower than the results for a single active sphere since the effective concentration driving force is less when all the spheres are exchanging mass with the fluid. Pfeffer [7] compared his analytical results with only this type of data and as his Figs. 2 and 3 indicate, most of the data fall below his predicted correlation (39). It is appropriate at this stage to compare the analytical results of this work and Pfeffer’s[7]. Table 1 compares the two estimates of g(c) if the overall mass transfer coefficient correlation is represented as Sh = g ( e)Pe1’3.
REMARKS
The analytical results of the proposed model of mass transfer to a single active sphere in a random homogeneous inactive spherical particle cloud agree quite well with the experimental data for higher values of void fraction of the particle cloud. The nonapplicability of the present work for low values of void fraction in packed beds follows from the use of the point force approximation and no particle-particle correlation in Tam’s derivation of the velocity profile through the particle cloud. For packed beds of high l and distended beds, the analytical result of this work is likely to be quite accurate. Pfeffer’s model is however, likely to be very good for packed beds of low E. More experimental data on overall mass transfer coefficient to a single active sphere in packed and distended beds of inactive spheres are needed at various values of E, for high Schmidt numbers in low Reynolds number flow. Further, models such as the present one and Pfeffer’s do not predict local mass transfer coefficients correctly at the location of the contact points of the active sphere with the inactive spheres surrounding it.
NOTATION
a A b B
defined by Eq. (2) defined by Eq. (1) radius of a spherical particle defined by Eq. (1) c volume concentration of spherical particles in the particle cloud ci concentration of ith species in the fluid Ci0 bulk concentration of ith species d diameter of active sphere Di diffusivity of the ith species D, defined by Eq. (4) L? defined by Eq. (26) k mass transfer coefficient
Creeping flow mass transfer 4 4 m3
I
moments defined by Eq. (5)
number density distribution of particle sizes number of particles per unit volume of uniform particles of radius d/2 Pe Peclet number, (U,d/D,) r distance from origin of coordinates Re Reynolds number, (U&/v) Sh Sherwood number, (kd/Di) SC Schmidt number (v/DJ the fluid velocity vector in a particle cloud 0 UO mean fluid velocity in the positive x-direction in the particle cloud before the addition of the active particle Us superficial velocity in the particle cloud UT fluid velocity in the radial direction around the active particle Us fluid velocity in the direction of positive 0 coordinate around the active particle W defined by Eq. (32)
n(b)
0 $ F Y
869
spherical angle defined by Eq. (8), the stream function dynamic viscosity kinematic viscosity
N
Greek symbols y (1 -#‘3
E void volume fraction
REFERENCES
HI ACRIVOS A. and GODDARD J. D., J. Fluid Mech.
1965 23 273.
r21 HAPPEL J., A.Z.Ch.E. J 19.584 197. [31 HAPPEL J. and BRENNER H., Low Reynolds Number Hydrodynamics. Prentice-Hall, Englewood Cliffs, N.J. 1965. [41 JOLLS K. R. and HANRATI’Y T. J., A.1Ch.E. J 1969 15 199. [51 KARABELAS A. J., WEGNER T. H. and HANRATTY T. J..Chem.En~n~Sci. 197126 1.581. K51 LEVICH V. G., Physic&hYemical Hydrodynamics. Prentice-Hall, Englewood Cliffs, N.J. 1962. [71 PFEFFER R., Ind. Engng Chem. Fundls 1964 3 380.
[8] PFEFFER R. and HAPPEL J., A.Z.Ch.E. J 1964 10 605. [9] TAM C. K. W., .I. Fluid Mech. 1969 38 537. [lo] THOENES D. and KRAMERS H., Chem. Engng Sci. 19588291.
[l l] WEGNER T. H., KARABELAS A. J. and HANRATTY T. J., Chem. Engng Sci. 197126 59.