Ind. Eng. Chem. Res. 2004, 43, 3439-3450
3439
Analytical Study of Drag and Mass Transfer in Creeping Power Law Flow across Tube Banks Jose M. Ferreira Departamento de Fı´sica, Universidade de Tra´ s-os-Montes e Alto Douro, Apartado 1013, 5000-911 Vila Real, Portugal
Rajendra P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India 208016
The Happel and Kuwabara cell models are combined with the Stokes flow approximation (creeping flow) and the Levich thin boundary layer assumption to investigate theoretically the creeping flow and convective mass transfer of power law fluids across banks of cylinders. A well-known perturbation technique involving the use of the Newtonian flow field as a first approximation is employed to determine the closed-form stream function for power law fluids. Analytical expressions are derived for the drag coefficient and Sherwood (or Nusselt) number and applied over a wide range of power law indices (0.6 e n e 1) and voidages (0.4 e e ∼1). The behavior of these expressions in the Newtonian limit, i.e., n ) 1, is also investigated. The analytical results reported herein for the drag coefficient are validated for the power law index and voidage ranges specified above using the numerical results available in the literature. Unfortunately, outside the Newtonian limit, there are only limited numerical results available for heat transfer to test the validity of the analytical expressions derived in the present study. Introduction The cross-flow of fluids across a collection of cylinders and tube banks (i.e., arrays of long cylinders) is encountered in many industrial processes, which include the use of fibers in adsorbers, electrolyzers, hollow-fiber reactors, membrane separation processes, tubular heat exchangers, the autoclave process of producing fiberreinforced resins, filtration of polymer melts, sewage sludges using screens and mats, etc.1-3 Additional applications where this geometry is of direct relevance include interstitial flow in ligaments and tendons4 and processing of high-consistency paper fiber suspensions in fluidized beds5 and in aerosol filters.6 Because of such important applications of these processes in the chemical, petroleum, polymer, paper, and food processing industries, considerable effort has been directed toward the determination of the main engineering parameters associated with such flow, namely, the drag coefficient (or the pressure drop) and the convective heat- and mass-transfer rates. The resulting voluminous body of work has been analyzed in earlier papers, and extensive reviews are available in the literature6-9 and the references contained therein. On the basis of a combination of analytical/numerical simulations coupled with experimental results, it is perhaps fair to say that satisfactory methods are now available, enabling the prediction of gross-engineering parameters such as the friction factor and Nusselt and/or Sherwood number for the flow of Newtonian fluids in these systems,7,10,11 in both the low and high Reynolds number regimes. It is readily acknowledged that for many materials encountered in some of the aforementioned industrial settings, notably in polymer, food, and process engineering applications, the liquid phase exhibits a range of non-Newtonian flow characteristics including shear thinning, viscoelasticity, time-dependent behavior, yield * To whom correspondence should be addressed. E-mail:
[email protected].
stress, etc. Despite such a wide occurrence of nonNewtonian behavior, very little work has been reported in the literature on the cross-flow and heat transfer to non-Newtonian liquids from tube banks. This work sets out to fill this gap existing in the currently available body of knowledge. In particular, consideration is given to the prediction of the frictional pressure drop and the rate of interphase heat/mass transfer at low Reynolds numbers in such systems. Owing to the generally high viscosity levels of non-Newtonian substances (such as polymeric solutions and melts, food suspensions, etc.), the flow regime often relates to low Reynolds numbers, and this work thus deals with the so-called creepingflow conditions. It is, however, instructive and desirable to first briefly recount the currently available scant literature on this topic. From a theoretical standpoint, because of the complex nature of the flow field across assemblages of cylinders, most of the work in this field relies on the use of modeling to describe the cylinder-cylinder interactions, which must be combined with the governing equations of continuity, momentum, and energy to obtain the cylinder drag and/or heat- and mass-transfer rates. Though several surface interaction models simulating the flow field across cylinder assemblages have been described in the literature, there are mainly two approaches available to approximate these interactions. In the first approach, the tube banks are seen as constituting a fibrous porous medium with tortuous flow passages present in it. By analogy with the granular porous media, some investigators have used the socalled capillary models to develop the flow rate/pressure drop law for tube bundles. The works of Adams and Bell,12 Kyan et al.,13 and Prakash et al.14 exemplify the applicability of this approach for the flow of Newtonian and power law fluids across bundles of cylinders. In the second approach, the flow in tube banks is seen as that
10.1021/ie030812e CCC: $27.50 © 2004 American Chemical Society Published on Web 05/29/2004
3440 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004
over submerged cylinders, and the resulting pressure drop is ascribed to the drag on the assemblage of solid rods of large aspect ratios. Within the framework of this submerged-objects model, one can discern two distinct approaches. In the first approach, the governing equations are applied for a periodic array of long cylinders with known geometrical configurations, e.g., the inline or staggered arrays, square or triangular, pitch, etc. The utility of this approach has been demonstrated among others by Sangani and Acrivos,15 Wung and Chen,16,17 and Martin et al.18 for Newtonian fluids and by Skartsis et al.19 and Bruschke and Advani8 for the creeping flow of power law liquids past hexagonal arrays of circular cylinders. Skartsis et al.19 employed the finite element method to calculate the permeabilities for the cross-flow of power law liquids past inline and staggered arrays of long cylinders encompassing the values of porosity in the range of 0.4-0.95 or so. With the increasing degree of shear-thinning behavior, the dependence of the Kozeny constant on the porosity was reported to weaken. They also reported a reasonable match between their predictions and experimental results. On the other hand, Bruschke and Advani8 numerically studied the creeping flow past hexagonal arrays of cylinders at moderate values of porosity, and these results were supplemented by the use of a cell model for sparse systems (high porosity). Similarly, Nieckele et al.20 have numerically studied the creeping flow of Bingham fluids past a staggered array of circular cylinders and reported detailed information on the yielded/unyielded regions and on the pressure drop. Obviously, such studies tend to be computationally intensive, and each geometrical configuration must be treated as a specific case, even for the flow of Newtonian fluids. In other words, the extrapolation of results from one arrangement to even a slightly different geometrical arrangement is neither justified nor possible. The second approach to the modeling of the cylinder-cylinder hydrodynamic interactions hinges on the use of the so-called cell models.21 However, only two such models, namely, Happel’s free surface model22 and Kuwabara’s zero-vorticity model,23 seem to have gained wide acceptance in predicting macroscopic transport phenomena in cylinder assemblages. While the latter approach is less rigorous than the one based on periodic arrays, it has been shown to yield good predictions of gross-engineering parameters for the flow of Newtonian and inelastic non-Newtonian fluids.24,25 Thus, for instance, both the aforementioned cell models and their counterparts for sphere assemblages have been shown to yield satisfactory predictions of frictional losses for the flow and heat transfer of Newtonian and inelastic non-Newtonian fluids through such assemblages of spheres25-30 and of long circular cylinders.31-40 In both the Happel22 and Kuwabara23 cell models, the viscous interactions between the fluid and the cylinder assemblage are accounted for by enclosing each cylinder by a hypothetical coaxial cylinder of the fluid of a size such that the voidage of each cell is equal to the overall mean voidage of the array. The two cell models are exactly identical with each other except for one boundary condition at the surface of the hypothetical outer cylinder. For Happel’s free surface model, zero shear stress is imposed on the outer surface, whereas a zerovorticity condition is used for the case of the Kuwabara model. Besides these two models, Spielman and Goren41 employed the Brinkman model to obtain the local flow
field around a fiber to evaluate the removal efficiency of fibrous filters. However, this model has a singularity at a porosity of 50%, thereby casting some doubt about its validity. Similarly, Yu and Soong42 have developed a random cell model for the prediction of the pressure drop in fibrous filters. Extensive discussions on the relative merits and demerits of all of these models are available in the literature.21,24,38,39,41-43 While it is not possible to offer a sound theoretical justification for any of these models, especially with regard to the boundary conditions at the cell boundary, the fact that the two cell models (i.e., the free surface and zero vorticity) are not inconsistent with the somewhat more rigorous and physically realistic model43 inspires confidence in the use of such simple cell models to approximate the hydrodynamic interactions in such complex flows, especially the free surface cell model. This assertion is further supported by the fact that the experimental values of the pressure drop and Nusselt number for Newtonian and power law liquids in granular beds and in tube bundles lie somewhere between the predictions of these two models and are also in reasonable agreement with the predictions based on the use of arrays.9,25,28,30,37,38,40,44,45 Furthermore, most of the work available in the literature is numerical, although for the case of creeping Newtonian flow across an assemblage of cylinders some analytical studies of both heat-46 and mass-transfer rates47,48 have been undertaken. However, an analytical study of the creeping flow of power law liquids across a cylinder assemblage has not been reported to date and is thus justified because of its obvious applications in the polymer processing industry. Moreover, from the engineering standpoint, an analytical solution is also a necessary first step toward a better understanding of the relationship between the non-Newtonian flow characteristics, kinematic parameters (drag and rates of heat or mass transfer), and design parameters (voidage of the system). Also, such analytical treatments are useful in benchmarking numerical studies. As far as we know, no such analytical study has been carried out for heat or mass transfer and only Bruschke and Advani8 have calculated the permeability of a bundle of cylinders as a function of the voidage and power law index n. However, their expression seems to be affected by the inadvertent omission of a factor of n from their stream function differential equation and corresponding solution. Fortunately, this omission does not have a significant impact on their results close to the Newtonian limit of n ) 1. The only other analytical attempt is that of Tripathi and Chhabra,31,32 who employed the velocity and stress variational principles to obtain upper and lower bounds on the drag coefficient under creeping-flow conditions for power law and Carreau model fluids. The two bounds coincide only in the limit of Newtonian fluid behavior (n ) 1) and, in fact, diverge appreciably with the increasing extent of shear-thinning fluid behavior. In view of the paucity of such analytical work in the literature, the present study thus endeavors to determine analytical expressions for the creeping flow of power law fluids and to evaluate drag, as well as heatand mass-transfer rates, as a function of the voidage and power law index for creeping flow across an ensemble of cylinders, using both the free surface and zero-vorticity cell models. Because the equivalent problem for sphere ensembles has been solved by Kawase and Ulbrecht,26 the procedure used in the present study
Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3441
where
γ)a j /b h
(2)
and the cylinder length L h . a j and b h . A cylindrical coordinate system (rj, θ, zj) fixed at the center of the cylinder is chosen so that the angle θ is measured from the front stagnation point, as shown in Figure 1b. Because the flow is assumed to be two-dimensional, all flow variables are independent of the zj coordinate; i.e., they are functions of rj and θ alone at each point rj within the cell. Under the aforementioned conditions of steady, incompressible, creeping, and axisymmetric flow, the equation of continuity and the rj and θ components of the equations of motion are nondimensionalized as
Figure 1. (a) Schematic representation of flow. (b) Cell model idealization.
to determine the stream function for the flow of power law fluids closely parallels their study, and the range of voidage (0.4 e < ∼1) and power law index (0.6 e n e 1) values consistent with their approximations is also implicit in the present study. The resulting stream function, in turn, has been used to derive the cylinder drag and to solve the species continuity equation. The cylinder-to-fluid mass (or heat)-transfer rates are determined in the present study using the thin boundary layer approximation,49 as has been employed by other investigators46,47 for Newtonian liquids. A similar procedure has been used for cylinder assemblages with Newtonian fluids and for sphere assemblages in Newtonian50 and non-Newtonian power law fluids,26,51 all under conditions of creeping flow and large Peclet numbers. Thus, under creeping-flow conditions, the large Peclet number condition is usually satisfied when the Schmidt number is high. This situation is quite common in masstransfer applications, e.g., mass transfer across the walls of hollow membranes, thereby justifying the use of the high Peclet number condition in the present study. In heat-transfer studies, the Prandtl number is used instead of the Schmidt number, and high Prandtl numbers can also occur in heat-transfer applications such as those involving viscous oils, polymer solutions, melts, etc.52 In this analysis, consideration is given to the mass-transfer problem, but the results are equally applicable to the heat-transfer problem with the proviso that the Schmidt and Sherwood numbers are replaced respectively by the Nusselt and Prandtl numbers. Statement of the Problem and Solution Procedure
πa j 2L h ) 1 - γ2 πb h 2L h
() [
Π ∂p ) ∂r 2
(n-1)/2
1 ∂∆rθ ∆θθ 1 ∂ (r∆rr) + + r ∂r r ∂θ r 1 ∂Π ∆rθ 1 ∂Π n-1 ∆rr + 2 Π ∂r r Π ∂θ
(
() [
Π 1 ∂p ) r ∂θ 2
(n-1)/2
1 ∂ 2 1 ∂∆θθ (r + ∆ ) + rθ r ∂θ r2 ∂r 1 ∂Π ∆θθ 1 ∂Π n-1 + ∆rθ 2 Π ∂r r Π ∂θ
(
(1)
(3)
)]
(4a)
)]
(4b)
where p is the pressure, r is the radial coordinate, and v, vr and vθ are the r and θ components of fluid velocity b and it has been assumed that all stress tensor components τij are related to those of the rate of deformation tensor ∆ij through the power law model, i.e.
(Π2 )
τij ) -
(n-1)/2
∆ij
(5)
with n the power law index and Π the second invariant of the rate of the deformation tensor. These dimensionless variables are obtained from their dimensional counterparts as
p)
p j - Fjgj rj m j (V h 0/a j )n
(6a)
r ) rj/a j
(6b)
vj rj vj θ , b v ) (vr, vθ) ) V h0 V h0
(6c)
τij )
Consider an incompressible power law fluid in steady, creeping, and axisymmetric flow normal to an ensemble of long solid cylinders (far away streaming velocity V h 0), as shown schematically in Figure 1a. Both the free surface and zero-vorticity cell models envisage each solid cylinder of radius a j to be surrounded by a hypothetical coaxial cylindrical envelope of fluid (Figure 1b). The radius of this envelope, b h , is chosen so that the voidage of each cell is equal to the average voidage of the assemblage, i.e.
)1-
∂vθ ∂ (rv ) + )0 ∂r r ∂θ
( )
∆ij )
a j V h0
( )
n
jτij m j
a j ∆ h V h 0 jı jj
Π ) (a j /V h 0)2Π h
(6d) (6e) (6f)
where the dimensional parameters Fj and m j are the power law fluid’s density and consistency index and gj is the acceleration due to gravity. The dimensionless continuity equation (3) is satisfied by the stream function ψ defined in the usual way as
vr ) -
1 ∂ψ r ∂θ
(7a)
3442 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004
vθ ) ∂ψ/∂r
(7b)
The three usual boundary conditions that are common to both the free surface and zero-vorticity cell models are
vr ) vθ ) 0 vr ) -cos θ
at r ) 1
(8)
at r ) γ-1
(9)
obtained using each cell model, in the analysis that follows equations “a” refer to the free surface model and equations “b” to the zero-vorticity model. From eqs 11 and 12, the components of the rate of the deformation tensor in cylindrical coordinates are given as
∆rr ) 2 )
The fourth boundary condition that applies to the free surface cell model is
∆rθ ) r
()
∂ vθ 1 ∂vr )0 + ∂r r r ∂θ
at r ) γ-1 (10a)
[
]
vθ )
[
[
]
4 1 2 (2 - γ2) 3 - + γ2r cos θ K1 r r
(
)
∆rθ )
vθ )
[
]
]
32 1 (1 + γ4)2 2 cos2θ (H1)2 r
Π≈
(16a)
(16b)
32 4 cos2 θ (K1)2 r2
(17a)
(17b)
1 ∂Π ∆rθ 1 ∂Π e ) ∆rr + Π ∂r r Π ∂θ ≈
[
)
(15a,b)
and
1 1 -(2 - γ2) 2 + 2(1 - γ2) - 4 ln r + γ2r2 cos θ K1 r (11b)
vr )
(
4 1 - γ4r sin θ H1 r3
[
≈
(13a)
For the zero-vorticity model, it is shown that
()
∂ vθ 1 ∂vr + ∂r r r ∂θ
4 1 (2 - γ2) 3 - γ2r sin θ K1 r
with
H1 ) -(1 - γ ) - 2(1 + γ )ln γ.
)
Π ) ∆rr2 + ∆θθ2 + 2∆rθ2
]
4
(14b)
In the analysis that follows, it is assumed that γ is smaller than 1 (maximum γ ) 0.775), and because γ4r < γ2r < γ, both γ2r and γ4r were neglected and only the dominant terms were retained in eqs 14-16. These dominant terms give contributions of order r-2 to the second invariant of the rate of the deformation tensor, whereas the next dominant term gives contributions of order r-4 and r-6 and was therefore neglected. Using the aforementioned approximations, which parallel those used for sphere ensembles by Kawase and Ulbrecht,26 gives
1 1 - + (1 + 3γ4) + 2(1 + γ4) ln r - 3γ4r2 sin θ H1 r2 (12a)
4
(14a)
1 ∂vθ vr ) -∆rr + r ∂θ r
∆rθ ) r
(10b)
1 1 - + (1 - γ4) - 2(1 + γ4) ln r + γ4r2 cos θ H1 r2 (11a)
vr )
]
∆θθ ) 2
at r ) γ-1
For mildly shear-thinning fluids, i.e., n not too different from unity, the term (Π/2)(n-1)/2 and the curved bracketed terms in eq 4 that are multiplied by (n - 1)/2 can be evaluated by using the Newtonian stream function as a first approximation, a perturbation technique that has been widely used because of its success.8,26,53-55 The Newtonian flow solution for both the free surface and zero-vorticity models is determined using a wellestablished procedure, and these are as such available in the literature:21 the boundary conditions specified in eqs 8-10 are used together with eq 7 to determine the Newtonian stream function and thus obtain vr and vθ. For the free surface model, it can be shown that
[
4 1 1 - (1 + γ4) + γ4r cos θ H1 r3 r
∆rr )
whereas for the zero-vorticity cell model, the fourth boundary condition is
1 ∂vr ∂vθ vθ |∇ + + )0 v || ) |B ∧b r ∂θ ∂r r
∂vr ∂r
]
1 1 -(2 - γ2) 2 + 2(1 + γ2) + 4 ln r - 3γ2r2 sin θ K1 r (12b)
1 8 (1 + γ4) 2 cos θ H1 r e≈
16 1 cos θ K1 r 2
(18a)
(18b)
1 ∂Π ∆θθ 1 ∂Π + f ) ∆rθ Π ∂r r Π ∂θ
with
K1 ) -(3 + γ4) + 4(γ2 - ln γ)
(13b)
To facilitate a comparison between the expressions
≈-
8 1 (1 + γ4) 2 sin θ H1 r
(19a)
Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3443
f≈-
16 1 sin θ K1 r2
(19b)
Inserting eqs 18 and 19 into eq 4 gives
() ( () (
Π ∂p ) ∂r 2
(n-1)/2
1 ∂p Π ) r ∂θ 2
Qn -
(n-1)/2
Ln r2
Sn +
cos θ
Ln r2
) )
(20)
sin θ
(21)
En ) -
2(n + 1)
(30)
It is also appropriate to mention here that the factor of n on the right-hand side of eqs 28 and 30 is missing from the corresponding equations of Bruschke and Advani.8 Inserting the non-Newtonian stream function into eq 7 and using boundary conditions (8)-(10) results in four equations for each cell model, with the first three being common to both cell models:
where
1 ∂ 1 ∂∆rθ 1 v )r ) (r∆rr) + - ∆θθ Qn ) (∇2b r ∂r r ∂θ r
nLn
(22)
An + Bn + Dn ) 0
(31)
-An + Bn + Cn + (n + 2)Dn ) 0
(32)
(23)
γ2An + Bn - (ln γ)Cn + γ-(n+1)Dn + (ln γ)2En ) 1 (33)
and for the free surface and zero-vorticity models Ln is given respectively by
The fourth boundary condition for the free surface and zero-vorticity cell models is given respectively by
Sn ) (∇2b v )θ )
Ln ) -
1 ∂ 2 1 ∂∆θθ (r ∆rθ) + 2 ∂r r ∂θ r
4 (n - 1)(1 + γ4) H1
8 Ln ) - (n - 1) K1
(24a)
4γ3An + (n + 1)2γ-nDn + 2γEn ) 0
(24b)
2γCn + (n + 1)(n + 3)γ-nDn + 2γ(1 - 2 ln γ)En ) 0 (34b)
Now eliminating pressure between eqs 20 and 21 by the usual method of cross-differentiation and using eq 17 gives
The solution obtained from the free surface model (eqs 31-33 and 34a) is
An )
∂Qn ∂ - (rSn) + (n - 1)(Sn - Qn tan θ) ) ∂θ ∂r 2nLn - 2 sin θ (25) r For cylindrical coordinates with vz ) 0 and no z dependence, the following identities always hold:
∂ 2 ∂ (∇ b v )r - [r(∇2b v )θ] ) -r∇4ψ ∂θ ∂r v )θ - tan θ(∇2B v )r ) (∇2b
(26)
(∂r∂ + tanr θ ∂θ∂ )∇ ψ 2
(
∇4ψ - (n - 1)
)
2nLn 1 ∂ tan θ ∂ 2 ∇ ψ ) 3 sin θ + 2 r ∂r r ∂θ r (28)
The non-Newtonian stream function is determined from eq 28 by standard mathematical techniques,56 giving
ψ)
2 -(n-1) ξn - ξ 1 (n + 1) γ + E 4 Hn Hn n
1 γ4 - (n + 1)2γ-(n-1) ξn 4 E Bn ) Hn Hn n
(35a)
(36a)
1 1 γ4 + (n + 1)γ-(n-1) ξn - (n + 3)ξ 2 2 +2 En Cn ) (n + 1) Hn Hn (37a)
(27)
From the definitions of Qn and Sn in eqs 22 and 23, a comparison of eqs 26 and 27 with eq 25 immediately gives
(34a)
Dn ) -
γ4 ξ + E Hn Hn n
(38a)
with
ξ)
γ4 γ2 + γ2 ln γ - + γ4(ln γ)2 2 2
(39a)
1 1 1 ξn ) γ-(n-1) + (n + 3)γ2 ln γ - γ4 + γ4(ln γ)2 2 2 2 1 (n + 1)2γ-(n-1)(ln γ)2 (40a) 4
An + Bnr + Cnr ln r + Dnrn+2 + Enr(ln r)2 sin θ r (29)
1 1 Hn ) γ4 - (n + 1)2γ-(n-1) + (n - 1)(n + 3)γ-(n-3) 4 4 1 (n + 1)γ4 ln γ - (n + 1)2γ-(n-1) ln γ (41a) 2
where An, Bn, Cn, and Dn are obtained for each cell model using the corresponding boundary conditions, and
The solution obtained from the zero-vorticity model (eqs 31-33 and 34b) is
[
]
3444 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004
1 An ) 2
1 Bn ) 2
1 (n + 1) γ2 - (n + 3)γ-(n-1) Fn - F 2 + En Kn Kn (35b)
[
]
1 (n + 3) γ2 - (n + 1)γ-(n-1) Fn 2 - En (36b) Kn Kn
[
]
1 Fn - (n + 3)F 2 En Kn (37b)
-(n-1) 1 (n + 1)(n + 3)γ +2 Cn ) 2 Kn
Dn ) -
γ2 F + E Kn Kn n
(38b)
with
F)
2
4
γ γ - + γ4 ln γ - γ2(ln γ)2 2 2
(39b)
1 1 1 Fn ) γ-(n-1) + (n + 3)γ2 ln γ - γ4 2 2 2 1 γ-(n-1) ln γ - (n + 3)γ2(ln γ)2 + γ4 ln γ 2 1 (n + 1)(n + 3)γ-(n-1)(ln γ)2 (40b) 4 1 1 Kn ) (n + 3)γ2 - 1 + (n + 1)(n + 3) γ-(n-1) 2 4 1 1 (n + 1)(n + 3)γ-(n-1) ln γ + (n + 1)(n + 3)γ-(n-3) 2 4 1 (n + 1)γ4 (41b) 2
[
]
Sn ) (∇2b v )θ )
Evaluation of the Drag Coefficient As mentioned earlier, the so-called cell models envision each solid cylinder in the ensemble to be surrounded by a hypothetical envelope of fluid (Figure 1b). The drag force exerted by the fluid on each cylinder can be determined from the dimensionless drag coefficient CD, which is written as
CD ) )
2n [Re n
2 Re
∫
π
(p)r)1 cos θ dθ + -π
∫
()
τrθ ) -
Π 2
(n-1)/2
() ( ()
∆rθ ) -
() (
Sn +
(n-1)/2
r
)
∂ vθ 1 ∂vr + ∂r r r ∂θ (44)
Inserting eq 7 into eq 44 and using eqs 21 and 43 gives
∂p + τrθ ) ∂θ Π (n-1)/2 ∂3ψ 1 ∂2 ∂ψ 1 ∂2ψ Ln sin θ (45) r 3 + - 2 2+ 2 2 r ∂θ ∂r r ∂r r ∂θ
() [
]
( )
The integrand of eq 42 is determined through the insertion of the non-Newtonian stream function (29) into eq 45 to obtain
(∂p∂θ + τ ) rθ
r)1
) Gn
[(Π2 )
(n-1)/2
]
r)1
sin θ
(46)
with
Gn ) -4An - 2Cn + (n + 1)[n(n + 2) - 1]Dn + Ln (47) From eqs 42 and 46,
CD )
2n G Re n
∫-ππ[(Π2 )
(n-1)/2
]
r)1
sin2 θ dθ
(48)
On the other hand, from eqs 17, 24, and 30
[(Π2 )
]
r)1
) Mn(cos2 θ)(n-1)/2 > 0
(49)
with
Mn )
[
]
2(n + 1)En n(n - 1)
n-1
>0
(50)
Inserting eq 49 into eq 48 and integrating by parts give
CD )
2n 4 G M Z Re nn n n
(51)
where
(τ ) sin θ dθ] -π rθ r)1 (42)
where the Reynolds number Re ) Fj(2a j )n/m j (V h 0)n-2. The first term in the integrand of eq 42 is obtained using eq 21, i.e. (n-1)/2
Π 2
π
+ τrθ) sin θ dθ ∫-ππ(∂p ∂θ r)1
Π ∂p )r ∂θ 2
(43)
The second term in the integrand of eq 42 is determined from eq 5, i.e.
(n-1)/2
Equations 35-41 complete the characterization of the non-Newtonian stream function under the conditions of the present analysis and, for n ) 1 (En ) 0, Hn ) H1, and Kn ) K1), reduce to the known expressions of the Newtonian stream function for both cell models.
∂ 2 (∇ ψ) ∂r
Ln r2
)
sin θ
(21)
where for cylindrical coordinates with vz ) 0 and no z dependence the following identity holds:
Zn )
∫0
π/2
(cos θ)
n+1
xπ dθ ) 2
(n2 + 1) n+3 Γ( 2 ) Γ
(52)
and Γ is the Euler gamma function. It is appropriate to mention here that, except at n ) 1, eq 51 for the drag coefficient is at variance with the corresponding equation derived by Bruschke and Advani.8 Mass-Transfer Analysis As mentioned before, the analysis that follows is based on an application of the thin boundary layer approximation49 to ensembles of cylinders. It is ap-
Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3445
propriate to mention here that the procedure that is outlined below closely parallels that already used in heat- and mass-transfer analysis for cylinder ensembles in Newtonian fluids with the necessary adaptations to the non-Newtonian situation.46-48 Consider the creeping power law flow across an assememblage of cylinders, which is described within each cell by the non-Newtonian stream function derived in eq 29, i.e.
ψ)
[
]
An + Bnr + Cnr ln r + Dnrn+2 + Enr(ln r)2 sin θ r (29)
Under these conditions, although no hydrodynamic boundary layer is formed, a diffusional boundary layer does exist near the solid cylinder surface. For a system with large Peclet numbers, mass transfer from the surface of the solid cylinder into the surrounding envelope of fluid is characterized by a change in concentration that is confined to a thin diffusional boundary layer of thickness yj, which is related to the radial coordinate rj and the solid cylinder radius a j as
rj ) a j + yj, a j e rj e b h
(
)
(54)
where D h is the mass diffusivity and the concentration h bh at points rj ) a j and rj ) (C h ) takes the values C h aj and C b h , respectively. Inserting eq 53 as well as the dimensionless variables
y ) yj/a j C)
1 1 (n + 2)Dn]y + An + Cn + (n + 1)(n + 2)Dn + 2 2
[
where
1 1 Pn ) An + Cn + (n + 1)(n + 2)Dn + En 2 2
(55b)
vr ) vθ )
(
)
where the Peclet number Pe is defined as
h Pe ) 2a jV h 0/D
(57)
r)1+y
(58)
For a very thin boundary layer (y , 1), the nonNewtonian stream function (29) can be expanded in a Taylor series using eq 58. Retaining terms in the expansion up to the second order in y gives
∂ψ ≈ 2Pny sin θ ∂r
(61b)
(62)
Equation 59 can be used to write eq 62 as a function of ψ and θ, giving
(∂C ∂θ )
ψ
)
4xPn ∂ ∂C xsin θ xψ Pe ∂ψ ∂ψ θ
{ [ ( ) ]}
θ
(63)
where Pn is a function of both power law index n and voidage , and eq 63 is subject to the three usual boundary conditions, namely
at rj ) a j , or equivalently
(i) C h )C h aj
C)0 (ii) C h )C h bh
at ψ ) 0
(64a)
at rj ) b h , or equivalently
C ) 1 at ψ ) Pn
(1 -γ γ)
2
sin θ
(64b)
(iii) At the point of incidence (ψ ) 0, θ ) 0), the incident flow has not yet been depleted by diffusion and the concentration is equal to that at the bulk of the solution, i.e.
at ψ ) 0 and θ ) 0
(64c)
The solution to eq 63 follows (with minor variations) a well-established procedure pioneered by Levich,49 and to avoid redundancy, only the relevant steps associated with these variations are outlined below. The change of variable
tn )
and
(61a)
∂C 1 ∂2C y ∂C 1 ≈ cos θ + ∂θ (Pn)(Pe) y ∂y2 2 ∂y
sin θ
C)1 (56)
1 ∂ψ ≈ -Pny2 cos θ r ∂θ
for vr and vθ can be inserted into eq 56, resulting in
vr, and vθ into eq 54 gives
vθ ∂C Pe ∂C ∂2C + vr - 2 ≈0 2 ∂y 1 + y ∂θ ∂y
(60)
and the zeroth- and first-order terms in y are identically zero because of boundary conditions (31) and (32), which are common to both cell models. Under very thin boundary layer conditions (y , 1), the approximations
(55a)
C h -C h aj C h bh - C h aj
]}
En y2 sin θ ) Pny2 sin θ (59)
(53)
Because under steady creeping-flow conditions the convective diffusion equation for mass transfer has the same structure for both Newtonian and non-Newtonian fluids, the approximations that have previously been used in the Newtonian context46-48 are equally valid for the present non-Newtonian analysis. Thus, for a very thin boundary layer (yj/a j , 1) and steady, creeping, power law flow conditions, the convective diffusion equation is approximated as
h ∂C h vj θ ∂C h ∂2C 1 + vj rj - 2 )0 D h ∂rj rj ∂θ ∂rj
{
ψ ≈ An + Bn + Dn + [-An + Bn + Cn +
4xPn xsin θ dθ + W1 Pe (W1: integration constant) (65)
∫
is introduced into eq 63, giving
∂ ∂C ∂C xψ ) ∂tn ∂ψ ∂ψ
(
)
(66)
3446 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004
and hence
∫
C(zn) ) W2 exp(-4zn3/9) dzn + W3
(67)
with
zn )
xψ
[
4xPn xsin θ dθ + W1 Pe
∫
]
(68)
1/3
and W2 and W3 are two additional constants to be determined using boundary conditions (64). These boundary conditions, in conjunction with eq 59, are applied to eqs 67 and 68, resulting in the following expressions:
C(zn) ) (12)1/3 1 Γ 3
()
(
n
)
3
∫0z exp - 4R9
(
∫0z exp - 4R9
dR ) 0.855
n
)
3
dR (69)
where
( ) [∫
Pn zn ) Pe 4
1/3
y
(sin |θ|)3/2 |θ|
0
]
xsin β dβ
1/3
, -π e θ e π
(70)
The Sherwood number (NSh) is defined as NSh ) -2 (∂C/ ∂y)y)0, and its surface-averaged value (Sh) is, therefore,
Sh )
1 2π
dθ ∫-ππNSh dθ ) - π1∫-ππ(∂C ∂y )y)0
(71)
Because (∂C/∂y)y)0 is an even function of θ, eqs 6971 result in 1/3
Sh 34/3 Pn )[ 1/3 π 1 Pe Γ 3
()
∫0πxsin β dβ]2/3 ) -0.921Pn1/3 (72)
which gives the average Sherwood number as a function of the Peclet number and is valid only for cylinder assemblages (voidage of 0.4 e < 1) in fluids of power law indices n in the range of 0.6 e n e 1, where Pn is a function of both n and . Results and Discussion At the outset, it is useful to consider the behavior in limiting situations of the analytical expressions derived in the present study. For instance, in the Newtonian limit (n ) 1), it can readily be shown that eq 72 for the Sherwood number is identical with the equivalent expression for Newtonian fluid flow reported by Govindarao47 and Ishimi et al.48 using the free surface model. Likewise, at n ) 1, eq 51 for the drag coefficient is in complete agreement with the expressions derived in the literature using both the free surface and zero-vorticity cell models; e.g., see work by Satheesh et al.,33 Dhotkar et al.,35 and Vijaysri et al.34 Moreover, for voidages close to unity, i.e., ∼ 1, to simulate the single cylinder limit, our analytical results for variation with the power law index of the drag parameter g(n) ) CDRe/[2n × 3(n+1)/2] are in close agreement with numerical results reported by Whitney and Rodin57 under creeping-flow conditions for a single cylinder (Figure 2). However, both the pres-
Figure 2. Drag parameter g(n) vs power law index n. Analytical results for γ ) 1/400 (i.e., voidage ≈ 1), using the free surface model (solid line) and zero-vorticity model (dashed line). Numerical results of Whitney and Rodin57 (empty circles) and Tanner58 (solid circles) for a single cylinder.
ent drag results and those of Whitney and Rodin57 differ significantly (up to 22% higher) from the numerical results obtained by Tanner58 for creeping flow, although this difference becomes less pronounced as the degree of shear-thinning decreases, i.e., as the value of n approaches unity (Figure 2). According to Whitney and Rodin,57 Tanner’s results are believed to be the most accurate, which suggests that the maximum 22% discrepancy between the present analytical results and Tanner’s48 numerical results is probably due to a decrease in the accuracy of the present analytical model at the very high voidage values that simulate the single cylinder condition. Such a trend is consistent with a comparison between the present analytical results and previous numerical results of Dhotkar et al.35 and Vijaysri et al.,34 which shows that for the lowest power law index (n ) 0.6) and extreme voidages ( g 0.9) the present analytical values of CDRe are 20-30% higher than their numerical counterparts, although excluding extreme voidage values, the accuracy of the present analytical results are in close agreement with the numerical data. Admittedly, D’Alessio and Pascal59 have also reported the values of the drag coefficient for the steady flow of power law fluids across a cylinder, but these values relate to Re g 5 and, therefore, are beyond the limits of applicability of the present analysis, which is restricted to Re f 0. Before leaving this section, it is appropriate to mention here that the single cylinder case was simulated by setting the cell boundary at 400 cylinder radii away. This value has been reported to be adequate for approximating the creeping flow of a Newtonian fluid past a single cylinder,60 and in view of the fact that, for n < 1, there is no Stokes paradox,58 this value of 400 cylinder radii is believed to be adequate for power law fluids. The variation of the drag coefficient with voidage () was also investigated for two extreme values of the power law index (n ) 0.9 and 0.6), using both the free surface and zero-vorticity cell models (Figure 3). The analytical results obtained for the drag using the free surface model (Figure 3a) were compared with the corresponding numerical results obtained under creeping-flow conditions (Re ) 0.01) by Dhotkar et al.35 It was found that at n ) 0.9 the present results deviated from those of the aforementioned authors by an average of 3.2% (range of 1.7-4.2%), while at n ) 0.6 this deviation increased to 15% (range of 0.74-31%). A similar tendency was also observed with the analytical results for the drag obtained herein using the zero-
Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3447
Figure 4. CDRe vs n for the free surface model (solid line) and zero-vorticity model (dashed line): bottom two intercepts (γ ) 1/400; i.e., ≈ 1), next two intercepts ( ) 0.8), next two intercepts ( ) 0.6), and top two intercepts ( ) 0.4).
Figure 3. (a) Plot of CDRe vs using the free surface model. Present analytical results for n ) 0.9 (dashed line) and n ) 0.6 (solid line) compared to numerical results of Dhotkar et al.35 at Re ) 0.01 for n ) 0.9 (empty circles) and n ) 0.6 (solid circles). (b) Plot of CDRe vs using the zero-vorticity model. Present analytical results for n ) 0.9 (dashed line) and n ) 0.6 (solid line). Numerical results of Vijaysri et al.34 at Re ) 0.01 for n ) 0.9 (empty circles) and n ) 0.6 (solid circles). Analytical results from the drag expression of Bruschke and Advani8 for n ) 0.9 (empty triangles) and n ) 0.6 (solid triangles).
vorticity model (Figure 3b). In this case, at n ) 0.9 the present analytical results deviated from the numerical results of Vijaysri et al.34 for Re ) 0.01 by an average of 3.8% (range of 0.057-7.9%), while at n ) 0.6 this deviation increased to 14% (range of 4.1-19%). Such a gradual increase in deviation away from the Newtonian limit of n ) 1 is consistent with the mildly shear-thinning approximation used in the present analysis, and under these conditions, the comparison between the present analytical results for the drag and the more accurate numerical results of Dhotkar et al.35 and Vijaysri et al.34 showed reasonable overall agreement down to n ) 0.6 (Figure 3). In contrast to this, both the present analytical results and the more accurate numerical results of Vijaysri et al.34 showed strong divergence from the analytical results obtained with the drag expression of Bruschke and Advani8 away from the Newtonian limit (Figure 3b). Such a strong divergence is probably due to the inadvertent omission of a factor of n from the latter authors’ non-Newtonian stream function differential equation and corresponding solution, which is expected to have a significant impact on their drag results away from the Newtonian limit of n ) 1. The present study of cylinder assemblages also showed that, as voidage increased from 0.4 to 0.8, the increase with the power law index of the value of CDRe became less pronounced and that, toward ≈ 1, CDRe became a decreasing function of the power law index (Figure 4). A similar anomaly had previously been reported for sphere assemblages by Kawase and Ulbrecht.26 For
Figure 5. CDRe vs for the free surface model (solid line) and zero-vorticity model (dashed line) with n ) 0.9 (top two intercepts) and n ) 0.6 (bottom two intercepts).
cylinder assemblages, this anomaly was found to occur at ≈ 0.95, corresponding to a constant value of CDRe ≈ 30, which was independent of the power law index, whereas for sphere assemblages Kawase and Ulbrecht26 reported a similar feature at the higher voidage of ≈ 0.97. However, there appears to be no obvious physical reason for such an anomaly, which has not been reported in previous numerical work carried out on cylinder and sphere arrays using both cell models. Because both of the aforementioned analytical and numerical works were carried out using cell models, it is unlikely that the anomaly is an artifact of these models, although it is probably due to a decrease in the accuracy of both the present analytical model and that of Kawase and Ulbrecht26 at very high voidage values ( > 0.90). The present study also showed that, as expected because of additional dissipation at the cell boundary, the values of CDRe obtained using the zerovorticity model were much larger (log scale) than those obtained using the free surface model, although for large-porosity values both were found to converge (Figures 4 and 5). It is also appropriate to compare the present analytical predictions of drag with the limited experimental results available in the literature.12,14,19,44,45,61 While the corresponding comparisons for Newtonian fluids are available elsewhere,9,32,33 the available scant results for power law liquids are compared here. Both Adams and Bell12 and Prakash et al.14 have reported pressure drop values in tube bundles at high Reynolds numbers, which are beyond the scope of this study. Table 1 shows a comparison between the present results and the experi-
3448 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 Table 1. Comparison between Experiments and Theory (Values of CDRe) n
free surface model
zero-vorticity model
experimental
0.54 0.56 0.72 0.81 0.84 0.38 0.48 0.52 0.62 0.70 0.72 0.33
0.78 0.78 0.78 0.78 0.78 0.87 0.87 0.87 0.87 0.87 0.87 0.434 0.455 0.68 0.434 0.455 0.434 0.455 0.68
48.42 49.28 58.19 64.80 67.27 36.54 36.37 36.62 37.82 39.22 39.62 89.67 85.09 52.68 104.15 97.76 159.19 145.65 65.21
60.89 62.24 76.31 86.93 90.95 44.64 44.86 45.41 47.60 50.05 50.74 99.58 95.00 62.32 117.46 110.88 186.23 171.49 81.09
61.39a 62.89a 83.75a 92.99a 103.67a 38.86a 44.08a 45.36a 46.62a 49.89a 54.29a 97.64b 92.8b 41.8b 187.7b 148.1b 347.00b 281.60b 74.37b
0.39 0.53
a
Reference 44. b Reference 19.
Figure 6. Sh/Pe1/3 vs for n ) 1 (solid line) and n ) 0.6 (dashed line), with the free surface model (bottom two intercepts) and zerovorticity model (top two intercepts). Sherwood number ) Sh; Peclet number ) Pe.
ments of Skartsis et al.19 and of Prasad and Chhabra.44 The results of Prasad and Chhabra44 deviate from the present predictions by up to 15% for the zero-vorticity cell model and by up to 35% from the free surface cell model. Except for the cases of n ) 0.53, ) 0.434 and 0.455 and n ) 0.39, ) 0.434, a similar sort of correspondence is seen for the results of Skartsis et al.,19 though some of these liquids might have been viscoelastic. However, there appears to be a weak trend for the correspondence to deteriorate for smaller values of n and/or of . This is perhaps not surprising owing to the assumptions inherent in the present analysis. Overall, the experimental values are seen to be somewhat closer to the predictions of the zero-vorticity cell model, and the correspondence seen in Table 1 is about as good as can be expected in this type of work. Unfortunately, the results of Sadiq et al.61 for power law liquids could not be included in Table 1 owing to the lack of details available, but their results for Newtonian liquids are within 30% of the present analytical results. The present study of mass transfer revealed that the parameter Sh/Pe1/3 was strongly dependent on voidage (Figure 6), and its value was found to decrease by about 80% from ) 0.4 to ≈ 1. In contrast, the dependence of Sh/Pe1/3 on the power law index was much weaker, although it became more pronounced as voidage increased (Figure 7). For example, at ) 0.4 the value of
Figure 7. Sh/Pe1/3 vs n for the free surface model (solid line) and zero-vorticity model (dashed line): bottom two intercepts (γ ) 1/400; i.e., ≈ 1), next two intercepts ( ) 0.8), next two intercepts ( ) 0.6), and top two intercepts ( ) 0.4).
Sh/Pe1/3 decreased from n ) 0.6 to n ) 1 by about 0.7%, whereas at ≈ 1 this decrease was about 18%. Furthermore, at ) 0.4 the change in the value of Sh/Pe1/3 with n was insignificant in comparison with that resulting from using a different cell model, whereas at ≈ 1 the reverse behavior occurred. The values of Sh/Pe1/3 obtained with the free surface model were always lower than those obtained with the zero-vorticity model (Figures 6 and 7). The present mass-transfer results for cylinder assemblages are in line with the previous results for sphere assemblages as reported by Kawase and Ulbrecht26 and by Satish and Zhu.51 Unfortunately, for power law fluids, there are no numerical results available to test the validity of the present masstransfer predictions for cylinder assemblages. However, when the usual analogy between mass and heat transfer, i.e., Sh ≡ Nu, is invoked, the present analytical values of Sh/Pe1/3 are in excellent agreement with the predictions based on the numerical solution9,40 of the complete energy equation without viscous dissipation for Pe ) 500 and Re ) 1 over limited ranges of the bed porosity and power law index. It is also noted that such small enhancements in mass transfer are consistent with the experimental results for beds of spherical and nonspherical particles with power law liquids at low Reynolds numbers.30,62,63 However, enhancements in heat transfer up to 30-35% are possible at high Reynolds numbers in both granular and fibrous beds.9,30,40 Before the discussion is closed, intuitively it appears that the traditional capillary bundle approach is perhaps more appropriate in dense systems (low porosity), whereas the cell and periodic array models might be more relevant at high values of porosity. However, it is not really possible to characterize this transition easily to pinpoint a value of beyond which the cell models will apply, albeit this transition appears to occur at about > ∼0.55-60 for granular beds.64 It is, however, not at all clear whether this will apply to tube bundles and fibrous beds. However, obviously, these two approaches complement each other rather than being mutually exclusive. Also, on the basis of the comparisons shown in Table 1 here and elsewhere,9,24,32 it appears that the values of macroscopic quantities such as drag and Nusslet (Sherwood) number are not too sensitive to the detailed geometry of the tube banks and are governed primarily by the mean value of the porosity, at least for > ∼0.55-0.60.
Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3449
Conclusions In this study, the creeping non-Newtonian flow across an assemblage of cylinders was investigated analytically for a wide range of power law indices (0.6 e n e 1) and voidages (0.4 e e ∼1) using the free surface and zerovorticity cell models. The analytical expressions derived for the non-Newtonian stream function and drag coefficient were found to be in reasonable agreement with the numerical data available in the literature. The drag coefficient was found to increase with the power law index for voidages in the range of 0.4 e < 0.94 and to decrease for > 0.94. An analytical expression for the Sherwood number as a function of the power law index, voidage, and Peclet number was also derived for high Peclet numbers. The dependence of the Sherwood number on the power law index was found to increase with voidage, although it was always found to be much weaker than the dependence of the Sherwood number on voidage. Both the drag coefficient and Sherwood number values were found to be higher for the zerovorticity model than for the free surface model. In the Newtonian limit of n ) 1, both of the aforementioned parameters were found to reduce to the known analytical results from the literature. The results presented herein are believed to have potential modeling applications and are likely to be of interest to researchers in the fields of fluid mechanics and rheology. List of Symbols a j ) radius of the cylinder (m) An ) constant, eqs 29, 31-33, 34a, 35a,b, 47, and 60 b h ) radius of the cell (m) Bn ) constant, eqs 29, 31-33, and 36a,b C h ) concentration (kg/m3) CD ) drag coefficient Cn ) constant, eqs 29, 32, 33, 34b, 37a,b, 47, and 60 D h ) molecular diffusivity (m2/s) Dn ) constant, eqs 29, 31-33, 34a,b, 38a,b, 47, and 60 e ) constant, eq 18a,b En ) constant, eqs 29, 30, 33, 34a,b, 35a,b, 36a,b, 37a,b, 38a,b, and 60 f ) constant, eq 19a,b b gj ) acceleration due to gravity (m/s2) g(n) ) function Gn ) constant, eqs 46-48 and 51 H1 ) constant, eqs 11a, 12a, 13a, 14a, 16a, 17a, 18a, and 19a Hn ) constant, eqs 35a, 36a, 37a, 38a, and 41a K1 ) constant, eqs 11b, 12b, 13b, 14b, 16b, 17b, 18b, and 19b Kn ) constant, eqs 35b, 36b, 37b, 38b, and 41b L h ) length of the cylinder (m) Ln ) constant, eqs 20, 21, 24a,b, 25, 28, 30, 45, and 47 m j ) power law consistency coefficient (Pa‚sn) Mn ) constant, eqs 49-51 n ) flow behavior index NSh ) Sherwood number p ) dimensionless pressure Pe ) Peclet number Pn ) constant, eqs 59, 60, 61a,b, 62, 63, 64b, 65, 68, 70, and 72 Qn ) constant, eqs 20, 22, and 25 r ) dimensionless radial coordinate Re ) Reynolds number Sh ) surface-averaged Sherwood number Sn ) constant, eqs 21, 23, 25, and 43 vj rj, vj θ ) rj and θ components of velocity (m/s) vz ) z component of velocity
V h 0 ) free stream velocity (m/s) W1 ) constant, eqs 65 and 68 W2 ) constant, eq 67 W3 ) constant, eq 67 yj ) concentration boundary layer thickness (m), eqs 53 and 55a z ) dimensionless coordinate zn ) constant, eqs 67-70 Zn ) constant, eqs 61 and 62 Greek Letters γ ) ratio (a j /b h) ∆ij ) component of the rate of the deformation tensor Π ) second invariant of the rate of the deformation tensor ) voidage ξ ) constant, eqs 35a, 37a, 38a, and 39a ξn ) constant, eqs 35a, 36a, 37a, and 40a θ ) cylindrical coordinate F ) constant, eqs 35b, 37b, 38b, and 39b Fn ) constant, eq 35b, 36b, 37b, and 40b Fj ) fluid density (kg/m3) τij ) components of the extra stress tensor Ψ ) stream function Subscripts a j, b h ) surface of the solid cylinder and the cell surface, respectively - ) overbar denoting dimensional quantity
Literature Cited (1) Mauret, E.; Renaud, M. Transport Phenomena in Multiparticle SystemssI. Limits of Applicability of Capillary Model in High Voidage BedssApplication to Fixed Beds of Fibers and Fluidised Beds of Spheres. Chem. Eng. Sci. 1997, 52, 1807. (2) Kiljanski, T.; Dziubinski, M. Resistance to Flow of Molten Polymers Through Filtration Screens. Chem. Eng. Sci. 1996, 51, 4533. (3) Abrate, S. Resin Flow in Fiber Preforms. Appl. Mech. Rev. 2002, 55, 579. (4) Chen, C.-T.; Malkus, D. S.; Vanderby, R., Jr. A Fiber Matrix Model for Interstitial Fluid Flow and Permeability in Ligaments and Tendons. Biorheology 1998, 35, 103. (5) Chen, K. F.; Chen, S.-M. Fluidization Properties of High-Consistency Fiber Suspensions. Exp. Therm. Fluid Sci. 1997, 14, 149. (6) Brown, R. C. Air Flow Through FilterssBeyond Single Fiber Theory. In Advances in Aerosol Filtration; Spurny, K. R., Ed.; Lewis Publishers: London, 1998; p 153. (7) Jackson, G. W.; James, D. F. The Permeability of Fibrous Porous Media. Can. J. Chem. Eng. 1986, 64, 364. (8) Bruschke, M. V.; Advani, S. G. Flow of Generalized Newtonian Fluids across a Periodic Array of Cylinders. J. Rheol. 1993, 37, 479. (9) Mandhani, V. K.; Chhabra, R. P.; Eswaran, V. Forced Convection Heat Transfer in Tube Banks in Cross Flow. Chem. Eng. Sci. 2002, 57, 379. (10) Zukauskas, A. Convective Heat Transfer in Cross Flow. Handbook of Single Phase Convective Heat Transfer; Wiley: New York, 1987; Chapter 6. (11) Bejan, A.; Kraus, A. D. Heat Transfer Handbook; Wiley: New York, 2003. (12) Adams, D.; Bell, K. J. Fluid Friction and Heat Transfer for Flow of Sodium Carboxymethyl Cellulose Solutions across Banks of Tubes. Chem. Eng. Prog. Symp. Ser. 1968, 64, 133. (13) Kyan, C. P.; Wasan, D. T.; Kintner, R. C. Flow of SinglePhase Fluids through Fibrous Beds. Ind. Eng. Chem. Fundam. 1970, 9, 956. (14) Prakash, O.; Gupta, S. N.; Mishra, P. Newtonian and Inelastic Non-Newtonian Flow Across Tube Banks. Ind. Eng. Chem. Res. 1987, 26, 1365. (15) Sangani, A.; Acrivos, A. Slow Flow Past Periodic Arrays of Cylinders with Application to Heat Transfer. Int. J. Multiphase Flow 1982, 8, 193. (16) Wung, T.-S.; Chen, C. J. Finite Analytic Solution of Convective Heat Transfer for Tube Arrays in Cross Flow. Part I. Flow Field Analysis. J. Heat Transfer 1989, 111, 633.
3450 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 (17) Chen, C. J.; Wung, T.-S. Finite Analytic Solution of Convective Heat Transfer for Tube Arrays in Cross Flow. Part II. Heat Transfer Analysis. J. Heat Transfer 1989, 111, 641. (18) Martin, A. R.; Saltiel, C.; Shyy, W. Frictional Losses and Convective Heat Transfer in Sparse, Periodic Cylinder Arrays in Cross Flow. Int. J. Heat Mass Transfer 1998, 41, 2383. (19) Skartsis, L.; Khomami, B.; Kardos, J. L. Polymeric Flow Through Fibrous Media. J. Rheol. 1992, 36, 589. (20) Nieckele, A. O.; Naccache, M. F.; Souza Mendes, P. R. Cross Flow of Viscoplastic Materials Through Tube Bundles. J. NonNewtonian Fluid Mech. 1998, 75, 43. (21) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: The Hague, The Netherlands, 1983. (22) Happel, J. Viscous Flow Relative to Arrays of Cylinders. AIChE J. 1959, 5, 174. (23) Kuwabara, S. The forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers. J. Phys. Soc. Jpn. 1959, 14, 527. (24) Chhabra, R. P. Bubbles, Drops and Particles in NonNewtonian Fluids; CRC Press: Boca Raton, FL, 1993. (25) Chhabra, R. P. Fluid Flow, Heat and Mass Transfer in Non-Newtonian Fluids: Multiphase Systems. Adv. Heat Transfer 1993, 23, 187. (26) Kawase, Y.; Ulbrecht, J. Drag and Mass Transfer in NonNewtonian Flows Through Multi-Particle Systems at Low Reynolds Numbers. Chem. Eng. Sci. 1981, 36, 1193. (27) Kawase, Y.; Ulbrecht, J. Motion of and Mass Transfer from an Assemblage of Solid Spheres Moving in a Non-Newtonian Fluid at High Reynolds Numbers. Chem. Eng. Commun. 1981, 8, 233. (28) Chhabra, R. P.; Comiti, J.; Machac, I. Flow of NonNewtonian Fluids in Fixed and Fluidized Beds. Chem. Eng. Sci. 2001, 56, 1. (29) Mao, Z.-S.; Wang, Y. Numerical Simulation of Mass Transfer of Spherical Particle Assemblage with the Cell Model. Powder Technol. 2003, 134, 145. (30) Shukla, R.; Dhole, S. D.; Chhabra, R. P.; Eswaran, V. Convective Heat Transfer for Power Law Fluids in Packed and Fluidized Beds of Spherical Particles. Chem. Eng. Sci. 2004, 59, 645. (31) Tripathi, A.; Chhabra, R. P. Slow Power Law Fluid Flow Relative to an Array of Infinite Cylinders. Ind. Eng. Chem. Res. 1992, 31, 2754. (32) Tripathi, A.; Chhabra, R. P. Transverse Laminar Flow of Non-Newtonian Fluids over a Bank of Long Cylinders. Chem. Eng. Commun. 1996, 147, 197. (33) Satheesh, V. K.; Chhabra, R. P.; Eswaran, V. Steady Incompressible Flow Over a Bundle of Cylinders at Moderate Reynolds Numbers. Can. J. Chem. Eng. 1999, 77, 978. (34) Vijaysri, M.; Chhabra, R. P.; Eswaran, V. Power Law Fluid Flow across an Array of Infinite Circular Cylinders. A Numerical Study. J. Non-Newtonian Fluid Mech. 1999, 87, 263. (35) Dhotkar, B. N.; Chhabra, R. P.; Eswaran, V. Flow of NonNewtonian Polymeric Solutions Through Fibrous Media. J. Appl. Polym. Sci. 2000, 76, 1171. (36) Chhabra, R. P.; Dhotkar, B. N.; Eswaran, V.; Satheesh, V. K.; Vijaysri, M. Steady Flow of Newtonian and Dilatant Fluids over an Array of Long Circular Cylinders. J. Chem. Eng. Jpn. 2000, 33, 832. (37) Shibu, S.; Chhabra, R. P.; Eswaran, V. Power Law Fluid Flow over a Bundle of Cylinders at Intermediate Reynolds Numbers. Chem. Eng. Sci. 2001, 56, 5545. (38) Le Clair, B. P.; Hamielec, A. E. Viscous Flow Through Particle Assemblages at Intermediate Reynolds Numbers: Heat or Mass Transport. Inst. Chem. Eng. Symp. Ser. 1968, 30, 197. (39) Masliyah, J. H. Viscous Flow across Banks of Circular and Elliptic Cylinders: Momentum and Heat Transfer. Can. J. Chem. Eng. 1973, 51, 550. (40) Mangadoddy, N.; Prakash, R.; Chhabra, R. P.; Eswaran, V. Forced Convection in Cross Flow of Power Law Fluids Over a Tube Bank. Chem. Eng. Sci. 2004, 59, 2213. (41) Spielman, L.; Goren, S. L. Model for Predicting Pressure Drop and Filtration Efficiency in Fibrous Media. Environ. Sci. Technol. 1968, 2, 279.
(42) Yu, C. P.; Soong, T. T. A Random Cell Model for Pressure Drop Prediction in Fibrous Filters. J. Appl. Mech.sTrans. ASME 1975, 42, 301. (43) Neale, G.; Masliyah, J. H. Flow Perpendicular to Mats of Randomly Arranged Cylindrical Fibers (Importance of Cell Models). AIChE J. 1975, 21, 805. (44) Prasad, D. V. N.; Chhabra, R. P. An Experimental Investigation of the Cross-Flow of Power Law Liquids past a Bundle of Cylinders and in a Bed of Stacked Screens. Can. J. Chem. Eng. 2001, 79, 28. (45) Malleswara Rao, T. V.; Chhabra, R. P. A Note on Pressure Drop for the Cross-Flow of Power Law Liquids and Air/Power Law Liquid Mixtures past a Bundle of Circular Rods. Chem. Eng. Sci. 2003, 58, 1365. (46) Hwang, T. H.; Yao, S. C. Cross-flow Heat Transfer in Tube Bundles at Low Reynolds Numbers. J. Heat Transfer 1986, 108, 697. (47) Govindarao, V. M. H. Mass Transfer with Surface Reaction in Arrays of Cylinders at Low Reynolds at High Peclet Numbers: Flow Perpendicular to the Axis of the Cylinders. J. Indian Inst. Sci. 1978, 60A, 1. (48) Ishimi, K.; Koroyasu, S.; Hikita, H. Mass Transfer in Creeping Flow Past Periodic Arrays of Cylinders. J. Chem. Eng. Jpn. 1987, 20, 492. (49) Levich, V. G. Physiochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (50) Pfeffer, R. Heat and Mass Transport in Multiparticle Systems. Ind. Eng. Chem. Fundam. 1964, 3, 380. (51) Satish, M. G.; Zhu, J. Flow Resistance and Mass Transfer in Slow Non-Newtonian Flow through Multiparticle Systems. J. Appl. Mech. 1992, 59, 431. (52) Chhabra, R. P.; Richardson, J. F. Non-Newtonian Flow in the Process Industries; Butterworth-Heinemann: Boston, 1999. (53) Hirose, T.; Moo-Young, M. Bubble Drag and Mass Transfer in Non-Newtonian FluidssCreeping Flow with Power Law Fluids. Can. J. Chem. Eng. 1969, 47, 265. (54) Bhavaraju, S. M.; Mashelkar, R. A.; Blanch, H. W. Bubble Motion and Mass Transfer in Non-Newtonian Fluids. Part 2. Swarm of Bubbles in a Power Law Fluid. AIChE J. 1978, 24, 1070. (55) Rodrigue, D.; DeKee, D.; Chan Man Fang, C. F. A Note on the Drag Coefficient of a Single Bubble in a Power Law Fluid. Can. J. Chem. Eng. 1999, 77, 766. (56) Kreyszig, E. Advanced Engineering Mathematics; Wiley: New York, 1993. (57) Whitney, M. J.; Rodin, G. R. Force-Velocity Relationships for Rigid Bodies Translating through Unbounded Shear-Thinning Power Law Fluids. Int. J. Nonlinear Mech. 2001, 36, 947. (58) Tanner, R. I. Stokes Paradox for Power Law Flow around a Cylinder. J. Non-Newtonian Fluid Mech. 1993, 50, 217. (59) D’Alessio, S. D. J.; Pascal, J. P. Steady Flow of a Power Law Fluid past a Cylinder. Acta Mech. 1996, 117, 87. (60) Zdravkovich, M. M. Flow Around Circular Cylinders Vol. 2: Applications; Oxford University Press: Oxford, U.K., 2003. (61) Sadiq, T. A. K.; Advani, S. G.; Parnas, R. S. Experimental Investigation of Transverse Flow Through Aligned Cylinders. Int. J. Multiphase Flow 1995, 21, 755. (62) Comiti, J.; Montillet, A.; Seguin, D.; Hilal, M. Modelling of Power Law Liquid-Solid Mass Transfer in Packed Beds in Darcy Regime. Chem. Eng. J. 2002, 89, 29. (63) Peev, G.; Nikolova, A.; Todorova, D. Mass Transfer from Solid Particles to Power Law Non-Newtonian Fluid in Granular Bed at Low Reynolds Numbers. Chem. Eng. J. 2002, 88, 119. (64) Ciceron, D.; Comiti, J.; Chhabra, R. P.; Renaud, M. NonNewtonian Fluidization of Spherical Particles. Chem. Eng. Sci. 2002, 57, 3225.
Received for review November 4, 2003 Revised manuscript received April 21, 2004 Accepted April 21, 2004 IE030812E
