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GENERAL RESEARCH A Semianalytical Expression for the Drag Force of an Interactive Particle Due to Wake Effect Jian Zhang* Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Liang-Shih Fan Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 43210
The drag forces of particles aligned in a line parallel to the direction of relative motion between the fluid stream and the particles are investigated. The particle-particle interactions and the wake effect should be taken into account for quantitatively describing the drag force of the interactive particle. Under the circumstances of the Reynolds number (Re) being around 100, the drag force of the interactive trailing particle is analyzed based on the velocity distribution in the far wake region downstream of the leading particle. As a result, a semianalytical expression for the drag force of the trailing particle is formulated. It is utilized to calculate the drag force ratio of the trailing particle for Re of 54-154. Agreement between the calculation and the measured data is achieved. 1. Introduction The behavior of a particle in a multiparticle flow field is different from that of an isolated particle in an unbounded flow field due to prevalent particle-particle and particle-fluid interactions. Particles aligned in a linear array or moving in a one-dimensional trajectory along the direction of relative motion with the fluid stream represent one of the basic forms of particle interaction. In such a form of particle flow, the wake of the upstream leading particle has significant effects on the interactive behavior of the downstream trailing particle. The particle wake could affect significantly the changes in the drag force of the trailing particle, which in turn affects the collision or separation process of the particles. The particle heat and mass transfer rates may be greatly influenced by the particle wake; for example, the presence of particle wake may affect the particle ignition time and burning rate and cause incomplete reaction and pollutant emission in a reaction or combustion system. The investigation of particle wake effects on the trailing particle behavior is thus of importance from both fundamental and applied aspects. Previous studies on the drag forces of interactive particles fixed or moving in a line in a flow field have employed experimental, analytical, and numerical approaches. The experimental measurements have covered a wide range of the particle Reynolds number (Re), from about 30 to 104, in which particle wake effects are significant. Rowe and Henwood1 performed measurements of the drag forces on two spheres in line for Re of 32-96. Lee2 measured the drag force on an interactive sphere at Re around 104. Tsuji et al.3 conducted * Corresponding author. Telephone: 010-62772932. Fax: 010-62781824. E-mail:
[email protected].
measurements of the drag forces with Re varying from 125 to less than 103. Zhu et al.4 and Liang et al.5 measured the drag forces of two or three spheres aligned in line in the Re range 40-150. They found that the drag force of the downstream trailing particle is much less than that of the upstream leading particle because of dynamic interactions between the trailing particle and the leading particle wake. The analytical studies on the drag force of interactive particles have been conducted only for sufficiently low Reynolds numbers (Re < 1). Stimson and Jeffery6 used bispherical coordinates to solve for the stream function of axisymmetric creeping flow past two spheres and derived a correction factor, which accounts for the particle interaction effects, to the Stokes’ drag formula. Other approaches, such as the method of reflections,7 dynamic simulation,8 and strong-interaction theory,9 were also developed and employed to solve for the Stokes flows past two-sphere or multisphere systems and to give the drag forces of interactive spherical particles. It is difficult, however, to obtain the analytical expressions for the drag forces of interactive particles in the intermediate flow regime. The numerical simulations of the flows past spherical particles in tandem and the resultant particle drag forces were conducted mainly in the intermediate flow regime. Tal et al.10 numerically solved the NavierStokes equations for a pair of spheres in tandem at Re ) 40 by using bispherical coordinates and obtained the drag coefficients of the spheres. Ramachandran et al.11 numerically simulated the laminar axisymmetric flow past a linear array of three spheres for the range of 1 e Re e 200. Wang and Liu12 simulated the flow fields around solid spheres in tandem for Re around 103 by using an algebraic turbulent viscosity model. Liang et al.5 performed a numerical study of the flow fields for
10.1021/ie011045r CCC: $22.00 © 2002 American Chemical Society Published on Web 08/30/2002
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5095
the three-coaligned particle configuration at Re of 53 and 106 along with their experimental measurements. The numerical results reveal flow and wake structures around spheres and indicate that the drag coefficients of interacting spheres deviate significantly from that of the isolated sphere at given Reynolds numbers. Since the particle Reynolds numbers are around several hundred for most process applications in multiphase flows, it is of interest to obtain a simple theoretical equation to evaluate the drag force of an interactive particle with wake effects in the intermediate flow regime. For this purpose, a semianalytical expression is developed that can account for the drag force of an interactive particle aligned in line for the Re on the order of 100 [Re ∼ O(100)]. Its formulation is based on the form of the drag force expression of an isolated particle. The particle wake effect is taken into consideration in terms of the wake velocity profiles in the formulation. The drag forces calculated from the present formulation are compared with the experimental data. 2. Velocity Distribution of the Far Wake Consider a far wake downstream of an axisymmetric body placed in a uniform stream parallel to its axis. The Re range considered is from 20 (after flow separation) to 200 (before wake shedding). In this range, the wake flow is laminar, axisymmetric, and in steady state. Since the difference between the incoming uniform stream velocity u0 and the wake flow velocity u(x,r) is small compared with u0 in the far wake region, the governing equation for the far wake flow can be simplified into the following asymptotic linear form13
∂u ν ∂ ∂u u0 ) r ∂x r ∂r ∂r
( )
(1)
where x is the axial distance from the base of the body and r is the radial distance from the symmetric axis. The above equation satisfies the boundary conditions as: ∂u/∂r ) 0 at r ) 0 and u ) u0 at r ) ∞. Its solution can be obtained by considering that the total momentum loss of the fluid is equal to the drag force on the body. Thus, the velocity distribution of the far wake is obtained as
( )
Cd0d u0r2 u(x,r) )1{Re}0 exp u0 32 x 4νx
(2)
where {Re}0 ) u0d/ν; d is the diameter of the frontal area of the body; Cd0 is the drag coefficient of the body. If the body moves along the axial direction with a velocity up, the relative velocity between the wake flow and the body can be expressed as
(
Cd0pd |u0 - up|r u(x,r) - up {Re}0p exp )1u0 - up 32 x 4νx
)
2
(3)
where {Re}0p ) |u0 - up|d/ν. The absolute magnitude of the relative velocity between the local wake flow and the body is seen to be always less than that of the relative velocity between the incoming stream and the body. 3. Drag Force of an Interactive Particle Consider a two-particle system fixed or moving in line along the direction of relative motion with the fluid
stream. The fluid stream has a uniform incoming velocity u0. The particles are of a spherical shape with an identical diameter d. The leading and trailing particles move at velocities up1 and up2, respectively. In the analysis, it is assumed that (1) the far wake solution of the upstream leading particle constitutes the inflow condition for the downstream trailing particle, (2) the presence of the downstream particles has negligible effects on the upstreamflow conditions, and (3) the drag force of an interactive particle takes a similar mathematical form as that of an isolated particle but is evaluated by the local fluid velocity. Using the wake velocity profiles expressed in eq 3, the averaged wake velocity experienced by the downstream trailing particle can be obtained as
u j - up1 4 ) u0 - up1 πd2
u-u
∫0d/2u0 - up1p12πr dr
(4)
or
[
(
)]
Cd01 {Re}01d u j - up1 )11 - exp u0 - up1 2 16x
(5)
where x is the separation distance between the rear surface of the leading particle and the frontal surface of the trailing particle. 3.1. Upstream Leading Particle. The leading particle is encountered with a uniform incoming stream of velocity u0. Hence, the drag force of the leading particle can be expressed as
π Fd1 ) Cd01 Fd2(u0 - up1)2 8
(6)
which is identical to the drag force of an isolated particle of the same diameter moving at the same relative velocity with the fluid stream. It means that the existence of the trailing particle has no effects on the drag force of the leading particle. This result is confirmed by the experiments of Tsuji et al.3 and Zhu et al.4 Their measurements show that the drag force of the leading particle is almost unchanged in comparison with that of the isolated particle. 3.2. Downstream Trailing Particle. The trailing particle is located in the far wake region of the leading particle. The local fluid velocity encountered by the trailing particle can be equated to the averaged wake velocity given in eq 5. On the basis of the aforementioned assumptions, the drag force of the trailing particle can be written as
π j - up2)2 Fd2 ) Cd2 Fd2(u 8
(7)
where Cd2 is the drag coefficient of the trailing particle. If the trailing particle is isolated in the same fluid stream and moves with the same velocity, its drag force will be
π Fd02 ) Cd02 Fd2(u0 - up2)2 8
(8)
where Cd02 is the drag coefficient of the isolated trailing particle. From eqs 8 and 9, the ratio of the drag force of the trailing particle to that of the isolated particle is given as
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Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002
(
Cd2 u Fd2 j - up2 ) Fd02 Cd02 u0 - up2
)
2
(9)
The drag coefficient of the trailing particle is not known; however, it should depend on the particle Reynolds number. Thus it is reasonable to assume that
( )
Cd2 {Re}02 ) Cd02 {Re}2
|
|
(10)
(11)
|
u0 - up1 u j - up1 + up1 - up2 Fd2 ) Fd02 u0 - up2 u0 - up1
|
2-R
(12)
Substituting eq 5 into the above equation yields
{
(
[
)]
Cd01 {Re}01d u0 - up1 Fd2 ) 11 - exp + Fd02 u0 - up2 2 16x up1 - up2 2-R (13) u0 - up1
}|
The above expression gives the drag force ratio of the trailing particle. 3.3. Drag Force Ratio for a Specific Case. For a specific case where the leading and trailing particles move at the same velocity, i.e., up1 ) up2 ) up, eq 13 is further expressed as
{
(
[
)]
(17)
This expression is seen to be a function of only the particle Reynolds number and the dimensionless separation distance between the two particles (x/d). 4. Validation of the Semianalytical Expression The drag force ratio of the trailing particle varying with the separation distance is calculated by the semianalytical expression, i.e., eq 17 for different particle Reynolds numbers in the intermediate flow regime. The following correlation14 for Cd0 is used in the calculation:
Cd0 )
2-R
or
|
(
R
where R is the exponential factor whose magnitude needs to be determined, {Re}02 ) |u0 - up2|d/ν, and {Re}2 ) |u j - up2|d/ν. They are the particle Reynolds numbers of the isolated trailing particle and the interactive trailing particle, respectively. Thus, it gives
u j - up2 Fd2 ) Fd02 u0 - up2
[
Fd2 Cd0 {Re}0d )11 - exp Fd02 2 16x
)]}
Cd0 {Re}0d Fd2 ) 11 - exp Fd02 2 16x
24 (1 + 0.15{Re}00.687) {Re}0
(18)
Figures 1-3 show the results at Re values of 54, 106, and 154, respectively. The experimental data of Zhu et al.4 and Liang et al.5 are also plotted in these figures for comparison. The former were measured from a twoparticle system fixed in a flow at Re of 54, 106, and 154, while the later were from a three-particle system fixed in a flow at Re of 54 and 106. However, both of their data used in the figures are taken from the measurements of the drag force of the trailing particle immediately downstream of the leading particle. The figures show that the drag force ratio of the trailing particle calculated from the present correlation agree well with both sets of the measured data for all the Re studied. Both calculations and measurements illustrate that the drag force of the trailing particle
2-R
(14)
where Cd0 ) Cd01 ) Cd02 and {Re}0 ) {Re}01 ) {Re}02. The above expression can be written in another form as
{
( )]}
Fd2 d ) 1 - a 1 - exp -b Fd02 x
[
2-R
(15)
where a ) a({Re}0) ) Cd0/2, and b ) b({Re}0) ) {Re}0/ 16. It is noted that Zhu et al.4 obtained an experimental correlation for the drag force ratio of the trailing particle for a two-particle system fixed in a flow stream. Their correlation has the following form:
Fd2 x ) 1 - (1 - A) exp -B Fd02 d
( )
Figure 1. Drag force ratio of the trailing particle at Re of 54.
(16)
where A ) A({Re}0) and B ) B({Re}0). To be consistent with the form of the above experimental correlation, it is seen that the exponential factor, R, in eq 15 should take the value of 1. Therefore, for the case of identical particle diameter and moving velocity in a flow stream, the drag force ratio of the trailing particle can be calculated by the following semianalytical expression as
Figure 2. Drag force ratio of the trailing particle at Re of 106.
Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5097 {Re}0 ) Reynolds number of an isolated particle {Re}2 ) Reynolds number based on the relative velocity between the local wake flow and the trailing particle u ) wake flow velocity (ms-1) u0 ) incoming stream velocity (ms-1) up ) particle velocity (ms-1) u j ) averaged wake velocity experienced by the trailing particle (ms-1) x ) axial distance from the base of a body; separation distance between the rear surface of the leading particle and the frontal surface of the trailing particle (m) Greek Letters R ) exponential factor ν ) kinematic viscosity of fluid (m2s-1) F ) fluid density (kgm-3) Figure 3. Drag force ratio of the trailing particle at Re of 145.
decreases with decreasing separation distance and is smaller than that of the isolated particle. Thus, the reduction in the drag force observed by the previous experiments is correctly predicted by the present semianalytical expression. On the basis of the present analysis, it is seen that the drag force reduction is mainly caused by the reduction in the relative velocity between the trailing particle and the local flow stream when the trailing particle is located in the wake region of the leading particle. This relative velocity becomes smaller as the trailing particle approaches its leading particle. It should be noted that the comparisons between the calculations and the measurements are presented for the separation distance x/d > 2 in Figures 1-3, since the present derived expression for the drag force is suitable for the far wake region. 5. Concluding Remarks A semianalytical expression is developed to account for the drag force of an interactive particle due to wake effect in the intermediate flow regime [Re ∼ O(100)]. The drag force ratio given by this equation varies as a function of the particle Reynolds number and the dimensionless separation distance between two particles under the conditions of identical particle diameter and moving velocity. The calculated drag force of the trailing particle decreases with decreasing separation distance and is smaller than that of the isolated particle. The calculated drag forces for both leading and trailing particles are found to agree well with the experimental data. The present analysis indicates that the reduction in the drag force of the trailing particle is primarily due to the reduction in the relative velocity between the particle and the local wake flow compared to that between the particle and the incoming fluid stream. Nomenclature Cd ) drag coefficient of an interactive particle Cd0 ) drag coefficient of an isolated particle d ) particle diameter (m) Fd ) drag force of an interactive particle (N) Fd0 ) drag force of an isolated particle (N) r ) radial distance from the symmetric axis (m) Re ) Reynolds number based on the relative velocity between the incoming stream and the particle
Subscripts 1 ) leading particle 2 ) trailing particle p )particle
Literature Cited (1) Rowe, P. N.; Henwood, G. A. Drag Forces in a Hydraulic Model of a Fluidised Bed-Part I. Trans. Inst. Chem. Eng. 1961, 39, 43-54. (2) Lee, K. C. Aerodynamic Interaction Between Two Spheres at Reynolds Numbers Around 104. Aerodyn. Q. 1979, 30, 371385. (3) Tsuji, Y.; Morikawa, Y.; Terashima, K. Fluid-Dynamic Interaction Between Two Spheres. Int. J. Multiphase Flow 1982, 8, 71-82. (4) Zhu, C.; Liang, S.-C.; Fan L.-S. Particle Wake Effects on the Drag Force of an Interactive Particle. Int. J. Multiphase Flow 1994, 20, 117-129. (5) Liang, S.-C.; Hong, T.; Fan L.-S. Effects of Particle Arrangements on the Drag Force of a Particle in the Intermediate Flow Regime. Int. J. Multiphase Flow 1996, 22, 285-306. (6) Stimson, M.; Jeffery, G. B. The Motion of Two Spheres in a Viscous Fluid. Proc. R. Soc. London 1926, A111, 110-116. (7) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff Publishers: Dordrecht, The Netherlands, 1983. (8) Durlofsky, L.; Brady, J. F.; Bossis, G. Dynamic Simulation of Hydrodynamically Interacting Particles. J. Fluid Mech. 1987, 180, 21-49. (9) Hassonjee, Q.; Ganatos, P.; Pfeffer, R. A Strong-Interaction Theory for the Motion of Arbitrary Three-Dimensional Clusters of Spherical Particles at Low Reynolds Number. J. Fluid Mech. 1988, 197, 1-37. (10) Tal, R.; Lee, D. N.; Sirignano, W. A. Heat and Momentum Transfer Around a Pair of Spheres in Viscous Flow. Int. J. Heat Mass Transfer 1984, 27, 1953-1962. (11) Ramachandran, R. S.; Wang, T.-Y.; Kleinstreuer, C.; Chiang, H. Laminar Flow Past Three Closely-Spaced Monodisperse Spheres or Nonevaporating Drops. AIAA J. 1991, 29, 4351. (12) Wang, B.-X.; Liu, T. Research on Hydrodynamics and Heat Transfer for Fluid Flow Around Heating Spheres in Tandem. Int. J. Heat Mass Transfer 1992, 35, 307-317. (13) Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, 1979. (14) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978.
Received for review December 30, 2001 Revised manuscript received June 25, 2002 Accepted June 27, 2002 IE011045R
