Effect of Particle Size on Particle Eddy Diffusivity Gerard P. Lilly Westinghouse Electric Corp.,' Pittsburgh, Pa. 15230
To investigate the effect of particle size on particle eddy diffusivity, concentration profiles were measured for three particle sizes in a plane turbulent air jet: (1) a carbon monoxide tracer, (2) a particle much larger than a molecule but small enough to completely follow the turbulent fluctuations, and (3) a large particle which could be expected to deviate significantly from completely following the turbulent fluctuations. Eddy diffusivities for the molecule and small particle were the same, indicating that molecular diffusion was not important in the turbulent diffusion process. Eddy diffusivity for the large particle was twice that of the small particle. Turbulent Schmidt numbers were 0.34 for the molecule and small particle and 0.1 7 for the large particle.
L i t t l e experimental work has been done with the object of establishing the relationship between particle size and particle eddy diffusivity in any given flow field. I n the absence of other information, particle and momentum eddy diffusivities are assumed equal, Le., SCT = €/ep = 1. 9 survey of turbulent pipe and channel flow data by Householder and Goldschmidt (1969) resulted in a mean value of SCT = 1.2 for diffusion of molecular species. The value for particles aproached 1.2 as the particles became very small. For other turbulent flow fields and for large particles, the turbulent Schmidt number can differ substantially from unity. Data are presented here demonstrating the significant change in particle eddy diffusivity which can occur with particle size. Details of all aspects of the present study can be found in the dissertation of Lilly (1971). Turbulent Diffusion of Particles
Semiempirical treatment of the turbulent diffusion of momentum and matter is most easily handled by the concept of a n eddy diffusivity coefficient. Turbulent shear stress is defined by 7 = -
dU
PE
-
bY
and the turbulent mass flux can be represented as
I n a given flow field (given E), two factors related to particle size may influence ep. For particles of molecular size, molecular diffusion may affect the transport by fluid eddies. Also, as a particle becomes very large, particle inertia can be important. Models and theoretical prediction of the effect of molecular processes on turbulent transport have been given by Jenkins (1951) and -4zer and Chao (1960). Both studies predict that PrT decreases with increasing Pr. The effects are small, however, when Pr is near or greater than unity and turbulent eddy viscosity, e, is much larger than fluid kinematic viscosity, Y. Marchello and Toor (1963) predict the opposite Work done at School of Engineering and Applied Science, University of California, Los Angeles, Calif. 268
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
trend in a turbulent field near a wall where e and Y are of the same order. Several authors have measured the mass eddy diffusivity for various tracer gases in air. Towle and Sherwood (1939) and Flint, et al. (1960), measured SCTfor hydrogen and carbon dioxide in a turbulent air stream (molecular Schmidt numbers of 0.22 and 0.94, respectively). They found no difference in SCTfor the two tracers. Goldman and illarchello (1969) measured eddy diffusivities for three gases in a turbulent air stream: helium (Sc = 0.22), carbon dioxide (Sc = 0.94), and n-octane (Sc = 2.5). These authors report a significant variation of the eddy diffusivity with molecular Schmidt number. Thus, experimentally, the effect of molecular diffusion on the turbulent transport of matter is not well established. Particle inertia could be expected to influence particle eddy diffusivity if it becomes large enough such t h a t a particle will no longer completely follow the turbulent velocity fluctuations. The degree to which a particle will follow the turbulent fluctuations depends on the particle size and density and the characteristics of the turbulent floiv field. Analytical studies of the motion of discrete particles in homogeneous, isotropic turbulence have been made by a number of authorsSo0 (1956), Liu (1956), Hinze (1959a), Friedlander (1957), and Tchen (1947). They are similar, beginning with a n equation of motion for the particle and developing expressions for the turbulent motion of the particle in terms of the fluid turbulence parameters. The basic parameter which determines the behavior of a particle in turbulent flow is the ratio T / T ~Particle . relaxation time, T , is defined by the equation v/vo
=
exp(-ft/T)
which represents the decay with time ( t ) of particle velocity (v) when i t is injected into a quiescent fluid a t a n initial velocity (vo). For a spherical particle in the Stokes flow regime, and p 1) it can be inferred t h a t molecular diffusion is unimportant for Sc > 0.77. The studies noted earlier of Towle and Sherwood (1939) and Flint, et al. Ind. Eng. Chem. Fundom., Vol. 12, No. 3, 1973
273
(1960), support this conclusion and indicate it applies for Sc > 0.2. Contrary to these results the data of Goldman and Marchello (1969) show a dependence of SCT on Sc in the range 0.2 < Sc < 2.5. Also from data presented here and the data correlation of Householder and Goldschmidt (1969) we must conclude that particle eddy diff usivity increases with particle inertia parameter r/T1. This increase starts a t a value of T / T near ~ 0.02. Two of the analytical studies mentioned above present results for the variation of SCTwith T / T ~Hinze . (1959a) shows an increase of SCTwith T / T ~The . simplified treatment by So0 (1956) does show a very slight decrease in SCTwith T / T Iin agreement with the experimental data. Soo's results indicate that the turbulent intensity, ut, of the particle motion decreases with increasing values of T / T I .At the same time the scale of the turbulent particle motion increases. Both these effects are expected since a large heavy particle will have less tendency to follow the turbulent velocity fluctuations, but once given a velocity it will persist with movement in this direction. Quantitatively So0 shows that the increase in scale is greater than the decrease in turbulent intensity. Eddy diffusivity, being proportional to the product of these quantities, increases with particle size. This may explain in part the increase of particle eddy diffusivity with size. Further experimental definition of the variation of particle eddy diffusivity with particle inertia parameter and development of simple models to explain this variation would certainly enhance the understanding of turbulent diffusion processes. Appendix A
Estimation of Lagrangian Turbulence Scales. I n analogy to molecular diffusion, eddy diffusivity can be defined as
Table A-I (€/UX)/(V'Z/W)
E= 0 0.12 0.07 0.15 0.10
lnvertiga tor
Heskestad Miller and Comings v.d. Hegge Zijnen Goldschmidt and Eskinazi
= 0.15
0.13 0.10 0.16 0.33
minimum value of 2'1. This occurs a t the jet centerline in all cases. An average value
($$)
=
0.12
has been used to calculate the value of the inertia parameter r/T1. I t should be noted that the Lagrangian macro-time scale, T I ,according to eq A-4 tends to zero for x -+ 0 if the parameter in parentheses is kept constant. The data of Heskestad (1965) indicate that turbulence parameters change quite rapidly below x/2a = 30 and the above estimate of T I should probably not be used any nearer to the slot. Appendix B
Simplified Method for Reduction of Velocity and Concentration Field Data in the Plane Turbulent Jet. T h e Gaussian profile suggested by Reichardt (1941) provides a good fit to experimental velocity and concentration field data. T h e profile for velocity, mass fraction, and mass flux, each with a single empirical constant, are: for velocity
E = exp( U
&)
for mass fraction
>
where is the variance of fluid particle displacement and 1 is time. The Lagrangian macro-time scale T1 is a measure of the average time over which a fluid particle moves in the same direction. The relation between T Iand ;is given by y2 = 2v"Tlt
for mass flux ratio (13-3)
(A-2)
where u' is the root-mean-square turbulent velocity fluctuation. This equation is developed in Hinze (1959). Combining (A-1) and (-4-2) we get
Assuming that
the relationship between the various empirical constants is The quantity 0' can be measured and E can be calculated from the jet velocity profile. I n terms of more readily available parameters (A-4)
e/Cx is a constant related to the empirical constant in the velocity profile (eq 4) and (v'/C)2 is a commonly measured turbulence parameter. The quantity ( ( e / I ; x ) / ( u t 2 /V))was calculated from the turbulence data of Heskestad (1965), Miller and Comings (1957), v.d. Hegge Zijnen (1958), and Goldschmidt and Eskinazi (1966). The results differ widely, principally due to differences in the measured value of v ' / U . The results are presented in Table -4-1. For the purpose of this study me are interested in the maximum value of the inertia parameter r/T1 and hence the 274
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
Most investigators give the rate of increase of the profile "half width," b I / J x , where bl/, is defined as the value of 2y where the profile is '/zof its centerline value. Since the spread is linear in x, bIl2/x = constant. From this constant the empirical constants in (B-1) (B-2) and (B-3) can be readily evaluated from the expressions c =
0.848 bu1,. -' X
C,
=
0.848 bml/. -' X
c,
=
1 . 2bn0 9. X
(B-4) (55)
The form of the momentum equation satisfied b y eq B-1 does not contain eddy diffusivity explicitly. However, Goertler’s velocity profile U
- = sechz ( u t )
(B-7)
bu by
(B-8)
u
satisfies the equation
au bx
u-+u-=€-
b2u by2
d,
particle diameter, cm
Pr PrT Sc ScT
= = = = U = u, v = x, y =
air) Prandtl number turbulent Prandtl number Schmidt number turbulent Schmidt number jet centerline velocity, ft/sec axial and lateral jet velocity components, ft/sec axial and lateral jet coordinates, ft
GREEKLETTERS
and
E
E
-
ux
1 4aZ
ep p
v
If the profiles (B-7) and (B-1) are set equal at u/U = ‘/zthe relationship between the constants u and c is u = 0.749/c and the two profiles agree within 2% except at the extreme fringe of the jet. Thus eddy diffusivity can be calculated from the equation .-
ux
- 0.448~~
(B-10)
If (13-1) is substituted into (B-8) and (B-2) is put into the species conservation equation
t
(B-11) a n expression for turbulent Schmidt number is found to be SCT
; = c,2 e
=
c2
(B-12)
As pointed out, the empirical constants for the finite slot width velocity and concentration profiles, eq 7 and 9, are equal to those in eq B-1 and B-2, respectively. Thus, the constants c and c, can be evaluated from data near the slot by trial and error fitting of eq 7 and 9 to the experimental velocity and concentration data. This method is particularly useful to evaluate e, when the contaminant is introduced into the entrainment air on one side of the jet. The distribution of contaminant then becomes, according to Niitsu and Katoh (1960)
.It very large distances from the slot (B-13) approaches (B-14) Equations 7 and 9 approach their asymptotes rapidly, but eq U-13 must be used for large distances from the slot. Nomenclature
b,l
= =
c
=
,c
=
one-half jet slot width, it value of 2y a t u/C7 = l/. empirical constant, velocity profiles, dimensionless empirical constant, coiicentration profiles, diniensionless
= = = = = =
momentum eddy diffusivity, ft2/sec particle eddy diffusivity, ft2/sec fluid viscosity, P fluid kinematic viscosity, ftz/sec
Y/X
fluid density, lb/ft3 particle density, g/cm3 = empirical constant = 0.749/c 2‘1 = Lagrangian macro-time scale of turbulence, sec 7 = particle relaxation time, eq 1, sec
p pp u
=
SUBSCRIPTS c f p 0 m
a
=
ml = mass fraction of contaminant (introduced in jet) m2 = mass fraction of contaminant (introduced in entrained
= = = = =
jet centerline fluid particle jet origin jet, y + m
literature Cited
Azer, N. S., Chao, B. T., Int. J . Heat Mass Transfer 1, 121 (1960). Baron, T., Alexander, L. G., Chem. Eng. Progr 47, 181 (1951). Flint, D. L., Kada, H., Hanratty, T. J., A.I.Ch.E. J . 6 , 325 (196Oi.
FlOra,-J: J., Jr., Goldschmidt, V. W., A.I.A.A. J . 7, 2344 (1969). Friedlander, S. K., AZ.Ch.E. J . 3, 381 (1957). Fuchs, N. A., “The Mechanics of Aerosols,” revised and enlarged ed, pp 83-87, 257-64, Pergammon Press, Oxford, 1964. Goertler, H., 2. Angew. Mat. Mech. 22, 244 (1942). Goldman, I. B., Marchello, J. M., Znt. J . Heat Mass Transfer 12, 797 (1969). Goldschmidt, V. W., Eskinazi, S., Trans. A S M E , Ser. E 88, 735 (1966). v.d. Hegge Zijnen, B. G., Appl. Sci. Res., Sec. A 7, 256 (1958). Heskestad. G.. Trans. A S M E . Ser. E 87. 721 (1965). Hinze, J. ’O., ’“Turbulence,”’ pp 352-364, hlcGraw-Hill, New York, N. Y., 1959a. Hinze, J. O., “Turbulence,” pp 42-49, McGraw-Hill, Yew York, N. Y , 1959b. Householder. h!I. K.. Ph.D. Thesis, Purdue Universitv. “ , Lafavette. “ , Ind., 1969.‘ Householder, 11. K., Goldschmidt, V. W., J . Eng. Mech. Diu. ASCE Proc. 95, 1345 (1969). Jenkins, R., “Heat Transfer and Fluid Mechanics Institute Preprints,’’ pp 147-158, Stanford University Press, Stanford, Calif., 1951. Lilly, G. P., Ph.D. Dissertation, University of California, Los Angeles, Calif. 1971. Liu, V. C., J . Meteorol. 13, 399 (1956). 2, 8 Marchello, J. M.,Toor, H. L., IND.ENG.CHEM.,FUNDAM. (1963). Miller, D. R.,Comings, E. W., J . Fluid Mech. 3, 1 (1957). Kiitsu, Y., Katoh, T., J . SOC.Sanitary Domestic Eng. (Japan)34 (12), 17 (1960). Reichardt, H., 2. Angew. Mat. Mech. 21, 257 (1941). Reichardt, H., VDZ Forschungsh. No. 414 (1942). Schlichting, H., “Boundary Layer Theory,” 4th ed, p 605, McGran-Hill, New York, S . Y., 1960. Soo, S. L., Chem. Eng. Sei. 5,57 (1956). Tchen, C . M., Dissertation, Delft, hlartinus Nijhoff, The Hague, 1447
ToGC,’W. L., Sherwood, T. K., Ind. Eng. Chem., 31, 457 (1939). RECEIVED for review Januar 24, 1972 ACCEPTED Marc{ 14, 1973
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