si
S 5” ttm t,O
tcm Vi V v, z
= = = = = = = = = =
stoichiometric coefficient in reference electrode reaction linear coefficient relating a, and 00 temperature, OK ion transference number in the rejecting layer ion transference number in the bulk solution water transport parameter partial molar volume of species i, cm3/mole volume of the transference number half-cell, cm3 velocity of species i, cm/sec distance from solution-membrane interface, cm
GREEKLETTERS 6 A K~ K’
= = = =
Y,
= = =
9
=
I.(
pi
rejecting layer thickness, cm backing structure thickness, cm electrical conductivity in the rejecting lager, mho/cm electrical conductivity in the backing structure, mho/cm viscosity, CP chemical potential of species i, ergs/mole number of ions i per molecule of electrolyte potential of a reference electrode, V
SUBSCRIPTS
e
i 0
+ -
= = = = =
electrolyte as a neutral species a n y arbitrary species solvent cation anion
SUPERSCRIPTS a
p y s
m
= = =
= =
adjacent to tight membrane (upstream side) inside membrane, at interface between rejection layer and backing structure adjacent t o backing structure (downstream side) implies the quantity is relative t o backing structure implies the quantity is relative to rejecting layer
literature Cited
Agrawal, J. P., Sourirajan, S., Ind. Eng. Chem. 61 ( l l ) , 62 (1969). Allgood, R . W., LeRoy, D. T., Gordon, A. R.,J . Chem. Phys. 10,124 (1942). Banks, W., Sharples, A., J.Appl. Chem. 16,28 (1966). Bennion, D. N., Department of Engineering, University of California, Los Angeles, Rept. 66-17 (1966). 8, Bennion, D. N., Rhee, B. w., Ira. ENG.CHEM.,FUNDAM. 36 (1969). Diamond, J. hl., Wright, E. AI., Ann. Rev. Physiol. 31, 581 (1969). Erickson, D. L., Department of Engineering, University of California, Los Angeles, Rept. 66-7 (1966). Erickson, D. L., Glater, J. McCutchan, J. W., Ind. Eng. Chem. Product Res. Develop. 5,205 (1966). Jagur-Grodzinski, J., Kedem, O., Desalination 1,327 (1966). Johnson, J. S., Bennion, D. K.,Chem. Eng. Prog. Symp. Ser. 64,N o . 90,270 (1968). Johnson, J. 6. McCutchan, J. W., Bennion, D. N., School of Engineering and Applied Science, University of California, Los Angeles, Rept. 7139 (1971). Kedem, O., Katchalsky, 9., Trans. Faraday Soc. 59, 1918 (1963). Loeb, S.,“Desalination by Reverse Osmosis,” U. Merten, Ed., Press, Cambridge, Mass., 1966. pp 55-90, M.I.T. Lonsdale, H. K., “Progress in Separation and Purification,” Vol. 3, p 202, Wiley-Interscience, Yew York, N.Y., 1970. Newman, J., Advan. Electrochem.Electrochem. Eng. 5,87 (1967). 7,514 (1968). Newman, J., IND.ENG.CHEM.,FUKDAM. Kewman, J., Bennion, D. N., Tobias, C. W., Ber. Bunseges. Phys. Chem. 69,608 (1965). Osborn, J. C., Bennion, D. N., IND.ENG.CHEM.,FUNDAM. 10, 273 (1971). Re, hl., M.S.Thesis, School of Engineering and Applied Science, University of California, Los Angeles, Calif., Sept 1970. Reid, C. E., Breton, E. J., J . Appl. Polym. Sci. 1, 133 (1959). Remy, H., Trans. Faraday Soc. 23,381 (1927). Riley, R. J., Gardner, J. O., hlerten, U., Science 143, 801 (1964). IkChIvm for review December 30, 1971 ACCEPTED October 4, 1972 This work was supported by the State of California through the Universit? of California Statewide Water Resources Center.
Fully Developed Turbulent Boundary-Layer Flow of a Fine Solid-Particle Gaseous Suspension Rockley G. Boothroyd” and Peter J. Walton CSIRO Division of Environmental Afechanics, Canberra, Australia
An examination has been made of the fully developed turbulent gas boundary layer in a 3-in. diameter pipe conveying fine solid particles. Results indicate that fluid turbulence is markedly reduced b y the particles near thewallwhereas earlier tests showedthe effect to b e of no practical significance nearer the center of the pipe. The mean velocity gradient of the gas near the wall is reduced considerably by the presence of particles, and impact of the particles themselves must account for a large fraction of the shear stress a t the wall. The results account for the observation that particles can reduce the friction factor and wall Nusselt number of a gas flowing in a pipe.
D e n s e gaseous suspensions of fine solid particles flowing in pipes are quite common in a number of industrial processes. If suspension flow is examined from a one-phase flow viewpoint, overall measurements of friction and heat transfer often yield some very surprising results. Figure 1 shows data (which also cover the flow conditions studied in more detail in the present paper) for the friction factor f, = 2X,/pfi7’,* and wall
Nusselt number Xu, = h,D/kf in three different sizes of pipe. - _ X is the wall shear stress due t o friction of the fluid, h is the wall convective heat transfer coefficient, and D is the pipe density PI, and diameter. The gas has mean velocity thermal conductivity k,. Suffixes s and f (or 0) refer to the suspension and the gas flowing alone, respectively. Thus near the pipe wall, but sufficiently far away for
of,
Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
75
Figure 2. Layout of tracer injection and sampling equipment: A, graphite injection collar; B, suspension flowing upward in 3-in. pipe; C, sampling probe; D, dust filter; E, katharometer by-pass; F1, Ff, gas humidifiers; G, katharometer; H, flowrator; I, vent to atmosphere; J, COZ-He tracer supply; K, flowrator; L, pressure gauge; M, purge air to prevent graphite “blinding” when no tracer gas is flowing; N, purge air to clean probe
molecular transport effects to be numerically insignificant ~
A8
AT0
- ss
- PdSUS,VI’
so
+
PlUf8)VfS~
(1)
P,UfOtV,0’
and the wall heat fluxes are related by
The suffixfs refers to the gas phase when particles are present, whereas suffixfo refers to the gas when i t is _ flowing alone. This _ _ ufs’ vis' # ufo’ vlo’ distinction is necessary because in general - _ _ and v,,’ TI,’ # V,O’ Trot. In these expressions both phases are treated as separate fluid continua and u and v are the axial and radial velocities in the pipe; C and T are the specific heat and temperature; a prime ( I ) refers to a turbulent fluctuation from the mean value, and a n overscore (-) refers to a time-mean average. Equations 1 and 2 are simplified in that fluctuations in the dispersed density of the solid-particle ( p d s ) and gaseous phases have been ingored in their derivation (Hinze, 1961). Taking the idealized one-phase view of a suspension where the particles are infinitely small, eq 1 and 2 simplify to (3) and
-
XU, ~-1 + Nu0
csw C ,m,
Earlier Turbulence Studies
I n a n earlier study (Boothroyd, 1967), the observed low values of fs/fo were further investigated by injecting a gas tracer in the center of the pipe. Results showed that the gas eddy diffusivity in the central parts of the pipe was not changed by the presence of particles by a sufficient amount to explain the results shown in Figure 1. It was surmised, therefore, that a reduction in eddy transfer effects might well take place much closer to the wall where the tubulent wave numbers are much higher. I n the present investigation a gas tracer was injected through a porous section of the pipe wall and its dispersion downstream was measured by sampling. Apparatus
(4)
where 7 6 , and W’, are the solids and gas mass flow rates in the duct. I n this case the suspension would behave simply as a homogeneous fluid of greater density with an altered specific heat. I n practice, particle agglomeration is always prevalent in dense suspensions and this places a lower limit on the particle 76 Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
diameter d which can exist. A comparison of eq 3 and 4 with the experimental data in Figure 1 indicates that either or both of two phenomena may occur. (1) Particles do not follow the - fluctuations closely so that: fluid - _ _eddies _ _ and temperatures u,’vs’< U,~‘V,,’ and v,‘T,‘ < v,,’T,’. This would be expected from the results of a study of particle behavior in homogeneous and isotropic turbulence (Hjelmfelt and Mockros, 1966). (2) Alternatively, because the data in Figure 1 indicate that Nu,/Nuo and fs/jo can both have values less than 1, the presthe gas turbuence of particles must also partially - _ _ _ _suppress _ lence (Saffman, 1962) so that uls’v,,’< U~O‘V,O’and vjs’Tls’ < vlo’T,o/. Our problem is to examine where these effects occur ~ in the flow.
General Layout. T h e apparatus is illustrated in Figure 2. All experiments were for fully developed flow carried out a t t h e end of a loiig, smooth, vertical 3-in. bore brass tube a t a Reynolds number R e = 3.5 x lo4.Test’section conditions were held very nearly constant a t a pressure of 19 psia and temperature 70°F. The solids flow rate was varied in t h e range 0 < W,/W, < 20 but almost all the results reported here are for one condition only (TV,/W, = 3). The flow rig has been
Table I. Typical Example of the Effect of Aspiration Rate on the Measured Tracer Concentrationa
Measured tracer concentration, yG 7 7 7 6 Sampling rate, om3, min 250 I140 a Tlie normal aspiration rate is 500 cind 'niin.
6 5 2400
descrihed b y 'Il-altoii (1071a, b). '1'2ie zinc particles (0-40 pni) \\-ere identical with those used in earlie 1966, 1967, 19i0). Tracer Injection. --I iionbuoyniit 62yc C02-3SyGHe inistiire was injected through a porous graphite p w t of the test, section and sampled a t several positions downstream using a radially traversing probe shown in Figure 3. ide-sampling technique, first' used b y Chao and Miii (1966), allows accurate gas samples to be taken with relat,ively lit'tle coiit'amination from t h e particles. T h e tracer gas concent'ration was measured with a katharometer after 100% humidification. Katharonieter calibration lvas almost linear for lo^ volumetric concentrations c of this particular tracer gas i n air. I n most of the present tests c lay in the range 0 < c < 0.07. With a nonbuoyant tracer gas the measured concentration was always independent' of the sampling rate over a very wide range, as indicated in Table I. Similar esperimental techniques have been used in esperiments with air flowing alone (Caseau and Deniau, 1969; Hall and Hashimi, 1964; Quarmby and . h a n d , 1969), but this is the first such study using a dusty gas. Tlie main additional esperimental difficulty in the present work lies in ensuring t h a t the dust does not "blind" the porous section so as to impede the tracer gas f l o ~and thus cause nonuniform t'racer injection. X high-permeability porous bronze (Grade Porosint made by Sintered Products Ltd.) was most unsatisfactory in this respect. Positive going pressure pulsations in the flow cause reverse floiv into the porous material and this prompted deposition of dust within the porous material. This deposit cannot be removed by air purging and resulted in nonuniform injection of the tracer gas. This problem was soon apparent from routine tests in which the injection collar was rotated with respect to the sampling section. A relatively low-permeability graphite (Grade 3780 manufactured by Le Cai,bone Ltd.) was satisfactory in every way. This has a mean pore size of the order 0.1 pm, which compares favorably wit'li the mean diameter of t'he particles ( 2 1 1 pm). Also a high injection pressure differential (= 25 psi) was needed over a relatively small thickness of the graphite and this is considerably larger than ariy of bhe pressure pulsations in the flow.
Probe and Grapnite Collar
Figure 3. Detail of test section at injection and sampling points. Dimensions: D1, 7 in.; 02, '/J in.; D3, '/z in.; D4, '/z in.; D5, j/16 in. B.S.W.; D6, 1 in., 2 in., and 3 in.; D7, 1 in. A, tracer; B, 4 bolts to hold collar together; C, dial 0-50 thousand; E, 20 T.P.I. in. B.S.F.; F, scale; G, probe column; H, adjustable stops; J, air and tracer; K, 4 bolts to hold collar to test section
so that for the purposes of this study eq 5 may be used in the form
l b
(D+ E , , ) ?bC-
r ar
br
--
A \
=
a f bC bX
I n order to confirm requirement (6) i t suffices to assume t h a t e,, is of the same order of magnitude as the measured value of e,, for any value of r. Treatment of Data. T h e experimental evaluation of e,? directly from eq 7 involves some error due to the uncertainty in measurements of the concentration gradient. However, there iq no reliable alternative but to evaluate E , ~directly from eq 7 . This was done mostly by integration from the wall (radius r,) so that
for positions doivnqtream of the injection source. Alternatively for longer distances z from the source, ecr may be evaluated fro111
Tracer Dispersion
Basic Equation. Ilescribiiig eddy dispersion ab a "diffusion-type" process, the coiiceiitrat'ion c of the t'racer iz giveii by l a
rir
(D +
dc eCT)r - = br
fif
6c -02
- (D +
62C
a22
(5I
for Lisisyniinetric tracer injectioii. D is the molecular diffuiivity of the tracer gas ant1 eCr :~iideCT are the air t u r h l e n t diffnsivity iii the radial ( r ) a n d a s i d ( , E ) directioiis, respectively. Kscept for conditioiis very close to the ring source of illjectioii
(9) Attention was concentrated on obtaining several setb of nieasurenieiits of c ( r , 2 ) for one set of flow coiiditiow (TI., \i',.= 3 a t R e = 3.5 x lo4)).In principle, e C r may be evaluated a t several axial positions for coni1)arisoii purposes. 111practice, measurenieiits a t several axial positioiis are vitally neceesai'y t'o miiiiinize uncertainty from experimental errors. Thus for the iiiost accurate measurements of ecr a t low i d i i e s of y(y = r, - r ) ) iiit,egratioiis a t low values of s are necessary well before the point where
fif(dc/d.z) r'dr' starts t,o cliange its
sign. Ind. Eng. Chem. Fundorn., Vol. 12, No. 1 , 1973
77
O'OU
OllU
Qstance f r ~ mwall
0 ' 4
4
6
8
Ib
15
rlr
Ib
ia
W
io
Figure 5. Effect of particles on radial concentration profiles Re = 3.5 X 1 04; D = 3 in.; x / D = 0.5; 0,W,/W, = 0;
Figure 4. Effect of solids loading on tracer concentration near the wall; c is the measured Concentration; c, is the fully mixed concentration ( 1 % in all tests): 0,y = 0; Q Y = 0.100 in.; X, y = 0.150 in.; A, y = 0.200in. Results
Some General Observations. Figure 4 shows measurements of the tracer concentration a t a distance 2 = 1.468 in. downstream f r o m the center of the injection collar a t four different values of y near the wall. The tracer dispersion from the wall appears t o be reduced as the solid particle content increases. The greater scatter in the results a t higher values of W J W f is associated with the much more unstable flow which occurs a t higher solids loadings. The evidence for turbulence suppression by the particles is quite convincing from the results in Figure 4 alone. Figure 5 shows another important change which has taken place due t o the addition of particles to the flow. I n all cases where solids were present
lrw > tipcrdr
Q
-
27r
(10)
where Q is the volume injection rate of the tracer. By contrast in all cases where particles were absent
The difference shown in Figure 5 is very marked and there is no reason to suppose that the tracer gas was insufficiently accelerated by the fluid stream solely on account of the presence of particles. The only feasible explanation seems to be that near the wall a,, < Q,O. Further support for this viewpoint lies in the axial variation of the discrepancy
which is shown in Figure 6. Any significant reduction in the gas velocity by the particles would be expected to occur only very near the wall, so that this would increase the value of CY more for low values of x / D where the tracer dispersion from the wall is quite small. These values of Q based on velocity can be used to construct a new velocity profile so that
Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
Y dW
i
%
78
0'10
Ob7
X ,W,/ W,
=
0.3;0, W,/ W,
=
3; A, W,/ W,
=
20
for all values of x / D . This procedure is illustrated in Figures 6 and 7 . This method can only be used to estimate a,, a t small distances from the wall (A to B in Figure 7). The velocity profile nearer the center of the pipe (from C to D in Figure 7 ) was estimated from an adaptation of the velocity defect law which is described as follows. Estimation of Velocity Profile near t h e Pipe Center. The mean velocity profile f i f o a t distances well away from the pipe wall is given by the velocity defect law
where u, is the pipe center line velocity, u* is the friction velocity ( U f d 5 / 2 ) ,and is the eddy diff usivity of momentum. The modifications of this velocity profile by the presence of particles depends on the extent to which they follow the larger eddies in the central part of the pipe. Separate tests of Arundel, et al. (1971), for the same flow conditions studied in this paper indicate that only slight agglomeration of particles occurs except quite near the wall. As the Stokes law relaxation to time for 5-40-km zinc particles lies in the range 5 X 3.2 X 10-2 sec, particles would be expected to follow the larger eddies quite closely in the center of the pipe for the present experimental conditions. Thus the shear stress a t y acting on the suspension in the axial direction is given by
where separate eddy diffusivities of momentum el, and eS are attributed to each phase. On the basis that particles follow the larger mainstream eddies, it is also reasonable to assume that
‘I *
X
X
0
.6
.4
Uro
X
b
A
0
A
Y
h
0
0
!
and U,, are the velocity profiles shown in Figure 7
10
kw
Gas mean velocity profiles Figure 7. Gas mean velocity profiles
i n the central part of the pipe so t h a t using eq 1 and 14
efo
f”
”.
I
bY For R e = 3.5 X lo4, D = 3 in., and P d s / P f W,/’W, = 3, the data in Figure 1 show that js/fo % 2 . However, the relationship betweeii eIs, e,o and e, cannot be determined esactly. Previous tracer measurements (Boothroyd, 1967) near the pipe ceiiter for the same conditions indicated that EC,fS -.-E
‘C,fO
1.3
2
Tracer injection Reference
Symbol
o Caseau and Deniau Quarmby and Anand A This study x
D, in. 2 . 9 X lo5 12 2 X lo4 3’18 3 3 . 5 x 104 Re
length, in.
7.8 X 5 75 1
E f0
as there is no serious reason to dispute the aiialogy between mass aiid momentum eddy transfer (Jenkins, 1951) of the gas phase merely on account of the presence of particles. Thus bethat e, < efs, the most reasonable numcause u,’v,’ < ufs’z~fs’so erical wtiniate we can make of the velocity gradient ratio in eq 15 is simply t’o assume efs N efo N e s . This approximation ~
Figure 8. Comparison of results for flow of air alone with other studies. x/D = 0.5 where x i s measured from the center of the injection length. Note that injection length/pipe diameter ratios are not the same in these studies
cannot be greatly in error for these particular conditions of = 3, although this simplification may be inappropriate at other solids loadings. With these approximations eq 15 gives
Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
79
Figure 9. Examples of tracer injection profiles for flow of air alone: Re = 2;A, 2.6;V, 4;0,8;0,16
+,
3.5 X 1 04;D
=
3 in. Values of x / D : 0,0.5;
x,
0.83;.,1,16; A, 1.5;
C
4 K 12-
2.0
0, 0
1
4
6
9
lb
10.Tracer concentration profiles for flow of 0,0.133;O f 0.2;X, 0.33;A, 0.6
Figure
h
14
%
4fo
-
IL1 + PIJI
(17)
,E
where c,
=
Pfcfs Pj
+
+
PdSes
(18)
Pds
Equation 18 is the region C-D shown in Figure 7 . The profiles in Figure 7 are then scaled for direct use by the relationship (19) 80
Ind. Eng. Chem. Fundom., Vol. 12, No. 1 , 1973
b
air alone at Re =
More specifically, the velocity defect law with particles is
fsu*2D
lb
i
1
3.5 X 1 04;D
4 =
i
__
%
3 in. Values of y / r z o : 0 , O ;
V f 0.067;
Flow of Air Alone. conlparison of tracer concentration tests in the absence of particles gave reasonable agreement with the results of Quarmby and h a n d (1969) and Caseau and Deniau (1969) as is illustrated in Figure 8. However, although the Reynolds number depeiidence of the concentration fields is quite small, a different injection length (L) to pipe diameter ( D ) ratio was used in each of these studies. Flow with Particles Present. Comparison of data in Figures 9-12 shows the effect of reduced tracer dispersioii when particles are present,. Using the velocity profiles shown in Figure 7 , ccr was evaluated from these data using eq 8 and 9. Calculated results are shown in Figure 13 both for the flow of air alone a n 3 for the suspension. The extent of turbulence suppression in the wall region is clearly shown in Figure 13, which is representative of several sets of data recorded in the thesis of Walton (1971b).
O b
1
A
b
B
i
O
h
f
4
0'
5
i
1
1
I
4
Figure 1 1 . Tracer concentration profiles with particles present: Re = 3.5 X 1 04; D = 3 in.; W,/Wf 0,O.z; X, 0.33; A, 0.6
%
= 3. Values of y / r w : 0 , 0;
' I ,0.067; 0,0.133;
1
Figure 12. Comparison of tracer concentration profiles for flow of air alone and with particles a t Re = 3.5 X lo4; D = 3 in.: X I W,/W, = 0 for x / D = 1.5; 0 , W,/W, = 3 for x / D = 1.5; W,/Wf = 0 for x / D = 16; 0, W,/Wf = 3 for x / D = 16
+,
Ailthoughthe accuracy of this met'hod of measurement is not high, certain features of the results were \Tell pronounced. Firstly, the gas eddy diffusivity is reduced by the particles up to quite a considerable distance from the wall (see curve 111). Secondly, with air flowing alone, t f a was always abnormally high near the wall when compared wit,h well established earlier results (Laufer, 1954). The same observation was also made iii pipes of 1 and 2-in. bore. Particular care x a s taken to ensure that this effect did not originate from unsatisfactory surface junctions in the test section. Other possible sources of error such as streamline deflection in the vicinity of the probe were also iiisufficient to cause this effect. Thus the possibility of the effect' being a real one cannot be discounted, in which case the suspected cause is the effect, of bracer gas injection. Ailthoughthe ratio of tracer radial injection veloci,ty (vi) to mean axial velocity 0,is only 5 X in this study, the radial displacement of the boundary layer by the tracer over
Figure 13. Variation of radial eddy diffusivity for Re = 1 04;D = 3 in. Curve 1. Theoretical results of Laufer 3.5 and Quarmby and Anand: 0,flow of air alone (this work); A, flow with solids (W,/W, = 3) using ulo (curve 11); X I flow with solids (W,/W, = 3) using Us0 (curve 111); V I experimental results of Quarmby and Anand (scaled for effect of of Re); 0 , results of Caseau and Deniau (scaled for effect of Re)
x
the injection length L1is of the same order of size as the thickness of the viscous sublayer. An examination of some of the earlier literature also casts some uncertainty over the effect of tracer injection on the flow. Thus, for example, Olson and Eckert's (1966) results for c C were also about 607, too high a t y/r, = 0.1, and their injection velocity ratio lay in the range 2.1 X < < 1.2 X Also in a more recent study of Kays, et al. (1970), it was shown that although acceleration of a boundary layer caused reduction of wall eddy transfer, the effect could be offset by gas injection even t o the extent of improving eddy heat transfer. On the other hand, neither Quarmby and ilnand (1969) nor Caseau and Deniau (1969) noted any such effect. However, in the former study their unquoted value of v i / c r , was likely to be very low. I n the latter study vi/^, was very
ui/ul
Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
81
high at, 3.3 X lo-*, but on the other hand the injection length
L1 \vas very small indeed a t 2 mm in a 30.3-cni diameter pipe. Despite a n est'eiisite literature on transpiration in boundary layers, there seems to be little reference to any possible influence of gas injection on eddy viscosity, although more recently ail equation of Wasan, et al. (1969), predict's a considerable increase in ef when y* > 10 for the conditions used in this study. In short, it is difficult to dismiss any suggestion that the act of tracer injection might interfere substantially \vit,h the eddy struct'ure near t'he wall. Severtheless, i t seems reasonable enough to conclude from curve I11 in Figure 13 that the gas eddy diffusivit,y near the wall is reduced very markedly solely 011 account of the presence of particles. Wall Shear Stress. For the flow conditions ill this work the wall shear stress acting 011 the fluid is 1.06 X lo-' pdl/ft' for air flowing alone and from the data in Figure 1 about 2.12 x 10-1 pdl/ft' when particles are present. However, from the velocity profile in Figure i the stress a t the wall which can be at,tribnted t,o shearing of the gas is much less than this. I t seems, therefore, that' most of the rvall shear stress is due to impact of t'he particles with the wall. Conclusions
I t has been shown esperimentally that small particles cause strong suppression of gas turbulence near the wall of a pipe. The particles also cause a much reduced gas velocity gradient near the wall. Tliw most of the wall shear stress can be attributed to particle impacts and not to viscous shearing by
the gas. The reduction in gas turbulence near the wall is sufficient to account for the low Nusselt number and friction factor often observed in this class of fluid. literature Cited
Arundel, P. A., Bibb, S. D., Boothroyd, R. G., Powder Techno/. 4 , 302 (1971). Boothroyd, R. G., Trans. Inst. Chem. Eng. 44,306 (1966). Boothrovd. R. G.. Trans. Inst. Chem. Ena. 45. 297 (1967). Boothroird, R. G.,'Haque, H., J . 3lech. Egg. Sit. 12, 191 (i970). Caseau, P., Deniau, R., Houzllc Blanche 24, 259 (1969). Chao, B. T., Min, K., -Vue/. Sci. Eng. 26, 534 (1966). Hall, W. B., Hashinii, J. A,, Proc. Inst. Jlech. Eng. Pt. 31, 178, 1 /10AAJ \ - V Y I , .
Hinze. J. 0..Aaol. Sci. Res. A l l . 33 11961). Hjelmfelt, A. T.: IIockros, L. F.; A p p l . Sei. Res. A16, 149 (1966). Jenkins, It., Proc. Heat Transfer and Fluid X w h . Inst. 147 (1951). Kays, W. M.,Noffat, R. J., Thielbahr, W. H., J . Heat Transfer 92C, 499 (1970). Laufer, J., aTat.Adv. Comm. Aeronaut. Tech. Rpt. 1174 (1954). Olson. R. 31.. Eckert. E. R. G.. J . A m . Jfech. 33. 7 11966). Quarmby, ii.; h a n d , R.K., J.'Fluid'-?fech. 38,433, 457 (1969). Saffman, P. G., J. Fluid Mech. 13, 120 (1962). Walton, P. J., Gammon, L. N., Boothrovd, R. G. Powder Technol. 4 , 293 (1971a). Walton, P. J., Ph.D. Thesis, University of Birmingham, England, 1971b. Wasan, D. T., Randhava, S. S.,Babu, P. S., Chem. Eng. Sei. 24, 595 (1969). RECEIVED for review January 10, 1972 ACCEPTEDOctober 4, 1972 This work was carried out at the Department of 11echanical Engineering, University of Birmingham, L-. K., and was supported by the Science Research Council.
A Theoretical Study of Pressure Drop and Transport in Packed Beds at Intermediate Reynolds Numbers Mohamed M. El-Kaissy and George M. Homsy* Department of Chemical Engineering, Stanford Cniversity, Stanford, Calif. 94305
Previously proposed theoretical cell models for transport in packed beds have been limited, with one exception, to the creeping flow regime. Two popular models a t e reviewed and extended to finite particle Reynolds numbers b y regular perturbation techniques. The central goal i s to develop a theoretical framework and methodology b y which cell models may b e rationally extended using analytical representations. The predictions of pressure drop for spherical cells demonstrate the failure of these models to predict deviations from creeping flow conditions with any accuracy. One possible extension, that of distorted cells, i s briefly treated and i s shown to b e capable of representing experimental behavior. Transport in beds a t high Peclet number is then treated in some generality and it i s shown that predictions of the transport rates a t high Peclet number are quite insensitive to Reynolds number, thus offering theoretical confirmation for this well-known empirical fact.
D e h p i t e the widespread occurrence of fluid-particle ststems in n-liicli the ~.olurnefractioii of solids iz aljpreciable, the ability to describe these tenis by the use of models is still in a n early stage of dev pment. 'The main obstacle to be ovei come in the descriptioii of such tenis is that of satisfactorily treating pnrticle-liarticle iilteractio~is.One such \\-ell kiioivti model which reeks to surmouiit this Ijroblem is the cell 82
Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
model due to Happel (1958). In its formulation, the difficult many-body problem is replaced by a simple and coiicept~ua~lS more attractive coiitiiiuous one involvi~~g only one particle. K a l l effect,sand/or ent,ry and exit effects are neglected. The assembly of particles in t8hefluid is assumed to be uniform and each sljhere is fixed in space x i t h equal spacing separating: them. The inteiaction of a particular sphere viith its neigh-
