Pressure drop during the flow of stokesian fluids through granular beds

Chemrcal

Enguzerrrng Srrence, 1974,

Vol

29, pp 2 13-223

Pergamon Press

Prmted in Great Brmm

PRESSURE DROP DURING THE FLOW OF STOKESIAN FLUIDS THROUGH GRANULAR BEDS Z KEMBLOWSKI

and J MERTL

Institute of Chemical Engmeermg. todi (Recerved28

September

Techmcal Umverslty,t6di,

1972, accepted6

Poland

March 1973)

Abstract-The

resistance to flow of StokesIan flulds (I e time-Independent fluids with no yield stress) through granular beds IS dlscussed A defimtlon of fnctlon factor A and generahzed Reynolds number ReLK IS proposed for flulds obeymg the “power-law” shear stress-shear rate relation The generahzed Ergun equation, derived m this paper, gives the dependence of the fnctlon factor on the generahzed Reynolds number and flow behavlour mdex n The vahdlty of the generahzed Ergun equation was proved expenmentally In the case of Newtoman flmd (for n = 1 0) a more exact form of the classlcal Ergun equation IS obtamed INTRODUCTION

The flow of fluids through granular beds 1s an operation often encountered m the processes of chemical technology As far as smglephase flow 1s concerned one can mention the followmg examples (a) the flow of filtrate dunng filtration of mdustrial suspensions, (b) the flow through a porous bed m order to equalize temperature or velocity field, (c) mixing, by means of flow through the porous bed, to get a uniform composltlon of the fluid, (d) the flow through a porous bed m underground hydraulics The smglephase flow of Newtonian fluids through beds was a SubJect of numerous researches They resulted m getting some useful correlations On the other hand there is, so far, no general solution co the problem of non-Newtonian flow through such beds Taking into account the constant mcrease m production and processing of hlghmolecular compounds, there 1s a need to carry out research m this field

The modified contmulty Eq (1) and the Darcy law (2), together with a suitable equation of state and vlscoslty equation, describe completely an isothermal flow of fluid through a porous bed In the literature of the SubJect only a few solutions of the simplest cases are given [2-51 In many systems of technical importance, especially when the vlscoslty of fluid flowing through a bed is not a constant quantity, Eqs (1) and (2) have not been solved so far In these cases the correlation equations, denved from expenmental data, must be used The most frequently quoted correlation equation, descnbmg the dependence between pressure drop and fluid velocity, 1s the Blake-Kozeny equation Apd,2

v”=-F1507)

8

(l-e)2

(3)

The effective diameter of particles 1s defined by the equation d, =p

LITERATURE SURVEY Newtonranjlurds

The basic equations of fluid mechanics- the contmulty equation and equation of motion-can be substituted, m the case of fluid flow through porous bed, by following relations [ 11

V 0- -

-;

(VP-f%)

where a, = specific surface of granular filhng The Blake-Kozeny equation (3) describes lamrnar flow of Newtonian fluids m a column of constant cross-section It was derived from a capillary model of porous bed, at some assumptions concemmg properties of the bed [6-81 It was found[9] that lammar motion occurs, when the modified Reynolds number defined by Blake [6]

(2)

13

214

Z

KEM~OWSKI and J MER-CL

1s less than 10 Definmg the frlctlon factor A (four times greater than the Fanmng factor f) as for the flow m straight tubes

one obtains the followmg dependence for lammar flow of Newtonian fluid through a porous bed

*

l3 (l--E)

300 =?&

A more general equation applicable for larnmar, transition as well as for turbulent motion of Newtonian fluids was given by Ergun [9] --Ap d, -= l3 !@v,* 1 (1-E) Substltutmg obtams

relations

30077(1-4+3 v+&p

,jO

Ostwald-de

Waele TYX =

(10)

(8)

(5) and (6) mto Eq (8) one 300

RQ~+~ 5o

(9)

A discussion of ObJections to the use of the power-law model 1s given for example by Remer [l 11and Christopher and Middleman [ 123 But It 1s an expenmental fact, that this model describes the rheological properties of many non-Newtonian fluids m a wide range of shear rate Application of the power-law (9) made It possible to generalize the Blake-Kozeny equation for lammar flow of Stokesian fluids through porous beds The modified Blake-Kozeny equation was given by Chnstopher and Middleman [ 121

where J, =;

-=- E3 ‘(1-e)

k(j)”

(9+~)1(150sc)“-““’

(11)

@a) and

The above dlscussed correlative equattons are represented graphically m Fig 1 The diagram shows dependence of product of fnctlon factor A and modulus l3/(1 -E) on the modified Reynolds number ReBK, defined by Eq (5) Non-Newtonlanjutds

The simplest and most frequently used rheologlcal model describing simple shear of Stokesian fluids [IO] (1 e time-mdependent fluids with no yield stress), is the well-known “power-law” of

(12) The quantity s, the so called permeablhty of the bed, IS independent of the character of the fluld flowing through it The value of Eq (10) to correlate expenmental data was proved by some authors [ 12-151 There are, as well, other correlation equations denved on the basis of the power-law (9) Equations of this kmd were given, for example, by Gregory and Gnskey [ 161 and Yu et al [ 171 They concern, like Eq (lo), only the range of lammar flow Practically no one has dealt, previously, with the transition and turbulent flow region of non-Newtoman Stokesian fluids The non-Newtoman flulds are characterized, m general, by high apparent vlscoslty, so the transltlon and turbulent flow regions occur less frequently than m the case of Newtonian fluids In mdustnal problems however, particularly m apparatus of high output, this kmd of motion can play an important part GENERALIZED REYNOLDSNUMBER

Fig

I Correlation between the value A [e3/(1 --E)] and Reynolds number Resx for Newtoman flulds

Our own expenmental research, concernmg the flow resistance of Stokesian flmds through granular beds, showed the need of re-analyzmg the problem of generalized Reynolds number As a result of consideration, (nmdar to that carried out by

Flow of StokesIan flulds through granular beds

Metzner and Reed [ 181 for the flow of “power-law” fluids m pipes), the following definition of generalized Reynolds number for the flow through porous beds of fluids obeying the power-law has been proposed by us [ 191 Re;,

=

L 2-“dpp

number and fnctlon factor are the only ones that satisfy the followmg three condltlons (a) the defimtlon of fnctlon factor IS correct from the pomt of view of the general rules of fluid mechamcs, (b) the definition of generahzed Reynolds number Re;, (13) slmphfies for a particular case of Newtonian fluid exactly to the Reynolds number defimtlon (5) quoted by Blake and used m classical papers, concerning this SllbJect [6,8,93, (c) cntrcal value of the generalized Reynolds number Re;, IS approximately the same as for Newtonian fluid

(13)

$(I -E) Usmg the denved defimtlon (13) one IS able to write the modified Blake-Kozeny equation (10) m the followmg form e3 (l-e)==

x

300

(14) GENERALIZED

Table 1 Defimtlons of fnctlon factor and generalized

Reynolds number for flulds obeying the “power-law” rate relation (9)

stress-shear

Defimtlon of fnctlon factor

Reference

Defimtlon of Reynolds number ReLM =

1

Chnstopher, Mtddleman

f

= %W CM Wo*I(l--E)

Gregory,

Gnskey iI61

Badle

4

Author’s proposal

c3

fCM= w,V(l--E)

Apd&ae3 puJ( 1 -E)

wodp

=

%Kdl -e)

Appd&

f, _ 3

&-

150 (l--E)2

fCM= g

&

c

= _gy_L-)I’“‘[-$_eJ’)-’

Re; =

Yu, Wen,

fCM=

(150SE)“-““*

CM

Re; 2

lSOJI(l-•E)

JI = $I+iJ

12Re,( 1 -e) C*-*n(9n+3)(1-4) [

4.n

1 ”

fP=

150(1-e) Re’

Re+=!&!

[I71

&.&P

A = +pvp

shear

Frlctlon factor as a function of Reynolds number for lammar flow

~P’4pn--l

SC-!-dZ

iI21

ERGUN EQUATION

The general dlmenslonless cntenon dependence concerning the pressure drop durmg the flow of Stokesian fluid through porous bed, can be denved using the method of dlmenslonal analysis Let us consider, for this purpose, the flow through porous bed of a fluid, satlsfymg the power-law (9) The functional dependence of pressure drop from other

The fnctlon factor h IS defined m Eq (14) m the same way as for Newtonian flulds (see Eq 6) In Table 1 different defimtlons of Reynolds number and fnctlon factor for “power-law” fluids are compared From a detailed analysis of the relatlons shown m Table 1 [ 191, it can be seen that our proposed definitions of the generahzed Reynolds

No

215

*

-=-E3

(1-e)

”

300 Reb,

Z KEMB~OWSKI and J MERTL

216

parameters can be put down m the form AP =

where

F(f,, 4, v, P, k", n)

(15)

1, = average length of the capdlary ducts m the bed dk = average diameter of the capillary ducts v = mean linear flow of the fluid m capillary ducts K’ = apparent fluid vlscoslty

consldered problem for all slmdar processes, regardless of the character of motion of the fluid m porous bed In order to make use of the Eq (24) It IS necessary to determine the rheologlcal parameters n and k of the flowmg fluid The parameters should be determined for the proper range of shear rate This can be calculated for the flow of “powerlaw” flmd m a porous bed from the formula [ 121 (3n+

12w, “=pV%iG

From the work of Metzner and coworkers [ 18, 201 there results the followmg definmon of apparent flmd vlscosny k” =

8”‘-’ k

(16)

where k’, n’-parameters m the Metzner-Rabmowltsch equation [21] Usually m the discussed range of shear rate one can assume n’ = const

= n

4n

(25)

Equation (24) gives a general functional dependence Let us take note of the slmllanty exlstmg between the Blake-Kozeny equation for Newtonian fluids (7) and the generahzatlon of this equation for Stokesian fluids (14) It IS possible to assume that the slmdanty will occur m case of the Ergun equation, too Taking this mto account one can assume the followmg form of generalized Ergun equation for Stokesian flmds

(17)

In this case Eq (16) can be transformed

1)

*

to

e2 (l--E)

c

=Reb,+czfl

where

(18) According to the Buckmgham II-theorem the dImensIona analysis will give a cntenon function of the form

a=

F(ReL,,,

n)

We propose to call the dependence generalized Ergun equation

(27)

(26) the

EXPERIMENTAL

The Raylelgh method of dlmenslonal analysis leads to the followmg dlmenslonless numbers

rI,=Eu=$ (21) r&=-

k”

(22)

dknV2-np

(23)

II4 = n

After suitable transformation of Eqs (19)-(23), using the capillary model of the bed, one obtams

iI91 d X(l-E)=F Equation

1 s’n

(

(24) 1s a dlmenslonless

>

(24)

solunon of the

The correctness of Eq (26) was proved expenmentally The tests were carned out m an expenmental apparatus, shown schematically m Fig 2 As expenmental media aqueous solutions of starch, poly (vinyl) alcohol (Vmavllol produced by Montecatml) and polyacrylamlde (Separan of Dow Chemical Corp ) were used Besides these were used starch suspension m aqueous solution of ethylene glycol with an addmon of polyphosphate and a suspension of kaolin m water with an addmon of polyphosphate In Table 2 the charactenstlc of packed beds IS given and m Table 3 the data showmg the range of experiments are given One can see, that the experiments were carried out for Newtonian fluids, generalized Newtonian fluids as well as for dllatant fluids The results of measurement of pressure drop dunng the flow of Newtonian fluids through granular beds are presented m Figs 3 and 4 m the form E? *(l-E)

= F(ReBK)

(28)

217

Flow of StokesIan flmds through granular beds 16

,TK

Fig 2 Schematic diagram of expenmental apparatus I-Nitrogen cyhnder, 2,5-Valves, 3-Working manometer, 4-Pressure tank, 5-Preclslon manometer, 7-Workmg tank, I-Pipe, 9-Column with porous bed, IO-Adapter of column, 1 I-Recelvmg tank, UT-Ultra-thermostat, TK-Contact thermometer, t-Thermometer, P2-Vacuum-pump Table 2 Characterlstlcs

of packed beds

No

Material

Symbol

dp 103 (m)

1 2 3 4

Quartz glass spheres Quartz glass spheres Polystyrene beads Quartz sand

gs 1 gs 2 ps b qs

2 77 3 246 1 175 0 265

3 Correlation between the value A [@/(l-e)] ana Reynolds number ReBx for flow of glycerine solutions through porous beds A-GL 1 (dp = 1 175 lo-?rn, I=OlOm), A-GL 2 (dp=O265 10m3m, l=OOSm), O-GL 2 (dp = 3 246 10e3 m, I = 0 1325 m) l -GL 3 (&=2 77 lOearn, 1=0 08m)DGL4(@=2 77 10-3m l=O 1325m)m-GL4(&=3 246 10-3m,1=008m)

E 0 0 0 0

387 383 350 392

0 0 0 0

09455 09106 06596 09907

7 890 x 10 36 x 0 9340 x 7 629 x

1O-9 1O-9 1O-g lo-”

Fig 4 Correlation between the value A [E~/(I--E)] and Reynolds number Resx for flow poly (vmyl)-alcohol solution PAW 30 (Newtonian fluid) through porous media dp = 0 265 10-j m, 1 = 0 0755 m

218

Z

KEM~OWSKI and J MERTL

Table 3 Range of experiments relatmg to flow of StokesIan flmds through porous media

No

Fhud system

1 2 3 4 5 6 7

GLl CL2 CL2 CL3 CL4 CL4 PAW30

psb qs gs 2 gs 1 gs 1 gs 2 qs

0 100 0 080 0 1325 0 080 0 1325 0080 00755

1 1 ::

8 9 10 11 12 13 14 15 16

PAA PAA s5 s5 s55 s55 S6 S6 S6

qs psb gs 1 gs 2 gs 1 gs 2 gs 1 gs 2 gs 2

0 080 0130 0 080 0 1325 0 1325 0 080 0 080 0 080 0 1325

17 18 19 20 21

SGK46 SGK46 SGK46 K71 K7l

gs gs gs gs gs

1 2 2 1 2

0080 0 080 0 1325 0 1325 0 1325

Granular bed Material Height m

Workmg pressure max mm atm atm

Reynolds number Ress or Rek, max mm

98 8 15 2 1.5 21 24 30 16 0

0 0 0 0 0 0 0

10 09 14 2 65 3 75 12 75 59 31 13 8

14 0 84 51 0 72 4 68 4 690 49 7 57 9 55 5

0 0294 0 922 0 230 0 741 0 132 200 0 354 0 168 2 21

6 3 19 2 34 2 9 6 0 35

90 98 89 89 80

0 0 0 0 0

0 0 0 2

1 2 2 3

9 5 6 1 3

0488 00105 0382 0236 0319 0463 256 1OV

346 817 958 780 353

Flow behavlour mdex n

Remarks

0404 0 00792 0 780 0606 0 489 1012 1 665 lo+

10 10 10 10 10 10 10

n=lO

0 708 12 68 80 35 93 5 52 25 97 3 603 115 3 79 1

0 0 0 0 0 0 0 0 0

n
1 345 1 908 1 567 1 484 1 980

91 91 75 75 62 62 51 51 51

1 41 1 41 1 41 1 58 1 58

n>lO

The expenmental points show a satisfactory correspondence with the Blake-Kozeny equation for a wide range of Reynolds numbers This correlation proved a proper methodology of measurements The measurements served also for the determmatlon of permeablhtles of some granular beds The rheologlcal parameters of expenmental media were determmed from the flow curves, which were obtamed by means of a rotary rheometer (Rheotest) with coaxial cyhnders, and by means of a capillary rheometer RK-2 ofour own constructlon WI RESULTS AND DISCUSSION

From the graphs

for non-Newtoman flmds, presented m Figs 5-7, one can find, that m the range of low Reynolds numbers Re;, the proposed modlficatlon of the Blake-Kozeny equation (14) describes satlsfactonly the expenmental pomts This proves the apphcablhty of the mtroduced defimtlon of generalized Reynolds number Rek, (13) and fnctlon factor h (6) It can be seen from this diagram, that for a

Fig 5 Correlation between the value and generahzed Reynolds number Re;, (acrylamlde) solution PAA 2 5 through l-dp = 0 265 10m3m, I = 0 08 m, 0-dp = I=0 13m

A [c3/(l --E)] for flow polyporous media 1 175 10m3m,

series of measurements, characterized by lower numerical value of mdex n, one obtams higher cntlcal values of Reynolds number This IS m accordance with literature data concemmg the flow m pipes [ 18,23,24] Extrapolation of the curves, crossing the experimental pomts for the successive

Flow of StokesIan fluids through granular beds

10-I

100

IO’

102

219

103

%,n

Fig 6 Correlation between the value A [es/( 1 --E)] and generahzed Reynolds number Reh, for generalized Newtoman flmds O-n = 0 75, dp = 2 77 10m3m, 1 = 0 080 m, m-n = 0 75, dp = 3 246 10-3m, 1 = 0 1325 m, A-n = 0 62, dp = 2 77 10m3m, 1 = 0 1325 m, A-n = 0 62, dp = 3 246 lo+ m, I=0 1325m, 0-n=O51, dp=277 lo-lm, 1=0080m, Cn=O51, dp=3246 10-3m, l= 0080m,@-n=O51,dp=3 246 10-3m,l=0 1325m

I

On this basis the followmg values of the coefficients C1 and C2 were taken m Eq (26) c,=300 c, = 3 50 1

(31)

and by an ehmmatlon method the followmg analytical form of the R function m Eq (27) was chosen WZ

a=

l)*+

Vp2(~z-

K~

(32)

Substltutmg dependencies (3 1) and (32) into the Eq (26) we get the following form of the generahzed Ergun equation Fig 7 Correlation between the value A [e3/ ( 1 -E)] and generahzed Reynolds number Rea,, for dllatant flulds 0-n=141, dp==277 W3m, I=O080m, O-n= 1 41, dp = 3 246 10m3m, 1 = 0 080 m, O-n = 1 41, dp=3246 10-3m, 1=01325m, A-n=l58, dp= 2 77 lO+?m, I=0 1325m, V-n = 1 58, dp = 3 246 lOVm,l=O 1325m

values of the flow behavlour index n, allows a supposltlon that m the range of turbulent motion (for Re;, > 103- 103 the curves ~111tend asymptotically to the lme given by equation l3 ho=350

WZ +(K’-

(33) l)‘+K’

It was found, that Eq (33) describes satlsfactonly the analyzed data, when the parameter K IS combmed with Reynolds number Rek, m followmg simple way K =

5

Re;,

(34)

The quantities I_Land 5 m Eqs (33) and (34) are functions of flow behavlour index n only Values of the parameters TVand 5, for n = const , were determmed using the method of least squares

Z KEMBEOWSKIand J MERTL

220 Because

of the comphcated form of the dependencies a numerical method was used Dependence of the parameters /I and 5 on the flow behavlour mdex n 1s shown graphically m Figs 8 and9 The dependencies

log/L = F(n) log 5 = F(n) I

(35)

are described by polynomial function of third and fifth order given m Table 4 One obtams an accurate fit through the expenmental pomts with the polynomial of fifth order In a particular case of flow behavlour mdex n = 1 0 the parameters p and [take the values /_L=lO

(36)

5=0041 The generahzed Ergun equation case of Newtonian flulds to 8

300 -+3

yiq=

K*

50

Fig 9

Dependence of e-parameter on flow behavlour

Index n

(37)

1)2+K2

v(K*-

Rt&

(33) 1s reduced m

From a comparison of the curves given by Eq (37) with the ongmal Ergun equation (8) or (8a) one can find, that the difference between the curves occurs only for the values of Reynolds number Rei, < 30 and 1s of the order of a few percent

6-

4-

Fig 10 Dependence of value n on generalizedReynolds number Re;, for tlow behavlourmdex n S 1 0

02

04

06

0.3

IO

12

14

16

n

Fig 8 Dependence of p-parameter on flow behavlour mdex n

Equation (37) seems to be a more exact form of the Ergun equation for Newtonian fluids In this case the quantity 3 50 on the right hand side of Eq (8), correspondmg to the range of turbulent motion, 1s not added as a constant value Its value depends on the value of Reynolds number, I e on the character of fluld motion It 1s clear that m the range of lammar motion the quanttty takes values near zero (according to the Blake-Kozeny equation (7)) In Fig 10 the dependence of R function (32) on generahzed Reynolds number Reh, IS shown for some values of flow behavlour index n

Flow of StokesIan fluids through granular beds Table 4 Dependence Shape of function

of p and 5 parameters on flow behawour mdex n

log /.&= n, + a1n + u2n2+ u3n3+ 114124 + u5n5

Degree of polynomial function Coefficients of polynomial function

V

111

a, = 0 16054 al=-2 882 uz=4 114 a3 = - 1 254

a, = - 1 7838 al = 5 219 up=-6239 03 = 1 559 u4 = 2 394 a,=-1 120

b, = - 1 9218 b, = - 1 363 bp = 2 895 bz=-0824

050
11 Dependence

log 5 = b, + bin + bg? + b& + ban4 + bg5

III

The generalized Ergun equation (33) indicates, that for high values of the Reynolds number R4,, the fnctlon factor A depends on it and on the flow behavlour mdex II m a small degree The value of fnctlon factor depends then, m practice, on the porosity of the granular bed This statement, apphcable to generalized Newtonian fluids as well as to dllatant flmds, indicates a turbulent character of flow at Reynolds numbers Re;, exceedmg values from lo3 and lo4 (depending on the value of index n)-Ag 11 The proposed form of generalized Ergun equatlon was checked expenmentally for the range of flow behavlour mdex

Fig

221

V b,,=-49035

bl = 10 91 bp=-12

29

b,=2364 b, = 4 425 b, = - 1 896

the quantity A(E~(1 - E)) are shown The pomts are located along the diagonal High values of correlatlon coefficient (r = 0 999) as well as high level of probablhty of the correlation cr = 0 001 prove the correct formulation of the Eq (33) Finally, it 1s worth mentlomng, that the denved dependencies concern flulds showing no flow anomaly such as vlscoelastlc effects and effective slip on the wall CONCLUSIONS

l A new defimtlon of generahzed Reynolds number Rek, (13) 1s proposed It was found, that a good correlation of the expenmental data for StokesIan fluids 1s obtained using the fnctlon factor X, given by Eq (6), and the generalized Reynolds number (13) 2 A generahzed Ergun equation (33), descnbmg the dependence of the fnctlon factor coefficient A

of value A [8/(1--E)] on generahzed Ergun equation)

Reynolds number Rek,

(generahzed

Z KEMWOWSKIand J MERTL

4 average length of the capillary ducts m the bed, m flow behavlour index n’ flow behavlour index m Eq (16) P pressure, N/m2 coefficient of correlation S bed permeability, m2 t time, set V mean linear flow of the fluid m capillary ducts, m/set vo mean linear velocity of flmd, related to an empty cross-sectlon of the column, mlsec wo mass flow rate, kg/m2sec n

V

100

I

I

I

IO'

IO'

103

Greek symbols a

P&il..,

Y

Fig

12 Correlation between the value [A (EY (l--E~)leomDutand [A (es/( 1-e))lexP for StokesIan flulds 0-n=O51, A=n=O62, q -n=O75, ln=141,V=n=158

YP

on the generalized Reynolds number Re;, and flow behavlour index n, is denved The valtdlty of the equation was proved expenmentally 3 Dependence of the parameters p and goccurrmg m Eq (33)-can be described by means of polynomial function of the flow behavlour index n 4 In the case of Newtoman fluids (n = 1 0), the generalized Ergun equation (33) reduces to Eq (37) Equation (37) seems to be a more exact form of the classical Ergun equation

CI, c2

4 4

f fCM fP g k k’

k”

1

NOTATION specific surface of granular filling, m2/m3 filling constans m Eq (27) particle diameter, m average diameter of the capillary ducts, m effective dameter of particles as defined by Eq (4), m Fanning fnctlon factor fnction factor, defined m Table 1 fnction factor, defined m Table 1 acceleration of gravity, m/se? fluid consistency factor, Nsecfl/m2 fluid consistency factor m Eq (16), Nsecfl’/m2 apparent fluid viscosity, Nsecn’/mz height of bed, m

V A EU

&I ResK Rep

4,

Re; Rel

level of probablhty of the correlation shear rate, see-’ shear rate m a porous bed, defined by Eq (29, see-’ porosity parameter, defined by Eq (34) coefficient of dynamic fluid vlscoslty, Nsec/m2 modified vlscoslty, defined m Table 1 parameter m Eq (3 2) fnction factor parameter m Eq (32) dlmenslonless numbers fluid density, kg/m3 bed permeablhty, m2 shear stress, N/m2 reciprocal of shape factor parameter, defined by Eq (1 l), Nsecn/ml+n function m generalized Ergun equation (26) differential vector operator operator mdlcatmg final minus mltlal or outlet minus inlet Euler number, defined by Eq (20) geometrical slmllanty number Reynolds number, defined by Eq (5) Reynolds number, defined m Table 1 generalized Reynolds number, defined by Eq ( 13) generalized Reynolds number, defined m Table 1 generalized Reynolds number, defined m Table 1 generalized Reynolds number, defined m Table 1

Flow of Stokesian fluids through granular beds REFERENCES

[12] CHRISTOPHER

[l] BIRD R B , STEWART W E and LIGHTFOOT J? N , Transport Phenomena, pp 149-150 Wdey, A Media,

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Toronto Press (1960) [3] DE WIEST R J M , Flow Through Porous Medta, pp l-50 Academic Press, New York (1969) [4] GLUSHCHENKO A A, Some Three-dlmenslonal [5]

[6] [7] [8] [9] [lo] [1 I]

Problems

of

Theory

of

Fdtratlon,

Kiev

Umverslty Pub1 , Kiev 1970 (Russ ) FOSTER W R, McMILLAN J M and ODEH A S , Sot Petrol Engrs Jll967 333 BLAKE F E Trans Am Instr Chem Engrs 1922 14415 KOZENY J , Sltzber Akad WISS Wlen, Math naturw Klasse 1927 136271 CARMAN P C , Trans Instn Chem Engrs (London) 1937 15 150 ERGUN S , Chem Engng Progr 1952 48 89 FREDRICKSON A G , Prrnclples and Applrcatlons of Rheology, p 65 Prentice-Hall, Englewood Chffs, New Jersey 1964 REINER M , Deformation, Strarn and Flow, p 250 H K Lewis, London 1960

R H and MIDDLEMAN

Ind Engng Chem

[13] GAITONDE Engng

New York (1962) [2] SCHEIDEGGER Through Porous

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N Y and MIDDLEMAN

Chem

S , Ind

Fundls 1967 6 145

[14] MARSHALL Engng

S,

Fundls 1965 4 422

R .I and METZNER

A B, Ind

Fundls 1967 6 393

[15] SISKOVIC N , GREGORY D R and GRISKEY RG,AIChEJ/l97117281 [ 161 GREGORY D R and GRISKEY R G ,A I Ch E _I/ 1967 13 122 [17] YUY H,WENC Y andBAILIER Chem Engng 1968 46,149 [18] METZNER A B and REED J C ,

C,Can

J

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