The. Chenucal Engmeermg Journal 60 (1995) 5542

THE CHEMICAL ENf;;EW;pG

A theoretical study of transient cross-flow filtration using force balance analysis

Wen Wang *, Xiaodong Jia, Graham A. Davies Chemxal Engmeerrng Department, UMIST, PO Box 88, Manchester M60 1QD. UK

Received 10 March 1994, accepted 14 December 1994

Abstract

A model of transient cross-flow filtration m a permeable cyhndncal tube based on a force balance analysis IS presented Particle layer (or cake) formation on the tube IS simulated A slip boundary condltlon for cross-flow IS apphed at the interface between the flow and the porous surface of the tube, which represents the hydrodynamic effect of the porosity of the tube wall and cake on the cross-flow For a given particle Size chstnbution m the fluid, the mean radius of particles at different depths m the cake and at different posltlons along the tube IS calculated It IS found that when a slip boundary conchtlon IS applied, the cntical radius of particles IS reduced, resultmg m a slower growth of cake on the surface of the tube In the model, the ttuckness of the cake can erther increase or decrease along the axial direction of the tube depending on the condltlons, pressures, permeablhty of the tube etc This differs from results of concentration polanzahon models, which predtct a continuously increasing particle layer

Keywords Cross-flow filtration, Force balance analysis, Particle size chstnbution, Shp boundary con&on

1. Introduction

Cross-flow filtration is commonly used in separation proc- esses such as the recovery of metal precipitates from waste water, the removal of fine particles from fluid m electronic industry, the separation of radioactive particles from coolmg water m the nuclear industry, the harvesting of bloreactlon products from fermentation broths and the separation of red cells from blood when collectmg plasma from donors Com- pared with dead-end filtration, cross-flow filtration has the advantages of a thinner particle layer, a higher filtration rate and a longer operation time before back flushing 1s required Most theoretical models of cross-flow filtration have assumed a constant mass transfer at the permeable wall along the axial direction (see the review by Belfort and Nagata [l] ) or a steady state filtration which falls to address the effect of cake bmld-up m the tube. [ 2-41

Recent works have addressed the problem of transient fil- tration and modelled cake formatlon along a porous wall [5,6] The concentration polanzatlon model [7,8] was applied by Romero and Davis [ 51 and shear-induced dlffu- slvity was used to replace the Browman dlffuslvlty of col-

* Present address Centre for Biolog& and Medical Systems and Phys- lology and Biophysics Department, Impenal College, London SW7 2BY, UK

0923-0467/95/$09 50 0 1995 Elsevier Science S A All nghts reserved SSDl0923-0467(95)02989-3

loldal particles [ 3,9] In this approach, particles of different sizes m the flow contribute sirndarly to the cake build-up However, Fischer and Raasch [ 101 found that most of the particles deposited came from the finer particles m the mltlal slurry and that higher cross-flow velocities led to finer particle layers They further suggested that a selective cut diameter of the deposited particles exists and proposed a model to descnbe this cut diameter by analysmg forces exerted on a single spherical particle [ 111 In their approach, a particle can be prevented from rolling along the wall when it comes mto contact with the porous surface A torque balance deter- mines whether deposltlon of the particle occurs Most pre- vious models using the force balance analysis have concentrated on the relatlonshlp between the cut diameter and the surface shear stress under a given flow condlhon [ 1 l- 151 They studled the problem locally focusmg at a single point on the membrane and failed to address the particle layer formation m the whole flow field In the paper by Lu et al [ 61, a numerical analysis was caed out to model the tran- sient cake build-up on a porous plate m a channel flow They predicted the variation m flow and cake formation at different posltlons m the dlrectlon of flow However, m their study, as m most of previous studies, an unreahsttc no-shp boundary condltlon was applied at the porous surface

Studies on shear flow over a porous layer show that a shp velocity exists at the porous interface [ 16-181 which has a

56 W Wang et al /The Chenucal Engrneerrng Journal 60 (1995) 55-62

strong local influence on the flow In the modellmg of particle

deposltlon, boundary condltlons will have important effects on the motion of particles when they come close to the surface of the porous tube

In this paper, we consider cross-flow filtration m a cylm- drlcal porous tube usmg a force balance analysis Lammar flow 1s assumed to simplify the derivation and results The pressure outside the tube 1s assumed to be constant In the problem that we consider, the permeate velocity 1s much smaller than the cross-flow velocity, so the pressure vanatlon and velocity m the radial direction of the tube can be neglected The flow m the tube 1s assumed to be axially symmetric and quasi-steady and variables are updated after each time step A slip boundary condltlon for the cross-flow 1s applied at the porous surface which links the velocity at the porous interface to the local shear rate The roughness parameter of the porous surface 1s used, which 1s a measure of the microstructure of the tube surface and has a dimension of length Particles m the solution are of mlcrometres m radu and neutrally buoyant In this paper, we focus our attention on the simulation of the transient, full field cake formation m the tube, and especially the influence of the slip boundary condltlon Two forces are assumed to dominate the motion of a particle near the porous surface forces m the axial and radial dlrectlons of the tube due to local flows m those dlrec- tlons, with mterpartlcle forces negligible compared with these forces The critical radius 1s calculated Particles of smaller radu m the permeate flux deposit on the tube surface and contribute to the build-up of the cake Particles of larger size form a moving particle rich layer near the surface of the tube For a given particle size distribution m the solution, the mean particle size m the cake 1s calculated numerically The Car- man-Kozeny relationship 1s applied to calculate the resls- tance of the cake to the permeate flow

2. Basic equations of flow in a porous tube

Flow containing small particles 1s driven by a pressure difference over a permeable cylmdrlcal tube As shown m Fig 1, the length of the tube 1s L, the mner radius R,, and the

PO

P*

Rg 1 Geometry of the porous tube L IS the length of the tube, R,, IS the

mner radms of the clean tube, E IS the tube wall thckness. x and r are the ax~al and radial coordmates of the tube and pO, p, and p2 are pressures at the outslde, the inlet and the outlet of the tube

porous wall thickness E Pressures at the inlet, the outlet and

the outside of the tube are pl, pz and p. We assume that Ro/ L IS small (about 10e3) and the Reynolds number 1s of the order of 103, so that entrance length 1s negligible compared with L The permeate flow velocity 1s everywhere small com- pared with the axial velocity, so the velocity and pressure gradients m the radial direction of the tube can be neglected

Theoretical and experimental work on flow over a rigid porous medium suggests a boundary condltlon at the surface of dlscontmurty that matches the flow m the fluid to the fully developed Darcy flow mslde the porous matenal [ 16,19-2 1 ]

Recent work [ 221 on a shear flow over a porous layer applied the Brmkman equation for the flow mslde the porous medium [ 231 and suggested a slip boundary condltlon at the porous interface to represent the hydrodynamic influences of the porous layer The slip boundary condltlon links the fluid velocity to tbe local shear rate by aconstant, 1 e u = - k&Jar The slip parameter

k= 42p l/2

-tanh K E [(3 1 (‘%)1’2 IJ

where E 1s the thickness of the porous layer, 4 1s its porosity, p 1s the fluid vlsconty, pa 1s the apparent vlscoslty m the porous layer and K IS the drag coefficient of the porous layer

For micron-sized particles, the slip velocity could have an important influence on their deposltlon when they come to near contact with the porous surface In the followmg anal- ysis, a slrp boundary condltlon 1s applied at the tube wall Its influence on the critical radius and the cake growth are dls- cussed m Section 5 where comparisons are made between different slip parameters The axial velocity m the tube with the above slip boundary condltlon 1s

u= e (R2-?+2Rk) (1)

where G = - ?$J/&x 1s the pressure gradlent, p IS the pressure m the tube, x is the axial position of the tube and R IS the

distance from the axis of the tube to the surface of the cake The volume flow rate can be determined by integrating u over the cross-sectional area of the tube

(2)

Darcy’s law is apphed for the permeate flow across the porous wall of the tube

P-P0 v= - a

where (Y = cu,,, + (Y, is the sum of the drag coefficients from the tube wall and the cake The drag coefficient cu, = p/K of the tube wall is taken as a constant, where K IS the wall permeability [ 241 cr, increases with t as the cake builds up and vanes with x

W Wang et al /The Chemical Engmeenng Journal 60 (1995) 5.5-62 51

3. Analytical solutions for a clean tube

For a clean tube, R and (Y are independent of time and

position (R = R, and (Y = cu,) and analytlcal solutions can be obtained From mass conservation m the tube, we have

where q = 21rRv 1s the permeate flux per unit length along the axis of the tube From Eqs (2) and (4)

a2P -= a.3 RzntiT 4k) (P-PO) (5)

with the followmg boundary condltlons at the inlet and outlet

of the tube

x=0, p=p*

x=L, p=p2

p can be expressed as

P=Po+ (PI -Po) smh[@“(L-x)]

smh( p”‘L)

+ (P2 -PO> smh( p”‘x)

smh( @“L.)

where the intermediate parameter

P= 16~

R2a(R+4k)

The Reynolds number based on the size of particles 1s assumed to be much less than 1 so a simple linear relationship exists between the drag force exerted on a spherical particle and the relative flow velocity [ 25,261 In the axial direction, the drag force 1s

F,=K,61rpuu, (7)

where a 1s the radms of the particle, U, 1s the axial velocity at the centre of the sphere and K, is the tube wall correction factor K, = 1 1s the Stokes relatlonshlp which corresponds to a sphere moving m a boundless fluid In the presence of a porous wall, K, > 1 and its value depends both on the distance between the sphere and the surface and on the porosity of the wall Slmllarly, the drag force on the particle m the radial direction 1s

F, = K,6npa u

with K, the wall correction factor m r direction

(8)

The roughness parameter 6 of the porous surface 1s a meas- ure of the average roughness of the tube surface and has a dimension of length As shown m Fig 2, the torque exerted by F, and F, on the particle about the pomt of contact A 1s

T=F,(2u6-62)1’2-FX(u-S) (9)

The critical radius u*(x) 1s reached when T=O It can be derived from the above equation that

Rg 2 Forces on a spherical particle on a porous surface and the torque about the pomt of contact A a IS the radius of the parttcle, 6 1s the roughness parameter of the porous surface and F, and F, are drag forces parallel and perpendicular to the porous surface

‘*=

1

‘+ [1+(FX/F,)2]“2-1 > ’ (10)

which reveals a simple linear relationship between S and u* The axial velocity at the centre of a sphere of radius a* 1s

u,= G [2R(u*+k) -.*‘I 4P

(11)

Since a* lR GC 1, the higher-order term a*’ 1s negligible and, usmgEqs (7), (8) and (9), wehave

u*4+2(k-6)u*3+(k2-4k~+62)u*2

+26(k&kZ-~2)u*+(k2+~2)S2=0 (12)

where

(= 2 2PW;Po)

X

which can be solved for a*

4. Modelling the cake build-up

In cross-flow filtration, as particles deposit onto the surface of the tube, the permeablhty and the effective radius of the tube decrease In the followmg analysis the filtration process 1s simulated numerically m small discrete time steps At dur- mg which variables are assumed to be time independent Their values are updated at the end of each time step

4 I Pressure dmtrrbutron tn the tube

The length of the tube 1s equally divided mto N sections with the length interval Ax= L/N From Eq (4), the con- servation of mass gives

a2P ax2+

R+3k aRap --- R(R/4+k) ax ax Rza($4+k) (P-PO) =’

(13)

58 W Wang et al /The Chemrcal Engmeenng Journal 60 (1995) 55-62

wherep, R and (Y are variables of x From the finite difference

equation of Eq (13), the pressure p(,) at point 1 can be expressed as

1

“‘) = 2 + w<,~Ax 2

[ P(r+l, +P(r-l)

+ (R,,,+3k)(R,,+,,-R,,-,,)

Rw W,,, + 4k)

x (P<,+,) -pw,,) +wwAx*~o 1 (14)

where I = 2, 3,4, , N, and

4P W

(‘) = R2<,,(Rc,,/4 +k)cq,,

pcrj IS calculated iteratively from z = 2 to z = N with boundary condltlonsp(,, =pl andp(,+,, =p2 At each tlmestep At the calculation of the pressure dlstrlbutlon terminates when CpZ 1 t?p(,) 1 lp,~ 10P5, where t3pcr) 1s the difference of pcl, between two iterations

The surface roughness parameter 6 is assumed to be mde- pendent of time and posltlon to simplify the problem Follow- mg the same denvatlons m Section 2, the critical radius U*(I) can be calculated Fluid coming through the tube wall IS assumed to be particle free so that particles of radu smaller than a* m the permeate flux deposit on the surface of the tube and contibute to the growth of the cake Particles of larger radu m the permeate flux are separated at the boundary and carried away by the cross-flow

4 2 Thickness of the cake

The thickness h of the cake takes different expressions depending on the way that a*(x) varies with x As shown m Fig 3, m region 1 where a*(x) decreases with x, particles m the moving particle rich layer above the cake do not contnbute to the cake build-up downstream, because particles m the moving layer have larger radu than the local a* The particle size dlstrlbutlon m the permeate flow 1s effectively not changed by this moving particle rich layer Assuming the volume fraction of particles m the solution 1s C and particles have a volume-weighted size dlstnbutlon f( a), from volume conservation the growth rate of the cake m region 1 can be written as

ah - --L cfv

at- 1 - 4c

where 4,~ the porosity of the cake and

(15)

a*

f= If(a) da -0

In the followmg calculations, 4c 1s assumed to be a constant to slmphfy solutions and A = 10 1s used

IS the fraction of particles whose radn are smaller than a* Like h, r,,, takes different expressions dependmg on a*(x) When a*(x) increases with x, as m region 2 of Fig 3, When a*(x) decreases with x, the formula 1s relatively

some of the particles m the moving particle rich layer have simple

2 n E

-3 = b

t

X’ X

atal dlrectlon of the tube

Fig 3 Diagram showmg the possible vanauon in the cntxal radms a* with

x along the tube When a* decreases with the distance along the tube as m

region 1, the moving particle nch layer on top of the cake does not contnbute

to the cake bmld-up When a* increases along the tube as m region 2,

particles whose radn are smaller than the local a* m the moving ptircle

nch layer contnbute to the cake build-up

smaller radu than the local a* and contribute to the cake build-up at downstream posltlons Since Pe ZS= 1, the moving particle-rich layer 1s thm and close to the surface of the cake The contrlbutlon from particles m the moving layer above the cake gives an additional term m the volume conservation equation and the growth rate of the cake can be wntten as

ah - -&(fv+ k ?’ f(a)dnjudx) at- (16) a’(~-Ax) x’

4 3 Drag coeficlent LX~ of the cake

For a packing of uniform coefficient can be expressed tlonshlp

180p(l-cp)*h cu,=

4q3a2

spheres of radius a, the drag by the Carman-Kozeny rela-

(17)

where cp IS the porosity of the packmg, d 1s the radius of particles and h the depth of the packmg In the followmg analysis, we apply the Carman-Kozeny relationship to cal- culate the increment A ac of the cake drag coefficient m each time interval At, but with two modlficatlons, 1 e the mean radius r,,, of particles m Ah (the increment of cake thickness m At) 1s used to replace a and a correction parameter A 1s used to represent the effect of non-uniform particle packmg

Aa

c =A 1801*(1-4c)*Ah

4&r’, (18)

W Wang et al /The Chemical Engrneerrng Journal 60 (1995) 55-62 59

(19)

When a*(x) increases with x, r, has an addltlonal term in ‘he formula owmg to the deposlt~on of particles from the novmg particle rich layer

5. Results and discussion

For a sphere on a sohd Impermeable plane wall m a uniform velocity gradient, the axial wall correction factor K, given by O’Nelll [2.5] is about 1 7 For a porous wall, the correction factor could be smaller because of the slip velocity at the porous interface but m the followmg calculations we ~111 use K,= 1 7 The wall correction factor K, given by Chang and Acrlvos [26] 1s

K = 1(x1-+) r 43

(21)

where 4 1s the porosity of the wall In the following calcu-

lations, we take $ = 0 3 to calculate K, In the calculations, the length Z, of the tube is 1 m and 1s

used to normalize the other length parameters The pressure pa outside the tube 1s lo5 Pa and 1s used to normalize the other pressures It 1s assumed that the clean tube radius R,/ L= 0 001 and thetubewallthlckness E/L= 0 001 Therough- ness parameter S/L for the porous tube is assumed to be 3 X 1O-6 and the tube wall drag coefficient (Y, = 7 X lo7 N s mV3 The vlscoslty f~ of the fluid 1s taken to be 10e3 N s m-’ The slip parameter k/L for the axial flow 1s 10e5 con- sldenng that pl~~ = 1 0 [ 23,27,28] We assume the volume fraction C of particles m the solution 1s 1 0% and the particle size has a normal dlstrlbutlon with a mean of 6 pm and a standard devlatlon of 2 pm

Fig 4 shows the cross-flow flux Q and the permeate flux q per unit length at different positIons along the tube when t = 0, 1, 2, ,6 s In the calculation, plIpo = 2 0 and p2/ pa = 1 5 In Fig 4(a), Q IS seen to decrease from the inlet to the outlet of the tube At t = 0, the clean tube has the maximum Q along most of the tube from the inlet Q decreases with time because the cake build-up decreases both the permea- b&y and radius of the tube There 1s a small region near the outlet of the tube where Q increases with t In Fig 4 (b) , q IS seen to decrease umformly with x The most rapid variation in q with time occurs at t = 0 when the tube 1s clean

The critical radius a*(x) IS plotted in Fig 5 At I = 0, the clean tube has the maximum value of a* As the cake builds up with time, a* becomes smaller Whenp, Ipa = 2 5 andp,l

I I I I I

00 02 04 06 06 10

(a) x/L

x lo6

I I I I 1

00 02 04 06 06 10

(b) X/L

Fig 4 Volume flow rate at different posmons along the tube (p,/pO= 2 0 and pJp,, = 1 5, curves from top to bottom are results at t = 0, 1,2, ,6 s) (a) cross-flow flow rate Q, (b) permeate flux q per utut length

pa = 2 0 (Fig 5 (a) ) , a* IS seen to increase with x and, as the cake builds up, it becomes less dependent on x Under dlf- ferent flow conditions, a* can vary differently with x When lower pressures are applied at both ends of the tube, p,/ pa= 15 and p21po= 1 1 (Fig 5(b)), the values of u* are reduced and now decrease with x

Fig 6 shows the thickness h of the cake at t= 1,2, ,6 s At the beginning of the filtration, h increases rapidly with t because of the high permeability of the tube The cake growth rate decreases as the cake builds up For p,Ipo = 2 0 and pressure p21po = 1 8 (Fig 6 (a) ) , h increases with x from the inlet to the outlet of the tube For the same inlet pressure but a smaller outlet pressure, p2/po= 1 6, so that the pressure difference (pl - p2) /pa increases from 0 2 to 0 4 (Fig 6 (b) ) , h decreases with x and a thinner cake forms on the surface of the tube

60 W Wang et al /The Chetmcal Engmeenn& Journal 60 (1995) 55-62

______ _- -. - _- _- - * ________----- 5

/

_________---- ____________------- -_---_ ---__- ----- ______--

______- _______ ____.________________..-.-- __________ _________ ___-___

G

4 1 I I I I 1

00 02 04 06 08 10

(a) X/L

55

1

30 1 I I I I I

00 02 04 06 06 10

(b) x/L

Rg 5 Cntrcal radms a* at different posmons m the tube (curves from top to bottomrepresenta* at t=O, 1,2, ,6 s) (a) pI/p0=2 Oandp,/p,= 1 5,

(b) p,lpO= 1 5 andp,lp,= 1 1

The increment Ah of the cake thickness m each time step and the mean particle radius r, m Ah are both functions of time In Fig 7, r,,, 1s plotted against tat the inlet, the midpomt and the outlet of the tube when p1 lpo = 2 5 and p2/po = 2 0 As the cake builds up, r,,, decreases rapidly with f This cor-

responds well to the experimental findmgs by Lu and Ju [ 131 that particles m the cake become finer from the tube wall to the top of the cake It 1s also seen that the difference m r,,,

between the inlet and outlet of the tube decreases with t A shp velocity at the porous boundary has strong effects

when particles come close to the surface of the tube In Fig 8, the effect of the slip parameter on the cntical radius a* 1s shown at the inlet, the nudpomt and the outlet of the tube under the same condltlons as m Ag 7 It is seen that a* 1s reduced when shp boundary conditions are applied The dlf- ference m a* between the inlet and the outlet of the tube becomes smaller with shp boundary conchtlons

60

1 6 _____________-_.____.__________

___________--_--------- 5

40 __--- _____--_----____________ ______----- 4

20- 2

t=1

I I I I I

00 02 04 06 06 10

(a) XIL

40-I

30 --___

I-

-w------_______ ---__ ----__. 6

----_ 2 -----____ ----s---- - _

----..-________ ----_ - _ t

----_ c 20- --------- 4 -------- ----___ -..

;---------- ----.

2

lo-

t=l

00 00 02 04 06 06 10

(b) X/L

Fig 6 Cake thxkness h at different positIons m the tube (curves from bottom to top represent h at t= 1,2, ,6 s) (a) p,/p,,=2 0 andp,lp,= 1 8, (b) p,lp,,=2 0 andp,/p,,= 1 6

In Fig 9, the effect of slip boundary condltlons on the growth of the cake 1s shown The same conditions as m Fig 8 are applied It 1s seen that, with the no-&p boundary con- &ion, h increases much faster with time When slip boundary condltlons are apphed, the cake forms more slowly on the surface of the tube Results from Figs 8 and 9 show effects of boundary con&ions on the modelhng of the cross-flow filtration

6. Conclusions

A full field simulation of transient cross-flow filtration 1s presented based on a force balance analysis Shp boundary conditions at the surface of the porous tube are applied which lmk the shp velocity at the boundary to the shear rate We find that a slip velocity reduces the cntical radms and leads to a more slowly growing cake m the tube Dependmg on the

W Wang et al /The Chemical Engrneerrng Journal 60 (1995) 55-62 61

61

5 2 t + k

31 0 1 2 3 4 5 6

t

Fig 7 The vanatlon m the mean partuzle radms r,,, m each mcrement Ah of the cake thickness with time (p, lpO = 2 5 and pz/p, = 2 0) A, the inlet of the tube, CI, the nudpomt of the tube, 0, the outlet of the tube

2 ; I I I I I I

0 1 2 3 4 5 6

t

Rg 8 Effect of the shp parameter on the crmcal radms a* (p,/p,= 2 5 and p21p0=20) ---, klL=O,--, klL=10-5, , k/L=5 x 1O-5, A, the inlet of the tube, Cl, the nudpomt of the tube, 0, the outlet of the tube

properties of the porous tube and the imposed flow condl- tlons, the cake thickness can either increase or decrease along the tube This differs from results of concentration polanza-

tlon models, where the cake thickness always increases with distance along the tube regardless of the flow condltlons and

the membrane properties The concentration polarlzatlon model works well m ultrafiltratton where particle sizes are small Its extension to mlcrofiltratlon may not be stralghtfor- ward because of the weak diffusive motion of particles The force balance analysis 1s effective m modellmg the separation

of micron-sized particles Interesting problems remam when there are both large and small particles m the fluid or when intermediate-sized particles are to be modelled In those clr- cumstances, convective and dlffuslve motions of the particles

2. t c

il 1 2 3 4 5 6

t

Fig 9 Effect of the shp parameter on the cake thickness h (p, lpO = 2 5 and pzlp0=20) ---, klL=O,--. klL=10-5, , k/L=5 x 10-5, A, the mlet of the tube, Cl. the mldpomt of the tube, 0, the outlet of the tube

are equally important and it would be mterestmg to bmld a

model which considers both effects The parameters used m the calculations, porosity of the

cake, the slip parameter at the boundary, wall correction fac- tors etc , are reasonable m their magnitudes, but they should not be taken as definitive values These calculations could be refined when more accurate values are avallable Neverthe-

less, this work provides a means to study transient full field microfiltration which considers the partlcle size dlstnbutlon m the fluid and shows the importance of proper boundary condltlons when modelhng cross-flow filtration

Appendix A: Nomenclature

a

a*

f” Fx, Fr G h k K

Km K, L

P PO Pl P2 Pe

4 Q r

radius of a particle critical radius volume fraction of particles m the solution volume-weighted particle size dlstrlbutlon m the solution drag forces on a particle m the x and r directions pressure gradient thickness of the cake slip parameter permeability of the tube wall wall correction factors m Eqs (7) and (8) length of the tube pressure pressure on the outside of the tube pressure at the mlet of the tube pressure at the outlet of the tube P&let number permeate flux per unit tube length cross-flow flow rate radial coordinate

62

r,

R

W Wang et al /The Chermcal Engmeenng Journal 60 (1995) 55-62

mean radius of particles m each increment of cake thickness distance from the axis of the tube to the surface of the cake inner radius of the tube Reynolds number time torque on a sphere axial velocity permeate flow velocity axial coordinate drag coefficient from the tube wall and the cake

surface roughness of the tube wall thickness of the tube wall drag coefficient m a porous medium correction parameter m Eq (18) fluid vlscoslty apparent vlscoslty m the porous tube wall porosity of the tube wall

porosity of the cake porosity of a uniform packing of spheres

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