permeability of complex porous media

ISSN 1061-933X, Colloid Journal, 2009, Vol. 71, No. 1, pp. 31–45. © Pleiades Publishing, Ltd., 2009.Original Russian Text © S.I. Vasin, A.N. Filippov, 2009, published in Kolloidnyi Zhurnal, 2009, Vol. 71, No. 1, pp. 32–46.

31

INTRODUCTION

The study of flows in porous media formed by par-ticles covered with porous layers is important forunderstanding both natural and technological pro-cesses. Examples of such processes are the flow of oiland subsoil water, the filtration of solutions throughmembranes, the sedimentation of colloidal dispersions,etc. [1]. These flows can be rather correctly describedwithin the framework of the cell method, which is oneof the most efficient in the study of concentrated dis-perse systems and porous membranes. The essence ofthe cell method consists of replacing a system of parti-cles randomly arranged in a concentrated dispersion bya periodic lattice composed of identical spheres con-fined within liquid spherical shells at which surfacespecial boundary conditions accounting for the influ-ence of adjacent particles are set.

The goal of this work is to review results of our andother studies of the flow of liquid in porous media per-formed within the framework of cell model for the setof solid spherical particles covered with porous layer.The cell approach makes it possible to calculate thehydrodynamic permeability of the lattice of such parti-cles modeling porous membrane at different boundaryconditions on the cell surface (Happel, Kuwabara,Kvashnin, and Cunningham (Mehta–Morse) models).

The presence of porous shell on particles enables usto characterize the membrane by additional internalstructural parameters. The necessity of introducing newparameters is related particularly to the fact that thestructure of initial membrane in the course of filtrationcan be substantially changed due to both the dissolutionof the surface of constituting grains and fibers (loosen-ing) [3] and the adsorption of polymers (contamina-tion) [4]. Note that, in some cases, the removal offormed porous layers requires the application of con-siderable efforts [3]. In the case surface loosening, thepermeability of membrane grows due to an increase in

its porosity, while, as a rule, the selectivity drops. Onthe contrary, in case of membrane contamination,porosity decreases, permeability drops, while as a rulethe selectivity increases. The presence of porous layerson surfaces leads to changes in the drag force betweensolid surface and liquid [3, 5–8], as well as to changesin the diffusion coefficients of colloidal particlesin membrane [8] and the permeability of membraneitself [9].

Hydrodynamic effects caused by the adsorption ofpolymers during the flow around the single sphericalparticle have been considered in [4] under the assump-tion of the smallness of characteristic thickness ofporous layer compared to the particle size. Specialcases of flow in porous media have been studied previ-ously by us [10–14]. The flow around completelyporous particle was considered in [10, 11] within theframework of the cell model, the algebraic expressionfor the permeability of the set of such particles (mem-brane) was derived, and different limiting cases werestudied. Hydrodynamic permeability of membranecomposed of solid spherical particles covered withporous layers has been calculated in [12, 14]. In [10–12, 14], the Cunningham (Mehta–Morse) boundarycondition was used for this purpose. The condition wasproposed for the first time by Cunningham [15] in1910. Later, Mehta and Morse [16] used Cunningham’sboundary condition; however, they made a mistake incalculations and did not obtain the correct limiting casefor the permeability of membrane composed of the setof completely impenetrable particles (see formula (71)below, which was first derived by Cunningham [15]).However, due to the known Happel and Brenner mono-graph [2], the names of Mehta and Morse were alsoaffixed to the Cunningham condition. Here, we will useboth of these titles.

Note also that, in [13], the problem of the motion ofsolid particle covered with porous layer in the infinite

Permeability of Complex Porous Media

S. I. Vasin and A. N. Filippov

Moscow State University of Food Production, Volokolamskoe sh. 11, Moscow, 125080 Russia

Received January 22, 2008

Absract

—Hydrodynamic permeability of membrane composed of a set of porous spherical particles with rigidimpenetrable cores is calculated. The cell method proposed by Happel and Brenner is used in calculations. Allknown boundary conditions on the cell surface, such as the Happel, Kuwabara, Kvashnin, and Cunningham(Mehta–Morse) models, are considered. The flow of liquid is described by the Brinkman equations. The prob-lem of flow around a single spherical particle covered with porous layer by the uniform flow of viscous incom-pressible liquid is solved. Theoretical and empirical results are compared. Different limiting cases for which thederived formulas lead to results known from published literature are considered.

DOI:

10.1134/S1061933X09010049

32

COLLOID JOURNAL

Vol. 71

No. 1

2009

VASIN, FILIPPOV

incompressible liquid was solved for the case of differ-ent viscosities of the Brinkman medium and pure liq-uid.

FORMUALTION OF THE PROBLEM

In accordance with the ideology of the cell method[2], we will simulate disperse system (or membrane) bythe periodic lattice composed of identical solid and

impenetrable spherical particles with radius cov-

ered with porous layer of thickness (Fig. 1). Weassume that particles are confined to spherical shells

with radius whose value is chosen so that the ratioof the volume of partially porous particle to the volumeof cell was equal to volume fraction

Ò

=

γ

3

of particlesin membrane (Fig. 1)

(1)

where

ε

is the porosity of medium and

= +

is thetotal radius of particle with porous layer.

Let us introduce a spherical system of coordinates

θ

, and

ϕ

with the origin in the center of particle and

the axis directed along the uniform flux

which is set at the cell boundary in case ofMehta–Morse model (Fig. 1).

The motion of liquid at small Reynolds numbers

(creep flow) outside the porous layer (

≤

≤

) willbe described by the Stokes equations

(2)

whereas, in the porous layer (

≤

≤

), it will bedescribed by Brinkman equations [17]

(3)

R,

δ

b,

1 ε– γ 3 a

b---⎝ ⎠

⎛ ⎞ 3

,= =

a R δ

r,

z U

U U=( ),

a r b

∇ po µo∆vo,=

∇.vo 0,= ⎭⎬⎫

R r a

∇ pi µi∆vi kvi,–=

∇.˜ vi 0,= ⎭⎬⎫

where the tilde denotes the dimensional quantities;indices

Ó

and

i

correspond to external liquid and liquid

in the porous layer, respectively;

and are corre-

sponding coefficients of liquid viscosity;

and

are pressures;

and are velocity vectors; and isthe Brinkman constant which is proportional to the spe-cific permeability of porous layer.

The use of the Brinkman equation to describe theflow of liquid inside the porous layer means that realporous shell is efficiently replaced by a liquid with the

coefficient of viscosity The rigidity of the shell istaken into account by introducing a friction force pro-portional to the local velocity of liquid acting on liquidflowing through the shell.

The problem of streamlining the particle coveredwith a porous layer by the uniform flow of viscous liq-uid was solved in [7]. As well as in our study, the flowin the porous layer was described using the Brinkmanequation; however, in contrast to our works [10–14], itwas suggested in [7] that viscosities of surrounding liq-

uid and Brinkman medium are equal (

=

). Thismakes it possible for the authors [7] to substantiallysimplify the solution of the problem. Masliyah et al. [7]explained this choice by the fact that, in initial Brink-man’s paper [18], these viscosities were assumed to beequal to achieve better correspondence between theo-retical predictions and experimental data on the perme-ability of the set of hard spheres. In addition, Lundgren[19] demonstrated that, upon the dilution of suspen-sion, viscosities of Brinkman medium and pure liquid

are getting closer (

). However, we did notfind any rigorous arguments in the published literaturethat were in favor of the equality or difference in viscos-ities inside and outside the porous layer. For example,Koplic et al. [20] assumed that aforementioned viscos-ities are not equal and, in case of high porosity,obtained effective viscosity

(4)

which decreases with an increase in the concentrationof solid phase

c

, whereas the Einstein formula for adilute suspension gives the effect different in sign

(5)

The dependence

(6)

was obtained using the differential method [12]. Here,

c

i

and

A

i

are parameters that are determined by theporosity and structure of the porous medium and areequal to unity in the first approximation.

The controversy between formulas (5), (6), and (4)is considered in [20] to be imaginary and is explainedby the fact that a hydromechanical system described by

µo, µi

po, pi

vo, vi k

µi.

µo µi

µo µi

µi/µo 1 c/2,–=

µi/µo 1 5c/2.+=

µi/µo 1

1 c/ci–( )2.5Ai

-------------------------------,=

U~ U

~

z~

R~

δ~

a~

b~ vr

o~

vθo~

θ

Fig. 1. Schematic representation of spherical cell with rigidparticle covered with porous layer.

COLLOID JOURNAL Vol. 71 No. 1 2009

PERMEABILITY OF COMPLEX POROUS MEDIA 33

formulas (5) and (6) is presented by spherical particlesthat move freely in a liquid, while, in [20], as well as inour work, rigid porous medium composed of particlesfixed in the space is considered. It is also indicated [22]that the flow of liquid is slowed down by the porous

skeleton and, hence, should be lower than i.e.,condition (4) is fulfilled.

Based on the aforementioned, in this work, we willstudy the general case of unequal viscosities in theporous layer and surrounding liquid and analyze whatresults follow from this difference.

Let us now set boundary conditions closing the sys-tem of differential equations (2)–(3). It is natural torequire the fulfillment of stick conditions

(7)

on the surface of rigid core of a particle.

At the = interface, we set the continuity condi-tion for the velocity, as well as for tangential and

normal stresses [20] as follows:

(8)

Physical substantiation of boundary conditions (8)is given in [20]. Note that other types of boundary con-ditions at the liquid–porous medium interface are usedin the aforementioned work [22], which are speciallysubstantiated and characterize the slip between thesemedia.

The problem of the formulation of boundary condi-

tions on the surface = of cell is worth special con-sideration. Four variants of these conditions are knownsuch as the Happel, Kuwabara, Kvashnin, and Cun-ningham (Mehta–Morse) models. In all four models, it

is assumed that radial velocity on the surface = ofcell is continuous as follows:

(9)

Let us also consider additional conditions that aretypical for each of indicated models.

The Happel model [2] suggests the absence of tan-

gential stresses on the surface = of cell

(10a)

In the Kuwabara model [23], the vorkxes on the sur-

face = of cell are considered to be equal to zero (theflow potentiality) as shown below:

(10b)

µi µo,

vi 0, r R= =

r aσrθ

σrr

vo vi,=

σrro σrr

i ,=

σrθo σrθ

i .=

r b

r b

v ro U θ.cos=

r b

σrθo 0.=

r b

rot vo( ) 0.=

The Kvashnin model [24] suggests the cell sym-metry

(10c)

In the Cunningham (Mehta–Morse) model [15], the

uniformity of flow on the surface = of cell is intro-duced

(10d)

There are no serious physical arguments in favor ofsome or other model in the published literature. More-over, for the case of flow in the flat slit (the limiting caseof particles with infinite radius), in the center of slit thatcorresponds to the cell surface, boundary conditionscorresponding to all aforementioned models are ful-filled [25]. In this work, we consider all four models.

METHOD OF THE SOLUTION

For convenience of analysis, let us pass to dimen-sionless operators, variables, and quantities

where = is the characteristic thickness of fil-

tration layer (the Brinkman radius).

Systems of governing equations (2) and (3) in thedimensionless forms are expressed as

(11)

(12)

v θo∂

r∂--------- 0, r b.= =

r b

v θo U θ.sin–=

ba---

1γ---, r

ra---, ∇ ∇ a, ∆⋅ ∆ a

2,= = = =

δ δa---, R

Ra--- 1 δ, v–

vU----;= = = =

ppp0-----, p0

U µo

a------------, m

µi

µo-----,= = =

s0a

Rb

-----, ss0

m--------,= =

Rbµo

k-----

po∇ ∆vo,=

∇ vo⋅ 0,=⎩⎨⎧

1 r1γ---≤ ≤⎝ ⎠

⎛ ⎞ ,

pi∇ m∆vi s02vi– ,=

∇ vi⋅ 0,=⎩⎨⎧

R r 1≤ ≤( ).

34


VASIN, FILIPPOV

In a spherical coordinate system, Eqs. (11), whichdescribe the flow of liquid outside the porous layer, arereduced to the following form:

(13)

(14)

(15)

Equations (12) describing the flow of liquid in theBrinkman layer in spherical coordinate system have theform

(16)

(17)

(18)

Boundary conditions (7)–(10) in dimensionlessvariables are written as follows:

(19)

(20)

(21)

po∂r∂

--------v

2 or∂

r2∂----------

1

r2----

v2 o

r∂θ2∂

---------- 2r---v r

o∂r∂

--------- θcot

r2-----------

v ro∂

θ∂---------+ + +=

–2

r2----

v θo∂

θ∂---------

2v ro

r2---------–

2 θcot

r2--------------v θ

o,–

1r--- po∂

θ∂--------

v2 o

θ∂r2∂

----------1

r2----

v2 o

θ∂θ2∂

---------- 2r---v θ

o∂r∂

--------- θcot

r2-----------

v θo∂

θ∂---------+ + +=

+2

r2----

v θo∂

θ∂---------

v θo

r2 θsin---------------,–

v ro∂

r∂---------

1r---v θ

o∂θ∂

---------2v r

o

r---------

θcotr

-----------v θo+ + + 0.=

pi∂r∂

------- mv

2 ir∂

r2∂----------

1

r2----

v2 i

r∂θ2∂

---------- 2r---v r

i∂r∂

-------- θcot

r2-----------

v ri∂

θ∂--------+ + +

⎝⎜⎛

=

–2

r2----

v θi∂

θ∂---------

2v ri

r2---------–

2 θcot

r2--------------v θ

i–⎠⎟⎞

s02v r

i ,–

1r--- pi∂

θ∂------- m

v2 i

θ∂r2∂

----------1

r2----

v2 i

θ∂θ2∂

---------- 2r---v θ

i∂r∂

--------- θcot

r2-----------

v θi∂

θ∂---------+ + +

⎝⎜⎛

=

+2

r2----

v θi∂

θ∂---------

v θi

r2 θsin---------------–

⎠⎟⎞

s02v θ

i ,–

v ri∂

r∂--------

1r---v θ

i∂θ∂

---------2v r

i

r---------

θcotr

-----------v θi+ + + 0.=

v ri 0, v θ

i 0 at r R;= = =

v ro

v ri , v θ

ov θ

i ,= =

po 2v r

o∂r∂

---------+– pi 2mv r

i∂r∂

--------,+–=

1r---v r

o∂θ∂

---------v θ

o∂r∂

---------v θ

o

r------–+ m

1r---v r

i∂θ∂

--------v θ

i∂r∂

---------v θ

i

r------–+⎝ ⎠

⎛ ⎞=⎭⎪⎪⎪⎬⎪⎪⎪⎫

at r 1;=

v ro θ, at rcos 1/γ ;= =

(22‡)

(22b)

(22c)

(22d)

Taking into account the axial symmetry of a prob-lem, we will seek solutions to Eqs. (13)–(15) and (16)–(18) in the form of first-order spherical harmonics [26]

(23)

(24)

Substituting expressions (23) and (24) into Eqs.(13)–(15) and (16)–(18), respectively, we arrive at thesystem of ordinary differential equations for findingunknown functions fo(r), Φo(r), ψo(r), f i(r), Φi(r), andψi(r):

(25)

(26)

(27)

(28)

(29)

and

(30)

Let us first find the solution to Eqs. (25)–(27). It fol-lows from Eq. (27) that

(31)

1r---v r

o∂θ∂

---------v θ

o∂r∂

---------v θ

o

r------–+ 0, at r 1/γ ;= =

1r---–v r

o∂θ∂

---------v θ

o∂r∂

---------v θ

o

r------+ + 0, at r 1/γ ;= =

v θo∂

r∂--------- 0, at r 1/γ ;= =

v θo θ, at rsin– 1/γ .= =

v ro f o r( ) θ, v θ

ocos Φo r( ) θ,sin= =

po ψo r( ) θ,cos=

v ri f i r( ) θ, v θ

icos Φi r( ) θ,sin= =

pi mψi r( ) θ.cos=

d2 f o

dr2-----------

2r---d f o

dr-------- 4 f o Φo+( )

r2--------------------------–+

dψo

dr---------,=

d2Φo

dr2------------

2r---dΦo

dr---------- 2 f o Φo+( )

r2--------------------------–+ ψo

r------– ,=

d f o

dr-------- 2 f o Φo+( )

r--------------------------+ 0,=

d2 f i

dr2----------

2r---d f i

dr-------- 4 f i Φi+( )

r2------------------------- s2 f i––+

dψi

dr--------,=

d2Φi

dr2-----------

2r---dΦi

dr--------- 2 f i Φi+( )

r2------------------------- s2Φi––+ ψi

r-----– ,=

d f i

dr-------- 2 f i Φi+( )

r-------------------------+ 0.=

Φo r2---d f o

dr-------- f o.––=



Substituting expression (31) into Eq. (26), wearrive at

(32)

The substitution of expressions (31) and (32) intoEq. (25) yields the Euler equation for unknown func-tion fo(r)

(33)

The search for the solution to the last equation in theform fo = const rn leads to the following four differentvalues of exponent n: n1 = 0, n2 = 2, n3 = –1, and n4 =−3. Thus, we find expression for function fo

(34)

where b1, b2, b3, and b4 are arbitrary constants of inte-gration.

The substitution of expression (34) into Eqs. (31)and (32), makes it possible to determine functions Φo

and ψo

(35)

(36)

Substituting expressions (34)–(36) into Eqs. (23),we obtain, respectively,

(37)

(38)

(39)

Now, we find distributions of velocity and pressureinside the porous layer. Performing operations analo-gous to aforementioned, we arrive at

(40)

(41)

ψo 12---r2d3 f o

dr3----------- 3r

d2 f o

dr2----------- 3

d f o

dr--------.+ +=

r3d4 f o

dr4----------- 8r2d3 f o

dr3----------- 8r

d2 f o

dr2----------- 8

d f o

dr--------–+ + 0.=

f o r( )b1

r3-----

b2

r----- b3 b4r2,+ + +=

Φo b1

2r3-------

b2

2r-----– b3– 2b4r2,–=

ψo b2

r2----- 10b4r.+=

v ro b1

r3-----

b2

r----- b3 b4r2+ + +⎝ ⎠

⎛ ⎞ θ,cos=

v θo b1

2r3-------

b2

2r-----– b3– 2b4r2–⎝ ⎠

⎛ ⎞ θ,sin=

po b2

r2----- 10b4r+⎝ ⎠

⎛ ⎞ θ.cos=

Φi r2---d f i

dr-------- f i,––=

ψi 12---r2d3 f i

dr3---------- 3r

d2 f i

dr2---------- 2

d f i

dr-------- r2s2

2---------d f i

dr-------- rs2 f i.––+ +=

The substitution of expressions (40) and (41) intoEq. (28) leads to the equation for f i

(42)

Performing substitution u(r) ≡ we reduce the

order of the last equation

(43)

We will seek for a solution to Eq. (43) in the form

u = where y(r) is the new unknown function;

then, Eq. (43) can be rewritten as

r2y"' – 4ry" + 4y' – y'r2s2 = 0. (44)

Reducing once again the order of the last equationby the substitution z = y', we obtain the Bessel equationas follows:

r2z" – 4rz' + (4 – r2s2)z = 0. (45)

The solution of Eq. (45) is expressed as follows viathe modified first-order Bessel functions of 3/2 order,which are reduced to hyperbolic functions:

(46)

Performing previous substitutions in the reverseorder, we obtain

(47)

From Eqs. (40) and (41), we have

(48)

r3d4 f i

dr4---------- 8r2d3 f i

dr3---------- 8r s2r3–( )d2 f i

dr2----------+ +

– 8 4s2r2+( )d f i

dr-------- 0.=

d f i

dr--------,

r3d3u

dr3-------- 8r2d2u

dr2-------- 8r s2r3–( )du

dr------+ +

– 8 4s2r2+( )u 0.=

y r( )r4

----------,

z c1 r2 sr( )coshrs-- sr( )sinh–⎝ ⎠

⎛ ⎞=

+ c2 r2 sr( )sinhrs-- sr( )cosh–⎝ ⎠

⎛ ⎞ .

f i c1sr( )cosh

r2s2--------------------- sr( )sinh

r3s3--------------------–⎝ ⎠

⎛ ⎞=

+ c2sr( )sinh

r2s2-------------------- sr( )cosh

r3s3---------------------–⎝ ⎠

⎛ ⎞ c3

r3---- c4.+ +

Φi c1sr( )cosh

2r2s2--------------------- sr( )sinh

2r3s3-------------------- sr( )sinh

2rs--------------------––⎝ ⎠

⎛ ⎞=

+ c2sr( )sinh

2r2s2-------------------- sr( )cosh

2r3s3--------------------- sr( )cosh

2rs---------------------––⎝ ⎠

⎛ ⎞ c3

2r3------- c4,–+

36


VASIN, FILIPPOV

(49)

Substituting expressions (47)–(49) into Eqs. (24),we obtain, respectively

(50)

(51)

(52)

After the substitution of expressions (37)–(39) and(50)–(52) into boundary conditions (19)–(22), wederive the system of algebraic equations for determin-ing unknown constants bj and cj. Depending on the cho-sen model, we use one of conditions (22a), (22b), (22c),or (22d). In general, the solution of algebraic systemhas a cumbersome form and is not reported here.

ψi s2 c3

2r2------- rc4–⎝ ⎠

⎛ ⎞ .=

v ri c1

sr( )cosh

r2s2--------------------- sr( )sinh

r3s3--------------------–⎝ ⎠

⎛ ⎞⎩⎨⎧

=

+ c2sr( )sinh

r2s2-------------------- sr( )cosh

r3s3---------------------–⎝ ⎠

⎛ ⎞ c3

r3---- c4+ +

⎭⎬⎫

θ,cos

v θi c1

sr( )cosh

2r2s2--------------------- sr( )sinh

2r3s3-------------------- sr( )sinh

2rs--------------------––⎝ ⎠

⎛ ⎞⎩⎨⎧

=

+ c2sr( )sinh

2r2s2-------------------- sr( )cosh

2r3s3--------------------- sr( )cosh

2rs---------------------––⎝ ⎠

⎛ ⎞ c3

2r3------- c4–+ θ ,sin

pi ms2 c3

2r2------- rc4–⎝ ⎠

⎛ ⎞ θ.cos=

RESULTS AND DISCUSSION

An important characteristic of the problem under

consideration is force which acts from the side ofthe liquid on a particle [2]

(53)

where

b

2

is the constant determined from the solution ofsystem (19)–(22) and integration is performed withrespect to the external surface of porous layer.

The hydrodynamic permeability of membrane, which represents one of the elements of Onsager matrix[27], is determined as the ratio of the cell flux of liquid

to the cell pressure gradient [2, 10–12] asshown below:

(54)

where

=

is the cell volume.

Substituting expression (53) for the force into for-mula (54), let us present hydrodynamic permeability asfollows:

(55)

where

(56)

F,

F σrro θcos σrθ

o θsin–( ) sdS∫∫ 4πb2aµoU ,–= =

L11,

U F/V

L11U

F/V----------,=

V43---πb3

L111

3b2γ 3------------- a2

µo-----–

2

9γ 3-------- 1

Ω---- a2

µo----- L11

a2

µo-----,≡= =

L112

9γ 3-------- 1

Ω---- 1–

3γ 3b2

-------------,= =

0.80.60.40.20–6

–4

–2

0

2

4

6

8

ln(L11)

1−3 4

γ

Fig. 2. Dependences of the natural logarithm of the dimensionless hydrodynamic permeability of membrane L11 composed of par-ticles covered with porous layer on parameter γ at m = 0.8, s0 = 8, and δ = 0.5 for different models: (1) Happel, (2) Kvashnin,(3) Kuwabara, and (4) Cunningham (Mehta–Morse).



is the dimensionless hydrodynamic permeability of

membrane and Ω is the ratio of force to Stokes force

Ω = –2b2/3. (57)

Hydrodynamic permeability L11(δ, γ, m, s0) is a func-tion of four arguments. Parameters δ and γ characterizethe fraction of porous phase in the particle and the cell;parameters m and s0, internal structure of porous layer.We derived exact analytical expression for hydrody-namic permeability L11 in a general form; however, it iscumbersome and, hence, is not reported here.

Figure 2 shows the dependences of the natural loga-rithm of the dimensionless hydrodynamic permeabilityof membrane L11 on parameter γ for different cell mod-els at δ = 0.5, s0 = 8, and m = 0.8. The largest possiblevalue γ = 0.905 is achieved for identical cells in the caseof their rhombohedral packing; for simple cubic pack-ing, γ = 0.806. The permeability of the membrane isreduced with an increase in γ; i.e., with an increase inthe fraction of solid phase. In this case, the rate ofreduction in permeability is higher at small values of γ.Note that, at γ 0, the permeability of membraneincreases infinitely. It can be seen from the figure thatvalues of permeability calculated within the frameworkof the Happel, Kuwabara, and Kvashnin models nearlycoincide with one another. Small deviation (decrease)of permeability from values corresponding to the afore-mentioned models is observed for the Cunningham(Mehta–Morse) model in the case of less porous media.From a mathematical point of view, this fact can be

F

Fst 6πaµoU:=

explained as follows. The same terms in various combi-nations are used in Happel, Kuwabara, and Kvashninmodels, while the Cunningham (Mehta–Morse) condi-tion substantially differs from other models. From aphysical point of view, the Cunningham (Mehta–Morse) condition is the joining of flow in the cell witha uniform flow at its boundary, which is the most “hara”condition out of four considered types of boundary con-

0.2 0.4 0.6 0.8 1.0δ

4

321

01.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

ln(L11)

Fig. 3. Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11 com-posed of particles covered with porous layer on parameter δat m = 1, s0 = 5, and γ = 0.3 for different models: (1) Happel,(2) Kvashnin, (3) Kuwabara, and (4) Cunningham (Mehta–Morse).

50 100 150 200 250 300 350 m0–6.0

–5.5

–5.0

–4.5

–4.0

–3.5

ln(L11)

4

321

Fig. 4. Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11 com-posed of particles covered with porous layer on parameterm at γ = 0.8, s0 = 5, and δ = 0.5 for different models:(1) Happel, (2) Kvashnin, (3) Kuwabara, and (4) Cunning-ham (Mehta–Morse).

2 4 6 8 10 12 14 16 18 s00–5

–4

–3

–2

ln(L11)

321

4

Fig. 5. Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11 com-posed of particles covered with porous layer on parameters0 at m = 1, γ = 0.8, and δ = 0.5 for different models:(1) Happel, (2) Kvashnin, (3) Kuwabara, and (4) Cunning-ham (Mehta–Morse).

38


VASIN, FILIPPOV

ditions. This condition should lead to the increased dis-sipation of energy in a system and, hence, to a morenoticeable decrease in the hydrodynamic permeabilityof membrane.

Figure 3 illustrates the dependence of the naturallogarithm of the dimensionless permeability of mem-brane on the thickness of porous layer δ for differentmodels at γ = 0.3, s0 = 5, and m = 1. The dependencesfor all models are monotonically increasing functions.As in Fig.2, the permeability increases in the order ofCunningham (Mehta–Morse), Kuwabara, Kvashnin,and Happel models. Parameter s0 characterizes thedepth of the penetration of flow inside the porous layer;the larger the s0 value, the lower the depth. At large s0values, the flow occurs not in the entire porous layer butonly in the layer with thickness almost equals the valueof Brinkman radius. Therefore, when the thickness ofporous layer becomes larger than the Brinkman radius,the permeability ceases to depend on the radius of rigidcore that is depicted in the presence of horizontal partson the curves in Fig. 3.

Figure 4 demonstrates the effect of the ratio of vis-cosities m on the behavior of the natural logarithm ofthe dimensionless hydrodynamic permeability of mem-brane L11 for different models at fixed values of otherparameters γ = 0.8, s0 = 5, and δ = 0.5. It is seen that anincrease in the internal viscosity leads to a drop in themembrane permeability. The permeability first sharplydrops and then achieves its limiting value correspond-ing to high viscosity inside the porous layer when thereis almost no flow inside this layer.

Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11on parameter s0 for different models at γ = 0.8, m = 1,and δ = 0.5 are shown in Fig. 5. As was mentioned pre-viously, dimensionless parameter s0 characterizes thedegree of the porosity of a Brinkman medium; thelower the porosity, the larger the s0 values. At s0 ∞,the particle becomes absolutely impenetrable. The pat-tern of all curves in Fig. 5 demonstrates that, as s0increases, the permeability monotonically lowers to itslimiting value that is dependent on the fraction of solidphase γ3 (see formulas (68)–(71)); the same is observedfor curves shown in Fig. 4 at m ∞. As before, theCunningham (Mehta–Morse) model gives much lowervalues of permeability compared to other models.

It should be emphasized that the new structure of thecomplex medium (membrane) considered here has oneadditional degree of freedom compared to the set ofcompletely porous particles treated in [10–12], namely,internal rigidity R is equal to the ratio of the radius ofrigid core to the radius of entire particle. As comparedto the set of absolutely rigid particles considered in [2,16, 23, 24], the new structure of membrane is character-ized by three additional degrees of freedom such asinternal rigidity R, the resistance of porous layer to fil-tration, s0, and the dimensionless viscosity of liquidinside the porous layer, m. We believe that the introduc-

tion of additional degrees of freedom makes it possibleto describe the influence of the processes of internalcontamination or dissolution of membrane on thechange in its permeability in the course of filtration.

Let us consider the most significant limiting cases ofa problem.

Set of Porous Particles

At δ = 1, particles are completely porous. Expres-sions for permeability L11(γ, m, s) have the followingforms:

for the Happel model,

(58)

for the Kuwabara model,

(59)

for the Kvashnin model,

(60)

L11 2γ 6 3γ 5– 3γ 2–+( )m2s2ω3=

+ 18 γ 5 1–( ) ω1 2ω2–( ) 3m 2s2γ 6 ω1 2ω2–( )[–

+ γ 5 ω1s2 4ω2s2 4ω1 8ω2 4ω3 10+ +–+ +–( )

– 2s2γ ω1 2ω2–( ) ω1s2 4ω1 4ω2s2– 8ω2+–+

+ ω3 10 ) ] / 3γ 3ms2 6ω1 12ω2+–( ) γ 5 1–( )[––

+ mω3 2γ 5 3+( ) ] ,

L11 2 γ 6 5γ 5– 9γ 5–+( )m2s2ω3=

– 90 ω1 2ω2–( ) 3m 2s2γ 6 ω1 2ω2–( )[–

+ 5γ 3 ω1s2 4ω1 8ω2 2ω3 10+ +–+( )

– 12s2γ ω1 2ω2–( ) 5 ω( 1s2 4ω2s2 4ω1– 8ω2+ + +

+ ω3 10 ) ] / 45γ 3ms2 2ω1 4ω2– mω3+( ) ,––

L11 m2s2ω3 γ 1–( )3 8γ 3 15γ 2 21γ 16+ + +( )=

+ 18 3γ 5 8–( ) ω1 2ω2–( ) 3m 8s2γ 6 ω1 2ω2–( )[–

– 3γ 5 ω1s2 4ω2s2– 4ω1– 8ω2 4ω3– 10–+( )

+ 5γ 3 ω1s2 4ω1 8ω2 2ω3 10+ +–+( )

– 18s2γ ω1 2ω2–( ) 8 ω1s2 4ω2s2 4ω1––(+

+ 8ω2 ω3 10 ) ] / 18γ 3ms2––+

× 8 3γ 5–( )ω1 6γ 5 16–( )ω2 mω3 γ 5 4+( )+ +[ ] ,



and, for the Cunningham (Mehta–Morse) model,

(61)

Here, new designations

(62)

are introduced.

Note that the limiting transition s 0 correspond-ing to the absence of solid phase in the particle gives thefollowing results:

(63)

and the solution of a problem is transformed into theAdamar–Rybchinskii general solution for liquid sphere[2]. In this case, Ω 0 and the viscous drag is nulli-fied, as it should be in the case of completely mixableliquids.

At 0 (vanishing internal viscosity m 0),the Brinkman medium is transformed into the Darcy fil-tering medium (viscous term in the right-hand side ofEq. (3) vanishes); then, the force acting on singleporous particle is always lower than the Stokes force.From formulas (58)–(61), in this important limitingcase, we derive the following expressions for hydrody-namic permeability:


(64)


(65)

L11 m2s2ω3 γ 1–( )4 4γ 2 7γ 4+ +( )=

+ 18 3γ 5 2+( ) ω1 2ω2–( ) 3m 4s2γ 6 ω1 2ω2–( )[–

– 3γ 5 ω1s2 4ω2s2– 4ω1– 8ω2 4ω3– 10–+( )

– 5γ 3 ω1s2 4ω1 8ω2 2ω3 10+ +–+( )

+ 6s2γ ω1 2ω2–( ) 2 ω1s2 4ω2s2 4ω1––(–

+ 8ω2 ω3 10 ) ] / 18γ 3ms2––+

× 3γ 5 2+( )–( )ω1 6γ 5 4+( )ω2 mω3 γ 5 1–( )+ +[ ] .

ω1 30 s( )cosh

s4------------------- s( )sinh

s5------------------ 1

3s2-------––⎝ ⎠

⎛ ⎞ ,=

ω2152------– s( )cosh

s4-------------------

s( )sinh

s5------------------ 1 s2+( )– 2

3s2-------+⎝ ⎠

⎛ ⎞ ,=

ω3 90–s( )cosh

s4------------------- 1 s2

6----+⎝ ⎠

⎛ ⎞ s( )sinh

s5------------------ 1 s2

2----+⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞ .=

ω1s 0→lim ω2

s 0→lim 1; ω3

s 0→lim 3–= = =

µi

L111

3γ 3-------- 1

3γ 2--------

1

γ 3s02

---------,+–=

L111

3γ 3-------- 2

5γ 2-------- γ 3

15------

1

γ 3s02

---------,+ +–=

for the Kvashnin model,

(66)

and, for the Cunningham (Mehta–Morse) model,

(67)

Note that the Darcy law is successfully fulfilled inmany practical situations. This supports the fact that the

viscosity of the Brinkman medium in membrane is

always lower than that of filtering liquid, Figure 6 shows the dependences of the natural loga-

rithm of the dimensionless hydrodynamic permeabilityof membrane, L11, composed of completely porous par-ticles on parameter γ for different models at m = 1 ands0 = 5. The obtained curves demonstrate the drop in per-meability with an increase in the concentration of solidphase for all models. Note that, in this case, the differ-ences between all models are less substantial than in thegeneral case.

Set of Impenetrable Particles

At δ = 0, s ∞, or m ∞, we have the systemof solid particles. The expressions for permeabilityL11(γ) in different models have the following form:


(68)

which coincides with expression reported in [2];

L111

3γ 3-------- 4

9γ 2-------- 5

9γ 2 8 3γ 5–( )------------------------------

1

γ 3s02

---------,+ +–=

L111

3γ 3-------- 3 2γ 5+

γ 2 6 9γ 5+( )---------------------------

1

γ 3s02

---------.+–=

µi

µo.

L112γ 6 3γ 5 3γ– 2+ +–

6γ 8 9γ 3+-----------------------------------------------,=

0.80.60.40.20–4

–2

0

2

4

6

8

ln(L11)

1−3 4

γ

Fig. 6. Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11 com-posed of porous particles on parameter γ at m = 1 and s0 = 5for different models: (1) Happel, (2) Kvashnin, (3) Kuwa-bara, and (4) Cunningham (Mehta–Morse).

40


VASIN, FILIPPOV


(69)

which coincides with expression reported in [2];for the Kvashnin model,

(70)

which coincides with expression reported in [24];and, for the Cunningham (Mehta–Morse) model,

(71)

which was derived previously in [10–12, 14, 15].Semi-empirical Kozeny–Karman formula [2] gives

the following expression for the permeability of the setof absolutely hard spheres:

(72)

Dependences of the hydrodynamic permeability ofmembrane composed of a set of impenetrable particlescorresponding to different models on parameter γ areshown in Fig. 7. It is seen that curves calculated by thecell method are almost identical. In this case, as before,small deviation toward a decrease in permeability isobserved for the Cunningham (Mehta–Morse) model.Semi-empirical Kozeny–Karman formula leads to

L112 γ 6 5γ 3 9γ 5–+–( )–

45γ 3----------------------------------------------------,=

L11γ 1–( )3 8γ 3 15γ 2 21γ 16+ + +( )–

18γ 3 γ 5 4+( )---------------------------------------------------------------------------------,=

L111 γ–( )3 4γ 2 7γ 4+ +( )

18γ 3 γ 4 γ 3 γ 2 γ 1+ + + +( )--------------------------------------------------------------,=

L111 γ 3–( )3

45γ 6--------------------.=

higher values of permeability than all of the cell mod-els; moreover, differences in models is most substantialin case of low concentrations of solid phase anddecreases with an increase in concentration. For highlyconcentrated media (γ > 0.7), this difference nearly van-ishes.

THE PARTICLE IN UNIFORM FLOWAt γ = 0, from general expressions (37)–(39) and

(50)–(52), we obtain the solution of a problem ofstreamlining around the single particle covered withporous layer by uniform flow of viscous incompressibleliquid. Note that, in this case, all boundary conditions(10a), (10b), (10c), and (10d) are fulfilled; i.e., as waspreviously mentioned in [25], the difference betweenmodels vanishes. This effect can also be observed inFig. 2. Thus, we can conclude that these models identi-cally describe weakly concentrated media.

Changes in parameters of a system (radii of capillar-ies or colloidal particles) due to the presence of aporous layer are often expressed by introducing effec-tive thickness [28] determined as the thickness ofimpenetrable layer on the particle surface, whose pres-ence on the particle surfaces causes the same hydrody-namic effect as the porous layer. The value is anexperimentally measured parameter that acts to find thecharacteristic thickness of porous layer. For determin-ing this parameter, we can use the Stokes formula forthe hydrodynamic force acting on a rigid particle withradius moving at constant velocity

(73)After the adsorption of the polymer and the formationof a porous layer on the particle surface, the hydrody-namic force acting on particle increases. Hydrody-namic thickness is determined by changes in parti-cle size, which takes into account the aforementionedincrease in drag force

(74)Passing to dimensionless values, we obtain

Lh = Ω – R, (75)

where Lh = and R = are the dimensionlesshydrodynamic thickness and radius of a rigid particle,respectively; Ω = is the dimensionless force cal-

culated by formula (57); and is the Stokes force act-

ing on a particle with radius After rather cumbersome algebraic manipulations,

the expression for Ω acquires the following form,which coincides with that derived previously in [13]:

Lh

Lh

R U:

Fsolid 6πµoRU .=

Lh

F 6πµo R Lh+( )U .=

Lh/a, R/a

F/Fst

Fst

a R δ.+=

(76)Ω 1m 6sR 3sR2 3s+( ) sδ( )cosh– 3R s2R3 3– 2s2+ +( ) sδ( )sinh–[ ] P–

3sRm 1 m+( ) 3m 1 s2R2m–( ) sδ( )sinh sm 3 m 3R s2R3 2s2+ +( )+[ ] sδ( ) T+cosh–+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------,–=

0.80.60.40.20–8

–4

0

4

8

12

16

ln(L11)

1−3 4

γ

5

Fig. 7. Dependences of the natural logarithm of the dimen-sionless hydrodynamic permeability of membrane L11 com-posed of rigid particles on parameter γ for different models:(1) Happel, (2) Kvashnin, (3) Kuwabara, (4) Cunningham(Mehta–Morse), and (5) Kozeny–Karman.



where

(77)

(78)

At m = 1 (the case of equal viscosities = ),the expression for Ω forces acquires the simpleform

(79)

In this case, dimensionless hydrodynamic thickness Lh is equal to

(80)

In the general case, the expression for Lh can bederived from Eq. (76); however, due to its cumbersomeform, it is not reported here.

Note that, although the problem analogous to thatconsidered in this work for the case of equal viscositieswas solved in [7], Masliyah et al. did not succeed inderiving the force (79) in explicit form and they com-puted this force using the set of equations. However, inthis approach, we deal with the counting instability dis-closed in [7] and confirmed by us. Note that this prob-lem vanishes in calculations using final formulas (76)or (79).

INFLUENCE OF THE POROUS LAYER ON THE MOTION OF A SINGLE PARTICLE

Let us consider some limiting cases of the expres-sion for the force acting on single particle covered withporous layer (formula (76)).

(1) If the porous layer is absent (δ = 0, R = 1), then,according to formula (76), Ω = 1, i.e., we obtain impen-etrable particle with the acting force equals the Stokesforce.

(2) At s ∞, we deal with infinite resistance to fil-tration through the porous medium so that the porouslayer becomes impenetrable and, from formula (76),we also arrive at Ω = 1; i.e., the force acting on particleis equal to the Stokes force.

(3) In the absence of rigid core (R = 0, δ = 0), theparticle becomes completely porous and, from expres-sions (76) or (58)–(61), we obtain

(81)

where

(82)

Note that, in the case of equal viscosities (m = 1), theknown expression for force

(83)

which coincides with that derived previously in [1, 10–12, 18], follows from Eq. (81).

(4) At m ∞ (i.e., ∞, the case of infiniteinternal viscosity), we obtain a rigid impenetrable par-ticle and hence, as in aforementioned cases 2 and 3, wehave Ω = 1.

(5) At 0 (i.e., m 0, the case of infinites-imal internal viscosity), and from expressions (76) or(64)–(67) at γ = 0, we obtain

(84)

P 18 1 m–( ) 6Rs

------- 1 3

s2---- 3R

s2------- R3––+ sδ( )sinh+

⎩⎨⎧

=

–3 1 R2+( )

s----------------------- (sδ )cosh

⎭⎬⎫

,

T 12 1 m–( ) 3 1 m–( )2

---------------------⎩⎨⎧

=

× 3Rs

-------3R s2R3+

s2------------------------ sδ( ) 3R2

s--------- sδ( )cosh+sinh+–

+ mR4s----- 7s2 18+( )– 9R

2s------- 3sR3

2------------ s

3sr2

4----------–+ +⎝ ⎠

⎛ ⎞+

× sδ( ) 9R2

2--------- 1– 3R

4------- s2R3

4---------- s2

2----–––⎝ ⎠

⎛ ⎞+cosh sδ( )sinh

+92s----- sδ( )cosh

9 3s2+

2s2----------------- sδ( ) 9R

2s-------–sinh–

⎭⎬⎫

.

µi µo

Ω 16sR 3sR2 3s+( ) sδ( ) 3R s2R3 3– 2s2+ +( ) sδ( )sinh–cosh–

6sR 3 1 s2R2–( ) sδ( )sinh 3s 3Rs s3R3 2s3+ + +( ) sδ( )cosh–+--------------------------------------------------------------------------------------------------------------------------------------------------------.–=

Lh δ 6sR 3sR2 3s+( ) sδ( ) 3R s2R3 3– 2s2+ +( ) sδ( )sinh–cosh–

6sR 3 1 s2R2–( ) sδ( )sinh 3s 3Rs s3R3 2s3+ + +( ) sδ( )cosh–+--------------------------------------------------------------------------------------------------------------------------------------------------------.–=

Ω 1 9

2ms2------------ 1

2 1 m–( ) m/Q–--------------------------------------+ +

1–,=

Q 1 3

s2---- 1 stanh

s-------------–⎝ ⎠

⎛ ⎞ 1–

.–+=

Ω 1 stanhs

-------------–⎝ ⎠⎛ ⎞ 1– 3

2s2-------+

1–

,=

µi

µi

Ω2s0

2

3s02 9+

-----------------.=

42


VASIN, FILIPPOV

In this case, the Brinkman medium is transformedinto the Darcy medium (the first viscous term in theright-hand side of Eq. (3) vanishes) and, hence, the Ωvalue depends only on dimensionless parameter s0.

Figure 8 demonstrates the dependences of the ratioof force acting on rigid particle with porous layer to theStokes force acting on rigid impenetrable particle of thesame radius on the dimensionless thickness of porouslayer δ at different values of parameter s0. Since thepresence of porous layers always decreases the resis-tance to the motion of particle compared to rigid parti-cle of the same general radius, Ω ≤ 1. Curves in Fig. 8are monotonically decreasing functions. Furthermore, adecrease occurs more quickly at small values of param-eter s0 (large resistance to filtration). At δ = 0, theporous layer is absent and, naturally, Ω = 1. As theporous layer becomes thicker, the force acting on theparticle will drop due to the effect of filtration of liquidthroughout the overall porous layer until its thickness δremains smaller than that of the Brinkman layer. As

approaches , this effect will be exhibited to a lesser

extent. Upon a further increase in ( > ), the fil-tration only occurs in the part of the porous layer whose

thickness is on the order of ; i.e., in the ( ≤ ≤ +

– ) region, the liquid is almost quiescent. Conse-quently, the relative effect of porous layer lowers withan increase in its thickness and the force acting on par-ticle approaches the limiting value that is displayed inthe appearance of horizontal parts at curves 2–4 in Fig.8. Curve 1 is a descending curve within the entire rangeof thickness variations, since, in this case, the thickness

of the Brinkman layer > Note that an analogousresult was obtained in [7] at m = 1. In the same work,physical picture of the liquid flow in porous layer has

δRb

δ δ Rb

Rb R r R

δ Rb

Rb δ.

been studied thoroughly, and streamlines have beenconstructed at different values of dimensionless param-eters.

Figure 9 shows the dependences of the ratio offorces Ω on parameter s0. An increase in s0 implies a

decrease in ; i.e., the thinning of the part of porouslayer (the Brinkman layer) in which liquid is filtered (at

fixed and values). There is almost no liquid flowin the porous layer outside the Brinkman layer. There-fore, in accordance with limiting transition 2 (at s ∞), viscous drag tends to the Stokes force for rigid

impenetrable particle with radius + (Ω 1).Hence, the substitution of porous layer for rigid impen-etrable layer decreases the resistance to particle

motion; the smaller the resistance coefficient , the

Rb

δ R

R δ

k

00.4

0.6

0.8

1.0Ω

0.2 0.4 0.6 0.8 1.0δ

1

2

3

4

Fig. 8. Dependences of the Ω ratio on parameter δ at m = 1and s0: (1) 2, (2) 5, (3) 8, and (4) 11.

2 4 6 8 10 12 14 s0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Ω

1

2

3

4

Fig. 9. Dependences of the Ω ratio on parameter s0 at m = 3and δ: (1) 0.4, (2) 0.6, (3) 0.8, and (4) 1.

20 40 60 80 100m

0

0.6

0.8

1.0Ω

321

4

Fig. 10. Dependences of the ratio Ω on parameter m at s0 =3 and δ: (1) 0.1, (2) 0.37, (3) 0.63, and (4) 0.9.



more this occurs. A rise in the thickness of porous layer

at large values of its permeability (s0 ≤ 1) alsodecreases the force. Note that, in the case of an ideally

permeable porous layer, = 0 and s0 = 0, the limitingtransition leads to the Stokes force acting on rigid par-

ticle with radius : as follows: = Hence, at s0 = 0, we have

(85)

The character of changes in the ratio Ω of forceswith parameter m at different values of parameter δ ands0 = 3 is demonstrated in Fig. 10. All curves originatefrom the same point, which is determined by formula(84); in our case, this is Ω = 0.5. As the internal viscos-ity rises, the drag force also increases and, according tolimiting case 4, Ω 1 at m ∞. In this case, thelimiting value is achieved more rapidly at smaller val-ues of the thickness of porous layer δ.

COMPARISON OF THEORETICAL AND EXPERIMENTAL DATA

The sedimentation of spherical particles made ofnylon with glued flexible polyester fibers modeling dif-ferent porous structures were experimentally studied in

[7]. Nylon particles with radius = 0.318 cm to which

polyester fibers with radius = 0.065 and length

= 1.6 were glued were considered. The number offibers varied from one to 20, which corresponded tochanges in the porosity of medium covering the parti-cle. In the course of the experiment, the sedimentationvelocity of particles with glued fibers was measuredand the resistance force acting on composite particlewas calculated. Figure 11 demonstrates experimentalvalues of dimensionless force θ acting on compositeparticle depending on the fraction Ò of solid phase in the

porous layer. Dimensionless force θ = is equalto the ratio of force acting on composite particle with

general radius = + to the Stokes force acting on

a rigid particle with radius

For theoretical calculations of dimensionless forceθ, we employed formula θ = Ω/R, where Ω was calcu-lated by formulas (76)–(78). Force Ω(R, δ, s, m)depends on four dimensionless parameters introduced

previously, e.g., δ = R = = 1 – δ, m = s =

According to experimental data, = 2.6

and = 1.6 and, hence, R = 5/13 and δ = 8/13.

δ

k

R Fsolid 6πµRU .

Ω R

δ R+------------- 1

δ 1+------------.≡=

R

Rcyl R

δ R,

F/Fsolid

a R δR.

δa---,

Ra--- µi

µo-----,

a k

mµo--------------. a R,

δ R,

For the calculation of coefficient which is

inversely proportional to permeability ( = 1/ ) ofa porous medium composed of cylindrical fibersarranged perpendicular to the flow, we used the follow-

ing values calculated by the cell method in [2, 29]:

(86)

(87)

(88)

(89)

Coefficients of permeability were calculated by the

cell method using boundary conditions: , for Hap-

pel; , for Kuwabara; , for Kvashnin; and for Cunningham models.

For the coefficient of permeability of aporous medium composed of cylindrical fibersarranged along the flow, we used expression [2, 29]:

(90)

Coefficient m equals the ratio of viscosity in theporous layer to that of bulk liquid was calculated by oneon the formulas (4)–(6).

k,

L k L

L,

LHP1– c2 1 c2+( ) c( )ln–+

8c 1 c2+( )--------------------------------------------------------

Rcyl

µo---------,=

LKU3– 4c c2 c( )ln––+

16c-------------------------------------------------

Rcyl

µo---------,=

LKV2 c 1–( ) 0.5 3 c2+( ) c( )ln–

4c 1 c3+( )-----------------------------------------------------------------

Rcyl

µo---------,=

LCU1– c 0.5 1 c+( ) c( )ln–+

4c 1 c+( )------------------------------------------------------------

Rcyl

µo---------.=

LHP

LKU LKV LCU

Lalong

Lalong1

8c------ c2 4c– 2 c( ) 3+ln+( ) Rcyl

µo---------.–=

321 4

5

0.0080.0060.0040.002 c1.0

0

1.2

1.4

1.6

1.8

2.0

2.2θ

Fig. 11. Dependences of dimensionless force θ on the frac-tion Ò of solid phase in porous layer for different models:(1) Cunningham (Mehta–Morse), (2) Kuwabara, (3) Kvash-nin, and (4) Happel; (5) model with longitudinal arrange-ment of fibers, and ( ) experimental data [7].

44


VASIN, FILIPPOV

Figure 11 shows dependences θ(R, δ, s, m) on thefraction of solid phase at R = 5/13, δ = 8/13, m = 1, and

s = where coefficient was calculated byone on the formulas (86)–(90) depending on the chosenmodel of the porous medium. At zero concentration ofsolid phase in the porous layer, θ = 1; i.e., the force act-ing on particle is equal to the Stokes force (see Fig. 11).When the fraction of solid phase tends to unity, the θtends to its limiting value which, in the case underconsideration, is equal to 2.6 (see Fig. 11). This meansthat the force acting on composite particle is equal tothe Stokes force acting on rigid particle with radius

= 2.6 An analogous dependence for the Happelmodel was determined previously in [7].

For each model, root-mean-square deviation σ wascalculated as equal to

(91)

where θti and θei are theoretical and empirical depen-dences of θ(c) function and n is the number of experi-mental points. The root-mean-square deviation is equalto 0.1168 for Happel model, 0.1015 for Kuwabara,0.106 for Kvashnin, 0.0941 for the Cunningham model,and 0.2487 for the model with a longitudinal arrange-ment of fibers.

It follows from the cited rms deviations and depen-dences shown in Fig. 11 that the Cunningham modelbetter corresponds to experimental data than Happel,Kuwabara, and Kvashnin models. The model with lon-gitudinal arrangements of fibers with respect to thedirection of flow much worse describes experimentaldata.

When considering other models of porous medium,the character of dependences of permeability on thefraction of solid phase remains the same as in Fig. 11.

The values of rms deviations for different modelsare listed in the table. Columns with numbers 1–4 referto models with transversal arrangement of fibers withrespect to flow: Ha corresponds to the Happel model,

a/ mµoL, L

a/R

a R.

σ

θti θei–( )2

i 1=

n

∑n 1–

----------------------------------,=

Kv corresponds to Kvashnin, Ku corresponds to Kuwa-bara; and Cu corresponds to the Cunningham model.The fifth column contains the values of errors for themodel with longitudinal arrangements of fibers. Col-umns 6–13 correspond to the models with randomarrangements of fibers and columns six through nineare models containing transversal and longitudinalfibers in 1 : 1 ratio, while 10–13 columns refer to thesame mixed arrangements of fibers in 2 : 1 ratio. Flowswith constant (fourth line) and variable (5th–7th lines)viscosities were considered. The Cunningham model(σ = 0.0941) describing the flow in a porous mediumcomposed of cylindrical fibers arranged perpendicularto the flow is characterized by the minimal error amongall of considered models. Maximal error (σ = 0.2487)has the model with longitudinal arrangements of fibers.Note that the porous layer with a very low content ofsolid phase was considered and, therefore, the viscosityratios m calculated by formulas (4)–(6) slightly differfrom unity and, correspondingly, errors for models withdifferent viscosities nearly coincide with one another.As was mentioned above, it should be assumed that, forhighly porous media, viscosities of liquid in the bulkand the layer are equal.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundationfor Basic Research, project no. 06-03-90575-BNTS_a.

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