dusty gas flow through porous media

il NOgIIt- HOUAND

D u s t y G a s F l o w T h r o u g h P o r o u s M e d i a

M. H. Hamdan

Department of Mathematics Statistics and Computer Science The University of New Brunswick P.O. Box 5050 Saint John, New Brunswick, Canada E2L ~L5

and

K. D. Sawalha, Ph.D.

Department of Mechanical and Electrical Engineering Aston University Aston Triangle Birmingham, United Kingdom

Transmitted by Melvin Scott

ABSTRACT

A set of partial differential equations describing the motion of a dusty fluid in porous media are developed. The governing equations are derived using intrinsic volume averaging and are based on Saffman's dusty gas model. The effect of the porous microstructure on the flowing mixture is analyzed via the concept of representative unit cell (RUC), and distinction is made between flow in consolidated and granular porous media.

1. INTRODUCTION

Gas particulate flow, dusty fluid flow, and the flow of suspension through porous media have a host of practical applications, including the operation and design of filters and liquid-dust separators. Applications of the above types of flow in the environment is exhibited in solid-waste and pollutant dispersal and storage in the ground layers.

APPLIED MATHEMATICS AND COMPUTATION 75:59-73 (1996) © Elsevier Science Inc., 1996 0096-3003/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)00104-P

60 M.H. HAMDAN AND K. D. SAWALHA

The above and many other applications emphasize two fundamental aspects of the study of gas particulate flow in porous media: (i) The modelling phase, which emphasizes the need for models that accurately take into account the effect of the porous media quantities on the flow constituents, and the influence of the constituents on each other, and (ii) solution of boundary and initial value flow problems, which should accurately predict the flow patterns and the nature of the flow fields of the constituents involved.

In a previous work, a model describing the flow of a dusty gas in porous media was developed [1] and is based on the differential equations approach. It incorporates the factors affecting the gas-particulate mixture in the type of porous media where Brinkman's equation is applicable, and thus inertial effects were ignored under the assumption of creeping motion in a high- porosity medium. Models that give account to the effects of inertia but do not take into account the porous microstructure have also been derived [2].

In the current work, an attempt is made to derive differential equations governing the gas-particulate flow through porous media of variable porosity. The models are intended to parallel the existing single-phase models of flow in porous media that take into account the effect of the porous microstructure. Distinction is made in these models between consolidated and isotropic porous media through the concept of representative unit cell, which was introduced by Du Plessis and Masliyah [3]. In the developed models an attempt is made to analyze the dispersion terms.

2. PRELIMINARY CONSIDERATIONS

Consider the flow of a fluid in an isotropic porous medium, and assume that the fluid contains a small bulk volume of dust particles. This same flow in free-space is governed by Saffman's dusty gas equations [4], which are considered here for steady, incompressible fluid flow in the following form: For the dust-phase:

Continuity equation:

V. (Nv) = 0 (1)

Linear momentum equation:

V . ( w ) -- ( k / m ) ( u - v ) (2)

For the fluid-phase:

Continuity equation:

Linear momentum equation:

V . u = 0 (3)

pv . (uu) = - v p + , V ~ u + kN(v - u) (4)

1/

where u, v are the fluid- and dust-phase velocity vectors, respectively, N is the dust particle number density, k is the Stokes' coefficient of resistance, m is the mass of a dust particle, and p is the fluid pressure.

In the following analysis, we base our derivation on intrinsic volume- averaging of the above equations over a representative elementary volume (REV), the definition of which is given by Bachmat and Bear [5] as a control volume 7/" that contains fluid and porous matrix in the same proportion as the whole porous medium. In other words, it is a control volume whose porosity is the same as that of the whole porous medium, where porosity 4) is defined as the ratio of the pore volume to the bulk volume of the medium. Letting ~ be the effective pore volume, then the porosity is defined in terms of the REV as:

4) = % / 7 / . (2)

In terms of microscopic and macroscopic length scales, t and L, respectively, the REV is chosen such that

t ~ .~ ~ . ~ L ~. (6)

Figure 1 represents a sketch of a typical REV, consisting of a solid matrix volume and a pore space. The pore volume ~ contains the fluid and dust

Dusty Gas Flow Through Porous Media 61

FIG. 1. A typical representative elementary volume.

62 M. H. HAMDAN AND K. D. SAWALHA

phases. If ~ is the volume that contains the fluid-phase and ~ is the volume containing the particle (dust-) phase, then:

% = 3 + %. (7)

It is within ~ that the effect of the phases on each other takes place, and on the surface area ~ ' bounding ~ the porous matrix interacts with the flowing phases.

It is assumed here that a no-slip condition holds for the fluid-phase on all solid matrix boundaries and a slip condition holds for the dust-phase (although it is possible to carry out the analysis based on a no-slip condition on both of the phases). It is further assumed that the dust particles are of large-enough size such that diffusion by brownian motion can be ignored.

3. THE AVERAGING RULES

Let F be a volumetrically additive quantity of the fluid-particle mixture within the REV. The phase average of F is defined as

( F) - ( 1 / 7 / ) fT/F dV. (8)

The corresponding intrinsic phase average (i.e., the volumetric average of F over the effective pore volume ~ ) is defined in rule (i) below. Letting F and G be two volumetrically additive scalar quantities and F a vector quantity, the following set of rules have been established, and given else- where [5] or [3]:

(i) ( F> = ~b( F>,~ = CF, (ii) ( F + G}•= ( f )¢_+ (G)4~ (iii) ( F G ) 4, = ( F)4,( G) 4, + ( F'G°)¢ (iv) ( VF ) = V ( F ) + ( 1 / 7 / ) f ~ n F d S , where n is a unit normal directed

into the solid matrix• (v) <VF> = CV(<f> e) + ( 1 / ~ ) f s ~ n b -~ dS (vi) ( V . F> = V. ( F ) + ( 1 / ~ ) f s ~ n • F dS (vii) Because of the no-slip condition on the fluid-phase, the surface

integral over the fluid-solid matrix interface vanishes if the integrand contains the fluid-phase velocity explicitly.

(viii) If c is a constant, ( c F ) = c(F) . (ix) The relationship between the true quantity F and its intrinsic

average is given by F = ( F ) ~ + b -~ where F ' is the deviation from F.


4. AVERAGING THE DUSTY GAS EQUATIONS

In the Saffman dusty fluid model, the effects of the flowing phases on each other are represented by a drag term proportional to the relative velocity of the phases involved. When the same dusty fluid flows in a porous medium, the other interactions that occur are consequences of the introduction of the porous matrix in the flow domain. We will therefore assume at the onset of the analysis that when a dusty fluid flows in a porous medium, the effects of the phases on each other are represented by drag terms proportional to the relative velocity. This assumption eliminates considera- tion of the interfacial surface area between the fluid and dust phases.

To derive the required set of equations, we adopt the intrinsic phase averaging procedure over the REV and apply the averaging rules (i)-(ix) to the governing equations (1-4), as follows.

Dust-Phase Continuity. Applying rules (vi), (i), (iii), and (vii), respectively, to equation (1), we obtain

V. (¢N~v¢) + V- (¢(N°v°)¢) = O. (9)

Dust-Phase Momentum. Applying rules (ii), (vi), (vii), (v), (i), (iii), and (viii) to equations (2), we obtain

V- (¢vcv¢) + V- (¢<v°v°>¢ = ( ¢ k / m ) ( u ¢ - v¢). (10)

Fluid-Phase Continuity. Applying rules (vi), (vii), and (i), respectively, to equation (3) we get

V- ( ¢ u ¢ ) = 0. (11)

Fluid-Phase Momentum. The fluid-phase momentum equation (4) is averaged term-wise, in accordance with rule (ii), as follows:

( pV- ( u u ) ) = - ( V p ) + ( t t V 2 u ) + ( k N v ) - ( k N u ) (12)

where, using rules (viii), (vi), (vii), and (iii), respectively, we get

( pV" (uu)> = pV. (¢uCu4,) + pV. (th(u°u°>,, (13)


using rule (v) we get

<Vp> = ~Vp, + (1 /T/ ) f~yp ° dS, (14)

and using rules (viii), (i), and (iii), respectively, we get

(kNv) = k~bN~v~ + k~b(N°v°)~ (15)

and

( k N u ) = k~bNou, + k~b(N°u°)6. (16)

To average the term /zV2u, it is written first in the form geV • Vu and thus, using rules (viii) and (vi) we have

</~V2u> =/z<V. Vu> =/zV. <Vu> + ( i.e/Yz')f5~. (Vu) dS. (17)

where, using rules (v) and (vii), we have

V. <Vu> =/zV2(~bu,). (18)

Hence,

( /zVeu) = /zV2(¢u6) + ( i.elT/')fs, n. (Vu) dS. (19)

Upon substituting equations (13-19) in equation (12) and rearranging, we get

pv- (¢u~u~) + pV. (4,<u°u%)

= - ¢ V p ~ +/zV2(~bu~) + k~bN~(v~ - u~)

+ k~b((N°v ° - N°u°)~) + ( 1 / ~ ) f ( - n p ° + / z n . Vu) dS.

(20)


Equations (9), (10), (11), and (20) represent the intrinsic averaged equations governing the flow of a dusty fluid in a porous medium. The deviation terms in these equations, and the surface integral in equation (20), contain the necessary information on the interactions between the phases present and the porous medium. In particular, the surface integral represents the effect of the porous matrix on the flowing mixture.

5. ANALYSIS OF THE DEVIATION TERMS

The terms V. (~b(v°v°)~) and V. (~b(u°u°)~) in equations (10) and (20) are related to the hydrodynamic dispersion of the average dust-phase and fluid-phase velocities. In the absence of high velocity and porosity gradients, [3], both of these terms may be assumed negligible. Equations (10) and (20) thus take the following forms, respectively,

v . ( ¢ v , v , ) = ( ¢ k / m ) ( u , - (21)

pV. ( ¢ u , u ¢ ) = -~bVpf, + ttV2(~bu¢) + kCN~(v¢ - u , )

+ k¢ ( (N°v ° - N°u°)~)

+ ( 1 / ~ / ) f ( - n p ° + /zn. Vu) dS. (22)

The term (N°v°) , appearing in equation (9) and the terms (N°v°)¢ and (N°u°) , appearing in equation (22) are dispersion vectors of the average number density due to the influence of average phase velocity vectors. These vectors vanish individually if the particle distribution is uniform. However, if N is not constant, then these terms can be sufficiently quantified by modelling the dispersion vectors as a diffusion mechanism only (cf. [6]).

The term V-(~b(N°v°),) appearing in the dust-phase continuity equation (9) represents the mass transfer, F, between the dust phase and the porous medium in the form of settling of the dust particle on the solid matrix boundaries and the sedimentation and resuspension of particles of the solid porous matrix in the flow field. It should be noted that there is no mass transfer between the fluid phase and the dust phase. To quantify F, we let 6 be the mass transfer coefficient, then in light of the channeling concept of the porous medium it may be argued that the mass transfer between the dust-phase and the solid matrix is dependent on the wetted surface area of


the REV, and thus the mass transfer F is taken as:

F = V. (¢(N°v°)4,) = c 5 '~ (23)

and thus the dust-phase continuity equation (9) takes the form

v . ( NCv ) = (24)

It should be emphasized at this point that determination of the mass transfer coefficient ~, which is postulated here to be a function of some quantities including the porosity and rigidity of the medium, should be experimental. When the particle distribution is uniform, i.e., dust particle number density is taken to be constant throughout, or when the dispersion of the dust phase is not taken into account, then F = 0, and the dust-phase continuity equation reduces to

V. (¢NCv¢) = O. (25)

The term ((NOv ° - N°u°)¢), which appears in equation (22), is the difference between two dispersion vectors and arises from averaging the relative velocity vector v - u. Under some conditions, provided in what follows, this dispersion term can be considered negligible. To illustrate this, we apply averaging rule (ii) to the above term and obtain:

(NOv ° - N°u°)¢ = (N°v°)¢ - (N°u°)4,. (26)

Using rule (iii), we have

(N°v°)4, = (N°)o(v°)¢ + ((N°v°)°)¢ (27)

and

(N°u°)~ = (N°)¢ (u° ) , + ( (N°u° ) ° ) , (28)

Equation (26) thus becomes

(NOv ° - N°u°)¢ = (N°)¢(v°)¢ + ((N°v°)°)¢

- (N°),~(u°)¢ - ((N°u°)°)4,. (29)


The terms ((N°v°)°)~ and ((N°u°)°)~ represent fluctuations of the average deviations of the averaged products over the REV. Because of the smoothing action of the averaging procedure these terms are individually negligibly small. Alternatively, they may be taken to be of the same order, and hence their net effect ((N°v°)°)~ - ((N°u°)°)~ is zero. In either ease, with the help of averaging rule (ii), equation (29) takes the form

(NOv ° - N°u°) , = (N°)e , ( (v°)~- (u°)~,). (30)

To analyze the right-hand side of equation (30), we consider the following cases:

1. If the dust particle distribution is uniform, and hence N is constant, then (N°)~ - 0 and the term (N°v ° - N°u°)¢ vanishes. Furthermore, if dispersion is negligible then clearly the dispersion vectors (N°v°)~ and (N°u°)~ both vanish.

2. If there are no abrupt changes in the averaged relative velocity vector in the porous medium, and (v)~ and (u)~ are sufficiently well-behaved, then the difference (v°)~ - (u°)~ is sufficiently small and hence negligible.

3. Since hydrodynamic dispersion (or the mixing of the phases in each other) is negligible in dusty gases, it is possible to drop out the relative dispersion vector under the premise that the fluid and dust phases only exert a drag force on each other.

4. If none of the above conditions is satisfied, then we treat dispersion as a diffusion process, expressed in terms of the product of a diffusion coefficient ~ and a number density driving differential ((N)¢ - (Ne)~) , and thus we obtain:

(N°v ° - N°u°)~ = 8((N)~ - (Nd)~). (31)

Further analysis in support of equation (31) is required in order to shed some light on the nature of the relative dispersion term.

Upon substituting equation (31) in (22), we get

pv. = -¢vp +

+ k~b[N~(v~- u~) + ~ ( N ~ - Net) ]

+ ( 1 / V / ) f ( - n p ° + p~n. Vu) dS. (32)


To summarize, the up-to-date averaged equations governing the flow at hand are (11), (21), (24), and (32).

6. ANALYSIS OF THE SURFACE INTEGRAL

The interactions that take place within ~ are the effects of the flowing phases on each other and the effect of the porous matrix on the flowing phases. The former occurs through the interfacial area between the fluid and dust phases and is represented at the onset by a drag force proportional to the relative velocity of the flowing phases. The effects of the porous matrix on the flowing phases occur through the portion ~ of the surface area of that is in contact with the dusty fluid at hand. The surface integral appearing in equation (32) contains the necessary information on the effect of the porous matrix on the flowing phases. Its accurate evaluation depends on the knowledge of the porous matrix structure and its geometric description. However, in the absence of a dust-phase partial pressure and a dust-phase viscosity, the effect of the porous matrix on the flowing mixture arises in the form of a surface integral that is expressed in terms of the fluid-phase quantities only.

In order to evaluate the surface integral [ ~ ( - n p° + lz c~ u / ~ n) dS, it is imperative to provide an accurate description of the porous medium. Evalu- ation of a similar integral was addressed by Du Plessis and Masliyah [3, 7] for the case of single phase flow in porous media, where they distinguish between consolidated and granular isotropic porous media. To take into account the effect of the porous microstructure on the flow they introduced the concept of an RUC, which is defined as the minimal REV in which the average properties of the porous medium are embedded, i.e., such that

= ¢ / 3 (33)

with the mean pore cross-sectional area, ~¢p, being the same in all directions by virtue of isotropy of the medium. The length / is the microscopic length, defined in equation (6), which represents the average distance between pores. Accordingly, the overlapping void volume available for flow is given by ~ p / .

The above authors represented the RUC for consolidated media by three short square duct sections, oriented mutually perpendicular, and showed that the total wetted surface within the RUC is given by

3 ( 1 - 2 ( 3 4 )


where T is the tortuosity of the medium that is related to the porosity ~b by

4, = 1) : / (4¢ 3) (35)

and is defined as

T = l i t (36)

and fe is the total tortuous path length within an RUC, given by

fe = ~/'J~'p- (37)

For granular porous media they represented the RUC by a cube of linear dimension f , aligned centrally within it is the solid element that is also of linear dimension [7]. The total wetted surface within this RUC is given by

~ = 6(1 - ~b)2/3f 2 (38)

where ~b is related to ~- by:

T = [1 - (1 - ~b)2/3]Ab. (39)

Du Plessis & Masliyah [3, 7] evaluated the surface integral in terms of the average wall shear stress over the streamwise flow length, and expressed it in terms of the product of the Reynolds number and the apparent friction factor for: (1) fully developed flow in a square straight duct (for consolidated media) and (2) fully developed flow between parallel plates (for granular media).

Following a similar approach, and letting in the current analysis 9 - to be the product of the Reynolds number and the friction factor associated with the flow of a dusty fluid in a square duct (for consolidated media) or between parallel plates (for granular media), then the surface integral in (32)

7O

takes the form:

M. H. HAMDAN AND K. D. SAWALHA

Substituting (34) in (40) and (40) in (32), we obtain after some manipula- tion the following fluid-phase momentum equation that is valid in consolidated porous media:

, v . (6u~u~) = -¢vp~o + , v ~ ( 6 . , )

+ k~b[N¢(v~ - u~) + (~(N~ - Nd~)]

- [ 3 ~ 1 - ~ ) ( 3 T - 1 ) f 2 / T 3]/zu~. (41)

Upon substituting (38) in (40) and (40) in (32), we obtain after some simplification the following fluid-phase momentum equation that is valid in granular media:

pv. ( ¢ u ~ u , ) = - ~ v , ~ + , v 2 ( + u ~ )

+ kcb[N¢(v¢ - u~) + tt(N¢ - N ~ ) ]

- [C~(1 - T )2 /3 / ( f~ ) ] ~u~. (42)

The dust-phase continuity equation (24) can be written in the following forms with the help of (34) and (38): For consolidated porous media, (24) takes the form

V. ( ~ N , v ¢ ) = - 3 s ( 1 - T)(3T -- 1 ) f2 /~ "2, (43)

while for granular media it takes the form

v . (¢N~v~) = - 6 , ( 1 - ~)2/3/2. (44)

To summarize thus far, the equations governing the dusty fluid flow in porous media are given by (11), (21), (41), and (43) for consolidated porous media, and by (11), (21), (42), and (44) for granular porous media.

( 1 / ~ ) f s ~ ( - n p p + t~n .Vu) dS = - ~ u ~ J / ~ ' ~ . (40)


7. THE FINAL FORM OF THE GOVERNING EQUATIONS

For computational purposes the above equations may be cast in terms of the volumetric flow rates of the phases involved, by substituting: ql = ~bu~ and q2 = ~bvt, and the macroscopic number densities n = ~bN,, n d = ~bN~¢. The following equations are thus obtained: For consolidated media:

Dust-phase: Equations (43) and (21) take the following forms, respectively:

V. (nq2/~b) = - 3 e ( 1 - T ) (3T- 1 ) f2 /72 (45)

V" (q2q2 /~) = ( k / m ) ( q l - q2)- (46)

Fluid-phase: Equations (11) and (41) can be cast in the following forms, respectively:

V - ( q l ) - 0 (47)

pV" (qlql/~b) = -- ~b Vp, + /.~V2(ql)

+ ( k l d p ) [ n ( q 2 - ql)] + 8 (n - n d ) ]

- 3Dl-l~f2q,[(1 - ~ ) / ( 3 T - 1)]. (48)

For granular media: Dust-phase: Equations (44) and (21) take the following forms, respec-

tively:

V . ( n q J d ~ ) = - 6 6 ( 1 - (b~')/2. (49)

V" (q2q2/~b) - - ( k / m ) ( q l - q2). (50)

Fluid-phase: Equations (11) and (42) can be cast in the following forms, respectively:

V- (ql) = O. (51)

pV. (qlql/~b) = -~bVp~ + ttV2(ql)

+ ( k / ~ ) [ n ( q 2 - q l ) ] + ~ ( n - nD]

- a . ~ - ~ q , [ ( 1 - ~ , - ) / ( / ~ 4 > ) ] . ( 5 2 )


Finally, the hydrodynamic permeability of the flow may be defined in terms of the tortuosity terms that appear in equations (45) and (48), which can easily be expressed in terms of the linear dimensions of the RUC depending on the type of flow and the type of porous medium.

CONCLUSIONS

In this work, an at tempt has been made to develop a set of partial differential equations describing the flow of a dusty fluid in variable porosity media. The developed equations take into account the effect of the porous microstructure on the flowing phases, but leave undetermined diffusion and mass transfer coefficients, which are worthy of further investigation. The developed models make a distinction between flow in consolidated and in granular porous media in the manner in which the fluid-phase momentum equations and the dust-phase continuity equations are expressed. The difference in the types of media clearly plays a role in the mass transfer between the dust-phase and the porous matrix, as witnessed by the dust-phase continuity equations. It also plays a role in distinguishing the kind of resistance that the porous structure is exerting on the fluid, and the inertial effects caused by the porous microstructure, as witnessed in the fluid-phase momentum equations. Depending on the type of flow and the porous microstructure, the developed equations may be cast in different forms that parallel the equations governing single-phase flow in porous media.

l---- I I l I

!

! I Y~ I

I I t_ _ ___.I

FIG. 2. A typical representative unit cell.


REFERENCES

1 M.H. Hamdan and R. M. Barron, A dusty gas flow model in porous media, Comput. AppL Math. 30:21-37, 1990.

2 M.H. Hamdan and R. M. Barron, On the Darcy-Lapwood-Brinkman-Saffman dusty fluid flow models through porous media. Part I: Models Development, Appl. Math. Comput. 54:65-79, 1993.

3 J .P . Du Plessis and J. H. Masliyah, Mathematical modelling of flow through consolidated isotropic porous media, Transport Porous Media 3:145-161, 1988.

4 P.G. Saffman, On the stability of laminar flow of a dusty gas, J. Fluid Mec]u 13 (Part 1):120-129, 1962.

5 Y. Bachmat and J. Bear, Macroscopic modelling of transport phenomena in porous media, I: The Continuum Approach, Transport Porous Media 1:213-240, 1986.

6 W.G. Gray, A derivation of the equations for multi-phase transport, Chem. Eng. Sei. 30:229-233, 1975.

7 J. P. Du Plessis and J. H. Masliyah, Flow through isotropic granular porous media, Transport Porous Media 6:207-221, 1991.

dusty gas flow through porous media

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