Chapter 2

BASIC EQUATIONS,

APPROXIMATIONS, BOUNDARY

CONDITIONS AND

DIMENSIONLESS PARAMETERS

In this chapter we present the required basic equations, approximations, boundary condi­

tions and dimensionless parameters. The problem considered in the dissertation is con­

cerned with solute transfer in Newtonian fluids through porous media.

2.1 BASIC EQUATIONS

A fluid which exhibits linear relationship between the shear stress and the rate of shear

is known as a Newtonian fluid. A porous medium is an assemblage of solid particles or

grains which encloses a system of interconnected pores each of which is saturated with

a fluid. The exact form of the structure, however, is highly complicated and differs from

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medium to medium. Therefore such a medium is particulate in nature and any attempt

to describe them in detail is fraught with virtually insurmountable complexity. It is not

possible to know the nature of fluid flow through each pore because of its microscopic

nature and the many degrees of freedom. Also the solution to the problem in a porous

medium as in any other dynamical system requires a statement of boundary geometry.

Due to the many intermediate boundary conditions of the flow geometry in porous media,

a complete mathematical solution of microscopic flow through pores is highly complex.

The prospect of representing exactly the flow of a fluid through a porous medium is there­

fore remote. Some of the examples of porous medium are towers packed with pebbles,

porous rocks such as limestone, pumice and dolomite, bricks, wood and fiber aggregate

such as cloth, filter paper etc.

In investigating the problems the transport of momentum, heat and mass through a porous

medium, it has been achieved remarkably by neglecting the complexity of the internal ge­

ometry and adopting the concept of equivalent continuum. Usually a representative sam­

ple of this continuum is considered and average values of the required physical properties

are obtained ([30], [66], [91], [64], [138]. Before going to basic equations, we present

below a relation due to Dupit-Forchheimer. If q and qJ are the average velocity of the

fluid respectively over a volume element Vm of the medium (incorporating both solid and

fluid materials) and volume element VJ consisting only the fluid then, these two velocities

are related by the following Dupit-Forchheimer relation

(2.1.1)

where ¢ is the porosity of the porous medium and is independent of time. The required

basic equations are the conservation of mass, momentum and species.

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2.1.1 Continuity Equation (Conservation of Mass)

The conservation of mass states that the amount of fluid flowing into a volume must be

equal to the amount of fluid flowing out of that volume which is expressed mathematically

by the continuity equation

where p is the density of the fluid. Using the Dupit-Forchheimer relation (2.1.]), the

above equation becomes

8pc/J 8t + \l.(pq)) = O.

For an incompressible fluid equation. (2.1.2) reduces to

\l.q = O.

(2.1.2)

(2.1.3)

2.1.2 Momentum Equation Through Fluid Saturated Porous Medium

In the literature one can find a considerable amount of discussion on the choice of con-

servation of momentum equation to study convection in a fluid-saturated porous medium.

When a fluid permeates a porous material the actual path of an individual fluid particle

cannot be followed analytically. The gross effect, as the fluid slowly percolates the pores

of the medium, must be represented by a macroscopic law, which is applicable to fluid

with large mass compared with the dimension of the porous structure of a medium and

this is the basis for the Darcy [34] law. According to this law, the driving force necessary

to move a specific volume of fluid at a certain speed through a porous medium is in equi-

librium with the resistance generated by internal friction between the fluid and the pore

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structure. Mathematically the Darcy law takes the form

Kq = -\lp,

J-lf(2.1.4)

where q is the filter velocity, K is the permeability of the porous medium andJ-lf is the

dynamic coefficient of viscosity. The constant of proportionality J( in the Darcy equation/1-f

(2.1.4) was shown experimentally to be what it is by Muskat [88] . He also showed that

the permeability K of the porous medium has the dimension of d2 , where d is generally

the diameter of the spherical balls which make the porous medium. In the presence an

extemal forceF, equation (2.1.4) can be written as

Kq = --(\lp - F),

J-lf(2.1.5)

where F = pg and 9 is the acceleration due to gravity. This form of Darcy law is still

being used in many practical problems. Dewiest [35] showed that Darcy's law is the em-

pirical equivalent of Navier-Stokes equations. In the case of a homogeneous, isotropic

porous medium, the flow governed by this modified Darcy law is of potential type rather

than a boundary layer type. This law is valid for a densely packed porous medium whose

permeability is very low. The Darcy model takes into account only the frictional force

offered by the solid particles to the fluid rather than the usual viscous shear.

Beavers et al. [15] experimentally demonstrated the existence of shear within the porous

medium, near the boundaries, thus forming a zone of shear influenced by fluid flow. The

Darcy equation cannot predict the existence of such a boundary zone as no macroscopic

shear term is included in the equation. Slattery [124]and Tam [132] generalized equation

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(2.1.5) to account for the shear in the form

111 I 2- K q + II, \l q = \lp - F, (2.1.6)

where f.i is the effective viscosity of the fluid in the porous medium and is a function of

the porosity, cP. This is the most suitable governing equation for an incompressible creep-

ing flow of a Newtonian fluid within an isotropic, homogeneous porous medium. Later,

Saffman[109], Lundgren [75] and Childress [31] presented elaborate statistical justifica-

tion of equation (2.1.6) which was originally proposed by Brinkman [17], [18], [19]. The

Brinkman equation (2.1.6) is physically consistent with the previously mentioned experi-

mentally observed boundary shear zone on account of the usual viscous shear force. The

Brinkman model is valid for a sparsely packed fluid-saturated porous medium wherein

there is more window for a fluid to flow so that the distortion of velocity gives rise to the

usual viscous shear force.

In many practical problems, when the porosity is close to unity, the flow of fluid through a

porous medium is curvilinear and curvature of the path gives rise to inertia effect. As the

inertia force increases relative to the viscous force, the streamlines become more distorted

and drag increases more rapidly than linearly with velocity. At present there are several

different views as to how the Darcy model should be generalized to include the inertia

effect. Lapwood [69]gave a mathematical form incorporating the convective inertial term

in the momentum equation. Subsequently many papers have appeared on this 'nonlinear'

non-Darcy model (Beavers and Sparrow [14], Vafai and Tien [137], Joseph et al. [66],

Nield[90]).

Now equation (2.1.6) using equation (2.1.1) can be written, with the usual inertial term

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p(qj.'V)qj incorporated, as

(2.1.7)

where <jJ is absorbed into J( in the Darcy term.

Since equation (2.1.7) does not take care of possible unsteady nature of velocity, one has

to consider the local acceleration term oqj as the flow pattern in certain region may be. ot

unsteady. Incorporating this, equation (2.1.7) can be written as

[18q 1] f-L j I 2P -- + -(q.'V)q = -'Vp + F - -q + f-L 'V q.<jJm ~ J(

(2.1.8)

This equation is known as Darcy-Lapwood-Brinkman equation. In the absence of viscous

shear equation (2.1.8) is known as the Darcy-Lapwood equation. If quadratic drag is

incorporated in the system, then the above equation becomes

[1oq 1] f-L j pCb I 2P -- + -(q.'V)q = -'Vp + F - -q - -Iqlq + f-L 'V q

<jJ ot <jJ2 K VR (2.1.9)

This equation is known as Darcy-Lapwood-Brinkman-Forchheimer model. If the Reynolds

number Re (defined in section 2.3) is very small, then the quadratic drag can be neglected.

If Re = 0 (1), then the quadratic drag law holds. If Re < 0 (1), then we will have to use

a cubic drag law.

2.1.3 Diffusion Equation (Conservation of Species)

Let C(x, y, z, t) be the concentration of a solute or the amount of the solute per unit

volume at the point(x, y, z) at the time t. Due to the concentration gradient, grad C, there

is a flow of solute given by the current density vector J, which according to Pick's first

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law of diffusion, is given by

J = -DgradC = -D\JC, (2.1.10)

whereD is the coefficient of diffusion or diffusivity. The negative sign in equation (2.1.10)

indicates that the flow takes place in the direction of decreasing concentration. D can

vary with x, y, z but, for a moment, we take it to be constant. Its value for some common

biological solutes in water lies between 0.05 x 10-6 andlO x 1O-6c:m2 / sec.. In the

presence of convective diffusion Fick's first law is given by

J = -D\JC + qC, (2.1.11)

where q is the velocity of the solvent. Now, consider a volume V with surface S. The rate

of change of the amount of the solute is given by

:t i C(x, y, z, t)dxdydz.

The amount of the solute which comes out of the surface S per unit time is given by

1J.hdS,

where h is the unit normal vector to the surface. If there is no source or sink inside the

volume, we have

:t i C(x, y, z, t)dxdydz = - 1J.hdS

= 1(D\JC).hdS -1(qC).hdS

= r \J.(D\JC)dV - r [q.\JC + C(\J.q)] dV../V ./V

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For an incompressible fluid, we have

:t.l CdV = [ \7.(D\7C)dV -[ q.\7CdV.

The above can be rewritten as

if [~~ -\7.(D\7C) + q.\7C] dV = O.

Since this holds good for all volumes, we get the Fick's second law of diffusion, with

convective diffusion, in the form

(2.1.12)

In deriving the above equation it has also been assumed that there is no chemical reaction

in the isothermal fluid system. The governing equations documented earlier cannot be

solved in its most general form. Many a time the problems are simplified, without com­

promising the qualitative picture, by resorting to certain approximations. The following

approximations have been used in this dissertation.

(i) The Newtonian fluid is incompressible.

(ii) Buoyancy is assumed to have a negligible effect on the dynamics of the system.

(iii) The mass diffusivity in the vertical direction and the horizontal direction are the

same (solutal isotropy).

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2.2 BOUNDARY CONDITIONS

2.2.1 Boundary Conditions on Velocity

The boundary conditions on velocity are obtained from mass-balance, the no-slip condi-

tions and the stress principle of Cauchy depending on whether the fluid layer is bounded

by rigid or free surfaces. In the case of flow through a parallel-plate channel bounded

by rigid boundaries, the no-slip condition implies that the velocity components vanish

identically at the boundaries. This condition can be expressed in the form

'U = 'U = 'W = 0 at Z= ±h, (2.2.1)

2.2.2 Boundary Conditions on Concentration

The diffusion of solute can be approximated as the direct analogue of the diffusion of

heat. Therefore on the surface

aaC = Sh(C - Cwall ) ,y

(2.2.2)

whereSh is the Sherwood number. The impermeable boundary conditions are obtained

from the above condition in the limitSh ---7 O. The limit Sh ---7 0 gives rise to Neumann

type of boundary condition which specifies the concentration flux at the boundaries (total

rejection of solute at the walls). For impermeable boundary condition, we have

C=O and ac = 0ay . (2.2.3)

The limit Sh ---7 0 gives rise to a Dirichlet type of boundary condition which represents

specifying concentration at the bounding walls for 0 < S h < 00, it is an imperfect

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rejection of solute at the walls.

We note here that in the case of problems involving heterogeneous chemical reaction (i.e.,

interphase mass transfer) the reaction rate parameter takes the place of S'h. Therefore for

permeable boundary condition, we have

BCD-=-KcC.By .

This type of boundary condition is used in Chapter-6.

2.3 DIMENSIONLESS PARAMETERS

(2.2.4)

The fluid flows are governed by a set of partial differential equations. There is no general

method of finding the solution to these equations and therefore to get some understand-

ing of these complex flow phenomena, two approaches can be used. One is to simplify

these fundamental equations according to some physical considerations so that the result-

ing equations may be solved. The second is based on an experimental investigation of the

flow problem under conditions which are similar to the actual case (dynamical similarity).

For both the approaches it is desirable to find the important parameters in a given flow

problem. These parameters can be found by reducing the governing differential equations

to non-dimensional form.

The dimensionless groups that appear in the thesis are given below:

(i) Reynolds number (Re)

Reynolds number represents the ratio of the inertia to viscous force

Re = puRh ,J.Lf

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where UR is the characteristic velocity, h is the thickness of the porous layer and other

quantities are as defined earlier. From the definition of Re it is clear that the viscous forces

are dominant for small Reynolds number and as the Reynolds number Re is increased,

the inertia force becomes dominant and small disturbances in the fluid may be amplified

to cause the transition from laminar to turbulent flow.

(ii) Peelet number (Pe)

The dimensionless parameter measures the amount of mass advection to the amount of

mass diffusion. The Peelet number Pe is defined as

Pe = uRhD'

The Peclet number for mass expresses the relative rates of propagation of momentum and

mass within the systems.

(iii) Damkohler number (f3)

Damkohler number represents the relative importance of wall reaction to mass diffusion.

In Chapter-6 it is defined as

(3 = KchDO

A non zero value of (3 signifies, heterogeneous chemical reaction.

(iv) Brinkman number (A)

The Brinkman number A is defined as the ratio of viscosities, i.e. ,

A=~"f-L

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where I-" is the fluid viscosity and 1-'" is the viscosity of the ~uid in the presence of porous

medium. Brinkman [17], [18], [19] setl-'" to be equal to p,. Recent experiments show that

A. varies from 0.5 to 10.9 ([70], [49]).

(v) Darcy number (Da)

The Darcy number Da is defined as

hDa = .jK'

whereK is the permeability ofporous medium. The permeability is indicative of the case

with which the fluid can flow through the porous matrix. This dimensionless parameter

represents the scale factor which describes the extent of the division of the structure of

the porous medium as compared to the horizontal or vertical extent of the porous layer.

(vi) Forchheimer number (F)

The Forchheimer number F is defined as

where C'b is the quadratic drag coefficient and Da and A. are the Darcy number and Brink

mann number respectively.

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