Mathematical models for mass and heat transport in porous media

About This Presentation

Title:

Mathematical models for mass and heat transport in porous media

Description:

In fact, Brenner showed how Darcy's experimental law and the permeability tensor ... 1.The Darcy's law (3.13) gives the global flow field in the periodic porous ... – PowerPoint PPT presentation

Number of Views:549

Avg rating:3.0/5.0

Slides: 60

Provided by: Semp5

Category:

more less

Transcript and Presenter's Notes

Title: Mathematical models for mass and heat transport in porous media

1
Mathematical models for mass and heat transport
in porous media

Stefan Balint and Agneta M.Balint
West University of Timisoara, RomaniaFaculty of
Mathematics- Computer ScienceFaculty of Physics
balint_at_balint.uvt.ro balint_at_physics.uvt.ro

The presentation is focused on the mathematical
modeling of mass and heat transport processes in
porous media.
Basic concepts as porous media, mathematical
models and the role of the model in the
investigation of the real phenomena are
discussed.
Several mathematical models as ground water flow,
diffusion, adsorption, advection, macro transport
in porous media are presented and the results
with respect to available experimental
information are compared.

3
TOPICS

MATHEMATICAL MODELING
POROUS MEDIA
FLUID FLOW IN A POROUS MEDIA
GROUNDWATER FLOW
MASS TRANSPORT IN POROUS MEDIA
COMPUTATIONAL RESULTS TESTED AGAINST EXPERIMENTAL
RESULTS
HEAT TRANSPORT IN POROUS MEDIA
COMPUTED CONDUCTIVITY FOR THE HEAT TRANSPORT IN
POROUS MEDIA TESTED AGAINST EXPERIMENTAL RESULTS

4
1.MATHEMATICAL MODELING

Mathematical modeling is a concept that is
difficult to define. It is first of all applied
mathematics or more precisely, in physics applied
mathematics.
According to A.C. Fowler Mathematical Models in
the Applied Sciences, Cambridge University Press,
1998
Since there are no rules, and an understanding
of the right way to model there are few texts
that approach the subject in a serious way, one
learns to model by practice, by familiarity with
a wealth of examples.

Applied mathematicians have a procedure, almost a
philosophy that they apply when building models.
First, there is a phenomenon of interest that one
wants to describe and to explain. Observations of
the phenomenon lead, sometimes after a great deal
of effort, to a hypothetical mechanism that can
explain the phenomenon. The purpose of a
mathematical model has to be to give a
quantitative description of the mechanism.
Usually the quantitative description is made in
terms of a certain number of variables (called
the model variables) and the mathematical model
is a set of equations concerning the variables.
In formulating continuous models, there are three
main ways of presenting equations for the model
variables.
The classical procedure is to formulate exact
conservation laws. The laws of mass, momentum and
energy conservation in fluid mechanics are
obvious examples of these.
The second procedure is to formulate constitutive
relations between variables, which may be based
on experiment or empirical reasoning (Hook law).
The third procedure is to use hypothetical laws
based on quantitative reasoning in the absence of
precise rules (Lotka-Volterra law).
.

The analysis of a mathematical model leads to
results that can be tested against the
observations.
The model also leads to predictions which, if
verified, lend authenticity to the model
It is important to realize that all models are
idealizations and are limited in their
applicability. In fact, one usually aims to
over-simplify the idea is that if the model is
basically right, then it can subsequently be made
more complicated, but the analysis of it is
facilitated by having treated a simpler version
first.

7
2. POROUS MEDIA

Example 1. The soil
Soil consists of an aggregation of variously
sized mineral particles (MP) and the pore
spaces (PS) between the particles. ( MP green,
PS red)
Figure 1.
When the pore space is completely full of
water, then the soil is saturated.
When the pore space contains both water and
air, then the soil is unsaturated.
In exceptional circumstances, soil can become
desiccated. But, usually there is some water
present.

Exemple 2. The column of the length 2m, filled
with a quasi uniform quartz sand, of mean
diameter 1.425 mm used by Bues and Aachib in the
experiment reported in
Influence of the heterogeneity of the
solutions on the parameters of miscible
displacements in saturated porous medium,
Experiments in fluids, 11, Springer Verlag,
25-32, (1991).
is a porous media.
Definition 1. A porous media is an array of a
great number of variously sized fixed solid
particles possessing the property that the volume
concentration of solids is not small. (Soil).
Often it is characterized by its porosity F (i.e.
the pore volume fraction) and its grain size d.
The latter characterizes the coarseness of the
medium.
Definition 2. A periodic porous media is a
porous media having the property that the fixed
solid particles are identical and the whole media
is a periodic system of cells which are replicas
of a standard (representative) cell (experimental
column).

Figure 2. Periodic porous media

10
3. FLUID FLOW IN A POROUS MEDIA

An incompressible viscous fluid moves in the
pore space of a porous media, according to the
Navier-Stokes equations
(3.1)
satisfying the incompressibility condition
(3.2)
where is the fluid flow velocity p
is the pressure, ? is the fluid density, µ is the
fluid viscosity and is the density of the
volume force acting in the fluid.
It is important to realize, that eqs. (3.1),
(3.2) are valid only in the pore space (x
belongs to the pore space). They can be
obtained from the mass and momentum conservation
laws and constitutive relations characterizing
viscous fluids.

On the boundary of the fixed solid particles the
fluid flow velocity has to satisfy the non slip
condition
(3.3)
Unfortunately, the boundary value problem (3.1),
(3.2), (3.3) even in the case of a very slow
stationary flow (Stokes flow), can not be solved
numerically in a real situation due to the great
number of the boundaries of fixed solid
particles.
Consequently, the mathematical model defined by
the eqs. (3.1), (3.2), (3.3) can not be analyzed
numerically in a real case and can not be tested
against the observations.
Several models for flow through porous media are
based on a periodic array of spheres.
Hasimoto H. in On the periodic fundamental
solution of the Stokes equations and their
application to viscous flow passed a cubic array
of spheres, J.Fluid Mech.,5, 317-328 (1959).
obtained the periodic fundamental solution to
the Stokes problem by Fourier series expansion,
and applied the results analytically to a dilute
array of uniform spheres.
Sangani A.S., Acrivos A. in Slow flow
through a periodic array of spheres, Int.J.
Multiphase Flow, 8(4), 343-360 (1982)
extended the approximation of Hasimoto to
calculate the drag force for higher concentration.

- Zick A.A. and Homsy G.M. in Stokes flow
through periodic arrays of spheres, J.Fluid
Mech.,115, 13-26 (1982). use Hasimotos
fundamental solution to formulate an integral
equation for the force distribution on an array
of spheres for arbitrary concentration. By
numerical solution of the integral equation,
results for packed spheres were obtained, for
several porosity values.
- Continuous variation of porosity was examined
only when the particles are in suspension.
- Strictly numerical computations have been made
earlier, based on series of trial functions and
the Galerkin method, for cubic packing of spheres
in contact
Snyder L.J. and Stewart W.A. Velocity and
pressure profiles for Newtonian creeping flow in
regular packed beds of spheres, A.I.Ch.J.,12(1),
167-173 (1966).
Sorensen J.P. and Stewart W.E. Computation of
forced convection in slow flow through ducts and
packed beds. II. Velocity profile in a simple
cubic array of spheres, Chem.Engng.Sci., 29,
819-825 (1974).
A general model for the flow through periodic
porous media has been advanced by Brenner in an
unpublished manuscript cited in
Adler P.M. Porous Media Geometry and Transports,
Butterworth-Heinemann. London (1992)
Brenner H. Dispersion resulting from flow
through spatially periodic porous media,
Phil.Trans.R.Soc. London, 297 A, 81-133 (1980).
In fact, Brenner showed how Darcys experimental
law and the permeability tensor can in principle,
be computed from a canonical boundary value
problem in a standard (representative) cell.

In the following we will present briefly this
model.
Consider a periodic porous media which is a union
of cells (cubes) of dimension l which are
replicas of a standard (representative) cell. Let
P0 the characteristic variation of the global
pressure P which may vary significantly over the
global size L of the porous media. Thus the
global pressure gradient is of order O(P0 /L).
Let the two size scales be in sharp contrast, so
that their ratio is a small parameter e l/Lltlt1.
Limiting to creeping flows, the local gradient
must be comparable to the viscous sheers so that
the local velocity is UO(P0 l 2/µL), where µ
is the viscosity of the fluid. Denoting physical
and dimensionless variables respectively by
symbols with and without asterisks, the following
normalization may be introduced in the
Navier-Stokes equations (3.1), (3.2)
(3.4)
with i 1,2,3.
Two dimensionless parameters would then appear
the length ratio
e l/L and the Reynolds number
(3.5)
which will be assumed to be of order O(e).
By introducing fast and slow variables, xi and Xi
e xi and multiple-scale expansions, it is
then found that the leading order p(0) pore
pressure depends only on the global scale (slow
variables), p(0) p0(Xi).

By expressing the solution for in
the following form
(3.6)
(3.7)
where depends on Xi only, the
coefficients kij(xi, Xi) and Sj(xi, Xj) are
found to be governed by the following canonical
Stokes problem in the standard (representative)
cell O
in ?

(3.8)
in ?

(3.9)
with
kij 0 on G

(3.10)
kij, Sj are periodic on ??
(3.11)
Here G and ?? are respectively the fluid-solid
interface and the boundary of the standard cell.

Equations (3.8)-(3.11) constitute the first cell
problem. For a chosen granular geometry, the
numerical solution of (3.8)-(3.10) replaced in
(3.6), (3.7) gives the local velocity and
pressure fluctuation in terms of the global
pressure gradient
Let the volume average over the standard cell be
defined by
(3.12)
where Of is the fluid volume in the cell.
Then the average of eq.(3.6) gives the law of
Darcy
(3.13)
where lt kij gt is the so called hydraulic
conductivity tensor, which is the permeability
tensor lt Kij gt divided by µ.
For later use, we note that in physical variables
(marked by ) the symmetric hydraulic
conductivity tensor is given by
(3.14)

Comments
1.The Darcys law (3.13) gives the global flow
field in the periodic porous media in function of
the global pressure field acting on the media. It
is important to realize that this field exists
not only in the pore space, but everywhere in
the media, i.e. also in the space occupied by the
solid fixed particles. The answer to the question
What represents this flow in the space occupied
by the solid and fixed particles? can be found
in
Tartar L. Incompressible Fluid Flow in a Porous
Medium. Convergence of the Homogenization Process
in Non-Homogeneous Media and Vibration Theory
Lecture Notes in Physics, Vol.127, 368-377
Springer Verlag, Berlin 1980.
where it is shown that for tending to zero,
the flow field in the pore space prolonged by
zero in the space occupied by the solid and fixed
particles tends to the global flow field given by
the Darcys law (3.13).
The Darcys law is written in the form
(3.15)
and it is shown that the flow is incompressible
i.e. .
Therefore, if the hydraulic conductivity tensor
is constant (constant permeability), then we have

(3.16)
2. The particularities of the porous media
porosity, shape of the solid and fixed particles
are incorporated in the permeability tensor lt Kij
gt. Numerical results for permeability were
obtained by
Lee C.K., Sun C.C., Mei C.C. Computation of
permeability and dispersivities of solute or heat
in a periodic porous mediaInt.J.Heat Mass
Transfer,39,4 661-675 (1996)
The computed values for the Wigner-Seitz grain
(grain is shaped as a diamond) are compared with
those given by the empirical Kozeny-Carman
formula
(3.17)
which is an extrapolation of measured data.
Within the range of porosities 0.37lt F lt 0.68
the computed results are consistent and in trend
with. Outside this range of porosities the
deviation increases.
The computed results for uniform spheres of
various packing agree remarkably well with those
obtained by Zick and Homsy, when the porosity is
high.
3. The method, used for the deduction of the new
model (eqs.(3-15), (3-16)) of the fluid flow in a
porous media is called the method of
homogenization. Basically, the two phase non
homogeneous media is substituted by a homogeneous
fluid, which flow is not anymore governed by
the Navier-Stokes equation.

18
4. GROUNDWATER FLOW

The groundwater flow is one that has immense
practical importance in the day-to-day management
of reservoirs, flood prediction, description of
water table fluctuation.
Although there are numerous complicating effects
of soil physics and chemistry that can be
important in certain cases, the groundwater flow
is conceptually easy to understand.
Groundwater is water that lies below the surface
of the Earth. Below a piezometric (constant
pressure) surface called the water table, the
soil is saturated, i.e. the pore space is
completely full of water. Above this surface, the
soil is unsaturated, and the pore space
contains both water and air.
Following precipitation, water infiltrates the
subsoil and causes a local rise in the water
table. The excess hydrostatic pressure thus
produced, leads to groundwater flow.
The flow satisfies the Darcys law presented
above
(4.1)

is the pressure gradient in the groundwater and
satisfies
19

(4.2)
which is the incompressibility condition in the
case of groundwater flow
k is the permeability tensor for simplicity has
the form
(4.3)
with k gt 0. The constant k is called permeability
too and has the dimension of (length)2.
Typical value of the permeability of several
common rock and soil types
Eqs. (4.1) (4.2) define the simplest model of the
incompressible groundwater flow through a rigid
porous medium.

Consider now the problem of determining the rate
of leakage through an earth fill dam built on an
impermeable foundation. The configuration is as
shown in Fig.3 where we have illustrated the
(unrealistic) case of a dam with vertical walls
in reality the cross section would be
trapezoidal.
Figure 3. Geometry of dam seepage problem
A reservoir of height h0 abuts a dam of width L.
Water flows through the dam between the base y
0 and a free surface (called phreatic surface) y
h, below which the dam is saturated and above
which it is unsaturated. We assume that this free
surface provides an upper limit to the region of
groundwater flow.

We therefore neglect the flow in the unsaturated
region, and the free boundary must be determined
by a kinematics boundary condition, which
expresses the idea that the free surface is
defined by the fluid elements that constitute it,
so that the fluid velocity at y h is the same
as the velocity of the interface itself
(4.4)
where d/dt is the material derivative for the
fluid flow.
In the two-dimensional configuration, shown in
Fig.3, we therefore have to solve
(4.5)
(4.6)
where with boundary conditions
that
(4.7)

These conditions describe the impermeable base at
y0, the free surface at y h, hydrostatic
pressure on x 0 and atmospheric pressure at x
L (the seepage face). The free boundary is to be
determined as part of the solution.
In order to solve the problem (4.5), (4.6), (4.7)
we nondimensionalize the variables by scaling as
follows
(4.8)
all for obtain various obvious balances in the
equations and boundary conditions. The
Dupuit-Forchheimer approximate solution is
obtained when h0 ltlt L.
In this case we define and the
equations become

with
(4.7)
Since we proceed by expanding
The leading order approximation for p is
just
(4.9)
This fails to satisfy the condition at x 1,
where the boundary layer is necessary to bring
back the x derivatives of p, unless there is no
seepage face, that is h(L) 0.
However, we also note that if
, then , which suggests that
constant

Alternatively, we realize that
simply indicates that the timescale of relevance
to transient problems is longer than our initial
guess , so
that we rescale t with .
Putting (and subsequently omitting
the over bar) we rewrite the kinematical boundary
condition as
(4.10)
Now we seek expansions
(4.11)
and we find successively
(4.12)
and
(4.13)

whence
(4.14)
so that eq. (4.10) gives
(4.15)
dropping the subscript, we obtain the nonlinear
diffusion equation
(4.16)
Notice, that this equation is not valid to x 1,
because we require p 0 at x 1, in
contradiction to eq. (4.12). We therefore expect
a boundary layer there, where p changes rapidly.
Eq. (4.16) is a second order equation, requiring
two boundary conditions. One is that
(4.17)
but it is not so clear what the other is. It can
be determined by means of the following trick.
Define

(4.18)
and note that the flux q is given by
(4.19)

Furthermore
(4.20)
and therefore we have the exact result
(4.21)
In a steady state, , so q is
constant, and therefore
(4.22)
The steady solution (away from x 1) is
therefore
(4.23)
And there is (to leading order) no seepage face
at x 1.
In fact, the derivation of eq. (4.22) applies for
unsteady problem also. If we suppose that q does
not jump rapidly near x 1, then we can use
Dupuit-Forchheimer approximation
in eq. (4.21) and an integration yields
(4.24)
as the general condition.
The boundary layer structure near x 1 can be
described as follows
near x 1 we have
and so we put

(4.25)
and we choose
(4.26)
to bring back the x derivatives in Laplaces
equation, we get
(4.27)
with
(4.28)
Exact solutions of this problem can be found
using complex variables, but for many purpose the
D-F approximation is sufficient, together with a
consistently scaled boundary layer problem.

28
5. MASS TRANSPORT IN POROUS MEDIA

We present the mass transport in porous media as
it is described by
Auriault I.L. and Lewandowska J. in Diffusion,
adsorption, advection, macrotransport in soils,
Eur.J.Mech. A/Solids 15,4, 681-704, 1996.
The pollutant transport in soils can be studied
by means of a model in which the real
heterogeneous medium is replaced by the
macroscopic equivalent (effective continuum) like
in the case of the fluid flow. The advantage of
this approach is the elimination of the
microscopic scale (the pore scale), over which
the variables such as velocity or the
concentration are measured.
In order to develop the macroscopic model the
homogenization technique of periodic media may be
employed. Although the assumption of the periodic
structure of the soil is not realistic in many
practical applications, it was found reasonably
model to real situations. It can be stated that
this assumption is equivalent to the existence of
an elementary representative volume in a non
periodic medium, containing a large number of
heterogeneities. Both cases lead to identical
macroscopic models as presented in
Auriault I.L., Heterogeneous medium, Is an
equivalent macroscopic description possible?
Int.J.Engn,Sci.,29,7,785-795, 1995.

The physical processes of molecular diffusion
with advection in pore space and adsorption of
the pollutant on the fixed solid particles
surface can be described by the following mass
balance equation
(5.1)
(5.2)
where c is the concentration (mass of pollutant
per unit volume of fluid), Dij is the molecular
diffusion tensor, t is the time variable is
the flow field and is the unit vector normal
to G. The coefficient a denotes the adsorption
parameter (a gt 0). For simplicity it is assumed
that the adsorption is instantaneous, reversible
and linear.
The advective motion (the flow) is independent of
the diffusion and adsorption. Therefore the flow
model (Darcys law and the incompressibility
condition)
(5.3)
(5.4)
which has been already presented in the earlier
sequence, will be directly used.

The derivation of the macroscopic model is
accomplished by the application of homogenization
method using the double scale asymptotic
developments. In the process of homogenization
all the variables are normalized with respect to
the characteristic length l of the periodic cell.
The representation of all the dimensional
variables, appearing in eqs. (5.1) and (5.2)
versus the non-dimensional variables is
where the subscript c means the characteristic
quantity (constant) and the superscript
denotes the non-dimensional variable.
Introducing the above set of variables into eqs.
(5.1)-(5.2) we get the following dimensionless
equations

In this way three dimensionless numbers appear
the
Péclet number
the
Damköhler number
The Péclet number measures the convection/diffusio
n ratio in the pores.
The Damköhler number is the adsorption/diffusion
ratio at the pore surface.
Pl represents the time gradient of concentration
in relation to diffusion in the pores.
In practice, Pel and Ql are commonly used to
characterize the regime of a particular problem
under consideration.
In the homogenization process their order of
magnitude must be evaluated with respect to the
powers of the small parameter .
Each combination of the orders of magnitude of
the parameters Ql , Pl , and Pel corresponds to a
phenomenon dominating the processes that take
place at micro scale and different regime
governing migration at the macroscopic scale.

32
i). Moderate diffusion, advection and adsorption

the case of
The process of homogenization leads to the
traditional phenomenological dispersion equation
for an adsorptive solute
(5.8)
where -the effective diffusion tensor Dij is
defined as
(5.9)
and the vector field is the solution of
the standard (representative) O cell problem
is periodic

(5.10)
(5.11)
(5.12)
(5.13)

-the coefficient Rd, called the retardation
factor, is defined as
(5.14)
with the total volume of the
periodic cell
the volume of the fluid in
the cell
Sp the surface of the solid in
the cell
In terms of soil mechanics
(5.15)
with F the porosity
as the specific surface of the
porous medium defined as the global surface of
grains in a unit volume of soil .
-the effective velocity is given by
the Darcys law.

34
ii) Moderate diffusion and adsorption, strong
advection

the case
The process of homogenization leads to two
macroscopic governing equations that give
succeeding order of approximations of real
pollutant behavior.
(5.16)
(5.17)
where - the macroscopic dispersion tensor is
defined as
(5.18)
and the vector field is the
solution of the following cell problem
(5.19)
(5.20)

is periodic
(5.21)
(5.22)
-the coefficient Rd is given by (5.14) or
(5.15)
-the effective velocity is given
by the Darcys law.
In order to derive the differential equation
governing the average concentration lt c gt,
equation (5.16) is added to equation (5.17)
multiplied by e. after transformations the final
form of the dispersion equation is obtained that
gives the macroscopic model approximation within
an error of O(e2).
(5.23)
In this equation the dispersive term as well as
the transient term is of the order e.

36
iii) Very strong advection

the case
The process of homogenization applied to this
problem leads to the following formulation
obtained at e -1 order
(5.24)
(5.25)
Eq. (5.24) rewritten as
(5.26)
shows that there is no gradient of concentration
c0 along the streamlines. This means that the
concentration in the bulk of the porous medium
depends directly on its value on the external
boundary of the medium. Therefore, the
rigorous macroscopic description, that would be
intrinsic to the porous medium and the phenomena
considered, does not exist. Hence, the problem
can not be homogenized. This particular case will
be illustrated when analyzing the experimental
data.

37
iv) Strong diffusion, advection and adsorption

the case
The homogenization procedure applied to this
problem gives for the first order approximation
the macroscopic governing equation which does not
contain the diffusive term. Indeed, it consists
of the transient term related to the microscopic
transient term as well as the adsorption and the
advection terms
(5.27)
where Rd is given by (5.14).
The next order approximation of the macroscopic
equation is
(5.28)

where the symbol lt gt G means
(5.29)
The local boundary value problem for determining
the vector field is the following
(5.30)
(5.31)
is periodic

(5.32)
(5.33)
Remark that depends not only on the
advection, as it was in the case of , but
also on the adsorption phenomenon. Moreover, in
this case the pollutant is transported with the
velocity ltvgt equal to the effective fluid
velocity divided by the retardation factor.

The tensor D is expressed as
(5.34)
and depends on the adsorption coefficient a too.
Therefore D may be called the
dispersion-adsorption coefficient.
Remark that the second term in (5.27) represents
the additional adsorption contribution defined as
the interaction between the temporal changes of
the averaged concentration field lt c0gt and the
surface integral of the macroscopic vector field
lt gtG .
Finally, the equation governing the averaged
concentration lt c gt can be found by adding
eq.(5.27) to eq.(5.28) multiplied by e.
(5.35)
where
(5.36)
If eq. (5.35) is compared with eq.(5.23) it can
be concluded that the increase by one in the
order of magnitude of parameters Pl and Ql
causes that the transient term
in the macroscopic
equation becomes of the order one.

40
v). Large temporal changes

the case
This is also a non-homogenizable case and in this
case the rigorous macroscopic description, that
would be intrinsic to the porous medium and the
phenomena considered, does not exist.

41
6. COMPUTATIONAL RESULTS TESTED AGAINST
EXPERIMENTAL RESULTS

Experimental results obtained when the sample
length is L150 cm, the solid particle diameter
is dp0.35 cm and the porosity F0.41 are
reported in
Auriault J.L., Heterogeneous medium, Is an
equivalent macroscopic description possible?
Int.J.Engn,Sci.,29,7,785-795, 1995.
If the characteristic length associated with the
pore space in the fluid-solid system is defined
(after Whitaker 1972) as
(6.1)
then, the small homogenization parameter is
(6.2)
According to the theoretical analysis presented
in sequence 5, a rigorous macroscopic model
exists if the Péclet number, which characterizes
the flow regime, does not exceeds
In terms of the order of magnitude, this
condition can be written as
(6.3)

Therefore a dispersion test
through a sample of the length
L150 cm (dp0.35 cm) is correct from the
point of view of the homogenization approach,
provided the maximum Péclet number is much less
than . If the Péclet number
approaches , then the problem
becomes non-homogenizable and the experimental
results are limited to the particular sample
examined.
In the case considered by
Neung -Wou H., Bhakta J, Carbonell R.G.
Longitudinal and lateral dispersion in packed
beds effect of column length and particle size
distribution, AICHE Journal, 31,2,277-288 (1985)
the range of the Péclet number was 102 -104 which
is practically beyond the range of the
homogenizability.
In order to make the problem homogenizable, the
flow regime should be changed, namely the Péclet
number should be decreased. If however, we want
the Péclet number to be, for example Pe 103,
then the sample length L should be greater than
240 cm. Moreover, almost all the previous
experimental measurements quoted in the above
paper exhibit the feature of non-homogenizability.
For this reason the results obtained can not be
extended to size conditions.

Bues M.A. and Aachib M. studied in 1991 in the
paper
Influence of the heterogeneity of the solutions
on the parameters of miscible displacements in
saturated porous medium, Experiments in fluids,
11, Springer Verlag, 25-32, (1991).
the dispersion coefficient in a column of length
2 m, filled with a quasi uniform quartz sand of
mean diameter 1.425 mm. The investigated range of
the local Péclet number was 102-104.
Concentrations were measured at intervals of 20
cm along the length of the column. The
corresponding parameter e (ratio of the mean
grain diameter to the position x) for each
position was 1.3610-2 4.6710-3 2.810-3
2.0210-3 1.5710-3 1.2910-3 1.0910-3
1.0110-3 8.910-4 7.910-4 respectively.
The order of magnitude O(e-1) corresponds to 75
214 357 495 636 775 917 990 1123 1266
respectively.
It can be seen that the condition Pel ltltO(e-1) is
roughly fulfilled at the end of the column when
the flow regime is Pel 240.
The experimental data presented in the above
paper show the asymptotic behavior of the
dispersion coefficient that reaches its constant
value for
x 180.5 cm.
Thus, one can conclude that the required sample
length for the determination of the dispersion
parameter in this sand at Pel 200 is at least
2 m.

44
7. HEAT TRANSPORT IN POROUS MEDIA

An interesting example of heat transport in
porous media by convection and conduction
represents the relatively recent discovered
black smokers on the ocean floor. They are
observed at mid-ocean ridges, where upwelling in
the mantle below leads to the partial melting of
rock and the existence of magma chambers. The
rock between this chambers and the ocean floor is
extensively fractured, permeated by seawater, and
strongly heated by magma below. Consequently, a
thermal convection occurs, and the water passing
nearest to the magma chamber dissolves sulphides
and other minerals with ease, hence the often
black color. The upwelling water is concentrated
into fracture zones, where it rises rapidly.
Measured temperatures of the ejected fluids are
up to 3000C.

45
(No Transcript)
46

Another striking example of heat transport in
porous media is offered by geysers, such as those
in Yellowstone National Park. Here meteoric
groundwater is heated by subterranean magma
chamber, leading to thermal convection
concentrated on the way up into fissures. The
ocean hydrostatic pressure prevents boiling from
occurring, but this is not the case for geysers,
and boiling of water causes the periodic eruption
of steam and water that is familiar to tourists.

FAMOUS GEYSERS

48
(No Transcript)
49
(No Transcript)
50
(No Transcript)
51
(No Transcript)
52

The heat transport in soil can be studied by
means of a model in which the real heterogeneous
medium is replaced by the macroscopic equivalent
(effective continuum) like in the case of mass
transport.
In order to develop the macroscopic model the
homogenization technique of periodic media may be
employed. It can be shown that the assumption of
periodic media is equivalent to the existence
of an elementary representative volume in a non
periodic medium, containing a large number of
heterogeneities.
The starting basic equations for diffusion and
convection of heat according to
Mei C.C. Heat dispersion in porous media by
homogenization method, Multiphase Transport in
Porous Media, ASME Winter Meeting, FED
vol.122/HTD vol.186 11-16 (1991).
Lee C.K., Sun C.C., Mei C.C. Computation
of permeability and dispersivities of solute or
heat in periodic porous media Int.J.Heat and
Mass Transfer 19,4 p.661-675 (1996).
are given by
(7.1)
(7.2)

where
denote respectively the
temperatures, densities, thermal conductivities,
specific heats and partial volumes of the fluid
and solid in the O standard (representative)
cell. On the solid fluid interface G, the
temperatures and heat flux must be continuous
(7.3)
(7.4)
where nk represent the components of the unit
normal vector pointing out of the fluid. In eqs.
(7.1) and (7.2) energy dissipation by viscous
stress has been neglected, which is justifiable
for low Reynolds numbers.
It was assumed that the flow is independent of
the temperature. Therefore, in
eq. (7.1) represents the Darcys flow.
The derivation of the macroscopic model is
accomplished by the application of homogenization
method.
The macroscopic model is defined by the equation
(7.5)

where-the macroscopic conductivity tensor K is
(7.6)
-the function a(y) is given by
(7.7)
-the functions ?j belong to HY defined
as
(7.8)
and satisfy
(7.9)
The function space which appears in relation
(7.8) is the Sobolev space used in
Sanchez-Palencia E., Lecture Notes on
Physics.,vol.127, Springer, Berlin, 1980.

Equation (7.5) is similar to the equation
obtained in
Prasad V., Convective Heat and Mass Transfer in
Porous Media, Kluwer Academic Publishers,
Dodrecht, 1991, p.563
and for is similar to the eq.
presented in
Mei C.C., Auriault J.L., Ng C.O. Advances in
Applied Mechanics vol.32, Academic Press, New
York, 1996 p.309.

56
8. COMPUTED CONDUCTIVITY FOR THE HEAT TRANSPORT
IN POROUS MEDIA TESTED AGAINST EXPERIMENTAL
RESULTS

Lee C.C., Sun C.C. and Mei C.C. Computation of
permeability and dispersivities of solute or heat
in periodic porous media Int.J.Heat Mass
Transfer col.39,4 p.661-676 (1996).
compute and compare conductivity for heat
transport in porous media with experimental
results. In the following we will present these
results.
With the mean flow directed along the x-axis, the
longitudinal and transverse conductivities KL and
KT for heat were computed for Péclet numbers Pe
up to 300 for two porosities F 0.38 and F
0.5 the thermal properties for fluid and solid
phases were assumed to be equal kf ks and ?s
cs ?f cf. They were compared with some
experimental results for randomly packed uniform
glass spheres in water with roughly comparable
thermal properties reported in
Levec J. and Carbonell R.G. Longitudinal and
lateral thermal dispersion in packed beds. II.
Comparison between theory and experiment
A.I.Ch.E.J. 31, 591-602 (1985)
Green D.W., Perry R.H. and Babcock R.E.
Longitudinal dispersion of thermal energy
through porous media with a flowing fluid
A.I.Ch.E.J. 10,5, 645-651 (1960).

In the limit Pe 0, both (KL, KT) approach unity
because the composite medium is homogeneous and
there is no distinction between Of and Os for
pure diffusion.
For a simple cubic packing of spheres with F
0.48 and ks kf 2
Sangani A.S. and Acrivos A. The effective
conductivity of a periodic array of spheres
Proc.R.Soc. Lond. A.386, 262-275 (1983)
give KT 1.46.
As a check Lee et al. have also calculated the
effective conductivities with F 0.5 and the
same ratio of conductivities ks, kf and obtain
KT 1.458. The small discrepancy is again due to
different grain geometries.
Computation in the relatively high Pe region show
that the dispersivities KL, KT increase with
decreasing porosity as in the case of passive
solute. This is again due to increased micro
scale mixing in the pore space caused by
increased velocity gradient for smaller porosity
value. The same trend has been observed for 2D
array of cylinders in
Sahrani M. and Kavary M., Slip and no slip
temperature boundary conditions at the interface
of porous media convection, Int.J.Heat Mass
Transfer 37, 1029-1044 (1994)

The experimental data for KL show a growth of
(Pe) m , where m has been estimated to be 1.256
by
Levec J. and Carbonell R.G. Longitudinal and
lateral thermal dispersion in packed beds. II.
Comparison between theory and experiment
A.I.Ch.E.J. 31, 591-602 (1985)
and 1.4 by
Green D.W., Perry R.H. and Babcock R.E.
Longitudinal dispersion of thermal energy
through porous media with a flowing fluid
A.I.Ch.E.J. 10,5, 645-651 (1960).
The discrepancy between theory and experiments
must be again attributed to the difference in
packing.
To see the effect of ks/kf, were calculated KL
and KT for two porosities F 0.38 and F 0.5
and two conductivity ratios, ks/kf 0 and 1. At
the higher Péclet number, the longitudinal
conductivity KL is greater, although the
difference is small. This increase is due to heat
diffusion in the solid phase. When the thermal
gradient is in the direction of the mean flow,
diffusion through the solid phase augments
dispersion Kxx in the fluid when ks/kf ? 0. But
for Kyy which is associated with the thermal
gradient normal to the flow, transverse
dispersion is weakened by the loss of heat into
solid. Quantitatively, the effect of ks/kf1 on
either KL and KT appears to be significant only
at relatively low Péclet number. This result is
reasonable since for high Pe dispersion by
convection must be dominated and diffusion in the
solid must become immaterial.

Thank you for your attention

Write a Comment

User Comments (0)

About PowerShow.com

Mathematical models for mass and heat transport in porous media - PowerPoint PPT Presentation

Mathematical models for mass and heat transport in porous media

In fact, Brenner showed how Darcy's experimental law and the permeability tensor ... 1.The Darcy's law (3.13) gives the global flow field in the periodic porous ... – PowerPoint PPT presentation