Title: Mathematical models for mass and heat transport in porous media II'
1Mathematical models for mass and heat transport
in porous media II.
- Agneta M.Balint and Stefan Balint
- West University of Timisoara, RomaniaFaculty of
Mathematics- Computer ScienceFaculty of Physics - balint_at_balint.uvt.ro balint_at_physics.uvt.ro
-
2TOPICS
- 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
35. 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.
4- 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.
5- 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
6- 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.
7i). 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)
8- -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.
9ii) 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) -
10- 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.
11iii) 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.
12iv) 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)
13- 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. -
14- 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.
15v). 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.
166. 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)
17- 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.
18- 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.
197. 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.
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21- 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.
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27- 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)
28- 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)
29- 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.
30- 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.
318. 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).
32- 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)
33- 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.
34- Thank you for your attention