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Mathematical models for mass and heat transport in porous media II'

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


1
Mathematical 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

2
TOPICS
  • 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

3
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.

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.

7
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)

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.

9
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)

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.

11
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.

12
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)

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.

15
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.

16
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)

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.

19
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.

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  • 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.

22
  • FAMOUS GEYSERS

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  • 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.

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  • 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.

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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).

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)

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  • 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.

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  • Thank you for your attention
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