summer university, vrnjacka banja, october 2007 1 mathematical models for mass and heat transport in...

Summer University, Vrnjacka Banja, October 2007

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 Science

Faculty of [email protected]; [email protected]

mailto:[email protected]

mailto:[email protected]


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


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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 Γ. The coefficient α denotes the adsorption parameter (α > 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 (Darcy’s law and the incompressibility condition)

• (5.3)• (5.4)

• which has been already presented in the earlier sequence, will be directly used.

0

cvx

cD

xt

ci

jij

i

ont

c

x

cDN

jiji

vN

pku 0u


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:

*;*

;*;*;;*

cc

ccc

DDD

vvvtttylXccc

0***

**

*2

cv

D

lv

y

cD

yt

c

tD

li

c

c

jij

icc

*

**

**

t

c

tD

l

y

cDN

cc

c

jiji


6

• In this way three dimensionless numbers appear:• the Péclet number

• the Damköhler number

• The Péclet number measures the convection/diffusion 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.

c

cl D

lvPe

cc

cl tD

lQ

ccl tD

lP

2

Ll


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) Ω cell problem:

• is periodic (5.10)• (5.11)

• (5.12)

• (5.13)

.,, 122 OPeandOQOP lll

0000

*0

i

ijij

id vc

xx

cD

xt

cR

f

dy

IDDk

jkjikij

1*

j

j

01

f

d

0

j

kjkjk

i yID

y

.0

ony

IDNj

kjkiji


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 Φ = 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 Darcy’s law.

pf

d

SR

f

sd aR

/ps Sa

0iv


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)


000

cvx ii

01001

0**

0

cvcv

xx

cD

xt

cR ii

ijij

id

f f

dkx

pd

yIDD ji

kk

jkjikij

11

01** 11

1j

001

01 1

kki

ki

j

kjkij

i

vvy

vy

IDy

ony

IDNj

kjkiji 0

1


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 Darcy’s law.

• In order to derive the differential equation governing the average concentration < c >, equation (5.16) is added to equation (5.17) multiplied by ε. after transformations the final form of the dispersion equation is obtained that gives the macroscopic model approximation within an error of O(ε2).

• (5.23)•

• In this equation the dispersive term as well as the transient term is of the order ε.

1j

f

d 01 11

0iv

)( 2** Ocvxx

cD

xt

cR i

ijij

id


11

iii) Very strong advection• the case:

• The process of homogenization applied to this problem leads to the following formulation obtained at ε -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.


000

cvy ii

ony

cDN

jiji 0

0

00

i

o

i y

cv

V


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)


0000

cv

xt

cR i

id

0

'

0***

10120

jij

i

oii

ii

id

x

cD

x

cvcvxxt

c

t

cR


13

• where the symbol < > Γ 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 <v*> equal to the effective fluid velocity divided by the retardation factor.

•

dSii

22 1 2

0*02

02

kki

ki

j

kjkij

i

vvy

vy

IDy

onvy

TDN kj

kjkiji

*02

2i

f

d 01 22

21


14

• The tensor D*** is expressed as:• (5.34)

• and depends on the adsorption coefficient α 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 < c0> and the surface integral of the macroscopic vector field < >Γ .

• Finally, the equation governing the averaged concentration < c > can be found by adding eq.(5.27) to eq.(5.28) multiplied by ε.

• (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 P l and Ql

causes that the transient term in the macroscopic equation becomes of the order one.

f

dR

vvy

IDD id

jjik

jkjikij

20202

*** 1

2

)( 2**** Ocvxx

cD

xt

cR i

ijij

id

2*

ii

dd xRR

tcRd *


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.



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6. COMPUTATIONAL RESULTS TESTED AGAINST EXPERIMENTAL RESULTS

• Experimental results obtained when the sample length is L=150 cm, the solid particle diameter is dp=0.35 cm and the porosity Φ=0.41 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)

1pdl

3106.1150

24.0 L

l

)()( 10 OOPel

31 106.0)( OPel


17

• Therefore a dispersion test through a sample of the length L=150 cm (dp=0.35 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.

3106.0 3106.0


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• 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 ε (ratio of the mean grain diameter to the position x) for each position was: 1.36·10-2; 4.67·10-3; 2.8·10-3 ; 2.02·10-3; 1.57·10-3 ; 1.29·10-3 ; 1.09·10-3 ; 1.01·10-3 ; 8.9·10-4 ; 7.9·10-4 respectively. The order of magnitude O(ε-1) corresponds to 75; 214; 357; 495; 636; 775; 917; 990; 1123; 1266 respectively.

• It can be seen that the condition Pel <<O(ε-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.


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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.


20


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.


22

• FAMOUS GEYSERS


23


24


25


26


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)

fffj

fj

fff xTk

x

Tu

t

Tc

ssss

ss xTkt

Tc


28

• where denote respectively the temperatures, densities, thermal conductivities, specific heats and partial volumes of the fluid and solid in the Ω standard (representative) cell. On the solid fluid interface Γ, 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 Darcy’s 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)

sfsfsfsfsf andcckkTT ,,,,,,,,

xTT sf

xn

x

Tkn

x

Tk k

k

ssk

k

ff

321 ,, uuuu

000TKTu

t

Tc

css

ff

ssf


29

• where:-the macroscopic conductivity tensor K is: • (7.6)

• -the function a(y) is given by• (7.7)

• -the functions Ψj 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.

3,2,1,1

jidyy

yaKi

jijij

sm

s

f

yifkk

yifya

1

01 dyyandperiodicHHY

j

j yyaya )()(


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.

00u


31

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 Φ = 0.38 and Φ = 0.5; the thermal properties for fluid and solid phases were assumed to be equal kf = ks and ρs · cs= ρf · 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 Ωf and Ωs for pure diffusion.

• For a simple cubic packing of spheres with Φ = 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 Φ = 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: Φ = 0.38 and Φ =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/kf=1 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

summer university, vrnjacka banja, october 2007 1 mathematical models for mass and heat transport in...

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