Transcript Mathematical models for mass and heat transport in porous

Mathematical models for mass
and heat transport in porous
media II.
Agneta M.Balint and Stefan Balint
West University of Timisoara, Romania
Faculty of Mathematics- Computer Science
Faculty of Physics
[email protected]; [email protected]
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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.
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• 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:

c  
c
•
(5.1)

 Dij 
 vi  c   0

t xi 
x j

•
(5.2)



c

c
  
 N i  Dij 
on 



x j 
t
• where c is the concentration (mass of pollutant per unit volume of fluid),
Dij is the molecular diffusion tensor, t is the time variable v is the flow
field and N 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)
u  k  p
•
(5.4)
u  0
• which has been already presented in the earlier sequence, will be
directly used.
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• 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
c  cc  c *; X  l  y ; t  tc  t *; v  vc  v *;
D  Dc  D *;    c   *
• 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:

l2
c *  
c * vc  l


 Dij * 

 vi * c *  0


Dc  tc t * yi 
y j
Dc


c *   c  l
c *

 N i  Dij * 

 *

y j  Dc  tc
t *

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• In this way three dimensionless numbers appear:
v l
•
the Péclet number
Pel  c
Dc
•
•
•
•
•
•
Ql 
c  l
Dc  tc
the Damköhler number
l2
Pl 
Dc  tc
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   l L .
• 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.
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i). Moderate diffusion, advection and adsorption
– the case of:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  1 .
• The process of homogenization leads to the traditional
phenomenological dispersion equation for an adsorptive solute:
c 0
  * c 0   0
•
(5.8)

Rd 

Dij 

c   vi0    0
t xi 
x j  xi
• where: -the effective diffusion tensor Dij* is defined as:
 j 

1
*
•
(5.9)


Dij 
D

I

d

ik
kj

yk 
 f

• and the vector field  j is the solution of the standard (representative)
Ω cell problem:
 j is periodic
•
(5.10)
1
•
(5.11)
  
  d  0

•
•
f
 
 k 

D
I

 jk  jk
0

y j 
 
 



k
  0 on .
 Ni  Dij  I jk 

y j 
 
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

yi
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(5.12)
(5.13)
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• -the coefficient Rd, called the retardation factor, is
defined as
•
•
•
•
•
•
•
•
Rd 
 f   Sp

(5.14)
with  = the total volume of the periodic cell
 f = the volume of the fluid in the cell
Sp = the surface of the solid in the cell
In terms of soil mechanics
Rd      as
(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
as.  S p / 
of soil;
0
• -the effective velocity  vi  is given by the Darcy’s law.
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ii) Moderate diffusion and adsorption, strong advection
• the case:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  0 .
• The process of homogenization leads to two macroscopic governing
equations that give succeeding order of approximations of real pollutant

behavior.
 vi0  c0  0
xi
•
(5.16)
0
0
1
c
  ** c  
Rd 

Dij 

 vi1  c 0   vi0  c  0
t xi 
x j  xi
•
(5.17)
• where: - the macroscopic dispersion tensor is defined as:
1
0




1

p
1
j
•
(5.18)
**
d 
D 
D I 

k   1 d




ij
•
•
•


f
ik


and the vector field 
kj
1
j
yk 
xk


i1
j
f
is the solution of the following cell problem:
  
 k1   0  k1 1
(5.19)

Dij I jk 
 vi 
  vk0  vk0
yi  
y j  
yi 



1 



 N i  Dij  I jk  k   0 on 

y j 

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(5.20)
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•
•
•
•
•
 1j is periodic
 1  
1

(5.21)
1

 d  0
(5.22)
f
-the coefficient Rd is given by (5.14) or (5.15)
-the effective velocity  vi0  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).
 **   c   


c


•

 vi    c   O( 2 ) (5.23)
  Rd 
 
  Dij 
•
t
xi 
x j  xi
• In this equation the dispersive term as well as the transient
term is of the order ε.
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iii) Very strong advection
• the case:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  1 .
• The process of homogenization applied to this problem leads to the
following formulation obtained at ε -1 order:
•
(5.24)
 0 0

v
y
i
•
•

c  0
i
0 


c
  0 on 
 N i   Dij 

y j 

Eq. (5.24) rewritten as
(5.25)
o

c
vi0 
0
•
(5.26)
yi
• 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 V 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.
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iv) Strong diffusion, advection and
adsorption
 
 
 
• the case: Pl  O  , Ql  O  , and Pel  O  .
•
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:
0
c

•
(5.27)
R 

 v0  c0  0
1
d
1
t
0

x
i

i
• where Rd is given by (5.14).
•
The next order approximation of the macroscopic equation is:
  c'    c 0  
 1
2 
0
o
1
R










v



c



v



c

i

i
i
• d
t
t xi
xi




(5.28)
 ***   c 0 
 Dij 
0
xi
x j
 
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• where the symbol < > Γ means
1
2
•
  i  

2

 i dS

(5.29)
2
• The local boundary value problem for determining the vector field
is the following:
2 
2

 




0
k
k
  vi 

 vk0 * vk0
 Dij   I jk 
yi 
y j 
yi
•
(5.30)

•
•
2 




 Ni   Dij   T jk  k      vk0 * on 

y j 


i2
is periodic
 2 
1
2

d  0

 f
(5.31)
(5.32)
•
(5.33)
2
• Remark that  depends not only on the advection, as it was in the
1
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.
•
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• The tensor D*** is expressed as:
2





•
1

j
***


Dij 


f



v



v





Dik   I kj 
  d


y
R


k 
d



0
i
2
j
0
j

2
i
(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 <  2 >Γ .
• Finally, the equation governing the averaged concentration < c > can
be found by adding eq.(5.27) to eq.(5.28) multiplied by ε.
•
c
  ***   c   

R 
 
Dij 

 vi    c    O( 2 )
t
xi 
x j  xi
*
d
(5.35)

• where
*
2
R

R







d
d
i 
•
(5.36)
xi
• 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 Rd*    c  t  in the macroscopic
equation becomes of the order one.

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
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v). Large temporal changes
• the case:
 
 
Pl  O , Ql  O  , and Pel  O  .
2
1
• 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)

l  d p 
1 
• then, the small homogenization parameter is:
l 0.24
•
(6.2)
 
 1.6  103
L 150
• 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 Pel  O( 0 )  O( 1 )
• In terms of the order of magnitude, this condition can be written as:
Pel  O( 1 )  0.6 103
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(6.3)
16
•
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 0.6 103 . If the Péclet number approaches0.6 103 ,
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 nonhomogenizability. For this reason the results obtained can not be
extended to size conditions.
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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.
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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.
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•
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:
•
•
 T
T 
 f  c f   f  u j  f   k f  T f
x j 
 t
 s  cs 
Ts
 k s  Ts
t
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xf
x  s
(7.1)
(7.2)
27
•
•
•
•
•
•
•
•







Tf ,Ts ,  f , s , k f , ks , c f , cs and  f , s
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
Tf  Ts
x 
(7.3)
T f
Ts
(7.4)
 nk
x
xk
xk
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, u  u1, u2 , u3  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:
kf 
f 
 nk  ks 
 s  cs
 s
 f  cf
T s


t
0

 u  T   K  T
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0
0

(7.5)
28
• where:-the macroscopic conductivity tensor K is:
 j 

1
•
dy
K ij 
a y     ij 
i, j  1,2,3


y
 
i 

•
-the function a(y) is given by
•
if y  
1

a y   k
s

 km
(7.6)
(7.7)
f
if
y  s
-the functions Ψj belong to HY defined as:


(7.8)
1
HY    H    periodicand   y dy  0



• and satisfy
(7.9)

 a( y)   j     a( y)  y j
• 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
Summer University, Vrnjacka
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.
29
• 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
•
0
u  0 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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30
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).
Summer University, Vrnjacka
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31
• 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)
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32
•
The experimental data for KL show a growth of (Pe)
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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m
, where m has been
33
Thank you for
your attention
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34