INTRODUCTION
It is known that the difficulty of solution of the problem about the transport of substance and charge in a porous environment is caused by the presence of the complex interphase boundary Σ of pore space, which for a chaotic environment is random. In the most realistic case, the very notion of the pores is not defined, which is intuitive. Therefore, in applied researches resort to the simplified representations, replacing real porous by system modeling. Modeling is achieved by a precise definition of the pores and becomes necessary to solving the relationship between such integral quantities as porosity , specific internal surface , the average pore radius , and etc. However, in this case arbitrary assumptions are inevitable, the loss of statistical information always remains an opened question about the adequacy of environment and model, since by its criterion it can serve only the degree of the correspondence of the integral parameters of model and actually measured. Unfor tunately, not all the integral characteristics are measured, and the results of calculations depend on an arbitrariness in the choice of the internal geometry of the environment or the assumptions of its probabilistic properties.
Transport Processes in Environments with Irregular Structure
SAIF A. MOUHAMMAD
Department of Physics, University College - Um Alqura University, Makkah (Saudi Arabia).
(Received: May 10, 2011; Accepted: June 12, 2011)
ABSTRACT
In this work we consider a principally new statistical approach to the theory of processes of transport in a two-phase condensed environment with randomly distributed non-uniform surface structure. Taking this approach as a base, we considered diffusion in a chaotic porous environment. Such a structure is described with the aid of a curvilinear orthogonal coordinates system natural for geometrical porous surface. The method of averaging the diffusion equation is developed. The equations for average diffused concentration in a porous surface of a solution of ionic components are obtained. These equations take into account the local characteristics of the structure of the environment. In general, this approach is applicable to other equations describing the transport of substance, charge and electromagnetic field in the environment with random interior.
Key words: Transport process theory, Environment, Irregular structures.
In this paper, a proposed method for calculating the above- indicated quantities for an arbitrary environment, principally different from the commonly used. It allows us to consider the geometry of the pore space and to investigate transport processes in it without the modeling approaches. The structure of space is assumed to be chaotic, i.e., random. This method is based on the use of an apparatus of orthogonal curvilinear cylindrical coordinates, moreover in each
i
pore is determined by its eigenvalue cylindrical curvilinearcoordinates .
We define at first the average value of
some local quantities by volume ΔV,
sufficiently small relative to the size of the environment
L
, but large in comparison with the characteristic scale
R
of surface change( i.e., )
Where - the local radius-vector ,
corresponds to the center . The average Ar
can be identified with the observable quantities (since measurement inevitably accompanies averaging) and simultaneously integrated characteristics of environment and processes (at presence of processes, random in time of eq. (1) must be averaged and for some time interval δt, exceeded the period of temporary time fluctuations, since with the reproducible measurements of the fluctuations also averaged). The problem of transport theory in porous environments consists in averagings of the true (microscopic) equations
for local and the solution of the equations
obtained in this way for those are observed . Thus equivalence between obtained average and true integrated properties of system is automatically observed.
The mathematical apparatus, which allows describing the geometry of the pores and average necessary value, in this paper, is based on the choice of appropr iate systems of or thogonal curvilinear coordinates, which is naturalized for the pore geometr y. With this purpose, we consider the flow of some physical quantity (for determining electric field) in the liquid phase, which does not contain volumetric sources. It is obvious that, the current lines and its equipotential (related to the solution of Laplace equation ) are determined by the geometry of the pore space Ω and the additional conditions on its boundary . If the surface Σ is non-conducting, then the current lines will be located along Σ and its equipotential is orthogonal to it. Due to randomness of the str ucture, some equipotential necessarily tangent to the walls Σ. Points (or curved) of contact of tangency will deter mine simultaneously the places of the branching of current. Such equipotentials will divide entirely the pore space Ω into the separate elements, in each of which branch point will be absent. single- valued regions Ωi in general case are simply individual pores.
In each i pore let us consider – instead of orthogonal curvilinear coordinate- a system of lines
, coinciding with the current lines, and surfaces i
q
, coinciding with the equipotentials. On the surfaces, we introduce a lines, orthogonal to the
walls of the pores, and lines , closed and
orthogonal to lines.
Reference index and intervals change of coordinates we will define them as follows: 1)
is counted from initial, i.e. passing through branching point of equipotential plane. Then, final equipotential corresponds to maximum value ;2) is counted from an axial line, unique in
each pore. The maximum value of is achieved at the wall of the pore; 3) the coordinate
is cyclic, measured in radians , and the
beginning of its index is arbitrary..
Coordinates for each of the
pore are Eigenvalue. Each one q should provide with an index , for simplicity can be omitted. With their aid we can get an expression for the conductivity and other characteristics of individual pores.
Actually, the potential ϕ, is defined by the boundary-value problem
-normal to the walls of pores, in their eigenvalues coordinates depends only on
1
q
,
moreover ( - Lame’s parameters)
...(2)
Hence, taking into account , we have
...(3)
The resistance of pores
...(4)
lateral surface area and volume of
individual pore analogously are determined. For
an element, allocated by surfaces
1 q=q
and
, we have:
...(5)
...(6)
Cyclicity allows to consider as the
generalized cylindrical coordinates, and below it is convenient to redesignate:
...(7)
are measured in the arbitrary units of the
length (which correspond to the limiting values
and dimensionless coefficients
r
α
). Elementary
arcs Using this, it is possible to connect
with the average section of pores, not
depend on z. Since for the rectilinear cylindrical coordinates
that the difference from unit i
α assigns the measure
for the deviation of generalized coordinates
from the usual cylindrical.
on the transition the
characteristics of individual pore (4)-(6) after utilizing dimensionless coordinates with the aid of the scales
will be transformed to the form
...(8)
consider, for example, the diffusion of the components of the solution with the concentration in the liquid pores, determined in the general case by boundary-value problem
...(9)
Where - the density of the
volumetric and surface sources, - normal to the interphase boundary Σ. The equation (9) is now to carry on an individual i pore, assuming Σ its lateral
surface. In Σ its eigenvalue coordinates , are made dimensionless using the scale
The problem (9) is written in the form of
(index and other variables, are omitted
for brevity, it is assumed):
...(10)
The conjugation conditions at the border with neighboring pores is expected to be met. If all
the ter ms are multiplied by a factor
integrated by the coordinates and take into account the boundary conditions , then (10) is reduced to use in the future as an integral form
...(11)
In the equation (10), the parameter ε is equal to the ratio of the characteristic scales of
absorption flow on the boundary surface and
diffusion,
But L is inter preted as the depth of penetration of diffusion into the volume of the environment. Since it is assumed that the substance is transferred through entire pore space, then and
are small
0
q
and ε respectively (Otherwise,
the limit within one pore, absorption almost complete). Therefore in (10) it is possible to use a
perturbation theory and to represent
with a series in powers of small parameter . Then in the zero by approximation, on the basis of (10), we have
...(10a)
Physically necessar y solution of the equation (10 a) is an arbitrar y function
of time and eigenvalues coordinate
z. It simultaneously satisfies the equation (11) which is due to the independence of C on r and φ is simplified:
Coefficients (11a) coincide with the
integrals (8), and if we make
dimensionless with the aid of the scales then (11) is led to the formula:
or in the dimensional form:
...(12)
Thus, it is possible to establish, that the equation (12), which expresses the balance of substance in the volume of a certain pore, connected the transport process with its local characteristics. This gives the possibility to consider the problems, complicated by the presence of mobile interphase boundary and to express the effective transport coefficients t h r o u g h t h e i n t e g r a l c h a r a c t e r i s t i c s o f environment. Let’s proceed with expressions (8) for local parameters
Preliminarily let us pass, using determination of sinuosity
...(13)
from z to the homogeneous coordinate x , on which all depend macroscopical, i.e. the averaged values. In a case of plane-parallel medium considered below the axis is perpendicular to its surface and
...(14)
Where - dimensional, and therefore Z is
absent.
For the calculation it is
necessary to average (14) with the aid of the rule
(1), and since are volume and surface of
i
pores, then as a result of their averagings we will
obtain and As the volume of averaging
let us select infinitely thin layer
with the area of base and will
act on the equation (2) by the operator
Summation over all ΔN(x) pores of the layer. Then on the left side we obtain:
To the right in (14) it is convenient to represent
...(16)
It is obvious, that is equal the density of
pores, which intersect layer
At formula
determines mean-static on the infinite
ensemble of pores of section
x
. From this point of view, by the force of limitation use (16) gives, strictly speaking, an average on a certain samplefrom this ensemble, and the density
ν
can
fluctuate. However, since the probability of
deviations from is proportional , and
, then the below noted difference is ignored.
From (14), (15) we find .
...(17)
Calculating , taking into account that
they characterize the curvatures of arcs
in the generalized coordinate system relative to the
appropriate arcs
, ,
z r
dl dl dlϕ
of the usual cylindrical
system (where ).
In the chaotic porous space, these curvatures are independent for different directions,
so that
At the same time, as a result of uniformity of medium
along the section and complete chaos in the geometry of pores, any deviations from one are equally probable and consequently
...(18)
If dispersions and are small,
then
...(19)
And by the force of (17)-(19), we obtain
...(20)
Thus, five integral character istics
, are bound by three relations
(20). From them, in particular, follow the equalities
..(21)
which in the general case of arbitrary environment were not obtained. For the problems of electrical conductivity the diffusion coefficient
in (20) is substituted by the coefficient of electrical conductivity χ. Sinuosity β is found from the
determination of arc length . Actually,
since
Where
Then, averaging this equation over an ensemble of pores, we have
Since
(Where- the Directing angles of
the arcs ) and averagings over the ensemble
and all orientations of arcs relative to axis
x
are equivalent, then
2 2 2
2 2
2 2
1 4
cos cos , .
4
i dx
d d
dl
π π
π π
π
α α β
π − − π
⎛ ⎞
= Θ Θ × = =
⎜ ⎟
⎝ ⎠
∫
∫
With the aid of that used above for calculating (20) the procedure it is possible to average the equation of diffusion in the separate pore. In the beginning in (12) we will pass from eigenvalues to the homogeneous coordinate
. Then we obtain taking into account (13)
...(13)
On both parts of (20), let us act by the operator of summation
and let us consider that the concentration
must weakly depend on index i, i.e., the
number of pore in the section ΔΣ. This is a consequence of connectivity and cross-pore space. Owing to the fulfillment of conditions for conjugation in the points of small intersections of the
root-mean-square spread σ of values , caused by difference in the geometry of the pores of layer
and their previous branchings (see
below). Therefore all values, connected with
it is possible to carry out from the summation sign and then from (22) we find
Where and have already the
sense of averages by volume density of
sources. Hence, taking into account (15) we finally obtain the required equation for the averages
...(23)
The presented method makes it possible to derive in the subsequently equations, analogous (23), in the case of mobile interphase boundary. The diffusion of the components of the solution in the porous medium, accompanied by their crystallization on the pore walls, can serve as its example.
For this it is expedient to consider the simplest model, which assumes all intersections with those
(!!!!) located in the planes , where
- the average length of the pores.
Furthermore, let us accept their radii , sinuosity
as that being independent of , and the
probability of branching and confluence (merging) in each plane equal. Then, using expressions for
the concentrations at the nodal points ( )n i
C which
appear from equations of diffusion, also of conjugating conditions, it is possible to show that
Summing up is here produced with the aid of , On the Boundary , (corresponding
) all concentrations are equal and,
consequently . The parameter
K
in the
expression defines the flow to the surface of the pore
And the depth of penetration of diffusion into the porous environment is assigned.
And since it is great in comparison with
the length of pore , then σ is small and,
therefore, is weakly depends on i . Further let
us pause in greater detail at the determination of sinuosity . Formula (13) assumes that, β - the single-valued function of homogeneous coordinate . However, this condition is disrupted, if a certain part of the pores had turning points, so that they
intersect layer several times in
opposite directions (in the chaotic pore space the probability of this it is certain, it is small). Then procedure presented above necessar y to generalize, since the flow of diffusion current for
n
-branching pore corresponds to the series, but
not parallel connection of its elements of those
belonging to the allocated layer , and this influences the calculation of , moreover sinuosity
. Considering for simplicity the case
of absence the sources, it is easy to show that the average for all
n
-branches of the pore
concentration value
n
C
satisfies the equation
Where
n
i
- the current through the
n
pore. According to its meaning n
χ
is the conductivity of the layer of unit thickness, containing a single pore. Therefore it is clear that effective conductivity
Acting by the operator of summing up Σ to both par ts of (12) again, we come to the homogeneous equation
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