Transport Processes in Environments with Irregular Structure

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.


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.
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 curvilinear coordinates .
We Where -the local radius-vector , corresponds to the center .The average A r 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 appropriate systems of orthogonal curvilinear coordinates, which is naturalized for the pore geometry.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 nonconducting, then the current lines will be located along Σ and its equipotential is orthogonal to it.Due to randomness of the structure, some equipotential necessarily tangent to the walls Σ. Points (or curved) of contact of tangency will determine 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 Ω i 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 The method of eigenvalue coordinates of pores allows to obtain the transport equations in a single pore, expressing its coefficients through the local parameters (8).Let us for this purpose 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 terms 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 interpreted 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 ... (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 through the integral character istics of 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 In the chaotic porous space, these curvatures are independent for different directions,  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- 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.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 necessary to generalize, since the flow of diffusion current for Acting by the operator of summing up Σ to both parts of ( 12) again, we come to the homogeneous equation analogous (11).Summarizing the state that the above obtained relations connecting the effective transport coefficients with the structure of an arbitrary porous environment.A general method is developed, which allows to obtain in different cases the homogeneous transport equation.

Let us return to evaluation quantity
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., ) ...(1) 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) Next, we use the expression of the total current through the pore, equal to 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 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) of the equation (10 a) is an arbitrary 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: ...(11a) 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:

( 1 )From
, 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: ...(15) 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 sample from 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.
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) 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 D 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 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)

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 iC 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 , flow to the surface of the pore And the depth of penetration of diffusion into the porous environment is assigned.

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 of the layer of unit thickness, containing a single pore.Therefore it is clear that effective conductivity