ВОДОРОДНАЯ ЭНЕРГЕТИКА И ТРАНСПОРТ
Газоаналитические системы и сенсоры водорода
HYDROGEN ENERGY AND TRANSPORT
Gas analytical systems and hydrogen sensors
THE NON-LOCAL FORMULATION OF ELECTROSTATIC PROBLEMS FOR SENSORS HETEROGENEOUS TWO- OR THREE PHASE MEDIA, THE TWO-SCALE SOLUTIONS AND MEASUREMENT APPLICATIONS
V. S. Travkin , A. T. Ponomarenko* It11
1 Member of International Editorial Board -11 Member of International Editorial Advisory Board
Hierarchical Scaled Physics and Technologies (HSPT) 10431 Larwin Ave., Chatsworth, CA 91311, USA E-mail: travkin@iname.com
Enikolopov Institute of Synthetic Polymeric Materials, Russian Academy of Sciences Profsoyuznaya, 70, Moscow, 117393, Russia
The main idea of performing transport and field description of electrostatic problems in heterogeneous media of sensors on the two scales through the spatial non-local theorems of the volume averaging theory (VAT) is to provide the means to account for multiple description scale characteristics and requirements. At present time it is the only consistent and reliable theory available for multiscale either 1D or 2D, 3D, linear or nonlinear statement tasks. The VAT approach has been successfully applied in the last two decades to a number of difficult problems in fluid mechanics, thermal physics, environmental science in heterogeneous media and in porous media. The present work suggests the mathematical formulation and few solutions for heterogeneous media electrostatic problems based on the VAT averaged homogeneous and inhomogeneous Maxwell's equations along with the potential field equations.
Three types of heterogeneous media are addressed for solution: 1D layered, capillary media and globular media. The results for capillary type of media were for media made up of random size capillaries in both dilute approximation and with conjugate interacting particulate fields. New types of mathematical models and governing equations for conductive dielectric composites are suggested. Various simplifications have been analyzed and some applications and comparisons with available theoretical approaches are made.
Nomenclature
» C — mass fraction concentration;
<
<c B — magnetic flux density [Wb/m2]
^ cd — mean drag resistance coefficient in the REV;
| cd — mean skin friction coefficient over the turbu-
| lent area of dSw;
| cp — specific heat, J/(kgK);
* ds — interface differential area in porous medium,
° m2;
i dS12 — internal surface in the REV, m2;
S D — molecular diffusion coefficient, m2/s;
o 2
q D — electric flux density, C/m2; E — electric field, V/m;
f ={fi} — VAT intrinsic phase averaged over AQ; value f ;
(f)f — VAT phase averaged value f, averaged over AQ; in a REV;
f — VAT morpho-fluctuation value of f in a
— time averaged value f ; j — current density, A/m2;
k1 = kf — fluid phase thermal conductivity, W/(mK);
k2 = ks — homogeneous thermal conductivity of solid
phase, W/(mK);
H — magnetic field, A/m;
m — porosity;
(m) — averaged porosity;
n — refraction index;
p — pressure (Pa) and phase function;
Q — electrical power from heater dissipated through
the specimen;
Статья поступила в редакцию 04.01.2005 г. The artisle has entered in publishing office 04.01.2005
— solid phase fraction;
12 — specific surface of a porous medium dS12/ÀO,
1/m;
T — temperature, K;
T2 = Ts — solid phase temperature, K;
T — interface surface temperature when i is in
upward direction, K; U — vector component in x-direction; V — vector component in y-direction; W — vector component in z-direction;
Subscripts
c — charge; e — effective; eff — effective; ex — experimental; f = 1 — fluid phase; 1 — first phase;
i — component of vector variable; L — laminar; r — roughness; s = 2 — solid phase; T — turbulent; w — wall.
Superscripts
--value in fluid phase averaged over the
phase REV AQ„;
* — equilibrium values at the assigned surface and complex conjugate variable; — fluctuation value in a phase.
Greek letters
12
12
a 21 — averaged heat transfer coefficient over dS W/(m2K);
a C — averaged mass transfer coefficient over dS m/s;
ed, em — dielectric permittivity, Fr/m; (xm — magnetic permeability, H/m; v — frequency (Hz) and kinematic viscosity, m2/s; pc — electric charge density, C/m3; p — density, kg/m3;
o — medium specific electric conductivity, A/V/m; O — electric scalar potential, V; ra — angular frequency, rad/s; AQ — representative elementary volume (REV), m3; AQ1 = AQf — pore or phase 1 volume in a REV, m3; AQ2 = AQQ — second or phase 2 volume in a REV, m3; Tw — wall shear stress, N/m2.
Introduction
There are quite a few methods used to describe the field modeling in electrostatic and general electrodynamics problems. In the literature starting from fifties with significant rise in research activity in seventies — eightieths actually given a number of analysis and approaches as by, for example, [1—3] etc. to name just a few. The
initial theoretical efforts were suffering from simplifications of physical as well as of mathematical nature and mostly focused on two or three techniques: 1) power series, perturbation expansions; 2) effective medium theories; and 3) geometrical consideration to separate and describe transport phenomena. Sufficient number of studies review the subject as, for example [1, 4—8]. Dul'nev and Zarichnyak [9-10] reviewed and analyzed different approaches to calculate the generalized (meaning thermal and electrical conductivities, electrical and magnetic inductions, dielectric permittivity, viscosity and diffusion) effective conductivity coefficient in heterogeneous systems making great use of geometrical considerations.
Because of the greatly increased computing power in the last two decades the interest of researches turned to different techniques in heterogeneous sciences that use this power.
The most common way became to treat such problems has been to seek a solution by doing numerical experiments over more or less the exact morphology of interest — what can be called the Detailed Micro-Modeling (DMM) which is often conducted via Direct Numerical Modeling (DNM).
This leads to heavy use of large computers to solve large algebraic statements. The treatment and analysis of the results of such a conventional Direct Numerical Modeling (DNM) is both difficult and limited in analysis. Performing DNM without proper theory is like performing experiments, often very challenging and expensive without data analysis and modeling tools — gives the data, but not the all needed results.
A good example of DMM-DNM for the linear problem of electrical field distribution in two-phase dielectric composites was demonstrated by Cheng and Torquato [11]. Their detailed analysis of numerical simulation results — "show that in general the probability density function for disks and squares exhibits a double-peak character, ... Not surprisingly, therefore, the variance or second moment of the field is generally inadequate in characterizing the field fluctuations in the composite". So, the pure statistical analysis of the DMM-DNM data revealed a clear demand for non-statistical tools for description of phenomena.
It is obvious that DMM-DNM can not meet the entire needs of Heterogeneous Media description and Modeling (HMM). What is the difference between DMM-DNM and Heterogeneous Media Modeling, why can't DMM be self-sufficient in heterogeneous media transport phenomena description? The answers arise via analysis of the issues:
1) There is a basic principal mismatching at the boundaries — boundary conditions problems. Meaning that for the DMM and for the bulk (averaged characteristics) materials' fields the boundary conditions are principally different.
2) The spatial scaling of heterogeneous problems with the chosen Representative Elementary
5
2
Volumes (REV) (for DMM) arises if one needs to address large or small deviation in the elements considered, with different underlying physics for some of them. Which denotes, that when the spatial heterogeneities of characteristics or morpholo-» gy are evolving along the coordinates then there is <c no chance to use DMM for that problem. I 3) Random morphologies treatment — as long 1 as numerical experiments provided with DMM-DNM -g need to be translated to the form which implies
I that the overall spatially bulk characteristics moda;
y eled. Which is not so simple, because of the ques-| tion — what kind of equations are considered as ^ governing equations? And what are the variables | being compared? As in the case of the local poros-
0 ity theory [12, 13], for example, when the results of real porous medium digitized images morphological analysis are used for calculation of effective dielectric constant, assuming applicability of homogeneous medium governing equations.
4) Discrete — continuum gap closure or matching, as long as DMM-DNM actually unrealistically claims on the description and simulation of continuum phenomena. This is the most fundamental drawback when DMM is being used to apply as the most exact reflection of the real physical phenomena. That means, that as soon as the developed solution of the DMM problem exists it needs to be matched to the correspondent HMM. Otherwise, DMM is only valid for the scales in which the problem was stated and in which the experimental confirmation only to be sought. So, no conclusion or generalization could be developed for the next or higher levels of hierarchy of the matter description. Meaning, there is no co-junction in the description of the different physical scales.
5) Interpretation of the results is always a problem. If results are presented for a heterogeneous continuum, then see the above point. If the results are being used as a solution for some discrete problem — the question is how to relate that one to the continuum problem of interest or even to a slightly different problem. Usually, in both ways, at some
^ point there appears a continuum model in the form
1 of some differential eq
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