This is the great topic of tremendous importance. It was spoken about through the volumes. One thing is obvious, when effective coefficients problem is razed for assessment in a heterogeneous medium. And this is - because the effective coefficients search is the problem of the Upper scale medium consideration, there is no other way to go around the VAT governing equations for formulation the effective coefficient models
Few more remarks on correctness of the definition for "effective coefficient" can be found in the discussion
Discussion: "Alternative Models of Turbulence in a Porous Medium, and Related Matters"( 70K)
which was driven by recent publications and by my desire to speak more on importance of not to profane some principal definitions.
There are hundreds of papers on effective characteristics evaluation that we could use as examples of - HOW NOT TO DO, What is incorrect in the homogeneous "effective" coefficients determinations.
Unfortunately, the most frustrating feature of the paper by Eidsath et al. (1983) is that the founder of scaled averaging technique and theories for few disciplines (S.Whitaker) did not found other way to compare characteristics of the VAT with the homogeneous characteristics of the lower scale problem statement but as to follow the same lower scale physics procedures for calculation of the such "effective" coefficients.
Of course, following this approach at that time they got to the same range of outcome results as in others published homogeneous treatment of Heterogeneous problems.
The closure was stated as for functions that is the fluctuation of the concentration field and for the dispersion tensor in the fluid phase determined as in our notations
where the mathematical statement for was taken (we read in the page 1809) as for
The solution of this closure problem was sought as for the problem's geometry which used in all homogeneous statements for this kind of tasks
Here the numerical method and simulation procedure were made as for the GO Theorem (GOT) - the homogeneous physics analysis and assessments, in spite that the whole problem was stated and supposed to be solved as for the Upper scale and according to the heterogeneous WSAM Theorem idea!?
The outcome could be predicted.
So - everybody did ask at that time and continue asking now - For what reasons we need to use these complicated, integro-differential, not affordable mathematical non-local averaged statements (governing equations) for the solution of our problems if the effective characteristics - those mostly we are looking after, are actually the same as with the traditional statements!! Really - for what reason?
That was a terrible, misleading mistake done by authors of this paper, as well as by others in the same style.
Because these Effective coefficients are DIFFERENT. Well, sometimes, when the problem is linear (constant coefficients) and we look for overall composite's effective coefficients - then these effective coefficients - Heterogeneous Media Effective and Homogeneous media presentation effective can be equal, as to one can see in the first pure analytical Two Scale HSP--VAT solution for the most tought and used diffusion, conductivity problem -
Meanwhile, if we are looking for separate, each phase effective coefficients, and for the components of these effective coefficients, as in the case of High Temperature Superconductors (HTSC), for example, and for the effective coefficients in the very most of the problems in heterogeneous media, then we need to use the correct mathematics and numerical procedures of the HSP-VAT, see these features in the known problem example again in -
Thus, there is no need to return to the homogeneous media methods and algorithms. This is incorrect. Look at our elaborative studies with the exact solutions of Local and Heterogeneous Non-Local the same almost globular medium problems on the Both scales in the subsection on Electrostatics -
The Fluxes (Displacement fields) are equal for some steady-state statements, yes, but not the effective coefficients.
Reading on fundamentals of such things as closure and calculation of effective coefficients presented, for example, by Quintard (1998) at the 11th Int. Conf. on Heat Transfer, one might find in the page 280 the algorithm for computation of this kind "homogeneous--like" effective coefficients:
"- take the average of the pore-scale equations;
- express the pore-scale equations in terms of temperature deviations;
- seek an approximate solution of the system of coupled equations, i.e., the pore-scale and averaged equations. This is obtained through a representation of the temperature deviations in terms of the macroscopic quantities such as averaged temperatures, their gradients, ....;
- introduce these representations into the averaged equations to get the macroscopic equations in a closed form involving only averaged temperatures. This also provides an explicit link between the macroscopic effective properties and the pore-scale characteristics.
It can be said that almost all in this program is correct. But the proverb says - " the devil is in details."
In another work-study an accepting of the idea that the phase temperature variables in each of the subvolumes and can be presented as sums of overall averaged temperature and local fluctuations (Nozad et al., 1985)
is not frustrating due to introduction of the two new variables and This can bring the the equation for the composite medium temperature follows (Nozad et al., 1985) to the following definition
which has five (5) variable temperatures. If later on all the assumptions and constraints given in Nozad et al. (1985) all satisfied then the final equation with the only 3 different temperatures resumes
which means that the neglect of global deviations terms still not removes the requirement or the two temperature solution.
One of the first closure procedures were developed by Whitaker and co-authors (see, for example, Nozad et. al., 1985)
those later on were accepted as
also there is need for requirement
The closure equations for fluctuations functions - see in the page 847 are
A number of accepted assumption make possible to write the fluctuation closure equations in this simplified form. Important among them are
The fluctuation closure variables found to be evaluated through the solution of the problem
where the mathematical solution (p.849) with the note - "It should be clear at this point that the closure problem is nearly as complex as the original problem stated by eqns. (2.1) - (2.6); however we have no intention of solving for and over the macroscopic region. Instead we want to solve for and in some representative region and use the results to predict the effective thermal conductivity for that region. Such a representative region is illustrated in Fig. 2 "
The figure is done as for the one phase intersected by the REV's boundary surface, and the same is done in the Fig. 3 page 850. They compared these homogeneous assessments for the heterogeneous problem with the same kind of Homogeneous experiments in the Fig. 6, page 853.
We did analysis of this common for the technology, science, and industry mistake for presentation of effective assessments and coefficients for heterogeneous media, see in -
We need to say again - that in this paper the GO Theorem (GOT) - and the homogeneous physics analysis and assessments were used in the Heterogeneous problem and instead of the Upper Scale treatment was used the Lower Scale treatment! With the predictable outcome.
By this mathematics alone and by that REV in Figs. 2,3 on p. 849 and p.850, authors of the paper deadly hurt the great idea of importance and difference of the effective coefficients for the Upper scale VAT statement and physics. Because they presented them as of no big difference (or equal) as to the conventionally calculated homogeneous "effective" coefficients!? It seems, they did afraid to be different.
The usual procedure for experimental measurement of effective coefficient of thermal conductivity we can find in the paper by Orain et al. (2001). For the three scale medium - as in their Fig. 1, they used the just homogeneous technique we spoke few times in our texts.
As we read in the page 256: "A thermal conductivity measurement method was developed for the thermal characterization of submicron dielectric thin films . The technique, based on analysis of the transient temperature response induced by a laser pulse, uses a thin gold electrical resistance (#150 nm) as a temperature sensor deposited on the dielectric material.
The thermal conductivity is identified, simultaneously with a thermal boundary resistance between the sensor and the film, by fitting experimental and theoretical responses with a genetic algorithm . "
That can mean ONLY that the Homogeneous GOT (theorem) helped to determine the "effective" transient "bulk" coefficient of conductivity in this study. But not the Effective Multiscaled Heterogeneous Coefficient of Conductivity as they want for their products - silicone and zirconia oxides, polymeric films. No surprise that in every industry and every company doing these measurements for themselves again and again. Then getting those coefficient tables as the internal trade secret - still the wrong data.
In their numerical experimental study Veyret et al. (1993) used to calculate the 2D effective coefficient in globular morphology composite as, for example, for 2D arrangements with cylindrical discontinuous phases using the formula
where and are the conductivity coefficients in continuous and discontinuous phases.
From page 867 we read: "The calculation is carried out in a plane parallel to the isothermal boundaries. Figure 4 shows the effective thermal conductivity variations for square-in-square and circle-in-square systems for different conductivity ratio and concentration. "
We need to understand by this - that there was the 1D, integration of the flux over the line as
So, obviously it is not the averaged over the composite volume dependency for the heat transfer flux. Yes, in this problem the integration over the plane surface parallel to the isothermal boundaries gives the same heat flux as needed for energy conservation. But this is not the Effective Volumetrically averaged heat flux and effective coefficient, and this neither correct averaging nor correct VAT heterogeneous averaging.
Also - the very important point is - That these boundary conditions were the setup for the one cell, they are not representing the real BC for the cell in the composite medium.
The 3d like methodology for calculation of "effective" coefficient by Veyret et al. (1993) were based on the same idea
where surface of the () element, and the surface of the horizontal cross-section of the cell are
In continuous formulation this formula means
That means again no close match by the definition to the proclaimed volumetrically averaged "effective" coefficient, not talking about the heterogeneous Upper scale effective coefficient of conductivity for the 3d globular medium.
The general usage of homogeneous formulae based on the homogeneous GO theorem for numerical, physical experiment and theoretical procedures give not surprisingly the close, reasonably close curves for such "effective" coefficents, see in this paper figures as
That is why all the body of these results is from the same flock - we are saying that the "birds of a feather flock together". And there is nothing offensive or insulting tone should be found in my these words.
People who read my studies, or my co-authors know that I am saying this for a lot of years.
The problem is with the scientific and technological needs. They are of not great demanding needs in most of the old established industries, so this kind of homogeneous results satisfying all folks.
Or, the some new technology won't getting into reality unless the new more accurate tools will be used for their development and design. First of all this should happen probably in the Nanotechnologies and in Biotech-Medicine -
Through the last more than ten years, mostly because I was affiliated with UCLA, I was careful when emphasized my opinion and mathematics with regard of homogeneous "effective" coefficients for heterogeneous media. The common knowledge and use coefficients. We published few works with the two scale VAT analysis of the problem, as those referred above and below, but no strong message was sent?
Nowadays, especially when after 2002 the few more common textbooks known problems were solved as they should be - on the two scales, see in -
there is the belief that then we serve to the good purpose stressing out following to the all above information and proofs, as well as to the mathematical analysis in -
1) The all effective coefficients obtained for heterogeneous media with mathematics based on the homogeneous GO theorem give incorrect results!
The actions ought to be corresponding to this message.
Eidsath, A., Carbonell, R.G., Whitaker, S., and Herrmann, L.R., "Dispersion in Pulsed Systems - III. Comparison Between Theory and Experiments For Packed Beds," Chemical Engineering Science, Vol.38, No. 11, pp. 1803-1816, (1983).
Quintard, M., "Modelling Local Non-Equilibrium Heat Transfer in Porous Media," in Proc. 11th Intern. Conf. on Heat Transfer, Korea, Vol. 1, pp. 279-285, (1998).
Nozad, I., Carbonell, R.G. and Whitaker, S., "Heat Conduction in Multiphase Systems I: Theory and Experiment for Two-Phase Systems," Chem. Engnr. Sci., Vol.40, No. 5, pp. 843-855, (1985).
Orain, S., Scudeller, Y., and Brousse, T., "Investigation of Structural Effects on Thin Film Thermal Conductivity," in 7th International Workshop on Thermal Investigation of IC's and Sytems - THERMINIC, Paris, Laboratoire TIMA, pp. 256-259, (2001).
Veyret, D., Cioulachtjian, S., Tadrist, L., and Pantaloni, J., "Effective Thermal Conductivity of a Composite Material: A Numerical Approach," J. Heat Transfer, Vol. 115, pp. 866-871, (1993).
Travkin, V.S., "Discussion: "Alternative Models of Turbulence in a Porous Medium, and Related Matters," (D.A. Nield, 2001, ASME J. Fluids Eng., 123, pp. 928-931)," J. Fluids Eng., 123, pp. 931-934, (2001).
Travkin, V.S. and Catton, I. Porous Media Transport Descriptions - Non-Local, Linear and Non-linear Against Effective Thermal/Fluid Properties//Advances in Colloid and Interface Science. 1998. V.76-77. P.389-443.
Travkin, V.S. and Catton, I. Transport phenomena in heterogeneous media based on volume averaging theory// Advances in Heat Transfer. (New York, Academic Press, 2001. Vol. 34.). P.1-144.
Travkin, V. S. and Ponomarenko, A. T., "The Non-local Formulation of Electrostatic Problems for Sensors Heterogeneous Two- or Three Phase Media, the Two-Scale Solutions and Measurement Applications -1," Journal of Alternative Energy and Ecology, No. 3, pp. 7-17, (2005).
Travkin, V. S. and Ponomarenko, A. T., "The Non-local Formulation of Electrostatic Problems for Sensors Heterogeneous Two- or Three Phase Media, the Two-Scale Solutions and Measurement Applications - 2," Journal of Alternative Energy and Ecology, No. 4, pp. 9-22, (2005).
Travkin, V. S. and Ponomarenko, A. T., "The Non-local Formulation of Electrostatic Problems for Sensors Heterogeneous Two- or Three Phase Media, the Two-Scale Solutions and Measurement Applications - 3," Journal of Alternative Energy and Ecology, No. 5, pp. 34-44, (2005).