Computation of effective thermal conductivities of aircraft materials using the action variable

Applied Mathematics, Mechanics and Physics


Аuthors

Garibyan B. A.*, Spirin G. G.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: bagarib@yandex.ru
**e-mail: spirinas@mail.ru

Abstract

A variety of inhomogeneous materials currently can be estimated in millions and annually increases. All kinds of composites, compounds, alloys, fibrous, porous and granular materials, and bounded media are now widely used in aerospace engineering. The studying of thermophysical properties of aircraft materials, including thermal conductivity coefficient, is a very topical problem due to the application for new coatings of structural elements, thermal insulators, plastics, material compounds, rubbers etc.
Here the definition of effective thermal conductivity of binary matrix type inhomogeneous materials is proposed. We use the theory of generalized conductivity [1], and the environment structure modeling is based on the unit cell method. Using the ergodic hypothesis [1, 2], the volume averaging is through the «action» functional [3] defined for steady temperature fields


                                                        (1)

This functional defines an integral characteristic of the field [3, 4].

The linearization of temperature field within the unit cell is carried out by Rayleigh-sections method [2]. The «blurring» of the material properties is by the equality , introduced; here is the effective «action» (1) with the thermal conductivity factor .

Depending on the sequence of sections and the geometry of formed areas, i. e. sections schemes, we obtain different expressions for the effective thermal conductivity of cell.

The proposed method allows one to confirm the well-known formulae and get new ones for the relative effective thermal conductivity coefficient inhomogeneous materials (having isolated inclusions, interpenetrating components, and flat-fibrous). Also we have to establish the relationships between them, to estimate the discrepancies of values, including the most unfavorable case of very inhomogeneous environment: .

To compute numerically the effective thermal conductivity value within unit cell we formulate transient initial boundary value problem of heat transfer having no artificial restrictions adopted in approximate analytic approach.

This problem was solved for all types of unit cells of considered inhomogeneous materials on the basis of the rectangular regular grid by fractional steps method [5]. We use here an implicit difference scheme obtained by the control volumes method [6]. Due to the transient statement of the problem the pseudoviscosity method [5] was used; for the quasi-steady thermal regime onset in the unit cell we take the time at which the equality of heat flows at the end faces  is true with the given confidence level p. The distribution of the temperature field in the cell corresponding to the time being a grid function is considered as approximate numerical solution of the steady problem. Applying the general «action» procedure by equating we can compute the effective thermal conductivity coefficient as follows:

                                              (2)

Here is the base value (1), are the mean temperatures on the cell end faces that are numerically computed on the known grid , and L is a cubic cells edge length.

The effective thermal conductivity for various unit cells of inhomogeneous materials mentioned above that were computed on the groundwork of the formula (2) agree with the existing as well as new combined approximate analytic formulae.

A practical implementation was shown by computing the effective thermal conductivity of various inhomogeneous materials for aircrafts, such as the CFRP. The technique of restoring of effective thermal conductivities was developed for binary metal alloys with limited solubility of components. The efficiency of the proposed approach was shown for the binary system Al- Mg at the temperatures between 20...100...200(С. Also the effective thermal conductivities of triple-component alloy AMg5P (1557) were obtained using some different methods.

Keywords:

aircraft material, «action», Rayleigh- sections method, sections method, sections scheme of a cell, thermal conductivity of inhomogeneous material, effective thermal conductivity coefficient

References

  1. Dul’nev G.N., Novikov V.V. Protsessy perenosa vneodnorodnykh sredakh (Transfer processes innon- uniform environments), Leningrad, Energoatomizdat, 1991, 248p.
  2. Dul’nev G.N., Zarichnyak Yu.P. Teploprovodnost’ smesei i kompozitsionnykh materialov (Heat conductivity ofmixes and composite materials), Leningrad, Energiya, 1974, 264p.
  3. Garibyan B.A., Spirin G.G. Nauchnoe obozrenie, 2013, no.7, pp. 92-98.
  4. Zarubin V.S., Rodikov A.V. Teplofizika vysokikh temperatur, 2007, vol.45, no.2, pp. 277-288.
  5. Kireev V.I., Panteleev A.V. Chislennye metody v primerakh iz adachakh (Numerical methods inexamples and tasks), Moscow, Vysshaya shkola, 2008, 480p.
  6. Dul’nev G.N., Parfenov V.G., Sigalov A.V. Primenenie EVM dlya resheniya zadach teploobmena (Use ofthe computer for the solution ofproblems ofheat exchange), Moscow, Vysshaya shkola, 1990, 207p.

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