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A METHOD OF RESEARCHING THE THERMAL CONDUCTIVITY COEFFICIENT OF DISPERSION COMPOSITE MATERIALS

Alexei I. Sharapov, Vladimir D. Korshikov, Anton А. Chernykh, Anastasia V. Peshkova

ABSTRACT

The paper considers several recently developed models to calculate the thermal conductivity coefficient for two-phase systems with dispersion inclusions. They are compared with another model which takes into account the influ-ence of thermal resistance of the phase boundary. In the paper are compared the results obtained from the derived theoretical dependence, which takes into account the influence of thermal resistance on the “material-inclusion” boundary and the results of computer simulation in the ANSYS program. Empirical dependencies are derived which make it possible to use the researched model to obtain the results of a change in the thermal conductivity coefficient over the entire range of ratios λ2/λ1.

Keywords: thermal conductivity coefficient, composite materials, heat flux.

Received 05 May 2019Accepted 27 Septenber 2019

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Department of Industrial Heat EngineeringLipetsk State Technical University30 Moskovskaya Str., Lipetsk 398055, RussiaE-mail: sharapov-lipetsk@yandex.ru

INTRODUCTION

Recently, it has been carried out an extensive re-search on materials, which contain various kinds of dis-persion inclusions. This is of interest for many industries such as the production of composite materials or durable materials for the aircraft and missile industry to with-stand relatively high temperatures with no loss in their functional characteristics. Such interest also originates from a number of problems in determining the thermal conductivity of such materials in calculation methods, which would take into account all factors: dispersion inclusion size, shape, porosity, thermal resistance at the boundary of two phases, etc. [1, 2].

The application of high-strength and high modulus fibers facilitates the improvement of the mechanical char-acteristics of such materials. However, in heat stressed structural elements which operate under simultaneous intense action of both mechanical and thermal loads, an important role is played by both the mechanical character-istics and the thermophysical properties of the composite material, in particular, thermal conductivity [2].

Currently, data on thermophysical measurements or analytical dependences are commonly used to determine the effective characteristics of composite materials. Analytical formulas for calculating effective thermal conductivity coefficients at various concentrations of inclusions, as well as thermal conductivity ratios were first presented in the works by Maxwell, Rayleigh, Bruggeman, Meredith [3 - 6], etc. Over the past century, analytical formulas have become considerably com-plicated and made it possible to take into account the various shapes and orientation of inclusions, though in a fairly narrow range. It should be noted, however, that any of these formulas uses a relatively crude model of a heterogeneous environment, which affects the accu-racy of determining effective coefficients. Experimental methods for determining the effective thermal conduc-tivity coefficients also have limitations [7 - 9] (sample size, matrix – inclusion properties contrast, etc.).

Recently, composite materials with metallic inclu-sions as aggregates have been widely studied. Such attention is explained by little information on thermal conductivity for such materials. Some shortcomings in

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determining thermal conductivity for such materials raise the question of creating more advanced calcula-tion methods.

MAIN RECENT MODELS

In order to make the point clear, the calculation of the “dispersion inclusions – material” model is first given, when the inclusions are located at the same set distance from each other. The main assumptions that are introduced in this model are: the temperature gradi-ent change Δt is small, the thermal conductivity change at the ends of the unit cell of inclusions is also small.

Dispersion inclusions are located at a distance of from their neighboring surfaces. Then, the main heat flux is represented as a ratio:

(1)

where is the gas thermal conductivity coefficient inside the inclusion (or it is the thermal conductivity of the metal from which the ball is made - another ex-ample); is the thermal conductivity of the metal in which the spherical aggregate is located; the length of the surface element inside the inclusion; is the length between the surface elements of adjacent inclusions.

Such a model was first mentioned by A.S. Lya-likov in [2]. Using the formula (1), we have:

(2)

where , is the thermal conductivity coef-ficient of dispersion inclusions, l is the distance between adjacent inclusions, r is their radius.

In recent years, a sufficient number of theories have been proposed to adequately assess thermal conductivity change in various materials with dispersion inclusions,

Fig. 1. Dependences of the ratio of the effective thermal conductivity coefficient to the thermal conductivity of metal balls at their different volume concentration: 1 - Meredith model from [1]; 2 - calculation by the authors of [1]; 3 - calculation according to the formula (3); 4 - experimental data by the authors of [1]; 5 - calculation according to the formula (2) at l = 0.02 mm.

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e.g. Meredith and Bruggerman models. Finally, using the results of [5, 6], it is possible to construct compara-tive dependencies according to different theories Fig. 1.

ELEMENTARY FINITE ELEMENT CALCULA-TION METHOD

The model under research is an elementary prismatic volume, selected from the cube model, Figs. 2 and 3, with sides a = 1 m and thickness δ = 0.01 m. The thermal conductivity coefficient of the sample material λ1 is 50 W·(m·K)-1. The sample’s lateral faces are isolated. The temperatures of the lateral faces of the cube and the plate are maintained constant and equal to 1000 and 300 K.

We divide the entire cube into thin surface elements with a thickness of dy. Such elements look like prisms with a spherical inclusion, Fig. 4. Next, we consider only

that area of the ball which touches the lines shown in Fig. 4, the area being limited only by the inclusion diameter 2R. Fig. 4 is the top view of the surface element shown in Fig. 3. The consideration of this area will make the heat flux integral more convenient for calculation. Now we extract another elementary unit of volume from the surface element of the model as shown in Fig. 4. As a result, we obtain an elementary prismatic volume shown in Fig. 5 which shows the radius drawn from the center of the spherical inclusion; we also calculate the lengths of different areas of the elementary volume. Since the temperature for each selected elementary volume of the lateral sides will be the same, then the temperatures of the side planes of the prism are . Now it is possible to find a Fourier law ratio for this model. Fourier law for an elementary prismatic volume Fig. 5

Fig. 2. Cube model with a cavity. Fig. 3. Model surface element.

Fig. 4. Model surface element. Fig. 5. Elementary prismatic volume (top view).

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will then be written as:

We integrate the ratio within the limits

(3)

Complete thermal resistance for an elementary prismatic volume will then be written as:

We integrate the eq. (3) with conditions indicated above:

By canceling terms in the overall thermal resistance, we obtain the unit volume for the unit element shown in Fig. 5 equaling

(4)

In order to calculate the heat flux of the entire 3-D model Q, we multiply (4) by , because it is nothing but the area of lateral surface elements the tem-peratures of which are equal to , respectively, and their product equals the complete heat flux:

The final integral ratio (3) will turn into a differential equation

Integrating the ratio, we obtain the formula:

(5)

For the sake of convenience, we assume that all ele-ments have a thickness δ. This does not tell on the solu-tion of the integral (5), but makes it one-dimensional:

(5*)

The limits in the right interval are taken because we consider two parts of the cube from 0 to 2R. In order

Fig. 6. Heat conductivity coefficient change depending on the spherical inclusion radius according to eq. (6).

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to make the integral less cumber some to calculate, we have the right to replace the integration limits from –R to Ror from 0 to R, but multiplying it by 2. After solving the interval (5*), which is quite cumbersome and which we thus leave out, we have a formula for calculating the thermal conductivity coefficient of a thin plate model with a spherical inclusion:

(6)

The formula (6) has a geometric interpretation shown in Fig. 6. It can be seen that the applicability limit of the formula is already limited under condition λ2/λ1 < 0.5. Further, the formula gets imaginary values of the thermal conductivity coefficient. Therefore, it is neces-sary to obtain universal ratios for a large range of change λ2/λ1 on the basis of numerical and theoretical data.

The eq. (6) is rather cumbersome to calculate, there-fore we derive an empirical dependence for the thermal conductivity coefficient of this model. It will have the form of a parabola with coefficients to be found by the

fitting method. The most interesting here will be the equation:

(7)The main difficulty is to select constants C and n.

The obtained data make it possible to draw a conclusion on the applicability of the given empirical ratio, as well as on the limit of its applicability:if λ2/λ1 > 1, then1.1 < С < 1.5, and n = 2;if λ2/λ1 < 1, then 1.2 < C < 1.3, and n = 2.

Hence another consequence for the correct selection of the coefficient C. Under the condition λ2/λ1 > 1, the closer the straight line λ1, the more the value of C has to be, but within the limits indicated above. The applicabil-ity of the obtained eq. (7) varies in the range of ratios of thermal conductivity coefficients 0.4 < λ2/λ1 < 10, over an interval of radius 0 < R < 0.5 m, but the rq. (7), un-like (6), can be used in a wider limit of the ratio λ2/λ1.

RESULTS OBTAINED BY NUMERICAL AND THEORETICAL METHODS

Distributions of heat and temperature fields along the entire length of a flat sample were obtained using the ANSYS software package. An example of such contours is given for the case of thin plastics with a spherical inclusion diameter of 0.5 m in Figs. 7 and 8.

Fig. 7. Specific heat flux distribution contours over the entire volume of a flat sample with isolated side walls with a spherical inclusion of 0.5 m in diameter, at given temperatures of the upper and lower faces (top view).

Fig. 8. Temperature distribution contours over the entire volume of a flat sample with isolated side walls with a spherical inclusion of 0.5 m in diameter, at given tempera-tures of the upper and lower faces (top view).

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In order to select the necessary ratios for calculat-ing the thermal conductivity coefficient of the model, it is necessary to compare the theoretical model with the results obtained in the

ANSYS software package. By approximating the numerical results of the specific heat fluxes, it is pos-sible to obtain the average value of the specific heat flux along the entire length of the sample as a single point. The obtained points are plotted along with the theoretical curve. As a result, we get Fig. 9.

Since the initial error between the numerical and theoretical results exceeds 10%, the value of the cor-rection coefficient C with an increase in the ratio λ2/λ1 will be in the range of 1.1 - 1.5:

(8)

The ratios C and n are selected on the basis of the geometry of the curves in Fig. 5 on the basis of the convergence of theoretical and numerical data. As a result, we obtain curves for the ratio (8) for different ratios of thermal conductivity coefficients λ2/λ1 which are given in Fig. 10.

RESULTS AND DISCUSSIONDistributions of thermal fields along the entire

length of a flat sample were obtainedusing the ANSYS software package. In order to select the necessary ratios for calculating the thermal conductivity of the model, it is necessary to compare the theoretical model with the results obtained in the ANSYS software package.

By approximating the numerical results of the spe-cific heat flux, it is possible to obtain the average value of the specific heat flux along the entire length of the sample as a single point; the obtained points are plotted along with the theoretical curve.

The ratios of C and n are selected on the basis of the geometry of the curves in Fig. 6. As a result, we obtain the curves of the thermal conductivity coefficients λ2/λ1 which are presented in Fig. 9 and 10. Noteshould be made of good agreement between the empirical ratios and the numerical experiment data in Fig. 10. Such results can be applied and developed further for more complex models, where the shapes of dispersion inclu-sionsare not spherical. The obtained empirical ratiocan be considered universal for the considered type of mod-els with a different interval of λ2/λ1 variation.

Fig. 9. Comparison of dependences of thermal conductivity coefficient change according to (6) with numerical experi-ment results.

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CONCLUSIONS

Some shortcomings in determining the thermal conductivity of such materials raise the question of creating more advanced calculation methods. Taking into account the influence of thermal resistance at the “material-cavity” boundary, a model was developed for such materialsto take into account the ratio of the thermal conductivity coefficients of the material and dispersion inclusions.

This method can be used in designing various ther-mal protection composite materials in the aircraft indus-try, in designingmaterials which can be used to insulate building facades, etc. The technique makes it possible to perform the necessary calculations by obtaining the criteria of the material thermal conductivity with a small fraction of the error from the actual values.

In comparing the results of determining the equiva-lent thermal conductivity coefficient, the relative devia-tion of the value calculated according to the formula (8) does not exceed 4-5% of the value obtained as a result of computer simulation in the ANSIS software pack-age, which indicates the applicability of the developed method to engineering calculations.

Fig. 10. Comparison of dependences of thermal conductivity coefficient change on the spherical inclusion radius with an empirical dependence (8).

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1. M.I. Epov, V.I. Terekhov, M.I. Nizovtsev, E.L. Shurina, N.B. Itkina, E.S. Ukolov, Effective Thermal Conductivity of Dispersion Materials with Contrast Inclusions, ТVТ, 53, 1, 2015, 48-53, (in Russian).

2. A.S. Lyalikov, Calculation of the Thermal Conductivity Coefficient of a Dry Bulk Grain Material, Review of ТPI, 110, 1962, (in Russian).

3. J.C. Maxwell, A Treatise on Electricity and Magnetism, Oxford, Oxford Univ. Press, 1904, 435.

4. L. Rayleigh, On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium, Phil. Mag., 34, 1892, 481.

5. D.A.G. Bruggeman, Berechnung verschiedener phys-ic kalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper ausisotropen Substanzen, Ann. Phys., 416, 7, 1935, 636.

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