Application of an improved three-phase model to calculate effective characteristics for a composite with cylindrical inclusions

Application of an improved three-phase model to calculate effective characteristics for a composite with cylindrical inclusions

Igor V. Andrianov^I; Jan Awrejcewicz^II; Galina A. Starushenko^III

^IInstitute of General Mechanics, RWTH Aachen University, Templergraben 64, D-52056, Aachen, Germany

^IITechnical University of Łódź, Department of Automation and Biomechanics, 1/15 Stefanowski St., PL-90-924, Łódź, Poland

^IIIDnipropetrovs'k Regional Institute of State Management of National Academy of State Management at the President of Ukraine, Gogolya 29, UA-49631, Dnipropetrovs'k, Ukraine

ABSTRACT

A modified three-phase composite model yielding reliable effective characteristics of composite structures has been proposed. In particular, the problem of effective heat transfer coefficient of the composite structure with periodically located inclusions of circular cross-sections located on a square net is solved. Advantages of the proposed model in comparison to the classical three-phase model are illustrated and discussed.

Keywords: Composite, three-phase model, periodic inclusions, heat transfer

1 INTRODUCTION

The three-phase model of a composite (TPhM) has been used in references [2-5,10,11,15-17] in order to define effective characteristics of composite structures. The physical meaning of the idealization introduced by TPhM relies on substitution of the periodic structure being studied by its counterpart homogeneous homogenized structure with the equivalently reduced parameters (to be defined) except of only one characteristic structure cell. Further step of the solution requires derivation of mathematical formulas regarding homogenized coefficients through the application of physical understanding and mainly via either the energy principle [4,5,10,11,17] or structure geometry [2,3,15,16].

In particular, in the problem devoted to determination of the equivalent heat transfer coefficient of a two-phase composite structure with periodic cylindrical inclusions of circular cross-sections located on a square net the analysis revealed the following essential features:

(i) For small inclusions size heat transfer parameter is validated for its arbitrary values;

(ii) For large inclusion size the TPhM model yields reliable results assuming that heat transfer of the inclusions is of the order of structure matrix ;

(iii) TPhM may yield even a qualitatively wrong result for either large inclusion size or large/small conductivity properties, i.e. for and .

A construction of the first order TPhM approximation using the method of variations of boundary shapes of the studied structure does not allow us to overcome the earlier mentioned problems, in particular with respect to the possible wide spectrum of applications. Furthermore, a solution being limited to the first inclusion approximation does not describe the structure properties adequately for large values of the inclusion size. Namely, it does not allow achieving even a qualitative picture of the processes which occur in the composite (for example, a validation of the infinite cluster occurrence). The latter commentary can also be expressed mathematically. Namely, the first terms of the asymptotic series do not influence the further asymptotic sequence being defined through a zero order approximation.

In this work novel algorithms associated with the application of the composite TPhM are developed. The main idea and advantages of our proposal are illustrated and discussed taking as an example a solution to the heat transfer problem of the composite structure with cylindrical inclusions of circular cross-sections.

2 SOLVING THE HEAT TRANSFER PROBLEM WITH THE USE OF THE IMPROVED COMPOSITE TPHM

The problem of determination of effective coefficients of the micro non-homogeneous material consisting of a continuous matrix and periodically located cylindrical inclusions with circular cross-sections is solved in this work. In our study structure size in the direction of the fibre length essentially exceed the remaining size, i.e. (Figure 2.1).

We assume that the studied structure is two-periodic with the same period in both directions and the inclusions are located within the square net. Period is small in comparison with the characteristic diameter of the composite cross-section, i.e. .

Phases of the composite have different heat transfer coefficients and in the matrix (area ) and inclusions (area ), respectively, where . The characteristic of the periodically repeated cell composite is shown in Figure 2.2.

Material behaviour in the area of and is governed by the Poisson equations of the following form

where , are the functions of temperature distributions regarding the mentioned areas; stands for the density of heat sources.

In the interface of the matrix with inclusions the following compatibility conditions hold:

where denotes a contour normal to the inclusion.

Solution to the boundary value problems (2.1)-(2.4), owing to the homogenization method [12], can be presented in the form of the asymptotic series with respect to a small parameter of the following form:

,

where are slow variables, having a measurement in the interval of the whole composite structure space; are fast variables, describing the problem on the structure cell and is the small parameter characterizing the composite periodicity.

1. Solution to the problem is constructed using the modernized three-phase composite model (ТPhMM) being characterised by the following properties: the whole composite structure , except one cell, is substituted by the equivalent homogeneous medium having the known (to be found) heat transfer coefficient . Furthermore, we introduce two circles to describe the square matrix cell contour with the following radii (see Figure 2.3):

The cell problem in polar coordinates can be cast to the following form [16]:

The solution to the boundary value problem (2.6)-(2.11) is as follows:

where

Observe that during the homogenization procedure subjected to the equation

the integration is carried out using Eq. (2.5), i.e. in formula (2.13) we take

In Figure 2.4 a quadrant of the three phase area is shown, where the integration via the TPhMM is carried out.

After integration using Eqs. (2.13), (2.14) and the following formula

,

where

, , ,

and after satisfying the necessary conditions, the following transcendental equation is derived defining heat transfer coefficient :

where

Transcendental Eqs. (2.15), (2.16) can be solved numerically in spite of the limiting values of the heat conductivity parameter: and . In the latter case it is worthwhile to use the asymptotic representations, which can be obtained on the basis of relations (2.15) and (2.16).

3 ASYMPTOTIC RELATIONS FOR THE EQUIVALENT HEAT TRANSFER PARAMETER

1) The equivalent heat transfer parameter obtained from the transcendental equations (2.15)-(2.16) satisfies Keller's theorem [9]:

Indeed, we have

where

,

and hence (3.1) has been proved.

Furthermore, the structure of equations (2.15) and (2.16) allows for a direct use of as a natural small parameter for arbitrary values of and .

Now, having this parameter one may define and investigate further the limiting transitions.

2) Let us investigate a composite having heat transfer of the matrix and inclusions of the same order, i.e.

.

In this case for arbitrary values of the inclusion size we have:

.

It implies that transcendental Eqs. (2.15) and (2.16) can be reduced to the following forms:

2.1) For inclusions of small geometric size :

.

2.2) For inclusions of large geometric size :

.

3) Let us study the case of the absolute heat transfer, i.e. when .

3.1) If size of the inclusions are small, i.e. when , we have

,

and consequently

.

3.2) We study the case of inclusions of large geometric size .

The physical meaning of the problem implies that in the case of inclusions with infinitely high heat transfer properties and geometric size close to the limiting large ones of the order , the homogenized heat transfer coefficient is also infinitely large, i.e.

.

Therefore, transcendental Eqs. (2.15) and (2.16) are transformed to the following form

and for one obtains

It should be emphasized that the main term of the series development (3.3) coincides with the asymptotic representation , reported in [13] (with accuracy of the normalisation introduced in [13]) for the effective heat transfer of a composite with circular cross-sections of large cylindrical inclusions having absolute heat transfer properties.

4) Below, we study the entirely resistant heat transfer inclusions, i.e. these with .

4.1) For small additives we have , and consequently

Formula (3.4) for coincides (with accuracy up to the terms of the order of ) with the results obtained in [1] for the effective heat transfer of the composite consisting of inclusions without heat transfer property.

4.2) Large inclusions .

Proceeding in the way analogous to that presented in Section 3.2, we may trace the limiting transition exhibited by relations of the equivalent heat transfer parameter defined by Eqs. (2.15), (2.16) for the non-conductive heat inclusions of large sizes: ; . In this case the homogenized heat transfer coefficient will be infinitely small, i.e. .

As a result we obtain

and for we get

5) Next, we consider a composite with small size inclusions, i.e. .

Motivated by physical properties one may conclude that for the case of arbitrary heat transfer values , including the infinite large and infinitely small values, the homogenized heat transfer is:

.

Therefore we finally obtain:

5.1) For inclusions of infinitely large heat transfer property we have

.

5.2) For inclusions of infinitely small we have

.

5.3) For inclusions with we have

.

6) In the case of the composite structure with inclusions of large size and for the equivalent homogenized heat transfer satisfies the following relation

,

which means that

for ,

for ,

for .

Then the transcendental Eq. (2.15) yields a relation to determine where the formula for in Eq. (2.16) takes the form

In particular we have:

6.1) for inclusions of large heat transfer :

,

6.2) for inclusions of small heat transfer :

,

6.3) for inclusions of heat transfer of the order of the matrix heat transfer :

.

4 RESULTS OF COMPUTATION OF THE EFFECTIVE HEAT TRANSFER COEFFICIENT FOUND USING TPHMM

1) In Figures 3.1 to 3.10 graphs of the homogenized heat transfer coefficient yielded by the TPhMM for various values of the heat transfer inclusions are reported:

, see Figures 4.1 and 4.2.

, see Figures 4.3 and 4.4.

, see Figures 4.5 and 4.6.

, see Figures 4.7 and 4.8.

, see Figures 4.9 and 4.10.

In order to compare the obtained results, the Hashin-Shtrickman (H-S) boundaries are also reported [6, 7, 18] which are defined through the following relations

It is worth noting that the solution obtained via the TPhM identically coincides with the lower boundary of the H-S estimation for and regarding the upper boundary for .

2) Graphs related to the cases of geometrically large inclusions , and either large or small heat homogenized transfer coefficients and that computed by the TPhMM as well as those obtained in [9,13] are shown in Figures 4.11 - 4.14. The asymptotic solution obtained in [13] is as follows

whereas that obtained in [9, 13] has the following form

While constructing the dependencies shown in Figures 4.1 to 4.10, a solution to the transcendental Eqs. (2.15) and (2.16) has been found numerically, whereas dependencies in the vicinity of the limiting values of parameters and (Figures 4.11 to 4.14) have been constructed with the help of asymptotic formulas (2.2) and (2.4), respectively.

3) For small size of the inclusions and various values of heat transfer the equivalent heat transfer parameter obtained through TPhMM in comparison to the Schwarz alternating method [1] are given in Figures 4.15 to 4.22.

4) Figures 4.23 to 4.26 give the dependencies of the homogenized heat transfer coefficient obtained through the TPhMM for various values of the inclusions size a. Namely, we have small inclusions: (Fig. 4.15), inclusions of average size: ; (Figs. 4.16 and 4.17), large inclusions: (Fig. 4.18).

Upper and lower Hashin-Shtrickman boundaries are marked, and the TPhM solution coinciding with for and with for is given.

5) Graphs shown in Figures 4.27 to 4.32 illustrate the behaviour of equivalent heat transfer coefficient versus conductivity of the inclusions of large size . While constructing the graphs asymptotic formulas (2.15) and (3.7) have been used. In addition, the dependencies and , constructed on the basis of (4.3) and (4.4), are added [9,13].

6) For the case of a composite structure with absolute heat transfer and of large size a comparison of the results regarding the effective computations of heat transfer coefficient found via the TPhMM with those reported by other authors are given in Tables 4.1 and 4.2.

7) Table 4.3 gives computational results of the homogenized heat transfer coefficient obtained using the TPhMM and their comparison with the analytical solution given in reference [13] for the case of the inclusion size and large heat conductivity .

8) In reference [1] for small inclusion size the formula for equivalent heat transfer coefficient obtained using the Schwarz method of successive approximations is reported. Table 4.4 gives the computational results of homogenized heat transfer coefficient using ThPM [16] as well as the Schwarz method [1] and the TPhMM for the case of various heat transfer and small values of the inclusion size .

Table 4.5 gives the mean absolute discrepancy of the computation of effective heat transfer coef-ficient using the TPhMM versus known results obtained by other authors.

5 CONCLUSIONS

One may conclude from the data reported in Table 4.1 and from the paper text body that our results constructed through the proposed improved TPhM allow us to improve the results obtained via the classical TPhM approach regarding the estimation of effective heat transfer parameter of the composite structure with periodically located cylindrical inclusions of circular cross-sections on the square net.

Finally, let us emphasise that the TPhMM is practically validated and gives reliable results in the whole area of both composite parameters variation, i.e. with respect to:

(i) geometric parameter regarding the size of inclusions : ;

(ii) physical parameter regarding the inclusion conductivity : , including the limiting cases:, and , .

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Publication Dates

• Publication in this collection
  25 Feb 2013
• Date of issue
  Jan 2013