# 2-Maxwell-两相导热率比例0.5比204.docx

Izmir Composites Science and Technology 63 2003 113–117 Thermal conductivity of particle ﬁlled polyethylene composite materials . Dilek Kumlutas¸ a,*, Ismail H. Tavmana, M. Turhan C¸ obanb aDokuz Eylu¨l University, Mechanical Engineering Department, Bornova, . 35100, Turkey bTUBITAK Marmara Research Cent., Energy Syst. Environmental Res. Inst., Gebze, Turkey Received 17 January 2002; received in revised 28 August 2002; accepted 28 August 2002 Abstract In this study, the eﬀ ective thermal conductivity of aluminum ﬁlled high-density polyethylene composites is investigated numeri- cally as a function of ﬁller concentration. The obtained values are compared with experimental results and the existing theoretical and empirical models. The thermal conductivity is measured by a modiﬁed hot-wire technique. For numerical study, the eﬀ ective thermal conductivity of particle-ﬁlled composite was calculated numerically using the micro structural images of them. By identi- fying each pixel with a ﬁnite diﬀ erence equation and accompanying appropriate image processing, the eﬀ ective thermal conductivity of composite material is determined numerically. As a result of this study, numerical results, experimental values and all the models are close to each other at low particle content. For particle content greater than 10, the eﬀ ective thermal conductivity is expo- nentially ed. All the models fail to predict thermal conductivity in this region. But, numerical results give satisfactory values in the whole range of aluminum particle content. 2002 Elsevier Science Ltd. All rights reserved. Keywords Composite material; Thermal conductivity; Image processing; Finite diﬀ erence 1. Introduction Knowing physical properties of the composite mate- rials has gained signiﬁcant importance in the design of new systems. For many materials applications, infor- mation is needed on their thermal properties. Deter- mining the thermal conductivity of composite materials is crucial in a number of industrial processes. The tem- perature ﬁelds in composite materials cannot be deter- mined unless the thermal conductivities of the media are known. Despite the importance of this material prop- erty and the considerable number of studies that have been carried out, the determination of eﬀ ective thermal conductivity of a composite material is partially under- stood. The eﬀ ective thermal conductivity of a composite material is a complex function of their geometry, the thermal conductivity of the diﬀ erent phases, distribution within the medium, and contact between the particles. * Corresponding author. Tel. 90-232-3883138-121; fax 90- 232-3887868. E-mail address dilex.kumlutasdeu.edu.tr D. Kumlutas¸ . Numerous theoretical and experimental approaches have been developed to determine the precise value of this parameter. Maxwell [1] studied the eﬀ ective thermal conductivity of heterogeneous materials. By solving Laplace’s equa- tion, he determined the eﬀ ective conductivity of a ran- dom suspension of spheres within a continuous medium. The model developed by Maxwell [1] assumes that the particles are suﬃ ciently far apart that the potential around each sphere will not be inﬂuenced by the presence of other particles. Russell [2] developed one of the early model systems using the electrical analogy, assuming that the discrete phase is isolated cubes of the same size dispersed in the matrix material and that the isothermal lines are planes. Based on Tsao’s [3] prob- abilistic model, Cheng and Vachon [4] assumed a para- bolic distribution of the discontinuous phase. Lewis and Nielsen [5] derived a semi-theoretical model by a modi- ﬁcation of the Halpin-Tsai equation [6,7] to include the eﬀ ect of the shape of the particles and the orientation or type of packing for a two-phase system. Baschirow and Selenew [8] developed the equation for the case when 0266-3538/02/ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII S0 266 -3 538 0 2001 94 -X 114 D. Kumlutas¸ et al. / Composites Science and Technology 63 2003 113–117 2 m the particles are spherical and the two phases are iso- tropic. In real composite material, the isothermal sur- and for series conduction model 1 1 Þ faces present a very complex shape and cannot be analytically determined. The models used to calculate thermal conductivities are thus highly simpliﬁed models kc ¼ km þ kf ð of the real media. Veyret et al. [9] characterized con- ductive heat transfer through composite, granular, or In the case of geometric mean model, the eﬀ ective thermal conductivity of the composite is given by ﬁbrous materials by using the ﬁnite element . ð1 Þ Terada et al. [10] generated the ﬁnite element model by identifying each pixel with a ﬁnite element and accom- panying appropriate image processing. Deissler and Boegli [11] carried out the numerical studies. They pro- posed a solution to Laplace’s equation for a cubic array of spheres presenting a single point of contact. Deiss- ler’s [11] works were extended by Wakao and Kato [12] for a cubic or orthorhombic array of uni spheres in contact. Shonnard and Whitaker [13] have investigated the inﬂuence of contacts on two-dimensional models. They have developed a global equation with an integral for heat transfer in the medium. In this study, the eﬀ ective thermal conductivity of aluminum ﬁlled high-density polyethylene composites is investigated numerically as a function of ﬁller con- centration. The obtained values are compared with experimental results and the existing theoretical and empirical models. kc ¼ kf km ð3Þ Tsao [3] derived an equation relating the two phase solid mixture thermal conductivity to the conductivity of the individual components and to two parameters which describe the spatial distribution of the two pha- ses. By assuming a parabolic distribution of the dis- continuous phase in the continuous phase Fig. 1, Cheng and Vachon [4] obtained a solution to Tsao’s [3] model that did not require knowledge of additional parameters. The constants of this parabolic distribution were determined by analysis and presented as a function of the discontinuous phase volume fraction. Thus, the equivalent thermal conductivity of the two phase solid mixture was derived in terms of the distribution func- tion, and the thermal conductivity of the constituents. For, kf km The eﬀ ective thermal conductivity of high-density 1 1 polyethylene containing particulate ﬁllers is obtained kc ¼ pﬃCﬃﬃﬃﬃﬃkﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃk ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃkﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃBﬃﬃﬃﬃkﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃﬃﬃkﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃ ﬃﬃ numerically at several ﬁller concentrations. A numerical approach was used to determine the eﬀ ective thermal conductivity of particle composites in this study. The ð f m Þ½ m þ ð f B m Þ ð4Þ eﬀ ective thermal conductivity of the material was determined using the Laplace equation, as were the ln pﬃðﬃkﬃﬃﬃmﬃﬃﬃﬃþ ﬃ ﬃ ﬃ ﬃBﬃﬃﬃðﬃkﬃﬃﬃfﬃﬃﬃ ﬃﬃﬃﬃkﬃﬃﬃmﬃﬃﬃÞﬃ þ 2 pﬃCﬃﬃﬃðﬃ ﬃkﬃ ﬃfﬃﬃﬃ ﬃ ﬃ ﬃ ﬃ ﬃkﬃ ﬃmﬃﬃﬃÞﬃ B 1 B þ k temperature and ﬂux ﬁelds within the control volume. A ﬁnite diﬀ erence was used in this study. Calcu- lation is carried out on two-dimensional geometric spaces. The results obtained from this calculation were compared to results found in prior literature. pﬃðﬃkﬃﬃﬃmﬃ ﬃ ﬃþﬃ ﬃ ﬃ ﬃBﬃﬃﬃðﬃkﬃﬃﬃfﬃﬃﬃ ﬃﬃﬃﬃkﬃﬃﬃmﬃﬃﬃÞﬃ 2 pﬃCﬃﬃﬃðﬃ ﬃk ﬃfﬃﬃﬃ ﬃﬃﬃﬃﬃkﬃ ﬃmﬃﬃﬃÞﬃ where for both equations B ¼ pﬃ3ﬃ ﬃ ﬃﬃﬃﬃ ﬃ2ﬃﬃ C ¼ 4pﬃ2ﬃ ﬃﬃ ﬃðﬃ ﬃ3ﬃ ﬃ ﬃﬃﬃÞﬃﬃ 2. Thermal conductivity models In this section are listed several models and a brief description of their basis. Many theoretical and empirical models have been proposed to predict the eﬀ ective ther- mal conductivity of two-phase mixtures. Comprehensive review articles have discussed the applicability of many of these models that appear to be more promising [14,15]. For a two-component composite, the simplest alter- natives would be with the materials arranged in either parallel or series with respect to heat ﬂow, which gives the upper or lower bounds of eﬀ ective thermal con- ductivity. For the parallel conduction model kc ¼ð1 Þkm þ kf c composite; m matrix; f filler; volume fraction of filler ð1Þ Lewis and Nielsen [5] modiﬁed the Halpin-Tsai equa- tion [6,7] to include the eﬀ ect of the shape of the parti- Fig. 1. Parabolic distribution of the discontinuous phase for the Cheng and Vachon [4] model. D. Kumlutas¸ et al. / Composites Science and Technology 63 2003 113–117 115 6 23 þ 23 5 m cles and the orientation or type of packing for a two- phase system. 1 þ AB Assuming the pores are cubes of the same size and the isothermal lines are planes, Russell [2] obtained the conductivity using a series parallel network kc ¼ km 1 B ð5Þ 2 kc ¼ km 6 km 31 kf 7 7 ð7Þ where kf 1 B ¼ km ¼ 1 þ 1 m 6 4 23 þ km 1 23 7kf þ kf A 2 km þ The values of A and m for many geometric shapes and orientation are given in the Tables 1 and 2. Maxwell [1], using potential theory, obtained an exact solution for the conductivity of randomly distributed and non-interacting homogeneous spheres in a homo- geneous medium 3. Experimental study The matrix material is a high-density polyethylene in powder , with a density of 0.968 g/cm3 and a melt index of 5.8 g/10 min. Its thermal conductivity at 36 C is 0.543 W/mK. The metallic ﬁller is aluminum in the of ﬁne powder, with particles approximately spherical in shape and particle size in the range of 40–80 microns for aluminum. The solid density of aluminum is 3 kf þ 2km þ 2 ðkf km Þ 2.7 g/cm and its thermal conductivity 204 W/mK. kc ¼ km Table 1 kf þ 2km ðkf km Þ ð6Þ Samples are prepared by the mold compression process. HDPE and aluminum powders are mixed at various volumetric concentrations. The mixed powder is then melted under pressure in a mold and solidiﬁed by air- Value of A for various systems Type of dispersed phase Direction of A heat ﬂow Cubes Any 2.0 Spheres Any 1.50 Aggregates of spheres Any 2.5/ n-1 cooling. The process conditions are molding temperature of 185 C, pressure of 4 MPa The resulting samples for thermal conductivity measurements are rectangular in shape of 100 mm length, 50 mm width, and 17 mm thick- ness because of the measuring probe. Homogeneity of the samples is examined using a light microscope. Aluminum particles are found to be unily distributed in high- Randomly oriented rods Aspect ratio2 Randomly oriented rods Aspect ratio4 Randomly oriented rods Aspect ratio6 Randomly oriented rods Aspect ratio10 Randomly oriented rods Aspect ratio15 Any 1.58 Any 2.08 Any 2.8 Any 4.93 Any 8.38 density polyethylene matrix with no voids in the structure. 4. Numerical modeling of the problem Two-dimensional numerical analysis was carried out for the conductive heat transfer in the composite mate- rial Fig. 2. The temperature ﬁeld in the composite Uniaxially oriented ﬁbers Parallel to ﬁbers Uniaxially oriented ﬁbers Perpendicular to ﬁbers 2L/D 0.5 material was deﬁned by solving Laplace’s equation numerically using a ﬁnite diﬀ erence ulation. The Table 2 Value of m for various systems Shape of particle Type of packing m Spheres Hexagonal close 0.7405 Spheres Face centered cubic 0.7405 Spheres Body centered cubic 0.60 Spheres Simple cubic 0.524 Spheres Random close 0.637 Rods or ﬁbers Uniaxial hexagonal close 0.907 Rods or ﬁbers Uniaxial simple cubic 0.785 Rods or ﬁbers Uniaxial random 0.82 Rods or ﬁbers Three dimensional random 0.52 Fig. 2. Two-dimensional model of the composite material. 116 D. Kumlutas¸ et al. / Composites Science and Technology 63 2003 113–117 X y Laplace‘s equation was solved by imposing the follow- ing boundary conditions 1. The horizontal sides perpendicular to the direc- tion of the heat ﬂow are isothermal at the entrance to and the exit from the cell. 2. The vertical sides parallel to the direction of the heat ﬂow are adiabatic. The heat ﬂow moving into or out of the cell reaches its peak at the center of the ﬁller particles. For an ele- mentary two-dimensional cell with the dimensions of Lx along the x axis and Ly along the y axis, the thermal conductivity is determined using the following relation from the numerical analysis are compared with the results obtained from the experimental study, it is seen that the numerical results are higher. It can be said that, since the quality of the pictures used in image processing carried out on the picture ﬁles in numerical study are k Lx Tcell c Ly Ti ¼ kixi ð8Þ i with Pixi ¼ Lx and kikm in the continuous phase, kf in the inclusions In the considered heat conduction problem the tem- peratures at the nodes along the boundaries y0 and yLy are prescribed and known as T1 and T2, but the temperatures at the nodes in the interior region and on the adiabatic boundaries are unknown. Therefore, the problem involves many unknown temperatures. The equations needed for the determination of these tem- peratures are obtained by writing the appropriate ﬁnite- diﬀ erence equation for each of these nodes. 5. Results and discussion In this study, a numerical approach is used to deter- mine the eﬀ ective thermal conductivity of particle ﬁlled composite materials. The temperature ﬁeld for the composite material is deﬁned by solving Laplace’s equation using a ﬁnite diﬀ erence ulation. The tem- perature ﬁeld for the composite material consisting of 4 by volume aluminum particles ﬁlled high-density polyethylene is shown in Fig. 3. It may be seen from this ﬁgure that the isotherms are deed at the location of aluminum particles represented by white inclusions. The ﬂux lines are directed toward inclusions due to the much higher thermal conductivity value of aluminum 204 W/ mK with respect to high-density polyethylene 0.543 W/mK. In Fig. 4, the thermal conductivity values obtained from the experimental study for aluminum ﬁl- led high-density polyethylene are compared with several thermal conductivity models and with the numerical results obtained in this study. As seen from this ﬁgure, Russel [2] and Cheng and Vachon [4] models predict fairly well thermal conductivity values up to 10 by volume of aluminum particles whereas beyond 10 of particle content all models underestimate the thermal conductivity of the composite. If the results obtained Fig. 3. Isothermal curves for the composite material composed of 4 aluminum ﬁlled high-density polyethylene. Fig. 4. Comparison of the numerical results, models and experimental values for aluminum ﬁller. D. Kumlutas¸ et al. / Composites Science and Technology 63 2003 113–117 117 not so good, the procedure was not so successful. Objects deﬁned as noise in the picture will be accepted as Al particles added into the high-