HEAT CONDUCTION IN ISOTROPIC HETEROGENEOUS MEDIA

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1 PAMUKKALE ÜNİVERSİTESİ MÜHENDİSLİK FAKÜLTESİ PAMUKKALE UNIVERSITY ENGINEERING COLLEGE M Ü H E N D İ S L İ K B İ L İ M L E R İ D E R G İ S İ J O U R N A L O F E N G I N E E R I N G S C I E N C E S YIL CİLT SAYI SAYFA : 1999 : 5 : -3 : HEAT CONDUCTION IN ISOTROPIC HETEROGENEOUS MEDIA Dile KUMLUTAŞ Douz Eylül University, Engineering Faculty, Departent o Mechanical Engineering, Bornova-İzir Geliş Tarihi : ABSTRACT In this study, the eective theral conductivity o aluinu or tin illed high-density polyethylene coposites is investigated nuerically as a unction o iller concentration. The obtained values are copared with experiental results and the existing theoretical and epirical odels. The theral conductivity is easured by a odiied hotwire technique. For nuerical study, the eective theral conductivity o particle-illed coposites were calculated nuerically using the icrostructural iages o the. By identiying each pixel with a inite dierence equation and accopanying appropriate iage processing, the eective theral conductivity o coposite aterial is deterined nuerically. As a result o this study, nuerical results, experiental values and all the odels are close to each other at low particle content. For particle content greater than 10%, the eective theral conductivity is exponentially ored. All the odels ail to predict theral conductivity in this region. But, nuerical results give satisactory values in the whole range o aluinu particle content. Key Words: Coposite aterial, Theral conductivity, Iage processing, Finite dierence ethod İZOTROPİK HETEROJEN ORTAMLARDA ISI İLETİMİ ÖZET Bu çalışada, alüinyu ya da alay ilave ediliş yüse yoğunlulu polietilen opozitlerin eeti ısı ileti atsayısı nüeri etodla dolgu alzeesi onsantrasyonunun bir onsiyonu olara araştırılıştır. Nueri çalışada, partiül atılı opozitlerin ısı iletenliği, bu opozitlerin iro görüntülerinden yararlanılara nüeri olara hesaplanıştır. Kopozit alzeenin resi üzerine uygun görüntü işlei yapıldıtan sonra, resin her pixeline sonlu ar denlei tanılanara ısı iletenliği saptanıştır. Bu çalışanın sonucunda, düşü partiül onsantrasyonlarında nüeri sonuçlar, deneysel değerler ve tü odeller birbirine yaındır. %10 dan daha büyü partiül onsantrasyonu için eeti ısı iletenliği exponansiyel şeil alatadır. Bu bölgede tü odeller iyi sonuç vereetedir. Aa, nüeri sonuçlar tü bölgelerde tatin edici sonuçlar veretedir. Anahtar Kelieler : Kopozit alzee, Isı iletenliği, Görüntü işlee, Sonlu arlar etodu 1. INTRODUCTION The eective theral conductivity o high-density polyethylene containing particulate illers is obtained nuerically at several iller concentrations. A nuerical approach was used to deterine the eective theral conductivity o particle coposites in this study. The eective theral conductivity o the aterial was deterined using the Laplace equation, as were the teperature and lux ields within the control volue. A inite dierence ethod was used in this study. Calculation is carried out on two-diensional geoetric spaces. The 105

2 results obtained ro this calculation were copared to results ound in prior literature. Knowing physical properties o the coposite aterials have gained signiicant iportance in the design o new systes. For any aterials applications, inoration is needed on their theral properties. Deterining the theral conductivity o coposite aterials is crucial in a nuber o industrial processes. The teperature ields in coposite aterials cannot be deterined unless the theral conductivities o the edia are nown. Despite the iportance o this aterial property and the considerable nuber o studies that have been carried out, the deterination o eective theral conductivity o a coposite aterial is partially understood. The eective theral conductivity o a coposite aterial is a coplex unction o their geoetry, the theral conductivity o the dierent phases, distribution within the ediu, and contact between the particles. Nuerous theoretical and experiental approaches have been developed to deterine the precise value o this paraeter. Maxwell was studied the eective theral conductivity o heterogeneous aterials. By solving Laplace's equation, he deterined the eective conductivity o a rando suspension o spheres within a continuous ediu. The odel developed by Maxwell assues that the particles are suiciently ar apart that the potential around each sphere will not be inluenced by the presence o other particles (Maxwell, 1954). Russell developed one o the early odel systes using the electrical analogy. Assuing that the discrete phase are isolated cubes o the sae size dispersed in the atrix aterial and that the isotheral lines are planes (Russell, 1935). Based on Tsao s probabilistic odel (Tsao, 1961), Cheng and Vachon assued a parabolic distribution o the discontinuous phase (Cheng and Vachon, 1969). Lewis and Nielsen derived a sei-theoretical odel by a odiication o the Halpin-Tsai equation (Ashton et all., 1969) to include the eect o the shape o the particles and the orientation or type o pacing or a two-phase syste (Lewis and Nielsen, 1970). Baschirow and Selenew developed the equation or the case when the particles are spherical and the two phases are isotropic (Baschirow and Manuian, 1974). In real coposite aterial, the isotheral suraces present a very coplex shape and cannot be analytically deterined. The odels used to calculate theral conductivities are thus highly sipliied odels o the real edia. Veyret, Cioulachtjian, Tadrist and Pantaloni characterized conductive heat transer through coposite, granular, or ibrous aterials by using the inite eleent ethod (Veyret et all., 1993). Terada, Miura and Kiuchi generated the inite eleent odel by identiying each pixel with a inite eleent and accopanying appropriate iage processing (Terada et all., 1997). Deissler and Boegli carried out the nuerical studies. They proposed a solution to Laplace s equation or a cubic array o spheres presenting a single point o contact (Deissler and Boegli, 1958). Deissler s wors were extended by Waao and Kato or a cubic or orthorhobic array o unior spheres in contact (Waao and Kato, 1968). Shonnard and Whitaer have investigated the inluence o contacts on twodiensional odels. They have developed a global equation with an integral ethod or heat transer in the ediu (Shonnard and Whitaer, 1989).. THERMAL CONDUCTIVITY MODELS In this section are listed several odels and a brie description o their basis. Many theoretical and epirical odels have been proposed to predict the eective theral conductivity o two phase ixtures. Coprehensive review articles have discussed the applicability o any o these odels that appear to be ore proising (Progelho, et all., 1976; Ott, 1981).. 1. Parallel, Series and Geoetric Mean Models For a two-coponent coposite, the siplest alternatives would be with the aterials arranged in either parallel or series with respect to heat low, which gives the upper or lower bounds o eective theral conductivity. For the parallel conduction odel: c (1 ).. (1) c : coposite, : atrix, : iller, : volue raction o iller and or series conduction odel: 1 1 () c In the case o geoetric ean odel, the eective theral conductivity o the coposite is given by: c ( 1). (3) Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

3 .. Cheng and Vachon Theoretical Model Starting with Tsao's basic probabilistic odel. Cheng and Vachon assued the discontinuous phase had a parabolic distribution. Based upon this assued distribution. Tsao's constants were evaluated and a closed or expressions or eective theral conductivity o a two phase ixture as a unction o discontinuous phase volue raction were obtained (Cheng and Vachon, 1969): c B ( B( ) C( ) B ln B (4) C( ) B( ) ( B( ) C( ) where or both equations: B 3 C 4 (3). 3. Lewis and Nielsen Sei-Theoretical Model Lewis and Nielsen odiied the Halpin-Tsai equation to include the eect o the shape o the particles and the orientation or type o pacing or a two-phase syste (Lewis and Nielsen, 1970). 1 AB c (5) 1 B where 1 A c 1 B 1 1 A The values o A and or any geoetric shapes and orientation are given in the ollowing tables: Table 1. Value o A or Various Systes Type o dispersed phase Direction o Heat Flow A Cubes Any.0 Spheres Any 1.50 Aggregates o spheres Any.5 1 n Any 1.58 Randoly oriented rods Aspect ratio = Randoly oriented rods Any.08 Aspect ratio = 4 Randoly oriented rods Any.8 Aspect ratio = 6 Randoly oriented rods Any 4.93 Aspect ratio = 10 Randoly oriented rods Any 8.38 Aspect ratio = 15 Uniaxially oriented ibers Parallel to ibers L/D Uniaxially oriented ibers Perpendicular to ibers 0.5 Table. Value o or Various Systes Shape o Type o Pacing Particle Spheres Hexagonal close Spheres Face centered cubic Spheres Body centered cubic 0.60 Spheres Siple cubic 0.54 Spheres Rando close Spheres Rando close Rods or ibers Uniaxial hexagonal close Rods or ibers Uniaxial siple cubic Rods or ibers Uniaxial rando 0.8 Rods or ibers Three diensional rando Maxwell Theoretical Model Maxwell using potential theory obtained an exact solution or the conductivity o randoly distributed and non-interacting hoogeneous spheres in a hoogeneous ediu (Maxwell, 1954): ( ) c (6) ( ). 5. Russell Theoretical Model Assuing the pores are cubes o the sae size and the isotheral lines are planes, Russell obtained the conductivity using a series parallel networ: / 3 / 3 (1 ) c (7) / 3 / 3 (1 ) 3. IMAGE PROCESSING An iage is a spatial representation o an object (Haralic and Shapiro, 1994). Necessary inoration in engineering is usually given by an iage or a set o several iages that is obtained in a easureent Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

4 or experiental procedure. The inoration is converted into a screen iage on a digital coputer. The digital iage-based (DIB) inite eleent odel (FEM) was originated by Hollister and Kiuchi, (1994) or studying a bone icrostructure by the hoogenization ethod. Although there are any operations in iage processing such as iltering, edge deduction, thresholding, and etc., we need only a ew o the or the present study. Filtering and thresholding, which are the ost iportant operations, in the irst step are the operations directly related to the inite dierence odel. When an iage is acquired by an iaging syste, the vision syste or which it is intended is unable to use it directly. The iage ay be corrupted by rando variations in intensity or poor contrast that ust be dealt with in the early stages o vision processing Size Filter It is very coon to use thresholding or inding a binary iage. In ost cases, there are soe regions in an iage that are due to noise. Usually, such regions are sall. In any applications, it is nown that objects o interest are o size greater than T o pixels. In such cases one ay use a size ilter to reove noise ater coponent labeling. All coponents below T o in size are reoved by changing the corresponding pixels to 0. This siple iltering echanis is very eective in reoving noise. Figure 1 shows an exaple o the application o a size ilter to a noisy character iage (Jain et all., 1995). 3.. Thresholding The thresholding is deined as an iage operation which produces a binary iage ro a gray scale iage. Furtherore, the thresholding can produce a binary one on the output digital iage whenever a pixel value on the input digital iage is below a speciied threshold level. A binary zero is produced otherwise. Although a coposite aterial is not always coposed o two phases, we shall consider two-phase coposites or siplicity. Nonetheless, the deinition can be easily extended by increasing the nuber o thresholding values. The threshold values ay be deterined interactively by the operator. While the video display aords direct coparison o the thresholded iage with the woring iage, the sotware enables us to odiy or ix the thresholded iages. Since we have assued that the phases o the coposite could be distinctive in the original iage, the thresholding is done by reerring to the histogra on the video screen. I the original iage has enough resolution and little noise, then the histogra can provide ost o the inoration needed to choose the threshold value required or generating approxiated geoetry. Furtherore, i the volue raction o each phase in the two diensional iage is given, the sotware ay provide a unction to calculate the ratio between pixel values, by which the threshold value is deterined. (a) Figure 1. A noisy binary iage and the resulting iage ater the application o a size ilter (right). T 0 = 10 (b) (a) Original iage (b) Thresholded iage Figure. Thresholding the digital iage Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

5 Once a threshold value has been chosen, it is easy to convert the selected iage into a binary iage. I the obtained iage does not see to be a satisactory representation o the original, new threshold values ay be chosen until we are satisied with the binary iage (Terada et all., 1997), (Figure ). 4. EXPERIMENTAL STUDY The atrix aterial is a high-density polyethylene in powder or, with a density o g/c 3 and a elt index o 5.8 g/10 in. Its theral conductivity at 36 C is W/.K. The etallic iller is aluinu in the or o ine powder, with particles approxiately spherical in shape and particle size in the range o icrons or aluinu. The solid density o aluinu is.7 g/c 3 and its theral conductivity 04 W/.K. Saples are prepared by the old copression process. HDPE and aluinu powders are ixed at various voluetric concentrations (Figure 3). The ixed powder is then elted under pressure in a old and solidiied by air cooling. The process conditions are olding teperature o 185 C, pressure o 4 MPa. The resulting saples or theral conductivity easureents are rectangular in shape o 100 length, 50 width, and 17 thicness because o the easuring probe. Hoogeneity o the saples is exained using a light icroscope. Aluinu particles are ound to be uniorly distributed in high-density polyethylene atrix with no voids in the structure. aterial. The teperature ield in the coposite aterial was deined by solving Laplace s equation nuerically using a inite dierence orulation. The Laplace s equation was solved by iposing the ollowing boundary conditions: (a) The vertical sides perpendicular to the direction o the heat low are isotheral at the entrance to and the exit ro the cell. (b) The horizontal sides parallel to the direction o the heat low are adiabatic. The heat low oving into or out o the cell reaches its pea at the center o the isotheral aces. For an eleentary two-diensional cell with the diensions o L x (along the x axis) and L y (along the y axis), the theral conductivity is deterined using the ollowing relation: L ytcell T i c i yi (7) L x i x with y i L y and i in thecontinuous phase i in theinclusions Q T 1 y Adiabatic ace L y T y x x L x Adiabatic ace Figure 4. Two-diensional odel o the coposite aterial Figure 3. Microscopic photograph o HDPE illed with 4 volue percent o Al 5. NUMERICAL MODELING OF THE PROBLEM Two-diensional nuerical analysis was carried out or the conductive heat transer in the coposite In the considered heat conduction proble the teperatures at the nodes along the boundaries x = 0 and x = LX are prescribed and nown as T 1 and T, but the teperatures at the nodes in the interior region and on the adiabatic boundaries are unnown. Thereore, the proble involves any unnown teperatures. The equations needed or the deterination o these teperatures are obtained by writing the appropriate inite-dierence equation or each o these nodes. Then, the inite-dierence or o the energy equation without heat generation is written as; Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

6 i1,j 1 i1,j T T T T T T 0 i1,j a1 i1,j 1 a a4 1 i1,j a3 i1,j (8) a1 a a3 a4 T a1 T a a4 T a3 T 0 a5 i1,j The inite-dierence equation or coposite aterial on an adiabatic boundary (9) i =, 3, 4,.N-1 j=1 (N, M) : Size o iage 1 i1, j Siilarly, the inite-dierence or o the energy equation or j = M is written as i1,j 1 i1,j T T T T T T 0 i1,j a1 i1,j 1 a a4 1 i1,j a3 i1,j (10) a1 a a3 a4 T a1 T a a4 T a3 T 0 a5 i1,j 1 i1, j The inite-dierence equation or coposite aterial on an adiabatic boundary (11) i =, 3, 4,.N-1 j = M (N, M) : Size o iage Siilarly, the inite-dierence or o the energy equation or interior nodes is written as i1,j 1 T T T T i1,j a1 i1,j 1 T T T T 0 i1,j a3 i1,j i1,j 1 a 1 a4 1 1 (1) 1 a a3 a4 T a1 T a T a3 T a4 T 0 a i1, j 1 i1, j 1 a5 The inite-dierence equation or coposite aterial in the interior nodes (13) i =, 3, 4,.N-1 j =, 3, 4,...M-1 (N, M) : Size o iage 6. RESULTS AND DISCUSSION In the Figure 5, the teperature distribution on the coposite aterial coposed o 4 % Aluinu illed high-density polyethylene is seen. Results obtained ro the experiental studies o Aluinu illed high-density polyethylene, several odels and nuerical studies are given in the Figure 6. As seen ro the Figure 6, soe odels give results approxiate with the experiental data. Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

7 photographed o the piece taen ro the specien. Since the Al particles are not well ixed into that region, they ay appear ore dense. It is clear that this increases the theral conductivity o the coposite. All the odels ail to predict theral conductivity or particle content greater than 10 %. Figure 5. Isotheral curves or the coposite aterial coposed o 4 % aluinu illed highdensity polyethylene Theral Conductivity (W/K 4,5 4 3,5 3,5 1,5 1 0,5 Nuerical Experiental Maxw ell Russel Baschirow &Selenew Lew is&nielsen Cheng&Vachon Üstel (Experiental) A=3, =0.637 A=3, =0.637 y = 0,5443e 0,0601x R = 0, Voluetric Ratio Al % Figure 6. Coparison o the nuerical results, odels and experiental values or aluinu iller I the results obtained ro the nuerical analysis are copared with the results obtained ro the experiental study, it is seen that the nuerical results are higher. It can be said that, since the quality o the pictures used in iage processing carried out on the picture iles in nuerical study are not so good, the procedure was not so successul. Objects deined as noise in the picture will be accepted as Al particles added into the high-density polyethylene and in return, these additional objects will increase the theral conductivity o the coposite. Another reason or this value to be high can be the nonhoogeneous distribution o the Al particles into high-density polyethylene during the preparation o the speciens which eect the picture 7. REFERENCES Ashton, J., Halpin, J., Petit, P Prier on Coposite Materials: Analysis. Technoic Pub. Co. Staord. Baschirow, A. B., Manuian, A. M Theral Conductivities o Polyers at Various Teperatures and Pressures. Mech. Poli. 3, 564. Cheng, S. C., Vachon, R.I The Prediction o the Theral Conductivity o two and Three Phase Solid Heterogeneous Mixtures. Int. J. Heat Mass Transer. 1, 49. Deissler, R. G., Boegli, J. S Investigation o Eective Theral Conductivites o Powders in Various Gases. Trans. ASME. 80, Haralic, R. M., Shapiro, L Glossary o Coputer Vision Ters. In E.R. Dougherty (Eds.), Digital Iage Processing Methods. Marcel Deer Inc. Hollister, S. J., Kiuchi, N Hoogenization Theory and Digital Iaging: a Basis or Studying the Mechanics and Design Principles o Bone Tissue. Biotechnol. Bioeng. 43 (7), Jain, R., Kasturi, R., Schunc, B. G Machine Vision. McGraw-Hill, Inc. New Yor Lewis, T., Nielsen, L Dynaic Mechanical Properties o Particulate-Filled Polyers. J. Appl. Poly. Sci. 14, Maxwell, J. C A Treatise on Electricity and Magnetis. Dover ( 3 rd Ed.), New Yor, Ch.9. Ott, H. J Theral Conductivity o Coposite Materials. Plastic and Rubber Processing and Applications. 1, 9-4. Progelho, R. C., Throne, J. L., Ruetsch, R. R Methods o Predicting the Theral Conductivity o Coposite Systes. Polyer Eng. And Sci. 16, Russell, H. W Principles o heat low in Porous Insulation. J. A. Cera. Soc. 18, 1. Shonnard, D. R., Whitaer, S The Eective Theral Conductivity or a Point-Contact Porous Mediu: an Experiental Study. Int. J. Heat Mass Tr. 3, Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)

8 Terada, K., Miura, T., Kiuchi, N Digital Iage-Based Modeling Applied to the Hoogenization Analysis o Coposite Materials. Coputational Mechanics. 0, Tsao, T. N. G Theral Conductivity o two Phase Materials. Industrial and Engineering Cheistry. 53, Waao, N., and Kato, K Eective Theral Conductivity o Paced beds. J. o Cheical Eng. o Japan., Veyret, D., Cioulachtjian, S., Tadrist, L., Pantaloni J Eective Theral Conductivity o a Coposite Material: A Nuerical Approach. J. o Heat Tr. 115, 866. Mühendisli Bilileri Dergisi (-3) Journal o Engineering Sciences (-3)