In-Plane Effective Thermal Conductivity of Plain-Weave Screen Laminates

Transcription

1 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 25, NO 4, DECEMBER In-Plane Effective Thermal Conductivity of Plain-Weave Screen Laminates Jun Xu and Richard A Wirtz Abstract A simple-to-fabricate woven mesh, consisting of bonded laminates of two-dimensional plain-weave conductive screens is described Geometric equations show that these porous matrices can be fabricated to have a wide range of porosity and specific surface area, A heat transfer model is developed It shows that the laminates can have a highly anisotropic thermal conductivity vector, with in-plane effective thermal conductivities ranging up to 785% of base material values A technique to measure the laminate in-plane effective thermal diffusivity is described Measurements of the in-plane effective thermal diffusivity of copper plain-weave laminates are used to benchmark the model Index Terms Plain-weave, porosity, screen-laminates, thermal conductivity NOMENCLATURE Specific heat Compression factor Diameters for the - and -direction filaments Diameter for sphere Effective thermal conductivity in -direction Thermal conductivity of base material Thermal conductivity of second phase material Heat flux Mesh numbers along - and -directions Wire filament lengths in unit cell Thickness of screen laminates with layers thickness Effective thermal resistance Thermal diffusivity Specific surface area Porosity Time Manuscript received January 16, 2002; revised July 26, 2002 This work was supported by the Missile Defense Agency through the Air Force Office of Scientific Research, USAF, under Contract F This work was recommended for publication by Guest Editors S V Garimella and Y K Joshi upon evaluation of the reviewers comments J Xu was with the Mechanical Engineering Department, University of Nevada, Reno, NV USA He is now with the School of Mechanical Engineering, Purdue University, West Lafayette, IN USA ( richardxujun@yahoocom) R A Wirtz is with the Mechanical Engineering Department, University of Nevada, Reno, NV USA ( rawirtz@unredu) Digital Object Identifier /TCAPT I INTRODUCTION ATHERMALLY conductive, open cell porous matrix such as an unconsolidated bed of small particles, or foamed metal such as foamed aluminum, make excellent heat exchanger surfaces due to their large specific surface area, Also, when used as a fill material, these matrices are effective solid composite conductivity enhancers However, due to high porosity, coupled with the tortuosity effect, the effective thermal conductivity of these materials, is relatively small, so that much of the gain in performance obtained by having a large is lost by having a relatively small Typical values of in fused spherical particle packed beds are 10% 15% of the particle thermal conductivity, Commercially available metal foam such as aluminum foam, has an effective thermal conductivity that ranges from 2% to 6% of the base metal value [1] An anisotropic porous matrix having a large specific surface area and high effective thermal conductivity in a particular direction will result in a very effective heat exchange device In this paper, we show that stacked laminates of conductive screening can be configured to have these characteristics Such a porous matrix can be incorporated into the design of a flow-through module or cold plate heat exchanger, resulting in a compact, high-flux device with reasonable pressure-drop characteristics [2] The earliest model for the effective thermal conductivity of a porous material is attributed to Rayleigh [3] The model assumes that the porous material is isotropic More recently, Alexander [3], Koh and Fortini [4] and Chang [5] have reported models and empirical correlations specifically directed at the cross-plane effective thermal conductivity component of screen material We have found no reference in the technical literature for the in-plane component of the effective thermal conductivity II PHYSICAL MODEL Fig 1 summarizes the development of the model of a plainweave screen The figure shows plan and edge views of a section of screen The region enclosed within dashed lines defines the unit cell of the plane weave Serpentine wire filaments have diameter and, and corresponding mesh numbers and The wire filament pitches in the - and -directions are and, respectively In the absence of crimping, the screen has thickness Filament lengths in the unit cell are, [6] (1) /02$ IEEE

2 616 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 25, NO 4, DECEMBER 2002 Fig 1 Plane weave unit cell (2) Fig 2 shows two possible arrangements of stacked plainweave screens The upper view shows the situation where successive wire filaments are aligned so that the thickness of a laminate consisting of screens is The lower element of the figure shows the situation where successive wire filaments are not necessarily in line so there is some interleaving of wire filaments when the screens are stacked In this case, the thickness of the laminate is, where is the compression factor The magnitude of the compression factor depends on the weave pattern and stacking arrangement of the screen layers If we restrict our attention to screens with, then Consideration of Figs 1 and 2 leads to expressions for the porosity, and specific surface area, of plain-weave screen laminates (3) Fig 2 Screen stacking configuration where is the reduced metal fraction SF in (4) is the shape factor for the particular weave pattern and stacking arrangement (4) (5) If we introduce the mesh number and wire filament diameter ratios and, then it can be shown that the reduced metal fraction is a function of and is a function of, [7] Following Chang [5], we transform the segment of screen unit cell shown in Fig 1 to the layered, rectangular cross section segment shown in the Fig 3 Each rectangular wire filament has thickness and width so that the cross section area of each filament is The geometric index, is a measure of the contact between wire filaments at intersections The in-plane effective thermal conductivity in the -direction may be determined by considering the thermal circuit for conduction across the transformed unit cell, shown in Fig 4 Fig 3 Transformed unit cell The figure shows three heat-flow paths: is the thermal resistance for conduction along the axis of the -wire filaments; is the thermal resistance across -wire filaments; is the thermal resistance to conduction in the -direction, across the intervening second phase material lying between -wire filaments; and, in the thermal resistance of the second phase

3 XU AND WIRTZ: IN-PLANE EFFECTIVE THERMAL CONDUCTIVITY 617 Fig 4 Thermal circuit Fig 5 Screen laminate reduced metal fraction -wire filaments These re- material slab that lays between the sistances are given as where is the thermal conductivity of screen filament material, and is the thermal conductivity of second phase material We further require that the volume of the unit cell be preserved across the geometric transformation shown in Fig 3 Then The -direction conductance across the transformed unit cell is (6) (7) (8) (9) (10) where is the overall thermal resistance of the thermal circuit (Fig 4) and is the -direction in-plane effective thermal conductivity of the screen Rearranging (10) gives where (11) (12) (13) and,, are the dimensionless effective conductivity and second phase material conductivity, respectively Since most available commercial woven mesh products are isotropic structures, ie, and, the above general results can be simplified to the following: (14) (15) (16) Equation (14) shows that the reduced metal fraction of isotropic screen laminates, is solely a function of the -product of the mesh We note that designates a tightly woven screen In actuality, there is a physical limit on the magnitude of For isotropic plain-weave screens where the thickness is limited to, so that the porosity is limited such that Equation (15) shows that the specific surface area, of isotropic screen laminate follows the functional form of other porous media, with By contrast, spheres for an unconsolidated bed of spherical particles, is the sphere diameter Finally, (16) shows that the effective conductivity is solely a function of, and Fig 5 plots the reduced metal fraction as a function of -product The case for an isotropic screen laminate is shown as a solid line This case always produces screen laminates having the highest reduced metal fraction (lowest porosity) The reduced metal fraction for two anisotropic plain-weave laminate cases, are also shown in the figure Fig 6 plots -product as a function of reduced metal fraction The case for an isotropic screen laminate is shown as a bold solid line The shape factor for this condition is (the slope of the plotted line) This is true for all conditions where The case, is shown superimposed on the isotropic case All conditions with, lead to larger values of at fixed Therefore, the condition always produces the smallest specific surface area Fig 6 compares of screen laminates with a bed of fused spheres and metal foams The theoretical porosity of a packed bed of unconsolidated spheres depends on the packing arrangement [8] It can range from for face centered cubic packing to for simple cubic However, the achievable porosity of a packed bed is difficult to control A typical porosity for an unconsolidated bed will range from 037 to 041 As a consequence, the metal fraction can range from 059 to 063 and will then range from 354 to 378 The range of the specific sur-

4 618 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 25, NO 4, DECEMBER 2002 Fig 6 Screen laminate specific surface area Fig 8 Test configuration value of more than 03, with the reduced metal fraction of about 04 Comparing with a fused bed of spheres, which is expected to have sphere with metal fraction,, this represents an approximate two-fold increase in thermal performance relative to fused sphere heat transfer matrices [8] The range of values of effective conductivity and metal fraction of metal foam [9] are also shown in the figure By virtue of their higher effective thermal conductivity, the screen laminates structures offer a two-to four-fold improvement in performance relative to low-porosity exchange matrices fabricated from metallic foam Fig 7 Screen laminate conductivity face area achievable with metallic foams [1] is also displayed Depending on the origin of the data for foams, screen laminates provide either a significant increase in specific surface area, or they are comparable to the foams In any case, screen laminates extend the range of metal fraction (or porosity) beyond what appears to be achievable with metal foam Fig 7 plots the reduced effective conductivity as a function of reduced metal fraction The case for an isotropic screen laminate with is shown as a bold solid line An additional case, isotropic screen laminate with is also shown (solid line) Under these conditions, the effective conductivity is shifted upward approximately proportional to the value of In many applications where the second phase material is a hydrocarbon with the first phase a good conductor, Two additional, anisotropic screen laminate cases are shown in the figure Cases with or always produce effective conductivities that are greater than obtained with isotropic plain weaves It is easy to find the maximum possible value of When, and, then, (11) gives (17) The dimensionless effective thermal conductivity of anisotropic screen laminates structure could be easily configured with a III MODEL VERIFICATION WITH EXPERIMENT A Measurement of Thermal Diffusivity Consider a uniform cross section, long and slender test article with well-insulated periphery, as shown in Fig 8 The test article is initially at a uniform temperature, when a time-varying heat flux is applied to one end We measure the temperature response at three locations, as shown in the figure Assume constant thermophysical properties and onedimensional transient conduction in the domain between points 1 and 3 The temperature response at point 2 will be given as (18) The temperature response at point 1 describes the heat input to the test domain; that at point 3 describes the heat outflow from the test domain Then, the temperature response at point 2 can be used to determine the thermal diffusivity,, [7] B Measurement Calibration Three bar-shaped oxygen-free copper test articles (Alloy C10100, m /s) were tested Each test was repeated six times The mean measured thermal diffusivity for each of the test articles was m /s, m /s and m /s, giving an overall average value of m /s The 95% confidence interval for all 18 measurements (2 about mean values) is m /s

5 XU AND WIRTZ: IN-PLANE EFFECTIVE THERMAL CONDUCTIVITY 619 Fig 10 Benchmark measurements of k in air Fig 9 Bonded screen laminate and the implied offset error, based on the handbook value of thermal diffusivity of alloy C10100, is 52% C Screen Laminates Test Articles Isotropic, plain-weave copper screens are stacked and bonded together (using 95/5 Sn/Antimony solder) to form the screen laminates Various screen laminate samples were fabricated with this method: the mesh numbers range from 63 cm (16 inch ) to 1575 cm (40 inch ), the bare copper wire filament diameter ranges from 028 mm (11/1000 inch) to 046 mm (18/1000 inch), and the number of layers ranges from 3 to 10 Measured compression factors range from 070 to 100 Fig 9 shows a typical copper plain-weave screen-laminate The wire filaments have an effective diameter of 048 mm, and the mesh number is 63 cm (16 inch ) The sample has, m (After the bonding, the effective thickness of solder layer over the screen filament ranges from 0006 mm to 0013 mm) Twenty (20) test articles of the above mentioned fabricated isotropic screen laminate specimens are tested In the calculation, the effective density/specific heat product, of the screen laminate is calculated based on the measured weight ratios of copper and solder material in the composites The copper wire properties are: kg/m, J/(kg K), W/(mK) The Sn Atimony solder material s properties are: kg/m, J(/kg K), W/(mK) As different specimens have different weight ratios of copper and solder material, so the calculated effective densities for the laminates range from 8694 kg/m to 8796 kg/m, the specific heats range from 3691 J/kg K to 3790 J/kg K and, the thermal conductivity range from 3497 W/mK to 3711 W/mK The thermal conductivity of air (00278 W/mK) at temperature of 323 K, is used D Test Results The average test result of dimensionless reduced in-plane effective conductivity for each specimen is shown with 2 error bars in Fig 10 Each data point shown represents six or more test runs of the same sample The model prediction for the isotropic plain-weave screen laminate saturated with air is also shown in the figure It shows that the prediction of the present model is within 10% of all of the measurements With consideration of the nonuniform solder layer s thickness on the filaments, and nonuniform properties of the high porosity screens, the prediction of the present model is quite accurate IV CONCLUSION Simple porous media structures can be fabricated from laminates of conductive screening The resulting structures can be configured to have a wide range of porosity, specific surface area and in-plane effective thermal conductivity The reduced effective conductivity is found to range up to 23% and 78% of the filament material s conductivity for isotropic and anisotropic plain-weave screen laminates, respectively The in-plane effective thermal conductivity of screen laminates, in particular for anisotropic screen laminates, is much greater than can be achieved with other porous media configurations The material can be configured to have highly anisotropic thermal properties while at the same time high porosity is maintained The simple universal model for effective in-plane thermal conductivity and porosity has been verified with benchmark experiments A technique for thermal conductivity (diffusivity) is set up and calibrated REFERENCES [1] M Ashby, A Evans, N Fleck, L Gibson, J Hutchinson, and H Wadley, Metal Foams, A Design Guide New York: Butterworth Heinemann, 2000 [2] J-W Park, D Ruch, and R A Wirtz, Thermal/fluid characteristics of plain-weave laminates as heat exchanger surfaces, in fproc 40th AIAA Aerosp Sci Meeting Exhibit, Reno, NV, 2002, Paper # [3] E G Alexander, Jr, Structure property relationships in heat pipe wicking materials, PhD dissertation, Dept of Chem Eng, North Carolina State Univ, Raleigh, NC, 1972 [4] J C Y Koh and A Fortni, Prediction of thermal conductivity and electrical resistivity of porous metallic materials, Int J Heat Mass Transfer, vol 16, pp , 1973 [5] W S Chang, Porosity and effective thermal conductivity of wire screens, J Heat Transfer, vol 112, pp 5 9, Feb 1990 [6] S Y Luo and A Mitra, Finite elastic behavior of flexible fabric composite under biaxial loading, J Appl Mech, vol 66, pp , 1999

6 620 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL 25, NO 4, DECEMBER 2002 [7] J Xu, In-plane effective thermal conductivity of plain weave screen laminates, MS thesis, Mech Eng Dept, Univ of Nevada, Reno, NV, 2001 [8] M Kaviany, Principals of Heat Transfer in Porous Media, 2nd ed New York: Springer, 1995 [9] V V Calmidi and R L Mahajan, The effective thermal conductivity of high porosity fibrous metal foams, J Heat Transfer, vol 121, pp , May 1999 Jun Xu received the BE degree in refrigeration and cryogenics from Shanghai Jiao Tong University, China, in 1991, the MS degree in mechanical engineering from the University of Nevada, Reno, in 2001, and is currently pursuing the PhD degree in the School of Mechanical Engineering, Purdue University, West Lafayette, IN He has worked in both academia and industry He spent more than four years in Shanghai Jiao Tong University as R&D Faculty He moved to Singapore in 1996, as a Mechanical Engineer for P&T consultants Pte, Ltd and Honeywell Southeast Asia His interests are electronics cooling and heat transfer His current research is in micro-scale heat transfer Richard A Wirtz is a Professor of mechanical engineering at the University of Nevada, Reno He is also founder (in 1995) of Sierra-Nevada Research and Development, Inc (a company specializing in thermally related product, process and intellectual property development and marketing) He and serves on the editorial board of the ASME Journal of Electronic Packaging Dr Wirtz received the ASME Electronic Packaging Division Award for Outstanding Contributions to the Field of Thermal Management of Microelectronics Equipment and Systems, and the Lemelson Award for Innovation and Entrepreneurship He is a Fellow of the American Society of Mechanical Engineers and a member of the American Institute of Aeronautics and Astronautics