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Finite element simulation of the thermal conductivity of perforated hollow sphere structures (PHSS): Parametric study
Materials Letters 63 (2009) 1135–1137
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Finite element simulation of the thermal conductivity of perforated hollow sphere structures (PHSS): Parametric study Seyed Mohammad Hossein Hosseini a, Markus Merkel b, Andreas Öchsner a,c,⁎ a b c
Department of Applied Mechanics, Technical University of Malaysia, 81310 Skudai, Malaysia Department of Mechanical Engineering, University of Applied Sciences Aalen, 73430 Aalen, Germany Centre for Mass and Thermal Transport in Engineering Materials, The University of Newcastle, Callaghan, New South Wales 2308, Australia
a r t i c l e
i n f o
Article history: Received 24 September 2008 Accepted 6 October 2008 Available online 14 October 2008 Keywords: Thermal properties Composite materials Cellular metals Computer simulation Porosity
a b s t r a c t This paper investigates the thermal properties of a new type of hollow sphere structures. For this new type, the sphere shell is perforated by several holes in order to open the inner sphere volume and surface. The effective thermal conductivity of perforated sphere structures in a primitive cubic arrangement is numerically evaluated for different hole diameters and different dimensions of the joining elements. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Hollow sphere structures (HSS) are novel lightweight materials within the group of cellular metals (such as metal foams) which are characterised by high specific stiffness, the ability to absorb high amounts of energy at a relatively low stress levels, potential for noise control, vibration damping and thermal insulation. A combination of these different properties opens a wide field of potential multifunctional applications e.g. in automotive or aerospace industry. Typical functional applications of cellular metals in the scope of heat transfer are heat exchangers [1–3], fire retardance systems [4] or thermal insulation. The high thermal insulation capability of hollow sphere structure has been addressed in a US Patent [5] by Schneider and co-workers. Baumeister and colleagues [6] investigated the linear thermal expansion coefficient of corundum based hollow sphere composites (HSC) using thermomechanical analysis. They found that the thermal behaviour of HSC is mainly governed by the epoxy resin. Lu and Kou [7] conducted a comprehensive numerical and experimental study based on unit cells of homogeneous hollow sphere structures. However, only automatically generated finite element meshes were used and the applied approach does not allow the consideration of different material combinations. In recent paper [8], the thermal conductivity of adhesively bonded and sintered HSS was numerically investigated while in paper [9], the influence of material parameters and geometrical properties of ⁎ Corresponding author. Department of Applied Mechanics, Technical University of Malaysia, 81310 Skudai, Malaysia. E-mail addresses: [email protected] (S.M.H. Hosseini), [email protected] (M. Merkel), [email protected] (A. Öchsner) 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.10.009
syntactic (hollow spheres completely embedded in a matrix) and partially bonded HSS (spheres joined in localised contact points) was analysed. A comparison between analytical, numerical and experimental approaches is given in [10]. Fiedler et al. reported in [11] recent advances in the prediction of the thermal properties of syntactic metallic hollow sphere structures. They described the application of the Finite Element and Lattice Monte Carlo Method in the case of syntactic periodic and random hollow sphere structures. Manufacturing techniques for perforated hollow sphere structures are actually under development [12] and we use an idealised model structure to clarify basic effects of the perforation on physical properties. In the present study, perforated hollow spheres arranged in a primitive cubic periodic pattern are numerically investigated based on the finite element method (unit cell approach) and the effective thermal conductivity is compared to structures without holes. The influence of the hole diameter and the geometry on the effective conductivity is analysed within a parametric computational study. 2. Perforated hollow sphere structure The major idea of introducing a perforation, i.e. holes of circular cross section in the sphere shells, is to make the inner sphere surface and volume usable. In this paper, the holes are defined in such a way that the largest possible hole can be located between the linking elements in a primitive cubic arrangement. The modeling approach is restricted to the simple case where the holes do not intersect with the link area. The additional inner surface may be used for chemical reactions in the case of catalysts or the additional inner volume may have a positive influence on the dissipation of e.g. acoustic waves. A simplified arrangement of a
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S.M.H. Hosseini et al. / Materials Letters 63 (2009) 1135–1137
Fig. 1. Schematic representation of a primitive cubic arrangement of perforated hollow spheres.
perforated hollow sphere structure in a primitive cubic pattern is shown in Fig. 1. The darker grey volumes represent the connecting elements (‘links’) which are formed in the case of liquid phase processes (e.g. formed by adhesives or soldering agents). The connecting material is during a certain period of the joining process in the liquid state which forms later a solid neck region between the spheres. In the case of diffusion bonding (sintering), such neck regions do not form and the spheres are in direct contact (consider Fig. 1 without the dark grey volumes where neighboring spheres are touching). Looking at a specific example where perforated spheres (outer diameter 2.7 mm and shell thickness of 0.1 mm, outer hole radius 0.68 mm) are considered, the available free surface increases by ~ 57% and the free volume increase by ~259% while the bulk volume or the weight is reduced by ~33%. In addition, we can state that the porosity is only increased by ~ 3.9%. 3. Finite element analysis In order to reduce the size of the system of equations, the so-called unit-cell approach is chosen where the computational structure is reduced from a larger or infinite amount of randomly arranged spheres to a single unit-cell which is commonly based on typical space groups as in crystallography [13]. Common unit cell approaches are based on primitive cubic, face-centered or hexagonal arrangements.
Fig. 3. Normalised conductivity of HSS as a function of the hole diameter (a fraction of 0 corresponds to the sphere without holes); markers (○,Δ) denote finite element results, lines (–) indicate the trend.
Since the aim of this research is to highlight the difference between perforated and entire spheres and to investigate the influence of several geometrical parameters, we restrict our computational approach to the primitive cubic arrangement because this is a first estimation for a finite element analysis that can give reasonable results. A typical finite element mesh including boundary conditions is shown in Fig. 2. Constant temperature boundary conditions (Ti = const.) are prescribed at the faces of the connective elements on the left and right side of the model. In order to generate a heat flux through the structure, only T1 ≠ T2 must be fulfilled. According to the assumed symmetry and simplifications (no radiative or convective effects), the ratio of the heat flux perpendicular to all remaining surfaces is zero. The thermal properties of the considered base materials are for steel ksteel = 0.05 W/mm K (AISI 8000) and for epoxy kepoxy = 0.000214 W/mm K. The evaluation of the effective thermal conductivities within the finite element approach is based on Fourier's law where the arearelated conductivity is defined by k=
: Q Δy : A ΔT
ð1Þ
The area A of the unit cell and the spatial distance Δy are given by the geometry (cf. Fig. 2), respectively the temperature gradient ΔT =T2 −T1
Fig. 2. Finite element mesh and boundary conditions of a perforated primitive cubic unit cell (R = 1.25 mm, a = 0.12 mm, b = 0.4 mm, d = 1.36 mm).
S.M.H. Hosseini et al. / Materials Letters 63 (2009) 1135–1137
1137
were done with the commercial finite element package MSC.Marc® (MSC Software Corporation, Santa Ana, CA, USA). 4. Results Fig. 3 summarises the effective thermal conductivities of entire and perforated spheres for both, homogeneous (sphere shell and connecting element assumed to be made of the same material) and non-homogeneous spheres (steel and connecting element assumed to be made of different material, i.e. steel and epoxy). The case of homogeneous sphere relates to diffusion bonded structures and a clear drop in the thermal conductivity can be seen when the material property of the connecting element is changed to epoxy (i.e. non-homogenous sphere). It can be seen that the shape of both curves is the same. However, the non-homogeneous sphere rather acts as an isolator. In addition, it can be seen that the consideration of holes significantly decreases the thermal conductivity. Fig. 4 shows the influence of the joining element on the effective conductivity for homogeneous and non-homogeneous sphere structures. The effect of a changing geometry is much less pronounced than the influence of the holes in the case of homogeneous spheres. However in the case of the non-homogeneous spheres the behaviour is contrary: The shape of the curve is dominated by the geometry of the joining element while the influence of the hole is marginal. It should be noted here that the values of the conductivity were normalised in all cases, i.e. homogeneous and nonhomogeneous, with the sphere shell material (= steel).
5. Outlook In the scope of this research project, the effective thermal conductivity of perforated hollow sphere structures has been numerically estimated and compared to configurations without holes. Introducing holes in the sphere shells clearly reduces the thermal conductivity of the HSS. The numerical approach has been based on a primitive cubic unit cell arrangement. Future investigations must clarify the influence of the hole arrangement and the geometrical modifications such as sphere geometry or the spatial pattern of the spheres. References Fig. 4. Normalised conductivity of HSS as a function of link geometry (a = 0.12 mm, b = 0.4 mm) for several hole diameters ranging from 1d = 1.36 mm (biggest hole) hole till 0d (without hole), cf. Fig. 2; markers (Δ) denote finite element results, lines (–) visualise the trend.
· by the boundary conditions, only the heat flux Q remains to be determined. This is done by summing up all nodal values of the reaction heat flux with a user subroutine at the left or right face where a temperature boundary condition is prescribed. Within the relevant temperature range, the contribution of thermal radiation to the heat transfer is low [14]. Furthermore, contributions from gaseous conduction and convection are neglected. The whole model consists of 51,056 elements in the case of the largest hole with a radius of 0.68 mm. Subsequent models where the hole diameter was reduced to 75, 50 and 25% were generated. In addition the geometrical parameters of the joining element (cf. parameters a and b in Fig. 2) where modified to investigate the influence of a changing link geometry. All the simulations
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14]
Lu W, Zhao CY, Tassou SA. Int J Heat Mass Transfer 2006;49:2751–61. Lu TJ, Stone HA, Ashby MF. Acta Mater 1998;46:3619–35. Boomsma K, Poulikakos D, Zwick F. Mech Mater 2003;35:1161–76. Lu TJ, Chen C. Acta Mater 1999;47:1469–85. L. Schneider, A. Boehm, C. Korhammer , R. Scholl, B. Voigtsberger, G. Stephani, US Patent 6501784. (2002). Baumeister E, Klaeger S, Kaldos A. J Mater Process Technol 2004;155:1839–46. Lu KT, Kou HS. Int Commun Heat Mass 1993;20:489–500. Fiedler T, Öchsner A. Mat Sci Forum 2007;553:39–44. Fiedler T, Öchsner A. Mat Sci Forum 2007;553:45–50. Fiedler T, Solórzano E, Öchsner A. Mater Lett 2008;62:1204–7. Fiedler T, Öchsner A, Belova IV, Murch GE. Adv Eng Mater 2008;10:361–5. Glatt GmbH, Binzen, Germany, private communication (2007). de Graef M, McHenry M. Structure of Materials. An Introduction to Crystallography, Diffraction and Symmetry. Cambridge: Cambridge University Press; 1997. Lu T, Chen G. Acta Mater 1999;47:1469–85.
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Finite element simulation of the thermal conductivity of perforated hollow sphere structures (PHSS): Parametric study Seyed Mohammad Hossein Hosseini a, Markus Merkel b, Andreas Öchsner a,c,⁎ a b c
Department of Applied Mechanics, Technical University of Malaysia, 81310 Skudai, Malaysia Department of Mechanical Engineering, University of Applied Sciences Aalen, 73430 Aalen, Germany Centre for Mass and Thermal Transport in Engineering Materials, The University of Newcastle, Callaghan, New South Wales 2308, Australia
a r t i c l e
i n f o
Article history: Received 24 September 2008 Accepted 6 October 2008 Available online 14 October 2008 Keywords: Thermal properties Composite materials Cellular metals Computer simulation Porosity
a b s t r a c t This paper investigates the thermal properties of a new type of hollow sphere structures. For this new type, the sphere shell is perforated by several holes in order to open the inner sphere volume and surface. The effective thermal conductivity of perforated sphere structures in a primitive cubic arrangement is numerically evaluated for different hole diameters and different dimensions of the joining elements. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Hollow sphere structures (HSS) are novel lightweight materials within the group of cellular metals (such as metal foams) which are characterised by high specific stiffness, the ability to absorb high amounts of energy at a relatively low stress levels, potential for noise control, vibration damping and thermal insulation. A combination of these different properties opens a wide field of potential multifunctional applications e.g. in automotive or aerospace industry. Typical functional applications of cellular metals in the scope of heat transfer are heat exchangers [1–3], fire retardance systems [4] or thermal insulation. The high thermal insulation capability of hollow sphere structure has been addressed in a US Patent [5] by Schneider and co-workers. Baumeister and colleagues [6] investigated the linear thermal expansion coefficient of corundum based hollow sphere composites (HSC) using thermomechanical analysis. They found that the thermal behaviour of HSC is mainly governed by the epoxy resin. Lu and Kou [7] conducted a comprehensive numerical and experimental study based on unit cells of homogeneous hollow sphere structures. However, only automatically generated finite element meshes were used and the applied approach does not allow the consideration of different material combinations. In recent paper [8], the thermal conductivity of adhesively bonded and sintered HSS was numerically investigated while in paper [9], the influence of material parameters and geometrical properties of ⁎ Corresponding author. Department of Applied Mechanics, Technical University of Malaysia, 81310 Skudai, Malaysia. E-mail addresses: [email protected] (S.M.H. Hosseini), [email protected] (M. Merkel), [email protected] (A. Öchsner) 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.10.009
syntactic (hollow spheres completely embedded in a matrix) and partially bonded HSS (spheres joined in localised contact points) was analysed. A comparison between analytical, numerical and experimental approaches is given in [10]. Fiedler et al. reported in [11] recent advances in the prediction of the thermal properties of syntactic metallic hollow sphere structures. They described the application of the Finite Element and Lattice Monte Carlo Method in the case of syntactic periodic and random hollow sphere structures. Manufacturing techniques for perforated hollow sphere structures are actually under development [12] and we use an idealised model structure to clarify basic effects of the perforation on physical properties. In the present study, perforated hollow spheres arranged in a primitive cubic periodic pattern are numerically investigated based on the finite element method (unit cell approach) and the effective thermal conductivity is compared to structures without holes. The influence of the hole diameter and the geometry on the effective conductivity is analysed within a parametric computational study. 2. Perforated hollow sphere structure The major idea of introducing a perforation, i.e. holes of circular cross section in the sphere shells, is to make the inner sphere surface and volume usable. In this paper, the holes are defined in such a way that the largest possible hole can be located between the linking elements in a primitive cubic arrangement. The modeling approach is restricted to the simple case where the holes do not intersect with the link area. The additional inner surface may be used for chemical reactions in the case of catalysts or the additional inner volume may have a positive influence on the dissipation of e.g. acoustic waves. A simplified arrangement of a
1136
S.M.H. Hosseini et al. / Materials Letters 63 (2009) 1135–1137
Fig. 1. Schematic representation of a primitive cubic arrangement of perforated hollow spheres.
perforated hollow sphere structure in a primitive cubic pattern is shown in Fig. 1. The darker grey volumes represent the connecting elements (‘links’) which are formed in the case of liquid phase processes (e.g. formed by adhesives or soldering agents). The connecting material is during a certain period of the joining process in the liquid state which forms later a solid neck region between the spheres. In the case of diffusion bonding (sintering), such neck regions do not form and the spheres are in direct contact (consider Fig. 1 without the dark grey volumes where neighboring spheres are touching). Looking at a specific example where perforated spheres (outer diameter 2.7 mm and shell thickness of 0.1 mm, outer hole radius 0.68 mm) are considered, the available free surface increases by ~ 57% and the free volume increase by ~259% while the bulk volume or the weight is reduced by ~33%. In addition, we can state that the porosity is only increased by ~ 3.9%. 3. Finite element analysis In order to reduce the size of the system of equations, the so-called unit-cell approach is chosen where the computational structure is reduced from a larger or infinite amount of randomly arranged spheres to a single unit-cell which is commonly based on typical space groups as in crystallography [13]. Common unit cell approaches are based on primitive cubic, face-centered or hexagonal arrangements.
Fig. 3. Normalised conductivity of HSS as a function of the hole diameter (a fraction of 0 corresponds to the sphere without holes); markers (○,Δ) denote finite element results, lines (–) indicate the trend.
Since the aim of this research is to highlight the difference between perforated and entire spheres and to investigate the influence of several geometrical parameters, we restrict our computational approach to the primitive cubic arrangement because this is a first estimation for a finite element analysis that can give reasonable results. A typical finite element mesh including boundary conditions is shown in Fig. 2. Constant temperature boundary conditions (Ti = const.) are prescribed at the faces of the connective elements on the left and right side of the model. In order to generate a heat flux through the structure, only T1 ≠ T2 must be fulfilled. According to the assumed symmetry and simplifications (no radiative or convective effects), the ratio of the heat flux perpendicular to all remaining surfaces is zero. The thermal properties of the considered base materials are for steel ksteel = 0.05 W/mm K (AISI 8000) and for epoxy kepoxy = 0.000214 W/mm K. The evaluation of the effective thermal conductivities within the finite element approach is based on Fourier's law where the arearelated conductivity is defined by k=
: Q Δy : A ΔT
ð1Þ
The area A of the unit cell and the spatial distance Δy are given by the geometry (cf. Fig. 2), respectively the temperature gradient ΔT =T2 −T1
Fig. 2. Finite element mesh and boundary conditions of a perforated primitive cubic unit cell (R = 1.25 mm, a = 0.12 mm, b = 0.4 mm, d = 1.36 mm).
S.M.H. Hosseini et al. / Materials Letters 63 (2009) 1135–1137
1137
were done with the commercial finite element package MSC.Marc® (MSC Software Corporation, Santa Ana, CA, USA). 4. Results Fig. 3 summarises the effective thermal conductivities of entire and perforated spheres for both, homogeneous (sphere shell and connecting element assumed to be made of the same material) and non-homogeneous spheres (steel and connecting element assumed to be made of different material, i.e. steel and epoxy). The case of homogeneous sphere relates to diffusion bonded structures and a clear drop in the thermal conductivity can be seen when the material property of the connecting element is changed to epoxy (i.e. non-homogenous sphere). It can be seen that the shape of both curves is the same. However, the non-homogeneous sphere rather acts as an isolator. In addition, it can be seen that the consideration of holes significantly decreases the thermal conductivity. Fig. 4 shows the influence of the joining element on the effective conductivity for homogeneous and non-homogeneous sphere structures. The effect of a changing geometry is much less pronounced than the influence of the holes in the case of homogeneous spheres. However in the case of the non-homogeneous spheres the behaviour is contrary: The shape of the curve is dominated by the geometry of the joining element while the influence of the hole is marginal. It should be noted here that the values of the conductivity were normalised in all cases, i.e. homogeneous and nonhomogeneous, with the sphere shell material (= steel).
5. Outlook In the scope of this research project, the effective thermal conductivity of perforated hollow sphere structures has been numerically estimated and compared to configurations without holes. Introducing holes in the sphere shells clearly reduces the thermal conductivity of the HSS. The numerical approach has been based on a primitive cubic unit cell arrangement. Future investigations must clarify the influence of the hole arrangement and the geometrical modifications such as sphere geometry or the spatial pattern of the spheres. References Fig. 4. Normalised conductivity of HSS as a function of link geometry (a = 0.12 mm, b = 0.4 mm) for several hole diameters ranging from 1d = 1.36 mm (biggest hole) hole till 0d (without hole), cf. Fig. 2; markers (Δ) denote finite element results, lines (–) visualise the trend.
· by the boundary conditions, only the heat flux Q remains to be determined. This is done by summing up all nodal values of the reaction heat flux with a user subroutine at the left or right face where a temperature boundary condition is prescribed. Within the relevant temperature range, the contribution of thermal radiation to the heat transfer is low [14]. Furthermore, contributions from gaseous conduction and convection are neglected. The whole model consists of 51,056 elements in the case of the largest hole with a radius of 0.68 mm. Subsequent models where the hole diameter was reduced to 75, 50 and 25% were generated. In addition the geometrical parameters of the joining element (cf. parameters a and b in Fig. 2) where modified to investigate the influence of a changing link geometry. All the simulations
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14]
Lu W, Zhao CY, Tassou SA. Int J Heat Mass Transfer 2006;49:2751–61. Lu TJ, Stone HA, Ashby MF. Acta Mater 1998;46:3619–35. Boomsma K, Poulikakos D, Zwick F. Mech Mater 2003;35:1161–76. Lu TJ, Chen C. Acta Mater 1999;47:1469–85. L. Schneider, A. Boehm, C. Korhammer , R. Scholl, B. Voigtsberger, G. Stephani, US Patent 6501784. (2002). Baumeister E, Klaeger S, Kaldos A. J Mater Process Technol 2004;155:1839–46. Lu KT, Kou HS. Int Commun Heat Mass 1993;20:489–500. Fiedler T, Öchsner A. Mat Sci Forum 2007;553:39–44. Fiedler T, Öchsner A. Mat Sci Forum 2007;553:45–50. Fiedler T, Solórzano E, Öchsner A. Mater Lett 2008;62:1204–7. Fiedler T, Öchsner A, Belova IV, Murch GE. Adv Eng Mater 2008;10:361–5. Glatt GmbH, Binzen, Germany, private communication (2007). de Graef M, McHenry M. Structure of Materials. An Introduction to Crystallography, Diffraction and Symmetry. Cambridge: Cambridge University Press; 1997. Lu T, Chen G. Acta Mater 1999;47:1469–85.