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Thermal conductivity of metallic hollow sphere structures: An experimental, analytical and comparative study
Materials Letters 63 (2009) 1128–1130
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
Thermal conductivity of metallic hollow sphere structures: An experimental, analytical and comparative study E. Solórzano ⁎, M.A. Rodríguez-Perez, J.A. de Saja CellMat Group, Condensed Matter Physics Department, Faculty of Science, University of Valladolid 47011, Valladolid, Spain
a r t i c l e
i n f o
Article history: Received 2 October 2008 Accepted 28 November 2008 Available online 6 December 2008 Keywords: Thermal properties Thermal conductivity Hollow spheres Composite materials Cellular metals
a b s t r a c t The thermal conductivity of metallic hollow sphere structures is analyzed in terms of their main structural parameters (sphere diameter, shell thickness and constituent metal) by using analytical/empirical models to estimate the relative density and admitted scaling laws to determine the thermal conductivity. These results are compared with experimental values of such structures determined by the transient plane source method. In addition, a thermal conductivity map comparing these results with the typical thermal conductivity of other cellular metallic materials is provided. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Hollow spheres structures (HSS) are an interesting kind of cellular metallic materials constituted by regular basic entities (spheres), random connected to form a 3D lattice of such units [1]. They can be made of ceramics or metals [1–3] and the spheres can be joint by different methods (adhesion, sintering/diffusion bonding, brazing, etc.) [1,3,4]. In addition, these structures can be used as a basis to produce syntactic structures by casting any liquid/molten material on their primary configuration [5,6]. The typical porosities are over 85% in case of conventional hollow spheres structures and lower than 60% if we are dealing with syntactic ones (SHSS). Structures produced by all these diverse techniques and raw materials offer unique multifunctional properties interesting in a wide number of applications. Among others, we can cite their high specific stiffness, optimum acoustical properties, interesting piezoelectric characteristics, adaptable thermal properties and the structural lightening capability [2,7–9]. The thermal conductivity is an interesting property of these materials that has been addressed experimentally, numerically and analytically in previous works but only for specific regular structural arrangements [10,11]. The purpose of this work is different since it tries to provide an overview on the thermal conductivity for hollow spheres-based structures giving an estimation of this property as a function of the different evolving parameters in the manufacturing processes and comparing the results with experimental data as well as with other types of cellular metals. The assumptions considered in this
⁎ Corresponding author. Tel.: +34983423572; fax: +34983423192. E-mail address: [email protected] (E. Solórzano). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.11.051
work are new compared to previous publications taking in account that they are based on random sphere arrangements in contrast with lattice-structured spheres already used to obtain symmetry simplifications in the later numerical analysis. 2. Materials and models 2.1. Materials’ main characteristics As mentioned, the different constituent materials, structural parameters and processing techniques can give as a result thousands of combinations to produce HSS that obviously cannot be considered in this study. Therefore we have focused our attention on a representative selection of metallic HSS. The parameters to study the thermal conductivity of these materials have been selected as follows: - Shell material: HS can be made of many different materials (ceramics, metals, etc.). As we are studying cellular metals, different alloys used typically to produce HSS have been selected. Namely these are: high carbon steel (AISI1055), stainless steel (AISI316), aluminium (AA6061), pure molibdenum and titanium (TiAl6V4). The thermal conductivities for the cited materials are respectively 50, 13.8, 180, 138 and 6.9 W/m K. - Sphere diameter: the elemental entities forming the HSS have been chosen to be monosized with diameters ranging from 1.5 to 10 mm. - Shell thickness: spheres can be produced with a thickness independent of the diameters. Nevertheless, in practice the shell thickness must be increased with increasing sphere diameter (a large sphere with a thin shell is weak). In our study it has been
E. Solórzano et al. / Materials Letters 63 (2009) 1128–1130
considered a linear dependency of thickness (t) with sphere radius (R), as other authors have proposed [12], including different constants of proportionality between these two parameters (t/ R = 0.02, 0.033, 0.054, 0.1013). These different constants express the thickness independency of the sphere diameter. The t/R ratio of samples which experimental thermal conductivity has been determined is in the range 0.02–0.054. - Spatial arrangement: the spheres are assumed to be random dispersed and in contact — zero distance between spheres. Both random packing efficiency and distance have important influence in the models as exposed in the next section. We can consider that these assumptions are not far from experimental findings [4] (spheres very near from each other and without any specific spatial arrangement). - Processing technique: in all cases we have selected the powder metallurgical – binder assisted – process [13] to produce sintered spheres. Although spheres produced by using this process present a certain micro-porosity we have assumed that shells are fully dense and that the sintering process has been carried out on specific atmospheres that avoid the carbon diffusion to the metal matrix – thermal properties have not been modified – with exception of high carbon steel produced from Fe powders that are carbon-enriched during the sintering process giving as a result this expected alloy. - Type of structures: three types of structures are studied in the frame of this work. Firstly, sintered-bonded hollow spheres structures (SBHSS) (a single solid metallic phase) have been studied as the simplest structure. In addition two kind of composite structures have been considered: adhesive-bonded (ABHSS) (metal and polymer solid phases), with the adhesive forming necks connecting the spheres and metal-polymer syntactic structures (SHSS), with the adhesive filling the whole space around the HS arrangement. In the models we assume that the bonding phase is a polymeric adhesive and therefore it presents a low thermal conductivity (i.e. lower than 0.5 W/m K). Then we have omitted other existing syntactic materials like aluminium castings on steel HSS. 2.2. Models for volume fractions In contrast with the predictions carried out by other authors based on structured packing (CP, BCC and FCC) to predict the mechanical and/or thermal behaviour [11,12,14], we have considered a random sphere arrangement – namely a dense random packing, DRP – to first estimate the porosity and then to apply well known analytical models. In this way, the DRP sphere-filling efficiency has been predicted to be 0.64 [15] in contrast with values 0.52, 0.68 and 0.74 for CP, BCC and FCC sphere arrangements respectively. The value 0.64 can be considered as the maximum efficiency when spheres are subjected to mechanical vibration and under other conditions probably this value trends to be lower. Nevertheless, this assumption seems to be more realistic to estimate both porosity and thermal conductivity in comparison with CP, BCC and FCC arrangements (regular arrangements offer a good prediction of properties using numerical analysis but are not able to be accurate in the estimation of relative densities). In case we considered a lower packing efficiency the overall thermal conductivity would be slightly lower. To estimate the fraction of metal in the whole volume, the previous value is multiplied by the factor C = 1 − (R − t/R)3. This factor represents the fraction of metal in each single sphere and it is calculated by comparing the external and internal sphere volumes. The previous assumptions for the relation t/R can be applied over the calculated solid fraction to simplify it. This model has been compared to the experimental relative density (mass and dimensions) by introducing in the model the values for R and t experimentally obtained. The results for SBHSS are
1129
in close agreement with a deviation lower than 7% in most of cases. Moreover, these differences can be explained if we consider that the shell is not fully dense and that spheres are slightly deformed (flattened) near the contact point for this kind of processing technique [12]. Both factors could be corrected, i.e. if factor C is multiplied by the densification factor and the value of sphere-filled volume (0.64) is increased considering the sphere deformation. On the other hand, the amount of adhesive for ABHSS has been modelled considering an average sphere coordination number of 6.2 for a random packing [16]. This value needs to be multiplied by the amount of adhesive in a single neck (Eq. (1)) calculated by Sanders and Gibson’s [12] – for a null distance among spheres– as: 1
VT
3
ad
= 2R 2:74 10
−7
θ
3:41
ð1Þ
with θ the contact angle between sphere and adhesive. Applying a contact angle adhesive-sphere of 25° and the mentioned average number of connections (6.2), the total amount of adhesive in a sphere is: VT
3
ad
= 6:2 0:032 R = 0:20 R
3
ð2Þ
Finally, to calculate the fraction of adhesive it is necessary to normalize this volume to the total volume of the representative unit cell (2R)3. Nevertheless, the volume fraction of a single sphere in a cubic unit cell is 0.52 but the real volume fraction is 0.64. Therefore a factor 1.23 is used to correct this difference and a final value for the fraction of adhesive is obtained: Vad i0:031
ð3Þ
The calculation for SHSS is easier by simply considering that the external free volume (0.36) is occupied by the embedding material. 2.3. Models for thermal conductivity To estimate the thermal conductivity (λ), two different models have been considered. In the particular case of SBHSS (only metallic and air phases present) it can be assumed that thermal conductivity of SBHSS (λ) depends only on the relative density (ρ/ρs) and the solid thermal conductivity (λs). Therefore the following scaling relation has been used: 1:55
λ = λs ðρ=ρs Þ
ð4Þ
This equation has been demonstrated to be consistent with the experimental determination of the thermal conductivity for a high diversity of cellular metals [17–19]. On the other hand, the Maxwell equation has been applied twice to estimate the thermal conductivity when 3 different phases – metal, air and adhesive – are present using the model developed by Fiedler et al. for the particular case of ABHSS and SHSS [11] (Eq. (5) λ, Vad, met, air: thermal conductivity and relative volume of adhesive, metallic and air phases respectively). This equation has been proven as valid to estimate the thermal conductivity of these materials and is in close agreement with numerical methods and experimental data [11]. λ=
λad ðð3 − 2Vad Þλm a + 2Vad λad λmet Vmet = λm − a = 2 Vad λm a + ð3 − Vad Þλad 3Vair + 2Vmet
ð5Þ
3. Results The effective thermal conductivity for SBHSS, based on the five alloys selected in this study, is presented in Fig. 1 versus the relative density of samples — i.e. the different sphere geometries (t/R) considered. In the same figure the experimental values for molybdenum and stainless steel samples, determined by the transient plane source method [20,21], are presented.
1130
E. Solórzano et al. / Materials Letters 63 (2009) 1128–1130
Fig. 3. Thermal conductivity map for different cellular metallic materials. Fig. 1. Effective thermal conductivity plotted versus the solid fraction for different alloys.
It is observed that thermal conductivities present values from 0.1 to 10 W/m K within the analyzed solid fractions range. Nevertheless, important differences exist according to the different raw alloys present in the shell and its thermal conductivity. In addition, it can be appreciated an acceptable estimation of the thermal properties when results are compared with experimental values. In both cases experimental values are over the theoretical curves and it is probably related to the mentioned microporosity and flattened regions. To determine the effective thermal conductivity of both SHSS and ABHSS we have assumed that hollow spheres are made of carbon steel 1055 (one of the most typical for this kind of materials) and an epoxy adhesive as bonding phase. The considered thermal conductivity of the resin was 0.21 W/m K. Fig. 2 shows the results obtained versus the absolute fraction of air within the spheres (i.e. percentage of closed porosity). It is important to mention that ABHSS present an additional interconnected porosity since the adhesive does not fill the whole space between the spheres. It can be appreciated a lower thermal conductivity of ABHSS for the same closed porosity fraction that is explained in terms of the higher absolute air fraction and its lower bulk density. The comparison with the experimental results reveals a minor overestimation of the thermal conductivity that is probably related to the assumption of a null distance between steel spheres (actually not realistic). Thus, it is expected that the consideration of the real average distance in the models would lead to a better congruence. Nevertheless, the results are in close agreement and the differences are, in both cases, lower than 7%. Due to the difficulty of presenting normalized results for a 3-phase-material, we have considered the effective thermal conductivity and the absolute density to build a comparative thermal conductivity map shown in Fig. 3. These maps are more useful to select a material having as a reference both absolute properties and densities. In the graph the experimental values obtained in the last years by different authors for diverse metal foams and cellular metals are showed. Data for lotus copper [22], ERG™ (AA6201) [23], Alporas™ [24], magnesium (AZ91) integral [18], low porosity infiltrated AlSi9Cu3 [17] as well as Zn and AlSi7 (produced within the powder metallurgical (PM) route [17]) foams are presented comparing their typical densities and thermal conductivities with selected HSS analyzed in this work.
It can be appreciated that only SHSS made of AA6061 are in the trend of the thermal conductivities of the previous studied foams — most based on the same solid constituent: aluminium. In contrast with those materials, ABHSS and SHSS based on carbon steel spheres and epoxy adhesive present much lower thermal conductivities (1–2 orders of magnitude) than other cellular metals with similar density. In addition, SBHSS based on carbon steel – with no adhesive – present lower thermal conductivity than other metal foams due to the high density and low thermal conductivity of the solid constituent in comparison with other metals, i.e. lower solid fraction contributing to the solid heat conduction.
4. Conclusions The analytic study of thermal conductivity for HSS, assuming random sphere arrangement, seems to be valid to predict this property with a realistic estimation of density. In addition, the present model can be reasonably improved if the shell densification, sphere deformation (flattened unions in SBHSS) and the real distance between spheres for adhesive-bonded and syntactic structures are considered. Nevertheless, the models seem to be enough accurate since experimental results are similar to the predicted ones. The thermal conductivity map provided, allows for a comparison between the studied and other metallic foams useful for materials selection in engineering applications and shows the differences between typical metal foams and the HSS. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Fig. 2. Thermal conductivity for ABHSS and SHSS using a low conductive phase as a bonding material.
[20] [21] [22] [23] [24]
Andersen O, Waag U, Schneider L, Stephani G, Kieback B. Adv Eng Mater 2000;2:192–5. Cochran JK. Curr Opin Solid State Mater Sci 1998;3:474–9. Bonino JP, Rousset A, Rossignol C, Blottière Y. Mater Tech 1990;78:25–8. Fiedler T, Öchsner A, Grácio J, Kuhn G. Mech Compos Mater 2005;41:559–70. Rickles SA, Cochran JK, Sanders TH. Ceram Eng Sci Proc 1989;10:1472–84. Palmer RA, Gao K, Doan TM, Green L, Cavallaro G. Mater Sci Eng A 2007;464:85–92. Tressler JF, Alkoy S, Dogan A, Newnham RE. Composites A 1999;30:4–482. Gasser S, Paun F, Cayzeele A, Bréchet Y. Scripta Mater 2003;48:1617–23. Lim TJ, Smith B, McDowell DL. Acta Mater 2002;50:2867–79. Fiedler T, Solórzano E, Öchsner A. Mater Lett 2008;63:1204–7. Fiedler T, Öchsner A, Belova IV, Murch GE. Adv Eng Mater 2008;10:361–5. Sanders WS, Gibson LJ. Mater Sci Eng A 2003;347:70–85. German Patent, DE 3902032 1989. Sanders WS, Gibson LJ. Mater Sci Eng A 2003;352:150–61. Jaeger HM, Nagel SR. Science 1992;255:1524–31. Lochmann K, Oger L, Stoyan D. Solid State Sci 2006;8:1397–413. Solórzano E, Rodriguez-Perez MA, de Saja JA. Adv Eng Mater 2008;10:371–7. Solórzano E, Hirchmann M, Rodriguez-Perez MA, Corner C, de Saja JA. Mater Lett 2008;62:3960–2. Solórzano E, Reglero JA, Rodriguez-Perez MA, Lehmhus D, Wichmann M, de Saja JA. Int J Heat Mass Tran; in press. Log T, Gustafsson SE. Fire Mater 1995;19:43–9. Gustafsson SE. Rev Sci Instrum 1991;62:797–804. Ogushi T, Chiva H. J Appl Phys 2004;95:5843–7. Bhattacharya A, Calmidi VV, Mahajan RL. Int J Heat Mass Transfer 2002;45:1017–31. Babcsán N, Mészáros I, Herman N. Materialwiss Werkst 2003;34:391–4.
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
Thermal conductivity of metallic hollow sphere structures: An experimental, analytical and comparative study E. Solórzano ⁎, M.A. Rodríguez-Perez, J.A. de Saja CellMat Group, Condensed Matter Physics Department, Faculty of Science, University of Valladolid 47011, Valladolid, Spain
a r t i c l e
i n f o
Article history: Received 2 October 2008 Accepted 28 November 2008 Available online 6 December 2008 Keywords: Thermal properties Thermal conductivity Hollow spheres Composite materials Cellular metals
a b s t r a c t The thermal conductivity of metallic hollow sphere structures is analyzed in terms of their main structural parameters (sphere diameter, shell thickness and constituent metal) by using analytical/empirical models to estimate the relative density and admitted scaling laws to determine the thermal conductivity. These results are compared with experimental values of such structures determined by the transient plane source method. In addition, a thermal conductivity map comparing these results with the typical thermal conductivity of other cellular metallic materials is provided. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Hollow spheres structures (HSS) are an interesting kind of cellular metallic materials constituted by regular basic entities (spheres), random connected to form a 3D lattice of such units [1]. They can be made of ceramics or metals [1–3] and the spheres can be joint by different methods (adhesion, sintering/diffusion bonding, brazing, etc.) [1,3,4]. In addition, these structures can be used as a basis to produce syntactic structures by casting any liquid/molten material on their primary configuration [5,6]. The typical porosities are over 85% in case of conventional hollow spheres structures and lower than 60% if we are dealing with syntactic ones (SHSS). Structures produced by all these diverse techniques and raw materials offer unique multifunctional properties interesting in a wide number of applications. Among others, we can cite their high specific stiffness, optimum acoustical properties, interesting piezoelectric characteristics, adaptable thermal properties and the structural lightening capability [2,7–9]. The thermal conductivity is an interesting property of these materials that has been addressed experimentally, numerically and analytically in previous works but only for specific regular structural arrangements [10,11]. The purpose of this work is different since it tries to provide an overview on the thermal conductivity for hollow spheres-based structures giving an estimation of this property as a function of the different evolving parameters in the manufacturing processes and comparing the results with experimental data as well as with other types of cellular metals. The assumptions considered in this
⁎ Corresponding author. Tel.: +34983423572; fax: +34983423192. E-mail address: [email protected] (E. Solórzano). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.11.051
work are new compared to previous publications taking in account that they are based on random sphere arrangements in contrast with lattice-structured spheres already used to obtain symmetry simplifications in the later numerical analysis. 2. Materials and models 2.1. Materials’ main characteristics As mentioned, the different constituent materials, structural parameters and processing techniques can give as a result thousands of combinations to produce HSS that obviously cannot be considered in this study. Therefore we have focused our attention on a representative selection of metallic HSS. The parameters to study the thermal conductivity of these materials have been selected as follows: - Shell material: HS can be made of many different materials (ceramics, metals, etc.). As we are studying cellular metals, different alloys used typically to produce HSS have been selected. Namely these are: high carbon steel (AISI1055), stainless steel (AISI316), aluminium (AA6061), pure molibdenum and titanium (TiAl6V4). The thermal conductivities for the cited materials are respectively 50, 13.8, 180, 138 and 6.9 W/m K. - Sphere diameter: the elemental entities forming the HSS have been chosen to be monosized with diameters ranging from 1.5 to 10 mm. - Shell thickness: spheres can be produced with a thickness independent of the diameters. Nevertheless, in practice the shell thickness must be increased with increasing sphere diameter (a large sphere with a thin shell is weak). In our study it has been
E. Solórzano et al. / Materials Letters 63 (2009) 1128–1130
considered a linear dependency of thickness (t) with sphere radius (R), as other authors have proposed [12], including different constants of proportionality between these two parameters (t/ R = 0.02, 0.033, 0.054, 0.1013). These different constants express the thickness independency of the sphere diameter. The t/R ratio of samples which experimental thermal conductivity has been determined is in the range 0.02–0.054. - Spatial arrangement: the spheres are assumed to be random dispersed and in contact — zero distance between spheres. Both random packing efficiency and distance have important influence in the models as exposed in the next section. We can consider that these assumptions are not far from experimental findings [4] (spheres very near from each other and without any specific spatial arrangement). - Processing technique: in all cases we have selected the powder metallurgical – binder assisted – process [13] to produce sintered spheres. Although spheres produced by using this process present a certain micro-porosity we have assumed that shells are fully dense and that the sintering process has been carried out on specific atmospheres that avoid the carbon diffusion to the metal matrix – thermal properties have not been modified – with exception of high carbon steel produced from Fe powders that are carbon-enriched during the sintering process giving as a result this expected alloy. - Type of structures: three types of structures are studied in the frame of this work. Firstly, sintered-bonded hollow spheres structures (SBHSS) (a single solid metallic phase) have been studied as the simplest structure. In addition two kind of composite structures have been considered: adhesive-bonded (ABHSS) (metal and polymer solid phases), with the adhesive forming necks connecting the spheres and metal-polymer syntactic structures (SHSS), with the adhesive filling the whole space around the HS arrangement. In the models we assume that the bonding phase is a polymeric adhesive and therefore it presents a low thermal conductivity (i.e. lower than 0.5 W/m K). Then we have omitted other existing syntactic materials like aluminium castings on steel HSS. 2.2. Models for volume fractions In contrast with the predictions carried out by other authors based on structured packing (CP, BCC and FCC) to predict the mechanical and/or thermal behaviour [11,12,14], we have considered a random sphere arrangement – namely a dense random packing, DRP – to first estimate the porosity and then to apply well known analytical models. In this way, the DRP sphere-filling efficiency has been predicted to be 0.64 [15] in contrast with values 0.52, 0.68 and 0.74 for CP, BCC and FCC sphere arrangements respectively. The value 0.64 can be considered as the maximum efficiency when spheres are subjected to mechanical vibration and under other conditions probably this value trends to be lower. Nevertheless, this assumption seems to be more realistic to estimate both porosity and thermal conductivity in comparison with CP, BCC and FCC arrangements (regular arrangements offer a good prediction of properties using numerical analysis but are not able to be accurate in the estimation of relative densities). In case we considered a lower packing efficiency the overall thermal conductivity would be slightly lower. To estimate the fraction of metal in the whole volume, the previous value is multiplied by the factor C = 1 − (R − t/R)3. This factor represents the fraction of metal in each single sphere and it is calculated by comparing the external and internal sphere volumes. The previous assumptions for the relation t/R can be applied over the calculated solid fraction to simplify it. This model has been compared to the experimental relative density (mass and dimensions) by introducing in the model the values for R and t experimentally obtained. The results for SBHSS are
1129
in close agreement with a deviation lower than 7% in most of cases. Moreover, these differences can be explained if we consider that the shell is not fully dense and that spheres are slightly deformed (flattened) near the contact point for this kind of processing technique [12]. Both factors could be corrected, i.e. if factor C is multiplied by the densification factor and the value of sphere-filled volume (0.64) is increased considering the sphere deformation. On the other hand, the amount of adhesive for ABHSS has been modelled considering an average sphere coordination number of 6.2 for a random packing [16]. This value needs to be multiplied by the amount of adhesive in a single neck (Eq. (1)) calculated by Sanders and Gibson’s [12] – for a null distance among spheres– as: 1
VT
3
ad
= 2R 2:74 10
−7
θ
3:41
ð1Þ
with θ the contact angle between sphere and adhesive. Applying a contact angle adhesive-sphere of 25° and the mentioned average number of connections (6.2), the total amount of adhesive in a sphere is: VT
3
ad
= 6:2 0:032 R = 0:20 R
3
ð2Þ
Finally, to calculate the fraction of adhesive it is necessary to normalize this volume to the total volume of the representative unit cell (2R)3. Nevertheless, the volume fraction of a single sphere in a cubic unit cell is 0.52 but the real volume fraction is 0.64. Therefore a factor 1.23 is used to correct this difference and a final value for the fraction of adhesive is obtained: Vad i0:031
ð3Þ
The calculation for SHSS is easier by simply considering that the external free volume (0.36) is occupied by the embedding material. 2.3. Models for thermal conductivity To estimate the thermal conductivity (λ), two different models have been considered. In the particular case of SBHSS (only metallic and air phases present) it can be assumed that thermal conductivity of SBHSS (λ) depends only on the relative density (ρ/ρs) and the solid thermal conductivity (λs). Therefore the following scaling relation has been used: 1:55
λ = λs ðρ=ρs Þ
ð4Þ
This equation has been demonstrated to be consistent with the experimental determination of the thermal conductivity for a high diversity of cellular metals [17–19]. On the other hand, the Maxwell equation has been applied twice to estimate the thermal conductivity when 3 different phases – metal, air and adhesive – are present using the model developed by Fiedler et al. for the particular case of ABHSS and SHSS [11] (Eq. (5) λ, Vad, met, air: thermal conductivity and relative volume of adhesive, metallic and air phases respectively). This equation has been proven as valid to estimate the thermal conductivity of these materials and is in close agreement with numerical methods and experimental data [11]. λ=
λad ðð3 − 2Vad Þλm a + 2Vad λad λmet Vmet = λm − a = 2 Vad λm a + ð3 − Vad Þλad 3Vair + 2Vmet
ð5Þ
3. Results The effective thermal conductivity for SBHSS, based on the five alloys selected in this study, is presented in Fig. 1 versus the relative density of samples — i.e. the different sphere geometries (t/R) considered. In the same figure the experimental values for molybdenum and stainless steel samples, determined by the transient plane source method [20,21], are presented.
1130
E. Solórzano et al. / Materials Letters 63 (2009) 1128–1130
Fig. 3. Thermal conductivity map for different cellular metallic materials. Fig. 1. Effective thermal conductivity plotted versus the solid fraction for different alloys.
It is observed that thermal conductivities present values from 0.1 to 10 W/m K within the analyzed solid fractions range. Nevertheless, important differences exist according to the different raw alloys present in the shell and its thermal conductivity. In addition, it can be appreciated an acceptable estimation of the thermal properties when results are compared with experimental values. In both cases experimental values are over the theoretical curves and it is probably related to the mentioned microporosity and flattened regions. To determine the effective thermal conductivity of both SHSS and ABHSS we have assumed that hollow spheres are made of carbon steel 1055 (one of the most typical for this kind of materials) and an epoxy adhesive as bonding phase. The considered thermal conductivity of the resin was 0.21 W/m K. Fig. 2 shows the results obtained versus the absolute fraction of air within the spheres (i.e. percentage of closed porosity). It is important to mention that ABHSS present an additional interconnected porosity since the adhesive does not fill the whole space between the spheres. It can be appreciated a lower thermal conductivity of ABHSS for the same closed porosity fraction that is explained in terms of the higher absolute air fraction and its lower bulk density. The comparison with the experimental results reveals a minor overestimation of the thermal conductivity that is probably related to the assumption of a null distance between steel spheres (actually not realistic). Thus, it is expected that the consideration of the real average distance in the models would lead to a better congruence. Nevertheless, the results are in close agreement and the differences are, in both cases, lower than 7%. Due to the difficulty of presenting normalized results for a 3-phase-material, we have considered the effective thermal conductivity and the absolute density to build a comparative thermal conductivity map shown in Fig. 3. These maps are more useful to select a material having as a reference both absolute properties and densities. In the graph the experimental values obtained in the last years by different authors for diverse metal foams and cellular metals are showed. Data for lotus copper [22], ERG™ (AA6201) [23], Alporas™ [24], magnesium (AZ91) integral [18], low porosity infiltrated AlSi9Cu3 [17] as well as Zn and AlSi7 (produced within the powder metallurgical (PM) route [17]) foams are presented comparing their typical densities and thermal conductivities with selected HSS analyzed in this work.
It can be appreciated that only SHSS made of AA6061 are in the trend of the thermal conductivities of the previous studied foams — most based on the same solid constituent: aluminium. In contrast with those materials, ABHSS and SHSS based on carbon steel spheres and epoxy adhesive present much lower thermal conductivities (1–2 orders of magnitude) than other cellular metals with similar density. In addition, SBHSS based on carbon steel – with no adhesive – present lower thermal conductivity than other metal foams due to the high density and low thermal conductivity of the solid constituent in comparison with other metals, i.e. lower solid fraction contributing to the solid heat conduction.
4. Conclusions The analytic study of thermal conductivity for HSS, assuming random sphere arrangement, seems to be valid to predict this property with a realistic estimation of density. In addition, the present model can be reasonably improved if the shell densification, sphere deformation (flattened unions in SBHSS) and the real distance between spheres for adhesive-bonded and syntactic structures are considered. Nevertheless, the models seem to be enough accurate since experimental results are similar to the predicted ones. The thermal conductivity map provided, allows for a comparison between the studied and other metallic foams useful for materials selection in engineering applications and shows the differences between typical metal foams and the HSS. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Fig. 2. Thermal conductivity for ABHSS and SHSS using a low conductive phase as a bonding material.
[20] [21] [22] [23] [24]
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