Thermal Conductivity of Powder Aggregates and Porous Compacts
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NTRODUCTION
IN a previous work, the authors reported an equation to calculate the effective electrical conductivity of a porous sintered compact as a function of r0, the electrical conductivity of the bulk or completely solid material, and the relative porosity HR, defined as the ratio of the sample porosity H to the maximum porosity of the system HM, which can be assimilated to the tap porosity of the starting powder ðHR ¼ H=HM Þ. The proposed equation was expressed as follows: [1]
rE ¼ r0 ð1 HR Þ
u
½1a
equation proposed therein, expressed in terms of conductivity, was as follows: 1 6 u 1 ½2a rE ¼ r0 ð1 þ aÞð1 HR Þ þa p where r0 and u are the aforementioned parameters, and a is a parameter that depends on HR as follows: a ¼ aM expðmð1 HR Þn Þ
where m and n are two empirical parameters (with values approximately 15 and 0.1, respectively), and aM is a dimensionless collection of parameters
where the exponent u, which usually ranges between 1 and 2, is given by 4
u ¼ 1 þ ð1 HM Þ5
½1b
Equations [1a] and [1b] satisfy the desired boundary conditions rE fi r0 as H fi 0 and rE fi 0 as H fi HM because in this last situation, interparticle contacts are points. Equations [1a] and [1b] are applicable not only to powder-sintered compacts (powdered materials) but also to aggregates of nonoxidized particles because the metallic phase exhibits connectivity. In another study,[2] the authors studied the electrical behavior of an oxidized metallic powder mass under compression. This is generally the case because metallic powder particles are usually covered with a nanoscale oxide layer (hydroxides can also be present). The
J.M. MONTES and J. CINTAS, Associate Professors, and S. MUN˜OZ, Assistant Professor, are with the Department of Mechanical and Materials Engineering, Escuela Te´cnica Superior de Ingenierı´ a, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain. Contact e-mail: [email protected] F.G. CUEVAS, Associate Professor, is with the Department of Chemistry and Materials Science, Escuela Te´cnica Superior de Ingenierı´ a, Universidad de Huelva, Campus La Ra´bida, Carretera Palos s/n, 21819 Palos de la Frontera, Spain. Manuscript submitted October 19, 2011. Article published online June 13, 2012 4532—VOLUME 43A, DECEMBER 2012
½2b
aM ¼
r0 d rX r 0
½2c
where rX and d are the electric conductivity and the thickness of the oxide film coating the metallic powder particles, respectively. Moreover, the parameter r0 represents the mean radius of the particles. As can be observed, when HfiHM, that is, when HRfi1, a assumes a finite value: a = aM. Moreover, in the limit of very low porosities (Hfi0), a abruptly decays to a residual value a ¼ aM expðmÞ. It is easy to verify that, as expected, when dfi0, Equations [2a] through [2c] are transformed into Equations [1a] and [1b]. This is because the null oxide layer thickness indicates that either the powder particles are deoxidized or they are sintered, producing, in any of the cases, metal–metal contacts. The aim of this work was to prop
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