Effective thermal conductivity of sintered porous media

International Journal of Heat and Mass Transfer 66 (2013) 868–878

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Effective thermal conductivity of sintered porous media: Model and experimental validation Juan Pablo M. Florez, Marcia B.H. Mantelli ⇑, Gustavo G.V. Nuernberg Federal University of Santa Catarina, Department of Mechanical Engineering, Heat Pipe Laboratory, Florianópolis, Santa-Catarina, 888040-900, Brazil

a r t i c l e

i n f o

Article history: Received 4 March 2013 Received in revised form 17 July 2013 Accepted 28 July 2013 Available online 23 August 2013 Keywords: Porous media Effective thermal conductivity Heat pipes Sinterization

a b s t r a c t A model to estimate the effective thermal conductivity of sintered porous media for heat pipes is proposed in this paper. An elementary cell of a porous media is physically modeled as two metallic hemispheres in contact with a fluid film around them. The electrical circuit analogy is employed to determine the heat leaving the top and reaching the bottom of the cell. The thermal circuit consists of two parallel resistance paths, one for the solid spheres in contact and the other for the heat transfer in the fluid. A literature model is employed to calculate the thermal resistance within the hemispherical particles. Also, literature models are used for the determination of the geometry of the neck produced between particles during the sintering process. The neck dimensions are used to estimate the neck thermal resistance, which is in series with the hemisphere resistances. Effective thermal conductivity experimental data were obtained for porous materials produced with atomized copper powder, with particle diameters ranging from 20 to 50 lm. The comparison between present model and data is good. Statistics of the particle size distribution is employed to determine average particle dimensions. The porosity and permeability of the material tested was characterized in the laboratory. The samples were tested in three conditions: vacuum and saturated with distilled water or methanol. Literature models for the effective thermal conductivity for bed packed (not sintered) porous media were also compared with the present model and data results. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Effective properties are usually used to define the physical characteristics of the porous materials. The effective thermal conductivity, for instance, is determined considering the thermal conductivity of the constituent phases of the material, i.e. the solid phase (matrix) and the fluid phase (liquid or gas). Much of the heat transfer work in the literature treats all porous media equally, independently of the technology employed for their fabrication. Actually, there is a large difference among porous media produced by different technologies, as observed in sintered and bed packed metal powder. Usually, a distinction is made between the thermal conductivity of densified (or sintered) and not densified porous media. Carson et al. [1] and Atabaki and Baliga [2] verified that the thermal conductivity of a non-sintered material is much smaller than that of sintered materials. The models proposed by Chi [3], Faghri [4] and Peterson [5], are simple arrangements in series or parallel of the porous media phases, using the analogy with electrical circuits. The values estimated by the series and par⇑ Corresponding author. E-mail addresses: [email protected] (Juan Pablo M. Florez), [email protected], [email protected] (M.B.H. Mantelli), [email protected] (G.G.V. Nuernberg). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.088

allel models are used as upper and lower limits of the effective thermal conductivity. Among his works conducted for non-sintered materials, Handley [6] presented a theoretical and experimental study concerning the effective thermal conductivity of a bed packed particulate material. This researcher proposed a theoretical model based on energy balance equations averaged in the volume, following the hypothesis adopted by Whitaker [7]. In this model, an experimental factor for the evaluation of the consolidation degree of particles during sinterization is adopted. Recently, Bahrami et al. [8] studied the effective thermal conductivity of two rough steel packed spheres in contact, surrounded by air, using the thermal circuit analogy. In this work, an expression for the determination of the constriction and spreading thermal resistances for solid and hollow spheres in contact were obtained, based on an exact solution previously developed by Yovanovich et al. [9]. Sintered porous media can be considered as sphere particles connected by necks, which are created by heating process and diffusion mass phenomena. Therefore, its effective thermal conductivity depends on the number of created necks. Birboim et al. [10] considered the influence of necks for modeling the effective thermal conductivity of porous media made of sinterization of zinc oxide (ZnO). They predicted the effective thermal conductivity

J.P.M. Florez et al. / International Journal of Heat and Mass Transfer 66 (2013) 868–878

869

Nomenclature A a b c d df Ds k kB ke kr l L md mw Q qz Q Qs R RL Rp R1,R2 R r rm rp T

area inner sphere radius outer sphere radius radius of the chord subtended by contact area particle mean diameter driving force surface diffusion coefficient thermal conductivity Boltzmann’s constant effective thermal conductivity reference thermal conductivity sample length cell length dry mass wet mass quartile heat flux rate average heat flux rate activation energy for surface diffusion thermal resistance constriction/spreading resistance thermal resistance of neck thermal resistance of sample case dimensionless resistance R⁄ = ks RL particle radius radius of curvature of the neck radius of the neck temperature

using a numerical solution of the Fourier Equation proposed by Atabaki and Baliga [2], which take into account a factor of consolidation degree of sintered powder wick. An interesting model for the effective thermal conductivity based on experimental data was proposed in the PhD thesis of Alexander (apud Atabaki and Baliga [2]) for metal felts, sintered powders, layers of wire cloth and unconsolidated beads. Sintering is also used in screens to produce heat pipe capillary structures. From this literature review, it is possible to observe that the effective thermal conductivity of sintered porous media has not yet been fully modeled. Therefore, the main goal of this work is to propose an effective thermal conductivity model for sintered porous media made from metal powders, based on two literature models: the phenomenological analyses of Birnboin et al. [10], which uses the fabrication process and the geometry of the metal powder spheres, including particle size, as input parameters, and the thermal conduction between spheres in contact model, presented by Bahrami et al. [8] and Yovanovich et al. [9]. The analogy between thermal and electrical circuits is employed.

TM x x_ x

melting temperature radius of neck in the plane of contact of two particles rate of neck growth position of temperature sensors

Greek symbols a contact half-angle cs Surface energy ds effective surface thickness Dos coefficient of the surface diffusion n radii ratio b/a q theoretical density qa apparent density qair air density ql liquid density qs solid density / polar angle X atomic volume Subscripts c hot h cool s solid l liquid r reference t total

2.1. Contacting sphere physical model The physical model adopted for determining the theoretical effective thermal conductivity of a sintered porous media is displayed in Fig. 1. The model is based on the thermal analyses of a unit cell, which thermally represents the whole porous media. The actual porous media can be reproduced by stacking several of these cells in a vertical arrangement and replicating this pile in a horizontal array, forming a three dimensional structure. The cell is formed by two metal hemispheres of radius r1 and r2, which represent the powder from which the porous media is made. These radius are considered equal and equivalent to the powder mean radius. The hemispheres are joined by means of a neck formed during the sintering process. A stationary liquid film, in a hollow hemi-

2. Model This section presents the model developed in this work for the effective thermal conductivity for sintered porous media. The conduction in the solid phase (matrix) is modeled using a literature conduction heat transfer model between two solid hemispheres in contact. The model for the liquid phase is based on a conduction heat transfer within hollow spherical shells model, available in the literature. The analogy between electrical and thermal circuits is employed to link these models. Following, details of the modeling are shown.

Fig. 1. Elementary cell physical model.

870


sphere shape, is considered around the hemispheres. x is the radius of the neck, given by the distance between the sphere contacting point and the neck outer radius, in the plane tangent to the contacting point. rm is the radius of the meniscus formed by the neck, in the plane which contains the center of the spheres (see Fig. 1). The rectangular coordinate system origin is located in the hemispheres contacting point: the abscissa crosses both the sphere centers and the ordinate pass through the neck radius center. The empty space between adjacent cells is filled with stagnant liquid film in such a temperature level which allows one to neglect the heat exchanged with the liquid film. All the geometric parameters must be determined as a function of r1 and r2, which represent the size of the particles that compose the powder used for the porous media fabrication. These diameters are obtained from the statistical treatment of the distribution dimensions of particles. The radius of neck is calculated from the sintering curve theory. The effective thermal conductivity of porous media is calculated from a heat balance, using Fourier Equation. For this heat balance, the following hypotheses were adopted:

the lower resistance path (solid hemispheres and neck) while the small arrow symbolizes the smaller amount of heat transferred through the larger resistance (liquid). The overall resistance of this circuit is determined by:

Steady state. Planes perpendicular to the plane shown in Fig. 1 are isothermal. Thermal equilibrium in the solid–fluid interface. Solid and fluid constant properties. The fluid layer surrounds and wets completely the particles. The fluid layer is considered stagnant, therefore, convective effects are neglected. Prescribed temperatures T2 and T1, where T2 > T1, are imposed to the two borders of the cell. All particles of powder which makes the porous media have spherical shape, have the same radius and are statistically represented by the mean diameter. The neck has circular shape in the plane perpendicular to the heat flux. Compaction external forces are not considered.

2.2.1. Spherical shell conduction model The constriction thermal resistances of the full (solid) and shell (liquid) hemispheres (RsL and RlL respectively), are calculated employing the model presented by Yovanovich et al. [9], developed for the conduction heat transfer between solid or hollow spheres in contact. In this model, the heat flux enters the elementary cell by the left flat circular area of radius r1 maintained at temperature T1 and is released to the right area maintained at temperature T2 (see Fig. 1). The heat delivered to the hot hemisphere is constricted to the neck region when passing to the other hemisphere and then spread in the cold hemisphere. The same happens to the liquid shell. The lateral walls of the elementary cell are considered insulated. The solid and hollow sphere constriction thermal resistances are obtained from the exact temperature distribution solution of the heat equation in spherical coordinates, as a function of directions r and u, solved by Yovanovich et al. [9]. This solution is given by the equation:

2.2. Electrical circuit analogy model The analogy between thermal and electrical circuits is used to model the effective thermal conductivity of the elementary cell just described. The resulting thermal circuit is present in Fig. 2. The heat flux, due to the difference of temperatures between T1 and T2, must cross two resistances in parallel. The first path is associated with the resistance due to the conduction heat transfer through the hemispheres (constriction and spreading resistances RsL ), in series with the neck Rsp resistance. The other path is associated with the static fluid surrounding the spheres (RlE ) and includes the resistance of the disk formed by the fluid in the neck (Rlp ). The large arrow refers to the larger amount of heat transferred through

Fig. 2. Thermal equivalent circuit modeling the effective thermal conductivity.

Rt ¼

1 1 RsL þRsL þRsp 1

2

ð1Þ

þ R1l

In this expression, s superscript means solid while l means liquid. Through Eq. (1), one is able to estimate the effective thermal conductivity, based on the linear approximation of the Fourier equation. For a unit cell, one gets:

ke A

ðT 2 T 1 Þ ðT 2 T 1 Þ ¼ L Rt

ð2Þ

which gives:

ke ¼

L ARt

kcR ¼

ð3Þ

sena

1 X

Eðb; n; nÞ

p½1 cosa2 n;impar ½cosna cosðn þ 1Þa½Pn1 cosa Pnþ1 cosa

nð2n þ 1Þ

ð4Þ

where

Eðr; n; nÞ ¼

rn b

n þ nþ1 n2nþ1 r

1 n2nþ1

ðnþ1Þ

b

ð5Þ

In these equations, k is the thermal conductivity of the media (metal or liquid), c is the radius of the contacting area between hemispheres, assumed as the neck radius in the present model, r is radius of solid hemispheres, b is internal radius of hollow hemisphere, n is ratio between internal and external radius of the hollow hemisphere, a1 is the neck contacting angle (one of the angle edges connects the hemisphere center and contacting point and the other the hemisphere center and the neck external interface radius in the contacting plane, see Fig. 1). In the present case, a1 = a2. In this work, three hundred terms of Eq. (4) series were employed. MapleÒ was used in the calculations. 2.2.2. Neck thermal resistance Sintering is one of the techniques used to process particulate materials. Among other purposes, sintering can be used for the fabrication of porous media. According to Coble [11], during sintering and as a direct effect of the material heating, the shape of particles changes as the grains grow, reducing the pore size. Thümmler and Oberacker [12] comment that, actually, there is not an exact defi-


nition for sintering. They describe sintering as thermally activated material transport, which happens in a bulk of compacted powder, where the specific surface area is reduced by the growth of the contact between the particles. The neck size depends on parameters such as: rate of heating, time, temperature of sintering, particle material and mechanisms of mass transport. The grain growth and the densification occur by means of the mass transport mechanism. Swinkles and Ashby [13] classify the sintering process according to these mechanisms: surface diffusion, diffusion in the grain boundary and evaporation. The grain boundary diffusion occurs due to defects in grain (vacancies, gaps and other inconsistencies). According to these same authors, the mass transport contribute with the neck growth, but only the boundary and lattice diffusion contribute to the densification of the material. The sintering process can be divided in the following stages: formation and growing of the necks, densification and particulate material grain growth. Thümmler and Oberacker [12], state that micro weldings in the contacting areas within particulates form micro bridges (necks), which size depends on the compression pressure. During the sintering, the shape of the sintered material does not change very much after compression. Kuczynski apud Ashby [14], affirm that the superficial diffusion is the dominant mechanism for copper powder sintering and propose _ the following expression to determine the neck growing rate, or X:

x_ ¼ 2Ds ds

cs X kB T

3

df

ð6Þ

where Qs

Ds ¼ ds Dos e RT

ð7Þ

and where cs is surface energy, Ds is surface diffusion, Dos is the coefficient of the superficial diffusion, Qs is energy of activation of the superficial diffusion, ds is effective superficial thickness, X is the atomic volume, kB is the constant of Stefan–Boltzmann e T is temperature. df is considered as the driving force of the sintering process and can be calculated from the expression:

df ¼

1 1 2 x þ 1 rm x r1 r1

ð8Þ

The factor (1 x/r1) is inserted to guarantee the validity of the model when x value approaches the sphere radius [12]. The general geometric relation between the radius of spherical particles r1 and r2, the meniscus radius (neck curvature rm) and neck radius x, is given by the following equation:

ðr1 þ r2 Þðx þ r m Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðr m þ r 1 þ r 2 Þr m r 1 r 2 2

ð9Þ

For the cases where r1 = r2, this equation becomes:

rm ¼

1 x2 2 r1 x

ð10Þ

The angles a1, a2 and a3 of Fig. 1 must be determined as a function of geometric parameters. As already noted, these angles correspond to the internal angles of the triangle formed by the center of the particle contacting circumferences and of the meniscus. The following expressions where obtained from the geometry analysis: 2

2

ðr1 þ r2 Þ ¼ ðr1 þ rm Þ þ ðr 2 þ r m Þ 2ðr1 þ r m Þðr 2 þ r m Þcosa3 sin a1 sin a3 ¼ r2 þ rm r1 þ r2 sin a2 sin a3 ¼ r1 þ rm r1 þ r2

Fig. 3. Neck geometry model.

In these equations, z1 and z2 are parameters related to the size of the neck meniscus. They correspond to the horizontal coordinates of the points where the circumference of neck and spheres intersect. These parameters are measured from the center of the particle 1, as one can observe in Fig. 3. The radius x is calculated from Eq. (6). Eqs. (10)–(13) are also used for the fluid phase. The one-dimensional Fourier Equation, with variable transversal area, is employed for the calculation of the thermal resistances of the neck region, for both liquid and solid phases. Assuming that qz is constant, one gets.

qz ¼ kAðzÞ

dT dz

qz

Z

z z0

dz ¼ k AðzÞ

ð13Þ

Z

T

dT

ð15Þ

T0

This integration results in:

!

k

qz ¼

Rz

dz z0 AðzÞ

ðT 2 T 1 Þ

ð16Þ

where A(z) represents the transversal area, which is a function of length z. The term in parentheses represents the inverse of neck thermal resistance:

Rz R¼

dz z0 AðzÞ

ð17Þ

k

Based on the geometry, the following expression for the radius as a function of the geometric characteristics of the cell is proposed for the solid neck:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rsp ðzÞ ¼ r 2m fz ½ðr 1 þ r m Þ cos a1 g2 þ r m þ x

ð18Þ

and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

2 rlp ðzÞ ¼ r 2m z ½ðr I þ rm þ rI 1 sin a1 1 þ r m Þ cos a1

ð19Þ

Then, the thermal resistance for neck of solid phase becomes:

1 pks

Z

z2

z1

ð11Þ ð12Þ

ð14Þ

This expression can be integrated in the form:

Rsp ¼

2

871

dz 2 s r p ðzÞ

ð20Þ

where z1 = r1 cos a1 and z2 = r1(2 cosa1) are the lengths in z measured from sphere center (see Fig. 3). Following the same procedure, the fluid phase neck resistance is expressed by:

Rlp ¼

1 pkl

Z

z2

z1

dz 2 2 l r p ðzÞ rsp ðzÞ

ð21Þ

872


Fig. 4. Electronic microscopy image 400x: PAC powder (left) and PAM powder (right).

3. Experimental work 3.1. Fabrication of the porous media In the present work, wick structures are produced by loose sintering metallurgy process. Atomized copper powder is employed to the porous media fabrication. These selected powders have commercial names of PAC and PAM. PAC has a fine and PAM a coarse average particle size powders, according to Fig. 4. Different porosity porous media, made of powder PAC and PAM, where fabricated. They were tested in vacuum conditions and saturated with water or methanol. For reproducibility testing, two similar samples (named tests 1 and 2), were produced. Physical properties of copper, provided by Swinkles and Ashby [13] and shown in Table 1, where employed in the present work. A schematic of the tested samples are shown in Fig. 5. The tested samples have parallelepiped geometry, with dimensions of 200 mm 30 mm 10 mm. The porous media is sintered directly over the flat base of a cooper case. 95% purity atomized commercial cooper powders are used. Tests were conducted to determine the

Table 1 Physical properties of copper [13]. Properties Surface energy [J/m2] Atomic volume [m3] Pre-exp, surface diffusion [m3/s] Surface diffusion activation energy [kJ/mol] Melting temperature [K] Boltzmann’s constant [J/N]

Fig. 5. Details of sample geometry.

1.72 2.56E10 6.00E10 205 1356 1.38E23

content of copper in the composition of the particulate material and a percentage of 95.45 ± 1% of copper were observed. According to the ASM [15] tables for pure copper and its alloys, the composition of the powder is similar to the copper alloy C2100, which has the thermal conductivity of 234 W/((m K)) at 20 °C. The casing is built of copper blades of 0.3 mm of thickness, in two parts: a thin parallelepiped shape case and a cover sheet. The case (housing) is filled with a controlled amount of powder and the array is sintered. The sintering heating rate is 5 °C/min and the system is left for 50 min after it reached the 850 °C level. After the porous media is sintered, the cover sheet is welded to the case edges with oxyacetylene, closing the case. A capillary tube is provided to allow the sample filling with the select working fluid. 3.2. Design and fabrication of the experimental set up An apparatus was built to measure the effective thermal conductivity of porous media, according to procedure described by Florez et al. [16]. Fig. 6 illustrates a scheme of the experimental apparatus. The experimental set up is divided in three main sections: heater, sample holder and cooler. The heater section is composed by two electric cartridge resistances of 50 W power each, placed inside an aluminum block. The heat is transferred by conduction to the sample, using two pure copper blades, of known thermal conductivity, of 85 mm 10 mm 2 mm, that work as flux meters. The porous sample to be tested is placed in series between the flux meters. Two pairs of aluminum plates work as clamping mechanisms and connect the flux meters to the testing plate, so that the heat that flows through the sample is conducted by the flux meters. The sample heat flux is, therefore, measured as an average of both flux meters measurements. The heat flux, in turn is determined using the heat conduction Fourier equation, using the measured temperature distribution and the known thermal conductivity of the flux meter material. The cooling system is kept at a prescribed temperature by means of water recirculation, which temperature is controlled by a thermal bath. Two copper guard heats, of total lengths equal to the flux meter lengths plus the sample length, are thermally connected to the heat source and sink and installed in parallel to the measuring set up. Therefore, the temperature distribution in the guard heat plates are very similar to that observed along the lower flux meter, the sample and upper flux meter and so, the plates work as efficient radiation shields. Convection heat transfer from the testing sample is


873

1 the top flange;2 the bottom flange;4 cooler; 5 flux meter; 6 sample; 7 guard heater; 8 polycarbonate tube; 9 filling through; 10 filling tube; 11 water recirculation; 12 vacuum tube

Fig. 6. Scheme of the experimental apparatus and details of sample assembling.

avoided, placing the system inside an evacuated polycarbonate tube. During tests, the device is located in a vertical arrangement. Both samples for each configuration were tested in vacuum (0, 001 mbar) or filled with saturated distilled water or methanol. The testing temperature range is 20 to 80 °C. The temperatures are measurement by means of type T (copper-constantan) thermocouples, which are connected to the data acquisition system, which, in turn, sends the signal to the computer, where it is processed using a software. 3.3. Effective thermal conductivity measurement The heat flux was calculated as the average of the fluxes measured by the flux meters connected in series with the sample. The heat flux is defined as an average of the temperature data collected for both heat flux meters weighted by the distances between the measuring points as:

Q¼

kr T 2;c T 1;c T 2;h T 1;h þ 2 x2;c x1;c x2;h x1;h

ð22Þ

In this equation, Q is the average heat flux, kr represents the thermal conductivity of the flux meter material, T2,c, T1,c and T2,h, T1,h are the flux meter temperatures, which sensor positions are given by x2,c, x1,c and x2,h, x1,h respectively. Using the Fourier model and the well-known thermal resistance concept, the porous media effective thermal conductivity, can be calculated as:

ke ¼

l 1 1 Q DT þ R1 R2 DTA

ð23Þ

where R1 and R2 represent the conduction resistances of the base and cover sheets of the parallelepiped case, DT is the temperature difference measured in the sample and l is the sample length. The welding thermal resistance between the case and the cover is neglected, because the case and the cover are considered to be in the same temperature. Uncertainties associated with the temperature measurement were taken into account and calculated according to INMETRO [17]. 3.4. Porosity measurement Many researchers have proposed correlations for the prediction of the effective thermal conductivity as a function of the fluid and solid thermal conductivities and of the porous structure geometrical parameters. Maybe the most important geometric parameter is the porosity. Therefore, in this work, the porosity of the sintered porous media tested was carefully measured. As already mentioned, two types of particulate materials (PAM and PAC) were employed to produce two different porous media. Their porosity was measured by Arquimedes method and images analysis. Cylindrical shape samples of these porous media were built for this purpose. The Arquimedes method measuring device consists of a balance Mettler Toledo Model VS 205 DU Dual Range,


qa ¼

md ðq qa Þ þ qair md mw l

ð24Þ

Finally, the porosity e is calculated trough the expression:

q e¼ 1 a qs

100

ð25Þ

The porosity is also measured by means of image analysis using the ImagoÒ software. The image processing technique can be divided basically in four steps: image acquisition, preprocessing, segmentation, pattern recognition and quantification. Samples of the porous media which porosity is to be measured are embedded in acrylic resin by vacuum impregnation to preserve the porous structure. The resulting specimen is polished with sandpaper with grain size varying from 150 to 2000 mesh. After that step, the samples are polished with alumina of 0.3 lm grain size, followed by diamond paste polishing with 0.25 lm grain size. The digital images are obtained with a reflected light microscope, for 20 different regions of each sample, using magnifications of 100 and 200 times. Finally, the images are processed by software ImagoÒ, which provides the porosity of the sample. 4. Analysis 4.1. Geometrical parameters Particle diameter mean values, statistically representative of each powder used in the fabrication of porous media, were determined to be applied as input parameters for the present paper models. The mean particle diameters are characterized based on 10.000 powder particles, which, according to Thümmler and Oberacker [12], comprise a sample with sufficient particles so that the statistical treatment shows a 1% of error. The distributions of powder PAM and PAC particle diameters, obtained by diffraction laser methodology, are presented in Figs. 7 and 8, respectively. The experimental diameter data is shown in histograms and cumulative frequency curves in logarithmic scale. From these figures one can see that PAC shows less dispersion compared to the PAM powder. The particle mean diameter d and thermophysical properties of copper are used as input parameters in Eq. (6)for the prediction of the neck diameter.

100

80

Histogram (x10)

with resolution of 0.0001 g and of an automatic system for determination of the sample densities, according to method proposes by Paiva [18]. The following equation, which input data consists of: dry mass md, wet mass mw, air density qair and liquid density ql, is used to calculate the apparent density qa:

Cumulative frequency %

874

60

40

20

0 1

10 Diameter particle (microns)

100

Fig. 8. PAC powder particle diameter distribution.

As already mentioned, the neck radius (x), for sintering process, is calculated by the model consisting of Eqs. (6)–(8). The neck radius is plotted against the average particle diameter ðdÞ in Fig. 9. This figure shows that the neck diameter is very sensible to the size of the particle, especially for small particle diameters. Also, the curve tends to a linear behavior for diameters larger than 20 lm. Actually, this happens because the concentration of the energy in the particle surfaces is higher for smaller particles, enabling the contact formation and growth of necks. Theoretical non dimensional total thermal resistance is determined for pairs of hemispheres of the same ðdÞ diameter (see Fig. 1), using Eqs. (4) and (5) and is plotted as a function of the non-dimensional neck parameter ðx=dÞ in Fig. 10. In this same graph, non dimensional constriction and expansion thermal resistances (the formulation is the same, as already mentioned) and also the neck resistance are plotted. The spherical liquid film resistance values are very high and, as a consequence, small amount of heat crosses this resistance, which can be removed from the thermal circuit shown in Fig. 2 with negligible error. Therefore, the liquid film resistance is not plotted in Fig. 10. The thermal resistances are non-dimensionalized using the expression:

R ¼ ks x RL

ð26Þ

Fig. 10 was generated considering that the process variables: heating rate, sinterization temperature and oven atmosphere, were kept constant, which means that the size of the neck formed be-

100

8

60

40

Radius of neck µm

6

Histogram ( x 10)

Cumulative frequency %

80

4

2

20

0

0 1

10

100

Diameter particle (microns) Fig. 7. PAM powder particle diameter distribution.

0

10

20

30

40

50

60

70

Mean particle diameter (microns) Fig. 9. Radius of neck against mean particle diameter.

80

90


Non-dimensional resistance R Ksd

12

Thermal constriction/expansion resistance Thermal neck resistance Total thermal resistance

11 10 9 8 7 6 5 4 3 2 1 0 0.1

0.2

0.3

0.4

x/d Fig. 10. Non-dimensional resistance against mean particle diameter.

tween powder particles are approximately the same. Therefore small x=d values are obtained for larger particle diameters. For the present case, the value of the neck and the particle dimensions approximates to each other for particles diameters smaller less than 2 lm. As shown in Fig. 10, the model predicts a higher variation of the non-dimensional constriction and expansion resistances for small particle diameters. Actually the constriction/expansion are the major resistances for most of the actual applications, which usual range is: 0:5 6 x=d 6 0:2. The model adopted considers that, for large x=d parameters, where the particle and neck diameters tend to the same value, the contacting surfaces between spheres increases and so, the constriction and spreading resistances decrease. In the limit, when the contacting surface is very large relatively to the sphere diameter ðx=d ! 1=2Þ, the contacting spheres tend to have a shape of a square cross section cylinder, of diameter and length of d. On the other hand, larger neck diameters imply in the rm radius in Fig. 1 tends to infinity. In this case, the neck also tends to a square cross section cylinder of length and diameter of d. Therefore, the neck and the constriction/spreading resistance curves tends to the same asymptote, y = 4/p, that corresponds to the thermal resistance of a cylindrical bar of length and diameter equal do the mean particle diameter d. Actually, the temperature levels needed to reach this limit, where both resistances contribute with half of the total resistance, is very high; the particles would melt before this conditions was reached and the porosity would vanish, forming a solid block.

875

4.1.1. Porosity The average porosities of the tested samples made from copper powders PAC and PAM, of mean diameters of 20 and 50 lm, were measured by the two methods described (image analyses and Arquimides) and showed to be 51.85% e 42.42%, respectively. The differences obtained by both methods are of 2.7% for PAC and 2.2% for PAM. This comparison is very good, showing that the porosity has been appropriately measured. Fig. 11 left side shows a schematic of the tridimensional arrangement of the unit cells (see Fig. 1) within the porous media. The solid spheres, which represent the cooper particles, are considered in contact in the vertical direction. In the horizontal plane, the spheres are surrounded by liquid layers. The central drawing in Fig. 11 shows a schematic of a horizontal cut of the porous media physical model, which tries to reproduce the actual configuration, shown in the right side of this figure. The horizontal distance between the spheres is calculated according to the porosity of the media, represented by the black spaces in the microscope pictures (right side of the figure). In other words, the black space area is equivalent to the open spaces between spheres (middle scheme). Therefore, the overall model consists of the three dimensional arrangement of vertical piles of a series of contacting spheres, which are in parallel arrangements. The distance between vertical piles is filled with the fluid. Actually the thermal resistances of the solid spheres are much smaller than the resistances of the liquid shell film, which in turn, work as solid sphere thermal insulators in the horizontal direction. One should note that, although the constriction and spreading resistances in hemispheres are tridimensional, in the present paper the heat is considered to be transferred only in the vertical straight direction. So, the tortuosity effect of the heat flux path is not considered. Actually, the fabrication process, which consists of dry loose powder spread along the plate and heated up to the sintering temperatures, may result in displays of the metallic powder not best represented by the cubic arrangement. For the proposed cubic arrangement shown in Fig. 11, supposing that, in the limit, the solid spheres touch each other in both horizontal and vertical directions (no liquid layers between spheres) the packing density, determined as the volume occupied by the solid spheres divided by the total volume of an elementary cubic cell, is 52%, and so, the porosity is around 48%. This value is close to the porosity measurements obtained in the laboratory. Fig. 4 shows that the size and shape of the particles are neither uniform nor spherical and so their arrangement is not regular, resulting in structures with larger porosity than predicted. The grid of spheres proposed in the model provides only one heat transfer path for each sphere (which may be not true) overpredicting the overall thermal resistance. On the other hand, the lengths of the

Fig. 11. Packing arrangements, (a) Proposed model; (b) Optical microscopy image of the sample PAC copper powder.


heat transfer paths modeled are shorter than the actual ones underpredicting the total resistance, as the tortuosity of the heat transfer path is not considered in the model. It is expected that these effects may compensate each other so that the model can be precise in the estimative of the effective thermal conductivity of the porous media. The present authors suggest that further work must be conducted to evaluate the packing, improving the present physical model, including the consideration that the powder materials are composed by particles with a distribution of sizes. Actually, small diameter particles are able to be located within the void spaces among larger porous, reducing the porosity. Also, a model able to predict the porosity from the particle geometry would be very interesting to be developed. In the present paper, the porosities used in the estimative of the liquid layer thicknesses were obtained from laboratory measurements.

160

Effective thermal conductivity W/m*K

876

140

Proposed model Experimental data metanhol 1 Experimental data metanhol 2

120 100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

Particle size µm

4.2. Comparison between theoretical results and data Present model theoretical and experimental effective thermal conductivities as function of the particle size for porous media in vacuum is presented in Fig. 12. The effective thermal conductivity of two samples made of PAC and two made of PAM were tested.


160 Proposed model Experimental data vacuum 1 Experimental data vacuum 2

140 120 100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

Particle size µm Fig. 12. Comparison between proposed model and data for porous media in vacuum.


160 Proposed model Experimental data water 1 Experimental data water 2

140 120 100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

Particle size µm Fig. 13. Comparison between proposed model and data for porous media filled with water.

Fig. 14. Comparison between proposed model and data for porous media filled with methanol.

The data showed to be very similar to each other (the symbols are superposed in the graph), showing that the measurements are very reproducible. This same reproducibility is also observed in the following graphs (Figs. 13 and 14). Fig. 12 also shows that the comparison between present model results and data is very good. The differences of the average values of effective thermal conductivity and model are around 6.7% for PAC and 23.9% for PAM. The vertical error bars shows the uncertainty range for each case tested. For the PAC case, the difference between data and model is within the range of uncertainty of the experimental results. This figure also shows that both model and data show the same trends. The same plot, but with both samples filled with distilled water is presented in Fig. 13. This case shows the best comparison between data and model, where the differences between averaged data and model are 4.6% for PAC and 16.9% for PAM. The model results are within the experimental error bars for three of the four tested samples. The same trends are also observed in both cases. Finally, Fig. 14 shows the comparison between average effective thermal conductances of methanol filled samples with model. Again, the comparison is quite good, of 8.8% for PAC and 18.5% for PAM samples. Observing Figs. 12–14, one can observe that, for the present data, the difference between model and experimental data increases with the particle size increase. This can be explained by the particle size distribution obtained for each of the copper powder used. Comparing Figs. 7 and 8 for PAM and PAC respectively one can see that PAC particle size distribution is narrower, or, in other words, the variation of the particle diameter is smaller and so the morphology of the resulting porous media fits better to the uniform size particle hypotheses assumed for the mathematical model. Also, from Figs. 12–14, one sees that the influence of the filling liquid in the effective thermal conductivity is very small, as the model curves and experimental results are very similar for all cases tested. It is important to note that this is true for the present case, where the thermal conductivity of the powder is much higher than that of the liquid. This observation was already made in Fig. 10 analysis, when the liquid heat transfer path was removed from the thermal circuit. ANOVA test, for 95% of confidence level, was performed, showing that the influence of the liquid (or vacuum) is statistically negligible for the present case. One should note that, in this work, only two powders (PAC and PAM) were used for fabrication of the samples tested, so that all the experimental data were grouped around two powder average

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4.3. Comparison between models and data Figs. 15–17 present plots of theoretical effective thermal conductivities, obtained from literature models, against porosity, for porous media saturated with water, methanol and in vacuum, respectively. These theoretical results are compared with data and the present proposed model results, for PAC and PAM sintered porous media. One should remember that, in the present model, the input parameter is the average diameter of the powder employed for the fabrication of the porous media, while the input parameter for the literature models is the porosity. Therefore, to be possible to compare the present with literature models for several porosity porous media, a model of the porosity as a function of the diameter of the powder should be provided. This is another suggestion for future work. In Fig. 15, the experimental data shows a difference to the Alexander’s model (apud Atabaki and Baliga [2]) that lies between 4% and 6.8%, for the PAC and PAM powder porous media, respectively. This comparison is within the range of data experimental uncertainty and therefore, can be considered very good. The data, in turn, compare very well with the model developed in this study. The model of Atabaki and Baliga [2] also shows a good comparison with the data. On the other hand, the parallel and series models are too simplified and do not show good results when compared with data and other models, including the developed in the present work. This case (porous media filled with water) presents, among all tested cases, the best comparison between data and models. One should note that the present proposed model also compares very well with the Alexander´s model. The effective thermal conductivity experimental results for porous material saturated with methanol are show in Fig. 16. The experimental results with methanol show that the effective thermal conductivity is not very sensitive to the fluid, although, the thermal conductivity of the methanol is about 10 times lower than that of the water. The average difference between data and Alexander´s model, which is the literature model that compares better with data, is about 11.4% and 21.6% for PAC and PAM samples. These differences are larger than those observed for water. In Fig. 17, the effective thermal conductivity of porous media data and model results are presented for vacuum conditions. In


240 Alexander's model Atabaki and Baliga model Parallel model Series model Experimental data water 1 Experimental data water 2 Proposed model

210 180 150

240 Alexander's model Atabaki and Baliga model Parallel model Series model Experimental data methanol 1 Experimental data methanol 2 Proposed m odel

220 200 180 160 140 120 100 80 60 40 20 0 0.0

0.2

0.4

0.6

0.8

1.0

Porosity Fig. 16. Effective thermal conductivity vs. porosity for experimental data and values calculated through literature models to porous media saturated with methanol.

240

Effective thermal conductivity W/(m*K)

diameters and/or two porosity values. Obviously it would be interesting to have more data to compare with model, which is a time and cost consuming task, which is suggested to be performed later.

Effective thermal conductivity W/(m*K)


Alexander`s model Atabaki and Baliga model Parallel model Series model Experimental data vacuum 1 Experimental data vacuum 2 Proposed model

220 200 180 160 140 120 100 80 60 40 20 0

0.0

0.2

0.4

0.6

0.8

1.0

Porosity Fig. 17. Effective thermal conductivity vs. porosity for experimental data and values calculated through literature models to porous media in vacuum condition.

Table 2 Differences between effective thermal conductivity determined form theoretical models and experimental data. Differences %

Water

Models Proposed Alexander Atabaki and Baliga Parallel Series

PAC 4.6 4 23.7 166 97.2

PAM 16.9 6.8 40.8 235.4 96.6

Methanol

Vacuum

PAC 8.8 11.4 18 181 99

PAC 6.7 52.1 6.2 175.3 99

PAM 18.5 21.6 25.7 234.6 98.8

PAM 23.9 62.1 6.1 254.6 98

120 90 60 30 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Porosity Fig. 15. Effective thermal conductivity vs. porosity for experimental data and values calculated through literature models for porous media saturated with water.

this case, the experimental data also better estimated by the model proposed by Atabaki and Baliga. One can find, in Table 2, a summary of the differences observed among the data and the models (present work and literature) used to predict the overall thermal resistance of the porous media. From this table one can see that the model proposed by Alexander and the present model compares very well with data. As already mentioned, the major characteristic of the present model is that the input parameter is the diameter of the particle used in the sintering fabrication of the porous media, while, in the literature models, the input parameter is the porosity. The lit-

878


erature models available for evaluating the porosity, which depends on the fabrications process employed, are fundamentally based on experimental work and are not precise for cases when the physical parameters differs from the tested conditions. Therefore, considerable errors could be made in the present work if these models were employed for the porosity estimation. So, it was decided not to use literature porosity models and, instead of having in Figs. 15–17 present model theoretical curves for porosities varying from zero to one, only the porosities measured where correlated with the particle powder mean diameters and used as input parameters. One should note that, although the porosity quantifies the ratio between voids and total volumes of the porous material, it does not provide information about the interconnection between the void spaces, which defined the permeability of the material, a very important parameter for some porous media applications, such as heat pipes.

compared with literature model results. The model of Alexander´s presented a very good comparison with the present model and with the present paper experimental data. The quality of the generated data is quite good and the thermal behavior of the samples was very reproducible. Proposal of future works includes a development of a more precise model for the porosity prediction taking the sintering conditions and the diameter of the powder particle as input parameters. Other porous media constructed from different powders are proposed to be fabricated and tested.

5. Conclusion

References

In this paper, a model for the prediction of the effective thermal conductivity of porous media formed by sintering of loose metal powder over flat surfaces is proposed. Differently from the literature models, the present model is based on heat transfer phenomenological aspects and uses the mean diameter of the powder particles, which are considered solid spheres, as input parameters. The present study shows clearly that the geometry of the porous media solid matrix has large influence on the effective thermal conductivity. Therefore, a physical model is proposed, consisting of a unitary cell, formed by two hemispheres of diameter equivalent to the mean diameter of the powder particles. The spheres geometry was obtained from statistical analysis. The hemispheres were considered surrounded by a spherical layer of liquid. Several stacks of these unitary cells, in parallel arrangement, composes the adopted three dimensional arrangement of the porous media studied. The stack consists of vertically aligned cells, where half of one sphere belongs to one cell and the other half to the next superior or inferior neighbor cell. The spheres of one pile to not touch the horizontal neighbor spheres, as a liquid film, of much smaller thermal conductivity, insulates the vertical piles from each other. The equivalent thermal circuit model is proposed in this work. The neck thermal resistances are much smaller while the liquid layer resistance is much larger than the constriction and spreading resistances, so that both can be removed from the thermal circuit with small errors, for most of the actual application cases. An experimental set up was designed and built for the present work. Several samples made from two different copper powders were fabricated and the porosity of the resulting porous media was carefully measured. The effective thermal conductivity testing conditions included evacuated and water and methanol filled samples as well. The present model and data showed a very good agreement. For most cases, the model results are within the experimental uncertainty ranges. The present model and data were also

Acknowledgments The authors acknowledge the National Council for Scientific and Technological Development (CNPq) for financial support through Universal Project and scholarships. They also acknowledge CAPES for the financial support via scholarships.

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