Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams

Applied Thermal Engineering 71 (2014) 536e547

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams Prashant Kumar*, Frederic Topin IUSTI, CNRS UMR 7343, Aix-Marseille University, Marseille, France

h i g h l i g h t s
CAD modeling of different strut shapes and porosity.
3-D pore scale numerical calculations to determine effective thermal conductivity.
Development of two analytical models.
Detailed discussion of importance of intrinsic solid phase thermal conductivity.
Validation of analytical models against numerical and experimental data.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 October 2013 Accepted 27 June 2014 Available online 12 July 2014

The relationship of geometrical parameters with thermal properties such as effective thermal conductivity is critical for conductive heat transfer in metal foams. We have realized different virtual isotropic structures of tetrakaidecahedron shape having circular, square, hexagon, diamond and star strut shapes with various orientations using CAD modeling. The range of solid to fluid phase conductivity ratios (ls/lf) studied is from 10 to 30,000 for porosity range 60e95%. At pore scale, 3-D direct numerical simulations were performed to calculate effective thermal conductivity, leff in local thermal equilibrium condition. A database of 2000 values of effective thermal conductivity is generated. Two models are derived in order to predict simultaneously intrinsic solid phase thermal conductivity, ls and effective thermal conductivity, leff. A modified correlation factor, F is introduced in analytical resistor model in order to take thermal conductivities of constituent phases into account and a new PF model based on Lemlich model is derived. An excellent agreement has been observed between the predicted results against numerical and experimental data. We have also shown the importance of intrinsic solid phase thermal conductivity (ls) in determining effective thermal conductivity (leff). © 2014 Elsevier Ltd. All rights reserved.

Keywords: Metal foams Strut diameter Porosity Effective thermal conductivity Solid and fluid phase conductivity

1. Introduction Transport phenomena in porous media have been the focus of many industrial and academic investigations. The majority of the literature studies dealing with low porosity media such as granular materials and packed beds. Recently, high porosity open cell porous media such as metal or ceramic foams and fibrous media have started to receive more attention because of their relatively low cost, ultra-low density, high surface area to volume ratio, and most importantly, their ability to mix the passing fluid. This makes them

* Corresponding author. 5, Rue Enrico Fermi, Technopole de Chateau Gombert, 13453 Marseille Cedex 13, France. Tel.: þ33 (0) 04 91 10 68 36. E-mail addresses: [email protected] (P. Kumar), frederic.topin@ univ-amu.fr (F. Topin). http://dx.doi.org/10.1016/j.applthermaleng.2014.06.058 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

excellent candidates for a variety of unique thermo-fluid applications and devices [1,2]. The widespread range of applications of metal foams has led to increase in the interest of modeling the heat transfer phenomena in such porous media. It is pointed out that, the precise calculation of leff is required for accurate modeling of thermal transport through open cell foams as well as foam heat exchangers when there are large differences in the thermal conductivities of the solid and fluid phases (2e3 orders of magnitude) as well as the high porosity of the medium. There are several kinds of models in the literature to determine leff by using analytical approach. One group of studies focuses on asymptotic bound approach while the other studies deal with micro-structural approach. Asymptotic bound approach studies are performed in the case of spherical cells or packed bed spheres and

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

micro-structural approach studies show up the importance of various geometrical parameters of foam matrix in determining leff. In asymptotic approach, the very simple existing models are parallel and series models that assume fluid and solid phases parallel or perpendicular to the heat flow direction; provide the highest and lowest bounds for the effective thermal conductivity of a porous medium. They are given by Eqs (1) and (2) as:




lparallel ¼ εs ls  lf

1 lseries

( ¼

εs



1 1  ls lf

!

1 þ lf

(1) ) (2)

Maxwell-Eucken [3] has developed a model for discontinuous phase in a medium to determine effective thermal conductivity by using Eqs. (1) and (2). Maxwell-Eucken upper and lower bound models provide more tight limits resulting more close values to the true thermal conductivity (see Zimmermann [4]). These models assume that the inclusions of the dispersed phase (fluid phase) do not encounter with similar neighboring inclusions of continuous solid phase [5,6]. Hamilton and Crosser [7] extended the MaxwellEucken upper and lower bound models to include non-spherical particles and developed an empirical-based model. Since the pore structure of most of the stones (especially Sander sandstone) consists of large and small pores interconnected through capillary tubes, therefore, Maxwell-Eucken upper and lower models do not provide satisfactory results. In the case of metallic or ceramic foams, both solid and fluid phases are fully connected. Thus, these upper and lower bound approach are not applicable in case of such open cell foams. The Landauer's effective medium theory model (EMT) [8] uses a similar approach to the Maxwell-Eucken models to establish a relationship for the effective thermal conductivity of the medium. However, it assumes a completely random distribution of each phase. EMT is a statistical approach that is often used to model thermal conductivity of random mixtures of component materials, particularly when one component has higher thermal conductivity than the other component [9]. EMT is also applicable for the estimation of electrical resistances for a network of resistors. Unlike the Maxwell-Eucken models, EMT does not have any continuous and dispersed phases. According to this theory, the effective thermal conductivity of a two-phase system can be estimated using Eq. (3) (see also Nait-Ali et al. [10]):

lEMT eff ¼


 1 lf 3Vf  1 þ ls ð3Vs  1Þ 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
 o2 lf 3Vf  1 þ ls ð3Vs  1Þ þ 8lf ls þ

(3)

where, Vf and Vs are the volumes fractions contained by pores/fluid and solid matrix, respectively. By combining these structural models or from heuristic approach, several other models have been developed. For example, Krischer [11] proposed a weighted harmonic mean of the series and parallel models for the effective thermal conductivity of heterogeneous materials as:

lKeff ¼

1

sensitive to the weighting parameter s which must be set for each material and porosity range. Lemlich [12] developed an analogy to predict the electrical conductivity of polyhedral liquid foam of high porosity and is given by the Eq. (5):

leff ¼ þ lf

. s=lseries þ ð1  sÞ lparallel

(4)

The Krischer's model [11] is often used in drying studies and provides a rough estimate for the thermal conductivity. It is

537

1 ls εs 3

(5)

The limitation of using Lemlich model [12] is that it does not predict an approximate value of leff when water is used as fluid medium but it works well with air as a fluid medium. In fact, this model takes into account only heat conduction in solid phase. When fluid phase conductivity is the same order of magnitude of solid phase conductivity, this model is no more valid because of significant heat exchange between foam ligament and interstitial fluid and therefore, Eq. (5) is not appropriate in determining leff. In the case of micro-structural analysis of leff in open cell foams, a unit cell approach has been generally taken to represent the foam microstructure [13e15], and it is assumed that this unit cell can be repeated throughout the medium by virtue of periodicity. The unit cell approach breaks the problem into distinct conduction paths in solid and fluid phases and calculates the conductivity of the medium as a series/parallel combination of the individual resistances for those paths. Applying the energy equation to the suggested unit cell, the effective thermal conductivity can be found analytically or numerically depending on the complexity of the unit cell. Proposed unit cells studied includes 2-D hexagon, 3-D dodecahedron, 3-D truncated octahedron, and polyhedra. There are various one, two and three dimensional conduction modeling of porous media that exist in the literature. Hsu et al. [16] introduced a one-dimensional conduction model based on in-line touching cubes, and carried out an elegant analysis to show good agreement with the experimental data for the case of packed beds. Ashby [17] proposed a model for cellular structures by adding two terms to the Lemlich model [12]. This model considers conduction in both the solid and fluid phases and is suitable for a medium with a small solid to fluid thermal conductivity ratio (e.g., RVC foam-water). This model provides good estimate for the effective thermal conductivity when lf/ls z 0 (e.g. RVC foamwater), but highly overestimate the effective thermal conductivity for lf/ls z 102. Calmidi and Mahajan [13] proposed one dimensional conductivity model considering the porous medium to be formed by a two dimensional array of hexagonal cells with square lumps at nodes for high porosity metal foams. They described a non-dimensional parameter of value 0.09 which was obtained through fitting of experimental data. Bhattacharya et al. [14] extended the model of Calmidi and Mahajan [13] with square and circular lumps at nodes and obtained non-dimensional parameter of value 0.19. Both the models can accurately predict the effective thermal conductivity for Al foam, but they overestimate the effective thermal conductivity for other foam structures. One of the probable reason is that they did not take into consideration the strut shape (convex triangular, concave triangular, triangular or even circular) while determining leff analytically. Moreover, the lumps at the node of a regular network are specific for a given set of foams and probably due to manufacturing process. Ozmat et al. [2] proposed compact analytical model using regular dodecahedron structure having 12 pentagon-shaped faces with triangular strut cross-section ligaments. They considered no lumped materials at the intersections of ligaments and found close agreement with their experimental data for low thermal conductivity ratios. This model underestimates the effective thermal conductivity values, because it does not include heat conduction in

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the fluid phase. Edouard [18] also predicted effective thermal conductivity using the family of regular dodecahedron structure and proposed to use 'slim' and 'fat' description of the structure according to strut shape variation namely cylindrical struts for porosities below 90% and triangular struts above 90%. The model of Edouard [18] (using 'slim' and 'fat' foams) predicts effective thermal conductivity like lower and upper bound of asymptotic approach. Importantly, regular dodecahedron is not a space filling shape and thus, could not be used as periodic unit cell as fully discussed by Steinhaus [19] and Wells [20]. On the other hand, the rhombic dodecahedron, slightly elongated dodecahedron and squashed dodecahedron appearing in sphere packing can be used as spacefilling periodic unit cell. There are various 3-D foam geometries discussed in the literature which are based on tetrakaidecahedron structure. Boomsma and Poulikakos [21] proposed 3-D tetrakaidecahedron (truncated octahedron) geometry with cubic nodes at the intersections of ligaments. Using experimental fitting of the data, they proposed a non-dimensional geometrical parameter, e ¼ 0.16 but the model becomes unrealistic when ε < 0.9. Schmierer and Razani [22] presented tetrakaidecahedron geometry with spherical nodes at the intersections of ligaments. These authors performed image and geometrical analyses of the microstructure to find node size. Numerical finite element analysis was performed to calculate the effective thermal conductivity. Many authors [13,14,16,21,22] used a unit cell composed of cylinder as struts and spheres or cubes as lumps at nodes which is not applicable to the majority of foams that possess triangular strut cross-section. Kanaun and Tkachenko [23] proposed a mathematical model that includes all the necessary geometric parameters to describe metallic foams based on tetrakaidecahedron geometry to calculate effective thermal conductivity. They focused on the influence of strut cross sections on effective thermal conductivity. De Jaeger et al. [24] introduced HW factor and take into account strut size variation along its axis. These authors characterized the convex triangular struts with no lumps formation at nodes which were manufactured in-house for porosity range 0.88 < ε < 0.98. Their analytical geometrical model includes the change in the strut formation from convex triangular to circular and is based on the measurement of actual dimensions of foams. However, these authors did not study the effective thermal conductivity. The 3-D numerical simulation tool has also been used to determine effective thermal conductivity on the actual foam structure. The 3-D foam geometry is usually obtained using X-ray mCT and calculations to determine effective thermal conductivity are performed either on simplified resistor network model [25,26] or on full foam geometry [27,28]. Vicente et al. [25] measured directional tortuosity of the solid matrix and correlate them to the cell shape and orientation. They quantify the dependence of the effective thermal conductivity with tortuosity. Bodla et al. [26] performed numerical simulations using resistor network model on three samples of grades 10, 20 and 40 PPI of very similar values of porosity. The effective thermal conductivity is estimated through a 1-D conduction model, representing individual ligaments as an effective thermal resistance using the topological information from the scan data. Body-Centered-Cubic (BCC) structure used by Krishnan et al. [27] proposed a numerical model to determine the effective thermal conductivity. Their results were in agreement with experimental data only when the porosity is greater than 0.94 because of geometry limitations. Hugo [28] showed that the porosity is not the alone parameter to determine effective thermal conductivity which has been widely quoted by many authors. It was shown using pore scale numerical simulations in the thesis of Hugo [28] that the ratios of solid to fluid phase thermal conductivity impact strongly the

local heat conduction. It was also shown that for slightly elongated foams by keeping the same porosity, effective thermal conductivity changes. Thus, porosity alone cannot be identified as a function of effective thermal conductivity. One group of studies also exists in the literature that deals with the asymptotic approach method and direct measurements of effective thermal conductivity on micro-structural foams for analytical modeling. Singh and Kasana [29] established the correlation only for high porosities (ε > 0.9) where the fitting curve followed a straight line. Their correlation has involved the solid and fluid phase conductivities where fluid phase is the order of 0.02e0.7 (air or water) and solid phase constitutes Al or RVC. The correlation proposed by them was derived for highly porous open cell foams with very high solid to fluid thermal conductivity ratios (>15,000), only for Al-Air, Alwater, RVC-air and RVC-water. Many authors [13e17,21,29] have rigorously used the solid phase thermal conductivity of parent material (e.g. pure Al/Al 6101 T alloy) for predicting effective thermal conductivity correlations. Manufacturing processes greatly impact the solid phase thermal conductivity of parent material when transformed into foams. As different commercially available foams employ different manufacturing techniques; that lead to significant changes in intrinsic solid phase thermal conductivity of foams compared to the same parent material one and none of these authors have measured the intrinsic value or actual value of solid phase thermal conductivity of foam materials. In this work, we use parent material solid conductivity to assign the solid conductivity of material by nature (e.g. Al). We use intrinsic solid phase thermal conductivity of the material that constitutes the phase or form of the foam (e.g. thermal conductivity of strut of Al) which usually differs from the parent material one. In general, intrinsic solid phase thermal conductivity of strut or foams is unknown when performing experiments to determine effective thermal conductivity. Dietrich et al. [30] measured intrinsic solid phase thermal conductivity of their ceramic foams. They proposed a two dimensional model to determine effective thermal conductivity of their ceramic foams using asymptotic approach of Krischer and Wast model [31]. These authors determined effective thermal conductivity using porosity and two fitting parameters from experiments. To determine effective thermal conductivity using experimental data, several authors [13,14,16,21,22] proposed circular struts with cubes or spheres as lumps at nodes and used porosity, ε as an input parameter to their proposed correlations. It was shown using 3-D numerical simulations in the thesis of Hugo [28] that the ratios of conductivity between phases impact greatly on the thermal conductivity under condition of local thermal equilibrium (LTE). In fact, average heat exchange between phases is zero, but locally there is strong exchange which depends on conductivity of each material. When the fluid phase conductivity has the same order of the solid phase, then fluid phase starts to play an important role in determining effective thermal conductivity and the correlations derived by several authors [13,14,16,21,22,29] will not hold and induce high error as shown in the work of Edouard [18] and Dietrich et al. [30]. All these methods do not take into account the real morphology of the foam but assume an idealized periodic pattern. To the best of our knowledge, none of the studies have identified an effective thermal conductivity correlation that encompasses geometrical parameters of open cell foams and intrinsic solid to fluid thermal conductivity ratios. Generally, intrinsic solid phase thermal conductivity (ls) is not measured and is critical (see Section 4.4) to predict effective thermal conductivity analytically or numerically (see also Dietrich et al. [30]). This remark brings two unknowns namely, ls and leff while fluid phase thermal conductivity (lf) is known when performing experiments. We present two

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

539

Fig. 1. Representation of different 3-D strut and ligament shapes (a) circular (b) square (c) diamond (double equilateral triangle) (d) hexagon (e) star (regular hexagram) (f) rotated square (g) rotated hexagon. The characteristic dimensions and equivalent radius, Req of different struts are also presented that are detailed in the Section 4 and Appendix A for analytical solutions.

models namely, resistor model and PF model (based on Lemlich model) that includes both unknowns, ls and leff. They can be used in either ways when ls is known to determine leff or vice versa. These models can also be used to solve a problem simultaneously as a system of two linear equations where both ls and leff are unknowns but the fluid phase thermal conductivity is known. Moreover, recent developments in micro or nano porous media, it is very difficult to measure the intrinsic solid phase thermal conductivity and consequently, other thermal properties linked to heat conduction. In such type of porous media, these models can be used as a powerful tool where identification of geometrical properties and intrinsic solid phase thermal conductivity would be very critical. These two models can also be used directly for any engineering application involving heat conduction in foam like materials. As heat conduction is invariant by homothetic transformation, these models could be applied to any pore size.

In our work, we have used a mixture of asymptotic and microstructural approaches to derive an effective thermal conductivity correlation in LTE condition for isotropic foams which is valid for a large range of solid to fluid thermal conductivity ratio (ls/ lf ¼ 10 ~ 30,000) for different strut shapes and porosity in the range of 60e95%. Our work focus on generation of a new database which is applicable to wide range of pore sizes. This also allows us to understand the geometrical parameters and its influence on both intrinsic solid phase thermal conductivity (ls) and effective thermal conductivity (leff). Note that, we have mainly focused on constant cross section of the ligament. 2. Geometrical modeling We have realized the different strut cross sections of Kelvin-like cell using truncated octahedron structure in a periodic unit cell. The reason to choose truncated octahedron is that it is a generic structure that can be tailored and can be produced easily in reality. Kelvin cell is the simplest structure that encompasses all specificity of foams and geometrically convenient. Using CAD modeling, we have created different strut shapes namely, circular, diamond (double equilateral triangle), square, hexagon and star (regular hexagram) for constant ligament cross section and porosity, 0.60 < ε < 0.95 (see Fig. 1). We have also generated rotated square and rotated hexagon by changing the relative orientation of the strut with respect to the cell axis. Our construction methodology is based on fixed cell diameter (dcell) as shown in Fig. 2 that allows us to vary strut shape and porosity, mainly as a control parameter. Using our construction method, we can generate foams of chosen porosity for any strut shape. We have measured all geometrical parameters of 45 virtual Kelvin like foams using classical approach and presented them in Table 1. 3. Effective thermal conductivity numerical modeling

Fig. 2. Presentation of tetrakaidecahedon model of Kelvin-like cell of hexagon strut shape in apcubic unit cell. Strut length (Ls), node to node length (L) and cubic unit cell ffiffiffi length (2 2L) are clearly shown.

We have performed numerical simulations using volume mesh generated from actual solid surface using commercial software

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P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

StarCCMþ. The mesh is composed of core polyhedral meshes in both solid and fluid phases and energy balance equation is solved on both phases with a coupled solver as shown in Fig. 3. The calculations were performed until effective thermal conductivity value between two consecutive mesh sizes differs by less than 1%. Mesh size is chosen in order to optimize results reliability and computational time (see Hugo [28]). We consider here only the pure conduction in both phases in an established stagnant steady state. We do not study the radiative transfer and there is no mass transfer at the interface or chemical reaction or heat source or phase change. The equations of energy conservation are solved numerically over the entire volume of the test sample and allow us to obtain the temperature fields and their gradients at any point of the two phases. We used the methodology and model proposed by Topin [32] to determine effective properties. We have checked that LTE is reached for all calculations. We can write Ts ¼ Tf. Thus, we can use one-temperature model to determine V < T > and F. Whitaker [33]

Fig. 3. Boundary conditions and unit cell model: Heat flux in the main direction, no fluxes in other direction (adiabatic walls). Temperature difference is imposed in heat flux direction.

Table 1 Representation of CAD data, analytical values of geometrical parameters of foam matrix and numerically obtained effective thermal conductivities (leff) of different strut shapes for various ls/lf. Shape

CAD measurement ε (%)

Circular

Square

Rotated square

Diamond

Hexagon

Rotated hexagon

Star

60 65 70 75 80 85 90 95 60 65 70 75 80 85 90 95 80 85 90 95 80 85 90 95 60 65 70 75 80 85 90 95 60 65 70 75 80 85 90 95 75 80 85 90 95

ds or A (mm)

1.212 1.110 1.006 0.900 0.789 0.669 0.534 0.367 1.075 0.984 0.891 0.797 0.698 0.591 0.472 0.325 0.694 0.589 0.470 0.324 0.752 0.637 0.508 0.349 0.665 0.609 0.552 0.494 0.432 0.367 0.292 0.201 0.664 0.608 0.551 0.493 0.432 0.366 0.292 0.201 0.347 0.305 0.258 0.206 0.142

Numerical data of leff (W m1 K1)

Analytical ac (m

982 979 960 926 873 796 686 515 1092 1091 1074 1037 979 895 772 580 996 906 779 583 1070 974 838 627 1025 1023 1005 970 915 835 720 540 1033 1029 1009 973 917 837 721 541 1399 1314 1195 1027 769

1

)

aeq ¼ Req/L

0.429 0.392 0.356 0.318 0.279 0.236 0.189 0.130 0.760 0.696 0.630 0.563 0.493 0.418 0.334 0.229 0.491 0.416 0.332 0.229 0.532 0.450 0.359 0.247 0.470 0.431 0.390 0.349 0.306 0.259 0.207 0.142 0.469 0.430 0.390 0.348 0.305 0.259 0.207 0.142 0.246 0.215 0.183 0.146 0.100

b ¼ Ls/L

0.341 0.393 0.447 0.504 0.563 0.628 0.702 0.794 0.341 0.393 0.447 0.504 0.563 0.628 0.702 0.794 0.563 0.628 0.702 0.794 0.563 0.628 0.702 0.794 0.341 0.393 0.447 0.504 0.563 0.628 0.702 0.794 0.341 0.393 0.447 0.504 0.563 0.628 0.702 0.794 0.504 0.563 0.628 0.702 0.794

ls/lf 25

100

1000

5000

30,000

leff

leff

leff

leff

leff

73 63 53 44 36 28 21 15 74 63 53 44 36 28 22 15 36 28 21 15 36 29 22 15 73 63 53 44 36 28 21 15 73 62 53 44 36 28 21 15 45 36 29 22 15

262 219 180 144 111 81 54 30 265 222 182 145 112 81 54 30 111 81 54 30 112 82 55 31 261 219 180 144 111 81 54 30 260 218 179 143 110 80 54 30 146 113 82 55 30

2526 2099 1703 1342 1011 712 445 209 2559 2124 1720 1352 1018 716 445 209 1016 716 446 209 1024 724 451 211 2518 2097 1699 1339 1008 711 444 208 2506 2083 1694 1330 1004 708 442 208 1360 1025 720 448 210

12,588 10,451 8473 6665 5011 3517 2183 1005 12,753 10,576 8558 6717 5046 3537 2191 1005 5036 3535 2189 1006 5075 3576 2213 1013 12,547 10,440 8449 6650 4996 3511 2177 1000 12,487 10,373 8425 6606 4974 3495 2167 998 6753 5080 3556 2199 1009

75,476 62,652 50,782 39,934 30,011 21,048 13,043 5977 76,446 63,404 51,294 40,248 30,220 21,167 13,090 5981 30,159 21,154 13,078 5983 30,395 21,400 13,225 6028 75,232 62,586 50,640 39,846 29,923 21,012 13,008 5951 74,867 62,182 50,495 39,581 29,792 20,917 12,950 5936 40,463 30,424 21,279 13,142 6004

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

has shown that F ¼ leff $V < T > when local thermal equilibrium is reached and can be written using an effective thermal conductivity tensor defined symmetric and positive as:

2

F ¼ leff $V < T > with leff

lxx ¼ 4 lxy lxz

lxy lyy lyz

3 lxz lyz 5 lzz

(6)

where, F is the macroscopic heat flux, leff is 2nd order symmetric tensor of effective thermal conductivity and V < T > is macroscopic temperature gradient. The definition of effective conductivity requires knowledge of the entire distribution of temperature and heat flux inside the sample. However, the volume integral can be replaced by surface integral [34]. For example, the average temperature gradient in the X direction is the scalar product of the averaged gradient by the unit normal vector i, in the X direction (Eq. (7)). Integrating by parts allows replacement of the volume integral by a surface integral in Eq. (8):

V < Tx > ¼

1 V

Z VTðxÞ$nx dV

(7)

V

< VTx > ¼

1 V

Z TðxÞi$nx dSint ¼ Sint

D < Tx > Dx

(8)

where, Sint is the boundary of the sample and nx is the unit vector normal to the elementary surface of integration dSint. Knowing the geometry of the sample and the distribution of temperature on its surface is therefore sufficient to calculate the average temperature gradient inside the sample. Similarly, the averaged specific heat flux in the X direction is defined by a volume integral. However, integration by parts shows that it can be replaced by a surface integral under steady state conditions.

4x ¼

1 V

Z x4$nx dSint ¼ Sint

Px Sintx

(9)

where, Px is the heat flux in Watts Again, this integral can be easily evaluated knowing the geometry of the sample and measuring the distribution of the heat fluxes on the boundary of the sample. Numerically, volume averaged and surface averaged quantities could be obtained. We have previously checked that both method lead to the same values (see Hugo [28]). Eq. (7) provides a system of three equations with six unknowns: the components of the tensor. This linear system is underdetermined. The following boundary conditions are used: prescribe temperature difference between two opposite face and null fluxes across the other faces as shown in Fig. 3. We successively impose temperature difference along each direction on the two opposite faces and null flux on the four other faces. We can then determine the components of the averaged specific heat flux vector and the components of the average temperature gradient for each flux experiment using Eqs. (8) and (9). We then calculate thermal conductivity tensors which verify at best, to the least square criteria, Eq. (9) for all flux direction. We obtain 6 unknowns, 9 equations underdetermined system to solve at the least square sense. As we are studying the isotropic open cell foam, the diagonal components of the matrix shown in Eq. (6) are equal while the non-diagonal terms are zero. We have thus created a database of more than 2000 values for entire range of solid to fluid thermal conductivity ratios for all strut

541

shapes of different porosities. The fluid phase conductivity (lf) used in the present work is 10 W m1 K1. Using direct numerical simulations, we can precisely calculate the effective thermal conductivity for known input properties: geometrical parameters of foams and intrinsic solid to fluid thermal conductivity ratios (ls/lf). We have presented some of the effective thermal conductivity (leff) values of different strut shapes in Table 1. In this work, we have mainly focused on ls/lf > 10 where heat conduction is mainly due to solid phase of foam. In Figs. 4 and 5, we have shown the temperature fields of various porosities and different strut shapes in LTE condition. According to strut shape, there is a slight difference in the values of leff. This difference is linked to the local difference in temperature field contours. 4. Analytical approach We introduce complex strut shapes to study their impact and characterize their effective thermal conductivities using microstructural and asymptotic approach of resistor modeling. The node junction at different porosities possesses complex shape and is difficult to understand. To make our analytical approach clear and user-friendly, we have approximated the strut shapes at the junctions because different strut shapes behave differently at the node. As circular shape is easy to visualize at the node and do not possess complex geometry compared to other strut shapes, we have defined an equivalent radius, Req for different shapes as provided in Fig. 1. For each porosity and different strut shapes, we have assumed an equivalent radius, Req which is the radius of the circle of same area than the strut cross section. Obviously, for a given Req, node volume is the same and independent of the strut shape. It is the most important hypothesis in our derivation. Using equivalent radius (Req), one can determine volume of ligament and node and consequently, correlate the geometrical parameters of foam matrix to numerically obtained effective thermal conductivity (see Section 3). We chose to base our node volume calculation given by Kanaun and Tkachenko [23]. Volume of node at the junction of 4 struts of equivalent circular strut shape is given as:

Vnode ¼

4 3 pR 3 eq

(10)

Volume of the ligament of equivalent circular strut shape is given as:

Vligament ¼ pR2eq Ls

(11)

Total volume of a truncated octahedron is given as

pffiffiffi VT ¼ 8 2L3

(12)

At the junction, we can approximate node by using geometrical interpretation based on our construction methodology (see Kanaun and Tkachenko [23]):

1:6Req þ Ls ¼ L

(13)

In non-dimensional form, we can rewrite Eq. (13) as

1:6aeq þ b ¼ 1

(14)

where, aeq ¼ Req/L and b ¼ Ls/L Using our analytical approach of same node volume irrespective of strut shapes, b is found to be independent of strut shape and vary only with the change in porosity (see Table 1). On the other hand, aeq depends strongly on strut shape and porosity.

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P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

Fig. 4. Temperature field in LTE at different porosities of circular strut shape.

Fig. 5. Temperature field in LTE of different strut shapes at 80% porosity. This figure shows that there is no strong impact of strut shape on temperature field.

4.1. Development of analytical resistor model We use the resistor model approach (see Singh and Kasana [29]) and apply it to the unit cell in order to incorporate varying individual geometries and non-linear flow of heat flux lines generated by the difference in the thermal conductivity of the constituent phases. The effective thermal conductivity lies between the parallel model and series model of a two phase system and can be found by incorporating a correlation factor F. This relationship is given by Eq. (15) as:

leff ¼ lFparallel $l1F series

F  0; 0  F  1

(15)

where, Fth fraction of the material is oriented in the direction of heat flow and remaining (1  F)th friction is oriented in the perpendicular direction. In a truncated octahedron structure (see Fig. 2), there are 36 ligaments and 24 nodes but only 1/3rd of both, volume of ligament

and volume of node will be considered because of periodic characteristics of Kelvin cell. On substitution of Eqs. (10)e(12) and (14) in Eqs. (1) and (2) of parallel and series conductivity models, we get:

lparallel ¼

1 3




36Vligament þ 24Vnode VT

 lparallel ¼


 ls  lf

3 12pa2eq b þ 32 3 paeq pffiffiffi 8 2




ls  lf

þ lf

(16)

 þ lf

(16a)

Similarly,

1 lseries

"  !#  1 36Vligament þ 24Vnode 1 1 1 ¼  þ 3 ls lf lf VT

(17)

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

20

1 lseries

1 3 ! 2 b þ 32 pa3 12pa 1 1 eq eq C 6B 7 1 pffiffiffi 3 ¼ 4@  A 5þ l l lf s 8 2 f

(17a)

Knowing precisely geometrical parameters of metallic foams and the strut shape, one can obtain the parallel and series combination of thermal conductivity of foams given by Eqs. (18) and (19).

lparallel ¼ 1 lseries

¼




za2eq b

(


þ

ca3eq




za2eq b þ ca3eq

ls  lf



 1 1  ls lf

þ lf

! þ

1 lf

(18) ) (19)

where, z ¼ 3.33216 and c ¼ 2.961922 are numerical values from Eqs. (16a) and (17a) In simple form, we can rewrite Eqs. (18) and (19) as


lparallel ¼ j ls  lf þ lf (

1 lseries

¼

1 1 j  ls lf

!

(20)

1 þ lf

) (21)

where, j ¼ za2eq b þ ca3eq Note that, Eqs. (20) and (21) represents the parallel and series thermal conductivities of equivalent circular strut shape of radius Req. Solid phase porosity (εs) appeared in Eqs. (1) and (2) can be easily derived for different strut shapes studied in this work and is presented in Appendix A. Eq. (15) is solved for F in terms of lparallel, lseries and leff that contains geometric function, j (see Singh and Kasana [29]). The solution is:

" ln ð1  F¼

"

# jÞ llefff

ln 1 þ jð1  jÞ

þ ls lf

j lleffs

þ

lf ls

!#

(22)

2

For a known ε, one can have only one lparallel and lseries which leads to only one value of leff for a given value of F. Thus, using this approach imply that influence of solid matrix geometry (as well as thermo-physical properties) will be taken into account only through F value. For the same porosity, one can obtain different effective thermal conductivity values as shown in the work of Dietrich et al. [30] which implies the necessity of additional geometrical parameters to be correlated in determining leff. So, the correlation factor, F proposed by Singh and Kasana [29] will not hold for the cases where the solid to fluid conductivity ratio are of the same order and its relation as a function of porosity only. In order to determine more precise correlation compared to the one proposed by Singh and Kasana [29] which is valid for wider porosity range and solid to fluid thermal conductivity ratios, we have performed numerical experiments to better support our model. From our database of 2000 values of numerically obtained leff, values of F using Eq. (22) are extracted and we have found the best correlation which includes geometrical parameters of open cell foam of different shapes and ratio of constituent phases. We have plotted F as a function of S ¼ ln(j2ls/lf) and is shown in Fig. 6. One can notice that all the values of F in relation with S are collapsed on a single curve for all the different strut shapes. The values obtained for leff and in turn, F are independent of strut

543

shape. This behavior was expected due to similar temperature field in LTE condition (see Figs. 4 and 5). The parameter j is function of strut shape and size which is an indirect function of porosity. It is observed from the graph that F increases roughly parabolically with increasing S. From Fig. 6, we have estimated a numerical approximation of F given by Eq. (23).

F ¼ 0:0039S2 þ 0:0593S þ 0:704971

(23)

From the Eq. (23) and Fig. 6, it is evident that F is a function of geometrical parameters and ratio of thermal conductivities of constituent phases and is applicable for wide range of thermal conductivity ratios (ls/lf). The expression of analytical resistor model is rather complex but validity range is explicit. All hypotheses could be checked by the knowledge of geometry and solid to fluid phase thermal conductivity ratio in the form of F function. The importance of this analytical model is that it can be extended and generalized to anisotropic nature of foams. For instance, in the case of an anisotropic media (stretching in one direction and compression in other directions to maintain the constant porosity), one has to determine aeq and b by measuring geometrical parameters of foams. This analytical model is useful in the cases when two foams acquire same porosity and are independent of the nature of the foam. 4.2. Development of PF model Eq. (5) with exponent of 1 on solid porosity, εs has been directly used as a check by several authors [13,14,21,29]. They have used the same value of parent material thermal conductivity as its intrinsic value of thermal conductivity in deriving their analytical correlations in order to fit their curves for foams of pure Al/Al 6101 T alloy (ls ¼ 218240 W/m K) as fluid phase does not play significantly an important role in predicting an approximate value of leff. On contrary, ls ¼ 218240 W/m K is not true thermal conductivity of fabricated foams and is far from actual intrinsic solid phase thermal conductivity of foam and should not be considered in determining effective thermal conductivity either analytically or numerically as fully discussed by Dietrich et al. [30]. Thus, there is a need to find an empirical correlation which incorporates intrinsic value of solid phase thermal conductivity of

Fig. 6. Plot of F (non-dimensional) and S (non-dimensional) to determine leff. The uniqueness of this curve is due to the similarity of local temperature field contours at LTE.

544

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

found but other geometrical properties of foams obviously control this exponent value. PF model has rather simple expression than resistor model but validity range is not explicit as other geometrical parameters combination impact the exponent in an unknown way. The effect of generalized behavior of isotropic and anisotropic foams (keeping the same porosity) is not taken into account in the Eq. (24). 4.3. Validation

where, l*eff ¼ leff/lf, l*s ¼ ls/lf and and h ¼ 0.89. The Eq. (24) is obtained by curve fitting and is obtained as best fit for n ¼ 1.3. No physical interpretation of exponent has been yet

From Fig. 8, it is found that the solution given by Eqs. (15) and (23) underestimates and overestimates the actual thermal conductivity but the error lies within ±6% which clarifies that the thermal conductivity can be precisely calculated using resistor model approach if the geometrical parameters or the relations between them are known. The errors are seen only at low porosities where node exhibits very complex shape. For the porosities ε > 0.70, we have identified one group for which node approximation for any strut shape do not introduce significant error and we have obtained accuracy in the range of ±3%. Another group of complex node shapes are found in the case of hexagon, rotated hexagon and square strut shapes for porosities ε < 0.70. In this case, the node approximation could not be geometrically accurate as for low porosity, node volume is significant and thus accuracy decreases. This introduces a loss in accuracy only at low porosity (0.60 < ε < 0.70) where the error as high as ±6% is obtained. The validation states that the resistor model gives precise results for the whole range of porosity (0.60 < ε < 0.95) as it can encompass all the strut shapes and even the complex ones depict model's robustness. The validation of PF model is presented in Fig. 7. The fit works very well for all known intrinsic ls. We have observed an error of ±3% in entire range of leff values that are associated to slight difference in temperature field contours. Dietrich et al. [30] measured intrinsic solid phase thermal conductivity of the ceramic foams (Alumina, Mullite and OBSiC) for different porosities (0.75e0.85) which have been taken directly in our empirical correlations for validation. They did not provide uncertainty on their experimental data of intrinsic solid phase conductivity values. We have validated the two correlations (resistor model and PF model) to the experimental values of effective thermal conductivity measured by Dietrich et al. [30] on ceramic foams. The validation

Fig. 8. Comparison of numerical and analytical leff. The error is mainly due to node approximation for strut shapes at low porosity (ε < 0.70).

Fig. 9. Validation of resistor and PF analytical models with experimental values of effective thermal conductivity (leff) of ceramic foams. Experimental data are taken from the work of Dietrich et al. [30].

Fig. 7. PF model and its relation is found as l*eff ¼ h$l*s ðjÞ1:3 .

foams and fluid phase to replace Lemlich model [12]. In comparison to the resistor model discussed in Section 4.1, we have tried to develop another model based on scattering of leff values obtained using direct numerical simulations very similar to Lemlich model [12]. In Eq. (5), there is an exponent of 1 on solid porosity. We propose to use this model [12] by introducing an exponent, n and replacing solid porosity, εs by a function of geometrical parameters, j. This exponent (n) takes into account the structural impact on effective thermal conductivity. We have tried several combinations between leff, ls, lf and j (for constant ligament cross-section). Using our database of leff, the best fit for all porosities and strut shapes in the studied range of ls/lf (10e30,000) collapsed on a single curve and is presented in Fig. 7. We have presented PF model for isotropic nature of foams in Eq. (24). From Fig. 7, the relation is observed as a straight line and is given by:

l*eff ¼ h$l*s $ðjÞ1:3

(24)

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

545

Table 2 Comparison of experimental and calculated leff and their associated errors (in %). Authors

Bhattacharya et al. [14]

Paek et al. [35]

Takegos-hi et al. [36]

ε

0.905 0.949 0.909 0.906 0.937 0.938 0.92 0.959 0.936 0.908 0.905

Exp.

Analytical using Eqs. (22) and (23)

Calculated using Eq. (24)

leff

ls

leff (calc.)

6.7 3.9 6.7 6.9 4.5 4.25 5.8 2.75 4.4 6.7 6.7

166 222 178 176 191 183 181 205 182 174 166

6.93 4.13 7.01 7.22 4.66 4.39 6.02 2.88 4.54 6.97 6.93

Average Deviation

(see Fig. 9) shows that the calculated values are in excellent agreement within an error limit of ±4% considering circular strut nature of ceramic foams as reported by Dietrich et al. [30]. A slight difference in prediction of leff between resistor and PF model is observed. This slight difference accounts the node approximation at lower porosity. This uncertainty is comparable to the estimated experimental data (±5%). 4.4. Intrinsic solid phase thermal conductivity determination In order to validate our analytical models with experimental data where no prior information of intrinsic solid phase thermal conductivity (ls) is known, we have enlisted various experimental effective thermal conductivity (leff) values reported by a few authors [14,35,36] in Table 2. To increase the scope of analytical models' validity over wide range of different strut shapes of different materials and different manufacturing techniques, we have first calculated intrinsic solid phase thermal conductivity (ls) from experimental values of [14,35,36] using Eqs. (22) and (23) of analytical resistor model and then from Eq. (24) of PF model (see Table 2) considering quasiequilateral triangular nature of the strut (as visualized by authors [14,35,36]). Intrinsic solid phase thermal conductivity (ls) obtained from resistor model (Eqs. (22) and (23)) is used in PF model (Eq. (24)) to determine effective thermal conductivity (leff) analytically. Similarly, intrinsic solid phase thermal conductivity (ls) obtained from PF model (Eq. (24)) is used in resistor model (Eqs. (22) and (23)) to determine effective thermal conductivity (leff) analytically (see Table 2). Both the models have predicted an approximate decrease of 20% in intrinsic solid phase thermal conductivity (ls) compared to the thermal conductivity of the parent material. Depending upon the manufacturing process employed and constituent materials, we have observed that the intrinsic solid phase conductivity values are varying from 160180 W/m K for most of porosities in the case of pure Al/Al 6101 T alloy foams. This is one of the reasons that analytical correlations reported in the literature do not hold with the experimental results or with other correlations for different set of foams types and materials. The change from parent to intrinsic solid phase thermal conductivity has not been considered while deriving analytical models by many authors [13,14,21,29]. In Table 2, intrinsic values of ls from our models are varying by 4% which confirms the validity of two models over wide range of porosity and material properties. We have also presented average deviations in leff to our proposed models and are found to be 4.07% and 7.89% for different models.

Error (%)

3.43 5.90 4.63 4.64 3.56 3.29 3.79 4.73 3.18 4.03 3.43 4.07

Analytical using Eq. (24)

Calculated using Eqs. (22) and (23)

ls

leff(calc.)

161 210 170 168 184 177 174 196 176 167 161

7.11 4.31 7.15 7.34 4.90 4.62 6.24 3.03 4.78 7.14 7.11

Error (%)

ls ¼ 170 (from Fig. 10)

Error (%)

leff(calc.) 6.12 10.51 6.72 6.38 8.89 8.71 7.59 10.18 8.64 6.57 6.12 7.89

7.09 3.16 6.71 7.00 4.16 4.07 5.67 2.38 4.24 6.80 7.09

5.9 19.0 0.1 1.4 7.6 4.2 2.2 13.5 3.5 1.6 5.9 3.18

We have plotted experimental values of effective thermal conductivity (leff) of samples provided by different authors [14,35,36] against their h.(j)1.3 (using PF model) to determine intrinsic solid phase thermal conductivity (ls) as traced in Fig. 10 and a value of ls z 170 (W/mK) is observed. Using this value of ls from Fig. 10, we have calculated leff using resistor model and the average deviation associated with all the porosities is 3.18%. Deviation is least for our empirical models to calculate effective thermal conductivity analytically. All the methodologies are equivalent in terms of precision. In determining effective thermal conductivity, prior knowledge of intrinsic solid phase thermal conductivity (ls) becomes crucial. By developing two analytical models (resistor and PF models), one does not need to measure intrinsic solid phase thermal conductivity (ls). Eqs. (22) (and 23) and Eq. (24) form a system of two linear equations as intrinsic solid phase thermal conductivity (ls) and effective thermal conductivity (leff) are unknown parameters. This approach expands the potential use of our analytical models. The calculated values of effective thermal conductivity are in excellent agreement and are shown in Figs. 7e9 and Table 2. This comparison suggests one to use the proposed analytical models in this work for ceramic, Al-ERG and casted foams to determine effective thermal conductivity analytically.

Fig. 10. Plot of experimental values of leff from authors [14,35,36] against h.(j)1.3 to determine an approximate value of ls for ERG (Al 6101 T alloy) foams. An approximate value of ls ¼ 170 W/m K is obtained using PF model.

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P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

5. Conclusion

lKeff

We created a database of 2000 values of effective thermal conductivity of open cell isotropic solid foams using 3-D pore scale direct numerical simulations. The simulations were performed on CAD modeled geometries for different strut shapes inside a wide porosity range (0.60 < ε < 0.95) and for various solid to fluid conductivity ratios (ls/lf ¼ 10 ~ 30,000). We derived two effective thermal conductivity models independently. The first one is based on resistor approach while the second one is PF model and is based on Lemlich model. Both models are validated against our numerical database and literature data and are found to predict accurate effective thermal conductivity values in the error range of ±6%. The final outcome of the analytical models is very effective. As both models are independent, they could be used simultaneously to predict both, intrinsic solid phase thermal conductivity (ls) and effective thermal conductivity (leff) of open cell foams. For a known foam geometry and fluid phase conductivity, this fact is extremely useful in regards with all engineering applications as intrinsic solid phase conductivity is usually unknown. It should be noted that experimental and numerical data for the effective thermal conductivity of metal foams for variable cross section of the ligament having wide range of porosity are still unavailable to our knowledge. However, further experimental data of different variable strut cross sections would be welcome to give better support to the proposed model, and this will be the subject of future article.

lparallel

effective thermal conductivity (by Krischer approach model, Eq. (4)), W m1 K1 effective parallel thermal conductivity (Eq. (1)), W m1 K1 effective series thermal conductivity (Eq. (2)), W m1 K1 heat flux, W m2 dimensionless geometrical parameter (Eqs. (20) and (21)), e constant (Eq. (24)), e

lseries F j h

Abbreviation BCC body centered cubic cell LTE local thermal equilibrium Appendix A. Solid phase porosity of different strut shapes For a circular strut shape, Req ¼ Rc

 εs ¼

1 3

4 36pR2c Ls þ 24: pR3c 3 pffiffiffi 3 8 2L

 ¼ 12pa2c b þ

32 3 pac 3

where, ac ¼ Rc/L and b ¼ Ls/L pffiffiffi For a square strut shape, Req ¼ As = p

 1 3

 4 .pffiffiffi p 36A2s Ls þ 24: A3s pffiffiffia3 12a2s b þ 332 3 p s pffiffiffi pffiffiffi ¼ 8 2L3 8 2

Acknowledgements

εs ¼

The authors would like to thank the ANR (Agence Nationale de la Recherche) for financial support in the framework of FOAM project and all project partners for their assistance.

where, as ¼ As/L and b ¼ Ls/L pffiffiffi For a rotated square strut shape, Req ¼ Ars = p



Nomenclature

εs ¼ Latin symbols dcell cell diameter, mm F correlation factor (Eq. (15)), e Ls strut length, mm L node to node length, mm P power, W Req equivalent radius, mm S constant (Eq. (23)), e T temperature,  C V〈T〉 temperature gradient,  C m1 Vf total fluid inside the octahedron, mm3 Vs total solid volume of the octahedron, mm3 Vligament volume of one ligament, mm3 Vnode volume of one node, mm3 VT volume of octahedron, mm3 Greek symbols εs solid phase porosity, e ε fluid (or open) porosity, e aeq ratio of equivalent radius to node to node length, e b ratio of strut length to node to node length, e ls intrinsic solid phase conductivity, W m1 K1 lf fluid phase conductivity, W m1 K1 leff effective thermal conductivity, W m1 K1 lEMT eff

effective thermal conductivity (by EMT model, Eq. (3)), W m1 K1

(A.1)

1 3

 4 .pffiffiffi p 36A2rs Ls þ 24: A3rs pffiffiffia3 12a2rs b þ 332 3 p rs pffiffiffi pffiffiffi ¼ 3 8 2L 8 2

(A.2)

(A.3)

where, ars ¼ Ars/L and b ¼ Ls/L qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi For a diamond strut shape, Req ¼ Adet : 3=2p

0

1 qffiffiffiffiffi pffiffiffiffi 4 pffiffi3ffi 3 3 þ 24: $ 2 $ 2p Adet A 3 pffiffiffi εs ¼ 3 8 2L qffiffiffiffiffi pffiffiffiffi pffiffiffi 2 16 6 3adet b þ pffiffiffi 2p3a3det pffiffiffi3 ¼ 8 2 pffiffiffi 1 @36 3A2 L 3 2 det s

(A.4)

where, adet ¼ Adet/L and b ¼ Ls/L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi For a hexagon strut shape, Req ¼ Ah : 3 3=2p

0

1 qffiffiffiffiffiffiffi pffiffiffiffi 4 3pffiffi3ffi 3 3 3 þ 24: $ 2 $ 2p Ah A 3 pffiffiffi εs ¼ 8 2L3 pffiffiffiffi pffiffiffi pffiffiffiqffiffiffiffiffiffiffi 18 3a2h b þ 16 3 32p3a3h pffiffiffi ¼ 8 2 pffiffiffi 1 @36 3 3A2 L 3 2 h s

where, ah ¼ Ah/L and b ¼ Ls/L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi For a rotated hexagon strut shape, Req ¼ Arh : 3 3=2p

(A.5)

P. Kumar, F. Topin / Applied Thermal Engineering 71 (2014) 536e547

0

1

qffiffiffiffiffiffiffi pffiffiffiffi 4 pffiffiffi þ 24: $3 2 3$ 32p3A3rh A 3 pffiffiffi εs ¼ 8 2L3 pffiffiffiffi pffiffiffi pffiffiffiqffiffiffiffiffiffiffi 18 3a2rh b þ 4 3 32p3a3rh pffiffiffi ¼ 8 2 pffiffiffi 1 @36 3 3A2 L 3 2 rh s

(A.6)

where, ah ¼ Arh/L and b ¼ Ls/L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi For a star (regular hexagram) strut shape, Req ¼ Ast : 3 3=p

0

1 pffiffiffiffi pffiffiffi qffiffiffiffiffiffiffi pffiffiffi 4 1 @36 3A2 L þ 24: $3 3$ 3 3A3 A st s st p 3 3 pffiffiffi εs ¼ 8 2L3 pffiffiffiffi pffiffiffi pffiffiffiqffiffiffiffiffiffiffi 36 3a2st b þ 32 3 3 p 3a3st pffiffiffi ¼ 8 2

(A.7)

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