Chai, John

Gu, Z.L.

Jin, L.W.

Yang, H.

Zhao, M.

English

H. Yang

M. Zhao

Z.L. Gu

L.W. Jin

J.C. Chai

Crossref

International Journal of Heat and Mass Transfer

A further discussion on the effective thermal conductivity of metal foam: An improved model

University of Huddersfield Repository

University of Huddersfield RepositoryYang, H., Zhao, M., Gu, Z.L., Jin, L.W. and Chai, JohnA further discussion on the effective thermal conductivity of metal foam: An improved modelOriginal CitationYang, H., Zhao, M., Gu, Z.L., Jin, L.W. and Chai, John (2015) A further discussion on the effective thermal conductivity of metal foam: An improved model. International Journal of Heat and Mass Transfer, 86. pp. 207-211. ISSN 0017-9310 This version is available at http://eprints.hud.ac.uk/23831/The University Repository is a digital collection of the research output of theUniversity, available on Open Access. Copyright and Moral Rights for the itemson this site are retained by the individual author and/or other copyright owners.Users may access full items free of charge; copies of full text items generallycan be reproduced, displayed or performed and given to third parties in anyformat or medium for personal research or study, educational or not-for-profitpurposes without prior permission or charge, provided:• The authors, title and full bibliographic details is credited in any copy;• A hyperlink and/or URL is included for the original metadata page; and• The content is not changed in any way.For more information, including our policy and submission procedure, pleasecontact the Repository Team at: E.mailbox@hud.ac.uk.http://eprints.hud.ac.uk/- 1 -  A further discussion on the effective thermal conductivity of metal foam: an 1 improved model 2 H. Yanga, M. Zhaob, Z.L. Gub, L.W. Jinb*, J.C. Chaic 3 aSchool of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China 4 bSchool of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China 5 cDepartment of Engineering and Technology, School of Computing and Engineering, University of Huddersfield, 6 Queensgate, Huddersfield, HD1 3DH, U.K. 7 Abstract 8 In this study, we explain the causes and effects of the geometrical impossible result 9 encountered in the widely adopted tetrakaidecahedron model (Boomsma and Poulikakos, 2001; 10 Dai et al., 2010) for the effective thermal conductivities (ETCs) of metal foam. The geometrical 11 impossible result is successfully eliminated by accounting for the size variation of the node with 12 porosity. The improved model provides predictions of ETCs that are more precise than available 13 models. For aluminum foams ( s 218k  W -1 -1m K ) using water and air as fluid media, the relative 14 root-mean-square (RMS) deviation of the present predictions from the experimental data is about 15 5.3%; for the reticulated vitreous carbon (RVC) foams ( s 8.5k  W -1 -1m K ), the relative RMS 16 deviation is about 7.4%. 17 Key words: effective thermal conductivity; foam structure; node size; deviation; improved model. 18 Nomenclature 19 a  foam ligament radius (m) d  dimensionless foam ligament radius e  dimensionless cubic node length f  function k  thermal conductivity (W m-1 K-1) L  ligament length (m) R  thermal resistance (m2 K W−1) r  cubic node length (m) V  volume (m3) Subscripts 20 A unit cell layer B unit cell layer C unit cell layer D unit cell layer eff effective f fluid - 2 -  i A, B, C, D s solid Greek symbol 21   porosity 1. Introduction 22 High-porosity metal foams are promising materials for thermal management applications. 23 Since the effective thermal conductivity (ETC) is one of their most important thermal properties, 24 an accurate evaluation of it becomes especially important. Studies on modeling the ETC of metal 25 foams have been carried out numerically [1-3], experimentally [4-6] and analytically [5-12]. 26 Among these approaches, the analytical approaches are less time consuming but more universal, 27 and have attracted the attention of investigators. A review of the analytical approaches for 28 prediction of the ETC has been conducted by Coquard and Baillis [13, 14], and Randrianalisoa 29 and Baillis [15]. 30     One of the most widely used analytical approaches was developed by Boomsma and 31 Poulikakos [10] who first used the idealized three dimensional tetrakaidecahedron model to 32 predict the metal foam ETC. Predictions were reported to accurately match the experimental data. 33 However, Dai et al. [11] pointed out a few problems in their work. Dai et al. [11] extended the 34 model by accounting for the ligament orientation. Predictions of the extended model were 35 compared with the experimental data [5], and a relative RMS deviation of about 12% was 36 observed. The deviation indicated that there was still room for improvement. In addition, results 37 obtained in Ref. [11] showed that, as the porosity decreased, the diameter of the ligament became 38 longer than the length of the node, which leaded to a geometrical impossible result. The diameter 39 of the ligament should be shorter than the length of the node (see Fig. 1), which was a basic 40 assumption in the development of the model.  41 - 3 -      In this paper, the tetrakaidecahedron model originally proposed by Boomsma and Poulikakos 42 [10] and later extended by Dai et al. [11] is first discussed. The causes and effects of the 43 geometrical impossible results are examined and explained. The model is further improved by 44 accounting for the size variation of the nodes. We then show that the geometrical impossible 45 results are eliminated. Lastly, predictions of our improved model are compared with several other 46 analytical solutions as well as experimental data available in literature. It is shown that the current 47 model has a steadily high precision in predicting the ETC of high porosity foams with a wide 48 range of phase conductivity ratios ( s f/k k ). 49 2. Calculation of the Effective Thermal Conductivity  50     It is important to note that, for a better understanding of the present discussion, reader should 51 be familiar with the analytical approaches developed by Boomsma and Poulikakos [10], and Dai 52 et al. [11]. Therefore, in this part of the discussion, we give a brief review of how the ETC is 53 calculated using their approaches. For more detailed discussions, reader may refer to Refs. [10, 54 11]. 55 2.1. The Unit Cell of the Tetrakaidecahedron Model 56  Cubic nodes and cylindrical ligaments were used to represent the actual components of the 57 foam network, as is shown in Fig. 1(a). As the lump shape at the ligament intersection varies with 58 the foam porosity, the simplified spherical geometry is adopted for various porosities based on the 59 fact that the lump volume has more significant effect on ETC than its shape does. The length of 60 the node is r , the radius of the ligament is a  and its length is L  (from node center to node 61 center). Based on the symmetry of the idealized model and one-dimension heat conduction along 62 the z  axis, a representative unit cell which contains all geometrical characteristics of the 63 - 4 -  tetrakaidecahedron model was selected. The height of the unit cell in the z  direction is 64 2 / 2L .While the length of the other two sides in the x y  plane are both 2L . It can be 65 proved that the ETC of the selected unit cell is equal to that of the tetrakaidecahedron model.  66 2.2. Effective Thermal Conductivity 67  In order to calculate the ETC, the unit cell is divided into four distinctive vertical layers along 68 the z  axis, namely A, B, C and D, as is shown in Fig. 1(b). The heights of the four layers are: 69 AL a , B / 2L r a  , C 2 / 2L L r   and D / 2L r  . 70  According to the extend model (accounting for the ligament orientation) proposed by Dai et al. 71 [11], the thermal resistance of each layer is  72 A 2 2s f4[2 (1 )] {4 [2 (1 )]}dLRe d e k e d e k         (1a) 73 B 2 2s f( 2 )(2 )e d LRe k e k    (1b) 74 C 2 2s f2( 2 2 )2 2(2 2)e LRd k d k     (1c) 75 D 2 2s f2(4 )eLRe k e k    (1d) 76 where sk  is the thermal conductivity of the solid and fk  is thermal conductivities of the fluid, 77 d  and e  are non-dimensional parameters, defined as: /d a L  and /e r L . 78  The overall thermal conductivity is calculated by assuming that the thermal resistances of the 79 layers are connected in series. Based on the Fourier law of heat conduction, the ETC can be 80 written as 81 A B C DeffA B C DL L L LkR R R R       (2) 82  Eq. (2) can be written as 83 eff 1( , )k f d e  (3) 84 here, 1f  is a known function. 85 - 5 -   The porosity , which is defined as the ratio of the solid volume to the total volume can be 86 calculate based on d  and e  as 87 2 32 2 221 [ (1 ) ( ) (1 2) ]2 2 2 4d e ede e d e d e           (4) 88 Solving for d in Eq. (4) gives 89 1233 22(2 2 )4(3 2 2)ede e          (5) 90 Substituting Eq. (5) into Eq. (3) gives 91 eff 2( , )k f e  (6) 92 where 2f  is another known function. Here, once the value of e  is given, the ETC can be 93 calculated purely by porosity.   94 3. Improved Model 95    We present a discussion on the model of Dai et al. [11]; highlighting the possible area of 96 improvement. We then discuss the reason for the appearance of the geometrical impossible result. 97 Our proposed model is then presented. 98 3.1. Revisit Dai et al.’s Model 99  Precision: In order to use Eq. (6) to predict the ETC, the value of e  should first be calibrated. 100 According to Ref. [11], a value of 0.198e   was found to minimize the relative RMS deviation 101 of predictions from the experimental data [5]. This deviation is about 12%, which indicates that 102 there is still room for improvement. 103  The Geometrical Impossible Result: It should be true that r/a > 2 to ensure that the length of 104 the node is larger than the diameter of the ligament. However, as has been mentioned in Ref. [11], 105 this requirement can hardly be fulfilled with e = 0.198. The reason for the appearance of this 106 - 6 -  geometrical impossible result is explained next. 107 3.2. Causes and Effects of the geometrical impossible results 108  According to Eq. (4) or the structure of the tetrakaidecahedron model (see Figs. 1(a), 1(b)), a 109 decrease in porosity  , is attributed to an increase in d when e is held constant and vice-versa. 110 Here, d and e can be considered to represent the diameter of the ligament and the length of the 111 node. In the model of Boomsma and Poulikakos [10] and Dai et al. [11], the parameter e  was set 112 as a constant value. Therefore, according to Eq. (4), a decrease in porosity can only be realized by 113 an increase in the diameter of the ligament d. As a result, as the porosity decreases, the diameter of 114 the ligament increases while the length of the node remains constant, and eventually, the former 115 exceeds the latter, leading to geometrical impossible results. In fact, the smaller the value of e , 116 the more likely the geometrical impossible result occurs. As a result, geometrical impossible 117 results were encountered more frequently in Ref. [11] than in Ref. [10].  118 3.3. Current Model  119     We improve on the model of Dai et al. [10] and eliminate the geometrical impossible result 120 by accounting for the changing foam structure with porosity through the variable e. Since the 121 experimental data contains information of the foam structure, using the experimental data, we can 122 find how e  varies with porosity. For a given porosity, we calibrate the e  value by comparing 123 the predictions made by Eq. (6) against experimental data (Ref. [5]). As a result, values of e  for 124 ten given porosities are obtained, as is shown in Fig. 2. The parameter e can be fitted by a third 125 order polynomial function of the porosity  (Fig. 2) as 126 2 3e a b c d                                                            (7) 127 where a = 327.25811, b = -1075.55645, c = 1182.83207 and d = -434.55535. The fitting error is 128 - 7 -  less than 1%. 129 4. Model Validation 130  After obtaining the function between e and porosity, the ETC can be predicted as a function of 131 porosity. Substituting Eq. (7) into Eq. (6), gives: 132 eff ( )k f                                                                  (8) 133 where f is a known function. 134    When Eq. (8) is used to compute the ETCs for the experimental data in Ref. [5], the relative 135 RMS deviation is about 5.0% for water-saturated foams, and 5.6% for air-saturated foams. The 136 ratios of the node length to the ligament radius are also predicted using the current approach. As a 137 result, within the porosity of 0.905 to 0.978, the r/a ratios are always > 2.0 (decreasing from 6.33 138 to 2.71). Thus, the geometrical impossible results are eliminated. 139     Figure 3 shows comparisons of our model with selected models. As a result of the changing e 140 values with porosity, our model is capable of capturing the non-linear variations in the ETCs as 141 function of porosity. The relative RMS deviations from water and air experimental data of the 142 present, Yang’s, Dai’s and Paek’s predictions are 5.3%, 11.7%, 13.2% and 13.3% respectively. 143 Since the e  value is calibrated from the experimental data in Ref. [5], the high precision against 144 the experimental data in Ref. [5] is expected. To validate our model, we use Eq. (8) to compute the 145 ETCs for the experimental data reported by Phanikumar and Mahajan [16]. They reported the 146 ETCs of air-saturated Al foams with porosity ranging from 0.899 to 0.959. Due to a similar foam 147 geometry, the present model is capable of accurately predicting the variation trend of the ETCs 148 with porosity; with a relative RMS deviation of about 12.1%. 149     To further assess the validity of our improved model as well as the fitted e  value (Eq. 7). 150 - 8 -  We use our model to predict the ETCs of high porosity RVC foams (which have similar structure 151 with high porosity metal foams), and the results are compared with experimental measurements 152 reported in Ref. [6]. As a result, for water-saturated RVC foams, all the analytical models can 153 accurately predict the ETCs (RMS deviation < 10%). When it comes to air-saturated RVC foams, 154 Yang’s model and Peak’s model are less accurate with a RMS deviation of more than 21.8%. The 155 relative RMS deviations of the present model are relative small; 7.2% and 7.6% for 156 water-saturated and air-saturated RVC foams respectively. These results indicate that our 157 improved model have a wider range of applicability. 158 5. Conclusions    159     We accounted for the size variation of the node with porosity and successfully eliminate the 160 geometrical impossible results. The improved model provides more accurate predictions of the 161 ETCs. Comparisons with other analytical models as well as experimental data validates that our 162 model has a steadily high precision in predicting ETCs of foams with a wide range of solid phase 163 to liquid phase conductivity ratios ( s f/k k ). 164 Conflict of interest 165 None. 166 Acknowledgments 167 This work was supported by Science and Technology Project of Shaanxi Province 168 (2014JZ2-002). The authors would like to thank sincerely Prof T.J. Lu and Dr X.H. Yang for their 169 extensive and constructive suggestions on the model analysis.  170 References 171 [1] M.A. Mendes, S. Ray, D. Trimis, An improved model for the effective thermal conductivity of 172 open-cell porous foams, International Journal of Heat and Mass Transfer 75 (2014) 224-230. 173 - 9 -  [2] M.A. Mendes, S. Ray, D. Trimis, A simple and efficient method for the evaluation of effective 174 thermal conductivity of open-cell foam-like structures, International Journal of Heat and Mass 175 Transfer 66 (2013) 412-422. 176 [3] M. Wang, N. Pan, Modeling and prediction of the effective thermal conductivity of random 177 open-cell porous foams, International Journal of Heat and Mass Transfer 51 (5) (2008) 1325-1331. 178 [4] N. Dukhan, K.C. Chen, Heat transfer measurements in metal foam subjected to constant heat 179 flux, Experimental Thermal and Fluid Science 32 (2) (2007) 624-631. 180 [5] V.V. Calmidi, R.L. Mahajan, The effective thermal conductivity of high porosity fibrous metal 181 foams, Journal of Heat Transfer 121 (2) (1999) 466–471.  182 [6] A. Bhattacharya, V.V. Calmidi, R.L. Mahajan, Thermophysical properties of high porosity 183 metal foams, International Journal of Heat and Mass Transfer 45 (5) (2002) 1017-1031. 184 [7] J.W. Paek, B.H. Kang, S.Y. Kim, J.M. Hyun, Effective Thermal Conductivity and Permeability 185 of Aluminum Foam Materials, International Journal of Thermophysics 21 (2) (2000) 453-464. 186 [8] X. Fu, R. Viskanta, J.P. Gore, Prediction of effective thermal conductivity of cellular 187 ceramics, International Communications in Heat and Mass Transfer 25 (2) (1998) 151-160. 188 [9] C.Y. Wang, C. Beckermann, A two-phase mixture model of liquid-gas flow and heat transfer in 189 capillary porous media-I. Formulation, International Journal of Heat and Mass Transfer 36 (1993) 190 2747-2747. 191 [10] K. Boomsma, D. Poulikakos, On the effective thermal conductivity of a three-dimensionally 192 structured fluid-saturated metal foam, International Journal of Heat and Mass Transfer 44 (4) 193 (2001) 827-836. 194 [11] Z. Dai, K. Nawaz, Y.G. Park, J. Bock, A.M. Jacobi, Correcting and extending the 195 Boomsma–Poulikakos effective thermal conductivity model for three-dimensional, fluid-saturated 196 metal foams, International Communications in Heat and Mass Transfer 37 (6) (2010) 575-580. 197 [12] X.H. Yang, J.X. Bai, H.B. Yan, J.J. Kuang, T.J. Lu, T. Kim, An analytical unit cell model for 198 the effective thermal conductivity of high porosity open-cell metal foams, Transport in Porous 199 Media 102 (3) (2014) 403-426. 200 [13] R. Coquard, D. Baillis, Numerical investigation of conductive heat transfer in high-porosity 201 foams, Acta Materialia 57 (18) (2009) 5466-5479. 202 [14] R. Coquard, D. Rochais, D. Baillis, Conductive and radiative heat transfer in ceramic and 203 metal foams at fire temperatures, Fire Technology 48 (3) (2012) 699-732. 204 [15] J. Randrianalisoa, D. Baillis, Thermal conductive and radiative properties of solid foams: 205 Traditional and recent advanced modelling approaches, Comptes Rendus Physique, 2014, in press. 206 [16] M.S. Phanikumar, R. L. Mahajan, Non-Darcy natural convection in high porosity metal 207 foams, International Journal of Heat and Mass Transfer 45 (18) (2002) 3781-3793. 208  209 - 10 -   210 Fig. 1 (a) The tetrakaidecahedron model and (b) four distinctive layers for the unit cell 211  212  213  214  215 - 11 -   216 Fig. 2 Calibrated e values 217  218  219  220  221  222  223  224  225  226  227  228  229  230 - 12 -   231 Fig. 3 Comparisons between the analytical models and the experimental data of (a) air-aluminum 232 and (b) water-aluminum 233 

A simple and efficient method for the evaluation of effective 174 thermal conductivity of open-cell foam-like structures,

A two-phase mixture model of liquid-gas flow and heat transfer in 189 capillary porous media-I.

An analytical unit cell model for 198 the effective thermal conductivity of high porosity open-cell metal foams,

An improved model for the effective thermal conductivity of 172 open-cell porous foams,

Conductive and radiative heat transfer in ceramic and 203 metal foams at fire temperatures,

Correcting and extending the 195 Boomsma–Poulikakos effective thermal conductivity model for three-dimensional, fluid-saturated 196 metal foams,

Effective Thermal Conductivity and Permeability 185 of Aluminum Foam Materials,

Heat transfer measurements in metal foam subjected to constant heat 179 flux,

Modeling and prediction of the effective thermal conductivity of random 177 open-cell porous foams,

Non-Darcy natural convection in high porosity metal 207 foams,

Numerical investigation of conductive heat transfer in high-porosity 201 foams,

On the effective thermal conductivity of a three-dimensionally 192 structured fluid-saturated metal foam,

Prediction of effective thermal conductivity of cellular 187 ceramics,

The effective thermal conductivity of high porosity fibrous metal 181 foams,

Thermal conductive and radiative properties of solid foams: 205 Traditional and recent advanced modelling approaches, Comptes Rendus Physique,

Thermophysical properties of high porosity 183 metal foams,

In this study, we explain the causes and effects of the geometrical impossible result encountered in the widely adopted tetrakaidecahedron model (Boomsma and Poulikakos, 2001; Dai et al., 2010) for the effective thermal conductivities (ETCs) of metal foam. The geometrical impossible result is successfully eliminated by accounting for the size variation of the node with porosity. The improved model provides predictions of ETCs that are more precise than available models. For aluminum foams (ks=218Wm-1K-1) using water and air as fluid media, the relative root-mean-square (RMS) deviation of the present predictions from the experimental data is about 5.3%; for the reticulated vitreous carbon (RVC) foams (ks=8.5Wm-1K-1), the relative RMS deviation is about 7.4%.</p

Yang, Hui Yan

Gu, Z. L.

Jin, L. W.

Chai, J. C.

Huddersfield Research Portal

A further discussion on the effective thermal conductivity of metal foam:An improved model

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A further discussion on the effective thermal conductivity of metal foam: An improved model

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