Issue 
Mechanics & Industry
Volume 21, Number 4, 2020



Article Number  410  
Number of page(s)  31  
DOI  https://doi.org/10.1051/meca/2020028  
Published online  15 June 2020 
Regular Article
Effective factors on thermal conductivity of stochastic structures open cell metal foams
Mechanical Engineering Department, Hormozgan University, Bandar Abbas, Iran
^{*} email: miladsaljooghi10@gmail.com
Received:
7
May
2019
Accepted:
12
March
2020
Effective thermal conductivity (ETC) is a considerable thermophysical property in design, manufacturing, and usage of multifunctional equipment that benefit cellular structures such as opencell metal foams. An accurate understanding of key parameters effecting on ETC is classified by: Analytical, Numerical and Experimental approaches. In this study, a comprehensive investigation based on mentioned approaches is presented and a comparison between various factors affecting ETC including porosity, pore size, temperature, pressure and shape factor is made. Porosity and pore size, as main morphological features of opencell metal foams, indicate structural characterization of them. Increase of porosity and pore size resulted decrease of ETC. The temperature effects on ETC in case of temperatures lower than 250 °C is ignorable although for temperature higher than 500 °C with change of heat transfer mechanism temperature plays a primary role in determining ETC. Few studies have been made on pressure parameter that illustrated its effect on ETC is insignificant. Multiple manufacturing methods produce different topological structures so; the influence of shape factor on ETC requires more efforts to reach a better understanding. Finally, applicable techniques for measuring ETC are briefly discussed.
Key words: Effective thermal conductivity / open cell / metal foam / pore size / porosity / shape factor
© AFM, EDP Sciences 2020
1 Introduction
In the last decade of the 20th century, miniaturization of hightech industries, creating tools and equipment required multifunctional materials. This revolutionary concept is called Multifunctional materials or structures. Engineering sciences introduced a new class of materials that have different properties from their shows. Significant efforts in incorporating the Multifunctional structures concept into materials engineering have led to an entirely new category of materials. Cellular solids are a group of materials with multifunctional attributes, which have tailorable structures to achieve systemlevel performance as materials that combine mechanical, thermal, electrical, acoustical and possibly other functionalities. Multifunctionality of cellular solids is an interdisciplinary research area that requires a concurrentengineering approach. One critical application area is ultralight multifunctional heat exchangers or heat sinks in integrated circuit where cellular solids give off an impression of being more appealing than the ordinary heat dissipation media. As the heat dissipation material, it also required to support large structural loads. In this manner, exceptionally viable and robust thermal management via these cellular solids is crucial. With the prerequisites on the capability of carrying both mechanical and thermal loads in mind, the challenges are to establish relationships between topology and properties to optimize the geometric parameters applicable to various thermomechanical applications. Beginning with the spearheading work of Maxwell (1891), heat conduction in completely immersed porous material (e.g., sand, packed beds of cylinders and spheres, fibrous insulations, etc.) has been considered in detail in recent decades.(1)
Figure 1 demonstrates a photo of the metal foam medium. It has an opencelled structure made out of dodecahedronlike cells, which have 12–14 pentagonal or hexagonal countenances. The edges of these cells are made out of the strands, commonly, there is a lumping of material (intersection) at focuses where the strands converge [1,2]. A further issue with these materials is that the repeatability of the morphology is not consistent, notwithstanding when similar assembling conditions are utilized, bringing about an intrinsic disperses in the material properties unless substantial examples are tried [3–5] (Fig. 2).
“When modern man builds large load bearing structures, he uses dense solids: steel, concrete, glass. When nature does the same, she generally uses cellular materials: cork, wood, coral. There must be good reason for it.”
Michael F. Ashby
As a matter of fact, they are particularly interesting for applications requiring multifunctionality. In most of the new fields of applications of metal foams, the knowledge of the thermal transport properties is of essential significance for the dimensioning of the structures. To evaluate the extent of heat conduction, one generally uses the ETC (keff) which implies that the thermal behavior of twophase materials can be wellmatched by a homogeneous conductive medium. The conductive heat transfer is then governed by the simple diffusion equation:
The structure of a porous material is very complex, consisting of different pore sizes and shape factors [6]. A detailed prediction of the ETC of heterogeneous media requires knowledge of the topology, size, location (distribution) and conductivity of each phase in the system together with interaction between particles [7].
Fig. 1 Open cell metal foam medium made. 
Fig. 2 Stochastic structures [Foam as seen through the camera lens of art photographer Michael Boran]. 
2 Effective parameters in the thermal conductivity open cell metal foams
Thermal conductivity is an intrinsic property of material, which has been influenced by numerous parameters. With the technology advancement, use of composite materials (composed of one or more substances) with cellular structures requires knowledge of their thermosphysical properties. Heat transfer in opencell metal foams is complex as it takes place in two phases. There is a network of solid ligaments of universally high thermal conductivity and a fluid with lower thermal conductivity [8,9]. In such circumstances, the ETC is no longer a property of a single material but depends on both the solid and fluid material properties, e.g. temperature, pressure, base material and also the topological structure of opencell metal foams; e.g. its porosity, pore size and shape factor.
2.1 Pore size
Number of pores per inch (PPI) or pore per millimeter or centimeter (PPMPPC) is a geometric feature of opencell metal foams that indicate number of pores on a porous surface. However, numbering technique is so elementary and not accurate but it is one of standardization methods among manufacturers and researchers. As more number of pores exist on a surface, smaller cell diameter resulted. Commercial range of pore size commonly varies from 5 PPI to 60 PPI. It is worthy to mention that increasing pore size is followed by decrease of ligament diameter but no significant change in conjunction angle is observed. For heat conduction in high porosity opencell metal foams, the experimental results on the influence of PPI upon effective thermal conductivity are analyzed below (Figs. 3,4).
Wang et al. [10] numerically analyzed ETC of open cell metal foam by pore scale approaches. They found pore size has no obvious influence on the ETC.
Yang et al. [11] studied the ETCs of aluminum foams with different porosities and pore sizes under various conditions. Pore size (PPI) is found to have little influence upon the ETC of the opencell metal foams.
Fig. 3 Effect of pore size on the ETC. 
Fig. 4 Effect of pore size on the ETC. 
2.2 Porosity
Based on etymology, porosity stems from Greek language and word “pore” meaning “passage” so things with porosity allow something through. Porosity includes different types but generally, it is ratio of void volume to total volume.(2)
Generally, for opencell metal foams, porosity varies from 50% to 98%. It is notable that porosity is a morphological property of opencell metal foams that has considerable impact on its structure. Because of wide range of porosity of metal foams in thermal and hydrodynamic application researcher's focus were attracted to it (Fig. 5).
The ETC has been observed to be exceedingly sensitive to the porosity, increasing as the porosity decreases [12–15] (Fig. 6).(3)
Fig. 5 Porosity of silver foam. 
Fig. 6 ETC of opencell metal foams decreasing with increasing porosity. 
2.3 Temperature and pressure
Thermal conductivity is one of critical material properties that defines capability of a material in case of heat transfer. Conductive heat transfer is expressed by Fourier law:
Heat transfer is categorized into various mechanisms: conduction, convection and radiation so, the total material thermal conductivity can be presented as equation (4) (4)
Process of heat transfer conduction is vibration energy by phonons and electrons in solid, liquid and gas medium and convection, radiation is transport of mass and heat electromagnetic energy by photons. Based on Figure 7, total conductivity is dependent on temperature. For temperature under 100 °C conduction is dominant mechanism. Zhao et al. [17] experimentally investigated metal foam ETC under vacuum and atmosphere condition in range of 300K–800K. The results showed that the ETC increased quickly as temperature raised. Also, the ETC in 800 K was reported to be 3 times higher than ETC in ambient temperature (300 K). They proposed an empirical correlation for the effective radiative conductivity as function of temperature.(5)
Sauerhering et al. [18] investigated influence of temperature on ETC of Inconel foams experimentally. They formulated equation (6) (6)
Porosity of samples varied from 55% to 85% and temperature range varies from ambient to 700 °C. Figures 8 and 9 showed ETC variation with temperature and porosity. Using Transient Plane Source Technique (or so called Hot Disk method) in case of temperature above 100 °C ETC significantly increased and above 300 °C radiation and convection were dominant heat transfer mechanisms.
Brendelberger et al. [19] depicted the temperature dependency of the ETC of the two samples of metal foams (Fig. 10). The inherent thermal conductivity of the fluid phase changes by less than 0.2% in the examined pressure range (1 bar–17 bar).
Also for the porous solid network, no influence of the pressure on the ETC in the examined pressure range is expected [20]. Therefore, the dependency of the ETC on the pressure is assumed to be trivial [19]. More studies in field of pressure effects on thermal conductivity of opencell metal foams need to be done. Moreever its role in determining ETC is not significant.
Fig. 10 ETC dependency on temperature. 
Fig. 11 Thermal conductivity of some known metals. 
2.4 Base material
A large number of studies demonstrated that base material is an important parameter that affects the ETCs of opencell metal foams. Aluminum, copper, nickel, iron and other metals and alloys are applied in manufacturing opencell metal foams. According to conventional thermal conductivity models like parallel and series, one may observe base material is important factor in calculation ETC. Higher the solid thermal conductivity, higher ETC is (Fig. 11).
2.5 Shape factor
Repeatability type is one of recognition features of opencell metal foams and their shape factor determination. Shape factor is dependent on various parameters including: ligament length, ligament conjunction angels, ligament cross sectional area, ligament diameter, node shape, node diameter, cell diameter, shape of faces (tetrahedral, pentagonal, hexagonal and so on) and etc. Researchers have investigated mentioned parameters and how they may affect ETC to reach an optimize geometry with highest ETC. Generally, increasing the number of pores per inch, decreases the ligament effective diameter. Also, decrease in porosity changed the shape of ligaments from triangular to circular. Many studies have been done conforming that a triangular cross section improved heat transfer for a given pore size.
Markos et al. [21] experimentally studied cross section shape factor and hydraulical parameters of opencell metal foams by various pore sizes and porosities. They found that by increasing the drag and velocity of flowing through the foam, cross section shape because of increasing porosity deformed from convex ligament to concave ligament. (Fig. 12).
Ozmat et al. [22] estimated an average measure of the ligament dimension of opencell metal foams. Figure 12 showed ligament cross section shapes by optical microscopic image. They calculated and measured value of ligament dimension by equation below and the results are available in Table 1.(7)
where μ_{ e } is the number of edges dodecahedron
μ_{ c } is the number of corners of dodecahedron
is the cross sectional area of a ligament
λ_{1}: a linear measure of the nodes
λ_{2}: nodal volume shape factor
ρ: density
S: area
V: volume
According to photographs they observed triangular cross section shape for pore size ranging from 10 PPI to 30 PPI and porosity 80%.
Bodla et al. [23] numerically analyzed opencell metal foam topology such as effective pore diameter and sphericity, ligament length, ligament area of cross section and node coordination number based on Xray micro tomography and resistance network model (Fig. 13).
Hutter et al. [24] numerically investigated effect of ligament shape pressure drop and turbulent kinetic energy. The results demonstrated the change of ligament shape from sharp edges to rounded edges caused pressure drop reduction to almost 30%.
Yang et al. [25] numerically investigated the effect of ratio of node length to ligament thickness in opencell metal foams with node scale approach. Results showed increasing the ratio by 4 times caused effective thermal conductivity reduction by 59%, also permeability and volumeaverage Nusslet number respectively increased by 16% and 27.6%.
Bock and Jacobi [26] analyzed Al opencell metal foams structures by tomography method in order to earn structuredcombinatorial data and characterization. Comprehensively compared ligament length, ligament diameter, ligament density, ligament cross sectional area, pore and face shape of real structure with existing models (WeairePhelan, Kelvin cell, P42a [27]). According to porosity range of used metal foam most of the cross sectional areas are triangular rather than circular.
Fig. 12 Illustrated different cross section shape of ligaments. 
3 Theoretical and experimental approaches for characterization thermal conductivity
Transport phenomena in opencell metal foams have been a requirement of large number of researches because of its different thermal and hydraulic properties. A wide range of analytical, numerical and experimental efforts to predict ETC of opencell metal foams were reviewed in the literature. Generally, applied methodology in studies is classified as categories below: (1) Asymptotic and Analytical Analysis, (2) Pore scale (Unit cell) approach, (3) Micro tomographyCFD Modeling, (4) Empirical correlation.
3.1 Asymptotic and analytical approach
Numerous efforts have been made by mathematicians and physicists using theoretical methodology to understand the nature of heat transfer features of composite structure materials. Upper and lower bounds were basic asymptotic analyses then many researchers with different approaches and sights upgraded them. Whether thermal resistances of the solid and fluid phases considered Series or Parallel, lower and upper bounds would be made respectively. MaxwellEucken using upper and lower bounds modified parallel and series equations as random distribution of small continues and discontinues solid spheres into fluid. Many researchers including Hamilton Crosser [28], HashinShtrikman [29], Hadley [30] and etc. developed correlations taking into account shape and other factors. Opencell metal foams are composed of two phase systems i.e. a solid structure and fluid. The ETC implies heat transfer through this composite system [31]. Figure 14 illustrated comparison of the ETC predicted by five basic structural models used in opencell metal foams, including the Series and Parallel models [20], Maxwell–Eucken models (two forms) [28] and Effective Medium Theory (EMT) equation [29]. According to Figure 14 there are large differences among calculated ETCs using above models, so it is clear that the thermal conductivity of opencell metal foams is influenced by other factors than porosity, like microstructure. Hence, selecting a proper model to predict the thermal conductivity of different types of opencell metal foams is challenging.
Boomsma and Poulikakos [32] derived a correlation to estimate ETC of opencell metal foams saturated by fluid based on a cellular structure composed of cells with equal sizes. In their research, term “e” which define dimensionless length of cubic node were calibrated by comparing calculations with existed experimental results of Calmidi and Mahajan [33] (in order to ensure reasonable ratio of node to ligament size). Finally, good agreement was reached comparing experimental results and analytical predictions of their model and it was shown for ε < 0.8 their model predicts: (a) lower value of ETC when and (b) higher value of ETC when than phasesymmetry model. Later, Dai et al. [34] tried to correct some errors in Boomsma and Poulikakos correlations but, results based on analytical predictions were still far from experimental results and reason was hidden in heat transfer direction assumption. When solid phase thermal conductivity is relatively much higher than fluid phase, in solid phase basically heat transfers through ligaments and ligaments orientation is not particularly parallel or perpendicular to heat transfer in fluid phase so by extending Boomsma and Poulikakos model into next level they proposed correlation considering ligaments orientation.
Zehner and Schlunder [35] considering an eighth of a cylinder as representative unit cell and assuming parallel path of heat transfer through fluid and solid phases, derived a correlation to estimate ETC which in addition of porosity, were function of shape factor too. They validated their correlation by comparing results with experimental data for case of porosity of 0.42 and establishing good agreement between them but, Kaviany [1] illustrated this correlation underestimated value of ETC in cases of . Hsu et al. [36] introduced two models by developing Zehner and Schlunder's model to overcome disagreement found by Kaviany. In first one (called AreaContact model) instead of point contact they consider area contact assumption between spheres and declared term “α” which indicates deformation. Second model (called PhaseSymmetry model) was provided for sponge like mediums with continuous phases but they could not determine second model accuracy because of lack of experimental data at the time. Behrang et al. [37] studied ETC of porous media saturated with two immiscible fluids. An analytical correlation was developed based on a hybrid approach deriving from Maxwell, Parallel and Series models. Using phonon theory, the model was extended to calculate heat transfer in nanoscale and based on it they figured out increase of surface roughness cause reduction of ETC. Kou et al. [38] focused on effective fractal parameters of porous materials and how they may affect dimension less ETC . They formulated a correlation to predict ETC of unsaturated porous media using thermalelectrical and selfsimilarity method under the assumption of onedimensional heat transfer. According to results, higher value of fractal dimension for tortuosity and pore diameter, porosity and lower value of saturation degree is followed by lower ETC. This approach led to a correlation consisting no empirical parameter.
Kumar and Topin [39] studied tetrakaidecahedron structures with different ligament cross section shapes including circle, square, hexagon, diamond and star. They investigated relation between geometrical parameters and thermal features of metal foams. Studied cases included porosity range from 0.6 to 0.95 and solid to fluid thermal conductivity ratio varying from 10 to 30 000. Finally, two models for estimating ETC were proposed. They introduced equivalent radius “R_{eq}” which is radius of a circle with area equal to ligament cross section area, to simplify and unite the model for all cases of ligament shapes. Main assumption in this model was based on R_{eq} and implied that for given equivalent radius, node volume is same and independent from ligament shape. T. Uhlířová and W. Pabst [40] analytically and numerically investigated relative thermal conductivity of porous material with cubic cells and compared quasilaminate solution with the upper HashinShtrikman bounds. Progelhof et al. [41] comprehensively investigated analytical models for effective thermal conductivity of porous media.
3.2 Pore scale (unit cell) approach
Since long time ago, filling space by unit volume cells having least surface area and highest adaptability with real structure of foams has led to new approach in simplifying metal foams structure. Lord Kelvin as one of wellknown precursors in 19th century proposed tetrakaidecahedron cell consisting of 14 faces (6 quadrilaterals and 8 hexagonals) and 24 vertices as a polyhedron conforming Plateau's laws and Euler principals (Fig. 15).
Gabbrielli [42] found out Swift–Hohenberg equation can be a proper candidate to Kelvin problem solution (8)
where r is real bifurcation parameter, and N(u) is some smooth nonlinearity. For almost a century Kelvin cell was known as applicable structure till discovery of the Weaire–Phelan model.
Weaire–Phelan model was proposed in 1993 in order to solve kelvin problem with this difference that it consisted of two types of cells with equal volume. As it is shown in Figure 16, first type is a 12facedpolyhedron with nonplan pentagonal faces and second type is a 14faced polyhedral with 2 flat hexagonal faces and 12 nonplans pentagonal faces. Two cells of first kind and six cells of second kind composed WeairePhelan packed model [43].
Haghighi and Kasiri [44] tried to estimate ETC focusing on geometrical characteristic of opencell metal foams using unit cell approach. They studied a nonisotropic tetrakaidecahedron unit cell as representative unit cell by dividing and repeating it in y and z directions. They found out with change of dimensionless height or radius of ligaments, dimensionless diameter of nodes and angle of face orientations it is possible to increase ETC to 33%. Edouard [45] focused on establishing an ideal periodic unit cell model and proposed a model consisting 12 cylindrical ligaments and 8 cubes. Each ligament divided between four cubic lattices and each cube divided between eight other cubes, finally he proposed a correlation capable of determining minimum and maximum values of ETC which later Zenner and Edouard [46] revised and derived an improved correlation. Huu et al. [47] investigated opencell metal foams using pore scale method. They came up with pentagonal dodecahedron model, which was based on material accumulation on nodes and ligaments. It was divided into two categories of flat and slim. Finally, results were compared to experimental data obtained from testing metal foams with porosities from 0.75 to 0.98 and good compromise was found. Researchers [48–50] used Cubic lattice cell model to investigate different features of opencell metal foams such as heat transfer. In this model, cylindrical ligaments construct cubic and it was assumed that cell size is equal to cubic lattices. Kumar and Topin [39] investigated ETC of opencell metal foams by unit cell approach. They proposed a model derivation, which were inspired by Singh and Kasana [51], they employed resistor model and applied it on unit cell. For another time importance of ligaments and heat flow orientation was highlighted and were taken into account by defining parameter “F”.
Fig. 13 (a) Distribution of ligament length Al foam with porosity 91%, (b) distribution of ligament area Al foam with porosity 91%, (c) distribution of node coordination number Al foam with porosity 91%, (d) distribution of effective pore diameter Al foam with porosity 91% [23]. 
Fig. 14 Predicted ETC by five different models . 
3.3 MicrotomographyCFD modeling approach
Modeling real structure of cellular materials as it was mentioned before has long history of challenging researchers' minds. In 20th century with technological revolution, Computed Tomography (CT) scan using computer process and Xray measurement from different angles provided a nondestructive approach and realistic 3D virtual image of an object without cutting. Lately, realistic 3D modeling using µCT scan imaging has been an innovative approach to understand and analyze metal foam structure. However, µCT scan is more expensive method, but it provides a very similar geometry to real structure that captures irregular and randomized cells. Number of researchers using this approach determined ETC of opencell metal foams. Mendes et al. [52] numerically evaluated ETC of opencell metal foams using µCT scan images to reconstruct precise 3D morphology and results showed a good validation between numerical and experimental data. Wulf et al. [53] experimentally and numerically determined ETC of opencell metal foams for various samples. They found good agreement between three dimensional LatticeBoltzmann method results obtained from real foam structure modeling based on 3D CTscan imaging and experimental data. Ranut et al. [54] simulated 3D structure of the foams by means of high resolution Xray micro tomography. The results of CFD analyze illustrated ETC is pertinent to morphological features. Bodla et al. [55] numerically studied a 3D complex geometrical nature of metal foams applying µCT scan to investigate thermal behavior. They found porosity as an important factor in determining ETC (Fig. 17).
AlAthel [56] summarized 3D realistic structure of metal foams by µCT scan method. As it is shown in Figure 18, after choosing suitable sample and taking photos from 0.7degree rotation each produced raw model and then by eliminating errors a clean RAW geometry was produced and finally using Solid work software he analyzed metal foam structure.
Amani et al. [57] experimentally obtained realistic 3D stochastic opencell metal foams structure by Xray tomography and then numerically calculated ETC using finite element method (Fig. 19).
However, benefits of µCT scan method and simulation approach is considerable unfortunately not many studies have been done and requires more efforts to improve simplicity and performance of this method. More studies are needed to improve this approach as a cheap, accessible and fast method to produce a fine 3D geometry image of opencell metal foams.
Fig. 15 Unit cell of Kelvin. 
Fig. 16 WeairePhelan cell: (left): type (a), (middle): type (b), (right) Orientation of WeairePhelan bubbles. 
3.4 Empirical correlation approach
Empirical method is an experimental process that results formula or curve fitting supported by measured data. In order to describe effective parameters of metal foam thermal conductivity an appropriate analysis can be made using experimental data. Calmidi and Mahajan [33] proposed an empirical correlation based on experimental data. They found thermal conductivity as function of k_{f} , k_{s} and ε. Bhattacharya et al. [13] comprehensively investigated thermophysical properties of various samples of opencell metal foams. They found ETC is affected by porosity and cross section ratio and intersection. Finally, a correlation was derived based on measured data from experimental analysis. Singh and Kasana [51], Chaudhary and Bhandari [58] established a correlation based on experimental data to estimate ETC easily. Heat transfer direction may be parallel or perpendicular to matrix orientation, so they introduce term F and weight mean correlation. Term F indicated parallel formation and 1F indicated perpendicular formation. Finally, they showed F rises linearly with increase of meaning most of heat transfer is parallel. Hamilton and Crosser [59] proposed an empirical correlation to predict ETC focusing on shape of particles by introducing term n as shape factor parameter. This factor indicated sphericity of particles that is ratio of surface area of particle to surface area of a sphere with same volume. Coquard et al. [60] investigated opencell metal foams in case of fire barrier applications. Radiative heat transfer is not negligible in high temperatures (>1000 °C) so they developed a correlation coupling conductive and radiative heat transfer mechanism based on results of numerical simulations and taking into account physic of structure. Fourie and Du Plessis [61] studied samples with porosity ranging from 0.8 to 0.98 and fluid to solid thermal conductivity ratio between 0.0001 and 0.1 in case of thermal nonequilibrium state. They found temperature distribution by solving 3D conduction equation and results showed insensitivity of ETC to pore size was in agreement with Takegoshi [62] and Peak [3] studies. Based on established data they proposed 4 empirical correlations as function of coupled thermal conductivities and porosity. Krisher [62] belived that since ETC of any sample is limited by values obtained from Parallel and Series models so mechanism structure should be a combination of these two models and based on this idea he proposed correlation introducing an empirical parameter showing share of parallel and perpendicular direction of heat flow to solid orientation. Sadeghi et al. [11] investigated ETCs of opencell metal foams. They proposed a new equation based on experimental data.
4 Measurement methods of thermal conductivities of metal foams
The ETC of porous structures is one of the important thermophysical parameters that affect heat transfer. Due to large number of analytical and numerical studies made on predicting the ETC an experimental validation is needed to evaluate their correctness. Plenty of researchers have reported various measurement techniques for studying apparent ETC of opencell metal foams.
The measurement techniques are generally divided into two types: steady state and transient techniques. Each method has its advantages and disadvantages: transient techniques are based on instantaneous measurement of temperature locally and in short time. However, experiment setup is easy to fabricate but data analysis, accuracy and uncertainty analysis require appropriate mathematical tools. In contrast to transient method, in case of steadystate measurement techniques data analysis is not much of challenge but collecting heat transfer data is a long time process in order to be sure of existence of steady state condition. The steady state experiment setup is a complex task and requires a wellengineered process and finally for porous structures ensuring 1D heat transfer, good contact between sample and heat source and accurate measurement could be additional problem. The steady state and transient methods mainly included guarded hot plate, comparative longitudinal, heat flow meter, laser flash, hot wire, hot disk method, temperature wave analysis and etc. Table 2 briefly summarized ETC measurement techniques.
Peak et al. [3] studied thermophysical properties of Al foams. Employing steady measurement technique while considering onedimensional heat conduction they determined effect of porosity and cell size on thermal conductivity and permeability. Results demonstrated that decrease of porosity was followed by increase of thermal conductivity but cell size changes with constant porosity simply did not affect it.
Calmidi and Mahajan [33] used similar technique to investigate ETC. They assumed onedimensional heat conduction in their theoretical studies and developed this assumption to experiments by heating sample from above and cooling it from bottom. Finally based on their experimental data an empirical correlation was derived. Other steadystate methods including comparative longitudinal method and panel test have been used by [17,64].
Measurement techniques and their field of application.
5 Conclusions
Effects of different parameters (temperature, pressure, pore size, porosity and shape factor) on ETC of opencell metal foams were discussed in this work. Additionally, analytical, numerical and experimental approaches were comprehensively reviewed; also, ETC measurement methods were summarized. Increase of pore size caused an ignorable decrease in ETC but decrease of porosity caused higher solid fraction and as a result higher ETC. Generally, higher temperature is followed by higher ETC meanwhile as long as temperature remains below 250 °C conduction would be dominant mechanism of heat transfer but while temperature raises to 500 °C convection and radiation are not ignorable anymore. Even though effect of pressure variation requires more studies but generally, its effect is not significant. Shape factor includes many parameters (such as ligament length, ligament conjunction angels, ligament cross sectional area, ligament diameter, node shape, node diameter, cell diameter and shape of faces (tetrahedral, pentagonal, hexagonal and so on)) even in same circumstances of manufacturing technique, morphological structure would be different and as a result it affects ETC. Analytical predictions showed good agreement with experimental data but absence of an universal model, which includes every effective parameter is noticeable. In last decades, numerical and experimental methods have had promising advancements but developing a cheap, fast and accurate model requires more efforts.
Author contribution statement
The manuscript was written by M.S. and advised by Y.B., S.N., J.K.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Appendix A
Summary of existing experimental studies on ETC of opencell metal foams.
Appendix B
Asymptotic models.
Analytical models.
Empirical correlations.
Unit cell models ETCs of opencell metal foams.
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Cite this article as: M. Saljooghi, A. Raisi, A. Farahbakhsh, Effective actors on thermal conductivity of stochastic structures open cell metal foams, Mechanics & Industry 21, 410 (2020)
All Tables
All Figures
Fig. 1 Open cell metal foam medium made. 

In the text 
Fig. 2 Stochastic structures [Foam as seen through the camera lens of art photographer Michael Boran]. 

In the text 
Fig. 3 Effect of pore size on the ETC. 

In the text 
Fig. 4 Effect of pore size on the ETC. 

In the text 
Fig. 5 Porosity of silver foam. 

In the text 
Fig. 6 ETC of opencell metal foams decreasing with increasing porosity. 

In the text 
Fig. 7 Effect of temperature on K_{ t }, K_{cond}, K_{conv}, K_{rad.} [16]. 

In the text 
Fig. 8 Variation of ETC with temperature [18]. 

In the text 
Fig. 9 Variation of ETC with porosity [18]. 

In the text 
Fig. 10 ETC dependency on temperature. 

In the text 
Fig. 11 Thermal conductivity of some known metals. 

In the text 
Fig. 12 Illustrated different cross section shape of ligaments. 

In the text 
Fig. 13 (a) Distribution of ligament length Al foam with porosity 91%, (b) distribution of ligament area Al foam with porosity 91%, (c) distribution of node coordination number Al foam with porosity 91%, (d) distribution of effective pore diameter Al foam with porosity 91% [23]. 

In the text 
Fig. 14 Predicted ETC by five different models . 

In the text 
Fig. 15 Unit cell of Kelvin. 

In the text 
Fig. 16 WeairePhelan cell: (left): type (a), (middle): type (b), (right) Orientation of WeairePhelan bubbles. 

In the text 
Fig. 17 Analyzing a realistic opencell metal foam structure created by µCT scan [55]. 

In the text 
Fig. 18 Steps of µCT scans process to produce 3D realistic image to analyze [56]. 

In the text 
Fig. 19 The 3D image of the stochastic open cell metal foam obtained by Xray tomography [57]. 

In the text 
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