High-Porosity Metal Foams: Potentials, Applications, and Formulations

Abstract

This chapter is aimed as a concise review, but well-focused on the potentials of what is known as “High-porosity metal foams,” and hence, the practical applications where such promising media have been/can be employed successfully, particularly in the field of managing, recovering, dissipating, or enhancing heat transfer. Furthermore, an extensive comparison is conducted between the formulations presented so far for the geometrical and thermal characteristics concerning the heat and fluid flow in open-cell metal foams.

1. Introduction

A porous medium can be defined as a material composed of a solid matrix consisting of interconnected voids. This solid matrix is usually assumed to be rigid, but sometimes it may undergo limited deformations. The interconnected pores allow for a single type of fluid or more to flow through the material. There are many examples of these permeable materials available in nature such as sand beds, limestone, sponges, wood, and so on.

Porous media have become one of the most important materials used in insulating, transferring, storing, and dissipating thermal energy. The benefits these characteristics confer have led to porous materials being widely used in practical applications such as thermal insulation, geothermal applications, cooling systems, recuperative/regenerative heat exchangers, and solar energy collection systems, in addition to chemical and nuclear engineering. Thus, convective flows in porous materials have been investigated widely over recent decades and various aspects have been considered for different applications so far.

High-porosity metal foams are usually porous media with low density and novel structural and thermal properties. This sort of media is mainly formed from multi-struts interconnected to each other at joint nodes to shape pores and cells (see the SEM image in Figure 1) (Liu et al. [1]). They offer very high porosity (ε ≥ 0.89), light weight, high rigidity and strength, and a large surface area, which make them able to recycle energy efficiently. This capacity to transport a large amount of heat is attributed to their superior thermal conductivity compared to ordinary fluids and high surface-area density (surface area per a given volume of metal foam) as well as enhanced convective transport (flow mixing) due to the tortuous flow paths existing within them, as shown in Figure 1 (Zhao [2]). Also, their open-cell structure makes them even less resistant to the fluids flowing through them, and hence, the pressure drop across them is much less than it is in the case of fluid flow via packed beds or granular porous materials.

Figure 1.

Open-cell metal foams: SEM image of the structure (left); mechanism of flow mixing (right).

Open-cell metal foams were first invented by the ERG Materials and Aerospace Corp. in 1967, and since then, they have been continuously developed. This invention was patented to Walz [3], where the manufacturing processes were based on an organic preformation cast. However, this invention was originally intended for only classified military and aerospace applications. Accordingly, nonclassified applications had not made use of this technology until the mid of 1990s, when it has become generally available for industrial applications. Since then, other manufacturers have joined the global competition in this industry. To name a few, M-Pore GmbH in Germany, the French company Alveotec, and Constellium from Netherlands are currently making open-cell metal foams on a large scale for a wide range of applications.

The traditional way of casting open-cell metal foams is still adopted by ERG Materials and Aerospace [4] as well as M-Pore GmbH [5], where the foams are cast with an investment process based on polyurethane preformation. As the fabrication process is affected by gravity, the foams resulted will be shaped from oval rather than spherical cells, as illustrated in Figure 2 (De Schampheleire et al. [6]). Alveotec [7] and Constellium, on the other hand, use a different way called leachable bed casting, in which metal is cast over a stack of soluble spheres to shape out the interconnecting open-cells desired. The spheres used are usually made out of either salt or sand plus a polymer bonding agent. After metal solidification, the spheres are dissolved through washing them simply with water. Using this casting technique results in foams having more uniformly spherical-shaped cells unlike those formed by the investment casting, as shown in Figure 2 (De Schampheleire et al. [6]).

Figure 2.

Open-cell metal foams formed by: Investment casting (left); (b) leachable bed casting (right).

2. Potentials

Open-cell metal foams possess unique characteristics, making them a promising candidate for plenty of practical and engineering applications. Among these potentials are the following:

1. Very high porosity, ranges usually from 80% and up to 97.5%.

2. The open-core structure makes them attractive for applications where lightweight is a crucial requirement.

3. The open-cell nature makes them even less resistant to the fluids flowing through them, resulting in a significant saving in the pressure drop resulted.

4. Very high effective thermal conductivity.

5. High surface-area density, roughly from 1000 to 3000 m²/m³. Therefore, exceptional heat transfer area per a given volume of metal foam is offered. The surface area density can be further increased through compressing them in a particular direction, where the specific surface area of such a compressed metal foam can reach up to 8000 m²/m³ [2].

6. The tortuous flow paths existing within them considerably enhance the convective transport (flow mixing).

7. They are efficient sound-absorption materials [8].

8. They have the potential to be used for radiation shielding [9].

9. Good-impact energy absorption [10], where the structure of metal foam makes it possible to ensure a constant stress throughout the deformation.

10. Have attractive stiffness/strength properties and can be processed in large quantity at low cost via the metal sintering route.

3. Applications

Due to their uniquely promising potentials, high-porosity metal foams have been increasingly utilized in a variety of engineering and industrial applications. Such applications are diversely increasing day by day, making it quite hard to categorize them into particular groups. Thus, it is aimed herein to present an overview of their most recent applications without intensely going into details.

Their ability to meet the increasing daily demands to effectively transfer, exchange, or dissipate heat has attracted researchers and manufacturers to utilize them as a successful alternative to traditional heat transport media. For example, the experiments conducted by Boomsma et al. [11] showed that the thermal performance offered by the compressed open-cell aluminum foam heat exchangers is usually two to three times higher than that achieved through the commercially available heat exchangers, while they require comparable pumping power. Similarly, Mahjoob and Vafai [12] pointed out that despite the potential increase in pressure loss, utilizing metal foams in heat exchangers leads to a substantial enhancement in heat transfer, which can compensate the increase in pressure drop.

Therefore, metal foam heat exchangers have emerged recently in various practical sectors. Among them, metal foams were used as alternative to find extended surfaces utilized in removing heat from geothermal power plants, where metal foam heat exchangers offer superior thermal performance compared to conventional finned surfaces, at no extra cost resulting from the pressure drop and/or material weight [13]. In this regard, wrapping a thin layer of foam around the surfaces of tubes was proposed to enhance the heat transferred from/to them with little increase in the pressure drop produced [14–16]. Despite the higher pressure loss resulted from the increase in foam layer thickness, it was observed that the exterior convective resistance is reduced significantly, and hence, a considerable transfer enhancement is achieved. Also, the overall performance attained through using the foam-covered tubes was comparable to that achieved by the helically finned tubes at the low levels of inlet velocities and far superior at the higher velocities.

In the context of HVAC&R applications, metal foams have been presented as a promising candidate to replace the conventionally finned heat exchangers. Dai et al. [17] compared the heat transfer performance of a flat-tube, louvered-fin heat exchanger with that obtained using an identical foam heat exchanger. The analytical results revealed that for the same fan power and heat transfer performance, the metal foam heat exchanger is significantly more economical in both size and weight for a wide range of design requirements. In another comparison study [18], it was observed that the heat transfer rate offered by metal-foam heat exchangers is up to six times better than that in the case of the bare-tube bundle with no extra fan power. Also, it was found that if the dimensions of the foamed heat exchanger are not fixed, that is, the frontal area can be manipulated, metal-foam heat exchangers outperform the louvered-fin heat exchanger. In other words, a smaller metal-foam heat exchanger can be used for the same thermal duty, and hence, a smaller fan can perform what is required.

Employing high-porosity metal foams to improve the thermal effectiveness of counterflow double-pipe heat exchangers has been the subject of increasing interest recently. Xu et al. [19] pointed out that to achieve high thermal effectiveness, that is, greater than 0.8, porosity and pore density should be in the range of (ε < 0.9) and (ω > 10 PPI), respectively. Furthermore, Chen et al. [20] observed that despite the increase occurred in the pressure drop, using metal foams results in a remarkable heat transfer enhancement (by as much as 11 times), which leads to a considerable improvement in the comprehensive performance, that is, up to 700%. More recently, an innovative double-pipe heat exchanger was proposed [21, 22] through using rotating metal foam guiding vanes fixed obliquely to force fluid particles to flow over the conducting surface while rotation. Furthermore, the conducting surface itself was covered with a metal foam layer to improve the heat conductance across it. To optimize the performance achieved, an overall performance system factor, that is, OSP, was introduced as the ratio of the heat exchanged to the total pumping power required. Overall, the negligibly small pumping power required compared to the amount of heat exchanged makes the overall performance of such heat exchangers incomparable, that is, OSP = O(10²) (Figure 3) (Alhusseny et al. [22]). It was also observed that while increasing the temperature difference from 30 to 300°C, the overall performance achieved can be improved up to 200–300% depending on the Re* value. This outcome indicates the promising prospects to utilize the proposed configuration as a recuperator in gas turbine systems.

Figure 3.

The change of the overall system performance OSP with the rotational speed Ω and characteristic temperature difference ΔT for ε = 0.9 and ω = 10 PPI.

Now, utilizing metal foam can offer as more as twice the cooling effectiveness obtained by the traditional finned heat exchangers. Thus, such a sort of heat exchangers is widely employed today in medical and medicinal products, defense systems, industrial power generation plants, semiconductor, manufacturing, and aerospace manned flight [4]. Similarly, they have been proposed as an effective way to enhance the heat dissipated from heavy-duty electrical generators through filling their rotating cooling passages either fully or partially with open-cell metal foams [23–25]. The value of the this proposal was inspected by introducing an enhancement factor as the ratio of heat transported to the pumping power required, that is, $\frac{\mathit{Nu}}{\mathit{Re}\left(p_{\mathit{\text{in}}}-p_e\right)/\rho u_{\mathit{\text{in}}}^2}$, and comparing it with the corresponding values from a previous work regarding turbulent flow in a rotating clear channel [26], as shown in Figure 4 [25], where it was confirmed that the proposed enhancement is practically justified and efficient.

Figure 4.

Influence of ΔT on the overall system performance; at S = 2, Ω = 500 rpm, and T_c1 = 20°C.

They have also been utilized effectively in electronic cooling, where various configurations of metal foam heat sinks have been suggested [27–33] as an effective alternative to the traditional heat sinks incorporated in electronic devices.

In similar context, utilizing metal foams to improve internal cooling of turbine blades is of increasing interest. Filling a radially rotating serpentine channel with open-cell aluminum foam is proposed as an effective way to improve the overall efficiency of the cooling process [34]. Recently, heat transport enhancement along a 180°. round channel was proposed through placing multiple aluminum foam blocks alternately along the flow path [35]. It was found that using discrete foam blocks increases the heat transported by 74–140% compared to what the empty channel yields. In addition, it was observed that staggering the foam blocks vertically is more desirable for improving the overall system performance.

Due to their capability to transport heat effectively, metal foams are shaped into rings placed between the combustor and the turbine section of a jet engine in order to homogenize the temperature profiles of the gases leaving the combustor and, hence, to improve the overall efficiency of turbojet engines [36]. To provide a stable isothermalized platform for the airborne laser communication systems, ERG Materials & Aerospace [4] have fabricated aluminum foam composite opticals having the ability to allow unprecedented heat transfer. In the experiments conducted by Williams et al. [37], a porous insert material (PIM) formed of high porosity foam was proposed to improve the swirl stabilization in LPM combustion systems. It was found that using a reticulated foam insert results in mitigating the thermoacoustic instability effectively as well as reducing the combustion noise over the entire frequency range for a wide range of the design parameters considered.

As most of the phase change materials (PCMs) used for latent heat thermal energy storage (LHTES) possess poor thermal conductivity, the charging/discharging rate achieved will be quite modest. To overcome this deficit, high porosity metal foams have been suggested by Zhao et al. [38] as an effective means to improve the PCMs’ overall thermal conductivity, leading to enhance the heat transported and, hence, promote the PCM melting and solidification, as can be seen in Figure 5 (Alhusseny et al. [39]). This concept has been extensively investigated later to further improve the performance achieved, to name a few, the works presented for low-temperature [40–42] and high-temperature LHTES systems [43–45].

Figure 5.

Time development of charging process of paraffin-copper foam composite vs. pure paraffin.

Open-cell aluminum foams are considered as promising lightweight materials for γ-ray and thermal neutron shielding materials. The data collected experimentally by Xu et al. [46] reveal that filling the foam with water results in improving the mass attenuation coefficients compared to the nonfilled samples. Overall, following such a proposal to achieve high shielding performance is still in progress [47, 48] and requires further optimizations.

Overall, there is a variety of applications where high-porosity metal foam can be utilized successfully. For further applications, ERG Materials & Aerospace [4] lists diverse sorts of applications whether in daily life or military industries. For example, open-cell metal foams are commonly used as energy absorber in aerospace and military applications, air/oil separators in aircraft engine gearboxes, baffles to prevent sudden surges in liquids while being penetrated by solid frames, and breather plugs in applications requiring fast equalization of pressure changes. Also, they are utilized in electrodes, fuel cells, bone researches, micrometeorite shields, optics/mirrors, windscreens and so on. Manufacturers’ data and the open literature can be further dug for much more applications where high-porosity metal foams have outperformed and/or achieved considerable savings in the expenses required.

4. Formulations

Open-cell metal foams are classified as high-porosity materials that consist of irregularly shaped and tortuous flow passages. Pressure drop and heat transfer through such media are significantly affected by their geometrical characteristics, namely the foam porosity ε, fiber size d_f, pore diameter d_p, pore density ω, and cell shape. Therefore, most aspects regarding granular porous media and packed beds need to be adjusted for metal foams [11].

4.1. Estimation of fiber and pore diameter

In practice, the ligament size, or in other words fiber diameter, is usually measured using a microscope. Alternatively, the mean pore diameter can be estimated by counting the number of pores that exist in a particular length of foam, which is usually provided by the manufacturer in terms of pore density (PPI), that is, number of pores per inch. Depending on the representative unit cell used (Figure 6), various models were proposed. Among them is the model proposed by Fourie and Du Plessis [49] for pore size estimation as a function of the width of a cubic representative unit cell and tortuosity (Eq. (1)). Another model was developed by Calmidi [50] to estimate the fiber- to pore-diameter ratio as a function of the porosity and shape function, G = 1 − e^{−(1 − ε)/0.04}, for both the cubic unit cell (Eq. (2)) and the three-dimensional structure of a dodecahedron unit cell (Eq. (3)).

Figure 6.

Models used to represent microstructural unit cells in metal foams.

Fourie and Du Plessis [49]:

$d_p=d\frac{2}{\left(3-\chi \right)},d=d_p+d_f$E1

Calmidi [50]:

$\frac{d_f}{d_p}=2\sqrt{\frac{\left(1-\epsilon \right)}{3\pi }}\frac{1}{G}$E2

$\frac{d_f}{d_p}=1.18\sqrt{\frac{\left(1-\epsilon \right)}{3\pi }}\frac{1}{G}$E3

4.2. Models developed for predicting tortuosity of high-porosity metal foams

The tortuosity, defined as the total tortuous pore length within a linear length scale divided by the linear length scale in the porous medium [49], was modeled by Du Plessis et al. [51] as a function of porosity only (Eq. (4)). However, experiments conducted by Bhattacharya et al. [52] indicated that the accuracy of tortuosity model proposed by Du Plessis et al. [51] is limited for higher levels of pore density; hence, a tortuosity formulation that accurately covers a wider range of porosity and pore densities was established in terms of porosity and shape function G (Eq. (5)). Recently, an analytical model was proposed by Yang et al. [53] (Eq. (6)), as a simple function of both foam porosity and a pore shape factor. The shape factor β is defined as the ratio of the representative pore perimeter to the perimeter of a typical reference circle with an area equal to that of the representative pore.

Du Plessis et al. [51]:

$\frac{1}{\chi }=\frac{3}{4\epsilon }+\frac{\sqrt{9-8\epsilon }}{2\epsilon }cos\left\{\frac{4\pi }{3}+\frac{1}{3}{\cos }^{-1}\left[\frac{8{\epsilon }^2-36\epsilon +27}{{\left(9-8\epsilon \right)}^{3/2}}\right]\right\}$E4

Bhattacharya et al. [52]:

$\frac{1}{\chi }=\frac{\pi }{4\epsilon }\left\{1-{\left(1.18\sqrt{\frac{\left(1-\epsilon \right)}{3\pi }}\frac{1}{G}\right)}^2\right\}$E5

Yang et al. [53]:

$\chi =\frac{\mathit{\beta \epsilon }}{1-{\left(1-\epsilon \right)}^{1/3}}$E6

4.3. Models developed for estimating pressure drop across open-cell metal foams

With regard to predicting the pressure drop produced in fluid flows across high-porosity metal foams, a variety of models have been developed, which can be classified into two main categories. The first encompasses those investigations interested in estimating the pressure drop by means of the foam friction factor. Among them is the model presented by Paek et al. [54] for the friction factor as a function of pore Reynolds number (Eq. (7)). Also, the empirical correlations established by Liu et al. [55] offer friction factor estimation for airflow via aluminum foams for a wide range of porosity and various flow regimes (Eq. (8)).

Paek et al. [54]:

$f=\frac{1}{{\mathrm{Re}}_K}+0.105f=\frac{\left(\mathit{\Delta P}/L\right)\sqrt{K}}{\rho u^2},\mathit{and}{\mathrm{Re}}_K=\frac{\rho u\sqrt{K}}{\mu }$E7

Liu et al. [55]:

$\begin{array}{l}\left.\begin{array}{l}30<{\mathrm{Re}}_{D_P}<300:f=\frac{22\left(1-\epsilon \right)}{{\mathrm{Re}}_{D_P}}+0.22\\ {\mathrm{Re}}_{D_P}>>300:f=0.22\end{array}\right\}\\ f=\frac{\left(\mathit{\Delta P}/L\right)D_P}{\rho u^2}\frac{{\epsilon }^3}{1-\epsilon },\mathit{and}{\mathrm{Re}}_{D_P}=\frac{\mathit{\rho u}\sqrt{K}}{\mu }\end{array}$E8

The other category of pressure drop models consists of those concerned with estimating the permeability and inertial coefficient according to Darcy-–Forchheimer’s equation:

$\frac{\mathit{dp}}{\mathit{dx}}=\frac{\mu }{K}u+\frac{\mathit{\rho F}}{\sqrt{K}}{u}^2$E9

Based on a cubic representative unit cell, a theoretical model for predicting permeability and inertial coefficient was derived by Du Plessis et al. [51] (Eqs. (10) and (11)), as functions of porosity, tortuosity, and the width of cubic representative unit cell. In the study conducted by Calmidi [50], mathematical models were developed for both permeability and inertial coefficient as functions of the fiber and pore diameters as well as the foam porosity (Eqs. (12) and (13)). A correlation was established by Bhattacharya et al. [52] for predicting the inertial coefficient (Eq. (14)), in terms of tortuosity, porosity, shape function, and form drag coefficient C_D(ε). Based on Ergun’s law $\left(\frac{\text{∆}p}{L}=\alpha \frac{{\left(1-\epsilon \right)}^2}{{\epsilon }^3{d}^2}\mathit{\mu u}+\beta \frac{\left(1-\epsilon \right)}{{\epsilon }^{3}d}\rho u^2\right)$, Tadrist et al. [56] developed an empirical model for estimating permeability and inertial coefficient as functions of foam porosity and fiber size (Eqs. (15) and (16)). In the experiments conducted by Dukhan [57], the pressure drop resulting from airflow across aluminum foam was correlated into a model predicting the permeability and inertial coefficient as functions of porosity only (Eqs. (17) and (18)). Recently, an analytical model for estimating the permeability of metal foams (Eq. (19)) was established by Yang et al. [53] according to the cubic representative unit cell. This model offers the capability of estimating the permeability for a wide range of foam porosities ε = 0.55 ~ 0.98 and pore densities ω = 5 ~ 100 PPI.

Du Plessis et al. [51]:

$K=\frac{{\epsilon }^2{d}^2}{36\chi \left(\chi -1\right)}$E10

$F=\frac{2.05\chi \left(\chi -1\right)}{{\epsilon }^2\left(3-\chi \right)}\frac{\sqrt{K}}{d}$E11

Calmidi [50]:

$K=0.00073d_p^2{\left(1-\epsilon \right)}^{-0.224}{\left(\frac{d_f}{d_p}\right)}^{-1.11}$E12

$F=0.00212{\left(1-\epsilon \right)}^{-0.132}{\left(\frac{d_f}{d_p}\right)}^{-1.63}$E13

Bhattacharya et al. [52]:

$\begin{array}{l}F=0.095\frac{C_{D\left(\epsilon =0.85\right)}}{12}{G}^{0.2}\sqrt{\frac{\epsilon }{3\left(\chi -1\right)}}{\left(1.18\sqrt{\frac{\left(1-\epsilon \right)}{3\pi }}\frac{1}{G}\right)}^{-1}\\ \left.\begin{array}{l}0.85<\epsilon <0.97:G=1-e^{-\left(1-\epsilon \right)/0.04}\\ \epsilon \ge 0.97:G=0.5831\end{array}\right\},\mathit{and},C_{D\left(\epsilon =0.85\right)}=1.2\end{array}$E14

Tadrist et al. [56]:

$K=\frac{{\epsilon }^3{d}_f^2}{\alpha {\left(1-\epsilon \right)}^2},\alpha :100\sim 865$E15

$F=\frac{\beta \left(1-\epsilon \right)}{{\epsilon }^3}\frac{\sqrt{K}}{d_f},\beta :0.65\sim 2.6$E16

Dukhan [57]:

$K=a_1{e}^{b_1\epsilon }$E17

$F=\left(a_2\epsilon +b_2\right)\sqrt{K}$E18

$a_1,a_2,b_1,\mathrm{and}{b}_2\mathrm{are}\mathrm{all}\text{constants}$

Yang et al. [53]:

$\frac{K}{d^2}=\frac{\epsilon {\left(1-{\left(1-\epsilon \right)}^{1/3}\right)}^2}{36\beta \left[{\left(1-\epsilon \right)}^{1/3}-\left(1-\mathit{\beta \epsilon }\right)\right]}$E19

4.4. Effective thermal conductivity models for high-porosity metal foams

Many investigations have been conducted to evaluate the effective thermal conductivity k_e as a key factor in the thermal analysis of such systems. Among them is the model established by Paek et al. [54] based on one-dimensional heat conduction through a cubic unit cell (Eq. (20)). This model indicates that foam porosity has a direct impact on the overall thermal conductivity unlike the pore size, which was found to have a marginal influence.

The theoretical model derived by Calmidi and Mahajan [58] for the effective thermal conductivity as a function of the foam porosity (Eq. (21)) was found to match well with the experiments for both air and water as fluid phase. Similarly, Bhattacharya et al. [52] examined a two-dimensional unit cell shaped as a hexagonal honey comb to estimate the effective thermal conductivity, but taking into account circular nodes at each intersection joint rather than the square nodes considered earlier by Calmidi and Mahajan [58]. Although this model showed excellent agreement with the experimental data obtained and its formulations detailed in Eq. (22) look mathematically simpler than that derived by Calmidi and Mahajan [58], it is in fact more complicated because this model is valid only for a limited case when the intersection size R approaches zero, which implies that r → ∞.

A tetrakaidecahedron unit cell was adopted by Boomsma and Poulikakos [59] to estimate the effective thermal conductivity. Despite the fact that taking tetrakaidecahedron topology into account can provide a better estimation, it leads to more complex formulae, detailed in Eq. (23). Moreover, it was found that this model includes some aspects need to be adjusted as found by Dai et al. [60] (Eq. (24)). Based on the tetrakaidecahedron unit cell as well but with assuming one-dimensional heat conduction along the highly tortuous ligaments, a quite simplistic model (Eq. (25)) was recently derived by Yang et al. [61] for effective thermal conductivity of metal foams saturated with low conducting fluids, for example, air. However, this model is limited to highly conducting foams, where heat conduction is assumed to occur only along the tortuous ligaments ignoring the heat conduction through the fluid phase. More recently, the 3D tetrakaidecahedron unit cell was considered by Yao et al. [62] to establish a more realistic formulation for effective thermal conductivity (Eq. (26)) through taking into account the concavity and orientation of the tri-prism ligaments and for four pyramids nodes. In addition to including no empirical parameters, this model has outperformed, in terms of accuracy, what were reported earlier in the literature [62].

Paek et al. [54]:

$k_e=k_f{\left(1-t\right)}^2+k_s{t}^2+\frac{2t\left(1-t\right)k_f{k}_s}{k_{f}t+k_s\left(1-t\right)},t=\frac{1}{2}+cos\left(\frac{1}{3}{\cos }^{-1}\left(2\epsilon -1\right)+\frac{4\pi }{3}\right)$E20

Calmidi and Mahajan [58]:

$\begin{array}{l}{k}_e={\left\{\frac{2}{\sqrt{3}}\left[\frac{r\left(\frac{b}{L}\right)}{k_f + \left(1 + \frac{b}{L}\right)\frac{\left(k_s - k_f\right)}{3}}+\frac{\left(1 - r\right)\left(\frac{b}{L}\right)}{k_f + \frac{2}{3}\left(\frac{b}{L}\right)\left(k_s - k_f\right)}+\frac{\frac{\sqrt{3}}{2} - \frac{b}{L}}{k_f + \frac{4r}{3\sqrt{3}}\left(\frac{b}{L}\right)\left(k_s - k_f\right)}\right]\right\}}^{-1}\\ \frac{b}{L}=\frac{-r+\sqrt{r^2+\frac{2}{\sqrt{3}}\left(1-\epsilon \right)\left[2-r\left(1+\frac{4}{\sqrt{3}}\right)\right]}}{\frac{2}{3}\left[2-r\left(1+\frac{4}{\sqrt{3}}\right)\right]},\mathit{and},r=0.09\end{array}$E21

Bhattacharya et al. [52]:

$k_e={\left\{\frac{2}{\sqrt{3}}\left[\frac{\frac{t}{L}}{k_f+\frac{\left(k_s-k_f\right)}{3}}+\frac{\frac{\sqrt{3}}{2}-\frac{t}{L}}{k_f}\right]\right\}}^{-1},\frac{t}{L}=\frac{-\sqrt{3}-\sqrt{3+\left(1-\epsilon \right)\left(\sqrt{3}-5\right)}}{1+\frac{1}{\sqrt{3}}-\frac{8}{3}}$E22

Boomsma and Poulikakos [59]:

$\begin{array}{l}{k}_e=\frac{\sqrt{2}}{\left(R_A+R_B+R_C+R_D\right)}\\ \left.\begin{array}{l}{R}_A=\frac{4d}{\left[2e^2+\mathit{\pi d}\left(1-e\right)\right]k_s+\left[4-2e^2-\mathit{\pi d}\left(1-e\right)\right]k_f}\\ R_B=\frac{{\left(e-2d\right)}^2}{\left(e-2d\right)e^2{k}_s+\left[2e-4d-\left(e-2d\right)e^2\right]k_f}\\ R_C=\frac{{\left(\sqrt{2}-2e\right)}^2}{2\pi d^2\left(1-2e\sqrt{2}\right)k_s+2\left[\sqrt{2}-2e-\pi d^2\left(1-2e\sqrt{2}\right)\right]k_f}\\ R_D=\frac{2e}{e^2{k}_s+\left(4-e^2\right)k_f}\\ d=\sqrt{\frac{\sqrt{2}\left[2-\left(5/8\right)e^3\sqrt{2}-2\epsilon \right]}{\pi \left(3-4e\sqrt{2}-e\right)}},\mathit{and},e=0.339\end{array}\right\}\end{array}$E23

Dai et al. [60]:

$\left.\begin{array}{l}{R}_C=\frac{\sqrt{2}-2e}{\sqrt{2}\pi d^2{k}_s+\left(2-\sqrt{2}\pi d^2\right)k_f}\\ d=\sqrt{\frac{\sqrt{2}\left[2-\left(3/4\right)e^3\sqrt{2}-2\epsilon \right]}{\pi \left(3-2e\sqrt{2}-e\right)}},\mathit{and},e=0.198\end{array}\right\}$E24

Yang et al. [61]:

$k_e=\frac{1}{3}\left(1-\epsilon \right)k_s$E25

Yao et al. [62]:

$\begin{array}{l}{k}_e=\frac{1}{\left(\lambda /k_A\right)+\left(\left(1-2\lambda \right)/k_B\right)+\left(\lambda /k_C\right)}\\ \left.\begin{array}{l}{k}_A=\frac{\sqrt{2}}{6}\mathit{\pi \lambda }\left(3-4\lambda \right)\frac{1+a_1^2}{a_1^2}{k}_s+\left[1-\frac{\sqrt{2}}{6}\mathit{\pi \lambda }\left(3-4\lambda \right)\frac{1+a_1^2}{a_1^2}\right]k_f\\ k_B=\frac{\sqrt{2}}{2}{\mathit{\pi \lambda }}^2\frac{1+a_1^2}{a_1^2}{k}_s+\left(1-\sqrt{2}{\mathit{\pi \lambda }}^2\frac{1+a_1^2}{a_1^2}\right)k_f\\ k_C=\frac{\sqrt{2}}{6}{\mathit{\pi \lambda }}^2\frac{1+a_1^2}{a_1^2}{k}_s+\left(1-\frac{\sqrt{2}}{6}{\mathit{\pi \lambda }}^2\frac{1+a_1^2}{a_1^2}\right)k_f\\ \epsilon =1-\frac{\sqrt{2}}{2}{\mathit{\pi \lambda }}^2\left(3-5\lambda \right)\frac{1+a_1^2}{a_1^2},\mathit{and},a_1=2.01\end{array}\right\}\end{array}$E26

4.5. Solid–fluid interstitial thermal exchange within open-cell metal foams

The condition of local thermal equilibrium (LTE) often occurs between the fluid and solid phases when a fluid flows across a permeable medium formed of comparably thermally conductive material. This makes the temperature difference between the two phases negligibly small. However, when the solid thermal conductivity is much higher than the corresponding value for the fluid phase, for example, metal foams, the assumption of LTE is no longer valid and usually results in an overestimation of the heat transported between the two phases. Hence, taking into account the local thermal nonequilibrium (LTNE) between the two phases becomes indispensable in metal foams, where two energy equations are coupled together to predict heat transfer in each phase separately.

Three principal heat transfer modes take place when a low conductive fluid flows across the ligaments of highly conductive foam: convection between the solid and fluid phases besides conduction via each one of the two phases. Thus, the three key parameters required for applying the LTNE approach are the effective thermal conductivity of the fluid k_fe and solid k_se phases in addition to the interstitial specific heat transfer rate between the two phases (a_sf h_sf), which depends on the foam structure and the flow regime across it.

The interstitial heat exchange rate depends on two individual quantities: the interfacial specific surface area a_sf and the solid-to-fluid interfacial heat transfer coefficient h_sf. By utilizing the dodecahedral structure of open-cell foams and taking into account the noncircular fiber cross section, the solid-to-fluid interfacial specific surface area a_sf was modeled by Calmidi and Mahajan [63] for arrays of cylinders that intersect in three mutually perpendicular directions (Eq. (27)), while Fourie and Du Plessis [49] established another model based on the cubic unit-cell representation (Eq. (28)). However, Schampheleire et al. [6] observed that the a_sf values estimated using Eq. (27) by Calmidi and Mahajan [63] deviates seriously from those obtained experimentally through a µCT scan with differences up to 233%, while the model of Fourie and Du Plessis [49] performs much better with up to 22% deviation from the experimental data of the full µCT scan.

Calmidi and Mahajan [63]:

$a_{\mathit{sf}}=\frac{3\pi d_f}{{\left(0.59d_p\right)}^2}\left[1-e^{-\left(1-\epsilon \right)/0.04}\right]$E27

Fourie and Du Plessis [49]:

$a_{\mathit{sf}}=\frac{3}{d}\left(3-\chi \right)\left(\chi -1\right)$E28

With regard to estimating the solid–fluid interfacial heat transfer coefficient in high-porosity metal foams, Calmidi and Mahajan [63] proposed a correlation for interfacial Nusselt number as a function of both the foam porosity and fiber diameter (Eq. (29)). Another model was established by Shih et al. [28] for Nusselt number as a function of the foam porosity and pore diameter (Eq. (30)), where a and b are constants depending on the geometrical characteristics of the foam samples used in the experiments conducted. The correlations developed by Zukauskas [64] for staggered cylinders are widely used as a model to predict the interfacial Nusselt number as a function of the foam porosity, fiber diameter, and the value of Reynolds number (Eq. (31)), which makes it more general than the one proposed by Calmidi and Mahajan [63] as it is valid for a limited range of Reynolds numbers (40–1000).

Calmidi and Mahajan [63]:

${\mathit{Nu}}_{\mathit{sf}}=\frac{h_{\mathit{sf}}{d}_f}{k_f}=C_T{\mathrm{Re}}_{d_f}^{0.5}{\mathrm{Pr}}^{0.37}=0.52\sqrt{\frac{{\mathit{ud}}_f}{\mathit{\epsilon \upsilon }}}{\mathrm{Pr}}^{0.37}$E29

Shih et al. [28]:

${\mathit{Nu}}_{D_P}=\frac{{\mathit{hD}}_P}{k_{\mathit{se}}}=a{\mathrm{Re}}_{D_P}^b=a{\left(\frac{\rho {\mathit{uD}}_P}{\mu }\right)}^b$E30

Zukauskas [64]:

$\begin{array}{l}{\mathit{Nu}}_{\mathit{sf}}=\frac{h_{\mathit{sf}}d}{k_f}=\left\{\begin{array}{l}0.76{\mathrm{Re}}_d^{0.4}{\mathrm{Pr}}^{0.37},\left(1\le {\mathrm{Re}}_d\le 40\right)\\ 0.52{\mathrm{Re}}_d^{0.5}{\mathrm{Pr}}^{0.37},\left(40\le {\mathrm{Re}}_d\le {10}^3\right)\\ 0.26{\mathrm{Re}}_d^{0.6}{\mathrm{Pr}}^{0.37},\left({10}^3\le {\mathrm{Re}}_d\le 2\times {10}^5\right)\end{array}\right\}\\ d=\left(1-e^{-\left(1-\epsilon \right)/0.04}\right)d_f\end{array}$E31