1. Introduction
Foam materials usually have disordered and interconnected network structures, tortuous flow paths, high porosity, light weight, low pressure drop and large specific surface areas [
1]. These unique physical properties make foam materials suitable for heat exchangers [
2], heat sinks [
3], volumetric solar air receiver [
4], thermal energy storage [
5], catalytic reactors [
6], radiation burners [
7] and hypersonic and re-entry vehicles [
8]. Specifically, the large specific surface areas and internal tortuous flow paths of foam materials make it ideal for the enhancement of convective heat transfer. Because of these two distinguishing features, the convective heat transfer flux in a porous matrix can be remarkably large, and the volumetric heat transfer coefficient can reach at 10
5–10
6 Wm
−3 K
−1 [
9] or even higher. The theory and detailed information of convective heat transfer inside the foam materials are crucial to the design and optimization of heat sinks, volumetric solar air receiver and so on. Unfortunately, the existing correlations have low accuracy in predicting the volumetric heat transfer coefficient within the foam materials. In fact, there are huge deviations [
10] between these correlations.
To search for a generalized correlation, and then to calculate the volumetric heat transfer coefficient within the foam materials, scientists worldwide have made many experimental and numerical studies. Additionally, there have been many experimental studies about the convective heat transfer inside foam materials. Younis [
11] experimentally measured the volumetric heat transfer coefficient between a stream of air and ceramic foam, alumina foams and cordierite foams. The sample thickness was 12–14 mm. The results show the volumetric heat transfer coefficient is mainly dependent on the superficial velocity and pore size. In addition, the heat transfer characteristic within the cordierite foams differed significantly from the alumina foams. The authors attributed this to the difference of the microstructure of the two materials. For the alumina foams, a correlation of Nusselt number about
was obtained through the changing of pore diameter. However, the effect of
was not investigated because the thickness is approximately fixed. On the same test rig, Fu [
12] measured the volumetric heat transfer coefficient between air stream and cellular ceramics, which were made of mullite, YZA, SiC and cordierite. The sample thickness used was 12–14 mm. The effects of pore length scale and specimen thickness on the volumetric heat transfer coefficient were presented and discussed. An empirical correlation of Nu was also proposed, which is a function of the material, PPI (Pores Per Linear Inch) and specimen thickness. Specifically, Fu [
12] concluded that the volumetric heat transfer coefficient increases with a decrease in the specimen thickness to the mean pore diameter ratio. Hwang [
13] studied the flow and convective heat transfer features of aluminum foams, with the sample thickness of 60 mm. Empirical correlations were proposed based on the experimental data. Based on the experimental data from several researchers and an equivalent strut diameter (from Dul’nev’s unit cell model [
14]) of open-cell foam as the characteristic length, Kamiuto [
15] derived a Nusselt versus Peclet number correlation where the porosity and the nominal cell number density were involved. Zhang [
16], Hernandez [
17] and Zhang [
18] also studied the volumetric heat transfer features of aluminum foams, the sample thickness were 50 mm, 50.8 mm and 110 mm respectively. Similarly, three empirical correlations were proposed. Dietrich [
19] used the transient single-blow method to obtain the volumetric heat transfer coefficient of three porous media of silicon carbide, alumina, and mullite with the sample thickness of 50 mm and the superficial velocity ranging from 0.5 m/s to 1.5 m/s. The volumetric heat transfer coefficient was established by fitting the experimental data. Vijay [
20] experimentally determined the volumetric heat transfer coefficient for alumina foams with different geometric parameters based on the transient heat transfer. To explore the volumetric solar air receiver, Fend [
21] and Xia [
22] experimentally investigated the convective heat transfer inside the ceramic foam (Xia [
22] also studied Cu and Ni foam), and empirical correlations were unsurprisingly proposed. In addition, Fend [
21] and Xia [
22] used the sample thickness of 70 mm and 120 mm respectively.
On the other hand, researchers worldwide have made many numerical simulations to investigate the convective heat transfer within the foam materials. Constrained by the knowledge of the precise structure of real foam material and limited computational power, most of the numerical studies used idealized geometries, such as a periodic array of square rods [
23], sphere pore [
24], square cylinders [
25], cubic model [
26] and octet truss lattice geometry [
27]. Wu et al. [
9] investigated the convective heat transfer characteristics of air flow through ceramic foams, which was approximated by the Kelvin tetrakaidecahedron model, and argued that it is not proper to use the “mean volumetric heat transfer coefficient” and recommended to use the local volumetric heat transfer coefficient. With the numerical results, a correlation involving porosity, mean cell size and Reynolds number was proposed. Iasiello et al. [
28] numerically investigated the thermally developed flow of air in an open-cell foam on a geometry of Kelvin tetrakaidecahedron foam, and presented a correlation involving the porosity and the Reynolds number. Saurish [
29] used an immersed boundary method and realistic random periodic structures, and proposed a Nusselt correlation for open-cell solid foams. This correlation includes the porosity, Reynolds number and Prandtl number.
With the progress of related technologies, more and more researchers use the computed tomography to get the real structure of foam materials, and then investigate the flow and heat transfer inside it on the high performance computational platform. Petrasch et al. [
30] obtained the real geometry of reticulate porous ceramics by using computed tomography and then studied the penetrability and interfacial heat transfer characteristics of this porous material by numerical simulation. Finally, a correlation involving Reynolds number and Prandtl number was proposed. Haussener [
31] and Suter [
32] did similar studies of porous ceramics, used the least-square fitting and obtained two different empirical correlations. Zafari [
33] conducted numerical simulations on open cell metal foams and presented a correlation through a curve-fitting procedure. The correlation of Nusselt number is a function of porosity and Reynolds number. Ambrosio [
34] conducted simulations on Kelvin’s ideal foam model and on tomography-based real foams, and concluded that the convection heat transfer coefficient slightly decreases with porosity, while the volumetric convection heat transfer coefficient markedly decreases. Liu [
35] investigated the heat transfer of porous media at the pore-scale with the double-population thermal lattice Boltzmann (LB) method. The studied sample porosity was about 0.6, and had a thickness of 5 mm. A correlation was also proposed, where the Nusselt number was a function of Reynolds number only. Meinicke [
36] numerically studied the single-phase hydrodynamics and conjugate heat transfer in cylinder shaped solid sponges, and literature data were used to validate the heat transfer coefficient. Iasiello [
37] concluded that the heat transfer characteristics were very close for a low Reynolds number for an ideal structure and a real structure, and the heat transfer coefficient and the Nusselt number were consistent. However, a larger Reynolds number would result in a larger difference between the two structures. Vijay [
38] concluded that a simplified foam structure could not represent real foam when considering a heat transfer via the foam tortuous structure. In fact, the exact use of geometric models and large computational demands can lead to limited numerical simulation. Nie [
39] investigated the pressure drop and heat transfer through open cell foams with a 3D Laguerre–Voronoi model. A correlation with a form of for
was proposed, where the coefficient
increased with the porosity. Based on a geometrical model generated through X-ray micro-computed tomography (CT), Dixit [
40] investigated the pressure drop and heat transfer of a small open-cell metal foam matrix with both experiment and simulation. Recent research articles on computational simulation of the digitized real open-cell metal foam structure were also listed. Based on the generalized Lévêque analogy [
41], Gancarczyk et al. [
42] derived a theoretical model describing heat transfer characteristics of the solid foams solely based on their geometric parameters, and the derived model was validated by the experimental data and within the accuracy of 25%.
To summarize the above-mentioned literatures, the interfacial heat transfer inside the foam materials has been extensively investigated both by experiments and numerical simulations. Additionally, a lot of empirical correlations were proposed. However, these correlations are so dispersed that none is universally reliable. By comparing with the data in the literature, we see that all of the correlations have the form of
. Furthermore, there are three features of these correlations. Firstly, there are huge differences between the two coefficients
and
. Within the correlations from the experimental studies, the coefficient “c” ranges from 0.13 to 2.43, and the exponent “m” ranges from 0.42 to 1.18. The convective heat transfer coefficient obtained from the correlations differs in two orders of magnitude and even more. The second feature is, except that they have a similar form of
, there is no other regularity among them. The third feature is that it seems the fidelity of these correlations is improving, because the coefficient
is no longer a constant, but a function of the porosity or pore size. However, due to the complexity of foam structures, the influence of porosity, pore size and thickness on the volumetric heat transfer coefficient within foam materials is still unclear. In addition, the local thermal equilibrium and local thermal nonequilibrium phenomenon within porous media have been widely discussed in the literature [
43,
44,
45,
46,
47,
48], but have not been experimentally verified. Therefore, the convective heat transfer inside the foam materials is still worthy of further studies.
The main purpose of this article is to investigate how the foam properties, namely the porosity, cell size (note, the pore density or sometimes PPI, is usually used in the material industry, and the pore size or cell size is usually used as the characteristic length when the quantitative analysis is needed) and thickness, affect the convective heat transfer inside the ceramic foam material. In this study, the single-blow technique is used to experimentally determine the volumetric heat transfer coefficient between ceramics foam and air stream in the temperature range from 283 K to 323 K, and the superficial air velocity from 0.58–1.76 m/s. The foam properties, including the porosity, cell size and sample thickness—which has never been investigated before—were all parametrically studied. Their influences on the volumetric heat transfer coefficient, and the local thermal equilibrium within the ceramic foam materials were analyzed. The results show that both the cell size and sample thickness have significant effects on the volumetric heat transfer coefficient, and the porosity’s influence is relatively weak. More importantly, the local thermal equilibrium was firstly verified with experimental data, as well as its influential factors have been analyzed. Further, the experimental data were correlated in dimensionless form to offer an equation for estimating the volumetric heat transfer coefficients at a given superficial air velocity for any types of foam material. This correlation is crucial to the field of convective heat transfer inside the porous media, and can be used to guide the design and optimization of volumetric solar air receiver, compact heat exchanger, heat sink, heat regenerator, packed bed reactor and so on.