A synthesis of fluid and thermal transport models for metal foam heat exchangers

https://doi.org/10.1016/j.ijheatmasstransfer.2007.12.012Get rights and content

Abstract

Metal foam heat exchangers have considerable advantages in thermal management and heat recovery over several commercially available heat exchangers. In this work, the effects of micro structural metal foam properties, such as porosity, pore and fiber diameters, tortuosity, pore density, and relative density, on the heat exchanger performance are discussed. The pertinent correlations in the literature for flow and thermal transport in metal foam heat exchangers are categorized and investigated. Three main categories are synthesized. In the first category, the correlations for pressure drop and heat transfer coefficient based on the microstructural properties of the metal foam are given. In the second category, the correlations are specialized for metal foam tube heat exchangers. In the third category, correlations are specialized for metal foam channel heat exchangers. To investigate the performance of the foam filled heat exchangers in comparison with the plain ones, the required pumping power to overcome the pressure drop and heat transfer rate of foam filled and plain heat exchangers are studied and compared. A performance factor is introduced which includes the effects of both heat transfer rate and pressure drop after inclusion of the metal foam. The results indicate that the performance will be improved substantially when a metal foam is inserted in the tube/channel.

Introduction

Metal foams are a class of porous materials with low densities and novel thermal, mechanical, electrical and acoustic properties [1]. The foams are lightweight, offering high strength and rigidity, nontoxic structure, high surface area and recyclable which improve energy absorption and heat transfer in thermal applications, such as heat exchangers. The rate of heat transfer is enhanced by conducting the heat to the material struts, which have a large accessible surface area per unit volume, along with high interaction with the fluid flowing through them [2], [3], [4], [5], [6], [7]. Normal foam ligaments in the flow direction results in boundary layer disruption and mixing. Turbulence and unsteady flow occur for pore-scale Reynolds number greater than 100 [8]. The effect of thermal dispersion is essential for a number of applications in the transport processes. Vafai et al. have shown that the effect of transverse dispersion is much more important than the longitudinal dispersion [9], [10]. The induced turbulence and dispersion cause further enhancement in heat transfer and increase performance and efficiency of the heat exchanger considerably [11], [12]. In addition, flow paths through the foam are interconnected, which makes the flow available in all areas. As such, utilizing the metal foam leads to smaller and lighter heat exchangers.

Metal foams have considerable applications in multi-functional heat exchangers [13], [14], [15], [16], [17], cryogenics [18], combustion chambers, cladding on buildings, strain isolation, buffer between a stiff structure and a fluctuating temperature field, geothermal operations, petroleum reservoirs, compact heat exchangers for airborne equipments, air-cooled condenser towers and cooling systems [19], high power batteries [20], compact heat sinks for power electronics and electronic cooling [17], [21], [22], [23], [24], heat pipes [25], [26] and sound absorbers [27], [28], [29], [30].

Metal foams can be classified as porous media with typically high porosity that consists of tortuous, irregular shaped flow passages. However, some aspects of the past studies on packed beds and granular porous media needs to be modified for metal foams [18]. Liu et al. [31] have indicated that the pressure drop resulting from foam matrices, is much lower than that by granular matrices at the same Reynolds number. Metal foams include small filaments that are continuously connected in an open-celled foam structure. Metal foam cells are usually polyhedrons of 12–14 faces in which each face has a pentagonal or hexagonal shape (by five or six filaments). Due to the geometric complexity and the random orientation of the solid phase of the porous medium, the solution of the transport equations inside the pores is difficult to obtain. As such, the foam cell geometric idealizations (such as cubic unit cell model) has been employed for analytical and computational studies which can be inaccurate especially for compressed metal foam modeling [32]. In an effort to simulate flow through foam filaments computationally, Boomsma et al. [33] modeled idealized open-cell metal foams based on a fundamental periodic unit of eight cells. Fluid flow was then modeled computationally utilizing a three-dimensional cellular unit along with periodic boundary conditions.

The strength of the foam depends mainly on the base material and the relative density of the foam. Other properties, such as pore size, pore density, area density, fiber size, and cell shape affect certain foam characteristics, such as pressure drop and heat transfer [34], [35]. Pore size and relative density affect the foam’s flexibility [8]. The pore size is specified by the diameter of the open space in each of the cell faces. Typically this open space varies between 0.3 mm and 4 mm [1]. The pore density is the number of pores per unit length of the material specified as PPI (pores per inch). The available pore densities vary based on the foam material. However, the typical overall uncompressed range is 5–100 PPI. The relative density is the volume of solid foam material relative to the total volume of metal foam. As such, an increase in the relative density improves the strength of the foam structure, since the filaments become larger in diameter and stronger. The area density is the ratio of the surface area of the foam to its volume.

Due to the high thermal conductivity and structural strength of aluminum, the open-cell aluminum foams have attracted the attention of researchers for heat exchanger design. The porosity of a foam metal can be estimated using the weight of a given volume of the sample and the density of the metal. The average fiber diameter, df, can be measured using a microscope, and the average pore diameter, dp, can be estimated by counting the number of pores in a given length of material. The pore density (PPI) is a nominal value supplied by the manufacturer [36].

Du Plessis et al. [37] established models for pore diameter estimation as a function of tortuosity, porosity, total volume and fluid–solid interface area of a cubic representative unit cell (CRUC), a foam cell geometric approximation, and also represented as a function of the width of the cubic representative unit cell (d = dp + df) (Table 1, Eq. (1)). Other models by Du Plessis et al. [37], [38] for pore and fiber diameters are presented as functions of porosity, turousity, and the width of the cubic representative unit cell (Table 1, Eqs. (2)–(4)). Also, Calmidi [39] developed a model for the fiber diameter estimation as a function of porosity, pore diameter and shape function for a cubic unit cell (Table 1, Eq. (5)). Calmidi [39] also presented a modified model utilizing three-dimensional dodecahedron unit cell structure (Table 1, Eq. (6)). The authors showed that this model has a maximum deviation of ±7% from measured values for pore and fiber diameters. Experimental investigation of Bhattacharya et al. [40] indicates that the tortuosity model by Du Plessis et al. [37] (Table 2, Eq. (1)) is accurate mainly for high pore densities. Bhattacharya et al. [40] established a model for tortuosity as a function of porosity and shape function (G), which can cover a wider range of pore densities and porosities [40] (Table 2, Eq. (2)). The shape function includes the fiber cross section variation with porosity.

An open-cell metal foam filled pipe is investigated by Lu et al. [41]. The results show that the overall Nusselt number of the pipe increases with an increase in the relative density or pore density (PPI), especially when the thermal conductivity of the solid is much higher than that of the fluid. Although metal foams with low porosity and small pore size (i.e. high pore density) are preferred for achieving high heat transfer performance, they lead to a significant increase in the pressure drop. However, larger pore size materials can also lead to larger Nusselt numbers at higher flow rates with relatively lower pumping power [12], [13]. Lu et al. [41] demonstrated that for low Reynolds numbers, the effects of the thermal conductivity of the foam on heat transfer is quite small. Therefore, cheaper and lower thermal conductivity foams can be used at lower flow rates. When low thermal conductivity foams are utilized, the effect of pore density would be quite small. As such, higher porosity foams can be employed for these cases resulting in lower pressure drops. Compared to plain tubes, metal foam filled tubes have significantly higher (up to 40 times) heat transfer performance [41].

Klett et al. [42] indicated that solid foam radiators can transfer heat an order of magnitude better than the fin radiators. Boomsma et al. [18] indicated that the thermal resistances generated by the compressed open-cell aluminum foam heat exchangers were two to three times lower than the commercially available heat exchangers, while requiring about the same pumping power. Bhattacharya and Mahajan [24] studied the finned metal foam heat sinks for electronic cooling in which the air gap between two adjacent longitudinal fin heat sinks was replaced by high porosity metal foams. The results display that the heat transfer was enhanced by a factor of 1.5–2. Kim et al. [43], [44] examined heat transfer through aluminum foams inserted between two isothermal plates. Their results show that the foam material have better heat transfer performance compared to the conventional array fins, but subject to a greater pressure drop. As such, metal foam heat exchangers provide a significant improvement over the commercially available heat exchangers, which operate under similar conditions.

Modeling or measurement techniques have been developed for effective thermal conductivities of solid–fluid structures [40], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54]. Dukhan and Quinones [55] have stated that the effective thermal conductivity of the aluminum foams can be up to four times larger than that of the solid aluminum thus substantially improving the heat transfer. Boomsma and Poulikakos [54] have established a correlation for the effective thermal conductivity. Their studies indicate that changing the fluid conductivity has a relatively small effect on increasing the effective thermal conductivity. As such, the thermal conductivity of the solid phase and porosity have the main effect on the overall effective thermal conductivity.

Calmidi and Mahajan [36] studied the solid-to-fluid heat transfer from a heated metal channel brazed to an aluminum metal foam. Their results indicate that thermal dispersion is quite low when air is passing through the metal foam. However, dispersion would be considerably larger in the case of water, as it was previously represented by Hunt and Tien [4]. The dispersion conductivity is usually dominant at high Reynolds numbers, especially if the effective thermal conductivity is low [36]. Phanikumar and Mahajan [56] investigated buoyancy driven flow through metal foams. Their results indicate that at high Rayleigh and Darcy numbers, the fluid and foam fiber temperature difference would be significant requiring the utilization of a local thermal nonequilibrium model (LTNE).

The assumption of local thermal equilibrium (LTE) has widely been used in analyzing transport processes through porous media. However, this assumption is not valid for applications where a substantial temperature difference exists between the solid phase and the fluid phase [57]. Amiri and Vafai [3], [58] employed the generalized model for the momentum equation and a two-phase model for the energy equation, including axial and transverse thermal dispersion to investigate forced convection and validated their findings utilizing experimental investigations. They presented detailed error maps for assessing the importance of various simplifying assumptions that are commonly used. In addition, they presented a comprehensive analysis of transient incompressible flow including inertia and boundary effects and the effects of thermal dispersion and local thermal nonequilibrium (LTNE) in the energy equation [59]. Alazmi and Vafai [60] comprehensively investigated the proper form of boundary conditions for constant wall heat flux in porous media under LTNE conditions. Effects of variable porosity and thermal dispersion were also analyzed. Lee and Vafai [61] classified the heat transfer characteristics through porous media within three regimes, each of which is dominated by one of three distinctive heat transfer mechanisms: fluid conduction, solid conduction and internal heat exchange between solid and fluid phases and presented pertinent analytical expressions for each regime. Marafie and Vafai [57] investigated analytically forced convection through a channel filled with a porous medium, utilizing Darcy–Forchheimer–Brinkman model and local thermal nonequilibrium energy transport model. Analytical solutions were obtained for both fluid and solid temperature fields incorporating the effects of various pertinent parameters such as Biot number, the thermal conductivity ratio, Darcy number and inertial parameters.

Section snippets

Pressure drop and heat transfer correlations for metal foam heat exchangers

Pressure drop and heat transfer coefficient are the two important factors to be considered in designing a heat exchanger. The available correlations in the literature for metal foam heat exchangers can be classified in terms of three pertinent categories. In the first category, the correlations are independent of heat exchanger geometry and function of microstructure of the foam. In this category the friction factor and pressure drop can be estimated directly through the correlations or through

Performance evaluation of metal foam heat exchangers

Two types of counter flow metal foam heat exchangers are investigated to evaluate the efficiency improvement via inserting an aluminum foam in a heat exchanger. These are metal foam tube and channel heat exchangers. This is done by considering the generated heating and required pumping power to overcome the pressure drop. For the tube case, the inner tube is filled with the metal foam and for the channel case, the channel is filled with a metal foam. The working fluids are cold water and hot

Conclusions

Metal foam heat exchangers have substantial advantages compared to commercially available heat exchangers under nearly identical conditions. They provide substantially more heat transfer surface area, and more boundary layer disruption and mixing resulting from foam filaments. This leads to considerably larger heat transfer rates. Metal foam microstructural properties, such as pore size, pore density, relative density, and porosity, control the heat transfer processes. Decreasing the pore size

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