Experimental based numerical approach for determination of volumetric heat transfer coefficients of modified graphite foams

Highlights

• Heat transfer coefficients between modified graphite foam and air are obtained.

• Transient single-blow technique (TSBT) is employed.

• Heat transfer coefficients are generalized by developing empirical correlations.

• Proposed correlations will be useful to reliably design new generation devices.

Abstract

Graphite-based porous materials are emerging as attractive alternatives to metals for use as heat dissipation elements in thermal management applications. While having several desirable features such as high thermal conductivity and low density, graphite foam heat sinks also tend to have low permeability that can limit transport of working fluid within the component and result in inefficient heat transfer. In order to improve their heat dissipation performance, graphite foams can be modified by channels drilled in various arrangements. However, the heat transfer characteristics of such modified graphite foams are not well characterized. In order to address this problem, we report novel empirical correlations for graphite foams modified in a specific configuration where circular channels with 2 mm diameter are drilled in graphite foam along the flow direction in a staggered arrangement. Then, volumetric heat transfer coefficients between the modified graphite foam and a stream of air are obtained by using transient single-blow technique (TSBT). The transient one-dimensional local thermal nonequilibrium (LTNE) model is employed for determination of the volumetric heat transfer coefficient from experimentally obtained data. Nine different modified graphite foam samples of various L/H ratios are studied in experiments and an empirical correlation of the form Nu_v = CRe^a for each sample is derived. Empirical correlations for three different sample lengths (L = 27 mm, 52 mm, 76 mm) at a fixed height are also developed in the form of Nu_v = CRe^a(L/H)^b . The novel empirical correlations in question are valid for the Reynolds (Re) number varying from approximately 1000 to 10000. Results show that Nu_v generally increases with the increasing value of Re and L at a fixed value of H and the uncertainties associated with Re and Nu_v are evaluated to be less than 1.3% and 3.6%, respectively. Consequently, we anticipate that the proposed correlations will be useful in reliable design of a new generation of electronic devices.

Introduction

Overheating in electronic components remains to be a major concern for the producers. Conventional cooling methods based on the natural or forced convection such as air-cooled metal finned heat sinks are often not adequate for effectively removing the heat dissipated from modern electronic components which have been decreasing in size and increasing in power in recent years [1]. In such applications, the open-cell metal porous heat sinks are considered to be an attractive solution to overcome overheating problems due to their high surface area to volume ratio and strong flow mixing capabilities [2]. Thus, a significant number of studies on metal porous heat sinks, many of which focus on aluminum, have been performed in recent years. Schampheleire et al. [3] experimentally studied the air-saturated buoyancy-driven flow in an open-cell aluminum foam heat sink with brazing and epoxy based bonding methods. Their results showed that brazed samples were superior in all cases and the influence of sample height on the heat transfer rate was most pronounced. Gong et al. [4] developed a micro-channel heat sink with a metallic porous/solid compound fin design and numerically examined the influence of metallic porous fins on the hydraulic and thermal performances of the proposed heat sink. The authors pointed out that the porous/solid compound fin showed a positive effect on both the hydraulic and thermal performances. Dukhan and Chen [5] proposed a simplified two-dimensional convection/conduction model to predict the heat transfer inside rectangular blocks of open-cell aluminum foam. Then, they conducted an experimental investigation on the samples for model validation. The authors found that the obtained results from the model only underestimated the temperature close to the inlet of the foam. Jeng and Tzeng [6] numerically analyzed the impinging cooling of the aluminum foam heat sink using the two-equation model. It was reported that heat transfer performance of impinging cooling was enhanced by implementing aluminum foam compared to plate fins. On another numerical study, the same findings regarding Jeng and Tzeng’s [6] study was reported by Ejlali and co-workers [7]. Leong and Jin [8] performed an experimental study to investigate the heat transfer performance of aluminum foam heat sinks of different pore densities subjected to oscillating flow. The obtained findings showed that heat sinks subjected to oscillating flow led to higher heat transfer rates. Kurtbas and Celik [9] experimentally investigated the mixed convective heat transfer characteristics of the open-cell metal foams of different pore densities and proposed new empirical correlations. Al-Athel [10] used μ-CT scan to develop an accurate 3D representation of the metal foam fins. The developed model was used to study the effects of assembly method, forced convection, fin orientation, and number of fins on the efficiency, effectiveness, and thermal resistance. It was also employed to find the heat transfer coefficient of the metal foam heat sinks. In contrast, although graphite based porous heat sinks offer significant advantages over their metal counterparts due to their unique properties such as high thermal conductivity, low density and high specific surface area, their mainstream use has been limited partly due to graphite foam's considerably lower permeability to the working fluid compared to metal foam, which results in significant pressure drops along the flow direction and limits heat transfer rates [11], [12], [13]. However, this fundamental hydrodynamic drawback of graphite foam can be eliminated substantially by using different configurations in the channel instead of bulk usage [14], [15], [16], [17].

Since direct temperature measurements of solid and fluid phases inside the porous heat sinks are very scarce due to experimental difficulties, theoretical investigations on the temperature distribution in porous heat sinks have great importance for performance evaluation. In literature, two different approaches to the modelling of energy transport are generally used. In the first approach, it is assumed that there is no temperature difference between the solid and fluid phases inside the porous medium i.e. convective heat transfer is negligible and thus local thermal equilibrium (LTE) is valid between the phases. Therefore, a single energy equation is sufficient for describing the temperature distribution. However, the results obtained from the theoretical solution based on the single energy equation may not be accurate, especially in the case of high fluid velocities and/or the large difference between the coefficients of thermal conductivity for the different phases [18]. In the second approach, it is considered that local thermal non equilibrium (LTNE) between the fluid and solid phases exist during the energy transport inside the porous medium on the acceptance that local temperature difference between the two phases throughout the computational domain is significant. Therefore, the two different energy conservation equations are developed to determine the individual temperature fields for the fluid and solid phases. However, if an individual energy equation is developed for each phase, the convective heat transfer between the phases must be expressed in these equations. In other words, it is necessary to know the volumetric convective heat transfer coefficient between the phases. The volumetric convective heat transfer coefficient is obtained by theoretical or experimental-theoretical studies. Jiang et al. [19] experimentally and numerically studied the heat transfer between sintered bronze particles with an average diameter of 0.2 mm and the air in a miniporous media. In that work, using the experimental data, one-dimensional and lumped capacitance numerical models were applied to determine the particle-to-fluid heat transfer coefficient. It is indicated that lumped capacitance approach can be used for simplicity since there is a good agreement in between the results of both models. Fu et al. [20] determined the volumetric heat transfer coefficients in the form of Nu_v = CRe^m for five different cellular ceramics of mullite, YZA, SiC, cordierite and cordierite with LS-2 coating and air pairs using the single-blow transient experimental technique in conjunction with an inverse analysis. Results revealed that the volumetric heat transfer coefficient increases with a decrease in specimen thickness to mean pore diameter ratio. Hwang et al. [21] experimentally investigated the combined effect of foam porosity and Re number on interstitial convective heat transfer and friction drag in aluminium foam inserted duct flow. The authors proposed empirical correlations for pore Nusselt number in terms of pore Re number under various foam porosities. Results showed that both the friction factor and the volumetric heat transfer coefficient increase with decreasing foam porosity when Re number is fixed. Ando et al. [22] carried out an experimental-numerical study for accurate determination of volumetric heat transfer coefficients of ceramic foams of 6, 9, 13, 20 PPI (Pores Per Inch) in forced convective flows using the single-blow method. The volumetric heat transfer coefficients obtained were presented for each sample in the form of Nu_v = CRe^m. Xia et al. [23] proposed a new simple correlation (NU_v = 0.34Ø⁻²Re^0.61Pr^1/3) for the prediction of volumetric heat transfer coefficients of porous foams of Cu, Ni and SiC with the structural parameters of 0.87–0.97 porosity and 10–40 PPI. It is clarified that volumetric heat transfer coefficients decrease with an increase in porosity and increase with an increase in the value of PPI. Yang et al. [24] experimentally determined the interstitial heat transfer coefficient in the packed beds including ellipsoidal or non-uniform spherical particles using an inverse method of TSBT. The study concluded that both the effects of packing form and particle shape are significant to the macroscopic hydrodynamic and heat transfer characteristics in structured packed beds. Vijay et al. [25] conducted an experimental study to determine the stagnant effective thermal conductivity, dispersion conductivity, effective radiative conductivity, and volumetric heat transfer coefficient for the forced convection of air through open-cell alumina foams. Obtained results proved the possibility of accurate prediction of the ranges of parameters. Wu et al. [26] proposed a correlation that presents local volumetric heat transfer coefficient between ceramic foam and the air. In the corresponding work, it is emphasized that the proposed correlation is valid for a wide range of porosity, velocity, cell size and temperature. In summary, studies in literature show that only a limited number of correlations are available for prediction of the volumetric heat transfer coefficients for certain porous materials and there is no single correlation that is applicable for all. In addition, the volumetric heat transfer coefficient of graphite foam has not been studied yet.

As mentioned before, distinct thermo-physical properties of graphite foam material make it an attractive candidate for the engineering applications. However, due to its low permeability value, it shows significant resistance to the fluid flow when it is used as block in a channel and this restricts its potential for use in heat sink applications. For that reason, in order to minimize the pressure drop, the block graphite foam material was modified by opening 2 mm cylindrical channels along the direction of flow as it has been done previously in the literature [14], [15]. Schematic and visual representations of the 2 mm cylindrical channels that were opened in a staggered manner along the flow direction are given in Fig. 4 for the sample having dimensions of 15 × 52 × 27 mm. Consequently, as a novel study based on strong theoretical fundamentals, the present experimental based numerical work aims to propose empirical correlations for the volumetric heat transfer coefficient between the modified graphite foam manufactured by POCO, Inc. now owned by Entegris, Inc. and a stream of air as a function of Re number and length/height ratio of the sample by using the method of TSBT [19], [20], [21], [22], [23], [24], [25]. Improved experimentation and the executed numerical procedure allowing to monitor not only the gas phase temperature but also the solid phase during the transient heat transfer resulting in improved accuracy and reliability is the other distinct originality of the current article.