Analysis of Local Thermal Equilibrium Assumption in Transient Forced Convection in a Graphite Foam C

Abstract

Keywords: Foam; forced convection; graphite; LTE; LTNE; porous media; transient.

1. Introduction

Overheating is one of the most critical criteria for the manufacturer of electronic components. Concerning the fact that in the components, the complete elimination of the dissipated heat is not possible by employing heat-sinks that utilize air-cooled metal fins, as the traditional cooling method, the graphite foam-based heat-sink has been introduced to solve this issue thanks to its high surface area to volume ratio, high thermal conductivity value, low density, and strong flow mixing potentials [1]. A two-phase porous medium is typically composed of solid and gas phases and the temperature distributions of those inside the computational domain are commonly assumed to be identical. In this situation, temperature fields of either phase are well demonstrated by a single *

Corresponding Author: e-mail: solmus@atauni.edu.tr, tel: (442) 231 48 45 2057

energy equation and this kind of approach is identified as Local Thermal Equilibrium (LTE) assumption. However, in most cases, the application of the LTE assumption contribute to insubstantial consequence of the simulation unless its validity is shown over the parameters investigated. If the LTE approach is not applicable, different energy conservation equations for solid and gas phases are needed for the sake of the calculation of the phases’ separate temperature fields in question. This approach is known as the Local Thermal Non-Equilibrium (LTNE) model. Several investigations have been carried out for the verification of the LTE supposition in forced convection flow in the porous channel. Abu-Hıjleh et al. [2] reported that the LTE supposition is valid in channels in which Nusselt number is significantly high and the thermal diffusivity ratio and Peclet number provide large values as well. On the other hand, the non-dimensional channel length along with the thermal capacity of solid to fluid brings low amounts where the LTE assumption is verified in the channels. Al-Nımr et al. [3] found that four dimensionless parameters such as porosity, volumetric Biot number, non-dimensional channel length, and solid to the fluid total thermal caspacity ratio control the local thermal equilibrium assumption. However, in this study, axial conduction in the domains of either fluid or solid is not considered. Rees et al. [4] investigated the impact of LTNE on the infiltration of hot fluid into a cold porous medium. They reported that LTE is always attained as time goes to infinite. Kim et al. [5] conducted a numerical analysis to describe the temperature distributions in the microchannel heat sink by using both one-equation and two-equation models for heat transfer. The obtained findings showed that an increase in the effective thermal conductivity ratio or a decrease in the Darcy number approaches the fluid temperature to the solid temperature, in which case the LTE assumption and the corresponding one-equation model is applicable. Kim and Jang [6] carried out a numerical study to present a general criterion for local thermal equilibrium in terms of parameters of engineering importance such as Darcy number, Prandtl number, and Reynolds number. The authors pointed out that as Preff,fRedpDa1/2ɛNu«1 criterion is satisfied, the LTE in a porous medium including convective and conductive heat transfer can be assumed. Khashani and Al-Nımr [7] performed a numerical study to assess the validity of the local thermal equilibrium assumption in the non-Newtonian forced convection flow through channels filled with porous media. Their findings revealed that LTE between the phases vanishes for the circumstances of a higher modified Peclet number, a lower modified Biot number, a lower fluid-to-solid thermal conductivity ratio, a lower power-law fluid index, and a lower microscopic and macroscopic frictional flow resistance coefficients. Haddad et al. [8] analytically investigated the validity of the LTE assumption in natural convection over a vertical flat plate embedded in porous medium. The authors reported that volumetric Biot number, modified Rayleigh number, modified Darcy number and the ratio of effective to dynamic viscosity manage the LTE assumption. They also proposed two correlation equations to determine the region where the LTE assumption is applicable. Zhang and Liu [9] presented a general criterion for LTE in forced convection flows through porous media under a constant heat flux boundary condition as function of boundary heat flux, the area of cross section, effective thermal conductivity ratio, characteristic pore length, porosity, Nusselt number, fluid conductivity and the heat source of solid phase. The presented criterion was also validated by the existing experimental and numerical results for convection heat transfer in a porous medium. The conditions, in which the LTE assumption is valid in porous media, have been studied extensively by several researchers up to now and some of them are summarized above. Graphite foam heat sink is a promising alternative for the thermal management of electronic components due to its unique thermal and physical properties as it is compared with the conventional heat sinks [11,12]. This study, unlike in the literature, attempts to provide a numerical investigation toward the impacts of fluid-to-solid Nusselt number (Nufs) and Reynolds (Re) number on the verification of the LTE supposition in transient forced convection in a rectangular channel filled with a block of graphite foam.

2. Mathematical Formulation

A schematic view of the physical problem is presented in Fig. 1. A block of graphite foam heat sink is inserted in a rectangular channel. Table 1 represents the graphite foam’s thermal and physical features which are taken into account in the present study. Air enters the channel with uniform velocity and temperature and flows through the block of foam. Initially, LTE is in existence between the solid graphite and fluid air phases and then, the fluid temperature at the channel inlet is suddenly raised to the desired value Ti. In addition to that, the channel walls are well insulated and hence, a transient one-dimensional LTNE approach is employed in the numerical study.

Ti x Insulated L Figure 1. A schematic view of the problem. Table 1. Thermal and physical properties of POCO graphite foam [10] ε 0.728

The porosity (ε), permeability (K) and pore dimameter (dp) of the graphite foam material are defined as the fraction of the volume of voids over the total material volume, measure of the resistance of the material to the fluid flow and the average diameter of each of the voids in the material, respectively. The following items are considered for providing more simplicity in the mathematical model:     

The foam porosity is spatially uniform and constant. The velocity and temperature of air at the channel inlet are assumed to be uniform. The fluid is incompressible. Thermo-physical properties of the solid and fluid phases are the constant. Radiative heat transfer and buoyancy forces are neglected [11-20].

3. Results And Discussion

The simulation results are given as a percentage deviation of the non-dimension solid-phase temperature from the non-dimension gas-phase temperature. The percentage deviation (η) is calculated as below:

In simulations, the length of the porous material is kept constant at 0.05 m. In the case where η is less than about 1%, it is figured out that the LTE assumption between the phases is valid. The dimensionless time required to attain LTE between the phases is labeled as τLTE and the real-time (σLTE) corresponding to τLTE is equal to τLTE*(L/ui). The impact of Reynolds number on the variation of η with τ at different axial locations is shown in Fig. 3 for a specified value of Nufs. It is obvious from this figure that τLTE is prolonged with the increasing value of the Reynolds number. However, σLTE for each Reynolds value is found to be about 4 sec. To justify this matter it is better to mention the increase of τLTE and ui with increasing value of Re . For this reason, σLTE remains almost constant in the considered Re interval . In other words, the effect of the Re number in the examined range on the LTE assumption is negligible for graphite foam. On the other hand, for small values of Re number, the change of the η with time especially in the region outside the inlet section of the graphite foam is almost the same. 100

Figure 3. The variation of η with τ as a function of Re number.

Fig. 4 shows the impact of Nufs on the variation of η with τ at different axial locations for a constant value of Reynolds number. In the simulations, Nufs is assumed to be constant and uniform throughout the computational domain. It is obvious from Fig. 4 that growth in Nufs contributes to a decrease in τLTE. The justification behind this is that the heat transfer between the phases is positively affected by the increase in Nufs and as a result, τLTE value is shortened. The σLTE value ranges from 1.5 to 5 seconds under the conditions investigated. The rate of decrease in σLTE increases with the increasing value of Nufs until a certain point. After that point, the rate of decrease in σLTE starts to reduce. Generally, the LTE assumption between the phases becomes valid after a few seconds, and the effect of Nufs and Re numbers on σLTE is negligible referring to the high values of the diffusivity ratio of solid to fluid phase. 100 80

Figure 4. The variation of η with τ as a function of Nufs number.

4. Conclusions

A numerical analysis is carried out to study the impact of Nufs and Re on the conditions under which the LTE assumption is verified in the transient forced convection in a rectangular channel filled with a block of graphite foam. Generally, the presumption of the thermal equilibrium between the graphite foam and air (solid and fluid phase) is valid a few seconds after the start and 2064

Nufs and Re have an insignificant effect on the σLTE referring to the high value of diffusivity ratio of solid to fluid phase. Consequently, it is found that it is reasonable to assume the LTE between the phases under the investigated conditions and thus, the mathematical model under the single energy equation could be appropriately employed to represent the temperature distribution for either phases in the computational domain.

Acknowledgements

The authors would like to thank the Scientific & Technological Research Council of Turkey (TÜBİTAK) for financially supporting this research under Project No. 315M136.

Nomenclature

inertial coefficient specific heat at constant pressure, J/kgoC pore diameter, m height of the porous insert, m heat transfer coefficient between solid and fluid phases, W/m3K thermal conductivity, W/mK permeability, m2 length of the porous insert, m fluid-to-solid Nusselt number, hv L2/kf Prandtl number, ʋ/α Reynolds number, ρf ui L/μ temperature, oC time, s velocity component in the x-direction, m/s darcy velocity, m/s dimensionless cartesian coordinate cartesian coordinates, m

dimensionless temperature dimensionless time porosity density, kg/m3

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