Numerical investigation of convective heat transfer in a plane channel ﬁlled with metal foam under local thermal non-equilibrium

– The present work consists on convective heat transfer modeling in a plate heat exchanger ﬁlled with metal foam under local thermal non-equilibrium (LTNE). The metal foam was inserted to ﬁll completely the studied channel, which is crossed by a ﬂuid. The modiﬁed Brinkman-Forchheimer extended Darcy model is used in the porous layer, while the macroscopic two-energy equation model is used for the thermal ﬁeld. The channel walls are maintained at a constant temperature and the velocity at the inlet is supposed uniform. A dimensionless formulation is developed to perform a parametric study in terms of certain dimensionless variables, and solved by the ﬁnite volume method (FVM). The results include the eﬀect of the interstitial heat transfer coeﬃcient and the solid to ﬂuid thermal conductivity of diﬀerent type of metal foams. The results were used to estimate the inﬂuence of the convective and conductive contributions using open-celled metal foams with high porosity. It has been shown that such supports can bring a signiﬁcant enhancement for the heat transfer.


Introduction
Metal foams are a type of porous materials with novel thermal, electrical and chemical properties which have considerable industrial applications, such as catalyst carriers, filters, biomedical applications, and spacecraft [1][2][3].The use of metal foams in heat exchangers and chemical applications can provide evident advantages due to their high specific interfacial area, their high thermal conductivity, low density and a good ability of mixing fluid (Fig. 1).Thus, the open cell metal foams may substantially enhance the heat transfer process into heat exchangers.The metal foams can also be used as catalyst support with superior heat and mass transfer due to their optimal thermal properties.Compared to the conventional heat exchangers, the performance of those filled with open-celled metal foams can be enhanced due to the increased fluid/solid contact specific area.The probability of the remarkable temperature gradients resulting in the occurrence of the hot spots, cold zones and decreasing the device efficiency is minimized.Although, transa Corresponding author: settarhakim@gmail.comport phenomena in porous media are studied for nearly two centuries, the studies about metal foams are still relatively few and recent [4][5][6][7].Thus, the study of heat transfer by forced convection in metal foams has been the subject of many studies [8][9][10], most of these works focus on the assumption of local thermodynamic equilibrium (LTE) [11][12][13].In certain configurations using metal foams, where the solid to fluid thermal conductivity is high, the net heat transfer between phases takes place such that the assumption of LTE is not satisfied.The need for two-energy equation model becomes necessary.In the last years, the problem of LTNE has received considerable attention due to its presence in a wide variety of engineering applications, especially when metal foams are used for the enhancement process.This fact is confirmed by the high amount of studies that focused on comprehension of behavior of metal foam based on the LTNE assumption, in particular, the plane channel configuration filled with this porous medium [1,14] because of its direct applicability in the plate heat exchangers and heat sinks.Zhao and Lu [8]   metal foam.They have examined the effect of morphology on the heat transfer.Their study showed that the heat exchangers filled with metal foams are more efficient than those with smooth tubes.Kim et al. [15] studied the effect of parameters such as the Darcy, Prandtl and Reynolds numbers in porous media.They showed that LTE assumption is valid when the convection heat transfer is dominant.Vafai and Kim [9] adopted the LTE assumption and conducted an analytical study of heat transfer by forced convection in porous media.Lee and Vafai [16] presented analytical solution for the parallel plates by considering the two-equation model.Similarly, Dukhan [17] conducted an analytical study of forced convection in a porous medium based on the LTNE assumption.The aim of this work is to study the dynamic and thermal behaviors of a plate heat exchanger provided with a metal foam.We present a numerical investigation of forced convection into a flat channel completely filled with metallic foam.The work is particularly focused in the thermal field by adopting the two-equation model.The results according to this assumption are analyzed and discussed, highlighting the effect of various heat transfer parameters.In this study, the dynamic field is obtained using the general model of Darcy-Brinkman-Forchheimer.

Physical and mathematical models 2.1 Physical model
The schematic diagram of the problem is presented in Figure 2: flow in a channel, of length L and height H, completely filled with porous material type metal foam and heated with a constant and uniform wall temperature T w .The flow is assumed to be steady, incompressible, laminar and two-dimensional.A stream of air with uniform velocity U 0 and temperature T 0 is considered at the inlet of the channel.For the thermal field, the thermophysical properties are considered constant and the effects of natural convection and the viscous dissipation are neglected.

Macroscopic transport equations
The channel is considered completely filled with homogeneous and isotropic metal foam.The fluid flow is modeled using a Brinkman-Forchheimer-extended Darcy model and is assumed to be in local thermal nonequilibrium with the porous medium.The wall temperature is higher than the inlet temperature of the fluid.

Flow equations
In the porous media, the macroscopic equations obtained after volume integration over a representative elementary volume (REV) are given as: a) Continuity In this case, it is considered that the assumption of LTE is not verified.Thus, the heat transfer is governed by an energy equation for each phase.
where K eff ,s and K eff ,f are the thermal conductivities for the solid and fluid phases respectively.In this work, the effective thermal conductivities of both phases are estimated by the correlation used by Calmidi et al. [18] for metal foams.They are given by: h sf is the interstitial heat transfer coefficient and a sf is the specific interfacial area, ūD = εū i is the intrinsic volume average velocity in macroscopic description.

d) Interstitial heat transfer coefficient
The interstitial heat transfer coefficient h sf for packed beds is typically calculated by using the correlation given by Wakao et al. [19].However, there is no general specific correlation for metal foams.Therefore, the following correlation developed by Zukauskas [20], is applied for copper and steel alloys metal foams (from 10 to 60 PPI); it is used in this work to estimate h sf :

e) Specific interfacial area
The specific area a sf is estimated by the correlation proposed by Zhao et al. [2].
where d p is the pore diameter expressed generally by the correlation proposed by Calmidi as function of pore density ω [2]: In order to generalize the results, the governing equations are transformed and reduced in dimensionless form by adopting the following non-dimensional variables: Then the governing equations become: K f , represent respectively the Reynolds number, the solid to fluid thermal conductivity ratio and the interstitial Biot number.

Dimensionless boundary conditions
• At the inlet of channel, the fluid enters the domain with uniform velocity and temperature which is different of wall temperature.• The downstream length of the channel is taken sufficiently long relatively to the high to ensure that fully developed conditions are applicable at the exit.
Then, the relevant dimensionless boundary conditions are summarized as follows:

Numerical methodology
The problem under investigation represented by the system of Equations ( 9) to ( 13) and boundary conditions ( 14) is solved by FVM [21].The SIMPLE algorithm is used for the pressure-velocity coupling.The nonlinear terms in the momentum equations, due to presence of the metal foam, are treated as source terms and linearized as described in Patankar [21].Due to the large variation of the physical quantities in certain regions of the channel, the grid distribution is non-uniform along both directions and concentrated at wall-fluid and metal foam-fluid interfaces.Convergence was monitored in terms of the relative residue, which was set to be lower than 10 −9 .The resulting system of algebraic equations is solved by the Line-by-Line method combined with TDMA.
The accuracy and the validity of the present calculation were checked by comparing the results with those

Results and discussion
The numerical calculations are conducted for fixed characteristic values of the studied channel filled completely with metal foam matrix, presented in Table 1.The effect of thermal conductivity ratio R k = K eff ,s /K eff ,f as well as interstitial heat transfer coefficient h s,f is discussed and analyzed.Other parameters are studied from their variation in specific ranges.Two R k are chosen, the  ratio of air (fluid) and two other materials often used in the manufacture of metal foams: FrCrAlloy and copper.Table 2 illustrates the thermal conductivity of the fluid (air) and the thermal conductivity of each material.The numerical results are represented especially for three transversal positions arbitrarily chosen: X = 0.1, X = 0.9, X = 2.
In the next subsections, we present first, the effect of R k , secondly the interstitial heat transfer coefficient, and finally, a discussion of how the metallic foam can give advantages to heat transfer process.

Effect of the thermal conductivity ratio R k
In this subsection, the effect of R k is presented with θ s = θ f in the inlet of channel.The variation of longitudinal non-dimensional temperatures is analyzed for three transversal positions: X = 0.1 (Fig. 4a), X = 0.9 (Fig. 4b) and X = 2 (Fig. 4c) respectively.R k used, are those about FrCrAlloy and copper foams.
Figure 4 shows that the effect of R k on dimensionless temperature profiles is important.The non-dimensional temperature difference between Air-Copper foam and Air-FrCrAlloy foam is increasing as a function of the nondimensional position X, and becomes large for X = 2. Comparing to the inlet of channel, the Air-Copper foam combination approaches quickly from the wall temperature, it is due to the large difference between solid thermal conductivities of both materials.This allows to the copper foam to transmit an amount of heat greater than that obtained using FrCrAlloy foam.Thus, as R k is higher, much the temperature profiles are rapidly approaching from the wall temperature, also, more this phenomenon is marked, much LTE is reached quickly.This equilibrium is reached more rapidly for most conductive foams such as copper metal foam.The non-dimensional temperatures are substantially reduced at X = 2; this diminution indicates the recovery of heat from the wall to the saturated porous medium.It means that temperature profiles tend to the wall temperature relatively to the used material.For the lower R k (FrCrAlloy foam), one observes that the saturated matrix by the fluid recuperates less amount of heat generated by the wall, unlike to materials with high R k such as copper foam.It should be noted that the difference between the temperatures of the two phases, solid and fluid, for the same materials, is very low.This is due to the significant capacity of foams in terms of

Effect of interstitial heat transfer coefficient h sf
To study the effect of h sf , a sensitivity analysis is conducted.It is expected that when h sf is high, the temperature difference between the fluid and solid phase becomes minimal, and LTE is reached quickly.The Nusselt number of two phases for conditions Da = 10 −4 , Re d = 100, varepsilon = 0.9, a sf = 1295 m 2 /m 3 , and θ s = θ f in the inlet of the channel, is shown in Figure 5 as a function of dimensionless coordinate X = x/H.In Figure 5, the nominal value of h sf is calculated using Zaukauskas correlation [20] which is valid for a pore Reynolds number between 40 and 10 3 .Effective values of h sf are artificially introduced by changing the nominal value: h eff = 0.01h sf , h eff = 0.1h sf , h eff = h sf .The Figure shows that the non-dimensional temperatures of solid phase are slightly lower compared to those of the fluid phase.The nominal value of h sf is used (Fig. 5c) and compared with two low h sf effective values (Figs.5a  and 5b).The local Nusselt number variation as function of X is very low, except at the entrance of the channel where the difference is remarkable, particularly for the fluid phase and for lowest effective h sf .By increasing h sf , the convergence of two-phase curves (solid and fluid) is becoming increasingly important to the point where the two phases take the same values for large h sf .Thus, it is found that from X = 1, the Nusselt number takes invariable values irrespective of h sf , this asymptotic value is equal to 9.78 for the fluid phase, and 9.83 for the solid phase.The study of h sf shows that the thermal equilibrium is reached rapidly for metallic foams; this, demonstrates the very significant increasing of performance in terms of heat transfer.
Note that the nominal h sf used in this paper is only an approximation because of the lack of a generalized correlation for this type of porous medium; however, further studies about the interstitial heat transfer coefficient would be welcome to give a better estimation.

Conclusion
The behavior of metallic foam support implemented into a flat channel was investigated.A numerical study was carried out to evaluate the following quantities: effect of solid to fluid thermal conductivity and the interstitial heat transfer.It has been shown that for the highest thermal conductivity ratios, the local thermal equilibrium is reached quickly, unlike to the lowest values, however, large values of thermal conductivity ratios, which are generally important, prove the significant contribution of these materials in terms of heat transfer, particularly by conduction.The results were used to estimate the influence of the convective contribution using metal insert structures with high porosity.It is relatively small compared to their conductive contribution between fluid and solid phases.It has been shown that such supports can bring a significant enhancement for heat transfer process due to their optimal properties.

Table 2 .
R k for different materials used in this work.