3D Numerical Modelling of Convective Heat Transfer through Two-sided Vertical Channel Symmetrically Filled with Metal Foams

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1 P Periodica Polytechnica Mechanical Engineering P 60(4), pp , 2016 DOI: /PPme.8511 Creative Commons Attribution b 3D Numerical Modelling o Convective Heat Transer through Two-sided Vertical Channel Symmetrically Filled with Metal Foams Bahareh Bidar 1, Farhad Shahraki 1*, Davod Mohebbi Kalhori 1 research article Received 19 August 2015; accepted ater revision 23 June 2016 Abstract A computational luid dynamics analysis o orced convective heat transer has been conducted numerically on the hydrodynamic and heat transer o airlow through vertical channel. The eects o airlow Reynolds number, metal oam porosity and thermal conductivity on the overall Nusselt number, pressure drop, maximum temperature and temperature distribution were considered. The novelty o the present study is the use o metal oams in a two-sided vertical channel and the quantiication o the heat transer enhancement compared to an empty channel or dierent oam material. Based on the generated results, it is observed that the heat transer rate rom the heated plate is the same or aluminium oam (porosity o 0.948) and copper oam (porosity o 0.877) against equal velocity range and heat lux conditions. Furthermore, it is noted that increasing the airlow velocity reduces the maximum temperature; however, the decrement is not linear. Results obtained rom the proposed model were successully compared with experimental data ound in the literature or rectangular metal oam heat exchangers. Keywords heat transer enhancement, numerical modelling, metal oam, orced convection, porous medium, Computational Fluid Dynamic (CFD) 1 Department o Chemical Engineering University o Sistan and Baluchestan P.O. Box Zahedan, Iran * Correspondig author, shahraki@eng.usb.ac.ir 1 Introduction Recently, there has been an increasing ocus on new class o materials with low densities and novel physical, mechanical, thermal, electrical and acoustic properties. Metal oams, which are a class o cellular materials improve eiciency and minimize the required weight and volume in energy producing or transerring systems. Due to their orms that have great strength to weight ratio, they are used in dierent engineering applications ranging rom mechanical to thermal [1-3]. Considering the possible implementation o air and metal oam in engineering systems, in the last decade, airlow through porous medium has been largely investigated by several authors; Bastawros [4], Dai et al. [5], Dukhan and Chen [6], Calmidi and Mahajan [7], Hwang et al. [8], Hsieh et al. [9], Kim et al. [10], Boomsma et al. [11], Poulikakos and Kazmierczak [12], Noh et al. [13], Kurtbas and Celik [14], Cavallini et al. [15], Mancin et al. [16-21], Tamayol and Hooman [22]. Bastawros [4] provided experimental measurements and modelling o the heat transer in porous metals subject to transverse airlow. Dai et al. [5] compared two conigurations; a metal oam and a louver ins as extended suraces or air cross low heat exchangers under the same thermal and hydraulic conditions. Dukhan and Chen [6] numerically and experimentally studied heat transer o air lowing through a block o aluminium oam, which is subjected to a constant heat lux on one side. He neglected conduction within the luid and applied the local thermal equilibrium assumption or solid and luid phases. Calmidi and Mahajan [7] experimentally and numerically studied orced convection in high-porosity metal oam over a range o porosities and pore densities using air as the working luid. In a similar study, Hwang et al. [8] showed that the volumetric heat transer coeicient increased with decreasing oam density at a given Reynolds number. Boomsma et al. [11] studied orced convective low in various compressed aluminium oams. They showed that these perormed well, not only in the heat transer enhancement, but they also made a signiicant improvement in the eiciency compared to conventional compact heat exchangers. Poulikakos and Kazmierczak [12] studied orced convection in a duct 3D Numerical Modelling o Convective Heat Transer

2 partially illed with a porous material. Mancin et al. [16-19] experimentally measured the heat transer coeicient and pressure drop values during air orced convection through a horizontal channel. They ound that oam thickness has no signiicant eect on the pressure drop. Moreover, Mancin et al. [20] investigated the heat transer and luid low behaviour o ive dierent 20 mm high copper oams. They also presented [21] the heat transer and luid low measurements during airlow through twenty-one dierent aluminium and copper oams. Two correlations or the heat transer coeicient and pressure drop calculations have been developed and validated. Tamayol and Hooman [22] numerically analysed orced convection heat transer through metal oam heat exchangersa using a thermal resistances approach and estimated the heat transer perormance o metal oam heat exchangers. Despite all the previously listed metal oam modelling and the high number o publications dealing with numerical and experimental analysis o metal oams, there are still some unaddressed issues o metal oam applications. The modelling investigations on metal oams as heat exchanger were conducted mainly or the unidirectional low condition whilst published modelling studies on metal oams subjected to multichannel low are rather scarce. The eect o Reynolds number, metal oam porosity and thermal conductivity on the heat transer and pressure drop have not been adequately investigated. In this work, a numerical model is established or a twosided channel illed with metal oam. The proposed model irst is presented and then is validated by comparing the velocity, pressure drop, temperature distribution and overall Nusselt number with an experimental study or rectangular metal oam heat exchangers under constant heat lux. Finally, the eect o airlow Reynolds number, metal oam porosity and thermal conductivity on the overall Nusselt number, pressure drop, maximum temperature and temperature distribution were considered to investigate their inluences on the heat transer through the porous channel compared to empty channel. In this work, more than 400% increase in the Nusselt number has been observed in some cases at high Reynolds numbers. 2 Numerical method 2.1 Description o heat transer system The schematic diagram o the heat transer system is shown in Fig. 1. The channel is well insulated and air is entered rom bottom and is withdrawn rom the top o channel. The channel is divided into two part by a lat heater with constant power output 100W. A 3 mm thick aluminum plate was inserted on either side o plate heater as can be seen in Fig. 1. Fig. 1 Schematic diagram and coordinate system o the problem The channel width 10 mm and height 150 mm on each side is illed with the metal oam block o dimensions 250(W) 150(L) 10(H) mm. In this work, we denote this section o heating system as test section. The metal oams used in the numerical study are aluminium and copper with 10 PPI and 10 mm thickness (Table 1). Table 1 Aluminium and copper oams characteristics Sample W L H/2 ε K p 10 7 C k s Aluminium 10 PPI Copper 10 PPI Studies have been done in dierent velocity o air. The inlet velocity is considered to be ully distributed in the transverse direction. In each case, the parabolic velocity proile is as ollows: x z V = V 2 2 max (1) where V max is in range o m/s. This velocity range usually encountered in electronic cooling such as tall printed circuit boards and or large-scale applications like data centers. It should be noticed that the ollowing assumptions are considered in our modelling: The luid low and heat transer conditions are ully developed. The thermo physical properties o the luid are constant. No luid crosses the solid material surace. Thermal radiation eect is neglected. Heat conduction through the luid is neglected. The eective properties o the porous medium are constant. Adiabatic wall condition is applied (heat loss is neglected). The porous medium is homogeneous and isotropic saturated. Local thermal equilibrium (LTE) condition is established. 194 Period. Polytech. Mech. Eng. B. Bidar, F. Shahraki, D. M. Kalhori

3 2.2 Governing equations The numerical modelling was perormed in the computational domain (Fig. 2). The incompressible, laminar, steady state luid low and heat transer in channel without metal oam (Region I and region III) is described by classical Navier Stokes equations, together with the continuity and corresponding energy equation. These equations are as ollows: on Lundgren experiment, the eective viscosity o saturated porous medium is equal to the viscosity o the luid [26]. Energy equation The heat transer in porous medium was modelled using the ollowing version o energy equation as the mathematical model or heat transer in porous medium C ( V T)= k 2 T (6) ( ) ρ p, e e where ρ is the luid density, C p, is the luid heat capacity at constant pressure, (ρ, C p, ) e is the eective volumetric heat capacity at constant pressure, k e is the eective thermal conductivity and v is the luid velocity ield. The velocity ield should be interpreted as the Darcy velocity vector, that is, the volume low rate per unit cross-section area. The average linear velocity, that is the velocity within pores, V, can be calculated as V= εv where ε is the porosity or luid s volume raction. The eective conductivity o porous medium, k e, is volume average o luid (k ) and solid (k s ) medium conductivities: ( ) k = εk + 1 ε k e s (7) Continuity equation Fig. 2 Computational domain V = 0 Momentum equations ρ ( V V)= p+ µ 2 V Energy equation C ( V T)= k 2 T ρ p where V = (u x, u y, u z ) is the low velocity vector. They are also used or region II when the empty channel was modelled. In illed channel, the low in porous medium (Region II) is governed by combination o continuity and momentum equations, which together orm Darcy-Brinkman-Forchheimer model [23-25]. These equations can be written as ollows: The continuity Eq. (2) and Momentum equations ρ µ ε V V µ ρ p V V c ε = + ερ 2 K K VV where μ is the dynamic viscosity o the luid, V is the Darcy velocity vector, ρ is the luid density, P is the pressure, ε is the porosity, K p is the permeability o porous medium and C is the dimensionless orm-drag coeicient. The apparent viscosity, μ in the Brinkman porous medium o the momentum equations should take a dierent value rom luid viscosity. Based p p (2) (3) (4) (5) where indices and s reer to luid and solid parts o porous medium, respectively. The equivalent volumetric heat capacity is calculated by: ( ρcp ) e = ε( ρ Cp, )+( 1 ε)( ρscps, ) In heat transer calculations, the average surace temperature o the aluminium plate is used. It is deined based on temperature excess, i.e. the dierence between wall temperature and inlet ambient temperature o air, as ollows: =( 1 ) T n T T ave Where n is the number o data points. The overall heat transer rate is calculated rom average heat transer coeicient deined in Eq. (10) or entire porous metal oam in the test section. h ave n i= 1 Q Qloss Q Q loss = have A T = 0 2 = 2A T heated w heated where Q is the power input, A heated is the heated area o aluminium plate (base area) and the actor 2 is to take into account the heat transer that takes place rom both sides o the aluminium plate. haveh Nu = (11) k where H is the oam thickness in the test section. In this work, the model neglects conduction in the luid and invokes the local thermal equilibrium (LTE) assumption or the solid and luid phases in the oam. In the absence o LTE, the single energy equation needs to be replaced with two energy (8) (9) (10) 3D Numerical Modelling o Convective Heat Transer

4 equations, one or the solid and another or the luid. Later Amiri and Vaai [27] investigated the validity o local thermal equilibrium (LTE) conditions or steady state, incompressible low through a porous medium. In addition, in Lee and Vaai [28], a theoretical investigation o orced convective low through a channel illed with a porous medium was presented. According to Minkowycz et al. [29], or most practical applications, the LTE is satisied when Sparrow number (Sp) > 100. In this work, a Sp > 100 is obtained or all numerical experiments, which satisies the assumption o LTE. 2.3 Computational domain and boundary conditions The channel is symmetrical in both x and z directions. Thereore, a quarter o that was selected as computational domain. It is divided to sub domain consists o test section (Region II), plate heater, aluminium plate and a pair o unheated sections located at upstream and downstream (Region I and region III) as shown in Fig. 2. Uniorm heat lux is applied to the heater. The air is entered to region I with constant temperature (T in =303K) and a parabolic velocity proile (V in ), then it passes through the porous zone along the plate heater and warms up. The static outlet pressure is ixed, and the remaining low variables are extrapolated rom the inlet o computational domain. The appropriate boundary conditions are given in Table 2 and Table 3. Table 2 Momentum boundary conditions or computational domain Boundary condition V(u x, u y, u z ) Inlet (mm) Aluminium plate wall (mm) Heater wall (mm) Heater + Aluminium plate bottom wall Heater + Al plate top wall Symmetry to x (mm) Symmetry to z (mm) Outlet y = y 160 x = y 160 x = 0.5 y = x 3.5 y = x y 170 x = y 170 z = 0 y = 170 u y = V in 0 (No slip) 0 (No slip) 0 (No slip) 0 (No slip) u x = 0 u = u x x = y z 0 u z = 0 u = u x y = 0 z z V = 0, p y = 0 Table 3 Energy boundary conditions or the computational domain Boundary condition Inlet (mm) Aluminium plate wall (mm) Heater wall (mm) Symmetry to x (mm) Symmetry to z (mm) Outlet Other walls y = y 160 x = y 160 x = y 170 x = y 170 z = 0 y = 170 T T in Q = 100 (W) Q = 100 (W) T x = 0 T z = 0 T y = 0 Adiabatic 2.4 Numerical Solution Procedure In the present work, the 3D channel has been modelled by CFD code. Equations (2) (4) and Eqs. (2), (5) (7) are solved separately using a inite volume code based on collocated grid system to discretize the governing equations. Figure 2 shows the grids in the studied section. The grid consists o hexahedral cells. A denser grid was used at the boundaries (solid-solid and solid-luid), in order to achieve greater numerical accuracy. The user deined unctions (UDF) are linked to model equations in order to apply the inlet parabolic velocity proile. Based on the inite volume approach, the SIMPLE algorithm [30] is employed to deal with the problem o pressure-velocity coupling and the second-order upwind calculation scheme is used to determinate momentum and energy balances. For each Reynolds number and each porous medium, the calculations were perormed with three dierent meshes; the sizes o grids were 99,280, 103,594, and 106,643. The results o Nusselt number and pressure drop were compared or all three meshes. It was ound that a grid with 106,643 cells is suicient to accurately predict the basic characteristics o low and heat transer with and without metal oam inserts. The relative error between two successive iterations was speciied by using a convergence criterion o 10 5 or each scaled residual component. To ensure the validity o numerical analysis, the numerical code was validated against the experimental results o Kamath et al. [31] or a vertical wind tunnel, containing symmetrically heated metal oams. 196 Period. Polytech. Mech. Eng. B. Bidar, F. Shahraki, D. M. Kalhori

5 3 Results and Discussion The study was conducted to discuss luid low and heat transer in a channel with and without o metal oams. In Section 3.1, the hydrodynamic analysis and in Section 3.2, the heat transer analysis have been explained. 3.1 Hydrodynamic analysis Hydrodynamic modelling was conducted with dierent inlet velocity and without energy calculations Velocity ield analysis Figure 3 (a) and 3 (b) show velocity distribution through aluminium oam or two maximum velocities (0.54 and 2.9 m/s). More mixing low corresponding to high velocity is seen at the inlet o the porous zone. By decreasing the cross-sectional area o channel at the inlet o test section, the maximum velocity o the luid increases signiicantly and shits towards aluminium plate, which can be called velocity ield rebuilding Pressure drop analysis The pressure drop results are compared in Fig. 4 (a) and 4 (b) or aluminium and copper oams with dierent airlow velocity. There is a qualitative agreement between the model and the experimental data. The greater airlow velocity made more variation in the local pressure drop, due to ΔP = (V2). It can be seen that local pressure drop or aluminium oam is lower than copper oam. It is due to lower orm drag to viscous drag ratio o high porosity metal oam than low porosity metal oam. (a) (b) Fig. 3 Velocity distribution or aluminium oam, (a) V max = 0.54 m/s (b) V max = 2.9 m/s (a) (b) Fig. 4 Model validation: (a) variation o pressure drop in aluminium oam with 10 PPI versus velocity (b) variation o pressure drop in copper oam with 10 PPI versus velocity 3D Numerical Modelling o Convective Heat Transer

6 3.2 Heat transer analysis Eect o velocity on temperature distribution through empty and metal oam illed channels Figure 5 (a) (c) presents temperature distribution o an empty channel along axial direction versus dierent inlet velocity. As the airlow velocity increases, the thermal boundary layer thickness on the aluminum plate decreases and the heat transer rom aluminum plate increases. However, the temperature distribution was decreased in the luid region due to high velocity. It shows that, a large amount o air leaves channel without temperature change. Figures 6 (a) () present temperature distribution o illed channel along axial direction versus dierent airlow velocity or both aluminium and copper oams. This is obvious that heat transer surace in the illed channel is much higher than empty channel. This is one o the advantages o employing metal oams that can provide a large surace area to volume ratio and it can intense mixing o low through metal oam in the heat transer systems. By using o metal oams, the surace temperature o the heated plate is signiicantly lower than that in the empty channel and temperature distribution gets smoother Eect o thermal conductivity o metal oams and Reynolds number on maximum temperature The heat removal mechanism rom the heater plate is conduction through the oam ligaments and convection through the oam. With increasing the airlow velocity, convective heat transer rate increases and aluminium plate temperature decreases. Maximum temperature decreases with increasing airlow velocity and consequently, Reynolds number as shown in Fig. 7. However, the decrement is not linear. As can be seen, ater Re = 3230 (V = 2.5 m/s), temperature change is negligible and urther increasing o velocity does not have a signiicant eect on the heat transer rate. It is evident that by using o metal oam with low thermal conductivity, the heated plate cannot transer heat well to metal oam and heat accumulation in the plate leads to increase surace temperature Eect o Reynolds number on the overall Nusselt number The most important parameters aecting the heat transer rom heater are Reynolds number, porosity and permeability o metal oams. It can be observed rom Fig. 8 and Fig. 9 that metal oams led to heat transer enhancement. Overall Nusselt number or illed channel is higher than empty channel or both investigated cases. The maximum heat transer rate was observed at Re=3970, which was signiicantly more than empty channel. The heat transer perormance o aluminium and copper oams is compared in Fig. 10. The higher thermal conductivity o copper oam did not contribute signiicantly to increase heat transer compared to aluminium oam. Consequently, copper oam with porosity o and aluminium oam with porosity o gave the same heat transer perormance or the investigated airlow velocity range and uniorm heat lux conditions. (a) (b) (c) Fig. 5 Temperature distribution along axial direction or empty channel (a) V max = 0.54 m/s, (b) V max = 1.24 m/s, (c) V max = 2.36 m/s 198 Period. Polytech. Mech. Eng. B. Bidar, F. Shahraki, D. M. Kalhori

7 (a) (b) (c) (d) (e) () Fig. 6 Temperature distribution o illed channel along axial direction (a) copper oam with V max = 0.54 m/s, (b) aluminium oam with V max = 0.54 m/s, (c) copper oam with V max = 1.24 m/s, (d) aluminium oam with V max = 1.24 m/s, (e) copper oam with V max = 2.36 m/s, () aluminium oam with V max = 2.36 m/s Fig. 7 Maximum temperature as a unction o Reynolds number or aluminium and copper oams Fig. 8 Overall Nusselt number as a unction o Reynolds number or empty channel and channel with copper oam 3D Numerical Modelling o Convective Heat Transer

8 Fig. 9 Overall Nusselt number as a unction o Reynolds number or empty channel and channel with aluminium oam Fig. 10 Heat transer perormance o aluminium and copper oams 4 Conclusions In the present work, heat transer enhancement o ully developed laminar low through a two-sided vertical channel which is illed with aluminium and copper oams is numerically investigated. The channel with and without metal oams were modelled using Darcy-Brinkman-Forchheimer, classical Navier-Stokes equations and corresponding energy equations. A inite volume method was utilized to solve the governing equations. The proposed model were successully validated with experimental data ound in the literature or rectangular metal oam heat exchangers. The eect o Reynolds number, porosity and thermal conductivity on surace temperature distribution and overall Nusselt number had been analysed. Based on the results, the pressure drop increased by either increasing the airlow velocity or decreasing the metal oam porosity. The heat transer actors, overall Nusselt number in the channel illed with metal oams are higher than empty channel. Thereore, a signiicant increase in heat transer is obtained in two-sided channel illed with metal oams. Maximum temperature decreased with increasing o Reynolds number. It is shown that at high Reynolds numbers, the convection heat transer has been dominated. Moreover, the heat transer enhancement through a metal oam illed channel increases with inlet velocity or studied ranges. Although, copper oam has higher thermal conductivity, but it could not contribute signiicantly to increase the heat transer compared to aluminium oam. It is recommended that aluminium oam is better than the more expensive copper oam with similar PPI in this system. It is revealed that the proposed numerical model can eiciently provide useul inormation or design o multi-channel heat transer system or a velocity range usually encountered in electronic cooling such as tall printed circuit boards and or large-scale applications like data centers. Nomenclature A Surace area o the aluminium plate [m 2 ] C Form-drag coeicient [mm -1 ] C p, Heat capacity o luid [J kg -1 K -1 ] C p,s Heat capacity o solid [J kg -1 K -1 ] H Foam thickness [mm] h Heat transer coeicient [W m -2 K -1 ] K p Permeability o porous medium [m 2 ] k Thermal conductivity o luid [W m -1 K -1 ] k s Thermal conductivity o solid [W m -1 K -1 ] k e Eective thermal conductivity [W m -1 K -1 ] L Foam height [mm] LTE Local Thermal Equilibrium Nu Overall Nusselt number n Number o temperature sample P Pressure [Pa] PPI Number o pores per linear inch[in -1 ] Q Heat input [W] Re Reynolds number Sp Sparrow number T Temperature [K] u Velocity component [m s -1 ] V Velocity vector[m s -1 ] V max Maximum velocity [m s -1 ] V in Inlet air velocity [m s -1 ] W Channel width [mm] x Coordinate [mm] y Coordinate [mm] z Coordinate [mm] ΔT Excess temperature [ C] Δp Pressure drop [Pa] 200 Period. Polytech. Mech. Eng. B. Bidar, F. Shahraki, D. M. Kalhori

9 Greek symbols ρ Density o luid [kg m -3 ] µ Dynamic viscosity o luid [kg m -1 s -1 ] µ Viscosity [kg m -1 s -1 ] ε Porosity o porous medium [ ] Subscripts ave e in loss p p s x y z w Reerences Average conditions Eective Fluid Inlet Heat loss through the insulation Permeability Constant pressure Solid Coordinate Coordinate Coordinate wall Ambient conditions [1] Ozmat, B., Leyda, B., Benson, B. "Thermal applications o open cell metal oams." Materials and Manuacturing Processes. 19(5), pp DOI: /AMP [2] Odabaee, M., Hooman, K. "Application o metal oams in air cooled condensers or geothermal power plants: an optimization study." International Communications in Heat and Mass Transer. 38(7), pp DOI: /j.icheatmasstranser [3] Ashby, M. F., Evans, A. G., Fleck, N. A., Gibson, L. J., Hutchingson, J. W., Wadley, H. N. G. "Metal oams: a design guide." Butterworth Heinemann, Woburn [4] Bastawros, F. "Eectiveness o open cell metallic oams or high power electronic cooling." 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10 [27] Amiri, A., Vaai, K. "Analysis o dispersion eects and non thermal equilibrium, non Darcian, variable porosity incompressible low through porous media." International Journal o Heat and Mass Transer. 37(6), pp DOI: / (94) [28] Lee, D.-Y., Vaai, K. "Analytical characterization and conceptual assessment o solid and luid temperature dierentials in porous media." International Journal o Heat and Mass Transer. 42(3), pp DOI: /S (98) [29] Minkowycz, W. J., Haji Sheikh, A., Vaai, K. "On departure rom local thermal equilibrium in porous media due to a rapidly changing heat source: the Sparrow number." International Journal o Heat and Mass Transer. 42(18), pp DOI: /S (99) [30] Van Doormall, J. P., Raithby, G. D. "Enhancements o the Simple Method or Predicting Incompressible Fluid Flow." Numerical Heat Transer. Part A: Applications. 7(2), pp DOI: / [31] Kamath, P. M., Balaji, C., Venkateshan, S. P. "Convection heat transer rom aluminium and copper oams in a vertical channel An experimental study." International Journal o Thermal Sciences. 64, pp DOI: /j.ijthermalsci Period. Polytech. Mech. Eng. B. Bidar, F. Shahraki, D. M. Kalhori