Numerical and experimental study of forced convection in graphite foams of different configurations

Applied Thermal Engineering 30 (2010) 520–532

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Numerical and experimental study of forced convection in graphite foams of different configurations K.C. Leong a,*, H.Y. Li a, L.W. Jin a, J.C. Chai b a b

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore Mechanical Engineering Department, The Petroleum Institute Abu Dhabi, United Arab Emirates

a r t i c l e

i n f o

Article history: Received 3 July 2009 Accepted 14 October 2009 Available online 20 October 2009 Keywords: Forced convection Graphite foams Structured foam Porous media

a b s t r a c t Forced convection heat transfer in a channel with different configurations of graphite foams is experimentally and numerically studied in this paper. The physical properties of graphite foams such as the porosity, pore diameter, density, permeability and Forchheimer coefficient are determined experimentally. The local temperatures at the surface of the heat source and the pressure drops across different configurations of graphite foams are measured. In the numerical simulations, the Navier–Stokes and Brinkman–Forchheimer equations are used to model the fluid flow in the open and porous regions, respectively. The local thermal non-equilibrium model is adopted in the energy equations to evaluate the solid and fluid temperatures. Comparisons are made between the experimental and simulation results. The results showed that the solid block foam has the best heat transfer performance at the expense of high pressure drop. However, the proposed configurations can achieve relatively good enhancement of heat transfer at moderate pressure drop. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The investigation of forced convective heat transfer in a channel filled with porous media is of practical interest in the industry. It has been demonstrated that heat sinks made of porous media with high thermal conductivity and large surface area improve heat transfer performance and thus are widely used in various industry applications such as heat exchangers, chemical reactors and pipes of different arrangements [1–3]. The fundamentals of heat transfer in porous media have been studied extensively in the past decades [4–9]. Recent investigations of heat transfer in porous media with spherical open cell microstructure such as metal and graphite foams indicate that such materials offer more promising prospects for use in heat sinks [10–12]. Fig. 1 shows the typical internal structure of ‘‘Poco” graphite foam developed at Oak Ridge National Laboratory (ORNL), USA in 1997 [13]. This material consists of predominantly spherical pores with small openings between the ligaments. The bulk thermal conductivity of graphite foam can be as high as 150 W/m K which is almost equivalent to that of dense aluminum alloys [14]. However, the mass density of graphite foams is only about 20% of aluminum. The very large surface area to volume ratio (20,000 m2/m3) of graphite foams results in high overall heat transfer. The prominent advantages of this material make it a promising material to be used * Corresponding author. Tel.: +65 6790 5596; fax: +65 6792 4062. E-mail address: [email protected] (K.C. Leong). 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.10.014

in many thermal management applications. Previous investigations showed that better heat transfer performance can be achieved by using open-cell graphite foams in conventional heat sinks although a larger pressure drop is incurred [10,12,15,16]. Comprehensive investigations have been performed by many investigators on the thermal properties of the graphite foams both experimentally and numerically [14,17–20]. The results of these investigations provide benchmarks for the study of convection heat transfer in graphite foams. Due to the high thermal conductivity of the ligaments and the large surface areas, graphite foams give a better heat transfer performance compared with traditional heat sinks. The ligament can entrain the heat from the heat source easily; and the heat is taken away by the fluid through convection in the large exposed surface area between the solid and fluid phases. However, the extremely low permeability of graphite foams results in the large flow resistance as fluid flows through the microstructure pores. High pressure drop requires substantial pumping power which is not favorable to thermal management systems. Researchers have experimented with different configurations of graphite foams to reduce their pressure drops [10,12]. The pioneer study using graphite foams of different configurations in thermal management systems was carried out by Gallego and Klett [10]. Some preliminary experimental data of pressure drop and heat transfer were provided. Their results showed that heat transfer as well as pressure drop was significantly affected by the configuration of graphite foams. Among all the tested samples, the solid block foam has the highest pressure drop. Williams and Roux

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Nomenclature A, B CE Cpf Da Dh dp hsf H K kf kfe kse L m, n Nu Pr P p Q q00 Re S T u

constants Forchheimer coefficient specific heat of fluid (J/kg K) Darcy number based on channel height hydraulic diameter of channel = 4 Ac/P (m) pore diameter of porous media (m) heat transfer coefficient between solid and fluid phases (W/m2 K) height of graphite foams (m) permeability of the porous medium (m2) thermal conductivity of fluid phase (W/m K) effective thermal conductivity of fluid phase (W/m K) effective thermal conductivity of solid phase (W/m K) length of the graphite foams (m) constants Nusselt number Prandtl number perimeter (m) pressure (Pa) heating power (W) heat flux (W/m2) pore diameter based Reynolds number source term temperature (°C) velocity component in the x-direction (m/s)

[12] performed an experimental study to evaluate the cooling effects of graphite foams and microfibrous material as heat sinks. Their results showed that the graphite foams with zig-zag configuration (Fig. 2b) achieved the highest heat transfer among the nine tested samples. Since it is impractical to measure the velocity and temperature distributions inside the graphite foams, numerical simulation plays an important role in investigating the transport phenomena in graphite foams. Straatman et al. [21] simulated the case of water flowing through the solid graphite foams block. The temperature distributions in the graphite foams were obtained. From the above literature survey, it is clear that the pressure drops in different configurations of graphite foam are significantly

V V W x, y, z

volume (m3) velocity vector (m/s) width of graphite foams (m) coordinates

Greek symbols specific surface area of porous media (m2/m3) density (kg/m3) d uncertainty e porosity l dynamic viscosity (kg/m s) u average quantity / dependent variable C diffusion coefficient

asf q

Subscripts ave average f fluid phase in inlet i, j direction of component s solid phase t total w wall x local

reduced compared to that of the solid block foams. However, there is no reported detailed study on both the pressure drop and heat transfer in the solid block and other configurations of graphite foams. In particular, there is no reported numerical study of forced convection in the different configurations of graphite foams. The objective of this paper is to study the pressure drops and heat transfer in different configurations of graphite foams. In the present work, two proposed configurations of graphite foams were fabricated based on the solid block graphite foam. An experimental facility was constructed to study the pressure drops and heat transfer in all the designed foams. To better understand the transport phenomena in all the graphite foams, numerical simulations were performed. The flow fields in all the foams as well as their associated heat transfer were investigated. Based on the experimental and numerical results, differences in the pressure drops and heat transfer in all the configurations of graphite foams were explored. 2. Experiments 2.1. Graphite foam configurations

Fig. 1. Typical internal structure of ‘‘Poco” graphite foam of 72.8% porosity.

The material used in this study is ‘‘Poco” graphite foam which is licensed by ORNL with a measured porosity of 72.8% [13]. Three different configurations as shown in Fig. 2 are studied in the present work. All the tested samples, namely, block foam (BLK) (Fig. 2), zig-zag foam (ZZG) (Fig. 2b) and baffle foam (BAF) (Fig. 2c), have the same external dimensions of 50 (L)  50 (W)  25 (H) mm. The designed configurations were fabricated by Electron Discharge Machining (EDM). Both the ZZG and BAF foams shown in Fig. 2b and c were designed in such a way that part of the fluid can flow through the channel without passing through the porous foams. The manufacturing constraints for achieving the minimum thickness of the foam walls were considered. In ZZG and BAF foams, the 4-mm-thick foam base is used to conduct heat effectively from

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K.C. Leong et al. / Applied Thermal Engineering 30 (2010) 520–532 Table 1 Parameters of designed structures. Parameters

BLK

BAF

External dimensions, W  L  H (mm) Foam wall thickness (mm) Foam wall number Foam wall length (mm) Slot gap width (mm) Base height (mm) Total surface area (mm2)

50  50  25 – 2–3 23 14/22 3.625 4 17,976

ZZG 2–3 10 41.5 3 4 17,913

areas on heat transfer is minimized. Important parameters of the designed configurations are given in Table 1. 2.2. Experimental setup

Fig. 2. Different configurations of graphite foams: (a) BLK; (b) ZZG; (c) BAF.

the heater surface to the designed configurations. The thickness of the foam walls is between 2 and 3 mm and these walls are aligned perpendicular to the inlet flow direction. Gaps in each of the foam wall allow the fluid to flow though. The slots between successive foam walls are connected together through the gaps. With these designs, the pressure drops would be reduced. Since it is rather complicated to determine the total surface area exposed to the fluid in the pore level in ZZG and BAF foams, the geometries of the configurations were determined based on the fact that the designed ZZG and BAF foams should possess approximately the same total wall surface area. Thus, the effect of different wall surface

The schematic diagram of the experimental facility is shown in Fig. 3a The graphite foams with different configurations were placed in a duct made of Teflon for insulation purpose. Two taps for pressure sensors (VALIDYNE DP15) are placed at the two ends of test section to measure the total pressure drop across the sample. Air flow was provided by an auto-balance compressor (SSR EP15SE). A hot-wire (DANTEC 55P11) sensor was mounted at the center of the two ends packed with 40 mesh woven screen discs. The packed screen provides a uniform velocity profile so that the measured velocity is the cross-section averaged velocity through the test section. A flow regulator and a valve were installed to adjust the flow velocity through the test section. The data measured by the thermocouples, pressure transducer and hot-wire were acquired by a data acquisition system, which consists of a signal board (NI CB-68LP), analog and digital I/O boards (NI PCI-6224) and an operating software (NI DAQmx 4.0) within a workstation. The temperature, pressure drop and velocity measured by the sensors were input into the data acquisition board through different channels on the signal board. A 32-channel data cable was used to connect the signal and I/O boards. The data acquisition system is compatible with VI Logger and ‘‘LabVIEW” software, which is easy to be programmed based on user-end applications. The analog inputs of PCI-6224 have calibration circuitry to correct gain and offset errors. It can be used to minimize AI and AO errors caused by time and temperature drift at run time. A three-dimensional close-up view of the test section is shown in Fig. 3b. A film heater was installed at the bottom of the channel and a 1-mm-thick copper plate with ten uniform narrow slots perpendicular to the flow direction was attached on the surface of the film heater. Ten K-type thermocouples were placed in the slots at 0.25, 0.75, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75, 4.25, 4.75 cm from the inlet to the test foam. Highly conductive thermal grease (k = 8.5 W/m K) was applied as filling material to reduce the contact resistances across the interfaces of the film heater, copper plate and test sample. By clamping the channel tightly, the temperatures of the graphite foam surface and copper plate at the bottom of the channel were assumed to be the same in the measurements. During the experiments, the temperatures were monitored for a period of three minutes to ensure that steady state has been reached. The thermocouples, pressure sensor and hot-wire were calibrated carefully before the commencement of the experiments. 2.3. Experimental procedures Before the commencement of the experiments, the physical properties of tested graphite foam were determined. The pore diameter and porosity were measured by a scanning electron microscope and Ultrapycnometer 100, respectively. The permeability and Forchheimer coefficient were determined on the facility

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Fig. 3. (a) Schematic diagram of test facility (b) three-dimensional view of test section.

described above at unheated condition. Details will be presented in the next section. To investigate the pressure drop and heat transfer characteristics in different configurations of graphite foams subjected to various heating powers, a series of experiments was conducted. For each configuration and during a typical experiment run, the heating power was set to a prescribed value and the inlet velocity was varied by adjusting the valve installed on the air supply line. The temperature readings were monitored every 3 min to ensure that steady state had been achieved. The steady state condition was assumed if the maximum temperature variation indicated in all the thermocouples is within 0.5 °C. This usually took about 10–15 min depending on the air velocity. The steady state velocity, pressure drop and temperatures were then recorded by the data acquisition board. The heating power was varied from 20 to 60 W. The same experimental procedure was repeated while the heating power is increased. The graphite foam in the channel was removed after completing the experiments for different

heating powers. The entire system was cooled to the ambient temperature before another configuration was studied. 2.4. Determination of properties Porosity is defined as the total void volume divided by the total volume including the solid matrix and void volume, which can be expressed as

e¼

Vt  Vs Vt

ð1Þ

where Vt and Vs are the total volume and the volume occupied by the solid phase, respectively. In this study, the porosity of the test graphite foam is measured by the ‘‘Ultrapycnometer 100” equipment, which can allow the density and the volume of the solid components in the porous medium to be determined. For the tested graphite foams, a sample with dimensions of 2 cm  2 cm  2 cm is measured. It is found that the solid density and the solid volume

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are 2.13 g/cm3 and 2.17 cm3. The porosity of tested sample is found to be 72.8%. The pore diameters of the tested graphite foam were measured by Image-Pro software based on photographs obtained by a Scanning Electron Microscope (SEM). With built-in functions, ImagePro is able to recognize clusters of brighter pixels as cells and to calculate the diameter of every cell. To further reduce the random effect of the pore diameter measurement, three samples were measured and the average value was used as the pore diameter. It is found that the average pore diameter of 72.8% porosity foam is 0.31 mm, which is close to the manufacturer’s data of 0.35 mm. The permeability K and Forchheimer coefficient CE of the graphite foams were calculated using experimentally measured pressure drop and velocity through the following equation [22]:

dp l CE ¼  u  qf pffiffiffiffi u2 dx K K

Dp ¼ Au þ Bu2 L

ð3Þ

where

A¼

l K

;

CE B ¼ qf pffiffiffiffi K

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 dq dq dq dq ¼ dv þ    þ dy þ    þ dz dv dy dz

ð5Þ

In the present experiment, the uncertainties of the measured data were assumed to be independent and random with normal distribution. Using Eq. (5), the uncertainties of Reynolds number and local Nusselt number were 3.9% and 5.7%, respectively. The uncertainties of permeability K and Forchheimer coefficient CE were determined similarly. The uncertainties of K and CE were 3.2% and 11.2%, respectively. 3. Numerical simulations

ð2Þ

Using Eq. (2), the permeability and the Forchheimer coefficient can be determined from a plot of pressure drop versus velocity. Eq. (2) can be simplified as



dv, . . . , dy, . . . , dz, and the measured values are used to compute the function q(v, . . . , y, . . ., z), then the uncertainty in q is given by

ð4Þ

A and B are the constants to be determined. Dp is the pressure drop and L is the length of the graphite foams. Fig. 4 shows the experimental results of pressure drop per unit length versus the velocity for air flow across the 72.8% porosity of graphite foams. A least-squares fit was performed to obtain the coefficients of A and B. The regression analysis showed a fit with a regression coefficient of R2 > 99.95. From the fitting results, the values of A and B can be obtained. The estimation of experimental errors is based on the uncertainties of the individual measured quantities. The accuracies of the thermocouple and pressure transducer readings were within ±0.5 °C and ±0.25% of full-scale, respectively. The accuracy of the velocity measured by the hot-wire anemometer was ±0.01 m/s. the uncertainties of temperature, velocity and pressure were estimated to be 3.0%, 2.0% and 2.0%, respectively. The uncertainties of the length and area were 1.0% and 1.5%, respectively. The uncertainties of the Reynolds number and local Nusselt number were calculated by the method introduced by Taylor [23]. If v, . . . , y, . . . , z are measured quantities with uncertainties

3.1. Statement of problem Numerical simulations were performed to better understand the transport phenomena in porous graphite foams. The schematic of the numerical domain is shown in Fig. 5. It consists of a channel with graphite foam placed in the middle of the channel. The lengths of L1 and L2 are consistent with the experimental setup. It removes both the inlet and outlet influence on the flow in the domain of interest, i.e. the graphite foams. Fluid flows into the channel at the inlet with a uniform velocity profile at a given temperature. A constant heat flux is imposed at the substrate of the graphite foam. The remaining walls of the channel are adiabatic. Heat is conducted from the foam base and through the foam to be carried away by the flowing fluid. 3.2. Governing equations Referring to the foam configurations proposed in Fig. 2, the computational domain contains both the porous and open regions (slots and gaps). For the open region, the traditional continuity and Navier–Stokes momentum equations can be used directly. For the porous region, the volume averaging method is usually used to derive the macroscopic momentum and energy equations [22]. Using the volume averaged method and for any quantity associated with the fluid or the solid, the volume averaged value is

hui ¼

1 V

Z

udV

ð6Þ

V

The symbol h i is the volume average. V is a representative elementary volume which is occupied by both fluid and solid phases. The intrinsic phase average is represented by

hui ii ¼

1 Vi

Z Vi

ui dV

ð7Þ

where i refers to either the solid or fluid phase. Since e = Vf /V, the quantities of the solid and fluid phases can be obtained by taking the average over the control volume V. With the assumptions of laminar flow and constant fluid properties and by using the volume averaged method, the steady state mass and momentum conservation equations for both fluid flows in the porous and open regions can be cast into a general form as

@hui i ¼0 @xi 

ð8Þ 





qf @hui i @hpi l @ @hui i hu i þ ¼ þ Sf @xi e2 j @xj e @xj @xj

Fig. 4. Pressure drop versus air velocity for 72.8% porosity graphite foam.

ð9Þ

where e is the porosity and i, j represents the component of the coordinate directions. Sf is the source term which in the porous region can be written as [24]

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Fig. 5. Schematic diagram of simulation domain.

Sf ¼ 

l K

qf C E hui i  pffiffiffiffi jVjhui i K

ð10Þ

where CE and K are the Forchheimer coefficient and permeability of the porous media, respectively. The two terms on the right hand side of Eq. (10) describe the microscopic viscous and inertial forces caused by the porous media, respectively. Eq. (9) with the source term shown in Eq. (10) is the Brinkman–Forchheimer momentum equation, which is an extension of the Darcy model. For the energy equation in the porous region, two different models can be employed. They are the local thermal equilibrium (LTE) and local thermal non-equilibrium (LTNE) models. The LTE model neglects the heat transfer between the solid and fluid phases, i.e. hTsi = hTfi at any location inside of the porous media. However, in the LTNE model, there is an additional term which accounts for the convection heat transfer between the solid and fluid phases. In this case, the temperature of the fluid phase is generally not equal to the solid phase. In the present study, the LTNE model is adopted due to the large difference of thermal conductivities between the solid and fluid phases [25]. Hence, the energy equation in the porous and open regions for both the solid and fluid phases can be written in the following forms, respectively:

  @ @hT s i  ST ¼ 0 kse @xi @xi ðqC p Þf hui i

  @hT f i @hT f i @ þ ST ¼ kfe @xi @xi @xi

ð11Þ

ð12Þ

where kse and kfe are the effective thermal conductivity for the solid and fluid phases, respectively. ST is the source term which accounts for the convection heat transfer at the solid and fluid interface. It is expressed mathematically as

ST ¼ hsf asf ðhT s i  hT f iÞ

ð13Þ

where asf is the interior surface area to volume ratio. It should be noted that by employing the volume averaging method, the velocity obtained from Eqs. (8)–(12) is the intrinsic average velocity which can be measured. The real velocity inside the porous medium can be determined by the Dupuit–Forchheimer relationship upi = e huii [22]. For simplicity, the symbol h i will be dropped from henceforth. With this formulation, only one set of governing equations is solved in the entire calculation domain. The additional source term Sf in Eq. (10) and ST in Eqs. (11)–(13) are non-zero only within the porous regions. Two different energy equations have to be solved separately and they are coupled together with the source term in Eq. (13). The appropriate properties such as the porosity, density, viscosity, specific heat and thermal conductivity are specified in the porous and

open regions, respectively. The value of the porosity in the open region was set to 1. Since the open region contains only the fluid phase, the solid temperature in this region is meaningless. Thus, an extremely small value of 1017 W/m K was prescribed for the thermal conductivity of the solid phase in the open region to avoid heat conduction to the porous region. The harmonic-mean method [26] was used to treat the thermophysical properties at the interface between the porous and open regions; no other special efforts are needed to deal with the interface between different regions. In this study, graphite foam is considered to be homogeneous. The effective thermal conductivities were calculated based on the model proposed by Tee et al. [17]

kse ¼ kf ð1  tÞ2 þ

2kst kf tð1  tÞ ksl ksj t2 þ ksl t þ ksj ð1  tÞ kf t þ kst ð1  tÞ

ð14Þ

where

t¼

  1 1 4 þ cos cos1 ð2e  1Þ þ p 2 3 3

ð15Þ

Here, t is the normalized thickness of the solid component which can be determined by the foam porosity e. ksl, ksj, kst stand for the thermal conductivities at different directions of the graphite foam, viz. longitudinal, strut juncture and transverse orientations, respectively. For simplicity, the thermal conductivities of the graphite foams in different directions were assumed to be the same. The interior surface area to volume ratio is obtained using [20]:

asf ¼

pdp l

3

ð3l  2dp Þ

ð16Þ

where l and dp are the length of cubic model and the diameter of the pore, respectively. Detailed information of the cubic model of the graphite foams can be found in the work of Yu et al. [20]. The interfacial heat transfer coefficient hsf is determined by an empirical equation proposed by Klett et al. [27]

Nusf ¼

hsf dp ¼ mRen Pr0:36 kf

ð17Þ

where m and n are the constants which are 0.0158 and 0.7225, respectively for air flow through ‘‘Poco” graphite foam with 75% porosity. It was reported that thermal dispersion is not significant when the difference of the thermal conductivity between the solid and fluid phases is large [25]. Therefore, the effect of thermal dispersion in graphite foams is neglected.

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3.3. Boundary conditions The numerical boundary conditions are prescribed according to the experimental conditions as stated in the foregoing section. A uniform velocity profile is specified at the inlet. The non-slip boundary condition is used for all the walls. Outflow condition is enforced at the outlet. The fluid enters the channel with a constant temperature of Tin. Zero normal gradient condition is imposed at the outlet for the fluid temperature field. The thermal condition of the solid phase is assumed to be insulated at both the inlet and outlet. Adiabatic conditions are imposed at y = 0, y = W, z = H for both the solid and fluid phases. A constant heat flux is imposed directly below the graphite foams. All remaining walls are insulated except for the heating wall. The boundary conditions for the two phases at the heated wall are mathematically expressed as [28]

  @T f @T s  kfe þ kse ¼ q00 ; @z @z

Ts ¼ Tf

ð18Þ

3.4. Numerical procedure The continuity (Eq. (8)), momentum (Eq. (9)) and energy equations (Eqs. (11) and (12)) can be written as a general convectivediffusive equation of the form

quj

  @/ @ @/ ¼ C þS @xj @xj @xj

ð19Þ

where /, C and S are commonly referred as the dependent variables, ‘‘diffusion” coefficient and source term in the FVM literature [26]. These equations were solved using the finite volume method [26]. A staggered grid with scalar and velocities defined respectively at the nodes and surfaces of the control volumes was employed. The power-law was used to treat the combined convection–diffusion effect. The velocity–pressure coupling was handled using the SIMPLER algorithm. The resulting algebraic equations were solved using a line-by-line tri-diagonal matrix algorithm. As the energy equations (for the solid and fluid) are decoupled from the continuity and momentum equations, they were solved after obtaining the converged velocity field. A relative error of less than 105 was prescribed for both the velocity and temperature fields between the successive iterations to satisfy the convergence criterion. 3.5. Code validation The present code was validated against the results of Alazmi and Vafai [29]. Laminar flow through a configuration with an infinite porous layer located above an open layer was considered. The third model of handling the interfacial velocity and temperature between the porous and open regions presented by Alazmi and Vafai [29] was adopted in the validation. Fig. 6 compares the velocity field and temperature distribution obtained from the current study and by Alazmi and Vafai [29]. The current results for the velocity and temperature distributions are almost identical to those obtained by Alazmi and Vafai. This demonstrates that the present approach can handle problems with coexisting open and porous regions. Further validation was carried out to test the ability of the present code in dealing with the coupling of velocity–pressure at high Reynolds number. The simulation case is based on fluid flow in the porous plug domain which has a porous region located between the two open regions [30]. The Reynolds number was chosen to be 1000. Fig. 7a and b show the dimensionless velocity profiles at x/H = 2.5, 7.5 and 12.5 from the results of Betchen et al. [30] and the current simulation, respectively. Compared with the two figures, the velocity profiles at different locations are similar. The ability of the code to deal with high Reynolds number is demonstrated.

Fig. 6. (a) Velocity profile and (b) temperature profile in the studied problem.

Fig. 7. (a) Results from Betchen et al. [30] and (b) present results in the porous plug domain.

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Fig. 8. Dimensionless velocity profiles in BAF foam at: (a) representative locations (b) x/Dh = 0.15; (c) x/Dh = 0.05; (d) x/Dh = 0.18; (e) x/Dh = 0.25.

4. Results and discussion Due to symmetry, only half of the BLK and BAF foams was simulated. Grid dependence tests were performed separately for each

configuration. For BLK, BAF and ZZG structures, mesh sizes of 82  22  22, 102  38  24 and 106  36  26 control volumes (CVs) are sufficient to achieve grid independent solutions, respectively.

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The dimensionless parameter x/Dh is introduced to express the non-dimensional locations of the temperatures along the tested sample. Dh is the hydraulic diameter of the channel which is defined as

Dh ¼

4Ac P

ð20Þ

where Ac and P are the cross-sectional area and perimeter of the channel, respectively. In the calculations of experimental and numerical data, the pore diameter based Re and Nux are defined as

Re ¼

qudp l

ð21Þ

and

Nux ¼

Q dp AðT x  T in Þ kf

ð22Þ

where Q is the heating power and A is the heated area on the substrate surface. Tin, Tx and kf are the inlet temperature, local temperature and fluid thermal conductivity, respectively. The pore diameter based Re and Nux are chosen in order to evaluate the ef-

Fig. 9. Dimensionless velocity profiles in ZZG foam at: (a) representative locations (b) x/Dh = 0.15; (c) x/Dh = 0.05; (d) x/Dh = 0.18; (e) x/Dh = 0.25.

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fects of the pores on the heat transfer in different configurations of graphite foams. Tx chosen from the simulation is at the same locations as the experimental study. The length-averaged Nusselt number is calculated as

RL Nuav e ¼

0

Nux dx L

ð23Þ

4.1. Velocity field Figs. 8 and 9 show the velocity distributions at different locations along the flow direction in the BAF and ZZG foams, respectively. For the periodic configurations in the BAF and ZZG foams, only the essential features in the first flow patterns are presented. Four chosen representative locations in the BAF foam of the velocity profiles are shown in Fig. 8a. Fig. 8b shows the velocity profile at x/Dh = 0.15. It is the position in the added empty channel where the fluid has not entered the BAF foams. The velocity profile is slightly affected by the first foam wall and the gap. The low permeability of the foam wall results in a large flow resistance. The fluid has difficulty penetrating the foam wall. Therefore, it is forced to flow through the gap. As fluid flows into the first foam wall i.e., x/Dh = 0.05, shown in Fig. 8c, one can observe that most of the fluid, referred to as the main flow henceforth, flows through the gap and only a small portion flows through the foam walls. The velocity in the gap is much larger than that in the porous foam walls. The velocity profile in the slot between the first and second foam walls is shown in Fig. 8d. The main flow is divided into three branches in an ‘‘E” manner guided by the gaps in the second foam wall. The main flow shown in Fig. 8c is distributed into three streams. When the fluid flows into the second foam wall as shown in Fig. 8e, the flow of the middle stream decreases as there is a foam wall directly in front of the gap. Guided by the foam wall, most of the fluid flows sideward into the next two gaps. The flow in the two side branches becomes dominant. The reduction of the flow rate in the foam walls decreases the heat transfer. Convection heat transfer occurs mainly in the foam wall surface rather than inside the foam walls. A careful examination of all the velocity profile shows that there is almost no fluid flow through the foam base. The slots and gaps above the foam base change the characteristics of fluid flow in the solid block foam which also modified the heat transfer. For the foam base, the heat absorbed by the solid ligament is mainly transferred by conduction to the upper ligament, rather than being convected away by the fluid directly. This is not favorable for heat transfer. However, the arrangements of the alternate gaps accelerate the velocity significantly, and fluid mixing is enhanced which is favorable to heat transfer. Fig. 9 shows the velocity profiles at different locations in ZZG foam. Fig. 9b–e give the velocity profiles according to the selected locations shown in Fig. 9a. The velocity profile before fluid flows into ZZG foam is shown in Fig. 9b. The main flow turns its direction to the gap due to the presence of the first foam wall in front. It results in the velocity difference between the orientation of the gap and the first foam wall. This velocity difference is higher as fluid flows into the first foam wall as shown as Fig. 9c. Most of the fluid flows through the gap due to the high flow resistance of the foam wall. Fig. 9d shows the velocity profile in the slot between the first and second foam walls. The main flow is forced to flow to the other y-direction before it hits the second foam wall. The inertia of the fluid flowing through the first gap is sufficiently high that the fluid turns its direction slowly. This results in the dominance of the main flow at the two ends along the y-direction. Fig. 9e shows the velocity profile in the second foam wall. Similar to the velocity profile in Fig. 9c, the velocity in the gap is larger than that in the foam wall. However, this velocity difference between the slot and

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the foam wall is not as large as in the first foam wall. The inertial force in the previous main stream is so large that the fluid has difficulty its direction in such a short slot before it hits the second foam wall. After that, the main stream flows in a zig-zag manner guided by the gaps and the foam walls. In the ZZG foam, the gaps are designed in the two ends along the width direction, where the gaps in the BAF foam are alternatively arranged in the middle and ends, respectively. The main flow in the ZZG foam would travel a long distance as it changes its direction to flow to another gap due to the presence of the foam walls in front. This would increase the fluid mixing, and enhance heat transfer at expense of the high pressure drop. There is almost no fluid flow through the foam base which is similar to fluid flow in the BAF foam. It is noted from Figs. 8d, 9c, d and e that the velocity near the foam base is slightly higher than that in the upper locations. In order to present this phenomenon clearly, the enlarged velocity vectors in the ZZG foam around the foam base at the middle plane of the width direction are shown in Fig. 10. The solid lines represent the joint section of the foam wall and foam base. An interesting finding is observed. The fluid is forced to flow downward to the foam base due to the foam wall in front. However, it is forced to flow upward from the foam base to the slot after passing through the foam wall. Therefore, the fluid velocity near the interface between the foam base and the slot increases slightly after the fluid penetrates the foam wall. 4.2. Heat transfer Fig. 11 shows the temperature distribution on the heated surface for different configurations of air flow at Re = 50. The dash line represents the location of the foam walls. For BLK foam (Fig. 11a), the temperature increases linearly along the flow direction with the lowest and highest temperatures at about 31 °C and 36 °C, respectively. Due to the structured passages in the BAF and ZZG configurations, the temperatures increase progressively and the lowest and the highest temperatures of these configurations are 35 °C and 45 °C, respectively (Fig. 11b and c). Therefore, the temperature distribution in the BLK foam is more uniform than those in the BAF and ZZG foams. As affected by the different local flow fields, the temperature distributions in the structured foams devi-

Fig. 10. Velocity vector in xz plane at y/Dh = 0.12 parallel to the flow direction.

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ate substantially from the BLK foam with higher temperature at the location where the flow changes its direction. A closer view shows that the ZZG foam has a generally lower temperature compared to

the BAF foam. Hence, if only heat transfer is considered, the BLK foam is still better than both the BAF and ZZG foams since it has lower temperatures and a more uniform temperature distribution.

Fig. 11. Temperature distributions at the heated surface with Re = 50 in: (a) BLK foam; (b) BAF foam; (c) ZZG foam.

K.C. Leong et al. / Applied Thermal Engineering 30 (2010) 520–532

The length-averaged Nusselt number Nuave calculated by Eq. (23) versus Re in different configurations of graphite foams is shown in Fig. 12a. Overall, the numerical results agree fairly well with the experimental data. It is obvious that Nuave in the BLK foam is the highest among all the configurations. In the calculation of Nuave, Nux is used to obtain Nuave, whereas Nux is inversely proportional to Tx. A high Nux corresponds to low substrate temperature. Hence, the highest Nuave in the BLK foam predicts the lowest substrate temperature among all the configurations. BLK foam possesses the largest surface area at pore level which is favorable to convection heat transfer between the solid and fluid phases. In ZZG and BAF foams, the velocity field is distorted by the design of the foam walls and gaps. Most of the fluid flows through the gaps. The flow rate in the graphite foams is dramatically reduced due to its low permeability. This weakens the convection heat transfer in the pore level compared with the BLK foam. The fluid is accelerated in the gap which improves the convection heat transfer between the solid and fluid phases. This improvement occurs only at the interface between the foam walls and the slots. The heat transfer area is greatly reduced compared with the contact area in the pore level. It is observed from the velocity field that there is little fluid flow through the foam base in the ZZG and

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BAF foams. Therefore, the heat transfer in the foam base is mainly conduction rather than conduction and convection. This is also not favorable for heat transfer. However, the designed slots and gaps in ZZG and BAF foams significantly reduced the pressure drop compared with BLK foam. The ZZG foam has slightly better heat transfer than BAF foam. Fluid flow in the ZZG foam is more distorted compared with the BAF foam which increases fluid mixing. This increased fluid mixing enhances the convection heat transfer, at the expense of high pressure drop. 4.3. Pressure drop The average pressure in the cross-section perpendicular to the flow direction is defined as

¼ p

1 Ac

Z

W

0

Z

H

pdydz

ð24Þ

0

To compare with the experimental data, the pressure drops in the channel are calculated by the difference between the average pressures across the graphite foams. Fig. 12b shows the pressure drops at different inlet velocities in all the configurations of graphite foams. As expected, the pressure drop increases with the increase of the inlet velocity for all the foams. The BLK foam has the highest pressure drop among the three foams. With the increase of velocity, the pressure drop in BLK foam increases significantly. Pressure drop in the designed ZZG and BAF foams reduces significantly. The pressure drop in the ZZG foam is higher than that in the BAF foam. For BLK foam, the low permeability results in high flow resistance as fluid flows. A large amount of the fluid flows through the designed gaps in ZZG and BAF foams which reduce the pressure drops significantly. In the ZZG foam, the main flow needs to flow pass from one end to the other along the width direction, whereas, in the BAF foam, the main flow only need to flow half the distance compared with the ZZG foam. Hence, the pressure drop in the BAF foam is lower than that in the ZZG foam. 5. Conclusions Fluid flow and heat transfer in the channel with different configurations of graphite foams were investigated both experimentally and numerically. The physical properties of the graphite foams were measured. The effects of the configurations on the flow fields were numerically revealed. The results showed that most of the fluid flows through the gaps in the ZZG and BAF foams and bypasses the contact surface between the solid and fluid phases. This is not favorable to heat transfer although the pressure drop is greatly reduced. The experimental and numerical results for heat transfer and pressure drops achieved good agreement for all configurations. The BLK foam gives the highest heat transfer rate with also the highest pressure drop. Acknowledgment The authors gratefully acknowledge the financial support provided under Defence Science and Technology Agency, Singapore Grant No. DSTA-NTU-DIRP/2005/01 for the work described in this paper. References

Fig. 12. Variations of: (a) Nuave with Re (b) pressure drop with inlet velocity in different configurations of graphite foams.

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