Experimental and Numerical Investigation of

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Experimental and Numerical Investigation of Combined Convection Heat Transfer and Fluid Flow around Circular Cylinder through Rectangular and Trapezoidal Open-Cell Aluminum Foams ab

c

a

d

Raed Abed Mahdi , H. A. Mohammed , K. M. Munisamy & N. H. Saeid a

Department of Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional, Selangor, Malaysia b

Department of Mechanical Maintenance, Doura Power Station, Ministry of Electricity, Baghdad, Iraq c

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Department of Thermo fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, Johor Bahru, Malaysia d

Department of Mechanical Engineering, Institute of Teknologi Brunei, Brunei, Darussalam Accepted author version posted online: 20 Jun 2014.Published online: 12 Jan 2015.

To cite this article: Raed Abed Mahdi, H. A. Mohammed, K. M. Munisamy & N. H. Saeid (2015) Experimental and Numerical Investigation of Combined Convection Heat Transfer and Fluid Flow around Circular Cylinder through Rectangular and Trapezoidal Open-Cell Aluminum Foams, Chemical Engineering Communications, 202:5, 674-693, DOI: 10.1080/00986445.2013.863188 To link to this article: http://dx.doi.org/10.1080/00986445.2013.863188

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Chemical Engineering Communications, 202:674–693, 2015 Copyright # Taylor & Francis Group, LLC ISSN: 0098-6445 print/1563-5201 online DOI: 10.1080/00986445.2013.863188

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Experimental and Numerical Investigation of Combined Convection Heat Transfer and Fluid Flow around Circular Cylinder through Rectangular and Trapezoidal Open-Cell Aluminum Foams RAED ABED MAHDI1,2, H. A. MOHAMMED3, K. M. MUNISAMY1, and N. H. SAEID4 1

Department Department 3 Department 4 Department 2

of of of of

Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional, Selangor, Malaysia Mechanical Maintenance, Doura Power Station, Ministry of Electricity, Baghdad, Iraq Thermo fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, Johor Bahru, Malaysia Mechanical Engineering, Institute of Teknologi Brunei, Brunei, Darussalam

Combined convection heat transfer and fluid flow around a circular cylinder surface placed in open-fcell aluminum foams and subjected to constant heat flux inside a rectangular, water-filled horizontal channel was numerically and experimentally studied. Two models (rectangular and trapezoidal open-cell aluminum foam shapes) made of 6101-T6 alloy with pore densities of 10 and 40 pores per linear inch (PPI) and 7–9% relative density were employed as test sections. The aluminum foam dimensions were 35.7 35.7 36.85 mm, the Reynolds number range was 60–2000, and the modified Grashof number range was 2 102–2.6 107. Governing equations (continuity, momentum, and energy) were solved using the finite-volume method (FVM). Effects of the porous characteristics of aluminum foams and mixed convection heat transfer parameters on buoyancy force, Nusselt number, friction factor, and pumping power values of the two models were investigated. The results show that high mixed convection occurred with the trapezoidal model. A high average Nusselt number value was obtained at 40PPI in the rectangular model. In the trapezoidal model, average Nusselt number decreased with increased aluminum foam pore density. Friction factor increased slightly with increasing modified Grashof number and decreased with increasing Reynolds number. Pumping power increased with increased pore density of aluminum foam and mixed convection parameters. The comparison shows good agreement between the numerical and experimental work and that the average results produced have an 8.02% deviation in average Nusselt number. Keywords: Experimental investigation; Flow around cylinder; Mixed convection; Open-cell aluminum foam; Trapezoidal shape

Introduction Any material that consists of a solid matrix with an interconnected void is described as a porous medium, such as rocks and open-cell aluminum foams (Nield and Bejan, 2006). There are two advantages to porous media. First, the dissipation area is greater than the conventional fins that enhance heat convection. Second the irregular motion of the fluid flow around the individual beads mixes the fluid more effectively. A wide range of porous media applications is Address correspondence to Raed Abed Mahdi, Department of Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional, Jalan IKRAM-UNITEN, 43000 Kajang, Selangor, Malaysia. E-mail: [email protected]; and H. A. Mohammed, Department of Thermo fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor Bahru, Malaysia. E-mail: hussein. [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/gcec.

found in many practical situations, such as aluminum foam applications in thermal management, including air-cooled condenser towers (Kaviany, 1995; Pop and Ingham, 2001; Ingham and Pop, 2002; Bejan et al., 2004; Vafai, 2005; Nield and Bejan, 2006; Vadasz, 2008). Boomsma et al. (2003) experimentally studied the heat performance of open-cell aluminum foam heat exchangers during forced convection heat transfer by using liquid coolant, and they investigated the effect of aluminum foam porosity on thermal performance. Their findings indicated that the pressure dropped significantly at the highest solid friction (lowest porosity), and Nusselt number increased with increasing coolant velocity. Jiang et al. (2004) experimentally studied forced air and water convection in porous plate channels. The effects of porous medium type (sintered or non-sintered), particle diameter, fluid velocity, and fluid properties on heat transfer enhancement and forced convection were investigated. Their findings indicate that heat transfer was enhanced with increased water and air flow rates. In a sintered porous medium, particle diameter had


Heat Transfer inside an Open Cell Aluminum Foam little effect on convection heat transfer. Convection heat transfer was lower in non-sintered porous plate channels than in sintered porous plate channels. Tzeng et al. (2005) experimentally studied mixed convection heat transfer in a rectangular porous channel with sintered copper beads. The effects of the average particle size of the sintered porous shot-copper and porosity with varying Reynolds number and heat flux were investigated. Their findings indicate that at a fixed porosity, higher flow rate increased the efficiency of heat exchange between the solid and fluid phases for the heat sink. Tzeng and Jeng (2006) experimentally studied forced convection and pressure drop in porous media channels with a 90 turned flow during isoflux heating on the bottom wall. The effects of aluminum foam pore density, entry width to porous sink height ratio, and Reynolds number were investigated. They found that an increase in Reynolds number increased the Nusselt number and that aluminum foam pore density and the ratio of entry width to porous sink height had negligible effects on Nusselt number. Mohammed and Salman (2007) experimentally studied mixed convection in a horizontal circular cylinder for hydrodynamically fully developed, thermally fully developed, and thermally developing laminar air flow. The effect of mixed convection parameters on heat transfer rate was investigated. They found that Nusselt number increased with heat flux increase in all entrance sections. In addition, the effects of natural convection tended to decrease the heat transfer results at low Reynolds number whereas the opposite was observed at high Reynolds number. Kurtbas and Celik (2009) experimentally analyzed mixed convective heat transfer and fluid flow through a horizontal aluminum foam plate in a rectangular channel with varying pore density. The effects of Richardson and Reynolds numbers and aspect ratio with laminar and turbulent flow were considered. Their findings indicate that average Nusselt number increased in proportion to pore density and rapidly increased at a critical Reynolds number. The local Nusselt number also increased to high levels at high Grashof and Reynolds numbers. Given an aspect ratio of a < 1, the local Nusselt number sharply decreased at the point where the metal foam ended. Guerroudj and Kahalerras (2010) numerically studied mixed convective heat transfer in a parallel plate channel with various porous blocks shapes. Blocks were heated from below and attached to the lower plate while the upper plate was thermally insulated. Varied shapes were studied, from rectangular (c ¼ 90 ) to triangular (c ¼ 50.1944 ). The effects of mixed convection parameter (Gr=Re2), Darcy number, porous block height, thermal conductivity ratio, and Reynolds number were investigated. Their findings indicate that the global Nusselt number increased with increases in mixed convection parameter, Reynolds number, and the thermal conductivity ratio. Rectangular is the optimal shape at high values of these parameters. Sivasamy et al. (2010) numerically studied twodimensional unsteady flow mixed convection with constant horizontal surface heat flux on jet impingement cooling immersed in a confined porous channel. They investigated the effects of modified Grashof, Reynolds, and Darcy

675 numbers as well as the jet width and distance between the heated portion and the jet. They found that average Nusselt number increased with increasing modified Grashof number at low Reynolds numbers, but became less significant compared with its high value when Reynolds number increased. Average Nusselt number increased with increased jet width and decreased at low Reynolds numbers (Re < 23), which resulted from the increase in Darcy number. Ahmed et al. (2011) numerically studied mixed convection heat transfer by using a thermal non-equilibrium approach in a vertical annular cylinder saturated with porous media. The inner annulus cylinder wall was heated at a constant temperature, Tw whereas the outer cylinder wall was kept at a constant temperature, T1, such that Tw > T1. The effects of Pećlet number, interphase thermal conductivity ratio, and heat transfer coefficient on Nusselt number for both solids and fluids were investigated. Their findings indicated that the Nusselt number for fluid remained constant with changes in thermal conductivity ratio in aiding flow and slightly decreased as Pećlet number increased. On the other hand, Nusselt number for solids increased with increase in thermal conductivity ratio. Wu and Wang (2010) numerically studied two-dimensional unsteady forced convection heat transfer with uniform heat generation on the bottom surface of a non-permeable cylinder with an incompressible laminar flow across a porous square cylinder in the middle of a channel. They investigated the effects of Reynolds and Darcy numbers, porosity, and cylinder-to-channel height ratio (B=H) and found that heat transfer increased with increase in Reynolds and Darcy numbers and porosity. Kamath et al. (2011) experimentally studied mixed convection in a vertical channel containing aluminum foams of varying pore density and porosity in the range 90–95%. In their study, the effects of Richardson and Reynolds numbers and porosity on heat transfer were investigated and their findings indicated that a porous medium in small inlet velocities does not enhance heat transfer, and porous matrix conduction marginally enhances heat transfer. The average surface temperature of the aluminum plate dropped and heat transfer coefficient increased with increasing power input. At high power input the strengthened buoyancy forces resulted in increased heat transfer. The literature review in this study shows that very limited data are available on mixed convection heat transfer using rectangular and trapezoidal aluminum foam with water as a working fluid. Therefore, this paper presents numerical simulations and experiments on combined convection heat transfer and fluid flow around the surface of a circular cylinder through rectangular and trapezoidal open-cell aluminum foams with constant heat flux inside a rectangular horizontal channel filled with water. The aims of the present study are to investigate the effects of the porous characteristics of aluminum foam and mixed convection parameters on buoyancy force, Nusselt number, friction factor, and pumping power, with rectangular and trapezoidal models in a rectangular horizontal channel. Studies similar to the present work are very limited. Therefore, the results were compared to the findings of studies that focused on flow-through


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porous media in a horizontal channel. The present results were compared to those obtained by Guerroudj and Kahalerras (2010), who studied mixed convective heat transfer in a parallel plate channel with various porous block shapes, including rectangular (c ¼ 90 ) and triangular (c ¼ 50.1944 ). The blocks were heated from below and attached to the lower plate while the upper plate was thermally insulated.

open-cell aluminum foam shapes is investigated in this paper with:

Physical Description of the Problem and Assumptions

The following assumptions are considered in the numerical study to simplify the problem:

The schematic diagram of the problem is presented in Figures 1(a) and (b). Open-cell aluminum foams in two models (rectangular and trapezoidal) inserted inside a horizontal rectangular channel and surrounded by horizontal circular cylinder surface as heat source shape are used. The fluid enters the rectangular horizontal channel with fully developed flow velocity (u1) and temperature (T1 ¼ 303 K), and all channel walls are thermally insulated. Both mixed convective heat transfer and fluid flow through open-cell aluminum foams filled with water depends on several parameters, such as buoyancy force, pore density (PPI), and modified Grashof and Reynolds numbers, and the shape of

1. two open-cell aluminum foam models (rectangular and trapezoidal); 2. an open-cell aluminum foam 6101-T6 alloy of pore densities 10 and 40 PPI; 3. Reynolds number range 60–2000; 4. modified Grashof number range 2 102–2.6 107.

1. The flow is a steady state, incompressible, and fully developing laminar regime (because with porous media when using liquids like water as working fluid, the buoyancy force effect does not appear with high Reynolds values because the liquids have low thermal expansion compared with gas). 2. The thermophysical properties of the water are constant with temperature (303 K). 3. No chemical reactions or internal heat generation occur, and viscous dissipation is neglected. 4. The aluminum foam is isotropic, homogeneous, and saturated with a single-phase fluid in local equilibrium with the solid matrix. 5. The effective viscosity equals the fluid viscosity in Brinkman viscous terms

Experimental Apparatus

Fig. 1. Open-cell aluminum foam shapes: (a) rectangular model, (b) trapezoidal model.

The experimental system (Figure 2) comprised two parts: (i) the channels of development and test sections and (ii) the data acquisition system. The development and test sections had a rectangular internal cross sectional area of 35.7 35.7 mm and were manufactured from 1.2 mm aluminum plate. The development and test sections were constructed as one unit to prevent misalignment. The water entered a long hydrodynamic development section (L=Dh 42) in order to establish a well-development velocity profile. The test section consisted of two aluminum foam models, one of which was rectangular and the other trapezoidal. The models measured 35.5 35.5 36.85 mm and were made from an aluminum foam 6101-T6 alloy metal with pore density of 10 and 40 PPI. Silicone material with a thickness of 0.2 mm and width of 1.2 mm was inserted between the aluminum foam and channel walls to prevent water leakage and ensure that an accurate pressure drop is obtained A main circular cylinder with 12 mm outlet diameter and a length of 55.7 mm in the middle of the aluminum foam was pressed into the foam to avoid thermal contact resistance. This was insulated from the channel wall by a 2 mm Teflon washer placed at the cylinder ends to reduce heat loss, as shown in Figure 3(a). At each 45 angle, eight K-type thermocouples (0.2 mm diameter) were attached to the internal surface of the main circular cylinder and around the middle of the circular cylinder length through a groove (1 mm depth and 27.85 mm length) and the thermocouple cylinder (9.5 mm diameter and 55.7 mm length), as shown in Figure 3(b).


Heat Transfer inside an Open Cell Aluminum Foam

677

Fig. 2. Schematic diagram of the overall experimental apparatus.

Fig. 3. Test section parts: (a) cross-section area for test specimen and positions of thermocouples; (b) thermocouple cylinder design and thermocouple locations.

678

R. A. Mahdi et al. available with high porosity and have an open-celled structure, as shown in Figure 4.

Data Analysis Method Porosity (e)


Porosity was calculated depending on the volume of both sample and solid (Kaviany, 1995): Vsolid e¼1 ð1Þ Vsample Permeability (K) and Inertia Coefficient (C) Permeability (K) and inertia coefficient (C) can be calculated in various ways (Antohe and Lage, 1997; Bhattacharya et al., 2002; Boomsma and Poulikakos, 2002), one of which is to extrapolate the form of pressure drop versus velocity data through the porous component (ANSYS Workbench# 14.0, 2011). Table I shows the porous characteristics of aluminum foams, such as porosity (e), permeability (K), and inertia coefficient (C). Thermophysical Properties of Aluminum Foam

Fig. 4. Pictures of aluminum foam samples with various PPI values: (a) 10PPI, (b) 40PPI.

A cartridge heater of diameter 6 mm was attached horizontally to the internal thermocouple cylinder surface, as shown in Figures 2 and 3(a). Two other thermocouples were used to monitor the ambient and water temperature at the test section inlet near the aluminum foam. A total of 10 K-type thermocouples (0.2 mm diameter) were connected to the data logger. The main water flow was supplied by a submersible centrifugal pump, and the flow rate was controlled by a control ball valve. A digital flow meter was installed in the inlet channel to measure water flow rate, and an inclined manometer was utilized to measure the pressure drop associated with water flow through the test section, as shown in Figure 2. Measurements were conducted for steady state conditions. The operation temperature was kept constant at approximately 303 K for each test run. The walls of the test section channel were carefully wrapped in a 50 mm foam insulation given the aluminized outer surface. Aluminum foams with pore density of 10 and 40 PPI are typically

The methods of defining the thermophysical properties of aluminum foam 6101-T6 alloy were discussed by ERG aerospace (2012) and Gibson and Ashby (1997): qaluminum foam ¼ ð1 eÞ qsðaluminium solidÞ

ð2Þ

cpaluminum foam ¼ ð1 eÞ cpsðaluminium solidÞ

ð3Þ

kaluminum foam ¼ ð1 eÞksðaluminum solidÞ 0:33

ð4Þ

0.33 ¼ represents the tortuosity factor (ERG aerospace, 2012). The thermophysical properties of aluminum metal 6101-T6 alloy are summarized in Table II (ERG aerospace, 2012), and those of aluminum foam 6101-T6 alloy in Table III. Effective Thermal Conductivity (Keff) Generally, effective thermal conductivity (keff) is described by the porosity (e) and thermal conductivities of the solid (ks) and fluid phases (kf) (Kaviany, 1995). The increase in effective thermal conductivity of the fluid and solid systems to a higher level than that predicted by equation (5) was shown by Calmidi and Mahajan (1999) in the

Table I. Porous characteristics of the aluminum foams samples Rectangular Porous characteristics Porosity (e) % Permeability (K)[m2] 08 Inertia coefficient (C)

Trapezoidal

10PPI

40PPI

10PPI

40PPI

91.64 3.35886 0.015148952

91.55 2.71622 0.023742184

92.16 16.9896 0.021877549

92.14 12.0218 0.043277071

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Heat Transfer inside an Open Cell Aluminum Foam Table II. Thermophysical properties for aluminum metal alloy 6101-T6 (ERG aerospace, 2012) Density Specific heat Thermal conductivity [J=kg.k] [W=m.k] [kg=m3]

Metal Aluminum alloy 6101-T6

2700

895

Governing Parameters The local Nusselt number based on the hydraulic diameter is calculated as follows (Nield and Bejan, 2006): Nuh ¼

218

ð10Þ

where hh is the local wall heat transfer coefficient at each angle on the cylinder’s surface:

two-dimensional conduction model:


hh Dh k

hh ¼

keff ¼ ekf þ ð1 eÞksðaluminum foamÞ

ð5Þ

ð11Þ

where Dh is the hydraulic diameter of the rectangular channel: Dh ¼ 4Ach =p

Data Reduction Method Simplified steps are used to analyze the heat transfer process for cross water flow over a circular cylinder inside a rectangular horizontal channel when its surface was subjected to a constant heat flux. The total input power supplied (Qtotal) to the cartridge heater inside the thermocouple and the main cylinders is obtained by the product of voltage and the supplied current, and can be calculated as Qtotal ¼ V I

ð12Þ

where Ach is the cross-sectional area of the channel (1.27449 103 m2) and p is the perimeter of the channel (1.428 101 m). The average Nusselt number is calculated as follows (Incropera et al, 2007): Nu ¼

i¼n 1 Z 2p Dh X Nu dh ffi Nuh h 2p 0 2p i¼1

Qconvection ¼ Qtotal Qloss

ð7Þ

where Qloss is the total conduction heat loss obtained by conduction heat transfer analysis across the foam insulation layers. The convection heat flux is represented by

where n is temperature measurement point’s number on the cylinder surface and Dh is the spacing between the two thermocouples. The modified Grashof, Reynolds, and Richardson numbers based on the rectangular channel hydraulic diameter are defined as Gr ¼

q2 gbqconv: D4h kl2

ð14Þ

qlDh l

ð15Þ

Gr Re2

ð16Þ

Re ¼

Qconvection ¼ Acylinder

ð8Þ Ri ¼

where Acylinder denotes the surface area of main cylinder in m2: Acylinder ¼ p Dcylinder Lcylinder

ð9Þ

The average values of the other parameters can be calculated as Friction factor(F):

where Dcylinder denotes the diameter of the main cylinder (12 mm) and Lcylinder the length of the main cylinder (35.7 mm).

ðFÞ ¼

Dp 0:5qv21

ð17Þ

Table III. Thermophysical properties for aluminum foam alloy 6101-T6 Rectangular Properties Density [kg=m3] Specific heat [J=kg k] Thermal conductivity [W=m k]

ð13Þ

ð6Þ

The convection heat transferred from the cylinder surface is

qconvection

qconvection ðT S T 1 Þ

Trapezoidal

10PPI

40PPI

10PPI

40PPI

225.8450604 74.8634552 6.01752

228.2223768 75.65149158 6.08086

211.7818128 70.20174906 5.64281

212.1546681 70.32534369 5.65274

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The thermophysical properties of the water in all non-dimensional parameters are based on the inlet temperature (303 K) of the test section. The heat convection performance of the heat system in any heat system design must be weighed against the energy required to operate the system, which is the pumping power in this configuration. For the aluminum foam heat system, the required pumping power is calculated at various water flow velocities according to equation (18):


_ Dp P_ ¼ M

ð18Þ

whereP_ is pumping power, Dp is pressure drop across the _ is the water volumetric flow rate aluminum foam, and M passing through the system

Uncertainty Analysis The accuracy of the experimental results depends on individual measuring instruments, aluminum foam manufacture, and main circular cylinder accuracy. The accuracy of any instrument is also limited by its sensitivity. The precision, error, and general validity of the experimental measurements are calculated. The probable errors in the experimental data refer to values with some uncertainty. Uncertainty varies greatly depending on experimental circumstances. The experimental error magnitude is permanently uncertain. In the present work, uncertainty in regard to the Nusselt, Reynolds, and modified Grashof numbers are estimated by following the differential approximation method reported by Holman (2001). To calculate particular results which are required, the measurements should be combined. Therefore, the uncertainty in the final result should be known due to the uncertainties in measurement. The differential approximation method is considered to evaluate the uncertainty in a result, Rs that is a function of the independent parameters X1, X2, X3. . ., Xn: Rs ¼ Rs ðX1 ; X2 ; X3 ; . . . . . . . . . Xn Þ

ð19Þ

At the same time, it may perturb the variables by DX1, DX2, DX3. . ., DXn and then Rs ðX1 þ DX1 Þ ¼ Rs ðX1 þ DX1 ; X2 ; X3 ; . . . . . . . . . Xn Þ; ð20Þ Rs ðXn þ DXn Þ ¼ ðX1 ; X1 ; X2 ; X3 ; . . . . . . . . . DXn Þ;

ð21Þ

Therefore, for sufficiently low values of the quantities DX1, DX2, DX3. . . , DXn the partial derivatives can be well approximated by @Rs Rs ðXi þ DXi Þ Rs ðXi Þ ; i ¼ 1; 2; 3; . . . . . . ; n DXi @Xi

given as (Holman, 2001) " WR ¼

@Rs W1 @X1

2 2 @Rs þ W2 @X2

@Rs þ W3 @X3

2

@Rs þ... þ Wn @Xn

2 #12

ð23Þ

Since the values of the partial derivative and the errors in the measuring parameters may be positive or negative, then the absolute values are considered to obtain the maximum absolute uncertainty in the result, WR. Measurements should be combined to calculate the required results. Data supplied by the instrumentation manufacturer indicated a 1% error in the pressure drop and flow velocity measurements. In the measured temperature, the uncertainty was 0.03 K. In the convective heat flux, the experimental uncertainty was estimated at 2%. The maximum uncertainties associated with the Reynolds and Nusselt numbers and friction factor were estimated to be 3.2, 7.4, and 2.24%, respectively.

Governing Equations Henry Darcy’s (1856) investigations into the hydrology of the water supply at Dijon, France and his experiments on steady state unidirectional flow in a uniform medium revealed a proportionality between flow rate and the applied pressure difference. In modern notation this is expressed, in refined form (Nield and Bejan, 2006), by rp ¼

lf ~ v K

ð24Þ

where rp is the pressure gradient in the flow direction and lf is the dynamic viscosity of the fluid. The coefficient K is independent of the nature of the fluid but it depends on the geometry of the medium Darcy’s equation (24) is linear in the Darcy velocity, u. It holds when u is sufficiently low. In practice, ‘‘sufficiently small’’ means that the Reynolds number, Rep of the flow, based on a typical pore or particle diameter, is of the order unity or smaller. As u increases, the transition to nonlinear drag is quite smooth; there is no sudden transition as Rep is increased in the range 1–10. Clearly this transition is not one from laminar to turbulent flow, since at such comparatively small Reynolds numbers the flow in the pores is still laminar. Rather, the breakdown in linearity is due to the fact that the form drag due to solid obstacles is now comparable to the surface drag due to friction. According to Joseph et al. (1982), the appropriate modification to Darcy’s equation is to replace Equation (24) by

ð22Þ

If there are uncertainties, W1, W2, W3, . . . , Wn in the independent variables and WR is the uncertainty in the result on the same odds, then the uncertainty in the result can be

rp ¼

l qcf ~ v þ pffiffiffiffi jvjvj K K

ð25Þ

where cf is a dimensionless form-drag constant (Forchheimer coefficient). Equation (25) is a modification of an equation

681

Heat Transfer inside an Open Cell Aluminum Foam associated with the names of Dupuit (1863) and Forchheimer (1901). For simplicity, Nield and Bejan (2006) called Equation (25) the Darcy–Forchheimer equation and referred to the latter term as the Forchheimer term. The flow was modeled using Darcy–Forchheimer’s model to combine the inertia effect in the aluminum foam region where qcf l pffiffiffiffi ~ rp ¼ v þ jvjvj ð26Þ K K |{z} |{z} 0


Darcy s term

The axial pressure drop in the homogeneous matrix with only a steady flow and no internal axial body forces under the assumption that the effective viscosity equals the fluid viscosity in Brinkman viscous terms is defined by FLUENT as @ðepÞ @ @ðesi Þ ¼ ðeqvi vi Þ þ ðeqvi vi Þ @x @x @x ffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl Convection acceleration

0

Forchheimer s term

0

Fluid domain was modeled by continuity, Navier–Stokes, and energy equations. Continuity: r:ðq~ vÞ ¼ 0

ð27Þ

Momentum: þ

r ðq~ v~ vÞ ¼ rp þ r t q~ gþ~ F

Viscous stress also called Brinkman Viscous term or bounding surface effect

B þ e~ gB Dlvi @ |ffl{zffl}

þ

Darcy0 s term

D¼

where l is the molecular viscosity and I is the unit tensor. In the porous region, assuming isotropic porous media porosity (e) and single-phase flow, the volume-averaged mass and momentum conservation equations to be solved by FLUENT become the following (ANSYS Workbench# 14.0, 2011). Continuity: r ðeq~ vÞ ¼ 0

ð30Þ

ð33Þ

C C A

Forchheimer0 s term

A comparison among Equations (26)–(33) shows that the viscous resistance term in Equation (32) is related to the permeability in Darcy’s term by

ð28Þ

where p is the static pressure, t is the stress tensor (described below), and q~ g and ~ F are the gravitational body force and external body forces, respectively. ~ F also contains other model-dependent source terms such as porous media. The stress tensor t is given by 2 t ¼ l r~ v þ r~ vT r ~ vI ð29Þ 3

c qjvjvi 2 |fflfflffl{zfflfflffl}

1

1 K

ð34Þ

The inertial resistance term is related to the Forchheimer coefficient and permeability in the Forchheimer term by 2cf C ¼ pffiffiffiffi K

ð35Þ

Energy: A local thermal equilibrium is assumed between the solid phase and the fluid within the porous medium. The energy equation takes then the following forms. Fluid region: @ vðqf Ef þ pÞÞ ðeq Ef þ ð1 eÞqs Es Þ þ r ð~ @x f ! X hj Jj þ ðs ~ vÞ þ Sfh ¼ r keff rT

ð36Þ

j

Momentum:

r ðeq~ v~ vÞ ¼ erP þ r et þ e~ gþ~ F

Porous region: ð31Þ

~ F denotes the gravitational body force and external body forces, accounting for viscous and inertial losses of the fluid within the porous media and is defined by FLUENT as 0 1 B C 3 3 BX C X 1 B C Fi ¼ B Dij lvj þ Cij qjvjvj C B j¼1 C 2 j¼1 @ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} A viscousresistance term or Darcy0 s term

ð32Þ

Inertialresistance term or For chheimer term

where Fi is the external body force term for the ith momentum equation, D is viscous resistance, and C is inertial resistance.

@ vðqf Ef þ pÞÞ ðeq Ef þ ð1 eÞqs Es Þ þ r ð~ @x f ! X hj Jj þ ðs ~ vÞ þ Sfh ¼ r keff rT

ð37Þ

j

where Ef is total fluid energy, Es is total solid media energy, keff is effective thermal conductivity of the media, Sfh is the fluid enthalpy source term, and Jj is the diffusion flux of species j. The effective thermal conductivity in the porous media keff is computed by FLUENT as the volume average of fluid and solid conductivity (ANSYS 14.0 Help, 2011): keff ¼ ekf þ ð1 eÞkporous media

ð38Þ

682

R. A. Mahdi et al.

The boundary conditions for the above set governing equations are as follows. . The flow is in steady state and incompressible with fully

developing laminar regimes imposed at the inlet: H @u : ¼ 0; v ¼ w ¼ 0 2 @x @T and T ¼ Tin ¼ 303 K; ¼ 0; x=l ¼ 0 @x

algorithm steps involved in arriving at a converged solution. The pressure-staggering option (PRESTO) scheme is used to solve pressure equations. The third-order MUSCL scheme is used to solve momentum and energy equations. When the normalized residual values reach 106, the solutions are considered to be converged for all variables.

x ¼ 0; 0 < y