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Materials Science
Procedía Materials Science 4 (2014) 323 - 328
www.elsevier.com/locate/procedia
8th International Conference on Porous Metals and Metallic Foams, Metfoam 2013
Thermal convection measurements inside aluminum foam and comparison to existing analytical solutions
Nihad Dukhan*, Muntadher A. Al-Rammahi, Ahmed S. Suleiman
Department of Mechanical Engineering, University of Detroit Mercy, 4001W. McNichols Rd., Detroit, MI 48221, USA
Abstract
Metal foams have high thermal conductivity and large surface area per unit volume. The internal structure of the foams promotes vigorous mixing of a moving fluid inside the foams. As such, metal foams are very suited for convection heat transfer designs. Clear models for forced convection heat transfer inside the foam, as well as reliable thermal measurements are indispensable for convection-based thermal system designs. This paper present direct experiment for Darcy airflow and fluid's temperature inside a heated aluminum cylinder filled with aluminum foam. The experimental fluid temperature is compared to the available analytical solutions for the two-equation model for fully-developed forced convection. Peculiar, physically-unexplainable behavior is displayed when plotting the existing analytical solution in the literature. An error was discovered and corrected. Good agreement is obtained with the correct solution.
©2014PublishedbyElsevierLtd. Thisis anopen access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of Scientific Committee of North Carolina State University Keywords: Convection; experiment; fluid temperature; analyis ; heat transfer
1. Introduction
Due to the complex internal structure of the foam, exact solutions of the complete transport equations are virtually impossible. Examples of numerical, experimental and analytical work are given in Calmidi et al. (2000), Hwang et al. (2002), Lee and Vafai (1999), and Vafai and Kim (1989).
Lu et al. (2006) analyzed forced convection in a tube filled with a porous medium subjected to constant heat flux using the two-equation model. Zhao et al. (2006) presented an analytical solution for a tube-in-tube heat exchanger,
* Corresponding author. Tel.: +1-313-993-3285; fax: +1-313-993-1187. E-mail address: nihad.dukhan@udmercy.edu
2211-8128 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of Scientific Committee of North Carolina State University doi:10.1016/j.mspro.2014.07.569
for which the inner tube and the annulus were filled with metal foam. Analytical solutions in porous media continue to be sought (Xu et al. (2011), Qu et al. (2012)). Direct comparisons to experimental values of key variables, e.g., the fluid and solid temperatures inside the foam, is lacking in the literature. In a recent comprehensive review, Zhao (2012) indicated that there is a lack of reliable experimental heat transfer data for open-cell metal foam.
Researchers typically measure substrate (wall) temperature, and/or the temperatures at the inlet and outlet of the foam. Average surface heat transfer coefficient, volumetric heat transfer coefficient and/or Nusselt number are determined and used for comparing analytical and numerical results to experimental data (Calmidi et al. (2000), Hwang et al. (2002), Bhattacharya et al. (2002), Boomsma et al. (2003), Kurbas and Celik (2009), Zhao et al. (2004), and Kim et al. (2000)).
1.1. Solution of Two-Equation Model
A cylindrical porous medium of radius r0 is bounded by a wall and heated by a constant heat fluxq», Fig. 1. There is fully-developed one-dimensional flow in the z-direction with a volume-averaged velocity component u. From the Brinkman-Darcy equation, the velocity is (Vafai and Kim (1989), Dukhan (2012)):
U = 1 -—---(1)
Where R = r/r0 ,Da = K/r02 (Darcy number), K is the permeability, y = ///e , / is the fluid viscosity, /e is the effective viscosity, U = u/u», u- is the velocity outside the boundary layer, O = and I0 is the modified
Bessel function of order zero.
^ + + +
^ ^ ^ ^ ^B ^ ^ ^
Fig. 1. Schematic of metal foam heated cylinder.
In the thermally fully-developed region, the volume-averaged two-equation model is (Calmidi and Mahajan (2000), Lee and Vafai (1999), Krishnan et al. (2004), and (Alazmi and Vafai 2002)):
1 ri "Af)
- — (R - Bi(ds-df) = 0 (2) R dR dR f
— d dOf
—- {R-1-) - Bi(Os -O) = 2U (3)
R dR dR f
At R = 0 dR d R
==i (5) At R = 1 dR dR and 0s= 0f=O
where 0f = (Tf - Tw) / q"r0 / ks , 0s = (Ts - Tw) / q"r0 / ks , Z = z / r0 , R = r / r0 , X = kf / ks and Bi = h a r02 / ks is the Biot number. Also, ks and kf are the effective thermal conductivities, Ts and Tf are temperatures of the solid and the fluid, respectively, h is the interstitial heat transfer coefficient, a is the surface area per unit volume and p and c are the density and the specific heat of the fluid is the wall temperature which is not known a priori (Lee and Vafai
(1999)). The equality of the temperatures at the heated wall as described in Eq. (5) has been used in the literature (Lu et al. (2006), Zhao et al. (2006), Xu et al. (2011)). 30f / dZ is constant for thermally fully-developed conditions, and is obtained by integrating the sum of Eqs. (2) and (3) over the cross section, which gives 30f / dZ = 2q"/(pcwmr0). Substituting for U from Eq. (1) in the sum yields:
I0(mR)
-1 + xef 11 =
R dR { dR L s f 1 J
10(ai)
which is an ordinary differential equation (ODE) with 0s +X0f as the dependent variable. The solution is
0 +Xdf =-
2 ( - >)-
2 I I„(mR) + ^
h(fl>)
Solving Eq. (7) for 0s in terms of 0f, substituting for 0s, 30f / dZ and U in Eq. (3) and rearranging:
d 2Of 1 dOf Bi(À +1)
dR2 R dR
-Of = -Bi(^j -1) - 2 + B-R2 + 2(
f À a> 2 2
a? - Bi) I0(mR)] m2 I0 (a)
The general solution of Eq. (8) is obtained by first rearranging the homogenous part in the general form of Bessel equation (Myers (1998)). The complete solution of Eq. (8) is
df = alI0 (pR) + a210 (wR) + a3 R2 + a4 (9)
dJJty) + d2 + d3 2(Bi -af) 1
where, a = 1 -2-3, a2- V ' " -
I0(ty) a>2I0(a)[Aa>2 -Bi(À +1)] 3 2(A +1)
2À 2 + Bi(l/2-2/to2) ^ _ ¡Bi(A +1)
4 Bi(À +1)2 Bi(À +1) v A
It is important to note that, while obtaining the homogenous solution, a complex root ±i +1) / A is
encountered. For this case, Bessel function Jo must be replaced by the modified Bessel function Io in the solution (Xu et al. (2011), Qu et al. (2012), and (Meyers 1998)) which was not done in Lu et al. (2006) and Zhao et al. (2006) and resulted in a physically-unexplained behavior of the fluid temperature as will be shown later.
2. Experiment
The experimental model was essentially an aluminum tube filled and brazed to an aluminum foam core. The tube had a length of 15.24 cm, inside diameter of 25.56 cm and a thickness of 6.4 mm. The 91%-porous 20-ppi (pores per linear inch) foam was obtained commercially (ERG).
To measure the fluid temperatures inside the foam, each thermocouple was shielded in a specially designed small perforated aluminum tube. A thermocouple was inserted in each perforated tube and fixed using epoxy, blocking as few holes as possible. The bead of each thermocouple did not extend out of the small tube, and remained shielded, as seen in Fig. 2a. As such, the bead would not touch the solid ligaments of the foam or the wall of the small tube.
A set of ten holes were drilled through the wall of the foam reaching 2.03, 3.30, 4.57, 5.84, 7.11, 8.38 9.56, 10.92, 12.19 and 13.46 cm, measured from the outer surface of the tube. The set was at a distance of 6.35 cm form the foam entrance. The holes were arranged around the cross section with an angle of 36o between each two
adjacent holes, and were organized in order to minimize interference with the air flow through the foam. The small perforated tubes with their thermocouples were inserted in these holes, Fig. 2 b.
Other small shallow holes were drilled in the wall of the cylinder for measuring the wall temperature at various locations in the flow direction. Thermfoil heaters were attached and covered the outside surface area. Each heater could provide up to 645 Watts. The assembly was insulated.
Two 7.9-mm holes were drilled at the top of the test section at a distance of 5.08 cm for pressure drop measurement using a differential pressure transmitter with a range of 0 to 746 Pa.
Experiments were performed in an open-loop wind tunnel, Fig. 3. A suction unit which could produce air flow to 17 m3/min inducted room air through the foam sample. A reducing nozzle connected the exit of the test section to a flow-measurement section. A gas flow meter that could measure speeds up to 35 m/s was used.
The pressure transducer and the flow meter were connected to a data acquisition system, which delivered the readings to a computer. The flow rate was varied to realize different velocities in the test section. For each velocity, the steady-state static pressure drop was measured.
The flow rate was adjusted again, such that the desired speed was realized inside the foam. The surface heaters were powered, and the power input was adjusted using a variac, so that the desired heat flux was achieved. The air temperatures inside the foam, as well as the wall temperature, were monitored until steady-state conditions were achieved, which took about 40 minutes. The steady-state air and wall temperatures were recorded, as was the ambient air temperature. The uncertainty in the non-dimensional temperature of the fluid was determined to be ±14.3% (Figliola and Beasley (2000)). The effective thermal conductivity of the foam was obtained from Calmidi and Mahajan (1999) and is reported in Table 1.
3. Results
For velocities below 0.3 m/s, the flow regime was Darcy, as was indicated by the behavior of the pressure drop. So, for heat transfer measurements, a velocity of 0.2 m/s was chosen. The permeability was determined from the Darcy-regime pressure-drop data points only (Dukhan and Ali (2012)). The surface area density of the foam was calculated form a correlation given by the manufacturer. The interstitial heat transfer coefficient inside the foam, h, was computed based on a correlation given in Kuwarahara et al. (2001).
Fig. 4 shows the fluid temperature as a function of radial distance. The solution given by Eq. (9) is plotted using a solid line, while the experimental data points are given using the black solid circles. In general, the two temperature curves decrease gradually as the distance from the heated wall increases. Also plotted (dashed line) is the solution of Lu et al. 2006. The fluctuations in this solution have no physical reason. They are due to using the wrong Bessel function (J0) in the solution of Lu et al. 2006, instead of using I0, since the roots of Bessel equation are imaginary.
Fig. 2. Experimental heat transfer model construction: a) Thermocouple in its small perforated tube, b) Themocouple assemblies inserted into
holes.
Fig. 3. Photograph of the experimental set-up.
Table 1: Parameters for the metal-foam filled model.
Parameter Value
ro 0.128 m
Um 0.2 m/s
K 1.16X10-07 m2
q" 8494.5 W/m2
ks 6.62 W/m.K
h 207.8 W/m2.K
Bi 682.9
o 1313.8 m2/m3
<a 340.8
¥ 303.1
I 0.0044
The experimental and the analytical temperature curves of the current study are relatively close to each other, with the analytical solution predicting a slightly lower temperature. In addition to experimental errors, other errors may have existed in the heat transfer coefficient and the effective thermal conductivity.
Another possible error may be the existence of a temperature slip at the wall, and physical non-uniformities near the wall, which alters the value of the effective conductivities of the solid and fluid (Sahraoui and Kaviany (1993)). The nature of the wall boundary condition when a constant heat flux is applied is still subject to debate, i.e., how the total heat flux is split between the solid and the fluid phases (Vafai and Yang (2013)).
The two points closest to the center of the foam are seen to produce higher physical temperature. The arrangement of the thermocouples caused blockage leading to higher temperatures at these two locations.
4. Conclusions
Direct measurements of the fluid temperature inside a heated metal-foam cylinder were conducted using a specially-designed technique. Scrutiny of literature revealed that such measurements have been lacking. The thermal-non-equilibrium two-equation model was revisited, and an error in a previously published solution was identified. Relatively good agreement between the analytical and experimental fluid temperature was displayed, with the analytical solution under-predicting the fluid temperature over most of the cross section. Nonetheless, the measurements give an opportunity to look into the issue of how volume-averaged values compare to physically-measured local values of the temperature inside porous media. The experimental technique can be of utility for heat transfer designs, and for validating complex numerical modes of the heat transfer in open-cell meso-porous media.
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