Characterization of the heat transfer in open-cell metal foam...Characterization of the heat...

Characterization of the heat transfer in open-cell metal foam

C. Briano-Calcagno, J. Fontánez-Delgado & N. Dukhan Department of Mechanical Engineering, University of Puerto Rico – Mayagüez, Mayagüez, P.R., U.S.A.

Abstract

The material characterization of open-cell aluminum foam in terms of heat transfer is presented. A one-dimensional heat transfer model for the combined convection and conduction in the foam is summarized. The model uses the foam parameters that are usually reported by the manufactures such as: the surface area, the relative densities, the ligament diameters and number of pores per inch. The model predicts the temperature profile in the foam. The model was applied successfully to a sample of aluminum foam having ten pores per inch and was verified by direct experiment. Excellent agreement between the predictions of the model and the experimental data was obtained. The assumption of a one-dimensional heat transfer was validated. The effect of the air flow rate on the heat transfer is also studied in order to further characterize the heat transfer behavior of the foam. The results for an aluminum foam sample of 10 pores per inch are presented at these flow rates. Keywords: heat transfer, open-cell metal foam, air flow rate.

1 Introduction

Metal foams are a relatively new class of materials with novel properties and a promise for significant performance gains. They have been used in aerospace geothermal operations, petroleum reservoirs, high-power batteries for lightweight cordless electronics and compact heat sinks for power electronics [1, 2, 3]. The open porosity, low relative density and high thermal conductivity of the cell edges, the large accessible surface area per unit volume, and the ability to mix the cooling fluid by promoting eddies [4]; all make metal foam heat exchangers efficient, compact and light weight. Current models for

High Performance Structures and Materials II, C.A. Brebbia & W.P. De Wilde (Editors)© 2004 WIT Press, www.witpress.com, ISBN 1-85312-717-5

packed beds are not applicable to high porosity metal foam due to the difference in porosity and geometry [3, 5]. The porous matrix consists of tortuous, irregularly shaped flow passages. The flow re-circulates at the back of the solid fibers, and for pore-scale Reynolds number greater than 100, turbulence and unsteady flows occur [6]. The geometric complexity prevents exact solutions of the transport equations inside the pores [3, 7, 8]. Due to their novelty metal foams are still incompletely characterized. Reliable models of the heat transfer in foam would certainly help the engineering research community. A number of studies were undertaken to fill this gap, [9, 10, 11]. The present work provides a macroscopic lumped-parameter treatment to determine the temperature distribution in open cell metal foams in a forced convective mode. The analysis uses the typical parameters reported by the foam manufacturers. The simplicity and applicability of the present approach eliminates the need for rigorous microscopic analytical or numerical modeling of the flow and the heat transfer in and around the pores. Another advantage is that the current model is easily verified by simple experiments, as described below. Once validated, the model can be used to study the effect of the flow rate on the temperature profile.

2 Heat transfer model

Consider a one-dimensional solid fin of length L, width W and thickness t as shown in fig. 1. Let the base temperature be Tb. The governing equation for the heat transfer through the fin is readily available [12]:

022

2

=− sss m

dxd θθ , (1)

where θs = T –T∞ and ms the fin parameter. For an insulated tip, dθs/dx = 0 at x = L, the solution to eqn. (1) is [12]:

LmxLm

s

s

b

s

cosh)(cosh −

=θθ , (2)

where θb = Tb –T∞ .

Figure 1: Schematic of a constant cross-sectional area fin.


244 High Performance Structures and Materials II

The study develops similar closed form results for a thin piece of porous foam having a constant cross-sectional area and an insulated tip. Unlike the solid fin case, the coolant flows through the foam, as shown in fig. 1. This condition gives rise to forced convection heat transfer inside the pores of the foam. The heat transfer coefficient for this convection, hfm, is availble [4, 8] and is different than hs. For the foam, consider now a control volume of thickness dx in the x-direction, as shown in fig. 2.

Figure 2: Control volume for the foam.

The conservation of energy is applied for steady-state conditions to this control volume and neglect radiation. The conduction terms are readily given by Fourier’s law; the convection term by Newton’s law of cooling:

+−=−+ dxxcondsxconds dx

dTAkdxdTAk )( ∞− TTAh convfm

(3)

where ks is the conductivity of the solid fibers and Acond is the area for conduction given by:

pccond AAA −= , (4) where Ac is the cross-sectional area and Ap is the area occupied by the pores. The area was approximated by [2]:

( )ε−= 1ccond AA , (5)

where ε is the porosity. For the control volume of fig. 2, the area for convection is simply given by:

dxAA cconv .σ= , (6) where σ is the surface area per unit volume, which is usually reported by the foam manufacturers. Substituting for the areas, simplifying and rearranging, eqn. (3) then becomes:

022

2

=− fmfmfm m

dxd

θθ , (7)

where θfm is analogous to θs above and mfm is a new foam parameter given by:


High Performance Structures and Materials II 245

( )εσ

−=

12

s

fmfm k

hm . (8)

For an adiabatic tip, the one-dimensional foam temperature distribution is given by eqn. (2) except for replacing ms with mfm:

Lm

xLm

fm

fm

b

fm

cosh

)(cosh −=

θθ . (9)

3 Experiment

To verify the model, one sample of commercially available aluminum, alloy 6101-T6, foam having ten pores per inch was tested. The foam parameters were σ = 803 m2/m3 and ε = 0.918, [13, 14]. The dimensions of the sample were W = 101.6 mm, t = 50.8 mm and L = 250.8 mm. The sample was glued to a 12.7-mm thick solid aluminum base using thermal epoxy, fig. 3.

Figure 3: Schematic of foam sample.

The porosity of the sample was calculated by using the mass, the volume and the density of aluminum alloy 6101-T6. Experiments were performed using a wind tunnel in the Porous Media Research Lab of the University of Puerto Rico at Mayagüez. The tunnel was a small open-loop tunnel shown schematically in fig. 4. The size of the tunnel’s test section was 14.9 cm by 30.2 cm. A thin thermfoil heater, with a surface area identical to the sample’s solid aluminum base, was placed at the base of the foam. The heater was connected to a DC power supply to provide a power of 197.6W. The four sides of the foam sample that constituted its outer perimeter were insulated using Styrofoam insulation, and the sample was placed in the tunnel’s test section. The other two sides were perpendicular to the flow direction and remained open to the airflow.



Thirteen thermocouples were used to measure the temperature at strategic locations in the foam as well as the ambient and the base temperatures. These locations are shown in fig. 3 by the small circles and are listed in table 1. Smaller spacing of the thermocouples was used closer to the base to capture the steep drop in temperature. These were attached to an automatic data acquisition system which was connected to a computer, as shown in fig. 4.

Figure 4: Schematic of the experimental set-up.

Table 1: Thermocouple locations.

Thermocouple No. Location, x, cm 1 0.64 2 1.27 3 1.91 4 2.54 5 3.18 6 3.81 7 4.44 8 5.08 9 6.35 10 8.89 11 13.97 12 19.05 13 Ambient

Table 2 shows the dynamic pressure of the air for 10ppi, after traveling through the foam, which was measured using the manometer. The pressure was used to calculate the average velocity and the actual flow rate. The temperature measurements for the sample were taken varying the exit opening of the tunnel at 25%, 50%, 75%, and 100%. The flow rates and corresponding average velocity are shown in table 2. The temperature measurements were taken at z = 0.64, 1.27, 2.54, 3.81 and 4.44 cm. The dependence of temperature on the y-direction was eliminated based on the symmetry of the boundary conditions, i.e., insulation at y = 0 and y = 10.16 cm.



The uncertainty in the location of the thermocouples was ± 0.8 mm. As for the temperature, the combined uncertainty was ± 1 oC [15].

Table 2: Dynamic pressure.

Flow Rate Dynamic Pressure (Pa) Average Velocity (m/s) Flow Rate (m3/s)

25 % 4.08 2.64 0.07 50 % 20.19 5.86 0.15 75 % 33.42 7.54 0.19 100 % 41.47 8.40 0.21

4 Results and discussion

The first part of this work was concerned with validating the one-dimensional assumption, which is fundamental to the development of the analytical model. Figs. 5 and 6 are plots of the temperature distribution as a function of the non-dimensional distance along the foam, at a flow rate of 0.21 m3/s and 0.07 m3/s, respectively. It is clear that, at a given flow rate, the temperature distribution for the five different z-locations 0.64, 1.27, 2.54, 3.81 and 4.44 cm are very similar. This proves that the temperature is not a function of the flow direction (z), but a function of the direction perpendicular to the flow direction (x). The third direction (y) is eliminated due to the symmetry of the boundary conditions, i.e., insulation at y = 0 and at y = 10.16 cm.

Figure 5: Temperature distribution (10ppi) at 100% flow rate.

Equation (9) was used to generate the analytical curves shown by the solid lines. The analytical model agrees well with the experimental results. The temperature distribution along the foam decreases in exponentially with the distance along the foam. It asymptotically reaches zero as the distance from the base increases. Therefore, most of the heat transfer takes place in the region close to the base. Both the analytical and the experimental results also agree on



predicting this phenomenon. A few points for the z = 3.81 mm case lie below the zero temperature line, i.e., the temperature in the foam is less than that of the ambient, which is physically impossible. Therefore these points are attributed to experimental errors.

Figure 6: Temperature distribution (10ppi) at 25% flow rate.

Figure 7: 25% Flow rate on temperature distribution.

The validation of the one-dimensional assumption was encouraging and it allows for the study of other phenomena that contribute to the characterization of the novel model. One of these is the effect of the flow rate. The effect of the flow rate on the temperature distribution can be seen in figs. 7 and 8 for the 25% and 100% flow, respectively. The experimental data points are those for the central location in the foam, i.e., y = 50.8 mm and z = 25.4mm. These were chosen as representative values for the one-dimensional temperature profile. The agreement between the experimental data and the analytical predictions is excellent. The experimental data shows various points that do not lie precisely on the analytical curve, especially around a dimensionless distance of 0.2. These



are due to random and systematic experimental errors. Even though these disturbances occur, the data remains within the uncertainty. Since these runs represent the maximum and the minimum flow rates, it is logical to conclude that the one-dimensional assumption and the model are valid for all flows in between. The next step is to assess the effect of the flow rate on the temperature profile.

Figure 8: 100% Flow rate on temperature distribution.

Figure 9 is a plot of the one-dimensional temperature profile according to the analytical model of the 10ppi sample for flow rates of 25%, 50%, 75%, and 100%. The temperature profiles decay in an apparent exponential fashion. The curve for the 25% case is higher than those for the 50%, 75%, and 100%; which indicates lower heat transfer. The temperature distributions for the 50, 75 and 100% flow cases are very similar, which shows a weak effect of the flow rate for The temperature of the foam reaches that of the ambient at a dimensionless distance of 0.6, approximately.

Figure 9: 10ppi Temperature distribution model.



Figure 10 shows the experimental results for 25%, 50%, 75%, and 100% flow. The trend can be identified among the first six points, x/l = 0.15, due to the high heat transfer rate that occurs near the base of the fin, which is shown in the small window on the figure. It can be seen that as the flow rate is increased, the level of the temperature distribution curves decreases. This indicates a higher heat transfer rate for higher flow rate. The same observations of fig. 9 can be made about fig. 10. The temperature distribution for the 25% flow case lie above all the other cases, with all other cases being similar.

Figure 10: Flow rate effect on temperature distribution.

5 Conclusions

The combined convection and conduction heat transfer in open-cell metal foam was investigated using a one-dimensional model. One advantage of the model is that it uses the typical parameters reported by the foam manufacturers. The temperature profile was determined for one aluminum foam, using air as the fluid. The model and the one-dimensional assumption were verified by direct experiment. Preliminary results showed excellent agreement between the model’s prediction and the experimental data. When the flow rate is raised, the convection in the foam sample increases, causing an increase in the heat transfer rate. The results are encouraging and provide basis for further verification of the validity of the analytical model in terms of the thickness of the foam in the flow direction.

References

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Characterization of the heat transfer in open-cell metal foam...Characterization of the heat...

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