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Strong wall and transverse size effects on pressure drop of flow through open-cell metal foam
International Journal of Thermal Sciences 57 (2012) 85e91
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Strong wall and transverse size effects on pressure drop of flow through open-cell metal foam Nihad Dukhan*, Mohamed Ali Department of Mechanical Engineering, University of Detroit Mercy, Detroit, MI 48221, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 July 2011 Received in revised form 30 January 2012 Accepted 15 February 2012 Available online 22 March 2012
In applications where a fluid flows through the open pores of metal foam, the foam is treated as an infinite porous medium for which the Darcy law and the Forchheimer equation are applied, in order to describe the pressure drop and to obtain the permeability and form drag coefficient. However, in many practical applications the foam is confined, and depending on the transverse size of the foam (perpendicular to the flow direction), the confining walls and the size may have a strong effect on the velocity field and the resulting pressure drop and its behavior. Actually, for small confined foam size, the above flow relations may not be applicable, or they may require modifications in order to account for the added pressure drop due to the confining walls and size effects. Little or no attention has been paid to the transverse size of the foam perpendicular to the confining wall, which may explain some of the divergence in reported pressure drop in the literature. For confined cylindrical foam systems, this paper experimentally establishes a minimum diameter necessary for the foam to have diameter-independent pressure drop, i.e., negligible wall and size effects and constant permeability and form drag coefficient. This minimum diameter is obtained for two types of open-cell aluminum foam subjected to fullydeveloped airflow in the Forchheimer regime. Below this diameter, values of the two key flow properties show strong dependence on diameter. The Reynolds number ranged from approximately 15,000 to 115,000, and the foam diameters ranged from five to forty five cells for 10- and 20- pore per inch aluminum foam. The intertwined wall and size effect is isolated and studied. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Metal foam Permeability Wall effect Size effect Pressure drop Aluminum foam
1. Introduction Metal foams have been used in various engineered applications. See for example Zhou et al. [1] and Azzi et al. [2]. Pressure drop of fluid flow through open-cell metal foam is a critical design parameter in filters, heat exchangers, catalysts and similar applications. The permeability and form drag coefficient of metal foam, which can be used to calculate the pressure drop, are usually determined experimentally, as in Seguin et al. [3], Tadrist et al. [4], Kim et al. [5], Paek et al. [6], Boomsma and Poulikakos [7], Lage et al. [8], Antohe et al. [9], Dukhan et al. [10], Noh et al. [11] and Dukhan and Patel [12]. To avoid formidable exact solutions [8,13] of the flow field and pressure drop in metal foam, some researchers relied on common geometrical shapes to construct models for the actual internal structure of metal foam, and used those models to study fluid flow in the foam. Examples of such studies are those of
* Corresponding author. Tel.: þ1 313 993 3285; fax: þ1 313 993 1187. E-mail address: [email protected] (N. Dukhan). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2012.02.017
Bhattacharya et al. [14], du Plessis et al. [15], Fourie and du Plessis [16], Despois and Mortensen [17] and Boomsma et al. [18]. Two important issues with a direct and strong impact on the pressure drop in metal foam have not been investigated in the published literature. These issues are the effects of confining walls and the transverse size of the foam perpendicular to the flow direction. The issue of the wall effect on pressure drop in traditional porous media received considerable study dating back to a pioneering work published in 1929, according to Cohen and Metzner [19]. This study was focused on fruits and vegetables (forming a porous medium when stacked for drying), as well as on other types of traditional porous media such as packed beds. Cohen and Metzner [19] developed a model that accounted for the effect of confining wall and porosity variation near the wall on the flow behavior in packed columns of spheres of uniform size. For Newtonian fluids, the wall effect became negligible for column-toparticle-diameter ratio greater than 30. Nield [20] provided an alternative model for including the wall effect through theoretical arguments. Both of the above models assumed Darcy flow.
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Mehta and Hawley [21] modified the famous Ergun equation (which is of the Forchheimer type) in order to account for the friction caused by the confining wall. For packed columns of glass beads of uniform size, the wall effect was present when the column-to-particle-diameter ratio was less than 50. The shape and cell structure of metal foams, as well as their high porosity, make them markedly different from traditional porous media. Due to the structure of the foam, a clear definition of a particle diameter that can be used in Ergun-like correlations is rather elusive, Dukhan and Patel [12]. Cheng and Yuan [22] presented a theoretical model for improving the Ergun equation by taking into account wall effects. Verboven et al. [23] investigated the effect of confining walls for airflow through beds of apples and chicory roots. They correlated the pressure drop using the Forchheimer equation with confinement parameters obtained by means of computational fluid dynamics. Eisfeld and Schnitzlein [24] studied the influence of confining walls on the pressure drop of packed beds. This influence depended on Reynolds number and was caused by two contradicting effects of increased friction and increased porosity near the wall. In an intriguing conceptual study, Lage et al. [25] attempted to systematically isolate the wall effect on the pressure drop through confined porous media, for the purpose of accurately determining the permeability K and form drag coefficient C. They used Darcy’s law and Newton’s law of flow around a bluff body as constitutive equations defining K and C, respectively. The results of Lage et al. [25] provided firm guidelines for accurately determining K and C by minimizing the secondary effect of confining walls. In this valuable study and in the studies of wall effects mentioned above, i.e., [19e24], the issue of the transverse size of the porous medium (perpendicular to the flow direction), or how the effect of size may be intertwined with the effect of confining walls, were not addressed. It can be inferred that it was indirectly assumed that the porous medium in these studies was large enough such that no size effects were present. Therefore, the findings of these researchers are not directly applicable to the current study, which deals with the presence of strong transverse size effect in addition to the wall effect. To the knowledge of the current authors, the presence and effect of confining walls and transverse size effect on the pressure drop in metal foam have not been published. These effects might have been present in some of the experimental pressure drop studies found in the literature. The ‘small’ metal foam sample’s dimension perpendicular to the confining wall used by different researchers is very likely one of the reasons causing discrepancies among the reported pressure drop data for similar foams. The variation in the hydraulic diameter of foam test samples among researches is likely to be dictated by the intended application of the foam, or by limitations of experimental set-ups. Moreover, standards for testing some porous materials such as polymeric materials ISO 4638, ISO 7231 [26,27]; textile fabrics ASTM D737 [28]; filtration media ASTM F778 [29] and urethane foams ASTM D3574-03 [30], do not specify the size of the test samples for rigid porous media. This paper presents results of a systematic experimental study targeting the effect of foam sample diameter and wall effects on the viscous and form contributions to the pressure drop for airflow through aluminum foam e the most common among metal foams. It also provides a clear indication of a minimum diameter of the foam for pressure drop to be practically free from any wall and transverse size effects.
shear forces on the surface of the internal structure of the medium, and the pressure drop is given by the famous Darcy’s law:
m dp ¼ vz dz K
(1)
where p is the static pressure, m is the fluid viscosity, K is the permeability of the porous medium and vz is the velocity component in the z-direction. For higher velocity, the flow departs from the Darcy regime to a second order regime. The form drag starts to become important, and the energy dissipation is represented by the sum of viscous and form drags, where the latter depends on the square of the velocity. In this case, the HazeneDupuiteDarcy (widely known as the Forchheimer) equation is used to relate the pressure drop to the velocity:
m dp c r ¼ vz pFffiffiffiffi jvz jvz K dz K
(2)
where cF is a dimensionless coefficient, r is the fluid density and jvz j is the magnitude of vz . In a macroscopic form, Eq. (2) is written in the following form, which is suitable for computing the pressure drop parameters from easily-measured quantities:
Dp L
m
¼ V rCV 2 K
(3)
where L is the length of the porous medium in the flow direction and C is a form drag coefficient, C ¼ cF =K. The seepage velocity V is an average value calculated by simply dividing the volumetric flow rate through the porous medium by the cross-sectional flow area. Both K and C are strongly dependent on the structure of the porous medium. Ergun [31] empirically related the permeability and the form drag coefficient to the porosity and the particle diameter of the porous medium. For the web-like structure of porous metals, the particle diameter is not easy to determine, Dukhan and Patel [12]. It should be noted that the Ergun equation, just like the above equations, assumes an infinite porous medium, where the pressure drop is purely due to the presence of the porous medium in the flow path, and no confining wall or size effects are present. 3. Model including wall effect To account for the pressure drop due to the presence of confining walls, a modified Brinkman’s equation that includes the Forchheimer term [32] is used:
m m d dp c r dvz r ¼ vz pFffiffiffiffivz jvz j þ e dz K r dr dr K
(4)
where me is the effective viscosity. The effective viscosity is a bulk property that is a function of the dynamic viscosity of the fluid and the geometry of the porous medium. It is used to match the Darcy or Darcy-extended law to the NaviereStokes equations at the boundary between the porous medium and a solid surface. Brinkman set me ¼ m, while others [32] said that me =m ¼ 7:5 for high porosity rigid foam. The last term of Eq. (4) is the Brinkman’s term which accounts for the wall effects. It should be noted here that in packed beds, the effect of the wall is due in part to an increase in porosity resulting from an inefficient packing, e.g. spheres, near the wall. 4. Experiment
2. Flow regimes and pressure drop relations For an infinite porous medium subjected to a one-dimensional Darcy flow in the z-direction, there is only viscous drag due to
Testing was conducted in an open-loop wind tunnel shown schematically in Fig. 1. A suction unit (Super Flow 600 Bench) capable of producing airflow rates reaching 17 m3/min was used to
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Fig. 1. Schematic of experimental set-up.
pull air through the tunnel. The unit had flow controls that could produce a desired air velocity through the tunnel. As shown in Fig. 1, the inlet of the suction unit, located after the test section, was square with a side length of 17.78 cm and was fastened to a main reducer that had a similar cross section. The inlet of the reducer was a square having a side length of 12.70 cm. The reducer was fastened to the outlet of the circular test section using another secondary reducer that gently changed from the circular cross section of the test section to the square inlet of the reducer. The secondary reducer changed depending on the diameter of the test section. It should be noted here that the changes in cross-sectional areas occurred downstream from the test section and did not affect the actual flow entering, or flow inside, the test section. The order of various sections of the test set-up can be ascertained by considering the flow direction shown and marked by an arrow in Fig. 1. The test section, along with the tunnel’s entrance region, were modular and fabricated from thin Plexiglas tubes. They had inner diameter that could vary from 1.27 to 12.07 cm (0.5e6 inches). The upstream tube representing the tunnel’s entry region always had a length greater than ten times the diameter, which insured nearly fully-developed flow condition at the test section. A foam sample could be inserted into the test section and be sealed disallowing air from escaping the tunnel. Two holes were drilled before and after the test section at a distance of 1.59 cm. The diameters of these holes were 0.79 cm and they held tubes for pressure measurements. The tubes were connected to an Omega differential pressure transmitter to measure the pressure drop. Two pressure transmitters, model PX653-10D5V, which had a range of 0e2487 Pa and model PX653-03D5V, which had a range of 0e746 Pa, were used depending on the actual pressure drop. The average velocity was measured using a velocity meter (Extech Heavy Duty CFM Metal Vane) with a range of 0.5e35 m/s. At the inlet of the tunnel, there was a plastic screen having a fine mesh (with a screen open-area ratio ¼ 0.53) to smooth out large scale flow structures. Fourteen samples of commercially-available aluminum foam (manufactured by ERG Materials and Aerospace [33]) were tested. Each sample was a cylinder having a length of 15.24 cm (6 inches) and diameter of 1.27, 2.54, 3.81, 5.08, 6.35, 7.62 and 8.89 cm (0.5, 1, 1.5, 2, 2.5, 3 and 3.5 inches). Seven of the samples had 10 pores per inch (ppi), while the remaining samples had 20 ppi. The foam’s pertinent parameters are given in Table 1. The three-dimensional building unit of the open-cell aluminum foam tested in this study was similar in shape to the geometric unit known as tetrakaidecahedron. This unit consists of eight hexagonal faces and six square faces, which make 24 vertices and 36 edges [16]. The cell diameter is an average equivalent distance
87
representing an imaginary sphere in which the tetrakaidecahedron can be fitted. The pore diameter is the average diameter of a window or aperture opening on the faces of a cell. Both the cell diameter and the pore diameter listed in Table 1 were obtained from manufacturer’s information. The samples carried an industrial designation referred to as ppi (number of pores per linear inch). A photograph of some of the 10-ppi samples that were tested is given as Fig. 2. The porosity of each sample was calculated form measurement of its volume and mass. Each sample was placed in the tunnel’s test section, as shown in Fig. 1, and sealed. The suction unit’s controls were adjusted to realize the desired velocities in the test section. For each velocity, the steady-state static pressure drop was measured. The measurements were repeated three times for each sample and the average of the three runs was reported. For gas flow in porous media, the pressure drop is usually significant such that the density changes. To account for variations in gas density, the pressure drop was computed using the following
p2i p2o 2pi
Dph
(5)
where pi and po were the inlet and exit pressures. Eq. (5) was obtained by integrating a recast form of Eq. (1) as a relationship between the mass flow rate and the pressure drop, and substituting for the air density from the ideal gas model. See Bear [34]. 4.1. Error and uncertainty analysis The uncertainty in the pressure measurements had a contribution from a fixed error (ef ¼ 10.8%) and a reading error (er ¼ 0.2%) as stated by the pressure transducer’s manufacturer. The reading error is related to the precision of the transducer, while the fixed error is related to the design of the instrument. The root-sum-squares method [35] states that the total uncertainty in the pressure values dp is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dp ¼ e2f þ e2r
(6)
This resulted in dp of 10.8%. The average air velocity in the tunnel was measured using a digital velocity meter. The reported uncertainties by the manufacturer of this device included a fixed error of 10:5% and an error of 0:5% of the reading. Using the root-sum-squares method, the maximum relative uncertainty in the air velocity was dV ¼ 10.5%. The uncertainty in the computed quantities is a result of propagation of uncertainties in the measured quantities that are used in their computation. The uncertainty in the reduced pressure drop, dRDp , is given by [35]:
Table 1 Foam parameters. ppi
Porosity ε %
Pore diameter Dpore (mm)
Cell diameter Dcell (mm)
10 20
89.27 90.00
2.55 1.27
4.24 2.12 Fig. 2. Photograph of two 10-ppi foam samples inside and outside test section.
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDP=LV 2 2 vDP=LV 2 2 vDP=LV 2 2 dRDp ¼ dDP þ dL þ dV vD P vL vV
The uncertainty in the density is too small and was ignored as was done for Eq. (11). Substituting in Eq. (14), the percentage uncertainty in the friction factor was obtained as df ¼ 23.6%.
(7) On a percentage basis, the uncertainty in this quantity is given by dRDp =ðDP=LVÞ. Performing the derivatives, substituting and arranging:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d DP 2 d L 2 d V 2 þ þ DP L V
dRDp ¼ DP=LV
(8)
Substituting numerical values of the various uncertainties in the pressure drop, length and velocity, the percent uncertainty in the reduced pressure drop is obtained as dRDp ¼ 15.0%. The uncertainty in the length measurement was 1 mm, and it was converted to a percentage by dividing it by the length of the foam sample. Further uncertainty analysis was performed in order to obtain the uncertainty in the computed values of K and C from curve fits of the experimental pressure drop and velocity data. According to the Forchheimer equation, the pressure drop per unit length of the porous medium is a second order polynomial of the velocity, i.e., Eq. (3). If we divide Eq. (3) by V, it becomes linear:
Dp LV
¼ A þ BV
(9)
where A and B are curve fit constants. By comparing this equation to Eq. (3), K ¼ m=A and C ¼ B=r. The percent uncertainty in K is a result of uncertainties in its constituents m and A, and is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dm d þ A ¼ m K A
dK
(10)
where dm and dA are the uncertainties in m and A, respectively. Similarly, the uncertainty in C is determined according to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dr dB þ ¼ r C B
dC
(11)
The uncertainty in m was estimated as 1 107 N s/m2 and in r as 0.001 kg/m3, both taken as the accuracy of the reported values in property tables [36]. These values are small enough to cause negligible impact on the overall uncertainties in K and C, therefore they were ignored as was done by Boomsma and Poulikakos [7]. Therefore, the uncertainties in K and C are simply given by:
dK K
dC C
dA
¼
A
dB
¼
B
100%
100%
(12)
5. Results The velocities that were used in this study were all high enough to be in the Forchheimer regime. As such Eq. (2) is used to present the pressure drop data. If this equation is divided by the Darcy velocity, a straight line results. The resulting quantity Dp=LV is referred to as the reduced pressure drop. The length of each foam sample in the flow direction, L, was sufficient to minimize entrance and exit effects. Figs. 3 and 4 are plots of the reduced pressure drop versus the Darcy or superficial velocity for the 10- and 20-ppi foam samples, respectively. The effect of diameter on the pressure drop is severe for the three small diameters 12.7, 25.4 and 38.1 mm (0.5, 1 and 1.5 inches) and diminishes for larger diameters. For the small diameters, the pressure drop behavior does not follow the Forchheimer equations, i.e., the reduced pressure drop is not a linear function of the Darcy velocity. When the diameter is greater than 50.8 mm (2 inches), the reduced pressure drop becomes independent of the diameter, and the pressure lines lie on top of each other for sample diameters 63.5, 76.2 and 88.9 mm (2.5, 3 and 3.5 inches). The linear relationship between the reduced pressure drop and the Darcy velocity indicates that the Forchheimer equation is sufficient for capturing the pressure drop behavior. The curves for the case of the intermediate diameter 38.1 mm (1.5 inches) for the 10- and 20-ppi foam warrants a closer look. For the 20-ppi foam, the pressure drop for this diameter is seen to obey the Forchheimer equation (straight line) and to have the same slope as the pressure drop for all larger diameters. This is not the case for the 10-ppi foam. For this case the pressure drop as a function of the Darcy velocity, for this same intermediate diameter, is not exactly a straight line. In addition, it has a different slope when compared to the cases of the other larger diameters. The behavior of the pressure drop for this case is most likely the result of interactions of multiple factors including wall, size and morphological effects. For a sample diameter of 38.1 mm (1.5 inches), there are approximately nine whole cells of the 10-ppi foam, while there are about 20 cells of the 20-ppi foam in the radial direction. See the average cell size of these foams as given in Table 1. This means that from the wall surface to the center of the cylindrical foam sample, there are only 4.5 cells of the 10-ppi foam compared to 10 whole cells of the 20-ppi foam. When the number of cells is small, the character of the foam based on its internal structure may not be established, and there is a dependence on the size of a given foam sample. Hence, one may argue that a size effect was present in the case of the 10-ppi foam
(13)
The average uncertainty in A and B were estimated from the leastsquare method to be 12.1 Pa s/m2 and 10.9 Pa s/m3 for the 10-ppi foam data; and 12.7 Pa s/m2 and 11.2 Pa s/m3 for the 20-ppi data. Using these values and Eqs. (12) and (13), the uncertainties in K and C were obtained and are as follows: for 10-ppi foam, the uncertainty in K is 13:4%; and in C 14:9%; while for 20-ppi, the uncertainty in K is 11:7%; and in C 14:4%. In a similar manner, the uncertainty in the friction factor given by f h2Dp=rV 2 is obtained as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dDP 2 dr 2 d 2 þ þ 2 V ¼ DP r f V
df
(14) Fig. 3. Reduced pressure drop vs. Darcian velocity for 10-ppi foam.
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
89
Fig. 6. Form drag coefficient as a function of sample diameter. Fig. 4. Reduced pressure drop vs. Darcian velocity for 20-ppi foam.
sample having a diameter of 38.1 mm (1.5 inches), which caused the pressure drop behavior to divert from the Forchheimer equation. The opposite is true for the 20-ppi case of the same diameter. One can attempt to compute the permeability and form drag coefficient for each sample by fitting straight lines to the reduced pressure drop data in the above plots. Curve fits constants can be used to calculate values for these two flow properties. This can be done without any difficulty for the case of large diameters, and meaningful values for the flow properties are obtained. However, when this is done for the small diameters, negative values of the permeability are obtained. Obviously, a negative permeability is physically meaningless, as the lowest conceivable permeability is zero for a solid material. The permeability and form drag coefficient are plotted as a function of number of cells in Figs. 5 and 6 for 10- and 20-ppi foam, respectively. In both cases, the permeability is seen to be evolving toward a constant value as the diameter increases. At about 30 cells, the permeability and the form drag coefficient seem to become independent of the diameter. It should be noted that in Eq. (4) the wall effect term and the Darcy term have opposite signs. Moreover, these two terms are viscous in nature. If the wall effect term is larger than the viscous term, the calculated permeability becomes negative and vice versa. This can be used as a mark for assessing the relative magnitude of the wall effect and in deciding whether to ignore it or not.
f ¼ aðReD Þn
(15)
f hDp=ðrV 2 =2Þ
where is the Darcy-Weisbach friction factor and ReD hrDV=m is the Reynolds number based on the tube diameter as a length scale. This correlation suffices when the diameter of the confined foam sample is greater than the critical diameter at which the wall and size effects become important. From the curve fits of Figs. 7 and 8, the following correlations can be given: For 10-ppi foam with diameter greater than 30 cells:
f ¼ 99:1ðReD Þ0:18
(16)
For 20-ppi foam with diameter greater than 30 cells:
f ¼ 152:3ðReD Þ0:21
(17)
In order to avoid calculating negative permeability values, the pressure drop data are correlated using the following:
It should be noted that the curve fits of Eqs. (16) and (17) are based on the data for the largest diameter which is 88.9 mm for both the 10- and 20-ppi foam. This choice is based on the fact that any size or wall effects are minimal (or absent) for this large diameter. For diameters less than the critical diameter, the friction factor dependence on the Reynolds number is not given by a power law. The friction factor depends on the Reynolds number as well as on the diameter. This dependence is more complex in the case of the 10-ppi foam. Due to the structure of the foam, it is very unlikely that the average porosity changes close to the wall, as in the case of packed beds. This is not to say that the structure of the foam does not change near the wall. In the wall region, the foam will have incomplete cells and ligaments that are randomly chopped. However, there are no packing issues in the case of foam. The nature of the wall effect in metal foam is viscous resulting from
Fig. 5. Permeability as a function of sample diameter.
Fig. 7. Friction factor as a function of Reynolds number for 10-ppi foam.
6. Friction factor
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N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Fig. 8. Friction factor as a function of Reynolds number for 20-ppi foam.
shear stresses, i.e., it is not due to form drag. It may be understood by considering the flow near the confining wall as a flow of fluid over the ligaments of the foam that are in contact with the wall. So the wall shear stress depends on the shape of the ligaments as well as the flow velocity or the Reynolds number. In an attempt to isolate the combined wall and size effects, the friction factor for each small diameter was subtracted from the friction factor for the largest diameter (88.9 mm), realizing that the wall/size effect was essentially absent for the latter. Once this is divided by the friction factor for the largest diameter, the percentage of the wall/size effect is obtained according to:
Wall=Size Effect ¼
f fN 100% fN
(18)
where fN is the friction factor for the case of 88.9-mm diameter. The two friction factors of Eq. (18), f and fN must be taken at the same Reynolds number. The wall/size effect is plotted against the Reynolds number in Figs. 9 and 10 for the 10- and 20-ppi foam, respectively. The wall effect is a function of Reynolds number and diameter. For both types of foam, the wall/size effect generally decreases as the diameter increases for high enough Reynolds numbers. The dependence on Reynolds number can be explained by the fact that as the Reynolds number increases, the thickness of the wall boundary layer decreases and the slope of the velocity decreases giving rise to a lower shear stress next to the wall. At high Reynolds numbers, there seems to be a rather weak dependence of the friction factor on Reynolds number. Such behavior is characteristics of a wide range of flows, including flow over bluff bodies
Fig. 9. Wall/size effect as a function of Reynolds number for 10-ppi foam.
Fig. 10. Wall/size effect as a function of Reynolds number for 20-ppi foam.
and rough walls. These two being relevant to flow through metal foam. For small diameters, the wall/size effect actually dominates and is seen to set at about 90% of the pressure drop. For the largest diameter, the wall/size effect accounts only for about 10% of the total pressure drop. The error associated with the wall/size effect as calculated by Eq. (18) is 23.6%, as was discussed in the error analysis subsection. Considering this level of error, there is a difference in behavior of the wall/size effect for the two types of foam in the case of the intermediate diameter of 38.1 mm. In the case of the 10-ppi foam, the wall effect is seen to start small and to increase with Reynolds number, while for the 20-ppi foam, it starts high and decays with Reynolds number. This difference is attributed to the difference morphologies of the two types of foams as given in Table 1. This diameter seems to be a critical diameter around which the wall/size effect changes character. 7. Conclusion Experimental data sets for the pressure drop of Forchheimer regime airflow through confined cylindrical samples of open-cell 10- and 20-ppi aluminum foam having different diameters have been presented. Both the permeability and form drag coefficient were functions of the diameter, and seemed to be approaching constant values as the diameter increased, signifying the weakening of the wall and diamter effects. For sample diameters equal or greater than 63.5 mm, the wall and size effects were absent; and the pressure drop followed the Forchheimer equation and became practically independent of the sample’s diameter. This critical diameter corresponds to 15 and 30 cell diameters for the 10- and 20-ppi foam samples. The value of 30 cell diameters for the 20-ppi foam is identical to the findings of one previous study for packed columns of spheres of uniform size, in which the wall effect was absent for column diameter greater than 30 particle diameters. In another previous study of packed columns of spheres, the wall effect was reported absent for columns of 50 particle diameters. Friction factor correlations were given for cases where the wall/size effect was weak. The wall/size effect was isolated and plotted as a friction factor against the tube Reynolds number. It became clear that the wall effect and the size effect may co-exist for small diameters, and it was rather difficult to isolate and investigate each of them separately.
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Nomenclature C form drag coefficient (m1) Forchheimer coefficient (dimensionless) cF D diameter of foam sample (m) f friction factor (dimensionless) K permeability (m2) L thickness of foam sample in the flow direction (m) p static pressure (Pa) inlet pressure (Pa) pi exit pressure (Pa) po ppi number of pores per inch Reynolds number based on diameter ReD velocity component in the flow z-direction (m/s) vz V Darcy velocity (m/s) Greek
d D m me r
uncertainty (%) change kinematic viscosity of air (kg/ms) effective viscosity (kg/ms) density of air (kg/m3)
References [1] J. Zhou, C. Mercer, W.O. Soboyejo, An investigation of the microstructure and strength of open-cell 6010 aluminum foam, Metall. Mater. Trans. 33A (5) (2002) 413e1427. [2] W. Azzi, W.L. Roberts, A. Rabiei, A study on pressure drop and heat transfer in open cell metal foams for jet engine applications, Mater. Des. 28 (2007) 569e574. [3] D. Seguin, A. Montillet, J. Comiti, Experimental characterization of flow regimes in various porous media-I: limit of laminar flow regime, Chem. Eng. Sci. 53 (21) (1998) 3751e3761. [4] L. Tadrist, M. Miscevic, O. Rahli, F. Topin, About the use of fibrous materials in compact heat exchangers, Exp. Therm. Fluid Sci. 28 (2004) 193e199. [5] S.Y. Kim, J.W. Paek, B.H. Kang, Flow and heat transfer correlations for porous fin in a plate-fin heat exchanger, J. Heat Trans. 122 (2000) 572e578. [6] J.W. Paek, B.H. Kang, S.Y. Kim, J.M. Hyun, Effective thermal conductivity and permeability of aluminum foam materials, Int. J. Thermophysics 21 (2) (2000) 453e464. [7] K. Boomsma, D. Poulikakos, The effect of compression and pore size variations on the liquid flow characteristics in metal foams, J. Fluids Eng. 124 (2002) 263e272. [8] J.L. Lage, B.V. Antohe, D.A. Nield, Two types of nonlinear pressure-drop versus flow-rate relation observed for saturated porous media, J. Fluids Eng. 119 (1997) 700e706. [9] B. Antohe, J.L. Lage, D.C. Price, R.M. Weber, Experimental determination of the permeability and inertial coefficients of mechanically compressed aluminum metal layers, J. Fluids Eng. 11 (1997) 404e412.
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[10] N. Dukhan, R. Picón-Feliciano, Á. Álvarez-Hernández, Air flow through compressed and uncompressed aluminum foam: measurements and correlations, J. Fluids Engin. 128 (2006) 1004e1012. [11] J.S. Noh, K.B. Lee, C.G. Lee, Pressure loss and forced convective heat transfer in an annulus filled with aluminum foam, Int. Comm. Heat Mass Trans. 33 (2006) 434e444. [12] N. Dukhan, P. Patel, Equivalent particle diameter and length scale for pressure drop in porous metals, Exp. Therm. Fluid Sci. 32 (2008) 1059e1067. [13] J.J. Hwang, G.J. Hwang, R.H. Yeh, C.H. Chao, Measurement of interstitial convective heat transfer and frictional drag for flow across metal foams, J. Heat Trans. 124 (2002) 120e129. [14] A. Bhattacharya, V.V. Calmidi, R.L. Mahajan, Thermophysical properties of high porosity metal foams, Int. J. Heat Mass Trans. 45 (2002) 1017e1031. [15] J.P. du Plessis, A. Montillet, J. Comiti, J. Legrand, Pressure drop prediction for flow through high porosity metallic foams, Chem. Eng. Sci. 49 (1994) 3545e3553. [16] J.G. Fourie, J.P. du Plessis, Pressure drop modeling in cellular metallic foams, Chem. Eng. Sci. 57 (2002) 2781e2789. [17] J.F. Despois, A. Mortensen, Permeability of open-cell microcellular materials, Acta Mat 53 (2005) 1381e1388. [18] K. Boomsma, D. Poulikakos, Y. Ventikos, Simulation of flow through open cell metal foams using an idealized periodic cell structure, Int. J. Heat Fluid Flow 24 (2003) 825e834. [19] Y. Cohen, A.B. Metzner, AIChE J. 27 (5) (1981) 705e715. [20] D.A. Nield, AIChE J. 29 (4) (1983) 688e689. [21] D. Mehta, M.C. Hawley, Wall effect in packed columns, EC Proc. Des. Dev. 8 (2) (1969) 280e282. [22] Z. Cheng, W.K. Yuan, Estimating radial velocity of fixed beds with low tube-toparticle diameter ratios, Am. Inst. Chem. Engineers J. 43 (5) (1997) 1319e1924. [23] P. Verboven, M.L. Hoang, M. Baelmans, B.M. Nicolai, Airflow through beds of apples and chicory roots, Biosys. Engin. 88 (1) (2004) 117e125. [24] B. Eisfeld, K. Schnitzlein, The influence of confining walls on the pressure drop in packed beds, Chem. Eng. Sci. 56 (2001) 4321e4329. [25] J.L. Lage, P.S. Krueger, A. Narasimhan, Protocol for measuring permeability and form coefficient of porous media, Phys. Fluids 17 (2005) 088101-1e088101088104. [26] ISO 4638, Polymeric Materials, Cellular Flexible Determination of Air Flow Permeability (1984). [27] ISO 7231, Polymeric Materials, Cellular Flexible Method of Assessment of Air Flow Value at Constant Pressure-drop (1984). [28] ASTM D737 Standard Test Method for Air Permeability of Textile Fabrics. [29] ASTM F 778 e 788, Standard Methods for Gas flow Resistance Testing of Filtration Media. [30] ASTM D3574eD3503 Standard Test Methods for Flexible Cellular MaterialsSlab, Bonded, and Molded Urethane Foams. [31] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (2) (1952) 89e94. [32] D.A. Nield, A. Bejan, Convection in Porous Media, second ed. Springer, New York, 1992, pp. 12, 13. [33] ERG Materials and Aerospace, Oakland, CA, http://www.ergaerospace.com, accessed 03.10. [34] J. Bear, Dynamics of Fluids in Porous Media, Dovers, New York, 1972, pp. 149. [35] R. Figliola, D. Beasly, Theory and Design for Mechanical Measurements, John Wiley and Sons, New York, 2000. [36] F.M. White, Fluid Mechanics, fourth ed. McGraw-Hill, New York, 1999.
Contents lists available at SciVerse ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Strong wall and transverse size effects on pressure drop of flow through open-cell metal foam Nihad Dukhan*, Mohamed Ali Department of Mechanical Engineering, University of Detroit Mercy, Detroit, MI 48221, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 July 2011 Received in revised form 30 January 2012 Accepted 15 February 2012 Available online 22 March 2012
In applications where a fluid flows through the open pores of metal foam, the foam is treated as an infinite porous medium for which the Darcy law and the Forchheimer equation are applied, in order to describe the pressure drop and to obtain the permeability and form drag coefficient. However, in many practical applications the foam is confined, and depending on the transverse size of the foam (perpendicular to the flow direction), the confining walls and the size may have a strong effect on the velocity field and the resulting pressure drop and its behavior. Actually, for small confined foam size, the above flow relations may not be applicable, or they may require modifications in order to account for the added pressure drop due to the confining walls and size effects. Little or no attention has been paid to the transverse size of the foam perpendicular to the confining wall, which may explain some of the divergence in reported pressure drop in the literature. For confined cylindrical foam systems, this paper experimentally establishes a minimum diameter necessary for the foam to have diameter-independent pressure drop, i.e., negligible wall and size effects and constant permeability and form drag coefficient. This minimum diameter is obtained for two types of open-cell aluminum foam subjected to fullydeveloped airflow in the Forchheimer regime. Below this diameter, values of the two key flow properties show strong dependence on diameter. The Reynolds number ranged from approximately 15,000 to 115,000, and the foam diameters ranged from five to forty five cells for 10- and 20- pore per inch aluminum foam. The intertwined wall and size effect is isolated and studied. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Metal foam Permeability Wall effect Size effect Pressure drop Aluminum foam
1. Introduction Metal foams have been used in various engineered applications. See for example Zhou et al. [1] and Azzi et al. [2]. Pressure drop of fluid flow through open-cell metal foam is a critical design parameter in filters, heat exchangers, catalysts and similar applications. The permeability and form drag coefficient of metal foam, which can be used to calculate the pressure drop, are usually determined experimentally, as in Seguin et al. [3], Tadrist et al. [4], Kim et al. [5], Paek et al. [6], Boomsma and Poulikakos [7], Lage et al. [8], Antohe et al. [9], Dukhan et al. [10], Noh et al. [11] and Dukhan and Patel [12]. To avoid formidable exact solutions [8,13] of the flow field and pressure drop in metal foam, some researchers relied on common geometrical shapes to construct models for the actual internal structure of metal foam, and used those models to study fluid flow in the foam. Examples of such studies are those of
* Corresponding author. Tel.: þ1 313 993 3285; fax: þ1 313 993 1187. E-mail address: [email protected] (N. Dukhan). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2012.02.017
Bhattacharya et al. [14], du Plessis et al. [15], Fourie and du Plessis [16], Despois and Mortensen [17] and Boomsma et al. [18]. Two important issues with a direct and strong impact on the pressure drop in metal foam have not been investigated in the published literature. These issues are the effects of confining walls and the transverse size of the foam perpendicular to the flow direction. The issue of the wall effect on pressure drop in traditional porous media received considerable study dating back to a pioneering work published in 1929, according to Cohen and Metzner [19]. This study was focused on fruits and vegetables (forming a porous medium when stacked for drying), as well as on other types of traditional porous media such as packed beds. Cohen and Metzner [19] developed a model that accounted for the effect of confining wall and porosity variation near the wall on the flow behavior in packed columns of spheres of uniform size. For Newtonian fluids, the wall effect became negligible for column-toparticle-diameter ratio greater than 30. Nield [20] provided an alternative model for including the wall effect through theoretical arguments. Both of the above models assumed Darcy flow.
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Mehta and Hawley [21] modified the famous Ergun equation (which is of the Forchheimer type) in order to account for the friction caused by the confining wall. For packed columns of glass beads of uniform size, the wall effect was present when the column-to-particle-diameter ratio was less than 50. The shape and cell structure of metal foams, as well as their high porosity, make them markedly different from traditional porous media. Due to the structure of the foam, a clear definition of a particle diameter that can be used in Ergun-like correlations is rather elusive, Dukhan and Patel [12]. Cheng and Yuan [22] presented a theoretical model for improving the Ergun equation by taking into account wall effects. Verboven et al. [23] investigated the effect of confining walls for airflow through beds of apples and chicory roots. They correlated the pressure drop using the Forchheimer equation with confinement parameters obtained by means of computational fluid dynamics. Eisfeld and Schnitzlein [24] studied the influence of confining walls on the pressure drop of packed beds. This influence depended on Reynolds number and was caused by two contradicting effects of increased friction and increased porosity near the wall. In an intriguing conceptual study, Lage et al. [25] attempted to systematically isolate the wall effect on the pressure drop through confined porous media, for the purpose of accurately determining the permeability K and form drag coefficient C. They used Darcy’s law and Newton’s law of flow around a bluff body as constitutive equations defining K and C, respectively. The results of Lage et al. [25] provided firm guidelines for accurately determining K and C by minimizing the secondary effect of confining walls. In this valuable study and in the studies of wall effects mentioned above, i.e., [19e24], the issue of the transverse size of the porous medium (perpendicular to the flow direction), or how the effect of size may be intertwined with the effect of confining walls, were not addressed. It can be inferred that it was indirectly assumed that the porous medium in these studies was large enough such that no size effects were present. Therefore, the findings of these researchers are not directly applicable to the current study, which deals with the presence of strong transverse size effect in addition to the wall effect. To the knowledge of the current authors, the presence and effect of confining walls and transverse size effect on the pressure drop in metal foam have not been published. These effects might have been present in some of the experimental pressure drop studies found in the literature. The ‘small’ metal foam sample’s dimension perpendicular to the confining wall used by different researchers is very likely one of the reasons causing discrepancies among the reported pressure drop data for similar foams. The variation in the hydraulic diameter of foam test samples among researches is likely to be dictated by the intended application of the foam, or by limitations of experimental set-ups. Moreover, standards for testing some porous materials such as polymeric materials ISO 4638, ISO 7231 [26,27]; textile fabrics ASTM D737 [28]; filtration media ASTM F778 [29] and urethane foams ASTM D3574-03 [30], do not specify the size of the test samples for rigid porous media. This paper presents results of a systematic experimental study targeting the effect of foam sample diameter and wall effects on the viscous and form contributions to the pressure drop for airflow through aluminum foam e the most common among metal foams. It also provides a clear indication of a minimum diameter of the foam for pressure drop to be practically free from any wall and transverse size effects.
shear forces on the surface of the internal structure of the medium, and the pressure drop is given by the famous Darcy’s law:
m dp ¼ vz dz K
(1)
where p is the static pressure, m is the fluid viscosity, K is the permeability of the porous medium and vz is the velocity component in the z-direction. For higher velocity, the flow departs from the Darcy regime to a second order regime. The form drag starts to become important, and the energy dissipation is represented by the sum of viscous and form drags, where the latter depends on the square of the velocity. In this case, the HazeneDupuiteDarcy (widely known as the Forchheimer) equation is used to relate the pressure drop to the velocity:
m dp c r ¼ vz pFffiffiffiffi jvz jvz K dz K
(2)
where cF is a dimensionless coefficient, r is the fluid density and jvz j is the magnitude of vz . In a macroscopic form, Eq. (2) is written in the following form, which is suitable for computing the pressure drop parameters from easily-measured quantities:
Dp L
m
¼ V rCV 2 K
(3)
where L is the length of the porous medium in the flow direction and C is a form drag coefficient, C ¼ cF =K. The seepage velocity V is an average value calculated by simply dividing the volumetric flow rate through the porous medium by the cross-sectional flow area. Both K and C are strongly dependent on the structure of the porous medium. Ergun [31] empirically related the permeability and the form drag coefficient to the porosity and the particle diameter of the porous medium. For the web-like structure of porous metals, the particle diameter is not easy to determine, Dukhan and Patel [12]. It should be noted that the Ergun equation, just like the above equations, assumes an infinite porous medium, where the pressure drop is purely due to the presence of the porous medium in the flow path, and no confining wall or size effects are present. 3. Model including wall effect To account for the pressure drop due to the presence of confining walls, a modified Brinkman’s equation that includes the Forchheimer term [32] is used:
m m d dp c r dvz r ¼ vz pFffiffiffiffivz jvz j þ e dz K r dr dr K
(4)
where me is the effective viscosity. The effective viscosity is a bulk property that is a function of the dynamic viscosity of the fluid and the geometry of the porous medium. It is used to match the Darcy or Darcy-extended law to the NaviereStokes equations at the boundary between the porous medium and a solid surface. Brinkman set me ¼ m, while others [32] said that me =m ¼ 7:5 for high porosity rigid foam. The last term of Eq. (4) is the Brinkman’s term which accounts for the wall effects. It should be noted here that in packed beds, the effect of the wall is due in part to an increase in porosity resulting from an inefficient packing, e.g. spheres, near the wall. 4. Experiment
2. Flow regimes and pressure drop relations For an infinite porous medium subjected to a one-dimensional Darcy flow in the z-direction, there is only viscous drag due to
Testing was conducted in an open-loop wind tunnel shown schematically in Fig. 1. A suction unit (Super Flow 600 Bench) capable of producing airflow rates reaching 17 m3/min was used to
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Fig. 1. Schematic of experimental set-up.
pull air through the tunnel. The unit had flow controls that could produce a desired air velocity through the tunnel. As shown in Fig. 1, the inlet of the suction unit, located after the test section, was square with a side length of 17.78 cm and was fastened to a main reducer that had a similar cross section. The inlet of the reducer was a square having a side length of 12.70 cm. The reducer was fastened to the outlet of the circular test section using another secondary reducer that gently changed from the circular cross section of the test section to the square inlet of the reducer. The secondary reducer changed depending on the diameter of the test section. It should be noted here that the changes in cross-sectional areas occurred downstream from the test section and did not affect the actual flow entering, or flow inside, the test section. The order of various sections of the test set-up can be ascertained by considering the flow direction shown and marked by an arrow in Fig. 1. The test section, along with the tunnel’s entrance region, were modular and fabricated from thin Plexiglas tubes. They had inner diameter that could vary from 1.27 to 12.07 cm (0.5e6 inches). The upstream tube representing the tunnel’s entry region always had a length greater than ten times the diameter, which insured nearly fully-developed flow condition at the test section. A foam sample could be inserted into the test section and be sealed disallowing air from escaping the tunnel. Two holes were drilled before and after the test section at a distance of 1.59 cm. The diameters of these holes were 0.79 cm and they held tubes for pressure measurements. The tubes were connected to an Omega differential pressure transmitter to measure the pressure drop. Two pressure transmitters, model PX653-10D5V, which had a range of 0e2487 Pa and model PX653-03D5V, which had a range of 0e746 Pa, were used depending on the actual pressure drop. The average velocity was measured using a velocity meter (Extech Heavy Duty CFM Metal Vane) with a range of 0.5e35 m/s. At the inlet of the tunnel, there was a plastic screen having a fine mesh (with a screen open-area ratio ¼ 0.53) to smooth out large scale flow structures. Fourteen samples of commercially-available aluminum foam (manufactured by ERG Materials and Aerospace [33]) were tested. Each sample was a cylinder having a length of 15.24 cm (6 inches) and diameter of 1.27, 2.54, 3.81, 5.08, 6.35, 7.62 and 8.89 cm (0.5, 1, 1.5, 2, 2.5, 3 and 3.5 inches). Seven of the samples had 10 pores per inch (ppi), while the remaining samples had 20 ppi. The foam’s pertinent parameters are given in Table 1. The three-dimensional building unit of the open-cell aluminum foam tested in this study was similar in shape to the geometric unit known as tetrakaidecahedron. This unit consists of eight hexagonal faces and six square faces, which make 24 vertices and 36 edges [16]. The cell diameter is an average equivalent distance
87
representing an imaginary sphere in which the tetrakaidecahedron can be fitted. The pore diameter is the average diameter of a window or aperture opening on the faces of a cell. Both the cell diameter and the pore diameter listed in Table 1 were obtained from manufacturer’s information. The samples carried an industrial designation referred to as ppi (number of pores per linear inch). A photograph of some of the 10-ppi samples that were tested is given as Fig. 2. The porosity of each sample was calculated form measurement of its volume and mass. Each sample was placed in the tunnel’s test section, as shown in Fig. 1, and sealed. The suction unit’s controls were adjusted to realize the desired velocities in the test section. For each velocity, the steady-state static pressure drop was measured. The measurements were repeated three times for each sample and the average of the three runs was reported. For gas flow in porous media, the pressure drop is usually significant such that the density changes. To account for variations in gas density, the pressure drop was computed using the following
p2i p2o 2pi
Dph
(5)
where pi and po were the inlet and exit pressures. Eq. (5) was obtained by integrating a recast form of Eq. (1) as a relationship between the mass flow rate and the pressure drop, and substituting for the air density from the ideal gas model. See Bear [34]. 4.1. Error and uncertainty analysis The uncertainty in the pressure measurements had a contribution from a fixed error (ef ¼ 10.8%) and a reading error (er ¼ 0.2%) as stated by the pressure transducer’s manufacturer. The reading error is related to the precision of the transducer, while the fixed error is related to the design of the instrument. The root-sum-squares method [35] states that the total uncertainty in the pressure values dp is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dp ¼ e2f þ e2r
(6)
This resulted in dp of 10.8%. The average air velocity in the tunnel was measured using a digital velocity meter. The reported uncertainties by the manufacturer of this device included a fixed error of 10:5% and an error of 0:5% of the reading. Using the root-sum-squares method, the maximum relative uncertainty in the air velocity was dV ¼ 10.5%. The uncertainty in the computed quantities is a result of propagation of uncertainties in the measured quantities that are used in their computation. The uncertainty in the reduced pressure drop, dRDp , is given by [35]:
Table 1 Foam parameters. ppi
Porosity ε %
Pore diameter Dpore (mm)
Cell diameter Dcell (mm)
10 20
89.27 90.00
2.55 1.27
4.24 2.12 Fig. 2. Photograph of two 10-ppi foam samples inside and outside test section.
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N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDP=LV 2 2 vDP=LV 2 2 vDP=LV 2 2 dRDp ¼ dDP þ dL þ dV vD P vL vV
The uncertainty in the density is too small and was ignored as was done for Eq. (11). Substituting in Eq. (14), the percentage uncertainty in the friction factor was obtained as df ¼ 23.6%.
(7) On a percentage basis, the uncertainty in this quantity is given by dRDp =ðDP=LVÞ. Performing the derivatives, substituting and arranging:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d DP 2 d L 2 d V 2 þ þ DP L V
dRDp ¼ DP=LV
(8)
Substituting numerical values of the various uncertainties in the pressure drop, length and velocity, the percent uncertainty in the reduced pressure drop is obtained as dRDp ¼ 15.0%. The uncertainty in the length measurement was 1 mm, and it was converted to a percentage by dividing it by the length of the foam sample. Further uncertainty analysis was performed in order to obtain the uncertainty in the computed values of K and C from curve fits of the experimental pressure drop and velocity data. According to the Forchheimer equation, the pressure drop per unit length of the porous medium is a second order polynomial of the velocity, i.e., Eq. (3). If we divide Eq. (3) by V, it becomes linear:
Dp LV
¼ A þ BV
(9)
where A and B are curve fit constants. By comparing this equation to Eq. (3), K ¼ m=A and C ¼ B=r. The percent uncertainty in K is a result of uncertainties in its constituents m and A, and is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dm d þ A ¼ m K A
dK
(10)
where dm and dA are the uncertainties in m and A, respectively. Similarly, the uncertainty in C is determined according to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dr dB þ ¼ r C B
dC
(11)
The uncertainty in m was estimated as 1 107 N s/m2 and in r as 0.001 kg/m3, both taken as the accuracy of the reported values in property tables [36]. These values are small enough to cause negligible impact on the overall uncertainties in K and C, therefore they were ignored as was done by Boomsma and Poulikakos [7]. Therefore, the uncertainties in K and C are simply given by:
dK K
dC C
dA
¼
A
dB
¼
B
100%
100%
(12)
5. Results The velocities that were used in this study were all high enough to be in the Forchheimer regime. As such Eq. (2) is used to present the pressure drop data. If this equation is divided by the Darcy velocity, a straight line results. The resulting quantity Dp=LV is referred to as the reduced pressure drop. The length of each foam sample in the flow direction, L, was sufficient to minimize entrance and exit effects. Figs. 3 and 4 are plots of the reduced pressure drop versus the Darcy or superficial velocity for the 10- and 20-ppi foam samples, respectively. The effect of diameter on the pressure drop is severe for the three small diameters 12.7, 25.4 and 38.1 mm (0.5, 1 and 1.5 inches) and diminishes for larger diameters. For the small diameters, the pressure drop behavior does not follow the Forchheimer equations, i.e., the reduced pressure drop is not a linear function of the Darcy velocity. When the diameter is greater than 50.8 mm (2 inches), the reduced pressure drop becomes independent of the diameter, and the pressure lines lie on top of each other for sample diameters 63.5, 76.2 and 88.9 mm (2.5, 3 and 3.5 inches). The linear relationship between the reduced pressure drop and the Darcy velocity indicates that the Forchheimer equation is sufficient for capturing the pressure drop behavior. The curves for the case of the intermediate diameter 38.1 mm (1.5 inches) for the 10- and 20-ppi foam warrants a closer look. For the 20-ppi foam, the pressure drop for this diameter is seen to obey the Forchheimer equation (straight line) and to have the same slope as the pressure drop for all larger diameters. This is not the case for the 10-ppi foam. For this case the pressure drop as a function of the Darcy velocity, for this same intermediate diameter, is not exactly a straight line. In addition, it has a different slope when compared to the cases of the other larger diameters. The behavior of the pressure drop for this case is most likely the result of interactions of multiple factors including wall, size and morphological effects. For a sample diameter of 38.1 mm (1.5 inches), there are approximately nine whole cells of the 10-ppi foam, while there are about 20 cells of the 20-ppi foam in the radial direction. See the average cell size of these foams as given in Table 1. This means that from the wall surface to the center of the cylindrical foam sample, there are only 4.5 cells of the 10-ppi foam compared to 10 whole cells of the 20-ppi foam. When the number of cells is small, the character of the foam based on its internal structure may not be established, and there is a dependence on the size of a given foam sample. Hence, one may argue that a size effect was present in the case of the 10-ppi foam
(13)
The average uncertainty in A and B were estimated from the leastsquare method to be 12.1 Pa s/m2 and 10.9 Pa s/m3 for the 10-ppi foam data; and 12.7 Pa s/m2 and 11.2 Pa s/m3 for the 20-ppi data. Using these values and Eqs. (12) and (13), the uncertainties in K and C were obtained and are as follows: for 10-ppi foam, the uncertainty in K is 13:4%; and in C 14:9%; while for 20-ppi, the uncertainty in K is 11:7%; and in C 14:4%. In a similar manner, the uncertainty in the friction factor given by f h2Dp=rV 2 is obtained as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dDP 2 dr 2 d 2 þ þ 2 V ¼ DP r f V
df
(14) Fig. 3. Reduced pressure drop vs. Darcian velocity for 10-ppi foam.
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
89
Fig. 6. Form drag coefficient as a function of sample diameter. Fig. 4. Reduced pressure drop vs. Darcian velocity for 20-ppi foam.
sample having a diameter of 38.1 mm (1.5 inches), which caused the pressure drop behavior to divert from the Forchheimer equation. The opposite is true for the 20-ppi case of the same diameter. One can attempt to compute the permeability and form drag coefficient for each sample by fitting straight lines to the reduced pressure drop data in the above plots. Curve fits constants can be used to calculate values for these two flow properties. This can be done without any difficulty for the case of large diameters, and meaningful values for the flow properties are obtained. However, when this is done for the small diameters, negative values of the permeability are obtained. Obviously, a negative permeability is physically meaningless, as the lowest conceivable permeability is zero for a solid material. The permeability and form drag coefficient are plotted as a function of number of cells in Figs. 5 and 6 for 10- and 20-ppi foam, respectively. In both cases, the permeability is seen to be evolving toward a constant value as the diameter increases. At about 30 cells, the permeability and the form drag coefficient seem to become independent of the diameter. It should be noted that in Eq. (4) the wall effect term and the Darcy term have opposite signs. Moreover, these two terms are viscous in nature. If the wall effect term is larger than the viscous term, the calculated permeability becomes negative and vice versa. This can be used as a mark for assessing the relative magnitude of the wall effect and in deciding whether to ignore it or not.
f ¼ aðReD Þn
(15)
f hDp=ðrV 2 =2Þ
where is the Darcy-Weisbach friction factor and ReD hrDV=m is the Reynolds number based on the tube diameter as a length scale. This correlation suffices when the diameter of the confined foam sample is greater than the critical diameter at which the wall and size effects become important. From the curve fits of Figs. 7 and 8, the following correlations can be given: For 10-ppi foam with diameter greater than 30 cells:
f ¼ 99:1ðReD Þ0:18
(16)
For 20-ppi foam with diameter greater than 30 cells:
f ¼ 152:3ðReD Þ0:21
(17)
In order to avoid calculating negative permeability values, the pressure drop data are correlated using the following:
It should be noted that the curve fits of Eqs. (16) and (17) are based on the data for the largest diameter which is 88.9 mm for both the 10- and 20-ppi foam. This choice is based on the fact that any size or wall effects are minimal (or absent) for this large diameter. For diameters less than the critical diameter, the friction factor dependence on the Reynolds number is not given by a power law. The friction factor depends on the Reynolds number as well as on the diameter. This dependence is more complex in the case of the 10-ppi foam. Due to the structure of the foam, it is very unlikely that the average porosity changes close to the wall, as in the case of packed beds. This is not to say that the structure of the foam does not change near the wall. In the wall region, the foam will have incomplete cells and ligaments that are randomly chopped. However, there are no packing issues in the case of foam. The nature of the wall effect in metal foam is viscous resulting from
Fig. 5. Permeability as a function of sample diameter.
Fig. 7. Friction factor as a function of Reynolds number for 10-ppi foam.
6. Friction factor
90
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Fig. 8. Friction factor as a function of Reynolds number for 20-ppi foam.
shear stresses, i.e., it is not due to form drag. It may be understood by considering the flow near the confining wall as a flow of fluid over the ligaments of the foam that are in contact with the wall. So the wall shear stress depends on the shape of the ligaments as well as the flow velocity or the Reynolds number. In an attempt to isolate the combined wall and size effects, the friction factor for each small diameter was subtracted from the friction factor for the largest diameter (88.9 mm), realizing that the wall/size effect was essentially absent for the latter. Once this is divided by the friction factor for the largest diameter, the percentage of the wall/size effect is obtained according to:
Wall=Size Effect ¼
f fN 100% fN
(18)
where fN is the friction factor for the case of 88.9-mm diameter. The two friction factors of Eq. (18), f and fN must be taken at the same Reynolds number. The wall/size effect is plotted against the Reynolds number in Figs. 9 and 10 for the 10- and 20-ppi foam, respectively. The wall effect is a function of Reynolds number and diameter. For both types of foam, the wall/size effect generally decreases as the diameter increases for high enough Reynolds numbers. The dependence on Reynolds number can be explained by the fact that as the Reynolds number increases, the thickness of the wall boundary layer decreases and the slope of the velocity decreases giving rise to a lower shear stress next to the wall. At high Reynolds numbers, there seems to be a rather weak dependence of the friction factor on Reynolds number. Such behavior is characteristics of a wide range of flows, including flow over bluff bodies
Fig. 9. Wall/size effect as a function of Reynolds number for 10-ppi foam.
Fig. 10. Wall/size effect as a function of Reynolds number for 20-ppi foam.
and rough walls. These two being relevant to flow through metal foam. For small diameters, the wall/size effect actually dominates and is seen to set at about 90% of the pressure drop. For the largest diameter, the wall/size effect accounts only for about 10% of the total pressure drop. The error associated with the wall/size effect as calculated by Eq. (18) is 23.6%, as was discussed in the error analysis subsection. Considering this level of error, there is a difference in behavior of the wall/size effect for the two types of foam in the case of the intermediate diameter of 38.1 mm. In the case of the 10-ppi foam, the wall effect is seen to start small and to increase with Reynolds number, while for the 20-ppi foam, it starts high and decays with Reynolds number. This difference is attributed to the difference morphologies of the two types of foams as given in Table 1. This diameter seems to be a critical diameter around which the wall/size effect changes character. 7. Conclusion Experimental data sets for the pressure drop of Forchheimer regime airflow through confined cylindrical samples of open-cell 10- and 20-ppi aluminum foam having different diameters have been presented. Both the permeability and form drag coefficient were functions of the diameter, and seemed to be approaching constant values as the diameter increased, signifying the weakening of the wall and diamter effects. For sample diameters equal or greater than 63.5 mm, the wall and size effects were absent; and the pressure drop followed the Forchheimer equation and became practically independent of the sample’s diameter. This critical diameter corresponds to 15 and 30 cell diameters for the 10- and 20-ppi foam samples. The value of 30 cell diameters for the 20-ppi foam is identical to the findings of one previous study for packed columns of spheres of uniform size, in which the wall effect was absent for column diameter greater than 30 particle diameters. In another previous study of packed columns of spheres, the wall effect was reported absent for columns of 50 particle diameters. Friction factor correlations were given for cases where the wall/size effect was weak. The wall/size effect was isolated and plotted as a friction factor against the tube Reynolds number. It became clear that the wall effect and the size effect may co-exist for small diameters, and it was rather difficult to isolate and investigate each of them separately.
N. Dukhan, M. Ali / International Journal of Thermal Sciences 57 (2012) 85e91
Nomenclature C form drag coefficient (m1) Forchheimer coefficient (dimensionless) cF D diameter of foam sample (m) f friction factor (dimensionless) K permeability (m2) L thickness of foam sample in the flow direction (m) p static pressure (Pa) inlet pressure (Pa) pi exit pressure (Pa) po ppi number of pores per inch Reynolds number based on diameter ReD velocity component in the flow z-direction (m/s) vz V Darcy velocity (m/s) Greek
d D m me r
uncertainty (%) change kinematic viscosity of air (kg/ms) effective viscosity (kg/ms) density of air (kg/m3)
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