Pressure drop prediction for flow through high porosity metallic foams

Pergnmon

Chemicd

En&eering

Science, Vol. 49, No. 21, pp. 3%5-3553, 1994 Copyright 8 1994 Elscvicr Scimcc Ltd Printed in Great Britain. All rights mcrvcd OOO!-2509/94 $7.00 + 0.00

ooo9-2509(!M)00170-7

PRESSURE DROP PREDICTION FOR FLOW THROUGH HIGH POROSITY METALLIC FOAMS Department

of Applied

PRIEUR DU PLESSIS Mathematics, University of Stellenbosch,

7600 Stellenbosch,

South Africa

and Laboratoire

AGNES MONTILLET, JACQUES COMITI and JACK LEGRAND? de Genie de Procedes, IUT Saint-Nazaire, B.P. 420,44606 Saint-Nazaire Cedex, France (First received 10 ApriL 1993; accepted

in revised form 21 May

1994)

Abstract-Experimental results for isothermal Newtonian flow through metallic foams are analysed in comparison with a theoretical model and the results interpreted. In both the Darcy and non-Darcy regimes the results show oromise for the accurate theoretical prediction of fluid dynamic phenomena in foamlike porous materials-of very high porosity.

INTRODUCTION Modern process engineering practice frequently requires very accurate predictive measures for optimal design of high cost reactor plants. In contrast to conventional packed beds formed by tight packing of granular material, the use of a metallic foam as bed material has become very interesting, with characteristics quite different from that of a conventional granular bed. Foams with very high specific surfaces may be advantageous in mass transfer processes and a detailed quantification of its hydrodynamic interaction with a fluid or gaseous phase is of paramount importance in, for example, chemical engineering design. Theoretical models of packed beds consisting of spherical or almost spherical particles abound, but modelling analyses of fluid discharge through foam or other spongelike material are scarce and still without claim of general success or applicability. In this paper a novel approach to porous media modelling is applied to experimental results for laminar flow of water and glycerol through high porosity foam materials. The results are discussed and the theoretical analysis shown to be adequately in harmony with the experimental work. A geometrical model characterizing the microstructure of foams by the rectangular distribution of solid material in a representative unit cell, as is shown in Fig. 1, was introduced by Du Plessis and Masliyah (1988). In th? present case of metallic foams of high porosity this structure is very appropriate, since it yields a totally three-dimensional winding path for the fluid through a consolidated solid matrix. It should be noted that the cell as shown in Fig. 1 conforms exactly

‘Author

to whom correspondence should be addressed. 3545

to the model proposed by Du Plessis and Masliyah (1988). This model performed well in the Darcy regime of very low intrapore Reynolds number flow, but its prediction of the Forchheimer inertial effect was not quantitatively correct. The former attempt (Du Plessis and Masliyah, 1988) to derive the Forchheimer nonlinearity by introduction of flow velocity redevelopment within subsequent pore sections did yield some deviation from the linear dependence of pressure gradient on velocity. The more severe deviation found experimentally and which suggests a square dependence of pressure gradient on the velocity, was however, still theoretically underpredicted and physically unresolved. Some models, such as the BurkePlummer part of the Ergun equation, imply classical turbulence as an explanation for the inertial effects, and do yield reasonable results because they are quantitatively tailored by adjusting the arbitrary coefficients (Bird et al., 1960). The physical aptness of such models was already questioned almost three decades ago by, e.g. Happel and Brenner (1965). In a recent modelling for high Reynolds number flow through granular porous media, considerable success was achieved towards quantitatively predicting the inertial effects by the introduction of local flow recirculation within transverse pore sections on the lee side of all solid material (Du Plessis, 1992b). Although not exactly the same, this relates to the well-known form drag for flow past an obstacle and is quite similarly introduced in this work on metallic foams of high porosity. MOMENTUM TRANSPORT Any intrapore variable may be averaged by volumetric integration over a representative elementary volume consisting of solid and void sections and of

3546

PRIEUR

Du

PLESSIS et al.

ture and is a measure of the tortuous pore length per height of porous medium. The quantitative evaluation of the surface integral in eq. (1) is subject to a description of the actual intrapore velocity gradients at the fluid-solid interfacial surfaces of the foam. This, as well as determination of the tortuosity, in turn warrants an accurate description of the porous microstructure, which will be addressed in the next s&&ion.

MODELLING

Fig. 1. Cubic representativeunit cell for high porosity.

which the size is subject to certain constraints. As this integration is only implied hypothetically it need not be elaborated on further for coherence of this paper. Volumetric averaging of the three-dimensional Navier-Stokes equation, which is assumed to govern the intrapore flow, yields the following one-dimensional pressure drop equation:

where SI. denotes the wetted surface interface between the fluid and the solid. The gravity term is implied as part of the pressure head p,-. The restriction to one-dimensional average flow merely simplifies the present analysis and does not imply any loss of general applicability of the results to follow. It also complies to the experimental conditions prevailing in this study, since the effect of the macroscopic boundaries are negligible in comparison to the internal friction. The aim of the analytical model is now to express the right-hand side of eq. (1) in terms of macroscopic variables such as the superficial velocity q. This velocity is a measure of the specific discharge and is defined as

q=’ G

vdV.

It produces a unique vector at each point in the porous domain and its direction at any point defines the local streamwise direction. In the present case this direction corresponds with the direction of eq. (1), which in turn also corresponds with the vertical average flow in the experimental work. The interstitial velocity, v,, which describes the average velocity at which a fluid particle proceeds streamwise through the porous medium, is related as follows to the superficial velocity q and the mean pore velocity up:

q=EVf=3

x In eq. (3) x is the tortuosity of the porous microstruc-

OF FOAM STRUCTURE

Following earlier work by Du Plessis and Masliyah (1988) the secondary modelling needed to provide closure of eq. (1) will be based on a rectangular arrangement of solid material within a rectangular representative unit cell (RUC). In this manner the basic phenomena occurring within an isotropic porous material may be conveniently approximated in a simple deterministic manner. The RUC is devised to provide a simple isotropic configuration of which the porosity may be changed without altering the nature of the configuration. It may thus be used in a similar manner for any porosity value, inclusive of very high porosity metallic foams. The assumption of geometrical isotropy “on average” for the presently considered foamlike medium allows the introduction of a cubic RUG of linear dimension d and volume Vo, so that its fluid filled “void” part can be written in terms of the porosity as v, =

E v.

=

Ed3.

(4)

It is furthermore assumed that the average geometrical properties of the foam structure within the RUC can be resembled by the three-dimensional staggered arrangement of three square duct sections, or equivalently solid sections, aligned with the RUC as is shown in Fig. 1. This representation is identical to the one proposed by Du Plessis and Masliyah (1988), in the present case the volume t: of solid material being shown to be much less than the void volume V,. Internally the relative positioning of material is subject to both maximal staggering and maximum interconnectivity of duct sections. Furthermore, the hypothetical arrangement of duct sections in neighbouring RUCs is also required to provide maximum possible staggering of resulting duct sections within the porous medium. This requirement ensures that isotropy is maintained and that fluid is forced through all transverse void sections. The specific assumption about the spatial distribution of the solid material within the RUC, which for the present case is shown in Fig. 1, also uniquely defines the tortuosity of the pore structure. The tortuosity x of the porous matrix is defined as the quotient of the total winding path length d,, which is available within the RUC for flow under a constant cross-section A,, and the basic streamwise dimension d of the porous medium microstructure. According to the specific arrangement proposed, its inverse may be

High porosity

expressed (Du Plessis, 1992a) as

4x 1 x cos 7 + gcos-l

3547

the shear stresses in transverse channel sections. Equation (1) then yields the following streamwise equation in case of very low Reynolds number (Darcy) flow:

II. (5)

8s2 - 36.5+ 27 (9 - 8~)~”

As shown in Fig. 1 the respective cross-sectional side lengths of the pore solid sections are denoted by d, and d, so that d, + d, = d. It now follows straightforwardly that the corresponding cross-sectional areas are given, respectively, by

A,=+:&

(10)

If the shear stress magnitude is taken to be constant at all surfaces and complies to fully developed flow, at mean velocity u,,, between parallel plates a distance d, apart, eq. (10) may be written as

. (11)

x

and (7) The cross-sectional pore-dimension d, may also be written as (Du Plessis and Masliyah, 1988)

Another subtle yet very important point is that at each of the six sides of the RUC a minimal pore opening A, and minimal prismatic solid connection A, is implied for coherence of the model. A further point of interest is the magnitude of the average pore diameter. Although it seems according to the present notation that d, might be a direct measure of the visible or experimentally observed pore diameter, the latter is quasi-three-dimensional and according to the schematic indication given in Fig. I, a better observable pore diameter is given approximately by Dp=$dp=

INTJZRSTITUL

metallic foams

(9)

FLOW MODELLING

The modelling of the interstitial flow proceeds in three parts. Firstly the drag effects of slow viscous flow through the foam is considered. Subsequently a modelling is done of the drag induced when the fluid is discharged at a high velocity and lastly the two respective asymptotic results are matched by a straightforward summation of the two terms. Viscous term

In the present model fully developed Poiseuille flow in each pore section will again be required at low Reynolds number flow to quantify the corresponding Darcy flow characteristics according to eq. (1). Investigation of the surface integral shows that shear stresses in transverse channels do not contribute to the viscous term of the integrand. Since, however, such shear stresses would cause axial pressure gradients over these channels, the pressure deviation term may be used to retrieve the corresponding effects of

Inertial term In the non-Darcy regime the inertial effects are accounted for by modelling the gradual increase with Reynolds number of flow recirculation on the streamwise lee-side surface of solid material. In the limit of reasonable high Reynolds numbers it then amounts to form drag locally within the pore structure. Acknowledging that viscous shear stresses become insignificant in comparison with form drag at “high” Reynolds number flow, eq. (1) may be written as ;vdS.

(12)

s,, On average, the only non-zero contribution to eq. (12) is due to the pressure differences caused by flow recirculation over surface sections normal of the streamwise direction. In case of the foam model of Fig. 1 the surface area subject to recirculation is one sixth of the total fluid-solid interface, yielding the following equation:

=s Sf. P4 -- dp =---= dx

cVO 6

2

Asymptotic matching Since each of the two limiting expressions for - (dp/dx) predominates in their respective regions of applicability, simple addition of the two results leads to the following general equation for the pressure gradient: dp --=dx

w 36X(x - 1) pg2 2.05x(x - 1) + 7 as(3 _ X) * d2 ~~

(14)

To obtain this equation use was made of the expressions derived formerly for the model geometry (Du Plessis and Masliyah, 1988) as well as a form drag coefficient of 2.05. The assumptions applied to quantify the two extremal expressions above are similar to those introduced with considerable success to a model for twodimensional transverse flow through a prismatic porous medium (Du Plessis and van der Westhuizen, 1993) and also to a granular porous medium (Du Plessis, 1992b).

3548

Transformation

PRIEUR DU PLEWS et al.

of equations for experimental

compari-

son

Experimental results may be conveniently presented in the form of straight lines on a linear-linear graph by writing eq. (14) in the following form: ---=1 dp

NfqM

4 dx

(15)

36x(x - I) d2 .E~

Grade

GlOO

GM)

G45

E K

0.973 1.198

0.975 1.189

0.978

9.019

8.510

7.943

0.2850

0.2675

0.2485

Nd2 -

( p > .”

where N and M are, respectively, defined as N ~ of.

Table 1. Analytical results obtained from porosity values (SI units)

&%”

(16)

1.179

and M ~ p 2.05x(x - I) d ~~(3 - x) . Given the porosity and the drag coefficient c,,for each foam, the model thus uniquely predicts for each foam the following two constant quantities for the viscous and the inertial terms of eq. (14), respectively:

an

= 36x(x 2

1)

in Table 1, suggests approximate pore diameters of 0.25, 0.42 and 0.87 mm, respectively. Pressure drop variations with superficial velocity were determined for a vertical test section of which the detail is described extensively in a former publication (Montillet et al., 1992). The cylindrical test section with a diameter of 60 mm is large enough to render the influence of the macroscopic boundaries negligible. The somewhat triangular appearance of the foam structure,

Md

(-> P

Bll

2.05x(x

-

1)

e2(3 - x)

(19)

The latter two equations will be utilised in the following section to facilitate analysis of the experimental results. The critical Reynolds number (Re,& gives the location of the centre of the transition region between the Darcy region, where N predominates, and the Forchheimer region, where non-linearity is introduced by the internal form drag causing a large Mvalue. It is therefore defined as that Reynolds number at which M = N and from eq. (14) it can be expressed explicitly as follows in terms of tortuosity:

(Jk&

=

W3 - xl cd

(20)

As a final note it may be added that the specific dynamic surface Avd is predicted by the model as (21) This is an important parameter for the analysis of the hydrodynamical properties of foams and the explicit expression given above for the functional dependence of the specific dynamic surface is therefore very helpful.

together

EXPERIMENTAL RESULTS

webs between

Table 2. Determination of pore-scale lengths from glycerol data (Sl units) Grade

Porosity measurement Three different grades, GlOO, G60 and G45, of metallic foam were used as porous media in the experimental programme. Each grade number, provided by the manufacturer of the foams, is an indication of the number of pores per inch and thus, due to the experimentally obtained high porosities as indicated

with the stretched

strands of solid, suggests a form drag coefficient midway between those for two-dimensional triangular and flat plates. An average drag coefficient of 1.8 was therefore assumed for all the three foams. Two working fluids were used in the test runs, namely, water and an aqueous solution of glycerol. For water the density was 997.8kgm-’ and the dynamic viscosity 0.00096Pas. The density for glycerol was determined as 1152 kg m- 3 and the dynamic viscosities for the three experimental sets are given in Table 2. Due to the high viscosity of the aqueous solution of glycerol these latter experimental results fall completely within the Darcy regime of purely viscous flow. This is also evident from the magnitude of the Reynolds numbers in comparison to the (Re,,,), values of approximately 30 as predicted by eq. (20) for the three foams. The model developed here necessitates knowledge about the microscale length parameter d for the particular foam under investigation. This may be obtained from experimental pressure drop results for

P’e

GlOO 0.00779

G60

0.00988

1O-3

4396

1231

x 10-6

564

125

dx103

0.1265

0.2609

A, x 1O-3

313

157

W&,x,x

G45 0.0103 643 62.4 0.3568 125

High porosity

a purely viscous discharge through the foam. Subsequently, through use of the equations derived above, pressure drops can be predicted for the discharge of any Newtonian fluid through the foam, irrespective of the discharge rate. In the present case the experimental results of the viscous discharge of an aqueous solution of glycerol will therefore be used to obtain an effective microstructure scale length d for each of the three foams. These values will then be used to predict the pressure gradients expected for water discharge through the same foams and the results compared to experimental findings. The much lower dynamic viscosity of water causes its discharge to be subject to inertial effects of the same order of magnitude as the viscous effects and will thus provide a good measure of the accuracy of the modelling over a wide discharge range. Aqueous solution of glycerol as working fluid In Fig. 2 the experimental data are given for a representative set of experiments with the three foams. Measurements were made for flow through stacked sheets of foam. Since the pressure gradient is linearly plotted against velocity, the slope of each graph corresponds to the coefficient N, for the specific foam according to eq. (15). These experimental values of N, may be used in conjunction with p’s and the left-hand side of eq. (18) to predict the magnitude of the RUC dimensions as follows:

(22) The stepwise numerical values for the intermediate calculations are presented in Table 2. Water as working fluid Equations (18) and (19) may now be used in the following forms, respectively, to calculate for each

3549

metallic foams

foam the predicted values for N, and M,: (23) and (24) The results are presented in Table 3 and are also compared in Fig. 3 with the actual experimental data. Grade G60 and G45 foams were shown to be quasiisotropic materials, but for the grade GlOO foam an anisotropy factor of 1.1 was measured (Montillet et al., 1992). Although experiments were performed on all foams for a flow-through configuration (i.e. discharge normal to sheets), the anisotropic foam was also subjected to a flow-by (i.e. discharge parallel to sheets) configuration relative to the stacking of foam sheets. The very good correlation between the experimental and the analytical results, even for the less isotropic foam, enhances confidence in the theoretical model. Microscopic determination of foam microstructure As a further test of the model applicability the value predicted for the characteristic length d may be compared to the porescale dimension of the microstructure as observed by microscope. It should be noted that this section is included as an extra and is included merely to show how easily the model may be adapted to conform with situations apparently very different from the rectangular arrangement adopted for the RUC. In Fig. 4 a typical micro-image of the three-dimensional foam structure is presented and in Table 4 the corresponding values of the pore diameter D, as given by the manufacturer are given for each grade. It is evident that the shape of the actual foam appears quite different from the conceptual square bar model displayed in Fig. 1. If the foam microstructure was better defined in relation to the presently used model, the pore structure may have been measured on beforehand and all predictions done without recourse to another set of experiments as was done in the reported procedure. Unfortunately the foam structures are usually as difficult to quantify geometrically as in case of very low porosity granular media with variable granule and pore sixes. Due to the weblike surface between the backbone strands of the foam the value of d will be underpredieted experimentally to compensate for extra friction

Table 3. Prediction of pressuregradient for water

experiments(SI units)

9

Fig. 2. Experimental results for glycerol. ES

49:21-B

Grade

GlOO

G60

G45

N,,,x 1O-3 M, x 10-a

541 2247

120 1023

60 695

-1

Cm * 1

High porosity metallic foams Table 4. Determination of shape factor from microscope images (all lengths in mm)

a:=

r

$D_

GIOO

G60

G45

D, (fab) 2.6D (mic)

0.254 0.07 0.047 0.1265 0.0125 3.7 0.47 0.74 2.3

0.423 0.08 0.054 0.2609 0.0246 2.2 0.57 0.90 0.68

0.564 0.08 0.054 0.3568 0.0318 1.7 0.61 0.97

The surface factor @, which is defined as the ratio of the actual surface area schematically shown in Fig. 5 and the model area S,, of Fig. 1, is then given by

0.17

The width L is considered to be the average extent of the web per cylindrical strand. Visual inspection of the foam structures under microscope suggests that the minimal diameter of the solid strand triplet of the foam as measured by microscope is approximately 2.60. For each foam grade these values are presented in Table 4 together with subsequent predictions for A L/D and D,. In this table the pore diameters of the manufacturer for the three foams are also given for comparison. It is evident that, although the interpretation of the three-dimensional foam structure is crudely oversimplified, the analysis seems to be in broad quantitative harmony with experimental evidence. A much more sophisticated treatment of this web-effect is not worth while at this stage, because it was shown that the analytical prediction does provide reasonable magnitudes of the microstructure.

D, (an) L/o (an)

induced. A physically more realistic valued’ = j?d may be obtained by correcting eqs (11) and (13) for the extra surface through the ‘multiplicative parameter B defined by s;, = I=,,.

(25)

The magnitude of B is thus directly coupled to a specific physical phenomenon of the foam and may be determined from micro-images as is discussed below. Equation (14) then reads as follows: -z=&J

B 2

0

36x(x - 1) +pg2 B E2 d’ (-1

ZOMx - 1) EZ(3 - x) *

(26) The

follows in terms of the strand diameter D:

Grade

d, (mic) d (an) d. (an) B d’ (an)

0

3551

remaining

question

is the quantification

of the

excess surface factor 8. A very simple approximation to the actual foam is to assume that each “bar” consists of three converging-diverging cylindrical strands with a massless film of solid spanned between them. It will be assumed that mass conservation applies in the comparison so that, equating the areas of the two models shown in Figs 1 and 5, it follows that the cross-sectional bar dimension d: may be expressed as

,=$y,+~).

(27)

(28)

COMPARtSQNWm CAPILLARY MODEL The only other analytical model known, which also presents modelling of the inertial effects for flow through high porosity foams, is the capillary model of Comiti and Renaud (1989). The predictions of their model were published earlier (Montillet et al., 1992) and only some aspects will be recapped here for coherence of the comparison. The model is based on the assumption of unconnected tortuous pores in the streamwise direction,

-

2.6 D -

Fig 5. Microscope image of a strand (G60, magnification: x 6CtO) and its scheme,

3552

PRIEUR

Du PLESSIS et aI.

with the mean diameter of each pore defined as the hydraulic diameter of the porous structure, 4V,/S,. in the present notation. In the Darcy flow regime, Poiseuille flow is assumed in the pore due to the pressure gradient. The inertial regime is modelled by assumption of the pore to be a rough conduit, with roughness of the same magnitude as the pore diameter. The empirical Nikuradse formula is used to obtain a friction factor for such a rough conduit and the corresponding kinetic energy loss due to the multiple changes in direction if derived. If the wall effect of the surrounding cylindrical capsule is omitted, as was done for the model described earlier, the model of Comiti and Renaud is given by

ldp ---= qdx

~(1 -dz 2XZAt,+QP(1 -4 &2 $2

sembles flow winding around a metallic wire in three dimensions. Except for the surface factor discussed, the basic modelling procedure is based only on physical principles with no adjustable parameters to fit experimental data. This is very important insofar its good performance suggests that the model captures the correct prevailing hydrodynamical conditions and this is a prerequisite for the study of dispersion and other important phenomena in foams. Acknowledgements - Financial and logistic support towards this project by the University of Nan&, the University of Stellenbosch and the South African Water Research Commission is gratefully acknowledged by the authors. NOTATION

cross-sectional pore area, m2 cross-sectional solid prism area, m* specific dynamic surface ( = S,JV,), m- ’ two-dimensional drag coefficient microscopic characteristic length, m pore section hydraulic diameter, m solid prism dimension, m streamwise gradient of fluid pressure ( = pf), Nm-’ apparent pore diameter, m inertial term coefficient, kgmm4 viscous term coefficient, kg m - ’ s - I intrapore pressure, N me2 intrinsic average fluid pressure

0.0968x' A.*.

(29) The present model, written in the same form, results in

---1 dp =--~0 - 4’ x2A&+ qP(l - 4 x2&i qdx

&=

X-1

--s----c

(30) The values obtained for A,,+from the latter equation are shown in Table 2. These values differ from the corresponding values obtained by the capillary model (Montillet et al., 1992) given in eq. (29) by 30-46%. Although this appears to be a significant difference it should be viewed against results from the BET method which are about five times as high (Montillet et al., 1992). Some difference with BET is to be expected since the latter involves gas adsorption at atomic level and thus include internal porosity, which is not needed in chemical engineering. It is evident that more comparisons of the models are needed against high porosity porous media, especially for cases where Avd is known to reasonable accuracy by another independent method. In contrast with the model of Comiti and Renaud, the tortuosity values of the present model are only a function of porosity, since the pore structure was built into the geometry. The present model nevertheless predicts values for tortuosity comparable to those predicted by the capillary model (Montillet et al., 1992).

4 M N p Pf

Tke,& S Sh Vf K v, V Vf

“P

Greek CONCLUSIONS

Experimental results for flow through high porosity foam materials were analysed with respect to a new model for the prediction of inertial effects in porous media. It was clearly shown that the modelling procedure is capable of accurately predicting the pressure gradients for both Darcy and non-Darcy flows. The RLJC concept used here in the modelling exercise is very convenient to study both high and low porosity porous media. In the present case of high porosity consolidated porous media the RUC re-

pressure deviation ( = p - p,), N m ’ magnitude of q, m s- ’ superficial velocity, m s- ’ critical Reynolds number surface area, m2 fluid solid interface in basic CRUC, rn’ fluid filled “void” volume within CRUC, m3 solid volume within CRUC, m3 total volume of CRUC, m3 actual fluid velocity field within V,, ms- ’ intrinsic phase average fluid velocity (= f), ms-l cross-sectional mean fluid speed in streamwise pore section, ms- ’

i 4

letters

B

E P Y P d X

surface excess factor porosity (void fraction) ( = V,lV,) fluid dynamic viscosity, Ns normal vector on solid surface and pointing into fluid fluid mass density, kgm- 3 generic variable pore structure tortuosity

Subscripts

an

analytical

High porosity metallic foams C

exp ; g mic

W

critical value experimental fluid phase fluid-solid interface glycerol microscoDe water 1

REFERENCES Bird, R. B., Stewart, W. E. and Lightfoot, E. N., 1960, Transport Phenomena. Wiley, New York. Comiti, J. and Renaud, M., 1989, A new model for determining mean structure parameters of fixed beds from pressure drop measurements: application to beds packed with parallelepipedal particles. Chem. Engng Sci. 44, 1539-1545. Du Plessis, 1992a, Pore-scale modelling for flow through different types of porous environments, in Heat and Mass

3553

Transfer in Porous Media (Edited by M. Quintard and M. Todorovic), pp. 249-262. Elsevier, Amsterdam. Du Plessis, 1992b, High Reynolds number flow through granular porous media, in Computational Methods in Water Resources IX, Vol. 2: Mathematical Modeling in Water Resources (Edited by T. F. Russel et al.), pp. 179-186. Computational Mechanics Publications, Southampton. Du Plessis, J. P. and Masliyah, J. H., 1988. Mathematical modelling of flow through consolidated isotropic porous media. Trans. Porous Media 3, 145-161. Du Plessis. J. P. and van der Westhuizen, I., 1993, Laminar crossflow through prismatic porous domains. R&D. J. South African Inst. mech. Engrs 9(2), 18-24. Happel, J. and Brenner, II., 1965, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice Hall, Englewood Cliffs, NJ. Montillet, A., Comiti, J. and Legrand, J., 1992, Determination of structural parameters of metallic foams from permeametry measurements. J. Mater. Sci. 27, 4460-4464.