Geometrical models for tortuosity of streamlines in three-dimensional porous media.

Yun, Meijuan ; Yu, Boming ; Xu, Peng 等

Based on the assumption that some particles in porous media are unrestrictedly overlapped and hence of different configurations, this paper presents three three-dimensional geometry models for tortuosity of streamlines in porous media with spherical, cubic and plate-like particles, respectively. The proposed models are expressed as a function of porosity with no empirical constant, and they are helpful for understanding the physical mechanism for tortuosity of flow paths in three-dimensional porous media. The model predictions are compared with those from the available correlations obtained by experimental data, and good agreement is found between them.

En emettant l'hypothese que certaines particules en milieux poreux se chevauchent sans restrictions et presentent donc des configurations differentes, on presente dans cet article trois modeles de geometrie tridimensionnelle pour la tortuosite des lignes de courant en milieux poreux avec des particules spheriques, cubiques et plates. Les modeles proposes sont exprimes en fonction de la porosite sans constante empirique et sont utiles pour comprendre le mecanisme physique expliquant la tortuosite des trajectoires d'ecoulement en milieux poreux tridimensionnels. Les predictions des modeles sont comparees a celles de correlations disponibles obtenues par des donnees experimentales, et un bon accord est trouve entre elles.

Keywords: tortuosity, stream tubes, porous media

INTRODUCTION

The characteristics of material transport in (single-phase and multiphase) porous media, such as the hydraulic permeability, electrical conductivity and diffusivity have steadily received much attention due to many unresolved problems in science and engineering applications. These macroscopic transport parameters are usually related to the tortuosity followed by transported materials (Wyllie and Gregory, 1955; Bear, 1972; Dullien, 1979; Comiti and Renaud, 1989; Koponen et al., 1996; Moldrup et al., 2001; Yu and Cheng, 2002; Yu et al., 2003; Zhang and Liu, 2003; Gerasimos et al., 2004; Yu and Li, 2004; Yun et al., 2005). However, it is known that the tortuous path a transported material follows is microscopically very complicated. Therefore, the tortuosity in porous media is conventionally determined by experiments (Wyllie and Gregory, 1955; Comiti and Renaud, 1989) and the experimental data are then correlated as correlations with one or more empirical constants. In addition, the tortuosity of streamlines in porous media can also be determined by numerical methods such as Lattice Gas (LG) cellular automaton (Koponen et al., 1996). In order to achieve a better understanding of the mechanisms for tortuosity through porous media, an analytical model for tortuosity is desirable and this is a challenging task. Recently, Yu and Li (2004) and Yun et al. (2005) proposed simple geometry models for tortuosities in two-dimensional porous medium with cubic and spherical particles. The tortuosity models are expressed as a function of porosity. However, the models are only suitable for two-dimensional porous media, and they are not applicable to three-dimensional porous media. So far, no analytical model has been reported for tortuosity of flow paths/stream tubes in three-dimensional porous media. This work is devoted to deriving the analytical models for tortuosities in three-dimensional porous media with spherical, cubic and plate-like particles.

The tortuosity is often defined by (Bear, 1972; Dullien, 1979):

[GAMMA] = [L.sub.e]/L (1)

where [L.sub.e] and L are the actual length of flow path and the straight length or thickness of a unit cell along the macroscopic pressure gradient, respectively. Some researchers define the tortuosity as [GAMMA] = [(L/[L.sub.e]).sup.2] (Bear, 1972). Carman gives the value L/[L.sub.e] = 0.71, and other values reported in the literature for L/Le vary in the range of 0.56-0.8 (Bear, 1972). In this work, Equation (1) is applied for the definition of tortuosity.

A correlation between the average tortuosity and the porosity was given by (Comiti and Renaud, 1989):

[GAMMA] = 1 + Cln(1/[phi]) (2)

which was obtained by the experiments on flow through beds packed with spherical and cubic particles. Here C is the empirical/ fitting constant obtained by fitting the experimental data, for spherical and cubic particles C = 0.41 and C = 0.63, respectively. [phi] is the porosity.

Koponen et al. (1996) applied the Lattice Gas cellular automaton method to solve numerically a creeping flow of Newtonian incompressible fluid in a two-dimensional porous substance constructed by randomly placed rectangles of equal size and with unrestricted overlap. They obtained a correlation between the average tortuosity of flow path and the porosity as:

[GAMMA] = 0.8(1-[phi]) + 1 (3)

This paper focuses on derivation of the tortuosity models for flow of Newtonian incompressible fluid in a three-dimensional porous media with spherical and cubic particles by applying the geometrical method. To this end, the ideal geometry models and representative flow paths are chosen, and then the tortuosity in three-dimensional porous media is derived by averaging the representative flow paths. The model predictions are finally compared with the existing experimental and numerical correlations.

GEOMETRY MODELS FOR TORTUOSITY OF STREAMLINES IN THREE-DIMENSIONAL POROUS MEDIA

For flow in real porous media, there may be numerous streamlines around the particles. The tortuosity of streamlines may be found by calculation of velocity field for flow in porous media. However, since the pore structure is very complicated for real porous media, the calculation of velocity field for tortuosity is a challenging task. In this work, for simplicity we apply a geometric method by averaging over some possible streamlines around particles for tortuosity. The average method used in this work is just a simple statistical average over all possible streamlines. The average tortuosity is thus expressed as:

[GAMMA] = 1/N [summation over (i)] [[GAMMA].sub.i] (4)

where N is the total number of flow paths/streamlines and [[GAMMA].sub.i] is the tortuosity for the [i.sup.th] flow path/streamline. Because of the diversity of flow paths/streamlines, we must choose some representative flow paths/streamlines and average over them to obtain the average tortuosity.

Figures 1 to 10 display the possible configurations for a Newtonian incompressible fluid flowing through porous media of three-dimensional spherical, cubic and plate-like particles. The particles in Figures 1, 3, 5, 6 and 7 are arranged in the form of an equilateral-triangle, while those in Figures 2, 4, 8, 9 and 10 are arranged in the square arrangements.

[FIGURES 1-10 OMITTED]

A Geometry Model for Tortuosity of Streamlines in Three-dimensional Porous Media with Spherical Particles

Figure 1a depicts the three-dimensional configuration for spherical particles arranged in an equilateral-triangle form. Figure 1b shows some streamlines which flow around spherical particles. Figure 1c is a top view of a representative flow path in Figure 1b. Figure 2a displays the unit cell when spherical particles are arranged in a square form and two representative flow paths. Figure 2b is side view of two representative flow paths in Figure 2a. R is the radius of spherical particles, and d is the gap size between spherical particles.

The length, width and height of the unit cell as shown in Figure 1a are [square root of (3(2R + d)/2)] 2R and 2R + d, respectively. So the total volume of the unit cell is [V.sub.t] = [square root of (3R[(2R + d).sup.2])] and the total pore volume in the unit cell is [V.sub.p] = [square root of (3R[(2R + d).sup.2])] - 4[pi][R.sup.3]/3. Thus, for Figure 1a the porosity is given by:

[phi] = [V.sub.p]/[V.sub.t] = 1 4[pi]/3[square root of (3[(2 + d/R).sup.2])] (5)

From Equation (5), we obtain d/R = 2/3 [square root of (3[pi]/1 - [pi])] - 2. Let the ratio d/R = p, then:

p = d/R = 2/3 [square root of (3[pi]/1 - [pi])] - 2 (6)

The streamline AB in Figure 1b is tangential to the particle, and the streamline BC flows along the surface of the particle. We can obtain the average tortuosity of a series of flow paths in Figure 1b by integrating flow paths in Figure 1c. In Figure 1c, [l.sub.AO] [square root of (3(2R + d)/2)], let [l.sub.BO] = x, then in the right-angled triangle AOB there is [l.sub.AB] = [square root of ([l.sup.2.sub.AO] - [l.sup.2.sub.BO])] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since the boundary layer thickness is very thin (<<R) on the surface of particles, we have the length of arc BC as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in Figure 1b, the actual length of a series of flow paths is [[integral].sup.R.sub.0] ([l.sub.AB] + [[??].sub.BC]) dx, and the corresponding total straight length is [[integral].sup.R.sub.0] [l.sub.AO] dx. According to the definition Equation (1) and Equation (6), we obtain the tortuosity for Figure 1 as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where p is determined by Equation (6).

Since particles in actual porous media are randomly distributed in the three-dimensional space, this means that some particles may overlap (i.e., cover/shield partially) each other, while the others do not. For spherical particles, however, overlapping does not influence fluid flow. This is different from the situation in two-dimensional porous media (Yun et al., 2005).

The side length of the unit cell as shown in Figure 2a is 2R+d. So the total volume of the unit cell is [V.sub.t] = [(2R + d).sup.3], and the total pore volume in this unit cell is [V.sub.p] = [(2R + d).sup.3] - 4[pi][R.sup.3]/3. Thus, for Figure 2a the porosity is:

[phi] = [V.sub.p]/[V.sub.t] = 1 - 4[pi]/3/[(2 + d/R).sup.3] (8)

Equation (8) indicates that when d/R = 0, [[phi].sub.min] = 1 - [pi]/6, this is the expected porosity when three spherical particles contact closely. From Equation (8), we obtain:

d/R = [[4[pi]/3(1-[phi])].sup.1/3] - 2 (9)

For streamline 1 in Figure 2b, since the boundary layer thickness is very thin (<<R) on the surface of particle, we have [l.sub.AB] = [l.sub.CD] = d/2, the arc length [[?].sub.BC] = [pi]R. According to Equation (1) and Equation (9), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

For streamline 2 in Figure 2b, the actual length is equal to the straight length of flow path in the unit cell, so its tortuosity is 1, i.e.,

[[GAMMA].sub.2-2] = 1 (11)

It is expected that the streamline is tortuous near the particle and is almost straight far away from the particle. It is also expected that the fraction of the straight flow path increases with the increase of porosity, but the fraction of the tortuous flow path decreases with the increase of porosity. Therefore, we take the weighted average tortuosity as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [R.sup.2]/[(R+d).sup.2] is defined as the weighted factor. Equation (12) indicates that when d[right arrow]0, [R.sup.2]/[(R+d).sup.2] [right arrow] 1 and [[GAMMA].sub.2][right arrow][[GAMMA].sub.2-1]. This means that the gap size between particles tends to be zero, and only tortuous streamlines exist. When d[right arrow][infinity], [R.sup.2]/[(R+d).sup.2][right arrow]0 and [[GAMMA].sub.2][right arrow][[GAMMA].sub.2-2] = 1, the gap d between the particles is infinity and the straight streamlines prevail the fluid field. These results are consistent with the general observation. Combining Equations (9) to (11), Equation (12) is expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Now we can obtain the averaged tortuosity for flow paths/ streamlines in three-dimensional porous media with spherical particles arranged in different configurations by averaging over Equations (7) and (13):

[[GAMMA].sub.av] = ([[GAMMA].sub.1] + [[GAMMA].sub.2])/2 (14)

Equation (14) is an approximate expression for tortuosity of flow paths/streamlines in three-dimensional porous media with spherical particles.

A Geometry Model for Tortuosity of Streamlines in Three-dimensional Porous Media with Cubic Particles

Figure 3a displays the three-dimensional configuration for cubic particles arranged in an equilateral-triangle form. Figure 3b is the side view of streamlines flowing around cubic particles arranged in an equilateral-triangle form. Figure 4a is a three-dimensional configuration for cubic particles arranged in a square form. Figure 4b is the side view of streamlines flowing around cubic particles arranged in square. In the figure, b is the side length of cubic particles, and d is the gap size between cubic particles.

We can obtain the averaged tortuosity in three-dimensional porous media with cubic particles arranged in different configurations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

See Appendix A for derivation of Equation (15).

Equation (15) indicates that when [phi][right arrow]1, [[GAMMA].sub.av][right arrow]1, and when [phi][right arrow]0, [[GAMMA].sub.av][infinity]. This is consistent with the physical situation. Equation (15) is an approximate expression for tortuosity of flow paths in three-dimensional porous media with cubic particles.

A Geometry Model for Tortuosity of Streamlines in Three-dimensional Porous Media with Plate-like Particles

A correlation between the average tortuosity of flow path/streamline and the porosity was given by (Comiti and Renaud, 1989):

[GAMMA] = 1 + 0.58exp(0.18a/e)1n(1/[phi]) (16)

which was obtained by the experiments on flow through beds packed with parallelepipedal particles.

From a set of experiments performed on wood chips for which the mean value of e/a was about 0.18, Pech (1984) proposed the correlation:

[GAMMA] = 1+1.61n(1/[phi]) (17)

Figures 5a, 6 and 7 display the three different three-dimensional configurations for plates arranged in an equilateral-triangle form. Figures 5b and 5c are the side view of two different streamlines flowing around the plates in Figure 5a, Figures 8 to 10 are the three different three-dimensional configurations for plates arranged in a square form and the corresponding unit cell. The length, width and thickness of plates are a, a and e, respectively, and d is the gap size between plates. Let a/e = s and d/e = p.

After many tests, we obtain the tortuosity of streamlines in three-dimensional porous media with plates:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

See Appendix B for derivation of Equation (18).

Equation (18) indicates that when [phi][right arrow]1, [[GAMMA].sub.av][right arrow]1, and when [phi][right arrow]0, [[GAMMA].sub.av][right arrow][infinity]. This is consistent with the physical situation. Equation (18) is an approximate expression for tortuosity in three-dimensional porous media with plates, wood chips and cubes.

RESULTS AND DISCUSSION

Figure 11 presents a comparison among the present model prediction (Equation (14)) for spherical particles, the available experimental correlation Equation (2) (with C = 0.41) and other experimental data at different porosities (Wyllie and Gregory, 1955 and Comiti and Renaud, 1989). It is seen that good agreement is found between the present model predictions and available experimental data, and this verifies the validity of the present model for tortuosity of flow paths/streamlines in three-dimensional porous media with spherical particles.

[FIGURE 11 OMITTED]

Figure 12 is a comparison among the present model (Equation (15)) predictions (see the solid line) for cubic particles, experimental correlation Equation (2) (see the solid with squares) (with C = 0.63), numerical correlation Equation (3) (see the solid line with stars) and experimental data (see open circles) (Wyllie and Gregory, 1955). It can be seen from the figure that the present model (Equation (15)) for cubic particles presents good agreement with the existing experimental correlation and experimental data. However, the significant deviation is also found between the numerical correlation Equation (3) (see the solid line with stars) and experimental data (see open circles) at porosity less than 0.40, see Figure 12. The reason is that the numerical correlation Equation (3) was obtained by simulating two-dimensional porous media, not three-dimensional porous media.

[FIGURE 12 OMITTED]

When e/a = 0.209, the tortuosity of the present model Equation (18), the experimental correlation Equation (16) and experimental data at porosity 0.52 (Comiti and Renaud, 1989) are shown in Figure 13. Figure 14 compares the present model predictions, experimental correlation Equation (16) and experimental data at porosity 0.31 (Comiti and Renaud, 1989). In Figure 15, we show the tortuosity by the present model Equation (18) and the two experimental correlation of plates Equation (16) and wood chips Equation (17) (Pech, 1984) when e/a = 0.18. Again it shows that they are in good agreement.

[FIGURES 13-15 OMITTED]

The proposed models (Equations (14), (15) and (18)) are expressed as a function of porosity and there are no empirical constant. Though experimental correlations look simple, they have one or more empirical constants which have no physical meaning and the mechanism behind the empirical constant is ignored. It is also worth pointing out that the correlation Equation (2) allows the porosity to approach zero, which is inconsistent with practical situation with spherical particles because for a porous medium with spherical particles, its minimum porosity should not be zero. However, the present model does not have such a weakness. This can be seen from Equations (6) and (9) since d/R must be positive. The minimum porosity, which cannot be zero for porous media with spherical particles, can be found by assuming d/R = 0 (meaning all particles touch each other). Therefore, the experimental correlation Equation (2) for spherical particles may present unreasonable results at lower porosity. The proposed models in this paper look somewhat complicated, but they are helpful for understanding the physical mechanism for tortuosity in three-dimensional porous media with spherical, cubic and plate-like particles.

CONCLUSION

In this paper, we have proposed three models for tortuosity in three-dimensional porous media with spherical, cubic and plate-like particles by assuming that some particles are unrestrictedly overlapped and hence of different configurations. The proposed models are expressed as a function of porosity with no empirical constant, and they might be helpful to understand the physical mechanism of tortuosity in three-dimensional porous media with spherical, cubic and plate-like particles. The model predictions are found to be in good agreement with the available experimental correlations and experimental data.

APPENDIX A

The length, thickness and height of the unit cell as shown in Figure 3a are b+d, b and b+d, respectively. So the total volume of the unit cell is [V.sub.t] = [b(b+d).sup.2], and the total pore volume in this unit cell is [V.sub.p] = [b(b+d).sup.2] - [b.sup.3]. Thus, for Figure 3a the porosity is:

[phi] = [V.sub.p]/[V.sub.t] = 1 - 1/[(1 + d/b).sup.2] (A-1)

From Equation (A-1), we obtain:

d/b = [square root of (1/1-[phi] - 1)] (A-2)

In Figure 3b, [l.sub.AB] = [l.sub.CD] = b/2, in the right-angled triangle BFC, [l.sub.BF] = d, [l.sub.CF] = b/2, so [l.sub.BC] = [square root of ([l.sup.2.sub.BF] + [l.sup.2.sub.CF])] = [square root of ([d.sup.2] + [b.sup.2]/4)], [l.sub.AE] = [l.sub.AB] + [l.sub.BF] + [l.sub.FE] = b+d. According to the definition Equation (1) and Equation (A-2), we obtain the tortuosity as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-3)

Considering that some particles may overlap each other, the streamlines AB and CD in Figure 3b might be sheltered from each other. Then, the tortuosity for the flow paths is in the pore scale. Therefore, the actual length of flow path is [l.sub.BC], the corresponding straight length is [l.sub.BF], and the tortuosity for the flow paths in the pore scale is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-4)

According to Equation (4), we obtain the tortuosity for Figure 3 as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-5)

The side length of the unit cell as shown in Figure 4a is b+d. So the total volume of the unit cell is [V.sub.t] = [(b+d).sup.3], and the total pore volume in the unit cell is [V.sub.p] = [(b+d).sup.3] - [b.sup.3]. Thus, for Figure 4a the porosity is:

[phi] = [V.sub.p]/[V.sub.t] = 1 - 4[pi]/3/[(2 + d/b).sup.3] (A-6)

From Equation (A-6), we obtain:

d/b = [[1/(1 - [phi])].sup.1/3] -1 (A-7)

For streamline 1 in Figure 4b, since the boundary layer thickness is very thin on the surface of particle, we have [l.sub.AB] = [l.sub.EF] = d/2, [l.sub.BC] = [l.sub.DE] = b/2 , [l.sub.CD] = b. According to Equation (1) and Equation (A-7), thus the tortuosity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-8)

For streamline 2 in Figure 4b, the actual length is equal to the straight length of flow path in the unit cell, so its tortuosity is 1, i.e.,

[[GAMMA].sub.4-2] = 1 (A-9)

It is expected that the streamline is tortuous near the particle and is almost straight far away from the particle. It is also expected that the fraction of the straight flow path increases with the increase of porosity, but the fraction of the tortuous flow paths decreases with the increase of porosity. Therefore, we take the weighted average tortuosity as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-10)

where [b.sup.2]/[(b+d).sup.2] is defined as the weighted factor. Equation (A-10) indicates that when d[right arrow]0, [b.sup.2]/[(b+d).sup.2][right arrow]1 and [[GAMMA].sub.4][right arrow][[GAMMA].sub.4-1]. This means that the gap size between particles tends to be zero, and only tortuous streamlines exist. When d[right arrow][infinity], [b.sup.2]/[(b+d).sup.2][right arrow]0 and [[GAMMA].sub.4][right arrow][[GAMMA].sub.4-2], the gap d between the particles is infinity and the straight streamlines prevail the fluid field. These results are consistent with the general observation.

Now we can obtain the averaged tortuosity for flow paths/ streamlines in three-dimensional porous media with cubic particles arranged in different configurations by averaging over Equations (A-5) and (A-10):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A-11)

which is Equation (15).

APPENDIX B

The length, width and thickness of the unit cell as shown in Figure 5a are a+d, a and e+d, respectively. So the total volume of the unit cell is [V.sub.t] = a(a+d)(e+d), and the total pore volume in this unit cell is [V.sub.p] = a(a+d)(e+d)-[a.sup.2]e. Thus, for Figure 5a the porosity is:

[phi] = [V.sub.p]/[V.sub.t] = 1 - [a.sup.2]e/a(a+d)(e+d) (B-1)

From Equation (B-1), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-2)

In Figure 5b, the tortuosity is controlled by flow paths in the pore scale when some particles overlap each other. Therefore, the actual length of flow path is [l.sub.BC], the corresponding straight length of flow path is [l.sub.BF], so the tortuosity is:

[[GAMMA].sub.5-1] = [L.sub.BC]/[L.sub.BF] = [square root of ([p.sup.2] + 1/4)]/p(B-3)

In Figure 5(c), [l.sub.AB] = [l.sub.CD] = a+d/2, [l.sub.BC] = e/2, the tortuosity is:

[[GAMMA].sub.5-2] = [l.sub.AB+BC+CD]/[l.sub.AB+CD] = 1 + 1/2(s+p) (B-4)

So the tortuosity in Figure 5 is:

[[GAMMA].sub.5] = [[GAMMA].sub.5-1] + [[GAMMA].sub.5-2]/2 (B-5)

The length, width and thickness of the unit cell as shown in Figure 6 are a+d, e and a+d, respectively. The length, width and thickness of the unit cell as shown in Figure 7 are e+d, a and a+d, respectively. According to the method in Figure 5, we can obtain the tortuosity in Figure 6 and Figure 7:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-6)

[[GAMMA].sub.6-2] [l.sub.AB+BC+CD]/[l.sub.AB+CD] = 1 + [square root of (1-[phi])]/2 (B-7)

[[GAMMA].sub.7-1] = [l.sub.BC]/[l.sub.BF] = [square root of (1 + [s.sup.2]/4[p.sup.2])] (B-8)

[[GAMMA].sub.7-2] = [l.sub.AB+BC+CD]/[l.sub.AB+CD] = 1 + s/2(1+p) (B-9)

So the tortuosity in Figures 6 and 7 are:

[[GAMMA].sub.6] = [[GAMMA].sub.6-1] + [[GAMMA].sub.6-2]/2 (B-10)

[[GAMMA].sub.7] = [[GAMMA].sub.7-1] + [[GAMMA].sub.7-2]/2 (B-11)

So the averaged tortuosity for flow paths in three-dimensional porous media with plates arranged in an equilateral-triangle form:

[[GAMMA].sub.av1] = [[GAMMA].sub.5] + [[GAMMA].sub.6] + [[GAMMA].sub.7] (B-12)

The length, width and thickness of the unit cell as shown in Figure 8 are a+d, a+d and e+d, respectively. So the total volume of the unit cell is [V.sub.t] = [(a+d).sup.2] (e+d), and the total pore volume in this unit cell is [V.sub.p] = [(a+d).sup.2] (e+d) - [a.sup.2]e. Thus, for Figure 8 the porosity is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-13)

In Figure 8, the tortuosity is:

[[GAMMA].sub.8-1] = [l.sub.AB+BC+CD+DE+EF]/[l.sub.AB+CD+EF] = 1 + e/d+a (B-14)

[[GAMMA].sub.8-2] = 1 (B-15)

Then we take the weighted average tortuosity as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-16)

In the same way, we can obtain the tortuosity for Figures 9 and 10 as:

[[GAMMA].sub.9-1] = [l.sub.AB+BC+CD+DE+EF]/[l.sub.AB+CD+EF] = 1 + a/d+a (B-17)

[[GAMMA].sub.9-2] = 1 (B-18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-19)

and

[[GAMMA].sub.10-1] = [l.sub.AB+BC+CD+DE+EF]/[l.sub.AB+CD+EF] = 1 + a/d+e (B-20)

[[GAMMA].sub.10-2] = 1 (B-21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-22)

So we obtain the averaged tortuosity for flow paths in three-dimensional porous media with plates arranged in a square form:

[[GAMMA].sub.av2] = [[GAMMA].sub.8] + [[GAMMA].sub.9] + [[GAMMA].sub.0]/3 (B-23)

Finally, we obtain the tortuosity of streamlines in three-dimensional porous media with plates by averaging over Equations (B-12) and (B-23):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B-24)

which is Equation (18).

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China through project number 10572052.

REFERENCES

Bear, J., "Dynamics of Fluids in Porous Media," American Elsevier Publishing Company, Inc. (1972).

Comiti, J. and M. Renaud, "A New Model for Determining Mean Structure Parameters of Fixed Beds from Pressure Drop Measurements: Application to Beds Packed with Parallelepipedal Particles," Chem. Eng. Sci. 44, 1539-1545 (1989).

Dullien, F. A. L., "Porous Media, Fluid Transport and Pore Structure," Academic Press, New York and London (1979).

Koponen, A., M. Kataja and J. Timonen, "Tortuous Flow in Porous Media," Phys. Rev. E 54, 406-410 (1996).

Moldrup, P., T. Oleson, T. Komatsu, P. Schjonning and D. E. Rolston, "The Tortuosity, Diffusivity, and Permeability in the Soil Liquid and Gaseous Phases," Soil Sci. Am. J. 65, 613-623 (2001).

Pech, D., "Etude de la permeabilite des lits compressibles constitues de copeaux de bois partiellement destructures", These de 3e cycle, INP Grenoble, France (1984).

Wyllie, M. R. J. and A. R. Gregory, "Fluid Flow through Unconsolidated Porous Aggregates," Ind. Eng. Chem. Proc. Des. Dev. 47, 1379-1399 (1955)

Yu, B. M. and P. Cheng, "A Fractal Model for Nucleate Pool Boiling Heat Transfer," J. Heat Transfer 124, 1117-1124 (2002).

Yu, B. M. and J. H. Li, "A Geometry Model for Tortuosity of Flow Path in Porous Media," Chin. Phys. Lett. 21, 1569-1571 (2004).

Yu, B. M., J. H. Li, Z. H. Li and M. Q. Zou, "Permeabilities of Unsaturated Porous Media," Int. J. Multiphase Flow 29, 1625-1642 (2003).

Yun, M. J., B. M. Yu, B. Zhang and M. T. Huang, "A Geometry Model for Tortuosity of Streamtubes in Porous Media with Spherical Particles," Chin. Phys. Lett. 22, 1464-1467 (2005).

Zhang, B. Q. and X. F. Liu, "Effects of Fractal Trajectory on Gas Diffusion in Porous Media," AICHE J. 44, 3037-3047 (2003).

Manuscript received December 6, 2005; revised manuscript received February 26, 2006; accepted for publication March 31, 2006.

* Author to whom correspondence may be addressed. E-mail address: yu3838@public.wh.hb.cn

Meijuan Yun, Boming Yu *, Peng Xu and Jinsui Wu

Department of Physics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China.