PHYSICS OF FLUIDS 18, 078103 共2006兲

Permeability of the fractal disk-shaped branched network with tortuosity effect Peng Xu, Boming Yu,a兲 Yongjin Feng, and Mingqing Zou Department of Physics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, People’s Republic of China

共Received 17 November 2005; accepted 19 June 2006; published online 11 July 2006兲 In this Brief Communication, the effective permeability of the fractal disk-shaped branched networks is derived and the relationship between the effective permeability and the geometry structures is analyzed in detail. It is found that the tortuosity has significant influence on transport properties of the network and that small variations in the geometrical structures can induce very large variations in the effective permeability. The effective permeability approaches zero as the diameter ratio ␤ ⬍ 0.5 for different length ratios ␥. A comparison of the fractal disk-shaped branched network with the traditional parallel net indicates that the fractal disk-shaped branched network has a much stronger capability of fluid transport. The conductivity scaling law for the fractal disk-shaped branched network is obtained to be Ke ⬃ Va, where the scaling exponent a = −共1 / 2兲ln共N / ␥兲 / ln共␥␤2兲, and the scaling exponent a depends on the microstructures of the network in a very wide range and a = 1 / 2 under the area-preserving condition. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2221861兴 The transport properties of the branched networks such as mammalian circulatory and respiratory systems, leaves, river basins, energy networks, world-wide webs, oilrecovery, Internet, and social networks have fascinated researchers in physics, chemistry, biology, physiology, petroleum engineering, and geology for decades. The investigations of the branched network began from 1926 due to Murray’s law1 for cardiovascular system. Numerous subsequent researches extended Murray’s work. Noteworthy was West et al.’s2,3 work, who discussed the origin of allometric scaling law 共of 3 / 4兲 in biology in a rather general and detailed fashion. Bejan et al.4–6 共1997–2000兲 developed “Constructal Theory” by optimizing the access between one point and a finite volume and applied the theory to the cooling of electronic devices and other engineering fields. In order to minimize the energy dissipated in the system, the network must be a self-similar fractal network that can be space filling.2,7,8 Point-circle flows are recognized as useful designs for electronics cooling.9 Wechsatol et al.10,11 presented a design between one point 共O兲 and many points situated on a circle centered at O in terms of botanical tree leaves and optimized the point-circle network. The pointcircle network can be employed in man-made systems, for example, in networks for distribution of electricity and water and the collection of city waste and rain water, grids for city traffic, oil recovery, etc. The permeability, which is considered as an important transport parameter, has received much attention due to practical application in science and engineering.12 In this brief communication, we focus on the effective permeability and its general properties of the fractal disk-shaped branched/pore/tree network without subtrees 共a relatively simple model兲 and compare it with the parallel a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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channel net. In addition, we also discuss the conductivity scaling law and its dependence on the microstructures of the network. On the other hand, the work by Winter and Tartakovsky13 focuses on the conductivity scaling of the pore network with subtrees 共a relatively complicated model兲. Figure 1共a兲 is a diagrammatic sketch of the fractal diskshaped branched/tree network with branching angle ␪ that can be extended to infinity. We assume that each branch of the network is a smooth cylinder and every channel is divided into N branches at the next level and the total number of branching levels is m. A typical branch at some intermediate level k 共k = 0 , 1 , 2 , . . . , m兲 has length lk and inner diameter dk. To characterize the branching structures, the scale factors ␥ = lk+1 / lk and ␤ = dk+1 / dk are introduced here. Thus, it is easy to get lk = l0␥k and dk = d0␤k, where l0 and d0 are the length and inner diameter of the zeroth branching level, respectively. According to the fractal characteristics of the structure,7 N = ␥−Dl = ␤−Dd, where Dl is the fractal dimension of channel length distribution and Dd is the fractal dimension of channel diameter distribution 共also called the diameter exponent兲. Fractal dimensions Dl and Dd are generally limited to between 1 and 3, and the value Dd = 3 is an optimized result known as Murray’s law. Then, for N = 2, ␥ and ␤ are in the range of 0.5– 0.7937, respectively. We can derive the effective permeability of the network by using a direct equivalent method. The fractal branched network can be equivalent to a tortuous channel with length m lk = l0共1 − ␥m+1兲 / 共1 − ␥兲, which is the actual length of Lt = 兺k=0 the streamline. According to the Hagen-Poiseuille equation, the pressure drop of a fully developed laminar flow in the kth level channel is ⌬pk = 32␮vklk / d2k , where ␮ is the liquid viscosity, and the velocity in the kth branching level channel can be obtained from the conservation of mass as vk = 共v0 / Nk兲共d0 / dk兲2. Thus, the total pressure drop of the network can be expressed as

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© 2006 American Institute of Physics

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Phys. Fluids 18, 078103 共2006兲

Xu et al.

FIG. 2. Tortuosity T vs m at ␥ = 0.6 and ␪ = 0.4, 0.6, 0.8, 1.0.

FIG. 1. Sketch of the fractal disk-shaped branched network model between a circle and its center 共N = 2 , m = 2兲.

m

⌬p = 兺 ⌬pk = k=0

32␮v0l0 1 − 共␥/N␤4兲m+1 . 1 − ␥/N␤4 d20

共1兲

The network is equivalent to a tortuous channel; thus, in terms of the Hagen-Poiseuille equation, the flow rate of the network can be also expressed as

␲d4e ⌬p ␲ Q = v0 d20 = . 4 128␮ Lt

共2兲

The equivalent diameter of the network de can be obtained by inserting Eq. 共1兲 into Eq. 共2兲 as de = d0

冋

1 − ␥m+1 1 − ␥/N␤4 1 − ␥ 1 − 共␥/N␤4兲m+1

册

1/4

共3兲

.

For the equivalent tortuous channel, according to Darcy’s law, the flow rate can be also written as Q=

冉冊

Ke de ␲ ␮ 2

2

⌬p , L0

共4兲

where Ke is the permeability of the equivalent tortuous channel, i.e., Ke is the effective permeability of the network; L0 is the straight length of the flow path. Combined Eqs. 共2兲 and 共4兲, the effective permeability of the network is obtained by Ke =

d2e 1 . 32 Lt/L0

共5兲

Due to the definition of tortuosity,12 i.e., T = Lt / L0, and Eq. 共3兲, we can get Ke =

冋

d20 1 − ␥m+1 1 − ␥/N␤4 32 1 − ␥ 1 − 共␥/N␤4兲m+1

册

1/2

1 . T

共6兲

The dimensionless effective permeability of the network is obtained as K+ =

冋

1 − ␥m+1 1 − ␥/N␤4 Ke = 1 − ␥ 1 − 共␥/N␤4兲m+1 d20/32

册

1/2

1 . T

共7兲

Equations 共6兲 and 共7兲 reveal that the effective permeability of the network depends on the geometrical structures. Equa-

tions 共6兲 and 共7兲 present the permeability versus the microstructures of the network and tortuosity. However, no quantitative expression on permeability with microstructures of the network was derived in Winter and Tartakovsky13 and Neuman’s work.14 2 2 According to the recursive relation L0k = L0共k−1兲 + l2k + 2L0共k−1兲lk cos ␪, where L0k 共k = 1 , 2 , 3 , . . ., and L00 = l0兲 is the straight length from the channel terminal of the kth branching level to the origin/center, we can get the straight length of the network L0 = L0m, we can then calculate the value of the tortuosity of the network by T = Lt / L0. It is clear that the tortuosity of the network is independent of the diameter ratio ␤. Further calculation indicates that the value of tortuosity of the network is always greater than 1.0 and increases with the increase of length ratio ␥, total number of branching levels m and branching angles ␪. Furthermore, the tortuosity reaches approximately an asymptotical value when m is greater than a certain value 共see Fig. 2兲. After the tortuosity is taken into account for, the relationships between the permeability and the microstructures are shown in Fig. 3. The effective permeability with tortuosity effect is lower than that without considering the tortuosity, for example, the permeability with tortuosity effect included decreases by more than 10% – 20% at m = 5, ␪ = 1.0. As shown in Fig. 3共a兲, the effective permeability increases as the increase of the diameter ratio ␤ and decreases with the increase of the length ratio ␥. We can see that the value of effective permeability is approximately zero as ␤ ⬍ 0.5 for different length ratios ␥. This is consistent with the actual situation because for a smaller diameter ratio 共⬍0.5兲 the resistance increases drastically, causing the permeability to reduce remarkably. Figure 3共b兲 displays the effective permeability, which decreases rapidly as the increase of the total number of branching levels m and approximately tends to an asymptotical value, and it can be seen clearly that the influence of branching angle on the effective permeability is remarkable and the effective permeability decreases as the increase of ␪. It is also interestingly found from Fig. 3 that the dimensionless effective permeability of the network is always less than 1.0, and this can be interpreted as that a network with slander branches increases the flow resistance and thus reduces the permeability compared to a single channel with the same total volume. We can draw the conclusion that tortuosity is an important geometrical parameter, which has a

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Phys. Fluids 18, 078103 共2006兲

Permeability of the fractal disk-shaped branched network

FIG. 4. The power scaling law between the permeability and the support volume at different length and diameter ratios.

FIG. 3. The dimensionless effective permeability K+ considering tortuosity of the network versus 共a兲 ␤ at m = 5, ␪ = 1.0 and ␥ = 0.5, 0.6, 0.7; 共b兲 m at ␥ = 2−1/2, ␤ = 2−1/3 for different ␪.

significant influence on transport properties of the network. It is also found that small variations in the network geometrical structures can induce very large variations in the effective permeability, which is similar to the net air flux in bronchial tree.15 In order to compare the permeability of the fractal diskshaped branched network 共b兲 with the traditional parallel net 共p兲, the flow rates are assumed to be the same and the velocity of the parallel channel net is also assumed to be the same value v0. The number of parallel channels is assumed to be an integer is Nm 共i.e., the number of parallel channels is equal to the total number of outlets in the fractal disk-shaped branched network兲; thus, d = N−m/2d0 is the diameter of the parallel channel net, and l is the length of the parallel channel net. For the parallel channel net composed of Nm channels, the equivalent permeability based on parallel model can be obtained by Kep =

d20

1 . 32 Nm

共8兲

Combining Eqs. 共6兲 and 共8兲 gives the dimensionless permeability

冋

Keb Nm 1 − ␥m+1 1 − ␥/N␤4 = Kep T 1 − ␥ 1 − 共␥/N␤4兲m+1

册

1/2

.

共9兲

The results of Eq. 共9兲 shows that the effective permeability of the fractal disk-shaped branched network is much larger than that of the parallel net under the condition of the same flow rate when the diameter ratio ␤ ⬎ 0.5, which are qualitatively consistent with those by Chen and Cheng.16

The scaling of conductivity in porous media is very important and has been studied extensively. Neuman14 presented that the hydraulic conductivity K⬘ scales with the support volume V by K⬘ ⬃ Va, and a 共=1 / 2兲 is the scaling exponent. Winter and Tartakovsky.13 demonstrated that when a pore network can be represented as a collection of hierarchical trees, scalability of the pore geometry also leads to the 1 / 2-power scaling law. In the following, we follow the assumption by West et al.2 that the terminal units 共capillaries兲 are invariant; i.e., the parameters of the mth channels are independent of body size. Then, compared with the permeability of the terminal channel Km, we can derive the dimensionless permeability of the network: K+ =

冋

1 − 共1/␥兲m+1 1 − N␤4/␥ Ke = Nm 1 − 1/␥ 1 − 共N␤4/␥兲m+1 Km

册

1/2

1 . T

共10兲

The total volume of the network can be expressed as m

V = 兺 Nk␲共dk/2兲2lk = VmNm k=0

1 − 共N␥␤2兲−共m+1兲 . 1 − 共N␥␤2兲−1

共11兲

If we neglect the tortuosity effect 共by assuming T = 1兲, for N␤4 / ␥ ⬍ 1 and m Ⰷ 1, the effective permeability of the network Ke is about proportional to 共N / ␥兲共m+1兲/2, while the total volume V is proportional to 共␥␤2兲−共m+1兲 for N␥␤2 ⬍ 1. Therefore, we can get the scaling law of permeability with the support volume as Ke ⬃ Va ,

共12兲

where the scaling exponent a = −共1 / 2兲ln共N / ␥兲 / ln共␥␤2兲. The employment of the area-preserving 共␤ = N−1/2兲 condition2,13 from one generation to the next results in Ke ⬃ V1/2, which means that the 1 / 2-power scaling law exists between the permeability and network volume. It is worth pointing out that our simple model yields the 1 / 2-power scaling law only under the area-preserving condition, whereas the same conductivity scaling arises under both the pore-area and the pore length preserving condition from one level to another in Winter and Tartakovsky’s complicated model with subtrees.13 Figure 4 shows the dependence of the scaling exponent a on microstructures of the network in a very wide range. For example, the scaling exponent a = 0.499共⬵1 / 2兲 as

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共N = 2 and兲 ␥ = 2−1/3 and ␤ = 2−1/2, a = 0.287 as ␥ = 2−1/3 and ␤ = 2−1, and a = 0.3999 as ␥ = 2−1 and ␤ = 2−1/3, respectively. The present results show that the scaling exponent a depends on the microstructures of the network.

This work was supported by the National Natural Science Foundation of China through Grant No. 10572052. 1

Phys. Fluids 18, 078103 共2006兲

Xu et al.

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A. Bejan, Shape and Structure, from Engineering to Nature 共Cambridge University, UK, 2000兲. 7 B. B. Mandelbort, The Fractal Geometry of Nature 共W. H. Freeman, New York, 1982兲. 8 B. J. West, “Fractal forms in physiology,” Int. J. Mod. Phys. B 4, 1629 共1990兲. 9 D. V. Pence, “Improved thermal efficiency and temperature uniformity using fractal-like branching channel networks,” in Heat Transfer and Transport Phenomena, edited by G. P. Celata, V. P. Carey, M. Groll, I. Tanasawa, and G. Zummo 共Begell House, New York, 2000兲, pp. 142–148. 10 W. Wechsatol, S. Lorente, and A. Bejan, “Optimal tree-shaped networks for fluid flow in a disc-shaped body,” Int. J. Heat Mass Transfer 45, 4911 共2002兲. 11 W. Wechsatol, S. Lorente, and A. Bejan, “Tree-shaped networks with loops,” Int. J. Heat Mass Transfer 48, 573 共2005兲. 12 J. Bear, Dynamics of Fluid in Porous Media 共Elsevier, New York, 1972兲. 13 C. L. Winter and Daniel M. Tartakovsky, “Theoretical foundation for conductivity scaling,” Geophys. Res. Lett. 28, 4367 共2001兲. 14 S. P. Neiman, “Generalized scaling of permeabilities: Validation and effect of support scale,” Geophys. Res. Lett. 21, 349 共1994兲. 15 B. Mauroy, M. Filoche, E. R. Weibel, and B. Sapoval, “An optimal bronchial tree may be dangerous,” Nature 共London兲 427, 633 共2004兲. 16 Y. P. Chen and P. Cheng, “Heat transfer and pressure drop in fractal treelike microchannel nets,” Int. J. Heat Mass Transfer 45, 2643 共2002兲. 6

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