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Optimal structure of tree-like branching networks for fluid flow
Physica A 393 (2014) 527–534
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Optimal structure of tree-like branching networks for fluid flow Jianlong Kou a,c,∗ , Yanyan Chen a , Xiaoyan Zhou a , Hangjun Lu a , Fengmin Wu a,∗ , Jintu Fan b,c,∗ a
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
b
Department of Fiber Science and Apparel Design, Cornell University, Ithaca, NY 14853-4401, USA
c
Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
highlights • We derive some dimensionless expressions of effective flow resistance. • The effects of structural parameters on the flow resistance are studied. • We give optimal design schemes when the flow resistance of the whole network is minimum.
article
info
Article history: Received 22 February 2011 Received in revised form 19 February 2013 Available online 20 September 2013 Keywords: Flow resistance Fractal tree-like branching network Optimal structure
abstract Tree-like branching networks are very common flow or transportation systems from natural evolution. In this study, the optimal structures of tree-like branching networks for minimum flow resistance are analyzed for both laminar and turbulent flow in both smooth and rough pipes. It is found that the dimensionless effective flow resistance under the volume constraint for different flows is sensitive to the geometrical parameters of the structure. The flow resistance of the tree-like branching networks reaches a minimum when the diameter ratio β ∗ satisfies β ∗ = N k , where, N is the bifurcation number N = 2, 3, 4, . . . and k is a constant. For laminar flow, k = −1/3, which is in agreement with the existing Murray’s law; for turbulent flow in smooth pipes, k = −3/7; for turbulent flow in rough pipes, k = −7/17. These results serve as design guidelines of efficient transport and flow systems. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Tree-like branching networks exist widely in nature (such as lungs and vascular systems in mammals, river basins and plants) and have received considerable attention [1–6]. It has been shown that natural tree-like branching networks tend to have a perfect structure [7,8] and often surpass man-made products, in terms of minimal resistance and optimal vascular diameter for driving the blood in mammals and water in plants. This provides inspiration for the design of transport or conversion systems in biological engineering [9,10], chemical engineering [11,12], textile engineering [13,14], microelectronic engineering [15,16] and energy sources recovery [17–19]. The tree-like branching structures of mammalian cardiovascular and respiratory systems are optimum for blood and gas flow, as discussed by Murray et al. in 1926 [20], who found an optimum relationship between the diameter of the parent vessel (Dk ) and that of two daughter branches (Dk+1 ) in the form of Dk+1 /Dk = 2−1/3 . The relationship has been verified in recent theoretical analysis and experimental observation [21–23]. And it is now known as Murray’s law. Recently, Bejan
∗
Corresponding authors. Tel.: +86 57982297912; fax: +86 57982297119. E-mail addresses: [email protected] (J. Kou), [email protected] (F. Wu), [email protected] (J. Fan).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.08.029
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J. Kou et al. / Physica A 393 (2014) 527–534
Fig. 1. Schematic of self-similar branch structure. The tree-like, spatial fractal (a) has self-similar branches, such that the small-scale structure (b) resembles the large-scale form (a), where the arrow represents the direction of fluid (Q ) flow, the self-similar branch structures (a) can be built by repeating a finite number of element structures (c).
[24–31] proposed a ‘‘constructal law’’ to explain the tree-like branching network in nature. The construct law states that, for a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it. Bejan et al. [24–26] showed that a tree-like branching network is an optimal configuration following the constructal law, as it generates minimum entropy for a flow between a source to a volume. Bejan et al. [27] applied the constructal law to the tree-like branching networks with local T-shaped and Y-shaped constructs, and showed that, for fixed total flow volume, there is an optimal diameter ratio of 2−1/3 and 2−3/7 , respectively. More recently, Xu and Chen et al. [28,29] proposed effective permeability of the whole networks and found tree-like networks could significantly increase the effective permeability of the composites compared to the traditional parallel networks under proper structural parameters. Although optimum tree-like networks have been studied in terms of pumping power requirements by Murray [20] and constructal law [24–37,30–39], the optimum design of the entire network system and the effects of various structural parameters under different types of flows have not been elucidated. In this paper, we have considered the optimal design of the entire tree-like network for both laminar and turbulent flow. Under the volume constraint of the whole network, we derive a dimensionless expression of effective flow resistance. We discuss the relationship between the dimensionless effective flow resistance and the geometrical parameters of the treelike branching networks (including diameter ratio, length ratio, branching number). Furthermore, we give an optimal design scheme when the flow resistance of the whole network is minimum. 2. Tree-like branching network A tree-like branching network is a complex structure. In order to minimize the energy dissipated in the system, the network must be a self-similar fractal network that can be space filling [40,41]. In this study, we consider a general Y-shaped tree-like branching network as shown in Fig. 1(a), which can be built by repeating a finite number of elements constructed in the shape as shown in Fig. 1(c). The structure satisfies the self-similar characteristic [40], it is space filling, significantly more variability tolerant than other structures and has an evolutionary advantage [42]. For the structure, every channel is divided into N branches at the next level (e.g., N = 2 in Fig. 1) and the branches are of the same geometries. In the present work, we consider that the thickness of the tube wall is sufficiently thin to be negligible; all the ducts are sufficiently slender and the losses at the junctions can be neglected. For a laminar flow through the network, each branch of the network is a smooth cylindrical tube. For a turbulent flow through the network, we consider turbulent flow through rough pipes and smooth pipes. In order to describe the branching structures, let the length and diameter of a typical branch at some intermediate level k (k = 0, 1, 2, 3 . . .) be lk and dk , respectively. We further introduce two scale factors, β = dk+1 /dk and γ = lk+1 /lk . Therefore, we can have dk = d0 β k
and lk = l0 γ k ,
where, l0 and d0 are the length and diameter of the 0th branching level, respectively.
(1)
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3. Optimal structure based on flow resistance In the fluid network introduced above, the Newtonian flow can be a laminar flow and a turbulent flow. In this section, we first discuss that the laminar flow through the network. We then extend the laminar flow to the turbulent flow with rough pipes and smooth pipes. 3.1. Laminar flow Firstly, we consider the case of incompressible and fully developed laminar flow. According to the Hagen–Poiseuille equation, the viscous resistance for flow in a single tube can be expressed as Rk = 128µlk /π d4k , where µ is the viscosity of the fluid. The flow resistance of the total network can be summed up in accordance with Ohm’s law as Rt =
k=m Rk
Nk
k=0
128µl0 1 − (γ /N β 4 )m+1
=
1 − γ /N β 4
π d40
.
(2)
For many natural systems or practical applications, such as water flow in botanical trees, airflow in bronchial trees, blood flow in the human cardiovascular system and gas supply systems, it is required to achieve minimum flow resistance of the entire network. It is also necessary to keep total volume and length as constraints. Otherwise, the structure would be infinitely large. In addition, space is expansive and limited, and different parts of the system compete for the available space [27,35,43,44]. It is hence essential to consider the flow resistance of the network to compare it with that of a single channel of the same length and volume. The flow resistance of the equivalent single channel of the same length and volume can be written as 128µls
Rs =
π d4s
,
(3)
where ls , ds are the equivalent length, and the equivalent cross-sectional diameter of the network, respectively. The branching structure total volume can be expressed as V =
k=m
N kπ
d2k 2
k=0
lk =
π d20 l0 1 − (N β 2 γ )m+1 . 4 1 − N β 2γ
(4)
Because the equivalent single channel length (ls ) is equal to the branching structure (L). The length of the equivalent single channel is ls = L =
m
lk =
l0 (1 − γ m+1 ) 1−γ
0
.
(5)
According to the relation between total volume and effective length, i.e. V = AL and A = π d2 /4, we can obtain the cross-sectional diameter of equivalent single channel as follows
ds = d0
1−γ
π d20
1 − (N γ β 2 )m+1
4 1 − γ m+1
12
1 − Nγ β2
.
(6)
Substituting expressions (5) and (6), into Eq. (3), the flow resistance of the equivalent single channel can be expressed as Rs =
128 µls
π d4s
=
128µl0
1 − γ m+1
3
1−γ
π d40
1 − N β 2γ 1 − (N β 2 γ )m+1
2
.
(7)
Combining Eqs. (2) and (7), we get the dimensionless effective flow resistance of the branching structure, R+ =
Rs Rt
=
1 − γ m+1 1−γ
3
1 − N β 2γ 1 − (N β 2 γ )m+1
2
1 − γ /N β 4 1 − (γ /N β 4 )m+1
.
(8)
Eq. (8) denotes that the laminar flow resistance of a network is a function of the total number of branching levels (m), branching number (N), ratio of the length (γ ) and ratio of the diameter of the channel (β ). 3.2. Turbulent flow 3.2.1. Rough pipes For the turbulent flow in rough pipes, the pressure drop over a kth branching tube of mean velocity υk yields [29]
∆pk = f
lk ρυk2 dk 2
,
(9)
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J. Kou et al. / Physica A 393 (2014) 527–534
where ρ is the density of fluid and f the Darcy friction factor, which is approximately constant for the fully rough turbulent flow, independent of Reynolds number or flow rate. From the pressure drop ∆pk and the flow rate m2 , we can obtain the flow resistance as [27] Rk =
8f lk ∆p k = , ˙2 m ρπ 2 d5k
(10)
˙ = ρυk Ak . The total flow resistance of the network can be written as where m RN =
m Rk
=
N 2k
k=0
8fl0
1 − (γ /N 2 β 5 )m+1
1 − γ /N 2 β 5
ρπ 2 d50
.
(11)
Using the same constraint with Section 3.1 (laminar flow) and Eq. (6), the flow resistance of the equivalent single channel can be expressed as Rs =
8f
l
ρπ 2 d5
8fl0
=
1 − γ m+1
7/2
1−γ
ρπ 2 d50
5/2
1 − γ Nβ2 1 − (γ N β 2 )m+1
.
(12)
Combining Eqs. (11) and (12), we get the dimensionless effective flow resistance of the branching structure, R+ =
Rs RN
=
1 − γ m+1
7/2
1−γ
5/2
1 − Nγ β2 1 − (N γ β 2 )m+1
1 − γ /N 2 β 5
1 − (γ /N 2 β 5 )m+1
.
(13)
Expression (13) presents a relationship between the turbulent flow resistance in rough pipes and the geometrical parameters (m, N, γ , β ) of the network under the constraint of total volume. 3.2.2. Smooth pipes For the turbulent flow in smooth pipes, the following empirical relation for the friction factor is commonly used [33]
f = 0.046
−1/5
˙ 4m µπ d
.
(14)
The pressure drop over a kth branching tube 9/5
˙ k lk 1.1156µ1/5 m
∆pk =
24/5
π 4/5 ρ
dk
.
(15) 9/5
˙ k , the flow resistance of the kth pipe is Since the pressure drop ∆pk is proportional to the flow rate m Rk =
∆pk 9/5 mk
=
1.1156µ1/5
lk
π 4/5 ρ
24/5 dk
˙
.
(16)
Considering the mass conservation law, the total flow resistance of the network can be written as RN =
m Rk k=0
N 9k/5
=
1.1156µ1/5 l0
1 − (γ /N 9/5 β 24/5 )m+1
24/5
1 − γ /N 9/5 β 24/5
π 4/5 d0
.
(17)
The constraint of the Section 3.1 (laminar flow) is again used. Submitting equations (6) into the flow resistance form of the equivalent single channel, flow resistance can be expressed as Rs =
1.1156µ1/5
π 4/5 ρ
ls 24/5
=
ds
1.1156µ1/5 l0 24/5
π 4/5 d0
1 − γ m+1
17/5
1−γ
1 − γ Nβ2
12/5
1 − (γ N β 2 )m+1
.
(18)
Combining Eqs. (17) and (18), we get the dimensionless effective turbulent flow resistance of branched structure in smooth pipes, R+ =
Rs RN
=
1 − γ m+1 1−γ
17/5
1 − γ Nβ2 1 − (γ N β 2 )m+1
12/5
1 − γ /N 9/5 β 24/5 1 − (γ /N 9/5 β 24/5 )m+1
.
(19)
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Fig. 2. For laminar flow, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, and (c) N at m = 3 and γ = 0.6, and (d) scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
4. Results and discussion In the previous section, a complete analysis of flow resistance in the network for laminar flow and turbulent flow is presented. From Eqs. (8), (13) and (19), it can be seen that geometrical parameters (m, N, γ , β ) are important to the design of tree-like branching network systems. Based on Eqs. (8), (13) and (19), we can obtain the relationship between flow resistance and geometrical parameters of the network, and the scaling laws of optimal diameter ratio for laminar flow and turbulent flow. For laminar flow, Fig. 2(a)–(c) plot the dimensionless effective flow resistance under the constraint of total volume against diameter ratios β in different m, γ , and N, respectively. From these figures, we can see that, with the increase of the diameter ratio β , the dimensionless effective flow resistance initially increases, then decreases. There is a maximum dimensionless effective flow resistance, at which there is an optimal diameter ratio β ∗ . In other words, the flow resistance of the entire network is minimum (see first part of equations (8)). Furthermore, from Fig. 2(a) and (b), the optimal diameter ratio β ∗ is independent of the number of branching levels m and the length ratio γ , but the flow resistance is sensitive to the number of branching levels m and the length ratio γ . It can be seen that the dimensionless effective flow resistance decreases with an increase of m and γ . It is easily explained since, the higher the branching levels m and bigger the length ratio γ under the constraint of volume conservation, leads to a smaller diameter of every pipe of the network, the greater the flow resistance of total networks, while the dimensionless effective flow resistance is decreased. However, it is found that the optimal diameter ratio depends on the bifurcation number N by comparing the optimal diameter ratios in different branching number N as shown in Fig. 2(c). As can be seen, the optimal diameter ratio β ∗ and the corresponding minimum flow resistance of network decrease with the increase of bifurcation number N. For example, at N = 3, the optimum diameter ratio is β ∗ = 0.693; at N = 2, the optimum diameter ratio is β ∗ = 0.793, which coincides with the optimal diameter ratio as determined by Murray’s law [20] and ‘‘constructal law’’ [27,35]. This verifies that the present method and model is adequate. By plotting the logarithm of optimal diameter ratio (β ∗ ) with the logarithm of the bifurcation number (N) as shown in Fig. 2(d), it is easy to find a scaling relation that β ∗ = N k , where k = −1/3, N = 2, 3, 4, . . .. Furthermore, we extend the laminar flow to the turbulent flow with rough pipes and smooth pipes. Comparing Figs. 2–4, we can find that for the turbulent flow, the trends are consistent with laminar flow not only rough pipers but also smooth pipes. We also find the scaling relation of turbulent flow as shown by logarithm in Figs. 3(d), 4(d) and summary in Table 1. It is noteworthy that the minimum flow resistance of network is different with the flow in the same bifurcation number N between laminar flow and turbulent flow. The algorithm for determination of the scaling relation of optimal diameter ratio is summarized as follows: 1. Given m, γ , and select a N (N = 2, 3, 4, . . .), find a optimal diameter ratio β ∗ from Eqs. (8), (13) and (18). 2. Continue to select an N, find an optimal diameter ratio β ∗ from Eqs. (8), (13) and (18).
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Fig. 3. For turbulent flow in rough pipes, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, (c) N at m = 3 and γ = 0.6, and (d) the scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
Fig. 4. For turbulent flow in smooth pipes, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, (c) N at m = 3 and γ = 0.6, and (d) scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
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Table 1 Scaling relation of optimal diameter ratio (β ∗ ) along branch number (N) in different flows. Scaling relation
Laminar flow
β∗ = Nk (N = 2, 3, 4, . . .)
k = − 13
Turbulent flow Rough pipes
Smooth pipes
k = − 37
7 k = − 17
Table 2 A comparison of the maximal dimensionless effective resistance of network under the constraint of volume conservation in different flows at m = 3, γ = 0.6 and optimal diameter ratio β ∗ . Branch number (N)
(R+ max ) laminar flow
2 3 4 5
0.490 0.297 0.200 0.143
(R+ max ) turbulent flow Rough pipes
Smooth pipes
0.718 0.582 0.498 0.438
0.669 0.515 0.423 0.360
3. By running procedures (1) and (2) repeatedly, we find a series of β ∗ and N. Fitting the relation of β ∗ and N, we find the scaling relation in Table 1. Table 2 presents a calculation of the maximal dimensionless effective resistance of network under the constraint of volume conservation in different flows at m = 3, γ = 0.6 and β ∗ . The turbulent flow has a bigger dimensionless effective flow resistance than the laminar flow, the bigger dimensionless effective flow resistance for the turbulent flow in smooth pipers. In other words, the laminar flow possesses minimum flow resistance in networks. This may be reason that most systems are laminar, such as plant systems and vessel systems. 5. Conclusions We have studied the dimensionless effective flow resistance of tree-like networks and the effect of geometrical parameters of tree-like networks on flow resistance of entire networks. The dimensionless effective flow resistance of tree-like networks is expressed as a function of ratio of diameter (β ), ratio of length (γ ), branch number (N), branching levels (m) including laminar flow and turbulent flow. The study shows that the dimensionless effective flow resistance decrease with an increase of the ratio of length, the branch number and the branching levels, and that the ratio of diameter scales as branch number, β ∗ = N k , where, N = 2, 3, 4, . . ., the laminar flow, k = −1/3, the turbulent flow in rough pipes, k = −3/7, the turbulent flow in smooth pipes, k = −7/17, respectively. The predicted optimal diameter ratio of the laminar flow of a treelike branching network obeys Murray’s law. The present model of tree networks explains many natural tree-like branching network systems have applications for the design of efficient transport systems. Acknowledgments This work was partially supported by the National Natural Science Foundation of China under Grant No’s 11005093, 11079029 and 61274099, the Zhejiang Provincial Natural Science Foundation under Grant No’s Z6090556 and Y6100384, the Research Fund of Department of Education of Zhejiang Provincial under Grant No’s. Y201223336 and Y201225965, and Zhejiang Provincial Science and Technology Key Innovation Team No’s 2011R50012. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Optimal structure of tree-like branching networks for fluid flow Jianlong Kou a,c,∗ , Yanyan Chen a , Xiaoyan Zhou a , Hangjun Lu a , Fengmin Wu a,∗ , Jintu Fan b,c,∗ a
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
b
Department of Fiber Science and Apparel Design, Cornell University, Ithaca, NY 14853-4401, USA
c
Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
highlights • We derive some dimensionless expressions of effective flow resistance. • The effects of structural parameters on the flow resistance are studied. • We give optimal design schemes when the flow resistance of the whole network is minimum.
article
info
Article history: Received 22 February 2011 Received in revised form 19 February 2013 Available online 20 September 2013 Keywords: Flow resistance Fractal tree-like branching network Optimal structure
abstract Tree-like branching networks are very common flow or transportation systems from natural evolution. In this study, the optimal structures of tree-like branching networks for minimum flow resistance are analyzed for both laminar and turbulent flow in both smooth and rough pipes. It is found that the dimensionless effective flow resistance under the volume constraint for different flows is sensitive to the geometrical parameters of the structure. The flow resistance of the tree-like branching networks reaches a minimum when the diameter ratio β ∗ satisfies β ∗ = N k , where, N is the bifurcation number N = 2, 3, 4, . . . and k is a constant. For laminar flow, k = −1/3, which is in agreement with the existing Murray’s law; for turbulent flow in smooth pipes, k = −3/7; for turbulent flow in rough pipes, k = −7/17. These results serve as design guidelines of efficient transport and flow systems. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Tree-like branching networks exist widely in nature (such as lungs and vascular systems in mammals, river basins and plants) and have received considerable attention [1–6]. It has been shown that natural tree-like branching networks tend to have a perfect structure [7,8] and often surpass man-made products, in terms of minimal resistance and optimal vascular diameter for driving the blood in mammals and water in plants. This provides inspiration for the design of transport or conversion systems in biological engineering [9,10], chemical engineering [11,12], textile engineering [13,14], microelectronic engineering [15,16] and energy sources recovery [17–19]. The tree-like branching structures of mammalian cardiovascular and respiratory systems are optimum for blood and gas flow, as discussed by Murray et al. in 1926 [20], who found an optimum relationship between the diameter of the parent vessel (Dk ) and that of two daughter branches (Dk+1 ) in the form of Dk+1 /Dk = 2−1/3 . The relationship has been verified in recent theoretical analysis and experimental observation [21–23]. And it is now known as Murray’s law. Recently, Bejan
∗
Corresponding authors. Tel.: +86 57982297912; fax: +86 57982297119. E-mail addresses: [email protected] (J. Kou), [email protected] (F. Wu), [email protected] (J. Fan).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.08.029
528
J. Kou et al. / Physica A 393 (2014) 527–534
Fig. 1. Schematic of self-similar branch structure. The tree-like, spatial fractal (a) has self-similar branches, such that the small-scale structure (b) resembles the large-scale form (a), where the arrow represents the direction of fluid (Q ) flow, the self-similar branch structures (a) can be built by repeating a finite number of element structures (c).
[24–31] proposed a ‘‘constructal law’’ to explain the tree-like branching network in nature. The construct law states that, for a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it. Bejan et al. [24–26] showed that a tree-like branching network is an optimal configuration following the constructal law, as it generates minimum entropy for a flow between a source to a volume. Bejan et al. [27] applied the constructal law to the tree-like branching networks with local T-shaped and Y-shaped constructs, and showed that, for fixed total flow volume, there is an optimal diameter ratio of 2−1/3 and 2−3/7 , respectively. More recently, Xu and Chen et al. [28,29] proposed effective permeability of the whole networks and found tree-like networks could significantly increase the effective permeability of the composites compared to the traditional parallel networks under proper structural parameters. Although optimum tree-like networks have been studied in terms of pumping power requirements by Murray [20] and constructal law [24–37,30–39], the optimum design of the entire network system and the effects of various structural parameters under different types of flows have not been elucidated. In this paper, we have considered the optimal design of the entire tree-like network for both laminar and turbulent flow. Under the volume constraint of the whole network, we derive a dimensionless expression of effective flow resistance. We discuss the relationship between the dimensionless effective flow resistance and the geometrical parameters of the treelike branching networks (including diameter ratio, length ratio, branching number). Furthermore, we give an optimal design scheme when the flow resistance of the whole network is minimum. 2. Tree-like branching network A tree-like branching network is a complex structure. In order to minimize the energy dissipated in the system, the network must be a self-similar fractal network that can be space filling [40,41]. In this study, we consider a general Y-shaped tree-like branching network as shown in Fig. 1(a), which can be built by repeating a finite number of elements constructed in the shape as shown in Fig. 1(c). The structure satisfies the self-similar characteristic [40], it is space filling, significantly more variability tolerant than other structures and has an evolutionary advantage [42]. For the structure, every channel is divided into N branches at the next level (e.g., N = 2 in Fig. 1) and the branches are of the same geometries. In the present work, we consider that the thickness of the tube wall is sufficiently thin to be negligible; all the ducts are sufficiently slender and the losses at the junctions can be neglected. For a laminar flow through the network, each branch of the network is a smooth cylindrical tube. For a turbulent flow through the network, we consider turbulent flow through rough pipes and smooth pipes. In order to describe the branching structures, let the length and diameter of a typical branch at some intermediate level k (k = 0, 1, 2, 3 . . .) be lk and dk , respectively. We further introduce two scale factors, β = dk+1 /dk and γ = lk+1 /lk . Therefore, we can have dk = d0 β k
and lk = l0 γ k ,
where, l0 and d0 are the length and diameter of the 0th branching level, respectively.
(1)
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3. Optimal structure based on flow resistance In the fluid network introduced above, the Newtonian flow can be a laminar flow and a turbulent flow. In this section, we first discuss that the laminar flow through the network. We then extend the laminar flow to the turbulent flow with rough pipes and smooth pipes. 3.1. Laminar flow Firstly, we consider the case of incompressible and fully developed laminar flow. According to the Hagen–Poiseuille equation, the viscous resistance for flow in a single tube can be expressed as Rk = 128µlk /π d4k , where µ is the viscosity of the fluid. The flow resistance of the total network can be summed up in accordance with Ohm’s law as Rt =
k=m Rk
Nk
k=0
128µl0 1 − (γ /N β 4 )m+1
=
1 − γ /N β 4
π d40
.
(2)
For many natural systems or practical applications, such as water flow in botanical trees, airflow in bronchial trees, blood flow in the human cardiovascular system and gas supply systems, it is required to achieve minimum flow resistance of the entire network. It is also necessary to keep total volume and length as constraints. Otherwise, the structure would be infinitely large. In addition, space is expansive and limited, and different parts of the system compete for the available space [27,35,43,44]. It is hence essential to consider the flow resistance of the network to compare it with that of a single channel of the same length and volume. The flow resistance of the equivalent single channel of the same length and volume can be written as 128µls
Rs =
π d4s
,
(3)
where ls , ds are the equivalent length, and the equivalent cross-sectional diameter of the network, respectively. The branching structure total volume can be expressed as V =
k=m
N kπ
d2k 2
k=0
lk =
π d20 l0 1 − (N β 2 γ )m+1 . 4 1 − N β 2γ
(4)
Because the equivalent single channel length (ls ) is equal to the branching structure (L). The length of the equivalent single channel is ls = L =
m
lk =
l0 (1 − γ m+1 ) 1−γ
0
.
(5)
According to the relation between total volume and effective length, i.e. V = AL and A = π d2 /4, we can obtain the cross-sectional diameter of equivalent single channel as follows
ds = d0
1−γ
π d20
1 − (N γ β 2 )m+1
4 1 − γ m+1
12
1 − Nγ β2
.
(6)
Substituting expressions (5) and (6), into Eq. (3), the flow resistance of the equivalent single channel can be expressed as Rs =
128 µls
π d4s
=
128µl0
1 − γ m+1
3
1−γ
π d40
1 − N β 2γ 1 − (N β 2 γ )m+1
2
.
(7)
Combining Eqs. (2) and (7), we get the dimensionless effective flow resistance of the branching structure, R+ =
Rs Rt
=
1 − γ m+1 1−γ
3
1 − N β 2γ 1 − (N β 2 γ )m+1
2
1 − γ /N β 4 1 − (γ /N β 4 )m+1
.
(8)
Eq. (8) denotes that the laminar flow resistance of a network is a function of the total number of branching levels (m), branching number (N), ratio of the length (γ ) and ratio of the diameter of the channel (β ). 3.2. Turbulent flow 3.2.1. Rough pipes For the turbulent flow in rough pipes, the pressure drop over a kth branching tube of mean velocity υk yields [29]
∆pk = f
lk ρυk2 dk 2
,
(9)
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where ρ is the density of fluid and f the Darcy friction factor, which is approximately constant for the fully rough turbulent flow, independent of Reynolds number or flow rate. From the pressure drop ∆pk and the flow rate m2 , we can obtain the flow resistance as [27] Rk =
8f lk ∆p k = , ˙2 m ρπ 2 d5k
(10)
˙ = ρυk Ak . The total flow resistance of the network can be written as where m RN =
m Rk
=
N 2k
k=0
8fl0
1 − (γ /N 2 β 5 )m+1
1 − γ /N 2 β 5
ρπ 2 d50
.
(11)
Using the same constraint with Section 3.1 (laminar flow) and Eq. (6), the flow resistance of the equivalent single channel can be expressed as Rs =
8f
l
ρπ 2 d5
8fl0
=
1 − γ m+1
7/2
1−γ
ρπ 2 d50
5/2
1 − γ Nβ2 1 − (γ N β 2 )m+1
.
(12)
Combining Eqs. (11) and (12), we get the dimensionless effective flow resistance of the branching structure, R+ =
Rs RN
=
1 − γ m+1
7/2
1−γ
5/2
1 − Nγ β2 1 − (N γ β 2 )m+1
1 − γ /N 2 β 5
1 − (γ /N 2 β 5 )m+1
.
(13)
Expression (13) presents a relationship between the turbulent flow resistance in rough pipes and the geometrical parameters (m, N, γ , β ) of the network under the constraint of total volume. 3.2.2. Smooth pipes For the turbulent flow in smooth pipes, the following empirical relation for the friction factor is commonly used [33]
f = 0.046
−1/5
˙ 4m µπ d
.
(14)
The pressure drop over a kth branching tube 9/5
˙ k lk 1.1156µ1/5 m
∆pk =
24/5
π 4/5 ρ
dk
.
(15) 9/5
˙ k , the flow resistance of the kth pipe is Since the pressure drop ∆pk is proportional to the flow rate m Rk =
∆pk 9/5 mk
=
1.1156µ1/5
lk
π 4/5 ρ
24/5 dk
˙
.
(16)
Considering the mass conservation law, the total flow resistance of the network can be written as RN =
m Rk k=0
N 9k/5
=
1.1156µ1/5 l0
1 − (γ /N 9/5 β 24/5 )m+1
24/5
1 − γ /N 9/5 β 24/5
π 4/5 d0
.
(17)
The constraint of the Section 3.1 (laminar flow) is again used. Submitting equations (6) into the flow resistance form of the equivalent single channel, flow resistance can be expressed as Rs =
1.1156µ1/5
π 4/5 ρ
ls 24/5
=
ds
1.1156µ1/5 l0 24/5
π 4/5 d0
1 − γ m+1
17/5
1−γ
1 − γ Nβ2
12/5
1 − (γ N β 2 )m+1
.
(18)
Combining Eqs. (17) and (18), we get the dimensionless effective turbulent flow resistance of branched structure in smooth pipes, R+ =
Rs RN
=
1 − γ m+1 1−γ
17/5
1 − γ Nβ2 1 − (γ N β 2 )m+1
12/5
1 − γ /N 9/5 β 24/5 1 − (γ /N 9/5 β 24/5 )m+1
.
(19)
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Fig. 2. For laminar flow, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, and (c) N at m = 3 and γ = 0.6, and (d) scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
4. Results and discussion In the previous section, a complete analysis of flow resistance in the network for laminar flow and turbulent flow is presented. From Eqs. (8), (13) and (19), it can be seen that geometrical parameters (m, N, γ , β ) are important to the design of tree-like branching network systems. Based on Eqs. (8), (13) and (19), we can obtain the relationship between flow resistance and geometrical parameters of the network, and the scaling laws of optimal diameter ratio for laminar flow and turbulent flow. For laminar flow, Fig. 2(a)–(c) plot the dimensionless effective flow resistance under the constraint of total volume against diameter ratios β in different m, γ , and N, respectively. From these figures, we can see that, with the increase of the diameter ratio β , the dimensionless effective flow resistance initially increases, then decreases. There is a maximum dimensionless effective flow resistance, at which there is an optimal diameter ratio β ∗ . In other words, the flow resistance of the entire network is minimum (see first part of equations (8)). Furthermore, from Fig. 2(a) and (b), the optimal diameter ratio β ∗ is independent of the number of branching levels m and the length ratio γ , but the flow resistance is sensitive to the number of branching levels m and the length ratio γ . It can be seen that the dimensionless effective flow resistance decreases with an increase of m and γ . It is easily explained since, the higher the branching levels m and bigger the length ratio γ under the constraint of volume conservation, leads to a smaller diameter of every pipe of the network, the greater the flow resistance of total networks, while the dimensionless effective flow resistance is decreased. However, it is found that the optimal diameter ratio depends on the bifurcation number N by comparing the optimal diameter ratios in different branching number N as shown in Fig. 2(c). As can be seen, the optimal diameter ratio β ∗ and the corresponding minimum flow resistance of network decrease with the increase of bifurcation number N. For example, at N = 3, the optimum diameter ratio is β ∗ = 0.693; at N = 2, the optimum diameter ratio is β ∗ = 0.793, which coincides with the optimal diameter ratio as determined by Murray’s law [20] and ‘‘constructal law’’ [27,35]. This verifies that the present method and model is adequate. By plotting the logarithm of optimal diameter ratio (β ∗ ) with the logarithm of the bifurcation number (N) as shown in Fig. 2(d), it is easy to find a scaling relation that β ∗ = N k , where k = −1/3, N = 2, 3, 4, . . .. Furthermore, we extend the laminar flow to the turbulent flow with rough pipes and smooth pipes. Comparing Figs. 2–4, we can find that for the turbulent flow, the trends are consistent with laminar flow not only rough pipers but also smooth pipes. We also find the scaling relation of turbulent flow as shown by logarithm in Figs. 3(d), 4(d) and summary in Table 1. It is noteworthy that the minimum flow resistance of network is different with the flow in the same bifurcation number N between laminar flow and turbulent flow. The algorithm for determination of the scaling relation of optimal diameter ratio is summarized as follows: 1. Given m, γ , and select a N (N = 2, 3, 4, . . .), find a optimal diameter ratio β ∗ from Eqs. (8), (13) and (18). 2. Continue to select an N, find an optimal diameter ratio β ∗ from Eqs. (8), (13) and (18).
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Fig. 3. For turbulent flow in rough pipes, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, (c) N at m = 3 and γ = 0.6, and (d) the scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
Fig. 4. For turbulent flow in smooth pipes, the dimensionless effective flow resistance (R+ ) versus β in different (a) γ at N = 2 and m = 3, (b) m at N = 2 and γ = 0.6, (c) N at m = 3 and γ = 0.6, and (d) scaling relation of optimal diameter ratio (β ∗ ) along branch number (N).
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Table 1 Scaling relation of optimal diameter ratio (β ∗ ) along branch number (N) in different flows. Scaling relation
Laminar flow
β∗ = Nk (N = 2, 3, 4, . . .)
k = − 13
Turbulent flow Rough pipes
Smooth pipes
k = − 37
7 k = − 17
Table 2 A comparison of the maximal dimensionless effective resistance of network under the constraint of volume conservation in different flows at m = 3, γ = 0.6 and optimal diameter ratio β ∗ . Branch number (N)
(R+ max ) laminar flow
2 3 4 5
0.490 0.297 0.200 0.143
(R+ max ) turbulent flow Rough pipes
Smooth pipes
0.718 0.582 0.498 0.438
0.669 0.515 0.423 0.360
3. By running procedures (1) and (2) repeatedly, we find a series of β ∗ and N. Fitting the relation of β ∗ and N, we find the scaling relation in Table 1. Table 2 presents a calculation of the maximal dimensionless effective resistance of network under the constraint of volume conservation in different flows at m = 3, γ = 0.6 and β ∗ . The turbulent flow has a bigger dimensionless effective flow resistance than the laminar flow, the bigger dimensionless effective flow resistance for the turbulent flow in smooth pipers. In other words, the laminar flow possesses minimum flow resistance in networks. This may be reason that most systems are laminar, such as plant systems and vessel systems. 5. Conclusions We have studied the dimensionless effective flow resistance of tree-like networks and the effect of geometrical parameters of tree-like networks on flow resistance of entire networks. The dimensionless effective flow resistance of tree-like networks is expressed as a function of ratio of diameter (β ), ratio of length (γ ), branch number (N), branching levels (m) including laminar flow and turbulent flow. The study shows that the dimensionless effective flow resistance decrease with an increase of the ratio of length, the branch number and the branching levels, and that the ratio of diameter scales as branch number, β ∗ = N k , where, N = 2, 3, 4, . . ., the laminar flow, k = −1/3, the turbulent flow in rough pipes, k = −3/7, the turbulent flow in smooth pipes, k = −7/17, respectively. The predicted optimal diameter ratio of the laminar flow of a treelike branching network obeys Murray’s law. The present model of tree networks explains many natural tree-like branching network systems have applications for the design of efficient transport systems. Acknowledgments This work was partially supported by the National Natural Science Foundation of China under Grant No’s 11005093, 11079029 and 61274099, the Zhejiang Provincial Natural Science Foundation under Grant No’s Z6090556 and Y6100384, the Research Fund of Department of Education of Zhejiang Provincial under Grant No’s. Y201223336 and Y201225965, and Zhejiang Provincial Science and Technology Key Innovation Team No’s 2011R50012. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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