The convective heat transfer of branched structure

International Journal of Heat and Mass Transfer 116 (2018) 813–816

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

The convective heat transfer of branched structure Liang Luo a,b, Fang Tian b, Jianchao Cai a,⇑, Xiangyun Hu a a b

Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, PR China College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414000, PR China

a r t i c l e

i n f o

Article history: Received 22 December 2016 Received in revised form 5 July 2017 Accepted 16 September 2017

Keywords: Convective heat transfer Branched structure Laminar flow Asymmetry

a b s t r a c t Branch network is a kind of structure that exists widely in nature, it can be found in diverse engineering systems from underground geothermal mining to fuel cell. In this paper, we study the convective heat transfer property in a basic unit of branch network. Based on the fully developed convective heat transfer model for laminar flow in a pipe, the theoretical expression of heat transfer rate is obtained. The effect of symmetry of branches on the heat transfer rate varies with the trunk diameter of the branched structure, and the effect of symmetry branched structure is not always the favorable. The heat transfer rate increases with the increase of pressure drop, trunk diameter and temperature difference, whereas the heat transfer rate decreases with the increase of the length of the branched structure. The presented work sheds some light on diverse applications of branch network for engineering system design. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The branch networks are very common in nature, such as lightning, river basins, trees, leaves, mammalian circulator and respiratory systems, neural dendrites, deltas, and flow in fractures in oil/ water reservoirs. The ubiquitous branch networks have unique property, which makes it a research topic whose relevance spans across multiple fields and applications. The heat and mass transport properties of branch networks have attracted a large interest from science and engineering, and this efficient transmission system also inspires people’s design inspiration [1–3]. At the beginning of the 19th century, Hess [4] and Thompson [5] studied the blood circulatory system, which they believed to be a perfect organization structure with the maximum transport efficiency, and explored the role of the branch network in blood circulation system. According to previous studies, Murray proposed the Murray’s law based on the minimum energy cost for driving and maintaining the blood circulation [6–8]. Murray’s law says that the optimal diameter of blood vessel is proportional to the cube of local blood velocity. Murray’s law was proposed for the cardiovascular system, but it can also be applied to non-living systems. Bejan [9,10] developed the Constructal Theory, a different optimization method compared to Murray’s Law, and it was applied to engineering applications such as electronic-cooling system, engineering thermodynamics, and fuel cell. Xu [11] extended the Murray’s Law and obtained an optimized branch structures. ⇑ Corresponding author. E-mail address: [email protected] (J. Cai). https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.056 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

Fluid flow in branch network has the advantages of high efficiency, good uniformity and robustness. The branch networks also demonstrated the superiority in pure heat conduction [12,13]. Besides, they are also used to design the shape and assembly of fins [14,15]. Recently, researchers have investigated micro- and nanoscale treelike structures with the purpose of high performance cooling systems in nanotechnology and small-scale electronics [16,17]. In order to enhance the cooling performance, the heat transfer efficiency needs to be enhanced, and in that regard the convective heat transfer is more efficient than the heat conduction. To address the issues of increased pumping power and non-uniformity of microchannel systems in the conventional parallel and serpentine micro-channels, the idea of a branch microchannel network was proposed and investigated by the experiment and numerical simulations. In order to enhance the heat convective performance of the disk heat sink, Xu [18] studied the thermal characteristics of microchannel electronic cooling system which is set up with tree-shaped networks. Senn [19] investigated the laminar convective heat transfer and pressure drop characteristics in tree-like microchannel nets of polymer electrolyte fuel cells. Their results also support the idea that branch networks result in better efficiency. The superiority of the branch network has been revealed by many researchers. The convective heat transfer performance of the branch network is of great significance in new energy fields. Earlier researches on branch networks are based either on numerical simulations or experiments, with very few studies on theoretical analysis. In this paper, we focus on the theoretical analysis part

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T s  T in T s  T out



of branch networks. Specifically, we study the basic unit of branch networks.

hAðT out  T in Þ=ln

2. Description of the branched structure

where q and c are density of fluid and specific heat capacity of fluid, respectively. For a single tube, A = pdL, and h is defined as:

The structure shown in Fig. 1 is the basic unit of a branch network. This basic unit of a branch network shows a trunk and two branches, where the branches are not symmetric. Here, l1 , l2 and 0 0 l2 represent the lengths, d1 , d2 and d2 represent the diameters, where the subscripts 1 and 2 denote the trunk and branch, respectively, and the dash in the superscript is used to distinguish the two branches. For the sake of convenience, we use the parameter b to denote the ratio of diameters of the two branches:

h¼

3. Mathematical model The laminar flow model is used in the branched structure, and it is given by Hagen-Poiseuille equation:

q¼

pd4 DP 128l L

ð2Þ

where DP and l represent the pressure drop and viscosity, respectively. For a tube whose volume is fixed, the diameters of branched structure should satisfy the following relationship as per Murray’s Law [11] that minimizes flow resistance: 3

3

03

d1 ¼ d2 þ d2

ð3Þ

The convective heat transfer equation Q e is defined as [20]:

Qe ¼

hAðT out  T in Þ
 T in ln TTssT out

ð4Þ

where h is convective heat transfer coefficient, A is heat transfer area. T s , T in and T out are tube wall temperature, initial fluid temperature, and final fluid temperature, respectively. According to the law of conservation of energy, the heat transfer rate should be equal to the heat absorbed by the fluid per unit of time. Therefore, we obtain the following equation:

Nuk d

ð6Þ



T out ¼ T s  ðT s  T in Þe

128klL2 Nu cd4 DPq

ð7Þ

And the heat transfer rate can be written as:

ð1Þ

where b ¼ 1 indicates that the branched structure is symmetrical. For simplicity, the length of the trunk and branches are set to be the same.

ð5Þ

where k is thermal conductivity and Nu is the Nusselt number. For a sufficiently developed laminar flow with a constant wall temperature, the Nusselt number is a constant [20]. By solving Eq. (5), the final temperature of the fluid is obtained:

0

d b¼ 2 d2

¼ qqcðT out  T in Þ



Qe ¼



128klL2 Nu  pd4 DP cq ðT s  T in Þ1  e qcd4 DP 128l L

 ð8Þ

where the symbols (d; DP; T in ) are variable, for e.g. d is different in trunk and branch. The final temperature of the fluid in trunk is the initial temperature in the branch (from Fig. 1), therefore:

T in 2 ¼ T out 1

ð9Þ

Combining Eqs. (1) and (3) to obtain:

d1 d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 b3 þ 1

ð10Þ

0

d2 ¼ bd2

ð11Þ

The pressure drop on trunk and branches can be written as Eq. (12), which is given by flow resistance analysis [21]:

DP 2 ¼ DP 1

1 þ b4 ð1 þ b3 Þ

ð12Þ

4=3

Substituting Eqs. (9)–(12) into Eq. (8), the heat transfer rates of trunk and branches is obtained. Eq. (13) gives the total heat transfer rate of the branched structure:

Q e total ¼ Q e1 þ Q e2 þ Q 0e2

ð13Þ

We analyze Eq. (13) in the following section. 4. Results and discussion According to Eqs. (13) and (8), the heat transfer rate is related to pressure drop, geometric structure of branched structure, temperature of fluid and temperature of tube wall. First, we analyze the effect of geometrical structure on heat transfer rate. The geometry parameters include the ratio of branch diameters, the diameter of trunk and length. The default parameters shown in Table 1 are in order to meet the fully developed laminar flow and heat transfer conditions, therefor the L/d ratio is larger than 50. The relationship between the heat transfer rate (Q e ) and the ratio of branch diameters (b) is plotted in Fig. 2. In Fig. 2a, the heat transfer rate reaches maximum when the branch is symmetric

Table 1 The default parameters.

Fig. 1. The schematic diagram of branched structure.

P (kg/m3)

C ðJ=kg=KÞ

k ðW=m=KÞ

l

ðPa  sÞ

Nu /

1000 L (m) 0.1

4200 T in (K) 293

0.6 T s ðKÞ 363

0.001 DP (Pa) 10

3.66 d (m) 0.001

L. Luo et al. / International Journal of Heat and Mass Transfer 116 (2018) 813–816

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Fig. 2. The relation between the heat transfer rate (Qe) and b with different d.

(b ¼ 1), but in Fig. 2b the symmetric branch network is most unfavorable for the convective heat transfer. Fig. 2c shows the more complex relationship between b and Q e . The heat transfer rate is minimum if the diameter ratio is about 0.4, where the difference between three figures is due to the trunk diameter (0:001 m;0:0034 m;0:004 m). Since the convective heat transfer process is closely related with the flow rate and heat transfer area, the heat exchange area and flow resistance are also the largest [21] under the condition of symmetrical branches. When the diameter is less than 0:001 m, the flow resistance is very large, such that the fluid and the tube wall are sufficiently heated and achieve the same temperature soon. At this point, the deciding factor is the flow rate. When the trunk diameter is within 0:001—0:004m, the flow and the heat transfer area are all important for the rate of heat transfer. When the trunk diameter is larger than 0:004 m, the importance of heat exchange area increases even further. In conclusion, the presented model for heat transfer rate shows that there is a tradeoff between the size of branched structure and branch symmetry, i.e., the use of symmetrical or asymmetrical branches can either strengthen or weaken the heat transfer rate depending on their size. Fig. 3 denotes the influence of trunk diameter on heat transfer rate with different diameter ratio of branches. The heat transfer rate increase with the increase of trunk diameter

Fig. 3. The relation between the heat transfer rate (Qe) and diameter of first branch (d).

for different b, and for the same trunk diameter, the heat transfer rate decreases if there is more deviation from a symmetric branch network. This conclusion is consistent with the observation from Fig. 2(a). Fig. 4 shows that the longer the length of a branched structure, the smaller the heat transfer rate. This is because increasing the length of a branched structure reduces the fluid flux. For the same pressure drop, a longer length of branched structure would result in a smaller pressure gradient and a smaller flow rate (from H-P equation). 5. Conclusions The convective heat transfer of a branched structure is analyzed and an expression for convective heat transfer rate is derived. This is a new theoretical expression that reflects the influence of various parameters on the heat transfer rate. For different trunk diameters of a branched structure, the heat transfer rate shows a different behavior for different branch symmetry. When trunk diameter is more than 0:004 m, the symmetrical branches results in the largest heat transfer rate. But when trunk diameter is less than 0:001 m, the symmetrical branches results in the smallest heat transfer rate. Also, the heat transfer rate decreases with the increase in the length of branched structure for the same pressure drop. The

Fig. 4. The relation between the heat transfer rate (Qe) and length of branch (L).

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increase of pressure drop and temperature difference between tube wall and fluid can increase the heat transfer rate. The results presented in this study reveal the connection between the convective heat transfer rate and trunk diameters. These results may have importance for experimental and industrial applications dealing with heat transfer and branched structures. This work only studies a simple theoretical model, and further research is needed to established behavior of convective heat transfer in a whole branch network, for e.g. the enhanced geothermal systems and the fuel cell pipeline. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 51206174, 41572116, 41722403) and the Scientific Research Fund of Hunan Provincial Education Department, China (Grant No. 17C0720). Conflict of interest We declare that we have no conflict of interest. References [1] J. Lee, J.Y. Yi, L.-H. Kim, C.B. Shin, Dynamic modeling of the hydraulic-thermal behavior of the buried pipe network for district heating, J. Energy Eng. 21 (2) (2012) 144–151. [2] H. Oh, W. Short, Optimal expansion planning for the deployment of wind energy, J. Energy Eng. 135 (3) (2009) 83–88. [3] J. Cai, E. Perfect, C.-L. Cheng, X. Hu, Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures, Langmuir 30 (18) (2014) 5142–5151.

[4] W. Hess, Über die periphere Regulierung der Blutzirkulation, Pflügers Arch. Eu. J. Physiol. 168 (9) (1917) 439–490. [5] D. Thompson, On growth and form. 793 pp, Cambridge, Eng, (1917). [6] C.D. Murray, The physiological principle of minimum work II. Oxygen exchange in capillaries, Proc. Natl. Acad. Sci. 12 (5) (1926) 299–304. [7] C.D. Murray, The physiological principle of minimum work applied to the angle of branching of arteries, J. Gen. Physiol. 9 (6) (1926) 835–841. [8] C.D. Murray, The physiological principle of minimum work I. The vascular system and the cost of blood volume, Proc. Natl. Acad. Sci. 12 (3) (1926) 207– 214. [9] A. Bejan, Constructal tree network for fluid flow between a finite-size volume and one source or sink, Revue Generale de thermique 36 (8) (1997) 592–604. [10] A. Bejan, M.R. Errera, Deterministic tree networks for fluid flow: geometry for minimal flow resistance between a volume and one point, Fractals 5 (04) (1997) 685–695. [11] P. Xu, Transport Properties of Fractal Tree-like Branching Network, Huazhong University of Science and Technology, Wuhan, China, 2008. [12] A. Bejan, Constructal-theory network of conducting paths for cooling a heat generating volume, Int. J. Heat Mass Tran. 40 (4) (1997) 799–816. [13] X.-Q. Wang, P. Xu, A.S. Mujumdar, C. Yap, Flow and thermal characteristics of offset branching network, Int. J. Therm. Sci. 49 (2) (2010) 272–280. [14] G. Lorenzini, L.A.O. Rocha, Constructal design of Y-shaped assembly of fins, Int. J. Heat Mass Tran. 49 (23) (2006) 4552–4557. [15] D. Calamas, J. Baker, Tree-like branching fins: performance and natural convective heat transfer behavior, Int. J. Heat Mass Tran. 62 (2013) 350–361. [16] L. Chen, H. Feng, Z. Xie, F. Sun, Constructal optimization for ‘‘disc-point” heat conduction at micro and nanoscales, Int. J. Heat Mass Tran. 67 (2013) 704–711. [17] H. Feng, L. Chen, Z. Xie, F. Sun, Constructal entransy dissipation rate minimization for triangular heat trees at micro and nanoscales, Int. J. Heat Mass Tran. 84 (2015) 848–855. [18] P. Xu, X. Wang, A.S. Mujumdar, C. Yap, B. Yu, Thermal characteristics of treeshaped microchannel nets with/without loops, Int. J. Therm. Sci. 48 (11) (2009) 2139–2147. [19] S. Senn, D. Poulikakos, Laminar mixing, heat transfer and pressure drop in treelike microchannel nets and their application for thermal management in polymer electrolyte fuel cells, J. Power Sources 130 (1) (2004) 178–191. [20] Y.A. Cengel, R.H. Turner, J.M. Cimbala, M. Kanoglu, Fundamentals of thermalfluid sciences, McGraw-Hill, New York, NY, 2008. [21] L. Luo, B. Yu, J. Cai, M. Mei, Symmetry is not always prefect, Int. J. Heat Mass Tran. 53 (21) (2010) 5022–5024.