October 15, 2018
16:15
0218-348X
1850073
Fractals, Vol. 26, No. 5 (2018) 1850073 (12 pages) c World Scientific Publishing Company DOI: 10.1142/S0218348X18500731
CONVECTION HEAT TRANSFER PERFORMANCE OF THE FRACTAL TUBE BANK UNDER CROSS FLOW DONGHUI ZHANG,∗,‡ YANLIN SHEN,∗ ZHIPING ZHOU,∗ JIAN QU,† HAIYANG XU,∗ WEI CAO,∗ DAN SONG∗ and FENGMEI ZHANG∗ ∗School of Energy and Power Jiangsu University of Science and Technology Zhenjiang 212003, P. R. China †School of Energy and Power Engineering, Jiangsu University Zhenjiang 212003, P. R. China ‡
[email protected] Received January 8, 2018 Accepted May 22, 2018 Published October 17, 2018
Abstract Convective heat transfer process between fractal tube bank and coolant water was simulated by using the finite element method. Both rectangular and circular tube banks were investigated in detail. For the relationship between Nusselt number and Reynolds number, the fractal circular tube bank shows the same trend as the rectangular tube one. Based on simulation results, the empirical heat transfer correlations for different stage fractals were regressed. The exponents of correlations increase with the fractal stage. Almost linear correlations between Nu and Re for fourth stage fractal were obtained, indicating a good heat transfer enhancement with the increase in fractal stages. Compared to the uniform tube bank, the fractal tube banks show better overall performance in enhanced heat transfer effect. This research would be helpful to understand the heat exchange mechanism of the multi-scale structure. Keywords: Multi-Scale; Fractal; Convective Heat Transfer; Pressure Drop; Tube Bank.
‡
Corresponding author. 1850073-1
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al.
NOMENCLATURE N R T Df Re Nu Pr D a p SL SD u, v x, y
fractal level tube radius temperature fractal dimension Reynolds number Nusselt number Prandtl number tube diameter thermal diffusivity pressure longitudinal pitch ratio transverse pitch ratio velocity components space coordinate.
GREEK SYMBOLS α β v
fractal dimension the ratio of the radius kinematic viscosity.
1. INTRODUCTION Tubular Heat exchangers are applied widely in the fields of energy production, electronics, heat recovery, air-conditioning and many other fields. The main idea of heat transfer enhancement is the balance between heat transfer performance and pumping consumption.1–3 In the heat exchanger design, increasing the heat exchange area is common but often is accompanied by a dramatic increase in flow resistance. It is of great importance to understand how to optimize the structure in a way that can reduce the tendency of increasing flow resistance. Appropriate multiscale structure can alleviate the contradiction between the two. Compared to a uniform structure, multiscale structure brings double benefits of: (i) high specific surface area per unit volume; (ii) more freedom for the system structure to be optimized. The fractal is a kind of multi-scale structure that shows self-similar characteristics at different scale stages.4,5 Of the fractal structures, the Sierpinski carpet caught wide attention, which was a candidate of the ideal porous model with nonuniform pores. In the past decade, the researches on transport processes (fluid flow,6 Diffusion,7 permeability,8 heat conduction,9 percolation,10 etc.) in fractal structures achieve great progress. The problem of convection heat transfer for the multiscale structure had been also concerned in recent years. For multi-scale structure, Cai and
Huai11 used the Lattice Boltzmann model to simulate fluid–solid coupling heat transfer process. Their numerical work disclosed almost a linear dependency of Nu on Re at low Re range (Re < 10). For the fractal distribution of nanoparticles, Xiao and Yu12 presented a model of convection heat transfer between nanoparticles and liquids. Both kinds of heat transfer mechanism, heat convection and natural convection, were considered. The effects of the concentration and size of nanoparticles on heat transfer performance could be predicted from this fractal model without any empirical constants. In order to reduce fouling rates of the heat exchanger, Mavridou and Bouris13,14 conducted light research on the influence of tube bank configurations with different scales. The simulated and experimental results showed that multi-scale tube bundles led to heat transfer enhancement by 30–40% with a negligible penalty in pressure drop, compared to uniform tube bank. The Reynolds numbers investigated in their work ranged from 3100–12500. The enhancement mechanism was attributed to the increase in wake width due to the impingement of vortices shed from the upstream small-scale tube. Bello– Ochende and Bejan15 used the constructal method to maximize heat transfer density in assemblies of multi-scale cylinders in cross flow. The optimal flow structure had multiple scales that are distributed nonuniformly through the available volume. Small scale cylinders were required to be placed in the wedge-shaped flow regions occupied by fluid that has not yet been used for heat transfer. The heat transfer density increased (with diminishing returns) as the optimized structure became more complex. All these studies suggest that appropriate multiscale design could really utilize multiple diameter scales to assist in increasing heat transfer rate of the thermal system of concern. It is for this reason that tube banks with fractal arrangements evoke our study interest. For uniform tube banks, Zukauskas’ group10 conducted extensive experimental tests and summarized a series of general correlations. Lately, Khan and Yovanovich16 extrapolated these correlations to the wider range of operating conditions often found in existing tubular heat exchangers. However, convention heat transfer mechanisms in the multi-scale tube bank are still lacking sufficient investigation. Cai and Huai11 discussed the convection heat exchange performance of fractal tube banks only in the low Re range from 1 to 10. Inspired by these studies, some of issues are
1850073-2
October 1850073
15,
2018
16:15
0218-348X
Convection Heat Transfer Performance of the Fractal Tube Bank Under Cross Flow
N=1
N=2
N=3
N=4
N=3
N=4
(a)
N=1
N=2
(b) Fig. 1 The construction process of a fractal tube bank (flow direction left to right): (a) rectangular tube bank and (b) circular tube bank.
naturally induced. Is the tube bank with the fractal arrangement really beneficial to the increase of overall heat transfer rate? Between the coolant and fractal structure, what factors are the heat transfer coefficient affected by? How is the heat transfer coefficient associated with the structural complexity of fractal structure, and so on. On these issues, we aim at the influence of structural complexity on heat transfer performance of the fractal tube bank in this study. Coolant flow across the tube bank configured as the fractal Sierpinski carpet is simulated by ANSYS Fluent software. Convection heat transfer process around the tube bank, as shown in Fig. 1, is carefully studied. Both rectangular and circular tube banks are investigated in detail. N denotes the construction stage of fractal structure. In addition, the performance comparisons of fractal tube bank with uniform one are presented to understand in depth the influence of hierarchical structure on flow behavior.
2. GENERAL DESCRIPTION OF FRACTALS Fractal structure is a kind of porous media with self-similarity properties. Different approaches can be used to generate the fractal structure. A kind of generic method in paper17 is proposed to create the fractal tube bank that satisfies with the Sierpinski Carpet. The model is constructed as follows. First, F1 tubes (the first generation) with radius
(or length) R0 , are placed into one region, in which an area S0 is occupied; second, F2 small tubes (the second generation) with a radius R1 are put in the surrounding region, and so on. The process is then repeated at successively higher generation. While the number of iteration increases, the geometrical structure becomes more and more complicated, consisting of different scales. By this method, a multiscale configuration was constructed in a kind of hierarchical distribution. Two principles should be followed in this model: (i) The ratio of the area of N th generating tubes to that of (N − 1)th generating tubes is constant P (0 < P < 1): SN /SN −1 = P.
(1)
(ii) The ratio of the radius of (N − 1)th generating structure to that of N th generating structure is constant β (β > 1): RN −1 /RN = β.
(2)
The cumulative numbers Fsum are a function of the radius Rk if the parameters R0 , N0 , P and β are given. It can be written as5,18 : Fsum ∝ R−α .
(3)
The slope α is exactly the fractal dimension Df of both structures5,18 : ln P . (4) α = Df = 2 + ln β
1850073-3
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al.
Water is assumed to be the working fluid in the model. A uniform flow is imposed at the inlet, driven by the pressure difference between the inlet and the outlet. The Re numbers investigated range from 10 to 600. In simulation, the following assumptions are utilized: (a) steady laminar flow, (b) constant fluid properties, and (c) gravity effect on liquid flow in porous media was not considered. The tube bank model, configured as the thirdstage Sierpinski Carpet, is presented in Fig. 2 and located in the middle part of the assumed straight channel. The computational domain, 6L in length and L in width, is divided into three sub-domains. The fractal tube bank is 2L and 3L away from the inlet and outlet of the test channel respectively. The length of the largest rectangular tube in the center is 1/3L. The pressure outlet condition is for the outlet boundary and the adiabatic boundary condition without slippage for the top and bottom boundaries. The isothermal boundary condition is adopted for each tube surface. The main objective of this study is to understand the effect of the structural complexity on the flow characteristics and heat transfer performance. The finite element method is utilized in simulation, based on the SIMPLE algorithm. The finite volume method is applied to the discretization of the Navier–Stokes equations set. The second-order upwind scheme is used for the convective terms and the semi-implicit method for the pressurelinked equations (SIMPLE) algorithm adopted for pressure correction. To make a comparison, uniform circular-tube banks with the same specific
For the rectangular-tube bank or circular-tube one discussed in this paper, P is equal to 8/9 and β equal to 3. According to Eq. (4), both have same fractal dimension of 1.893.
3. GOVERNING EQUATIONS AND NUMERICAL MODEL 3.1. Physical Model The governing equations for two-dimensional, unsteady, laminar, incompressible fluid flow with constant fluid properties in the Cartesian coordinate system can be written as follows1 : Continuity equation: ∂u ∂ν + = 0. (5) ∂x ∂y Momentum equations: 2 ∂u ∂u ∂u 1 ∂p ∂ u ∂2u +u +ν =− +ν + 2 , ∂t ∂x ∂y ρ ∂x ∂x2 ∂y (6) 2 2 ∂ ν ∂u ∂ν 1 ∂p ∂ ν ∂u + 2 . +u +ν =− +ν ∂t ∂x ∂y ρ ∂y ∂x2 ∂y (7) Energy equation: ∂T ∂T ∂T +u +ν =a ∂t ∂x ∂y
∂2T ∂2T + , ∂x2 ∂y 2
(8)
where u and v are the velocity components, p is the pressure, ν is the kinematic viscosity, ρ is the density, T is the temperature of the fluid, and a the thermal diffusivity of the fluid.
(a)
(b) Fig. 2
General view of the meshes for fractal tube bank: (a) rectangular tube bank and (b) circular tube bank. 1850073-4
15,
2018
16:15
0218-348X
Convection Heat Transfer Performance of the Fractal Tube Bank Under Cross Flow
area of the corresponding stage fractals are also in simulation.
50 Simulation Khan Zukauskas
40
3.2. Benchmarking Test
30
First, grid convergence tests are conducted to determine the optimal grid resolution and ensure the solver model. A non-uniform grid distribution is employed with a more refined grid generated around the tube wall, as shown in Fig. 3. The total number of elements and nodes is 407300. For the uniform circular tube bank, the grid numbers are 177520. The grid independence study has also been carried out for different Reynolds numbers. Then, the case of an in-line uniform tube bank under the cross flow of water (P r = 5.6) is investigated in order to validate our simulation process. The 27 rows and 27 columns circular tube bank is considered. Both the longitudinal pitch ratio (SL /D) and transverse pitch ratio (ST /D) is 2.7. On the isothermal boundary condition, the mean Nusselt numbers simulated were compared with the following correlations available in the literatures: Zhuauskas’ correlation (100 ≤ Re ≤ 600)10 : N u = F · 0.52 · Re0.5 P r0.36 ,
(9)
where F is correction factor, representing the effect of the row numbers (N ≤ 16). Khan’s correlation (1 ≤ Re ≤ 600)16 : 0.5
N u = C · Re
Pr
1/3
,
(10)
C = [0.2 + exp(−0.55 · ST )] · ST0.285 · SL0.222 ,
(11)
where the coefficients C is determined in terms of pitch ratios for different arrangements, ST is the longitudinal pitch ratio and SL is the transverse pitch ratio for in-line arrangements. The comparison results are shown in Fig. 4. We find that Khan’s correlation agrees well with the
(a)
(b)
Fig. 3 Typical grid structure around the rectangular and circular tube.
NuD
October 1850073
ST/D=2.7 SL/D=2.7
20
Pr=5.42
10 0
100
200
300
400
500
600
ReD
Fig. 4 Comparison of simulation result with Grimism’s correlation and Zukauskas’ one.
simulated results, but Zukauskas’ correlation differs considerably. Khan’s correlation considers the influence of longitudinal and transverse pitch ratio in detail, which could provide more accuracy in analyzing the heat transfer performance of tube bank. The testing results show that the numerical procedure has a reasonable accuracy and can be applied in the study on the fractal structure.
4. RESULTS AND DISCUSSION 4.1. Flow Fields of Fractal Tube Banks The velocity field has been non-dimensionalized, based on the inlet velocity. The Reynolds number is defined by the maximum inlet velocity and the maximum tube diameter of the fractal tube bank. The third-stage configuration (N = 3) includes three different scales of circular tubes. The instantaneous streamlines and velocity distribution at different Re values are illustrated in Figs. 5a and 5b for N = 3. At Re = 100, a pair of stable trapped vortices is observed between each tube in the center line. The velocity magnitude in the longitudinal gap between tubes of different rows is evidently larger than that in other regions, known as the preferential flow phenomenon. And at Re = 600, interacting with such a multi-scale structure, the inflow would be slowed down, accelerated and altered in direction vigorously, which yields higher overall flow resistance. The velocity distribution is approximately symmetrical. In Fig. 6, for the fourth stage structure, the velocity fields become more homogeneous, caused by a strong mixing effect. The introduction of additional small tubes alters the velocity distribution of the fractal structure significantly and breaks the trapped vortices into smaller ones.
1850073-5
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al.
(a) Fig. 5
(b)
The velocity fields of fractal circular-tube bank for N = 3: (a) Re = 100 and (b) Re = 600.
(a) Fig. 6
(b)
The velocity fields of fractal circular-tube bank for N = 4: (a) Re = 100 and (b) Re = 600.
Streamlines feature more crooked than those of the third stage configuration. Li and Yu18 derived a recursive model for the tortuosity in Sierpinski carpet in order to characterize the curvature of streamline. Higher tortuosity means strong transverse mixing effect, and also yield a large pressure loss. In another perspective, the fluid particles cost much pressure energy to go wandering across such a multiscale structure. For the fractal rectangular-tube bank, the velocity distribution is similar to that of the fractal circular-tube bank. Most of streamlines are more zigzag than those of the circular one due to shape
effect. The separation of the boundary layer is more severe.
4.2. Temperature Fields of Fractal Tube Banks In Figs. 7–9, temperature fields of both kinds of structures are compared at different Re. The fluid temperature distribution continues to increase from the inlet to the outlet. Especially for the fourth stage fractal at Re = 100, the high temperature region covers a large part, as shown in Fig. 8a.
1850073-6
October 1850073
15,
2018
16:15
0218-348X
Convection Heat Transfer Performance of the Fractal Tube Bank Under Cross Flow
(a) Fig. 7
(b)
The temperature fields of fractal circular-tube bank for N = 3: (a) Re = 100 and (b) Re = 600.
(a) Fig. 8
(b)
The temperature fields of fractal circular-tube bank for N = 4: (a) Re = 100 and (b) Re = 600.
Another interesting phenomenon merits our attention that the cold stream is distributed like some of fingers. It is attributed to the heating effect of multiscale circular tubes. Such a kind of phenomenon is very clearly in the third stage fractal structure. Thanks to this effect, the coolant could penetrate more deeply into multi-scale tube bank. As seen from the later calculation, the unique fingerlike phenomenon in fractal could promote the heat exchange rate efficiently.
4.3. The Pressure Drop and Heat Transfer Analysis of Tube Banks Both the pressure drop and heat transfer rate of the rectangular and circular tube bank are investigated
in detail. The variations of pressure drop with the Reynolds number are illustrated in Figs. 10a and 10b. The variation in the pressure drop trend is almost identical. For the fourth stage fractals, the pressure drop of the rectangular structure is almost twice as large as that of the circular one. According to the boundary layer theory, the boundary layer around a rectangular tube is easily separated from its surface as a fluid flow passes a rectangular tube. It yields high flow resistance. Whether the circular tube bank or rectangular tube bank one, the pressure drop increases gently with the Re for the first three stages but rapidly for the fourth stage. Just as the tube shape and Re Number, the influence of fractal stage on the flow resistance is
1850073-7
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al.
(a)
The temperature fields of fractal rectangular-tube bank at Re = 600: (a) N = 3 and (b) N = 4. 30 25
∆P(pa)
20
30
N=1 N=2 N=3 N=4
25 20
N=1 N=2 N=3 N=4
uniform 3×3 uniform 9×9 uniform 27×27
∆P(pa)
Fig. 9
(b)
15
15
10
10
5
5
0
0 -3
4.0x10
-3
-2
8.0x10 1.2x10 V(m/s)
4.0x10-3
-2
1.6x10
(a)
8.0x10-3 1.2x10-2 V(m/s)
1.6x10-2
(b)
Fig. 10 The pressure drop versus inflow velocity for different fractal stages: (a) fractal rectangular-tube bank and (b) fractal circular-tube bank.
very significant. The problem needs to be concerned that a multiscale structure may yield higher flow resistance. The tube banks discussed above could be also viewed as a kind of tubular heat exchanger. The inflow is assumed as the cool water that exchanges heat with the isothermal tube surface. It is an interesting question whether a fractal structure could enhance the heat dissipation effect or not. Figures 11a and 11b show the variation of total heat transfer rate versus Reynolds number. At the same Reynolds number, the total heat transfer increases with the fractal stage. Unexpectedly, the total heat transfer of rectangular structure and circular one differs slightly for the same fractal stage, though the overall heat transfer area of the former is larger than that of the latter. At small
Reynolds number, the global heat transfer rate of both increases only marginally with the fractal stage. However, for a larger Re flow, the heat transfer rate increases rapidly with the increasing fractal stage. This implies that the heat transfer argument effect of the fractal structure is pronounced in the range of high Re flow investigated. The uniform circular-tube banks, with the same specific area of the corresponding stage (N = 2, 3, 4), are plotted in Fig. 12. The overall heat transfer rate of uniform tube bank is also illustrated in Fig. 11b. It is evident that the fractal circular-tube bank shows better heat exchange performance than the uniform one. At Re = 100, the performance of the fractal structures and uniform ones differs slightly. With the increase of Re, the discrepancy becomes more and more significant. The fractal
1850073-8
15,
2018
16:15
0218-348X
Convection Heat Transfer Performance of the Fractal Tube Bank Under Cross Flow
4
8×10
10×104
N=1 N=2 N=3 N=4
8×104 Q(W)
10×104
6×104 Q(W)
October 1850073
4×104
N=1 N=2 N=3 N=4
uniform 3×3 uniform 9×9 uniform 27×27
6×104 4×104 2×104
2×104
0
0 -3
4.0x10
-3
8.0x10
-2
1.2x10 V(m/s)
4.0x10-3
-2
1.6x10
(a)
8.0x10-3 1.2x10-2 V(m/s)
1.6x10-2
(b)
Fig. 11 The total heat transfer quantity versus inflow velocity for different fractal stages: (a) fractal rectangular-tube bank and (b) fractal circular-tube bank.
(a) Fig. 12
(b)
(c)
Uniform tube banks: (a) N = 2, (b) N = 3 and (c) N = 4.
tube bank consists of different scales of tubes. As the coolant passing across such a multiscale structure, strong transverse mixing effect occurs that can boost up the heat transfer rate of system efficiently. With the increase in the fractal stage, the enhancing effect is more obvious at high Re flow investigated. By comprehensive comparisons of Figs. 10a, 10b and 11a, 11b, the heat transfer rate of rectangular structure and circular one is almost same, but the pressure drop of the former is twice as large as that of the latter. Hence, the fractal circular-tube bank shows a better overall performance. In other words, not all the fractal structures are favorable for the choice of the tubular heat exchanger. The optimal fractal structure (multiscale structure) is determined by the trade-off between the heat transfer rate and pressure drop. Moreover, above results indicate that the uniform tube bank is a practicable option for the low Re laminar flow while the fractal tube bank is worth considering for the high Re flow investigated. To understanding the fractal performance further, the relationships between Nu and Re, applicable to different stage fractals, are derived.
The general form is described as1 : ln N u = a · ln Re + b.
(12)
N u ∝ Rea ,
(13)
It follows where a is the exponent of the power function. The simulation results of Nu against Re on a logarithmic scale are presented in Figs. 13a and 13b for different fractal stages. The values of a and b of the heat transfer correlation, regressed by the leastsquare method, are listed in Tables 1 and 2. The Reynolds number exponent for the Nusselt number also increases with the increase in the fractal stage. Up to the fourth stage, a is about 0.8, indicating that Nu is almost linear with Re. This result is consistent with the work by Cai and Huai,1 but the Reynolds number investigated in their work ranged only from 1–10. From the point of view of the Field Synergy Principle,19,20 the intersection angle between the velocity vector and temperature gradient vector is important for the heat transfer performance. The convective heat transfer process would be enhanced by reducing the intersection angle,
1850073-9
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al. 7.0
7.0 N=1 N=2 N=3 N=4
6.0
5.0 LnNu
LnNu
5.0
6.0
4.0
3.0
2.0
2.0
5.0
5.5 LnRe
6.0
6.5
1.0 4.5
(a) Fig. 13
5.5 LnRe
6.0
6.5
Nu versus Re for different fractal stages: (a) fractal rectangular-tube bank and (b) fractal circular-tube bank.
N
Re
a
b
Correlation Coefficient
1 2 3 4
30 ≤ Re ≤ 600 30 ≤ Re ≤ 600 30 ≤ Re ≤ 600 30 ≤ Re ≤ 600
0.6808 0.5739 0.6126 0.9162
0.3146 1.2170 1.4181 −0.4982
0.9891 0.9904 0.9999 0.9992
Table 2 Heat Transfer Correlations Fitted for the Circular Tube Banks.
1 2 3 4 Uniform 27 × 27
5.0
(b)
Table 1 Heat Transfer Correlations Fitted for the Rectangular Tube Banks.
N
uniform 27×27
4.0
3.0
1.0 4.5
N=1 N=2 N=3 N=4
Re
a
b
Correlation Coefficient
30 ≤ Re ≤ 600 30 ≤ Re ≤ 600 30 ≤ Re ≤ 600 30 ≤ Re ≤ 600 30 ≤ Re ≤ 600
0.4569 0.4454 0.5514 0.8525 0.2189
1.6302 1.9666 1.8542 0.0429 0.4009
0.9914 0.9788 0.9984 0.9975 0.9809
defined as the synergy angle. It is observed from Figs. 8b and 9b that the velocity vector of the flow field is almost in the same direction as the temperature gradient of the fourth fractal tube bank at Re = 600. The best argument effect would be achieved. Actually, the Field Synergy Principle is a result, but not a cause. The strong flow distribution effect of the fourth stage fractal is just the essential mechanism. It needs further quantitative analysis in the future work. In Figs. 13a and 13b, the average Nu of the fourth stage fractal is even less than that of the second stage fractal at Re = 100, regardless of the tube shape. The temperature field of the fourth stage fractal tube bank at Re = 100 is shown in Fig. 8a.
There is a high temperature region in the left part of the graph. Because of the larger heat exchange area, the coolant would be easily heated at the low flow rate. Some of the heat exchange area will lose effectiveness. As a result, the average Nu of the fourth fractal structure is not very high at Re = 100. Up to Re = 300, the advantage of large heat exchange area for the fourth fractal brings into full play at high flow rate. Eventually, the average Nu of the fourth stage fractal starts to exceed that of the third stage at Re = 300. The finger-like phenomenon of thermal fluid also plays a role in the enhancement of heat transfer coefficient, especially for the third fractal structure. Due to this mechanism, more coolant can penetrate deeply into the tube bank and exchange heat with isothermal tube surfaces at different Re, as shown in Figs. 7a and 7b. The finger-like behavior will aggravate the flow disturbance that induces strong transverse mixing effect, and increase the heat transfer coefficient of the third stage tube bank greatly. However, for the fourth fractal tube bank, the finger-like phenomenon becomes weakened and appears only near the region of the entrance due to the heating effect of the fourth stage small tubes. At Re = 600, the large Nu is mainly attributed to the strong flow disturbance effect and the full play of the heat exchange area. In Fig. 13b, the Nusselt number of uniform tube bank is significantly lower than that of fractal one. It is due to the different definition of the Reynolds number. For a fractal tube bank, the characteristic size is defined by the maximum tube diameter while, for a uniform tube bank, the characteristic size defined by the tube diameter. As a result, it yields the large Nu discrepancy between the fractal tube bank and the uniform one. The difficulty of the Re definition of the fractal tube bank may make
1850073-10
October 1850073
15,
2018
16:15
0218-348X
Convection Heat Transfer Performance of the Fractal Tube Bank Under Cross Flow
comparison meaningless. To face this problem, the same inflow velocity is imposed as compared the heat transfer rate of both kinds of tube banks in the previous simulation.
5. CONCLUSION By the simulation, we analyze the flow characteristics and heat transfer performance of fractal tube banks. It is found that the heat transfer performance of a fractal tube bank is related to the fractal stage, the Reynolds number and tube shape. There are various mechanisms of heat transfer enhancement, including the strong flow disturbance of different scales of tubes, the finger-like phenomenon and the effective heat exchange areas. These mechanisms play different roles in the heat transfer performance in different conditions. Key findings from the study are as follows: (1) The velocity distribution varies significantly with the increase in the fractal stage. For the fourth stage fractal structure, The velocity fields become more homogeneous due to the strong fluid disturbance of different scale tubes. The cold stream is observed to be distributed like many fingers near the entrance, which is also attributed to the heating effect of multiscale tubes. (2) For the fourth stage fractals, the pressure drop of the rectangular structure is almost twice as large as that of the circular one. At small Reynolds numbers, the global heat transfer rate increases only marginally with the fractal stage. However, the heat transfer rate increases rapidly with an increasing fractal stage at larger Re investigated. The simulation suggests that the uniform tube bank is a practicable option at low Re while the fractal tube bank is worth considering for a high Re flow. (3) For the relationship between Nu and Re, the circular tube bank shows the same tendency as the rectangular tube one. The exponent of the Re term in the heat transfer correlation increases with the fractal stage. Almost the linear function of Nu on Re for the fourth stage fractal is found, which infers excellent heat exchange performance. This research is helpful to understand the heat transfer mechanism of the multi-scale structure in depth. The fractal tube bank could provide a better choice in the design of tubular heat exchanger
under certain conditions. In the future, quantitative analysis of temperature distribution and enhanced heat transfer efficiency need to be conducted to gain more insight into the heat transfer mechanism of the fractal tube bank. Some new kinds of methods applied in fractals might enlighten this problem.12,21 In these works, the transport process is related to structural parameters and the tortuosity dimension of fractal porous media. The Field Synergy conditions of different stage fractals are required to be investigated in detail. Moreover, the problem by what factors the heat transfer coefficient is affected is also very important and worth further effort for the fractal structure.
ACKNOWLEDGMENTS We gratefully acknowledge the financial support of the National Science Foundation of China (21406095), and the Jiangsu University of Science and Technology Foundation (35011001).
REFERENCES 1. W. Q. Tao, The Multi-Scale Numerical Simulation of Heat Transfer and Flow Problem (Science Press, Beijing, 2009). 2. Z. W. Huang and Y. H. Hwang, Review of natureinspired heat exchanger technology, Int. J. Refrig. 78 (2017) 1–17. 3. R. L. Webb and N. Kim, Principles of Enhanced Heat Transfer, 2nd edn. (Taylor & Francis, Roca, Boston, 2005). 4. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982). 5. B. M. Yu, P. Xu, M. Q. Zhou, J. C. Cai and Q. Zheng, Transport Physics in Fractal Porous Media (Science Press, Beijing, 2014), pp. 466–470. 6. P. M. Adler, Fractal porous media III: Transversal stokes flow through random and sierpinski carpets, Transp. Porous Media 3 (1988) 85–198. 7. P. Xu and S. X. Qiu, A novel analytical solution for gas diffusion in multi-scale fuel cell porous media, J. Power Sources 362 (2017) 73–79. 8. B. Q. Xiao, H. X. Chen, S. X. Xiao and J. C. Cai, Research on relative permeability of nanofibers with capillary pressure effect by means of Fractal–Monte Carlo technique, J. Nanosci. Nanotechnol. 17(7) (2017) 6811–6817. 9. B. Q. Xiao et al., Optimization of the fractal-like architecture of porous fibrous materials related to permeability, diffusion and thermal conductivity, Fractals 23(3) (2017) 1750030. 10. D. Chen et al., General regulation of air flow distribution characteristics within planar solid oxide
1850073-11
October 1850073
15,
2018
16:15
0218-348X
D. Zhang et al.
11. 12.
13.
14.
15.
16.
fuel cell stacks, ACS Energy Lett. 2 (2017) 319– 326. A. Zukauskas, Convective Heat Transfer in the Heat Exchanger (Science Press, 2014), pp. 466–470. J. Cai and X. L. Huai, Study on fluid–solid coupling heat transfer in fractal porous medium by Lattice Boltzmann method, Appl. Thermal Eng. 30 (2010) 715–723. B. Q. Xiao and B. M. Yu, A fractal model for heat transfer of nanofluids by convection in a pool, Phys. Lett. A 373 (2009) 4178–4181. S. G. Mavridou and D. G. Bouris, Numerical evaluation of a heat exchanger with inline tubes of different size for reduced fouling rates, Int. J. Heat Mass Transf. 55 (2012) 5185–5195. S. G. Mavridou and E. Konstandinidis, Experimental evaluation of pairs of inline tubes of different size as components for heat exchanger tube bundles, Int. J. Heat Mass Transf. 90 (2015) 280–290. T. Bello-Ochende and A. Bejan, Constructal multiscale cylinders in cross-flow, Heat Mass Transf. 48 (2005) 1373–1383.
17. W. Khan, J. Culham and M. Yovanovich, Convection heat transfer from tube banks in cross flow: Analytical approach, Int. J. Heat Mass Transf. 49 (2006) 4831–4838. 18. M. F. Mei and B. M. Yu, A hierarchical model for multi-fractal media, Fractals 18(1) (2010) 53–64. 19. J. H. Li and B. M. Yu, Tortuosity of flow paths through a Sierpinski carpet, Chin. Phys. Lett. 28 (2011) 034701. 20. X. W. Zhu and J. Q. Zhao, Improvement in field synergy principle: More rigorous application, better results, Int. J. Heat Mass Transf. 100 (2016) 347– 354. 21. Z. Y. Guo, W. Q. Tao and R. K. Shah, The field synergy (coordination) principle and its applications in enhancing single phase convective heat transfer, Int. J. Heat Mass Transf. 48 (2005) 1797–1807. 22. B. Q. Xiao et al., Fractal model for water flow through unsaturated porous rocks, Fractals 26(2) (2018) 1840015.
1850073-12
