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Numerical simulation of heat transfer and fluid flow characteristics of composite fin
International Journal of Heat and Mass Transfer 75 (2014) 414–424
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical simulation of heat transfer and fluid flow characteristics of composite fin Wu Xuehong ⇑, Zhang Wenhui, Gou Qiuping, Luo Zhiming, Lu Yanli School of Electromechanical Science and Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China
a r t i c l e
i n f o
Article history: Received 3 April 2013 Received in revised form 9 February 2014 Accepted 31 March 2014 Available online 26 April 2014 Keywords: Composite fin Entransy dissipation principle Enhanced heat transfer Numerical simulation
a b s t r a c t The composite fin is presented based on the advantage of longitudinal vortex generator and slit fin, respectively. The performance of air-side heat transfer and fluid flow is investigated by numerical simulation for Reynolds number ranging from Re = 304 to 2130. Stepwise approximation method is applied on the mesh generation for the irregular domains of delta winglets and slit fins. The mechanism for augmenting heat transfer is also analyzed based on the local fluid field, field synergy principle and entransy dissipation principle. The computational results show that some eddies are developed behind the X-shaped slit and delta winglet, which produce some disruptions to fluid flow and enhance heat transfer; compared with plain fin and slit fin, it shows the composite fin has better heat transfer performance. By applying on the field synergy principle and entransy dissipation principle to analyze the composite fin, the computational results show that composite fin can improve the synergy of temperature gradient and velocity fields, and its equivalent thermal resistance is smaller and its irreversibility of heat transfer is lower. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fin-and-tube heat exchangers are widely used in many fields, such as power engineering, petrochemical engineering, air-conditioning engineering and refrigeration system et al. The air-side thermal resistance generally is 5–10 times of the liquid, which significantly increases the energy consumption [1]. In order to improve the performance of fin-and-tube heat exchanger, enhanced fins surface including wavy fin, interrupted (slit) fin and dimpled fin have been developed. As an interrupted fin, slit fins are paid more attention because of the diversity of structure and complexity of principle. The experiment and numerical simulation had been widely used to investigate the heat transfer and fluid flow characteristics of slit fin. Nakayama and Xu experimentally studied the heat transfer coefficient for two-row staggered slit fin-and-tube heat exchanger, the result revealed that the heat transfer coefficient was about 78% higher than that of the plain fin [2]. Wang et al. also experimentally investigated the airside performance of slitfin-and-tube heat exchanger [3]. Yun and Lee applied the Taguchi method to systematically analyze the effect of various design parameters on the heat transfer and fluid flow characteristics of slit ⇑ Corresponding author. Tel.: +86 371 63556718. E-mail address: [email protected] (X. Wu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.087 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
fin. Kang and Kim [4] experimentally and Qu et al. [5] numerically investigated the location effect of X-strip fin on the heat transfer and pressure drop characteristics. Their investigated results indicated plain fin at front row and strip fin at rear row was more effective to enhance heat transfer than that of the whole strip fin under the same fan power. Li et al. analyzed the heat transfer and fluid flow characteristics of slit fin by numerical simulation, their conclusions showed that the proposed structure of slit fin based on ‘‘front coarse and rear dense’’ was optimized for the heat transfer and fluid flow characteristics [6]. In present, many studies focused on the longitudinal vortex generator (LVG, and different forms of wing-type longitudinal vortex generator had developed (shown in Fig. 1), which were used to mount on the fin surface with ‘‘common flow down’’ configuration or ‘‘common flow up’’ configuration, as shown in Fig. 2. In general, LVGs might involve three kinds of heat transfer mechanisms, such as destroying the boundary layers, producing swirl and flow destabilization, and lasting over long distances in the flow direction, so which had high heat transfer coefficient and low pressure loss. Fiebig et al. experimentally studied the effect of wing-type vortex generators on a three-row fin-and-tube heat exchanger, four configurations were tested for an inline and a staggered arrangement. The highest heat transfer enhancement due to the vortex generators was found with the inline tube arrangement, which increased 55–65% heat transfer and 20–45% the friction factor for Reynolds
X. Wu et al. / International Journal of Heat and Mass Transfer 75 (2014) 414–424
415
Nomenclature Ac Af Ao Cp Dc Fp f j h H l L Ls Nu Dp Pl Pt Pr Q ReDc Rh
area at minimum cross section, m2 fin surface area, m2 total area, m2 specific heat capacity, j/(kg k) tube outside diameter, mm fin pithc, mm friction factor Colbum j factor heat transfer coefficient,w/(m2 k);or of lvgs, mm height of slit, mm delta winglet length, mm fin length, mm width of slit, mm Nusselt number Pressure drop, Pa longitudinal tube pitch, mm transversal tube pitch, mm Prandtl number heat transfer rate ,w Reynolds number equivalent thermal resistance,(k m2)/w
Tw u um W
the
height
numbers from 600 to 2700. The corresponding increases for the staggered tube arrangement were found to be lower [7]. Zhu et al. [8] investigated influences of four types of longitudinal vortex generators (delta wing, rectangular wing, delta winglet pair and rectangular winglet pair) on heat transfer and flow loss, their results showed that the mean heat transfer rate could increase by 16–19% with the vortex generators for an area of channel wall which was 30 times larger than the vortex generator area. Torii et al. [9] experimentally investigated delta winglets in a plain fin-and-tube heat exchanger at a relatively low Reynolds number. The winglets were not only completely located on the upstream or downstream region of the tubes, but also placed more on the sides of the tubes. Their investigated results showed winglet type vortex generators could increase the overall heat transfer coefficient. Their results also revealed the augmentation was more important for the case of the common-flow-up configuration. The similar experimental study by Jaordar and Jacobi [10] and Sommers and Jacobi [11] also proved that delta winglet LVGs could improve the heat transfer rate in many thermal systems. Pesteei et al. [12] experimentally studied the effect of winglet location on heat transfer enhancement and pressure drop in the fin and tube heat exchangers. Allison and Dally [13] analyzed the effects of deltawinglet vortex generators on the performance of fin and tube radiator. Compared to a standard louver fin surface, experimental results showed that the winglet surface had 87% of the heat transfer capacity but only 53% of the pressure drop of the louver fin
tube wall temperature, K velocity, m/s velocity at minimum cross section, m/s the width of slit, mm
Greek symbols a attack angle of the delta winglet, deg. b field synergy angle, deg. df fin thickness, mm l dynamic viscosity, kg/(m s) h angle of slit, deg. gf fin efficiency go total efficiency k thermal conductivity, W/(m2 K) Subscripts a airside f fin in inlet m mean value out outlet
surface. Joardar and Jacobi also experimentally studied the effectiveness of a vortex-generator alternate-tube inline array of vortex generators. They found that the air-side heat transfer coefficient increased from 16.5% to 44% for the single-row winglet arrangement with a corresponding increase less than 12% in the pressure drop. For the three-row vortex generator array, the heat transfer coefficient increases with Reynolds number from 29.9% to 68.8% with a pressure drop penalty from 26% at Re = 960 to 87.5% at Re = 220. Their results indicated that vortex generator arrays could evidently enhance the performance of fin-and-tube heat exchanger [14]. Akbari et al. [15] studied the effects of two different configurations of delta-winglet pair vortex generators on heat transfer enhancement. Wu and Tao [16,17] applied the field synergy principle to analyze the mechanism of heat transfer enhancement in finand-tube heat exchanger in aligned arrangement with longitudinal vortex generator. Chu et al. [18] analyzed the heat transfer characteristics and fluid flow structure of fin-and-oval-tube heat exchangers with longitudinal vortex generators (LVGs). Their results showed that the average Nu for the three-row fin-andoval-tube heat exchanger with longitudinal vortex generators increased by 13.6–32.9% over the baseline case and the corresponding pressure loss increased by 29.2–40.6%. Lei et al. [19] investigated the effects of vortex generators on heat transfer and pressure drop of heat exchanger, they found that the delta-winglet vortex generator with an attack angle of 20° and an aspect ratio of 2 could provide the best integrated performance over the range of
Fig. 1. Different types of longitudinal vortex generators.
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(a) common flow dow configuration
(b) common flow up configuration Fig. 2. Configuration of winglet type vortex generator on the fin surface-tube bank.
Reynolds number computed. The Colburn j-factor of the optimal configuration increased by 35.1–45.2% with a corresponding increase of 19.3–34.5% in the friction factor. Tian et al. pointed that delta winglet generated a downstream main vortex and a corner vortex, which could enhance the heat transfer behind the tube and develop a short distance along the main-flow direction [20]. Chang et al. used the cross-averaged absolute vorticity flux in the main flow direction to specify the intensity of the secondary flow produced by vortex generators [21]. Wu and Tao [22] studied the fluid flow and heat transfer characteristics with two rows of tubes in different diameters using vortex generators. Zhou and Ye [23] experimentally investigated the performance of a pair of new vortex generators-curved trapezoidal winglet (CTW), their results showed that delta winglet pair was the best in laminar and transitional flow region, while curved trapezoidal winglet pair (CTWP) had the best thermo-hydraulic performance in fully turbulent region due to the streamlined configuration and then the low pressure drop, which indicated the advantages of using vortex generators for heat transfer enhancement. Min et al. [24] studied
turbulent flow and heat transfer characteristics in a channel with novel longitudinal vortex generators. Ahmed et al. [25] reviewed the application of vortex generators as approach of heat transfer augmentation. He and Zhang [26] reviewed the advance of heat transfer augmentation by longitudinal vortex generators. Wu et al. [27] analyzed two kinds of novel fin-tube surfaces with punched longitudinal vortex generator. Their investigated results showed that proposed fin could improve the heat transfer and reduce the pressure drop. Gong et al. [28] studied air-side heat transfer and fluid flow characteristics of wavy fin-and-tube heat exchanger punched with combined rectangular winglet pairs (CRWPs). Du et al. [29] investigated air-side flow and heat transfer characteristics of the wavy finned flat tube of direct air-cooled condensers in power plant by experimental method. Huisseune et al. [30,31] investigated the effect of punching delta winglet vortex generators into the louvered fin surface in the near wake region of each tube, their results showed that the compound heat exchanger could yield a higher heat transfer per unit volume. Simultaneously, they also investigated the effect of five design
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parameters on the thermal hydraulic performance of the compound heat exchanger [32]. The field synergy principle [16–18] and entropy production principle [33] were also used to analyze the longitudinal vortex generators, which discovered the enhancement mechanism primarily in theory. The aforesaid researches indicated that the slit fin could significantly increase heat transfer and different shapes of slit fins had different effect on heat transfer coefficient and pressure drop. However, these articles generally paid attention on shape of slit fin; there were not many works about composite fin with different shapes of slit. In this paper, a composite fin (Fig. 3) is proposed based on the advantage of longitudinal vortex generator and slit, which longitudinal vortex generators mounted around the first tube and slit mounted around the second tube. The performance of heat transfer and fluid flow is investigated by numerical simulation and the enhancement mechanism is analyzed by applying on the field synergy and entransy dissipation principle. 2. Physical and mathematical model 2.1. Physical model and computational domain The schematic diagram of the composite fin in staggered arrangement is shown in Fig. 4 with two rows of tubes along the flow direction. A pair of delta-winglets is punched on the surface behind the front tube. The strip slits are punched on the base sheet around the rear tube with the X-shape. Due to decrease the pressure loss, the height of delta winglet is smaller than half of the fin spacing [34]. The geometric dimensions of winglets and strips are listed in Table. 1. The material of tube and fin is copper and aluminium, respectively. In this paper, the x-direction is defined as flow direction, fin span wise direction is y-direction and fin thickness direction is z-direction. The computational domain is extended 1.5 times of the fin length at the entrance in order to obtain a uniform velocity distribution, and 9 times of the fin length at the outlet in order to ensure no recirculation at the outlet boundary. 2.2. Governing equations and boundary conditions
l
@uk @xi
¼0
ð2Þ
(3) Energy equation:
@ @ k @T ðqui TÞ ¼ @xi @xi C p @xi
ð3Þ
In this paper, the tube wall temperature is assumed constant because of the relatively high heat transfer coefficient on the inner wall and the high thermal conductivity of the cooper tube. The tube wall temperature is fixed at 318 K and the flow temperature of inlet is 308 K. At the inlet boundary, the uniform velocity and constant temperature is given. At the exit of downstream extension, the outflow boundary condition is set. The periodic boundary conditions are given at the upper and lower boundaries of the computational domain, and the front and back faces are defined as symmetric boundary conditions, no-slip conditions are given at fin surface region and tube surface region. The governing equations and boundary conditions are solved by a commercial CFD code. The SIMPLE algorithm is applied to deal with the coupling of pressure and velocity. The fin temperature can be obtained by coupling heat transfer amid fin and airflow. The second-order upwind scheme is used to discretize the convection terms, the central difference scheme is utilized to discretize the diffusion term. Numerical convergence can be accepted only when the residuals of the continuity, components of velocity are smaller than 106, and the residual of the energy is smaller than 108. For the composite fin, the fine grids in the coil are hardly generated due to the complexity of structure, especially in the deltawinglets and X-shape slits domain. The coarse grids and high skewing grids will affect the accuracy and convergence of numerical iterative process. Therefore, the stepwise approximation method is applied on the mesh generation for the irregular domains of delta winglets and slit fins to generate structured grids. The grids are shown in Fig. 5. The boundary layer grids (Fig. 6) are adopted because the velocity and temperature have drastic gradient around the tube wall. 3. Validation of grid independence and numerical model
The governing equations for three-dimensional, laminar, incompressible with constant properties, steady and no viscous dissipation can be expressed as following. (1) Continuity equation:
@ðqui Þ ¼0 @xi
(2) Momentum equation:
@p @ @ þ ðqui uk Þ @xk @xi @xi
ð1Þ
3.1. Grid independence In order to validate the accuracy of computational results, a careful check for the grid independence of the numerical solutions has been made. For this purpose, four different grids number are studied. The grids number are 894740, 1624920, 1782900 and 3998520 at ReDc = 1825 based on the tube outside diameter. The results of four sets of grid number are listed in Table. 2. The maximum relative errors in the Colbum factor j and friction factor f are less than 4.1% and 1.7%. Therefore, in order to obtain the accurate results and save computer resource, the grid of 1782900 is used for further computation in the current investigation. 3.2. Numerical model
Fig. 3. Composite fin.
In the present computation, ReDc number ranges from 314 to 2130 based on the tube outside diameter and the corresponding frontal air velocity ranges from 0.5 to 3.5 m/s. But if Re number is defined based on the Hydraulic diameter, ReDh ranges from 116 to 813. Where, Dh ¼ 4Ac L=A0 , Ac is area at minimum cross section, Ao is total area, L is depth of the heat exchanger in air flow direction. Certainly, due to large Re number, a question had produced as whether the steady state computational model was suitable. In
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Y
outlet
Dc
inlet
symmetry
l
Pt
L Pl
W
(a) Top View periodic Fp
Z H
f
h
X (b) Front view Fig. 4. Schematic of the composite fin.
Table 1 Geometric dimension of composite fin. Parameter
Symbol/unit
Value
Outside diameter of tube Longitudinal tube pitch Transverse tube pitch Number of tube row Fin length along flow direction Fin pitch Fin thickness Angle of slit pattern Slit width Slit height Slit length Attack angle of delta winglet Length of delta winglet Height of delta winglet LVGs position
Dc/mm Pl/mm Pt/mm n L/mm Fp/mm df/mm h/° W/mm H/mm LS/mm a/° l/mm h/mm Lp/mm
7 12.7 12 2 25.4 2 0.2 45 1 0.9 7 45 3.2 0.8 5.5
recent, some authors applied unsteady state model to compute heat transfer and fluid flow characteristics of fin and tube heat exchanger. Huisseune et al. [30,31] applied laminar and unsteady state model to compute heat transfer performance of a louvered fin heat exchanger by using delta winglet vortex generators and ReDh ranged from 140 to 1220. Xue and Min [35] compared the steady and unsteady model in corrugated channels; their investigated results showed that it was appropriate to apply steady models to predict the Nusselt number and friction factor when the flow reached periodic unsteady regime. He et al. [36] also compared the steady and unsteady model for plate fin and tube heat exchanger at
ReDc = 5000, they found that the difference of averaged Nusselt number of two models was only about 0.35%. Many numerical studies in the existing literatures adopted the steady and laminar models for plate fin-and-tube heat transfer surfaces when ReDc > 1000, and reasonably good results are obtained. In Liu et al. [37] the upper limit of ReDc is about 1800. Min and Webb [38], Comini and Groce [39] adopt laminar steady model, and the values of ReDc were both larger than 1000. Biswas et al. [40] adopted laminar steady model, the ReDh ranged from 400 to 800 and corresponding frontal air velocity ranges from 2 to 4 m/s. He et al. [41] applied laminar and steady state model to compute heat-transfer enhancement by punched winglet-type vortex generator arrays in fin-and-tube heat exchangers. Gong et al. [28] adopted the steady model for fin and tube heat exchanger, and the ReDc ranged from 1000 to 5000. So the steady model is adopted in present computation. To further enhance our consideration for the steady model computation, the present authors also performed the comparison between steady and unsteady computation for the case of ReDh = 1825 and 1782900 grids. In the unsteady model, second order time accuracy, 50 iterations per time step and 2 ms time steps are applied. At the same time, these time steps also allow the residuals to decrease below 106. It is found that the difference in the Nusslet number and friction factor f between these two models is only about 0.28% and 0.068%. 4. Parameter definition In order to present the simulation results, some characteristic and non-dimensional parameters defined as follows:
Fig. 5. Grid generation for composite fin.
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Fig. 6. Local grid.
Table 2 Results of different gird numbers. j
Relative error (%)
f
Relative error (%)
894740 1624920 1782900 3998520
0.0226 0.0222 0.0213 0.0214
1.8 4.1 0.47
0.115 0.115 0.113 0.114
0 1.7 0.8
ReDc ¼
qum Dc l
hDc k Q h¼ g0 A0 DT Nu ¼
Nu
Grid number
ð4Þ
0
ð7Þ
Af ð1 gf Þ A0 tanhðmr/Þ gf ¼ mr/
ð8Þ
1500
2000
2500
0.36 0.32 0.28 0.24 0.20 0.16 0.12 0.08 0.04
Plain fin Slit fin Composite fin
0
500
1000
1500 Re
2000
2500
Fig. 8. Friction factor f.
ð9Þ
where r = 0.5Dc, / ¼ ðReq =r 1Þ½1 þ 0:35 lnðReq =rÞ, Req =r ¼ 1:27X M =r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2t þ P 2l , m ¼ 2h=ðkf df Þ, Af is the
ðX L =X M 0:3Þ0:5 , X M ¼ Pt , X L ¼ 12
1000
Fig. 7. Nu number.
ð6Þ
where Tw is wall temperature, Tin is inlet temperature, Tout is outlet temperature. The total efficiency go and fin efficiency gf are described as:
g0 ¼ 1
500
Re
f
ðT w T in Þ ðT w T out Þ ln½ðT w T in Þ=ðT w T out Þ
Plain fin Slit fin Composite fin
ð5Þ
where q is air density, um is the mean velocity at the minimum flow cross-sectional area Ac, Dc is the outside diameter, l is the dynamic viscosity, k is the thermal conductivity, h is the heat transfer coefficient, Q is the heat transfer rate. The log-mean temperature difference is defined as:
DT ¼
44 40 36 32 28 24 20 16 12
fin surface area, A0 is the total surface area, kf is fin thermal conductivity. With Eqs. (6)–(9), an iterative procedure is needed to obtain the air-side heat transfer coefficient, h, and the surface efficiency, g0 .
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The Colbum factor j and friction factor f are expressed in the following equations to describe the heat transfer performance and pressure drop characteristics:
Nu
The Nu number and friction factor f at various Re number range from 304 to 2130 are shown in Figs. 7 and 8, respectively. It is seen that the Nu number and friction factor f of composite fin are 77.16–90.21% and 96%–176% higher than that of plain fin, respectively. At the same time, the Nu number of composite fin are 6–36% higher than that of slit fin, but the friction factor f of composite fin are 19–22% lower than that of slit fin.
2=3
hPr RePr1=3 qum cp Dp Ac 2Dp Ac ¼ f ¼1 2 A0 qu2m q u m A0 2
j¼
5.1. Heat transfer and flow friction of composite fin
¼
ð10Þ ð11Þ
where Pr is the Prandtl number, cp is the specific heat of air, Dp is the pressure drop.
5.2. Flow characteristics of composite fin Fig. 9 illustrate the isotherm patterns for plain fin and composite fin on the xy-plane (fin surface) at Re = 1521, respectively. The temperature of plain fin behind each tube is higher than that of composite fin, and the high temperature regions are bigger behind each tube than that of composite fin. Another, the temperature of plain fin is higher around the region of second tube. On the contrary, the temperature of composite fin is lower on these regions. These results show that the heat transfer rate of the composite fin is improved by the LVGs and X-shape slit in these regions of the first tube and second tube, respectively. In order to further understanding enhancement heat transfer mechanism of the composite fin, the local vector and streamlines are shown in Fig. 10. The Fig. 10(a) illustrate the local temperature contours and velocity vectors at the cross-sections. From the
5. Numerical results and discussions In this section, the fluid flow and heat transfer characteristic of composite fin are investigated by numerical simulation. The Nusselt number and friction factor of present fin are compared with the plain fin and slit fin. Slit fin is based on the ‘‘front coarse and rear dense’’ and four slits is set in the front and the slits in the rear are the same as composite fin. To further understanding the enhancement heat transfer mechanism of the composite fin, we focus on the local characteristic of fluid field; meanwhile, the field synergy principle and entransy dissipation principle are applied to validate the mechanism of enhancement heat transfer of composite fin.
(a)
(b) Fig. 9. Isotherm on the fin surface at Re = 1521.
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(b)
(a)
(c) Fig. 10. Local vector and streamlines.
5.3. The overall performance of composite fin The radio j/f1/3 called JF factor, which is an index in evaluating the overall performance of the fin [42], is presented in Fig. 12, where the superior performance of the composite fin can be clearly observed. Compare with the plain fin and slit fin, it is found that the overall performance of composite fin increases 26.18%–50.89% and
35
Composite Fin Plain Fin
30
ΔP/Pa
25 20 15 10 5 0
0
5
10
15
20
25
X Fig. 11. Pressure drop along the X direction at Re = 1521.
0.14 Plain fin Slit fin Composite fin
0.12 0.10
JF
Fig. 10(b), it is found that there are two longitudinal vortices behind the delta winglet, which rotating axes parallel to the main-flow direction. An acceleration flow is generated between the LVGs and tube because of the effect of LVGs. As a result, the fluid rushes into the wake zone and heat transfer is enhanced. Fig. 10(c) presents the temperature contours and velocity streamlines on the leading edge of the slit fin. Vortex is generated by flow separation along the side edge of slit fin, which enhances the heat transfer in the vicinity of tube and wake region. At the same time, the interrupted surfaces formed by the strip-slit can provide higher average heat transfer coefficients owing to destroy the development of boundary layer. These explain why the heat transfer performance of composite fin is improved by the LVGs in the wake regions and X-shape slit in the second tube domain. Fig. 11 presents the variation of pressure drop along the length. By carefully inspecting the figure, the following conclusions can be observed. Firstly, the difference of pressure drop at X 6 13 between the plain fin and composite fin is very small, which reveals that the LGVs can enhance the heat transfer accompanying with little pressure drop penalty. Secondly, the difference of pressure drop at X P 13 between the plain fin and composite fin is very large, which shows that the strip-slit fin can enhance the heat transfer accompanying with high pressure drop penalty. From the Figs. 9 and 11, it can be seen that the LVGs can decrease the temperature of fin but the pressure loss does not increase. However, the strip-slit fin can increase heat transfer but the pressure loss increase remarkably. These results indicate that the LGVs fin have better heat transfer rate under the same pressure drop than that of the strip-slit fin.
0.08 0.06 0.04 0
500
1000
1500 Re
2000
2500
Fig. 12. Comparison the overall performance.
45%–14%, respectively. Thus, it is doubtless that the composite fin is a better performance fin. 5.4. Analysis from field synergy principle The synergy angle is defined as an indication of the synergy degree between velocity and temperature field for the entire flow and heat transfer domain. It is found that for the ideal case, the
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β
Temperature/K
β
308 309 310 311 312 313 314 315 316 317 318 (a) Plain fin
β
β β Temperature/K
308 309 310 311 312 313 314 315 316 317 318 (b) Composite fin
50
92 90 88 86 84 82 80 78 76 74 72 70
45
Plain Fin Composite Fin
-1
35
2
40 30
Rh/K•m •W
β /°
Fig. 13. Isothermals and streamlines at Re = 1521.
Plain Fin Composite Fin
25 20 15 10
0
500
1000
1500
2000
2500
0
500
Re
1000
1500 Re
2000
2500
Fig. 15. Equivalent thermal resistance vs. Re number. Fig. 14. Comparison of field synergy angle.
angle should equal 0, and for most of the engineering heat transfer cases, its value is far more than 0, it shows a large room for the heat transfer enhancement study. In present study, the fin with smaller angle is better in the field synergy degree between velocity and temperature field, which denotes the better heat transfer enhancement [43–45].
For simplicity of the comparison, the local intersection angle, b’ is defined as follows: 0
b ¼
X
1
cos
! u gradT ! j u jjgradTj
! ð12Þ
And a domain averaged intersection angle of the fin area can be obtained by using numerical integration.
X. Wu et al. / International Journal of Heat and Mass Transfer 75 (2014) 414–424
P 0 bi;j;k dv i;j;k b¼ P dv i;j;k
ð13Þ
! where u is the velocity vector and dvi,j,k is the volume element of the control volume (i,j,k). To examine the synergy between the velocity and temperature gradient, the isothermals and streamlines of plain fin (a) and composite fin (b) at Re = 1521 are presented in Fig. 13. Compared to Fig. 13(a) and (b), the local velocities are more tend to parallel with isothermals and the intersection angle between the velocity and temperature gradient is smaller for composite fin behind the first tube and around the region of the second tube. Fig. 14 shows the computational results of the field synergy angle. From the figure, we can see that the composite fin has the smaller synergy angle than the plain one and can improve the synergy of temperature gradient and velocity fields. It is good to explain why the composite fin has higher heat transfer coefficients compared with the plain fin.
The concept of Entransy, Eh ¼ 12 Q v h T, has been proposed for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction. Entransy dissipation, /h ¼ q rT ¼ kðrTÞ2 , occurs during heat transfer processes and there exists the extremum principle of entransy dissipation as a measure of the heat transfer irreversibility. For a fixed boundary heat flux, the heat transfer process is optimized when the entransy dissipation is minimized; at the same time the heat transfer is optimized when the entransy dissipation is maximized for a fixed boundary temperature. The equivalent thermal resistance is defined based on the entransy dissipation, so that the minimum thermal resistance principle equals to the extremum principle of entransy dissipation. The equivalent thermal resistance can be used as a measure of the heat transfer irreversibility, the smaller the equivalent thermal resistance is, the lower the irreversibility is and the better the heat transfer is [46,47]. For heat exchanger problems, the steady energy conservation equation is:
qcp U rT ¼ r ðkrTÞ
ð14Þ
Multiplying Eq. (14) by the temperature T results in
T2 2
! ¼ r ðkT rTÞ kjrTj2
ð15Þ
The integration of Eq. (15) and by using Gauss’ theorem, we can get
Z Z
1 qcp T 2 Uds ¼ 2
A
Z Z
TkrTds
Z Z Z
kjrTj2 dV
/h ¼
v
kjrTj2 dV ¼
Z Z
TkrTds
A0
Z Z A
1 qcp T 2 Uds 2
ð17Þ
In the fin and tube heat exchanger, the terms on the right side of Eq. (17) can be defined as
Z Z
krTds ¼ Q ¼ qucp AðT out T in Þ
ð18Þ
1 1 qcp T 2 Uds ¼ qcp UAðT 2out T 2in Þ 2 2
ð19Þ
A0
Z Z
A
¼
/h Q
2
¼
T W qucp AðT out T in Þ þ 12 qucp AðT 2in T 2out Þ ½qucp AðT out T in Þ2
ðT W T in Þ þ ðT W T out Þ 2qucp AðT out T in Þ
ð21Þ
To further understanding the essence of enhanced heat transfer of composite fin, the equivalent thermal resistance versus Reynolds number is shown in Fig. 15. It can be seen from the figure that the equivalent thermal resistance decreases with the increase of Re number. The equivalent thermal resistance of composite fin is smaller than that of plain fin over a Reynolds number range of 314 6 Re 6 2130, which reveals that its irreversibility of heat transfer is lower. These explain why the composite fin can enhance the heat transfer.
In this paper, the composite fin is proposed, and the performance of heat transfer and fluid flow of composite fin is investigated by a three-dimensional numerical simulation. The numerical results are also analyzed from the view point of the field synergy principle and entransy dissipation principle. The conclusions are summarized as follows: (1) The composite fin is presented based on the advantage of longitudinal vortex generator and slit. Compared with plain fin and slit fin, the performance of heat transfer of composite fin can improve 77.16%–90.21% and 6–36%, and the Nu number increases observably with the increasing of Re number. The overall performance of composite fin can increase 26.18%–50.89% and 45%–14%, respectively. (2) The longitudinal vortices can be generated by delta-winglets and X-shaped slits, which can improve the heat transfer performance. The heat transfer and fluid flow characteristic of composite fin has outperformed the traditional slit fin and longitudinal vortex generator, which not only decreases the wake region size but also increases the flow velocity in the wake zone. (3) By applying on the field synergy principle and entransy dissipation principle to analyze the composite fin, the computational results show that composite fin can improve the synergy of temperature gradient and velocity fields, and its equivalent thermal resistance is smaller and its irreversibility of heat transfer is lower than that of plain fin. Conflict of interest
The second term on the right side of Eq. (16) is define as the entransy dissipation /h , so we can get
Z Z Z
Rh ¼
ð16Þ
v
A
According to the definition of the equivalent thermal resistance, the Eq. (21) can be obtained as follows
6. Conclusions
5.5. Analysis from entransy dissipation principle
qcp U r
423
Acknowledgements The present work is supported by the Project of National Natural Science Foundation of China (No. 21076200), Henan province and college cooperation projects (No. 092106000013), supported by Foundation for University Key Teacher of the Henan province (2012GGJS-115) and Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City (131PLJRC640). References
Substituting Eqs. (18) and (19) into Eq. (17) gives
1 /h ¼ T W qucp AðT out T in Þ þ qucp AðT 2in T 2out Þ 2
None declared.
ð20Þ
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical simulation of heat transfer and fluid flow characteristics of composite fin Wu Xuehong ⇑, Zhang Wenhui, Gou Qiuping, Luo Zhiming, Lu Yanli School of Electromechanical Science and Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China
a r t i c l e
i n f o
Article history: Received 3 April 2013 Received in revised form 9 February 2014 Accepted 31 March 2014 Available online 26 April 2014 Keywords: Composite fin Entransy dissipation principle Enhanced heat transfer Numerical simulation
a b s t r a c t The composite fin is presented based on the advantage of longitudinal vortex generator and slit fin, respectively. The performance of air-side heat transfer and fluid flow is investigated by numerical simulation for Reynolds number ranging from Re = 304 to 2130. Stepwise approximation method is applied on the mesh generation for the irregular domains of delta winglets and slit fins. The mechanism for augmenting heat transfer is also analyzed based on the local fluid field, field synergy principle and entransy dissipation principle. The computational results show that some eddies are developed behind the X-shaped slit and delta winglet, which produce some disruptions to fluid flow and enhance heat transfer; compared with plain fin and slit fin, it shows the composite fin has better heat transfer performance. By applying on the field synergy principle and entransy dissipation principle to analyze the composite fin, the computational results show that composite fin can improve the synergy of temperature gradient and velocity fields, and its equivalent thermal resistance is smaller and its irreversibility of heat transfer is lower. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fin-and-tube heat exchangers are widely used in many fields, such as power engineering, petrochemical engineering, air-conditioning engineering and refrigeration system et al. The air-side thermal resistance generally is 5–10 times of the liquid, which significantly increases the energy consumption [1]. In order to improve the performance of fin-and-tube heat exchanger, enhanced fins surface including wavy fin, interrupted (slit) fin and dimpled fin have been developed. As an interrupted fin, slit fins are paid more attention because of the diversity of structure and complexity of principle. The experiment and numerical simulation had been widely used to investigate the heat transfer and fluid flow characteristics of slit fin. Nakayama and Xu experimentally studied the heat transfer coefficient for two-row staggered slit fin-and-tube heat exchanger, the result revealed that the heat transfer coefficient was about 78% higher than that of the plain fin [2]. Wang et al. also experimentally investigated the airside performance of slitfin-and-tube heat exchanger [3]. Yun and Lee applied the Taguchi method to systematically analyze the effect of various design parameters on the heat transfer and fluid flow characteristics of slit ⇑ Corresponding author. Tel.: +86 371 63556718. E-mail address: [email protected] (X. Wu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.087 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
fin. Kang and Kim [4] experimentally and Qu et al. [5] numerically investigated the location effect of X-strip fin on the heat transfer and pressure drop characteristics. Their investigated results indicated plain fin at front row and strip fin at rear row was more effective to enhance heat transfer than that of the whole strip fin under the same fan power. Li et al. analyzed the heat transfer and fluid flow characteristics of slit fin by numerical simulation, their conclusions showed that the proposed structure of slit fin based on ‘‘front coarse and rear dense’’ was optimized for the heat transfer and fluid flow characteristics [6]. In present, many studies focused on the longitudinal vortex generator (LVG, and different forms of wing-type longitudinal vortex generator had developed (shown in Fig. 1), which were used to mount on the fin surface with ‘‘common flow down’’ configuration or ‘‘common flow up’’ configuration, as shown in Fig. 2. In general, LVGs might involve three kinds of heat transfer mechanisms, such as destroying the boundary layers, producing swirl and flow destabilization, and lasting over long distances in the flow direction, so which had high heat transfer coefficient and low pressure loss. Fiebig et al. experimentally studied the effect of wing-type vortex generators on a three-row fin-and-tube heat exchanger, four configurations were tested for an inline and a staggered arrangement. The highest heat transfer enhancement due to the vortex generators was found with the inline tube arrangement, which increased 55–65% heat transfer and 20–45% the friction factor for Reynolds
X. Wu et al. / International Journal of Heat and Mass Transfer 75 (2014) 414–424
415
Nomenclature Ac Af Ao Cp Dc Fp f j h H l L Ls Nu Dp Pl Pt Pr Q ReDc Rh
area at minimum cross section, m2 fin surface area, m2 total area, m2 specific heat capacity, j/(kg k) tube outside diameter, mm fin pithc, mm friction factor Colbum j factor heat transfer coefficient,w/(m2 k);or of lvgs, mm height of slit, mm delta winglet length, mm fin length, mm width of slit, mm Nusselt number Pressure drop, Pa longitudinal tube pitch, mm transversal tube pitch, mm Prandtl number heat transfer rate ,w Reynolds number equivalent thermal resistance,(k m2)/w
Tw u um W
the
height
numbers from 600 to 2700. The corresponding increases for the staggered tube arrangement were found to be lower [7]. Zhu et al. [8] investigated influences of four types of longitudinal vortex generators (delta wing, rectangular wing, delta winglet pair and rectangular winglet pair) on heat transfer and flow loss, their results showed that the mean heat transfer rate could increase by 16–19% with the vortex generators for an area of channel wall which was 30 times larger than the vortex generator area. Torii et al. [9] experimentally investigated delta winglets in a plain fin-and-tube heat exchanger at a relatively low Reynolds number. The winglets were not only completely located on the upstream or downstream region of the tubes, but also placed more on the sides of the tubes. Their investigated results showed winglet type vortex generators could increase the overall heat transfer coefficient. Their results also revealed the augmentation was more important for the case of the common-flow-up configuration. The similar experimental study by Jaordar and Jacobi [10] and Sommers and Jacobi [11] also proved that delta winglet LVGs could improve the heat transfer rate in many thermal systems. Pesteei et al. [12] experimentally studied the effect of winglet location on heat transfer enhancement and pressure drop in the fin and tube heat exchangers. Allison and Dally [13] analyzed the effects of deltawinglet vortex generators on the performance of fin and tube radiator. Compared to a standard louver fin surface, experimental results showed that the winglet surface had 87% of the heat transfer capacity but only 53% of the pressure drop of the louver fin
tube wall temperature, K velocity, m/s velocity at minimum cross section, m/s the width of slit, mm
Greek symbols a attack angle of the delta winglet, deg. b field synergy angle, deg. df fin thickness, mm l dynamic viscosity, kg/(m s) h angle of slit, deg. gf fin efficiency go total efficiency k thermal conductivity, W/(m2 K) Subscripts a airside f fin in inlet m mean value out outlet
surface. Joardar and Jacobi also experimentally studied the effectiveness of a vortex-generator alternate-tube inline array of vortex generators. They found that the air-side heat transfer coefficient increased from 16.5% to 44% for the single-row winglet arrangement with a corresponding increase less than 12% in the pressure drop. For the three-row vortex generator array, the heat transfer coefficient increases with Reynolds number from 29.9% to 68.8% with a pressure drop penalty from 26% at Re = 960 to 87.5% at Re = 220. Their results indicated that vortex generator arrays could evidently enhance the performance of fin-and-tube heat exchanger [14]. Akbari et al. [15] studied the effects of two different configurations of delta-winglet pair vortex generators on heat transfer enhancement. Wu and Tao [16,17] applied the field synergy principle to analyze the mechanism of heat transfer enhancement in finand-tube heat exchanger in aligned arrangement with longitudinal vortex generator. Chu et al. [18] analyzed the heat transfer characteristics and fluid flow structure of fin-and-oval-tube heat exchangers with longitudinal vortex generators (LVGs). Their results showed that the average Nu for the three-row fin-andoval-tube heat exchanger with longitudinal vortex generators increased by 13.6–32.9% over the baseline case and the corresponding pressure loss increased by 29.2–40.6%. Lei et al. [19] investigated the effects of vortex generators on heat transfer and pressure drop of heat exchanger, they found that the delta-winglet vortex generator with an attack angle of 20° and an aspect ratio of 2 could provide the best integrated performance over the range of
Fig. 1. Different types of longitudinal vortex generators.
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(a) common flow dow configuration
(b) common flow up configuration Fig. 2. Configuration of winglet type vortex generator on the fin surface-tube bank.
Reynolds number computed. The Colburn j-factor of the optimal configuration increased by 35.1–45.2% with a corresponding increase of 19.3–34.5% in the friction factor. Tian et al. pointed that delta winglet generated a downstream main vortex and a corner vortex, which could enhance the heat transfer behind the tube and develop a short distance along the main-flow direction [20]. Chang et al. used the cross-averaged absolute vorticity flux in the main flow direction to specify the intensity of the secondary flow produced by vortex generators [21]. Wu and Tao [22] studied the fluid flow and heat transfer characteristics with two rows of tubes in different diameters using vortex generators. Zhou and Ye [23] experimentally investigated the performance of a pair of new vortex generators-curved trapezoidal winglet (CTW), their results showed that delta winglet pair was the best in laminar and transitional flow region, while curved trapezoidal winglet pair (CTWP) had the best thermo-hydraulic performance in fully turbulent region due to the streamlined configuration and then the low pressure drop, which indicated the advantages of using vortex generators for heat transfer enhancement. Min et al. [24] studied
turbulent flow and heat transfer characteristics in a channel with novel longitudinal vortex generators. Ahmed et al. [25] reviewed the application of vortex generators as approach of heat transfer augmentation. He and Zhang [26] reviewed the advance of heat transfer augmentation by longitudinal vortex generators. Wu et al. [27] analyzed two kinds of novel fin-tube surfaces with punched longitudinal vortex generator. Their investigated results showed that proposed fin could improve the heat transfer and reduce the pressure drop. Gong et al. [28] studied air-side heat transfer and fluid flow characteristics of wavy fin-and-tube heat exchanger punched with combined rectangular winglet pairs (CRWPs). Du et al. [29] investigated air-side flow and heat transfer characteristics of the wavy finned flat tube of direct air-cooled condensers in power plant by experimental method. Huisseune et al. [30,31] investigated the effect of punching delta winglet vortex generators into the louvered fin surface in the near wake region of each tube, their results showed that the compound heat exchanger could yield a higher heat transfer per unit volume. Simultaneously, they also investigated the effect of five design
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parameters on the thermal hydraulic performance of the compound heat exchanger [32]. The field synergy principle [16–18] and entropy production principle [33] were also used to analyze the longitudinal vortex generators, which discovered the enhancement mechanism primarily in theory. The aforesaid researches indicated that the slit fin could significantly increase heat transfer and different shapes of slit fins had different effect on heat transfer coefficient and pressure drop. However, these articles generally paid attention on shape of slit fin; there were not many works about composite fin with different shapes of slit. In this paper, a composite fin (Fig. 3) is proposed based on the advantage of longitudinal vortex generator and slit, which longitudinal vortex generators mounted around the first tube and slit mounted around the second tube. The performance of heat transfer and fluid flow is investigated by numerical simulation and the enhancement mechanism is analyzed by applying on the field synergy and entransy dissipation principle. 2. Physical and mathematical model 2.1. Physical model and computational domain The schematic diagram of the composite fin in staggered arrangement is shown in Fig. 4 with two rows of tubes along the flow direction. A pair of delta-winglets is punched on the surface behind the front tube. The strip slits are punched on the base sheet around the rear tube with the X-shape. Due to decrease the pressure loss, the height of delta winglet is smaller than half of the fin spacing [34]. The geometric dimensions of winglets and strips are listed in Table. 1. The material of tube and fin is copper and aluminium, respectively. In this paper, the x-direction is defined as flow direction, fin span wise direction is y-direction and fin thickness direction is z-direction. The computational domain is extended 1.5 times of the fin length at the entrance in order to obtain a uniform velocity distribution, and 9 times of the fin length at the outlet in order to ensure no recirculation at the outlet boundary. 2.2. Governing equations and boundary conditions
l
@uk @xi
¼0
ð2Þ
(3) Energy equation:
@ @ k @T ðqui TÞ ¼ @xi @xi C p @xi
ð3Þ
In this paper, the tube wall temperature is assumed constant because of the relatively high heat transfer coefficient on the inner wall and the high thermal conductivity of the cooper tube. The tube wall temperature is fixed at 318 K and the flow temperature of inlet is 308 K. At the inlet boundary, the uniform velocity and constant temperature is given. At the exit of downstream extension, the outflow boundary condition is set. The periodic boundary conditions are given at the upper and lower boundaries of the computational domain, and the front and back faces are defined as symmetric boundary conditions, no-slip conditions are given at fin surface region and tube surface region. The governing equations and boundary conditions are solved by a commercial CFD code. The SIMPLE algorithm is applied to deal with the coupling of pressure and velocity. The fin temperature can be obtained by coupling heat transfer amid fin and airflow. The second-order upwind scheme is used to discretize the convection terms, the central difference scheme is utilized to discretize the diffusion term. Numerical convergence can be accepted only when the residuals of the continuity, components of velocity are smaller than 106, and the residual of the energy is smaller than 108. For the composite fin, the fine grids in the coil are hardly generated due to the complexity of structure, especially in the deltawinglets and X-shape slits domain. The coarse grids and high skewing grids will affect the accuracy and convergence of numerical iterative process. Therefore, the stepwise approximation method is applied on the mesh generation for the irregular domains of delta winglets and slit fins to generate structured grids. The grids are shown in Fig. 5. The boundary layer grids (Fig. 6) are adopted because the velocity and temperature have drastic gradient around the tube wall. 3. Validation of grid independence and numerical model
The governing equations for three-dimensional, laminar, incompressible with constant properties, steady and no viscous dissipation can be expressed as following. (1) Continuity equation:
@ðqui Þ ¼0 @xi
(2) Momentum equation:
@p @ @ þ ðqui uk Þ @xk @xi @xi
ð1Þ
3.1. Grid independence In order to validate the accuracy of computational results, a careful check for the grid independence of the numerical solutions has been made. For this purpose, four different grids number are studied. The grids number are 894740, 1624920, 1782900 and 3998520 at ReDc = 1825 based on the tube outside diameter. The results of four sets of grid number are listed in Table. 2. The maximum relative errors in the Colbum factor j and friction factor f are less than 4.1% and 1.7%. Therefore, in order to obtain the accurate results and save computer resource, the grid of 1782900 is used for further computation in the current investigation. 3.2. Numerical model
Fig. 3. Composite fin.
In the present computation, ReDc number ranges from 314 to 2130 based on the tube outside diameter and the corresponding frontal air velocity ranges from 0.5 to 3.5 m/s. But if Re number is defined based on the Hydraulic diameter, ReDh ranges from 116 to 813. Where, Dh ¼ 4Ac L=A0 , Ac is area at minimum cross section, Ao is total area, L is depth of the heat exchanger in air flow direction. Certainly, due to large Re number, a question had produced as whether the steady state computational model was suitable. In
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Y
outlet
Dc
inlet
symmetry
l
Pt
L Pl
W
(a) Top View periodic Fp
Z H
f
h
X (b) Front view Fig. 4. Schematic of the composite fin.
Table 1 Geometric dimension of composite fin. Parameter
Symbol/unit
Value
Outside diameter of tube Longitudinal tube pitch Transverse tube pitch Number of tube row Fin length along flow direction Fin pitch Fin thickness Angle of slit pattern Slit width Slit height Slit length Attack angle of delta winglet Length of delta winglet Height of delta winglet LVGs position
Dc/mm Pl/mm Pt/mm n L/mm Fp/mm df/mm h/° W/mm H/mm LS/mm a/° l/mm h/mm Lp/mm
7 12.7 12 2 25.4 2 0.2 45 1 0.9 7 45 3.2 0.8 5.5
recent, some authors applied unsteady state model to compute heat transfer and fluid flow characteristics of fin and tube heat exchanger. Huisseune et al. [30,31] applied laminar and unsteady state model to compute heat transfer performance of a louvered fin heat exchanger by using delta winglet vortex generators and ReDh ranged from 140 to 1220. Xue and Min [35] compared the steady and unsteady model in corrugated channels; their investigated results showed that it was appropriate to apply steady models to predict the Nusselt number and friction factor when the flow reached periodic unsteady regime. He et al. [36] also compared the steady and unsteady model for plate fin and tube heat exchanger at
ReDc = 5000, they found that the difference of averaged Nusselt number of two models was only about 0.35%. Many numerical studies in the existing literatures adopted the steady and laminar models for plate fin-and-tube heat transfer surfaces when ReDc > 1000, and reasonably good results are obtained. In Liu et al. [37] the upper limit of ReDc is about 1800. Min and Webb [38], Comini and Groce [39] adopt laminar steady model, and the values of ReDc were both larger than 1000. Biswas et al. [40] adopted laminar steady model, the ReDh ranged from 400 to 800 and corresponding frontal air velocity ranges from 2 to 4 m/s. He et al. [41] applied laminar and steady state model to compute heat-transfer enhancement by punched winglet-type vortex generator arrays in fin-and-tube heat exchangers. Gong et al. [28] adopted the steady model for fin and tube heat exchanger, and the ReDc ranged from 1000 to 5000. So the steady model is adopted in present computation. To further enhance our consideration for the steady model computation, the present authors also performed the comparison between steady and unsteady computation for the case of ReDh = 1825 and 1782900 grids. In the unsteady model, second order time accuracy, 50 iterations per time step and 2 ms time steps are applied. At the same time, these time steps also allow the residuals to decrease below 106. It is found that the difference in the Nusslet number and friction factor f between these two models is only about 0.28% and 0.068%. 4. Parameter definition In order to present the simulation results, some characteristic and non-dimensional parameters defined as follows:
Fig. 5. Grid generation for composite fin.
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Fig. 6. Local grid.
Table 2 Results of different gird numbers. j
Relative error (%)
f
Relative error (%)
894740 1624920 1782900 3998520
0.0226 0.0222 0.0213 0.0214
1.8 4.1 0.47
0.115 0.115 0.113 0.114
0 1.7 0.8
ReDc ¼
qum Dc l
hDc k Q h¼ g0 A0 DT Nu ¼
Nu
Grid number
ð4Þ
0
ð7Þ
Af ð1 gf Þ A0 tanhðmr/Þ gf ¼ mr/
ð8Þ
1500
2000
2500
0.36 0.32 0.28 0.24 0.20 0.16 0.12 0.08 0.04
Plain fin Slit fin Composite fin
0
500
1000
1500 Re
2000
2500
Fig. 8. Friction factor f.
ð9Þ
where r = 0.5Dc, / ¼ ðReq =r 1Þ½1 þ 0:35 lnðReq =rÞ, Req =r ¼ 1:27X M =r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2t þ P 2l , m ¼ 2h=ðkf df Þ, Af is the
ðX L =X M 0:3Þ0:5 , X M ¼ Pt , X L ¼ 12
1000
Fig. 7. Nu number.
ð6Þ
where Tw is wall temperature, Tin is inlet temperature, Tout is outlet temperature. The total efficiency go and fin efficiency gf are described as:
g0 ¼ 1
500
Re
f
ðT w T in Þ ðT w T out Þ ln½ðT w T in Þ=ðT w T out Þ
Plain fin Slit fin Composite fin
ð5Þ
where q is air density, um is the mean velocity at the minimum flow cross-sectional area Ac, Dc is the outside diameter, l is the dynamic viscosity, k is the thermal conductivity, h is the heat transfer coefficient, Q is the heat transfer rate. The log-mean temperature difference is defined as:
DT ¼
44 40 36 32 28 24 20 16 12
fin surface area, A0 is the total surface area, kf is fin thermal conductivity. With Eqs. (6)–(9), an iterative procedure is needed to obtain the air-side heat transfer coefficient, h, and the surface efficiency, g0 .
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The Colbum factor j and friction factor f are expressed in the following equations to describe the heat transfer performance and pressure drop characteristics:
Nu
The Nu number and friction factor f at various Re number range from 304 to 2130 are shown in Figs. 7 and 8, respectively. It is seen that the Nu number and friction factor f of composite fin are 77.16–90.21% and 96%–176% higher than that of plain fin, respectively. At the same time, the Nu number of composite fin are 6–36% higher than that of slit fin, but the friction factor f of composite fin are 19–22% lower than that of slit fin.
2=3
hPr RePr1=3 qum cp Dp Ac 2Dp Ac ¼ f ¼1 2 A0 qu2m q u m A0 2
j¼
5.1. Heat transfer and flow friction of composite fin
¼
ð10Þ ð11Þ
where Pr is the Prandtl number, cp is the specific heat of air, Dp is the pressure drop.
5.2. Flow characteristics of composite fin Fig. 9 illustrate the isotherm patterns for plain fin and composite fin on the xy-plane (fin surface) at Re = 1521, respectively. The temperature of plain fin behind each tube is higher than that of composite fin, and the high temperature regions are bigger behind each tube than that of composite fin. Another, the temperature of plain fin is higher around the region of second tube. On the contrary, the temperature of composite fin is lower on these regions. These results show that the heat transfer rate of the composite fin is improved by the LVGs and X-shape slit in these regions of the first tube and second tube, respectively. In order to further understanding enhancement heat transfer mechanism of the composite fin, the local vector and streamlines are shown in Fig. 10. The Fig. 10(a) illustrate the local temperature contours and velocity vectors at the cross-sections. From the
5. Numerical results and discussions In this section, the fluid flow and heat transfer characteristic of composite fin are investigated by numerical simulation. The Nusselt number and friction factor of present fin are compared with the plain fin and slit fin. Slit fin is based on the ‘‘front coarse and rear dense’’ and four slits is set in the front and the slits in the rear are the same as composite fin. To further understanding the enhancement heat transfer mechanism of the composite fin, we focus on the local characteristic of fluid field; meanwhile, the field synergy principle and entransy dissipation principle are applied to validate the mechanism of enhancement heat transfer of composite fin.
(a)
(b) Fig. 9. Isotherm on the fin surface at Re = 1521.
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(b)
(a)
(c) Fig. 10. Local vector and streamlines.
5.3. The overall performance of composite fin The radio j/f1/3 called JF factor, which is an index in evaluating the overall performance of the fin [42], is presented in Fig. 12, where the superior performance of the composite fin can be clearly observed. Compare with the plain fin and slit fin, it is found that the overall performance of composite fin increases 26.18%–50.89% and
35
Composite Fin Plain Fin
30
ΔP/Pa
25 20 15 10 5 0
0
5
10
15
20
25
X Fig. 11. Pressure drop along the X direction at Re = 1521.
0.14 Plain fin Slit fin Composite fin
0.12 0.10
JF
Fig. 10(b), it is found that there are two longitudinal vortices behind the delta winglet, which rotating axes parallel to the main-flow direction. An acceleration flow is generated between the LVGs and tube because of the effect of LVGs. As a result, the fluid rushes into the wake zone and heat transfer is enhanced. Fig. 10(c) presents the temperature contours and velocity streamlines on the leading edge of the slit fin. Vortex is generated by flow separation along the side edge of slit fin, which enhances the heat transfer in the vicinity of tube and wake region. At the same time, the interrupted surfaces formed by the strip-slit can provide higher average heat transfer coefficients owing to destroy the development of boundary layer. These explain why the heat transfer performance of composite fin is improved by the LVGs in the wake regions and X-shape slit in the second tube domain. Fig. 11 presents the variation of pressure drop along the length. By carefully inspecting the figure, the following conclusions can be observed. Firstly, the difference of pressure drop at X 6 13 between the plain fin and composite fin is very small, which reveals that the LGVs can enhance the heat transfer accompanying with little pressure drop penalty. Secondly, the difference of pressure drop at X P 13 between the plain fin and composite fin is very large, which shows that the strip-slit fin can enhance the heat transfer accompanying with high pressure drop penalty. From the Figs. 9 and 11, it can be seen that the LVGs can decrease the temperature of fin but the pressure loss does not increase. However, the strip-slit fin can increase heat transfer but the pressure loss increase remarkably. These results indicate that the LGVs fin have better heat transfer rate under the same pressure drop than that of the strip-slit fin.
0.08 0.06 0.04 0
500
1000
1500 Re
2000
2500
Fig. 12. Comparison the overall performance.
45%–14%, respectively. Thus, it is doubtless that the composite fin is a better performance fin. 5.4. Analysis from field synergy principle The synergy angle is defined as an indication of the synergy degree between velocity and temperature field for the entire flow and heat transfer domain. It is found that for the ideal case, the
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β
Temperature/K
β
308 309 310 311 312 313 314 315 316 317 318 (a) Plain fin
β
β β Temperature/K
308 309 310 311 312 313 314 315 316 317 318 (b) Composite fin
50
92 90 88 86 84 82 80 78 76 74 72 70
45
Plain Fin Composite Fin
-1
35
2
40 30
Rh/K•m •W
β /°
Fig. 13. Isothermals and streamlines at Re = 1521.
Plain Fin Composite Fin
25 20 15 10
0
500
1000
1500
2000
2500
0
500
Re
1000
1500 Re
2000
2500
Fig. 15. Equivalent thermal resistance vs. Re number. Fig. 14. Comparison of field synergy angle.
angle should equal 0, and for most of the engineering heat transfer cases, its value is far more than 0, it shows a large room for the heat transfer enhancement study. In present study, the fin with smaller angle is better in the field synergy degree between velocity and temperature field, which denotes the better heat transfer enhancement [43–45].
For simplicity of the comparison, the local intersection angle, b’ is defined as follows: 0
b ¼
X
1
cos
! u gradT ! j u jjgradTj
! ð12Þ
And a domain averaged intersection angle of the fin area can be obtained by using numerical integration.
X. Wu et al. / International Journal of Heat and Mass Transfer 75 (2014) 414–424
P 0 bi;j;k dv i;j;k b¼ P dv i;j;k
ð13Þ
! where u is the velocity vector and dvi,j,k is the volume element of the control volume (i,j,k). To examine the synergy between the velocity and temperature gradient, the isothermals and streamlines of plain fin (a) and composite fin (b) at Re = 1521 are presented in Fig. 13. Compared to Fig. 13(a) and (b), the local velocities are more tend to parallel with isothermals and the intersection angle between the velocity and temperature gradient is smaller for composite fin behind the first tube and around the region of the second tube. Fig. 14 shows the computational results of the field synergy angle. From the figure, we can see that the composite fin has the smaller synergy angle than the plain one and can improve the synergy of temperature gradient and velocity fields. It is good to explain why the composite fin has higher heat transfer coefficients compared with the plain fin.
The concept of Entransy, Eh ¼ 12 Q v h T, has been proposed for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction. Entransy dissipation, /h ¼ q rT ¼ kðrTÞ2 , occurs during heat transfer processes and there exists the extremum principle of entransy dissipation as a measure of the heat transfer irreversibility. For a fixed boundary heat flux, the heat transfer process is optimized when the entransy dissipation is minimized; at the same time the heat transfer is optimized when the entransy dissipation is maximized for a fixed boundary temperature. The equivalent thermal resistance is defined based on the entransy dissipation, so that the minimum thermal resistance principle equals to the extremum principle of entransy dissipation. The equivalent thermal resistance can be used as a measure of the heat transfer irreversibility, the smaller the equivalent thermal resistance is, the lower the irreversibility is and the better the heat transfer is [46,47]. For heat exchanger problems, the steady energy conservation equation is:
qcp U rT ¼ r ðkrTÞ
ð14Þ
Multiplying Eq. (14) by the temperature T results in
T2 2
! ¼ r ðkT rTÞ kjrTj2
ð15Þ
The integration of Eq. (15) and by using Gauss’ theorem, we can get
Z Z
1 qcp T 2 Uds ¼ 2
A
Z Z
TkrTds
Z Z Z
kjrTj2 dV
/h ¼
v
kjrTj2 dV ¼
Z Z
TkrTds
A0
Z Z A
1 qcp T 2 Uds 2
ð17Þ
In the fin and tube heat exchanger, the terms on the right side of Eq. (17) can be defined as
Z Z
krTds ¼ Q ¼ qucp AðT out T in Þ
ð18Þ
1 1 qcp T 2 Uds ¼ qcp UAðT 2out T 2in Þ 2 2
ð19Þ
A0
Z Z
A
¼
/h Q
2
¼
T W qucp AðT out T in Þ þ 12 qucp AðT 2in T 2out Þ ½qucp AðT out T in Þ2
ðT W T in Þ þ ðT W T out Þ 2qucp AðT out T in Þ
ð21Þ
To further understanding the essence of enhanced heat transfer of composite fin, the equivalent thermal resistance versus Reynolds number is shown in Fig. 15. It can be seen from the figure that the equivalent thermal resistance decreases with the increase of Re number. The equivalent thermal resistance of composite fin is smaller than that of plain fin over a Reynolds number range of 314 6 Re 6 2130, which reveals that its irreversibility of heat transfer is lower. These explain why the composite fin can enhance the heat transfer.
In this paper, the composite fin is proposed, and the performance of heat transfer and fluid flow of composite fin is investigated by a three-dimensional numerical simulation. The numerical results are also analyzed from the view point of the field synergy principle and entransy dissipation principle. The conclusions are summarized as follows: (1) The composite fin is presented based on the advantage of longitudinal vortex generator and slit. Compared with plain fin and slit fin, the performance of heat transfer of composite fin can improve 77.16%–90.21% and 6–36%, and the Nu number increases observably with the increasing of Re number. The overall performance of composite fin can increase 26.18%–50.89% and 45%–14%, respectively. (2) The longitudinal vortices can be generated by delta-winglets and X-shaped slits, which can improve the heat transfer performance. The heat transfer and fluid flow characteristic of composite fin has outperformed the traditional slit fin and longitudinal vortex generator, which not only decreases the wake region size but also increases the flow velocity in the wake zone. (3) By applying on the field synergy principle and entransy dissipation principle to analyze the composite fin, the computational results show that composite fin can improve the synergy of temperature gradient and velocity fields, and its equivalent thermal resistance is smaller and its irreversibility of heat transfer is lower than that of plain fin. Conflict of interest
The second term on the right side of Eq. (16) is define as the entransy dissipation /h , so we can get
Z Z Z
Rh ¼
ð16Þ
v
A
According to the definition of the equivalent thermal resistance, the Eq. (21) can be obtained as follows
6. Conclusions
5.5. Analysis from entransy dissipation principle
qcp U r
423
Acknowledgements The present work is supported by the Project of National Natural Science Foundation of China (No. 21076200), Henan province and college cooperation projects (No. 092106000013), supported by Foundation for University Key Teacher of the Henan province (2012GGJS-115) and Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City (131PLJRC640). References
Substituting Eqs. (18) and (19) into Eq. (17) gives
1 /h ¼ T W qucp AðT out T in Þ þ qucp AðT 2in T 2out Þ 2
None declared.
ð20Þ
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