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Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts
International Journal of Heat and Mass Transfer xxx (xxxx) xxx
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Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts Xilong Zhang, Bilong Liu ⇑, Jiang Liu, Xingang Wang, Hongbo Zhang School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao 266520, China
a r t i c l e
i n f o
Article history: Received 18 March 2019 Received in revised form 8 August 2019 Accepted 15 October 2019 Available online xxxx Keywords: Governing equations Parabolic duct Focal lengths Divergent type Field synergy theory
a b s t r a c t In this paper, the heat transfer and flow characteristics of parabolic ducts with different focal lengths (P = 120 mm, 140 mm, and 160 mm) are analyzed and compared with rectangular and trapezoidal ducts respectively. The governing equations in the curved ducts, including the mass conservation equation, momentum equation and energy conservation equation, are established. The results show that the order of the overall heat transfer performance of the three ducts from the best to worst is parabolic duct, trapezoidal duct and rectangular duct. The parabolic duct has the best overall enhanced heat transfer performance, which is 5.1% and 25.4% larger than that of the trapezoidal and the rectangular ducts respectively. This means that, although the divergent type structure reduces the flow velocity and thus weakens the intensity of convective heat transfer in the duct, it improves the downstream heat transfer area and reduces the pressure drop loss in the flow process, thus promoting the improvement of overall heat transfer performance. Besides, the temperature difference in the parabolic flow duct has the most uniform distribution, which is an important factor to improve and strengthen heat transfer performance. From the perspective of field synergy theory, the average field synergy angle decreases with the increase of focal length, and the field synergy becomes better. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Due to its advantages such as simple structure, low cost and high heat transfer performance, etc., the heat exchange surface of equal-section flow duct is widely used in plain fin-and-tube exchangers [1,2], and its structure diagram is shown in Fig. 1(a). However, there are many defects in the traditional equal-section flow duct structure: (1) the temperature difference is large at the entrance but small at the outlet, which is because the cooling medium is constantly heated in the flowing process. This limits the convective heat transfer capacity at the outlet region; (2) the pressure drop loss increases with the increase of the flow velocity of the cooling medium, and the loss increases parabola with the increase of flow velocity. In order to improve the above defects, a parabolic curved duct structure is proposed in the paper. The cross-sectional area of the flow duct perpendicular to the flow direction varies, and the flow duct from the inlet to the outlet is of a gradually expanding structure, as shown in Fig. 1(b). In our previous research [3], the heat transfer performance of trapezoidal flow duct had been discussed, and the trapezoidal duct is one of the special shapes of the curved flow duct. The results ⇑ Corresponding author. E-mail address: [email protected] (B. Liu).
showed that when the slope angle (b) in the trapezoidal duct changes from 0° to 40°, the overall heat transfer performance of the trapezoidal ducts are better than that of the rectangular duct. For different slope angle, the gohstda (Core Volume Goodness Factor) value of trapezoidal ducts is about 5–20% higher than that of rectangular duct. At the same time, the temperature difference in the trapezoidal duct is linearly distributed, which improves the heat transfer effect at the outlet region, as is shown in Fig. 1(c). A number of scientists have tested, simulated and analyzed many models through the studies that they have done in trapezoidal duct. For the first time, Falkner et al. [4] studied the flow field on the wedge surface by using similarity solution method and obtained the corresponding partial differential equation. Later, Hartree [5] and Stewartson [6] analyzed the solutions of the partial differential equations obtained by Falkner et al [4]. They found that when the angle of the wedge exceeds 0.199p, the separation of the boundary layer occurs on the wedge. A few years later, Schlichting [7] reviewed and analyzed various kinds of wedgeshaped ducts for the characteristics of boundary layer. Farhanieh et al. [8] numerically analyzed the heat transfer and flow characteristics of laminar convective heat transfer in trapezoidal cross-section flow ducts with different structural sizes. On the application side, Cur et al. [9] first proposed an Accelerated Flow Evaporator (AFE), which is a trapezoidal flow path structure with
https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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Nomenclature A cp dh E Fh h Sf Lf _ m P p DP tf T DT U
surface area [m2] specific heat capacity [J/kgK] hydraulic diameter [mm] fluid pumping power per unit surface area [W/m2] inlet height [mm] heat transfer coefficient [W/m2K] fin pitch [mm] fin length [mm] mass flux [kg/s] focal length [mm] pressure [Pa] pressure difference [Pa] fin thickness [m] temperature [K] temperature difference [K] velocity [m/s]
Dimensionless f friction factor proportionality constant in Newton’s second law of mogc tion, gc = 1 j colburn factor Nu Nusselt number
(a) Rectangular duct
Pr Re
Prandtl number Reynolds number
Greek letters heat transfer surface area to volume ratio [m2/m3] b slope angle [°] q air density [kg/m3] k heat conductivity [W/mK] l dynamic viscosity [kg/(ms)] go extended surface efficiency on one fluid side of the extended surface heat exchanger, dimensionless r ratio of free flow area to frontal area, dimensionless
a
Subscripts a air side A inlet air B outlet air C inlet water D outlet water Max maximum value w water side
(b) Cambered duct
(c) Trapezoidal duct [3] Fig. 1. Schematic of cambered duct and temperature difference distribution in trapezoidal duct. (a) rectangular duct; (b) cambered duct; (c) trapezoidal duct [3].
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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large inlets and small outlets. Under the influence of the contracted structure, the air velocity at the downstream part of the flow duct increases gradually, which improves the local convective heat transfer performance of the downstream region. However, the larger flow velocity also increased the pressure drop loss in AFE. Waltrich et al. [10] carried out experimental studies on nine different types of AFE samples with different structural parameters, and the results showed that the AFE required less pump power when compared with the uniform cross section evaporator at low heat transfer capacity. Therefore, in this application, the AFE has less volume and uses less material than the uniform cross section evaporator. However, the power consumption of the AFE increases exponentially at high heat transfer capacity, which limits the application of the accelerator. This is because the gradual contraction evaporator has large pressure drop loss at high Reynolds number (Re). Therefore, the divergent type curved flow duct proposed in this paper can better overcome the fact that the contraction duct cannot be applied to the occasion of high heat transfer capacity. Compared with the amount of research literature on trapezoidal duct, the research on cambered duct relatively less. Bahaidarah et al. [11] studied the volume of entropy production rates in sharp-edged corrugated channels. The results showed that the total entropy production increases with the increase of Re. However, when the Reynolds number ranges from 25 to 400, the total entropy production along the flow path direction is gradually reduced. Sarkar et al. [12] numerically studied the two-dimensional flow characteristics in a corrugated flow channel under different Reynolds numbers (100 < Re < 2123). In laminar flow and transition flow, Nusselt number, friction factor and Area Goodness Factor were used to compare and analyze the heat transfer and flow performance of the six structures at different amplitudes (0.05 mm, 0.075 mm and 0.1 mm) and wavelengths (0.5 mm and 1 mm). Ho et al. [13] experimentally studied the sinusoidal and pin fin surface runners of nine different structures, which was mainly used in the condenser to enhance the film condensation on the vertical plate. By comparing the two runner surfaces, it can be seen that the sinusoidal surface has the best heat transfer performance at the same p/l ratio. Wen et al. [14] used numerical methods to study the overall heat transfer performance of sinusoidal plate-fin heat exchangers based on fluid structure interaction (FSI) analysis. Analysis of the pressure distribution indicated that the maximum pressure value is mainly at the inlet and outlet of the fin, and the fluctuating pressure value reaches a peak value. However, the cambered ducts mentioned above all have equal sectional area and shape in the flow direction, which is different from the shape studied in this paper. As for the study of parabolic duct, it is basically blank. At the same time, in the wet condition in which the phase change occurs, the condensate in the cambered duct is more likely to flow out of the flow path, and the condensate is mainly affected by the component of gravity on the curved surface and the liquid film tensor. However, due to the limitations of the length of the article, research on phase change heat transfer will be explored in future research. Therefore, the significance of studying the cambered duct is that the duct can improve the temperature difference distribution, improve the efficiency of heat transfer, reduce the pressure drop and promote the discharge of condensate. In this paper, the parabolic duct is taken as an example to carry out experiments and simulation studies on the cambered duct. 2. Experimental system and data reduction 2.1. Physical model Experimental studies were carried out on three different ducts, namely, the rectangular duct, the trapezoidal duct and the parabolic duct. The three flow heat exchange units were all
manufactured by 6061 aluminum alloy, all of which had the same volume, the same width and the same inlet height. Fig. 2(a)–(c) are pictures of real products. Fig. 3(a)–(c) are schematic diagram of structural parameters for three different ducts respectively. Tables 1–3 are specific structural parameters of three different ducts. 2.2. Wind tunnel tests The wind tunnel testbed used to conduct the testing of three different ducts is shown in Fig. 4. The testbed was mainly composed of an air circulation system, a water circulation system, a test sample, a control unit and a data acquisition system. The range of Reynolds number in the experiment was from 1200 to 5100, which is mainly applicable to the service conditions of heavy vehicles. Water and air were adopted as cooling medium on the water and the air sides respectively in heat exchanger unit. The temperatures of inlet and outlet in the air circulation loop were measured using a T-type cooper-constantan thermocouple (calibrated accuracy of 0.1 °C) network. The number and placement of thermocouples followed the ASHRAE [15] standard recommended. The air flow rate was measured by an inclined pressure gauge and pitot tube. The pressure drop across the heat exchange unit was measured by a U-type pressure gauge. The hot water bath was heated by an equal power heating method [16]. This method can effectively reduce the heat balance time. The water pipe, test channel and test sample were covered with a layer of 10 mm insulation material for heat insulation. The mass flow rate on water side was measured using a pulse flow sensor with a test accuracy of 0.0025 l/s. Two pressure gauges were arranged on both sides of the heat exchange unit for measuring the pressure drop loss on the water side with an accuracy of 2 Pa. Two calibrated RTD sensors were used to measure temperature on water side with a measurement accuracy of ±0.05 °C. The uncertainty analysis for Colburn factor j and friction factor f can be obtained from the following general formula [17]:
dR ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 @R dv 1 @R dv 2 @R dv n þ þ þ @v 1 @v 2 @v n v1 v2 vn
ð1Þ In the above formula, R is a function of the independent variables vi (i = 1, 2, . . ., n). dvi (i = 1, 2, . . ., n) is the uncertainty of variable vi, and variables vi has the same probability. So, the uncertainty expressions for Colburn factor j and friction factor f can be written as:
@j ¼ j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 @h @q @u 2 @Pr þ þ þ q h u 3 Pr
@f ¼ f
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 @ DP @q @u þ þ 2 q u DP
ð2Þ
ð3Þ
By using the above error evaluation method, the uncertainty of Colburn factor j and friction factor f can be obtained to be 2.12% and 3.19% respectively. 2.3. Data reduction The rate of heat transfer (Q) was an average value of the heat transfer rate Qw and the heat transfer rate Qa, and was used to calculate the convective heat transfer coefficient (h) on the air side. The specific calculation formula as follows:
Q ¼ ðQ a þ Q w Þ=2
ð4Þ
_ a cp;a DT a Qa ¼ m
ð5Þ
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) Rectangular duct
(b) Trapezoidal duct
(c) Parabolic duct Fig. 2. Picture of real products for three ducts. (a) rectangular duct; (b) trapezoidal duct; (c) parabolic duct.
_ w cp;w DT w Qw ¼ m
ð6Þ
where DTw was water temperature difference, DTa was air temperature difference on air side. Convective heat transfer coefficient on air side can be described as
h ¼ Q=ðA DT ln Þ
ð7Þ
where DTln was mean logarithmic temperature difference, which can be expressed as
DT ln ¼
ðT C T B Þ ðT D T A Þ B ln TTDC T T A
ð8Þ
The turbulence intensity and flow condition in the three ducts were evaluated using the largest Reynolds number (Remax) in the duct, which can be described as
Remax ¼
Udh
l=q
ð9Þ
Nu ¼ Pr ¼
hdh k
ð13Þ
lc p
ð14Þ
k
JF factor, a comprehensive evaluation index, was adopted to evaluate the same flow duct and different structural parameters. The expression of JF factor is [19]
JF ¼
ji =j0 ðf i =f 0 Þ
1=3
ð15Þ
where the subscript ‘‘i” stands for different duct, for example, it stands for different focal lengths in the parabolic duct; ‘‘000 represents the reference duct with rectangular cross section. The larger the JF factor, the better the overall heat transfer performance. 3. Numerical model descriptions and numerical method
where
dh ¼ 4
where Nusselt number (Nu) and Prandtl number (Pr) can be described as
AC LC
ð10Þ
where AC was cross sectional area at any cross section, LC was wetted perimeter. The heat transfer and pressure drop were evaluated using j factor and f factor according to the structural parameters of different ducts. In plain fin-and-tube exchangers, j factor and f factor can be expressed by [18]
j¼
Nu Pr1=3 Re
ð11Þ
f ¼
dh DP 4Lf ð1=2Þqm u2
ð12Þ
3.1. Computational domain and boundary conditions Taking a parabolic duct as an example, the heat transfer calculation area and boundary conditions of the duct is as shown in Fig. 5. In the figure, the ‘‘Mass flow inlet” condition was applied to the inlet of the calculation area, and the initial surface temperature was 293.15 K; The ‘‘Pressure outlet” condition applied to the outlet; The ‘‘Wall” boundary condition applied to the surfaces of fins and parabolic duct, and the initial temperature of those surfaces were set as 320.15 K. The computation domain was extended 1.5 times of the fin length from the inlet so that a uniform velocity distribution can be assigned at the inlet. In order to form a fully developed boundary layer in the flow channel, computational domain was extended downstream 5 times of the fin length [20].
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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coordinate system (s, n, z), s was parallel to the tangent of the surface, and n was parallel to the normal of the surface. The positive directions of the x and s axes were in the same direction as the air flow; the directions of the y and n axes were in the same direction as the fin height; and the z axis was in the same direction as the fin pitch. Some assumptions were as follows: (1) When the air flows through the cambered duct, the physical properties of the air remained unchanged. (2) Treating the flow of air as an incompressible steady-state turbulent flow. (3) The surfaces of the cambered duct and fins were ideally smooth surfaces (no burrs and surface abrupt changes) and were evenly arranged in the heat exchanger. (4) Ignoring heat radiation and natural convection heat transfer. (5) The temperature of the cambered duct and the surface of the fin was considered to be equal temperature.
(a) Rectangular duct
Through the translation and rotation of the coordinate system, the relationship between the two coordinate systems can be expressed as: 0
f x ¼ xO0 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 1þf
(b) Trapezoidal duct
ð16Þ
0
f y ¼ yO0 þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 1þf
ð17Þ
z ¼ zO0
ð18Þ
In the above three equations, xO’, yO’ and zO’ were the displacement of the coordinate system (s, n, z) along with the coordinate system (x, y, z) in the x axis, y axis and z axis respectively. Therefore, the governing equations in the cambered duct can be obtained as follows: (1) Continuity equation:
(c) Parabolic duct Fig. 3. Schematic diagram of structural parameters for three different ducts. (a) rectangular duct; (b) trapezoidal duct; (c) parabolic duct.
@u @ v @w þ ¼0 þ @s @n @z
ð19Þ
In Eq. (17), u, v, w were velocities in direction of axes x, y and z respectively.
The calculation areas of the three ducts were set as the same boundary condition and initial condition. During the simulation and test, the air mass flow on the air side were set as 0.48 kg/h, 0.6 kg/h, 0.72 kg/s, 0.9 kg/h, 1.2 kg/h, 1.5 kg/h and 1.8 kg/h. respectively.
(2) Momentum equations: s direction: R Rn
@p @s
Rl @ R þ Rn ½@s ðRn
@us Þ @s
2
R un ðRnÞ 2
3.2. Governing equations A schematic diagram of cambered surface (y = f (x)) and its coordinate system is shown in Fig. 6. Two coordinate systems, coordinate systems (x, y, z) and (s, n, z), were created in the figure. In
þ Rn R
@ 1 ðÞ @s R
@ 2 us @n2
@ 2 us @z2
þ Rn R
2R ðRnÞ 2
@un @s
1R
@us @n
ð20Þ
us ðRnÞ 2 ¼ 0
n direction: 2
2
Rl @ @p @ un @ un R @un n qg @n þ Rn ½@s ðRn Þ þ Rn þ Rn 1R @u R @n2 R @z2 @s @n 2
us R þ ðRnÞ 2
Table 1 Specific structural parameters of rectangular ducts.
@ 1 ðÞ @s R
2 þ Rn
@us @s
ð21Þ
us ðRnÞR ¼0
z direction:
No.
Lf /mm
Sf /mm
Fh /mm
tf /mm
1 2 3 4 5 6 7 8 9 10 11
60 60 60 55 55 55 45 45 45 45 45
3.0 3.5 4.0 3.0 3.5 4.0 3.0 3.5 4.0 3.0 3.0
9.0 9.0 9.0 9.0 9.0 9.0 8.0 8.0 8.0 9.0 11.0
0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3
@p Rl @ R @uz R n @ 2 uz R n @ 2 uz 1 @uz þ þ ½ ð Þþ ¼0 @z R n @s R n @s R @n2 R @z2 R @n ð22Þ
(3) Energy equation qcp uR @T k
Rn @s
R2 þ v @T þ w @T ¼ ðRnÞ 2 @n @z
@2 T @s2
2
2
Rn þ @@nT2 þ @@z2T ðRnÞ 3 n @R @T RðRnÞ @n @n
R Rn
@R @T @s @s
1 @T R @n
þ
ð23Þ
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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Table 2 Specific structural parameters of trapezoidal ducts. No.
Fh /mm
b/°
Sf /mm
Lf /mm
tf /mm
1 2 3 4 5 6 7 8 9
9.5 10.5 11.5 9.5 10.5 11.5 9.5 10.5 11.5
20 20 20 30 30 30 40 40 40
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
60 60 60 60 60 60 60 60 60
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
Table 3 Specific structural parameters of parabolic ducts. No.
Fh /mm
P /mm
Sf /mm
Lf /mm
tf /mm
1 2 3 4 5 6 7 8 9
9.5 10.5 11.5 9.5 10.5 11.5 9.5 10.5 11.5
120 120 120 140 140 140 160 160 160
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
60 60 60 60 60 60 60 60 60
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
Fig. 4. The schematic diagram of testbed.
where R was the radius of curvature at any point on the surface of the cambered duct. The subscript s, n and z represented the components of the velocity along the axes s, n and z respectively. 3.3. Meshing and numerical methodology 3.3.1. Grid generation technique and numerical method Taking the parabolic duct as an example, the duct is meshed by hexagonal structured grids which are shown in Fig. 7. In order to accurately capture the temperature gradient changed at the boundary layer and improved the simulation results, a dense grid
was used near the wall and fins of the cambered duct, and the grid at these locations was exponentially growing. The grid was divided by ANSYS ICEM CFD 14.5 software [21]. The governing equations were solved by CFD software ANSYS FLUENT 14.5 [22], and the RNG k-e turbulence model was used for calculation. The SIMPLER algorithm was used to solve finite-difference equations. Second order upwind scheme was chosen to discretize these governing equations for the steady-state flow. In the iterative calculation process, when the error between the two adjacent iterations in the continuity equation, the momentum equation and the energy equation can be kept within 106, the solution was considered to be convergent.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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that the maximum j factor and f factor deviations were 3% and 5.38% respectively, as shown in Fig. 8(a) and (b). As a consequence, the experimental data and the simulated values were in good agreement, which verified the correctness and reliability of the numerical methods used in this paper. 4. Results and discussion 4.1. Analysis and comparison of the overall heat transfer performance for parabolic duct
Fig. 5. The calculation areas and boundary conditions.
Fig. 6. Coordinate systems in cambered duct.
Fig. 7. Hexahedron grid in the duct.
3.3.2. Grid independence test and code validation The parabolic duct was used to verify the independence of the grid. The structural parameters of the duct were Fh = 9.5 mm, P = 120 mm, Sf = 4.0 mm, Lf = 60 mm, tf = 0.4 mm, and the Reynolds number ranges from 790 to 6000. Four sets of different grid numbers were used, which were 15 100 150, 20 120 200, 30 150 220 and 35 160 240, respectively. The results showed that when the number of grids was 30 150 220 and 35 160 240, the corresponding errors of the j and f factors were 1% and 2% respectively. Therefore, by weighing the calculation time and accuracy, it can be seen that the simulation requirement can be met while the number of grids was 30 150 220. Similarly, this method was still used for grid independence verification in trapezoidal and rectangular ducts. In order to verify the simulation method used in this paper, the simulation data was compared with the experimental data of the parabolic duct. The results showed
4.1.1. Performance study with different focal lengths Fig. 9(a) and (b) plot j factor and f factor versus Re for three different focal lengths respectively. It can be seen from the figure that for any parabolic parabola, the j factor and the f factor are gradually reduced as Re increases. When Re changes, as the focal length increases (from 120 mm to 140 mm, then from 140 mm to 160 mm), the magnitude of the increase of j and f factors is basically the same, that is, increasing the focal length does not cause drastic changes in j factor and f factor. Under the same Re, both the j factor and the f factor decrease as the focal length decreases. That is to say, a larger focal length has a promoting effect on heat transfer, but a larger focal length also brings a relatively large pressure drop. And when the focal length P = 160 mm, the duct has the largest j factor and the largest f factor. Compared with the focal length P = 160 mm, the j factor is reduced by 8.1% and 14.3% respectively while the focal length P = 120 mm and P = 140 mm respectively, and the f factor is reduced by 26.5% and 51.5% respectively. The JF factor is used to analyze the overall heat transfer performance in parabolic duct with different focal lengths, as shown in Fig. 9(c). It can be seen from the figure that under the same Re, the focal length P = 160 mm has the best overall heat transfer enhancement capability. That is to say, the larger the focal length, the better the heat transfer performance of the parabolic duct. This is because with the increase of the focal length, the curvature radius of the parabolic duct surface becomes smaller. This leads to the gradual reduction of the flow cross-section area in the duct, thus increasing the flow velocity of air and improving the local heat transfer coefficient. 4.1.2. Comparison of overall heat transfer performance of three different ducts The Core Volume Goodness Factor was used to evaluate the overall heat transfer performance of three different ducts including rectangular duct, trapezoidal duct and parabolic duct. The significance of using the Core Volume Goodness Factor is that it can compare the overall heat transfer performance of different duct with different structures and heat transfer areas. The Core Volume Goodness Factor was proposed by Shah and Sekulic´ [23] and its expression is
go hstd a ¼ Estd a ¼
cp l 4r g j Re Pr 2=3 o D2h
l 3 4r 3 f Re 2g c q2 D4h
ð24Þ
ð25Þ
In the above equation, gohstda is energy consumption per unit volume while the temperature changes by 1 K in a heat exchanger; Estda is the frictional power consumed per unit volume. The subscript ‘‘std” stands for the benchmark and follows the ARI standard [24]. From the perspective of the volume of the heat exchanger, in the case of the same Estda, the larger the gohstda, the smaller the volume of the heat exchanger used at the same heat transfer capacity.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) j factor
(b) f factor
Fig. 8. Comparison of experimental values with simulated values. (a) j factor; (b) f factor.
(a) j factor
(b) f factor
(c) JF factor Fig. 9. Performance comparison with different focal lengths. (a) j factor; (b) f factor. (c) JF factor.
Fig. 10 shows the trend of ‘‘gohstda” with ‘‘Estda” for three different ducts. It can be seen from the figure that the ‘‘gohstda” of the three ducts all increases as the ‘‘Estda” increases. Under the same ‘‘Estda”, the parabolic duct has the best overall heat transfer performance, while the rectangular has the worst one. Comparing the overall heat transfer performance of the three ducts, it is found that the parabolic, trapezoidal and rectangular duct are in order from
good to bad. In the entire Estda interval, gohstda in the parabolic duct is about 6–25.4% larger than that of in the rectangular. In fact, when Estda is maximum, the gohstda of the parabolic duct is about 5.1% and 25.4% larger than the values of the trapezoidal duct and rectangular duct respectively. This means that the parabolic duct has better overall heat transfer performance than the linearly divergent flow duct.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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Fig. 10. Comparison of core volume goodness factors of three ducts.
4.2. The effect of different focal lengths on temperature and velocity fields 4.2.1. Temperature field Taking Re = 3140 as an example, the parabolic ducts with focal lengths of 120 mm, 140 mm and 160 mm are numerically analyzed, and the selected plane for exhibition is the middle position of two adjacent fins. The temperature field contour corresponding to different focal lengths are shown in Fig. 11(a). As can be seen
(a) Influence of different focal lengths on temperature field.
9
from the figure, for the three different focal lengths, the temperature in the main flow zone near the intermediate fluid domain is relatively low, and the temperature near the wall surface is higher. Therefore, in the parabolic duct, heat transfer near the wall surface plays a leading role. Comparing the three subgraphs in the figure, it can be found that the thickness of the thermal boundary layer gradually decreases during the increase of the focal length from 120 mm to 160 mm. This means that the larger the focal length, the smaller the thickness of the thermal boundary layer and the stronger the heat transfer performance. This is consistent with the results of the previous analysis. The main reason for this phenomenon is that the curvature radius of the parabolic channel surface decreases with the increase of the focal length. This makes the flow section area at the back of the channel gradually decrease, so that the flow rate of air becomes larger in the process of flow. The increase of the flow velocity in the duct improves the local convection heat transfer performance. Then, the velocity boundary layer becomes thinner with the increase of velocity. Also, the thermal boundary layer becomes thinner with the increase of focal length, which promotes the convection heat transfer. A comparison of the temperature difference between the inner wall surfaces and the average temperature of the fluid at the cross section for the three different ducts is shown in Fig. 11(b). It can be seen from the figure that the temperature difference in the rectangular duct varies greatly, and the change trend is parabolic. The temperature difference between the inlet and outlet is about 3.56 K. The temperature difference in the trapezoidal duct (b = 40°) has a certain degree of linearity, and the temperature difference between the inlet and the outlet is about 2.76 K. For the parabolic duct, the temperature difference distribution curve is a relatively straight line form and has better linearity than the trapezoidal duct. The temperature difference between the inlet and the outlet in the parabolic duct, taking P = 120 mm as an example, is 2.51 K. What’s more, with the increase of focal length, the temperature difference in the parabolic duct becomes more uniform. This indicates that there is a relatively more uniform temperature difference distribution in the parabolic duct, which has a positive effect on improving the heat transfer efficiency of the heat exchanger. 4.2.2. Velocity fields Fig. 12(a) is a velocity field contour corresponding to different focal lengths, and the selected plane for exhibition is the same as the temperature field. It can be seen from the figure that at three different focal lengths, the flow velocity in the main flow zone is relatively high, and the flow velocity near the wall surface is relatively low. This is because the air is affected by the viscous force on the wall when it flows at the wall surface. This is similar to the flow in a rectangular duct. In addition, by comparing the three subgraphs in the figure, it can be found that as the focal length increases, the thickness of the velocity boundary layer gradually becomes thinner. With the reduction of the focal length, the minimum focal length studied in this paper is P = 120 mm, there is no phenomenon of reflow or boundary layer separation (shown in Fig. 12(b)) inside the duct. 4.3. Analysis from the field synergy principle
f
(b) Comparison of temperature difference distribution of three different ducts Fig. 11. Temperature field and temperature difference distribution. (a) influence of different focal lengths on temperature field; (b) comparison of temperature difference distribution of three different ducts.
Using the field synergy theory [25], the synergy between the temperature field and the velocity field is quantitatively analyzed by calculating the angle between the velocity vector and the temperature gradient. The smaller the field synergy angle, the stronger the heat transfer enhancement performance. The relationship between the velocity vector and the temperature gradient, the local field synergy angle h, and the average field synergy angle (FSA) are defined as follows:
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) Velocity contour
(b) Velocity vector
Fig. 12. Influence of different focal lengths on velocity field. (a) velocity contour; (b) velocity vector. !
!
U rT ¼ j U j jgradTjcosh0 h ¼ cos1
ð26Þ
u @T þ v @T þ w @T @x @y @z
ð27Þ
!
j U j jgradTj
R 0 h dV FSA ¼ R dV
ð28Þ
In the above equations, u, v, w are the velocity of the fluid in the !
direction of the three coordinate axes x, y, z, U is the velocity vector, rT is the temperature gradient, and V is the tiny control body in the fluid region. Fig. 13(a) shows the local field synergy angle distribution in three parabolic ducts with different focal lengths at the same inlet air flow rate u = 4 m/s (Re = 3140), the selected plane position is the same as described above. It can be seen from the figure that the distribution of the local field synergy angles in the parabolic duct with three different focal lengths has a certain similarity. It can be concluded that different focal lengths do not change the similarity of the field coordination angle distribution in the duct. The local field synergy angle distribution in the parabolic duct is quite different from the distribution in the rectangular duct. Larger local
(a) The distribution of local field synergy angle
field synergy angles in the parabolic ducts occur near the wall surface, while smaller local field synergy angles are mainly distributed in the inlet and in the central main flow region. Moreover, the proportion of the smaller local field synergy angle area in the central mainstream area is also large, which indicates that the field synergy in the parabolic duct is much better than that of in the rectangular duct. It can be clearly seen from the figure that when the focal length is P = 120 mm, there is a large average field synergy angle, while the average field synergy angle at P = 140 mm and P = 160 mm is not easy to distinguish. Therefore, different focal length in the parabolic duct has a certain influence on the degree of field synergy. Fig. 13(b) shows the average field synergy angle in the entire calculation area for different Re and different focal lengths. It can be seen from the figure that under the same Re, as the focal length increases, the average field synergy angle gradually decreases. Therefore, the larger the focal length, the smaller the average field synergy angle and the better the field synergy. When Re < 3000, with the increase of Re, the average field synergy angle of three different focal lengths increases obviously. However, when Re > 3000, with the increase of Re, the increment of the average field synergy angle for three different focal lengths is relatively small. Therefore, increasing the Reynolds number to a certain extent does not have a large impact on the average field synergy angle.
(b) Comparison of average field synergy angle comparison
Fig. 13. Field synergy analysis. (a) the distribution of local field synergy angle; (b) comparison of average field synergy angle comparison.
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5. Conclusions In this paper, the heat transfer and flow characteristics in parabolic ducts are studied by means of experiment and simulation. Different focal lengths P (120 mm, 140 mm and 160 mm) in the ducts are analyzed. The Core Volume Goodness Factors are used to compare the ducts with rectangular, trapezoidal and parabolic structures. We conclude that: (1) The governing equation in cambered duct is established, including the mass conservation equation, the momentum equation and the energy conservation equation. The governing equations can be used to calculate the heat transfer and flow characteristics in the parabolic duct. (2) When the focal lengths range from 120 mm to 160 mm, the overall heat transfer performance is better as the focal length increases. Compared with the focal length of P = 160 mm, the j factor at 120 mm and 140 mm decreases by 8.1% and 14.3% respectively, and the f factor decreases by 26.5% and 51.5% respectively. (3) The Core Volume Goodness Factor is used to compare the overall heat transfer performance of three different ducts. It is concluded that the order of performance from superior to inferior is parabolic, trapezoidal and rectangular ducts. The gohstda in the parabolic duct is 25.4% larger than the value in the rectangular duct and 5.1% larger than the value in the trapezoidal duct. This indicates that the divergent structure reduces the flow velocity at the downstream region of the duct and thus reduces the intensity of convective heat transfer. However, the process of air flow is a supercharging process, which reduces the pressure drop loss and greatly improves the overall heat transfer performance. The divergent type curved flow duct proposes in this paper can better overcome the defects existing in contraction duct. (4) For the parabolic duct, the temperature difference distribution curve is a relatively straight line form and has better linearity than the trapezoidal duct. This indicates that there is a relatively more uniform temperature difference distribution in the parabolic duct, which has a positive effect on improving the heat transfer efficiency of the heat exchanger. (5) As the focal length increases, both the thermal boundary layer and the velocity boundary layer become thinner. Under the same Re, the larger the focal length, the smaller the average field synergy angle and the better the field synergy.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements
11
51575288), Taishan Scholar Program of Shandong (Grant No. ts201712054). References [1] H.K. Park, B.J. Chung, Optimal tip clearance in the laminar forced convection heat transfer of a finned plate in a square duct, Int. Commun. Heat Mass Transf. 63 (2015) 73–81. [2] H.Z. Yang, J. Wen, S.M. Wang, Y.Z. Li, Effect of fin types and Prandtl number on performance of plate-fin heat exchanger: Experimental and numerical assessment, Appl. Therm. Eng. 144 (2018) 726–735. [3] X. Zhang, Y. Wang, R. Jia, R. Wan, Investigation on heat transfer and flow characteristics of heat exchangers with different trapezoidal duct, Int. J. Heat. Mass Transf. 110 (2017) 863–872. [4] V.M. Falkner, S.W. Skan, Some approximate solutions of the boundary layer equations, Phil. Mag. 12 (1931) 865–896. [5] D.R. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equation of the boundary layer, Proc. Cambr. Phil. Soc. 33 (1937) 223–239. [6] K. Stewartson, Further solution of the Falkner-Skan equation, Proc. Cambr. Phil. Soc. 50 (1954) 454–465. [7] H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 1968. [8] B. Farhanieh, B. Sunden, Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts, Int. J. Numer. Meth. Fl. 13 (1991) 537– 559. [9] N.O. Cur, J.J. Anselmino, Evaporator for Home Refrigerator, US Patent No. 5157941, 1992. [10] P.J. Waltrich, J.R. Barbosa, C.L. Hermes, C. Melo, Air-side heat transfer and pressure drop characteristics of accelerated flow evaporators, Int. J. Refrig. 34 (2011) 484–497. [11] H.M. Bahaidarah, Entropy generation during fluid flow in sharp edge wavy channels with horizontal pitch, Adv. Mech. Eng. 8 (2016) 1–10. [12] M. Sarkar, S.B. Paramane, A. Sharma, Periodically fully developed heat and fluid flow characteristics in furrowed wavy channel, Heat transfer Eng. 38 (2016) 278–288. [13] J.Y. Ho, K.C. Leong, T.N. Wong, Filmwise condensation of steam on sinusoidal pin fin arrays: effects of fin height and fin pitch, Int. J. Heat. Mass Transf. 130 (2019) 1004–1015. [14] J. Wen, K. Li, C. Wang, X. Zhang, S. Wang, Optimization investigation on configuration parameters of sine wavy fin in plate-fin heat exchanger based on fluid structure interaction analysis, Int. J. Heat. Mass Transf. 131 (2019) 385– 402. [15] ASHRAE, Chapter 13, Handbook Fundamental, S1 Edition., ASHRAE, 1993, pp. 14–15. [16] X. Zhang, Y. Wang, P. Cang, R. Wang, Experimental investigation of thermal hydraulic performance of heat exchangers with different Reynolds numbers on both air-side and water-side, Appl. Therm. Eng. 99 (2016) 1331–1339. [17] R.J. Mofft, Describing the uncertainties in experiment results, Exp. Therm. Fluid Sci. 1 (1988) 3–17. [18] R.M. Manglik, A.E. Bergles, Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger, Exp. Therm. Fluid Sci. 10 (1995) 171–180. [19] K. Yakut, N. Alemdaroglu, I. Kotcioglu, C. Celik, Experimental investigation of thermal resistance of a heat sink with hexagonal fins, Appl. Therm. Eng. 26 (2006) 2262–2271. [20] Y.B. Tao, Y.L. He, J. Huang, Z.G. Wu, W.Q. Tao, Three-dimensional numerical study of wavy fin-and-tube heat exchangers and field synergy principle analysis, Int. J. Heat. Mass Transf. 50 (2007) 1163–1175. [21] ANSYS ICEM, CFD User Guide Release 14.5, ANSYS Inc., Canonsburg, PA, 2012. [22] ANSYS FLUENT, Solver Theory Guide Release 14.5, ANSYS Inc., Canonsburg, PA, 2012. [23] R.K. Shah, D.P. Sekulic´, Fundamentals of Heat Exchanger Design, John Wiley & Sons, New Jersey, 2003. [24] ARI Standard 410, Standard for Forced Circulation Air-Cooling and Air-Heating Coils, Air-conditioning and Refrigeration Institute, Arlington, VA, 1981. [25] Z.Y. Guo, D.Y. Li, B.X. Wang, A novel concept for convective heat transfer enhancement, Int. J. Heat. Mass Transf. 41 (1998) 2221–2225.
Financial support is provided by the National Natural Science Foundation of China (Grant No. 51806114, 11874034 and
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts Xilong Zhang, Bilong Liu ⇑, Jiang Liu, Xingang Wang, Hongbo Zhang School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao 266520, China
a r t i c l e
i n f o
Article history: Received 18 March 2019 Received in revised form 8 August 2019 Accepted 15 October 2019 Available online xxxx Keywords: Governing equations Parabolic duct Focal lengths Divergent type Field synergy theory
a b s t r a c t In this paper, the heat transfer and flow characteristics of parabolic ducts with different focal lengths (P = 120 mm, 140 mm, and 160 mm) are analyzed and compared with rectangular and trapezoidal ducts respectively. The governing equations in the curved ducts, including the mass conservation equation, momentum equation and energy conservation equation, are established. The results show that the order of the overall heat transfer performance of the three ducts from the best to worst is parabolic duct, trapezoidal duct and rectangular duct. The parabolic duct has the best overall enhanced heat transfer performance, which is 5.1% and 25.4% larger than that of the trapezoidal and the rectangular ducts respectively. This means that, although the divergent type structure reduces the flow velocity and thus weakens the intensity of convective heat transfer in the duct, it improves the downstream heat transfer area and reduces the pressure drop loss in the flow process, thus promoting the improvement of overall heat transfer performance. Besides, the temperature difference in the parabolic flow duct has the most uniform distribution, which is an important factor to improve and strengthen heat transfer performance. From the perspective of field synergy theory, the average field synergy angle decreases with the increase of focal length, and the field synergy becomes better. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Due to its advantages such as simple structure, low cost and high heat transfer performance, etc., the heat exchange surface of equal-section flow duct is widely used in plain fin-and-tube exchangers [1,2], and its structure diagram is shown in Fig. 1(a). However, there are many defects in the traditional equal-section flow duct structure: (1) the temperature difference is large at the entrance but small at the outlet, which is because the cooling medium is constantly heated in the flowing process. This limits the convective heat transfer capacity at the outlet region; (2) the pressure drop loss increases with the increase of the flow velocity of the cooling medium, and the loss increases parabola with the increase of flow velocity. In order to improve the above defects, a parabolic curved duct structure is proposed in the paper. The cross-sectional area of the flow duct perpendicular to the flow direction varies, and the flow duct from the inlet to the outlet is of a gradually expanding structure, as shown in Fig. 1(b). In our previous research [3], the heat transfer performance of trapezoidal flow duct had been discussed, and the trapezoidal duct is one of the special shapes of the curved flow duct. The results ⇑ Corresponding author. E-mail address: [email protected] (B. Liu).
showed that when the slope angle (b) in the trapezoidal duct changes from 0° to 40°, the overall heat transfer performance of the trapezoidal ducts are better than that of the rectangular duct. For different slope angle, the gohstda (Core Volume Goodness Factor) value of trapezoidal ducts is about 5–20% higher than that of rectangular duct. At the same time, the temperature difference in the trapezoidal duct is linearly distributed, which improves the heat transfer effect at the outlet region, as is shown in Fig. 1(c). A number of scientists have tested, simulated and analyzed many models through the studies that they have done in trapezoidal duct. For the first time, Falkner et al. [4] studied the flow field on the wedge surface by using similarity solution method and obtained the corresponding partial differential equation. Later, Hartree [5] and Stewartson [6] analyzed the solutions of the partial differential equations obtained by Falkner et al [4]. They found that when the angle of the wedge exceeds 0.199p, the separation of the boundary layer occurs on the wedge. A few years later, Schlichting [7] reviewed and analyzed various kinds of wedgeshaped ducts for the characteristics of boundary layer. Farhanieh et al. [8] numerically analyzed the heat transfer and flow characteristics of laminar convective heat transfer in trapezoidal cross-section flow ducts with different structural sizes. On the application side, Cur et al. [9] first proposed an Accelerated Flow Evaporator (AFE), which is a trapezoidal flow path structure with
https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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Nomenclature A cp dh E Fh h Sf Lf _ m P p DP tf T DT U
surface area [m2] specific heat capacity [J/kgK] hydraulic diameter [mm] fluid pumping power per unit surface area [W/m2] inlet height [mm] heat transfer coefficient [W/m2K] fin pitch [mm] fin length [mm] mass flux [kg/s] focal length [mm] pressure [Pa] pressure difference [Pa] fin thickness [m] temperature [K] temperature difference [K] velocity [m/s]
Dimensionless f friction factor proportionality constant in Newton’s second law of mogc tion, gc = 1 j colburn factor Nu Nusselt number
(a) Rectangular duct
Pr Re
Prandtl number Reynolds number
Greek letters heat transfer surface area to volume ratio [m2/m3] b slope angle [°] q air density [kg/m3] k heat conductivity [W/mK] l dynamic viscosity [kg/(ms)] go extended surface efficiency on one fluid side of the extended surface heat exchanger, dimensionless r ratio of free flow area to frontal area, dimensionless
a
Subscripts a air side A inlet air B outlet air C inlet water D outlet water Max maximum value w water side
(b) Cambered duct
(c) Trapezoidal duct [3] Fig. 1. Schematic of cambered duct and temperature difference distribution in trapezoidal duct. (a) rectangular duct; (b) cambered duct; (c) trapezoidal duct [3].
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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large inlets and small outlets. Under the influence of the contracted structure, the air velocity at the downstream part of the flow duct increases gradually, which improves the local convective heat transfer performance of the downstream region. However, the larger flow velocity also increased the pressure drop loss in AFE. Waltrich et al. [10] carried out experimental studies on nine different types of AFE samples with different structural parameters, and the results showed that the AFE required less pump power when compared with the uniform cross section evaporator at low heat transfer capacity. Therefore, in this application, the AFE has less volume and uses less material than the uniform cross section evaporator. However, the power consumption of the AFE increases exponentially at high heat transfer capacity, which limits the application of the accelerator. This is because the gradual contraction evaporator has large pressure drop loss at high Reynolds number (Re). Therefore, the divergent type curved flow duct proposed in this paper can better overcome the fact that the contraction duct cannot be applied to the occasion of high heat transfer capacity. Compared with the amount of research literature on trapezoidal duct, the research on cambered duct relatively less. Bahaidarah et al. [11] studied the volume of entropy production rates in sharp-edged corrugated channels. The results showed that the total entropy production increases with the increase of Re. However, when the Reynolds number ranges from 25 to 400, the total entropy production along the flow path direction is gradually reduced. Sarkar et al. [12] numerically studied the two-dimensional flow characteristics in a corrugated flow channel under different Reynolds numbers (100 < Re < 2123). In laminar flow and transition flow, Nusselt number, friction factor and Area Goodness Factor were used to compare and analyze the heat transfer and flow performance of the six structures at different amplitudes (0.05 mm, 0.075 mm and 0.1 mm) and wavelengths (0.5 mm and 1 mm). Ho et al. [13] experimentally studied the sinusoidal and pin fin surface runners of nine different structures, which was mainly used in the condenser to enhance the film condensation on the vertical plate. By comparing the two runner surfaces, it can be seen that the sinusoidal surface has the best heat transfer performance at the same p/l ratio. Wen et al. [14] used numerical methods to study the overall heat transfer performance of sinusoidal plate-fin heat exchangers based on fluid structure interaction (FSI) analysis. Analysis of the pressure distribution indicated that the maximum pressure value is mainly at the inlet and outlet of the fin, and the fluctuating pressure value reaches a peak value. However, the cambered ducts mentioned above all have equal sectional area and shape in the flow direction, which is different from the shape studied in this paper. As for the study of parabolic duct, it is basically blank. At the same time, in the wet condition in which the phase change occurs, the condensate in the cambered duct is more likely to flow out of the flow path, and the condensate is mainly affected by the component of gravity on the curved surface and the liquid film tensor. However, due to the limitations of the length of the article, research on phase change heat transfer will be explored in future research. Therefore, the significance of studying the cambered duct is that the duct can improve the temperature difference distribution, improve the efficiency of heat transfer, reduce the pressure drop and promote the discharge of condensate. In this paper, the parabolic duct is taken as an example to carry out experiments and simulation studies on the cambered duct. 2. Experimental system and data reduction 2.1. Physical model Experimental studies were carried out on three different ducts, namely, the rectangular duct, the trapezoidal duct and the parabolic duct. The three flow heat exchange units were all
manufactured by 6061 aluminum alloy, all of which had the same volume, the same width and the same inlet height. Fig. 2(a)–(c) are pictures of real products. Fig. 3(a)–(c) are schematic diagram of structural parameters for three different ducts respectively. Tables 1–3 are specific structural parameters of three different ducts. 2.2. Wind tunnel tests The wind tunnel testbed used to conduct the testing of three different ducts is shown in Fig. 4. The testbed was mainly composed of an air circulation system, a water circulation system, a test sample, a control unit and a data acquisition system. The range of Reynolds number in the experiment was from 1200 to 5100, which is mainly applicable to the service conditions of heavy vehicles. Water and air were adopted as cooling medium on the water and the air sides respectively in heat exchanger unit. The temperatures of inlet and outlet in the air circulation loop were measured using a T-type cooper-constantan thermocouple (calibrated accuracy of 0.1 °C) network. The number and placement of thermocouples followed the ASHRAE [15] standard recommended. The air flow rate was measured by an inclined pressure gauge and pitot tube. The pressure drop across the heat exchange unit was measured by a U-type pressure gauge. The hot water bath was heated by an equal power heating method [16]. This method can effectively reduce the heat balance time. The water pipe, test channel and test sample were covered with a layer of 10 mm insulation material for heat insulation. The mass flow rate on water side was measured using a pulse flow sensor with a test accuracy of 0.0025 l/s. Two pressure gauges were arranged on both sides of the heat exchange unit for measuring the pressure drop loss on the water side with an accuracy of 2 Pa. Two calibrated RTD sensors were used to measure temperature on water side with a measurement accuracy of ±0.05 °C. The uncertainty analysis for Colburn factor j and friction factor f can be obtained from the following general formula [17]:
dR ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 @R dv 1 @R dv 2 @R dv n þ þ þ @v 1 @v 2 @v n v1 v2 vn
ð1Þ In the above formula, R is a function of the independent variables vi (i = 1, 2, . . ., n). dvi (i = 1, 2, . . ., n) is the uncertainty of variable vi, and variables vi has the same probability. So, the uncertainty expressions for Colburn factor j and friction factor f can be written as:
@j ¼ j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 @h @q @u 2 @Pr þ þ þ q h u 3 Pr
@f ¼ f
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 @ DP @q @u þ þ 2 q u DP
ð2Þ
ð3Þ
By using the above error evaluation method, the uncertainty of Colburn factor j and friction factor f can be obtained to be 2.12% and 3.19% respectively. 2.3. Data reduction The rate of heat transfer (Q) was an average value of the heat transfer rate Qw and the heat transfer rate Qa, and was used to calculate the convective heat transfer coefficient (h) on the air side. The specific calculation formula as follows:
Q ¼ ðQ a þ Q w Þ=2
ð4Þ
_ a cp;a DT a Qa ¼ m
ð5Þ
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) Rectangular duct
(b) Trapezoidal duct
(c) Parabolic duct Fig. 2. Picture of real products for three ducts. (a) rectangular duct; (b) trapezoidal duct; (c) parabolic duct.
_ w cp;w DT w Qw ¼ m
ð6Þ
where DTw was water temperature difference, DTa was air temperature difference on air side. Convective heat transfer coefficient on air side can be described as
h ¼ Q=ðA DT ln Þ
ð7Þ
where DTln was mean logarithmic temperature difference, which can be expressed as
DT ln ¼
ðT C T B Þ ðT D T A Þ B ln TTDC T T A
ð8Þ
The turbulence intensity and flow condition in the three ducts were evaluated using the largest Reynolds number (Remax) in the duct, which can be described as
Remax ¼
Udh
l=q
ð9Þ
Nu ¼ Pr ¼
hdh k
ð13Þ
lc p
ð14Þ
k
JF factor, a comprehensive evaluation index, was adopted to evaluate the same flow duct and different structural parameters. The expression of JF factor is [19]
JF ¼
ji =j0 ðf i =f 0 Þ
1=3
ð15Þ
where the subscript ‘‘i” stands for different duct, for example, it stands for different focal lengths in the parabolic duct; ‘‘000 represents the reference duct with rectangular cross section. The larger the JF factor, the better the overall heat transfer performance. 3. Numerical model descriptions and numerical method
where
dh ¼ 4
where Nusselt number (Nu) and Prandtl number (Pr) can be described as
AC LC
ð10Þ
where AC was cross sectional area at any cross section, LC was wetted perimeter. The heat transfer and pressure drop were evaluated using j factor and f factor according to the structural parameters of different ducts. In plain fin-and-tube exchangers, j factor and f factor can be expressed by [18]
j¼
Nu Pr1=3 Re
ð11Þ
f ¼
dh DP 4Lf ð1=2Þqm u2
ð12Þ
3.1. Computational domain and boundary conditions Taking a parabolic duct as an example, the heat transfer calculation area and boundary conditions of the duct is as shown in Fig. 5. In the figure, the ‘‘Mass flow inlet” condition was applied to the inlet of the calculation area, and the initial surface temperature was 293.15 K; The ‘‘Pressure outlet” condition applied to the outlet; The ‘‘Wall” boundary condition applied to the surfaces of fins and parabolic duct, and the initial temperature of those surfaces were set as 320.15 K. The computation domain was extended 1.5 times of the fin length from the inlet so that a uniform velocity distribution can be assigned at the inlet. In order to form a fully developed boundary layer in the flow channel, computational domain was extended downstream 5 times of the fin length [20].
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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coordinate system (s, n, z), s was parallel to the tangent of the surface, and n was parallel to the normal of the surface. The positive directions of the x and s axes were in the same direction as the air flow; the directions of the y and n axes were in the same direction as the fin height; and the z axis was in the same direction as the fin pitch. Some assumptions were as follows: (1) When the air flows through the cambered duct, the physical properties of the air remained unchanged. (2) Treating the flow of air as an incompressible steady-state turbulent flow. (3) The surfaces of the cambered duct and fins were ideally smooth surfaces (no burrs and surface abrupt changes) and were evenly arranged in the heat exchanger. (4) Ignoring heat radiation and natural convection heat transfer. (5) The temperature of the cambered duct and the surface of the fin was considered to be equal temperature.
(a) Rectangular duct
Through the translation and rotation of the coordinate system, the relationship between the two coordinate systems can be expressed as: 0
f x ¼ xO0 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 1þf
(b) Trapezoidal duct
ð16Þ
0
f y ¼ yO0 þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 1þf
ð17Þ
z ¼ zO0
ð18Þ
In the above three equations, xO’, yO’ and zO’ were the displacement of the coordinate system (s, n, z) along with the coordinate system (x, y, z) in the x axis, y axis and z axis respectively. Therefore, the governing equations in the cambered duct can be obtained as follows: (1) Continuity equation:
(c) Parabolic duct Fig. 3. Schematic diagram of structural parameters for three different ducts. (a) rectangular duct; (b) trapezoidal duct; (c) parabolic duct.
@u @ v @w þ ¼0 þ @s @n @z
ð19Þ
In Eq. (17), u, v, w were velocities in direction of axes x, y and z respectively.
The calculation areas of the three ducts were set as the same boundary condition and initial condition. During the simulation and test, the air mass flow on the air side were set as 0.48 kg/h, 0.6 kg/h, 0.72 kg/s, 0.9 kg/h, 1.2 kg/h, 1.5 kg/h and 1.8 kg/h. respectively.
(2) Momentum equations: s direction: R Rn
@p @s
Rl @ R þ Rn ½@s ðRn
@us Þ @s
2
R un ðRnÞ 2
3.2. Governing equations A schematic diagram of cambered surface (y = f (x)) and its coordinate system is shown in Fig. 6. Two coordinate systems, coordinate systems (x, y, z) and (s, n, z), were created in the figure. In
þ Rn R
@ 1 ðÞ @s R
@ 2 us @n2
@ 2 us @z2
þ Rn R
2R ðRnÞ 2
@un @s
1R
@us @n
ð20Þ
us ðRnÞ 2 ¼ 0
n direction: 2
2
Rl @ @p @ un @ un R @un n qg @n þ Rn ½@s ðRn Þ þ Rn þ Rn 1R @u R @n2 R @z2 @s @n 2
us R þ ðRnÞ 2
Table 1 Specific structural parameters of rectangular ducts.
@ 1 ðÞ @s R
2 þ Rn
@us @s
ð21Þ
us ðRnÞR ¼0
z direction:
No.
Lf /mm
Sf /mm
Fh /mm
tf /mm
1 2 3 4 5 6 7 8 9 10 11
60 60 60 55 55 55 45 45 45 45 45
3.0 3.5 4.0 3.0 3.5 4.0 3.0 3.5 4.0 3.0 3.0
9.0 9.0 9.0 9.0 9.0 9.0 8.0 8.0 8.0 9.0 11.0
0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3
@p Rl @ R @uz R n @ 2 uz R n @ 2 uz 1 @uz þ þ ½ ð Þþ ¼0 @z R n @s R n @s R @n2 R @z2 R @n ð22Þ
(3) Energy equation qcp uR @T k
Rn @s
R2 þ v @T þ w @T ¼ ðRnÞ 2 @n @z
@2 T @s2
2
2
Rn þ @@nT2 þ @@z2T ðRnÞ 3 n @R @T RðRnÞ @n @n
R Rn
@R @T @s @s
1 @T R @n
þ
ð23Þ
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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X. Zhang et al. / International Journal of Heat and Mass Transfer xxx (xxxx) xxx
Table 2 Specific structural parameters of trapezoidal ducts. No.
Fh /mm
b/°
Sf /mm
Lf /mm
tf /mm
1 2 3 4 5 6 7 8 9
9.5 10.5 11.5 9.5 10.5 11.5 9.5 10.5 11.5
20 20 20 30 30 30 40 40 40
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
60 60 60 60 60 60 60 60 60
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
Table 3 Specific structural parameters of parabolic ducts. No.
Fh /mm
P /mm
Sf /mm
Lf /mm
tf /mm
1 2 3 4 5 6 7 8 9
9.5 10.5 11.5 9.5 10.5 11.5 9.5 10.5 11.5
120 120 120 140 140 140 160 160 160
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
60 60 60 60 60 60 60 60 60
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
Fig. 4. The schematic diagram of testbed.
where R was the radius of curvature at any point on the surface of the cambered duct. The subscript s, n and z represented the components of the velocity along the axes s, n and z respectively. 3.3. Meshing and numerical methodology 3.3.1. Grid generation technique and numerical method Taking the parabolic duct as an example, the duct is meshed by hexagonal structured grids which are shown in Fig. 7. In order to accurately capture the temperature gradient changed at the boundary layer and improved the simulation results, a dense grid
was used near the wall and fins of the cambered duct, and the grid at these locations was exponentially growing. The grid was divided by ANSYS ICEM CFD 14.5 software [21]. The governing equations were solved by CFD software ANSYS FLUENT 14.5 [22], and the RNG k-e turbulence model was used for calculation. The SIMPLER algorithm was used to solve finite-difference equations. Second order upwind scheme was chosen to discretize these governing equations for the steady-state flow. In the iterative calculation process, when the error between the two adjacent iterations in the continuity equation, the momentum equation and the energy equation can be kept within 106, the solution was considered to be convergent.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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7
that the maximum j factor and f factor deviations were 3% and 5.38% respectively, as shown in Fig. 8(a) and (b). As a consequence, the experimental data and the simulated values were in good agreement, which verified the correctness and reliability of the numerical methods used in this paper. 4. Results and discussion 4.1. Analysis and comparison of the overall heat transfer performance for parabolic duct
Fig. 5. The calculation areas and boundary conditions.
Fig. 6. Coordinate systems in cambered duct.
Fig. 7. Hexahedron grid in the duct.
3.3.2. Grid independence test and code validation The parabolic duct was used to verify the independence of the grid. The structural parameters of the duct were Fh = 9.5 mm, P = 120 mm, Sf = 4.0 mm, Lf = 60 mm, tf = 0.4 mm, and the Reynolds number ranges from 790 to 6000. Four sets of different grid numbers were used, which were 15 100 150, 20 120 200, 30 150 220 and 35 160 240, respectively. The results showed that when the number of grids was 30 150 220 and 35 160 240, the corresponding errors of the j and f factors were 1% and 2% respectively. Therefore, by weighing the calculation time and accuracy, it can be seen that the simulation requirement can be met while the number of grids was 30 150 220. Similarly, this method was still used for grid independence verification in trapezoidal and rectangular ducts. In order to verify the simulation method used in this paper, the simulation data was compared with the experimental data of the parabolic duct. The results showed
4.1.1. Performance study with different focal lengths Fig. 9(a) and (b) plot j factor and f factor versus Re for three different focal lengths respectively. It can be seen from the figure that for any parabolic parabola, the j factor and the f factor are gradually reduced as Re increases. When Re changes, as the focal length increases (from 120 mm to 140 mm, then from 140 mm to 160 mm), the magnitude of the increase of j and f factors is basically the same, that is, increasing the focal length does not cause drastic changes in j factor and f factor. Under the same Re, both the j factor and the f factor decrease as the focal length decreases. That is to say, a larger focal length has a promoting effect on heat transfer, but a larger focal length also brings a relatively large pressure drop. And when the focal length P = 160 mm, the duct has the largest j factor and the largest f factor. Compared with the focal length P = 160 mm, the j factor is reduced by 8.1% and 14.3% respectively while the focal length P = 120 mm and P = 140 mm respectively, and the f factor is reduced by 26.5% and 51.5% respectively. The JF factor is used to analyze the overall heat transfer performance in parabolic duct with different focal lengths, as shown in Fig. 9(c). It can be seen from the figure that under the same Re, the focal length P = 160 mm has the best overall heat transfer enhancement capability. That is to say, the larger the focal length, the better the heat transfer performance of the parabolic duct. This is because with the increase of the focal length, the curvature radius of the parabolic duct surface becomes smaller. This leads to the gradual reduction of the flow cross-section area in the duct, thus increasing the flow velocity of air and improving the local heat transfer coefficient. 4.1.2. Comparison of overall heat transfer performance of three different ducts The Core Volume Goodness Factor was used to evaluate the overall heat transfer performance of three different ducts including rectangular duct, trapezoidal duct and parabolic duct. The significance of using the Core Volume Goodness Factor is that it can compare the overall heat transfer performance of different duct with different structures and heat transfer areas. The Core Volume Goodness Factor was proposed by Shah and Sekulic´ [23] and its expression is
go hstd a ¼ Estd a ¼
cp l 4r g j Re Pr 2=3 o D2h
l 3 4r 3 f Re 2g c q2 D4h
ð24Þ
ð25Þ
In the above equation, gohstda is energy consumption per unit volume while the temperature changes by 1 K in a heat exchanger; Estda is the frictional power consumed per unit volume. The subscript ‘‘std” stands for the benchmark and follows the ARI standard [24]. From the perspective of the volume of the heat exchanger, in the case of the same Estda, the larger the gohstda, the smaller the volume of the heat exchanger used at the same heat transfer capacity.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) j factor
(b) f factor
Fig. 8. Comparison of experimental values with simulated values. (a) j factor; (b) f factor.
(a) j factor
(b) f factor
(c) JF factor Fig. 9. Performance comparison with different focal lengths. (a) j factor; (b) f factor. (c) JF factor.
Fig. 10 shows the trend of ‘‘gohstda” with ‘‘Estda” for three different ducts. It can be seen from the figure that the ‘‘gohstda” of the three ducts all increases as the ‘‘Estda” increases. Under the same ‘‘Estda”, the parabolic duct has the best overall heat transfer performance, while the rectangular has the worst one. Comparing the overall heat transfer performance of the three ducts, it is found that the parabolic, trapezoidal and rectangular duct are in order from
good to bad. In the entire Estda interval, gohstda in the parabolic duct is about 6–25.4% larger than that of in the rectangular. In fact, when Estda is maximum, the gohstda of the parabolic duct is about 5.1% and 25.4% larger than the values of the trapezoidal duct and rectangular duct respectively. This means that the parabolic duct has better overall heat transfer performance than the linearly divergent flow duct.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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Fig. 10. Comparison of core volume goodness factors of three ducts.
4.2. The effect of different focal lengths on temperature and velocity fields 4.2.1. Temperature field Taking Re = 3140 as an example, the parabolic ducts with focal lengths of 120 mm, 140 mm and 160 mm are numerically analyzed, and the selected plane for exhibition is the middle position of two adjacent fins. The temperature field contour corresponding to different focal lengths are shown in Fig. 11(a). As can be seen
(a) Influence of different focal lengths on temperature field.
9
from the figure, for the three different focal lengths, the temperature in the main flow zone near the intermediate fluid domain is relatively low, and the temperature near the wall surface is higher. Therefore, in the parabolic duct, heat transfer near the wall surface plays a leading role. Comparing the three subgraphs in the figure, it can be found that the thickness of the thermal boundary layer gradually decreases during the increase of the focal length from 120 mm to 160 mm. This means that the larger the focal length, the smaller the thickness of the thermal boundary layer and the stronger the heat transfer performance. This is consistent with the results of the previous analysis. The main reason for this phenomenon is that the curvature radius of the parabolic channel surface decreases with the increase of the focal length. This makes the flow section area at the back of the channel gradually decrease, so that the flow rate of air becomes larger in the process of flow. The increase of the flow velocity in the duct improves the local convection heat transfer performance. Then, the velocity boundary layer becomes thinner with the increase of velocity. Also, the thermal boundary layer becomes thinner with the increase of focal length, which promotes the convection heat transfer. A comparison of the temperature difference between the inner wall surfaces and the average temperature of the fluid at the cross section for the three different ducts is shown in Fig. 11(b). It can be seen from the figure that the temperature difference in the rectangular duct varies greatly, and the change trend is parabolic. The temperature difference between the inlet and outlet is about 3.56 K. The temperature difference in the trapezoidal duct (b = 40°) has a certain degree of linearity, and the temperature difference between the inlet and the outlet is about 2.76 K. For the parabolic duct, the temperature difference distribution curve is a relatively straight line form and has better linearity than the trapezoidal duct. The temperature difference between the inlet and the outlet in the parabolic duct, taking P = 120 mm as an example, is 2.51 K. What’s more, with the increase of focal length, the temperature difference in the parabolic duct becomes more uniform. This indicates that there is a relatively more uniform temperature difference distribution in the parabolic duct, which has a positive effect on improving the heat transfer efficiency of the heat exchanger. 4.2.2. Velocity fields Fig. 12(a) is a velocity field contour corresponding to different focal lengths, and the selected plane for exhibition is the same as the temperature field. It can be seen from the figure that at three different focal lengths, the flow velocity in the main flow zone is relatively high, and the flow velocity near the wall surface is relatively low. This is because the air is affected by the viscous force on the wall when it flows at the wall surface. This is similar to the flow in a rectangular duct. In addition, by comparing the three subgraphs in the figure, it can be found that as the focal length increases, the thickness of the velocity boundary layer gradually becomes thinner. With the reduction of the focal length, the minimum focal length studied in this paper is P = 120 mm, there is no phenomenon of reflow or boundary layer separation (shown in Fig. 12(b)) inside the duct. 4.3. Analysis from the field synergy principle
f
(b) Comparison of temperature difference distribution of three different ducts Fig. 11. Temperature field and temperature difference distribution. (a) influence of different focal lengths on temperature field; (b) comparison of temperature difference distribution of three different ducts.
Using the field synergy theory [25], the synergy between the temperature field and the velocity field is quantitatively analyzed by calculating the angle between the velocity vector and the temperature gradient. The smaller the field synergy angle, the stronger the heat transfer enhancement performance. The relationship between the velocity vector and the temperature gradient, the local field synergy angle h, and the average field synergy angle (FSA) are defined as follows:
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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(a) Velocity contour
(b) Velocity vector
Fig. 12. Influence of different focal lengths on velocity field. (a) velocity contour; (b) velocity vector. !
!
U rT ¼ j U j jgradTjcosh0 h ¼ cos1
ð26Þ
u @T þ v @T þ w @T @x @y @z
ð27Þ
!
j U j jgradTj
R 0 h dV FSA ¼ R dV
ð28Þ
In the above equations, u, v, w are the velocity of the fluid in the !
direction of the three coordinate axes x, y, z, U is the velocity vector, rT is the temperature gradient, and V is the tiny control body in the fluid region. Fig. 13(a) shows the local field synergy angle distribution in three parabolic ducts with different focal lengths at the same inlet air flow rate u = 4 m/s (Re = 3140), the selected plane position is the same as described above. It can be seen from the figure that the distribution of the local field synergy angles in the parabolic duct with three different focal lengths has a certain similarity. It can be concluded that different focal lengths do not change the similarity of the field coordination angle distribution in the duct. The local field synergy angle distribution in the parabolic duct is quite different from the distribution in the rectangular duct. Larger local
(a) The distribution of local field synergy angle
field synergy angles in the parabolic ducts occur near the wall surface, while smaller local field synergy angles are mainly distributed in the inlet and in the central main flow region. Moreover, the proportion of the smaller local field synergy angle area in the central mainstream area is also large, which indicates that the field synergy in the parabolic duct is much better than that of in the rectangular duct. It can be clearly seen from the figure that when the focal length is P = 120 mm, there is a large average field synergy angle, while the average field synergy angle at P = 140 mm and P = 160 mm is not easy to distinguish. Therefore, different focal length in the parabolic duct has a certain influence on the degree of field synergy. Fig. 13(b) shows the average field synergy angle in the entire calculation area for different Re and different focal lengths. It can be seen from the figure that under the same Re, as the focal length increases, the average field synergy angle gradually decreases. Therefore, the larger the focal length, the smaller the average field synergy angle and the better the field synergy. When Re < 3000, with the increase of Re, the average field synergy angle of three different focal lengths increases obviously. However, when Re > 3000, with the increase of Re, the increment of the average field synergy angle for three different focal lengths is relatively small. Therefore, increasing the Reynolds number to a certain extent does not have a large impact on the average field synergy angle.
(b) Comparison of average field synergy angle comparison
Fig. 13. Field synergy analysis. (a) the distribution of local field synergy angle; (b) comparison of average field synergy angle comparison.
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912
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5. Conclusions In this paper, the heat transfer and flow characteristics in parabolic ducts are studied by means of experiment and simulation. Different focal lengths P (120 mm, 140 mm and 160 mm) in the ducts are analyzed. The Core Volume Goodness Factors are used to compare the ducts with rectangular, trapezoidal and parabolic structures. We conclude that: (1) The governing equation in cambered duct is established, including the mass conservation equation, the momentum equation and the energy conservation equation. The governing equations can be used to calculate the heat transfer and flow characteristics in the parabolic duct. (2) When the focal lengths range from 120 mm to 160 mm, the overall heat transfer performance is better as the focal length increases. Compared with the focal length of P = 160 mm, the j factor at 120 mm and 140 mm decreases by 8.1% and 14.3% respectively, and the f factor decreases by 26.5% and 51.5% respectively. (3) The Core Volume Goodness Factor is used to compare the overall heat transfer performance of three different ducts. It is concluded that the order of performance from superior to inferior is parabolic, trapezoidal and rectangular ducts. The gohstda in the parabolic duct is 25.4% larger than the value in the rectangular duct and 5.1% larger than the value in the trapezoidal duct. This indicates that the divergent structure reduces the flow velocity at the downstream region of the duct and thus reduces the intensity of convective heat transfer. However, the process of air flow is a supercharging process, which reduces the pressure drop loss and greatly improves the overall heat transfer performance. The divergent type curved flow duct proposes in this paper can better overcome the defects existing in contraction duct. (4) For the parabolic duct, the temperature difference distribution curve is a relatively straight line form and has better linearity than the trapezoidal duct. This indicates that there is a relatively more uniform temperature difference distribution in the parabolic duct, which has a positive effect on improving the heat transfer efficiency of the heat exchanger. (5) As the focal length increases, both the thermal boundary layer and the velocity boundary layer become thinner. Under the same Re, the larger the focal length, the smaller the average field synergy angle and the better the field synergy.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements
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Financial support is provided by the National Natural Science Foundation of China (Grant No. 51806114, 11874034 and
Please cite this article as: X. Zhang, B. Liu, J. Liu et al., Experimental and numerical analysis of heat transfer and flow characteristics in parabolic ducts, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118912