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Analysis of heat transfer and flow characteristics in typical cambered ducts
International Journal of Thermal Sciences 150 (2020) 106226
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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Analysis of heat transfer and flow characteristics in typical cambered ducts Xilong Zhang, Yongliang Zhang *, Zunmin Liu, Jiang Liu School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao, 266520, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cambered duct Cosinoidal duct Peak region Temperature gradient Gradually expanding structure
The heat transfer and flow characteristics of the air-water cross flow over cambered ducts were experimental and numerical investigated. The sequence of their Core Volume Goodness Factor (CVGF) is cosinoidal, parabolic, circular, trapezoidal and rectangular ducts successively from superior to inferior. Cambered ducts have more uniform temperature difference distribution than the equal cross section duct, and it has the minimum tem perature difference in inlet and outlet of the cosinoidal duct. With the optimal overall heat transfer performance, the cosinoidal duct is superior to that of the rectangular duct by 7.3%–28.1%. In the cosinoidal duct, the smaller the amplitude is, the better the heat transfer performance is. The thickness of the thermal and velocity boundary layers adjacent to the wall surface decreases constantly with increased Reynolds number. In the near wall region, n ¼ 5um, the main heat transfer area is the peak and middle areas, but with weaker heat transfer performance in the trough region. Although gradually expanding cambered duct slows down the flow velocity, the structure form decreases the pressure drop loss during the flow process. While improving the convective heat exchange capa bility of the upstream, the heat transfer area of the downstream is also improved to boost the overall heat transfer performance.
1. Introduction Due to its simple structure, lower cost, superior heat transfer per formance and other advantages, the uniform cross section duct is applied widely to the plate-fin heat exchangers [1–3], with its structure sketch shown in Fig. 1(a). The flow pattern of the plate-fin heat ex changers is usually cross flow between two parallel plates with different constructions. This traditional duct structure has many defects: (1) Bigger temperature difference at the inlet but smaller one at the outlet. During cooling medium flowing, the temperature difference at the outlet decreases, which confines the convective heat exchange capability in the outlet area; (2) Accelerated flow velocity of the cooling medium in creases the pressure drop loss. With accelerated flow velocity, the pressure drop rises in the form of parabola. To improve the two main defects in uniform cross section ducts, this paper proposes the cambered duct structure, i.e. the transverse section area changes along the direc tion of cooling medium flow and the duct between the inlet and the outlet has the gradually expanding structure, with its sketch shown in Fig. 1(b). In our previous research investigation [4], relevant discussions have been conducted for the heat transfer performance of the trapezoidal duct which was a type of the cambered duct in a special shape. The results
indicated when the slope angle (β) in the trapezoidal duct ranges from 0 to 40� , the overall heat transfer performance of the trapezoidal duct was superior to that of the rectangular duct. Core Volume Goodness Factor (CVGF), which is proposed by Shah and Sekuli�c [5], is adopted to evaluate the overall heat transfer performance of five different ducts. The significance of adopting the CVGF is that it can compare the overall heat transfer performance of different duct structures under various heat transfer areas. The CVGF of trapezoidal duct was superior by 5–20% than that of rectangular duct. Meanwhile, the change of temperature difference between the inlet and outlet in the trapezoidal duct showed a linear distribution, which improved the heat transfer performance in the outlet area, with the schematic diagram of temperature difference shown in Fig. 1(c). Some analysis investigations were made in the theory, numerical and experiment for the analysis of the trapezoidal duct. During the appli cation, Cur and Anselmino [6] proposed an Accelerated Flow Evaporator (AFE) for the first time, which was a trapezoidal duct structure with a big inlet and a small outlet. During its flow in the evaporator, the flow ve locity of air at the downstream accelerated and improved the local convection heat transfer performance of the downstream. However, the accelerated flow evaporator increased more pressure drop loss. Waltrich et al. [7,8] conducted the experimental analysis of nine accelerated
* Corresponding author. E-mail address: [email protected] (Y. Zhang). https://doi.org/10.1016/j.ijthermalsci.2019.106226 Received 16 July 2019; Received in revised form 7 December 2019; Accepted 9 December 2019 Available online 20 December 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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evaporator had more pressure drop loss at high Reynolds number. Therefore, the gradually expanding cambered duct structure in the paper can overcome the practical application better when the reducing duct structure cannot be applied to the high heat transfer capability. There are relatively fewer analyses of the cambered duct. Poskas et al. [9] experimentally studied heat transfer performance for both convex and concave surfaces of helical ducts with curvature D/H ¼ 5–9 and width b/H ¼ 2–20. D (D ¼ 0.5(d1þd2)/sin2ϕ), b and H are mean curvature diameter, mean channel width and channel height
evaporator samples with different structural parameters. The results indicated that the accelerated evaporator, at a low heat transfer capa bility, required less pump power when comparing it with a reference evaporator (a straight evaporator). Therefore, in this practical applica tion, the accelerated evaporator had a smaller volume and consumes less material than the straight evaporator. However, with a high heat transfer capability, the accelerated evaporator showed exponential growth of power consumption, which confined practical applications of the accelerated evaporator. The reason behind it was that a reducing
Fig. 1. Schematic diagram of duct structure and distribution diagram of temperature difference. (a) Uniform cross section duct; (b) Cambered duct; (c) Temperature difference distribution in trapezoidal duct [4]. 2
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International Journal of Thermal Sciences 150 (2020) 106226
respectively, which is shown in Fig. 2. With both convex and concave surfaces heated, they found that the heat transfer from convex and concave walls increases by 20% than that of one sided heating. Bahai darah et al. [10] analyzed the volume entropy generation rate in the sharp edge wavy duct. The results indicated that the total entropy rises gradually with the increased Re. However, the total entropy production along the duct direction decreased gradually when Reynolds number ranged from 25 to 400. Sarkar et al. [11] and others analyzed the two-dimensional flow characteristics in the wavy duct under different Reynolds numbers (100 < Re < 2123). At the states of laminar and transient flows, the Nusselt number, friction factor and Area Goodness Factor were adopted to compare and analyze the heat transfer and flow performance in ducts of six structures under different amplitudes (0.05 mm, 0.075 mm and 0.1 mm) and wavelength (0.5 mm and 1 mm). In the research of Baik et al. [12], the thermal performance of the corrugated channel in a printed circuit heat exchanger (PCHE) and the effects of amplitude and periodic equivalence factors were numerically studied. The results indicated that the recirculating flow caused by the waviness can be ignored while the mass flow of CO2 was in the typical range. Matsubara et al. [13] numerically analyzed the heat transfer and flow characteristics in a curved duct with a radius ratio are 0.92. They concluded that the convection heat transfer performance on the outer wall of the curved duct was the strongest by the evaluation of the local Nusselt number. In addition, this enhanced heat transfer came from large-scale organized vortex, which exacerbated the lateral convection of the secondary flow, thereby increased the turbulent heat flux value. Direct Numerical Simulation (DNS) is a common and effective method in the various flow ducts, such as Vinuesa et al. [14], Marin et al. [15] and Vidal et al. [16] conducted numerical studies for rectangular, hexagonal and sinusoidal ducts respectively. As for cambered ducts studied in this paper, basically no relevant literature has been published. However, the cambered ducts are similar to a diffuser, which has been studied for years. Ghosh et al. [17]
optimized the shapes of diffusers with different lengths, and used a ge netic algorithm for the optimum design of plane symmetric diffusers. They found that not much difference has been noted in the optimum C values (pressure recovery factor) obtained by using different methods. � � The expression of the C is C ¼ Δp = 12ρf v2inl , where Δp is the gain in static pressure, ρf is the density of the fluid and υinl is the inlet velocity of the fluid flow. Chen et al. [18] used CFD method to study the effects of injectors with different structures and obtained optimal geometrical factors to maximize entrainment ratio. They concluded that the opti mum inclination of the mixing chamber is 14� ; the optimum diameter ratio between the mixing tube and the primary nozzle is 1.7. In fact, studies involving heat transfer in such diffusers are still limited. The meaning of analyzing the cambered duct is that the cambered duct can improve the distribution of temperature difference, raise the efficiency of heat transfer, reduce the pressure drop and accelerate the drainage of condensate liquid. Taking five different duct structures as an instance, this paper conducts experimental and numerical analysis of cambered ducts. 2. Experimental system and procedure 2.1. Physical model Experimental analysis has been carried out for five ducts including rectangular, trapezoidal, circular, parabolic and cosinoidal ducts respectively. Processed from 6061 aluminum alloy, the five heat ex change units do not have the same overall heat transfer area but have the same volume. Fig. 3 and Fig. 4 are physical and parameter drawings of five different duct structures respectively. Table 1, Table 2, Table 3, Table 4 and Table 5 are specific parameters of five different duct structures. A special processing method [19] was used to form the rectangular and cambered ducts. The processing method can obtain an integrated structure by drawing ducts in an aluminum alloy matter, thus can form a tube. And then squeeze fins on the tube instead of brazing and expanding. 2.2. Wind tunnel tests Fig. 5 shows the wind tunnel experiment bench for testing five different duct structures, which consists of the air circulating system, water circulating system, testing part, control unit and data collection system. During experiment, the Reynolds number on air side ranged from 1210 to 5080. The water flow rate is 1 kg/s, and the Reynolds numbers on water side are about 2200 in the five different ducts. Water and air served respectively as the work medium at the air and water sides. Temperature at the inlet and outlet in the air circulating loop adopted T cooper-constantan thermocouple (calibrated accuracy was 0.1 � C) network for measurement. The quantity and layout of the thermocouple follow the methods recommended by ASHRAE [20] standard. Air flux was measured by the inclined-tube manometer and pressure drops of the heat exchange unit measured by the U-type manometer. The equal power heating method [19] was adopted for heating the hot water, which can reduce heat balance time and improve heat balance. Water pipe, experimental channel and test unit adopted 10 mm heat insulation materials for insulation. The mass flux at the water side was measured by a pulse flux sensor, with 0.0025 l/s reso lution. Two manometers were arranged at both sides of the heat ex change unit for measuring the pressure drop loss at the water side, with 2Pa in its accuracy. Two calibrated thermal resistor sensors (RTDs, pt1000Ω) were used for measuring the temperature value at the water side, with �0.05 � C in its measurement accuracy. By adopting the de viation evaluation method [21], it was known that the relative de viations of the Colburn (j) and Friction (f) factors are 1.79% and 2.49% respectively.
Fig. 2. The definition of D, b and H (From Ref. [9]). 3
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Fig. 3. Physical drawing of five ducts. (a) Rectangular duct; (b) Trapezoidal duct; (c) Circular duct; (d) Parabolic duct; (e) Cosinoidal duct.
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Table 1 Parameter table of rectangular duct. No.
Lf (mm)
Sf (mm)
hf (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 9.0 110
0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3
Table 2 Parameter table of trapezoidal duct. No.
Fh (mm)
β (� )
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
2 2 2 3.5 3.5 3.5 5 5 5
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 Parameter table of circular duct. No.
Fh (mm)
R (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
200 200 200 250 250 250 300 300 300
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 4 Parameter table of parabolic duct. 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
Table 5 Parameter table of cosinoidal duct.
Fig. 4. Schematic diagram of structural. (a) Rectangular duct; (b) Trapezoidal duct; (c) Circular duct; (d) Parabolic duct; (e) Cosinoidal duct.
5
No.
Fh (mm)
2A (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
5 5 5 7 7 7 9 9 9
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
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International Journal of Thermal Sciences 150 (2020) 106226
Fig. 5. Sketch of experiment bench.
2.3. Data reduction
In Eq. (6) and Eq. (7), Rem is calculated by using mean velocity (um). Remax is calculated by using maximum velocity (uin) at the inlet. dh is hydraulic diameter of rectangular duct with the same form, but dh,in is hydraulic diameter at the inlet of the cambered ducts. In the formula,
The heat transfer capacity Q was used for calculating the convective heat transfer coefficient (h) at the air side, with its calculation method shown as follows: Q ¼ ðQa þ Qw Þ = 2
(1)
Qa ¼ m_ a cpa ΔTa
(2)
Qw ¼ m_ w cpw ΔTw
(3)
AC dh ¼ 4⋅ ; LC
where AC is the transverse section area in the duct and LC is the wetted perimeter in the duct. The heat transfer and flow resistance performance were evaluated by the Colburn (j) and Friction (f) factors based on structural parameters of different ducts. The j factor and f factor were defined as [22]:
In the above equations, the subscript “a” and “w” represent param eter at the air side and water side respectively. ΔTa and ΔTw are tem perature difference at the air and water sides respectively. The convective heat exchange coefficient at the air side is indicated as follows:
j¼
In Eq. (4), ΔTln is the logarithmic mean temperature difference, with its calculation method shown as follows: ΔTln ¼
ðTC
TB Þ ðTD � � ln TTDC TTBA
TA Þ
um dh μ=ρ
(5)
JF ¼
(10)
ji =j0 ðfi =f0 Þ1=3
(11)
In the formula, the subscript “i” represents different structural pa rameters, for instance, it represents different amplitude in the cosinoidal duct; “0” represents the rectangular reference duct. The bigger the JF factor is, the better the overall heat transfer performance is.
(6)
3. Numerical model descriptions and numerical method
The flow condition of cambered ducts can be evaluated by the maximum Reynolds number (Remax): uin dh;in Remax ¼ μ=ρ
(9)
The overall evaluation index JF factor was adopted to evaluate the same duct form and different structural parameters, with the expression of JF factor shown as follows [23]:
Two types of Reynolds numbers were used for the cambered ducts and rectangular duct. Reynolds number in rectangular duct is mean Reynolds number (Rem): Rem ¼
Nu Dh ΔP ; f¼ ⋅ Pr1=3 ⋅Re 4L ð1=2Þρm u2
The definitions of Nu number and Pr number are as follows: � Nu ¼ hDh =λ; Pr ¼ ucp λ
(4)
h ¼ Q = ðA ⋅ ΔTln Þ
(8)
3.1. Computational domain
(7)
Taking the parabolic duct as an instance for analysis, the boundary 6
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International Journal of Thermal Sciences 150 (2020) 106226
conditions in the calculation region is shown in Fig. 6. In the figure, “Mass flow inlet” condition is applied to the inlet surface of the calcu lation region and the initial temperature of the flow is 293.15 K which remains constant. “Mass flow inlet” boundary conditions provided mass flow rate distribution at an inlet, which is used when it is more impor tant to match a prescribed mass flow rate. And “Pressure outlet” con dition is applied to the outlet surface. It requires the specification of a static pressure at the outlet boundary, and a set of “backflow” condition is also specified should the flow reverse direction at the pressure outlet boundary. Initial temperatures of the heat transfer surfaces for five different ducts and fins are 320.15 K. To ensure well-distributed air flow velocity at the inlet, the calculation region is extended 1.5 times of the fin length ahead of the inlet. Similarly, the calculation region is extended 5 times of the fin length behind the outlet to prevent reverse flow at the outlet and ensure the sufficient development of the boundary layer. Due to the uniform distribution of the duct and fin, only one unit of the duct is analyzed in this paper. 3.2. Governing equations and boundary conditions The schematic diagram of a physical model and coordinate system of an arbitrary cambered duct is shown in Fig. 7 in which curve y ¼ f (x) represents the cambered surface. Two coordinate systems including Cartesian coordinate system (x, y, z) and the orthogonal coordinate system (s, n, z) on the surface are established. In the orthogonal coor dinate system on a camber, s is parallel to the tangent line of the camber, n is parallel to the normal vector of camber and z is parallel to z in the Cartesian coordinate. In the figure, x and s axes are the air flow direc tion; y and n axes are the fin height direction; z axis is the fin spacing direction. During the numerical analysis, the following assumed condi tions are also necessary:
Fig. 7. Coordinate systems of cambered duct.
0
f n qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ f0
(12)
f y ¼ y00 þ n qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ f02
(13)
z¼z
(14)
x ¼ x0
(1) When air flows in a cambered duct, its physical properties are constant; (2) The flow at the air side is deemed as incompressible turbulent flow; (3) Surfaces of a duct and fin are ideal and smooth (free from burr and surface mutation) and equally arranged in a heat exchanger; (4) Neglect the thermal radiation and natural convective heat exchange; (5) Temperatures of a cambered duct and fin surface are deemed as equitemperature.
In Eq. (12) and Eq. (13), xO’, yO’ and zO’ were the displacement of the coordinate system (s, n, z) along x axis, y axis and z axis of the coordinate system (x, y, z) respectively. The governing equations in a cambered duct can be obtained as follows: (1) Continuity Equation:
∂us ∂un ∂uz þ þ ¼0 ∂s ∂n ∂z
After translation and rotation of a coordinate system, the relation between the right angle coordinate system (x, y, z) and the cambered orthogonal coordinate system (s, n, z) can be obtained as follows:
(15)
(2) Momentum Equations: Along direction s: � 3 2 � ∂ R ∂us R n ∂2 us 1 ∂us þ 2 7 6 R ∂n R ∂n R ∂p Rμ 6 ∂s R n ∂s 7 þ 7 ¼0 6 � � 2 R n ∂s R n 4 R un ∂ 1 2R ∂un us 5 ðR nÞ2 ∂s ðR nÞ2 ðR nÞ2 ∂s R (16) Along direction n: 2
ρg
�
�
3
∂ R ∂un R n ∂2 un 1 ∂un þ 7 6 ∂ R n R ∂z2 R ∂n s ∂ s ∂p Rμ 6 7 þ 7¼0 6 � � 2 5 4 ∂n R n us R ∂ 1 2 ∂us us þ þ 2 R n R n ðR nÞR ðR nÞ ∂s R (17)
Along direction z: Fig. 6. Calculation region. 7
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International Journal of Thermal Sciences 150 (2020) 106226
�
�
�
∂p Rμ ∂ R ∂uz R n ∂2 uz þ þ R ∂z2 ∂z R n ∂s R n ∂s
� 1 ∂uz ¼0 R ∂n
(3) Energy Equation � � ρcp us R ∂T un ∂T uz ∂T R2 ∂2 T ∂2 T þ þ þ ¼ λ R n ∂s ∂n ∂z ðR nÞ2 ∂s2 ∂z2 n ∂R ∂T RðR nÞ ∂n ∂n
R RðR
its ε equation that improves the accuracy for flows. The SIMPLER al gorithm is used for solving the finite difference equation. During setup of solving, the second order upwind format is selected for discrete process of the control equation, which can improve the solving accuracy of flow at the steady state. During solving iteration, the solving is deemed as convergence when the deviation between two adjacent iteration results of continuity, momentum and energy equations is kept at 10 6.
(18)
∂R ∂T þ nÞ3 ∂s ∂s
Rn ðR
3.3.2. Grid independence test and code validation With Remax ¼ 2000, the parabolic duct is adopted to verify the in dependence of a grid; its structural parameter is Fh ¼ 11.5 mm, P ¼ 120 mm, Sf ¼ 4 mm, Lf ¼ 60 mm and tf ¼ 0.4 mm. The total number of grid points for four groups are 0.225 million (15 � 100 � 150), 0.48million (20 � 120 � 200), 0.99 million (30 � 150 � 220) and 1.344 million (35 � 160 � 240). The results indicate that the deviations of j factor and f factor are 0.6% and 0.8% when the grid numbers are 30 � 150 � 220 and 35 � 160 � 240 respectively. Therefore, it is relatively proper to adopt 30 � 150 � 220, considering the calculation time and accuracy comprehensively. This method is still adopted for the grid independence verification of the others ducts. In the meanwhile, the parabolic duct is taken as an instance and the numerical data are compared with the experimental data to verify the numerical method adopted in this paper. The comparison results indicate the average deviation of j factor is 2.56% and that of f factor is 3.6% as shown in Fig. 9(a) and (b) respectively. Therefore, the experimental and numerical values have better coincidence, which verifies the correctness and reliability of the numerical method adopted in this paper.
∂T nÞ3 ∂n (19)
In Eq. (15)–(19), us, un, uz are velocity components in “s”, “n” and “z” directions respectively. s and n are coordinates along and normal to the cambered surface. z is vertical coordinate. R is the curvature radius of a cambered duct surface. g is acceleration of gravity. All calculation regions of five ducts are set to identical boundary condition and initial condition. During numerical and experimental processes, the air mass duct at the air side is set respectively to 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. 3.3. Meshing and numerical methodology 3.3.1. Grid generation technique and numerical method The structured grids division method is adopted to divide five duct regions, which are divided by using software ANSYS ICEM CFD [24]. Taking the parabolic duct as an instance, the divided effect is shown in Fig. 8. To accurately seize the temperature gradient change at the boundary layer and improve the numerical results, a relatively dense grid is adopted around the wall surface and fin of the duct, which shows the exponential growth rule. When dealing with flows in the boundary layer, we adopt the grid of yþ>30. yþ is a meshed dimensionless mea sure value, which determines the limit range of the near-wall layer. The definition of yþ is described as yþ ¼
y⋅u* ⋅ρ
4. Results and discussion 4.1. Comparison of overall heat transfer performance of different cambered duct structures The Core Volume Goodness Factor (CVGF) is adopted to evaluate the overall heat transfer performance, which can be described as:
η0 hstd α ¼
(20)
μ
Estd α ¼
where, pffiffiffiffiffiffiffiffiffi u* ¼ τ0 =ρ
μ⋅yþ u* ⋅ρ
μ3 4σ 3 fRe 2gc ρ2 D4h
(23) (24)
In the formula, ηohstdα is the heat transfer power of the unit heat exchanger volume per unit temperature change; Estdα is the consumed friction power per unit heat exchanger volume. The subscript “std” is the comparison benchmark and follows the ARI standard [26]. For the same “Estdα”, a bigger ηohstdα value means only a smaller volume of a heat exchanger is needed under the same heat transfer capacity. Fig. 10 provides the change curve of “ηohstdα” value of five different duct structures as a function of “Estdα”. The figure shows that “ηohstdα” value of five structures rises with increased “Estdα”. For the same “Estdα”, the cosinoidal duct has the best overall heat transfer performance and the rectangular duct has the worst overall heat transfer performance. Within the entire Estdα section, ηohstdα value in the cosinoidal duct is bigger than ηohstdα value in the rectangular duct by about 7.3%–28.1%. Actually, when Estdα is the maximum, ηohstdα value of the cosinoidal duct is bigger than that of the rectangular duct by about 28.1%. When ranking the overall heat transfer performance of five ducts, it is obtained that the cosinoidal > the parabolic > circular > trapezoidal > rectan gular. This proves that the cambered duct has a better overall heat transfer performance than that of the trapezoidal and rectangular ducts. Fig. 11 is the comparison diagram of the temperature difference between the inner wall surfaces of five different ducts and the average temperature of the fluid at the transverse section. The figure indicates that the temperature difference change amplitude in the rectangular duct (β ¼ 0� ) is bigger than others ducts, and the temperature difference
(21)
So, y¼
cp μ 4σ η jRe Pr2=3 0 D2h
(22)
In the above equation, y is the height of grid which is closest to the wall; τ0 is the shear stress at the wall; u* is the shear velocity. A RNG k-ε turbulent model was used in the paper and the control equations are solved by using software ANSYS FLUENT CFD [25] based on the finite volume method. The RNG model has an additional term in
Fig. 8. Plot of grid-point distribution. 8
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International Journal of Thermal Sciences 150 (2020) 106226
Fig. 9. Comparison between experimental and numerical values. (a) Comparison of j factor; (b) Comparison of f factor.
1.8 K. This indicates the most well-distributed temperature difference exists in the cosinoidal duct, which influences the heat transfer effi ciency of the heat exchanger actively.
between the inlet and the outlet is 3.55 K, with the change tendency in parabolic shape. The temperature difference in the trapezoidal duct (β ¼ 40� ) has certain degree of linearity and the temperature difference be tween the inlet and the outlet is 2.8 K. For the cosinoidal, parabolic and circular duct structures, the temperature difference distribution curves are relatively standard straight line form and have better degree of linearity than that in the trapezoidal duct. The temperature difference between the inlet and the outlet of the cosinoidal duct (2A ¼ 5 mm)is 9
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Fig. 10. Comparison of Core Volume Goodness Factor of five different ducts.
Fig. 11. Temperature difference distribution of five different ducts.
4.2. Cosinoidal duct heat transfer and its characteristics analysis
proportional variation of j and f factors with the variation of amplitude from 5 mm to 7 mm and from 7 mm to 9 mm. And the best j factor and the maximum f factor appear at amplitude 2A ¼ 5 mm. When 1210 � Re � 5080, comparing with the duct structure with amplitude 2A ¼ 5 mm, j factor of the duct with amplitude 2A ¼ 7 mm and 2A ¼ 9 mm decreases 6.96% and 12.6% respectively and f factor decreases 29.8% and 50% respectively. Meanwhile, it can also be seen that the change trend of j and f factors in the cosinoidal duct is consistent with that of the trape zoidal and rectangular ducts. Moreover, the j factor in the cosinoidal
4.2.1. Influence of different amplitudes on heat transfer and flow performance Fig. 12 (a) and 12(b) show the relation of j and f factors of three cosinoidal ducts (2A ¼ 5 mm, 2A ¼ 7 mm and 2A ¼ 9 mm), trapezoidal duct (β ¼ 40� ) and rectangular duct (β ¼ 0� ) as a function of Re. The figure shows that for a cosinoidal duct with any amplitude, j and f factors decrease gradually with increased Re. With the increase of Re, there is a 10
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duct is greater than the other two ducts, but the f factor is between the rectangular and trapezoidal ducts. Although the gradually expanding duct slows down the flow veloc ity, this structure form decreases the pressure drop loss to a large extent during the fluid flow process. While improving the convective heat ex change capability of the upstream, the heat transfer area of the down stream is also improved to boost the improvement of the overall heat transfer performance. In Fig. 12(c), JF factor is adopted to compare the overall heat transfer performance in the cosinoidal duct with three different amplitudes. According to the results in the figure, under the same Re, the duct with amplitude 2A ¼ 5 mm has the best overall heat transfer enhancement capability. In other words, the smaller the amplitude is, the better the heat transfer performance of the cosinoidal duct is. 4.2.2. Influence of different Reynolds numbers on temperature and flow fields Taking amplitude 2A ¼ 5 mm as an instance, numerical analysis is conducted for the cosinoidal duct when the Reynolds number is 1210, 2170, 3140, 4110 and 5080. Fig. 13(a) and (b) are the distribution contour of temperature and velocity fields under different Re and the section of the contour plot is the middle plane between two fins (fin spacing direction). Fig. 13(a) indicates that the temperature in the main flow close to the middle fluid domain is relatively low and the temperature value close to the wall surface is relatively high whether under small Reynolds number or big Reynolds number. The thickness of the thermal boundary layer adjacent to the wall surface decreases constantly with increased Re. Therefore, the bigger Re is, the stronger the convective heat exchange performance in the cosinoidal duct is. Fig. 13(b) indicates that the thickness of the velocity boundary layer decreases gradually with increased Re. With increased Re and even at its highest value, the reverse flow or boundary layer separation has not occurred in the duct. The reason behind this is that the maximum slope angle in the duct fails to be up to the condition of boundary layer sep aration. The other obvious phenomenon is that the red high speed region in the middle expands forward with increased Re. The higher flow ve locity in the duct can also improve the convective heat transfer perfor mance. Taking uin ¼ 4 m/s (2A ¼ 5 mm, 7 mm and 9 mm; Re � 3140) as an example, the effect of local enlargement is shown as“A”in Fig. 13(c). As can be seen from the figure, there is no backflow or boundary sepa ration in the parabolic duct. 4.2.3. Distribution of temperature gradient on surface of cosinoidal duct Since the temperature distribution near the wall of the cambered duct along axis s cannot directly reflect the heat flux distribution which is related closely to the heat transfer efficiency, we chose temperature gradient to evaluate the heat transfer performance. Taking the cosinoi dal duct, 2A ¼ 9 mm, Lf ¼ 60 mm and n ¼ 5um (the surface of cosinoidal duct is the starting point of the coordinate (s, n, z), where n ¼ 0 μm) as an instance, the temperature gradient on the surface of the duct is analyzed. When the temperature gradient on the surface is bigger, the convective heat exchange performance on the surface is superior. Therefore, the temperature gradient is adopted as the standard to evaluate the heat flux is more beneficial than adopting the temperature value or the boundary layer thickness. In Fig. 14, the temperature gradient is in proportion relation with the heat flux which is perpendicular to the surface direc tion (n). Therefore, the analysis of the temperature gradient change is equivalent to analyze the heat transfer performance in different posi tions on the duct surface. Fig. 14 shows that the temperature gradient in the peak region (0 mm � s � 25 mm) is obviously bigger than that in the trough region (45 mm � s � 68 mm); in the area (25 mm < s < 45 mm) between the peak and trough regions, the gradient of the temperature changes slightly with increasing of s. It indicates that the convective heat exchange co efficient in the peak region is bigger than the convective heat exchange
Fig. 12. Performance comparison of different ducts with respect to Re. (a) j factor comparison; (b) f factor comparison; (c) Overall perfor mance comparison.
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Fig. 14. Temperature gradient distribution of cosinoidal duct (2A ¼ 9 mm, n ¼ 5um) along s direction.
thickness in this area is relatively thin, causing smaller thermal resis tance of heat conduction. It is noticeable that the temperature gradient in the middle area between the peak and trough regions changes slightly, which indicates relatively well-distributed convective heat ex change performance in this area. According to the analysis of Fig. 14, it is concluded that the main heat transfer area on the surface of the cosinoidal duct is the peak region and the middle area, with weaker heat transfer performance in the trough region. 5. Conclusions This paper analyzes the flow and heat transfer characteristics in five different cambered ducts through the experimental and numerical method. The Core Volume Goodness Factor (CVGF) is adopted for the comparative analysis of rectangular, trapezoidal, circular, parabolic and cosinoidal ducts. And the following conclusions can be reached: (1) The Core Volume Goodness Factor is adopted to compare five ducts. It is concluded that the sequence of their overall heat transfer performance is cosinoidal, parabolic, circular, trape zoidal and rectangular ducts successively from superior to inferior. (2) With the optimal overall heat transfer performance, the cosinoi dal duct is superior to that of the rectangular duct by 7.3%– 28.1%. The temperature difference distribution at the inlet and outlet of different ducts is analyzed, we can conclude that the uniformity distribution of temperature difference in the duct plays an active role in improving the heat transfer efficiency of the heat exchanger. (3) In the cosinoidal duct, the smaller the amplitude is, the better its overall heat transfer performance is. With Re increasing, the thicknesses of the temperature and velocity boundary layers decrease gradually and the reverse flow or boundary layer sep aration has not been occurred in the duct. (4) In the cosinoidal duct, the temperature gradient in the peak re gion is bigger than that in the trough region; however, the gradient of the temperature in the middle area between the peak and trough regions changes slightly. (5) With Reynolds number ranging from 1210 to 5080, the overall heat transfer performance of a gradually expanding duct is su perior to that of ducts with equal cross-sectional area.
Fig. 13. Influence of different Reynolds number on temperature and flow fields for cosinoidal duct. (a) Temperature field (2A ¼ 5 mm); (b) Flow field (2A ¼ 5 mm); (c) Velocity vector (2A ¼ 9 mm).
coefficients in the trough region and the middle area, i.e. the peak region has the maximum heat transfer efficiency. The reason behind this is that the air flow velocity adjacent to the peak region is higher; at the initial development stage of the boundary layer, the thermal boundary layer 12
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Acknowledgements
Foundation of China (51806114, 51874187) and China Postdoctoral Fund.
Financial support is provided by the National Natural Science
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijthermalsci.2019.106226. Nomenclature A b cp D Dh d1 , d 2 E Fh g H h L/Lf m_ P p Q R Sf T tf ΔT u
Heat transfer area [m2]; Amplitude [mm] Mean channel width Specific heat capacity [J/kg⋅K] Mean curvature diameter (mm) Hydraulic diameter [mm] Diameters of wetted lateral surfaces of the channel; inner and outer tubes of the annulus, respectively (mm) Fluid pumping power per unit surface area [W/m2] Inlet height [mm] Acceleration of gravity (m/s2) Channel height (mm) Heat transfer coefficient [W/m2⋅K] Fin length [mm] Mass flux [kg/s] Focal length [mm] Pressure [Pa] Average value of the heat flux [W] Circular radius [mm] Fin pitch [mm] Temperature [K] Fin thickness [mm] Temperature difference [K] Velocity [m/s]
Dimensionless f Fanning friction factor gc Proportionality constant in Newton’s second law of motion, gc ¼ 1 j Colburn factor Nu Nusselt number Pr Prandtl number Greek letters Ratio of total heat transfer area to the total volume of an exchanger [m2/m3] β Slope angle [� ] η0 Extended surface efficiency on one fluid side of the extended surface heat exchanger λ Heat conductivity [W/m⋅K] μ Dynamic viscosity [Pa⋅s] ρ Air density [kg/m3] σ Ratio of free flow area to frontal area ϕ Angle of channel swirling (� )
α
Subscripts A a B C D in ln max w
Inlet air Air side Outlet air Inlet water Outlet water inlet Logarithm Maximum value Water side
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References
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[1] S.M. Hoi, A.L. Teh, E.H. Ooi, I.M.L. Chew, J.J. Foo, Plate-fin heat sink forced convective heat transfer augmentation with a fractal insert, Int. J. Therm. Sci. 142 (2019) 392–406. [2] S. Rashidi, M. Akbarzadeh, N. Karimi, R. Masoodi, Combined effects of nanofluid and transverse twisted-baffles on the flow structures, heat transfer and irreversibilities inside a square duct - a numerical study, Appl. Therm. Eng. 130 (2018) 135–148. [3] S.W. Al, D.A. Al, H.M. Thompson, A numerical investigation of the thermalhydraulic characteristics of perforated plate fin heat sinks, Int. J. Therm. Sci. 121 (2019) 266–277. [4] 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. [5] R.K. Shah, D.P. Sekulic0 , Fundamentals of Heat Exchanger Design, John Wiley & Sons, New Jersey, 2003. [6] N.O. Cur, J.J. Anselmino, Evaporator for Home Refrigerator, 1992. US Patent No. 5157941. [7] P.J. Waltrich, J.R. Barbosa, C. Melo, C.L. Hermes, Air-side heat transfer and pressure drop in accelerated flow evaporators, Int. Refrig. Air Cond. Conf. 7 (2008) 2311–2319. [8] 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. [9] P. Poskas, V. Simonis, V. Ragaisis, Heat transfer in helical channels with two-side heating in gas flow, Int. J. Heat Mass Transf. 54 (2011) 847–853. [10] H.M. Bahaidarah, Entropy generation during fluid flow in sharp edge wavy channels with horizontal pitch, Adv. Mech. Eng. 8 (2016) 1–10. [11] M. Sarkar, S.B. Paramane, A. Sharma, Periodically fully developed heat and fluid flow characteristics in furrowed wavy channel, Heat Transf. Eng. 38 (2016) 278–288.
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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Analysis of heat transfer and flow characteristics in typical cambered ducts Xilong Zhang, Yongliang Zhang *, Zunmin Liu, Jiang Liu School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao, 266520, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cambered duct Cosinoidal duct Peak region Temperature gradient Gradually expanding structure
The heat transfer and flow characteristics of the air-water cross flow over cambered ducts were experimental and numerical investigated. The sequence of their Core Volume Goodness Factor (CVGF) is cosinoidal, parabolic, circular, trapezoidal and rectangular ducts successively from superior to inferior. Cambered ducts have more uniform temperature difference distribution than the equal cross section duct, and it has the minimum tem perature difference in inlet and outlet of the cosinoidal duct. With the optimal overall heat transfer performance, the cosinoidal duct is superior to that of the rectangular duct by 7.3%–28.1%. In the cosinoidal duct, the smaller the amplitude is, the better the heat transfer performance is. The thickness of the thermal and velocity boundary layers adjacent to the wall surface decreases constantly with increased Reynolds number. In the near wall region, n ¼ 5um, the main heat transfer area is the peak and middle areas, but with weaker heat transfer performance in the trough region. Although gradually expanding cambered duct slows down the flow velocity, the structure form decreases the pressure drop loss during the flow process. While improving the convective heat exchange capa bility of the upstream, the heat transfer area of the downstream is also improved to boost the overall heat transfer performance.
1. Introduction Due to its simple structure, lower cost, superior heat transfer per formance and other advantages, the uniform cross section duct is applied widely to the plate-fin heat exchangers [1–3], with its structure sketch shown in Fig. 1(a). The flow pattern of the plate-fin heat ex changers is usually cross flow between two parallel plates with different constructions. This traditional duct structure has many defects: (1) Bigger temperature difference at the inlet but smaller one at the outlet. During cooling medium flowing, the temperature difference at the outlet decreases, which confines the convective heat exchange capability in the outlet area; (2) Accelerated flow velocity of the cooling medium in creases the pressure drop loss. With accelerated flow velocity, the pressure drop rises in the form of parabola. To improve the two main defects in uniform cross section ducts, this paper proposes the cambered duct structure, i.e. the transverse section area changes along the direc tion of cooling medium flow and the duct between the inlet and the outlet has the gradually expanding structure, with its sketch shown in Fig. 1(b). In our previous research investigation [4], relevant discussions have been conducted for the heat transfer performance of the trapezoidal duct which was a type of the cambered duct in a special shape. The results
indicated when the slope angle (β) in the trapezoidal duct ranges from 0 to 40� , the overall heat transfer performance of the trapezoidal duct was superior to that of the rectangular duct. Core Volume Goodness Factor (CVGF), which is proposed by Shah and Sekuli�c [5], is adopted to evaluate the overall heat transfer performance of five different ducts. The significance of adopting the CVGF is that it can compare the overall heat transfer performance of different duct structures under various heat transfer areas. The CVGF of trapezoidal duct was superior by 5–20% than that of rectangular duct. Meanwhile, the change of temperature difference between the inlet and outlet in the trapezoidal duct showed a linear distribution, which improved the heat transfer performance in the outlet area, with the schematic diagram of temperature difference shown in Fig. 1(c). Some analysis investigations were made in the theory, numerical and experiment for the analysis of the trapezoidal duct. During the appli cation, Cur and Anselmino [6] proposed an Accelerated Flow Evaporator (AFE) for the first time, which was a trapezoidal duct structure with a big inlet and a small outlet. During its flow in the evaporator, the flow ve locity of air at the downstream accelerated and improved the local convection heat transfer performance of the downstream. However, the accelerated flow evaporator increased more pressure drop loss. Waltrich et al. [7,8] conducted the experimental analysis of nine accelerated
* Corresponding author. E-mail address: [email protected] (Y. Zhang). https://doi.org/10.1016/j.ijthermalsci.2019.106226 Received 16 July 2019; Received in revised form 7 December 2019; Accepted 9 December 2019 Available online 20 December 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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evaporator had more pressure drop loss at high Reynolds number. Therefore, the gradually expanding cambered duct structure in the paper can overcome the practical application better when the reducing duct structure cannot be applied to the high heat transfer capability. There are relatively fewer analyses of the cambered duct. Poskas et al. [9] experimentally studied heat transfer performance for both convex and concave surfaces of helical ducts with curvature D/H ¼ 5–9 and width b/H ¼ 2–20. D (D ¼ 0.5(d1þd2)/sin2ϕ), b and H are mean curvature diameter, mean channel width and channel height
evaporator samples with different structural parameters. The results indicated that the accelerated evaporator, at a low heat transfer capa bility, required less pump power when comparing it with a reference evaporator (a straight evaporator). Therefore, in this practical applica tion, the accelerated evaporator had a smaller volume and consumes less material than the straight evaporator. However, with a high heat transfer capability, the accelerated evaporator showed exponential growth of power consumption, which confined practical applications of the accelerated evaporator. The reason behind it was that a reducing
Fig. 1. Schematic diagram of duct structure and distribution diagram of temperature difference. (a) Uniform cross section duct; (b) Cambered duct; (c) Temperature difference distribution in trapezoidal duct [4]. 2
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respectively, which is shown in Fig. 2. With both convex and concave surfaces heated, they found that the heat transfer from convex and concave walls increases by 20% than that of one sided heating. Bahai darah et al. [10] analyzed the volume entropy generation rate in the sharp edge wavy duct. The results indicated that the total entropy rises gradually with the increased Re. However, the total entropy production along the duct direction decreased gradually when Reynolds number ranged from 25 to 400. Sarkar et al. [11] and others analyzed the two-dimensional flow characteristics in the wavy duct under different Reynolds numbers (100 < Re < 2123). At the states of laminar and transient flows, the Nusselt number, friction factor and Area Goodness Factor were adopted to compare and analyze the heat transfer and flow performance in ducts of six structures under different amplitudes (0.05 mm, 0.075 mm and 0.1 mm) and wavelength (0.5 mm and 1 mm). In the research of Baik et al. [12], the thermal performance of the corrugated channel in a printed circuit heat exchanger (PCHE) and the effects of amplitude and periodic equivalence factors were numerically studied. The results indicated that the recirculating flow caused by the waviness can be ignored while the mass flow of CO2 was in the typical range. Matsubara et al. [13] numerically analyzed the heat transfer and flow characteristics in a curved duct with a radius ratio are 0.92. They concluded that the convection heat transfer performance on the outer wall of the curved duct was the strongest by the evaluation of the local Nusselt number. In addition, this enhanced heat transfer came from large-scale organized vortex, which exacerbated the lateral convection of the secondary flow, thereby increased the turbulent heat flux value. Direct Numerical Simulation (DNS) is a common and effective method in the various flow ducts, such as Vinuesa et al. [14], Marin et al. [15] and Vidal et al. [16] conducted numerical studies for rectangular, hexagonal and sinusoidal ducts respectively. As for cambered ducts studied in this paper, basically no relevant literature has been published. However, the cambered ducts are similar to a diffuser, which has been studied for years. Ghosh et al. [17]
optimized the shapes of diffusers with different lengths, and used a ge netic algorithm for the optimum design of plane symmetric diffusers. They found that not much difference has been noted in the optimum C values (pressure recovery factor) obtained by using different methods. � � The expression of the C is C ¼ Δp = 12ρf v2inl , where Δp is the gain in static pressure, ρf is the density of the fluid and υinl is the inlet velocity of the fluid flow. Chen et al. [18] used CFD method to study the effects of injectors with different structures and obtained optimal geometrical factors to maximize entrainment ratio. They concluded that the opti mum inclination of the mixing chamber is 14� ; the optimum diameter ratio between the mixing tube and the primary nozzle is 1.7. In fact, studies involving heat transfer in such diffusers are still limited. The meaning of analyzing the cambered duct is that the cambered duct can improve the distribution of temperature difference, raise the efficiency of heat transfer, reduce the pressure drop and accelerate the drainage of condensate liquid. Taking five different duct structures as an instance, this paper conducts experimental and numerical analysis of cambered ducts. 2. Experimental system and procedure 2.1. Physical model Experimental analysis has been carried out for five ducts including rectangular, trapezoidal, circular, parabolic and cosinoidal ducts respectively. Processed from 6061 aluminum alloy, the five heat ex change units do not have the same overall heat transfer area but have the same volume. Fig. 3 and Fig. 4 are physical and parameter drawings of five different duct structures respectively. Table 1, Table 2, Table 3, Table 4 and Table 5 are specific parameters of five different duct structures. A special processing method [19] was used to form the rectangular and cambered ducts. The processing method can obtain an integrated structure by drawing ducts in an aluminum alloy matter, thus can form a tube. And then squeeze fins on the tube instead of brazing and expanding. 2.2. Wind tunnel tests Fig. 5 shows the wind tunnel experiment bench for testing five different duct structures, which consists of the air circulating system, water circulating system, testing part, control unit and data collection system. During experiment, the Reynolds number on air side ranged from 1210 to 5080. The water flow rate is 1 kg/s, and the Reynolds numbers on water side are about 2200 in the five different ducts. Water and air served respectively as the work medium at the air and water sides. Temperature at the inlet and outlet in the air circulating loop adopted T cooper-constantan thermocouple (calibrated accuracy was 0.1 � C) network for measurement. The quantity and layout of the thermocouple follow the methods recommended by ASHRAE [20] standard. Air flux was measured by the inclined-tube manometer and pressure drops of the heat exchange unit measured by the U-type manometer. The equal power heating method [19] was adopted for heating the hot water, which can reduce heat balance time and improve heat balance. Water pipe, experimental channel and test unit adopted 10 mm heat insulation materials for insulation. The mass flux at the water side was measured by a pulse flux sensor, with 0.0025 l/s reso lution. Two manometers were arranged at both sides of the heat ex change unit for measuring the pressure drop loss at the water side, with 2Pa in its accuracy. Two calibrated thermal resistor sensors (RTDs, pt1000Ω) were used for measuring the temperature value at the water side, with �0.05 � C in its measurement accuracy. By adopting the de viation evaluation method [21], it was known that the relative de viations of the Colburn (j) and Friction (f) factors are 1.79% and 2.49% respectively.
Fig. 2. The definition of D, b and H (From Ref. [9]). 3
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Fig. 3. Physical drawing of five ducts. (a) Rectangular duct; (b) Trapezoidal duct; (c) Circular duct; (d) Parabolic duct; (e) Cosinoidal duct.
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Table 1 Parameter table of rectangular duct. No.
Lf (mm)
Sf (mm)
hf (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 9.0 110
0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3
Table 2 Parameter table of trapezoidal duct. No.
Fh (mm)
β (� )
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
2 2 2 3.5 3.5 3.5 5 5 5
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 Parameter table of circular duct. No.
Fh (mm)
R (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
200 200 200 250 250 250 300 300 300
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 4 Parameter table of parabolic duct. 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
Table 5 Parameter table of cosinoidal duct.
Fig. 4. Schematic diagram of structural. (a) Rectangular duct; (b) Trapezoidal duct; (c) Circular duct; (d) Parabolic duct; (e) Cosinoidal duct.
5
No.
Fh (mm)
2A (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
5 5 5 7 7 7 9 9 9
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
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Fig. 5. Sketch of experiment bench.
2.3. Data reduction
In Eq. (6) and Eq. (7), Rem is calculated by using mean velocity (um). Remax is calculated by using maximum velocity (uin) at the inlet. dh is hydraulic diameter of rectangular duct with the same form, but dh,in is hydraulic diameter at the inlet of the cambered ducts. In the formula,
The heat transfer capacity Q was used for calculating the convective heat transfer coefficient (h) at the air side, with its calculation method shown as follows: Q ¼ ðQa þ Qw Þ = 2
(1)
Qa ¼ m_ a cpa ΔTa
(2)
Qw ¼ m_ w cpw ΔTw
(3)
AC dh ¼ 4⋅ ; LC
where AC is the transverse section area in the duct and LC is the wetted perimeter in the duct. The heat transfer and flow resistance performance were evaluated by the Colburn (j) and Friction (f) factors based on structural parameters of different ducts. The j factor and f factor were defined as [22]:
In the above equations, the subscript “a” and “w” represent param eter at the air side and water side respectively. ΔTa and ΔTw are tem perature difference at the air and water sides respectively. The convective heat exchange coefficient at the air side is indicated as follows:
j¼
In Eq. (4), ΔTln is the logarithmic mean temperature difference, with its calculation method shown as follows: ΔTln ¼
ðTC
TB Þ ðTD � � ln TTDC TTBA
TA Þ
um dh μ=ρ
(5)
JF ¼
(10)
ji =j0 ðfi =f0 Þ1=3
(11)
In the formula, the subscript “i” represents different structural pa rameters, for instance, it represents different amplitude in the cosinoidal duct; “0” represents the rectangular reference duct. The bigger the JF factor is, the better the overall heat transfer performance is.
(6)
3. Numerical model descriptions and numerical method
The flow condition of cambered ducts can be evaluated by the maximum Reynolds number (Remax): uin dh;in Remax ¼ μ=ρ
(9)
The overall evaluation index JF factor was adopted to evaluate the same duct form and different structural parameters, with the expression of JF factor shown as follows [23]:
Two types of Reynolds numbers were used for the cambered ducts and rectangular duct. Reynolds number in rectangular duct is mean Reynolds number (Rem): Rem ¼
Nu Dh ΔP ; f¼ ⋅ Pr1=3 ⋅Re 4L ð1=2Þρm u2
The definitions of Nu number and Pr number are as follows: � Nu ¼ hDh =λ; Pr ¼ ucp λ
(4)
h ¼ Q = ðA ⋅ ΔTln Þ
(8)
3.1. Computational domain
(7)
Taking the parabolic duct as an instance for analysis, the boundary 6
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conditions in the calculation region is shown in Fig. 6. In the figure, “Mass flow inlet” condition is applied to the inlet surface of the calcu lation region and the initial temperature of the flow is 293.15 K which remains constant. “Mass flow inlet” boundary conditions provided mass flow rate distribution at an inlet, which is used when it is more impor tant to match a prescribed mass flow rate. And “Pressure outlet” con dition is applied to the outlet surface. It requires the specification of a static pressure at the outlet boundary, and a set of “backflow” condition is also specified should the flow reverse direction at the pressure outlet boundary. Initial temperatures of the heat transfer surfaces for five different ducts and fins are 320.15 K. To ensure well-distributed air flow velocity at the inlet, the calculation region is extended 1.5 times of the fin length ahead of the inlet. Similarly, the calculation region is extended 5 times of the fin length behind the outlet to prevent reverse flow at the outlet and ensure the sufficient development of the boundary layer. Due to the uniform distribution of the duct and fin, only one unit of the duct is analyzed in this paper. 3.2. Governing equations and boundary conditions The schematic diagram of a physical model and coordinate system of an arbitrary cambered duct is shown in Fig. 7 in which curve y ¼ f (x) represents the cambered surface. Two coordinate systems including Cartesian coordinate system (x, y, z) and the orthogonal coordinate system (s, n, z) on the surface are established. In the orthogonal coor dinate system on a camber, s is parallel to the tangent line of the camber, n is parallel to the normal vector of camber and z is parallel to z in the Cartesian coordinate. In the figure, x and s axes are the air flow direc tion; y and n axes are the fin height direction; z axis is the fin spacing direction. During the numerical analysis, the following assumed condi tions are also necessary:
Fig. 7. Coordinate systems of cambered duct.
0
f n qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ f0
(12)
f y ¼ y00 þ n qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ f02
(13)
z¼z
(14)
x ¼ x0
(1) When air flows in a cambered duct, its physical properties are constant; (2) The flow at the air side is deemed as incompressible turbulent flow; (3) Surfaces of a duct and fin are ideal and smooth (free from burr and surface mutation) and equally arranged in a heat exchanger; (4) Neglect the thermal radiation and natural convective heat exchange; (5) Temperatures of a cambered duct and fin surface are deemed as equitemperature.
In Eq. (12) and Eq. (13), xO’, yO’ and zO’ were the displacement of the coordinate system (s, n, z) along x axis, y axis and z axis of the coordinate system (x, y, z) respectively. The governing equations in a cambered duct can be obtained as follows: (1) Continuity Equation:
∂us ∂un ∂uz þ þ ¼0 ∂s ∂n ∂z
After translation and rotation of a coordinate system, the relation between the right angle coordinate system (x, y, z) and the cambered orthogonal coordinate system (s, n, z) can be obtained as follows:
(15)
(2) Momentum Equations: Along direction s: � 3 2 � ∂ R ∂us R n ∂2 us 1 ∂us þ 2 7 6 R ∂n R ∂n R ∂p Rμ 6 ∂s R n ∂s 7 þ 7 ¼0 6 � � 2 R n ∂s R n 4 R un ∂ 1 2R ∂un us 5 ðR nÞ2 ∂s ðR nÞ2 ðR nÞ2 ∂s R (16) Along direction n: 2
ρg
�
�
3
∂ R ∂un R n ∂2 un 1 ∂un þ 7 6 ∂ R n R ∂z2 R ∂n s ∂ s ∂p Rμ 6 7 þ 7¼0 6 � � 2 5 4 ∂n R n us R ∂ 1 2 ∂us us þ þ 2 R n R n ðR nÞR ðR nÞ ∂s R (17)
Along direction z: Fig. 6. Calculation region. 7
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International Journal of Thermal Sciences 150 (2020) 106226
�
�
�
∂p Rμ ∂ R ∂uz R n ∂2 uz þ þ R ∂z2 ∂z R n ∂s R n ∂s
� 1 ∂uz ¼0 R ∂n
(3) Energy Equation � � ρcp us R ∂T un ∂T uz ∂T R2 ∂2 T ∂2 T þ þ þ ¼ λ R n ∂s ∂n ∂z ðR nÞ2 ∂s2 ∂z2 n ∂R ∂T RðR nÞ ∂n ∂n
R RðR
its ε equation that improves the accuracy for flows. The SIMPLER al gorithm is used for solving the finite difference equation. During setup of solving, the second order upwind format is selected for discrete process of the control equation, which can improve the solving accuracy of flow at the steady state. During solving iteration, the solving is deemed as convergence when the deviation between two adjacent iteration results of continuity, momentum and energy equations is kept at 10 6.
(18)
∂R ∂T þ nÞ3 ∂s ∂s
Rn ðR
3.3.2. Grid independence test and code validation With Remax ¼ 2000, the parabolic duct is adopted to verify the in dependence of a grid; its structural parameter is Fh ¼ 11.5 mm, P ¼ 120 mm, Sf ¼ 4 mm, Lf ¼ 60 mm and tf ¼ 0.4 mm. The total number of grid points for four groups are 0.225 million (15 � 100 � 150), 0.48million (20 � 120 � 200), 0.99 million (30 � 150 � 220) and 1.344 million (35 � 160 � 240). The results indicate that the deviations of j factor and f factor are 0.6% and 0.8% when the grid numbers are 30 � 150 � 220 and 35 � 160 � 240 respectively. Therefore, it is relatively proper to adopt 30 � 150 � 220, considering the calculation time and accuracy comprehensively. This method is still adopted for the grid independence verification of the others ducts. In the meanwhile, the parabolic duct is taken as an instance and the numerical data are compared with the experimental data to verify the numerical method adopted in this paper. The comparison results indicate the average deviation of j factor is 2.56% and that of f factor is 3.6% as shown in Fig. 9(a) and (b) respectively. Therefore, the experimental and numerical values have better coincidence, which verifies the correctness and reliability of the numerical method adopted in this paper.
∂T nÞ3 ∂n (19)
In Eq. (15)–(19), us, un, uz are velocity components in “s”, “n” and “z” directions respectively. s and n are coordinates along and normal to the cambered surface. z is vertical coordinate. R is the curvature radius of a cambered duct surface. g is acceleration of gravity. All calculation regions of five ducts are set to identical boundary condition and initial condition. During numerical and experimental processes, the air mass duct at the air side is set respectively to 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. 3.3. Meshing and numerical methodology 3.3.1. Grid generation technique and numerical method The structured grids division method is adopted to divide five duct regions, which are divided by using software ANSYS ICEM CFD [24]. Taking the parabolic duct as an instance, the divided effect is shown in Fig. 8. To accurately seize the temperature gradient change at the boundary layer and improve the numerical results, a relatively dense grid is adopted around the wall surface and fin of the duct, which shows the exponential growth rule. When dealing with flows in the boundary layer, we adopt the grid of yþ>30. yþ is a meshed dimensionless mea sure value, which determines the limit range of the near-wall layer. The definition of yþ is described as yþ ¼
y⋅u* ⋅ρ
4. Results and discussion 4.1. Comparison of overall heat transfer performance of different cambered duct structures The Core Volume Goodness Factor (CVGF) is adopted to evaluate the overall heat transfer performance, which can be described as:
η0 hstd α ¼
(20)
μ
Estd α ¼
where, pffiffiffiffiffiffiffiffiffi u* ¼ τ0 =ρ
μ⋅yþ u* ⋅ρ
μ3 4σ 3 fRe 2gc ρ2 D4h
(23) (24)
In the formula, ηohstdα is the heat transfer power of the unit heat exchanger volume per unit temperature change; Estdα is the consumed friction power per unit heat exchanger volume. The subscript “std” is the comparison benchmark and follows the ARI standard [26]. For the same “Estdα”, a bigger ηohstdα value means only a smaller volume of a heat exchanger is needed under the same heat transfer capacity. Fig. 10 provides the change curve of “ηohstdα” value of five different duct structures as a function of “Estdα”. The figure shows that “ηohstdα” value of five structures rises with increased “Estdα”. For the same “Estdα”, the cosinoidal duct has the best overall heat transfer performance and the rectangular duct has the worst overall heat transfer performance. Within the entire Estdα section, ηohstdα value in the cosinoidal duct is bigger than ηohstdα value in the rectangular duct by about 7.3%–28.1%. Actually, when Estdα is the maximum, ηohstdα value of the cosinoidal duct is bigger than that of the rectangular duct by about 28.1%. When ranking the overall heat transfer performance of five ducts, it is obtained that the cosinoidal > the parabolic > circular > trapezoidal > rectan gular. This proves that the cambered duct has a better overall heat transfer performance than that of the trapezoidal and rectangular ducts. Fig. 11 is the comparison diagram of the temperature difference between the inner wall surfaces of five different ducts and the average temperature of the fluid at the transverse section. The figure indicates that the temperature difference change amplitude in the rectangular duct (β ¼ 0� ) is bigger than others ducts, and the temperature difference
(21)
So, y¼
cp μ 4σ η jRe Pr2=3 0 D2h
(22)
In the above equation, y is the height of grid which is closest to the wall; τ0 is the shear stress at the wall; u* is the shear velocity. A RNG k-ε turbulent model was used in the paper and the control equations are solved by using software ANSYS FLUENT CFD [25] based on the finite volume method. The RNG model has an additional term in
Fig. 8. Plot of grid-point distribution. 8
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International Journal of Thermal Sciences 150 (2020) 106226
Fig. 9. Comparison between experimental and numerical values. (a) Comparison of j factor; (b) Comparison of f factor.
1.8 K. This indicates the most well-distributed temperature difference exists in the cosinoidal duct, which influences the heat transfer effi ciency of the heat exchanger actively.
between the inlet and the outlet is 3.55 K, with the change tendency in parabolic shape. The temperature difference in the trapezoidal duct (β ¼ 40� ) has certain degree of linearity and the temperature difference be tween the inlet and the outlet is 2.8 K. For the cosinoidal, parabolic and circular duct structures, the temperature difference distribution curves are relatively standard straight line form and have better degree of linearity than that in the trapezoidal duct. The temperature difference between the inlet and the outlet of the cosinoidal duct (2A ¼ 5 mm)is 9
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Fig. 10. Comparison of Core Volume Goodness Factor of five different ducts.
Fig. 11. Temperature difference distribution of five different ducts.
4.2. Cosinoidal duct heat transfer and its characteristics analysis
proportional variation of j and f factors with the variation of amplitude from 5 mm to 7 mm and from 7 mm to 9 mm. And the best j factor and the maximum f factor appear at amplitude 2A ¼ 5 mm. When 1210 � Re � 5080, comparing with the duct structure with amplitude 2A ¼ 5 mm, j factor of the duct with amplitude 2A ¼ 7 mm and 2A ¼ 9 mm decreases 6.96% and 12.6% respectively and f factor decreases 29.8% and 50% respectively. Meanwhile, it can also be seen that the change trend of j and f factors in the cosinoidal duct is consistent with that of the trape zoidal and rectangular ducts. Moreover, the j factor in the cosinoidal
4.2.1. Influence of different amplitudes on heat transfer and flow performance Fig. 12 (a) and 12(b) show the relation of j and f factors of three cosinoidal ducts (2A ¼ 5 mm, 2A ¼ 7 mm and 2A ¼ 9 mm), trapezoidal duct (β ¼ 40� ) and rectangular duct (β ¼ 0� ) as a function of Re. The figure shows that for a cosinoidal duct with any amplitude, j and f factors decrease gradually with increased Re. With the increase of Re, there is a 10
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duct is greater than the other two ducts, but the f factor is between the rectangular and trapezoidal ducts. Although the gradually expanding duct slows down the flow veloc ity, this structure form decreases the pressure drop loss to a large extent during the fluid flow process. While improving the convective heat ex change capability of the upstream, the heat transfer area of the down stream is also improved to boost the improvement of the overall heat transfer performance. In Fig. 12(c), JF factor is adopted to compare the overall heat transfer performance in the cosinoidal duct with three different amplitudes. According to the results in the figure, under the same Re, the duct with amplitude 2A ¼ 5 mm has the best overall heat transfer enhancement capability. In other words, the smaller the amplitude is, the better the heat transfer performance of the cosinoidal duct is. 4.2.2. Influence of different Reynolds numbers on temperature and flow fields Taking amplitude 2A ¼ 5 mm as an instance, numerical analysis is conducted for the cosinoidal duct when the Reynolds number is 1210, 2170, 3140, 4110 and 5080. Fig. 13(a) and (b) are the distribution contour of temperature and velocity fields under different Re and the section of the contour plot is the middle plane between two fins (fin spacing direction). Fig. 13(a) indicates that the temperature in the main flow close to the middle fluid domain is relatively low and the temperature value close to the wall surface is relatively high whether under small Reynolds number or big Reynolds number. The thickness of the thermal boundary layer adjacent to the wall surface decreases constantly with increased Re. Therefore, the bigger Re is, the stronger the convective heat exchange performance in the cosinoidal duct is. Fig. 13(b) indicates that the thickness of the velocity boundary layer decreases gradually with increased Re. With increased Re and even at its highest value, the reverse flow or boundary layer separation has not occurred in the duct. The reason behind this is that the maximum slope angle in the duct fails to be up to the condition of boundary layer sep aration. The other obvious phenomenon is that the red high speed region in the middle expands forward with increased Re. The higher flow ve locity in the duct can also improve the convective heat transfer perfor mance. Taking uin ¼ 4 m/s (2A ¼ 5 mm, 7 mm and 9 mm; Re � 3140) as an example, the effect of local enlargement is shown as“A”in Fig. 13(c). As can be seen from the figure, there is no backflow or boundary sepa ration in the parabolic duct. 4.2.3. Distribution of temperature gradient on surface of cosinoidal duct Since the temperature distribution near the wall of the cambered duct along axis s cannot directly reflect the heat flux distribution which is related closely to the heat transfer efficiency, we chose temperature gradient to evaluate the heat transfer performance. Taking the cosinoi dal duct, 2A ¼ 9 mm, Lf ¼ 60 mm and n ¼ 5um (the surface of cosinoidal duct is the starting point of the coordinate (s, n, z), where n ¼ 0 μm) as an instance, the temperature gradient on the surface of the duct is analyzed. When the temperature gradient on the surface is bigger, the convective heat exchange performance on the surface is superior. Therefore, the temperature gradient is adopted as the standard to evaluate the heat flux is more beneficial than adopting the temperature value or the boundary layer thickness. In Fig. 14, the temperature gradient is in proportion relation with the heat flux which is perpendicular to the surface direc tion (n). Therefore, the analysis of the temperature gradient change is equivalent to analyze the heat transfer performance in different posi tions on the duct surface. Fig. 14 shows that the temperature gradient in the peak region (0 mm � s � 25 mm) is obviously bigger than that in the trough region (45 mm � s � 68 mm); in the area (25 mm < s < 45 mm) between the peak and trough regions, the gradient of the temperature changes slightly with increasing of s. It indicates that the convective heat exchange co efficient in the peak region is bigger than the convective heat exchange
Fig. 12. Performance comparison of different ducts with respect to Re. (a) j factor comparison; (b) f factor comparison; (c) Overall perfor mance comparison.
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Fig. 14. Temperature gradient distribution of cosinoidal duct (2A ¼ 9 mm, n ¼ 5um) along s direction.
thickness in this area is relatively thin, causing smaller thermal resis tance of heat conduction. It is noticeable that the temperature gradient in the middle area between the peak and trough regions changes slightly, which indicates relatively well-distributed convective heat ex change performance in this area. According to the analysis of Fig. 14, it is concluded that the main heat transfer area on the surface of the cosinoidal duct is the peak region and the middle area, with weaker heat transfer performance in the trough region. 5. Conclusions This paper analyzes the flow and heat transfer characteristics in five different cambered ducts through the experimental and numerical method. The Core Volume Goodness Factor (CVGF) is adopted for the comparative analysis of rectangular, trapezoidal, circular, parabolic and cosinoidal ducts. And the following conclusions can be reached: (1) The Core Volume Goodness Factor is adopted to compare five ducts. It is concluded that the sequence of their overall heat transfer performance is cosinoidal, parabolic, circular, trape zoidal and rectangular ducts successively from superior to inferior. (2) With the optimal overall heat transfer performance, the cosinoi dal duct is superior to that of the rectangular duct by 7.3%– 28.1%. The temperature difference distribution at the inlet and outlet of different ducts is analyzed, we can conclude that the uniformity distribution of temperature difference in the duct plays an active role in improving the heat transfer efficiency of the heat exchanger. (3) In the cosinoidal duct, the smaller the amplitude is, the better its overall heat transfer performance is. With Re increasing, the thicknesses of the temperature and velocity boundary layers decrease gradually and the reverse flow or boundary layer sep aration has not been occurred in the duct. (4) In the cosinoidal duct, the temperature gradient in the peak re gion is bigger than that in the trough region; however, the gradient of the temperature in the middle area between the peak and trough regions changes slightly. (5) With Reynolds number ranging from 1210 to 5080, the overall heat transfer performance of a gradually expanding duct is su perior to that of ducts with equal cross-sectional area.
Fig. 13. Influence of different Reynolds number on temperature and flow fields for cosinoidal duct. (a) Temperature field (2A ¼ 5 mm); (b) Flow field (2A ¼ 5 mm); (c) Velocity vector (2A ¼ 9 mm).
coefficients in the trough region and the middle area, i.e. the peak region has the maximum heat transfer efficiency. The reason behind this is that the air flow velocity adjacent to the peak region is higher; at the initial development stage of the boundary layer, the thermal boundary layer 12
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Acknowledgements
Foundation of China (51806114, 51874187) and China Postdoctoral Fund.
Financial support is provided by the National Natural Science
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijthermalsci.2019.106226. Nomenclature A b cp D Dh d1 , d 2 E Fh g H h L/Lf m_ P p Q R Sf T tf ΔT u
Heat transfer area [m2]; Amplitude [mm] Mean channel width Specific heat capacity [J/kg⋅K] Mean curvature diameter (mm) Hydraulic diameter [mm] Diameters of wetted lateral surfaces of the channel; inner and outer tubes of the annulus, respectively (mm) Fluid pumping power per unit surface area [W/m2] Inlet height [mm] Acceleration of gravity (m/s2) Channel height (mm) Heat transfer coefficient [W/m2⋅K] Fin length [mm] Mass flux [kg/s] Focal length [mm] Pressure [Pa] Average value of the heat flux [W] Circular radius [mm] Fin pitch [mm] Temperature [K] Fin thickness [mm] Temperature difference [K] Velocity [m/s]
Dimensionless f Fanning friction factor gc Proportionality constant in Newton’s second law of motion, gc ¼ 1 j Colburn factor Nu Nusselt number Pr Prandtl number Greek letters Ratio of total heat transfer area to the total volume of an exchanger [m2/m3] β Slope angle [� ] η0 Extended surface efficiency on one fluid side of the extended surface heat exchanger λ Heat conductivity [W/m⋅K] μ Dynamic viscosity [Pa⋅s] ρ Air density [kg/m3] σ Ratio of free flow area to frontal area ϕ Angle of channel swirling (� )
α
Subscripts A a B C D in ln max w
Inlet air Air side Outlet air Inlet water Outlet water inlet Logarithm Maximum value Water side
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