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Natural convection of plate finned tube heat exchangers with two horizontal tubes in a chimney: Experimental and numerical study
International Journal of Heat and Mass Transfer 147 (2020) 118948
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection of plate finned tube heat exchangers with two horizontal tubes in a chimney: Experimental and numerical study Han-Taw Chen ⇑, Wei-Xuan Ma, Pei-Yu Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan
a r t i c l e
i n f o
Article history: Received 17 June 2019 Received in revised form 2 September 2019 Accepted 22 October 2019
Keywords: Natural convection Plate finned tube Horizontaly aligned tubes IHCA and CFD
a b s t r a c t This study uses a hybrid method of computational fluid dynamics (CFD) and inverse heat conduction analysis (IHCA) combined with experimental temperatures to investigate the natural convection of plate finned tube heat exchangers with two horizontal tubes located in a chimney. The influence of the position of the two heated tubes on the results obtained is considered. IHCA combined with experimental temper and h . Subsequently, the CFD commercial software combined with the atures is used to estimate h b obtained inverse results and experimental temperatures is employed to get the present results. The results reveal that the zero-equation turbulence model can be applied to determine more accurate results than the other two flow models. Therefore, the appropriate flow model for this study is the zero-equation turbulence model. The fringe pattern for each heated tube is mainly aligned in the vertical direction and the two plumes are slightly inclined toward the inward direction. The obtained velocity pattern and air temperature contour are consistent with existing interferometric images or isotherms. The proposed correlation between Ra and Nu agrees with the obtained inverse and CFD results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and numerical methods have been applied to investigate the natural convection of plate finned tube heat exchangers. It is seen from Refs. [1–4] that due to the air plume above the heated horizontal tubes, very complex threedimensional (3D) natural convection velocity patterns and air temperature contours are presented in these heat exchangers. Therefore, the interaction of fluid flow and heat transfer within these heat exchangers is noteworthy. Choosing a suitable flow model and determining more accurate results are an important task in designing high-performance heat exchanger devices. As shown in Refs. [1,2], there are two large natural circulations in the upper right and left corners near the exit of the box. This intense circulation flow is helpful for transferring heat from the fins to the surrounding environment. A disadvantage of the traditional inverse heat conduction method (IHCM) is that the velocity and temperature fields of the problem under study may not be obtained. Owing to the lack of reliable heat transfer coefficient estimates and experimental temperatures, CFD is also not easily used to select a suitable flow model and to obtain more accurate numerical results. It is found
⇑ Corresponding author. E-mail address: [email protected] (H.-T. Chen). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118948 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
from Refs. [1–4] that a specific flow model is selected in advance and grid independence is often assumed. Chen et al. [1–4] applied mixed IHCM and 3D CFD commercial software [5] in combination with the measured temperatures to study natural convection of plate fin and tube heat exchanger. It is seen that a suitable flow model and more accurate results can be obtained. The assumption of grid independence may not be appropriate. As far as we know, few researchers applies this hybrid approach to investigate the problems studied. Sajedi et al. [6] used FLUENT in conjunction with experiments to study the optimal number of fins for externally extended finned tube heat exchangers. Kumar et al. [7] applied the OpenFOAM2.2 software based on the finite volume method to study the 3D laminar transient natural convection of the air around the annular finned tube in a small chimney. The temperature of the fins and tube is assumed to be constant. Senapati et al. [8] studied the laminar natural convection on an annular finned horizontal cylinder using FLUENT. It can be found that the results obtained are assumed to be independent of the grid points. The comparison between experimental and numerical fin temperatures is not made due to the use of high thermal conductivity aluminum fins. Liu et al. [9,10] used ANSYS FLUENT 15.0 to study the twodimensional (2D) laminar natural convection of two horizontally and vertically connected horizontal cylinders. Stafford and Egan [11] used scale analysis and ANSYS FLUENT in combination with
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Nomenclature d0 DH gj h(x,y),h h b
Hf keff kf L LH Nus
tube diameter (mm) distance between centers of two tubes (mm) gravitational acceleration component (m/s2) local and average heat transfer coefficients (W/m2 K) heat transfer coefficient at T 0 (W/m2 K) distance from the center of the tube to the entrance surface of the chimney (mm) effective thermal conductivity of the air (W/m K) thermal conductivity of fin (W/m K) fin length (mm) distance between the centers of two tubes (mm) Nusselt number, Nus ¼ hS=k a
experimental measurements to study the heat transfer characteristics of 2D laminar natural convection and cylinder interactions for horizontal and vertical alignment of horizontal cylinder pairs. Pelletier et al. [12] used ANSYS FLUENT 14.0 along with the ReynoldsAveraged Navier Stokes method to investigate the unsteady natural convection from a pair of isothermally vertically aligned heated horizontal cylinders in water. Narayan et al. [13] conducted an interferometric study on the phenomenon of natural convective around an array of heated cylinders. The cylinder surface temperature is maintained at four different temperatures of 323.15 K, 333.15 K, 338.15 K and 343.15 K with the ambient temperature of 298.15 K. Interference measurements of natural convection around single cylinder, two cylinders with various tube-to-tube distances and three cylinders with a triangular configuration are presented. They found that the buoyancy plumes above the cylinders were almost independent of each other with a larger centerto-center distance. However, as the spacing between the cylinders decreases, a strong coupling of plumes can be seen. Laminar flow is assumed in Refs. [6–12]. As shown in Refs. [1–4], the hybrid method of IHCM and CFD commercial software [5] combined with experimental temperatures can be utilized to obtain more accurate results. Therefore, this hybrid method [1–4] combined with experimental temperatures is used to investigate the present problem. To validate the
Ra S T, Ta Tk T0,1,T0,2 T 0 ui b
gf meff q
4
1 ÞS Rayleigh number, Ra ¼ gbðT 0mT aL fin spacing (mm) fin and air temperatures (K) experimental temperature at the kth fin location (K) fin base temperatures on the 1st and 2nd tubes (K) average temperature of two tubes, T 0 = (T0,1 + T0,2)/2 air velocity in the i direction (m/s) volumetric thermal expansion coefficient fin efficiency effective kinematic viscosity (m2/s) air density (kg/m3)
accuracy of the present results, the obtained numerical results of h and T are compared with experimental temperatures h, b k and inverse results. The obtained velocity pattern and air temperature contours are compared to the isotherms given by Stafford and Egan [11] and the interferometric images given by Narayan et al. [13]. 2. Inverse heat conduction analysis Fig. 1 presents the experimental setup for this study. T0,1 and T0,2, Tk and ambient temperature T1 are measured from the experimental setup of this study. The adiabatic boundary condition is assumed at the tip of the fin [1–4]. The inverse heat conduction method (IHCM) combined with experimental temperatures is uti and h . The distribution of sub-fin region and lized to estimate h b measurement location is shown in Fig. 2. To avoid repetition, the experimental method can be found in Refs. [1–4]. The 2D steadystate heat conduction equation of thin fins and corresponding boundary conditions under the assumption of constant thermal properties can be written as
@2T @2T 2hðx; yÞ þ ¼ ðT T 1 Þ 2 @x @y2 kf t
Fig. 1. Experimental setup in this study.
ð1Þ
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
3
to the desired temperature. The ambient air temperature and the fin temperatures at the selected locations are measured by the Ttype thermocouples. However, too many thermocouples are used on the fins, the air temperature and velocity distributions between the two adjacent fins may be significantly disturbed. The heat transfer behavior on the fins may also be affected. Therefore, an appropriate number of T-type thermocouples are used. The fin base temperature T0,n is the average of the temperatures measured at the four different fin base positions of the nth tube. The middle fin is divided into seven sub-fin regions. That is, the value of N is 7. T-type thermocouples with 0.4% accuracy are fixed at the selected fin locations to obtain the fin temperature measurements, as shown in Fig. 2. Two thermocouples are mounted on the walls of
Fig. 2. Distribution of sub-fin regions and fin measurement locations.
@T ¼0 @x
at x ¼ 0 and L
ð2Þ
@T ¼0 @y
at y ¼ 0 and L
ð3Þ
T ¼ T 0;n
on St;n for n ¼ 1; 2
ð4Þ
where h(x,y) includes radiation and convective heat transfer coefficients. t is the fin thickness. St,1 and St,2 represent the outer surfaces of the 1st and 2nd tubes, respectively. The difference equation of Eq. (1) in the kth sub-fin region can be written as follows.
T iþ1;j 2T i;j þ T i1;j T i;jþ1 2T i;j þ T i;j1 þ l2 l2 2h k ¼ for i; j ¼ 1; 2; ; Nx T i;j T 1 kf t
ð5Þ
where Nx is the number of nodes in the x direction. ‘ is equal to is considered to be constant and will be estimated. L/(Nx 1). h k for The entire fin is devided into N sub-fin regions to estimate h k k = 1,2,. . .,N. It is worth mentioning that the T0,1 value cannot be with varequal to the T0,2 value. The details for the prediction of h k ious sub-fin regions can be found in Refs. [1–4]. The relative error of Tk between the measured and numerical temperatures is less than 10-6. 3. Experimental methods and procedures A schematic diagram of this experimental setup with downward and upward openings, 220 mm wide, 220 mm long and 270 high. t, L, emissivity and kf of the AISI 304 stainless steel fin are 1 mm, 126 mm, 0.17 and 14.9 W/mK is presented in Fig. 1. The outer diameter, thickness and length of the tube are respectively 27.3 mm, 1 mm and 500 mm. Hf are 90 mm, 135 mm and 180 mm, respectively. The centers of two heated horizontal tubes are located at (L/4, L/2) and (3L/4, L/2) of the fins, respectively. That is, LH is 63 mm. Therefore, the values of Af, L/d0 and LH/d0 are 0.0147 m2, 4.62 and 2.31, respectively. A heated coil of 200 W power input is placed in two circular tubes to heat the test fins
Fig. 3. Computational domain and physical model with dashed lines. (a) x-y plane, (b) y-z plane.
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and
kf
@T ¼ hðT T 1 Þ at z ¼ t=2 @z
ð9Þ
Since T1 is lower than T0,1 and T0,2, the buoyancy effect is considered. To select a suitable flow model, laminar flow, zeroequation turbulence model and RNG k-e model are introduced. It is found in Ref. [4] that the zero-equation turbulence model is chosen to get the desired results. This result is different from the RNG k-e model of Chen et al. [1] and the laminar flow of Kumar et al. [7], Liu et al. [9,10] and Stafford and Egan [11]. This implies that the selection of a suitable flow model can depend on the physical geometry of the problem being study. The 3D natural convection heat transfer of this study was investigated using ANSYS FLUENT 15.0 [5]. The 3D steady-state governing differential equations for air velocity and temperature under the assumption of constant thermal properties, incompressible and non-viscous dissipation can be written as follows. [1–4]
@ui ¼0 @xi
ð10Þ
Fig. 4. Distribution of grid points.
uj the chimney at the upper and lower openings to measure the ambient air temperature T1. T1 is the average of the two temperature measurements. Four thermocouples are attached to the four different fin base locations of each tube. T0,n is the average temperature measured from four thermocouples on the nth tube. Experimental method and procedures refer to Refs. [1–4]. 4. Numerical analysis A disadvantage of the inverse heat conduction problem is that the fluid velocity pattern and temperature contours of the problem under investigation cannot be obtained. To compensate for this shortcoming, ANSYS FLUENT 15 [5] combined with the obtained inverse results and measured air and fin temperatures is introduced into this study. Radiation heat transfer is included. The boundary conditions in the x and y directions are given in Eqs. (2)–(4). The 3D steady-state heat conduction equation of the fins and the corresponding boundary conditions in z directon can be expressed as follows.
r2 T ¼ 0
ð7Þ
@T ¼ 0 at z ¼ 0 @z
ð8Þ
@ui 1 @p @ 2 ui ¼ þ meff þ g j bdj2 ðT a T 1 Þ @xj q @xj @x2j
ð11Þ
and
cp uj
@T a keff @ 2 T a ¼ @xj q @x2j
ð12Þ
where ui, p, Ta, q, b, cp and keff are the velocity component, pressure, temperature, density, volumetric thermal expansion coefficient, specific heat and effective thermal conductivity of the air, respectively. gj represents the component of gravitational acceleration. x1, x2 and x3 respectively represent x, y and z. dj2 is the Kronecker delta function. The definition of the effective kinematic viscosity meff can be found in Ref. [4]. 4.1. Boundary conditions The viscous dissipation rate ein and the inlet turbulence kinetic energy kin are both given as 0.5 [1–4]. Half of the chimney is taken as the computational domain, as shown by the dashed line in Fig. 3. At the inlet boundary, the air temperature and air velocity are taken as the ambient temperature and zero, respectively. At the exit boundary, the gradient of air temperature and air velocity is assumed to be zero. The pressure is considered to be atmospheric pressure. The adiabatic boundary conditions at the tip of the fin
Table 1 Comparison of results obtained from three different flow models with S = 15 mm. Hf = 90 mm, T1 = 299.89 K T0,1 = 352.01 K, T0,2 = 351.84 K
Hf = 180 mm, T1 = 300.28 K T0,1 = 352.37 K, T0,2 = 351.79 K
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h
330.60 330.71 337.17 335.68 336.70 319.37 319.39 7.17
332.80 332.89 339.21 338.29 338.78 322.28 322.32 5.51
326.08 326.13 329.85 338.99 329.06 316.11 316.11 13.60
(W/m2K) h b Q (W)
3.74
3.19
5.73 52.1% 143,032
4.89 57.9%
gf Nt
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
331.61 331.99 338.97 337.01 338.85 321.24 321.19 6.49 (Inv.)
331.36 331.29 338.09 336.39 337.36 320.59 320.58 6.66
333.14 333.07 339.69 338.58 339.01 322.81 322.82 5.45
328.57 328.56 330.61 339.65 329.64 316.79 316.76 12.10
332.48 332.31 340.52 336.82 339.99 322.77 322.33 6.03 (Inv.)
6.04
3.68 (Inv.)
3.56
3.13
5.54
3.53 (Inv.)
9.26 44.4%
5.64 (Inv.) 56.7% –
5.44 53.5% 140,032
4.77 57.4%
8.46 45.8%
5.44 (Inv.) 58.5% –
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948 Table 2 Comparison of results obtained from three different flow models with S = 20 mm. Hf = 90 mm, T1 = 300.28 K T0,1 = 350.98 K, T0,2 = 350.03 K
Hf = 180 mm, T1 = 300.09 K T0,1 = 351.89 K, T0,2 = 351.94 K
CFD results
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h (W/m2K) h b Q (W)
gf Nt
Exp. data
Zero- Eq.
Laminar
RNG k-e
329.33 329.08 336.63 334.62 335.65 319.33 319.15 7.12
332.27 331.99 338.68 337.05 337.85 322.08 321.91 5.43
323.33 323.10 328.13 337.15 326.99 314.61 314.42 15.1
3.79
3.11
5.60 53.20% 143,812
4.59 57.18%
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
330.21 330.02 336.98 335.12 336.49 319.97 319.78 7.17 (Inv.)
329.70 330.32 337.04 335.53 337.42 320.04 320.48 6.66
332.76 332.94 339.40 338.09 339.21 322.47 322.67 5.41
325.74 326.31 328.50 338.17 328.41 314.76 315.08 13.3
330.00 329.62 338.07 336.73 338.35 320.33 320.77 6.49 (Inv.)
6.23
4.02 (Inv.)
3.59
3.12
5.88
3.72 (Inv.)
9.22 41.25%
5.94 (Inv.) 56.07% –
5.48 53.85% 143,812
4.77 57.71%
8.98 44.21%
5.68 (Inv.) 57.32% –
Fig. 5. Laminar velocity pattern on z = S/2 plane with S = 15 mm. Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
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Fig. 6. Velocity pattern on z = S/2 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
and the sidewalls of the chimney are assumed. No-slip conditions are imposed on the surface of all fins, circular tubes and sidewalls of the chimney. @u3/@z, @Ta/@z and @p/@z at the symmetry of the chimney are given as zero. The matching conditions of heat flux and temperature at the interface between the fin and the air are expressed as follows.
T ¼ Ta
and kf
@T @T a ¼ ka @z @z
ð13Þ
,h and g can be approximated as follows. [1–4] Q, h b f
Q 2
N X
and
ð15Þ 2
where Af is Af ¼ L2 pd0 =2. Q is the heat transfer rate dissipated by and Tave are the average the entire fin. gf is the fin efficiency. Tave k temperatures in the kth sub-fin region and the entire fin, respec are obtained by Eq. (14). The FLUENT results for h̅ and h tively. h b and Q can be obtained according to the definition in Ref. [4]. 5. Typical grid distribution
ðT av e T Þ ¼ 2A h Ak h 1 k f b ðT 0 T 1 Þ k
k¼1
¼ 2Af hðT av e T 1 Þ
h
gf ¼ b h
ð14Þ
An unstructured grid system with non-uniformly distributed grid points is utilized to obtain FLUENT results. This grid system is presented in Fig. 4. The number of grid points Nx, Ny and Nz in the x, y and z directions on the fins are 40, 40 and 4 with Hf = 90 mm, and 34, 34 and 4 with Hf = 180 mm, respectively. The
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
7
Fig. 7. Air temperature contour on z = S/2 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
number of grid points Nza in the z direction of the air region between two adjacent fins are 9 with Hf = 90 mm, and 7 with Hf = 180 mm, respectively. Nx and Ny on the fin are the same as those of the air region between two adjacent fins for FLUENT 15. Both Nx and Ny are taken as 29 for IHCM. Therefore, Nx and Ny of FLUENT 15 may differ from those of IHCM. The total number of grid points Nxt, Nyt and Nzt in the x, y and z directions are taken as 50, 61 and from 31 to 35 with Hf = 90 mm and 180 mm and 50, 68 and from 31 to 36 with Hf = 135 mm. As shown in Refs. [3,4], the total error involved in the finite difference calculation includes the discretization error plus the round-off error. Due to round-off error considerations, the choice of relative convergence criteria and number of iterations may require caution. The difference between the numerical results obtained by the convergence criteria of 10-4, 10-5 and 10-6 was found to be negligible. Therefore, in order to save computation time, the relative convergence criteria of the momentum and energy equations is given as 10-4. In other words, 10-4 is a good choice for the momentum and energy equations. The number of iterations was set to 750 to obtain the present result for each case. The appropriate choice of relative convergence criteria, the number
of iterations, and the number of grid points may need to be based and h.. on experimental temperatures and estimates of h b
6. Results and discussion Thermophysical properties of this study are determined by (T 0 +T1)/2, where T 0 is defined as T 0 ¼ ðT 0;1 þ T 0;2 Þ=2. Tables 1 and 2 present the comparison between the FLUENT results, the inverse results and experimental temperatures with Hf = 90 mm and 135 mm for S = 15 mm and 20 mm, respectively. It is found in Tables 1 and 2 that Tk obtained by the laminar flow and the zero-equation turbulence model is closer to the experimental tem h , Q and perature than that obtained by the RNG k-e model. But, h, b gf obtained by the zero-equation turbulence model are more consistent with the inverse results than those obtained by the laminar flow and the RNG k-e model. Therefore, the zero-equation turbulence model is chosen to get the desired results. The selection of an appropriate flow model may need to be based on more accurate and h and experimental temperatures. h b
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
Fig. 8. Air temperature contour on x = 0 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 180 mm.
The numerical results obtained from the zero-equation turbulence model are shown in Figs. 6–12 and Tables 3–5 with various S and Hf values. The velocity pattern obtained by laminar flow with S = 15 mm on the z = S/2 plane is presented in Fig. 5 with S = 15 mm. Fig. 6 displays the velocity pattern on the z = S/2 plane with S = 15 mm. It is found that the maximum and low velocity regions occur respectively in the intermediate region of the exit of the chimney and in the plume region. The maximum velocity decreases with increasing Hf value. The air velocity at the fin tip
with Hf = 180 mm is less than Hf = 90 mm. The faster velocity region with Hf = 90 mm is greater than Hf = 180 mm. It is expected that the fins cooled by cold air with Hf = 90 mm are earlier than Hf = 180 mm. In other words, buoyancy caused by the fins with Hf = 90 mm is stronger than Hf = 180 mm. This implies that the chimney effect with Hf = 90 mm is slightly stronger than Hf = 180 mm. The fins with larger Hf values exhibit greater air thermal resistance. The air velocity between the two plumes is faster than in the plume region. Thus, the velocity pattern for each heated cylinder is
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
9
Fig. 9. Fin surface temperature contour with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
mainly aligned in the vertical direction. The two plumes are slightly inclined towards the inward direction due to the pressure difference and appear to attract each other in the region above the cylinder. The direction of the convective plume looks approximately symmetrical. The velocity patterns of Figs. 5 and 6 are similar except for the intermediate region of the exit of the chimney and the outer regions of the fins near the two tubes. An important finding is that the velocity pattern in Figs. 5 and 6 coincides with interferometric image of the convective field in Fig. 4 of Ref. [13] in the fin region. Thus, the numerical results obtained have good accuracy. It is observed from Figs. 5 and 6 that the natural convective boundary layer flow is caused by two heated tubes and fins moves upwards. The two plumes above the two heated tubes belong to the low-velocity region. Due to the difference in velocity between the two plume regions and the central region surrounded by the two tubes, the boundary layer starts separating from the oblique line by about 30 degree from the vertical line. The cold air in the environment flows upward through the entrance of the chimney. This flow field differs from that in Ref. [1] for a box with only one top opening. In addition, there are two natural circulations found in the upper right and left corners near the exit of the
chimney. This phenomenon is similar to that of Ref. [4] with only a single tube. The 3D flow field and heat transfer phenomenon of this study is very complicated. Table 3 presents the influence of grid points on the results obtained with Hf = 180 mm and S = 15 mm and 20 mm. It is seen h and T values are more consistent with that the obtained h, b k the inverse results and experimental temperatures using Nx = 34, Ny = 34, Nzf = 4 and Nza = 7 with S = 15 mm and Nx = 34, Ny = 34, Nzf = 4 and Nza = 9 with S = 20 mm compared to other grid points. It is found from Table 5 that Nt for 5 mm S 30 mm is increased from 127,382 to 143,946 with Hf = 90 mm and 180 mm and from 139,176 to 163,016 with Hf = 135 mm, respectively. Nt can vary with S and Hf. Therefore, the assumption of grid independence may not be appropriate. for various H and S values. Table 4 gives the inverse results of h k f It is observed that h1 and h2 are the smallest in the plume regions 3 and h 5 are the largest in the compared to other sub-fin regions. h outer regions of the two tubes compared to the other sub-fin regions. The correlation between Ra and Nus proposed based on the results obtained in Table 5 can be expressed as follows for 18 < Ra < 22,000.
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
Fig. 10. Distribution of h(x,y) with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
with S. Fig. 11. Variation of h
with S. Fig. 12. Variation of h b
11
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948 Table 3 Effect of grid points on the results obtained with Hf = 180 mm. Nx Ny Nz Nza S = 15 mm
S = 20 mm
34 34 4 7
34 33 4 6
34 33 3 9
34 34 4 9
34 33 4 12
34 33 3 15
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h
331.36 331.29 338.09 336.39 337.36 320.59 320.58 6.66
330.75 330.63 337.48 335.75 336.73 320.30 320.28 6.67
331.39 331.22 338.02 336.30 337.24 320.50 320.40 6.77
329.70 330.32 337.04 335.53 337.42 320.04 320.48 6.66
329.58 330.22 337.80 335.84 337.45 320.47 320.58 6.74
329.49 330.25 337.73 335.80 337.46 320.37 320.49 6.78
(W/m2K) h b Q (W) Nt
3.56
3.57
3.62
3.59
3.63
3.65
5.44 140,032
5.45 158,180
5.53 186,916
5.48 143,812
5.54 162,552
5.58 192,068
Table 4 with various S and H values. Inverse results of h k f Hf = 90 mm
Hf = 180 mm
S = 5 mm
S = 20 mm
S = 5 mm
S = 20 mm
(W/m2K) h 1 (W/m2K) h 2 (W/m2K) h
1.73
3.84
2.20
3.85
1.58
3.74
1.97
4.10
11.23
15.15
4.49
11.90
(W/m2K) h 4 (W/m2K) h 5 (W/m2K) h
6.00
10.40
3.02
7.72
10.85
14.78
7.31
11.35
7.26
10.10
5.34
9.40
(W/m2K) h 7
7.02
9.99
5.74
8.96
3
6
Table 5 and h for various S and H values. Comparison of h b f (W/m2K) h b
(W/m2K) h
Hf(mm)
S (mm)
Ra
Nt
Inv.
CFD
Eq. (16)
Inv.
CFD
90
5 10 15 20 30
17.49 278.94 1406.38 4320.88 22051.16
127,382 135,306 143,032 143,812 148,964
4.49 5.28 6.49 7.17 8.22
3.23 6.57 7.17 7.12 7.14
3.22 6.35 7.10 7.36 7.84
2.99 3.33 3.68 4.02 4.37
2.26 3.72 3.74 3.74 3.73
135
5 10 15 20 30
17.65 281.10 1394.13 4446.81 21829.60
139,176 148,448 156,664 157,456 163,016
4.06 4.98 6.24 6.81 7.94
3.15 6.38 6.91 6.93 6.88
3.04 5.98 6.67 7.02 7.36
2.77 3.18 3.65 4.04 4.17
2.21 3.57 3.66 3.63 3.64
180
5 10 15 20 30
17.42 273.93 1394.74 4421.09 21542.29
127,382 135,306, 143,032 143,812 148,964
3.59 4.76 6.03 6.49 7.91
2.97 6.11 6.66 6.66 6.62
2.84 5.55 6.24 6.55 6.86
2.26 3.02 3.53 3.72 4.20
1.98 3.47 3.56 3.59 3.58
2d0 2d0 ð1 Þð0:255 þ 0:28Ra1=3 Þ½1 L L 1:4e200S ½0:78 þ 0:27ejð100S1j
Nus ¼ ð5:77 6:8Hf Þ
ð16Þ
gbðT 0 T 1 ÞS4 where Nus and Ra are defined as Nus ¼ hS=k . a and Ra ¼ maL To validate the availability and reliability of the desired results, obtained by FLUENT 15, IHCM and corre and h a comparison of h b lation (16) is presented in Table 5. It is found that the FLUENT and h are consistent with inverse results. h , h and results of h b b gf increase as S increases, and approach a constant. The optimum fin spacing is approximately 15 mm. Another interesting finding and h with H = 90 mm are higher those with H = 180 mm is that h b f f
due to the effect of buoyancy. gf decreases with increasing S. gf for 5 mm S 30 mm is reduced from 70.0% to 52.2% with Hf = 90 mm, from 66.7% to 52.9% with Hf = 135 mm and from 66.7% to 53.5% with Hf = 180 mm. But, it is not very sensitive to Hf. Because of the combination of the thermal boundary layer and h will become smaller for small between two adjacent fins, h b S values. The correlation (16) is close to the obtained FLUENT and inverse results. The maximum error between correlation (16) and FLUENT results is approximately 9%. Therefore, the correlation (16) has good reliability and availability. The air temperature contours on the z = S/2 plane with S = 15 mm and various Hf values is presented in Fig. 7. It is observed that the air temperature profile exhibits the non-
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
uniform distribution in the air region between two adjacent fins and a strong interaction with each other between the two tubes. The merging of the thermal boundary layers on the two tubes is distinguished in the region surrounded by them. The air temperature distribution between the two tubes is significantly influenced by the thermal plumes originating from them. This also causes an increase in the temperature of this region. In this region, the air temperature is higher and the air speed is lower. Another important finding is that the air temperature contour of Fig. 7 for DH/ d0 = 2.31 is consistent with Fig. 13 in Ref. [11] for DH/d0 = 0.28 and Fig. 7 in Ref. [13] for DH/d0 = 2.5 in the fin region. This comparison further confirms the accuracy of the present results. The air temperature between two heated tubes with Hf = 180 mm is lower than that with Hf = 90 mm. Air temperature is almost at 306 K for Hf = 180 mm and is approximately at 313 K for Hf = 90 mm. The air temperature contours on the = 0 plane with S = 15 mm and Hf = 90 mm and 180 mm are presented in Fig. 8. As the cold air moves upward from the entrance of the chimney, the thermal boundary layer gradually thickens from the bottom of the fin. The thermal boundary layers in the fin region is found to merge with each other. This phenomenon causes a decrease in the heat transfer coefficient on the fins above the tubes. As shown in Table 3, 4 is less than h 3 and h 5 . Air may not easily flow into the environh ment from the tip of the fin. The fin surface temperature distribution are presented in Fig. 9 for various Hf values and S = 15 mm. The uneven distribution of fin temperature is found. The interaction of two heated tubes causes an increase in the temperature of the air around them. Thus, it is found in Table 1 that the measured temperatures are higher from the 3rd to 5th sub-fin regions than in the other sub-fin regions. The measured temperatures are lower in the 6th and 7th fin regions than in the other sub-fin regions. In other words, the fin temperature in the lower region of the tube is lower than that in its upper region. T4 is lower than T3 and T5. Higher temperatures are also found near the base of the fin. The distribution of h(x,y) with S = 15 mm is presented in Fig. 10. Fig. 10 exhibits that the heat transfer coefficient is lowest in the plume region. However, it is not always highest in the bottom region of the tube. The above results are consistent with those in Table 3. This consistency further implies that the results obtained are reliable. and h obtained by FLUFigs. 11 and 12 show the variation of h b and h increase ENT 15 [5] with S, respectively. It can be seen that h b with increasing S and approach a value obtained at approximately and h decrease as H increases. S = 15 mm. h b f 7. Conclusions The hybrid method of IHCM and CFD combined with experimental temperatures is proposed to determine the desired results. and h obtained by the zero-equation turbulence It is found that h b model are more consistent with the inverse and FLUENT results and h than those of the RNG k-e model and laminar flow. h b increase with decreasing Hf and increasing S. gf decreases with increasing S. But, it is not very sensitive to Hf. There are two natural circulations in the upper right and upper left corners near the exit
of the chimney. The fringe pattern for each heated tube is mainly aligned in the vertical direction and the two plumes are slightly inclined toward the inward direction. The air temperature in the inner half of the two heated tubes displays a strong interaction with each other and the merging of the thermal boundary layers between them is prominent. It is worth mentioning that the obtained velocity pattern and air temperature contour are consistent with the isotherm of Ref. [11] or the interferometric image of Ref. [13] in the fin region. Nt can vary with S and Hf. This observation means that the present results have good reliability and accuracy, and the assumption of grid independence may not be appropriate. Appropriate selection of the relative convergence criteria for momentum and energy equations needs to be based on and h. The proexperimental temperatures and estimates of h b posed correlation between Ra and Nus agrees with the obtained FLUENT and inverse results in the range of 18 < Ra < 22,000. The maximum error between correlation (16) and FLUENT results is approximately 9%. Therefore, this correlation can be very useful to industrial designers. Declaration of Competing Interest The authors declared that there is no conflict of interest. References [1] H.T. Chen, Y.S. Lin, P.C. Chen, J.R. Chang, Numerical and experimental study of natural convection heat transfer characteristics for vertical plate fin and tube heat exchangers with various tube diameters, Int. J. Heat Mass Transfer 100 (2016) 320–331. [2] H.T. Chen, Y.J. Chiu, C.S. Liu, J.R. Chang, Numerical and experimental study of natural convection heat transfer characteristics for vertical annular finned tube heat exchanger, Int. J. Heat Mass Transfer 109 (2017) 378–392. [3] H.T. Chen, Y.L. Hsieh, P.C. Chen, Y.F. Lin, K.C. Liu, Numerical simulation of natural convection heat transfer for annular elliptical finned tube heat exchanger with experimental data, Int. J. Heat Mass Transfer 127 (2018) 541–554. [4] H.T. Chen, H.Y. Chou, H.C. Tseng, J.R. Chang, Numerical study on natural convection heat transfer of annular finned tube heat exchanger in chimney with experimental data, Int. J. Heat Mass Transfer 127 (2018) 483–496. [5] Fluid Dynamics Software, FLUENT 15, Lebanon, NH-USA, 2013. [6] R. Sajedi, M. Taghilou, M. Jafari, Experimental and numerical study on the optimal fin numbering in an external extended finned tube heat exchanger, Appl. Therm. Eng. 83 (2015) 139–146. [7] A. Kumar, J.B. Joshi, A.K. Nayak, P.K. Vijayan, 3D CFD simulations of air cooled condenser. Part II: natural draft around a single finned tube kept in a small chimney, Int. J. Heat Mass Transfer 92 (2016) 507–522. [8] J.R. Senapati, S.K. Dash, S. Roy, Numerical investigation of natural convection heat transfer over annular finned horizontal cylinder, Int. J. Heat Mass Transfer 96 (2016) 330–345. [9] J. Liu, H. Liu, Q. Zhen, W.Q. Lu, Numerical investigation of the laminar natural convection heat transfer from two horizontally attached horizontal cylinders, Int. J. Heat Mass Transfer 104 (2017) 517–532. [10] J. Liu, H. Liu, Q. Zhen, W.Q. Lu, Laminar natural convection heat transfer from a pair of attached horizontal cylinders set in a vertical array, Appl. Therm. Eng. 115 (2017) 1004–1018. [11] J. Stafford, V. Egan, Configurations for single-scale cylinder pairs in natural convection, Int. J. Therm. Sci. 84 (2014) 62–74. [12] Q. Pelletier, D.B. Murray, T. Persoons, Unsteady natural convection heat transfer from a pair of vertically aligned horizontal cylinders, Int. J. Heat Mass Transfer 95 (2016) 693–708. [13] S. Narayan, A.K. Singh, A. Srivastava, Interferometric study of natural convection heat transfer phenomena around array of heated cylinders, Int. J. Heat Mass Transfer 109 (2017) 278–292.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection of plate finned tube heat exchangers with two horizontal tubes in a chimney: Experimental and numerical study Han-Taw Chen ⇑, Wei-Xuan Ma, Pei-Yu Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan
a r t i c l e
i n f o
Article history: Received 17 June 2019 Received in revised form 2 September 2019 Accepted 22 October 2019
Keywords: Natural convection Plate finned tube Horizontaly aligned tubes IHCA and CFD
a b s t r a c t This study uses a hybrid method of computational fluid dynamics (CFD) and inverse heat conduction analysis (IHCA) combined with experimental temperatures to investigate the natural convection of plate finned tube heat exchangers with two horizontal tubes located in a chimney. The influence of the position of the two heated tubes on the results obtained is considered. IHCA combined with experimental temper and h . Subsequently, the CFD commercial software combined with the atures is used to estimate h b obtained inverse results and experimental temperatures is employed to get the present results. The results reveal that the zero-equation turbulence model can be applied to determine more accurate results than the other two flow models. Therefore, the appropriate flow model for this study is the zero-equation turbulence model. The fringe pattern for each heated tube is mainly aligned in the vertical direction and the two plumes are slightly inclined toward the inward direction. The obtained velocity pattern and air temperature contour are consistent with existing interferometric images or isotherms. The proposed correlation between Ra and Nu agrees with the obtained inverse and CFD results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and numerical methods have been applied to investigate the natural convection of plate finned tube heat exchangers. It is seen from Refs. [1–4] that due to the air plume above the heated horizontal tubes, very complex threedimensional (3D) natural convection velocity patterns and air temperature contours are presented in these heat exchangers. Therefore, the interaction of fluid flow and heat transfer within these heat exchangers is noteworthy. Choosing a suitable flow model and determining more accurate results are an important task in designing high-performance heat exchanger devices. As shown in Refs. [1,2], there are two large natural circulations in the upper right and left corners near the exit of the box. This intense circulation flow is helpful for transferring heat from the fins to the surrounding environment. A disadvantage of the traditional inverse heat conduction method (IHCM) is that the velocity and temperature fields of the problem under study may not be obtained. Owing to the lack of reliable heat transfer coefficient estimates and experimental temperatures, CFD is also not easily used to select a suitable flow model and to obtain more accurate numerical results. It is found
⇑ Corresponding author. E-mail address: [email protected] (H.-T. Chen). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118948 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
from Refs. [1–4] that a specific flow model is selected in advance and grid independence is often assumed. Chen et al. [1–4] applied mixed IHCM and 3D CFD commercial software [5] in combination with the measured temperatures to study natural convection of plate fin and tube heat exchanger. It is seen that a suitable flow model and more accurate results can be obtained. The assumption of grid independence may not be appropriate. As far as we know, few researchers applies this hybrid approach to investigate the problems studied. Sajedi et al. [6] used FLUENT in conjunction with experiments to study the optimal number of fins for externally extended finned tube heat exchangers. Kumar et al. [7] applied the OpenFOAM2.2 software based on the finite volume method to study the 3D laminar transient natural convection of the air around the annular finned tube in a small chimney. The temperature of the fins and tube is assumed to be constant. Senapati et al. [8] studied the laminar natural convection on an annular finned horizontal cylinder using FLUENT. It can be found that the results obtained are assumed to be independent of the grid points. The comparison between experimental and numerical fin temperatures is not made due to the use of high thermal conductivity aluminum fins. Liu et al. [9,10] used ANSYS FLUENT 15.0 to study the twodimensional (2D) laminar natural convection of two horizontally and vertically connected horizontal cylinders. Stafford and Egan [11] used scale analysis and ANSYS FLUENT in combination with
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Nomenclature d0 DH gj h(x,y),h h b
Hf keff kf L LH Nus
tube diameter (mm) distance between centers of two tubes (mm) gravitational acceleration component (m/s2) local and average heat transfer coefficients (W/m2 K) heat transfer coefficient at T 0 (W/m2 K) distance from the center of the tube to the entrance surface of the chimney (mm) effective thermal conductivity of the air (W/m K) thermal conductivity of fin (W/m K) fin length (mm) distance between the centers of two tubes (mm) Nusselt number, Nus ¼ hS=k a
experimental measurements to study the heat transfer characteristics of 2D laminar natural convection and cylinder interactions for horizontal and vertical alignment of horizontal cylinder pairs. Pelletier et al. [12] used ANSYS FLUENT 14.0 along with the ReynoldsAveraged Navier Stokes method to investigate the unsteady natural convection from a pair of isothermally vertically aligned heated horizontal cylinders in water. Narayan et al. [13] conducted an interferometric study on the phenomenon of natural convective around an array of heated cylinders. The cylinder surface temperature is maintained at four different temperatures of 323.15 K, 333.15 K, 338.15 K and 343.15 K with the ambient temperature of 298.15 K. Interference measurements of natural convection around single cylinder, two cylinders with various tube-to-tube distances and three cylinders with a triangular configuration are presented. They found that the buoyancy plumes above the cylinders were almost independent of each other with a larger centerto-center distance. However, as the spacing between the cylinders decreases, a strong coupling of plumes can be seen. Laminar flow is assumed in Refs. [6–12]. As shown in Refs. [1–4], the hybrid method of IHCM and CFD commercial software [5] combined with experimental temperatures can be utilized to obtain more accurate results. Therefore, this hybrid method [1–4] combined with experimental temperatures is used to investigate the present problem. To validate the
Ra S T, Ta Tk T0,1,T0,2 T 0 ui b
gf meff q
4
1 ÞS Rayleigh number, Ra ¼ gbðT 0mT aL fin spacing (mm) fin and air temperatures (K) experimental temperature at the kth fin location (K) fin base temperatures on the 1st and 2nd tubes (K) average temperature of two tubes, T 0 = (T0,1 + T0,2)/2 air velocity in the i direction (m/s) volumetric thermal expansion coefficient fin efficiency effective kinematic viscosity (m2/s) air density (kg/m3)
accuracy of the present results, the obtained numerical results of h and T are compared with experimental temperatures h, b k and inverse results. The obtained velocity pattern and air temperature contours are compared to the isotherms given by Stafford and Egan [11] and the interferometric images given by Narayan et al. [13]. 2. Inverse heat conduction analysis Fig. 1 presents the experimental setup for this study. T0,1 and T0,2, Tk and ambient temperature T1 are measured from the experimental setup of this study. The adiabatic boundary condition is assumed at the tip of the fin [1–4]. The inverse heat conduction method (IHCM) combined with experimental temperatures is uti and h . The distribution of sub-fin region and lized to estimate h b measurement location is shown in Fig. 2. To avoid repetition, the experimental method can be found in Refs. [1–4]. The 2D steadystate heat conduction equation of thin fins and corresponding boundary conditions under the assumption of constant thermal properties can be written as
@2T @2T 2hðx; yÞ þ ¼ ðT T 1 Þ 2 @x @y2 kf t
Fig. 1. Experimental setup in this study.
ð1Þ
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
3
to the desired temperature. The ambient air temperature and the fin temperatures at the selected locations are measured by the Ttype thermocouples. However, too many thermocouples are used on the fins, the air temperature and velocity distributions between the two adjacent fins may be significantly disturbed. The heat transfer behavior on the fins may also be affected. Therefore, an appropriate number of T-type thermocouples are used. The fin base temperature T0,n is the average of the temperatures measured at the four different fin base positions of the nth tube. The middle fin is divided into seven sub-fin regions. That is, the value of N is 7. T-type thermocouples with 0.4% accuracy are fixed at the selected fin locations to obtain the fin temperature measurements, as shown in Fig. 2. Two thermocouples are mounted on the walls of
Fig. 2. Distribution of sub-fin regions and fin measurement locations.
@T ¼0 @x
at x ¼ 0 and L
ð2Þ
@T ¼0 @y
at y ¼ 0 and L
ð3Þ
T ¼ T 0;n
on St;n for n ¼ 1; 2
ð4Þ
where h(x,y) includes radiation and convective heat transfer coefficients. t is the fin thickness. St,1 and St,2 represent the outer surfaces of the 1st and 2nd tubes, respectively. The difference equation of Eq. (1) in the kth sub-fin region can be written as follows.
T iþ1;j 2T i;j þ T i1;j T i;jþ1 2T i;j þ T i;j1 þ l2 l2 2h k ¼ for i; j ¼ 1; 2; ; Nx T i;j T 1 kf t
ð5Þ
where Nx is the number of nodes in the x direction. ‘ is equal to is considered to be constant and will be estimated. L/(Nx 1). h k for The entire fin is devided into N sub-fin regions to estimate h k k = 1,2,. . .,N. It is worth mentioning that the T0,1 value cannot be with varequal to the T0,2 value. The details for the prediction of h k ious sub-fin regions can be found in Refs. [1–4]. The relative error of Tk between the measured and numerical temperatures is less than 10-6. 3. Experimental methods and procedures A schematic diagram of this experimental setup with downward and upward openings, 220 mm wide, 220 mm long and 270 high. t, L, emissivity and kf of the AISI 304 stainless steel fin are 1 mm, 126 mm, 0.17 and 14.9 W/mK is presented in Fig. 1. The outer diameter, thickness and length of the tube are respectively 27.3 mm, 1 mm and 500 mm. Hf are 90 mm, 135 mm and 180 mm, respectively. The centers of two heated horizontal tubes are located at (L/4, L/2) and (3L/4, L/2) of the fins, respectively. That is, LH is 63 mm. Therefore, the values of Af, L/d0 and LH/d0 are 0.0147 m2, 4.62 and 2.31, respectively. A heated coil of 200 W power input is placed in two circular tubes to heat the test fins
Fig. 3. Computational domain and physical model with dashed lines. (a) x-y plane, (b) y-z plane.
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and
kf
@T ¼ hðT T 1 Þ at z ¼ t=2 @z
ð9Þ
Since T1 is lower than T0,1 and T0,2, the buoyancy effect is considered. To select a suitable flow model, laminar flow, zeroequation turbulence model and RNG k-e model are introduced. It is found in Ref. [4] that the zero-equation turbulence model is chosen to get the desired results. This result is different from the RNG k-e model of Chen et al. [1] and the laminar flow of Kumar et al. [7], Liu et al. [9,10] and Stafford and Egan [11]. This implies that the selection of a suitable flow model can depend on the physical geometry of the problem being study. The 3D natural convection heat transfer of this study was investigated using ANSYS FLUENT 15.0 [5]. The 3D steady-state governing differential equations for air velocity and temperature under the assumption of constant thermal properties, incompressible and non-viscous dissipation can be written as follows. [1–4]
@ui ¼0 @xi
ð10Þ
Fig. 4. Distribution of grid points.
uj the chimney at the upper and lower openings to measure the ambient air temperature T1. T1 is the average of the two temperature measurements. Four thermocouples are attached to the four different fin base locations of each tube. T0,n is the average temperature measured from four thermocouples on the nth tube. Experimental method and procedures refer to Refs. [1–4]. 4. Numerical analysis A disadvantage of the inverse heat conduction problem is that the fluid velocity pattern and temperature contours of the problem under investigation cannot be obtained. To compensate for this shortcoming, ANSYS FLUENT 15 [5] combined with the obtained inverse results and measured air and fin temperatures is introduced into this study. Radiation heat transfer is included. The boundary conditions in the x and y directions are given in Eqs. (2)–(4). The 3D steady-state heat conduction equation of the fins and the corresponding boundary conditions in z directon can be expressed as follows.
r2 T ¼ 0
ð7Þ
@T ¼ 0 at z ¼ 0 @z
ð8Þ
@ui 1 @p @ 2 ui ¼ þ meff þ g j bdj2 ðT a T 1 Þ @xj q @xj @x2j
ð11Þ
and
cp uj
@T a keff @ 2 T a ¼ @xj q @x2j
ð12Þ
where ui, p, Ta, q, b, cp and keff are the velocity component, pressure, temperature, density, volumetric thermal expansion coefficient, specific heat and effective thermal conductivity of the air, respectively. gj represents the component of gravitational acceleration. x1, x2 and x3 respectively represent x, y and z. dj2 is the Kronecker delta function. The definition of the effective kinematic viscosity meff can be found in Ref. [4]. 4.1. Boundary conditions The viscous dissipation rate ein and the inlet turbulence kinetic energy kin are both given as 0.5 [1–4]. Half of the chimney is taken as the computational domain, as shown by the dashed line in Fig. 3. At the inlet boundary, the air temperature and air velocity are taken as the ambient temperature and zero, respectively. At the exit boundary, the gradient of air temperature and air velocity is assumed to be zero. The pressure is considered to be atmospheric pressure. The adiabatic boundary conditions at the tip of the fin
Table 1 Comparison of results obtained from three different flow models with S = 15 mm. Hf = 90 mm, T1 = 299.89 K T0,1 = 352.01 K, T0,2 = 351.84 K
Hf = 180 mm, T1 = 300.28 K T0,1 = 352.37 K, T0,2 = 351.79 K
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h
330.60 330.71 337.17 335.68 336.70 319.37 319.39 7.17
332.80 332.89 339.21 338.29 338.78 322.28 322.32 5.51
326.08 326.13 329.85 338.99 329.06 316.11 316.11 13.60
(W/m2K) h b Q (W)
3.74
3.19
5.73 52.1% 143,032
4.89 57.9%
gf Nt
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
331.61 331.99 338.97 337.01 338.85 321.24 321.19 6.49 (Inv.)
331.36 331.29 338.09 336.39 337.36 320.59 320.58 6.66
333.14 333.07 339.69 338.58 339.01 322.81 322.82 5.45
328.57 328.56 330.61 339.65 329.64 316.79 316.76 12.10
332.48 332.31 340.52 336.82 339.99 322.77 322.33 6.03 (Inv.)
6.04
3.68 (Inv.)
3.56
3.13
5.54
3.53 (Inv.)
9.26 44.4%
5.64 (Inv.) 56.7% –
5.44 53.5% 140,032
4.77 57.4%
8.46 45.8%
5.44 (Inv.) 58.5% –
5
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948 Table 2 Comparison of results obtained from three different flow models with S = 20 mm. Hf = 90 mm, T1 = 300.28 K T0,1 = 350.98 K, T0,2 = 350.03 K
Hf = 180 mm, T1 = 300.09 K T0,1 = 351.89 K, T0,2 = 351.94 K
CFD results
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h (W/m2K) h b Q (W)
gf Nt
Exp. data
Zero- Eq.
Laminar
RNG k-e
329.33 329.08 336.63 334.62 335.65 319.33 319.15 7.12
332.27 331.99 338.68 337.05 337.85 322.08 321.91 5.43
323.33 323.10 328.13 337.15 326.99 314.61 314.42 15.1
3.79
3.11
5.60 53.20% 143,812
4.59 57.18%
CFD results
Exp. data
Zero- Eq.
Laminar
RNG k-e
330.21 330.02 336.98 335.12 336.49 319.97 319.78 7.17 (Inv.)
329.70 330.32 337.04 335.53 337.42 320.04 320.48 6.66
332.76 332.94 339.40 338.09 339.21 322.47 322.67 5.41
325.74 326.31 328.50 338.17 328.41 314.76 315.08 13.3
330.00 329.62 338.07 336.73 338.35 320.33 320.77 6.49 (Inv.)
6.23
4.02 (Inv.)
3.59
3.12
5.88
3.72 (Inv.)
9.22 41.25%
5.94 (Inv.) 56.07% –
5.48 53.85% 143,812
4.77 57.71%
8.98 44.21%
5.68 (Inv.) 57.32% –
Fig. 5. Laminar velocity pattern on z = S/2 plane with S = 15 mm. Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
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Fig. 6. Velocity pattern on z = S/2 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
and the sidewalls of the chimney are assumed. No-slip conditions are imposed on the surface of all fins, circular tubes and sidewalls of the chimney. @u3/@z, @Ta/@z and @p/@z at the symmetry of the chimney are given as zero. The matching conditions of heat flux and temperature at the interface between the fin and the air are expressed as follows.
T ¼ Ta
and kf
@T @T a ¼ ka @z @z
ð13Þ
,h and g can be approximated as follows. [1–4] Q, h b f
Q 2
N X
and
ð15Þ 2
where Af is Af ¼ L2 pd0 =2. Q is the heat transfer rate dissipated by and Tave are the average the entire fin. gf is the fin efficiency. Tave k temperatures in the kth sub-fin region and the entire fin, respec are obtained by Eq. (14). The FLUENT results for h̅ and h tively. h b and Q can be obtained according to the definition in Ref. [4]. 5. Typical grid distribution
ðT av e T Þ ¼ 2A h Ak h 1 k f b ðT 0 T 1 Þ k
k¼1
¼ 2Af hðT av e T 1 Þ
h
gf ¼ b h
ð14Þ
An unstructured grid system with non-uniformly distributed grid points is utilized to obtain FLUENT results. This grid system is presented in Fig. 4. The number of grid points Nx, Ny and Nz in the x, y and z directions on the fins are 40, 40 and 4 with Hf = 90 mm, and 34, 34 and 4 with Hf = 180 mm, respectively. The
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
7
Fig. 7. Air temperature contour on z = S/2 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
number of grid points Nza in the z direction of the air region between two adjacent fins are 9 with Hf = 90 mm, and 7 with Hf = 180 mm, respectively. Nx and Ny on the fin are the same as those of the air region between two adjacent fins for FLUENT 15. Both Nx and Ny are taken as 29 for IHCM. Therefore, Nx and Ny of FLUENT 15 may differ from those of IHCM. The total number of grid points Nxt, Nyt and Nzt in the x, y and z directions are taken as 50, 61 and from 31 to 35 with Hf = 90 mm and 180 mm and 50, 68 and from 31 to 36 with Hf = 135 mm. As shown in Refs. [3,4], the total error involved in the finite difference calculation includes the discretization error plus the round-off error. Due to round-off error considerations, the choice of relative convergence criteria and number of iterations may require caution. The difference between the numerical results obtained by the convergence criteria of 10-4, 10-5 and 10-6 was found to be negligible. Therefore, in order to save computation time, the relative convergence criteria of the momentum and energy equations is given as 10-4. In other words, 10-4 is a good choice for the momentum and energy equations. The number of iterations was set to 750 to obtain the present result for each case. The appropriate choice of relative convergence criteria, the number
of iterations, and the number of grid points may need to be based and h.. on experimental temperatures and estimates of h b
6. Results and discussion Thermophysical properties of this study are determined by (T 0 +T1)/2, where T 0 is defined as T 0 ¼ ðT 0;1 þ T 0;2 Þ=2. Tables 1 and 2 present the comparison between the FLUENT results, the inverse results and experimental temperatures with Hf = 90 mm and 135 mm for S = 15 mm and 20 mm, respectively. It is found in Tables 1 and 2 that Tk obtained by the laminar flow and the zero-equation turbulence model is closer to the experimental tem h , Q and perature than that obtained by the RNG k-e model. But, h, b gf obtained by the zero-equation turbulence model are more consistent with the inverse results than those obtained by the laminar flow and the RNG k-e model. Therefore, the zero-equation turbulence model is chosen to get the desired results. The selection of an appropriate flow model may need to be based on more accurate and h and experimental temperatures. h b
8
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
Fig. 8. Air temperature contour on x = 0 plane with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 180 mm.
The numerical results obtained from the zero-equation turbulence model are shown in Figs. 6–12 and Tables 3–5 with various S and Hf values. The velocity pattern obtained by laminar flow with S = 15 mm on the z = S/2 plane is presented in Fig. 5 with S = 15 mm. Fig. 6 displays the velocity pattern on the z = S/2 plane with S = 15 mm. It is found that the maximum and low velocity regions occur respectively in the intermediate region of the exit of the chimney and in the plume region. The maximum velocity decreases with increasing Hf value. The air velocity at the fin tip
with Hf = 180 mm is less than Hf = 90 mm. The faster velocity region with Hf = 90 mm is greater than Hf = 180 mm. It is expected that the fins cooled by cold air with Hf = 90 mm are earlier than Hf = 180 mm. In other words, buoyancy caused by the fins with Hf = 90 mm is stronger than Hf = 180 mm. This implies that the chimney effect with Hf = 90 mm is slightly stronger than Hf = 180 mm. The fins with larger Hf values exhibit greater air thermal resistance. The air velocity between the two plumes is faster than in the plume region. Thus, the velocity pattern for each heated cylinder is
H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
9
Fig. 9. Fin surface temperature contour with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
mainly aligned in the vertical direction. The two plumes are slightly inclined towards the inward direction due to the pressure difference and appear to attract each other in the region above the cylinder. The direction of the convective plume looks approximately symmetrical. The velocity patterns of Figs. 5 and 6 are similar except for the intermediate region of the exit of the chimney and the outer regions of the fins near the two tubes. An important finding is that the velocity pattern in Figs. 5 and 6 coincides with interferometric image of the convective field in Fig. 4 of Ref. [13] in the fin region. Thus, the numerical results obtained have good accuracy. It is observed from Figs. 5 and 6 that the natural convective boundary layer flow is caused by two heated tubes and fins moves upwards. The two plumes above the two heated tubes belong to the low-velocity region. Due to the difference in velocity between the two plume regions and the central region surrounded by the two tubes, the boundary layer starts separating from the oblique line by about 30 degree from the vertical line. The cold air in the environment flows upward through the entrance of the chimney. This flow field differs from that in Ref. [1] for a box with only one top opening. In addition, there are two natural circulations found in the upper right and left corners near the exit of the
chimney. This phenomenon is similar to that of Ref. [4] with only a single tube. The 3D flow field and heat transfer phenomenon of this study is very complicated. Table 3 presents the influence of grid points on the results obtained with Hf = 180 mm and S = 15 mm and 20 mm. It is seen h and T values are more consistent with that the obtained h, b k the inverse results and experimental temperatures using Nx = 34, Ny = 34, Nzf = 4 and Nza = 7 with S = 15 mm and Nx = 34, Ny = 34, Nzf = 4 and Nza = 9 with S = 20 mm compared to other grid points. It is found from Table 5 that Nt for 5 mm S 30 mm is increased from 127,382 to 143,946 with Hf = 90 mm and 180 mm and from 139,176 to 163,016 with Hf = 135 mm, respectively. Nt can vary with S and Hf. Therefore, the assumption of grid independence may not be appropriate. for various H and S values. Table 4 gives the inverse results of h k f It is observed that h1 and h2 are the smallest in the plume regions 3 and h 5 are the largest in the compared to other sub-fin regions. h outer regions of the two tubes compared to the other sub-fin regions. The correlation between Ra and Nus proposed based on the results obtained in Table 5 can be expressed as follows for 18 < Ra < 22,000.
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
Fig. 10. Distribution of h(x,y) with S = 15 mm. (a) Hf = 90 mm, (b) Hf = 135 mm, (c) Hf = 180 mm.
with S. Fig. 11. Variation of h
with S. Fig. 12. Variation of h b
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948 Table 3 Effect of grid points on the results obtained with Hf = 180 mm. Nx Ny Nz Nza S = 15 mm
S = 20 mm
34 34 4 7
34 33 4 6
34 33 3 9
34 34 4 9
34 33 4 12
34 33 3 15
T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) (W/m2K) h
331.36 331.29 338.09 336.39 337.36 320.59 320.58 6.66
330.75 330.63 337.48 335.75 336.73 320.30 320.28 6.67
331.39 331.22 338.02 336.30 337.24 320.50 320.40 6.77
329.70 330.32 337.04 335.53 337.42 320.04 320.48 6.66
329.58 330.22 337.80 335.84 337.45 320.47 320.58 6.74
329.49 330.25 337.73 335.80 337.46 320.37 320.49 6.78
(W/m2K) h b Q (W) Nt
3.56
3.57
3.62
3.59
3.63
3.65
5.44 140,032
5.45 158,180
5.53 186,916
5.48 143,812
5.54 162,552
5.58 192,068
Table 4 with various S and H values. Inverse results of h k f Hf = 90 mm
Hf = 180 mm
S = 5 mm
S = 20 mm
S = 5 mm
S = 20 mm
(W/m2K) h 1 (W/m2K) h 2 (W/m2K) h
1.73
3.84
2.20
3.85
1.58
3.74
1.97
4.10
11.23
15.15
4.49
11.90
(W/m2K) h 4 (W/m2K) h 5 (W/m2K) h
6.00
10.40
3.02
7.72
10.85
14.78
7.31
11.35
7.26
10.10
5.34
9.40
(W/m2K) h 7
7.02
9.99
5.74
8.96
3
6
Table 5 and h for various S and H values. Comparison of h b f (W/m2K) h b
(W/m2K) h
Hf(mm)
S (mm)
Ra
Nt
Inv.
CFD
Eq. (16)
Inv.
CFD
90
5 10 15 20 30
17.49 278.94 1406.38 4320.88 22051.16
127,382 135,306 143,032 143,812 148,964
4.49 5.28 6.49 7.17 8.22
3.23 6.57 7.17 7.12 7.14
3.22 6.35 7.10 7.36 7.84
2.99 3.33 3.68 4.02 4.37
2.26 3.72 3.74 3.74 3.73
135
5 10 15 20 30
17.65 281.10 1394.13 4446.81 21829.60
139,176 148,448 156,664 157,456 163,016
4.06 4.98 6.24 6.81 7.94
3.15 6.38 6.91 6.93 6.88
3.04 5.98 6.67 7.02 7.36
2.77 3.18 3.65 4.04 4.17
2.21 3.57 3.66 3.63 3.64
180
5 10 15 20 30
17.42 273.93 1394.74 4421.09 21542.29
127,382 135,306, 143,032 143,812 148,964
3.59 4.76 6.03 6.49 7.91
2.97 6.11 6.66 6.66 6.62
2.84 5.55 6.24 6.55 6.86
2.26 3.02 3.53 3.72 4.20
1.98 3.47 3.56 3.59 3.58
2d0 2d0 ð1 Þð0:255 þ 0:28Ra1=3 Þ½1 L L 1:4e200S ½0:78 þ 0:27ejð100S1j
Nus ¼ ð5:77 6:8Hf Þ
ð16Þ
gbðT 0 T 1 ÞS4 where Nus and Ra are defined as Nus ¼ hS=k . a and Ra ¼ maL To validate the availability and reliability of the desired results, obtained by FLUENT 15, IHCM and corre and h a comparison of h b lation (16) is presented in Table 5. It is found that the FLUENT and h are consistent with inverse results. h , h and results of h b b gf increase as S increases, and approach a constant. The optimum fin spacing is approximately 15 mm. Another interesting finding and h with H = 90 mm are higher those with H = 180 mm is that h b f f
due to the effect of buoyancy. gf decreases with increasing S. gf for 5 mm S 30 mm is reduced from 70.0% to 52.2% with Hf = 90 mm, from 66.7% to 52.9% with Hf = 135 mm and from 66.7% to 53.5% with Hf = 180 mm. But, it is not very sensitive to Hf. Because of the combination of the thermal boundary layer and h will become smaller for small between two adjacent fins, h b S values. The correlation (16) is close to the obtained FLUENT and inverse results. The maximum error between correlation (16) and FLUENT results is approximately 9%. Therefore, the correlation (16) has good reliability and availability. The air temperature contours on the z = S/2 plane with S = 15 mm and various Hf values is presented in Fig. 7. It is observed that the air temperature profile exhibits the non-
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H.-T. Chen et al. / International Journal of Heat and Mass Transfer 147 (2020) 118948
uniform distribution in the air region between two adjacent fins and a strong interaction with each other between the two tubes. The merging of the thermal boundary layers on the two tubes is distinguished in the region surrounded by them. The air temperature distribution between the two tubes is significantly influenced by the thermal plumes originating from them. This also causes an increase in the temperature of this region. In this region, the air temperature is higher and the air speed is lower. Another important finding is that the air temperature contour of Fig. 7 for DH/ d0 = 2.31 is consistent with Fig. 13 in Ref. [11] for DH/d0 = 0.28 and Fig. 7 in Ref. [13] for DH/d0 = 2.5 in the fin region. This comparison further confirms the accuracy of the present results. The air temperature between two heated tubes with Hf = 180 mm is lower than that with Hf = 90 mm. Air temperature is almost at 306 K for Hf = 180 mm and is approximately at 313 K for Hf = 90 mm. The air temperature contours on the = 0 plane with S = 15 mm and Hf = 90 mm and 180 mm are presented in Fig. 8. As the cold air moves upward from the entrance of the chimney, the thermal boundary layer gradually thickens from the bottom of the fin. The thermal boundary layers in the fin region is found to merge with each other. This phenomenon causes a decrease in the heat transfer coefficient on the fins above the tubes. As shown in Table 3, 4 is less than h 3 and h 5 . Air may not easily flow into the environh ment from the tip of the fin. The fin surface temperature distribution are presented in Fig. 9 for various Hf values and S = 15 mm. The uneven distribution of fin temperature is found. The interaction of two heated tubes causes an increase in the temperature of the air around them. Thus, it is found in Table 1 that the measured temperatures are higher from the 3rd to 5th sub-fin regions than in the other sub-fin regions. The measured temperatures are lower in the 6th and 7th fin regions than in the other sub-fin regions. In other words, the fin temperature in the lower region of the tube is lower than that in its upper region. T4 is lower than T3 and T5. Higher temperatures are also found near the base of the fin. The distribution of h(x,y) with S = 15 mm is presented in Fig. 10. Fig. 10 exhibits that the heat transfer coefficient is lowest in the plume region. However, it is not always highest in the bottom region of the tube. The above results are consistent with those in Table 3. This consistency further implies that the results obtained are reliable. and h obtained by FLUFigs. 11 and 12 show the variation of h b and h increase ENT 15 [5] with S, respectively. It can be seen that h b with increasing S and approach a value obtained at approximately and h decrease as H increases. S = 15 mm. h b f 7. Conclusions The hybrid method of IHCM and CFD combined with experimental temperatures is proposed to determine the desired results. and h obtained by the zero-equation turbulence It is found that h b model are more consistent with the inverse and FLUENT results and h than those of the RNG k-e model and laminar flow. h b increase with decreasing Hf and increasing S. gf decreases with increasing S. But, it is not very sensitive to Hf. There are two natural circulations in the upper right and upper left corners near the exit
of the chimney. The fringe pattern for each heated tube is mainly aligned in the vertical direction and the two plumes are slightly inclined toward the inward direction. The air temperature in the inner half of the two heated tubes displays a strong interaction with each other and the merging of the thermal boundary layers between them is prominent. It is worth mentioning that the obtained velocity pattern and air temperature contour are consistent with the isotherm of Ref. [11] or the interferometric image of Ref. [13] in the fin region. Nt can vary with S and Hf. This observation means that the present results have good reliability and accuracy, and the assumption of grid independence may not be appropriate. Appropriate selection of the relative convergence criteria for momentum and energy equations needs to be based on and h. The proexperimental temperatures and estimates of h b posed correlation between Ra and Nus agrees with the obtained FLUENT and inverse results in the range of 18 < Ra < 22,000. The maximum error between correlation (16) and FLUENT results is approximately 9%. Therefore, this correlation can be very useful to industrial designers. Declaration of Competing Interest The authors declared that there is no conflict of interest. References [1] H.T. Chen, Y.S. Lin, P.C. Chen, J.R. Chang, Numerical and experimental study of natural convection heat transfer characteristics for vertical plate fin and tube heat exchangers with various tube diameters, Int. J. Heat Mass Transfer 100 (2016) 320–331. [2] H.T. Chen, Y.J. Chiu, C.S. Liu, J.R. Chang, Numerical and experimental study of natural convection heat transfer characteristics for vertical annular finned tube heat exchanger, Int. J. Heat Mass Transfer 109 (2017) 378–392. [3] H.T. Chen, Y.L. Hsieh, P.C. Chen, Y.F. Lin, K.C. Liu, Numerical simulation of natural convection heat transfer for annular elliptical finned tube heat exchanger with experimental data, Int. J. Heat Mass Transfer 127 (2018) 541–554. [4] H.T. Chen, H.Y. Chou, H.C. Tseng, J.R. Chang, Numerical study on natural convection heat transfer of annular finned tube heat exchanger in chimney with experimental data, Int. J. Heat Mass Transfer 127 (2018) 483–496. [5] Fluid Dynamics Software, FLUENT 15, Lebanon, NH-USA, 2013. [6] R. Sajedi, M. Taghilou, M. Jafari, Experimental and numerical study on the optimal fin numbering in an external extended finned tube heat exchanger, Appl. Therm. Eng. 83 (2015) 139–146. [7] A. Kumar, J.B. Joshi, A.K. Nayak, P.K. Vijayan, 3D CFD simulations of air cooled condenser. Part II: natural draft around a single finned tube kept in a small chimney, Int. J. Heat Mass Transfer 92 (2016) 507–522. [8] J.R. Senapati, S.K. Dash, S. Roy, Numerical investigation of natural convection heat transfer over annular finned horizontal cylinder, Int. J. Heat Mass Transfer 96 (2016) 330–345. [9] J. Liu, H. Liu, Q. Zhen, W.Q. Lu, Numerical investigation of the laminar natural convection heat transfer from two horizontally attached horizontal cylinders, Int. J. Heat Mass Transfer 104 (2017) 517–532. [10] J. Liu, H. Liu, Q. Zhen, W.Q. Lu, Laminar natural convection heat transfer from a pair of attached horizontal cylinders set in a vertical array, Appl. Therm. Eng. 115 (2017) 1004–1018. [11] J. Stafford, V. Egan, Configurations for single-scale cylinder pairs in natural convection, Int. J. Therm. Sci. 84 (2014) 62–74. [12] Q. Pelletier, D.B. Murray, T. Persoons, Unsteady natural convection heat transfer from a pair of vertically aligned horizontal cylinders, Int. J. Heat Mass Transfer 95 (2016) 693–708. [13] S. Narayan, A.K. Singh, A. Srivastava, Interferometric study of natural convection heat transfer phenomena around array of heated cylinders, Int. J. Heat Mass Transfer 109 (2017) 278–292.