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Impacts of geometric structures on thermo-flow performances of plate fin-tube bundles
International Journal of Thermal Sciences 107 (2016) 161e178
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Impacts of geometric structures on thermo-flow performances of plate fin-tube bundles Y.Q. Kong, L.J. Yang*, X.Z. Du, Y.P. Yang Key Laboratory of Condition Monitoring and Control for Power Plant Equipments of Ministry of Education, School of Energy Power and Mechanical Engineering, North China Electric Power University, Beijing, 102206, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 November 2015 Received in revised form 6 April 2016 Accepted 11 April 2016
Continuous plate fin-tube heat exchangers are commonly applied to natural draft dry cooling system in power plants due to their excellent thermo-flow performances, but the effects of geometrical structures on the air-side flow and heat transfer characteristics have not been thoroughly clarified yet. In this study, the air-side flow and heat transfer models of plate fin-tube bundles are developed, and the numerical modeling and methods are validated by the experimental test. The thermo-flow characteristics of the plate fin-tube bundles at various parameters of air face velocity, fin oblique angle, fin spacing, fin thickness and tube external diameter are numerically simulated. The buoyancy effect due to natural convection is also investigated. The results show that the performance evaluation criterion (PEC) varies little with the fin oblique angle. As the fin spacing increases, the average PEC will increase despite of a considerable heat transfer reduction. The increased fin thickness and tube external diameter are both beneficial to the thermo-flow characteristics, but the buoyancy effect on the flow and heat transfer performances can be neglected. The fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube external diameter of 18 mm are recommended within the scope of investigation. The correlations between the Nusselt number, friction factor and the Reynolds number and geometrical parameters are recommended using the orthogonal experimental design method, which can be used to the optimal design of air-cooled heat exchanger in power plants. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Natural draft dry cooling system Air-cooled heat exchanger Finned tube bundles Plate fin Flow and heat transfer Performance evaluation criterion
1. Introduction Nowadays, natural draft dry cooling system with air-cooled heat exchangers and dry-cooling towers is commonly applied to power plants in the regions where water resource is of shortage or expensive [1e3]. Thanks to less initial costs and self-supporting nature, the heat exchanger cooling deltas are frequently arranged vertically around the circumference of the dry-cooling tower [4]. The ambient air flows through the air-cooled heat exchanger and dry-cooling tower due to the buoyancy force originating from the air density difference between the interior and exterior of the tower, taking away the heat rejection from the circulating water. The plate fin-tube heat exchanger is the core facility of natural draft dry cooling system, so it is helpful for the air-cooled heat exchanger design to investigate the thermo-flow performances of the plate fin-tube bundles.
* Corresponding author. E-mail address: [email protected] (L.J. Yang). http://dx.doi.org/10.1016/j.ijthermalsci.2016.04.011 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
The flow and heat transfer performances of finned tube heat exchangers have attracted great attentions in the past years. Yang et al. [5] investigated the effect of oblique configuration of wavefinned flat tube bundles on the thermo-flow characteristics, and obtained the flow and heat transfer correlations. S¸ahin et al. [6] investigated the pressure drop and heat transfer of seven plain fin-tube heat exchangers with different fin angles, concluding that the inclined fin with the angle of 30 is the optimum configuration. Mon and Gross [7] studied the effect of fin spacing on the four-row annular-finned tube bundles in staggered and in-line arrangements, finding that the boundary layer development and horseshoe vortices between the fins depend substantially on the fin spacing to height ratio and Reynolds number. Erek et al. [8] numerically studied the pressure drop and heat transfer of the plate fin-tube bundles at different fin geometries, finding that the pressure drop is closely related to the fin distance. Madi et al. [9] experimentally studied 28 heat exchanger samples in an open thermal wind tunnel, concluding that the fin type plays important roles in the flow and heat transfer, but the number of tube rows has almost no effect on the friction factor. Matos et al. [10] optimized the three-
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Nomenclature Ab Ac Af cp D Dh FS f GPC g H H1 H2 He h I k L m Nf Nu P Pc PEC Pl Pt Pr
base tube surface area (m2) minimum flow cross-sectional area (m2) fin surface area (m2) specific heat capacity(J kg1 K1) tube external diameter (mm) hydraulic diameter (mm) fin spacing friction factor global performance criterion gravitational acceleration (m s2) enthalpy (J kg1) upper slotted strip height (mm) lower slotted strip height (mm) height of fined tube (m) air side convection heat transfer coefficient (W m2 K1) turbulence intensity turbulence kinetic energy (m2 s2) flow depth (mm) mass flow rate (kg s1) fin number Nusselt number pressure (Pa) minimum flow cross-section perimeter (mm) performance evaluation criterion longitudinal tube pitch (mm) transverse tube pitch (mm) Prandtl number
dimensional finned tube bundles in staggered arrangements of circular and elliptic tubes, showing that the elliptical tube configuration has a better overall performance and a lower cost than the circular tube. For six types of plate fin-tube heat exchangers used in power plants, Yang et al. [11] pointed out that the Forgo-type aluminum finned tube bundle is superior to the steel finned ovaltube one. Kim et al. [12] tested 22 heat exchangers with various fin pitches, various numbers of tube row and tube alignment by experiments. As is well known, there are still some limitations to the air-side flow and heat transfer performances, and a clear understanding of air flow in the complex passages (such as slot, louvers and wave fin) of heat exchangers is needed, so that the heat transfer surface can be designed efficiently. For the plain plate fin and three types of radial slotted (full slotted, partially slotted, back slotted) fin surfaces, the numerical simulation showed that the full slotted fin surface has the highest heat flow rate with the biggest pressure drop penalty at the same face velocity [13]. Carija et al. [14] compared flat and louvered fin-and-tube heat exchangers, and no remarkable differences of Nusselt number and pressure drop were found. Li et al. [15] proposed a new wavy fin configuration, and pointed out that the periodic convergent-divergent channel formed by wavy fins results in a big pressure drop, which leads to a poor overall performance despite of the excellent heat transfer performance. The heat exchanger performance involves many parameters, among which the main geometrical parameters consist of fin oblique angle, fin pitch, fin thickness, tube pitch and tube outside diameter. Previous investigations were restricted to specific finned tube heat exchangers for the natural draft dry cooling system in power plants, while the influences of geometrical parameters,
Q Qe qv Re Ri S SW T Ti To Tb u ui um xi
heat flow rate (W) total heat flow rate of an entire finned tube volumetric flow rate (m3 s1) Reynolds number Richardson number slotted strip spacing (mm) slotted strip width (mm) temperature (K) air inlet temperature (K) air outlet temperature (K) base tube temperature (K) air face velocity (m s1) velocity in the i direction (m s1) minimum flow cross-sectional velocity (m s1) Cartesian coordinate (m)
Greek symbols fin oblique angle ( ) thermal expansion coefficient (K1) fin thickness (mm) ε turbulence dissipation rate (m2 s3) hf fin efficiency l thermal conductivity (W m1 K1) m dynamic viscosity (kg m1 s1) mt turbulent viscosity (kg m1 s1) y kinetic viscosity (m2 s1) r fluid density (kg m3) t shear stress (Pa) 4 dissipation function in energy equation (W m3)
a b d
except the fin pitch and tube pitch [16], have not been analyzed so far. The effects of air face velocity, oblique angle of fin, fin spacing, fin thickness and tube external diameter will be studied in this work so that the optimal geometrical parameters can be expected. The relative importance of various geometry parameters will be quantitatively assessed based on the performance evaluation criterion. Additionally, the flow and heat transfer correlations for the plate fin-tube bundles in air-cooled heat exchanger will be presented, which can be applied to the design of natural draft dry cooling system in power plants. 2. Models 2.1. Physical model The natural draft dry cooling system in power plants is schematically shown in Fig. 1. The finned tube bundles of air-cooled heat exchanger are arranged vertically around the circumference of the dry-cooling tower as shown in Fig. 1(a), and the ambient air flows across the air-cooled heat exchanger and through the dry-cooling tower sequentially, while the circulating water flows inside the round tubes in staggered configuration. Fig. 1(b) shows the six-row finned tube bundles, which are widely adopted in the air-cooled heat exchanger. The fin and tube are both made of aluminum, which are connected by the tube-expansion technology. In Fig. 2, the geometric parameters and computational domains for the six-row finned tube bundles are presented. For the oblique finned tube bundles, as the oblique angle increases, the total length of the fin will increase either, while the horizontal distance of the fin keeps constant. Table 1 shows the geometric details of the finned tube bundles. The dashed lines confine the computational
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Fig. 1. Schematic of natural draft dry cooling system. (a) air-cooled heat exchanger and dry-cooling tower, (b) six-row plate finned tube bundles.
domain, where 80 mm is extended for the entrance section to ensure a uniform velocity distribution and 300 mm is extended for the exit section to avoid the recirculation flows. The middle surfaces of the neighboring two fins are defined as the symmetric boundaries, as shown in Fig. 2(a) and (b), and the other two sides of the computational domain across the middle sections of the neighboring tubes are also set symmetric boundaries as shown in Fig. 2(c). Due to the relatively high convective heat transfer coefficient on the water side and high thermal conductivity of the tube, the outer wall of the tube is assumed to keep a constant temperature Tw ¼ 310.85 K equal to the average temperature of the water inlet and outlet temperatures. However, the temperature distribution on the fin surface has to be determined by solving the conjugate problem, in which both the temperatures on the solid fin surface and in the fluid need to be solved simultaneously. The computational domain inlet is defined as the velocity inlet
boundary condition with the velocity ranging from 0.5 m/s to 3.5 m/s and a constant inlet air temperature of 289.15 K. The turbulent intensity is defined by the widely used correlation I ¼ 0:16ReDh 1=8 . Due to the unchanging velocity and temperature fields, the outflow boundary condition is applied to the domain outlet. At the solid surfaces, the no-slip conditions for the velocity are specified. In addition, some assumptions are made as follows:
The flow is steady-state; The fluid is incompressible; The radiation effect is ignored; The thermal contact resistance between the base tube and fin is ignored due to the order with the small magnitude of 105 [17], which is far less than the air side convection thermal resistance of 102;
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Fig. 2. Geometric details, computational domain and boundary conditions of finned tube bundles. (a) front view of horizontal-plate finned tube bundles, (b) front view of obliqueplate finned tube bundles, (c) top view of finned tube bundles.
Y.Q. Kong et al. / International Journal of Thermal Sciences 107 (2016) 161e178 Table 1 Geometric dimensions of plate fin-tube heat exchanger. Oblique angle of fin (a/ ) Fin spacing (FS/mm) Fin thickness (d/mm) Tube external diameter (D/mm) Longitudinal tube pitch (Pl/mm) Transverse tube pitch (Pt/mm)
0, 10, 15, 20, 25, 30 2, 3, 4, 5, 6, 8, 10, 12, 14 0.1, 0.3, 0.6, 0.9, 1.2 18, 22, 25, 27, 32 35 30
The thermo-physical properties of the solid and fluid are considered constant.
2.2. Governing equations The governing equations for the three-dimensional, steadystate flow and heat transfer of air can be expressed as follows. Continuity equation:
vðrui Þ ¼0 vxi
(1)
For turbulent flows, the momentum equation takes the following form by using the Reynolds time averaging method [18,19]:
v vP vt v rui uj ¼ þ ij þ vxi vxi vxj vxj
ru0i u0j
(2) 2.4. Parameter definitions
vui vuj 2 vuk þ d vxj vxi 3 ij vxk
! (3)
By using the Boussinesq theorem, the turbulent shear stress can be obtained by the following form.
ru0i u0j
as shown in Fig. 3(b). The grid independence is tested for the heat transfer coefficient of the finned tube bundles (a¼0 , FS ¼ 3.2 mm, d ¼ 0.3 mm and D ¼ 25 mm) at the air velocity of 1.5 m/s. The results at different grid numbers are tabulated in Table 2. The relative errors of the convection heat transfer coefficients between the three sparse grids and the dense grid are 4.86%, 1.27% and 0.84% respectively, showing that the calculated results based on 1,216,107 cells are insensitive to further grid refinement, so the grid number of 1,216,107 is employed in this case. In other cases, similar works are also done to obtain the final grid numbers. The governing equations with the boundary conditions are solved by the commercial software FLUENT 6.3.26. The equations for the continuum, momentum, energy, turbulent kinetic energy and its dissipation rate are discretized using the second-order upwind scheme to obtain more accurate results. The SIMPLE algorithm is applied to the pressureevelocity coupling, which stores the discrete values of the scalar variables at the center of the control volume, as well as the interpolated values at the center of all the faces incorporated. The complex flow in the fin channel is not aligned with the grid. To control the update of the computed variables at each iteration, the under-relaxation factors vary between 0.3 and 1.0. A divergence-free criterion of 106 based on the scaled residual is prescribed for the continuity, momentum, k and ε equations, and 108 for the energy equation.
where
tij ¼ mt
165
¼ mt
vui vuj 2 vuk þ d vxj vxi 3 ij vxk
!
2 dij rk 3
(4)
where the symbols with apostrophes denote the fluctuation components. The energy equation in terms of enthalpy has the following form:
v v vT m vH ðrui HÞ ¼ l þ t vxi vxi vxi Prt vxi
þ ui
· vP þQ þ4 vxi
(5)
The turbulence effect on the fluid flow is included through the application of low Reynolds number k-ε models [20]. The physical properties of air are evaluated at the mean air temperature (Ti þ To)/ 2. When taking the natural convection into consideration, the air density is determined by the ideal gas law and the gravity is set at the momentum equation to provide the buoyancy effect. 2.3. Numerical methods The geometric model is created and meshed by using GAMBIT software. For the finned tube region, the hexahedral structured mesh with a minimal cell skewness is used to resolve the swirling flow, while for those in the extended regions, the coarse grids are adopted to reduce the computational cost as shown in Fig. 3(a). For turbulent convective heat transfer, the air velocity and temperature near the wall region have great gradients, so the boundary layer mesh are adopted at the near wall region for accurate simulations,
For the heat transfer performance, the characteristic nondimensional parameters are defined as follow:
Re ¼ rum Dh =m
(6)
Nu ¼ h$Dh =l
(7)
where, Re is the Reynolds number, Nu is the Nusselt number. r, m, l are the density, dynamic viscosity and thermal conductivity respectively. Dh is the hydraulic diameter of the finned tube and takes the following form.
Dh ¼ 4Ac =Pc
(8)
where Ac is the minimum free flow area, Pc is the perimeter at the minimum flow cross-section. um is the air velocity at the minimum flow area, h is the air side convection heat transfer coefficient and can be calculated with the following equation:
h¼
Q Ab þ hf Af DTm
(9)
where Ab is the base tube surface area. Af is the fin surface area. Q is the air-side heat transfer rate, Q ¼ mcp(To Ti). DTm is the logarithmic mean temperature difference between the tube wall and the air, which is defined as [21]:
DTm ¼
T b Ti T b To ðT Ti Þ ln b ðT b To Þ
(10)
The fin efficiency, hf, is calculated from the values of the simulated results [21]:
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Fig. 3. Grid system of computational models. (a) extended and fin-and-tube regions, (b) boundary layer mesh in the fin-and-tube domain.
Table 2 Grid independence results at various grid numbers.
hf ¼
Finned tube model
Grid number
Heat transfer coefficient (W/m2 K)
a¼0 , FS ¼ 3.2 mm, d ¼ 0.3 mm, D ¼ 25 mm
901,152 1,100,534 1,216,107 1,454,965
36.546 37.924 38.089 38.413
Tf Ta Tb Ta
(11)
where T a ¼ 1=2ðTi þ To Þ is theR average value of the inlet and outlet TdV temperatures of the air. T f ¼ V is the volume-weighted average fin temperature, which is computed by dividing the summation of the product of the temperature and cell volume by the total volume of the fin zone. T b is the base tube average temperature. For the flow performance, the friction factor f of the finned tube is defined as [16]:
f ¼
2DP Dh $ ru2m L
(12)
where L is the flow depth, Dp is the pressure drop across the finned
tube bundles. The performance evaluation index PEC, representing the heat transfer capability of a surface with a given flow resistance, is commonly defined as follows. Generally speaking, the higher the PEC is, the better the overall performance of the fin-and-tube bundles [21].
. PEC ¼ Nu f 1=3
(13)
The non-dimensional Richardson number, Ri, is used to analyze the influence of natural convection, which is defined as the ratio of the buoyancy force to inertia force.
Ri ¼
Gr Re2
¼
g bDtl3
y2 Re2
(14)
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Fig. 4. Experimental system for the thermo-flow performances of finned tube bundles. (a) photo of alternant slotted finned tube bundles in staggered configuration, (b) geometric structure of alternant slotted finned tube, (c) experimental setup and locations of measuring points, (d) wind tunnel experimental system.
Table 3 Geometric dimensions of the experimental sample.
a 0
D
N
Pt
Pl
d
FS
H1
H2
SW
S
25 mm
4
30 mm
35 mm
0.3 mm
3.2 mm
1.1 mm
0.5 mm
2.75 mm
2 mm
Table 4 Summary of estimated uncertainties. Measured variables
Derived quantities
Parameter
Uncertainties (%)
Parameter
Uncertainties (%)
mwater uair tin-air tout-air tin-water tout-water DP
4.77 5.45 0.18 0.67 0.15 0.38 4.46
Qair Qwater Re h Nu f
4.49 5.49 5.45 7.14 7.14 7.04
where Dt ¼ T f T a , T ft g is the gravitational acceleration. b is the thermal expansion coefficient. l is the feature size of finned tube and equals the fin spacing. As long as Ri is smaller than 0.01, the buoyancy force can be ignored, and the inertia force plays a dominant role in the air flows. In this work, a global performance criterion (GPC) is also adopted
to evaluate the comprehensive performance of the finned tube bundles with a given volume, which is defined as the total heat transfer rate of the entire finned tube bundles per unit flow power consumption, and takes the following form:
GPC ¼
Qe
DPqv
(15)
where Qe ¼ Q $Nf , is the total heat transfer rate of the entire finned tube. The fin number Nf of finned tube bundles is given by
Nf ¼
He Fs þ d
(16)
where He is the height of the finned tube. qv ¼ uAe is the volumetric flow rate of the finned tube bundles. u is the average value of the inlet and outlet air velocities. Ae ¼ He $Pt , is the face area of the finned tube. Pt is the transverse tube pitch.
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2.5. Verification of the computational model By comparing with the experimental results, the standard k-ε model, low-Re k-ε model, realizable k-ε model and laminar model are verified for their feasibility in the flow simulations. The flow and heat transfer experiments are made in a wind tunnel system. The alternant slotted finned tube bundles in the staggered configuration produced by a certain manufacturer of China are selected as the experimental sample as shown in Fig. 4(a). The geometric structure of the finned tube is schematically shown in Fig. 4(b), and the geometric dimensions of the experimental sample are listed in Table 3. The experimental system is schematically shown in Fig. 4(c) and 4(d), where the transversal surface has the size of 500 700 mm and the length is 1000 mm for the test section. The test section is insulated by the asbestos layer to minimize the heat loss. The experimental sample closely matches the test section in order to eliminate the contraction and expansion losses. The experimental system consists of two loops, an air loop and a water loop. Thermostatic water tank can be adjusted by transformers to obtain a uniform water inlet temperature during
Fig. 6. Variations of convection heat transfer coefficient with air face velocity at various fin oblique angles.
the test. The water flow rate inside the vertical tube is measured by a turbine flow meter with an accuracy of 0.5%. During the experiment, the water inlet temperature is held constant at 60 C with an error range of ±0.5 C, and the water flow rate is approximately 0.694 kg/s. Both the inlet and outlet tube wall temperatures are measured by setting six T-type thermocouples at the inlet and outlet parts of the tubes. The air flow rate over the test surface is adjusted by a digital blower speed controller. The inlet air velocity varies from 0.5 to 3.5 m/s, which is measured by a hotwire anemometer with the precision of ±2%. The inlet and outlet pressures of the test sample are obtained by two Pitot tubes at the center of both sides of the test section, and a differential pressure transmitter with the precision of ±0.15 Pa. The inlet and outlet air temperatures across the test section are measured by two T-type thermocouple meshes. The inlet measuring mesh consists of 12 thermocouples, while the outlet mesh contains 16 thermocouples due to the increased air disturbance after the finned tube bundles. All the thermocouples are pre-calibrated by the ice-water mixtures with an error range of ±0.1 C. Fig. 4(c) labels all the locations of the measuring points. The energy un-balance between the air side and
Fig. 5. Validation of numerical results with experimental data. (a) Nusselt number, (b) friction factor.
Fig. 7. Variations of pressure drop with air face velocity at various fin oblique angles.
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Fig. 8. Temperature and pressure contours for different fin oblique angles at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
water side is within 7.57%. The experimental data are recorded when the measured temperatures stabilize. All the data signals are collected and converted by a data acquisition system and a host computer. The experimental uncertainties are tabulated in Table 4 according to the analysis method used by Xi et al. [22]. Fig. 5 shows the comparisons between the experimental and computed Nusselt number and friction factor. It can be seen that the numerical results by the standard and realizable k-ε models both show substantial increases compared with the low Re k-ε model resulting from the strong turbulence intensity in the flow. While the laminar model predicts a smaller result than others due to the weaker turbulence intensity. The simulation results of the low Re k-ε model agree well with the experimental data with the highest deviation of the Nusselt number and friction factor as low as 5.3% and 8.3% respectively, and moreover, the low Re k-ε model can provide several modification schemes for the high Re number model, which will automatically adapt to different ranges of Re of the flow. Hence, the low Re k-ε model is selected as the turbulence model. Besides, the same turbulence model and numerical method
are adopted in the subsequent simulations for different fin shapes in this work. 3. Results and discussion The influences of various geometric parameters and the buoyancy force of warm air on the flow and heat transfer characteristics are investigated. In order to quantitatively assess the relative effects of different geometric parameters of finned tube bundles, the finned tube with the parameters of a ¼ 0 , FS ¼ 3 mm, d ¼ 0.3 mm and D ¼ 25 mm is set as a baseline one for the comparisons. 3.1. Effect of fin oblique angle Figs. 6 and 7 show the variations of the air-side convection heat transfer coefficient and pressure drop with air face velocity at various fin oblique angles (a ¼ 0 , 10 , 15 , 20 , 25 , 30 ) respectively. In general, the convection heat transfer coefficient and pressure drop vary dramatically with the face velocity. The higher
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the face velocity is, namely the higher the Reynolds number is, the higher the heat transfer coefficient and pressure drop are. It can also be seen that the effect of fin oblique angle on the convection heat transfer coefficient can be neglected. At the air face velocity of 2 m/s as an illustrative case, the convection heat transfer coefficient with the fin oblique angle of 30 only decreases by 0.22% in comparison with the baseline fin. It is because that as the fin oblique angle increases, the total fin surface area and heat flow rate both increase, which leads to an almost unchanging air-side convection heat transfer coefficient at the same face velocity. As shown in Fig. 7, the pressure drop also decreases little as the fin oblique angle goes up. At the air face velocity of 2 m/s, the pressure drop with the fin oblique angle of 30 decreases by 6.31% in comparison with the baseline fin. When the fin oblique angle increases, the flow across cylinder changes to the flow across elliptic cylinders, resulting in a reduced local flow loss, but the flow friction loss will increase due to the increased flow surface area. The former plays a dominant role in the total pressure drop, so the pressure drop decreases with increasing the fin oblique angle. For a better understanding of the fin oblique angle effects, Fig. 8 shows the temperature and pressure contours between the fin surfaces for a ¼ 0 and a ¼ 30 at the air face velocity of 1.5 m/s. From Fig. 8(a), it can be seen that the temperature of fin surface increases along the flow direction due to the increased air temperature. Moreover, the flow separates at the rear portion of a tube, and then forms a large region with a deteriorated heat transfer performance between the tubes. Owing to the similar thickness of the thermal boundary layer on the fin surface and the turbulence intensity, the two configurations possess an almost same temperature distribution. Fig. 8(b) shows that the increased oblique angle can effectively lower the pressure drop due to the reduced local resistance loss from the elliptical cross-section tube. Fig. 9 shows the variation of the performance evaluation criterion, PEC, with the air face velocity at various fin oblique angles. It is observed that the PEC always increases with increasing the air face velocity. However, the PEC varies little with the oblique angle. At the air face velocity of 2 m/s, the PEC with the fin oblique angle of 30 only increases by 1.43% in comparison with the baseline fin. 3.2. Effect of fin spacing The variations of the convection heat transfer coefficient with air face velocity at different fin spacing are shown in Fig. 10, from
Fig. 9. Effect of fin oblique angle on PEC of finned tube bundles.
Fig. 10. Variations of convection heat transfer coefficient with air face velocity at various fin spacings.
which can be seen that the convection heat transfer coefficient decreases as the fin spacing increases. But when the fin spacing is bigger than 3 mm, the convection heat transfer coefficient varies little with the fin spacing. At the face velocity of 2 m/s, the convection heat transfer coefficient with the fin spacing of 6 mm decreases by 3.96% in comparison with the baseline fin. This is because that as the fin spacing increases, the disturbance of cooling air is weakened and the thermal boundary layer becomes thick, which results in a reduced heat rejection from the fin-and-tube to the cooling air. When the fin spacing arrives at a certain high value, the boundary layer will have no longer impacts on the heat transfer, result in almost no changing convection heat transfer coefficients. As shown in Fig. 11, the pressure drop declines as the fin spacing increases, especially at high air face velocities, which is a result of the reduced friction loss for air flows. At the air face velocity of 2 m/ s, the pressure drop at the fin spacing of 6 mm decreases by 46.27% in comparison with the baseline fin. Fig. 12 shows the temperature and pressure contours between the fins for the fin spacing of 2 mm and 6 mm at the inlet air velocity of 1.5 m/s. As observed from Fig. 12(a), the average
Fig. 11. Variations of pressure drop with air face velocity at various fin spacings.
Y.Q. Kong et al. / International Journal of Thermal Sciences 107 (2016) 161e178
temperature of 309.96 K for the fin spacing of 2 mm is bigger than the average temperature of 309.45 K at the fin spacing of 6 mm, showing a better heat transfer from the fins to the air at a small fin spacing. However, Fig. 12(b) shows that the increased fin spacing can effectively lower the pressure drop due to the reduced flow friction loss. Fig. 13 shows the variation of PEC with the air face velocity at various fin spacing. It can be seen that the performance evaluation criterion increases with increasing the fin spacing. At the air face velocity of 2 m/s, PEC with the fin spacing of 6 mm increases by 59.77% in comparison with the baseline fin, showing that the big fin spacing benefits a lot to the thermo-flow performances of plate finned tube bundles. However, the increased fin spacing will reduce the fin number, thus lead to a reduced heat transfer surface area. 3.3. Effect of fin thickness Fig. 14 shows the variations of air-side convection heat transfer coefficient with the air face velocity at various fin thicknesses. With
171
increasing the fin thickness, the contact area between the base tube and fin goes up and more heat can be removed from the based tube to the fin and cooling air, so the convection heat transfer coefficient increases. At the air face velocity of 2 m/s, the average convection heat transfer coefficient with the fin thickness of 1.2 mm can increase by 19.11% in comparison with the baseline fin. The pressure drop versus the air face velocity at various fin thicknesses are shown in Fig. 15. It can be seen that the pressure drop also increases with increasing the fin thickness, due to the increased inlet local loss and increased flow friction loss. At the air face velocity of 2 m/s, the pressure drop with the fin thickness of 1.2 mm increases by 52.38% in comparison with the baseline fin. Fig. 16 shows the temperature and pressure contours between the fin surfaces for the fin thickness of 0.1 mm and 1.2 mm at the inlet air velocity of 1.5 m/s. From Fig. 16(a), it can be seen that the temperature gradients around the tubes as well as along the flow direction at the fin thickness of 0.1 mm are larger than those at the fin thickness of 1.2 mm, and the fin average temperatures at the fin thicknesses of 0.1 mm and 1.2 mm are 308.13 K and 310.49 K,
Fig. 12. Temperature and pressure contours for different fin spacing at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
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respectively. It shows that the thin fin results in a reduced heat transfer rate. Fig. 16(b) shows that the thick fin will result in an increased pressure drop due to the increased flow friction loss from the narrow channel. Fig. 17 shows the average PEC versus the air face velocity at various fin thicknesses. It can be seen that the PEC arrives at the maximum value at the fin thickness of 1.2 mm. At the air face velocity of 2 m/s, the average PEC with the fin thickness of 1.2 mm increases by 20.50% in comparison with the baseline fin. Since the extremely small fin thickness results in almost no heat rejection from the fin, which will deteriorate the comprehensive thermoflow performances of finned tube bundles. However, the excessive fin thickness will bring on the overweight of the air-cooled heat exchanger, and then increase the investment cost. In conclusion, the optimum fin thickness should be determined by taking both the thermo-flow performances and capital investment of finned tube bundles into consideration, which will be of benefit to the geometric optimization of air-cooled heat exchangers. Fig. 13. Effect of fin spacing on PEC of finned tube bundles.
Fig. 14. Variations of convection heat transfer coefficient with air face velocity at various fin thicknesses.
3.4. Effect of tube diameter The variations of the air-side convection heat transfer coefficient and pressure drop with the air face velocity at various tube external diameters are shown in Figs. 18 and 19 respectively. It can be seen that both the heat transfer coefficient and pressure drop increase with increasing the tube external diameter. As the tube diameter increases, the cross section of fluid flow narrows and the flow through the finned tube region is squeezed, so the boundary layers become thin and the swirl flows are generated due to the flow separation at the rear part of the tube. Consequently, the disturbed air flow leads to a well mixing of cooling air, and improves the heat transfer performance, but increases the pressure drop. At the air face velocity of 2 m/s, the average air-side heat transfer coefficient with the tube external diameter of 32 mm increases by 20.79% and the pressure drop increases by 60.43% in comparison with the baseline fin. Fig. 20 shows the temperature and pressure contours between the fins for tube diameters of 18 mm and 32 mm at the inlet air velocity of 1.5 m/s. From Fig. 20(a), it can be observed that the fin average temperature is 310.12 K for the tube diameter of 32 mm, while it is only 309.18 K for the tube diameter of 18 mm because the fin temperature distribution is more uniform and close to the tube wall temperature for a bigger tube diameter. As a result, the increased heat transfer rate will take place for the finned tube bundles with a big tube diameter. Fig. 20(b) shows that the pressure drop goes up with increasing the tube external diameter, resulting from the increased local resistance loss and flow friction loss. Fig. 21 shows the PEC versus the air face velocity at various tube external diameters. It can be observed that the PEC increases with increasing the tube external diameter, showing that the big tube diameter is of benefit to the comprehensive performance of plate finned tube bundles. Besides, the higher the air face velocity is, the more conspicuous the improvement of the PEC with increasing the tube diameter. At the air face velocity of 2 m/s, the average PEC with the tube external diameter of 32 mm increases by 14.20% in comparison with the baseline fin. In practical engineering, the big tube diameter will result in the reduced fin surface area for heat transfer, so it should be taken both the performance and investment into account for the determination of the tube external diameter. 3.5. Effect of natural convection
Fig. 15. Variations of pressure drop with air face velocity at various fin thicknesses.
Figs. 22e24 show the air-side convection heat transfer coefficient, pressure drop and heat transfer rate as a function of the air face velocity when the natural convection is taken into account or
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Fig. 16. Temperature and pressure contours for different fin thicknesses at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
ignored. It can be seen that the natural convection effects on the flow and heat transfer characteristics of plate finned tube bundles are negligible, although the buoyancy force results in a small increase of the pressure drop. For the plate finned tube bundles, the value of Ri is far less than 0.01, so the effect of natural convection can be ignored in the flow and heat transfer analyses. 3.6. Comprehensive performance of finned tube bundles For the finned tube bundles with a given volume, the global performance criterion (GPC) is used to evaluate the comprehensive performance in this work, which represents the total heat transfer rate per unit power consumption of the entire finned tube bundles. Fig. 25 shows the GPC versus the air inlet velocity at various fin oblique angles. It can be seen that the GPC increase slightly with the increase of fin oblique angle due to the reduced fin number and reduced flow power consumption. However, this change is so small that it can almost be neglected.
The GPC versus the air inlet velocity at various fin spacing from 2 to 14 mm is shown in Fig. 26. It can be observed that as the fin spacing increases, the GPC increases at first, but when the fin spacing is higher than 10 mm, it decreases yet, although this change is not conspicuous. The increased fin spacing will lead to a reduced pressure drop, thus a reduced power consumption. But the increase of fin spacing will also result in the reduced fin number and heat transfer surface area, so the heat transfer rate of finned tube bundles will decrease. At a small fin spacing, the decrease of the heat transfer rate is slower than the decrease of the power consumption, as a result, the GPC will increase with increasing the fin spacing. When the fin spacing further increases, the heat transfer rate decreases more conspicuously than the power consumption, so GPC decreases with increasing the fin spacing. At the fin spacing of 10 mm, the GPC arrives at its maximum value. Fig. 27 shows the GPC versus the air inlet velocity at various fin thicknesses. It can be seen that with increasing the fin thickness, the GPC increases at first but then decreases with the maximum
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Fig. 17. Effect of fin thickness on PEC of finned tube bundles.
value at the fin thickness of 0.3 mm, showing that the further increased fin thickness means the reduced fin number and increased local flow resistance, which will result in the deteriorated comprehensive thermo-flow performances. Fig. 28 gives the GPC versus the air inlet velocity at various tube external diameters. It can be seen that the GPC decreases as the tube external diameter increases due to the reduced heat transfer surface area and increased pressure drop of finned tube. Based on the global performance criterion (GPC), the optimal configurations of six-row plate finned tube bundles with the fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube diameter of 18 mm are recommended within the scope of investigation in this work for the Re ranging from 220 to 5500.
Fig. 19. Variations of pressure drop with air face velocity at various tube external diameters.
related data reduction method, was used to develop the physical models [23,24]. The details of the test samples are listed in Table 5, which can cover the whole range of geometric structures. With these 25 cases, the models of plate finned tube bundles are updated and meshed. Based on the simulation results, the correlations for the Nusselt number and friction factor can be achieved as follows, with the air face velocity ranging from 0.5 to 3.5 m/s and different geometrical parameters. The regression analysis coefficients are assessed with the help of the classical least square method.
Nu ¼ 0:3313Re0:575 ð1 aÞ0:0522
3.7. Flow and heat transfer correlations For the plate finned tube bundles, four geometric parameters, the fin oblique angle (0e30 ), fin spacing (2e14 mm), fin thickness (0.1e1.2 mm) and tube external diameter (18e32 mm) were investigated. The orthogonal experimental method also termed as a
D Dh
0:1002
f ¼7:6551Re0:4927 ð1aÞ0:1441
FS Dh
FS Dh
1:3204
1:7383
d
0:0351
Dh
d Dh
0:0004
(17)
D Dh
0:0922 (18)
The comparisons between the predicted results from the correlations and the numerical data are shown in Fig. 29. For the predicted and numerical Nusselt numbers as shown in Fig. 29(a), the average and maximum relative errors are 2.62% and 9.67% respectively. For the predicted and numerical friction factors as shown in Fig. 29(b), the average and maximum relative errors are 4.95% and 14.33% respectively. It shows that the flow and heat transfer correlation are of sufficient accuracy, which can be applied to the design and operation of air-cooled heat exchanger in power plants. The application ranges of the correlations are listed as follows:
0a
p
0:5476
0:013 Fig. 18. Variations of convection heat transfer coefficient with air face velocity at various tube external diameters.
(19)
6 FS 0:678 Dh
d Dh
0:332
(20)
(21)
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Fig. 20. Temperature and pressure contours for different tube external diameters at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
Fig. 21. Effect of tube external diameter on PEC of finned tube bundles.
Fig. 22. Effect of natural convection on convection heat transfer coefficient of finned tube bundles.
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Fig. 23. Effect of natural convection on pressure drop of finned tube bundles.
Fig. 24. Effect of natural convection on PEC of finned tube bundles.
Fig. 25. GPC versus air face velocity at various fin oblique angles.
Fig. 26. GPC versus air face velocity at various fin spacings.
Fig. 27. GPC versus air face velocity at various fin thicknesses.
Fig. 28. GPC versus air face velocity at various tube external diameters.
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Table 5 Schemes for the orthogonal experimental method. Case
Parameters
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2:228
d
FS
0 (1) 0 (1) 0 (1) 0 (1) 0 (1) 10 (2) 10 (2) 10 (2) 10 (2) 10 (2) 15 (3) 15 (3) 15 (3) 15 (3) 15 (3) 20 (4) 20 (4) 20 (4) 20 (4) 20 (4) 30 (5) 30 (5) 30 (5) 30 (5) 30 (5)
D 9:143 Dh
220 Re 5500
2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6
mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5)
0.1 0.3 0.6 0.9 1.2 0.3 0.6 0.9 1.2 0.1 0.6 0.9 1.2 0.1 0.3 0.9 1.2 0.1 0.3 0.6 1.2 0.1 0.3 0.6 0.9
D mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (2) (3) (4) (5) (1) (3) (4) (5) (1) (2) (4) (5) (1) (2) (3) (5) (1) (2) (3) (4)
18 22 25 27 32 25 27 32 18 22 32 18 22 25 27 22 25 27 32 18 27 32 18 22 25
mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (3) (4) (5) (1) (2) (5) (1) (2) (3) (4) (2) (3) (4) (5) (1) (4) (5) (1) (2) (3)
(22) (23)
4. Conclusions The impacts of the fin oblique angle, fin spacing, fin thickness and tube external diameter, as well as the natural convection on the flow and heat transfer characteristics of plate finned tube bundles are analyzed by numerical approach. The optimal geometric parameters are recommended and the flow and heat transfer correlations are developed. (1) The effect of fin oblique angle on the thermo-flow performances of plate finned tube bundles can be neglected. However, the increased fin oblique angle will lower the inlet air flow loss of air-cooled heat exchanger, which is of benefit to the operation of natural draft dry cooling system in power plants. The heat transfer coefficient and pressure drop both decrease with increasing the fin spacing. But when the fin spacing is bigger than 3 mm, the convection heat transfer coefficient varies little. (2) Increasing the fin thickness will result in an enhanced heat transfer, but lead to the overweight of heat exchanger and increased investment cost. Both the heat transfer coefficient and pressure drop increase as the tube diameter increases. (3) With a given heat exchanger volume, the optimal configurations of six-row plate finned tube bundles with the fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube diameter of 18 mm are recommended within the scope of investigation for the Re ranging from 220 to 5500. (4) The correlations between the Nusselt number, friction factor and the Reynolds number and geometrical parameters are
Fig. 29. Comparisons between predicted and numerical results (x-coordinate refers to the numerical data, y-coordinate stands for the predicted results from the correlations). (a) Nusselt number, (b) friction factor.
recommended using the orthogonal experimental design method, which can be used to the optimal design of aircooled heat exchanger in power plants. Acknowledgments The financial supports for this research, from the National Natural Science Foundation of China (Grant No. 51476055) and the National Basic Research Program of China (Grant No. 2015CB251503) are gratefully acknowledged. References [1] Al-Waked R, Behnia M. The performance of natural draft dry cooling towers under crosswind: CFD study. Int J Energy Res 2004;28(2):147e61. [2] Zhai Z, Fu S. Improving cooling efficiency of dry-cooling towers under crosswind conditions by using wind-break methods. Appl Therm Eng 2006;26(10):1008e17. [3] Viladkar MN, Karisiddappa, Bhargava P, Godbole PN. Static soilestructure interaction response of hyperbolic cooling towers to symmetrical wind loads. Eng Struct 2006;28(9):1236e51. [4] Verne E, Antonione C, Battezzati L, du Preez AF, Kroger DG. The effect of the
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Y.Q. Kong et al. / International Journal of Thermal Sciences 107 (2016) 161e178 heat exchanger arrangement and wind-break walls on the performance of natural draft dry-cooling towers subjected to cross-winds. J Wind Eng Ind Aerodyn 1995;58(3):293e303. Yang L, Tan H, Du X, Yang Y. Thermal-flow characteristics of the new wavefinned flat tube bundles in air-cooled condensers. Int J Therm Sci 2012;53: 166e74. S¸ahin HM, Dal AR, Baysal E. 3-D numerical study on the correlation between variable inclined fin angles and thermal behavior in plate fin-tube heat exchanger. Appl Therm Eng 2007;27(11e12):1806e16. Mon MS, Gross U. Numerical study of fin-spacing effects in annular-finned tube heat exchangers. Int J Heat Mass Transf 2004;47(8e9):1953e64. € _ Erek A, Ozerdem B, Bilir L, Ilken Z. Effect of geometrical parameters on heat transfer and pressure drop characteristics of plate fin and tube heat exchangers. Appl Therm Eng 2005;25(14e15):2421e31. Abu Madi M, Johns RA, Heikal MR. Performance characteristics correlation for round tube. Int J Refrig 1998;21:507e17. Matos RS, Laursen TA, Vargas JVC, Bejan A. Three-dimensional optimization of staggered finned circular and elliptic tubes in forced convection. Int J Therm Sci 2004;43(5):477e87. Yang L, Jia S, Bu Y, Du X, Yang Y. Numerical study on flow and heat transfer characteristics of finned tube bundles for air-cooled heat exchangers of indirect dry cooling systems in power plants. Proc CSEE 2012;32:50e7. Kim Y, Kim Y. Heat transfer characteristics of flat plate finned-tube heat exchangers with large fin pitch. Int J Refrig 2005;28(6):851e8. Zhou J, Tao W. Three dimensional numerical simulation and analysis of the airside performance of slotted fin surfaces with radial strips. Eng Comput 2004;22(7). 940e957(918). Carija Z, Frankovi c B, Per ci c M, Cavrak M. Heat transfer analysis of fin-andtube heat exchangers with flat and louvered fin geometries. Int J Refrig 2014:160e7.
[15] Li L, Du X, Yang L, Xu Y, Yang Y. Numerical simulation on flow and heat transfer of fin structure in air-cooled heat exchanger. Appl Therm Eng 2013;59:77e86. [16] Xie G, Wang Q, Sunden B. Parametric study and multiple correlations on airside heat transfer and friction characteristics of fin-and-tube heat exchangers with large number of large-diameter tube rows. Appl Therm Eng 2009;29(1): 1e16. [17] Yang S.M., Tao W.Q., Heat Transfer Higher Education Press, Beijing (2006) 67e68. [18] Tian L, He Y, Tao Y, Tao W. A comparative study on the air-side performance of wavy fin-and-tube heat exchanger with punched delta winglets in staggered and in-line arrangements. Int J Therm Sci 2009;48(9):1765e76. [19] Tao WQ. Numerical heat transfer. Xi'an Jiao Tong University Press; 2001. p. 333e88. [20] Mohseni M, Bazargan M. The effect of the low Reynolds number k-e turbulence models on simulation of the enhanced and deteriorated convective heat transfer to the supercritical fluid flows. Heat Mass Transf 2010;47(5):609e19. [21] Du X, Feng L, Yang Y, Yang L. Experimental study on heat transfer enhancement of wavy finned flat tube with longitudinal vortex generators. Appl Therm Eng 2013;50(1):55e62. [22] Xi XM, Song YF, Liu Y, Zhang XL, Du XZ, Yang LJ, et al. Study on freezing characteristics of direct air-cooling condenser based on heat transfer performance of wavy-finned flat tube. Proc CSEE 2014;34:4495e6. [23] Wang G, Ma X, Hu T, Zhang D. Experimental and analytical study on factors influencing biomimetic undulating fin propulsion performance based on orthogonal experimental design. Adv Robot 2013;27(8):597e609. [24] Wang Y, Wu C, Tang Z, Yang X, Deng Y, Su C. Optimization of fin distribution to improve the temperature uniformity of a heat exchanger in a thermoelectric generator. J Electron Mater 2014;44(6):1724e32.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Impacts of geometric structures on thermo-flow performances of plate fin-tube bundles Y.Q. Kong, L.J. Yang*, X.Z. Du, Y.P. Yang Key Laboratory of Condition Monitoring and Control for Power Plant Equipments of Ministry of Education, School of Energy Power and Mechanical Engineering, North China Electric Power University, Beijing, 102206, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 November 2015 Received in revised form 6 April 2016 Accepted 11 April 2016
Continuous plate fin-tube heat exchangers are commonly applied to natural draft dry cooling system in power plants due to their excellent thermo-flow performances, but the effects of geometrical structures on the air-side flow and heat transfer characteristics have not been thoroughly clarified yet. In this study, the air-side flow and heat transfer models of plate fin-tube bundles are developed, and the numerical modeling and methods are validated by the experimental test. The thermo-flow characteristics of the plate fin-tube bundles at various parameters of air face velocity, fin oblique angle, fin spacing, fin thickness and tube external diameter are numerically simulated. The buoyancy effect due to natural convection is also investigated. The results show that the performance evaluation criterion (PEC) varies little with the fin oblique angle. As the fin spacing increases, the average PEC will increase despite of a considerable heat transfer reduction. The increased fin thickness and tube external diameter are both beneficial to the thermo-flow characteristics, but the buoyancy effect on the flow and heat transfer performances can be neglected. The fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube external diameter of 18 mm are recommended within the scope of investigation. The correlations between the Nusselt number, friction factor and the Reynolds number and geometrical parameters are recommended using the orthogonal experimental design method, which can be used to the optimal design of air-cooled heat exchanger in power plants. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Natural draft dry cooling system Air-cooled heat exchanger Finned tube bundles Plate fin Flow and heat transfer Performance evaluation criterion
1. Introduction Nowadays, natural draft dry cooling system with air-cooled heat exchangers and dry-cooling towers is commonly applied to power plants in the regions where water resource is of shortage or expensive [1e3]. Thanks to less initial costs and self-supporting nature, the heat exchanger cooling deltas are frequently arranged vertically around the circumference of the dry-cooling tower [4]. The ambient air flows through the air-cooled heat exchanger and dry-cooling tower due to the buoyancy force originating from the air density difference between the interior and exterior of the tower, taking away the heat rejection from the circulating water. The plate fin-tube heat exchanger is the core facility of natural draft dry cooling system, so it is helpful for the air-cooled heat exchanger design to investigate the thermo-flow performances of the plate fin-tube bundles.
* Corresponding author. E-mail address: [email protected] (L.J. Yang). http://dx.doi.org/10.1016/j.ijthermalsci.2016.04.011 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
The flow and heat transfer performances of finned tube heat exchangers have attracted great attentions in the past years. Yang et al. [5] investigated the effect of oblique configuration of wavefinned flat tube bundles on the thermo-flow characteristics, and obtained the flow and heat transfer correlations. S¸ahin et al. [6] investigated the pressure drop and heat transfer of seven plain fin-tube heat exchangers with different fin angles, concluding that the inclined fin with the angle of 30 is the optimum configuration. Mon and Gross [7] studied the effect of fin spacing on the four-row annular-finned tube bundles in staggered and in-line arrangements, finding that the boundary layer development and horseshoe vortices between the fins depend substantially on the fin spacing to height ratio and Reynolds number. Erek et al. [8] numerically studied the pressure drop and heat transfer of the plate fin-tube bundles at different fin geometries, finding that the pressure drop is closely related to the fin distance. Madi et al. [9] experimentally studied 28 heat exchanger samples in an open thermal wind tunnel, concluding that the fin type plays important roles in the flow and heat transfer, but the number of tube rows has almost no effect on the friction factor. Matos et al. [10] optimized the three-
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Nomenclature Ab Ac Af cp D Dh FS f GPC g H H1 H2 He h I k L m Nf Nu P Pc PEC Pl Pt Pr
base tube surface area (m2) minimum flow cross-sectional area (m2) fin surface area (m2) specific heat capacity(J kg1 K1) tube external diameter (mm) hydraulic diameter (mm) fin spacing friction factor global performance criterion gravitational acceleration (m s2) enthalpy (J kg1) upper slotted strip height (mm) lower slotted strip height (mm) height of fined tube (m) air side convection heat transfer coefficient (W m2 K1) turbulence intensity turbulence kinetic energy (m2 s2) flow depth (mm) mass flow rate (kg s1) fin number Nusselt number pressure (Pa) minimum flow cross-section perimeter (mm) performance evaluation criterion longitudinal tube pitch (mm) transverse tube pitch (mm) Prandtl number
dimensional finned tube bundles in staggered arrangements of circular and elliptic tubes, showing that the elliptical tube configuration has a better overall performance and a lower cost than the circular tube. For six types of plate fin-tube heat exchangers used in power plants, Yang et al. [11] pointed out that the Forgo-type aluminum finned tube bundle is superior to the steel finned ovaltube one. Kim et al. [12] tested 22 heat exchangers with various fin pitches, various numbers of tube row and tube alignment by experiments. As is well known, there are still some limitations to the air-side flow and heat transfer performances, and a clear understanding of air flow in the complex passages (such as slot, louvers and wave fin) of heat exchangers is needed, so that the heat transfer surface can be designed efficiently. For the plain plate fin and three types of radial slotted (full slotted, partially slotted, back slotted) fin surfaces, the numerical simulation showed that the full slotted fin surface has the highest heat flow rate with the biggest pressure drop penalty at the same face velocity [13]. Carija et al. [14] compared flat and louvered fin-and-tube heat exchangers, and no remarkable differences of Nusselt number and pressure drop were found. Li et al. [15] proposed a new wavy fin configuration, and pointed out that the periodic convergent-divergent channel formed by wavy fins results in a big pressure drop, which leads to a poor overall performance despite of the excellent heat transfer performance. The heat exchanger performance involves many parameters, among which the main geometrical parameters consist of fin oblique angle, fin pitch, fin thickness, tube pitch and tube outside diameter. Previous investigations were restricted to specific finned tube heat exchangers for the natural draft dry cooling system in power plants, while the influences of geometrical parameters,
Q Qe qv Re Ri S SW T Ti To Tb u ui um xi
heat flow rate (W) total heat flow rate of an entire finned tube volumetric flow rate (m3 s1) Reynolds number Richardson number slotted strip spacing (mm) slotted strip width (mm) temperature (K) air inlet temperature (K) air outlet temperature (K) base tube temperature (K) air face velocity (m s1) velocity in the i direction (m s1) minimum flow cross-sectional velocity (m s1) Cartesian coordinate (m)
Greek symbols fin oblique angle ( ) thermal expansion coefficient (K1) fin thickness (mm) ε turbulence dissipation rate (m2 s3) hf fin efficiency l thermal conductivity (W m1 K1) m dynamic viscosity (kg m1 s1) mt turbulent viscosity (kg m1 s1) y kinetic viscosity (m2 s1) r fluid density (kg m3) t shear stress (Pa) 4 dissipation function in energy equation (W m3)
a b d
except the fin pitch and tube pitch [16], have not been analyzed so far. The effects of air face velocity, oblique angle of fin, fin spacing, fin thickness and tube external diameter will be studied in this work so that the optimal geometrical parameters can be expected. The relative importance of various geometry parameters will be quantitatively assessed based on the performance evaluation criterion. Additionally, the flow and heat transfer correlations for the plate fin-tube bundles in air-cooled heat exchanger will be presented, which can be applied to the design of natural draft dry cooling system in power plants. 2. Models 2.1. Physical model The natural draft dry cooling system in power plants is schematically shown in Fig. 1. The finned tube bundles of air-cooled heat exchanger are arranged vertically around the circumference of the dry-cooling tower as shown in Fig. 1(a), and the ambient air flows across the air-cooled heat exchanger and through the dry-cooling tower sequentially, while the circulating water flows inside the round tubes in staggered configuration. Fig. 1(b) shows the six-row finned tube bundles, which are widely adopted in the air-cooled heat exchanger. The fin and tube are both made of aluminum, which are connected by the tube-expansion technology. In Fig. 2, the geometric parameters and computational domains for the six-row finned tube bundles are presented. For the oblique finned tube bundles, as the oblique angle increases, the total length of the fin will increase either, while the horizontal distance of the fin keeps constant. Table 1 shows the geometric details of the finned tube bundles. The dashed lines confine the computational
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Fig. 1. Schematic of natural draft dry cooling system. (a) air-cooled heat exchanger and dry-cooling tower, (b) six-row plate finned tube bundles.
domain, where 80 mm is extended for the entrance section to ensure a uniform velocity distribution and 300 mm is extended for the exit section to avoid the recirculation flows. The middle surfaces of the neighboring two fins are defined as the symmetric boundaries, as shown in Fig. 2(a) and (b), and the other two sides of the computational domain across the middle sections of the neighboring tubes are also set symmetric boundaries as shown in Fig. 2(c). Due to the relatively high convective heat transfer coefficient on the water side and high thermal conductivity of the tube, the outer wall of the tube is assumed to keep a constant temperature Tw ¼ 310.85 K equal to the average temperature of the water inlet and outlet temperatures. However, the temperature distribution on the fin surface has to be determined by solving the conjugate problem, in which both the temperatures on the solid fin surface and in the fluid need to be solved simultaneously. The computational domain inlet is defined as the velocity inlet
boundary condition with the velocity ranging from 0.5 m/s to 3.5 m/s and a constant inlet air temperature of 289.15 K. The turbulent intensity is defined by the widely used correlation I ¼ 0:16ReDh 1=8 . Due to the unchanging velocity and temperature fields, the outflow boundary condition is applied to the domain outlet. At the solid surfaces, the no-slip conditions for the velocity are specified. In addition, some assumptions are made as follows:
The flow is steady-state; The fluid is incompressible; The radiation effect is ignored; The thermal contact resistance between the base tube and fin is ignored due to the order with the small magnitude of 105 [17], which is far less than the air side convection thermal resistance of 102;
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Fig. 2. Geometric details, computational domain and boundary conditions of finned tube bundles. (a) front view of horizontal-plate finned tube bundles, (b) front view of obliqueplate finned tube bundles, (c) top view of finned tube bundles.
Y.Q. Kong et al. / International Journal of Thermal Sciences 107 (2016) 161e178 Table 1 Geometric dimensions of plate fin-tube heat exchanger. Oblique angle of fin (a/ ) Fin spacing (FS/mm) Fin thickness (d/mm) Tube external diameter (D/mm) Longitudinal tube pitch (Pl/mm) Transverse tube pitch (Pt/mm)
0, 10, 15, 20, 25, 30 2, 3, 4, 5, 6, 8, 10, 12, 14 0.1, 0.3, 0.6, 0.9, 1.2 18, 22, 25, 27, 32 35 30
The thermo-physical properties of the solid and fluid are considered constant.
2.2. Governing equations The governing equations for the three-dimensional, steadystate flow and heat transfer of air can be expressed as follows. Continuity equation:
vðrui Þ ¼0 vxi
(1)
For turbulent flows, the momentum equation takes the following form by using the Reynolds time averaging method [18,19]:
v vP vt v rui uj ¼ þ ij þ vxi vxi vxj vxj
ru0i u0j
(2) 2.4. Parameter definitions
vui vuj 2 vuk þ d vxj vxi 3 ij vxk
! (3)
By using the Boussinesq theorem, the turbulent shear stress can be obtained by the following form.
ru0i u0j
as shown in Fig. 3(b). The grid independence is tested for the heat transfer coefficient of the finned tube bundles (a¼0 , FS ¼ 3.2 mm, d ¼ 0.3 mm and D ¼ 25 mm) at the air velocity of 1.5 m/s. The results at different grid numbers are tabulated in Table 2. The relative errors of the convection heat transfer coefficients between the three sparse grids and the dense grid are 4.86%, 1.27% and 0.84% respectively, showing that the calculated results based on 1,216,107 cells are insensitive to further grid refinement, so the grid number of 1,216,107 is employed in this case. In other cases, similar works are also done to obtain the final grid numbers. The governing equations with the boundary conditions are solved by the commercial software FLUENT 6.3.26. The equations for the continuum, momentum, energy, turbulent kinetic energy and its dissipation rate are discretized using the second-order upwind scheme to obtain more accurate results. The SIMPLE algorithm is applied to the pressureevelocity coupling, which stores the discrete values of the scalar variables at the center of the control volume, as well as the interpolated values at the center of all the faces incorporated. The complex flow in the fin channel is not aligned with the grid. To control the update of the computed variables at each iteration, the under-relaxation factors vary between 0.3 and 1.0. A divergence-free criterion of 106 based on the scaled residual is prescribed for the continuity, momentum, k and ε equations, and 108 for the energy equation.
where
tij ¼ mt
165
¼ mt
vui vuj 2 vuk þ d vxj vxi 3 ij vxk
!
2 dij rk 3
(4)
where the symbols with apostrophes denote the fluctuation components. The energy equation in terms of enthalpy has the following form:
v v vT m vH ðrui HÞ ¼ l þ t vxi vxi vxi Prt vxi
þ ui
· vP þQ þ4 vxi
(5)
The turbulence effect on the fluid flow is included through the application of low Reynolds number k-ε models [20]. The physical properties of air are evaluated at the mean air temperature (Ti þ To)/ 2. When taking the natural convection into consideration, the air density is determined by the ideal gas law and the gravity is set at the momentum equation to provide the buoyancy effect. 2.3. Numerical methods The geometric model is created and meshed by using GAMBIT software. For the finned tube region, the hexahedral structured mesh with a minimal cell skewness is used to resolve the swirling flow, while for those in the extended regions, the coarse grids are adopted to reduce the computational cost as shown in Fig. 3(a). For turbulent convective heat transfer, the air velocity and temperature near the wall region have great gradients, so the boundary layer mesh are adopted at the near wall region for accurate simulations,
For the heat transfer performance, the characteristic nondimensional parameters are defined as follow:
Re ¼ rum Dh =m
(6)
Nu ¼ h$Dh =l
(7)
where, Re is the Reynolds number, Nu is the Nusselt number. r, m, l are the density, dynamic viscosity and thermal conductivity respectively. Dh is the hydraulic diameter of the finned tube and takes the following form.
Dh ¼ 4Ac =Pc
(8)
where Ac is the minimum free flow area, Pc is the perimeter at the minimum flow cross-section. um is the air velocity at the minimum flow area, h is the air side convection heat transfer coefficient and can be calculated with the following equation:
h¼
Q Ab þ hf Af DTm
(9)
where Ab is the base tube surface area. Af is the fin surface area. Q is the air-side heat transfer rate, Q ¼ mcp(To Ti). DTm is the logarithmic mean temperature difference between the tube wall and the air, which is defined as [21]:
DTm ¼
T b Ti T b To ðT Ti Þ ln b ðT b To Þ
(10)
The fin efficiency, hf, is calculated from the values of the simulated results [21]:
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Fig. 3. Grid system of computational models. (a) extended and fin-and-tube regions, (b) boundary layer mesh in the fin-and-tube domain.
Table 2 Grid independence results at various grid numbers.
hf ¼
Finned tube model
Grid number
Heat transfer coefficient (W/m2 K)
a¼0 , FS ¼ 3.2 mm, d ¼ 0.3 mm, D ¼ 25 mm
901,152 1,100,534 1,216,107 1,454,965
36.546 37.924 38.089 38.413
Tf Ta Tb Ta
(11)
where T a ¼ 1=2ðTi þ To Þ is theR average value of the inlet and outlet TdV temperatures of the air. T f ¼ V is the volume-weighted average fin temperature, which is computed by dividing the summation of the product of the temperature and cell volume by the total volume of the fin zone. T b is the base tube average temperature. For the flow performance, the friction factor f of the finned tube is defined as [16]:
f ¼
2DP Dh $ ru2m L
(12)
where L is the flow depth, Dp is the pressure drop across the finned
tube bundles. The performance evaluation index PEC, representing the heat transfer capability of a surface with a given flow resistance, is commonly defined as follows. Generally speaking, the higher the PEC is, the better the overall performance of the fin-and-tube bundles [21].
. PEC ¼ Nu f 1=3
(13)
The non-dimensional Richardson number, Ri, is used to analyze the influence of natural convection, which is defined as the ratio of the buoyancy force to inertia force.
Ri ¼
Gr Re2
¼
g bDtl3
y2 Re2
(14)
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Fig. 4. Experimental system for the thermo-flow performances of finned tube bundles. (a) photo of alternant slotted finned tube bundles in staggered configuration, (b) geometric structure of alternant slotted finned tube, (c) experimental setup and locations of measuring points, (d) wind tunnel experimental system.
Table 3 Geometric dimensions of the experimental sample.
a 0
D
N
Pt
Pl
d
FS
H1
H2
SW
S
25 mm
4
30 mm
35 mm
0.3 mm
3.2 mm
1.1 mm
0.5 mm
2.75 mm
2 mm
Table 4 Summary of estimated uncertainties. Measured variables
Derived quantities
Parameter
Uncertainties (%)
Parameter
Uncertainties (%)
mwater uair tin-air tout-air tin-water tout-water DP
4.77 5.45 0.18 0.67 0.15 0.38 4.46
Qair Qwater Re h Nu f
4.49 5.49 5.45 7.14 7.14 7.04
where Dt ¼ T f T a , T ft g is the gravitational acceleration. b is the thermal expansion coefficient. l is the feature size of finned tube and equals the fin spacing. As long as Ri is smaller than 0.01, the buoyancy force can be ignored, and the inertia force plays a dominant role in the air flows. In this work, a global performance criterion (GPC) is also adopted
to evaluate the comprehensive performance of the finned tube bundles with a given volume, which is defined as the total heat transfer rate of the entire finned tube bundles per unit flow power consumption, and takes the following form:
GPC ¼
Qe
DPqv
(15)
where Qe ¼ Q $Nf , is the total heat transfer rate of the entire finned tube. The fin number Nf of finned tube bundles is given by
Nf ¼
He Fs þ d
(16)
where He is the height of the finned tube. qv ¼ uAe is the volumetric flow rate of the finned tube bundles. u is the average value of the inlet and outlet air velocities. Ae ¼ He $Pt , is the face area of the finned tube. Pt is the transverse tube pitch.
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2.5. Verification of the computational model By comparing with the experimental results, the standard k-ε model, low-Re k-ε model, realizable k-ε model and laminar model are verified for their feasibility in the flow simulations. The flow and heat transfer experiments are made in a wind tunnel system. The alternant slotted finned tube bundles in the staggered configuration produced by a certain manufacturer of China are selected as the experimental sample as shown in Fig. 4(a). The geometric structure of the finned tube is schematically shown in Fig. 4(b), and the geometric dimensions of the experimental sample are listed in Table 3. The experimental system is schematically shown in Fig. 4(c) and 4(d), where the transversal surface has the size of 500 700 mm and the length is 1000 mm for the test section. The test section is insulated by the asbestos layer to minimize the heat loss. The experimental sample closely matches the test section in order to eliminate the contraction and expansion losses. The experimental system consists of two loops, an air loop and a water loop. Thermostatic water tank can be adjusted by transformers to obtain a uniform water inlet temperature during
Fig. 6. Variations of convection heat transfer coefficient with air face velocity at various fin oblique angles.
the test. The water flow rate inside the vertical tube is measured by a turbine flow meter with an accuracy of 0.5%. During the experiment, the water inlet temperature is held constant at 60 C with an error range of ±0.5 C, and the water flow rate is approximately 0.694 kg/s. Both the inlet and outlet tube wall temperatures are measured by setting six T-type thermocouples at the inlet and outlet parts of the tubes. The air flow rate over the test surface is adjusted by a digital blower speed controller. The inlet air velocity varies from 0.5 to 3.5 m/s, which is measured by a hotwire anemometer with the precision of ±2%. The inlet and outlet pressures of the test sample are obtained by two Pitot tubes at the center of both sides of the test section, and a differential pressure transmitter with the precision of ±0.15 Pa. The inlet and outlet air temperatures across the test section are measured by two T-type thermocouple meshes. The inlet measuring mesh consists of 12 thermocouples, while the outlet mesh contains 16 thermocouples due to the increased air disturbance after the finned tube bundles. All the thermocouples are pre-calibrated by the ice-water mixtures with an error range of ±0.1 C. Fig. 4(c) labels all the locations of the measuring points. The energy un-balance between the air side and
Fig. 5. Validation of numerical results with experimental data. (a) Nusselt number, (b) friction factor.
Fig. 7. Variations of pressure drop with air face velocity at various fin oblique angles.
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169
Fig. 8. Temperature and pressure contours for different fin oblique angles at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
water side is within 7.57%. The experimental data are recorded when the measured temperatures stabilize. All the data signals are collected and converted by a data acquisition system and a host computer. The experimental uncertainties are tabulated in Table 4 according to the analysis method used by Xi et al. [22]. Fig. 5 shows the comparisons between the experimental and computed Nusselt number and friction factor. It can be seen that the numerical results by the standard and realizable k-ε models both show substantial increases compared with the low Re k-ε model resulting from the strong turbulence intensity in the flow. While the laminar model predicts a smaller result than others due to the weaker turbulence intensity. The simulation results of the low Re k-ε model agree well with the experimental data with the highest deviation of the Nusselt number and friction factor as low as 5.3% and 8.3% respectively, and moreover, the low Re k-ε model can provide several modification schemes for the high Re number model, which will automatically adapt to different ranges of Re of the flow. Hence, the low Re k-ε model is selected as the turbulence model. Besides, the same turbulence model and numerical method
are adopted in the subsequent simulations for different fin shapes in this work. 3. Results and discussion The influences of various geometric parameters and the buoyancy force of warm air on the flow and heat transfer characteristics are investigated. In order to quantitatively assess the relative effects of different geometric parameters of finned tube bundles, the finned tube with the parameters of a ¼ 0 , FS ¼ 3 mm, d ¼ 0.3 mm and D ¼ 25 mm is set as a baseline one for the comparisons. 3.1. Effect of fin oblique angle Figs. 6 and 7 show the variations of the air-side convection heat transfer coefficient and pressure drop with air face velocity at various fin oblique angles (a ¼ 0 , 10 , 15 , 20 , 25 , 30 ) respectively. In general, the convection heat transfer coefficient and pressure drop vary dramatically with the face velocity. The higher
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the face velocity is, namely the higher the Reynolds number is, the higher the heat transfer coefficient and pressure drop are. It can also be seen that the effect of fin oblique angle on the convection heat transfer coefficient can be neglected. At the air face velocity of 2 m/s as an illustrative case, the convection heat transfer coefficient with the fin oblique angle of 30 only decreases by 0.22% in comparison with the baseline fin. It is because that as the fin oblique angle increases, the total fin surface area and heat flow rate both increase, which leads to an almost unchanging air-side convection heat transfer coefficient at the same face velocity. As shown in Fig. 7, the pressure drop also decreases little as the fin oblique angle goes up. At the air face velocity of 2 m/s, the pressure drop with the fin oblique angle of 30 decreases by 6.31% in comparison with the baseline fin. When the fin oblique angle increases, the flow across cylinder changes to the flow across elliptic cylinders, resulting in a reduced local flow loss, but the flow friction loss will increase due to the increased flow surface area. The former plays a dominant role in the total pressure drop, so the pressure drop decreases with increasing the fin oblique angle. For a better understanding of the fin oblique angle effects, Fig. 8 shows the temperature and pressure contours between the fin surfaces for a ¼ 0 and a ¼ 30 at the air face velocity of 1.5 m/s. From Fig. 8(a), it can be seen that the temperature of fin surface increases along the flow direction due to the increased air temperature. Moreover, the flow separates at the rear portion of a tube, and then forms a large region with a deteriorated heat transfer performance between the tubes. Owing to the similar thickness of the thermal boundary layer on the fin surface and the turbulence intensity, the two configurations possess an almost same temperature distribution. Fig. 8(b) shows that the increased oblique angle can effectively lower the pressure drop due to the reduced local resistance loss from the elliptical cross-section tube. Fig. 9 shows the variation of the performance evaluation criterion, PEC, with the air face velocity at various fin oblique angles. It is observed that the PEC always increases with increasing the air face velocity. However, the PEC varies little with the oblique angle. At the air face velocity of 2 m/s, the PEC with the fin oblique angle of 30 only increases by 1.43% in comparison with the baseline fin. 3.2. Effect of fin spacing The variations of the convection heat transfer coefficient with air face velocity at different fin spacing are shown in Fig. 10, from
Fig. 9. Effect of fin oblique angle on PEC of finned tube bundles.
Fig. 10. Variations of convection heat transfer coefficient with air face velocity at various fin spacings.
which can be seen that the convection heat transfer coefficient decreases as the fin spacing increases. But when the fin spacing is bigger than 3 mm, the convection heat transfer coefficient varies little with the fin spacing. At the face velocity of 2 m/s, the convection heat transfer coefficient with the fin spacing of 6 mm decreases by 3.96% in comparison with the baseline fin. This is because that as the fin spacing increases, the disturbance of cooling air is weakened and the thermal boundary layer becomes thick, which results in a reduced heat rejection from the fin-and-tube to the cooling air. When the fin spacing arrives at a certain high value, the boundary layer will have no longer impacts on the heat transfer, result in almost no changing convection heat transfer coefficients. As shown in Fig. 11, the pressure drop declines as the fin spacing increases, especially at high air face velocities, which is a result of the reduced friction loss for air flows. At the air face velocity of 2 m/ s, the pressure drop at the fin spacing of 6 mm decreases by 46.27% in comparison with the baseline fin. Fig. 12 shows the temperature and pressure contours between the fins for the fin spacing of 2 mm and 6 mm at the inlet air velocity of 1.5 m/s. As observed from Fig. 12(a), the average
Fig. 11. Variations of pressure drop with air face velocity at various fin spacings.
Y.Q. Kong et al. / International Journal of Thermal Sciences 107 (2016) 161e178
temperature of 309.96 K for the fin spacing of 2 mm is bigger than the average temperature of 309.45 K at the fin spacing of 6 mm, showing a better heat transfer from the fins to the air at a small fin spacing. However, Fig. 12(b) shows that the increased fin spacing can effectively lower the pressure drop due to the reduced flow friction loss. Fig. 13 shows the variation of PEC with the air face velocity at various fin spacing. It can be seen that the performance evaluation criterion increases with increasing the fin spacing. At the air face velocity of 2 m/s, PEC with the fin spacing of 6 mm increases by 59.77% in comparison with the baseline fin, showing that the big fin spacing benefits a lot to the thermo-flow performances of plate finned tube bundles. However, the increased fin spacing will reduce the fin number, thus lead to a reduced heat transfer surface area. 3.3. Effect of fin thickness Fig. 14 shows the variations of air-side convection heat transfer coefficient with the air face velocity at various fin thicknesses. With
171
increasing the fin thickness, the contact area between the base tube and fin goes up and more heat can be removed from the based tube to the fin and cooling air, so the convection heat transfer coefficient increases. At the air face velocity of 2 m/s, the average convection heat transfer coefficient with the fin thickness of 1.2 mm can increase by 19.11% in comparison with the baseline fin. The pressure drop versus the air face velocity at various fin thicknesses are shown in Fig. 15. It can be seen that the pressure drop also increases with increasing the fin thickness, due to the increased inlet local loss and increased flow friction loss. At the air face velocity of 2 m/s, the pressure drop with the fin thickness of 1.2 mm increases by 52.38% in comparison with the baseline fin. Fig. 16 shows the temperature and pressure contours between the fin surfaces for the fin thickness of 0.1 mm and 1.2 mm at the inlet air velocity of 1.5 m/s. From Fig. 16(a), it can be seen that the temperature gradients around the tubes as well as along the flow direction at the fin thickness of 0.1 mm are larger than those at the fin thickness of 1.2 mm, and the fin average temperatures at the fin thicknesses of 0.1 mm and 1.2 mm are 308.13 K and 310.49 K,
Fig. 12. Temperature and pressure contours for different fin spacing at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
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respectively. It shows that the thin fin results in a reduced heat transfer rate. Fig. 16(b) shows that the thick fin will result in an increased pressure drop due to the increased flow friction loss from the narrow channel. Fig. 17 shows the average PEC versus the air face velocity at various fin thicknesses. It can be seen that the PEC arrives at the maximum value at the fin thickness of 1.2 mm. At the air face velocity of 2 m/s, the average PEC with the fin thickness of 1.2 mm increases by 20.50% in comparison with the baseline fin. Since the extremely small fin thickness results in almost no heat rejection from the fin, which will deteriorate the comprehensive thermoflow performances of finned tube bundles. However, the excessive fin thickness will bring on the overweight of the air-cooled heat exchanger, and then increase the investment cost. In conclusion, the optimum fin thickness should be determined by taking both the thermo-flow performances and capital investment of finned tube bundles into consideration, which will be of benefit to the geometric optimization of air-cooled heat exchangers. Fig. 13. Effect of fin spacing on PEC of finned tube bundles.
Fig. 14. Variations of convection heat transfer coefficient with air face velocity at various fin thicknesses.
3.4. Effect of tube diameter The variations of the air-side convection heat transfer coefficient and pressure drop with the air face velocity at various tube external diameters are shown in Figs. 18 and 19 respectively. It can be seen that both the heat transfer coefficient and pressure drop increase with increasing the tube external diameter. As the tube diameter increases, the cross section of fluid flow narrows and the flow through the finned tube region is squeezed, so the boundary layers become thin and the swirl flows are generated due to the flow separation at the rear part of the tube. Consequently, the disturbed air flow leads to a well mixing of cooling air, and improves the heat transfer performance, but increases the pressure drop. At the air face velocity of 2 m/s, the average air-side heat transfer coefficient with the tube external diameter of 32 mm increases by 20.79% and the pressure drop increases by 60.43% in comparison with the baseline fin. Fig. 20 shows the temperature and pressure contours between the fins for tube diameters of 18 mm and 32 mm at the inlet air velocity of 1.5 m/s. From Fig. 20(a), it can be observed that the fin average temperature is 310.12 K for the tube diameter of 32 mm, while it is only 309.18 K for the tube diameter of 18 mm because the fin temperature distribution is more uniform and close to the tube wall temperature for a bigger tube diameter. As a result, the increased heat transfer rate will take place for the finned tube bundles with a big tube diameter. Fig. 20(b) shows that the pressure drop goes up with increasing the tube external diameter, resulting from the increased local resistance loss and flow friction loss. Fig. 21 shows the PEC versus the air face velocity at various tube external diameters. It can be observed that the PEC increases with increasing the tube external diameter, showing that the big tube diameter is of benefit to the comprehensive performance of plate finned tube bundles. Besides, the higher the air face velocity is, the more conspicuous the improvement of the PEC with increasing the tube diameter. At the air face velocity of 2 m/s, the average PEC with the tube external diameter of 32 mm increases by 14.20% in comparison with the baseline fin. In practical engineering, the big tube diameter will result in the reduced fin surface area for heat transfer, so it should be taken both the performance and investment into account for the determination of the tube external diameter. 3.5. Effect of natural convection
Fig. 15. Variations of pressure drop with air face velocity at various fin thicknesses.
Figs. 22e24 show the air-side convection heat transfer coefficient, pressure drop and heat transfer rate as a function of the air face velocity when the natural convection is taken into account or
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Fig. 16. Temperature and pressure contours for different fin thicknesses at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
ignored. It can be seen that the natural convection effects on the flow and heat transfer characteristics of plate finned tube bundles are negligible, although the buoyancy force results in a small increase of the pressure drop. For the plate finned tube bundles, the value of Ri is far less than 0.01, so the effect of natural convection can be ignored in the flow and heat transfer analyses. 3.6. Comprehensive performance of finned tube bundles For the finned tube bundles with a given volume, the global performance criterion (GPC) is used to evaluate the comprehensive performance in this work, which represents the total heat transfer rate per unit power consumption of the entire finned tube bundles. Fig. 25 shows the GPC versus the air inlet velocity at various fin oblique angles. It can be seen that the GPC increase slightly with the increase of fin oblique angle due to the reduced fin number and reduced flow power consumption. However, this change is so small that it can almost be neglected.
The GPC versus the air inlet velocity at various fin spacing from 2 to 14 mm is shown in Fig. 26. It can be observed that as the fin spacing increases, the GPC increases at first, but when the fin spacing is higher than 10 mm, it decreases yet, although this change is not conspicuous. The increased fin spacing will lead to a reduced pressure drop, thus a reduced power consumption. But the increase of fin spacing will also result in the reduced fin number and heat transfer surface area, so the heat transfer rate of finned tube bundles will decrease. At a small fin spacing, the decrease of the heat transfer rate is slower than the decrease of the power consumption, as a result, the GPC will increase with increasing the fin spacing. When the fin spacing further increases, the heat transfer rate decreases more conspicuously than the power consumption, so GPC decreases with increasing the fin spacing. At the fin spacing of 10 mm, the GPC arrives at its maximum value. Fig. 27 shows the GPC versus the air inlet velocity at various fin thicknesses. It can be seen that with increasing the fin thickness, the GPC increases at first but then decreases with the maximum
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Fig. 17. Effect of fin thickness on PEC of finned tube bundles.
value at the fin thickness of 0.3 mm, showing that the further increased fin thickness means the reduced fin number and increased local flow resistance, which will result in the deteriorated comprehensive thermo-flow performances. Fig. 28 gives the GPC versus the air inlet velocity at various tube external diameters. It can be seen that the GPC decreases as the tube external diameter increases due to the reduced heat transfer surface area and increased pressure drop of finned tube. Based on the global performance criterion (GPC), the optimal configurations of six-row plate finned tube bundles with the fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube diameter of 18 mm are recommended within the scope of investigation in this work for the Re ranging from 220 to 5500.
Fig. 19. Variations of pressure drop with air face velocity at various tube external diameters.
related data reduction method, was used to develop the physical models [23,24]. The details of the test samples are listed in Table 5, which can cover the whole range of geometric structures. With these 25 cases, the models of plate finned tube bundles are updated and meshed. Based on the simulation results, the correlations for the Nusselt number and friction factor can be achieved as follows, with the air face velocity ranging from 0.5 to 3.5 m/s and different geometrical parameters. The regression analysis coefficients are assessed with the help of the classical least square method.
Nu ¼ 0:3313Re0:575 ð1 aÞ0:0522
3.7. Flow and heat transfer correlations For the plate finned tube bundles, four geometric parameters, the fin oblique angle (0e30 ), fin spacing (2e14 mm), fin thickness (0.1e1.2 mm) and tube external diameter (18e32 mm) were investigated. The orthogonal experimental method also termed as a
D Dh
0:1002
f ¼7:6551Re0:4927 ð1aÞ0:1441
FS Dh
FS Dh
1:3204
1:7383
d
0:0351
Dh
d Dh
0:0004
(17)
D Dh
0:0922 (18)
The comparisons between the predicted results from the correlations and the numerical data are shown in Fig. 29. For the predicted and numerical Nusselt numbers as shown in Fig. 29(a), the average and maximum relative errors are 2.62% and 9.67% respectively. For the predicted and numerical friction factors as shown in Fig. 29(b), the average and maximum relative errors are 4.95% and 14.33% respectively. It shows that the flow and heat transfer correlation are of sufficient accuracy, which can be applied to the design and operation of air-cooled heat exchanger in power plants. The application ranges of the correlations are listed as follows:
0a
p
0:5476
0:013 Fig. 18. Variations of convection heat transfer coefficient with air face velocity at various tube external diameters.
(19)
6 FS 0:678 Dh
d Dh
0:332
(20)
(21)
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Fig. 20. Temperature and pressure contours for different tube external diameters at air face velocity of 1.5 m/s. (a) temperature, (b) pressure.
Fig. 21. Effect of tube external diameter on PEC of finned tube bundles.
Fig. 22. Effect of natural convection on convection heat transfer coefficient of finned tube bundles.
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Fig. 23. Effect of natural convection on pressure drop of finned tube bundles.
Fig. 24. Effect of natural convection on PEC of finned tube bundles.
Fig. 25. GPC versus air face velocity at various fin oblique angles.
Fig. 26. GPC versus air face velocity at various fin spacings.
Fig. 27. GPC versus air face velocity at various fin thicknesses.
Fig. 28. GPC versus air face velocity at various tube external diameters.
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177
Table 5 Schemes for the orthogonal experimental method. Case
Parameters
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2:228
d
FS
0 (1) 0 (1) 0 (1) 0 (1) 0 (1) 10 (2) 10 (2) 10 (2) 10 (2) 10 (2) 15 (3) 15 (3) 15 (3) 15 (3) 15 (3) 20 (4) 20 (4) 20 (4) 20 (4) 20 (4) 30 (5) 30 (5) 30 (5) 30 (5) 30 (5)
D 9:143 Dh
220 Re 5500
2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6
mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5)
0.1 0.3 0.6 0.9 1.2 0.3 0.6 0.9 1.2 0.1 0.6 0.9 1.2 0.1 0.3 0.9 1.2 0.1 0.3 0.6 1.2 0.1 0.3 0.6 0.9
D mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (2) (3) (4) (5) (1) (3) (4) (5) (1) (2) (4) (5) (1) (2) (3) (5) (1) (2) (3) (4)
18 22 25 27 32 25 27 32 18 22 32 18 22 25 27 22 25 27 32 18 27 32 18 22 25
mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm
(1) (2) (3) (4) (5) (3) (4) (5) (1) (2) (5) (1) (2) (3) (4) (2) (3) (4) (5) (1) (4) (5) (1) (2) (3)
(22) (23)
4. Conclusions The impacts of the fin oblique angle, fin spacing, fin thickness and tube external diameter, as well as the natural convection on the flow and heat transfer characteristics of plate finned tube bundles are analyzed by numerical approach. The optimal geometric parameters are recommended and the flow and heat transfer correlations are developed. (1) The effect of fin oblique angle on the thermo-flow performances of plate finned tube bundles can be neglected. However, the increased fin oblique angle will lower the inlet air flow loss of air-cooled heat exchanger, which is of benefit to the operation of natural draft dry cooling system in power plants. The heat transfer coefficient and pressure drop both decrease with increasing the fin spacing. But when the fin spacing is bigger than 3 mm, the convection heat transfer coefficient varies little. (2) Increasing the fin thickness will result in an enhanced heat transfer, but lead to the overweight of heat exchanger and increased investment cost. Both the heat transfer coefficient and pressure drop increase as the tube diameter increases. (3) With a given heat exchanger volume, the optimal configurations of six-row plate finned tube bundles with the fin oblique angle of 30 , fin spacing of 10 mm, fin thickness of 0.3 mm and tube diameter of 18 mm are recommended within the scope of investigation for the Re ranging from 220 to 5500. (4) The correlations between the Nusselt number, friction factor and the Reynolds number and geometrical parameters are
Fig. 29. Comparisons between predicted and numerical results (x-coordinate refers to the numerical data, y-coordinate stands for the predicted results from the correlations). (a) Nusselt number, (b) friction factor.
recommended using the orthogonal experimental design method, which can be used to the optimal design of aircooled heat exchanger in power plants. Acknowledgments The financial supports for this research, from the National Natural Science Foundation of China (Grant No. 51476055) and the National Basic Research Program of China (Grant No. 2015CB251503) are gratefully acknowledged. References [1] Al-Waked R, Behnia M. The performance of natural draft dry cooling towers under crosswind: CFD study. Int J Energy Res 2004;28(2):147e61. [2] Zhai Z, Fu S. Improving cooling efficiency of dry-cooling towers under crosswind conditions by using wind-break methods. Appl Therm Eng 2006;26(10):1008e17. [3] Viladkar MN, Karisiddappa, Bhargava P, Godbole PN. Static soilestructure interaction response of hyperbolic cooling towers to symmetrical wind loads. Eng Struct 2006;28(9):1236e51. [4] Verne E, Antonione C, Battezzati L, du Preez AF, Kroger DG. The effect of the
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[5]
[6]
[7] [8]
[9] [10]
[11]
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