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Large eddy simulation of flow and heat transfer of the flat finned tube in direct air-cooled condensers
Applied Thermal Engineering 61 (2013) 75e85
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Large eddy simulation of flow and heat transfer of the flat finned tube in direct air-cooled condensers Juan Wen*, Dawei Tang, Zhicheng Wang, Jing Zhang, Yanjun Li Institute of Engineering Thermophysics, Chinese Academy of Sciences, No. 11, BeiSiHuanXi Road, Beijing 100190, China
h i g h l i g h t s LES method was used to the flat finned-tube in ACC. Transient flow fields in the wake region ware predicted more precisely by LES method. Wake interactions and the effects on heat transfer were considered. The effect of Re on vortex-shedding and mutual interaction was obtained.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2013 Accepted 18 July 2013 Available online 1 August 2013
The striking phenomenon during the flow past multiple wavy finned flat tubes is the generation of a complex flow structure as a consequence of the nonlinear interactions among the wakes behind the bodies. This study investigates the flow structures, heat transfer and vortex-shedding characteristics behind a single and two side-by-side flat finned-tubes at a gap ratio (spacing to diameter ratio) of 1.95. The three-dimensional numerical simulations are performed with the computational fluid dynamics code ANSYS FLUENT12.0 using large eddy simulation (LES) with the subgrid-scale model-Dynamic SmagorinskyeLilly model. The instantaneous properties of velocity, temperature, pressure and vorticity are obtained and analyzed. In addition, the time history of drag and lift coefficients, local Nusselt number and skin friction coefficients are determined and discussed at various Reynolds numbers. Special attention is paid to investigate the effect of the Reynolds number on the vortex-shedding characteristics and the mutual interaction of the wakes behind the two side-by-side finned-tubes. A discussion on the mechanism of vortex-shedding and wake interaction behind the flat finned-tube and their effects on the thermal histories are presented. 2013 Elsevier Ltd. All rights reserved.
Keywords: Large eddy simulation Flat finned-tube Vortex shedding Wake interaction Flow and heat transfer
1. Introduction An air-cooled steam condenser (ACC) is a cooling system using ambient air as the cooling medium. The mechanical draft ACCs are used extensively in direct cooled thermoelectric power plants for the areas rich in coal resources but poor in water due to economic and environmental considerations [1]. Air-cooled condenser is composed of an array of condenser cells. For each condenser cell, the flat finned-tube bundles are arranged in the form of an A-frame fitted with an axial flow fan below. Ambient cooling air is inhaled by the axial fans and then condenses the steam which flows in the flat finned-tubes [2,3]. In multi-finned tube arrays with many rows, the flow patterns are even more complicated. These wake
* Corresponding author. Tel.: þ86 10 82543092; fax: þ86 10 82543022. E-mail address: [email protected] (J. Wen). 1359-4311/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.07.020
interactions behind the finned tubes subsequently lead to the complex vortex shedding phenomena which will produce very complicated flow effects such as causing vibration [4]. Therefore, a thorough knowledge of the wake interaction and vortex shedding mechanism is required for better understanding heat transfer in the wakes which is essential for the development of air-cooled condensers. Flat tube heat exchangers are expected to have lower air-side pressure drop and better air-side heat transfer coefficients compared to circular tube heat exchangers, because of a smaller wake area. For the same reason, vibration and noise are expected to be less in flat tube heat exchangers [5]. The thermo-flow characteristics of flat finned-tube heat exchangers have been researched in recent years, but the literature on the unsteady flow over flat tubes are limited. Bahaidarah et al. [6] numerically investigated steady, two dimensional, laminar, incompressible flow over a bank of flat tubes for both staggered as
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Nomenclature L D
D l
d h s T U0 Nu Nux Re P u’ Cd Cl
flat tube length (m) flat tube diameter (m) flat tube thickness (m) fin length (m) fin thickness (m) fin height (m) fin spacing (m) temperature (K) free-stream velocity (m s1) Nusselt number local Nusselt number Reynolds number pressure (Pa) velocity fluctuations drag coefficient lift coefficient
well as inline configurations and determined an optimum configuration from the point of view of heat transfer. Benarji et al. [5] carried out the transient numerical simulations of fluid flow and heat transfer over a bank of flat tubes. Correlations are proposed for both pressure drop and Nusselt number and optimum configurations have been determined. Dong et al. [7] investigated the effects of fin pitch, height and length on the performance of heat transfer and pressure drop of the wavy fin and flat tube. Yang et al. [8] compared the thermo-flow characteristics of the oblique and perpendicular arrangement of the wavy finned flat tube. The influences of the fin pitch, height and the Reynolds number on the flow and heat transfer performances were investigated. Feng et al. [9] investigated the heat transfer enhancement of discontinuous short flat wave finned tube. The discontinuous short wave fin can enhances heat transfer obviously by breaking the development of boundary layer periodically. Flat finned-tube bundles are employed in heat exchangers, the design of which is still based on empirical correlations of heat transfer and pressure drop which have some limitations on the mechanism of fluid flow and heat transfer. Apart from heat transfer considerations, the design of flat finned-tubes should also account for wake interaction and vortex-shedding. The instability of the shear layers and, the large-scale vortex shedding as well as the different wake interaction mechanisms are factors that can lead to large-amplitude vibrations. Zdravkovich [10] observed that when more than one body is placed in the fluid flow, the resulting forces and vortex shedding pattern may be completely different from those found at the same Reynolds number. Rodi et al. [11] found that the LES approach was highly recommended because of its ability to resolve the unsteady large-scale turbulent motion. The numerical results can provide useful turbulent statistics of the cyclic formation of the vortices downstream. L. Chen [12] investigated the formation and the convection of vortices behind two side-byside cylinders at two different cylinder spacings by large eddy simulation. Multiple peaks of shedding frequencies were obtained due to the gap squeezing effect and amalgamation of the vortices generated behind the cylinders. Liang and Papadakis [13] used LES model to study the vortex shedding characteristics inside a staggered tube array with intermediate spacing at subcritical Reynolds. Two distinct and independent shedding frequencies are detected behind the first two rows, but the high-frequency component vanishes in the downstream rows. Yen and Liu [14] experimentally studied the characteristics of boundary-layer flow around one single and two side-by-side square cylinders using various
St f A yþ G t x, y, z us V Cp
Strouhal number vortex shedding frequency reference area (m2) wall units coordinate filter function (m1) time(s) Cartesian coordinates (m) friction velocity (m1) volume (m3) skin friction coefficient
Greek symbols thermal conductivity (W m1 k1) density (kg m3) stress tensor subgrid-scale stress molecular viscosity (kg m1 s1)
l r sij sij m
Reynolds numbers and gap ratios. The gap-flow mode exhibits the anti-phase vortex shedding and the couple vortex-shedding mode exhibits the in-phase vortex shedding. Chen et al. [15] experimentally investigated the flow characteristics, around a short uniform-diameter circular cylinder in crossflow. The spanwise variation of the mean velocity at any cross-stream location in the near-wake shows a typical boundary layer velocity profile. Hesam and Navid [16] numerically investigated the flow over two transverse cylinders in laminar and turbulent flows. Different Reynolds numbers and gap ratios have been used in order to investigate the effect of various parameters on flow characteristics and hydrodynamic forces acting on the bodies. For turbulent flow, LES method is becoming a very popular method. It is capable of simulating the complex turbulent flows behind the cylinders. The direct numerical simulation (DNS) method requires extreme grid resolution and computational expense that is not viable for realistic geometries and flow conditions. The Reynolds averaged NaviereStokes simulations (RANS) method attempts to model the turbulence by performing time or space averaging. The averaging process wipes out most of the important characteristics of a time dependent solution. Thus it limited the study of the mechanism of the turbulence on the subject. The LES technique is situated between the RANS and DNS in the sense that large scales eddies are calculated directly while the small scale eddies are modeled [17]. The major merit of LES lies in that it is good at identifying the large scale eddies encountered in local features, which are the most meaningful to the engineering problems, and tracing the time history of the motion [18]. The fact that in tube bundles momentum and heat transfer are, to a large extent, controlled by large scale vortices shed behind the tubes, makes this technique particularly attractive. From the above discussion, numerous attempts have been made for modeling the fluid flow and heat transfer over circular and square cylinders. However, the studies pertaining to the flat finnedtube bundles are actually very less, apart from some heat transfer and pressure drop considerations. Furthermore, to the best of the authors’ knowledge, there is a real scarcity in the literature and limited quantitative information for the different flow regimes evolving as a result of wake interactions and vortex shedding during flow over flat finned-tubes, in particular the Reynolds number effect on the flow patterns. In order to fill this gap, threedimensional LES calculations over flat finned-tube are reported in this paper, which predicts the flow around tubes more precisely. The flat tube induces various instantaneous unsteady effects such
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as separation, shear-layer instability, and vortex shedding when the flow moves around it. Vortices were generated and shed from alternate sides of the flat tubes. This phenomenon resulted in a shear-layer instability between the wake of the flat tubes. And transient-periodic behavior is induced. The flows are dominated by unsteady flow interactions because of flow separation behind the flat tubes. We are engaged, therefore, in a transient numerical simulation to understand fundamental mechanisms of unsteady interactions in air-cooled condensers and the effects on heat transfer. So it is important to know their transient fluid flow and heat transfer behavior for a better design of such heat exchangers. Accordingly, our aim of the present work is to numerically investigate the unsteady coupled fluid flow and heat transfer over single and two side-by-side flat finned- tubes by LES methods at different Reynolds numbers. Analysis of shedding process and flow behavior behind flat finned-tubes would provides a broad perspective toward understanding the flow characteristics, the mechanism of vortex shedding, wake interaction and their effects on the thermal histories of the finned tubes. 2. Computational models
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Table 1 Model dimensions of the flat finned-tube (unit in mm). Parameter
Value
Flat tube length (L) Flat tube diameter (D) Flat tube thickness ðDÞ Fin length (l) Fin thickness (d) Fin height (h) Fin spacing (s) Spacing between two tubes (H)
220 20 1.5 200 0.35 19 2.8 39
2.2. Governing equations The governing equations employed for LES are obtained by filtering the time-dependent NaviereStokes equations in either Fourier (wave-number) space or configuration (physical) space. The finite-volume discretization itself implicitly provides the filtering operation:
fðxÞ ¼
1 V
Z v fðx
0
Þdx0 ; x0 ˛v
(1)
where V is the volume of a computational cell. The filter function, G(x,x’) implied here is then
2.1. Physical model The aluminum fins are brazed in the outer surface of the flat tube, curved fins and the tube wall forming the air flow channel. The wavy finned flat tube used in direct air-cooled condensers is schematically shown in Fig. 1. Table 1 lists the model dimensions according to the actual sizes of the flat finned-tube bundles adopted by air-cooled condensers in a power plant. The mesh is generated by using GAMBIT software. Fig. 2 shows the computational domains and the corresponding boundary conditions for the two side-by-side flat finned-tubes. Two assistant computational zones, inlet and outlet, are setup to reduce the flow distortion at the entrance and exit of the flowing passage. The length of the two assistant computational zones is 50 mm and 150 mm respectively. Across the entrance of the computational domain, the velocity inlet boundary condition is specified. The fluid is assumed to enter with a uniform velocity U0 of 2 m/s, 3.5 m/s, and temperature T of 300 K respectively. The spectral synthesizer algorithm is selected to model the fluctuating velocity at velocity inlet boundaries. And the turbulent intensity is 10％.The pressure out boundary condition is appointed at the exit of the computational domain. For the interface of the neighboring tubes, the symmetric boundary conditions are appointed. Periodic boundary conditions are applied on the length direction of flat tube. The temperature of the tube wall maintains a constant 336 K which is equal to the saturated temperature of the exhaust steam. A no-slip boundary condition is prescribed at the surface of the tube wall.
Fig. 1. The flat finned-tube bundles.
Gðx; x0 Þ
¼
8 < 1=V; :
0;
x0 ˛v (2) x0 ;v
Filtering the NaviereStokes equations, one obtains
vr v rui ¼ 0 þ vt vxi
(3)
v v v vp vsij rui þ rui uj ¼ s vt vxj vxj ij vxi vxj
(4)
where sij is the stress tensor due to molecular viscosity defined by
"
sij h m
vui vxj
! þm
vuj vxi
!#
2 vu m i dij 3 vxi
(5)
and sij is the subgrid-scale stress defined by
sij hrui uj rui uj
(6)
In the present simulations, the dynamic SmagorinskyeLilly model is applied. The fluid mechanics are combined with the energy transport in the energy equation
Fig. 2. Schematic of the computational domains and the corresponding boundary conditions.
78
v v v rT þ rui T ¼ vt vxj vxj
J. Wen et al. / Applied Thermal Engineering 61 (2013) 75e85
l
vT vxj
!
vq vxi
(7)
where q represents sub-grid scale heat flux, which is modeled by the SGS model. 2.3. Computational methods In the present simulation, the finite-volume method applied on non-uniform grids is employed to calculate the 3D unsteady incompressible NaviereStokes equation with the computational fluid dynamics code ANSYS FLUENT12.0. Air is used as working fluid assuming constant properties. Because the physical parameters changes little when the inlet air was heated by the wall of flat finned tube. The steady flow data simulated with the standard ke3 model are used as the initial conditions for the unsteady LES. The number of grid points and their distribution is an important issue in such transient flows over finned tubes because of the associated complexity in the flow as well as the separation and vortex shedding. Fig. 3 shows the grid distribution in the computational domain. It is discretized accordingly by a non-uniform hexahedral grid with a finer grid distribution near the tube, with the help of a successive ratio scheme using the gridding software GAMBIT to capture the viscous boundary layer as well as the wake and the vortex street behind the tubes. The first point condition near the wall, in the wall unit of yþ(yþ ¼ rusyp/m), ensured that it was less than1. The accuracy of the computational results using LES is highly dependent on the mesh size and cell numbers. To show the independency of results from the grid size, the verification of mesh independency is done for different grid sizes in Fig. 4. According to this verification, for grids finer than 2,700,000 cells, no significant change is observed and there is just a 0.18% difference between this grid structure and 2,400,000 grid. So the solution is sufficiently grid independent. To ensure better results and capture detailed vortex structures, the finer grid number of 2,700,000 and 5,400,000 are adopted for the all simulations of the single and two side-by-side flat finned-tubes respectively. A bounded central-differencing scheme is used for momentum and energy discretization while a second-order implicit scheme is employed to advance the equations in time. The well-known pressure implicit method with splitting of operators (PISO) algorithm is used to deal with the pressureevelocity coupling between the momentum and the continuity equations. A convergence criterion of 1 103 was applied to the residuals of the continuity and the momentum equations and 1 106 to the residuals of the energy equation. In the numerical solution a time step size of 104 is used satisfying the above restrictions.
Fig. 4. Grid independency test of the numerical solutions for grid system.
Using more advanced modeling approach such as the LES model, which is often good at predicting local flow features around a body. There are scarcity experimental results reported in the published literature for the flow features in the wake region of a flat finnedtube. In order to validate the LES model, the flow at a subcritical Reynolds number ReD ¼ 3900 (based on cylinder diameter0.01 m and free-stream velocity) was chosen mainly because of the existence of the particle image velicometry (PIV) experimental data of Lourenco and Shih [19] and the hot-wire measurement results at the same Reynolds number by Ong and Wallace [20] which provided the mean flow data at several locations in the near wake of the cylinder. LES are carried out with a high-order accurate numerical method based on B-splines by Kravchenko and Moin Ref. [21]. The mean velocities and turbulent Reynolds stresses from the current numerical results are compared with the results of the experimental data and other numerical results of circular cylinder [19e22]. Figs. 5 and 6 show that the profiles of mean streamwise velocity along the centerline and Reynolds stresses in the wake of a circular cylinder obtained in the large eddy simulation are in well agreement with the experimental data. Therefore, we will use the LES model which represent the flow features around a body more precisely.
2.4. Validation of the numerical model The air flow around the flat finned-tube of ACC includes the flow around cylinder and the forced movement between the plates.
Fig. 3. The grid of the computational domains (a closer view around the flat tube).
Fig. 5. Mean streamwise velocity along the centerline.
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Fig. 6. Streamwise velocity fluctuations at three locations in the wake of a circular cylinder.
3. Results and discussion 3.1. Non-dimensional parameters The drag and lift are normalized by rU02 =2 to get the drag and lift coefficients Cd and Cl as:
Cd ¼ Fd = rAU02 =2 ; Cl ¼ Fl = rAU02 =2 where Fd and Fl are the total drag and lift forces respectively, U0 is the free-stream velocity, A is the reference area
The Strouhal number is defined by St ¼ fD=U0 where D is the tube diameter, f is the vortex shedding frequency.
Re ¼ rU0 D=m Thermal properties of air such as m and r is the molecular viscosity and density respectively. The local Nusselt number based on the flat finned-tube is calculated using the following equation:
Nu ¼ h$D=l where h is the convective heat transfer coefficient, l is the thermal conductivity of the fluid 3.2. The properties of flow field Fig. 7 shows the flow characteristics for single flat finned-tube by LES model and ke3 model [23] at the time tU$D1 ¼ 100. In Fig. 7(a), we can see that the velocity distribution values on the surface of fin channel is higher than other zones due to the area of the fin channel suddenly reduced. When the cooling air flow through the wake region of the finned-tube, the phenomenon of flow separation behind the tube is appeared. We can see that a great deal of vortex are identified, and the vortices emerged and shed in the wake region. Fig. 7(b) illustrates that the air isotherms in the fin channel have a forward protruding trend because of the velocity distribution values on the surface of fin channel is high. The temperature in the wake region of the finned-tube is increased due to the phenomenon of hot air circumfluence. Thus, there will be an attendant decrease in convection heat transfer in the wake region of the finned-tube. It is to be mentioned here that since both the
Fig. 7. Flow characteristics for single flat finned-tube calculated by LES model and ke3 model at Re ¼ 4006 and tU$D1 ¼ 100 (a) velocity field (b) temperature field (c) pressure field (d) streamlines (e) three-dimensional x-vorticity.
velocity and thermal energy are transported by the flow itself, the velocity contours as well as the isotherms exhibit similar features. From Fig. 7(c) we can find that the static pressure is reduced along
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the streamwise location. In the wake region, a negative pressure is showed. From the streamlines in Fig. 7(d), the essential features of
Fig. 9. Flow characteristics for two side-by-side flat finned-tubes at Re ¼ 7012 tU$D1 ¼ 100 and H/D ¼ 1.95 (a) velocity field (b) temperature field (c) pressure field (d) streamlines.
Fig. 8. Flow characteristics for two side-by-side flat finned-tubes at Re ¼ 4006 tU$D1 ¼ 100 and H/D ¼ 1.95 (a) velocity field (b) temperature field (c) pressure field (d) streamlines (e) three-dimensional x-vorticity.
the vortex shedding mechanism are the similar as those in the circular cylinder, except that the separation points of the boundary layers on the flat tube surface are different. From Fig. 7(e) we can take one pair of three-dimensional x-vorticity distribution, and the value of vorticity is 30 .The vortices are generated in pairs and the direction of rotation is opposite. In the wake region of the finned-tube the vortex and fluid mixed, momentum and energy exchanged. The comparison of these results with ke3 are presented to reveal that the LES approach can capture the complex turbulent
J. Wen et al. / Applied Thermal Engineering 61 (2013) 75e85
motions in detail, which leads to accuracies in the flow field prediction of the wake region behind the flat finned-tubes. Furthermore, the current works consider the wake interaction and the mutual interference of the adjacent flat finned-tubes, and their effects on the fluid flow and heat transfer. Fig. 8 illustrates the flow characteristics for two side-by-side flat finned-tubes at Re ¼ 4006 tU$D1 ¼ 100 and H/D ¼ 1.95. Fig. 8(a), (b) shows that the wakes behind the tubes interact in a complicated manner resulting in a variety of thermo-hydrodynamic regimes. It is worth mentioning that the results confirmed the dominant existence of anti-phase synchronized pattern in the flow. Two Karman vortex streets are observed in this figure. One can see that there is some evidence of merging of vortices at far downstream locations (close to the exit of the domain). The vortex shedding from one flat tube seems to have a definite phase relationship with the shedding from other tubes. The energy is transported in a similar fashion by the fluid flow to the downstream region. The vortices remain distinct throughout the computational domain without significant lateral spread. This can be attributed to weaker interactive flow pattern arising out of relatively larger spacing between the flat tubes. Fig. 8(c) represents that a high-pressure region developed in front of the tubes. The pressure in this region drops as the flow passes the gap. Fig. 8(d), the streamlines in the wakes of both flat finned-tubes follow an anti-phase asymmetric pattern induced from the interference between the two flat finned-tubes. Also, it can be seen that the separation points of flow are symmetric with respect to the centerline of gap spacing. And the separation points of the boundary layers on the flat tube surface are different from the single. When more than one flat finned tube is placed in the fluid flow, the resulting forces and vortex shedding pattern is different from those found at the same Reynolds number due to the wake interaction of the adjacent tubes. The difference between the flat tube bundles in Fig. 8 and single flat tube in Fig. 7 is the nonlinear wake interactions of the adjacent tubes and their effects on the flow pattern and heat transfer characteristics. The velocity, temperature, pressure and stream-function fields for Re ¼ 7012 are shown in Fig. 9.The interference effects and flow pattern around two side-by-side flat tubes are sensitive to the Reynolds number at this spacing. Comparing the flow behaviors for the counterpart gaps at Re ¼ 4006, we can clearly see the effect of the Reynolds number on flow characteristics. Fig. 9(a), the more complicated and unsymmetrical velocity field pertains to the fact that the flat tube at this Reynolds number has a shorter near-wake
Fig. 10. Mean streamwise velocity component u/U0 along the centerline axis of the flat tube1.
81
Fig. 11. Cross-stream velocity in the near wake region of a flat tube.
region than as described in the work for Re ¼ 4006 and same spacing. Due to the increase of the Reynolds number, at this gap pressure is distributed differently in comparison to the same gap at Re ¼ 4006, as shown in Fig. 9(c). The pressure field for Re ¼ 7012 suggests that both cylinders experience different base pressures at this time. Fig. 9(d) shows that the flow pattern becomes more unsymmetrical than before because interference effects increase as the Reynolds number increases to 7012. Variations of the time-averaged streamwise velocity component u/U0 along the centerline axis of the cases for Re ¼ 4006 and 7012 are shown in Fig. 10. The lengths of recirculation zones in the wake region of the flat tube1 for single tube is larger than in the two sideby-side flat finned-tubes, due to the wake interaction of the adjacent tubes. The size and the form of the recirculation region is directly related to the length of the vortex formation region and the dynamics of the downstream flow. As the Reynolds number is increased to 7012, these vortices are developed in a shorter distance and are placed close to the flat tube. The lengths of recirculation zones in the wake region at Re ¼ 7012 is shorter than in the wake region at Re ¼ 4006.
Fig. 12. Mean streamwise velocity at different locations in the wake of the two sideby-side flat finned-tubes.
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Fig. 12 shows the mean streamwise velocity at different locations in the wake of the two side-by-side flat finned-tubes at Re ¼ 7012. The V-shape profile develops farther downstream. The cross-stream location of the maximum velocity deficit is always located at y/D ¼ 0.5 and y/D ¼ 3.5. The velocity deficit decays with increasing x/D. The shape of the mean velocity profile is directly related to the level of velocity fluctuations and, consequently, to the transition in the shear layers. The spanwise variation of streamwise velocity fluctuations at x/ D ¼ 6 is shown in Fig. 13. It has an ‘M’ shape, with a local minimum (at y/D ¼ 0.5) and two local maxima (at y/D ¼ 0.85 and y/ D ¼ 0.15). 3.3. The properties of drag and lift of flat finned-tubes
Fig. 13. Streamwise velocity fluctuations in the near wake region of a flat tube.
Flow patterns on the endwall surface show that the flat tube has a large reverse flow zone. In Fig. 11, the velocity distributions are obtained across the crosseestream plane (2 < y/D < 1 and 0.07 z/D 0.07) at the downstream location x/D ¼ 6. The spanwise variation of the mean velocity at any cross-stream location in the near-wake shows a boundary layer velocity profile. The streamwise mean velocity showed a V-shape profile further downstream at x/D ¼ 6. At all spanwise locations, the maximum velocity deficit is found at the line of symmetry (y/D ¼ 0.5). The velocity deficit is largest at the spanwise location closest to the endwall. And the velocity (z/D ¼ 0) at the spanwise location far away from the endwall is smallest, because the fin is fixed at the plane of z/D ¼ 0 and therefore the velocity behind the fin is small. With the increasing of the distance from the plane of fin (z/D ¼ 0), the velocity at the spanwise location far away from the endwall is increased. The wake width stays almost constant between-0.07 z/ D 0.07.
Fig. 14(a)e(d) shows the time series of lift and drag coefficients of the two adjoining flat tubes for H/D ¼ 1.95 at Re ¼ 4006 and 7012. It can be observed from the figures that both lift and drag coefficients are sinusoidal in nature. The amplitude of CL remains constant with a zero mean, whereas the amplitude of CD is observed to oscillate. The drag coefficients vary irregularly with time. The frequency of oscillation of the instantaneous CL is half of that of the instantaneous CD. From Fig. 14(a), (c), we can also observe that lift coefficients at this gap are in anti-phase synchronization with respect to each other which confirms the synchronized behavior of vortex shedding for this gap spacing, whereas the drag coefficients in Fig. 14(b), (d) are approximately in-phase. We can observe that increasing the Reynolds number has some effect on the flow behavior. The time response of lift coefficients of flat finned-tube at the Re ¼ 7012 shows more chaotic behavior than those at Re ¼ 4006. And the mean drag coefficients of flat finnedtube at Re ¼ 7012 are lower than the ones obtained for Re ¼ 4006. The flow generates a vortex street in the wake region. The periodic shedding of the vortices from the surface of the body induces periodic pressure variation on the body-structure. In the transverse direction, the excitation force has a dominant frequency called
Fig. 14. The time history of lift and drag coefficients for flat finned tube1and tube2 at Re ¼ 4006 and 7012.
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Fig. 15. Amplitude spectrum of drag and lift coefficient. (a) drag (b) lift.
Table 2 Different flow parameters for flat finned-tubes in present work. Parameter
CD
St
Single
42.08 42.08 42.08 27.45 27.45
0.286 0.291 0.291 0.295 0.295
Re ¼ 4006 Re ¼ 7012
Tube1 Tube2 Tube1 Tube2
0.05 0.04 0.01 0.02
Karman vortex shedding frequency. In the drag direction the dominant frequency is at twice the Kaman frequency. The shedding frequencies are determined by selecting a long time trace and performing a Fourier transform operation on it. Fig. 15 shows the amplitude spectrum of drag and lift coefficient, respectively. The signal of the drag coefficient shows multiple frequencies, while that of the lift coefficient shows the presence of the primary vortex shedding frequency. Oppositely oriented vortices traveling above and below the midplane of the wake will result in a bimodal distribution of the streamwise velocity variation (oscillation) and the drag coefficient. Since the variation of the lift coefficient and that of
the transverse velocity are synonymous, the fluctuations display single peak. The power spectrum analysis for the lift coefficient signal reveals that the vortex shedding mechanism is dominated by the primary frequency (Strouhal number) at this separation ratio (H/D ¼ 1.9) and Reynolds number (Re ¼ 4006) combination. Furthermore, it is evident from the Fig. 15 that the vortex shedding frequency increases with an increase of the Reynolds number at this given gap spacing. The flow parameters of each flat finned-tubes at different Reynolds number is presented in Table 2. It is worth noting that at a given gap, the Strouhal number increases with the increase of the Reynolds number, and both flat tube has the same Strouhal number. The Strouhal number of the single flat tube is smaller than those of the two side-by-side at the same Reynolds number due to the interaction between the adjacent flat finned-tubes. We can observe from Table 2 that the mean drag coefficient at Re ¼ 7012 is smaller than those at the Re ¼ 4006. Fig. 16(a)e(d) illustrates the time series of drag and lift coefficients of the fins (for example fin1 and fin2). Fig. 16(a)(b) indicates that both lift and drag coefficients of the fins at Re ¼ 4006 are again sinusoidal in nature. The drag coefficients at this Reynolds
Fig. 16. The time history of drag and lift coefficients for different fins at Re ¼ 4006 and 7012.
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Fig. 17. Amplitude spectrum of lift coefficient for fin at Re ¼ 4006 and 7012.
number are in anti-phase synchronization, whereas the lift coefficients in are approximately in-phase. The lift coefficient of the fin surface on both sides of the finned-tube has the characteristic of mutually exclusive. It is evident from Fig. 16(c), (d) that the time response of lift and drag coefficients at Re ¼ 7012 show more chaotic behavior than those at Re ¼ 4006. Furthermore, Fig. 16(a), (c) reveals that the lift coefficients of fin2 for two side-by-side flat tubes are more irregular, and the lift coefficients become more chaotic with an increase Reynolds number. From Fig. 17 we can seen the primary frequency in the amplitude spectrum. Besides, the diffused peaks correspond to the combined frequencies which are culminations of the nonlinear interactions among the shed vortices of different finned tubes can be represented. This combined frequency is termed as the secondary or flat finned-tubes interaction frequency.
3.4. The properties of heat transfer The time responses of Nusselt number of fins (for two fins, e.g. fin1 and fin2) for the two side-by-side flat finned-tubes at different Reynolds number are shown in Fig. 18(a), (b). The Nusselt number of the fins is found to oscillate periodically in time due to vortex shedding and a sinusoidal variation is observed for the Reynolds number of 4006. However, the variation becomes chaotic for larger Reynolds number7012. Fig. 19 shows the variations of local Nusselt number Nux for the fins along the fin length direction under different Reynolds number. It is to be noted that the local Nusselt number decreases along the
Fig. 19. The local Nusselt number for the different fins at Re ¼ 4006 and 7012.
flow direction because the thermal boundary layer development increases the thermal resistance. As seen from Fig. 19 the fin1 and fin2 of the single flat finned-tube have the same distribution of local Nusselt number since they experience the same undisturbed fluid, whereas for the two side by-side flat finned-tubes at Re ¼ 4006 and 7012 the Nusselt number for the fin2 displays higher value than the fin1. Furthermore, the local Nusselt number of the same fin in single flat finned-tube is lower than in the two side-byside finned tubes at same Reynolds number. As a result of interactions in the wakes behind the flat tubes and the mutual interference of the fins, the local Nusselt number distribution on the fin1 and fin2 for two side-by-side flat finned-tubes at H/ D ¼ 1.95 are different. These interactions give rise to the finned tube interaction frequencies that are responsible for the variations of heat transfer for the fined tube. And there is perceptible variation in the local Nusselt number for different fins. Furthermore, the difference of the local Nusselt number between the two fins are larger with the increased Reynolds number. Fig. 20 illustrates the local skin friction coefficient distribution on the fin surfaces. It is clearly seen that the skin friction coefficient decreases along the flow direction, and increases with an increased Reynolds number. It is also worth noting that the skin friction coefficients of the fin1 and fin2 are obvious different at Re ¼ 7012,which is probably attributed to wake interactions behind the flat finnedtubes. This finding is consistent with the mechanism of the vortex shedding and nonlinear interactions among the wakes behind the flat tubes.
Fig. 18. The time history of Nusselt number for different fins at Re ¼ 4006 and 7012.
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design of ACC should also account for wake interaction and vortex-shedding. Acknowledgements This work was supported by the National Basic Research Program of China (973 Program) (Grant No. 2009CB219804) and the Supercomputing center of Chinese Academy of Sciences. References
Fig. 20. The local skin friction coefficient for the different fins at Re ¼ 4006 and 7012.
4. Conclusions The study of the nonlinear interactions among the wakes behind multiple flat finned-tubes and their effects on the fluid flow and heat transfer in direct air-cooled condensers (ACC) requires accurate simulations. More advanced modeling techniques, such as LES calculations on heat transfer and wake interaction behind flat finned-tubes have been performed at different Reynolds numbers. The following conclusions have been reached. (1) Using LES model for the direct air-cooled condensers, which captures the complex turbulent motions in detail, resolves most of the turbulent eddies, and the transient flow fields in the wake region of the flat finned-tubes are predicted more precisely than RANS approach. (2) When more than one flat finned-tube is placed in the fluid flow and the wake interaction is considered, it will lead to a notable deviation from the results obtained for a single in terms of wake pattern, heat transfer and vortex shedding from both tubes at the same Reynolds number. The local Nusselt number of the same fin in single flat finned-tube is lower than in the two side-by-side finned tubes at same Reynolds number and the local Nusselt number and skin friction coefficient distribution on different fins for two side-by-side flat finned-tubes are different, due to the interactions and the mutual interference of the flat finned-tubes. The interaction frequencies are responsible for the variations of heat transfer for the finned tubes. (3) At the spacing H/D ¼ 1.95 adopted by ACC in a power plant, the degree of nonlinear wake interaction depends on the Reynolds number, which is found to have substantial effect on the flow behavior and thermal histories especially at Re ¼ 7012. (4) The 3D LES calculations of transient fluid flow and heat transfer behavior, wake interactions and the effects on heat transfer will provide a foundation for better design of such heat exchangers. Apart from heat transfer and pressure drop considerations, the
[1] D.G. Kröger, Air-cooled Heat Exchangers and Cooling Towers, Penwell Corp., Tulsa, USA, 2004. [2] J.R. Bredell, D.G. Kröger, G.D. Thiart, Numerical investigation of fan performance in a forced draft air-cooled steam condenser, Applied Thermal Engineering 26 (2006) 846e852. [3] P.Q. Liu, H.S. Duan, W.l. Zhao, Numerical investigation of hot air recirculation of air-cooled condensers at a large power plant, Applied Thermal Engineering 29 (2009) 1927e1934. [4] D. Chatterjee, G. Biswas, S. Amiroudine, Numerical investigation of forced convection heat transfer in unsteady flow past a row of square cylinders, International Journal of Heat and Fluid Flow 30 (2009) 1114e1128. [5] N. Benarji, C. Balaji, S.P. Venkateshan, Unsteady fluid flow and heat transfer over a bank of flat tubes, Heat Mass Transfer 44 (2008) 445e461. [6] H.M.S. Bahaidarah, N.K. Anand, H.C. Chen, A numerical study of fluid flow and heat transfer over a bank of flat tubes, Numerical Heat Transfer, Part A 48 (2005) 359e385. [7] J.Q. Dong, J.P. Chen, Z.J. Chen, Y.M. Zhou, W.F. Zhang, Heat transfer and pressure drop correlations for the wavy fin and flat tube heat exchangers, Applied Thermal Engineering 27 (2007) 2066e2073. [8] L.J. Yang, H. Tan, X.Z. Du, Y.P. Yang, Thermal-flow characteristics of the new wave-finned flat tube bundles in air-cooled condensers, International Journal of Thermal Sciences 53 (2012) 166e174. [9] L.L. Feng, X.Z. Du, Y.P. Yang, L.J. Yang, Study on heat transfer enhancement of discontinuous short wave finned flat tube, Science China Technological Sciences 54 (2011) 3281e3288. [10] M.M. Zdravkovich, The effects of interference between two circular cylinders in cross flow, Journal of Fluids and Structures 1 (1987) 239e261. [11] W. Rodi, J.H. Ferziger, M. Breuer, M. Pourquie, Status of large eddy simulation: results of a workshop, ASME Journal of Fluids Engineering 119 (1997) 248e262. [12] L. Chen, J.Y. Tu, G.H. Yeoh, Numerical simulation of turbulent wake flows behind two side-by-side cylinders, Journal of Fluids and Structures 18 (2003) 387e403. [13] C. Liang, G. Papadakis, Large eddy simulation of cross-flow through a staggered tube bundle at subcritical Reynolds number, Journal of Fluids and Structures 23 (2007) 1215e1230. [14] S.C. Yen, J.H. Liu, Wake flow behind two side-by-side square cylinders, International Journal of Heat and Fluid Flow 32 (2011) 41e51. [15] S.B. Chen, S. Sanitjai, K. Ghosh, R.J. Goldstein, Three-dimensional vortex flow near the endwall of a short cylinder in crossflow: uniform-diameter circular cylinder, Applied Thermal Engineering 49 (2012) 73e78. [16] S.M. Hesam, N. Navid, Numerical simulation of flow over two side-by-side circular cylinders, Journal of Hydrodynamics 23 (6) (2011) 792e805. [17] Z.S. Zhang, G.X. Cui, C.X. Xu, The Theory and Application of Large Eddy Simulation, Tsing hua University Press, 2008.01. [18] Y.J. Wei, M.S. Zhu, Z.Y. He, Large eddy simulation and spectrum analysis of flow around two square cylinders arranged side by side, Applied Mathematics and Mechanics 25 (2004) 903e910. [19] L.M. Lourenco, C. Shihi, Characteristics of the Plan Turbulent Near Wake of a Circular Cylinder. A Particle Image Velocimetry Study[R]. Private Communication, 1993 (Data from Kravchenko (2000)). [20] L. Ong, J. Wallace, The velocity field of the turbulent very near wake of a circular cylinder, Experiments in Fluids 20 (1996) 441e453. [21] A.G. Kravchenko, P. Moin, Numerical studies of flow over a circular cylinder at ReD ¼ 3900, Physics of Fluids 12 (2) (2000) 403e417. [22] X.H. Jia, H. Liu, Large eddy simulation of flow around two circular cylinders, Chinese Journal of Hydrodynamics 23 (2008) 625e632. [23] J. Zhang, Study on Flow and Heat Transfer Characteristics of Spray Cooling System Used in Direct Air-cooled Unit, 2013. Doctor Dissertation of University of Chinese Academy of Sciences.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Large eddy simulation of flow and heat transfer of the flat finned tube in direct air-cooled condensers Juan Wen*, Dawei Tang, Zhicheng Wang, Jing Zhang, Yanjun Li Institute of Engineering Thermophysics, Chinese Academy of Sciences, No. 11, BeiSiHuanXi Road, Beijing 100190, China
h i g h l i g h t s LES method was used to the flat finned-tube in ACC. Transient flow fields in the wake region ware predicted more precisely by LES method. Wake interactions and the effects on heat transfer were considered. The effect of Re on vortex-shedding and mutual interaction was obtained.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2013 Accepted 18 July 2013 Available online 1 August 2013
The striking phenomenon during the flow past multiple wavy finned flat tubes is the generation of a complex flow structure as a consequence of the nonlinear interactions among the wakes behind the bodies. This study investigates the flow structures, heat transfer and vortex-shedding characteristics behind a single and two side-by-side flat finned-tubes at a gap ratio (spacing to diameter ratio) of 1.95. The three-dimensional numerical simulations are performed with the computational fluid dynamics code ANSYS FLUENT12.0 using large eddy simulation (LES) with the subgrid-scale model-Dynamic SmagorinskyeLilly model. The instantaneous properties of velocity, temperature, pressure and vorticity are obtained and analyzed. In addition, the time history of drag and lift coefficients, local Nusselt number and skin friction coefficients are determined and discussed at various Reynolds numbers. Special attention is paid to investigate the effect of the Reynolds number on the vortex-shedding characteristics and the mutual interaction of the wakes behind the two side-by-side finned-tubes. A discussion on the mechanism of vortex-shedding and wake interaction behind the flat finned-tube and their effects on the thermal histories are presented. 2013 Elsevier Ltd. All rights reserved.
Keywords: Large eddy simulation Flat finned-tube Vortex shedding Wake interaction Flow and heat transfer
1. Introduction An air-cooled steam condenser (ACC) is a cooling system using ambient air as the cooling medium. The mechanical draft ACCs are used extensively in direct cooled thermoelectric power plants for the areas rich in coal resources but poor in water due to economic and environmental considerations [1]. Air-cooled condenser is composed of an array of condenser cells. For each condenser cell, the flat finned-tube bundles are arranged in the form of an A-frame fitted with an axial flow fan below. Ambient cooling air is inhaled by the axial fans and then condenses the steam which flows in the flat finned-tubes [2,3]. In multi-finned tube arrays with many rows, the flow patterns are even more complicated. These wake
* Corresponding author. Tel.: þ86 10 82543092; fax: þ86 10 82543022. E-mail address: [email protected] (J. Wen). 1359-4311/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.07.020
interactions behind the finned tubes subsequently lead to the complex vortex shedding phenomena which will produce very complicated flow effects such as causing vibration [4]. Therefore, a thorough knowledge of the wake interaction and vortex shedding mechanism is required for better understanding heat transfer in the wakes which is essential for the development of air-cooled condensers. Flat tube heat exchangers are expected to have lower air-side pressure drop and better air-side heat transfer coefficients compared to circular tube heat exchangers, because of a smaller wake area. For the same reason, vibration and noise are expected to be less in flat tube heat exchangers [5]. The thermo-flow characteristics of flat finned-tube heat exchangers have been researched in recent years, but the literature on the unsteady flow over flat tubes are limited. Bahaidarah et al. [6] numerically investigated steady, two dimensional, laminar, incompressible flow over a bank of flat tubes for both staggered as
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Nomenclature L D
D l
d h s T U0 Nu Nux Re P u’ Cd Cl
flat tube length (m) flat tube diameter (m) flat tube thickness (m) fin length (m) fin thickness (m) fin height (m) fin spacing (m) temperature (K) free-stream velocity (m s1) Nusselt number local Nusselt number Reynolds number pressure (Pa) velocity fluctuations drag coefficient lift coefficient
well as inline configurations and determined an optimum configuration from the point of view of heat transfer. Benarji et al. [5] carried out the transient numerical simulations of fluid flow and heat transfer over a bank of flat tubes. Correlations are proposed for both pressure drop and Nusselt number and optimum configurations have been determined. Dong et al. [7] investigated the effects of fin pitch, height and length on the performance of heat transfer and pressure drop of the wavy fin and flat tube. Yang et al. [8] compared the thermo-flow characteristics of the oblique and perpendicular arrangement of the wavy finned flat tube. The influences of the fin pitch, height and the Reynolds number on the flow and heat transfer performances were investigated. Feng et al. [9] investigated the heat transfer enhancement of discontinuous short flat wave finned tube. The discontinuous short wave fin can enhances heat transfer obviously by breaking the development of boundary layer periodically. Flat finned-tube bundles are employed in heat exchangers, the design of which is still based on empirical correlations of heat transfer and pressure drop which have some limitations on the mechanism of fluid flow and heat transfer. Apart from heat transfer considerations, the design of flat finned-tubes should also account for wake interaction and vortex-shedding. The instability of the shear layers and, the large-scale vortex shedding as well as the different wake interaction mechanisms are factors that can lead to large-amplitude vibrations. Zdravkovich [10] observed that when more than one body is placed in the fluid flow, the resulting forces and vortex shedding pattern may be completely different from those found at the same Reynolds number. Rodi et al. [11] found that the LES approach was highly recommended because of its ability to resolve the unsteady large-scale turbulent motion. The numerical results can provide useful turbulent statistics of the cyclic formation of the vortices downstream. L. Chen [12] investigated the formation and the convection of vortices behind two side-byside cylinders at two different cylinder spacings by large eddy simulation. Multiple peaks of shedding frequencies were obtained due to the gap squeezing effect and amalgamation of the vortices generated behind the cylinders. Liang and Papadakis [13] used LES model to study the vortex shedding characteristics inside a staggered tube array with intermediate spacing at subcritical Reynolds. Two distinct and independent shedding frequencies are detected behind the first two rows, but the high-frequency component vanishes in the downstream rows. Yen and Liu [14] experimentally studied the characteristics of boundary-layer flow around one single and two side-by-side square cylinders using various
St f A yþ G t x, y, z us V Cp
Strouhal number vortex shedding frequency reference area (m2) wall units coordinate filter function (m1) time(s) Cartesian coordinates (m) friction velocity (m1) volume (m3) skin friction coefficient
Greek symbols thermal conductivity (W m1 k1) density (kg m3) stress tensor subgrid-scale stress molecular viscosity (kg m1 s1)
l r sij sij m
Reynolds numbers and gap ratios. The gap-flow mode exhibits the anti-phase vortex shedding and the couple vortex-shedding mode exhibits the in-phase vortex shedding. Chen et al. [15] experimentally investigated the flow characteristics, around a short uniform-diameter circular cylinder in crossflow. The spanwise variation of the mean velocity at any cross-stream location in the near-wake shows a typical boundary layer velocity profile. Hesam and Navid [16] numerically investigated the flow over two transverse cylinders in laminar and turbulent flows. Different Reynolds numbers and gap ratios have been used in order to investigate the effect of various parameters on flow characteristics and hydrodynamic forces acting on the bodies. For turbulent flow, LES method is becoming a very popular method. It is capable of simulating the complex turbulent flows behind the cylinders. The direct numerical simulation (DNS) method requires extreme grid resolution and computational expense that is not viable for realistic geometries and flow conditions. The Reynolds averaged NaviereStokes simulations (RANS) method attempts to model the turbulence by performing time or space averaging. The averaging process wipes out most of the important characteristics of a time dependent solution. Thus it limited the study of the mechanism of the turbulence on the subject. The LES technique is situated between the RANS and DNS in the sense that large scales eddies are calculated directly while the small scale eddies are modeled [17]. The major merit of LES lies in that it is good at identifying the large scale eddies encountered in local features, which are the most meaningful to the engineering problems, and tracing the time history of the motion [18]. The fact that in tube bundles momentum and heat transfer are, to a large extent, controlled by large scale vortices shed behind the tubes, makes this technique particularly attractive. From the above discussion, numerous attempts have been made for modeling the fluid flow and heat transfer over circular and square cylinders. However, the studies pertaining to the flat finnedtube bundles are actually very less, apart from some heat transfer and pressure drop considerations. Furthermore, to the best of the authors’ knowledge, there is a real scarcity in the literature and limited quantitative information for the different flow regimes evolving as a result of wake interactions and vortex shedding during flow over flat finned-tubes, in particular the Reynolds number effect on the flow patterns. In order to fill this gap, threedimensional LES calculations over flat finned-tube are reported in this paper, which predicts the flow around tubes more precisely. The flat tube induces various instantaneous unsteady effects such
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as separation, shear-layer instability, and vortex shedding when the flow moves around it. Vortices were generated and shed from alternate sides of the flat tubes. This phenomenon resulted in a shear-layer instability between the wake of the flat tubes. And transient-periodic behavior is induced. The flows are dominated by unsteady flow interactions because of flow separation behind the flat tubes. We are engaged, therefore, in a transient numerical simulation to understand fundamental mechanisms of unsteady interactions in air-cooled condensers and the effects on heat transfer. So it is important to know their transient fluid flow and heat transfer behavior for a better design of such heat exchangers. Accordingly, our aim of the present work is to numerically investigate the unsteady coupled fluid flow and heat transfer over single and two side-by-side flat finned- tubes by LES methods at different Reynolds numbers. Analysis of shedding process and flow behavior behind flat finned-tubes would provides a broad perspective toward understanding the flow characteristics, the mechanism of vortex shedding, wake interaction and their effects on the thermal histories of the finned tubes. 2. Computational models
77
Table 1 Model dimensions of the flat finned-tube (unit in mm). Parameter
Value
Flat tube length (L) Flat tube diameter (D) Flat tube thickness ðDÞ Fin length (l) Fin thickness (d) Fin height (h) Fin spacing (s) Spacing between two tubes (H)
220 20 1.5 200 0.35 19 2.8 39
2.2. Governing equations The governing equations employed for LES are obtained by filtering the time-dependent NaviereStokes equations in either Fourier (wave-number) space or configuration (physical) space. The finite-volume discretization itself implicitly provides the filtering operation:
fðxÞ ¼
1 V
Z v fðx
0
Þdx0 ; x0 ˛v
(1)
where V is the volume of a computational cell. The filter function, G(x,x’) implied here is then
2.1. Physical model The aluminum fins are brazed in the outer surface of the flat tube, curved fins and the tube wall forming the air flow channel. The wavy finned flat tube used in direct air-cooled condensers is schematically shown in Fig. 1. Table 1 lists the model dimensions according to the actual sizes of the flat finned-tube bundles adopted by air-cooled condensers in a power plant. The mesh is generated by using GAMBIT software. Fig. 2 shows the computational domains and the corresponding boundary conditions for the two side-by-side flat finned-tubes. Two assistant computational zones, inlet and outlet, are setup to reduce the flow distortion at the entrance and exit of the flowing passage. The length of the two assistant computational zones is 50 mm and 150 mm respectively. Across the entrance of the computational domain, the velocity inlet boundary condition is specified. The fluid is assumed to enter with a uniform velocity U0 of 2 m/s, 3.5 m/s, and temperature T of 300 K respectively. The spectral synthesizer algorithm is selected to model the fluctuating velocity at velocity inlet boundaries. And the turbulent intensity is 10％.The pressure out boundary condition is appointed at the exit of the computational domain. For the interface of the neighboring tubes, the symmetric boundary conditions are appointed. Periodic boundary conditions are applied on the length direction of flat tube. The temperature of the tube wall maintains a constant 336 K which is equal to the saturated temperature of the exhaust steam. A no-slip boundary condition is prescribed at the surface of the tube wall.
Fig. 1. The flat finned-tube bundles.
Gðx; x0 Þ
¼
8 < 1=V; :
0;
x0 ˛v (2) x0 ;v
Filtering the NaviereStokes equations, one obtains
vr v rui ¼ 0 þ vt vxi
(3)
v v v vp vsij rui þ rui uj ¼ s vt vxj vxj ij vxi vxj
(4)
where sij is the stress tensor due to molecular viscosity defined by
"
sij h m
vui vxj
! þm
vuj vxi
!#
2 vu m i dij 3 vxi
(5)
and sij is the subgrid-scale stress defined by
sij hrui uj rui uj
(6)
In the present simulations, the dynamic SmagorinskyeLilly model is applied. The fluid mechanics are combined with the energy transport in the energy equation
Fig. 2. Schematic of the computational domains and the corresponding boundary conditions.
78
v v v rT þ rui T ¼ vt vxj vxj
J. Wen et al. / Applied Thermal Engineering 61 (2013) 75e85
l
vT vxj
!
vq vxi
(7)
where q represents sub-grid scale heat flux, which is modeled by the SGS model. 2.3. Computational methods In the present simulation, the finite-volume method applied on non-uniform grids is employed to calculate the 3D unsteady incompressible NaviereStokes equation with the computational fluid dynamics code ANSYS FLUENT12.0. Air is used as working fluid assuming constant properties. Because the physical parameters changes little when the inlet air was heated by the wall of flat finned tube. The steady flow data simulated with the standard ke3 model are used as the initial conditions for the unsteady LES. The number of grid points and their distribution is an important issue in such transient flows over finned tubes because of the associated complexity in the flow as well as the separation and vortex shedding. Fig. 3 shows the grid distribution in the computational domain. It is discretized accordingly by a non-uniform hexahedral grid with a finer grid distribution near the tube, with the help of a successive ratio scheme using the gridding software GAMBIT to capture the viscous boundary layer as well as the wake and the vortex street behind the tubes. The first point condition near the wall, in the wall unit of yþ(yþ ¼ rusyp/m), ensured that it was less than1. The accuracy of the computational results using LES is highly dependent on the mesh size and cell numbers. To show the independency of results from the grid size, the verification of mesh independency is done for different grid sizes in Fig. 4. According to this verification, for grids finer than 2,700,000 cells, no significant change is observed and there is just a 0.18% difference between this grid structure and 2,400,000 grid. So the solution is sufficiently grid independent. To ensure better results and capture detailed vortex structures, the finer grid number of 2,700,000 and 5,400,000 are adopted for the all simulations of the single and two side-by-side flat finned-tubes respectively. A bounded central-differencing scheme is used for momentum and energy discretization while a second-order implicit scheme is employed to advance the equations in time. The well-known pressure implicit method with splitting of operators (PISO) algorithm is used to deal with the pressureevelocity coupling between the momentum and the continuity equations. A convergence criterion of 1 103 was applied to the residuals of the continuity and the momentum equations and 1 106 to the residuals of the energy equation. In the numerical solution a time step size of 104 is used satisfying the above restrictions.
Fig. 4. Grid independency test of the numerical solutions for grid system.
Using more advanced modeling approach such as the LES model, which is often good at predicting local flow features around a body. There are scarcity experimental results reported in the published literature for the flow features in the wake region of a flat finnedtube. In order to validate the LES model, the flow at a subcritical Reynolds number ReD ¼ 3900 (based on cylinder diameter0.01 m and free-stream velocity) was chosen mainly because of the existence of the particle image velicometry (PIV) experimental data of Lourenco and Shih [19] and the hot-wire measurement results at the same Reynolds number by Ong and Wallace [20] which provided the mean flow data at several locations in the near wake of the cylinder. LES are carried out with a high-order accurate numerical method based on B-splines by Kravchenko and Moin Ref. [21]. The mean velocities and turbulent Reynolds stresses from the current numerical results are compared with the results of the experimental data and other numerical results of circular cylinder [19e22]. Figs. 5 and 6 show that the profiles of mean streamwise velocity along the centerline and Reynolds stresses in the wake of a circular cylinder obtained in the large eddy simulation are in well agreement with the experimental data. Therefore, we will use the LES model which represent the flow features around a body more precisely.
2.4. Validation of the numerical model The air flow around the flat finned-tube of ACC includes the flow around cylinder and the forced movement between the plates.
Fig. 3. The grid of the computational domains (a closer view around the flat tube).
Fig. 5. Mean streamwise velocity along the centerline.
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Fig. 6. Streamwise velocity fluctuations at three locations in the wake of a circular cylinder.
3. Results and discussion 3.1. Non-dimensional parameters The drag and lift are normalized by rU02 =2 to get the drag and lift coefficients Cd and Cl as:
Cd ¼ Fd = rAU02 =2 ; Cl ¼ Fl = rAU02 =2 where Fd and Fl are the total drag and lift forces respectively, U0 is the free-stream velocity, A is the reference area
The Strouhal number is defined by St ¼ fD=U0 where D is the tube diameter, f is the vortex shedding frequency.
Re ¼ rU0 D=m Thermal properties of air such as m and r is the molecular viscosity and density respectively. The local Nusselt number based on the flat finned-tube is calculated using the following equation:
Nu ¼ h$D=l where h is the convective heat transfer coefficient, l is the thermal conductivity of the fluid 3.2. The properties of flow field Fig. 7 shows the flow characteristics for single flat finned-tube by LES model and ke3 model [23] at the time tU$D1 ¼ 100. In Fig. 7(a), we can see that the velocity distribution values on the surface of fin channel is higher than other zones due to the area of the fin channel suddenly reduced. When the cooling air flow through the wake region of the finned-tube, the phenomenon of flow separation behind the tube is appeared. We can see that a great deal of vortex are identified, and the vortices emerged and shed in the wake region. Fig. 7(b) illustrates that the air isotherms in the fin channel have a forward protruding trend because of the velocity distribution values on the surface of fin channel is high. The temperature in the wake region of the finned-tube is increased due to the phenomenon of hot air circumfluence. Thus, there will be an attendant decrease in convection heat transfer in the wake region of the finned-tube. It is to be mentioned here that since both the
Fig. 7. Flow characteristics for single flat finned-tube calculated by LES model and ke3 model at Re ¼ 4006 and tU$D1 ¼ 100 (a) velocity field (b) temperature field (c) pressure field (d) streamlines (e) three-dimensional x-vorticity.
velocity and thermal energy are transported by the flow itself, the velocity contours as well as the isotherms exhibit similar features. From Fig. 7(c) we can find that the static pressure is reduced along
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the streamwise location. In the wake region, a negative pressure is showed. From the streamlines in Fig. 7(d), the essential features of
Fig. 9. Flow characteristics for two side-by-side flat finned-tubes at Re ¼ 7012 tU$D1 ¼ 100 and H/D ¼ 1.95 (a) velocity field (b) temperature field (c) pressure field (d) streamlines.
Fig. 8. Flow characteristics for two side-by-side flat finned-tubes at Re ¼ 4006 tU$D1 ¼ 100 and H/D ¼ 1.95 (a) velocity field (b) temperature field (c) pressure field (d) streamlines (e) three-dimensional x-vorticity.
the vortex shedding mechanism are the similar as those in the circular cylinder, except that the separation points of the boundary layers on the flat tube surface are different. From Fig. 7(e) we can take one pair of three-dimensional x-vorticity distribution, and the value of vorticity is 30 .The vortices are generated in pairs and the direction of rotation is opposite. In the wake region of the finned-tube the vortex and fluid mixed, momentum and energy exchanged. The comparison of these results with ke3 are presented to reveal that the LES approach can capture the complex turbulent
J. Wen et al. / Applied Thermal Engineering 61 (2013) 75e85
motions in detail, which leads to accuracies in the flow field prediction of the wake region behind the flat finned-tubes. Furthermore, the current works consider the wake interaction and the mutual interference of the adjacent flat finned-tubes, and their effects on the fluid flow and heat transfer. Fig. 8 illustrates the flow characteristics for two side-by-side flat finned-tubes at Re ¼ 4006 tU$D1 ¼ 100 and H/D ¼ 1.95. Fig. 8(a), (b) shows that the wakes behind the tubes interact in a complicated manner resulting in a variety of thermo-hydrodynamic regimes. It is worth mentioning that the results confirmed the dominant existence of anti-phase synchronized pattern in the flow. Two Karman vortex streets are observed in this figure. One can see that there is some evidence of merging of vortices at far downstream locations (close to the exit of the domain). The vortex shedding from one flat tube seems to have a definite phase relationship with the shedding from other tubes. The energy is transported in a similar fashion by the fluid flow to the downstream region. The vortices remain distinct throughout the computational domain without significant lateral spread. This can be attributed to weaker interactive flow pattern arising out of relatively larger spacing between the flat tubes. Fig. 8(c) represents that a high-pressure region developed in front of the tubes. The pressure in this region drops as the flow passes the gap. Fig. 8(d), the streamlines in the wakes of both flat finned-tubes follow an anti-phase asymmetric pattern induced from the interference between the two flat finned-tubes. Also, it can be seen that the separation points of flow are symmetric with respect to the centerline of gap spacing. And the separation points of the boundary layers on the flat tube surface are different from the single. When more than one flat finned tube is placed in the fluid flow, the resulting forces and vortex shedding pattern is different from those found at the same Reynolds number due to the wake interaction of the adjacent tubes. The difference between the flat tube bundles in Fig. 8 and single flat tube in Fig. 7 is the nonlinear wake interactions of the adjacent tubes and their effects on the flow pattern and heat transfer characteristics. The velocity, temperature, pressure and stream-function fields for Re ¼ 7012 are shown in Fig. 9.The interference effects and flow pattern around two side-by-side flat tubes are sensitive to the Reynolds number at this spacing. Comparing the flow behaviors for the counterpart gaps at Re ¼ 4006, we can clearly see the effect of the Reynolds number on flow characteristics. Fig. 9(a), the more complicated and unsymmetrical velocity field pertains to the fact that the flat tube at this Reynolds number has a shorter near-wake
Fig. 10. Mean streamwise velocity component u/U0 along the centerline axis of the flat tube1.
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Fig. 11. Cross-stream velocity in the near wake region of a flat tube.
region than as described in the work for Re ¼ 4006 and same spacing. Due to the increase of the Reynolds number, at this gap pressure is distributed differently in comparison to the same gap at Re ¼ 4006, as shown in Fig. 9(c). The pressure field for Re ¼ 7012 suggests that both cylinders experience different base pressures at this time. Fig. 9(d) shows that the flow pattern becomes more unsymmetrical than before because interference effects increase as the Reynolds number increases to 7012. Variations of the time-averaged streamwise velocity component u/U0 along the centerline axis of the cases for Re ¼ 4006 and 7012 are shown in Fig. 10. The lengths of recirculation zones in the wake region of the flat tube1 for single tube is larger than in the two sideby-side flat finned-tubes, due to the wake interaction of the adjacent tubes. The size and the form of the recirculation region is directly related to the length of the vortex formation region and the dynamics of the downstream flow. As the Reynolds number is increased to 7012, these vortices are developed in a shorter distance and are placed close to the flat tube. The lengths of recirculation zones in the wake region at Re ¼ 7012 is shorter than in the wake region at Re ¼ 4006.
Fig. 12. Mean streamwise velocity at different locations in the wake of the two sideby-side flat finned-tubes.
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Fig. 12 shows the mean streamwise velocity at different locations in the wake of the two side-by-side flat finned-tubes at Re ¼ 7012. The V-shape profile develops farther downstream. The cross-stream location of the maximum velocity deficit is always located at y/D ¼ 0.5 and y/D ¼ 3.5. The velocity deficit decays with increasing x/D. The shape of the mean velocity profile is directly related to the level of velocity fluctuations and, consequently, to the transition in the shear layers. The spanwise variation of streamwise velocity fluctuations at x/ D ¼ 6 is shown in Fig. 13. It has an ‘M’ shape, with a local minimum (at y/D ¼ 0.5) and two local maxima (at y/D ¼ 0.85 and y/ D ¼ 0.15). 3.3. The properties of drag and lift of flat finned-tubes
Fig. 13. Streamwise velocity fluctuations in the near wake region of a flat tube.
Flow patterns on the endwall surface show that the flat tube has a large reverse flow zone. In Fig. 11, the velocity distributions are obtained across the crosseestream plane (2 < y/D < 1 and 0.07 z/D 0.07) at the downstream location x/D ¼ 6. The spanwise variation of the mean velocity at any cross-stream location in the near-wake shows a boundary layer velocity profile. The streamwise mean velocity showed a V-shape profile further downstream at x/D ¼ 6. At all spanwise locations, the maximum velocity deficit is found at the line of symmetry (y/D ¼ 0.5). The velocity deficit is largest at the spanwise location closest to the endwall. And the velocity (z/D ¼ 0) at the spanwise location far away from the endwall is smallest, because the fin is fixed at the plane of z/D ¼ 0 and therefore the velocity behind the fin is small. With the increasing of the distance from the plane of fin (z/D ¼ 0), the velocity at the spanwise location far away from the endwall is increased. The wake width stays almost constant between-0.07 z/ D 0.07.
Fig. 14(a)e(d) shows the time series of lift and drag coefficients of the two adjoining flat tubes for H/D ¼ 1.95 at Re ¼ 4006 and 7012. It can be observed from the figures that both lift and drag coefficients are sinusoidal in nature. The amplitude of CL remains constant with a zero mean, whereas the amplitude of CD is observed to oscillate. The drag coefficients vary irregularly with time. The frequency of oscillation of the instantaneous CL is half of that of the instantaneous CD. From Fig. 14(a), (c), we can also observe that lift coefficients at this gap are in anti-phase synchronization with respect to each other which confirms the synchronized behavior of vortex shedding for this gap spacing, whereas the drag coefficients in Fig. 14(b), (d) are approximately in-phase. We can observe that increasing the Reynolds number has some effect on the flow behavior. The time response of lift coefficients of flat finned-tube at the Re ¼ 7012 shows more chaotic behavior than those at Re ¼ 4006. And the mean drag coefficients of flat finnedtube at Re ¼ 7012 are lower than the ones obtained for Re ¼ 4006. The flow generates a vortex street in the wake region. The periodic shedding of the vortices from the surface of the body induces periodic pressure variation on the body-structure. In the transverse direction, the excitation force has a dominant frequency called
Fig. 14. The time history of lift and drag coefficients for flat finned tube1and tube2 at Re ¼ 4006 and 7012.
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Fig. 15. Amplitude spectrum of drag and lift coefficient. (a) drag (b) lift.
Table 2 Different flow parameters for flat finned-tubes in present work. Parameter
CD
St
Single
42.08 42.08 42.08 27.45 27.45
0.286 0.291 0.291 0.295 0.295
Re ¼ 4006 Re ¼ 7012
Tube1 Tube2 Tube1 Tube2
0.05 0.04 0.01 0.02
Karman vortex shedding frequency. In the drag direction the dominant frequency is at twice the Kaman frequency. The shedding frequencies are determined by selecting a long time trace and performing a Fourier transform operation on it. Fig. 15 shows the amplitude spectrum of drag and lift coefficient, respectively. The signal of the drag coefficient shows multiple frequencies, while that of the lift coefficient shows the presence of the primary vortex shedding frequency. Oppositely oriented vortices traveling above and below the midplane of the wake will result in a bimodal distribution of the streamwise velocity variation (oscillation) and the drag coefficient. Since the variation of the lift coefficient and that of
the transverse velocity are synonymous, the fluctuations display single peak. The power spectrum analysis for the lift coefficient signal reveals that the vortex shedding mechanism is dominated by the primary frequency (Strouhal number) at this separation ratio (H/D ¼ 1.9) and Reynolds number (Re ¼ 4006) combination. Furthermore, it is evident from the Fig. 15 that the vortex shedding frequency increases with an increase of the Reynolds number at this given gap spacing. The flow parameters of each flat finned-tubes at different Reynolds number is presented in Table 2. It is worth noting that at a given gap, the Strouhal number increases with the increase of the Reynolds number, and both flat tube has the same Strouhal number. The Strouhal number of the single flat tube is smaller than those of the two side-by-side at the same Reynolds number due to the interaction between the adjacent flat finned-tubes. We can observe from Table 2 that the mean drag coefficient at Re ¼ 7012 is smaller than those at the Re ¼ 4006. Fig. 16(a)e(d) illustrates the time series of drag and lift coefficients of the fins (for example fin1 and fin2). Fig. 16(a)(b) indicates that both lift and drag coefficients of the fins at Re ¼ 4006 are again sinusoidal in nature. The drag coefficients at this Reynolds
Fig. 16. The time history of drag and lift coefficients for different fins at Re ¼ 4006 and 7012.
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Fig. 17. Amplitude spectrum of lift coefficient for fin at Re ¼ 4006 and 7012.
number are in anti-phase synchronization, whereas the lift coefficients in are approximately in-phase. The lift coefficient of the fin surface on both sides of the finned-tube has the characteristic of mutually exclusive. It is evident from Fig. 16(c), (d) that the time response of lift and drag coefficients at Re ¼ 7012 show more chaotic behavior than those at Re ¼ 4006. Furthermore, Fig. 16(a), (c) reveals that the lift coefficients of fin2 for two side-by-side flat tubes are more irregular, and the lift coefficients become more chaotic with an increase Reynolds number. From Fig. 17 we can seen the primary frequency in the amplitude spectrum. Besides, the diffused peaks correspond to the combined frequencies which are culminations of the nonlinear interactions among the shed vortices of different finned tubes can be represented. This combined frequency is termed as the secondary or flat finned-tubes interaction frequency.
3.4. The properties of heat transfer The time responses of Nusselt number of fins (for two fins, e.g. fin1 and fin2) for the two side-by-side flat finned-tubes at different Reynolds number are shown in Fig. 18(a), (b). The Nusselt number of the fins is found to oscillate periodically in time due to vortex shedding and a sinusoidal variation is observed for the Reynolds number of 4006. However, the variation becomes chaotic for larger Reynolds number7012. Fig. 19 shows the variations of local Nusselt number Nux for the fins along the fin length direction under different Reynolds number. It is to be noted that the local Nusselt number decreases along the
Fig. 19. The local Nusselt number for the different fins at Re ¼ 4006 and 7012.
flow direction because the thermal boundary layer development increases the thermal resistance. As seen from Fig. 19 the fin1 and fin2 of the single flat finned-tube have the same distribution of local Nusselt number since they experience the same undisturbed fluid, whereas for the two side by-side flat finned-tubes at Re ¼ 4006 and 7012 the Nusselt number for the fin2 displays higher value than the fin1. Furthermore, the local Nusselt number of the same fin in single flat finned-tube is lower than in the two side-byside finned tubes at same Reynolds number. As a result of interactions in the wakes behind the flat tubes and the mutual interference of the fins, the local Nusselt number distribution on the fin1 and fin2 for two side-by-side flat finned-tubes at H/ D ¼ 1.95 are different. These interactions give rise to the finned tube interaction frequencies that are responsible for the variations of heat transfer for the fined tube. And there is perceptible variation in the local Nusselt number for different fins. Furthermore, the difference of the local Nusselt number between the two fins are larger with the increased Reynolds number. Fig. 20 illustrates the local skin friction coefficient distribution on the fin surfaces. It is clearly seen that the skin friction coefficient decreases along the flow direction, and increases with an increased Reynolds number. It is also worth noting that the skin friction coefficients of the fin1 and fin2 are obvious different at Re ¼ 7012,which is probably attributed to wake interactions behind the flat finnedtubes. This finding is consistent with the mechanism of the vortex shedding and nonlinear interactions among the wakes behind the flat tubes.
Fig. 18. The time history of Nusselt number for different fins at Re ¼ 4006 and 7012.
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design of ACC should also account for wake interaction and vortex-shedding. Acknowledgements This work was supported by the National Basic Research Program of China (973 Program) (Grant No. 2009CB219804) and the Supercomputing center of Chinese Academy of Sciences. References
Fig. 20. The local skin friction coefficient for the different fins at Re ¼ 4006 and 7012.
4. Conclusions The study of the nonlinear interactions among the wakes behind multiple flat finned-tubes and their effects on the fluid flow and heat transfer in direct air-cooled condensers (ACC) requires accurate simulations. More advanced modeling techniques, such as LES calculations on heat transfer and wake interaction behind flat finned-tubes have been performed at different Reynolds numbers. The following conclusions have been reached. (1) Using LES model for the direct air-cooled condensers, which captures the complex turbulent motions in detail, resolves most of the turbulent eddies, and the transient flow fields in the wake region of the flat finned-tubes are predicted more precisely than RANS approach. (2) When more than one flat finned-tube is placed in the fluid flow and the wake interaction is considered, it will lead to a notable deviation from the results obtained for a single in terms of wake pattern, heat transfer and vortex shedding from both tubes at the same Reynolds number. The local Nusselt number of the same fin in single flat finned-tube is lower than in the two side-by-side finned tubes at same Reynolds number and the local Nusselt number and skin friction coefficient distribution on different fins for two side-by-side flat finned-tubes are different, due to the interactions and the mutual interference of the flat finned-tubes. The interaction frequencies are responsible for the variations of heat transfer for the finned tubes. (3) At the spacing H/D ¼ 1.95 adopted by ACC in a power plant, the degree of nonlinear wake interaction depends on the Reynolds number, which is found to have substantial effect on the flow behavior and thermal histories especially at Re ¼ 7012. (4) The 3D LES calculations of transient fluid flow and heat transfer behavior, wake interactions and the effects on heat transfer will provide a foundation for better design of such heat exchangers. Apart from heat transfer and pressure drop considerations, the
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