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Numerical study of pool boiling heat transfer in a large-scale confined space
Applied Thermal Engineering 118 (2017) 188–198
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical study of pool boiling heat transfer in a large-scale confined space Yongsheng Tian, Keyuan Zhang, Naihua Wang ⇑, Zheng Cui, Lin Cheng Institute of Thermal Science & Engineering, Shandong University, Jinan 250061, China
h i g h l i g h t s A three-dimensional pool boiling coupling model in a confined space is constructed. The dynamic characteristics of nucleate boiling heat transfer are analyzed. The characteristics of nucleate boiling heat transfer rely on the height of the tube. Tube pitch effect is more significant with the height increase.
a r t i c l e
i n f o
Article history: Received 6 August 2016 Revised 12 January 2017 Accepted 26 February 2017 Available online 28 February 2017 Keywords: Confined space Pool boiling Volume of fluid method Numerical simulation
a b s t r a c t The article aims to numerically investigate the heat transfer characteristics of pool boiling on large-scale tube bundles, according to the VOF method. We build a numerical model of single tube in a rectangle confined space to simulate the tube bundles based on the Passive Residual Heat Removal Heat Exchanger, which is applied in nuclear power. The numerical model presents the solid substrate coupled heat transfer between the convection in tube and boiling outside tube. We analyze the dynamic and local characteristic of boiling heat transfer along the tube and the effect of the tube pitch on the heat transfer. Heat transfer characteristics vary along the tube vertical direction. The time-average heat fluxes outside surface increase along the tube. The numerical results are compared with the classical correlation and reliable to predict the pool boiling heat transfer for tubes in the intermediate region of bundle. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Because of better heat transfer efficiency associated with boiling, it seems to offer a solution to meet modern demand of industries for development of more efficient heat transfer equipment. Pool boiling has its application in power generation, heating and refrigeration processes, nuclear engineering and cooling high energy density electronic component [1]. Recently, pool boiling has been employed to design passive heat removal systems in nuclear power plants, including emergency core cooling system, passive residual heat removal system and cooling of spent fuel pool [2,3]. Passive Residual Heat Removal Heat Exchanger (PRHR HX) immerged in the In-containment Refueling Water Storage Tank (IRWST) is employed in pressurized water reactor (PWR). The function of the PRHR HX is to remove decay heat from the core in a passive manner during an emergency shutdown. Heat is rejected from
⇑ Corresponding author. E-mail address: [email protected] (N. Wang). http://dx.doi.org/10.1016/j.applthermaleng.2017.02.110 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
the tube-side to the IRWST and finally to the circumstances through natural convection and nucleate boiling of water. The pool boiling over a single tube and tube bundles have been investigated by numerous researchers [4–9]. Some important correlations have been proposed to explain the relationship between heat flux and the wall superheat [10–13]. However, as boiling heat transfer involves many parameters, the models and empirical correlations proposed have their own limitation and differ a lot [1,14]. For example, the heat flux predicted by the correlation developed by Corletti et al. is less one order than that by the correlation developed by Rohsenow [10,15]. The most possible explanation is the difference of the tubes’ geometry scale. Due to the complexity of the pool boiling heat transfer, the heat transfer mechanism is different if the scale of the heat exchanger is quite large. Most of the literature [16–18] available in the area of nucleate boiling seem to be for horizontal tubes. However, due to its typical applications such as in nuclear power plant, some amount of studies on vertical surfaces or vertical tubes have been carried out in the recent past. It is well known that the phenomena of boiling are complex because of large number of influential parameters like
Y. Tian et al. / Applied Thermal Engineering 118 (2017) 188–198
189
Nomenclature Cp D Fv g H h hfg L M p q R r S Sh t T v w Re Pr Gr
specific heat at constant pressure (J kg K1) diameter of heat exchanging tube (m) interface-induced volume force (N) gravity acceleration (m s2) energy (J kg1) heat transfer coefficient (W/m2 K1) latent heat (J kg1) vertical length (m) molecular weight of vapor pressure (Pa) heat flux (W/m2) universal gas constant (8.314 J mol1 K1) mass-transfer intensity factor volumetric mass source term heat source term time (s) temperature (K) velocity (m s1) inlet velocity in stream wise direction (m s1) Reynolds number Prandtl number Grashof number
heater surface characteristics, geometrical scale, thermodynamic and transport properties of fluid and bubble dynamics of pool boiling. The correlation of Jakob and Hawkins [19] on nucleate boiling suggested that the heat transfer coefficient on a vertical heating surface is larger than that of horizontal heating surfaces. But, van Stralen and Sluyter [20] reported that the heat transfer coefficient on horizontal wires in pure liquids is more than that on vertical wires. Aprin et al. [21] conducted the study to analyze the local heat transfer analysis integrating the two-phase flow dynamics. Bundle effect is seen along the height of the tube bundle. The lower tubes are under the nucleate boiling regime or the bubbly flow regime. With the increase in height the heat transfer rate increases as with convective boiling effect and annular dispersed flow. Kang [22] carried out an experimental study to determine the effect of the length of vertically installed heat exchanger tube on the nucleate pool boiling heat transfer. The results for a vertically installed tube show that the amount of the heat transfer rate gets decreased with increasing tube length. In addition, pool boiling is also affected by the heat transfer space. Kang [23] carried out an experimental study to identify effects of the outer tube length on pool boiling heat transfer in a vertical annulus (gap size = 6.35 mm) with closed bottoms. The author observed that as the outer tube length is increased the heat transfer coefficient decrease at a constant heat flux. Gupta et al. [9] investigated a bundle of vertical stainless steel tubes (L = 800 mm, D = 19.0 mm). The experimental results show the nucleate boiling heat transfer coefficient on a vertical tube bundle increases along the direction of bubbles flow. The reason is the turbulence generated due to onset of boiling in the lower half of the bundle. Nishikawa et al. [24] studied the effect of surface inclination on the heat flux. Their experimental results show that the heat transfer coefficient increases with the increase of angle of inclination under low heat flux conditions or in partial nucleate boiling regime. However, the heat transfer coefficient is independent of surface inclination under high heat flux conditions or in fully developed nucleate boiling regime. This was attributed to the presence of sliding bubbles which enhance heat transfer by cyclic disruption and reformation of the thermal boundary layer. The effect of bubbles sliding on the heating surface have been studies in some amount of literatures. Okawa et al. [25] studied
X, Y, Z
coordinates
Greek symbols volume fraction b thermal expansion coefficient e roughness j interface curvature k effective thermal conductivity t kinematic viscosity l dynamic viscosity q density r surface tension coefficient u correction factor
a
Subscripts l liquid phase v vapor phase w wall sat saturated temperature s surface temperature 1 bulk temperature
the bubble rise characteristics in vertical upflow boiling of subcooled water at low heat fluxes. Three different bubble rise paths were observed after the departure from nucleation sites: some bubbles slid upward the vertical wall for long distance, while other bubbles were detached from the wall after sliding for several millimeters and then migrated toward the bulk liquid. After the migration, some of the detached bubbles were collapsed in subcooled liquid, but others remained close to the wall and were reattached to the wall. Thorncroft and Klausner [26] showed experimentally that the heat transfer rate from the heating wall is enhanced by the existence of vapor bubbles sliding along the wall. Sateesh et al. [27] indicated that sliding bubble mechanism plays an important role in pool boiling on vertical surfaces, and argued that sliding bubbles will be present at lower heat flux and disappear at a higher heat flux due to bubble interaction. Zhang et al. [28] used the Ghost Fluid Method for sharp interface representation. The complete single bubble pool boiling process including the transient thermal response of the solid wall was simulated numerically. Bubble dynamics and local heat transfer influenced by thickness and material parameters of the solid wall are analyzed at constant temperature. The study of boiling heat transfer on vertical tube is of great important to the safety analysis and design of nuclear power plants. However, it is difficult to get proper insight into the boiling over tube in industrial application because of the expensive industrial scale experiment. Although the empirical correlations can be applied for the design of engineering systems, but they are valid only in the range according to the experimental data obtained for the correlations. They are right in the given applicable range but may not beyond the limit. The empirical correlations normally obtained by lots of experimental data which costs time and money. CFD can easily be implemented with less cost. Furthermore, numerical simulation has become a feasible and effective method to analyze the local heat transfer characteristics and mechanism. And the simulation results can be shown the detail features of boiling heat transfer including local heat transfer coefficient, temperature, phase distribution, flow field. Krepper et al. [29] have performed computational fluid dynamics (CFD) methodology to investigate the natural convective and the thermal stratification in a large pool. Krepper et al. [30] applied
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CFD approaches to investigate subcooled boiling and verified their capability to contribute to fuel assembly design. Ganguli et al. [31] and Gandhi et al. [32] carried out single tube experiment in a rectangular tank, and numerically investigated the thermal stratification, the velocity distributions, and the turbulent parameters, etc. These researches contributed to a better understanding of CFD methodology employed in the passive residual heat transfer process. In addition, Zhang et al. [33] utilized the commercial CFD software to investigate the heat transfer effect of PRHR HX and buoyancy-induced flow in the IRWST. They built the overall scaled IRWST and PRHR HX models. Pan [34], Xue et al. [35,36] also utilized the commercial CFD software to simulation the thermalhydraulics phenomena associated with PRHR HX. Although the empirical correlations or the research results have served us well in the design of engineered systems, their predictive ability becomes suspect when applied to new situations. Furthermore, some aspects are also not sufficient about the study of pool boiling heat exchanger. Firstly, the vital heat transfer mechanism of boiling involves bubble behaviors like formation of bubble, departure and sliding [27]. However, phase interface of liquid and vapor has not been tracked in the simulation, and the bubble dynamics has not been demonstrated successfully in most papers. Secondly, the local heat transfer on the vertical tube surface is not distributed uniformly and it will evolve dynamically over time. Therefore, the simulation of local heat transfer characteristic is necessary to understand the mechanism of pool boiling. Thirdly, most simulations decoupled the solid substrate by assuming constant heat flux or wall temperature, but boiling heat transfer on the fluid side are tightly coupled to the heat conduction in the solid wall [37]. The temperature at the solid wall varies as the time. Simulation coupled with the solid heat conduction provides a more realistic description of nucleate pool boiling. Therefore, boiling and heat transfer over tube bundles in a pool is very complex due to coupling of different heat transfer mechanisms including natural convection, boiling and bubbling motion. It is important to understand the flow and temperature pattern in vicinity of tube bundles external channel in a pool. It is also crucial to understand the mechanism of flow and heat transfer. In this paper, we employed the CFD approach to investigate the thermal hydraulic characteristics of pool boiling heat transfer in the confined space. We established a reasonable numerical model based on PRHR HX. In the initial operation stage, heat is mainly rejected from the tube-side fluid to the IRWST fluid with subcooled boiling. Saturated boiling occurs when water reaches its saturation temperature which is the focus of our investigation. Bubble dynamics and local characteristics of boiling heat transfer were presented in the model and the numerical results were verified with the empirical correlations. 2. Mathematical model The boiling flow was simulated by the volume of fluid model (VOF), with a User Defined Function (UDF) as a vapor-liquid phase change model. The VOF model can track the interface between the two phases and the large topological deformation of bubbles. It has been widely used in analyzing two phase systems [38–41]. In terms of the numerical model, liquid water and vapor phase are treated as the primary phase and the secondary phase respectively for VOF.
@ v Þ ¼ Sl ða q Þ þ r ðal ql ~ ql @t l l 1
1
qv
Tracking of the interface between the phases was achieved by solving continuity equations for the volume fractions of liquid and vapor phases. For the liquid and vapor phases, the volume fraction equations can be written as
@ ðav qv Þ þ r ðav qv~ v Þ ¼ Sv @t
ð1Þ ð2Þ
Momentum equation:
@ ðq~ v Þ þ r ðq~ v~ v Þ ¼ rp þ r ½lðr~ v þ r~ v T Þ þ q ~g þ F V @t where q ¼ ql al þ qv av , Energy equation:
ð3Þ
l ¼ ll al þ lv av .
1 @ v HÞ ¼ r ðkrTÞ þ Sh ðqHÞ þ r ðq~ q @t
ð4Þ
The VOF model treats energy, H, and temperature, T, as massaveraged variables:
H¼
al ql Hl þ av qv Hv al ql þ av qv
ð5Þ
where Hv and Hl are based on the shared temperature and the specific heat of vapor phase and liquid phase respectively. The properties k (effective thermal conductivity) are shared by the phases. Moreover, the continuum surface force (CSF) model [42] has been introduced to describe the surface tension as the pressure drop cross the surface. The surface tension force is treated as a volume force (FV) in the momentum equation as following:
FV ¼ r
al ql jv rav þ av qv jl ral 0:5ðql þ qv Þ
ð6Þ
where jl ¼ Dal =ral , jv ¼ Dav =rav . The geometric reconstruction scheme represents the interface between fluids using a piecewise-linear approach [43]. In CFD simulation, this scheme is the most accurate [44]. In the present study, water is selected as the heat transfer fluid. The zone inside the tube is single-phase forced-convection heat transfer with Reynolds number about 2.3 106. For the natural convection with boiling heat transfer in the pool outside the tube, the Grashof (Gr) number is used as a criterion for the transition of heat transfer law near the wall region. The Grashof (Gr) number is
Gr ¼
gbðT s T 1 ÞD3
t2
The transition to turbulent flow occurs in the range 109 < Gr < 1010 for natural convection from vertical tube [45]. In the present study, the boundary layer is turbulent (1.05 1010 < Gr < 2.38 1010). Therefore, the realizable j-e model is adopted due to its good performance for rotating flow, the flow separation and the secondary flow [46,47]. 2.2. Boiling model The mass and energy transfer is the key term in the process of evaporation and condensation. Sl and Sv refer to the mass transfer rate accompanied by evaporation and condensation for the liquid phase and vapor phase respectively. Sh represents the energy corresponding to the mass transfer in the phase change. The relationship is
S¼ 2.1. Governing equations
Sh hfg
ð7Þ
The energy Sh and the mass transfer rates Sl and Sv can be obtained by the temperature field or boundary condition. The simulation of boiling can be achieved by a phase change source terms. The mass flow rate at the gas liquid interface is as following based on Hertz–Knudsen equation [48]:
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rffiffiffiffiffiffiffiffiffi M pv p ðT l Þ 00 pffiffiffiffiffi sat pffiffiffiffiffi jS j ¼ u 2pR T v Tl
ð8Þ
In the saturated state, the relationship between the pressure and the temperature can be established with the ClausiusClapeyron equation [49]:
hfg dp ¼ dT Tð1=qv 1=ql Þ
ð9Þ
Assuming that the pressure and temperature of the fluid are close to the saturated state, the Eq. (9) can be written as
p psat ¼
hfg ðT T sat Þ Tð1=qv 1=ql Þ
ð10Þ
Accordingly, the Eq. (8) is written as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hfg M T T sat jS j ¼ u 2pRT sat Tð1=qv 1=ql Þ T sat 00
ð11Þ
Then the mass transport can be realized by the mass source term in the volume fraction equation of VOF model. The condensation and evaporation coefficients are assumed to be one, representing complete evaporation and condensation at the interface. The evaporation and condensation processes are simplified concerning mass transfer model as follows:
Sv ¼ rl al ql ðT l T sat Þ=T sat ;
T l > T sat
ð12Þ
Sl ¼ rv av qv ðT l T sat Þ=T sat ;
T l 6 T sat
ð13Þ
where mass-transfer intensity factor r is an empirical coefficient. Here, r is set as 0.1 as most of the literature adopted [50–53]. However, many scholars [54,55] have set the relaxation factors as 100. But, it is found that large values of r bring about numerical convergence problem in this investigation. 3. Numerical model and simulation strategies The numerical model is based on the passive residual heat removal heat exchanger (PRHR HX) with a bundle of 689 C-shape tubes which is applied in AP1000 PWR [56,57]. Pool boiling heat transfer is worse for tubes in the intermediate region than those in the marginal region. Then the results of intermediate tubes can be used to make conservative estimates of the overall performance. Here we focus on the vertical section of heat exchange tube in the intermediate region of bundle. To reduce the amount of computation, we build a single tube model to simulate the characteristic of tube bundles (Fig. 1), and a quarter of the computational domain is considered because of the symmetrical structure. The symmetry boundary can effectively reduce the computational complexity. Fig. 2 shows the numerical model. The diameter of
Fig. 2. Numerical model.
heat exchanging tube is D = 19.05 mm. The thickness of tube wall is 1.65 mm. The sectional dimension of the computation region is set as 1D 2D. The length of the tube is 270D, and the depth of water pool is 320D. The boundary conditions are shown in Table 1. In addition, wall roughness effect is not considered and negligible. GAMBIT is used for meshing. Hexahedral mesh is adopted. To facilitate meshing and speed up the simulation, the whole computational domain is divided into four blocks. Finer mesh is employed near wall region where the gradients of variables are likely to be very high. After a series of grid independent tests (Table 2), a meshing method with total elements of 7,205,488 is chosen. All the computational work has been carried out using the software FLUENT 15.0. The pressure-implicit with splitting of operators (PISO) scheme is used for the transient simulation. Using PISO allows for increased values of all under-relaxation factors, without loss of the solution stability, resulting in faster convergence [58]. The second order semi-implicit scheme is used for the time integration. The spatial discretization is performed on a standard collocated grid using the finite volume method. The discretization for pressure is the scheme of PRESTO!. The discretization for volume fraction is the scheme of GeoReconstruct. The second-order upwind scheme is adopted to discretize the convective terms in momentum, energy, k and x equa-
Fig. 1. Configuration of tube bundles and computational domain (a) the vertical tube of the PRHR HX, and (b) the computational region.
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Table 1 Boundary conditions. No.
Location
Boundary type
1 2 3 4 5 6
Tube inlet Tube outlet Tube wall Pool bottom Pool sides Pool outlet
Free stream velocity Pressure-outlet Coupled Wall Symmetry Pressure-outlet
Table 2 Mesh independent validation. No.
Number of mesh
Heat fluxes (W/m2)
1 2 3 4
4,249,537 5,325,682 7,205,488 8,863,956
75,125 80,510 83,545 83,913
tions. In the transient simulation by VOF with an explicit scheme, the time step is restricted by the global Courant Number. This means that the smaller the grid size is, the smaller the time step needs. The time step was set as 0.0001–0.00025 s. 4. Results and discussion The physical processes of pool boiling include complex interactions of heat and mass transfer, nature and force convection and conjugate heat transfer. Bubble formation, coalescence and movement have been successfully demonstrated in this simulation. Furthermore, the numerical results reveal the flow pattern and the distribution of local properties along the height of tube. 4.1. Validation of the code In order to validate the reliability of the numerical model, we simulate two conditions in nucleate boiling region where the temperature conditions of inlet are T1 (395.15 K) and T2 (420.15 K). For the T1 condition, the wall superheat DT is between 10 K and 15 K in different locations, and for T2 condition 20 < DT < 30. The simulation results are compared with that obtained from the conventional nucleate pool boiling correlations (Table 3) in Fig. 3.
Table 3 The conventional nucleate pool boiling correlations. Author
Correlations h
Remarks i1=2 h
i C p ðT s T sat Þ 3 C sf hfg Prsl
Rohsenow [10]
q ¼ ll hfg
Corletti et al. [15]
q ¼ ll hfg
Kang [22]
q ¼ 0:019e0:570 DT 4:676 =ðD1:238 L0:072 Þ
Parlatan and Rohatgi [59]
q ¼ 52:4DT 3:058
gðql qv Þ
r
h
i1=2 h
gðql qv Þ
r
i3
C p ðT s T sat Þ C sf hfg Prsl
C sf for the watermechanically polished stainless steel is about 0.013 C sf is identified as 0.034, the experimental data obtained from PRHR test facility of the Westinghouse An experimental study of nucleate pool boiling outside a vertical tube in a water tank (cross section (0.79 0.86 m and 1 m in height). The HX tube is simulated by resistance heater 6 6 DT 6 22 K, the length of tube is 2 m
Fig. 3. Comparison of present results with typical existing correlation and Corletti et al.’s PRHR test data.
As shown in Fig. 3, the difference is significant between the results predicted by the correlations of Rohsenow [10], Kang [22] and Parlatan and Rohatgi [59]. The mainly reason is the nucleate boiling surface has a great impact on boiling heat transfer. Although the factors affecting the boiling heat transfer are known, it is not realistic to add surface properties to the boiling heat transfer correlation. Therefore, the empirical correlations proposed have their own limitation and may be different because they may be obtained in different situations. Even in the same experimental situation, the results may be deviate significantly due to minor factors. Therefore, it is meaningful to develop a model to get an insight of boiling mechanism and to predict the boiling heat transfer for a new engineering. It is worth noting that the heat fluxes predicted by the correlations of Rohsenow [10], Kang [22] and Parlatan and Rohatgi [59] are significantly greater than that by correlation and PRHR HX test data of Corletti et al. [15]. The main reason for this difference may be attributable to the difference in heat exchanger tube geometries. The other correlations are based on the large pool boiling. While the tube length and the tube pitch of the PRHR heat exchanger test tubes are 5486.4 mm and 38.1 mm, respectively. Compared with the other correlations based on experiments, PRHR HX tube length is longer and tube pitch is smaller. It is a finite space of pool boiling. The mechanism of boiling heat transfer is different from the single tube or wire in large pool. For longer heated tubes with smaller pitches, the effect of bubble coalescence and the formation of large vapor slugs could be magnified, which could effectively reduce heat transfer from the tube surface. The larger vapor slugs formed on the tube surface decrease the number of active nucleation sites, which will also reduce the heat transfer rate. Chun and Kang [60] pointed out that the boiling is not true pool boiling but rather convective boiling. It is believed that the flow is so strong that it suppresses boiling such that the tube wall must reach a higher superheat to boil. Therefore, the Corletti et al. data is significantly less than the other correlations. In addition, comparing Corletti et al. experimental data [22] with Corletti et al. correlation (Fig. 3), the empirical correlation is only in good agreement with the experimental data in the range of 20–40 K of superheat, but not beyond the range. The significant deviation between the experimental data and correlation of Corletti et al. exist at low superheat. In general, the prediction errors for the remaining correlations increase with decreasing superheat [61]. It is well-known, in the various types of convection heat transfer correlations, there is a maximum deviation between the
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empirical correlation and the experimental data due to the complexity of the boiling heat transfer. And nucleate boiling is the most complex than transition and film boiling. Parlatan and Rohatgi [59] analyzed separately the saturated pool boiling of water in the range of 0–400 K of superheat and have summed up the pool boiling correlations. In the range of 3–30 K of superheat in nucleate boiling stage, three correlations are proposed in 3 DT 6, 6 DT 22, 22 DT 30, respectively. Similarly, the correlations obtained with Corletti et al.’s data should be proposed in different range of superheat. As shown in Fig. 3, for the T2 temperature condition, the superheat is in range of 20–30 K. The numerical results are in agreement with the experimental data. For the T1 condition, the superheat is in range of 10–20 K, the simulation results predict the experimental data of Corletti et al. [22] correctly even they deviate from the Corletti et al. correlation. Based on the Corletti et al.’s experimental data in the range of 10–20 K of superheat, a new correlation has been obtained:
q ¼ 78:35DT 2:38 ð10 6 DT 6 20Þ
Figs. 4–7 show the temperature, the phase and the velocity distribution in the pool boiling side of the tube at different height. The buoyancy-driven flow is induced by the temperature gradient and
PRHR test data.
ð14Þ
We validated our numerical results with Corletti et al.’s correlation (q = 7.6DT3) and Eq. (14) from the Corletti et al. data. The deviations between the correlations and numerical results are shown in Table 4. Comparison condition T1 with new correlation (q = 78.35DT2.38), the deviation is within 30%. For condition T2, the deviation compared with Corletti et al.’s correlation is within 20%. The error within 30% is reasonable between the numerical results and the correlation. Such as, in Ref. 22 of manuscript, the accuracy of the correlation is within ±20%. And the accuracy is +35%, 20% in Ref. 60 of manuscript. The error is related to the complex nature of the boiling heat transfer. In general, the simulation results of this manuscript are in agreement with the experimental data of Corletti et al. and the simulation results are reasonable to predict the pool boiling heat transfer for tubes in the intermediate region of bundle. And the simulation results are shown the detail features of boiling heat transfer including local heat transfer coefficient, temperature, phase distribution, flow field.
Temperature Void fraction Velocity Fig. 4. The physical field in lower part.
4.2. Physical field analysis The physical fields include flow field, temperature field and phase distribution. We mainly analyze the coupling among them. The physical parameters vary along the tube vertical direction. The velocity, the temperature and the vapor fraction in the pool side increase gradually as the tube height increases.
Temperature Void fraction Velocity Fig. 5. The physical field in middle part.
Table 4 Comparison of the results between Corletti et al. correlation and numerical simulation. Conditions
Numerical results
Correlations based on the Corletti et al. data
Correlations results
Deviation (%)
T1
31,054.7 33,812.5 40,560.1 52,023.3 56,900.4 62,250.2
q = 78.35DT2.38 10 DT 20
31,228.1 31,538.5 35,014.4 40,207.7 44,516.2 48,293.8
0.5 7.2 15.8 29.4 27.8 28.9
T2
72,922.6 79,911.1 90,516.7 96,220. 1 104,738.9 102,887.1
q = 7.6DT3 [15]
89,077.8 91,469.2 104,062.4 110,403.5 107,710.9 120,044.6
18.1 12.6 13.0 12.8 2.8 14.3
Temperature Void fraction Velocity Fig. 6. The physical fields in the upper part.
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5m
2.5m
1m
Temperature
Void fraction
Velocity
Fig. 7. The local physical field in different height.
the bubbles’ motion. Heat transfer in the pool is a combination of free convection and nucleate boiling. Moreover, the physical parameter fields along the tube length are obvious different. The flow pattern and heat transfer characteristics in subcooled boiling are formed in the bottom region. Bubbles formation and growing are observed. Meanwhile, they slide and merge upon the tube surface. Temperature on the tube surface is influenced by disturbance of the flow field induced by the slight bubbles agitation. The fluid temperature varies significantly from the vicinity of the heated tube wall to the main stream apart from it. Bubbles coalescence and collapse can be found in the middle region. Many bubbles diffuse to the main stream. The bubbles are elongated under the effect of the tension perpendicular to the wall, buoyancy force and tangential force of flow field. The enhanced stirring and agitation of the bubbles can be found. In upper region, water in the pool side reaches a global saturation state. Bubbles scatter completely in the pool. Sliding bubbles disappear at upper part due to boiling developed in the pool. Bubbles will coalesce and enlarge. The larger bubbles will reattach to the tube surface or collapse under the impact of the fluid.
Fig. 8 shows parts of the velocity vectors in the bottom, the middle and the upper regions. In lower part of the pool, under the weak action of phase change effect and thermal buoyancy, low velocity flow is formed near the wall. Low-speed flow nearby the wall cannot drive the fluid far away from the wall to form an obvious flow circulation. The thermal-driven buoyancy increases with height, and the bubbles rise faster. The fluid flows in reverse direction between vicinity the tube surface and the main stream in the middle and the upper regions. Thermal convection is induced consequently. Local forced convection arises under the effect of lateral suction caused by bubble formation, growing and detachment. Moreover, the intensity of convection increases with the increase of height. The Fig. 9 indicates that the temperature gradient from the inside to the outside the tube. The axial temperature of tube decreases gradually along the flow direction. Inner and exterior tube surface temperature also decrease. But the temperature fluctuations happen on the high positions, especially on outer wall. Bubble behavior like formation of bubble, departure and sliding affects the wall temperature distribution. Due to the spatial and
Fig. 8. The local velocity vector for the different heights (a) lower part, (b) middle part, and (c) upper part.
Y. Tian et al. / Applied Thermal Engineering 118 (2017) 188–198
Fig. 9. The temperatures along the tube.
temporal variation of boiling phenomenon, wall temperature will fluctuate. The volume fraction of vapor is high for the upper part. The accumulation and attachment of bubbles near the wall result in a strong temperature fluctuations. The temperature will be significantly increased as the steam bubble attached to the wall of tube. While the liquid touch the wall due to bubble migration, the temperature will be significantly decreased. As shown in Fig. 9, temperature peak occurs in the location Z1 (4.75 m) and temperature valley in the location Z2 (5 m). The phase distributions at the location Z1 and Z2 in Fig. 10 can explain the mechanism of the temperature peak and the temperature valley. As the bubble covers the exterior tube surface (Fig. 10(a)), heat transfer decline, and the temperature of the surface rise rapidly. As the bubble detaches from the tube surface (Fig. 10(b)), heat transfer enhance, and the temperature reduce effectively. The heat transfer coefficient occur the peak and valley values on the location Z1 and Z2, 1514.9 W/(m2 K) and 5750.6 W/(m2 K), respectively (Fig. 11). Fig. 11 shows the variation of local heat transfer coefficients along the tube. In the lower part of heat exchange tube (0 < L < 1.5), the heat transfer coefficient increases steadily with the altitude. In the upper part (1.5 L), the local heat transfer coefficient fluctuates more greatly with the height increase. The heattransfer rate is mainly influenced by the agitation caused by bubbles. The agitation of bubbles is weak in the lower part. Bubble agitation become stronger with the height increase.
The tube is divided into six sections (named as section one to six) to analyze the heat transfer performance. The height of the bottom section is 0.5, and the height of the other five sections is 1 m. As shown in Fig. 12, the heat flux is steady in the section one to three. However, the heat flux fluctuate in the section four to six. The time-average heat fluxes outside surface increase along the tube. They are 55315.73, 61172.27, 72707.03, 90786.84, 96189.14, 101158.72 W/m2 respectively for the section one to six. The heat flux of section five and section six between 2.9 s and 3.2 s are approximately equal. It should be noted that the heat flux in section five is larger than that in section six at time-point t2. Possible explanation for this phenomenon is the influence of vapor fraction. There are some corresponding relationships between the heat flux and the surface coverage fraction of the vapor (Figs. 13 and 14). Coverage fraction increases with the height. In the section one and two, the bubbles distribute uniformly, detach at low frequency and slide longer distance along the tube (Fig. 4). The bubble agitation is weak in these zones correspondingly. Heat transfer becomes worse when surface coverage fraction of the vapor increases. In section three and four, the bubble departure frequency and agitation increase with the height increase, resulting in the vapor transport of energy into the body of the liquid increase. Therefore, the increase of vapor fraction enhances the
Fig. 11. The instantaneous distribution of heat transfer coefficient.
4.3. Time-average characteristics of the heat transfer Boiling heat transfer is transient with time and space. The local heat transfer performances are variable at a given time, and for different instant are dynamic at the specific location. Accordingly, it is necessary to discuss the time-average characteristics of the boiling heat transfer.
(a) Z1=4.75
(b) Z2=5 m
Fig. 10. Local phase and velocity distribution at Z1 and Z2.
195
Fig. 12. Transient and time-average heat flux on different tube sections.
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Fig. 13. The heat flux on the surface of tube sections. Fig. 14. The coverage fraction of vapor on the tube surface.
Fig. 15. Phase distribution (a) longitudinal sectional view for 4D spacing, (b) longitudinal sectional view for 2D spacing, and (c) three-dimensional plots of vapor phase for the computational domain.
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and parametric trends in nucleate pool boiling of confined space are as follows. (1) The temperature field, phase distribution and flow field are obviously different in different height. (2) The flow pattern of boiling varies along the tube height. The bubbly flow and slug flow patterns appear in the lower part and upper part of tube, respectively. (3) The various heat transfer regimes lead to different heat transfer coefficients. The heat transfer rate increases along the tube height and fluctuates due to the bubble behaviors. (4) In addition, for the 4D 2D, Effect of tube pitch is not obvious on overall heat transfer performance. But the heat transfer performance on the tube surface of 2D cross section is more unsteady. And local wall temperature fluctuates more greatly. Fig. 16. The distribution of heat transfer coefficient along the tube length on the small tube pitch (2D) and large tube pitch (4D).
heat transfer. However, in the section five and six, the void fraction reaches the upper limit on the nucleate boiling regime. The bubbles are coalesced to slug flow on the tube surface. The void fraction further increasing results in the significant decline of the heat flux.
To study the effect of the tube length and large heat flux on the boiling flow and heat transfer characteristics, experimental and numerical investigations should be carried out in the future. Acknowledgements Funding for this research was provided by the Natural Science Foundation of Shandong Province (Project No.: ZR2016EEM38).
4.4. Effect of tube pitch
References
The equivalent tube pitches are 2D and 4D in the numerical model. The void fractions in the 2D and 4D cross sections are shown in Fig. 15. The void fraction distributions are identical for different tube pitch in lower region where weak boiling occurs but not in the upper region where developed nucleate boiling occurs. For 2D tube pitch, the large bubbles are generated easily, and an intermittent slug flow pattern is often encountered. However, for 4D tube pitch, the bubbles are elongated and split under the effect of shear force, and a slug flow pattern is not encountered. In addition, the area-average void fractions along the longitudinal cross sections of 2D and 4D are 0.32 and 0.12. The weightedaverage velocities along the longitudinal cross sections of 2D and 4D are 0.25 m/s and 0.11 m/s, respectively. High void fraction and velocity will lead the fluctuation of heat transfer intensely. The local heat transfer coefficients are different on the 2D side and 4D side in the region from 1.5 m to 5.5 m (Fig. 16). The oscillation of local heat transfer coefficients on the 2D side is more intense than that on the 4D side. The small space between relevant tubes will cause the fluctuation of heat transfer. However, the areaaverage heat transfer coefficients are almost the same (3050.8 W/ (m2 K) for 2D side tube surface, 2976.7 W/(m2 K) for 4D side). The small tube pitch can enhance the heat transfer despite of local high temperature. However, it is not beneficial for bubble dispersion causing bubble coalescence on the tube surface. Therefore, the local heat transfer deteriorates and the local tube surface temperature increases.
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical study of pool boiling heat transfer in a large-scale confined space Yongsheng Tian, Keyuan Zhang, Naihua Wang ⇑, Zheng Cui, Lin Cheng Institute of Thermal Science & Engineering, Shandong University, Jinan 250061, China
h i g h l i g h t s A three-dimensional pool boiling coupling model in a confined space is constructed. The dynamic characteristics of nucleate boiling heat transfer are analyzed. The characteristics of nucleate boiling heat transfer rely on the height of the tube. Tube pitch effect is more significant with the height increase.
a r t i c l e
i n f o
Article history: Received 6 August 2016 Revised 12 January 2017 Accepted 26 February 2017 Available online 28 February 2017 Keywords: Confined space Pool boiling Volume of fluid method Numerical simulation
a b s t r a c t The article aims to numerically investigate the heat transfer characteristics of pool boiling on large-scale tube bundles, according to the VOF method. We build a numerical model of single tube in a rectangle confined space to simulate the tube bundles based on the Passive Residual Heat Removal Heat Exchanger, which is applied in nuclear power. The numerical model presents the solid substrate coupled heat transfer between the convection in tube and boiling outside tube. We analyze the dynamic and local characteristic of boiling heat transfer along the tube and the effect of the tube pitch on the heat transfer. Heat transfer characteristics vary along the tube vertical direction. The time-average heat fluxes outside surface increase along the tube. The numerical results are compared with the classical correlation and reliable to predict the pool boiling heat transfer for tubes in the intermediate region of bundle. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Because of better heat transfer efficiency associated with boiling, it seems to offer a solution to meet modern demand of industries for development of more efficient heat transfer equipment. Pool boiling has its application in power generation, heating and refrigeration processes, nuclear engineering and cooling high energy density electronic component [1]. Recently, pool boiling has been employed to design passive heat removal systems in nuclear power plants, including emergency core cooling system, passive residual heat removal system and cooling of spent fuel pool [2,3]. Passive Residual Heat Removal Heat Exchanger (PRHR HX) immerged in the In-containment Refueling Water Storage Tank (IRWST) is employed in pressurized water reactor (PWR). The function of the PRHR HX is to remove decay heat from the core in a passive manner during an emergency shutdown. Heat is rejected from
⇑ Corresponding author. E-mail address: [email protected] (N. Wang). http://dx.doi.org/10.1016/j.applthermaleng.2017.02.110 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
the tube-side to the IRWST and finally to the circumstances through natural convection and nucleate boiling of water. The pool boiling over a single tube and tube bundles have been investigated by numerous researchers [4–9]. Some important correlations have been proposed to explain the relationship between heat flux and the wall superheat [10–13]. However, as boiling heat transfer involves many parameters, the models and empirical correlations proposed have their own limitation and differ a lot [1,14]. For example, the heat flux predicted by the correlation developed by Corletti et al. is less one order than that by the correlation developed by Rohsenow [10,15]. The most possible explanation is the difference of the tubes’ geometry scale. Due to the complexity of the pool boiling heat transfer, the heat transfer mechanism is different if the scale of the heat exchanger is quite large. Most of the literature [16–18] available in the area of nucleate boiling seem to be for horizontal tubes. However, due to its typical applications such as in nuclear power plant, some amount of studies on vertical surfaces or vertical tubes have been carried out in the recent past. It is well known that the phenomena of boiling are complex because of large number of influential parameters like
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Nomenclature Cp D Fv g H h hfg L M p q R r S Sh t T v w Re Pr Gr
specific heat at constant pressure (J kg K1) diameter of heat exchanging tube (m) interface-induced volume force (N) gravity acceleration (m s2) energy (J kg1) heat transfer coefficient (W/m2 K1) latent heat (J kg1) vertical length (m) molecular weight of vapor pressure (Pa) heat flux (W/m2) universal gas constant (8.314 J mol1 K1) mass-transfer intensity factor volumetric mass source term heat source term time (s) temperature (K) velocity (m s1) inlet velocity in stream wise direction (m s1) Reynolds number Prandtl number Grashof number
heater surface characteristics, geometrical scale, thermodynamic and transport properties of fluid and bubble dynamics of pool boiling. The correlation of Jakob and Hawkins [19] on nucleate boiling suggested that the heat transfer coefficient on a vertical heating surface is larger than that of horizontal heating surfaces. But, van Stralen and Sluyter [20] reported that the heat transfer coefficient on horizontal wires in pure liquids is more than that on vertical wires. Aprin et al. [21] conducted the study to analyze the local heat transfer analysis integrating the two-phase flow dynamics. Bundle effect is seen along the height of the tube bundle. The lower tubes are under the nucleate boiling regime or the bubbly flow regime. With the increase in height the heat transfer rate increases as with convective boiling effect and annular dispersed flow. Kang [22] carried out an experimental study to determine the effect of the length of vertically installed heat exchanger tube on the nucleate pool boiling heat transfer. The results for a vertically installed tube show that the amount of the heat transfer rate gets decreased with increasing tube length. In addition, pool boiling is also affected by the heat transfer space. Kang [23] carried out an experimental study to identify effects of the outer tube length on pool boiling heat transfer in a vertical annulus (gap size = 6.35 mm) with closed bottoms. The author observed that as the outer tube length is increased the heat transfer coefficient decrease at a constant heat flux. Gupta et al. [9] investigated a bundle of vertical stainless steel tubes (L = 800 mm, D = 19.0 mm). The experimental results show the nucleate boiling heat transfer coefficient on a vertical tube bundle increases along the direction of bubbles flow. The reason is the turbulence generated due to onset of boiling in the lower half of the bundle. Nishikawa et al. [24] studied the effect of surface inclination on the heat flux. Their experimental results show that the heat transfer coefficient increases with the increase of angle of inclination under low heat flux conditions or in partial nucleate boiling regime. However, the heat transfer coefficient is independent of surface inclination under high heat flux conditions or in fully developed nucleate boiling regime. This was attributed to the presence of sliding bubbles which enhance heat transfer by cyclic disruption and reformation of the thermal boundary layer. The effect of bubbles sliding on the heating surface have been studies in some amount of literatures. Okawa et al. [25] studied
X, Y, Z
coordinates
Greek symbols volume fraction b thermal expansion coefficient e roughness j interface curvature k effective thermal conductivity t kinematic viscosity l dynamic viscosity q density r surface tension coefficient u correction factor
a
Subscripts l liquid phase v vapor phase w wall sat saturated temperature s surface temperature 1 bulk temperature
the bubble rise characteristics in vertical upflow boiling of subcooled water at low heat fluxes. Three different bubble rise paths were observed after the departure from nucleation sites: some bubbles slid upward the vertical wall for long distance, while other bubbles were detached from the wall after sliding for several millimeters and then migrated toward the bulk liquid. After the migration, some of the detached bubbles were collapsed in subcooled liquid, but others remained close to the wall and were reattached to the wall. Thorncroft and Klausner [26] showed experimentally that the heat transfer rate from the heating wall is enhanced by the existence of vapor bubbles sliding along the wall. Sateesh et al. [27] indicated that sliding bubble mechanism plays an important role in pool boiling on vertical surfaces, and argued that sliding bubbles will be present at lower heat flux and disappear at a higher heat flux due to bubble interaction. Zhang et al. [28] used the Ghost Fluid Method for sharp interface representation. The complete single bubble pool boiling process including the transient thermal response of the solid wall was simulated numerically. Bubble dynamics and local heat transfer influenced by thickness and material parameters of the solid wall are analyzed at constant temperature. The study of boiling heat transfer on vertical tube is of great important to the safety analysis and design of nuclear power plants. However, it is difficult to get proper insight into the boiling over tube in industrial application because of the expensive industrial scale experiment. Although the empirical correlations can be applied for the design of engineering systems, but they are valid only in the range according to the experimental data obtained for the correlations. They are right in the given applicable range but may not beyond the limit. The empirical correlations normally obtained by lots of experimental data which costs time and money. CFD can easily be implemented with less cost. Furthermore, numerical simulation has become a feasible and effective method to analyze the local heat transfer characteristics and mechanism. And the simulation results can be shown the detail features of boiling heat transfer including local heat transfer coefficient, temperature, phase distribution, flow field. Krepper et al. [29] have performed computational fluid dynamics (CFD) methodology to investigate the natural convective and the thermal stratification in a large pool. Krepper et al. [30] applied
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CFD approaches to investigate subcooled boiling and verified their capability to contribute to fuel assembly design. Ganguli et al. [31] and Gandhi et al. [32] carried out single tube experiment in a rectangular tank, and numerically investigated the thermal stratification, the velocity distributions, and the turbulent parameters, etc. These researches contributed to a better understanding of CFD methodology employed in the passive residual heat transfer process. In addition, Zhang et al. [33] utilized the commercial CFD software to investigate the heat transfer effect of PRHR HX and buoyancy-induced flow in the IRWST. They built the overall scaled IRWST and PRHR HX models. Pan [34], Xue et al. [35,36] also utilized the commercial CFD software to simulation the thermalhydraulics phenomena associated with PRHR HX. Although the empirical correlations or the research results have served us well in the design of engineered systems, their predictive ability becomes suspect when applied to new situations. Furthermore, some aspects are also not sufficient about the study of pool boiling heat exchanger. Firstly, the vital heat transfer mechanism of boiling involves bubble behaviors like formation of bubble, departure and sliding [27]. However, phase interface of liquid and vapor has not been tracked in the simulation, and the bubble dynamics has not been demonstrated successfully in most papers. Secondly, the local heat transfer on the vertical tube surface is not distributed uniformly and it will evolve dynamically over time. Therefore, the simulation of local heat transfer characteristic is necessary to understand the mechanism of pool boiling. Thirdly, most simulations decoupled the solid substrate by assuming constant heat flux or wall temperature, but boiling heat transfer on the fluid side are tightly coupled to the heat conduction in the solid wall [37]. The temperature at the solid wall varies as the time. Simulation coupled with the solid heat conduction provides a more realistic description of nucleate pool boiling. Therefore, boiling and heat transfer over tube bundles in a pool is very complex due to coupling of different heat transfer mechanisms including natural convection, boiling and bubbling motion. It is important to understand the flow and temperature pattern in vicinity of tube bundles external channel in a pool. It is also crucial to understand the mechanism of flow and heat transfer. In this paper, we employed the CFD approach to investigate the thermal hydraulic characteristics of pool boiling heat transfer in the confined space. We established a reasonable numerical model based on PRHR HX. In the initial operation stage, heat is mainly rejected from the tube-side fluid to the IRWST fluid with subcooled boiling. Saturated boiling occurs when water reaches its saturation temperature which is the focus of our investigation. Bubble dynamics and local characteristics of boiling heat transfer were presented in the model and the numerical results were verified with the empirical correlations. 2. Mathematical model The boiling flow was simulated by the volume of fluid model (VOF), with a User Defined Function (UDF) as a vapor-liquid phase change model. The VOF model can track the interface between the two phases and the large topological deformation of bubbles. It has been widely used in analyzing two phase systems [38–41]. In terms of the numerical model, liquid water and vapor phase are treated as the primary phase and the secondary phase respectively for VOF.
@ v Þ ¼ Sl ða q Þ þ r ðal ql ~ ql @t l l 1
1
qv
Tracking of the interface between the phases was achieved by solving continuity equations for the volume fractions of liquid and vapor phases. For the liquid and vapor phases, the volume fraction equations can be written as
@ ðav qv Þ þ r ðav qv~ v Þ ¼ Sv @t
ð1Þ ð2Þ
Momentum equation:
@ ðq~ v Þ þ r ðq~ v~ v Þ ¼ rp þ r ½lðr~ v þ r~ v T Þ þ q ~g þ F V @t where q ¼ ql al þ qv av , Energy equation:
ð3Þ
l ¼ ll al þ lv av .
1 @ v HÞ ¼ r ðkrTÞ þ Sh ðqHÞ þ r ðq~ q @t
ð4Þ
The VOF model treats energy, H, and temperature, T, as massaveraged variables:
H¼
al ql Hl þ av qv Hv al ql þ av qv
ð5Þ
where Hv and Hl are based on the shared temperature and the specific heat of vapor phase and liquid phase respectively. The properties k (effective thermal conductivity) are shared by the phases. Moreover, the continuum surface force (CSF) model [42] has been introduced to describe the surface tension as the pressure drop cross the surface. The surface tension force is treated as a volume force (FV) in the momentum equation as following:
FV ¼ r
al ql jv rav þ av qv jl ral 0:5ðql þ qv Þ
ð6Þ
where jl ¼ Dal =ral , jv ¼ Dav =rav . The geometric reconstruction scheme represents the interface between fluids using a piecewise-linear approach [43]. In CFD simulation, this scheme is the most accurate [44]. In the present study, water is selected as the heat transfer fluid. The zone inside the tube is single-phase forced-convection heat transfer with Reynolds number about 2.3 106. For the natural convection with boiling heat transfer in the pool outside the tube, the Grashof (Gr) number is used as a criterion for the transition of heat transfer law near the wall region. The Grashof (Gr) number is
Gr ¼
gbðT s T 1 ÞD3
t2
The transition to turbulent flow occurs in the range 109 < Gr < 1010 for natural convection from vertical tube [45]. In the present study, the boundary layer is turbulent (1.05 1010 < Gr < 2.38 1010). Therefore, the realizable j-e model is adopted due to its good performance for rotating flow, the flow separation and the secondary flow [46,47]. 2.2. Boiling model The mass and energy transfer is the key term in the process of evaporation and condensation. Sl and Sv refer to the mass transfer rate accompanied by evaporation and condensation for the liquid phase and vapor phase respectively. Sh represents the energy corresponding to the mass transfer in the phase change. The relationship is
S¼ 2.1. Governing equations
Sh hfg
ð7Þ
The energy Sh and the mass transfer rates Sl and Sv can be obtained by the temperature field or boundary condition. The simulation of boiling can be achieved by a phase change source terms. The mass flow rate at the gas liquid interface is as following based on Hertz–Knudsen equation [48]:
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rffiffiffiffiffiffiffiffiffi M pv p ðT l Þ 00 pffiffiffiffiffi sat pffiffiffiffiffi jS j ¼ u 2pR T v Tl
ð8Þ
In the saturated state, the relationship between the pressure and the temperature can be established with the ClausiusClapeyron equation [49]:
hfg dp ¼ dT Tð1=qv 1=ql Þ
ð9Þ
Assuming that the pressure and temperature of the fluid are close to the saturated state, the Eq. (9) can be written as
p psat ¼
hfg ðT T sat Þ Tð1=qv 1=ql Þ
ð10Þ
Accordingly, the Eq. (8) is written as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hfg M T T sat jS j ¼ u 2pRT sat Tð1=qv 1=ql Þ T sat 00
ð11Þ
Then the mass transport can be realized by the mass source term in the volume fraction equation of VOF model. The condensation and evaporation coefficients are assumed to be one, representing complete evaporation and condensation at the interface. The evaporation and condensation processes are simplified concerning mass transfer model as follows:
Sv ¼ rl al ql ðT l T sat Þ=T sat ;
T l > T sat
ð12Þ
Sl ¼ rv av qv ðT l T sat Þ=T sat ;
T l 6 T sat
ð13Þ
where mass-transfer intensity factor r is an empirical coefficient. Here, r is set as 0.1 as most of the literature adopted [50–53]. However, many scholars [54,55] have set the relaxation factors as 100. But, it is found that large values of r bring about numerical convergence problem in this investigation. 3. Numerical model and simulation strategies The numerical model is based on the passive residual heat removal heat exchanger (PRHR HX) with a bundle of 689 C-shape tubes which is applied in AP1000 PWR [56,57]. Pool boiling heat transfer is worse for tubes in the intermediate region than those in the marginal region. Then the results of intermediate tubes can be used to make conservative estimates of the overall performance. Here we focus on the vertical section of heat exchange tube in the intermediate region of bundle. To reduce the amount of computation, we build a single tube model to simulate the characteristic of tube bundles (Fig. 1), and a quarter of the computational domain is considered because of the symmetrical structure. The symmetry boundary can effectively reduce the computational complexity. Fig. 2 shows the numerical model. The diameter of
Fig. 2. Numerical model.
heat exchanging tube is D = 19.05 mm. The thickness of tube wall is 1.65 mm. The sectional dimension of the computation region is set as 1D 2D. The length of the tube is 270D, and the depth of water pool is 320D. The boundary conditions are shown in Table 1. In addition, wall roughness effect is not considered and negligible. GAMBIT is used for meshing. Hexahedral mesh is adopted. To facilitate meshing and speed up the simulation, the whole computational domain is divided into four blocks. Finer mesh is employed near wall region where the gradients of variables are likely to be very high. After a series of grid independent tests (Table 2), a meshing method with total elements of 7,205,488 is chosen. All the computational work has been carried out using the software FLUENT 15.0. The pressure-implicit with splitting of operators (PISO) scheme is used for the transient simulation. Using PISO allows for increased values of all under-relaxation factors, without loss of the solution stability, resulting in faster convergence [58]. The second order semi-implicit scheme is used for the time integration. The spatial discretization is performed on a standard collocated grid using the finite volume method. The discretization for pressure is the scheme of PRESTO!. The discretization for volume fraction is the scheme of GeoReconstruct. The second-order upwind scheme is adopted to discretize the convective terms in momentum, energy, k and x equa-
Fig. 1. Configuration of tube bundles and computational domain (a) the vertical tube of the PRHR HX, and (b) the computational region.
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Table 1 Boundary conditions. No.
Location
Boundary type
1 2 3 4 5 6
Tube inlet Tube outlet Tube wall Pool bottom Pool sides Pool outlet
Free stream velocity Pressure-outlet Coupled Wall Symmetry Pressure-outlet
Table 2 Mesh independent validation. No.
Number of mesh
Heat fluxes (W/m2)
1 2 3 4
4,249,537 5,325,682 7,205,488 8,863,956
75,125 80,510 83,545 83,913
tions. In the transient simulation by VOF with an explicit scheme, the time step is restricted by the global Courant Number. This means that the smaller the grid size is, the smaller the time step needs. The time step was set as 0.0001–0.00025 s. 4. Results and discussion The physical processes of pool boiling include complex interactions of heat and mass transfer, nature and force convection and conjugate heat transfer. Bubble formation, coalescence and movement have been successfully demonstrated in this simulation. Furthermore, the numerical results reveal the flow pattern and the distribution of local properties along the height of tube. 4.1. Validation of the code In order to validate the reliability of the numerical model, we simulate two conditions in nucleate boiling region where the temperature conditions of inlet are T1 (395.15 K) and T2 (420.15 K). For the T1 condition, the wall superheat DT is between 10 K and 15 K in different locations, and for T2 condition 20 < DT < 30. The simulation results are compared with that obtained from the conventional nucleate pool boiling correlations (Table 3) in Fig. 3.
Table 3 The conventional nucleate pool boiling correlations. Author
Correlations h
Remarks i1=2 h
i C p ðT s T sat Þ 3 C sf hfg Prsl
Rohsenow [10]
q ¼ ll hfg
Corletti et al. [15]
q ¼ ll hfg
Kang [22]
q ¼ 0:019e0:570 DT 4:676 =ðD1:238 L0:072 Þ
Parlatan and Rohatgi [59]
q ¼ 52:4DT 3:058
gðql qv Þ
r
h
i1=2 h
gðql qv Þ
r
i3
C p ðT s T sat Þ C sf hfg Prsl
C sf for the watermechanically polished stainless steel is about 0.013 C sf is identified as 0.034, the experimental data obtained from PRHR test facility of the Westinghouse An experimental study of nucleate pool boiling outside a vertical tube in a water tank (cross section (0.79 0.86 m and 1 m in height). The HX tube is simulated by resistance heater 6 6 DT 6 22 K, the length of tube is 2 m
Fig. 3. Comparison of present results with typical existing correlation and Corletti et al.’s PRHR test data.
As shown in Fig. 3, the difference is significant between the results predicted by the correlations of Rohsenow [10], Kang [22] and Parlatan and Rohatgi [59]. The mainly reason is the nucleate boiling surface has a great impact on boiling heat transfer. Although the factors affecting the boiling heat transfer are known, it is not realistic to add surface properties to the boiling heat transfer correlation. Therefore, the empirical correlations proposed have their own limitation and may be different because they may be obtained in different situations. Even in the same experimental situation, the results may be deviate significantly due to minor factors. Therefore, it is meaningful to develop a model to get an insight of boiling mechanism and to predict the boiling heat transfer for a new engineering. It is worth noting that the heat fluxes predicted by the correlations of Rohsenow [10], Kang [22] and Parlatan and Rohatgi [59] are significantly greater than that by correlation and PRHR HX test data of Corletti et al. [15]. The main reason for this difference may be attributable to the difference in heat exchanger tube geometries. The other correlations are based on the large pool boiling. While the tube length and the tube pitch of the PRHR heat exchanger test tubes are 5486.4 mm and 38.1 mm, respectively. Compared with the other correlations based on experiments, PRHR HX tube length is longer and tube pitch is smaller. It is a finite space of pool boiling. The mechanism of boiling heat transfer is different from the single tube or wire in large pool. For longer heated tubes with smaller pitches, the effect of bubble coalescence and the formation of large vapor slugs could be magnified, which could effectively reduce heat transfer from the tube surface. The larger vapor slugs formed on the tube surface decrease the number of active nucleation sites, which will also reduce the heat transfer rate. Chun and Kang [60] pointed out that the boiling is not true pool boiling but rather convective boiling. It is believed that the flow is so strong that it suppresses boiling such that the tube wall must reach a higher superheat to boil. Therefore, the Corletti et al. data is significantly less than the other correlations. In addition, comparing Corletti et al. experimental data [22] with Corletti et al. correlation (Fig. 3), the empirical correlation is only in good agreement with the experimental data in the range of 20–40 K of superheat, but not beyond the range. The significant deviation between the experimental data and correlation of Corletti et al. exist at low superheat. In general, the prediction errors for the remaining correlations increase with decreasing superheat [61]. It is well-known, in the various types of convection heat transfer correlations, there is a maximum deviation between the
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empirical correlation and the experimental data due to the complexity of the boiling heat transfer. And nucleate boiling is the most complex than transition and film boiling. Parlatan and Rohatgi [59] analyzed separately the saturated pool boiling of water in the range of 0–400 K of superheat and have summed up the pool boiling correlations. In the range of 3–30 K of superheat in nucleate boiling stage, three correlations are proposed in 3 DT 6, 6 DT 22, 22 DT 30, respectively. Similarly, the correlations obtained with Corletti et al.’s data should be proposed in different range of superheat. As shown in Fig. 3, for the T2 temperature condition, the superheat is in range of 20–30 K. The numerical results are in agreement with the experimental data. For the T1 condition, the superheat is in range of 10–20 K, the simulation results predict the experimental data of Corletti et al. [22] correctly even they deviate from the Corletti et al. correlation. Based on the Corletti et al.’s experimental data in the range of 10–20 K of superheat, a new correlation has been obtained:
q ¼ 78:35DT 2:38 ð10 6 DT 6 20Þ
Figs. 4–7 show the temperature, the phase and the velocity distribution in the pool boiling side of the tube at different height. The buoyancy-driven flow is induced by the temperature gradient and
PRHR test data.
ð14Þ
We validated our numerical results with Corletti et al.’s correlation (q = 7.6DT3) and Eq. (14) from the Corletti et al. data. The deviations between the correlations and numerical results are shown in Table 4. Comparison condition T1 with new correlation (q = 78.35DT2.38), the deviation is within 30%. For condition T2, the deviation compared with Corletti et al.’s correlation is within 20%. The error within 30% is reasonable between the numerical results and the correlation. Such as, in Ref. 22 of manuscript, the accuracy of the correlation is within ±20%. And the accuracy is +35%, 20% in Ref. 60 of manuscript. The error is related to the complex nature of the boiling heat transfer. In general, the simulation results of this manuscript are in agreement with the experimental data of Corletti et al. and the simulation results are reasonable to predict the pool boiling heat transfer for tubes in the intermediate region of bundle. And the simulation results are shown the detail features of boiling heat transfer including local heat transfer coefficient, temperature, phase distribution, flow field.
Temperature Void fraction Velocity Fig. 4. The physical field in lower part.
4.2. Physical field analysis The physical fields include flow field, temperature field and phase distribution. We mainly analyze the coupling among them. The physical parameters vary along the tube vertical direction. The velocity, the temperature and the vapor fraction in the pool side increase gradually as the tube height increases.
Temperature Void fraction Velocity Fig. 5. The physical field in middle part.
Table 4 Comparison of the results between Corletti et al. correlation and numerical simulation. Conditions
Numerical results
Correlations based on the Corletti et al. data
Correlations results
Deviation (%)
T1
31,054.7 33,812.5 40,560.1 52,023.3 56,900.4 62,250.2
q = 78.35DT2.38 10 DT 20
31,228.1 31,538.5 35,014.4 40,207.7 44,516.2 48,293.8
0.5 7.2 15.8 29.4 27.8 28.9
T2
72,922.6 79,911.1 90,516.7 96,220. 1 104,738.9 102,887.1
q = 7.6DT3 [15]
89,077.8 91,469.2 104,062.4 110,403.5 107,710.9 120,044.6
18.1 12.6 13.0 12.8 2.8 14.3
Temperature Void fraction Velocity Fig. 6. The physical fields in the upper part.
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5m
2.5m
1m
Temperature
Void fraction
Velocity
Fig. 7. The local physical field in different height.
the bubbles’ motion. Heat transfer in the pool is a combination of free convection and nucleate boiling. Moreover, the physical parameter fields along the tube length are obvious different. The flow pattern and heat transfer characteristics in subcooled boiling are formed in the bottom region. Bubbles formation and growing are observed. Meanwhile, they slide and merge upon the tube surface. Temperature on the tube surface is influenced by disturbance of the flow field induced by the slight bubbles agitation. The fluid temperature varies significantly from the vicinity of the heated tube wall to the main stream apart from it. Bubbles coalescence and collapse can be found in the middle region. Many bubbles diffuse to the main stream. The bubbles are elongated under the effect of the tension perpendicular to the wall, buoyancy force and tangential force of flow field. The enhanced stirring and agitation of the bubbles can be found. In upper region, water in the pool side reaches a global saturation state. Bubbles scatter completely in the pool. Sliding bubbles disappear at upper part due to boiling developed in the pool. Bubbles will coalesce and enlarge. The larger bubbles will reattach to the tube surface or collapse under the impact of the fluid.
Fig. 8 shows parts of the velocity vectors in the bottom, the middle and the upper regions. In lower part of the pool, under the weak action of phase change effect and thermal buoyancy, low velocity flow is formed near the wall. Low-speed flow nearby the wall cannot drive the fluid far away from the wall to form an obvious flow circulation. The thermal-driven buoyancy increases with height, and the bubbles rise faster. The fluid flows in reverse direction between vicinity the tube surface and the main stream in the middle and the upper regions. Thermal convection is induced consequently. Local forced convection arises under the effect of lateral suction caused by bubble formation, growing and detachment. Moreover, the intensity of convection increases with the increase of height. The Fig. 9 indicates that the temperature gradient from the inside to the outside the tube. The axial temperature of tube decreases gradually along the flow direction. Inner and exterior tube surface temperature also decrease. But the temperature fluctuations happen on the high positions, especially on outer wall. Bubble behavior like formation of bubble, departure and sliding affects the wall temperature distribution. Due to the spatial and
Fig. 8. The local velocity vector for the different heights (a) lower part, (b) middle part, and (c) upper part.
Y. Tian et al. / Applied Thermal Engineering 118 (2017) 188–198
Fig. 9. The temperatures along the tube.
temporal variation of boiling phenomenon, wall temperature will fluctuate. The volume fraction of vapor is high for the upper part. The accumulation and attachment of bubbles near the wall result in a strong temperature fluctuations. The temperature will be significantly increased as the steam bubble attached to the wall of tube. While the liquid touch the wall due to bubble migration, the temperature will be significantly decreased. As shown in Fig. 9, temperature peak occurs in the location Z1 (4.75 m) and temperature valley in the location Z2 (5 m). The phase distributions at the location Z1 and Z2 in Fig. 10 can explain the mechanism of the temperature peak and the temperature valley. As the bubble covers the exterior tube surface (Fig. 10(a)), heat transfer decline, and the temperature of the surface rise rapidly. As the bubble detaches from the tube surface (Fig. 10(b)), heat transfer enhance, and the temperature reduce effectively. The heat transfer coefficient occur the peak and valley values on the location Z1 and Z2, 1514.9 W/(m2 K) and 5750.6 W/(m2 K), respectively (Fig. 11). Fig. 11 shows the variation of local heat transfer coefficients along the tube. In the lower part of heat exchange tube (0 < L < 1.5), the heat transfer coefficient increases steadily with the altitude. In the upper part (1.5 L), the local heat transfer coefficient fluctuates more greatly with the height increase. The heattransfer rate is mainly influenced by the agitation caused by bubbles. The agitation of bubbles is weak in the lower part. Bubble agitation become stronger with the height increase.
The tube is divided into six sections (named as section one to six) to analyze the heat transfer performance. The height of the bottom section is 0.5, and the height of the other five sections is 1 m. As shown in Fig. 12, the heat flux is steady in the section one to three. However, the heat flux fluctuate in the section four to six. The time-average heat fluxes outside surface increase along the tube. They are 55315.73, 61172.27, 72707.03, 90786.84, 96189.14, 101158.72 W/m2 respectively for the section one to six. The heat flux of section five and section six between 2.9 s and 3.2 s are approximately equal. It should be noted that the heat flux in section five is larger than that in section six at time-point t2. Possible explanation for this phenomenon is the influence of vapor fraction. There are some corresponding relationships between the heat flux and the surface coverage fraction of the vapor (Figs. 13 and 14). Coverage fraction increases with the height. In the section one and two, the bubbles distribute uniformly, detach at low frequency and slide longer distance along the tube (Fig. 4). The bubble agitation is weak in these zones correspondingly. Heat transfer becomes worse when surface coverage fraction of the vapor increases. In section three and four, the bubble departure frequency and agitation increase with the height increase, resulting in the vapor transport of energy into the body of the liquid increase. Therefore, the increase of vapor fraction enhances the
Fig. 11. The instantaneous distribution of heat transfer coefficient.
4.3. Time-average characteristics of the heat transfer Boiling heat transfer is transient with time and space. The local heat transfer performances are variable at a given time, and for different instant are dynamic at the specific location. Accordingly, it is necessary to discuss the time-average characteristics of the boiling heat transfer.
(a) Z1=4.75
(b) Z2=5 m
Fig. 10. Local phase and velocity distribution at Z1 and Z2.
195
Fig. 12. Transient and time-average heat flux on different tube sections.
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Fig. 13. The heat flux on the surface of tube sections. Fig. 14. The coverage fraction of vapor on the tube surface.
Fig. 15. Phase distribution (a) longitudinal sectional view for 4D spacing, (b) longitudinal sectional view for 2D spacing, and (c) three-dimensional plots of vapor phase for the computational domain.
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and parametric trends in nucleate pool boiling of confined space are as follows. (1) The temperature field, phase distribution and flow field are obviously different in different height. (2) The flow pattern of boiling varies along the tube height. The bubbly flow and slug flow patterns appear in the lower part and upper part of tube, respectively. (3) The various heat transfer regimes lead to different heat transfer coefficients. The heat transfer rate increases along the tube height and fluctuates due to the bubble behaviors. (4) In addition, for the 4D 2D, Effect of tube pitch is not obvious on overall heat transfer performance. But the heat transfer performance on the tube surface of 2D cross section is more unsteady. And local wall temperature fluctuates more greatly. Fig. 16. The distribution of heat transfer coefficient along the tube length on the small tube pitch (2D) and large tube pitch (4D).
heat transfer. However, in the section five and six, the void fraction reaches the upper limit on the nucleate boiling regime. The bubbles are coalesced to slug flow on the tube surface. The void fraction further increasing results in the significant decline of the heat flux.
To study the effect of the tube length and large heat flux on the boiling flow and heat transfer characteristics, experimental and numerical investigations should be carried out in the future. Acknowledgements Funding for this research was provided by the Natural Science Foundation of Shandong Province (Project No.: ZR2016EEM38).
4.4. Effect of tube pitch
References
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