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Numerical simulation of dryout and post-dryout heat transfer in a straight-pipe once-through steam generator
Applied Thermal Engineering 105 (2016) 132–141
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical simulation of dryout and post-dryout heat transfer in a straight-pipe once-through steam generator Jianxin Shi a, Baozhi Sun a,⇑, Wenjing Han b, Guolei Zhang a,⇑, Yanjun Li a, Longbin Yang a a b
College of Power and Energy Engineering, Harbin Engineering University, 145 Nantong Street, Harbin 150001, PR China Tianjin Center for patent review of the State Intellectual Property Office, Hongshun Street, Tianjin 300302, PR China
h i g h l i g h t s Two-fluid three-flow-field model is developed to predict dryout in steam generator. The empirical correlation is used to correct dryout criterion. The interactions between three-flow-fields and the wall are considered. Dryout and post-dryout heat transfer mechanisms are discussed through the results.
a r t i c l e
i n f o
Article history: Received 2 March 2016 Revised 19 May 2016 Accepted 24 May 2016 Available online 24 May 2016 Keywords: Once-through steam generator Vertical pipe Flow boiling Dryout Heat transfer characteristic
a b s t r a c t Accurately predicting dryout and post-dryout heat transfer characteristics is critical for proper design of once-through steam generators. This paper provides a reasonable and simple method for this prediction by introducing a two-fluid, three-flow-field mathematical model and improving the dryout criterioncritical quality, and conducts a numerical simulation of dryout and post-dryout heat transfer in a once-through steam generator to prove the model’s performance. The results show that the critical quality in a once-through steam generator is about 0.82, with the heat transfer capacity significantly reducing and the wall temperature sharply increasing in a non-linear form by approximately 30 K when dryout occurs. Part of the steam is superheated in the post-dryout region, resulting in a deviation from thermodynamic equilibrium between the vapor and liquid phases. Dryout and post-dryout heat transfer in the once-through steam generator operate between complete deviation from thermodynamic equilibrium and complete thermodynamic equilibrium. Therefore, the presence of droplets has a significant influence on the mass, momentum and energy transfer between the film and vapor phases. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Compared to traditional natural circulation steam generators, a once-through steam generator (OTSG) used in the nuclear power plant with pressurized water reactor (PWR) generates superheated steam at the outlet, which provides the design good economical efficiency, compactness, modular construction, high power generation efficiency and good thermal heating effect. Because of these advantages, more and more researchers have focused on studying OTSG in recent years. In a OTSG, the working fluid in the secondary side experiences a complex vapor-liquid two-phase flow and heat ⇑ Corresponding authors. E-mail addresses: [email protected] (J. Shi), [email protected] (B. Sun), [email protected] (W. Han), [email protected] (G. Zhang), liyanjunhrb@ 163.com (Y. Li), [email protected] (L. Yang). http://dx.doi.org/10.1016/j.applthermaleng.2016.05.145 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
transfer process that includes preheating and boiling until overheating. Deterioration in the heat transfer occurs when the steam quality reaches a certain value, which has a major influence on the safe and reliable operation of the equipment. This heat transfer deterioration can be divided into two types according to its cause: (1) DNB (departure from nucleate boiling): DNB occurs when the heat flux exceeds the critical heat flux such that the flow pattern changes from nucleate boiling to film boiling. This means that bubbles near the wall are generated too late to spread to the main stream, causing the wall to be covered with a vapor film. (2) Dryout: This phenomenon occurs when the steam quality reaches a sufficiently high value, that the adherent annular liquid film is torn by the steam—that is, the flow pattern changes from an annular flow to a mist flow [1,2].
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
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Nomenclature t
a q
r !
U
Cl Cd SE SD p g F vd F!v l F lift
s sRe h q qv l qv d
vc A qwv qlv qwl
time (s) volume fraction density (kg/m3) gradient velocity vector (m/s) mass transfer rate between liquid film and steam, kg/(m3s) mass transfer rate between droplets and steam (kg/(m3 s)) droplet entrainment rate (kg/(m3 s)) droplet deposition rate (kg/(m3 s)) pressure (MPa) gravity (m/s2) drag force between steam and droplets (N/m2) drag force between steam and liquid film (N/m2) buoyancy lift (N/m2) shear stress (N/m2) Reynolds stress (N/m2) enthalpy (kJ/kg) heat flux (W/m2) heat flux per unit volume between steam and its interface with the liquid film (W/m3) heat flux per unit volume between steam and its interface with the droplets (W/m3) heating perimeter (m) flow area (m2) heat flux between wall and steam (W/m2) heat flux per unit volume between liquid film and its interface with the steam (W/m3) heat flux between wall and liquid film (W/m2)
The heat transfer deterioration that occurs during operation of a OTSG is dryout. Liquid films disappear when dryout occurs as the wall is no longer wetted in this state (i.e., there is no liquid contact with the wall), which results in a significant reduction in the surface heat transfer coefficient and a sharp rise in wall temperature. If effective control of this situation is not taken, wall temperature may exceed the maximum allowable limit, posing a security threat to running equipment. To describe this phenomenon, researchers have conducted a great deal of research and proposed two different models: a liquid film thickness model and a two-fluid three-flowfield model. Whalley et al. [3] proposed a liquid film thickness model based on a flow analysis approach, used the model to calculate the liquid film flow rate of an unbalanced flow (such as evaporation flow) through a uniformly heated straight pipe, then extended the model to predict the dryout and post-dryout heat transfer phenomenon. Azzopardi [4] studied the impact of the initial entrainment fraction and mass flow rate on dryout position for an electrically heated vertical tube using the liquid film thickness model. Chong et al. [5] modified the model in [4] and used it to predict dryout in a serpentine reboiler passage. Adamsson and Anglart [6] combined the liquid film thickness model with a sub-channel model of a nuclear reactor fuel assembly and captured the effect of shaft power distribution on dryout position. However, it is worth noting that the liquid film thickness model only considers the steam and the liquid film when predicting the dryout and post-dryout heat transfer, and there is presently no mature film thickness measurement method. Thus, it is difficult to verify the model. At the same time, some scholars have instead turned to conducting OTSG simulations by simplifying the flow boiling process. Li et al. [7] adopted a moving boundary lumped parameter method to investigate the change of inlet and outlet
qdv qwd qwil qwid d u Cl f d
m Dh k Nu T Pr x z
heat flux per unit volume between droplets and their interface with the steam (W/m3) heat flux between wall and droplets (W/m2) heat flux between wall and the interface of the steam with the liquid film (W/m2) heat flux between wall and the interface of the steam with the droplets (W/m2) liquid film thickness (m) velocity (m/s) lift coefficient drag function diameter (m) kinematic viscosity (m2/s) hydraulic diameter (m) thermal conductivity (W/(m K)) Nusselt number temperature (K) Prandtl number steam quality the height of heat transfer tube, m
Subscripts
v
l d i q k w DO
steam liquid film droplet i ¼ v ; l; d q ¼ l; d k ¼ v; l wall dryout
parameters in HTR-10 that considered the preheating, boiling and superheating regions. Li and Ren [8] studied the influence of heating power on the outlet steam temperature and the length of each heat transfer region by using a lumped parameter method to divide the heat transfer region into preheating, boiling and superheating regions based on the moving boundary model. Zhu et al. [9] established a dynamic mathematical model of the preheating, boiling and superheating regions based on lumped parameters and a moving boundary, and then conducted both steady and dynamic simulations. Wang et al. [10] studied the flow and heat transfer characteristics of the preheating and nucleate boiling regions in a OTSG using a two-fluid model and an RPI thermal phase change model built with CFX software. However, this study was limited by the method itself, as the simulation was only concerned with the inlet and outlet parameters of the OTSG and ignored local changes in the axial direction, and there exists dryout, heat transfer deterioration and wall temperature increases in localized regions. Generally speaking, the lumped parameter method will invariably ignore these factors when determining the thermal-hydraulic characteristics of a OTSG. To predict dryout more accurately, Thurgood et al. [11] considered liquid films, droplets, steam flow-fields and the interactions between these states in 1983. Based on their analysis, the group proposed a two-fluid three-flow-field model that used the easily measured steam quality as the dryout criterion, then implemented the model with COBRA-TF software to predict dryout-induced leakage accidents in the primary side cooling system of a nuclear reactor. Hoyer [12] applied a one-dimensional two-fluid threeflow-field model to MONA code and simulated the flow boiling of an electrically heated vertical tube in an experimental database. The results of the simulation showed that the model could accurately predict dryout and post-dryout heat transfer. Hoyer and
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Stepniewski [13] then extended the model for the pipe flow in [12] to predict the dryout between rod bundles in a nuclear reactor. Jayanti and Valette [14] improved the closure correlations of the two-fluid three-flow-field model and extended the range of the model’s application to high pressure systems (3–20 MPa), then analyzed the impact of spacer grids on dryout position and wall temperature in nuclear reactor systems using the model [15]. Ha et al. [16] applied the three-dimensional two-fluid three-flowfield model to SPACE code to simulate the flow boiling process in an electrical heating tube—except for single-phase steam convection, observing trends in the wall temperature in the post-dryout region. The numerical simulations in the above studies were mostly conducted based on electric heating experiments. That is, they applied a constant heat flux boundary to the wall and could not simulate the actual heat transfer conditions for the flow boiling process, which in turn is closely related to dryout and postdryout heat transfer. To make up for this lack, we use a OTSG operated by B&W as a prototype, introduce a two-fluid three-flow-field model mathematical model and improve it so that it can properly predict dryout in an actual running OTSG, then conduct a numerical simulation to obtain the dryout and post-dryout heat transfer characteristics for a straight-pipe OTSG.
2. Model development 2.1. Mathematical model As flow boiling develops in the secondary side, steam quality gradually increases. When dryout occurs, the steam becomes the continuous phase, with liquid only existing in the form of a continuous liquid film and discrete droplets entrained in the steam. There is also a mass transfer between the liquid film and the steam due to droplet entrainment, droplet deposition and evaporation of the liquid film. The liquid film slowly thins under the influence of this entrainment, deposition and vaporization [17]. If the velocity with which the droplets deposit onto the liquid film is less than the rate of the evaporation and entrainment of the liquid film, the liquid film will partially break. If this occurs, the coverage of the liquid film on the wall will gradually decrease, with the steam eventually occupying the entire flow path and causing dryout. The existence of droplets has a significant influence on the mass, momentum and energy transfer between the liquid film and the continuous steam, so it is necessary to consider all interactions between the droplets and the other flow fields (steam and liquid film) and the heat transfer between the droplets and the wall. With this model, the heat can transfer from the wall to the fluid through six paths: (1) from the wall to droplets that hit the wall; (2) from the wall to droplets that enter into the thermal boundary layer but do not wet the wall; (3) from the wall through convection heat to the steam; (4) from the steam to droplets suspended in the steam core (convective heat transfer); (5) radiative heat transfer from the wall to droplets; and (6) radiative heat transfer from the wall to the steam. The 6 heat transfer processes is described in Fig. 1 schematically. Of these heat transfer paths, (1) and (2) are usually taken into account together, while (5) and (6) only work in the case of high wall superheat, which is usually negligible in a OTSG. A traditional two-fluid model is only able to describe the flow and heat transfer behavior of a continuous liquid and dispersed bubbles. However, as we can see from the breakdown of possible heat transfers, there exist interactions between three flow-fields in the dryout and post-dryout region. Therefore, this paper introduces a two-fluid three-flow-field mathematical model to describe the behavior of flow and heat transfer in the dryout and post-dryout region [10,11,13,14,18–20]. The model establishes mass, momentum
Fig. 1. 6 heat transfer processes in post-dryout region.
and energy balance equations for the liquid film, droplets and steam, respectively. Mass balance equation for the steam: ! @ ðav qv Þ þ r ðav qv U v Þ ¼ Cl þ Cd @t
ð1Þ
Mass balance equation for the liquid film: ! @ ðal ql Þ þ r ðal ql U l Þ ¼ Cl SE þ SD @t
ð2Þ
Mass balance equation for the droplets: ! @ ðad qd Þ þ r ðad qd U d Þ ¼ Cd þ SE SD @t
ð3Þ
Momentum balance equation for the steam: !
! ! ! ! @ðav qv U v Þ þ r ðav qv U v U v Þ ¼ av rp þ av qv ~ g F v d ðU v U d Þ @t !
!
!
F v l ðU v U l Þ F lift þ r av ðsv þ sRe v Þ !
!
!
!
þ Cl ðU l U v Þ þ Cd ðU d U v Þ ð4Þ Momentum balance equation for the liquid film: !
! ! ! ! @ðal ql U l Þ þ r ðal ql U l U l Þ ¼ al rp þ al ql~ g F v l ðU l U v Þ @t ! ! þ r al ðsl þ sRe l Þ Cl ðU l U v Þ !
!
þ SD ðU l U d Þ
ð5Þ
Momentum balance equation for the droplets: !
! ! ! ! @ðad qd U d Þ þ r ðU d U d Þ ¼ ad rp þ ad qd~ g F v d ðU d U v Þ @t ! ! ! F lift þ r ad ðsd þ sRe d Þ Cd ðU d U v Þ !
!
þ SE ðU l U d Þ
ð6Þ
Energy balance equation for the steam: ! @ @p ðav qv hv Þ þ r ðav qv hv U v Þ av @t @t
v
¼ Cl hv þ Cd hv þ qv l þ qv d þ c qwv A
ð7Þ
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Energy balance equation for the liquid film:
Energy transfer between wall and flow-field:
@ @p ðal ql hl Þ þ r ðal ql hl U l Þ al @t @t !
¼ Cl hv þ SD hd SE hl þ qlv þ
qwk ¼
vc A
ð8Þ
qwl
k
@ @p ðad qd hd Þ þ r ðad qd hd U d Þ ad @t @t A
ð9Þ
qwd
In the above equations, the volume fractions satisfy
X
ai ¼ 1; 0 6 ai 6 1
ð10Þ
i
2.2. Closure correlations There are numerous source terms in the above equations, so it is necessary to make the equations closed. The closure correlations mainly include the transfer between wall and flow-fields, and the transfer between three flow-fields. The mass transfer between the flow-fields includes thermal phase change terms (Cl and Cd ) and hydraulic terms (SE and Sd ). The thermal phase change terms are obtained by evaporation and heat transfer at the steam-liquid film interface or the steamdroplet interface. The droplet entrainment rate and droplet deposition rate in hydraulic terms is the droplet mass taken away by the steam when the annular liquid film is torn by the steam and the droplet mass that the annular liquid film deposits the droplets entrained in the steam, respectively. The thermal phase change correlations are as follows:
vc =A qlv qv l
Cl ¼
qwil
Cd ¼
qwid
ð11Þ
hv hl
vc =A qdv qv d
ð12Þ
hv hd
The droplet entrainment rate is calculated according to [21]:
SE ¼ 1:1 104 d2:25 qd
ð13Þ
The droplet deposition rate is calculated according to [22]:
Sd ¼ kd C H
ð14Þ CH ¼ a
where
ad qd ud 6
kd ¼ 1:1474 102 1:2854
and
d qd ud þav qv uv
10 C H þ 1:0150 10 4:2501 109 C 3H þ 6:819 1012 C 4H . The momentum transfer between the flow-fields mainly includes the lift force and the drag force [23]: 4
!
!
C 2H
!
!
F lift ¼ C l qv ad ðU v U d Þ ðr U v Þ
ð15Þ
F v q ¼ K v q ðU v U q Þ
ð16Þ
!
!
2
where K v q ¼ 6qsvvf dv Ai , sv ¼ 18dvmv . Energy transfer between the flow-fields:
qv l ¼
the above correlation is given by a forced convective heat transfer, !
!
vc
ð19Þ
0:4 jU k jDh where Nuwk ¼ 0:023Re0:8 . k Pr k , Rek ¼ m
Energy balance equation for the droplets:
¼ Cd hd SD hd þ SE hl þ qdv þ
kk Nuwk ðT w T k Þ Dh
1 kv Nuv l ðT sat T v Þ Dh
Note that the dryout standard is a key parameter that will decide when dryout occurs, except for the above key closure correlations. The soaring rate of the wall temperature is usually used as the dryout criterion in experiments, while the critical heat flux, mass flow rate of the liquid film and the steam quality are used as the dryout criterion in numerical research. The two-fluid three-flow-field mathematical model often determines the occurrence of dryout by steam quality, and the default value is constant. The critical quality is related to the category of device and its operating conditions in actual cases. Thus, this paper applies a correlation of critical quality suitable for the operating conditions of B&W’s steam generator to determine the occurrence of dryout. This is determined by [25]: 1
0:07 np
xDO ¼ mq8 ðquv Þ3 do 1
e
ð20Þ
with the following operating parameters: p = 7–14 MPa; m = 46.0; n = 0.0255; G = 350–700 kg/(m2 s); and q = (2.3–5.8) 105 W/m2. The inlet Reynolds number calculated by the given boundary conditions and the geometric parameters of the physical model is about 31,098. The Reynolds number will increase along the flow direction of the secondary side in a OTSG because of the rising velocity and the decreasing kinematic viscosity with the development of the flow boiling. We can find that the flow process belongs to turbulent flow. The corresponding turbulence model is described in [18]. 2.3. Physical model and grid system A OTSG has a huge volume and possesses a large number of heat transfer tubes that have a thin tube wall and small tube pitch. The flow path and heat transfer process of the secondary side in a running OTSG is extremely complex, making it difficult to achieve a full-sized numerical simulation. To reflect an actual working process, this paper uses B&W’s operational OTSG as the prototype, but simplifies that prototype by assuming that the heat absorbed by feed water in the annular channel ultimately is provided by the primary side. This simplifies the process in that the feed water flow into the region is from the lower tube sheet and is heated directly through the heating role of the primary coolant. Other key geometric parameters such as diameter, tube pitch, height and other operating conditions are the same as the prototype. The arrangement of the heat transfer tubes is an equilateral triangle, with a series of tube support plates arranged in the secondary side. The tube support plates only affect the local thermalhydraulic characteristics and do not affect the overall flow and heat transfer characteristics of the OTSG, so this paper does not consider the tube support plates [18]. However, this is something we expect to include into the model for subsequent studies. The simplified physical model is shown in Fig. 2, with the key geometric parameters for the model given in Table 1.
ð17Þ
pD h
2.4. Boundary condition
the above correlation is given by a forced convective heat transfer, !
0:4 jU v jDh where Nuv l ¼ 0:023Re0:8 v Pr v , Rev ¼ mv , and
qv d ¼
6ad kv Nuv d ðT sat T v Þ dd dd
ð18Þ
the above correlation is given by a convection law for flow around !
!
1=3 jU v U d jd spheres [24], where Nuv d ¼ 2 þ 0:6Re0:5 . v Pr v , Rev ¼ mv
The given boundary conditions are provided in Table 2 along with the actual operating parameters of B&W’s operating OTSG. The heat transfer partitions (single-phase liquid convection, twophase flow boiling, dryout and post-dryout) that the secondary side experiences all have different flow and heat transfer characteristics. The coupled heat transfer between the two loops in the OTSG, and the transition from two-phase flow boiling to dryout,
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Fig. 2. Physical model and grid system.
simulation is converged when the scaled residuals in the control volumes for each of above governing equations are smaller than 105.
Table 1 Geometric parameters. Name
Unit
Value
Tube diameter Tube pitch Height of the tube
mm mm mm
15.875 22.225 9000
4. Results and discussion 4.1. Model validation
Table 2 Boundary conditions. Name
Unit
Value
Inlet mass flow Inlet temperature Outlet pressure Outlet saturation temperature Heat flux
kg/s K MPa K W/m2
0.1312 510.95 6.38 552.77 Formula (21)
add significant difficulties to the numerical simulation. Therefore, this paper replaces the heating role of the primary side with the second boundary condition. In order to simulate the actual heat transfer process, we add the heat flux function (21) to the wall based on theoretical calculations.
8 > < 145200 þ 3900z; z 6 2 q ¼ 48400 þ 2728z; 2 6 z 6 7:5 > : 19535 þ 990z; z P 7:5
ð21Þ
For this work, we conducted a simulation using the experimental data obtained from [12]-who conducted dryout and postdryout heat transfer experiments in a uniform electrically heated vertical tube—to verify the rationality and reliability of the introduced mathematical model in predicting dryout and post-dryout heat transfer. The experimental conditions used for the test were: inside tube diameter 10 mm, pressure 7 MPa, mass flow 1495 kg/(m2 s) and heat flux 797 kW/m2. We can see from Fig. 3 that the maximum error for the model is about 4.68%, which occurs at 6.5 m. This shows that the feasibility and rationality of the mathematical model is sound. We first performed a grid independent solution verification, shown in Fig. 3, before attempting authentication of the physical model in [12]. We found that there was substantially no change in the output of the model when the number of cells in the grid went beyond 32,000 and so the number of grid computing cells used for the remainder of this work was set to 32,000. Fig. 3 also shows the dryout location, the experimental and numerical results for the inner
3. Numerical methods The above governing equations are used to predict the flow boiling process of the secondary side in a OTSG, which contains the single-phase liquid convective heat transfer (the inlet water is in a subcooled state at a given pressure and mass flow), twophase flow boiling, dryout and post-dryout heat transfer. The numerical simulation is conducted based on steady-state mode, the governing equations are discretized by the finite volume method and the Coupled Implicit algorithm [26] is used to solve all equations for phase velocity corrections and shared pressure correction simultaneously, the fully implicit scheme is used to deal with the transfer between flow-fields, the closed correlations. The flux difference splitting scheme is used in the spatial discretization. The local time stepping and the implicit residual smoothing technique are also used in steady-state calculation. The numerical
Fig. 3. Model validation.
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
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Fig. 4. Grid independent validation.
Fig. 6. The axial distribution of the steam quality.
Fig. 5. The axial distribution of the vapor volume fraction.
wall temperature distribution. Our calculations show that the maximum error in the axial tube, which occurred at approximately 6.5 m along the tube, was 4.68%, which was within the allowed range. These results show that the use of a two-fluid three-flowfield mathematical model and method can accurately predict dryout and post-dryout heat transfer characteristics within a thin-walled tube, substantiating the feasibility of this numerical simulation to model a OTSG’s dryout and post-dryout heat transfer. 4.2. Grid independent validation of a OTSG Since the numerical results of a simulation heavily depend on the quality of the mesh and the quantity of elements in the grid, it is necessary to verify the grid independence before attempting a numerical simulation of dryout in a OTSG. The results of this validation test, shown in Fig. 4, confirm that the results do not change as a function of number of elements in the grid beyond 344,000 elements. Therefore, we used a grid containing 410,000 elements for this work. 4.3. Heat transfer analysis of OTSG The dryout in steam generator is a risk for the nuclear fuel rods in the nuclear reactor, which affects the safe and reliable operation of the nuclear power plant with PWR and results in an outage of the plant. Therefore, an in-depth study of dryout in an actual running OTSG—i.e. a study that analyzes changes in steam quality,
Fig. 7. The distribution contour of the vapor volume fraction.
fluid temperature, wall temperature and surface heat transfer coefficient when dryout occurs—can provide a reference for identifying the burning problem of heat transfer tubes. 4.3.1. Variation of steam quality The steam in a heat transfer tube will tear the liquid film on the wall of the tube when the steam quality reaches a certain value during OTSG operation, such that the wall is in contact with the steam and will cause dryout. Figs. 5 and 6 show the axial distribution of the vapor volume fraction and the steam quality in the secondary side, while Fig. 7 depicts the corresponding distribution contour of the vapor volume fraction. As can be seen from Figs. 5 and 6, the steam quality first remains constant before gradually increasing in the axial direction of the heat transfer tubes. At the same time, steam quality is 0 and remains constant up to an axial height of 1.8 m. The theoretical calculated height of the singlephase liquid convective heat transfer region based on the given boundary conditions is 1.86 m, giving a relative error of 3.2%. The
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Fig. 8. The axial distribution of the fluid velocity.
fluid begins to boil as the heating process develops, causing the steam quality to gradually increase to a value of 0.856 at the outlet. The theoretical maximum calculated steam quality was 0.9, giving a relative error of 4.89%. 4.3.2. Fluid velocity of the secondary side Figs. 8 and 9 show the axial distribution of the fluid velocity and corresponding distribution contour. As can be seen from the figures, there is only liquid in preheating section, the liquid velocity is almost constant and approximately 0.25 m/s. As flow boiling develops, the bubbles generate in the secondary side and the number of bubbles increases gradually, which results in that the fluid density decreases, therefore the fluid velocity increases. The steam velocity is always greater than the liquid velocity since the vapor density is larger than the liquid density in the secondary side. 4.3.3. Fluid temperature of the secondary side Fig. 10 shows the axial distribution of the fluid temperature in the secondary side and the associated wall temperature. As can be seen from the figure, the fluid first flows into the region in a subcooled condition and is subsequently heated by single-phase heat convection, the fluid temperature rises gradually. Nucleate boiling then begins when the fluid temperature reaches saturation. In this region, bubbles generate sharply on the wall, and the temperature of the liquid and vapor phases are maintained at saturation temperature. With the development of flow boiling, dryout occurs at an axial height of 7.5 m, and part of the steam begins to be superheated. In the dryout and post-dryout region, the flow pattern transforms from an annular flow to a mist flow, while the liquid phase gradually transforms from a continuous liquid film to droplets as its temperature remains saturated. Meanwhile, the continuous vapor phase is heated directly by the wall so that part of the steam is superheated. Note that the steam temperature is still at saturation temperature at the dryout location. However, the steam begins to superheat in the post-dryout region, which results in that vapor-liquid two-phase deviates from thermodynamic equilibrium. This thermodynamic non-equilibrium phenomena does not disappear until the droplets completely evaporate. In this paper, the outlet steam quality is 0.856 and the heat transfer does not involve a single-phase vapor convection region. Thus, the temperature distribution in Fig. 10 agrees with that in Fig. 11 where xc < 1 [1], with the corresponding discussion indicating that there exists an essential difference between nucleate boiling and dryout and post-dryout heat transfer. Meanwhile the results prove the rationality and feasibility of the modified
Fig. 9. The distribution contour of the fluid velocity at different sections.
Fig. 10. The axial distribution of fluid temperature and wall temperature in the secondary side.
model used in this work to simulate the flow and heat transfer process from single-phase liquid convection to dryout in a OTSG. 4.3.4. Wall temperature analysis The wall temperature is a key parameter in OTSG safety. When dryout occurs, the local wall temperature will be much higher than the saturation temperature, causing a deviation from the equilibrium. There are two extreme cases:
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
Fig. 11. Variation in steam temperature and wall temperature when dryout occurs [1].
1. Complete deviation from thermodynamic equilibrium. This case generally occurs at low pressure and low mass flow rate. The heat is transferred to the continuous vapor phase contacted with the heated wall completely, which causes steam superheating. Assuming that the heat transfer rate from the steam to the droplets is very slow, we can ignore the existence of the droplets and assume that the heat transferred from the wall is completely used to superheat the steam. In this case, the increase in wall temperature is consistent with the case of single-phase vapor convection and its value can be calculated by the single-phase convection correlation and corresponding heat flux. 2. Complete thermodynamic equilibrium. This case occurs at high pressure and high flow rates. In this case, we assume that the heat transferred from the vapor phase to the droplets is sufficient; that is, the heat is transferred from the vapor phase to the droplets quickly enough that the steam and the droplets maintain the same temperature, one that is equal to the saturation temperature. In other words, the heat transferred through the wall is used to completely evaporate the droplets until all droplets are evaporated, and only then is the heat re-used to superheat the steam. In this case, the wall temperature may appear different non-linear changes, and will reach a maximum value before declining slightly. The main reason for this result is that the droplets hit the wall [27]. Fig. 12 shows the temperature variations for the two extreme cases. Note, however, that the actual change in wall temperature actually seen in application is usually between these two extreme cases. Therefore, it is difficult to determine and describe the temperature trends in the dryout and post-dryout regions and also complex to model the process. As such, it is important to note that this paper conducts a preliminary exploration of the changes in wall temperature when dryout occurs in a OTSG. The heat flux and fluid temperature in the secondary side have a significant effect on wall temperature. Fig. 10 shows that wall temperature rises as fluid temperature slowly rises in the single-phase liquid convective heat transfer region. As the fluid enters the nucleate boiling region, its temperature remains at the saturation temperature, while the wall temperature slowly rises and is higher than the saturation temperature of the secondary side fluid due to the increasing heat flux and heat transfer coefficient. As flow boiling develops, the flow pattern is converted from a slug flow to an annular flow, and the liquid film on the wall is torn under the multiple effects of deposition, entrainment and evaporation.
139
Fig. 12. Temperature variations for the complete deviation from thermodynamic equilibrium case and the complete thermodynamic equilibrium case.
This also causes a corresponding higher steam quality, causing dryout to occur at this position. The wall contacts directly with the steam, which greatly reduces heat transfer capacity and results in a sharp rise in wall temperature. The maximum amplitude of the temperature change is about 30 K in the simulation, whereas
Fig. 13. A comparison of steam quality and wall temperature along the heat transfer tube’s axial length.
Fig. 14. A comparison of wall temperature and heat transfer coefficient along the heat transfer tube’s axial length.
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Table 3 Heat transfer coefficient comparison between numerical results and empirical correlations. Parameter Surface heat transfer coefficient
Heat transfer partition Preheating Nucleate boiling Dryout and post-dryout
Empirical correlation Dittus-Boelter [29] Chen [30] Remizov [31]
the actual maximum amplitude of the temperature change in the B&W data is 27 K [28], giving a relative error of 11.1%. In the post-dryout region, the fluid exists in the form of vapor entrained droplets. Since the convective heat transfer coefficient is small and thermal resistance is large in this region, the wall temperature continues to increase, albeit at a slow rate. We can see by comparing Figs. 10 and 12 that dryout in the OTSG is between the complete deviation from thermodynamic equilibrium case and the complete thermodynamic equilibrium case. Therefore, the presence of droplets has a considerable influence on the momentum and energy transfer between the film and the steam phase. Thus, it is necessary to consider the existence of the droplets, all interactions between the droplets and other flow-fields (steam and liquid film) and the heat transfer between the droplets and the wall in the numerical simulation to achieve accurate results. 4.3.5. Comparison of key parameters Fig. 13 shows a comparison of steam quality and wall temperature along the axial direction, revealing the relationship between these key dryout parameters in a OTSG. As can be seen from the figure, the wall temperature begins to soar in a non-linear form when steam quality reaches about 0.82. It is at this position that dryout occurs, and the heat transfer transforms from two-phase flow boiling to dryout. Fig. 14 shows a comparison of wall temperature and heat transfer coefficient along the heat transfer tube’s axial length. Table 3 shows a comparison of the heat transfer coefficients derived from the numerical results against widely used empirical correlations. The numerical results were calculated by integrating the surface heat transfer coefficient along the whole heat transfer area of each heat transfer partition and then averaging the integral value. We can see from Fig. 14 and the table that the heat transfer in the preheating region results from a convective heat transfer between wall and liquid, which means that the surface heat transfer coefficient is small. When entering the nucleate boiling region, a large number of bubbles generate near the wall, and the disturbance caused by the bubbles greatly enhances heat transfer, causing the surface heat transfer coefficient to sharply increase at this region. At the same time, we can see that dryout occurs at an axial height of 7.5 m, where the liquid film on the wall disappears and there is no liquid wetting the wall at this position. Thus, the surface heat transfer coefficient shows a sharp decline at this point and wall temperature rises sharply. In the post-dryout region, the wall surface is covered with a continuous steam. Since the steam heat transfer coefficient is much smaller than the heat transfer coefficient of the liquid, the heat transfer capacity in this region is weak and the surface heat transfer coefficient is very small. We can see that the maximum relative error is 17.9%, which is within the allowable design range for engineering, and proves the accuracy of the numerical simulation. 5. Conclusion (1) This paper introduced a two-fluid three-flow-field mathematical model and used empirical correlation to correct the dryout criterion. The results from our simulation showed
Unit 2
W/(m K)
Numerical results
Empirical results
Relative error (%)
3693.27 67410.99 1893.39
3133.14 60060.12 2128.26
17.9 12.2 11.0
that the model can be used to accurately predict the dryout and post-dryout heat transfer phenomenon of an actual OTSG. (2) In the simulation, the critical quality was about 0.82, with dryout occurring at an axial height of 7.5 m. The wall temperature at this point rose by about 30 K, followed by superheating part of the steam in the post-dryout region. The average surface heat transfer coefficients in the heat transfer partitions were in good agreement with empirical results. (3) Part of the steam in the post-dryout region is superheated, resulting in a deviation from thermodynamic equilibrium between the vapor and liquid. This phenomenon does not disappear until all droplets are evaporated. Dryout in the OTSG exists between two extreme cases—complete deviation from thermodynamic equilibrium and complete thermodynamic equilibrium—with the presence of droplets having a considerable influence on momentum and energy transfer between the film and vapor phases. Thus, it is necessary to consider the interactions between the droplets and other flow-fields, between the droplets and the wall to produce accurate simulation results.
Acknowledgements This work was sponsored by the National Natural Science Foundation of China (No. 51479040, No. 51579048), the Natural Science Foundation of Heilongjiang Province of China (No. E201422, No. E201346), and Postdoctoral Research Startup Foundation of Heilongjiang Province of China (No. LBH-Q14036), which we gratefully acknowledge. References [1] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, Oxford University Press, New York, 1981. [2] S. Talebi, H. Kazeminejad, H. Davilu, Prediction of dryout and post-dryout wall temperatures using film thickness model, Nucl. Eng. Des. 244 (2012) 73–82. [3] P.B. Whalley, P. Hutchinson, G.F. Hewitt, Calculation of critical heat flux in forced convection boiling, in: Fifth International Heat Transfer Conference, Tokyo, Japan, 1974, pp. 290–294. [4] B.J. Azzopardi, Prediction of dryout and post-burnout heat transfer with axially non-uniform heat input by means of an annular flow model, Nucl. Eng. Des. 163 (1) (1996) 51–57. [5] L.Y. Chong, B.J. Azzopardi, D.J. Bate, Calculation of conditions at which dryout occurs in the serpentine channels of fired reboilers, Chem. Eng. Res. Des. 83 (4) (2005) 412–422. [6] C. Adamsson, H. Anglart, Influence of axial power distribution on dryout: filmflow models and experiments, Nucl. Eng. Des. 240 (6) (2010) 1495–1505. [7] H. Li, X. Huang, L. Zhang, A lumped parameter dynamic model of the helical coiled once-through steam generator with movable boundaries, Nucl. Eng. Des. 238 (7) (2008) 1657–1663. [8] Y.Q. Li, T.J. Ren, Moving boundary modeling study on supercritical boiler evaporator: by using enthalpy to track moving boundary location, in: Power and Energy Engineering Conference, Asia-Pacific, 2009, pp. 1–4. [9] J.Y. Zhu, Z.J. Zhang, Y. Guo, Dynamic real-time simulation research on oncethrough steam generator with concentric annuli tube, Atomic Energy Sci. Technol. 45 (8) (2011) 937–942 (in Chinese). [10] C. Wang, T. Cong, S. Qiu, et al., Numerical prediction of subcooled wall boiling in the secondary side of SG tubes coupled with primary coolant, Ann. Nucl. Energy 63 (2014) 633–645. [11] M.J. Thurgood, J.M. Kelly, T.E. Guidotti, et al., COBRA/TRAC – A ThermalHydraulics Code for Transient Analysis of Nuclear Reactor Vessels and Primary
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
[12] [13]
[14]
[15]
[16] [17] [18]
[19]
[20]
Coolant Systems: Equations and Constitutive Models, Nuclear Regulatory Commission, Washington DC, 1983, pp. 21–38. N. Hoyer, Calculation of dryout and post-dryout heat transfer for tube geometry, Int. J. Multiphase Flow 24 (2) (1998) 319–334. N. Hoyer, M. Stepniewski, Dryout prediction in bundle geometry, in: Ninth International Topical Meeting on Nuclear Reactor Thermal Hydraulics, San Francisco, California, 1999, pp. 378–389. S. Jayanti, M. Valette, Prediction of dryout and post-dryout heat transfer at high pressures using a one-dimensional three-fluid model, Int. J. Heat Mass Transfer 47 (22) (2004) 4895–4910. S. Jayanti, M. Valette, Calculation of dry out and post-dry out heat transfer in rod bundles using a three field model, Int. J. Heat Mass Transfer 48 (9) (2005) 1825–1839. S.J. Ha, C.E. Park, K.D. Kim, et al., Development of the space code for nuclear power plants, Nucl. Eng. Technol. 43 (1) (2011) 45–62. J. Weisman, B.S. Pei, Prediction of critical heat flux in flow boiling at low qualities, Int. J. Heat Mass Transfer 26 (10) (1983) 1463–1477. B. Sun, Y. Yang, Numerically investigating the influence of tube support plates on thermal-hydraulic characteristics in a steam generator, Appl. Therm. Eng. 51 (1) (2013) 611–622. H. Li, S.A. Vasquez, H. Punekar, et al., Prediction of boiling and critical heat flux using an Eulerian multiphase boiling model, in: ASME 2011 International Mechanical Engineering Congress and Exposition, Colorado, USA, 2011, pp. 463–476. T. Nishimura, Y. Ohori, Y. Kawamura, Flow characteristics in a channel with symmetric wavy wall for steady flow, J. Chem. Eng. Jpn. 17 (5) (1984) 466–471.
141
[21] V. Stevanovic, M. Studovic, A simple model for vertical annular and horizontal stratified two-phase flows with liquid entrainment and phase transitions: onedimensional steady state conditions, Nucl. Eng. Des. 154 (3) (1995) 357–379. [22] T. Saito, E.D. Hughes, M.W. Carbon, Multi-fluid modeling of annular two-phase flow, Nucl. Eng. Des. 50 (2) (1978) 225–271. [23] N.I. Kolev, Multiphase Flow Dynamics 2, Springer, Berlin, 2005. [24] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Lectures in Transport Phenomena, Wiley, New York, USA, 1969. [25] A.S. Konikov, A. Modnikova, Experimental study of the conditions under which heat exchange deteriorates when a steam-water mixture flows in heated tubes, Teploenergetika 10 (5) (1966) 49–52. [26] A. Ghobadian, S.A. Vasquez, A general purpose implicit coupled algorithm for solution of Eulerian multi-phase transport equations, in: International Conference on Multiphase Flow, Leipzig, Germany, 2007. [27] R.J. Mattson, K.G. Condie, S.J. Bengston, et al., Regression analysis of post-CHF flow boiling data, in: Fifth International Heat Transfer Conference, Tokyo, Japan, 1974, pp. 115–119. [28] The Babcock & Wilcox Company, Pressurized Water Reactor Babcock & Wilcox Technology Crosstraining Course Manual, America, 2011, pp. 241–248. [29] F.W. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of the tubular type, Int. Commun. Heat Mass Transfer 12 (1) (1985) 3–22. [30] J.C. Chen, Correlation for boiling heat transfer to saturated fluids in convective flow, Ind. Eng. Chem. Process Des. Dev. 5 (3) (1966) 322–329. [31] X.J. Chen, L.X. Chen, D.F. Zhou, et al., The Basis of Liquid-vapor Two-phase Flow and Heat Transfer, Science Press, Beijing, 1995, pp. 190–220 (in Chinese).
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical simulation of dryout and post-dryout heat transfer in a straight-pipe once-through steam generator Jianxin Shi a, Baozhi Sun a,⇑, Wenjing Han b, Guolei Zhang a,⇑, Yanjun Li a, Longbin Yang a a b
College of Power and Energy Engineering, Harbin Engineering University, 145 Nantong Street, Harbin 150001, PR China Tianjin Center for patent review of the State Intellectual Property Office, Hongshun Street, Tianjin 300302, PR China
h i g h l i g h t s Two-fluid three-flow-field model is developed to predict dryout in steam generator. The empirical correlation is used to correct dryout criterion. The interactions between three-flow-fields and the wall are considered. Dryout and post-dryout heat transfer mechanisms are discussed through the results.
a r t i c l e
i n f o
Article history: Received 2 March 2016 Revised 19 May 2016 Accepted 24 May 2016 Available online 24 May 2016 Keywords: Once-through steam generator Vertical pipe Flow boiling Dryout Heat transfer characteristic
a b s t r a c t Accurately predicting dryout and post-dryout heat transfer characteristics is critical for proper design of once-through steam generators. This paper provides a reasonable and simple method for this prediction by introducing a two-fluid, three-flow-field mathematical model and improving the dryout criterioncritical quality, and conducts a numerical simulation of dryout and post-dryout heat transfer in a once-through steam generator to prove the model’s performance. The results show that the critical quality in a once-through steam generator is about 0.82, with the heat transfer capacity significantly reducing and the wall temperature sharply increasing in a non-linear form by approximately 30 K when dryout occurs. Part of the steam is superheated in the post-dryout region, resulting in a deviation from thermodynamic equilibrium between the vapor and liquid phases. Dryout and post-dryout heat transfer in the once-through steam generator operate between complete deviation from thermodynamic equilibrium and complete thermodynamic equilibrium. Therefore, the presence of droplets has a significant influence on the mass, momentum and energy transfer between the film and vapor phases. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Compared to traditional natural circulation steam generators, a once-through steam generator (OTSG) used in the nuclear power plant with pressurized water reactor (PWR) generates superheated steam at the outlet, which provides the design good economical efficiency, compactness, modular construction, high power generation efficiency and good thermal heating effect. Because of these advantages, more and more researchers have focused on studying OTSG in recent years. In a OTSG, the working fluid in the secondary side experiences a complex vapor-liquid two-phase flow and heat ⇑ Corresponding authors. E-mail addresses: [email protected] (J. Shi), [email protected] (B. Sun), [email protected] (W. Han), [email protected] (G. Zhang), liyanjunhrb@ 163.com (Y. Li), [email protected] (L. Yang). http://dx.doi.org/10.1016/j.applthermaleng.2016.05.145 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
transfer process that includes preheating and boiling until overheating. Deterioration in the heat transfer occurs when the steam quality reaches a certain value, which has a major influence on the safe and reliable operation of the equipment. This heat transfer deterioration can be divided into two types according to its cause: (1) DNB (departure from nucleate boiling): DNB occurs when the heat flux exceeds the critical heat flux such that the flow pattern changes from nucleate boiling to film boiling. This means that bubbles near the wall are generated too late to spread to the main stream, causing the wall to be covered with a vapor film. (2) Dryout: This phenomenon occurs when the steam quality reaches a sufficiently high value, that the adherent annular liquid film is torn by the steam—that is, the flow pattern changes from an annular flow to a mist flow [1,2].
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
133
Nomenclature t
a q
r !
U
Cl Cd SE SD p g F vd F!v l F lift
s sRe h q qv l qv d
vc A qwv qlv qwl
time (s) volume fraction density (kg/m3) gradient velocity vector (m/s) mass transfer rate between liquid film and steam, kg/(m3s) mass transfer rate between droplets and steam (kg/(m3 s)) droplet entrainment rate (kg/(m3 s)) droplet deposition rate (kg/(m3 s)) pressure (MPa) gravity (m/s2) drag force between steam and droplets (N/m2) drag force between steam and liquid film (N/m2) buoyancy lift (N/m2) shear stress (N/m2) Reynolds stress (N/m2) enthalpy (kJ/kg) heat flux (W/m2) heat flux per unit volume between steam and its interface with the liquid film (W/m3) heat flux per unit volume between steam and its interface with the droplets (W/m3) heating perimeter (m) flow area (m2) heat flux between wall and steam (W/m2) heat flux per unit volume between liquid film and its interface with the steam (W/m3) heat flux between wall and liquid film (W/m2)
The heat transfer deterioration that occurs during operation of a OTSG is dryout. Liquid films disappear when dryout occurs as the wall is no longer wetted in this state (i.e., there is no liquid contact with the wall), which results in a significant reduction in the surface heat transfer coefficient and a sharp rise in wall temperature. If effective control of this situation is not taken, wall temperature may exceed the maximum allowable limit, posing a security threat to running equipment. To describe this phenomenon, researchers have conducted a great deal of research and proposed two different models: a liquid film thickness model and a two-fluid three-flowfield model. Whalley et al. [3] proposed a liquid film thickness model based on a flow analysis approach, used the model to calculate the liquid film flow rate of an unbalanced flow (such as evaporation flow) through a uniformly heated straight pipe, then extended the model to predict the dryout and post-dryout heat transfer phenomenon. Azzopardi [4] studied the impact of the initial entrainment fraction and mass flow rate on dryout position for an electrically heated vertical tube using the liquid film thickness model. Chong et al. [5] modified the model in [4] and used it to predict dryout in a serpentine reboiler passage. Adamsson and Anglart [6] combined the liquid film thickness model with a sub-channel model of a nuclear reactor fuel assembly and captured the effect of shaft power distribution on dryout position. However, it is worth noting that the liquid film thickness model only considers the steam and the liquid film when predicting the dryout and post-dryout heat transfer, and there is presently no mature film thickness measurement method. Thus, it is difficult to verify the model. At the same time, some scholars have instead turned to conducting OTSG simulations by simplifying the flow boiling process. Li et al. [7] adopted a moving boundary lumped parameter method to investigate the change of inlet and outlet
qdv qwd qwil qwid d u Cl f d
m Dh k Nu T Pr x z
heat flux per unit volume between droplets and their interface with the steam (W/m3) heat flux between wall and droplets (W/m2) heat flux between wall and the interface of the steam with the liquid film (W/m2) heat flux between wall and the interface of the steam with the droplets (W/m2) liquid film thickness (m) velocity (m/s) lift coefficient drag function diameter (m) kinematic viscosity (m2/s) hydraulic diameter (m) thermal conductivity (W/(m K)) Nusselt number temperature (K) Prandtl number steam quality the height of heat transfer tube, m
Subscripts
v
l d i q k w DO
steam liquid film droplet i ¼ v ; l; d q ¼ l; d k ¼ v; l wall dryout
parameters in HTR-10 that considered the preheating, boiling and superheating regions. Li and Ren [8] studied the influence of heating power on the outlet steam temperature and the length of each heat transfer region by using a lumped parameter method to divide the heat transfer region into preheating, boiling and superheating regions based on the moving boundary model. Zhu et al. [9] established a dynamic mathematical model of the preheating, boiling and superheating regions based on lumped parameters and a moving boundary, and then conducted both steady and dynamic simulations. Wang et al. [10] studied the flow and heat transfer characteristics of the preheating and nucleate boiling regions in a OTSG using a two-fluid model and an RPI thermal phase change model built with CFX software. However, this study was limited by the method itself, as the simulation was only concerned with the inlet and outlet parameters of the OTSG and ignored local changes in the axial direction, and there exists dryout, heat transfer deterioration and wall temperature increases in localized regions. Generally speaking, the lumped parameter method will invariably ignore these factors when determining the thermal-hydraulic characteristics of a OTSG. To predict dryout more accurately, Thurgood et al. [11] considered liquid films, droplets, steam flow-fields and the interactions between these states in 1983. Based on their analysis, the group proposed a two-fluid three-flow-field model that used the easily measured steam quality as the dryout criterion, then implemented the model with COBRA-TF software to predict dryout-induced leakage accidents in the primary side cooling system of a nuclear reactor. Hoyer [12] applied a one-dimensional two-fluid threeflow-field model to MONA code and simulated the flow boiling of an electrically heated vertical tube in an experimental database. The results of the simulation showed that the model could accurately predict dryout and post-dryout heat transfer. Hoyer and
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Stepniewski [13] then extended the model for the pipe flow in [12] to predict the dryout between rod bundles in a nuclear reactor. Jayanti and Valette [14] improved the closure correlations of the two-fluid three-flow-field model and extended the range of the model’s application to high pressure systems (3–20 MPa), then analyzed the impact of spacer grids on dryout position and wall temperature in nuclear reactor systems using the model [15]. Ha et al. [16] applied the three-dimensional two-fluid three-flowfield model to SPACE code to simulate the flow boiling process in an electrical heating tube—except for single-phase steam convection, observing trends in the wall temperature in the post-dryout region. The numerical simulations in the above studies were mostly conducted based on electric heating experiments. That is, they applied a constant heat flux boundary to the wall and could not simulate the actual heat transfer conditions for the flow boiling process, which in turn is closely related to dryout and postdryout heat transfer. To make up for this lack, we use a OTSG operated by B&W as a prototype, introduce a two-fluid three-flow-field model mathematical model and improve it so that it can properly predict dryout in an actual running OTSG, then conduct a numerical simulation to obtain the dryout and post-dryout heat transfer characteristics for a straight-pipe OTSG.
2. Model development 2.1. Mathematical model As flow boiling develops in the secondary side, steam quality gradually increases. When dryout occurs, the steam becomes the continuous phase, with liquid only existing in the form of a continuous liquid film and discrete droplets entrained in the steam. There is also a mass transfer between the liquid film and the steam due to droplet entrainment, droplet deposition and evaporation of the liquid film. The liquid film slowly thins under the influence of this entrainment, deposition and vaporization [17]. If the velocity with which the droplets deposit onto the liquid film is less than the rate of the evaporation and entrainment of the liquid film, the liquid film will partially break. If this occurs, the coverage of the liquid film on the wall will gradually decrease, with the steam eventually occupying the entire flow path and causing dryout. The existence of droplets has a significant influence on the mass, momentum and energy transfer between the liquid film and the continuous steam, so it is necessary to consider all interactions between the droplets and the other flow fields (steam and liquid film) and the heat transfer between the droplets and the wall. With this model, the heat can transfer from the wall to the fluid through six paths: (1) from the wall to droplets that hit the wall; (2) from the wall to droplets that enter into the thermal boundary layer but do not wet the wall; (3) from the wall through convection heat to the steam; (4) from the steam to droplets suspended in the steam core (convective heat transfer); (5) radiative heat transfer from the wall to droplets; and (6) radiative heat transfer from the wall to the steam. The 6 heat transfer processes is described in Fig. 1 schematically. Of these heat transfer paths, (1) and (2) are usually taken into account together, while (5) and (6) only work in the case of high wall superheat, which is usually negligible in a OTSG. A traditional two-fluid model is only able to describe the flow and heat transfer behavior of a continuous liquid and dispersed bubbles. However, as we can see from the breakdown of possible heat transfers, there exist interactions between three flow-fields in the dryout and post-dryout region. Therefore, this paper introduces a two-fluid three-flow-field mathematical model to describe the behavior of flow and heat transfer in the dryout and post-dryout region [10,11,13,14,18–20]. The model establishes mass, momentum
Fig. 1. 6 heat transfer processes in post-dryout region.
and energy balance equations for the liquid film, droplets and steam, respectively. Mass balance equation for the steam: ! @ ðav qv Þ þ r ðav qv U v Þ ¼ Cl þ Cd @t
ð1Þ
Mass balance equation for the liquid film: ! @ ðal ql Þ þ r ðal ql U l Þ ¼ Cl SE þ SD @t
ð2Þ
Mass balance equation for the droplets: ! @ ðad qd Þ þ r ðad qd U d Þ ¼ Cd þ SE SD @t
ð3Þ
Momentum balance equation for the steam: !
! ! ! ! @ðav qv U v Þ þ r ðav qv U v U v Þ ¼ av rp þ av qv ~ g F v d ðU v U d Þ @t !
!
!
F v l ðU v U l Þ F lift þ r av ðsv þ sRe v Þ !
!
!
!
þ Cl ðU l U v Þ þ Cd ðU d U v Þ ð4Þ Momentum balance equation for the liquid film: !
! ! ! ! @ðal ql U l Þ þ r ðal ql U l U l Þ ¼ al rp þ al ql~ g F v l ðU l U v Þ @t ! ! þ r al ðsl þ sRe l Þ Cl ðU l U v Þ !
!
þ SD ðU l U d Þ
ð5Þ
Momentum balance equation for the droplets: !
! ! ! ! @ðad qd U d Þ þ r ðU d U d Þ ¼ ad rp þ ad qd~ g F v d ðU d U v Þ @t ! ! ! F lift þ r ad ðsd þ sRe d Þ Cd ðU d U v Þ !
!
þ SE ðU l U d Þ
ð6Þ
Energy balance equation for the steam: ! @ @p ðav qv hv Þ þ r ðav qv hv U v Þ av @t @t
v
¼ Cl hv þ Cd hv þ qv l þ qv d þ c qwv A
ð7Þ
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Energy balance equation for the liquid film:
Energy transfer between wall and flow-field:
@ @p ðal ql hl Þ þ r ðal ql hl U l Þ al @t @t !
¼ Cl hv þ SD hd SE hl þ qlv þ
qwk ¼
vc A
ð8Þ
qwl
k
@ @p ðad qd hd Þ þ r ðad qd hd U d Þ ad @t @t A
ð9Þ
qwd
In the above equations, the volume fractions satisfy
X
ai ¼ 1; 0 6 ai 6 1
ð10Þ
i
2.2. Closure correlations There are numerous source terms in the above equations, so it is necessary to make the equations closed. The closure correlations mainly include the transfer between wall and flow-fields, and the transfer between three flow-fields. The mass transfer between the flow-fields includes thermal phase change terms (Cl and Cd ) and hydraulic terms (SE and Sd ). The thermal phase change terms are obtained by evaporation and heat transfer at the steam-liquid film interface or the steamdroplet interface. The droplet entrainment rate and droplet deposition rate in hydraulic terms is the droplet mass taken away by the steam when the annular liquid film is torn by the steam and the droplet mass that the annular liquid film deposits the droplets entrained in the steam, respectively. The thermal phase change correlations are as follows:
vc =A qlv qv l
Cl ¼
qwil
Cd ¼
qwid
ð11Þ
hv hl
vc =A qdv qv d
ð12Þ
hv hd
The droplet entrainment rate is calculated according to [21]:
SE ¼ 1:1 104 d2:25 qd
ð13Þ
The droplet deposition rate is calculated according to [22]:
Sd ¼ kd C H
ð14Þ CH ¼ a
where
ad qd ud 6
kd ¼ 1:1474 102 1:2854
and
d qd ud þav qv uv
10 C H þ 1:0150 10 4:2501 109 C 3H þ 6:819 1012 C 4H . The momentum transfer between the flow-fields mainly includes the lift force and the drag force [23]: 4
!
!
C 2H
!
!
F lift ¼ C l qv ad ðU v U d Þ ðr U v Þ
ð15Þ
F v q ¼ K v q ðU v U q Þ
ð16Þ
!
!
2
where K v q ¼ 6qsvvf dv Ai , sv ¼ 18dvmv . Energy transfer between the flow-fields:
qv l ¼
the above correlation is given by a forced convective heat transfer, !
!
vc
ð19Þ
0:4 jU k jDh where Nuwk ¼ 0:023Re0:8 . k Pr k , Rek ¼ m
Energy balance equation for the droplets:
¼ Cd hd SD hd þ SE hl þ qdv þ
kk Nuwk ðT w T k Þ Dh
1 kv Nuv l ðT sat T v Þ Dh
Note that the dryout standard is a key parameter that will decide when dryout occurs, except for the above key closure correlations. The soaring rate of the wall temperature is usually used as the dryout criterion in experiments, while the critical heat flux, mass flow rate of the liquid film and the steam quality are used as the dryout criterion in numerical research. The two-fluid three-flow-field mathematical model often determines the occurrence of dryout by steam quality, and the default value is constant. The critical quality is related to the category of device and its operating conditions in actual cases. Thus, this paper applies a correlation of critical quality suitable for the operating conditions of B&W’s steam generator to determine the occurrence of dryout. This is determined by [25]: 1
0:07 np
xDO ¼ mq8 ðquv Þ3 do 1
e
ð20Þ
with the following operating parameters: p = 7–14 MPa; m = 46.0; n = 0.0255; G = 350–700 kg/(m2 s); and q = (2.3–5.8) 105 W/m2. The inlet Reynolds number calculated by the given boundary conditions and the geometric parameters of the physical model is about 31,098. The Reynolds number will increase along the flow direction of the secondary side in a OTSG because of the rising velocity and the decreasing kinematic viscosity with the development of the flow boiling. We can find that the flow process belongs to turbulent flow. The corresponding turbulence model is described in [18]. 2.3. Physical model and grid system A OTSG has a huge volume and possesses a large number of heat transfer tubes that have a thin tube wall and small tube pitch. The flow path and heat transfer process of the secondary side in a running OTSG is extremely complex, making it difficult to achieve a full-sized numerical simulation. To reflect an actual working process, this paper uses B&W’s operational OTSG as the prototype, but simplifies that prototype by assuming that the heat absorbed by feed water in the annular channel ultimately is provided by the primary side. This simplifies the process in that the feed water flow into the region is from the lower tube sheet and is heated directly through the heating role of the primary coolant. Other key geometric parameters such as diameter, tube pitch, height and other operating conditions are the same as the prototype. The arrangement of the heat transfer tubes is an equilateral triangle, with a series of tube support plates arranged in the secondary side. The tube support plates only affect the local thermalhydraulic characteristics and do not affect the overall flow and heat transfer characteristics of the OTSG, so this paper does not consider the tube support plates [18]. However, this is something we expect to include into the model for subsequent studies. The simplified physical model is shown in Fig. 2, with the key geometric parameters for the model given in Table 1.
ð17Þ
pD h
2.4. Boundary condition
the above correlation is given by a forced convective heat transfer, !
0:4 jU v jDh where Nuv l ¼ 0:023Re0:8 v Pr v , Rev ¼ mv , and
qv d ¼
6ad kv Nuv d ðT sat T v Þ dd dd
ð18Þ
the above correlation is given by a convection law for flow around !
!
1=3 jU v U d jd spheres [24], where Nuv d ¼ 2 þ 0:6Re0:5 . v Pr v , Rev ¼ mv
The given boundary conditions are provided in Table 2 along with the actual operating parameters of B&W’s operating OTSG. The heat transfer partitions (single-phase liquid convection, twophase flow boiling, dryout and post-dryout) that the secondary side experiences all have different flow and heat transfer characteristics. The coupled heat transfer between the two loops in the OTSG, and the transition from two-phase flow boiling to dryout,
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Fig. 2. Physical model and grid system.
simulation is converged when the scaled residuals in the control volumes for each of above governing equations are smaller than 105.
Table 1 Geometric parameters. Name
Unit
Value
Tube diameter Tube pitch Height of the tube
mm mm mm
15.875 22.225 9000
4. Results and discussion 4.1. Model validation
Table 2 Boundary conditions. Name
Unit
Value
Inlet mass flow Inlet temperature Outlet pressure Outlet saturation temperature Heat flux
kg/s K MPa K W/m2
0.1312 510.95 6.38 552.77 Formula (21)
add significant difficulties to the numerical simulation. Therefore, this paper replaces the heating role of the primary side with the second boundary condition. In order to simulate the actual heat transfer process, we add the heat flux function (21) to the wall based on theoretical calculations.
8 > < 145200 þ 3900z; z 6 2 q ¼ 48400 þ 2728z; 2 6 z 6 7:5 > : 19535 þ 990z; z P 7:5
ð21Þ
For this work, we conducted a simulation using the experimental data obtained from [12]-who conducted dryout and postdryout heat transfer experiments in a uniform electrically heated vertical tube—to verify the rationality and reliability of the introduced mathematical model in predicting dryout and post-dryout heat transfer. The experimental conditions used for the test were: inside tube diameter 10 mm, pressure 7 MPa, mass flow 1495 kg/(m2 s) and heat flux 797 kW/m2. We can see from Fig. 3 that the maximum error for the model is about 4.68%, which occurs at 6.5 m. This shows that the feasibility and rationality of the mathematical model is sound. We first performed a grid independent solution verification, shown in Fig. 3, before attempting authentication of the physical model in [12]. We found that there was substantially no change in the output of the model when the number of cells in the grid went beyond 32,000 and so the number of grid computing cells used for the remainder of this work was set to 32,000. Fig. 3 also shows the dryout location, the experimental and numerical results for the inner
3. Numerical methods The above governing equations are used to predict the flow boiling process of the secondary side in a OTSG, which contains the single-phase liquid convective heat transfer (the inlet water is in a subcooled state at a given pressure and mass flow), twophase flow boiling, dryout and post-dryout heat transfer. The numerical simulation is conducted based on steady-state mode, the governing equations are discretized by the finite volume method and the Coupled Implicit algorithm [26] is used to solve all equations for phase velocity corrections and shared pressure correction simultaneously, the fully implicit scheme is used to deal with the transfer between flow-fields, the closed correlations. The flux difference splitting scheme is used in the spatial discretization. The local time stepping and the implicit residual smoothing technique are also used in steady-state calculation. The numerical
Fig. 3. Model validation.
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Fig. 4. Grid independent validation.
Fig. 6. The axial distribution of the steam quality.
Fig. 5. The axial distribution of the vapor volume fraction.
wall temperature distribution. Our calculations show that the maximum error in the axial tube, which occurred at approximately 6.5 m along the tube, was 4.68%, which was within the allowed range. These results show that the use of a two-fluid three-flowfield mathematical model and method can accurately predict dryout and post-dryout heat transfer characteristics within a thin-walled tube, substantiating the feasibility of this numerical simulation to model a OTSG’s dryout and post-dryout heat transfer. 4.2. Grid independent validation of a OTSG Since the numerical results of a simulation heavily depend on the quality of the mesh and the quantity of elements in the grid, it is necessary to verify the grid independence before attempting a numerical simulation of dryout in a OTSG. The results of this validation test, shown in Fig. 4, confirm that the results do not change as a function of number of elements in the grid beyond 344,000 elements. Therefore, we used a grid containing 410,000 elements for this work. 4.3. Heat transfer analysis of OTSG The dryout in steam generator is a risk for the nuclear fuel rods in the nuclear reactor, which affects the safe and reliable operation of the nuclear power plant with PWR and results in an outage of the plant. Therefore, an in-depth study of dryout in an actual running OTSG—i.e. a study that analyzes changes in steam quality,
Fig. 7. The distribution contour of the vapor volume fraction.
fluid temperature, wall temperature and surface heat transfer coefficient when dryout occurs—can provide a reference for identifying the burning problem of heat transfer tubes. 4.3.1. Variation of steam quality The steam in a heat transfer tube will tear the liquid film on the wall of the tube when the steam quality reaches a certain value during OTSG operation, such that the wall is in contact with the steam and will cause dryout. Figs. 5 and 6 show the axial distribution of the vapor volume fraction and the steam quality in the secondary side, while Fig. 7 depicts the corresponding distribution contour of the vapor volume fraction. As can be seen from Figs. 5 and 6, the steam quality first remains constant before gradually increasing in the axial direction of the heat transfer tubes. At the same time, steam quality is 0 and remains constant up to an axial height of 1.8 m. The theoretical calculated height of the singlephase liquid convective heat transfer region based on the given boundary conditions is 1.86 m, giving a relative error of 3.2%. The
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Fig. 8. The axial distribution of the fluid velocity.
fluid begins to boil as the heating process develops, causing the steam quality to gradually increase to a value of 0.856 at the outlet. The theoretical maximum calculated steam quality was 0.9, giving a relative error of 4.89%. 4.3.2. Fluid velocity of the secondary side Figs. 8 and 9 show the axial distribution of the fluid velocity and corresponding distribution contour. As can be seen from the figures, there is only liquid in preheating section, the liquid velocity is almost constant and approximately 0.25 m/s. As flow boiling develops, the bubbles generate in the secondary side and the number of bubbles increases gradually, which results in that the fluid density decreases, therefore the fluid velocity increases. The steam velocity is always greater than the liquid velocity since the vapor density is larger than the liquid density in the secondary side. 4.3.3. Fluid temperature of the secondary side Fig. 10 shows the axial distribution of the fluid temperature in the secondary side and the associated wall temperature. As can be seen from the figure, the fluid first flows into the region in a subcooled condition and is subsequently heated by single-phase heat convection, the fluid temperature rises gradually. Nucleate boiling then begins when the fluid temperature reaches saturation. In this region, bubbles generate sharply on the wall, and the temperature of the liquid and vapor phases are maintained at saturation temperature. With the development of flow boiling, dryout occurs at an axial height of 7.5 m, and part of the steam begins to be superheated. In the dryout and post-dryout region, the flow pattern transforms from an annular flow to a mist flow, while the liquid phase gradually transforms from a continuous liquid film to droplets as its temperature remains saturated. Meanwhile, the continuous vapor phase is heated directly by the wall so that part of the steam is superheated. Note that the steam temperature is still at saturation temperature at the dryout location. However, the steam begins to superheat in the post-dryout region, which results in that vapor-liquid two-phase deviates from thermodynamic equilibrium. This thermodynamic non-equilibrium phenomena does not disappear until the droplets completely evaporate. In this paper, the outlet steam quality is 0.856 and the heat transfer does not involve a single-phase vapor convection region. Thus, the temperature distribution in Fig. 10 agrees with that in Fig. 11 where xc < 1 [1], with the corresponding discussion indicating that there exists an essential difference between nucleate boiling and dryout and post-dryout heat transfer. Meanwhile the results prove the rationality and feasibility of the modified
Fig. 9. The distribution contour of the fluid velocity at different sections.
Fig. 10. The axial distribution of fluid temperature and wall temperature in the secondary side.
model used in this work to simulate the flow and heat transfer process from single-phase liquid convection to dryout in a OTSG. 4.3.4. Wall temperature analysis The wall temperature is a key parameter in OTSG safety. When dryout occurs, the local wall temperature will be much higher than the saturation temperature, causing a deviation from the equilibrium. There are two extreme cases:
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Fig. 11. Variation in steam temperature and wall temperature when dryout occurs [1].
1. Complete deviation from thermodynamic equilibrium. This case generally occurs at low pressure and low mass flow rate. The heat is transferred to the continuous vapor phase contacted with the heated wall completely, which causes steam superheating. Assuming that the heat transfer rate from the steam to the droplets is very slow, we can ignore the existence of the droplets and assume that the heat transferred from the wall is completely used to superheat the steam. In this case, the increase in wall temperature is consistent with the case of single-phase vapor convection and its value can be calculated by the single-phase convection correlation and corresponding heat flux. 2. Complete thermodynamic equilibrium. This case occurs at high pressure and high flow rates. In this case, we assume that the heat transferred from the vapor phase to the droplets is sufficient; that is, the heat is transferred from the vapor phase to the droplets quickly enough that the steam and the droplets maintain the same temperature, one that is equal to the saturation temperature. In other words, the heat transferred through the wall is used to completely evaporate the droplets until all droplets are evaporated, and only then is the heat re-used to superheat the steam. In this case, the wall temperature may appear different non-linear changes, and will reach a maximum value before declining slightly. The main reason for this result is that the droplets hit the wall [27]. Fig. 12 shows the temperature variations for the two extreme cases. Note, however, that the actual change in wall temperature actually seen in application is usually between these two extreme cases. Therefore, it is difficult to determine and describe the temperature trends in the dryout and post-dryout regions and also complex to model the process. As such, it is important to note that this paper conducts a preliminary exploration of the changes in wall temperature when dryout occurs in a OTSG. The heat flux and fluid temperature in the secondary side have a significant effect on wall temperature. Fig. 10 shows that wall temperature rises as fluid temperature slowly rises in the single-phase liquid convective heat transfer region. As the fluid enters the nucleate boiling region, its temperature remains at the saturation temperature, while the wall temperature slowly rises and is higher than the saturation temperature of the secondary side fluid due to the increasing heat flux and heat transfer coefficient. As flow boiling develops, the flow pattern is converted from a slug flow to an annular flow, and the liquid film on the wall is torn under the multiple effects of deposition, entrainment and evaporation.
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Fig. 12. Temperature variations for the complete deviation from thermodynamic equilibrium case and the complete thermodynamic equilibrium case.
This also causes a corresponding higher steam quality, causing dryout to occur at this position. The wall contacts directly with the steam, which greatly reduces heat transfer capacity and results in a sharp rise in wall temperature. The maximum amplitude of the temperature change is about 30 K in the simulation, whereas
Fig. 13. A comparison of steam quality and wall temperature along the heat transfer tube’s axial length.
Fig. 14. A comparison of wall temperature and heat transfer coefficient along the heat transfer tube’s axial length.
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Table 3 Heat transfer coefficient comparison between numerical results and empirical correlations. Parameter Surface heat transfer coefficient
Heat transfer partition Preheating Nucleate boiling Dryout and post-dryout
Empirical correlation Dittus-Boelter [29] Chen [30] Remizov [31]
the actual maximum amplitude of the temperature change in the B&W data is 27 K [28], giving a relative error of 11.1%. In the post-dryout region, the fluid exists in the form of vapor entrained droplets. Since the convective heat transfer coefficient is small and thermal resistance is large in this region, the wall temperature continues to increase, albeit at a slow rate. We can see by comparing Figs. 10 and 12 that dryout in the OTSG is between the complete deviation from thermodynamic equilibrium case and the complete thermodynamic equilibrium case. Therefore, the presence of droplets has a considerable influence on the momentum and energy transfer between the film and the steam phase. Thus, it is necessary to consider the existence of the droplets, all interactions between the droplets and other flow-fields (steam and liquid film) and the heat transfer between the droplets and the wall in the numerical simulation to achieve accurate results. 4.3.5. Comparison of key parameters Fig. 13 shows a comparison of steam quality and wall temperature along the axial direction, revealing the relationship between these key dryout parameters in a OTSG. As can be seen from the figure, the wall temperature begins to soar in a non-linear form when steam quality reaches about 0.82. It is at this position that dryout occurs, and the heat transfer transforms from two-phase flow boiling to dryout. Fig. 14 shows a comparison of wall temperature and heat transfer coefficient along the heat transfer tube’s axial length. Table 3 shows a comparison of the heat transfer coefficients derived from the numerical results against widely used empirical correlations. The numerical results were calculated by integrating the surface heat transfer coefficient along the whole heat transfer area of each heat transfer partition and then averaging the integral value. We can see from Fig. 14 and the table that the heat transfer in the preheating region results from a convective heat transfer between wall and liquid, which means that the surface heat transfer coefficient is small. When entering the nucleate boiling region, a large number of bubbles generate near the wall, and the disturbance caused by the bubbles greatly enhances heat transfer, causing the surface heat transfer coefficient to sharply increase at this region. At the same time, we can see that dryout occurs at an axial height of 7.5 m, where the liquid film on the wall disappears and there is no liquid wetting the wall at this position. Thus, the surface heat transfer coefficient shows a sharp decline at this point and wall temperature rises sharply. In the post-dryout region, the wall surface is covered with a continuous steam. Since the steam heat transfer coefficient is much smaller than the heat transfer coefficient of the liquid, the heat transfer capacity in this region is weak and the surface heat transfer coefficient is very small. We can see that the maximum relative error is 17.9%, which is within the allowable design range for engineering, and proves the accuracy of the numerical simulation. 5. Conclusion (1) This paper introduced a two-fluid three-flow-field mathematical model and used empirical correlation to correct the dryout criterion. The results from our simulation showed
Unit 2
W/(m K)
Numerical results
Empirical results
Relative error (%)
3693.27 67410.99 1893.39
3133.14 60060.12 2128.26
17.9 12.2 11.0
that the model can be used to accurately predict the dryout and post-dryout heat transfer phenomenon of an actual OTSG. (2) In the simulation, the critical quality was about 0.82, with dryout occurring at an axial height of 7.5 m. The wall temperature at this point rose by about 30 K, followed by superheating part of the steam in the post-dryout region. The average surface heat transfer coefficients in the heat transfer partitions were in good agreement with empirical results. (3) Part of the steam in the post-dryout region is superheated, resulting in a deviation from thermodynamic equilibrium between the vapor and liquid. This phenomenon does not disappear until all droplets are evaporated. Dryout in the OTSG exists between two extreme cases—complete deviation from thermodynamic equilibrium and complete thermodynamic equilibrium—with the presence of droplets having a considerable influence on momentum and energy transfer between the film and vapor phases. Thus, it is necessary to consider the interactions between the droplets and other flow-fields, between the droplets and the wall to produce accurate simulation results.
Acknowledgements This work was sponsored by the National Natural Science Foundation of China (No. 51479040, No. 51579048), the Natural Science Foundation of Heilongjiang Province of China (No. E201422, No. E201346), and Postdoctoral Research Startup Foundation of Heilongjiang Province of China (No. LBH-Q14036), which we gratefully acknowledge. References [1] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, Oxford University Press, New York, 1981. [2] S. Talebi, H. Kazeminejad, H. Davilu, Prediction of dryout and post-dryout wall temperatures using film thickness model, Nucl. Eng. Des. 244 (2012) 73–82. [3] P.B. Whalley, P. Hutchinson, G.F. Hewitt, Calculation of critical heat flux in forced convection boiling, in: Fifth International Heat Transfer Conference, Tokyo, Japan, 1974, pp. 290–294. [4] B.J. Azzopardi, Prediction of dryout and post-burnout heat transfer with axially non-uniform heat input by means of an annular flow model, Nucl. Eng. Des. 163 (1) (1996) 51–57. [5] L.Y. Chong, B.J. Azzopardi, D.J. Bate, Calculation of conditions at which dryout occurs in the serpentine channels of fired reboilers, Chem. Eng. Res. Des. 83 (4) (2005) 412–422. [6] C. Adamsson, H. Anglart, Influence of axial power distribution on dryout: filmflow models and experiments, Nucl. Eng. Des. 240 (6) (2010) 1495–1505. [7] H. Li, X. Huang, L. Zhang, A lumped parameter dynamic model of the helical coiled once-through steam generator with movable boundaries, Nucl. Eng. Des. 238 (7) (2008) 1657–1663. [8] Y.Q. Li, T.J. Ren, Moving boundary modeling study on supercritical boiler evaporator: by using enthalpy to track moving boundary location, in: Power and Energy Engineering Conference, Asia-Pacific, 2009, pp. 1–4. [9] J.Y. Zhu, Z.J. Zhang, Y. Guo, Dynamic real-time simulation research on oncethrough steam generator with concentric annuli tube, Atomic Energy Sci. Technol. 45 (8) (2011) 937–942 (in Chinese). [10] C. Wang, T. Cong, S. Qiu, et al., Numerical prediction of subcooled wall boiling in the secondary side of SG tubes coupled with primary coolant, Ann. Nucl. Energy 63 (2014) 633–645. [11] M.J. Thurgood, J.M. Kelly, T.E. Guidotti, et al., COBRA/TRAC – A ThermalHydraulics Code for Transient Analysis of Nuclear Reactor Vessels and Primary
J. Shi et al. / Applied Thermal Engineering 105 (2016) 132–141
[12] [13]
[14]
[15]
[16] [17] [18]
[19]
[20]
Coolant Systems: Equations and Constitutive Models, Nuclear Regulatory Commission, Washington DC, 1983, pp. 21–38. N. Hoyer, Calculation of dryout and post-dryout heat transfer for tube geometry, Int. J. Multiphase Flow 24 (2) (1998) 319–334. N. Hoyer, M. Stepniewski, Dryout prediction in bundle geometry, in: Ninth International Topical Meeting on Nuclear Reactor Thermal Hydraulics, San Francisco, California, 1999, pp. 378–389. S. Jayanti, M. Valette, Prediction of dryout and post-dryout heat transfer at high pressures using a one-dimensional three-fluid model, Int. J. Heat Mass Transfer 47 (22) (2004) 4895–4910. S. Jayanti, M. Valette, Calculation of dry out and post-dry out heat transfer in rod bundles using a three field model, Int. J. Heat Mass Transfer 48 (9) (2005) 1825–1839. S.J. Ha, C.E. Park, K.D. Kim, et al., Development of the space code for nuclear power plants, Nucl. Eng. Technol. 43 (1) (2011) 45–62. J. Weisman, B.S. Pei, Prediction of critical heat flux in flow boiling at low qualities, Int. J. Heat Mass Transfer 26 (10) (1983) 1463–1477. B. Sun, Y. Yang, Numerically investigating the influence of tube support plates on thermal-hydraulic characteristics in a steam generator, Appl. Therm. Eng. 51 (1) (2013) 611–622. H. Li, S.A. Vasquez, H. Punekar, et al., Prediction of boiling and critical heat flux using an Eulerian multiphase boiling model, in: ASME 2011 International Mechanical Engineering Congress and Exposition, Colorado, USA, 2011, pp. 463–476. T. Nishimura, Y. Ohori, Y. Kawamura, Flow characteristics in a channel with symmetric wavy wall for steady flow, J. Chem. Eng. Jpn. 17 (5) (1984) 466–471.
141
[21] V. Stevanovic, M. Studovic, A simple model for vertical annular and horizontal stratified two-phase flows with liquid entrainment and phase transitions: onedimensional steady state conditions, Nucl. Eng. Des. 154 (3) (1995) 357–379. [22] T. Saito, E.D. Hughes, M.W. Carbon, Multi-fluid modeling of annular two-phase flow, Nucl. Eng. Des. 50 (2) (1978) 225–271. [23] N.I. Kolev, Multiphase Flow Dynamics 2, Springer, Berlin, 2005. [24] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Lectures in Transport Phenomena, Wiley, New York, USA, 1969. [25] A.S. Konikov, A. Modnikova, Experimental study of the conditions under which heat exchange deteriorates when a steam-water mixture flows in heated tubes, Teploenergetika 10 (5) (1966) 49–52. [26] A. Ghobadian, S.A. Vasquez, A general purpose implicit coupled algorithm for solution of Eulerian multi-phase transport equations, in: International Conference on Multiphase Flow, Leipzig, Germany, 2007. [27] R.J. Mattson, K.G. Condie, S.J. Bengston, et al., Regression analysis of post-CHF flow boiling data, in: Fifth International Heat Transfer Conference, Tokyo, Japan, 1974, pp. 115–119. [28] The Babcock & Wilcox Company, Pressurized Water Reactor Babcock & Wilcox Technology Crosstraining Course Manual, America, 2011, pp. 241–248. [29] F.W. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of the tubular type, Int. Commun. Heat Mass Transfer 12 (1) (1985) 3–22. [30] J.C. Chen, Correlation for boiling heat transfer to saturated fluids in convective flow, Ind. Eng. Chem. Process Des. Dev. 5 (3) (1966) 322–329. [31] X.J. Chen, L.X. Chen, D.F. Zhou, et al., The Basis of Liquid-vapor Two-phase Flow and Heat Transfer, Science Press, Beijing, 1995, pp. 190–220 (in Chinese).