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Thermal-hydraulic analysis code development for sodium heated once-through steam generator
Annals of Nuclear Energy 127 (2019) 385–394
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Thermal-hydraulic analysis code development for sodium heated once-through steam generator Rongshuan Xu, Dalin Zhang ⇑, Wenxi Tian, Suizheng Qiu, G.H. Su State Key Laboratory of Multiphase Flow in Power Engineering, Shaanxi Key Lab. of Advanced Nuclear Energy and Technology, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 14 August 2018 Received in revised form 16 December 2018 Accepted 18 December 2018
Keywords: Once-through steam generator Sodium cooled fast reactor Computer code Thermal-hydraulic model Transient analysis
a b s t r a c t Sodium heated once-through steam generator (OTSG) is an essential component of Sodium cooled Fast Reactor (SFR). It transfers the heat from hot sodium to water and converts water to superheat steam. Steam can roll the turbine and generate electricity. Therefore, the safety of the steam generator is vital to the operation of the plant. China is developing the China Demonstration SFR (CDSFR) after successful construction of China Experiment Fast Reactor (CEFR). In order to make sure the safety of the steam generator under various transient conditions and support the design of the steam generator for CDSFR, a Transient analysis Code of Once-through Steam generator for Sodium cooled fast reactor (TCOSS) has been developed. To benchmark the developed physical model, the calculated results of the code have been compared with some design data of OTSGs and they are found in a good agreement. In addition, the transient results of the code have been compared with the experimental results of a shutdown experiment for the validation of the code. The predictions of the code agree well with the experimental data. Hence, the code can be utilized to predict the variations of thermal-hydraulic parameters in steam generator under different transient conditions. In addition, predictions can support the design of the steam generator for CDSFR. Ó 2018 Published by Elsevier Ltd.
1. Introduction Sodium heated once-through steam generator plays a critical role in Sodium-cooled Fast Reactor (SFR) plant. SFR is one of six potential reactor candidates for generation Ⅳ nuclear systems. It adopts liquid sodium as coolant. In sodium cooled fast reactors, the heat generated from the core is absorbed by primary sodium and transferred to secondary sodium through the intermediate heat exchanger. Then the secondary sodium transfers heat to water/steam in the Steam Generator (SG). The once through steam generators of SFR can be classified into two kinds, the integral type, and the modular type. They are all counter current, with oncethrough-type shell and tube-heat-exchanger having straight vertical tubes (Nandakumar et al., 2012). The modular type steam generator contains two vertical counter-current shell and tube heat exchangers, one called evaporator, the other called superheater. In modular type steam generator, sodium flows in the shell side while water/steam flows in the tube side. Heated sodium from intermediate heat exchanger enters into the bottom of superheater. Then the sodium flows upwards in ⇑ Corresponding author. E-mail address: [email protected] (D. Zhang). https://doi.org/10.1016/j.anucene.2018.12.027 0306-4549/Ó 2018 Published by Elsevier Ltd.
the shell side then enters into the top of evaporator. After that, it flows downwards in the shell side and exits the evaporator via the bottom outlet plenum in the end. Feed water enters into the bottom of evaporator. The water flows upwards in the tubes, then enter into the top of superheater. After that, it flows downwards in the tubes and exits the superheater via the bottom outlet plenum. As showed in Fig. 1 (Zhong et al., 2017), the steam generator of China Experiment Fast Reactor (CEFR) is modular type steam generator. The integral type steam generator is a vertical countercurrent shell and tube heat exchanger. In integral type steam generator, hot sodium enters into the top of SG, flows downwards in the shell side and feed water enters into the bottom of SG, flows upwards in the tubes. Feed water gets heated to transfer into superheated steam and is to roll the turbine. As Fig. 2 shows, the steam generator of India Commercial Fast Breeder Reactor (CFBR) is an integral type steam generator (Nandakumar et al., 2012). China is developing the China Demonstration SFR (CDSFR) after successful construction of CEFR. And the steam generator of CDSFR is a modular type once-through steam generator. Steam generator is a critical component for the operation and safety of SFR. The thermal-hydraulic analysis of steam generator in various transient conditions is crucial in the design stage of steam generator. The large change of thermodynamic parameters in transient conditions
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Fig. 1. Once-through steam generator of CEFR. Fig. 2. Once-through steam generator of CFBR.
would be harmful to the safety of steam generator. In order to study the thermodynamic parameters variations of different conditions in the steam generator and make the design of SFR steam generator safer, the transient analysis code is needed for sodium heated once-through steam generator. Vaidyanathan had developed the DESign Operational (DESOPT) code for the thermal-hydraulic design of sodium heated oncethrough steam generators (Vaidyanathan et al., 2010). And this code had used to support the design of the Prototype Fast Breeder Reactor (PFBR) (Vinod et al., 2014). Kim and Baek had developed a steady-state code to analyze the thermal-hydraulic performance of once-through steam generator using helically coiled tubes (Kim and Baek, 2011). Sun had developed a thermal-hydraulic design and analysis code for modular type steam generator (Sun et al., 2016). Berry and Tzanos had developed a model of once-through
steam generator with moving boundaries and variable nodes and had presented some typical results (Berry, 1983; Tzanos, 1988). The above design codes could not be used to analyze the transient conditions of the steam generator, and the existing transient codes are not applicable for steam generator with two modules of CDSFR. Therefore, a thermal-hydraulic analysis and transient calculation code for steam generator of CDSFR is needed. In this paper, a one-dimensional transient analysis code TCOSS (Transient analysis Code of Once-through Steam generator for Sodium cooled fast reactor) is developed to simulate sodium heated once-through steam generator. The simulation results were compared with the design data and experimental data of the OTSGs of SFRs, to benchmark the code. This paper will show thermal-hydraulic models and basic verification of TCOSS based on the analysis of numerical
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experiment B1-B and of Energy Technology Engineering Center (ETEC) steam generator shutdown experiment (Tzanos, 1988). The results can support the design of SG for CDSFR.
where x is the equilibrium quality. The equilibrium quality is calculated by the following equation:
x¼ 2. Thermal-hydraulic model In the TCOSS code, the steam generator is simulated by a onedimensional characteristic tube. Water flows inside the tube and sodium flows outside the tube. Along the direction of fluid flowing, the tube can be specified into arbitrary numbers of control volumes averagely. The length and location of the control volume are constant in the calculation. According to the thermal properties of the water/steam in the control volumes, the tube can be divided into four regions: subcooled region, nucleate boiling region, film boiling region and superheated region. The boundaries of the four region are adjusted after every time step. In addition, some assumptions are made as follows in order to simplify the models of the steam generator. (1) Neglecting the axial heat conduction of tube wall and fluid. (2) No phase change in sodium side and sodium is considered as incompressible fluid. (3) Homogenous fluid in each control volume. (4) Neglecting the effect of subcooled boiling. (5) Thermal equilibrium between water and steam phase in boiling regions. (6) Neglecting the energetic changes caused by the changes of gravity and kinetic energy. 2.1. Governing equations Conservation equations for the mass, momentum and energy are used for both sides of the tube. Based on the above assumptions, the one-dimensional conservation equations can be described as follows. Mass conservation equation:
@q @ W ¼0 þ @t @z A
h hf hfg
ð6Þ
where hf is the specific enthalpy of saturated water, J/kg; hfg is the latent heat, J/kg. The following equation is used to simplify the conservation equations:
@ q @ q @h @ q @p þ ¼ @t @h @t @p @t
ð7Þ
Therefore, the simplified conservation equations can be obtained:
@p ¼ 1 @t
1
@h ¼ 1 @t
1
@q @q q @h þ @p
@q @q q @h þ @p
W @ q @h 1 @W @ q qU qA @h @z A @z @h qA
qU @ q W @ q @h 1 @W qA @p qA @p @z qA @z
In the TCOSS code, the steam generator can be represented by one characteristic tube as schematically illustrated in Fig. 3. The subcooled feedwater enters the tube, flows upwards in the tube, gets heated, and finally leaves the tube in a superheated steam state. The hot sodium flows outside the tube in the opposite direction. In the sodium side of the steam generator, there is no phase change but phase change occurs in the water/steam side of the steam generator. Therefore, the tube can be divided into four regions according to the water/steam state and different heat
where q is the density, kg/m3; t is the time, s; W is the mass flow, kg/s; z is the axial coordinate, m; A is the flow area of fluid at the flow direction, m2. Momentum conservation equation:
ð2Þ
where p is the pressure, Pa; f is the friction coefficient; De is the hydraulic diameter, m; g is the acceleration of gravity, m/s2. Energy conservation equation:
q
@h W @h qU @p þ ¼ þ @t A @z A @t
ð3Þ
where h is the specific enthalpy, J/kg; q is the heat flux, W/m2; U is the heated perimeter, m. Homogenous model is used for the two-phase flow. The governing equations are the same as the above equations. The homogenous density is calculated as,
q ¼ qf ð1 - aÞ þ qg a
ð4Þ 3
where qf is the density of liquid water, kg/m ; qg is the density of steam, kg/m3; a is the void fraction. The void fraction is calculated by the following equation:
a¼
x
qg
1 1 ¼ 1x þ 1x 1 þ q x f
qg qf
ð5Þ
ð9Þ
2.2. Heat transfer correlations
ð1Þ
! @ W @ W2 @p fW jW j þ ¼ qg @t A @z qA2 @z 2De qA2
ð8Þ
Fig. 3. Characteristic tube model of SG.
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where Q is the heat transfer rate, W; K is the overall heat transfer coefficient, W/(m2K); A is the heat transfer area, m2; DT is the log mean temperature difference, K. DT and K can be calculated by the following equations,
transfer mechanisms: subcooled region, nucleate boiling region, film boiling region and superheated region. The regions are separated by (1) (2) (3) (4)
H 6 Hf , subcooled region. H > Hf , x 6 xCHF , nucleate boiling region. xCHF < x < 1, film boiling region (Carey, 1992). H P Hg , superheated region.
DT ¼
K¼
1 h1
1 þ h12 ddo þ 2kdow ln ddo i
ð13Þ
i
where h is the convective heat transfer coefficient, W/(m2K); do and di are the outer and inner diameter of the tube, m; kw is the tube thermal conductivity, W/(m2K); the subscript 1 and 2 are represent the sodium side and water side respectively.
ð10Þ
where Nu is the Nusselt number; P is the tube pitch, m; Do is the tube outer diameter, m; Pe is the Peclet number. The heat transfer correlations for water/steam flow in different regions are shown in Table 1. In Sider-Tate and Collier correlation, where Nu is the Nusselt number; Re is the Reynolds number; Pr is the Prandtl number; Gr is the Grashof number; subscript w means wall temperature as qualitative temperature. In Chen correlation, where h is the heat transfer coefficient, W/(m2K); G is the mass flux, kg/(m2s); x is the quality; De is the hydraulic diameter, m. l is the dynamic viscosity, Pas; C p is the specific heat at constant pressure, J/(kgK); k is the thermal conductivity, W/(mK); r is the surface tension, N/m; Hfg is the latent heat of vaporization, J/kg; T is the temperature, K; P is the pressure, Pa; subscripts f, g mean saturated water and saturated steam respectively. In Biasi correlation, where q is heat flux, W/m2; P is the pressure, bar. In Mcadams correlation, where b is the coefficient of volume expansion, K1; subscript vf means the steam at the temperature of ðT w þ T g Þ=2. According to the basic heat transfer equation, the heat transfer rate can be obtained:
Q ¼ KADT
ð12Þ
where DT max is the maximum temperature difference between sodium and water in a control volume, K; DT min is the minimum temperature difference between sodium and water in a control volume, K.
where H is the water/steam enthalpy, J/kg; Hf is the enthalpy of the saturated water, J/kg; x is the steam quality; xCHF is the steam quality at the location of critical heat flux; Hg is the enthalpy of the saturated steam, J/kg. The critical heat flux is calculated by the Biasi correlation (Weisman and Ying, 1985). The heat transfer correlation for sodium flow is as follow (Mikityuk, 2009),
Nu ¼ 0:047 1 e3:8ðP=Do 1Þ Pe0:77 þ 250
DT max DT min ln DDTTmax min
2.3. Pressure drop correlations The total pressure drop comprises gravitational, frictional, and accelerational pressure drop
Dp ¼ Dpg þ Dpa þ Dpf
ð14Þ
where Dp is the total pressure drop, Pa; Dpg is the gravitational pressure drop, Pa; Dpa is the accelerational pressure drop, Pa; Dpf is the frictional pressure drop, Pa. The gravitational pressure drop and accelerational pressure drop can be calculated by the following equations respectively,
Dpg ¼ qg ðz2 z1 Þ
ð15Þ
Dpa ¼ qV ðV 2 V 1 Þ
ð16Þ
The single-phase frictional pressure drop can be calculated by Darcy equation:
Dpf ¼ f
ð11Þ
l qV 2 De 2
ð17Þ
Table 1 Heat transfer correlations in different regions. Region
Name of correlation
Correlation
Subcooled region
Sider-Tate (Bird et al., 1960; Incropera et al.) Collier (Collier and Thome, 1995) Chen (Campolunghi and Cumo, 1977)
Nu ¼ 0:023Re
Miropolskiy (Miropolskiy, 1963)
Nu ¼ 0:021Re0:8 Pr 0:43 Y f h 0:4 i0:8 q q Y f ¼ 1 0:1 qf 1 ð1 xÞ0:4 x þ qf ð1 xÞ
Biasi
qCHF1 ¼ 1:283 102nþ7 Dn G1=6 ½0:681F ðP ÞG1=6 x
Nucleate boiling region Film boiling region
Condition 0:8
Pr
0:33
Nu ¼ 0:17Re0:33 Pr 0:43
0:14
l lw
Pr Prw
0:25
Re > 2500 Re < 2500
Gr 0:1
0:79 0:45 0:49 h i0:8 hl C i0:4 kf C pf qf kf ÞDe 0:24 f pf h ¼ 0:023F Gð1:0x ðP w P s Þ0:75 l De þ 0:00122S r0:5 l0:29 H0:24 q0:24 ðT w T s Þ k f
f
f
g
Sider-Tate (Bird et al., 1960; Incropera et al.) McAdams (Mcadams et al., 1949)
1 HðP Þ ¼ 1:159 þ 0:149P expð0:019P Þ þ 8:89 10 þ P 2
0:4; De P 0:01 n¼ 0:6; De < 0:01
maxðqCHF1 ; qCHF2 Þ; G P 300 kg=ðm2 sÞ qCHF ¼ qCHF2 ; G < 300 kg=ðm2 sÞ 0:14 Nu ¼ 0:023Re0:8 Pr 0:33 ll
h ¼ 0:13kv f
g
g
qCHF2 ¼ 4:47 102nþ6 Dn G0:6 HðP Þð1 xÞ F ðP Þ ¼ 0:7249 þ 0:099P expð0:032P Þ
Superheated region
fg
D = 0.003–0.0375 m; P = 2.7– 140 bar; G = 100–6000 kg/(m2s)
Re > 2500
w
1=3 C l 1=3 pk
q2v f gbv f ðT w T s Þ l2v f
vf
Re < 2500
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where f is the Darcy friction factor; l is the length of control volume, m; De is the hydraulic diameter, m; q is the density of fluid, kg/m3; V is the flow velocity of fluid, m/s. According to the value of Reynold number, the friction factor can be obtained by different equations (Genic´ et al., 2011; Kandlikar et al., 2005),
(
f ¼
64=Re 0:25 þe 0:11 68 Re
Re 6 2000
ð18Þ
Re P 3000
where e is the roughness height, m. For the region between Re = 2000 and Re = 3000, the friction factors are obtained by the approximation method of linear interpolation. Moreover, in TCOSS code, the roughness height is neglected. Therefore, the friction factor equation can be described as follow: f¼
8 > < 64=Re
Re2000 ½f ðRe ¼ 3000Þ f ðRe ¼ 2000Þ þ f ðRe ¼ 2000Þ 30002000 > : 0:25
0:316=ðReÞ
Re 6 2000 2000 < Re < 3000 Re P 3000 ð19Þ
For the two-phase region, the frictional pressure drop is multiplied by the two-phase frictional pressure drop multiplier /2f 0 , and it is calculated by
"
/2f 0
qf qg ¼ 1:0 þ x qg
!#"
lf lg 1:0 þ x lg
!#0:25 ð20Þ
where x is the quality of the fluid; qf is the density of saturated fluid, kg/m3; qg is the density of the saturated steam, kg/m3; lf is the dynamic viscosity of saturated fluid, Pas; lg is the dynamic viscosity of saturated steam, Pas. 2.4. Heat conduction model Heat conduction is considered in the tube wall. The heat conduction model is shown in Fig. 4. Only the radial heat conduction is considered with axial heat conduction ignored. The basic equation for the heat conduction can be described as follow:
qw Aw cp
@T w ðsÞ ¼ h1 A1 ðT 1 T w1 Þ h2 A2 ðT w2 T 2 Þ @t
ð21Þ
where T w is the average wall temperature, K; T w1 is the outer wall temperature, K; T w2 is the inner wall temperature, K. By using the energy balance equations between heat conduction and convection:
k
T w T w1 ¼ h1 ðT w1 T 1 Þ d=2
ð22Þ
k
T w T w2 ¼ h2 ðT w2 T 2 Þ d=2
ð23Þ
where d is the thickness of the tube wall, m. The inner and outer wall temperature can be obtained:
T w1 ¼
T w2 ¼
k T þ h1 T 1 d=2 w k þ h1 d=2 k T þ h2 T 2 d=2 w k þ h2 d=2
ð24Þ
Fig. 4. Heat conduction model in tube wall.
the Eqs. (8) and (9), the governing equations for every control volume can be described as:
dhi 1 ¼ Al dt
½ðW i1 hi1 W i hi þ qi U i li Þ þ ðh 1= @@pqÞ ðW i W i1 Þ ðq þ @@hq = @@pqÞ
i
i
ð25Þ
2.5. Numerical method Based on the staggered grid as shown in Fig. 5, the conservation equations are discretized by using the finite difference method. Using the Eqs. (1)–(3) (7) and finite difference method to simplify
ð26Þ dpi W i1 W i @ q @ q dhi ð = Þ ¼ dt @h @p i dt Alð@@pqÞ
ð27Þ
dW i ðpi piþ1 Þ ½Dpf þ Dpg þ Dpa þ Dpc i l ¼ dt A
ð28Þ
i
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Fig. 5. Staggered mesh of the SG model.
where Dpc is the local resistance pressure drop, Pa; subscripts i and i-1 are the control volume numbers. Both Eqs. (8) and (27) can be used to solve for dp . However using Eq. (27) to solve for dp costs more dt dt time than Eq. (8). Therefore, in order to speed up the calculation, the dp dt
Eq. (27) was adapted to solve for instead of Eq. (8) in TCOSS code. For incompressible fluid, the mass flow is the constant. The partial derivatives of density can be neglected. Therefore, the above equations can be simplified as follow:
dhi ðW i1 hi1 W i hi þ qi U i lÞ ¼ dt qAl
ð29Þ
pi ¼ piþ1 þ Dpf þ Dpg þ Dpa þ Dpc i
ð30Þ
W i ¼ W in
ð31Þ
where Win is the mass flow at the inlet. In TCOSS code, the sodium was accounted as incompressible fluid. There is an option inputted by users to decide water is incompressible fluid or not. When Eqs. (26)–(28) are applied to water, it takes several hours to obtain hundreds seconds output for TCOSS. When Eqs. (29)–(31) are applied to water, it only takes a few minutes to obtain the same results for TCOSS. Therefore, in order to obtain the results quickly, the Eqs. (29)–(31) are applied in most of conditions. Firstly, based on the initial (h, p, W), the (h, p, W) of each control volume can be obtained. The state of boundary volume is defined based on the fluid enthalpy in the volume. For the boundary between the subcooled region and nucleate boiling region, if the fluid enthalpy is greater than the enthalpy of saturated water, then we define it as nucleate boiling state and the heat transfer coefficient and pressure drop are calculated by correlations of nucleate boiling region. If the fluid enthalpy of boundary between subcooled region and nucleate boiling region is lower than the enthalpy of saturated water, then we define it as subcooled state, the heat transfer coefficient and pressure drop are calculated by correlations of subcooled region. For the boundary between the film boiling region and superheated region, if the fluid enthalpy is greater than the enthalpy of saturated vapor, then we define it as superheated state and the heat transfer coefficient and pressure drop are calculated by correlations of superheated region. If the fluid enthalpy of boundary between film boiling region and superheated region is lower than the enthalpy of saturated vapor, then we define it as film boiling state, the heat transfer coefficient and pressure drop are calculated by correlations of film boiling region. Based on the conservation equations and other auxiliary equations, we can obtain the equation set which can describe the thermal-hydraulic of the steam generator for sodium-cooled fast reactor, and the basic form of the equation set is
@y y0 ¼ f t; y; y0 ; @Z
ð32Þ
@y By discretizing space derivative @Z with difference methods, the equation can turn to
8 < d! y
¼ f ðt; y; y0 Þ dt :! ! y ðoÞ ¼ y 0
ð33Þ
Using the Gear method (Feng, 1982) to solve the Eq. (33), the enthalpy, pressure and mass flow of every control volume at any time step can be obtained. In TCOSS code, before starting the GEAR method, the derivatives of enthalpy, pressure and mass flow should be calculate firstly. Enthalpy, pressure, mass flow and their derivatives are assumed as y and y’. With the initial value inputted by users, the differential equation set can be solved by GEAR method. Before starting the transient calculation, a steady state needs to be obtained. In order to obtain the steady state, the boundary conditions of steady state should be inputted. These boundary conditions include inlet mass flow, inlet temperature and outlet pressure both of water and sodium. Based on these conditions, TCOSS code can start calculation. In the calculation, by observing the variations of temperature, pressure and other parameters, when these parameters do not change with the increase of time step, then we can define these parameters as steady state parameters. Using these steady state parameters as the initial conditions of the transient calculation, and add the boundary conditions of transient, then we can start the transient calculation. 3. Benchmark calculations In order to verify the accuracy of the TCOSS code, some benchmark calculations have been carried out, including steady calculations and transient calculations. Steady calculation results are compared with the design data of the design code SG-33 of Russia (Sun et al., 2016). The design data of India Commercial Fast Breeder Reactor (CFBR) (Athmalingam, 2011) and the experiment data of Fast Sodium Reactor 300 (SNR-300) (Unal, 1981; Ünal et al., 1977) are compared as well. A further comparison of transient calculations has been made with B1-B transient (Berry, 1983) and with the ETEC shutdown experiment data (Tzanos, 1988). 3.1. Steady calculations To verify the accuracy of TCOSS code for calculating the modular type and integral type steam generators, the design data of CFBR are applied. And a design code SG-33 is also applied. Steam generators of SG-33 are the modular type steam generator with one evaporator and one superheater. The steam generators of CFBR and SNR-300 are the integral type steam generator. The design parameters of CFBR and SG-33 are shown in Table 2. The results are presented in Table 3 and Fig. 6.
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For the calculation of TCOSS, the temperature of sodium at the inlet and the temperature of feedwater are given. The flow rates of sodium and water are given as well. Therefore, in order to compare the calculations with the design data, the authors choose the temperature of sodium at the outlet and the temperature of the steam outlet. The results of SG-33 are shown in Table 3, the errors are almost zero. Fig. 6 displays the temperature distribution of CFBR. The beginning of the tube length is the inlet of feedwater and the outlet of sodium. The end of the tube length is the steam outlet and sodium inlet. As shown in Fig. 6, water flows from 0 m to 30 m, while sodium flows from 30 m to 0 m. The temperature of water is increasing along its flow direction until it reaches the saturated temperature. In saturated conditions, the temperature of water/steam is constant. After the saturated region, the water/ steam becomes superheat steam. Its temperature is increasing along its flow direction. In addition, the sodium temperature is decreasing by transferring the heat to water along its flow direction. The small peak of sodium temperature at around 8 m indicates that the heat transfer region of water has turned into film boiling region from nucleate boiling. The heat transfer coefficient at this location would decrease sharply. Therefore, the heat transferred to water from sodium would decrease. Hence, the decreasing velocity of sodium temperature would become lower. Then with the increase in temperature difference between sodium and water, the transferred heat would increase. The decreasing velocity of sodium temperature would become higher. It can be observed that the temperature calculated by TCOSS agree well with the design data of CFBR. The maximum temperature deviation is within 5%. From the comparisons, it can be concluded that the TCOSS code is able to predict the thermal-hydraulic behavior both of modular type and of integral type steam generator to a satisfactory degree of accuracy. Deviation is within 1% of the SG-33 design temperature. In addition, the predicted temperature distribution of CFBR agrees well with the design value. A further comparison between TCOSS and SNR-300 experiment data has been made. SNR-300 was a 50 MW liquid-sodium heated steam generator, and it was an integral type steam generator. It was comprised of 139 tubes of 12.6 mm ID and 17.2 mm OD. The tubes arranged in a shell side in a triangular pattern of 27.5 mm pitch, and the effective heated length was 18.64 m. Some conditions of SNR-300 have been chosen to validate the accuracy of the TCOSS code. The calculation results are shown in Table 4.
Table 2 Steam generator design parameters of CFBR and SG-33. Parameters
CFBR
SG-33
Thermal power, MW Sodium temperature, K
210.5 628.15/798.15
Water/steam temperature, K
508.15/766.15
Steam pressure, MPa Sodium flow, kg/s Water flow, kg/s Tube outer/inner diameter at the evaporator, m Tuber outer/inner diameter at the superheater, m Tube pitch at the evaporator, m Tube pitch at the superheater, m Tube length at the evaporator, m Tube length at the superheater, m Number of tubes at the evaporator Number of tubes at the superheater Tube material
18.2 973.53 93.67 0.015/0.0126
32.75 583.15/ 768.15 463.15/ 753.15 14.0 137.92 13.35 0.016/0.011 0.016/0.011
0.0324 30.0 433
Modified 9Cr-1Mo steel
0.032 0.036 12.26 7.72 187 138 2.25Cr-1Mo
Table 3 Comparison of TCOSS code with SG-33 design data. Parameters
TCOSS code
SG-33
Error (%)
Outlet sodium temperature, K Outlet steam temperature, K
582.83 752.84
583.15 753.15
0.05 0.04
Fig. 6. Temperature distribution of the sodium and water/steam.
Fig. 7 shows the error between the calculation and experiment. It can be seen that the errors both of steam temperature and of exit quality are within 2%. Hence, it can be concluded that the TCOSS code can be used to predict the thermal-hydraulic behavior of sodium heated once-through steam generator. These calculation results preliminarily verify the reasonability of TCOSS. 3.2. Transient calculations The transient calculations include the B1-B transient and ETEC shutdown experiment. The geometric design parameters of steam generators in both transients are shown in Table 5. The B1-B transient is a benchmark task for the sodium-heated once-through steam generator with the following characteristics: Fast reduction of feedwater flow; Slow increasing of feedwater temperature; Steam outlet pressure varied during the transient in the range of 15.8–17.2 MPa; Slow reduction of sodium flow; Variations in the feedwater temperature and sodium inlet temperature. For the calculation of the TCOSS code, the variations of the thermal-hydraulic parameters are needed to be inputted. The variations of sodium mass flow, feedwater mass flow and feedwater pressure are shown in Table 6. The variations of sodium inlet temperature and feedwater temperature are shown in Fig. 8. Fig. 9 shows the comparison of steam outlet temperature and sodium outlet temperature between the calculated results of TCOSS and PSM-W. It can be observed that sodium outlet temperature of two codes agree well with each other. The maximum deviation of sodium outlet temperature and steam outlet temperature between two codes is within 3%. Although the agreement of steam outlet temperature between two codes is good overall, there exists a discrepancy at the beginning of the transient. The steam outlet temperature calculated by TCOSS shows a decreasing trend in the
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Table 4 Comparison of TCOSS code with SNR-300 experiment data. Steam pressure, MPa
Feedwater subcooling, K
Mass flow, kg/s
17.0 17.0 16.7 16.7
66 66 49 90
617.4 496.2 663.5 623.1
Steam temperature, K
Exit quality
Exp
TCOSS
Exp
TCOSS
701.15 678.15 700.15 702.15
702.15 685.15 696.15 703.15
1.573 1.460 1.550 1.559
1.565 1.482 1.519 1.552
Fig. 7. Comparison of steam temperature and exit quality.
Table 5 Design parameters of steam generator. Parameters
B1-B
ETEC
Tube length, m Tube outer diameter, m Tube inner diameter, m Pitch, m Number of tubes
20.5 0.02 0.011 0.0305 804
14.021 0.015875 0.010338 0.03168 757
Table 6 Input data for B1-B transient. Time, s
Sodium mass flow, kg/s
Feedwater mass flow, kg/s
Feedwater pressure, MPa
0 10 30 50 100 1100
335.0 100.0 – 26.8 24.8 24.8
32.02 – 1.6 – 1.6
15.78 17.17 16.58 – 16.58
beginning while the steam outlet temperature calculated by PSMW is not. The reason that causes the discrepancy is supposed that the difference of thermal-hydraulic models used in the two codes. In order to make further efforts to verify the TCOSS code, the actual data of ETEC shutdown experiment have been used to compare with the calculation of the TCOSS. Fig. 10 shows the variations of feedwater flow and sodium inlet flow under the shutdown transient. The feedwater flow decreases in the transient while sodium flow increases in the transient overall. At about 1.3 h of the transient, a sharply loss of feedwater flow occurred and then it quickly increases. Fig. 11 shows the variations of the feedwater temperature and sodium inlet temperature under the shutdown transient. The feedwater temperature is staying at
Fig. 8. Variations of sodium inlet temperature and feedwater temperature.
480 K with slight oscillation. The sodium inlet temperature decreases in the former 3 h and then increases. These variations are the boundary conditions of shutdown transient for the calculation of TCOSS. The results are shown in Figs. 12 and 13. Fig. 12 shows the comparison of the sodium outlet temperature between the calculated results of TCOSS and the experimental data under the shutdown transient. It can be observed that although the calculation results agree well with the experimental data, there exists a great difference in the peak of the sodium outlet temperature of two results. It is pointed out by Tzanos (1988) that the experimental data were recorded every 5 min, the peak of the sodium outlet temperature was missed. Therefore, the maximum sodium outlet temperature of the experiment is actually not the peak of the sodium outlet temperature. Thus the maximum sodium
R. Xu et al. / Annals of Nuclear Energy 127 (2019) 385–394
Fig. 9. TCOSS and PSM-W predictions for the B1-B transient.
Fig. 12. Sodium outlet temperature during the shutdown transient.
Fig. 10. Feedwater and sodium inlet flow during the shutdown transient.
Fig. 13. Steam outlet temperature during the shutdown transient.
393
computed steam outlet temperature and measured steam outlet temperature at steady state and the very early part of the transient. In the later part of the transient, the computed and measured temperature are practically identical. The difference between TCOSS and experiment at steady state is about 10 K. It is pointed out by Tzanos (1988) that the energy change of water-side was about 0.6% less than that of sodium side at steady-state conditions. Therefore, in the steady state and the very early part of the transient there exists a discrepancy between computed steam outlet temperature and measured steam outlet temperature. Moreover, at steady-state conditions, 1-r values of up to 16 K were determined for the steam outlet temperature (Tzanos, 1988). Thus, the difference of the steam outlet temperature is well within the error of the experimental measurements. From the comparison, it can be concluded that the TCOSS code can be used to calculate the transient of sodium heated once-through steam generator with a good accuracy. Fig. 11. Feedwater and sodium inlet temperature during the shutdown transient.
4. Conclusion outlet temperature calculated by TCOSS is larger than the experiment. Fig. 13 shows the comparison of the steam outlet temperature between the calculated results of TCOSS and the experimental data under the shutdown transient. The computed temperature agrees well with the measured temperature. There is a difference between
A one-dimensional code TCOSS for the transient analysis of sodium heated once-through steam generator has been developed based on the conservation equations of mass, momentum and energy. Some steady calculations and transient calculations are carried out in order to verify the TCOSS code. The steady calculations involve the design results of SG-33 and CFBR and the exper-
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iment data of SNR-300. The transient calculations involve the B1-B numerical experiment and ETEC shutdown experiment. Calculated results are compared with other codes and experimental data. In general, the TCOSS calculated results both of modular type and of integral type steam generators have a good agreement with the design data and experimental data. These calculation results preliminarily verify the reasonability. Thus, the TCOSS code can not only be used to calculate the thermal-hydraulic responses of modular type steam generator but also of integral type steam generator under various transient conditions. The conceptual design of CDSFR steam generator is carried out. Moreover, the steam generators of CDSFR are modular type steam generator with one evaporator and one superheater. Therefore, this code can be used to help the design work of steam generator. For various design concept of steam generator, it can assess the safety of steam generator under different conditions. And the designers can choose a better design concept based on the calculation results of TCOSS. Acknowledgements The authors gratefully acknowledge the supports from Natural Science Foundation of China (Grant No. 11675127) and K. C. Wong Education Foundation. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.anucene.2018.12.027. References Athmalingam, S., 2011. Steam generator for PFBR and future FBR. Berry, G., 1983. Model of a once-through steam generator with moving boundaries and a variable number of nodes. Bird, R.B., Stewart, W.E., Lightfoot, E.N., Spalding, D.B., 1960. Transport Phenomena. John Wiley & Sons, pp. 338–359. Campolunghi, F., Cumo, M., 1977. Subcooled and bulk boiling correlations for thermal design of steam generators.
Carey, V.P., 1992. Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment. Hemisphere Pub, Corp. Collier, J., Thome, J., 1995. 95/03868 Convective boiling and condensation. Fuel & Energy Abstr. 36, 277. Feng, B.P., 1982. The gear program for solving initial value problems in general or stiff ordinary differential equations. J. Numer. Meth. Computer Appl., 12–23 Genic´, S., Arandjelovic´, I., Kolendic´, P., Jaric´, M., Budimir, N., Genic´, V., 2011. A review of explicit approximations of Colebrook’s equation. FME Trans. 39 (2), 67–71. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., Foundations of Heat Transfer, sixth ed. International Student Version. Kandlikar, S.G., Schmitt, D., Carrano, A.L., Taylor, J.B., 2005. Characterization of surface roughness effects on pressure drop in single-phase flow in minichannels. Phys. Fluids 17, 133–198. Kim, E.-K., Baek, B.-J., 2011. Thermal hydraulic performance analysis of a double tube bundle steam generator for a liquid metal reactor. Ann. Nucl. Energy 38, 2625–2633. Mcadams, W.H., Addoms, J.N., Kennel, W.E., 1949. Heat transfer to superheated steam at high pressures. Trans.am.nuclear Soc 4: no. 1. Mikityuk, K., 2009. Heat transfer to liquid metal: review of data and correlations for tube bundles. Nucl. Eng. Des. 239, 680–687. Miropolskiy, Z.L., 1963. Heat transfer in film boiling of a steam-water mixture in steam generating tubes. Teploenergetika 10, 49–52. Nandakumar, R., Selvaraj, P., Athmalingam, S., Balasubramaniyan, V., Chetal, S.C., 2012. Thermal simulation of sodium heated once through steam generator for a fast reactor. Int. J. Adv. Eng. Sci. Appl. Math. 4, 127–137. Sun, P., Wang, Z., Zhang, J., Su, G., 2016. Thermal–hydraulic design and analysis code development for steam generator of CFR600. Ann. Nucl. Energy 90, 256–263. Tzanos, C.P., 1988. A movable boundary model for once-through steam generator analysis. Nucl. Technol. 82, 5–17. Unal, H.C., 1981. Some aspects of two-phase flow, heat transfer and dynamic instabilities in medium and high pressure steam generators. Ünal, H.C., Van Gasselt, M.L.G., Ludwig, P.W.P.H., 1977. Dynamic instabilities in tubes of a large capacity, straight-tube, once-through sodium heated steam generator. Int. J. Heat Mass Transfer 20, 1389–1399. Vaidyanathan, G., Kothandaraman, A.L., Kumar, L.S.S., Vinod, V., Noushad, I.B., Rajan, K.K., Kalyanasundaram, P., 2010. Development of one-dimensional computer code DESOPT for thermal hydraulic design of sodium-heated once through steam generators. Int. J. Nucl. Energy Sci. Technol. 5, 143–161. Vinod, V., Sivakumar, L.S., Kumar, V.A.S., Noushad, I.B., Padmakumar, G., Rajan, K.K., 2014. Experimental evaluation of the heat transfer performance of sodium heated once through steam generator. Nucl. Eng. Des. 273, 412–420. Weisman, J., Ying, S.H., 1985. A theoretically based critical heat flux prediction for rod bundles at PWR conditions. Nucl. Eng. Des. 85, 239–250. Zhong, X., Jiyang, Y.U., Saeed, M., Yang, C., Zhang, X., 2017. Analysis of hydrodynamic characteristic in straight-tube sodiumheated once-through steam generator. International Conference on Nuclear Engineering.
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Thermal-hydraulic analysis code development for sodium heated once-through steam generator Rongshuan Xu, Dalin Zhang ⇑, Wenxi Tian, Suizheng Qiu, G.H. Su State Key Laboratory of Multiphase Flow in Power Engineering, Shaanxi Key Lab. of Advanced Nuclear Energy and Technology, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 14 August 2018 Received in revised form 16 December 2018 Accepted 18 December 2018
Keywords: Once-through steam generator Sodium cooled fast reactor Computer code Thermal-hydraulic model Transient analysis
a b s t r a c t Sodium heated once-through steam generator (OTSG) is an essential component of Sodium cooled Fast Reactor (SFR). It transfers the heat from hot sodium to water and converts water to superheat steam. Steam can roll the turbine and generate electricity. Therefore, the safety of the steam generator is vital to the operation of the plant. China is developing the China Demonstration SFR (CDSFR) after successful construction of China Experiment Fast Reactor (CEFR). In order to make sure the safety of the steam generator under various transient conditions and support the design of the steam generator for CDSFR, a Transient analysis Code of Once-through Steam generator for Sodium cooled fast reactor (TCOSS) has been developed. To benchmark the developed physical model, the calculated results of the code have been compared with some design data of OTSGs and they are found in a good agreement. In addition, the transient results of the code have been compared with the experimental results of a shutdown experiment for the validation of the code. The predictions of the code agree well with the experimental data. Hence, the code can be utilized to predict the variations of thermal-hydraulic parameters in steam generator under different transient conditions. In addition, predictions can support the design of the steam generator for CDSFR. Ó 2018 Published by Elsevier Ltd.
1. Introduction Sodium heated once-through steam generator plays a critical role in Sodium-cooled Fast Reactor (SFR) plant. SFR is one of six potential reactor candidates for generation Ⅳ nuclear systems. It adopts liquid sodium as coolant. In sodium cooled fast reactors, the heat generated from the core is absorbed by primary sodium and transferred to secondary sodium through the intermediate heat exchanger. Then the secondary sodium transfers heat to water/steam in the Steam Generator (SG). The once through steam generators of SFR can be classified into two kinds, the integral type, and the modular type. They are all counter current, with oncethrough-type shell and tube-heat-exchanger having straight vertical tubes (Nandakumar et al., 2012). The modular type steam generator contains two vertical counter-current shell and tube heat exchangers, one called evaporator, the other called superheater. In modular type steam generator, sodium flows in the shell side while water/steam flows in the tube side. Heated sodium from intermediate heat exchanger enters into the bottom of superheater. Then the sodium flows upwards in ⇑ Corresponding author. E-mail address: [email protected] (D. Zhang). https://doi.org/10.1016/j.anucene.2018.12.027 0306-4549/Ó 2018 Published by Elsevier Ltd.
the shell side then enters into the top of evaporator. After that, it flows downwards in the shell side and exits the evaporator via the bottom outlet plenum in the end. Feed water enters into the bottom of evaporator. The water flows upwards in the tubes, then enter into the top of superheater. After that, it flows downwards in the tubes and exits the superheater via the bottom outlet plenum. As showed in Fig. 1 (Zhong et al., 2017), the steam generator of China Experiment Fast Reactor (CEFR) is modular type steam generator. The integral type steam generator is a vertical countercurrent shell and tube heat exchanger. In integral type steam generator, hot sodium enters into the top of SG, flows downwards in the shell side and feed water enters into the bottom of SG, flows upwards in the tubes. Feed water gets heated to transfer into superheated steam and is to roll the turbine. As Fig. 2 shows, the steam generator of India Commercial Fast Breeder Reactor (CFBR) is an integral type steam generator (Nandakumar et al., 2012). China is developing the China Demonstration SFR (CDSFR) after successful construction of CEFR. And the steam generator of CDSFR is a modular type once-through steam generator. Steam generator is a critical component for the operation and safety of SFR. The thermal-hydraulic analysis of steam generator in various transient conditions is crucial in the design stage of steam generator. The large change of thermodynamic parameters in transient conditions
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Fig. 1. Once-through steam generator of CEFR. Fig. 2. Once-through steam generator of CFBR.
would be harmful to the safety of steam generator. In order to study the thermodynamic parameters variations of different conditions in the steam generator and make the design of SFR steam generator safer, the transient analysis code is needed for sodium heated once-through steam generator. Vaidyanathan had developed the DESign Operational (DESOPT) code for the thermal-hydraulic design of sodium heated oncethrough steam generators (Vaidyanathan et al., 2010). And this code had used to support the design of the Prototype Fast Breeder Reactor (PFBR) (Vinod et al., 2014). Kim and Baek had developed a steady-state code to analyze the thermal-hydraulic performance of once-through steam generator using helically coiled tubes (Kim and Baek, 2011). Sun had developed a thermal-hydraulic design and analysis code for modular type steam generator (Sun et al., 2016). Berry and Tzanos had developed a model of once-through
steam generator with moving boundaries and variable nodes and had presented some typical results (Berry, 1983; Tzanos, 1988). The above design codes could not be used to analyze the transient conditions of the steam generator, and the existing transient codes are not applicable for steam generator with two modules of CDSFR. Therefore, a thermal-hydraulic analysis and transient calculation code for steam generator of CDSFR is needed. In this paper, a one-dimensional transient analysis code TCOSS (Transient analysis Code of Once-through Steam generator for Sodium cooled fast reactor) is developed to simulate sodium heated once-through steam generator. The simulation results were compared with the design data and experimental data of the OTSGs of SFRs, to benchmark the code. This paper will show thermal-hydraulic models and basic verification of TCOSS based on the analysis of numerical
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experiment B1-B and of Energy Technology Engineering Center (ETEC) steam generator shutdown experiment (Tzanos, 1988). The results can support the design of SG for CDSFR.
where x is the equilibrium quality. The equilibrium quality is calculated by the following equation:
x¼ 2. Thermal-hydraulic model In the TCOSS code, the steam generator is simulated by a onedimensional characteristic tube. Water flows inside the tube and sodium flows outside the tube. Along the direction of fluid flowing, the tube can be specified into arbitrary numbers of control volumes averagely. The length and location of the control volume are constant in the calculation. According to the thermal properties of the water/steam in the control volumes, the tube can be divided into four regions: subcooled region, nucleate boiling region, film boiling region and superheated region. The boundaries of the four region are adjusted after every time step. In addition, some assumptions are made as follows in order to simplify the models of the steam generator. (1) Neglecting the axial heat conduction of tube wall and fluid. (2) No phase change in sodium side and sodium is considered as incompressible fluid. (3) Homogenous fluid in each control volume. (4) Neglecting the effect of subcooled boiling. (5) Thermal equilibrium between water and steam phase in boiling regions. (6) Neglecting the energetic changes caused by the changes of gravity and kinetic energy. 2.1. Governing equations Conservation equations for the mass, momentum and energy are used for both sides of the tube. Based on the above assumptions, the one-dimensional conservation equations can be described as follows. Mass conservation equation:
@q @ W ¼0 þ @t @z A
h hf hfg
ð6Þ
where hf is the specific enthalpy of saturated water, J/kg; hfg is the latent heat, J/kg. The following equation is used to simplify the conservation equations:
@ q @ q @h @ q @p þ ¼ @t @h @t @p @t
ð7Þ
Therefore, the simplified conservation equations can be obtained:
@p ¼ 1 @t
1
@h ¼ 1 @t
1
@q @q q @h þ @p
@q @q q @h þ @p
W @ q @h 1 @W @ q qU qA @h @z A @z @h qA
qU @ q W @ q @h 1 @W qA @p qA @p @z qA @z
In the TCOSS code, the steam generator can be represented by one characteristic tube as schematically illustrated in Fig. 3. The subcooled feedwater enters the tube, flows upwards in the tube, gets heated, and finally leaves the tube in a superheated steam state. The hot sodium flows outside the tube in the opposite direction. In the sodium side of the steam generator, there is no phase change but phase change occurs in the water/steam side of the steam generator. Therefore, the tube can be divided into four regions according to the water/steam state and different heat
where q is the density, kg/m3; t is the time, s; W is the mass flow, kg/s; z is the axial coordinate, m; A is the flow area of fluid at the flow direction, m2. Momentum conservation equation:
ð2Þ
where p is the pressure, Pa; f is the friction coefficient; De is the hydraulic diameter, m; g is the acceleration of gravity, m/s2. Energy conservation equation:
q
@h W @h qU @p þ ¼ þ @t A @z A @t
ð3Þ
where h is the specific enthalpy, J/kg; q is the heat flux, W/m2; U is the heated perimeter, m. Homogenous model is used for the two-phase flow. The governing equations are the same as the above equations. The homogenous density is calculated as,
q ¼ qf ð1 - aÞ þ qg a
ð4Þ 3
where qf is the density of liquid water, kg/m ; qg is the density of steam, kg/m3; a is the void fraction. The void fraction is calculated by the following equation:
a¼
x
qg
1 1 ¼ 1x þ 1x 1 þ q x f
qg qf
ð5Þ
ð9Þ
2.2. Heat transfer correlations
ð1Þ
! @ W @ W2 @p fW jW j þ ¼ qg @t A @z qA2 @z 2De qA2
ð8Þ
Fig. 3. Characteristic tube model of SG.
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where Q is the heat transfer rate, W; K is the overall heat transfer coefficient, W/(m2K); A is the heat transfer area, m2; DT is the log mean temperature difference, K. DT and K can be calculated by the following equations,
transfer mechanisms: subcooled region, nucleate boiling region, film boiling region and superheated region. The regions are separated by (1) (2) (3) (4)
H 6 Hf , subcooled region. H > Hf , x 6 xCHF , nucleate boiling region. xCHF < x < 1, film boiling region (Carey, 1992). H P Hg , superheated region.
DT ¼
K¼
1 h1
1 þ h12 ddo þ 2kdow ln ddo i
ð13Þ
i
where h is the convective heat transfer coefficient, W/(m2K); do and di are the outer and inner diameter of the tube, m; kw is the tube thermal conductivity, W/(m2K); the subscript 1 and 2 are represent the sodium side and water side respectively.
ð10Þ
where Nu is the Nusselt number; P is the tube pitch, m; Do is the tube outer diameter, m; Pe is the Peclet number. The heat transfer correlations for water/steam flow in different regions are shown in Table 1. In Sider-Tate and Collier correlation, where Nu is the Nusselt number; Re is the Reynolds number; Pr is the Prandtl number; Gr is the Grashof number; subscript w means wall temperature as qualitative temperature. In Chen correlation, where h is the heat transfer coefficient, W/(m2K); G is the mass flux, kg/(m2s); x is the quality; De is the hydraulic diameter, m. l is the dynamic viscosity, Pas; C p is the specific heat at constant pressure, J/(kgK); k is the thermal conductivity, W/(mK); r is the surface tension, N/m; Hfg is the latent heat of vaporization, J/kg; T is the temperature, K; P is the pressure, Pa; subscripts f, g mean saturated water and saturated steam respectively. In Biasi correlation, where q is heat flux, W/m2; P is the pressure, bar. In Mcadams correlation, where b is the coefficient of volume expansion, K1; subscript vf means the steam at the temperature of ðT w þ T g Þ=2. According to the basic heat transfer equation, the heat transfer rate can be obtained:
Q ¼ KADT
ð12Þ
where DT max is the maximum temperature difference between sodium and water in a control volume, K; DT min is the minimum temperature difference between sodium and water in a control volume, K.
where H is the water/steam enthalpy, J/kg; Hf is the enthalpy of the saturated water, J/kg; x is the steam quality; xCHF is the steam quality at the location of critical heat flux; Hg is the enthalpy of the saturated steam, J/kg. The critical heat flux is calculated by the Biasi correlation (Weisman and Ying, 1985). The heat transfer correlation for sodium flow is as follow (Mikityuk, 2009),
Nu ¼ 0:047 1 e3:8ðP=Do 1Þ Pe0:77 þ 250
DT max DT min ln DDTTmax min
2.3. Pressure drop correlations The total pressure drop comprises gravitational, frictional, and accelerational pressure drop
Dp ¼ Dpg þ Dpa þ Dpf
ð14Þ
where Dp is the total pressure drop, Pa; Dpg is the gravitational pressure drop, Pa; Dpa is the accelerational pressure drop, Pa; Dpf is the frictional pressure drop, Pa. The gravitational pressure drop and accelerational pressure drop can be calculated by the following equations respectively,
Dpg ¼ qg ðz2 z1 Þ
ð15Þ
Dpa ¼ qV ðV 2 V 1 Þ
ð16Þ
The single-phase frictional pressure drop can be calculated by Darcy equation:
Dpf ¼ f
ð11Þ
l qV 2 De 2
ð17Þ
Table 1 Heat transfer correlations in different regions. Region
Name of correlation
Correlation
Subcooled region
Sider-Tate (Bird et al., 1960; Incropera et al.) Collier (Collier and Thome, 1995) Chen (Campolunghi and Cumo, 1977)
Nu ¼ 0:023Re
Miropolskiy (Miropolskiy, 1963)
Nu ¼ 0:021Re0:8 Pr 0:43 Y f h 0:4 i0:8 q q Y f ¼ 1 0:1 qf 1 ð1 xÞ0:4 x þ qf ð1 xÞ
Biasi
qCHF1 ¼ 1:283 102nþ7 Dn G1=6 ½0:681F ðP ÞG1=6 x
Nucleate boiling region Film boiling region
Condition 0:8
Pr
0:33
Nu ¼ 0:17Re0:33 Pr 0:43
0:14
l lw
Pr Prw
0:25
Re > 2500 Re < 2500
Gr 0:1
0:79 0:45 0:49 h i0:8 hl C i0:4 kf C pf qf kf ÞDe 0:24 f pf h ¼ 0:023F Gð1:0x ðP w P s Þ0:75 l De þ 0:00122S r0:5 l0:29 H0:24 q0:24 ðT w T s Þ k f
f
f
g
Sider-Tate (Bird et al., 1960; Incropera et al.) McAdams (Mcadams et al., 1949)
1 HðP Þ ¼ 1:159 þ 0:149P expð0:019P Þ þ 8:89 10 þ P 2
0:4; De P 0:01 n¼ 0:6; De < 0:01
maxðqCHF1 ; qCHF2 Þ; G P 300 kg=ðm2 sÞ qCHF ¼ qCHF2 ; G < 300 kg=ðm2 sÞ 0:14 Nu ¼ 0:023Re0:8 Pr 0:33 ll
h ¼ 0:13kv f
g
g
qCHF2 ¼ 4:47 102nþ6 Dn G0:6 HðP Þð1 xÞ F ðP Þ ¼ 0:7249 þ 0:099P expð0:032P Þ
Superheated region
fg
D = 0.003–0.0375 m; P = 2.7– 140 bar; G = 100–6000 kg/(m2s)
Re > 2500
w
1=3 C l 1=3 pk
q2v f gbv f ðT w T s Þ l2v f
vf
Re < 2500
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where f is the Darcy friction factor; l is the length of control volume, m; De is the hydraulic diameter, m; q is the density of fluid, kg/m3; V is the flow velocity of fluid, m/s. According to the value of Reynold number, the friction factor can be obtained by different equations (Genic´ et al., 2011; Kandlikar et al., 2005),
(
f ¼
64=Re 0:25 þe 0:11 68 Re
Re 6 2000
ð18Þ
Re P 3000
where e is the roughness height, m. For the region between Re = 2000 and Re = 3000, the friction factors are obtained by the approximation method of linear interpolation. Moreover, in TCOSS code, the roughness height is neglected. Therefore, the friction factor equation can be described as follow: f¼
8 > < 64=Re
Re2000 ½f ðRe ¼ 3000Þ f ðRe ¼ 2000Þ þ f ðRe ¼ 2000Þ 30002000 > : 0:25
0:316=ðReÞ
Re 6 2000 2000 < Re < 3000 Re P 3000 ð19Þ
For the two-phase region, the frictional pressure drop is multiplied by the two-phase frictional pressure drop multiplier /2f 0 , and it is calculated by
"
/2f 0
qf qg ¼ 1:0 þ x qg
!#"
lf lg 1:0 þ x lg
!#0:25 ð20Þ
where x is the quality of the fluid; qf is the density of saturated fluid, kg/m3; qg is the density of the saturated steam, kg/m3; lf is the dynamic viscosity of saturated fluid, Pas; lg is the dynamic viscosity of saturated steam, Pas. 2.4. Heat conduction model Heat conduction is considered in the tube wall. The heat conduction model is shown in Fig. 4. Only the radial heat conduction is considered with axial heat conduction ignored. The basic equation for the heat conduction can be described as follow:
qw Aw cp
@T w ðsÞ ¼ h1 A1 ðT 1 T w1 Þ h2 A2 ðT w2 T 2 Þ @t
ð21Þ
where T w is the average wall temperature, K; T w1 is the outer wall temperature, K; T w2 is the inner wall temperature, K. By using the energy balance equations between heat conduction and convection:
k
T w T w1 ¼ h1 ðT w1 T 1 Þ d=2
ð22Þ
k
T w T w2 ¼ h2 ðT w2 T 2 Þ d=2
ð23Þ
where d is the thickness of the tube wall, m. The inner and outer wall temperature can be obtained:
T w1 ¼
T w2 ¼
k T þ h1 T 1 d=2 w k þ h1 d=2 k T þ h2 T 2 d=2 w k þ h2 d=2
ð24Þ
Fig. 4. Heat conduction model in tube wall.
the Eqs. (8) and (9), the governing equations for every control volume can be described as:
dhi 1 ¼ Al dt
½ðW i1 hi1 W i hi þ qi U i li Þ þ ðh 1= @@pqÞ ðW i W i1 Þ ðq þ @@hq = @@pqÞ
i
i
ð25Þ
2.5. Numerical method Based on the staggered grid as shown in Fig. 5, the conservation equations are discretized by using the finite difference method. Using the Eqs. (1)–(3) (7) and finite difference method to simplify
ð26Þ dpi W i1 W i @ q @ q dhi ð = Þ ¼ dt @h @p i dt Alð@@pqÞ
ð27Þ
dW i ðpi piþ1 Þ ½Dpf þ Dpg þ Dpa þ Dpc i l ¼ dt A
ð28Þ
i
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Fig. 5. Staggered mesh of the SG model.
where Dpc is the local resistance pressure drop, Pa; subscripts i and i-1 are the control volume numbers. Both Eqs. (8) and (27) can be used to solve for dp . However using Eq. (27) to solve for dp costs more dt dt time than Eq. (8). Therefore, in order to speed up the calculation, the dp dt
Eq. (27) was adapted to solve for instead of Eq. (8) in TCOSS code. For incompressible fluid, the mass flow is the constant. The partial derivatives of density can be neglected. Therefore, the above equations can be simplified as follow:
dhi ðW i1 hi1 W i hi þ qi U i lÞ ¼ dt qAl
ð29Þ
pi ¼ piþ1 þ Dpf þ Dpg þ Dpa þ Dpc i
ð30Þ
W i ¼ W in
ð31Þ
where Win is the mass flow at the inlet. In TCOSS code, the sodium was accounted as incompressible fluid. There is an option inputted by users to decide water is incompressible fluid or not. When Eqs. (26)–(28) are applied to water, it takes several hours to obtain hundreds seconds output for TCOSS. When Eqs. (29)–(31) are applied to water, it only takes a few minutes to obtain the same results for TCOSS. Therefore, in order to obtain the results quickly, the Eqs. (29)–(31) are applied in most of conditions. Firstly, based on the initial (h, p, W), the (h, p, W) of each control volume can be obtained. The state of boundary volume is defined based on the fluid enthalpy in the volume. For the boundary between the subcooled region and nucleate boiling region, if the fluid enthalpy is greater than the enthalpy of saturated water, then we define it as nucleate boiling state and the heat transfer coefficient and pressure drop are calculated by correlations of nucleate boiling region. If the fluid enthalpy of boundary between subcooled region and nucleate boiling region is lower than the enthalpy of saturated water, then we define it as subcooled state, the heat transfer coefficient and pressure drop are calculated by correlations of subcooled region. For the boundary between the film boiling region and superheated region, if the fluid enthalpy is greater than the enthalpy of saturated vapor, then we define it as superheated state and the heat transfer coefficient and pressure drop are calculated by correlations of superheated region. If the fluid enthalpy of boundary between film boiling region and superheated region is lower than the enthalpy of saturated vapor, then we define it as film boiling state, the heat transfer coefficient and pressure drop are calculated by correlations of film boiling region. Based on the conservation equations and other auxiliary equations, we can obtain the equation set which can describe the thermal-hydraulic of the steam generator for sodium-cooled fast reactor, and the basic form of the equation set is
@y y0 ¼ f t; y; y0 ; @Z
ð32Þ
@y By discretizing space derivative @Z with difference methods, the equation can turn to
8 < d! y
¼ f ðt; y; y0 Þ dt :! ! y ðoÞ ¼ y 0
ð33Þ
Using the Gear method (Feng, 1982) to solve the Eq. (33), the enthalpy, pressure and mass flow of every control volume at any time step can be obtained. In TCOSS code, before starting the GEAR method, the derivatives of enthalpy, pressure and mass flow should be calculate firstly. Enthalpy, pressure, mass flow and their derivatives are assumed as y and y’. With the initial value inputted by users, the differential equation set can be solved by GEAR method. Before starting the transient calculation, a steady state needs to be obtained. In order to obtain the steady state, the boundary conditions of steady state should be inputted. These boundary conditions include inlet mass flow, inlet temperature and outlet pressure both of water and sodium. Based on these conditions, TCOSS code can start calculation. In the calculation, by observing the variations of temperature, pressure and other parameters, when these parameters do not change with the increase of time step, then we can define these parameters as steady state parameters. Using these steady state parameters as the initial conditions of the transient calculation, and add the boundary conditions of transient, then we can start the transient calculation. 3. Benchmark calculations In order to verify the accuracy of the TCOSS code, some benchmark calculations have been carried out, including steady calculations and transient calculations. Steady calculation results are compared with the design data of the design code SG-33 of Russia (Sun et al., 2016). The design data of India Commercial Fast Breeder Reactor (CFBR) (Athmalingam, 2011) and the experiment data of Fast Sodium Reactor 300 (SNR-300) (Unal, 1981; Ünal et al., 1977) are compared as well. A further comparison of transient calculations has been made with B1-B transient (Berry, 1983) and with the ETEC shutdown experiment data (Tzanos, 1988). 3.1. Steady calculations To verify the accuracy of TCOSS code for calculating the modular type and integral type steam generators, the design data of CFBR are applied. And a design code SG-33 is also applied. Steam generators of SG-33 are the modular type steam generator with one evaporator and one superheater. The steam generators of CFBR and SNR-300 are the integral type steam generator. The design parameters of CFBR and SG-33 are shown in Table 2. The results are presented in Table 3 and Fig. 6.
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For the calculation of TCOSS, the temperature of sodium at the inlet and the temperature of feedwater are given. The flow rates of sodium and water are given as well. Therefore, in order to compare the calculations with the design data, the authors choose the temperature of sodium at the outlet and the temperature of the steam outlet. The results of SG-33 are shown in Table 3, the errors are almost zero. Fig. 6 displays the temperature distribution of CFBR. The beginning of the tube length is the inlet of feedwater and the outlet of sodium. The end of the tube length is the steam outlet and sodium inlet. As shown in Fig. 6, water flows from 0 m to 30 m, while sodium flows from 30 m to 0 m. The temperature of water is increasing along its flow direction until it reaches the saturated temperature. In saturated conditions, the temperature of water/steam is constant. After the saturated region, the water/ steam becomes superheat steam. Its temperature is increasing along its flow direction. In addition, the sodium temperature is decreasing by transferring the heat to water along its flow direction. The small peak of sodium temperature at around 8 m indicates that the heat transfer region of water has turned into film boiling region from nucleate boiling. The heat transfer coefficient at this location would decrease sharply. Therefore, the heat transferred to water from sodium would decrease. Hence, the decreasing velocity of sodium temperature would become lower. Then with the increase in temperature difference between sodium and water, the transferred heat would increase. The decreasing velocity of sodium temperature would become higher. It can be observed that the temperature calculated by TCOSS agree well with the design data of CFBR. The maximum temperature deviation is within 5%. From the comparisons, it can be concluded that the TCOSS code is able to predict the thermal-hydraulic behavior both of modular type and of integral type steam generator to a satisfactory degree of accuracy. Deviation is within 1% of the SG-33 design temperature. In addition, the predicted temperature distribution of CFBR agrees well with the design value. A further comparison between TCOSS and SNR-300 experiment data has been made. SNR-300 was a 50 MW liquid-sodium heated steam generator, and it was an integral type steam generator. It was comprised of 139 tubes of 12.6 mm ID and 17.2 mm OD. The tubes arranged in a shell side in a triangular pattern of 27.5 mm pitch, and the effective heated length was 18.64 m. Some conditions of SNR-300 have been chosen to validate the accuracy of the TCOSS code. The calculation results are shown in Table 4.
Table 2 Steam generator design parameters of CFBR and SG-33. Parameters
CFBR
SG-33
Thermal power, MW Sodium temperature, K
210.5 628.15/798.15
Water/steam temperature, K
508.15/766.15
Steam pressure, MPa Sodium flow, kg/s Water flow, kg/s Tube outer/inner diameter at the evaporator, m Tuber outer/inner diameter at the superheater, m Tube pitch at the evaporator, m Tube pitch at the superheater, m Tube length at the evaporator, m Tube length at the superheater, m Number of tubes at the evaporator Number of tubes at the superheater Tube material
18.2 973.53 93.67 0.015/0.0126
32.75 583.15/ 768.15 463.15/ 753.15 14.0 137.92 13.35 0.016/0.011 0.016/0.011
0.0324 30.0 433
Modified 9Cr-1Mo steel
0.032 0.036 12.26 7.72 187 138 2.25Cr-1Mo
Table 3 Comparison of TCOSS code with SG-33 design data. Parameters
TCOSS code
SG-33
Error (%)
Outlet sodium temperature, K Outlet steam temperature, K
582.83 752.84
583.15 753.15
0.05 0.04
Fig. 6. Temperature distribution of the sodium and water/steam.
Fig. 7 shows the error between the calculation and experiment. It can be seen that the errors both of steam temperature and of exit quality are within 2%. Hence, it can be concluded that the TCOSS code can be used to predict the thermal-hydraulic behavior of sodium heated once-through steam generator. These calculation results preliminarily verify the reasonability of TCOSS. 3.2. Transient calculations The transient calculations include the B1-B transient and ETEC shutdown experiment. The geometric design parameters of steam generators in both transients are shown in Table 5. The B1-B transient is a benchmark task for the sodium-heated once-through steam generator with the following characteristics: Fast reduction of feedwater flow; Slow increasing of feedwater temperature; Steam outlet pressure varied during the transient in the range of 15.8–17.2 MPa; Slow reduction of sodium flow; Variations in the feedwater temperature and sodium inlet temperature. For the calculation of the TCOSS code, the variations of the thermal-hydraulic parameters are needed to be inputted. The variations of sodium mass flow, feedwater mass flow and feedwater pressure are shown in Table 6. The variations of sodium inlet temperature and feedwater temperature are shown in Fig. 8. Fig. 9 shows the comparison of steam outlet temperature and sodium outlet temperature between the calculated results of TCOSS and PSM-W. It can be observed that sodium outlet temperature of two codes agree well with each other. The maximum deviation of sodium outlet temperature and steam outlet temperature between two codes is within 3%. Although the agreement of steam outlet temperature between two codes is good overall, there exists a discrepancy at the beginning of the transient. The steam outlet temperature calculated by TCOSS shows a decreasing trend in the
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Table 4 Comparison of TCOSS code with SNR-300 experiment data. Steam pressure, MPa
Feedwater subcooling, K
Mass flow, kg/s
17.0 17.0 16.7 16.7
66 66 49 90
617.4 496.2 663.5 623.1
Steam temperature, K
Exit quality
Exp
TCOSS
Exp
TCOSS
701.15 678.15 700.15 702.15
702.15 685.15 696.15 703.15
1.573 1.460 1.550 1.559
1.565 1.482 1.519 1.552
Fig. 7. Comparison of steam temperature and exit quality.
Table 5 Design parameters of steam generator. Parameters
B1-B
ETEC
Tube length, m Tube outer diameter, m Tube inner diameter, m Pitch, m Number of tubes
20.5 0.02 0.011 0.0305 804
14.021 0.015875 0.010338 0.03168 757
Table 6 Input data for B1-B transient. Time, s
Sodium mass flow, kg/s
Feedwater mass flow, kg/s
Feedwater pressure, MPa
0 10 30 50 100 1100
335.0 100.0 – 26.8 24.8 24.8
32.02 – 1.6 – 1.6
15.78 17.17 16.58 – 16.58
beginning while the steam outlet temperature calculated by PSMW is not. The reason that causes the discrepancy is supposed that the difference of thermal-hydraulic models used in the two codes. In order to make further efforts to verify the TCOSS code, the actual data of ETEC shutdown experiment have been used to compare with the calculation of the TCOSS. Fig. 10 shows the variations of feedwater flow and sodium inlet flow under the shutdown transient. The feedwater flow decreases in the transient while sodium flow increases in the transient overall. At about 1.3 h of the transient, a sharply loss of feedwater flow occurred and then it quickly increases. Fig. 11 shows the variations of the feedwater temperature and sodium inlet temperature under the shutdown transient. The feedwater temperature is staying at
Fig. 8. Variations of sodium inlet temperature and feedwater temperature.
480 K with slight oscillation. The sodium inlet temperature decreases in the former 3 h and then increases. These variations are the boundary conditions of shutdown transient for the calculation of TCOSS. The results are shown in Figs. 12 and 13. Fig. 12 shows the comparison of the sodium outlet temperature between the calculated results of TCOSS and the experimental data under the shutdown transient. It can be observed that although the calculation results agree well with the experimental data, there exists a great difference in the peak of the sodium outlet temperature of two results. It is pointed out by Tzanos (1988) that the experimental data were recorded every 5 min, the peak of the sodium outlet temperature was missed. Therefore, the maximum sodium outlet temperature of the experiment is actually not the peak of the sodium outlet temperature. Thus the maximum sodium
R. Xu et al. / Annals of Nuclear Energy 127 (2019) 385–394
Fig. 9. TCOSS and PSM-W predictions for the B1-B transient.
Fig. 12. Sodium outlet temperature during the shutdown transient.
Fig. 10. Feedwater and sodium inlet flow during the shutdown transient.
Fig. 13. Steam outlet temperature during the shutdown transient.
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computed steam outlet temperature and measured steam outlet temperature at steady state and the very early part of the transient. In the later part of the transient, the computed and measured temperature are practically identical. The difference between TCOSS and experiment at steady state is about 10 K. It is pointed out by Tzanos (1988) that the energy change of water-side was about 0.6% less than that of sodium side at steady-state conditions. Therefore, in the steady state and the very early part of the transient there exists a discrepancy between computed steam outlet temperature and measured steam outlet temperature. Moreover, at steady-state conditions, 1-r values of up to 16 K were determined for the steam outlet temperature (Tzanos, 1988). Thus, the difference of the steam outlet temperature is well within the error of the experimental measurements. From the comparison, it can be concluded that the TCOSS code can be used to calculate the transient of sodium heated once-through steam generator with a good accuracy. Fig. 11. Feedwater and sodium inlet temperature during the shutdown transient.
4. Conclusion outlet temperature calculated by TCOSS is larger than the experiment. Fig. 13 shows the comparison of the steam outlet temperature between the calculated results of TCOSS and the experimental data under the shutdown transient. The computed temperature agrees well with the measured temperature. There is a difference between
A one-dimensional code TCOSS for the transient analysis of sodium heated once-through steam generator has been developed based on the conservation equations of mass, momentum and energy. Some steady calculations and transient calculations are carried out in order to verify the TCOSS code. The steady calculations involve the design results of SG-33 and CFBR and the exper-
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iment data of SNR-300. The transient calculations involve the B1-B numerical experiment and ETEC shutdown experiment. Calculated results are compared with other codes and experimental data. In general, the TCOSS calculated results both of modular type and of integral type steam generators have a good agreement with the design data and experimental data. These calculation results preliminarily verify the reasonability. Thus, the TCOSS code can not only be used to calculate the thermal-hydraulic responses of modular type steam generator but also of integral type steam generator under various transient conditions. The conceptual design of CDSFR steam generator is carried out. Moreover, the steam generators of CDSFR are modular type steam generator with one evaporator and one superheater. Therefore, this code can be used to help the design work of steam generator. For various design concept of steam generator, it can assess the safety of steam generator under different conditions. And the designers can choose a better design concept based on the calculation results of TCOSS. Acknowledgements The authors gratefully acknowledge the supports from Natural Science Foundation of China (Grant No. 11675127) and K. C. Wong Education Foundation. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.anucene.2018.12.027. References Athmalingam, S., 2011. Steam generator for PFBR and future FBR. Berry, G., 1983. Model of a once-through steam generator with moving boundaries and a variable number of nodes. Bird, R.B., Stewart, W.E., Lightfoot, E.N., Spalding, D.B., 1960. Transport Phenomena. John Wiley & Sons, pp. 338–359. Campolunghi, F., Cumo, M., 1977. Subcooled and bulk boiling correlations for thermal design of steam generators.
Carey, V.P., 1992. Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment. Hemisphere Pub, Corp. Collier, J., Thome, J., 1995. 95/03868 Convective boiling and condensation. Fuel & Energy Abstr. 36, 277. Feng, B.P., 1982. The gear program for solving initial value problems in general or stiff ordinary differential equations. J. Numer. Meth. Computer Appl., 12–23 Genic´, S., Arandjelovic´, I., Kolendic´, P., Jaric´, M., Budimir, N., Genic´, V., 2011. A review of explicit approximations of Colebrook’s equation. FME Trans. 39 (2), 67–71. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S., Foundations of Heat Transfer, sixth ed. International Student Version. Kandlikar, S.G., Schmitt, D., Carrano, A.L., Taylor, J.B., 2005. Characterization of surface roughness effects on pressure drop in single-phase flow in minichannels. Phys. Fluids 17, 133–198. Kim, E.-K., Baek, B.-J., 2011. Thermal hydraulic performance analysis of a double tube bundle steam generator for a liquid metal reactor. Ann. Nucl. Energy 38, 2625–2633. Mcadams, W.H., Addoms, J.N., Kennel, W.E., 1949. Heat transfer to superheated steam at high pressures. Trans.am.nuclear Soc 4: no. 1. Mikityuk, K., 2009. Heat transfer to liquid metal: review of data and correlations for tube bundles. Nucl. Eng. Des. 239, 680–687. Miropolskiy, Z.L., 1963. Heat transfer in film boiling of a steam-water mixture in steam generating tubes. Teploenergetika 10, 49–52. Nandakumar, R., Selvaraj, P., Athmalingam, S., Balasubramaniyan, V., Chetal, S.C., 2012. Thermal simulation of sodium heated once through steam generator for a fast reactor. Int. J. Adv. Eng. Sci. Appl. Math. 4, 127–137. Sun, P., Wang, Z., Zhang, J., Su, G., 2016. Thermal–hydraulic design and analysis code development for steam generator of CFR600. Ann. Nucl. Energy 90, 256–263. Tzanos, C.P., 1988. A movable boundary model for once-through steam generator analysis. Nucl. Technol. 82, 5–17. Unal, H.C., 1981. Some aspects of two-phase flow, heat transfer and dynamic instabilities in medium and high pressure steam generators. Ünal, H.C., Van Gasselt, M.L.G., Ludwig, P.W.P.H., 1977. Dynamic instabilities in tubes of a large capacity, straight-tube, once-through sodium heated steam generator. Int. J. Heat Mass Transfer 20, 1389–1399. Vaidyanathan, G., Kothandaraman, A.L., Kumar, L.S.S., Vinod, V., Noushad, I.B., Rajan, K.K., Kalyanasundaram, P., 2010. Development of one-dimensional computer code DESOPT for thermal hydraulic design of sodium-heated once through steam generators. Int. J. Nucl. Energy Sci. Technol. 5, 143–161. Vinod, V., Sivakumar, L.S., Kumar, V.A.S., Noushad, I.B., Padmakumar, G., Rajan, K.K., 2014. Experimental evaluation of the heat transfer performance of sodium heated once through steam generator. Nucl. Eng. Des. 273, 412–420. Weisman, J., Ying, S.H., 1985. A theoretically based critical heat flux prediction for rod bundles at PWR conditions. Nucl. Eng. Des. 85, 239–250. Zhong, X., Jiyang, Y.U., Saeed, M., Yang, C., Zhang, X., 2017. Analysis of hydrodynamic characteristic in straight-tube sodiumheated once-through steam generator. International Conference on Nuclear Engineering.