Atomic Energy, Vol. 111, No. 3, January, 2012 (Russian Original Vol. 111, No. 3, September, 2011)

SODIUM BOILING: ONE-DIMENSIONAL TWO-LIQUID MODELING USING THE SOKRAT-BN COMPUTER CODE

I. G. Kudashov,1 V. N. Semenov,1 A. L. Fokin,1 R. V. Chalyi,1 S. I. Lezhnin,2 and E. V. Usov3

UDC 621.039.534,621.039.526

The results of testing the thermohydraulic module of the SOKRAT-BN computing code for analyzing accidents with boiling of sodium coolant in fast reactors are presented. The computational results are compared with experimental data. It is shown that the thermohydraulic module of the SOKRAT-BN code models stationary sodium boiling well. Using as a basis the results obtained by modeling sodium boiling in a vertical heated channel, a system of closure relations for calculating two-phase sodium flow regimes, including the interphase velocity, was modified and checked. Modeling sodium boiling in a vertical annular channel also showed that the closure relations incorporated in the thermohydraulic module of the SOKRAT-BN code are suitable for calculating heat-exchange with a wall.

Recent years have seen an increasingly wider use of the methods of computational fluid dynamics (CFD methods) for calculating flow in reactor facilities. However, because of their complexity and resource-intensiveness three-dimensional models have still not fully entered computational practice even for single-phase flows. For this reason, quasi-one-dimensional thermohydraulic modules remain an integral part of the unified system of codes for validating the safety of nuclear power plants with thermal and fast reactors with liquid metal coolant, especially in the two-phase region. Since sodium boiling accompanies most scenarios of unanticipated accidents involving fast reactors, the quality of sodium boiling models is a condition for validating the safety of such reactors. The two-fluid approximation for modeling boiling in channels is widely used because it is easy to implement and the calculations and experiments are in agreement with one another. In addition, there are a large number of methods for circumventing the limitations inhering in this approximation. In the two-fluid approximation for two-phase flow in a channel, the equations of conservation of mass, momentum, and energy are written down for each phase. An important component of the approximation is the choice of the system of closure relations for the exchange of mass, momentum, and energy between phases. The closure relations in the SOKRAT-BN code [1] are given by the interphase interactions, such as interphase heat and mass transfer and friction, interaction of the phases with the channel walls (friction against the channel walls, local resistance, and heat exchange with the walls), and heat release. The system of closure relations in the two-phase region is directly related with the sodium flow regimes. The flow regimes were split up on the basis of an analysis of the experimental work and modern codes simulating sodium boiling. 1

Institute of Problems in the Safe Development of Nuclear Energy, Russian Academy of Sciences (IBRAE RAN), Moscow. Kutateladze Institute of Thermal Physics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk. 3 Novosibirsk State University. 2

Translated from Atomnaya Énergiya, Vol. 111, No. 3, pp. 136–139, September, 2011. Original article submitted April 13, 2011. 1063-4258/12/11103-0179 ©2012 Springer Science+Business Media, Inc.

179

Fig.1. Pressure difference calculated with the SOKRAT-BN code using the correlation of [7] (B) and [10] (b) and the experimental data (c) with coolant temperature at the entrance into the heated channel and heat flux from the walls of the heated channel, respectively, 838 K, 1.38 MW/m2 (a), 843 K, 1.27 MW/m2 (b), 847 K, 1.09 MW/m2 (c), 852 K, 1.75 MW/m2 (d), 858 K, 1.57 MW/m2 (e).

Four flow regimes are considered in the SOKRAT-BN code. Two regimes correspond to single-phase flow – flow of liquid and gaseous sodium. Two regimes are separated in the two-phase region – subcritical with specific volume vapor content to 0.957 and supercritical, which transitions into a flow regime of gaseous sodium with specific volume vapor content >0.957. Here, the transcritical regime is a regime with partial drying of the channel surface. In the subcritical flow regime, only the annular regime is considered as being most likely, which is confirmed by experimental and theoretical studies of boiling of alkali metals in channels at pressure close to atmospheric [2–4]. Similar flow regimes are also singled out in the codes THERMIT-6S [5] and GRIF-SM [6]. As calculations have shown, this model gives a satisfactory description of stationary boiling in sodium. For calculating the interphase surface velocity, which is used to obtain the closure relations for the equations of conservation of energy and momentum, it is proposed for two-phase flow regimes that this velocity equals the velocity of the fluid phase [7]. The friction force of the liquid and gas with the interphase surface is calculated using the cor-

180

Fig. 2. Computed and measured [7] pressure differences with heat flux 1.38 (B), 1.27 (b), 1.09 (a), 1.75 (A), and 1.58 MW/m2 (×).

Fig. 3. Position of the boiling onset point, calculated with the SOKRAT-BN code (B), and the experimental data of [7] (c).

relation proposed in [8]. It should be noted that when calculating nonstationary boiling processes it is desirable to take account of the presence of bubble and projectile flow regimes [7]. A method for calculating the friction of two-phase flow against the walls has been proposed in [9]. This model is most suitable for describing the motion of vapor–liquid mixtures in a separate flow regime of the phases. To calculate the pressure difference in a two-phase flow, this model uses a parameter ϕ2 that represents the two-phase quality and takes account of the increase of the friction resistance in two-phase flow as compared with fluid flow. To calculate this parameter, the correlation ϕ2 = 1/(1 – α)2, where α is the true volume vapor content, is chosen in the SOKRTA-BM on the basis of analysis of various relations [10].

181

Fig. 4. Temperature (B) and velocity (——) at the exit from a heated channel, calculated with the SOKRAT-BN code, and the experimental data for a nonstationary sodium boiling regime [10]: a) temperature; - - - ) velocity.

Fig. 5. Wall temperature along the channel axis in a stationary regime (b) and at boiling onset (a), calculated with the SOKRAT-BN code, experimental data of [11] (B) and (A) for these regimes, respectively, and calculation using the BLOW3 code (——, - - -, respectively).

The experiments of [11, 12] were used to test the thermohydraulic module of the SOKRAT-BN code on the basis of the closure relations for two-phase flow. Because their internal geometry is simple, these experiments make it possible to analyze the closure relations used in their modeling. The first experiment studies a flow of a flow of boiling liquid sodium in a heated channel with the pressure differential and the position of the point of boiling being measured [11]. Analysis of the experimental data showed that wall friction makes the main contribution to the pressure differential. Thus, the experiment makes it possible to analyze the adequacy of the closure relations, incorporated in the thermohydraulic program module, for the wall friction (Fig. 1). Two calculations were performed using two different correlations for the two-phase quality parameter – the correlation in [10] and [11]: logϕ = = 0.1046(log X)2 – 0.5098logX + 0.6252, where X is a parameter [9]. The correlation of [11] was obtained by fitting the exper182

imental data. As one can see in Fig. 1, the relation of [10] is better suited for calculating the wall friction in the two-phase region with high vapor content. Figure 2 shows that the data are in agreement with one another. The computed pressure difference in the region of high heat fluxes is understated by 15%. The overstatement of the experimental data could be due to the migration of drops from the surface of the film into the vapor nucleus, which is neglected in the map of the sodium flow regimes that is presently used in the SOKRAT-BN code. The dependences presented in Fig. 3 show that the results of the numerical calculation are in good agreement with experiment. Thus, the friction model and the proposed boiling regimes of sodium coolant make it possible to calculate, to good accuracy, the integral characteristics of vapor-liquid sodium flow under the condition of steady boiling. The thermohydraulic module of the SOKRAT-BN code for nonstationary regimes was tested in an experiment on sodium boiling onset in an annular channel with the flow rate decreasing linearly [12, 13]. This experiment, where the processes characteristic for an accident with cessation of coolant flow through the core and failure of the emergency protection of the reactor are studied, makes it possible to check the possibility of using the chosen closure relations to calculate interphase heat and mass transfer as well as heat exchange with the walls. It is evident in Figs. 4 and 5 that the computational results are in agreement with the experimental data of [12, 13]. Figure 5 displays, in addition to the experimental data, the computational results obtained with the BLOW3 code (Germany) [13]. The calculations were performed for the stationary state and for boiling onset. It can be concluded on the basis of the results obtained that the simplified flow regimes can be used to model qualitatively correctly nonstationary processes as well. The thermohydraulic module of the code is being modified, using an expanded map of the boiling regimes, in order to calculate nonstationary processes more accurately. In closing, we note that the results of testing the thermohydraulic module of the SOKRAT-BN code on the experimental data of [11–13] showed this code holds promise for validating the safety of sodium-cooled reactor facilities. This study was supported by the Russian Foundation for Basic Research (Grant No. 09-08-13758-ofi_ts).

REFERENCES 1.

2. 3. 4. 5. 6.

7. 8. 9. 10.

I. S. Vozhakov, V. O. Ivanitskii, D. I. Kachulin, et al., “Development of a program system for modeling accident regimes of sodium-cooled fast reactors,” in: Scientific Works of the All-Russia Scientific-Applications Conference on Thermophysical Foundations of Power-Generation Technologies, Tomsk (2010), pp. 138–142. Yu. A. Zeigarnik and V. D. Litvinov, “On boiling of liquid alkali metals in tubes,” Teplofiz. Vys. Temp., 7, No. 2, 374–376 (1969). Yu. A. Zeigarnik, V. P. Kirillov, P. A. Ushakov, and M. N. Ivanovskii, “Heat exchange of liquid metals during boiling and condensation,” Teploenergetika, No. 3, 2–8 (2001). N. S. Grachev, V. N. Zelenskii, P. L. Kirillov, et al., “Heat transfer and fluid dynamics during boiling of potassium in tubes,” Teplofiz. Vys. Temp., 6, No. 4, 682–690 (1968). H. No and M. Kazimi, “An investigation of the physical foundations of two-fluid representation of sodium boiling in the liquid-metal fast breeder reactor,” Nucl. Sci. Eng., 97, 327–343 (1987). A. V. Volkov, I. A. Kuznetsov, and Yu. E. Shvetsov, Calculation of Sodium Boiling during a Fast-Reactor Accident Taking Account of the Distributed Nature of the Parameters over a Fuel Assembly Cross Section, Preprint FEI-2787 (1999). Yu. A. Zeigarnik and V. D. Litvinov, Boiling of Alkali Metals, Nauka, Moscow (1983). G. B. Wallis, One-Dimensional Two-Phase Flows [Russian translation], Mir, Moscow (1972). R. Lockhart and R. Martinelli, “Proposed correlation of data for isothermal two-phase, two-component flow in pipes,” Chem. Eng. Prog., 45, No. 1, 39–48 (1949). S. Levy, “Steam-slip theoretical model prediction from momentum model,” J. Heat Transfer, 82, No. 3, 113–124 (1960). 183

11. 12. 13.

184

H. Kotowski and V. Savtteri, “Fundamentals of liquid metal boiling thermohydraulics,” Nucl. Sci. Design, 82, 281–304 (1984). A. Kaiser and W. Peppler, “Sodium boiling experiments in an annular test section under flow rundown conditions,” in: KfK 2389 (1977), pp. 1–18. P. Wirtz, “Ein Beitrag zur theoretischen Beschreibung des Siedens unter Störfallbedingungen in natriumgekühlten schnellen Reaktoren,” in: KfF 1858 (1973), pp. 1–107.

SODIUM BOILING: ONE-DIMENSIONAL TWO-LIQUID MODELING USING THE SOKRAT-BN COMPUTER CODE

I. G. Kudashov,1 V. N. Semenov,1 A. L. Fokin,1 R. V. Chalyi,1 S. I. Lezhnin,2 and E. V. Usov3

UDC 621.039.534,621.039.526

The results of testing the thermohydraulic module of the SOKRAT-BN computing code for analyzing accidents with boiling of sodium coolant in fast reactors are presented. The computational results are compared with experimental data. It is shown that the thermohydraulic module of the SOKRAT-BN code models stationary sodium boiling well. Using as a basis the results obtained by modeling sodium boiling in a vertical heated channel, a system of closure relations for calculating two-phase sodium flow regimes, including the interphase velocity, was modified and checked. Modeling sodium boiling in a vertical annular channel also showed that the closure relations incorporated in the thermohydraulic module of the SOKRAT-BN code are suitable for calculating heat-exchange with a wall.

Recent years have seen an increasingly wider use of the methods of computational fluid dynamics (CFD methods) for calculating flow in reactor facilities. However, because of their complexity and resource-intensiveness three-dimensional models have still not fully entered computational practice even for single-phase flows. For this reason, quasi-one-dimensional thermohydraulic modules remain an integral part of the unified system of codes for validating the safety of nuclear power plants with thermal and fast reactors with liquid metal coolant, especially in the two-phase region. Since sodium boiling accompanies most scenarios of unanticipated accidents involving fast reactors, the quality of sodium boiling models is a condition for validating the safety of such reactors. The two-fluid approximation for modeling boiling in channels is widely used because it is easy to implement and the calculations and experiments are in agreement with one another. In addition, there are a large number of methods for circumventing the limitations inhering in this approximation. In the two-fluid approximation for two-phase flow in a channel, the equations of conservation of mass, momentum, and energy are written down for each phase. An important component of the approximation is the choice of the system of closure relations for the exchange of mass, momentum, and energy between phases. The closure relations in the SOKRAT-BN code [1] are given by the interphase interactions, such as interphase heat and mass transfer and friction, interaction of the phases with the channel walls (friction against the channel walls, local resistance, and heat exchange with the walls), and heat release. The system of closure relations in the two-phase region is directly related with the sodium flow regimes. The flow regimes were split up on the basis of an analysis of the experimental work and modern codes simulating sodium boiling. 1

Institute of Problems in the Safe Development of Nuclear Energy, Russian Academy of Sciences (IBRAE RAN), Moscow. Kutateladze Institute of Thermal Physics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk. 3 Novosibirsk State University. 2

Translated from Atomnaya Énergiya, Vol. 111, No. 3, pp. 136–139, September, 2011. Original article submitted April 13, 2011. 1063-4258/12/11103-0179 ©2012 Springer Science+Business Media, Inc.

179

Fig.1. Pressure difference calculated with the SOKRAT-BN code using the correlation of [7] (B) and [10] (b) and the experimental data (c) with coolant temperature at the entrance into the heated channel and heat flux from the walls of the heated channel, respectively, 838 K, 1.38 MW/m2 (a), 843 K, 1.27 MW/m2 (b), 847 K, 1.09 MW/m2 (c), 852 K, 1.75 MW/m2 (d), 858 K, 1.57 MW/m2 (e).

Four flow regimes are considered in the SOKRAT-BN code. Two regimes correspond to single-phase flow – flow of liquid and gaseous sodium. Two regimes are separated in the two-phase region – subcritical with specific volume vapor content to 0.957 and supercritical, which transitions into a flow regime of gaseous sodium with specific volume vapor content >0.957. Here, the transcritical regime is a regime with partial drying of the channel surface. In the subcritical flow regime, only the annular regime is considered as being most likely, which is confirmed by experimental and theoretical studies of boiling of alkali metals in channels at pressure close to atmospheric [2–4]. Similar flow regimes are also singled out in the codes THERMIT-6S [5] and GRIF-SM [6]. As calculations have shown, this model gives a satisfactory description of stationary boiling in sodium. For calculating the interphase surface velocity, which is used to obtain the closure relations for the equations of conservation of energy and momentum, it is proposed for two-phase flow regimes that this velocity equals the velocity of the fluid phase [7]. The friction force of the liquid and gas with the interphase surface is calculated using the cor-

180

Fig. 2. Computed and measured [7] pressure differences with heat flux 1.38 (B), 1.27 (b), 1.09 (a), 1.75 (A), and 1.58 MW/m2 (×).

Fig. 3. Position of the boiling onset point, calculated with the SOKRAT-BN code (B), and the experimental data of [7] (c).

relation proposed in [8]. It should be noted that when calculating nonstationary boiling processes it is desirable to take account of the presence of bubble and projectile flow regimes [7]. A method for calculating the friction of two-phase flow against the walls has been proposed in [9]. This model is most suitable for describing the motion of vapor–liquid mixtures in a separate flow regime of the phases. To calculate the pressure difference in a two-phase flow, this model uses a parameter ϕ2 that represents the two-phase quality and takes account of the increase of the friction resistance in two-phase flow as compared with fluid flow. To calculate this parameter, the correlation ϕ2 = 1/(1 – α)2, where α is the true volume vapor content, is chosen in the SOKRTA-BM on the basis of analysis of various relations [10].

181

Fig. 4. Temperature (B) and velocity (——) at the exit from a heated channel, calculated with the SOKRAT-BN code, and the experimental data for a nonstationary sodium boiling regime [10]: a) temperature; - - - ) velocity.

Fig. 5. Wall temperature along the channel axis in a stationary regime (b) and at boiling onset (a), calculated with the SOKRAT-BN code, experimental data of [11] (B) and (A) for these regimes, respectively, and calculation using the BLOW3 code (——, - - -, respectively).

The experiments of [11, 12] were used to test the thermohydraulic module of the SOKRAT-BN code on the basis of the closure relations for two-phase flow. Because their internal geometry is simple, these experiments make it possible to analyze the closure relations used in their modeling. The first experiment studies a flow of a flow of boiling liquid sodium in a heated channel with the pressure differential and the position of the point of boiling being measured [11]. Analysis of the experimental data showed that wall friction makes the main contribution to the pressure differential. Thus, the experiment makes it possible to analyze the adequacy of the closure relations, incorporated in the thermohydraulic program module, for the wall friction (Fig. 1). Two calculations were performed using two different correlations for the two-phase quality parameter – the correlation in [10] and [11]: logϕ = = 0.1046(log X)2 – 0.5098logX + 0.6252, where X is a parameter [9]. The correlation of [11] was obtained by fitting the exper182

imental data. As one can see in Fig. 1, the relation of [10] is better suited for calculating the wall friction in the two-phase region with high vapor content. Figure 2 shows that the data are in agreement with one another. The computed pressure difference in the region of high heat fluxes is understated by 15%. The overstatement of the experimental data could be due to the migration of drops from the surface of the film into the vapor nucleus, which is neglected in the map of the sodium flow regimes that is presently used in the SOKRAT-BN code. The dependences presented in Fig. 3 show that the results of the numerical calculation are in good agreement with experiment. Thus, the friction model and the proposed boiling regimes of sodium coolant make it possible to calculate, to good accuracy, the integral characteristics of vapor-liquid sodium flow under the condition of steady boiling. The thermohydraulic module of the SOKRAT-BN code for nonstationary regimes was tested in an experiment on sodium boiling onset in an annular channel with the flow rate decreasing linearly [12, 13]. This experiment, where the processes characteristic for an accident with cessation of coolant flow through the core and failure of the emergency protection of the reactor are studied, makes it possible to check the possibility of using the chosen closure relations to calculate interphase heat and mass transfer as well as heat exchange with the walls. It is evident in Figs. 4 and 5 that the computational results are in agreement with the experimental data of [12, 13]. Figure 5 displays, in addition to the experimental data, the computational results obtained with the BLOW3 code (Germany) [13]. The calculations were performed for the stationary state and for boiling onset. It can be concluded on the basis of the results obtained that the simplified flow regimes can be used to model qualitatively correctly nonstationary processes as well. The thermohydraulic module of the code is being modified, using an expanded map of the boiling regimes, in order to calculate nonstationary processes more accurately. In closing, we note that the results of testing the thermohydraulic module of the SOKRAT-BN code on the experimental data of [11–13] showed this code holds promise for validating the safety of sodium-cooled reactor facilities. This study was supported by the Russian Foundation for Basic Research (Grant No. 09-08-13758-ofi_ts).

REFERENCES 1.

2. 3. 4. 5. 6.

7. 8. 9. 10.

I. S. Vozhakov, V. O. Ivanitskii, D. I. Kachulin, et al., “Development of a program system for modeling accident regimes of sodium-cooled fast reactors,” in: Scientific Works of the All-Russia Scientific-Applications Conference on Thermophysical Foundations of Power-Generation Technologies, Tomsk (2010), pp. 138–142. Yu. A. Zeigarnik and V. D. Litvinov, “On boiling of liquid alkali metals in tubes,” Teplofiz. Vys. Temp., 7, No. 2, 374–376 (1969). Yu. A. Zeigarnik, V. P. Kirillov, P. A. Ushakov, and M. N. Ivanovskii, “Heat exchange of liquid metals during boiling and condensation,” Teploenergetika, No. 3, 2–8 (2001). N. S. Grachev, V. N. Zelenskii, P. L. Kirillov, et al., “Heat transfer and fluid dynamics during boiling of potassium in tubes,” Teplofiz. Vys. Temp., 6, No. 4, 682–690 (1968). H. No and M. Kazimi, “An investigation of the physical foundations of two-fluid representation of sodium boiling in the liquid-metal fast breeder reactor,” Nucl. Sci. Eng., 97, 327–343 (1987). A. V. Volkov, I. A. Kuznetsov, and Yu. E. Shvetsov, Calculation of Sodium Boiling during a Fast-Reactor Accident Taking Account of the Distributed Nature of the Parameters over a Fuel Assembly Cross Section, Preprint FEI-2787 (1999). Yu. A. Zeigarnik and V. D. Litvinov, Boiling of Alkali Metals, Nauka, Moscow (1983). G. B. Wallis, One-Dimensional Two-Phase Flows [Russian translation], Mir, Moscow (1972). R. Lockhart and R. Martinelli, “Proposed correlation of data for isothermal two-phase, two-component flow in pipes,” Chem. Eng. Prog., 45, No. 1, 39–48 (1949). S. Levy, “Steam-slip theoretical model prediction from momentum model,” J. Heat Transfer, 82, No. 3, 113–124 (1960). 183

11. 12. 13.

184

H. Kotowski and V. Savtteri, “Fundamentals of liquid metal boiling thermohydraulics,” Nucl. Sci. Design, 82, 281–304 (1984). A. Kaiser and W. Peppler, “Sodium boiling experiments in an annular test section under flow rundown conditions,” in: KfK 2389 (1977), pp. 1–18. P. Wirtz, “Ein Beitrag zur theoretischen Beschreibung des Siedens unter Störfallbedingungen in natriumgekühlten schnellen Reaktoren,” in: KfF 1858 (1973), pp. 1–107.