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A theoretical study of cyclon-effect in PWR downcomer during a LOCA blowdown
Annals of Nuclear Energy 36 (2009) 1294–1297
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical Note
A theoretical study of cyclon-effect in PWR downcomer during a LOCA blowdown F.J. Arias a,*, F. Reventós a,b a b
Department of Physics and Nuclear Engineering, Technical University of Catalonia, (UPC), Spain Institute of Energy Technologies, Technical University of Catalonia, (UPC), Spain
a r t i c l e
i n f o
Article history: Received 10 February 2009 Received in revised form 27 March 2009 Accepted 3 April 2009 Available online 9 May 2009
a b s t r a c t The current strategy based on the use of so-called cross-junction, allows partial modeling during blowdown episodes driven by larger-scale (flow features such as helical profile) in downcomers nuclear reactors. However a subtle but significant effect may appear by the combined action of two factors: on the one hand high azimuthal flow, on the other hand the intrinsic curvature of downcomer, and additionally in presence of a two-phases (vapor–liquid) a cyclon-effect can manifest. The present paper is a theoretical analysis of a possible cyclon-effect during blowdown episodes that allows a qualitative estimate of the impact on the calculations. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
(b) The flow is turbulent. (c) In agreement with assumption b a turbulent profile approximation is used for the velocity. (d) In agreement with assumption b, a bubble diffusion coefficient Db is defined using the turbulent core approach (the called ‘‘Prandtl law”). (e) It is assumed that at a given radial position y liquid and vapor velocities, ul and uv , are equal, in other words an homogeneous model is considered. This simplification, although unusual, is not far from reality in the first stages of this kind of scenarios and, in any case, it is sufficiently accurate to provide a good picture of the degree of stratification. (f) Vapor is in form of spherical bubbles of radius a, all of them having the same size. (a) In the same way, according to assumption f, bubble coalescence phenomena are not considered.
Helical flow appears as a result of transversal pressure gradients. Such situations are of interest in industrial applications like chemical, and biological technology. However in PWR nuclear reactor the effect can be appear during LOCA transients in the downcomer geometry. in current 2D thermohydraulic codes only gravitational stratification (vertical and/or horizontal) is considered (Relap, Trace, Cathare, etc.) where the current strategy of use of cross-junction in the downcomer simulation in blowdown episodes has been widely adopted by the research community. However a subtle but significant cyclon-effect may appear by the combined action of high azimuthal flow and the intrinsic curvature of downcomer. Here an attempt is done to estimate qualitatively the importance of a possible cyclon-effect in the blowdown episodes in a Loss of Coolant Accident LOCA in a typical PWR nuclear reactor. 2. Theoretical approach Consider a duct of annular geometry, of internal and external radio r1 and r2 , respectively. The coordinate system used is represented in Fig. 1b; the coordinate origin is placed in the external wall; the three spatial directions are represented y (radial), z (axial) and h (azimuthal). In order to simplify the stratification analysis, only azimuthal velocity (^ eh direction) is considered in the fluid. The main assumptions will be described below. (a) The annular geometry may be represented by a cylindrical pipe with a hydraulic diameter associated. This approximation is sufficiently accurate for engineering calculations. * Corresponding author. Tel.: +34 934016047; fax: +34 934017148. E-mail address: frariasm7@fis.ub.edu (F.J. Arias). 0306-4549/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2009.04.003
2.1. Equations The momentum equation in cylindrical coordinates can be write as
@ur @ur uh @ur @ur u2h þ ur þ þ uz @t @r r @h @z r " # @p 1 @ @ur 1 @ 2 ur @ 2 ur ur 2 @uh þ 2 2 2 ¼ qg z þl r þ 2 @r r @r r @h2 r @h @r @z r
q
ð1Þ taking into account that the relaxation time t r (full stratification time) may be calculate in first approximation as the time spend by a single small bubble of radius a and velocity v b (deduced below, Eq. (6)) traveling at a distance of Dr ¼ r 2 r 1 ; tr in first approximation yields the following relation:
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
1295
Nomenclature a c g J r s u um
bubble radio local bubble concentration acceleration due to gravity bubble flux in direction normal to wall radio y=r h , dimensionless distance from wall velocity of two-phase mixture at distance y from wall velocity of two-phase mixture at tube centerline transverse bubble velocity distance from wall
vb y
Greek symbols void fraction void fraction at tube centerline qv =ql b j universal constant l dynamic viscosity
a am
tr
9 Dr r2 ml 2 a2 u2h
ð2Þ
hence, for typical values in a LOCA with uh ¼ 90 m=s; r 2 r 1 ¼ 0:25 m and r2 ¼ 1 m and a characteristic blowdown time t b 5 s, where is clear that ttbr 1, the assumption of steady state h ¼ 0. Then, in a typical LOCA as menis justified, i.e, ur ¼ 0, and @u @h tioned above, Eq. (1) becomes
u2h @p ¼ @r r
ð3Þ
The above equation represents the centripetal acceleration, which is directed towards the center of curvature of the streamline and is associated with the change in direction of the velocity of a bubble. Consider a bubble of radius a and density qv , surrounded by liquid of density ql and viscosity ll ; the azimuthal velocity being uh . The body force experienced by the bubble is given by Eq. (3), considering a spherical bubble, it is easy shown, that the centripetal force F c acting upon the bubble is given by
Fc ¼
4 u2 pql a3 ð1 bÞ h 3 r
ð4Þ
m q s
kinematic viscosity density of two-phase mixture wall shear stress in two-phase system circumferential component kinetic energy correction factor Martinelly–Nelson parameter constant defined by Eq. (15)
h
c UM W
Subscripts 1, 2 inside, outside h hydraulic l,v liquid, vapor sp single phase tp two phase z axial component r radial component
where r is the distance to the center of the pipe (see Fig. 1b) and b ¼ qv =ql . The centripetal force, in first approximation, is equal to drag forces. According to Stokes’law in spherical approximation:
4 u2 pq a3 ð1 bÞ h ¼ 6pll a v b 3 l r
ð5Þ
where v b is the radial velocity of the bubble in ^ey direction. hence, from Eq. (5):
vb ¼
2 a2 v2 ð1 bÞ h 9 ml r
ð6Þ
where ml is the kinematic viscosity ðml ¼ ll =ql Þ. According to assumption (c), a turbulent velocity profile is considered. The azimuthal velocity profile in the radial direction is given by
uh ¼ um þ
rffiffiffiffiffi
s ln s j ql 1
ð7Þ
where um is the velocity of the two-phase mixture at the centerline of the equivalent pipe, j is a universal constant, s is the two-phase wall shear stress, and s is the dimensionless distance defined as (see Fig. 1b)
s¼
y r2 r1
ð8Þ
In general, the drag force for a turbulent fluid flow surrounded by a solid can be expressed as the product of a friction factor and the mean kinetic energy per unit volume. In circular pipes
s¼
ql u2m c f 2
ð9Þ
where c is the kinetic energy correction factor (in circular pipes c ’ 1) and f is the friction factor. Following the Blasius relationship
f ¼ 0:079 Re0:25
ð10Þ
The bubbles concentration, c, in a given sector of the angular cylinder can be represented by a simple equation:
d dc d @c þ ðcv b Þ þ Ssource ¼ Db dy dy dy @t
Fig. 1. (a) Flow pattern in the downcomer during LOCA. (b) Physical model and coordinates system.
ð11Þ
where Db is the bubbles (turbulent) diffusion coefficient and v b is the velocity of the bubbles driven by inertial forces (Eq. (6)). Ssource is the source (evaporation) term. At this point, due to t b t r , the source term may be neglected, hence, Eq. (11) is transformed as
1296
Db
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
dc ¼ cv b dy
ð12Þ
which represents that the bubble flow in the inside direction (due to inertial forces) is equal to the flux in the outside direction (turbulent dispersion forces). According to assumption (d), the bubbles diffusion coefficient is given by
dul Db ¼ j2 y2 dy
ð13Þ
Combining Eqs. (7), (12) and (13), and using the Reynolds analogy (Eq. (9)), and taking into account that the concentration of bubbles, c, is proportional to the void fraction a, finally the following expression is obtained:
1 da 1 dc W a2 um 1 ¼ ¼ 1þ a dy c dy r y j
!2 rffiffiffi f y ln 2 rh
ð14Þ
where
W¼
2 ð1 bÞ 9 ml j
sffiffiffi 2 f
ð15Þ
In Eq. (14), r is the real distance, i.e., r ¼ r2 y. At this point, to simplify the equations, in an actual downcomer geometry it can be considered valid the approximation: r2 y ! r r 2 . Hence, integrating Eq. (14) between ða; sÞ and ðam ; 1Þ, one obtains
"
rffiffiffi
a W a2 um f 1 f 2 3 ln s þ ln s ln ¼ ln s þ am 6 j2 r2 j 2
#
Fig. 3. Stratification effects on the mass flow according to Eq. (17).
W ¼ W l U2M a2l
ð17Þ
where UM is the Martinelly–Nelson parameter and al is the void fraction liquid (i.e., al ¼ 1 a). Fig. 3 shows the shape of the curve predicted by Eq. (7) for steam–water system.
ð16Þ
Fig. 2 compares the variation in the shape of the curves predicted by Eq. (16) with the bubble radius for typical values in a steam–water system: j ¼ 0:4; ml ¼ 0:0158 m2 =s; central velocity um ¼ 90 m=s; and ra2 ¼ 3 103 ; ra2 ¼ 1:5 102 ; ra2 ¼ 1:0 102 with a friction factor f ¼ 0:0055. Fig. 2 shows the strong sensitivity to bubble size. This influence of the bubble radius can be found in other.
3. Comparison with numerical simulation: RELAP-3DÒ To demonstrate the application of the analytical solution, there are two groups of methodologies in investigation of the two-phase flow and of the boiling process.
2.2. Mass flow drop in radial stratification The effect of radial stratification can be calculated by making some considerations on the pressure drop. Pressure drop in stratified two-phase flow systems was calculated by Levi (1952), where it is easy shown the following relation between the mass flow nonstratified (W) and stratified W l .
Fig. 2. Radial void-fraction distribution calculated from Eq. (16) as function of dimensionless distance s for various values of a=r2 .
Fig. 4. Nodalization of the multi-dimensional model of the downcomer with RELAP-3DÒ.
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
In the first group are the experiments that are conducted with full scales apparatus. These experiments are costly and time-consuming but they yield results that can be applied directly to a design of a nuclear reactor. However, theses experiments, because the complexity of the phenomenon, because of incomplete instrumentation, and other experimental difficulties, very seldom provide detailed information which is needed for integrating the results and the understanding of the process. for example in the problem being considered in the present paper, the experiment performed at the full-scale called ‘‘Upper Plenum Test Facility (UPTF)” to investigate the flow phenomena in the primary system of a pressurized water reactor (PWR) during a loss-of-coolant accident (LOCA), the helicoidal two-phase flow is clearly observed, however the UPTF instrumentation was generally not sufficient to measure multi-dimensional phenomena in the downcomer and nothing the radial stratification was discussed. For an exhaustive review the reader can refer to the reference (CSNI, 1999) of the nuclear energy agency. The second methodologies can be performed on thermohydraulics computer programs, these simulations are relatively cheap a require a short time to provide de desired information. Although the results cannot be applied directly to a system design or validity of analytical expressions, these codes yield basic information and they clearly indicate the basic characteristic of two-phase flow. A 3D model with RELAP-3DÒ of full vessel model of a 1000 MWe three-loop PWR was developed using two multidimensional components to simulate a blowdown. In order to clarify the analysis and to underline the effect of radial stratification the simulated scenario consists in an isolated full PWR vessel, initially at nominal pressure (about 16 MPa) and with liquid at full power inlet temperature (about 290°C), that suffers a 100% break in a cold leg. The conditions at the break are maintained at 1 bar. The downcomer component (11,000) was divided into three 120° azimuthal sectors and six radial rings. The three azimuthal sectors corresponded to the three nozzles connecting the loops and vessel. The axial nodalization of each multidimensional component was of six levels. The actual shape and dimensions of the downcomer nodalization (component 11,000) are shown in Fig. 4. The broken loop is connected to sector loop 1 in sub-component 11,700, with a diameter of 1.96 m. Three representatives sectors have been analysed: in loop 1: sub-component 11,702; and in loop 3: sub-component 11,902 and 11,904 (loop 2 is symmetrically equal to loop 3). Fig. 5 compares the radial stratification (variation of the void fraction) as function of radial position s for the subcomponents 11,702, 11,902 and 11,904 calculated with RELAP3DÒ and the results from Eq. (16). Referring to Fig. 5, it is seen that the radial stratification is stronger for the sub-component 11,902 and 11,904 in loop 3, compares with sub-component 11,702 in loop 1. The above behavior is easily explained because in loop 1 the azimuthal velocity (circumferential) is less than loop 3 being practically full axial velocity uz uh , then, the cyclon-effect is strongly reduced. In the same manner, is expected that radial stratification in sub-component 11,902 to be higher than sub-component 11,904 for loop 3, and effectively this situation is also observed in the simulation, where
1297
Fig. 5. Radial void-fraction distribution calculated from Eq. (16) and simulation data.
Eq. (16) agree with the simulation within 10%, in good agreement, taking into account the simplifying assumptions. 4. Summary and conclusions Radial stratification due to centrifugal forces in downcomer during a typical LOCA and its significance with regard to mass flow prediction was discussed. Utilizing a simplified geometrical model, an analytical expression (Eq. (16)) for the void fraction profile was derived. The hydraulic responses with the 3D input model with RELAP3D were generally in reasonable agreement with Eq. (16). The present work provides added importance in the need of 3D simulation for the downcomer, where, for example the use of cross-junction in 2D-thermohydraulics codes is clearly not sufficient. Acknowledgements The authors wishes to thanks the anonymous referees, for helpful comments. The assistance of X. Sabater in the simulation is gratefully acknowledged. FJA presents this work as a thesis to the Department of Physics and Nuclear Engineering at the Technical University of Catalonia, in partial fulfillment with the requirements for the Ph.D. degree. Portion of this work was performed under auspices and financial support of the Council for Nuclear Safety of Spain-CSN. References CSNI Code Validation Matrix Version 2 (August 1999), UPTF-Separate Effects Test Data, Nuclear Energy Agency (NEA). Levi, S., 1952. Theory of pressure drop and heat transfer for two-phase-component annular flow in pipes. In: Proceedings of Second Midwestern Conference of Fluids Mechanics. Ohio State University, Engineering Experimental Station Bulletin No. 149, p. 337.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical Note
A theoretical study of cyclon-effect in PWR downcomer during a LOCA blowdown F.J. Arias a,*, F. Reventós a,b a b
Department of Physics and Nuclear Engineering, Technical University of Catalonia, (UPC), Spain Institute of Energy Technologies, Technical University of Catalonia, (UPC), Spain
a r t i c l e
i n f o
Article history: Received 10 February 2009 Received in revised form 27 March 2009 Accepted 3 April 2009 Available online 9 May 2009
a b s t r a c t The current strategy based on the use of so-called cross-junction, allows partial modeling during blowdown episodes driven by larger-scale (flow features such as helical profile) in downcomers nuclear reactors. However a subtle but significant effect may appear by the combined action of two factors: on the one hand high azimuthal flow, on the other hand the intrinsic curvature of downcomer, and additionally in presence of a two-phases (vapor–liquid) a cyclon-effect can manifest. The present paper is a theoretical analysis of a possible cyclon-effect during blowdown episodes that allows a qualitative estimate of the impact on the calculations. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
(b) The flow is turbulent. (c) In agreement with assumption b a turbulent profile approximation is used for the velocity. (d) In agreement with assumption b, a bubble diffusion coefficient Db is defined using the turbulent core approach (the called ‘‘Prandtl law”). (e) It is assumed that at a given radial position y liquid and vapor velocities, ul and uv , are equal, in other words an homogeneous model is considered. This simplification, although unusual, is not far from reality in the first stages of this kind of scenarios and, in any case, it is sufficiently accurate to provide a good picture of the degree of stratification. (f) Vapor is in form of spherical bubbles of radius a, all of them having the same size. (a) In the same way, according to assumption f, bubble coalescence phenomena are not considered.
Helical flow appears as a result of transversal pressure gradients. Such situations are of interest in industrial applications like chemical, and biological technology. However in PWR nuclear reactor the effect can be appear during LOCA transients in the downcomer geometry. in current 2D thermohydraulic codes only gravitational stratification (vertical and/or horizontal) is considered (Relap, Trace, Cathare, etc.) where the current strategy of use of cross-junction in the downcomer simulation in blowdown episodes has been widely adopted by the research community. However a subtle but significant cyclon-effect may appear by the combined action of high azimuthal flow and the intrinsic curvature of downcomer. Here an attempt is done to estimate qualitatively the importance of a possible cyclon-effect in the blowdown episodes in a Loss of Coolant Accident LOCA in a typical PWR nuclear reactor. 2. Theoretical approach Consider a duct of annular geometry, of internal and external radio r1 and r2 , respectively. The coordinate system used is represented in Fig. 1b; the coordinate origin is placed in the external wall; the three spatial directions are represented y (radial), z (axial) and h (azimuthal). In order to simplify the stratification analysis, only azimuthal velocity (^ eh direction) is considered in the fluid. The main assumptions will be described below. (a) The annular geometry may be represented by a cylindrical pipe with a hydraulic diameter associated. This approximation is sufficiently accurate for engineering calculations. * Corresponding author. Tel.: +34 934016047; fax: +34 934017148. E-mail address: frariasm7@fis.ub.edu (F.J. Arias). 0306-4549/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2009.04.003
2.1. Equations The momentum equation in cylindrical coordinates can be write as
@ur @ur uh @ur @ur u2h þ ur þ þ uz @t @r r @h @z r " # @p 1 @ @ur 1 @ 2 ur @ 2 ur ur 2 @uh þ 2 2 2 ¼ qg z þl r þ 2 @r r @r r @h2 r @h @r @z r
q
ð1Þ taking into account that the relaxation time t r (full stratification time) may be calculate in first approximation as the time spend by a single small bubble of radius a and velocity v b (deduced below, Eq. (6)) traveling at a distance of Dr ¼ r 2 r 1 ; tr in first approximation yields the following relation:
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
1295
Nomenclature a c g J r s u um
bubble radio local bubble concentration acceleration due to gravity bubble flux in direction normal to wall radio y=r h , dimensionless distance from wall velocity of two-phase mixture at distance y from wall velocity of two-phase mixture at tube centerline transverse bubble velocity distance from wall
vb y
Greek symbols void fraction void fraction at tube centerline qv =ql b j universal constant l dynamic viscosity
a am
tr
9 Dr r2 ml 2 a2 u2h
ð2Þ
hence, for typical values in a LOCA with uh ¼ 90 m=s; r 2 r 1 ¼ 0:25 m and r2 ¼ 1 m and a characteristic blowdown time t b 5 s, where is clear that ttbr 1, the assumption of steady state h ¼ 0. Then, in a typical LOCA as menis justified, i.e, ur ¼ 0, and @u @h tioned above, Eq. (1) becomes
u2h @p ¼ @r r
ð3Þ
The above equation represents the centripetal acceleration, which is directed towards the center of curvature of the streamline and is associated with the change in direction of the velocity of a bubble. Consider a bubble of radius a and density qv , surrounded by liquid of density ql and viscosity ll ; the azimuthal velocity being uh . The body force experienced by the bubble is given by Eq. (3), considering a spherical bubble, it is easy shown, that the centripetal force F c acting upon the bubble is given by
Fc ¼
4 u2 pql a3 ð1 bÞ h 3 r
ð4Þ
m q s
kinematic viscosity density of two-phase mixture wall shear stress in two-phase system circumferential component kinetic energy correction factor Martinelly–Nelson parameter constant defined by Eq. (15)
h
c UM W
Subscripts 1, 2 inside, outside h hydraulic l,v liquid, vapor sp single phase tp two phase z axial component r radial component
where r is the distance to the center of the pipe (see Fig. 1b) and b ¼ qv =ql . The centripetal force, in first approximation, is equal to drag forces. According to Stokes’law in spherical approximation:
4 u2 pq a3 ð1 bÞ h ¼ 6pll a v b 3 l r
ð5Þ
where v b is the radial velocity of the bubble in ^ey direction. hence, from Eq. (5):
vb ¼
2 a2 v2 ð1 bÞ h 9 ml r
ð6Þ
where ml is the kinematic viscosity ðml ¼ ll =ql Þ. According to assumption (c), a turbulent velocity profile is considered. The azimuthal velocity profile in the radial direction is given by
uh ¼ um þ
rffiffiffiffiffi
s ln s j ql 1
ð7Þ
where um is the velocity of the two-phase mixture at the centerline of the equivalent pipe, j is a universal constant, s is the two-phase wall shear stress, and s is the dimensionless distance defined as (see Fig. 1b)
s¼
y r2 r1
ð8Þ
In general, the drag force for a turbulent fluid flow surrounded by a solid can be expressed as the product of a friction factor and the mean kinetic energy per unit volume. In circular pipes
s¼
ql u2m c f 2
ð9Þ
where c is the kinetic energy correction factor (in circular pipes c ’ 1) and f is the friction factor. Following the Blasius relationship
f ¼ 0:079 Re0:25
ð10Þ
The bubbles concentration, c, in a given sector of the angular cylinder can be represented by a simple equation:
d dc d @c þ ðcv b Þ þ Ssource ¼ Db dy dy dy @t
Fig. 1. (a) Flow pattern in the downcomer during LOCA. (b) Physical model and coordinates system.
ð11Þ
where Db is the bubbles (turbulent) diffusion coefficient and v b is the velocity of the bubbles driven by inertial forces (Eq. (6)). Ssource is the source (evaporation) term. At this point, due to t b t r , the source term may be neglected, hence, Eq. (11) is transformed as
1296
Db
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
dc ¼ cv b dy
ð12Þ
which represents that the bubble flow in the inside direction (due to inertial forces) is equal to the flux in the outside direction (turbulent dispersion forces). According to assumption (d), the bubbles diffusion coefficient is given by
dul Db ¼ j2 y2 dy
ð13Þ
Combining Eqs. (7), (12) and (13), and using the Reynolds analogy (Eq. (9)), and taking into account that the concentration of bubbles, c, is proportional to the void fraction a, finally the following expression is obtained:
1 da 1 dc W a2 um 1 ¼ ¼ 1þ a dy c dy r y j
!2 rffiffiffi f y ln 2 rh
ð14Þ
where
W¼
2 ð1 bÞ 9 ml j
sffiffiffi 2 f
ð15Þ
In Eq. (14), r is the real distance, i.e., r ¼ r2 y. At this point, to simplify the equations, in an actual downcomer geometry it can be considered valid the approximation: r2 y ! r r 2 . Hence, integrating Eq. (14) between ða; sÞ and ðam ; 1Þ, one obtains
"
rffiffiffi
a W a2 um f 1 f 2 3 ln s þ ln s ln ¼ ln s þ am 6 j2 r2 j 2
#
Fig. 3. Stratification effects on the mass flow according to Eq. (17).
W ¼ W l U2M a2l
ð17Þ
where UM is the Martinelly–Nelson parameter and al is the void fraction liquid (i.e., al ¼ 1 a). Fig. 3 shows the shape of the curve predicted by Eq. (7) for steam–water system.
ð16Þ
Fig. 2 compares the variation in the shape of the curves predicted by Eq. (16) with the bubble radius for typical values in a steam–water system: j ¼ 0:4; ml ¼ 0:0158 m2 =s; central velocity um ¼ 90 m=s; and ra2 ¼ 3 103 ; ra2 ¼ 1:5 102 ; ra2 ¼ 1:0 102 with a friction factor f ¼ 0:0055. Fig. 2 shows the strong sensitivity to bubble size. This influence of the bubble radius can be found in other.
3. Comparison with numerical simulation: RELAP-3DÒ To demonstrate the application of the analytical solution, there are two groups of methodologies in investigation of the two-phase flow and of the boiling process.
2.2. Mass flow drop in radial stratification The effect of radial stratification can be calculated by making some considerations on the pressure drop. Pressure drop in stratified two-phase flow systems was calculated by Levi (1952), where it is easy shown the following relation between the mass flow nonstratified (W) and stratified W l .
Fig. 2. Radial void-fraction distribution calculated from Eq. (16) as function of dimensionless distance s for various values of a=r2 .
Fig. 4. Nodalization of the multi-dimensional model of the downcomer with RELAP-3DÒ.
F.J. Arias, F. Reventós / Annals of Nuclear Energy 36 (2009) 1294–1297
In the first group are the experiments that are conducted with full scales apparatus. These experiments are costly and time-consuming but they yield results that can be applied directly to a design of a nuclear reactor. However, theses experiments, because the complexity of the phenomenon, because of incomplete instrumentation, and other experimental difficulties, very seldom provide detailed information which is needed for integrating the results and the understanding of the process. for example in the problem being considered in the present paper, the experiment performed at the full-scale called ‘‘Upper Plenum Test Facility (UPTF)” to investigate the flow phenomena in the primary system of a pressurized water reactor (PWR) during a loss-of-coolant accident (LOCA), the helicoidal two-phase flow is clearly observed, however the UPTF instrumentation was generally not sufficient to measure multi-dimensional phenomena in the downcomer and nothing the radial stratification was discussed. For an exhaustive review the reader can refer to the reference (CSNI, 1999) of the nuclear energy agency. The second methodologies can be performed on thermohydraulics computer programs, these simulations are relatively cheap a require a short time to provide de desired information. Although the results cannot be applied directly to a system design or validity of analytical expressions, these codes yield basic information and they clearly indicate the basic characteristic of two-phase flow. A 3D model with RELAP-3DÒ of full vessel model of a 1000 MWe three-loop PWR was developed using two multidimensional components to simulate a blowdown. In order to clarify the analysis and to underline the effect of radial stratification the simulated scenario consists in an isolated full PWR vessel, initially at nominal pressure (about 16 MPa) and with liquid at full power inlet temperature (about 290°C), that suffers a 100% break in a cold leg. The conditions at the break are maintained at 1 bar. The downcomer component (11,000) was divided into three 120° azimuthal sectors and six radial rings. The three azimuthal sectors corresponded to the three nozzles connecting the loops and vessel. The axial nodalization of each multidimensional component was of six levels. The actual shape and dimensions of the downcomer nodalization (component 11,000) are shown in Fig. 4. The broken loop is connected to sector loop 1 in sub-component 11,700, with a diameter of 1.96 m. Three representatives sectors have been analysed: in loop 1: sub-component 11,702; and in loop 3: sub-component 11,902 and 11,904 (loop 2 is symmetrically equal to loop 3). Fig. 5 compares the radial stratification (variation of the void fraction) as function of radial position s for the subcomponents 11,702, 11,902 and 11,904 calculated with RELAP3DÒ and the results from Eq. (16). Referring to Fig. 5, it is seen that the radial stratification is stronger for the sub-component 11,902 and 11,904 in loop 3, compares with sub-component 11,702 in loop 1. The above behavior is easily explained because in loop 1 the azimuthal velocity (circumferential) is less than loop 3 being practically full axial velocity uz uh , then, the cyclon-effect is strongly reduced. In the same manner, is expected that radial stratification in sub-component 11,902 to be higher than sub-component 11,904 for loop 3, and effectively this situation is also observed in the simulation, where
1297
Fig. 5. Radial void-fraction distribution calculated from Eq. (16) and simulation data.
Eq. (16) agree with the simulation within 10%, in good agreement, taking into account the simplifying assumptions. 4. Summary and conclusions Radial stratification due to centrifugal forces in downcomer during a typical LOCA and its significance with regard to mass flow prediction was discussed. Utilizing a simplified geometrical model, an analytical expression (Eq. (16)) for the void fraction profile was derived. The hydraulic responses with the 3D input model with RELAP3D were generally in reasonable agreement with Eq. (16). The present work provides added importance in the need of 3D simulation for the downcomer, where, for example the use of cross-junction in 2D-thermohydraulics codes is clearly not sufficient. Acknowledgements The authors wishes to thanks the anonymous referees, for helpful comments. The assistance of X. Sabater in the simulation is gratefully acknowledged. FJA presents this work as a thesis to the Department of Physics and Nuclear Engineering at the Technical University of Catalonia, in partial fulfillment with the requirements for the Ph.D. degree. Portion of this work was performed under auspices and financial support of the Council for Nuclear Safety of Spain-CSN. References CSNI Code Validation Matrix Version 2 (August 1999), UPTF-Separate Effects Test Data, Nuclear Energy Agency (NEA). Levi, S., 1952. Theory of pressure drop and heat transfer for two-phase-component annular flow in pipes. In: Proceedings of Second Midwestern Conference of Fluids Mechanics. Ohio State University, Engineering Experimental Station Bulletin No. 149, p. 337.