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Vapor explosions: Prediction of a critical liquid–corium velocity in vapor destabilization mechanism of corium melt in the fuel–coolant interactions – FCI
Annals of Nuclear Energy 37 (2010) 1236–1240
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Vapor explosions: Prediction of a critical liquid–corium velocity in vapor destabilization mechanism of corium melt in the fuel–coolant interactions – FCI F.J. Arias * Department of Physics and Nuclear Engineering, Technical University of Catalonia (UPC), Spain
a r t i c l e
i n f o
Article history: Received 14 September 2009 Received in revised form 25 January 2010 Accepted 30 March 2010 Available online 2 June 2010 Keywords: Vapor explosions Nuclear safety Corium Debris Fuel–coolant interaction FCI
a b s t r a c t In this paper, utilizing a simplified physical model an analytical expression for the prediction of most probable corium fragment size as function of relative velocity corium–coolant is derived. Combining this equation with the available experimental data an important consequence for the nuclear safety result: the existence of a critical velocity corium–coolant which maximizes the destabilization of film boiling when an external pressure wave is supplied. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A vapor explosion (also called a littoral explosion, or fuel–coolant interaction – FCI) is a violent boiling or flashing of water into vapor, occurring when water is either superheated, rapidly heated by fine hot debris produced within it, or the interaction of molten metals (e.g., fuel–coolant interaction of molten nuclear-reactor fuel rods with water in a nuclear reactor core following a core-meltdown, see Fig. 1). Pressure vessels (e.g., Pressurized-Water (nuclear) Reactors) that operate at above atmospheric pressure can also provide the conditions for a rapid boiling event which can be characterized as a steam explosion. The water changes from a liquid to a gas with extreme speed, increasing dramatically in volume. Events of this general type are also possible if, under extreme circumstances, the fuel of a liquid-cooled nuclear reactor becomes molten. Such explosions are known as fuel–coolant interactions or FCI. In these events the passage of the pressure wave through the pre dispersed material creates flow forces which further fragment the melt, resulting in rapid heat transfer, and thus sustaining the wave. Much of the physical destruction in the Chernobyl disaster, a graphite-moderated, light-water cooled reactor (an RBMK-1000 reactor), is thought to have been due to such a steam explosion. As mentioned above, it is not surprising that after sadly famous Chernobyl accident, vapor explosion have received a considerable
* Present address: Department of Nuclear Engineering, University of California, Berkeley, CA 94720, USA. E-mail address: frariasm7@fis.ub.edu 0306-4549/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2010.03.018
attention from the research community in application of a possible nuclear melting. Physical analysis of explosive, magma–water interactions or fuel coolant interaction is complicated by several reasons (Wohletz, 1986): (1) the initial geometry and location of the contact between corium (or magma) and water; (2) the process by which thermal energy is transferred from the magma to the water; (3) the degree to and manner by which the corium and water become intermingled prior to explosion; (4) the thermodynamic equation of state for mixtures of corium fragments (debris) and water; (5) the dynamic metastability of superheat water; and (6) the propagation of shock waves through the system.
1.1. Explosion mechanism As mentioned before, vapor explosions, or fuel–coolant interactions, has been the subject of considerable investigation (Wohletz, 1986) and the most recent works applied to corium (Song et al., 2006; Journeau et al., 2009). The modern models propose an explosive scenario that includes: (1) preliminary mechanically induced coarse mixing of fuel and coolant (premixing); (2) the formation of a vapor film at the fuel–coolant interface; (3) destabilization of the film, leading to direct contact between the two liquids; and (4) fine-scale fragmentation of the fuel and explosive vaporization of the coolant. Theoretical and experimental studies have shown that fine-scale fragmentation is required to produce the interactive surface area necessary to reconcile the observed energy release with known
F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
1237
Nomenclature a al c m n P r R* u
corium thickness liquid depth wave speed wave number wave frequency pressure radial distance the most probable fragment size velocity
k
g x x0 l
Subscripts m corium-melt l liquid v liquid
Greeks
q r
wavelength distance perpendicular to liquid–corium interface imaginary part of frequency equal to the growth parameter real part of frequency dynamical viscosity
density surface tension
Fig. 2. Taylor and Taylor–Helmholtz instabilities at the interface corium–water. Oscillation of the interface transmits sufficient momentum to the corium so that its surface distorts into waves that can grow until they detach to form small corium fragments (debris).
Fig. 1. Illustration of melt pool of corium in the lower plenum, and corium stream during nuclear melting.
heat transfer rates (for a more complete review, Takashima and Yoshihiro (2001)). In the present paper , we are interesting in the effect of corium– coolant velocity during the reflooding episodes in the destabilization mechanism of film boiling covered the corium fragments (debris).
n ¼ x0 þ ix
1.2. Taylor–Helmholtz instabilities In the present physical model we consider Taylor and Taylor– Helmholtz instabilities at the interface corium–water, where the oscillation of the interface transmits sufficient momentum to the corium so that its surface distorts into waves that can grow until they detach to form small corium fragments (debris), see Fig. 2. The behavior of an interface separating two different fluids of different densities when subjected to perturbations, such a liquid–vapor interface in film boiling can be studied into the framework of first-order perturbations. Taylor‘s analysis of the problem assumes potential flow. The form of the interface disturbance is given by the following equation:
g ¼ g0 expint cos mx
where n is the wave frequency, m the wave number, t the time and g the distance perpendicular to liquid–vapor interface. In above equation, it is seen that if n is real , the disturbance is periodic in time , and therefore stable. However, if n is imaginary, the disturbance grows exponentially with time. In general, the classical models assume that n is imaginary and, hence the disturbance grows with time. However, it is plausible that we can have a complex frequency, i.e., the frequency is the superposition of a part imaginary (x) and real x0
ð1Þ
ð2Þ
in this case, Eq. (1) can be re-written as
g ¼ g0 expix0 t expxt cos mx |fflfflfflfflffl{zfflfflfflfflffl} oscilatory
|fflffl{zfflffl}
ð3Þ
conv ectiv e
it is evident from Eq. (3) that with a complex frequency, the form of the interface disturbance is the superposition of a oscillators instability ðexpix0 t Þ and a disturbance grows exponentially with time or convective instability (expxt). 1.3. Stability equation The general solution of the irrotational flow kinematic equation yield the following relation (Milne-Thompson, 1968)
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F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
mqv ðum cÞ2 coth ma þ mql ðul cÞ2 coth mal 2
¼ g 0 rm þ gðqm ql Þ
ð4Þ
where the meaning of the various terms in the above equation are defined in the nomenclature. On the other hand, we took account that in most cases
coth mal ¼ 1
ð5Þ
and taking into account that wave speed is defined as
c¼
n m
ð6Þ
Combining Eqs. (5) and (6) and inserting in Eq. (4) and resolving for n result in the following expression
n ¼ x0 þ ix
ð7Þ
where according with Eq. (3), the parameter of interest in predicting the growth of the two phase boundary is the coefficient of t x which is given below and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uq q ðu þ u Þ2 m2 gðq q Þm rm3 m l x ¼ t m l m ql 2l ql ql qm þ ma qm þ ma ðqm þ ma Þ ma
ð8Þ
and, when the corium is in free fall, i.e., with vertical velocity, in this case the acceleration towards the melt corium surface is zero, in this situation, the equation for the corium stream is obtained making g = 0 in Eq. (8), and yield the following result.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uq q ðu þ u Þ2 m2 rm3 x ¼ t m l m ql 2l ql q ðqm þ ma Þ ma m þ ma
ð9Þ
where of course, it is see in Eq. (9) that, the case of a vertical free fall corium stream, the only stabilizer force is due to surface tension. Now, the appropriate wavelength is obtained differentiating Eq. (8) whit respect to m and solving for the value which maximizes x i.e., @ x=@mjm ¼ 0. Fig. 3 is a plot of Eqs. (8) and (9) as a function of m, for a typical parameter of corium: a ffi 100 mm, r ffi 0.35 N m1, qm ffi 8.0 103 kg m3 and water conditions qw ffi 954 103 kg m3, and g = 9.8 m s2 for pool corium melt and g = 0 for vertical corium
stream and different values of um + ul. Referring to Fig. 3, it is seen that it is reasonable to neglect the effect of gravity. 2. Critical fragment size for vapor film destabilization in a spherical geometry Takashima and Yoshihiro (2001), investigated experimentally the destabilization process of a vapor film formed around a hemispherical heated surface made of a copper or stainless-steel rod 30 mm in diameter and immersed in R-113 was, where the vapor film is destabilized by an external pressure wave, which is produced by a magnetic hammer. The pressure change in the vapor film when the external pressure wave is supplied was measured. The measured result for the effect of diameter of heat transfer surface on the maximum vapor film pressure when an external pressure above DPl = 0.15 MPa is applied is plotted in Fig. 4. Referring to Fig. 4, it is clear the existence of a critical diameter, which maximizes DPv i.e., a critical diameter for vapor destabilization. 3. Effect of the corium–liquid velocity The most probable radio for the corium fragments may be obtained taking into account the following considerations: It is well know, that in film boiling, the vapor flowing to any one bubble is generated in an area equal to k2/2. The results of stability analysis applied to three dimensions require this condition (Allred and Blunt, 1954). The two-dimensional stability analysis may be applied to three-dimensional problem. The experimental investigations of Taylor Instability have verified in the past century this assumption. Then, as mentioned above, the maximum radial distance i.e., fragment radius, is given by
k2 2 pffiffiffiffiffiffiffi 2p k r ¼ pffiffiffiffiffiffiffi ¼ m 2p
pr2 ¼
then, the most probable radius of corium fragment R*, correspond to the most probable wavelength, i.e., the value m* which maximizes x.
R ffi
Fig. 3. Plot of x as a function of m for corium melting for pool corium-melting Eq. (8) and a vertical corium stream, Eq. (9).
ð10Þ
pffiffiffiffiffiffiffi 2p m
ð11Þ
Fig. 4. Effect of diameter of heat transfer surface on the maximum vapor film pressure. (Reproduced from Takashima and Yoshihiro (2001).)
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F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
The above equation for the most probable fragment size taking into account Eq. (8) for the calculation of m*, expresses it as function of relative velocity um + ul. Finally, considering the experimental data for the effect of diameter on the maximum vapor film pressure response when an external pressure is applied (see Fig. 4) and combining with Eqs. (11) and (8) we obtain Fig. 5 (the typical values for physical parameters are the same as those in Fig. 3), where a critical relative velocity ul + uv for the corium–water system is predicted. It is interesting to see, that in the absence of forced convection i.e., ul = 0, and in the case of a vertical corium stream in free fall 4. Experimental data Fig. 6 shows the explosion pressure versus the hot liquid (water) injection velocity for three different cold liquid depths from Park and Park (1994) made on water/R22 under various conditions depth (8 12 cm), water temperature (70–97 °C) and water injection velocity (2–4 m s1), where it is clear that the explosions pressure has a very dependency upon the hot liquid
Table 1 Measured explosions pressure for three different coolant depth from Park and Park (1994). R22 depth (cm)
Water Vel. (m s1)
Peak pressure (MPa)
8.0 10.0 12.0
2.57 3.26 3.26
0.24 0.46 0.26
velocity, and on the other hand the existence of a critical velocity which maximize the peak pressure in good agreement with the theoretical prediction in the present paper. Table 1 is the measured explosions pressure for three different coolant depth and the associated velocity according with Fig. 6. We can see that the critical velocity obtained from theoretical considerations predict a critical velocity between 2 and 2.5 m s1 (see Fig. 5) very close to the experimental data (see Fig. 6 and Table 1) the above results is in very good agreement with experimental data not only in the prediction of a critical velocity but also quantitatively considering the complexity of theoretical analysis of the phenomena, and in this way constitute a strong evidence between the critical size fragments and critical velocity within framework of Taylor–Helmholtz instabilities. 5. Appendix 5.1. Corium in free fall For the case of a corium stream in free fall and in absence of forced convection, we have, that the maximum velocity is determined by stokes equation
u1 ¼
Fig. 5. Prediction of a critical velocity um + ul. The destabilization of film boiling and pressure change in vapor film when an external pressure wave is supplied, where is predict a critical value for velocity reflooding.
mg 6plR
ð12Þ
Then, considering a experimental values for the corium (Journeau et al., 2006). l = 35 Pa s, a corium melt density of 7140 kg/ m3, surface tension of r = 0.5 N m1 and a radius approximately of 0.1 m we have for the corium stream velocity approximately above ffi4.5 m s1. Hence, taking into account Fig. 5, where the critical velocity is above ffi2 m s1 it is plausible to think that the enhancement of destabilization of film boiling controlled by fragments size could be present in the corium stream. 6. Conclusions (a) Within the framework of Taylor–Helmholtz instabilities combining with available experimental data, a critical relative velocity corium-water is predicted, which maximizes the destabilization of film boiling when an external pressure wave is supplied. (b) The above result constitutes perhaps the first connection between critical size fragments and critical velocity within framework of Taylor–Helmholtz instabilities.
Acknowledgements The author wishes to thank Professor, Perez Madrid of Department of Fundamental Physics of the University of Barcelona-Spain. References Fig. 6. Explosion pressure vs. hot velocity. Experimental data from Park and Park (1994).
Allred, J.C., Blunt, G.H., 1954. Experimental studies of Taylor instability. University of California, Los Alamos Laboratory, Report LA-1600.
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F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
Christophe Journeau, et al., 2006. The VULCANO VE-U/corium spreading benchmark. Progress in Nuclear Energy, 215–234. Christophe Journeau, et al., 2009. Two-dimensional interaction of oxidic corium with concretes: the VULCANO VB test series. Annals of Nuclear Energy. doi:10.1016/j.anucene.2009.07.006. Milne-Thompson, L., 1968. Theoretical Hydrodynamics, 5th ed. The Macmillan Company, New York, NY. Park, I.K., Park, G.C., 1994. Experimental investigation on the vapor explosions with water/R22. Journal of Korean Nuclear Society. 26 (2), 257–264.
Song, Jin Ho, Kim, Jong Hwan, Hong, Seong Wan, Min, Beong Tae, Kim, Hee Dong, 2006. The effect of corium composition and interaction vessel geometry on the prototypic steam explosion. Annals of Nuclear Energy 33 (17–18), 1437–1451. Takashima, Takeo, Yoshihiro, Iida, 2001. Destabilization of film boiling and pressure change in vapor film when an external pressure wave is supplied. Heat Transfer – Asian Research 30 (8). Wohletz, Kenneth H., 1986. Explosive magma–water interactions: thermodynamics, explosion mechanisms, and field studies. Bulletin of Volcanology 48, 254–264.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Vapor explosions: Prediction of a critical liquid–corium velocity in vapor destabilization mechanism of corium melt in the fuel–coolant interactions – FCI F.J. Arias * Department of Physics and Nuclear Engineering, Technical University of Catalonia (UPC), Spain
a r t i c l e
i n f o
Article history: Received 14 September 2009 Received in revised form 25 January 2010 Accepted 30 March 2010 Available online 2 June 2010 Keywords: Vapor explosions Nuclear safety Corium Debris Fuel–coolant interaction FCI
a b s t r a c t In this paper, utilizing a simplified physical model an analytical expression for the prediction of most probable corium fragment size as function of relative velocity corium–coolant is derived. Combining this equation with the available experimental data an important consequence for the nuclear safety result: the existence of a critical velocity corium–coolant which maximizes the destabilization of film boiling when an external pressure wave is supplied. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A vapor explosion (also called a littoral explosion, or fuel–coolant interaction – FCI) is a violent boiling or flashing of water into vapor, occurring when water is either superheated, rapidly heated by fine hot debris produced within it, or the interaction of molten metals (e.g., fuel–coolant interaction of molten nuclear-reactor fuel rods with water in a nuclear reactor core following a core-meltdown, see Fig. 1). Pressure vessels (e.g., Pressurized-Water (nuclear) Reactors) that operate at above atmospheric pressure can also provide the conditions for a rapid boiling event which can be characterized as a steam explosion. The water changes from a liquid to a gas with extreme speed, increasing dramatically in volume. Events of this general type are also possible if, under extreme circumstances, the fuel of a liquid-cooled nuclear reactor becomes molten. Such explosions are known as fuel–coolant interactions or FCI. In these events the passage of the pressure wave through the pre dispersed material creates flow forces which further fragment the melt, resulting in rapid heat transfer, and thus sustaining the wave. Much of the physical destruction in the Chernobyl disaster, a graphite-moderated, light-water cooled reactor (an RBMK-1000 reactor), is thought to have been due to such a steam explosion. As mentioned above, it is not surprising that after sadly famous Chernobyl accident, vapor explosion have received a considerable
* Present address: Department of Nuclear Engineering, University of California, Berkeley, CA 94720, USA. E-mail address: frariasm7@fis.ub.edu 0306-4549/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2010.03.018
attention from the research community in application of a possible nuclear melting. Physical analysis of explosive, magma–water interactions or fuel coolant interaction is complicated by several reasons (Wohletz, 1986): (1) the initial geometry and location of the contact between corium (or magma) and water; (2) the process by which thermal energy is transferred from the magma to the water; (3) the degree to and manner by which the corium and water become intermingled prior to explosion; (4) the thermodynamic equation of state for mixtures of corium fragments (debris) and water; (5) the dynamic metastability of superheat water; and (6) the propagation of shock waves through the system.
1.1. Explosion mechanism As mentioned before, vapor explosions, or fuel–coolant interactions, has been the subject of considerable investigation (Wohletz, 1986) and the most recent works applied to corium (Song et al., 2006; Journeau et al., 2009). The modern models propose an explosive scenario that includes: (1) preliminary mechanically induced coarse mixing of fuel and coolant (premixing); (2) the formation of a vapor film at the fuel–coolant interface; (3) destabilization of the film, leading to direct contact between the two liquids; and (4) fine-scale fragmentation of the fuel and explosive vaporization of the coolant. Theoretical and experimental studies have shown that fine-scale fragmentation is required to produce the interactive surface area necessary to reconcile the observed energy release with known
F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
1237
Nomenclature a al c m n P r R* u
corium thickness liquid depth wave speed wave number wave frequency pressure radial distance the most probable fragment size velocity
k
g x x0 l
Subscripts m corium-melt l liquid v liquid
Greeks
q r
wavelength distance perpendicular to liquid–corium interface imaginary part of frequency equal to the growth parameter real part of frequency dynamical viscosity
density surface tension
Fig. 2. Taylor and Taylor–Helmholtz instabilities at the interface corium–water. Oscillation of the interface transmits sufficient momentum to the corium so that its surface distorts into waves that can grow until they detach to form small corium fragments (debris).
Fig. 1. Illustration of melt pool of corium in the lower plenum, and corium stream during nuclear melting.
heat transfer rates (for a more complete review, Takashima and Yoshihiro (2001)). In the present paper , we are interesting in the effect of corium– coolant velocity during the reflooding episodes in the destabilization mechanism of film boiling covered the corium fragments (debris).
n ¼ x0 þ ix
1.2. Taylor–Helmholtz instabilities In the present physical model we consider Taylor and Taylor– Helmholtz instabilities at the interface corium–water, where the oscillation of the interface transmits sufficient momentum to the corium so that its surface distorts into waves that can grow until they detach to form small corium fragments (debris), see Fig. 2. The behavior of an interface separating two different fluids of different densities when subjected to perturbations, such a liquid–vapor interface in film boiling can be studied into the framework of first-order perturbations. Taylor‘s analysis of the problem assumes potential flow. The form of the interface disturbance is given by the following equation:
g ¼ g0 expint cos mx
where n is the wave frequency, m the wave number, t the time and g the distance perpendicular to liquid–vapor interface. In above equation, it is seen that if n is real , the disturbance is periodic in time , and therefore stable. However, if n is imaginary, the disturbance grows exponentially with time. In general, the classical models assume that n is imaginary and, hence the disturbance grows with time. However, it is plausible that we can have a complex frequency, i.e., the frequency is the superposition of a part imaginary (x) and real x0
ð1Þ
ð2Þ
in this case, Eq. (1) can be re-written as
g ¼ g0 expix0 t expxt cos mx |fflfflfflfflffl{zfflfflfflfflffl} oscilatory
|fflffl{zfflffl}
ð3Þ
conv ectiv e
it is evident from Eq. (3) that with a complex frequency, the form of the interface disturbance is the superposition of a oscillators instability ðexpix0 t Þ and a disturbance grows exponentially with time or convective instability (expxt). 1.3. Stability equation The general solution of the irrotational flow kinematic equation yield the following relation (Milne-Thompson, 1968)
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F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
mqv ðum cÞ2 coth ma þ mql ðul cÞ2 coth mal 2
¼ g 0 rm þ gðqm ql Þ
ð4Þ
where the meaning of the various terms in the above equation are defined in the nomenclature. On the other hand, we took account that in most cases
coth mal ¼ 1
ð5Þ
and taking into account that wave speed is defined as
c¼
n m
ð6Þ
Combining Eqs. (5) and (6) and inserting in Eq. (4) and resolving for n result in the following expression
n ¼ x0 þ ix
ð7Þ
where according with Eq. (3), the parameter of interest in predicting the growth of the two phase boundary is the coefficient of t x which is given below and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uq q ðu þ u Þ2 m2 gðq q Þm rm3 m l x ¼ t m l m ql 2l ql ql qm þ ma qm þ ma ðqm þ ma Þ ma
ð8Þ
and, when the corium is in free fall, i.e., with vertical velocity, in this case the acceleration towards the melt corium surface is zero, in this situation, the equation for the corium stream is obtained making g = 0 in Eq. (8), and yield the following result.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uq q ðu þ u Þ2 m2 rm3 x ¼ t m l m ql 2l ql q ðqm þ ma Þ ma m þ ma
ð9Þ
where of course, it is see in Eq. (9) that, the case of a vertical free fall corium stream, the only stabilizer force is due to surface tension. Now, the appropriate wavelength is obtained differentiating Eq. (8) whit respect to m and solving for the value which maximizes x i.e., @ x=@mjm ¼ 0. Fig. 3 is a plot of Eqs. (8) and (9) as a function of m, for a typical parameter of corium: a ffi 100 mm, r ffi 0.35 N m1, qm ffi 8.0 103 kg m3 and water conditions qw ffi 954 103 kg m3, and g = 9.8 m s2 for pool corium melt and g = 0 for vertical corium
stream and different values of um + ul. Referring to Fig. 3, it is seen that it is reasonable to neglect the effect of gravity. 2. Critical fragment size for vapor film destabilization in a spherical geometry Takashima and Yoshihiro (2001), investigated experimentally the destabilization process of a vapor film formed around a hemispherical heated surface made of a copper or stainless-steel rod 30 mm in diameter and immersed in R-113 was, where the vapor film is destabilized by an external pressure wave, which is produced by a magnetic hammer. The pressure change in the vapor film when the external pressure wave is supplied was measured. The measured result for the effect of diameter of heat transfer surface on the maximum vapor film pressure when an external pressure above DPl = 0.15 MPa is applied is plotted in Fig. 4. Referring to Fig. 4, it is clear the existence of a critical diameter, which maximizes DPv i.e., a critical diameter for vapor destabilization. 3. Effect of the corium–liquid velocity The most probable radio for the corium fragments may be obtained taking into account the following considerations: It is well know, that in film boiling, the vapor flowing to any one bubble is generated in an area equal to k2/2. The results of stability analysis applied to three dimensions require this condition (Allred and Blunt, 1954). The two-dimensional stability analysis may be applied to three-dimensional problem. The experimental investigations of Taylor Instability have verified in the past century this assumption. Then, as mentioned above, the maximum radial distance i.e., fragment radius, is given by
k2 2 pffiffiffiffiffiffiffi 2p k r ¼ pffiffiffiffiffiffiffi ¼ m 2p
pr2 ¼
then, the most probable radius of corium fragment R*, correspond to the most probable wavelength, i.e., the value m* which maximizes x.
R ffi
Fig. 3. Plot of x as a function of m for corium melting for pool corium-melting Eq. (8) and a vertical corium stream, Eq. (9).
ð10Þ
pffiffiffiffiffiffiffi 2p m
ð11Þ
Fig. 4. Effect of diameter of heat transfer surface on the maximum vapor film pressure. (Reproduced from Takashima and Yoshihiro (2001).)
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F.J. Arias / Annals of Nuclear Energy 37 (2010) 1236–1240
The above equation for the most probable fragment size taking into account Eq. (8) for the calculation of m*, expresses it as function of relative velocity um + ul. Finally, considering the experimental data for the effect of diameter on the maximum vapor film pressure response when an external pressure is applied (see Fig. 4) and combining with Eqs. (11) and (8) we obtain Fig. 5 (the typical values for physical parameters are the same as those in Fig. 3), where a critical relative velocity ul + uv for the corium–water system is predicted. It is interesting to see, that in the absence of forced convection i.e., ul = 0, and in the case of a vertical corium stream in free fall 4. Experimental data Fig. 6 shows the explosion pressure versus the hot liquid (water) injection velocity for three different cold liquid depths from Park and Park (1994) made on water/R22 under various conditions depth (8 12 cm), water temperature (70–97 °C) and water injection velocity (2–4 m s1), where it is clear that the explosions pressure has a very dependency upon the hot liquid
Table 1 Measured explosions pressure for three different coolant depth from Park and Park (1994). R22 depth (cm)
Water Vel. (m s1)
Peak pressure (MPa)
8.0 10.0 12.0
2.57 3.26 3.26
0.24 0.46 0.26
velocity, and on the other hand the existence of a critical velocity which maximize the peak pressure in good agreement with the theoretical prediction in the present paper. Table 1 is the measured explosions pressure for three different coolant depth and the associated velocity according with Fig. 6. We can see that the critical velocity obtained from theoretical considerations predict a critical velocity between 2 and 2.5 m s1 (see Fig. 5) very close to the experimental data (see Fig. 6 and Table 1) the above results is in very good agreement with experimental data not only in the prediction of a critical velocity but also quantitatively considering the complexity of theoretical analysis of the phenomena, and in this way constitute a strong evidence between the critical size fragments and critical velocity within framework of Taylor–Helmholtz instabilities. 5. Appendix 5.1. Corium in free fall For the case of a corium stream in free fall and in absence of forced convection, we have, that the maximum velocity is determined by stokes equation
u1 ¼
Fig. 5. Prediction of a critical velocity um + ul. The destabilization of film boiling and pressure change in vapor film when an external pressure wave is supplied, where is predict a critical value for velocity reflooding.
mg 6plR
ð12Þ
Then, considering a experimental values for the corium (Journeau et al., 2006). l = 35 Pa s, a corium melt density of 7140 kg/ m3, surface tension of r = 0.5 N m1 and a radius approximately of 0.1 m we have for the corium stream velocity approximately above ffi4.5 m s1. Hence, taking into account Fig. 5, where the critical velocity is above ffi2 m s1 it is plausible to think that the enhancement of destabilization of film boiling controlled by fragments size could be present in the corium stream. 6. Conclusions (a) Within the framework of Taylor–Helmholtz instabilities combining with available experimental data, a critical relative velocity corium-water is predicted, which maximizes the destabilization of film boiling when an external pressure wave is supplied. (b) The above result constitutes perhaps the first connection between critical size fragments and critical velocity within framework of Taylor–Helmholtz instabilities.
Acknowledgements The author wishes to thank Professor, Perez Madrid of Department of Fundamental Physics of the University of Barcelona-Spain. References Fig. 6. Explosion pressure vs. hot velocity. Experimental data from Park and Park (1994).
Allred, J.C., Blunt, G.H., 1954. Experimental studies of Taylor instability. University of California, Los Alamos Laboratory, Report LA-1600.
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