- Email: [email protected]
Decontamination factors and release rates of UO2 particles from boiling pools of sodium
Nuclear Engineering and Design 74 (1982) 127-132 North-Holland Publishing Company
127
DECONTAMINATION FACTORS AND RELEASE RATES OF UO 2 PARTICLES FROM BOILING POOLS OF SODIUM A b r a h a m D A Y A N a n d Silvia Z A L M A N O V I C H Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Israel Received 30 August 1982
A semi-mechanistic model for calculating solid radionuclide release rates from boiling or bubbling pools of sodium was developed. The influence of particle spacial and size distributions on the decontamination of the releases was analysed and found significant. Decontamination factors are shown as a function of pool depth, bubbling characteristics and particle size distribution. The calculation of a decontamination factor for estimating the source term of large scale hypothetical core disruptive accidents is presented. The decontamination factor for a large scale accident was found to be two orders of magnitude greater than results obtained from small scale experiments conducted with uniform particle distributions.
1. Introduction Determination of safety margins for protection of the public from hypothetical core disruptive accidents (HCDA's) requires evaluation of radiological source terms. The present work is concerned with the release rate of solid radionuclide from a boiling or bubbling pool of contaminated coolant. Formation of such a pool within a reactor cavity could be a consequence of a breached primary containment. Hence, this is a subject of important consideration in safety assessments of major accidents. A semi-mechanistic model is presented for determination of release rates following HCDA's in sodium cooled reactors. It is intended to reduce some of the uncertainty and arbitrariness existing in calculations of radiological source terms. For instance, it is demonstrated that the NRC requirement that source terms would include one percent of the plutonium inventory is overly conservative. Large scale experiments with hot ceramic fuel and liquid sodium revealed violent interactions with massive fuel fragmentation [1]. Measured fuel particle size distributions indicate that a few percent of the fuel inventory could be fragmented into particle sizes smaller than 10 /~m in diameter. If present in a boiling or bubbling pool, mechanical release and subsequent carry-over of such particles from the pool is expected. Decay of fission products and chemical reactions between pool constituents and containment materials are enormous heat sources capable of promoting boiling conditions. Bub-
bling of gases through the coolant pool could result from the vapor released by the heated containment (concrete) and from gas-producing chemical reactions. Little information exists on particulate release rates from boiling pools. An experimental study by O'Connel and Pettyjohn [2] of liquid droplets carry over from evaporators revealed dependence on heat fluxes. In that experiment solutions of sodium chloride, sodium sulphate and magnesium sulphate were used and droplet emission from the pool was primarily due to foaming and splashing. Heat flux was influencial through bubble size and frequency. The release of radioactive materials from coater evaporators, in treatment of nuclear waste disposal, was also studied experimentally by Manowitz et al. [3]. The ratio between the still pot activity to the product activity was the main concern of the work and was defined as a decontamination factor. In a recent small scale experimental investigation by Jordan and Ozawa with sodium and fuel [4], the decontamination factor was defined somewhat differently, as: DF = ( mf/rnNa)pool/( rnf/mNa)rel . . . . d,
(1)
where m f and m Na are the masses of fuel and sodium, respectively. This experiment revealed magnitudes of decontamination factors ranging between 103 to 10 4. The test results, however, are believed to represent conservative figures for application in studies of large scale accidents. The experiment lacked two features which affect realistic decontamination factors, the pool
0 0 2 9 - 5 4 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
128
A. Dayan, S. Zalmanovich / Decontamination factors and release rates
depth and the particle size distribution. Reduced surface concentrations of particles are expected in deeper pools owing to the relatively high specific gravity of the fuel. Particle discharge rates, being dependent on surface concentrations, would be smaller in deeper pools. Release rates would be further diminished if a larger portion of the fuel is in particle size too large to be suspended in and above the pool. In contrast to known particle size distributions, as produced by fuel-coolant interactions, mono-disperse particles were used in the test. Both pool depth and particle size are included in the model described below.
2. Physical model Consider a bubbling pool of coolant containing a dispersion of ceramic fuel particles. Release of particles from the pool is assumed to be caused solely by the sparging action of the gas bubbles. Predictions of particle release rates would depend on the ability of defining the pool surface conditions. This, in turn, requires modeling of the particles spacial distribution based on bubble characteristics and frequency, gas holdup, pool depth and particle size distribution. Bubbling could be caused by boiling and by gas-producing chemical reaction. Two major heat sources could exist in HCDA's of liquid metal fast breeder reactors. One is the fission products decay heat, the other is the reaction heat from chemical interaction between sodium and water vapor. It is well known that a considerable amount of water vapor is released from the containment concrete structure as a result of heating [5]. The two heat sources are similar in magnitude and are powerful enough to rapidly raise the pool temperature to the boiling point of sodium (880°C). At this temperature heat conduction across the containment walls and into the surrounding soil could dissipate more than 90% of the total heat generation [5,6]. This would leave only a few percent for conversion to latent heat, nevertheless representing a considerable rate of boiling. The total gas evolution is described by the following expression for the gas superficial velocity %s
(qd + qc)'q + vos, 08 h f~
(2)
where qd and q¢ are the heat flux per unit floor area due to decay heating and chemical reactions, respectively, 08 the gas density, h f8 the latent heat of vaporization, v~ the superficial gas velocity resulting from chemical reactions, and r/the ratio of latent heat to total heat. Should
the steel liner of the reactor cavity containment remain intact, chemical reactions would be prevented in large and eq. (2) would account for decay heating only. According to Stokes equation the drag force exerted by the gas stream on a suspended particle above the pool is D = 6~r/~gr(vp - %),
(3)
where/~s is the gas viscosity, r the particle radius and Vp the particle velocity. A particle will remain suspended if the upward drag force is larger than the particle weight. For a given bubbling rate there exists a critical particle size above which particle escape is impossible. The critical particle radius is obtained by equating the drag and buoyancy forces to the particle weight subject to vp = 0, which gives
9/.~g~)g r~r=
]1/2
2 ( 0 ~ - 0-g)g ]
'
(4)
where Or, is the particle density and g the gravitational acceleration. Eqs. (2) and (4) provide the size criterion in terms of heat fluxes and gas production rate. It is demonstrated in a later section that only a small portion of the particle population could remain airborne if ejected from the pool surface. The surface concentration of particles depends strongly on bubbling intensity and characteristics. Boiling of liquid metals is known to produce large bubbles [7]. X-ray photography revealed bubble diameters exceeding 1.5 cm. After formation and detachment, bubbles of that size rise in the form of spherical caps, carrying behind spherical wakes. Experimental investigations showed that spherical cap bubbles and their wakes are nearly spherical in shape [8,9]. In a recent study, considerable entrainment of particles within bubble wakes was observed [9]. A model was developed to describe the vertical distributions of particles resulting from such entrainment. The model was tested with experimental data and axial dispersion models of bubble columns. By balancing the forces acting on a particle within a bubble wake, the depletion rate of the particle population of the wake was derived. The forces considered were: inertial, gravitational, buoyancy and frictional. After integration, the particle population of a given radius r in a wake was found to be
RP -- P£ 2 N(r,t)= N(r, O)exp(-O.573~gr t),
N(r,
(5)
where 0) is the initial population of these particles within the wake at the time of detachment, Pe the liquid
A. Dayan, S. Zalmanovich / Decontaminationfactors and release rates density, R the spherical cap bubble radius of curvature, / ~ the liquid viscosity, and t the time that elapsed since the moment of detachment. The fluid surrounding the bubbles and wakes was defined as external fluid. On the average, this fluid would have a downward velocity to compensate for the fluid dragged upward in the form of bubble wakes. It is assumed that the bubble velocity relative to the external fluid is identical to Taylor's bubble velocity (i.e. solitary spherical cap bubbles moving in an infinite fluid [8]), which is
VT= 2(gR ) 1/2.
(6)
To obtain the bubble velocity relative to stationary coordinates, the gas porosity % is defined. This is the volume fraction of the pool occupied by gas bubbles. Similarly the wakes and external fluid porosities are defined as Ew and %, respectively. Clearly the following relationship among the three porosities exists
Vw/Vb) =1,
%+%+,w=%+%(1+
(7)
where Vw/V b is the averaged ratio of wake volume to gas bubble volume. Spherical cap bubbles on the average show a ratio of Vw/V b = 7.55 [9]. For continuity to hold, the upward flow rate of fluid in wakes should equal the downward flow of the external fluid, thus
V~sVw/Vb= vo,o.
(8)
The external fluid velocity v~ can be expressed as the difference between the gas bubbles velocity in moving and stationary coordinates which is
v~ = v v - Vgs/% .
129
where z is the vertical distance from the pool floor, and vs is the settling velocity of a fuel particle in a quiescent pool [9], which is
2 (Pp - P.e)r2g Vs
9
(12)
/~e
Neglecting interaction effects between bubbles and particles, the average downward velocity of particles in the external fluid would be w = vs + ve = vs + v T - v ~ J % .
(13)
To satisfy continuity, a balance should exist between the rising particles in the wakes and the settling particles in the external fluid, thus
w % n ( r , z ) = vg--Z~ N ( r , z ) (g (w Vw ,
(14)
where n (r, z) designates number density of particles of size r within the external fluid at a height z. Substitution of eq. (11) into (14) gives
n(r,z)
Vbw%
exp
V~s ~
.
The population of particles of radius r at the time of detachment, N(r, 0) is to be determined. This is done by conservatively assuming that all particles are suspended within the pool, hence
dd(r)sr=Am(r)foH[~,w+n(r,z)%]dz,
(9) (16)
Combining eqs. (7) through (9) yields / V - r - vgs//1 - %~1
%Jr
Vb]J
=
--.
(10)
%svb
This is a quadratic equation in terms of %. Based on the knowledge of V~s from eq. (2) and on the value of Vw/V b, it can be solved. The roots correspond to two different conditions for discharging the generated gases. However, only the solution of lower gas holdup % is physically possible, the other yields overlapping bubbles with lower bubble velocity. Eq. (9) provides the bubble velocity (= vss/% ) is stationary coordinates. The particle population in a wake of a rising bubble, described by eq. (5), can be expressed as a function of position and bubble velocity, i.e.
N(r,z)=N(r,O)exp
(15)
(
-2.58
t~gs
t
,
(ll)
where re(r) is the mass of a particle with radius r, r e ( r ) = 4~rrapp/3, and M(r) is the total mass of particles having a radius smaller than r. H and A are the pool height and cross sectional area, respectively. Note that the original pool height, h, expands due to gas bubble holdup, H = h/(1 - % ) . Integration of eq. (16) and substitution in eq. (15) yields 2.58 vs
,,(r, z) .4R,o(w vj,~) =
+
[
{
vs%
1] - ' f ( r )
wheref(r) 8r is the size distribution function and denotes mass of fuel particles with radii in the range between r and r + ~r. Clearly, f(r) 8rim(r) is the number of these particles and f ( r ) = d M ( r ) / d r . The size-distribution
130
A. Dayan, S. Zalmanovich / Decontaminationfactors and releaserates
function can be readily estimated from existing experimental data [1]. Eqs. (14) and (17) provide the basis for calculating the average mass concentration of particles of radius r at a height z of the pool, which is per unit volume of liquid
x(r, z)Sr=
%n(r,z)+,wN(r,z)/Vw
re(r)
I[ e "1- C w
bf(r)Sr [ e x p ( b R ) _ 11 - ' AR(1 - %)
(18)
where b is a dimensionless parameter proportional to the ratio of particle slip velocity to bubble velocity, b = 2.58 vs%/vss. A typical value of the parameter for 5 /~m particles and a bubble volume of 2 cm3 would be 0.01. It is later demonstrated that this parameter is large enough to significantly reduce the surface concentration of fuel in the sodium, x(r, H)Sr. To calculate particulate release rates the pool surface conditions are analysed. Owing to the excellent wetting characteristics of fuel and sodium at 880°C, all fuel particles are expected to be in the liquid phase. Expulsion of fuel particles from the pool is thus assumed to be possible only through the release of sodium droplets containing such particles. Gas bubbles leaving the pool are likely to be free of solid fuel contaminants. Determination of the relative amount of liquid sodium and vapour are evidently of major importance for calculating release rates and decontamination factors. It is interesting to note that hot gases leaving the pool have the capability of entraining a mist of tiny droplets weighing about one third of the cloud mass. This can be verified by equating the density of the hot gases and droplets to the gas density of the surroundings. Hence, it can be concluded that the lifting capability of the gases leaving the pool is not a limitation on release rates. Under conditions of extremely low boiling rates, bubbles reaching a pool surface would not disintegrate immediately. Drainage of liquid and formation of thin lamellas above the bubbles would characterize the surface conditions. The bursting of such thin lamellas, typically 1 to 10 ~m thick, could produce droplets with diameters of a few microns. This is a consequence of the rigid property of thin films [10]. In regular boiling conditions, formation of small droplets would be of smaller probability. Agitation of the pool surface is likely to minimize the formation of thin lamellas and small droplets. Though expected small, the amount of liquid that
can escape the pool in the form of airborne droplets needs to be estimated. Small scale test results are used for this purpose. In the test, mono-disperse systems were used and measured decontamination factors ranged in magnitude from 103 to 104. It is conceivable that particle spacial distributions were also uniform due to the small size of the test apparatus. Therefore, and in accordance with the above discussion, the decontamination factors of the test should be equal to
D Ftest
x(r)Sr = 1+--, mzx(r)-Sr~me+ mv) . . . . me
where x(ra)Sr is the mass concentration of particles in liquid, 8r is a small interval describing the size spread of the mono-dispersed particles, and m v and m2 are the released masses of sodium vapour and liquid, respectively. Note that the concentration of fuel at the pool surface and in the released droplets are assumed identical. The measured decontamination factors and eq. (19) indicate that a ratio of at least 103 exists between mass of vapour release to mass of liquid release. This conclusion is in agreement with experimental data of droplet carry-over from boilers [2]. Based on the foregoing analyses, the release rate of particles from a pool surface can be readily written as
mp = Avs s m£ f r , x(r, H)dr. rng o
(20)
Likewise, the decontamination factor of a bubbling pool is obtained as M(oo)(1 + mg/rn,) DF-
Ah--rCrx(rJ'o
H)dr
(21)
Eqs. (20) and (21) reveal the influencing factors affecting solid radionuclide release from a boiling pool. These factors are: (a) deeper pools yield smaller releases due to reduced surface contaminations, x( r, H)Sr; (b) smaller gas production rates result in decreased surface contaminations and weaker entrainment capability above the pool, i.e. smaller rcr; (c) particle size distribution with smaller portions of fuel in the size range of 0 < r < rcr would show as decreased surface contaminations (smaller integral of x(r, H)Sr relative to the fuel load M(~)). Notice that a decontamination factor is not influenced by the amount of fuel M(oo). The ratio x(r, H ) / M ( ~ ) eliminates this dependence. This fact was confirmed in reported experimental data [4].
A. Dayan, S. Zalmant~ich / Decontaminationfactors and releaserates
[M(oe)/f(r)Sr] designates the ratio of the total mass of
3. Results and discussion The significance of the equations developed could probably be best demonstrated through the calculation of a decontamination factor. In a large commercial size breeder reactor the sodium and UO 2 inventories are about 900 and 25 tons, respectively. Penetration of these masses into the reactor cavity could form a pool 15 m wide and over 5 m high. However, it could take hours for the penetration process to be completed and for pool temperatures to reach the boiling point of sodium. During this time the fission-products decay heating could drop to a level of 10 MW, reflecting a heat flux of 5.7 W per cm2 of floor area. Natural convection and heat conduction into the outer containment would account for most of this heat [5,6], leaving about 1 MW (0.57 W / c m 2) for boiling vaporization of sodium. Gas production due to chemical reactions between pool constituent and containment materials is ignored in the calculation. As mentioned previously, a structurally intact steel liner (properly vented from behind [11]) would suppress chemical reactions. The maximum diameter of fuel particles that can be entrained by vapour, according to eq. (4), is about 5 #m (see table 1 for material properties). Experimental data show that only 10% of the total mass of fuel would be in the form of particles smaller than 10 #m. In the present analysis it will be assumed that all these particles have the average diameter of 5/~m. To calculate the decontamination factor, eq. (18) is substituted into eq. (21) which gives
[
DF=[1 +m -~v ]l[ ~M(o0) ( - ~ ]]
fuel to the mass of the fuel in the form of small particles. The third dimensionless group ( b h / R ) reflects the ratio of gravitational settling to upward entrainment velocities. Evaluation of this parameter can be made only after solving eq. (10) for %. In addition, the following data are incorporated: (i) the radius of curvature of the spherical cap is R = 1.56 cm, (ii) the pool depth is h = 5 m, and (iii) the mass ratio of sodium vapour releases to sodium droplet releases is 5 x 103. The resultant value of bH/R is 3.4. The expression in the last brackets of eq. (22) is the ratio of the overall mass concentration to the surface concentration of small particles. Evaluation of this expression shows that the surface concentration of small particles could be 8.5 times smaller than the average concentration in the pool. This ratio, exponentially dependent on pool height, may decrease substantially following sodium losses. The combined effects of particle size distribution, particle spacial distribution, and surface release phenomena yield a decontamination factor of 3 x 105. This value lies between the figures measured in the small scale test [4] and the figure of 107 measured in evaporation tests [2]. Therefore, when considering releases of plutonium particles from a boiling pool, decontamination factors on the order of 104 to 106 would be more realistic for calculations of maximum concentrations than the conservative NRC requirement of 102.
4. Concluding remarks
[ exp( bH / R ) bH/g
l ] r=ra (22)
The decontamination factor is clearly dependent on three dimensionless groups. The first group (rnv/rn~e) represents the mass ratio of the total sodium releases to the contaminated releases. The second group
Table 1 Material properties of sodium at 880°C vapour density vapour viscosity liquid density liquid viscosity latent heat
131
2.7 x 10-4 g/cm3 1.8 X 10 - 4 g/cm s 0.75 g/cm3 1.54x l0 -3 g/cm s 3.89x l0 m erg/g
Solid UO2 density at 880°C is 10.3 g/cm3.
In the analyses it was assumed that all the fuel inventory would be delivered to the sodium pool and be suspended by bubble action. This is a conservative assumption that deserves attention. Likewise, the assumption that the pool is bottom heated rather than volume heated is also conservative (yielding larger suspensions). Some uncertainty exists with regard to the accuracy of the single bubble model in describing the particle spacial distribution. This distribution could be affected by bubble interaction and convective fluid motion at large heat fluxes. Interaction of bubbles would reduce the suspension of particles, while convective currents would tend to increase it. The presence of large debris within the pool, on the other hand, would suppress convective currents. Therefore, the model used should be looked upon as a simplified approximation of a complex situation. Release of UO 2 by vaporization of fuel and sodium uranate was analysed and found improbable. Both subs-
A. Dayan, S. Zalmanovich / Decontamination factors and release rates
132
tances are in the solid form at 880 ° and have negligible v a p o u r pressures. The analyses can be extended to the. study of source terms for water-cooled reactors. This, however, requires incorporation of other e n t r a i n m e n t models for spherical bubbles.
Nomenclature A b D DF f(r) H h h fg M(r) m
m(r) N ( r , z) n(r, z) qd
q¢ R r
r~r t 15 T
V t~
v~s W
x(r)
cross sectional area of the pool dimensionless p a r a m e t e r drag force d e c o n t a m i n a t i o n factor, defined by eq. (1) size distribution function height of b u b b l i n g pool quiescent, pool height latent heat of vaporatization total mass of particles having a radius smaller than r mass mass of a particle with radius r p o p u l a t i o n of particles with radius r in the wake at a height z average n u m b e r density of particles that have a radius r in the external fluid at a height z decay heat flux chemical reaction heat flux b u b b l e radius of curvature radius of fuel particles radius of largest particle that would be susp e n d e d in vapor stream time Taylor b u b b l e velocity volume velocity superficial gas velocity resulting from chemical reactions particles d o w n w a r d velocity in the external fluid average mass c o n c e n t r a t i o n of particles with radius r at a height z of the pool per unit volume of liquid vertical distance from the pool floor
Greek symbols ratio of latent heat to total heat porosity
/L p
viscosity density
Subscripts a b e g gs f £ p s v w
average bubble external fluid gas gas superficial fuel liquid particle settling vapor wake
References [1] E.S. Sowa, J.D. Gabor, J.R. Pavlik, J.C. Cassulo, C.J. Cook and L. Baker, Jr., Molten core debris-sodium interactions: M-series experiments, Proceedings of the International Meeting on Fast Reactor Safety Technology, Seattle, Washington, August 19-23 (1979) 733-741. [2] H.E. O'Connel and E.S. Pettyjohn, Liquid carry over in a horizontal tube evaporator, AIChE 42 (1946) 795-814. [3] B. Manowitz, R.H. Bretton and R.V. Horrigan, Entrainment in evaporators, Chemical Engineering Progress 51 (7) (1955) 313-319. [4] S. Jordan and Y. Ozawa, Fuel particle and fission product release from LMFBR core catchers, International Meeting Fast Reactor Safety and Related Physics, Chicago, October 5-8, 1976. [5] A. Dayan and E.L. Gluekler, Heat and mass transfer within an intensely heated concrete slab, Internat. J. of Heat and Mass Transfer 25 (10) (1982) 1461-1467 . [6] L.S. Tong, Boiling Heat Transfer and Two-Phase Flow (Robert E. Krieger Publ. Comp., New York, 1975) pp. 5-47. [7] O.E. Dwyer, Boiling liquid-metal heat transfer (American Nuclear Society, 1976) pp. 135-224. [8] G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Press, 1967) pp. 474-477. [9] A. Dayan and S. Zalmamovich, Axial dispersion and entrainment of particles in wakes of bubbles, Chem. Engrg. Sci. 37 (1982) 1253-1257. [10] J.J. Bikerman, Foams (Springer-Verlag, New York, 1973) pp. 1-32. [11] m. Dayan and E.L. Gluekler, Heat and mass transfer behind a heated reactor cell liner, Trans. of the American Nuclear Society 26 (1977) 401-402.
127
DECONTAMINATION FACTORS AND RELEASE RATES OF UO 2 PARTICLES FROM BOILING POOLS OF SODIUM A b r a h a m D A Y A N a n d Silvia Z A L M A N O V I C H Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Israel Received 30 August 1982
A semi-mechanistic model for calculating solid radionuclide release rates from boiling or bubbling pools of sodium was developed. The influence of particle spacial and size distributions on the decontamination of the releases was analysed and found significant. Decontamination factors are shown as a function of pool depth, bubbling characteristics and particle size distribution. The calculation of a decontamination factor for estimating the source term of large scale hypothetical core disruptive accidents is presented. The decontamination factor for a large scale accident was found to be two orders of magnitude greater than results obtained from small scale experiments conducted with uniform particle distributions.
1. Introduction Determination of safety margins for protection of the public from hypothetical core disruptive accidents (HCDA's) requires evaluation of radiological source terms. The present work is concerned with the release rate of solid radionuclide from a boiling or bubbling pool of contaminated coolant. Formation of such a pool within a reactor cavity could be a consequence of a breached primary containment. Hence, this is a subject of important consideration in safety assessments of major accidents. A semi-mechanistic model is presented for determination of release rates following HCDA's in sodium cooled reactors. It is intended to reduce some of the uncertainty and arbitrariness existing in calculations of radiological source terms. For instance, it is demonstrated that the NRC requirement that source terms would include one percent of the plutonium inventory is overly conservative. Large scale experiments with hot ceramic fuel and liquid sodium revealed violent interactions with massive fuel fragmentation [1]. Measured fuel particle size distributions indicate that a few percent of the fuel inventory could be fragmented into particle sizes smaller than 10 /~m in diameter. If present in a boiling or bubbling pool, mechanical release and subsequent carry-over of such particles from the pool is expected. Decay of fission products and chemical reactions between pool constituents and containment materials are enormous heat sources capable of promoting boiling conditions. Bub-
bling of gases through the coolant pool could result from the vapor released by the heated containment (concrete) and from gas-producing chemical reactions. Little information exists on particulate release rates from boiling pools. An experimental study by O'Connel and Pettyjohn [2] of liquid droplets carry over from evaporators revealed dependence on heat fluxes. In that experiment solutions of sodium chloride, sodium sulphate and magnesium sulphate were used and droplet emission from the pool was primarily due to foaming and splashing. Heat flux was influencial through bubble size and frequency. The release of radioactive materials from coater evaporators, in treatment of nuclear waste disposal, was also studied experimentally by Manowitz et al. [3]. The ratio between the still pot activity to the product activity was the main concern of the work and was defined as a decontamination factor. In a recent small scale experimental investigation by Jordan and Ozawa with sodium and fuel [4], the decontamination factor was defined somewhat differently, as: DF = ( mf/rnNa)pool/( rnf/mNa)rel . . . . d,
(1)
where m f and m Na are the masses of fuel and sodium, respectively. This experiment revealed magnitudes of decontamination factors ranging between 103 to 10 4. The test results, however, are believed to represent conservative figures for application in studies of large scale accidents. The experiment lacked two features which affect realistic decontamination factors, the pool
0 0 2 9 - 5 4 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
128
A. Dayan, S. Zalmanovich / Decontamination factors and release rates
depth and the particle size distribution. Reduced surface concentrations of particles are expected in deeper pools owing to the relatively high specific gravity of the fuel. Particle discharge rates, being dependent on surface concentrations, would be smaller in deeper pools. Release rates would be further diminished if a larger portion of the fuel is in particle size too large to be suspended in and above the pool. In contrast to known particle size distributions, as produced by fuel-coolant interactions, mono-disperse particles were used in the test. Both pool depth and particle size are included in the model described below.
2. Physical model Consider a bubbling pool of coolant containing a dispersion of ceramic fuel particles. Release of particles from the pool is assumed to be caused solely by the sparging action of the gas bubbles. Predictions of particle release rates would depend on the ability of defining the pool surface conditions. This, in turn, requires modeling of the particles spacial distribution based on bubble characteristics and frequency, gas holdup, pool depth and particle size distribution. Bubbling could be caused by boiling and by gas-producing chemical reaction. Two major heat sources could exist in HCDA's of liquid metal fast breeder reactors. One is the fission products decay heat, the other is the reaction heat from chemical interaction between sodium and water vapor. It is well known that a considerable amount of water vapor is released from the containment concrete structure as a result of heating [5]. The two heat sources are similar in magnitude and are powerful enough to rapidly raise the pool temperature to the boiling point of sodium (880°C). At this temperature heat conduction across the containment walls and into the surrounding soil could dissipate more than 90% of the total heat generation [5,6]. This would leave only a few percent for conversion to latent heat, nevertheless representing a considerable rate of boiling. The total gas evolution is described by the following expression for the gas superficial velocity %s
(qd + qc)'q + vos, 08 h f~
(2)
where qd and q¢ are the heat flux per unit floor area due to decay heating and chemical reactions, respectively, 08 the gas density, h f8 the latent heat of vaporization, v~ the superficial gas velocity resulting from chemical reactions, and r/the ratio of latent heat to total heat. Should
the steel liner of the reactor cavity containment remain intact, chemical reactions would be prevented in large and eq. (2) would account for decay heating only. According to Stokes equation the drag force exerted by the gas stream on a suspended particle above the pool is D = 6~r/~gr(vp - %),
(3)
where/~s is the gas viscosity, r the particle radius and Vp the particle velocity. A particle will remain suspended if the upward drag force is larger than the particle weight. For a given bubbling rate there exists a critical particle size above which particle escape is impossible. The critical particle radius is obtained by equating the drag and buoyancy forces to the particle weight subject to vp = 0, which gives
9/.~g~)g r~r=
]1/2
2 ( 0 ~ - 0-g)g ]
'
(4)
where Or, is the particle density and g the gravitational acceleration. Eqs. (2) and (4) provide the size criterion in terms of heat fluxes and gas production rate. It is demonstrated in a later section that only a small portion of the particle population could remain airborne if ejected from the pool surface. The surface concentration of particles depends strongly on bubbling intensity and characteristics. Boiling of liquid metals is known to produce large bubbles [7]. X-ray photography revealed bubble diameters exceeding 1.5 cm. After formation and detachment, bubbles of that size rise in the form of spherical caps, carrying behind spherical wakes. Experimental investigations showed that spherical cap bubbles and their wakes are nearly spherical in shape [8,9]. In a recent study, considerable entrainment of particles within bubble wakes was observed [9]. A model was developed to describe the vertical distributions of particles resulting from such entrainment. The model was tested with experimental data and axial dispersion models of bubble columns. By balancing the forces acting on a particle within a bubble wake, the depletion rate of the particle population of the wake was derived. The forces considered were: inertial, gravitational, buoyancy and frictional. After integration, the particle population of a given radius r in a wake was found to be
RP -- P£ 2 N(r,t)= N(r, O)exp(-O.573~gr t),
N(r,
(5)
where 0) is the initial population of these particles within the wake at the time of detachment, Pe the liquid
A. Dayan, S. Zalmanovich / Decontaminationfactors and release rates density, R the spherical cap bubble radius of curvature, / ~ the liquid viscosity, and t the time that elapsed since the moment of detachment. The fluid surrounding the bubbles and wakes was defined as external fluid. On the average, this fluid would have a downward velocity to compensate for the fluid dragged upward in the form of bubble wakes. It is assumed that the bubble velocity relative to the external fluid is identical to Taylor's bubble velocity (i.e. solitary spherical cap bubbles moving in an infinite fluid [8]), which is
VT= 2(gR ) 1/2.
(6)
To obtain the bubble velocity relative to stationary coordinates, the gas porosity % is defined. This is the volume fraction of the pool occupied by gas bubbles. Similarly the wakes and external fluid porosities are defined as Ew and %, respectively. Clearly the following relationship among the three porosities exists
Vw/Vb) =1,
%+%+,w=%+%(1+
(7)
where Vw/V b is the averaged ratio of wake volume to gas bubble volume. Spherical cap bubbles on the average show a ratio of Vw/V b = 7.55 [9]. For continuity to hold, the upward flow rate of fluid in wakes should equal the downward flow of the external fluid, thus
V~sVw/Vb= vo,o.
(8)
The external fluid velocity v~ can be expressed as the difference between the gas bubbles velocity in moving and stationary coordinates which is
v~ = v v - Vgs/% .
129
where z is the vertical distance from the pool floor, and vs is the settling velocity of a fuel particle in a quiescent pool [9], which is
2 (Pp - P.e)r2g Vs
9
(12)
/~e
Neglecting interaction effects between bubbles and particles, the average downward velocity of particles in the external fluid would be w = vs + ve = vs + v T - v ~ J % .
(13)
To satisfy continuity, a balance should exist between the rising particles in the wakes and the settling particles in the external fluid, thus
w % n ( r , z ) = vg--Z~ N ( r , z ) (g (w Vw ,
(14)
where n (r, z) designates number density of particles of size r within the external fluid at a height z. Substitution of eq. (11) into (14) gives
n(r,z)
Vbw%
exp
V~s ~
.
The population of particles of radius r at the time of detachment, N(r, 0) is to be determined. This is done by conservatively assuming that all particles are suspended within the pool, hence
dd(r)sr=Am(r)foH[~,w+n(r,z)%]dz,
(9) (16)
Combining eqs. (7) through (9) yields / V - r - vgs//1 - %~1
%Jr
Vb]J
=
--.
(10)
%svb
This is a quadratic equation in terms of %. Based on the knowledge of V~s from eq. (2) and on the value of Vw/V b, it can be solved. The roots correspond to two different conditions for discharging the generated gases. However, only the solution of lower gas holdup % is physically possible, the other yields overlapping bubbles with lower bubble velocity. Eq. (9) provides the bubble velocity (= vss/% ) is stationary coordinates. The particle population in a wake of a rising bubble, described by eq. (5), can be expressed as a function of position and bubble velocity, i.e.
N(r,z)=N(r,O)exp
(15)
(
-2.58
t~gs
t
,
(ll)
where re(r) is the mass of a particle with radius r, r e ( r ) = 4~rrapp/3, and M(r) is the total mass of particles having a radius smaller than r. H and A are the pool height and cross sectional area, respectively. Note that the original pool height, h, expands due to gas bubble holdup, H = h/(1 - % ) . Integration of eq. (16) and substitution in eq. (15) yields 2.58 vs
,,(r, z) .4R,o(w vj,~) =
+
[
{
vs%
1] - ' f ( r )
wheref(r) 8r is the size distribution function and denotes mass of fuel particles with radii in the range between r and r + ~r. Clearly, f(r) 8rim(r) is the number of these particles and f ( r ) = d M ( r ) / d r . The size-distribution
130
A. Dayan, S. Zalmanovich / Decontaminationfactors and releaserates
function can be readily estimated from existing experimental data [1]. Eqs. (14) and (17) provide the basis for calculating the average mass concentration of particles of radius r at a height z of the pool, which is per unit volume of liquid
x(r, z)Sr=
%n(r,z)+,wN(r,z)/Vw
re(r)
I[ e "1- C w
bf(r)Sr [ e x p ( b R ) _ 11 - ' AR(1 - %)
(18)
where b is a dimensionless parameter proportional to the ratio of particle slip velocity to bubble velocity, b = 2.58 vs%/vss. A typical value of the parameter for 5 /~m particles and a bubble volume of 2 cm3 would be 0.01. It is later demonstrated that this parameter is large enough to significantly reduce the surface concentration of fuel in the sodium, x(r, H)Sr. To calculate particulate release rates the pool surface conditions are analysed. Owing to the excellent wetting characteristics of fuel and sodium at 880°C, all fuel particles are expected to be in the liquid phase. Expulsion of fuel particles from the pool is thus assumed to be possible only through the release of sodium droplets containing such particles. Gas bubbles leaving the pool are likely to be free of solid fuel contaminants. Determination of the relative amount of liquid sodium and vapour are evidently of major importance for calculating release rates and decontamination factors. It is interesting to note that hot gases leaving the pool have the capability of entraining a mist of tiny droplets weighing about one third of the cloud mass. This can be verified by equating the density of the hot gases and droplets to the gas density of the surroundings. Hence, it can be concluded that the lifting capability of the gases leaving the pool is not a limitation on release rates. Under conditions of extremely low boiling rates, bubbles reaching a pool surface would not disintegrate immediately. Drainage of liquid and formation of thin lamellas above the bubbles would characterize the surface conditions. The bursting of such thin lamellas, typically 1 to 10 ~m thick, could produce droplets with diameters of a few microns. This is a consequence of the rigid property of thin films [10]. In regular boiling conditions, formation of small droplets would be of smaller probability. Agitation of the pool surface is likely to minimize the formation of thin lamellas and small droplets. Though expected small, the amount of liquid that
can escape the pool in the form of airborne droplets needs to be estimated. Small scale test results are used for this purpose. In the test, mono-disperse systems were used and measured decontamination factors ranged in magnitude from 103 to 104. It is conceivable that particle spacial distributions were also uniform due to the small size of the test apparatus. Therefore, and in accordance with the above discussion, the decontamination factors of the test should be equal to
D Ftest
x(r)Sr = 1+--, mzx(r)-Sr~me+ mv) . . . . me
where x(ra)Sr is the mass concentration of particles in liquid, 8r is a small interval describing the size spread of the mono-dispersed particles, and m v and m2 are the released masses of sodium vapour and liquid, respectively. Note that the concentration of fuel at the pool surface and in the released droplets are assumed identical. The measured decontamination factors and eq. (19) indicate that a ratio of at least 103 exists between mass of vapour release to mass of liquid release. This conclusion is in agreement with experimental data of droplet carry-over from boilers [2]. Based on the foregoing analyses, the release rate of particles from a pool surface can be readily written as
mp = Avs s m£ f r , x(r, H)dr. rng o
(20)
Likewise, the decontamination factor of a bubbling pool is obtained as M(oo)(1 + mg/rn,) DF-
Ah--rCrx(rJ'o
H)dr
(21)
Eqs. (20) and (21) reveal the influencing factors affecting solid radionuclide release from a boiling pool. These factors are: (a) deeper pools yield smaller releases due to reduced surface contaminations, x( r, H)Sr; (b) smaller gas production rates result in decreased surface contaminations and weaker entrainment capability above the pool, i.e. smaller rcr; (c) particle size distribution with smaller portions of fuel in the size range of 0 < r < rcr would show as decreased surface contaminations (smaller integral of x(r, H)Sr relative to the fuel load M(~)). Notice that a decontamination factor is not influenced by the amount of fuel M(oo). The ratio x(r, H ) / M ( ~ ) eliminates this dependence. This fact was confirmed in reported experimental data [4].
A. Dayan, S. Zalmant~ich / Decontaminationfactors and releaserates
[M(oe)/f(r)Sr] designates the ratio of the total mass of
3. Results and discussion The significance of the equations developed could probably be best demonstrated through the calculation of a decontamination factor. In a large commercial size breeder reactor the sodium and UO 2 inventories are about 900 and 25 tons, respectively. Penetration of these masses into the reactor cavity could form a pool 15 m wide and over 5 m high. However, it could take hours for the penetration process to be completed and for pool temperatures to reach the boiling point of sodium. During this time the fission-products decay heating could drop to a level of 10 MW, reflecting a heat flux of 5.7 W per cm2 of floor area. Natural convection and heat conduction into the outer containment would account for most of this heat [5,6], leaving about 1 MW (0.57 W / c m 2) for boiling vaporization of sodium. Gas production due to chemical reactions between pool constituent and containment materials is ignored in the calculation. As mentioned previously, a structurally intact steel liner (properly vented from behind [11]) would suppress chemical reactions. The maximum diameter of fuel particles that can be entrained by vapour, according to eq. (4), is about 5 #m (see table 1 for material properties). Experimental data show that only 10% of the total mass of fuel would be in the form of particles smaller than 10 #m. In the present analysis it will be assumed that all these particles have the average diameter of 5/~m. To calculate the decontamination factor, eq. (18) is substituted into eq. (21) which gives
[
DF=[1 +m -~v ]l[ ~M(o0) ( - ~ ]]
fuel to the mass of the fuel in the form of small particles. The third dimensionless group ( b h / R ) reflects the ratio of gravitational settling to upward entrainment velocities. Evaluation of this parameter can be made only after solving eq. (10) for %. In addition, the following data are incorporated: (i) the radius of curvature of the spherical cap is R = 1.56 cm, (ii) the pool depth is h = 5 m, and (iii) the mass ratio of sodium vapour releases to sodium droplet releases is 5 x 103. The resultant value of bH/R is 3.4. The expression in the last brackets of eq. (22) is the ratio of the overall mass concentration to the surface concentration of small particles. Evaluation of this expression shows that the surface concentration of small particles could be 8.5 times smaller than the average concentration in the pool. This ratio, exponentially dependent on pool height, may decrease substantially following sodium losses. The combined effects of particle size distribution, particle spacial distribution, and surface release phenomena yield a decontamination factor of 3 x 105. This value lies between the figures measured in the small scale test [4] and the figure of 107 measured in evaporation tests [2]. Therefore, when considering releases of plutonium particles from a boiling pool, decontamination factors on the order of 104 to 106 would be more realistic for calculations of maximum concentrations than the conservative NRC requirement of 102.
4. Concluding remarks
[ exp( bH / R ) bH/g
l ] r=ra (22)
The decontamination factor is clearly dependent on three dimensionless groups. The first group (rnv/rn~e) represents the mass ratio of the total sodium releases to the contaminated releases. The second group
Table 1 Material properties of sodium at 880°C vapour density vapour viscosity liquid density liquid viscosity latent heat
131
2.7 x 10-4 g/cm3 1.8 X 10 - 4 g/cm s 0.75 g/cm3 1.54x l0 -3 g/cm s 3.89x l0 m erg/g
Solid UO2 density at 880°C is 10.3 g/cm3.
In the analyses it was assumed that all the fuel inventory would be delivered to the sodium pool and be suspended by bubble action. This is a conservative assumption that deserves attention. Likewise, the assumption that the pool is bottom heated rather than volume heated is also conservative (yielding larger suspensions). Some uncertainty exists with regard to the accuracy of the single bubble model in describing the particle spacial distribution. This distribution could be affected by bubble interaction and convective fluid motion at large heat fluxes. Interaction of bubbles would reduce the suspension of particles, while convective currents would tend to increase it. The presence of large debris within the pool, on the other hand, would suppress convective currents. Therefore, the model used should be looked upon as a simplified approximation of a complex situation. Release of UO 2 by vaporization of fuel and sodium uranate was analysed and found improbable. Both subs-
A. Dayan, S. Zalmanovich / Decontamination factors and release rates
132
tances are in the solid form at 880 ° and have negligible v a p o u r pressures. The analyses can be extended to the. study of source terms for water-cooled reactors. This, however, requires incorporation of other e n t r a i n m e n t models for spherical bubbles.
Nomenclature A b D DF f(r) H h h fg M(r) m
m(r) N ( r , z) n(r, z) qd
q¢ R r
r~r t 15 T
V t~
v~s W
x(r)
cross sectional area of the pool dimensionless p a r a m e t e r drag force d e c o n t a m i n a t i o n factor, defined by eq. (1) size distribution function height of b u b b l i n g pool quiescent, pool height latent heat of vaporatization total mass of particles having a radius smaller than r mass mass of a particle with radius r p o p u l a t i o n of particles with radius r in the wake at a height z average n u m b e r density of particles that have a radius r in the external fluid at a height z decay heat flux chemical reaction heat flux b u b b l e radius of curvature radius of fuel particles radius of largest particle that would be susp e n d e d in vapor stream time Taylor b u b b l e velocity volume velocity superficial gas velocity resulting from chemical reactions particles d o w n w a r d velocity in the external fluid average mass c o n c e n t r a t i o n of particles with radius r at a height z of the pool per unit volume of liquid vertical distance from the pool floor
Greek symbols ratio of latent heat to total heat porosity
/L p
viscosity density
Subscripts a b e g gs f £ p s v w
average bubble external fluid gas gas superficial fuel liquid particle settling vapor wake
References [1] E.S. Sowa, J.D. Gabor, J.R. Pavlik, J.C. Cassulo, C.J. Cook and L. Baker, Jr., Molten core debris-sodium interactions: M-series experiments, Proceedings of the International Meeting on Fast Reactor Safety Technology, Seattle, Washington, August 19-23 (1979) 733-741. [2] H.E. O'Connel and E.S. Pettyjohn, Liquid carry over in a horizontal tube evaporator, AIChE 42 (1946) 795-814. [3] B. Manowitz, R.H. Bretton and R.V. Horrigan, Entrainment in evaporators, Chemical Engineering Progress 51 (7) (1955) 313-319. [4] S. Jordan and Y. Ozawa, Fuel particle and fission product release from LMFBR core catchers, International Meeting Fast Reactor Safety and Related Physics, Chicago, October 5-8, 1976. [5] A. Dayan and E.L. Gluekler, Heat and mass transfer within an intensely heated concrete slab, Internat. J. of Heat and Mass Transfer 25 (10) (1982) 1461-1467 . [6] L.S. Tong, Boiling Heat Transfer and Two-Phase Flow (Robert E. Krieger Publ. Comp., New York, 1975) pp. 5-47. [7] O.E. Dwyer, Boiling liquid-metal heat transfer (American Nuclear Society, 1976) pp. 135-224. [8] G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Press, 1967) pp. 474-477. [9] A. Dayan and S. Zalmamovich, Axial dispersion and entrainment of particles in wakes of bubbles, Chem. Engrg. Sci. 37 (1982) 1253-1257. [10] J.J. Bikerman, Foams (Springer-Verlag, New York, 1973) pp. 1-32. [11] m. Dayan and E.L. Gluekler, Heat and mass transfer behind a heated reactor cell liner, Trans. of the American Nuclear Society 26 (1977) 401-402.