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The radii of surface nucleation sites which initiate sodium boiling
NUCLEAR ENGINEERING AND DESIGN 24 (1973) 388-392. NORTH-HOLLAND PUBLISHING COMPANY
THE RADII OF SURFACE NUCLEATION
SITES
WHICH INITIATE SODIUM BOILING P.K. HOLLAND* and R.H.S. W1NTERTON~ Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, England Received 18 December 1972
Values for the radii of nucleation sites are calculated from incipient boiling superheat measurements obtained in several different experiments. The radii predicted by applying the pre-boiling pressure history theory to a new cavity model are in agreement with the experimental results.
1. Introduction The temperature at which sodium boiling starts in a fast reactor overheating incident is of interest in safety studies. Boiling is nucleated when e v - e = (20/0
- eg
(1)
and a major problem is the prediction of r, the radius of the liquid meniscus at nucleation. In this paper, the pre-nucleation pressure history theory (Holtz [ 1], Apfel [2]) is applied to a surface nucleation site model proposed earlier (Holland and Winterton [3]). The predicted radii are in good agreement with those deduced from superheat experiments.
2. Theory It is known that steels are wetted by sodium at high temperatures, i.e. a drop of sodium placed on the surface will spread out, with a low or zero contact angle. A consequence of this is that the sodium will tend to penetrate and fill any cracks or cavities in the * Present address: CEGB NW Region scientific services department, 825 Wilmslow Rd., Manchester 20, England. :~Present address: Department of Mechanical Engineering, University of Birmingham, P.O. Box 363, Birmingham 15, England.
surface. However, a cavity filled with liquid cannot act as a nucleation site, so it.is necessary to assume that locally the surface is non-wetted. These non-wetted regions are probably non-metallic inclusions, which are known to be present in steels in large numbers (Kiessling [4]). They consist mostly of stable oxides, such as MnO, SiO 2, A1203, their size is from 0.1 to 10 tam and there are about 107/cm 2 of surface. Similar stable oxides (MgO, BaO, CaO) are not wetted by sodium (Bradhurst and Buchanan [5]). A model cavity on a steel surface is sketched in fig. la under the pressure conditions before boiling, the liquid-vapour interface is pushed down until it reaches an inclusion, where it has an advancing contact angle 0A > 90 ° (non-wetting). The meniscus will proceed further into the cavity until the increasing excess pressure due to surface tension across the more highly curved surface just balances the external pressure. Of course, the liquid might penetrate beyond the non-wetted inclusion before this equilibrium state is reached. However, the surface probably contains many cavities and it may be expected that there will be at least one cavity which has a non-wetted inclusion at a position appropriate to any preboiling pressure. The position of the meniscus is shown in fig. lb. The condition for mechanical equilibrium of the interface is
P.K. Holland, R.H.S. Winterton, The radii o f surface nucleation sites which initiate sodium boiling
(fig. lc). Boiling will nucleate once this critical radius is reached, since further growth of the vapour region results in increased r and hence a reduction in the 2air term in eq. (1) which opposes bubble growth. If the cavity sides are steep and the non-wetted regions small, the radius at nucleation, r, will not be significantly different to rA. Substituting from (4) into (1), with r = rA, gives
(a)
- uou,o soo,oM /
389
wETTEoMETA.
PI / I / I / t------W ~12¢//
+ o ( e ' - ei -
/
(5>
For cases where Pg and Pg' = 0 this reduces to P = P + o (P' - P~) / o'7.
(6)
I /
b/ (b)
(¢)
Fig. 1. (a) Typical surface cavity; (b) position of liquid meniscus under pre-boiling conditions; (c) critical position of liquid meniscus at nucleation. P'-
P' - P' = 2a'/r',
v
g
(2)
where P' is the external pressure, Pv, Pg and o' the vapour pressure, gas pressure and surface tension at the temperature in question. It follows from simple geometry that
The advantage of the new cavity model used here is that eq. (5) has been obtained without the physically implausible assumption of a surface with simultaneously very large advancing contact angle and very small receding contact angle. Most sodium systems are filled with sodium above its melting point (100°C) and then the temperature is increased to T' (typicaUy 500°C). The solubilities of inert gases in sodium increase so rapidly that such a filling procedure gives P~ = 0 initially. It takes several days for P~ to increase to the cover gas pressure, because in a stagnant system, diffusion from the free surface through the sodium to the surface cavities is very slow. The radius, r, is given by the highest pressurisation. This normally occurs in the early stages when Pg = 0. Therefore r = rA = 7 2 o ' / ( P ' - Pv)"
(7)
rA = - r' cos (0A - c). For equilibrium to be possible at all, the meniscus must be concave on the liquid side, i.e. 0A - c > 90 °, and the value of cos (0A - c) will be negative. Defining 3' = - cos (0A - c) = [ cos (0A - c)[
(3)
we then have rA = 7 / substituting in (2) gives rA = 7 2 a ' / ( P ' - Pv - P~g)
(4)
(c.f. Apfel [2] eq. 3). When the vapour pressure increases on approaching boiling, the meniscus is pushed back until it reaches a wetted region, where it stops and the shape of the meniscus changes. The minimum radius the meniscus has to achieve is the radius of the cavity at this point
3. Experimental results Unless Pg is known, an experimental value for the cavity radius cannot be found from superheat measurements. Holland and Winterton [3] show that gas from surface nucleation sites dissolves during heating and Pg in eq. 1 decreases continuously to zero. The time scales for argon to dissolve from surface cavities are days at 500°C, minutes at 700°C and seconds at 900°C. Thus Pg ~ 0 at high saturation temperatures ( > 800°C) in experiments heated at a moderate rate. The radius of the surface nucleation site is then from (1) r = 2o/(P v - t).
(8)
390
P.K. Holland, R.H.S. Winterton, The radii of surface nucleation sites which initiate sodium boiling
Four distinct sets of experimental data are available from which cavity radii may be calculated. In the first three, reasonable precautions were taken to ensure that Pg = 0. In the other, older, work, while there was little or no control of gas content, it is possible to choose data for which Pg was probably zero. (a) Holtz et al. [6] used no cover gas; the saturation pressure was produced electromagnetically. Thus Pg = 0 in all runs, and they found 7 = 0.34 from their results at values of P' from 0.07-0.7 atm. (b) Holland and Winterton [3] studied loss of argon from surface cavities, and found an asymptotic value of superheat corresponding to Pg = 0. (c) Holland [7] did some experiments with helium cover gas so that Pg = 0,can be attained quickly at low saturation temperatures because helium dissolves much more rapidly than argon. (d) Le Gonidec et al. [8], Pinchera et al. [9], Holtz and Singer [10], employed argon cover gas. Since the time scale for argon to dissolve at the highest temperatures used by these workers is of order of several seconds, it is likely that for some of the results the argon had time to dissolve completely. Consequently, the highest superheat at a high saturation temperature is assumed to correspond to P_ = 0. Also Smidt [ 111 describes some experiments without cover gas which fall on a line of constant nucleation site radius (2/lm). However, P' and T' are not specified. Tile results for r calculated from eq. (8) are given in table 1, with a theoretical r/7 calculated from eq. (7), giving a value for 3' which correlates theory and experiment. Tile Holtz and Singer [10] result may be neglected because the free gas surface was near heated region, and natural convection could have maintained a high Pg (Singer and Holtz [12]). The weighted mean of the other five sets of results is 3' = 0.35. Tile calculated error on the mean is -+ 0.01, but this is perhaps misleading, since under constant conditions individual workers find a scatter of typically -+ 30°C in the superheat measurements. This corresponds to an uncertainty of -+ 20% in 7, if used to predict the result of just one boiling event.
4. Discussion Using eq. (3), this value of 7 means 0A/> 110 ° which is consistent with advancing contact angles measured on oxide surfaces. There are two published experiments at pressures above 1.35 atm (Holtz and Singer [10] and Chen [13]) but in both cases convection causes unpredictable gas behaviour. It is noted that Le Gonidec used lnconel, Pinchera and Holtz used AISI304, and Holland and Winterton used AIS1321 ; no systematic difference is observed from one alloy to another. A characteristic feature of all superheat data is the large scatter, which is much larger than the error in measurement. This type of behaviour is to be expected if the walls of the cavity are only locally non-wetted. Quite small variations in the pre-boiting conditions would be sufficient to ensure that the meniscus sticks sometimes at non-wetted inclusion and sometimes at another, since it cannot reach equilibrium while in contact with the wetted surface. These small variations might be caused by surface active impurities, or by pressure pulses produced during previous boiling. A further prediction of this cavity model does not follow from the earlier pressurisation theories is that once tile predicted cavity radius becomes appreciably less than tile size of the non-wetted inclusions there is a possibility of the cavity being wholely within an inclusion. At this point, with r perhaps about 0.1 #m and P' about 10 arm., the theory will fail, and further pressurisation will have no effect. Tile results of table 1 all apply to natural surfaces. Tlaere is evidence that surfaces with artificially drilled cavities do not respond to pressurisation (Peppier et al. [ 14]). This can be explained if it is assumed that the drilling process itself introduced non-wetted material into tile surface (Holtz and Singer [15]). Although we have not used a lot of the experimental data, because of the unknown but almost certainly appreciable Pg term, it should be possible to bring these results into line with the theory by the assumption of plausible Pg values. This fact is the case. In particular, the lowest superheats observed at low saturation temperatures could reasonably be assumed to
P.K. Holland, R.H.S. Winterton, The radii of the surface nucleation sites which initiate sodium boiling
391
Table 1 Workers
P' [atm]
T' [°C]
r [um]
Holtz et al. [6] Holland and Winterton [ 3 ] Holland [7] (Tsat = 634-815°C) Le Gonidec et al. [8] Pinchera et al. [9] Holtz and Singer [ 10]
< 0.7 1.0 1.0 1.35 1.0 1.08
540 390 390 500 861 370
1.1 1.3 0.72 2.8 3.5
correspond to no gas loss during heating. Assumption of a Pg value close to the cover gas pressure yields the same value of 7 as found when Pg = 0. One point that perhaps requires further discussion is the three-dimensional nature of the cavity, of which fig. 1 is only a two-dimensional simplification. It is not suggested that the non-wetted inclusion neccessarily extends in a ring right round the c~vity. Provided one or two inclusions extend most of the way round, the shape of the meniscus will be much as in fig. 1, except that near the cavity wall there will be a complicated dimpling by the wetted regions, with the effective radius of curvature remaining constant but the radii in the two perpendicular directions no longer being equal. The experimental results imply c < 0A - 110 °, which is consistent with a very steep-sided cavity, so it is quite possible for an inclusion to occupy most of the distance round the cavity, and be roughly the same dimension in very direction, without invalidating the assumption r = rA.
5. Application to the fast reactor core Typically the pressures in the core range from 3 - 6 atm., and the corresponding temperatures from 6 0 0 400°C. If there are insufficient gas bubbles in the flow, nucleation will occur at surface cavities, and the highest superheats would result ifP~ in eq. (5) was zero. Winterton [16] has shown that if there is no entrainment of bubbles at the free surface the number of bubbles produced by dissolved gas coming out of solution will not be enough to guarantee nucleation by gas bubbles, and also that the sodium in the core will be very undersaturated with dissolved gas, leading to very low values of P~. Under these conditions the vapour pressure needed to nucleate boiling can be calculated from eq. (6), since P~ = 0 and hence
r/~/[uml 3.3 3.3 2.25 11.5 3.3
3'
Number of points
0.34 0.33 0.4 0.32 0.24 1.06
18 12 7 1 1 1
1300%
1200"C / f 1
/ /
/ f
11000C
J
J
J
1000"C 3
J
J
J
J
T sa~
J
[ 4 ATMOSPHERES
Fig. 2. Predicted nucleation temperature, T, as a function o f the pressure in different parts o f the fast reactor core.
Pg = 0 (Pg would be zero regardless of P~ since at the high temperatures required for nucleation (over 1000°C the gas will dissolve in a couple of seconds or less). The nucleation temperatures shown as a solid line in fig. 2 are calculated from eq. (6) with 7 = 0.35. A scatter band of-+ 30 °, typical of experimental resuits, is added. At these high temperatures the fuel cladding fails rapidly, so there is a strong possibility that boiling will be nucleated by the fission gas bubbles released.
6. Conclusions The superheat needed to nucleate sodium boiling from natural surface cavities, in cases where the gas
392
P.K. Holland, R.H.S. Winterton, The radii o f the surface nucleation sites which initiate sodium boiling
partial pressure in the cavity is zero, may be predicted from ev : e +
(P'
ev),
where 7 = 0.35 (-+ 20%). This equation satisfactorily predicts the data of different workers. The value of 7 is consistent with the theoretical derivation of the equation, which is based upon a new and physical more reasonable cavity model. In the absence of entrained gas bubbles the e q u a t i o n can be used to correlate the superheat in a fast reactor boiling incident. It predicts values a r o u n d 150°C, leading to nucleation temperatures above likely cladding failure temperatures.
Nomenclature C= half-angle of equivalent conical cavity, p = external pressure at nucleation (saturation pressure), I p = external pressure before boiling, Pv, Pg = vapour, gas pressure in cavity at nucleation,
ev,+,'g
: vapour, gas pressure in cavity u n d e r preboiling conditions, r = radius of liquid meniscus at nucleation, r' = radius of liquid meniscus under pre-boiling conditions, rA = radius of cavity (see fig. 1), T = nucleation temperature, T' = pre-boiling temperature, 7 = l COS (0A C) ], O = surface tension at nucleation, o' = surface tension during pre-boiling, 0A = advancing contact angle.
Acknowledgement This paper is published by permission of the Central Electricity Generating Board.
References [ 1 ] R. Holtz, The effect of pressure temperature history upon incipent boiling superheat in liquid metals, ANL 7184 (1969). [2] R. Apfel, J. Acoust. Soc. Amer., 48 (1970) 1179-1186. [3] P.K. Holland and R.H.S. Winterton, Int. J. Heat and Mass Transfer (to be published). [4] Kiessling, Non-metallic inclusions in steel. Part 1II. The origin and behaviour of inclusions and their influence on the properties of steel, The Iron and Steel Institute, ISI 115. I51 D. Bradhurst and A. Buchanan, Austral. J. Chem., 14 (1961) 397-408. [6] R. Holtz, H. Fauske and D. Eggen, Paper 23, Trogdir seminar on liquid metal heat transfer, International cen tre for Heat and Mass Transfer (1971). [71 P.K. Holland, The nucleation of boiling in sodiumcooled fast reactor incidents, CNAA PhD thesis (1971). [8] Le Gonidec, Rouvillios, R. Semeria, N. Lions and M. Robin Simon, Proc. Int. Conf. Safety of Fast Reactors, Aix-en-Provence CEA ed, Denielon (1967). [9] G. Pinchera, G. Tomassetti, G. Gambardella, and G. Farrello Proc. Int. Conf. Safety of Fast Reactors, Aixen-Provence CEA ed, Denielon (1967). [10] R. Holtz and R. Singer, Chem. Eng. Prog. Symp. Series. Tenth National Heat Transfer Conference (1968) 121129. [ 11 ] D. Smidt, P. Fette, W. Peppier, E. Schlechtendahl and G. Schultheiss, KFK 790. EUR 3960e (1968). [12] R.M. Singer and R.E. Holtz, Trans. Am Nuc. Soc. (1970) 820. [13] J.C. Chen, J. Heat Transfer. Trans. ASME (c) 90 (1968) 303-312. [14] W. Peppier, E. Schlechtendahl and G. Schultheiss, Nuc. Eng. and Design, 14 (1970) 23-42. [15] R.E. Holtz and R.M. Singer, Heat Transfer 1970, X, Elsevier, Amsterdam (1971) 250. [16] R.H.S. Winterton, Nuc. Eng. and Design, 22 (1972) 262-271.
THE RADII OF SURFACE NUCLEATION
SITES
WHICH INITIATE SODIUM BOILING P.K. HOLLAND* and R.H.S. W1NTERTON~ Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, England Received 18 December 1972
Values for the radii of nucleation sites are calculated from incipient boiling superheat measurements obtained in several different experiments. The radii predicted by applying the pre-boiling pressure history theory to a new cavity model are in agreement with the experimental results.
1. Introduction The temperature at which sodium boiling starts in a fast reactor overheating incident is of interest in safety studies. Boiling is nucleated when e v - e = (20/0
- eg
(1)
and a major problem is the prediction of r, the radius of the liquid meniscus at nucleation. In this paper, the pre-nucleation pressure history theory (Holtz [ 1], Apfel [2]) is applied to a surface nucleation site model proposed earlier (Holland and Winterton [3]). The predicted radii are in good agreement with those deduced from superheat experiments.
2. Theory It is known that steels are wetted by sodium at high temperatures, i.e. a drop of sodium placed on the surface will spread out, with a low or zero contact angle. A consequence of this is that the sodium will tend to penetrate and fill any cracks or cavities in the * Present address: CEGB NW Region scientific services department, 825 Wilmslow Rd., Manchester 20, England. :~Present address: Department of Mechanical Engineering, University of Birmingham, P.O. Box 363, Birmingham 15, England.
surface. However, a cavity filled with liquid cannot act as a nucleation site, so it.is necessary to assume that locally the surface is non-wetted. These non-wetted regions are probably non-metallic inclusions, which are known to be present in steels in large numbers (Kiessling [4]). They consist mostly of stable oxides, such as MnO, SiO 2, A1203, their size is from 0.1 to 10 tam and there are about 107/cm 2 of surface. Similar stable oxides (MgO, BaO, CaO) are not wetted by sodium (Bradhurst and Buchanan [5]). A model cavity on a steel surface is sketched in fig. la under the pressure conditions before boiling, the liquid-vapour interface is pushed down until it reaches an inclusion, where it has an advancing contact angle 0A > 90 ° (non-wetting). The meniscus will proceed further into the cavity until the increasing excess pressure due to surface tension across the more highly curved surface just balances the external pressure. Of course, the liquid might penetrate beyond the non-wetted inclusion before this equilibrium state is reached. However, the surface probably contains many cavities and it may be expected that there will be at least one cavity which has a non-wetted inclusion at a position appropriate to any preboiling pressure. The position of the meniscus is shown in fig. lb. The condition for mechanical equilibrium of the interface is
P.K. Holland, R.H.S. Winterton, The radii o f surface nucleation sites which initiate sodium boiling
(fig. lc). Boiling will nucleate once this critical radius is reached, since further growth of the vapour region results in increased r and hence a reduction in the 2air term in eq. (1) which opposes bubble growth. If the cavity sides are steep and the non-wetted regions small, the radius at nucleation, r, will not be significantly different to rA. Substituting from (4) into (1), with r = rA, gives
(a)
- uou,o soo,oM /
389
wETTEoMETA.
PI / I / I / t------W ~12¢//
+ o ( e ' - ei -
/
(5>
For cases where Pg and Pg' = 0 this reduces to P = P + o (P' - P~) / o'7.
(6)
I /
b/ (b)
(¢)
Fig. 1. (a) Typical surface cavity; (b) position of liquid meniscus under pre-boiling conditions; (c) critical position of liquid meniscus at nucleation. P'-
P' - P' = 2a'/r',
v
g
(2)
where P' is the external pressure, Pv, Pg and o' the vapour pressure, gas pressure and surface tension at the temperature in question. It follows from simple geometry that
The advantage of the new cavity model used here is that eq. (5) has been obtained without the physically implausible assumption of a surface with simultaneously very large advancing contact angle and very small receding contact angle. Most sodium systems are filled with sodium above its melting point (100°C) and then the temperature is increased to T' (typicaUy 500°C). The solubilities of inert gases in sodium increase so rapidly that such a filling procedure gives P~ = 0 initially. It takes several days for P~ to increase to the cover gas pressure, because in a stagnant system, diffusion from the free surface through the sodium to the surface cavities is very slow. The radius, r, is given by the highest pressurisation. This normally occurs in the early stages when Pg = 0. Therefore r = rA = 7 2 o ' / ( P ' - Pv)"
(7)
rA = - r' cos (0A - c). For equilibrium to be possible at all, the meniscus must be concave on the liquid side, i.e. 0A - c > 90 °, and the value of cos (0A - c) will be negative. Defining 3' = - cos (0A - c) = [ cos (0A - c)[
(3)
we then have rA = 7 / substituting in (2) gives rA = 7 2 a ' / ( P ' - Pv - P~g)
(4)
(c.f. Apfel [2] eq. 3). When the vapour pressure increases on approaching boiling, the meniscus is pushed back until it reaches a wetted region, where it stops and the shape of the meniscus changes. The minimum radius the meniscus has to achieve is the radius of the cavity at this point
3. Experimental results Unless Pg is known, an experimental value for the cavity radius cannot be found from superheat measurements. Holland and Winterton [3] show that gas from surface nucleation sites dissolves during heating and Pg in eq. 1 decreases continuously to zero. The time scales for argon to dissolve from surface cavities are days at 500°C, minutes at 700°C and seconds at 900°C. Thus Pg ~ 0 at high saturation temperatures ( > 800°C) in experiments heated at a moderate rate. The radius of the surface nucleation site is then from (1) r = 2o/(P v - t).
(8)
390
P.K. Holland, R.H.S. Winterton, The radii of surface nucleation sites which initiate sodium boiling
Four distinct sets of experimental data are available from which cavity radii may be calculated. In the first three, reasonable precautions were taken to ensure that Pg = 0. In the other, older, work, while there was little or no control of gas content, it is possible to choose data for which Pg was probably zero. (a) Holtz et al. [6] used no cover gas; the saturation pressure was produced electromagnetically. Thus Pg = 0 in all runs, and they found 7 = 0.34 from their results at values of P' from 0.07-0.7 atm. (b) Holland and Winterton [3] studied loss of argon from surface cavities, and found an asymptotic value of superheat corresponding to Pg = 0. (c) Holland [7] did some experiments with helium cover gas so that Pg = 0,can be attained quickly at low saturation temperatures because helium dissolves much more rapidly than argon. (d) Le Gonidec et al. [8], Pinchera et al. [9], Holtz and Singer [10], employed argon cover gas. Since the time scale for argon to dissolve at the highest temperatures used by these workers is of order of several seconds, it is likely that for some of the results the argon had time to dissolve completely. Consequently, the highest superheat at a high saturation temperature is assumed to correspond to P_ = 0. Also Smidt [ 111 describes some experiments without cover gas which fall on a line of constant nucleation site radius (2/lm). However, P' and T' are not specified. Tile results for r calculated from eq. (8) are given in table 1, with a theoretical r/7 calculated from eq. (7), giving a value for 3' which correlates theory and experiment. Tile Holtz and Singer [10] result may be neglected because the free gas surface was near heated region, and natural convection could have maintained a high Pg (Singer and Holtz [12]). The weighted mean of the other five sets of results is 3' = 0.35. Tile calculated error on the mean is -+ 0.01, but this is perhaps misleading, since under constant conditions individual workers find a scatter of typically -+ 30°C in the superheat measurements. This corresponds to an uncertainty of -+ 20% in 7, if used to predict the result of just one boiling event.
4. Discussion Using eq. (3), this value of 7 means 0A/> 110 ° which is consistent with advancing contact angles measured on oxide surfaces. There are two published experiments at pressures above 1.35 atm (Holtz and Singer [10] and Chen [13]) but in both cases convection causes unpredictable gas behaviour. It is noted that Le Gonidec used lnconel, Pinchera and Holtz used AISI304, and Holland and Winterton used AIS1321 ; no systematic difference is observed from one alloy to another. A characteristic feature of all superheat data is the large scatter, which is much larger than the error in measurement. This type of behaviour is to be expected if the walls of the cavity are only locally non-wetted. Quite small variations in the pre-boiting conditions would be sufficient to ensure that the meniscus sticks sometimes at non-wetted inclusion and sometimes at another, since it cannot reach equilibrium while in contact with the wetted surface. These small variations might be caused by surface active impurities, or by pressure pulses produced during previous boiling. A further prediction of this cavity model does not follow from the earlier pressurisation theories is that once tile predicted cavity radius becomes appreciably less than tile size of the non-wetted inclusions there is a possibility of the cavity being wholely within an inclusion. At this point, with r perhaps about 0.1 #m and P' about 10 arm., the theory will fail, and further pressurisation will have no effect. Tile results of table 1 all apply to natural surfaces. Tlaere is evidence that surfaces with artificially drilled cavities do not respond to pressurisation (Peppier et al. [ 14]). This can be explained if it is assumed that the drilling process itself introduced non-wetted material into tile surface (Holtz and Singer [15]). Although we have not used a lot of the experimental data, because of the unknown but almost certainly appreciable Pg term, it should be possible to bring these results into line with the theory by the assumption of plausible Pg values. This fact is the case. In particular, the lowest superheats observed at low saturation temperatures could reasonably be assumed to
P.K. Holland, R.H.S. Winterton, The radii of the surface nucleation sites which initiate sodium boiling
391
Table 1 Workers
P' [atm]
T' [°C]
r [um]
Holtz et al. [6] Holland and Winterton [ 3 ] Holland [7] (Tsat = 634-815°C) Le Gonidec et al. [8] Pinchera et al. [9] Holtz and Singer [ 10]
< 0.7 1.0 1.0 1.35 1.0 1.08
540 390 390 500 861 370
1.1 1.3 0.72 2.8 3.5
correspond to no gas loss during heating. Assumption of a Pg value close to the cover gas pressure yields the same value of 7 as found when Pg = 0. One point that perhaps requires further discussion is the three-dimensional nature of the cavity, of which fig. 1 is only a two-dimensional simplification. It is not suggested that the non-wetted inclusion neccessarily extends in a ring right round the c~vity. Provided one or two inclusions extend most of the way round, the shape of the meniscus will be much as in fig. 1, except that near the cavity wall there will be a complicated dimpling by the wetted regions, with the effective radius of curvature remaining constant but the radii in the two perpendicular directions no longer being equal. The experimental results imply c < 0A - 110 °, which is consistent with a very steep-sided cavity, so it is quite possible for an inclusion to occupy most of the distance round the cavity, and be roughly the same dimension in very direction, without invalidating the assumption r = rA.
5. Application to the fast reactor core Typically the pressures in the core range from 3 - 6 atm., and the corresponding temperatures from 6 0 0 400°C. If there are insufficient gas bubbles in the flow, nucleation will occur at surface cavities, and the highest superheats would result ifP~ in eq. (5) was zero. Winterton [16] has shown that if there is no entrainment of bubbles at the free surface the number of bubbles produced by dissolved gas coming out of solution will not be enough to guarantee nucleation by gas bubbles, and also that the sodium in the core will be very undersaturated with dissolved gas, leading to very low values of P~. Under these conditions the vapour pressure needed to nucleate boiling can be calculated from eq. (6), since P~ = 0 and hence
r/~/[uml 3.3 3.3 2.25 11.5 3.3
3'
Number of points
0.34 0.33 0.4 0.32 0.24 1.06
18 12 7 1 1 1
1300%
1200"C / f 1
/ /
/ f
11000C
J
J
J
1000"C 3
J
J
J
J
T sa~
J
[ 4 ATMOSPHERES
Fig. 2. Predicted nucleation temperature, T, as a function o f the pressure in different parts o f the fast reactor core.
Pg = 0 (Pg would be zero regardless of P~ since at the high temperatures required for nucleation (over 1000°C the gas will dissolve in a couple of seconds or less). The nucleation temperatures shown as a solid line in fig. 2 are calculated from eq. (6) with 7 = 0.35. A scatter band of-+ 30 °, typical of experimental resuits, is added. At these high temperatures the fuel cladding fails rapidly, so there is a strong possibility that boiling will be nucleated by the fission gas bubbles released.
6. Conclusions The superheat needed to nucleate sodium boiling from natural surface cavities, in cases where the gas
392
P.K. Holland, R.H.S. Winterton, The radii o f the surface nucleation sites which initiate sodium boiling
partial pressure in the cavity is zero, may be predicted from ev : e +
(P'
ev),
where 7 = 0.35 (-+ 20%). This equation satisfactorily predicts the data of different workers. The value of 7 is consistent with the theoretical derivation of the equation, which is based upon a new and physical more reasonable cavity model. In the absence of entrained gas bubbles the e q u a t i o n can be used to correlate the superheat in a fast reactor boiling incident. It predicts values a r o u n d 150°C, leading to nucleation temperatures above likely cladding failure temperatures.
Nomenclature C= half-angle of equivalent conical cavity, p = external pressure at nucleation (saturation pressure), I p = external pressure before boiling, Pv, Pg = vapour, gas pressure in cavity at nucleation,
ev,+,'g
: vapour, gas pressure in cavity u n d e r preboiling conditions, r = radius of liquid meniscus at nucleation, r' = radius of liquid meniscus under pre-boiling conditions, rA = radius of cavity (see fig. 1), T = nucleation temperature, T' = pre-boiling temperature, 7 = l COS (0A C) ], O = surface tension at nucleation, o' = surface tension during pre-boiling, 0A = advancing contact angle.
Acknowledgement This paper is published by permission of the Central Electricity Generating Board.
References [ 1 ] R. Holtz, The effect of pressure temperature history upon incipent boiling superheat in liquid metals, ANL 7184 (1969). [2] R. Apfel, J. Acoust. Soc. Amer., 48 (1970) 1179-1186. [3] P.K. Holland and R.H.S. Winterton, Int. J. Heat and Mass Transfer (to be published). [4] Kiessling, Non-metallic inclusions in steel. Part 1II. The origin and behaviour of inclusions and their influence on the properties of steel, The Iron and Steel Institute, ISI 115. I51 D. Bradhurst and A. Buchanan, Austral. J. Chem., 14 (1961) 397-408. [6] R. Holtz, H. Fauske and D. Eggen, Paper 23, Trogdir seminar on liquid metal heat transfer, International cen tre for Heat and Mass Transfer (1971). [71 P.K. Holland, The nucleation of boiling in sodiumcooled fast reactor incidents, CNAA PhD thesis (1971). [8] Le Gonidec, Rouvillios, R. Semeria, N. Lions and M. Robin Simon, Proc. Int. Conf. Safety of Fast Reactors, Aix-en-Provence CEA ed, Denielon (1967). [9] G. Pinchera, G. Tomassetti, G. Gambardella, and G. Farrello Proc. Int. Conf. Safety of Fast Reactors, Aixen-Provence CEA ed, Denielon (1967). [10] R. Holtz and R. Singer, Chem. Eng. Prog. Symp. Series. Tenth National Heat Transfer Conference (1968) 121129. [ 11 ] D. Smidt, P. Fette, W. Peppier, E. Schlechtendahl and G. Schultheiss, KFK 790. EUR 3960e (1968). [12] R.M. Singer and R.E. Holtz, Trans. Am Nuc. Soc. (1970) 820. [13] J.C. Chen, J. Heat Transfer. Trans. ASME (c) 90 (1968) 303-312. [14] W. Peppier, E. Schlechtendahl and G. Schultheiss, Nuc. Eng. and Design, 14 (1970) 23-42. [15] R.E. Holtz and R.M. Singer, Heat Transfer 1970, X, Elsevier, Amsterdam (1971) 250. [16] R.H.S. Winterton, Nuc. Eng. and Design, 22 (1972) 262-271.