Nuclear Engineering and Design 53 (1979) 347354 © NorthHolland Publishing Company
GAS BUBBLE ADHESION TO HEAT TRANSFER SURFACES R.H.S. WINTERTON, A.B.H. CHEVALIER and G.A. FOWLES * Mechanical EngineeringDepartment, University of Birmingham, P.O. Box 363, Birmingham B15 2TT, England Received 10 May 1977
A model of gas bubble adhesion to heat transfer surfaces is developed, by considering the drag, surface tension and buoyancy forces acting on a gas bubble attached to a solid wall with liquid flowing past. The model is consistent with the fuel pin failures that were observed in the irradiation tests of fuel subassemblys for the British Prototype Fast Reactor, and predicts that changing to upward flow will not in itself necessarily prevent such failures in the future.
mental results for air bubbles in water [3], and in explaining overall reductions in heat transfer to liquid metals [4]. This present work uses a computer program to calculate the three forces involved  surface tension, drag, and buoyancy  which are proportional to different powers of the bubble radius. The balance of forces parallel to the surface is considered to determine whether or not the bubble remains in contact with the surface.
1. Introduction A large programmme of irradiation testing of fuel pins for the British Prototype Fast Reactor (PFR) has been carried out in the Dounreay Fast Reactor (DFR). Most o f the pin failures that were observed in the initial tests have been attributed to gas bubbles adhering to the fuel pin surfaces and causing local hot spots. The origin of these gas bubbles was probably entrainment at the free surface above the core. The failed fuel pins have longitudinal cracks in the cladding, surrounded b y a tearstain marking [1]. Similar marks also appear on intact fuel pins, and in a number of cases paired tearstains on neighbouring fuel pins were noticed, suggesting that a bubble had spanned the gap between the pins. Nearly all the failures occurred at positions where the clad temperature was comparatively low, i.e. in the range 250450°C. Also the failures were associated with low coolant velocities, around 1 m/s [2]. This paper uses a mathematical model o f the forces acting on a gas bubble to predict the maximum size of bubble which could remain in contact with the vertical surface of a pin. • Using a similar model of gas bubble adhesion, limited success has been possible in explaining experi
2. Surface tension force A net surface tension force arises from the fact that the contact angles will be different upstream and downstream of the bubble. Upstream o f the bubble the liquid is tending to advance over the surface and the advancing contact angle, 0 a, will be displayed. Similarly, downstream of the bubble where the liquid is tending to recede from the bubble the receding contact angle, Or, will be displayed (fig. 1). These are the
J
/,
SOLID SURFACE
* Present address: Performance and Safety Department, Nuclear Power Company, Cambridge Road, Whetstone, Leicestershire.
• Fig. 1. Gas bubble adhering to surface with different contact angles upstream and downstream. 347
348
R.H.S. Winterton et aL / Gas bubble adhesion to heat transfer surfaces
limiting values of contact angle and will apply just before the bubble breaks away from the surface. For smaller bubbles the contact angles will be closer to the equilibrium value on each side, and the net surface tension force will just balance the resultant drag and buoyancy forces. The surface tension force is exactly analogous to the frictional force, increasing as the bubble grows and maintaining equilibrium up to a maximum value. It will also change direction as a result of the other forces changing direction. Obviously with different contact angles on each side the bubble is not spherical, but nonetheless we assume that the line of contact between the bubble and the surface remains circular, and the same as that for a spherical bubble with the equilibrium contact angle 0e = (0a + 0r)/2. For a smooth variation of 0 round the bubble, between the limits 0 a and 0r, the integrated effect of the surface tension, resolved in the flow direction, gives a net surface tension force [3] Fs = ~Tr3'r sin 0e(cos Or  cos 0a) ,
(l)
where r is the radius of the bubble. An alternative approach is to use the work of Macdougall and Ockrent [5 ] on the stability of a liquid drop on an inclined plane. The appearance of the drop is just as though fig. 1 were tilted through an angle a. By equating the surface tension and gravitational force on a thin lamina through the centre of the drop in the plane o f the illustration they showed that the condition for equilibrium is "),(cos Or  cos 0a) = A o g s i n a ,
and A =r20r  0 e +~ sin 20e).
(4)
To calculate the other forces the bubble is treated simply as a truncated sphere with contact angle 0 e.
3. Drag f o r c e In general the drag force will be given by Fd = 1 CdpO2A ,
where C d is a function of the bubble Reynotds number. However, measured values of the drag coefficient Cd for bubbles on a surface are not available. Instead, expressions for bubbles rising freely under buoyancy are used. Table 1 shows empirical correlations recommended by Peebles and Garber [6]. Before these results can be applied to the present problem, however, we have to consider what linear scale to use in forming the bubble Reynolds number, and what is an appropriate velocity o. For the characteristic linear dimension in the bubble Reynolds number we decided to use the distance the bubble adhering to the wall projects into the flow (fig. 2), in place of the bubble diameter used for the free bubble. However, the effect of keeping to the bubble diameter was also investiagted. Unlike the freely rising bubble the velocity of the flow some distance upstream of the bubble is not constant but varies with distance from the wall. Accordingly the cross section presented to the flow was divided into elementary strips of area ZL4, as shown in fig. 2.
where A is the area of the lamina. Equating total surface tension and gravitational forces on the drop gives Table 1 Drag coefficient equations which hold for separate regions
F s = V p g sin c~,
and combining these two equations gives F s = VT(cos Or  cos Oa)/A .
(2)
If V, the drop volume, and A are directly measured then these equations are exact for the case where only the two forces are acting. For our application we consider eq. (2) only as an alternative to eq. (1), and for Vand A use the expressions for a spherical bubble, i.e. V =~Irr3(2 + 3 cos 0e  cos30e)
(3)
Region
Drag coefficient Cd
Range of applicability
1
2 3
24/Re b 18.7 Reff 0"68 0.0275G 1 Re~
4
0.82G~ 25 Re b
Reb < 2 2 < Reb < 4.02G~0"214 4.02G~ T M < Reb < 3.1G~025 3.1G~ 0.25 ~ Reb
G 1 = gtt4/p7 3, a dimensionless group.
R.H.S. lCinterton et al. / Gas bubble adhesion to heat transfer surfaces
coel
AA
///////////// SOLID SURFACE
Fig. 2. Projected area of the bubble facing the flow. The drag force is obtained by summation over elementary strips of area AA.
The drag force is now given by F d = l c d p ~ V ( X ) 2 ALl,
(5)
with AA = Ax2 (x + z) 1/2(2r  x  z) 1/2 z =r(1

349
and the bubble radius used is taken as a lower bound to the value of the critical radius on detachment. The trial radius is now doubled and F recalculated. This process is repeated until a radius is reached which gives a negative value of F and this radius is taken as the upper bound. For the next step the trial radius is the average of the upper and lower bounds, and depending on the value of F either the upper or lower bound is then replaced by a better estimate. In this way the iteration converges on the desired critical radius, which corresponds to the largest bubble that can remain stationary in contact with the wall. For downward flow with moderately high velocities in narrow channels the bubbles will tend to detach downwards and a similar procedure is used with (8)
F = Fs + F b  F d .
and
cos 0e).
The velocity v ( x ) a t a distance x from the wall is found from the universal velocity profile for a smooth circular pipe, e.g. [3]. Cd is worked out using the velocity at half the maximum distance the bubble projects from the wall.
4. Buoyancy force
For the main series of results to be described in this paper this approach is adequate, but in principle at least a number of problems can arise. In downward flow the buoyancy force may dominate and the bubbles detach upwards; also there may be more than one critical bubble radius at which the forces on the bubble exactly balance. If it is suspected that these problems may be encountered the program can be used to plot the regions of bubble stability for a range of values of radius.
This is given by Fb = g V o ,
(6)
where V is the bubble volume given by eq. (3).
5. Calculation of critical radius The construction of the program to calculate the critical radius is quite straightforward. An initial guessed value of the bubble radius on detachment from the surface is fed in and the surface tension, drag, and buoyancy forces calculated, as already described. For upward flow, where the drag and buoyancy forces always act in the opposite direction to the surface tension force, a residual force F is calculated as follows: F =Fs Fd  Fb .
(7)
If F i s positive the bubble will adhere to the surface
6. Sodium properties These were taken from the L i q u i d M e t a l H a n d b o o k [8], which gives equations for density and surface tension. For the viscosity a standard form of equation was fitted to the H a n d b o o k data. The following equation was used to determine the density, p, of liquid sodium: #= 949  0.223T  1.75 10ST 2 (kg/m3),
(9)
where T is the temperature in degrees centigrade. The viscosity, g, of liquid sodium was evaluated according to the relation: logda = 9.3714 +
796.68 (kg/s  m ) . T + 273.16
(10)
The surface tension, 7, was obtained by the expression: 3' = 0.202  104T (N/m).
(11)
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R.H.S. Winterton et al. / Gas bubble adhesion to heat transfer surfaces
Table 2 Contact angles of a bubble adhering to a surface of abraded M316 steel in a flow of sodium Temperature (°C)
Advancing angle (degrees)
Receding angle (degrees)
200 250 300 350 400 450 500
140 115 90 75 60 50 40
50 45 40 35 30 30 3O
Note: The times after which the angles were measured varied from a few minutes to a number of hours.
B
EXCEEDED
~\\\/
m o ,<
._ ...... I.~
CEEDED
1
g
7. Contact angles Values of contact angles were found in refs. [9] and [10]. The first of these measured the advancing angle by the sessile drop technique, and the other the receding angle by the vertical plate method. A summary of these results is given in table 2 for the wetting of M316 stainless steel by liquid sodium over the temperature range of 200500°C.
F L O W VELOCITY
ms'
Fig. 3. Radii of the largest bubbles that can remain in contact with the surface as a function of flow velocity, for downward flow of sodium in PFR equivalent channel. The temperatures should be read in conjunction with the contact angles in table 2. Criterion A gives 2 mm coverage of the surface, criterion B means the bubble can span the minimum pin to pin gap.
8. Results The main results presented in this paper are for the fuel pin geometry of the Prototype Fast Reactor, using an equivalent circular diameter of 4.36 mm to represent the flow passages between the fuel pins. To investigate the effect of different degrees of wetting of the fuel can surface the contact angle data in table 2 were used. It is not though that the values in table 2 would necessarily be reproduced if the measurements were repeated under slightly different conditions, so where our model predicts sizes of bubbles adhering to a surface at a particular temperature, the results are for the associated contact angles in the table, rather than a definite prediction for that temperature. Since we could not find contact angle data for NaK most of the results are for sodium. As will be shown later, the effect of changing to NaK properties, but keeping the contact angle values given in table 2, is very small. Fig. 3 shows bubble radius versus flow velocity for
downward flow. These are the largest bubbles that can remain stationary in contact with the surface. Apart from the use of sodium properties these results can be compared with the results of testing PFR fuel pins in the DFR. The size of bubble that would be required to cause failure of the can is not clear. A 40 ° coverage of the perimeter of the can has been suggested [ 11 ]. This corresponds to 2 mm on the can surface, which is con. sistent with the shorter dimension of the tearstain markings [1 ]. We will call this criterion A. Another possible mechanism for failure is that bubbles can span the narrow gap between neighbouring fuel pins, and thus achieve an enhanced stability. The closest gap is 1.52 mm, so criterion B is that the bubble projects more than 1.52 mm from the surface. It is clear from fig. 3 that for the comparatively poor wetting at 300 and 400°C, both criteria are likely to be
351
R.H.S. Winterton et al. / Gas bubble adhesion to heat transfer surfaces
B EXCEEDED I I
I
,,\!\ \ e~
u~
\"X,; \
n~
,.n
rn rn
en
FLOW V E L O C I T Y
ms "i
Fig. 4. As fig. 3, but upward flow.
exceeded once the velocity falls much below 1 m/s. This is consistent with the conditions under which failed fuel pins were observed in the irradiation tests. The very poor wetting corresponding to the temperature of 200°C in table 2 is unlikely to be relevant to fast reactor core design, but might be applicable in cold trap circuits. Fig. 4 shows the largest bubbles that can adhere to the surface in upward flows, i.e. PFR conditions. In fig. 5 the effect of small changes in the model is investigated. All the results are for the contact angles corresponding to 300°C, as given in table 2, and for downward flow. The top full curve is the prediction of the basic model, as shown in fig. 3. The effect of changing to the alternative surface tension force [eq. (2)] is so small that for practical purposes the curves coincide. Using values of density, viscosity and surface tension appropriate to NaK at 300°C also makes very little difference. The Dounreay raector uses 70% sodium, 30% potassium by weight. The Liquid Metal Handbook [8] gives density and viscosity values for NaK, but we could not find a value for the surface tension at 300°C. Two sources [12,13] agree that the surface tension of
FLOW
VELOCITY
ms I
Fig. 5. Effect of changes in the model. All predictions for downward flow at 300°C. Curve a is the prediction of the basic model for sodium, as infig. 3. Curveb shows the effect of changing to eq. (2) for the surface tension force. For c NaK physical properties were used instead of those of sodium, but with contact angles still as in table 2. Curve d comes from changing the characteristic length used to calculate the bubble Reynolds number. The dashed curve results from an arbitrary 50% reduction in the drag coefficient. %
potassium at this temperature is 46% lower than that of sodium and if the properties of the mixture are assumed to be proportional to the molar fractions present, then the surface tension of the reactor NaK is 9% lower than that of sodium. The bottom curve in fig. 5 shows the largest change from the basic model. This is a result of working out the bubble Reynolds number in the drag force calculation using as the characteristic dimension the bubble diameter rather than the distance the bubble sticks out into the flow. From this point of view the predictions of the basic model are pessimistic. However, in view of the considerable uncertainty that exists over the value of the drag coefficient, it is of interest to study the effect of an arbitrary 50% reduction in its value. This gives the dashed surve in fig. 5.
352
2:
2
R.H.S. Winterton et al. / Gas bubble adhesion to heat transfer surfaces
jSTABLE ISTABLE
O"
2
36. c3 ~8 10
.2
./*
.6
BUBBLE EADIUS
.B
1.0
1
1.2
mm
Fig. 6. An illustration of the difficulties that can sometimes arise with the model. These results are for downward flow of sodium, with a weak surface tension force and in a much larger channel. The net force on the bubble is positive, indicating that the bubble will stick to the surface of the channel, for three distinct ranges of bubble size. The third region of stability arises because when the bubble Reynolds number enters region 3 of table 1 the drag force increases faster than the buoyancy force as the bubble gets larger. As mentioned earlier, under certain circumstances it is possible for the computer program to produce more than one real solution for the maximum size of bubble that can adhere to the surface. For example, for downward flow of sodium at 400°C and 0.6 m/s in a 20 mm internal diameter circular pipe, with advancing and receding contact angles o f 20 and 10 °, respectively, the results o f fig. 6 are obtained. Here the residual force on the bubble, defined as F =F s 
1F d  Fbl ,
(12)
is plotted against bubble radius. A positive value o f F means that a bubble o f the associated radius will adhere to the wall, a negative value that it will detach either upwards or downwards. The maximum size of bubble that might adhere to the wall in practice would depend on the origin of the bubbles. If they were growing slowly by the dissolution of dissolved gas, then they would detach while still very small. If larger bubbles were present in the flow of the right size for stability then they might adhere to the wall directly. 9. Discussion The predicted bubble sizes for downward flow in the PFR fuel pin geometry (fig. 3) are in good agree
ment with the observed failures in the irradiation tests, which occurred at flows o f around I m/s and for temperatures in the range 2 5 0 M 5 0 ° C . The agreement is particularly satisfactory because there are no empirical parameters in the model or the data which have been altered to obtain a good fit. However, three iinportant assumptions can be identified, two in the model and one in the data. The first assumption is that the drag coefficient will be the same as for free bubbles. Some confirmation that this is valid is given by the fact that the computer program correctly predicts experimental results that have been published [3] for gas bubbles in water. The predicted bubble radii are about 10% smaller than the experimentally observed values. However, the bubble Reynolds numbers in the water tests were lower than those o f the larger bubbles in fig. 3. Because of the high surface tension and low viscosity of sodium it is possible to get large bubbles with very high bubble Reynolds numbers, which are difficult to model adequately with nonmetallic liquids. The second assumption is that the complicated flow in the subassembly, with grids to support the fuel pins at intervals, can be adequately handled using fully developed flow and the equivalent hydraulic diameter. The importance o f this is hard to assess, but it seems reasonable that if the computer program predicts large bubbles and dangerous hotspots, then the real situation will display them too. The computer model events out variations, whereas in the real flow there will be regions where bubble adhesion is less likely than the model, but other regions where it will be more likely. Fig. 3 suggests in fact that failures would be encountered once the flow velocity falls below around 0.9 m/s, whereas the irradiation tests had failures at around 1 m/s. The last assumption is that contact angles for sodium on stainless steel can be used for NaK on stainless steel, and that advancing and receding contact angle data from different sources can be combined. The justification for this is that no better information was available. However, because of the rapid way that the predicted bubble size increases as the velocity is reduced, quite large uncertainties in the contact angles and in the model itself will not reduce the velocity at which local overheating occurs very much. Looking at the predictions for PFR, we see that upward flow in itself does not change the situation sig
R . H . S . W i n t e r t o n e t aL / Gas b u b b l e a d h e s i o n to h e a t transfer surfaces
nificantly and the assumption that similar failures cannot occur with upward flow is invalid. However, the average flow velocity in PFR is much higher, at 6 m/s for full load operation, and only near the edge o f the core during part load operation would velocities fall below 1 m/s. A typical heat flux for one of the more highly rated channels at full power would be about 2.5 MW/m 2, much the same as used in the irradiation tests. It is difficult to see how the heat generation and mass flow rates could locally get so out of balance to combine near the maximum heat flux with near the minimum flow rate, but possibly this could occur in some transient or accident situation. In principle gas bubble adhesion could provide a mechanism for propagation of fual pin failure. Fission gas released from a first failed pin could attach to other pins downstream and cause them to fail. If flow velocities were to drop much below 1 m/s, perhaps in a radial breeder assembly, then very large areas could be blanketed b y adhering gas bubbles, possibly large enough to cause trouble even with the greatly reduced heat generation rates. Of course, before there is any possibility o f gas bubbles sticking to surfaces and causing hot spots a source of gas and nonzero contact angles are essential. Since at the present time there is no way of guaranteeing the absence of either of these in liquid metal heat transfer equipment, high liquid velocities appear desirable.
353
Nomenclature Z
area of lamina through centre of bubble
C~ bubble deag coefficient F
= net force on bubble
F b   buoyancy force on bubble F d   drag force on bubble Fs = surface tension force on bubble g r Re Re b V o x z ol 7
Oa Oe Or p P
= acceleration due to gravity = bubble radius = pipe Reynolds number = bubble Reynolds number 2 r o / u mean flow velocity = bubble volume = Liquid velocity friction velocity = distance from wall = r(1  cos 0e) = angle of tilt of plate to horizontal = surface tension dimensionless distance from wall = advancing contact angle = equilibrium contact angle = ~(0 a + Or) = receding contact angle = kinematic viscosity = density = dimensionless velocity.
References 10. Conclusions The failures observed in the irradiation tests on PFRtype fuel pins are consistent with a simple model of gas bubble adhesion. The influence of the buoyancy force is small, and a change to upward flow will not in itself prevent a recurrence of this type of failure. A source o f gas, nonzero contact angles and low flow velocities are all necessary before large bubbles can adhere to heat trasnfer surfaces. The only parameter over which the designer has reasonable control is the flow velocity. Low velocities should be avoided, particularly in constant heat flux situations with high heat generation rates.
[ 1 ] H. Lawton et al., Irradiation Testing of Fuel for the British Prototype Fast Reactor, 4th Int. Conf. Peaceful Uses of Atomic Energy 1971, vol. 10 (UN and IAEA), pp. 3951. [2] J.F.W. Bishop and E.F. Kemp, Nucl. Eng. Int. 16 (1971) 643. [3] R.H.S. Winterton, Chem. Eng. Sci. 27 (1972) 1223. [4] R.H.S. Winterton, Int. J. Heat Mass Transfer 17 (1974) 549. [5] G. MacdougaUand C. Ockrent, Proc. Roy. Soc. A180 (1942) 151. [6] F.N. Peebles and H.J. Garber, Chem. Eng. Progr. 49 (1953) 88. [7] R.H.F. Pao, Fluid Mechanics (Wiley, New York, 1961), pp. 235269. [8] Liquid Metal Handbook, SodiumNaK supplement (US Atomic Energy Commission, 1955).
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R.H.S. Winterton et al. / Gas bubble adhesion to heat transfer surfaces
[9] E.N. Hodkin, D.A. Mortimer and M. Nicholas, The Wetting of Some Ferrous Materials by Sodium, Conference on Liquid Alkali Metals, 1974 (British Nuclear Energy Society, 1974), pp. 167170. [10] B. Longson and J. Prescott, Some Experiments on the Wetting of Stainless Steel, Nickel and Iron in Liquid Sodium, Conference on Liquid Alkali Metals, 1974 (Brit
ish Nuclear Energy Society, 1974), pp. 171~176. [ 11 ] J. Graham, Fast Reactor Safety (Academic Press, New York, 1971). [12] D.O. Jordan and J.E. Lane, J. Chem. 18 (1965) 1711. [13] A.N. Solov'ev and O.P. Markarova, Temp. Akad. Nauk. SSSR 4 (1966) 189.