J. Aerosol $ci., Vol. 20, No. 3, pp. 29~302, 1989. Printed in Great Britain. MAXIMUM 0021~8502/89 $3.00+0.00 Pergamon Press plc AEROSOL DENSITIES FROM EVAPORATIONCONDENSATION PROCESSES C. F. CLEMENTand I. J. FORD Theoretical Physics Division, UKAEA Harwell Laboratory, Oxon OX11 0RA, U.K. (Received 10 May 1988; an21in final form 20 October 1988) Almtraet--The coupled equations of heat and mass transfer are used to develop a model of aerosol generation in fluid flow over liquid surfaces. Maximum aerosol densities'achieved in turbulent flow of an initially saturated gas/vapour mixture along a heated or cooled pipe are calculated for water-vapour/alr and for sodium-vapour/argon. The model is also applied to flows near the heated surface in free convective situations. An analysis of an experiment in which sodium aerosols were generated in a cavity over a hot sodium pool indicates that the droplets which formed grew by passing many times through the boundary layer over the pool. NOMENCLATURE A Cross-sectional area of pipe c, Saturated vapour concentration cp Specific heat at constant pressure ~p Mean specific heat of mixture Cn~ Surface condensation number clS Element of surface area dV Volume element D Vapour-gas diffnsivity h Specific ¢a thalpy j~ Diffusive aerosol current relative to gas j, Diffusive vapour current relative to gas k Therm/d conductivity of vapour/gas mixture 1 Perimeter of pipe L Latent heat of vaporization .Z Loading ~h~ Aerosol growth rate per unit volume n Outward normal from surface N Aerosol number density Conductive heat current R Mean radius of aerosol droplets S Saturation t Time T Temperature of bulk flow T, Roof temperature of cover gas space T, Temperature of sodium pool v Velocity x Distance normal to wall z Distance along pipe Dimensionless parameter [equation (33)] 6a Dimensionless ratio [equation (29)] e Temperature difference between rising and falling currents in the cover gas space 4o Dimensionless ratio [equation (28)] Heat transfer quantity [equation (41)] p Total density Subscripts a e Pertaining to Pertaining to g Pertaining to v Pertaining to w Pertaining to aerosol equilibrium gas vapour wall Superscript Derivative with respect to T © Crown. 293 294 c.F. CLEMENTand I. J. FORD 1. I N T R O D U C T I O N There are two main areas of nuclear power technology where an understanding of aerosol physics can contribute. Firstly, in the design of the fast reactor cover gas space, where a sodium aerosol can develop, and secondly for studying the consequences of a release of radioactive material as aerosol in a hypothetical accident. Sodium aerosol in the cover gas space can affect the heat transfer properties of the cavity and can freeze in deposits on the roof. It is important to understand these phenomena, not least for reasons of safety. As for accident studies the interest lies in the processes of formatfon and removal of possibly dangerous suspended particles in a reactor containment atmosphere. In practical cases the formation and growth of aerosols is complicated, although the basic physics is clear. The complications arise from the coupling together of several distinct processes: fluid dynamics, heat transfer and droplet growth. Fluid flow has an influence upon heat dissipation, which in turn is linked with mass transfer and aerosol growth dynamics. These latter processes can affect the characteristics of the flow, especially in cases of free convection. In this paper we simplify the problem by imposing a flow. We then show how maximum aerosol densities resulting from heat and mass transfer into or out of the flow can be calculated. Much work has been done on the relevance to aerosol physics of the coupling between heat and mass transfer (Clement, 1985). There have, however, been few experiments carried out expressly to test these predictions. An exception is a study currently being performed at Harwell which is concerned with the behaviour of aerosol in a turbulent flow along a vertical pipe with cooled walls. This is a situation where to a first approximation the fluid flow is not coupled to the aerosol physics, and can be taken to be a well-mixed turbulent flow (for high enough Reynolds number) in the bulk, and a boundary layer at the wall through which heat and mass pass out of the bulk flow. The physics of the system, and that for the analogous heating case in which heat and mass pass into the flow are shown in Fig. 1. In the next section, we develop a model to describe how aerosol density changes in the flow are related to the temperature changes. In Section 3 we perform some example calculations for aerosols formed by heating and cooling water-vapour/air and sodium-vapour/argon mixtures. The results are discussed in Section 4. In Section 5 we apply the model to free flow over an open pool in a convective situation. Finally, in Section 6, we give our conclusions on the results obtained. 2. T H E M O D E L The basic conservation equations for the separate phases of the three component vapour-gas-aerosol system and the total enthalpy are: dpg ~t t-V.(pgv~) = 0, (1) ~ - + v . ( p v v ~ ) + v . i , = -,~v, (2) t3pv dpa dt t - V ' ( p , v , ) + V ' j o = rh~, (3) ~(p~hg + p~h,, + p,,ho) + V'(q + pghgv a + p~hvvv + p,,hovo) = 0. (4) d The subscripts g, v, and a of the density p, the velocity v, and the specific enthalpy h, refer to the gas, vapour, and aerosol respectively. The aerosol growth rate, or mass transfer rate per unit volume from the vapour to the aerosol is rho, and the currents Js, Jo and q are defined below. In accordance with common practice in fluid problems, any sources to the enthalpy from friction have been neglected in equation (4). The equation then applies to pipe flow even Maximumaerosol densitiesfrom evaporation-condensationprocesses Ti-Pai Well-mixed flow " •TI>TL . )" . Tw Pc,, Aerosol Tw I ~ (M°ss Evoporotion ) Tw iHeot (o) Heeling TL Pa~. ",f Well- mixed flow ~/ 295 • • Moss (Condensotion) ~ Heel • • "If < T i. "', F • Po~ Tw /////////////////////// (b) Cooling Fig. 1. Changes in the total aerosol density from po~ to P,r produced by a well-mixed flow being (a) heated by a hot pool and (b) cooled at a wall. when the pressure is changing moderately, the energy from this going into changes in the bulk velocity, v. In equation (2), the vapour current has been written as pvvv = pvvg + j , , (5) where j, is the vapour current relative to the gas. Similarly, the aerosol current is written as pore = p,v~+j,, (6) where jo is the aerosol current relative to the gas. Finally, the conductive heat current is q = -kVT. (7) Consider the steady state flow through a pipe of cross-sectional area A. The time derivatives vanish in equations (1)-(4) and the equations can be integrated over a volume element of length dz along the flow, obtaining, for example V" (pg vg) d V = ~ dS pg %" n = dz d (pgvgA), (8) where the outward normal n is taken to be in the gas flow direction in which there is a uniform velocity v9. The equations are averaged over possible realizations of the flow which is taken to be turbulent. It is assumed that the streamwise turbulent fluctuations in flow velocity are much smaller than the mean velocity along the pipe, so that the contribution of fluctuations to the integrals can be neglected. It is assumed that for a turbulent flow in the pipe the densities etc. within the bulk flow are constant across the cross-section, outside the boundary layers. Only the contribution of the vapour current against the wall is important, i.¢. Sis" dS = IAdz, (9) where I is the perimeter of the area A andjs is the magnitude of j,. A positivej~ then indicates a loss of vapour to the wall and condensation. Since the vapour currentj, is only required at the wall, the effects of turbulent transport are irrelevant and js is given by (Clement, 1985) j,= -D(Vpv-~Vp,), where D is the vapour-gas molecular diffusivity. (10) 296 C.F. CLEMENTand 1. J. FORD Under these particular conditions, equations (1)-(3) become d dz (PgvgA) = O, il 1) d ~z(pvvaA ) = -- Ij~ -- A thv, (I 2) d ~z(povgA) = - lja + Arhv. (13) Likewise the enthalpy equation (4) becomes d -~zz(pghgvgA + pvh~vgA + pahavgA ) = - lhvwjs - lhawja - lq, (14) where the subscript w denotes wall value. The enthalpies are functions of temperature and pressure and if the temperature is changing along the tube, then for the flow conditions considered here dhg dT dz = cpg dz' (15) etc., where cH is the specific heat at constant pressure, and T is the bulk flow temperature. Using equations (11)-(13), equation (14) can be rewritten as dT _ pCpVoA-~z = lj,(hv - h~w) + lja(ha - haw) + Ath~(hv - ha) - lq, (16) where the mean specific heat 6p is given by p~p = pgc~ + p~cp~ + pacp,. (17) The first two terms on the right-hand side of equation (16) are correction terms arising from any change in temperature of the vapour or aerosol in going from the bulk flow to the wall. They will generally be small in comparison to the conductive flux (fourth term) and the aerosol condensation term (third term), which is specified by the latent heat of vaporization L ( T ) = h , , - ha. (18) Using the definitions of h, etc., equation (16) can be rewritten as _ dT p c p v g A ~ z = ALth~ - lq + I ( T - Tw)(%~js + c~ja), (19) where Tw is the wall temperature. T h e heat current at the wall can be related to the m a s s current using a surface condensation number (Clement, 1985) Cns = q/(L(Tw)js). (20) From equation (11), the rate of change of pg is related to that of Avg by d (Avo) = - vgA 1 dp. (21) Po dz ' and so equations (12) and (13) reduce to voA v°A ( d~; pv dp3 "~ - ljs - Ath~, Po dz ] = Po dz J = (22, Maximum aerosol densities from evaporation-condensation processes 297 Next my can be eliminated between equations (22) and (23) and between equations (19) and (22) to obtain and L az az p. az j = -- IL(Tw)j,(1 + C n s ) + l ( T - Tw)ct,,,(j,,+j, ), (25) where equation (20) has been used and also the relation between the latent heats at different temperatures: L ( T ) - L ( T , , ) = (cpv - c n , , ) ( T - Tw). (26) Finally, the ratio of equation (24) to (25) provides an expression for the rates of change along the path: '- dp~ L ( T ' ) [ I + Cn'--(~°] d (pv + P°) = (I + 2")LL(T)-d~-z + -Pg {L(Tw)(pv+po)[1 + C n , - f o ] - L ( T ) ( 1 + P~'-~z dT1 + 2o)po}, (27) where (28) 2o = jo/j,, and (~° is a small correction term that can often be neglected: t~° = %,(1 + 2a) ( T - Tw)/L(T~). (29) If it is assumed that the vapour density in the main flow is kept close to saturation by aerosol formation or evaporation, p~ takes on its equilibrium value, p,e(T), so that dpv dz = p',,,(T) (30) . If this were true for the boundary layer as well, then the surface condensation number would be given by k(1 -- ce(T)) Cns = D L p c ' e ( T ) ' (31) where k is the thermal conductivity of the mixture and the equilibrium vapour concentration is c~(T) = p ~ , ( T ) / ( p v , ( T ) + Pg), (32) i.e. the vapour concentration is defined as the mass fraction of aerosol. The derivative of c~ with respect to T is taken at constant pressure. Otherwise, an amended value should be used (Clement, 1987), corresponding to a maximum supersaturation (S- 1) in the boundary layer Cns(S) - Cns(1) (1 4- ~t(S- 1)½) _ Cns(1) (33) 1 + (4nN/~)½ c; where at = (cTc,)½/c', ~ 1 and the plus and minus signs refer to the cooling and heating cases, respectively. In the second expression S - 1 has been replaced by a quantity specified by the aerosol number density, N, its mean radius, R, and the temperature gradient d T / d x normal to the wall at the surface of the tube (Clement, 1987). All spatially dependent quantities are evaluated at the surface. 298 C . F . CLEMENT and I. J. FORt) With the neglect of 6a and any aerosol current at the wall the first term on the RHS of equation (27) reduces to the previous result (Clement and Julien Dolias, 1987) d [- 1 pep -]dT dz (P,,+ Pa) - 1 + Cn, ~_P'e(T) + L-(T~ J-d-z ' ~34) which can easily be integrated using equation (30) to obtain changes in density as a function of temperature, i.e. ,o,-po,- 1 p(p F (35) Here, the density p and the surface condensation number Cn~ are taken at some representative bulk temperature, e.g. ½(T~+ TI). The size of the final term in equation (27) depends upon the physics of the situation which determines dpg/dz. In terms of the total pressure p, the gas density is, assuming an ideal gas law, Pg _ IJgPg _ #o ~(p_pve(T)), R~ T (36) R G7 where #g is the molar mass of the gas, Po and Pve the gas and equilibrium vapour pressures respectively, and R~ the gas constant. From this expression, and using an ideal gas law for the vapour density, 1 dp. podT 1 T five(T) p-p,,e(T) 11 - #,Poe] 1+ #~Pol !aoP've (37) #. Pg' where #~ is the molar mass of the vapour. It is therefore possible to east equation (27) in the form of a differential equation in T: d(P + Pa) - - l+)'a [ LL{TT)P,ve(T) . LT\ U,P,, 1+ On,- aa L PCP ] . } - _ _ I~ Po ] L(T ) L( Tw) 1 + Cn~- 6a " With a knowledge of Cn, and 2a, equation (38) can be integrated to determine the aerosol density change along the path from an initial temperature T~ to a final temperature TI. 3. EXAMPLE CALCULATIONS The integration of equation (38) is easy to perform numerically with backward Euler extrapolation over a temperature step which is small enough such that the results do not depend upon its size. Calculations have been performed for water-vapour/air and sodfumvapour/argon mixtures using the data specified in Appendix A. Any differences between the results of integrating equations (38) and (34) arise from a feedback from the vapour and aerosol already present in the mixture, corresponding to the second term on the RHS of equation (38). This term tends to enhance aerosol growth for a cooling case, and to reduce it in the case of heating, relative to the results of equation (34). This is shown clearly in Fig. 2 which gives the final aerosol densities obtained, starting with Pal = 0, for a heating and a cooling case of a saturated water-vapour/air mixture. Note that as maximum densities are being considered here, it is assumed that the mixture reaches the temperature of the walls before emerging from the tube. Also, it is assumed that the boundary layers are saturated, and that no aerosol is lost by deposition onto the walls, i.e. 2a = 0. Although easily calculable and for this reason attractive, the expression in equation (34) does not yield a result very close to the exact behaviour for this mixture (shown as a Maximumaerosol densitiesfrom evaporation--condensationprocesses x.~. 10 x -- "~-~ x \ × \\ x .......--0~.~ x\ -2 - - ...o ~ J 163 -- ao-~ 299 O-- ~ ( _ ",, /J / \\ I t I i 10 20 30 ho I 50 Tw °C 1 I I 60 70 ~,0 90 Fig. 2. Exact (solid lines) and approximate (dashed lines) increases p, in water aerosol density on heating a well-mixedwater-vapour/air mixture at 1 atm pressure from I0°C to Tw(circles) and cooling from 90°C to Tw(crosses). continuous line) for all choices of initial and final temperatures. However, for a sodium/ argon mixture similar example calculations display much closer agreement, as shown in Fig. 3. The behaviour of p,f 'for different wall temperatures in the heating case is explained in terms of competing effects. The rise in temperature of the mixture tends to thin out the aerosol as the carrier gas expands; against this has to be set the increased rate of evaporation into the flow from the walls for larger temperature differences. This can be seen in another way by considering the evolution of po with temperature along a pipe, an example of which is given in Fig. 4. The increase of mass of airborne water in this case is reflected in the plot of loading La, which is defined as L# _ po+p~,(r) po(T ) (39) This quantity rises approximately exponentially. 4. DISCUSSION The results make it clear that it is physically possible to produce quite dense aerosols by heating or cooling a saturated vapour, especially for large coolings in the water example. The densities that have been calculated, however, are upper limits to those which would be produced in less than ideal experimental conditions. Unless the tube is very long, it is unlikely that the mixture would emerge from it with a bulk temperature equal to that of the walls. This poses no difficulty for the model, and only requires a different upper limit to the integration of equation (33), for instance. The experimental results are more likely to differ 300 C. F, CLEMENT a n d I. J. FORt) lO-2~ - E 10-4t '~ -5/ I 10 300 I 350 400 I I 450 500 1 I 550 I 600 650 Tw/*C Fig. 3. As for Fig. 2, b u t for s o d i u m - v a p o u r / a r g o n . T h e initial t e m p e r a t u r e s for the h e a t i n g ( O ) a n d cooling ( x ) cases are 300 and 600°C, respectively. 0.014 - Loading 0.012 0.010 i 0.008 0.006 0.004 o.oo2/ 0.000 10 ~ I 20 t 30 I I \Pa I 40 50 00 Bulk ternperature/°C 1 70 I 80 I 90 Fig. 4. Evolution of water aerosol density p, and loading due to heating of a saturated watervapour/air mixture, initially at IO°C, in a flow along a pipe with walls at 90°C. from the predictions given above due to deviations of the vapour density in the boundary layers from saturation. An exact theoretical description of the development of such deviations would have to incorporate the dynamics of fluid exchange between the boundary layer and the bulk flow, as well as droplet growth kinetics to determine/~ in equation (33). The essential ingredient of such a treatment would be the timeseale of residence of a fluid packet within the boundary layer before ejection into the main flow, but such compheations are beyond the scope of the simple picture being drawn here. The calculations described above have neglected aerosol deposition upon the walls, in accordance with the expressed intention to predict maximum aerosol densities. If it is desired to extend the model to include the effects of turbulent or thermophoretic deposition then it would be necessary to estimate the parameter 2a. Max/mum aerosol densitiesfrom evaporation-condensation processes 301 5. CONVECTIVE F L O W SITUATIONS It is instructive to apply the aerosol growth model to convective flow situations which develop within the cover gas space. A passage of a packet of gas/vapour mixture over the hot sodium pool is analogous to the pipe flow situation considered in this paper, except that the packet is in contact with the liquid surface only over a portion of its surface. Consider an incremental slice of the packet, perpendicular to the overall flow direction, with a perimeter l and length dz. Vapour enters diffusively only through a portion of the area ldz; that part which contacts the hot pool. To a first approximation it is possible to neglect the loss of vapour through the remaining portion as the packet warms. This remaining portion resides in the turbulently mixed region of the cavity, where gradients of concentration and temperature are small. The density of aerosol in the packet increases according to equation (38), where 2= is largely due to a gravitationally settling aerosol current. The temperature to which the packet is heated is determined by the fluid dynamics of free convection. It is then carried away as a thermal plume into the bulk mixture. Similar cooling processes occur on the roof of the cover gas space, which also result in aerosol formation. This picture of the mechanism of loading the cover space with aerosol can provide an estimate of the number of circuits of the cavity taken by a particular droplet before falling out. This in turn can give an indication of the distribution of aerosol density within the cavity. Figure 5 shows the maximum fractional increase in aerosol density to be expected from one passage of a packet of gas close to the surface of a sodium pool at temperature T, with a roof temperature T, of 120°C. Steady state bulk aerosol densities as a function of T, are taken from Japanese experimental work (Himeno and Takahashi, 1978). The boundary layer is assumed to be saturated. The calculations are for an initial packet temperature of Tb -- el2, and a final temperature of Tb + el2, using two typical values of e. Tb is the average bulk temperature and is determined (Clement, 1985) by the relation: ~(rb) = ½(~(T,) + ~(T,)), (40) ?,(T) = T+ L(T): ce(T) (41) where Cp At the wall and roof - k V ¢ = q,o,, (42) where qtot is the total heat transfer rate due to conduction and evaporation/condensation 0.2 t I I I 60. l 0.1 ~ ~ x - - . . . . 0 I 350 I 400 1 450 I 500 --x E = 2&C ~ £ = 10eC I 550 TsI°C Fig. 5. The maximumfractionalincreasein sodiumaerosoldensityachievedon heatinga packetof a saturated mixture of sodium-vapour/~gon from a temperature ~-~/2 to T~+8/2. ~ is the convectivemean temperaturefor each pool temperature T,, with a roof temperature of 120°C.The bulk aerosol densitiesare taken from Himenoand Takahashi(1978). 302 C.F. CLEMENTand I. J. FORD between the p o o l a n d the roof. E q u a t i o n s (40)-(42) are n o t general relations, but hold for this case since the Lewis n u m b e r for s o d i u m - v a p o u r in a r g o n is close to unity. The t e m p e r a t u r e difference e w o u l d increase as Ts increases, so that a m o r e accurate t r e a t m e n t w o u l d p r o d u c e a plot flatter t h a n those shown in Fig. 5. A greater accuracy in the c o n s i d e r a t i o n of this p r o b l e m is unnecessary however, since it is clear that the bulk aerosol density in the cavity is insensitive to a passage of a constituent p a c k e t of mixture close to the heated pool. A b o u t 10-50 such passages are required to grow a e r o s o l in the p a c k e t to the observed steady state density. These explicit calculations confirm a c o m m o n supposition; that a e r o s o l g r o w t h can be a v e r a g e d o u t over the cover gas cavity (due to the fast t u r b u l e n t mixing) a n d s m o o t h e d out over time (due to the small change possible from a pass over the pool). A e r o s o l g e n e r a t e d by e v a p o r a t i o n / c o n d e n s a t i o n in the b o u n d a r y layer a b o v e a h o t s o d i u m p o o l will pass into the bulk cavity virtually u n d e p l e t e d by i m m e d i a t e fallout. 6. C O N C L U S I O N S A m o d e l of the g e n e r a t i o n of an a e r o s o l density in well-mixed fluid flow situations close to heated o r cooled liquid surfaces has been presented. The e v o l u t i o n p r o c e e d s simply by solution of the c o u p l e d e q u a t i o n s of h e a t a n d m a s s transfer. C e r t a i n ideal c o n d i t i o n s allow a c a l c u l a t i o n of the m a x i m u m possible a e r o s o l density generated. The m o s t i m p o r t a n t of these c o n d i t i o n s is the s a t u r a t i o n of the b o u n d a r y layer between the well-mixed b u l k flow a n d the walls. A better u n d e r s t a n d i n g of the d e v e l o p m e n t of d e v i a t i o n s from s a t u r a t i o n w o u l d be r e q u i r e d in o r d e r to calculate realistic g e n e r a t i o n rates of a e r o s o l in pipe flows, b u t ideal case calculations have been presented which give u p p e r limits to these rates for v a r i o u s conditions. Convective flow in the fast reactor cover gas space has been c o n s i d e r e d in terms of the passage of p a c k e t s of a r g o n gas over a hot s o d i u m pool, d u r i n g which they are h e a t e d t h r o u g h a small t e m p e r a t u r e difference. C a l c u l a t e d m a x i m u m densities of s o d i u m a e r o s o l g e n e r a t e d in such packets p r o v i d e lower limits to the n u m b e r of circuits a r o u n d the cavity m a d e by a d r o p l e t before g r o w i n g sufficiently to fall out. This n u m b e r is fairly large for a range of typical s o d i u m p o o l t e m p e r a t u r e s suggesting t h a t a e r o s o l g r o w t h can indeed be viewed as being a v e r a g e d out over the whole cavity. REFERENCES Barrett, J. C. and Clement, C. F. (1985) J. Aerosol Sci. 17, 129. Clement, C. F. (1985) Proc. R. Soc. A398, 307. Clement, C. F. (1987) The supersaturation in vapour-gas mixtures condensing into aerosols. Harwell Report TP.1223. Clement, C. F. and Julien Dolias, M. (1987) Sodium aerosol formation in an argon flow over hot sodium. Harwell Report AERE-R.12489. Himeno, Y. and Takahashi, J. (1978) PNC N941 78-42. APPENDIX Water-vapour/air data: Sodium-vapour/argon data: Latent heat: Equilibrium vapour pressure: Specific heat capacities: A summarized in Barrett and Clement (1985) (T in Kelvin) L = 4.1993 x 1 0 6 - 985.58 T J kg- 1 P~e = 1.01325 x 10s ( - - 12818 ) xexp~ T 0.5 In T+ 14.6306 Pa cpg = 521 J kg- 1K- 1 = 900 J kg -1K-1 P~e = 1 0 3 exp ( - 1/z) kg m- 3 z = 1.91711 x 10- ~ T2 + 8.4563 x 10- s T - 7.3053 x 10- 4 D = 7.215 x 10- 9(T - 149)3/2 m 2 s cw Equilibrium vapour density: Sodium-vapour diffusion coeflieient (at 1 atm. pressure) Thermal conductivity k -~kg= 2.378 x 10- a + 5.561 x 10- s T-- 1.558 x 10- s T 2 W m - t K - These fits are sufficient for the range of temperatures relevant to the cover gas space.

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