J. Aerosol $ci., Vol. 20, No. 3, pp. 29~302, 1989.
Printed in Great Britain.
0021~8502/89 $3.00+0.00
Pergamon Press plc
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.
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
Pertaining to
Pertaining to
g Pertaining to
v Pertaining to
w Pertaining to
Derivative with respect to T
© Crown.
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
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:
~t t-V.(pgv~) = 0,
~ - + v . ( p v v ~ ) + v . i , = -,~v,
dt t - V ' ( p , v , ) + V ' j o = rh~,
~(p~hg + p~h,, + p,,ho) + V'(q + pghgv a + p~hvvv + p,,hovo) = 0.
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
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>TL
. Tw
I ~ (M°ss
Evoporotion )
(o) Heeling
TL Pa~.
Well- mixed flow
• •
"If < T i.
(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 , ,
where j, is the vapour current relative to the gas.
Similarly, the aerosol current is written as
pore = p,v~+j,,
where jo is the aerosol current relative to the gas.
Finally, the conductive heat current is
q = -kVT.
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),
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 =
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)
where D is the vapour-gas molecular diffusivity.
Under these particular conditions, equations (1)-(3) become
dz (PgvgA) = O,
il 1)
~z(pvvaA ) = -- Ij~ -- A thv,
(I 2)
~z(povgA) = - lja + Arhv.
Likewise the enthalpy equation (4) becomes
-~zz(pghgvgA + pvh~vgA + pahavgA ) = - lhvwjs - lhawja - lq,
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
dz = cpg dz'
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
pCpVoA-~z = lj,(hv - h~w) + lja(ha - haw) + Ath~(hv - ha) - lq,
where the mean specific heat 6p is given by
p~p = pgc~ + p~cp~ + pacp,.
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.
Using the definitions of h, etc., equation (16) can be rewritten as
p c p v g A ~ z = ALth~ - lq + I ( T - Tw)(%~js + c~ja),
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).
From equation (11), the rate of change of pg is related to that of Avg by
d (Avo) = - vgA
1 dp.
Po dz '
and so equations (12) and (13) reduce to
( d~;
pv dp3 "~ - ljs - Ath~,
Po dz ] =
Po dz J =
Maximum aerosol densities from evaporation-condensation processes
Next my can be eliminated between equations (22) and (23) and between equations (19) and
(22) to obtain
p. az j
= -- IL(Tw)j,(1 + C n s ) + l ( T -
Tw)ct,,,(j,,+j, ),
where equation (20) has been used and also the relation between the latent heats at different
L ( T ) - L ( T , , ) = (cpv - c n , , ) ( T - Tw).
Finally, the ratio of equation (24) to (25) provides an expression for the rates of change
along the path:
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
+ 2o)po},
2o = jo/j,,
and (~° is a small correction term that can often be neglected:
t~° = %,(1 + 2a) ( T - Tw)/L(T~).
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
dz = p',,,(T)
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 ) '
where k is the thermal conductivity of the mixture and the equilibrium vapour concentration
c~(T) = p ~ , ( T ) / ( p v , ( T ) + Pg),
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) -
(1 4- ~t(S- 1)½)
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.
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)
pep -]dT
dz (P,,+ Pa) - 1 + Cn, ~_P'e(T) + L-(T~ J-d-z '
which can easily be integrated using equation (30) to obtain changes in density as a function
of temperature, i.e.
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
_ IJgPg _ #o ~(p_pve(T)),
R~ T
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.
#. 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:
+ Pa) - -
[ LL{TT)P,ve(T) .
1+ On,- aa L
. } - _ _
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.
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
- - ...o
163 --
O-- ~ (
Tw °C
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 )
This quantity rises approximately exponentially.
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
C. F, CLEMENT a n d I. J. FORt)
lO-2~ -
10 300
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 -
i 0.008
Bulk ternperature/°C
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.
aerosol densitiesfrom evaporation-condensation processes
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,)),
?,(T) = T+ L(T): ce(T)
At the wall and roof
- k V ¢ = q,o,,
where qtot is the total heat transfer rate due to conduction and evaporation/condensation
0.1 ~
~ x
- . . . .
E = 2&C
£ = 10eC
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).
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
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.
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
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.
Water-vapour/air data:
Sodium-vapour/argon data:
Latent heat:
Equilibrium vapour pressure:
Specific heat capacities:
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
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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