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Growth and redistribution in a droplet cloud interacting with radiation
J. AerosolSci., Vol. 21, No. 6, pp. 761 776, 1990.
0021-8502/90 $3.00+0.00 Pergamon Press pie.
Printed in Great Britain.
GROWTH AND REDISTRIBUTION IN A DROPLET CLOUD INTERACTING WITH RADIATION J. C . BARRETT a n d C . F . CLEMENT* Theoretical Physics Division, Harwell Laboratory, Didcot, O x o n OX11 0RA, U.K.
(Received 15 January 1990) Abstract Previous work on the growth and redistribution of a droplet distribution interacting with a radiation field has been extended. With no other macroscopic heat and mass transfer processes the s u p e r s a t u r a t i o n is s h o w n to be non-zero when the Lewis n u m b e r , Le, of the s u r r o u n d i n g v a p o u r - g a s mixture differs from unity. Redistribution of m a s s with e v a p o r a t i o n of the smallest droplets then occurs for radiative cooling a n d Le < 1, a condition satisfied for water aerosols. Exact and approximate expressions are given for the wavelength-averaged absorption efficiencies of water droplets. Calculations of changes to water droplet distributions were performed using exact methods and simplified m o m e n t equations. The latter are adequate when radiative redistribution leads to a c o m m o n radius and there is no initial transient. The conditions for no mass transfer to the aerosol a n d ' a constant total water density and temperature are discussed. In the latter case with radiative cooling, which could possibly apply to reactor situations, a considerable widening of the size distribution takes place on a long timescale.
NOMENCLATURE A c ?p
Cn D eba f F g
G(R) G1 G2 Ia /k L Le th, M M,
n(R) N p Q~ QR R /~ R~ Rg RG /~, s S S.,~ t T u t,
constant in growth rate equation vapour mass concentration mean specific heat at constant pressure condensation n u m b e r vapour-gas diffusivity Planck black body function dimensionless n u m b e r [see e q u a t i o n (58)] fraction of aT'* numerical fraction dimensionless radiative growth function coefficient of R z specifying G(R) for small drops coefficient of R specifying G(R) for large d r o p s radiation intensity radiation intensity averaged over wavelength thermal conductivity latent heat of vaporization Lewis number total local aerosol condensation rate total water mass density nth m o m e n t of the size distribution aerosol n u m b e r density total aerosol number density total pressure efficiency factor for absorption of radiation total radiative heat flux from droplet droplet radius droplet growth rate critical radius p a r a m e t e r of log-normal distribution gas constant nth m o m e n t of the aerosol size distribution path distance coordinate vapour saturation asymptotic saturation in time time temperature difference between vapour concentration and its equilibrium value velocity
* Author to w h o m correspondence should be addressed. As~:~-0
761
762
i ~, BARRH~and C F, Ct.EMENI weighting for integration x distance in 1-D radiation field; /~; change of variable [_seeequation (b3!~ abbreviation for/~2 Y dimensionless function of °s coefficient in exponent fl coefficientgiving temperature derivative of Pve(T) b small length ). wavelength # cosine of angle in radiation field /~ (with a subscript) molecular weight p total density o Stefan-Boltzmann constant ag width parameter of log-normal distribution f~ angular direction of radiation field AT temperature difference specifyingg
Subscripts c d e g i 1 L s S v o
pertaining to critical size pertainingto droplet pertainingto equilibrium pertainingto gas pertaining to set of points pertainingto liquid pertaining to large droplets pertaining to surface pertainingto small droplets pertainingto vapour pertainingto initial value
Superscripts derivative with respect to T average over the size distribution 1. I N T R O D U C T I O N The interaction between radiation and a cloud of droplets can lead to its heating or cooling and result in evaporation from, or condensation onto, the droplets. For water clouds these effects are certainly important in radiation fogs (Roach, 1976; W o b r o c k et al., 1986) and on droplet growth near cloud tops (Barkstrom, 1978; Guzzi and Rizzi, 1980). They can also occur in water droplets which might be formed in the containment of a pressurized water reactor ( P W R ) following a possible accident, where water mass m a y be redistributed, as has previously been pointed out (Barrett and Clement, 1987). The aim of this paper is to extend the previous work and explore in general the effect on droplet size distributions of the radiative interaction. The results are reported of a number of calculations performed for water clouds. In section 2 a general result for the growth rate, R, of a droplet valid for a saturation not too much above unity is quoted (Barrett and Clement, 1988). The radiative term in/~, which arises from net radiative heat loss or gain, is expressed in terms of a temperature-averaged efficiency factor, and the general form of the growth rate is given in the continuum growth regime valid for larger droplets. This rate is re-expressed in terms of the local total mass transfer rate to the aerosol, and it is shown how it can be specified by an equation for the supersaturation. In section 2.2 it is shown that a pure radiation field with no vapour or heat currents can result in mass redistribution with evaporation of small droplets if the Lewis number of the mixture is less than unity. This is the case for water vapour in air, and the temperature dependence of the effect is discussed. In section 2.3 results and approximations for the absorption efficiency factors for water, drawing upon previous work of Barrett (1985a, b) and Roach (1976) are given. In section 3 methods for solution of the aerosol equation for water droplets when radiation induces growth or redistribution of the size distribution are examined. The cases of constant supersaturation and of a constant vapour plus aerosol water density at constant temperature, and how these might arise in practice, are discussed. Calculations starting
Radiative redistributionin a cloud
763
from log-normal size distributions are performed, and the adequacy of using moment equations as an approximation are explored. The procedure for obtaining such equations is described in section 3.2, and could be adapted to other aerosol problems. Finally, in section 4 conclusions from this work are drawn, and an outline of what is needed to extend it and incorporate it into calculations for water aerosols in PWR containments is given. 2. THE G R O W T H OF C L O U D D R O P L E T S IN A R A D I A T I O N F I E L D Droplets in a cloud with a size distribution specified by n(R), where R is the droplet radius, are considered. This distribution is also generally a function of position, _r, and time, t. Previous results (Barrett and Clement, 1988) on general growth rates for such droplets are reiterated and how the quantities needed for their calculation can be obtained is discussed. These include the supersaturation, and an equation for this (Clement, 1987, 1988a), which includes a radiative term, is given. The case of growth in a pure radiation field is considered in section 2.2 and it is shown that for vapour-gas mixtures, whose Lewis number, Le, is nonzero, the supersaturation does not vanish. This means that radiation acting alone can cause the redistribution of mass in the size distribution. Finally, in section 2.3, the results of calculations of absorption efficiency factors for water and sodium droplets are given. These factors are the basic quantities needed to specify the dependence of the growth rates on R. 2.1 General orowth rates
Only the case when the vapour supersaturation is fairly small is considered, so that the general growth rate is given by (Barrett and Clement, 1988); /~_ 1 S - 1 + [RAD] p~ [ M A S S ] + [ H E A T ] '
(1)
where Pl is the droplet density and the saturation S is given in terms of the actual and equilibrium vapour densities by: S = pv/pve(T).
(2)
The resistances to growth from mass and heat transfer, [MASS] and [HEAT], respectively, take different forms (Barrett and Clement, 1988) according to the droplet size and temperature. Here, the continuum growth regime, which is a good approximation at normal pressures for R > 1/~m, will be considered. It would be straightforward to extend all the results obtained here to the transition and molecular regimes which apply to a smaller R. The main difference is that the dependence on R of [MASS] and [HEAT] changes from R to a constant at small R. In terms of the total pressure, p, the vapour-gas diffusivity, D, the thermal conductivity, k, and the latent heat, L, an excellent approximation (Barrett and Clement, 1988) is provided by: R(p - Pve(T)) [MASS] = , (3) pDpv¢(T) [HEAT] = RflL/(kT2),
(4)
where Pve is the vapour density corresponding to p,, and fl is given by: fl = #vL/RG,
(5)
where #v is the vapour molecular weight and R 6 is the gas constant; fl is the exponent in the Clausius-Clapeyron relation for p,~ (T). The ratio of [MASS] to [HEAT] is given by the condensation number (Clement, 1985), either in the bulk (Cn) or surface (Cn~), [MASS] - - (1 - c e ( T ) ) C n ( T ) = Cns(T), [HEAT] where c~(T) is the equilibrium vapour concentration.
(6)
764
i
In the continuum Clement, 1988 t:
( i B A R I ~ t l i a n d C. F. C L t : M E N I
region the radiation
term in equation(li
is l Barre~.~ and
4 nflQ~ k R T 2'
[RAD]
i::
where QR is the net radiative heat flux from the droplet given by (Ozisik, ~973: Barrett, 1985):
c
QR = ~R2
d2 Q~(R, 2)[4eba(Ta) --
., 0
f
I~(s, ~ ) d ~ ]
4"~
= 7rg 2 < Q , ( R ,
,~) > [4~rr~ -
f,
[(x, ~ ) d ~ ] .
i8)
~ i
The first expression refers to the absorption of radiation of intensity I~(s, ~ ) depending on position s along a path with ~ as direction of propagation, and emission of radiation where eba(Ta) is the Planck black body function at the droplet's temperature. In practice, in nearly all cases, Ta may be taken as T, the local temperature of the medium. The second expression in equation (8) is obtained in the grey medium approximation for a 1-D radiation field along distance x with # as the cosine of the angle to the x direction. Both expressions are specified by the droplet's efficiency factor, Q~(R, 2), for the absorption of radiation of wavelength 2; the brackets in the second expression denote an average over the black body spectrum. For any grey medium, Q~ = 47rR2F ( Q . ( R , ,t)),
(9)
can be written, where F is a fraction, g, of aT a given by 1/4 of the square bracketed expression in equation (8). From equations (3)-(9) R in the continuum region can be expressed as: A = R (S- 1 +G(R)),
(101
where A . 1 _ ptflL [1 + C n ( T ) ( 1 - c ~ ) ] , kT z
(11)
fl F R (Q.(R)).
(12)
G(R)=
The general behaviour of ( Q a ( R ) ) is an increase with R which levels out to a constant at
large R. Thus, the two terms in equation (10) will always have a different R dependence, RADIATIVE COOLING ...............
b.~
Droplets evaporate .....
C00!
.
.
.
.
.
.
mo~mu-m-[or10#m "drop~e-t -
.
.
All droplets grow
/o. X wa,er ciouds
AIt droplets evaporate .
...... Gt(.!~ i \'\ Dropie,s I "~grow I ~ j p, ~x
:. 6 . 6 , , ~
0.01 S -! I:q [,1 ~ ~" I DroDlets'x~ Droplets grow }evop.onrate4~" to R c
i ,o ~c /
. . . . . . . . .
\
~
. . . . . . . .
RADIAT!VF HEATING Fig. I. Behaviour of a droplet cloud subject to a supersaturation, S - I, and radiative growth term, G(R), depending on droplet radius R. The hatched regions show the behaviour expected for a cloud of water droplets subject to radiative heat transfer alone.
Radiative redistribution in a cloud
765
implying the general behaviour of k in relation to heating and cooling which was pointed out recently (Barrett and Clement, 1987). This is summarized in Fig. 1, which is augmented by showing possible regions for a cloud of water droplets subject to radiative heat transfer alone (see next section). T o explore possibilities for general b e h a v i o u r it is necessary to calculate the supersaturation, S - 1, or the overall mass transfer rate to the aerosol per unit volume, vhv. Relations between these quantities in terms of moments of the size distribution can be obtained: t* N
= jn(R)dR,
(13)
= fRn(R)dR/N,
(14)
OR = f QRn(e)dR/X = 4rckT 2 f R G ( R ) n ( R ) d R / f l N .
(15)
Integrating equation (10) over the size distribution gives:
rhv = 4npl ( R EIln( R )dR iJ
= 4npl(S-- 1)NRA +
NQR
(16)
L [ 1 +Cn(1-Ce)]"
The relation between thv and S - 1 enables the individual growth rate to be rewritten as:
lt=4rcp~N~R ~
Q~-
.
(17)
The final term, which gives a radiative redistribution of mass in a distribution, is more general than that given previously [final term in equation(l l) of Barrett and Clement (1988)], where only the limiting case for large R, where ( Q a ( R ) ) was independent of R, was considered. This implied that the previously defined Grad = G(R) was independent of R and QR ~ R2. Other terms in the general form for/~ representing Ostwald ripening from the Kelvin effect, and hygroscopicity in a water droplet are omitted in this paper for simplicity, but they could easily be inserted from our previous expressions (Barrett and Clement, 1988). The quantity, rhv, occurs in the original heat and mass transfer equations for vapour-gas mixtures (Clement, 1985), but is not easy to obtain from them except in the saturated (S = 1) limit. This limit is only reached slowly at high aerosol concentrations (Clement and Taylor, 1986), and it is better to use directly an equation which has recently been obtained (Clement, 1987, 1988) for S - 1 or,
u = C-Ce(T) = [/~v - (#v -- #g)c] Ce(T)(S - 1)/#v.
(18)
This form of u is appropriate for changes occurring at constant pressure, and it is an excellent approximation to put c = c c on the right hand side (RHS) in equation (18). One accurate form for an equation for u is (Clement, 1988a) p
~+v'Vu
= Dp[c.(VT)2
+V'(--DpVu) (Cpv-
Cpg) c'cVc'VT] +
Le
- c',( 1 - Le)V. ( - DpVT) - rhv( 1 - c + Le/Cn),
-NQrt (19)
766
J ( ]~ARREI'Tand C, F, CLEMEN'I
where p is the total density, a prime denotes a derivative with respect to temperature, ~p,. a n d Cog are specific heats at constant pressure for vapour and gas with c~ the density-
averaged specific heat, and the Lewis number, Le, is given by [,C ::
k/D~;p p.
(2())
Amongst the source terms which give rise to non-zero values of u, and hence droplet evaporation or growth, is the radiative term containing QR. In general, when this term is non-zero, there will also be spatial gradients in temperature, especially in boundary layers through which heat and mass transfer is occurring. However, if the spatial gradients are small in the bulk of the fluid through turbulent mixing there can still be a strong radiative interaction, a case now considered in detail. 2.2 Growth in a radiation.field When a turbulently well-mixed aerosol is interacting with a field passing through it, or is radiating to walls, the spatial temperature and concentration gradients in equation (19) will vanish. Transients have a short time-scale for this equation (Clement, 1988a) so that the solution is obtained by setting the radiative source term equal to the aerosol loss term proportional to rhv: Le N QR rhv = L [ ( I - c ) C n + L e ] "
(21)
Neglecting any second order terms arising from differences between c and c e, equations (16) and (17) give:
S-l=
flQa ( 1 - c e ) C n ( L e - 1 ) 47tkT2 R [ 1 - ce)Cn + Le] '
1
(22)
[~ Q.a(1-e~)Cn(Le-1)]
[~ = 4np, LR[1 + ( 1 - G ) C n ]
+ ~- [ ( 1 - - - c ~ - C n + T e ] J '
(23)
Neither of these results depends explicitly on the aerosol number density, N. Their most interesting feature is the dependence on L e - 1. For Le = 1 the supersaturation is zero: physically the rates of conductive heat transfer and diffusive mass transfer to and from droplets are the same, so that the effect of radiant loss or gain of heat is to leave the gas-vapour medium exactly in balance. For Le ~ 1 an imbalance arises from the different rates of heat and mass transfer. Thus, the conditions for radiative redistribution to occur are: A. Le < 1 i24) B. QR is not proportional to R.
(25)
If these conditions are satisfied the two terms in equation (23) have opposite signs and, when Qa increases more rapidly than R, the smallest droplets evaporate when most of the distribution grows, and vice versa. These phenomena should occur for clouds of water droplets in air at normal pressures where Le = 0.85. The processes involved are illustrated in Fig. 2, and will be more important at low T t h a n at high T because Cn is unity at T = 4°C at 1 atm pressure and falls to be much less than unity at high temperatures (Clement, 1985). Numerically, for water droplets at 4°C with fl = 5390 K, k = 0.024 Wm -1 K -~ , T = 277 K, Le = 0.85, Cn(1 - c ¢ ) = 1, the radiative heat flux is QR=4.2 x 10 -9 g R 2 (/tm) ( O a ) IV, and with its limiting value for large R of ( Q, ) = 1.2, S-- I = - 9 . 5 10 -4 g R 2 / R ,
(26)
where the radius averages are measured in/~m. At the top of the atmosphere g ~ 1 for radiation to outer space could be used, but a more realistic value for engineering applications would be g ~ 0.1. The corresponding value for R
Radiative redistributionin a cloud RADIATIVE
EVAPORATION
767
COOLING
/4 CONDENSATION
CONDENSATION
#~EVAPORATION
Fig. 2. Expectedphysical behaviour of large and very small water droplets in a cloud interacting with radiation alone.
would be /~=0.16g
1-0.081~-
~ m s -t.
(27)
For log-normal size distributions with a S = 1.5 and 2, k would change sign at
R/R < 0.095, 0.13, respectively. However, this does not take account of the behaviour of QR with R for water droplets which will now be examined. 2.3 Absorption efficiency factors In a radiation field with an effective black body spectrum the quantity describing its interaction with a spherical droplet, radius R, is the wavelength averaged absorption efficiency, (Qa(R)), contained in equations (9) and (12). For water droplets at moderate temperatures the approximations (with R in #m) can be used:
(Qa(R)) = 0.2R, small R = 1.2, large R,
(28)
whereas, over the whole size range, a form suggested by Roach (1976) can be used:
(Q,(R)) = 1.211 - e x p ( - 0 . 3 g ) ] .
(29)
In Fig. 3 these expressions are compared with the results of exact Mie theory calculations (Barrett, 1985a) averaged over the black body spectrum at 280 K. An exponent of 0.167 to replace 0.3 in equation (29) gives better agreement with Mie theory as R -, 0, but significantly underestimates ( Q , ) in the important range of 3 < R < 10 #m. For atmospheric applications it may be more appropriate to use a clear sky emission spectrum; however, Fig. 1 of Roach (1976) indicates that the difference is not large.
3. RADIATIVE R E D I S T R I B U T I O N OF WATER D R O P L E T S The problem addressed is to solve the aerosol equation with a growth term
~n c~(nR) t~t
Jr - - 0, t~R
(30)
where, for water droplets, R is specified by equation (10), with
G(R) = GIR 2, small drops R ~ 5 #m
(31)
= G2R, large drops R >~ 10/~m
(32)
G 2 R [ 1 - e x p ( - - ~ t R ) ] , all R.
(33)
=
768
J. C, BARREl')andC. F. CLEMEN'I ----I-
I
........ r - - T
:
~--
:
/!
!....
0 2 R /:
.
¢"
r
Mie
/+:,J~
1.0!-
°,b
. . . . . . ] ....
.
.
- - ! ......
i
....
] ..... T ....
~
theory
.
'
-r
" ,:~ .- . . . :
~ 2(l_e-O.3R)
,f
//7
0.2
-i
L._.L__._L___L_~
I
I
__l___J_
I
[
5
I
J
I
10 R
I
_L.__L__.J___L___
15
_: 2 : "~
,um
Fig. 3. Accurate(Mie theory-crosses),approximate(dashes) and asymptotic(solid lines)curves for the wavelengthaveragedabsorption efficiency,(Qa(R)), for water droplets at 280 K as functionsof droplet radius R.
From equations (12), (28) and (29), good approximations are given by :t = 0.3 vm ' ~
G1 = 1.9 x 1 0 - ' ( T ) G2 = 1.2 x 10 .3
2 9 #m-2
9l~m - 1 ,
(34) (35)
where g is likely to be a small fraction. For radiation across a small temperature difference, AT, to an absorbing surface .q = 4 A T / T .
(361
Methods of solution of equation (30) can now be discussed, although, to include coagulation and other terms in the aerosol equation only numerical methods are generally applicable. It is, however, useful to have analytic results or simple methods of solution available where possible to gain an understanding of the behaviour of water mists. First, the case of constant S will be considered, and then, in section 3.2, how approximate solutions for moments of the distribution are obtained will be described. These are applied to the case of constant total water density in section 3.3. Exact numerical results are also obtained. 3.1 Growth with constant S
The condition of constant S corresponds to constant u as a solution of equation (19). F r o m equations (9) and (22) it can be seen that this cannot be achieved by radiation alone, because in no circumstances for water droplets is Qa ~/~. Any transport term with fluid velocity, v, will affect the aerosol too, so that the only chance of achieving constant isis via the diffusive and nonlinear terms in equation (19). However, these will usually only be large in boundary layers where the radiative terms in the equation are small. It would therefore be probable that an experimental situation to realize this case would have to be designed: it will not be naturally important. However, the solutions are easy to obtain and are quoted for completeness.
Radiative redistributionin a cloud
769
As previously noted (Barrett and Clement, 1987), the solution of equation (30) may be written down in terms of the initial distribution no(Ro), (Clement, 1978):
R [ S - 1 +G(R0) ] n(R, t) = no(Ro) Ro[S _ 1 + G ( R ) ] '
(37)
where Ro(R, t) is specified by the trajectory fRRo S - 1 +R'G(R')
dR' = At.
(38)
With G(R) given by equation (33), the integral must be performed numerically, but for small and large drops results obtained from equations (31) and (32) are:
G1 R2 + S - 1 = ( G1R 2 + S-- 1)exp(-2AG~t),
(39)
( S - 1 + G2Ro)exp[ - G2Ro/(S - 1)] = ( S - 1 + G z R ) e x p [ - G z R / ( S - 1) + A G 2 t / ( S - 1)].
(40)
For no radiative interaction with G = 0,
R 2 - R g = 2A(S-- 1)t.
(41)
In the first quadrant in Fig. i, where G(R), S - 1 > 0, growth is unconstrained and the results indicate that for large t R increases exponentially for small droplets, R increases as t for large droplets, but as x/~ when there is no net radiation. Growth with radiation is significantly faster than growth from supersaturation alone. When G(R), S - 1 < 0 the droplets evaporate completely in a time given by putting R = 0 in equation (39) for small droplets: large droplets go into the small droplet region and, unless R 0 is very small, the time spent in the region where molecular processes limit evaporation and the denominator in equation (10) goes to a constant, can be neglected. More accurate results could be obtained by replacing R by 6 + R in the denominator where, for water molecular sticking probabilities not much less than unity, 6 is about 0.2 pm (Barrett and Clement, 1988). In the fourth quadrant in Fig. 1, where S - 1 > 0 and G(R) < 0, the drops grow or evaporate to asymptotic radii given for small and large droplets by: R s = [ ( S - 1)/JGI[] ~/2,
(42)
RL = ( S - 1)/1621.
(43)
Numerical values can be obtained using equations (34) and (35). Unless S - 1 < 0.01, R s > 10/~m and small droplets would not remain in the small droplet regime. Finally, if S - 1 < 0 and G(R) > 0 (second quadrant in Fig. 1) droplets with initial radii satisfying S - 1 + G(Ro) < 0 evaporate completely, whereas the rest grow without limit. In the general case with equation (33) the critical radius R c satisfies: Rc [ 1 - e x p ( - ~R~)] + ( S - 1)/G2 = 0.
(44)
The quantity ~ is a constant independent of T and g, so that this equation can be used to plot ( S - 1 ) / G 2 as a function R c and to predict the future behaviour in the second and fourth quadrants for any radiative flux and supersaturation (see Fig. 4). 3.2 Moment equations From an experimental point of view it is usually only possible to observe the first few moments of a size distribution, n(R, t). This is an additional reason, other than simplicity, for attempting to obtain a closed set of equations for the moments rather than solving the full aerosol equation (30). The moments are defined in general by Mn(t) = f ~ Rnn(R, t)dR/N(t), n>_ l, do
(45)
770
,~ {: BA~krr~ and C. F, ()_EMEN-i i i;-;5 i
[
/
iI....
/ J
t
.
.
.
,/
.
/
// 2o--
J-g : 7
/
,
/
7
~g=I.5 /
i
,
i
/
,"
15~--
//
,,
/
L
/
i
I .11
,,
/ //I // C ]
//
5~--
,-. ,,' ,'.Y
/
// // .
%"1V/ ~////
//%=',
'
Rc for Rg = 8 # m O-g : 1,5
7 I
o ~
L_
2
._L ......
A
6
I ____k_ ..... L__
8
10
12
.. E
t
14
t6
...... J ......
18
Rg #m
Fig. 4. Variation of - ( S - 1 ) / 6 3 with R, for various values of trg specifying log-normal size distributions changing with rhv = 0. Given R, and ag the procedure to find the critical radius, R e, is indicated by arrows.
where, for the special cases ofn = 1, 2, M1 =/~ = x and M z = R 2 = y will often be written for brevity. It is possible to show from the general solution of equation (30) that (Clement, 19781
M. =
f dRono(Ro)Rn(R0, t)/N(t),
(46)
where R along a trajectory is a function of Ro and t. The general procedure to obtain moment equations then has the steps: (a) Differentiate equation (46) with respect to t, and use dR~& =/~ and equation (10) for/~ to relate the derivative of Mn to M,_ 1, and M. + 1 if necessary. This is possible with the small and large drop forms of equations (31) and (32) for G(R). (b) Close the equations for M t and M2 by expanding R-1 about 1//~, a procedure expected to be a good approximation for narrow distributions:
=~ =
-
!-[
dRono(Ro)~ +(R_l~) {y--x2-)R3--3xy+2x 3 ] 1+\ )+..
1 ( 4y
R3
x
x 3
x~
) 2
. /
(47)
Radiative redistribution in a cloud
771
(C) Use any condition on rhv, or the total water density, to express S - 1 in terms of moments M l, M 2 and M a. In practice moment equations have only been used when N = constant and the condition rhv = 0, the next case considered, then gives: S - 1 = - G 1R 3/R, small droplets
(48)
= --G2 R 2 / R , large droplets. The moment equations obtained for large droplets in this case are dX=AG2(1 --
- -y R - 1 )
dt
x
(49)
'
whereas those for small droplets are dx ( ~f = AG 1 x
RaR -1 ) -x
'
(51)
dy ( Ra ) d t = 2AG1 y - ~ •
(52)
3.3 Behaviour with a constant vapour plus aerosol amount The second case considered is for a constant water mass density, M, M = Spvc(T)+~-pi
Ran(R,t)dR.
(53)
If the mixture is gaining or losing heat by radiation generally the temperature will be changing, and this includes the pure radiative case considered in section 2.2. As it is not wished to include any additional equations here, but just the aerosol equation, the discussion is restricted to the case when T = constant. This does not correspond to any local region where the supersaturation satisfies equation (19). However, it does correspond to a well-mixed region whose radiative input or output of heat is exactly matched by the opposite output or input from other means to make the net heat input zero. The heat includes any from net condensation, but this occurs in a very short time scale. The whole case is perfectly realizable in a P W R containment where there is no net water input and the temperature is constant from net heat input from conduction, convection or radioactive decay and heat loss by radiation. For constant Pvc(T), equation (53) is rewritten as: S(t) = S o + f
Raono(Ro)dRo o
fo
R3n(R, t)dR
]
.
(54)
For an initial log-normal distribution,
N
n(R)=(2 )rln ,exp
{__,21_r,n.
,,s,
this can be written S(t)= So-fN
(
R a ( t ) - R a g y9
)
,
(56)
where Y = exp[0.5(ln %)2],
(57)
f = 4zrpl/(3p,¢(T)).
(58)
r72
~i ( BAR~L:t-rand C. F. Ci.EMENi
Substituting for S in equation (10) gives moment equations as before, by multiplying b) R~no(Ro) and integrating over Ro, In the small droplet case this leads Io d.~: = [ S o .... f N ~ R-~ - - R-~ {ot~] R ::~ + G~ x~
d~
.
.
.
.
.
d:~ =. 2 [ s o --/'74{ k 3 -- R 3 (o)}] + 2G l y ,
dt
"
dR 3 A t : 3 [ S 0 - - f N ( R ~ - R ~ ( o ) ) ] / ~ + 3GxR 3.
~9i
60)
t611
In the large droplet case the final terms in equations 158)-(60) are replaced by G~, 2Gzx, and 3G2y, respectively. To solve this problem numerically the method developed by Barrett and Fissan (1988) was used. First, the third moment N R 3 appearing in equation (56) was expressed as an integral over the size in the initial distribution Ro that grew to R at time t:
NR 3 =
R3(Ro, t)no(Ro)dRo.
(62}
O
The transformation
R o = R,(I +x)/(1 - x )
163)
was then used to transform the integral over an infinite interval to one over a finite interval, approximated by a quadrature sum over m points dx NR....... 3 = 2R, f'_ R3(Ro(x), t)no(Ro(x))~(-i-.~-i2
R3(t)wl i-i--~_--~--~-, i=1
(64)
.
where x~, wi are the quadrature points and weights, respectively, and R~(t) = R(Ro(x~), t). The Rg are found by integrating the m trajectory equations
dR i .... dt
A ( S ( t ) - 1 + G(R,)) , R~
(65)
with initial conditions Ri(0) = Ro(xi). The unknown S(t) in equation (65) can be expressed in terms of the R~ using equations (56) and (64). Once the R i are known the moments can be calculated from:
~, R~(t)wi M . ~ 2R, ,=, i 1 Z x ~ '
(66)
and the distribution itself n(R~, t) can be found from (Williams, 19841:
n,R,,t),
=
no(Rolx,))exp(-f'° dR,'_ dt ~ ) .
,67)
In the convergent case S - 1 >0, G[R)< O, 24 Gaussian quadrature points and weights in equation (64) were used. The 24 equations (65) for the R~ were solved using a fourth order Runge-Kutta method. Figures 5 and 6 show the results of calculations with the small and large growth laws and the interpolation growth law of equation (33), starting with a log-normal size distribution. The initial saturation, S o = 1.1, is such that there is, in effect, a transient with rh~ ¢ 0 for the initial 1 or 2 s during which the mean radius rapidly increases. The moment calculations with large droplets then give poor agreement with exact calculations, although the large drop growth law treated exactly gives accurate results.
Radiative redistribution in a cloud
773
4.5
Smoll
/ 4.0
/
~+.~'~
"~ 'Exact"
~
ff
/
3.5
z-Large drop
exact
-0 ~''- O ~ 0 ~ 0
[
#m
drop
0
/
o/°
/
-- O-- O
0 ~ L ~ a r o e droo Lmaor grneen( ° P
0
3.0
I 25
0
I 50 t
I 75
100
s
Fig. 5. The mean,/~, as a function of time for an initial log-normal size distribution with R , = 3 l~m, ag = 1.5, N = 101° m - 3 c o n v e r g i n g to a final m o n o d i s p e r s e distribution. The results are for S O = 1.1 with small, large and 'exact' growth laws calculated by the moment and exact m e t h o d s .
1.5
\
\ \ 1
o
Large drop moment
-
o
IT- ~2/'2
\ ~-
E I-
/J
"~o
\ \
"~o~
,. -o
I0 L.
Large drop exact ~
~
~Smo[~
I 25
:2
drop
--0"""~
I_ 50 t
- - ' Exact' 0~0~-'~"~ ~
--1--
75
"
lOO
s
Fig. 6. As for Fig. 5, for the standard deviation
(j~2 1~2)1/2. _ _
After the transient S hardly changes, and liquid is redistributed among the droplets. For Rg = 3 #m, ag = 1.5, the final monodisperse radius is (R 3 )1/3 = 3.839 #m, so that the final equilibrium saturation satisfies:
S~o -- 1 "I- G([ R3 ] 1/3)
-- 0,
(68)
which gives Soo = 1.002917. Finally, some 'exact' calculations for the radiative cooling case of S < 1, G(R)>O were performed. In this case the change of variable equation (63) was not used, but instead the integral for N R 3 w a s evaluated by Simpson's rule for 99 points from Rmi n = 0.3 /~m to Rmax =30/~m. With these values the initial.distribution was integrated accurately to five significant figures, but the accuracy of calculated values of S deteriorated at later times.
TI4
J, ( i BAI~RI~'Trand C, IF. (].EMENT
The size distribution after 5 and 10 rain was calculated using equation (67), and previously calculated values of S used as input with linear interpolation for S(t) between these values. The results (Fig. 7) show a large increase in width of the distribution. Three radu are shown on the curves: R 1 is the radius of a droplet in the initial distribution that evaporates to the minimum radius shown. Smaller droplets in the initial distribution evaporated completely. The radius R2 is that for which a droplet is at its initial radius at tirnc ! ~hesc droplets grew at first but then evaporated back to their original radius. The radius R :~is tl~a~ for which S(t)-- 1 + G ( R 3 ) = 0 at time t. These droplets have grown to their maximum radiu~ R 3 and at later times evaporate. Integrating the trajectory equation for droplets of radius R2 gives: i" S .... I + G ( R t Q )
A i
......~
......
t69l
dr=O,
U0
The values of R2 arise from the cancellation of large positive and negative quantities, and are thus very sensitive to accurate values of S. This is also true of values near R2 (note that R1 and R: are very close together), but for radii greater than R 3 the integrand in equation (69) is always positive, so that data points should be fairly accurate, despite uncertainties in S. The closeness of R~ and R2 also explains the difficult), in evaluating S accurately; the first non-zero quadrature point at t = 10 min is Ro ~ 9.09 #m, so the volume less than this is poorly approximated. In general, for radiative cooling with S < 1 it is possible for S to rise or fall initially. A condition may be found by differentiating equation (56), and using equation (10); for 1~, when it is found that S increases initially if: j'~ [ ( S o - - 1 ) S o + G(Ro)Ro]n(Ro)dRo
<
~70)
0.
For G(Ro) given by the small or large growth laws this reduces to conditions on the moments of the distribution. For example, with So = 0.99 both the conditions are satisfied for log-normal distributions with ~g = 2 and Rg < 2.7/zm, but are both violated for R g > 4.5 pm.
..............
3
15
,4! I
\,=0 \
I °glon
15 5#rn
74~zm~'f ~ ~-~.
13~-
/
"/ ~2 i I
I,\ R~
\
3.
-.'~
't...J'\
L 1o
2 5.3/,zrn
\
\ I
R1
'R \
I
11 7,231xm
t=5min
"~.
\ \
t:lOmr ~\ 30
20
40
-~o
R ~m
Fig. 7. Aerosol size distributions, log n, against radius R at three times. The significance of the radii shown, RI, R2 and R 3, is explained in the text.
Radiative redistribution in a cloud
775
The case where large droplets grow at the expense of small ones, which would also apply with a different dividing line in the pure radiation field case of section 2.2, is clearly very interesting in that distributions grow much wider and some droplets evaporate completely. However, in comparison to other cases, accurate calculations are not easy to perform, particularly, as would be the case in practical applications, when other terms such as gravitational settling and coagulation are included in the aerosol equation (30). 4. C O N C L U S I O N S The behaviour of a cloud of droplets in a vapour-gas mixture in a radiation field has been examined. Illustrative calculations have been carried out for water droplets. The conclusions may be summarized as follows: (a) The net radiative heat transfer from a droplet of radius R is described in terms of the wavelength averaged absorption efficiency, ( Q a ( R ) ) . Exact and approximate expressions for this quantity have been given for water droplets, agreeing with previous results of Roach (1976). (b) The growth rate of a droplet is expressed in terms of a radiative quantity, G(R), obtained from ( Q a ( R ) ) and properties of the radiation field, and the supersaturation, S - 1, which represents the effect of heat and mass transfer to the surrounding vapour-gas medium. Because G(R) depends on R the net growth rate has the four possible regions shown in Fig. 1, in two of which there is mass redistribution between small and large droplets. (c) An alternative expression, equation (17), has been obtained for /~ in which the dependence on S - 1 is replaced by that on the overall growth rate. This has enabled equation (23) to be derived for growth in a pure radiation field. For water droplets, the case considered in detail, this could apply to a water cloud in a PWR containment. The supersaturation has been shown to be proportional to L e - 1, and some redistribution from the smallest water droplets then occurs in the radiative cooling case. (d) Some exact and approximate methods have been described for solution of the aerosol equation for water droplets with a growth term only. For redistribution on radiative heating, when the aerosol distribution tends to a common radius, simple moment equations give an adequate description of the process for the large droplet or exact growth laws except when there is an initial transient. (e) For moderate aerosol densities fast transients from saturations not close to unity have time scales of seconds or less (see also Clement, 1988a). Growth and redistribution will then occur for most of the time when the saturation is very close to unity with a value controlled by relatively slow heat transfer processes such as radiative transfer. Calculations have been performed for the case when there is a constant density of water mass and the temperature is also kept constant. For radiative cooling this implies a corresponding heat source and this case is also possibly applicable to a PWR containment. For radiative cooling, it has been shown that large growth is possible in the width of a water droplet size distribution in a time scale at least of the order of minutes, but that accurate numerical calculations to describe the size distribution are not easy to perform. An extensive exploration of the effects of the interaction between radiation and a water droplet cloud when the droplet radii are in the continuum regime has been performed. Unless the molecular sticking probability is small, this should be adequate for R > 1/~m. The case of smaller droplets is not expected to be important because most water clouds involve R > 1/~m and, in any case, the radiative interaction is relatively smaller for small droplets. However, the case can be explored on the lines of this paper by replacing R in the denominator in equation (10) for R by R +6, where 6 is temperature dependent and expected to be of the order of 0.2/~m, provided that the molecular sticking probability is not small (Barrett and Clement, 1988).
776
$ ( BARI~ i"T and C. F'. ('t.EMI~NT
F o r p r o b l e m s i n v o l v i n g h i g h t e m p e r a t u r e w a t e r mists, s u c h as in the P W R c o n t a i n m e r v . , it will be e s s e n t i a l to i n c l u d e t h e r a d i a t i v e i n t e r a c t i o n w i t h walls. T h i s is b e c a u s e o n l y a s m a i l i n t e r a c t i o n is n e c e s s a r y to i n d u c e net g r o w t h o r significant r e d i s t r i b u t i o n w i t h i n ~ ct¢)ud. T h e crucial q u a n t i t y to c a l c u l a t e is the s u p e r s a t u r a t i o n , S - - l . In ~he case (~i ~'.patiai d e p e n d e n c e o n e e q u a t i o n has b e e n q u o t e d [ e q u a t i o n ( 1 9 ) ] w h i c h c a n be used ik:;t th~s p u r p o s e . H o w e v e r , for a w e l l - m i x e d r e g i o n subject to i n p u t s of h e a t a n d wateE m~:~,,, i~ i~ p r e f e r a b l e 1o a d o p t the a p p r o a c h used ~'ecently in c o n n e c t i o n w i t h h y g r o s c o p i c ac~-osoi.,,. ( C l e m e n t , 1988) to o b t a i n an e q u a t i o n for the s u p e r s a t u r a t i o n . A d e s c r i p t i o n of h,,w t~ i n c l u d e the n e c e s s a r y r a d i a t i v e t e r m s will be g i v e n in a n e x t e n s i o n of that w o r k
This work was funded as part of the General Nuclear Safety Research Programme undertaken by AEA Technology. Acknowledgement
REFERENCES Barkstrom, B. R. (1978) J. Atmos. Sci. 35, 665. Barrett, J. C. (1985a) J. Phys. D 18, 753. Barrett, J. C. (1985b) The interaction of radiation with aerosol-vapour-gas mixtures. Thesis, Queen Mary College; London and Harwell Report AERE-TP.1171. Barrett, J. C. and Clement, C. F. (1985) The influence of radiation on aerosol formation by vapour condensation. Harwell report AERE-TP. 1120 and Proc. Int. Syrup Workshop on Particulate and Multiphase Processes and 16rh Annual Meeting of the Fine Particle Society, Miami Beach, April 1985. Proc. of Condensed Papers 35-12. Barrett, J. C. and Clement, C. F. (1986) The effect of sodium aerosols on total heat transfer and temperature in the cover gas space. Harwetl report AERE-TP.1159 and Specialists Meeting on Heat and Mass Transfer in the Cover Gas Space, Harwell 1985, IAEA IWGFR-57 2/1-1. Barrett, J. C. and Clement, C. F. (1987) The behaviour of water droplet size distributions interacting with radiation. Aerosols in Science, Medicine and Technology, GAeF Conference, Hannover, Sept. 1987. J. Aerosol Sci. 18, 647-9. Barrett, J. C. and Clement, C. F. (1988) J. Aerosol Sci. 19, 223. Barrett, J. C. and Fissan, H. J. (1989) J. Aerosol Sci. 20, 279. Clement, C. F. (1978) Proc. R. Soc. A 364, 107. Clement, C. F. (1985) Proc. R. Soc. A 398, 307. Clement, C. F. (1987) The supersaturation in vapour-gas mixtures condensing into aerosols. Harwell report AERE-TP.1223. Clement, C. F. (1988a) The formation of nuclear aerosols by evaporation-condensation processes. Harwell Report AERE-TP.1285. Clement, C. F. (1988b) Water condensation on hygroscopic aerosols. The 1988 European Aerosol ConJerence, Lund, Sweden. J. Aerosol Sci. 19, 875. Clement, C. F. and Taylor, A. J. (1986) Nuct. Engmj. and Design 92, 109. Guzzi, R. and Rizzi, R. (1980) Contr. Arm. Physics 53, 351. ()zisik, M. N. (1973) Radiative Transfer and Interaction with Conduction and Convection. John Wiley, New York. Roach, W. T. (1976) Q. J. K Meteorol. Soc. 102, 361 Truelove, J. S. (1984) Int. J. Heat Mass Transfer 27, 2085. Williams, M. M. R (1984) J. Colloid Interface Sci. 101, 19. Wobrock, W., Kramm, G. and Herbert, F. (1986) The influence of the aerosol distribution and composition on the formation and evolution of radiation fog. Aerosols: Formation and Reactivity. Proc. 2nd Int. Aerosol Conf, Berlin. Pergamon Press, Oxford
0021-8502/90 $3.00+0.00 Pergamon Press pie.
Printed in Great Britain.
GROWTH AND REDISTRIBUTION IN A DROPLET CLOUD INTERACTING WITH RADIATION J. C . BARRETT a n d C . F . CLEMENT* Theoretical Physics Division, Harwell Laboratory, Didcot, O x o n OX11 0RA, U.K.
(Received 15 January 1990) Abstract Previous work on the growth and redistribution of a droplet distribution interacting with a radiation field has been extended. With no other macroscopic heat and mass transfer processes the s u p e r s a t u r a t i o n is s h o w n to be non-zero when the Lewis n u m b e r , Le, of the s u r r o u n d i n g v a p o u r - g a s mixture differs from unity. Redistribution of m a s s with e v a p o r a t i o n of the smallest droplets then occurs for radiative cooling a n d Le < 1, a condition satisfied for water aerosols. Exact and approximate expressions are given for the wavelength-averaged absorption efficiencies of water droplets. Calculations of changes to water droplet distributions were performed using exact methods and simplified m o m e n t equations. The latter are adequate when radiative redistribution leads to a c o m m o n radius and there is no initial transient. The conditions for no mass transfer to the aerosol a n d ' a constant total water density and temperature are discussed. In the latter case with radiative cooling, which could possibly apply to reactor situations, a considerable widening of the size distribution takes place on a long timescale.
NOMENCLATURE A c ?p
Cn D eba f F g
G(R) G1 G2 Ia /k L Le th, M M,
n(R) N p Q~ QR R /~ R~ Rg RG /~, s S S.,~ t T u t,
constant in growth rate equation vapour mass concentration mean specific heat at constant pressure condensation n u m b e r vapour-gas diffusivity Planck black body function dimensionless n u m b e r [see e q u a t i o n (58)] fraction of aT'* numerical fraction dimensionless radiative growth function coefficient of R z specifying G(R) for small drops coefficient of R specifying G(R) for large d r o p s radiation intensity radiation intensity averaged over wavelength thermal conductivity latent heat of vaporization Lewis number total local aerosol condensation rate total water mass density nth m o m e n t of the size distribution aerosol n u m b e r density total aerosol number density total pressure efficiency factor for absorption of radiation total radiative heat flux from droplet droplet radius droplet growth rate critical radius p a r a m e t e r of log-normal distribution gas constant nth m o m e n t of the aerosol size distribution path distance coordinate vapour saturation asymptotic saturation in time time temperature difference between vapour concentration and its equilibrium value velocity
* Author to w h o m correspondence should be addressed. As~:~-0
761
762
i ~, BARRH~and C F, Ct.EMENI weighting for integration x distance in 1-D radiation field; /~; change of variable [_seeequation (b3!~ abbreviation for/~2 Y dimensionless function of °s coefficient in exponent fl coefficientgiving temperature derivative of Pve(T) b small length ). wavelength # cosine of angle in radiation field /~ (with a subscript) molecular weight p total density o Stefan-Boltzmann constant ag width parameter of log-normal distribution f~ angular direction of radiation field AT temperature difference specifyingg
Subscripts c d e g i 1 L s S v o
pertaining to critical size pertainingto droplet pertainingto equilibrium pertainingto gas pertaining to set of points pertainingto liquid pertaining to large droplets pertaining to surface pertainingto small droplets pertainingto vapour pertainingto initial value
Superscripts derivative with respect to T average over the size distribution 1. I N T R O D U C T I O N The interaction between radiation and a cloud of droplets can lead to its heating or cooling and result in evaporation from, or condensation onto, the droplets. For water clouds these effects are certainly important in radiation fogs (Roach, 1976; W o b r o c k et al., 1986) and on droplet growth near cloud tops (Barkstrom, 1978; Guzzi and Rizzi, 1980). They can also occur in water droplets which might be formed in the containment of a pressurized water reactor ( P W R ) following a possible accident, where water mass m a y be redistributed, as has previously been pointed out (Barrett and Clement, 1987). The aim of this paper is to extend the previous work and explore in general the effect on droplet size distributions of the radiative interaction. The results are reported of a number of calculations performed for water clouds. In section 2 a general result for the growth rate, R, of a droplet valid for a saturation not too much above unity is quoted (Barrett and Clement, 1988). The radiative term in/~, which arises from net radiative heat loss or gain, is expressed in terms of a temperature-averaged efficiency factor, and the general form of the growth rate is given in the continuum growth regime valid for larger droplets. This rate is re-expressed in terms of the local total mass transfer rate to the aerosol, and it is shown how it can be specified by an equation for the supersaturation. In section 2.2 it is shown that a pure radiation field with no vapour or heat currents can result in mass redistribution with evaporation of small droplets if the Lewis number of the mixture is less than unity. This is the case for water vapour in air, and the temperature dependence of the effect is discussed. In section 2.3 results and approximations for the absorption efficiency factors for water, drawing upon previous work of Barrett (1985a, b) and Roach (1976) are given. In section 3 methods for solution of the aerosol equation for water droplets when radiation induces growth or redistribution of the size distribution are examined. The cases of constant supersaturation and of a constant vapour plus aerosol water density at constant temperature, and how these might arise in practice, are discussed. Calculations starting
Radiative redistributionin a cloud
763
from log-normal size distributions are performed, and the adequacy of using moment equations as an approximation are explored. The procedure for obtaining such equations is described in section 3.2, and could be adapted to other aerosol problems. Finally, in section 4 conclusions from this work are drawn, and an outline of what is needed to extend it and incorporate it into calculations for water aerosols in PWR containments is given. 2. THE G R O W T H OF C L O U D D R O P L E T S IN A R A D I A T I O N F I E L D Droplets in a cloud with a size distribution specified by n(R), where R is the droplet radius, are considered. This distribution is also generally a function of position, _r, and time, t. Previous results (Barrett and Clement, 1988) on general growth rates for such droplets are reiterated and how the quantities needed for their calculation can be obtained is discussed. These include the supersaturation, and an equation for this (Clement, 1987, 1988a), which includes a radiative term, is given. The case of growth in a pure radiation field is considered in section 2.2 and it is shown that for vapour-gas mixtures, whose Lewis number, Le, is nonzero, the supersaturation does not vanish. This means that radiation acting alone can cause the redistribution of mass in the size distribution. Finally, in section 2.3, the results of calculations of absorption efficiency factors for water and sodium droplets are given. These factors are the basic quantities needed to specify the dependence of the growth rates on R. 2.1 General orowth rates
Only the case when the vapour supersaturation is fairly small is considered, so that the general growth rate is given by (Barrett and Clement, 1988); /~_ 1 S - 1 + [RAD] p~ [ M A S S ] + [ H E A T ] '
(1)
where Pl is the droplet density and the saturation S is given in terms of the actual and equilibrium vapour densities by: S = pv/pve(T).
(2)
The resistances to growth from mass and heat transfer, [MASS] and [HEAT], respectively, take different forms (Barrett and Clement, 1988) according to the droplet size and temperature. Here, the continuum growth regime, which is a good approximation at normal pressures for R > 1/~m, will be considered. It would be straightforward to extend all the results obtained here to the transition and molecular regimes which apply to a smaller R. The main difference is that the dependence on R of [MASS] and [HEAT] changes from R to a constant at small R. In terms of the total pressure, p, the vapour-gas diffusivity, D, the thermal conductivity, k, and the latent heat, L, an excellent approximation (Barrett and Clement, 1988) is provided by: R(p - Pve(T)) [MASS] = , (3) pDpv¢(T) [HEAT] = RflL/(kT2),
(4)
where Pve is the vapour density corresponding to p,, and fl is given by: fl = #vL/RG,
(5)
where #v is the vapour molecular weight and R 6 is the gas constant; fl is the exponent in the Clausius-Clapeyron relation for p,~ (T). The ratio of [MASS] to [HEAT] is given by the condensation number (Clement, 1985), either in the bulk (Cn) or surface (Cn~), [MASS] - - (1 - c e ( T ) ) C n ( T ) = Cns(T), [HEAT] where c~(T) is the equilibrium vapour concentration.
(6)
764
i
In the continuum Clement, 1988 t:
( i B A R I ~ t l i a n d C. F. C L t : M E N I
region the radiation
term in equation(li
is l Barre~.~ and
4 nflQ~ k R T 2'
[RAD]
i::
where QR is the net radiative heat flux from the droplet given by (Ozisik, ~973: Barrett, 1985):
c
QR = ~R2
d2 Q~(R, 2)[4eba(Ta) --
., 0
f
I~(s, ~ ) d ~ ]
4"~
= 7rg 2 < Q , ( R ,
,~) > [4~rr~ -
f,
[(x, ~ ) d ~ ] .
i8)
~ i
The first expression refers to the absorption of radiation of intensity I~(s, ~ ) depending on position s along a path with ~ as direction of propagation, and emission of radiation where eba(Ta) is the Planck black body function at the droplet's temperature. In practice, in nearly all cases, Ta may be taken as T, the local temperature of the medium. The second expression in equation (8) is obtained in the grey medium approximation for a 1-D radiation field along distance x with # as the cosine of the angle to the x direction. Both expressions are specified by the droplet's efficiency factor, Q~(R, 2), for the absorption of radiation of wavelength 2; the brackets in the second expression denote an average over the black body spectrum. For any grey medium, Q~ = 47rR2F ( Q . ( R , ,t)),
(9)
can be written, where F is a fraction, g, of aT a given by 1/4 of the square bracketed expression in equation (8). From equations (3)-(9) R in the continuum region can be expressed as: A = R (S- 1 +G(R)),
(101
where A . 1 _ ptflL [1 + C n ( T ) ( 1 - c ~ ) ] , kT z
(11)
fl F R (Q.(R)).
(12)
G(R)=
The general behaviour of ( Q a ( R ) ) is an increase with R which levels out to a constant at
large R. Thus, the two terms in equation (10) will always have a different R dependence, RADIATIVE COOLING ...............
b.~
Droplets evaporate .....
C00!
.
.
.
.
.
.
mo~mu-m-[or10#m "drop~e-t -
.
.
All droplets grow
/o. X wa,er ciouds
AIt droplets evaporate .
...... Gt(.!~ i \'\ Dropie,s I "~grow I ~ j p, ~x
:. 6 . 6 , , ~
0.01 S -! I:q [,1 ~ ~" I DroDlets'x~ Droplets grow }evop.onrate4~" to R c
i ,o ~c /
. . . . . . . . .
\
~
. . . . . . . .
RADIAT!VF HEATING Fig. I. Behaviour of a droplet cloud subject to a supersaturation, S - I, and radiative growth term, G(R), depending on droplet radius R. The hatched regions show the behaviour expected for a cloud of water droplets subject to radiative heat transfer alone.
Radiative redistribution in a cloud
765
implying the general behaviour of k in relation to heating and cooling which was pointed out recently (Barrett and Clement, 1987). This is summarized in Fig. 1, which is augmented by showing possible regions for a cloud of water droplets subject to radiative heat transfer alone (see next section). T o explore possibilities for general b e h a v i o u r it is necessary to calculate the supersaturation, S - 1, or the overall mass transfer rate to the aerosol per unit volume, vhv. Relations between these quantities in terms of moments of the size distribution can be obtained: t* N
= jn(R)dR,
(13)
= fRn(R)dR/N,
(14)
OR = f QRn(e)dR/X = 4rckT 2 f R G ( R ) n ( R ) d R / f l N .
(15)
Integrating equation (10) over the size distribution gives:
rhv = 4npl ( R EIln( R )dR iJ
= 4npl(S-- 1)NRA +
NQR
(16)
L [ 1 +Cn(1-Ce)]"
The relation between thv and S - 1 enables the individual growth rate to be rewritten as:
lt=4rcp~N~R ~
Q~-
.
(17)
The final term, which gives a radiative redistribution of mass in a distribution, is more general than that given previously [final term in equation(l l) of Barrett and Clement (1988)], where only the limiting case for large R, where ( Q a ( R ) ) was independent of R, was considered. This implied that the previously defined Grad = G(R) was independent of R and QR ~ R2. Other terms in the general form for/~ representing Ostwald ripening from the Kelvin effect, and hygroscopicity in a water droplet are omitted in this paper for simplicity, but they could easily be inserted from our previous expressions (Barrett and Clement, 1988). The quantity, rhv, occurs in the original heat and mass transfer equations for vapour-gas mixtures (Clement, 1985), but is not easy to obtain from them except in the saturated (S = 1) limit. This limit is only reached slowly at high aerosol concentrations (Clement and Taylor, 1986), and it is better to use directly an equation which has recently been obtained (Clement, 1987, 1988) for S - 1 or,
u = C-Ce(T) = [/~v - (#v -- #g)c] Ce(T)(S - 1)/#v.
(18)
This form of u is appropriate for changes occurring at constant pressure, and it is an excellent approximation to put c = c c on the right hand side (RHS) in equation (18). One accurate form for an equation for u is (Clement, 1988a) p
~+v'Vu
= Dp[c.(VT)2
+V'(--DpVu) (Cpv-
Cpg) c'cVc'VT] +
Le
- c',( 1 - Le)V. ( - DpVT) - rhv( 1 - c + Le/Cn),
-NQrt (19)
766
J ( ]~ARREI'Tand C, F, CLEMEN'I
where p is the total density, a prime denotes a derivative with respect to temperature, ~p,. a n d Cog are specific heats at constant pressure for vapour and gas with c~ the density-
averaged specific heat, and the Lewis number, Le, is given by [,C ::
k/D~;p p.
(2())
Amongst the source terms which give rise to non-zero values of u, and hence droplet evaporation or growth, is the radiative term containing QR. In general, when this term is non-zero, there will also be spatial gradients in temperature, especially in boundary layers through which heat and mass transfer is occurring. However, if the spatial gradients are small in the bulk of the fluid through turbulent mixing there can still be a strong radiative interaction, a case now considered in detail. 2.2 Growth in a radiation.field When a turbulently well-mixed aerosol is interacting with a field passing through it, or is radiating to walls, the spatial temperature and concentration gradients in equation (19) will vanish. Transients have a short time-scale for this equation (Clement, 1988a) so that the solution is obtained by setting the radiative source term equal to the aerosol loss term proportional to rhv: Le N QR rhv = L [ ( I - c ) C n + L e ] "
(21)
Neglecting any second order terms arising from differences between c and c e, equations (16) and (17) give:
S-l=
flQa ( 1 - c e ) C n ( L e - 1 ) 47tkT2 R [ 1 - ce)Cn + Le] '
1
(22)
[~ Q.a(1-e~)Cn(Le-1)]
[~ = 4np, LR[1 + ( 1 - G ) C n ]
+ ~- [ ( 1 - - - c ~ - C n + T e ] J '
(23)
Neither of these results depends explicitly on the aerosol number density, N. Their most interesting feature is the dependence on L e - 1. For Le = 1 the supersaturation is zero: physically the rates of conductive heat transfer and diffusive mass transfer to and from droplets are the same, so that the effect of radiant loss or gain of heat is to leave the gas-vapour medium exactly in balance. For Le ~ 1 an imbalance arises from the different rates of heat and mass transfer. Thus, the conditions for radiative redistribution to occur are: A. Le < 1 i24) B. QR is not proportional to R.
(25)
If these conditions are satisfied the two terms in equation (23) have opposite signs and, when Qa increases more rapidly than R, the smallest droplets evaporate when most of the distribution grows, and vice versa. These phenomena should occur for clouds of water droplets in air at normal pressures where Le = 0.85. The processes involved are illustrated in Fig. 2, and will be more important at low T t h a n at high T because Cn is unity at T = 4°C at 1 atm pressure and falls to be much less than unity at high temperatures (Clement, 1985). Numerically, for water droplets at 4°C with fl = 5390 K, k = 0.024 Wm -1 K -~ , T = 277 K, Le = 0.85, Cn(1 - c ¢ ) = 1, the radiative heat flux is QR=4.2 x 10 -9 g R 2 (/tm) ( O a ) IV, and with its limiting value for large R of ( Q, ) = 1.2, S-- I = - 9 . 5 10 -4 g R 2 / R ,
(26)
where the radius averages are measured in/~m. At the top of the atmosphere g ~ 1 for radiation to outer space could be used, but a more realistic value for engineering applications would be g ~ 0.1. The corresponding value for R
Radiative redistributionin a cloud RADIATIVE
EVAPORATION
767
COOLING
/4 CONDENSATION
CONDENSATION
#~EVAPORATION
Fig. 2. Expectedphysical behaviour of large and very small water droplets in a cloud interacting with radiation alone.
would be /~=0.16g
1-0.081~-
~ m s -t.
(27)
For log-normal size distributions with a S = 1.5 and 2, k would change sign at
R/R < 0.095, 0.13, respectively. However, this does not take account of the behaviour of QR with R for water droplets which will now be examined. 2.3 Absorption efficiency factors In a radiation field with an effective black body spectrum the quantity describing its interaction with a spherical droplet, radius R, is the wavelength averaged absorption efficiency, (Qa(R)), contained in equations (9) and (12). For water droplets at moderate temperatures the approximations (with R in #m) can be used:
(Qa(R)) = 0.2R, small R = 1.2, large R,
(28)
whereas, over the whole size range, a form suggested by Roach (1976) can be used:
(Q,(R)) = 1.211 - e x p ( - 0 . 3 g ) ] .
(29)
In Fig. 3 these expressions are compared with the results of exact Mie theory calculations (Barrett, 1985a) averaged over the black body spectrum at 280 K. An exponent of 0.167 to replace 0.3 in equation (29) gives better agreement with Mie theory as R -, 0, but significantly underestimates ( Q , ) in the important range of 3 < R < 10 #m. For atmospheric applications it may be more appropriate to use a clear sky emission spectrum; however, Fig. 1 of Roach (1976) indicates that the difference is not large.
3. RADIATIVE R E D I S T R I B U T I O N OF WATER D R O P L E T S The problem addressed is to solve the aerosol equation with a growth term
~n c~(nR) t~t
Jr - - 0, t~R
(30)
where, for water droplets, R is specified by equation (10), with
G(R) = GIR 2, small drops R ~ 5 #m
(31)
= G2R, large drops R >~ 10/~m
(32)
G 2 R [ 1 - e x p ( - - ~ t R ) ] , all R.
(33)
=
768
J. C, BARREl')andC. F. CLEMEN'I ----I-
I
........ r - - T
:
~--
:
/!
!....
0 2 R /:
.
¢"
r
Mie
/+:,J~
1.0!-
. . . . . . ] ....
.
.
- - ! ......
i
....
] ..... T ....
~
theory
.
'
-r
" ,:~ .- . . . :
~ 2(l_e-O.3R)
,f
//7
0.2
-i
L._.L__._L___L_~
I
I
__l___J_
I
[
5
I
J
I
10 R
I
_L.__L__.J___L___
15
_: 2 : "~
,um
Fig. 3. Accurate(Mie theory-crosses),approximate(dashes) and asymptotic(solid lines)curves for the wavelengthaveragedabsorption efficiency,(Qa(R)), for water droplets at 280 K as functionsof droplet radius R.
From equations (12), (28) and (29), good approximations are given by :t = 0.3 vm ' ~
G1 = 1.9 x 1 0 - ' ( T ) G2 = 1.2 x 10 .3
2 9 #m-2
9l~m - 1 ,
(34) (35)
where g is likely to be a small fraction. For radiation across a small temperature difference, AT, to an absorbing surface .q = 4 A T / T .
(361
Methods of solution of equation (30) can now be discussed, although, to include coagulation and other terms in the aerosol equation only numerical methods are generally applicable. It is, however, useful to have analytic results or simple methods of solution available where possible to gain an understanding of the behaviour of water mists. First, the case of constant S will be considered, and then, in section 3.2, how approximate solutions for moments of the distribution are obtained will be described. These are applied to the case of constant total water density in section 3.3. Exact numerical results are also obtained. 3.1 Growth with constant S
The condition of constant S corresponds to constant u as a solution of equation (19). F r o m equations (9) and (22) it can be seen that this cannot be achieved by radiation alone, because in no circumstances for water droplets is Qa ~/~. Any transport term with fluid velocity, v, will affect the aerosol too, so that the only chance of achieving constant isis via the diffusive and nonlinear terms in equation (19). However, these will usually only be large in boundary layers where the radiative terms in the equation are small. It would therefore be probable that an experimental situation to realize this case would have to be designed: it will not be naturally important. However, the solutions are easy to obtain and are quoted for completeness.
Radiative redistributionin a cloud
769
As previously noted (Barrett and Clement, 1987), the solution of equation (30) may be written down in terms of the initial distribution no(Ro), (Clement, 1978):
R [ S - 1 +G(R0) ] n(R, t) = no(Ro) Ro[S _ 1 + G ( R ) ] '
(37)
where Ro(R, t) is specified by the trajectory fRRo S - 1 +R'G(R')
dR' = At.
(38)
With G(R) given by equation (33), the integral must be performed numerically, but for small and large drops results obtained from equations (31) and (32) are:
G1 R2 + S - 1 = ( G1R 2 + S-- 1)exp(-2AG~t),
(39)
( S - 1 + G2Ro)exp[ - G2Ro/(S - 1)] = ( S - 1 + G z R ) e x p [ - G z R / ( S - 1) + A G 2 t / ( S - 1)].
(40)
For no radiative interaction with G = 0,
R 2 - R g = 2A(S-- 1)t.
(41)
In the first quadrant in Fig. i, where G(R), S - 1 > 0, growth is unconstrained and the results indicate that for large t R increases exponentially for small droplets, R increases as t for large droplets, but as x/~ when there is no net radiation. Growth with radiation is significantly faster than growth from supersaturation alone. When G(R), S - 1 < 0 the droplets evaporate completely in a time given by putting R = 0 in equation (39) for small droplets: large droplets go into the small droplet region and, unless R 0 is very small, the time spent in the region where molecular processes limit evaporation and the denominator in equation (10) goes to a constant, can be neglected. More accurate results could be obtained by replacing R by 6 + R in the denominator where, for water molecular sticking probabilities not much less than unity, 6 is about 0.2 pm (Barrett and Clement, 1988). In the fourth quadrant in Fig. 1, where S - 1 > 0 and G(R) < 0, the drops grow or evaporate to asymptotic radii given for small and large droplets by: R s = [ ( S - 1)/JGI[] ~/2,
(42)
RL = ( S - 1)/1621.
(43)
Numerical values can be obtained using equations (34) and (35). Unless S - 1 < 0.01, R s > 10/~m and small droplets would not remain in the small droplet regime. Finally, if S - 1 < 0 and G(R) > 0 (second quadrant in Fig. 1) droplets with initial radii satisfying S - 1 + G(Ro) < 0 evaporate completely, whereas the rest grow without limit. In the general case with equation (33) the critical radius R c satisfies: Rc [ 1 - e x p ( - ~R~)] + ( S - 1)/G2 = 0.
(44)
The quantity ~ is a constant independent of T and g, so that this equation can be used to plot ( S - 1 ) / G 2 as a function R c and to predict the future behaviour in the second and fourth quadrants for any radiative flux and supersaturation (see Fig. 4). 3.2 Moment equations From an experimental point of view it is usually only possible to observe the first few moments of a size distribution, n(R, t). This is an additional reason, other than simplicity, for attempting to obtain a closed set of equations for the moments rather than solving the full aerosol equation (30). The moments are defined in general by Mn(t) = f ~ Rnn(R, t)dR/N(t), n>_ l, do
(45)
770
,~ {: BA~krr~ and C. F, ()_EMEN-i i i;-;5 i
[
/
iI....
/ J
t
.
.
.
,/
.
/
// 2o--
J-g : 7
/
,
/
7
~g=I.5 /
i
,
i
/
,"
15~--
//
,,
/
L
/
i
I .11
,,
/ //I // C ]
//
5~--
,-. ,,' ,'.Y
/
// // .
%"1V/ ~////
//%=',
'
Rc for Rg = 8 # m O-g : 1,5
7 I
o ~
L_
2
._L ......
A
6
I ____k_ ..... L__
8
10
12
.. E
t
14
t6
...... J ......
18
Rg #m
Fig. 4. Variation of - ( S - 1 ) / 6 3 with R, for various values of trg specifying log-normal size distributions changing with rhv = 0. Given R, and ag the procedure to find the critical radius, R e, is indicated by arrows.
where, for the special cases ofn = 1, 2, M1 =/~ = x and M z = R 2 = y will often be written for brevity. It is possible to show from the general solution of equation (30) that (Clement, 19781
M. =
f dRono(Ro)Rn(R0, t)/N(t),
(46)
where R along a trajectory is a function of Ro and t. The general procedure to obtain moment equations then has the steps: (a) Differentiate equation (46) with respect to t, and use dR~& =/~ and equation (10) for/~ to relate the derivative of Mn to M,_ 1, and M. + 1 if necessary. This is possible with the small and large drop forms of equations (31) and (32) for G(R). (b) Close the equations for M t and M2 by expanding R-1 about 1//~, a procedure expected to be a good approximation for narrow distributions:
=~ =
-
!-[
dRono(Ro)~ +(R_l~) {y--x2-)R3--3xy+2x 3 ] 1+\ )+..
1 ( 4y
R3
x
x 3
x~
) 2
. /
(47)
Radiative redistribution in a cloud
771
(C) Use any condition on rhv, or the total water density, to express S - 1 in terms of moments M l, M 2 and M a. In practice moment equations have only been used when N = constant and the condition rhv = 0, the next case considered, then gives: S - 1 = - G 1R 3/R, small droplets
(48)
= --G2 R 2 / R , large droplets. The moment equations obtained for large droplets in this case are dX=AG2(1 --
- -y R - 1 )
dt
x
(49)
'
whereas those for small droplets are dx ( ~f = AG 1 x
RaR -1 ) -x
'
(51)
dy ( Ra ) d t = 2AG1 y - ~ •
(52)
3.3 Behaviour with a constant vapour plus aerosol amount The second case considered is for a constant water mass density, M, M = Spvc(T)+~-pi
Ran(R,t)dR.
(53)
If the mixture is gaining or losing heat by radiation generally the temperature will be changing, and this includes the pure radiative case considered in section 2.2. As it is not wished to include any additional equations here, but just the aerosol equation, the discussion is restricted to the case when T = constant. This does not correspond to any local region where the supersaturation satisfies equation (19). However, it does correspond to a well-mixed region whose radiative input or output of heat is exactly matched by the opposite output or input from other means to make the net heat input zero. The heat includes any from net condensation, but this occurs in a very short time scale. The whole case is perfectly realizable in a P W R containment where there is no net water input and the temperature is constant from net heat input from conduction, convection or radioactive decay and heat loss by radiation. For constant Pvc(T), equation (53) is rewritten as: S(t) = S o + f
Raono(Ro)dRo o
fo
R3n(R, t)dR
]
.
(54)
For an initial log-normal distribution,
N
n(R)=(2 )rln ,exp
{__,21_r,n.
,,s,
this can be written S(t)= So-fN
(
R a ( t ) - R a g y9
)
,
(56)
where Y = exp[0.5(ln %)2],
(57)
f = 4zrpl/(3p,¢(T)).
(58)
r72
~i ( BAR~L:t-rand C. F. Ci.EMENi
Substituting for S in equation (10) gives moment equations as before, by multiplying b) R~no(Ro) and integrating over Ro, In the small droplet case this leads Io d.~: = [ S o .... f N ~ R-~ - - R-~ {ot~] R ::~ + G~ x~
d~
.
.
.
.
.
d:~ =. 2 [ s o --/'74{ k 3 -- R 3 (o)}] + 2G l y ,
dt
"
dR 3 A t : 3 [ S 0 - - f N ( R ~ - R ~ ( o ) ) ] / ~ + 3GxR 3.
~9i
60)
t611
In the large droplet case the final terms in equations 158)-(60) are replaced by G~, 2Gzx, and 3G2y, respectively. To solve this problem numerically the method developed by Barrett and Fissan (1988) was used. First, the third moment N R 3 appearing in equation (56) was expressed as an integral over the size in the initial distribution Ro that grew to R at time t:
NR 3 =
R3(Ro, t)no(Ro)dRo.
(62}
O
The transformation
R o = R,(I +x)/(1 - x )
163)
was then used to transform the integral over an infinite interval to one over a finite interval, approximated by a quadrature sum over m points dx NR....... 3 = 2R, f'_ R3(Ro(x), t)no(Ro(x))~(-i-.~-i2
R3(t)wl i-i--~_--~--~-, i=1
(64)
.
where x~, wi are the quadrature points and weights, respectively, and R~(t) = R(Ro(x~), t). The Rg are found by integrating the m trajectory equations
dR i .... dt
A ( S ( t ) - 1 + G(R,)) , R~
(65)
with initial conditions Ri(0) = Ro(xi). The unknown S(t) in equation (65) can be expressed in terms of the R~ using equations (56) and (64). Once the R i are known the moments can be calculated from:
~, R~(t)wi M . ~ 2R, ,=, i 1 Z x ~ '
(66)
and the distribution itself n(R~, t) can be found from (Williams, 19841:
n,R,,t),
=
no(Rolx,))exp(-f'° dR,'_ dt ~ ) .
,67)
In the convergent case S - 1 >0, G[R)< O, 24 Gaussian quadrature points and weights in equation (64) were used. The 24 equations (65) for the R~ were solved using a fourth order Runge-Kutta method. Figures 5 and 6 show the results of calculations with the small and large growth laws and the interpolation growth law of equation (33), starting with a log-normal size distribution. The initial saturation, S o = 1.1, is such that there is, in effect, a transient with rh~ ¢ 0 for the initial 1 or 2 s during which the mean radius rapidly increases. The moment calculations with large droplets then give poor agreement with exact calculations, although the large drop growth law treated exactly gives accurate results.
Radiative redistribution in a cloud
773
4.5
Smoll
/ 4.0
/
~+.~'~
"~ 'Exact"
~
ff
/
3.5
z-Large drop
exact
-0 ~''- O ~ 0 ~ 0
[
#m
drop
0
/
o/°
/
-- O-- O
0 ~ L ~ a r o e droo Lmaor grneen( ° P
0
3.0
I 25
0
I 50 t
I 75
100
s
Fig. 5. The mean,/~, as a function of time for an initial log-normal size distribution with R , = 3 l~m, ag = 1.5, N = 101° m - 3 c o n v e r g i n g to a final m o n o d i s p e r s e distribution. The results are for S O = 1.1 with small, large and 'exact' growth laws calculated by the moment and exact m e t h o d s .
1.5
\
\ \ 1
o
Large drop moment
-
o
IT- ~2/'2
\ ~-
E I-
/J
"~o
\ \
"~o~
,. -o
I0 L.
Large drop exact ~
~
~Smo[~
I 25
:2
drop
--0"""~
I_ 50 t
- - ' Exact' 0~0~-'~"~ ~
--1--
75
"
lOO
s
Fig. 6. As for Fig. 5, for the standard deviation
(j~2 1~2)1/2. _ _
After the transient S hardly changes, and liquid is redistributed among the droplets. For Rg = 3 #m, ag = 1.5, the final monodisperse radius is (R 3 )1/3 = 3.839 #m, so that the final equilibrium saturation satisfies:
S~o -- 1 "I- G([ R3 ] 1/3)
-- 0,
(68)
which gives Soo = 1.002917. Finally, some 'exact' calculations for the radiative cooling case of S < 1, G(R)>O were performed. In this case the change of variable equation (63) was not used, but instead the integral for N R 3 w a s evaluated by Simpson's rule for 99 points from Rmi n = 0.3 /~m to Rmax =30/~m. With these values the initial.distribution was integrated accurately to five significant figures, but the accuracy of calculated values of S deteriorated at later times.
TI4
J, ( i BAI~RI~'Trand C, IF. (].EMENT
The size distribution after 5 and 10 rain was calculated using equation (67), and previously calculated values of S used as input with linear interpolation for S(t) between these values. The results (Fig. 7) show a large increase in width of the distribution. Three radu are shown on the curves: R 1 is the radius of a droplet in the initial distribution that evaporates to the minimum radius shown. Smaller droplets in the initial distribution evaporated completely. The radius R2 is that for which a droplet is at its initial radius at tirnc ! ~hesc droplets grew at first but then evaporated back to their original radius. The radius R :~is tl~a~ for which S(t)-- 1 + G ( R 3 ) = 0 at time t. These droplets have grown to their maximum radiu~ R 3 and at later times evaporate. Integrating the trajectory equation for droplets of radius R2 gives: i" S .... I + G ( R t Q )
A i
......~
......
t69l
dr=O,
U0
The values of R2 arise from the cancellation of large positive and negative quantities, and are thus very sensitive to accurate values of S. This is also true of values near R2 (note that R1 and R: are very close together), but for radii greater than R 3 the integrand in equation (69) is always positive, so that data points should be fairly accurate, despite uncertainties in S. The closeness of R~ and R2 also explains the difficult), in evaluating S accurately; the first non-zero quadrature point at t = 10 min is Ro ~ 9.09 #m, so the volume less than this is poorly approximated. In general, for radiative cooling with S < 1 it is possible for S to rise or fall initially. A condition may be found by differentiating equation (56), and using equation (10); for 1~, when it is found that S increases initially if: j'~ [ ( S o - - 1 ) S o + G(Ro)Ro]n(Ro)dRo
<
~70)
0.
For G(Ro) given by the small or large growth laws this reduces to conditions on the moments of the distribution. For example, with So = 0.99 both the conditions are satisfied for log-normal distributions with ~g = 2 and Rg < 2.7/zm, but are both violated for R g > 4.5 pm.
..............
3
15
,4! I
\,=0 \
I °glon
15 5#rn
74~zm~'f ~ ~-~.
13~-
/
"/ ~2 i I
I,\ R~
\
3.
-.'~
't...J'\
L 1o
2 5.3/,zrn
\
\ I
R1
'R \
I
11 7,231xm
t=5min
"~.
\ \
t:lOmr ~\ 30
20
40
-~o
R ~m
Fig. 7. Aerosol size distributions, log n, against radius R at three times. The significance of the radii shown, RI, R2 and R 3, is explained in the text.
Radiative redistribution in a cloud
775
The case where large droplets grow at the expense of small ones, which would also apply with a different dividing line in the pure radiation field case of section 2.2, is clearly very interesting in that distributions grow much wider and some droplets evaporate completely. However, in comparison to other cases, accurate calculations are not easy to perform, particularly, as would be the case in practical applications, when other terms such as gravitational settling and coagulation are included in the aerosol equation (30). 4. C O N C L U S I O N S The behaviour of a cloud of droplets in a vapour-gas mixture in a radiation field has been examined. Illustrative calculations have been carried out for water droplets. The conclusions may be summarized as follows: (a) The net radiative heat transfer from a droplet of radius R is described in terms of the wavelength averaged absorption efficiency, ( Q a ( R ) ) . Exact and approximate expressions for this quantity have been given for water droplets, agreeing with previous results of Roach (1976). (b) The growth rate of a droplet is expressed in terms of a radiative quantity, G(R), obtained from ( Q a ( R ) ) and properties of the radiation field, and the supersaturation, S - 1, which represents the effect of heat and mass transfer to the surrounding vapour-gas medium. Because G(R) depends on R the net growth rate has the four possible regions shown in Fig. 1, in two of which there is mass redistribution between small and large droplets. (c) An alternative expression, equation (17), has been obtained for /~ in which the dependence on S - 1 is replaced by that on the overall growth rate. This has enabled equation (23) to be derived for growth in a pure radiation field. For water droplets, the case considered in detail, this could apply to a water cloud in a PWR containment. The supersaturation has been shown to be proportional to L e - 1, and some redistribution from the smallest water droplets then occurs in the radiative cooling case. (d) Some exact and approximate methods have been described for solution of the aerosol equation for water droplets with a growth term only. For redistribution on radiative heating, when the aerosol distribution tends to a common radius, simple moment equations give an adequate description of the process for the large droplet or exact growth laws except when there is an initial transient. (e) For moderate aerosol densities fast transients from saturations not close to unity have time scales of seconds or less (see also Clement, 1988a). Growth and redistribution will then occur for most of the time when the saturation is very close to unity with a value controlled by relatively slow heat transfer processes such as radiative transfer. Calculations have been performed for the case when there is a constant density of water mass and the temperature is also kept constant. For radiative cooling this implies a corresponding heat source and this case is also possibly applicable to a PWR containment. For radiative cooling, it has been shown that large growth is possible in the width of a water droplet size distribution in a time scale at least of the order of minutes, but that accurate numerical calculations to describe the size distribution are not easy to perform. An extensive exploration of the effects of the interaction between radiation and a water droplet cloud when the droplet radii are in the continuum regime has been performed. Unless the molecular sticking probability is small, this should be adequate for R > 1/~m. The case of smaller droplets is not expected to be important because most water clouds involve R > 1/~m and, in any case, the radiative interaction is relatively smaller for small droplets. However, the case can be explored on the lines of this paper by replacing R in the denominator in equation (10) for R by R +6, where 6 is temperature dependent and expected to be of the order of 0.2/~m, provided that the molecular sticking probability is not small (Barrett and Clement, 1988).
776
$ ( BARI~ i"T and C. F'. ('t.EMI~NT
F o r p r o b l e m s i n v o l v i n g h i g h t e m p e r a t u r e w a t e r mists, s u c h as in the P W R c o n t a i n m e r v . , it will be e s s e n t i a l to i n c l u d e t h e r a d i a t i v e i n t e r a c t i o n w i t h walls. T h i s is b e c a u s e o n l y a s m a i l i n t e r a c t i o n is n e c e s s a r y to i n d u c e net g r o w t h o r significant r e d i s t r i b u t i o n w i t h i n ~ ct¢)ud. T h e crucial q u a n t i t y to c a l c u l a t e is the s u p e r s a t u r a t i o n , S - - l . In ~he case (~i ~'.patiai d e p e n d e n c e o n e e q u a t i o n has b e e n q u o t e d [ e q u a t i o n ( 1 9 ) ] w h i c h c a n be used ik:;t th~s p u r p o s e . H o w e v e r , for a w e l l - m i x e d r e g i o n subject to i n p u t s of h e a t a n d wateE m~:~,,, i~ i~ p r e f e r a b l e 1o a d o p t the a p p r o a c h used ~'ecently in c o n n e c t i o n w i t h h y g r o s c o p i c ac~-osoi.,,. ( C l e m e n t , 1988) to o b t a i n an e q u a t i o n for the s u p e r s a t u r a t i o n . A d e s c r i p t i o n of h,,w t~ i n c l u d e the n e c e s s a r y r a d i a t i v e t e r m s will be g i v e n in a n e x t e n s i o n of that w o r k
This work was funded as part of the General Nuclear Safety Research Programme undertaken by AEA Technology. Acknowledgement
REFERENCES Barkstrom, B. R. (1978) J. Atmos. Sci. 35, 665. Barrett, J. C. (1985a) J. Phys. D 18, 753. Barrett, J. C. (1985b) The interaction of radiation with aerosol-vapour-gas mixtures. Thesis, Queen Mary College; London and Harwell Report AERE-TP.1171. Barrett, J. C. and Clement, C. F. (1985) The influence of radiation on aerosol formation by vapour condensation. Harwell report AERE-TP. 1120 and Proc. Int. Syrup Workshop on Particulate and Multiphase Processes and 16rh Annual Meeting of the Fine Particle Society, Miami Beach, April 1985. Proc. of Condensed Papers 35-12. Barrett, J. C. and Clement, C. F. (1986) The effect of sodium aerosols on total heat transfer and temperature in the cover gas space. Harwetl report AERE-TP.1159 and Specialists Meeting on Heat and Mass Transfer in the Cover Gas Space, Harwell 1985, IAEA IWGFR-57 2/1-1. Barrett, J. C. and Clement, C. F. (1987) The behaviour of water droplet size distributions interacting with radiation. Aerosols in Science, Medicine and Technology, GAeF Conference, Hannover, Sept. 1987. J. Aerosol Sci. 18, 647-9. Barrett, J. C. and Clement, C. F. (1988) J. Aerosol Sci. 19, 223. Barrett, J. C. and Fissan, H. J. (1989) J. Aerosol Sci. 20, 279. Clement, C. F. (1978) Proc. R. Soc. A 364, 107. Clement, C. F. (1985) Proc. R. Soc. A 398, 307. Clement, C. F. (1987) The supersaturation in vapour-gas mixtures condensing into aerosols. Harwell report AERE-TP.1223. Clement, C. F. (1988a) The formation of nuclear aerosols by evaporation-condensation processes. Harwell Report AERE-TP.1285. Clement, C. F. (1988b) Water condensation on hygroscopic aerosols. The 1988 European Aerosol ConJerence, Lund, Sweden. J. Aerosol Sci. 19, 875. Clement, C. F. and Taylor, A. J. (1986) Nuct. Engmj. and Design 92, 109. Guzzi, R. and Rizzi, R. (1980) Contr. Arm. Physics 53, 351. ()zisik, M. N. (1973) Radiative Transfer and Interaction with Conduction and Convection. John Wiley, New York. Roach, W. T. (1976) Q. J. K Meteorol. Soc. 102, 361 Truelove, J. S. (1984) Int. J. Heat Mass Transfer 27, 2085. Williams, M. M. R (1984) J. Colloid Interface Sci. 101, 19. Wobrock, W., Kramm, G. and Herbert, F. (1986) The influence of the aerosol distribution and composition on the formation and evolution of radiation fog. Aerosols: Formation and Reactivity. Proc. 2nd Int. Aerosol Conf, Berlin. Pergamon Press, Oxford