3 Drop Interaction in a Spray
Downloaded by NANYANG TECHNOLOGICAL UNIV on May 12, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/ba-1978-0166.ch003
A L L E N L. WILLIAMS—R. R. 1, Box 90, Edgewood, IL 62426 JOHN C. CARSTENS—Physics Department, Graduate Center for Cloud Physics Research, University of Missouri—Rolla, Rolla, MO 65401 JOSEPH T. ZUNG—Chemistry Department, University of Missouri—Rolla, Rolla, MO 65401
An analytic formalism describing the direct interaction between growing or evaporating drops in the continuum regime is developed. This formalism is modified for the case in which one of the drops is replaced by an inert sphere. Emphasis is placed on the exposition of appropriate boundary conditions and general solution. There is less than a 10% reduction in the growth/evaporation rate if drop separations exceed about 10 times the average radius. When gravity is the predominant force, relative fall velocities reduce the overall effect of the interaction so that only drops of equal radii suffer prolonged interaction. Thus, the brevity of this interaction between drops of different size tends to minimize any size distribution spreading.
The effects of drop interaction in a spray or cloud are conventionally taken into account by evaluating the influence of the drop assemblage on the overall temperature and vapor concentration or vapor pressure fields as if these were uniform and the drops were otherwise isolated. It follows that in this treatment, the drops interact solely by means of their mutual effect on the assumed uniform vapor and temperature fields. As drop concentration increases, one expects a direct interaction involving the localfieldsof closely spaced drops; such spatial interaction manifests itselffirstin terms of drop pairs, then triplets, and so on. In this chapter we develop formulas describing local vapor concentration and temperature fields for two juxtaposed drops in a gas-vapor mixture of infinite extent. Instantaneous growth or evaporation rates can be calculated straightforwardly from knowledge of the vapor concentration fields in the vicinity of the drop. Our primary and fundamental assumptions are that continuum transport theory is a valid approxima0-8412-0383-0/78/33-166-054$05.00/0
©
1978 American Chemical Society
Zung; Evaporation—Combustion of Fuels Advances in Chemistry; American Chemical Society: Washington, DC, 1978.
3.
WILLIAMS ET AL.
55
Drop Interaction in a Spray
tion, that the vapor is dilute with respect to the noncondensible carrier gas, that the calculated profiles are adequately represented by steadystate solutions to the relevant transport equations, and that, assuming a gravitational field, fall velocity can be decoupled from the transport processes herein contemplated. Further discussion as well as identification of secondary assumptions appears below.
Downloaded by NANYANG TECHNOLOGICAL UNIV on May 12, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/ba-1978-0166.ch003
Basic Equations The gas mixture under consideration is composed of an indifferent carrier gas and dilute condensate. The total molar concentration of the mixture is c, the binary diffusion coefficient is D, and the thermal conductivity is K. As we are considering an evaporation/growth process that is predominantly diffusion controlled, we adopt Fick's and Fourier s laws in the following form (1): (1)
N — -cDVx + x(N + N ) g
7 = -KVT
+ iVCp (T - Too)
(2)
in which N is the molar vaporfluxand N that of the carrier gas, x is the vapor mole fraction, c is the heatflux,T is the temperature ( t h a t at r = oo, i.e., ambient), and C is the molar specific heat at constant pressure. Our first approximation neglects both connective terms (far righthand terms). Two major contributions to the convective flux can be identified: the convective field resulting from drop fall and that resulting from the production (evaporation) or adsorption (condensation) of vapor at the surface of the drop. The former we neglect since drop sizes of interest have less than a 30-/* radius so that the effects of fall are likely to contribute only a few percent to the total flux (2). The latter contrig
p
bution affects only N so that with drop fall negelected (N = 0), i.e.: g
N= Now
-[cD/(l ln(l -
Vz = c£>Vln(l - x)
-x)]
x) — x -
(x /2) + 2
(3)
...,
and we regard the condensate dilute enough that: N=
-cDVx
Zung; Evaporation—Combustion of Fuels Advances in Chemistry; American Chemical Society: Washington, DC, 1978.
(4)
56
EVAPORATION-COMBUSTION OF FUELS
is satisfactory for our purposes. The vaporfluxis likewise neglected in Equation 2. (See for example Bird et al. (1)). It follows that: (5)
7= -KVT
Our next approximation considers the transport process as quasisteady state. This implies that the characteristic diffusion and conduction times, F/D and l cc /K (in which I is a characteristic length), are much smaller than the characteristic time appropriate for relative fall. A very conservative characteristic time would be that required for a drop to fall a distance of one diameter, i.e., by using Stake's law:
Downloaded by NANYANG TECHNOLOGICAL UNIV on May 12, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/ba-1978-0166.ch003
2
v
T - (9 / g) (1/a) Vg
(6)
Pl
in which a is the drop radius, pi is the liquid density (assumed to be much greater than that of the gas), g is the acceleration of gravity, and rj is the gas viscosity. Actually, we are more concerned with drops that are nearly the same size so that we can entertain rather larger values of T given by: g
r ' = (9/2)
( / g) (1/Aa) Vg
Pl
(7)
in which Aa is the difference in radii. With the above approximations, the equations to be solved are: Vz= 0
(8)
V r = 0
(9)
2
2
Solutions: Boundary Conditions
The two-sphere geometry suggests use of the bispherical coordinate grid. We adopt the notation of Morse and Feshbach (3) to whom we refer for details. The two independent spatial variables describing the axisymmetric geometry are /JL and rj. In particular fx± describes the surface of drop 1 and — p that of drop 2. (The minus sign associated with fi is carried along explicitly so that fx is a positive number.) The exterior solutions of Equations 8 and 9 are: 2
2
2
00
T — T + J '* £ e
K
1
{An exp [- (n + 1/2) p.]
n=0
+ B exp [(n + 1/2),»]} P (w) n
n
Zung; Evaporation—Combustion of Fuels Advances in Chemistry; American Chemical Society: Washington, DC, 1978.
(10)
3.
Drop Interaction in a Spray
WILLIAMS ET AL.
z — x« + f * 2
57
{Cn exp [-(n +1/2) /*]
1/
n =0
+ D exp[(rz + l/2) p]}P»(w)
(11)
n
in which
—fi
2
