Ind. Eng. Chem. Res. 1994,33, 411-416
411
Evaporation of a Droplet on a Surface W. Douglas Baines' and David F. James Department of Mechanical Engineering, University of Toronto, Toronto, Canada M5S 1A4
An equation is developed for the rate of evaporation of a liquid droplet sitting on or adhering to an impermeable wall bounding a gas flow. The equation applies to droplets which are small enough to be spherical segments and to lie within the linear portion of the velocity profile. A fundamental approach is taken, starting with the conservation equation for droplet vapor and applying the boundary layer approximation. For a two-dimensional droplet of small thickness, the analysis yields a solution which gives the Sherwood number, the dimensionless mass transfer rate, as a function of the Peclet number, the ratio of convection to diffusion transfer rates. Two approximations are made to apply the exact solution t o a droplet: the mass transfer rate is the same for all shapes and the mass transfer rate from a disk is equal t o that of a square of the same area. Predictions from the equation are compared with experimental measurements of droplet evaporation. There is broad but reasonable agreement. A better assessment of the equation cannot be made because of the scatter and discrepancies in the experimental data. Introduction The evaporation of a droplet on a surface exposed to a gas flow occurs by a combination of diffusion and convection. This type of evaporation arises routinely in ink-jet printing, in demisting operations, and in the many forms of spray coating, including agricultural spraying. In these situations, one generally wants to know the evaporation rate or the time for complete evaporation. If the droplets are hazardous because they are the result of an accidental release or explosion of a liquid chemical, the quantity of interest is generally the distribution of the vapor downwind. The downwind concentration levels depend, of course, on atmospheric dispersion, but fundamentally these depend on the rate of the evaporation at the source. Hence the rate of evaporation of a single droplet is still the primary quantity of interest. In the above situations, the basic physical problem is the same: a small droplet, generally about 2 mm in size, rests on or is attached to a surface exposed to the wind. Whether the surface is a leaf, a wall, or the ground, the surface is flat on the scale of the droplet. At this scale, the shape of the droplet is a spherical segment because of surface tension, with the particular segment shape being determined by the contact angle. Because the area of the surface is relatively large, the flow of the gas around the droplet is along the plane of the surface, independent of the orientation of the surface to the bulk or external flow. That is, even for the situation where the droplet is adhering to an inclined blade of grass, the local flow is a boundary layer flow and the situation is qualitatively the same as that for a droplet on the roof of a car. Figure 1is a definition sketch of the problem; the droplet adheres to a flat impermeable surface and is exposed to a shear flow. The shear rate is nearly constant on the scale of the droplet. That is, in a turbulent boundary layer, the velocity increases linearly with distance from the wall in the region closest to the wall, i.e., in the viscous sublayer. This sublayer is about 1mm thick in atmospheric boundary layers, according to Monin (1970). More concrete evidence about the thickness of the sublayer comes from the measurementsin the atmosphericboundary layer by Andre et al. (1978),who showed that the mean friction velocity at the surface is about 0.1 m/s; this magnitude translates to a viscous sublayer thickness of 1mm. Hence, the viscous sublayer thickness is comparable to the size of the droplet and the velocity field around the droplet is 0SSS-5~S5/9~/2633-0411$04.50/0
Figure 1. Definition sketch of an evaporating droplet in uniform shear.
therefore close to linear. In the present work, the profile is assumed to be linear and so the rate of evaporation depends only on the shear rate, i.e., on the velocity gradient at the wall. The dependence on this flow variable is in contrast to prior work in which the evaporation rate was assumed to be a function of velocity. The evaporation rate is certainly related to a characteristic velocity, but the rate depends much more directly on conditions at the wall, i.e., on the velocity gradient. Hence the shear rate is the appropriate flow variable in the present work. Prior work on this problem consists primarily of two experimental studies conducted in wind tunnels. Cooper et al. (1983) carried out a set of tests following a bifactorial design to identify which variables (of large number) most affect the mass transfer process. The experiment achieved the objective, but the data cannot be generalized and used for other fluids, surfaces, and wind conditions. The data, however, may find use in checking the accuracy of any predictive model and that use will raise later in this work. A more fully documented experiment was carried out by Coutant and Penski (1982). The flow conditions in their wind tunnel were better defined, and the range of air velocity included both laminar and turbulent flows. Rates of evaporation were measured using a microbalance with its weigh pan flush with the floor of the tunnel. A single droplet was placed on the pan, and its weight was monitored continuously for a fixed air flow rate. Measurements were gathered for five liquids, from which mass transfer rates were calculated. Coutant and Penski correlated these rates in an empirical equation containing the zero-flow evaporation rate, the Reynolds number, and the instantaneous mass. Their equation is not in terms of the normal mass transfer dimensionlessgroups such as the Sherwood number of Schmidt number and, as such, is not useful for design purposes. The authors correctly state that the zero-flow evaporationrate which is necessary 1994 American Chemical Society
412 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994
for their empirical equation can be found from the theoretical results of Picknett and Bexon (1977). However, their zero-flow data scatter widely about the theoretical values, and so the reliability of their measurements is uncertain. It is evident that the two experimental studies suffer from a lack of theoretical underpinning. The data are not presented in an appropriate dimensionless form, as mentioned, nor are theoretical solutions developed which could be compared with the data. The objective of the present work, therefore, is to find a solution of the governing equations and from it develop a model accurate enough for engineering applications.
Sh = F(8) (4) The function F(8) has been calculated by Picknett and Bexon (1977)when the droplet has the shape of aspherical segment. Their solution is a complex function which is approximated by the following relations:
Formulation of the Problem
On many common surfaces the segments are thin, in which case 8 is small and Sh = 0.32. Evaporation in Moving Air. A general solution of eq 1is difficult because analytical techniques are not available and a numerical approach is a massive undertaking for this inherently three-dimensional problem. As a consequence, the approach taken here is to develop an approximate solution of eq 1for larger Peclet numbers, and to use eqs 4 and 5 for small Peclet numbers. The solution for large Peclet numbers, set out below, is based on the use of boundary layer techniques for the analogous planar problem. By adapting the results from this two-dimensional solution, we obtain an approximate solution for the three-dimensional problem. Similarity Solution of the Planar Problem. An analytical solution of eq 1 is sought for the following conditions. The evaporative surface is assumed to be flat, so that u = KY everywhere in the flow field. The vapor concentration C(x,y) is COasy -, and the concentration is C, for y = 0, x 1 0. Conservation of mass is given by
The definition sketch of Figure 1illustrates the geometry of the droplet and the relevant parameters. The radius at the base of the spherical segment is R , and the contact angle is 8. The linear velocity field surrounding the spherical segment is given by the shear rate K , so that u = KY away from the droplet. The vapor concentration C is the mass of droplet vapor per unit mass of ambient gas. Far from the droplet the concentration is CO,and at the spherical surface it is C,, the saturation concentration at the droplet temperature. This temperature is equal to the ground temperature because heat transfer between the ground and liquid is much greater than that between the liquid and air. The governing equation for this problem is the steadystate transport equation V-VC = DV2C (1) where the V is the velocity field and D is the diffusivity of the vapor in the surrounding gas, which is air for most applications. This equation has been analyzed extensively for heat and mass transfer problems. Once a solution of eq 1is found, the mass transfer rate m is found from the gradient of C. For dimensional consistency, the Sherwood number, which is the dimensionless mass transfer rate, must have the following form in the present problem:
where p is the gas density (mass per unit volume), Re is the Reynolds number, and Sc is the Schmidt number. In this formulation, 8 is the parameter representing droplet shape. By definition, the Reynolds number involves a characteristic length and a characteristic velocity. In the present problem, the characteristic length is obviously the base diameter 2R, but there is not characteristic velocity because of the linear velocity field. However, since the velocity is given by u = KY, on appropriate characteristic velocity is ~ R K Hence . the Reynolds number for this problem is 4R2Klv, where v is the kinematic viscosity. Notwithstanding this choice for Re, the flow field in the present problem depends only weakly on v. If 8 is much less than 1,then the velocity distribution in the immediate vicinity of the droplet can be approximated as linear. With a fixed flow field, the fluid property Y does not enter the problem and this variable is eliminated by combining Re and Sc to simplify eq 2 to
Sh = f(e,Pe) (3) where Pe is the Peclet number 4KR2/D, Evaporation in Still Air. When there is no motion of the surrounding air, evaporation proceeds by diffusion only and then
F(e) = (0.63668 - 0.09598~- 0.06144@)/2sin e (0< 8 < 0.175) (5a)
+
+ 0.63338 o.ii6e2 - 0.088788~+ 0.0133#)/2 sin 8 (0.175 < 8 < T) (5b)
F(e) = (8.957 x iob
-
when d2C/dx is small compared to the transverse diffusion d2C/dy2. This approximation is equivalent to assuming that the concentration boundary layer thickness S is thin compared with distances in the streamwise direction. When the dimensionless concentration difference
c - co
J/=-
C, - CO is substituted i eq 6, the differential equation is unchanged and the boundary conditions become $(x,-) = 0, and $(x,O) = 1 for x 1 0. A similarity solution is sought by assuming J,=F ( 4 ,
q=u2 Xn
If n = 113and if u3= SKID,then eq 6 reduces to the ordinary differential equation
F" + q2Ft = 0 The solution for this equation is
(7)
where CI and Cp are constants. Inserting the boundary conditions, F(0) = 1 and F(-) = 0, yields F(7)= 1- 0.776ge-Pi3 d(
(8)
The mass transfer rate at the surface per unit area, m", is
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 413
m’’= -pD ac -(x,O) aY
-)
= 0.538pAC( &2
113
X
(9)
where p is the mass density of air and AC = C, - CO. This two-dimensional planar solution may be used to approximate the mass transfer rate when evaporation is from a disk or from the curved surface of a thin spherical segment. To apply the above theory to a disk, the disk is equated to a square of the same area. If the disk radius is R, then the side length L of the square is 1.773R. Integrating eq 9 over the square, the total mass transfer rate m is
m =L$oLm” dx
(ii)constant contact area-the radius R remains constant as the droplet becomes flatter; (iii) a mixture of the two cases-as time progresses, the mode changes from one to the other. In all cases the height and surface area decrease with time. This area variation is applied to the result from eq 10 as noted above. It is not known which pattern applies to which surfaces and liquids, nor is it known if these patterns apply when evaporation is by convection. However, it is evident that the two most important cases are (i) and (ii) and so the integrations to predict mass loss with time were carried out assuming constant 0 and constant R. In the case of constant contact angle, the result is the explicit relation
and in the case of constant contact area, the result is the integral relation
= 2.095pAC(~D~)’/~R~/~
Hence the Sherwood number is Sh = 0.105(Pe)1/3 (10) When evaporation takes place from the curved surface of a spherical segment, the right-hand side of eq 10 is multiplied by the correction factor A$rR2, where A, is the area of the spherical cap and rR2 is the base area. Equation 10 bears some similarity to the corresponding equation for mass transfer from a flat plate in laminar boundary layer flow. For a flat plate of length L, the corresponding relation is Sh = 0.66(ReL)1/sPe1/3
as given by equation 32-10 in Bennett and Myers (1982). This relation has a 113-power dependence on the Peclet number and a very weak dependence on the Reynolds number, which is similar to the relation derived above when the flow field is constant shear. Another check on the above work is that eq 10 is consistent with the results of Cain (1990),who solved eq 1 numerically for the problem of a two-dimensional droplet on a flat plate. In the next section, initial evaporation rates calculated from eq 10 will be compared with experimental values when convection dominates, for Pe >> 1. Also, still-air experimental data will be compared with the exact solution, as expressed by eq 4 and 5. Mass Loss with Time. In the two experimental studies, the mass of a droplet was measured as a function of time, and/or the times were recorded for complete and halfcomplete evaporation. Equation 10 can be used to predict these quantities, and comparison of these to the measured quantities will provide a further assessment of the accuracy of this engineering model. To follow the progress of evaporation with time, the mass of a spherical segment must be related to R and 8, and this relation is
where p’ is the mass density of the droplet liquid. To find mass as a function of time, eq 10 is integrated using the definition of eq 2 and substituting eq 11. The integration requires knowledge of R and 6 during evaporation. Three patterns have been observed in a laboratory study in which evaporation was by diffusion (Picknett and Bexon, 1977): (i) constant contact angle-the shape remains constant as the droplet shrinks;
where 0 is related to m (and A) through eq 11. In these equations the dimensionless mass and time are
and the subscripts 0 and f refer to initial and final conditions, respectively. Comparison with Experiments Equations 10, 12, and 13 were used to calculate mass transfer rates and evaporation times which will be compared to the experimental data of Coutant and Penski (1982) and Cooper et al. (1983). From a search of the literature, these two studies are the only ones for which comparisons can be made because initial droplet size and shape are sufficiently well documented in them. Data of Coutant and Penski (1982). In this study evaporation measurements were made for single droplets of five liquids: dimethylformadine, ethylbenzene, mesitylene, butyl alcohol, and water. For the first liquid, the measurements were made on Teflon and glass surfaces and, for the other four, on Teflon only. Each droplet weighed between 5 and 20 mg, which corresponds to an initial spherical diameter between 2.6 and 3.4 mm. Their wind tunnel had a square cross section 70 mm on a side and an inlet length sufficient to generate fully-developed flow. Since Coutant and Penski presented their data in dimensional form, for comparison purposes it was necessary to recast their data in terms of the Sherwood and Peclet numbers, as defined in eqs 2 and 3. The properties and parameters which were required to calculate these values of Sh and Pe were found in the following way: (i) The initial mass transfer rate, m, was determined from the experimental data which were given as droplet mass versus time. (ii) The density of air, p, was calculated using the ideal gas law and the given wind tunnel temperature. (iii) The saturation concentration, C,, for each liquid was given in the paper but, except for water, these values were not those listed in the CRC Handbook of Chemistry and Physics (1993) and thus Handbook values were used. (iv) The far-field concentration COwas taken to be zero, except when the liquid was water and then COwas found from the measured relative humidity.
414
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a
1.2 I
1
0.4
1
0.2
0.0 0
331
20
60
40
80
c
100
'
'
'
a00
8000
'2000
Pe
Pe
d
2.0 1 .a
I
16-
0'4iL-,..L 0 0.0 2
0
500
1000 1500 2000 2500 3000 Pe
4
1
0.2
: 800 1600 2400 3200
0.0 0
Pe
Figure 2. Sherwood number as a function of the Peclet number. The points are the data of Coutant and Penski (1982); the solid line is eq 10, and the dashed line is eq 4 for still air. (a) Water on Teflon. (b) Dimethylformadineon glass. (c) Dimethylformadine on Teflon. (d) The squares are for mesitylene, the circles are for ethylbenzene, and the triangles are for butyl alcohol, all on Teflon.
(v) The diffusivity, D, was approximate as ~12.6,which is the mean value of gases of comparable molecular weight. This method has to be adopted because D is not given in the CRC Handbook for the test liquids. The product pAC, used in this analysis and the values quoted by Coutant and Penske are, respectively, for cyanoethylformamide, 0.059 and 0.174 mg/(cm s); for ethylbenzene, 0.208 and 0.396 mg/(cm s); for mesitylene, 0.0616 and 0.0140 mgl (cm s); and for butyl alcohol, 0.084 and 0.025 mg/(cm s). (vi) The initial radius R was found from eq 5 using the initial mass of the droplet, the stated contact angle 0, and the liquid density p' found from the CRC Handbook. The values of I3 are crucial for determining evaporation times, but Coutant and Penski give few details about the measurements of I3 and thus the given values are thought to have a significant band. (vii) The wall shear rate K was found from the Reynolds number and from established flow equations (Schlichting, 1960). Since the flow rates covered both laminar and turbulent conditions, two formulas were needed. For laminar flow, the formula used was K = 8Rev/B2,where B is the side length of the square tunnel, and for turbulent flow, the formula was K = 0.040 Re7I4v/B2,which is based on Blasius friction law, f = 0.316/Re114. The Coutant and Penski data are plotted in terms of Sherwwd and Peclet numbers in Figure 2. The four plots are for different combinations of fluid and surface; each graph contains a plot of eq 10, indicating mass transfer by convection, as well as a short horizontal line, indicating F(I3) for pure diffusion.
In reviewing the four plots, it is evident that there is considerable scatter about the line given by eq 10. However, there is no evident pattern in the discrepancies because the data lie above and below the theoretical curves. A number of features should be noted, however: (1) In Figure 2b,c, there is considerable scatter for the three data points at Pe = 0, i.e., when there was no air flow in the tunnel. The scatter indicates the general difficulty in making mass transfer measurements because this simple static case should have yielded data which coincide with theory. (2) In Figure 2b,d, values for Sh do not increase with Pe at high Peclet numbers; i.e., the mass transfer rate does not increase with air speed, which is not physically credible. (3) The scatter in Figure 2c for Pe > 0 is considerable but it is consistent with the scatter at Pe = 0. The scatter is probably a reflection of the variation of 8 for individual droplets. (4) The contact angle I3 varies from 13 to 99 for the five liquids and two surfaces. Since eq 10 is based on a flat evaporative surface, with an arbitrary correction for surface area, one might expect agreement between the data and eq 10 to be best when I3 is small, i.e., when no correction is necessary. However, agreement seems to be unrelated to the value of 8. (5) C, is a difficult property to determine, as mentioned earlier, partly because it depends significantly on temperature. We calculated the error in Sh due to the largest
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 415
a
b
2.8
1
1.0,
0.9
1
" I 1.2
a 0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
0.0
0.2
0.4
0.6
0.8
1.0
1.2
I
t
d
t
C
?
I
m
00
01
02
-t
03
-
8
-
0.4
.
1 0.5
0.00
0.04
-
0.08
0.12
0.1 6
t
Figure 3. Dimensionless mass as a function of dimensionlesstime for evaporation of an individual droplet. The points are the data of Coutant and Penski (1982),the solid line is eq 13for constant R evaporation, and the dotted line is eq 12for constant-8 evaporation. The Peclet numbers and contact angles given for each graph are their initial values. (a) Water on Teflon, Peo = 75.3,80 = 99. (b)Dimethylformadine on Teflon, Peo = 162, 80 = 52. (c) Dimethylformadine on Teflon, Peo = 2833, 80 = 52. (d) Dimethylformadine on glass, Peo = 3182,Oo = 13.
possible error in temperature and found that this error cannot account for the discrepancies in Figure 2. (6) It should be noted that the magnitude of O.lO~iPel/~ is comparable to that of F(B), which demonstrates that the rate of evaporation in a high wind is not that much greater than that in still air. This is an unexpected result in view of the large values of the Peclet number, but the data support this finding. From the comparisons in Figure 2 one cannot judge whether eq 10 is inadequate or the measurements are inaccurate. There is no pattern in the discrepancies, so far as we can determine, and so it is not evident what needs to be done to improve eq 10. As for the measurements, it is evident that these are difficult to make because of the considerable scatter in the results when the data are plotted in dimensionless form. The scatter at Pe = 0 suggests that the error in the measurements is about 30%. The scatter for this condition should be minimal because the flow (or K ) is not a variable in this case; consequently, when Pe > 0 and K is a parameter, the scatter may be even larger. The 30% error should therefore be considered a minimum. Withsuch a wide error band, these data cannot provide a better assessment of the accuracy of eq 10. Further comparisons are in order, particularly with regard to mass loss with time. However, any comparison of this type is useful only if the initial conditions are the same, i.e., if the predicted evaporation rate agrees with the experimental rate at t = 0. Hence the mass loss comparisons are made only for those cases where the
experimental point lies on or close to the theoretical curve in one of the four graphs of Figure 2. Comparisons were carried out for all available cases and typical results are shown in Figure 3. Each graph shows the experimental data and two predictions, one assuming that the radius R is constant during evaporation and the other assuming that the contact angle B is constant. From the four plots, one concludes that the constant4 mode is appropriate for water, and that the constant-R mode is appropriate for the other fluids. The cases not presented here show the same behavior. Hence it appears that eq 10 is useful for predicting m(t)if the initial evaporation rate is known and if the evaporation mode is known. Data of Cooper, Edwards, and Hardaway (1983). The evaporation measurements by Cooper et al. (1983) were made with multiple droplets distributed on Teflon or aluminum plates in a rectangular duct (which.from photographs appears to be about 0.4 m wide and 0.2 m high). Two liquids were tested, dimethyl malonate (DMA) and methyl salicylate (MES). For both liquids, droplet weights and contact areas (spread factors) were measured for two droplet sizes, a t two temperatures, for two wind speeds, and at two times after deposit. There were small variations in the contact area for each liquid, except for MES on aluminum where large variations were found. In all cases, the mean contact area was calculated and from it the base radius R was found for substitution in the formulas. The fluid properties given in the paper were used for the calculations of Sherwood number and Peclet
416 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994
2 .' 2 0
7i '
t
'
0
3
'*t
0
.
n
0.4
0.0 02 0 102
6 1 o3
1 1 o4
1 o5
Pe
Figure 4. Ratio of the mass transfer rate calculated from eq 10 to the rate measured by Cooper et al. (1983), as a function of the Peclet number. Open symbols, dimethyl malonate; shaded symbols, methyl salicylate; circles, Teflon substrate; squares, aluminum substrate. Table 1. Comparisons to Measurements of Cooper et al. (1983)
droplet init contact air temp, air speed, r+J diam,mm anele. dee OC mls mmDimethyl Malonate, DMA Teflon 5 46 15.6 1.34 1.74 52 37.8 1.34 1.07 48 15.6 4.38 1.68 50 37.8 4.38 1.00 aluminum 5 12 15.6 1.34 1.87 14 37.8 1.34 1.19 13 15.6 4.38 1.90 15 37.8 4.38 1.37 Teflon 2.2 48 37.8 1.35 1.47 67 37.8 4.38 1.14 aluminum 2.2 20 37.8 4.38 1.11 Methyl Salicylate, MES Teflon 2.2 67 15.6 1.43 0.93 0.86 70 37.8 1.43 65 15.6 4.92 0.82 65 37.8 4.25 0.85 aluminum 2.2 24 15.6 1.43 1.08 24 37.8 1.43 0.83 24 15.6 4.92 0.92 24 37.8 4.47 0.83 surface ~
number, and the wall shear rate K was determined from the turbulent-flow formula given earlier in (vii). Comparison between measurements and eq 10 are presented in Table 1and Figure 4. The comparisons are in terms of the ratio of the calculated initial evaporation and the table rate to the measured rate, rbc~c/mmeas, supplies the experimental conditions. It is seen that the ratios are above 1.00 for the most part, which indicates that eq 10 generally overpredicts the evaporation rate. The table shows that the ratios for DMA at 15.6 O C are about 50 76 higher than those at 37.8 "C. Since temperature affects m only through material properties, this disparity suggests that material properties are not reliably known as functions of temperature for this fluid. It is apparent that agreement in the table between theory and experiment is good for MES. To determine if there is any other pattern of agreement in the table, the m ratios are plotted versus Peclet number in Figure 4, with different symbols to identify the two fluids and the two surfaces. The filled symbols for MES are close to the 1.00 line, as expected, but agreement does not improve the Peclet number, although one might anticipate otherwise because of the boundary-layer approach of the theory. Nor is agreement
better for the aluminum surface and its smaller contact angle, also as one might expect from the theory. This pattern of broad agreement is similar to that found earlier for the data of Coutant and Penski. Cooper et al. also measured the times for complete and half-complete evaporation. These were computed from eqs 1 2 and 13 for constant 6 and constant R, respectively. The comparisons are not presented here because, again, the scatter is large and without a pattern. The results do show, however, that the ratio of predicted to measured times is close to 1.0 for the constant4 mode and about 0.8 for the constant-R mode (both with standard deviations of about 0.4). Hence the results are consistent with those of Figure 3: in general, an evaporating droplet maintains its shape as it shrinks. Conclusions
The comparisons in the Figures and Table 1show that our model, eq 10, is useful in predicting droplet evaporation, but its accuracy is unclear. It predicts the evaporation rate well for some combinations of fluids and surfaces, but it overpredicts for others and by as much as 90%. The model has known limitations, but discrepancies with the data are not explained by these limitations; nor do the discrepancies suggest how else the model might be improved. It is evident from the organization of the experimental data that these data are not self-consistent and thus cannot be used to provide a more accurate assessment of the model. The lack of consistency in the data is probably not due to faulty measurement techniques, but more likely due to variations in the contact angle Bfor individual droplets. Acknowledgment
The authors gratefully acknowledgethe primary support of this work by the Defense Research Establishment Suffield in Alberta, as well as support by the National Science and Engineering Research Council of Canada. Literature Cited Andr6, J.-C.; De Moor, G.; Lacardre, P.; Therry, G.; du Vachet, R. J. Atmos. Sci. 1978, 35, 1861-1883. Bennett, C. 0.;Myers, J. E. Momentum, Heat and Mass Transfer; McGraw-Hill: New York, 1982; p 556. Cain, B. 'A 2-D Analysis of Evaporation in Laminar Flow"; Report No. 1093,Defence ResearchEstablishment Ottawa, 18pages, 1991. Carlslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Oxford University Press: Oxford, UK, 1959. Cooper, W.; Edwards, L.; Hardaway, F. 'Vapour/Liquid Hazards Associated with Persistent Liquid Drops on Non-Porous Surfaces"; ARCSL-TR-82092, Chemical System Laboratory, Aberdeen Proving Ground, Maryland, 1983. Coutant, R. W.; Penski, E. C. Ind. Eng. Chem. Fundam. 1982,21, 250-254. Additional data are on file with the American Chemical Society. CRC Handbook of Chemistry and Physics, 74th ed.; CRC Press: West Palm Beach, FL, 1993. Monin, A. S. The Atmospheric Boundary Layer. In Annu. Rev. Fluid Mech. 1970,2, 225-250. Picknett, R. G.; Bexon, R. Evaporation of Sessile Drops. J. Colloid Interface Sci. 1977, 61, 336-350. Schlichting, H. Turbulent Flow Through Pipes. In Boundary Layer Theory; McGraw-Hill: New York, 1960.
Received f o r review June 15, 1993 Revised manuscript received September 16, 1993 Accepted October 13, 1993' Abstract published in Advance ACS Abstracts, December 15, 1993. @
