turn is determined by the conductivity of the electrolyte and frequency of the bridge used. For example, for measurements using a 0.1-mm electrode in tap water, the frequency must be about 20 kHz. (d) Particles touching the electrode are not deformed upon impact. For liquid drops this is true if they move slower than 100 mm/sec and if their diameter is less than 2 mm; otherwise devices to slow down the velocity are necessary. For rigid particles there is no upper size limit other than the condition that the small electrode must be much smaller than the larger one. Spheres with a diameter of 15 mm were measured successfully with a 1-mm electrode. (e) The statistical evaluation may be used to calculate the mean drop diameter and its standard deviation provided that the latter is less than 40%. Nomenclature a = radius of dispersed phase particle a = meanradius b = distance between center of small electrode and that of particle k = ratio of exposed to total surface area of electrode N = parameter defined by 1 - p f / p p q = defined by eq 12c R = resistance
Ro = total resistance of cell in absence of dispersed phase AR = change in resistance of cell due to presence of dispersed phase r = radial distance measured from center of small electrode rl = radius of small electrode r2 = radius of large electrode X = dimensionless parameter b / a Y = dimensionless parameter AR/Ro Y = mean value of Y 2 = dimensionless parameter a/rl Greek Letters standard deviation of a uy = standard deviation of Y p f = specific resistance of continuous phase pp = specific resistance of dispersed phase cp = angle subtended a t electrode by that part of spherical shell occupied by particle Literature Cited ua =
Mlynek, Y., Resnick, W.,A.bCh.E. J., 18, 122 (1972). Nehls, R., Verfahrenstechnik, 5, 1 (1971). Pilhofer, T. H.. Miller, H. D., Chem. lng. Tech., 44, 295 (1972). Princen. L. H., Kwolek, W. F., Rev. Sci. Instrum., 36, 646 (1965). Towell, G. D., Strand. C. P., Ackermann. G. H., A./.Ch.E., 1. Chem. E. Symp. Ser., No. 10, (1965).
Received for review July 16,1973 Accepted April 11,1974
The Measurement of Evaporation Rates of Submicron Aerosol Droplets E. James Davis* and Edward Chorbajian Department 01 Chemical Engineering. Clarkson College of Technology. Potsdam. New York 13676
Evaporation rates of single submicron dioctyl phthalate aerosol droplets have been measured using laser light scattering from droplets suspended in a Millikan cell. Results at atmospheric pressure are in good agreement with the Maxwell equation for diffusion-controlled evaporation. The experimental method should be applicable to measure evaporation rates of very small droplets in the free-molecule and intermediate Knudsen aerosol regimes.
Recently instrumentation has been developed (Wyatt and Phillips, 1972a) which makes it possible to capture and hold a charged particle in an electric field and to measure the light-scattering characteristics of the droplet or solid particle. Wyatt and his colleagues have used the instrument they developed to study light scattering by bacteria (Berkman and Wyatt, 1970; Wyatt and Phillips, 1972b), by polystyrene latex (Wyatt, e t al., 1970) and by aerosol particles (Phillips and Wyatt, 1972). The technique, described more fully by Wyatt and Phillips (1972a), utilizes a modified Millikan cell together with laser optics to hold the charged specimen precisely in the path of the laser beam. The intensity of the scattered light is measured as a function of angle by means of a photomultiplier system that traverses the light scattering cell in a horizontal plane. We have adapted the Science Spectrum Corp. instrument to measure evaporation rates of single liquid aerosol particles at various pressures to examine the limits of applicability of continuum theory and to test theoretical, semitheoretical, and empirical analyses of evaporation in the Knudsen aerosol regime. This paper is concerned with 272
Ind. Eng. Chem., Fundarn., Vol. 13, No. 3, 1974
the experimental technique and its application to the study of the evaporation of submicron aerosol particles a t atmospheric pressure. Evaporation Theory Maxwell (1879) considered the simplest case of droplet evaporation, that is, isothermal, quasi-stationary, quasisteady-state evaporation of a spherical droplet into an infinite medium of constant vapor concentration C,. In that case the evaporation is controlled by the rate of diffusion of the evaporating species in the gaseous medium, and the evaporation rate per unit area is given by eq 1, which has been attributed variously to Stefan, Maxwell, and Langmuir . j , = D-(Co AB
- c,>
a
Fuchs (1959) modified the quasi-steady-state theory of Maxwell to take into account the transient period associated with the sudden introduction of a droplet into an unsaturated medium. The resulting equation for the mass flux is
Fuchs’ analysis is not valid for rapidly evaporating droplets nor for droplets with a n initial temperature significantly different than the surrounding gas temperature because the interfacial concentration CO is a function of the interfacial temperature, and the interfacial temperature can be expected to vary during the transient period. Recently, Chang and Davis (1974) have solved the unsteadystate problem taking into account the temperature variations inside the droplet and temperature and concentration variations outside the droplet together with the evaporation process. For liquids having a relatively high vapor pressure a t the interfacial temperature and a large heat of vaporization, the rigorous analysis and eq 2 show significant disagreement. However, for low vapor pressure species such as dibutyl phthalate, a material which has been widely used in aerosol studies, the evaporation rate is so slow that the transient period is extremely short, and the effects of the nonisothermal evaporation are not significant. In the latter case eq 1 and 2 are good approximations to the rigorous analysis. We shall refer to this point again in the discussion of the experimental results. Equations 1 and 2 and the analysis of Chang and Davis apply to diffusion-controlled evaporation in the continuum regime. When the mean free path X of the escaping molecules is of the same order as the droplet radius a, that is, when the Knudsen number Kn = X/a is of order one, continuum theory is invalid in the vicinity of the particle surface, and it is necessary to take into account the molecular nature of the evaporation process. As Hidy and Brock (1970) have extensively discussed the theoretical problems associated with Knudsen aerosols (Kn = 0 (1)) and evaporation in the free molecule regime (Kn >> I), we need not review that literature here. Recently, Fuchs and Sutugin (1971) developed an interpolation formula for the mass flux which is in reasonable agreement with free molecule theory for Kn >> 1 and converges on the Maxwell equation for Kn
