The Evaporation Coefficient

To determine the monomer escape frequency from an i-mer, assume that the t -mer is in equilibrium with the surrounding vapor. Then the t-mer vapor pressure equals the system vapor pressure. By (11.48), we obtain 

At equilibrium the escape frequency equals the collision frequency. Thus using (11.19) 

is thus a function of p, mu a, and v and is independent of the actual species partial pressure. Thus (11.62) holds at any value of the saturation ratio. 

We see from (11.64) that subcritical i-mers (/ / ) tend to evaporate since their evaporation frequency exceeds their collision frequency. Critical size i-mers (i = / ) are in equilibrium with the surrounding vapor (y, = p,). Supercritical clusters (/ f) grow since monomers tend to condense on them faster than monomers evaporate. 

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For the majority of metals, the evaporation coefficient is found to be unity, but, as mentioned before, the coefficient of many non-metallic elements with a complex vaporization mechanism such as... 

Finally, it is to be expected that the evaporation coefficient of a very stable compound, such as alumina, which has a large heat of sublimation resulting from the decomposition into the elements, will be low. Since the heat of evaporation must be drawn from the surface, in die case of a substance widr a low thermal conductivity such as an oxide, the resultant cooling of the surface may lead to a temperature gradient in and immediately below the surface. This will lower die evaporation rate compared to that which is calculated from the apparent, bulk, temperature of the evaporating sample as observed by optical pyromeuy, and thus lead to an apparently low free surface vaporization coefficient. This is probably die case in the evaporation of alumina in a vacuum. 

Offringa, J.C.A., de Kruif, C.G., Van Ekeren, P.J., Jacobs, M.H.G. (1983) Measurement of the evaporation coefficient and saturation vapor pressure of fraras -diphenylethene using a temperature-controlled vacuum quartz-crystal microbalance. J. Chem. Ther-modyn. 15, 681-690. 

General Mass Burning Considerations and the Evaporation Coefficient... 

In order to obtain the solution desired, a value of Ts is assumed, the vapor pressure of A is determined from tables, and mAs is calculated from Eq. (6.98). This value of mAs and the assumed value of Ts are inserted in Eq. (6.97). If this equation is satisfied, the correct Ts is chosen. If not, one must reiterate. When the correct value of Ts and mAs are found, BT or BM are determined for the given initial conditions Tx or mAco. For fuel combustion problems, mAcc is usually zero however, for evaporation, say of water, there is humidity in the atmosphere and this humidity must be represented as mAco. Once BT and BM are determined, the mass evaporation rate is determined from Eq. (6.87) for a fixed droplet size. It is, of course, much preferable to know the evaporation coefficient (5 from which the total evaporation time can be determined. Once B is known, the evaporation coefficient can be determined readily, as will be shown later. 

It is now possible to calculate the burning rate of a droplet under the quasisteady conditions outlined and to estimate, as well, the flame temperature and position however, the only means to estimate the burning time of an actual droplet is to calculate the evaporation coefficient for burning, f3. From the mass burning results obtained, f3 may be readily determined. For a liquid droplet, the relation... 

Note that this suggests that the term (a/HRT), the evaporation coefficient for dissolved gases moving from the liquid to the gas phase, is analogous to a, which reflects the efficiency of uptake across the inter-... 

The Temperature-Dependence of) the Evaporation Coefficient for Evaluating Cooling Towers... 

Wetting of BN. At equilibrium, the partial pressure of N2 for BN dissociation lies between the values for AIN and Si3N4 (Table 7.5). In practice, even in a high vacuum the stability of BN is much higher than that predicted by equilibrium thermodynamics because the evaporation coefficient is as low as 10-3 (Table 7.5). BN is also an oxidizable ceramic but boron oxide, B203, is volatile with a partial pressure at 1000°C Pb203 = 10 7 atm (Chase et al. 1985). Moreover,... 

It can be seen that, once an assumption is made for the value of M, the only quantity still unknown in the above equation is a, the evaporation coefficient, which must have a finite value equal to or less than 1. If it is assumed that a is constant but unknown, then the vapor pressure at any given temperature is proportional to the vaporization rate, and the enthalpy of vaporization may be found from the Clausius-Clapeyron type treatment. If a value is assigned to a, then vapor pressure values and the entropy of vaporization can be calculated as well. If the entropy of vaporization found in this way is a reasonable value, then the assumed value of a receives support. The latter procedure has been adopted here, and a value of unity has been taken for a. The reasons for choosing this value are ... 

The use of the term evaporation coefficient comes about from mass and heat transfer experiments without combustion—i.e., evaporation, as generally used in spray-drying and humidification. Basically, the evaporation coefficient p is defined by the following expression, which has been verified experimentally ... 

The effect of volatility in fractionating elements is due to the ratio of the saturation vapor pressures, but as shown by Equation (6), the relative masses of the gas species and possible differences in the evaporation coefficients also affect the degree of chemical fractionation produced by evaporation. When Equation (6) is used in connection with isotope fractionation, it is generally assumed that isotopes of the same element have the same evaporation coefficient and that the ratio of the saturation vapor pressures is equal to the isotopic ratio at the surface of the evaporating material (i.e., no equilibrium fractionation). This results in the following equation for the relative flux of the isotopes ... 

Hashimoto (1988, 1989) and Hashimoto et al. (1989) used a vacuum furnace to perform evaporation experiments on a number of simple oxides and found that evaporation coefficients (Equation (3)) ranged from 1 to 0.03. Experiments on simple oxides have also been done by Wang et al. (1994), who showed that the evaporation coefficient for molten FeO is 1, i.e., that there is no kinetic inhibition of evaporation, and that the iron isotopic fractionation factor, a, is that expected for evaporation of iron atoms. Similar experiments on other compositions have been done (e.g.. Young et al., 1998), but it is difficult to derive evaporation coefficients from the limited data given in these reports. 

Figure 9 Evaporation coefficients for magnesium and silicon for forsterite and CAI-like liquids. The evaporation coefficients depend strongly on temperature, but are not strongly dependent on evaporating composition or ambient...

Their partial pressure (J ) should be free of large error from Error does arise from uncertainties in the evaporation coefficients, ionization cross sections and total flux of species ( ). An extreme example of this error is the formation of o-AlgOg, which is biased by -9 kcal mol" if calculated from the reported ( ) pressures of Al and 0. Our assigned uncertainty includes contributions from this error and that of the Gibbs-energy function. 

Vaues of a may be very much less than unity and be temperature dependent. Somoijai and Lester [40] comment that "all the kinetic information is contained in the evaporation coefficient and its variation with conditions of vaporization", and they recommend the avoidance of the use of ot, in describing the rates of evaporation of solids under non-equilibrium conditions. The rate of sublimation is dependent on the attaimnent of sufficient energy by suitably disposed siuface molecules (possibly accompanied by electron or proton transfer in ionic solids). The overall rate of reactant removal is sensitive to the presence of impurities at the surface. The reverse reaction may be significant if the volatile material is not immediately removed from the vicinity of the reactant particles. Arrhenius parameters measured for sublimation processes may include a term which represents a temperature dependent concentration of surface intermediates [42]. The observation that measured evaporation rates are lower than those estimated from equilibrium vapour pressures suggests that the kinetics may be determined by a surface dissociation that precedes evaporation. This view is supported by evidence that, in selected systems, specific additives can considerably promote evaporation rates. For example [40], the evaporation rate of red phosphorous between 550 and 675 K was found to be increased by three orders of magnitude by the presence of thallium. 

The above equation expresses what should happen theoretically. The problem is that few molecules go directly from the evaporating surface to the condenser in a straight line and at a 90° angle. A correction factor, f, called the evaporative coefficient, developed by Burrows Molecular Distillation, Oxford Clarendon Press, 1960) considers three factors (1) the fraction of molecules reaching the condenser without collision, (2) the fraction that collides first and how many reach the condenser, and (3) those that have collided and reach the condenser by random motion. 

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