Mechanism of desolvation of sample droplets in flame spectrometry

In the flame, the droplets are desolvated, i.e. dried to form tiny solid particles which can then liberate free atoms by vaporiza- tion and dissociati...
0 downloads 0 Views 959KB Size
Download PDF
Investigation into the Mechanism of Desolvation of Sample Droplets in Flame Spectrometry N. C. Clampitt and G. M. Hieftje’ Department of Chemistry, Indiana University, Bloomington, Ind. 47401 A discrete droplet generator is used to measure the rates of desolvation of isolated droplets of water, methanol, ethanol, isopropanol, carbon tetrachloride, and cyclohexane in an air-acetylene flame. A comparison of the measured rates to those predicted by existing theories indicates that droplet desolvation in an analytical flame is a thermally controlled process, which depends on the thermal properties and composition both of the droplet and of the flame gases. Through use of statistical thermodynamic considerations, it is shown that the necessary parameters can be calculated for use in predicting droplet desolvation rates under specified conditions. The model which has been developed for this prediction portrays droplet evaporation to be controlled by conduction of heat to the droplet surface through the flame and through a film of solvent vapor surrounding the droplet. Errors in this model and in the resulting calculated desolvation rates are considered, as are practical consequences of application of the model to desolvation in analytical flames, in spray chambers, and in other thermal sources. IN FLAME EMISSION, absorption, and fluorescence spectrometric methods of analysis, a number of important steps must take place to convert a portion of the sample existing in the form of a solution to the free atoms which take part in the absorption or emission of radiation. Generally, a sample solution is nebulized and the resulting droplet spray sent into the flame. In the flame, the droplets are desolvated, i.e. dried to form tiny solid particles which can then liberate free atoms by vaporization and dissociation. Because each of these steps has a very pronounced effect upon the quantitative results obtained in the analysis, it is obviously advantageous to thoroughly characterize each step so that the appropriate experimental variables can be intelligently optimized. This characterization is most readily obtained when each step can be separated and studied independently from the others. Until recently, temporal and spatial separation of the individual steps in the flame was not possible because of the difficulty involved in controlling and observing the events which occur in a typical nebulizer-burner system. By employing a unique sample introduction device, however, it is now possible to introduce isolated droplets of sample solution into a flame and to resolve in time and space the events which transpire between introduction of each droplet into the flame and the atomization of the species to be analyzed ( I ) . With this new technique, it has been possible to make a detailed investigation of the desolvation of droplets as they move through the flame. Previous studies using this isolated droplet technique ( I ) have served to develop the methods for obtaining experimental values for the rate of desolvation of droplets of a number of solvents commonly used in flame spectrometry. Comparison of these experimental rates to those calculated from a pro-

’ Author to whom correspondence should be sent. (1) G. M. Hieftje and H. V. Malmstadt, ANAL.CHEM., 40, 1860 (1968).

posed theory (2) unfortunately showed only qualitative agreement. In those studies, the agreement was suggested to be limited by a lack of reliable values for thermal parameters which were required in the theoretical calculations. Therefore, irrefutable evidence for the proposed desolvation modei could not be presented. Although no other experimental investigation of droplet desolvation in a flame has yet been made, several articles have been written concerning the evaporation as it applies to the combustion of fuel drops under various conditions (3-6). Although good agreement has been reported between theoretical models of combustion and experimental observations, consistency among the various studies is lacking. Other workers have compared experimental and theoretical rates of desolvation and combustion and have found that they could obtain better agreement and consistency by adding arbitrary constants to “corrected” theoretical expressions for the desolvation rate (7). However, no theory has been proved satisfactory when applied to the desolvation of droplets under the conditions encountered in flame spectrometry. In this report a coherent mechanism for desolvation is proposed whicli is derived from both experimental and theoretical considerations. To enable the theoretical prediction of desolvation rates, a method will be described for the calculation of high temperature thermal parameters which are required for examination of the desolvation mechanism of flame spectrometric solvent droplets. Using these relationships, an extended model for the desolvation processes will be developed which provides excellent correlation between theoretical and experimental rates of desolvation. Assumptions and limitations of the theory will be discussed along with its practical implications to conventional routine flame spectrometry.

DROPLET DESOLVATION IN A FLAME Experimentally, the desolvation rate is found to be a linear function of the time spent by the droplet in the flame (1). This function can be expressed by the relationship : D 2 = Do2 - kt

(1)

where D is the droplet diameter at any time t after its entry into the flame, D o is the initial droplet diameter, and k is the rate of desolvation. Existing theories have attempted to relate the rate of desolvation ( k ) to various properties of the droplet, the liquid, and the flame ( 4 , 5 , 8 , 9 ) . The most prom(2) F. A. Williams in “Eighth Symposium on Combustion,” Williams and Wilkens, Baltimore, Md., 1962, p 50. (3) G. S. Bahn in “Literature of the Combustion of Petroleum,” Adt:an. Chem. Ser., 20, 104 (1958). (4) E. G. Masdin, Fuel SOC.J., 12, 17 (1961). ( 5 ) T. W. Hoffman and W. H. Garvin, Can. J. Chem. Enn., - 38, 129 (1960).

(6) C. E. Polymeropoulos and R. L. Peskin, Combust. Flame, 13, 166 (1969). (7) R. H. Sioui and L. H. S. Roblee, Jr., ibid., p 447.

( 8 ) C. L. Pritchard and S. K. Bismas, Indian Chem. Eng., 8, 93 (1966). (9) E. F. M. van der Held, Z . Phyzik, 77,459 (1932). ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

1211

F l a m e Gases

Burning F

eters for most solvents have been reported only for temperatures ranging from near room temperature to several hundred degrees Centigrade. Because the calculation of the desolvation rate must be made at flame temperatures of 2350 OK and above, these low-temperature values for heat capacity ,and thermal conductivity obtained from the literature are of little value. Some workers ( I , 3) have attempted to estimate the true values of the parameters but were not totally successful. Various methods or operations have also been employed (3, 11) to extrapolate low temperature values to obtain values appropriate for flame temperatures. Because most extrapolation techniques provide very poor approximations at such high temperatures, these methods have met with only partial success. In order to solve this problem and to critically compare a desolvation model with experimental observations, we have applied statistical thermodynamic considerations to the evaluation of the heat capacity and thermal conductivity of solvent vapors at various representative flame temperatures. Because these considerations are expected to be of considerable importance in extending this work and in application to other solvents, they will be discussed in some detail.

:er

u Diffurlon M i xture

Figure 1. Schematic representation of the processes involved in heat-transfer controlled desolvation of a droplet in an analytical flame T , = Temperature at the droplet surface T, = Flame temperature XI = A,, = Xu =

Thermal conductivity of flame gases Thermal conductivity of diffusion mixture Thermal conductivity of solvent vapor

THEORETICAL DETERMINATION OF THERMAL PARAMETERS

ising theory as applied to the flame portrays the droplet to be surrounded by a thin film of solvent vapor which spreads by diffusion away from the droplet (2). This model, portrayed in Figure 1, predicts that the rate of desolvation of a droplet in a flame is limited by the rate at which heat can be conducted from the flame to the droplet surface. Because heat conduction through the vaporized solvent surrounding the droplet is much slower than the mass transfer process of the vapor leaving the droplet, the diffusion of the vapor away from the droplet is not predicted to affect desolvation. An expression for this heat-transfer controlled evaporation is given by the equation :

8XM k = __ l n ( l B ) CPPl where X is the thermal conductivity of the vapor surrounding the drop in cal/sec cm OK, C, is the heat capacity of the vapor at constant pressure in cal/mole-OK, p1 is the density of the liquid in g/cm3, M is the molecular weight of the solvent, and B is the transfer number (IO). The transfer number can be expressed as :

+

(3)

where Tq is the temperature of the flame in OK, T, is the temperature of the droplet surface (which can be assumed to be at the boiling point of the liquid), H , is the heat of vaporization of the solvent in cal/mole, y is the ratio of the amount of oxygen present to that needed for stoichiometric combustion (see Figure l), and Q is the heat of combustion (in cal/mol) for flammable solvents. Q is taken to be zero for noncombustible liquids such as carbon tetrachloride and water. A problem which is encountered in applying this theory to an actual calculation of the desolvation rate for a droplet in a flame arises from the fact that, for a given solvent vapor, the heat capacity and thermal conductivity are functions of temperature. Experimentally determined values of these param(10) D. B. Spalding in “Fourth Symposium on Combustion,” Williams and Wilkens, Baltimore, Md., 1953, p 847. 1212

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

Heat Capacity. Methods of statistical mechanics have dealt in some detail with the problem of calculating the heat capacity and thermal conductivity of vapors as a function of temperature. From the equipartition of energy, the heat capacity at constant volume can be expressed by the following equation (12):

where R is the gas constant, k is the Boltzmann constant, h is Planck’s constant, N is the number of atoms in the molecule of interest, and T i s the absolute temperature. This equation shows the heat capacity of a molecule to arise from its ability to store energy in various excited states. At flame temperatures and for the solvents used in this study, of course, electronic excitation will not contribute significantly to this ability. Also, it can be confidently assumed that the translational and rotational energy of the molecule can each be approximated by a value of 3/2 kT to give a total of 3 kT. Because heat capacity is the change in energy storage which occurs with a change in temperature, Equation 4 reflects the first derivative of the energy with temperature. This produces a total rotational and translational contribution to energy storage of 3 k per molecule or 3 R per mole of the vapor under study. Determination of the vibrational contribution to the total energy of the molecule requires a summation over the fundamental vibrational frequencies of the molecule as indicated by Equation 4. Theoretically, a nonlinear polyatomic molecule should have 3N - 6 fundamental frequencies. However, for many molecules, not all of the vibrational energy levels are populated sufficiently at reasonable temperatures to contribute significantly to the heat capacity. Therefore little error is introduced by omitting these from the calculation of C,. Because the desolvation model (Equation 2 ) requires knowledge of the heat capacity at constant pressure (Cp), (11) K. Annamalai, V. K. Rao, and A. V. Sreenath, Combust. Flume, 16, 287 (1971). (12) T. L. Hill, “An Introduction to Statistical Thermodynamics,” Addison-Wesley Publishing Co., Reading, Mass., 1960, Chap. 9.

Temp,

X,,,(cal/cm 0.621 x 0.853 X 1.109 X 1.367 x 1.505 x 1 ,625 x 1.744 x 1.864 X

O K

400 500

600 700 750 800 850

900 a

% error

=

Mor XZ)

Table I. Calculated and Theoretical Values for the Thermal Conductivity of H20as a Function of Temperature XI = calculated value using Eucken correction (19) - see text hz = calculated value using “ffactor” (20) - see text XZ (cal/cm. sec OK) XI (cal/cm sec O K ) % errora in XI sec OK) 0.641 X lo-‘ 0.814 X 31.1 lo-‘ 21.9 0.838 X 1.04 x 10-4 16.2 1.06 x 10-4 1.29 x 10-4 13.4 1.29 x 10-4 1.55 x 10-4 11.6 1.41 x 10-4 1.68 x 10-4 10-4 12.0 1.54 x 10-4 10-4 1.82 x 10-4 11.8 1.67 x 10-4 1.95 x 10-4 10-4 12.7 1.81 x 10-4 2.10 x 10-4

errora in X P 1.6 -1.8 -4.4 -5.6 -6.3 -5.2 -4.2 -2.9

- A,,

Xexp

rather that at constant volume (C,) it is necessary to use the following expression (13) to modify Equation 4:

tivity and viscosity (?) given by (18):

which is, in turn, equivalent to the expression Here Bz* is the second derivative of the first virial coefficient ( 1 3 ) and can be found tabulated as a function of the reduced temperature T* (14). In turn, the reduced temperature is given by T*

=

kTjt

(6)

where T is the flame temperature and B is a force constant giving the effective attraction potential between two solvent molecules (15). In Equation 5 , V* is the reduced molar volume which can be found from the molar volume at the flame temperature:

v* =

V/b,

(7)

The factor bo accounts for the fact that the molecules being studied are not points of zero volume but can be approximated by rigid spheres of effective diameter a (15). To determine b,, listed values of u (see Table 11) can be employed in the following equation

bo

=

z//3

aNa3

(8)

where Nis Avogadro’s number and a 3is in liters. Thermal Conductivity. The thermal conductivity of vapors can also be calculated using statistical mechanics (16,17). The final, reduced relationship used in this investigation is ,-

xx

107 =

66.737 C u v T / M (9 CPICU u2i12’z*(T*)

- 5)

(9)

where u is given in A, M is the molecular weight in g/mole, C, and C , are in cal/mole O K and Q 2 ~ **(T*)is a dimensionless collision integral to be described later (18). This equation was developed from the relationship between thermal conduc(13) J. 0. Hirschfelder, C. 0. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N.Y., 1954, p 231. (14) Ibid., p 1114. (15) Ibid., p 1110. (16) Ibid., p 534. (17) L. Monchick and E. A. Mason, J. Chern. Phys., 35, 1676 (1961). (18) J. 0.Hirschfelder, C. 0. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N.Y., 1954, p 526.

__

where m is the mass of a solvent molecule. Qualitatively, Equations 10 and 11 relate the thermal conductivity of a vapor to the successive storage and transfer of thermal energy from molecule to molecule as the heat travels through the vapor. Thestorage of energy is, of course, specifically related to the heat capacity ( C , ) (cf:Equation 10). The collisions between molecules which control energy transfer, however, are related to the mean velocity of a molecule ( d 8 k T l a r n ) and to its collisional cross section (uP)>.The collision integral ( Q 2 ~ 2 * ) , seen also in Equation 9, accounts for the fact that for real molecules in free flight, an attraction exists which can promote energy transfer beyond the assumed limits defined by the collisional cross section. To obtain Equation 9 from Equation 11, we have applied the Eucken correction (19) which is more exact than Equation 10:

This correction is intended to account for the internal energy that can be stored by polyatomic molecules in addition to the translational, rotational, vibrational, and electronic degrees of freedom. This internal energy becomes an important factor in determining the thermal conductivity during inelastic collisions between molecules. Some difficulty arises in applying this simplified theory to polar molecules because of the attractive forces which arise from their dipole moments. For these molecules, special tables have been prepared (17) which list values for E , u, and Qz$2. It was apparent that this more rigorous treatment significantly improved the calculation of thermal conductivity for polar molecules. To confirm this, calculated values of thermal conductivity for water vapor at elevated temperatures were compared to those experimental values which were available. The agreement between experimental ),A,(, and calculated results (Al), shown in Table I, is seen to improve at higher temperatures because of the greater dilution and enhanced (19) Ibid., p 498. ANALYTICAL CHEMISTRY, VOL. 44,

NO. 7, JUNE 1972

1213

Flome Temporature

( O K )

Figure 2. Calculated variation in heat capacity with absolute temperature for cyclohexane vapor “ideality” of the vapor. However, a 10% error still exists at 900 “C because of the difficulty involved in accurately predicting the effect of polarity on the transfer of heat. As a further improvement, another theory (20) has been proposed for calculating the thermal conductivity of polar vapors. This method replaces the Eucken correction with an “f factor” which results in the following relationship for the thermal conductivity :

A = f - vc, M The theoretical basis for calculating the “ffactor” (20) lies in molecular parameters such as the self-diffusion coefficient and approximations for factors related to inelastic collisions. Because of the complexity of the theoretial calculation, a tabulated value for fwas used in Equation 13 to calculate the thermal conductivity for water in the temperature range from 400 to 900 OK. The results of calculations for thermal conductivity using thisffactor (A,) are listed in Table I for comparison with those obtained without this correction (A,). The improvement observed using this correction cannot, unfortunately, be applied to all solvents used in this study because of the unavailability of suitable “f factors.” Therefore, for consistency, this correction was not applied to any of the calculations pertaining to the desolvation rate. Because conditions at the flame temperature (2350 OK) are expected to be more ideal than those found at the lower temperatures listed in Table I, the compromise is expected to provide satisfactory results and, as observed in Table I, can be expected to cause an

(20) E. A. Mason andL. Monchick, J . Cltem. Phys., 36,1622 (1962). 1214

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

error of less than 10% for water, which is the most polar of the solvents studied. Errors in Calculated Values. The total error involved in calculating the heat capacity is expected to be small at elevated temperatures such as those found in a flame. From Equation 4, it can be seen that C,, and therefore, C , (cf. Equation 5 ) will approach a limiting value at high temperatures. This can be observed in Figure 2, where the calculated variation in heat capacity with temperature is plotted for cyclohexane vapor. It is seen that at flame temperatures (i,e., above 2000 OK), the heat capacity changes little with temperature so that errors from this source are expected to be small. In addition, the greater kinetic energy of the vapor molecules above 2000 OK will cause the vapor to behave in a manner closer to that expected for an ideal gas, thereby enhancing the validity of the calculations. Unlike heat capacity, the thermal conductivity of cyclohexane vapor does not approach a constant value but increases almost linearly with temperature as shown in Figure 3. The behavior in Figure 3, calculated from Equation 9 for cyclohexane vapor, is typical of that found for all solvents studied. According to the relationship predicting the rate of desolvation (Equation 2), this will consequently produce an almost equivalent change in the calculated rate. Therefore, flame temperature and variations thereof will be very significant in determining the predicted rate and the accuracy of the value obtained. To minimize this problem, a flame of constant temperature and fixed composition was employed throughout this study. Also, the greater ideality of the vapor at the flame temperature should provide increased accuracy for thermal conductivity calculations just as for those of the heat capacity, even though the Eucken correction is used in Equation 2 instead of the more complex “f factor.” EXPERIMENTAL

Desolvation rates for the various solvents were measured using techniques and apparatus similar to those employed earlier ( I ) . In these measurements, isolated droplets of the selected solvent were sent into a stable air-acetylene flame provided with an inert nitrogen sheath. Their size was then measured as a function of the residence time in the flame by collecting impressions of the droplets in a thick bed of magnesium oxide. Through measurement and calibration of the impressions, information was obtained on the droplet size at a point in space and time within the flame. A sequence of impressions taken at various points in the flame then enabled calculation of the desolvation rate. To find the experimental desolvation rate, the square of the diameter of the droplet was plotted cs. the time that had elapsed between the point of entry of the droplet into the flame and the point where the diameter was measured. The slope of the line obtained from the graph was then determined by a first order least squares analysis of the data points. Initial droplet sizes used here ranged between 40 and 80 micrometers. The overall accuracy of the desolvation rate measurement is estimated to be = t 7 Z . A typical desolvation plot is shown in Figure 4. In this study, experimental rates of desolvation were measured for water, methanol, isopropanol, and cyclohexane. For water and methanol, excellent agreement was obtained between these values and those which were measured in an earlier study ( I ) . For this reason, the desolvation rates for carbon tetrachloride and ethanol from the previous work ( I ) were not remeasured but were compared directly to theoretical values in this investigation. These six solvents were chosen for this study to encompass a broad range of characteristics, including polarity, molecular shape, molecular weight, and flammability. All solvents were of analytical reagent grade.

For use in desolvation rate calculations, the air-acetylene flame temperature was measured at several vertical locations using the sodium line reversal technique (21). Because the flame temperature varied significantly with composition, fixed gas flows were chosen to provide optimum stability. At gas flows of 1.75 lpm CzHzand 12.7 lpm air, corrected to STP, the flame temperature was 2380 OK with a variation of 1 2 0 OK over the vertical region of droplet size measurement. To enhance stability and flame temperature uniformity, a flowing nitrogen sheath, supplied at 1.9 lpm, was used to surround the flame during all measurements. Purified gases (Matheson Co., Joliet, Ill.) were supplied from single or double stage regulators (Matheson Co., Joliet, Ill.) and controlled by needle valves (type B4M, Nuclear Products Co., Cleveland, Ohio). Gas flows were monitored using dual-float rotameters (type 592, F. W. Dwyer Mfg. Co., Michigan City, Ind.) which had been calibrated with a precision wet test meter (Precision Scientific Co., Chicago, 111). To facilitate the theoretical calculation of the desolvation rate, a computer program was written in Fortran IV for use in an XDS Sigma 2 computer. All available and tabulated and B2* were read into memory for maximum values for computation flexibility. This program is available upon request from the authors. Q 2 m 2

L

1000

2000 FlOmr

A MODIFIED THEORY OF DESOLVATION

To develop a coherent model for the desolvation of droplets in a flame, measured rates of desolvation will be compared to rates calculated using Equations 2, 3, 4, 5, and 9. Resulting discrepancies will then be used to develop a modified desolvation model which can be used to accurately predict desolvation rates for solvents not yet studied. Calculation of Theoretical Desolvation Rates. In order to calculate the rates of desolvation, the temperature profile of the droplet and of the gases surrounding it must be defined in detail. No advantage is gained by being able to calculate the values of the various parameters as a function of temperature if the correct temperature cannot be determined. The temperature of the droplet surface, of course, can reasonably be expected to be at the boiling point of the liquid at one atmosphere of pressure. Therefore, the values for liquid density and heat of vaporization can be evaluated at this temperature and can be found conveniently tabulated for all common solvents (22). However, because of the large temperature differential which exists between the droplet surface and the flame (cf. Figure l ) , a question arises concerning the temperature at which the heat capacity and thermal conductivity should be evaluated. In order to obtain “effective” values for these parameters which are applicable over an appreciable distance from the droplet surface, it is apparent that the temperature gradient between the surface and the flame must be known. From the variation in X and C , with temperature (Figures 2 and 3), the effective parameters can then be calculated. A number of workers have considered the temperature gradient which surrounds a burning fuel droplet ( 7 , I I , 23) and have made proposals ranging from a linear increase in temperature with distance from the droplet center (23) to an extremely nonlinear gradient (11). One approach has been to (21) W. Snelleman in “Flame Emission and Atomic Absorption Spectrometry, vol. I-Theory,” J. A. Dean and T. C. Rains, Ed., Marcel Dekker, New York, N.Y., 1969, p 213. (22) “Handbook of Chemistry and Physics,” 50th ed., R. C. Weast, Ed., Chemical Rubber Publishing Co., Cleveland, Ohio, 1970. (23) S. Kotake and T. Okazaki, Int. J. Heat Mass Transfer, 12, 595 (1969).

401

3000

Temprraturo

(*K)

Figure 3. Variation in thermal conductivity of cyclohexane vapor with absolute temperature, calculated using Equation 9

1.8

-

%

-

-

16-

1.4

-

E

m

12-

0 x N

10-

L

W

W

E 080

04 0.6

02-

I 2.0

3.0 TI me

4.0

!3

( Milliseconds)

Figure 4. Typical desolvation plot showing rate of decrease of the diameter* of methanol droplets in a nitrogen-shielded air-acetylene flame then calculate values for the heat capacity and thermal conductivity which are the arithmetic means of the values calculated at the flame temperature and those obtained at the liquid surface temperature (3,24). Another method suggests a single evaluation of C, and X at the mean temperature (3) Because neither of these methods has proved satisfactory, an attempt has been made to improve them by accounting for inhomogeneity of the vapor surrounding the droplet (3, 11). By using elaborate calculations, the composition of the vapor (24) F. A. Williams, Combust. Flame, 3, 529 (1959). ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

1215

Table 11. Values for Physical Parameters Used in Equations 5 and 9 for Calculation of Heat Capacity and Thermal Conductivity of Solvent Vaporsc Molecule u in Ab elk in O K b Ref. for Ji Watera 2.56 (17) 775 (17) (12) Carbon tetrachloride 5.881 (15) 327 (15) (25) Methanolu 3.69 ( 1 7 ) 417 (17) (26) Ethanol. 4.31 (17) 431 (17) (26) Isopropanola 4.64 (17) 518 ( 1 7 ) (27) Cyclohexane 6.093 (15) 324 (15) (28) a Use Table I, page 1685 in reference I7 for values for W 2 * , * References in which values can be found. All other constants were found in reference 29.

Figure 5. Schematic diagram showing the predicted variation in temperature between the surface of an evaporating droplet and the flame surrounding a burning fuel droplet was determined so that an “effective” thermal conductivity could be found. Unfortunately, these conclusions have limited application to the situation found in an analytical flame. Whereas a fuel droplet receives thermal energy from its burning vapor, a droplet in a flame depends largely on the hot flame gases for its thermal supply. In the flame, the thermal conductivity of the flame gases and of the solvent vapor surrounding the droplet both increase greatly with temperature (Figure 3), so that heat will flow increasingly rapidly toward the droplet as its surrounding temperature increases. This increased heat flow will in turn heat up the vapor and gases nearer the droplet for a further increase in thermal conductivity. Simple extension of this cyclic process leads to the argument that the temperature profile near the droplet will appear qualitatively as shown in Figure 5. From Figure 5 and using the model from which Equation 2 was derived (2),the resulting model indicates that the solvent is vaporized at the surface of the liquid and then rapidly heated to approximately flame temperature. This produces a thin, stagnant film of gas composed almost entirely of solvent vapor which surrounds the liquid droplet (see Figure 1). The temperature of this film is approximately equal to that of the flame except for a layer of vapor only a few molecular diameters thick across which exists the greater part of the temperature difference between the droplet surface and the flame. From this argument, it is apparent that heat conduction must occur across a region of solvent vapor which is almost completely at the flame temperature. Therefore, for use in the desolvation rate equations, the “effective” value for thermal conductivity can be safely approximated by that which is evaluated at the flame temperature. This approximation has been employed throughout this study. Like the thermal conductivity, the effective heat capacity values must be calculated for all solvent molecules which are present across a broad range of temperatures. Conveniently, the heat capacity of a vapor reaches a limiting value at a temperature (cf. Fig. 2) which is usually below the flame temperature. Also, the rapid rise in heat capacity with temperature below the limiting value argues that most of the solvent vapor will have that value. Therefore, in this study, the limiting value of heat capacity, evaluated at the flame temperature, was employed in all calculations of the desolvation rate. 1216

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

For droplets of combustible solvents, it is necessary to estimate the stoichiometric fraction of oxidizer (y) which is available to support combustion. In the case of droplets evaporating in a fuel-rich flame, such as that used in this study, this estimation becomes difficult. On first consideration it can even be argued that no appreciable droplet combustion should occur in this flame, although this is contradictory to visual and spectral observations which we have made. In the region beyond the stagnant film of solvent vapor which surrounds each droplet, solvent molecules and flame gas molecules are beginning to diffuse together (Figure 1). It is in this region that the combustion boundary appears to be located. Although the exact reactions which occur at this boundary are difficult to postulate, it can be suggested that an exothermic equilibrium will be established between the solvent vapor and the flame gases. Because the stoichiometric fraction of oxidizer available at this point is certainly less than that existing in the bulk of the flame gases (which for this flame is 0.568), a somewhat lower value must be taken for y . An empirical fit has shown that a value of 0.5 provides adequate agreement for all solvents examined in this study. To calculate theoretical rates of desolvation, the necessary physical constants for each solvent were obtained from the literature. Incorporation of the constants into Equations 5 and 9 then enabled calculation of heat capacity and thermal conductivity at the specified flame temperature (2350’ or 2380 OK). The resulting values for C, and X were then employed in Equation 2 to determine the theoretical desolvation rate assuming a value of y = 0.5 for combustible solvents. Values for various physical constants used in Equation 2, 5, and 9 have been listed in Table I1 along with their sources. Comparison of Theoretical and Experimental Desolvation Rates. After each calculation of the theoretical desolvation rate (kcale) a comparison of this value was made with the experimentally determined rate (k)exp. This comparison for the six solvents is shown in Table 111. As can be seen from Table 111, carbon tetrachloride and cyclohexane exhibit very good agreement between theory and experiment, while the other four solvents show significant differences. Because carbon tetrachloride and cyclohexane are less polar and have (25) G. Herzberg, “Infrared and Raman Spectra of Polyatomic Molecules,” D. Van Nostrand Co., New York, N.Y., 1945. (26) K. Krishnan, Proc. Indian Acad. Sei., Sect. A , 53, 151 (1961). (27) S . C. Schumann and J. G. Aston, J. Chem. Phys., 6,485(1938). (28) K. B. Wiberg and A. Shrake, Spectrochim. Acta, 27A, 1139

(1971). (29) N. A. Lange, “Handbook of Chemistry,” Handbook Publishers, Inc., Sandusky, Ohio, 1956.

Table 111. High-Temperature Thermal Parameters and Desolvation Rates for Droplets of Various Organic Solvents Thermal Heat capacity (C,) conductivity (A) Solvent (cal/mole-"K) (cal/sec cm "K) keXpc(mrnzisec) k,,~d (mm2/sec) k x = x , d (mmzisec) 12.4 6.43 x 10-4 0.38 0.981 0.488 Water" 29.3 5.83 x 10-4 0.77 1.87 1.03 Methanol. 45.6 5.39 x 10-4 1.04 1.85 1.10 Ethanola 1.08 1.76 1.05 62.6 5.36 x 10-4 Isopropanol5 0.61 0.62 Carbon tetrachloride" 25.7 0.978 x 58.7 2.70 x 10-4 1.77 1.76 Cyclohexaneb a Taken at 2350 OK. a Taken at 2380 OK. c k,,, = experimentally determined desolvation rate constant. kcslo = calculated desolvation rate constant using thermal conductivity of solvent vapor. kx =xxz = calculated desolvation rate constant using thermal conductivity of flame gases.

more regularly shaped molecules than do the other solvents studied, they would be expected to behave in a more ideal fashion and conform more closely to theoretical predictions. However, the magnitude of error involved in calculating C , and A, which can be estimated using the data in Table I, is found to be much smaller than the deviation between experimental and theoretical desolvation rates seen in Table 111. This implies that the previously described model of desolvation is in error and must be extended to provide a model which can explain the desolvation of a broader range of solvents. Extended Model of Droplet Desolvation. In the simplified desolvation model expressed by Equation 2, the effect of the finite thermal conductivity of the flame gases has not been considered. The flame has therefore been assumed to be an infinite heat reservoir which furnishes thermal energy to each droplet by conduction through a thin film of solvent vapor surrounding the droplet. Although the thermal supply of the flame is indeed very large compared to that required for droplet desolvation, the rate of transfer of the heat is limited by the thermal conductivity of the flame gases. Because the thermal conductivity of the flame gases has a finite value which can even be less than that of the solvent vapor surrounding each droplet, it can have an important and even controlling effect on heat transfer to the droplet and on its resulting desolvation. In order to ascertain the effect of these considerations on desolvation rates, it is necessary to estimate the thermal conductivity of the flame gases. At the flow rates used in this study, the flame gas mixture will contain predominantly nitrogen and the combustion products of acetylene, especially carbon dioxide. Because of the difficulty involved in accurately predicting the composition of this mixture and because nitrogen is present in considerable excess (N), the total thermal conductivity of the mixture to a first approximation can be assumed to be that of nitrogen, which is 3.20 X 10-4 cal/cm sec OK (22). It is important to note that this value is less than the thermal conductivities calculated for all solvents except carbon tetrachloride and cyclohexane, all of which are shown in Table 111. For the other solvents, therefore, the rate-limiting process controlling desolvation is the conduction of heat through the flame gases rather than through the vapor film. In these cases, replacement of the thermal conductivity of the solvent vapor in Equation 2 by that of nitrogen should therefore improve the accuracy of the calculation. (30) T. Hollander, Ph.D. Thesis, University of Utrecht, The Netherlands, 1964.

When new rates of desolvation are calculated using the thermal conductivity of nitrogen, the improved results(kx = x.v2) in Table I11 are obtained. The excellent agreement with experimental results for ethanol, isopropanol, and even highly polar water argue strongly for adoption of this expanded model for the desolvation process. No explanation can presently be offered for the remaining disparity in the results for methanol. It is interesting to note, however, that if the value for y, the stoichiometric fraction of oxidizer, is taken to be zero for methanol in Equation 2, Le., if it is assumed that combustion does not occur, the resulting theoretical value is 0.732 compared to the experimentally obtained value of 0.769. It is not suggested here that this fact is more than a coincidence, but is mentioned to assist others in offering an explanation to this unexpected result. CONCLUSIONS

The data presented in Table I11 provide strong evidence in support of the heat transport theory of desolvation expressed by Equations 2 and 3. The scarcity of values for thermal constants, which has previously been the primary limitation to the use of this theory, has been largely overcome by the methods of calculation described in this paper. Therefore, it is now possible to confidently predict the effect of specific variables (e.g., flame temperature) on the rate and extent of desolvation of solvent droplets in a flame. Also, rates of desolvation can be calculated for solvent systems which have not been and cannot conveniently be studied experimentally. For example, many solvents burn with a sooty flame, thereby making accurate desolvation rate measurements difficult. These rates can now be calculated using the methods described. With additional knowledge about the specific flame and nebulizer being used, the desolvation model presented herein can be used to predict the extent of total evaporation which can be expected for a droplet dispersion sent into the flame. Because the desolvation rate is independent of the initial droplet size, the extent of desolvation of droplets in a spray will depend upon and can be calculated from the droplet size distribution of the spray (31) and the average time spent by the droplets within the flame. By knowing the average residence time of droplets within the flame and the droplet desolvation rate, the maximum droplet size which will completely evaporate can be calculated. From the droplet size distribution, the fraction of droplets smaller than this size can then be used to determine the total desolvation efficiency (31) J. A. Dean and W. J. Carnes, ANAL.CHEM., 34, 192 (1962). ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

1217

I

I

I

I

2000

1000 Flame

Tmperolure

I

3000

I

I

4000

(‘K I

Figure 6. Calculated dependence of the desolvation rate ( k )on absolute temperature for cyclohexane droplets (32). Investigations are presently under way in our laboratory to determine the utility and feasibility of these calculations for conventional nebulizer-burner systems. In applying the results of this study to other solvents, several difficulties can arise. In many cases, it is difficult and timeconsuming to obtain all the necessary physical parameters required by Equations 4, 5, and 9 for calculation of thermal conductivity and heat capacity. A literature search is presently being conducted for these constants. Also, for those solvents for which calculations can be made, it is important to compare the calculated thermal conductivity of the solvent with that expected for the flame gases. If the solvent thermal conductivity is greater than that of the flame gases, the latter must be used in Equation 2 in order to obtain an accurate estimate of the desolvation rate. Examples of this situation which have been examined in this study are droplets of water, ethanol, and isopropanol evaporating in an air-acetylene flame. Since X, in Figure 1 obviously must have a value between the two extreme values of X, and A,, it can never be the limiting factor, In some cases, it may be necessary when applying the desolvation theory to employ a correction for convective heat transfer to the droplets ( I , 2). This correction, which will be especially important in turbulent flames, has been ignored in this study. Nonetheless, the acceleration of the droplets within the flame which occurs in our system ( I ) is capable of producing a small change in desolvation rate (33). This effect is possibly responsible for the additional error in the calculated rates in Table 111. As an example of the utility of these results in predicting desolvation characteristics, Figure 6 shows the rate of desolvation of cyclohexane droplets calculated as a function of flame temperature. Because of the temperature dependence of heat capacity and thermal conductivity (Equations 4, 5, and 9), the variation of desolvation rate with temperature is -

(32) J. D. Winefordner and T. J. Vickers, ANAL.CHEM.,35, 1607 (1963). (33) S. Kumagai, Jet Propd., 26,786 (1956). 1218

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

seen to be a complex relationship. Although the magnitude of the desolvation rate in Figure 6 will be different for other solvents, the curve shape will be approximately the same, thereby suggesting the use of the highest temperature flame available in order to increase the efficiency of droplet desolvation. However, from Figure 6, it is seen that, for cyclohexane, the desolvation rate increases by a factor of only 4 for a temperature change from 500 to 3000 OK. Therefore, for a droplet dispersion sent into a heated spray chamber (34, 3 9 , it can be expected that most of the droplets will be desolvated before reaching the burner, because of the relatively long time spent by the droplets within the chamber. When the thermal desolvation theory is applied to a spray chamber, of course, combustion is not expected so that the value of y can be taken to be zero. For conventional, unheated spray chambers, however, an entirely different evaporation mechanism holds (36, 37), so that comparisons with the theory considered in this study are difficult to make unless conditions are specified completely. The rather ideal conditions provided by the shielded laminar flame used in this investigation have served to simplify somewhat the theoretical considerations and experimental determination of droplet desolvation characteristics. It is to be expected that the conditions in many practical analytical flames will be more complex. Although this work may not be directly applicable to all these situations, it represents an important step in approaching the problem of understanding the basic processes which occur in the flame. Using this information as a basis, extensions of’the theory can be made which would apply to turbulent and inhomogeneous flames. This work is presently being pursued in our laboratory. Although the droplet sizes employed in this study are larger than found in most conventional nebulizer-burner systems (3Z), it is expected that the desolvation rates measured herein and the proposed theoretical model will nevertheless be applicable to these other systems. In previous studies on conventional fuels (36),it has been found that droplets below a critical diameter exhibit similar desolvation characteristics which are thermally controlled. Both the droplets employed in this study and those produced by common nebulizerburners are below this diameter. For solvents such as water, ethanol, and isopropanol, the possibility of a new technique is suggested from having examined the mechanism of desolvation in this investigation. Because the thermal conductivity of the flame is important in determining the rate and extent of evaporation of droplets of these solvents, these characteristics would be expected to be enhanced by the addition to the flame of gases such as helium or hydrogen which have a high thermal conductivity. Other researchers have already noted that variation in the flame gas does affect the rate of desolvation (38). Investigations are being initiated to determine the extent of this effect. It is also expected that addition of these high thermal conductivity gases to a heated spray chamber (34, 35) would enhance desolvation.

(34) A. Hell, W. F. Ulrich, N. Shifrin, and J. Ramirez-Mufioz, Appl. Opt., 7 , 1317 (1968). (35) C. Veillon and M. Margoshes, Spectrochim. Acta, 23B, 553

(1968). (36) I. Langmuir, Phys. Rer;.,12, 368 (1918). (37) C. Th. J. Alkemade in “Flame Emission and Atomic Absorption Spectrometry, vol. I-Theory,” J. A. Dean and T. C . Rains, Ed., Marcel Dekker, New York, N.Y., 1969, p 101. (38) H. Wise and G. A. Agoston in “Literature of the Combustion of Petroleum,” Adcan. Chem. Ser., 20, 116 (1958).