CRITICAL S T A T E S OF M I X T U R E S A N D EQUATIONS OF S T A T E R. R. SPEAR,' R. L. ROBINSON, JR.,
AND K. C . CHAO2
School of Chemical Engineering, Oklahoma State University, Stillwater, Oklar
74074
The vapor-liquid critical states of binary mixtures are determined by a procedure that follows the boundary curve of material instability in search of an extremum pressure. A computer program is developed to apply the procedure to simple mixtures for which the extremum pressure is a maximum. Results are reported for calculatims based on the use of the Redlich and Kwong equation of state. Agreement of the calculated results with data is quantitative for binary mixtures of hydrocarbons, carbon dioxide, hydrogen sulfide, carbon monoxide, nitrogen, and others, with the use of an empirical binary interaction parameter in the equation of state. RITICAL states of mixtures are of interest for a number of C r e a s o n s . They delineate the regions of homogeneous and heterogeneous phases. They are used in numerous correlations of properties of mixtures. They are intricately connected with retrograde phenomena. I n spite of these interests, quantitative studies of the critical states of mixtures remain relatively scanty. Empirical correlations for either the critical temperature or the critical pressure of mixtures, usually restricted to mixtures of paraffins or light hydrocarbons, have been presented by Mayfield (1 942), Organick and Brown (1952), Eilerts (1957), Grieves and Thodos (1960), and Etter and Kay (1961). One of the very few critical density correlations has been presented by Grieves and Thodos (1963). Chueh and Prausnitz (1967) correlated critical temperature and critical volume as quadratic functions of the surface fractions and then substituted into the Redlich and Kwong equations of state to obtain the critical pressure. The theory of critical state of mixtures was well established by the pioneering work of Gibbs (1928). Since that time only limited attempts have been made to obtain quantitative predictions of the critical state from the rigorous thermodynamic relations that define the critical state of mixtures. Redlich and Kister (1962) expressed the defining relations of the critical state of binary mixtures in terms of fugacity coefficients of the components. These two equations were then transformed into two equivalent equations which could be evaluated using a suitable equation of state. Although Redlich and Kister claimed that quantitative results could be obtained without serious difficulties, they employed a short-cut method for predicting the critical locus of binary mixtures. The limiting slopes of the critical locus were calculated as the mixture approached its two pure component limits and hyperbolic interpolation was used to predict the critical locus over the entire composition range. Redlich and Kister's results were of basic importance in showing that critical states are an integral part of the larger equation of state problem. This interrelationship makes it possible for improvements in equations of state to be directly applicable to studies of critical state?. Conversely, equation of state studies stand to be enriched by information on critical states. Joffe and Zudkevitch (1967) performed calculations using
Present address, International PetroData, Inc., Calgary, Alberta, Canada. Present address, School of Chemical Engineering, Purdue University, Lafayette, Ind. 2
l&EC FUNDAMENTALS
the Redlich and Kwong equation of state in conjunction with the critical state equations developed by Redlich and Kister. A graphical procedure was used to solve the two simultaneous equations that defined the critical state. Quantitative results were reported for the ethane-carbon dioxide and n-butanecarbon dioxide systems. I n the present investigation the thermodynamic definitions of the critical state of binary mixtures developed by Gibbs (1928) and Kuenen (1906) are shdwn to provide a suitable basis for computer calculations with the use of a n equation of state. Critical State of Binary Mixtures Defined
From consideration of incipient material instability of a homogeneous phase, the critical state of a binary mixture can be defined (Rowlinson, 1959) as follows:
Equation 1 defines the material instability boundary which separates the unstable states from the stable and metastable states of a homogeneous phase. Equation 2 expresses the coalescence of the two branches of the instability boundary a t the critical state. Equations 1 and 2 are not suitable for use with equations of state because of the appearance of T , P, and x as the independent variables. Practically all equations of state employ the independent variables T , V , and x in the form
P
= P(T,
v,x )
(3)
Upon suitable transformation, Equation 1 becomes
Equation 4 involves the proper set of independent variables for use with equations of state. Equation 2 can be similarly transformed into an equation in terms of Helmholtz free energy. However, for binary mixtures, the complicated transformation can be avoided in the manner described below. Figure 1 shows a n isothermal volume-composition diagram for a binary mixture, including the phase boundary curve, AKD, and the material instability boundary curve, BKC,
described by Equation 4. The critical point, K , is the point of tangency of the two curves. Three curves of constant pressure are shown in Figure 1. The three pressures shown in the figure are in the ascending order Pi < P2 < P,. As pressure is increased from a low value, the corresponding isobaric curve approaches the critical point, K. This is the point of maximum pressure on curve BKC, and can be located therefore as such by searching along the conditions defined by Equation 4. The critical state of simple binary mixtures is known to be the point of maximum pressure along the material instability boundary (curve B K C ) of a constant temperature system. Kuenen (1906) deduced that at the critical point ( a P / d x z ) ~ , , = 0, where u denotes the material instability boundary. I n addition to pressure being a maximum, this equation can also be satisfied by a minimum pressure condition. However, such condition arises only in azeotropic systems, which are not considered in this work.
/
STABLE REGION
Calculations Using Redlich and Kwong Equation of State
A numerical search procedure and computer program have been prepared to carry out the necessary calculations to determine the critical pressure, volume, and composition at the specified critical temperature. This program was written so that the search procedure is generally applicable with any equation of state that is employed to express the partial derivatives in Equation 4. Referring to Figure 1, the search a t a fixed critical temperature starts with an initial estimate of critical composition x,. Values of molal'volume V are tried until Equation 4 is satisfied. Equation 4 assumes a negative value in the unstable region, and a positive value in the other regions. I t is equal to zero at the boundary. The pressure at the solution point is stored. The composition is changed slightly, and V is again searched to satisfy Equation 4. The pressure at the new x and V is compared with the stored value. Further changes are introduced in x in the direction of increasingp until a maximum value ofp is found. The search is preferably conducted along the lower branch of the instability boundary curve, CK, that is adjacent to the metastable liquid region, as no extremum value of x exists on this branch. The results of application of the Redlich and Kwong equation in this approach are reported below. The equation of state of Redlich and Kwong (1949) is a two-parameter equation of the form
p =
RT -V-b
a TO.'V(V
+ b)
0
XC
MOLE FRACTION OF COMPONENT 2
Figure 1. Isothermal volume-composition diagram of a simple binary mixture
Redlich and Kwong used the geometric average for a12, equivalent to 0 = 1, and the arithmetic average for b12, equivalent to 4 = ( 6 1 b2)/(2dblbz). The terms in Equation 4 are evaluated using the RedlichKwong equation of state. Thus,
+
RTb' (12)
( g ) ~ , ~ +~+ 4 2 V
RT
b)
= T0.'V2(V
(V
6)'
(13)
- b)'
/ P A1 \G)T,V
-
RT
( b 'Y (V - b)2
1 In (1 bRT1.'
(5) bRT1,6(V
+
+--(V b"- b )
+ b/V) [a"
+--"I-
- 2T a'b' 2 a(b')2 b
ab"
a(b')2 + 2 a'b' - 2b
b)
b
a(b')2 ~
( V + b)
]+
For pure substances the equation of state parameters a and b are usually expressed as a = 0.4278 R2T,2.6/P,
(6)
b = 0.0867RTC/P,
(7)
where
For binary mixtures parameters a and b used in this investigation were obtained from the mixing rules
+ 2 a1zx1xz + biixiz + 2 bizxixz +
a =a m 2
~ 2 2 x 2 ~
(8)
b =
bzz~2'
(9)
where a12 and blz are binary cross-interaction parameters and can be related to the pure component parameters by a12
= OdG2
and at' =
(10)
b" =
VOL. 8
(g) (g) NO. 1 F E B R U A R Y 1 9 6 9
3
System
Table I. Comparison of Calculated and Experimental P, at Specified T, R-K Eq. with e = 7 R-K Eq. with Optimum 0 Av. .ifax. Optimum AD. .\lax. error, 70 error, 70 e error, error, %
Rejerence
I. SYSTEMS OF PARAFFIN-PARAFFIN MIXTURES Methane-ethane Methane-propane Methane-n-butane Methane-n-pentane Methane-n-heptane Ethane-propane Ethane-n-butane Ethane-n-pentane Ethane-n-heptane Propane-n-butane Propane-n-pentane Propane-n-hexane Propane-n-heptane Propane-n-octane n-Butane-n-pentane n-Butane-n-heptane n-Butane-n-octane Av. Ethane-propene Ethane-benzene Ethane-cyclohexane Propane-benzene n-Butane-ethene n-Hexane-toluene n-Heptane-ethene Ethene-propene Av.
2.0 5.1 11 .o 9.0
18.5 1.8
1.5 5.4 7.9
1.9
4.4 9.2 17.2 16.1 26.1 2.4 2.5 10 5 12.2 2.8
0,948 0.939 0.911 0.964 0.939 1.036 0.973 0.875 0.875 1.050 1.005 0.975 0.962
0.5 3.4 4.3 7.9
0.983 1.020 0.992
1.5 1.9 6.8 8.3 17.3 1.3 1.2 3.8 3.5 1.3
3.0 3.5 11.7 14.8 23.0 2.5 2.0 5.6 6.6 2.0
0.2
0.4
0.970
0.3 1.9
2.4 4.6
1.8
2.3 3.6
3.4 4.1 6.0
Bloomer et d. (1953) Reamer et al. (1950) Sage et d . (1940) Sage et d . (1942) Reamer et al. (1956) Matschke and Thodos (1962) Cota and Thodos (1962) Reamer et al. (1960) Kay (1938) Nysewander et d . (1940) Sutton (1965) Sutton (1965) Sutton (1965) Sutton (1965) Sutton 11965) Kay-(l941)
’
Sutton (1965)
SYSTEMS CONTAINING NONPARAFFIN HYDROCARBONS 11. HYDROCARBON 0.5 0.8 0.5 0.8 Lu et d . (1941) 1.000 Kay and Nevens (1952) 7.4 11.6 0.898 3.2 5.1 Kay and Albert (1956) 9.1 13.8 0.867 3.9 7.2 2 0 3.5 Glanville et al. (1950) 3.9 5.8 0,927 4.6 10.1 2.9 7.3 Sutton 11965) 0.911 tt’atson‘and Dodge (1952) 1.086 1.3 2.1 3.3 4.8 10.5 15.3 0,898 5.3 12.0 Kav (1948) Haielden et al. (1945) 4.0 5.4 0.948 3.3 4.4 5.4 8.5 2.8 5.3
Methane-nitrogen Methane-carbon monoxide Methane-carbon dioxide Methane-hydrogen sulfide Ethane-hydrogen sulfide Propane-carbon monoxide Propane-carbon dioxide n-Butane-carbon dioxide n-Pentane-carbon dioxide Ethene-carbon dioxide Av.
111. SYSTEMS OF HYDROCARBON-NONHYDROCARBON MIXTURES Cines et d . (1953) 3.0 2.1 4.9 0.961 1.8 . Toyma et ~ l (1962) 0.5 2.0 3.6 0.952 0.4 Donnelly and Katz (1954) 12.1 8.2 9.2 0.864 4.1 Reamer et al. (1951a) 12.9 19.7 0,873 1.9 7.8 Kay and Price (1953) 2.8 5.6 0.889 1.2 2.5 TViddoes and Katz (1948) 1.095 10.9 16.6 Reamer et d . (1951b) 3.0 4.4 Olds et d . (1949) Poettman and Katz (1945) 11 .o 18.1 Sutton (1965) 2.7 3.2 0.845 0.7 1.1 6.8 11.6 3.2 6.4
IV. SYSTEMS OF NONHYDROCARBON MIXTURES Carbon dioxide-sulfur dioxide Carbon dioxide-hydrogen sulfide Hydrogen chloride-krypton AV.
Total averages
1.9 3.5 6.0 3.8 5.3
3.6 6.7 8.5 6.3 8.8
Joffe and Zudkevitch (1966) showed that substantial improvement in the representation of fugacity of gas mixtures could be obtained by treating e and $ as empirical parameters. This latter approach was adopted in this study. Two separate calculations were carried out for each mixture system studied. I n the first set of calculations the simple mixing rules with 9 = 1 and 4 = 1 were used. The general character of the variation of P,, T,, and V , with composition was qualitatively represented. However, considerable deviations were observed for many systems, particularly those containing a hydrocarbon and a ndnhydrocarbon component. Some empirical adjustment of the binary interaction parameters, 0 and #, seemed to be indicated in these latter systems. In the second calculation the value of e was varied in search for the best representation of experimental data. Since the critical pressure was found to be the most difficult to predict and most sensitive to e, the optimum value of 0 was obtained 4
I&EC FUNDAMENTALS
0.967 0,875 0,848
1.7 1.7 4.8
2.7 3.3
2.4 2.8 12.6
Caubet (1902) Bierlein and Kay (1953) Glockler et ~ l (1933) .
5.9
6.0
for each system a t the minimum of the sum of the squares of the deviations of the predicted critical pressures from the experimental values. $ was kept a t a constant value of 1. Results and Discussion
The general behavior of the calculated and experimental critical pressure, temperature, and volume of several typical systems is presented in Figures 2 to 10. Comparisons of the experimental and calculated critical pressures a t the specified critical temperatures appear in Table I. In general, qualitative results were obtained for the critical properties of binary mixtures using the Redlich and Kwong equation of state with no empirical adjustment of the interaction parameters, 0 and $, The average error in the critical pressure for the four general classes of mixtures was 5.3%. The largest deviations in the prediction of the critical state properties occur in the paraffin-nonparaffin hydrocarbon and
' 600 t 14C
a m
a
w- lZ( LT 3 (0
m
W
LL
a _I
2a
-
IO(
[r
0
8C
600.
6
O0.2
0.4 O
0.6 1
j
0.8
M O L E FRACTION PROPANE
Figure 2. Critical pressure for methanepropane system
---
Data of Reamer, Sage, and Lacey (1 950) R-K equation with 8 = 1 R-K equation with 8 = 0.939
-
40("
1
0.2
1
1
04
1
i
06
I
'
I
08
paraffin-nonhydrocarbon mixtures. These two classes of mixtures include the ethane-benzene, ethene-n-butane, methane-carbon dioxide, and methane-hydrogen sulfide binaries. Large deviations were not totally unexpected, since the difference between the molecular species tends to be the greatest in these two classes of mixtures. An adjustment of the interaction
400 0
0.2 0.4 06 00 MOLE FRACTION n - H E P T A N E
1.0
Figure 4. Critical pressure for ethanen-heptane system data of Kay (1 938) - - - Experimental R-K equation with 8 = 1 - R-K equation with 8 = 0.875
parameters will be necessary if the critical state predictions of these mixtures are to have the same accuracy as those having more similar molecules. The errors in the prediction of the critical pressure for the hydrocarbon-nonhydrocarbon mixtures were surprisingly low. The average errors for the nonhydrocarbon mixtures were comparable to those for hydrocarbon mixtures. The optimum value of the interaction parameter, 0, was determined from a least squares criterion on the deviation of the predicted from experimental critical pressure for each system. The results of these calculations appear in Tzble I. Values of the optimum 0 ranged from 1.095 to 0.775. The average error in the critical pressure was reduced from 5.3% for 0 = 1 to 3.3% for the optimum values of 8. A summary of the average and maximum errors in the critical pressure for the Redlich-Kwong equation with 0 = 1 and with the optimum values of 8 appears in Table I. I n general, the greater the difference between the components of the binary, the more the optimum value of 8 deviates from 1.0. Use of the optimum 0 to improve the critical pressure representation also improves the representation of the critical temperature (Figures 3, 5, and 7). Critical volumes were calculated simultaneously with the critical pressure and composition a t any specified critical temperature. T h e agreement between the calculated and experimental critical volumes was rather poor (Figure 8). Large deviations in the critical volume were expected, since the Redlich and Kwong equation of state is known to fail to reproduce critical density of pure substances. Nevertheless, Figure 8 is encouraging in showing that the predicted mixture critical volumes are no less accurate than those of the pure VOL. 8
NO. 1
FEBRUARY 1969
5
4 Figure 5. Critical temperature for ethane-n-heptane system data of Kay (1 938) -- - Experimental R-K equation with 8 = 1 R-K equation with
8 = 0.895
s ln a
wa' 3 ln ln W
a 0
-I
a 2
b Figure 6. Critical pressure for n-butane-carbon dioxide system
5 0 0 h ' 0.2
MOLE
I
0.4
'
I 0.6 ' 0.8 I
Experimental data of Oldr, Reamer, Sage, and Lacey ( 1 949) R-K equation with 8 = 1 R-K equation with 8 = 0.775 '
---
FRACTION n-HEPTANE
~
1.0
0.2 0.4 06 08 MOLE FRACTION n-BUTANE
4000
.
O
-
~
G
~
X
I
'
p
4 Figure 7. Critical temperature for n-butane-carbon dioxide system Experimental data of Oldr, Reamer, 4.0 Sage, and Lacey ( 1 949) -I R-K equation with 8 = 1 0 R-K equation with 8 = 0.n5 m
---
3
1
2 LL 2
0 W
E
J
0
>
b MOLE FRACTION n-BUTANE
Figure 8. Critical volume for n-butane-carbon dioxide system Experimental data of Olds, Reamer, Sage, and Lacey (1 949) R-K equation with 8 = 1 R-K equation with 8 = 0.775
--6
I&EC FUNDAMENTALS
0 t-
1.0 0
'
r
'
'
'
t
'
0.2 0.4 0.6 0.8 MOLE FRACTION n - B U T A N E
I
I
I
I
1600/
t
1
substances. The lesson seems clear: Choose a n equation of state that reproduces pure component critical volume in order to predict the critical volume of mixtures correctly. Binary interaction parameters determined in this study can be expected to be useful in several ways. They should be used in future studies on the calculation of critical states of multicomponent mixtures. Their usefulness in the prediction of phase equilibrium in the critical region should be investigated with particular reference to retrograde phenomena. An advantage of the equation of state approach to the predicting of critical states of mixtures is that a single calculation yields values for T,, Pc,and V , simultaneously. Such is not the case in several previous investigations, in which a separate empirical equation was required for each of the critical properties. Nomenclature
A = Helmholtz free energy per mole a = parameter of Redlich and Kwong equation of state b = parameter of Redlich and Kwong equation of state G = Gibbs free energy per mole
P R
=
= T= V = x =
600A
’
’ ’ 0.2
“
“
’ ’ 0.8
0.4 0.6 MOLE FRACTION B E N Z E N E
Figure 9. Critical pressure for benzene system
’
I,0
ethane-
data of Kay and Nevens ( 1 952) -- - Experimental R-K equation with 8 = 1 R-K equation with
pressure universal gas constant temperature volume mole fraction
GREEKSYMBOLS
e
= interaction parameter associated with a b u = material instability boundary
@ = interaction parameter associated with
SUBSCRIPTS i = component reference c
= critical state property
8 = 0.898
literature Cited
600h
I
1
1
1
1
1
I
1
J
0.2 0.4 0.6 0.8 1.0 MOLE FRACTION HYDROGEN S U L F I D E
Bierlein, J. A., Kay, W. B., Ind. Eng. Chem. 45,618 (1953). Bloomer, 0. T., Gami, D. C., Parent, J. D., Znst. Cas Technol. Res. Bull. 22. 1 (Julv 19531. Caubet, F., 2.Phys. &em. 40; 257 (1902). Chueh, P. L., Prausnitz, J. M., A.I.Ch.E. J . 13,1107 (1967). Cines, M. R., Roach, J. T., Hogan, R. J., Roland, C. H:, Cham. Eng. Progr., Symp. Ser. 49,6 , l (i953). Cota, H. M., Thodos, G., J . Chem. Eng. Data 7 , 62 (1962). Donnelly, H. G., Katz, D. L., Znd. Eng. Chem. 46,511 (1954). Eilerts, C. K., “Phase Relations of Gas-Condensate Fluids,” Monograph 10, Bureau of Mines, p. 101, American Gas Association, New York, 1957. Etter, D. D., Kay, W. B., J . Chem;,Eng. Data 6, 409 (1961). Gibbs, J. W., “Collected Works, Vol. 1, p. 55, Yale University Press, New Haven, Conn., 1928. Glanville, J. W., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 42, 508 (1950). Glockler, G.,’Fuller, D. L., Roe, C. P., J . Chem. Phys. 1, 714 (1933). Grieves, R. B., Thodos, G., A.I.Ch.E. J . 6, 561 (1960). Grieves, R. B., Thodos, G., A.I.Ch.E. J . 9,25 (1963). Haselden, G. G., Holland, F. A,, King, M. B., Strickland-Constable, R. F., Proc. Roy. Sic. 240,l (1945). Joffe, J., Zudkevitch, D., Chem. Eng. Progr. Symfi. Ser. 63 (81), 43 (1967). Joffe,’J., Zudkevitch, D., IND.END. CHEM.FUNDAMENT ‘ALS 5 45r3 (lroo). ..”‘ Kay, W. B.. Ind. Enp.. Chem. 30. 459 11938). Kay, W. B.; Ind. E G . Chem. 33; 590 (1941j. Kay, W. B., Ind. Eng. Chem. 40, 1459 (1948). Kay, W. B., Albert, R. E., Ind. Eng. Chem. 48, 422 (1956). Kay, W. B., Nevens, T. D., Chem. Eng. Progr. Symp, Ser., 48, No. 3, inn
,.,--,.
/IO&?)
Figure 10. Critical pressure for methanehydrogen sulfide system
K av. W. B.. Price. D. B.. Znd. Enp; Chem: 45. 615 (1953). Kuknen, J.’P., “Theorie’der Vepdampfungund Verflussingung von Gemischen und der Fractionierten Destillation,” Barth, Leipzig, 1906. Lu, H., Newitt, D. M., Ruhemann, M., Proc. Roy. SOC. 178A, 506
Experimental data of Reamer, Sage, and Lacey 119511. . . .. , R-K equation with 8 = 1 R-K equation with 8 = 0.873
Matschke, E., Thodos, G., J . Chem: Eng: Data 7 , 232 (1962). Mayfield, F. D., Ind. Eng. Chem. 34, 843 (1942). Nysewander, C. N., Sage, B. H., Lacey, W. N., Znd. Eng. Chem. 32,118 (1940).
I1 941-,. \. \ - -
.
.-.
-
VOL 8
NO. 1
FEBRUARY 1 9 6 9
7
Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 41,475 (1949). Organick, E. I., Brown, G. G., Chem: Eng. Progr. Symp. Ser. 48 (2), 97 (1952). Poettman, F. H., Katz, D. L., 2nd. Eng. Chem: 37, 847 (1945). Reamer, H. H., Sage, B. H., Lacey, W. N., Znd. Eng: Chem: 42, 534 (1950). Reamer, H. H., Sage, B. H., Lacey, W. N., 2nd. Eng: Chem: 43, 976 (1951a). Reamer, H. H., Sage, B. H., Lacey, W. N., 2nd. Eng. Chem. 43, 2515 (1951b). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem., Chem. Eng. Data Ser. 1, 29 (1956). Reamer, H. H., Sage, B. H., Lacey, W. N., J . Chem. Eng. Datu 5 . 44 (1960). Redlich; O., Kister, A. T., J.Chem. Phys. 36, No. 8 , 2002 (1962). Redlich, O., Kwong, J. N. S., Chem. Rev. 44,233 (1949).
Rowlinson, J. S., “Liquids and Liquid Mixtures,” p. 163, Academic Press, New York, 1959. Sage, B. H., Hicks, B. L., Lacey, W.N., 2nd. Eng. Chem. 32, 1085 (1940). Sage, B. H., Reamer, H.H., Olds, R. H., Lacey, W. N., Znd. Eng. Chem. 34, 1108 (1942). Sutton, J. R., “Critical Pressure of Multicomponent Mixtures,” Third Symposium on Thermophysical Properties, ASME, Purdue University, March 22-25, 1965. Toyma, A., Chappelear, P. S., Leland, T. W., Kobayashi, R., Advan. Cryog. Eng. 7, 125 (1962). Watson, I. M., Dodge, B. F., Chem. Eng. Progr., Symp, Ser. 48, No. 3, 73 (1952). Widdoes, L. c.,Katz, D. L., 2nd. Eng. Chem. 40, 1742 (1948). RECEIVED for review June 9, 1967 ACCEPTED September 30, 1968
EFFECTS OF ELECTROSTATIC FORCE, RELATIVE HUMIDITY, HEATING SURFACE TEMPERATURE, AND SIZE AND SHAPE ON DROPLET EVAPORATION RATE DON AYLOR AND W. S. BRADFIELD State University of New York at Stony Brook, Stony Brook, N . Y.
An empirical comparison was made between the evaporation rates of film boiling droplets with and without applied electrostatic voltage. Only small increases were observed and this compared favorably with the prediction of a simple parallel plate capacitor model. Qualitative descriptions based on empirical observations of the effects of liquid-solid contact, arcing, and forced convection inside the droplet on heat flow are given. Informationconcerning droplet shape as a function of volume is important for the prediction of droplet evaporation rates. A semi-empiricalcorrelation is presented which fits the rate data in the spheroidal segime within 15%.
boiling there is a barrier to the heat flux from a hot solid surface to a liquid substance by the interposition of a gaseous phase. Bradfield (1966) has shown that intermittent liquid-solid contact may occur in stable film boiling, but for the most part a quasi-stable gaseous film completely separates the liquid from a smooth solid surface. Markels and Durfee (1964) tried to enhance liquid-solid contact to increase heat flux in their pool film boiling apparatus by applying an electrostatic field across the vapor layer between the hot surface and the “boiling” liquid. They found that large increases in heat flux could be effected by applying large voltages to their system, attributing the increase to two effects: WRING film
1. Dielectrophoresis (or the action of a nonuniform electric field on an induced dipole within the field). 2. Capacitive effect (or the coulombic attraction between the charged plates of a condenser). 8
I&EC FUNDAMENTALS
Later (1965) Markels and Durfee estimated these two effects for the case of a radial electric field in a pool boiling experiment. The shape of the film boiling droplet is particularly good for this investigation because of the direct visual access to the liquid-gas interface and also because the phenomenon is localized for easy study. Figure 1 shows the droplet and its environment and indicates the possible heat and mass transfer processes which may occur. I t was thought that by applying a n electrostatic field across the vapor layer one could reduce the vapor layer thickness under the droplet by the action of the capacitive effect and thus increase the conduction heat transfer to the droplet. This process is limited only by possible electrical breakdown of the vapor layer. Breakdown, in this work, means either dielectric breakdown of the gas layer or the creation of large areas of liquid-solid contact which act as shunt conduction paths across
