Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
r32 = mean radius of bubbles by third moment over second moment S = surface area of bubbles T = absolute temperature t = effective average thickness of lamella Greek Letters y = surface tension A = differencein = dimensionless frequency distribution function of bubble radii based on the number of bubbles present initially p = radius of fictitious bubble of zero holdup T = time T = dimensionless time, K7/rC2or Kr/rlo2(0) $ = dimensionless frequency distribution function of bubble radii based on the number of bubbles actually present Literature Cited Abe, T., Pap. Meterol. Geophys., 5 (3,4), 243 (1955). Bayens, C. A., as reported by Gal-Or, B.. Hoelscher, H. E., AlChEJ., 12, 499
(1966).
93
Chang, R. C., Schoen, H. M., Grove, C. S . , Jr., lnd. Eng. Chem., 48, 2035
(1956). Clark, N. O., Blackman, M., Trans. Faraday Soc., 44, l(1948). de Vries, A. J., "Foam Stability", Rubber-Stichting, Delft, 1957. de Vries, A. J., in "Adsorptive Bubble Separation Techniques", pp 7-31, R. Lemlich, Ed., Academic Press, New York, N.Y., 1972. Jashnani, I. L., Lemiich, R., lnd. Eng. Chem. Process Des. Dev. 12, 312
(1973). Jashnani, i. L., Lemlich, R., lnd. Eng. Chem. fundam., 14, 131 (1975). Lemlich, R., Ind. Eng. Chem., 60(10), 16 (1968). Lemlich, R., J. Cosmet. Chem., 23,299 (1972). Lemlich, R., in "Chemical Engineers Handbook", Vol. 17, 29-34,R . H. Perry and C. H. Chiiton, Ed., McGraw-Hill, New York, N.Y., 1973. Leonard, R. A., Lemlich, R., AlChEJ., 11, 18(1965). Manegold, E.,"Schaum", Strassenbau, Chemie und Technik Verlag, Heidelberg,
1953. New, G. E., Proc. lnt. Congr. Surface Active Substances 7964, 2, 1167
(1967).
Received for review March 15,1977 Accepted February 6,1978
The author thanks Ajit Ranadive for writing and running the computer program, and the Herman Schneider Laboratory for financial support for this assistance.
The Latent Heat of Vaporization Prediction for Binary Mixtures Dragoljub Santrach and Janis Llelmers' Chemical Engineering Department, The University o f British Columbia, Vancouver, British Columbia, Canada
Using the dimensionless mixture coordinates SM"(SM* = (A~/M/TM)/(A~/BM/TBM)), where Am/M = heat of vaporization of the binary mixture at the given mixture temperature, TM (K), while Am/BM = heat of vaporization at the normal boiling point temperature of mixture, TBM (K), and TM* ( TM* = ( TcM/ TM I)/( TcM/TBM 1) where TCM= the critical mixture temperature (K), an empirical equation, SM* = ( TM* -k TM* P)/(1 4- TM* g), has been established to calculate the latent heats of vaporization for binary mixtures along the mixture liquid-vapor saturation curve over the entire liquid range from the normal melting to the critical point for the given mixture. The proposed method (this work) compares very well in overall accuracy with other prediction methods over the complete range of investigation (15 binary systems, 135 data point pairs).
-
Introduction and the Proposed Method T o describe the coexistence behavior for binary mixture along its liquid-vapor saturation curve, a modified form of the differential Clausius-Clapeyron equation may be used
(2) S,X
M V M - TM(VGM - VLM)
(1)
where M V M is the heat of vaporization of the binary mixture at the given mixture temperature TM(K), and the term ( VGM - VLM) is the associated molar volume change between gaseous (G) and liquid (L) phases along the binary mixture liquid-vapor saturation curve. The subscripts S and X refer to this liquid-vapor pressure equilibrium state and the mixture concentration X of component (l),respectively. For the given binary mixture the latent heat of vaporization may be calculated from eq 1 as the product between the differential mixture pressure-temperature ratio and the temperature times the corresponding mixture volume change. However, eq 1 is difficult to apply directly to calculate the latent heat of vaporization for binary mixtures. There is an inherent inaccuracy in obtaining the slopes of the experimental PM-TM curves since accurate binary mixture saturated vapor pressure data are available only a t a few temperatures for the given 0019-7874/78/1017-0093$01.00/0
-
concentration X of the binary system. This inaccessibility of needed experimental data has led to other considerations in calculation of latent heat of vaporization for binary mixtures (compare with Reid and Sherwood, 1966). Noting the overall accuracy of the recently introduced Lielmezs-Fish latent heat temperature functions for calculation of the vaporization heat of pure substances (Fish and Lielmezs, 1975), it was felt that these functions could be extended for latent heat prediction of binary systems. To do this, the previously reduced dimensionless coordinates, S*and T* (valid for pure substances (Fish and Lielmezs, 1976)) are rewritten to describe the binary mixtures as s ~and *TM*.
where AHVM = heat of vaporization of the binary mixture at = mixture the given mixture temperature (cal/mol), MVBM heat of vaporization at the normal boiling point temperature of this binary mixture (cal/mol), and TM,TCM,and TBMare the mixture, critical mixture, and normal boiling point of mixture temperatures, respectively (K). Then for this coor0 1978 American Chemical Society
94
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
LEGEND 0
11
PRRRF -PRRRFFIN
A
PRRRF - C X L PRRRF
0
PRRRF - O L E F I N
+
ARON HE-PRRRFFIN
X
RROH HC-CYCL H
+
PRRRF -HC HRLIOE
0
RROH HC-HC HRLIDE
K
PRRRF -NRPHTHRLENE
C
a
RLCOHOL-RROH H C
+
RLCOHOL-YRTER
1.
TI IM
Figure 1. SM*-TM*relation (eq 5 ) for binary mixtures.
dinate system (eq 2 and 3) we can write the following empirical relation (Figure 1, Table 11)
( T M * -k T M * P ) / ( l
+ TM*q)
(4) where p and q are constants obtained from the experimental data sets (Table I and Figure 1)by means of nonlinear leastsquares regression methods (Streal and Moore, 1976). The behavior of the plotted data indicates (Figure 1,Table 11) that eq 4 yields values which within the indicated accuracy limits (Table 11)are valid over the entire liquid range from the melting point to the critical point. Equation 4 written in the dimensionless SM*-TM*coordinate system predicts the latent heat of vaporization values for binary systems without direct reference to the nature of the vaporization process, the magnitude and type of intermolecular forces, and the molecular structure. However, it requires the knowledge of the binary SM* =
Table I. Summarv of Data Binary system groups Paraffin-paraffin
Paraffin-cycloparaffin Paraffin-olefin Aromatic hydrocarbonparaffin Aromatic hydrocarboncyclohydrocarbon Paraffin-halide Aromatic hydrocarbonhalide Paraffin-naphthalene Alcohol-aromatic hydrocarbon Alcohol-water
Literature citations 2,3,4,5,6,7,8,9,10,11,12,14 17,18,19,21,22,23,24,27,28, 29,30,31,39,40,43,45,46,49, 56,57,58,59 17,18,32,40,41,46,49,56,59
17,18,32,42,44,46,49,53,56,59 17,18,31,32,33,36,38,40,46,49 56,59 17,18,19,20,32,35,37,46,49,56,59
I ,15,18,25,26,32,46,47,49,56,59 15,18,32,46,48,49,56,59 17,18,32,34,46,49,56,59 17,18,32,46,48,49,51,56,59
16,17,18,32,46,49,54,56,59
mixture vapor-liquid critical point and normal boiling point temperatures and latent heat of vaporization value at the normal boiling point, all taken as functions of the binary mixture component concentration. The general thermodynamic restrictions through the principle of corresponding states when applied to binary interacting mixtures limit the range of applicability of eq 4 to nonelectrolyte binary systems containing only one liquid phase. That is, for any partially miscible binary solution the proposed heat of vaporization calculation method is applicable only when the solution is outside the solution critical temperature range (temperature a t which composition of the two liquid phases becomes identical (Dodge, 1948)).
Results Using the nonlinear least-squares regression methods (Streal and Moore, 1976), it was possible from the given data (Figure 1and Table I) to determine the values of constants p and q in eq 4. Inserting the calculated constant values in eq 4, the correlating equation now becomes sM* = (TM*+ TM*0.28468)/(1+ TM*0.049521) (5) where p = 0.28468 and q = 0.049521. The error mean square for the total input of 135 experimental data point pairs representing 15 binary systems is 5 2 = 0.002330, while the standard deviation (Streal and Moore, 1976) for coefficient p is 0.018507 and for coefficient q is 0.021829. In use in this work, we have defined the error mean square as
S2=
f: (Yi - -Yi2) weight/(n - m )
i= 1
-
(6)
where Yi is the input data point “ i ” , Yi is calculated point ‘5’’ value, n is the number of data point pairs, and m is the number of program variables such that the curve (Figure 1) is forced through the point SM*= 1.0; TM*= 1.0. Equation 5, although mainly derived from the contributions of hydro-
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978 95
Table 11. Comparison between This Work, That of Shettigar et al. (1969), and Experimental Data“
Binary system
Total Temp no-of range, points, K N N
n-Pentane- 33310 trans502 decalin n-Pentane- 20018 cyclo450 pentane Toluene3508 1,2-dichlo450 roethane n-Pentane- 20010 n-nonane 450
This work Simplified method, Eq 5 eq 5 and 7 > 2% A v % Max% N > 2% A v % Max% N
Shettigar et al. (1969) > 2% A v % Max%
Parameters needed This Shettigar work et al. (1969)b
5
3.85
8.5
8
5.30
20.1 AHVBM, AHvl, TCM,TBM U v 2
-3.39
7
2.05
14.0
6
3.5
-6.75
3.26
10.00
5
4.25
11.3
3
1.81
5.0
2.41
10.39
5
3.15
11.4
4
2.45
8.9
5
2.78
6
1.62
4 3
7.11
A@,
cpl, cp,, TB
ATDEW, YI
18 2.33%c 22 3.06 21 3.37 Total: 46 Over the whole range of concentrations, from X 1 = 0 to X I= 1.0. AHE= excess enthalpy, TDEW= DEW point temperature, Y 1 = mole fraction. c This average error compares favorably with the overall average error of 4.5496, Lielmezs and Fish method (Fish and Lielmezs, 1975) for pure substances.
carbon binary systems, nevertheless predicts the latent heat of vaporization for highly nonideal systems such as alcoholwater, alcohol-hydrocarbon, and similar oxygen derivative, hydrogen-bonded organic binary systems. Table I presents the specific binary mixture groupings used to establish eq 5. This collection of 15 binary systems represents 135 data point pairs. Table I1 presents an evaluation of the goodness and generality of the proposed correlation (eq 5) for the given sets of binary mixtures through comparison of the results of this work with those obtained by Shettigar et al. (1969). The results of this comparison (Table 11)indicate that the proposed method (this work) may be thought to be of general validity.
Clausius-Clapeyron equation (eq 1)in conjunction with the known (or predicted, if not known) mixture vapor pressures and molar volumes along the liquid-vapor saturation curve (Spencer and Danner, 1973). Table I1 presents comparison between the results of this work (eq 5 and the Simplified Method-eq 7 combined with eq 5) and the results obtained by the Shettigar-ViswanathKuloor (1969) method taken with respect to the available experimental data. Table I1 shows that for the four selected binary test systems (46 data point pairs) the proposed method (either as eq 5 or as the Simplified Method when eq 5 is combined with eq 7) compares excellently with the method of Shettigar et al. (1969) both in the overall accuracy (average accuracy 2.33% (eq 5 ) and 3.06% (Simplified Method) as compared to 3.37% obtained by the Shettigar-ViswanathDiscussion Kuloor method (1969)) and in the relative simplicity of the While the dimensionless SM*-TM*coordinate system (eq input parameters. For instance, in the proposed method (this 2, 3, and 5) predicts the binary system latent heat of evapowork) we do not need to have the vapor composition and ration without direct reference (Table 11) to the nature of specific heat over a wide range of temperatures (Table 11). vaporization process, the physical and structural property To illustrate the validity of the proposed heat of vaporizainfluence on the evaporation of the binary mixture may be tion prediction method for binary mixtures (eq 4 and 5), we thought to be detected through the presence of the charactake the n-pentane-trans-decalin system. Experimental teristic group constants p and q (eq 4 and 5 ) , and the experivapor pressure-temperature data for this system have been mentally obtained mixture input data L W ~TM B M,and , TCM. obtained by Lenoir et al. in 1971. Using this set of data, we However, if the needed experimental mixture input data are obtain for 0.725 mole fraction of n-pentane in the liquid phase not readily available or are entirely missing, then these mixa t 395.206 K the latent heat of vaporization of the binary ture input parameters may be estimated using appropriate mixture to be 10.594 kcal/mol. The proposed method (this correlation methods (Reid and Sherwood, 1966). In view of work, eq 4 and 5 ) predicts, however, this heat of vaporization this we simplified our proposed method (eq 4 and 5) by exto be 9.974 kcal/mol; that is, we have -5.9% deviation from panding the T Mcoordinate (eq 3) as a first approximation the latent heat of vaporization value obtained using the into available experimental data. However, if we use the Shettigar-Viswanath-Kuloor method (1969) (ideal mixing of pure TM*= Xi2T1* xZ2T2* 2~1~2T12* (7) components corrected by excess enthalpy and made temperwhere Ti* ( i = 1 , 2 ) are dimensionless reduced temperatures ature dependent using heat capacities, Table I1 and also for pure substances such that Ti* = (Tci/Ti - l ) / ( T c L / T ~ ,previous discussion), we obtain that the latent heat of evap- l ) ,X 1 is the liquid mole fraction of pure substance 1,and oration for this mixture a t 395.206 K is 12.633 kcal/mol; that T12* = Zi=l2 xiTi* is the molar average of the ‘‘i” reduced is, the obtained value deviates by +19.2% from the experidimensionless temperatures, T1*and Tz*.Substituting eq 7 mental data. Finally, using the mole fraction-weighted average into eq 5, we obtain the so-called Simplified Method (Table of the pure-component heat of vaporization calculation 11). method (the pure component ideal mixing case), we calculate To predict the missing mixture input parameters (Simplithat this heat of vaporization is 6.117 kcal/mol. This value fied Method, eq 5) we used the following estimation methods: deviates from the experimental latent heat of vaporization for mixture critical temperature prediction, Li method value by -42.3%. This simple illustration strengthens the (Spencer and Danner, 1973); for mixture normal boiling point overall averaged validity of the proposed method (this work, estimation, the bubble point method (Strel’tsov, 1975); and eq 4 and 5 and Table 11). for the mixture heat of vaporization prediction at the normal In analyzing additional application possibilities of the boiling point of the binary system, we used the general proposed method, we note that although the input parameters
+
+
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Ind. Eng. Chem. Fundarn., Vol. 17, No. 2, 1978
may be different, yet SM*as given by eq 5 is almost identical with S* (Fish and Lielmezs, 1975) for the same TM*(or T* as given by Fish and Lielmezs, 1975),thus in effect defining an equivalent "pure" compound for the given concentration X. This observation may be explained if we assume that the introduced coordinate set for mixtures, SM*-TM*(eq 2 and 3) is in the corresponding states with the reduced parameters S*-T* representing pure substances (Fish and Lielmezs, 1975) and that by necessity there results a shift along the given SM*-TM* curve because of the concentration X change. Following this it is feasible to believe that this proportionality (or shift along SM*-TM*curve) may also hold for other than binary mixtures thus permitting possible extension of eq 5 to higher order mixture systems. As the results of comparison indicate (Table 11),the proposed method in this work is indeed of general validity.
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