Ind. Eng. Chem. Res. 1988, 27, 1737-1743
1737
An Equation of State for Electrolyte Solutions. 2. Single Volatile Weak Electrolytes in Water G a n g Jin and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218
An equation of state is used for predictions of vapor-liquid equilibria of aqueous systems containing single volatile weak electrolytes. This equation of state is derived by adding contributions for charge-charge interactions and charge-molecule interactions to the Perturbed-Anisotropic-Chain theory (PACT) of Vimalchand and Donohue. The equation of state contains four parameters: s, the number of segments per particle; q, the normalized surface area per particle; E , a characteristic energy per unit external surface area of a particle; and c, one-third the number of external degrees of freedom. For neutral molecules, these parameters have been determined by using PACT fitting simultaneously experimental vapor-pressure and liquid-density data. For ions, the parameters are calculated using literature values of ionic radius and polarizability. Our previous calculations for a large number of aqueous solutions containing strong electrolytes show the great promise of this equation of state. Here, preliminary calculations for volatile weak electrolytes are presented. The theory shows remarkable agreement with experimental data over wide ranges of temperature, pressure, and composition without any adjustable parameters.
I. Introduction Thermodynamic properties of volatile weak electrolytes in aqueous solutions play an important role in separation and purification processes in numerous industries. For example, there is renewed interest in vapor-liquid equilibria of ammonia-water systems (at both low and high concentrations) in the fertilizer industry, solubilities of carbon dioxide in aqueous solutions are important for process design in food and chemical industries, and calculations in environmental engineering require thermodynamic properties of industrial wastewater which often contains volatile solutes such as hydrogen sulfide and sulfur dioxide. In contrast to the extensive amount of work reported in the literature for strong electrolyte solutions, limited work has been done on weak electrolyte systems: Edwards et al. (1975,1978) have developed a correlation based on Pitzer's equation (1973) to predict vapor-liquid equilibria of aqueous solutions containing volatile weak electrolytes; Cruz and Renon (1978,1979), Chen and Evans (1986), and Chen et al. (1979, 1980) modeled systems of strong and weak electrolytes by introducing charge-charge and charge-molecule interactions into the NRTL theory; and Daumn et al. (1986) extented the perturbed-hard-chain equation of state to describe the vapor-liquid equilibrium behavior of weak electrolyte systems. While the utility of these empirical or semiempirical models is well established (Horvath, 1985; Reid et al., 1987), in all these models, the user must regress experimental data to determine values of the required parameters in the equations. Most recently, an equation of state for electrolyte solutions has been developed by using perturbation theory (Jin and Donohue, 1988). By use of this equation of state, the predictions of thermodynamic properties for a large number of aqueous strong electrolyte systems were made. Here, we extend this work to aqueous solutions containing volatile weak electrolytes. In this paper, we apply this equation of state to predictions of vapor-liquid equilibria of four aqueous systems each of which contains a single volatile weak electrolyte: ammonia-water, carbon dioxide-water, sulfur dioxidewater, and hydrogen sulfide-water. Unlike the other models, our calculations are carried out without any ad-
justable parameters determined by fitting the equation to experimentaldata. When we compare our predictions with experimental data over wide ranges of temperatures (from 0 to 150 " C ) ,pressures (from 0.007 to 37.5 bars) and concentrations of weak electrolytes (from 0.01 to 25 M), the relatively small average absolute errors show the usefulness of this equation of state for binary weak electrolyte aqueous solutions (see Table I). Aqueous systems containing more weak electrolytes will be treated in a subsequent publication. 11. Thermodynamic Analysis Consider an aqueous solution containing a single volatile weak electrolyte as shown in Figure 1. The vapor-liquid equilibria for both the solvent, water, and the solute, molecular electrolyte, can be described in thermodynamics as
fi(v)(T,P)= f,(')(T,P) i = w, el (1) where f i @ ) ( T , P is)the fugacity of component i in phase 0 at a given temperature, T, and pressure, P, and w and el denote water and electrolyte, respectively. Since vaporphase dissociation is appreciable only at very high temperatures, all components in the vapor phase are considered to be neutral molecules. As depicted in Figure 1, the ionic dissociation of a weak electrolyte takes place only in the liquid phase and can be described by a dissociation equilibrium constant, Kw Those concentrations of H+and OH- ions in the liquid phase which result from dissociation of water are neglected here because the ion concentrations resulting from dissociation of the weak electrolyte are of much greater magnitude. Since a molecular weak electrolyte in the liquid phase is only partially dissociated, the liquid concentration of the weak electrolyte can be defied based on two forms: a bulk (macroscopic) form, me@which is the concentration of the weak electrolyte before the dissociation, and a molecular (microscopic) form, mel,mwhich is the concentration of the nondissociated electrolyte remained after the dissociation. The fugacities of the weak electrolyte must be in terms of the molecular species, i.e., fel,m(')
= Yel4eP)P
fel,m(l) = mel,mYel,mHet
* Author to whom correspondence should be addressed.
~
(2) (3)
where mel,mis molarity of electrolyte (mole of electrolyte
0888-588~/88/2627-1~37$01.50/0 0 1988 American Chemical Society
1738 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 Table I. Average Absolute Errors of the Vapor-Pressure Calculations for Weak Electrolyte Aqueous Solutions T,K P, bar mal,mol/kg of H,O % (error) in Pel data source NHq-HoO 333.2-423.2 0.440-18.84 3.10-25.1 13.96 Clifford and Hunter (1933) 0.01-1.5 1.80 Houghton et al. (1957) 273.2-373.2 1.020-37.48 COzIH,b 0.01-2.8 7.10 Rabe and Harris (1963) 273.2-373.2 0.007-2.18 S02-HzO 0.03-0.2 2.01 Clarke and Glew (1971) 273.2-323.2 0.467-0.95 HZS-HZO V I P O R PHASE
p WATER
T. P
I
relationship between the concentrations in these two forms can be written by an overall mass balance for the liquid phase: m, + mmel,b = mel,m + (5)
MOLECULAR ELECTROLYTE
IT
V
where m+and m- are concentrations (molarities) of cations and anions in solution, and v is a total stoichiometric coefficient (=u+ + v-). The relationship between the concentrations of cations and anions can be obtained from a charge balance:
W4TER
I
‘OLECULAR E L E C T R O L Y T E +------
CATION
+
ANION
LIQUID PHASE
CZimion,l =
0
(6)
1
Figure 1. Schematic illustration of vapor-liquid equilibria for an aqueous solution containing single volatile weak electrolyte at temperature T and pressure P.
where zi is ionic charge number of ion i. Also, chemical equilibria give the relationship between molecular concentration of undissociated electrolyte and concentrations of ions formed by the partial dissociation:
0
Keq=
m+’+m-’- y*” -
~
meI,m
(7)
Yel,m
where the mean ionic activity coefficient, y+,is defined as y*“= y+v+y-v-
Unfortunately, m,, m-, and mel,mcannot be obtained simply by solving eq 5-7 since calculations of y+ and yel,m also require the concentrations of all components in the system. Edwards et al. (1975) have developed an effective iteration scheme to calculate melm for an aqueous system containing single weak electrolyte by setting initial values of y+ and yelF as unity. This iteration scheme is used here. To carry out phase equilibrium calculations, the thermodynamic properties, y+, yelF, and Hel,in eq 2 , 3 , and 7 can be evaluated by using an equation of state developed for both vapor and liquid including electrolyte solutions by Jin and Donohue (1988). This equation of state takes different interactions between the species (molecules and ions) in the system into account using perturbation theory. A brief description is given below.
00
MOLARlTl
OF Y H , (molNH,/kgH,O)
Figure 2. Calculated vapor pressures of ammonia in water a t VLE compared with experimental data of Clifford and Hunter (1933).
-
per kilogram water) in molecular form, and ~ ~ is the 1 , activity coefficient of electrolyte defined by ~ ~ 11 as, mel,m 0 (y, 1 as x , 1). Since ionic dissociation is negligible in the vapor phase, yel, the mole fraction of electrolyte in vapor phase, and the fugacity coefficient of electrolyte, are equivalent in bulk and molecular forms. The Henry’s constant, Hel,in eq 3 is defined by
- - -
Notice that since both the bulk and molecular concentrations of a weak electrolyte in an infinitely dilute solution approach zero, the values of He, obtained in either form, bulk or molecular, are equivalent. Since industrial interest is primarily in the stripping of solute (weak electrolyte) from aqueous streams, the bulk concentrations of electrolyte in the liquid phase are usually given. In order to carry out a phase equilibrium calculation (bubble point calculation) by using eq 1,2, and 3, one must evaluate mel,mfrom a given bulk concentration, mel,b. The
(8)
~ 111. ~
Equation of State An equation of state can be obtained from the Helmholtz free energy by using the simple thermodynamic relation P = -( dA / a V ) T (9)
Recently, Jin and Donohue (1988) derived a perturbation expansion for the Helmholtz free energy of electrolyte solutions by adding contributions for charge-charge and charge-molecule interactions to Perturbed-AnisotropicChain theory (PACT) (Vimalchand, 1986; Vimalchand et al., 1985). In this perturbation expansion, the Helmholtz free energy, A , is calculated by corrections to ideal gas behavior for repulsions and attractions: A = A’G + AREP + AAW
(10)
The f i t term in the equation is the ideal gas contribution, the second is a repulsive term obtained from the expression of Carnahan and Starling (1972), and the last term, the attractive term, is given by
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1739 AATT
=
(11) where AMMaccounts explicitly for molecule-molecule interactions in which the following interactions are considered: Lennard-Jones (LJ), dipole-induced-dipole (DID), dipole-dipole (DD), quadrupole-quadrupole (QQ), and dipole-quadrupole (DQ) interactions, i.e., AMM= AM ADID ADD AQQ + A D Q (12)
Table 11. Values of Parameters, 8 , q , c , and clk, Given by Vimalchand (1986) molecule S 4 C elk
The derivation and detailed form of each term in the above equation is given by Vimalchand (1986) and Vimalchand et al. (1985). The second term in eq 11is the contribution of long-range charge-charge interactions and is derived as a perturbation expansion about a hard-sphere reference system. The third term, ACM,is a charge-molecule interaction term which includes contributions for chargedipole (CD) and charge-induced-dipole (CID)interactions:
eter, c, defined as one-thud the number of external degrees of freedom. This takes spatial complexity of large particles into account. For simple molecules (argon and methane), the parameter c is unity. For neutral molecules, these four parameters have been evaluated for PACT by fitting simultaneously experimental vapor-pressure and liquid-density data. Their values are given by Vimalchand (1986) and listed in Table 11. For ions, since the ions considered in this work are not very large, we choose the parameter c as unit for all ions. The other three parameters for ions can be estimated from their ionic radii and mean polarizabilities by using the following equations suggested by Jin and Donohue (1988):
+
’
+ A C C + ACM
AMM
+
=
ACM
ACD
HZO NH3 COP
so2
+
+ ACm
(13) The three terms involving charges, Acc, ACD,and ACID,are described in a previous publication (Jin and Donohue, 1988). By use of eq 9-13, an equation of state for electrolyte solutions can be obtained as follows: p = P I G + P R E P + P L J + P D I D + P D D + PQQ + P D Q + Pcc + PcD PcID (14)
H2S
1.1029 1.3266 2.0817 2.3147 2.0278
4.3029 4.0701 1.8936 2.9977 0.0267
Sion
=
+
The chemical potential of component i can be determined from the thermodynamic relation pi = ( d A / d r ~ ~ ) ~ , and ~,,,,,, then the activity coefficient, y, fugacity coefficient, 6,and Henry’s constant, H , can be calculated by &(V)
=R T exp($)
. = w, el
6
(15)
UW
(
-)
p$1)(me1
RT
”dip)
uoH,
-
= RT
0)
i#w
i#w
(17)
(18)
and Yw =
RT
p:
qion
= pi(T,V,xior yi) - p?G(T,V,x;or yi)
(20)
IV. Determination of Parameters for Ions Calculations of thermodynamic properties for a mixture using this equation of state are based on the following considerations. First, each particle (molecule or ion) in an electrolyte mixture can be divided into s equal-size segments. Interactions between two particles can be considered as a sum of the segmental interactions of the two particles. Second, each particle in a system has a surface area, q , and interaction energies per unit surface area, E. For ions, this suggests that, while the charge on an ion is located at its center, the effects of the charge can be distributed over its surface. In addition to these parameters, s, q, and c, the PACT contains another param-
3
=
“ion
2
“CH;
and ‘ion _ -- 356 ~~ion~’~(ne,ion)O.‘ “ion
ti
where rbnis the crystal ionic radius in cm, rch is the radius of CH2group which is chosen 8s a single segment in PACT (=1.46435 X lo3 cm), nejonis the number of electrons on an ion, and aionis the mean polarizability of ions in cm3. In eq 21, C,is an adjustable parameter defied by Jin and Donohue (1988). However, a value of unity is used here for C, for all the systems, and therefore, there are not any adjustable parameters used in the calculations in this paper. For the ions whose literature values of crystal ionic radii and mean polarizabilities are not available, a group contribution method can be used to estimate rionand qOn. For example, assuming that UHCO,- = uco, + uOH-, the radii of HC03- can be calculated by ~Hc0,-=
In the above equations, the residual chemical potential of component i is defined by
110.0 105.0 120.0 140.0 150.0
“CH?
12
d”)P
Hi = RT exp
cs-“ion
1.2890 1.5048 1.1788 1.5037 1.1181
(~co,”
+ rOH-3)1/3
(24)
The values of rm0, and rm- can be estimated in the same way. Similarly, the mean polarizability of HCO, ion can be treated as a sum of the contributions from its constitutional groups, C=O, C-OH and C-0-; i.e., “HC09-
=
%=O
+ %--OH + %4-
(25)
where the values of a m ,HC,a+and %+- can be obtained from CYCO,, CYCH,OH (Prausnitz et al., 1986), aC0,- (Nightingale, 1959), and C Y C - ~ (Hirschfelder et al., 1985). The values of CYHSO,- and CVHS-are determined by fitting experimental data because of lack of necessary data for the polarizabilities of the constituent groups. Because of the zero value of a H + and the difficulty of finding a correct value for rH+,eq 23 cannot be used for the hydrogen ion H+. In our calculations, the value of eH+/k is chosen as 105, and the radius of hydrated H+ is used in determinations of SH+ and qH+.
1740 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 Table IV. Dissociation Equilibrium Constants at 0, 25, and 100 "C
Table 111. Physical Properties of Ions
ion H+ NH4+ OHHCO; HSO; HS-
ionic radii, X108 cm
mean polarizability, x1oZ4cm3
0.2800° 1.4800" 1.5300b 2.1629' 2.2138' 1.939lC
1.709* 1.830* 3.199' 2.600d 1.890d
ne 0 10 10 32 42 18
"Nightingale (1959). b X and ~ Hu (1986). cEstimated by using a group contribution method. Obtained from fitting experimental data.
e
/
I
'
The literature values of crystal ionic radii and mean polarizabilities for all ions involved in our calculations are listed in Table 111. There are not any other adjustable parameters introduced in the equation. The assumptions discussed above for the determination of ionic parameters lead to fairly good predictions of experimental data. V. Calculations and Results Vapor-liquid equilibrium calculations involving vapor pressures of volatile weak electrolytes are carried out for four binary aqueous electrolytic solutions: ammonia-water, carbon dioxide-water, sulfur dioxide-water, and hydrogen sulfide-water. The calculations require knowledge of ionic dissociations of the weak electrolytes in the systems. The dissociation equilibrium constant, Keq,defined in eq 7 is temperature dependent and can be described by the van't Hoff equation as d(ln Keq) A H R =dT RT2
d m ~ / d T= Cp,ion - Cp,molecularelectrolyte = ACp (27) where Cp is the heat capacity. First, taking a second-order Taylor expansion for AHRbased on 298 K and, second, if eq 26 is integrated from 298 K to temperature T (Edwards et al., 1975, 19781, an expression of K,, can be obtained as a function of temperature: In K,, = A1/T + Az In T + A3T + A4 (28) where
AHR298
A 2 -
R
+ 298-
(29)
ACp298 (298)2 dACPzg8 - -- (30) R 2R dT
ACpzgS 298 dACPzg8 AB=---R R dT
7
/,++
t
l
(26)
(31)
and
Updated values of coefficients, Al, A2,A3 and A,, have been reported by Kawazuishi and Prausnitz (1987). Since these values are determined by fitting thermodynamic data and equilibrium-constant data, it is not surprising that Keq values obtained from the above correlation work well not only for Edwards' model but also for the calculations carried out in this paper. As shown in Table IV, the dissociation equilibrium constants for the second dissociations to HC03-, HS03-,
I
i
0
/
0 30
In the equation, the enthalpy of dissociation, AHR is defined by
Al = integration constant
3 0 8 2.h 3 2 3 2"h 3402'K X
7
I
CO, - H,O
E x p data
0 60
I
,
l
0 90
,
,
120
,
I50
M O L 4 R I T I OF (0, (molCO,/kgH,OI
Figure 3. Calculated vapor pressures of carbon dioxide in water at VLE compared with experimental data of Houghton et al. (1957).
and HS- ions in the systems of C02-H20, S02-H20, and H2S-H20 are much less than those of the first dissociations, respectively. Thus, the second dissociations in these three systems are neglected. For the first dissociations considered in our calculations, all values of K,, at given temperatures are determined by using eq 28. Having determined parameters, s, q , c , and t, and the dissociation equilibrium constant, Keq,for single-solute systems, vapor pressures of the volatile weak electrolytes at their fixed liquid concentrations can be calculated by using eq 1-3 and 15-20. In the calculations, determinations of molecular concentrations of the undissociated electrolytes in the solutions are carried out following the iteration scheme developed by Edwards et al. (1975). Figures 2-5 show our results compared with experimental data. One can see that this equation of state gives quite satisfactory predictions of vapor-liquid equilibrium behavior of a single weak electrolyte aqueous solution over wide ranges of temperature, pressure, and liquid concentrations. Because of a very wide range of liquid concentrations of ammonia in water from 3.1 to 25.1 m, the vapor-pressure calculations for this system give the largest error of 13.96% (see Table I). To demonstrate the usefulness of this equation of state, we also compare our results with those obtained from the model of Edwards et al. (1978) (see Figure 6). For very low liquid concentrations (less than 0.5 M), the predictions obtained from both our equation of state and their model show good agreement with the experimental data. However, since the model of Edwards is based on compensating the Debye-Huckel law and cannot predict ionic activity coefficients at higher electrolyte concentrations, our calculations give much better results for concentrations of the
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1741 M O L A R I T Y OF E L E C T R O L Y T E ( m o l / k g H , O )
c. 0 1
'
1
'
9.0
,O.O
'
NH, -T h i s
I
0.07
0.u
0.08
'
(T=353OK) Model
__ E d_ w a r d. s. e t (1978)
3332'K 3532OK 373Z'K
27.0 0.05
18.0
' H,O
Exp
,'
al.
data
,'
CO,
-
H,O ( T = 2 9 3 W )
?
5
M O L A R I T Y OF SO, ( m o l S O , / k g H , O )
Figure 4. Calculated vapor pressures of sulfur dioxide in water at VLE compared with experimental data of Rabe and Harris (1963).
04
12
08
3.0
, OB
,
/ 1.2
,
jP
IW
M O L A R I T Y OF E L E C T R O L Y T E (mol/kgH,O)
Figure 6. Comparisons of vapor pressures for four single solute systems, C02-H20, S02-H20, NH3-H20, and H2S-H20, calculated by using this EOS and the model of Edwards et al. (1978) with experimental data. 1
N
SO,
- H,O
'
1
'
1
I
This m o d e l ..... Edwards, et al. (1978) Beutler k Renon (1978) Exp. Data ( R a b e & H a r r i s ) ~
-_
m = 0702
+4
2
'
0:05
'
0'08
'
M O L A R I T Y OF H,S
0'11
'
0'14
'
_.
(molH,S/kgH,OI
Figure 5. Calculated vapor pressures of hydrogen sulfide in water at VLE compared with experimental data of Clarke and Glew (1971).
weak electrolytes higher than 0.5 M. Beutler and Renon (1978) have indicated the limitations of Edwards' model and modified it by taking ion-molecule interactions into account in a better representation of activity coefficients. With three adjustable parameters, Beutler's model extends the validity of Edwards' model to higher electrolyte concentrations. Figure 7 shows comparisons of calculated vapor pressures of sulfur dioxide in water using our model and Beutler's model. One can see that our model (without any adjustable parameters) and Beutler's model (with two adjustable parameters) are of similar accuracy except at highest concentrations. We also have compared Henry's constants calculated from this theory (eq 17) with values estimated from the correlation of Edwards et al. (1978). They determined Henry's constant by regressing data at fiiite concentrations for Hi and the two parameters in the activity coefficient expression of Pitzer. The results are shown in Figure 8. Given the difficulties in determining unique parameter values when regressing so many parameters from the ex-
T E M P E R A T U R E ("C)
Figure 7. Comparisonsof vapor pressures of sulfur dioxide in water calculated by using this model, the model of Edwards et al. (1978), and the model of Beutler and Renon (1978) with experimental data of Rabe and Harris (1963).
perimental data and given that our theory has no adjustable binary parameters, the agreement between our predictions and their correlations is remarkable. Although the calculations presented in this work are preliminary, they demonstrate that this equation of state shows considerable promise for predicting vapor-liquid equilibrium behavior of weak electrolyte aqueous solutions even without any adjustable parameter. To extent this work to systems containing more than one electrolytic solute, it might be necessary to introduce a single binary interaction parameter. Future work also will include electrolytic mixtures containing large molecules and polyions.
1742 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988
D = dipole ID = induced dipole IG = ideal gas L J = Lennard-Jones M = molecule Q = quadrupole r = residure REP = repulsion (1) = liquid phase (v) = vapor phase Subscripts + = cation - - anion el = electrolyte i = component indexes w = water
I
Literature Cited
From Correlaian of Edwards. e t a ] . (1978) From this theory [equation (17)) IO+
”
50.0
1 ‘ 100.0
1
150.0
” ” ’ 200.0 250.0
I 300.0
100
TEMPERATURE (”C)
Figure 8. Calculated values of Henry’s constants for NH3 and C02 for temperature range of 0-300 K from this theory (eq 17) compared with those obtained from the correlation of Edwards et al. (1978).
Acknowledgment
The support of this research b y the Division of Chemical, Biological and Thermal Engineering of the National Science Foundation under Grant CBT-8513434is gratefully acknowledged. We also thank Dr. G . D. Ikonomou for his assistance in manuscript preparation. Nomenclature A = Helmboltz free energy Ai = coefficient in e q 28 C, = h e a t capacity C, = parameter in e q 21 3c = total number of external degrees of freedom per molecule f = fugacity, bar H = Henry’s constant, ( b m k g of H 2 0 ) / m o l A H R = enthalpy of dissociation K,, = dissociation equilibrium constant, mol/kg of H 2 0 k = Boltzmann’s constant m = molarity, mol/ kg of H 2 0 ni = number of component i in system ne = number of electrons o n an ion P = pressure, bar q i = normalized surface area of particle i R = gas constant, (bar.L)/(moLK) rion = ionic radii, cm rCH, = radius of CH2 segment, cm si = number of segments of species i T = absolute temperature, K u = molar volume, L / m o l V = total volume of system, L x = mole fraction in liquid phase y = mole fraction in vapor phase zi = charge number of ion i
Greek Symbols a = average polarizability, cm3 y = activity coefficient Y~ = mean ionic activity coefficient c = characteristic energy per u n i t external surface area w = chemical potential v = stoichiometric coefficient Superscripts
ATT = attraction C = charge
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Vimalchand, P.; Donohue, M. D.; Celmins, I. “Thermodynamics of Dipolar Molecules: The Perturbed-Anisotropic-Chain Theory”. In Equations of State: Theories and Applications; Chao, K. C., Robinson, Jr., R. L., Eds.; American Chemical Society: Washington, DC, 1985; pp 297-313. Xu, Y. N.; Hu, Y. “Prediction of Henry’s Constants of Gases in Electrolyte Solutions”. Fluid Phase Equilib. 1986, 30, 221-228.
Received for review February 2, 1988 Accepted April 21, 1988
CORRESPONDENCE Comments on “Ultrasonic Technique for Dispersed-Phase Holdup Measurements” Sir: Your journal recently published a paper by Bonnet and Tavlarides (1987) concerning the ultrasonic technique for dispersed-phaseholdup measurements, where our work on this topic (HavliEek and Sovovd, 1984) was quoted; however, it was not quoted quite correctly. Bonnet and Tavlarides report that we have correlated experimental data by a linear relationship between the velocities of the sound in the dispersion and in the individual phases and the fractional holdup. In fact, we have found a linear dependence of the sound pulse transmission time on the holdup, described by eq 6 in our paper: L/V = L X / V , + L(1- X ) / V , (1) Substituting 4 for X , t+fcir L / V , t2 for L / V l , and tl for L / V,, one obtains, after rearrangement,
their experiments with the transducers located outside the vessel and the shaft in the middle of the accoustic path, in spite of a significant distortion of the signal from the receiving transducer. For the on-line measurement with automatic evaluation, a sufficient quality of the signal is achieved far easier with the transducers immersed in the dispersion. Using the ultrasonic technique for holdup measurement, one can choose from both alternatives-either the noninvasive technique with the output on the oscilloscope, which is suited for the measurements on the laboratory scale, or the technique with the sensing probe immersed in the liquid and with the automatic evaluation of the transmission time, which can be used in larger equipment up to industrial scale for holdup monitoring and control.
Acknowledgment This is relationship 8 used by Bonnet and Tavlarides to correlate the results of their measurements, and it is called, according to Kuster and Toksoz (1974), the time-average model. In this way, Bonnet and Tavlarides confirmed our finding that the velocity of ultrasound in a dispersion under conditions typical for extraction is independent of the drop size and is described by eq 1 or 2. We would like to add that we used the frequency of ultrasound as 2 MHz and the size of drops as 0.1-2 mm. Accordingly, the maximum ratio of the sound wavelength to the drop diameter was equal to 10, as in the work by Bonnet and Tavlarides. Kuster and Toksoz concluded in their investigation of the velocity of sound in suspensions of solid particles that the time-averaged model was valid only when the ratio of the sound wavelength to the particle diameter was smaller than 0.03, as indicated in Figure 8 of their paper. This shows that the propagation of sound through liquid dispersions differs significantly from that through the suspensions of solids. The ultrasonic technique with a sensing probe immersed in the dispersion and with an automatic evaluation of holdup has been patented and used in our laboratory for continuous holdup measurement since 1982. It enabled us to measure the transient behavior of holdup in a reciprocating plate column extractor (Sovovd and HavliEek, 1986) and to control the holdup automatically. Bonnet and Tavlarides measured the pulse travel times directly on the oscilloscope screen. This method does not require as sharp and distinct signals as the automatic evaluation. This fact permitted them to perform part of OSSS-5SS5/SS/2627-1743$01.50/0
The technique for continuous measurement of the volume ratio of two unmiscible or partially miscible liquids in dispersion is given in Czech. Patent A 0 247 852.
Nomenclature L = acoustic path length, m t+ = sound pulse transmission time through the dispersion, S
tl = sound pulse transmission time through the continuous phase, s t2 = sound pulse transmission time through the dispersed phase, s V = velocity of sound in the dispersion, m/s Vl = velocity of sound in the dispersed phase, m/s V2= velocity of sound in the continuous phase, m/s X,4 = fractional volume dispersed-phaseholdup Literature Cited Bonnet, J. C.; Tavlarides, L. L. Znd. Eng. Chem. Res. 1987,26, 811. HavliEek, A,; Sovovi, H. Collect. Czech. Chem. Commun. 1984.49, 378. Kuster, G. T.; Toksoz, M. N. Geophysics 1974,39, 607. Sovovi, H.; HavliEek, A. Chem. Eng. Sci. 1986, 41, 2579.
H. SovovP,* A. HavliEek Institute of Chemical Process Fundamentals Czechoslovak Academy of Sciences Prague, Czechoslovakia 0 1988 American Chemical Society
