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Holleran, E. M. A Dimensionless Constant Characteristic of Gases, Equations of State, and Intermolecular Potentials. J. Phys. Chem. 1969, 73, 167-173. Holleran, E. M. Accurate Virial Coefficients from PVT Data. J. Chem. Thermodyn. 1970,2, 779-786. Holleran, E. M. The Overall Unit CompressibilityLines for Real and Simulated Fluids. Znd. Chem. Eng. Res. 1990, 29, 632-636. Holleran, E. M.; Hammes, J. P. A Three-Parameter Equation of State for Gases. Cryogenics (Feb) 1975,95-102. Holleran, E. M.; Walker, R. E.; Ramos, C. M. A Correlation of Critical Points. Cryogenics (April) 1975, 210-216. Kleinrahm, R.; Duschek, W.; Wagner, W.; Jaeschke, M. Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Methane in the Temperature Range from 273.15 K to 323.15 K at Pressures up to 8 MPa. J.Chem. Thermodyn. 1988, 20,621-631.
Levelt, J. M. H. The Reduced Equation of State, Internal Energy and Entropy of Argon and Xenon. Physica 1960,26, 361-377. Michels, A.; van Straaten, W.; Dawson, J. Isotherms and Thermodynamical Functions of Ethane at Temperatures Between 0 OC and 150 OC and Pressures up to 200 Atm. Physica 1954, 20, 17-23.
Morsy, T. Ideal Curves. Dissertation Technische Hochschule Karlsruhe, 1963.
Powles, J. G. The Boyle Line. J. Phys. C Solid State Phys. 1988, 16,503-514.
Pry&, R.; Straty, G. C. PVT Measurements, V i Coefficients, and Joule-Thomson Inversion Curve of Fluorine. J. Res. Natl. Bur. Stand. 1970, 74A, 747-760. Schmidt, R.; Wagner, W. A New Form of the Equation of State for Pure Substances and Ita Application to Oxygen. Fluid Phaee Equilib. 1985, 19, 175-200. Sherwood, A. E.; Prausnitz, J. M. Third Virial Coefficient for the Kihara, Exp-6, and Square-Well potentials. J. Chem. Phys. 1964a, 41,413-428.
Sherwood, A. E.; Prausnitz, J. M. Intermolecular Potential Functions and the Second and Third Virial Coefficients. J. Chem. Phys. 1964b, 41, 429-437.
Thomas, R. H. P.; Harrison, R. H. Pressure-VolumeTemperature Relations of Propane. J. Chem. Eng. Data 1982,27, 1-11. Wagner, W.; Ewers, J.; Schmidt, R. An Equation of State for Oxygen Vapour-Second and Third Virial Coefficients. Cryogenics (Jan) 1984, 175-200.
Younglove, B. A. Thermophysical Properties of Fluids. J. Phys. Chem. Ref. Data 1982, 11 (Suppl 11, 1-347.
Receiued for reuiew October 15, 1990 Accepted February 4, 1991
RESEARCH NOTES Prediction of the McAllister Model Parameters from Pure Component Properties for Liquid Binary n -Alkane Systems A new method for predicting the McAllister viscosity model parameters from pure component properties for binary n-alkane liquid systems is reported. The resulta of this method are compared with experimental data.
Introduction Solution of many engineering problems requires the knowledge of the dependence of kinematic viscosities on composition. Moreover, Viscosities of liquid mixtures help in elucidating the fundamental behavior of liquid systems. However, a general and reliable theory for predicting the kinematic viscosities of liquid mixtures from pure component properties is not available yet. Consequently, information on the dependence of viscosity on composition continues to depend on costly and time-consuming experimental measurements. Several models (McAllister, 1960; Auslander, 1964; Heric, 1966; Wei and Rowley, 1984, 1985; etc.) for the prediction of the dependence of viscosities of liquid mixtures on composition have been reported in the literature. McAllister’s model is based on Eyring’s absolute rate theory assuming three-body or four-body interactions. For three-body interactions, the equation reported by McAllister is In b’ = XA3 In b’A + 3xA2xB In b’m+ 3xAxB2 In b’BA + xB3 In b’B - In [xA + x$MB/MA] + 3xA2xB In [(2 + MB/MA)/3] + 3xAxB2 In [(I + 2MB/MA)/3] + xB3
[MB/MAI (1)
where XA and xg are the mole fractions of components A and B, respectively, MAand MB are their respective molecular weights, and U A , uB, and Y are the kinematic viscosities of the pure components and the liquid mixture,
respectively. The model given by (1) contains two adjustable parameters um and YBA. These adjustable parameters are determined by fitting experimental kinematic viscosity-composition data to (1). The McAllister four-body interaction model is In Y = x A 4 In VA + 4xA3xBIn b’m+ ~ X A ~ XhB b ~ ’m~ 4xAxB3 In b’BBBA + xB4 ln YB ln [xA + x$MB/MA] + 4xA3xB In [(3 + MB/MA)/4] + ~ ~ A In ~ [(I X + B MB/MA)/2] ~ + 4XAxB3 In [(I + 3MB/MA)/4] + xB4 In (k?B/MA) (2) This contains three adjustable parameters, v u , vmB, and VBBBA, which again are determined from kinematic viscosity-composition data. The major drawback of both of the McAllister models is the presence of the adjustable parameters. This is bec a w the determination of these parameters requires costly experimental data. Therefore, the development of a technique for predicting the values of the McAuister model parameters from pure component properties would be a significant improvement. The unsatisfactory state of the art with respect to the structure of liquids led one of the present authors to break liquid solutions into three classes, viz., regular solutions, n-alkane solutions, and associated solutions (Asfour, 1980). Such a classification led to success in tackling molecular diffusion problems in liquids, for example (Asfour, 1985; Dullien and Asfour, 1985; Asfour and Dullien, 1986). Such
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+
Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991 1667 Table I. McAllister's ThresBody Model Parameters and Percent Absolute Average Errors Committed in Predicting Kinematic Viscosity Values; Data Obtained in Our Laboratory
system n-octane (Ah-undecane (B)
n-tridecane (A)-n-pentadecane (B)
n-decane (A)-n-tridecane (B)
n-undecane (A)-n-tridecane (B)
temp, K 293.15 298.15 308.15 313.15 293.15 298.15 308.15 313.15 293.15 298.15 308.15 313.15 293.15 298.15 308.15 313.15
calcd from exptl data 10%106vBA 1.017 1.293 0.963 1.199 0.857 1.055 0.811 0.992 2.859 3.260 2.587 2.937 2.173 2.426 1.980 2.224 1.627 2.024 1.507 1.854 1.300 1.583 1.216 1.469 1.867 2.150 1.718 1.967 1.469 1.673 1.365 1.550
a classification can be extended to treating viscosity of liquid mixture problems. The objective of this Research Note is to report a new technique for predicting the values of the McAllister three-body and four-body model parameters from pure component properties in the case of liquid n-alkane binary systems. It should be noted here that viscosity-composition data on n-alkane binary liquid systems are scarce in the literature. Consequently, we resorted to collecting our own data at several temperature levels and used them to validate the proposed technique. Work is currently progressing in our laboratory to develop similar expressions for regular solutions. Preliminary results obtained so far seem to be promising.
Development and Discussion of the Technique for Predicting the McAllister Model Parameters (i) The McAllister Three-BodyModel. Experimental kinematic viscosity-composition data reported by Cooper (1988)on the following n-alkane binary systems were fitted to (1)to determine the values of uAB and uBA: n-hexanen-heptane, n-hexane-n-octane, n-heptane-n-octane, nheptane-n-decane, n-octane-n-decane, and n-tetradecane-n- hexadecane. The kinematic viscosity-composition data employed here were obtained at 293.15,298.15,308.15,and 313.15 K. Having determined um and YBA for each of the above systems at each temperature level, the values of these parameters were plotted versus their corresponding temperature. This showed that these parameters were temperature dependent. Since the parameter YAB involves, according to McAllister (1960),two A- and one B-type interactions, then assume that
calcd from (5)and (6) 10%Wu, 1.026 1.304 0.963 1.215 0.856 1.066 0.810 1.003 2.857 3.271 2.947 2.587 2.434 2.156 2.230 1.983 2.040 1.631 1.871 1.507 1.597 1.306 1.220 1.484 1.865 2.160 1.716 1.977 1.472 1.679 1.557 1.371
to sL
4.04
-
P. -
0.4 0.3 0.2 0.2 0.07 0.1 0.15 0.1 0.02 0.02 0.03 0.03 0.1 0.1 0.15 0.2
0 0
3
%a Y
46 a h av error
1.02--.
ha I
8
-.
m 0
u
R 0 I
I
P
1
The slope of the straight line is kland the intercept = 1.0. This is logical since at N B - NA = 0, NA = N B ; i.e., components A and B are identical and have a viscosity equal to that of the pure substance. The value of kl was found to be 0.044for the above-named systems where I N B - N A l Y A B OC (vAvAvB)''' (3) I3. Equation 5 allows the calculation of the value of the where UA involves only A-type interactions and UB involves McAllister parameter, vu, from pure component viscosities B-type interactions. The power 1/3 is required for diand the number of carbon atoms. mensional consistency. Using the same rationale Dividing (4)by (3)gives vBA a (vAvBvB)'/3 (4) VBA = vAB(vB/vA)'/' (6) Plotting vm/(vAvI\vB)1/3 versus the inverse of the absolute Equation 6 permits the calculation of the McAllister patemperature, 1/TIgave horizontal lines, thus indicating rameter, uBA, from the value of v u , obtained from (5), and independence of the lumped parameter U ~ / ( U A , V A Uof~ ) ~ ~ ~the pure component kinematic viscosities. temperature (Figure 1). The lumped parameter YBA/ Experimental kinematic viscosity data obtained in our ( U A U B U # ~ ~ behaves in an identical manner. laboratory for the systems octane-undecane, tridecaneA plot of the lumped parameter vAg/(vAvAvB)''' versus pentadecane, decane-tridecane, and undecane-tridecane [ ( N B - NA)2/(NA2NB)1/8], where NA and NB are the number in the temperature range 293.15-313.15 K were used to teat
1668 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
Table 11. McAllirter’r Four-Body Model Parameters and Percent Absolute Average Errors Committed in Predicting Kinematic Viscosity Valuer; Data Obtained in Our Laboratory ~~
temp, K 293.15 298.15 308.15 313.15 n-octane (A)-n-pentadecane (B) 293.15 298.15 308.15 313.15 n-decane (A)-n-pentadecane (B) 293.15 298.15 308.15 313.15 n-undecane (A)-n-pentadecane (B) 293.15 298.15 308.15 313.15
system n-octane (Ah-tridecane (B) . . . .
10%1.196 1.055 0.948
l@vwB
1.435 1.388 1.195 1.136 1.963 1.811 1.531 1.428 2.338 2.122 1.793 1.661 2.539 2.299 1.944 1.791
0.884 1.324 1.248 1.120 1.052 1.750 1.628 1.402 1.310 2.046 1.891 1.597 1.490
1OavBBBA 1.972 1.792 1.542 1.428 2.761 2.507 2.111 1.946 2.962 2.688 2.238 2.057 3.090 2.795 2.316 2.125
10%1.109 1.038 0.919 0.867 1.296 1.209 1.063
l@vwB
1.481 1.374 1.196 1.120 1.917 1.764 1.515 1.411 2.288 2.090 1.774 1.643 2.512 2.287 1.926 1.780
1.000 1.748 1.612 1.393 1.299 2.033 1.866 1.594 1.482
10B5BBA 1.972 1.818 1.555 1.447 2.834 2.574 2.158 1.990 2.996 2.711 2.259 2.077 3.104 2.804 2.328 2.138
error 0.3 0.4 0.5 0.5 0.7 0.8 0.9 1.0 0.3 0.4 0.2 0.3 0.3 0.3 0.15 0.15
actions and two B-type interactions; and the third parameter involves three B-type interactions and one A-type interactions. Again, plotting v m B / ( V A v B ) 1 / 4 versus 1/T gives horizontal lines similar to those shown in Figure 1, thus indicating independence of vMBB of temperature. Now, using the rationale employed earlier in the case of the three-body model, one obtains
. I
4 . 0
I
0.5
Figure 2. Variation of lumped parameter
1
i.o
- NA)*/(NA*NB)’/*] for 8y8bmS for which INB- NAI
I
I
!.5
Plotting v m B / (vA2vB2)1/4 versus ( N B - NA)2/(NA2NB2)’I4 yields, again,a straight line with kz as the slope. The value of k2 = 0.03 is obtained by least squares from data on the systems n-hexane-n-decane, n-heptane-n-dodecane, nheptane-n-tetradecane, n-octane-n-tetradecane, n-hexane-n-tetradecane, and n-hexane-n-hexadecane at different temperature levels. Now, assume that
with [(NB 3. SYIllbOh
V A A A B a (vA3vB)’14
are the same as in Figure 1.
(5) and (6). Table I shows a comparison between the valuea of v u and ~ B determined A from fitting experimental data to (1)using least squares and those calculated from (5) and (6). The close agreement is very obvious. Moreu and YBA calculated from (5) and (6) pver, the values of v were substituted in (1)and the kinematic viscosities of the mixture were calculated over the entire composition range and then compared with the experimental kinematic viscosity data. The percent average absolute error was calculated by the equation
T h e percent average absolute error for each system is reported in Table I. (ii) The McAlliclter Four-Body Model. For binary n-alkane systems with Irv, - Nbl 1 4, the error committed in predicting the kinematic viscosity of a mixture using the values of the McAllieter model parameters v u and ~ B A wlculated according to the proposed procedure from (5) and (6) was in the range of 2-7%. Therefore, it was decided to develop a modified calculation technique valid for the four-body model. This, as shown later, resulta in significantly reduced errors. The four-body model given earlier by (2) involves three interaction parameters, viz., vu, U ~ B and , Q B B ~ The first parameter involves three A-type interactions and one 8-type interaction; the second involves two A-type inter-
(9)
2 2 114 VAABB a (vA vB
(10)
VBBBA a (flAvB3)1/4
(11)
Dividing (9) by (10) and (11)by (10) yields V
W
=V
m B (vA / VB)
(12)
and
= vAABB(vB/vA)1/4 (13) Consequently, (€9,(121, and (13) can be used for the calculation of the three parameters of the McAllister fourbody interaction model from pure component kinematic viscosities and the number of carbon atoms of each component in a binary n-alkane system. Again the comparison procedure outlined earlier in the case of the three-body model is followed here where data obtained in our laboratory in the temperature range 293.15-313.15 K on the systems octane-tridecane, octane-pentadecane, decane-pentadecane, and undecanepentadecane and the resulta are reported in Table 11. The close agreement is quite evident. VBBBA
Conclusions A technique for predicting the values of the McAllister three-body and four-body model parameters for n-alkane binary syetems from pure component properties ie reported here. The McAllister parameters for the cases of the three-body and four-body models are calculated from the pure component kinematic viscosities and the number of
1669
Znd. Eng. Chem. Res. 1991,30, 1669-1671
carbon atoms of each component. The reported technique predicts values of the parameters that are in a very close agreement with those calculated from experimental data. Acknowledgment We acknowledge with thanks an operating grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). Nomenclature kl = constant in ( 5 ) k2 = constant in (8) n = number of experimental points N = number of carbon atoms per molecule M = molecular weight x = mole fraction Creek Letters v = kinematic viscosity, m2/s
McAllister three-body model interaction parameter = McAllister three-body model interaction parameter vM = McAllister four-body model interaction parameter YBBBA = McAllister four-body model interaction parameter u ~ = McAllister B four-body model interaction parameter vm = YBA
Subscripts
A = component A in a binary mixture B = component B in a binary mixture Registry No. Octane, 111-65-9;decane, 124-18-5; undecane, 1120-21-4;tridecane, 629-50-5; pentadecane, 629-62-9.
Literature Cited Asfour, Abdul-Fattah A. Mutual and Intra-(Self-)Diffusion Coefficients and Viscosities of Binary Liquid Solutions at 25.00
OC.
Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, Canada, 1980. Asfour, Abdul-Fattah A. Dependence of Mutual Diffueivities on Composition in Regular Solutions: A Rationale for a New Equation. Znd. Eng. Chem. Process Des. Dev. 1985, 24, 1306-1308. &four, Abdul-Fattah A.; Dullien, Francis A. L. Dependence of Mutual Diffusivities on Concentration in Liquid n-Alkane Binary Mixtures at 25 O C : A Modification of the Asfour-Dullien Equation. Chem. Eng. Sci. 1986, 41, 1891-1894. Auslander, G. The Properties of Mixtures. Br. Chem. Eng. 1964,9, 610-618. Cooper, Elizabeth, F. Density and Viscosity of n-Alkane Binary Mixtures 88 a Function of at Several Temperatures. M.A.Sc. Thesis, University of Windsor, Windsor, Ontario, Canada, 1988. Dullien, Francis A. L.; Asfour, Abdul-Fattah A. Concentration Dependence of Mutual Diffusion Coefficients in Regular Binary Solutions: A New Predictive Equation. Znd. Eng. Chem. Fundam. 1985,24, 1-7. Heric, E. L. On the Viscosity of Ternary Mixtures. J. Chem. Eng. Data 1966,11,66-68. McAUister, R. A. The Viscosity of Liquid Mixtures. AIChE J. 1960, 6,427-431. Wei, I. C.; Rowley, R. L. Binary Liquid Mixture Viscosities and Densities. J. Chem. Eng. Data 1984,29, 332-335. Wei, I. C.; Rowley, R. L. A Local Composition Model for Multicomponent Liquid Mixture Shear Viscosity. 1985,40, 401-408.
*Author to whom correspondence should be addreeaed.
Abdul-Fattah A. Asfour,* Elizabeth F. Cooper Jiangning Wu Chemical Engineering Department, University of Windsor Windsor, Ontario, Canada N9B 3P4
Rouchdy R.Zahran Chemical Engineering Department, Alexandria University Alexandria, Egypt Received for review June 26, 1990 Revised manuscript received January 2, 1991 Accepted January 24,1991
Kinetic versus Thermodynamic Control in Chlorination of Imidazolidin-4-one Derivatives The monochlorination of 2,2,5,5-tetramethylimidazolidin-4-one (compound P) in chloroform at ambient temperature has been studied by use of 'H NMR and UV measurements. It has been (compound MC3) is the kinetically established that 3-chloro-2,2,5,5-tetramethylimidazolidin-4-one controlled product of this reaction. The second-order rate constant for formation of MC3 was ca. 4.2 X M-' s-l. Following formation of MC3, rearrangement occurred to produce l-chloro2,2,5,5-tetramethylimidazolidin-4-one (compound MC1). This reaction occurred slowly through a second-order process (k = 1.4 X M-' s-l ) and probably involves the formation of C1+ with subsequent reaction with P to produce MC1. This work is relevant to the formation of a new class of biocidal N-halamine compounds. Over the past decade a number of N-halamine compounds in the oxazolidinone and imidazolidinone classes have been synthesized and tested in these laboratories for use as stable biocides in aqueous solution and for hard surfaces. The experimental parameters for many of these biocides were summarized in a recent review by Worley and Williams (1988). The most recent work in these laboratories has resulted in the preparation of a new series of imidazolidinone derivatives that are inexpensive to synthesize, quite stable in water, and generally have greater biocidal efficacies than the previous compounds discussed by Worley and Williams (1988). Data concerning these new biocides have been presented recently in a communication (Tsao et al., 1990) and in an extensive research paper (Tsao et al., 1991).
The parent compound 1,3-dichloro-2,2,5,5-tetramethylimidazolidin-4-one (structure DC in Figure 1) is produced by adding 2 equiv of free chlorine in aqueous alkaline solution to 2,2,5,5-tetramethylimidazolidin-4-one (structure P in Figure 1). If only 1equiv of Clz were used, a monochloramine product should result. Chlorination of the l-position on the imidazolidinone ring to produce 1chloro-2,2,5,5-tetramethylimidazolidin-4-one (compound MC1 in Figure 1)would be expected given the presence of the two electron donating methyl substituents on the ring carbons adjacent to the l-nitrogen which should greatly stabilize the N-Cl bond (Williams and Worley, 1988). Compound MC1, which was used by Toda and co-workers (1972) as a source of amino radicals in an ESR experiment, has been isolated and shown to be biocidal and
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