Ind. Eng. Chem. Process Des. Dev.
performed at the highest temperature (242 “C) with low excess of oxygen in run 2 and high excess in run 6. The oscillations are very complicated and almost aperiodic in run 2. In run 6 at the high oxygen excess, the frequency is very high and the oscillations are almost periodic. High excess of oxygen, together with low temperature (runs 3 and 4), will obviously favor the suppression of the emission of sulfur trioxide. It has been known for the past 10 years (Scheintuch and Schmitz, 1977) that the oxidation of carbon monoxide can oscillate under isothermal conditions and under conditions where the surface reactions do not interact with physical transport steps. Various explanations of this behavior have been proposed and tested by mathematical simulations (Scheintuch and Schmitz, 1977; Olsson, 1984). Of special interest for the present investigation are the observations made by Mukesh et al. (1983),who found that carbon monoxide began to oscillate under isothermal conditions when 1-butene was added to the feed. Unlike the present study, their reactor was a CSTR and the content of 1-butene was as high as 1%, i.e., half the content of carbon monoxide. In our study, the content of carbon monoxide was about 1% and the content of sulfur dioxide only 0.002%. Moreover, the temperature was rather low (150 OC) in their experiments, resulting in relaxation oscillations with low frequency. Cutlip and Kenney (1978) considered that 1-butene competed with carbon monoxide for the same active sites and thus delayed the ignition of the rapid oxidation of carbon monoxide. During the ignition, carbon monoxide is removed both from the gas phase and from the catalyst surface. Consequently, 1butene will be further adsorbed followed by some oxidation, and this increased adsorption eventually extinguishes the oxidation of carbon monoxide. Carbon monoxide begins again to adsorb and accumulates on the surface until the next ignition starts. A similar explanation may be valid for the initiating effect of sulfur dioxide on the oscillatory carbon monoxide oxidation. Since the oscillations are very rapid, it is possible that only the most active sites take part in the oxidation of sulfur dioxide. These sites are rapidly deactivated by adsorbed sulfur trioxide which quenches the
1988,25, 531-536
531
carbon monoxide oxidation. The content of carbon monoxide thus increases rapidly and reduces the adsorbed sulfur trioxide to sulfur dioxide which, on rapid desorbing, gives a fresh active surface. Subsequently, the carbon monoxide oxidation ignites and sulfur dioxide is oxidized to sulfur trioxide, until the active sites are deactivated by adsorbed sulfur trioxide again. The proposed explanation given here is partly based on a study by Yao et al. (1981). They found that the decreased oxidation rate of carbon monoxide on platinum in the presence of sulfur dioxide was attributed to the effect of surface sulfates formed by sulfur dioxide adsorption and oxidation, on the chemisorption capacity for carbon monoxide. As shown from infrared adsorption studies, the surface sulfates inhibited the associative chemisorption of carbon monoxide, which resulted in a lower activity for carbon monoxide oxidation. The effect of the supporting material on the oscillations was discussed by Mukesh et al. (1983). We found that the supporting material had little influence on the oscillatory behavior of the reaction system. The same type of oscillations as in the present study was found when the oxidation was performed in an unsupported platinum catalyst tube used in a previous study (Olsson and Schoon, 1985). Acknowledgment
The financial support of the Swedish Board for Technical Development is gratefully acknowledged. Registry No. SO3, 7446-11-9; SO2, 7446-09-5; CO, 630-08-0; 02,
7782-44-7.
L i t e r a t u r e Cited Cutlip, M. 0.; Kenney, C. N. ACS. Symp. Ser. 1978, No. 65, 475. HiavZek, V.; van Rompay, P. Chem. Eng. Sci. 1981, 36, 1587. Mukesh, D.; Kenney, C. N.; Morton, W. Chem. Eng. Sci. 1983, 38, 69. Nayfeh, A. H.; Mook, D. T. “Nonlinear Oscillations”; Wiley: New York, 1979. Oisson, P.; SchMn, N.-H. Chem. Eng. Scl. 1985, 4 0 , 1123. Oisson, P. Doctoral Thesis, Chaimers University of Technology, Gothenburg, 1984. Pierson, W. R. Chemtech 1978, 6, 333. Scheintuch, M.; Schmitz, R. A. Catal. Rev. Sci. Eng. 1977, 15, 107. Yao, H. C.; Stepien, H. K.; Gandhi, H. S . J. Cafal. 1981, 6 7 , 231.
Received for review July 27, 1984 Revised manuscript received August 19, 1985 Accepted August 27, 1985
A Cubic Equation of State for Polar and Apolar Fluids Hosseln Toghlanl and Dablr S. Vlswanath’ Department of Chemical Engineering, University of Missouri-Columbia,
Columbia, Missouri 652 1 1
A generalized cubic equation of state is presented which utilizes the associating parameter, x,of Halm and Stiel. The substancedependent critical compressibility factor, as well as the adimensional factor, a(T J , was obtained by minimizing the deviatlon in calculated liquid volume, while maintaining the fugacity equivalence of the two phases in equilibrium. These parameters were generalized as functions of Pitzer’s acentric factor, w , and the associating parameter, Thus, for pure substances, calculations require knowledge of only the critical temperature and pressure, w , and x. This incorporation extends the applicability of the equation to highly polar substances, such as ammonia, water, acetic acid, and methanol. This equation may also be used for aromatics including naphthalene and isoquinoline and for high molecular weight paraffins such as hexadecane and eicosane. Ninety-six substances of different chemical nature were tested and gave average absolute percentage deviations in vapor pressure of 1.38%, 2.14%, and 2.08%, respectively, for the proposed, Soave, and Peng-Robinson equations. The deviations in liquid volume are 3.9%, 17.61 % , and 7.71 % , respectively, for the three equations.
re,
x.
Equations of state have proven to be useful tools in industrial applications, allowing prediction of equilibrium phenomena and sizing of process equipment from a rela0196-4305/86/1125-0531$01.50/0
tively small amount of input data. Although the complexity of any equation of state no longer presents a computational problem, the majority of authors proposing 0 1986 American Chemical Society
532
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
equations of state have attempted to retain the simplicity found in the van der Waals’ cubic equation while trying to improve its accuracy through modifications. In spite of its simplicity, the van der Waals’ equation is tremendously successful in representing thermodynamic properties qualitatively. Most of the semiempirical equations of state set forth in recent years have retained the van der Waals’ thermal pressure term. The differences lie in the expression of the attractive pressure term. One of the earliest attempts to modify this attractive term was by Redlich and Kwong (1949). Several authors have modified this term, assuming a temperature dependence for a and replacing the denominator by a quadratic polynomial in volume. Two important modifications which have enjoyed widespread acceptance are the Soave (1972) and the Peng-Robinson (1976) equations of state. Both the Soave and Peng-Robinson equations have been widely tested to determine their predictive capabilities in the liquid region and for vapor-liquid equilibria (VLE) for a large number of systems. Both perform reasonably well in VLE calculations, but the Soave equation typically overestimates liquid volumes, whereas the Peng-Robinson equation does better in the prediction of liquid densities for normal fluids. However, the use of these equations for systems containing heavy hydrocarbons and polar fluids leads to discrepanciesbetween calculated and experimental liquid densities. The major drawback in both the Soave and Peng-Robinson equations is that the critical compressibility takes on values of 0.333 and 0.307, regardless of the substance. In an effort to overcome this fixed critical compressibility factor, Abbott (1973) anticipated that for many applications, the fixed value should be replaced by a substance-dependent adjustable parameter, CC. Typically, this calculated critical compressibility factor is larger than the experimental critical compressibility factor. Many authors such as Harmens and Knapp (1980), Schmidt and Wenzel (19801, and Patel and Teja (1982) have incorporated this adjustable factor in their models, which resulted in improved predictive capabilities for both vapor pressure and liquid volume. All the above equations can be generated from the most general cubic equation given by Martin (1979) and also can be obtained from the equation of Schmidt and Wenzel(l980) with an appropriate choice of the parameters. In this work, we have used the general cubic equation of Schmidt and Wenzel (1980) along with the constraint U + W = 1. However, instead of setting W = -3w, we set it equal to -c, where c is one of the equation parameters. We will also incorporate the polar factor of Halm and Stiel (1967) in our modification so that polar and associating substances can be treated. These choices result in the predicted critical compressibility factor being substancedependent. Proposed Cubic Equation The functional form of the proposed cubic equation of state is aa(Tr) RT = 0 (1) G(P,u,T,a)= -P + --b u2 + bv + cbu - cb2 The three pure component parameters a, b, and c are determined based on the observation of the pressurevolume behavior at the critical point, where a(T,) is equal to unity. Three constraints are imposed on eq 1 at the gas-liquid critical point. The standard requirement of the disappearance of the first two pressure-volume derivatives a t the critical point comprises the first two of these three
constraints. The third constraint arises from incorporation of Abbott’s hypothesis (1973) and defies the apparent or calculated critical compressibility. The constraints imposed are
(3) p ,,,calcd
(4) The third constraint results in a P-V-T surface which has a calculated critical compressibility factor different from the experimental critical compressibilityfactor. This leads to better quantitative prediction of properties in regions removed from the critical point while sacrificing predictive capabilities in the region very near the critical point. This does not lead to the impasse that it would appear to, because methods of treating the critical region and its nonanalytic behavior are under investigation such as that described by Fox (1983), which may well provide a powerful tool in the future. The parameters obtained from applying these three constraints to the equation of state after algebraic manipulation are (5)
1 - 3lc cc = -
(7)
Lvc
where nb(w,x)
=
vctc
fia(u,x) = (1- t c ( 1 - vc)I3
(8) (9)
The value of qc may be determined analytically or numerically by using the equation 17;
+ v2(2/lc - 3) + 3vc - 1 = 0
(10)
once the value of lcis set. If the value of c is set to 3w in eq 7, the above results are identical with those of Schmidt and Wenzel (1980). The critical properties used in applying the constraints at the critical point were taken from Reid et al. (1977). Now the problem arises of selecting the optimal value of the calculated critical compressibility factor, lc. The optimal value of Cc was determined by following the path taken previously by Patel and Teja (1982) and Adachi et al. (1983). The following steps are taken to evaluate the optimum value. 1. A value of l, in the neighborhood of the experimental critical compressibility factor is assumed. 2. The critical parameters are evaluated by using eq 5-10. 3. For each data point along the saturation dome, a value of a( T,) is determined by using the fugacity equivalence of the two phases in equilibrium.
f=t“
(11)
Utilizing the proposed equation of state in the relation
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 A
1 n [ i ] = ~ " [ ~ - & ] d u + Z - l - l n Z (12) the fugacity coefficient takes the form
PROPANOL PENTANOL
where B = bP/RT, H = ((b + cb)' + 4 ~ b ' ) ~FI ~=, P(b + cb - H)/(2RT), and G = P(b + cb + H)/(2RT). 4. The average absolute deviation in the liquid volume is calculated. 5. The assumed value of Cc is incremented by 0.001, and steps 2-4 are repeated until a minimum in the average absolute deviation in the liquid volume is found. The value of lCwhich provided this minimum is used in subsequent calculations. The temperature dependence of a ( T ) is established by following an approach identical with that of Soave by assuming that a ( T ) is composed of two parts: a(T) = aca(Tr) (14) Thus, like Soave's equation, the term a(Tr)must equal unity along the critical isotherm. This provides some insight into the functional form of a(T,), but after many attempts to find a suitable form for a(Tr),it was decided to use the following two expressions given by Soave (1972) and Mathias and Copeman (1983), respectively.
a(Tr)= (1+ m(1 - T,1/2))2 a'(Tr) = (1+ C1(l - Trl/')
533
1.2
\
-
A
A A
A
14
A A
.
E 1.0
t-
L
C
I
A IV
CACETlC
A
a
0.6
(15)
A
LACETONE
1
A A A
+ C2(l - Tr1/2)2+ C3(l - Tr1/2)3)2 (16)
Mathias and Copeman (1983) found it necessary to include higher order terms in the expansion in order to correlate the vapor pressures of highly polar substances. We determined the constants in both of these correlations for each substance by using the Britt-Luecke (1973) generalized least-squares regression algorithm. The coefficient m in eq 15 is plotted vs. the acentric factor in Figure 1. It is seen that the associating and highly polar substances do not follow the same trend as the normal fluids. This has been corrected as shown in the next section. Generalization of a ( T , ) and Cc In order to better predict liquid properties, investigators such as Zudkevitch and Joffe (1970) who modified the original Redlich-Kwong equation, Fuller (1976) who modified the Redlich-Kwong-Soave equation, and Heyen (1980) who modified the Peng-Robinson equation have made the covolume parameter as well as the attraction parameter temperature-dependent. However, as stated by Bazua (1983), this approach may lead to difficulties in calculating thermodynamic properties that require derivatives of the equation of state. Although this approach has proved moderately successful in the prediction of liquid densities for polar substances, we have taken a somewhat different approach, similar to that of Pate1 and Teja (1982). Their generalized equation allowed prediction of both vapor and liquid properties of nonpolar substances. In their work, they also studied polar substances such as water, ammonia, and alcohols. However, these substances could not be included in their generalization as was expected. Thus, in our approach, we include a fourth temperature-independent parameter, in an attempt to include the associating and highly polar substances in our generalization of CC and m. As a possible fourth parameter, we examined the parachor developed by Quayle (1953), the radius of gyration
0 DIPOLE MOMENT > 0 . 8 A DIPOLE MOMENT 0.8
