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An equation of state for hydrogen fluoride
Fluid Phase Equilibria, 86 (1993) 47-62 Elsevier Science Publishers B.V., Amsterdam
47
An equation of state for hydrogen fluoride Chorng
H. Twu
Simulation (Received
*, John E. Coon
and John R. Cunningham
Sciences, Inc. 1051 W. Bastanchury June 23, 1992; accepted
Road, Fullerton,
in final form October
CA 92633 (USA)
11, 1992)
ABSTRACT Twu, C.H., Coon, J.E. and Cunningham, J.R., fluoride. Fluid Phase Equilibria 86: 47-62
1993. An equation
of state‘for
hydrogen
Twu et al. (1992) [Twu, C.H., Coon, J.E. and Cunningham, J.R., 1992. An equation of state for carboxylic acids. Sixth International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, July 19-24, 1992 in Italy. Fluid Phase Equilibria, 82: 379-3881 applied a generic cubic equation of state to carboxylic acids by taking association into account. The incorporation of chemical theory for carboxylic acids into the cubic equation of state was based on a monomerdimer chemical equilibrium model. Hydrogen fluoride strongly associates by hydrogen bonding and strong evidence indicates that the vapor exists primarily as monomer and hexamer (Long et al., 1943 [Long, R.W., Hilderbrand, J.H. and Morrell, W.E., 1943. The polymerization of gaseous hydrogen and deuterium fluorides. J. Am. Chem. Sot.. 6.5:182- 187.1). Using the same approach as Twu et al. (1992) a monomer-hexamer chemical equilibrium model is built into a cubic equation of state to account for association of hydrogen fluoride. A closed-form equation of state is derived, which is written in terms of monomer parameters and the monomer-hexamer chemical equilibrium constant. This paper provides a new method for calculating the properties of HF. The calculated fugacity coefficient, vapor compressibility factor, heat of vaporization, and enthalpy departure of hydrogen fluoride exhibit significant deviations from ideal behavior. Failure to take this chemical association into account can lead to serious errors in vapor-liquid and liquiddliquid phase equilibrium and energy balance calculations.
INTRODUCTION
Hydrogen fluoride is an important chemical in the chemical industry. It is used in the HF alkylation process and in the manufacture of refrigerants and other halogenated compounds. Because of the toxic and corrosive nature of HF, very few data are available in the literature. The ability to describe the properties of HF at any temperature and pressure using an equation of state is an important step in the proper design and operation of
* Corresponding 0378-3812/93/$06.00
author. 0
1993 - Elsevier
Science Publishers
B.V. All rights reserved
48
C.H. Twu et al. / Fluid Phase Equilibria 86 (1993) 47-62
the industrial processes mentioned above. Modeling the properties of HF properly is especially important in the final separation and purification of hydrogen fluoride in these industrial processes. There have been many conflicting models proposed for the description of the self-association of HF gas. Models such as monomer-hexamer, various monomer-n -mer, monomer-trimer-hexamer, monomer-n-mer-m-met-, monomer-dimer-tetramer-hexamer, and indefinite self-association have been proposed. Owing to evidence showing that the vapor exists primarily as monomer and hexamer (Long et al., 1943) a monomer-hexamer model has been adopted in this work. However, the results from this work indicate that a simple monomer-hexamer model may not be quite proper for HF. There might be polymers with a higher and/or lower degree of association present in HF. Nevertheless, a method is proposed to correct the monomer-hexamer model for this deficiency so that the vapor pressure of HF can be accurately reproduced and other thermodynamic properties such as the vapor compressibility factor, heat of vaporization and enthalpy departure can be correctly predicted. HF is polar and hydrogen bonded and is therefore a highly associated compound. Hydrocarbons, on the other hand, are non-polar and nonassociated. As such, HF and hydrocarbons are immiscible in each other at low temperatures. The available theories for liquid solutions do not adequately deal with such mixtures. The method proposed here should be more appropriate for dealing with systems like this. This will be demonstrated in a separate paper by extending it to mixtures and using it to model the HF alkylation process. CHEMICAL
A generic equation
RT P=(v-
THEORY
two-parameter
cubic equation
a(T) u* + ubv + Wb2
of state can be expressed
by the
(1)
where P is the pressure, T is the absolute temperature, u is the molar volume and R is the gas constant. The constants u and w are typically integers. The parameters a and b at the critical temperature, a, and b,, are found by setting the first and second derivatives of pressure with respect to volume to zero at the critical point. Equation (1) refers to one mole of substance. We rewrite the equation in terms of V/nT, where V is the volume of the total number of moles, nT.
nTRT ‘=(V--n,b)-
n$a(T) V*+zq-bV+wnGb*
(2)
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
49
The total number of moles, nr, is not a constant because, for an associating fluid, it depends on the temperature and the density. That is, the “pure” hydrogen fluoride is considered here to be a mixture of monomers and hexamers. Therefore a and b depend on the “true” composition of the fluid. The deviation of hydrogen fluoride from the phase behavior of common polar or non-polar components can be explained by chemical theory. On the basis of chemical theory, hydrogen fluoride is treated as a mixture of monomers and hexamers which are in chemical equilibrium. The formation of associated hydrogen fluoride is relatively stable, since the six monomers are bound together by hydrogen bonds and formed a benzene-like structure. Let Zi stand for the mole fraction of species i (monomer and hexamer) in the mixture. Equation (2) can be extended to the calculation of mixture properties if the constants a and b are replaced by any usual set of mixing rules, such as the classical quadratic one a =
1 C z,z~(u,u,)‘/~ 1
b =
J
c z,bi
Heidemann
and Prausnitz (1976) that assumed for i-mers
a, = i2al
(5)
b, = ib,
(6)
where 1 and i refer to the monomer and i-mer, respectively. They then derived the mixing rule for a pure fluid a =
(7)
(%/nT>2al
(8)
b = hlnT)b,
The total number of moles of monomer and hexamer, nT, and the number of moles that would exist in the absence of association, no, are related by @O/nT>
=
1
(9)
izl
True mole fractions in terms of association species are defined as: z, = 12,/nT
(10)
where ~1~is the number of moles of monomer and n6 the number of moles of hexamer. Substituting eqns. (7) and (8) into (2) and introducing the quantity ?z,= +/no, the equation of state incorporating association becomes RT ___’ = nr (u _ b)
a(T) u’+ ubu + wb2
(11)
50
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
where p&=-n,; no
DE-
V
(12)
no
where a and b refer to the monomer (the subscript 1 has been dropped for convenience). Except for the factor n, appearing in the first term, eqn. (11) is identical to eqn. (1). This indicates that the term ~1,is the only contribution of association to the equation of state and the attractive term becomes independent of association. This simplified result follows from the choice of pure-fluid mixing rules, eqns. (7) -( 8). The quantity of interest emerging from the chemical theory is the factor II,, the extent of association. It is 1.0 when there is no association and approaches l/6 when there is complete hexamerization. In order to solve for the extent of association and for the true mole fractions, the chemical equilibrium is considered. The chemical theory postulates that there is a hexamerization equlibrium of the type (13)
6( HF) ++ (HF)h
where six HF monomers form a hexamer, (HF&. To describe this chemical equilibrium quantitatively, a chemical equilibrium constant is defined as (14) where 4 is the fugacity coefficient of the true species (monomer or hexamer), z is the true mole fraction and P is the total pressure. The equilibrium constant is a function of temperature only. The fugacity coefficients are found from the equation of state, using classical thermodynamics. To solve for n,, the ratio of the fugacity coefficients in eqn. (14) is evaluated and eqns. (9) and (14), and the material balance (2, = 1) are solved. The resulting expression using the generic cubic equation of state, eqn. ( 1 l), is K*~:=(6-52,)~(1
-2,)
where K* is the reduced K*=K
[
RT (u-b)
(15) equilibrium
1 5
(16)
Equation (15) can be solved fraction of monomer, z,, by nT
constant,
for z~, and n, is related
mole
1 (17)
%=(6-k,) Equation
to the true
(17) is derived
from eqns. (9) and (14).
51
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
The power of z1 in eqn. (IS) can be up to 6. The solution of eqn. ( 15) is sensitive to the initial value of zI . To solve eqn. (15) for zI, the simple equation, which is derived from the monomer-dimer model, 3-(1+X*)“*
z, =
with
2(1 -K*)
RT
K* = K (v _ b) [
1
(18)
can be used to give a starting value. The z1 can then be solved within a few iterations. Equation (11) is the closed form of the equation of state with the chemical equilibrium monomer-hexamer model. ~1,is given by eqn. (17). The expression for the fugacity coefficient of “pure” hydrogen fluoride using the equation of state incorporating chemical association is derived from eqn. (11). In this work, the Redlich-Kwong equation of state is used (i.e. setting u = 1 and w = 0 in eqn. (11)). f In F 0
=ln(n,z,)
+(Z-n,)
-ln(Z-b*)
-$ln
l+$ (
>
(19)
where the true mole fraction of monomer, z, , and IZ,are calculated from the equilibrium constant and volume at the system temperature and pressure (eqns. (15)-( 17)). Z, a * and b* in eqn. ( 19) are defined as z=---
PV
(20)
n,RT
a* =-
Pa
(21)
R*T*
b*=Rp RT
(22)
As mentioned previously, a and b refer to the monomer (the subscript 1 has been dropped). Z in eqn. (19) is the compressibility factor without reference to the degree of association between monomers. Z is the solution of the following non-cubic (in Z) equation: Z3 - n,Z* + (a* - n,b* - b**)Z - a*b* = 0 CRITICAL
CONSTRAINTS
FOR HYDROGEN
(23)
FLUORIDE
Although hydrogen fluoride consists of monomer and hexamer species in our model, the monomer is the only independent species. Therefore, the parameters appearing in the associated cubic equation of state (CEOS) for the hydrogen fluoride can be treated in the same way as in the common CEOS. The monomer parameters a and b in the associated CEOS have been found, as usual, from the critical point and from setting the first and second derivatives
52
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
of pressure with respect to volume to zero at the critical point. The resulting expressions derived from the associated CEOS are more involved than those from a non-associated CEOS, because it may not be assumed that an associating fluid becomes monomeric at its critical point. As in the case of the common CEOS, the critical constraints result in three expressions for three unknowns a,, b, and 2,. Actually, a, and b, are the only true unknowns appearing in these equations, because T,, PC, and V, are properties of a substance, having numerical values independent of any equation of state. In solving these three equations, V, is in fact treated as a third unknown. The procedures for solving for these three unknowns from the associated CEOS are almost the same as for the common CEOS. The chemical equilibrium constant given by Long et al. (1943) is used to calculate the equilibrium constant at the critical temperature for hydrogen fluoride. For hydrogen fluoride the equilibrium constant is given by the equation, log,, K = -43.65 + 8910/T
(24)
where K is in mmHgP5 and Tin K. The critical temperature, T, = 461.15 K, and the critical pressure, PC = 64 atm, for hydrogen fluoride are taken from Daubert and Danner (1990). At the critical point, the “pure” hydrogen fluoride is a mixture of monomer and hexamer. Applying eqn. ( 11) at the critical point for the hydrogen fluoride, we have solved equations for three unknowns in terms of T, and PC, a(T,)
=
R=T=
0.110894 p
c
(25)
(26) (27) where the subscript “c” denotes the critical point. It is worth noting that the calculated critical compressibility with association is slightly more realistic than the one without association (0.3213 in this work versus 0.3333 from the Redlich-Kwong CEOS (Soave, 1972)). Twu et al. (1992) report that the calculated critical compressibility with association for acetic acid is 0.2912. This implies that the HF molecules are not associating as strongly as acetic acid molecules at the higher temperatures. The critical molar volume is then calculated from Z,, T, and PC and these values are subsequently used in calculations for the true mole fraction of monomer and the extent of association. The results are z1(Tc) = 0.7006
(28)
n,( T,) = 0.4005
(29)
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
53
The extent of hexamerization in the liquid should decrease with increasing temperature but the value of n, is still far different from unity at the critical point. This indicates that there is a considerable amount of hexamer even at the critical point. Since Heidemann and Prausnitz (1976) developed their elegant approach for associating fluids many authors have proposed ways to fit the monomer parameters to the pure-component properties, instead of deriving the exact solution (eqns. (25) to (27)) relating the monomer parameters to experimental critical temperature and critical pressure in a way consistent with that of the common CEOS. One of the advantages of this procedure for finding these parameters is that when the chemical association disappears, the parameters are restored back to those of the non-associated RedlichKwong CEOS as follows: R2T2 a(T,)= 0.42748Op' c
(30)
RTc b =0.0866403p c
(31)
z,=o.3333
(32)
In this way, all the pure component parameters, regardless of associating or non-associating components, are calculated in a consistent manner which is crucial for mixture property prediction. This consistency becomes important in mixture calculations, especially for systems involving both associating and non-associating components. The value of a(T) at other temperatures can be calculated from a(T) = a(T
(33)
where cc(T) is a temperature-dependent function which takes into account the attractive forces between molecules. The real gas approaches an ideal gas at high temperature, thus requiring that c1 goes to a finite value as temperature becomes infinite. In our model there are two parameters in the associated CEOS, i.e. a(T) and b, for hydrogen fluoride. Since the parameters a, and b have been found from the critical point, all that remains is to evaluate a(T). a(T) is found from vapor pressure data. THE ct FUNCTION
As mentioned previously, many authors use vapor pressure and liquid density data to fit the parameters in the CEOS. However, the ability of a
54
C.H. Twu et al. / Fluid Phase Equilibria
86 (1993) 47-62
CEOS to correlate phase equilibria of mixtures depends upon the accurate prediction of pure-component vapor pressures and mixture properties, not the liquid density. Since a CEOS is notorious for not being able to correlate liquid density, parameters fitted to the liquid density data usually will create inconsistent problems in the prediction of other properties. Therefore it is proposed that only the vapor pressure data from the triple point to the critical point as reported from Daubert and Danner ( 1990) should be used to find the CIfunction. As shown by Twu et al. ( 1991), a properly chosen temperature dependent CI function permits a CEOS to match vapor pressure data with high accuracy. During the calculations, the equilibrium constant at a specified temperature is calculated from eqn. (24). The following procedure is used to find the value of the cc function. The first step is to solve the associated CEOS, eqn. (23), for the compressibility factor. Equation (23) which is no longer cubic in volume, can be solved numerically. At each specified temperature and pressure, the liquid volume and vapor volume can be found numerically from eqn. (23) by giving proper initial guesses. For the liquid compressibility factor, a proper initial value is slightly greater than b* and for the vapor compressibility factor, a proper initial value is 1.O + b*. After solving for the densities, these values are used to compute the value of a which yields equal vapor and liquid fugacities at each temperature and its saturated vapor pressure. The a values depend on the temperature. The sequence of c( calculations starts from the critical point down to the triple point. At the critical point, the exact value of a is unity. This c( value is used as the initial guess for the next lower temperature. This procedure is repeated until the CIfunction is found for all temperatures from the critical point down to the triple point. The resulting CIversus temperature curve can indicate whether the current model is adequate for hydrogen fluoride or not.
RESULTS AND DISCUSSION
Calculations in this work are based on the CEOS incorporating monomer-hexamer chemical theory. The CI values for hydrogen fluoride corresponding to the saturated vapor pressure data are shown in Fig. 1. The a function represents the physical and chemical interaction forces between molecules. The expected characteristics of this CI function are that it decreases monotonically with increasing temperature and equals unity at the critical point. Figure 1 does not exhibit the first of these characteristics. Instead, it exhibits extrema in the c( behavior. This unusual behavior can probably be explained by the presence of one or more polymers with higher and/or lower
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
190
240
290
340
Temperature Fig. 1. Hydrogen
fluoride
390
440
55
490
(K)
alpha function
along the vapor-liquid
saturation
curve.
degrees of association. This implies that the monomer-hexamer chemical equilibrium model might not adequately describe hydrogen fluoride behavior. Examination of the a values, however, shows that their values are all quite close to unity. From a practical point of view, therefore, it would seem that the monomer-hexamer model accounts reasonably well for the HF association. The method proposed in this work is to assume that the monomer-hexamer association model can account for the HF association and let the calculated a values reflect the inadequacy of the model. As long as these real c(values are used, the vapor pressure and critical point of hydrogen fluoride can be exactly reproduced. The question that remains is how to correlate these a values? None of the existing CIforms proposed in the literature can correlate these values as shown in Fig. 1. Because of this, a well-known special function, the cubic spline, is adopted for correlating the a values for HF. Cubic splines have several useful properties: (1) They pass exactly through each data point, while still maintaining a smooth curve; (2) With cubic polynomial splines, further mathematical manipulations (e.g. differentiation) are usually simple; (3) For thermodynamic properties like enthalpy and heat capacity, first and second derivatives are required. Cubic splines provide functional relationships for these derivatives. However, cubic splines can only be used to interpolate between data, they cannot be used to extrapolate data. Because of this, the inclusion of some estimated a data at supercritical conditions as a skeleton is highly desirable. The agreement of
C.H. Twu et al. /Fluid Phase Equilibria 86 (1993) 47-62
56
the vapor pressure, and other properties discussed below with the monomer-hexamer equilibrium model indicates that the model is a reasonable method of accounting for the HF association. The fugacity coefficient for hydrogen fluoride along the vapor-liquid saturation curve from the triple point to the critical point is shown in Fig. 2. The figure shows that although the saturated pressure is far below atmospheric at the low temperatures, the fugacity coefficient for hydrogen fluoride is well below unity. Because of the strong hexamerization between hydrogen fluoride molecules, deviations from ideality are large, much larger than one might expect at these low pressures. However, it was found that at sufficiently high temperatures all of the associated molecules tend to dissociate into single molecules of HF. At higher temperatures, HF will be expected to obey the normal (non-associating) gas laws fairly closely. This trend is also indicated by the fact that the calculated critical compressibility (0.3213) is close to that for a non-associating component (0.3333). Figure 3 illustrates the change of the true mole fraction of hexamer with temperature at saturated conditions. The extent of hexamerization is greater in the liquid phase than in the vapor phase. The saturated liquids at low temperatures are almost pure hexamer. Hydrogen fluoride has a strong tendency to hexamerize with itself in the gas phase even at low temperatures; at a temperature of 244.15 K, the mole fraction of hexamer is found experimentally to be 0.6858 (Gillespie et al., 1985). Figure 3 shows that the calculated value at this temperature from this work matches the data well.
V i
o.lr,“,I,,,,I,,,,I,,,,I,‘,,I’,,,I 190
240
290
340
390
440
490
Temperature(K)
Fig. 2. Hydrogen
fluoride
fugacity
coefficient
along the vapor-liquid
saturation
curve.
C.H. Twu et al. 1Fluid Phase Equilibria 86 (1993) 47-62
190
240
290
340
Temperature
390
440
490
(K)
Fig. 3. Hydrogen fluoride hexamer mole fraction along the vapor-liquid saturation curve. Gillespie et al. f 1985) found that the hexamer mole fraction to be 0.6858 at a temperature of 244.1 S K.
The mole fraction of hexamer decreases with increasing temperature along the coexistence curve in both the vapor and liquid phases. In this region, it seems that the effect of temperature on the true mole fraction dominates the effect of the density. However, close to the critical point, the hexamerization in the vapor phase reaches a minimum, then increases with temperature until reaching the critical point. This trend seems to be due to a sharp increase in density in the critical region as shown in Fig. 5. This phenomenon leads to a maximum close to the critical point in the fugacity coefficient along the saturation curve, as shown in Fig. 2. This phenomenon also leads, in the lower temperature range, to an increase with temperature in the difference of the hexamer mole fraction of liquid and vapor, and in the higher temperature range to a decrease with temperature in this quantity. It follows that the enthalpy required to break the hydrogen bonds during evaporation increases with increasing temperature. The normal decrease in the heat of vaporization with increasing temperature is thus slowed or even fully compensated so that an increase is observed. Figure 4 illustrates that the HF heat of vaporization increases with increasing temperature to a maximum at a temperature close to the critical temperature, then decreases rapidly with temperature to zero at the critical temperature. Figure 5 portrays the coexistence curves for the vapor and liquid compressibility factors as functions of the temperature. It is interesting to note
C.H. Twu et al. 1 Fluid Phase Equilibria
58
86 (1993) 47-62
a, ; 3000 D ;
2500
c‘ g
2000
.N 5 a
1500
P 5
1000
E I”
500
0
190
,I
240
290
340
390
440
490
Temperature(K)
Fig. 4. Hydrogen fluoride heat of vaporization along the vapor-liquid saturation curve. Symbols are data from SimSci databank. The uncertainty of the data is about 10%.
that the compressibility factor increases with temperature for both the coexisting vapor and liquid phases. The normal behavior of the vapor compressibility factor for non-polar or even polar systems is to decrease with increasing temperature. In the region close to the critical point, the vapor phase compressibility reaches a maximum, then decreases with temperature and coincides with the liquid phase at the critical point. If HF were completely hexamerized at low temperatures, then one mole of HF vapor would become only one-sixth of a mole of HF actually present, and the vapor compressibility factor would become l/6. In the liquid phase, at low temperatures, the molecules are already closely packed. As a result, the hexamerization has little effect on the liquid volume. HF has a strong tendency to hexamerize with itself in the gas phase even at low temperatures, the vapor compressibility factor being close to l/6 at low tempera( 1924) report the tures as shown in Fig. 5. Simons and Hildebrand measured vapor compressibility factor of hydrogen fluoride to be 0.223 at a temperature of 234 K and a pressure of 56.2 mmHg. The measured data fall close to the line shown in Fig. 5. In a separation process, phase equilibrium calculations (i.e. K-value) are used to determine the mass balance. In addition to the mass balance, a heat balance is also required. Enthalpy is not only used for the heat balance, but for adiabatic flash and column calculations as well. The enthalpy calculated from an equation of state represents the departure from the ideal gas
C.H. Twu et al. / Fluid Phase Equilibria
190
240
390
340
290
86 (1993) 47-62
59
490
440
Temperature(K) Fig. 5. Hydrogen fluoride compressibility factor along the vapor-liquid saturation curve. Simons and Hildebrand (1924) report that vapor compressibility factor of HF to be 0.223 at 234 K and 56.2 mmHg.
behavior. The enthalpy of the ideal gas shown in Fig. 6 was found from the zero-pressure heat capacity taken from Daubert and Danner (1990). The integration constant of the ideal gas enthalpy was determined arbitrarily by setting the saturated liquid enthalpy of hydrogen fluoride equal to 0.0 at its
a,
6400
ii L P y 3900
1
I
I
I
/f
I
\ I \
Temperature(K) Fig. 6. Hydrogen
fluoride
enthalpy
graphical
view.
60
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
normal boiling point. The saturated liquid enthalpies were obtained by adding the ideal gas enthalpy to the saturated liquid enthalpy departure from the equation of state implemented in this work. The saturated vapor enthalpies are obtained by adding the heat of vaporization, which was given in Fig. 4, to the saturated liquid enthalpy. The complete enthalpy diagram for hydrogen fluoride is shown in Fig. 6. This indicates that the HF enthalpy diagram is entirely different from a normal one, especially for the vapor enthalpy departure and the heat of vaporization. (The heat of vaporization has already been discussed.) The normal behavior of the vapor enthalpy departure for normal components is to increase with increasing temperature, and the vapor enthalpy departure for normal components is almost negligible at the triple point temperature. However, this is not the case for HF. At the triple point, the vapor enthalpy departure for HF is more than 5 times larger than the heat of vaporization. The properties calculated from the proposed method for HF, as shown in Figs. 1-6, are all internally consistent with each other.
CONCLUSION
The degree of non-ideality of a component depends on the strength of molecular interactions. The fugacity coefficient is a measure of non-ideality and a departure of the fugacity coefficient from unity is a measure of the extent to which a molecule interacts with its neighbors. While fugacity coefficients are usually close to unity for non-polar or even polar molecules at low pressures, this is not the case for HF. It exhibits significant deviations from ideal behavior. These deviations are significant in vaporliquid and liquid-liquid equilibria and failure to take them into account can lead to serious errors. In order to predict successfully the fugacity and enthalpy behavior for HF over the entire range of system conditions using chemical theory, an internally consistent method is used to derive the parameters in the associated CEOS. The parameters obtained from this method properly reflect the physical and chemical forces without losing their theoretical background, and the results of the predicted properties are internally consistent. Using the method proposed, the properties of hydrogen fluoride can be represented adequately by assuming that only monomer and hexamer exist in the system. The significant deviations from ideal behavior at low temperatures can be explained by the hypothesis that the monomer and cyclic benzene-like hexamer are the only appreciable species existing in HF.
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
61
LIST OF SYMBOLS
Redlich-Kwong equation of state parameters a, b fugacity of pure component f HF, (HF)e hydrogen fluoride monomer and hexamer, respectively K n0 4 nT
P
R T %w V
V ;
chemical equilibrium constant the number of moles that would exist in the absence of association extent of association total number of moles of monomer and dimer total pressure gas constant temperature volume function parameters molar volume total volume true mole fraction of species i compressibility factor
Greek letters
temperature-dependent fugacity coefficient
function in the parameter a
Subscripts C
i,j 176
critical property property of species i, j monomer, hexamer property, respectively
Superscripts *
reduced property
REFERENCES Daubert, T.E. and Danner, R.P., 1990. DIPPR project 801 Data Compilation, Tables of Physical and Thermodynamic Properties of Pure Compounds. equilibrium Gillespie, P.C., Cunningham, J.R. and Wilson, G.M., 1985. Vapor-liquid measurements for the hydrogen fluoride/hydrogen chloride system. AIChE Symp. Ser., Phase Equilibrium Measurement (DIPPR), 81 (244): 41-48. Heideman, R.A. and Prausnitz, J.M., 1976. A van der Waals-type equation of state for fluids with associating molecules. Proc. Natl. Acad. Sci. USA, 73: 1773-1776. Long, R.W., Hilderbrand, J.H. and Morrell, W.E., 1943. The polymerization of gaseous hydrogen and deuterium fluorides. J. Am. Chem. Sot., 65:182- 187.
62
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Phase Equilibria
86 (1993) 47-62
Redlich, 0. and Kwong, N.S., 1949. On the thermodynamics of solutions. V: an equation of state. Fugacities of gaseous solutions. Chem. Rev., 44: 233. Simons, J. and Hildebrand, J. H., 1924. The density and molecular complexity of gaseous hydrogen fluoride. J. Am. Chem. SOL, 46: 2183-2191. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197. Twu, C.H., Bluck, D., Cunningham, J.R. and Coon, J.E.. 1992. A cubic equation of state: relation between binary interaction parameters and infinite dilution activity coefficients. Fluid Phase Equilibria, 72: 25-39. Twu, C.H., Bluck, D., Cunningham, J.R. and Coon, J.E.. 1991. A cubic equation of state with a new alpha function and a new mixing rule. Fluid Phase Equilibria, 69: 33-50. Twu, C.H., Coon, J.E. and Cunningham, J.R., 1992. An equation of state for carboxylic acids. Sixth International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, July 19-24. 1992 in Italy. Fluid Phase Equilibria, 82: 379-388.
47
An equation of state for hydrogen fluoride Chorng
H. Twu
Simulation (Received
*, John E. Coon
and John R. Cunningham
Sciences, Inc. 1051 W. Bastanchury June 23, 1992; accepted
Road, Fullerton,
in final form October
CA 92633 (USA)
11, 1992)
ABSTRACT Twu, C.H., Coon, J.E. and Cunningham, J.R., fluoride. Fluid Phase Equilibria 86: 47-62
1993. An equation
of state‘for
hydrogen
Twu et al. (1992) [Twu, C.H., Coon, J.E. and Cunningham, J.R., 1992. An equation of state for carboxylic acids. Sixth International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, July 19-24, 1992 in Italy. Fluid Phase Equilibria, 82: 379-3881 applied a generic cubic equation of state to carboxylic acids by taking association into account. The incorporation of chemical theory for carboxylic acids into the cubic equation of state was based on a monomerdimer chemical equilibrium model. Hydrogen fluoride strongly associates by hydrogen bonding and strong evidence indicates that the vapor exists primarily as monomer and hexamer (Long et al., 1943 [Long, R.W., Hilderbrand, J.H. and Morrell, W.E., 1943. The polymerization of gaseous hydrogen and deuterium fluorides. J. Am. Chem. Sot.. 6.5:182- 187.1). Using the same approach as Twu et al. (1992) a monomer-hexamer chemical equilibrium model is built into a cubic equation of state to account for association of hydrogen fluoride. A closed-form equation of state is derived, which is written in terms of monomer parameters and the monomer-hexamer chemical equilibrium constant. This paper provides a new method for calculating the properties of HF. The calculated fugacity coefficient, vapor compressibility factor, heat of vaporization, and enthalpy departure of hydrogen fluoride exhibit significant deviations from ideal behavior. Failure to take this chemical association into account can lead to serious errors in vapor-liquid and liquiddliquid phase equilibrium and energy balance calculations.
INTRODUCTION
Hydrogen fluoride is an important chemical in the chemical industry. It is used in the HF alkylation process and in the manufacture of refrigerants and other halogenated compounds. Because of the toxic and corrosive nature of HF, very few data are available in the literature. The ability to describe the properties of HF at any temperature and pressure using an equation of state is an important step in the proper design and operation of
* Corresponding 0378-3812/93/$06.00
author. 0
1993 - Elsevier
Science Publishers
B.V. All rights reserved
48
C.H. Twu et al. / Fluid Phase Equilibria 86 (1993) 47-62
the industrial processes mentioned above. Modeling the properties of HF properly is especially important in the final separation and purification of hydrogen fluoride in these industrial processes. There have been many conflicting models proposed for the description of the self-association of HF gas. Models such as monomer-hexamer, various monomer-n -mer, monomer-trimer-hexamer, monomer-n-mer-m-met-, monomer-dimer-tetramer-hexamer, and indefinite self-association have been proposed. Owing to evidence showing that the vapor exists primarily as monomer and hexamer (Long et al., 1943) a monomer-hexamer model has been adopted in this work. However, the results from this work indicate that a simple monomer-hexamer model may not be quite proper for HF. There might be polymers with a higher and/or lower degree of association present in HF. Nevertheless, a method is proposed to correct the monomer-hexamer model for this deficiency so that the vapor pressure of HF can be accurately reproduced and other thermodynamic properties such as the vapor compressibility factor, heat of vaporization and enthalpy departure can be correctly predicted. HF is polar and hydrogen bonded and is therefore a highly associated compound. Hydrocarbons, on the other hand, are non-polar and nonassociated. As such, HF and hydrocarbons are immiscible in each other at low temperatures. The available theories for liquid solutions do not adequately deal with such mixtures. The method proposed here should be more appropriate for dealing with systems like this. This will be demonstrated in a separate paper by extending it to mixtures and using it to model the HF alkylation process. CHEMICAL
A generic equation
RT P=(v-
THEORY
two-parameter
cubic equation
a(T) u* + ubv + Wb2
of state can be expressed
by the
(1)
where P is the pressure, T is the absolute temperature, u is the molar volume and R is the gas constant. The constants u and w are typically integers. The parameters a and b at the critical temperature, a, and b,, are found by setting the first and second derivatives of pressure with respect to volume to zero at the critical point. Equation (1) refers to one mole of substance. We rewrite the equation in terms of V/nT, where V is the volume of the total number of moles, nT.
nTRT ‘=(V--n,b)-
n$a(T) V*+zq-bV+wnGb*
(2)
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
49
The total number of moles, nr, is not a constant because, for an associating fluid, it depends on the temperature and the density. That is, the “pure” hydrogen fluoride is considered here to be a mixture of monomers and hexamers. Therefore a and b depend on the “true” composition of the fluid. The deviation of hydrogen fluoride from the phase behavior of common polar or non-polar components can be explained by chemical theory. On the basis of chemical theory, hydrogen fluoride is treated as a mixture of monomers and hexamers which are in chemical equilibrium. The formation of associated hydrogen fluoride is relatively stable, since the six monomers are bound together by hydrogen bonds and formed a benzene-like structure. Let Zi stand for the mole fraction of species i (monomer and hexamer) in the mixture. Equation (2) can be extended to the calculation of mixture properties if the constants a and b are replaced by any usual set of mixing rules, such as the classical quadratic one a =
1 C z,z~(u,u,)‘/~ 1
b =
J
c z,bi
Heidemann
and Prausnitz (1976) that assumed for i-mers
a, = i2al
(5)
b, = ib,
(6)
where 1 and i refer to the monomer and i-mer, respectively. They then derived the mixing rule for a pure fluid a =
(7)
(%/nT>2al
(8)
b = hlnT)b,
The total number of moles of monomer and hexamer, nT, and the number of moles that would exist in the absence of association, no, are related by @O/nT>
=
1
(9)
izl
True mole fractions in terms of association species are defined as: z, = 12,/nT
(10)
where ~1~is the number of moles of monomer and n6 the number of moles of hexamer. Substituting eqns. (7) and (8) into (2) and introducing the quantity ?z,= +/no, the equation of state incorporating association becomes RT ___’ = nr (u _ b)
a(T) u’+ ubu + wb2
(11)
50
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
where p&=-n,; no
DE-
V
(12)
no
where a and b refer to the monomer (the subscript 1 has been dropped for convenience). Except for the factor n, appearing in the first term, eqn. (11) is identical to eqn. (1). This indicates that the term ~1,is the only contribution of association to the equation of state and the attractive term becomes independent of association. This simplified result follows from the choice of pure-fluid mixing rules, eqns. (7) -( 8). The quantity of interest emerging from the chemical theory is the factor II,, the extent of association. It is 1.0 when there is no association and approaches l/6 when there is complete hexamerization. In order to solve for the extent of association and for the true mole fractions, the chemical equilibrium is considered. The chemical theory postulates that there is a hexamerization equlibrium of the type (13)
6( HF) ++ (HF)h
where six HF monomers form a hexamer, (HF&. To describe this chemical equilibrium quantitatively, a chemical equilibrium constant is defined as (14) where 4 is the fugacity coefficient of the true species (monomer or hexamer), z is the true mole fraction and P is the total pressure. The equilibrium constant is a function of temperature only. The fugacity coefficients are found from the equation of state, using classical thermodynamics. To solve for n,, the ratio of the fugacity coefficients in eqn. (14) is evaluated and eqns. (9) and (14), and the material balance (2, = 1) are solved. The resulting expression using the generic cubic equation of state, eqn. ( 1 l), is K*~:=(6-52,)~(1
-2,)
where K* is the reduced K*=K
[
RT (u-b)
(15) equilibrium
1 5
(16)
Equation (15) can be solved fraction of monomer, z,, by nT
constant,
for z~, and n, is related
mole
1 (17)
%=(6-k,) Equation
to the true
(17) is derived
from eqns. (9) and (14).
51
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
The power of z1 in eqn. (IS) can be up to 6. The solution of eqn. ( 15) is sensitive to the initial value of zI . To solve eqn. (15) for zI, the simple equation, which is derived from the monomer-dimer model, 3-(1+X*)“*
z, =
with
2(1 -K*)
RT
K* = K (v _ b) [
1
(18)
can be used to give a starting value. The z1 can then be solved within a few iterations. Equation (11) is the closed form of the equation of state with the chemical equilibrium monomer-hexamer model. ~1,is given by eqn. (17). The expression for the fugacity coefficient of “pure” hydrogen fluoride using the equation of state incorporating chemical association is derived from eqn. (11). In this work, the Redlich-Kwong equation of state is used (i.e. setting u = 1 and w = 0 in eqn. (11)). f In F 0
=ln(n,z,)
+(Z-n,)
-ln(Z-b*)
-$ln
l+$ (
>
(19)
where the true mole fraction of monomer, z, , and IZ,are calculated from the equilibrium constant and volume at the system temperature and pressure (eqns. (15)-( 17)). Z, a * and b* in eqn. ( 19) are defined as z=---
PV
(20)
n,RT
a* =-
Pa
(21)
R*T*
b*=Rp RT
(22)
As mentioned previously, a and b refer to the monomer (the subscript 1 has been dropped). Z in eqn. (19) is the compressibility factor without reference to the degree of association between monomers. Z is the solution of the following non-cubic (in Z) equation: Z3 - n,Z* + (a* - n,b* - b**)Z - a*b* = 0 CRITICAL
CONSTRAINTS
FOR HYDROGEN
(23)
FLUORIDE
Although hydrogen fluoride consists of monomer and hexamer species in our model, the monomer is the only independent species. Therefore, the parameters appearing in the associated cubic equation of state (CEOS) for the hydrogen fluoride can be treated in the same way as in the common CEOS. The monomer parameters a and b in the associated CEOS have been found, as usual, from the critical point and from setting the first and second derivatives
52
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
of pressure with respect to volume to zero at the critical point. The resulting expressions derived from the associated CEOS are more involved than those from a non-associated CEOS, because it may not be assumed that an associating fluid becomes monomeric at its critical point. As in the case of the common CEOS, the critical constraints result in three expressions for three unknowns a,, b, and 2,. Actually, a, and b, are the only true unknowns appearing in these equations, because T,, PC, and V, are properties of a substance, having numerical values independent of any equation of state. In solving these three equations, V, is in fact treated as a third unknown. The procedures for solving for these three unknowns from the associated CEOS are almost the same as for the common CEOS. The chemical equilibrium constant given by Long et al. (1943) is used to calculate the equilibrium constant at the critical temperature for hydrogen fluoride. For hydrogen fluoride the equilibrium constant is given by the equation, log,, K = -43.65 + 8910/T
(24)
where K is in mmHgP5 and Tin K. The critical temperature, T, = 461.15 K, and the critical pressure, PC = 64 atm, for hydrogen fluoride are taken from Daubert and Danner (1990). At the critical point, the “pure” hydrogen fluoride is a mixture of monomer and hexamer. Applying eqn. ( 11) at the critical point for the hydrogen fluoride, we have solved equations for three unknowns in terms of T, and PC, a(T,)
=
R=T=
0.110894 p
c
(25)
(26) (27) where the subscript “c” denotes the critical point. It is worth noting that the calculated critical compressibility with association is slightly more realistic than the one without association (0.3213 in this work versus 0.3333 from the Redlich-Kwong CEOS (Soave, 1972)). Twu et al. (1992) report that the calculated critical compressibility with association for acetic acid is 0.2912. This implies that the HF molecules are not associating as strongly as acetic acid molecules at the higher temperatures. The critical molar volume is then calculated from Z,, T, and PC and these values are subsequently used in calculations for the true mole fraction of monomer and the extent of association. The results are z1(Tc) = 0.7006
(28)
n,( T,) = 0.4005
(29)
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
53
The extent of hexamerization in the liquid should decrease with increasing temperature but the value of n, is still far different from unity at the critical point. This indicates that there is a considerable amount of hexamer even at the critical point. Since Heidemann and Prausnitz (1976) developed their elegant approach for associating fluids many authors have proposed ways to fit the monomer parameters to the pure-component properties, instead of deriving the exact solution (eqns. (25) to (27)) relating the monomer parameters to experimental critical temperature and critical pressure in a way consistent with that of the common CEOS. One of the advantages of this procedure for finding these parameters is that when the chemical association disappears, the parameters are restored back to those of the non-associated RedlichKwong CEOS as follows: R2T2 a(T,)= 0.42748Op' c
(30)
RTc b =0.0866403p c
(31)
z,=o.3333
(32)
In this way, all the pure component parameters, regardless of associating or non-associating components, are calculated in a consistent manner which is crucial for mixture property prediction. This consistency becomes important in mixture calculations, especially for systems involving both associating and non-associating components. The value of a(T) at other temperatures can be calculated from a(T) = a(T
(33)
where cc(T) is a temperature-dependent function which takes into account the attractive forces between molecules. The real gas approaches an ideal gas at high temperature, thus requiring that c1 goes to a finite value as temperature becomes infinite. In our model there are two parameters in the associated CEOS, i.e. a(T) and b, for hydrogen fluoride. Since the parameters a, and b have been found from the critical point, all that remains is to evaluate a(T). a(T) is found from vapor pressure data. THE ct FUNCTION
As mentioned previously, many authors use vapor pressure and liquid density data to fit the parameters in the CEOS. However, the ability of a
54
C.H. Twu et al. / Fluid Phase Equilibria
86 (1993) 47-62
CEOS to correlate phase equilibria of mixtures depends upon the accurate prediction of pure-component vapor pressures and mixture properties, not the liquid density. Since a CEOS is notorious for not being able to correlate liquid density, parameters fitted to the liquid density data usually will create inconsistent problems in the prediction of other properties. Therefore it is proposed that only the vapor pressure data from the triple point to the critical point as reported from Daubert and Danner ( 1990) should be used to find the CIfunction. As shown by Twu et al. ( 1991), a properly chosen temperature dependent CI function permits a CEOS to match vapor pressure data with high accuracy. During the calculations, the equilibrium constant at a specified temperature is calculated from eqn. (24). The following procedure is used to find the value of the cc function. The first step is to solve the associated CEOS, eqn. (23), for the compressibility factor. Equation (23) which is no longer cubic in volume, can be solved numerically. At each specified temperature and pressure, the liquid volume and vapor volume can be found numerically from eqn. (23) by giving proper initial guesses. For the liquid compressibility factor, a proper initial value is slightly greater than b* and for the vapor compressibility factor, a proper initial value is 1.O + b*. After solving for the densities, these values are used to compute the value of a which yields equal vapor and liquid fugacities at each temperature and its saturated vapor pressure. The a values depend on the temperature. The sequence of c( calculations starts from the critical point down to the triple point. At the critical point, the exact value of a is unity. This c( value is used as the initial guess for the next lower temperature. This procedure is repeated until the CIfunction is found for all temperatures from the critical point down to the triple point. The resulting CIversus temperature curve can indicate whether the current model is adequate for hydrogen fluoride or not.
RESULTS AND DISCUSSION
Calculations in this work are based on the CEOS incorporating monomer-hexamer chemical theory. The CI values for hydrogen fluoride corresponding to the saturated vapor pressure data are shown in Fig. 1. The a function represents the physical and chemical interaction forces between molecules. The expected characteristics of this CI function are that it decreases monotonically with increasing temperature and equals unity at the critical point. Figure 1 does not exhibit the first of these characteristics. Instead, it exhibits extrema in the c( behavior. This unusual behavior can probably be explained by the presence of one or more polymers with higher and/or lower
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
190
240
290
340
Temperature Fig. 1. Hydrogen
fluoride
390
440
55
490
(K)
alpha function
along the vapor-liquid
saturation
curve.
degrees of association. This implies that the monomer-hexamer chemical equilibrium model might not adequately describe hydrogen fluoride behavior. Examination of the a values, however, shows that their values are all quite close to unity. From a practical point of view, therefore, it would seem that the monomer-hexamer model accounts reasonably well for the HF association. The method proposed in this work is to assume that the monomer-hexamer association model can account for the HF association and let the calculated a values reflect the inadequacy of the model. As long as these real c(values are used, the vapor pressure and critical point of hydrogen fluoride can be exactly reproduced. The question that remains is how to correlate these a values? None of the existing CIforms proposed in the literature can correlate these values as shown in Fig. 1. Because of this, a well-known special function, the cubic spline, is adopted for correlating the a values for HF. Cubic splines have several useful properties: (1) They pass exactly through each data point, while still maintaining a smooth curve; (2) With cubic polynomial splines, further mathematical manipulations (e.g. differentiation) are usually simple; (3) For thermodynamic properties like enthalpy and heat capacity, first and second derivatives are required. Cubic splines provide functional relationships for these derivatives. However, cubic splines can only be used to interpolate between data, they cannot be used to extrapolate data. Because of this, the inclusion of some estimated a data at supercritical conditions as a skeleton is highly desirable. The agreement of
C.H. Twu et al. /Fluid Phase Equilibria 86 (1993) 47-62
56
the vapor pressure, and other properties discussed below with the monomer-hexamer equilibrium model indicates that the model is a reasonable method of accounting for the HF association. The fugacity coefficient for hydrogen fluoride along the vapor-liquid saturation curve from the triple point to the critical point is shown in Fig. 2. The figure shows that although the saturated pressure is far below atmospheric at the low temperatures, the fugacity coefficient for hydrogen fluoride is well below unity. Because of the strong hexamerization between hydrogen fluoride molecules, deviations from ideality are large, much larger than one might expect at these low pressures. However, it was found that at sufficiently high temperatures all of the associated molecules tend to dissociate into single molecules of HF. At higher temperatures, HF will be expected to obey the normal (non-associating) gas laws fairly closely. This trend is also indicated by the fact that the calculated critical compressibility (0.3213) is close to that for a non-associating component (0.3333). Figure 3 illustrates the change of the true mole fraction of hexamer with temperature at saturated conditions. The extent of hexamerization is greater in the liquid phase than in the vapor phase. The saturated liquids at low temperatures are almost pure hexamer. Hydrogen fluoride has a strong tendency to hexamerize with itself in the gas phase even at low temperatures; at a temperature of 244.15 K, the mole fraction of hexamer is found experimentally to be 0.6858 (Gillespie et al., 1985). Figure 3 shows that the calculated value at this temperature from this work matches the data well.
V i
o.lr,“,I,,,,I,,,,I,,,,I,‘,,I’,,,I 190
240
290
340
390
440
490
Temperature(K)
Fig. 2. Hydrogen
fluoride
fugacity
coefficient
along the vapor-liquid
saturation
curve.
C.H. Twu et al. 1Fluid Phase Equilibria 86 (1993) 47-62
190
240
290
340
Temperature
390
440
490
(K)
Fig. 3. Hydrogen fluoride hexamer mole fraction along the vapor-liquid saturation curve. Gillespie et al. f 1985) found that the hexamer mole fraction to be 0.6858 at a temperature of 244.1 S K.
The mole fraction of hexamer decreases with increasing temperature along the coexistence curve in both the vapor and liquid phases. In this region, it seems that the effect of temperature on the true mole fraction dominates the effect of the density. However, close to the critical point, the hexamerization in the vapor phase reaches a minimum, then increases with temperature until reaching the critical point. This trend seems to be due to a sharp increase in density in the critical region as shown in Fig. 5. This phenomenon leads to a maximum close to the critical point in the fugacity coefficient along the saturation curve, as shown in Fig. 2. This phenomenon also leads, in the lower temperature range, to an increase with temperature in the difference of the hexamer mole fraction of liquid and vapor, and in the higher temperature range to a decrease with temperature in this quantity. It follows that the enthalpy required to break the hydrogen bonds during evaporation increases with increasing temperature. The normal decrease in the heat of vaporization with increasing temperature is thus slowed or even fully compensated so that an increase is observed. Figure 4 illustrates that the HF heat of vaporization increases with increasing temperature to a maximum at a temperature close to the critical temperature, then decreases rapidly with temperature to zero at the critical temperature. Figure 5 portrays the coexistence curves for the vapor and liquid compressibility factors as functions of the temperature. It is interesting to note
C.H. Twu et al. 1 Fluid Phase Equilibria
58
86 (1993) 47-62
a, ; 3000 D ;
2500
c‘ g
2000
.N 5 a
1500
P 5
1000
E I”
500
0
190
,I
240
290
340
390
440
490
Temperature(K)
Fig. 4. Hydrogen fluoride heat of vaporization along the vapor-liquid saturation curve. Symbols are data from SimSci databank. The uncertainty of the data is about 10%.
that the compressibility factor increases with temperature for both the coexisting vapor and liquid phases. The normal behavior of the vapor compressibility factor for non-polar or even polar systems is to decrease with increasing temperature. In the region close to the critical point, the vapor phase compressibility reaches a maximum, then decreases with temperature and coincides with the liquid phase at the critical point. If HF were completely hexamerized at low temperatures, then one mole of HF vapor would become only one-sixth of a mole of HF actually present, and the vapor compressibility factor would become l/6. In the liquid phase, at low temperatures, the molecules are already closely packed. As a result, the hexamerization has little effect on the liquid volume. HF has a strong tendency to hexamerize with itself in the gas phase even at low temperatures, the vapor compressibility factor being close to l/6 at low tempera( 1924) report the tures as shown in Fig. 5. Simons and Hildebrand measured vapor compressibility factor of hydrogen fluoride to be 0.223 at a temperature of 234 K and a pressure of 56.2 mmHg. The measured data fall close to the line shown in Fig. 5. In a separation process, phase equilibrium calculations (i.e. K-value) are used to determine the mass balance. In addition to the mass balance, a heat balance is also required. Enthalpy is not only used for the heat balance, but for adiabatic flash and column calculations as well. The enthalpy calculated from an equation of state represents the departure from the ideal gas
C.H. Twu et al. / Fluid Phase Equilibria
190
240
390
340
290
86 (1993) 47-62
59
490
440
Temperature(K) Fig. 5. Hydrogen fluoride compressibility factor along the vapor-liquid saturation curve. Simons and Hildebrand (1924) report that vapor compressibility factor of HF to be 0.223 at 234 K and 56.2 mmHg.
behavior. The enthalpy of the ideal gas shown in Fig. 6 was found from the zero-pressure heat capacity taken from Daubert and Danner (1990). The integration constant of the ideal gas enthalpy was determined arbitrarily by setting the saturated liquid enthalpy of hydrogen fluoride equal to 0.0 at its
a,
6400
ii L P y 3900
1
I
I
I
/f
I
\ I \
Temperature(K) Fig. 6. Hydrogen
fluoride
enthalpy
graphical
view.
60
C.H. Twu et al. 1 Fluid Phase Equilibria
86 (1993) 47-62
normal boiling point. The saturated liquid enthalpies were obtained by adding the ideal gas enthalpy to the saturated liquid enthalpy departure from the equation of state implemented in this work. The saturated vapor enthalpies are obtained by adding the heat of vaporization, which was given in Fig. 4, to the saturated liquid enthalpy. The complete enthalpy diagram for hydrogen fluoride is shown in Fig. 6. This indicates that the HF enthalpy diagram is entirely different from a normal one, especially for the vapor enthalpy departure and the heat of vaporization. (The heat of vaporization has already been discussed.) The normal behavior of the vapor enthalpy departure for normal components is to increase with increasing temperature, and the vapor enthalpy departure for normal components is almost negligible at the triple point temperature. However, this is not the case for HF. At the triple point, the vapor enthalpy departure for HF is more than 5 times larger than the heat of vaporization. The properties calculated from the proposed method for HF, as shown in Figs. 1-6, are all internally consistent with each other.
CONCLUSION
The degree of non-ideality of a component depends on the strength of molecular interactions. The fugacity coefficient is a measure of non-ideality and a departure of the fugacity coefficient from unity is a measure of the extent to which a molecule interacts with its neighbors. While fugacity coefficients are usually close to unity for non-polar or even polar molecules at low pressures, this is not the case for HF. It exhibits significant deviations from ideal behavior. These deviations are significant in vaporliquid and liquid-liquid equilibria and failure to take them into account can lead to serious errors. In order to predict successfully the fugacity and enthalpy behavior for HF over the entire range of system conditions using chemical theory, an internally consistent method is used to derive the parameters in the associated CEOS. The parameters obtained from this method properly reflect the physical and chemical forces without losing their theoretical background, and the results of the predicted properties are internally consistent. Using the method proposed, the properties of hydrogen fluoride can be represented adequately by assuming that only monomer and hexamer exist in the system. The significant deviations from ideal behavior at low temperatures can be explained by the hypothesis that the monomer and cyclic benzene-like hexamer are the only appreciable species existing in HF.
C.H. Twu et al. 1 Fluid Phase Equilibria 86 (1993) 47-62
61
LIST OF SYMBOLS
Redlich-Kwong equation of state parameters a, b fugacity of pure component f HF, (HF)e hydrogen fluoride monomer and hexamer, respectively K n0 4 nT
P
R T %w V
V ;
chemical equilibrium constant the number of moles that would exist in the absence of association extent of association total number of moles of monomer and dimer total pressure gas constant temperature volume function parameters molar volume total volume true mole fraction of species i compressibility factor
Greek letters
temperature-dependent fugacity coefficient
function in the parameter a
Subscripts C
i,j 176
critical property property of species i, j monomer, hexamer property, respectively
Superscripts *
reduced property
REFERENCES Daubert, T.E. and Danner, R.P., 1990. DIPPR project 801 Data Compilation, Tables of Physical and Thermodynamic Properties of Pure Compounds. equilibrium Gillespie, P.C., Cunningham, J.R. and Wilson, G.M., 1985. Vapor-liquid measurements for the hydrogen fluoride/hydrogen chloride system. AIChE Symp. Ser., Phase Equilibrium Measurement (DIPPR), 81 (244): 41-48. Heideman, R.A. and Prausnitz, J.M., 1976. A van der Waals-type equation of state for fluids with associating molecules. Proc. Natl. Acad. Sci. USA, 73: 1773-1776. Long, R.W., Hilderbrand, J.H. and Morrell, W.E., 1943. The polymerization of gaseous hydrogen and deuterium fluorides. J. Am. Chem. Sot., 65:182- 187.
62
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Phase Equilibria
86 (1993) 47-62
Redlich, 0. and Kwong, N.S., 1949. On the thermodynamics of solutions. V: an equation of state. Fugacities of gaseous solutions. Chem. Rev., 44: 233. Simons, J. and Hildebrand, J. H., 1924. The density and molecular complexity of gaseous hydrogen fluoride. J. Am. Chem. SOL, 46: 2183-2191. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197. Twu, C.H., Bluck, D., Cunningham, J.R. and Coon, J.E.. 1992. A cubic equation of state: relation between binary interaction parameters and infinite dilution activity coefficients. Fluid Phase Equilibria, 72: 25-39. Twu, C.H., Bluck, D., Cunningham, J.R. and Coon, J.E.. 1991. A cubic equation of state with a new alpha function and a new mixing rule. Fluid Phase Equilibria, 69: 33-50. Twu, C.H., Coon, J.E. and Cunningham, J.R., 1992. An equation of state for carboxylic acids. Sixth International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, July 19-24. 1992 in Italy. Fluid Phase Equilibria, 82: 379-388.