Getting a Handle on Advanced Cubic Equations of State

More documents

Recommendations

Info

wide range of components, which can be obtained by contacting the authors directly. Twu’s α(T) function has been found to be more accurate than other α(T) functions found in literature for predicting vapor pressures for polar and non-polar components at temperatures ranging from the triple point to the critical point. Using the component-dependent parameters, the accuracy of reproducing the vapor pressure from Eq. 17 is within experimental error given in DIPPR. Twu’s α(T) function has some unique features. Fluids such as hydrogen exhibit a maximum alpha at low temperatures (about 19 K). Twu’s α(T) function accurately describes this behavior and is the only function so far that shows this maximum at low temperature. In addition, it is continuous at the critical point for all components and extrapolates very well to supercritical conditions without exhibiting abnormal characteristics, such as increasing with increasing temperature. Due to the accuracy and wide temperature range of application of the advanced Twu α(T) functions (Eqs. 10, 11, 12 and 17) for predicting vapor pressures when used in the SRK, PR and TST equations of state, these equations have been selected for use in the SRK, PR and TST models as follows. For petroleum fractions and components that do not have vaporpressure data available, Eqs. 10, 11 and 12 should be used. For all components that do have vapor-pressure data available, Eq. 17 can be used. van der Waals mixing rules In addition to using an appropriate α(T) function for the accurate prediction of vapor pressure, the ability of a CEOS to correlate and predict phase equilibria of mixtures depends strongly upon the mixing rule applied. The most commonly used method to extend equations of state to a non-polar mixture is to use the van der Waals one-fluid mixing rules: a = ∑ ∑ x x a a 1−k i where kij is the binary interaction parameter that is obtained from the regression of vapor-liquid equilibrium data. The van der Waals mixing rules are capable of accurately representing vapor-liquid equilibria using only one binary-interaction parameter for non-polar or slightly polar systems. However, many mixtures of interest in the chemical industry contain strongly polar or associating components. One way to modify the van der Waals mixing rules so that they can better describe polar systems is to use the asymmetric kij for the a parameter, as developed by Twu et. al. (17): a = ∑ ∑ xixj aa i j ( 1−kij)+ i j b = x x ⎡1 ∑ ∑ i j b + b i j ⎣⎢ 2 j i j ( ) i j ij ( i j) ⎤ ⎦⎥ ( ) ( − ) ⎡ ∑ x ⎢∑ x aa k k i ⎣ j i j i j ji ij ⎤ ⎥ ⎦ 16 13 3 / ( 18) ( 19) ( 20) Eq. 20 has two adjustable parameters, k ij and k ji . Generally, they are not equal. The use of an asymmetric definition of k ij in the van der Waals mixing rule of Eq. 20 is similar to the asymmetric parameters A ij and A ij used in the excess Gibbs energy (G E ) mixing rule of the NRTL model. If k ij = k ij , Eq. 20 reduces to the van der Waals mixing rule (Eq. 18). The modified van der Waals mixing rule (Eq. 20) provides accurate correlation of complex mixtures, including highly non-ideal systems that previously could only be correlated by activity-coefficient models. However, Eq. 20 no longer satisfies the second virial coefficient. Huron-Vidal infinite-pressure mixing rules Since the van der Waals mixing rules are applicable only to mixtures whose excess Gibbs energy can be approximated by the regular solution theory, quite a few new mixing rules have been recently developed based on appropriate excess energy models. Huron and Vidal (18) successfully formulated a new EOS parameter a by assuming that the excess Gibbs energy at infinite pressure, GE ∞ , which is derived from a CEOS, is equal to GE derived from a liquid-activity-coefficient model. They also assumed that the liquid volume at infinite pressure equals the EOS co-volume, b. The Huron- Vidal derivation led to the following equation for a * : The parameters a * and b * a b x in Eq. 21 are defined as: a * * ⎡ * i = ⎢∑ i * ⎣ i bi 1 G + ⎛ ∞ ⎞ ⎤ C ⎝ RT ⎠ ⎥ 1 ⎦ ( 21) a * = Pa/R 2 T 2 and b * = Pb/RT (21a, b) The Huron-Vidal approach uses the conventional linear mixing rule for the b parameter: b = xx ⎡1 ∑ ∑ i j b i + b ⎤ ( j) i j ⎣⎢ 2 ⎦⎥ The C1 in Eq. 21 is a constant, and is defined as: E ( 22) 1 w C =− ⎛1+ ⎞ 1 ln ( 23) ( w−u) ⎝ 1 + u ⎠ where u and w are EOS-dependent constants used to represent a particular two-parameter CEOS. For example, u = –0.5 and w = 3.0 for the TST CEOS. Although Huron and Vidal presented their model more than two decades ago, their mixing rule is not widely used due to lack of parameters available in literature for the excess Gibbs energy at infinite pressure and the inability to accurately describe nonpolar hydrocarbon mixtures and to satisfy the quadratic composition dependence of the second virial coefficient. Wong-Sandler infinite-pressure mixing rules The Huron-Vidal approach requires the excess volume, V E , to be zero and EOS parameter b to be calculated via a linear mixing rule. To allow the use of a different mixing rule for parameter b, Wong and Sandler (19) developed a model equating CEP November 2002 www.cepmagazine.org 61
Measurement & Control the excess Helmholtz energy at infinite pressure from a CEOS to excess Helmholtz energy calculated from an activity coefficient model, and they constrained the EOS parameters a and b to satisfy the second-virial-coefficient condition: a b x a E * * ⎡ * i 1 A = ∑ i * + ⎛ ∞ ⎞ ⎤ ⎢ ( ) ⎣ i biC ⎝ RT ⎠ ⎥ 24 1 ⎦ n n a xx ⎛ ij ∑ ∑ i j bij − ⎞ j ⎝ RT ⎠ i b = a* 1 − * b The Wong-Sandler model introduced a binary interaction parameter, k ij , to correct the assumption that the excess Helmholtz free energy at infinite pressure can be approximated by the excess Gibbs free energy at low pressure: b ij a a a ij 1 ⎡ i j − = ⎛ bi − ⎞ ⎛ ⎞ ⎤ ⎢ + bj − 1 k RT ⎝ RT ⎠ ⎜ ⎟ ⎣ ⎝ RT ⎠ ⎥ − 2 ⎦ Wong and Sandler demonstrated that parameters in the activity coefficient models correlated at low temperatures can be used to extrapolate to higher temperatures. Michelsen and Heidemann (20) suggest that this success is fortuitous. In an effort to reduce the Wong-Sandler mixing rules to the conventional van der Waals mixing rules, Orbey and Sandler (21) modify the G E model of NRTL differently. Consequently, the mixing rules used in their model cannot employ the NRTL parameters reported in the DECHEMA Chemistry Data Series. Twu-Coon infinite-pressure mixing rules Twu and Coon (22) used a van der Waals mixture as a reference to derive the mixing rules for their EOS. These mixing rules re-define a and b as: * E E * * ⎡avdw 1 A A vdw a = b * + ⎛ ∞ ∞ − ⎞ ⎤ ⎢ ( ) b C ⎝ RT RT ⎠ ⎥ 27 ⎣ vdw 1 ⎦ * * * bvdw − avdw b = ( 28) * E E ⎡avdw 1 A A vdw − * + ⎛ ∞ ∞ − ⎞ ⎤ 1 ⎢ ⎣ bvdw C ⎝ RT RT ⎠ ⎥ 1 ⎦ C1 in Eqs. 27 and 28 is defined in Eq. 23. The asymmetric van der Waals mixing rule is applied to the avdw parameter, and the linear mixing rule is used for the bvdw parameter: avdw = ∑ ∑ xixj aa i j ( 1−kij)+ i j ( ) ( − ) ⎡ ∑ x ⎢∑ x aa k k i ⎣ j i j i j ji ij ( j ) bvdw = x x ⎡1 ∑ ∑ i j b + b i j ⎣⎢ 2 ⎤ ⎦⎥ ⎤ ⎥ ⎦ 16 13 3 / / ( ij ) ( 25) ( 26) ( 29) ( 30) Subscript vdw in AE in Eqs. 27 and 28 denotes that the proper- ∞ vdw ties are evaluated from the CEOS using the van der Waals mixing 62 www.cepmagazine.org November 2002 CEP rule for its a and b parameters as given by Eqs. 29 and 30. The Twu-Coon mixing rules reduce to the van der Waals mixing rules if AE ∞ = AE . It is extremely desirable that the ∞ vdw composition-dependent mixing rules reduce to the classical mixing rules because the latter work very well for non-polar systems. The use of asymmetric parameters, k and k ,im- ij ij proves the accuracy of the reproduction of the liquid-activity coefficients. Although it may be more accurate to simply refit the GE parameters in the mixing rules instead of introducing two new parameters, k and k , to match the results of the ac- ij ij tivity coefficient model, doing so would sacrifice the extrapolation ability of the infinite-pressure approach. Furthermore, the original idea of using directly published parameters for activity coefficient models in the equation of state will be lost. Twu-Sim-Tassone infinite-pressure mixing rules It is a challenge to find a general expression for AE ∞ that can be simplified to AE in the Twu-Coon infinite-pressure ∞ vdw mixing rules (Eqs. 27 and 28). As mentioned previously, Orbey and Sandler modified the GE model of NRTL to a different form to reduce the Wong-Sandler mixing rules to the conventional van der Waals mixing rules. Twu et. al. (23) developed a new excess Gibbs energy function, which reduces to the van der Waals one-fluid mixing rule (Eqs. 18 and 19). The incorporation of this new excess Gibbs function into a CEOS allows the Twu-Coon infinite-pressure mixing rules to smoothly transition to the conventional van der Waals onefluid mixing rules. A general multi-component equation for a liquid activity coefficient model is proposed by Twu et. al. (23) for incorporation in the infinite-pressure mixing rules: E ∑ x τ G G n j = ∑ xi n RT i ∑ xG n k j ji ji k ki ( 31) Eq. 31 appears to be similar to the NRTL equation, but there is a fundamental difference between them. NRTL assumes A ij , A ij , and α ij are the parameters of the model, but the TST excess Gibbs energy model assumes τ ij and G ij are the binary-interaction parameters. More importantly, any appropriate temperature-dependent function can be applied to τ ij and G ij . For example, to obtain the NRTL model, τ ij and G ij are calculated as usual from the NRTL parameters A ij , A ij and α ij : τ ij = A ij /T (32) G ij = exp[–α ij τ ij ] (33) In this way, the NRTL parameters reported in the DECHEMA Chemistry Data Series can be used in Eqs. 32 and 33, and both the NRTL model and the TST model (Eq. 31) are equally accurate for use in phase equilibrium calculations. Eq. 31 reduces to the conventional van der Waals mixing rules when the following expressions are used for τ ij and G ij :
Page 1 and 2: Measurement & Control Getti
Page 3: Measurement & Control Unfortunately
Page 7 and 8: Measurement & Control pendent throu

Getting a Handle on Advanced Cubic Equations of State

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?