Accuracy of vapour ^ liquid critical points computed from cubic equations of state

Transcription

1 High Temperatures ^ High Pressures 2000 volume 32 pages 449 ^ ECTP Proceedings pages 433 ^ 443 DOI: /htwu303 Accuracy of vapour ^ liquid critical points computed from cubic equations of state Juha-Pekka Pokki Juhani Aittamaa Laboratory of Chemical Engineering and Plant Design Helsinki University of Technology POB 6100 FIN HUT Finland; fax: ; jpokki@cc.hut.fi Kari I Keskinen Neste Engineering Oy POB 310 FIN Porvoo Finland Presented at the 15th European Conference on Thermophysical Properties Wu«rzburg Germany 5 ^ 9 September 1999 Abstract. The calculation methods for the critical point of vapour ^ liquid mixtures can be classified into empirical and rigorous methods. Both methods are briefly presented and calculated critical points are compared to measured ones. The rigorous methods are found to be more accurate. The rigorous methods are divided into indirect and direct methods and they utilise some equation of state. In the indirect method the whole phase envelope is calculated and the critical point is interpolated. In this work the direct method of Heidemann and Khalil is studied with three equations of state: Soave ^ Redlich ^ Kwong (SRK) Peng ^ Robinson (PR) and Adachi ^ Lu ^ Sugie (ALS) together with volume translation added to SRK and PR. These equations of state are accurate in critical temperature and pressure but PR and ALS are more accurate in critical molar volume than SRK. 1 Introduction The critical point of vapour ^ liquid mixtures is a special point where the liquid and vapour phases become similar. The critical point of a pure component is a fixed temperature pressure and molar volume. Pure components can be identified according to their critical points and these values are tabulated for many common components. The critical temperature (T ) pressure (p) and molar volume (v) of a mixture are strongly dependent on the components and the composition of the system. Because there are a very large number of components and an infinite number of compositions only a very small number of mixture critical points are tabulated in the literature. The critical point of a mixture is of great interest in those processes that operate near the critical point. To investigate the behaviour of the process a reliable and accurate method to determine the critical point of a mixture is needed. The methods to calculate the critical point of mixtures can be divided into empirical and rigorous methods. The empirical methods provide the pseudocritical point of the mixture by relating the critical properties of individual components with empirical composition-dependent mixing rules. The rigorous methods can be divided into indirect and direct methods and they both need a thermodynamic model. In the indirect methods the entire phase envelope is constructed while the direct method provides the critical point of a mixture without construction of the phase envelope. The development of computers has made the rigorous methods more attractive compared to empirical methods because of their better accuracy and consistency. The critical point of the mixture calculated with the aid of a direct method is consistent with the calculation of vapour ^ liquid equilibrium if the same equation of state is used. The pseudocritical point does not necessarily hit the phase boundary generated with rigorous thermodynamics. The greatest benefit of the empirical method is its mathematical simplicity.

2 450 J-P Pokki J Aittamaa K I Keskinen 15 ECTP Proceedings page Empirical method The empirical methods provide a pseudocritical point of the mixture. The critical properties of the individual components are combined with empirical relations. There are several complex empirical methods that are claimed to be accurate. The simplest empirical correlation for the critical temperature T c is (Reid et al 1987 p. 76): T cm ˆ XN C z i T c i (1) where subscript m denotes mixture i are the components z i is the mole fraction and N C is the total number of components. If the same analogy is applied to pressure and molar volume then: p cm ˆ XN C v cm ˆ XN C z i p c i (2) z i v c i. (3) These linear correlations give satisfactory results if the critical properties of the individual components are similar or the mole fraction of one component is near to unity. The main emphasis of this work is on the direct method. 3 Rigorous direct method One of the early works in the development of the direct method is the work of Peng and Robinson (1977). The method was superior to previous methods at that time because it used the two-parameter cubic equation of state proposed by Peng and Robinson (1976). Later Heidemann and Khalil (1980) formulated the problem differently and made it less complex. Michelsen (1980) simplified some higher order derivatives needed for the method of Heidemann and Khalil (1980). By combining these two works an effective procedure for calculating the critical point of mixtures rigorously was established. Stockfleht and Dohrn (1998a 1998b) developed further the transformation of variables from volume and temperature to pressure and temperature which is the most common set of variables in commercial simulators. In this work the set of variables is volume and temperature. Heidemann and Khalil (1980) organised the problem of calculating the critical point of mixtures of known composition as two equations in the unknown temperature and volume. It applies for multicomponent mixtures in which all fluid phases are described by an equation of state. It provides the initial guesses for temperature and volume for ordinary vapour ^ liquid systems. The following is a short presentation of the most important equations; detailed information can be found in the articles of Heidemann and Khalil (1980) and Michelsen (1980). According to Heidemann and Khalil (1980) the first condition of the critical point is that the determinant of the matrix Q is zero: det (Q) ˆ 0 (4) where q ij ˆ (T=K) 100 n q ln fi qn j (5) T;V where n is the number of moles f i is the fugacity n j is the number of moles of component j and V is total volume. There should also be a vector Dn which satisfies: QDn ˆ 0 Dn ˆ (Dn 1 Dn 2... Dn NC ) T (6) where Dn i is the variation in the amount of component i.

3 Accuracy of vapour ^ liquid critical points ECTP Proceedings page 435 The second condition for the critical point which is simplified by Michelsen (1980) is that: C ˆ Dn T Q Dn ˆ 0 (7) where Q ˆ 1 2e Q(n Dn T V) Q(n Dn T V)Š O(2 ) (8) where is an increment for numerical differentiation. The initial points T ini and V ini for the iterative solution of this method can be obtained by taking a sufficiently high initial temperature: n i T in:5 XN C n T c i (9) and the initial volume V ini ˆ 4B (10) where B is the scaled b parameter of the cubic equation of state. In the inner loop V is fixed and T is found to satisfy the zero determinant. The vector Dn is determined and normalised. In the outer loop V is found to satisfy zero C. The fugacity coefficient j i of component i when the variables are T V and n is presented conveniently by Mollerup and Michelsen (1992): qf ln j i ˆ ln (Z) qn i T;V (11) where F is the reduced residual Helmholtz energy and Z is the compressibility factor. The derivative of F is computed from the residual Helmholtz energy A r : F ˆ Ar (T V n) ˆ 1 V p nrt dv RT RT 1 V (12) where R is the gas constant. The fugacity f i of component i is then obtained from the relationship: j i (T V n) ˆ fi (T V n) (n i =n)p (13) where pressure p is the result from a pressure-explicit equation of state. Hence the thermodynamic model is called with the `natural' variables T V and n instead of T p and n which can be considered `unnatural' in this case. 4 Equations of state included in the comparison Three equations of state are included namely SRK (Soave 1972) PR (Peng and Robinson 1976) and ALS (Adachi et al1983). These are shown in Appendix A. Pëneloux-type volume translation (Pëneloux et al 1982) is implemented in SRK and PR in order to increase the accuracy of the saturated liquid volumes. They are abbreviated as SRK-P and PR-P respectively. The volume translation has been under active research. There are many volume translations as given by Soave (1984) Mathias et al (1989) Chou and Prausnitz (1989) Ji and Lempe ( ) Zabylon and Brignole (1997) Tsai and Chen (1998) and Monnery et al (1998). The basic idea of Pëneloux translation is to match saturated liquid molar volume at reduced temperature T r ˆ 0:7. The translation is a component-dependent constant and applied also for vapour. In this work Pe neloux-type translation is adjusted for PR as Pe neloux et al (1982) suggested in their work. The coefficients of Pëneloux translation in

4 452 J-P Pokki J Aittamaa K I Keskinen 15 ECTP Proceedings page 436 PR are optimised based on ten light hydrocarbons as Pëneloux et al did. The volume translation for SRK and PR is below where the coefficients a 1 and a 2 are equation-ofstate dependent: RTc i c i ˆ a 1 (a 2 z RA i ) (14) P c i where z RA is the Rackett compressibility factor. These coefficients for SRK are optimised by Pëneloux et al (1982): a 1SRK ˆ 0:40768 and a 2SRK ˆ 0: These coefficients for PR are optimised in this work: a 1PR ˆ 0: and a 2PR ˆ 0: The critical temperature and pressure acentric factor o and saturated liquid volume needed in optimisation are taken from the same source to avoid discrepancy between critical and volume data of saturated liquid. 5 Results and discussion The scope of this work was to compare the critical points of mixtures computed with rigorous and empirical methods to the reported critical points of mixtures. Reported critical temperature pressure and molar volume were collected from the articles of Hicks and Young (1975) and of Sadus (1992) where the accuracy of the collected data is not mentioned. However the accuracy is believed to be much better than the computational methods can predict. The results are summarised in tables B1 to B8 of Appendix B. The calculated critical points are compared to reported critical points. Relative error is defined as: 100% (x calc x rep ) (15) x rep where x is critical temperature pressure or molar volume. Maximum (max=%) and minimum (min=%) relative error are the greatest and the smallest values of relative error in tables B1 to B8. Average relative (aver=%) error is defined as: 1 X NP 100% (x calc i x rep i ) (16) NP x rep;i where NP is number of points per group. The method of Heidemann and Khalil (1980) may not converge which is the reason NP depends on the equation of state. The pseudocritical method indicates the number of collected points per group. The critical properties of individual components are taken from Reid et al (1987) as tables B1 to B8 of Appendix B were computed. The binary interaction parameters k ij of SRK and PR are non-zero for the same binary pairs but the values are different naturally. The hydrocarbon ^ hydrocarbon binary interactions of SRK and PR are zero. Binary interaction parameters for ALS are zero for all components. The group of ternary hydrocarbon mixtures in table B1 contains only critical temperature and pressure values. All equations of state give good accuracy. The pseudocritical method gives a great absolute average relative error. The group of binary hydrocarbon mixtures in table B2 is the biggest group containing 387 data points. The errors in calculated critical temperature and pressure are usually low but the calculated critical molar volumes are inaccurate. This is in accordance with the knowledge that cubic equations of state cannot model critical molar volume accurately. The best accuracy is provided by PR then ALS PR-P SRK-P SRK and the pseudocritical method in order of decreasing accuracy. It can be seen that Pëneloux translation of SRK improves the

5 Accuracy of vapour ^ liquid critical points ECTP Proceedings page 437 modelling of critical molar volume. The group of binary mixtures of nitrogen and simple molecules in table B3 contains only nine data points. The simple molecules are carbon monoxide and carbon dioxide. Calculated critical temperature and pressure seem to be accurate. ALS is very accurate in critical molar volume in this particular case. The group of binary mixtures of nitrogen and hydrocarbon in table B4 contains some large errors and some calculations did not converge. The average relative error in critical pressure of carbon monoxide ^ hydrocarbon systems in table B5 is high compared to previous systems. The Pe neloux translation improves the modelling of critical molar volume of SRK but not PR in carbon dioxide ^ hydrocarbon systems in table B6. The same behaviour can be found in critical molar volume of hydrogen sulphide ^ hydrocarbon pairs in table B7. One reason for this behaviour is the liquid molar volume of the pure component at T r ˆ 0:7. PR gives for light hydrocarbons like methane and ethane too small a liquid molar volume but SRK gives too large a molar volume. For heavy hydrocarbons like n-nonane and n-decane both SRK and PR give too large a molar volume at T r ˆ 0:7. ThePëneloux translation of SRK shifts the molar volume to smaller molar volume for both light and heavy hydrocarbons. The Pëneloux translation of PR shifts the molar volume to larger molar volume for light hydrocarbons but to smaller molar volume for heavy hydrocarbons. Because the Pe neloux translation is constant it drives the critical molar volume computed from PR in the wrong direction. Finally all groups are collected in table B8. It is clear that rigorous methods are superior to simple empirical methods. The empirical method can be made more accurate with complex mixing rules and binary interaction parameters but still the critical properties obtained are inconsistent. The binary interaction parameters of the empirical method can be obtained only from the critical properties of the mixture but the binary interaction parameters for a cubic equation of state are much more general because they are obtained from the vapour ^ liquid equilibrium measurements. The rigorous method based on the equation of state uses the same binary interaction parameters as needed in vapour ^ liquid equilibrium calculation. 6 Conclusions The computer implementation of the method of Heidemann and Khalil (1980) is relatively easy; the method is robust fast and converges with high probability to the correct vapour ^ liquid critical point of simple mixtures. The rigorous method has several benefits over the pseudocritical method. The greatest benefit is the consistency of critical point to vapour ^ liquid equilibrium calculation ie the critical point hits the phase boundary constructed with vapour ^ liquid equilibrium calculation. Another important benefit is better accuracy. The critical temperature and pressure are accurate but the critical molar volume is not accurate. The accuracy of PR is better than SRK and SRK-P. The volume translated PR does not provide significant benefit over the original PR. ALS is not as accurate as PR but ALS gives better results than SRK. The accuracy of these three cubic equations of state is good for hydrocarbon mixtures. If there is a non-hydrocarbon molecule in the mixture then the errors in mixture critical properties increase. The empirical method presented in this work does not give reliable results. Acknowledgements. Financial support from the Foundation for Financial Aid at Helsinki University of Technology to one of the authors (JPP) is gratefully acknowledged. References Adachi Y Lu B C-L Sugie H 1983 Fluid Phase Equilib ^ 48 Bian B G Wang Y R Shi J 1992 Fluid Phase Equilib ^ 334 Chou G F Prausnitz J M 1989 AIChE J ^ 1496 Heidemann R A Khalil A M 1980 AIChE J ^ 779 Hicks C P Young C L 1975 Chem. Rev ^ 175 Ji W-R Lempe D A 1997 Fluid Phase Equilib ^ 63 Ji W-R Lempe D A 1999 Fluid Phase Equilib

6 454 J-P Pokki J Aittamaa K I Keskinen 15 ECTP Proceedings page 438 Mathias P M Naheiri T Oh E M 1989 Fluid Phase Equilib ^ 87 Michelsen M L 1980 Fluid Phase Equilib. 4 1^10 Mollerup J M Michelsen M L 1992 Fluid Phase Equilib. 74 1^15 Monnery W D Svrcek W Y Satyro M A 1998 Ind. Eng. Chem. Res ^ 1672 Pe neloux A Rayzy E Frëze R 1982 Fluid Phase Equilib. 8 7^23 Peng D-Y Robinson D B 1976 Ind. Eng. Chem. Fundam ^ 64 Peng D-Y Robinson D B 1977 AIChE J ^ 144 Reid R C Prausnitz J M Poling B E 1987 The Properties of Gases and Liquids fourth edition (New York: McGraw-Hill) Sadus R J 1992 High Pressure Phase Behaviour of Multicomponent Fluid Mixtures (Amsterdam: Elsevier) Soave G 1972 Chem. Eng. Sci ^ 1203 Soave G 1984 Chem.Eng.Sci ^ 369 Stockfleht R Dohrn R 1998a Fluid Phase Equilib ^52 Stockfleht R Dohrn R 1998b Fluid Phase Equilib ^ 382 Tsai J-C Chen Y-P 1998 Fluid Phase Equilib ^215 Zabylon M S Brignole E A 1997 Fluid Phase Equilib ^ 95 APPENDIX A The equations of state included in this comparison are given below. Pe neloux volume-translated SRK equation of state original SRK if C ˆ 0: nrt p ˆ V C B D (V C)(V C B). (A1) Pe neloux volume-translated PR equation of state original PR if C ˆ 0: nrt p ˆ V C B D (V C)(V C B) B(V C B) (A2) where the parameters of attraction D covolume B and volume translation C must be expressed with mole numbers as given below. D ˆ n 2 a(t ) ˆ XN C X N C j ˆ 1 n i n j a ij (T ) (A3) B ˆ nb ˆ XN C C ˆ nc ˆ XN C n i b i (A4) n i c i (A5) a ij (T ) ˆ a c;i a c; j a i (T )a j (T )Š 1=2 (1 k ij ) 1=2 T a 0:5 m i 1 T c;i (A6) (A7) m i SRK ˆ 0:48 1:574o i 0:176o 2 i (A8) m i PR ˆ 0: :54226o i 0:26992o 2 i (A9) R 2 T 2 c i a c i ˆ O a (A10) p c i RT c i b i ˆ O b (A11) p c i

7 Accuracy of vapour ^ liquid critical points ECTP Proceedings page O a SRK ˆ 9( 3p and O b SRK ˆ ( 3p 2 1) 2 1) 3 O a;pr ˆ 0: and O b;pr ˆ 0: In this work the maximum computer digit precision of coefficients O a and O b was used. Often the coefficients are truncated after five digits. For further information see Bian et al (1992). c i SRK ˆ 0:40768 RT c i (0:29441 z p RA i ) (A12) c i c i PR ˆ 0: RT c i (0: z p RA i ). (A13) c i The ALS equation of state is: p ˆ nrt D V B 1 (V B 2 )(V B 3 ) ; (A14) for D see equations (A3) (A6) and (A7). B k ˆ nb k ˆ XN C n i b ki k ˆ ; (A15) a c i ˆ A 0 R 2 T 2 c i (A16) p c i b ki ˆ B krt 0 c i k ˆ ; (A17) p c i A 0 ˆ 0: :04024o i 0:01111o 2 i 0:00576o 3 i (A18) B 0 1 ˆ 0: :03452o i 0:00330o 2 i (A19) B 0 2 ˆ 0: :00405o i 0:01073o 2 i 0:00157o 3 i (A20) B 0 3 ˆ 0: :14122o i 0:00272o 2 i 0:00484o 3 i (A21) m i;als ˆ 0:4070 1:3787o i 0:2933o 2 i. (A22)

8 456 J-P Pokki J Aittamaa K I Keskinen 15 ECTP Proceedings page 440 APPENDIX B Table B1. Relative error in ternary hydrocarbon mixtures. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo Table B2. Relative error in binary hydrocarbon mixtures. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo in v c : SRK SRK-P PR PR-P ALS pseudo

9 Accuracy of vapour ^ liquid critical points ECTP Proceedings page 441 Table B3. Relative error in binary mixtures of nitrogen and simple molecules. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo in v c : SRK SRK-P PR PR-P ALS pseudo Table B4. Relative error in binary mixtures of nitrogen and hydrocarbon. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo

10 458 J-P Pokki J Aittamaa K I Keskinen 15 ECTP Proceedings page 442 Table B5. Relative error in binary mixtures of carbon monoxide and hydrocarbon. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo Table B6. Relative error in binary mixtures of carbon dioxide and hydrocarbon. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo in v c : SRK SRK-P PR PR-P ALS pseudo

11 Accuracy of vapour ^ liquid critical points ECTP Proceedings page 443 Table B7. Relative error in binary mixtures of hydrogen sulphide and hydrocarbon. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo in v c : SRK SRK-P PR PR-P ALS pseudo Table B8. Relative error in all groups. SRK SRK-P PR PR-P ALS pseudo SRK SRK-P PR PR-P ALS pseudo in v c : SRK SRK-P PR PR-P ALS pseudo

12 ß 2000 a Pion publication printed in Great Britain