Microsoft Word - 425simasatitkul.doc CHEMICAL ENGINEERING TRANSACTIONS VOL. 74, 2019 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Sauro Pierucci, Jiří Jaromír Klemeš, Laura Piazza Copyright © 2019, AIDIC Servizi S.r.l. ISBN 978-88-95608-71-6; ISSN 2283-9216 Smart Implementation of Bender Equation of State Filippo Bisottia, Alessandro di Pretoroa,b, Anna Dell’Angeloa, Daniele Previtalia, Andre Furtado Amarala, Ecem Muge Andogluc, Flavio Manentia,* a Dipartimento CMIC “Giulio Natta”, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy b Laboratoire de Génie Chimique, Université de Toulouse, CNRS/INP/UPS, Toulouse, France c Chemical Engineering Department, Bilecik Seyh Edebali University, Gülümbe, 11230, Bilecik, Turkey flavio.manenti@polimi.it In this paper matrix and vector products are exploited to reformulate Bender Equation of State. Finally, the new formulation is used to generate results which has been compared with experimental data sets available in literature and the analogous findings coming from different thermodynamic packages commonly used in Aspen Hysys for Air Separation Unit. 1. Introduction In the last years, the need for more powerful and detailed thermodynamic tools to be used in process simulators has exponentially increased. A more precise prediction allows to save energy, to have a better control and to optimize of each unity (Lasala et al., 2018). For cryogenic separations it is very difficult to have robust and reliable thermodynamic packages; generally, mixtures are at low temperatures (so very far from ideal behaviour) or close to critical point for one or more compounds. In such operative conditions, for example, Cubic Equations of State (CEoS) exhibit some convergence and oscillation problems that are solved by using fitting-interaction parameters (Lasala et al., 2018). This numerical solution however is suitable only for specific mixtures. In 60’s and 70’s of the last century, Bender developed an Equation of State tailored on Air Separation Unit, however, his methodology in the development of the Equation of State can be easily translated to other compounds as already done by several researchers (Bühner et al., 1981; Platzer et al., 1993; Cibulka et al., 2001; Ghazouani et al., 2005). In any case, Bender Equation of State was originally developed to predict thermodynamic behaviour and properties in cryogenic conditions for nitrogen, argon and oxygen both pure or in mixture. However, in the original doctorate thesis by Bender, units of measure and univocal definitions of mixing rules for parameters and additional coefficients are missing. In this work, a smart and complete implementation of Bender Equation of State is proposed exploiting vector and matrix products to make the code more compact and to optimize the calculations. 2. Bender Equation of State for mixtures Bender Equation of State is an evolution of the Benedict – Webber – Rubin EoS. As stated by Bender, among the many empirical equations of state published, only two have been largely applied for predicting phase equilibria in multicomponent systems. It is well known that BWR Equation is not suitable for use in the liquid region of both pure fluid and mixtures. Whereas, RKS and PR equation, despite their capacity to easily fit experimental PVT data at low density gas region or those of the liquid region, have unique sets of coefficients that, even if these are temperature dependent, are not able to provide good accuracy and reliability on the whole range of temperature and pressure. Bender EoS tries to achieve a good representation of the gas region as well as of the condensed phase including the two-phase region with one single function that has an expansive but still rational form (Bender, 1973). Bender EoS contains twenty fitting parameters (for each pure compound , ) and six virial terms. As shown in equation (1), Bender Equation of State relates pressure and temperature with the density of the system (therefore the reciprocal of the molar volume) and with prominent nonlinearity related to density: DOI: 10.3303/CET1974120 Paper Received: 20 June 2018; Revised: 16 September 2018; Accepted: 17 December 2018 Please cite this article as: Bisotti F., Di Pretoro A., Dell'Angelo A., Previtali D., Furtado Amaral A., Andoglu E.M., Manenti F., 2019, Smart Implementation of Bender Equation of State, Chemical Engineering Transactions, 74, 715-720 DOI:10.3303/CET1974120 715 = + − − + + + + + ( + ) ∙ exp(−a ) (1) Parameters present in equation (1) are directly provided with both linear and nonlinear mixing rules. The mixing parameters are calculated starting from the twenty fitting parameters of pure compound and the phase composition. Indeed • for mixing , and geometrical mixing rule is applied = (2) • for , , , , and linear mixing rules = (3) • for cubic mixing rules are necessary = (4) The only exception to standard mixing rules is given by mixing parameter. Bender found out that linear mixing rule alone was not correct due to binary mixture deviations. The additional term in the mixing rule, therefore, accounts for these deviations and it is the result of nonlinear regression on experimental data sets: = + 100 + (5) Bender also estimated and provided , and values, that here are proposed in matrix form: = 0 −0.0072 0.0057−0.0072 0 0.00950.0057 0.0095 0 = 0 0.007 00.007 0 0.0040 0.004 0 = 0 6 86 0 48 4 0 (6) The elements appearing in expressions (2-5) are functions of the twenty fitting parameters , (obviously, is function of the , and the same is for and with the right correspondences): = , + , + , (7) = , + , + , (8) = , + , (9) = , + , (10) = , (11) = , + , + , (12) = , + , + , (13) Moreover, starting from equation (1) and applying the definition of fugacity in mixture, it is possible to obtain an analytical expression. The result shows a dependence on temperature, pressure and phase compositions: ∗ = ∙ exp (14) = + 2 − 2 + 32 + + + + − − ( + ) , (15) As for the pressure, the coefficients appearing in the fugacity equation (14) are function of both the parameters and mixing ones (2-4). Their expressions are provided below: 716 = , + , (16) = ,/ ∙ ,/ (17) = / ∙ / (18) = / ∙ / (19) = + 3 (20) = + 4 (21) = + 5 (22) = 4 + (23) While parameter has slightly more complex expression that depends on the considered species: = 2 + + −0.0072 100 + 0.007 (1 − ) + 0.0057 100 (1 − ) (24) = 2 + + −0.0072 100 + 0.007 (1 − ) + 0.0095 100 + 0.004 (1 − ) (25) = 2 + + 0.0057 100 (1 − ) + 0.0095 100 + 0.004 (1 − ) (26) Finally = + /2 (27) = exp(− ) 2 + 2 1 + 1 (28) = [1 − exp(− ) ∙ ( + 1)] (29) = ( ) 2 − exp(− ) ∙ [( + 1) + 1] (30) 3. Matrix and vector product to estimate Bender Equation of State coefficients The equations of the previous section show a very high complexity degree. In order to speed up the calculations, it is worth to exploit matrixes and vectors products especially for vapour-liquid equilibria. Indeed, looking at expressions (7-13) it is evident that coefficients are elements of vectors whose length is ; moreover, they are composition independent. Therefore, defining a composition matrix whose dimensions are × ( number of phase and number of compounds) as follows: = = ̅ = (31) Is straightforward to compute all the parameters defined in expressions (2-4) through matrix products. These operations will provide as results vectors of length NP whose first elements are related to the liquid phase, while the second ones to the vapour: [ × 1] = [ × ] ∙ [ × 1] (32) 717 = ∙ = ∙ = = (33) Obviously, for linear mixing rule (3) = 1 and = 1, while according to nonlinear combination rules (2) and (4) the power exponents are = 0.5, = 1 and = 1/3, = 3 respectively. The evaluation is fairly more complex for the fugacity coefficients (16-30). In this case, instead of vectors, the results are matrixes with dimensions × . Neglecting powers, which have to be applied as explained before element-by-element, the results are: [ × ] = [ × ] ∙ [ × 1] ∙ [1 × ] (34) = ∙ = ∙ [ ] = (35) analogously, [ × ] = [ × ] ∙ [ × 1] + [1 × ] (36) = + = + [ ] = + + ++ + + (37) Calculation are furtherly speeded up by decomposing the third additive terms in expressions (24-26). These can be collected inside a vector by previously setting: = −0.0072 100 + 0.007; = 0.0057 100 ; = 0.0095 100 + 0.004 (38) = ( + )(1 − )( + )(1 − )( + )(1 − ) (39) = 1 − 0 00 1 − 00 0 1 − ∙ ( + )( + )( + ) (40) = 1 − 0 00 1 − 00 0 1 − ∙ 0 0 0 ∙ = ∙ ∙ (41) Matrix products have to be performed for each phase present in the system. Finally, a matrix is obtained: = ( = ̅)( = ) (42) [ × ] = 2 ̅[ × 1] + [1 × ] + [ × ] (43) 4. Comparison with largely-used Equation of State in ASU At this point it is useful and interesting to test and compare Bender Equation of State (B) performance with respect to two among the most commonly used thermodynamic packages in Air Separation Unit: Peng- Robinson (PR) and Benedict-Webber-Rubin (BWR). For simplicity, the bubble and dew problems in pressure are performed on fixed dry air composition. This means that the composition vector ̅ is equal to = 0.7812, = 0.2096 and = 0.0092. In the case of bubble problem, the assigned composition corresponds to that 718 of the liquid, while in the dew one to that of the vapour phase. Experimental data set is directly taken from NIST Database published in several specialized papers (Jacobsen at al., 2000). Bender Equation of State has been implemented in a MatLab® code and in a Visual Studio C++ 2013 source in which BzzMath Library© was embodied, while, concerning BWR and PR Equation of State, the thermodynamic packages available in Aspen Hysys V10 ware used. The results are compared in Figure 1 and Figure 2. Figure 1: comparison of bubble pressure (left) and liquid phase density (right) relative errors of dry air at different temperatures Figure 2: comparison of dew pressure (left) and vapour phase density (right) relative errors of dry air at different temperatures 5. Conclusions Despite its complexity, Bender Equation of State shows a very good accuracy and reliability both in the bubble and dew problem in the region of interest for Air Separation Unit. With respect to PR Equation of State, Bender model exhibit more stability near the critical point due to special nonlinear fitting and regression performed in this region by Bender in order to a very robust response of his Equation of State. Whereas, both thermodynamics models show lower accuracy and precision near the triple point. Considering that, generally, Air Separation Unit works far from both, extremely low temperature and the critical point of the mixtures, due to control and technical reasons, the temperature range is limited to moderate temperature. Therefore, future developments may concern on how to increase Bender Equation of State prediction capacity in this range of interest in order to improve its performance and reliability comparing to another thermodynamic tools already available in simulation software. 719 Notation and Unit of Measure Symbol Meaning Unit of measure / notes System Pressure [ ] using original , (Bender, 1973) Molar density [ / ] using original , (Bender, 1973) Ideal gas constant = 8.314472 [ ∙ /( ∙ )] Temperature [ ] absolute temperature Compound Index Nitrogen (1), argon (2) and oxygen (3) following (Bender, 1973), Bender coefficients = 1,2,3… . 20 are provided in Bender thesis (Bender, 1973) Generic molar fraction Generic composition vector ̅ , are respectively vapour liquid composition vectors Number of phases Number of compounds Composition matrix [ × ] First row for liquid phase and second one for vapour phase Pure virial coefficients Reference expressions are (7-13) Vector of pure virial coefficients Vector collecting the generic elements previously defined Mixing coefficient in equation (1) , , , , , , , , , . Mixing rules are classified and given in equations (2-5); they are vectors of length [ × 1] , , Matrix element for estimation of mixing parameter in (5) ∗ Fugacity coefficient in mixture [ ] fugacity in mixture defined in equation (15) Generic coefficients for fugacity is a matrix element in multiphase system Matrix of fugacity parameters Reference equations (16-28), matrix dimensions [ × ] , , Elements of matrixes defined in (38) References Bender E., 1972, Die Berechnung der Verdampfungsgleichgewichte von Mehrstoffsystemen bei hohen Drücken, Chemie Ingenieur Technik, 44, 576-582. Bender E., 1973, An equation of state for predicting vapour-liquid equilibria of system N2 - Ar - O2, Cryogenics, 13, 11-18. Bender E., 1973, The Calculation of Phase Equilibria from a Thermal Equation of State Applied to the Pure Fluids Argon, Nitrogen, Oxygen and their Mixtures, Karlsruhe, Verlag Müller Bühner K., Maurer G., Bender E., 1981, Pressure-enthalpy diagrams for methane, ethane, propane, ethylene and propylene, Cryogenics, 21, 157-164. Cibulka I., Kováčiková J., Hnědkovský L., Novák J.P., 2001, A simple method for evaluation of parameters of a Bender equation of state from experimental data, Fluid Phase Equilibria, 180, 27-40. Ghazouani J., Chouaieb O., Bellagi A., 2005, Evaluation of the parameters of the Bender equation of state for low acentric factor fluids and carbon dioxide, Thermochimica Acta, 432, 10-19. Lasala S., Privat R., Jaubert J.N., Arpentinier P., 2018, Modelling the thermodynamics of air-component mixtures (N2, O2 and Ar): Comparison and performance analysis of available models, Fluid Phase Equilibria, 458, 278-287. Lemmon E.W., Jacobsen R.T, Penoncello S.G., Friend D.G., 2000, Thermodynamics Properties of Air and Mixtures of Nitrogen, Argon and Oxygen From 60 to 2000K at Pressures to 2000MPa, Journal of Physical Chemical Reference Data, 29, 331-385. Platzer B., Maurer G., 1993, Application of a generalized Bender equation of state to the description of vapour-liquid equilibria in binary systems, Fluid Phase Equilibria, 84, 79-110. 720