Non-zero Binary Interaction Parameters Research Articles

On the density prediction of multicomponent hydrocarbon mixtures at near- and supercritical conditions

This paper presents the prediction capability of the density of near- and super-critical hydrocarbon mixtures, especially diesel fuel, by three cubic equations of state (EoS), which include PPR78, PSRK and RK-PR, two virial EoS that are BWR and SBWR, and PC-SAFT EoS. Comparison with published measured density of three different hydrocarbon mixtures, which are methane-propane-pentane, heptane-octane, and RP3 aviation kerosene, revealed that all the equations of state predict well the density of the gas phase. On the other hand, a comparison with RP3 experimental data showed that all EoS over-predict the density of gas-like supercritical phase. However, to ascertain the findings of this comparison, experimental data for other hydrocarbon mixtures at such a state are required. The results showed also that PC-SAFT is a suitable choice for liquid and liquid-like density calculations with an AAD of 2.9%, closely followed by SBWR and BWR with AAD values of 3.7% and 4.0%, respectively, while RK-PR and PPR78 produce reliable prediction of the density of liquid and liquid-like supercritical phases of hydrocarbons with an AAD of 5.5% and 4.4%, respectively. RK-PR and SBWR are found as the most accurate EoSs for calculating the density of hydrocarbons mixtures at the transition from liquid-like to gas-like supercritical phases with an AAD of 11.2% and 12.8%, respectively. Using zero and non-zero binary interaction parameters with PPR78, PSRK, and PC-SAFT showed negligible effect on the density calculation. Comparison of the predictions with the experimental diesel density showed that BWR has the best results, followed by PC-SAFT. RK-PR is found to deviate by less than 5% in predicting the diesel density for all examined surrogates. The predictions of diesel density by different EoS at near- and super-critical conditions are compared with each other because of the lack of published experimental data. This comparison showed that RK-PR can be used, with a reasonable accuracy and computational cost, for predicting the density of both the liquid-like phase and transition interval, and PPR78 for the gas-like supercritical phase. On the other hand, more accurate predictions of the liquid-like phase density can be achieved using BWR, and that of the transition interval and gas-like phase can be realized by SBWR but with a higher computational cost.

An improved reduction method for phase stability testing in the single-phase region

AbstractA new reduction method for mixture phase stability testing is proposed, consisting in Newton iterations with a particular set of independent variables and residual functions. The dimension of the problem does not depend on the number of components but on the number of components with nonzero binary interaction parameters in the equation of state. Numerical experiments show an improved convergence behavior, mainly for the domain located outside the stability test limit locus in the pressure–temperature plane, recommending the proposed method for any applications in which the problematic domain is crossed a very large number of times during simulations.

Modeling of the Asphaltene Onset Pressure from Few Experimental Data: A Comparative Evaluation of the Hirschberg Method and the Cubic-Plus-Association Equation of State

Asphaltene onset pressure (AOP) is a key parameter to determine the flow assurance of live oils. In this study, the capabilities of the cubic-plus-association (CPA) equation of state (EoS) and the Hirschberg method of calculating the AOP of five oils are compared to a new approach using few experimental data. Two experimental data points of AOP and only one of the bubble pressure (BP) are required for adequate parametrization of both models. The number of non-zero binary interaction parameters needed for the CPA EoS is reduced to only four binary pairs. In the Hirschberg method, the BP calculations were performed with the Soave–Redlich–Kwong EoS. For all oils, the CPA EoS provided lower deviations for both AOP and BP. To achieve a good correlation, the Hirschberg method requires the use of AOP experimental data to calculate the difference between the liquid-phase solubility parameter and the asphaltene solubility parameter. Similarly, CPA EoS requires the calculation of the cross-associating energy parame...

Multiphase equilibrium calculations using a reduction method

Phase equilibrium problems (phase stability testing and multiphase flash) have to be solved repeatedly, sometimes a very large number of times, in process simulators and in compositional reservoir simulation. The computational effort can be significant, particularly when a detailed description (with a large number of components nc) of the mixture is required. The reduction method represent an attractive technique in the attempt to reduce the computational time, by significantly reducing the dimensionality of the problem, from nc×(np−1) in the conventional methods to a maximum of (2c+3)×(np−1), where np is the number of equilibrium phases and c the number of components with non-zero binary interaction parameters (BIP). The number of independent variables does not depend on nc, but only on c, and the computation time increases linearly with nc. Traditionally, the reduction approaches use the reduction parameters and the phase mole fractions as independent variables. In this work, a recent improvement of two-phase reduced flash calculations (Nichita and Graciaa, Fluid Phase Equilib. 302, 2011, 226–233) is extended to multiphase equilibrium calculation with any number of phases. Two approaches are proposed: (i) a direct extension of the reduction method for two-phase flashes and (ii) a constrained minimization Gibbs free energy with respect to a specific set of variables and constraints (taking advantage of symmetry). The proposed algorithms are tested for several multiphase systems (with up to four phases and exhibiting complex phase envelopes) containing hydrocarbon components, carbon dioxide and hydrogen sulfide. Numerical experiments show that the reduction methods for multiphase flash calculations are robust and they become faster than the conventional methods when (i) the number of components increases, (ii) the number of equilibrium phases increases and (iii) the number of BIP families decreases. For mixtures with many components and few BIP families, the reduction method may be at least one order of magnitude faster than conventional methods.

A comparison of conventional and reduction approaches for phase equilibrium calculations

A major part of the computational effort in chemical process simulation and compositional reservoir simulation is spent by phase equilibrium calculations: phase split and phase stability analysis. The calculation algorithms must be efficient and highly robust. The reduction methods offer an attractive alternative to conventional methods. Several recent papers have questioned on the efficiency of the reduction methods as compared to conventional methods, concluding that reduction methods become faster than conventional ones starting from a certain number of components. In this work, besides evaluating the computational effort required by various conventional and reduction formulations, the efficiency (per iteration and global) of stability testing and flash calculations is evaluated separately, the condition of the linear system of equations is analyzed, as well as the influence of symmetry properties and of the linear solver. Some important links are formally established between conventional and reduced methods and between different reduction methods. Numerical experiments carried out with several mixtures confirm that flash calculations using the reduction methods are more efficient than conventional methods only for mixtures with many components (more than 20) and few non-zero binary interaction parameters; this means that reduction methods for phase split are suitable for process simulation (mixtures may contain hundreds of components), but there are not suitable for compositional simulation, where the number of components is limited, typically to a dozen. On the contrary, for phase stability testing, it is shown that reduction methods are more efficient than conventional ones even for mixtures with few components, thus they are suitable for compositional simulation. This observation is very important since, in most compositional reservoir simulators, no explicit phase split is required, flash calculations being part of a larger problem (coupled with flow equations), while phase stability testing must be performed in most grid blocks.

Improved Delumping of Compositional Simulation Results

A recently developed method for pseudo-component delumping is tested for delumping the results of compositional reservoir simulation. The procedure, based on the reduction concept, is analytical and consistent and it accurately takes into account non-zero binary interaction parameters in cubic equations of state. The delumping method was implemented in a postprocessor to the commercial Eclipse reservoir simulator and successfully tested on several cases: oil and gas condensate reservoirs for depletion, lean gas re-injection, and carbon dioxide injection. The analytical delumping method based on reduction gives systematically better results than the regression-based delumping method used previously.

Reservoir fluid applications of a pseudo-component delumping new analytical procedure

Compositional reservoir simulations require a huge number of flash calculations. The problem dimensionality (and thus the computational effort) is usually reduced by lumping several (often many) individual components into pseudo-components. Typically, less than 10 pseudo-components are used for full-scale field simulations. However, the detailed fluid phase split is important for surface process simulations. The detailed phase compositions resulting from a flash calculation performed on a lumped mixture can be estimated using a delumping (inverse lumping) procedure. In fact, the delumping acts like an interface between reservoir and surface simulations. We have recently proposed a delumping procedure based on the reduction method for phase equilibrium calculations, which is (i) analytical, (ii) consistent, (iii) applicable for equations of state (EoS) with non-zero binary interaction parameters (BIPs), and (iv) its applicability holds for a large variety of EoS, provided suitable mixing rules are used. In this paper, the new delumping method is tested for several reservoir fluids and reservoir processes (such as gas injection), focusing on how severe changes in composition are affecting its accuracy. We examine a series of flashes, considered to be representative of reservoir processes (differential expansion and constant volume depletion, multistage separations), and swelling tests. Non-zero BIPs between hydrocarbon components and classical contaminants, as well as between methane and heavier hydrocarbon components are considered in all cases. For all test examples, phase mole fractions and the vapor mole fraction of the delumped mixture are in excellent agreement with the values obtained by flashing the original mixture. Advancing to the next pressure step, the detailed composition obtained by delumping preserves a high accuracy in predicting detailed fluid properties even for drastic compositional changes.

An analytical consistent pseudo-component delumping procedure for equations of state with non-zero binary interaction parameters

For mixtures with many components, some or most of the components are grouped into pseudo-components in order to reduce the dimensionality of the problem for phase equilibrium calculations, and therefore the computational effort. However, knowing the detailed fluid phase split may be important for a variety of applications. The detailed phase compositions resulting from a flash calculation performed on a lumped mixture can be predicted using a delumping (inverse lumping) procedure [C.F. Leibovici, E.H. Stenby, K. Knudsen, Fluid Phase Equilibr. 117 (1997) 225–232]. If the mixture parameters of an equation of state (EoS) can be expressed as a linear combination of pure component parameters and the phase mole fractions, then the component fugacity coefficients can also be expressed as a linear combination of pure component parameters with coefficients only depending on mixture properties. As a result, the equilibrium coefficients are related only to component properties and EoS coefficients, independently on phase compositions. In this work, we show using a reduction method how to effectively obtain such an expression of the equilibrium constants even for non-zero binary interaction parameters (BIPs) in the EoS, and based on these results, we propose a totally consistent analytical procedure for the estimation of equilibrium constants of detailed mixtures from lumped information, which is an extension of Leibovici's delumping method. For several examples with non-zero BIPs between hydrocarbon components and classical contaminants, phase mole fractions and the vapor mole fraction of the delumped mixture are in excellent agreement with the exact values obtained by flashing the original mixture. The delumping procedure has multiple applications, mainly for reservoir simulation and distillation problems.

Consistent delumping of multiphase flash results

An analytical and consistent delumping method, recently formulated and implemented for processes involving two-phase flash calculations, is extended to flash calculations involving any number of coexisting phases. The equation of state considered is a two-parameter cubic equation of state (EoS) with van der Waals mixing rules and non-zero binary interaction parameters (BIPs). The method is used for calculating three- and four-phase equilibria in a series of hydrocarbon mixtures containing different amounts of carbon dioxide, nitrogen, hydrogen sulphide and water. In all examples, the location, compositions and proportions of the coexisting phases obtained by the delumping procedure are extremely close to those obtained from calculations with the detailed fluid, even in cases where the multi-phase region is very tiny. To the best of our knowledge, this is the first time that a delumping procedure is formulated and successfully used for multiphase equilibria.

A new method for critical points calculation from cubic EOS

AbstractDevelopment of accurate and efficient methods for direct calculation of critical points is valuable for phase equilibrium calculations in chemical and petroleum engineering. A new method for critical points calculation is developed. For critical point calculation methods, the spinodal equation is given by setting to zero a determinant of nc (number of components) or nc‐1 order, containing second‐order derivatives of thermodynamic potentials. In this approach, this determinant has a lower order, which does not depend on the number of components in the mixture, but on the number of reduction parameters, with no restrictions on the binary interaction parameters (BIPs). The new method has been tested for several mixtures having various phase diagram shapes, and proved its reliability. The computation effort for each step for correcting temperature and molar volume can be significantly reduced. The proposed method is particularly efficient for mixtures with many components and few nonzero BIPs. Cubic two‐parameter EOS are used; however, the proposed method can be applied for any EOS that observes the restrictions of the Reduction Theorem. © 2005 American Institute of Chemical Engineers AIChE J, 2006

Calculation of critical points using a reduction method

Efficient and accurate calculation of critical points is an important aspect of general phase equilibrium calculations. When calculating critical points, the computationally expensive part is to solve the spinodal equation. For Heidemann and Khalil formulation the dimensionality of the problem is equal to the number of components (nc). In this paper we describe a new procedure enabling the reduction of problem dimensionality.The reduction is effectively achieved by spectral decomposition of the matrix C¯ with elements (1−Cij), where Cij denotes the binary interaction parameters (BIP). The dimensionality of the problem is given by the number of reduction parameters, M=m+1, where m is the number of nonzero (or non-negligible) eigenvalues of C¯. The spinodal equation consists in equating to zero a determinant of order M (for naturally occurring hydrocarbon mixtures, M is much less than nc). The procedure suggested by Michelsen and Heidemann for zero BIP is a particular case of our approach.For the spectral decomposition of C¯, the method is particularly efficient for mixtures with many components and few nonzero BIP. The proposed method proved its reliability when tested for several mixtures having various phase diagram shapes. Finally, we show that critical points of mixtures with more than 50 components, with a detailed paraffin-naphthene-aromatic (PNA) distribution of the heavy fraction, can be calculated by only performing linear algebra operations with a small dimensionality. Application of the reduction method leads to important savings in computer time.