Rachford-Rice Equation Research Articles

Accelerating Flash Calculation using Compositional Space for Compositional Simulation

Compositional simulation is often required for modeling complex oil recovery processes, such as CO2 enhanced oil recovery (Chen and Reynolds, 2016; Dai et al., 2014). Flash calculation using Equation of State (EOS) is very CPU expensive in compositional simulation. Conventional approaches for initial guess of K-values usually have several levels of limitations. This paper reports our research motivated to provide trustworthy initial guess for flash calculation and further improve flash performance in isothermal processes. Reservoir fluid composition changes are categorized into two different scenarios: (a) slightly varying or constant composition changes in depletion process; and (b) sharp varying composition in gas injection process. Scenario (a) is reasonably parameterized as a compositional space of K-value and vapor mole fraction varied with pressure. Considering the actual simulation of gas injection process, a compositional space is defined via introducing a key pseudo composition as a controlling variable in addition to pressure. Flash calculation is formulated through direct solution of those non-linear equations in phase-split calculation, and solved by combining successive substitution iteration (SSI) and Newton-Raphson (NR) method. Using interpolation from the constructed compositional space, it is found that the initial guess of K-values for flash calculation are always obtained with safeguard. A variety of fluid samples has been applied for testing this new flash technique. The results show that initial guesses of K-values based on Wilson’s correlation results cause much more flash iterations and even failures in some gas injection examples. On the other hand, the proposed method shows significant advantages in reducing the number of flash iterations in all examples. In addition, the SSI procedure in flash can even be bypassed and only NR procedure and a simple Rachford-Rice Equation Preconditioning is sufficient.

On relationship between two ancient Chinese algorithms and their application to flash evaporation

This paper gives a brief introduction to two ancient Chinese methods, Ying Buzu Shu and He Chengtian average. Their relationship is revealed, and it is proved that the two methods are equivalent under some special cases, that means He Chengtian average can be derived from the method of Ying Buzu Shu. Rachford-Rice equation arising in flash evaporation is used as an example for their simple application.

Prediction of CO2-CH4-H2S-N2 gas mixtures solubility in brine using a non-iterative fugacity-activity model relevant to CO2-MEOR

Numerical simulations of carbon dioxide-microbial enhanced oil recovery (CO2-MEOR) would require computationally rigorous iterative methods to solve resulting system of flow, transport and kinetic reaction equations. This includes additional iterative procedures to account for the solubility of gas mixtures in the aqueous phase. This work proposes a new non-iterative fugacity-activity thermodynamic model to predict the solubility of CO2-CH4-H2S-N2 gas mixtures in brine. This model can readily be implemented in MEOR simulation programs to account for mass transfer and kinetics of microbial reactions in CO2-MEOR operations. Fugacity coefficients (ϕi) in the proposed model were calculated using Predictive Peng-Robinson 78 (PPR78) and Peng-Robinson (PR) equation of state (EOS). The proposed model was successful in predicting CO2 solubility in the aqueous phase with ϕi calculated either using PPR78 EOS or PR EOS. Comparison showed that at temperature and pressure conditions relevant to MEOR, using calibrated binary interaction parameters (PR EOS) leads to more accurate predictions than binary interaction parameters estimated from the group contribution expression (PPR78 EOS). Comparison of predicted CO2-CH4-H2S gas mixtures solubility in the aqueous phase obtained with the proposed non-iterative fugacity-activity model and an iterative fugacity-activity model, confirmed that proposed model with ϕi calculated using PR EOS can be used as substitute for iterative fugacity-activity models that relies on the solution of Rachford-Rice equation.

Modelling and dynamic simulation of a large MSF plant using local phase equilibrium and simultaneous mass, momentum, and energy solver

Abstract Seawater desalination is an important method for producing fresh water, a critical resource in many areas of the globe. Multi-stage flash distillation (MSF) is a leading technology within the thermal desalination field. Mathematical modelling and simulation provides a means to enhance engineering and operation of complicated production plants. This paper presents a new method for one-dimensional modelling and dynamic simulation of thermal desalination processes. The approach combines the simultaneous mass, momentum, and energy solution, local phase equilibrium by Rachford-Rice equation, and rigorous calculation of the seawater properties as function of temperature, pressure and salinity. A brine recycling MSF plant was modelled as a case study, presenting advanced and unpublished simulated features and transients. The successful results suggest that the method presented is a competent approach for dynamic simulation of thermal desalination processes.

On a choice of independent variables in Newton iterations for multiphase flash calculations

Multiphase split calculations consist in the minimization of the Gibbs free energy. If the second-order Newton method is used, either component mole numbers or the logarithms of equilibrium constants (lnK) are selected as independent variables. Several recent papers advocated the use of the phase mole fractions together with lnK as independent variables in Newton iterations (lnK-θ). Unlike in the lnK approach, there is no explicit resolution of the Rachford-Rice (RR) equations, which are solved implicitly during the Newton update. The resulting linear system is non-symmetric and is solved by Gaussian elimination. In this study, this latter choice of variables is analyzed in detail, and an efficient formulation is proposed in which algebraic transformations are used to recast the lnK-θ method in a way that enables using the symmetry efficiently, both for constructing the Jacobian matrix and for solving the linear system of equations using a Cholesky factorization. Taking advantage of the block structure of the Jacobian matrix, it is revealed that the matrix of the linear system is formally identical to the Jacobian of lnK (evaluated at a lower implicitness level) and phase mole fraction can be updated separately using the partial solution of this system. Numerical experiments carried out on several multiphase mixtures exhibiting complex phase envelopes show the robustness and the efficiency of the proposed approach as compared to the original lnK-θ methodology, with the observed computational time savings up to 25%.

Multiphase isenthalpic flash integrated with stability analysis

Isobaric, isenthalpic (PH) flash is challenging for multiphase non-isothermal flow simulation using an equation of state (EOS). This is because the number of equilibrium phases is unknown in temperature and composition space, and because the system of equations in PH flash becomes nearly degenerate for narrow-boiling fluids. The term “narrow-boiling” is used in the literature to refer to enthalpy that is sensitive to temperature.The primary objective of this research is to develop the multiphase PH-flash algorithm integrated with stability analysis that resolves the two technical challenges mentioned above. The secondary objective is to present a new analysis of narrow-boiling behavior by coupling energy and phase behavior equations through the temperature dependency of K values. The thermodynamic model used is the Peng-Robinson EOS with the van der Waals mixing rules.PH flash in this research is formulated by use of the tangent-plane-distance function, in which phase-split computation is integrated with phase-stability analysis. The formulated PH flash is solved by the direct-substitution algorithm with an arbitrary number of sampling compositions (NS), at which phase stability is measured during the iteration. The number of equilibrium phases is not required to be fixed in the new algorithm.Results in case studies show that the new algorithm can robustly handle phase appearance/disappearance with narrow-boiling behavior, including the case of one degree of freedom. The algorithm becomes more robust with increasing NS because the possibility of finding all stationary points of the tangent-plane-distance function increases. However, the number of iterations required tends to increase with increasing NS because the algorithm with more sampling compositions may take more iterations for merging and adding some of the sampling compositions.The general condition presented for narrow-boiling behavior is that the interplay between energy balance and phase behavior is significant. Two subsets of the condition are derived by analyzing the convex function whose gradient vectors consist of the Rachford-Rice equations; (i) the overall composition is near an edge of composition space, and (ii) the solution conditions (temperature, pressure, and overall composition) are near a critical point, including a critical endpoint. A special case of the first specific condition is the fluids with one degree of freedom. These conditions for narrow-boiling behavior are demonstrated in case studies.

Biodiesel Production by Reactive Flash: A Numerical Simulation

Reactive flash (RF) in biodiesel production has been studied in order to investigate steady-state multiplicities, singularities, and effect of biodiesel quality when the RF system approaches to bubble point. The RF was modeled by an index-2 system of differential algebraic equations, the vapor split (ϕ) was computed by modified Rachford-Rice equation and modified Raoult’s law computed bubble point, and the continuation analysis was tracked on MATCONT. Results of this study show the existence of turning points, leading to a unique bubble point manifold,(xBiodiesel,T)=(0.46,478.41 K), which is a globally stable flashing operation. Also, the results of the simulation in MATLAB® of the dynamic behavior of the RF show that conversion of triglycerides reaches 97% for a residence time of 5.8 minutes and a methanol to triglyceride molar flow ratio of 5 : 1.

A new parametric algorithm for isothermal flash calculations by the Adomian decomposition of Michaelis–Menten type nonlinearities

We develop a robust algorithm for isothermal flash calculations of multi-component mixtures. The algorithm provides an explicit, parametric solution for a recent variation of the Rachford–Rice equation by employing a powerful analytical technique, namely the Adomian decomposition method. The computational speed of the algorithm is shown to be further enhanced by applying the iterated Shanks transformation. Unlike previous methods, our algorithm obviates the troublesome need for an initial guess, guarantees an excellent rate of convergence and is demonstrated to be more computationally economic than prior art. Several reliable examples from the literature, including the flash calculation for a mixture with nineteen components, as well as an extensive set of hypothetical, randomly-generated problems are presented to illustrate the exceptional efficacy of our new approach.

On solving the Rachford-Rice equation with higher order methods

The Rachford-Rice equation calculates the amounts and compositions of two equilibrium phases at constant K-factors. This single variable equation is a perfect candidate for applying higher order iteration methods because the additional cost for evaluating the derivatives with order higher than two is relatively low. This study compares the performance of Householder's high order iterations up to order seven. In addition, a method to improve the initial estimate is proposed for the situation where the Rachford-Rice function shows abrupt change close to the asymptotes and overshooting happens. The proposed method can largely reduce repeated use of the bisection adjustment in the subsequent iterations. The comparison shows that the Householder's iteration with order three or four gives the best performance on average.

A rapid and robust method for solving the Rachford–Rice equation using convex transformations

The Rachford–Rice equation must be solved in an inner loop of flash calculations. It consists in a sum of hyperbolas, and the desired root is located between two asymptotes which define the negative flash window. First, a simple procedure to remove the adjacent poles by changing the variable is proposed. Then, a class of convex transformations is identified, and several elementary functions that respect the general convexity condition are recognized. A new extremely simple solution window is proposed (depending only on compositions of the lightest and heaviest components), as well as an initialization procedure. Based on several convex formulations, two solution algorithms are proposed, based on the selection of the appropriate convex function (such that the monotonic convexity condition is fulfilled), and several Newton methods (including higher order) can be used without any control (it is guaranteed that an overshoot of the solution cannot occur). The transformations to make the functions convex within the search region, enable the use of Newton-type updates without the need for bounds checking and interval reduction lead to significant increase in speed of solution. The proposed algorithms are simple, very easy to implement, robust and rapid; as shown by various test examples (for selected very difficult cases and for a very large number of randomly generated flashes), only few iterations are required for convergence and the proposed method compares favorably to previous solution methods.

An efficient method to calculate multisolid-(wax) precipitation

Based on the multisolid-wax precipitation assumption for petroleum mixtures, a monotonic equation is proposed for the calculation of liquid and multisolid equilibria. Owing to the fact that the number of precipitating species can be determined instantly at fixed temperature and pressure through a stability analysis, the computation is fast and reliable. Although the developed iteration procedure is focus to liquid mixtures it is shown that can be easily incorporate into existing vapor–liquid iteration procedures based on the Rachford–Rice equation to get a vapor–liquid–multisolid-(wax) flash calculations. Numerical experimentations are shown and even more the proposed iteration method is used in the construction of phase boundaries without any problem.

Initialization of phase fractions in Rachford–Rice equations for robust and efficient three-phase split calculation

A key element of efficient and robust three-phase split calculation is the initialization of phase mole fractions. While the initial guess of equilibrium ratios is made from two-phase split calculation and two-phase stability testing, initializing phase fractions through solving three-phase Rachford–Rice equations can be a challenge especially close to the bicritical points and phase boundaries. In this research, we examine three different methods: the improved two-dimensional bisection method, the minimization method, and the direct Newton method. We present a large number of three-phase examples of various degrees of complexity to demonstrate both robustness and efficiency of all these methods. The direct Newton method combined with the starting guess of phase fractions from two-phase split calculation and two-phase stability testing is the most efficient approach.

An Iterated Eigenvalue Algorithm for Vapor–Liquid Split in Flash Calculations

Flash calculations are used extensively in computer-aided process simulation programs. Inherent to flash calculations is the determination of the vapor–liquid split. Currently, to calculate the vapor–liquid split, the Rachford–Rice equation is the most accepted method. In this work, it is shown that the Rachford–Rice equation can be posed as a generalized eigenvalue problem, where each eigenvalue of the resulting system is exactly a root of the Rachford–Rice equation. In this way, an ad hoc strategy based on linear algebra is proposed for finding the vapor–liquid split. The main advantage of the proposed strategy is that the resulting linear system has an almost-diagonal structure and generates a discrepancy function with a more linear shape than that of the Rachford–Rice equation. In this way, the solution of the system is simple and computationally inexpensive. Numerical experiments show the reliability of the proposed solution strategy for calculating the vapor–liquid split.

A general framework of model functions for rapid and robust solution of Rachford–Rice type of equations

For most phase equilibrium calculations, a non-linear function in one unknown, known as the Rachford–Rice equation for vapor–liquid equilibrium problems, needs to be solved for determining the molar fractions of the co-existing phases required for achieving the desired feed separation. Two issues appear when solving this type of equations. Firstly, the existence of as many asymptotes as the number of components present in the mixture splits these equations to a set of n−1 branches, each one containing a different solution. Therefore, care should be taken to ensure that the obtained solution lies within the branch of interest. Secondly, conventional solution methods often exhibit slow convergence rates which is an issue when repeated solutions are required as in the case of compositional reservoir simulation and pipeline flow.In this work, a general framework for developing new, rapid and robust solution methods is presented utilizing function classes, the members of which exhibit behavior similar to that of the equation to be solved. Fitting such functions, instead of a simple straight line at each iteration of the solution process, as is the case of the Newton–Raphson approach, leads to better subsequent estimates and thus to faster convergence. Moreover, a new initial value estimation method is presented for cases in which the Rachford–Rice function exhibits a very abrupt shape and it is shown that, for such cases, the number of iterations required by the conventional methods can be reduced to only one or two.

Calculating the vapor-liquid phase equilibrium for multicomponent systems using the Soave-Redlich-Kwong equation

An algorithm for calculating the vapor-liquid phase equilibrium for multicomponent gases is presented. A six-component gas mixture is considered. The algorithm for calculating phase equilibrium is based on the solution to the Soave-Redlich-Kwong equation of state for real gases. An iteration algorithm for updating phase equilibrium constants is used. The total vapor fraction is determined using the solution to the Rachford-Rice equation. The difficulties of developing such algorithms are considered in full detail. The derived solution is compared with results obtained in the commercial package HYSYS (version 3.2) when the Soave-Redlich-Kwong equation is also used.

A rigorous approach to the solution of phase equilibrium of geothermal fluids

The phase-split problem for a geothermal multicomponent mixture is considered. The mixture is mainly composed by water (or brine) and other gas species. Solving the phase-split problem require the minimization of the Gibbs free energy of the system. This is addressed by means of an iterative procedure which is stopped whenever the equality of chemical potentials (i.e. the thermodynamic equilibrium condition) is met within some fixed tolerance. At any iteration, the mole fraction of each phase is evaluated according to the Rachford-Rice equation and excess chemical potentials are calculated from Peng-Robinson equation of state. Results from numerical simulations are shown.

Efficient and robust three‐phase split computations

AbstractThe combination of successive substitution and the Newton method provides a robust and efficient algorithm to solve the nonlinear isofugacity and mass balance equations for two‐phase split computations. The two‐phase Rachford–Rice equation may sometimes introduce complexity, but the Newton and bisection methods provide a robust solution algorithm. For three‐phase split calculations, the literature shows that the computed three‐phase region is smaller than measured data indicate. We suggest that an improved solution algorithm for the three‐phase Rachford–Rice equations can address the problem. Our proposal is to use a two‐dimensional bisection method to provide good initial guesses for the Newton algorithm used to solve the three‐phase Rachford–Rice equations. In this work, we present examples of various degree of complexity to demonstrate powerful features of the combined bisection‐Newton method in three‐phase split calculations. To the best of our knowledge, the use of the bisection method in two variables has not been used to solve the three‐phase Rachford–Rice equations in the past. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Three-phase free-water flash calculations using a new Modified Rachford–Rice equation

A novel Modified Rachford–Rice equation is developed for three-phase equilibrium calculations in hydrocarbon–water systems, based on the free-water assumption, i.e., the water-rich liquid phase consists of pure water only. In the inner loop of the flash algorithm, the three-phase problem (consisting in a system of two non-linear equations) is replaced by a pseudo-two-phase problem (consisting in a non-linear equation). Unlike previous formulations, the new Modified Rachford–Rice function is guaranteed to monotonically decrease between two adjacent asymptotes. The negative flash concept is used, and a search window is proposed for the vapor fraction. The new free-water flash method proved to be robust, and excellent agreement between full three-phase flash and pseudo-two-phase free-water flash was obtained for various test problems. The proposed method is very useful in compositional reservoir simulation of certain oil recovery methods and in process simulation.

A New Algorithm for Rachford-Rice for Multiphase Compositional Simulation

Summary Flash calculations for use in compositional simulation are more difficult and time-consuming as the number of equilibrium phases increases beyond two. Because of its complexity, many simulators do not even attempt to incorporate three or more hydrocarbon phases, even though such cases are important in many low-temperature gasfloods or for high temperatures where hydrocarbons can partition into water. Multiphase flash algorithms typically use successive substitution (SS) followed by Newton's method. For NP-phase flash calculations, (NP–1) Rachford-Rice (RR) equations are solved in every iteration step in SS and, depending on the choice of independent variables, in Newton's method. Solution of RR equations determines both compositions and amounts of phases for a fixed overall composition and set of K-values. A robust algorithm for RR is critical to obtain convergence in multiphase compositional simulation and has not been satisfactorily developed, unlike the traditional two-phase flash. In this paper, we develop an algorithm for RR equations for multiphase compositional simulation that is guaranteed to converge to the correct solution independent of the number of phases for both positive and negative flash calculations. We derive a function whose gradient vector consists of RR equations. This correct solution to the RR equations is formulated as a minimization of the nonmonotonic convex function using the independent variables of (NP–1) phase mole fractions. The key to obtaining a robust algorithm is that we specify nonnegative constraints for the resulting equilibrium phase compositions, which are described by a very small region with no poles. The minimization uses Newton's direction with a line-search technique to exhibit superlinear convergence. We show a case in which a previously developed method cannot converge while our algorithm rapidly converges in a few iterations. We implement the algorithm both in a standalone flash code and in UTCOMP (Chang et al. 1990), a multiphase compositional simulator, and show that the algorithm is guaranteed to converge when a multiphase region exists as indicated by stability analysis.

Una Solución de la Ecuación de Rachford-Rice para Sistemas Multifases Aplicando el Método Newton-Raphson, un Parámetro de Broyden y el Flash Negativo

A solution is obtained for the generalized Rachford-Rice equation (GR-R) for multiple phases and multicomponent systems by using a numeric approach that coupled a modified Newton-Raphson’s method, a Broyden-type damping parameter (obtained from the Euclidian vector norm of the GR-R residual functions) and the Negative Flash, which conditioned the positive nature of the component fractions in the resulting phases. This solution was tested on hypothetical systems of P phases and N components randomly generated, and the same was done with the initial value vectors. It was found that the proposed solution is highly stable and converges for any type of the initial value vector and that the number of iterations is critically affected by this vector and not necessarily for the amount of phases or the number of components.