Successive Substitution Method Research Articles

A robust and efficient solution for COSMO-based activity coefficient models

COSMO-based activity coefficient models, such as COSMO-RS and COSMO-SAC, offer a powerful tool that bridges quantum chemical calculations with macroscopic phase equilibrium calculations. These models utilize a group contribution methodology similar to UNIFAC but require an expensive numerical solution of the self-consistency equation for interactions among all surface segments, representing a primary bottleneck in their application to analyses that involve extensive phase equilibrium calculations. This work demonstrates that the self-consistency equation can be derived by minimizing the system energy from all pairwise segment interactions. The derivation leads to an optimization-based second-order solution procedure that ensures convergence and enhances efficiency. We demonstrate the robustness and efficiency of the proposed solution using various examples, including a comparison with the classical successive substitution method, and illustrate that COSMO-based models can be coupled with modern second-order convergent algorithms for phase equilibrium calculations. Furthermore, this work reveals the similarity between COSMO-based models and the SAFT method.

Modification of Quasi-Newton Successive Substitution Method for Calculating Phase Equilibria of Hydrocarbon Mixtures Taking into Account the Capillary Pressure Jump

Modification of Quasi-Newton Successive Substitution Method for Calculating Phase Equilibria of Hydrocarbon Mixtures Taking into Account the Capillary Pressure Jump

Modeling transcranial magnetic stimulation coil with magnetic cores

Objective. Accurate modeling of transcranial magnetic stimulation (TMS) coils with the magnetic core is largely an open problem since commercial (quasi) magnetostatic solvers do not output specific field characteristics (e.g. induced electric field) and have difficulties when incorporating realistic head models. Many open-source TMS softwares do not include magnetic cores into consideration. This present study reports an algorithm for modeling TMS coils with a (nonlinear) magnetic core and validates the algorithm through comparison with finite-element method simulations and experiments. Approach. The algorithm uses the boundary element fast multipole method applied to all facets of a tetrahedral core mesh for a single-state solution and the successive substitution method for nonlinear convergence of the subsequent core states. The algorithm also outputs coil inductances, with or without magnetic cores. The coil–core combination is solved only once i.e. before incorporating the head model. The resulting primary TMS electric field is proportional to the total vector potential in the quasistatic approximation; it therefore also employs the precomputed core magnetization. Main results. The solver demonstrates excellent convergence for typical TMS field strengths and for analytical B–H approximations of experimental magnetization curves such as Froelich’s equation or an arctangent equation. Typical execution times are 1–3 min on a common multicore workstation. For a simple test case of a cylindrical core within a one-turn coil, our solver computed the small-signal inductance nearly identical to that from ANSYS Maxwell. For a multiturn rodent TMS coil with a core, the modeled inductance matched the experimental measured value to within 5%. Significance. Incorporating magnetic core in TMS coil design has advantages of field shaping and energy efficiency. Our software package can facilitate model-informed design of more efficiency TMS systems and guide selection of core material. These models can also inform dosing with existing clinical TMS systems that use magnetic cores.

Robustness and efficiency of phase stability testing at VTN and UVN conditions

Beyond the well-documented conventional phase equilibrium calculations at pressure and temperature (PT) specifications, other specifications, such as volume, temperature and moles (VTN) and internal energy, volume and moles (UVN) have received an increasingly interest in the last years. Whatever the selected set of independent variables, phase stability at UVN conditions can be asserted by solving simpler VTN or PT stability problems (a non-linear equation is solved only once at UVN specifications for temperature). In VTN stability, modified Newton iterations were previously used to achieve convergence, since successive substitution iterations (SSI) may exhibit problematic convergence. A mathematical analysis of convergence properties of the successive substitution (SSI) method for VTN stability reveals its potential instability, unlike for PT stability, in which SSI is very robust. This problem can be theoretically overcome by using a damping factor, but severe damping may lead to unacceptable slow convergence rates. It is shown that the SSI method should never be used alone, combined undamped SSI-standard Newton method is neither robust nor efficient as claimed previously and must be avoided; moreover, damped SSI can be used with caution only in early iteration stages. The convergence behavior of several methods for VTN stability is analyzed for a variety of mixtures and the numbers of iterations are compared with those reported in the literature. Several domains in the molar density-temperature plane were identified, where SSI is either systematically not detecting a phase split (converging to the trivial solution from all initial guesses) or systematically diverges. Our calculation procedures are highly robust and systematically faster than previous methods (for some test points up to one or even two orders of magnitude) and are able to find the global minimum of the TPD function in all test cases, unlike previously proposed SSI/Newton methods, which either are strongly attracted by local minima or they diverge. This paper gives for the first time a detailed mathematical and numerical analysis of the convergence behavior of SSI and combined SSI-Newton methods in VTN stability testing.

Matrix Approach for Analyzing n-Site Generalized ASIP Systems: PGF and Site Occupancy Probabilities

The Asymmetric Simple Inclusion Process (ASIP) is an n-site tandem stochastic network with a Poisson arrival influx into the first site. Each site has an unlimited buffer with a gate in front of it. Each gate opens, independently of all other gates, following a site-dependent Exponential inter-opening time. When a site’s gate opens, all particles occupying the site move simultaneously to the next site. In this paper, a Generalized ASIP network is analyzed where the influx is to all sites, while gate openings are determined by a general renewal process. A compact matrix approach—instead of the conventional (and tedious) successive substitution method—is constructed for the derivation of the multidimensional probability-generating function (PGF) of the site occupancies. It is shown that the set of (2nn) linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of 2n−1n equations, and then further reduced to a set of 2n−n+1 equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an n+1-site network can be derived from the corresponding PGF of an n-site system. Explicit results for networks with n=3 and n=4 sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site k k=1,2,…,n is occupied. We show that, in the case where arrivals occur to the first site only, these probabilities are functions of both the site’s index and the arrival flux and not solely of the site’s index. Consequently, refined formulas for the latter probabilities and for the mean conditional site occupancies are derived. We further show that in the case where the arrival process to the first site is Poisson with rate λ, the following interesting property holds: Psite k is occupied | λ=1=Psite k+1 is occupied | λ→∞. The case where the inter-gate opening intervals are Gamma distributed is investigated and explicit formulas are obtained. Mean site occupancy and mean total load of the first k sites are calculated. Numerical results are presented.

An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model

The efficiency and accuracy of phase equilibrium calculations are essential in compositional reservoir models. Usually, a significant part of the computational effort in compositional reservoir simulations is spent on phase equilibrium calculations. The nonlinear nature of phase equilibrium calculations requires an iterative solution procedure. Although the successive substitution method (SSM) is robust and simple to implement, it suffers from slow convergence, especially near the critical point of the mixture. The general dominant eigenvalue method (GDEM) has been widely used to accelerate SSM, but its stability and efficiency deteriorate as the temperature and pressure approach the critical point. This paper proposes a modified form of GDEM to improve its performance in the near-critical region. The modifications have two aspects. First, the liquid phase fraction in the mixture is added as a variable when performing GDEM acceleration, improving both stability and efficiency. The second modification is a post-calibration step imposed to replace the conventional criterion, which is applied before triggering GDEM. With the help of the post-calibration step, the stability of the modified GDEM is ensured, and more importantly, the calculation efficiency can be improved. Numerical tests of three hydrocarbon mixtures, including different numbers of components, show that the stability of the modified GDEM is almost the same as SSM and that its calculation efficiency is much higher than SSM and the conventional GDEM. Cited as: Wang, X., Wei, D., Wang, X., Zhao, X., Li, J., Noetinger, B. An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model. Advances in Geo-Energy Research, 2022, 6(3): 241-251. https://doi.org/10.46690/ager.2022.03.07

Optimal MEA/DIPA/water blending ratio for minimizing regeneration energy in absorption-based carbon capture process: Experimental CO2 solubility and thermodynamic modeling

A new CO2 absorption solution was developed by blending monoethanolamine (MEA) and diisopropanolamine (DIPA) in H2O. The solubility of CO2 in MEA/DIPA aqueous solutions with various blending ratios was measured at a temperature range of 323.15–383.15 K under CO2 partial pressures of up to 400 kPa. The successive substitution method was introduced to calculate the molar fractions of all 12 species, and the electrolyte–UNIQUAC model was applied to consider non-idealities. Then, the optimal blending ratio was obtained for four different targets: 1) max ΔαCO2 (total amine-based CO2 cyclic capacity), 2) max ΔβCO2 (total solution-based CO2 cyclic capacity), 3) min − Δhabs (heat of CO2 absorption under stripping condition), and 4) min Qregen (regeneration energy for the carbon capture process). Here, ΔαCO2, ΔβCO2, and − Δhabs are the properties of solution, while Qregen is a performance parameter of the process. The Qregen, the sum of sensible heat, latent heat, and heat of reaction, was calculated according to the amine blending ratio using the “shortcut method” [1]. In the process, besides the cyclic capacity and heat of absorption, the effect of sensible heat and latent heat must be considered simultaneously. Therefore, the most reliable method to determine the amine blending ratio is Qregen minimization. The obtained minimum Qregen of the process was 3.44 GJ/t CO2 using MEA/DIPA/H2O solutions of 4.01/25.99/70 (w/w/w), which was 8.7% and 2.6% lower than those of the single MEA and DIPA aqueous solutions due to the synergistic effect of MEA and DIPA.

Rigorous Phase Equilibrium Calculation Methods for Strong Electrolyte Solutions: The Isothermal Flash

New algorithms for vapor – liquid (VLE) and liquid - liquid (LLE) equilibrium calculations at constant temperature, pressure and feed phase composition (PT flash), with application to mixtures containing fully dissociating electrolytes are presented. A new approach is proposed and is rigorously derived by treating phase equilibrium as a minimization problem of the Gibbs free energy, while utilizing the electro-neutrality condition as a constraint. In this way, an augmented function (Lagrange function) is formulated which serves as the basis for the equations that govern phase equilibrium. It is shown that the Lagrange multiplier which corresponds to the electro-neutrality constraint has theoretical meaning and direct relation to the electrostatic potential difference between inhomogeneous phases. Based on the new approach, named Electrochemical Ionic Approach (EIA), two new numerical methods are presented; a successive substitution method similar to the classical Rachford-Rice method and a second-order one, which is based on Newton's method. The new algorithms are general and can be readily applied to mixtures of multiple solvents and one or multiple salts without any modifications. Finally, the Mean Ionic Approach (MIA) for mixtures with multiple solvents and only one salt is discussed in detail and a successive substitution and a second-order method are formulated. Moreover, the MIA is formally extended and presented for mixtures of multiple solvents and multiple salts. All the resulting methods are applied to mixtures with multiple solvents and one or multiple strongly dissociating electrolytes. The performance and technical details of each method are discussed and compared. The calculations are performed with the electrolyte Statistical Associating Fluid Theory with the Mie potential of variable range (eSAFT-VR Mie) equation of state but the proposed methods are general and could be used with any other thermodynamic model.

Computational Model Development for Hybrid Tilting Pad Journal Bearings Lubricated with Supercritical Carbon Dioxide

Fluid film bearings lubricated with supercritical carbon dioxide (sCO2) eliminate the infrastructural requirement for oil lubricant supply and sealing in turbomachinery for sCO2 power systems. However, sCO2’s thermohydrodynamic properties, which depend on pressure and temperature, pose a challenge, particularly with computational model development for such bearings. This study develops a computational model for analyzing sCO2-lubricated tilting pad journal bearings (TPJBs) with external pressurization. Treating sCO2 as a real gas, the Reynolds equation for compressible turbulent flows solves the pressure distribution using the finite element method, and the Newton−Raphson method determines the static equilibrium position by simultaneously calculating forces, moments, flow rates of externally pressurized sCO2, and pressure drop due to flow inertia. The finite difference method solves the energy equation for temperature distribution. The density and viscosity of sCO2 are converged using the successive substitution method. The obtained predictions agree with the previous and authors’ computational fluid dynamics predictions, thus validating the developed model. Hybrid lubrication increases the minimum film thickness and stiffness up to 80% and 65%, respectively, and decreases the eccentricity ratio by up to 65% compared to those of pure hydrodynamic TPJB, indicating significant improvement in the load capacity. The bearing performance is further improved with increasing sCO2 supply pressure.

CO2 absorption mechanism in aqueous ternary solutions of alkanolamines: Experimental and thermodynamic modeling approaches

The absorption mechanism of CO2 in an aqueous solution containing three alkanolamines was analyzed experimentally and theoretically. The vapor–liquid equilibrium of a CO2–monoethanolamine (MEA)–diisopropanolamine (DIPA)–2-amino-2-methyl-propanol (AMP)–H2O system was evaluated experimentally over a wide temperature range (323.15–393.15 K) at several MEA:DIPA:AMP:H2O blending ratios (15:10:5:70, 10:10:10:70, 7.5:7.5:15:70, and 5:15:10:70 wt%). The successive substitution method was used to calculate the concentrations of five molecules (CO2, MEA, DIPA, AMP, and H2O) and nine electrolytes (four cations and five anions) in the liquid phase by solving eight equilibrium equations, four mass balance equations, and one charge balance equation. The Deshmukh–Mather model, which is based on an activity coefficient approach, and the fugacity coefficient model were used to evaluate the nonideality of the liquid and vapor phases, respectively. Thereafter, the effect of the MEA:DIPA:AMP blending ratio was evaluated using the triangular diagrams of the carbamate, bicarbonate and carbonate molar fractions in liquid phase, CO2 loading ratio, CO2 cyclic capacity, and heat of CO2 absorption.

Poroelastic response of spherical indentation into a half space with an impermeable surface via step displacement

A fully coupled poroelastic solution for spherical indentation into a half space with an impermeable surface when the indenter is subjected to step displacement loading is presented. The solution is obtained within the framework of Biot’s theory using the McNamee–Gibson displacement function method. Mathematical difficulties associated with solving poroelastic contact problems are overcome by the use of a series of special functions such as the modified Struve and Bessel functions for evaluating integrals with kernels that oscillate rapidly and the method of successive substitution for solving Fredholm integral equation of the second kind. Expressions for the poroelastic fields on the surface and inside the half space are derived. Effect of poroelasticity on incipient failures in form of tensile crack initiation and plastic deformation are discussed. The theoretical analysis shows that the normalized indentation force relaxation has a relatively weak dependence on material properties through a single derived material constant ω only and the asymptotic behaviors at ω=0 at both early and late times can be expressed in closed-form. Master curves of indentation force relaxation can be constructed by fitting the full solution with an elementary function for convenient use of poroelasticity characterization in the laboratory. In addition, excellent agreement is achieved between the theoretical solution and numerical results from FEM simulations using a hydromechanically coupled algorithm that we had previously benchmarked rigorously.

Amine blending optimization for maximizing CO2 absorption capacity in a diisopropanolamine – methyldiethanolamine – H2O system using the electrolyte UNIQUAC model

Experimental data on CO2 solubility in diisopropanolamine (DIPA) and methyldiethanolamine (MDEA) blended aqueous solutions were measured at different amine blending ratios and working temperatures. The successive (iterative) substitution method was implemented to calculate the molar fractions of all chemical species, including molecules and electrolytes, from equilibrium along with four material balances and one electroneutrality equation. The electrolyte universal quasi-chemical (electrolyte UNIQUAC) model was used to consider the nonideality in the liquid phase. The partial pressures of CO2 in the gas phase and molar fractions of all components in the liquid phase were recalculated using thermodynamic models. In addition, the effect of the blending ratio of DIPA, MDEA, and H2O was investigated and expressed using the newly applied triangular diagrams of pH, heat of absorption, and cyclic capacity of CO2 according to the absorption and stripping conditions.

Effect of Blending Ratio and Temperature on CO2 Solubility in Blended Aqueous Solution of Monoethanolamine and 2-Amino-2-methyl-propanol: Experimental and Modeling Study Using the Electrolyte Nonrandom Two-Liquid Model.

This paper reports the newly measured experimental data for CO2 solubility in a blended aqueous solution of monoethanolamine (MEA) and 2-amino-2-methyl-propanol (AMP) at different amine mixing ratios (MEA/AMP/H2O = 9:21:70, 15:15:70, and 21:9:70 wt %) and working temperatures (323.15, 373.15, and 383.15 K). The successive substitution method was used for calculating the mole fractions of all molecules (four molecules) and electrolytes (three cations and four anions) from the equilibrium along with the material and charge balance equations (11 equations). The electrolyte nonrandom two-liquid (e-NRTL) model was used to investigate nonideality in the liquid phase. Using the abovementioned thermodynamic models, the partial pressures of CO2 in the gas phase, mole fractions of all components in the liquid phase, pH variations, heats of absorption, and cyclic capacities of CO2 according to the absorption/desorption temperature and the blending ratio of MEA/AMP were estimated.

Solución de la ecuación de Rachford – Rice por homotopía diferencial

En el presente trabajo se desarrolla una rutina de cálculo numérico para resolver la ecuación de Rachford-Rice Generalizada (RR-G) en sistemas de tres fases y múltiples componentes, fundamentado en el acople del método de sustitución sucesiva y el de continuación por homotopía diferencial, el cual se aplicó a diferentes tipo de mezclas en equilibrios de Vapor-Liquido-Liquido (EVLL), Vapor-Liquido-Solido (EVLS) y Liquido-Liquido-Liquido (ELLL). El algoritmo propuesto se probó con tres mezclas distintas a condiciones de temperatura, presión, composición y distintos componentes, encontrándose que la solución propuesta es estable y convergente para cualquier tipo de vector de inicio. Los resultados predicen de forma satisfactoria las fases en equilibrio, siendo el error mínimo del 1.9% en ELLL y el máximo igual a 15.47 % en EVLL.

Formulation of equation error estimator using measured displacement de-noised by temporal–spatial filter for system identification of elastic solids

ABSTRACT This paper presents a new class of the equation error estimator using de-noised displacement by the temporal–spatial filter for the system identification in an elastic solid. The finite element method is employed to discretize the equation of motion for an elastic solid, and the Young’s modulus of each finite element is estimated by the proposed approach. The temporal–spatial filter is applied to de-noise measured nodal displacements and to reconstruct nodal accelerations. The equation of motion modified for the de-noised dynamic responses is derived, and the equation error estimator (EEE) for the system identification is defined with the modified equation of motion. The proposed EEE represents the residual error of the modified equation of motion over a measurement interval, and results in a nonlinear equation, which is solved by the successive substitution method. The sensitivity of the temporal–spatial filter required to solve the nonlinear equation is derived by the direction differentiation of the analytical expressions of the filter. The validity and accuracy of the proposed approach are tested through numerical simulation studies. It is shown that the proposed method yields accurate and stable solutions without regularization for all cases tested in this study.

Improved Multiple-Mixing-Cell Method for Accelerating Minimum Miscibility Pressure Calculations

Summary An improved algorithm for accelerating minimum miscibility pressure (MMP) computation using the multiple-mixing-cell (MMC) methods is presented. The MMC method is widely used to accurately calculate the MMP. In this study, we proposed an acceleration algorithm toward original MMC method to directly locate the shortest key tie-line (TL) after a certain amount of contacts through the adjustment of the gas/oil mixing ratio during the calculation process. The algorithm contains the following key components: (1) mixing cell cutoff strategy to avoid unnecessary flash calculations; (2) gas/oil mixing ratio adjustment to prevent lost information on the shortest key TL during the cell cutoff process; (3) a search algorithm for pressure to improve the next step pressure estimate; (4) the fast and reliable two-phase flash implementation by combining full Newton method with recently proposed iteration variables and conventional successive substitution method. The improved MMC model is shown to be faster than the original MMC method in computing MMP.

SOLUTIONS IN EXPLICIT FORM FOR DETERMINING THE HYDRAULIC RESISTANCE COEFFICIENT FOR TURBULENT FLOW

A comparative analysis of explicit solutions of the Colebrook-White equation is carried out. The median values of relative deviations, coefficients of determination and computational complexities for each approximation were obtained. The results of the iterative solution of the Colebrook-White equation by successive substitution method were used as the intrinsic solution. Approximations by B. Eck and A.R. Vatankhah were identified as the most effective in terms of computational complexity. It was shown that widely used approximations by P.R.H. Blasius, A.D. Altshul and J. Nikuradze although simple, provide acceptable accuracy only within restricted ranges of Reynolds and relative roughness.

Numerical Modeling of CO2, Water, Sodium Chloride, and Magnesium Carbonates Equilibrium to High Temperature and Pressure

In this work, a thermodynamic model of CO2-H2O-NaCl-MgCO3 systems is developed. The new model is applicable for 0–200 °C, 1–1000 bar and halite concentration up to saturation. The Pitzer model is used to calculate aqueous species activity coefficients and the Peng–Robinson model is used to calculate fugacity coefficients of gaseous phase species. Non-linear equations of chemical potentials, mass conservation, and charge conservation are solved by successive substitution method to achieve phase existence, species molality, pH of water, etc., at equilibrium conditions. From the calculated results of CO2-H2O-NaCl-MgCO3 systems with the new model, it can be concluded that (1) temperature effects are different for different MgCO3 minerals; landfordite solubility increases with temperature; with temperature increasing, nesquehonite solubility decreases first and then increases at given pressure; (2) CO2 dissolution in water can significantly enhance the dissolution of MgCO3 minerals, while MgCO3 influences on CO2 solubility can be ignored; (3) MgCO3 dissolution in water will buffer the pH reduction due to CO2 dissolution.

Multiphase flowback rate-transient analysis of shale gas reservoirs

Multi-fractured horizontal well (MFHW) is commonly used in the development of unconventional reservoirs. Flowback data after hydraulic fracturing is of critical importance in the characterization of hydraulic fracture, stimulation evaluation, and reservoir simulation. This study presents a two-phase diagnostic plot and a semi-analytical flowback model for the early-time flowback period when fluid influx from matrix remains insignificant and production is mainly from the fracture network. The developed model considers the changes of water saturation in hydraulic fracture (HF) under fracture depletion mechanism and pressure-dependent permeability and porosity, and is able to predict HF attributes, such as fracture half-length, initial fracture permeability, and initial pore volume of fracture network. A new method of calculating average pressure in the fracture is developed based on the solution of water-phase diffusivity equation and is compared with the traditional material balance approach based on the tank model. Three iterative workflows are proposed using the successive substitution method to calculate fracture attributes under different production conditions. Validation of the developed flowback model, average pressure calculation method, and the analysis workflow, as well as their application in an MFHW drilled in Horn River Shale show the applicability of this study to interpret flowback data quickly and estimate HF properties accurately.

Efecto del proceso de combustión sobre la eficiencia total de una central térmica de vapor

In the present investigation, a simulation model of the thermal calculation of the steam generator's steam generator 350 MW, operating under a Rankine cycle with superheat and regeneration, was developed. An analysis was carried out, modifying some parameters of the combustion (air and fuel flow) and the temperatures reached by the products of the combustion at the exit of the steam generator's home were determined, using the method of successive substitutions, applied to the thermal calculation methodology, with a stopping criterion of the approximate relative error of ε_ra <1 x10-8. It was found that the temperatures reached by the combustion gases at the steam generator's outlet for the 100%, 75%, 50% and 25% regimes are: 1492.4 ° C, 1379.3 ° C, 1238.0 ° C and 980.0 ° C, respectively. Regarding the effect of air dosing on the thermal efficiency of the cycle, it is observed that it increases slightly with the increase of the dosage. Being the maximum thermal efficiency of the cycle of 35.8%