Rate-Controlled Constrained-Equilibrium Method Research Articles

A Reformulation of Degree of Disequilibrium Analysis for Automatic Selection of Kinetic Constraints in the Rate-Controlled Constrained-Equilibrium Method

Abstract The rate-controlled constrained equilibrium (RCCE) is a model reduction scheme for chemical kinetics. It describes the evolution of a complex chemical system with acceptable accuracy with a number of rate controlling constraints on the associated constrained-equilibrium states of the system, much lower than the number of species in the underlying detailed kinetic model (DKM). Successful approximation of the constrained-equilibrium states requires accurate identification of the constraints. One promising procedure is the fully automatable Approximate Singular Value Decomposition of the Actual Degrees of Disequilibrium (ASVDADD) method that is capable of identifying the best constraints for a given range of thermodynamic conditions and a required level of approximation. ASVDADD is based on simple algebraic analysis of the results of the underlying DKM simulation and is focused on the behavior of the degrees of disequilibrium (DoD) of the individual chemical reactions. In this paper, we introduce an alternative ASVDADD algorithm. Unlike the original ASVDADD algorithm that require the direct computation of the DKM-derived DoDs and the identification of the set of linearly independent reactions, in the alternative algorithm, the components of the overall degree of disequilibrium vector can be computed directly by casting the DKM as an RCCE simulation considering a set of linearly independent constraints equaling the number of chemical species in size. The effectiveness and robustness of the derived constraints from the alternative procedure is examined in hydrogen/oxygen and methane/oxygen ignition delay simulations and the results are compared with those obtained from DKM.

Study of the Constraint Selection Through ASVDADD Method for Rate-Controlled Constrained-Equilibrium Modeling on Ethanol Oxidation Without PLOG Reactions

Abstract Reduction of the detail chemical kinetic mechanism is important in solving complex combustion simulation. In this work, a model reduction scheme rate-controlled constrained-equilibrium (RCCE) is considered in predicting the oxidation of ethanol. A detail kinetic mechanism by Merinov from Lawrence Livermore National Laboratory (LLNL) is used in modeling this reduction technique. The RCCE method considers constrained equilibrium states which subjected to a lower number of constraints compared to the number of species. It then has to solve a smaller number of differential equations compared to the number of equations required in solving the detailed kinetic model (DKM). The accuracy of this solution depends on the selection of the constraint. A systematic procedure which will help in identifying the constraint at an optimal level of accuracy is an essential for RCCE modeling. A fully automated Approximate Singular Value Decomposition of the Actual Degrees of Disequilibrium (ASVDADD) method is used in this study to derive the constraint for RCCE simulation. ASVDADD uses an algorithm which follows the simple algebraic analysis on results of underlying DKM to find the degree of disequilibrium (DoD) of the individual chemical reactions. The number of constraints which will be used in RCCE simulation can be selected to reduce the number of equations required to solve. In the current work, this ASVDADD method is applied on ethanol oxidation to select the constraint for RCCE simulation. Both DKM and RCCE calculations on ethanol fuel are demonstrated to compare the result of temperature distribution and an ignition delay time for validating the method.

Review of Applications of Rate-Controlled Constrained-Equilibrium in Combustion Modeling

AbstractDeveloping an effective model for non-equilibrium states is of great importance for a variety of problems related to chemical synthesis and combustion. Rate-Controlled Constrained-Equilibrium (RCCE), a model order reduction method that originated from the second law of thermodynamics, assumes that the non-equilibrium states of a system can be described by a sequence of constrained-equilibrium states kinetically controlled by a relatively small number of constraints within acceptable accuracy. The full chemical composition at each constrained-equilibrium state is obtained by maximizing (or minimizing) the appropriate thermodynamic quantities, e. g., entropy (or Gibbs functions), subject to the instantaneous values of RCCE constraints. Regardless of the nature of the kinetic constraints, RCCE always guarantees a correct final equilibrium state. This paper reviews the fundamentals of the RCCE method, its constraints, constraint potential formulations, and major constraint selection techniques, as well as the application of the RCCE method to combustion of different fuels using a variety of combustion models. The RCCE method has been proven to be accurate and to reduce computational time in these simulations.

The Rate-Controlled Constrained-Equilibrium Combustion Modeling of n-Pentane/Oxygen/Diluent Mixtures

Rate-controlled constrained equilibrium (RCCE) is a reduction technique used to describe the time evolution of complex chemical reacting systems. This method is based on the assumption that a nonequilibrium system can reach its final equilibrium state by a series of RCCE states determined by maximizing entropy or minimizing relevant free energy. Those constraints are imposed by some small number of slow reactions. Much research has been done on this method and many RCCE models of C1−C4 hydrocarbon fuel combustion have been established by the previous researchers. Those models show good performance compared with the result of detailed kinetic model (DKM). In this study, RCCE method is further developed to model normal pentane (n-C5H12) combustion with least number of constraints. The chemical mechanism for DKM contains 133 species and 922 reactions. Two sets of constraints were found during the study: (1) 16 constraints for the normal pentane and pure oxygen mixture and (2) 14 constraints for the mixture of normal pentane and oxygen with argon as diluent. Results of the first constraint set were compared with result of DKM and results of the second constraint set were compared with those of DKM and experimental data by calculating their ignition delay times. Comparisons showed that the first set of constraints had relatively good accuracy and the second set of constraints agreed very well with the experimental data.

The Rate-Controlled Constrained-Equilibrium combustion modeling of n-butane/oxygen/diluent mixtures

Rate-Controlled Constrained-Equilibrium (RCCE) is a model order reduction method which assumes that the non-equilibrium states of a system can be described by a sequence of constrained-equilibrium states subject to a small number of constraints. It can be used to predict ignition delay time with good accuracy and low computational cost. In this paper RCCE approach has been further developed for applying to the oxidation of n-butane for ignition study and prediction of a constant volume, constant internal energy system over a wide range of initial temperatures, pressures and equivalence ratios. The USC-Mech II (109 species and 781 reactions, without nitrogen chemistry) is chosen as chemical kinetic mechanism for n-butane oxidation for Detailed Kinetic Model (DKM). The constraint selection for n-butane/oxygen mixture starts from the eight universal constraints for carbon-fuel oxidation. Additional species constraints are selected based on researchers’ experience to have the best performance with the minimum number of constraints. The selected 17 constraints have been used to predict ignition delay times for butane combustion. The results of RCCE method are compared with those of detailed kinetic model and experimental data to verify the effectiveness of constraints and the efficiency of RCCE. Rate-Controlled Constrained-Equilibrium results show good agreements with DKM results under different initial temperatures, pressures and equivalence ratios. Even better performance than DKM has been achieved by the selected 17 constraints when the results are compared with shock tube experimental data from literature with initial temperatures 1200–1500 K and initial pressures 2–20 atm. It has been applied to predict the ignition delay time of butane/air mixture over a wide range of initial temperatures, initial pressures and equivalence ratios.

Combustion Simulation of Propane/Oxygen (With Nitrogen/Argon) Mixtures Using Rate-Controlled Constrained-Equilibrium

The rate-controlled constrained-equilibrium (RCCE), a model order reduction method, has been further developed to simulate the combustion of propane/oxygen mixture diluted with nitrogen or argon. The RCCE method assumes that the nonequilibrium states of a system can be described by a sequence of constrained-equilibrium states subject to a small number of constraints. The developed new RCCE approach is applied to the oxidation of propane in a constant volume, constant internal energy system over a wide range of initial temperatures and pressures. The USC-Mech II (109 species and 781 reactions, without nitrogen chemistry) is chosen as chemical kinetic mechanism for propane oxidation for both detailed kinetic model (DKM) and RCCE method. The derivation for constraints of propane/oxygen mixture starts from the eight universal constraints for carbon-fuel oxidation. The universal constraints are the elements (C, H, O), number of moles, free valence, free oxygen, fuel, and fuel radicals. The full set of constraints contains eight universal constraints and seven additional constraints. The results of RCCE method are compared with the results of DKM to verify the effectiveness of constraints and the efficiency of RCCE. The RCCE results show good agreement with DKM results under different initial temperature and pressures, and RCCE also reduces at least 60% CPU time. Further validation is made by comparing the experimental data; RCCE shows good agreement with shock tube experimental data.

A kinetics-based method for constraint selection in rate-controlled constrained equilibrium

The rate-controlled constrained-equilibrium (RCCE) method is a thermodynamics-based dimension reduction method that enables representation of chemistry involving ns chemical species in terms of a smaller number, nr, of constraints, and thus reduces the computational burden imposed by detailed chemical kinetics. The accuracy in the reduced description by RCCE strongly depends on the specification of constraints. In this work, the connection between the RCCE method and the conventional partial equilibrium assumption (PEA) method is revisited. It is shown that with the constraints being specified based on rate-controlling slow reactions, the RCCE method is equivalent to PEA. A criterion based on time scale analysis is then employed for the identification of fast reactions. Then the importance index, which represents the contribution of individual species to the slow dynamics of a reactive system, is proposed for the selection of species constraints in RCCE. This makes the thermodynamics-based constrained equilibrium manifolds (CEMs) a good approximation to the actual kinetics-controlled slow invariant manifolds (SIMs) in reactive systems. The method is demonstrated in the reduced descriptions of perfectly stirred reactors (PSRs) burning stoichiometric methane/air mixtures. Sample reaction states ranging from near-equilibrium conditions to the extinction turning points on the S-curve are examined for the identification of fast reactions and species constraints. Reduced descriptions of transient PSRs by RCCE are subsequently performed with their accuracy assessed against the full descriptions, in which all the chemical species are transported. The study shows that the dynamics predicted by RCCE agrees well with that from the full description and that near-limit reaction states are important for the selection of species constraints in RCCE.

Degree of Disequilibrium analysis for automatic selection of kinetic constraints in the Rate-Controlled Constrained-Equilibrium method

The Rate-Controlled Constrained-Equilibrium (RCCE) model reduction scheme for chemical kinetics provides acceptable accuracies with a number of differential equations much lower than the number of species in the underlying Detailed Kinetic Model (DKM). To yield good approximations, however, the method requires accurate identification of the rate controlling constraints. So far, a drawback of the RCCE scheme has been the absence of a fully automatable and systematic procedure that is capable of identifying the best constraints for a given range of thermodynamic conditions and a required level of approximation. In this paper, we propose a new methodology for such identification based on a simple algebraic analysis of the results of a preliminary simulation of the underlying DKM, which is focused on the behaviour of the degrees of disequilibrium (DoD) of the individual chemical reactions. The new methodology is based on computing an Approximate Reduced Row Echelon Form of the Actual Degrees of Disequilibrium (ARREFADD) with respect to a preset tolerance level. An alternative variant is to select an Approximate Singular Value Decomposition of the Actual Degrees of Disequilibrium (ASVDADD). Either procedure identifies a low dimensional subspace in the DoD space, from which the actual DoD traces do not depart beyond a fixed distance related to the preset tolerance (ARREFADD methodology) or to the first neglected singular value of the matrix of DoD traces (ASVDADD methodology). The effectiveness and robustness of the method is demonstrated for the case of a very rapid supersonic nozzle expansion of the products of hydrogen and methane oxycombustion and for the case of methane/oxygen ignition. The results are in excellent agreement with DKM predictions. For both variants of the method, we provide a simple Matlab code implementing the proposed constraint selection algorithm.

A Study of Interactions between Mixing and Chemical Reaction Using the Rate-Controlled Constrained-Equilibrium Method

AbstractThe rate-controlled constrained-equilibrium (RCCE) method is employed to study the interactions between mixing and chemical reaction. Considering that mixing can influence the RCCE state, the key objective is to assess the accuracy and numerical performance of the method in simulations involving both reaction and mixing. The RCCE formulation includes rate equations for constraint potentials, density and temperature, which allows taking account of mixing alongside chemical reaction without splitting. The RCCE is a dimension reduction method for chemical kinetics based on thermodynamics laws. It describes the time evolution of reacting systems using a series of constrained-equilibrium states determined by RCCE constraints. The full chemical composition at each state is obtained by maximizing the entropy subject to the instantaneous values of the constraints. The RCCE is applied to a spatially homogeneous constant pressure partially stirred reactor (PaSR) involving methane combustion in oxygen. Simulations are carried out over a wide range of initial temperatures and equivalence ratios. The chemical kinetics, comprised of 29 species and 133 reaction steps, is represented by 12 RCCE constraints. The RCCE predictions are compared with those obtained by direct integration of the same kinetics, termed detailed kinetics model (DKM). The RCCE shows accurate prediction of combustion in PaSR with different mixing intensities. The method also demonstrates reduced numerical stiffness and overall computational cost compared to DKM.

Development of the Rate-Controlled Constrained-Equilibrium Method for Modeling of Ethanol Combustion

The rate-controlled constrained-equilibrium method (RCCE) has been further developed to model the combustion process of ethanol air mixtures. The RCCE is a reduction technique based on local maximization of entropy or minimization of a relevant free energy at any time during the nonequilibrium evolution of the system subject to a set of kinetic constraints. An important part of RCCE calculation is determination of a set of constraints that can guide the nonequilibrium mixture to the final stable equilibrium state. In this study, 16 constraints have been developed to model the nonequilibrium ethanol combustion process. The method requires solution of 16 differential equations for the corresponding constraint potentials. Ignition delay calculations of ethanol oxidizer mixtures using RCCE have been compared to those of detailed chemical kinetics using 37 species and 235 reactions. Agreement between the two models is very good. In addition, ignition delay of C2H5OH/O2/Ar mixtures using RCCE has been compared with the experimental measurements in the shock tube and excellent agreement has been reached validating the RCCE calculation.

A Comparison of Constraint and Constraint Potential Forms of the Rate-Controlled Constraint-Equilibrium Method

A comparative assessment is made of two implementations of the rate-controlled constrained-equilibrium (RCCE) method. These are the constraint and constraint potential formulations in which rate equations are solved for the RCCE constraints and constraint potentials, respectively. The two forms are equivalent mathematically; however, they involve different numerical procedures and thus show different computational performance. The main objective of this study is to compare the accuracy and numerical efficiency of the two formulations to attain the most effective implementation of the RCCE in turbulent combustion simulations. The RCCE method is applied to study methane oxygen combustion in an adiabatic, isobaric well stirred reactor. Simulations are carried out over a wide range of initial temperatures and equivalence ratios. Performance studies are conducted and RCCE results are compared with those obtained by direct integration of detailed chemical kinetics. The results show that both methods provide very accurate representation of the kinetics. It is also demonstrated that while the constraint form involves less numerical stiffness, the constraint potential implementation results in more saving in computation time.

Constrained-Equilibrium Modeling of Methane Oxidation in Air

Rate-controlled constrained-equilibrium method has been further developed to model methane/air combustion. A set of constraints has been identified to predict the nonequilibrium evolution of the combustion process. The set predicts the ignition delay times of the corresponding detailed kinetic model to within 10% of accuracy over a wide range of initial temperatures (900 K–1200 K), initial pressures (1 atm–50 atm) and equivalence ratios (0.6–1.2). It also predicts the experimental shock tube ignition delay times favorably well. Direct integration of the rate equations for the constraint potentials has been employed. Once the values of the potentials are obtained, the concentration of all species can be calculated. The underlying detailed kinetic model involves 352 reactions among 60 H/O/N/C1-2 species, hence 60 rate equations, while the RCCE calculations involve 16 total constraints, thus 16 total rate equations. Nonetheless, the constrained-equilibrium concentrations of all 60 species are calculated at any time step subject to the 16 constraints.

A study of the rate-controlled constrained-equilibrium dimension reduction method and its different implementations

The Rate-Controlled Constrained-Equilibrium (RCCE) method is a thermodynamic based dimension reduction method which enables representation of chemistry involving n s species in terms of fewer n r constraints. Here we focus on the application of the RCCE method to Lagrangian particle probability density function based computations. In these computations, at every reaction fractional step, given the initial particle composition (represented using RCCE), we need to compute the reaction mapping, i.e. the particle composition at the end of the time step. In this work we study three different implementations of RCCE for computing this reaction mapping, and compare their relative accuracy and efficiency. These implementations include: (1) RCCE/TIFS (Trajectory In Full Space): this involves solving a system of n s rate-equations for all the species in the full composition space to obtain the reaction mapping. The other two implementations obtain the reaction mapping by solving a reduced system of n r rate-equations obtained by projecting the n s rate-equations for species evaluated in the full space onto the constrained subspace. These implementations include (2) RCCE: this is the classical implementation of RCCE which uses a direct projection of the rate-equations for species onto the constrained subspace; and (3) RCCE/RAMP (Reaction-mixing Attracting Manifold Projector): this is a new implementation introduced here which uses an alternative projector obtained using the RAMP approach. We test these three implementations of RCCE for methane/air premixed combustion in the partially-stirred reactor with chemistry represented using the n s=31 species GRI-Mech 1.2 mechanism with n r=13 to 19 constraints. We show that: (a) the classical RCCE implementation involves an inaccurate projector which yields large errors (over 50%) in the reaction mapping; (b) both RCCE/RAMP and RCCE/TIFS approaches yield significantly lower errors (less than 2%); and (c) overall the RCCE/TIFS approach is the most accurate, efficient (by orders of magnitude) and robust implementation.

Rate-Controlled Constrained-Equilibrium Modeling of H/O Reacting Nozzle Flow

Amethodology based on the degree of disequilibriumof chemical reactions is proposed to select kinetic constraints in the method of rate-controlled constrained equilibrium (RCCE). Predictions of chemical relaxation of the combustion products within a supersonic nozzle are also made under the constraints identified using the proposed methodology. The quasi-one-dimensional steady Euler equations are employed. The RCCE method provides a general framework, which enables studying shifting equilibrium, frozen equilibrium, as well as nonequilibrium chemical kinetics under different sets of constraints, either a single or a linear combination of species. It is shown that the previously identified generalized constraints of the total number of moles, free valence, and free oxygen are implied as a consequence of this methodology for the kinetics considered and for the relaxation studied. The kinetic scheme involves anH=O reactingmixture with 24 reactions and eight species. The constraints identified areH H2, HO2 H2O2, andO OH H2O, which result in less than 0.06% local errors in temperature compared with the detailed kinetic model calculations throughout the nozzle.

The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics

The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A generalized mathematical formulation is presented here that allows including nonlinear constraints in non-local equilibrium systems characterized by the existence of a non-increasing Lyapunov functional under the system’s internal dynamics. The generalized formulation of RCCE enables to clarify the essentials of the method and the built-in general feature of thermodynamic consistency in the chemical kinetics context. In this paper, we work out the details of the method in a generalized mathematical-physics framework, but for definiteness we detail its well-known implementation in the traditional chemical kinetics framework. We detail proofs and spell out explicit functional dependences so as to bring out and clarify each underlying assumption of the method. In the standard context of chemical kinetics of ideal gas mixtures, we discuss the relations between the validity of the detailed balance condition off-equilibrium and the thermodynamic consistency of the method. We also discuss two examples of RCCE gas-phase combustion calculations to emphasize the constraint-dependent performance of the RCCE method.

Rate-Controlled Constrained-Equilibrium Theory Applied to the Expansion of Combustion Products in the Power Stroke of an Internal Combustion Engine

Rate-controlled constrained-equilibrium method, firmly based on the second law of thermodynamics, is applied to the expansion of combustion products of methane during the power stroke of an internal combustion engine. The constraints used in this study are the elemental oxygen, hydrogen, carbon and nitrogen together with other four dynamic constraints of total number of moles, moles of DCO (CO+HCO), moles of free valence and moles of free oxygen. Since at chemical equilibrium, the mixture composition is dominated by H/O, CO/CO2, and a few other carbon-containing species, almost independent of the fuel molecule, the set results in accurate predictions of the kinetic effects observed in all H/O and CO/CO2 compounds and temperature history. It is shown that the constrained-equilibrium predictions of all the species composed of the specified atomic elements can be obtained independent of a kinetic path, provided their Gibbs free energies are known.

A more accurate projection in the rate-controlled constrained-equilibrium method for dimension reduction of combustion chemistry

The rate-controlled constrained-equilibrium (RCCE) method for dimension reduction of combustion chemistry is revisited from a geometric viewpoint. A constrained equilibrium manifold (CEM) is defined as all compositions that satisfy the maximum-entropy or minimum-free energy conditions of the gas mixture, subject to specified constraints. The RCCE method is based solely on thermodynamics, and it is shown that this method contains a hidden assumption of an orthogonal projection that projects the rate equation of the chemical system onto the CEM. An extension of the RCCE method is constructed by making an alternative projection based on the conjecture that, near the CEM, there is a close parallel inertial manifold (CPIM). The CPIM assumption introduces the chemical kinetics directly through the local Jacobian and hence leads to greater accuracy than RCCE. The comparison between the RCCE method and its extension is made in the test calculations of hydrogen–air and methane–air autoignition.