Reaction Source Terms Research Articles

One-dimensional turbulence-based closure with extinction and reignition

Scalar statistics from stand-alone one-dimensional turbulence (ODT) simulations are constructed to develop a doubly conditioned table based on the mixture fraction and temperature for the prediction of extinction and reignition in piloted methane–air jet diffusion flames. The ODT-based closure approach is formulated to predict scalar statistics coupled with the Reynolds-averaged Navier–Stokes (RANS) approach. Comparison with experimental correlations of reactive scalars with the two conditioning variables show that double conditioning may be adequate to prescribe scalar statistics in the jet diffusion flames. The results also show that the ODT model may be used to construct these statistics. The 2D conditioning table is coupled with RANS to compute Sandia flames D, E, and F, which exhibit increasing rates of extinction followed by reignition as the Reynolds numbers are increased. The coupling also requires the transport of the means and variances of the mixture fraction and temperature. Closure terms in the temperature mean and variance equations are obtained by using the 2D table for reaction source terms and by assuming a presumed PDF shape for the temperature PDF. Comparisons show adequate predictions of axial and radial profiles of the mixture fraction, the streamwise velocity, and the reactive scalars for flames D and E and mixed results for flame F. Nonetheless, qualitative trends of increasing the jet Reynolds numbers resulting in more pronounced extinctions are obtained with the RANS-ODT approach. The discrepancy between computation and experiment may be attributed primarily to the closure for the temperature and its variance and to the presumed PDF shape for the temperature.

Multi-environment probability density function method for modelling turbulent combustion using realistic chemical kinetics

A computational fluid dynamics (CFD) tool for performing turbulent combustion simulations that require finite-rate chemistry is developed and tested by modelling a series of bluff-body stabilized flames that exhibit different levels of finite-rate chemistry effects ranging from near equilibrium to near global extinction. The new modelling tool is based on the multi-environment probability density function (MEPDF) methodology and combines the following: the direct quadrature method of moments (DQMOM); the interaction-by-exchange-with-the-mean (IEM) mixing model; and realistic combustion chemistry. Using DQMOM, the MEPDF model can be derived from the transport PDF equation by depicting the joint composition PDF as a weighted summation of a finite number of multi-dimensional Dirac delta functions in the composition space. The MEPDF method with multiple reactive scalars retains the unique property of the joint PDF method of treating chemical reactions exactly. However, unlike the joint PDF methods that typically must resort to particle-based Monte-Carlo solution schemes, the MEPDF equations (i.e. the transport equations of the weighted delta-peaks) can be solved by traditional Eulerian grid-based techniques. In the current study, a pseudo time-splitting scheme is adopted to solve the MEPDF equations; the reaction source terms are computed with a highly efficient and accurate in-situ adaptive tabulation (ISAT) algorithm. A 19-species reduced mechanism based on quasi-steady state assumptions is used in the simulations of the bluff-body flames. The modelling results are compared with the experimental data, including mixing, temperature, major species and important minor species such as CO and NO. Compared with simulations using a Monte-Carlo joint PDF method, the new approach shows comparable accuracy.

Conditional source-term estimation with laminar flamelet decomposition in large eddy simulation of a turbulent nonpremixed flame

Conditional source-term estimation (CSE) is a method to close the mean chemical reaction source term based on the conditional moment closure hypothesis. It has been shown previously to be successful in a priori tests against direct numerical simulation data and in the large eddy simulation (LES) of a nonpremixed flame using reduced chemistry. Laminar flamelet decomposition (LFD) is a method to incorporate more complex chemistry into CSE by using laminar flamelets as basis functions and inverting an integral equation for a basis function coefficient vector which describes the linear combination of flamelets that best approximates the conditional average of certain scalar fields within an ensemble of points in a flow field. This coefficient vector is used to obtain the conditional average of the chemical source term for this ensemble of points which can then be transformed into the unconditional average chemical source term in the transport equations of reactive scalars in the flow. This study focuses on the application of CSE with LFD in LES of the Sandia D-flame. The simulation results show that LFD is able to predict temperature and major species well with both steady and unsteady flamelet libraries. However, in order to predict NO well, it is necessary to use a mixed library that includes both steady and unsteady flamelets. The computational cost of this method is low because very few transport equations need to be solved (specifically, equations for the filtered continuity, momentum, mixture fraction and its variance, temperature and the mass fraction of CO are solved) while other species mass fractions can be obtained directly from the flamelet library.

On the numerical solution of diffusion–reaction equations with singular source terms

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.

A quadrature closure for the reaction-source term in conditional-moment closure

A Gaussian-quadrature closure is introduced for the conditional reaction-source term in conditional-moment closure, motivated by the desire to accurately model the dynamics of homogeneous extinction and re-ignition in turbulent non-premixed flames. A priori analysis of the quadrature model was performed using results of direct numerical simulations (DNS) of isotropic turbulence which included a mixture fraction and reaction-progress variable with a reversible chemical-source term. The DNS calculations for three Damköhler numbers, each exhibiting a different degree of extinction and all exhibiting re-ignition, were used to demonstrate the shortcomings of the Taylor-series closure. A quantitative validation was completed, and the quadrature closure showed significantly less error than the Taylor-series closures for all of the conditions considered.

Lattice Boltzmann simulations of phase separation in chemically reactive binary fluids

We use a lattice Boltzmann method to study pattern formation in chemically reactive binary fluids in the regime where hydrodynamic effects are important. The coupled equations solved by the method are a Cahn-Hilliard equation, modified by the inclusion of a reactive source term, and the Navier-Stokes equations for conservation of mass and momentum. The coupling is twofold, resulting from the advection of the order parameter by the velocity field and the effect of fluid composition on pressure. We study the evolution of the system following a critical quench for a linear and for a quadratic reaction source term. Comparison is made between the high and low viscosity regimes to identify the influence of hydrodynamic flows. In both cases hydrodynamics is found to influence the pathways available for domain growth and the eventual steady states.

Substantiating a fractal-based algebraic reaction closure of premixed turbulent combustion for high pressure and the Lewis number effects

A comprehensive set of nearly 100 atmospheric and high-pressure flame data of Kobayashi et al. are a good source for numerical analysis to address two main aspects in premixed turbulent combustion—high-pressure influence and effects of fuel type on the reaction rate. The present work deals with the lucid and realizable fractal-based reaction rate closure from Lindstedt and Váos (LV model) for premixed flames in the thin-flame limit. In this study, the reaction source term is customized on the eddy viscosity closure of turbulent transport, for practical reasons. Computed results from the LV model show the right qualitative trends with the experimental findings, as a function of turbulence. However, quantitative predictions of the original model are partly too low, and preclude the effects of pressure and fuel type on the reaction rate. With an extensive parametric study, based on numerical findings as well as on theoretical argumentation, the LV model is substantiated for these two effects. Results from the proposed tuned LV model are found to be in very good agreement with most of the measured data.

Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport

An efficient projection scheme is developed for the simulation of reacting flow with detailed kinetics and transport. The scheme is based on a zero-Mach-number formulation of the compressible conservation equations for an ideal gas mixture. It relies on Strang splitting of the discrete evolution equations, where diffusion is integrated in two half steps that are symmetrically distributed around a single stiff step for the reaction source terms. The diffusive half-step is integrated using an explicit single-step, multistage, Runge---Kutta---Chebyshev (RKC) method. The resulting construction is second-order convergent, and has superior efficiency due to the extended real-stability region of the RKC scheme. Two additional efficiency-enhancements are also explored, based on an extrapolation procedure for the transport coefficients and on the use of approximate Jacobian data evaluated on a coarse mesh. We demonstrate the construction in 1D and 2D flames, and examine consequences of splitting errors. By including the above enhancements, performance tests using 2D computations with a detailed C1C2 methane-air mechanism and a mixture-averaged transport model indicate that speedup factors of about 15 are achieved over the starting split-stiff scheme

MHD Flow and heat transfer of non-Newtonian power-law fluid with diffusion and chemical reaction on a moving cylinder

The problem of flow and heat transfer of an electrically conducting non-Newtonian fluid over a continuously moving cylinder in the presence of a uniform magnetic field is analyzed for the case of power-law variation in the temperature and concentration at the cylinder surface. A diffusion equation with a chemical reaction source term is taken into account. The governing non-similar partial differential equation are solved numerically by employing shooting method. The effects of various parameters on the velocity, temperature and concentration profiles as well as the heat and mass transfer rate from the cylinder surface to the surrounding fluid are presented graphically and in tabulated form.

Development of an algebraic reaction rate closure for the numerical calculation of turbulent premixed methane, ethylene, and propane/air flames for pressures up to 1.0 MPa

In this work a simplified but effective closure for the numerical simulation of turbulent premixed flames is developed, being applicable for pressures up to 1.0 MPa. Here the reaction source term of the reaction progress variable is modeled with an algebraic relation for the flame-wrinkling ratio A T / A ¯ . The closure is based on a three-parameter description, including an explicit pressure term. In order to determine the fit parameters, an extensive numerical optimization study is performed, where the calculated and measured flame angles of 101 different Bunsen-type flames are compared for operating pressures between 0.1 and 1.0 MPa. Experimental data on 20-mm Bunsen-type flames for lean methane/air, ethylene/air, and propane/air mixtures for different flow and turbulence inlet conditions were provided by the group of Kobayashi (Japan). For each fuel the three parameters of the algebraic relation are varied such that a minimum least-square deviation between computed and measured flame cone angles is achieved. It is found that for all the fuels investigated this relation collapses to a unique equation, in which the Lewis number of the fuel/air mixture is included. It has been proposed recently that differential molecular transport effects, as described by the Lewis number, may be related not only to laminar flame instability effects at low turbulence but also to visible effects at higher degrees of turbulence. This fits to our finding of the explicit Lewis number dependency in the source term. It is possible to substantiate the applicability of this algebraic closure for methane/air flames for higher pressures up to 3 MPa in reasonable agreement with experimental data, as described in Appendix A.

Large‐Eddy simulation of a turbulent reacting liquid flow

AbstractA subgrid‐scale (SGS) model for the filtered reaction source term is presented to develop the large‐eddy simulation (LES) of a nonpremixed, turbulent liquid flow with a moderately fast reaction. The SGS model is based on the SGS probability density function (PDF) and SGS conditional expectation. The SGS probability density function (SGS–PDF) is assumed to follow a beta distribution and a simple algebraic model for the SGS conditional expectation is developed using the filtered data obtained from the direct numerical simulation (DNS) of stationary isotropic liquid turbulence with a second‐order chemical reaction. For a rapid reaction, the SGS–PDF model based on the conserved scalar is used as the SGS model. The LES based on these SGS models is applied to a liquid mixing layer flow downstream of a turbulence‐generating grid with a chemical reaction, and the LES predictions of the mean concentration and concentration variance are directly compared with the previous measurements for both moderately fast and rapid reactions to examine the proposed SGS models. The results show that the predictions by the LES are in good agreement with the measurements and the present LES can well describe the diffusive–reactive process in a turbulent reacting liquid flow. © 2004 American Institute of Chemical Engineers AIChE J, 50: 2705–2720, 2004

Assessment of a flame surface density-based subgrid turbulent combustion model for nonpremixed flames of wood pyrolysis gas

A flame surface density (FSD) model for closing the unresolved reaction source terms is developed and implemented in a large eddy simulation (LES) of turbulent nonpremixed flame of wood pyrolysis gas and air. In this model, the filtered reaction rate ω¯α of species α is estimated as the product of the consumption rate per unit surface area mα and the filtered FSD Σ¯. This approach is attractive since it decouples the complex chemical problem (mα) from the description of the turbulence combustion interaction (Σ¯). A simplified computational methodology is derived for filtered FSD Σ¯, which is approximated as the product of the conditional filtered gradient of mixture fraction and the filtered probability density function. Two models for flamelet consumption rate mα are proposed to consider the effect of filtered scalar dissipation rate. The performance of these models is assessed by direct numerical simulation (DNS) database where a laminar diffusion flame interacts with a decaying homogeneous and isotropic turbulent flow field. The chemistry is modeled by a four-step reduced mechanism that describes the oxidization process of gaseous fuel released from high temperature pyrolysis of wood occurring in a wildland fire. Two-dimensional (2D) and 3D LES computations based on the FSD models are conducted for the same conditions as the DNS. The comparative assessments confirm the applicability of the proposed FSD model to describe the filtered reaction rate and the time evolution of temperature and species concentration in the turbulent nonpremixed flame.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Numerical modelling of the pressure dependent reaction source term for premixed turbulent methane/air flames

Effects of pressure are studied numerically for premixed turbulent flames under varied turbulence conditions up to 1.0 MPa. In order to model the pressure dependency of the reaction source term, a set of available experimental Bunsen flame data for methane/air mixtures from Kobayashi et al. are used as reference for a numerical optimisation study, where three parameters of a generalized algebraic closure relation for a reaction progress variable approach are determined. This approach is based on the algebraic turbulent flame speed closure model proposed by Zimont, which is subjected to modifications for influence of pressure and for low turbulence conditions. The net pressure dependency of the turbulent flame speed is slightly above zero, similar to the experimental fit of Kobayashi. Dependency of turbulent velocity is found proportional to u'0.55, similar to several experimental data fits by Bradley et al. If the resulting flame angles are represented in terms of sT/sL vs. u'/sL, then the fitted curve of the calculated results is closely related to that of the experiments. Both are showing a nonlinear bending behaviour especially at low turbulence intensities, which seems to be related to the influence of laminar flame instabilities.

PRESSURE-BASED UNSTRUCTURED-GRID FINITE-VOLUME METHOD FOR SIMULATING LAMINAR REACTING FLOWS

A pressure-based, unstructured, finite-volume method has been developed to resolve accurately the physically and geometrically complex reacting flows. A pressure-correction algorithm together with a conventional operator-splitting procedure was employed to handle the pressure-velocity coupling and the stiff reaction source terms. The conservative forms of the governing equations are integrated over a cell-centered control volume with collocated storage for all transport variables. Computations using detailed chemistry and variable transport properties were performed for a counterflow hydrogen-air diffusion flame and a lifted methane-air triple flame. Numerical results indicate that the present numerical and physical models are quite capable of predicting the essential features and complex structure of laminar nonpremixed and partially premixed flames in terms of flame location, temperature, mass fraction of species, triple flame structure, and lift-off height.

Multicomponent unsteady-state film model: a general analytical solution to the linearized diffusion–reaction problem

In this paper, a classical film model is extended to account for unsteady-state multicomponent diffusion and reaction coupling. The governing matrix-form partial differential equation is complemented by the initial and boundary conditions formulated as general vector functions. An additional linearization of the reaction source term results in the formulation which allows an analytical handling of the problem. The latter is found to be successful even with the generally set initial and boundary conditions. The solution approach combines the superposition principle and the method of the separation of variables extended for the matrix operations. The exact analytical matrix solution obtained is a generalization of many simpler problems and either can be employed by itself or provides suitable preset values for relevant numerical simulations of industrial-scale reactive separation operations.

CFD Analysis of Premixed Methane Chlorination Reactors with Detailed Chemistry

With the implementation of efficient algorithms for the accurate calculation of reaction source terms, computational fluid dynamics (CFD) is now a powerful tool for the simulation and design of chemical reactors with complex kinetic schemes. The example studied in this work is the methane chlorination reaction for which the detailed chemistry scheme has 152 reactions and 38 species. The adiabatic, jet-stirred chlorination reactor used for the CFD simulations is an insulated right cylinder with a coaxial premixed feed stream at one end. In order for this reactor to remain lit, recirculation of hot products is crucial, and hence, reactor stability is sensitive to both macroscale and microscale mixing. By neglecting density variations, a Lagrangian composition probability density function (PDF) code with a novel chemistry tabulation algorithm (in-situ adaptive tabulation or ISAT) for handling complex reactions is used to simulate the species concentrations and temperature field inside of the reactor. In addition, a reduced mechanism with 21 reactions and 15 species is tested for accuracy against the detailed chemistry scheme, a simplified CSTR model is used to illustrate the shortcomings of zero-dimensional models, and a pair-wise mixing stirred reactor (PMSR) model is used to show the stabilizing effect of micromixing on reactor stability. The CFD simulations are generally in good agreement with results from pilot-scale reactors for the outlet temperature and major species.

A numerical study of wedge-induced detonations

The structure of detonations generated by the flow of a combustible mixture over a wedge is examined numerically. A recently designed, unsplit algorithm which integrates the convective and reaction source terms simultaneously, is employed for the simulations. The chemical kinetics is assumed to follow a simple one-step Arrhenius law. Two different geometrical configurations are considered. In the first one, the wedge is assumed to be long enough so that its top corner can not affect the structure of the reaction zone. In the second one, the wedge is short so that this corner can influence the reaction zone. It is observed that for moderate wedge angles the leading shock curves smoothly until it reaches a final angle. A stable detonation and a steady flow-field is established near the wedge. For higher wedge angles, an explosion occurs at the leading shock and the detonation becomes unstable. Grid-convergence studies for both stable and unstable detonations are presented and comparisons with earlier results are made.

Comparison of numerical and analytical solutions of a multicomponent reaction-mass-transfer problem in terms of the film model

In this paper, a film-model-based description of multicomponent reactive separation is considered and two different approaches to the solution of the reaction–mass transfer problem in the liquid film are studied and compared. The first approach is based on a numerical treatment of the resulting nonlinear boundary value problem. The second approach requires a simplification of the differential transport equation by linearizing its reaction source term, and further, takes advantage of an analytical technique. To realize numerical solution of a complex matrix-form boundary value problem, a special method has been developed and tested. Both approaches have strengths and weaknesses which are analyzed and discussed. For different systems and process conditions, a comparison of simulation results obtained by both methods is carried out.

A Semi-implicit Numerical Scheme for Reacting Flow: I. Stiff Chemistry

An additive semi-implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed. The scheme relies on a zero-Mach number formulation of the compressible conservation equations with detailed chemistry. The governing equations are discretized in space using second-order differences and integrated in time using a semi-implicit approach. Time integration of the evolution equations for species mass fraction, thermodynamic pressure, and density is performed using a semi-implicit, nonsplit scheme that combines a second-order predictor–corrector treatment of convection and diffusion terms, and a stiff integrator for the reaction source terms. Meanwhile, the momentum equations are integrated using a second-order projection scheme. The projection scheme is based on a predictor–corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations. A pressure Poisson equation is inverted following both the predictor and corrector steps using a fast solver. The advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor–corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion. The comparison in based on both one-dimensional (1D) unsteady tests of a premixed methane–air flame, and unsteady two-dimensional tests of the same flame interacting with a counterrotating vortex pair. In both cases, the GRImech1.2 reaction mechanism with 32 species and 177 elementary reactions is used. Computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations. For the present reaction mechanism and flame conditions, speedup factors of about 10 are achieved in the 1D tests and about five in two dimensions. Possible extensions of the present scheme to further improve efficiency are also discussed.