A Maximum Entropy-Inspired Interpolative Closure for the Prediction of Radiative Heat Transfer in Laminar Co-Flow Diffusion Flames

ABSTRACT

An interpolative-based first-order maximum entropy (M₁) moment closure for providing approximate solutions to the equation of radiative transfer in non-gray participating media is proposed and described. This newly developed non-gray moment closure technique results in significant computational savings compared to an approach that makes use of the direct numerical solution of the optimization problem for entropy maximization. Its predictive capabilities are also assessed by comparing its solutions to those of the more commonly adopted first-order spherical harmonics (P₁) moment closure technique, as well as the popular discrete ordinates method (DOM), which is used as a benchmark for the model comparisons. The evaluations are performed for sooting co-flow laminar diffusion flames for blends of ethanol and methane fuels at atmospheric as well as at elevated pressures and include comparisons to available experimental data for soot volume fraction and flame temperature. The strong spectral dependence exhibited by the absorption coefficient of real gases is treated herein using the statistical narrow-band correlated-k method, and the chemical kinetics of the underlying species are modeled using a reduced mechanism for methane and ethanol fuels. Theoretical details of the proposed interpolative M₁, along with a description of the proposed Godunov-type finite-volume scheme developed for the numerical solution of the resulting system of hyperbolic moment equations are briefly discussed. The finite-volume method makes use of limited second-order solution reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton–Krylov–Schwarz (NKS) iterative method for solution of the resulting nonlinear algebraic equations arising from the finite-volume discretization procedure. The numerical results for laminar co-flow flames show that the flame solutions and predictions of soot formation of the non-gray M₁ maximum-entropy moment closure are very promising and in very good agreement with those of both the DOM and P₁ spherical harmonics model, while offering a substantial reduction in the number of dependent solution variables, compared to the DOM.

KEYWORDS:

• Laminar diffusion flames
• radiative heat transfer
• moment closures
• maximum entropy
• optimization
• efficiency

Acknowledgments

This research was funded by grants and contracts from the Green Aviation Research and Development Network (GARDN), Southern Ontario Smart Computing for Innovation Platform (SOSCIP), as well as Pratt & Whitney Canada. The first author also received support in the form of a scholarship from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Additionally, the computational resources for performing the numerical simulations reported herein were provided by the SOSCIP program as well as the SciNet High Performance Computing Consortium at the University of Toronto and Compute/Calcul Canada (the latter are funded by the Canada Foundation for Innovation (CFI) and Province of Ontario, Canada).