Ankan Kumar

Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes

The method of spherical harmonics (or P N ) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P 3 equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the P 1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P 3 results increase as the optical thickness is decreased. The P 3 approximation is found to be more accurate than the P 1 approximation for optically thick cases. Finally the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is,found that the solution of the four P 3 equations requires 14.5% additional CPU time, while the solution of the single P 1 equation requires 9.3% additional CPU time over the case without radiation.

Assessment of various diffusion models for the prediction of heterogeneous combustion in monolith tubes

Abstract In the case of heterogeneous reactions, diffusion is the only mechanism, locally, of transport of species to and from a surface. Thus, accurate prediction of diffusive transport is a prerequisite for accurate prediction of the operation of devices in which heterogeneous reactions occur. Three different diffusion models are examined from the standpoint of both accuracy and efficiency. Two of these models, namely the dilute approximation (DA) model and the Schmidt number (SN) model, are approximate models, and are compared against a rigorous multi-component diffusion (MCD) model derived from the Stefan–Maxwell equation. Both hydrogen–air and methane–air combustion in a monolith channel are studied. Inlet equivalence ratio, Reynolds number (flow rate), and wall temperature are considered as parameters. The results show that both the DA model and the SN model are accurate within 2% irrespective of the equivalence ratio or fuel—the worst accuracy being for hydrogen combustion. The DA model and the SN model produce almost identical results. In comparison to the MCD model, the DA model is approximately twice as computationally efficient, while the SN model is 2–16 times more efficient. The accuracy and efficiency of the SN model, in conjunction with its simplicity, makes it an attractive choice for the treatment of diffusion in catalytic combustion calculations.

Ionic and Biomolecular Transport in Nanochannels

In this work, both steady and transient ionic and biomolecular transport in nanochannels is considered. Electroosmotic flow (EOF) has been analyzed for both steady and transient two and three ionic components in a nanochannel. The sudden introduction of a species at the inlet of a channel generates a short transient regime followed by fully developed and steady-state EOF in which the concentrations, potential, and velocity are independent of the streamwise coordinate. The flux of any species is composed of Fickian diffusion, electrophoresis, and bulk convection and the mutual balance between these driving forces determines the direction of the movement of the species as well as its transit time. In a channel with negatively charged walls and the cathode on the upstream side, a negatively charged species may move in a direction opposite to the direction of bulk fluid flow. A positively charged species is transported in the direction of fluid flow and there is a significant decrease in transit time as compared to an uncharged or negatively charged species. Results for concentration and species flux are presented for both charged and uncharged species. The steady-state model is compared with a number of experimental results and the comparisons are extremely good.

A Low-Memory Block-Implicit Solver for Coupled Solution of the Species Conservation Equations on an Unstructured Mesh

Many reacting flow applications mandate coupled solution of the species conservation equations. A low-memory coupled solver was developed to solve the species transport equations on an unstructured mesh. The first step was the decomposition of the domain into sub-domains comprised of geometrically contiguous cells—a process termed internal domain decomposition (IDD). This was done using the binary spatial partitioning (BSP) algorithm. Following this step, for each subdomain, the discretized equations were set up, written in block implicit form, and solved using two different solvers: a direct solver using Gaussian elimination and an iterative solver based on Krylov sub-space iterations, i.e., the Generalized Minimum Residual (GMRES) solver. Overall (outer) iterations were then performed to treat explicitness at sub-domain boundaries and non-linearities in the governing equations. The solver is demonstrated for a simple two-dimensional multi-component diffusion problem involving 5 species. Sample calculations show that the solver with direct solution for each block is most efficient if the number of cells in each block is small—typically tens of cells, while the solver with iterative solution for each block is most efficient if the number of cells is relatively large—typically hundreds of cells. Overall the iterative solution based solver performed best, with maximum efficiency gain of a factor of seven over a block Gauss-Seidel (GS) solver and was found to be comparable or better in efficiency than a block-implicit Alternating Direction Implicit (ADI) solver. The gain in efficiency was found to increase with increase in cell aspect ratios.Copyright

Investigation of Approximate Diffusion Models for the Prediction of Heterogeneous Combustion in Monolith Tubes

In the case of heterogeneous reactions, diffusion is the only mechanism, locally, of transport of species to and from a surface. Thus, accurate prediction of diffusive transport is a prerequisite for accurate prediction of the operation of devices in which heterogeneous reactions occur. Three different diffusion models are examined from the standpoint of both accuracy and efficiency. Two of these models, namely the Dilute Approximation (DA) model and the Schmidt Number (SN) model, are approximate models, and are compared against a rigorous Multi-Component Diffusion (MCD) model derived from the Stefan-Maxwell equation. Both hydrogen-air and methane-air combustion in a monolith channel are studied. Inlet equivalence ratio, Reynolds number (flow rate), and wall temperature are considered as parameters. The results show that both the DA model and the SN model are accurate within 2% irrespective of the equivalence ratio or fuel—the worst accuracy being for hydrogen combustion. The DA model and the SN model produce almost identical results. In comparison to the MCD model, the DA model is approximately twice as computationally efficient, while the SN model is 3–4 times more efficient. The accuracy and efficiency of the SN model, in conjunction with its simplicity, makes it an attractive choice for the treatment of diffusion in catalytic combustion calculations.Copyright

Transient Electroosmotic Flow in Nano-Channels

One dimensional transient electroosmotic o w(EOF) in a rectangular nano-channel under the presence of an impulsive electric eld is studied numerically. We consider aqueous electrolyte of three species, consisting of , and a third species being a bio-molecule with a net negative charge. The parameters of interest are the mole fractions, velocity, potential, o w rate and current in the channel. The problem is a multiple time scale problem in the sense that the velocity eld responds on a very short time scale while the mole fractions of the electrolyte species respond to the long time scale. The results have application to bio-molecular sensing processes. Flow behavior due to a sudden change in wall potential is also studied.

The In Situ Adaptive Tabulation (ISAT) Algorithm for Reacting Flow Computations With Complex Surface Chemistry

The In Situ Adaptive Tabulation (ISAT) procedure, originally developed for the efficient computation of homogeneous reactions in chemically reacting flows, is adapted and demonstrated for reacting flow computations with complex heterogeneous (or surface) reactions. The treatment of heterogeneous reactions within a reacting flow calculation requires solution of a set of nonlinear differential algebraic equations at boundary faces/nodes, as opposed to the solution of an initial value problem for which the original ISAT procedure was developed. The modified ISAT algorithm, referred to as ISAT-S, is coupled to a three-dimensional unstructured reacting flow solver, and strategies for maximizing efficiency without hampering accuracy and convergence are developed. These include use of multiple binary tables, use of dynamic tolerance values to control errors, and periodic deletion and/or re-creation of the binary tables. The new procedure is demonstrated for steady-state catalytic combustion of a methane-air mixture on platinum using a 24-step reaction mechanism with 19 species, and for steady-state three-way catalytic conversion using a 61-step mechanism with 34 species. Both reaction mechanisms are first tested in simple 3D channel geometry with reacting walls, and the impact of various ISAT parameters is investigated. As a final step, the catalytic combustion mechanism is demonstrated in a laboratory-scale monolithic catalytic converter geometry with 57 channels discretized using 354,300 control volumes (4.6 million unknowns). For all of the cases considered, the reduction in the time taken to perform surface chemistry calculations alone was found to be a factor of 5–11.Copyright

Finite-Volume Solution of the P3 Equations of Radiative Transfer and Coupling to Reacting Flow Calculations

The method of spherical harmonics (or PN ) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled second-order partial differential equations. The FV procedure, presented here, is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the Binary Spatial Partitioning algorithm, followed by fully implicit coupled solution within each cell group using a pre-conditioned Generalized Minimum Residual (GMRES) solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for a benchmark problem. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the 4 P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the 10 equations that are solved for the reacting flow calculations.© 2009 ASME

An Unstructured Reacting Flow Solver With Coupled Implicit Solution of the Species Conservation Equations

Many reacting flow applications mandate coupled solution of the species conservation equations. A low-memory coupled solver was developed to solve the species transport equations on an unstructured mesh with implicit spatial as well as species-to-species coupling. First, the computational domain was decomposed into sub-domains comprised of geometrically contiguous cells—a process termed internal domain decomposition (IDD). This was done using the binary spatial partitioning (BSP) algorithm. Following this step, for each sub-domain, the discretized equations were developed using the finite-volume method, written in block implicit form, and solved using an iterative solver based on Krylov sub-space iterations, i.e., the Generalized Minimum Residual (GMRES) solver. Overall (outer) iterations were then performed to treat explicitness at sub-domain interfaces and non-linearities in the governing equations. The solver is demonstrated for a laminar ethane-air flame calculation with five species and a single reaction step, and for a catalytic methane-air combustion case with 19 species and 22 reaction steps. It was found that the best performance is manifested for sub-domain size of about 1000 cells, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block Gauss-Seidel procedure.Copyright