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At least 541 records · Page 30

Solid Particle Erosion of a Plasma Spray – Physical Vapor Deposition Environmental Barrier Coating in a Combustion Environment

Environmental barrier coatings (EBCs) were developed to reduce the susceptibility of SiC-based composites to the rapid volatilization and surface recession that occurs in the presence of water vapor. Extensive research and development has been performed on EBCs, and their corresponding failure mechanisms under different environmental conditions. However, one key failure mechanism, solid particle erosion (SPE), has not been thoroughly investigated. As a result, the present work investigates the SPE behavior of a ytterbium disilicate (Yb2Si2O7) environmental barrier coating (EBC) deposited via plasma spray-physical vapor deposition (PS-PVD). Erosion testing was performed at elevated temperature (1,200 °C) in a simulated combustion environment at the NASA Glenn Research Center Erosion Burner Rig Facility. Alumina (Al2O3) particles were used as the eroding media. A range of particle kinetic energies and impingement angles were investigated along with the effect of coating surface roughness. The post-erosion damage morphology was characterized using scanning electron microscopy (SEM). The SPE behavior of the PS-PVD EBC was shown to be comparable to other EBC systems reported in the literature.

Solid particle erosion; environmental barrier coat↗

Effect of fuel composition and EGR on spark-ignited engine combustion with LPG fueling: Experimental and numerical investigation

This paper presents an experimental and numerical investigation of a spark-ignited (SI) cooperative fuel research (CFR) engine fueled with different liquefied petroleum gas (LPG) fuels and exhaust gas recirculation (EGR). Here, the effects of LPG fuel composition on engine combustion characteristics are initially evaluated at two different compression ratios (CR). Results show normal combustion at CR 7 and heavy knocking combustion at CR 10 for all the tested fuels, with a more substantial impact for the LPG fuel with high proportions of n-butane species. The Livengood-Wu (LW) integral method is then used to analyze the knock occurrence risk of individual fuel based on the reactivity of the tested fuels. The introduction of EGR then demonstrates the potential of knock intensity reduction below the borderline knock limit. A zonal-based kinetic interactions study is also performed to understand the knock mitigation effectiveness of EGR over the pressure–temperature domain relevant to SI engine operation. Finally, a multidimensional, computational fluid dynamics (CFD) simulation model is shown to predict the LPG combustion characteristics and presents the evolution of in-cylinder temperature and chemical species to demonstrate the development of end-gas autoignition events without and with EGR.

02 PETROLEUM↗

The effects of sooting in droplet combustion

The study of the burning of a single droplet is an ideal problem from which to gain fundamental understanding of diffusion flame characteristics. Droplet combustion is a complex physico-chemical process that involves a chemically-reacting two-phase flow with phase changes and yet simple experiments and analysis can be used to attain important insights into the burning rate, flame dynamics, kinetic extinction and disruption processes. It is a subject that has been actively studied for the past 40 years with most of the fundamental experiments being performed under reduced-gravity conditions for direct comparisons with theoretical/computational analyses that invoke spherical symmetry assumptions. In the earlier studies, the effects of sooting on the overall burning characteristics were not considered. However, recent microgravity investigations performed at the NASA-LeRC droptowers (Droplet Combustion Experiment) and others indicate that effects of soot and sootcloud formation may be significant during the lifetime of the droplet and therefore must be included in the analysis.

Lee, Kyeong-Ook↗

Computational study on the impact of gasoline-ethanol blending on autoignition and soot/NO x emissions under low-load gasoline compression ignition conditions

Here, in the present work, computational fluid dynamics (CFD) simulations of a single-cylinder gasoline compression ignition (GCI) engine are performed to investigate the impact of gasoline-ethanol blending on autoignition, nitrogen oxide (NO x ), and soot emissions under low-load conditions. In order to represent the test gasoline (RD5-87), a four-component toluene primary reference fuel (TPRF)+ethanol (ETPRF) surrogate (with 10% ethanol by volume; E10) is employed. A three-dimensional (3D) engine CFD model employing finite-rate chemistry with a skeletal kinetic mechanism (including NO x sub-mechanism), adaptive mesh refinement (AMR), and hybrid method of moments (HMOM) is adopted to capture the in-cylinder combustion phenomena and soot/NO x emissions. The engine CFD model is validated against experimental data for three gasoline-ethanol blends: E10, E30 and E100, with varying ethanol content by volume. Model validation is carried out for a broad range of start-of-injection (SOI) timings (−21, −27, −36, and −45 crank angle degrees (°CA) after top-dead-center (aTDC)) with respect to in-cylinder pressure, heat release rate, combustion phasing, NO x and soot emissions. For relatively later injection timings (−21 and −27 °CA aTDC), E30 yields higher amount of soot than E10; while the trend reverses for early injection cases (−36 and −45 °CA aTDC ). On the other hand, E100 yields the lowest amount of soot among all fuels irrespective of SOI timing. Further, E10 shows a non-monotonic trend in soot emissions with SOI timing: SOI-36>SOI-45>SOI-21>SOI-27, while soot emissions from E30 exhibit monotonic decrease with advancing SOI timing. NO x emissions from various fuels follow a trend of E10>E30>E100. On the other hand, NO x emissions increase as SOI timing is advanced for all fuels, with an anomaly for E10 and E100 where NO x decreases when SOI is advanced beyond −36 °CA aTDC. Detailed analysis of the numerical results is performed to investigate the soot/NO x emission trends and elucidate the impact of chemical composition and physical properties on autoignition and emissions characteristics.

Computational fluid dynamics↗

Methanol mixing−controlled compression ignition enabled via homogeneous charge compression ignition of dimethyl ether through catalytic decomposition of methanol

Methanol is a potentially attractive fuel for marine and off−road engines owing to its availability at bunkering and global distribution locations. Although methanol is well−distributed worldwide, its fuel chemistry and ignition properties make it poorly suited as a direct drop−in replacement for diesel fuel in compression−ignition engines. However, industrial processes are regularly used to convert methanol, via catalytic dehydration, to dimethyl ether (DME) over nonprecious metal catalysts. This chemical conversion can occur at relatively low pressures, temperatures, and catalyst space velocities, highlighting a potential opportunity to generate DME via onboard catalytic dehydration of methanol. DME’s fuel kinetic and ignition properties for compression ignition are much more favorable than those of methanol or even diesel fuel, but DME is more challenging than diesel fuel or methanol to pump, store, and deliver through conventional diesel fueling injection hardware. Thus, a potential opportunity exists to use the ignition and kinetic properties of DME, with the transportation and delivery advantages of methanol, in a methanol−fueled mixing−controlled compression−ignition engine. The present work explores performance, combustion behavior, and emissions reduction opportunities for methanol mixing−controlled combustion, enabled by a HCCI of DME that represents a small fraction of the total fuel energy that can be generated onboard via catalytic dehydration of methanol.

Splitter, Derek [ORNL] (ORCID:0000000174044047)↗

A physics-constrained neural ordinary differential equations approach for robust learning of stiff chemical kinetics

The high computational cost associated with solving for detailed chemistry poses a significant challenge for predictive computational fluid dynamics (CFD) simulations of turbulent reacting flows. While deep learning techniques have been explored to develop faster surrogate models, they often fail to integrate reliably with CFD solvers. This instability arises because traditional deep learning approaches optimize for training error without ensuring compatibility with ordinary differential equation (ODE) solvers, resulting in accumulation of errors over time. Recently, neuralODE (NODE) based approaches have been shown to be a promising technique to emulate and accelerate detailed chemistry computations. Here, in the present work, we extend this NODE framework for stiff chemical kinetics by incorporating mass conservation constraints directly into the loss function during training. This ensures that the total mass as well as the individual elemental species masses are conserved in an a-posteriori manner. Proof-of-concept studies are performed with the novel physics-constrained NODE (PC-NODE) approach for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions. It is demonstrated that the PC-NODE framework not only improves the physical consistency of the resulting data-driven model with respect to mass conservation criteria, but also improves training efficiency. PC-NODE is shown to achieve 2–100× speedup relative to the hydrogen-air detailed chemical mechanism depending on the type of the ODE solver (implicit or explicit) used during autoregressive inference tests. Lastly, a-posteriori studies are performed wherein the trained PC-NODE model is coupled with a CFD solver. It is shown that higher accuracy is achieved with PC-NODE relative to the purely data-driven NODE approach. Moreover, PC-NODE also exhibits robustness and generalizability to unseen initial conditions from within (interpolative capability) as well as outside (extrapolative capability) the training regime.

computational combustion↗

Octane Requirements of Lean Mixed-Mode Combustion in a Direct-Injection Spark-Ignition Engine

Here, this study investigates the octane requirements of a hybrid flame propagation and controlled autoignition mode referred to as mixed-mode combustion (MMC), which allows for strong control over combustion parameters via a spark-initiated deflagration phase. Due to the throughput limitations associated with both experiments and 3-D computational fluid dynamics calculations, a hybrid 0-D and 1-D modeling methodology was developed, supported by experimental validation data. This modeling approach relied on 1-D, two-zone engine simulations to predict bulk in-cylinder thermodynamic conditions over a range of engine speeds, compression ratios, intake pressures, trapped residual levels, fueling rates, and spark timings. Those predictions were then transferred to a 0-D chemical kinetic model, which was used to evaluate the autoignition behavior of fuels when subjected to temperature–pressure trajectories of interest. Finally, the predicted autoignition phasings were screened relative to the progress of the modeled deflagration-based combustion in order to determine if an operating condition was feasible or infeasible due to knock or stability limits. The combined modeling and experimental results reveal that MMC has an octane requirement similar to modern stoichiometric spark-ignition engines in that fuels with high research octane number (RON) and high octane sensitivity (S) enable higher loads. Experimental trends with varying RON and S were well predicted by the model for 1000 and 1400 rpm, confirming its utility in identifying the compatibility of a fuel’s autoignition behavior with an engine configuration and operating strategy. However, the model was not effective in predicting (nor designed to predict) operability limits due to cycle-to-cycle variations, which experimentally inhibited operation of some fuels at 2000 rpm. Putting the operable limits and efficiency from MMC in the context of a state-of-the-art engine, the MMC showed superior efficiencies over the range investigated, demonstrating the potential to further improve fuel economy.

33 ADVANCED PROPULSION SYSTEMS↗

Influence of Natural Gas Addition on Combustion Instabilities on a Lean-Premixed Hydrogen Flame

The influence of fuel composition on combustion instabilities in a high-pressure combustor operated with hydrogen and natural gas fuels is characterized. A micromix multi-stage injector with 19 elements is used to introduce hydrogen and natural gas premixed with heated air into an optically accessible combustor. As the equivalence ratio and flame temperature is varied along with change in hydrogen fraction in the fuel, self-excited longitudinal instabilities are observed in the combustor. The instability amplitude reduces considerably as hydrogen fraction is increased. This sensitivity differs at higher flame temperature, with large amplitude instabilities attenuating at lower hydrogen fractions for flames with more thermal power. Increasing hydrogen concentration is shown, through kinetic modeling, to decrease the bulk reaction timescale linearly with observed instability amplitude. Spatial distribution of heat release in the combustor evaluated from OH* chemiluminescence imaging reveals excitation of primarily the 2L mode in the combustion chamber at ~1100 Hz, at all equivalence ratios and fuel compositions. Increasing equivalence ratio and hydrogen addition decreases the length of the positively correlated region, which leads to instability dampening.

08 HYDROGEN↗

Effects of non-thermal termolecular chemistry on detonation development in hydrogen (H2) / methane (CH4) - air mixtures

In combustion simulations, it is usually assumed that highly-energized collision complexes (either radical or stable species) formed in exothermic reactions are always under thermal equilibrium as they evolve through competing networks of reactions. However, in practical flames, non-trivial amounts of reactive radicals such as H, O and OH are present apart from O2. As a result, collisions of these reactive species with the energized collision complexes have been shown in recent studies to induce non-thermal reactivity. These studies have also demonstrated that such non-thermal reactions can be suitably represented in macroscopic kinetics models as chemically termolecular reactions. The present work was focused on identifying and quantifying the effects of including such reactions on the evolution of an initial deflagration front to a developing detonation in H2/CH4-air mixtures under boosted internal combustion (IC) engine conditions. Specifically, fully resolved simulations, with and without non-thermal reactivity, were performed for a constant volume reactor containing stoichiometric H2/CH4-air mixture. It was found that inclusion of termolecular chemistry resulted in a delayed onset of end-gas auto-ignition. Concurrently, the developing detonation intensity was observed to be significantly higher. Chemical explosive mode analysis (CEMA) was performed to identify the dominant species/reactions responsible for the observed variation in the results.

Detonation↗

Characterization of Swirl-Venturi Lean Direct Injection Designs for Aviation Gas-Turbine Combustion

Injector geometry, physical mixing, chemical processes, and engine cycle conditions together govern performance, operability and emission characteristics of aviation gas-turbine combustion systems. The present investigation explores swirl-venturi lean direct injection combustor fundamentals, characterizing the influence of key geometric injector parameters on reacting flow physics and emission production trends. In this computational study, a design space exploration was performed using a parameterized swirl-venturi lean direct injector model. From the parametric geometry, 20 three-element lean direct injection combustor sectors were produced and simulated using steady-state, Reynolds-averaged Navier-Stokes reacting computations. Species concentrations were solved directly using a reduced 18-step reaction mechanism for Jet-A. Turbulence closure was obtained using a nonlinear κ-ε model. Results demonstrate sensitivities of the geometric perturbations on axially averaged flow field responses. Output variables include axial velocity, turbulent kinetic energy, static temperature, fuel patternation and minor species mass fractions. Significant trends have been reduced to surrogate model approximations, intended to guide future injector design trade studies and advance aviation gas-turbine combustion research.

Combustion Chamber↗

Selective Chemical Looping Combustion of Terminal Alkynes in Mixtures with Alkenes

The selective combustion of terminal alkynes in mixtures with alkenes is demonstrated during anaerobic reduction half-cycles on bulk bismuth oxide (Bi 2 O 3 ) as an approach to remove alkynes, which act as inhibitors in olefin polymerization. Bi 2 O 3 combusts phenylacetylene in styrene, 3-methylphenylacetylene in 3-methylstyrene, propyne in propylene, 1-hexyne in 1-hexene, and 1-octyne in 1-octene, with alkyne combustion selectivities exceeding 96%. Near unity reaction orders for hydrocarbon consumption during reduction half-cycles are consistent with combustion pathways initiated by rate-determining initial C–H activation, which drive selective alkyne combustion through intrinsic differences in the first-order rate constants for alkyne and alkene combustion rather than preferential adsorption of alkynes on Bi 2 O 3 surfaces. Computational assessments of initial C–H activation pathways for alkynes and alkenes on (010) α-Bi 2 O 3 surfaces using density functional theory illustrate that heterolytic transition states which form proton-carbanion pairs on Bi–O sites kinetically favor the activation of alkynes rather than alkenes due to differences in C–H bond acidity, and the barrier for heterolytic C–H activation is dictated in part by the sum of the molecular deprotonation energy and the energy to bind an R – carbanion to a Bi site in its transition-state geometry. Finally, these heterolytic reactivity channels during selective chemical looping combustion present novel routes for purifying olefin gas streams containing alkyne impurities.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Numerical Simulations of High-Speed Chemically Reacting Flow

The Essentially NonOscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗

Numerical Simulations of High-Speed Chemically Reacting Flow

The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗

Comparison of High Aspect Ratio Cooling Channel Designs for a Rocket Combustion Chamber

An analytical investigation on the effect of high aspect ratio (height/width) cooling channels, considering different coolant channel designs, on hot-gas-side wall temperature and coolant pressure drop for a liquid hydrogen cooled rocket combustion chamber, was performed. Coolant channel design elements considered were: length of combustion chamber in which high aspect ratio cooling was applied, number of coolant channels, and coolant channel shape. Seven coolant channel designs were investigated using a coupling of the Rocket Thermal Evaluation code and the Two-Dimensional Kinetics code. Initially, each coolant channel design was developed, without consideration for fabrication, to reduce the hot-gas-side wall temperature from a given conventional cooling channel baseline. These designs produced hot-gas-side wall temperature reductions up to 22 percent, with coolant pressure drop increases as low as 7.5 percent from the baseline. Fabrication constraints for milled channels were applied to the seven designs. These produced hot-gas-side wall temperature reductions of up to 20 percent, with coolant pressure drop increases as low as 2 percent. Using high aspect ratio cooling channels for the entire length of the combustion chamber had no additional benefit on hot-gas-side wall temperature over using high aspect ratio cooling channels only in the throat region, but increased coolant pressure drop 33 percent. Independent of coolant channel shape, high aspect ratio cooling was able to reduce the hot-gas-side wall temperature by at least 8 percent, with as low as a 2 percent increase in coolant pressure drop. The design with the highest overall benefit to hot-gas-side wall temperature and minimal coolant pressure drop cooling can now be done in relatively short periods of time with multiple iterations.

Wadel, Mary F.↗

Comparison of High Aspect Ratio Cooling Channel Designs for a Rocket Combustion Chamber with Development of an Optimized Design

An analytical investigation on the effect of high aspect ratio (height/width) cooling channels, considering different coolant channel designs, on hot-gas-side wall temperature and coolant pressure drop for a liquid hydrogen cooled rocket combustion chamber, was performed. Coolant channel design elements considered were: length of combustion chamber in which high aspect ratio cooling was applied, number of coolant channels, and coolant channel shape. Seven coolant channel designs were investigated using a coupling of the Rocket Thermal Evaluation code and the Two-Dimensional Kinetics code. Initially, each coolant channel design was developed, without consideration for fabrication, to reduce the hot-gas-side wall temperature from a given conventional cooling channel baseline. These designs produced hot-gas-side wall temperature reductions up to 22 percent, with coolant pressure drop increases as low as 7.5 percent from the baseline. Fabrication constraints for milled channels were applied to the seven designs. These produced hot-gas-side wall temperature reductions of up to 20 percent, with coolant pressure drop increases as low as 2 percent. Using high aspect ratio cooling channels for the entire length of the combustion chamber had no additional benefit on hot-gas-side wall temperature over using high aspect ratio cooling channels only in the throat region, but increased coolant pressure drop 33 percent. Independent of coolant channel shape, high aspect ratio cooling was able to reduce the hot-gas-side wall temperature by at least 8 percent, with as low as a 2 percent increase in coolant pressure drop. ne design with the highest overall benefit to hot-gas-side wall temperature and minimal coolant pressure drop increase was the design which used bifurcated cooling channels and high aspect ratio cooling in the throat region. An optimized bifurcated high aspect ratio cooling channel design was developed which reduced the hot-gas-side wall temperature by 18 percent and reduced the coolant pressure drop by 4 percent. Reductions of coolant mass flow rate of up to 50 percent were possible before the hot-gas-side wall temperature reached that of the baseline. These mass flow rate reductions produced coolant pressure drops of up to 57 percent.

Wadel, Mary F.↗

Highly Selective Methane to Methanol Conversion on Inverse SnO 2 /Cu 2 O/Cu(111) Catalysts: Unique Properties of SnO 2 Nanostructures and the Inhibition of the Direct Oxidative Combustion of Methane

Direct methane to methanol (CH 4 → CH 3 OH) conversion in heterogeneous catalysis has been a long-standing challenge due to the difficulties in equalizing the activation of methane and protection of the methanol product at the same reaction conditions. Here, we report an inverse catalyst, consisting of small structures of SnO 2 (0.5-1 nm in size) dispersed on Cu 2 O/Cu(111), for highly selective CH 3 OH production from CH 4 . This system was investigated by combining theoretical [density functional theory calculations (DFT), kinetic Monte Carlo simulations (KMC)] and experimental methods [scanning tunneling microscopy (STM), ambient-pressure X-ray photoelectron spectroscopy (AP-XPS)]. The DFT and AP-XPS studies showed that on SnO 2 /Cu 2 O/Cu(111) the conversion of CH 4 by oxygen (O 2 ) preferred complete combustion to carbon dioxide (CO 2 ). The addition of water (H 2 O) enhanced the production of CH 3 OH to nearly 100% selectivity in KMC simulations. This trend was consistent with results of AP-XPS. The presence of water in the reaction environment rendered an extremely high amount of methoxy species (*CH 3 O), a precursor for CH 3 OH production. Further, the high CH 3 OH selectivity of SnO 2 /Cu 2 O/Cu(111) reflected the unique atomic and electronic structure of the supported SnO 2 nanoparticles. As a result, the O 2 adsorption and dissociation, and thus the full combustion of CH 4 to CO 2 , was completely suppressed; while the H 2 O dissociative adsorption was still feasible, providing active hydroxyl species for a truly selective CH 4 to CH 3 OH conversion.

03 NATURAL GAS↗

Group additivity values for the heat of formation of C 2 –C 8 alkanes, alkyl hydroperoxides, and their radicals

A set of 58 group additivity values (GAV) for the calculation of the heat of formation is derived from an extensive and accurate database of 192 ab initio heats of formation. The ab initio values are from companion calculations at the CCSD(T)–F12/cc-pVTZ-F12//B2PLYPD3/cc-pVTZ level of theory employing second order connectivity based hierarchy reference reactions with ANL energies for the reference species. This database of 192 species consists of alkanes (RH), alkyl radicals ($\dot{\text {R}}$), alkyl hydroperoxides (RO 2 H), alkyl-peroxy (R$\dot{\text {O}}$ 2 ) radicals, and hydroperoxy-alkyl ($\dot{\text {Q}}$OOH) radicals of all of the isomers of C 2 –C 5 alkane fuels and a select set of C 6 –C 8 isomers. The GAV and ab initio based enthalpies for this dataset show excellent agreement, with an estimated 2σ uncertainty of 0.9 kcal mol –1 . Among the 58 GAV terms, 40 are refinements of previously reported terms, while 18 are newly developed. These new GAV terms are mainly non-next-nearest neighbor interactions (NNI) and β-corrections. The inclusion of these new groups significantly improves the accuracy of the GAV estimates of the heats of formation. Finally, the updated GAVs can be used with increased confidence to estimate the heats of formation of combustion relevant hydrocarbons and oxygenated hydrocarbons and their corresponding radicals, which are important in predicting low-temperature chemistry and are critical in the development of accurate chemical kinetic models.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Heat Transfer to a Thin Solid Combustible in Flame Spreading at Microgravity

The heat transfer rate to a thin solid combustible from an attached diffusion flame, spreading across the surface of the combustible in a quiescent, microgravity environment, was determined from measurements made in the drop tower facility at NASA-Lewis Research Center. With first-order Arrhenius pyrolysis kinetics, the solid-phase mass and energy equations along with the measured spread rate and surface temperature profiles were used to calculate the net heat flux to the surface. Results of the measurements are compared to the numerical solution of the complete set of coupled differential equations that describes the temperature, species, and velocity fields in the gas and solid phases. The theory and experiment agree on the major qualitative features of the heat transfer. Some fundamental differences are attributed to the neglect of radiation in the theoretical model.

Bhattacharjee, S.↗