Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

Combustion and microexplosion of collision-merged methanol/alkane droplets

The combustion characteristics of freely falling droplets, individually generated by the merging of colliding methanol and alkane droplets, were investigated and compared with those for pure methanol and alkanes. The merging of the nominally immiscible methanol and alkanes was manifested in an apparently adhesive, but unmixed, manner in all test conditions. An air bubble was found to be trapped at the colliding interfaces where they were “adhered,” with the trapping favored for head-on or near head-on collision orientations. The trapped air bubble ostensibly induced heterogeneous nucleation of the methanol, being facilitated by the relatively low limit of superheat of methanol. Consequently, the droplet exploded almost immediately upon ignition, leading to an extremely short overall lifetime. For collision orientations that were more off-centered, bubble trapping and thereby heterogeneous nucleation were not favored. However, delayed, albeit strong, microexplosion occurred through homogeneous nucleation of methanol at the contacting interface. The global burning rate was therefore again augmented. In general, microexplosion was facilitated for high-boiling-point alkanes such as hexadecane and tetradecane. The co-vaporization of methanol and alkane from their respective hemispherical segments constituting the adhered droplet also led to flame colors that were more bluish than yellowish, indicating the reduction of soot from alkane burning in the presence of methanol vapor. In light of the difficulty of forming stable methanol/oil emulsions, the potential of separate injection of oil and methanol in opposed jet arrangement, in direct-injection engines to facilitate collision, is suggested.     

Structure of propagating edge diffusion flames in hydrocarbon fuel jets

Direct numerical simulations with a C₃-chemistry model have been performed to investigate the transient behavior and internal structure of flames propagating in an axisymmetric fuel jet of methane, ethane, ethylene, acetylene, or propane in normal earth gravity (1g) and zero gravity (0g). The fuel issued from a 3-mm-i.d. tube into quasi-quiescent air for a fixed mixing time of 0.3 s before it was ignited along the centerline where the fuel–air mixture was at stoichiometry. The edge of the flame formed a vigorously burning peak reactivity spot, i.e., reaction kernel, and propagated through a flammable mixture layer, leaving behind a trailing diffusion flame. The reaction kernel broadened laterally across the flammable mixture layer and possessed characteristics of premixed flames in the direction of propagation and unique flame structure in the transverse direction. The reaction kernel grew wings on both fuel and air sides to form a triple-flame-like structure, particularly for ethylene and acetylene, whereas for alkanes, the fuel-rich wing tended to merge with the main diffusion flame zone, particularly methane. The topology of edge diffusion flames depend on the properties of fuels, particularly the rich flammability limit, and the mechanistic oxidation pathways. The transit velocity of edge diffusion flames, determined from a time series of calculated temperature field, equaled to the measured laminar flame speed of the stoichiometric fuel–air mixtures, available in the literature, independent of the gravity level.     

Gas-phase and catalytic combustion in heat-recirculating burners

An experimental study of a spiral counterflow “Swiss roll” burner was conducted, with emphasis on the determination of extinction limits and comparison of results with and without bare-metal Pt catalyst. A wide range of Reynolds numbers (Re) were tested using propane–air mixtures. Both lean and rich extinction limits were extended with the catalyst, though rich limits were extended much further. With the catalyst, combustion could be sustained at Re as low as 1.2 with peak temperatures as low as 350 K. A heat transfer parameter characterizing the thermal performance of both gas-phase and catalytic combustion at all Re was identified. At low Re, the “lean” extinction limit was actually rich of stoichiometric, and rich-limit had equivalence ratios exceeded 40 in some cases. No corresponding behavior was observed without the catalyst. Gas-phase combustion, in general, occurred in a “flameless” mode near the burner center. With or without catalyst, for sufficiently robust conditions (high Re, near-stoichiometric) not requiring heat recirculation, a visible flame would propagate out of the center, but this flame could only be re-centered if the catalyst were present. Gas chromatography indicated that at low Re, even in extremely rich mixtures, CO and non-propane hydrocarbons did not form. For higher Re, where both gas-phase and catalytic combustion could occur, catalytic limits were slightly broader but had much lower limit temperatures. At sufficiently high Re, catalytic and gas-phase limits merged. It is concluded that combustion at low Re in heat-recirculating burners greatly benefits from catalytic combustion with the proper choice of mixtures that are different from those preferred for gas-phase combustion. In particular, the importance of providing a reducing environment for the catalyst to enhance O₂ desorption, especially at low Re where heat losses are severe thus peak temperatures are low, is noted.     

Numerical study of influence of molecular diffusion in the Mild combustion regime

In this paper, the importance of molecular diffusion versus turbulent transport in the moderate or intense low-oxygen dilution (Mild) combustion mode has been numerically studied. The experimental conditions of Dally et al. [Proc. Combust. Inst. 29 (2002) 1147–1154] were used for modelling. The EDC model was used to describe the turbulence–chemistry interaction. The DRM-22 reduced mechanism and the GRI 2.11 full mechanism were used to represent the chemical reactions of an H₂/methane jet flame. The importance of molecular diffusion for various O₂ levels, jet Reynolds numbers and H₂ fuel contents was investigated. Results show that the molecular diffusion in Mild combustion cannot be ignored in comparison with the turbulent transport. Also, the method of inclusion of molecular diffusion in combustion modelling has a considerable effect on the accuracy of numerical modelling of Mild combustion. By decreasing the jet Reynolds number, decreasing the oxygen concentration in the airflow or increasing H₂ in the fuel mixture, the influence of molecular diffusion on Mild combustion increases.     

Microgravity droplet combustion: effect of tethering fiber on burning rate and flame structure

Droplets tethering on fibers has become a well established technique for conducting droplet combustion experiments in microgravity conditions. The effects of these supporting fibers are frequently assumed to be negligible and are not considered in the experimental analysis or in numerical simulations. In this work, the effect of supporting fibers on the characteristics of microgravity droplet combustion has been investigated numerically; a priori predictions have then been compared with published experimental data. The simulations were conducted using a transient one-dimensional spherosymmetric droplet combustion model, where the effect of the supporting fiber was implicitly taken into account. The model applied staggered convective flux finite volume method combined with high-order implicit time integration. Thermal radiation was evaluated using a statistical narrow band radiation model. Chemical kinetics and thermophysical properties were represented in rigorous detail. Tether fiber diameter, droplet diameter, ambient pressure and oxygen concentration were varied over a range for n-decane droplets in the simulations. The results of the simulations were compared to previously published experiments conducted in the Japan Microgravity Center (JAMIC) 10 second drop tower and the NASA Glenn Research Center (GRC) 5.2 second drop tower. The model reproduces closely nearly all aspects of tethered n-decane droplet burning phenomena, which included droplet burning history, transient and average burning rate, and flame standoff ratio. The predictions show that the presence of the tethering fiber significantly influences the observed burning rate, standoff ratio, and extinction.     

Large Eddy Simulations of a piloted lean premix jet flame using finite-rate chemistry

A Large Eddy Simulation (LES) model capable of accurately representing finite-rate chemistry effects in turbulent premixed combustion is presented. The LES computations use finite-rate chemistry and implicit LES combustion modelling to simulate an experimentally well-documented lean-premixed jet flame stabilized by a stoichiometric pilot. The validity of the implicit LES assumption is discussed and criteria are expressed in terms of subgrid scale Damköhler and Karlovitz numbers. Simulation results are compared to experimental data for velocity, temperature and species mass fractions of CH₄, CO and OH. The simulation results highlight the validity and capability of the present approach for the flame and in general the combustion regime examined. A sensitivity analysis to the choice of the finite-rate chemistry mechanism is reported, this analysis indicates that the one and two-step global reaction mechanisms evaluated fail to capture the reaction layer with sufficient accuracy, while a 20-species skeletal mechanism reproduces the experimental observations accurately including the key finite-rate chemistry indicators CO and OH. The LES results are shown to be grid insensitive and that the grid resolution within the bounds examined is far less important compared to the sensitivity of the finite-rate chemistry representation. The results are analyzed in terms of the flame dynamics and it is shown that intense small scale mixing (high Karlovitz number) between the pilot and the jet is an important mechanism for the stabilization of the flame.     

Numerical study of the effect of turbulence on rate of reactions in the MILD combustion regime

In this paper, the importance of fluctuations in flow field parameters is studied under MILD combustion conditions. In this way, a turbulent non-premixed CH₄+H₂ jet flame issuing into a hot and deficient co-flow air is modeled using the RANS Axisymmetric equations. The modeling is carried out using the EDC model to describe the turbulence-chemistry interaction. The DRM-22 reduced mechanism and the GRI2.11 full mechanism are used to represent the chemical reactions of H₂/methane jet flame. Results illustrate that although the fluctuations in temperature field are small and the reaction zone volume are large in the MILD regime, the fluctuations in temperature and species concentrations are still effective on the flow field. Also, inappropriate dealing with the turbulence effect on chemistry leads to errors in prediction of temperature up to 15% in the present flame. By decreasing of O₂ concentration of hot co-flow air, the effect of fluctuations in flow field parameters on flame characteristics are still significant and its effect on species reaction rates does not decrease. On the other hand, although decreasing of jet inlet Reynolds number at constant inlet turbulence intensity addresses to smaller fluctuations in flow filed, it does not lead to lower the effect of turbulence on species distribution and temperature field under MILD combustion conditions.     

The role of cool-flame chemistry in quasi-steady combustion and extinction of n-heptane droplets

Experiments on the combustion of large n-heptane droplets, performed by the National Aeronautics and Space Administration in the International Space Station, revealed a second stage of continued quasi-steady burning, supported by low-temperature chemistry, that follows radiative extinction of the first stage of burning, which is supported by normal hot-flame chemistry. The second stage of combustion experienced diffusive extinction, after which a large vapour cloud was observed to form around the droplet. In the present work, a 770-step reduced chemical-kinetic mechanism and a new 62-step skeletal chemical-kinetic mechanism, developed as an extension of an earlier 56-step mechanism, are employed to calculate the droplet burning rates, flame structures, and extinction diameters for this cool-flame regime. The calculations are performed for quasi-steady burning with the mixture fraction as the independent variable, which is then related to the physical variables of droplet combustion. The predictions with the new mechanism, which agree well with measured autoignition times, reveal that, in decreasing order of abundance, H₂O, CO, H₂O₂, CH₂O, and C₂H₄ are the principal reaction products during the low-temperature stage and that, during this stage, there is substantial leakage of n-heptane and O₂ through the flame, and very little production of CO₂ with no soot in the mechanism. The fuel leakage has been suggested to be the source of the observed vapour cloud that forms after flame extinction. While the new skeletal chemical-kinetic mechanism facilitates understanding of the chemical kinetics and predicts ignition times well, its predicted droplet diameters at extinction are appreciably larger than observed experimentally, but predictions with the 770-step reduced chemical-kinetic mechanism are in reasonably good agreement with experiment. The computations show how the key ketohydroperoxide compounds control the diffusion-flame structure and its extinction.     

A normalised residence time transport equation for the numerical simulation of combustion with high-temperature air

This study concerns the numerical simulation of turbulent non-premixed combustion in highly preheated air streams. One of the objectives is to settle an efficient computational procedure to proceed with the numerical simulation of large-scale industrial devices. It is also expected that the availability of such a computational framework may facilitate comprehensive sensitivity analyses as well as the development of mathematical models able to represent turbulence-chemistry interactions (TCI) in such conditions. Based on the salient physical ingredients that characterise scalar mixing, propagation, and self-ignition processes, a turbulent combustion modelling framework is thus introduced and applied to the numerical simulation of well-documented laboratory flames. In the corresponding geometries, the bulk flow velocities of the reactants streams can reach rather large values, which lead the flame to lift from the burner rim. Partially premixed flame edges thus stabilise the whole flame structure and the temperature of the oxidising stream can be increased by vitiation with burned gases so as to promote the corresponding flame-stabilisation processes. For sufficiently large values of the vitiated airstream temperature, self-ignition mechanisms may be triggered thus leading to a competition between mixing, propagation, and ignition processes. In this context, the ratio of the residence time to the self-ignition delay is thought to be a relevant variable to delineate the possible influence of ignition phenomena. Therefore, a modelled transport equation for this normalised residence time is considered. The performance of the corresponding modelling proposal is analysed with special emphasis placed on its ability to reproduce ‘memory’ or ‘lagrangian’ effects related to thermal aging processes. In this respect, it is noteworthy that the present set of computations makes use of tabulated quantities associated to (i) steady laminar one-dimensional diffusion flamelets, so as to describe the composition of combustion products, (ii) steady laminar one-dimensional premixed flamelets, to describe the flame brush propagation, and (iii) temporal evolution of zero-dimensional homogeneous mixtures to account for the possible occurrence of self-ignition phenomena. In particular, the tabulated self-ignition time value is used to evaluate the increase in the normalised residence time. Finally, two modelling parameters are put into evidence and studied through a detailed sensitivity analysis.