Y. Tambour

On the origins of spray sectional conservation equations

Abstract In this note it is demonstrated how Tambours previously discretely derived, sectional conservation equations for the description of polydisperse sprays can be deduced directly from Williamss spray equation for a droplet number density probability function. The analysis also produces a new compact integral formulation of spray source terms, involving sectional vaporization and coalescence coefficients.

A Lagrangian sectional approach for simulating droplet size distribution of vaporizing fuel sprays in a turbulent jet

Abstract A new Lagrangian sectional approach is used to analyze spray vaporization in a turbulent air jet flow. The fuel is considered to be in the form of discrete liquid droplets which have an arbitrary range of sizes and differ in their rates of vaporization. Assuming that the droplets follow the flow streamlines, the “residence time” of droplets as a function of the spatial coordinates is computed via the fluid Eulerian velocity field. Then, following a “group” of vaporizing droplets along a streamline, the Lagrangian approach is used. To avoid the dimensionality problem associated with the discrete form of the droplet population balance equations, “sectional conservation equations” are employed. Simulation of the downstream changes in volume distribution of fuel droplets of a vaporizing spray produced by an air-kerosene jet atomizer is presented. The theoretically predicted results of this simulation show good agreement with experimentally reported data.

Vaporization of polydisperse fuel sprays in a laminar boundary layer flow: A sectional approach☆

Abstract The effect of the presence of a cold wall on the downstream changes in size distribution of a spray of fuel droplets undergoing vaporization and combustion is theoretically analyzed. The fuel is considered to be in the form of discrete liquid droplets which have an arbitrary range of sizes and differ in their rates of vaporization. In fact, the total number of discrete droplet sizes needed to simulate actual fuel sprays can be immense. To avoid the dimensionality problem associated with the discrete form of population balance equations of an ensemble of individual burning or evaporating particles, “sectional conservation equations” are used. The method, based on dividing the droplet size domain into sections and dealing only with one integral quantity in each section (e.g., number, surface area of droplets, or volume), has the advantage that the integral quantity is conserved within the computational domain and the number of conservation equations required is simply equal to the number of sections. Employing known solutions for the boundary layer flow field, the “sectional size conservation equations” are solved assuming that droplets follow streamlines. New solutions for the changes in size distributon of droplets as a function of temperature and distance from the wall are presented. Since the present analysis uses an arbitrary droplet size distribution as an initial condition, it may be used to evaluate the performance of various atomizers, as demonstrated in the present study.

Stoichiometry and polydisperse effects in premixed spray flames

An analysis of a one-dimensional liminar premixed polydisperse spray flame is presented, for the whole range of fuel rich/lean mixtures. The spray of droplets is modeled using a sectional approach, thus making it possible to account for droplet-droplet interaction within the spray. A dimensional analysis of the mathematical model for the spray flame reveals its structure as consisting of a primary evaporation zone, a preheat zone, a homogeneous reaction zone, and a postreaction zone. If droplets survive the flame they may continue to burn or evaporate in the post reaction zone, depending on the fuel vapor-to-air ratio in the homogeneous reaction zone. Detailed analytic solutions are developed and matched for each of the aforementioned zones. Computed results show that the initial size distribution of the droplets in the spray has a profound influence on the flames characteristics. Although the use of the Sauter mean diameter (SMD) may be sufficient for the treatment of monodisperse sprays it is demonstrated here that its use to characterize the average evaporation rate of polydisperse sprays may lead to erroneous results, since the local SMD may actually decrease or increase depending on the initial droplet size distribution.

A theoretical study of polydisperse liquid sprays in a shear‐layer flow

A shear layer formed by two unidirectional gas streams of different velocities with a multisize (polydisperse) spray of evaporating droplets suspended in one of the gas streams is considered here. Similarity solutions are presented for the evolution in droplet size distributions across the shear layer and the effects of various initial droplet size distributions on the profiles of vapor concentrations are examined. A qualitative comparison between the present results for typical computed total mass distributions of the liquid phase and experimental data reported by Lazaro and Lasheras [Phys. Fluids A 1, 1035 (1989); Proceedings of the 22nd Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1988), pp. 1991–1998; J. Fluid Mech. 235, 143 (1992)] shows strong similarity between the two sets of profiles. This supports the assumptions and boundary conditions employed in the present theoretical study. The general behavior of the theoretical SMD (Sauter mean diameter) distribution of t...

Phenomenological theory of thermodynamic coupling in multicomponent compressible laminar boundary layers

Thermodynamic couplings due to thermal diffusion, diffusion‐thermo, and ob aliam diffusion effects in multicomponent compressible laminar boundary layers and in stagnation‐point flows with blowing or suction, are analyzed and compared with other available analyses for simpler specific cases. The mathematical treatment includes new definitions of generalized Prandtl, Lewis, and Schmidt numbers for coupled heat and multicomponent mass transfer. Using matrix notation and new tranformations, the energy and the species conservation equations are decoupled and transformed into a widely useful single compact ’’energy‐species’’ equation. For frozen multicomponent flows, the new ’’energy‐species’’ equation is of the same form as the energy equation for single‐component flows in the absence of thermodynamic coupling. Hence, solutions for generalized flows, which are presented here in transformed forms are obtainable from known solutions of ’’simpler’’ cases.

On Droplet Enhancement of the Burning Velocity of Laminar Premixed Spray Flames

The analyses of spray flames always predicted a burning velocity that was less than that of an equivalent single-phase gaseous premixed flame. This note reports a correction factor that can be applied to previously developed spray burning velocity formulae

Derivation of near-field sectional equations for the dynamics of polydisperse spray flows: An analysis of the relaxation zone behind a normal shock wave

Abstract A gas flow field behind a normal shock wave inflicts a step function in velocity, temperature, and pressure upon a polydisperse spray suspended in the flow. As a result, there are large differences in velocity and temperature between the carrying gas and the droplets of the spray immediately after the shock. These differences gradually decrease in a relaxation zone which ends at a distance, where the so-called far-field begins, at which the droplets eventually travel at a velocity which is close to the host fluid velocity. Thus, the investigation of the relaxation zone may be regarded as the analysis of a near-field problem in contradistinction to far-field problems that have been analyzed by Greenberg, Tambour and their coworkers employing a sectional approach. Treatment of a near-field problem via a sectional approach requires derivation of new generalized sectional-equations that extend the previous sectional-equations that were derived by Tambour for far-field problems. Such new generalized sectional-equations are derived in the present paper. They rigorously account for the heritage of momentum and enthalpy that droplets carry with them as they “move” from one size section to another due to evaporation and coalescence processes, for example, a large droplet formed by coalescence (of two small droplets) will differ from the originally large droplets since the properties of the newly formed droplet depend on the properties of the small droplets which participate in the coalescence process, and generally in the near-field , small droplets differ in velocity and enthalpy from large ones. These effects, as well as the influence of the velocity lag between droplets and the most fluid, between droplets of various sizes, and between droplets of the same size on the evolution in pointwise droplet size distributions along the axial direction are presented here. The computed results also include changes of velocities and temperatures of each droplet size-section in the relaxation zone.

Transport coupling theory for multicomponent nonequilibrium chemically reacting boundary layers

A general analysis of thermodynamic coupling effects in flames and other nonequilibrium gas‐phase chemically reacting multicomponent boundary‐layer flows is presented. Changes in characteristic diffusion time relative to characteristic reaction time due to thermal‐diffusion, diffusion‐thermo‐ and multicomponent‐diffusion effects are considered and a new generalized gas‐phase Damkohler number which encompasses these effects is defined. Using a transformed total enthalpy‐concentration field and the newly defined Damkohler number, compressible laminar reacting boundary layers were analyzed and compared with other available analyses for simpler specific cases. The applicability of the present theory is demonstrated by new solutions for the problem of combustion of injected premixed fuel. Comparison between these results and uncoupled solutions reported in the literature indicate that thermodynamic coupling effects are significant, especially when Damkohler numbers are smaller than unity.

Theory of thermodynamic coupling in surface reacting boundary layers

Thermodynamic coupling effects become important in such applications as injection of low molecular gases into boundary‐layer flows, ablative thermal protection processes, chemically reacting flows, and membrane transport processes which involve large gradients of temperature or concentrations. Thermodynamic couplings due to thermal diffusion, diffusion‐thermo and ob aliam diffusion effects in multicomponent compressible laminar boundary layers with nonequilibrium surface chemical reactions are analyzed and compared with other available analyses for simpler specific cases. A generalized mathematical description of the boundary conditions associated with surface reactions and thermodynamic coupling contributions to the various fluxes is also presented. The boundary conditions are transformed into total enthalpy‐concentration fields, and a new generalized Damkohler number for surface reactions is defined. The applicability of the present theory is demonstrated by new solutions to surface catalytic reactions ...