scholarly journals A New Mathematical Framework for Describing Thin-Reaction-Zone Regime of Turbulent Reacting Flows at Low Damköhler Number

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 109
Author(s):  
Vladimir A. Sabelnikov ◽  
Andrei N. Lipatnikov
Keyword(s):  
Molecular Transport ◽  
Characteristic Functions ◽  
Constant Density ◽  
Mathematical Framework ◽  
Damkohler Number ◽  
Field Equations ◽  
Conditioned Reaction ◽  
Reaction Progress ◽  
Progress Variable ◽  
Damköhler Number

Recently, Sabelnikov et al. (2019) developed a phenomenological theory of propagation of an infinitely thin reaction sheet, which is adjacent to a mixing layer, in a constant-density turbulent flow in the case of a low Damköhler number. In the cited paper, the theory is also supported by Direct Numerical Simulation data and relevance of such a physical scenario to highly turbulent premixed combustion is argued. The present work aims at complementing the theory with a new mathematical framework that allows for appearance of thick mixing zones adjacent to an infinitely thin reaction sheet. For this purpose, the instantaneous reaction-progress-variable c ( x , t ) is considered to consist of two qualitatively different zones, that is, (i) mixture of products and reactants, c ( x , t ) < 1 , where molecular transport plays an important role, and (ii) equilibrium products, c ( x , t ) = 1 . The two zones are separated by an infinitely thin reaction sheet, where c ( x , t ) = 1 and | ∇ c | is fixed in order for the molecular flux into the sheet to yield a constant local consumption velocity equal to the speed of the unperturbed laminar reaction wave. Exact local instantaneous field equations valid in the entire spaceare derived for the conditioned (to the former, mixing, zone) reaction progress variable, its second moment, and instantaneous characteristic functions. Averaging of these equations yields exact, unclosed transport equations for the conditioned reaction-progress-variable moments and Probability Density Function (PDF), as well as a boundary condition for the PDF at the reaction sheet. The closure problem for the derived equations is beyond the scope of the paper.

On the Stability of Planar Premixed Flames Under Nonadiabatic Conditions and Preferential Diffusion

2017 ◽  
Vol 9 (3) ◽  
Author(s):  
Eman Al-Sarairah ◽  
Bilal Al-Hasanat ◽  
Ahmed Hachicha
Keyword(s):  
Heat Loss ◽  
Numerical Study ◽  
Lewis Number ◽  
Premixed Flame ◽  
Constant Density ◽  
Damkohler Number ◽  
Damköhler Number ◽  
Number Ratio ◽  
Preferential Diffusion ◽  
The Stability

In this paper, we provide a numerical study of the stability analysis of a planar premixed flame. The interaction of preferential diffusion and heat loss for a planar premixed flame is investigated using a thermodiffusive (constant density) model. The flame is studied as a function of three nondimensional parameters, namely, Damköhler number (ratio of diffusion time to chemical time), Lewis number (ratio of thermal to species diffusivity), and heat loss. A maximum of four solutions are identified in some cases, two of which are stable. The behavior of the eigenvalues of the linearized system of stabilty is also discussed. For low Lewis number, the heat loss plays a major role in stabilizing the flame for some moderately high values of Damköhler number. The results show the effect of increasing or decreasing Lewis number on adiabatic and nonadiabatic flames temperature and reaction rate as well as the range of heat loss at which flames can survive.

scholarly journals Assessment of Extrapolation Relations of Displacement Speed for Detailed Chemistry Direct Numerical Simulation Database of Statistically Planar Turbulent Premixed Flames

2021 ◽  
Author(s):  
Nilanjan Chakraborty ◽  
Alexander Herbert ◽  
Umair Ahmed ◽  
Hong G. Im ◽  
Markus Klein
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Premixed Flames ◽  
Turbulent Premixed Flames ◽  
Reaction Progress ◽  
Progress Variable ◽  
Stretch Rate ◽  
Non Linear ◽  
Displacement Speed ◽  
Curvature Dependence

AbstractA three-dimensional Direct Numerical Simulation (DNS) database of statistically planar $$H_{2} -$$ H 2 - air turbulent premixed flames with an equivalence ratio of 0.7 spanning a large range of Karlovitz number has been utilised to assess the performances of the extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed $$S_{d}^{*}$$ S d ∗ . It has been found that the correlation between $$S_{d}^{*}$$ S d ∗ and curvature remains negative and a significantly non-linear interrelation between $$S_{d}^{*}$$ S d ∗ and stretch rate has been observed for all cases considered here. Thus, an extrapolation relation, which assumes a linear stretch rate dependence of density-weighted displacement speed has been found to be inadequate. However, an alternative extrapolation relation, which assumes a linear curvature dependence of $$S_{d}^{*}$$ S d ∗ but allows for a non-linear stretch rate dependence of $$S_{d}^{*}$$ S d ∗ , has been found to be more successful in capturing local behaviour of the density-weighted displacement speed. The extrapolation relations, which express $$S_{d}^{*}$$ S d ∗ as non-linear functions of either curvature or stretch rate, have been found to capture qualitatively the non-linear curvature and stretch rate dependences of $$S_{d}^{*}$$ S d ∗ more satisfactorily than the linear extrapolation relations. However, the improvement comes at the cost of additional tuning parameter. The Markstein lengths LM for all the extrapolation relations show dependence on the choice of reaction progress variable definition and for some extrapolation relations LM also varies with the value of reaction progress variable. The predictions of an extrapolation relation which involve solving a non-linear equation in terms of stretch rate have been found to be sensitive to the initial guess value, whereas a high order polynomial-based extrapolation relation may lead to overshoots and undershoots. Thus, a recently proposed extrapolation relation based on the analysis of simple chemistry DNS data, which explicitly accounts for the non-linear curvature dependence of the combined reaction and normal diffusion components of $$S_{d}^{*}$$ S d ∗ , has been shown to exhibit promising predictions of $$S_{d}^{*}$$ S d ∗ for all cases considered here.

A static generalization of the Einstein universe

1958 ◽  
Vol 245 (1241) ◽  
pp. 202-212
Keyword(s):  
General Problem ◽  
Constant Density ◽  
De Sitter ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Einstein Universe

Solutions of the Einstein field equations are found for the problem of a sphere of constant density surrounded by matter of different constant density. The solutions are discussed and particular attention paid to the topology of the surrounding matter. The Schwarzschild, de Sitter, and Einstein solutions emerge as particular cases of the general problem.

A Generalized FGM Progress Variable Weight Optimization Using HEEDS

2017 ◽  
Author(s):  
Graham Goldin ◽  
Yongzhe Zhang
Keyword(s):  
Constant Pressure ◽  
Operating Conditions ◽  
Weighted Sum ◽  
Weight Optimization ◽  
Reaction Progress ◽  
Optimization Software ◽  
Progress Variable ◽  
Flamelet Generated Manifold ◽  
Variable Weight ◽  
Mass Fractions

The Flamelet Generated Manifold (FGM) model requires a reaction progress variable which is usually defined as a weighted sum of species mass fractions. This progress variable should increase monotonically as flamelet states progress from unburnt to chemical equilibrium. A favorable attribute of the progress variable is that the flamelet species should change gradually with the progress variable, which reduces sensitivity of these species to any predicted errors in the progress variable. Previous publications have presented optimization algorithms for specific flamelet operating conditions, including fuel and oxidizer compositions and temperatures, and pressures. This work applies the HEEDS optimization software to find optimal species weights for a range of fuels and operating conditions. The fuels included are methane, methane-hydrogen, n-dodecane and n-heptane, at fuel-oxidizer temperatures of 293K and 1000K, and pressures of 1 and 30 atmospheres. For manifolds modeled by constant pressure ignition reactors, the optimal progress variable weights using four species weights are {αCO2 = 1, αCO = 0.91, αH2O = 0.52, αH2 = 1}, and for eight species weights are {αCO2 = 1, αCO = 0.91, αH2O = 0.51, αH2 = 1, αC2H2 = 0.16, αOH = −0.66, αH = −0.38, αO = 0.4}.

A Two-Dimensional Cellular Automaton Crystal with Irrational Density

2003 ◽  
Author(s):  
David Griffeath ◽  
Dean Hickerson
Keyword(s):  
Cellular Automaton ◽  
Asymptotic Density ◽  
Coarse Grain ◽  
Constant Density ◽  
Mathematical Framework ◽  
Two Dimensional ◽  
Empty Cell ◽  
Asymptotic Shape ◽  
Conway’S Game Of Life ◽  
Asymptotic Densities

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

Evaluation of the One-Well Uranium Leaching Test: Restoration

1982 ◽  
Vol 22 (01) ◽  
pp. 141-150 ◽  
Author(s):  
Muhammad I. Kabir ◽  
Larry W. Lake ◽  
Robert S. Schechter
Keyword(s):  
Acid Leaching ◽  
Leach Solution ◽  
Damkohler Number ◽  
Damköhler Number ◽  
Well Spacing ◽  
Number Of Factors ◽  
Well Pattern ◽  
Mineralized Zone ◽  
Selection Of

Abstract In-situ leach mining for uranium is an emerging technology. Currently, the selection of a well pattern designed to recover mineral values is governed primarily by arguments based on hydrological considerations. The effects of well pattern and well spacing on uranium recovery and oxidant utilization are considered in this paper. As expected, formation permeability heterogeneities and anisotropies are found to be important issues requiring careful consideration, however, it also is shown that chemical factors cannot be ignored. In particular, it is shown that the oxidant efficiency and the produced uranium solution concentrations are sensitive to the presence of other minerals competing with uranium for oxidant. If the Damkohler number for competing minerals, which measures the speed of the reaction, exceeds that for uranium, the competing mineral will have to be oxidized completely to recover a large proportion of the uranium. If the Damkohler number is smaller, it may be possible to achieve considerable selectivity for uranium by adjusting the well spacing. It also is shown that the oxidant efficiency is generally highest for well patterns that give streamlines of roughly equal length and that there is a minimum distance between injection and production wells to utilize oxidant most advantageously. Introduction In-situ solution mining is a process whereby uranium is recovered from permeable sandstone bodies by injecting and producing a leach solution through an array of wells penetrating the mineralized zone. It appears to have broad application and in many situations offers both economic and environmental advantages. The processes may be classified generally as acid or alkaline, but the general features of both are the same. The insoluble uranium in the mineralized zone is in the +4 state of oxidation. To be mobilized, the uranium must be oxidized to the +6 state and complexed either with sulfate in the case of acid leaching or carbonate in the case of alkaline leaching to form highly soluble uranyl sulfates or carbonates. The leach solutions, therefore, contain an oxidant (oxygen, hydrogen peroxide, ferric cations, sodium hyperchlorite, etc.) together with a complexing agent (anion). The choice of leach solution depends on a number of factors including selectivity and injectivity. For example, formations containing more than 1 wt% carbonates are not likely to be candidates for acid leaching because of the large acid requirement and because of permeability loss due to precipitation of calcium sulfate. It is the purpose of this paper to consider the technical factors (as opposed to economic) that govern the choice of well pattern to be used for leaching. The discussion is structured so that the conclusions apply to both alkaline and acid lixiviants and to any oxidant, although an occasional reference to a particular oxidant may appear. Considerable use is made of the computer simulator previously reported. The computational details are available in that paper. A number of factors that pertain to the selection of a well pattern are considered. It is shown that the effectiveness of the oxidant - i.e., the uranium recovered per unit of oxidant injected - is related to the well pattern, to the reaction rates, and to the permeability variations, especially if the formation is anisotropic. Furthermore, the spacing between wells is related to reactions with oxidizable minerals that compete for oxidant. These considerations can be quantified to some extent by studying linear systems. Linear Flow Systems SPEJ P. 132^

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