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Governing Equations
Published in Dalia E. E. Khalil, Essam E. Khalil, Sprinklers and Smoke Management in Enclosures, 2020
Dalia E. E. Khalil, Essam E. Khalil
Given a volume containing a mixture of gas species, a mixture fraction can be defined as the ratio of the mass of a subset of the species to the total mass present in the volume. During combustion, the mixture fraction is a conserved quantity traditionally defined as the (mass) fraction of the gas mixture that originates in the fuel stream. Thus, at a burner surface, the mixture fraction is 1 and in fresh air it is 0. In a region where combustion has occurred, this fraction will comprise any unburned fuel and the portion of the combustion products that came from the fuel. The mixture fraction is a function of space and time, commonly denoted Z(x, t). If it can be assumed that, upon mixing, the reaction of fuel and oxygen occurs rapidly and completely, the combustion process is referred to as “mixing-controlled.” This implies that all species of interest can be described in terms of the mixture fraction alone. The correspondence between the mass fraction of an individual species and the mixture fraction is called its “state relation.”
Laminar Non-Premixed Flames
Published in Achintya Mukhopadhyay, Swarnendu Sen, Fundamentals of Combustion Engineering, 2019
Achintya Mukhopadhyay, Swarnendu Sen
A schematic illustration of the counterflow flame experimental configuration is shown in Figure 7.11. Experiments are performed with fuel introduced from the bottom duct and oxidiser introduced from the top duct. An annular nitrogen gas curtain is provided to isolate the flow from the atmosphere. The opposing jets of oxidiser and fuel create a stagnation plane where axial velocity (vx) is equal to zero. The relative magnitude of oxidiser and fuel jet initial momentum flux determines the location of the stagnation plane. The stagnation plane resides at the midpoint of the separation gap between the nozzles for equal momentum flux of both streams. The location of the flame is determined by the mixture fraction value. The flame establishes itself where mixture fraction attains stoichiometric conditions. For hydrocarbon fuel (methane, propane and so on) burning in air, flame lies in the oxidiser side as it requires more air than fuel at stoichiometry ( fstoic = 0.06). The flame lies in the fuel side for hydrogen–air flame where more fuel is required than air in stoichiometric conditions.
Flameless Combustion with Liquid Fuels for Ultra-Low Emissions from Combustion Systems
Published in Debi Prasad Mishra, Advances in Combustion Technology, 2023
Saurabh Sharma, Sudarshan Kumar
The modelling uses a mixture-fraction approach to calculate different combustion parameters such as species mass fraction, density, temperature, etc. using different chemistry models such as equilibrium, flamelet model, and non-equilibrium model. For turbulent flow, the average values of these fluctuating parameters are calculated. It is the turbulence-chemistry interaction which defines the accuracy of the instantaneous value prediction of these parameters. A probability density function (PDF) approach is used to calculate the average values of the fluctuations in the combustion parameters. A PDF table is generated in which time-averaged values of density, temperature, and mass fractions are derived from the mixture-fraction (f) values at all the points.
A Comprehensive Study on Flame Length under Oxygen Enhanced Laminar Non-premixed Combustion
Published in Combustion Science and Technology, 2022
Anurag Mishra, Pushan Sharma, Bisrat Yoseph Gebreyesus, Mayank Kumar, Anjan Ray
In this work, the mixture fraction has been chosen as the parameter to determine the numerical flame length. Mixture fraction is defined as the ratio of the mass of the mixture having its origin in the fuel stream with the mass of the mixture in the control volume. It is the variable that couples local mass fractions of the fuel and oxidizer. The most intense reaction zone (stoichiometric mixture fraction iso-contour) was identified by calculating the stoichiometric mixture fraction value () (Thierry Poinsot 2005). An iso-contour for stoichiometric mixture fraction corresponds to where the mass fraction of fuel and oxidizer almost approaches zero. Figure 6 (a) shows the temperature iso-contours along with the contour corresponding to the fuel composition of 65% CH4 in jet and 30% O2 in co-flow for a 9-mm diameter burner. The flame length is shown corresponding to the stoichiometric mixture fraction iso-contour. Similarly, Figure 6 (b) represents the same iso-contours for the same fuel and oxidizer composition for the 4-mm diameter burner. Numerically is evaluated as follows in Eqn. 2..
Examination of probability distribution of mixture fraction in LES/FDF modelling of a turbulent partially premixed jet flame
Published in Combustion Theory and Modelling, 2022
Haifeng Wang, Pei Zhang, Jie Tao
The mixture fraction is one of the most fundamental quantities for the study of non-premixed and partially-premixed combustion. It quantifies the degree of mixing of two streams with a single parameter, and it is utterly important for the modelling of mixing-controlled combustion problems. The probability density function (PDF) of the mixture fraction is commonly assumed as the Beta-PDF. This presumed-PDF has been widely used in turbulent combustion modelling such as in the flamelet models [1]. Studying the PDF of the mixture fraction is critically important for providing support to the presumed-PDF used in many existing modelling as well as providing theoretical validation basis to other turbulent combustion models that do not rely on the presumed-PDF such as the transported PDF methods [2] and the transported filtered-density function (FDF) methods [3–5].
A generalised spray-flamelet formulation by means of a monotonic variable
Published in Combustion Theory and Modelling, 2021
D. O. Maionchi, F. P. Santos, J. Melguizo-Gavilanes, M. A. Endo Kokubun
Besides enabling a more computationally efficient solution in composition space compared to the physical-space solution, the mixture fraction concept is widely used in turbulent combustion models, since it allows the turbulent flame to be described in terms of simple one-dimensional elements called flamelets [11]. Extending this formulation to spray-flames would in principle enable the analysis tools developed for gaseous flames to be applied. However, a direct extrapolation of the classical mixture fraction to spray-flames is not possible because Z becomes non-monotonic due to the presence of vaporisation sources [7,12–14]; the constraint of monotonicity is required to guarantee that the solution is single-valued. In addition to Z, other composition spaces have been proposed and analysed in previous studies such as the total mixture fraction [15–17] and the conserved mixture fraction [7]. The aforementioned alternatives do not have their monotonicity guaranteed mainly due to differential diffusion and the relative velocity that exists between the liquid and gaseous phases. An effective composition variable combining the gaseous mixture fraction and the liquid-to-gas mass ratio was applied to the analysis of counterflow spray-flames in [7]. This variable was then employed to derive the governing equations for a spray-flamelet formulation. Although this formulation was found to reproduce the response of the flame structure to variations in the droplet diameter and strain rate, it required the use of a closure model for the scalar dissipation rate, χ [18].