General form of conservation equations Let f( ,t) be a general variable (e.g., density, specific energy) that satisfies a conservation principle (mass, momentum, energy), defined per unit volume, and F(t) the integral of that variable over an arbitrary control volume V F(t) may change in time due to the convective and/or diffusive flux, , across the boundary of the control volume, and due to the production or consumption of F within the control volume (let sf be the amount of F generated or consumed per unit of volume and time). The rate of change of F is then given by Conservation equation: Conservation equations Combustion

Mass conservation equation In the mass conservation or continuity equation f( ,t) stands for the density and is the mass flux per unit volume, while the source term is zero f = r sf = 0 Conservation equations Combustion

Mass conservation of a chemical species In the mass conservation equation of a chemical species f( ,t) stands for the mass concentration of that species, is the mass flux of that species per unit volume, while the source term is the reaction rate of the species f = r yi - absolute velocity of the mixture - absolute velocity of species i - diffusion velocity of species i Ri [kg/m3.s], [kmol/m3.s] The mass diffusion flux of species i is directly proportional to the diffusion velocity of the species, , yielding the following conservation equation Conservation equations Combustion

Mass conservation of a chemical species It can be demonstrated that so that the sum of the mass conservation equations for all species yields the continuity equation The diffusion mass flux is due to three different contributions: - Ordinary diffusion due to species concentration gradient - Soret effect due to temperature gradient - Diffusion originated from pressure gradient Conservation equations Combustion

Mass conservation of a chemical species [kg m-1 s-1] – Thermal diffusion coefficient [m2 s-1] – Multicomponent ordinary diffusion coefficients The diffusion due to pressure gradients is negligible in combustion processes The Soret effect is also negligible for most species, except for species with low molar weight (H, H2, He) and at low temperatures The multicomponent ordinary diffusion coefficients for species i may be approximated by an average diffusion coefficient for species i in the mixture of the other species: Conservation equations Combustion

Mass conservation of a chemical species With the previous three approximations, the mass diffusion flux is approximated by Fick’s law: Hence the mass conservation equation for a chemical species is written as: The average diffusion coefficient for species i in the mixture is generally determined from the binary diffusion coefficients Dij using the following approximation In a mixture with a dominant species (e.g., the N2 for combustion in air), may be approximated by the binary diffusion coefficient of species i in N2 Conservation equations Combustion

Momentum conservation equation In the momentum conservation equation, vector stands for the momentum per unit volume, the flux comprises a convective and a diffusion term, with both viscous and pressure forces included in the diffusion term, and the source term is the gravity force per unit volume sij = -p dij + tij sij: molecular stress tensor tij: viscous stress tensor or deviatoric stress tensor for a Newtonian fluid Conservation equations Combustion

Energy conservation equation In the energy conservation equation, f may stand for the total energy per unit volume (other choices are possible), the flux comprises a convective term, a term due to the work done by pressure and viscous forces and the heat flux term, and the source term is the radiative heat transfer rate per unit volume f = r e e [J/kg] : total specific energy u [J/kg] : internal specific energy G [J/kg]: potential energy per unit mass Conservation equations Combustion

Energy conservation equation Energy conservation equation expressed in terms of the specific internal energy Energy conservation equation expressed in terms of the specific enthalpy The heat flux vector comprises three different contributions: - Heat conduction due to temperature gradient - Dufour effect due to species concentration gradient - Energy flux due to mass diffusion Conservation equations Combustion

Energy conservation equation l [W/m.K]: Thermal conductivity The Dufour effect is negligible in combustion processes. For a mixture of ideal gases, we have Hence: Conservation equations Combustion

Energy conservation equation Expressing the diffusive flux according to Fick´s law Schmidt number: ratio of the rate of momentum transport to the rate of mass transport Prandtl number: ratio of rate of momentum transport to the rate of energy transport Lewis number: ratio of rate of energy transport to the rate of mass transport Energy equation (negligible Dufour effect, mixture of ideal gases, Fick´s law) Conservation equations Combustion

Energy conservation equation The pressure terms are negligible for low Mach number flows The viscous dissipation terms are also negligible for low Mach number flows The Lewis number does not change much from species to species, being close to one for all chemical species. Hence, it is common to assume Le = 1 for all chemical species, especially for turbulent flows. In some problems, radiative heat losses are small compared to the heat release rate, especially in small combustion chambers and for small concentration. It is then assumed that With the above simplifications, the energy equation is written as Conservation equations Combustion

Energy conservation equation In the energy equation the temperature may be taken as the dependent variable. Neglecting the Dufour effect, the following equation holds for a mixture of ideal gases The third term of the right hand side is generally small, and is identically zero if the specific heats of all species are assumed to be equal. If that term is neglected, as well as the viscous dissipation term, and the spatial gradients of cp, the following equation is obtained: Conservation equations Combustion

Conserved scalar Conservation equations Consider the following global, single-step, irreversible reaction: Fuel + oxygen  products The conservation equations for fuel, oxygen and products mass fractions are given by: s: mass of oxygen required to fully burn one unit of mass of fuel under stoichiometric conditions Conservation equations Combustion

Conserved scalar Conservation equations Assuming that the mass diffusion coefficients of all species are equal, i.e., , it follows that the scalars defined as satisfy the following transport equation, without source term: Scalars that are not influenced by chemical reaction, but only by convective and diffusive transport, satisfying a transport equation without source term, are referred to as conserved scalars, strictly conserved scalars, passive scalars or Shvab-Zeldovich functions Conservation equations Combustion

Mixture fraction Conservation equations Both the fuel and the oxidizer may include inert species. In that case, yfu denotes the mass fraction of pure fuel, and s is the ratio of mass of oxygen to the mass of pure fuel in stoichiometric conditions. In a reactive mixture between two homogeneous but distinct streams, namely a fuel stream denoted by subscript 1 and an oxidizer stream identified by subscript 2, the mixture fraction is defined as the element mass fraction of matter originating from the fuel stream The mixture fraction ranges from 0 (in the case only oxidant is present) to 1 (in case only fuel is present), and it is a conserved scalar Conservation equations Combustion

Mixture fraction Conservation equations Any conserved scalar b may be related to mixture fraction as follows: b = Z b1 + (1-Z) b2 Consider again the global, single-step, irreversible reaction between a fuel and an oxidizer, and take the conserved scalar bFO. Then, and, since For stoichiometric conditions, , yielding: Conservation equations Combustion

Mixture fraction Conservation equations Z < Zesteq - fuel-lean mixture Z > Zesteq - fuel-rich mixture Relation between mixture fraction and equivalence ratio: The specific enthalpy is a conserved scalar when the assumptions leading to the following energy conservation equation hold: Conservation equations Combustion