Shvab–Zeldovich formulation. The Shvab–Zeldovich formulation is an approach to remove the chemical-source terms from the conservation equations for energy and chemical species by linear combinations of independent variables, when the conservation equations are expressed in a common form. Expressing conservation equations in common form often limits the range of applicability of the formulation. The method was first introduced by V. A. Shvab in 1948 and by Yakov Zeldovich in 1949.

Method
For simplicity, assume combustion takes place in a single global irreversible reaction

where  is the ith chemical species of the total  species and  and  are the stoichiometric coefficients of the reactants and products, respectively. Then, it can be shown from the law of mass action that the rate of moles produced per unit volume of any species  is constant and given by

where  is the mass of species i produced or consumed per unit volume and  is the molecular weight of species i.

The main approximation involved in Shvab–Zeldovich formulation is that all binary diffusion coefficients  of all pairs of species are the same and equal to the thermal diffusivity. In other words, Lewis number of all species are constant and equal to one. This puts a limitation on the range of applicability of the formulation since in reality, except for methane, ethylene, oxygen and some other reactants, Lewis numbers vary significantly from unity. The steady, low Mach number conservation equations for the species and energy in terms of the rescaled independent variables

where  is the mass fraction of species i,  is the specific heat at constant pressure of the mixture,  is the temperature and  is the formation enthalpy of species i, reduce to

where  is the gas density and  is the flow velocity. The above set of  nonlinear equations, expressed in a common form, can be replaced with  linear equations and one nonlinear equation. Suppose the nonlinear equation corresponds to  so that

then by defining the linear combinations  and  with , the remaining  governing equations required become

The linear combinations automatically removes the nonlinear reaction term in the above  equations.

Shvab–Zeldovich–Liñán formulation
Shvab–Zeldovich–Liñán formulation was introduced by Amable Liñán in 1991 for diffusion-flame problems where the chemical time scale is infinitely small (Burke–Schumann limit) so that the flame appears as a thin reaction sheet. The reactants can have Lewis number that is not necessarily equal to one.

Suppose the non-dimensional scalar equations for fuel mass fraction  (defined such that it takes a unit value in the fuel stream), oxidizer mass fraction  (defined such that it takes a unit value in the oxidizer stream) and non-dimensional temperature  (measured in units of oxidizer-stream temperature) are given by

where  is the reaction rate,  is the appropriate Damköhler number,  is the mass of oxidizer stream required to burn unit mass of fuel stream,  is the non-dimensional amount of heat released per unit mass of fuel stream burnt and  is the Arrhenius exponent. Here,  and  are the Lewis number of the fuel and oxygen, respectively and  is the thermal diffusivity. In the Burke–Schumann limit,  leading to the equilibrium condition

.

In this case, the reaction terms on the right-hand side become Dirac delta functions. To solve this problem, Liñán introduced the following functions

where ,  is the fuel-stream temperature and  is the adiabatic flame temperature, both measured in units of oxidizer-stream temperature. Introducing these functions reduces the governing equations to

where  is the mean (or, effective) Lewis number. The relationship between  and  and between  and  can be derived from the equilibrium condition.

At the stoichiometric surface (the flame surface), both  and  are equal to zero, leading to , ,  and , where  is the flame temperature (measured in units of oxidizer-stream temperature) that is, in general, not equal to  unless . On the fuel stream, since , we have . Similarly, on the oxidizer stream, since , we have .

The equilibrium condition defines

The above relations define the piecewise function  

where  is a mean Lewis number. This leads to a nonlinear equation for . Since  is only a function of  and , the above expressions can be used to define the function 

With appropriate boundary conditions for , the problem can be solved.

It can be shown that  and  are conserved scalars, that is, their derivatives are continuous when crossing the reaction sheet, whereas  and  have gradient jumps across the flame sheet.

References

Combustion
Fluid dynamics