Shvab–Zeldovich formulation

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

${\displaystyle \sum _{i=1}^{N}\nu _{i}'\Re _{i}\rightarrow \sum _{i=1}^{N}\nu _{i}''\Re _{i}}$

where ${\displaystyle \Re _{i}}$ is the ith chemical species of the total ${\displaystyle N}$ species and ${\displaystyle \nu _{i}'}$ and ${\displaystyle \nu _{i}''}$ 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 ${\displaystyle \omega }$ is constant and given by

${\displaystyle \omega ={\frac {w_{i}}{W_{i}(\nu _{i}''-\nu _{i}')}}}$

where ${\displaystyle w_{i}}$ is the mass of species i produced or consumed per unit volume and ${\displaystyle W_{i}}$ is the molecular weight of species i.

The main approximation involved in Shvab-Zeldovich formulation is that all binary diffusion coefficients ${\displaystyle D}$ 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[3]

${\displaystyle \alpha _{i}=Y_{i}/[W_{i}(\nu _{i}''-\nu _{i}')]\quad {\text{and}}\quad \alpha _{T}={\frac {\int _{T_{ref}}^{T}c_{p}\,\mathrm {d} T}{\sum _{i=1}^{N}h_{i}^{0}W_{i}(\nu _{i}'-\nu _{i}'')}}}$

where ${\displaystyle Y_{i}}$ is the mass fraction of species i, ${\displaystyle c_{p}=\sum _{i=1}^{N}Y_{i}c_{p,i}}$ is the specific heat at constant pressure of the mixture, ${\displaystyle T}$ is the temperature and ${\displaystyle h_{i}^{0}}$ is the formation enthalpy of species i, reduce to

${\displaystyle {\begin{aligned}\nabla \cdot [\rho {\boldsymbol {v}}\alpha _{i}-\rho D\nabla \alpha _{i}]=\omega ,\\\nabla \cdot [\rho {\boldsymbol {v}}\alpha _{T}-\rho D\nabla \alpha _{T}]=\omega \end{aligned}}}$

where ${\displaystyle \rho }$ is the gas density and ${\displaystyle {\boldsymbol {v}}}$ is the flow velocity. The above set of ${\displaystyle N+1}$ nonlinear equations, expressed in a common form, can be replaced with ${\displaystyle N}$ linear equations and one nonlinear equation. Suppose the nonlinear equation corresponds to ${\displaystyle \alpha _{1}}$ so that

${\displaystyle \nabla \cdot [\rho {\boldsymbol {v}}\alpha _{1}-\rho D\nabla \alpha _{1}]=\omega }$

then by defining the linear combinations ${\displaystyle \beta _{T}=\alpha _{T}-\alpha _{1}}$ and ${\displaystyle \beta _{i}=\alpha _{i}-\alpha _{1}}$ with ${\displaystyle i\neq 1}$, the remaining ${\displaystyle N}$ governing equations required become

${\displaystyle {\begin{aligned}\nabla \cdot [\rho {\boldsymbol {v}}\beta _{i}-\rho D\nabla \beta _{i}]=0,\\\nabla \cdot [\rho {\boldsymbol {v}}\beta _{T}-\rho D\nabla \beta _{T}]=0.\end{aligned}}}$

The linear combinations automatically removes the nonlinear reaction term in the above ${\displaystyle N}$ equations.