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^[1] and by Yakov Zeldovich in 1949.^[2]

Method

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.

Shvab–Zeldovich–Liñán formulation

Shvab–Zeldovich–Liñán formulation was introduced by Amable Liñán in 1991^[4][5] 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 ${\displaystyle Y_{F}}$ (defined such that it takes a unit value in the fuel stream), oxidizer mass fraction ${\displaystyle Y_{O}}$ (defined such that it takes a unit value in the oxidizer stream) and non-dimensional temperature ${\displaystyle T}$ (measured in units of oxidizer-stream temperature) are given by^[6]

${\displaystyle {\begin{aligned}\rho {\frac {\partial Y_{F}}{\partial t}}+\rho \mathbf {v} \cdot \nabla Y_{F}&={\frac {1}{Le_{F}}}\nabla \cdot (\rho D_{T}\nabla Y_{F})-\omega ,\\\rho {\frac {\partial Y_{O}}{\partial t}}+\rho \mathbf {v} \cdot \nabla Y_{O}&={\frac {1}{Le_{O}}}\nabla \cdot (\rho D_{T}\nabla Y_{O})-S\omega ,\\\rho {\frac {\partial T}{\partial t}}+\rho \mathbf {v} \cdot \nabla T&=\nabla \cdot (\rho D_{T}\nabla T)+q\omega \end{aligned}}}$

where ${\displaystyle \omega =Da\,Y_{F}Y_{O}e^{-E/RT}}$ is the reaction rate, ${\displaystyle Da}$ is the appropriate Damköhler number, ${\displaystyle S}$ is the mass of oxidizer stream required to burn unit mass of fuel stream, ${\displaystyle q}$ is the non-dimensional amount of heat released per unit mass of fuel stream burnt and ${\displaystyle e^{-E/RT}}$ is the Arrhenius exponent. Here, ${\displaystyle Le_{F}}$ and ${\displaystyle Le_{O}}$ are the Lewis number of the fuel and oxygen, respectively and ${\displaystyle D_{T}}$ is the thermal diffusivity. In the Burke–Schumann limit, ${\displaystyle Da\rightarrow \infty }$ leading to the equilibrium condition

${\displaystyle Y_{F}Y_{O}=0}$.

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

${\displaystyle {\begin{aligned}Z={\frac {SY_{F}-Y_{O}+1}{S+1}},&\qquad {\tilde {Z}}={\frac {{\tilde {S}}Y_{F}-Y_{O}+1}{{\tilde {S}}+1}},\\H={\frac {T-T_{0}}{T_{s}-T_{0}}}+Y_{F}+Y_{O}-1,&\qquad {\tilde {H}}={\frac {T-T_{0}}{T_{s}-T_{0}}}+{\frac {Y_{O}}{Le_{O}}}+{\frac {Y_{F}-1}{Le_{F}}}\end{aligned}}}$

where ${\displaystyle {\tilde {S}}=SLe_{O}/Le_{F}}$, ${\displaystyle T_{0}}$ is the fuel-stream temperature and ${\displaystyle T_{s}}$ is the adiabatic flame temperature, both measured in units of oxidizer-stream temperature. Introducing these functions reduces the governing equations to

${\displaystyle {\begin{aligned}\rho {\frac {\partial Z}{\partial t}}+\rho \mathbf {v} \cdot \nabla Z&={\frac {1}{Le_{m}}}\nabla \cdot (\rho D_{T}\nabla {\tilde {Z}}),\\\rho {\frac {\partial H}{\partial t}}+\rho \mathbf {v} \cdot \nabla H&=\nabla \cdot (\rho D_{T}\nabla {\tilde {H}}),\end{aligned}}}$

where ${\displaystyle Le_{m}=Le_{O}(S+1)/({\tilde {S}}+1)}$ is the mean (or, effective) Lewis number. The relationship between ${\displaystyle Z}$ and ${\displaystyle {\tilde {Z}}}$ and between ${\displaystyle H}$ and ${\displaystyle {\tilde {H}}}$ can be derived from the equilibrium condition.

At the stoichiometric surface (the flame surface), both ${\displaystyle Y_{F}}$ and ${\displaystyle Y_{O}}$ are equal to zero, leading to ${\displaystyle Z=Z_{s}=1/(S+1)}$, ${\displaystyle {\tilde {Z}}={\tilde {Z}}_{s}=1/({\tilde {S}}+1)}$, ${\displaystyle H=H_{s}=(T_{f}-T_{0})/(T_{s}-T_{0})-1}$ and ${\displaystyle {\tilde {H}}={\tilde {H}}_{s}=(T_{f}-T_{0})/(T_{s}-T_{0})-1/Le_{F}}$, where ${\displaystyle T_{f}}$ is the flame temperature (measured in units of oxidizer-stream temperature) that is, in general, not equal to ${\displaystyle T_{s}}$ unless ${\displaystyle Le_{F}=Le_{O}=1}$. On the fuel stream, since ${\displaystyle Y_{F}-1=Y_{O}=T-T_{0}=0}$, we have ${\displaystyle Z-1={\tilde {Z}}-1=H={\tilde {H}}=0}$. Similarly, on the oxidizer stream, since ${\displaystyle Y_{F}=Y_{O}-1=T-1=0}$, we have ${\displaystyle Z={\tilde {Z}}=H-(1-T_{0})/(T_{s}-T_{0})={\tilde {H}}-(1-T_{0})/(T_{s}-T_{0})-1/Le_{O}+1/Le_{F}=0}$.

The equilibrium condition defines^[7]

${\displaystyle {\begin{aligned}{\tilde {Z}}<{\tilde {Z}}_{s}:&\qquad Y_{F}=0,\,\,\,Y_{O}=1-{\frac {\tilde {Z}}{{\tilde {Z}}_{s}}}=1-{\frac {Z}{Z_{s}}},\\{\tilde {Z}}>{\tilde {Z}}_{s}:&\qquad Y_{O}=0,\,\,\,Y_{F}={\frac {{\tilde {Z}}-{\tilde {Z}}_{s}}{1-{\tilde {Z}}_{s}}}={\frac {Z-Z_{s}}{1-Z_{s}}}.\end{aligned}}}$

The above relations define the piecewise function ${\displaystyle Z({\tilde {Z}})}$

${\displaystyle Z={\begin{cases}{\tilde {Z}}/Le_{m},\quad {\text{if}}\,\,{\tilde {Z}}<{\tilde {Z}}_{s}\\Z_{s}+Le({\tilde {Z}}-{\tilde {Z}}_{s})/Le_{m},\quad {\text{if}}\,\,{\tilde {Z}}>{\tilde {Z}}_{s}\end{cases}}}$

where ${\displaystyle Le_{m}={\tilde {Z}}_{s}/Z_{s}=(S+1)/(S/Le_{F}+1)}$ is a mean Lewis number. This leads to a nonlinear equation for ${\displaystyle {\tilde {Z}}}$. Since ${\displaystyle H-{\tilde {H}}}$ is only a function of ${\displaystyle Y_{F}}$ and ${\displaystyle Y_{O}}$, the above expressions can be used to define the function ${\displaystyle H({\tilde {Z}},{\tilde {H}})}$

${\displaystyle H={\tilde {H}}+{\begin{cases}(1/Le_{F}-1)-(1/Le_{O}-1)(1-{\tilde {Z}}/{\tilde {Z}}_{s}),\quad {\text{if}}\,\,{\tilde {Z}}<{\tilde {Z}}_{s}\\(1/Le_{F}-1)(1-{\tilde {Z}})/(1-{\tilde {Z}}_{s}),\quad {\text{if}}\,\,{\tilde {Z}}>{\tilde {Z}}_{s}\end{cases}}}$

With appropriate boundary conditions for ${\displaystyle {\tilde {H}}}$, the problem can be solved.

It can be shown that ${\displaystyle {\tilde {Z}}}$ and ${\displaystyle {\tilde {H}}}$ are conserved scalars, that is, their derivatives are continuous when crossing the reaction sheet, whereas ${\displaystyle Z}$ and ${\displaystyle H}$ have gradient jumps across the flame sheet.

References

1. Shvab, V. A. (1948). Relation between the temperature and velocity fields of the flame of a gas burner. Gos. Energ. Izd., Moscow-Leningrad.
2. Y. B. Zel'dovich, Zhur. Tekhn. Fiz. 19,1199(1949), English translation, NACA Tech. Memo. No. 1296 (1950)
3. Williams, F. A. (2018). Combustion theory. CRC Press.
4. A. Liñán, The structure of diffusion flames, in Fluid Dynamical Aspects of Combustion Theory, M. Onofri and A. Tesei, eds., Harlow, UK. Longman Scientific and Technical, 1991, pp. 11–29
5. Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.
6. Linán, A. (2001). Diffusion-controlled combustion. In Mechanics for a New Mellennium (pp. 487-502). Springer, Dordrecht.
7. Linán, A., Orlandi, P., Verzicco, R., & Higuera, F. J. (1994). Effects of non-unity Lewis numbers in diffusion flames.