Materials and methods
1. Therapeutic NHEJ-mediated Gene Editing in Fanconi Anemia A Lymphoblastic Cell Lines
A very important quantity for the description of nonpremixed combustion is the mixture fraction Z (sometimes also denoted by f or ξ) which plays a role similar to that of the scalar G in premixed combustion in determining the flame surface. Therefore, before going into the description of current modeling approaches, we will present various definitions of the mixture fraction.
The definition of the mixture fraction is best derived for a homogeneous system in the absence of diffusion. Then, by writing the global reaction equation for complete combustion of a hydrocarbon fuel CmHn, for instance, as
νF CmHn+ νO2O2→ νCO 2CO2+ νH2OH2O (3.1) one defines the stoichiometric coefficients of fuel and oxygen asνFandνO2= (m+ n/4)νF, respectively. The reaction equation relates the changes of mass fractions of oxygen dYO2and fuel dYFto each other by
dYO2
νO2WO2 = dYF
νFWF, (3.2)
where the Wis are the molecular weights. For a homogeneous system this equation may be integrated to obtain
νYF− YO2 = νYF,u− YO2,u, (3.3) whereν = νO2WO2/νFWFis the stoichiometric oxygen-to-fuel mass ratio and the subscript u denotes the initial conditions in the unburnt mixture. The mass fractions YFand YO2correspond to any state of combustion between the unburnt and the burnt state. If the diffusivities of fuel and oxidizer are equal, (3.3) is also valid for spatially inhomogeneous systems such as a diffusion flame.
A mixture is called stoichiometric if the fuel-to-oxygen ratio is such that both fuel and oxygen are entirely consumed after combustion to CO2and H2O
3.2 The mixture fraction variable 173 is completed. The condition for a stoichiometric mixture requires that the ratio of the concentrations [Xi]= ρYi/Wiof oxygen and fuel in the unburnt mixture is equal to the ratio of the stoichiometric coefficients:
[XO2]u
or in terms of mass fractions YO2,u
In a two-feed system,† where subscript 1 denotes the fuel stream with mass flux ˙m1and subscript 2 denotes the oxidizer stream with mass flux ˙m2into the system, the mixture fraction Z is defined at any location in the system as the local ratio of the mass flux originating from the fuel feed to the sum of both mass fluxes:
Z = m˙1
˙ m1+ ˙m2
. (3.6)
Both fuel and oxidizer streams may contain inert substances such as nitrogen.
If the system is homogeneous or if equal diffusivities of fuel, oxygen, and inert substances are assumed in an inhomogeneous system, the local mass fraction YF,uof fuel in the unburnt mixture is related to the mixture fraction Z as
YF,u= YF,1Z, (3.7)
where YF,1denotes the mass fraction of fuel in the fuel stream. Similarly, since 1− Z represents the mass fraction of the oxidizer stream locally in the unburnt mixture, one obtains the local mass fraction of oxygen as
YO2,u = YO2,2(1− Z), (3.8) where YO2,2 represents the mass fraction of oxygen in the oxidizer stream (YO2,2 = 0.232 for air). Introducing (3.7) and (3.8) into (3.3) and integrat-ing between the unburnt and any other state of combustion, one can relate the mass fractions of fuel and oxygen to the mixture fraction as
Z =νYF− YO2+ YO2,2
νYF,1+ YO2,2 . (3.9)
†If more than two feeds enter into a combustion chamber, the concept of a single mixture fraction can no longer be used. It can formally be extended to multiple mixture fraction variables, but then it becomes less attractive for modeling because the resulting flamelet equations are defined in a multi-dimensional mixture fraction space.
For a stoichiometric mixture the r.h.s of (3.3) vanishes by definition, since both fuel and oxygen are zero, such that
νYF− YO2= 0. (3.10)
Therefore the stoichiometric mixture fraction is Zst=
For pure fuels (YF,1= 1) mixed with air, the stoichiometric mixture fraction is, for instance, 0.0284 for H2, 0.055 for CH4, 0.0635 for C2H4, 0.0601 for C3H8, and 0.072 for C2H2. It is worth keeping in mind that these numbers are all much smaller than unity, which implies that in terms of mass ratios, a large amount of oxidizer is needed to completely consume the fuel.
The mixture fraction can be related to the commonly used equivalence ratio φ, which is defined as the fuel-to-air ratio in the unburnt mixture normalized by that of a stoichiometric mixture:
φ = YF,u/YO2,u
YF,u/YO2,u
st
=ν YF,u
YO2,u. (3.12)
Introducing (3.7) and (3.8) into (3.12) leads with (3.11) to
φ = Z
1− Z
(1− Zst) Zst
. (3.13)
This suggests that the mixture fraction can be interpreted as a normalized fuel-to-air equivalence ratio.
There is a more general way to define the mixture fraction, namely as a quantity related to chemical elements, rather than to the equivalence ratio.
While the mass of chemical species may change due to chemical reactions, the mass of elements is conserved. Denoting ai jas the number of atoms of element j in a molecule of species i , and Wjas the molecular weight of that atom, one may write the mass of all atoms j in the system as
mj =
The mass fraction of element j is then Zj =mj
Adding Equations (1.62) for the mass fractions Yiin the same way as in (3.15)
3.2 The mixture fraction variable 175
where no chemical source term appears, since n
i=1
ai jWiνi k= 0 (3.17)
for each element j in any reaction k. This shows that the element mass fraction is conserved during combustion. If a binary diffusion flux as in (1.63) is used and if all diffusivities are equal, Di = D, the balance equation for the element mass fraction becomes
ρ∂ Zj
∂t + ρv · ∇ Zj = ∇ · (ρ D∇ Zj). (3.18) Let ZC, ZH, and ZOdenote the element mass fractions of C, H, and O, and WC, WH, and WOtheir molecular weights, respectively. Setting, for simplicity of notation, the stoichiometric coefficientνF of the global reaction (3.1) equal to unity, one gets the element mass fractions
ZC
mWC = ZH
nWH =YF,u
WF, ZO= YO2,u. (3.19) From (3.5) it follows that the coupling function
β = ZC
vanishes at stoichiometric conditions. This corresponds to the original definition of Burke and Schumann (1928) of a conserved scalar. It can be normalized to vary between 0 and 1 as
Z = β − β2
β1− β2
(3.21) to obtain Bilger’s (1988) definition of the mixture fraction:
Z = ZC/(mWC)+ ZH/(nWH)+ 2
This formula is often used to determine the mixture fraction from experimental or numerical data of mass fractions that are available. However, in experiments mass fractions of minor species are usually not available and in most numerical
calculations the diffusivities are not all equal to each other. Therefore the defini-tion (3.22) implicitly contains unspecified addidefini-tional influences. Its advantage is, however, that it reduces to (3.11) if the elements C, H, and O are in stoichio-metric proportions.
If all diffusivities Diare equal to D, it follows from (3.18), (3.20), and (3.21) that Z satisfies the balance equation
ρ∂ Z
∂t + ρv · ∇ Z = ∇ · (ρ D∇ Z). (3.23) This equation, together with its boundary conditions Z= 1 in the fuel stream and Z = 0 in the oxidizer stream, can also be postulated, independently of any combinations of element mass fractions, to provide a further and alternative definition of the mixture fraction. The diffusion coefficient D in (3.23) then is arbitrary, but since the maximum temperature determines the location of the reaction zone, diffusion of enthalpy is the most important transport process in mixture fraction space. It has been discussed in Chapter 1 that under certain simplifying conditions the enthalpy equation (1.75) takes the same form as (3.23), where D is the thermal diffusivity. Therefore we choose the thermal diffusivity as the diffusion coefficient in the mixture fraction equation.