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We assume the mixture fraction field to be given in the flow field as a function of time by the solution of (3.23). Then the surface of the stoichiometric mixture

3.4 Nonequilibrium flames 179 can be determined from

Z (x, t) = Zst. (3.28)

Since the temperature is the highest in the vicinity of that surface, combustion chemistry is fast. Therefore there will be a thin reaction zone in the vicinity of the surface of stoichiometric mixture, where the fuel and the oxygen are depleted and radicals and products are formed. In that region also the highest temperatures will occur.

In Section 1.11, the location of the flame surface was defined as that of the inner layer. For simplicity, we approximate this by the iso-surface of the stoichiometric mixture. Then Equation (3.28) defines the flame surface in non-premixed combustion in a similar way in terms of the mixture fraction field as G(x, t) = G0 defines the flame surface in premixed combustion in terms of the G-field. An important difference, however, is that the mixture fraction Z is well defined in the entire flow field, while G is not.

The surface of the stoichiometric mixture is shown schematically in Figure 3.3 for a laminar candle flame. The flow entraining the air into the flame is driven by buoyancy rather than by forced convection as in a jet flame.

The paraffin of the candle first melts because of the radiative heat flux received from the flame; it mounts by capillary forces into the wick where it then evapo-rates to become paraffin vapor, a gaseous fuel. As the fuel vapor is transported toward the surface of the stoichiometric mixture it heats up and begins to py-rolize and to form soot under fuel rich conditions. Soot particles strongly radiate at a typical yellow color. This region is also shown in Figure 3.3. The soot par-ticles are convected to the surface of the stoichiometric mixture where they are

Z = Z_st

air air yellow luminous region

of soot radiation

thin blue layer of chemiluminescent radiation dark region of

evaporated fuel

Figure 3.3. The candle flame as the classical example of a laminar diffusion flame.

abruptly oxidized within the thin reaction zone, mainly by OH radicals, which have a strong maximum concentration in that zone. Therefore there is typically a sharp boundary to the yellow luminous region in the laminar candle flame. If, however, flame stretch or radiative heat loss causes the temperature of the reac-tion zone to fall below a temperature of typically 1,300 K, which corresponds to the crossover temperature of chain-branching and chain-breaking reactions, identified as the inner layer temperature in the footnote in Section 1.6, soot particles can break through the surface of the stoichiometric mixture, generat-ing a sootgenerat-ing candle flame. Kent and Wagner (1984), for instance, found that soot was emitted from gaseous laminar diffusion flames if the particle tem-perature dropped below about 1,300 K. At the leading edge of the surface of the stoichiometric mixture shown in Figure 3.3 there appears a thin blue layer, which is caused by chemiluminiscence, mainly of CH and C^∗₂ radicals. This provides visual evidence of the existence of a thin reaction zone in a candle flame.

As was already discussed in Section 1.11, an important quantity in non-premixed combustion is the instantaneous scalar dissipation rate defined by

χ = 2D|∇ Z|². (3.29)

It has the dimension 1/s and may be interpreted as the inverse of a characteristic diffusion time. Once the solution of (3.23) is known,χ can be calculated at each location in the flow field. The instantaneous local value of the scalar dissipation rate at stoichiometric mixture will be denoted by

χst= 2Dst|∇ Z|²_st. (3.30)

Two Example Solutions of the Mixture Fraction Equation

Example 1: The Counterflow Diffusion Flame. The counterflow geometry is very often used in experimental and numerical studies of diffusion flames because it leads to an essentially one-dimensional diffusion flame structure.

Figure 3.4 shows a flame that has been established between an oxidizer stream from above and a gaseous fuel stream from below.

We consider a steady two-dimensional (planar or axially symmetric) coun-terflow configuration. There exists an exact solution in terms of a similarity coordinate, which results in a set of one-dimensional equations (cf. for instance Dixon-Lewis et al., 1984 or Peters and Kee, 1987). Here we will not introduce a similarity coordinate, but use the y coordinate directly. We introduce the ansatz

3.4 Nonequilibrium flames 181

Figure 3.4. A schematic illustration of the experimental configuration for counterflow diffusion flames for gaseous fuels.

u= U x to obtain the following governing equations:

Continuity

Here j= 0 applies for the planar configuration and j = 1 for the axially sym-metric configuration. The velocity in the y direction is denoted byv and the gradient of the velocity u in the x direction by U . The parameter P represents the axial pressure gradient and is related to the strain rate a by

P = ρ_∞a². (3.35)

The inverse of the strain rate a is the characteristic time scale of the problem.

It is prescribed if two potential flows coming from y = +∞ and y = −∞

are considered. If, however, the oxidizer and the fuel streams issue from a counterflow burner closer to the flame, nonslip boundary conditions U= 0 must be imposed on both sides. This is called the plug flow configuration. The continuity and momentum equations, forming a third-order system of ordinary differential equations, must then satisfy four boundary conditions. This is only possible if P and consequently the strain rate a is calculated as an eigenvalue of the problem.

As an introduction to the flamelet concept to be presented in Sections 3.11–

3.13, it is useful to show that, for a one-dimensional system, flamelet equations can easily be derived by replacing the spatial coordinate y by the mixture fraction as a new independent variable. Introducing the transformation

d For simplicity we will consider here the special case Di = D, leaving the more general case Di = D, known as differential diffusion, to Section 3.12. Then it is immediately seen that in view of (3.33) the term on the l.h.s. in (3.37) and the first term on the r.h.s. cancel. One then obtains the steady state flamelet equation

which is equivalent to (1.139) for Lei = 1, since the term in front of the second derivative can be identified asρχ/2. For a one-dimensional system the scalar dissipation rateχ is a function of the spatial coordinate y, which again can be replaced by the mixture fraction coordinate Z , by inverting the solution Z (y), as will be shown in the following.

If one assumes that the flow velocities of both streams are sufficiently large and sufficiently removed from the stagnation plane, the flame is embedded between two potential flows, one coming from the oxidizer and one from the fuel side. Prescribing the potential flow velocity gradient in the oxidizer stream by a= −∂v∞/∂y, the boundary conditions in the oxidizer stream are

y→ ∞: v_∞= −ay, U_∞= a. (3.39)

3.4 Nonequilibrium flames 183 Equal stagnation point pressure for both streams requires the boundary condi-tions in the fuel stream to be

y→ −∞: v_−∞= −(ρ_∞/ρ_−∞)^1/2ay,

U_∞= aρ_∞/ρ_−∞. (3.40)

By definition, the boundary conditions of the mixture fraction equation are Z = 0 for y → +∞ and Z = 1 for y → −∞.

An integral of the mixture fraction equation may be obtained by separation of variables. If Z^denotesρ D d Z/dy, (3.33) may be transformed to d(lnZ^)/dy = v/D. Integrating this twice leads to

Z = c1I (y)+ c2, (3.41)

and c₁ and c₂are constants of integration. Applying the boundary conditions Z = 0 at y → ∞ and Z = 1 at y → −∞ one obtains

Z = I (∞) − I (y)

I (∞) − I (−∞). (3.43)

An approximate solution is obtained by assuming a constant density in the momentum equation. Then U = a satisfies (3.32) and the integral I (y) becomes

I (y)=

0 ρ dy. An often used approximation in analytic studies of laminar flames is to setρ²D= ρ²_∞D_∞= const. This approximation is sometimes called the Chapman gas approximation. It is justified by the observation that D varies with the temperature typically as T^1.7, whileρ is inversely proportional to the temperature. With this approximation the nondimensional coordinate

η =

may be introduced and one obtains from (3.43) with (3.44) the solution Z =1

2erfc(η/√

2), (3.46)