Turbulent Combustion, Conservation Equations Closure

1. Turbulent Combustion, Conservation Equations Closure • Turbulence and its effects on mass, momentum, species and energy transport • Practical devices involve turbulent flows specifically promoted by the design engineer to obtain efficient mixing, preheating and volumetric reaction rates. • Practical device sizes and speeds desired by human beings and their environment automatically lead to turbulent flows. • First principle solution of turbulence and turbulent combustion is an unsolved grand challenge problem. • Turbulent Non-premixed Flames • Length and time scales of turbulence and their influence on combustion • Turbulent Premixed Flames

2. Turbulence • See Figure 11.1 pg. 429 of Turns. vXis shown as a • function of time. Imagine that in a three dimensional flow vYand vZwould also fluctuate with time but together the three components and density must satisfy the conservation of mass equation and the equation of state. • Now consider the simpler problem of a two dimensional boundary layer over a flat plate (see pp. 438-449 of Turns). • Eq. 11.13 is the axial momentum equation with term (1) : transient mass flux in the axial direction, (2) : advection of axial mass flux by the axial velocity, term (3): advection of axial mass flux by the radial velocity, and term (4): the effect of molecular viscosity on the axial momentum. 3/20

3. Turbulence • Equation 11.14 splits the axial velocity into its mean and fluctuating components but does not recognize the fluctuations in density that are omnipresent in combustion. Therefore, equation 11.14 is highly limiting. • In class, we will derive a version that recognizes the fluctuating density and the fact that its product with fluctuating velocity may not necessarily have a zero mean (If you get a chance, practice this exercise on your own). • Ignoring the density fluctuations and defining a stationary flow (time derivative of mean velocity is zero), the highly limiting equation 11.14 results.

4. Turbulence Take averages and substitute zero and non zero values In variable density flows, density weighted or Favre averaging is used to avoid the Complications arising from correlations involving density fluctuations.

5. Turbulence • The second and the third term of equation 11.13 upon separation of stream-wise (x) and cross-stream (y) velocity components into their mean and fluctuating values and averaging yield terms that involve gradients of non-zero quantities called “Reynolds Stresses.” • Reynolds Stress involving stream-wise velocity fluctuations multiplied by themselves will generally have positive and negative values cancelling each other in magnitude and therefore “tau-xx” defined in equation 11.18a is negligible compared to “tau-xy” defined in equation 11.18b resulting in equation 11.19. Turns mentions the nine Reynolds stress components out of which only one is retained in an axisymmetric jet mixing problem.

6. Turbulence Continuity equation involving average density and average velocity has a source Coupling between conservation of mass and conservation of energy

7. Turbulence and Combustion • Equation 11.20 describes momentum transfer with an axisymmetric (fuel or premixed fuel and air) jet problem with boundary conditions allowing for a co-flow of air or stagnant atmospheric air or “stagnant,” products surrounding the axisymmetric jet. • For describing any reacting flow (premixed or non-premixed) corresponding conservation of mass, species and energy equations are necessary. • The conservation of mass equation results in transient of mean density being driven by divergence of mean mass flux and the divergence of fluctuating mass flux represented by the density-velocity correlation. The density-velocity correlation is a very important quantity in turbulent combustion studied over a couple of decades with models like Bray Moss Libby (BML) model covered in ME 625. 6/20

8. Turbulence and Combustion • The conservation of species equation results in the transient of mean mass fraction of every species “i” being driven by the mean and fluctuating advection flux, mean and fluctuating diffusion flux, and mean and fluctuating components of the reaction rate. • The conservation of energy equation results in the transient of the mean chemical plus sensible energy being driven by the mean and fluctuating advection, mean and fluctuating thermal conduction/diffusion flux, and mean and fluctuating radiation heat loss. • Fluctuating temperature and species concentrations lead to the mean reaction rate and the mean radiation loss being different than the quantities calculated based on mean temperature and mean species concentrations. • The significance of these effects is still discussed. 7/20

9. Non-premixed Jet Flame Heights:Turbulent Flame Analog • For a laminar jet, the flame length is determined by the time that it takes for the co-flow fluid to diffuse to the jet centerline: • The mean squared displacement of the co-flow molecules due to turbulent diffusion during that time must be equal to the square of the jet radius: 8/20

10. Non-premixed Jet Flame Heights • For turbulent flows, the molecular viscosity is replaced by the turbulent eddy viscosity (see discussion of turbulent mixing length, Turns, p. 427-450): 9/20

11. Non-premixed Turbulent Jet Flame Heights:Effects of Buoyancy • For buoyant turbulent flames, relative importance of initial jet momentum and buoyancy is characterized by Froude number: where: 10/20

12. Nonpremixed Turbulent Jet Flame Heights: Kalghati Correlations

13. Nonpremixed Turbulent Jet Flame Heights: Kalghatgi Correlations • The dimensionless flame length can be correlated with the Froude number:

14. Non-premixed Turbulent Jet Flame Radiation • Non-premixed flames are often used in furnaces and the radiation flux emitted to a load is an important design parameter. • The radiation heat flux normal to a flame enclosing load integrated over the load surface area yields the energy received by the load that can be divided by the fuel flow rate multiplied by the heating value of the fuel (see denominator of equation 13.13) to yield the radiant fraction (see LHS of equation 13.13). • The energy received by the load is equal to the net energy emitted by the flame and this energy is used to define the Planck mean absorption coefficient, effective radiation temperature and effective radiating volume of the flame in equations 13.14 and 13.15.

15. Non-premixed Turbulent Jet Flame Liftoff and Blowout: Kalghatgi Correlations • Kalghatgi view of turbulent jet diffusion flame stabilization: x = h = liftoff height u(r) umax(x=h) rstoich rstoich Oxidizer Oxidizer Fuel ST = u(rstoich) ST r Rich Limit Lean Limit Rich Limit Lean Limit

16. Non-premixed Turbulent Jet Flame Liftoff : Kalghatgi Correlation • For flame liftoff, Kalghatgi (CST 41, 17-29, 1984) has developed the following correlation: where h = flame liftoff height

17. Non-premixed Turbulent Jet Flame Blowout: Kalghatgi Correlation • For flame blowout, Kalghatgi (CST 26, 233-239, 1981) proposes the following correlation: where

18. Non-premixed Turbulent Jet Flame Blowout: Kalghatgi Correlation • From isothermal turbulent jet mixing theory, H is the distance along the jet centerline where the mixture fraction drops to the stoichiometric value fstoich. Flame blowout occurs when the liftoff height h is approximately equal to 0.7H. • Also see Pitts (Combust. Flame 76, 197-212, 1989).

19. Turbulent Premixed Flames • Laminar premixed flames are limited by the amount of fuel + air mixture they can burn per unit area. Turbulence increases the area over which the mixture can be burnt by wrinkling the flame. • The increase in the mass burning rate by flame wrinkling leads to increase in the volumetric combustion rate even if the flame speed remains fixed at the value given by the laminar flame speed. • The increase in flame surface area per unit volume is one way to keep track of the effect of turbulence on the premixed flame fuel consumption and heat release rates. • Another way is to define an equivalent turbulent flame speed over a fixed mean surface area (see equation 12.1 on page 457 and Example 12.1 on page 458).

20. Turbulent Premixed Flame Regimes • Premixed jet flames wrinkled by turbulence (see Figure 12.6 page 459) define a flame brush. • In spark ignition engine, a spark kernel may be small and laminar at the time of ignition but may grow to be turbulent and have significantly higher surface area than that of an equivalent spherical surface and may lead to very high propagation speed allowing high engine RPM. • Visually compare the mass burned by the flame in the first 10 pictures and the last ten pictures in Figure 12.7. At higher speeds the wrinkling of the flame may finally lead to breaking into many spherical wrinkled pockets that burn with even higher overall speed and lead to complete combustion.

21. Turbulent Premixed Flame Regimes • Flame thickness less than Kolmogorov scale leads to wrinkled laminar flame regime. (Reactions are very fast and the flame is so thin that any fluctuation in local properties does not affect the flames effects on the reactants, its speed, its temperature etc.) • Flame thickness greater than Kolmogorov scale but less than integral scale leads to flamelets in eddies. Flame thickness is greater than the scales of the smallest fluctuations but less than the scale of the larger fluctuations. The preheating and reaction rate processes inside of the flame are affected by the scalar fluctuations and a correction to the overall volumetric reaction rate is needed on the basis of the effects of these fluctuations. • Flame thickness greater than integral scale leads to distributed reactions (Reactions are very slow)