Lecture 29: Review for Comprehensive Final Examination

1. Lecture 29: Review for Comprehensive Final Examination Final Examination on April 29, 1-3 pm, ME2061 Topics covered since Examination 2

2. Lecture 21: Simplified Descriptions of Laminar Diffusion Flames • Simplified Descriptions of Laminar Diffusion Flames • Burke-Schuman simplified description • Roper solution • Constant density solution • Variable density approximate solution by Fay • Numerical solutions

3. Laminar Jet Diffusion Flames (Non-premixedJet Flames)

4. Simplified Theoretical Description of Laminar Jet Diffusion Flame • Assume: 1. Laminar, steady, axisymmetric flow, vertical flame axis, axial diffusion is neglected 2. Equal diffusivity, unity Lewis number, conserved scalar approximation 3. Radiation heat transfer treated using radiation heat loss fraction 4. Pressure gradient assumed to be hydrostatic

5. Conservation Equations: Cylindrical Coordinates, Thin Flame Conservation of Mass Conservation of Axial Momentum Conservation of Species Mass Fractions Conservation of Energy Constitutive Relationships: Ideal Gas Law, Lewis Number etc.

6. Conserved Scalar Equations for Laminar Jet Flame • Boundary Conditions At the jet exit plane Count Unknowns:

7. Non-dimensional Laminar Jet Diffusion Flame • A dimensionless enthalpy is defined: • The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f. • Heat Loss Fraction

8. Description of Laminar Flame State-relationships • Laminar Flame State-Relationships imply that all species concentrations are solely functions of the mixture fraction. • These functions can be determined from experiments involving careful laminar flame measurements that are yield plots of species mass fractions or species mole fractions as functions of mixture fraction. • Laminar flamelet state relationships do not assume that the chemistry is fast. All that they assume is that the reaction rates are known and are functions solely of mixture fraction. • Once the reaction rates are defined solely as a function of mixture fraction, the capability to have transient processes is lost. • This capability can be partially restored by defining the transient combustion processes to occur between one state relationship to the other.

9. 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

10. 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.

11. 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.

12. 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.

13. 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.

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

15. Characterization in terms of mean and fluctuating properties Introduction (Brief) to Turbulent Flow • Length Scales • Characteristic width of the flow, the macroscale: L • Integral scale or Taylor macroscale: ℓ0Mean size of the large eddies in a flow. Distance between two points in the flow where the correlation between the fluctuating velocities at two location goes to zero.

16. Characterization in terms of mean and fluctuating properties Introduction (Brief) to Turbulent Flow R(r,t) 1 0 r Buoyant Flame Movies

17. Introduction (Brief) to Turbulent Flow • Length Scales (cont.) • Kolmogorov microscale: ℓK The scale at which molecular dissipation effects are important. Turbulent kinetic energy is dissipated into fluid internal energy at this scale.

18. Introduction (Brief) to Turbulent Flow • At the Kolmogorov microscale, the time for an eddy of size ℓK to rotate is equal to the momentum diffusion time across the eddy. • Different scales can be related by the turbulence Reynolds numbers:

19. Introduction (Brief) to Turbulent Flow • Eddy break-down occurs through interaction (deforming, folding, squeezing) between eddies of multiple scales • Large eddies feed energy to the smaller eddies and so on until viscous dissipation • The inertial subrange is characterized by a turbulent flow that is roughly isotropic, homogeneous and inviscid Eddies are dissipated into heat at the Kolmogorov length scale Eddies are generated at the Taylor length scale Energy input (Taylor Scale) Inertial subrange (Integral Scale) Viscous dissipation (Kolmogorov Scale) Energy Frequency or wave-number (1/L)

20. Effect of Turbulence on the Rates of Chemical Reactions • Consider a bimolecular reaction between species A and B. At some point in the flow field we measure the mass fractions of A and B and the temperature T. Are mean measurements adequate for determining the reaction rate between the species?

21. Effect of Turbulence on the Rates of Chemical Reactions • The mean reaction rate is given by:

22. Effect of Turbulence on the Rates of Chemical Reactions • What is the mean value of k? • Example from Warnatz, Maas, and Dibble, Combustion: Sinusoidal variation of temperature.

23. Effect of Turbulence on the Rates of Chemical Reactions T (K) 2000 Temporal dependence of T: 1250 500 t

24. Combusting Drop: Combined Evaporation and Burning Known: Unknown: Assume: , no dissolved gases in liquid. Liquid: r < rs Inner Region: rs < r < rf Outer Region: rf < r < “∞” YOX~Yox,∞ T~T∞, Free Stream: rs Ts Tf rf YF,s Conservation of Mass:

25. Combusting Droplet: Energy Conservation • For constant specific heat, spherical coordinates, Le = 1, get same energy equation as for evaporating droplet in both inner and outer regions:

26. Combusting Droplet: Energy Conservation Solving the conservation of energy equation in a similar manner as the conservation of fuel species equation and applying the boundary conditions: Inner Region: Outer Region:

27. Combusting Droplet: Energy Balance It can be verified that these relations fulfill the temperature boundary conditions: Energy Balances: At this point assume that TsTboiland that all heat conducted into the droplet goes into vaporizing liquid. This simplifies the energy balance at the droplet surface. At the droplet surface:

28. Combusting Droplet: Energy Balance • Substituting for dT/dr gives a relation between • Energy Balance at the flame sheet:

29. Combusting Droplet: Energy Balance Energy Balance at the flame sheet:

30. Combusting Droplet: Energy Balance • Evaluating the temperature gradients on each side for the flame sheet and rearranging we obtain: • This is our fourth equation involving the unknowns. • The Clausius-Clapeyron relation is generally used to relate Tsand :

31. Combusting Droplet: The D2 Law • Solving Eqns. 2,3,4:

32. Combusting Droplet: The D2 Law • Using transient mass loss analysis, D2law can be derived: • The droplet lifetime is the time it takes for the droplet to evaporate completely: