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Lecture 27: Droplet Evaporation and Burning

Lecture 27: Droplet Evaporation and Burning. Model for drop evaporation and burning. Evaluation of model constants. Correction factors. Combusting Drop: Combined Evaporation and Burning . Known: Unknown: Assume:. , no dissolved gases in liquid. Liquid: r < r s.

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Lecture 27: Droplet Evaporation and Burning

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  1. Lecture 27: Droplet Evaporation and Burning Model for drop evaporation and burning. Evaluation of model constants. Correction factors.
  2. 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:
  3. Combusting Drop: Combined Evaporation and Burning Assumptions I: 1. Droplet is enveloped by a spherical flame in quiescent, infinite medium 2. Quasi-steady process (droplet radius changes slowly in time) 3. Fuel is single-component, no dissolved gases in droplet (including product gases), no other condensed phases 4. Droplet temperature is uniform: Tdis appropriately less than Tboil 5. Pressure is uniform and constant.
  4. Combusting Drop: Combined Evaporation and Burning Assumptions II: 6. Le = 1 Z = ZF = ZT 7. kg, cPg, and rg are constants evaluated at some mean temperature One step reaction, three species: fuel, product, oxidizer Radiation heat transfer is neglected Buoyancy is neglected At the droplet surface, PF,s = Pvap(Ts)
  5. Combusting Droplet: Species Conservation, Inner Region rs < r < rf •Species Conservation: If species A is the fuel and species B is the product:
  6. Combusting Droplet: Species Conservation, Inner Region rs < r < rf In spherical coordinates: Apply boundary conditions: Inner Region r rs rf Outer Region
  7. Combusting Droplet: Species Conservation, Inner Region rs < r < rf The mass fraction of fuel at the droplet surface follows thermodynamic phase equilibrium. Partial pressure of fuel vapor is equal to the saturation pressure at the surface temperature. Inner Region r rs Outer Region rf The temperature of the liquid at the drop surface is slightly less than the boiling point and allows heat transfer from the gas phase to the liquid surface. The product gases and inert diffuse into the inner region and to the fuel surface. The fuel diffuses outward to the flame.
  8. Combusting Droplet: Species Conservation, Inner Region rs < r < rf Separating variables and integrating: Applying the first boundary condition:
  9. Combusting Droplet: Species Conservation InnerRegion rs < r < rf Applying the second boundary condition yields a relation between: ; applying overall species conservation inside the flame surface yields
  10. Combusting Droplet: Species Conservation, Outer Region r > rf At the flame front: Outer Region rs Inner Region rf
  11. Combusting Droplet: Species Conservation, Outer Region r > rf In the outer region:
  12. Combusting Droplet: Species Conservation, Outer Region r > rf Integrating this expression gives us:
  13. Combusting Droplet: Species Conservation, Outer Region r > rf Apply the flame surface boundary condition: Apply the boundary condition:
  14. 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:
  15. 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:
  16. 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:
  17. Combusting Droplet: Energy Balance Substituting for dT/dr gives a relation between Energy Balance at the flame sheet:
  18. Combusting Droplet: Energy Balance Energy Balance at the flame sheet:
  19. 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.
  20. Combusting Droplet: The D2 Law The Clausius-Clapeyron relation is generally used to relate Tsand :
  21. Combusting Droplet: The D2 Law Solving Eqns. 2,3,4:
  22. 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:
  23. Combusting Droplet: The D2 Law Accuracy of calculated droplet lifetimes depends on property evaluations. The following property evaluation rules gave good results for the burning rates and lifetimes of burning alkane droplets (Law and Williams, Combust. Flame, Vol. 19, 393-406, 1972):
  24. Combusting Droplet: The D2 Law However, the above relations for Tf and rf do not give good agreement with experiment. Law and Williams assume that: Experimentally, it is found that the ratio rf /rs is much less than would be predicted from the above relations. Law and Williams also propose correction factors for finite rate chemical kinetics, buoyancy, and forced convection:
  25. Combusting Droplet: The D2 Law
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