ME 525: Combustion Lecture 27: Carbon Particle Combustion

1. ME 525: CombustionLecture 27: Carbon Particle Combustion • Carbon surface reactions. • One-film model. • Two-film model.

2. Combusting Carbon Particle: Surface Reactions

3. Combusting Carbon Particle: One-Film Model • Temperature and Species Profiles, One-Film Model

4. Combusting Carbon Particle: One-Film Model Assume: • 1. Particle burns in quiescent, infinite medium containing O2 and inert N2 initially. • 2. Burning process is quasi-steady. • 3. Reaction 1 is dominant at the surface (not a good assumption). • Particle has uniform temperature, radiates as a gray body. • No diffusion of gas-phase species into particle. • Le = 1 in the gas phase • kg, cPg, Dg, rg are constants evaluated at some mean temperature.

5. Combusting Carbon Particle: One-Film Model rs r

6. Combusting Carbon Particle: One-Film Model • Following the same procedures as for the evaporating droplet :

7. Combusting Carbon Particle: One-Film Model • From a consideration of the chemical kinetics at the carbon particle surface we obtain: YO2,s Ts T∞ YCO2,s YO2,∞ rs

8. Combusting Carbon Particle: One-Film Model, Diffusion and Kinetic Resistances • Burning of the carbon particle is controlled by matching of the reaction rate and diffusion rate of oxygen to the surface. • To analyze these effects we develop expressions for the kinetic and diffusive resistances

9. Combusting Carbon Particle: One-Film Model, Diffusion and Kinetic Resistances • The kinetic resistance is : • For the diffusive resistance, note that the typical value of the transfer number BO,m is typically <<1 [ln (1+x) ≈ x for x<<1]:

10. Combusting Carbon Particle: One-Film Model, Diffusion and Kinetic Resistances • Combining these two relations:

11. Combusting Carbon Particle: Two-Film Model • Temperature and Species Profiles, Two-Film Model

12. Combusting Carbon Particle: Two-Film Model Assume: • 1. Particle burns in quiescent, infinite medium containing O2 and inert N2 initially. • 2. Burning process is quasi-steady. • 3. Reaction 3 is dominant at the surface. • 4. Particle is surrounded by a flame sheet. • 5. At the flame sheet, CO reacts in stoichiometric proportion with O2 to produce CO2. • Particle has uniform temperature, radiates as gray body. • No diffusion of gas-phase species into particle. • Le = 1 in the gas phase • 9. kg, cPg, Dg, rg are constants evaluated at some mean temperature

13. Combusting Carbon Particle: Two-Film Model rs rf

14. Combusting Carbon Particle: Two-Film Model • At the particle surface:

15. Combusting Carbon Particle: Two-Film Model • At the flame sheet:

16. Combusting Carbon Particle: Two-Film Model In this problem analysis: Knowns: Unknowns: Tf YCO2,s Ts YI,f T∞ YCO2,f YO2,∞ rs rf YI,∞

17. Combusting Carbon Particle: Two-Film Model • Regarding the surface temperature Ts and flame temperature Tf as known quantities for the moment, we obtain the following four relations by considering the species balances:

18. Combusting Carbon Particle: Two-Film Model Tf YCO2,f YCO2,s Rearranging these expressions gives us a relation between the burning rate and Ts YI,f T∞ rs rf YO2,∞ YI,∞

19. Combusting Carbon Particle: Two-Film Model • Another relation between and is provided by the chemical kinetic equation for the surface reaction:

20. Combusting Carbon Particle: Two-Film Model • The rate coefficient k3 is given by • For reaction 3 the enthalpy of combustion is given by

21. Combusting Carbon Particle: Two-Film Model The enthalpy of combustion is negative - the reaction is endothermic! The energy needed to drive the reaction comes from the reaction of CO and O2 at the flame sheet. Tf YCO2,s Ts YI,f YCO2,f T∞ rs rf YO2,∞ YI,∞

22. Combusting Carbon Particle: Two-Film Model • Often assumed that the particle burning process is in the diffusion-controlled regime: • In the diffusion-controlled regime, surface chemical kinetics are fast compared to diffusion times and the CO2 reacts as soon as it gets to the particle surface. • For burning carbon particles, the process will be diffusion controlled at high pressures, when the surface temperature is high, and/or when the particle is large.

23. Combusting Carbon Particle: Two-Film Model • Up to now we have assumed that the surface and flame temperatures are known. Two more equations to determine these temperatures are provided by energy balances at the flame sheet and at the particle surface. Solution for the whole problem is found by iteration in the quantities

24. Combusting Carbon Particle: Two-Film Model • At the particle surface: rs

25. Combusting Carbon Particle: Two-Film Model • The temperature gradient is found from the same temp profile expression that we obtained for the burning liquid droplet:

26. Combusting Carbon Particle: Two-Film Model • The energy equation at the surface becomes:

27. Combusting Carbon Particle: Two-Film Model, Diffusion and Kinetic Resistances • We can develop expressions for the diffusive and kinetic resistances for the two-film model in analogy with the one-film model. Recall the two expressions that we developed previously for the burning rate:

28. Combusting Carbon Particle: Two-Film Model, Diffusion and Kinetic Resistances • The kinetic resistance is found immediately: • For the diffusive resistance, we note that the typical value of the transfer number BCO2,m is typically 0.1-0.2:

29. Combusting Carbon Particle: Two-Film Model, Diffusion and Kinetic Resistances • The diffusive resistance is thus given by: