On averaging approximations of a poly-disperse fuel spray in the auto-ignition problem

1. On averaging approximations of apoly-disperse fuel spray in the auto-ignition problem V. Bykov Institut für Technische Thermodynamik Universität Karlsruhe (TH), Germany Collaboration: BGU: V. Gol’dshtein, I. Goldfarb Technion: J.B. Greenberg Queens Hotel, Brighton, UK, July 16-18, 2007

2. Overview Motivation: Modelling of poly-disperse spray combustion Parcel approach, a simplified mathematical model and standard analysis Averaging and approximation of the system dynamics Modification, analysis and comparisons Conclusions

3. Cell and parcel approach Lagrange approach is combined with cell method to model and simulate the spray combustion phenomena Parcels are used to replace the actual size distribution of the spray radiuses by its discrete approximation (Example: Ignition of a Cluster of Heptane Droplets by Oliver Desjardins, http://www.stanford.edu/group/pitsch/Research/Multiphase.htm)

4. Simplified model Energy Mass Concentration

5. Non-dimensional form Energy Mass Concentration

6. Reduction Mass conservation equations can be integrated Energy integral Reduced model

7. Preliminary analysis Slow curve on the dimensionless maximal radius/temperature plane is Critical regimes and classification explosion limit of Semenov’s type! Explosive regime Delayed regime

8. Time histories System profiles of the delayed regime

9. Phase plane Typical behavior of delayed regime in projection to the phase plane of dimensionless temperature and maximal radius

10. Approximate mono-disperse model Approximate mono-disperse system Overall volume Equilibrium value Idea: Replace the poly-disperse model by a mono-disperse one in such a way that it reproduces same dynamical scenarios for considered parametric range!

11. Comparative analysis Quantitative similarity of critical regimes: Critical parameters can be applied for construction of a qualitative test of an approximation!

12. Standard definitions Standard parametrical averaging In dimensionless form it reads This modification guaranties the same equilibrium point of both poly-disperse and its approximate mono-disperse models!

13. Standard definitions 1 - full poly-disperse system, 2 – approximating mono-disperse (standard SMD) system

14. Modified averaging Dynamically consistent definition (Auto-ignition of a polydisperse fuel spray, 31st Combustion Symposium, Bykov et. al.) This modification guaranties the same equilibrium point of both poly-disperse and its approximate mono-disperse models!

15. Modified definitions 1 - full poly-disperse system, 2 – approximating mono-disperse (conserved SMD) system D32

16. Further comparisons and thoughts Digits correspond to the averages (conserved): 1 – D31, 2 – D32, 3 – D21, 4 - D20: Depending on the regime different parameters can be used in order to insure the best approximation!

17. Conclusions A poly-disperse spray model in a combustible gas medium has been considered and compared with the dynamics of an “equivalent” mono-disperse spray based on different approximations of an average droplet diameter; The Sauter mean diameter (SMD) has been taken and analyzed for this purpose; Preliminary computed results suggest that the use of the usual SMD-based mono-disperse spray leads to quite a significant over-estimate of the ignition time! An alternative modified definition of the SMD is suggested. It reduces significantly the discrepancy between the ignition time for the poly-disperse spray and that of the equivalent mono-disperse spray. Further studies: Parametric analysis… More realistic models… Extension to models with transport… Algorithms and implementation schemes for DNS…