MATHEMATICAL MODELS OF FUEL SPRAY AUTOIGNITION

1. MATHEMATICAL MODELS OF FUEL SPRAY AUTOIGNITION Vladimir Gol’dshtein Ben-Gurion University of the Negev Department of Mathematics P.O.B. 653, Beer-Sheva 84105 ISRAEL

2. IN COOPERATION WITH • V.BIKOV • J.B.GREENBERG • I.GOLDFARB • D.KATZ • S.SAZHIN • E.SAZHINA UK-Israel workshop Brighton, UK

3. Methodology. • Physico-chemical processes with sufficiently different time scales are generally involved in spray combustion problems. • Existence of a dispersion on time scale leads to specific type of a system of governing equations (singularly perturbed systems) and justifies an application of various asymptotical tools. • A geometrical asymptotic method of integral manifolds, accepted by the authors, permits conduct analytical investigation of the considered systems. UK-Israel workshop Brighton, UK

4. Invariant Manifolds Method Semenov model of self-ignition: Initial conditions: Small parameters (due to the high activation energy) UK-Israel workshop Brighton, UK

5. Invariant Manifolds Method - fastvariable - slow variable The slow surface (curve in two-dimensional case) : Turning points: Fast motion (slow variable does not change): UK-Israel workshop Brighton, UK

6. Trajectory Analysis I1UO - slow trajectory, I1U is afastpart, UO isa slowone (“sitting” on the attractive part of the slow curve OTP). I3V - fast trajectory, it does not approach the slow curve I2TPO - critical trajectory, it moves via turning point T. UK-Israel workshop Brighton, UK

7. Content • Thermal explosion in mono-disperse sprays. • Classification of main regimes. • Long time delay before auto-ignition. • Discussion on poly-disperse spray UK-Israel workshop Brighton, UK

8. Thermal explosionin sprays Main physical assumptions - evaporating fuel droplets in combustible gaseous mixture comprise a mono-disperse spray- droplets are on the saturation line; the mixture is placed in a thermally insulated enclosure (adiabatic approximation)- chemical reaction is modeled as a one step highly exothermic first order reaction UK-Israel workshop Brighton, UK

9. MONODISPERSE SPRAY • Energy and concentration equations • Initial conditions UK-Israel workshop Brighton, UK

10. Dimensionless System • Dimensionless variables are introduced along the lines of classical Semenov’s approach • Dimensionless system reads as UK-Israel workshop Brighton, UK

11. Dimensionless Parameters • 3 dimensionless parameters describe competition between reaction and evaporation processes: y - ratio of reaction heat to latent heat of evaporation e1is defined by competition between combustion and evaporation. e2represents the ratio between the potential energy of the combustible gas and the evaporation energy UK-Israel workshop Brighton, UK

12. Slow Curve (zero approximation of invariant manifold) The slow curve for the current system is given by the equation • The shape and position of the slow curve in the plane depend on the combination of the five parameters of the system. Any combination of the parameters dictates the relative location of the slow curve to the initial point. • Here qis afast, and r is a slowvariable. UK-Israel workshop Brighton, UK

13. Possible Scenarios Delayed trajectory ABTCS Explosive trajectory DES UK-Israel workshop Brighton, UK

14. Possible positions of the slow curve and corresponding trajectories • Delayed trajectory EFTG, the slow part FT belongs to the integral manifold. UK-Israel workshop Brighton, UK

15. Summary of dynamical regimes • Conventional fast explosion Delayed explosion, concentration increases Slow regime Freeze delay Delayed explosion, concentration decreases UK-Israel workshop Brighton, UK

16. Delay Time (Case ) • The suggested type of analysis permits to evaluate the main characteristic times of the process: tc1 - time of fast motion (evaporation and cooling), tc2 - time of slowmotion UK-Israel workshop Brighton, UK

17. Time History:Thermal Explosion with Delay UK-Israel workshop Brighton, UK

18. Theory versus Simulations • Relative error (%) in the delay time upper bound versus the number of droplets per unit volume:(a) n-decane fuel, Cf0 = 10-4 kmol/m3, Rd0 = 10-5 m;(b) tetralin fuel, Cf0 = 10-4 kmol/m3, Rd0 = 10-6 m. UK-Israel workshop Brighton, UK

19. OXIDIZER • Dimensionless system includes four ODEs and reads as • Initial conditions UK-Israel workshop Brighton, UK

20. Reduced System The linear integrals exist due to the adiabatic approach The number of the variables can be reduced UK-Israel workshop Brighton, UK

21. Classification of Variables Variables can be subdivided according to their rates of change: Case A. radius is fast variable, temperature is intermediate, and concentrations are slow ones. Case B. temperature is fast, radius and two concentrations are slow. UK-Israel workshop Brighton, UK

22. Slow Curves • The slow curves of the considered system are determined by the equations • Frank-Kamenetskii approximation (b=0) is used. • This simplifies sufficiently the slow curve equation, as well as the further system analysis on the slow curve. UK-Israel workshop Brighton, UK

23. Case A: Slow Curves and Trajectory Analysis • The slow curve QOU of the considered system (radius is a fast variable). • Part PO describes fast decrease of the droplets’ radius. At the point O the radius vanishes and the model is no longer valid. • Part OU corresponds to conventional gaseous thermal explosion. UK-Israel workshop Brighton, UK

24. Case B: Slow Curves and Trajectory Analysis • The slow curve QT2T1U of the considered system. • Parts P1A and PB describe fast change of the gas temperature. • Parts AT2 and BU represent slow motion along the slow curve. • Turning point T2 serves as initial point of final thermal explosion. UK-Israel workshop Brighton, UK

25. Criterion of Explosion • A - parametric region of conventional thermal explosion (chemical reaction dominates the evaporation process, the system explodes). • B - parametric region of delayed explosion (chemical reaction and evaporation are balanced on the slow curve). UK-Israel workshop Brighton, UK

26. Evaporation Time Case A. Evaporation time represents a time of the fast motion from the initial point to the slow curve (radius changes from the unity to zero). UK-Israel workshop Brighton, UK

27. The Delay Time Case B. Delay time is defined as a time of trajectory’s motion along the slow curve. Analytical estimations for the delay time are derived. hypergeometric function of the second kind UK-Israel workshop Brighton, UK

28. Theory versus Numerics Left - dimensionless delay time (103delay) versus dimensionless parameter 1 for various initial gas temperatures (red - 600 K, green- 650K, blue - 700K). Black circles - numerical results, solid lines - theoretical predictions. Right - corresponding relative error (%). UK-Israel workshop Brighton, UK

29. Theory versus Numerics(lean mixture) Left - delay time (sec) versus initial dimensionless fuel concentration for various initial gas temperatures (red - 600 K, green- 650K, blue - 700K). Black circles - numerical results, solid lines - theoretical predictions. Right - corresponding relative error (%). UK-Israel workshop Brighton, UK

30. RADIATION,FAST GAS TEMPERATURE UK-Israel workshop Brighton, UK

31. UK-Israel workshop Brighton, UK

32. , UK-Israel workshop Brighton, UK

33. Following figure shows the dependence of the delay time on the droplets number for different values of droplet radii: 1 (mkm), 1.1 (mkm), 1.2 (mkm), 1.3 (mkm). UK-Israel workshop Brighton, UK

34. CONCLUSIONS FOR MONODISPERCE SPRAY • Possible types of dynamical behaviour of the system are classified and parametric regions of their existence are determined analytically. • It is demonstrated that the original problem can be decomposed into two sub problems, due to the underlying hierarchical timescale structure. • The first sub problem relates to the droplet heat up period, for which a relatively rapid time scale is applicable. The second subproblem begins at the saturation point. • For the latter more significant second stage, it is found that there are five main dynamical regimes: • slow regimes, conventional fast explosive regimes, fast explosion regimes with two different types of fast explosion with delay (the concentration of the combustible gas decreases or increases). UK-Israel workshop Brighton, UK

35. POLY-DISPERSE SPRAY Energy equation Dimensional System Mass equations Concentration equation UK-Israel workshop Brighton, UK

36. Problem formulation Non-dimensional variables (Semenov type) Dimensionless system UK-Israel workshop Brighton, UK