1. Thermal explosion of particles with internal heat generation in turbulent temperature of surrounding fluid Igor Derevich, Vladimir Mordkovich, Daria Galdina Bauman Moscow State Technical University (BMSTU) Moscow, Russia Federal State Institution “Technological Institute for Superhard and Novel Carbon Materials” (FSBI TISNCM), Troitsk, Moscow, Russia 8th International Symposium on Turbulence, Heat and Mass Transfer

2. Contents  Technical applications  The classical theory of thermal explosion. Semenov’s diagram  Explosion in random temperature field. Lagrange approach  PDF of temperature fluctuations of particles  Equations for moments  Pontryagin equation for the first passage time  Numerical simulation of thermal explosion  Induction time of thermal explosion  Summary points 2

3. Technical applications  Ignition of dispersed fuel in turbulent flows in the combustion chambers of power apparatus  Estimation the probability of thermal explosion in storages and transportation lines of dispersed combustible materials  Thermal stability of porous catalyst pellets in the synthesis of macromolecular compounds by Fischer-Tropsch technology 3

4. Main assumptions  We study the thermal explosion of particles that is mean the exponential increase in temperature.  We investigated the effect of temperature fluctuations of the fluid on thermal parameters of particles with an exothermic chemical reaction.  Concentration of reagents is constant; temperature is considered in the quasistationary approximation.  The chemical reaction rate is given by the Arrhenius law.  Fluid temperature fluctuations are modeled as Gaussian random process with a finite decay time of the autocorrelation function. 4

5. History of the problem Classical theory of thermal explosion Zeldovich Ya.B., Barenblatt G.I., Librovich V.B., Makhviladze G.M. The mathematical theory of combustion and explosions. Consultants Bureau: New York. 1985 Frank-Kamenestkii D.A. Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press: New York. 1969 Random noise in combustion science Warnatz J., Maas U., Dibble R.W. Combustion. Physical and chemical fundamentals, modeling and simulations, experiments, pollutant formation. Springer. 2001 Fedotov S.P., Tretyakov S.P. J. Chemical Physics. 1988. Т. 7, № 11. С. 1533-1537 Fedotov S.P., Tretyakov S.P. J. Chemical Physics. 1991. Т. 10, № 2. С. 238-241 Random noise in Physics Horsthemke W., Lefever R. Noise-induced transitions. Theory and applications in physics, chemistry, and biology. Springer. 1984 Lindnera B., Garcia-Ojalvo J., Neiman A., Schimansky-Geier L. Physics Reports. 2004. V. 392. P. 321–424 Klyatskin, V. I., 2005. Stochastic equations through the eye of the physicist: basic concepts, exact results and asymptotic approximations. Elsevier...... 5

6. Particles in turbulence Equation for particles motion Equation for particles temperature Fluid temperature along the particle trajectory Fluid temperature fluctuations along the particle trajectory 6

7. Equation for temperature of particles Time of temperature growth due to exothermic reaction Time of heat diffusion in the particle Particle thermal relaxation time Heat transfer Heat generation time of heat diffusion in the particle volume is less than the characteristic time of temperature rise due to chemical reactions 7

8. The classical theory of thermal explosion dimensionless time dimensionless temperature of the particles Semenov’s diagram Critical temperature 8

9. Main problem A random process of temperature fluctuations with finite probability leaves the any temperature interval. For sufficiently long period of time the temperature fluctuations of particles exceeds the critical value, and will happen thermal explosion. 9

10. Random temperature of the fluid actual temperature of the fluid, averaging over an ensemble of random realization temperature fluctuations of temperature of particles, stationary Gaussian random process autocorrelation function of temperature fluctuations of the fluid integral time scale of temperature fluctuations of the fluid parameter of thermal inertia of the particles exponential approximation 10

11. PDF of particles temperature PDF and characteristic function Unclosed equation for PDF Furutsu-Novikov’s formula (Klyatskin, 2005) Integral equation for the functional derivative 11

12. Closed equation for PDF Direct Kolmogorov equation response function The ratio of time scales Response function for exponential approximation of fluid correlation Drift Diffusion 12

13. Response function Parameter of thermal inertia of particles 0.5; 1; 1.3 Dashed lines are response function without heat generation dimensionless time dimensionless temperature of particles 0.5; 1; 1.3 13

14. Equations for first and second moments Equation for averaged temperature of particles Influence variation of particles temperature on the rate of chemical reaction The equation for dispersion of temperature fluctuations of particles 14

15. Heat explosion from equations for moments Crossing by particles temperature the critical value Complete picture of thermal explosion 15

16. Average delay time of thermal explosion Equation for transition PDF of temperature of particles Drift and diffusion initial condition The probability that random particle temperature will be localized within the interval 16

17. Pontryagin equation for first passage time Average time of first crossing by the temperature of particles the boundaries of specified interval initial temperature of the particles Pontryagin equation Boundary conditions thermal explosion 17

18. The delay time of thermal explosion Particles relaxation times Dispersion of fluid temperature The average time of crossing by temperature of particles the maximum value Dimensionless time Dimensionless initial temperature 18

19. DNS of thermal explosion seeded Gaussian random process - white noise equation for fluid temperature fluctuations equation for particles temperature dimensionless temperature fluctuations of particles and fluid 19

20. Validation the numerical algorithm of stochastic simulation Example of actual temperature fluctuations Analytical results on the base of spectral theory Particles without thermal explosion 20

21. Numerical simulation thermal explosion Example before the onset of thermal explosion Example with heat explosion Averaging over a large number of random trajectories in temperature phase space gives the average value of the first crossing time by particles temperature the critical value. This period of time is treated as delay time of thermal explosion. 21

22. Delay time of thermal explosion Effect of level of temperature fluctuations of fluid on the delay time of thermal explosion Points are data of numerical simulations; lines are solution of the Pontryagin equation 23%, 2 – 26%, 3 – 27%, 4 – 29% Effect of thermal inertia of particles on delay time of thermal explosion 1; 2 – 1.2; 3 – 1.5 Dimensionless time Dimensionless initial temperature 22

23. Summary points  Closed equation for PDF for actual temperature of the particles with exothermal chemical reaction was derived  Coupled equations for average temperature and dispersion of particles temperature was obtained  Temperature fluctuations lead to the onset of thermal explosion regardless of the initial temperature of the particles  On the basis of the Pontryagin equation was found that the average waiting time of thermal explosion is finite regardless of the initial temperature of particles  From the results of DNS the average delay time of thermal explosion it should be concluded that the temperature fluctuation of the fluid always leads to a loss of thermal stability 23

24. Acknowledgements This work was supported by RFBR grant 14-08-00970. This work was also partly supported by the Ministry of education and science of Russian Federation, project №3553. 24

25. Thank you for your attention 25