Method for numerical simulation of the temperature of
granule with internal heat release in a medium with random temperature
fluctuations is proposed. The method utilized the solution of a system of
ordinary stochastic differential equations describing temperature fluctuations
of the surrounding and granule. Autocorrelation function of temperature fluctuations
has a finite decay time. The suggested method is verified by the comparison
with exact analytical results. Random temperature behavior of granule with
internal heat release qualitatively differs from the results obtained in the
deterministic approach. Mean first passage time of granules temperature
intersecting critical temperature is estimated at different regime parameters.

A. P. Steynberg, M. E. Dry, B. H. Davis and B. B. Breman, “Chapter 2—Fischer-Tropsch Reactors,” Studies in Surface Science and Catalysis, Vol. 152, 2004, pp. 64-195. http://dx.doi.org/10.1016/S0167-2991(04)80459-2

A. G. Merzhanov and E. N. Rumanov, “Nonlinear Effects in Macroscopic Kinetics,” Uspekhi Fizicheskikh Nauk, Vol. 151, 1987, pp. 553-593.
http://dx.doi.org/10.3367/UFNr.0151.198704a.0553

J. Warnatz, U. Maas and R. W. Dibble, “Combustion. Physical and Chemical Fundamentals, Modeling and Simulations, Experiments, Pollutant Formation,” Springer, 2001.

I. V. Derevich and R. S. Gromadskaya, “Rate of Chemical Reactions with Regard to Temperature Fluctuations,” Theoretical Foundations of Chemical Engineering, Vol. 31, No. 4, 1997, pp. 392-397.

V. G. Medvedev, V. G. Telegin and G. G. Telegin, “Statistical Analysis of Kinetics of an Adiabatic Thermal Explosion,” Combustion, Explosion, and Shock Waves, Vol. 45, No. 3, 2009, pp. 274-277.
http://dx.doi.org/10.1007/s10573-009-0036-2

I. V. Derevich, “Temperature Oscillation in a Catalytic Particle of Fischer-Tropsch Synthesis,” International Journal of Heat and Mass Transfer, Vol. 53, No. 1-3, 2010, pp. 135-153.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.09.045

I. V. Derevich, “Effect of Temperature Fluctuations of Fluid on Thermal Stability of Particles with Exothermic Chemical Reaction,” International Journal of Heat and Mass Transfer, Vol. 53, No. 25-26, 2010, pp. 5920-5932.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.07.031

I. V. Derevich, “Influence of Temperature Fluctuations on the Thermal Explosion of a Single Particle,” Combustion, Explosion, and Shock Waves, Vol. 47, No. 5, 2011, pp. 538-547. http://dx.doi.org/10.1134/S0010508211050078

V. I. Klyatskin, “Stochastic Equations Eyes of the Physicist: Substantive Provisions, Exact Results and Asymptotic Approaches,” FIZMATHLIT, Moscow, 2001.

G. Y. Liang, L. Cao and D. J. Wu, “Approximate Fokker-Planck Equation of System Driven by Multiplicative Colored Noises with Colored Cross-Correlation,” Physica A, Vol. 335, No. 3-4, 2004, pp. 371-384.
http://dx.doi.org/10.1016/j.physa.2003.12.023

D. T. Gillespie, “Exact Numerical Simulation of the Ornstein-Uhlenbeck Process and Its Integral,” Physical Review E, Vol. 54, No. 2, 1996, pp. 2084-2091.
http://dx.doi.org/10.1103/PhysRevE.54.2084

S. Ilie and A. Teslya, “An Adaptive Stepsize Method for the Chemical Langevin Equation,” Journal of Chemical Physics, Vol. 136, No. 18, 2012, Article ID: 184101.

C. W. Gardiner, “Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,” Springer, 1983. http://dx.doi.org/10.1007/978-3-662-02377-8