Simulation of Thermal Explosion of Catalytic Granule in Fluctuating Temperature Field

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.


Introduction
The catalytic synthesis processes are generally accompanied by heat release.Synthesis of heavy hydrocarbons in the Fischer-Tropsch process (GTL technology) is associated with essential heat generation [1].GTL technology can solve a number of environmental and economic problems.
In the Technological Institute for Superhard and Novel Carbon Materials (Troitsk, Russian Federation), industrial reactor is developed with a capacity of 5000 Nm 3 /h of synthesis gas with a production of 500 kg/h stabilized liquid hydrocarbons.The reactor used fixed bed of catalyst granules.
Exothermic heat of reaction is transferred from the volume of catalytic granules to the boundary of the granules.At the boundary heat is removed to the liquid products of the synthesis.Exceeding heat generation over heat transfer leads to uncontrolled growth temperature (thermal explosion).Loss of thermal stability of catalyst granules is responsible of thermal explosion of the reactor.Therefore, investigation of critical conditions of thermal explosion is an important practical problem.
Reasons leading to thermal explosion in deterministic situation have been well studied [2][3][4][5][6].There is a critical temperature, the excess of which causes a significant increase in temperature of granules.The situation drasti-cally changes when the temperature of the environment is a random process.In this case there is always a non-zero probability for a temperature fluctuation, the magnitude of which exceeds a critical value, which may lead to the loss of thermal stability.Study of the effect of random noise is dedicated to the behavior of systems with explosive features, for example, [7][8][9][10][11].The results of this study can also be applicable in modeling of ignition conditions of dispersed fuel in aircraft and rocket engines, and power stations.Main trends obtained in the paper are helpful for the estimation of the probability of thermal explosion in storages and transportation lines of dispersed combustible materials.
Investigation on effect of noise is devoted to the behavior of systems with explosive behavior [7][8][9][10][11].Study of random temperature fluctuations was carried out in the framework of probability density function approach [12].This approach requires the use of modern methods of stochastic processes and functional analysis and yields results which have practical importance.However, the method of the probability density function does not take into account some important details of the complicated chemical kinetics.In this situation, it is appropriate to use the methods of modeling of temperature dynamics which is based on direct numerical solutions of stochastic ordi-nary differential equations [13][14][15][16][17].
In this paper we propose a method for direct numerical modeling of a random temperature of granule with internal heat generation with accounted temperature fluctuations in the surrounding.We construct temperature fluctuations with internal temporal structure.The autocorrelation function of temperature fluctuations of the surrounding has a finite decay time.This approach can be used in future for modeling stochastic behavior in not only temperature, but also reactant concentration inside the granule with detailed complex kinetics.Verification of the proposed algorithm is based on a comparison with exact analytical solutions.We illustrate various scenarios of the loss of thermal stability of catalytic granule.Calculations on results of the average waiting time of thermal explosion are presented.

Equation for Temperature of Granule with Internal Heat Release. Semenov's Diagram
In this section we write down the equation for the temperature of the granule with internal heat source and perform the analysis of Semenov's diagram.

Equation for Temperature of the Catalytic Granule
We investigate spherical granule with diameter p d , which is placed in liquid products with temperature Here p m is mass of the granule; area of the granule surface; 3 6 is volume of the granule; A is the frequency factor; is the universal gas constant.

R 
The equation for the granule temperature can be rewritten in the relaxation form where is thermal relaxation time of the granule.
Temperature of the surrounding liquid is given as Angular brackets denote the results of averaging over an ensemble of random realization of fluid temperature.
Equation (1) in dimensionless variables has the form Here   is parameter of thermal inertia of the granule.

Semenov's Diagram
Based on the analysis of Semenov's diagram we show the existence of critical temperature.Infinitely small excess above the critical temperature leads to uncontrolled increase of temperature of the granule (thermal explosion).
Analysis of Semenov's diagram is provided for steadystate temperature of the liquid medium.Looking for a stationary temperature of the granule from the following equation We introduce dimensionless power of heat transfer to the liquid phase  At the tangential lines b and c in Figure 1 the temperature of the granule returns to a steady state with low or high temperatures, respectively.To study the types of stationary temperature we performed numerical integration of the nonlinear Equation (2) without taking into account fluctuations in the temperature of the medium.
Figure 2 illustrates the dynamics of change of temperature of the granule, if the initial temperature is close to the second root on the Semenov's diagram.It can be seeing, that infinitely small disturbance above II  give a loss of thermal stability of the granule.
If initial temperature of the granule is infinitively less than the value II , the temperature of the granule proceeds to low value close to ambient temperature.The second root at the Semenov's diagram may be regarded as critical value .

Autocorrelation Function of Temperature Fluctuations. Exact Results
In this section, we obtain some exact results for comparison with data of numerical simulation.Exact solutions exist for linear equations.We consider the equation for the fluctuations of temperature of the granule (1) without the chemical heat source Temperature fluctuations of fluid is statisticcally stationary random process with correlation We use the relationship between the autocorrelation function and its spectrum Solution of Equation ( 4) has the form Correlation function of temperature fluctuations of the granule is written as With the help of spectrum of the fluid temperature autocorrelation function ( 5) and ( 6) we write down expression for granule autocorrelation Square of dispersion of the granule temperature fluctuations is follows from expression (8) Let us consider two special cases of the autocorrelation function of the temperature fluctuations of the fluid.

Delta-Correlated in Time Random Process
Temperature fluctuations is delta-correlated in time random process.The autocorrelation function Here   is integral time scale Spectrum of autocorrelation function ( 9) is found from expression ( 7) Substitution expression for the spectrum into formula 2 , e Delta-correlation approach is correct for granule with high thermal inertia.Autocorrelation functio perature fluctuations of the granule has exponential form w orrelation Function ation of fluid n of temith integral temporary scale equal to the granule relaxation time.

Exponential Approximation of Autoc
Second approach is exponential approxim temperature Spectrum of the autocorrelation function (11) follows from formula (7) Correlation of the granule temperature fluctuation is obtained from formula ( 11) Calculation of the above integral under theory of futions with complex variables leads to the result Square of dispersion of the granule tempera tuations is follows from Equation ( 12) at t = 0 ture fluc- Autocorrelation function of the gra fluctuation also obtained from Equati nule temperature on ( 12)  (see, also Equation ( 10)).Obtained exact results will be used for testing numerical al of simulation of temperature of granule in a random t ndings.

System of Stochastic Differential Equations
Analytical results show that modeling autoco function with finite relaxation time is possible on base of stochast Write down system of differential equations for temperature fluctuations of fluid and the granule with heat release     lta-correlated is seeded Gaussian random process with de function Integration of the system of Equations ( 15) and ( 16) is carried out by explicit Euler method Here n is the number of temporary steps; rando ncrement of seeded process is modeled as where   n  is random realization of the normalized Gaus ess (white noise) with zero mean and unit dispersion.
re 3 nclude that increasing the thermal inertia redu sian proc

Figu
illustrates the effect of thermal inertia of the granules on temperature fluctuations without heat source.It can be co ces the amplitude of temperature fluctuations of the granule.
Figure 4 shows influence of thermal inertia of the granule on dispersion of temperature fluctuations.The increasing thermal inertia decreases the intensity of tempe s of numerical simulations satisfactory agree w g time of Explosion scenaral heat rature fluctuations of the granule.From the Figure 4 is also evident a satisfactory agreement between the results of calculations by the exact formula (13) and numerical data obtained by averaging random realizations of temperature.
Autocorrelation function of the granule temperature fluctuations are shown in Figure 5.It can be seen that the result ith obtained exact results.The growth of thermal inertia increases the damping region of the autocorrelation function of the granules.

Simulation of Thermal Explosion. Average Waitin
This section presents results showing the various ios of behavior of granule temperature with intern generation with account temperature fluctuatuation of the fluid.Figure 6 shows the behavior of the actual temperature of the granules with heat generation.On the figure is actual temperature of surrounding fluid.It can be seen that fluctuations of magnitude of chemical reactions make a significant contribution to the value of random temperature of the granule.
On all illustrations following next the initial temperature of the granule is less than the critical value corresponding to the second root II gram).
Random process with nonzero probability may exceed any level.After some random ti the actual temperature of  (see Semenov's diame the granule will be over the critical value cr II    and there will be a loss of thermal stability.This scenario is illustrated by Figure 7.
The waiting time of a thermal explosion we define as the average time of first crossing by random temperature of the granule the critical level cr  .Waiting time of    As initial te e approaches to the critical value, the average waiting time of thermal explosion dramatically reduced.The critical temperature essentially depends on parameter of thermal inertia granule.From the Figure 8 it is evident that average delay time of thermal explosion depends on the parameter of thermal relaxation of the granule.Based on direct numerical simulations, the average waiting time of thermal explosion is investigated.Effect of stochastic drift of the granule temperature to its critic lue is found.Further research in the area of numerical simulation is possible to be carried out in two directions.Firstly, it is the use of the actual kinetic schemes, the Fischer-Tropsch synthesis, on cobalt catalysts.The second direction of research focuses on the accounting of the random medium temperature with intermittency, which is characterized by the log-normal distribution.

Conclusions
exothermal reaction inside the granule is Q.Rate of chemical reactions is modeled as Arrhenius law with activation energy E. Heat transfer coefficient is α.Equation for the volume-averaged temperature of the granule p  has the following form

Figure 1
Figure 1 represent Semenov's diagram.It is evident that there is a region with three stationary temperatures of the granule.This region with three roots of Equation (3) is bounded by the tangential lines, whose position is determined by the values of thermal relaxation parameter of the granule.

Figure 2 .
Figure 2. Temperature of the granule with initial value near second root on Semenov's diagram.

( 8 ).
leads to autocorrelation function of the granule temperature fluctuations Intensity of temperature fluctuations and autocorrelation function of granule are   2 2 about existence o two granules types.Granule with small thermal inertia with thermal relaxation time much smaller than integral time scale of flu f id temperature autocorrelation function E T    .In that case dispersion of temperature fluctuations of the granule and fluid is close

Figure 3 .
Figure 3. Random temperatures of surroundings and granule.

Figure 4 .
Figure 4.The ratio between dispersions of temperature fluctuations of granule and the fluid: points are simulation results; curve is the formula (13).

Figure 6 .
Figure 6.Example of granule temperature without heat explosion.

Figure 7 .
Figure 7. Example of temperature of the granule with heat explosion.thermal explosion cr  is function of initial temperature of the granule p   mp (Fi eratur gure 8).As initial te e approaches to the critical value, the average waiting time of thermal explosion dramatically reduced.The critical temperature essentially depends on parameter of thermal inertia granule.From the Figure8it is evident that average delay time of thermal explosion depends on the parameter of thermal relaxation of the granule.
Method of numerical simulation of random temperatu e of granules with internal heat source in surrounding lrelease is described by the Arrhenius law.flu ss explosion for various valerature relaxation times, initial te and dispersion of temperature fluc-fluctuations is designed.The intenity of heat s For temperature ctuations, a numerical generation of random Gau ian process with an exponentially decaying autocorrelation function is suggested.Autocorrelation function and dispersions of temperature fluctuations without heat generation obtained by the numerical simulation are compared with the exact formulas, found by spectral analysis of stochastic processes.Analysis of the influence of the fluid temperature fluctuations on the process of thermal explosion is carried out.Dynamics of thermal ues of granules temp perature of granules, mtuations are investigated.

Figure 8 .
Figure 8.Average waiting time of thermal explosion.