Simulation of Thermal Explosion of Catalytic Granule in Fluctuating Temperature Field

doi:10.4236/jamp.2013.15001

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Journal of Applied Mathematics and Physics, 2013, 1, 1-7

Published Online November 2013 (http://www.scirp.org/journal/jamp)

http://dx.doi.org/10.4236/jamp.2013.15001

Open Access JAMP

Simulation of Thermal Explosion of Catalytic Granule

in Fluctuating Temperature Field

Igor Derevich, Daria Galdina

Department of Applied Mathematics, Faculty of Fundamental Sciences,

Moscow State Technical University by N.E. Bauman (BMSTU), Moscow, Russian Federation

Email: [email protected], [email protected]

Received July 29, 2013; revised August 29, 2013; accepted September 15, 2013

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

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 b ehavior of granule with internal heat release qualitatively differs fro m the results obtained in the

deterministic approach. Mean first passage time of granules temperature intersecting critical temperature is estimated at

different regime parameters.

Keywords: Stochastic Ordinary Differential Equation; Autocorrelation Function; Heat Explosion; Semenov’s Diagram;

Temperature Fluctuations

1. Introduction

The catalytic synthesis processes are generally accompa-

nied by heat release. Synthesis of heavy hydrocarbons in

the Fischer-Tropsch process (GTL technology) is associ-

ated with essential heat generation [1]. GTL techno logy can

solve a number of environmental and economic problem s.

In the Technological Institute for Superhard and Nov el

Carbon Materials (Troitsk, Russian Federation), industrial

reactor is developed with a capacity of 5000 Nm3/h of syn-

thesis gas with a production of 500 kg/h stabilized liquid hy-

drocarbons. 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 gran-

ules. At the boundary heat is removed to the liquid prod-

ucts 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 reac-

tor. 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 [2-6]. 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. Stud y of the effect of

random noise is dedicated to the behavior of systems

with explosive features, for example, [7-11]. The results

of this study can also be applicable in modeling of igni-

tion 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 be-

havior of systems with explosive behavior [7-11]. Study

of random temperature fluctuations was carried out in the

framework of probability density fun ction 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 situ ation, it is appropriate to us e

the methods of modeling of temperature dynamics which

is based on direct numerical solutions of stochastic ordi-

I. DEREVICH, D. GALDINA

nary differential equations [13-17].

In this paper we propose a method for direct numerical

modeling of a random temperature of granule with inter-

nal heat generation with accounted temperature fluctua-

tions in the surrounding. We construct temperature fluc-

tuations with internal temporal structure. The autocorre-

lation function of temperature fluctuations of the sur-

rounding 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 algo rithm is based on a comparison with

exact analytical solutions. We illustrate va rious scenarios

of the loss of thermal stability of catalytic granule. Cal-

culations on results of the average waiting time of ther-

mal explos ion are pre sented.

2. Equation for Temperature of Granule

with Internal Heat Release. Semenov’s

Diagram

In this section we write down the equation for the tem-

perature of the granule with internal heat source and per-

form the analysis of Semenov’s diagram.

2.1. Equation for Temperature of the Catalytic

Granule

We investigate spherical granule with diameter

which is placed in liquid products with temperature



Thermal effect of 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

has the following fo rm



ppp fpp

mc SVQA







 .

Here

m is mass of the granule; 2

Sd area of

the granule surface; 36

Vd is volume of the gran-

ule; A is the frequency factor; is the universal gas

constant. R

The equation for the granule temperature can be re-

written in the relaxation form

pfp











, (1)

where





 is thermal relaxation time of the

granule.

Temperature of the surrounding liquid is given as

 

fff



 ,

where

 is averaged temperature of the fluid;

is temperature fluctuations;









.

Angular brackets denote the results of averaging over

an ensemble of random realization of fluid temperature.

Equation (1) in dimensionless variables has the form





















. (2)

Here



 is dimensionless temperature of

the granule;





is dimensionless time; TE is inte-

gral time scale of fluid temperature autocorrelation func-

tion;





 is a dimensionless temperature

fluctuation of fluid surrounding;

EER



 is

dimensionless activation energy;



pp f

QQA c





is dimensionless heat of exothermal reaction;







 is parameter of thermal inertia of the

granule.

2.2. Semenov’s Diagram

Based on the analysis of Semenov’s diagram we show

the existence of critical temperature. Infinitely small ex-

cess above the critical temperature leads to uncontrolled

increase of temperature of the granule (thermal explo-

sion).

Analysis of Semenov’s diagra m is provided for steady-

state temperature of the liquid medium. Looking for a

stationary temperature of the granule from the following

equation

0, e















. (3)

We introduce dimensionless power of heat transfer to

the liquid phase











  and dimensionless

power of heat release



exp

WQ E

 



.

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 1. Semenov’s diagram.

Open Access JAMP

I. DEREVICH, D. GALDINA 3

At the tangential lines b and c in Figure 1 the tem-

perature of the granule returns to a steady state with low

or high temperatures, respectively. To study the types of

stationary temperature we performed numerical integra-

tion of the nonlinear Equation (2) without taking into

account fluctuations in the temperature of the medium.

Figure 2 illustrates the dynamics of change of tem-

perature 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



give

a loss of thermal stability of the granule.

If initial temperature of the granule is infinitively less

than the value

I, the temperature of the granule pro-

ceeds to low value close to ambient temperature. The

second root at the Semenov’s diagram may be regarded

as critical value .



cr II



3. Autocorrelation Function of Temperature

Fluctuations. Exact Results

In this section, we obtain some exact results for com-

parison with data of numerical simulation. Exact solu-

tions exist for linear equations. We consider the equation

for the fluctuations of temperature of the granule (1)

without the chemical heat source

  

pfp









t

. (4)

Temperature fluctuations of fluid is statistic-

cally stationary random process with correlation





 

ff ff

tt tt

 

 







. (5)

We use the relationship between the autocorrelation

function and its spectrum

  













, (6)

 











. (7)

Figure 2. Temperature of the granule with initial value near

second root on Semenov’s diagram.

Solution of Equation (4) has the form

 

1ed













s.

Correlation function of temperature fluctuations of the

granule is written as









 

de de

pp pp

tst s

tt tt



 







  









 





 

 .

With the help of spectrum of the fluid temperature

autocorrelation function (5) and (6) we write down ex-

pression for granule autocorrelation

 













 





. (8)

Square of dispersion of the granule temperature fluc-

tuations is follows from expression (8) at t = 0













 



.

Let us consider two special cases of the autocorrela-

tion function of the temperature fluctuations of the fluid.

3.1. Delta-Correlated in Time Random Process

Temperature fluctuations is delta-correlated in

time random process. The autocorrelation function





(5)

has the form





ftt tt







 

 

. (9)

Here



 is integral time scale

 

fss ss









 





Spectrum of autocorrelation function (9) is found from

expression (7)

 

2e d2

fss



 











Substitution expression for the spectrum into formula

(8) leads to autocorrelation function of the granule tem-

perature fluctuations



pp f













 

.

Intensity of temperature fluctuations and autocorrela-

tion function of granule are



222

pffp















. (10)

Open Access JAMP

I. DEREVICH, D. GALDINA

Delta-correlation approach is correct for granule with

high thermal inertia. Autocorrelation functio

perature fluctuations of the granule has exponential form

orrelation Function

ation of fluid

n of tem-

ith integral temporary scale equal to the granule relaxa-

tion time.

3.2. Exponential Approximation of

Autoc

Second approach is exponential approxim

temperature





autocorrelation function





 (11)

Spectrum of the autocorrelation function (11) follows

from formula (7)

 

itTE









.



 

Correlation of the granule temperature fluctuation is

obtained from formula (11)



22e d



















 









Calculation of the above integral under theory of fu-

tions with complex variables leads to the result

.



22 2

pp f















 .



(12)

Square of dispersion of the granule tempera

tuations is follows from Equation (12) at t = 0 ture fluc-









. (13)

Autocorrelation function of the gra

fluctuation als o obtai ned from Equatinule temperature

on (12)













 .

E

(14)

Integral time scale of the granule temp

tion is erature fluctua-



TttT









.

One can conclude about existence o two granules

types. Granule with small thermal inertia with thermal

relaxation time much smaller than integral time scale of

flu

id temperature autocorrelation function



. In

that case dispersion of temperature fluctuations of the

granule and fluid is close 22



, and integral time

scale of granule temperature fluctuations is





. For

granule with high thermal inertia



 dispersion of

granule temperature flucess then fluid tuations is l











. Integral time scale of tem-

perature fluctuations is close to ure relaxation

time T

granule

temperat







, and granule autocorrelation function de-

cays as









ptext







gorithm

emperature of surrou

rrelation

ly on

theic ordinary equations.

 (see, also Equation (10)).

Obtained exact results will be used for testing numeri-

cal al of simulation of temperature of granule in a

random tndings.

4. 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 tem-

perature fluctuations of fluid and the granule with heat

release



 









 , (15)

 







ptt















Here







. (16)









lta-correlatedis seeded Gaussian random process with

de function









2tt tt

 



 





.

Integration of the system of Equations (15) and (16) is

carried out by explicit Euler method





11nn nn

ff f

 



 ,



 







1*e

pp Q











nn









 











Here n is the number of temporary steps; randon-

crement of seeded process is modeled as m i

 



 





,

where





is random realization of the normalized

Gausess (white noise) with zero mean and unit

dispersion.

re 3

nclude that increasing the thermal inertia re-

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

Open Access JAMP

I. DEREVICH, D. GALDINA 5

increasing thermal inertia decreases the intensity of tem-

s of numerical simulations satisfactory agree

g time of Explosion

scenar-

al 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 tem-

perature.

Autocorrelation function of the granule temperature

fluctuations are shown in Figure 5. It can be seen that

the result

ith obtained ex act results. The growth of thermal inertia

increases the damping region of the autocorrelation func-

tion of the g ranules.

5. Simulation of Thermal Explosion.

Average Waitin

This section presents results showing the various

ios of behavior of granule temperature with intern

Figure 3. Random temperatures of surroundings and gran-

ule.

generation with account temperature fluctuatuation of the

fluid. Figure 6 shows the behavior of the actual tem-

perature of the granules with heat generation. On the

figure







 is actual temperature of sur-

rounding fluid. It can be seen that fluctuations of magni-

tude of chemical reactions make a significant contribu-

tion to the value of random temperature of the granule.

On all illustrations following next the initial tempera-

ture of the granule is less than the critical value corre-

sponding to the second root

gram).

Random process with nonzero probability may exceed

any level. After some random ti the actual temperature

 (see Semenov’s dia-

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

Figure 5. Autocorrelation functions of temperature fluctua-

tions fluid and granule. Points are numerical simulations

lines are the formulas (11) and (14). Dashed line is exponen,

tial approximation (10).

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. Example of granule temperature without heat

explosion.

Open Access JAMP

I. DEREVICH, D. GALDINA

Figure 7. Example of temperature of the granule with heat

explosion.

thermal explosion cr



is function of initial temperature

of the granule



mp (Fi

eratur

gure 8).

As initial tee approaches to the critical value,

the average waiting time of thermal explosion dramati-

cally reduced. The critical temperature essentially

depends on parameter of thermal inertiagranule.

From the Figure 8 it is ev ident that averag e delay time

of thermal explosion depends on the parameter of ther-

mal relaxation of the granule.

6. Conclusions

Method of numerical simulation of random temperatue

of granules with internal heat source in surrounding l-

release is described by the Arrhenius law.

flu

explosion for various val-

erature relaxation times, initial te

and dispersion of temperature fluc-



of the

uid with temperature fluctuations is designed. The inten-

ity of heatsFor temperature ctuations, a numerical generation

of random Gauian process with an exponentially de-

caying autocorrelation function is suggested. Autocorre-

lation function and dispersions of temperature fluctua-

tions 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 car-

ried out. Dynamics of thermal

ues of granules temp

perature of granules, m-

tuations are investigated.

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

Figure 8. Average waiting time of thermal explosion.

research focuses on the accounting of the random me-

dium temperature with intermittenc y, which is character-

ized by the log-normal distribution.

7. Acknowledgements

This work was supported by the Russian Foundation for

Basic Research (RFBR), grant number 11-08-00645-a.

REFERENCES

[1] A. P. Steynberg, M. E. Dry, B. H. Davis and B. B. Bre-

man, “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

[2] nd

[4] A. G. Merzhanov and E. N. Rumanov, “Nonlinear Effects

Uspekhi Fizicheskikh Nauk,

Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich a

G. M. Makhviladze, “Mathematical Theory of Combus-

tion and Explosion,” Nauka, Moscow, 1980.

[3] D. A. Frank-Kamenetskii, “Diffusion and Heat Transfer

in Chemical Kinetics,” Plenum, New York, 1969.

in Macroscopic Kinetics,”

Vol. 151, 1987, pp. 553-59

http://dx.doi.org/10.3367/UFNr.0151.198704a.0553

[5] J. Warnatz, U. Maas and R. W. Dibble, “Combustion.

Physical and Chemical Fundamentals, Modeling and Si-

mulations, Exmation,” Springer,

2001.

al Ex-

periments, Pollutant For

[6] W. Horsthemke and R. Lefever, “Noise-Induced Transi-

tions,” Theory and Applications in Physics, Chemistry and

Biology, Springer, 1984.

[7] I. V. Derevich and R. S. Gromadskaya, “Rate of Chem

cal Reactions with Regard to Temperature Fluctuations,”

Theoretical Foundations of Chemical Engineering, Vol.

31, No. 4, 1997, pp. 392-397.

[8] V. G. Medvedev, V. G. Telegin and G. G. Telegin, “Sta-

tistical Analysis of Kinetics of an Adiabatic Therm

plosion,” Combustion, Explosion, and Shock Waves, Vol.

45, No. 3, 2009, pp. 274-277.

http://dx.doi.org/10.1007/s10573-009-0036-2

Open Access JAMP

I. DEREVICH, D. GALDINA 7

[9] I. V. Derevich, “Temperature O

Particle of Fischer-Tropsch Synthesi s,” International J

scillation in a Catal

our-

ytic

nal of Heat and Mass Transfer, Vol. 53, No. 1-3, 2010,

pp. 135-153.

http://dx.doi.org/10.1016/j.ijheat masst ransfer.200 9.09.045

[10] I. V. Derevich, “Effect of Temperature Fluctuations of

.ijheatmasstran sfer.2010.07.03 1

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

34/S0010508211050078

[11] 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/10.11

[12] V. I. Klyatskin, “Stochastic Equations Eyes of the Physi-

cist: Substantive Provisions, Exact Results and Asymp-

totic Approaches,” FIZMATHLIT, Moscow, 2001.

[13] G. Y. Liang, L. Cao and D. J. Wu, “Approximate Fok-

ker-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

[14] D. T. Gillespie, “Exact Numerical Simulation of the Orn-

stein-Uhlenbeck Process and Its Integral,” Physical Re-

view E, Vol. 54, No. 2, 1996, pp. 2084-2091.

http://dx.doi.org/10.1103/PhysRevE.54.2084

[15] 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

[16] C. W. Gardiner, “Handbook of Stochastic M

: 184101.

ethods for

Physics, Chemistry and the Natural Sciences,” Springer,

1983. http://dx.doi.org/10.1007/978-3-662-02377-8

[17] P. Lévy, “Processus Stochastiques et Mouvem

nien,” Gauthier-Villars, Paris, 1965. ent Brow-

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