HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 30. pp. 281-284 (2002) 

STOCHASTIC MODEL OF FIRST ORDER CHEMICAL REACTIONS 
TAKING PLACE IN PARTICLES 

T. BUCKLE, A. BEZEGH1 and C. MIHALYK62 

(Research Institute of Chemical and Process Engineering, University of Kaposvar, 
Egyetem u. 2., Veszprem, H-8200, HUNGARY 

1Department of Environmental Economics and Technology, Budapest University of Economic Sciences and 
Public Administration, Fovam ter 8., Budapest:, H-1093, HUNGARY 

2Department of Mathematics and Computing, University ofVeszprem, Veszprem, P.O. Box 158, H-8201, HUNGARY) 

Received: October 17, 2002 

Experimental observation had been made in the 1950s, that the application of the fluidisation method increases the 
formation rate and homogeneity of the iron oxide red product. Results of the interpretation of this observation by the 
recently developed mathematical models had been presented. The interpretation is based on the stochastic treatment of 
the virtual rate constant due to the random temperature differences of the particles at heating processes. The stochastic 
model is derived from the first order kinetic equations assuming uniform distribution for the rate constant and includes 
the momentum equations. Discussion of the model is described for different sets of parameters. 

Keywords: stochastic model, particle dispersion processes, iron oxide red production, conversion rate, product 
homogeneity, momentum equation, particulate system 

Introduction 

During the years of the '50s, an intensive research was 
lead by Prof. K. Polinszky, to study the properties of 
inorganic pigments and their production methods. This 
had been summarised elsewhere [1]. 

It is a crucial question what result can be obtained, if 
we treat the virtual rate constant as a stochastic 
parameter, due to the random temperature differences of 
the particles in thermal processes? The models of the 
first order processes are presented in the following: 

A group of inorganic pigments are produced by heat 
treatment of solid particles. For example: 

- minium, from lead oxide, 
chromium oxide green, from potassium dichromate, 
iron oxide red from iron oxide black and yellow and 
from ferrous sulphate heptahydrate. 

At that time one of the authors, T. Blickle, dealt with 
mathematical modelling and the application of 
fluidisation, respectively. As it was experienced, in a 
fluidised bed the conversion rate was higher than that in 
a rotating kiln, or in a steady layer. It was impossible to 
explain these phenomena by the available mathematical 
models. 

The mathematical models of particle dispersion 
processes, including stochastic processes as well, had 
been studied in the recent and past decades [2-7]. Here 
we present the results of the interpretation of 
observations, gained by the recently developed 
mathematical model. 

Stochastic Model 

The kinetic equations: 

de~ * 
-

1-=-k.c. 
dt J J 

(l) 

where c ~ is the concentration of the initial component, 
J 

ci is the concentration of the final component in the j4.11 
particle, A. is the stochiometric factor. 

The virtual rate constant, k. is a stochastic variable 
with the probability density function, 8( ld and its ki 
values are assigned randomly to the J-dt particle by 
drawing according to this density function, i.e. ~ is a 
realisation of k. In the moment of the start, the 



282 

concentration frequency density function of the particles 
in the system is given by: 

n[c(O), c * (0),0] (3) 

From Eqs.(l) and (2), the transition functions can be 
obtained: 

(4) 

(5) 

Density function n[c(O),c*(O),O] and Eqs.(4) and (5) 

yield the following model of the integral transformation 
equation: 

k [c(t)- Ac * (t)(ekt -1),] 
n~(t),c * (t),t ]= Jn c * (t)ekt ,0 . (6) 

k., • ekt B(k)dk 

The momentum equation is: 

1 1 

M 1 (t) = J J c1 (t)n[c(t),c * (t),t]dc(t)dc * (t) (7) 
00 

Using the Eq.(6), one can get: 

1 1 k 1 ) [c(t)- Ac * (t)(ekt -1),] 11 c (t n · 
M 1 (t) = J J J c* (t)ekt ,0 (8) 

OOk .. kt * 
· e B(k )dkdc(t)dc (t) 

By applying Eqs.(4) and (5), and after 
transformation: 

1 1 ku[c(O) + c * (O)il(l- e-kt)]1 • 
M (t) = If I (9) 1 

0 0 t .. · n[c(O).c * (O),OJB(k)dkdc(O)dc * (0) 

Assuming that the distribution of k values are 
uniform in the range of km - ku: 

k- k'" +ku AI. k l1k ,. ak - 2 _,OJ{.= .y-k,n, ak = .Jli, a;;= k 
Values k.t and ku ar~ the lower and upper limits of a 

uniform distribution, k is the average value of the 
uniformly distributed virtual rate constant, l:Jc is the 
width and tY~; is the standard deviation of its range. 

while a 1 is the coefficient of variation (relative 
standard deviation} of the uniformly distributed virtual 
rate constant. 

Until now it was assumed, that the values of ki,, 
which had been drawn at t = 0 stay unchanged during 
the entire process. However, in many cases there are 
such effects in the system which make it necessary to 
draw again periodically, after each e. Here, when the 
elapsed time, r between two drawings is 0 < r 50 and t 
time is: 

t=l·E>+-r (12) 

where lis the number of drawing. 

If c(O) = 0 and -r * 0, then: 

1- e-k't 2 . 

fe .JUak [

sinh fe .Jl2a k ]
1 

c(t) = ilc * (0) 2 (13) 

·[sinhf.r~l 
k
- Jiiak 
7:--

2 

If c(O) = 0 and -c = 0, then: 

[ [

sinh fe ..JYia k ]
1

] 

c(t) = ilc * (0) 1- e -kt - ..fij? (13a) 
ke 12o-" 

2 

If ti(O) = u*2(0) = 0 and -c * 0, then: 

If rl(O) = u*2(0) = 0 and 7: = 0, then: 



o.s J.() 1.5 2.0 25 3.0 3.5 4.0 - 45 5.0 
kt 

Fig. I Plot of conversion against time, at different relative 
ranges of the virtual rate constant 

(
sinh 'keJUa- k 'Jt 

keJlia-k 
-2 - 'J/ 

CJz (t) = k c * (O)e-zkt [ . - .J12 " ]~ 
smhkG--a 

2 k 
(14a) 

Because limx-tO sinh x = 1 if r and 0 converges to 0, 
X 

while t is constant: 

(15) 

(16) 

Discussion of the Stochastic Model 

Eqs.(13a) and (14a) are used. 

r = 0; l = 1; c(O) = 0 (in this case t = 8 ). While 

discussing the function c(t) +kt let ak be 0; 0.25; 
..1-c* (0) 

0.5. (See Fig.l.) 

t = 0; l = 1, c(O) = 0, tT\0) = 0'*2(0) = 0. Here we 

discuss cr(t) = ~(t) +kt relationship, if 8k = 0.25; 0.5. 
c(t) 

(Fig.2.) 

i 
rr(IJ 

f(ll 

OJJ 1.0 2.0 3JJ 4.0 

283 

I 
I 
I 

I 
I 

! 
kt s.u i 

Fig.2 Plot of relative concentration variance against time, at 
different relative ranges of the virtual rate constant 

I 
8,0 ' 

I 

I I 
7,0 

i 6/.J 
I 

~) l 
i.e • (0.) ~ 

4.'J 

i 
I 2/! 

Fig.3 Plot of conversion against the relative range of the 
virtual rate constant, at different times 

Here the c(t) is studied at different (0.8, 0.9, 0.95) 
AC * (0) 

values of the necessary timet (if r= 0, l = 1, ~(0) = 0 J 
as a function of 8k. (See Fig.3.) 

We studied c and a 2 (t) at different l values. Let 
r = 0 ; and for example kt = 1, &a = 2 . Eq.( 13aj is 
then: 

-; . -1 • 1 
[ 

(l 

C(t) = Ac (0) 1-e (smh/) J (17) 
and Eq.( 14a): 



284 

i 0.7 .-·--·-- ·--'"··~><······-· 
0.65 1------1"----+----t-----;J 

1 

0.6v i 
~ o.ss 1------1"----+----t-----; 

OS '----....-L--~--'----...,.1---_...; 

1 

Fig.4 Plot of conversion against the number of drawings 

-
Fig.4 shows the c(t) +l relationship according 

llc* (0) 

to Eq.( 17), while the [ <r~t) ]
2 

+ l function according 
llc (t) 

to Eq.(l8) is plotted in Fig.5. 

Conclusions 

On the basis of the above discussions it could be 
concluded: 

i. The conversion time and the relative standard 
deviation of the product concentration are 
increasing as the relative standard deviation of 
the virtual rate constant increases. (Fig. I, Fig.2 
andFig.3.) 

ii. At a given e a~ l increases the conversion 
increases and the variance decreases. (Fig.4 and 
Fig.5.) 

iii. In steady layer system 7: -t 0 and l = 1; in 

rotating kiln when e is a given value: l = !._ (that 
e 

is the time of one revolution) and cr is less then in 
a steady layer; while in fluidised layer e -t 0 , 
l -t oo and a~ 0. 

iv. These all, in our opmwn, support the 
experimental observation: at the ferric oxide red 
production the fluidisation increased the rate of 
conversion and the homogeneity of the product. 

Summary 

It is an experimental observation made in the years of 
the '50s. that the application of the ftuidisation method 

i 
[ 

11(1) ]' 
).7(t) 

Fig.5 Plot of concentration variance against the number of 
drawings 

has increased the formation rate and homogeneity of the 
iron oxide red product. It could be explained by the 
models developed later that, the virtual rate constant is a 
stochastic variable due to the variation of the 
temperature in the fluidised bed. This stochastic model 
showed the most important tendencies as well. 

REFERENCES 

1. POLINSZKY K.: Further examinations on the 
chemistry and technology of iron oxide pigments. 
(in Hungarian) Veszprem, 1960 

2. BLICKLE T., MffiALYK6 C. and LAKATOS B.: 
Mathematical models of particle dispersion systems 
and their processes.(in Hungarian). Proceedings of 
the Conference of Technical Chemistry Days, 
Veszprem, 7-13, 2000 

3. BLICKLE T., MffiALYK6 C. and LAKATOS B.: 
Stochastic and deterministic models of processes 
taking place in operation units (in Hungarian) 
Proceedings of the Conference of Technical 
Chemistry Days, Veszprem, 25-30, 20CH 

4. CHEN X. -Q. and Pereira J. C. F.: International 
Journal of Heat and Mass Transfer, 1997,40, 1727-
1741 

5. MARCHISIO D. L., BARRESI A. A. and GARBERO 
M.: AICHE Journal, 2002, 48, 2039-2050 

6. HILL P. J. and NG K. M.: Chern. Eng. Sci., 2002, 
57' 2125-2138 

7. JONES A. G., HOSTOMSKY. J. and WACHI. S.: 
Chern. Eng. Comm., 1996, 146, 105-130 


	Page 283 
	Page 284 
	Page 285 
	Page 286