CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Cellular Automata Application for Simulation of Uranium 

Crystallization Process 

Anton. O. Ochoa Bique*a, Dmitry A. Serikova, Alexey G. Goryunova, 

Flavio Manentib 

aTomsk Polytechnic University, Dept. of Electronics and Automation of Nuclear Plants, Lenina St, 30, Tomsk 634050, 

Russia 
bPolitecnico di Milano, Dept. di Chimica, Materialie Ingegneria Chimica „Giulio Natta“, Piazza Leonardo da Vinci 32, 20133 

Milano, Italy. 

anthonob@tpu.ru 

The paper addresses the dynamic modelling of the UNH crystal growth by means of cellular automata(CA) 

method for nuclear fuel reprocessing as the alternative to traditionally used extraction method. Model-based 

CA approach is considered to predict dependence of product’s properties on input changes: uranium and nitric 

acid concentration, solution temperature. CA approach is preferable for crystallization process to get a high 

purity of reprocessed materials. C++ application is developed as a result of the research. The application 

allows monitoring of dynamics changes of solution temperature, uranium concentration and shape of crystals. 

1. Introduction 

Advantages of nuclear energy compared to other alternative energy sources and fossil fuels lead to increase 

of its share in electricity generation. However, problem of spent nuclear fuel handling restrain growth of 

nuclear share in world’s energetics. Current system or nuclear fuel cycle lead to unacceptable growth of spent 

nuclear fuel handling cost. Crystallization process is considered in this paper as a complement or an 

alternative to traditionally applied extraction process for high material purification purpose. Crystallization 

application provide reduction of organic substances and liquid radioactive wastes volume during fuel 

reprocessing (Veselov et al., 2015). As a result, crystallization allows getting desired purity of output materials. 

Crystals purity depends on accuracy of control of the process. In seeded cooling crystallization, that is 

researched in this paper, supersaturation is generated by external cooling of crystallizer. High purity of 

reprocessing is achieved holding supersaturation on desirable level that corresponds with growth of crystals 

and non-appearance of new undesirable seeds. Using cellular automata it is possible to monitor the structure 

of crystals and change adjustments of crystallization process keeping desirable purity of output material up. 

2. Cellular automata for crystallization process 

Cellular automata are discrete dynamic models whose behaviour is completely specified in a local 

dependence terms.  Cellular automata may be considered as a space divided with a regular grid into cells that 

keep some information about their state (Toffoli, 1990). Time is discrete and set with a number of iterations. A 

new state of each cell is determined from the current state of the cell and states of cells in its neighbourhood 

according to the fixed rules of evolution. CA are successfully applied as the alternative to Partial Differential 

Equations (PDEs) when their solving requires extremely high computational power (Prieto and Gonçalves, 

2014). 

CA crystallization process description include next steps: 

1) determination of rules of the system evolution; 

2) determination of processes which occur during crystallization such as latent heat release during 

crystallization and the decrease of solution concentration in neighbourhood of the crystallized cell; 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652064 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Ochoa Bique A. O., Serikov D. A., Goryunov A. G., Manenti F., 2016, Cellular automata application for simulation of 
uranium crystallization process, Chemical Engineering Transactions, 52, 379-384  DOI:10.3303/CET1652064   

379



3) acceptance of assumptions that simplify implementation of the model but do not affect results of the 

research. 

Hexagonal grid is used in the model and each cell of this grid determines the unit of nitrate solution of uranyl 

nitrate. The cause of hexagonal grid use is its high isotropy compared with triangular and rectangular grids 

(Figure 1.а). Hexagonal grid is transformed (Figure 1.b) for its implementation. 

 

Figure 1: A - hexagonal grid, b - transformed hexagonal grid 

Abasheva (2007) has offered a method of CA crystallization modelling: cells of CA have two possible states 
( )
, {0,1}
n

i j  where 0 corresponds to a liquid state, 1 – solid state, index n determines a number of iteration 

(time) and indices i,j specify coordinates of cells in the grid. Each cell contain information about its state  

concentration 
( )
,
n

i jC  and temperature 
( )
,
n

i jT  whose initial values are set before simulation. The process of 

crystal growth is going on under certain rules and conditions of evolution. 

Conditions of cell’s state transition from a liquid to a solid state: 

1) Cell changes its state only from liquid to solid one, there is no reverse transition; 

2) Cell is able to crystallize if only there is a solid cell in its neighbourhood; 

3) Cell changes its state from liquid to solid one if following condition is satisfied: 

( ) ( )
, ,, ( , )

* 3 ,
n n

i j k lk l O i j



       
   

(1) 

where
( )
,
n

i j  – supersaturation of {i,j}-cell; 

* – limit of supersaturation; 
( )

,
n

k l – state of {k,l}-cells of  {i,j}-cell neighbourhood;  

 – a weight coefficient that specify influence of local curvature of interstate surface to the state transition. 

State transition of {i,j}-cell from liquid to solid state processes with a latent heat release (dT) of the cell and 

liquid neighbours concentration decrease
( )

,
n

k l : 

( 1) ( )
, , ,
n n

i j i jT T dT


      (2) 

( 1) ( ) ( )
, , , ,
n n n

k l k l k lC C q


    (3) 

where q – coefficient, that is an analog to kinetic constant of crystal growth rate. 

Following equations of heat transfer and diffusion are extremely important for crystallization process: 

 

 ( 1) ( ) ( ) ( ), , ,, ,n n n ni j i j i jk l
D

C C C C
m


    

(4) 

 ( 1) ( ) ( ) ( ), , ,, ,n n n ni j i j i jk l
a

T T T T
p


    (5) 

where 

380



( ) ( )
, ,, ( , )

1
,

n n
k l k lk l O i j

l

C C
n


   

(6) 

( ) ( )
, ,, ( , )

1
,

6

n n
k l k lk l O i j

T T


   (7) 

nl is a number of liquid cells in neighbourhood of the {i,j}-cell (nl ≤ 6); 
( )
,
n

k lC and 
( )
,
n

k lT – average 

concentration of liquid cells and average temperature of all neighbours of the {i,j}-cell; D, а, m, р – parameters, 

that depend on diffusion coefficient, thermal conductivity and space-time discretization – Eq.(6). 

Saturation of uranium solution is specified by the following function of temperature and nitric acid 

concentration obtained by Chikazawa et al (2012). 

10 ,UC


  (8) 

where 

2 2 4 2 3
5,504 1, 411 0,0263 0,913 10 0,088 10 3,586 10 .H H abs abs H absC C T T C T

  
               (9) 

Tabs – solution temperature (K), CH and CU – concentrations of nitric acid (mol\L) and uranium (g\L). Eq(8) is 

obtained from empirical solubility graphs (Figure 2). 

 

Figure 2: Uranium solubility graphs 

 

3. Results 

Next figures represent results of CA simulation of crystallization process. Developed application provides user 

interface for crystallization process adjusting and displays results of simulation: crystal shape, concentration 

distribution and temperature distribution (Figure 3). Figure 4 represents crystal growth dynamics for n=10, 20, 

30 iterations. 

381



 

Figure 3: Crystal shape, concentration distribution and temperature distribution 

 

 

Figure 4: Crystal growth dynamics (n = 10, 20, 30 iterations) 

 

Initial conditions influence analysis and varying of these conditions allow searching for crystallization optimal 

parameters that provide desirable crystal purity (Figures 5 and 6). 

 

 

Figure 5: Initial conditions influence (solution concentration increase) 

 

382



 

Figure 6: Initial conditions influence (solution temperature decrease) 

 

High influence of diffusion and heat transfer are obvious from Figures 7 and 8. Thus, it is necessary to choose 

values of their coefficients with a high accuracy. 

 

 

Figure 7: Diffusion influence (diffusion coefficient increase) 

 

 

Figure 8: Heat transfer influence (heat transfer coefficient increase) 

Table 1 demonstrates dependences of crystal purity on crystallization parameters variation. Purity is 

calculated as the ratio of solid cells of the crystal to all cells. 

  

383



Table 1:  Purity dependence 

Concentration Temperature Diffusion coefficient Heat transfer coefficient  

Variation Purity Variation Purity Variation Purity Variation Purity 

C0 0.4382 T0 0.4476 (D/m)0 0.7723 (a/p)0 0.6465 

1.2C0 0.6171 0.9T0 0.5905 1.2(D/m)0 0.8169 1.5(a/p)0 0.6877 

1.4C0 0.8606 0.8T0 0.7923 1.4(D/m)0 0.9576 2.0(a/p)0 0.9030 

4. Conclusion 

Application for crystallization process simulation is developed. Presented application allows monitoring of 

crystallization process dynamics and obtaining information about crystallization output material. Time of 

simulation is specified with a number of iterations and depends on chosen time-space discretization. 

Importance of diffusion and heat transfer coefficient’s values determination for crystals purity is demonstrated. 

Crystallization dynamics may be controlled via initial uranium concentration and solution temperature. 

Uranium saturation for crystallization conditions is calculated as a function of nitric acid concentration and 

solution temperature.  

References 

Abasheva E.R., Chan H.K., Koltsova E.M., 2006, Cellular automata for simulation of producing process of Fe - 

Nanothreads. In CHISA 2006 - 17th International Congress of Chemical and Process Engineering. 

Chikazawa T., Kikuchi T., Shibata A., Koyama T., Homma S., 2008, Batch crystallization of uranyl nitrate. 

Journal of Nuclear Science and Technology, 45, 582-587. 

Gonçalves M.A., Prieto M., 2014, Development of Compositional Patterns during the Growth of Solid Solutions 

from Aqueous Solutions: A Cellular Automaton Simulation, Crystal Growth & Design, 14, 2782−2793 

DOI:10.1021 /cg500010p 

Toffoli T., Margolus N.H., 1990, Invertible cellular automata: A review. Physica D: Nonlinear Phenomena, 45, 

229-253. 

Veselov S., Volk V., Kasheev V., Podimova T., Posenitskiy E., 2015, Mathematic Simulation of Crystallization 

Refining Process of Spent Nuclear Fuel Reprocessing Desired Products in Linear Crystallizer, Advanced 

Materials Research, 1084, 666-672, Trans Tech Publications, Switzerland, DOI:10.4028 /AMR.1084.666 

 

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