CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 57, 2017 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Sauro Pierucci, Jiří Jaromír Klemeš, Laura Piazza, Serafim Bakalis 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN 978-88-95608- 48-8; ISSN 2283-9216 

Modeling of the Actinide Oxides Reduction Process in the 
Rotary Kiln 

Igor S. Nadezhdina, Aleksey G. Goryunova, Flavio Manenti*,b 
a National Research Tomsk Polytechnic University, Institute of Physics and Technology, Department of Electronics and 
Automation of Nuclear Plants, Tomsk 634050, Russia  
b Politecnico di Milano, Dept. di Chimica, Materiali e Ingegneria Chimica „Giulio Natta“, Piazza Leonardo da Vinci 32, 20133 
Milano, Italy  
flavio.manenti@polimi.it  

The experimental research in general is needed in order to develop an efficient control system for a process. 
However, the high radioactivity of the spent nuclear fuel limits the possibility of direct experiments, because it 
is necessary to ensure a high level people and equipment security he only way to avoid this situation is to 
implement a virtual experiment on the recovery process of actinide oxide in the rotary kiln. This article is 
devoted to the development of a mathematical model of the process that generates actinide oxides in a rotary 
kiln.  

1. Introduction 

Nowadays, the enterprise of the fuel industry could produce great damages to the environment like the 
release of CO2 into the atmosphere. In this regard, the aim of reduce CO2 emissions is gaining more attention 
and, consequently, the number of publications on this topic is growing every year. Zarogiannis et al. (2016) 
described a systematic approach proposed for the preliminary screening of binary amine mixtures as a CO2 
capture candidates. Peirce et al. (2015) studied the improvement of Carbon Capture and Storage processes 
(CCS) due to biocatalyst stability at the typical operating conditions (high temperature, alkaline pH, high salt 
concentration). Another example comes from the work of Bassani et al. (2016) in which CO2 emissions is 
reduced (not stored as in the previous case) by means of an oxi-reduction reaction with H2S taking advantage 
of a novel technology, called AG2STM. 
Nuclear energy is one of the main energy sources in the future and the number of nuclear plants are 
increasing all over the world (Taylor, 2004). However, the disposal of extinguish nuclear fuel remains a 
pressing issue. Therefore, the development of novel technologies, that are able to reuse the spent nuclear 
fuel, is necessary. Toumanov et al. (1991) describe some of these techniques that are being developed from 
the 80s to the present. Oshima et al. (1982) and Koisumi et al. (1983) describe research aimed at the use of 
microwave energy for denitration of nitrate solutions of uranium and plutonium. One of the stages for 
reprocessing spent nuclear fuel is the calcination of actinide oxides in a rotary kiln. The high radioactivity of 
the spent nuclear fuel limits the possibility of direct experiments because it is necessary to ensure a high level 
of protection of the equipment and of the researchers. Therefore need develop mathematical model of the 
process in order to implement a virtual experiment avoiding the actual experimental procedure. For these 
reasons, the aim of this study is to develop a mathematical model of the process related to obtain a mixture of 
actinide oxides in the rotary kiln. This model will be use in future study for setting an optimal control system for 
rotary kiln. 

2. Description of the process 

Toumanov et al. (1991) presents the technological scheme of the thermal denitration process of actinide 
nitrate in a microwave oven. This technological scheme includes the tank for preparation of feed solution, the 
microwave oven, the rotary kiln for the recovery of actinide oxides of UO3 and UO2, a balloon with a mixture of 
argon-hydrogen and the capacitor-absorber. In this paper, only the processes occurring in the rotary kiln are 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1757144

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Nadezhdin I., Goryunov A.G., Manenti F., 2017, Modeling of the actinide oxides reduction process in the rotary kiln, 
Chemical Engineering Transactions, 57, 859-864  DOI: 10.3303/CET1757144 

859



considered. The rotary kiln has a screw conveyor with a constant step on which the mixture of oxides is 
moved. Argon-hydrogen mixture is supplied into the rotary kiln in a countercurrent configuration. Moreover, in 
order to maintain the temperature of about 700 °C, the rotary kiln is equipped with a  tubular electric heating 
elements. When the inlet kiln temperature reach 650 ºС, the chemical reactions of uranium dioxide (UO2) 
formation occurs. The process follows this chemical reaction:  

         3 2 2 2 UO sol H gas UO sol H O gas  (1) 

As mention before, it is necessary to maintain the temperature in the kiln at a certain level in order to have a 
stable chemical process. In addition, it is necessary to regulate the rotation speed of the screw to reach the 
optimum residence time of the process. After this brief introduction of the process, a conceptual model of the 
process is presented in order to develop the mathematical model. The conceptual model, shown in Figure 1.  

 

Figure 1: A conceptual model of recovery process of the oxides of actinides 

The input variables are the mass of the oxide mixture (moxid), the concentration of oxides in the mixture (CUO3, 
CUO2, CPuO2), the initial temperature of the mixture (Tin), the electric power used by heating tube (Poven), the 
number of the revolutions of the screw conveyor (Nauger), the volumetric flow rate (GAr-H2) and the temperature 
(TAr-H2) of the hydrogen-argon mixture and the concentration of argon (CAr) and hydrogen (CH2). The output 
variables are mass of produced oxides (moxid), the volume of the off-gas (Voff-gas), the concentration of 
components (CAr, CH2, CH2O) and oxides (CUO3, CUO2, CPuO2) in the off-gas and the oxides temperature in the 
rotary kiln (Toxid). 

3. Mathematical model 

In order to evaluate the speed of rotation of the screw conveyor it is necessary to calculate the time for 
reducing oxides in the rotary kiln. The required time is calculated following this expression: 

 heat reaction.t t t  (2) 

where theat is time necessary to heat the solution up to the operating temperature and treaction is related to the 
chemical reactions rate at operating temperature. These variables are defined as follows: 

   




     



1
heat . .

1
reaction.

0.24

proc

oxid oxid proc in oven

T

t с m T Т Р

t k
 (3) 

where, coxid is the heat capacity of oxides actinides mixture, 0.24 is the thermal equivalent of work, Tproc is the 
operating temperature of the recovery process of oxides in the rotary kiln, 

procT
k  is the rate constant of 

chemical reduction reaction. The rotational speed of the screw conveyor is evalueted according to the 
expression: 

 


  
1

auger oven auger
N l t l  (4) 

where, loven is the length of the rotary kiln, Δlauger is the step of blades on screw conveyor. During one rotation 
of the screw auger, the mixture of oxides moves in the kiln by an amount equal to the step blades (Δlauger).  
The oxides are moved into the kiln by means of a use of a screw auger. Based on the design features of kiln, 
it is supposed that an ideal mixing of actinide oxides loaded occurs in the interval Δlauger and so the rotary kiln 
was divided into 40 cells of width 25 mm. The principle of the rotary kiln partitioning is shown in Figure 2. 

Voff-gas, l
CH2O, 
vol.%

CAr 
vol.%

CH2, 
vol.%

Poven, 
kWh

moxid, kg

CUO3, g/kg

CPuO2, g/kg

Tin, ºС

GAr-H2, 
l/h

CAr, 
vol.%

CH2, 
vol.%

moxid, kg

CUO2, g/kg

CPuO2, g/kg

Toxid, ºС

TAr-H2, 
ºС

Nauger, 1/min

860



 

Figure 2: Cell model of the rotary kiln 

The diameter of the loading neck is 100 mm. the feed loading is carried out in the cells 4-6 and partially in the 
cells 3 and 7 (Figure 2). Unloading of the obtained oxides occurs from cells 35 and 36, since the auger has 
blades in the opposite direction of cells 37 and 38. Therefore, the obtained oxides do not enter into the cells 
37-40. Each cell has a different contact area with the neck of the kiln and so the distribution of loading and 
unloading oxides is not evenly in the cells. The distribution of oxides mass in the cells is determined according 
to the following expression: 

 
  
 

i i
oxid oxid

neck

S
m m

S
 (5) 

where, Si is area of crossing neck with the i-th cell, Sneck is area of the neck, moxid is mass of actinides oxides 
entering the kiln, i

oxid
m is mass of actinides oxides entering the i-th cell. Si is defined as an area of a circle 

segments. The weight of cells is calculated by correlating the areas Si and Sneck. The distribution of oxides 
mass, which is loaded into the kiln, is in accordance with these weights. The system of differential equations 
that describe the distribution of mass of the actinides oxides along the length kiln is reported below: 

   

 

 

 

  

 

 

    






  
   





  




3 2 2 3 2 2

3 33
3 2

2 22

1 1 1

1 1

2

1

3

1

2

2

,

2

,,

2 proc

i i i i i ii
UO PuO UO UO PuO UOoxid

auger
i i

k k
i

oxid
i i i ii
UO oxid UO oxid i iUO

T UO H
auger

i i i ii
UO oxid UO oxidUO

aug

m m m m m mdm
dt T

dC m
dt m

C m C mdm
k m m

dt

k UO PuO UO

T

C m C mdm

dt T

 

 

 









  

  




 

 
  

 

 
  
  
 

3 2

2 22

2

2 2

2 2

22

2

2

1 1

2

100

100

proc

i i

T UO H
er

i i i ii
PuO oxid PuO oxidPuO

auger
i

iAr
Ar Ar H

i ii
i H m H O moff gas

Ar
H H O

i i

Ar Ar
i

off gas

ii
H mH

i

H off gas

i

H O

k m m

C m C mdm

dt T

dV
C G

dt
m V m VdV

V
dt M M

dC V
dt V

m VdC

dt M V

dC  

 

 





 





























 
  
  
 
      
          

 
      

 

2

2

2 2 2 2 22
3 2

2

2
2 2 3 2

2

11

1
1

100

proc

proc

i

H O m

i

H O off gas

i ii
H H Ar H H Ar H i iH cell

T UO H
m Ar H

i
i i i iH O cell
H O H O T UO H

Ar H

m V

dt M V

M C G C Gdm V
k m m

dt V G

dm V
m m k m m

dt G





















 

(6) 

C
el

l  
2

C
el

l  
3

C
el

l  
1

C
el

l  
 

C
el

l  
 

C
el

l  
 

C
el

l  
 

C
el

l  
 

C
el

l  
 

C
el

l  
10

C
el

l  
40

Oxides of actinides
UO2, PuO2

Oxides of actinides
UO3, PuO2 Argon-

hydrogen 
mixture 
(Ar+H2)

C
el

l  
3
 

C
el

l  
3
 

C
el

l  
3
 

C
el

l  
3
 

C
el

l  
3
 

C
el

l  
3
 

Off-
gases

861



where, i
oxid

m  is mass of mixture of oxides actinides in the i-th cell, i
k

m  is the mass of the k-th component in 

the i-th cell, 
procT

k  is the rate constant of chemical reaction, which depends on temperature according to the 

Arrhenius equation, iArV  is the volume of argon in i-th cell, Vm is the molar volume of an ideal gas, Mk is the 
molar mass of the k-th component, Vcell is the volume of cell, which is calculated according to the following 
formula: 

   cell oven auger augerV S S l  (7) 

where Soven is sectional area of the rotary kiln, Sauger is area of the screw, located in the rotary kiln. Tauger is the 
time constant of screw conveyor (s). The dependence between the time constant the screw rotation speed 
was obtained experimentally and is represented by this following relation: 

 



60

4auger
auger

T
N

 (8) 

The developed mathematical model, based on a cell model, allows to evaluate the temperature change in a 
rotary kiln with a countercurrent heat exchange between the gas and the particulate material. This model is 
also based on the theory of Markov chains (Mizonov et al. (2002)). The process conditions are described by 
vectors m, T, Q. These vectors represent the distribution in the cells, i.e. along the kiln, respectively of mass, 
temperature, and heat. Thus, the obtained system of equations (from (9) to (14)) of model is: 

  
i i i i

oxid oxid oxid oxid
Q T m c  (9) 

  
i i i i

gas gas gas gas
Q T m c  (10) 

      i i i ioxid gasQ S T T t  (11) 

where coxid and cgas are the heat capacity of oxides and argon-hydrogen mixture (J/(kg·K)); α is the heat 
transfer coefficient, S is the surface of heat exchange in the cell. 


  

1i i i i
oxid oxid EHT

Q Q Q Q  (12) 

  

1i i i
gas gas

Q Q Q  (13) 

In Eq.(12), QEHT is the quantity of heat transmitted to oxides from the tubular heating elements in the rotary 
kiln (J). 

   


 

   
 

 

11
1 1

1 1 1 1and
iI

i i gasoxid
oxid gasI i i i

oxid oxid gas gas

QQ
T T

c m c m
 (14) 

4. Results and discussions 

The mathematical model presented before, was implemented in MatLab. The simulation was carried out with 
the following parameters and input variables: Poven = 20 kWh, GAr-H2 = 2500 l/h, CAr = 90 vol %, CH2 = 10 vol.%, 
TAr-H2 = 25 ºC, moxid = 2200 g, CUO3 = 760 g/kg, CPuO2 = 240 g/kg, Tin = 100 ºC, Soven = 0.23 m

2, Sauger = 0.13 
m2. 
The first result, obtained from equations (2)-(4), is that the recovery process of uranium trioxide takes 1.8 h. 
Therefore, the screw rotation speed should not exceed Nauger = 0.1 min

-1. Then, the distribution of the oxides 
mass along the kiln, the variation of oxides temperature and of argon-hydrogen mixture over the kiln length 
were calculated. The results are shown in Figure 3.  

862



    

Figure 3: The modelling results 

As seen from the simulation results (Figure 3), the oxide mixture moves along the rotary kiln by means of 
screw conveyor. At the same time, oxides are heated up to 700 °С and the heat exchange takes place with 
the argon-hydrogen mixture that enters the kiln. The variations in the composition of oxides mixture and of the 
off-gas are due to the chemical reactions (1). As a result, the progress of the chemical process is obtained and 
is shown in Figure 4.  

    
Changing of the mass and concentration of actinide oxides during processing 

    
Changing of the volume and concentrations of off-gases 

Figure 4: The modelling results 

Figure 4 shows that the mass of the oxides is decreased during heat treatment in the rotary kiln. Furthermore, 
the concentrations of plutonium dioxide (PuO2) are increased and uranium trioxide (UO3) is converted to 
uranium dioxide (UO2). The interaction between hydrogen (H2) and oxygen (O2) generates water, which 

863



comes out with the off-gases. The concentration of argon (Ar) in the off-gases is decreased. The obtained 
actinide oxides start to be unloaded from the kiln after 1.8 h (Figure 5).  

 

Figure 5: Mass of the actinide oxides at the outlet of the rotary kiln 

As seen from the transition process (Figure 5), the unloading of the oxides takes more than 3.5 h, if  Nauger = 
0.1 min-1. In the future, it is necessary to develop a control system for rotary kiln, which will reduce the time of 
unloading of actinide oxides from the kiln.  

5. Conclusions 

This paper presents the results of a development of mathematical model related to the process for obtaining 
uranium dioxide from uranium trioxide in a rotary kiln. The mathematical model is based on a cell model and 
describes both the material balance and the heat balance which lead to the evaluation of the oxides 
temperature variation in the rotary kiln. The obtained simulation results show that it is necessary to develop a 
robust control system for rotary kiln. The development of a control system should allow to reduce the time of 
unloading of actinide oxides from the kiln.  

Acknowledgments  

This work was funded as a part of the project  .30  .201 /ПЧ of the Federal government-sponsored program 
«Science».  

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