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 

Mathematical Modeling of the Thermal Denitration of 
Actinides in a Microwave Oven 

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. The only way to avoid this situation is to 
implement a virtual experiment on denitration of nitrates actinides. The mathematical model of the process is 
necessary for implementation the virtual experiment. This article is devoted on the development of a 
mathematical model of the thermal denitration of nitrates actinides in a microwave oven.  

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. For instance, lithium 
orthosilicate-based pellets were developed and characterized as potential regenerable high-temperature CO2 
sorbents (Puccini et al. 2016). Molina et al. (2015) studied an alternative way to absorb CO2 in ammonia 
aqueous solution with a membrane contactor in order to improve the capture processes and to intensify the 
gas-liquid transfer. Another example comes from the work of Bassani et al. (2016) in which CO2 emissions is 
reduced 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. One of these (Oshima et al. (1989)) studied the methods for obtaining MOX fuel using 
microwave energy. Another authors (Imai et al. (2011)) investigate the electric power absorption of the solid 
Pu/U mixed nitrate medium in a cylindrical oven cavity. Therefore, it is important to underline the fact that one 
of the principal stages for reprocessing spent nuclear fuel is to obtain actinide oxides by using thermal 
denitration of actinide nitrate in a microwave oven. Unfortunately, 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 researchers and also to those working close to the area near the experimental apparatus. Thus, the aim of 
this work is to develop a mathematical model of the thermal denitration of nitrates actinides in a microwave 
oven in order to implement a virtual experiment avoiding the actual experimental procedure. 

2. Description of the process 

In this section is presented the description of the global process starting from the difference between the 
conventional process using a resistance furnace and the novel using microwave oven. In the novel process, 
heating takes place simultaneously over the entire volume, providing a higher heating rate with activation of 
chemical reactions by microwave. However, the violation of the required temperature or residence time of 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1757145

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Nadezhdin I., Goryunov A.G., Manenti F., 2017, Mathematical modeling of the thermal denitration of actinides in a 
microwave oven, Chemical Engineering Transactions, 57, 865-870  DOI: 10.3303/CET1757145 

865



thermal denitration, may lead to product formation that are refractory to the action of the microwave field. This 
can be avoided by pre-calculating the optimum temperature and residence time of microwave oven 
denitration. The technological scheme of the thermal denitration process is shown in Figure 1.  

 

Figure 1: The technological scheme of the thermal denitration process  

The process flow diagram includes the tank for preparation of feed solution, microwave oven, rotary kiln to 
convert the oxides actinides, such as UO3 to UO2, balloon with mixture of argon-hydrogen and capacitor-
absorber. The preparation of feed solution, carried out in the tank, consists in a mixture, diluted with water, of 
uranium-plutonium ligature and the uranyl nitrate solution. Then, the solution is sent to the microwave oven in 
which the thermal denitration generates a mixture of off-gases and actinide oxides. The off-gases are treated 
using the capacitor-absorber and the actinide oxides are reduced in a rotary kiln to obtain a uranium dioxide 
(UO2) from uranium trioxide (UO3). In this work, only the processes that take place in the microwave oven are 
considered. The first stage, that takes place at 120 °С, consist in the evaporation of free water and nitric acid 
according to the following chemical reactions (1) that lead to crystal hydrate uranyl nitrate formation: 

         
         
   
   





 

 

3 3

2 3 2 2 3 22 2

2 3 2 2 3 22 2

2 2

;  6    ·6  
  6  ;

;
;

  ·6  
  

 

UO NO liq H O liq UO NO H O liq

PuO NO liq H O liq PuO

gas

HNO

NO H O liq

H O liq H O

li HNO gasq

 (1) 

As mentioned before, the control of temperature and residence time is crucial for this stage. Then, in second 
stage of the process, which is carried out at 350 °С, occurs the decomposition of crystalline hydrate uranyl 
nitrate on components. The stage process takes place according to the following chemical reactions: 

   
   

   

   
2 3 2 3 2 2 22

3 2 2 2 2 24

( ) ( ) ( ) ( );·6      2   0.5   6 
·5      4    ( ) ( ) ( )  ;5 ( )

UO NO H O liq UO sol NO gas O gas H O gas

Pu NO H O liq PuO sol NO gas O gas H O gas
 (2) 

After this brief introduction of the process, a conceptual model of the process is presented in order to develop 
the mathematical model of the denitration process. The conceptual model, shown in Figure 2, describes the 
relationship between the input and output variables. 

 

Figure 2: The conceptual model of the thermal denitration process in a microwave oven 

Argon-
hydrogen 
mixture 
(Ar+H2)

Microwave oven

UO2(NO3)2 + 6H2O;
PuO2(NO3)2 + 6H2O;

Off-gases

Oxides of actinides
UO3, PuO2

Water (Н2O);

Nitric acid 
(HNO3)

Rotary kiln
Oxides of 
actinides

UO2, PuO2

Capacitor-

absorber

Solution of 

nitrates 

actinides

Off-
gases

Voff-gas, 
l

CH2O, 
vol.%

CHNO3, 
vol.%

CNO2, 
vol.%

CO2, 
vol.%

moxid, kg

CUO3, g/kg

CPuO2, g/kg

Tsol, ºС

Vsol, l

CU, g/l

CPu, g/l

Poven, 
kWh

Tin., ºС

V*sol, l

T*sol, ºС

CPu, g/l
CU, g/l

C*H2O, g/l
C*HNO3, g/l

The thermal denitration
(the second stage)

Evaporating the H2O 
and HNO3

(the first stage)

2

in
H OC , g/l

3

in
HNOC , g/l

866



The input variables of the model are the volume of feed solution (Vsol), the concentrations of the components 
in the feed solution (

2

in
H OC , 3

in
HNOC , CU, CPu), and the power of the microwave oven (Poven). The output variables 

are the mass of the produced oxides (moxid), the volume of the formed mixture off-gas (Voff-gas), the 
concentrations of components in the mixture off-gas mixture (CH2O, CHNO3, CNO2, CO2) and in the solid products 
(CUO3, CPuO2) and the temperature of  the solution (Tsol) in the microwave oven. The mathematical model 
includes both stages of the process. So, it is necessary to define also the output variables of the first stage. 
These variables are the volume of solution (V*sol), the concentrations of the components in the solution (C

*
H2O, 

C*HNO3, CU, CPu), and the temperature of solution (T
*
sol) in the microwave oven. These variables are at the 

same time the input variables of the second stage. 

3. Mathematical model 

In order to maintain optimal temperature of the thermal denitration in the microwave oven is necessary to 
control the residence time of the process. The optimal residence time for thermal denitration is defined as 
follows: 

2

1
 

i

stage

i

t t  (3) 

In Eq.(3), i
stage

t  is the time required for each stage of the thermal denitration. Time for each stage is defined as 

follows: 

heat reaction. 
i

stage
t t t  (4) 

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: 

    






. .
heat

reaction.

0.24
1

proc

sol sol sol proc in

oven

T

V с T Т
t

Р

t
k

 (5) 

where ρsol is the density of the feed solution, csol is the specific heat of feed solution, 0.24 is thermal equivalent 
of work; Tin is the initial temperature of the feed solution, Tproc is the temperature of the feed solution in the 
microwave oven during the denitration process, 

procT
k  is the rate constant of the chemical reactions of 

denitration process. The variation of the feed solution temperature in the microwave oven is evaluated using 
the expression presented by Didenko (2003). Temperature of the feed solution is defined as follows: 

  * 4* 1  


     


 

oven eff solsol

sol sol sol

P S TdT

dt V c
 (6) 

where S is the area of heated surface, σ is the Stefan-Boltzmann constant, η is the coefficient of transmission 
of microwave power to the heating solution, γeff is effective coefficient of the reflection of power. It is important 
to underline the fact that the the solution is heated uniformly over the entire volume in microwave ovens and, 
as a result the chemical reactions takes place uniformly. Thus, the developed mathematical model is a 
lumped-parameters model. At the first stage of the process, Dalton law determines the evaporation of water 
and nitric acid. The system of differential equations describing the material balance of the processes occurring 
at the first stage is as follows: 

867



 

 

2 3

2 2

2

2

3 3

3

3

2 2 2

2

3

3

1

*

1

0

0
* 1

*

*

*

1

1





 

 


   




   



 
    

 
 

   

off gas

H O HNO

sol
sol off gas

H O H O

Par SH O

m

H O

HNO HNO

Par SHNO

m

HNO

H O H O H Oin

H O sol

sol m

HNO Hin

HNO sol

sol

dV
V V

dt
dV

V V
dt

P PdV
V C S

dt M P

P PdV
V C S

dt M P

dС V M
C V

dt V V

dС V
C V

dt V

3 3

2 2

3 3

1

100

100






















 
 
 


 





 


NO HNO

m

H O H O

off gas

HNO HNO

off gas

M

V

dC V

dt V

dC V

dt V

 
(7) 

where *
sol

V  is volume of the feed solution that remains after the first stage, 
2

*
H O

m  and 
3

*
HNO

m  are mass of water 

and nitric acid, which remain in solution after the first stage (g), 
2

1
H O

V  and VHNO3 are volume of water and nitric 

acid after the evaporation process, Mk is the molar mass of the k-th component, Vm is molar volume of an ideal 
gas, C is a coefficient in order to take into account of the blowing air speed and density (С = 0.95·10-4 
(g/(m2·h) because is not blowing), Р0 is the barometric pressure of the atmosphere; kSP  is the pressure of 

steam at a temperature of evaporating liquid, k
Par

P  is partial pressure of k-th component vapor.  
The dependence of water and nitric acid vapor pressure on temperature was obtained by analysis of 
experimental data by means of a regression model by using cftool tool included in MatLab. The dependences 
obtained are as follows: 





 

 

*
2

*
3

0.04143

0.01586
12.47
1.757

sol

sol

TH O

S

HNO T

S

P e

P e
 (8) 

where *
sol

T  is the temperature of the solution in a microwave oven at the first stage. 
At the end of the first stage of the process is formed a crystals of hydrate uranyl nitrate that have to be sent at 
the second stage of the process. In this stage, microwave oven power is increased in order to rise up the 
temperature of the solution. As a result, the exothermic chemical reactions, reported in equation (2), take 
place. This stage is described by a system of differential equations of material balance and equation of heat 
balance: 



           

   
  

   

  
    

3 2 2 3 2 3

2 3

2 3 2 2

2 32

2

* * * * * *

* * *

3 2

* * *

, ( , ; , )

2 4

oxid
UO PuO U sol U H O HNO Pu sol Pu H O HNO

i sol i H O HNOj

oxid

off gas

H O HNO NO O

m sol H O HNONO

U U

NO

dm
m m k V C C C k V C C C

dt
k V C C CdС

j UO PuO i U Pu
dt m

dV
V V V V

dt
V V C CdV

k C
dt M

 

 

 

 


 

  
     

  
       

 

        




2 32

2

2 32

2

2

2 3

* * *

* * *
1

*
*

2 2

*

2

*
*

0.5

6 5

10  ,  ,,  0

Pu Pu

m sol H O HNOO

U U Pu Pu

O

m sol H O HNOH O

U U Pu Pu H O

HO

k k

off gas

chem sol
U U U H O HNO

sol sol

k C

V V C CdV
k C k C

dt M

V V C CdV
k C k C V

dt M

dC V

dt V

dT V
H k C C C

dt m c

k H O NO O

 






















    


2 3

* *
Pu Pu Pu H O HNO

H k C C C

 
(9) 

868



where moxid is the mass of actinide oxides, ki is the rate constant of the i-th chemical reaction, which depends 
on the temperature according to the Arrhenius law, VH2O, VHNO3, VNO2, VO2 are volume of water, nitric acid, 
nitrogen oxide and oxygen generated as a result chemical reactions, Mk is the molar mass of the k-th 
component, ΔHi is enthalpy of the i-th chemical reaction.  
Due to the fact that the reactions are exothermic, the equation that allows to calculate the change of the 
solution temperature in the second stage, has the following form:  

 
 

2 3 2 3

4 *
* * * *

1 2* * * * *

1  


     
             

  

oven eff solsol sol
U U H O HNO Pu Pu H O HNO

sol sol sol sol sol

P S TdT V
H k C C C H k C C C

dt V c m c
 (10) 

4. Results and discussions 

The developed mathematical model of the denitration process was implemented in MatLab package using 
following input values: Vsol = 3.5 l, 

2

in
H OC = 340 g/l, 3

in
HNOC = 210 g/l, CU = 150 g/l, CPu = 90 g/l, Tin = 25 °C, 

Poven = 5 kWh.  
First of all, was calculated the necessary time for the process. The duration of the first stage of the process is 
1.6 hours and of the second stage is 3.6 h. The result variation of the solution temperature in the microwave 
oven is shown in Figure 3. 
 

 

Figure 3: Changing of the solution temperature in the microwave oven during thermal denitration 

In order to maintain the solution temperature at the desired level, the power of the microwave oven is modify 
during the process. As reported in Figure 3, at time 1.6 h the solution temperature increases since the second 
stage of denitration process (300 – 350 °С). The results of simulations are shown in Figure 4.  

    
Changing of the mass and concentrations actinide oxides during the thermal denitration 

869



    
Changing of the volume and concentrations mixture of off-gas during the thermal denitration  

Figure 4: The modelling results 

As shown in Figure 4, the formation of oxides actinides begins at time of 2 h, when the temperature of the 
solution in the microwave oven reaches 300°С and when the chemical processes, described in equations (2), 
are activated. Initially there is the formation of uranium trioxide (UO3), and then of plutonium dioxide (PuO2). 
This is due to the different concentrations of uranium and plutonium in the feed solution, as well as because of 
the different values of the rate constants of chemical reactions.  
In addition, Figure 4 reports the change of volume and concentrations of the off-gas. As seen from the graphs, 
the evaporation of water and nitric acid occurs in the first stage of the process. Nitric acid starts to evaporate 
before, since its boiling point is 80 °С. The formation of nitric acid is terminated at the second stage of the 
process. But begins formation nitrogen dioxide and oxygen according to the chemical reactions (2). Thus, the 
concentration of nitric acid and water is rather reduced in the off-gas 

5. Conclusions 

This article is devoted to the development of a mathematical model of a thermal denitration of actinides 
nitrates in a microwave oven. The conceptual model has been developed as a result of analysis of the 
process. The paper presents a mathematical model that allows to calculate the optimal time of the denitration 
process, as well as the variations of the solution temperature. The developed system based on differential 
equations, allow to estimate productivity of the denitration method. The modeling results will be compared to 
experimental data in the near future. At present, experimental apparatus is develops. In the future, it is 
planned to use this model in order to optimize the process and to develop an efficient control system for the 
microwave oven. 

Acknowledgments  

This work was funded as a part of the project 8.3079.2017/ПЧ of the Federal government-sponsored program 
«Science». 

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