**3.1. Fuel 238U + 235U**

High fission nuclides reproduction in thermal reactors is possible if amount of capture acts in fission nuclides is close to amount of absorption acts in raw nuclides.

Dependence of multiplication factor and fission nuclides reproduction coefficient for reactors with different fuel types and no neutron loss is shown on figure 1.

Three types of fuel are shown. Blue lines show data for mix of 238U and 239Pu, red lines for mix of 238U and 235U, brown lines for mix of 232Th and 233U. Continuous lines show multiplication factor, dotted lines – reproduction coefficient. Left lines of each type show data of neatly thermal reactor without epithermal neutron absorption. Right lines show data for reactor with 10% epithermal neutron absorption from total absorptions.

Multiplication factor for such reactor with mix 238U and 235U is calculated as:

$$K = \mathbf{n}^\* \cdot \sigma\_{5\ell} \stackrel{\*}{\rightharpoonup} \nu \;/\; \langle (1 - \mathbf{n}) \stackrel{\*}{\rightharpoonup} \sigma\_{8\mathfrak{a}} + \mathbf{n} (\sigma\_{5\ell +} \sigma\_{5\mathfrak{a}}) \rangle ; \tag{5}$$

Where:

n – portion of isotope 235U in fuel; σ5f – fission cross-section of isotope 235U; σ5a – absorption cross-section of isotope 235U; σ8a – absorption cross-section of isotope 238U; ν – neutron amount at fission of 235U.

**Figure 1.** Dependence of multiplication factor and fission nuclides reproduction coefficient for reactors with different fuel types and no neutron loss.

Data for cross-sections and number of secondary neutrons, used in calculations, are taken from [5].

Fission materials reproduction coefficient in initial fuel is calculated:

$$\mathbf{KB} = \begin{pmatrix} \mathbf{1} - \mathbf{n} \end{pmatrix} \stackrel{\ast}{\ } \sigma\_{8\mathbf{a}} \text{ / \ n \ } ^ {\ast} \sigma\_{5\mathbf{i}'} \text{\.} \tag{6}$$

From data, which shown at figure 1 for hypothetical reactor, it can be seen that there is diapason of uranium enrichment (from 0.46 up to 0.66 %) in which reproduction coefficient and multiplication factor are more than unity simultaneously. For each of shown variants at reproduction coefficient equal unity multiplication factor is close to 1.1.

To estimate possibility of real reactor work at this diapason of enrichment is necessary to take into account two factors – presence of additional neutron losses in construction materials and neutron leakage, and production influence of secondary fission materials, actinides and fission products, which are sufficient neutron absorbers.

Estimation of these factors in point model of reactor is possible by introduction of neutron loss in construction materials and leakage term in system of equations, describing accumulation and neutron absorption in initial fuel nuclides and additional nuclides produced during reactor work.

Sufficiently precise estimation of reactor campaign characteristics with different fuel types can be made taking into account following nuclides:


182 Nuclear Power – Practical Aspects

usage.

ways simultaneously is possible.

spent fuel reprocessing.

**of fission materials** 

**3.1. Fuel 238U + 235U** 

Where:

n – portion of isotope 235U in fuel; σ5f – fission cross-section of isotope 235U; σ5a – absorption cross-section of isotope 235U; σ8a – absorption cross-section of isotope 238U;

ν – neutron amount at fission of 235U.

The shortage, connected with impossibility of cheap extraction of fission isotope 235U from raw isotope 238U during spent fuel reprocessing, can be overcome by two ways. The first way is in increasing of initial 235U burn-up, so at the end of campaign its contents is negligibly small. Another way is in using fuel with different nucleus charge – using fission isotope of uranium in thorium, fission isotopes of plutonium in raw 238U. Usage of both

Because of fission isotopes absence in raw thorium there is no problem with raw thorium

Known stocks of thorium are bigger than known stocks of uranium. It will be ideal if nuclear power industry use both elements in its work. Basic raw material in the present time is uranium. Spending cheap stocks of uranium will lead to necessity of depleted uranium

One of the problems with thorium fuel is 232U production, which is source of high energy gamma-ray quanta [4]. One way of this problem solution is usage of automatics at thorium

**3. Characteristics of hypothetical thermal reactors with high reproduction** 

High fission nuclides reproduction in thermal reactors is possible if amount of capture acts

Dependence of multiplication factor and fission nuclides reproduction coefficient for

Three types of fuel are shown. Blue lines show data for mix of 238U and 239Pu, red lines for mix of 238U and 235U, brown lines for mix of 232Th and 233U. Continuous lines show multiplication factor, dotted lines – reproduction coefficient. Left lines of each type show data of neatly thermal reactor without epithermal neutron absorption. Right lines show data

5f 8a 5f 5a *К* n \* \* / ((1 n) \* n( ));

 

 (5)

in fission nuclides is close to amount of absorption acts in raw nuclides.

reactors with different fuel types and no neutron loss is shown on figure 1.

for reactor with 10% epithermal neutron absorption from total absorptions.

Multiplication factor for such reactor with mix 238U and 235U is calculated as:

 

reprocessing from dumps of enrichment plants and spent fuel and thorium usage.


In general, for each of nuclides the equation is solved:

$$\frac{d\mathbf{Nz}}{dt} = \lambda\_{z-1} \cdot \mathbf{N}\_{z-1} - \lambda\_z \cdot \mathbf{N}\_z - \sigma\_z \cdot \mathbf{N}\_z \cdot \Phi;\tag{7}$$

Thermal Reactors with High Reproduction of Fission Materials 185




100



13 1.42 0.002 0.054 29.5 63.6 5.43 2.15 no

9+13 12 1.80 0.009 0.034 37.26 4.08 44.26 14.40

1.7 44 5.27 0.001 0.017 5.05 0.16 12.70 67.69 14.40 5.27

27 3.13 0.001 0.012 4.10 0.10 72.83 22.99

25 2.52 0.013 0.018 17.52 68.37 14.10 3.81

29 3.38 0.006 0.024 19.78 67.31 12.90 3.38

15 1.79 0.0003 0.009 73.4 26.60 - 100

29 3.51 0.001 0.017 1.89 0.04 4.69 71.22 22.16 14.28

24 3.34 0.005 0.025 8.35 69.24 22.41 7.61

10 1.26 0.001 0.085 45.13 50.91 3.95 1.26

34 4.12 0.001 0.01 4.79 0.13 16.23 65.66 13.19 4.12

№ С FM F M R M Mode AKM Φ T Y Rmin R SZ U3 U53 U55 Pu9 Pu1 Qu op Qu sh

16 4.09 0.001 0.02

2 0,37 Pu9 13 1.63 0.008 0.021 76.6 23.40

4 0,35 Pu 9, 1 U8 13 1.56 0.002 0.03 73.5 26.50

8 5+13 25 2.15 0.008 0.030 40.54 3.98 42.33 13.14

10 0,712 34 3.92 0.010 0.027 17.07 69.05 13.87 3.92

13 2.8 24 2.84 0.008 0.029 23.52 64.91 11.55 2.84 14 5.2 19 2.41 0.002 0.026 27.72 62.22 10.05 2.41

16 0,47 0.0 34 3.47 0.012 0.027 5.27 0.13 12.70 64.30 17.60 5.26

19 ,35+,35 34 4.22 0.005 0.024 2.12 0.06 7.78 69.14 20.90 8.58 20 ,2+,41 U8(1.2) 24 3.49 0.004 0.014 2.28 0.04 5.52 69.62 22.54 12.12

22 ,35+,35 U8 24 3.08 0.016 0.037 10.69 67.46 21.84 6.27

25 1.65 U3, 5 Th 5.0 6+13 24 4.35 0.015 0.044 92.58 7.42 - 100

Reactor power with constant neutron flux increases during campaign. Power has peak of 28 % and at the end of campaign is 20 % greater than initial value. It is caused by 239Pu accumulation, which has bigger fission cross-section than 235U, and 239Pu contents

Contents of 239Pu becomes stable during campaign but its value is less than initial contents of 235U in fuel. It shows necessity of taking into account characteristics difference between fission nuclides at calculation of reproduction coefficient. With use of formula (2) and 235U fission characteristics reproduction coefficient equal to unity is calculated. At the same

1 0,47 U5

1,65 U3, 5 Th

0,68 U + Pu U8+Th

U5

U5

U5+Pu 9, 1

0,712 U5 U8

**Table 1.** Campaign characteristics

21 ,31+,39 U8(1.1) Super-

5

7

12

15

23

24

9 0,47

11 0,35 Pu 9, 1

0,712

17 0,35 Pu 9, 1

18 ,17+,35

U8

U8

Line

Superposition

Super-Positio n + U3 generation

position

Line

S P G

stabilization with decreasing portion of 235U.

0.0

1.7

1.7

5.2 9+13

conditions reproduction coefficient for 239Pu is less unity (0.786).

9+13

3 1,65 U3 Th 39 8.78 0.019 0.028 90.9 9.10

6 5+13 38 4.89 0.028 0.046 90.4 9.60

9+13

where:

Nz – number of nuclei with charge z in fuel;

λ – decay constant;

σ – neutron cross-section absorption;

Ф – neutron flux in fuel

Calculations of campaign of point reactor model with these conditions are made with the program [6]. The program takes into account fission possibility for 238U, 232Th, 233Pa, 234U, 236U, 237U, 239Np, 240Pu, 242Pu [5].

Results of reactor campaign calculation with 0.47 % initial contents of 235U in fuel and absence of epithermal neutrons absorption in 238U are shown at figure 2. Here (and everywhere in analogous cases) contents of fission nuclides (235U и 239Pu) is normalized on initial contents of 235U in fuel, and fuel power – on its initial value.

For representative comparison of campaigns is made Table 1. It includes:

C FM - the content of the base fission materials at the campaign beginning, %;

F M – Fission materials in fuel;

R M – raw fuel nuclides;

Mode – campaign conducting features:

Line – the simplest flow of the campaign (without fuel replacements);

Sup Poz – campaign with the joint work of fuel with different burn-up (look at i.5);

Sup Poz +233U Gen – campaign with generation of 233U (look at i. 6);

AKM – the absorption of neutrons in construction materials and leakage, %;

Φ – neutron flux, sm-2s-1;

Т – campaign duration, hours;

Y – fuel burn-up, %.

Rmin – minimum operational reactivity during the campaign, %;

R SZ – integral reactivity at the end of campaign, %;

U3 – portion of 233U fissions from total fissions;

U53 – portion of 235U fissions, which formed from 233U, from total fissions;

U55 – portion of 235U fissions, which is from natural uranium, from total fissions;

Pu9 – portion of 239Pu fissions from total fissions;

Pu1 – portion of 241Pu fissions from total fissions;

Qu op –portion of raw uranium usage in open fuel cycle;

Qu sh – portion of raw uranium usage in closed fuel cycle;


#### **Table 1.** Campaign characteristics

184 Nuclear Power – Practical Aspects

where:

λ – decay constant;

Ф – neutron flux in fuel

Eu); 155 (Sm, Eu, Cd); 157 (Eu, Cd).

Nz – number of nuclei with charge z in fuel;

σ – neutron cross-section absorption;

236U, 237U, 239Np, 240Pu, 242Pu [5].

F M – Fission materials in fuel; R M – raw fuel nuclides;

Φ – neutron flux, sm-2s-1; Т – campaign duration, hours;

Y – fuel burn-up, %.

Mode – campaign conducting features:

In general, for each of nuclides the equation is solved:

*dt*

initial contents of 235U in fuel, and fuel power – on its initial value.

For representative comparison of campaigns is made Table 1. It includes:

C FM - the content of the base fission materials at the campaign beginning, %;

Line – the simplest flow of the campaign (without fuel replacements);

Sup Poz +233U Gen – campaign with generation of 233U (look at i. 6); AKM – the absorption of neutrons in construction materials and leakage, %;

U53 – portion of 235U fissions, which formed from 233U, from total fissions; U55 – portion of 235U fissions, which is from natural uranium, from total fissions;

Rmin – minimum operational reactivity during the campaign, %;

R SZ – integral reactivity at the end of campaign, %; U3 – portion of 233U fissions from total fissions;

Pu9 – portion of 239Pu fissions from total fissions; Pu1 – portion of 241Pu fissions from total fissions;

Qu op –portion of raw uranium usage in open fuel cycle; Qu sh – portion of raw uranium usage in closed fuel cycle;



*dNz N NN*

Calculations of campaign of point reactor model with these conditions are made with the program [6]. The program takes into account fission possibility for 238U, 232Th, 233Pa, 234U,

Results of reactor campaign calculation with 0.47 % initial contents of 235U in fuel and absence of epithermal neutrons absorption in 238U are shown at figure 2. Here (and everywhere in analogous cases) contents of fission nuclides (235U и 239Pu) is normalized on

Sup Poz – campaign with the joint work of fuel with different burn-up (look at i.5);

1 1 ; *z z zz zz*

(7)

Reactor power with constant neutron flux increases during campaign. Power has peak of 28 % and at the end of campaign is 20 % greater than initial value. It is caused by 239Pu accumulation, which has bigger fission cross-section than 235U, and 239Pu contents stabilization with decreasing portion of 235U.

Contents of 239Pu becomes stable during campaign but its value is less than initial contents of 235U in fuel. It shows necessity of taking into account characteristics difference between fission nuclides at calculation of reproduction coefficient. With use of formula (2) and 235U fission characteristics reproduction coefficient equal to unity is calculated. At the same conditions reproduction coefficient for 239Pu is less unity (0.786).

**Figure 2.** Reactor campaign characteristics with initial contents 235U in fuel 0.47 % and absence of absorption in 238U on epithermal neutrons. (string 1 of Attachment Table 1)

Thermal Reactors with High Reproduction of Fission Materials 187

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2

**K-1**

Portion of 239Pu is 0.37 % from total mass of these nuclides and in 238U there is no absorptions

Contents of 239Pu during campaign is stable enough, but not equal to initial one and decreasing for ~25 %. Stabilization of 239Pu is reached earlier than in the previous campaign. The role of 241Pu increases. Its amount increases to ~ 1/3 of 239Pu at stationary level. Value of reactivity margin during this campaign is less, but its fluctuation is also less. Despite less contents of fission material in initial fuel slightly higher burn-up is reached after the same

Decreasing of 239Pu amount and reactivity margin in the first hours of campaign is caused by delay of transformation of 239U into 239Pu, and significant neutron losses in the chain of 241Pu. It is important to make comparison with fuel characteristics on the base of mix 233U

For the fuel on the base of mix 232Th + 233U the region with reproduction coefficient equal to unity relocates to higher contents of fission materials. Multiplication factor in this region

Reactor campaign characteristics with 233U + 232Th in initial fuel is shown at figure 4. Contents change of 233U during campaign is not big. Reactor power change is also not big. But power is decreasing at the campaign beginning and after that returns to its initial value. Power decrease at the campaign beginning is caused by 233U contents decrease, and return is caused by 235U accumulation. Comparatively small accumulation of 235U is well explained by

**Figure 4.** Reactor campaign characteristics with initial contents of 233U in uranium-thorium fuel equal to

U 233 U 235 Pa 233 W K-1 SZ K-1 oper

0 5000 10000 15000 20000 25000 30000 35000 **time, h**

also shows increase comparing to variants with fuel on the base of 238U and 235U.

small neutron absorption cross-section of 233U, from which produces 235U.

in epithermal region.

**3.3. Fuel 232Th + 233U** 

1.55 % (string 3 of application's table 1).

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

**C FM**

work time.

+ 232Th.

### **3.2. Fuel 238U + 239Pu**

Region with reproduction coefficient equal unity for fuel on the base of mix 238U + 239Pu relocates to less value of fission materials contents comparing to fuel on the base of 238U and 235U.

Multiplication factors for RC=1 is increased.

Reactor campaign characteristics with initial fuel containing 238U and 239Pu are shown at figure 3.

**Figure 3.** Reactor campaign characteristics with initial contents of 239Pu in uranium-plutonium fuel 0.37 % and absence of absorption in 238U of epithermal neutrons (string 2 of application's table 1).

Portion of 239Pu is 0.37 % from total mass of these nuclides and in 238U there is no absorptions in epithermal region.

Contents of 239Pu during campaign is stable enough, but not equal to initial one and decreasing for ~25 %. Stabilization of 239Pu is reached earlier than in the previous campaign. The role of 241Pu increases. Its amount increases to ~ 1/3 of 239Pu at stationary level. Value of reactivity margin during this campaign is less, but its fluctuation is also less. Despite less contents of fission material in initial fuel slightly higher burn-up is reached after the same work time.

Decreasing of 239Pu amount and reactivity margin in the first hours of campaign is caused by delay of transformation of 239U into 239Pu, and significant neutron losses in the chain of 241Pu. It is important to make comparison with fuel characteristics on the base of mix 233U + 232Th.
