# Thermal-Group Neutron Fluxes

When only the diffusion coefficients (D_{1} , D_{2}) are fuzzy and other parameters are crisp, then in the region 0-40 cm, left, centre and crisp values are close to each other. From Figure 8.8, it is noticed that there is a deviation of the centre and crisp values from 40 to 160 cm. After that, the centre and crisp values converge. We see that there is a sudden deviation (increase in the uncertain width) of right values in 80-160 cm and 160-240 cm. Similarly, there is sudden deviation of left values in 40-120 cm and 120-200 cm.

In the case where only the neutron interaction coefficients (?_{r1}, *~L _{a2},* ?

_{12}) are fuzzy, the distribution of uncertain thermal-group flux is shown in Figure 8.10. It is seen that the left, centre and crisp values are the same in the region 0-40 cm. From 40 to 120 cm, there is a sudden deviation of the left value. Then, from 120 to 160 cm the values are the same, and then there is a deviation. Similarly, the right values show deviation throughout the domain except in the region 200-240 cm, where the right, centre and crisp values are the same. Finally, it may be seen that the distribution of centre and crisp thermal-group fluxes is almost the same along the domain.

Next, we have considered the group fission constants (vf *vL _{f2})* as only fuzzy, and the uncertain thermal-group fluxes are depicted in Figure 8.12. Here, the centre and right values are the same in 0-80 cm and 160-240 cm. The centre and left values are almost the same in 0-80 cm. In Figure 8.14, we see a continuous fluctuation of the centre and crisp values throughout the domain.

Finally, all the parameters are taken as fuzzy and the obtained results are depicted in Figure 8.4, where the crisp and centre values are very close. There is a sudden left and

**Neutron Diffusion: Concepts and Uncertainty Analysis for Engineers and Scientists**

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right deviation between 40-120 cm and 80-160 cm, respectively. From 0 to 80 cm, the right, centre and crisp values almost overlap.

Further, to check the sensitiveness of the used parameters, the deviation of left and right values from the centre and crisp values is shown graphically in Figures 8.14 through 8.17. In view of these four figures, it is concluded that when the group fission constants are fuzzy, the system is sensitive. A slight change in the parameter drastically changes the distribution of thermal-group fluxes. The width of the uncertain thermal-group fluxes increases with larger width values of ?_{r1}, *~L _{a2},* S

_{12}.