Nodal diffusion method
The finite difference method is widely used in the design calculation of fast reactors, the analysis of critical assembly experiments, and so on. For fast reactors, convergence of the outer iteration is fast due to the long mean free paths of neutrons, and moreover, the nuclear and thermalhydraulic coupled core calculation is not needed. Hence, the computation speed required for the design calculation can be achieved even with fine meshes in the finite difference method. For conventional LWRs as shown in Fig. 2.22, however, a fine meshing of 12 cm is necessary to obtain a highaccuracy solution because the neutron mean free path is short.
For example, a 3D fine meshing of a PWR core of over 30,000 liters leads to a formidable division of several tens of millions of meshes even though
Fig. 2.22 Comparison of conventional LWRs and critical assembly (TCA) in size
excluding the reflector region. Furthermore, the neutron diffusion equation is repeatedly solved in the nuclear and thermalhydraulic coupled core calculation, core burnup calculation, spacedependent kinetics calculation, and so on. Hence, a calculation using the 3D finite difference method with such a fine meshing is extremely expensive even on a current highperformance computer and therefore it is impractical. The nodal diffusion method [18] was, therefore, developed to enable a highspeed and highaccuracy calculation with a coarse meshing comparable to a fuel assembly pitch (about 20 cm). It is currently the mainstream approach among LWR core calculation methods [19].
The numerical solution of the nodal diffusion method is somewhat complicated and is not discussed here. The main features of the approach are briefly introduced instead.
(i) Since the coarse spatial mesh (node) is as large as a fuel assembly pitch, the number of unknowns is drastically reduced compared with that of the finite difference method.
(ii) The 3D diffusion equation in a parallelepiped node (k) is integrated over all directions except for the target direction and then is reduced to a 1D diffusion equation including neutron leakages in its transverse directions. For example, the diffusion equation in the x direction is expressed as
Fuel Type 1
Fuel Type 2
Reflective Boundary
Fig. 2.23 3D benchmark problem of IAEA
—D (.х^+ХІфІ (x) = Sk Gc)—^~^kLy (л) + "“]Г^ (x) (291)
where Ay and Akz are the mesh widths in y and z directions, respectively. Lky (x) and Lkz (x) represent the neutron leakage in each direction. These unknown functions are provided by a secondorder polynomial expansion using the transverse leakages of two adjacent nodes.
(iii) Typical solutions to the 1D diffusion equation of Eq. (2.91) are (a) the analytic nodal method [20] for twogroup problems, (b) the polynomial expansion nodal method [18, 21] to expand фкх(x) into a about fourth — order polynomial, and (c) the analytic polynomial nodal method [22, 23] to expand SX(x) into a secondorder polynomial and to express фХ(x) as an analytic function.
A 3D benchmark problem [24] given by the IAEA as shown in Fig. 2.23 is taken as a calculation example for the suitability of the nodal diffusion method. A PWR core model is composed of two types of fuels and control rods are inserted at five locations of the quadrant inner fuel region. At one location, control rods are partially inserted to 80 cm depth from the top of the core. The meshing effect on the power distribution is relatively large in this case and hence this benchmark problem has been widely employed to verify diffusion codes.
The calculation results using MOSRALight code [21] which is based on the fourthorder polynomial nodal expansion method are shown in Fig. 2.24 for two
pr = 1.02903) 
0.597 

•SRALight 
1.31 

02897) 
F 
0.51 

>SRALight 
0.476 
0.700 
0.611 

02909) 
— 0.73 
0.37 
— 1.02 

0.01 
0.15 
0.32 

У 
1.178 
0.972 
0.923 
0.866 

0.39 
0.33 
— 0.40 
— 0.94 

P 
0.16 
0.13 
0.03 
0.22 

1.368 
1.311 
1.181 
1.089 
l.(KK) 
0.711 

Position 
0.46 
0.53 
0.30 
0.16 
0.79 
— 1.29 

— 0.26 
— 0.24 
0.16 
0.06 
0.03 
0.26 

1.397 
1.432 
1.291 
1.072 
1.055 
0.976 
0.757 

0.47 
0.50 
0.39 
0.35 
0.09 
0.72 
— 1.24 

0.32 
— 0.34 
0.29 
0.22 
0.09 
— 0.03 
0.16 

0.729 
1.281 
1.422 
1.193 
0.610 
0.953 
0.959 
0.777 

0.80 
0.52 
0.43 
0.54 
0.18 
0.16 
0.5 
— 0.99 

0.26 
0.31 
0.29 
0.22 
0.07 
0.05 
0.07 
0.13 
Fig. 2.24 Comparison of effective multiplication factor and assembly power distribution by nodal diffusion method [21] 
mesh sizes (20 and 10 cm). The reference solution has been taken by an extrapolation to zero size from the five calculations with different mesh sizes using a finite difference method code. For the effective multiplication factor, the discrepancy with the reference value is less than 0.006 % Ak in either case and thus the meshing effect can be almost ignored. For the assembly power, the discrepancy in the case of 20 cm mesh is as small as 0.6 % on average and 1.3 % at maximum. The discrepancy becomes smaller than 0.5 % on average in the case of a mesh size of 10 cm or less. It is noted that the finite difference method requires a mesh size smaller than 2 cm and more than 100 times longer computation time to achieve the same accuracy as that in the nodal diffusion method.
Thus the IAEA benchmark calculation indicates the high suitability of the nodal diffusion method to LWR cores which have fuel assemblies of about 14— 21 cm size. An idea of the nodal diffusion method is its approach to decompose the reactor core into relatively large nodes and then to determine the neutron flux distribution within each node to maintain the calculation accuracy. For example, the polynomial nodal expansion method introduces the weighted residual method to obtain highorder expansion coefficients. It leads to an increased number of equations to be solved.
In other words, the high suitability of the nodal diffusion method to practical LWR calculation results is because the computation cost reduction due to a substantial decrease in the number of meshes surpasses the cost rise due to an increase in number of equations. Conversely, if it is possible to reach sufficient accuracy with the same meshing, the finite difference method will be effective
Enthalpy rise Void fraction Pressure drop Inlet flow rate
Fig. 2.25 Nuclear and thermalhydraulic coupled core calculation of LWR
because it is not necessary to solve extra equations. Hence, the nodal diffusion method does not need to have an advantage over the finite difference method for the analysis of fast reactors or small reactors.