Previous inflow transport approximation

The previous inflow transport approximation was developed by Choi, which is based on the moderator region and normalized fission source. In this method, the multi-group form of the one-dimensional P_N transport equation is derived as follows: $\mu \frac{\partial {\psi }_g\left(z,\mu \right)}{\partial z}+{\Sigma }_{\text{t},g}\left(z,\mu \right){\psi }_g\left(z,\mu \right)=\frac{1}{2k_{\text{eff}}}{\chi }_g{\displaystyle \sum _{g=1}^G\upsilon {\Sigma }_{\text{f},g}\left(z\right){\phi }_g^0\left(z\right)}+{\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}{P}_{\text{l}}\left(\mu \right)}{\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^l\left(z\right){\phi }_{g\text{'}}^l\left(z\right)}$ (1) where ${\Sigma }_{\text{t},g}\left(z,\mu \right){\psi }_g\left(z,\mu \right)={\displaystyle {\int }_g{\Sigma }_{\text{t}}\left(E\right)\psi \left(z,\mu ,E\right)}\text{d}E={\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}{P}_l\left(\mu \right){\Sigma }_{\text{t},g}^l\left(z\right)}{\varphi }_g^l\left(z\right)$ (2) ${\Sigma }_{\text{t},g}^l\left(z\right)=\frac{{\displaystyle {\int }_g{\Sigma }_{\text{t}}\left(E\right){\varphi }_l\left(z,E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(z,E\right)\text{d}E}}$ (3) ${\Sigma }_{s,g\text{'}\to g}^l\left(z\right)=\frac{{\displaystyle {\int }_{g\text{'}}\text{d}E\text{'}}{\displaystyle {\int }_g{\Sigma }_{\text{s}}\left(E\text{'}\to E\right){\varphi }_l\left(z,E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(z,E\right)\text{d}E}}$ (4) ${\psi }_g\left(z,\mu \right)={\displaystyle {\int }_g\psi \left(z,E,\mu \right)\text{d}E}={\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}{P}_{\text{l}}\left(\mu \right)}{\varphi }_g^l\left(z\right)$ (5) where ${\psi }_g$ is the angular flux in group g, $\mu$ is the cosine angle in the z-direction, $P_{\text{l}}\left(\mu \right)$ is the Legendre polynomial, l is the order of the Legendre polynomial, ${\Sigma }_{x,g}^l$ is the l-th moment macroscopic XS of x reaction (f: fission, t: total, s: scattering), and ${\varphi }_g^l$ is the l-th flux moment.

It should be noted that the distinction between ${\Sigma }_{\text{t},g}^0$ and ${\Sigma }_{\text{t},g}^n$ (n=1,2…) disappears in the absence of self-shielding (i.e., infinity dilution) [16].

The multigroup transport equation in Eq. (1) can be extended using Eqs. (2)–(5), as follows: $\begin{array}{l}\mu \frac{\partial }{\partial z}{\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}}{P}_{\text{l}}\left(\mu \right){\varphi }_g^l\left(z\right)+{\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}}{P}_{\text{l}}\left(\mu \right){\Sigma }_{\text{t},g}^l\left(z\right){\varphi }_g^l\left(z\right)\\ =\frac{1}{2k_{\text{eff}}}{\chi }_g{\displaystyle \sum _{g=1}^G\upsilon {\Sigma }_{\text{f},g}\left(z\right){\phi }_g^0\left(z\right)}\\ +{\displaystyle \sum _{l=0}^{\infty }\frac{2l+1}{2}{P}_{\text{l}}\left(\mu \right)}{\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^l\left(z\right){\phi }_{g\text{'}}^l\left(z\right)}.\end{array}$ (6)

Multiplying Eq. (6) by the Legendre polynomial $\left(2n+1\right)P_n\left(\mu \right)$ and integrating $\mu$ from–1 to +1, and using the following Bonnet’s recursion formula and the orthogonality of the Legendre polynomials: $\left(2n+1\right)\mu P_n\left(\mu \right)=(n+1)P_{n+1}\left(\mu \right)+n{P}_{n-1}\left(\mu \right),$ (7) ${\displaystyle {\int }_{-1}^1{P}_n\left(\mu \right)P_l\left(\mu \right)\text{d}\mu =\frac{2}{2l+1}{\delta }_{nl}}.$ (8)

Inserting Eqs. (7) and (8) into Eq. (6) will lead to the following equation: $\begin{array}{l}\left(n+1\right)\frac{\partial {\varphi }_g^{n+1}\left(z\right)}{\partial z}+n\frac{\partial {\varphi }_g^{n-1}\left(z\right)}{\partial z}+\left(2n+1\right){\Sigma }_{\text{t},g}^n\left(z\right){\varphi }_g^n\left(z\right)\\ =\frac{\left(2n+1\right){\delta }_{0n}{\chi }_g}{k_{\text{eff}}}{\displaystyle \sum _{g=1}^{G}v{\Sigma }_{\text{f},g}\left(z\right){\varphi }_g^0\left(z\right)}\\ +\left(2n+1\right){\displaystyle \sum _{g=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n\left(z\right){\varphi }_{g\text{'}}^n\left(z\right)}.\end{array}$ (9)

Approximating the spatial shape using buckling B: ${\varphi }^n\left(z,E\right)={\varphi }^n\left(E\right)\times e^{iBz}.$ (10)

Inserting Eq. (10) into Eq. (9) will produce the following equation: $\begin{array}{l}\left(n+1\right)\frac{\partial {\varphi }_g^{n+1}\left(z\right)}{\partial z}+n\frac{\partial {\varphi }_g^{n-1}\left(z\right)}{\partial z}+\left(2n+1\right){\Sigma }_{\text{t},g}^n\left(z\right){\varphi }_g^n\left(z\right)\\ =\frac{\left(2n+1\right){\delta }_{0n}{\chi }_g}{k_{\text{eff}}}{\displaystyle \sum _{g=1}^{G}v{\Sigma }_{\text{f},g}\left(z\right){\varphi }_g^0\left(z\right)}\\ +\left(2n+1\right){\displaystyle \sum _{g=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n\left(z\right){\varphi }_{g\text{'}}^n\left(z\right)}.\end{array}$ (11)

Additionally, another approximation was introduced for Choi’s method. The fission source is forced to normalize, and the value of k is set to 1, leading to: ${\displaystyle \sum _{g=1}^{G}v{\Sigma }_{\text{f},g}(z){\varphi }_g^0}=1.$ (12)

In Choi’s method, Eq. (11) is then converted to the following equation: $\begin{array}{l}\left(n+1\right)iB{\varphi }_g^{n+1}+niB{\varphi }_g^{n-1}+\left(2n+1\right){\Sigma }_{\text{t},g}^n\left(z\right){\varphi }_g^n\\ =(2n+1){\delta }_{0n}{\chi }_g\\ +\left(2n+1\right){\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n\left(z\right){\varphi }_{g\text{'}}^n}.\end{array}$ (13)

Solving Eq. (13), the ${\varphi }_g^n$ then leads to the following: ${\varphi }_g^n=\frac{(2n+1){\delta }_{0n}{\chi }_g-\left(n+1\right)iB{\varphi }_g^{n+1}-niB{\varphi }_g^{n-1}}{\left(2n+1\right){\Sigma }_{\text{tr},g}^{n-1}\left(z\right)}.$ (14) where the transport XS ${\Sigma }_{\text{tr},g}^{n-1}(z)$ is written as: ${\Sigma }_{\text{tr},g}^{n-1}\left(z\right)={\Sigma }_{\text{t},g}^n\left(z\right)-\frac{{\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n\left(z\right){\varphi }_{g\text{'}}^n}}{{\varphi }_g^n}.$ (15)

When n is an even number and n-1 and n+1 are odd numbers.

For n-1: ${\varphi }_g^{n-1}=\frac{-niB{\varphi }_g^n-(n-1)iB{\varphi }_g^{n-2}}{\left(2n-1\right){\Sigma }_{\text{tr},g}^{n-2}\left(z\right)}.$ (16)

For n+1: ${\varphi }_g^{n+1}=\frac{-(n+2)iB{\varphi }_g^{n+2}-(n+1)iB{\varphi }_g^n}{\left(2n+3\right){\Sigma }_{\text{tr},g}^n\left(z\right)}.$ (17)

Inserting Eqs. (16) and (17) into Eq. (14) leads to the following: ${\varphi }_g^n=\frac{{\delta }_{0n}{\chi }_g-{\alpha }_{2,g}^n{B}^2{\varphi }_g^{n-2}-{\alpha }_{3,g}^n{B}^2{\varphi }_g^{n+2}}{{\Sigma }_{\text{tr},g}^{n-1}\left(z\right)-{\alpha }_{1,g}^n{B}^2}.$ (18) where ${\alpha }_{1,g}^n=-\frac{n+1}{2n+1}\frac{n+1}{2n+3}\frac{1}{{\Sigma }_{\text{tr},g}^n\left(z\right)}-\frac{n}{2n+1}\frac{n}{2n-1}\frac{1}{{\Sigma }_{\text{tr},g}^{n-2}\left(z\right)}$ (19) ${\alpha }_{2,g}^n=\frac{n-1}{2n-1}\frac{n}{2n+1}\frac{1}{{\Sigma }_{\text{tr},g}^{n-2}\left(z\right)}$ (20) ${\alpha }_{3,g}^n=\frac{n+1}{2n+1}\frac{n+2}{2n+3}\frac{1}{{\Sigma }_{\text{tr},g}^n\left(z\right)}$ (21)

If the order of n is the maximum value, ${\alpha }_{3,g}^n$ becomes 0. For the odd n, Eq. (14) becomes the following: ${\varphi }_g^n=-iB\frac{\left(n+1\right){\varphi }_g^{n+1}+n{\varphi }_g^{n-1}}{\left(2n+1\right){\Sigma }_{\text{tr},g}^{n-1}\left(z\right)}.$ (22)

The high-order flux moments and P₀ transport XS are then generated. Furthermore, the P₀ scattering matrix is calculated using the transport XS, and can be used in the transport computation. Only the corrected diagonal elements are as follows: $\Sigma {\text{'}}_{\text{s},g\to g}^n\left(z\right)={\Sigma }_{\text{s},g\to g}^n\left(z\right)-\left({\Sigma }_{\text{t},g}^n\left(z\right)-{\Sigma }_{\text{tr},g}^n\left(z\right)\right).$ (23)

The transport equation is related to the geometric value z. It was calculated for a dominated moderator material in each problem case, and showed a problem-dependent result.

Improved inflow transport approximation

From the above derivation, a key point in the inflow transport approximation is applying the accurate flux moment from a one-dimensional P_N transport equation, considering the leakage effect and anisotropic scattering effect. This is the same as the assembly homogenization techniques [17]. However, there are two terms (multi-group data and fission source) that are inconsistent with the theoretical model through the derivation of the previous inflow transport approximation.

When applying the flux shape with buckling B, the multigroup XS and scattering matrix in Eqs. (3) and (4) become the following: $\begin{array}{c}{\Sigma }_{\text{t},g}^l\left(z\right)=\frac{{\displaystyle {\int }_g{\Sigma }_{\text{t}}\left(E\right){\varphi }_l\left(z,E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(z,E\right)\text{d}E}}=\frac{{\displaystyle {\int }_g{\Sigma }_{\text{t}}\left(E\right){\varphi }_l\left(E\right)e^{iBz}\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(E\right)e^{iBz}\text{d}E}}\\ =\frac{{\displaystyle {\int }_g{\Sigma }_{\text{t}}\left(E\right){\varphi }_l\left(E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(E\right)\text{d}E}}={\Sigma }_{\text{t},g}^l\end{array}$ (24) $\begin{array}{l}{\Sigma }_{\text{s},g\text{'}\to g}^l\left(z\right)=\frac{{\displaystyle {\int }_{g\text{'}}\text{d}E\text{'}}{\displaystyle {\int }_g{\Sigma }_{\text{s}}\left(E\text{'}\to E\right){\varphi }_l\left(z,E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(z,E\right)\text{d}E}}\\ =\frac{{\displaystyle {\int }_{g\text{'}}\text{d}E\text{'}}{\displaystyle {\int }_g{\Sigma }_{\text{s}}\left(E\text{'}\to E\right){\varphi }_l\left(E\right)e^{iBz}\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(E\right)e^{iBz}\text{d}E}}\\ =\frac{{\displaystyle {\int }_{g\text{'}}\text{d}E\text{'}}{\displaystyle {\int }_g{\Sigma }_{\text{s}}\left(E\text{'}\to E\right){\varphi }_l\left(E\right)\text{d}E}}{{\displaystyle {\int }_g{\varphi }_l\left(E\right)\text{d}E}}={\Sigma }_{\text{s},g\text{'}\to g}^l\end{array}$ (25)

This shows that applying the flux shape with buckling B eliminates the geometrical value z in the derivation. Therefore, the previous one-dimensional P_N transport equation becomes a homogenized system. The cross-section and scattering matrix data are homogenized in the entire region, such as fuel, cladding, and moderator regions.

Excluding the homogenized system, the accurate fission source is also treated using homogenization techniques. Therefore, the transport equation in Eq. (11) is replaced by the following: $\begin{array}{l}\left(n+1\right)iB{\varphi }_g^{n+1}+niB{\varphi }_g^{n-1}+\left(2n+1\right){\Sigma }_{\text{t},g}^n{\varphi }_g^n\\ =\frac{(2n+1){\delta }_{0n}{\chi }_g}{k}{\displaystyle \sum _{g=1}^{G}v{\Sigma }_{\text{f},g}{\varphi }_g^0}\\ +\left(2n+1\right){\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n{\varphi }_{g\text{'}}^n}\end{array}$ (26)

The solution of flux moment then leads to the following: ${\varphi }_g^n=\frac{Q_{\text{f},g}-\left(n+1\right)iB{\varphi }_g^{n+1}-niB{\varphi }_g^{n-1}}{\left(2n+1\right){\Sigma }_{\text{tr},g}^{n-1}}$ (27) where $Q_{\text{f},g}=\frac{(2n+1){\delta }_{0n}{\chi }_g}{k}{\displaystyle \sum _{g=1}^{G}v{\Sigma }_{\text{f},g}{\varphi }_g^0}$ (28) ${\Sigma }_{\text{tr},g}^{n-1}={\Sigma }_{\text{t},g}^n-\frac{{\displaystyle \sum _{g\text{'}=1}^G{\Sigma }_{\text{s},g\text{'}\to g}^n{\varphi }_{g\text{'}}^n}}{{\varphi }_g^n}$ (29)

When n is an even number and n-1 and n+1 are odd numbers.

For n-1: ${\varphi }_g^{n-1}=\frac{-niB{\varphi }_g^n-(n-1)iB{\varphi }_g^{n-2}}{\left(2n-1\right){\Sigma }_{\text{tr},g}^{n-2}}.$ (30)

For n+1: ${\varphi }_g^{n+1}=\frac{-(n+2)iB{\varphi }_g^{n+2}-(n+1)iB{\varphi }_g^n}{\left(2n+3\right){\Sigma }_{\text{tr},g}^n}.$ (31)

Inserting Eqs. (30) and (31) into Eq. (27) leads to the following: ${\varphi }_g^n=\frac{Q_{\text{f},g}-{\alpha }_{2,g}^n{B}^2{\varphi }_g^{n-2}-{\alpha }_{3,g}^n{B}^2{\varphi }_g^{n+2}}{{\Sigma }_{\text{tr},g}^{n-1}-{\alpha }_{1,g}^n{B}^2}.$ (32) where ${\alpha }_{1,g}^n=-\frac{n+1}{2n+1}\frac{n+1}{2n+3}\frac{1}{{\Sigma }_{\text{tr},g}^n}-\frac{n}{2n+1}\frac{n}{2n-1}\frac{1}{{\Sigma }_{\text{tr},g}^{n-2}}$ (33) ${\alpha }_{2,g}^n=\frac{n-1}{2n-1}\frac{n}{2n+1}\frac{1}{{\Sigma }_{\text{tr},g}^{n-2}}$ (34) ${\alpha }_{3,g}^n=\frac{n+1}{2n+1}\frac{n+2}{2n+3}\frac{1}{{\Sigma }_{\text{tr},g}^n}$ (35)

If the order of n is the maximum value, ${\alpha }_{3,g}^n$ becomes 0. For the odd n, Eq. (27) becomes the following: ${\varphi }_g^n=-iB\frac{\left(n+1\right){\varphi }_g^{n+1}+n{\varphi }_g^{n-1}}{\left(2n+1\right){\Sigma }_{\text{tr},g}^{n-1}}.$ (36)

Therefore, the flux moment and transport XS were obtained using the improved inflow transport approximation. Compared with the form of the previous inflow transport equation, the improved inflow transport equation was based on the homogenization XS and accurate fission source, which is consistent with the homogenization techniques.

Implementation of improved inflow transport approximation

In Choi’s parametric study, the inflow transport approximation is insensitive to the buckling and order of the Legendre polynomial. For the value of buckling, even with a ten-fold difference in buckling, the influence of k-effective is less than 2 pcm error. Therefore, the typical buckling value (B²=0.0001) was used for the calculation. For the order of the Legendre polynomial, the P₁ transport equation is sufficient to calculate the converged P₀ transport XS, which has the same impact as the P₅ transport equation [9].

In the improved inflow transport approximation, the typical buckling value and P₁ transport equation were used. The detailed iteration process is as follows.

(a) Obtain the homogenization data based on the outflow transport approximation in an actual problem. The DRAGON code produces the following homogenization data: ${\overline{\varphi }}_g^0=\frac{{\displaystyle \sum _i{\varphi }_{i,g}^0{V}_i}}{{\displaystyle \sum _i{V}_i}},{\overline{\Sigma }}_{x,g}^0=\frac{{\displaystyle \sum _i{\Sigma }_{x,g,i}^0{\varphi }_{i,g}^0{V}_i}}{{\displaystyle \sum _i{\varphi }_{i,g}^0{V}_i}},{\overline{\Sigma }}_{\text{s},g\text{'}\to g,i}^0=\frac{{\displaystyle \sum _i{\Sigma }_{\text{s},g\text{'}\to g,i}^0{\varphi }_{i,g}^0{V}_i}}{{\displaystyle \sum _i{\varphi }_{i,g}^0{V}_i}}$ (37) ${\overline{\varphi }}_g^n={\overline{\varphi }}_{g,\mathrm{mod}}^n\text{, }{\overline{\Sigma }}_{x,g}^n={\overline{\Sigma }}_{x,g,\mathrm{mod}}^n,{\overline{\Sigma }}_{\text{s},g\text{'}\to g}^n={\overline{\Sigma }}_{\text{s},g\text{'}\to g,\mathrm{mod}}^n,\text{}\left(n=1,2\right)$ (38) where i is used for every region, mod is the moderator region, x is represented as the reaction (f: fission, t: total, s: scattering), and n is the order of the Legendre polynomial (the anisotropic scattering effect for non-light nuclides is ignored).

(b) Initial flux moments and transport XS. $B^2={10}^{-4},{\overline{\varphi }}_g^n=0\text{ (n=1,2), }{\overline{\Sigma }}_{\text{tr},g}^n={\overline{\Sigma }}_{\text{t},g}^n\left(n=0,1,2\right)$ (39)

(c) Update the P₀ and P₂ flux moments using Eq. (32), and the flux moments converge. ${\overline{\varphi }}_g^n=\frac{Q_{\text{f},g}-{\alpha }_{2,g}^n{B}^2{\overline{\varphi }}_g^{n-2}-{\alpha }_{3,g}^n{B}^2{\overline{\varphi }}_g^{n+2}}{{\overline{\Sigma }}_{\text{tr},g}^{n-1}-{\alpha }_{1,g}^n{B}^2}\left(n=0\text{ and }2\right)$ (40)

(d) Update the P₁ flux moments. ${\overline{\varphi }}_g^1=\frac{-2iB{\overline{\varphi }}_g^2-iB{\overline{\varphi }}_g^0}{3{\overline{\Sigma }}_{\text{tr},g}^0}$ (41)

(e) Update the P₀ transport XS. ${\overline{\Sigma }}_{\text{tr},g}^0={\overline{\Sigma }}_{\text{t},g}^1-\frac{{\displaystyle \sum _{g\text{'}=1}^G{\overline{\Sigma }}_{\text{s},g\text{'}\to g}^1{\overline{\varphi }}_{g\text{'}}^1}}{{\overline{\varphi }}_g^1}$ (42)

(f) Repeat steps (b)–(e) until the flux moments and transport XS converge.

In the above procedure, the flux moments and transport XS may be negative. The iterative equation is as follows:. ${\overline{\varphi }}_g^{n,(k+1)}=0.2{\overline{\varphi }}_g^{n,(k)}+0.8{\overline{\varphi }}_g^{n,(k-1)}$ (43) ${\overline{\Sigma }}_{\text{tr},g}^{n,(k+1)}=0.2{\overline{\Sigma }}_{\text{tr,}g}^{n,(k)}+0.8{\overline{\Sigma }}_{\text{tr},g}^{n,(k-1)}$ (44) where k is the iterative number.

In this study, the investigation focused only on the dominant moderator material (H₂O). Therefore, the transport XS for H₂O was selected for the improved transport approximation. A detailed discrepancy analysis between these transport approximations is presented in the next section.

General description of verification

The lattice code DRAGON5.0.5 was used for verification. It was developed by École Polytechnique de Montréal for lattice calculations, solving the neutron transport equation with a deterministic approach, and can read different types of multi-group libraries, such as the versions of WIMS-D and MATXS. In this study, the WIMS-D 70 library [20] based on ENDF/B-VII.0 [21] was selected to study the influence of the anisotropic scattering effect. Different transport approximations are implemented in the WIMS-D 70 library through the transport XS and scattering matrix. Resonance self-shielding is an equivalence theory (SHI+LJ+LEVEL2) with the generalized Stamm’ler method [22], Livolant and Jeanpierre normalization factor, and Riemann integration [23]. The neutron transport calculation method was the interface current method (SYBILT). cosRMC was used as the reference code. The results of the k-infinity, k-effective, and P₀ flux moments are compared for various verification problems.

Typical LWR fuel cell and fuel assembly cases

A typical LWR fuel cell and fuel assembly were used to verify the application of different transport approximations. The factors of the fuel cell, fuel assembly, boron concentration, fuel enrichment, control rod material, and temperature in the fuel assembly with a control rod were selected as verification cases.

The single fuel cell and 17×17 fuel assembly benchmark were extracted from problem-1A and 2A in VERA [24], which contained 3.1 wt% UO₂ fuel, 0.743 g/cc moderators with a 1,300 ppm boron concentration, and Zr-4 for cladding. The temperature was 300 K for all regions. The geometric shapes for 1A and 2A are shown in Fig. 3, and a detailed description of each case is given in Table 2.

Fig. 3Full-screen View

(Color online) Geometry shape of 1A(left) and 2A(right)

Table 3 shows the k_inf results for each case with different transport approximations. Compared with the k_inf in cosRMC, the k_inf error in DRAGON for inflow transport approximations was less than 100 pcm. Contrarily, the error of k_inf in DRAGON for outflow transport approximation was less than 100 pcm in the fuel cell and fuel assembly, but could cause approximately 150 to 250 pcm in fuel assembly with the control rod. This discrepancy is consistent with that of previous research [9].

Compared with the k_inf in the previous inflow transport approximation, the results of the improved transport approximation were less than 10 pcm in cases 1A and 2A. Moreover, the results of the improved transport approximation were underestimated by 11 to 49 pcm for 2H-typical, 2H-0 ppm, 2H-4.8%, and 2G, and was also overestimated by 46 pcm for 2H-600 K. This tendency indicates that the transport approximations have less influence on the fuel cell and fuel assembly, but they have some effect on the fuel assembly with control rod cases. For the 300 K cases, the application of homogenization techniques in the improved transport approximation is based on the homogenous XS and accurate fission source, considering the influence of the resonance self-shielding effect in fuel materials. Compared with the previous inflow transport approximation, the improved transport approximation can absorb neutrons in different cases, which finally decreases the k_inf results. During the 600 K case, the impact of the resonance self-shielding effect was weakened in the improved inflow transport approximation with an increase in temperature [25], leading to an increase in k_inf compared to that in the previous transport approximation.

As a result, the different inflow transport approximations reached a higher accuracy than the outflow transport approximation in reactivity calculations, and the error of k_inf was less than 50 pcm for different inflow transport approximations.

Figures 4 and 5 show the comparison of the normalized P₀ flux moment with different inflow transport approximations for these cases. Specifically, compared with the results of cosRMC, the P₀ flux moment in the improved inflow transport approximation had a better agreement than the previous method. Furthermore, the P₀ flux moment of the previous inflow transport approximation was slightly overestimated by approximately 1 MeV and underestimated in the thermal energy range. Particularly, the previous inflow transport approximation made the P₀ flux moment more difficult than the improved method.

Fig. 4Full-screen View

(Color online) Comparison of normalized P₀ flux moment with the previous method. (a) 1A fuel cell; (b) 2A fuel assembly; (c) 2H-typical fuel assembly with B₄C; (d) 2H-0 ppm fuel assembly with B₄C at 0 ppm; (e) 2H-4.8% fuel assembly with B₄C in 4.8% fuel enrichment; (f) 2G fuel assembly with AIC; (g) 2H-600 K fuel assembly with B₄C in 600 K

Fig. 5Full-screen View

(Color online) Comparison of normalized P₀ flux moment in the improved method. (a) 1A fuel cell; (b) 2A fuel assembly; (c) 2H-typical fuel assembly with B₄C; (d) 2H-0 ppm fuel assembly with B₄C at 0 ppm; (e) 2H-4.8% fuel assembly with B₄C in 4.8% fuel enrichment; (f) 2G fuel assembly with AIC; (g) 2H-600 K fuel assembly with B₄C in 600 K.

In addition to the above investigation, the results of the improved inflow transport approximation in different LWR cases were in good agreement with the cosRMC, which is more versatile than the previous method. It can achieve higher accuracy for the cases of the fuel cell, fuel assembly, different boron concentrations, fuel enrichment, control rod material, and temperature in fuel assembly with the control rod. Therefore, the improved inflow transport approximation is accurate, universal, and suitable for LWR analysis.

Analysis of influence factors

In a previous investigation, the inflow transport approximation was problem-dependent. However, transport approximations are problem-independent data in most nuclear data libraries. Hence, we need to study the influence of the transport approximation and produce a nuclear data library with a general transport approximation.

Figure 6 shows the macroscopic transport XS and relative error of 2A for H₂O in the different inflow transport approximations. The macroscopic transport XS for H₂O was similar in most LWR cases, but the biggest difference existed in the thermal energy group in the 600 K case. The relative error of transport XS in the 600 K case was approximately 30% to 40% in the 2A case in the different inflow transport methods, but was less than 20% in other cases. Therefore, the most significant factor influencing inflow transport approximations is temperature through the comparison of transport XS.

Fig. 6Full-screen View

(Color online) Macroscopic transport XS (left) and relative error (right) of 2A for H₂O. (a) Results of previous inflow transport approximation; (b) Results of the improved inflow transport approximation.

Moreover, for the previous inflow transport approximation, the transport XS was based on the moderator region, and its relative error to 2A was less than 1% with the same moderator XS. Therefore, the previous inflow transport approximation was only related to the moderator region. For the improved inflow transport approximation, the transport XS was based on the homogeneous system, and its relative error to 2A was over 5% in 0.28 eV to 27.7 eV. Specifically, in a homogeneous system, the resonance self-shielding effect for fuel and absorber nuclides was considered in the derivation of transport XS. Then, the neutron spectrum was changed according to the different nuclide density ratios in the fuel and absorber nuclides. Thus, the transport XS in the improved inflow transport approximation was different from the previous inflow method, which is consistent with the actual LWR model.

The transport XS had a strong correlation with temperature and had less difference in the different LWR operation factors, such as fuel assembly, boron concentration, fuel enrichment, and control rod materials in the fuel assembly with the control rod. There are two different WIMS-D 70-group libraries, based on the improved inflow transport approximation, to test the influencing factors of transport approximations. The first is a problem-dependent library. The transport XS and scattering matrix data for ¹H and ¹⁶O are problem-dependent, which are developed by the improved inflow method in actual cases. The second is a problem-independent library. The transport XS and scattering matrix data for H-1 and O-16 are problem independent, developed by the improved inflow method, from the typical 2H case (B₄C, 300 K, 1,300 ppm, and 3.1 wt%).

Table 4 shows the results of the k_inf in the improved transport approximation. Compared to the k_inf results between the problem-dependent and problem-independent libraries, the k_inf error was less than 20 pcm for most cases. However, the k_inf error was 37 pcm for the 2H case at 600 K. This indicates that the inflow transport approximation considering fuel and absorber nuclides has less influence on k_inf calculations, but the factor of temperature can influence the k_inf results. This is consistent with the difference in the transport XS in Fig. 6. Although the largest difference of transport XS was in the resonance energy range, the magnitude of transport XS in the resonance energy range was comparatively smaller than the thermal energy range, and its influence on the k_inf error was less than 20 pcm in each case. Therefore, the improved inflow transport approximation is only related to the temperature factor, and it can be easily implemented in a temperature-dependent nuclear data library.

Application of inflow transport approximation to core cases

The Babcock & Wilcox (B&W) critical experiment benchmarks [26] selected to verify the application of different transport approximations in core cases. The B&W 1484 core 1 and core 3 cases are selected in this section, and the geometry shape is given in Fig. 7. The B&W 1484 core 1 is the 48×48 case, and the B&W 1484 core 3 is the 68×68 case.

Fig. 7Full-screen View

(Color online) Geometry shape of B&W 1484 Core 1 (left) and Core 3 (right)

Table 5 shows the results of the B&W 1484 core 1 and core 3 cases (2D and 3D). The results indicated that the outflow transport approximation can introduce an error of over 1,000 pcm in these cases, and both inflow transport methods were close to the reference values. For the B&W 1484 core 1 (2D and 3D) cases, the results of k_inf in the improved inflow transport approximation were overestimated by approximately 364 pcm and 291 pcm, respectively. For the B&W 1484 core 3 (2D and 3D) case, the results of k_inf in the improved inflow transport approximation were overestimated by approximately 115 pcm and 121 pcm, respectively. The k_inf error in the B&W 1484 core 1 and core 3 cases was inversely proportional to the cases of the LWR assembly. In addition, the different transport approximations were sensitive to the small core cases, which can cause over 1,000 pcm for the k_inf calculations in core 1 and approximately 600 pcm in core 3.

Although the previous inflow method is close to the reference results, the improved inflow method is close to the actual system, which is consistent with the theoretical model. The reason for this tendency may come from the limitation of the WIMS-D library, such as ignorance of scattering resonance integral, lumped fission spectrum, etc.

In summary, the outflow transport approximation can introduce an error of over 1,000 pcm in the B&W 1484 core 1 and core 3 cases, but the inflow transport methods are relatively accurate in the core cases.