Alternative scattering kernels for the first estimates of a reactor: diffusion length

NUCLEAR ENERGY SCIENCE AND ENGINEERING

Alternative scattering kernels for the first estimates of a reactor: diffusion length

Hakan Öztürk，

Ayça Şimşek Yapar

Nuclear Science and Techniques

Vol.29, No.3

Article number 37

Published in print 01 Mar 2018

Available online 19 Feb 2018

DOI：10.1007/s41365-018-0380-6

90700

Diffusion length calculations of neutrons are performed using the Chebyshev polynomials of the second kind. The neutrons are assumed to move with constant energy in a uniform homogeneous slab. An alternative scattering kernel called an Anlı-Güngör (AG) phase function and a traditional Henyey-Greenstein (HG) phase function are used for the scattering function in the stationary neutron transport equation. First, analytic expressions and then numerical results are obtained for the diffusion length for various values of the scattering and cross-section parameters. Numerical results obtained from both scattering kernels for the diffusion length of the neutrons are given in tables side by side for comparison. The applicability of the method is easily demonstrated by these results.

Scattering kernelDiffusion lengthChebyshev polynomialsTransport equation

1 Introduction

First estimates of a reactor are very important, to establish and operate it safely, as these studies provide details for the subsequent developments for the design of a reactor. In the solution algorithm of the approximate methods, higher order approximations are generally preferred to obtain a closer description of the real system. In these methods, the angular part of the neutron flux is expanded in terms of orthogonal polynomials. The arguments of these polynomials supply the definition interval of the cosine of the travelling direction of the neutron with respect to the x axis. However, it is still worth considering the first order approximation since it is valid for the first estimates of the system. The first order approximation of these polynomial expansion–based methods is called a diffusion approximation. It provides convenient results regarding the transport and energy spectrum of the neutrons if the number of secondary neutrons per collision, c, is close to unity [1].

As the Legendre polynomials (spherical harmonics) approximation is not the unique and valid method for all problems, the Chebyshev polynomials of both kinds can be used for the solutions in neutron transport theory because these polynomials have the same definition interval and are in the same polynomial family, i.e., Jacobi polynomials. In some recent studies, Chebyshev polynomials of the second kind, i.e., U_Nmethod, have been successfully applied to the one-dimensional transport equation in slab and spherical geometries for the solution of the diffusion length and criticality problems [2-5].

It is generally difficult to improve a solution algorithm for the solution of the transport equation because of its integro-differential form. Therefore, the integrand term, including the scattering function, is usually converted to a sum using an integral transformation. By doing so, this difficulty is overcome and the analytic, semi analytic, or numerical solution of the problem in transport theory is found. In those methods, the scattering function is expanded in terms of the orthogonal polynomials [6,7]. However, the structure of the equation allows only a few terms representing the degrees of anisotropy. Therefore, these approximate functions have serious restrictions on the exact solution of the transport equation. A scattering (phase) function, such as Henyey-Greenstein (HG), is preferred for the solution of problems in the field of particle and photon transport, radiative transfer, and light scattering [8-14]. An alternative scattering function derived by Anlı et al. is used for specifying the eigenvalue spectrum of the neutrons in slab geometry [15]. This scattering function has been successfully applied to the transport equation in a few studies, using Legendre and Chebyshev polynomials. Results consistent with the literature were obtained [16,17]. However, this present work is the first to consider the Chebyshev polynomials of the second kind and the HG and AG phase functions were used together for the solution of the transport equation.

In this work, we apply the U_N method in the calculation of the diffusion lengths for one-speed neutrons in a uniform homogeneous slab. The alternative scattering function previously derived and applied to transport equation [16,17] together with the well-known Henyey-Greenstein phase function is used. Therefore, by examining the derivations of the equations and analyzing the results given in the tables, the method can be extended to other problems in applied science and engineering.

2 U₁approximation with Henyey-Greenstein (HG) Phase Function

The time-independent linear transport equation in one-dimensional geometry for one-speed neutrons without a source can be written as,

μ \frac{\partial ψ (x, μ)}{\partial x} + σ_{T} ψ (x, μ) = \int_{0}^{2 π} \int_{- 1}^{1} ψ (x, μ^{'}) σ_{S} (μ_{0}) d μ^{'} d φ^{'}, - a \leq x \leq a, - 1 \leq μ \leq 1,

(1)

where $ψ (x, μ)$ is the angular flux or flux density of neutrons at position x traveling in direction µ, μμ′ is the direction cosine of the neutron after a collision, and σ_T is the macroscopic total cross section [18]. µ₀ is the cosine of the scattering angle between the the directions of neutron velocity before $Ω$ and after $Ω^{'}$ a collision, and is obtained by the scalar product $μ_{0} = Ω \cdot Ω^{'}$ ;

μ_{0} = μ μ^{'} + \sqrt{1 - μ^{2}} \sqrt{1 - {μ^{'}}^{2}} \cos (ϕ - ϕ^{'}),

(2)

and $σ_{S} (μ_{0})$ is the scattering kernel which describes all neutron interactions with the materials inside the system.

One of the most popular scattering functions used in the solutions of the problems regarding radiative transfer is the Henyey-Greenstein (HG) phase function. In the first part of this study, the HG phase function is used for the scattering function in the transport equation;

σ_{S}^{H G} (μ_{0}) = \frac{σ_{S} (1 - t^{2})}{4 π {(1 - 2 μ_{0} t + t^{2})}^{3 / 2}},

(3)

where σ_S is any non-negative coefficient and t represents all kinds of scattering, such as, isotropic, forward, backward, and anisotropic, and is in the range $- 1 \leq t \leq 1$ [19]. The integral of the HG phase function is needed on the right hand side of Eq. (1) and can be obtained with the addition theorem of the Legendre polynomials [15],

\int_{0}^{2 π} σ_{S}^{H G} (μ_{0}) d ϕ^{'} = \frac{σ_{S}}{2} \sum_{n = 0}^{\infty} (2 n + 1) t^{n} P_{n} (μ) P_{n} (μ^{'}) .

(4)

Using Eq. (4) in Eq. (1) with a definition of a dimensionless constant, σ_Tx/ν→x, we obtain;

μ \frac{\partial ψ (x, μ)}{\partial x} + v ψ (x, μ) = \frac{v c}{2} \sum_{n = 0}^{N} (2 n + 1) t^{n} P_{n} (μ) \int_{- 1}^{+ 1} ψ (x, μ') P_{n} (μ') d μ',

(5)

where ν denotes the eigenvalues and $c = σ_{S} / σ_{T}$ . In the solution algorithm of Eq. (5), the neutron angular flux is expanded in terms of the Chebyshev polynomials of the second kind as [3,4],

ψ (x, μ) = \frac{2}{π} \sqrt{1 - μ^{2}} \sum_{n = 0}^{N} Φ_{n} (x) U_{n} (μ), - a \leq x \leq a, - 1 \leq μ \leq 1.

(6)

The orthogonality and recurrence relations of the Chebyshev polynomials of the second kind are needed for obtaining the U_N moments of the angular flux [20],

\int_{- 1}^{1} U_{n} (μ) U_{m} (μ) \sqrt{1 - μ^{2}} d μ = {\begin{cases} π / 2, n = m \\ 0, n \neq m \end{cases}

(7)

2 μ U_{n} (μ) = U_{n + 1} (μ) + U_{n - 1} (μ), - 1 \leq μ \leq 1.

(8)

In order to obtain the U_N moments of the angular flux, Eq. (6) is substituted in Eq. (5). After multiplying by U_m(μ) and integrating over μ in the interval $- 1 \leq μ \leq 1$ ,

\frac{d Φ_{1} (x)}{d x} + 2 v Φ_{0} (x) = 2 v c Φ_{0} (x),

(9)

\frac{d Φ_{2} (x)}{d x} + \frac{d Φ_{0} (x)}{d x} + 2 v Φ_{1} (x) = 2 v c t Φ_{1} (x) .

(10)

for n = 0 and n = 1, respectively. Equation (10), corresponding to of n = 1, is generally referred to as the diffusion approximation and the last contribution of the flux is negligible by setting $d Φ_{N + 1} / d x = 0$ , as for the spherical harmonics (P_N) method [18]. Then for the U₁ approximation, the coupled Eqs. (9) and (10) can be solved for Φ₀(x),

\frac{d^{2} Φ_{0} (x)}{d x^{2}} - 4 v^{2} (1 - c) (1 - c t) Φ_{0} (x) = 0.

(11)

The diffusion length is defined as the inverse square root of the coefficient of Φ₀(x) in Eq. (11),

L^{HG} = \frac{1}{2 v \sqrt{(1 - c) (1 - c t)}} .

(12)

3 U₁ approximation with Anlı-Güngör (AG) Phase Function

In the second part of this study, the Anlı-Güngör (AG) phase function is used in the transport equation with the defined parameters in the previous section [15],

σ_{S}^{A G} (μ_{0}) = \frac{σ_{S}}{4 π {(1 - 2 μ_{0} t + t^{2})}^{1 / 2}}

(13)

The integral of the AG phase function is needed and is replaced in the right hand side of the one-dimensional transport equation given in Eq. (1),

\int_{0}^{2 π} σ_{S}^{A G} (μ_{0}) d ϕ^{'} = \frac{σ_{S}}{2} \sum_{n = 0}^{\infty} t^{n} P_{n} (μ) P_{n} (μ^{'}) .

(14)

The transport equation with the integral transformation of the AG phase function given in Eq. (14) can then be written as,

μ \frac{\partial ψ (x, μ)}{\partial x} + ν ψ (x, μ) = \frac{ν c}{2} \sum_{n = 0}^{\infty} t^{n} P_{n} (μ) α_{n} (x),

(15)

where α_n(x) is defined as

α_{n} (x) = \int_{- 1}^{1} P_{n} (μ^{'}) [\frac{2}{π} \sqrt{1 - {μ^{'}}^{2}} \sum_{n = 1}^{\infty} Φ_{n} (x) U_{n} (μ^{'})] d μ^{'} .

(16)

The U_N moments of the angular flux can easily be obtained by following the same procedure described for Eq. (9) and Eq. (10):

\frac{d Φ_{1} (x)}{d x} + 2 v Φ_{0} (x) = 2 v c Φ_{0} (x),

(17)

\frac{d Φ_{2} (x)}{d x} + \frac{d Φ_{0} (x)}{d x} + 2 v Φ_{1} (x) = \frac{2}{3} v c t Φ_{1} (x) .

(18)

for n = 0 and n = 1, respectively. Then, Eqs. (17) and (18) can be solved simultaneously for Φ₀(x) by following the same procedure for Eq. (11)::

\frac{d^{2} Φ_{0} (x)}{d x^{2}} - \frac{4}{3} v^{2} (1 - c) (3 - c t) Φ_{0} (x) = 0.

(19)

The diffusion length in the case of the AG phase function can then be obtained by

L^{AG} = \frac{1}{2 v} \sqrt{\frac{3}{(1 - c) (3 - c t)}} .

(20)

4 Numerical Results

The diffusion approximation was performed for one-speed neutrons in a homogeneous finite slab using both conventional HG and AG phase functions. In the solution algorithm, the neutron angular flux was expanded in a series of the Chebyshev polynomials of the second kind. This problem has been investigated by Öztürk and Anlı using the AG phase function with Legendre and Chebyshev polynomials of the first kind [16]. Then, analytic expressions for the diffusion lengths were derived using the U₁ approximation with the HG and AG phase functions and the numerical results were calculated for various values of the cross-section parameters c and t. Finally, these were tabulated side by side for comparison. The parameter ν relating to the total macroscopic cross-section was assumed to be ν = 1 cm^-1. Maple software was used for all of the calculations.

In Tables 1 and 2, the diffusion lengths were calculated from a weakly absorbing medium (c = 0) to a highly scattering medium (c = 1) for t in the interval $- 1 \leq t \leq 1$ . The same results for the diffusion length were obtained for a weakly absorbing medium (c = 0) in the case of both HG and AG phase functions (0.50000) for all values of the parameter t. As can be seen from these tables, the calculated diffusion lengths obtained from both HG and AG phase functions increased with increasing values of c and t.

Diffusion lengths L (cm) as calculated by the U₁ approximation for c = 0.99, 0.98 and 0.95.

t	c = 0.99		c = 0.98		c = 0.95
	HG	AG	HG	AG	HG	AG
-1.00	3.5444	4.3356	2.5126	3.0696	1.6013	1.9487
-0.99	3.5533	4.3409	2.5188	3.0733	1.6052	1.9511
-0.75	3.7878	4.4767	2.6841	3.1686	1.7087	2.0101
-0.50	4.0893	4.6324	2.8964	3.2780	1.8412	2.0776
-0.25	4.4766	4.8057	3.1686	3.3995	2.0101	2.1525
0.00	5.0000	5.0000	3.5355	3.5355	2.2361	2.2361
0.25	5.7639	5.2200	4.0689	3.6894	2.5607	2.3302
0.50	7.0360	5.4718	4.9507	3.8653	3.0861	2.4373
0.75	9.8533	5.7639	6.8680	4.0689	4.1703	2.5607
0.99	35.4441	6.0935	20.4808	4.2982	9.1670	2.6988
1.00	50.0000	6.1085	25.0000	4.3086	10.0000	2.7050

Diffusion lengths L (cm) as calculated by the U₁ approximation for c = 0.90, 0.80 and 0.50 and 0.00.

t	c = 0.90		c = 0.80		c = 0.50		c = 0.00
	HG	AG	HG	AG	HG	AG	HG	AG
-1.00	1.1471	1.3868	0.8333	0.9934	0.5774	0.6547	0.5000	0.5000
-0.99	1.1498	1.3884	0.8352	0.9945	0.5783	0.6551	0.5000	0.5000
-0.75	1.2217	1.4286	0.8839	1.0206	0.6030	0.6667	0.5000	0.5000
-0.50	1.3131	1.4744	0.9449	1.0502	0.6325	0.6794	0.5000	0.5000
-0.25	1.4286	1.5250	1.0206	1.0825	0.6667	0.6928	0.5000	0.5000
0.00	1.5811	1.5811	1.1180	1.1180	0.7071	0.7071	0.5000	0.5000
0.25	1.7961	1.6440	1.2500	1.1573	0.7559	0.7223	0.5000	0.5000
0.50	2.1320	1.7150	1.4434	1.2010	0.8165	0.7386	0.5000	0.5000
0.75	2.7735	1.7961	1.7678	1.2500	0.8944	0.7559	0.5000	0.5000
0.99	4.7891	1.8858	2.4515	1.3032	0.9950	0.7738	0.5000	0.5000
1.00	5.0000	1.8898	2.5000	1.3056	1.0000	0.7746	0.5000	0.5000

A comparison table for the calculated diffusion lengths obtained from the present U₁ and those obtained from the T₁ and P₁ approximations for selected values of the scattering parameters are given in Table 3. The diffusion lengths obtained from T₁ and P₁ approximations tabulated in Table 3 are quoted from Öztürk and Anlı [16,17]. A general agreement, except for the case of highly scattering medium, was demonstrated between the results obtained from the HG and AG phase functions. However, in the case of a highly scattering medium and in particular for the forward peaked scattering of the neutrons, the diffusion lengths calculated from the HG phase function differed with the results calculated from the AG phase function. In addition, as expected the numerical results obtained from all methods were similar for all cases, since they all belong to the same polynomial family, i.e., Jacobi polynomials.

Comparison of the results (cm) with the literature for selected values of the scattering parameters.

t	c = 0.99						c = 0.90						c = 0.50
	HG			AG			HG			AG			HG			AG
	U₁	T₁ [16]	P₁ [16,17]	U₁	T₁ [16]	P₁ [16,17]	U₁	T₁ [16]	P₁ [16,17]	U₁	T₁ [16]	P₁ [16,17]	U₁	T₁ [16]	P₁ [16,17]	U₁	T₁ [16]	P₁ [16,17]
-1.00	3.5444	5.0125	4.0927	4.3356	6.1314	5.0063	1.1471	1.6222	1.3245	1.3868	1.9612	1.6013	0.5774	0.8165	0.6667	0.6547	0.9258	0.7559
-0.99	3.5533	5.0251	4.1029	4.3409	6.1390	5.0125	1.1498	1.6261	1.3277	1.3884	1.9634	1.6031	0.5783	0.8179	0.6678	0.6551	0.9265	0.7565
-0.75	3.7878	5.3567	4.3737	4.4767	6.3309	5.1692	1.2217	1.7277	1.4107	1.4286	2.0203	1.6496	0.6030	0.8528	0.6963	0.6667	0.9428	0.7698
-0.50	4.0893	5.7831	4.7219	4.6324	6.5512	5.3490	1.3131	1.8570	1.5162	1.4744	2.0851	1.7025	0.6325	0.8944	0.7303	0.6794	0.9608	0.7845
-0.25	4.4766	6.3309	5.1691	4.8057	6.7963	5.5491	1.4286	2.0203	1.6496	1.5250	2.1567	1.7609	0.6667	0.9428	0.7698	0.6928	0.9798	0.8000
0.00	5.0000	7.0711	5.7735	5.0000	7.0711	5.7735	1.5811	2.2361	1.8257	1.5811	2.2361	1.8257	0.7071	1.0000	0.8165	0.7071	1.0000	0.8165
0.25	5.7639	8.1514	6.6556	5.2200	7.3821	6.0275	1.7961	2.5400	2.0739	1.6440	2.3250	1.8983	0.7559	1.0690	0.8729	0.7223	1.0215	0.8341
0.50	7.0360	9.9504	8.1244	5.4718	7.7382	6.3182	2.1320	3.0151	2.4618	1.7150	2.4254	1.9803	0.8165	1.1547	0.9428	0.7386	1.0445	0.8528
0.75	9.8533	13.9347	11.3776	5.7639	8.1514	6.6556	2.7735	3.9223	3.2026	1.7961	2.5400	2.0739	0.8944	1.2649	1.0328	0.7559	1.0690	0.8729
0.99	35.4441	50.1255	40.9273	6.0935	8.6175	7.0362	4.7891	6.7729	5.5300	1.8858	2.6669	2.1775	0.9950	1.4072	1.1490	0.7738	1.0944	0.8935
1.00	50.0000	70.7107	57.7350	6.1085	8.6387	7.0535	5.0000	7.0711	5.7735	1.8898	2.6726	2.1822	1.0000	1.4142	1.1547	0.7746	1.0954	0.8944

The diffusion lengths calculated from the HG phase function were very similar to the results obtained from the AG phase function if the neutrons moved in a weakly absorbing medium, i.e., c~0. However, if the neutrons moved in a highly scattering medium, i.e., c ~1, the results for the diffusion lengths calculated by the HG phase function differed from those from the AG phase function. This situation was especially observed for t> 0.75, corresponding to the case of strongly forward scattering [15]. The discrepancies between the diffusion lengths were not observed when these calculations were obtained by using the AG phase function. This was also observed in the study by Öztürk and Anlı, which used the AG phase function with the Legendre and Chebyshev polynomials of the first kind [16]. In addition, the diffusion length corresponding to t = 0 is equivalent to the case of isotropic scattering.

5 Conclusion

In this work, the AG phase function was used in the scattering function of the transport equation to calculate the diffusion lengths for one-speed neutrons in a uniform homogeneous slab. The neutron angular flux was expanded in terms of the Chebyshev polynomials of the second kind and the diffusion lengths were calculated for various values of c and t.

The HG and/or AG phase functions were examined to determine if they can be applied to other problems in science and engineering, and to further suggest new phase functions and methods to be developed for more accurate solutions of the transport equation.

From the tabulated numerical results, we concluded that both the HG and AG phase functions can be applied to transport theory to calculate the diffusion lengths for one-speed neutrons in a slab, except in the case of a highly scattering medium with forward peaked scattering. In this case while the diffusion lengths obtained from the HG phase function were unexpectedly high, the diffusion lengths obtained from the AG phase function were realistic. Therefore, the AG phase function is more applicable than the HG phase function for this specific case. In addition, it was seen from the provided tables, that realistic results were always obtained from the AG phase function, independently of the scattering properties of the medium.

References

J.R. Lamarsh, A.J. Baratta, Introduction to nuclear engineering, 3rd edn. (Prentice Hall, Inc., 2001), pp. 52-110

S. Yabushita,

Tschebyscheff polynomials approximation method of the neutron transport equation

. J. Math. Phys. 2, 543-549 (1961). doi: 10.1063/1.1703739

Alternative scattering kernels for the first estimates of a reactor: diffusion length

1 Introduction

2 U1approximation with Henyey-Greenstein (HG) Phase Function

3 U1 approximation with Anlı-Güngör (AG) Phase Function

4 Numerical Results

5 Conclusion

2 U₁approximation with Henyey-Greenstein (HG) Phase Function

3 U₁ approximation with Anlı-Güngör (AG) Phase Function