Post on 25-Mar-2018
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24.505 Lecture 17 Numerical Solution of the Multigroup
Diffusion Equations II
Prof. Dean Wang
1. The Analytic Nodal method in 2D Cartesian Geometry The 2D Cartesian heterogeneous reactor configurations correspond to the case where the neuron flux is a function of two spatial variables. These cases cannot be solved analytically and the analytic nodal method (ANM) is an attempt to find the solution with the smallest possible approximation. Here we limit our investigations to a 2D Cartesian domain made from the assembly of many 𝑥 − 𝑦 rectangular nodes which are infinite in the 𝑧 direction. In this case, the nuclear properties of the reactor are only a function of the independent variables x and y. The multigroup diffusion equation in 2D Cartesian geometry can be written as
− !!"𝐷! 𝑥,𝑦
!!! !,!!"
− !!"𝐷! 𝑥,𝑦
!!! !,!!"
+ Σ!,! 𝑥,𝑦 𝜙! 𝑥,𝑦 = 𝑄! 𝑥,𝑦 (1a) where 𝑄! 𝑥,𝑦 = Σ!!→! 𝑥,𝑦 𝜙!! 𝑥,𝑦
!!!!! + !!
!!""νΣ!,!! 𝑥,𝑦!
!!!! 𝜙!! 𝑥,𝑦 (1b)
Each node is assumed to be homogeneous, so that the corresponding nuclear properties 𝐷! 𝑥,𝑦 , Σ!,! 𝑥,𝑦 , Σ!!→! 𝑥,𝑦 and νΣ!,!! 𝑥,𝑦 are piecewise continuous as shown in Fig. 1, the reactor is divided into 𝐼×𝐽 regions of indices 1 ≤ 𝐼 ≤ 𝐼 and 1 ≤ 𝐽 ≤ 𝐽, in such a way that the nuclear properties in node 𝑖, 𝑗 are constant and equal to 𝐷!,!,! , Σ!,!,!,! , Σ!!→!,!,! , and νΣ!,!!,!,! .
Fig. 1. 2D discretization. The linear transformation technique is applied on each node, leading to the linear transformation 𝐺×𝐺 matrix 𝑇!,! and to a set of 𝐺 eigenvalues 𝜆!,!,! on 𝑥!!!/! < 𝑥 <
(i, j-‐1)
Δxi
(i, j) (i-‐1, j) (i+1, j)
(i, j+1)
Δyi
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𝑥!!!/! and 𝑦!!!/! < 𝑦 < 𝑥!!!/!. The transformation process is repeated for each node, leading to 𝐼×𝐽 matrix equations written as !!
!!!Φ 𝑥,𝑦 + !!
!!!Φ 𝑥,𝑦 + FΦ 𝑥,𝑦 = 0 (2)
with
Φ 𝑥,𝑦 =𝜙! 𝑥,𝑦
⋮𝜙! 𝑥,𝑦
(3a)
F =
𝑓!! 𝑓!" ⋯ 𝑓!!𝑓!" 𝑓!! ⋯ 𝑓!!⋮ ⋮ ⋮ ⋮𝑓!! 𝑓!! ⋯ 𝑓!!
(3b)
where 𝑓!,!! =
!!! !,!
−Σ!,! 𝑥,𝑦 δ!!! + Σ!!→! +!!!!""
νΣ!,!! 𝑥,𝑦!!!!! (3c)
The next step consists in finding all eigenvectors 𝑡! of matrix F with the associated eigenvalues 𝜆!. We build a matrix 𝑇 as 𝑇 = 𝑡! 𝑡! ⋯ 𝑡! (4) So that 𝐹𝑇 = 𝑇𝑑𝑖𝑎𝑔( 𝜆!) (5) So we have
Φ 𝑥,𝑦 = 𝑇𝜓 𝑥,𝑦 =
𝑡!! 𝑡!" ⋯ 𝑡!!𝑡!" 𝑡!! ⋯ 𝑡!!⋮ ⋮ ⋮ ⋮𝑡!! 𝑡!! ⋯ 𝑡!!
𝜓! 𝑥,𝑦𝜓! 𝑥,𝑦
⋮𝜓! 𝑥,𝑦
(6)
Substituting (6) into (2) gives !!
!!!TΨ 𝑥,𝑦 + !!
!!!TΨ 𝑥,𝑦 + FTΨ 𝑥,𝑦 = 0 (7)
Multiplying (7) with T!! yields !!
!!!Ψ 𝑥,𝑦 + !!
!!!Ψ 𝑥,𝑦 + diag 𝜆! Ψ 𝑥,𝑦 = 0 (8)
Each equation is uncoupled in energy, and can be written as G differential equations of the form !!
!!!𝜓! 𝑥,𝑦 + !!
!!!𝜓! 𝑥,𝑦 + 𝜆!,!,!𝜓! 𝑥,𝑦 = 0 (9)
where 𝑔 = 1,2, . .𝐺. Unfortunately, it is impossible to find the analytical solution of (9) because its dependent variable 𝜓! 𝑥,𝑦 is generally not separable. The ANM is based on transverse integration of (9). Transverse integration along the Y axis leads to 𝑑𝑦 !!
!!!𝜓! 𝑥,𝑦
!!!!/!!!!!/!
+ 𝑑𝑦 !!
!!!𝜓! 𝑥,𝑦
!!!!/!!!!!/!
+ 𝜆!,!,! 𝑑𝑦𝜓! 𝑥,𝑦!!!!/!!!!!/!
= 0 (10) which can be rewritten as !
!
!!!𝜓!,!! 𝑥 + 𝜆!,!,!𝜓!,!
! 𝑥 = !!!!
𝐹!,!! 𝑥 (11)
where Δ𝑦! = 𝑦!!!/! − 𝑦!!!/!,
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𝜓!,!! 𝑥 = !
!!!𝑑𝑦𝜓! 𝑥,𝑦
!!!!/!!!!!/!
(12)
and where we introduce the X-‐directed transverse leakage term as 𝐹!,!
! 𝑥 = − 𝑑𝑦 !!
!!!𝜓! 𝑥,𝑦
!!!!!
!!!!!
= − !!"𝜓! 𝑥,𝑦
!!!!/!
!!!!/! (13)
Similarly, the transverse integration along the 𝑋 axis leads to !
!
!!!𝜓!,!! 𝑦 + 𝜆!,!,!𝜓!,!! 𝑦 = !
!!!𝐹!,!! 𝑦 (14)
where Δ𝑥! = 𝑥!!!/! − 𝑥!!!/!, 𝜓!,!! 𝑦 = !
!!!𝑑𝑥𝜓! 𝑥,𝑦
!!!!/!!!!!/!
(15) and where we introduce the Y-‐directed transverse leakage term as 𝐹!,!! 𝑦 = − !!
!!!𝜓! 𝑥,𝑦
!!!!/!!!!!/!
= − !!"𝜓! 𝑥,𝑦
!!!!/!
!!!!/! (16)
(11) and (14) can be solved analytically, provided that the x and y variation of the transverse leakage terms 𝐹!,!
! 𝑥 and 𝐹!,!! 𝑦 are known. This is where we introduce the unique approximation of the ANM. Many possibilities exist to predict this variation, and have been investigated in the seventies. Shober initially assumed that the transverse leakages and the 1D fluxes had the same shape as 𝐹!,!
! 𝑥 = 𝐵!,!,!! 𝜓!,!
! 𝑥 (17a) 𝐹!,!! 𝑦 = 𝐵!,!,!! 𝜓!,!! 𝑦 (17b) This buckling-‐type approximation would be exact if the dependent variable 𝜓! 𝑥,𝑦 were spatially separable within node 𝑖, 𝑗 . However, Shober found that the use of the buckling approximation led to large errors in highly nonseparable cases [1]. As an alternative to the buckling approximation, Shober proposed to use a flat leakage approximation in which the transverse leakage shape is spatially flat over the each node, leading to 𝐹!,!
! 𝑥 = 𝐹!,!,!! = 𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/! (18a)
𝐹!,!! 𝑦 = 𝐹!,!,!! = 𝐽!,!! 𝑥!!!/! − 𝐽!,!
! 𝑥!!!/! (18b) where the transformed current are defined as 𝐹!,!
! 𝑥 ≡ − !!"𝜓!,!! 𝑥 (19a)
𝐹!,!! 𝑦 ≡ − !!"𝜓!,!! 𝑦 (19b)
Later, Smith introduced a quadratic leakage approximation in the ANM [2]. The expansion coefficients of the leakage fit are calculated by assuming that the quadratic polynomial extends over the two neighboring nodes and satisfies the average leakages in the central and two neighboring nodes. The quadratic leakage fit does not rely on the diffusion equation itself and can only be justified if the transverse leakages vary smoothly across the three nodes. Such an approximation can be constructed for node 𝑥,𝑦 , in the X-‐direction, using 𝐹!,!!!,!
! , 𝐹!,!,!! and 𝐹!,!!!,!
! , the transverse leakage terms without linear transformation. Under these conditions, (11) is rewritten as !
!
!!!𝜓!,!! 𝑥 + 𝜆!,!,!𝜓!,!
! 𝑥 = !!!!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/! (20)
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Integration of (20) over node 𝑥,𝑦 leads to the transformed nodal balance equation, written as 𝜓!,!,! =
!!!!!!,!,!
𝐽!,!! 𝑥!!!/! − 𝐽!,!
! 𝑥!!!/!
+ !!!!!!,!,!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/! (21)
Let us first consider the case where 𝜆!,!,! ≥ 0. In energy group g and in node 𝑥,𝑦 , (20) has an analytical solution of the form 𝜓!,!
! 𝑥 = !!!!!!,!,!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/!
+𝐴!,!,!cos 𝜆!,!,!𝑥 + 𝐵!,!,!sin 𝜆!,!,!𝑥 (22) Integrating (22) over the node leads to 𝜓!,!,! =
!!!!!!,!,!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/! + !!,!,!
!!! !!,!,!sin 𝜆!,!,!𝑥 !!!!/!
!!!!/!
− !!,!,!
!!! !!,!,!cos 𝜆!,!,!𝑥 !!!!/!
!!!!/! (23)
Differentiating (22) over the node gives 𝐽!,!
! 𝑥 = 𝐴!,!,! 𝜆!,!,!sin 𝜆!,!,!𝑥 − 𝐵!,!,! 𝜆!,!,!cos 𝜆!,!,!𝑥 (24) If 𝜆!,!,! < 0, we have 𝜓!,!
! 𝑥 = !!!!!!,!,!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/!
+𝐶!,!,!cosh −𝜆!,!,!𝑥 + 𝐸!,!,!sin −𝜆!,!,!𝑥 (25) 𝜓!,!,! =
!!!!!!,!,!
𝐽!,!! 𝑦!!!/! − 𝐽!,!! 𝑦!!!/! + !!,!,!
!!! !!!,!,!sinh −𝜆!,!,!𝑥 !!!!/!
!!!!/!
− !!,!,!
!!! !!!,!,!cosh −𝜆!,!,!𝑥 !!!!/!
!!!!/! (26)
𝐽!,!! 𝑥 = −𝐶!,!,! −𝜆!,!,!sinh −𝜆!,!,!𝑥 − 𝐸!,!,! −𝜆!,!,!cosh −𝜆!,!,!𝑥 (27)
References 1. R. A. Shober and A. F. Henry, “Nonlinear Methods for Solving the Diffusion
Equations,” M.I.T. Report MITNE-‐196, 1976. 2. K. S. Smith, “An Analytic Nodal Method for Solving the Two-‐Group,
Multidimensional, Static and Transient Neutron Diffusion Equation,” Nuclear Engineer’s Thesis, MIT, Department of Nuclear Engineering, 1979.
3. N. Z. Cho, “Fundamentals and Recent Developments of Reactor Physics Methods,” Nucl. Eng. Tech. 37 (1): 25-‐78.