K. Kobayashi and S. Tsumura

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Using Eqs.(53) and (54) in Eqs.(44) and (45), the kinetics equations for iodine and xenon become dδIm(t) =ˆγ Iδsm(t) − λIδIm(t), (56) dt dδXm(t) dt = ˆγ X − βm − ˆσ Xf XmδXm(t) δsm(t)+λIδIm(t) − λX(1 + ηm)δXm(t). (57) For simplicity, we assume that the reactor is symmetric such that k0 11 = k0 22 ,k0 12 = k0 21 . In this case, from Eq.(38), we obtain 1 k k0 11 − 1 − ˆσG00 X1 X0 1 1 + k k0 12 s 0 1 =0. (58) In order to be valid for s0 1 = 0, the term in the bracket of the above equation must vanish. Then, using Eq.(55), the criticality factor k must satisfy the following equation k = (k0 11 + k0 12 ) 1+ˆσ G00 X1 X0 =(k 1 0 11 + k 0 12) for the existence of a steady state solution. 2..4 LINEAR APPROXIMATION 1+ (ˆγ X +ˆγ I)ˆσ G00 X1 s01 λX +ˆσ 00 X1s0 −1 1 , (59) In the practical case of xenon spatial oscillations in PWRs, the amplitude of flux oscillations is small and the linear approximation neglecting the nonlinear terms is known to be a good approximation 4) . Retaining only the first order terms in Eqs.(43), we obtain the linearized equations for neutron productions as follows; dδsm(t) lm dt 1 = k ks mm − ˆσG0f XmX0 m − 1 δsm(t)+ 1 k ks mnδsn(t) − ˆσ GX0 Xm s0 mδXm(t) −α Cφ m δΣCGm (t), for m = n, m, n =1, 2. (60) The linearized equations of Eqs.(57) for xenon are dδXm(t) dt =(ˆγ X − β m) δsm(t)+λIδIm(t) − λX(1 + η m)δXm(t). (61) In a steady state, Eqs.(3), (13) and (14) can be written A + σXGδgGX 0 (r) φ 0 g (r) =1 k χgs 0 (r), (62) γ IΣfG(r)φ 0 G(r) − λII 0 (r) =0, (63) γ XΣfG(r)φ 0 G(r)+λII 0 (r) − λXX 0 (r) − σXGX 0 (r)φ 0 G(r) =0, (64) where I 0 (r) is the iodine density at a steady state. 8
Solving the multi-group diffusion equations of Eqs.(62) together with Eqs.(63) and (64), the flux for the steady state can be obtained. The importance function of Eq.(4) can be easily obtained by using a conventional multi-group diffusion program where the usual source term is replaced by the fission cross section as input quantity for the right hand side of Eq.(4). Using these flux and importance functions in Eqs.(29) to (36), the kinetics parameters used in Eqs.(56), (60) and (61) can be obtained numerically. 2..5 ANALYTICAL SOLUTION Let us solve Eqs.(56), (60) and (61) using the Laplace transformation. Using the transformation parameter ω, the Laplace transform of δsm(t) is defined by δ¯sm(ω) = ∞ 0 δsm(t)e −ωt dt. (65) Laplace transforms of δXm(t) and δIm(t) are defined also by similar equations. We assume as initial condition that the system is at a steady state at t = 0 and then the initial values of δXm(t) and δIm(t) are zero. From Eq.(56), we obtain δĪm(ω) = ˆγ I δ¯sm(ω). ω + λI (66) Substituting this equation into the Laplace transformed equation of Eq.(61), we obtain δ ¯ Xm(ω) = ˆγ X − β m + ˆγ IλI ω+λI ω + λX(1 + η m) δ¯sm(ω). (67) Substituting these equations into the transformed equations of Eqs.(60), we obtain ⎛ ⎜ ⎝l1ω λXη + ∆1 + G 1 ˆγ X − β1 + ˆγ ⎞ IλI ω+λI ⎟ ω + λX(1 + η ⎠ 1) δ¯s1(ω)−∆12δ¯s2(ω) where = l1δs1(0) − α Cφ 1 δ ¯ Σ C G1 (ω), (68) η G m = ˆσGX0 Xm s λX 0 m, ∆m =1− 1 k ks mm +ˆσ G0f XmX 0 m, ∆mn = 1 k ks mn. (69) Equation (59) can be written as If we use the approximations ˆσ G00 X1 symmetrical system becomes 1+ˆσ G00 X1 X0 1 1 = k (k0 11 + k0 12 ). (70) ≈ ˆσG0f X1 ,k0 11 ≈ ks 11 and use Eq.(70), ∆1 of Eq.(69) for a ∆1 ≈ 1 k (k0 11 − ks 11 + k0 1 12 ) ≈ k k0 12 . (71) 9
Page 1 and 2: AN ANALYSIS OF XENON OSCILLATIONS U
Page 3 and 4: We use the boundary condition that
Page 5 and 6: Using the assumption that the fissi
Page 7: 2..3 KINETICS EQUATIONS FOR IODINE
Page 11 and 12: which can be written in a form wher
Page 13 and 14: 2..6 CONTROL OF XENON OSCILLATIONS
Page 15 and 16: where κi = Σai/Di, i = c, r and
Page 17 and 18: Importance Function G1(x) G1(x) 1 0
Page 19 and 20: AOI-AOX 0.002 0 -0.002 B C A D E -0
Page 21: CONCLUSIONS It has been shown that

K. Kobayashi and S. Tsumura

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