Generalized Lambda-Mode Xenon Stability

PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future The Westin Miyako, Kyoto, Japan, September 28 - October 3, 2014, on CD-ROM (2014)

A GENERALIZATION OF λ-MODE XENON STABILITY ANALYSIS Justin M. Pounders and Jeffery D. Densmore Bettis Atomic Power Laboratory PO Box 79, West Mifflin, PA 15122-0079, USA [email protected] [email protected] ABSTRACT A new method for analyzing xenon transients in nuclear reactors based on λ-mode stability analysis is developed. Like previous work, the new analysis method is based on a linearization of the coupled diffusion, iodine, and xenon equations, but the approach is generalized here to permit flux perturbations that include a component in the fundamental spatial mode and thus a complete expansion of any perturbation. This extension also allows power to be explicitly constrained. The typical constant-reactivity approach to λ-mode analysis appears as a special case of the new generalized theory.

Key Words: xenon oscillations, reactor transients, stability analysis, perturbation theory

1. INTRODUCTION

Xenon-induced power oscillations in large thermal reactors are well known phenomena that have been extensively studied [1–4]. A common approach for analyzing these transients is the λ method [1] in which the perturbation of a steady-state reactor is expanded in the eigenfunctions of the steady-state neutron diffusion equation (i.e., λ modes). A linearized analysis of the quasi-static diffusion equation coupled with transient iodine and xenon equations then provides a criterion for reactor stability. A central (and justifiable) assumption of the λ method is that reactivity remains constant. There are, however, a number of mathematical consequences associated with this assumption that have never been investigated. First, the constant reactivity assumption in the linearized dynamic equations necessitates that any perturbation to the neutron flux be orthogonal to the steady-state flux distribution. This fact manifests in the λ method by starting all perturbation expansions with the first harmonic mode under the assumption that the fundamental mode would be held constant by external control. Second, in general the power associated with each λ mode is not zero, and it is thus impossible to enforce a constant-power constraint with this method. In this work we develop a generalized λ method that makes it possible to include perturbations to the fundamental mode and explicitly constrain power level. As will be shown, including these two extensions makes it impossible to simultaneously hold reactivity constant. The degree to which reactivity deviates from its steady-state value indicates the amount of feedback or control action

J. Pounders & J. Densmore

that would be necessary to keep the reactor critical. One could argue that the power deviation in the usual λ method provides the same indication, but with the current method the inclusion of the fundamental mode enables a complete expansion of perturbations that has not previously been possible. In a broader sense, the generalized method described in the current work encompasses both the constant reactivity and the constant power cases. This generalization is made possible by recognizing that the system of linearized dynamic equations is singular: because the quasi-static diffusion equation is an eigenvalue problem a solution is not guaranteed under arbitrary perturbation. Explicitly acknowledging this fact exposes two options. Either limit flux perturbations to those that are orthogonal to the fundamental mode (the constant-reactivity approach) or permit general flux perturbations and allow reactivity to vary as a function of the perturbation (the constant-power approach). The constant-reactivity approach is equivalent to the usual λ method, while the constant-power option becomes apparent only when the singularity of the diffusion equation is explicitly addressed. Throughout the remainder of this paper, these two options will be referred to as “constant reactivity” and “constant power,” respectively, but the difference between these two constraints is more than superficial because it determines the permissible space of flux perturbations. In the following section, we derive the new constant-power approach within the context of general λ-mode expansions and indicate the point-of-departure for previous constant-reactivity methods. Comparing these two methods is complex, but some insight can be gained by viewing the constantreactivity method as embedded within the constant-power method. We present such a formalism, and use it to draw some qualitative conclusions about the appearance of dynamic modes. Finally, a series of simple example problems are shown with numerical results compared against an explicit transient calculation.

2. THEORY

2.1. Governing Equations The basis of the λ method is the quasi-static diffusion equation coupled to transient iodine and xenon equations, 0 =

∂ ∂φ D − [Σa,0 + σXe NXe ] φ + λνΣf φ , ∂x ∂x

∂NI = γI Σf φ − λI NI , ∂t ∂NXe = λI NI + γXe Σf φ − λXe NXe − σXe NXe φ , ∂t

(1) (2) (3)

where D(x) is the diffusion coefficient, Σa,0 (x) is the base macroscopic absorption cross section that does not include absorption due to xenon, 2/18

PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014

Generalized Lambda-Mode Xenon Stability

Σf (x) is the macroscopic fission cross section, νΣf (x) is the macroscopic neutron production cross section, σXe is the microscopic xenon absorption cross section, −1 λ(t) is the inverse multiplication factor, keff ,

φ(x, t) is the neutron scalar flux, NXe (x, t) is the xenon atom density, NI (x, t) is the iodine atom density, γI is the fission yield of iodine, λI is the iodine decay constant, γXe is the fission yield of xenon, and λXe is the xenon decay constant. Note that λ(t) would be a constant (usually equal to unity) if the time-derivative of the flux were retained, but because a static form of the diffusion equation is used in this case λ(t) must be allowed to vary in time to guarantee the existence of a non-trivial solution for all t. This assertion can be verified by viewing Eq. (1) as an eigenvalue problem that is parameterized in time, i.e., once NXe (t) is specified, both φ(t) and λ(t) are determined.

2.2. Linearization and Expansion As in previous work, we seek a solution to the nonlinear Eqs. (1)-(3) by linearizing the sytem about the steady-state equilibrium given by the equations ∂ ∂φ0 D − [Σa,0 + σXe NXe,0 ] φ0 + λ0 νΣf φ0 , ∂x ∂x γI Σf φ0 = , λI (γI + γXe ) Σf φ0 = . λXe + σXe φ0

0 = NI,0 NXe,0

(4) (5) (6)

Eq. (4) is a regular Sturm-Liouvile problem, and thus there are an infinite set of eigenpairs (λi , ϕi ), i = 0, 1, 2, ...∞, that give non-trivial solutions to this equation. These eigenfunctions are the λ modes and are orthogonal with respect to the weight function νΣf . They can be normalized to satisfy hϕm , ϕn iνΣf = δm,n , (7) where

Z hf, giw ≡

f (x)w(x)g(x)dx .


(8) 3/18


The physically relevant solution of Eq. (4) at steady state corresponds to the smallest eigenvalue λ0 and the corresponding eigenfunction ϕ0 (the fundamental mode), which is equivalent to φ0 up to a constant scaling. To proceed with the linearization, the time-dependent variables are perturbed about the steady-state equilibrium, φ(t) NI (t) NXe (t) λ(t)

= = = =

φ0 + δφ(t) , NI,0 + δNI (t) , NXe,0 + δNXe (t) , λ0 + δλ(t) .

(9) (10) (11) (12)

Equation (12) marks the first real departure from previous analyses in which the eigenvalue is held constant at λ0 . The importance of perturbing the eigenvalue can be seen from the linearized system, which is obtained by substituting these expansions into Eqs. (1)-(3), using the equilibrium equations, and removing products of perturbations, 0 =

∂ ∂δφ(t) D − (Σa,0 + σXe NXe,0 ) δφ(t) + λ0 νΣf δφ(t) ∂x ∂x +δλ(t)νΣf φ0 − σXe δNXe (t)φ0 ,

∂δNI (t) = γI Σf δφ(t) − λI δNI (t) , ∂t ∂δNXe (t) = λI δNI (t) − (σXe φ0 − λXe ) δNXe (t) ∂t −σXe NXe,0 δφ(t) + γXe Σf δφ(t) .

(13) (14)

(15)

Inspection of Eq. (13) reveals that the operator acting on δφ(t) is singular: any component of δφ(t) in the span of ϕ0 will be annihilated. Because of this singularity, Eq. (13) will only have a solution if σXe hϕ0 , δNXe (t)φ0 i1 δλ(t) = , (16) hϕ0 , φ0 iνΣf which is confirmed by the Friedholm Alternative Theorem. Assuming that reactivity remains constant throughout the transient [δλ(t) = 0] results in a situation in which Eq. (16) can not generally be satisfied, and thus the perturbed diffusion equation fails to have a solution. This problem has been (implicitly) avoided in previous analyses by concurrently assuming that the the flux perturbation has no component in the span ϕ0 , effectively restricting the solution of Eq. (13) to the subspace absent the singularity. The constant reactivity assumption is not necessary, however. Rather, general flux perturbations, including perturbations to the fundamental mode, can be permitted if reactivity is allowed to vary according to Eq. (16). Thus we can write δφ(t) = α(t)ϕ0 +

∞ X

an (t)ϕn ,

(17)

n=1

where α(t) is an arbitrary function of time representing the component of the solution along the “singular dimension,” and the an coefficients can be determined by substituting Eq. (17) into Eq. 4/18



(13) and using the orthogonality of the eigenfunctions: an (t) =

σXe hϕn , δNXe (t)φ0 i1 . λ0 − λn

(18)

Because the coefficient α(t) is arbitrary, we require an additional constraint to uniquely determine the solution. Here we choose to hold the total fission rate constant (in lieu of power), Z Z Σf (x)φ(x, t)dx = Σf (x)φ0 (x) = constant , (19) which implies Z Σf (x)δφ(x, t)dx = 0 .

(20)

This constraint leads to the definition α(t) =

∞ X σXe hϕn , δNXe (t)φ0 i

(λn − λ0 )

n=1

R 1 R Σf (x)ϕn (x)dx . Σf (x)ϕ0 (x)dx

(21)

Again note that it is not possible to use the constraint δλ(t) = 0 to generate a definition of α(t) as this would invalidate the condition for the existence of a solution, (16). Regardless of condition (16) or the value of δλ(t), however, the flux perturbation has a unique solution in the space orthogonal to ϕ0 as shown by Eq. (18). Thus by using the definition of the an provided by (18) and setting α(t) = 0 and δλ(t) = 0 we arrive at the usual constant-reactivity result, although the solution is only valid in the subspace orthogonal to the fundamental mode. Substituting the definitions of α(t) and an (t) into the expansion (17) and defining R Σf (x)ϕn (x)dx , βn ≡ R Σf (x)ϕ0 (x)dx

(22)

leads to a general expression for the flux perturbation in terms of the xenon perturbation as a function of time ∞ X σXe hϕn , δNXe (t)φ0 i1 δφ(t) = (ϕn − βn ϕ0 ) . (23) (λ − λ ) 0 n n=1 We can now eliminate the flux perturbation from the linearized iodine and xenon perturbation equations to obtain ∞

X σXe hϕn , δNXe (t)φ0 i ∂δNI (t) 1 = γI Σf (ϕn − βn ϕ0 ) ∂t (λ0 − λn ) n=1 −λI δNI (t) , ∂δNXe (t) = λI δNI (t) − (σXe φ0 − λXe ) δNXe (t) ∂t ∞ X σXe hϕn , δNXe (t)φ0 i1 −σXe NXe,0 (ϕn − βn ϕ0 ) . (λ0 − λn ) n=1 PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014

(24)

(25)

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At this point we have assumed that the fission yield of xenon is zero for brevity, although its inclusion would not alter our approach or results. Now let δNI (t) =

∞ X

bn (t)νΣf ϕn ,

(26)

cn (t)νΣf ϕn .

(27)

n=0

δNXe (t) =

∞ X n=0

Previous analyses have begun these expansions with n = 1, in a manner analogous with the flux expansion. The fundamental mode is retained here for completeness of the expansion. The factor of νΣf is included so that orthogonality of the eigenfunctions can be used more easily to simplify subsequent expressions. Substituting these expansions into Eqs. (24)-(25), multiplying by one of the eigenfunctions, ϕm , and integrating over x yields ∂bm (t) = −λI bm ∂t ∞ X ∞ X σXe hϕn , ϕn0 φ0 iνΣf +γI cn0 (t) hϕm , Σf ϕn i1 − βn hϕm , Σf ϕ0 i1 , (28) (λ0 − λn ) n=1 n0 =0 ∞ X ∂cm (t) cn (t) hϕm , ϕn φ0 iνΣf = λI bm − λXe cm − σXe ∂t n=0

−σXe

∞ ∞ X σXe hϕm , NXe,0 ϕn i X 1

n=1

(λ0 − λn )

cn0 (t) hϕn , ϕn0 φ0 iνΣf

n0 =0

∞ ∞ X σXe βn X +σXe hϕm , NXe,0 ϕ0 i1 cn0 (t) hϕn , ϕn0 φ0 iνΣf (λ0 − λn ) n0 =0 n=1

.

(29)

Finally note the possible (but not necessary) simplification that if ν is constant wherever Σf > 0 then 1 1 hϕm , Σf ϕn i1 = hϕm , ϕn iνΣf = δm,n . ν ν Therefore the iodine equation can be simplified slightly to ∞ ∞ X σXe βn X ∂b0 (t) = −λI b0 − γI hϕ0 , Σf ϕ0 i1 cn0 (t) hϕn , ϕn0 φ0 iνΣf ∂t (λ − λ ) 0 n 0 n=1 n =0

and

∞ σXe hϕm , Σf ϕm i1 X ∂bm (t) cn0 (t) hϕm , ϕn0 φ0 iνΣf = −λI bm + γI ∂t (λ0 − λm ) 0 n =0

.

,

(30)

(31)

for m > 0. Equations (29)-(31) and are the constant-power dynamic equations. If we had used the more customary assumptions of the constant-reactivity approach and excluded the fundamental

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mode from all expansions then Eqs. (31) and (29) become, respectively, ∞ σXe hϕm , Σf ϕm i1 X ∂bm (t) = −λI bm + γI cn0 (t) hϕm , ϕn0 φ0 iνΣf ∂t (λ0 − λm ) 0 n =1

,

(32)

,

(33)

∞ X ∂cm (t) = λI bm − λXe cm − σXe cn (t) hϕm , ϕn φ0 iνΣf ∂t n=1

−σXe

∞ ∞ X σXe hϕm , NXe,0 ϕn i X 1

n=1

(λ0 − λn )

cn0 (t) hϕn , ϕn0 φ0 iνΣf

n0 =1

for m > 0.

2.3. Discussion In the previous section both the traditional constant-reactivity and the new constant-power λ methods were derived. Up to this point the constant-power Equations (29)-(31) are exact with respect to the governing linearized equations, while the constant-reactivity equations are approximate with an error proportional to ϕ0 . In reality the modal expansions must be truncated at some finite N > 0, resulting additional error for both methods. The constant-reactivity approach thus consists of 2N linear equations that generate a perturbation expressed in terms of the spatial basis (ϕ1 , ϕ2 , . . . , ϕN ). The constant-power approach, on the other hand, consists of 2N + 2 linear equations that generate a perturbation in terms of the spatial basis (ϕ0 , ϕ1 , . . . , ϕN ). The constant-power system of equations can be written more compactly as ~b ∂ ~b = M , ~c ∂t ~c

(34)

where ~b = [b0 , b1 , . . . , bN ]T ~c = [c0 , c1 , . . . , cN ]T

, ,

and M is a (2N + 2) × (2N + 2) matrix. The solution can be written in terms of the eigenvalues, γn , and eigenvectors, ~vn , of M, 2N +2 X ~b(t) = dn~vn eγn t , ~c(t) n=1

(35)

where the dn coefficients are obtained using the initial condition equation ! 2N +2 X h~ ϕ, δNI (x, 0)iνΣf = dn~vn , h~ ϕ, δNXe (x, 0)iνΣf

(36)

n=1

where ϕ ~ = [ϕ0 , ϕ1 , ϕ2 , . . . , ϕN ]T PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014

.

(37) 7/18


Once the xenon perturbation coefficients (~c) have been calculated from Eq. (35), the flux perturbation can be evaluated using δφ(x, t) =

2N +2 X

T ¯ vn eγn t , dn ϕ ~ (x)A~

(38)

n=1

¯ = [0 A] with 0 the (N + 1) × (N + 1) zero matrix and A constructed using Eq. (23) so where A that ~a = A~c. Note that in the following discussion T reference will be made to dynamic modes, which are the cou¯ vn eγn t , not to be confused with the N space-only modes pled 2N + 2 space-time modes ϕ ~ (x)A~ ϕi . The dynamic modes provide a linearly independent basis for describing transient behavior. As seen in Eq. (38), each dynamic mode is separable in space time with its space component described as a fixed linear combination of the N + 1 spatial modes, and its time component described by an exponential with a possibly complex exponent. The constant-reactivity equations, Eqs. (32) and (33), can likewise be written ~ ∂ ~b ⊥ b = M . ~c ∂t ~c

(39)

and as before the corresponding solution to the flux perturbation is δφ⊥ (x, t) =

2N X

T ¯ ⊥~v ⊥ eγn⊥ t . d⊥ ~ (x)A n ϕ n

(40)

n=1

The ⊥ symbol has been included as a superscript in these constant-reactivity equations to distinguish them from the constant-power equations and as a reminder that these solutions are always orthogonal to the fundamental spatial mode, i.e., δφ ⊥ ϕ0 . M⊥ is a rank-2N matrix, but for simplicity of notation we write it here as a (2N + 2) × (2N + 2) matrix with zeros in rows and columns 1 and N + 2. The matrix thus has a degenerate zero eigenvalue of multiplicity two that is omitted from Eq. (40). This approach also permits a more direct analysis of the differences between M and M⊥ , which we now consider. The operator M is associated with 2N +2 dynamic modes, each of which includes the fundamental spatial mode in their basis. In contrast, previous work could represent only 2N dynamic modes associated with M⊥ , all of which are orthogonal to the fundamental spatial mode. The exclusion of the fundamental spatial mode not only eliminates two dynamic modes, but the remaining 2N modes are altered because they are constrained to a lower dimensional space. One could equivalently say that these 2N modes are perturbed into a smaller space. Thus the 2N dynamic modes associated with M⊥ are not generally the same as the 2N + 2 modes associated with M but neither are they independent. To see this write M(α) = (1 − α)M⊥ + M.

(41)

The addition on the right-hand-side is permissible because we have defined M⊥ with the additional trivial rows and columns corresponding to the zeroth expansion order. Thus the spectrum of M(0) 8/18



is equivalent to the spectrum of M⊥ and the spectrum of M(1) is equivalent to the spectrum of M. Because two of the eigenvalues of M⊥ are always trivially zero, we can partition the spectrum as σ (M(0)) = σ2 (M(0)) ∪ σ2N (M(0)) where σ2 (M(0)) = {0, 0}. Because eigenvalues are a continuous function of their operator, we may fix this partition under a continuation of α from 0 to 1. At the end points we then have ⊥ σ(M⊥ ) = γ1⊥ , γ2⊥ , . . . , γ2N ∪ {0, 0} , and σ(M) = {γ1 , γ2 , . . . , γ2N } ∪ {γ2N +1 , γ2N +2 } , where the eigenvalues of the σ2N set have been ordered so that γi⊥ → γi as α goes from 0 to 1. We have thus partitioned the spectrum of the constant-power method into two “new” dynamic modes corresponding to the inclusion of the fundamental spatial mode and a set of 2N dynamic modes that can be paired with dynamic modes of the constant-reactivity method. We can, in fact, write γi⊥ u γi .

(42)

How close this correspondence is depends primarily on kM − M⊥ k [5]. Figure 1 shows two examples (one stable, one unstable) of spectra highlighting this pairing.

Figure 1. Two examples of dynamic mode spectra for a symmetric problem. Open red circles are constant-power eigenvalues; filled blue circles are constant-reactivity eigenvalues. The scattering ¯ is 1.5 cm; and the total fission rate, F , is 1012 (left) and 1014 ratio, c, is 0.96; the mean-free-path, λ, −3 −1 (right) cm s .

From a stability point of view, we are usually interested in the dynamic modes that persist for a long time. This set of modes are associated with the eigenvalues with largest real part: Λ(M) = η : η ∈ σ(M), Re[η] = max Re[λ] . (43) λ∈σ(M)

If this set has more than one element, they will be unique only in their imaginary part. Based on the preceding discussion if Λ(M) ⊂ σ2N then the two methods will likely yield similar stability PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014

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predictions because the dominant dynamic mode is not strongly dependent on the fundamental spatial mode. If however, Λ(M) ⊂ σ2 then the dominant mode is likely to be qualitatively different between the two methods because it contains a substantial contribution from the fundamental spatial mode. The latter situation is typically observed when the system is stable and there are slowly decaying fundamental mode contributions. Finally, because the eigenvalues are a function of the expansion order, it is possible for the dominant eigenvalue of M to transition between σ2 and σ2N as N changes. This type of behavior is observed in the following section, and we will refer to it as dynamic mode crossing.

3. NUMERICAL EXPERIMENTS

Four problems are now investigated numerically using the constant-power and constant-reactivity methods. Both a symmetric and an asymmetric slab problem are considered under stable and unstable configurations. Table I shows the parameters that are fixed in this investigation. The meanTable I. Fixed Parameters. Description

Parameter

Value

γI γX λI λXe σXe f ν

0.061 0.000 2.9173 × 10−5 hours 2.1158 × 10−5 hours 2.6 × 10−18 cm2 0.415 2.4

Fission yield of iodine-135 Fission yield of xenon-135 Decay constant of iodine-135 Decay constant of xenon-135 Xenon-135 microscopic cross section Fission-to-absorption ratio Fission neutron yield

¯ the scattering ratio (c), and the total fission rate (F ) are treated as variable parameters free-path (λ), to control stability and symmetry properties. The diffusion coefficient, absorption cross section, and fission cross section are calculated by 1 ¯, 3λ 1−c = ¯ , λ f (1 − c) = . ¯ λ

D = Σa Σf

At any given time the total fission rate is calculated Z F (t) = Σf (x)φ(x, t)dx.

(44)

The symmetric geometry consists of a 500-cm homogeneous slab centered at x = 0. The stability ¯ ∈ [0.5, 1.5] cm, c ∈ [0.96, 0.9999], and F ∈ [1012 , 1014 ] of the slab was investigated for values of λ 10/18



¯ because the coupling cm−2 s−1 . When F = 1012 cm−3 s−1 , the slab is stable for all values of c and λ between the flux and xenon distributions is relatively weak. When F = 1014 cm−3 s−1 , on the ¯ At the intermediate value of F = 1013 other hand, the slab is unstable for all values of c and λ. cm−3 s−1 the slab exhibits both unstable and stable behavior, primarily for small and large values of ¯ respectively. Two representative points, shown in Table II, were selected to investigate in more λ, detail below. The asymmetric geometry is formed by increasing the absorption cross section in the symmetric slab by a factor of 1.005 for x ∈ [50, 100], but the slabs are otherwise identical. The same two problems analyzed for the symmetric slab are investigated for the asymmetric case, as seen in Table II. In addition to simply looking for a global stability criterion by determining the location of the dominant eigenvalue, we will consider linearized predictions of flux redistribution using Eqs. (38) and (40). The former analysis requires knowledge of only the base (steady-state) configuration, while the latter requires an initial perturbation to be specified. Thus Table II lists functions, ε(x), that provide a perturbation to the base system by Σperturbed (x) = ε(x)Σa (x). a

(45)

[The unstable symmetric case is perturbed asymmetrically specifically to excite the (asymmetric) unstable mode.] The perturbed absorption cross section is used in a coupled calculation to generate the perturbed xenon distribution (details are provided in the following paragraph). One could arbitrary impose a xenon perturbation, but performing the coupled calculation based on perturbed cross sections guarantees that the xenon perturbation is physically realizable. Table II. Description of test problems. Problem 1 2 3 4

c

¯ λ

F

0.960 1.5 1012 0.995 0.5 1013 0.960 1.5 1012 0.995 0.5 1013

Geometry

Stable?

Symm Symm Asymm Asymm

Yes No Yes No

Perturbation ε(x) = 0.990 for −50 ≤ x ≤ 50 ε(x) = 0.999 for −60 ≤ x ≤ 50 ε(x) = 0.950 for −50 ≤ x ≤ 50 ε(x) = 0.999 for −50 ≤ x ≤ 50

The total sequence of calculations performed to obtain results in this work is summarized below, including step 5 that generates reference transient solutions for comparison. 1. Determine the equilibrium: Solve the nominal steady-state coupled diffusion and xenon equations for (φ0 , λ0 , NI,0 , NXe,0 ). 2. Get spatial modes: Calculate N eigenpairs (ϕi , λi ) for i = 1, 2, . . . , N . 3. Get dynamic modes: Calculate the eigenvalues and eigenvectors of the the linearized system to obtain the dynamic modes using the ϕi eigenfunctions as the spatial basis.


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4. Determine the perturbation: Solve the perturbed steady-state coupled diffusion, iodine and ˜I , N ˜Xe and calculate xenon equations for the perturbed xenon and iodine distributions N the dn and d⊥ n expansion coefficients. ˜I , N ˜Xe . 5. Explicilty solve the full transient: Solve the full transient using the initial conditions N Steps 1-3 form the fundamental stability analysis in which the system is linearized and the spatial and dynamic modes are determined. After completing these steps we know the global stability of the system and what dynamic modes are possible, but step 4 is required to determine what modes are actually present and to what degree. After completion of step 4 the coupled xenon/flux transient can be approximated from the linearized system using Eq. (38) or Eq. (40). Step 5, which solves the fully coupled transient system over a given time interval, is performed only to check the accuracy of the linearized predictions. Each step in the above sequence requires a numerical calculation. The goal of this work is to further the theory and explore the performance of generalized xenon stability methods. For simplicity, all calculations were performed in Matlab using built-in algorithms for matrix inversion and eigenvalue/eigenvector computations. In all cases, the diffusion equation was discretized using cell-centered, three-point finite differences. All explicit transients were solved using Newton’s method with implicit time differencing.

3.1. Convergence of the Dominant Eigenvalue Figure 2 shows the convergence of the dominant eigenvalue (i.e., the eigenvalue with the largest real part) for problems 1 (symmetric, stable), 2 (symmetric, unstable), and 4 (asymmetric, unstable). Problem 3 (asymmetric, stable) is not shown because it demonstrates the same behavior as problem 1. The convergence of problem 1 shows what was referred to in the previous section as dynamic mode crossing. For small values of N the constant-power method predicts a dominant eigenvalue associated with the σ2 spectral subset (which is degenerate in the constant-reactivity method), so the two methods show qualitatively different behavior. For symmetric geometries, it was observed that the odd and even order spatial modes will generally decouple, thus the dominant eigenvalue for the constant-power method in this case corresponds to an eigenfunction spanning the even-order spatial modes for small N while the dominant eigenvalue for the constant-reactivity method corresponds to an eigenfunction spanning the odd-order spatial moments. This is also why, as seen in the middle plot of Figure 2, the dominant eigenvalues in the symmetric, unstable configuration are identical between the two methods: the dominant (unstable) mode lies exclusively in the span of the odd spatial modes for all N and thus is not affected by the inclusion of the fundamental spatial mode. For asymmetric geometries the decoupling between odd and even spatial modes does not generally occur, and the spatial distributions of the dynamic modes are not as simple to describe, but similar behavior is observed. In the case of problem 3

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(not shown), for example, the convergence of the dominant eigenmode exhibits behavior identical to that of problem 1. For problems 1-3 we see similar behavior for large N between the two methods: they both predict a dominant eigenvalue belonging to the set σ2N representing paired dynamic modes. Problem 4 also converges to one of the paired dynamic modes, but in this case there is an observable difference between the eigenvalues of the two modes. This example underscores the observation that inclusion or exclusion of the fundamental spatial mode has the potential to impact all dynamic modes, although the impact is generally felt on only a subset of the modes.

Figure 2. Dominant eigenvalue convergence with respect to expansion order. Only the real part of the dominant eigenvalue is plotted.


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3.2. Predicting Transient Behavior The preceding discussion focused exclusively on predictions of the dominant eigenvalue. This eigenvalue indicates the growth or decay of dynamic modes (and hence global system stability). Whether any of these dynamic modes actually are present in a transient depends on the perturbation initiating the transient. In this section the prediction of actual transient behavior for the same four problems is investigated. This type of prediction would be useful, for example, in a stable reactor that nonetheless exhibits significant power oscillations upon certain perturbations. In addition to flux or power oscillations, the constant-power method can also predict the change in reactivity through the transient (assuming no feedback or control action). This type of capability could be used to generate estimates of how much control action would be required to maintain criticality. Table III shows the differences in fission rate, F and multiplication factor, k, at the beginning of each transient for each of the four problems being considered. Because the fission rate is always constant in the constant-power method there is never any difference from the exact solution. For the symmetric problems (1 and 2) with an expansion order of 1 there is no fission rate deviation with the constant-reactivity method because the only transient mode is antisymmetric about the slab midpoint and thus integrates to zero. For expansion orders greater than one, these deviations are very uniform for all of the problems considered, so only the average (over N > 1) is shown in the table. The multiplication factor is always constant with the constant-reactivity method so the δk(CR) row indicates the difference between the steady-state equilibrium and the perturbed state at the beginning of the transient, i.e. the reactivity change induced by the perturbation which cannot be accounted for in the constant-reactivity method. The constant-power method method predicts a time varying multiplication factor with some error resulting from the expansion truncation and the linearization. These errors are shown in the last two rows of the table. Again for expansion orders greater than one, these errors are very uniform for all of the problems considered, so only the average is shown in the table. Table III. Fission rate and multiplication factor error summary*. 1 ∆F/F (CR) [N = 1] 0.0% ∆F/F (CR) [N > 1] 1.4% δk(CR)** -1.2 δk(CP) [N = 1] -0.4 δk(CP) [N > 1] 0.2

2

3

4

0.0% 0.8% -0.2 -0.2 -0.1

0.6% -2.1% 3.2 1.3 0.5

0.1% 0.8% -0.2 -0.1 0.1

*“CR” is the constant-reactivity method; “CP” is the constant-power method. **δk = kapprox − kexact × 104

Tables IV-VII show various metrics pertaining to how well the flux perturbation is predicted rela-

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tivel to the full transient simulation. For each time step, a relative error is defined as RE(x) =

φapprox (x) − φexact (x) , φexact (x)

(46)

and a weighted error is defined as |RE(x)|φexact (x) W E(x) = R , φ (x0 )dx0 V exact

(47)

where V is the volume of the slab. The norms shown in the tables are taken with respect to space, and only the largest value over all time is shown. Thus kW Ek1 indicates the average weighted error, which obtains its largest value at the time shown as t∗ , and kREk∞ indicates the space-wise maximum relative error, which obtains it largest value at the time shown as t∗ . The ratio of the local to the average flux (i.e. the peaking factor), P ∗ , is also shown at the position and time of the largest relative flux error as an indication of how significant that relative error is. Table IV. Problem 1 flux error summary. Order 1 2 4 6 499

Constant Reactivity kW Ek1 t∗ kREk∞ P ∗ t∗ 5.25% 0.0 8.77% 0.01 0 1.44% 0.5 -3.53% 1.43 0 1.40% 0.5 -1.92% 1.41 0 1.40% 0.5 -1.77% 1.41 0 1.40% 0.5 -1.75% 1.40 0

kW Ek1 5.25% 0.96% 0.12% 0.01% 0.01%

Constant Power t∗ kREk∞ P ∗ t∗ 0.0 8.77% 0.01 0 0.0 -2.51% 0.01 1 2.5 0.30% 0.01 0 2.0 -0.15% 0.01 3.5 2.0 0.17% 1.40 2.5

For problem 1 the maximum errors for both methods tend to occur very early in the transient, which is not surprising because this is a stable configuration–the maximum deviation from equilibrium occurs either very close to or at the initial perturbation. Also because this is a symmetric problem being perturbed symmetrically, it is unsurprising that the fundamental spatial mode plays an important role in the transient. Thus the constant-power method demonstrates significantly improved accuracy relative to the constant-reactivity method. Table V. Problem 2 flux error summary. Order 1 2 4 499

kW Ek1 5.88% 2.11% 2.84% 2.96%

Constant Reactivity t∗ kREk∞ P ∗ t∗ 4.0 -10.45% 1.29 4.0 13.0 4.33% 0.01 20.5 20.5 5.56% 0.01 20.5 20.5 5.39% 0.22 20.5

kW Ek1 5.88% 2.07% 2.87% 2.99%

Constant Power t∗ kREk∞ 4.0 -10.45% 13.0 4.29% 20.5 5.50% 20.5 5.33%

P∗ 1.29 0.01 0.01 0.01

t∗ 4 20.5 20.5 20.5

Prolem 2 is symmetric but unstable, and it is perturbed slightly asymmetrically specifically to excite the unstable mode. The errors shown in the table were calculated only over the first period of the unstable oscillation, which is 20.5 hours. The unstable mode is antisymmetric in this case and can thus be well described using odd spatial modes. It is therefore unsurprising that both methods predict the transient behavior with similar accuracy. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014

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Table VI. Problem 3 flux error summary Order 1 2 4 6 8 499

kW Ek1 9.13% 2.62% 2.22% 2.19% 2.17% 2.16%

Constant Reactivity t∗ kREk∞ P ∗ 0.0 -21.07% 0.01 0.0 7.39% 0.01 0.0 3.51% 0.01 0.0 2.97% 1.81 0.0 2.17% 1.01 0.0 2.17% 1.13

∗

t 0.0 1.0 1.0 0.0 1.0 1.0

kW Ek1 9.22% 1.81% 0.73% 0.43% 0.38% 0.37%

Constant Power t∗ kREk∞ 0.0 -23.72% 0.0 10.94% 0.0 2.27% 0.0 3.33% 0.0 1.94% 0.0 2.02%

P∗ 0.01 0.01 0.61 0.01 0.35 0.01

t∗ 0.0 0.5 1.0 1.0 1.0 1.0

Problem 3, which is asymmetric and stable, shows improvements in accuracy with the constantpower method similar to problem 1. Convergence with respect to expansion order is slightly slower because the spatial dynamics are more complicated because of the asymmetry. Table VII. Problem 4 flux error summary. Order 1 2 4 6 499

kW Ek1 6.16% 1.29% 1.76% 1.92% 1.92%

Constant Reactivity t∗ kREk∞ P ∗ 4.0 -11.64% 1.28 12.0 -3.61% 0.01 13.0 -2.96% 0.01 13.5 -3.34% 0.01 13.5 -3.33% 0.26

∗

t 4.0 4.5 19.5 13.5 12.5

kW Ek1 6.16% 2.42% 2.34% 2.18% 2.27%

Constant Power t∗ kREk∞ 4.0 -11.38% 19.5 5.80% 19.5 -5.26% 19.5 4.77% 19.5 4.87%

P∗ t∗ 1.28 4.0 0.01 19.0 0.01 19.5 0.01 19.5 0.01 19.5

Finally, in Problem 4 both methods perform quite similarly over the first period of the unstable oscillation because the unstable mode is again largely an odd function about the slab origin. The table indicates a degradation in accuracy with the constant-power method, but direct comparison is complicated by the fact the maximum errors between the methods occur at different times, and for an unstable problem such as this one, the linearization approximations become a factor late into the transient. Figure 3 shows the L∞ norm of the weighted errors as a function of a time for all four problems with an expansion order of six. These four figures indicate that even for the unstable problems, the constant-power method generally better predicts flux perturbation, particularly early in the transient.

4. CONCLUSION

A new method for the analysis of xenon transients in nuclear reactors based on generalized λ-mode expansions has been derived and numerically demonstrated. This new approach has several notable differences from previous work:

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Figure 3. Weighted relative errors as a function of time for problems 1-4.

• The flux perturbation can include components in the fundamental equilibrium spatial mode; • The power level can be explicitly constrained;


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• Reactivity is allowed to vary to accommodate the quasistatic nature of the transient. Numerical results seem to indicate that the main practical advantage of the new method is improved prediction of the flux perturbation. Thus for small perturations one can better predict local power changes without explicitly simulating the transient. In reality, reactors undergoing a xenon transient will be operated at both constant power level and constant reactivity. Therefore both of the methods approximate reality in some sense because the effects of thermal-hydraulic feedback and direct operator control action have been neglected. It is possible to include passive feedback effects in modal stability analyses ([1, 4, 6], for example), but it is impossible to include operator-driven control actions in the type of linearized analysis presented here. Pragmatically, the best such an analysis can offer is the prediction of transient behavior in regimes where nonlinear feedback or external actions are not significant. An example of such a regime is a small perturbation to a stable reactor, which is the case where the new constant-power method seems to excel. In such situations the new method is expected to accurately predict (to first order) the response of the reactor including spatial power redistribution and associated changes in reactivity.

REFERENCES [1] W. Stacey. “Linear analysis of xenon spatial oscillations.” Nuclear Science and Engineering, 30: p. 453 (1967). [2] W. Stacey. “A nonlinear xenon stability criterion for a spatially dependent reactor model.” Nuclear Science and Engineering, 35(3): pp. 395–396 (1969). [3] G. Lellouche. “Space dependent xenon oscillations.” Nuclear Science and Engineering, 12(4): pp. 482–489 (1962). [4] T. R. England, G. L. Hartfield, and R. K. Deremer. Xenon Spatial Stability in Large SeedBlanket Reactors. Technical Report WAPD-TM-606, Bettis Atomic Power Laboratory, Pittsburgh, PA (1967). [5] F. L. Bauer and C. T. Fike. “Norms and exclusion theorems.” Numerishce Mathematik, 2(1): pp. 137–141 (1960). [6] S. Kaplan and J. B. Yasinsky. “Natural modes of the xenon problem with flow feedback–an example.” Nuclear Science and Engineering, 25(4): pp. 430–438 (1966).

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Generalized Lambda-Mode Xenon Stability

Generalized Lambda-Mode Xenon Stability

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