Supposing that the control system keeps constant the total reactor power output, it is still possible to have local power variations in which the power output is increasing in some part of the reactor while decreasing in other parts. There are two possible forms of spatial instability: temperature and Xe instability.

Temperature instability can occur for sufficiently large values of a positive moderator temperature coefficient. To a local flux increase a temperature increase corresponds which through the positive temperature coefficient tends to enhance further the flux

Fig. 12.5. THTR reactivity ramp of 6.43%r in 0.002 sec. Influence of the prompt neutron lifetime on the zero­dimensional calculation.

disturbance. This effect must compete with the negative fuel coefficient and the increase in leakage corresponding to the distortion of the flux distribution. Depending on the reactor size and the fuel temperature coefficient there is a threshold value for the moderator coefficient above which spatial temperature instability occurs. While the temperature instability has a monotone divergent character, the Xe instability gives rise

to oscillations which can have divergent amplitude. A local flux increase causes a local reduction of Xe concentration because of neutron absorption in Xe. This Xe reduction will, in turn, produce a further flux increase, and at the same time increases the 135I production from fission, so that after a few hours the decay of 135I will overcompensate the 135Xe loss and the flux will start to decrease, reversing the trend.

A positive moderator temperature coefficient will emphasize this phenomenon, while a damping effect comes from the increase in leakage due to the flux distortion.

The threshold beyond which the oscillations diverge is determined by the reactor size, equilibrium Xe concentration (power level) and temperature coefficient.

Power-flattening usually tends to give a lower stability.

The high migration length of HTRs is a stabilizing factor in comparison to LWRs, but the lower power density of HTRs requires bigger dimensions for a given power level, so that these two reactor types have, for the same power, approximately similar stability problems. On the other hand, HTRs are more stable than other graphite reactors (e. g. Magnox). Because of their very long period these spatial instabilities do not usually pose a safety problem, but rather a control problem.

Xenon and temperature instabilities can be counteracted by providing different regions with independent control systems (sector controllers) which respond to the temperature of each considered region and not to the overall reactor temperature or flux.

A problem is posed by the measurement of the flux and temperature distribution which is, in HTRs, more difficult than in other reactor types. Because of the high temperature, in-core instrumentation is either impossible or unreliable, so that one must rely on informations obtained from measurements made outside the core (e. g. flux in outer reflector).

Xenon and temperature instabilities can be numerically studied with space dependent dynamics codes for slow transients of the type described in the preceding paragraphs.

A first investigation on the instability of a reactor can be performed with simplified analytical methods.

The system is described by the Xe and I equations (12.24) together with the one group diffusion equation in the form (§8.11)

V2(/> + В 2ф — 0.

Simplified solutions can be obtained assuming only small perturbations about the steady state119’ so that [see eqn. (8.41)]

Фіг, t) = ф'(г) + (f (r, f),

Br, t) = B2r) + ^2= B2r) + jpATm(r, t) + j^-2ATt(r, t) + fpx(r, t),

Xe(r, f) = Xe'(r) + x(r, f),
l(r, t) = l'(r)+i(r, t),

where Ф*(г), B2′(r), Xe*(r) and I*(r) are steady-state values, ax is a Xe reactivity coefficient, ©/ and 0m the fuel and moderator temperature coefficients, AT/ and ДTm the fuel and moderator temperature perturbations. (In ref. 19 the Xe and I concentra­
tions are expressed as relative values to the saturation Xe concentration which is reached at very high flux levels.)

The perturbations are then assumed to have a time dependence of the form

<p(r, t) = <pt(r) e“",

x(r, t) = Xi(r) e“”, (12.39)

і (r, t) = 11 (r) e“

and the space dependence is expanded as sum of a truncated infinite series of the orthonormal eigenfunctions f of the boundary-value problem that has the unperturbed flux shape as its fundamental solution

<Pi(r) = 2 a‘f‘

x,(r) = 2V< (12.40)

fi(r) = 2 cf,

(V2+B2*)/i + p2f =0 (12.41)

with this modal expansion it is possible to linearize the system of eqns. (12.24) and (12.26) and the associated heat-transfer equations, neglecting the product of the perturbations. If coefficient like 0m, 0/, ax. . . can be considered as space — and time-independent, it is possible to obtain a characteristic equation in w, for each mode i, using the orthogonality property:

ffdV = 8„.

The mode і is stable if all roots of its characteristic equation have a negative real part.

A further simplification consists in neglecting the heat-transfer equations, defining a temperature coefficient aT expressed in terms of reactivity per unit flux

0mATm(r, t) + QfATf(r, t) = aT(p(r, t). (12.43)

With these assumptions091 it is possible to obtain the following stability criterion:
The threshold for oscillations in the jth mode of the flux distribution occurs when

1

M2B2 д aT , _ 1 + Ах! сгхф ^x ax 1 ax + (Xx + Хі)Ісгхф

where Л, is the difference between the eigenvalue of the jth mode of the flux distribution and the eigenvalue of the fundamental mode, divided by the geometrical buckling.

В2 = is the geometrical buckling,

A„ Ax = decay constant of l35I and 135Xe,

yx = fraction of l35Xe formed directly from fission rather than through 135I, crx — l35Xe cross-section,

a*=is the reactivity held by saturated 135Xe (corresponding to the limit concentration Xesal for very high flux) «’,= a, Xe“t,

M2 = migration area.

In this way it is possible to assess the stability of each mode independently.

The smallest Л occurs usually for the first azimuthal mode, which is then the first mode to become unstable.

The simplest eigenfunction shapes are the least stable because they have fewer nodal lines, which determine the increase in leakage relative to the fundamental mode.

The form of some of the radial-azimuthal and axial eigenfunctions is shown schematically in Fig. 12.7.<20) The form of the imperturbed flux has an important influence on stability, so that it is important to repeat the investigation with different control-rod configurations (which in the simplified treatment appear through the term B2′(r)). The solutions are usually studied for a range of values of the moderator temperature coefficient and of power levels. As the temperature coefficient gets positive and bigger the period of the Xe oscillation increases and the oscillatory Xe instability tends toward the monotone temperature instability. With this simplified treatment it is possible to check the stability margin for each mode. The stability margin is usually defined as the difference between the actual moderator temperature coefficient and the value of the coefficient for which instability occurs. Independent sector controllers are required not only in case of instability, but also in case of too-small a stability margin when the natural damping of external perturbations is insufficient.

The number and position of the sector controllers is dictated by the form of the modes which need stabilizing. The reactor operation should be possible also in case of failure of one controller.

It is possible to include the feedback of the reactor controller in simplified analytical methods, but usually only numerical codes allow an accurate investigation of the effect

Fig. 12.7. Xe instability: shape of the most important azimuthal radial and axial modes.120′

of power distribution and control-rod position on stability and the optimization of the sector control system.

Two-dimensional (г, в) codes are necessary to study azimuthal and radial instability, while axial instability may be studied with one-dimensional codes.

Codes to study Xe spatial oscillations are described in refs. 21, 22 and 23. About the optimization of the control strategy see ref. 24. Investigations carried out for HTRs<20> show that axial instability is not likely to occur for core heights < 10 m (the present values are around 5 m). Also radial modes appear to be very stable. The most unstable mode is the first azimuthal and its natural damping might become critically small already for reactor sizes of the order of 1000 MWe, depending from core composition, power-flattening and height-to-diameter ratio. The control-rod requirement necessary to damp Xe oscillations is, in any case, very small.