An application of the one-group first-order perturbation theory is discussed here.

A bare cylindrical reactor of extrapolated radius R and height H is considered, in which a central control rod of radius a is partially inserted, as shown in Fig. 1.15. The insertion depth of the control rod from the origin in the top of the cylindrical reactor is denoted by x. If the control rod is a relatively weak absorber of neutrons and the control rod insertion has a small effect on the change in neutron flux distribution, then the first-order perturbation theory can be applied to obtain the reactivity worth of the partially inserted control rod.

The macroscopic absorption cross section is assumed to increase by 8 X a in the region of 0 ^ z ^ x and 0 ^ r S a • In this coordinate system, the unperturbed neutron flux is

(1.107)

where A is a constant. Introducing Eq. (1.107) into Eq. (1.106) and noting that the differential volume element d3r is 2nrdrdz, the reactivity change due to the rod insertion of x can be obtained as Eq. (1.108).

Comparing this with the reactivity change p(H) when the control rod is fully inserted, the relative reactivity worth of the control rod is given by Eq. (1.109) which is illustrated in Fig. 1.16.

Fig. 1.16 Relative reactivity worth of control rod as a function of its insertion depth

This is called the S-curve of control rod worth. The maximum change in the reactivity occurs when the end of the control rod is at the center of the reactor.

This case shows that the perturbation theory can be used to provide a satis­factory estimate of the reactivity worth as a function of location in the perturbation region.

Exercises of Chapter 1

1. Consider the hypothetical case of an infinite-sized thermal reactor initially fueled with enriched uranium which operates at a constant neutron flux. For a constant atomic density of 238U: (a) derive the equations that determine the time

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dependence of the U and Pu concentrations in the reactor and (b) plot the behavior as a function of time.

2. (a) Derive the equations that determine the transient behavior of the 135I and 135Xe concentrations in an infinite-size thermal reactor which operates at a neutron flux ф0 when the flux is changed to ф1 and (b) plot the behavior as a function of time.

3. Explain the main reactivity feedback effects and reactivity control methods for each system of the BWR, PWR, high-temperature gas-cooled reactor (HTGR), and liquid-metal fast breeder reactor (LMFBR).

4. Consider a light water-cooled graphite-moderated thermal reactor fueled with very low-enriched uranium which operates with burnup. Explain the causes of the results in the cases of (a) a positive coolant void coefficient and (b) a positive moderator temperature coefficient [27].

5. Consider a 235U fueled thermal reactor which is a homogeneous cubic reactor of side length L. Assume that the temperature suddenly increases by the amount AT
throughout a cubic region of side length L/6 along the center of the reactor. Compute the reactivity change introduced into the reactor by this local temper­ature rise using the one-group first-order perturbation theory. Ignore the change in neutron leakage due to the temperature rise [26].