Reactor Kinetics and Point Kinetics

Abstract

The point kinetics model can be obtained directly from the space- and time-dependent transport equations. However, these equations are too complicated to be of any practical application. The diffusion approximation, obtained by keeping only the PI terms of the spherical harmonic expansion in the angular variable of the directional flux, is frequently used in neutronic analysis. This chapter discusses reactor characteristics that change because of changing reactivity. A basic approach using a minimum of mathematics has been followed. Emphasis has been placed on distinguishing between prompt and delayed neutrons and showing relationships among reactor variables, k_eff, period, neutron density, and power level.

Problems

Problem 11.1

Thermal Reactor Example

Consider a thermal reactor example. The following results were obtained for a six-group model of the Advanced Reactivity Measurement Facility reactor: a small 93% enriched aluminum, slab core reactor. The model is based on a bare core version of this reactor shaped into a cube with a 51.09 cm side dimension. This gave a buckling of B² = 0.011344/cm². Note that this is a fairly small reactor.

Consider the following results:

Group	Upper energy	Velocity	vΣf	Flux	Adjoint	χ p	χ d	χ
1	17.33 MeV	4.24(+9) cm/s	0.000643	0.03691	0.122862	0.05258	1.27(−6)	0.05221
2	4.97.MeV	2.24(+9) cm/s	0.000492	1.00383	0.531294	0.52040	0.050525	0.51711
3	1.35 MeV	8.66(+8) cm/s	0.000433	2.03885	0.695888	0.41086	0.803196	0.41361
4	111.0 keV	1.93(+8) cm/s	0.000854	0.59341	0.849079	0.01608	0.145344	0.01699
5	3.35 keV	5.95(+6) cm/s	0.01863	1.89784	0.875391	8.0(−5)	0.000933	8.6(−5)
6	0.100 eV	2.20(+5) cm/s	0.18219	1.00000	1.000000	0.0	0.0	0.0

• Calculate the normalization factor.

• Calculate β_effective and β_physica for this problem.

Problem 11.2

The thermal absorption cross section for Xe-135 at 2200 meters is 2.7E+6 barns. Based on the bounds developed in class, what is the maximum absorption cross section that could be expected at 0.0253 eV for this nuclide? Estimate the total cross section at 2.0 MeV also.

Problem 11.3

Determine the thermal lifetime or diffusion time of neutron in the bare critical reactor where L² = 57.3 and B² = 0.0051 cm^-2 and consisting of beryllium and uranium-235 in the atomic ratio of 10⁴ to 1.

Problem 11.4

The reactivity in a steady-state reactor, in which the neutron generation time is 10⁻³ s, is suddenly made 0.0022 positive; assuming one group of delayed neutrons, determine the subsequent change of neutron flux with time. Compare the stable period with that which would result had no delayed neutrons. Use the following equation for variation of the neutron density with time:

$$ n\approx {n}_0\left[\frac{\beta }{\beta -\rho }{e}^{\frac{\lambda \rho t}{e^{\beta }-\rho }}-\frac{\rho }{\beta -\rho }{e}^{-\frac{\left(\beta -\rho \right)t}{l^{\ast }}}\right] $$

(11.74a)

In addition, for the stable reactor period, assuming one (average) group of delayed neutron is given by:

$$ {T}_p\approx \frac{\beta -\rho }{\lambda \rho} $$

(11.75b)

Assume on average, for one group of delayed neutron, λ is 0.08 s⁻¹; ρ is given as 0.0022 and neutron generation time l^∗ as 10⁻³ s; β for uranium-235 is 0.0065.

Problem 11.5

The prompt neutron lifetime in a reactor moderated by heavy water is 5.7 × 10⁻⁴ s. For a reactivity of 0.00065, express the reactor period (a) in second, (b) in hours (Ih), using the relationship between reactor period T_p, and in hours (Ih) as T_p ≈ 3600/Ih. What is the reactivity in dollar units? Assume for very long periods and very small reactivity the following equation applies and (β/λ) = 0.084:

$$ {T}_p\approx \frac{\beta }{\lambda \rho} $$

(11.76a)

where:

• T_p is reactor period

• β is total fraction of delayed neutrons = 0.0065

• λ is radioactive decay constant

• ρ is reactivity

In addition, the unit that is called “the dollar,” is defined by:

$$ \mathrm{Reactivity}\ \mathrm{in}\ \mathrm{dollars}\equiv \frac{\rho }{\beta } $$

Problem 11.6

As part of one-group point kinetics equation, if we assume a case for when only one group of delayed neutron is assumed, the results of solution provide a polynomial in the denominator as part of general solution, when we assume source term is constant and equal to S₀ with two roots of s₁ and s₂ of the following mathematical notation:

$$ {s}_{1,2}=\frac{-\left(\frac{\beta }{\Lambda}-\frac{\rho_0}{\Lambda}+\lambda \right)\pm \sqrt{{\left(\frac{\beta }{\Lambda}-\frac{\rho_0}{\Lambda}+\lambda \right)}^2+4\frac{{\lambda \rho}_0}{\Lambda}}}{2} $$

(11.77a)

where:

• β is fractional yield of the delayed neutrons

• Λ is average neutron generation time

• ρ₀ is reactivity for constant source S₀

• λ is decay constant.

Then the neutron flux can be obtained as:

$$ {\displaystyle \begin{array}{l}n(t)={n}_ex(t)+{n}_e\\ {}\kern1.5em ={n}_e\left\{1+\left(\frac{\rho_0}{\Lambda}+\frac{S_0}{n_e}\right)\left[\frac{\lambda }{s_1{s}_2}+\frac{s_1+\lambda }{s_1\left({s}_1+{s}_2\right)}{e}^{s_1t}+\frac{s_2+\lambda }{s_2\left({s}_2-{s}_1\right)}{e}^{s_2t}\right]\right\}\end{array}} $$

(11.78b)

where:

• n(t) is neutron density as function of time t

• n_e is neutron density at equilibrium

Assuming one group of delayed neutrons, determine the subsequent change of neutron flux with time. Additionally, the reactivity is taking place in a steady-state thermal reactor with no external neutron source, in which the neutron generation time Λ = 10⁻³ s is suddenly made 0.0022 positive and the following data applies: λ = 0.08 s⁻¹ and β = 6.5 × 10⁻³.

Problem 11.7

A critical reactor operated during a long period of time with the mean weighted number of neutrons equal to n_e = 10⁶. Suddenly a source of neutrons with mean weighted yield equal to S₀ = 10⁶ s⁻¹ was introduced into the reactor. Find the number of neutrons as a function of time, n(t). In calculations assume one group of delayed neutrons with λ = 0.1 s⁻¹ and β = 6.4 × 10⁻³ and the mean generation time Λ = 10⁻³ s.

Problem 11.8

An under-critical reactor operated during a long period of time with the reactivity equal to ρ₀ =  − 9β. A source of neutrons with a mean weighted yield equal to S₀ = 10⁶ s⁻¹ was present in the reactor. Suddenly the source of neutrons was removed from the reactor. Find the number of neutrons as a function of time, n(t). In calculations assume one group of delayed neutrons with λ = 0.1 s⁻¹ and β = 6.5 × 10⁻³ and the mean generation time Λ = 10⁻³ s. Note that the equilibrium number of neutrons before the removal of the source is given by n_e =  − S₀Λ/ρ₀.