Reactor Dynamics

Abstract

In order for nuclear fission power to operate at a constant power level, the rate of neutron production via fission reactions must be exactly balanced by neutron loss via absorption and leakage. If we deviate from this simple balancing role, it would cause time dependence of neutron population and therefore the power level of the reactor. Such situation may take place, for a number of reasons, such as reactor operator may have a requirement to change the reactor power level by temporarily altering the control fuel rod so it will change the core or source multiplication or there may be long-term changes in core multiplication due to fuel depletion and isotopic buildup. Other examples may also be encountered that require attention and adjustment to day-to-day operation of reactor, such as unforeseen accident or failure of primary coolant pump system, etc. The topic of nuclear kinetic reactor as we have learned in the previous chapter is handling this situation by allowing us to predict the time behavior of the neutron population in a reactor core driven by changes in reactor multiplication, which are not circumstances that are totally controlled by the operator of power plant and reactor core. Furthermore, variables such as indirect accessibility to control such as the fuel temperature or coolant density distribution throughout reactor do have impact to the situation. However, these variables depend on the reactor power level and hence the neutron fluxes itself. Additionally, study of the time dependence of the related process, which is involved in determining the core multiplication as a function of power level of the reactor multiplication, is subject of our study in this chapter, and it is called nuclear reactor dynamics. This usually involves detailed modeling of the entire nuclear steam supply system, which is part of feedback system as well.

Problems

Problem 12.1

The AGN-201 is operating at 5 watts. It is then placed on a positive transient by inserting a reactivity of 0.00375. After 30 s, the reactor scrams automatically. At what power level is the scram point set? Use a One Delay Family model for the reactor’s behavior. The following data apply.

$$ {\beta}_{\mathrm{effective}}=0.0075\kern2em \Lambda =0.00015\ \mathrm{s}\kern2em \lambda -\mathrm{bar}-\mathrm{inv}=0.07675/\mathrm{s} $$

Problem 12.2

Given the following solution for half of a cell in slab geometry, calculate the fast and thermal self-shielding factors for the cross sections in this cell.

Z Dimension	0.0	Fuel	0.2	Moderator	0.3
Fast Flux	1.0		0.85		0.85
Thermal Flux	0.15		0.9		1.0

Fig. 12.5

Problem 12.2

Problem 12.3

Consider a hypothetical reactor in which all of the materials have the same volumetric coefficient of thermal expansion. Thus, all of the nuclide densities decrease according to the same ratio: N^'/N = constant <1. Additionally, consider that relationship between leakage effects and infinite medium is given by the following equation:

$$ \frac{dk}{k}=\frac{dk_{\infty }}{k_{\infty }}-\frac{M^2{B}^2}{k_{\infty }}\left(\frac{dM^2}{M^2}+\frac{dB^2}{B^2}\right) $$

(12.34)

Moreover, since the leakage probability P_L for large power reactors usually is quite small, so we can write:

$$ {P}_L=\frac{M^2{B}^2}{1+{M}^2{B}^2}\ll 1 $$

(12.35)

In addition, the effective multiplication k_∞ can be expressed as Eq. (12.35) below, using energy-averaged cross section. This is based on examination of neutron spectra, assuming that all the constituents of a reactor core are exposed to the same energy-dependent flux ϕ(E):

$$ {k}_{\infty }=\frac{\int_0^{\infty }v{\Sigma}_f(E)\phi (E) dE}{\int_0^{\infty }{\Sigma}_a(E)\phi (E) dE} $$

(12.36)

• (a) Show that the expansion with increased temperature has no effect on k_∞.

• (b) Using the facts that the core mass, NV, remains constant and that M ∝ N⁻¹, show from Eq. (9.4) that the reactivity change from expansion is negative, with a value of $\frac{dk}{k}=-\frac{4}{3}{P}_L\frac{dV}{V}$.

Problem 12.4

Heat transmission in system with internal sources.

If the heat transfer balance for steady state is defined as below for a thin slab of thickness dx at x:

$$ {\displaystyle \begin{array}{l}\mathrm{Heat}\ \mathrm{Conducted}\ \mathrm{out}\ \mathrm{of}\ Adx-\mathrm{Heat}\ \mathrm{Conducted}\ \mathrm{in}\mathrm{to}\ Adx\\ {}\kern11em =\mathrm{Heat}\ \mathrm{Generated}\ \mathrm{in}\ Adx\end{array}} $$

where A is the heat conduction area and Adx is the volume of the section A normal to the direction of x, then (a) define the flow of heat by conduction, and describe your parameters in the equation defining the heat flow. This is what we know as Fourier law. (b) Expand the Fourier law to a volumetric thermal source that is expressed by Q(x), which is heat generated per unit time per unit volume at x with dimension of Btu/(hr)(ft²) in British system.

Problem 12.5

Using the part (b) of solution for Problem 12.4:

• Part (a): Provide the appropriate differential equation for a situation, where we have heat transmission in shields and pressure vessels, i.e., slab with exponential heat source Q(x) = Q₀e^−μx, where again Q(x) is volumetric heat source strength rate of heat release per unit volume, Q₀ is constant for heat source, and μ is the linear attenuation coefficient or macroscopic cross section of the radiations.

• Part (b): Solve the differential equation of Part (a), by firstly finding the general solution and secondly giving the boundary condition as T(x)∣_x = 0 = T₁ and T(x)∣_x = L = T₂, and find the particular solution of the general solution in the first part. Thirdly draw the depiction of heat transmission in slab with exponential source and the given boundary conditions.

• Part (c): Under certain condition, one can find the maximum for the particular solution in second step of part (b) and then find that maximum expression.

Problem 12.6

A water-cooled and water-moderated power reactor is contained within a thick-walled pressure vessel. This vessel is protected from excessive irradiation and thus excessive thermal stress by a series of steel thermal shields between the reactor core and the pressure vessel. One of these shields, 2 in. thick, whose surfaces are both maintained at 500 °F, receives a gamma-ray energy flux of 10¹⁴ MeV/(cm²)(s). Calculate the location and magnitude of the maximum temperature in this shield. The linear attenuation coefficient of the radiation in the steel may be taken to be 0.27–1 cm and the thermal conductivity as 23 Btu/(hr)(ft²)(⁰F/ft). Hint: Use the results of solution that you found in Part (c) of Problem 15.5 and Figure in Part (b). Additionally, assume that for steel Q₀ = ϕEμ_e, where μ_e is energy absorption coefficient, which is 0.164 for steel, and ϕE is gamma-ray energy flux of gamma-ray energy E.

Problem 12.7

The Fuchs-Nordheim model predicts the shape and magnitude of the transient. We do not really solve analytical solution of the model but instead some characteristic from it. If we write the first part of Eq. 12.18 as the following form for point kinetic equation (PKE), we have to:

• (a) Argue under what assumption Eq. 12.18 (i.e., first part) reduces to Eq. 12.25 and why historically this assumption was made by Manhattan Project weapon designers.

• (b) Assume that if transient is so rapid that no heat is transferred from the fuel (i.e., The time constant for heat to be removed from UO₂ fuel is about 5 min) with heat capacity, C_P is given by the following relation:

$$ {T}_{fuel}(t)={T}_{fuel}^0+\frac{1}{C_p}\int P(t) dt $$

(12.37)

In addition, assume a Doppler feedback coefficient independent of temperature (recall that we calculate that PWRs have a Doppler coefficient which is about −3 pcm/K), and we can write:

$$ \rho (t)={\rho}_{rod}-\alpha \left({T}_{fuel}-{T}_{fuel}^0\right) $$

(12.38)

Then calculate the peak temperature characteristics for power distribution P(t).

• (c) Calculate asymptotic characteristics for power distribution P(t).

• (d) Finally, yet importantly, show that asymptotic temperature is independent of the reactivity insertion rate.

Problem 12.8

Solve Eq. 12.29.