2.1.1 Fundamental Neutron Transport Equation

The collective behavior of neutrons in a reactor core is described by the neutron transport equation presented in Eq. (2.1) which is also referred to as the Boltzmann equation.

Y. Oka (ed.), Nuclear Reactor Design, An Advanced Course in Nuclear Engineering 2, DOI 10.1007/978-4-431-54898-0_2, © Authors 2014

— ~g~ф(г, Й, E, t)=S(r Й, E, t)—QeV(/)G’, Ї2, i£, О /о

2.1)

—> —> —> 4 7

—X*(r, 1£, t)<p(r, І2, 1£, 0

Here, S is the neutron source, £t is the macroscopic total cross section, and ф is the angular neutron flux being calculated. This equation represents the balance between gain and loss in the unit volume of neutrons that are characterized by a specific kinetic energy E (velocity u) and are traveling in a specific direction ~ at a time t and a position r. That is, the time change of the target neutrons (the first term in the LHS) is given by the balanced relation among: the gain of neutrons appearing from the neutron source S (the first in the RHS), the net loss of neutrons traveling (the second term in the RHS), and the loss of neutrons due to nuclear collisions (the third term in the RHS). It should be noted that the changes in angle and energy of neutrons are also included in the gain and loss.

The target neutrons are gained from three mechanisms: scattering, fission, and external neutron sources. Each gain is represented in

S(r, Й, E, t)=f0°°dE’finbs(r, E’^E, t)<j>(r, Ъ’, E t)d£2′

+ Z^-/0oodE’/inx(E)v’Lf(r, E E’, t)dQ’

—Sex(r, І2, E, £)

(2.2)

where £s and are the macroscopic scattering and fission cross sections, respec­tively. v is the average number of neutrons released per fission and the product vlf is treated as a production cross section. The cross sections are described in the next section.

The first term in the RHS of Eq. (2.2) is called the scattering source and it totals the number of the target neutrons scattering into E and ~~ from another energy E0 and

direction ~~ by integrating the number for E0 and ~~ . The second term is the fission source and it indicates that the neutrons produced by fission over the whole range of energies are distributed with the isotropic probability in direction (1/4n) and the probability x (E) in energy. x (E) is called the fission spectrum and it is dependent on the nuclide undergoing fission and the energy of incident neutrons. For instance, the fission spectrum in an enriched uranium-fueled LWR is well described by the function of Eq. (2.3).

%(E) = sinh>J2.29E exp (-£/0.965) (2.3)

Since x(E) is a probability distribution function, it is normalized so that

The third term in the RHS of Eq. (2.2), Sex, expresses the external neutron source for reactor startup which may be such species as 252Cf or Am-Be. Therefore, it is not used in the nuclear design calculation of a reactor in operation.

More details of the neutron transport equation are not handled here. If the cross section data (£t, £s, and v£f) in Eqs. (2.1) and (2.2) are provided, the angular neutron flux ф (r, £2, E, t), which depends on location, traveling direction, energy, and time, can be calculated by properly solving the equation.

The information on traveling direction of neutrons is finally unimportant in the nuclear design calculation. The scalar neutron flux integrated over the angle is rather meaningful.

(2.5)