7: Neutron Balance

7: Neutron Balance B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2013 Sept.-Dec.

Contents • The Neutron-Transport Equation • The Neutron-Diffusion Equation • Stages of practical neutronics calculations: • lattice calculations • full-core calculations

Reactor Statics: Neutron Balance • In reactor statics we study time-independent phenomena. • Independence of time means that there is (or is assumed to be) neutron balance everywhere. • Therefore, in reactor statics, all phenomena which involve neutrons must result altogether in equality between neutron production and neutron loss (i.e., between neutron sources and sinks) at every position r in the reactor and for every neutron energy E. • These phenomena are: • Production of neutrons by induced fission • Production of neutrons by sources independent of the neutron flux • Loss of neutrons by absorption • Scattering of neutrons to other energies or directions of motion • Leakage of neutrons into or out of each location in the reactor

Neutron-Transport (Boltzmann) Equation • Neutron balance is expressed: • essentially exactly, by the neutron-transport (Boltzmann) equation – see Section 4.II in Duderstadt & Hamilton, ending with Eq. (4-43) • to some degree of approximation, by the neutron-diffusion equation -see Section 4.IV.D in Duderstadt & Hamilton, ending with Eq. (4-162)

Neutron Balance • Both the Transport and the Diffusion time-independent equations express the neutron balance at a point (actually, in a differential volume, but since this can be assumed as small as desired, it’s really at a point) • The Transport equation expresses the balance in the angular flux, whereas • The Diffusion equation expresses the balance in the total flux • To write down the balance, terms for all the events that can take place are included (these were listed 2 slides back)

Neutron Production Rate By Source Production of neutrons at in a given direction of motion and at a given energy E (per differential volume, solid angle and energy interval): • From an external (independent) source (assumed isotropic) = • From fission where (E) = the fission neutron spectrum (fraction of fission neutrons born with energy E)

Production Rate of Neutrons By Scattering In • The rate of neutrons entering the differential volume (of space, energy, and direction of motion) by scattering from other neutron directions of motion or other neutron energies =

Loss Rate of Neutrons Loss rate of neutrons: • by absorption and scattering: where = Total cross section • by physical leakage out:

Neutron-Transport Equation • We can now write the time-independent neutron-transport equation, which expresses the balance between the production and loss rates of neutrons: • The left-hand-side of Eq. (1) gives, per differential volume at r, direction of motion  and energy E, the total production of neutrons minus the total loss of neutrons. • This is the integro-differential form of the equation.

Neutron-Transport Equation (cont.) • Note how complicated the transport equation is: • It involves both derivatives and integrals of the flux • It involves integrals in energy, over very large ranges in energy (from several MeV to small fractions of 1 eV), with quantities (cross sections) which are very complex functions of energy, especially in the resonance range • It involves 6 independent variables: 3 for space (r), 2 for the neutron’s direction of motion (), and 1 for energy E. • Note: it is first-order in terms of derivatives.

Neutron-Transport Equation (cont.) • The transport equation is the most accurate (essentially exact) representation of neutronics in the reactor. • Therefore, ideally, it should be the equation to solve for all problems in reactor physics. • However, because of its complexity, it is very difficult, or extremely time-consuming, to apply the transport equation to full-core calculations. • Because cross sections do not depend on the initial angle of motion , it would be “nice” if  could be removed as a variable.

Integrating over Angle • We can try to remove the angle  by integrating the equation [Eq. (1)] over it, to see if we can obtain an equation in the angle-integrated flux only. • However, integration of over presents a challenge to our plan, since the angle-integrated current bears in general no algebraic relationship to the angle-integrated current.

Fick’s Law • But we can use an approximation often used in diffusion problems, Fick’s Law. • This is an approximate relationship between the neutron flux and the neutron current: where is called a “diffusion coefficient”. • Physically, Fick’s law says that the overall neutron current (at a given neutron energy) is in the direction of maximum decrease of the total flux of neutrons of that energy.

Significance of Fick’s Law • Fick’s Law expresses the expectation/fact that in regions of totally free neutron motion the overall net neutron current will tend to be from regions of high density to regions of low density. • Mathematically speaking, the net overall current should flow along the direction of greatest decrease in the neutron density (or, equivalently, of flux), i.e., it will be proportional to the negative of the gradient of the flux. • This is a consequence of the greater number of collisions in regions of greater density, with collisions allowing neutrons to go off freely in all directions.

Breakdown of Fick’s Law • The approximation inherent in Fick’s Law breaks down near regions of strong sources or strong absorption, or near boundaries between regions with large differences in properties, or near external boundaries, because the motion of neutrons is biased in or near such regions. • Here “near” a region or boundary means within, say, 2 or 3 neutron mean free paths of the region or boundary.

Neutron-Diffusion Equation • By integrating the transport equation over angle, and making use of Fick’s Law, we get the (here, time-independent) diffusion equation [I will leave you to study the full derivation in Duderstadt & Hamilton]: • Identify and make sure you understand each term in the neutron-diffusion equation. • Why is there a + sign in front of the ?

Neutron-Diffusion Equation • The neutron-diffusion equation is much simpler than the transport equation, because it removes the neutron direction of motion from consideration, i.e., the dependent variable is the total flux at each energy rather than the angular flux. • However, it is based on an approximate relationship between the angle-integrated neutron current and flux.

Discretizing the Energy • The equation 2 slides back is over continuous energy E. • To simplify the equation further, we discretize the energy variable (i.e., subdivide the range [0, ) into a number of G of subintervals. • All the neutrons of any energy in subinterval g (g = 1,…,G) are considered to be in the same “energy group” g and the nuclear properties are uniform over energy in any single energy group. • By convention, group 1 is the group with highest energy, and group G is the one with lowest energy (the thermal group) 0 EG EG-1 E1 Group: G G-1 G-2 … 1

Multigroup Neutron-Diffusion Equation • The diffusion equation in the discretized energy is called the multigroup diffusion equation. It is actually a set of equations, one for each energy group g. Time-independent equation: External source in group g Total cross section for group g, including scattering out of g Leakage from group g Fission from all groups; g = fraction of fission neutrons appearing in group g. Scattering into group g

Solution of Neutronics Problem • The neutron-diffusion equation cannot be used to calculate the flux in the basic lattice cell (see figure in next slide), because the fuel itself is a strong neutron absorber and the cell is very heterogeneous. • Therefore, the overall neutronics problem is solved in 2 stages, as explained further below.

CANDU BASIC-LATTICE CELL WITH 37-ELEMENT FUEL Face View of a Bundle in a Fuel Channel

2-Stage Solution of Neutronics Problem • Stage 1: The transport equation is applied to the basic lattice cells: • to find the detailed flux in space and energy (a large number of energy groups) in a basic cell, and • to derive “homogenized” (average) properties over each cell (therefore weakening absorption, on the average) • and “collapse” onto a very small number of energy groups (often 2 groups). • Stage 2: These homogenized lattice-cell properties are then applied in full-core reactor models using diffusion theory. See a simplified diffusion model in the next slide. • This is the strategy used most frequently (and successfully) in the design and analysis of nuclear reactors.

Face View of Diffusion Reactor Model Legend Each square is a homogenized lattice cell. Different-colour cells have different properties, mostly on account of different fuel ages (burnups).

Interface & Boundary Conditions • To solve the transport or diffusion equation, we generally subdivide (as described earlier) the overall domain into regions within which the coefficients in the equations (i.e., the nuclear properties) are constant (homogenized). • The equation is then solved over each region, and the solutions must be connected by interface conditions at the interfaces (infinitely thin virtual surfaces) between regions. • We also generally need boundary conditions at the external boundary of the domain.

Interface & Boundary Conditions for Transport • The neutron-transport equation has derivatives of first order  we need one interface condition at each interface, and one boundary condition • At interfaces the angular flux must be continuous (since there are no sources or scatterers at an infinitely thin virtual interface): where r+ and r- are the two sides of the interface • At rv, an outer boundary (assumed convex) with a vacuum, no neutrons can enter, since the vacuum has no neutron sources or scatterers:

Interface & Boundary Conditions for Diffusion • Interface conditions at each interface: The total flux and the total current must be continuous (since they are integrals of the angular flux, which is continuous):

Boundary Condition for Diffusion • The boundary condition with a vacuum, in plane geometry and in 1 energy group,is written as a relation between the flux and its gradient at the boundary xv: • tr is called the “transport cross section”.

Extrapolation Distance • The boundary condition Eq.(8) can be interpreted geometrically as follows. • If one were to extrapolate the diffusion flux linearly away from the boundary, it would go to zero at an extrapolation point xexbeyond the boundary: • Note that the flux does not actually go to zero, but the boundary condition is mathematically equivalent to flux = 0 at xex. • 0.71*tr is therefore called the “extrapolation distance”. • The boundary condition can be applied as is in Eq. (8), i.e., as a relationship between the flux and its derivative at the physical boundary xv, but it is also often applied by “extending” the reactor region to a new boundary at xex+tr, and forcing the flux to be zero there. (This represents an approximation - usually small - since it means assuming the reactor is slightly larger than it really is.)

1-Energy-Group Neutron-Diffusion Equation • Diffusion theory is applied mostly in 1 or 2 energy groups, or at most a few energy groups. • So let’s start with the simplest case – 1 energy group. • In this case, the energy ranges in Eq. (4) are reduced to a single distinct energy value, and therefore the energy label can simply be removed. • If we assume that all neutrons have the same energy (or speed), Eq. (4) reduces to the following : • [Exercise: Where do the  and the scattering terms go?]

Derivation of Eq. (12) from Eq. (4) Solution to Exercise:

Operator Formulation • From Eq. (12) we can see that for the 1-group diffusion equation, the flux “vector” and the operators take the form • and the diffusion equation in operator form is

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7: Neutron Balance