Neutron Balance

1. Neutron Balance B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr.

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

3. Reactor Statics: Neutron Balance • In this course we study time-independent phenomena • Independence of time means that there is (or is assumed to be) neutron balance everywhere. • Therefore, 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

4. 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)

5. Neutron Balance • Both the Transport and the Diffusion time-independent equations express the neutron balance at a point (actually, in a differential space, 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)

6. Differential Volume for Transport Leakage of “angular-flux” neutrons at solid angle , per unit volume: The reaction rates for the other phenomena are of the form

7. Neutron-Transport Equation • This is the neutron-transport equation in its integro-differential form. See if you can identify the meaning of each term in the equation: • where • (E) is the fission-neutron spectrum, i.e., the fraction of neutrons which are born with energy E. • If we understand each term, we can see that the equation, at each position r and for each energy E, equates the summed loss of neutrons to the summed production of neutrons.

8. 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 7 independent variables: 3 for space (r), 3 for the neutron’s direction of motion (), and 1 for energy.Because the rates of absorption and induced fission do not depend on , it would be “nice” if  could be removed as a variable. • Note: it is first-order in terms of derivatives.

9. Application of Neutron-Transport Equation • 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. • So the neutronics problem is divided into stages, as explained in the next slide.

10. Application of Neutron-Transport Equation • For practicality, the transport equation is applied to small regions of the reactor (“lattice basic cells”): • to find the detailed flux in space and energy in these cells, and • to derive “homogenized” properties (cross sections), uniform over each lattice cell and which are collapsed onto a very small number of energy groups (as small as 2 groups), for application over full-core models with diffusion theory. • This is actually the strategy used most frequently, and successfully, in the design and analysis of nuclear reactors.

11. Neutron-Diffusion Equation • The neutron-diffusion equation is an approximation to the neutron-transport equation. • It 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, and • it is based on an approximate relationship between the neutron flux and the neutron current, Fick’s Law:

12. Neutron-Diffusion Equation (cont.) • Note: Fick’s Law expresses the fact that in regions of totally free neutron motion the net neutron current will be 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. • 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 external boundaries, because the motion of neutrons is biased in or near such regions. • This is why diffusion theory cannot be used in lattice physics, as the fuel itself is a strong neutron absorber. Transport theory must be used to homogenize properties (and therefore weaken absorption, on the average) over (relatively large) lattice cells.

13. Leakage in Neutron-Diffusion Equation

14. Neutron-Diffusion Equation (cont.) • Do the same exercise for the diffusion equation as you did for the transport equation, i.e., identify the meaning and structure of each term:

15. 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.

16. Interface & Boundary Conditions for Transport • The Boltzmann 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:

17. 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):

18. 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”.

19. Extrapolation Distance • The boundary condition Eq.(8) can be interpreted geometrically as follows. • If one extrapolates the diffusion flux linearly away from the boundary, it would go to zero at an extrapolation point xex beyond the boundary: • Note that the flux does not actually go to zero, but the boundary condition is mathematically equivalent to flux = 0 at xex. • 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.)

20. Application of the Neutron-Diffusion Equation • As previously indicated, the neutron-diffusion equation is applied mostly in full-core calculations, because of its much greater simplicity than the transport equation. • Full-core models (see example in next slide) consist of homogeneous (uniform) properties over lattice cells, or large portions of cells, for a small number of neutron energy groups. • The flux distribution (or flux shape) in the reactor (one value per parallelepiped and energy group) is then obtained by solving the diffusion equation - often in its finite-difference form.

21. Full-Core Diffusion Model The parallelepipeds (cells) over which the flux is calculated are defined by the intersections of the horizontal and vertical mesh lines, shown on the left and top axes. Note - Illustration credit: document 20040502 on CANTEACH website, “Introduction to Reactor Physics”, by B. Rouben, p. 96

22. 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 :

23. Interactive Discussion/Exercise • Derive Eq. (12) from Eq. (4); in particular, explain how the a arises, and why  disappears.

24. Derivation of Eq. (12) from Eq. (4)

25. 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

26. END