Until now in the treatment of spectrum calculations we have only considered an homogeneous reactor. In all practical cases we are confronted with reactor systems having various levels of heterogeneity. All power reactors consist of at least two
regions, a core and a reflector. For reasons of power shaping the core is normally radially and axially subdivided in regions of different composition. Usually a spectrum calculation has to be performed independently for each of the reactor regions. The coupling between these regions is taken into account through the use of energy — dependent bucklings, obtained for each region from a few group two — or three­dimensional calculation over the complete reactor.

As has already been mentioned in the proceding sections an iteration is then necessary. With a guess on the bucklings first spectrum calculations are performed producing condensed few-group constants for two — or three-dimensional calculations, out of which new energy-dependent bucklings are obtained for each region. This iterative procedure is continued until the desired accuracy is achieved. A second level of heterogeneity is given by the fine structure of the reactor cells.

Even if HTRs are rather homogeneous compared with many other reactor types, heterogeneity in the fuel structure cannot be neglected. All reactors consist of rather regular lattices of cells containing fuel pins, coolant channels, moderator and absorbers.

In spectrum calculations one has the choice between a multi-group space-dependent transport treatment, or a calculation on a homogenized cell. The codes we have examined so far follow this second procedure. The homogenization of the lattice cell implies a previous few-group transport calculation on the cell. Once the fine structure of the flux within the cell is known, the homogenization is performed multiplying the cross-sections by energy-dependent self-shieldings (sometimes called disadvantage factors). These self-shieldings Sk(E) are defined for each material к in such a way as to give the same reaction rate in the homogenized calculation as in the real cell. The homogenized calculation deals only with a reference flux фя(Е) while in reality the flux ф(г, E) depends on the position within the cell. As a reference flux one can take the flux at any point of the cell, e. g. at its centre, or at the outer boundary, or an average value

фн{Е) —

where V is the cell volume and the integral is extended over the whole cell. The spectrum which is obtained from the flux calculation depends, of course, on the reference which has been chosen, although reaction rates and reactivity are indepen­dent of this choice.

In order to obtain the proper reaction rates the self-shielding Sk (here we omit, for simplicity, the indication of the energy-dependence, or of the group index) must satisfy the equation

where Nk(r) space-dependent concentration of isotope k,

— average concentration of isotope k, crk cross-section of isotope k.

The self-shieldings must then be defined as

(8.8)

The cross-section of each material must be multiplied by its self-shielding. Very often the average cell flux is used as фя, but this might give rise to some difficulty in imposing boundary conditions, so that sometimes the flux at the outer cell boundary is used as reference. This is anyhow not very important in HTR calculations, because self- shieldings are usually very near to unity. Cell transport calculations are performed with one of the methods described in Chapter 4.

In general simplifications are made in order to avoid three-dimensional cell calcula­tions. The cell is usually considered as infinitely long and the cell outer boundary, which has usually a polygonal shape, is replaced by a cylindrical boundary giving the same cell volume (Wigner-Seitz approximation). In this way one-dimensional transport calcula­tions are often sufficient. Reflective conditions are required at the external cell boundary since this procedure assumes an infinite lattice of identical cells. In the case of the cell cylindrization these reflective conditions tend to give too high a flux at the outer boundary, while a diffuse reflection (white boundary condition) gives better results. If these conditions are not available in the transport code being used, it is often possible to simulate them by surrounding the cell with a pure scatterer.

In order to save computer time transport cell calculations are performed in a limited number of energy groups, otherwise this procedure would not differ from a space — dependent spectrum calculation in a heterogeneous cell. The few-group self-shieldings obtained in this way must then be adapted to the higher number of groups of the spectrum calculation. It is usually sufficiently accurate to use the same self-shielding for all the fine groups included within each broad group of the transport calculation. For the few-group transport cell calculation few-group constants must be obtained with a code for spectrum calculations for which a guess on the self-shieldings is needed. Here again we have an iterative procedure. Actually these iterative procedures are simplified by the fact that in most practical cases good guesses are available. The reactor design is a slow evolution from one version to the next and very seldom is a calculation performed without having previous experience of similar cases. This, of course, applies also to the buckling iteration.

A third level of heterogeneity is given in HTRs by the presence of coated particles. A homogenization by means of self-shieldings as in the case of cell calculations is possible. In the case of resonance absorption this grain structure can become very important (see §7.13).