Neutron, balance equation

The thermal-neutron balance equation for the reference design condition states that the rate of production of thermal neutrons equals the rate of consumption ... 

For an operating reactor, the terms of the neutron balance will differ from Table 3.9 and the neutron balance equation will differ from Eq. (3.25) because the fuel charged to the reactor may differ from the reference fuel and because the composition of fuel will change as a result of reaction with neutrons. The terms in the neutron balance of an operating reactor are listed in Table 3.10. 

The adjoint source in Eq. (259) represents some detector distribution which, when integrated over the reactor space (after weighting with the neutron distribution), yields a single measure of the reactor behavior. Physically, there is no reason why such a detector distribution should be positive definite. The existence condition [Eq. (260)] for a solution to Eq. (259) in a critical system, however, as in the neutron balance equation, requires the orthogonality of and V and since is (physically) positive definite, must be part positive, part negative. Physically, this orthogonality condition, therefore, expresses the idea that the acceptable virtual adjoint sources are detector distributions which, when reacting to the distribution of neutrons in the fundamental mode, lead to no total or net effect. 

In changing the neutron balance equation, we transform the adjoint equation. For example, if we have transformed the critical equation by means of an operator, B, say, we compare... 

This expression is the generalization of the form (5.17) to a neutron flux which is a function of all three space coordinates. We will use this analytical expression for the net leakage in (5.6). The neutron-balance equation is then written... 

The integral equation (7.180) is the solution to the differential equation (7.175), and this general result will be very useful in the work which follows. As will be shown shortly, the neutron-balance equations for other, more complex systems may be reduced to the form (7.175) consequently, their solutions can be written down immediately from (7.180). Before proceeding with these calculations, however, we will develop an alternate expression for the solution (7.180) which is obtained by transforming the line integral to a volume integral. This form will be especially convenient for describing the more complex systems. 

The function Pfi(r r ) defined here will be the kernel in the integral-equation formulation of the neutron-balance equation note that it is independent of energy in this particular model in which the cross sections are energy independent. It should be noted that this definition differs from that given the function K r P) used in the preceding section. Whereas the kernel K represented the flux at r due to a unit source at r the kernel Pb implies a function of the type K multiplied by 2r. This relationship may be seen more clearly if we consider, for example, the quantity 2rif(r r ) dr, a physical interpretation of which is... 

The neutron-balance equation may be expressed in terms of this function. If as before we define a source S(r E) according to (8.272), then the removal rate is given by... 

The two basic results obtained above, (8.303) and (8.308), are generally valid and may be applied directly to the physical system described by (8.296). Our only purpose in introducing the fictitious system was to offer some aid in grasping the essential ideas and analytical tools which may be used to solve the neutron-balance equation. Let us now apply these results to Eq. (8.296). The substitution of (8.304) into (8.296) yields... 

The application of the diffusion theory to the calculation of the eigenfunctions and eigenvalues for the Feynman-Wei ton method requires a restatement of the neutron-balance equation in terms of the removal cross section and the neutron yield per removal It is obvious that the appropriate form is [cf. equation preceding (8.242) and Eq. (8.238)]... 

We begin with the neutron-balance equations, which we write in the form... 

The pre.sent treatment of the heterogeneous method develops the principal concepts obtained in the general theory. The analysis begins with a derivation of the neutron-balance equations. These results are then used to obtain the criticality condition and to demonstrate the basis of the homogenization method used with the unit-cell model. The section is concluded with an application of the method to a reactor with four fuel rods and a numerical example. 

We can now write down the neutron balance equation for the element considered since, in the steady state, we must have... 

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