Reactor bare homogeneous

Let the rate of energy per unit volume generated in a solid cylinder or a solid sphere be u" (r) = u , the radius and the thermal conductivity of the cylinder or the sphere be R and k(T) = kr (alternate notations ur and kr are used for convenience in the following formulation). Under steady conditions, the total energy generated in the cylinder or sphere is transferred, with a heat transfer coefficient h, to an ambient at temperature Too. This cylinder could be one of the fuel rods of a reactor core, or one of the elements of an electric heater, and the cylinder or sphere could be a bare, homogeneous reactor core. We wish to determine the radial temperature distribution. 

The conditions to be satisfied by j on the outer boundary of the reactor cannot really be deduced in a logical fashion from the condition (2.2) satisfied by (f> since it is not possible for a mildly anisotropic satisfying equation (3.1) to comply also with (2.2). The condition on usually employed is used because it is correct for bare homogeneous slabs and spheres and so is not expected to be very inaccurate for any convex region. It is that (f)j = 0 at a distance zo = 0.7104//Ltrr outside the reactor. For a fuller discussion of this point see Davison [1]. 

The preceding paper by H. Soodak is devoted primarily to problems of reactor kinetics associated with the bare homogeneous reactor. By a homogeneous reactor is meant a reactor the small-scale composition of which is uniform and isotropic. By a bare reactor is meant one which is not surrounded by any reflector thus, all neutrons which once leave the reactor are lost from the chain process. On the other hand, this paper is concerned chiefly with the inhomogeneous and reflected reactor and consequently with problems of greater mathematical complexity than those previously considered. 

The expression (6) may be called the generalized reactivity since it reduces to the conventional definition of reactivity for a bare homogeneous reactor, ( eff — 1)/A eff- Thus, (6) may be regarded as a generalization of the notion of reactivity in a mathematical sense. However, (6) may be identified with... 

The secondary equations (9) and (10) are conventionally called the reactor kinetics equations, rather than the primary equations (1) and (2). They are of the same/orm as the bare homogeneous reactor kinetic equations. The definition of reactivity in (6) and the reactor kinetics equations (9) and (10), or more general formulations of the same concepts, are often used in reactor physics whether or not physical significance can be associated with the formalism. They do however represent generalizations in a mathematical sense. Thus, in this paper, (6) has been termed the generalized reactivity and (9) and (10) the generalized kinetic equations. 

The equations (19), (20), and (21) are the generalized kinetic equations for the theoretical model under consideration. Note that these equations, because of the related definitions which have been made, bear strong resemblance to the kinetic equations for a bare homogeneous reactor. 

Solution of space-independent kinetic equations. In this section the kinetic equations for a bare (or equivalent bare) homogeneous reactor are studied in the form... 

The problem is separable for a bare homogeneous reactor. However, only the case of a step input of reactivity, i.e., the case of a constant value of p, is easily solved. In this case, the kinetic equations are readily reduced to a second order (for the case of one delayed neutron group) homogeneous linear differential equation with constant coefficients. For an input of positive reactivity two solutions arise, of the form and where o>i > 0 and 0)2 < 0. The first solution controls the persisting exponential rise of the flux, where it is recalled that T = l/o>i is the reactor period, and the second solution which rapidly becomes small is called the transient solution. 

The critical equation for the bare homogeneous reactor with mono-energetic fission neutrons may be developed from the solutions (6.72). By the usual definition of criticality, we require that the number of neutrons introduced into the system be equal to the number produced by fissions in the subsequent generation. Thus the source function (6.55) may be used to establish the criticality condition. The lethargy integral of (6.55) yields... 

The solutions to the case of the bare homogeneous reactor with a distributed fission spectrum may be summarized in terms of the above results ... 

The temperature dependence of the material density and reactor size are easily defined in the case of bare homogeneous reactors. A change in the temperature produces, of course, a change in the various nuclear densities which are involved in the macroscopic cross sections. Moreover, the thermal expansions which accompany increases in temperature influence directly the physical size of the system and therefore affect the neutron leakage. 

Consider a bare homogeneous spherical reactor with moderator and fuel cross sections as specified in Prob. 4.8 and Prob. 5.12. 

The result (9.315) is now applied to the criticality calculation of a bare homogeneous reactor operating at constant power. The appropriate relation for the multiplication constant is... 

S The flux in a bare homogeneous one-velocity reactor is time dependent. Find this time behavior under the following two (different) physical assumptions (a) all neutrons are prompt b) a fraction /8 of the neutrons from fission are delayed time to. Assume that the form... 

A bare homogeneous thermal reactor is critical and operating in steady state with flux o(r) At time / 0, a small positive reactivity is introduced and held constant until t h. At time t — U the reactivity is reduced to zero. 

A bare homogeneous thermal reactor is operating in steady state up to time / = 0, when some additional multiplication 6ko is introduced into the system. Elxam-ine the time behavior of this system on the basis of the following model ... 

If the homogeneous reactor is bare and uniform then its flux distribution can be estimated rather well by considering the actual reactor to be replaced by an infinite reactor of the same composition. One then asks for that solution of the infinite reactor equation which is positive inside the reactor and extrapolates to zero on a surface which surrounds the actual reactor and is about 0.7 mean free paths beyond the physical surface. Such a solution in every case satisfies the wave equation (22) however, its analytical continuation outside the reactor is alternately positive and negative. 

Homogeneous bare reactor. First, it should be appreciated that the case of a homogeneous bare reactor is atypical. For example, consider the simple one-group diffusion approximation [7, pp. 192-193 11, p. 235] ... 

This assumption is apparently made in all time independent diffusion codes which are accelerated by means of Chebyshev polynomials. (See, for example, [46].) It has not been proved to be true for general heterogeneous reactor models in n dimensions except for the trivial cases of only one lethargy group and homogeneous bare problems. 

D. K. Trubey, H. Moran and A. M. Weinberg, An Estimate of the Non-Leakage Probability for Bare Aqueous Homogeneous U-235 Reactors, ORNL memorandum CF 57-11-15 (195.7). 

These results are entirely consistent with those of our previous analysis of the bare reactor using the Fermi age model (refer to Sec. 6.3). In this formulation, Eq. (8.281a) describes the neutron-energy spectrum and is easily recognized as the integral equation for the collision density in an infinite homogeneous system. If we select, for example [cf. Eq. (4.36)],... 

---