A bare slab reactor has thickness and extends effectively to infinity in the other…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Partial differential equations and boundary-value problems

Energy from Physics, Ch The critical reactor: neutron diffusion, buckling and the size of a core

Problem

A bare slab reactor has thickness 2a2a and extends effectively to infinity in the other two directions, so the flux ϕ\phi depends only on the coordinate xx measured from the mid-plane. The medium is a single homogeneous lattice with diffusion coefficient DD, absorption cross section Σa\Sigma_a, and a fission source νΣfϕ\nu\Sigma_f\phi per unit volume, so that k∞=νΣf/Σak_\infty = \nu\Sigma_f/\Sigma_a. Neutrons are lost only by leaking out through the two faces at x=±ax=\pm a, where the extrapolated flux is taken to vanish. ((a)) Consider a thin slice of the slab between xx and x+dxx+dx and of unit cross-sectional area. Write a steady-state neutron balance for the slice in terms of the neutron current J(x)J(x), the absorption rate and the fission source, then use Fick's law J=−D dϕ/dxJ = -D\,d\phi/dx to obtain the one-group diffusion equation Dd2ϕdx2−Σaϕ+νΣfϕ=0D\dfrac{d^2\phi}{dx^2} - \Sigma_a\phi + \nu\Sigma_f\phi = 0. ((b)) Write the equation as d2ϕdx2+Bm2ϕ=0\dfrac{d^2\phi}{dx^2} + B_m^2\phi = 0 with Bm2=(k∞−1)/L2B_m^2 = (k_\infty - 1)/L^2, where L2=D/ΣaL^2 = D/\Sigma_a is the lattice's diffusion area; as in the chapter, a fuller treatment then replaces L2L^2 by the migration area M2=L2+τM^2 = L^2 + \tau. Solve it subject to ϕ(±a)=0\phi(\pm a) = 0, keeping only the solution that is positive throughout the slab, and show that criticality requires the geometric buckling Bg2=(π/2a)2B_g^2 = (\pi/2a)^2 to equal Bm2B_m^2. ((c)) A slab lattice has k∞=1.15k_\infty = 1.15 and migration area M2=100 cm2M^2 = 100\,\mathrm{cm^2}. Using M2M^2 in place of L2L^2, find its critical thickness 2a2a.

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