# Extend the considerations of the preceding problem to particle diffusion, and assume that there is a net particle generation rate gu that is proportional to the local particle concentration, gu = n/t0, where t0 is a characteristic Unit Constant, Such

Extend the considerations of the preceding problem to particle diffusion, and assume that there is a net particle generation rate gu that is proportional to the local particle concentration, gu = n/t0, where t0 is a characteristic Unit Constant, Such behavior describes the neutron generation in a nuclear reactor. The value of t0 depends on the concentration of 235U nuclei, if no surface losses took place, the neutron concentration would grow as exp(t/t0). Consider a reactor in the shape of a cube of volume L3 and assume that surface losses pin the neutron surface concentration at zero. Show that Eq(3), if augmented by a generation term gu = n/t0, has solutions of the form

n(x, y, z, t) ∞ exp (t/t1) cos(kxx) cos (kyy) cos (kzz),

where kxL, kyL and kzL are integer multiples of π. Give the functional dependence of the net time constant t1 on kx, ky, kz and t0, and show that for at least one of the solutions of the form (67) the neutron concentration grows with time if L exceeds a critical value Lcrit. Express Lcrit as a function of Du and t0. In actual nuclear reactors this increase is ultimately halted because the neutron generation rate gu decreases with increasing temperature.

n(x, y, z, t) ∞ exp (t/t1) cos(kxx) cos (kyy) cos (kzz),

where kxL, kyL and kzL are integer multiples of π. Give the functional dependence of the net time constant t1 on kx, ky, kz and t0, and show that for at least one of the solutions of the form (67) the neutron concentration grows with time if L exceeds a critical value Lcrit. Express Lcrit as a function of Du and t0. In actual nuclear reactors this increase is ultimately halted because the neutron generation rate gu decreases with increasing temperature.

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