Question 22.7: Suppose that we have solved the neutron transport equation i......
Suppose that we have solved the neutron transport equation in a Cartesian coordinate system using the S_N or P_N methods. The results reveal that the neutrons are moving at an angle of 30° with respect to the z-axis and 60° with respect to the x-axis. If the neutron flux ϕ(r, E, Ω, t) at the origin is 1 × 10^{15} neutrons/cm²/s, what is the value of the neutron current J(r, E, Ω, t) along the x-axis, the y-axis, and the z-axis at r = 0?
From our previous discussion, we know that the directional neutron current is given by J(r, E, Ω, t) = [i sin θ cos φ + j sin θ sin φ + k cos θ] · ϕ(r, E, Ω, t). The x component is J_x = i sin θ cos φ · ϕ(r, E, Ω, t), the y component is J_y = j sin θ sin φ · ϕ(r, E, Ω, t), and the z component is J_z = k cos θ · ϕ(r, E, Ω, t). In this problem, θ = 30° and φ = 60°. The neutron current along the x, the y, and the z directions is therefore J_x = i sin 30° cos 60° · ϕ(r, E, Ω, t), J_y = j sin 30° sin 60° · ϕ(r, E, Ω, t), and J_z = k cos 30° · ϕ(r, E, Ω, t). Plugging in the appropriate values for the sine and cosine, we find that sin 30° = 0.5, sin 60° = 0.866, cos 30° = 0.866, and cos 60° = 0.5. The directional currents are therefore J_x = i0.5 × 0.5 · ϕ(r, E, Ω, t), J_y = j0.5 × 0.866 · ϕ(r, E, Ω, t), and J_z = k0.866 · ϕ(r, E, Ω, t) or J_x = 0.25 × 10^{15} neutrons/cm²/s, J_y = 0.433 × 1015 neutrons/cm²/s, and J_z = 0.866 × 10^{15} neutrons/cm²/s.