Steadystate scalar neutron flux in a spherical reactor of ra

Steady-state, scalar neutron flux in a spherical reactor of radius R is given by phi(r) = phi_0 sin (pi r/R)/(r/R) Where phi_0 = 15.708 times 10^13 n/cm^2-s, R = 50 cm, D= 1 cm, and macroscopic absorption and fission cross sections are 0.113/cm and 0.0732/cm. (Assume all neutrons have energy of .0253 eV.) [When you are asked to \"estimate\" a quantity, you need to make a justifiable assumption and simplify your calculation. Usually it means that you evaluate the flux at a point and assume it to not vary much over the domain of interest.] What is the maximum flux in the sphere? Plot the flux as a function of r. What is the total number of neutrons in the sphere at any given time? Estimate the number of neutrons at any given time in one cm^3 at the center of the sphere, and the number of fissions taking place in one second in that volume. Estimate the amount of energy (in J/s) generated due to fission in a sphere of volume 1 cm^3 at the center of the sphere? (What is the radius of the sphere of volume 1 cm^3?) Estimate the amount of energy (in J/s) generated due to fission in a sphere of volume 1 cm^3 at r = R/2? How much energy (in J/s) is generated due to fission in the sphere in one second? Estimate the number of neutrons absorbed in one cm^3 at the center of the sphere. What is the net current (J) at a point on the surface of this sphere? Plot J(r) as a function of r. Estimate the total number of neutrons being leaked out from the surface of the sphere. What is the net number of neutrons crossing the spherical surface at r = R/2.

Solution

Neutron Flux and Current v = v, ( r, E ) vn ( r, i , E ) = i vn ( r, , E ) d Rx = x 4 J ( r, E ) v i n ( r, i , E ) = i mv 2 E= 2 vn ( r, , E ) d 4 dn = J dA J dA HT2008 Diffusion equation

Balance of Neutrons Change rate in number Production Absorption Leakage of neutrons in V = rate in V rate in V rate from V Change rate in number d n of neutrons in V = dt n(r, E, t )dV = t dV V V Production rate in V = stot dV V Absorption rate in V = a dV V Leakage rate from V = J dA = divJ dV A V HT2008 Diffusion equation

Continuity Equation n stot + a + divJ dV = 0 t V Exact equation, two unknowns Diffusion theory: Diffusion equation: HT2008 1 = s + f a divJ v t J = Dgrad = D (Fick\'s law) 1 = s + f a + D2 v t Diffusion equation