Consider a one-dimensional cylinder of radius R of a non-multiplying medium with a line source of strength S¬0 located at r=0.
- Solve this problem analytically. Provide an expression for the flux as a function of r. Plot the flux between r=0 and r=R. Assume a source strength S0=1x10^10 #/s-cm, and a variety of radii R (1cm, 10cm, 100cm).
- Assuming a 5-node numerical discretization (one central node at r=0, three internal nodes, and one edge node at r=R, as shown below), derive:
a. The finite-difference equation for an arbitrary internal node i
b. The finite-difference equation for the central node (r=0) assuming the same boundary condition as the analytical solution
Assume a zero-flux boundary condition at the right-most node (r=R).
- Assuming the 5-node specification above, define the matrix system of equations that would have to be solved. Carefully define all matrix entries and vectors involved. Describe the primary unknowns? (i.e., what are you trying to solve for?).
Deliverables:
- Using the above one-group constants representative of a light water reactor (the average values at the bottom of the table, ignore fission), write a program to solve the numerical problem. Plot the numerical versus analytical solutions side by side, and compute the difference (error) of the numerical flux at each node.
- Initially, use the 5-node representation provided, however, subsequently adjust your computer program to handle an arbitrary number of nodes (N) and compare your numerical results versus your analytical results as you increase the number of nodes from N=5, 10, 20, 50, etc. Provide comparisons of the numerical solution versus the analytical solution (plot the differences/errors) as you increase the number of nodes.