Calculating Time-Integral Quantities in Depletion Calculations

A method referred to as tally nuclides is presented for accurately and efficiently calculating the time-step averages and integrals of any quantities that are weighted sums of atomic densities with constant weights during the step. The method allows all such quantities to be calculated simultaneously as a part of a single depletion solution with existing depletion algorithms. Examples of results that can be extracted include step-average atomic densities and macroscopic reaction rates, the total number of fissions during the step, and the amount of energy released during the step. The method should be applicable with several depletion algorithms, and the integrals or averages should be calculated with an accuracy comparable to that reached by the selected algorithm for end-of-step atomic densities. The accuracy of the method is demonstrated in depletion calculations using the Chebyshev rational approximation method. As an example of a possible use, we demonstrate how the ability to calculate energy release in depletion calculations can be used to determine the accuracy of the normalization in a constant-power burnup calculation during the calculation without a need for a reference solution.