A Heat Conduction and Convection Analytical Benchmark for Adjoint Solution Verification of Computational Fluid Dynamics Codes Used in Reactor Design

This work presents a heat transport benchmark problem when modeling the steady-state radial conduction in a fuel rod coupled to the axial heat convection in a coolant surrounding the rod and flowing along it. This benchmark problem admits exact analytical solutions for the spatially dependent temperature distributions within the rod and the surrounding coolant. The adjoint sensitivity analysis methodology (ASAM) is applied to compute the analytical expressions of the adjoint state functions for this benchmark problem. In turn, these adjoint state functions are used to compute exactly the first-order sensitivities of the various temperature distributions to the benchmark’s thermal-hydraulics parameters. Locations of particular importance are those where the rod, the rod surface, and the coolant temperatures attain their maxima. The analytical expressions of the benchmark sensitivities thus obtained are subsequently used to compute numerical values of the sensitivities of the various temperature distributions that would arise in the preliminary design of the G4M Reactor to thermal-hydraulics parameters characteristic of this reactor.

The exact benchmark sensitivities are used for verifying the numerical results produced by the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulics processes within the G4M Reactor. This solution verification process indicates that the current FLUENT Adjoint Solver cannot compute any sensitivities for the temperature distribution within the solid rod. However, the FLUENT Adjoint Solver is capable of computing the sensitivities of fluid temperatures to boundary parameters (e.g., boundary temperature, boundary velocity, and boundary pressure), but yields accurate results only for the sensitivities of the fluid outlet temperature and the maximum rod surface temperature to the inlet temperature and inlet velocity, respectively. Even for these sensitivities, the FLUENT Adjoint Solver typically needed over 20 000 iterations to converge to the correct solution. In fact, if the exact sensitivity results had not been known a priori, employment of a user-defined iteration-stopping criterion would have likely produced an erroneous result, which would have been noticed by the user only if the user had had the foresight of computing the respective sensitivities independently, via finite-differences using FLUENT recomputations. Several other important sensitivities, including sensitivities to the boundary heat transfer coefficient and sensitivities to material properties (thermal conductivity and specific heat), cannot be obtained from the current FLUENT postprocessing output.

Ideally, the solution verification of the adjoint functions produced by the FLUENT Adjoint Solver would be performed by directly comparing these to the exact expressions of the adjoint functions for the benchmark problem. Such a direct comparison and, hence, a direct solution verification of the FLUENT Adjoint Solver, is currently not possible, because the current FLUENT Adjoint Solver does not provide access to the adjoint functions it computes. Therefore, the results produced by the FLUENT Adjoint Solver can only be verified indirectly, by comparing temperature sensitivities computed using the FLUENT Adjoint Solver to the exact results obtained from the analytical expression of the corresponding benchmark sensitivities. This situation further underscores the need for developing additional thermal-hydraulics benchmark problems that admit exact solutions.