This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2 nd -ASAM) to a paradigm nonlinear heat conduction benchmark, which models a conceptual experimental test section containing heated rods immersed in liquid lead-bismuth eutectic. This benchmark admits an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2 nd -ASAM indicates that, for a physical system comprising N parameters, the computation of all of the first-and secondorder response sensitivities requires (per response) at most N "large-scale" computations using the first-level and, respectively, second-level adjoint sensitivity systems (1 st -LASS and 2 nd -LASS). For this illustrative problem, six "large-scale" adjoint computations sufficed to compute exactly all five 1 st -order and fifteen distinct 2 nd -order derivatives of the temperature response to the five model parameters. The construction and solution of the 2 nd -LASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the rightsides of the heat conduction differential operator needed to be modified; the left-side of the differential equations (and hence the "solver" in large-scale practical applications) remains unchanged.Since the test section is insulated on its radial surface and since the length of the cylindrical test section is much greater than its radius, the temperature variation in the radial direction can be neglected by comparison to the temperature variations in the axial direction, for the purposes of this illustrative problem. Hence, the axial temperature distribution, T z , in the LBE can be modeled by the following nonlinear heat conduction model: