This contribution focuses on the conjugate heat transfer computation of a rib-roughened internal turbine blade cooling channel with a blockage ratio of 0.3 and a Reynolds number of 40,000. The work considers the coupling between two numerical tools: a Large Eddy Simulation flow solver and a solid conduction solver. While interested in the thermal steady states of such problems, both solvers provide time dependent de-synchronized solutions. A novel weak coupling strategy, the hFFB method, is applied to solve the challenge of varying fluid and solid time scales. This method first computes the fluid domain until it reaches a statistically steady state, from which the mean temporal solution is obtained. Then, it solves the solid domain with the mean boundary conditions imposed from the fluid leading to an updated fluid boundary condition. This process is repeated until convergence of the coupled problem. The coupled convective-conductive solution shows that conjugate heat transfer is important for rib-roughened internal cooling ducts and has a large impact on the heat transfer at the solid-fluid interface. The results were validated against experimental data provided by the von Karman Institute for Fluid Dynamics.