A novel method to predict a local Nusselt number profile on the hemispherical surface under air jet impingement cooling

This research focuses on accurately estimating the local Nusselt number profile ($N{u}_s$) on the hot surface of a convex hemispherical block under air jet impingement cooling using the inverse technique such as Bayesian inverse approach with the Metropolis Hastings–Markov Chain Monte Carlo (MH-MCMC) algorithm, which is critical for applications like thermal treatment of materials, electronics cooling, cooling of turbine blade leading-edge, rocket launcher cooling, rotary cement kiln shell cooling, casting industry processes, etc. To accurately evaluate the local Nusselt number profile on the hot hemispherical surface, the unknown parameters $a$, $b$, and $c$ of the $N{u}_s$ profile were predicted using the proposed inverse technique combined with artificial neural networks and steady-state temperatures measured on the bottom surface of the hemispherical block. The local Nusselt number profile was predicted as $N{u}_s=0.0703R{e}^{0.78}exp[-0.1747{(s/d)}^{1.001}]$ for the $L/d$ ratio of 6 and Reynolds numbers ranging from 23 000 to 50 000. Surrogate and synthetic temperature data were initially employed to assess the effectiveness of the inverse method. The estimated parameters closely matched the target values with low percentage deviations, proving the robustness of the inverse method. The local Nusselt number predicted was then compared against local Nusselt number obtained from experiments and $v^2-f$ turbulence model simulations, showing strong agreement with the $N{u}_s$ profile estimated using the Bayesian with MH-MCMC approach. Simulated temperature distributions were also analyzed to understand the thermal behavior on the hemispherical surface under various Reynolds numbers. The findings highlight that the proposed inverse methodology accurately predicts the local Nusselt number profile on the hot hemispherical surface under air jet impingement conditions, with potential applications in optimizing cooling processes in various industrial systems.