Mathematical Modeling and Error Estimation for the Thermal Dunking Problem: A Hierarchical Approach

Abstract

We consider the thermal dunking problem, in which a solid body is suddenly immersed in a fluid of different temperature, and study both the temporal evolution of the solid and the associated Biot number -- a non-dimensional heat transfer coefficient characterizing heat exchange across the solid-fluid interface. We focus on the small-Biot-number regime. The problem is accurately described by the conjugate heat transfer (CHT) formulation, which couples the Navier-Stokes and energy equations in the fluid with the heat equation in the solid through interfacial continuity conditions. Because full CHT simulations are computationally expensive, simplified models are often used in practice. Starting from the coupled equations, we systematically reduce the formulation to the lumped-capacitance model, a single ordinary differential equation with a closed-form solution, based on two assumptions: time scale separation and a spatially uniform solid temperature. The total modeling error is decomposed into time homogenization and lumping contributions. We derive an asymptotic error bound for the lumping error, valid for general heterogeneous solids and spatially varying heat transfer coefficients. Building on this theoretical result, we introduce a computable upper bound expressed in measurable quantities for practical evaluation. Time scale separation is analyzed theoretically and supported by physical arguments and simulations, showing that large separation yields small time homogenization errors. In practice, the Biot number must be estimated from so-called empirical correlations, which are typically limited to specific canonical geometries. We propose a data-driven framework that extends empirical correlations to a broader range of geometries through learned characteristic length scales. All results are validated by direct numerical simulations up to Reynolds numbers of 10,000.