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

TL;DR

This paper delivers a rigorous analysis of the thermal dunking problem by contrasting the full conjugate heat transfer (CHT) model with reduced models, notably the autonomous Robin heat equation (RHE$_a$) and the lumped capacitance model (LCM). It derives a sharp asymptotic bound for the lumping error in terms of geometry and material heterogeneity via a coefficient $ ext{phi}$, and provides a computable upper bound that can be evaluated with scalar geometric quantities. The work shows that, under time-scale separation and small Biot number conditions, the LCM can accurately predict the solid temperature, while also developing a data-driven framework to extend Nusselt-number correlations across general geometries through learned characteristic length scales. Validations against high-Reynolds-number simulations (up to ${ m Re}=10^4$) demonstrate the method’s accuracy and reveal regimes where the ISO and Nu-learning approach remain reliable, guiding engineers in rapid, geometry-aware heat-transfer estimations. The framework thus enables robust, scalable predictions for heterogeneous solids in diverse geometries without resolving the full fluid–solid system every time.

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.

Figures (36)

• Figure 1.1: (a) Nusselt number at the leading edge (LE), trailing edge (TE), side edge (SE), and the spatial average over the surface of a circular cylinder in cross-flow. The superscripts "CHT" are omitted. (b) Evolution of the non-dimensional average solid temperature, which closely follows a simple exponential decay.
• Figure 2.1: Schematic of the fluid-solid system. The solid domain is denoted by $\mkern1mu\underline{\mkern-1mu \Omega\mkern-1mu}\mkern1mu_{\mathrm{s}}$ and the fluid domain by $\mkern1mu\underline{\mkern-1mu \Omega\mkern-1mu}\mkern1mu_{\mathrm{f}}$. The interface between the two domains is denoted by $\mkern1mu\underline{\mkern-1mu \Gamma\mkern-1mu}\mkern1mu$.
• Figure 2.2: Cross-flow over a circular cylinder of diameter $\mkern1mu\underline{\mkern-1mu D\mkern-1mu}\mkern1mu$ in two dimensions (not to scale). The flow enters the domain from the left, as illustrated in \ref{['fig:setup']}.
• Figure 2.3: Temperature fields from the CHT solution at different time steps.
• Figure 2.4: (a) Nusselt number at LE, TE, SE and spatial average over time. (b) Nusselt number measured in angular coordinate $\theta$ at different time steps and temporal average. The superscripts "CHT" are omitted for brevity.

Theorems & Definitions (11)

• Proposition 4.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof
• Proposition 4.2: Lumping error
• proof
• Lemma 4.3
• proof
• Proposition 4.3