Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem

TL;DR

This work analyzes the small-Biot limit of the conduction-dunking problem for solids with heterogeneous material properties under Robin boundary conditions. It develops a systematic asymptotic framework that yields a first-order classical lumped model and introduces a novel second-order lumped approximation that relies on a sensitivity-derived quantity $\phi$, which also governs error estimates. By connecting $\phi$ to an effective conduction length scale and a corrected Biot number ${\text{Bi}}'_{dunk}=\phi\, {\text{Bi}}_{dunk}$, the authors provide rigorous asymptotic and non-asymptotic error bounds for domain-mean quantities and the domain-boundary mean difference, complemented by accurate computational evaluation of $\phi$ via the sensitivity equation. The paper also offers closed-form results for canonical geometries, adaptive finite-element verification, and practical strategies to estimate $\phi$ for complex domains using dictionaries and geometry features. Collectively, the results yield a robust, geometry-aware extension of the lumped-approximation paradigm, enabling reliable error control in small-$B$ regimes for heterogeneous materials and complex geometries.

Abstract

We consider the dunking problem: a solid body at uniform temperature $T_{\text i}$ is placed in a environment characterized by farfield temperature $T_\infty$ and spatially uniform time-independent heat transfer coefficient. We permit heterogeneous material composition: spatially dependent density, specific heat, and thermal conductivity. Mathematically, the problem is described by a heat equation with Robin boundary conditions. The crucial parameter is the Biot number -- a nondimensional heat transfer (Robin) coefficient; we consider the limit of small Biot number. We introduce first-order and second-order asymptotic approximations (in Biot number) for several quantities of interest, notably the spatial domain average temperature as a function of time; the first-order approximation is simply the standard engineering `lumped' model. We then provide asymptotic error estimates for the first-order and second-order approximations for small Biot number, and also, for the first-order approximation, alternative strict bounds valid for all Biot number. Companion numerical solutions of the heat equation confirm the effectiveness of the error estimates for small Biot number. The second-order approximation and the first-order and second-order error estimates depend on several functional outputs associated to an elliptic partial differential equation; the latter is derived from Biot-sensitivity analysis of the heat equation eigenproblem in the limit of small Biot number. Most important is $φ$, the only functional output required for the first-order error estimates; $φ$ admits a simple physical interpretation in terms of conduction length scale. We investigate the domain and property dependence of $φ$: most notably, we characterize spatial domains for which the standard lumped-model error criterion -- Biot number (based on volume-to-area length scale) small -- is deficient.

Figures (20)

• Figure 1: Fin attached to a square block.
• Figure 2: 3-node thermal circuit.
• Figure 3: Illustrative domains $\Omega^1$ and $\Omega^2$.
• Figure 4: Geometry Small-Angle Right Triangle, denoted in short as SART: solution $\psi'^0$ and final (adaptively refined) finite element mesh. The geometry is given by \ref{['eq:rtriangledef']} with $W=1/4$. Note we may construct from $\Omega^{2\mathrm{D}}$ associated extruded domains in $\mathbb{R}^3$.
• Figure 5: Geometry Small-Angle Right Triangle with Fillet (SART with a $10^{-4}$ radius fillet), denoted in short as SARTF: solution $\psi'^0$ and final (adaptively refined) finite element mesh. Note we may construct from $\Omega^{2\mathrm{D}}$ associated extruded domains in $\mathbb{R}^3$.