Certified Lumped Approximations for the Conduction Dunking Problem

TL;DR

The paper analyzes the dunking problem for a solid body in the small-$B$ limit, introducing a PDE-based sensitivity framework governed by the field $\xi$ and the quadratic outputs $\phi$, $\chi$, and $\Upsilon$ to certify first- and second-order lumped approximations of the domain-average temperature. It develops Padé-type second-order lumped models for $u_{avg}$ and $u_{\Delta}$, with rigorous asymptotic error estimates that depend on the geometry through $\phi$ and related functionals, and provides non-asymptotic bounds for the first-order model. Numerical finite-element results validate the theory and illustrate how geometry controls $\phi$, including closed-form values for canonical domains and tensorization rules. The work also frames a physical interpretation via body-average resistance and a conduction-length scale $L_{cond}=\mathcal{L}_{cond}$, enabling a thermal-circuit view that clarifies when lumped models are reliable and when geometry can undermine the standard small-$B$ criterion.

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 time-independent spatially uniform heat transfer coefficient; we permit heterogeneous material composition. The problem is described by a heat equation with Robin boundary conditions. The crucial parameter is the Biot number, a nondimensional heat transfer coefficient; we consider the limit of small Biot number. We introduce first-order and second-order asymptotic approximations (in Biot number) for the spatial domain average temperature as a function of time; the first-order approximation is the standard `lumped model'. We provide asymptotic error estimates for the first-order and second-order approximations for small Biot number, and also, for the first-order approximation, non-asymptotic bounds valid for all Biot number. We also develop a second-order approximation and associated asymptotic error estimate for the normalized difference in the domain average and boundary average temperatures. 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 with an elliptic partial differential equation; the latter can be derived from Biot-sensitivity analysis of the heat equation eigenproblem in the limit of small Biot number. Most important is the functional output $φ$, the only functional output required for the first-order error estimate and also the second-order approximation; $φ$ admits a simple physical interpretation in terms of conduction length scale. We characterize a class of spatial domains for which the standard lumped-model criterion -- Biot number (based on volume-to-area length scale) small -- is deficient.

Figures (3)

• Figure 1: Thermal circuit: three temperature nodes and two thermal resistances.
• Figure 2: Spatial domains for illustration of stability: a) SART-1, b) SARTC, c) RECT, and d) RECTSART. In all cases we present the geometry, the final adapted finite element mesh, and the field $\xi$ (on the basis of which $\phi$ is evaluated).
• Figure 3: A "finned-block" domain $\Omega \subset \mathbb{R}^2$ for illustration of our feature $\mathbb{F}$ and associated sufficient condition. The domain comprises two subdomains: a $4\times 2$ rectangle (the block, to the right) and an $L\times H$ rectangle (the fin, to the left).