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 $\phi$, the only functional output required for the first-order error estimate and also the second-order approximation; $\phi$ 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.

翻译：我们研究浸没问题：一个初始温度为$T_\text{i}$的均匀固体被置于远场温度为$T_\infty$、具有与时间无关且空间均匀的对流换热系数的环境中；我们允许材料成分非均匀。该问题由带有Robin边界条件的导热方程描述。关键参数是无量纲换热系数——毕渥数；我们考虑小毕渥数极限。我们针对空间域平均温度随时间的变化，分别引入了一阶和二阶（关于毕渥数的）渐近近似；一阶近似即标准的“集总模型”。我们给出了小毕渥数下一阶和二阶近似的渐近误差估计，并对一阶近似提供了适用于所有毕渥数的非渐近误差界。我们还针对域平均温度与边界平均温度的归一化差值，建立了二阶近似及其渐近误差估计。对导热方程的伴随数值解验证了小毕渥数下误差估计的有效性。二阶近似以及一阶和二阶误差估计依赖于一个椭圆型偏微分方程的若干泛函输出；后者可通过小毕渥数极限下导热方程特征值问题的毕渥数敏感性分析导出。最重要的是泛函输出$\phi$，它是一阶误差估计和二阶近似所需的唯一泛函输出；$\phi$可根据传导长度尺度给出简单的物理解释。我们刻画了一类空间域，对于这些域，基于体积-表面积长度尺度的毕渥数小这一标准集总模型判据是不充分的。