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 $\phi$, the only functional output required for the first-order error estimates; $\phi$ admits a simple physical interpretation in terms of conduction length scale. We investigate the domain and property dependence of $\phi$: 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.

翻译：我们考虑浸没问题：一个初始温度为 $T_{\text i}$ 的均匀温度固体被置于一个远场温度为 $T_\infty$、且具有空间均匀且与时间无关的对流换热系数的环境中。我们允许非均质的材料组成：即密度、比热容和热导率可随空间变化。该问题在数学上由带有 Robin 边界条件的导热方程描述。关键参数是毕奥数——一个无量纲的对流换热（Robin）系数；我们考虑小毕奥数的极限情况。我们针对几个关注的物理量，特别是空间域平均温度随时间的变化，引入了一阶和二阶（关于毕奥数的）渐近近似；一阶近似即为标准的工程“集总”模型。随后，我们给出了小毕奥数下一阶和二阶近似的渐近误差估计，并且针对一阶近似，还提供了适用于所有毕奥数的替代性严格误差界。对导热方程的伴随数值解证实了这些误差估计在小毕奥数下的有效性。二阶近似以及一阶和二阶误差估计依赖于与一个椭圆型偏微分方程相关的多个泛函输出；该方程源于小毕奥数极限下导热方程特征值问题的毕奥数敏感性分析。其中最重要的是 $\phi$，它是一阶误差估计所需的唯一泛函输出；$\phi$ 可以根据导热长度尺度给出简单的物理解释。我们研究了 $\phi$ 对求解域和材料属性的依赖性：最值得注意的是，我们刻画了那些使得标准集总模型误差判据——即基于体积与表面积之比的长度尺度定义的毕奥数很小——失效的空间域。