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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.12047 |
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| _version_ | 1866913612668338176 |
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| author | Kaneko, Kento Bris, Claude Le Patera, Anthony T. |
| author_facet | Kaneko, Kento Bris, Claude Le Patera, Anthony T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem Kaneko, Kento Bris, Claude Le Patera, Anthony T. Numerical Analysis 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. |
| title | Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2406.12047 |