Optimizing Beer Glass Shapes to Minimize Heat Transfer -- New Results
This paper addresses the problem of determining the optimum shape for a beer glass that minimizes the heat transfer while the liquid is consumed, thereby keeping it cold for as long as possible. The proposed solution avoids the use of insulating materials. The glass is modeled as a body of revolutio...
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Zusammenfassung: | This paper addresses the problem of determining the optimum shape for a beer
glass that minimizes the heat transfer while the liquid is consumed, thereby
keeping it cold for as long as possible. The proposed solution avoids the use
of insulating materials. The glass is modeled as a body of revolution generated
by a smooth curve, constructed from a material with negligible thermal
resistance, but insulated at the base. The ordinary differential equation
describing the problem is derived from the first law of Thermodynamics applied
to a control volume encompassing the liquid. This is an inverse optimization
problem, aiming to find the shape of the glass (represented by curve $S$) that
minimizes the heat transfer rate. In contrast, the direct problem aims to
determine the heat transfer rate for a given geometry. The solution obtained
here is analytic, and the resulting function describing the relation between
height ans radius of the glass, is in closed form, providing a family of
optimal glass shapes that can be manufactured by conventional methods. Special
attention is payed to the dimensions and the capacity of the resulting shapes. |
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DOI: | 10.48550/arxiv.2410.12043 |