Optimizing Beer Glass Shapes to Minimize Heat Transfer During Consumption

Abstract

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 modelled as a body of revolution generated by a smooth curve S, constructed from a material with negligible thermal resistance at the revolution surface but insulated at the bottom. 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 is analytic, and the resulting expression for S is in closed form, providing a family of optimal glass shapes that can be manufactured using conventional methods.