Optimizing Beer Glass Shapes to Minimize Heat Transfer -- New Results

TL;DR

This paper derives an analytical method to optimize beer glass shapes, minimizing heat transfer without insulation, to keep the beverage cold longer, and provides a family of manufacturable optimal designs.

Contribution

It introduces a novel inverse optimization approach with an analytic solution for designing heat-efficient beer glasses without insulation.

Findings

Analytic family of optimal glass shapes derived in closed form.

Designs effectively minimize heat transfer during consumption.

Results are practical for manufacturing and improve beverage cooling duration.

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 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.