Optimizing Beer Glass Shapes to Minimize Heat Transfer During Consumption

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.


Introduction
An effective pedagogical strategy in mathematical and physics education involves demonstrating to students how the theories taught in the classroom can be used to address everyday problems.Nevertheless, this proves to be a difficult task more often than not.
Few everyday problems in physics possess a level of simplicity that allows for analytical treatment, while still capturing all phenomena involved.Intro-ducing too many simplifications may render the problem unrealistic 1 .If too few simplifications are emploied, the problem may become unsuitable for analytical treatment 2 .
In the field of mathematics, finding practical problems that are easy to understand but can be solved by straightforward techniques, is an even more challenging task.The four-color theorem 3 ([1] and [2]) and the hairy ball theorem 4  [3] are traditional examples.Both require considerable expertise to under-stand the solution.More recently, ideas about how to divide a pizza into an arbitrary number of unconventionally divided parts [4] have also gained interest, but is equally challenging to understand.
Motivated by the last problem, the authors envisioned an analysis related to the beverage that arguably (or not) pairs best with pizza: beer.In tropical countries like Brazil, a recurring problem is how to keep it cold during consumption, especially in coastal regions.
This paper considers the problem of finding the optimum shape of a beer glass such that the heat transfer rate is minimized in order to keep the beverage cold for as long as possible, while it is consumed.This is an inverse optimization problem, in the sense that the objective is to find the geometry that minimizes the heat transfer rate as opposed to the direct problem, where the objective would be to find the heat transfer rate associated with a given geometry.The intention is to encourage engineering students to develop a rational approach to problems in physics, abandoning the fairly common concept that "theory in practice is different".As a final contribution, nevertheless, the study also aspires to help improve the palatability of our beers.
A brief search on the literature shows literally hundreds of articles focused on analysing practical problems across diverse areas of physics, but nothing was found on heat transfer, except for [5], which addresses the heating of cheese.
The problem of keeping a liquid in a reservoir at the lowest possible temperature may be solved, as will be shown later, by finding a surface that minimizes the area-to-volume ratio of the reservoir.The Greeks knew, even though they could not prove this rigorously, that the answer to the two-dimensional version of this problem was the circle.Later findings showed that the solid possessing the lowest surface-to-volume ratio was the sphere.However, formal proofs for both cases had to wait until the 19th century.In contemporary times, this problem is addressed through the application of the isoperimetric inequality5 in three dimensions (for which there is a variety of proofs) or through variational calculus concepts.
Applications of surface-to-volume ratio optimization are numerous and extend to practically all areas.In chemistry, there are studies involving reactions of all types, such as combustion (in engines or fires), drying and humidification of particles.In biology, there are studies involving exchanges through the skin of living beings and the membrane of cells, microorganisms or organelles.In engineering, there are studies of heat transfer in reservoirs and heat and mass transfer in systems subject to phase change.In atmospheric sciences, there are studies involving the formation of raindrops, hail and snowflakes, as well as evaporation in vegetables and bodies of water.In pharmacology, there are studies on the absorption of medications.It is unnecessary to point out the existence of a series of multidisciplinary studies, with an interface between different areas of study.
A common factor in all the above mentioned studies, is the fact that the heat exchange surface does not change shape during the process.This, however, is not the case when considering the heat exchanged by a glass of liquid with the surroundings during consumption.Even in a cylindrical glass, the total exchange surface undergoes changes in shape during consumption.As the liquid level lowers, the side area reduces while the upper area remains preserved.
Therefore, the question we intend to answer here is: what is the optimal shape-varying surface that minimizes the heat transfer by convection on a glass of liquid with its surroundings?In other words, what is the ideal beer glass, considering potential variations in shape as the liquid is consumed?

Mathematical model
The process is quite straightforward here: a request is made for a beer, the waiter delivers it, it is served, it is consumed.Repeat.
Once poured in the glass, the beer begins to exchange heat with its surroundings, a process that lasts until it attains thermal equilibrium with of the environment (including the glass), a result that essentially nobody wants.Depending on the initial temperature difference between the beer and the surroundings, within a short period of time the drink may become unsuitable for consumption.In the most favourable case, the environment at 10 °C and the beer6 at 4 °C, personal experience shows that as long as 30 minutes may pass before the beer gets warm.In the most critical scenario, such as at the beach on a 38 °C windy day, as few as 3 minutes may be sufficient (again based on personal experience, exhaustively repeated) to render the beer undrinkable.
There are several established practical methods to decrease the beer's heat transfer with the environment.The use of insulation tubes made of expanded polystyrene (EPS or Styrofoam ® ) is probably the most common and it is also used for beer bottles in Brazil.The use of handles on mugs is also a common method to isolate the consumer's hand heat from the drink inside the container.The habit of maintaining a layer of foam on the top of the beer acts as a thermal insulator due to its low conductivity.In addition, it also prevents excessive loss of CO 2 .All those methods have in common the fact that they are passive, i.e., they do not depend on any heat transfer device or substance.That is exactly the approach we shall use here.
To state the most general version of the problem, consider a container holding a liquid initially not in thermal equilibrium with its surroundings, as shown in Fig 1 .Assume that the vessel is a body of revolution, generated by the rotation of a curve S, differentiable of class C 1 , around the vertical axis, y.Such a geometry describes all commercially available containers, except for some nonaxisymmetric non-returnable beer bottles.However, S is not entirely unrestricted: it must contain exactly one opening and one impermeable bottom.Therefore, let 0 ≤ h(t) ≤ H be the variable height of the liquid in the container at time t, and let 0 ≤ r(t) be the corresponding radius, where r(0) = R b is the radius of the base and r(H) = R o is the radius of the opening.For obvious reasons, r = 0 may only occur at h = 0. Let CV be a variable shape control volume encompassing the liquid, as depicted in Fig. 1.
To begin particularising the problem, now assume that body of the vessel has negligible thermal resistance, whereas its bottom is thermally insulated.This is a reasonable approximation for beer glasses, where the body is made of thin glass for comfort, and the bottom of thick glass for mechanical resistance.Also contributes for making the bottom insulation a reasonable hypothesis the habit of letting beer glasses rest on insulating surfaces, such as tables, counters and cardboard disks (the famous "wafer").
Furthermore, consider that the temperature of the beer remains uniform throughout the glass while it is being consumed.This holds true when the rate of temperature change over time is small, enabling the liquid to reach instantaneous thermal equilibrium.Under such circumstances, the initial temperature difference between the liquid and the environment must not be excessively large, which is always the case.
Finally, assume the liquid to be homogeneous.This characteristic holds true for most filtered beers, that do not form accumulations at the bottom, including Weizenbier.A few craft beers present small amounts of yeast sediment and are not considered here.
The law of conservation of energy ([6] for example) for the CV chosen is q ac = q in − q out + q gen (1)   where q = dQ/dt is the heat transfer rate and Q is heat.In Eqn.(1), q ac represents the accumulated heat within the CV, q gen is the heat generated inside the CV, and q in and q out denote the heat entering and leaving the CV, respectively.
In general, while drinking beer and most beverages, there is no internal heat generation.Possible exceptions would be liquids undergoing fermentation.Thus, in our case, q gen = 0. Assuming that the environment is the only external source of heat, Eqn.(1) yields where q in enters through the body of the glass and the foam at the opening.
Using the concept of thermal resistance, Eqn.(2) can be rewritten as where T = T(t) is the spatially uniform temperature of the beer, T ∞ is the ambient temperature, R side and R foam are the thermal resistances of the glass and of the foam respectively.Each resistance is composed by two other resistances associated in series: a conductive resistance through the materials and a convective resistance on their external surfaces.However, to simplify the problem, the thermal conductivity of the side glass and the foam are neglected7 .This may not be completely realistic but nevertheless represents the most critical situation8 .Equation (3) may then be rewritten as From the definitions of specific heat and density for an homogenous liquid it follows that q ac = ρVc p (dT/dt).Thence, Substituting R cv = 1/(h cv A) finally yields where A tot is the total heat transfer area, i.e., the side and opening areas of the glass.Equation ( 6) suggests some strategies for minimizing the rate of temperature change, dT/dt, thereby extending the drinkability window: 1. Reduce (T − T ∞ ) by keeping the glass in a cool environment and avoiding radiant heat from the sun9 ; 2. Increase the sum of resistances in the denominator of Eqn. ( 6) by keeping a tick, generous foam over the beer; 3. Increase the conductivity resistance of the sides of the glass (the vessel) by substituting the glass (the material) by a more insulating material, such as ceramic10 ; 4. Keep the glass away from drafts, avoiding forced convection, which is far more efficient than natural convection in transfering heat.
The considerations above show why the beach is the most challenging environment for beer drinking: the air temperature is high, the wind is persistent, the sun shines, and ceramic mugs are not welcome.
Rearranging Eqn. ( 6) yields Equation ( 7) is the ODE that governs the problem.It can be easily solved for a known geometry, since it is of the separable type.However, this would be the direct problem which is not the goal here.
For any fixed instant of time t and any given value of h cv /(ρc p ), Eqn.(7) shows that dT/dt gets smaller as A tot /V is reduced.Therefore, minimizing the heat transfer reduces to This is the subject of the next section.

Solution
Consider Fig. 2, where the glass and its generating curve, here identified as r = r(h), are depicted The minimum for the surface-tovolume ratio follows from d dh Here, the height of the liquid, h, is used as the independent variable as the beer is consumed.Differentiation of Eqn.(9) yields Simplifying results in for V ̸ = 0 and A tot ̸ = 0. Integration gives ln A tot = ln V + C 0 and, thus, where C 1 is a dimensional constant to be examined later.
From Calculus, for bodies of revolution the side area is and the volume is Therefore, Eqn.(12) becomes where Simplifying and rearranging, for r ̸ = 0, except possibly at h = 0. Squaring both sides and simplifying, This is a separable ODE.Thus which can be integrated by substitution Recalling that C 1 = 2C 2 and rearranging yields which can be rewritten to make r explicit as This equation solves the problem.
Substituting Eqn. ( 22) into Eqns.( 13) and ( 14) and integrating from 0 to H yields the total area and volume of the glass:

Discussion
Equation ( 22) shows an exponential dependence of r with √ h.One should therefore expect a rapid growth of r with h, affected by the parameter C 1 which cannot be chosen arbitrarily, as demonstrated subsequently.
From Eqn. (21), and therefore Equation ( 22) also shows that the glass must have a small base and a large opening, with its radius continually increasing from the base up.Indeed, putting r ′ = 0 in Eqn.(18) implies that C 1 r = 2, which is false.Therefore, that is no maximum or minimum points in function r.Additionally, from Eqn. (18) and C 1 = 2C 2 it follows that r ′ = (1 − C 2 1 r 2 /4)/C 1 r which is always positive, showing that r(h) is a monotonically increasing function.
Before Eqn. ( 22) is sent to the factory, and the optimized glasses start to be mass produced, it is necessary to establish adequate values for C 1 , limited by the restriction C 1 R > 2 as C 1 is a free parameter in Eqn.(21).
In Eqn.(21), as C 1 R → 2 from the right for fixed values of r, then h → ∞ for any finite C 1 .In words, as C 1 R → 2 it is only possible to obtain the optimal solution for very tall glasses.This should not be an issue as one could simply chose a large value of C 1 R to avoid the problem.However, as C 1 R multiplies exp(C 1 h/2), large values of C 1 R will result in very large values of r.Thus as C 1 R ≫ 2 it is only possible to obtain the optimal solution for very large openings.Both situations suggest that there exists a suitable range of values of C 1 R that results in practically viable glass shapes.
To obtain C 1 first Eqn.( 21) is rewritten as Then, based on practical considerations, values of Equations ( 29) and ( 30) provide an approximate way to obtain C 1 : 1. Values of R o and R b are chosen; 2. An initial approximation for C 1 is obtained through C 1 R b = 2.This value is not final because it makes the denominator of Eqn.(30) go to zero; 3. This value is then substituted into Eqn.(29) to calculate ε; 4. This value obtained for ε is substituted into Eqn.(30) to give a final estimate for C 1 As C 1 is used only as a shape parameter, there is no need for further iterations or a convergence criterion.
The influence of C 1 and R b on the shape of the glass is illustrated in Figs. 3  and 4, for glasses with H = 20.0 cm.The numerical results are presented in Tables 1 and 2.
Figure 3 shows the results obtained for three glasses with a base radius R b = 3.0 cm and C 1 varying around 0.667 cm -1 , the value calculated by Eqn.(29) with λ = 2.Only glass 1, with C 1 = 0.677 cm -1 is commercially acceptable.Glasses 2 and 3 are certain to have equilibrium issues due to their large relation R o /R b .Additionally, their volumes are so large that it is very unlikely the beer final temperature is acceptable, despite the fact they are thermally optimized12 .Figure 4 and Table 2 shows the results obtained for three glasses with C 1 = 0.677cm -1 and R b varying.Again, only glass 3 is acceptable with a volume of 7.87 litres.As seen on Tab. 2 the volumes of glasses 4 and 5 are absolutely ridiculous.Due to the simplicity of the calculations required for obtaining optimized glasses, many alternative configurations were explored.Of all resulting glasses, two were considered feasible.Their dimensions are provided in Table 3, alongside those of glass 3. Figure 5 shows a comparison between these three glasses, numbered 3, 6 and 7.All of them have total volumes considered high by most non-alcoholic consumers and considerably large radius of the opening, requiring a wide foot to keep balance.Smaller radius of the base would make C 1 increase, leading to even wider openings.Larger radius of the base, on the other hand, would yield small values of C 1 , resulting in almost cylindrical bodies, which would require very tall glasses to be optimized.Figure 6 presents an in scale shape comparison between glasses 3 and 7.An image of a commercial glass similar to glass 3 is presented in figure 7.  Here, as in the case of cutting pizzas, the mathematically optimized solution to the problem may prove to be somewhat complicated to implement in practice.

Conclusions
In this work, a method has been proposed to optimize the shape of beer containing glasses.The optimizations criterion was to minimize heat transfer, thus maintaining a low temperature for as long as possible while the liquid is consumed.The analysis yielded a family of shapes that can be easily manufactured by traditional methods and used in dayto-day life, as long as some attention is pay to the radii of the base and of the opening.
Throughout the analysis several hypothesis were used.The glass was considered to be a body of revolution generated by the rotation of a smooth curve S around the vertical axis.Additionally, the thermal resistance on the body of the glass was neglected, whereas the bottom was considered insulated.The liquid's temperature was supposed to be spatially uniform, the liquid was considered to be homogeneous and the thermal resistance of the foam was disregarded.Finally, no radiative heat transfer was considered, as well as the conduction heat transfer due to hand contact with the body of the glass.
While those hypotheses may appear restrictive, the analytical result obtained still hold didactic and practical interest despite the fact that a complete formulation including all effects disregarded earlier, can be effectively addressed by numerical methods.Nevertheless, given that the primary focus of the investigation conducted here is didactic, analytical solutions are preferable.Closed analytical solutions are generally welcomed in Physics, even though they usually represent the result of simplified analyses.They offer a comprehensive view of the problem, explicitly depicting the influence of all parameters involved.
Moreover, an analytical solution is almost always a general conclusion about the problem addressed and rarely constitutes a case study.Analytical solutions also clarify the conditions under which the obtained results are valid.These statements may seem somewhat obvious, but in times of such intense and careless use of computer simulation, they are particularly timely.
Potential areas of further investigation include allowing the base of the glass and its body to exchange heat with the ambient, including radiative heat transfer and/or conduction heat transfer through the foam.Some of these analysis may require differential methods volume or an iterative procedure.
In conclusion, this paper applied basic concepts of heat transfer and extreme values of functions to an everyday, yet relevant topic -beer drinking.The primary goal, of course, was to enhance engineering students' interest in Physics and Mathematics.However, a secondary yet crucial application of the present results was to safeguard the quality of our beers.

Figure 7 :
Figure 7: Over-the-shelf glass similar to glass 3.

Table 1 :
Optimized glass shapes.H = 20.0 cm and R b = 3.0 cm