The perfect shape

A liquid container designed to keep its content cool as long as possible would be spherical in shape, since the sphere is the closed surface that has the minimum area to enclosed volume ratio, thus minimizing convection heating. When the container is also meant to drink from, the sphere is no longer the optimum shape for coolness (leaving aside the fact that one should have to pierce it to access the liquid) because the surface of the remaining liquid changes as the container is being emptied.

A volume V of liquid at temperature T inside a glass is heated by convection on two different surfaces, one in direct contact with the environment and the other as enclosed by the glass wall.

These two surfaces contribute to the increase in temperature, given by Newton's law of cooling (here heating):

dT = (1/c_vV)(h_open A_open + h_closed A_closed)(T_env - T) dt,

where c_v is the volumetric heat capacity of the liquid, h_open and h_closed the overall heat transfer coefficients of the open and closed interfaces, respectively, and T_env the temperature of the environment. If the liquid is drunk at a constant rate of ϕ units of volume per unit of time, we have

dT = (1/c_vϕV)(h_open A_open + h_closed A_closed)(T - T_env) dV,

with A_open and A_closed varying as functions of V; this differential equation implicitly gives T as a function of (A_open, A_closed, V), although its analytical solution is only feasible for the simplest shapes. We define the optimum container as that for which the average temperature of the liquid during consumption

T_avg = ∫_[Vinit,0]TdV

is minimum.

Let us calculate this shape numerically. Candidate glasses are restricted to surfaces of revolution generated by nonnegative functions y = f(x) between 0 and some maximum height h.

We fix the rest of parameters: the contained liquid is one liter of water (or beer, for that matter) brought out of the fridge at 5 ºC and drunk in 3 minutes in a very hot summer afternoon at 40 ºC; the glass wall (made of, well, glass) is set to be 3 mm thick. All of this gives us:

V_init = 1,000 cm³,
ϕ = 1,000/180 cm³/s,
T_init = 5 ºC,
T_env = 40 ºC,
c_v = 4.1796 J/cm³/K,
h_water = 50 W/m²/K,
h_air = 100 W/m²/K,
θ_glass = 3 mm,
k_glass =  0.85 W/m/K,
h_open = 1/((1/h_water)+(1/h_air)) = 33.3 W/m²/K,
h_closed = 1/((1/h_water)+(θ_glass/k_glass)+(1/h_air)) = 29.8 W/m²/K.

We will use a genetic algorithm to solve the problem:

• Candidate solutions are codified as arrays with the values f(0), f(0.5 cm), ... , f(50 cm) (that is, the maximum height of the glass is h = 50 cm, much higher than really needed —excess height will simply manifest as a tiny column of radius ≈ 0).
• The initial pool of candidates is populated with values from random Hermite interpolation polynomials of degree 3 normalized so that the enclosed volume is V_init.
• Individual fitness is simply its T_avg calculated figure: the lower T_avg the fitter the solution.
• At each generation, the fittest 25% of the pool is kept and the rest replaced by random breeding of the former. Crossover of f₀ and f₁ is implemented by randomly selecting two cutoff points c₁ and c₂, putting f as:
  • f(x) = f₀(x) for x < c₁,
  • f(x) = f₀(x)·(c₂- x)/(c₂- c₁) + f₁(x)·(x - c₁)/(c₂- c₁) for c₁ ≤ x < c₂,
  • f(x) = f₁(x) for x ≥ c₂,
  and normalizing f.
• Bred individuals are mutated (before normalization) by multiplying some of their values (mutation rate = 1%) by a random factor in the range [0,2).

The calculation program has been implemented in C++ (Boost, and in particular Boost.Units, is used). Trials with different initial populations and algorithm parameters (mutation rates, pool sizes and so on) yield basically the same result, which is depicted in the figure:

Optimum glass profile.

The area of this glass is 537 cm² and its associated average temp T_avg is 6.40 ºC (quite cool!). As predicted, the excess height from ~25 cm up shows as a tiny filament whose actual capacity is totally negligible: trimming the glass cap off at a height of 18.5 cm produces a more practical glass with an opening diameter of 4.2 cm and almost the same performance (we only dropped 2% of the initial content and the temperature of the liquid at that point, 3.6 seconds after beginning to drink, is only slightly higher than T_init). The glass thus obtained is, perhaps surprisingly, very reminiscent of real-life glasses:

Optimum glass trimmed at 18.5 cm.

In general, trimming the optimum glass f(x) for a given T_init at height h_cut produces the optimum solution of the problem for V_init = V(height = h_cut), T_init = T(height = h_cut) with the additional constraint that the opening of the glass is precisely 2f(h_cut): we can then play with this property to obtain solutions to the modified problem

minimize T_avg while maintaining a given opening diameter δ

by starting with V_init+ ΔV and T_init -  ΔT and adjusting ΔV and ΔT until the volume and temperature at the desired opening hit V_init and T_init simultaneously (experimenting with this is left as an exercise for the reader). 

Parameter sensitivity

An informal analysis reveals that the optimum glass shape is quite independent of changes in most problem parameters. In particular:

• T_init does not affect significantly the solution (as long as T_init < T_env).
• Changing V_init simply produces a scaled version of the same shape.
• Decreasing ϕ (that is, taking longer to drink the beer up) yields only slightly different shapes, with a wider lower segment.

The one aspect that truly changes the shape is the h_open/h_closed ratio. As this increases, the optimum glass gets thinner at the base and longer, approaching a cylinder as h_open/h_closed → ∞. This has practical implications: for instance, making the beer grow a thick foam topping reduces h_open, which allows for shorter, wider, rounder glasses.

Postscript

Some readers have asked why we haven't used calculus of variations to solve the problem. In its classical one-dimensional formulation (Euler-Lagrange equation), this calculus provides tools for finding the stationary values of a functional J(f) defined as

J(f) = ∫_[a,b] L(f(x), f'(x), x) dx,

for some function L : ℝ³ → ℝ with appropriate smoothness properties. The important aspect to note here is that L must depend on the point values of f and f' at x alone: by contrast, in our case the integrand T(V) is a function of h_open A_open(v) + h_closed A_closed(v) for all values of v in [V, V_init], so the conditions for the application of Euler-Lagrange equation do not hold.