The Perfect Martini Stem “Bowl” Shape!

Have you ever sat there with a Martini comfortably cradled in your hand and wonder what is the optimum bowl shape for a Martini?  Is it a very wide flaring bowl?  A very narrow tall bowl, almost a flute?

stem-wide                 or                         stem-tall

I’m not talking aesthetics here, that is an individual’s preference.  Some people like the crooked stem or the stem that connects at the top of the bowl, and some the bulbous stem.  (I’ve expressed my personal opinion on these in the past:  To Stem or Not to StemThoughts on Stemware, Part Three).  Others like colored glass stemware, especially in the base.

No, I’m thinking more from an engineering point of view.  Is there an optimum geometry to the fundamental inverted cone shape of a Martini stem?  What’s the optimum height?  Cross section?  And how is optimum defined?

stem-geometry      stem-equations

Alright, I’m sure most people never worry about such things.  I guess this is the curse of being an Engineer.  But there is one important consideration that came from my pondering.  From a strictly geometric point of view, what shape will keep my Martini coldest the longest?

The liquid enters the stem at a given temperature, hopefully very well chilled.  Then it gains heat through both the glass of the stem and the surface area of the liquid.  For the sake of my geometric analysis I made a couple of assumptions:
1.  The heat transfer rate is the same between the glass of the stem and the air above the liquid.  This is dependent on assumption 2.   (I am ignoring heat transfer from the drinker’s hands or the bar top through the stem.  I’m also ignoring radiated heat, such as from the lights.)
2.  Any heat transfer from the glass bowl to the liquid would be transferred via the glass from the air surrounding the bowl.   So, in essence all heat enters the Martini from the air, either directly from above or via the glass sides.   (It isn’t strictly true, but that’s another analysis for another day.)
3.  The optimum shape will be that which maximizes the ration of liquid volume (V) to surface area (SA).

Before I jumped into a bunch of algebra, I did a couple of mental experiments so I would know what to expect from the algebra, and if those answers made sense.   If I keep the radius constant and vary the height, how does the volume vary?  How does the surface area vary?  Contrarily, if I keep the height constant and varied the radius, what happens.  While that proved interesting, it really didn’t help  But if you want to see what came out of it, I’ve included a few observations at the very end of this post.

Clearly this was going to be a multi dimensional analysis.  In order to keep the algebra, and potentially calculus, manageable I made one more assumption:
4.  Set the Martini volume to 1.
(It could be anything, since I’m looking for the optimum cone, and cones are proportional, I can scale the volume after I find the best geometry.)

So what’s the answer?  42!  No wait, that’s something else.  The ‘right’ answer for a Martini stem is,  interestingly, 45 degrees.  Specifically, the cone half angle (α) should be 45 degrees.  The full angle is therefore 90 degrees.  To convert that to an appropriately sized stem, lets pick a reasonable bowl height, such as 2 inches.  Then r would also be 2 inches and the diameter of the bowl would be 4 inches.  So what does that geometry look like?  Something like this:

stem-shape-final

Well it turns out that “typical” martini stem we all know and love is actually pretty darn close to optimum shape.  Sure stems vary from manufacturer to manufacturer, but the traditional martini stem is very close to perfect!  Now I just wonder if the manufacturers really designed this for thermodynamic optimization?  Or because is just looks pretty?

 

 

Random thoughts:

If the radius is constant, then the maximum volume to surface area occurs with an infinitely tall Martini “column”.  Not a useful stem, though the Martini volume would be great.  The volume to surface area (V/SA) approaches 1/3 in the extreme.

If the height is constant, then the maximum volume to surface area occurs with an infinitely wide Martini ‘disk’.  Difficult to lift and unwieldy, but again a nice large Martini.   Here the volume to surface area (V/SA) approaches 1/6 in the extreme.