back of the envelope calculations

soya latte at the coffee jar camden — Not one made by me! But instead a soya-latte at the Coffee Jar a couple of years ago.

During these strange times of working from home, perhaps you, like me, have been preparing a lot more coffee. For me this has included, not just my regular V60s, but a type of cafe-au-lait for someone who used to regularly drink lattes outside. My previous-latte-drinker turns out to be a little bit discerning (the polite way of phrasing it) and so prefers the coffee made in a similar way each day. Which is why I’ve been weighing the (oat) milk I’ve been using.

So, each morning to prepare a coffee, I’ve been using a V60 recipe from The Barn and then, separately, weighing out 220g of refrigerated oat milk into a pan that I then heat on the stove. Generally I heat the milk for just over 5 minutes until it is almost simmering whereupon I pour it into a mug (with 110 – 130g of coffee inside – depending on the coffee). Being naturally lazy, I keep the cup on the scales so that it is easier to pour the milk in and then, completely emptying the pan into the coffee, the scales register an increase of mass (of milk) in the cup of 205-210g. Which means about 10-15g of milk goes missing each morning.

Now clearly it is not missing as such, it has just evaporated, but it does prompt a question: can this tell us anything about the physics of our world? And to pre-empt the answer, it actually tells us a great deal. But to see how, we need to go on an historical diversion to just over three hundred years ago, when Edmond Halley was presenting an experiment to the Royal Society in London. The experiment shares a number of similarities with my heated oat milk pan. It was later written into a paper which you can read online: “An estimate of the quantity of vapour raised out of the sea by the warmth of the Sun; derived from an experiment shown before the Royal Society at one of their late meetings: by E Halley“.

lilies on water, rain on a pond, droplets — Coffee, evaporation, clouds, rain, rivers, seas, evaporation. Imagining the water cycle by making coffee.

Halley heated a pan of water to the temperature of “the Air in our hottest summers” and then, keeping the temperature constant, placed the pan on a set of scales to see how much water was lost over 2 hours. The temperature of the air in “our hottest summers” cannot have been very high, perhaps 25-30C and there was no evaporation actually seen in the form of steam coming from the pan (unlike with my milk pan). Nonetheless, Halley’s pan lost a total of 13.4g (in today’s units) of water over those two hours.

Halley used this amount to estimate, by extrapolation, how much water evaporated from the Mediterranean Sea each day. Arguing that the temperature of the water heated that evening at the Royal Society was similar to that of the Mediterranean Sea and that you could just treat the sea as one huge pan of water, Halley calculated that enough water evaporated to explain the rains that fell. This is a key part of the water cycle that drives the weather patterns in our world. But Halley took one further step. If the sea could produce the water for the rain, and the rain fed the rivers, was the flow of the rivers enough to account for the water in the Mediterranean Sea and, specifically, how much water was supplied to the sea compared to that lost through the evaporation? Halley estimated this by calculating the flow of water underneath Kingston Bridge over the Thames. As he knew how many (large) rivers flowed into the Mediterranean, Halley could calculate a very rough estimate of the total flow from the rivers into the Mediterranean.

Grecian, Devereux, Coffee house London — A plaque outside the (old) Devereux pub, since refurbished. The Devereux pub is on the site of the Grecian Coffee House which was one of the places that Halley and co used to ‘retire’ to after meetings at the Royal Society.

The estimates may seem very rough, but they were necessary in order to know if it was feasible that there could be a great water cycle of rain, rivers, evaporation, rain. And although Halley was not the first to discuss this idea (it had been considered by Bernard Palissy and Pierre Perrault before him), this idea of a quantitative “back of the envelope” calculation to prompt more thorough research into an idea, is one that is still used in science today: we have an idea, can we work out, very roughly, on the back of an envelope (or more often on a serviette over a coffee) if the idea is plausible before we write the research grant proposal to study it properly.

So, to return to my pan of oat milk simmering on the stove. 15g over 5 minutes at approaching 100C is a reasonable amount to expect to lose. Only, we can go further than this now because we can take the extra data (from the thermostats we have in our house and the Met Office observations for the weather) of the temperature of your kitchen and the relative humidity that day and use this to discover how these factors affect the evaporative loss. Just as for Halley, it may be an extremely rough estimate. However, just as for Halley, these estimates may help to give us an understanding that is “one of the most necessary ingredients of a real and Philosophical Meteorology” as Halley may have said before he enjoyed a coffee at one of the Coffee Houses that he, Newton and others would retire to after a busy evening watching water evaporate at the Royal Society.

Back of the envelope calculations with coffee

coffee at Watch House — Coffee is generally a great help for reading, but to properly see the clouds in your coffee, it may help if you prepared yourself a brew now.

To read this post it will help if you have a cup of lovely, hot, freshly prepared coffee or tea with you.

Got it? Ok, let’s begin.

A few weeks ago, there was a talk given by Prof. Paul Williams of the University of Reading about the Mathematics of turbulence and climate change. An entertaining talk about the importance of, and the effort of comprehension required to, use mathematics in order to understand climate change. There were several thought provoking comments through the talk that demanded further reflection. But one, almost throw-away comment has been bugging me since. Although I’ve forgotten the exact words, they went along the lines of

Of course mostly we think about the impact of climate change on the weather, after all, we live in the bottom few metres of the atmosphere and so that is what mostly affects us. What I would like to talk about is the effect of climate change on airplane turbulence…

The bottom few metres of the atmosphere? It’s true. The bit we’re most experienced with is just a tiny portion of it. It’s about perspective. To us, it seems the atmosphere is very big, we pump all sorts of exhaust fumes into it and they disappear. We have expressions such as “the sky is the limit” that suggests that the atmosphere is a huge volume of gas. We all know it is not really limitless, but day to day, on our human scale, it seems enormous.

Now the mathematics that Prof Williams uses to calculate the effect of changing temperature and carbon dioxide levels on the jet stream (and consequently the turbulence felt by planes) is way beyond the sort of back of the envelope calculation that we can do with a cup of tea (or coffee). Understandably, to even start to comprehend these mathematical models requires years of training in maths and physics. However, assuming that we are not ourselves atmospheric physicists, there are things that we can do to help us to see our atmosphere in a more realistic way. And this is where your coffee comes in.

Earth from space, South America, coffee — Clouds swirling above our common home. But if the atmosphere is represented by the white mists on the surface of a cup of coffee, what size coffee are we drinking?
The Blue Marble, Credit, NASA: Image created by Reto Stockli with the help of Alan Nelson, under the leadership of Fritz Hasler

Take a close look at that coffee. Assuming it is not cold brew, hopefully your coffee or tea is still fairly warm. Watch the surface of the coffee. You may start to see movement such as convection in the mug, perhaps you can see a film of oil on the surface. But do you see something else? In very hot tea or coffee, you should be able to see what appear as white mists hovering over the surface of the cup*. It is easy to miss them, but as you watch, cracks suddenly appear in the mists and then there is a re-organisation of them which allows you to start to see them dancing over the surface of your drink*.

These mists are the result of the levitation of many thousands of droplets of water just above the surface of the coffee. I have written about them elsewhere. No one knows quite how they levitate above the surface, but what is known is that they are at a distance of up to 100 μm (0.1mm) from the surface of the coffee.

Let’s construct a scale model of our coffee as the Earth and its atmosphere. These mists can then do a fairly good job of representing the atmosphere with its drifting clouds. So, assuming that the mists are the atmosphere and the coffee is the Earth (on the same scale), what size of coffee would you have to have? Would you be drinking:

a) an espresso

b) a long black

c) a venti

d) a ristretto

Think you know the answer? Let’s work it out with a “back of the envelope” calculation. The easy bit is deciding the radius of the Earth, it’s just under 6400 km, our first problem comes with the estimate of the thickness of the atmosphere. There are several layers in the atmosphere. The one that we are most familiar with, the one closest to us is the troposphere. This extends for the first 16 km above the surface of the Earth (though this varies with latitude, it is only 8 km at the poles). Most of our weather happens in this region and it is also the layer of the atmosphere that planes fly in. Above the troposphere is the stratosphere which extends until about 50 km. Beyond that, things get very rarified indeed though the boundary between our atmosphere and “space” does not happen for several hundred km (indeed, the orbit of the International Space Station is in this bit of our extended atmosphere).

Coffee Corona — Look carefully around the central (reflected) white light. Can you see a rainbow like spreading of the colours? Another manifestation of the white mists on the coffee surface.

As we are mostly concerned with the weather (and airplane flight etc) though, it seems sensible to define the atmosphere height to be the top of the troposphere. After all, most of us will tend to think that the Space Station is in, well, space. This definition is further justified by the fact that about 75% of the mass of the atmosphere is found within this region (the atmosphere gets thinner as you go higher).

What size coffee would we be drinking if the white mists (0.1 mm above the coffee surface) represent the 16 km of the Earth’s atmosphere? We’ll call the coffee height, hc. Our first step is quite easy, we can just use the ratios of the heights to calculate the coffee size:

(height of troposphere)/(radius of Earth) = (white mist height)/(height of coffee)

A bit of rearrangement:

height of coffee = (white mist height)*(radius of Earth)/(height of troposphere)

hc = (0.1) * (6400)/16

hc = 40 mm (4cm)

So for the mists to represent the atmosphere in your coffee, you would need to be drinking a 4cm tall coffee which is probably a smallish long black. I would leave it to you to calculate the coffee size for the atmosphere defined as outer space (beyond the orbit of the International Space Station). But perhaps this perspective gives us another way of looking at our atmosphere. Vast indeed, but fragile too.

*As I was writing this, I had a warm, very drinkable, cup of coffee but it wasn’t steaming and so showed no white mists over the surface. The mists are best seen in freshly made, very hot drinks.