As part of my (3 year!) probation as a lecturer at Birmingham, there is a substantial compulsory teaching course. A part of that course is to write a teaching blog; I’ve always seen blogs as a way to promote research, advertise positions/yourself, or describe some fun science, so this is a rather new experience to me.

I can’t imagine anyone particularly wants to hear my opinions about teaching (others are more erudite and motivated), and I’m not really one for introspection – I find it rather counter-productive when there is so much to enjoy, experience, and study – so I thought I’d use the opportunity to discuss one of the more fun/interesting topics that I taught in my 4th-year (Masters) lecture course in Mathematical Biology, and why I think I my students weren’t interested.

Transport processes are fundamental to all life – feeding, growing, breeding, avoiding death always involves transport in every species, from bacteria, to humans, to trees. We transport blood/oxygen around our body, trees transport water from root to leaf, bacteria transport themselves, and so on. Two fundamental methods of transport are advection and diffusion. Advection is an active pumping via fluid flow (as Oxygen in the bloodstream is transported to tissue), while diffusion is a passive process relying on random “spreading out” motion of molecules. Advection requires input of energy, whereas diffusion does not.

Whether or not a natural system has evolved to exploit advection, diffusion, or a combination of the two, turns out to depend on a parameter known as the Péclet number Pe of the system (after 19th century physicist Jean Claude Eugène Péclet), which can be thought of as the ratio of advective to diffusive transport rates. Thus, calculating the Péclet number for a system can help us understand its design.

I asked my students to estimate the Péclet number at the top and the bottom of our lungs (they could use google for values). They were not into it. Yet this is probably the exercise closest to actual research that I’ve ever given out (after I calculated it, I found a paper online from this side of 2010 measuring it), and on the surface only requires multiplication and division – primary school operations. The calculation is as follows:

For a given system, Pe = vL/D for a typical velocity v, length scale L, and molecular diffusivity D. We are interested in how Oxygen gets into our blood, so we need D = diffusivity of oxygen in air = 0.2 squared cm per second. The typical length scale is very roughly 1-2 cm at the top of the trachea (its diameter), and 0.02 cm at an alveolus, where the oxygen ends up.

At this point, it would be easy to think that Pe changes by a factor of 100 from top to bottom, but in fact air slows down a lot between the top and bottom of our lungs. Estimating this slow down is the tricky part. Our lung capacity is about half a litre, and we can breath-in in about a second. So 500 cubic cm of air passes through the top of the trachea – an area of about 2 squared cm – in a second, giving a typical velocity of v = 500/2 = 250 cm/s (about 5 miles an hour in old money). So the Péclet number at the top of our trachea is about 2×250/0.2 = 2500, which is much bigger than 1 – so advection is the dominant process.

But the area per alveolus through which this same flow passes is more like 0.02×0.02 = 0.0004 squared cm. There are around 700 million alveoli in typical adult lungs, so a total area of 0.0004×700,000,000 = 280,000 squared cm, so a typical velocity of flow entering an alveolus is 500/280,000 = 0.0018 cm/s – very slow. The Péclet number is then Pe = 0.00018, much less than 1, and diffusion dominates.

The point is, diffusion is required in order to selectively exchange oxygen and carbon dioxide between our blood and the air, but diffusion is a painfully slow process over large distances and small surface areas, so the branching structure of our lungs has evolved to draw in a large amount of air quickly into a massive number of very small spaces, and slow it down so diffusion can take over.

Insects don’t have this, by the way – they have an open network of trachea delivering oxygen via diffusion directly to their tissues. Because of the limits of diffusion, this means that insects can’t grow too large (which is good). I’ve gone on too long, this was supposed to be 500 words.

I think my students didn’t like this exercise because they haven’t really done much in their course to prepare them for this kind of open-ended question (that can be understood through fairly simple mathematics). I don’t think it’s a bad exercise (it is testing a different skill), but next time I think I need to execute it differently – ease them into it more somehow – or perhaps set it as part of a formative assignment. Either way, I’m not quite done with the experiment of bringing my research to my teaching.

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