Monday, July 21, 2014

Calculus of Steam Engines and Steam Boats

I had the good fortune of taking a ride on a classic steam boat over the weekend on the Mississippi River while listening to some great Dixieland music. Being curious about these classic industrial-revolution-era designs, Geoff and I went down to the steam room to see how the steam engine works. The most interesting part to me was that the engine has only one incoming steam pipe (connected to the boiler room), which means that in order for the piston to move both forwards and backwards, there is a sliding valve that determines which chamber adjacent to the piston is being filled with steam, and therefore which direction the piston will be pushed.

It looks something like this (although I couldn't find a diagram exactly similar to the design on the particular boat. I am pretty sure the boat we were on had an engine whose slide valve moves more symmetrically than this site indicates).

Anyhow, it made me think about the Calculus that must be involved in steam engines, since the shape is changing dynamically. With a little research, here is an authentic physics problem (adapted from since I needed a little physics refresher) that I can see giving to my Calculus students next year:

You know that within a steam engine, both the pressure and volume of the chamber are changing constantly. But, the pressure and volume are related at any given moment by PV^1.4 = k, where k is a constant. In order to calculate the amount of physical work put out by the steam engine, you need to know that Work = constant pressure times change in volume, or Work = constant volume * change in pressure. Since in this case both are changing, we will need to use Calculus to determine the total work done as volume changes, via integrating the equation dW = p*dv, where p is the steam pressure as a function of the instantaneous volume v inside the steam engine.

For any given steam engine, we can measure a starting volume and pressure to give us something to work with. For simplicity, we'll say that the original steam engine conditions are volume V= 100in^3 and pressure P = 160 lb/in^2. It follows then that any new combination of (v, p) is pv^1.4 = 160(100)^1.4. 

a.) Assuming that the volume of 100 in^3 is the smallest engine chamber that exists in this engine (ie. when the piston is fully compressed towards the starting side), and that the chamber can expand to 800 in^3, can you find some pairings of volume and pressure that the engine will necessarily experience during its movement? 

b.) Sketch the curve from Part A. Is the curve continuous and differentiable? Explain why or why not.

c.) In order to find the physical work done as the initial chamber expands (and pushes on the piston), we will need to find the formula that describes p as a function of v, and then integrate dW = p*dv from v = 100 in^3 to v = 800 in^3. Do that, and carefully write down the resulting units for your answer.

d.) In physics, it is easiest to relate parts of simple machines using the SI unit "Joules", which is equivalent to lb*ft. 1 Joule is approximately the same as the energy required to lift a small apple by 1 meter. Can you figure out how many Joules this steam engine will complete in one complete cycle (through expansion and then through compression, or expansion of the opposite chamber)?

d.) An early steam boat may be powered by an engine of several hundred horsepower. (See for some sample points.) One horsepower is the same as 745.7 Joule/second. If your engine is the size described in this problem, then how long (in seconds) does it have to complete 1 full cycle, in order to achieve 200 horsepower?

e.) Is this a reasonably sized steam engine for a fair-sized steam boat? If so, explain. If not, find more reasonable specs and justify your choice through calculations!

What do you think? Is this authentic? Is it rigorous? Is it interesting? Help me improve it!

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