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Chapter 1 Interactive Exercise

December 22, 2009
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So I'm reading ahead over the holidays because while I am one of those students who shows up to class every day and does all the assignments (or tries to do all the assignments) I'm not one of those overachieving, naturally smart kids. I'm a C's pass classes kind of working-slave right now. So to get into the holiday spirit, I'll do a little "design problem".
 
Let's suppose all of Santa's reindeer get Swine Flu. So he's got to figure out some other way to propel his sleigh. We'll assume he's got a whole group of elves/interns/grad-students to build this thing once he decides on a design. We'll leave out the more complex decisions Santa might have to make and assume for today he's trying to narrow it down to what type of engine might be best.
 
We'll say he's got X quantity of reindeer poo that he can use as combustion fluid. However, he still wants his sleigh to be fairly fuel efficient if it can be as his magical reindeer poo (remember, they fly, so don't try burning the normal stuff at home folks) is what he also uses to heat his ridiculously cold and poorly insulated home during the next several months. We'll say he works as kind of a paper boy and need only toss gifts in the right direction as he whizzes by. So we'll reduce his circuit to three circumferential passes of the earth to hit the major population centers, or approximately 120,000 km. We'll also say he's got the rising sun behind him at all times and give him 24 hours (or 86,400 s) to make this route. That means he needs to have an average speed of 1388 m/s, which would be a mach 4 speed. Well that makes this kind of boring, but let's carry on.
 
Santa probably looked at a basic chemical rocket to start with (that's what his elves asked for anyways) and dismissed it straight away. An engine with a propeller can use the airflow passing through the prop to increase efficiency. This allows your propeller engine to have about ten times the thrust (per energy use) as your chemical rocket. But wait! While a chemical rocket can cruise along at maybe 5000 m/s max, your propeller engine can only max out at subsonic speeds, maybe up to 130 m/s. Why's that? Well, whatever speed you're flying at increases the propeller tip speed which increases the velocity of air that's flying at the propeller blade. This can't be very high or you end up with fluid separation and shock waves on the surface of your blades. Therefore, anything approaching the speed of sound is out for the prop.
 
So let's move on. I'll add in a picture here of the turbojet, turbofan and turboprop engines. The turboprop has an increased airflow that gives it a fuel efficiency advantage over the turbojet at lower air speeds. Your turbofan engine could be seen as a compromise between these two designs. It has an enclosed fan, allowing it to achieve much higher mach numbers, around 0.85. This is still not fast enough for our Santa though. Your turbojet efficiency is limited by the fuel to air ratio, which given how fast Santa plans to cruise shouldn't be a problem. Because he's going so fast, it turns out he doesn't even need a compressor in his engine. This leaves the ramjet design. With a maximum mach number of 6, and a compressor-free ability achieved at around mach 3.5 or mach 4, the ramjet becomes the most efficient engine at these speeds.
 
Great so we're done, right? Wrong! Ramjets can't operate at a steady state at subsonic mach numbers, so we'll need something to get us up there. Because the limitations of the turboprop, we might be able to get away with using a turbofan to get us up to mach 0.85 and then let the ramjet kick in or we might have to resort to a turbojet for the initial speed up.
 
This is pretty lame, as Santa is a toymaker not an elite R&D facility for hybrid engines. He's pretty PO'd. Because Santa's been supported by some US Government TARP funds (he had branded himself as a financial institution, and was heavily invested in some risky mortgage derivatives), we'll say he decides to stick to the US this year and outsource the rest of his route. We'll give him three passes over the US at an approximate length of 4000 km, and twelve hours of nighttime to get it done in (I know this is probably inaccurate, I'm not a time-expert here people). This means he needs an average speed of 279 m/s or mach 0.82. Perfect! Didn't we already say the turbofan was capable of speeds up to mach 0.85 and more fuel efficient than a turbojet? Woohoo!
 
Sounds like Santa's got some great options. Clearly, magical reindeer will continue to be the best option until our technology can catch up with it, but for now there are some viable solutions to world travel. And thank you, dear reader (the one of you who got this far, or the three of you who skipped to the last paragraph in utter frustration) for following along with me in Chapter 1. And feel free to correct Santa's calculations if you see any gross errors, after all he's still studying this stuff.
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