I attempt to understand things better by removing magic constants, and Captain Phillips brings us an emotional satisfying experience.

Power availability relationships

I was working through the power availability relationships today, which include:

  1. \text{Ideal maximum rate of climb:} R_{C,max} = 33000 \frac{BHP}{W}
  2. \text{Reference prop airspeed:} V_{prop} = \sqrt[3]{41.8 \frac{BHP}{D_p}^2}
  3. \text{Propeller tip mach number:} M_p = \frac{RPM \cdot D_p}{21008}

Removing magic constants

I’m dead keen to remove those magic constants though, such as 33000, 41.8 and 21008. For the most part they are rounded-off approximations, and I want the math to not only be accurate but to also reflect the core fundamentals of where those numbers actually come from.

The 33000 is an easy one to resolve. From the current definitions of horsepower, 1 horsepower is 550 ft lb/s, which when multiplied per minute is 33000 ft lb/s.

the 41.8 I have a good idea of where that one comes from, and will work through that one tomorrow.

The 21008 is the most interesting one that I resolved today, thanks to this propeller Performance page. The figure comes from how fast the propeller tip is moving, versus and the speed of sound.

  • The RPM is converted to seconds: 1/60
  • the prop diameter is taken as a radius: 1/2
  • letting us get the circumference of that radius: 2*PI, or TAU (a full turn)
  • and lastly we divide by the speed of sound: 1100 feet per second

1/60 * 1/2 * TAU * 1/1100 = 1/21008.452… which is how we end up at that figure of 21008.

So now I can use the more accurate figure instead, and also gain a better insight about what that mach speed relates to.

Captain Phillips movie

Captain Phillips (2013) is based on the real man’s memoirs of the six days he spent in 2009 as they tried to avoid a gang of Somali pirates, and as their captive. It’s a compelling tale that attempts to explore things from the Somali point of view as well.

Most touching though was near the end of the movie, where Tom Hanks brings us a pitch-perfect emotionally real moment that is better than anything I’ve seen from him before.