Friday, September 11, 2009

Did it hit Mach, or not?

There's another thread on TRF discussing whether a model rocket can break Mach.  The discussion covers the OOP FSI Mach-1 Dart and the OOP RocketVision Machbuster (mine disappeared on a G55).  The discussion wouldn't be complete without mentioning rocketjunkie's 'EX' 13mm D40 that pushes small modrocs to 1000 mph.  The most interesting item is  Bob Krech's post on the requirements to hit Mach, which I'll quote below. 

There are 2 criteria required for a rocket to reach mach 1.

1.) The rocket motor must develop enough thrust to overcome the drag forces at mach 1.

Drag = 0.5 * Cd * rho * A * V^2

where Cd is the drag coefficient at mach 1, rho is the density of air which at sea level is ~1.3 kg/m^3, A is the cross-sectional area of the rocket, and V is the velocity of the rocket at mach 1 = 340.3 m/s.

Below is the calculated thrust required for a minimum diameter rocket with a given motor size to reach mach 1 assuming a sea level air density, 15 C air temperature and Cd = 1.

10 mm - 7.6 N
13 mm - 11.6 N
18 mm - 20.5 N
24 mm - 37.2 N
29 mm - 55.5 N

If you know your altitude, the air temperature, and the actual Cd of your rocket at mach 1 you can get a more exact value, however this simple number can be used to throw out motors that simply don't have the thrust to make the claim.

2.) Besides have enough thrust to overcome the drag forces at mach 1, the motor must have sufficient total impulse to accelerate from 0 to 340.3 m/s. This requires a rather complicated multi-parameter fit. But there are some rules thumb form minimum diameter rockets based on the above minimum thrust criteria.

29 mm rockets - Average thrust must exceed 56 N and T/W must exceed 120 to stay below 160 NS

24 mm rockets - Average thrust must exceed 38 N and T/W must exceed 80 to stay below 160 NS, T/W must exceed 160 to stay below 80 NS, T/W must exceed 320 to stay below 40 NS

18 mm rockets - Average thrust must exceed 21 N and T/W must exceed 44 to stay below 160 NS, T/W must exceed 88 to stay below 80 NS, T/W must exceed 176 to stay below 40 NS, T/W must exceed 352 to stay below 20 NS

13 mm rockets - Average thrust must exceed 12 N and T/W must exceed 50 to stay below 80 NS, T/W must exceed 100 to stay below 40 NS, T/W must exceed 200 to stay below 20 NS

10 mm rockets - Average thrust must exceed 8 N and T/W must exceed 65 to stay below 40 NS, T/W must exceed 130 to stay below 20 NS, T/W must exceed 260 to stay below 10 NS

Again these numbers assume sea-level altitude, temperature = 15 C and Cd = 1. If the altitude is higher, or the Cd is lower, the average thrust and total impulse numbers are less.

But you can't beat the laws of Physics.

Bob