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defining $ teff = { T \over { n+1 } } = { V_{max} \over { a ( n+1) } } $ defining $ teff = { T \over { n+1 } } = \large { V_{max} \over { a ( n+1) } } $

Hypervelocity Drag

Based on Trajectory Optimization for an Apollo-type Vehicle under Entry Conditions Encountered During Lunar Returm by John W. Young (famous astronaut) and Robert E. Smith Jr., May 1967, NASA TR-R-258, Langley Research Center.

Equations on Page 5 in Foot-second-slug-BTU :

  • 1a convective power: ~ ~ \dot Q_c = 20 \rho^{1/2} \left( V \over 1000 \right)^3 Btu/ft2-s

  • 1b radiative power: ~ ~ \dot Q_r = 6.1 \rho^{3/2} \left( V \over { 10 000 } \right)^{20} Btu/ft2-s

  • Equations assume an effective nose radius of 1 foot
  • Equations from Shock Layer Radiation During Hypervelocity Re-Entry by Robert M. Nerem and George H. Stickford, AIAA Entry Technology Conference, CP-9, American Institute of Aeronautics and Astronautics, Oct. 1964, pp 158-169. (not downloaded yet)

Density in slugs/ft3: multiply kg/m3 by 1.9403203e-3

Power in Btu/ft2-s: multiply by 11350.54 to get W/m2

Velocity in ft/s: divide m/s by 0.3048

Metric equations:

  • Metric 1a convective power: ~ ~ \dot Q_c = 3.53e-4 \rho^{1/2} ~ V^3 Watts

  • Metric 1b radiative power: ~ ~ \dot Q_r = 1.24e-69 \rho^{3/2} ~ V^{20} Watts

These are for a 1 foot diameter nose, and scale by {r_n}^{-1/2} according to equation 4B-4 on page 520 of Part 4B (Entry Heat Transfer) of the SAE Aerospace Applied Thermodynamics Manual. That sites reference 1, A study of the motion and aerodynamic heating of missiles entering the earth's atmosphere at high supersonic speeds, H. Julian Allen and A. J. Eggers, Jr, NACA TN 4047, 1957. If r_n is in meters, scale by 0.552 {r_n}^{-1/2} .

If we scale these for a half-spherical nose, area \pi {r_n}^2 , we get:

  • Total nose convective power: ~ ~ \dot Q_c = 6.1e-4 ( {r_n}^3 ~ \rho )^{1/2} ~ V^3 Watts

  • Total nose radiative power: ~ ~ \dot Q_r = 2.2e-69 ( {r_n} ~ \rho )^{3/2} ~ V^{20} Watts

Effective time:

Assume constant acceleration for the vehicle, v = a t , to a maximum velocity V = a T . If the drag power \dot Q = k v^n = k a^n t^n , then the time integrated power:

Q = k a^n T^{n+1} / (n+1) = k a^n T^n ( T/(n+1) ) = k V^n ( T/(n+1) ) = \dot Q_{max} ( T/(n+1) ) = \dot Q_{max} teff

defining teff = { T \over { n+1 } } = \large { V_{max} \over { a ( n+1) } }

There will also be additional exit or climb-out time for the launch loop, TBD.

Examples:

For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, and density at 80, 100, and 120 km:

altitude km

80

100

120

density kg/m3

1.846e-5

5.604e-7

2.222e-8

\dot Q_c W

\dot Q_r W

accel. time

374

total time

Q_c J

94

|| Q_r J || 18 || || || ||

HypervelocityDrag (last edited 2017-03-01 00:19:49 by KeithLofstrom)