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| '''Examples:''' | |
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| For a 1 meter diameter nose, V=11 km/s and density at 80, 100, and 120 km: | ==== Effective time: ==== |
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| || altitude km || 80 || 100 || 120 || || density kg/m^3^ || 1.846e-5 || 5.604e-7 || 2.222e-8 || || $ \dot Q_c $ W || || || || || $ \dot Q_c $ W || || || || |
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 } } = { 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/m^3^ || || 1.846e-5 || 5.604e-7 || 2.222e-8 || || $ \dot Q_c $ W || || || || || || $ \dot Q_r $ W || || || || || || accel. time || 374 ||<-3> total time || || $ Q_c $ J || 94 || || || || || $ Q_r $ J || 18 || || || || |
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:
Qc Btu/ft2-s
=20
1
2
V1000
3 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 } } = { 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 |
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|
|
\dot Q_r W |
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accel. time |
374 |
total time |
||
Q_c J |
94 |
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|| Q_r J || 18 || || || ||