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| 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: |
Assume constant acceleration for the vehicle, $ v = a t $, to a maximum velocity $ V = a T $. |
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| $ 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 $ | define $ t_{eff} = {\Large { T \over { n+1 } } } = { \Large { V \over { a ( n+1) } } } $ |
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| defining $ teff = { T \over { n+1 } } = \large { V_{max} \over { a ( n+1) } } $ | If the drag power $ \dot Q = k v^n = k a^n t^n $, then the time integrated power: |
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| There will also be additional exit or climb-out time for the launch loop, TBD. | $ Q=k a^n{\Large {T^{n+1}\over {n+1}}}=k a^n T^n{\Large {T\over{n+1}}} = k V^n t_{eff} = \dot Q_{max} t_{eff} $ There will also be additional exit or climb-out time for the launch loop added to $ t_eff $, TBD. |
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| For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, and density at 80, 100, and 120 km: | For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, and density at 80, 100, and 120 km: |
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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_r $ W || || || || || || accel. time || 374 ||<-3> total time || || $ Q_c $ J || 94 || || || || || $ Q_r $ J || 18 || || || || |
|| altitude km || || 80 || 100 || 120 || || density kg/m^3^ || || 1.85e-5 || 5.60e-7 || 2.22e-8 || || $ \dot Q_c $ W || || 3.49E-6 || 6.08E+5 || 1.21E+5 || || $ \dot Q_r $ W || || 1.17E+5 || 6.21E+2 || 4.90E+0 || || ||$t_{eff}$|| || $ Q_c $ J || 94 || 3.26E+8 || 5.69E+7 || 1.13E+7 || exponent n = 3 || || $ Q_r $ J || 18 || 2.09E+6 || 1.11E+4 || 8.73E+1 || exponent n = 20 || || $ Q_total $ J || || 3.28E+8 || 5.69E+7 || 1.13E+7 || || loss fraction || || 1.09E-3 || 1.88E-4 || 3.74E-5 || |
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 .
define t_{eff} = {\Large { T \over { n+1 } } } = { \Large { V \over { a ( n+1) } } }
If the drag power \dot Q = k v^n = k a^n t^n , then the time integrated power:
Q=k a^n{\Large {T^{n+1}\over {n+1}}}=k a^n T^n{\Large {T\over{n+1}}} = k V^n t_{eff} = \dot Q_{max} t_{eff}
There will also be additional exit or climb-out time for the launch loop added to t_eff , TBD.
Examples:
For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, and density at 80, 100, and 120 km:
altitude km |
|
80 |
100 |
120 |
|
density kg/m3 |
|
1.85e-5 |
5.60e-7 |
2.22e-8 |
|
\dot Q_c W |
|
3.49E-6 |
6.08E+5 |
1.21E+5 |
|
\dot Q_r W |
|
1.17E+5 |
6.21E+2 |
4.90E+0 |
|
|
t_{eff} |
||||
Q_c J |
94 |
3.26E+8 |
5.69E+7 |
1.13E+7 |
exponent n = 3 |
Q_r J |
18 |
2.09E+6 |
1.11E+4 |
8.73E+1 |
exponent n = 20 |
Q_total J |
|
3.28E+8 |
5.69E+7 |
1.13E+7 |
|
loss fraction |
|
1.09E-3 |
1.88E-4 |
3.74E-5 |