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| . 1a: $ \dot Q_c = 20 \rho^{1/2} \left( V \over 1000 \right)^3$ Btu/ft^2^-s . 1b: $ \dot Q_r = 6.1 \rho^{3/2} \left( V \over { 10 000 } \right)^{20} $ Btu/ft^2^-s |
. 1a convective power: $ ~ ~ \dot Q_c = 20 \rho^{1/2} \left( V \over 1000 \right)^3$ Btu/ft^2^-s . 1b radiative power: $ ~ ~ \dot Q_r = 6.1 \rho^{3/2} \left( V \over { 10 000 } \right)^{20} $ Btu/ft^2^-s |
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| . 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. | . Equations from [[ http://arc.aiaa.org/doi/abs/10.2514/6.1964-1313 | 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)'' |
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| . Metric 1a: $ \dot Q_c = $3.53e-4$ \rho^{1/2} V^3 $ | . 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 |
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| . Metric 1b: $ \dot Q_r = $3.32e-71$ \rho^{3/2} V^{20} $ | 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, [[ http://hdl.handle.net/2060/19930084817 | 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/m^3^ || || 1.846e-5 || 5.604e-7 || 2.222e-8 || || $ \dot Q_c $ W || || || || || || $ \dot Q_r $ W || || || || || || ||$t_{eff}|| || $ 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: ~ ~ \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 |
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80 |
100 |
120 |
density kg/m3 |
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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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$t_{eff} |
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Q_c J |
94 |
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Q_r J |
18 |
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