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=== Drag on an accelerating launch loop vehicle with a hemispherical nose ===

I am not an aeronautical engineer and probably misunderstand the sources. In any case, the numbers are approximate, and should be treated skeptically. The drag and heating is acceptable at all altitudes; however, incoming debris impactors in decaying orbits will be more unpredictable when the drag is higher, and more difficult to shield or dodge. However, perhaps most may be intercepted a few orbit earlier, reducing flux at launch loop track level.
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'''Equations on Page 5 in Foot-second-slug-BTU :''' ==== Equations on Page 5 in Foot-second-slug-BTU : ====
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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 = 3.05 \rho^{1/2} V^3 $
 
 . Metric 1b: $ \dot Q_r = 3.32e-71 rho^{3/2} V^{20} $
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'''Metric equations: ''' ==== 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, [[ 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. This additional time will be proportionally larger for the radiation fraction, but that will remain small, especially in thinner, higher altitude atmosphere.

The drag losses are much higher; most of the lost energy ends up heating the upper atmosphere (where it radiates efficiently into space, not to the ground). The drag power is $ P = C_D \rho Area V^3 $ and the drag loss is $ P = C_D \rho Area V^3 T/4 $

==== Examples: ====

For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, C,,D,, = 2.0 and density at 80, 100, and 120 km:

|| 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 ||
||<-4> ||$t_{eff}$|| exponent ||
|| $ Q_c $ J || 3.26e+8 || 5.69e+7 || 1.13e+7 || 94 || 3 ||
|| $ Q_r $ J || 2.09e+6 || 1.11e+4 || 8.73e+1 || 18 || 20 ||
|| $ Q_{total}$ J || 3.28e+8 || 5.69e+7 || 1.13e+7 ||
|| heat fraction || 1.09e-3 || 1.88e-4 || 3.74e-5 ||
|| drag loss J || 1.44e+10 || 4.38e+8 || 1.74e+7 ||
|| drag fraction || 4.77e-2 || 1.45e-3 || 5.75e-5 ||
|| heat/drag || 0.023 || 0.130 || 0.651 ||

Hypervelocity Drag

Drag on an accelerating launch loop vehicle with a hemispherical nose

I am not an aeronautical engineer and probably misunderstand the sources. In any case, the numbers are approximate, and should be treated skeptically. The drag and heating is acceptable at all altitudes; however, incoming debris impactors in decaying orbits will be more unpredictable when the drag is higher, and more difficult to shield or dodge. However, perhaps most may be intercepted a few orbit earlier, reducing flux at launch loop track level.

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. This additional time will be proportionally larger for the radiation fraction, but that will remain small, especially in thinner, higher altitude atmosphere.

The drag losses are much higher; most of the lost energy ends up heating the upper atmosphere (where it radiates efficiently into space, not to the ground). The drag power is P = C_D \rho Area V^3 and the drag loss is P = C_D \rho Area V^3 T/4

Examples:

For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, CD = 2.0 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}

exponent

Q_c J

3.26e+8

5.69e+7

1.13e+7

94

3

Q_r J

2.09e+6

1.11e+4

8.73e+1

18

20

Q_{total} J

3.28e+8

5.69e+7

1.13e+7

heat fraction

1.09e-3

1.88e-4

3.74e-5

drag loss J

1.44e+10

4.38e+8

1.74e+7

drag fraction

4.77e-2

1.45e-3

5.75e-5

heat/drag

0.023

0.130

0.651

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