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Parametrically determining the Performance of Anti-Surface Missiles through the Breguet Equation
Revised 20 Aug 2023
(Created 21 Jun 2011)
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References:
Subsonic and Supersonic Anti-Ship Missiles: An
Effectiveness and Utility Comparison
[January 1997 Naval
Engineers Journal (LINK) ]
Brief Description
You can parametrically determine the major performance characteristics of interest for anti-surface missiles such as land attack cruise missiles (BGM-109 Tomahawk, V-1 Buzz Bomb) or anti-tank missiles (BGM-71 TOW) with the Breguet equation, because these aerodynamic vehicles fly flight profiles which are tuned for maximum range, since their targets are either immobile or moving so slowly (15.433 m/sec = 30 knots) that the attacking weapon can use the small amount of remaining specific excess energy they have to successfully attack their targets.
The Breguet Equation
R = 2.3 (L/D) (V/C)
(LOG10(w1/w2))
Where:
R: Range of Missile or Air Vehicle (Nautical
Miles)
L/D: Lift to Drag Ratio of Missile or Air
Vehicle.
V:
Velocity of the Missile or Air Vehicle (knots)
C: Specific
fuel consumption (SFC) rate in lb/lb/hr.
w1:
Launch/Takeoff
Weight of Missile or Air Vehicle
w2:
Fuel
Exhaustion Weight of Missile or Air Vehicle.
EXAMPLE
A typical subsonic turbojet anti-ship missile has the following
characteristics:
L/D: 2.5
V: 530 knots (0.8 Mach)
C:
1.5 lb/lb/hr
w1/w2: 1.13
It's range would then be:
2.3 (2.5) (530/1.5)
(LOG10(1.13)) = 107.8377~ nm
A typical supersonic liquid fuelled ramjet powered anti-ship missile built with the same structural technology as the prior subsonic missile has the following
characteristics:
L/D: 0.5
V: 1322 knots (2.0 Mach)
C:
3 lb/lb/hr
w1/w2: 1.13
NOTE: Supersonic Ramjet missile designs (as of mid 1990s) in the 300 to 900 kg weight class can achieve a w1/w2 ratio of about 1.3 with the best technology available at the time.
Advanced Use of the Equation
You could split up an aero vehicle's flight trajectory into
multiple phases and then calculate the range for each phase, such as
a Lo-Hi-Lo profile to perform more detailed analysis.
Notes on Specific Technologies
Propulsion TSFCs
18.67~ TSFC: 185~ ISP - Me 163 T-Stoff/C-Stoff Motor
17.47~ TSFC: 206.5~ ISP (Double Base) Solid Motor (1950s Sidewinder-9B)
15.78~ TSFC: 228.6~ ISP (Composite HTPB/AP) Solid Motor (1980s HARM)
3.6~ TSFC: 1,000~ ISP (Boron/HPTB) Solid Fuel Ramjet (Theoretical)
2.9~ TSFC: Simple Pulsejet (1940s V-1 Buzz Bomb, etc)
1.2~ TSFC: Expendable Turbojet (1970s J402-CA-400, Harpoon)
1.12~ TSFC: Reusable Turbojet (1940s J33)
0.77~ TSFC: Reusable Turbojet (1950s J57)
0.68~ TSFC: Expendable Turbofan using JP-5 Jet Fuel (Early 1980s F107-WR-400, Tomahawk) [30 kg mass, 276.6 kgf thrust]
0.486 TSFC: Reusable Turbofan (2000s FJ33 for Business Jets)
0.37~ TSFC: Reusable Turbofan (1970s TF34-GE-100, A-10A)
0.34~ TSFC: Expendable Turbofan (Late 1980s F121, AGM-136 Tacit Rainbow) [22.2 kg mass, 31.75 kgf thrust] (LINK TO PDF TESTING IT)
0.25~ TSFC: Expendable Turbofan using JP-10 Boron Slurry (1990s F112-WR-100, AGM-129) [73 kg mass, 332 kgf thrust]
Lift/Drag Ratios for Tactical Missiles
According to Koorosh Goudarzi & Mehdi Jamali (2016): Study of Subsonic Flow over a
TOW 2B Missile; the TOW missile has the following approximate L/D Ratios:
AOA 1°: 0.44 L/D
AOA 4°: 1.26 L/D
AOA 8°: 1.93 L/D
This aligns well with Chun-Chi Li & Wei-Chan Hong (2017): A Study of Aerodynamic Characteristics of an Anti-tank Missile's summation (LINK) that an eight-fin ATGM (the TOW is a 4-fin design) can achieve a L/D of approximately 3.0 at an AOA of 5°.
It appears the true limitation of ATGM range is the guidance/homing technology of the era; if you're limited to wire-guidance, you have to stay at shallow angles of attack, instead of flying an optimized lofted flight profile, to avoid breaking the wires, explaining the TOW's relatively low 0.9 L/D ratio when aerodynamically the missile can do much better.
The following list is not exhaustive but will provide you with the knowledge of basic missile configurations and their L/D ratios:
Early 1970s 4-Fin ATGM (Image) - 1.93 (BGM-71 TOW)
Pop-Out Simple Wings, 283+ lb/ft2 loading (Image) - 2.425 (TLAM)
Advanced 2000s 8-Fin ATGM (Image) - 3.0 (Conceptual)
Fixed Simple Wings, 86+ lb/ft2 loading (Image) - 4.6 (V-1 Buzz Bomb)
Fixed Complex Wings, 131.4+ lb/ft2 loading (Image) - 9 (SM-62 Snark)
w1/w2 Weight Ratios
1.098 (8.96% of Gross Mass as Fuel) (BGM-71 TOW ATGM) (27.1% Gross Mass in Warhead)
1.100 (9.12% of Gross Mass as Fuel) (A/R/UGM-84 Harpoon) (42% Gross Mass in Warhead)
1.137 (12.06% of Gross Mass as Fuel) (IM-99A BOMARC)
1.217 (17.84% of Gross Mass as Fuel) (AIM-7 AAM) (17.5% Gross Mass in Warhead)
1.287 (22.29% of Gross Mass as Fuel) (Fi-103/V-1 Buzz Bomb) (39.5% Gross Mass in Warhead)
1.333 (25% of Gross Mass as Fuel) (AIM-9B AAM) (6.3% Gross Mass in Warhead)
1.425 (29.82% of Gross Mass as Fuel) (BGM-109C/D Conv. Tomahawk) (34.4% Gross Mass in Warhead)
1.852 (46% of Gross Mass as Fuel) (SM-62 Snark with 6230 lb [2825.8 kg] MK39 Warhead and no Drop Tanks)
2.025 (50.61% of Gross Mass as Fuel) (BGM-109A/G Nuke Tomahawk) (10% Gross Mass in Warhead)
2.274 (56% of Gross Mass as Fuel) (SM-62 Snark with 290 lb [130 kg] W80 Warhead and no Drop Tanks) (Hypothetical)
NOTE: The Snark W80 hypothetical was calculated using known warhead volumes of 1.679 m3 for the MK39 and 0.0565 m3 for the W80, leaving 1.6225 m3 of volume available for fuel. At 775 kg/m3 for Jet Fuel, this gives us 1257.4 kg (2,772 lb) of extra fuel available within the Snark Outer Mold Line.