An additional note is necessary for technical readers of this article, regarding the drag computed for the Tomahawk cruise missile.
The question may reasonably be asked whether the guaranteed cruise thrust given by Williams for the F107R engine necessarily equals the thrust required to fly at that speed and atmospheric condition.
Our analysis has assumed that the cruise drag does equal the guaranteed engine thrust at the given condition by the engine manufacturer [0 ft altitude; M 0.7 flight speed]. But this assumption may not necessarily be valid. For instance, we see on the engine performance data from Williams, that a much higher thrust, 464 lbf is available at the same flight condition under full engine power, as seen in the chart.
We see here that the cruise thrust in yellow is 318 lbf, while the maximum continuous engine power at the same flight condition [0 ft altitude; M 0.7] produces 464 lbf of thrust [green].
It means the actual engine thrust required to push that T-Hawk through the air at the given flight condition could, conceivably, be somewhere between those two parameters. Now it is necessary to point out that at its cruising condition, the missile must have reserve engine power in store—in order to climb to avoid obstacles or rising terrain, and in order to turn, since turning flight requires greater engine thrust if airspeed in the turn is to be maintained.
[The parameter highlighted in red is the exhaust gas temperature at the respective engine power settings. We see that the lower EGT in cruise indicates lower engine rpm.]
It is therefore necessary to cross-check our assumption of cruise thrust being equal to the missile drag, by means of accurate data for the actual drag of the missile.
Such data is generally obtained only after actual flight testing, and has in fact been published. In Technology Advances in Cruise Missiles, published by the AIAA in 1981 and authored by B. J. Kuchta, an engineer with General Dynamics, the original developer and manufacturer of the Tomahawk, a drag polar is given.
As well, a comprehensive technical analysis of the T-Hawk was published in 1992, in the journal Global Security, authored by MIT physicists Theodore A. Postol and George N. Lewis, Long Range Nuclear Cruise Missiles and Stability.
In that paper the authors compute their own coefficient of drag estimate on the Tomahawk, based on a rigorous aerodynamic analysis known as a drag build-up technique.
Below is a graph from that paper [page 78] showing the results compared to the Kuchta data:
We see that the respective drag polar curves match very closely, with the Postol-Lewis estimate giving a slightly higher drag and hence slightly lower lift to drag ratio, in flight regimes up to a lift coefficient [CL] of about 0.7 in comparison to the Kuchta data.
Professor Postol has sent me two additional graphs based on plots of their drag computation from the 1992 paper:
The above plot shows the thrust required for the T-Hawk at flight speeds from just below M 0.2, to just above M 0.8—at a flight weight of 2,610 lbm, altitude of 1,000 ft, and air density of 0.0023 slugs per cubic foot [1.183 kg/m^3]. This air density indicates a temperature 3 degrees C above ISA standard.
We see from the plot that the missile drag [and therefore thrust required] at flight speed M 0.7 is about 350 lbf, about 30 lbf higher than the thrust required used in my analysis based on the published engine thrust data.
This plot is based on the coefficient of drag [CD] estimate computed by profs Lewis and Postol in their 1992 paper, and expressed in the standard form [page 78 from above paper]:
CD = 0.034 + 0.071 CL^2 [Equation 1]
Where the first term on the right-hand side is the zero lift drag coefficient [CD0] and the second term is the lift induced drag, which is a function of lift coefficient squared.
Using the Lewis-Postol data for missile weight and CD, and a flight speed of M 0.7, 0 ft altitude, and standard ISA conditions [as in our reference analysis], we arrive at the following drag numbers:
Flight Altitude: 0.0 ft
Temperature Above or Below ISA: 0 C
Flight Mach Number: 0.7 [463 KTAS; 533 mph; 782 ft/s]
Dynamic Pressure: 726 lbf/ft^2
Wingspan: 8.5 ft [2.59 m]
Wing Area: 12 ft^2 [1.11 m^2]
Aspect Ratio: 6.04
Lift Coefficient CL: 0.301
Drag Coefficient CD: 0.040
Lift to Drag Ratio L/D: 7.45
Total Drag: 351 lbf
Drag Area: 0.483 ft^2
This result gives a slightly higher drag than our assumed drag based on the published engine data. The resulting drag area is thus greater by about 10 percent, so our drag area of the Quds, which we scaled to the T-Hawk is also higher by a similar amount.
Assuming again that the Quds-1 has half the drag of the T-Hawk [instead of 0.43, as scaled linearly by fuselage section area], it means the Quds-1 would have a drag area of 0.24 ft^2, instead of 0.22 ft^2.
This assumption of drag, based on the Lewis-Postol drag computation does not change the end result significantly—the Quds-1 still has more than enough range to strike Abqaik from Houthi territory.
For instance, we note that the flight distance from the northern Houthi city of Saada to Buqayq, Saudi Arabia is actually 1,165 km. The Houthis control considerable territory to the north and east of Saada, a city of one million, and could thus launch missiles from even closer range, about 1,100 km.
Quds-1 Fuel Capacity
There are some additional aspects of the Quds analysis to consider, specifically with regards to the fuel capacity. Below is a cutaway illustration of the Tomahawk cruise missile, showing the fuel tank sections:
We see that the T-Hawk Fuel tank comprises three fuselage sections, fully 59 percent of the total missile length, 10.7 ft. Our initial assessment of the fuel tank length of the Quds-1 of 6.5 ft may be considerably less than its actual length.
At least part of the aft-body section on which the TJ-100 engine is mounted could in fact be part of the fuel tank. We will need to perform a more precise scaling from the Quds photos. The result could be more fuel capacity than originally estimated.
Additional Reports From Media
Military experts believe Iranian-made drones set off from Yemen to strike AramcoRai al-Youm daily, 17 Sept 2019
Former British diplomat Alastair Crooke has conveyed this report, which has been picked up by Colonel Pat Lang’s blog.
The military commanders in the USA are keeping silent with regards to the source of the attacks that targeted the Aramco oil facilities last Saturday and that caused some serious damages there. It seems that the Houthis have used Iranian-made modern drones and that a commando unit from Saudi land served to steer the drones at the last few minutes prior to the strikes.
The odd part consists of the silence of the American military commanders. Indeed, there has been no official report released by the Pentagon that accuses Iran. Furthermore, no senior military officers made any statements to accuse Iran.
The senior officers and military experts are actually cautious and not taking the risk of making any incorrect statements. They realize that the arsenal of the American radars at the Arabian Gulf including the land and the ship-mounted radars have recorded at least half the drones’ flight, which is enough data to tell the source.
I had noted in my original analysis below:
No actual expert in aerodynamics and jet propulsion could possibly arrive at such a ridiculous figure [700 km range for Quds-1]. It would require an engine fuel consumption of DOUBLE that of the actual TJ-100 engine.
In an earlier note to Prof Postol, I wrote:
I have to believe that serious technical people within the defense establishment are cognizant of the fact that the Quds has twice the range being cited by bogus experts, and that it did indeed come from Yemen.
Some Technical Notes on Drag Analysis
In Equation 1 we see the numbers arrived at by Profs Lewis and Postol for the Tomahawk.
CD = 0.034 + 0.071 CL^2 [Equation 1]
As noted already, this is the result of a drag build-up procedure that is widely used in industry by aerodynamicists to predict the drag of an aircraft or other flight vehicles, during the design process.
The techniques employed in industry are quite sophisticated [and often proprietary trade secrets] and can achieve results within 99 percent accuracy of the finished and flight tested aircraft.
The above numbers for the Tomahawk represent a computed figure for zero lift drag coefficient CD0, plus lift-induced drag coefficient CDi—thereby representing the entire drag coefficient of the vehicle, CD.
CD = CD0 + CDi
We note that CDi is the total lift-induced drag coefficient, and includes additional parasite drag due to lift [such as the extra profile drag on both wing and fuselage that results from an aircraft pitched up to a high angle of attack at high lift, thus presenting a greater sectional area to the airstream].
The lift-induced drag coefficient CDi can be further broken down into its constituent components and the above equation can be stated as:
CD = CD0 + CL^2 / π / e / AR
Where CL is the lift coefficient
e is the Oswald Efficiency Factor, which takes into account the extra parasite drag due to lift.
And AR is the wing aspect ratio, which is the ratio of the wing span squared, divided by wing area.
Since we know the aspect ratio of 6, the value of pi, and the value of the second constant in Equation 1, we can solve for the Oswald efficiency factor that is implicit in the constant in the second term of equation 1.
Setting the CL to unity, the Oswald efficiency factor is then:
e = 1 / 3.14 / 6 / 0.071 = 0.742
Where 0.071 is the constant in the second term of equation 1.
Looking at these results of the Lewis and Postol drag build-up, as expressed in Equation 1, we see that the zero lift-drag coefficient of 0.034 as well as the Oswald factor implicit in the constant of 0.071 appear to be in close proximity to what flight tests of the Tomahawk-type flight vehicle would be likely to confirm.