Why does this calculation show Gustave Whitehead's propellers were more than 100% efficient?

I have tried to calculate the efficiency of Gustave Whitehead's propellers using the information he provided to the American Inventor Magazine in 1902 (see the citation below).

Number of propellers = 2

Engine power per propeller, $$P$$ = 20 hp

Propeller diameter, $$D$$ = 6 ft

Static thrust per propeller, $$T$$ = 254 lbf

Propeller efficiency, $$\eta_{prop}$$

Gear efficiency, $$\eta_{gears}$$ < 100%

Air density, $$\rho=1.2 \frac{kg}{m^3}$$

Doing the calculations: $$\eta_{prop} > \sqrt{\frac{T^3}{(P^2 \cdot \pi \cdot \eta_{gears} \cdot \frac{D^2}{2} \cdot \rho )}} = 101.4\%$$ which is impossible.

Is there a mistake? If yes, where?

Has the End been Finally Attained, and is the Dirigible Balloon to Go?

Editor, American Inventor

Dear Sir: Replying to your recent letter, I take pleasure in sending you the following description of my flying machine No. 22, the latest that I have constructed:

This machine was built in four months with the aid of 14 skilled mechanics and cost about $1,700 to build. It is run by a 40 horse-power kerosene motor of my own design, especially constructed for strength, power and lightness, weighing but 120 pounds complete. It will run for a week at a time if required, without running hot, stopping, or in any possible manner troubling the operator. No electrical apparatus is required for ignition purposes. Ignition is accomplished by its own heat and compression; it runs about 800 revolutions per minute, has five cylinders and no fly-wheel is used. It requires a space 10 inches wide, 4 feet long and 10 inches high. ... The propellers are 6 feet in diameter and have a projecting blade-surface of 4 square feet each. They are made of wood and are covered with very thin aluminum sheeting. The propellers run about 600 revolutions per minute under full power and turn in opposite directions. When running at full speed they will exert a thrust of 508 pounds. I measured this thrust by attaching the machine to a post by means of a dynamometer and running the engines at full speed. ... I have no photographs taken yet of No. 22, but send you some of No. 21, as these machines are exactly alike, except the details mentioned. No. 21 has made four trips, the longest one and a half miles, on August 14, 1901. The wings of both machines measure 30 feet from tip to tip, and the length of the entire machine is 32 feet. It will run on the ground 50 miles an hour, and in air travel at about 70 miles. I believe that if wanted it would fly 100 miles an hour. The power carried is considerably more than necessary. ... Trusting this will interest your readers, I remain, Very truly yours, GUSTAVE WHITEHEAD Bridgeport, Conn. • The value you use for air density is probably too low. Whitehead is reporting on an actual experiment probably performed in cold weather. Also he makes no mention how he measured P. or indeed if he did. It is possible he just worked back from the thrust by assuming 100% efficiency. Also you did note the letter is dated April 1? Oct 28, 2015 at 4:17 • 1) Even if I use the air density for -25 C, 1.42 kg/m^3, I still get an efficiency of 93.2% which is too big. 2) Whitehead says in the article that he was the one who designed the engine. If he was capable to design such a motor it is unbelievable he was incapable to determine its power. In order to have worked back the power from the thrust he must have known the formula I mentioned and in this case he would not have used an efficiency of 100% as long as the propellers of the time were around 50% efficient. Oct 28, 2015 at 5:32 • Basically, I find it very hard to believe that the power of an experimental, self designed and largely built (?) engine would just happen to be a round number. And it does not actually say he meant shaft horsepower. In fact given that he designed the whole system, what reason would he have for reporting the power of engines separate from the transmission and propellers they were built for? Oct 28, 2015 at 6:21 • He could have meant effective power converted to thrust, I suppose. It would match the numbers pretty good and solve your puzzle. Oct 28, 2015 at 6:27 • Ville Niemi, It is hard to follow you. What do you mean by "effective power converted to thrust"? This seems equivalent to using an engine much powerful than 40 hp, around 80 hp, weighing just 120 pounds, in 1902?! If you have an explanation try to find a mathematical form for it. Oct 28, 2015 at 7:41 5 Answers The equation you referenced shows prop diameter squared, divided by 2, not 4. Even then, the calculation shows that marketing is to experimentation as about 2:1 :) • Streak, Yes you are right. It is D^2/2 not D^2/4. With your correction the efficiency gets lower, 101.4%, but it still stays above 1 which is not possible. I will correct the question. Thank you for your answer. Oct 28, 2015 at 3:23 As you present it, you are right and Whitehead's numbers are impossible. I doubt that he could measure engine power with enough precision, and who knows what definition for horsepower he used. Power measurement needs some heavy machinery, and Whitehead most likely did not own a suitable dynamometer. It would be interesting to know the fuel consumption of this engine - that could help to check his numbers better. • What I know is that a previous engine of Whitehead, used in 1901, delivered 30 HP and needed 60 pounds of fuel to run for 6 hours (see: The flight apparatus of Gustave Whitehead) page 6-7. Oct 29, 2015 at 6:46 • @RobertWerner: Thank you, that helps. A standard gasoline engine with carburetor consumes 280 g of fuel per KW and hour. 10 pounds for 30 HP are 202 g per KW - either Gustave did not run the engine on full power, or his consumption figure is unbelievably low. Now we have the opposite situation where the power output is too high for the given conditions. Could it be that Gustave rounded the numbers to his advantage? Oct 29, 2015 at 10:50 • @RobertWerner: Now I read that these figures are for the calcium carbide-acetylene engine, so a direct comparison to a gasoline engine is not possible. Oct 29, 2015 at 11:07 • Heat of combustion for acetylene = 49.9 MJ/kg, for gasoline = 47 MJ/kg (see: Heat of combustion). The difference is small. Oct 29, 2015 at 19:20 Example of two ~static thrust tests 1) Engine: C-85-12 Continental (85hp, redline 2575rpm) Propeller: McCauley 71×46 Met-L, aluminum (This is a climb prop for a 120) Wind: 4-9mph headwind Performance -- Thrust: 340 pounds, RPM: 2350 Propeller efficiency: ~ 37.5% 2) Engine: Continental O-200, 100hp, 2750rpm redline Propeller: McCauley Clip Tip 68″ diameter, aluminum, standard pitch Wind: 5-7mph headwind Performance -- Thrust: 335 pounds, RPM: 2332 Propeller efficiency: ~ 32.5% Note: The efficiency was calculated using the formula:$\eta_{prop} = \sqrt{\frac{T^3}{(P^2 \cdot \pi \cdot \eta_{gears} \cdot \frac{D^2}{2} \cdot \rho )}}$Conclusion: If the formula for$\eta_{prop}\$ is reliable it seems that a modern propeller has a rather poor efficiency during a static thrust measurement. Gustave Whitehead might have optimized his aerial screws for supplying maximum static thrust possible but even in this case he could not have obtained an efficiency above 90%.

Whitehead's claims for his engine are difficult to believe. According to his description it was a five-cylinder Diesel (compression-ignition) running on kerosene. It would have needed a high compression ratio with fuel injection. That in turn means strong, heavy cylinders. It makes interesting comparison with the successful spark-ignition Antoinette V8 of a few years later:

• Length: Whitehead 48 in, Antoinette 44 in
• Width: Whitehead 10 in, Antoinette 25 in
• Height: Whitehead 10 in, Antoinette 21 in
• RPM: Whitehead 800, Antoinette 1,100

It is clear that Whitehead's cylinders could not have been significantly bigger than the 1.0 L of the Antoinette's, and there were only 5 instead of 8 of them. And his engine revolved more slowly. Yet he claims 80% of the power output. This is perfectly possible if his compression ratio is much higher. But here we come to the crunch:

• Weight: Whitehead 120 lb, Antoinette 209 lb.

As a crude comparison based on cylinder count, 5/8 of 209 = 130.6. So his claim implies a cylinder weight no greater than the Antoinette's, yet capable of withstanding far higher pressures.

Junkers later built two-stroke Diesels, developing high output per cylinder as the compression was not only high but you got a bang every revolution instead of every other. The 12-piston (in 6 cylinders!) Jumo 204 produced 590 hp at 1,600 rpm. At Whitehead's 800 rpm, half the 204's revs, that corresponds crudely to half the power, or around 300 hp. It weighed 1,653 lb giving, at 800 rpm, a mere 22 hp per 120 lb, half Whitehead's claim - and that not until 1929.

So either Whitehead was a total genius at Diesel engine design, with access to remarkably high-quality materials and advanced manufacturing technology, or somebody in the food chain has embellished the truth. He did earn his living at the time, and paid for his experiments, by making and selling aero engines, and in this he appears to have had a reputation for first-class workmanship, but even so....

I used the formulas from my answer to What is the correct formula to calculate propeller efficiency? and arrived at the following:

we have:

• $$\beta$$=0,
• $$P$$= 20 hp=14,914 W,
• $$D$$=6 ft=1.8±0.3 m,
• $$T$$= 254 pounds=1130 N,
• $$\rho=1.2 \rm\ kg/m^3$$, yielding

$$T_0=\left[\pi\times 1.2\times 14,914^2\times 1.8^2\right]^{1/3}\rm N=1,395±232\ N,$$ with a propeller efficiency of $$\eta_{\rm P}$$=(1,130/1,395)=81±13%, which seems to me to be a reasonable efficiency.

• Mar 15, 2021 at 15:29
• Arthur Tayler, welcome to aviation.stackexchange. I can't see the other answer you refer to, also not as deleted question (I have access to part of the moderator tools ). Something must have gone wrong when you submitted it. As it stands, this answer is a 'link-only' answer, which will likely be deleted by the community. Please include the calculation you did in your answer here (you can keep the link; it is fine as support). Mar 15, 2021 at 16:16
• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Mar 15, 2021 at 16:17
• I made the edit so that the part relevant to this question is included in here, the formula and every else remain in the original question. This way, 1) your work doesn't get lost, 2) your answer here will likely survive. If you disagree you may always rollback my edits. Mar 15, 2021 at 20:21
• What happens when an owner decides to delete their linked answer, totally within their right, but leaves the other one? Link-only applies to on- and off-site equally. See: meta.stackexchange.com/a/327882/392799 -- "Link-only answers don't stop being link-only answers if the link is local" -- maybe the owner doesn't delete the answer, but the asker deletes the whole post, again within their right. Keeping it simple is future-proofing.
– user14897
Mar 15, 2021 at 20:24