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

Question

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?

Note: I used the formula that relates static thrust to efficiency, power and propeller diameter.

The Whitehead Flying Machine

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.

Source: The American Inventor Magazine , 1 April 1902 .

Answer 1

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 :)

Answer 2

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.

Answer 3

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%

see: Thrust testing 85 and 100 hp engines

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%.

Answer 4

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....

Answer 5

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.