### Buoyancy-Differential Propulsion

Solomon Andrews was the first person to propel a manned airship without an engine, by exploiting the action of a net buoyant force on an inclined airship. Andrews built and flew two such craft.

The first (Aereon I) had three hulls, and was flown on four dates in 1863. It demonstrated the ability to make headway against the wind, although the speed estimates that were made for it are open to question. It was this ship that popped up in A. Bertram Chandler’s alternate history of Australia, Kelly Country (1985), as the breakthrough weapon of the Australian rebels led by Ned Kelly.

Andrews’ 1866 airship (Aereon II) was a monohull of at least double the lift, and it was flown twice. It experienced difficulty making headway against the wind.

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When a spherical balloon drops ballast or vents gas, it ascends or descends vertically. Andrews’ proposal was to give a lighter-than-air craft a “flattened and elongated shape” and shift weight (pilot or ballast) so as to incline it. He predicted that it would then travel in the direction the nose was pointing (this is almost correct), which would be the path of least resistance.

Andrews controlled the pitch (the angle of the airship’s longitudinal axis makes with the horizontal) by moving forward and backward, achieving an pitch of as much as 15 degrees.

Dropping ballast to create positive net buoyancy and bringing the nose-up, the Aereon will glide upward until it reaches an altitude at which the air density is low enough so that the static lift is reduced to zero and the frictional and profile drag forces decelerate it to a stop (upon which it will behave like a free balloon). By venting gas to create negative net buoyancy and bringing the nose down, the Aereon will glide downward until the air density is high enough so that the net buoyancy is brought back to zero. The process (a “yo-yo”) can then be repeated as needed; the Aereon thus progresses by virtue of a cyclic variation in net buoyancy. In still air, the airship will follow an essentially sawtooth path in the vertical plane.

While Andrews did not work further on airships, his “buoyancy engine” is equally applicable to movement through water. “Underwater gliders” using the basic concept (but typically in winged form) have been in active oceanographic use for several decades, and the underlying hydrodynamic theory is well established. More recently, there has been preliminary work on unmanned aerial vehicles with the same propulsion concept; the underlying aerodynamic principles are analogous. (See Notes).

### Analysis of the Motion of the Aereon

In still air, the Aereon is acted upon by the following forces: gravity, buoyancy (static lift), dynamic lift (hereafter, just lift) and drag. Since both gravity and buoyancy act vertically, it’s convenient to combine them into net buoyancy (buoyant force minus gravitational force), which may be positive or negative. Lift is perpendicular to the line of flight and drag is backward along the line of flight.

In steady (constant speed) flight, the forces must be in balance; the vertical components must add up to zero and the horizontal ones too. (The flight will actually be quasi-steady state because the Aereon isn’t propelled unless it’s ascending or descending, and air density and thus equilibrium speed changes with density. The density changes are slow, however; air at 100 feet is 99.71% the density at sea level.)

The net buoyancy (in newtons) is the ballast mass dropped or the lift lost by venting gas, in kilograms, multiplied by the gravitational acceleration, 9.8 meters/second. Roughly speaking, dropping nine pounds of ballast produces 40 newtons of net buoyant force, upward. Venting 126.4 cubic feet of hydrogen produces the same force, downward.

The propulsive force provided by the “buoyancy engine” is the component parallel to the glidepath; multiply the net buoyancy by the sine of the glidepath angle (Krawetz).

The glidepath angle (the angle that the direction of motion makes with the horizontal) will be close but not quite equal to the pitch. Why? Because the net buoyancy also has a component (net buoyancy * cosine of glidepath angle) perpendicular to the glidepath, thus altering the direction of motion (contrary to the steady flight assumption) unless it is opposed by some other force. And plainly, that other force must be lift.

There are essentially two ways of generating lift. First, a symmetrical airfoil—like the bare hull of an airship, or even a flat plate—will generate lift (positive or negative) if it is inclined to the airflow, i.e., it is at a positive or negative angle of attack. Second, you can have camber, an asymmetry between the top and bottom surface, as on a conventional airplane wing. If so, you will generate lift at a zero angle of attack.

The Aereon is essentially a symmetrical airfoil. When ascending (positive glidepath angle) at constant speed, it must have a negative angle of attack so it generates negative lift—balancing the perpendicular component of the net buoyancy. And to have a negative angle of attack, its pitch must be a bit less than the glidepath angle. The pitch will still be positive, however. The reverse, of course, is true when the Aereon is descending at constant speed.

The drag experienced by a symmetrical airfoil increases as the angle of attack deviates from zero; aerodynamicists calculate total drag as being the sum of the “zero lift drag” and the “lift-induced drag.”

The pilot determines the net buoyancy and the pitch angle, and the direction of motion and airspeed change until the forces are in balance. The drag must equal the propulsive force and the lift must equal the perpendicular component of the net buoyancy.

Please remember that the resulting airspeed is the airspeed along the glide path, which is itself inclined to the ground. To obtain the horizontal speed, you must multiply the resulting glide speed by the cosine of the glide path angle.

As the glidepath angle (for ascent) is decreased, the angle of attack becomes more negative, and the pitch angle decreases. There is a minimum glidepath angle that is determined by the angle of attack at which the lift/total drag ratio is maximized. Also, there is a limit to how negative (for ascent) or positive (for descent) the angle of attack can be; when it’s too far from zero, there is flow separation (“stall”) and the lift (negative or positive) rapidly returns to zero.

It is possible to make some generalizations about performance if a simpler aerodynamic model is used (Graver; Purandare). If the total drag coefficient is assumed to be a constant (thus, independent of air speed), and the lift coefficient to be proportional to angle of attack (per “thin airfoil” theory), and thus the lift-induced drag coefficient to be proportional to the squared angle of attack, then

1) any two of the glidepath angle (G), angle of attack, and pitch determine the third, regardless of air speed;

2) the air speed is dependent just on the net buoyancy; and

3) the horizontal component of the airspeed is proportional to (Graver 234ff)

cos G * (sin G)^{0.5},

this has a maximum at a glidepath angle of about 35.26^{o}; the required lift/drag ratio is 1.4.

However, with a shallower glidepath angle (for which a higher lift/drag ratio is required), the airship will take longer to reach an altitude at which it has to change its net buoyancy once again; this might be advantageous on recon missions, for which time aloft is important.

A pitch of 15^{o} was the highest one Andrews recommended in his patent, and the relative dimensions of his car and envelopes suggests that it was the maximum achievable by the Aereon I (see Notes). In general, with my aerodynamic model, a 15^{o} pitch corresponds to a glidepath angle of ~20^{o} (glide ratio—horizontal/vertical distance—of 2.74:1). A lower pitch would result in a lower glidepath angle and thus a higher glide ratio.

Buoyancy-driven airships in the new universe of course can be designed to permit a greater pitch and therefore a greater glidepath angle (and speed). A closed control car would be advisable if higher pitch operations are contemplated. Airships can be designed to fly at high pitch safely; FAA requires that an airship be able to recover from 30^{o} nose down or nose up, and the USS Los Angeles survived a wind-driven inclination to 85^{o}.

### Performance Calculations

I have used standard aircraft/airship preliminary design methods (see Appendix 3) to calculate the performance of the Aereon with various combinations of ballast drop and pitch. (For the aerodynamic model and its accuracy, see Appendix 3 and the Notes; for comparison with eyewitness accounts, see Appendix 4; for the effect of model variations, see Appendix 5.)

^{ It appears from my calculations that the Aereon I can’t achieve a glidepath angle much lower than about 15}^{ degrees (glide ratio 3.66), regardless of its pitch.}

As it obliquely ascends after a ballast drop, the Aereon‘s speed will decline (see Appendix 5). Likewise, its speed will decline as it obliquely descends after venting gas.

Naturally, it is not very safe for 0 feet to be the “base” altitude. I would imagine that in routine use, the Aereon would first ascend like an ordinary balloon to a “lower cruising altitude” of say 100 feet, and then yo-yo between that altitude and an “upper cruising altitude” of say 1200 feet.

In landing, if the pilot thought the rate of descent was too high, he could adopt a shallower pitch, thus reducing air speed; if the pitch were zero, the airship would descend vertically like a balloon, and the large planform area would make the descent quite slow.

As on a balloon, the pilot could toss out handfuls of ballast to reduce speed, and it might be helpful to use a drag rope.

****

An interesting question is whether it’s better to drop a lot of ballast at one time, or make several smaller drops that add up to the same weight. Let’s say we dropped 240 pounds instead of 60, and set pitch to 35^{o}. The speed at 0 feet would be 30.48 mph, about twice the speed (14.59) achieved by a 60 pound drop at the same pitch. That’s just as we’d expect, since drag is roughly proportional to speed squared. The Aereon will rise to the altitude whose density is 86.47% that at 0 feet. This would be, approximately, 4,883 feet. It’s thus rising 4.4 times as far. So it will also travel horizontally 4.4 times as far. But if we carried out four 60 pound drops, we would travel 4 times as far as with just one. If pitch is 15^{o}, the speed increases from 10.11 to 20.97 mph, but the altitude change is the same.

So, from a range standpoint, one big drop is superior to one small one, and of course, it’s faster too.

### Emergency Buoyancy-Driven Propulsion

It’s worth noting that an airship doesn’t need any special equipment in order to use buoyancy-driven propulsion. Thus, if a conventionally powered airship runs low on fuel, and there’s no wind, it might “crawl” to its destination by this method.—if it knows about it, and has enough ballast and lift gas. Even if a conventional airship is not low on fuel, it might want to use buoyancy-driven propulsion to close silently on a target when the air is still so it can’t “free balloon.”

Of course, the ballast drop must be in due proportion to the gross lift of the vessel. For the hydrogen-filled Aereon, 60 pounds is 3.21%.

### A Faster Aereon

Based on Andrews’ 1866 “First Aereon” illustration, I assumed that it lacked the “membrane” (webbing between the tapered ends) referred to in his patent. Putting in this membrane slightly worsens the area-volume ratio, thus frictional drag, and reduces speed.

Reducing the number of hulls substantially improves that ratio, and increases speed, as shown in Table 4-2.

A more modest improvement is obtained by replacing the pointed (tangent ogive) ends with rounded (ellipsoidal) ends. Please note that the shape of the ends is considered by the model only to the extent that it affects surface and planform area. The pointed end of the Aereon hulls would produce less pressure and wave drag in high speed flight. However, at the low subsonic speeds typical of airships, the rounded forebody is more aerodynamically efficient; pointed forebodies are used in rockets and supersonic fighters.

*60 pound drop at 0 feet; 15^{o} pitch. ** tentative calculation.

Andrews in fact contemplated a catamaran; his patent (Fig. 7) also depicts a hypothetical two-hulled “war-aerostat”, and says that the fineness ratio (length/diameter) is 1.5.

If Monohull #3 is engineered to be able to achieve a pitch of 35^{o}, then the airspeed would be 19.70 mph and horizontally 15.29 mph.

****

Of course, the Aereon is a rather small airship. In my database of historical airships, the only smaller ones were the Star (72′ x 47.9′) and the Santos Dumont (72′ x 28′). In general, bigger is better. If we increase the envelope volume, keeping the shape the same, we increase gross lift. We also increase structural mass, but that should be proportional to the envelope surface area for a nonrigid airship unless increases less rapidly. Thus, we are increasing useful lift. With more useful lift, we can carry more ballast, and with more gross lift, we can vent more gas as needed. We thus increase endurance (more cycles) or propulsive force per cycle.

The larger airship will experience more drag, but the frictional drag will scale with the surface area whereas the propulsive force available scales with the volume. So the achievable speed will be greater. Double the length and diameter of the Aereon hulls, and you have eight times the volume and four times the surface area; octuple the ballast drop, and you get 50% more air speed.

The table made the Aereon II look bad, but that’s because we kept the ballast drop the same even though the volume is four times that of the Aereon I. If the ballast drop were increased proportionately, the Aereon II‘s air speed would be 11.73 mph.

One catch is that Andrews method of pitch control wouldn’t scale up very well; for a larger ship, you would either need movable ballast and a pulley system for hauling it forward or aft (as Andrews suggested), or if the ballast were water, it could be pumped between a forward tank and an aft tank. Or the trimming could be carried out in the envelope directly, by inflation or deflation of a fore or aft internal air bag.

### Winged Aereon

What about wings? The advantage of wings is that they provide a much better lift/drag ratio than does a streamlined body, which means that you have a lower minimum glidepath angle. That’s good for endurance but not for speed. Note that all of the commonly used underwater gliders, and also the aerial designs of Krawetz, Purandare and Wu (Appendix 3), have wings. Note that the developed lift is proportional to speed squared.

A quick calculation shows that with wings of 90 foot span, 10 feet chord at root and 5 at tip, 20% thick, AR 12, and dropping 60 pounds, the Aereon can achieve a glidepath angle of 7.8^{o} with airspeed of 5.18 mph (5.14 horizontal). This isn’t necessarily the best wing dimension or placement, or the lowest possible glidepath angle. The angle of attack is -8.28^{o}, so pitch is slightly negative (-0.48^{o}).

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With wings, you have two basic choices: a monohull with “outboard” wings, and a multihull with “inboard” (between the hulls) wings. A few “twin fuselage” aircraft, with a partially inboard wing, have been built, notably the F-82 Twin Mustang.

If you are willing to accept the disadvantages of a second hull (higher surface area to volume ratio, therefore higher construction cost and more zero-lift drag), the inboard wing has both structural and aerodynamic advantages. Structurally, since the wing is suspended between two hulls, there is less bending moment for a given span and the wing may be more lightly built than if it were cantilevered out from a single fuselage. Also, the fuselages are presumably half as long and so they have greater stiffness.

Secondly, wind tunnel experiments suggest that wingtip tanks on conventional aircraft increase effective aspect ratio slightly and thus reduce lift-induced drag, if they are attached so as to increase the span. (Hoerner 7-7).

Preliminary model testing has shown that addition of airship-like bodies to a low aspect ratio wing (forming, in effect, a “catamaran” airship) eliminates spanwise flow over the wing, increasing its lifting efficiency by about 20%, and also reducing lift-induced drag (Spearman; Orr). ( “Low” is relative; Orr’s wing was AR 2, whereas the effective AR of the Aereon was 0.56, and with two hulls it would have been lower.) Whether this benefit would be seen in a full-scale airship remains to be determined.

### The Problem of Buoyancy Control

Purandare points out that an airship can change its net buoyancy either by varying its weight or by varying its volume. We may think of the Aereon as a “mixed” operation device which decreases weight (dropping ballast) to ascend and decreases volume (venting lift gas) to descend.

The basic problem that to go forward, Andrews has to repeatedly drop ballast then vent gas. He has only a finite supply of each, so the endurance of the craft is very limited. The Andrews patent admits this.

Nor is that the only problem. While it can remain aloft as long as its envelope remains gas-tight, its range and endurance for controlled flight are limited. By my calculations (Appendix 6), with a 60 pound ballast drop at 15^{o} pitch, a fully inflated Aereon ascends about 1100 feet, with average vertical speed 1.31 fps, and this takes about 841 seconds. Its average air speed over the ascent is 5.53 mph, and the average horizontal speed is 7.6 fps (5.18 mph). So a 60 pound drop carries it horizontally 6393 feet (1.2 miles). And by venting the equivalent amount of lift gas, it goes forward another 1.2 miles.

That certainly gives it significant advantages for say battlefield surveillance over an ordinary observation balloon, which is completely at the mercy of the wind. Especially if the air is still. And for short-range stealth missions, it has the advantage of complete silence.

However, buoyancy propulsion remains inferior to engine propulsion. If the Aereon were equipped with a 30% efficient internal combustion engine and a 65% efficient propeller, and was traveling at that same 5.18 mph in level flight, by the same drag formulae it would consume about 0.0048 pounds M85 fuel (energy content of 24,000 KJ/kg) per mile or 0.0254 pounds/hour. So 60 pounds of fuel would carry it a very long distance!

If the Aereon instead takes off only 90% inflated, it will have an essentially constant speed of 9.49 mph (6.47 fps) horizontally and 2.38 fps vertically, up to its pressure altitude of 2,762 feet, which it will reach in about 30 minutes. In the same time it covers 4.75 miles horizontally. It can double this by venting the equivalent amount of lift gas. That’s better but still not competitive with a conventional engine.

Plainly, it’s better to use the available lift to carry fuel for a conventional engine than ballast for buoyancy propulsion operations, unless there’s a reason you can’t use an engine.

However, there are alternatives—using the weight not as ballast but as fuel to operate an improved buoyancy control system, as I’ll discuss in part 5, together with other engineless propulsion systems.

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To be continued . . .

Appendices 1-6 follow. The Notes, and the Bibliography for the whole Airship Propulsion series, will be posted to the Gazette Extras section of www.1632.org eventually.

### Appendix 1: Aereon I Design and Performance

Our best sources for the geometry of the first Aereon are Andrews’ U.S. Patent 43,449 (issued July 5, 1864) and his 1865 and 1866 books (available at the Library of Congress Rare Book Room). There were three envelopes, each 80 feet long, 13 feet in diameter, and made of linseed oil-varnished linen. These were cylindrical over the central 48 feet, and the fore- and aftbodies tapered symmetrically to a point. The three cylindroids were aligned in parallel (like a trimaran), but with sides touching (see figs. 5-6).

Each cylindroid had, on each side, a longitudinal strip of wood, 2.5 inches wide by 3/8 inch thick in the center and one inch wide at the ends. Where the strips of adjacent cylindroids touched, they were screwed together, and at the six ends the strips were attached to a cork nose cone, 5 inches in base diameter.

The cylindroids were also held together by a net of cotton twine, attached to the control car by 120 cords (4 rows of 30 apiece, spaced 20 inches apart), each running to one of four wooden “hoops” above the car. There were another 24 heavier cords that passed from these hoops under the car.

There is also reference to a flat two inch thick membrane of cambric muslin extending between the strips and filling up the otherwise empty space between them. However, this membrane is not apparent in the figure depicting the “first Aereon” in Andrews’ book and it may have been a post-flight afterthought.

The car, suspended 16 feet below the center envelope, was 12 feet long, with a gently curved bottom. Inside this car was an “inner car,” on runners, where the pilot sat, and the ballast was placed there, too. The weight of the cars was 58 pounds, and of the equipment in the cars, about 130 pounds. Since the car was attached to the envelopes by a network of cords, the inclination of the car caused a pitching of the envelopes.

The airship had a triangular rudder, made of cambric muslin stretched over a light “reed” frame, and of 17 sf. There is no reference to fins of any kind.

The Aereon said to have a gas volume of 26,000 cubic feet and to carry 600 pounds (Andrews1866, 5), which covered the aeronaut, the ballast, and, I believe, the cars and their accouterments. (I calculate that if it had tangent ogive ends, it would have a total volume of 26,221 cubic feet and, completely filled with hydrogen, a gross lift at sea level of 1,967 pounds.)

****

The Aereon I is said to have undergone four trials, in June-September 1863. However, the only newspaper account (New York Herald) was for its last trial, on Sept. 4, 1863, over Perth Amboy, New Jersey. In this trial, it carried Andrews (172 pounds) and 256 pounds of ballast.

According to one witness (Hamilton Fonda, factory foreman and former Ordnance Sergeant), it first flew north, against the wind, returned and landed. Then it took off again westward, returned, and landed. Witnesses asserted that it was able to make headway against a wind “blowing . . . not less than 10 mph” (Fonda), or even “10-15 mph” (architect Ellis White). Four other witnesses (merchant and ex-postmaster John Manning, constable and tax collector N.H. Tyrrell, Justice of the Peace Isaac Ward, and hotelkeeper James Allen) agreed that he had made headway against the wind, but without attempting to guess at the speed.

We don’t what altitudes it favored in this trial, but in the second (July) trial, it didn’t ascend above 200 feet (bank cashier SVR Patterson).

After these two manned flights, it was released unmanned, with a locked rudder and inner car, to spiral upward and destroy itself; its net buoyancy was that generated by relieving it of Andrews’ weight and “several 7-pound bags of ballast” (Fonda statement), amounting to “over 200 pounds” (Andrews patent). Presumably, the rest of the ballast had already been expended, since the goal was to give it maximum speed; Andrews asserted that it peaked at 200 mph but I very much doubt that (see Notes).

Andrews was not entirely satisfied with this design; on Aug. 26 he wrote to Lincoln that she was “too much of the clipper . . . She can only be balanced when she is full of gas . . . .” Note that balloons were usually inflated only to 50-75% capacity, as the gas would expand as the balloon gained altitude. Thus, the design limited the altitude range, and Andrews wanted the ship to be able to ascend to a height “out of reach of bullets.” (Whitman).

### Appendix 2: Aereon II Flight Record

The information concerning the Aereon II‘s geometry, other than that it was shaped like a “long lemon” (SciAm) or a “large fish” (NY Herald), is hopelessly contradictory (see Notes).

Aereon II was flown twice. On May 25, 1866, she flew with four aeronauts from its base at the corner of Houston and Green Streets in New York City, rising at an angle of “less than 45^{o}“, north to Harlem, then was driven by a southwest wind across the East River, passing over part of Blackwell’s Island and brought to a standstill by a contrary wind over Hunter’s Point. She was then put about, and flew “slowly” toward Ravenswood, landing near Astoria. Its maximum altitude was 2,000 feet. The rudder was found to be too small to hold the ship head to wind, and the car not long enough to permit the ship to be pitched to the angle desired. (NY Times, May 25, 1866; Scientific American, May 26 and June 2). Total time aloft was 25 minutes. (Niagara Falls Gazette, 12/24/1928).

For its June 5 flight (NY Times, June 7, 1866) the airship was given a larger rudder. It was inflated at 5:30 pm (NY Herald says it was already aloft at 5), and carried four hundred pounds ballast (Herald said 450) and two aeronauts. Its initial ascent was too slow relative to its horizontal movement, risking collision with a high building, and the copilot threw out an additional twenty five pounds of ballast, which allegedly gave the ship an “ascending power of thirty-nine pounds”, implying that the original takeoff was effectuated by dropping fourteen pounds. (Andrews must have also increased the pitch, as otherwise the additional “ascending power” would just have caused them to collide sooner.)

Further problems ensued. First the rudder tackle got entangled, and they lost rudder control, moving in circles in the general direction (S or SW toward Canal St.) the wind was blowing. They freed the tackle, threw out more ballast, and returned, heading up against a “gentle” wind. Then, when they reached an altitude of 1200–1500 feet, they lost the ability to control the inclination of the cylindroids, as a result of envelope expansion against a control band, and they were at the mercy of the winds, blowing toward NE. After 20 minutes, they had risen to 6000 feet, and were over the Long Island Sound.

They vented gas and, dropping to 4000 feet, regained inclination control. They turned and “slowly” descended, ascended, and descended again. The wording of the article is ambiguous but it sounds as though this was 8 miles of controlled flight. If wind direction hadn’t changed, then they were probably heading 4–8 points off the wind. They landed by the village of Brookville (ENE of starting point) on Long Island at close to 7 pm, after a journey of about 30 miles. (New York Times, June 7, 1866).

Andrews concluded that the Aereon II needed a fifty-foot car, so he could shift his weight further. In addition, he decided that the airship should be equipped with “two lateral wing-shaped appendages.”

### Appendix 3: Geometric and Aerodynamic Modeling of the Aereon

I assumed that the Aereon had tangent ogive ends because those are pointed, look like the pictures of the Aereon, and yield a volume close to that reported for the Aereon. Since the membranes mentioned in the patent do not appear in the 1866 illustration of the first Aereon, I assumed that they were an afterthought and did not include them in the model. (For their aerodynamic effect, see Notes.)

In part 1, I described the standard (“build-up”) methodology used for calculating drag force-air speed relationships in preliminary design of aircraft fuselages and airships. They calculate the frictional drag for a flat plate, multiply by a “shape factor” based on length/diameter ratio to get the total drag (including pressure drag) for a bare hull, and then multiply by a rigging factor to get the drag for a fully rigged airship.

Despite the computational simplicities of the simpler aerodynamic model used by Purandare (and of Graver’s hydrodynamic counterpart), in which the drag coefficient is unaffected by airspeed, I decided to keep the drag formulae the same as for part 1, except that instead of using the Dorringon rigging factor, which is dependent on Reynolds Number, I assume here that the fully rigged airship experienced twice the drag as did its bare hull (Durand 1934 reported that bare hull drag was 47% total for the Bodensee; 53.2%, Los Angeles, 57.1%, Macon). That’s in fact more favorable to the Aereon, as the Dorrington rigging factor for it is 2.52–2.53 for speeds of 10–20 mph. While the Aereon doesn’t have fins or as many cars as the airships studied by Durand, it does have a very non-aerodynamic car suspension system (every rope will be “seen” as a narrow cylinder at a large angle of attack) and its car is not streamlined.

I also needed a dynamic lift model, and I chose to use Saedraey’s empirical equation for the 2D lift coefficient and the Helmbold correction (an empirical blend of the low AR/slender body and high AR/lifting line formulae) to adjust for the finite span of the airship as airfoil. In calculating lift-induced drag, I assumed an Oswald efficiency of 0.8.

For the drag model, I am most confident about the frictional drag coefficient, least about the rigging factor, with the shape factor in-between. For lift-induced drag, the Oswald efficiency is open to debate.

The drag buildup method has been estimated as having a probable error of 10% (Jobe Fig. 3); the probable error here is likely to be greater because I used a rigging factor rather than individually calculating the total drag on each of the non-hull elements. If these methods estimate drag (nominally proportional to speed squared) with an accuracy of 20%, then speed is 89–110% predicted; if to 50%, speed is 71–122% predicted.

Buildup methods typically underestimate drag. In Graver’s study of underwater gliders, he found (169) that the experimentally calculated drag was 75% higher at zero angle of attack and 150% higher at 3^{o} than the his drag estimate.

### Appendix 4: Comparison of Model Predictions to Eyewitness Accounts

We know that when given “over two hundred pounds ascending power” (Andrews patent p2c1), it flew with a “streamer thirty feet long on the tail . . . [that] stood out in a straight line behind her . . . ” (Fonda). This ascending power was the result of it being relieved of Andrews’ own weight (172 pounds; Herald) and “several 7 pound bags of ballast.”

The wind speed needed to cause a flag to fly straight out depends on the weight of the flag, but the “modern” (1906) version of the Beaufort scale says that force 3, gentle breeze, results in light flags extended, and corresponds to a wind speed of 8–12 mph. (For a longer, heavier flag, it could be somewhat more.)

By my calculation, starting from sea level with 15^{o} pitch, if four bags had been tossed (net buoyancy 200 pounds), the glidepath angle would be 19.87^{o} and the air speed 19.05 mph (on the low end of Beaufort Force 5). That seems consistent with the reported streamer behavior, which is the only objective (albeit ambiguous) evidence of the Aereon‘s speed. It’s still way short of the 200 mph that Andrews claimed, but that number was preposterous (see Notes). High speed winds would damage or destroy the streamer!

For the manned flights, the witnesses agreed that the Aereon was able to make headway against the wind, and I believe them. One witness thought the wind speed was 10 mph; a second, 10–15; the others didn’t say. Since quantitative wind speeds were not reported by newspapers in the 1860s, and it’s unlikely that any of the witnesses had the means to accurately estimate wind speed, I am disinclined to trust their estimates. I am sure they could tell that there was a wind. If they actually felt the wind, that would imply that it was at least a light breeze (4–7 mph) and if they merely saw smoke drifting, that it was at least a light air (1–3 mph).

Nor is it likely that they were able to accurately estimate the speed of the Aereon. Loftus, Eyewitness Testimony 29 (1996) comments, ” . . . many investigators produced evidence of marked inaccuracies in the reporting of details such as time, speed and distance. The judgment of speed is especially difficult, and practically every automobile accident results in huge variations from one witness to another as to how fast a vehicle was actually traveling (Gardner 1933). In one test administered to air force personnel who knew in advance that they would be questioned about the speed of a moving automobile, estimates ranged from ten to fifty miles per hour. The car they watched was actually going only twelve miles per hour (Marshall 1966/1969, p. 12).”

Andrews did not claim that the Aereon flew a set course before observers trained and prepared to estimate its speed. Rather, it seems that the witnesses were simply curious about whether Andrews would manage to fly, and they weren’t questioned about the Aereon‘s performance until a month after the flight was observed. This plainly reduces the reliability of the estimates even further. (For further discussion of visual speed estimation, see notes).

How close the predicted still air performance of the Aereon comes to the actual speeds as estimated by the observers depends, clearly, on how much of a ballast drop is assumed (see Appendix 6). With a 15^{o} pitch, we can move against light air with a drop of just 7 pounds, against light breeze with one of 14–40 pounds. We may reach a horizontal speed very close to 10 mph (middle of Beaufort Force 3) with a drop of 60 pounds. To reach 15 mph horizontal is much harder; for 15^{o} pitch, about 147 pounds is needed.

I believe that given the limitations of the eyewitness testimony, the predictions here are reasonably consistent with the eyewitness reports.

Andrews claimed that there was a linear 1:2 or even 1:1 relationship between the number of pounds dropped and the horizontal air speed in mph. With 15^{o} pitch, to achieve 10 mph with just a 20 pound drop would require that the drag was just 29% of that predicted by my standard model. To achieve 20 mph, the drag would have to be just 7%. And it’s well known in aerodynamics that drag actually increases nonlinearly with speed V (V^{1.5} for laminar flow, perhaps V^{1.86} for turbulent, often approximated as V^{2}).

### Appendix 5: Effect of Modeling Variations

That said, I did have to make choices in designing the model that are open to question. In the next table, I look at the sensitivity of the results for a single flight profile to certain modifications of the “standard” geometric and aerodynamic model. Please note that these modifications aren’t equally plausible!

(* based on average of Zahm implied frictional drag (total drag divided by Dorrington shape factor for each model hull) for speed of 20 mph, multiplied by Dorringon shape factor for Aereon).

** depends on fineness ratio but not on speed.

***speed dependent; 2.54 at indicated airspeed.

****Zero-lift bare hull drag by Purandare-Young, rigging factor=1, 2d lift coefficient for thin airfoil, 3D airfoil correction per Jones, Oswald Efficiency=1.

While the model’s predictions are consistent with the behavior of the streamer during the unmanned flight, and with the lower end (10 mph) of the eyewitness speed estimates for the manned flights, the Aereon would have to have quite extraordinary aerodynamic properties to reach 15 mph with a “normal” ballast drop.

### Appendix 6: Ascent Profile

If the Aereon (tangent ogive ends model) is 100% filled with hydrogen at neutral buoyancy at an altitude of 0 feet, it will have a gross lift of 1868 pounds. As it rises, density decreases, so net buoyancy, drag and dynamic lift will all decrease. If it then drops 60 pounds ballast (3.21% of the gross lift), it will rise to the altitude at which the air density is (1808/1868=) 96.79% that at the initial altitude.

Since the Aereon is fully inflated, it’s a “constant gas volume” system. As it rises, its hydrogen assumes the temperature and pressure of the ambient air. The hydrogen expands, and since the airship is already fully inflated, hydrogen is continually vented off. As a result, the lift gas density declines and since the gas volume is constant, the gross lift and therefore the net buoyancy decreases. Also, its speed decreases even though the loss of positive buoyancy is partially compensated for by the decreased air resistance. At about 1112 feet, the air speed is essentially zero.

If it then vents lift gas (867 ft3) such that its gross lift at 0 feet will equal the current weight of the airship, and pitches downward by the same angle it previously pitched upward, it will continue moving forward while ultimately descending back to that altitude. That completes one cycle of movement.

As it descends, because of the changing ambient air temperature and pressure, the lift gas volume should shrink, rendering the Aeron flabby. To avoid this, Andrews allowed air to enter the envelope at the bottom, to preserve the form, and he blithely asserts (in his patent) that it will not quickly and readily mix with the hydrogen.(If it did, a flammable or even explosive mixture could form.)

Calculating the length of a cycle is somewhat tortuous because of the change in speed with altitude, but by averaging the initial and final speeds for each of the 100 foot increments, I calculate an approximate ascent time of 841 seconds to 1100 feet (an average vertical speed of 1.31 fps). The complete cycle would be twice that, i.e., 1682 seconds (28 minutes).

The behavior of the Aereon would be different if the Aereon, at takeoff, wasn’t fully inflated. It could be a hydrogen envelope with an inner air ballonet, or multiple hydrogen cells inside a larger air filled envelope. In essence, if there is no superheat (air and gas are equal temperature) and no superpressure (air and gas are equal pressure), then, as the airship ascends, the hydrogen cells expand and the air ballonet shrinks, the gross lift remaining constant.

Consequently, the net buoyancy created by a ballast drop will remain constant, and the quasi-steady state airspeed will increase slightly as the airship ascends (decreased air resistance), from 10.11 mph at 0 feet (ballast drop) to 10.25 mph at 1000 feet. Purandare asserts (57ff) that a buoyancy-propelled airship with a partially inflated air ballonet will lose buoyancy and therefore speed as it ascends to more rarefied air; I think he misses the point that while the specific lift is less, the volume is greater, keeping static lift the same (ZSG-2 Notes).

This “variable gas volume” behavior will continue until the Aereon reached its pressure altitude ) at which point, if it rose further, it would be fully inflated and thus would switch over to the “constant gas volume” behavior previously described. If it started 90% inflated, the pressure altitude would be 2762 feet. If the average speed of ascent was 1.56 fps (quasi-steady state, 60 pound drop, 0 feet) it would reach pressure altitude in about 30 minutes. And it would spend the same time descending.

Just to complicate matters further, a non-rigid airship would have a slight superpressure (say, 0.005 atmospheres) to maintain its shape. The speed at 1000 feet would be 10.22 mph, and the gross lift would be 3% higher. But at 2700 feet the speed would be 10.43 mph and the gross lift only 0.05% higher than at sea level.