Airship Propulsion, Part One: Thrust and Drag


If it were not for the wind, we could use engines of quite low power. The first airship, with engines of a mere 3 hp, was able to make 5 mph in still air. While that isn’t impressive to the up-timers, bear in mind that if you can fly all day and night at that speed, you are making 120 miles a day. A sailing ship would probably average 60 miles a day, and a pack train more like 15-30. See Cooper, “Hither and Yon: Transportation Modes, Costs and Infrastructure in 1632 and after,” Grantville Gazette 11.

However, if the wind is blowing, and your engines aren’t powerful enough to let you make headway upwind, then the directions that you can fly in are limited. More limited, in fact, than for a sailing ship, as tacking isn’t possible. Cooper, “Untying the Wind,” Grantville Gazette 35, talks about what the characters will know or can readily learn about prevailing winds at different altitudes,and Cooper, “Highways of the Sky,” Grantville Gazette 36, about the effect of wind direction and speed on the airspeed for a particular course.

Winds vary. At Plymouth, England, the mean wind speed is 10.2 knots. However the wind speed distribution is skewed so the median wind speed is lower. (Windpower) At Amsterdam, Netherlands, the average wind speed for 2001-11 was 11 knots. The wind force was greater or equal to Beaufort Force 4 (Moderate Breeze, 11-15 knots) 47% of the time. (Windfinder).

The strength of wind that an airship must be able to make headway against depends on several factors: (1) where it’s flying (some areas have stronger winds than others); (2) how long and how often it’s flying (the more it’s in the air, the more of the strong winds it will encounter); (3) how urgent is it that it get to a particular place at a particular time (scheduled passenger or perishable merchandise service or military missions have stiffer requirements than bulk cargo or pleasure craft). Count Zeppelin declared in 1899, “it may be laid down as an axiom that a navigable balloon to be a practicable success must be able to attain a speed of at least 20 miles per hour.” (AJ 77). An airship (LZ4) with maximum speed of about 30 mph (26 knots) was available in 1908 (Brooks).

Milestones in Airship Propulsion

Steam Propulsion. In 1852, Henry Giffard created the first airship, that, is, an aerial vehicle that had aerostatic lift (like a balloon) but could engage in powered flight. There is a conflict of authority as to some of its specs. It had a three horsepower steam engine, weighing 100 (or 250) pounds; the boiler added 100 pounds. This drove a cigar-shaped airship of 144 feet length and 40 feet maximum diameter, holding 88,000 (or 113,000) cubic feet of gas. In still air, it achieved 4-5 (or 6) mph. (Smith 51; USCoFC).

Buoyancy-Differential Propulsion. In 1863, Solomon Andrews took to the air in the Aereon, a trilobal hydrogen-lift airship trimmed to obliquely direct the net buoyant force created by ballast drop or hydrogen venting to generate horizontal motion. He did it again in 1866 in the Aereon II, a single-hulled airship. These should not be confused with the late-twentieth century Aereon 26, which used an old two-cycle drone aircraft engine (McPhee 86, 165).

Battery-Powered Propulsion. Albert and Gaston Tissandier’s 1883 airship, of 37,500-cubic feet, was driven by a 1.5 hp (180 rpm) electric motor, with a bichromate battery, at 3 mph. The powerplant weight was over 500 pounds. The next year, the La France (165 feet long, 27 feet maximum diameter, 66,000 cubic feet) took to the air; an 8.5 (or 8 or 9) hp, 212 (or 220) pound motor turned a 23 feet, 88 pound, two-bladed propeller at 50 rpm; the engine used a 580 (or 1500) pound chromium chloride battery. On its first flight, it attained a speed of nearly 15 mph. (Delacombe 105ff; Sinclair; Boyne 23; American School).

Gasoline Engine Propulsion. Beginning in 1898, Alberto Santos-Dumont built and successfully flew a series of small airships. His “No. 1” was 82 feet by 11.5 feet, with 6,454 cubic feet gas (450 pounds lift; hydrogen purchased at 1 franc/cubic meter, $1,270 total). He coupled two 1.75 hp single cylinder De Dion tricycle (!) engines together, thereby obtaining 3.5 hp for only 66 pounds, to drive a 6.6 foot diameter prop. (Hoffman 53ff).

Diesel Propulsion. The 1929 British R101 had 600 hp Tornado diesels of railroad origin; these were expected to be safer than gasoline engines, especially for tropical service (Castle 37). I believe that the diesels of the 1931 Hindenburg were the first designed for airship use.

Thrust and Drag

A drag force is created by air resistance when an airship attempts to move. The airship will accelerate, under the influence of its engines, until it reaches an air speed at which the thrust (propulsive force) it is generating equals the drag force upon it. At that point, the airship is in steady flight (constant speed).

There is in truth no simple formula that exactly reveals, for all airship geometries, the drag force that it will experience at each possible air speed. At present, the “gold standard” for predicting aerodynamic behavior is computational fluid dynamics (CFD), which assumes that the motion of the fluid is described by the Navier-Stokes equations, often simplified to make the calculations faster. But even our authors, let alone our characters, are unlikely to have access to CFD software (or to a supercomputer to run it on).

However, for practical purposes, we can as a first approximation assume that the drag force follows Newton’s Drag Law:

F = ½ *ρ*C*A*V2 where

F is the drag force

ρ is the density of the fluid (air)

C is the drag coefficient

A is the reference area for the calculation of C

V is the airspeed

If an airship model is placed in a wind tunnel, we can control the density of the gas (usually air), and the speed with which it flows past the model, and measure the force on the model. This allows us to solve for C*A, which is sometimes called the “drag area.” The drag coefficient C is referenced to a particular reference area A, which could be the cross-sectional area, the exposed surface area, or the “volumetric area” (V2/3). It doesn’t matter which reference area you choose, as long as you are consistent.

We can also determine the drag coefficient for a full-sized airship, by what is called a “deceleration” test; we stop the engines and, if the airship is in still air, it will eventually grind to a halt as a result of the drag force upon it (which will decline as the airspeed declines). The time that this takes can be related mathematically to the drag coefficient.

If Newton’s law were accurate, the drag coefficient would be a constant for a particular airship, determined by its shape alone. However, because it doesn’t capture all the complex behavior of fluids, the drag coefficient is itself dependent on airspeed. (See “Drag Computation (Advanced),” below.)

Power, Propulsive Efficiency and Speed

Of course, what we really care about is the relationship between the power of the engine and the speed of the airship. So how do we go from force to engine power?

Power is work/time. Work is force times distance. And speed is the rate at which distance is covered, distance/time. That’s all Physics 101. So, we can infer that power equals force times speed (P=F*V).

Now, according to Newton’s Drag Law, if the density and area are kept constant, the drag force is proportional to the square of the airspeed (V2). Since, at steady speed, the propulsive force equals the drag force, that must also be true of the propulsive force. Which means that the propulsive power to provide that force must be proportional to V2 times V, i.e., to V3.

The engine power is the power (“brake horsepower”) exerted by the engine at its output, the crankshaft. To achieve a given propulsive power, the engine power must be higher, because of energy losses in the transmission system and at the propeller (e.g., drag forces on the propeller blades). The propulsive efficiency is the ratio of the propulsive power to the engine power.

So, this gives us the extremely important information that as long as the drag coefficient and the propulsive efficiency remain constant, the engine power must be proportional to the cube of the speed; to double the speed, you must increase the power eight-fold.

Later in this article, we’ll look more closely at propulsive efficiency, and see how the variation in drag coefficient with airspeed can be estimated.

Propulsion Scaling Laws

Let’s explore some of the implications of Newton’s Drag Law for airship design. Suppose that for some airship we know the speed achieved in still air at a particular power. We may use the following general scaling law if we keep the airship shape the same but change the airship power P, size (as quantified by a reference area, surface area, cross-sectional area or volumetric area A) or air speed V:

(Pnew/Pold) = (Anew/Aold) * (Vnew3/Vold3)

The Graf Zeppelin had a maximum speed of 80 mph (Wikipedia). If our ship is just like the Graf Zeppelin, except that it only has half the power, then we can simplify the scaling law and solve for Vnew:

Vnew = Vold * (Pnew/Pold)1/3

so the new maximum speed is

Vnew=80*0.51/3 = 80*0.79= 63.5 mph.

Chances are, of course, that a 163x airship is not as big as the Graf Zeppelin. But let’s say that it’s the same shape as the Graf Zeppelin, only smaller. If so, the three reference areas “surface area, cross-sectional area, and volumetric area” will all scale the same way. So if the new ship is half the length and half the maximum diameter of the Graf Zeppelin, the new area is one-quarter the extent of the old one, and the ship speed potential (power unchanged) increases by 41/3, or 1.59 fold.

The catch, of course, is that such assumes that the new ship has the same power as the old one. But with half the length and half the maximum diameter, we have one-eighth the gas volume, and thus one-eighth the lift capacity, and if engine power is proportional to engine weight and engine weight a fixed proportion of total weight, the maximum power of the engines is one-eighth the original’s. If so, then we have 80*(1/8*4)1/3 = 80*(0.5)1/3 = 63.5 mph as the maximum speed.

Now, if the new ship is not merely of a different size than the old one, but a different shape as well, these scaling rules won’t work, because the drag coefficient will change. The fineness ratio (length/maximum diameter) was shown to have a significant effect on drag; Zahm reported that the lowest drag was at 4.62:1. More recently, it has been shown that the position of the maximum diameter (the relative length of the forebody) is significant.

Airship Size and Power

Think of an airship as having a characteristic linear dimension “R” “its length, perhaps, or the cube root of its volume. The total mass (vehicle, fuel, ballast, crew, passengers, cargo) it can lift will be proportional to R3 and to the air density. The power required will be the drag force times the airspeed. The drag force will be proportional to R2 (frictional drag being proportional to the surface area, and profile drag to the cross-sectional area, both of which are proportional to R2), and to the square of the airspeed.

Since payload is proportional to R3, but drag force to R2, you’d expect larger to be better.

After a bit of mathematical jiu-jitsu, we find that for a fully-loaded airship, the required power is proportional to the two-thirds power of the total mass, the one-third power of the air density, and the cube of the airspeed. Plainly, the ideal airship is big and slow. (Lorenz).

In contrast, the corresponding “law” for an aircraft, which generate lift aerodynamically, is that the required power is linearly proportional to both the total mass and the speed, and inversely proportional to the lift/drag ratio. So aircraft pay less for speed, but more for weight. (The “real-life” dependence for propeller-driven aircraft is on mass0.9 speed0.8; Lorenz).

Engines in the 163x Universe

Propulsive power, of course, comes from engines (with the exception of certain exotic propulsion systems I will discuss in part 3). During the 1630s, we have the following engine choices for airships:

(1) converted up-time diesel engines;

(2) converted up-time gasoline engines (truck, auto, powerboat, motorcycle, lawnmower, garden tractor, golf cart, wood chipper, pressure washer, compactor, hydraulic lift, tiller, etc.);

(3) newly-built steam engines; and

(4) newly-built gasoline engines.

In part 2, I will outline the general considerations in selecting and designing engines, and then give in-depth consideration to internal combustion (gasoline and diesel engines). In part 3, I will apply the same principles to steam engines.

Power Transmission

Power must be transmitted from the engine crankshaft to the propeller. (I am assuming that we aren’t we building jet engines in the 1630s.) Save for a few unusual early aircraft/airship designs, power is transmitted mechanically (as opposed to say the electric transmission found on diesel-electric locomotives).

If the aircraft or airship has direct drive, the propeller is directly attached to the crankshaft, and the propeller rpm equals the engine rpm. If an auto engine is converted to aircraft use, you will probably need to reduce the rotary speed by as much as 2.5:1. This can be achieved with a belt drive in which the sheave on the prop is bigger than the one on the crankshaft. However, belt drives are limited in terms of how much power they can transmit (Childs, Mechanical Design 155), they have a useful life of only a few hundred hours, and the belts can shrink or expand with temperature (playing havoc with the belt tension). (EPI).

On the Hindenburg, a 2:1 reduction was achieved with a geared drive, and geared transmissions are used, despite the weight they add, on many aircraft. With turbine engines (typically reaching 30,000 rpm), a large rpm reduction is needed, and so we need to step down the rpm using either a series of gears, or more commonly planetary gearing. Planetary gearing may be used with slower engines; it was used on the L49’s Liberty 12 engines, which were only 1700 rpm. (Robinson 54).

For the Akron, the transmission system was 13,096 pounds, and the propellers 2,000 pounds. (Burgess).

Propeller Design

Propellers may be of the pusher type, as on the Hindenburg, or the tractor type, as on the Graf Zeppelin II.

Propellers for airships were initially made of wood, and later of metal. Propellers typically have anywhere from two to four blades. The blades have an asymmetrical cross-section, like the wing of a plane, so as to create “lift” “which is converted by the propeller into thrust. They also, again like a wing, create drag, which is the resistance that must be overcome by the engine. The blades are twisted so as to maintain a constant angle of attack (“blade pitch”) with the airflow around the propeller.

Propellers for airships were big. For the L49 (1.97 mcf), the two centerline engines had geared 16.5 foot diameter props, and flank engines had directly driven 11.5 foot diameter ones. (Robinson 50). The USS Shenandoah (ZR1) (2.6 mcf) had four wooden two-bladed 18 foot diameter geared propellers, and two 10 foot direct drive ones. (73, 211). The USS Akron and Macon had metal, three-bladed 16.3 diameter propellers with ground-adjustable pitch. (186, 211). The Hindenburg had laminated wood propellers, three- (aft port) and four-bladed (other three), of 19.7 feet diameter, ground-adjustable. (Dick 87, 141). For historical airships, propeller size can be estimated by assuming that for a envelope volume of 5,000 m2, propeller is 2m, and that the diameter is proportional to volume0.25. (Dorrington).

The propellers on airships were often staggered so that the forward propellers didn’t disturb the air felt by the rear propellers. However, on the Akron, the propellers were disposed “single file.”

It is possible for a single engine to drive two propellers that are on the same axis but rotate in opposite directions (contra-rotation). The rear propeller actually takes advantage of the disturbed airflow created by the forward propeller.

It is also possible for propellers on different engines and parallel axes to rotate in opposite directions, e.g., clockwise on the left engine and counter-clockwise on the right, or the other way around (“counter-rotation”). This minimizes the torque imparted by the propeller to the airship.

A fan is a propeller with a large number of blades, which increases the thrust provided for a given diameter, and thus allows uses of shorter blades at a given rotational speed. However, the propeller aerodynamics become more complex. The fan may be mounted within a shroud (duct) to increase efficiency, although the shape of the duct should be optimized for a particular air speed.

The USS Akron was equipped with pivotable propellers. The advantage of such propellers is that they can provide vertical as well as horizontal thrust, thus compensating for the airship being statically light or heavy, or allowing it to ascend or descend even when at neutral buoyancy. However, the transmission is necessarily of a more complex nature.

Propulsive Efficiency

The propulsive efficiency is the product of the transmission efficiency and the propeller efficiency. If the propeller is driven directly by the engine, the transmission efficiency should be nearly 100%. For geared transmissions, Torenbeek estimates the efficiency as 100 less 13 times the common log of the reduction ratio, and reported values for airships are 92-97%. (Dorrington).

Consequently, propulsive efficiency is mostly controlled by the propeller efficiency. Some of the engine’s power must go to overcoming the aerodynamic drag on the propeller blades as they spin, rather than just providing the propulsive thrust. The lower the propeller efficiency, the more engine power you need to achieve a given thrust.

Propeller efficiency itself depends on the airspeed. More precisely, it depends on the advance ratio (the ratio of the airspeed to the product of the propeller rpm and the propeller diameter), which relates propeller motion to airship motion.

Generally speaking, it’s better to catch a large amount of air (with a big propeller) and accelerate it a little bit (low propeller rpm) than a small amount of air and try to accelerate it a lot. (Hepperle). However, for a given propeller rpm, the larger the propeller, the higher the tip speed, and performance degrades significantly once that tip speed exceeds Mach 0.85 (1 being the speed of sound). Consequently, with a large propeller diameter, you make the propeller rpm smaller, which means that the transmission reduction ratio is probably greater, so while you gain propeller efficiency you lose a little transmission efficiency. (Dorrington).

The peak efficiency of a two-blader is higher than that of a three-blader but comes at a lower engine power. (Hartzell). Yes, that means you can use a lower power engine but lower power implies lower maximum speed. If you must achieve a particular speed with a particular airship you might need to use more blades.

For a fixed-pitch propeller, if you increase the advance ratio, then propeller efficiency initially increases, reaches a maximum, and then decreases, as airspeed increases. The greater the pitch of the blade, the higher the airspeed at which maximum efficiency is achieved. The efficiency climbs gradually with the advance ratio up to that peak, then falls off much more sharply. You still get more thrust by increasing the advance ratio above that corresponding to maximum efficiency but the engine power is used less efficiently, which affects fuel consumption.

Ideally, the airship has an in-flight adjustable variable pitch propeller. The pitch of the propeller blades (relative to the plane in which they rotate) can then be adjusted for efficient operation at the desired airspeed. However, even the Hindenburg had a fixed pitch propeller.

Dorrington suggests that average propeller efficiency was 85% of the maximum.

Maximum propeller efficiency is variously quoted as 75% (Thurston 140; Leyensstter 11), 80-85% (Konstantinov), 85% (Loftin), 85-87% (EPI), 87% (FAA 3-22), and 87% for cruise and 75% for climb (McCormick 19). It’s very easy to end up with poorer performance through ignorance. Langley’s nineteenth-century propeller tests yielded an efficiency of only 52%, most likely because he thought the most efficient propeller “is that which sets in motion the least” amount of air. (Anderson 178). The Wright brothers did better; their 1903 propeller peaked at 82% and averaged 75% (Ash).

Here are some reported propeller efficiencies (average? maximum?) for airships:

USS Shenandoah: (ZR-1): 40% @ 50 knots

USS Los Angeles: (ZR-3): 44%, ditto

USS Akron: (ZRS-4): 60%, ditto

Hindenburg: 73% with new blades, was 68%

Thunder & Colt GA-42: 50%

American Blimp A-60: 59%

SSE3: 66%

Zeppelin LZ-N07: 70%

Goodyear ZS2G1: 70%

Beardmore R29: 65%

Short Brothers R32: 68%

Armstrong Whitworth R33: 67%

Zeppelin LZ-129: 72%

(Althoff 153; Dick 141; Dorrington Tables 1-2). A 1941 War Department tech manual assumes an airship propeller efficiency of 60% (TM1-320, s17).

Airspeed Selection

As was explained above, selecting the desired airspeed determines the required engine power. We don’t have good models for choosing the cruising speed of airships.

The total drag on an aircraft includes components that increase with speed, and ones that decrease with speed, and thus there is a speed (the “best range”) speed at which drag is a minimum. (Carson).

An airship, unlike an aircraft, can stay airborne without generating aerodynamic lift, and therefore can fly without experiencing lift-induced drag. (I talked about dynamic lift and drag in Cooper, “Sitting on Cloud Nine: Airship Lift and Altitude Control,” Grantville Gazette 39. According to Newton’s Drag Law, there’s no “sweet spot”; the higher the airspeed, the higher the drag.

We are venturing into virgin territory here, but it seemed to me that the solution could be to consider (1) revenue, and/or (2) non-fuel costs. We’re worrying about drag, obviously, because it increases fuel consumption, and fuel costs money.

Suppose we calculate a net revenue rate, dollars/flight hour. That will equal the gross revenue rate, less the cost rate. If we charge a flat fare per ton-mile or passenger-mile, then the faster the ship is, the more mileage we can get in, and so the gross revenue rate will be of the form A*V, with A being a constant (fare times number of passengers or tons of cargo per trip) and V the speed. If the cost is just fuel, the cost rate is the fuel consumption rate and its proportional to the power, so we can write it as B*V3, with B as a constant taking into account the drag coefficient, air density, overall efficiency, energy density of fuel, and fuel price. So the net revenue rate is A*V-B*V3. The first derivative with respect to V is zero at V2=A/(3*B)), and the second derivative is -6*B*V. So the second derivative is negative, which means that we found a maximum. (A good thing since there’s not much demand for a method of minimizing revenue.)

Non-fuel costs fall into two categories, those that are constant per unit flight hour (lubricating oil consumption, lift gas leakage during flight, burner fuel for hot air or heated gas-lifted airships, provisions, replacement parts) and those that are constant per unit chronological hour (interest payments, facilities costs, wages, etc.). These don’t shift the sweet spot, they just reduce the net revenue.

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I don’t know the values of A and B, so I can’t calculated the most economical speed V, but it’s possible that it will feel too slow. Economists quantify the preference for getting things done more quickly as the “value of time,” and airship passengers and freight customers are going to be willing to pay some kind of premium to go faster. So the fare, instead of being flat, is itself a function of airspeed.

However, the value of time depends on what’s being shipped. In 2007, USAID calculated, for different types of goods, that “from a consumer’s perspective, each day that a firm saves by air shipping rather than ocean shipping the good is equivalent to lowering the good’s price by a particular percentage.” They assumed that air shipping always took one day and compared it to the actual ocean shipping times.

Here are some of their findings:

crude oil, coal, fertilizers: 0 (don’t care about time at all)

medicines: 0.3%

textiles: 0.6%

cereals: 0.8%

vegetables and fruit: 0.9%

coffee, tea, cocoa, spices: 1.1%

road vehicles: 2%

There’s no guarantee, of course, that the time sensitivity of price would be the same in the “new” 1630s that they were in 2007. For example, our ability to preserve perishable foods may be less, so they would have greater time sensitivity. But the USAID data is still interesting.

We could throw a price-equivalent-of-speed (say, 0.8%/day’s delay) into the net revenue rate equation and see how it affects the “sweet spot”. Bear in mind that if a transatlantic airship makes the passage in one week, and a sailing ship in eight weeks, the seven-week difference then translates to a sailing ship fare 39% less than the airship price.

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Even though the required power will be increasing generally as the cube of the airspeed, there will be a bump in the power-airspeed curve as a result of the variation in engine and propulsive efficiency (hopefully, the engine and propeller are matched so they are both at maximum efficiency at the same airspeed). Chances are that we will pick a cruising power that benefits from that bump.

Sizing the Engine

Suppose that on a particular route the desired airspeed requires 1200 hp at the engine output. You could use an engine set that totals 1200 hp rated power, but then all of the engines must run at 100% maximum power. Or you could use an engine set totaling 1600 hp, and run each engine at 75% maximum power (but that engine set would probably weigh a third more, too.) Or assemble 1800 hp worth of engine, and run at two-thirds maximum power. And so on.

In general, of course, the more powerful the engines, the more they weigh and the more they cost. But engine performance is affected by the power setting.

In general, your engine needs to be sized so that it spends most of the time operating at significantly less than maximum power. Manufacturers of light aircraft engines typically recommend that cruising power be not more than 75% of maximum power, but it isn’t made clear how much this is to conserve engine life and how much for fuel economy.

For aircraft, typical power settings are 45% maximum power for best endurance, 55% for best range (flown at lowest altitude at which throttle is fully open for that speed), and 75% for best cruise (Brandon, 2.10), although I have found cruising settings as low as 65% ( Jupner 25) and as high as 85% (11, 14, 19, etc.) Carson calculated that the “best cruise” would require 52% more power than best range, and 55%*1.52=83.6%.

Are aircraft power settings appropriate for airships, too? Not necessarily. Those settings are inspired, more or less, by the nature of the total drag curve for aircraft, which is quite different that the total drag curve for an airship.

Of course, Carson’s assumption of constant efficiency isn’t really correct. For any internal combustion engine, there will be an engine speed at which the brake-specific fuel consumption (BSFC), a measure of engine efficiency, is at a minimum. At low speeds, the problem is heat loss through the walls of the cylinders. At high speeds, the problem is friction. BSFC will eventually increase as the throttle is closed. This is because of the extra work (“pumping loss”) the engine must do to draw the air past the throttle. (Edgar; Corson).

I have looked at quite a few performance curves, and I have found that auto engines typically have lowest BSFC at 50% power, modern light aircraft engines at 65-75% power, and WW I vintage military aircraft engines at 85-95% power. The effect of moving from lowest BSFC power to max power on BSFC was perhaps 25-35% for the auto and light aircraft engines, and 4-6% for the military aircraft. (See Appendix 3).

Secondly, there’s the matter of propeller efficiency, which is affected by both airspeed and engine rpm. At too low an airspeed, you might have such a low propeller efficiency as to counterbalance the benefits of drag reduction vis-a-vis fuel consumption. Particularly for a fixed-pitch prop, there’s an ideal airspeed for maximum propeller efficiency.

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In sizing the engines for an airship, you also need to take into account the altitude that it’s likely to be flying at, since normally aspirated internal combustion engines deliver less power (for a given air-fuel mixture and throttle setting) at altitude than at sea level. The engine setting resulting in 75% power at sea level might only yield 55% of the maximum power at sea level when at 10,000 feet. (Langley). (For “ordinary” engines, the percentage decline in power is essentially the percentage decline in air density.) It follows that if you intend to cruise at 10,000 feet, you will need a more powerful engine to hit the same “sweet spot.” The inverse is true for steam locomotives, where the same engine runs BETTER as the altitude increases, as it’s exhausting steam against a lower pressure. However, steam powerplants for airships must recycle their water, so altitude doesn’t affect them except insofar as it affects the efficiency of the condenser.

The situation for turbocharged or supercharged engines is a bit different; these maintain sea level power up to the critical altitude, above which they, too, suffer from reduced air density. (Gardner).

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The Hindenburg‘s four engines each had a maximum power of 1050-1600 hp (sources are inconsistent), but at cruise setting generated 850 hp. Taking the lower figure for max power as most likely, that means it cruised at 81% power. The ZR2 (R35) was designed to cruise at 37% power (655/1755). (Robinson 205).

Drag Computation (Advanced)

The other aspect of sizing the engines is making sure that their maximum (or cruising) power is such that engine power * propulsive efficiency / designed maximum (or cruising) airspeed equals the expected drag force at that airspeed. For the benefit of those readers who are actually trying to design airships (for a story!), I am going to explain how to do this with greater accuracy than that provided by Newton’s Drag Law. This is going to be math-heavy, and if you aren’t interested in the how-to, just skip down to Table 3!

The drag force experienced by an airship at zero angle of attack and zero elevator is the sum of the frontal (pressure) drag the airship has to push air out of its way and to its rear and the frictional drag a boundary layer is formed as the air closest to the airship is forced to move with the airship.

Early work on airships was purely empirical, the effect of speed on the drag coefficient was simply determined by fitting a power law (drag~Vn) equation to the wind tunnel data. If you changed the shape, the parameters changed.

A more modern approach is semi-empirical. The standard preliminary design methodology for aircraft is what’s called a “build-up”. For each component (wings, fuselage, tail, engine nacelles, cockpit for aircraft; hull, cars, fins, etc. for airships), we

(1) calculate the Reynolds number and use this to determine the flow regime (laminar, transitional, turbulent over smooth surface, turbulent over rough surface, etc.) ,

(2) determine the frictional drag for a flat plate having the same exposed area as the component, and

(3) Multiply the frictional drag by a shape factor (typically a function of fineness ratio, or of the ratio of surface area to cross-sectional area) to get the total drag. (Konstantinov; Dorrington; Kale).

We then sum the total drag for all components, and estimate the interference drag between the components (based on statistical data) and add this to the nominal total drag.

Now, this can be done for airships. The components would be the hull, cars, fins, etc. But the problem is that it requires the designer to determine the dimensions of all of those components. So, we are going to use a shortcut.

We carry out steps (1)-(3) above, but just for the bare hull. Then we multiply by a “rigging factor” that estimates the ratio of the total drag of a fully-rigged airship to that of its bare hull.

Reynolds Number

The Reynolds number is used to determine whether flow is laminar (smooth) or turbulent; more drag is experienced if flow is turbulent. The Reynolds number is the airspeed times “characteristic length” of the component in airflow direction, times the ratio of the air’s density (itself dependent on altitude) to its dynamic viscosity (dependent on temperature).

For aircraft wings, we use the “chord length” of the wing. For airships, the characteristic length may be the geometric length or the cube root of the hull volume, the corresponding Reynolds numbers are symbolized by ReL or ReV respectively.

 

Table 1: Sample Reynolds Numbers for Airships(100′ altitude, air density 99.7% sea level)

Airship

Length, L

(ft)

Diameter, D

(ft)

Airspeed

(mph)

ReL

ReV

Giffard’s

144

40

5

6.70e6

2.30e6

Aereon(#triple hulled)

80

13

10

7.45e6

#2.58e6

Hindenburg

804

135

70

5.24e8

1.29e8

84

6.29e8

1.54e8

Sandterne *

222

51

20

4.13e7

1.25e7

Upwind*

Swordfish*

150

60

35

4.89e7

2.14e7

Royal Anne *

650

70

35

2.12e8

3.86e7

50

3.03e8

5.52e7

70

4.24e8

7.73e7

*canon

When the dimensionless Reynolds number is low, the flow in the boundary layer is laminar (orderly), and when it’s high, it’s turbulent.

The transition from laminar to turbulent flow is likely to begin somewhere between ReL 1e5 and 1e6 (Hoerner 6-3). In studies of a 20-foot long model of the USS Akron, transition occurred at 8.14e5 (Freeman, NACA Report 430). As the Reynolds number increases beyond this “critical” value, the transition point moves aft, and the flow is fully turbulent when it’s at the tail. The transition region may be fairly broad, extending to perhaps 1e7 (Hoerner) or 2e7. (Dorrington2007).

Note how high the Reynolds numbers are for the 1632 universe airships at the indicated airspeeds; they are in the range for fully turbulent flow. At a sufficiently low speed, they will experience transitional flow, and the drag on them is less predictable (you may thus rationalize doing a bit better than what my formulae predict).

Frictional Drag

Boundary layer theory (which simplifies the Navier-Stokes equations) allows one to predict the drag experienced by a flat plate moving edgewise; this is pure frictional (viscous) drag.

Equations can be derived that state the drag coefficient as a function of Reynolds number for various flow regimes. (Anderson; Theodorsen). These are semi-empirical; the form of the equation is determined by theory but then the parameters are determined by fitting to wind tunnel or other data. The most popular such equation, the Von Karman-Schoenherr line, is awkward to solve (requires guesswork), so simpler equations have been fit to it.

I have found over a dozen frictional drag equations in the scientific literature; the most commonly cited are those in the table below. Note how much lower the laminar flow value is than its turbulent flow counterparts, and the spread of the latter for a given Reynolds number. I have italicized the values that I know to be outside the formula’s validity range.

 

Table 2: Frictional Drag Coefficients vs. Reynolds Number (ReL), by Various Formulae(referenced to surface area)

Formula

1e6

1e7

5e7

1e8

5e8

Blasius laminar flow1.328*ReL-0.5

0.00133

0.00042

0.00019

0.00013

0.00006

Prandtl-Schlichting transitional flow formula for critical 5e5:0.455*(log(ReL))-2.58-1700/ReL

0.00277

0.00283

0.00232

0.00211

0.00171

turbulent flow

Prandtl0.455*(log(ReL))-2.58

0.00447

0.00300

0.00235

0.00213

0.00171

Von Karman0.074*ReL-1/5

valid 5e5-1e7

0.00467

0.00295

0.00214

0.00186

0.00135

Hoerner:0.044 * ReL-1/6

valid 1e6-1e8

0.00440

0.00300

0.00229

0.00204

0.00156

Hoerner0.030*ReL-1/7

valid 1e7-1e9

0.00417

0.00300

0.00238

0.00216

0.00172

White-Christoph0.43*(log(ReL))-2.56

0.00438

0.00295

0.00231

0.00210

0.00169

Shape Factor

The shape factor is also dependent on the flow regime but we’ll concentrate on those for turbulent flow around an ellipsoid, which is a decent model for the airship hull. The more your airship departs from an ellipsoid, the easier you can justify performing better than predicted here.

The simplest airship hull formula I found was Young’s, which is a simplified fit to the result of theoretical calculations for a range of fineness ratios, distances of the transition point from the nose, and Reynolds numbers. His shape factor is just

(L/D)/((L/D)-0.4), using surface area as the reference.

The equation usually found in aerodynamics texts is Hoerner’s, which is based on experimental data:

1+1.5 (D/L)1.5+7(D/L)3,

based on surface area, or

4(L/D)1/3+6(D/L)1.2+24(D/L)2.7,

based on volumetric area; the latter shape factor formula has a minimum at D/L =0.217 (Khoury 32).

I have found several others, too. To give you an idea of how much they vary for the same airship, for the Royal Anne, I get the following different values (referenced to surface area):

Dorrington: 1.11

Hoerner (my preference): 1.06

Boeing-Vertol: 1.20

DATCOM: 1.10

Young 1.05

It’s clear from CFD analysis (Kale; Kanikdale) that one can get more accurate results by taking the location of the maximum diameter into account, but I don’t know of any simple formula that does this.

Rigging Factor

The workaround I promised is to multiply the computed total drag for the bare hull by an empirically determined “rigging” factor to obtain the total drag (including the individual drag from of cars and fins, and interference drag) for a fully-rigged airship. (Abbott, Dorrington). For large airships (Bodensee, USS Los Angeles,USS Macon), a factor of two is about right, as hull drag was 47-57% of total drug. (Khoury 34).

Dorrington recommends the speed-dependent rigging factors

2.55 – (ReL/4.65e8)

or

0.298 – 0.0342*log(ReV),

both for airship and bare hull drag coefficients referenced to volumetric area. I have assumed that the same ratio of drag coefficients would apply if the coefficients were referenced to surface area.

There is obviously a range of variation in rigging factors for airships depending on how smooth they are.

****

Just how good are these formulae? According to Hoerner (14-2), the standard build-up underestimates total drag on a full-sized airship (per pre-1918 deceleration tests), most likely because they fail to take into account the drag from the struts and cables used to mount the old-style cars, the extra drag from old open-type radiators, and perhaps also surface roughness. Rivets increased the drag on an airfoil by 6% for countersunk rivets and 27% for protruding rivet heads. (Hood).

The flip side is that it might be possible to do a bit better than formulae based on old airship data suggest. Dorrington reports that cleaning up the Goodyear ZS2G-1 hull (“smoothing the nose battens and removing other unnecessary excrescences”) reduced drag by 20% at ReL=1.88e8.

That means that there’s a bit of wiggle room if an airship in canon doesn’t meet the power requirements suggested by the formulae in this article. Of course, if it does meet them, all the better.

****

Drag would be proportional to the second power of the speed, and required propulsive power to the third power, if the drag coefficient were a constant (Newton’s Drag Law). The drag calculated by the advanced method are considerably lower.

The following are my spreadsheet results for several real and 1632 universe airships, with particular choices for the frictional drag, shape factor, and rigging factor equations. Other choices would lead to somewhat different results.

 

Table 3: Sample Drag Force and Required Propulsive Power Estimates for Airships

Airship (5) and Engine Power

Speed

ReL

(altitude

1000′) (1)

frictional

drag coeff,

hull (2)

shapefactor

(3)

riggingfactor

(4)

total

drag,

airship

(N)

Propulsv

Power

needed

(hp)

Giffard’s (3 hp)

5

6.56e6

0.00296

1.370

2.54

42

0.12

Hindenburg (3400 hp cruise, 4200 hp max)

70

5.13e8

0.00171

1.137

1.45

41,219

1,730

84

6.15e8

0.00167

1.23

49,167

2,476

Upwind (80 hp)

Swordfish (120 hp)

20

2.73e7

0.00251

1.827

2.49

1,187

14

30

4.10e7

0.00238

2.46

1394

45

35

4.78e7

0.00233

2.45

3,317

70

Sandterne

20

4.05e7

0.00238

1.25

2.46

925

11

Royal Anne

50

2.96e8

0.00183

1.062

1.91

11,584

347

70

4.16e8

0.00175

1.66

18,847

791

84

4.98e8

0.00171

1.48

23,665

1,192

(1) air is 97% sea level density. (2) Prandtl-Schlichting for ReCrit 5e5 (3) Hoerner (4) Dorrington (5) treated as ellipsoid.

Remember, the required engine power is the propulsive power divided by the propulsive efficiency, with altitude adjustments for internal combustion engines without turbochargers or superchargers.

The Swordfish, with 120 hp, can easily reach its announced top still air speed of 35 mph; to do this, it only requires a propulsive efficiency of 58%. In fact, it could probably go a bit faster. The Upwind, on the other hand, with just 80 hp, would need a propulsive efficiency of 87.5% to reach that speed, but could easily travel at 30 mph.

Remember, the drag is density- and therefore altitude-dependent, and so the top speed will increase at higher altitudes (if engine power is maintained).

Boundary Layer Control

Looking back at table 2, there’s a big difference in drag coefficient between laminar flow and turbulent flow at a given Reynolds number. So, is there any way to delay transition to turbulent flow?

In theory, yes, if we can prevent flow separation. Air can be sucked in through the hull, pulling the airflow toward the hull surface, and then exhausted in back for additional thrust. Or air can be blown into the boundary layer near the leading edge to energize it. There is no free lunch, energy is needed to power the suction pumps or blowers, and the equipment adds weight and requires maintenance.

Ideally, instead of sucking air in through discrete slots, the entire skin would be permeable, and provide continuous suction. However, this permeability would have to be combined with a high order of surface smoothness.

Suction may be combined with the Griffith airfoil. This has a bullet-like shape for most of its length, then a tail cone, and finally a pencil-like tail boom. The airflow over the tail cone would be unfavorable, but a suction slot is placed just upstream of it. (Goldschmeid).

This is all pie-in-the-sky stuff, even today, and I doubt anyone in Grantville would know about it, but if airships themselves aren’t weird enough tech for you, you might want to read up on it.

Conclusion

Prior to World War II, the airship was far superior to the aircraft as a long-distance carrier. In 1929, the Graf Zeppelin circumnavigated the world in twenty-one days, five hours and thirty-one minutes, starting at Lakehurst, New Jersey and stopping only at Friedrichshafen, Tokyo and Los Angeles. The total flight time was twelve days, eleven minutes. In contrast, the first aircraft circumnavigation, in 1924, took 175 days, and required sixty-nine stops. (Dick).

The speed of the airship was inferior to that of the aircraft, but not hugely so, and the airship was of course faster than any sailing ship or even a steamer.

This suggests that the airship may have an important role to play in long-distance trade.

What is not so clear is how the airships of the new universe will be powered. I will explore that question further in subsequent parts of this article.

****

To be continued . . .

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About Iver P. Cooper

Iver P. Cooper, an intellectual property law attorney, lives in Arlington, Virginia with his wife and two children. Two cats and a chinchilla rule the household with iron paws. Iver has received legal writing awards from the American Patent Law Association, the U.S. Trademark Association, and the American Society of Composers, Authors and Publishers, and is the sole author of Biotechnology and the Law, now in its twenty-something edition. He has frequently contributed both fiction and nonfiction to The Grantville Gazette.

 

When not writing (or trying to get an “orange blob” off his chair so he can start writing), he has been known to teach swing dancing and folk dancing, or to compete in local photo club competitions. Iver adds, “I can’t get my wife to read my fiction, but she has no trouble cashing the checks.”

Iver’s story “The Chase” is in Ring of Fire II