What factors influence the maximum speed of an airship?

Question

I'm interested in learning about the physics of airships; in particular, how you determine an airships maximum speed.

For example, the Hindenburg topped out at about 80 mph. The ship itself weighed about 130 tons, not including cargo, passengers, etc.

If it weighed less could it have achieved a higher top speed, or would 80 mph have been about the fast it could have gone regardless, for other reasons such as drag/air resistance?

If it had twice as many engines, could it have gone twice as fast?

These are oddly specific, noodly questions, and I'm not looking for the answers themselves, but rather I'd like to be pointed in the direction of the math/physics fundamentals for getting the answers. I'm working on designing some fantasy airships for an RPG, and of course while they're not real and very fantastical, I do want them to have some semblance of verisimilitude, and imposing real world physics (with some wiggle-room for the fantasy part) I think helps a lot.

Thanks so much for your time and answers!

Answer 1

I'll do this without most of the math since your target audience won't want to read equations in your story.

In the simplest terms, the maximum speed of an airship occurs when the maximum thrust generated by its engines is equal to the drag it experiences while being pushed through the air at that speed. That drag depends on the diameter and length of the airship (more specifically, the projected frontal area and the total "wetted area" of the entire vehicle's outer skin).

This is a complicated business because the relationships between all of those variables are strongly nonlinear- for example, the drag varies with the speed squared and the power required at a given speed varies with the cube of the speed.

So in your story: doubling the power of the engines will increase the airship speed only slightly, at the cost of doubling the fuel burn rate and therefore the fuel tank capacity- and thereby reducing the useful payload.

Similarly, doubling the length of the airship will roughly double the wetted-area drag, which means you have to double the horsepower of the engines or their number in order to achieve the same maximum airspeed as the shorter airship.

Answer 2

The top speed depends on the type of the airship. While the first designs were non-rigid, it became soon obvious that useable speeds could best be achieved with rigid designs because the higher dynamic pressure at higher speeds required more internal pressure to maintain the hull's shape. Given the low strength of early hull materials, the internal pressure puts a firm limit on non-rigid airship speeds.

Both designs produce drag mostly from friction and here size helps. A larger airship has a higher Reynolds number at the same speed. Its drag, being proportional to the wetted surface, only grows with less than the square of the linear size increase while lifting capacity is proportional to volume or the cube of the linear size. Next, the additional drag of the car, fins, gondolas and their rigging is a substantial factor: In case of LZ 126, S. Hörner (page 14-1) gives the drag coefficient as $c_D = 0.023$ for the hull alone and $c_D = 0.071$ for the complete ship, including nacelles, fins and all. The reference area here is the airship's frontal area. A similar result is given in NACA Report 394 which documents the tests done on models of Goodyear Zeppelins in the NACA variable density wind tunnel in 1932.

As others have pointed out, speed does not increase in proportion to thrust, but increased thrust will result in an increase a bit larger than the square root of the thrust ratios due to the Reynolds number effect (friction drag changes approximately in proportion to $\text{Re}^{-0.2}$). Depending on the means of propulsion, thrust itself drops with speed for a given power with propeller or bypass engines. Only with rockets will thrust be constant over speed.

Drag coefficient over Reynolds number for an airship hull from NACA Report 397.

In order to calculate the top speed of airships, first wind tunnel tests on models were performed in order to find the drag coefficient of the general shape. Next, that result was scaled up to the real size and then the effects of the car, fins, gondolas, struts and bracing wires were added. The results were calibrated with deceleration tests on real airships. By measuring the change in airspeed of the unpowered airship, the drag over speed can be determined, and those results have been collected in NACA Technical Report 117 by Max Munk (1921) and NACA Report 397.

In order to make existing Zeppelins larger and faster, hull extensions were added together with additional engine gondolas. For your RPG, therefore, you should also have modular designs where the number of engines can be selected. Each one adds some thrust but also increases drag a bit.

Actually, the dynamics of airships are much more interesting than their top speed. In order to pitch up, the horizontal fins need to produce a downforce. If speed is low, this will force the airship down. Increase speed, and there is a point where no altitude gain is possible with elevator deflection. That is called the critical speed. Only when the airship flies faster than its critical speed will control effectiveness be restored. In case of the Cargolifter CL-160, power had to be increased by 40% from the initial design in order to stay well above its critical speed.

Also, maneuvering is an art in itself: If the ship sinks into warmer air, the adiabatic heating of the lifting gas might not be enough to maintain lift and the ship picks up sink speed. Now dynamic lift needs to be added in order to avoid a crash unless plenty of water ballast is available for dropping to lighten the ship. Size is now a disadvantage: Since aerodynamic lift is proportional to the hull area but lifting force proportional to its volume, larger ships can create less aerodynamic lift as a proportion of their static lift. While small ships can easily add or subtract 10% of their lift by means of pitching up or down, the Cargolifter flight envelope allowed only for ±4% of aerodynamic lift.

Answer 3

As for the weight vs. speed: a rigid or semirigid airship has a max takeoff weight, which depends on it's size (because the size pretty much determines the max lift). If this airship was to fly, say, only half of the max weight, it would not go any faster, as the drag would be the same because the size does not change.

An airship may lift a load heavier than the buyoancy it has, by using engine power to create lift. This is by no means good for fuel consumption, but some modern airships have a lifting body shape, so part of the lift is created by the airships forward motion through the air. Such a ship uses steerable engines to produce extra lift on takeoff and landing. This kind of airship is faster than a traditional "blimp" of same weight and power.

Also, it matters what "floats" the airship: helium, hydrogen, or, being a fantasy game, even vacuum maybe? Hydrogen gets you better buoyancy than helium, so an airship filled with hydrogen would be able to lift x tonnes more than the same ship filled with helium, or lift the same weight with a smaller volume of body and therefore be faster with same engine(s). We of course know the downside of hydrogen, it's an accident waiting to happen.

My ad-hoc tongue-in-cheeck proposition of vacuum airship (vacship?) Would be most efficient (per unit of volume), but in a non-fantasy world, next to impossible to construct.

As for the scaling, niels gave you the basic formulas for speed vs. drag vs. power. If you will, dig up some formulas and find relations between volume, diameter, height (or rather length in this case), and skin area of cylinder to roughly optimize the size of an airship.

Some other aspects to consider in the game: Heating the buyoant gas would increase the lift a bit, but again, not a good idea with hydrogen. A hydrogen filled airship could use the hydrogen also as a source of energy for the engines... Being a fantasy game, you of course have the possibility to strech the limits of the real world as you please, just google some historical and contemporary examples and sprinkle them with some magic :)

Answer 4

Comparison of two giants from the golden age of airships LZ 127 Graf Zeppelin and LZ 129 Hindenburg provides some useful information on your proposed scaling.

Both airships were around 800 feet long and cruised at 80 - 85 mph. Hindenburg was 35 feet wider, with more than double the lifting capacity, but required 4 x 1200 hp compared with 5 x 550 hp of the Graf Zeppelin. The increase in girth of the Hindenburg resulted in an 82 % increase in its frontal area, which matches well the increased power required.

Would increasing length and decreasing diameter be a better way to make it faster? Absolutely yes, as racing scull design, Concorde, and this answer imply, with one caveat, many airships that were lost in turbulence, due to their great size, literally put one end in the updraft and the other in the downdraft, resulting in overstressing and failure of structural components.

Putting the Hindenburg engines on the Graf Zeppelin would result in a 40% increase in speed, as WindSoul points out, but again structural stresses would have to be evaluated.

Reducing weight would affect speed only if it results in a more aerodynamic shape that produces less drag, as compared with heavier than air craft that can reduce their wing AOA or speed to lower drag with a lighter load.

So, at the expense of more power, the stockier design would be much safer, and be able to carry more load. Apparently this is what the designers of Hindenburg had in mind, as well as helium, which they were unable to obtain.

For a modern hybrid approach you may also wish to check out the Hybrid Air Vehicles Airlander 10 for more ideas.

Answer 5

The drag increases with the square of speed. Since thrust only slightly overcomes drag, doubling the thrust only results in a 40% increase of speed.

In the case of a dirigible, doubling the thrust won’t double the speed. It will only increase the speed by 40%.

There are other factors to account. Once the flow approaches transonic speeds meaning the speed of sound might be reached on locations on the surface of the ship, then the drag increases considerably due to initiation of the shockwaves. At that time the drag formula needs adjusted.

To wrap it up: while subsonic, doubling the thrust only increases the speed by 40% for the same aerodynamic shape. If the increase of thrust reached transonic flow of air around the aerodynamic shape, then the schckwaves will increase the drag considerably and put more stress on the airframe. Increasing the speed beyond supersonic will require the study of how shockwaves develop on the airframe. The pressure increases, the temperature increases as well. At a point in hypersonic flight the shockwave detaches completely from the airframe and forms ahead, leaving the airframe engulfed in a pressurized bubble where the friction of pressurized air flowing on the surface creates heat.