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I'm currently writing a military science fiction story set in a post cyberpunk world I've been building. The story focuses on a tank crew, but it also prominently features an enormous airship that functions as a flying airfield or aircraft carrier. I'm trying to get a sense of scale such a ship would have. The nice thing about the square cube law is that it means that doubling the size of an airship octuples its volume.

  • The airship needs to be large enough to carry at least 40 strike fighters with a maximum takeoff weight of roughly 20 tonnes each, and at least two helicopters with a maximum takeoff weight of 10 tonnes each. Plus extra fuel, munitions, and other expendables.
  • The airship uses hydrogen as a lifting gas. I realize this is very flammable, but I think it can be made safe enough through the use of compartmentalization, blow off panels, and lightweight graphene armor to protect against stray shell fragments. I'm also considering helium, it would an even bigger airship, but the reduction in required safety features may make up for this.
  • The airship is propelled using ducted electric fans powered by an onboard anuetronic fusion reactor (less than 1% of the reactor's output is in the form of high energy neutrons.)
  • The airship needs to have 360 degree turret coverage for its point defense weapons systems. It's a big ship, so it will need at least a dozen of these turrets (let's use the Phalanx CIWS as a reference at 6.2 tonnes each)
  • The airship's flight deck should ideally be long enough to land the above mentioned fighters without arresting cables. If they overshoot, they can just fall until they regain enough airspeed to try again. Likewise, there is no need for catapult launches. I'm thinking the flight deck will be suspended below the airship's envelope, with the hangers within the envelope.
  • This airship is not designed to land on the ground. Instead it can be docked to a tower like traditional airships were and resupplied using its own cargo winches.

So, my question is: What would the length, width, and height of airship like this be for it to contain enough hydrogen (or helium) to lift its fully loaded weight of approximately 50,000 tonnes? Assume a prolate spheroid (typical cigar shape) for any volume calculations.

Edit: My sleep deprived brain took the name "square-cube law" too literally. Someone pointed out that burying the reactor and the ammunition magazines deep within the ship would provide adequate protection since if anything penetrates that far, the ship is doomed anyways. My initial reason for using hydrogen was that it is a better lifting gas and is currently far more abundant on Earth than helium. However, given the prevalence of fusion power in my world, helium may be readily available in large quantities. At this point, I'm preferring helium.

I am aware that 50,000 tonnes is half the displacement of a modern super carrier, but those are predominantly made from steel. My airship is made predominantly from woven graphene fiber which is a couple hundred times stronger than steel by weight. The lack of steam catapults, arresting cables, and dramatic reduction in the need for radiation shielding compared to a fission reactor also translates into significant weight savings.

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    $\begingroup$ IMHO this is in-topic here, but you'd have a higher chance of getting an answer somewhere else. $\endgroup$ – o0'. Sep 3 '15 at 10:55
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    $\begingroup$ You might want to reconsider a lot of things. The USS Macon was a huge Zeppelin which could carry only six very small biplane fighters and had no armour or on board weapons at all. Instead of an air wing, you might carry a squadron, and instead of fighters, drones or UAV's which are much smaller and lighter in size. $\endgroup$ – Thucydides Sep 3 '15 at 11:31
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    $\begingroup$ I want to know why you want to be lifted by hydrogen when you have a far more powerful lifting power in the form of anuetronic fusion power plants? That's like building a Tesla Model S then insisting it be pulled by horses. $\endgroup$ – Green Sep 3 '15 at 11:55
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    $\begingroup$ You need (50,000 tons * 1,000,000 g/T) of air displacement. If we approximate 1/22g=L of hydrogen, and air at 1.19g/L each liter of displacement gives you (1.19 - 1/22) grams of air. So you need: 50,000*1,000,000*(1.19-1/22) = 57227272727.3 liters of hydrogen displacement of air to get your lift. This results in a sphere with a radius of 250 meters. However, I expect that your 50,000 ton estimate is VERY low for a functional airship like this, a Nimitz class carrier displaces 100k tons and this is nearly guaranteed to be more. $\endgroup$ – enderland Sep 3 '15 at 13:06
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    $\begingroup$ Jet fighters and a gasbag of Hydrogen leaking out all around are a very ...volatile....combination. $\endgroup$ – Oldcat Sep 3 '15 at 16:52
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You need (50,000 tons * 1,000,000 g/T) of air displacement. If we approximate 1/22g=L of hydrogen, and air at 1.19g/L each liter of displacement gives you (1.19 - 1/22) grams of air. So you need: 50,000*1,000,000 grams / (1.19-1/22 liter/gram) = 43,685,464,654 liters of hydrogen displacement of air to get your lift.

This results in a sphere with a radius of about 250 meters, give or take a few. Or a diameter of 500 meters (about 1500 feet).


The story is not over though.

This is all assuming a few key things:

  • Changes in air densities don't matter as you climb
  • 50k tons is the right amount

I expect that your 50,000 ton estimate is VERY low (perhaps by multiple orders of magnitude) for a functional airship with what you are describing. A Nimitz class carrier displaces 100k tons and this is nearly guaranteed to be more. Your airship is basically an entire aircraft carrier plus a ton more equipment.

Since you are going to be flying an insanely huge floating bomb, combined with all sorts of things that can go wrong, you are going to need large amounts of protection equipment/etc.

Some additional insights as to why it's going to be heavy:

The airship needs to be large enough to carry at least 40 strike fighters with a maximum takeoff weight of roughly 20 tonnes each, and at least two helicopters with a maximum takeoff weight of 10 tonnes each.

So these strike fighters are slightly heavier than F-18's, so you can guesstimate that the flight deck characteristics of this airship will be similar to a carrier's.

Keep in mind that you will need lots of supporting... personnel and living quarters and other amenities for life for this ship. It's probably ever harder to refuel/rearm than a carrier.

An aircraft carrier services less than 100 planes and yet has a crew of many thousand. This airship will be no different.

The airship uses hydrogen as a lift gas with an inert gas (possibly helium) surrounding each hydrogen gas bladder.

How does it land? Does it land? If so, you need a hugely complex internal support structure to land. It's not trivial to just drop a 50k ton object that is about the size of most skyscrapers (or larger) onto ground somewhere to land/rearm/refuel. The internal structure is going to have to support this, which means more weight and more volume.

A carrier gains benefit because water also provides support. Ships need much less structural integrity as a result, because the water is a very firm foundation/base. Air is not the same, so you again are going to need to take more care to ensure the structure is self supporting.

You are also going to need an incredibly complex altitude regulating system. Carriers just displace enough water that a bunch of stuff sticks out, so as they gain/lose 10,000 tons of weight all that happens is the ship sits lower/higher. An airship... with not have this advantage, at all. So your airship will need some way to regulate its altitude, by either compressing hydrogen (and pumping oxygen in/out) or otherwise having insane fans/lift.

The airship's envelope is made from graphene able to withstand hits from shell fragments and 30mm autocannon fire.

Lets just use that 25mm steel as a frame of reference. The volume for 2.5cm around a 250M radius structure is:

  • 3.14*(250^3 -(250-0.025)^3) = 14700 cubic meters

Steel weighs about 7850 kg/meter^3. Or about 4 tons/meter cubed. So just in your protective shell you have about 50,000 metric tons of steel.

Add more if you are doing a non-sphere shape as a sphere is the most efficient volume for a shape.

The airship is heavily compartmentalized to prevent the spread of fire and uses automated gaseous fire suppression equipment throughout.

Fire isn't your primary problem, explosions are your problem. Electrical shortages, battle damage (one missile => boom?), sabotage, etc. Fire onboard your ship is death through explosion.

The airship's reactor and ammunition magazines are encased in at least 25mm of face hardened ballistic steel on top of any required radiation shielding.

This is probably less needed since you have a ship the diameter of 5 football fields, so you can just put all the explosive stuff in the middle and figure if something gets there you're already screwed.

The airship needs to have 360 degree turret coverage for its point defense weapons systems.

This is going to be super weight intensive since your ship is... huge. Keep in mind the more weight you add, the more volume you need, and the more volume you need, the more coverage/etc you need.

The airship's flight deck should ideally be long enough to land the above mentioned fighters without arresting cables. If they overshoot, they can just fall until they regain enough airspeed to try again. Likewise, there is no need for catapult launches.

This might work, but if you are high enough for this to happen that also means your air density is very low. This means your ship is... going to be even larger.

Notice that in this picture from here:

enter image description here

air density decreases pretty significantly even only going up 1000 meters. All those calculations above are considered at sea level, as you go up, you need even more displacement for your ship to float.

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I realize that I'm not only answering your question (enderland posted some numbers in his comment, and see my first bullet point), but here are my two cent, anyway ...

  • Square cube means that area scales with $length^2$ and volume scales with $length^3$. Scaling this one up, you would get 0.75 miles length.
  • Those strike fighters will need fuel and ordnance. Say 20 tons takeoff weight are actually 10 tons of fighter and 10 tons of consumables. Call it five missions where all fuel and ordnance are consumed and ten missions where five tons of fuel are expended. Each strike fighter gives you 110 tons. Most of that is fuel and missiles, which require damage-proof storage.
  • Is this hydrogen/helium combo really worth the complication? Hydrogen doesn't burn if there is no oxygen, so it will only burn if the gas cells are ruptured. If the internal partitions are splinter-proof, the airship gets too heavy.
  • Armor against 30mm autocannon is about the level of a Bradley IFV. Getting that airborne is a tall order. Attack helicopters are sometimes billed as being able to come home after a few autocannon hits, but that's not the same.
  • Face-hardening 25mm of armor won't make much of a difference.

Summarized, for a this gets downchecked.

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