The mistake you're making is to assume the two types of craft have similar-enough shapes that you can mix up surface area and volume. But even though a heavier than air craft has a lot less volume, it doesn't have to have less surface area. Planes are pretty flat, and have a much higher ratio of surface area to volume.
A small delicate plane can weigh less than 50 grams per square meter.
Or we could look at planes designed for human-powered flight. Those are ruthlessly optimized so you know they have no more surface area than necessary, and they weigh well under a kilogram per square meter, even if you added a motor on top.
The balloons google was using to lift mini cell towers, at 50 feet wide, had about 2 cubic meters of helium per square meter of surface area. So about two kilograms of payload per square meter. And if you made it bigger you could turn that into five or ten kilograms per square meter without even trying.
Is it extremely hard for a blimp to beat an airliner, which even for a plane has a small surface area? Yes. But lots and lots of other plane designs lose to a big blimp. Some of them even lose to a small blimp. Especially slow planes. And this is only talking about reasonable plane designs.
Is it possible to make a heavier than air aircraft with terrible enough surface area to weight and power ratios that it will make a blimp seem easy to control in bad weather? Sure, I guess. I wasn't saying you couldn't if you read my comment - I was saying you can make a heavier than air aircraft with a LOWER surface area to weight/power, unlike lighter than air aircraft, so you can avoid being knocked around as much, have less drag, etc.
Heavier than air craft are far more versatile in general.
You can never get a lighter than air craft to an overall density higher than air by definition, and that is hugely limiting.
A 50 foot wide balloon (r=25ft), would have a surface area of 7853 square feet (729 m^2) if an ideal sphere. If you add up all of the wing and control surfaces on a 757 [https://www.b757.info/boeing-757-200-specifications/], you get 3992 square feet. Add another several thousand for the fuselage, and you're probably in the same ballpark.
The balloon you're talking about has a volume (assuming a perfect sphere, r=25ft) of 65,449 cubic feet (1853 m^3). Per [https://www.airships.net/helium-hydrogen-airships/], that seems to pencil out at around 4000 lbs of lift for helium, and 4500 lbs of lift for hydrogen (in 'real world' situations), add 20% to be closer to ideal. Or 2.5m^3 of gas per square meter of surface area. But the literal maximum amount of lifting force you can get is 1.01kg/m^3 with helium and 1.2kg/m^3 with hydrogen.
That really isn't much lift for something that big. You could scale it up, but then you're talking more surface area no? a LOT more surface area? We'll figure that out later.
Said 757 weight will vary from 130,000 lbs-255,000 lbs (empty to max takeoff weight), or 59k-72k lbs of payload if configured as a freighter. Each engine produces 36,000-43,000 lbs of thrust depending on model.
So for the 757, it is lifting (payload alone, on top of it's own weight, fuel, etc.) 3.4kg/m^2, and empty, is lifting 7.5kg/m^2. If you look at max takeoff weight, it's hitting 32kg/m^2. Way more if you care about just the airfoils of course. And to hit that takeoff, it is likely going over 200+km/h.
For a balloon to lift the same weight as the 757 at max takeoff weight, you need one with a volume of at least 114520 m^3 (for helium, ideal) or 96388 m^3 (for hydrogen, ideal), which is a minimum of r=30m for helium and r=28m for hydrogen (ideal). That is a sphere approximately 95-100ft in diameter.
That comes out to a surface area of 11309m^2 for helium and 9852m^2 for hydrogen (assuming perfect spheres, which don't happen).
That is 6.1x the surface area for helium, and 5.3x for hydrogen, assuming everything is perfect - and there is zero way you could drag that through the atmosphere or control it in any way like you can a 757 (or a Cessna, even), even if you used the same engines.
And even if you use a balloon big enough to literally lift a 757 at max takeoff weight, you're weight to surface area ratio is just hitting 10kg/m^2. 1/3 of the 757, and that means you have 3x more 'surface' to drag through the air for the same available weight (aka power/airframe) budget.
so you need to be talking multiple max-takeoff-weight-of-a-757 worth of ballon lifting capacity before you start getting in the same ballpark from a raw 'surface exposed vs weight' perspective. If we use weight as a raw proxy for power (roughly probably correct), you get the same setup.
And from a air resistance/drag perspective (what we care about here), it still isn't even all that close due to airfoil shape vs giant spheres. If you're using a blimp/zepplin shape, you're trading off airframe weight for aerodynamics, but it doesn't help as much - you end up having to spend a lot of your weight budget structuring it more like an airfoil, because the density still has to be low, so the shape has to be much bigger, and you have less budget for engines - so the heavier than air craft actually have an even bigger advantage. But even doing these very basic comparisons show it pretty clearly enough.
If you need to move through air faster than the air itself is moving, density helps - by reducing the surface area (and hence impact of these winds) and allowing you to have more engines, or a fancier airframe, or whatever. If you need to resist weather and similar forms of strong, high speed wind currents and changes, you need to be able to move through the air fast, and preferably have a strong frame.
Lighter than air craft are hindered in this by having a cap on their density, and for buildable/practical sized craft, high surface areas to weight ratios (which is a proxy for strength of airframes and available power).
> I was saying you can make a heavier than air aircraft with a LOWER surface area to weight/power, unlike lighter than air aircraft, so you can avoid being knocked around as much, have less drag, etc.
You didn't say that you "can" make a heavier than air craft with a better ratio. You said that "by definition" lighter than air craft will have a worse ratio than any heavier than air craft. That's a very different statement! (And I'm being fair, I'm interpreting "any" as "any reasonable".)
> You can never get a lighter than air craft to an overall density higher than air by definition, and that is hugely limiting.
That's true, but the statement I objected to was that weight:surface-area is worse by definition, not any statement about volume.
> 757 stuff
The problem with that chain of logic is that you're starting with some of the best planes around for surface area vs. weight, and then trying to make a blimp that beats them.
Of course that's super hard to do!
But if you take a slow ultralight plane instead, you'll see that it's not very hard to beat with a blimp. The kind of plane that cruises at 35mph and not 500mph.
The truth isn't that [reasonable] planes automatically beat [reasonable] blimps. It's that planes similar to a 757 beat reasonable blimps. That's a much weaker statement.
There are lots of reasonable plane designs that might only hit 5kg/m^2, and it's easy to make a blimp that beats that. Or the 10kg/m^2 in your math, that's not something that takes unreasonable materials to reach in a blimp.
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tl;dr: If you demand a blimp beat 30kg/m^2, it probably won't happen. But in the 2-10kg/m^2 range, sometimes planes beat blimps and sometimes blimps beat planes, using reasonable designs for both. "[an airship] always has to have a much larger surface area in proportion to the weight it is carrying than any heavier than air craft, by definition." is a false statement.
A small delicate plane can weigh less than 50 grams per square meter.
Or we could look at planes designed for human-powered flight. Those are ruthlessly optimized so you know they have no more surface area than necessary, and they weigh well under a kilogram per square meter, even if you added a motor on top.
The balloons google was using to lift mini cell towers, at 50 feet wide, had about 2 cubic meters of helium per square meter of surface area. So about two kilograms of payload per square meter. And if you made it bigger you could turn that into five or ten kilograms per square meter without even trying.
Is it extremely hard for a blimp to beat an airliner, which even for a plane has a small surface area? Yes. But lots and lots of other plane designs lose to a big blimp. Some of them even lose to a small blimp. Especially slow planes. And this is only talking about reasonable plane designs.