Airship to Orbit [Was: Cities in the sky]

Hi Andy,

thanks for your reply. Your line of reasoning is not bad, and basically
what JPAerospace
suggests. But many experts don't buy it.

Also note that my line of reasoning was based on your posting which said

>The article points to the use of an ion-engine to slowly accelerate the
>craft once lift=0, in the very thin high atmosphere. I guess as speeds
>increase the dynamic lift increases and you can s-l-o-w-l-y fly the last
>bit.
>
So I left out all the buoyance parts.

Andrew Goddard wrote:

>Hi Axel,
>
>Ignoring the stuff on electrostatic lifters (as I can't see these as
>more than toys for the amateur scientist - but don't mind being proved
>wrong in the future ;-) )
>
Well, in my opinion that is the interesting part. Maybe we can get back
to that later.
(I may be wrong, but I would like to know why.)

> the flight into orbit /is/ surely practical
>for ion engines. Here's why:
>
>A balloon floats when its mass is equal to the mass of the air its
>volume displaces. This buoyancy is called static lift. If the balloon's
>mass is less than the mass of the air in this volume, it will rise. It
>will only stop rising when the air pressure (and therefore the air
>density) drops to the point where the mass of volume of the air it
>displaces masses the same as the balloon. Currently, NASA high altitude
>balloons peak at about 50km. At this altitude the atmospheric pressure
>is around 0.75mb, and the density about 1.25 grammes per cubic metre,
>about a thousandth of that at sea level.
>
Ok.
But there are some very practical limits on the height you can achieve.
Buoyancy force is the weight of the gas displaced by the balloon. Minus
the weight of the balloon.
So close to vacuum, the weight of the displaced gas is effectively zero.
Theoretically you could make the balloon larger and larger to make it
lighter, but the relation is
not very favorable.
[As soon as your ballon get's larger than earth, you are definitely on
the unrealistic side ;*) ]

Also the main problem to be solved is not to go to orbital height, but
to go to orbital speed.
(I guess you know that, do you?)

>But there's another form of lift: dynamic lift. A 747 doesn't have the
>power to take off vertically (like our balloon) - it only has a thrust
>to weight ratio of around 35%, and it therefore has to rely on dynamic
>lift to fly. This is produced by high-speed air flow over its wings;
>creating, during take-off, enough lift to exceed the mass.
>
I know. But unfortunately this high lift works best for low speeds,
speeds of a not too big
fraction of the speed of sound. Wings for supersonic aircrafts already
look different, and
get a lot less lift. And beyond say fife times supersonic speed not much
dynamic lift due to
the shape of the wing is left. (This is basically what some people on
the net claim to be
the opinion of experts.)

You might try to tell me that speed of sound at high altitudes is much
higher, but from
kinematic gas theory an average speed of 5 km/s of oxygen O2 molecules
is equivalent
to about 40000 Kelvin temperature. I'd say that speed of sound is
roughly related to
the speed of gas molecules. So even if you tell me temperature of high
atmosphere
is 1000 degree Celsius, i.e. 1300 degree Kelvin, 5 km/s is very, very
much supersonic speed.

You might also tell me that a certain amount of this lift is not related
to the shape of the
wing (bernoulli gas stuff) but only depends on the angle of attack of
onstreaming air.
Which is a point I've not finally settled upon what the correct answer is.
This part might be available at high speeds.
But somehow I think it is just a way to convert forward momentum into
upward momentum.

We are looking for a situation where both forward speed and height increase.

>Imagine a properly designed vehicle (one that is buoyant, capable of
>reaching 50km, capable of creating dynamic lift, and carrying
>low-air-density propellers and an ion engine). Assuming its bulky (full
>of drag) let's say it couldn't travel at more than 10m/s at sea level.
>But drag is proportional to the square of the velocity and to the
>density of the air around it, so at 50km it could speed up to around
>320m/s.
>
Slowly, please. We are discussing the buoyant part now, right?
Then the vehicle buoyant at see level is much smaller than the one at 50 km.
So area for drag is increased too.

What's the general behavior of this relation?
Atmoshere density reduces exponentially with height: density ~ e^(-h)
So volume of vehicle grows exponentially with height: volume ~ e^(h)
[This neglects structural mass of the ship which for technical reasons
tends to grow with size.]
Effective area of drag grows a little less: sqare of the cubic root of
volume. (Diameter is about cubic root,
area is approx. sqare of diameter, assuming shape of ship stays the
same, i.e. optimal for
minimal drag.)
area ~ volume^(2/3)
Drag is basically proportional to square of velocity, as you say (but
note exceptions near
sound barrier and corrective factors for supersonic flight). And linear
with air density, which
decreases exponentially with height.

drag ~ area*density*v^2

Idea of JPAerospace is to choose height so that drag remains small
enough to continue accelerating.

So let's set drag to be constant (as you did in your example), and since
we care only for the relation
and not for factors we say it's 1.
Then, putting it all together:

v^2 ~ 1/(area*density)

v ~ (1/(area*density))^(1/2)
~ (1/((volume^(2/3))*density))^(1/2)
~ (1/(((e^(h))^(2/3))*(e^(-h))))^(1/2)
= ((e^(h))/(((e^(h))^(2/3))))^(1/2)
= ((e^(h))*(((e^(h))^(-2/3))))^(1/2)
= (((e^(h))^(1/3))^(1/2)
= (e^(h))^(1/6)

So if I didn't make any mistakes, your estimate is missing a 6th root.
I assume your computation of air density is right (maybe you could give
us your calculation?).
Then yes, the airship could go faster, but instead of your estimated
32 times faster, it would go only 1.8 times faster, or let's say twice
as fast.

Putting in my own numbers I get a speedup of 3.

For your example of a sea-level speed of 10 m/s, we would need a speedup
of about 800
to go to orbit. So the airship would have to go to 300 km on buoyance.

And on the other side of this concept: ion engines can't push an airship
with 10 m/s
on sea level!!! So you need to add many zeros to the speedup, which
results in even
more unrealistic heights of buoyance.

So I don't think this is the road to success. At least not with standard
ion engines.

> If well-designed, with enough dynamic lift it could be flown to
>higher altitudes. As the air thins, it can go faster, eventually
>requiring ion engines to take over when propellers become inefficient.
>Of course, the faster it goes, the less the effect of gravity on the
>vehicle and therefore the less reliant on dynamic lift it becomes. For
>altitudes around the 100km mark:
>
>1000 2000 3000 4000 5000 6000 7000 V m/s
>1.6 6.5 14.6 26.0 40.6 58.4 79.5 % gravity
>"savings"
>
I know.

First let me point out that gravity cancels out as buoyance is concerned.
But I guess our discussion is now back on the dynamic part of the flight?

My feeling is that a speed of around 5 to 6 km/s is most critical.
We still have most of the weight, but drag is already very high.
Convince me that it works for this speed, and I'll believe everything.

>I've seen thrust to weight ratios as low as 1:100000 for ion engines,
>
Yes, and engine weight must be a very, very small fraction of the
airships weight. So add some
zeros for the thrust to weight of the airship.

>but the reaction mass is low and energy readily available, so I can see
>
In fact I feel that energy is a critical point. Why do you say there is
(more than) enough?

P.S.: Andrew, you got me to think a few new thoughts about it! Some real
good points in your post. Thanks.