[The Following is an excerpt from a letter to Allen Meece]
[Updated 4 September 2002]
I've gotten a better equation fit for air pressure at altitude
h.
P(h) = P.o exp( -a
h^2 - b h )
where P.o is the air
pressure at sea level
a
& b are decay constants, a = 1.62e-9 1/m^2, b = 0.000111 1/m at the equator
This intertwined exponential function will give air pressure
to within +/-3mb at all altitudes, year round. Considering that yearly
variation in air pressure at a given altitude is as much as +/-2mb, I think
that's a pretty good fit.
Air Density and Unit Bouyancy (lift per cubic meter of
lift cell volume) are similarly well behaved, though they do vary slightly over
the course of the year due to air pressure.
Essentially, Unit Bouyancy is just the air density minus the
density of the lift gas, assuming a massless aerostat. As you can see, the
lift gas density (and thus the difference between the air density and unit
bouyancy) becomes vanishingly small at high altitudes. Thus, larger
balloons are required. The yearly variation in bouyancy and air density
with altitude also drops with altitude, as changes due to the
dropping pressure become greater than yearly temperature
fluctuations. The variations become quite small by the top of the
troposphere, and it takes the dramatic tewmperature variations above the
thermocline to make them grow large again.
These variations above the thermocline can decrease our
bouyancy by as much as 10% over the course of a year, assuming we deploy to
around 20km. This corresponds to a drop in altitude of less than 1km over
the course of the year due to temperature variation alone. This would put
our minimum deployment altitude at 18.5km, to keep us over the
tropopause even without active bouyancy control.
Daily variations are not as dramatic.
I've been comparing other airships to the raw size
required of the VBP. If we try to minimize its size during the
descent so that we can get around the drag problem, it can be fairly small, but
upon reaching the upper levels it's going to be huge, comparing in size to
Lindstrand's proposed CargoLifter 160 airship, which when constructed will be
the largest airship ever flown. A 20T VBP platform will require 270000m^3
of lift cells, compared to 430000m^3 for the CL160. However, if we try to
realize a smaller cross section by extending the length, a platform with an
equivalent cross section radius of 20m or less would have to
be longer than the CL160. Increasing the equivalent radius of
the VBP to 25m would only reduce this length by a third. To go to 30m
or more will require a much larger rope to pull us through the upper level
winds, decreasing our payload.
The VBP leaving the ground isn't going to look like a balloon
on a string. It's going to look like a stick or a pancake! :)
(At least, compared to its final size.)
