[The following is an excerpt from a letter to Allen Meece]
[Updated 4 September 2002]
If we start our ascent with a fraction of the lift cells fully
inflated, these will have to vent air during the ascent to keep from
bursting.
The rate at which they must vent pressure will vary with
altitude. The ambient air pressure at any given altitude is
roughly
P(h) = P.o exp( -r.h h )
where P(h) is the pressure at altitude
h
P.o is the pressure at sea level (1013mb)
r.h is the atmospheric pressure decay rate (0.00013
1/m)
Thus, to avoid an increase in its internal air pressure, the
balloon must vent at least as much pressure as the decrease in ambient air
pressure. The rate of change of ambient air pressure is:
P'(h,v.y) = -P.o
r.h v.y
exp( -r.h h )
where P'(h,v.y) is the time derivative of P(h), for a
given height and speed of ascent
v.y is the vertical
velocity
Being an exponential function, P' will be greatest when h =
0.
P'(0,v.y) = (0.132mb*s/m)
v.y
This doesn't look like much, until you realize that a free
balloon of the size we're considering can reach ascent speeds of 10m/s. So
can the VBP platform, with no rope hanging under it. At this rate of
increase, at full volume and without venting, the lift cells will exceed 20mb
overpressure (and likely burst) in about 15 seconds. Reducing the
ascent rate by putting tension in the tether will decrease the rate of
overpressure increase and allow time for venting. Also, P' is lower at
higher altitudes for the same velocity.
Note that some upward velocity is needed to reach the target
altitude in a safe amount of time.
How fast can the balloon vent? Well, at any given point
in the ascent:
P.B = n R T / V.B
where P.B
and V.B are the internal pressure and volume of the
balloon
n is the number of moles of gas in the
balloon
T is the internal temperature of the balloon
R is the universal gas constant.
If the volume of the balloon is constant, the time derivative
of P.B is:
P'.B = n' R T / V.B + n R T' / V.B
where n' is the rate of venting in moles of gas
T'
is the rate of temperature change of the balloon
If the gas is vented adiabatically through a nozzle,
conservation of energy says:
E.B + H.B =
E.vent + H.vent
where E.B and E.vent are the kinetic energy per unit volume of
the gas inside the balloon and just after venting, respectively
H.B and
H.vent are the enthalpy of the gas inside the balloon and just after
venting.
If E.B = 0,
H.B
= 0.5 W n' v.e^2+
H.vent
v.e^2 = 2 ( H.B
- H.vent ) / W
n'
where W is the
molecular weight of the lift gas
v.e is the exhaust velocity at the vent
For adiabatic expansion of a gas:
H = u + P / r J
( H.B - H.vent )
= c.p ( T.B - T.vent
)
c.p
= g R / ( g - 1)
J
( P.B / P.vent )
^ ( g / ( g - 1 ) ) = T.B / T.vent
where c.p is the specific heat of the lift
gas under constant pressure
T.B and T.vent are the
temperatures of the gas before and after venting
g is the ratio of specific heats of the lift gas
J is Joule's
Constant for the Heat Equivalent of Work
u is the
internal energy of the gas
r is the gas
density
Note that P.vent = P(h), but, because
there is no heat exchange between the inside and outside gases during venting,
T.vent <> T.air.
Fortunately, we can just solve for it using the pressure to
temperature relation for an adiabatic gas.
These four equations can be solved for every variable in
the equation for exhaust velocity.
v.e^2 = 2 g R T.B ( 1
- ( P.vent / P.B ) ^ ( g / ( g - 1 ) ) ) / W / ( g -
1)
Actually, this equation more closely applies to the exhaust
from a rocket engine than to the venting from a low pressure, low temperature
balloon, because a rocket engine comes a lot closer to true adiabatic
behavior. But at low speeds of ascent it's close enough for our
needs.
The volume of gas vented is:
V.vent' =
A sqrt( 2 g R T.B ( 1 - ( P.vent / P.B ) ^ ( g / ( g - 1 ) ) ) / W / ( g - 1)
)
The rate of venting necessary is related to the overpressure
pressure increase.
V' / V.B = P' / P.B
V' = P' V.B / P.B
= P' V.B^2 / n / R / T.B
= P.o r.h v.y V.B^2 exp( -r.h h
) / n / R / T.B
If V' > V.vent', then pressure builds
inside the balloon. Now, as it turns out, in order for
v.y to be any reasonable speed and still have
V.vent' satisfy this
relationship at the surface takes one of two things: A very high overpressure in
each lift cell, in excess of 100mb, or a very large vent on each lift
cell. Too large a vent, in fact, to be easily handled by remote control,
IMHO.
For example, if the overpressure desired is just 5mb (enough
to hold the balloon's shape under most foreseeable wind forces) and we want to
ascend at 1m/s (enough to get us above the tropopause in 4.5 hours), then the
vent area required is more than 8m^2 at the surface. That's the size of
the mouth on your average hot air balloon. That's ridiculous. If
that's all we want (and it really is all we need), it's far better to start off
only partially pressurized and allow the expanding lift gas to take up the
slack. Venting is problematic at best.
We will be unable to use superpressure cells during the entire
ascent. We can't vent lift cells that large fast enough.
We can vent fast enough with smaller nozzles at higher
altitudes because the ambient pressure is so low. But we must get there
first. We are unlikely to reach those altitudes without passing through
some dramatic winds.
The trouble is, we will need something during our ascent to
a) prop the lift cells firm against the wind forces and keep them from
being compressed and b) hold the shroud in place.
At full inflation, the shroud should cover all of the balloons
and hold them firmly in place. (Preferably it should cover all of the
platform that we'll need access to for changing lift cells, as well.) This
implies some relationship between the exposed surface area of the lift cells and
the surface area of the shroud in contact with them.
As long as it still contains gas to hold it erect, the
perimeter of a lift cells' vertical cross section will not change regardless of
its internal pressure, much as the surface area of a tube doesn't change if you
mash it flat. If the axis of the lift cells can remain the same, then
their exposed area along the side facing the shroud won't change either.
This means that the lift cells won't drag along the shroud when they inflate as
long as their axis can be kept roughly the same.
However, the exposed area along the side facing other lift
cells is likely to change with pressure, as is the exposed area on top of the
lift cell array. Without pressure from other lift cells to hold them in
place, each lift cell would be free to move toward other lift cells in the
array. This could only be accomplished in one of two ways: a) moving
relative to the shroud (read "sliding against it") or b) folding the
shroud. Case a is bad. It causes abrasion, which causes holes.
Case b is pretty much neutral, and as such is preferable to case
a. If we allow them to happen randomly, a is just as likely to happen
as b. But we can intervene and change this.
I had considered propping the lift cell array internally using
pressurized air cells, thereby preventing both case a and b. If we make
them small, we could reduce the required venting to something manageable.
But then we would need some means of emptying them completely for the final leg
of the ascent. It is also possible to arrange for the lift cells to fully
pressurize at some point later in the ascent when P' has dropped to some
reasonable level and we become capable of venting them fast enough. But
this only reduces the duration of the problem we face at launch.
Another alternative is to add tension to the shroud to ensure
that it folds instead of dragging along the lift cells. Afixing the lift
cells to the shroud adds additional strain to the fabric of each lift cell and
creates a point where the fabric is likely to rip.
That leaves tie downs. We could pull the length of
shroud between each lift cell down and force it to fold by strapping it with a
length of rope. Or we could strap the cells together. Either way, we
could release the straps at the appropriate time during flight to keep the
shroud taut while still allowing the lift cells to expand inside. Such a
scheme woud give us some of the strength of a superpressure design.
A system of tie downs can be designed which can be released by
cutting a series of ties at only one key point. This
is already better than remotely controlled vents because of its
simplicity. If the remote for the tie-downs fails, the repair requires
nothing more than a sharp knife.
