Mathematical
Models For Candle Balloon Design  overflite

All of the Overflite Balloon Designs
 (2)
are
based on the concept of a Theoretical
Universal Balloon. Here, "Virtual
Balloons" are imagined to exist
mathematically. They can be any size.
They can be powered by any number of
candles. Their burntimes are flexible.
The results are assumed to be a Reasonable
Proxy of Reality. Even though the
actual reality is known to be somewhat different,
the calculations are close enough to design
balloons, and to get predictable results. As the basis for theoretical design, different
sized balloons are imagined, all heating to the
same temperature. As balloon volume
increases, the bag, frame and engine weight
increase too, but by less than the increase in
volume and gross lift. Hence, the relative
net lift goes up, and the engine burntime can be
increased, by adding on more wax. See Design
Synopsis Contents of Page: Geometric Mathematics General Heating
Properties of Candle Powered Balloons Geometric
Mathematics The mathematics of the
physical world is Geometric. Here,
the only number with a static reality is One.
All of the other numbers can be viewed as
multiples of one. Their realities are Dynamic.
The
simplest dynamic number is Two. It
can be created geometrically from one, by Doubling
. Each doubling of one is a Step,
or Magnitude. The number of steps is
called an Exponential Power. The
simplest description for how the physical world
works is the Base Two Geometric Sequence:
Base Two Geometic Sequence and Exponent Table
General
Heating Properties of Candle Powered Balloons
Candle powered balloons heat up
quickly at first, then more slowly. Finally,
if the air is perfectly calm, balloons reach a Peak
Equilibrium Heat Rise, over the ambient
temperature. Here, the heat input from the
candles is exactly equal to the heat that is
liberated by the balloon. Hence, the rate of
heat loss is a highly accurate proxy for the
heating rate. The Maximum Sustainable Heat Rise depends
on the wind conditions. It is obviously less
than the Peak Equilibrium Heat Rise. Heat
loss by conduction is less. Heat loss by
convection is more. Heat rise can still be
viewed as an equilibrium though, since long
heating times are assumed. Heat loss can
still be viewed as an accurate proxy for the
heating rate. The Minimum Sustainable Heat Rise is more
important. When balloons are launched, they
need to be able to recover quickly from wind
blasts. Hence, the assumed basis is how hot
balloons get in a reasonably short heating
time. Here the heating rate is greater than
the heat loss. Technically this means that
the heat loss is no longer a highly accurate proxy
for the heating rate. For modelling purposes
though, it is still close enough to the actual
reality to design balloons. To compare balloons with different volumes and
candlepowers, the mathematical model assumes Comparably
Equivilent Heating Times. Here, large
balloons are "allowed" more time to heat than
small balloons. In a sense though, this
works as a builtin selffulfilling prophecy,
since at some point in the heating time, the heat
rise will match the prediction. The model also assumes that different sized
balloons will have similar ratios between the
different measurements for heat rise. But
this is not strictly true. As balloons get larger
the different measurements would seem to
converge. Hence, larger balloons probably
have a higher Minimum Sustainable Heat Rise than
predicted by the model. With more candles, or less volume, the heat rise
is more, and the heating rate is faster.
With fewer candles, or more volume, the heat rise
is less, and the heating rate is slower.
Basically what happens is that balloons store up
heat energy, over time, until they reach a maximum
capacity, where the rate of heat loss becomes
equal to the rate of heat input. If the candles are increased, then the heating
per candle goes down. This demonstrates that
the heating is less efficient. Heat loss
goes up, both through conduction and
convection. But the rate is less than the
candle increase. More heat energy is stored,
and the gross lift is more. Alternately, if the volume is increased, with the
same number of candles, then the heat rise will go
down, but by less than the volume increase.
Hence, the gross lift is more. This suggests
that the heating is more efficient. But more
so, what is really happening is that the balloon
is storing up more heat energy, over a longer
length of time. In cold weather, the heat rise goes down,
slightly. This is because there is more air to
heat. Equivilently, in hot weather,
or at high elevations, the heat rise goes up,
since there is less air to heat. The
effect is assumed to be around 1 % per 10 degree
change in ambient temperature, and around 1 1/2 %
per thousand feet of elevation. The
assumption here is that the heat rise changes by
around the square root of the change in the mass
of the ambient air displaced by the balloon. Geometric
Model for Equivilent Heat Rise Different sized balloons are
imagined, all in scale with each other. See Scaling.
Each balloon is heated by the number of candles
that makes it heat to the same temperature as all
of the other balloons. In comparing the
volume and surface area of any one balloon against
the others, the following equations are true: Change in Volume = Change in
Dimensions ^3 Change in Volume = ( Change in
Surface Area ^1/2 ) ^3 = Change in
Surface Area ^3/2 Since the heated temperature is the same for all
of the balloons, the heat loss through conduction
changes by the difference in surface area.
This is equal to the change in volume^2/3.
This suggests that the difference in candles
should probably also equal around the change in
volume^2/3. If so, then the airflow into and
out of the balloon will change by this rate too,
and the heat loss through convection will probably
also change at around this rate. All this
suggests the following formula: Change in Candles to get Equivilent Heat
Rise = Change in Volume ^2/3 In reality though, larger balloons have less
relative air turnover than small balloons.
This means that the air flowing out of the bag has
more time to cool, so less heat is lost than
predicted. Hence, larger balloons should get
roughly equivilent heating with somewhat fewer
candles than predicted by this simplified
mathematical model. For simpler balloons
this doesn't make much of a difference. For
advanced balloon designs the exact rate of roughly
equililent heating has an effect on Balloon Duration.
Change in Candles to Get Roughly Equivilent Heating  Approximated at Change in Volume ^ 2/3
Suggested
Number of Candles for Different Sized Balloons
The mathematical model is based
on a five cubic foot, 4 1/2 foot tall Dry Cleaner Bag Balloon,
heated by 20 birthday candles. At ambient 69
Fahrenheit (529 Absolute), the air weighs 635/529,
or 1.20 ounces per cubic foot. The heat rise
is liberally assumed to be 139 degrees.
Here, at 208 Fahrenheit (668 Absolute), the air
weighs 668/635, or .95 ounces per cubic
foot. Hence the gross lift is around .25
ounce per cubic foot. See How Hot Air Balloon
Lift is Calculated. At 28 Fahrenheit (488 Absolute), the air weighs
635/488, or 1.30 ounces per cubic foot. The
heat rise goes down by a few degrees as described,
and is assumed to be 134 degrees. Here, at
162 Fahrenheit (622 Absolute), the air weighs
635/622, or 1.02 ounces per cubic foot.
Hence the gross lift is around .28 ounces per
cubic foot, or around 12% more. As a practical matter, especially where the lift
is significantly more than the weight, it does not
matter all that much exactly how many candles are
used. Fewer candles will work out perfectly
okay. In hot weather though, it is not a
good idea to use more than the suggested number of
candles, since the balloons could heat up past the
melting point of the plastic. Estimated # of Candles for Equivilent Heating of +130 to 140 degrees Farenheit = Around 7 * (Volume ^ 2/3)
Arithmetic
Model for Candle and Volume Changes
Two experimental balloons are
constructed. Both are in scale with each
other. One has exactly twice the volume of
the other. Both are heated by the same
number of candles, and the different heat rises
are estimated. The doublevolume balloon
appears to get around 70% of the heat rise of the
singlevolume balloon. So now the doublevolume balloon gets a
doublecandlepower engine. This heat rise is
estimated. It appears to be around 170% of
the heat rise from the singleengine, or 119% of
the heat rise of the singlevolume balloon, ie.
170% of 70%. To make the doublevolume balloon get the same
heat rise as the singlevolume balloon, the 70%
heat rise needs to increase by around 43%.
This should happen with 60% more candles, since
the heat rise "yield" is around 70% of the candle
increase. So now the doublevolume balloon gets
around the same heat rise as the original
balloon. This process can be repeated, to
imagine larger or smaller scale balloons, or it
can be fractionalized. The results are reasonably
close to what actually happens. But
mathematically, the arithmetic model only works
precisely when the volume is doubled. Geometric
Model for Candle and Volume Changes If the arithmetic model
is converted to a geometric model, it can predict
theoretical balloon heating over a wide range of
volumes and candlepowers. Here the estimates
are compared with the Base Two Exponent Table:
Base Two Exponent Table
The estimated heating for the doublevolume
balloon, ie. 70% equates to 1 / 1.41,
ie. 1 / ( 2 ^1/2). Change in Heat Rise = Change in
Candles ^3/4 / Change in Volume ^1/2
Candle Equalizer Power, ie. 2/3 =
Volume Power, ie. 1/2 / Candle Power, ie.
3/4 The Candle Power is more efficient if it is
higher. The Volume Power is more efficient
if it is lower. The Candle Equalizer Power
is more efficient if it is lower. As
demonstrated by the Geometric Model for Equivilent
Heat rise, it makes sense that it should be around
2/3. As long as the ratio between the Volume
and Candle Powers is 2/3, then the Candle
Equalizer Power will also be 2/3. As example, the Candle Power could become less
efficient, while the Volume Power becomes more
efficient. With a small balloon, or with a
balloon with a relatively high candle to volume
ratio, more candles would seem to increase the
heat rise by less than expected, while more volume
would seem to reduce the heat rise by less than
expected. Hence the Candle and Volume Powers
would both seem to drift down. Alternately, the Candle Power could become
more efficient, while the Volume Power becomes
less efficient. With a large balloon, or
with a balloon with a relatively low candle to
volume ratio, more candles would seem to increase
the heat rise by more than expected, while more
volume would seem to reduce the heat rise by more
than expected. Hence, the Candle and Volume
Powers would both seem to drift up. In reality, the Candle Equalizer Powers should
tend to be less than 2/3. As described,
large balloons lose relatively less heat by
convection than small balloons. In addition,
they will lose less heat when they get pummelled
by the wind. For comparison purposes, the
Candle Equalizer Power can be idyllically imagined
as drifting down towards around 3/5. Another
adjustment is to be liberal rather than
conservative when making assumptions about the
Minimum Sustainable Heat Rise. Balloon
Duration Candle balloons are fuel
consumption machines. The weight of the
engine can be viewed as the multiple of its
consumption rate and its "minutes of burntime."
Similarly, the bag and frame weight, the expected
gross lift, and the expected net lift can be
divided by the consumption rate too, to be
converted into "equivilent or theoretical
minutes of burntime." As example, if a balloon volume is doubled, then
60% more candles should double the lift too.
Meanwhile the total weight increases by around
60%, if the engine burntime stays the same.
Hence, the total balloon weight can be increased
by up to around 25%, by adding extra wax onto the
engine, thereby increasing the burntime of the
balloon. Duration can be analyzed in terms of the Candle
Equalizer Power. For modelling
purposes it is assumed to be 2/3. As an
idyllic comparison, it is also presented by what
would happen if it were 3/5 instead:
As point of inquiry, the question is: How
much does the volume have to increase to double
the Theoretical Minutes of Burntime? Here,
if we assume the Candle Equalizer Power to be 2/3,
the answer is 8 times. The balloon lift
increases by 8 times. The weight increases
by 4 times. Hence the balloon weight can be
doubled. This leads to the following
formulas: If the CEP = 2/3, then the Change in Balloon
Duration = Change in Volume ^1/3 Generically, through some calculus process, the
equation can be written as follows: Change in Balloon Duration = 1 / (
Change in Volume ^ ( Candle Equalizer Power  1)
) Change in Balloon Duration =
Change in Volume ^ (  Candle Equalizer
Power  1  ) Calculations
for Balloon Duration Both the 36/pack and the
24/pack birthday candles weigh approximately an
ounce per pack. For modelling purposes, a
36/pack candle is assumed to burn for
approximately 10 minutes, while a 24/pack candle
is assumed to burn for approximately 15
minutes. Hence, in both cases a single
candle burns approximately 1/360 of an ounce per
minute. Twenty birthday candles burn
approximately 1/18, or .0556 of an ounce per
minute. The twenty candle five cubic foot dry cleaner bag
balloon is assumed to lift approximately 1.25
ounces at 69 degrees Fahrenheit, and approximately
1.40 ounces at 28 degrees Fahrenheit. Hence
it would take 22.50 minutes for the candles to
burn through 1.25 ounces of wax, and 25.20 minutes
to burn through 1.40 ounces of wax. These theoretical
minutes of burntime are allocated between
the different components of the balloon. For
the bag and frame, at 7/10 of an ounce, 12.60
minutes of burntime is accounted for. The
engine accounts for another 10.00 minutes.
In the case of the 28 degree ambient balloon, the
net lift of .15 ounces accounts for another 2.70
minutes of burntime. Theoretical Minutes of Burntime  If Candle Equalizer Power = 2/3  At 69 Fahrenheit Ambient Sea Level
How does the table get used?
Basically the weight of the bag and frame gets
converted into theoretical burntime, then gets
subtracted from the total theoretical
burntime. Also, for larger balloons a
certain number of minutes should get subtracted
for net lift, since the heating takes
longer. The number of minutes left is how
long the candles can burn for. If the bag size increases in scale, its
theoretical burntime will stay the same. In
reality though it typically would get fatter,
which would reduce its theoretical burntime.
But as a practical matter the difference isn't
much. With the 1/3 mil material, being lighter, 9
minutes of burntime is a reasonable
subtraction. For designing paper balloons,
though, which are heavier, 20 to 30 minutes of
theoretical burntime is probably called for.
Hence 30 to 40 cubic feet would apparently be
needed to get a birthday candle powered paper
balloon off the ground. Stay tuned for more
information here. The table assumes the Candle Equalizer Power to
be 2/3. In reality though it is less.
Hence the numbers are conservative... Stay Tuned For Advanced
Calculations on Model hot Air Balloon
Design

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