Volume
Calculations
for Cylinder Shaped Model Hot air
Balloons The simplest way to calculate areas for circles
and ovals is to compare them with squares.
The perimeter is the most elegant geometric
measurement.The quarter-perimeter though is
usually more practical to use. NOTE: If the Perimeter of a circle, ie.
its circumference, is divided by its diameter,
the result is 3.1416..., or Pi. So, the Perimeter
/ 2 Pi equals the radius.
Similarly, 2 * Quarter-Perimeter / Pi
also
equals the radius. Area of a Square =
Quarter-Perimeter ^ 2 (ie. the
square of one of the sides) Area of a Square = Perimeter
^2 / 16 Area of a Circle = Pi * Radius ^2
= Pi * ( 2 * Quarter-Perimeter / Pi )
^2 = 1.273 * Quarter-Perimeter ^
2 ( ie. 4/Pi) Area of a Circle = Pi * (
Perimeter / 2 * Pi ) ^2 =
Perimeter ^2 / 12.566 ( ie. 4 *
Pi ) ( ie. 16 / 1.273) So, with equal perimeters, circles have around
27% more area than squares.Moderate ovals have
around 20% more area than squares, and wide
ovals have around 15% more area. So, ovals
have around 5 - 10% less area than
circles. What if the scale changes? Twice the
perimeter gets four times the area. Four times
the perimeter gets sixteen times the area.
So the change in area equals the square of the
change in perimeter. Basically scale and
measurement are the same thing. Table
for Approximating Balloon Volumes The following table assumes
that balloons are round. To account for
the material at the top of the balloon, and
derive an "Effective Height," the
presumed radius is subtracted from the "Material
Height." In reality, this
overcompensates. But it does offset for a
small amount of tapering and ovalness. It
would be more accurate to subtract around 80% of
the radius. NOTE: The table is set up to be
mathematically consistent, not to correspond
directly to actual balloon volumes. To
adjust for highly oblong balloons, volumes can
be discounted by 5-10%. To adjust for
tapering, half-perimeters can be
discounted. So, in general, assume the
calculations to over-approximate actual finished
balloon volumes by up to several cubic feet. Starting
Approximations for Volume, Material Area and
Weight for Different Sized Balloons
Table
for
Suggested Number of Candles for Different
Sized Balloons *Classic
Dry
Cleaner Bag Balloon (Typically
powered by around twenty birthday candles) As described, a balloon's "Effective Height" is
approximated by subtracting its presumed radius
from its "Material Height." Since more
width increases the radius, the different
balloons are not really in scale with each
other. Blimp shaped balloons can be calculated as
either tipped over cylinders, or as short fat
cylinders.Either way the volume calculations are
fairly close to each other. In both cases,
to account for the extra ovalness and tapering,
the volume should be discounted more than for
rounder balloons. To summarize, if a balloon is made wider, and
its effective height stays the same, the volume
changes by the square of the increased
width. If a balloon is made taller, and
the width stays the same, the volume change is
the same as the increased height. NOTE: The Area to Volume Ratio is
used to approximate a balloon's weight per cubic
foot, based on its material weight. For
lightweight Plastic Balloons, this is usually
not much of an issue. For heavier Paper
Balloons though, this ratio is very
important in evaluating a balloon's overall
aerostatic potential. Geometry of
Scaling -- For Dimension, Volume and Surface
Area
Cylinder shaped balloons are
in scale with each other if the relative changes
in the height and the width are the same.
The following formulas can be used to compare
scaled balloons, or any other scaled
three-dimensional objects: Change in Volume = Change in
Dimensions ^ 3 Change in Volume = Change in Surface
Area ^ 1/2 ^ 3 =
Change in Surface Area ^ 3/2 NOTE: Mathematically, "Powers are
Powered" by multiplying the exponents times each
other. The formulas can be demonstrated, by
alternately doubling the dimensions, the surface
area and the volume: Base Two Geometric
Sequence and Exponent Table -- Useful for a
Doubling the dimensions increases the surface
area by four times (ie. 2^2) and increases the
volume by eight times (ie. 2^3). Doubling the surface area increases the
dimensions by 1.414 times (ie. 2^1/2) and
increases the volume by 2.828 times (ie. 2^1/2^3
or 2^3/2). Doubling the volume increases the dimensions by
1.26 times ( ie. 2^1/3) and increases the
surface area by 1.587 times Summary: A range of changes
in dimensions, area and volume can be calculated
from the following table:
As described "Powers are
Powered" by multiplying the exponents times each
other. For balloon mathematics this
appears to be the only application.As examples,
to square a number, the exponent is
doubled. To cube a number, the exponent is
tripled. Equivilently, the square root
gets 1/2 the exponent, and the cube root gets
1/3 the exponent, etc. Try it. There
are many examples on the exponent table.
One example is that 4^3/2 = 8. Another is
that 1.682^2 = 2.828 Also, numbers can be multiplied times each
other by adding their exponents. They can
be divided by subtracting their exponents.
For balloon mathematics this application is not
really used. But it is important to know
anyway, to understand how exponents work.
As examples, 1.189 times1.682 and 1.26 times
1.587 are both equal to one. Other numbers besides two can also be used as
the base for a geometric sequence,
including those on the Two Scale.
As example, if four is used, each
exponent gets half its previous power. If
the square root of two is used, ie. 1.414,
then each exponent gets double its previous
power, etc. Hence scales can be shrunk or
enlarged and still be compared to the Two Scale.
Alternately, any other number, definite or
indefinite, except for zero, can be set up as
the base, with the changes calculated relative
to it.This can be done simply by declaring that
a certain value is now to be considered as
equivilent to one "unit." NOTES ON
TERMINOLOGY: 1) Saying 2^1/2 (power) is the same as
saying the square root of two. Saying
2^1/3 (power) is the same as saying the cube
root of two, etc. Technically the term
"root" refers to the specific number which
when multiplied by itself a certain number of
times yields the starting number. Hence
saying the half root of two would technically
seem to mean 2^2 (power). Hence, the term
"root" is avoided here, except as a shorthand
way of referring to the 1/2 and 1/3 powers. 2) As a general concept, or in groups,
the term "power" is usually called an
"exponent." Similarly, a "powered function" is
usually referred to as an "exponential
equation." This wording convention is
probably used because using the word "power"
outside of the specific context of raising a
number to a power might tend to get confusing. NOTES ON
SCALING CONCEPTS: Sometimes model designers use a Base Three
Scale, ie. 3, 9, 27, 81, etc. From
here combinations of the two and three scales
can be used to create a Double Scale.
Hence, using a Single or Double Scale
System the simplest scales are: ( 1.414 )23468912 16
18242732 36 48 54
64 7281 96 108
128 144 ... People who build models might recognize some of
these numbers. On a practical level
though, only the smaller scales would actually
be used. But why bother to use numbers
along the so-called alternate scales
sequence?. Basically because the
calculations tend to work out more evenly, and
comparisons are easier to make, so the models
are simpler to design, test and manufacture. By Thomas Taylor -- balloons@overflite.com |
www.overflite.com balloons@overflite.com - - - - - - |
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