Aerostatic Design Principles for Paper Hot Air Balloons -- Overflite

    Aerostatics is the science of gases in equilibrium.  An Aerostat is a lighter-than-air aircraft.  Well heated hot air balloons can potentially lift around a 1/4 ounce per cubic foot. See Lift Tables.   Paper Balloons though, should ideally weigh less than an 1/8th ounce per cubic foot -- in order to get a reasonable amount of flight time, from a reasonable amount of heating.  Alternately, the inverse of  Weight per Cubic Foot can be used instead.  This is the Number of Cubic Feet per Ounce.

    Aerostatic Design Principles are revealed by the geometric properties of three-dimensional space.  The most important concept here is Scaling, ie. what happens to equivilently shaped objects, when the size changes.  The objective here is to model the mathematical relationships between Surface Area and Volume, for different shapes, at various sizes.  Good luck!!  With this information, plus the weight of the envelope material, the approximate weight per cubic foot can be calculated.

    The Cube is the simplest geometric shape.  It demonstrates the common properties of all three-dimensional objects.  It can also serve as a "proxy," or substitute, for more complex shapes.  So, to design round Paper Balloons, this essay starts with theoretical "Square Balloons," which will then serve to help explain the aerostatics of more complex shapes.

Hot Air Balloon Envelope Materials

    Material Weight Per Square Foot can be approximated, by weighing a quantity of material, then dividing by its square footage.  Alternately, the inverse of this calculation can be used instead.  This is the Number of Square Feet to an Ounce.
So -- Here are some materials that can be used to make hot air balloons:

    Lightweight Dry Cleaner Bag Material, ie. 1/2 Mil Plastic Sheeting...  weighs an Ounce per 30 Square Feet.  (Calculation --  A 4 1/2 foot tall, 2 foot wide, 1/2 mil dry cleaner bag has 18 square feet of surface area, and weighs around 6/10 of an ounce.  So each square foot weighs 1/30th of an ounce, ie. 6/180.)

    Lightweight High Density 1/3 Mil Plastic Sheeting... weighs an Ounce per 45 Square Feet.
(Calculation --  The material weighs 2/3 as much as dry cleaner bag material.  So each square foot weighs 1/45th of an ounce.)

    Tissue Paper... depending on the brand, weighs around an Ounce per 15 Square Feet.

    Tracing Paper... depending on the brand, weighs around an Ounce per 10 Square Feet.

Geometric Properties of Cubes, and Three-Dimensional Space

    Squares and Cubes are the most basic building blocks of geometry.  To compare different shapes, the simplest and most practical common measurement to use is the Quarter-Perimeter, ie. a "Side." Meanwhile though, the Perimeter is also a standard common measurement, and is considered to be more elegant.  Hence, both formats are presented:

Area of a Square  =  Quarter-Perimeter^2
Area of a Square  =  Perimeter^2  / 16

Area of a Cube  =  6 Sides * Quarter-Perimeter^2
Surface Area of a Cube  =  6 * ( Perimeter^2  / 16 )  =  3/8 * Perimeter^2  =  Perimeter^2  / 2.666

Volume of a Cube  =  Quarter-Perimeter^3
Volume of a Cube  =  ( Perimeter / 4 )^ 3  = Perimeter^3  / 64

Area to Volume Ratio  =  ( 6 * QP^2 ) / QP^3 =  6 / Quarter-Perimeter
Area to Volume Ratio  =  ( 6 * (Perimeter^2  / 16 ) )  / ( Perimeter^3  / 64 )  =  24 / Perimeter
 

Quarter-Perimeter  =  6 * Volume / Area
Perimeter  =  24 * Volume / Area

Quarter-Perimeter  =  ( Area / 6 )^1/2  = Area^1/2  / 2.449
Perimeter  =  ( Area / 2.666 )^1/2  =  Area^1/2  / 1.633

Quarter-Perimeter  = Volume^1/3
Perimeter  =  ( 64 * Volume )^1/3  = 4 * Volume^1/3
 

Area of a Cube  =  6 * QP^2  =  6 * ( Volume^1/3 )^2  =   6 * Volume^2/3

Volume of a Cube  =  QP^3  =  ( ( Area  / 6 )^1/2 )^3  =  ( Area / 6 )^3/2  =  Area^3/2  / 14.697
    ( Note that  14.697  is equal to  6^3/2 )

    So, as the size of  a balloon increases, more volume is contained per unit of surface area, and it becomes "lighter," per unit of volume.  As example, if the dimensions are doubled, then the surface area increases by 4 times, ie. 2^2.  Meanwhile the volume increases by 8 times, ie. 2^3.  Hence, the Weight Per Cubic foot becomes 1/2, ie 4/8.    Equivilently, if the Material Weight is doubled, it takes 2 times the dimensions, with 8 times the volume, to get the same Weight Per Cubic Foot as before.

    These formulas demonstrate Scaling, the most important principle in geometry, for ballooning.  What happens here, as dimensions increase, is that Surface Area increases by a square function, while Volume increases by a cube function.  So, Surface Area increases by the change in Volume ^2/3, while Volume increases by the change in Surface Area^3/2:
 
 Scale for Changes   One   Two   Three   Four   Five   Six   Seven   Eight   Nine   Ten
 Area = Volume.^2/3    1    1.587    2.080    2.520    2.924    3.302    3.659    4    4.327    4.642
 Volume = Area^3/2    1    2.828    5.196    8  11.180  14.697   18.520  22.627  27  31.623

Geometric Properties of Spheres

    Whoever said that The Circle and/or The Sphere were the simplest most elegant shapes obviously did not have to work out the mathematics!  Warm up your calculators and prepare for Brain Damage!! Please note that Pi is equal to 3.1416.

Area of a Circle  =  Pi * Radius^2  =  Pi * ( 2 * Quarter-Perimeter / Pi )^2  =  1.273* Quarter-Perimeter^2
     ( Note that  1.273 represents  4 / Pi )
Area of a Circle  =  Pi * ( Perimeter / ( 2 * Pi ) )^2  =  Perimeter^2  / 12.566
    ( Note that  12.566  represents  4 * Pi, or equivilently,  16 / 1.273 )

Area of a Sphere =  4 * Pi * Radius^2 =  4 * Pi * ( 2 * Quarter-Perimeter / Pi )^2  =  5.093 * Quarter-Perimeter^2
    ( Note that  5.093  represents  16 / Pi )
Area of a Sphere =  4 * Pi * Radius^2  =  4 * Pi * ( Perimeter / ( 2 * Pi ) )^2  =  Perimeter^2  / Pi

Volume of a Sphere =  4/3 * Pi * Radius^3 =  4/3 * Pi * ( 2 * QP / Pi  )^3  =  1.081 * Quarter-Perimeter^3
    ( Note that 1.081  represents  ( 32/3 ) / Pi^2, or equivilently  10.667 / 9.870 )
Volume of a Sphere  =  4/3 * Pi * ( Perimeter / ( 2 * Pi ) )^3 =  Perimeter^3  / 59.218
    ( Note that  59.218  represents  6 * Pi^2 )

Area to Volume Ratio  =  5.093 * QP^2  / ( 1.081 * QP^3  )  =  4.712 / Quarter-Perimeter
     ( Note that  4.712  represents  3/2 * Pi )
Area to Volume Ratio  =  ( Perimeter^2 / Pi ) / ( Perimeter^3 / 59.218 )  = 18.850 / Perimeter
    ( Note that  18.850  represents  6 * Pi )
 

Quarter-Perimeter  =  4.712 * Volume / Area
Perimeter  =  18.850 * Volume / Area

Quarter-Perimeter  =  ( Area / 5.093 )^1/2  = Area^1/2  / 2.257
    ( Note that 2.257  represents  4 / pi^1/2, or equivilently,  4 / 1.772 )
Perimeter  =  ( Area / pi )^1/2  =  Area^1/2 / 1.772

Quarter-Perimeter  =  ( Volume / 1.081 )^1/3  =  Volume^1/3  / 1.026
    ( Note that  1.026  represents  10.667^1/3  / Pi^2/3, or equivilently,  2.201 / 2.145 )
Perimeter  =  ( Volume / 59.218 )^1/3  =  Volume^1/3  / 3.898
    ( Note that  3.898  represents  6^1/3 *  Pi^2/3, or equivilently,  1.817 * 2.145 )
 

Area of a Sphere  =  5.093 * QP^2  =  5.093 * ( Volume^1/3  / 1.026 )^2  =  4.836 * Volume^2/3
    ( Note that  4.836  represents  4 * Pi / ( ( 4/3 *  Pi )^2/3 ), or equivilently, 6^2/3 * Pi^1/3, or 3.302 * 1.465

Volume of a Sphere  =  1.081 * QP^3  =  1.081 * ( Area^1/2  / 2.257 )^3  =  Area^3/2  / 10.635
    ( Note that  10.635  represents  6 * P^1/2, or equivilently,  6 * 1.773

    So, common formats can be used for both cubes and spheres.  The formulas have two basic modes.  Dimensions can be related to area and volume.  Or area and volume can be related directly to each other.  This mode is more theoretical.
 
Comparison of Cubes and Spheres
  Comparison of Formulas      Cube      Sphere     Comments:
  Area by QP     6 * QP ^ 2    5.093 * QP ^ 2  cube: 18% more area 
  Volume by QP     QP ^ 3    1.081 * QP ^ 3  sphere: 8% more volume
 Area/Vol Ratio by QP    6 / QP    4.712 / QP  cube: 1.273 ( ie. 4/Pi) more
  Area by Volume    6 * Volume 2/3    4.836 * Volume ^ 2/3  cube:: 24% more area
  Volume by Area    Area ^ 3/2  / 14.697   Area ^ 3/2  / 10.635  sphere:: 38% more volume

    Summary -- Imagination Exercises:  Six squares are formed into a cube.  The cube then "morphs" into a sphere, preserving the same perimeter.  It now has 85% of its previous surface area, and holds 8% more volume.  Its Area to Volume ratio decreases by 1 / 1.273, which is equivilent to Pi / 4.

    The original cube then "morphs" again into a sphere, this time preserving the same surface area.  It now holds 38% more volume.  To hold that same volume as a cube would require 24% more surface area.  If the additional surface area was instead applied towards making the sphere larger, then the sphere's new volume would be 1.24 ^ 3/2 or 1.382, ie. 38% more.

    Hence, cubes and spheres can be compared in two basic ways, relative to the perimeter or quarter-perimeter dimension, or relative to the relationship between area and volume.

Introduction to "Shape Factors"

    Shape Factors are mathematical shortcuts.  They are the numerical portions of formulas that are otherwise the same.  By thinking of geometric objects in terms of their Shape Factors, the mathematical differences between them can be simplified.  In addition, by looking at the changes in the Shape Factors between different geometric objects, "proxies" can be approximated for geometric objects whose exact mathematics are difficult to figure out.

For Cubes --  Area  =  6 * QP^2
For Spheres --  Area  =  5.093 * QP^2
Hence we see -- Shape Factors -- For Area by Quarter-Perimeter

For Cubes -- Volume  =  1 * QP^3
For Spheres -- Volume  =  1.081 * QP^3
Hence we see --  Shape Factors -- For Volume by Quarter-Perimeter

For Cubes --  Area/Volume Ratio  =  6 / QP
For Spheres -- Area/Volume Ratio  =  (5.093 / 1.081) / QP  =   4.712 / QP
Hence we see --  Shape Factors -- For A/V Ratio by QP  =  Area by QP Shape Factor  / Volume by QP Shape Factor

For Cubes  -- Area  =  6 * Volume^2/3
For Spheres -- Area  =  4.836 * Volume^2/3

    So, what does 4.836 represent?  Geometrically it represents  4* Pi / ( ( 4/3 * Pi )^2/3 ), or equivilently, 6^2/3 * Pi^1/3,
or 3.302 * 1.465.   It also represents 5.093 / (1.081^2/3).  So, generically, this means that :

Shape Factors -- For Area by Volume  =  Area by QP Shape Factor  / ( Volume by QP Shape Factor)^2/3

For Cubes  -- Volume  =  1/ 14.697 * Area^3/2
For Spheres -- Volume  =  1 / 10.635 * Area^3/2

    What do these numbers represent?  Besides being the inverses of the Area by Volume Shape Factors^3/2 ?  Or their other origins?  1/10.635 represents 1.081 / ( 5.093)^3/2.  So, generically, this means that:

Shape Factors -- For Volume by Area  =  Volume by QP Shape Factor / (Area by QP Shape Factor)^3/2

Weight Calculation Formulas for Theoretical Balloons

    In real life, a balloon will almost never be shaped like a cube.  The balloon will inflate so it is round.  But The Cube is still the simplest shape.  Hence, it is the natural proxy, and basis of comparison, for all of the different shapes.  Meanwhile, The Sphere is the most efficient shape.  So, in the mathematical range betweeen The Cube and The Sphere, a broad range of possible balloon shapes are potentially covered.  In addition, by using Shape Factors, the formulas can be expressed generically.  Algebraic substitution builds the following sequences of formulas:

Weight Formulas for Cubes:

Weight per Cubic Foot  =  Weight per Square Foot * Area / Volume

Weight per Cubic Foot  =  Weight per Square Foot * ( 6 * Quarter-Perimeter^2 ) / Quarter-Perimeter^3

Weight per Cubic Foot  =  6 * Weight per Square Foot  / Quarter-Perimeter

Quarter-Perimeter  =  6 * Number of Cubic Feet per Ounce / Number of Square Feet per Ounce
 

Number of Cubic Feet per Ounce  =  Quarter-Perimeter * Number of Square Feet per Ounce / 6

Number of Cubic Feet per Ounce  =  Volume^1/3 * Number of Square Feet per Ounce / 6

Volume =  ( 6 * Number of Cubic Feet per Ounce  / Number of Square Feet per ounce )^3
 

Weight Formulas for Spheres:

Weight per Cubic Foot  =  Weight per Square Foot * Area / Volume

Weight per Cubic Foot  =  Weight per Square Foot * ( 5.093 * Quarter-Perimeter^2 ) / ( 1.081 * Quarter-Perimeter^3

Weight per Cubic Foot  =  4.712 * Weight per Square Foot  / Quarter-Perimeter

Quarter-Perimeter  =  4.712 * Number of Cubic Feet per Ounce / Number of Square Feet per Ounce
 

Number of Cubic Feet per Ounce  =  Quarter-Perimeter * Number of Square Feet per Ounce / 4.712

Number of Cubic Feet per Ounce  =  Volume^1/3 * Number of Square Feet per Ounce / 4.712 * 1.026

Volume =  ( 4.836 * Number of Cubic Feet per Ounce  / Number of Square Feet per ounce )^3

Generic Weight Formulas:

Quarter-Perimeter  =  Area/Volume by QP Shape Factor * #CF per oz.  / #SF per oz.
    ( Note that the Area/Volume by QP S.F. is equal to the Area by QP S.F. / Volume by QP S.F. )

Number of Cubic Feet per Ounce  =  QP * #SF per oz. / Area/Volume by QP Shape Factor

Volume  =  ( Area by Volume Shape Factor * #CF per oz. / #SF per oz. )^3
    ( Note that the Area by Volume S.F. is equal to the Area by QP S.F.  / ( Volume by QP S.F. )^2/3  )
 

Designing Theoretical Cube and Sphere Shaped Balloons

    So, now we have Formulas, to design Theoretical Cube or Sphere Shaped Balloons.  For each material, the Weight per Cubic Foot can be calculated, based on the Quarter-Perimeter or the Volume.  Equivilently, calculations can be based upon a specific weight objective... like 1/8th Ounce per Cubic Foot.  But remember -- Glue-Weight is not included!!

    As stated, The Cube is a relatively inefficient shape.  The Sphere is perfectly efficient.  Hence, most other shapes will range in efficiency somewhere between these two shapes.  So, by making a balloon at least as efficient as an inefficient shape, it should definitely fly, if reasonably heated.  So, here are minimum specifications for 1/8 ounce per cubic foot balloons:
    ( Note:  The simplest equations are used.  With the more complex equations the results should work out the same.)

    With High-Density 1/3 mil Plastic Sheeting, weighing an Ounce per 45 Square Feet:

For Cubes:
Quarter-Perimeter  =  6 * 8 Cubic feet per ounce  / 45 SF per oz.  =  48 / 45  =  1.067 Feet
Volume  =  1.067^3  =  1.215 Cubic Feet
Surface Area  =  6 * 1.067^2  =  6 * 1.138  =  6.829 Square Feet
Weight  =  6.829 SF / 45 SF per oz.  =  .152 Ounces
Weight per Cubic Foot  = .152 oz. / 1.215 CF  =  .125 Ounces  =  1/8th Ounce per Cubic Foot

For Spheres:
Quarter-Perimeter  =  4.712 * 8 CF per oz.  / 45 SF per oz.  =  37.696  / 45  =  .838 Feet
Volume  =  1.081 * .838^3  =  1.081 * .589  =  .636 Cubic Feet
Surface Area  =  5.093 * .838^2  =  5.093 * .702  =  3.576 Square Feet
Weight  =  3.576 SF / 45 SF per oz.  =  .079 Ounces
Weight per Cubic Foot  =  .079 oz. / .636 CF  =  .124 Ounces  =  1/8th Ounce per Cubic Foot

    With 1/2 mil Lightweight Dry Cleaner Bag Material, weighing an Ounce per 30 Square Feet

For Cubes:
Quarter-Perimeter  =  6 * 8 CF per oz. / 30 SF per oz.  =  48 / 30  =  1.60 Feet
Volume  =  1.60^3  =  4.096 Cubic Feet
Surface Area  =  6 * 1.60^2  =  6 * 2.56  =  15.36 Square Feet
Weight  =  15.36 SF / 30 SF per oz.  =  .512 Ounces
Weight per Cubic Foot  =  .512 oz. / 4.096 CF  =  .125 Ounces  =  1/8th Ounce per Cubic Foot

For Spheres:
Quarter-Perimeter  =  4.712 * 8 CF per oz. / 30 SF per oz.  =  37.696 / 30  =  1.257 Feet
Volume  =  1.081 * 1.257^3  =  1.081 * 1.986  =   2.147 Cubic Feet
Surface Area  =  5.093 * 1.257^2  =  5.093 * 1.580  =  8.047 Square Feet
Weight  =  8.047 SF  / 30 SF per oz.  =  .286 Ounces
Weight per Cubic Foot  =  .286 oz. / 2.147 CF  =  .125 Ounces  =  1/8th Ounce per Cubic Foot

    With Tissue Paper, weighing an Ounce per 15 Square Feet

For Cubes:
Quarter-Perimeter  =  6 * 8 CF per oz. / 15 SF per oz.  =  48 / 15  =  3.2 Feet
Volume  =  3.2^3  =  32.768 Cubic Feet
Surface Area  =  6 * 3.2^2  =  6 * 10.24  =  61.44 Square Feet
Weight  =  61.44 SF / 15 SF per oz.  =   4.096 Ounces
Weight per Cubic Foot  =   4.096 oz / 32.768 CF  = .125 Ounces  =  1/8th Ounce per Cubic Foot

For Spheres:
Quarter-Perimeter  =  4.712 * 8 CF per oz. / 15 SF per oz.  =  37.696 / 15  =  2.513 Feet
Volume  =  1.081 * 2.513^3  =  1.081 * 15.871  =  17.157 Cubic Feet
Surface Area  =  5.093 * 2.513^2  =  5.093 * 6.315  =  32.162 Square Feet
Weight  =  32.162 SF / 15 SF per oz.  =   2.144 Ounces
Weight per Cubic Foot  =   2.144 oz. / 17.157 CF  = .125 Ounces  =  1/8th Ounce per Cubic Foot

    With Tracing Paper, weighing an Ounce per 10 Square Feet

For Cubes:
Quarter-Perimeter  =  6 * 8 CF per oz. / 10 SF per oz.  =  48 / 10  =  4.8 Feet
Volume  =  4.8^3  =  110.592 Cubic Feet
Surface Area  =  6 * 4.8^2  =  6 * 23.04  = 138.24 Square Feet
Weight  = 138.24 SF / 10 SF per oz.  = 13.824 Ounces
Weight per Cubic Foot  =  13.824 oz. / 110.592 CF  = .125 Ounces  =  1/8 Ounce per Cubic Foot

For Spheres:
Quarter-Perimeter  =  4.712 * 8 CF per oz. / 10 SF per oz.  =  37.696 / 10  =  3.770 Feet
Volume  =  1.081 * 3.770^3  =  1.081 * 53.582  =  57.922 Cubic Feet
Surface Area  =  5.093 * 3.770^2  =  5.093 * 14.213  = 72.386 Square Feet
Weight  = 72.386 SF / 10 SF per oz.  = 7.239 Ounces
Weight per Cubic Foot  =  7.239 oz. / 57.922 CF  = .125 Ounces  =  1/8 Ounce per Cubic Foot

    The examples demonstrate the scaling concepts quite clearly.  For any given shape, to keep the weight per cubic foot the same, 2 times the material weight requires 2 times the quarter-perimeter, resulting in 8 times the volume.  Similarly, 3 times the material weight requires 3 times the quarter-perimeter, resulting in 27 times the volume.  So, the dimension change is directly proportional to the material weight change.  The volume change is then the cube of the dimension change.

    Meanwhile, it is noticed that the dimensions for the cubes are 1.273 times those for the spheres.  This is a geometric curiousity, proportional to 6 / 4.712, or equivilently to 4 / Pi.  It shows that the changes in dimensions are directly proportional to the changes in the efficiency of the shape.  It also means that shape efficiency and material weight are somewhat equivilent to each other in their effects.

    Finally, it is noticed that the volume requirements for the cubes are 1.91 times those for spheres.  What is this the result of?  It is the same as the cube of the proportion between Area by Volume Shape Factors.  ( 6 / 4.836)^3 = 1.241^3 = 1.91.

    So, in the end, boiling down all the geometry, what appears most important for approximating a balloon's weight per cubic foot is its Area by Volume Shape Factor.  As seen, this factor relates volume and area directly to each other, without considering the actual dimensions.  This is quite reasonable to do, since the varieties of different shapes will not have dimensions that correlate directly to each other anyway.

    As stated, the Area by Volume Shape Factor will range from 4.836, for a perfect sphere, to around 6.000, for a perfect cube.  Somewhere between these two amounts most of the other shapes will range.

    As example, a Dry Cleaner Bag has an Area of 18 Square Feet and a Volume of 5 Cubic Feet.  It weighs 6/10 Ounce.  So, it weighs 30 Square Feet per Ounce.  It holds 8.333 Cubic Feet per Ounce.  This leads to the following calculations:

Volume  =  ( Area by Volume Shape Factor * #CF per oz. / #SF per oz. )^3

Volume^1/3  =  Area by Volume Shape Factor * #CF per oz. / #SF per oz.

Area by Volume Shape Factor  =  Volume^1/3 * #SF per oz. / #CF per oz.

Area by Volume Shape Factor  =  5^1/3 * ( 30 / 8.333)  =  1.71 * 3.60  =  6.156

    Okay, okay, so a dry cleaner bag is less efficient than a cube!  But not by much, even though it would be even less efficient if we factored in a bottom for the balloon.  But there you go anyway.  If you know the statistics on a balloon you can calculate its shape factor from that, and forget about the geometry.  Then, any other shape that is the same as the one you calculated for will have the same shape factor.  And onward...

    Where do we go from here?  The next step is to to consider different shapes!  And to see how they compare to The Cube and The Sphere:

    To start, the assumed first shape will be a "cube" whose height is either increased or decreased, presumably by some unknown amount at first, then by different proportions of the quarter-perimeter.  This will provide a format and direction in estimating the effects of height changes on the Area by Volume Shape Factor.

    The assumed second shape will be a cylinder where the height is equal to the quarter-perimeter.  This, in effect, will become the substitute for The Cube.  Finally the cylinder's height will be either increased or decreased, by unknown amounts then by proportions of the quarter-perimeter.  This will approximate real balloons that people are likely to build.

    In the meantime you might be wondering why the bottom of the balloon is being considered, when presumably it will be open.  There are two reasons for this.  First is that not having a bottom changes the shapes away from their classic geometric characteristics, which can screw up the formulas for no good reason.  Second is that the balloons will presumably have frames of some type anyway, so including the bottom of the balloon helps make an allowance for frame weight.

    Finally the question arises -- What about the Glue? and the Overlap?  But to include allowances for these screws up the equations.  So it is easier to just figure on the paper being a little heavier than calculated, or to get overkill, making the balloon somewhat bigger than you thought it needed to be.  Good luck everyone, with your ballooning projects!!  Bye for now, Overflight.

 Stay Tuned For Advanced Calculations on Model hot Air Balloon  Design

By Thomas Taylor -- balloons@overflite.com

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