The following is a suggested Hot Air Balloon Laboratory Lesson Plan:

1)  How  Basic Gas Laws work, and how they apply to hot air balloons.  How to calculate the  Weight of Air, under different conditions, and how to calculate Hot Air Balloon Lift.

2)  Construction of one or more hot air balloons, possibly with different volumes.  Calculations for balloon volume and weight.  Setting up of a Hot Air Balloon Laboratory.  Determination of current Ambient Air Conditions.

3)  Demonstration of hot air balloon lift, with weight and temperature measurements.  Calculations to compare measurements with theory.   Discussion of observations.  Discussion of the effects and limitations of measurements and assumptions.  Calibration of  the "Apparent Balloon Volume," based on the information and results gathered in the laboratory.

4)  Series of  "Balloon Exercises," based on different specifications for balloon weight and/or volume.  Calculation of balloon weight based on temperature.  Calculation of balloon temperature based on weight.  Calculation of the apparent weight of air, at different temperatures, based on knowing the temperature of Absolute Zero.  Calculation of the apparent temperature of Absolute Zero, based on knowing the weight of the ambient air.

5)  (Optional)  Advanced experiments and calculations, to "discover" the apparent temperature of Absolute Zero, and the apparent weight of air, at different temperatures, based on knowing only the experimental data.

Possible advanced topics:  a)  Three-dimensional geometry and properties of scale. b)  Mathematical designs for manned hot air balloons.  c)  Construction of a computerized spreadsheet that can function as a "Balloon Computer."  d)  Analysis of atmospheric pressure, based on "weighing" a 14.7 pound square inch "column" of atmosphere.

Other possible advanced topics:  a)  Calculations for weights of air, hydrogen, helium and other gases based on their molecular weights.  Calculations of the lifting characteristics of different gases.  b)  Analysis and calculations for the lifting characteristics and design of Rozier type balloons, such as the Breitling Explorer.

Materials and Tools:

2)   Scotch Tape --  seals the top of the bag.  Attaches the frame.  Also has other assorted uses.

3)   Flexible Drinking Straws -- are made into a frame, at the bottom of the balloon.  Alternately, the frame can be made with thin wire, thin cardboard, layers of scotch tape, or by bunching up the plastic with a needle and thread.

4)   Small Paperclips -- are used for weights, and are made into hooks, to moor the balloon, and to attach the weights.  Alternately, Christmas tree hangers or wire can be used for the hooks and small washers can be used for the weights.  To keep the balloon even, the weights should be added in pairs, to each side of the balloon.

5)  Heavy thread -- tethers the buoyant balloon, and functions as an equilibrium guage.

6)  Strip of Notebook Paper (Optional) -- holds the tether in position, and functions as an equilibrium guage.

7)  Hot Air Popcorn Popper - (b) - (c) -- heats the balloon and maintains its buoyancy.  For advanced experiments, either two popcorn poppers, or a can of sterno, can be used to heat the balloon to higher temperatures.

NOTE:  Hairdryers don't work very well for the balloon laboratory.  They blow too much air, and the heat is too dissipated to get good results.  On low fan, with one of the intakes blocked, they might work, but will likely overheat and break too.

8)  Pocket Dial Thermometer -  (b) -  (c) -- The temperature range should be 0 - 220 F.  Try to use at least three thermometers, if possible, to accurately average the readings.  The correct thermometers may be difficult to find.  Look in hardware stores and kitchen supply stores.  Prices should range from $5 - $10 each.

NOTE:  Try to avoid using oven thermometers as a substitute.  They won't measure the ambient temperature and they are not very accurate.  Also do not use digital thermometers.  They will break.  Their electronic components are not designed to withstand temperatures above 140 degrees Fahrenheit.

9)Laboratory Scale -- measures balloon weights.  Gram measurements can be converted into ounces.  Alternately, some or all of the various calibrations can be converted to metric.

10)  Barometer (Strongly Recommended) -- helps in calculating a reasonably precise weight for the ambient air.

11)  Thin Wooden Strip, around 1/2"x 1", around 5 to 5 1/2 feet tall --  Drilled holes, angled slightly downward, hold the thermometers at different levels, to approximate the average temperature of the heated air.  Look in a building supply store.

12)  Wide Thin Board, around 1"x 8", around 1 1/2 feet long -- holds up the wooden strip.  A rectangular hole is made to one side of the board, using a drill and a woodcutting tool.   Alternately the strip can be nailed to the board.  NOTE:  To make a platform for two popcorn poppers, a second board can be placed on top of the first.

13)  5x7" Notecards, cut lengthwise -- Notecards are scotch taped to the wooden strip, under each thermometer, to baffle the hot air, and improve the accuracy of the thermometers.  Notecards are also useful to channel the hot air, if two popcorn poppers are used to heat the balloon.

14)  Other Assorted Items -- Scissors, measuring tape, ruler, marking pen, pencil, pin and wirecutter, drill and drillbits, saw, woodcutting tool (or hammer and nails), handheld calculators, worksheets, etc.

Setting up the Hot Air Balloon Laboratory

To calculate the approximate bag volume -- imagine it as a cylinder, or stack of circles.  Some of the material is used up at the top of the balloon though, and the shape is distorted too.  Hence the "effective height" is less than the "material height."

If the area of the top of the imaginary cylinder is subtracted from the material height, the result will be an over-estimate of the volume, since the shape is distorted too.  If the radius of the cylinder is subtracted, the result will be an under-estimate of the volume, since it only accounts for a little over half of the material.  A compromise approximation is to subtract around 3/4 of the radius from the material height.  The result appears to be reasonably close to accurate.

For a 24 inch wide bag, each foot of effective height holds 1.27 cubic feet of volume.  Each inch of effective height holds a little over 1/10 of a cubic foot.   See Volume.   The radius is 7.64 inches.  3/4 of the radius is 5.73.  Hence, the 54 inch tall bag is approximated to have an effective height of 48.27 inches, and a volume of  5.11 cubic feet.  But, since the volume calculation is only an approximation, it makes sense to assume the volume to be exactly 5 cubic feet.

NOTE:  A 25 inch wide bag has around 8% more volume.  A 23 inch wide bag has around 8% less volume.

To make a simple frame, with mooring hooks -- stick together the flexible ends of two drinking straws.  Make two sets.  Bend two paperclips into hooks.  Optionally, cut off any excess wire.  Poke holes through the middle of the straws, and stick in the hooks.  Scotch tape the drinking straws to the bottom of the balloon bag, centered on the middle creases.  Make sure the hooks are angled correctly.

Weigh the finished balloon.  Optionally, to get the finished balloon to weigh an exact amount, like say 3/4 of an ounce, add scotch tape or other material.  While you are at it, weigh a bunch of paperclips too, and figure out how much each one weighs.

Alternately the frame can be set up another way, or made with strips of thin cardboard, wire, scotch tape or other material.  Its purpose is simple, not elaborate: to hold open the bag and to provide a place to attach the mooring hooks.

To make the thermometer holder -- Decide how tall you want the wooden strip.  5 to 5 1/2 feet sounds about right, for a 4 1/2 foot bag.  Saw the wooden strip to length.  Then mark it, at six inch intervals, starting at the top.  Drill a series of holes, at slight downward angles.  This way you can experiment with the thermometer placement, to see which ones work best.

To make the thermometer holder base -- outline the rectangular dimension of the wooden strip onto the wooden board.  Place the hole almost to one end, so the thermometers will point towards the middle.  Drill a series of large, closely spaced holes inside the outline.  Cut away the extra material with a woodcutting tool, so the thermometer holder fits in tightly, and is reasonable vertical.  Alternately the wooden strip can be attached to the base with small nails or screws.

Setting up the Equipment

Attach the tether to the balloon --  Cut several lengths of thread, around four feet long.  Optionally, make a slipknot in the middle, to adjust the length of the thread.  Get an idea about how high you want the balloon to climb.  Around 6 inches or so above the top of the popcorn popper seems about right.  Plan to adjust the length of the tether accordingly, either by adjusting the slipknot, or by cutting off any extra thread.  Tie the ends of the string to the mooring hooks.

Get everything ready -- Place the balloon over the thermometer holder.  Optionally, cut a strip of notebook paper and make a small cut in each end.  Place the tether underneath the paper, and into the cuts.  Position the popcorn popper.  Make adjustments so everything is even.  Turn on the popcorn popper so the balloon becomes buoyant.  See that it climbs evenly and to a suitable height.  Adjust the various components, as necessary, so the setup seems reasonably likely to get good results.  Plan to make additional adjustments, as you go along, to improve the performance and accuracy of your readings.

NOTE:  If you want to run two popcorn poppers, you should probably place a second board crossways on top of the first, and you should attach notecards to the top of the popcorn poppers to direct the heat towards the inside of the balloon.

Operating the Hot Air Balloon Laboratory

1)  Your approximate elevation, the current barometric pressure, and the current ambient temperature.  With this information you can determine the weight of the ambient and heated air.

2)  The approximate volume of the hot air balloon, its basic weight, and the weight of any additional weights to be added.  With this information you can determine the weight of the air displaced by the balloon.

As you run your experiments, keep in mind the following technical considerations:

1)  Pocket thermometers are generally accurate to within around 1% of their range, in a static temperature environment.  Hence the best possible readings may be inaccurate by up to around 2 degrees Fahrenheit.

2)  For small temperature changes, the thermometers have a time delay of several seconds, before reaching an accurate reading.  For large temperature changes, the time delay may be longer.

3)  While the balloon is being heated, hot air collects under the balloon bag and increases the lift.  In effect, the apparent volume of the balloon becomes greater than the actual volume.  One possible adjustment is to take the apparent volume as the actual volume.  Alternately, more accurately, the lift can be assumed to be slightly less than it appears to be.

4)  When the balloon heating is stopped, hot air escapes out of the bottom of the bag.  Cold air mixes in and reduces the lift.  This phenomona can be viewed as either a decrease in the apparent volume of the balloon, or as a decrease in its average temperature.  One option is to take the apparent volume as the actual volume.  Alternately, more accurately, the average temperature can be assumed to be slightly less than it appears to be.

SUMMARY:  To get the most accurate readings, the balloon should be maintained at a constant temperature and neutral buoyancy for a fairly long period of time.  The best way to achieve this seems to be by  intermittently turning the popcorn popper on and off.  Other options include adjusting the heights of the thermometers, and adjusting the length of the tether.

Modes of Operation:

2)  Off Mode:  Here the balloon is heated to a maximum temperature, then the popcorn popper is turned off, and the balloon is allowed to cool, until it loses bouyancy.  Because of the time delay, the temperature readings will probably be too high.  But they might be close enough anyway, and in the meantime observations can be made about how the balloon operates.

3)  Intermittent Mode:  Here the balloon is heated to buoyancy, allowed to cool, heated to buoyancy again, etc.  The objective is to achieve both neutral buoyancy and a constant temperature, and to maintain this condition for a reasonably long period of time.  This should provide fairly accurate comparisons between the temperature readings and the lift.

How Gas Laws Work

Boyle's Law states, for a given mass of gas, if the temperature stays the same, that a relative change in pressure will change the volume by an inverse proportion:

Change in Volume = 1 / Change in Pressure

Change in Pressure = 1 / Change in Volume

For hot air balloons, a relative change in pressure will change the mass of air inside the balloon by a direct proportion:

Change in Mass = Change in Pressure

Charles' Law states, for a given mass of gas, if the pressure stays the same, that a relative change in the temperature will change the volume by a direct proportion, compared to Absolute Zero, which is equal to minus 460 degrees Fahrenheit.:

Change in Volume = Change in Temperature

For hot air balloons, a relative change in the temperature of the air inside the balloon will change the mass of the air by an inverse proportion.  Equivilently a change in the mass means the temperature has changed, by an inverse proportion:

Change in Mass = 1 / Change in Temperature

Change in Temperature = 1 / Change in Mass

Change in Temperature * Change in Mass  =  1

Calculating the Weight of Sea Level Air, using the "635 Factor"

Mathematically, with two variables,  if one goes down at the same rate the other one goes up, the variables can be multiplied times each other, and their product will be constant.  As prime example  519 * 1.224 = 635.  So, at standard normal pressure sea level, ie. 29.92 inches of mercury, the following equations are true:

Absolute Air Temperature * Cubic Foot Air Weight  =  "635" (or other specific number, adjusted for air pressure)

Cubic Foot Air Weight =  "635"  / Absolute Air Temperature

Absolute Air Temperature = "635"  / Cubic Foot Air Weight

As prime examples, normal sea level air at 175 degrees Fahrenheit (635 Absolute) weighs an ounce per cubic foot, ie 635/635, and air at 48 degrees Fahrenheit (508 Absolute) weighs 1 1/4 ounces per cubic foot, ie. 635/508.  Both numbers are divisible by 127, so they make very good examples.  So, sea level air heated from 48 degrees to 175 degrees heats by 5/4, the air weighs 4/5 of its original amount, and the gross lift is 1/5 the weight of the original air, or 1/4 ounces per cubic foot.

Formulas for Model Hot Air Balloon Lift:

Gross Lift =  ( "635" / Ambient Temperature )  -  (  "635" / Heated Air Temperature )

Gross Lift =  ( "635" / Ambient Temperature )  -  (  "635" / ( Ambient Temperature + Heat Rise  )  )

Gross lift can also be expressed as a fraction, as follows:

Gross Lift =  "635" * (  Heated Air Temp.  -  Ambient Temp  )  /  (  Ambient Temp. * Heated Air Temp. )

Gross Lift = (  "635" * Heat Rise )  /  ( Ambient Temperature * (  Ambient Temperature + Heat Rise )  )

Alternately, the heated air temperature and the heat rise can be expressed in terms of gross lift, as follows:

Heated Air Temperature =  (  "635" * Ambient  )  /  (  "635" - ( Gross Lift * Ambient  )  )

Heat Rise =  ( Gross Lift * Ambient ^ 2 )  /  (  "635" - ( Gross Lift * Ambient  )  ) 

Adjusting the "635 Factor" to Account for Weather and Elevation

So, covering most weather conditions, the temperature where sea level air weighs exactly an ounce per cubic foot will vary from around 160 F (620 A) to around 185 F (645 A).  This means that the sea level  "635 factor" will vary from around 620 to around 645.  For lift calculations though, these changes are not very significant, and can reasonably be ignored.

Elevation reduces air pressure, air weight, and the temperature where air weighs exactly an ounce per cubic foot by around 3% per thousand feet, slightly more for lower elevations, and slightly less for higher elevations.  This reduces the "635 factor" by around 20 degrees per thousand feet.  Adjustments for altitudes of up to 25,000 feet, that are over 99% accurate, can be calculated using the following formula:

Air Pressure Adjustment  =  1  -  (  (  Elevation -- in thousands ) / (  27 + ( Elevation -- in thousands / 2  ) )  )

NOTE:  For altitudes of up to 35,000 feet, adjustments that are close to 99% accurate can be calculated by increasing the addition factor to 28; for up to 40,000 feet by increasing the factor to 29, and for up to 45,000 feet by increasing the factor to 30, etc. See Standard Atmospheric Pressure Tables - (2).
 

To calculate the "635 factor" precisely, use the following formulas:

"635 Factor" =  635 * (  Barometric Reading in Inches of Mercury / 29.92  )

"635 Factor" =  635 * (  Barometric Reading in Millibars / 1013.25  )

NOTE:  Use a barometer to get accurate readings.  Weather service readings are generally NOT the actual air pressure.  Instead, to simplify the information, they typically "normalize" the air pressure to sea level 29.92 inches of mercury.  Regional and seasonal air pressure differences are presumably factored out too.  As an exercize, you could call up your regional airport.  Ask about your local standard pressures, both "normalized" to sea level, and at the actual elevation of the airport.  With this information, you can try to convert the weather service readings to the actual barometric readings.

The elevation is sea level.  The pressure is 29.92 inches of mercury.  The ambient air temperature is 69 degrees Fahrenheit (529 Absolute Fahrenheit).  Hence the ambient air weighs 635/529, or 1.20 ounces per cubic foot.  The heated balloon will displace exactly 6 ounces of ambient air.

At perfect neutral buoyancy, the average temperature is recorded.  The readings are also perfect, with no apparent effects from either the lag time of the thermometers, or from hot air collecting under the balloon, or from cold air displacing the air at the bottom of the balloon.

The heated air temperature is 145 degrees Fahrenheit (605 Absolute Fahrenheit).  It weighs 635/605, or 1.05 ounces per cubic foot.  The weight of the air inside the balloon is 5.25 ounces.

To calculate the lift -- On a per cubic foot basis, the ambient air weighs 1.20 ounces, and the heated air weighs 1.05 ounces.  Hence the lift is .15 ounces per cubic foot.  On a five cubic foot basis, the ambient air weighs 6 ounces, and the heated air weighs 5.25 ounces.  Hence the lift is .75 ounces.

To calculate the heated air temperature -- The ambient air, at 69 degrees Fahrenheit (529 Absolute), is known to weigh 1.20 ounces per cubic foot, and the balloon is known to displace 6 ounces of air.  Since the balloon weighs 3/4 of an ounce, the heated air weighs 5.25 ounces, or 1.05 ounces per cubic foot.  Hence, the average heated air temperature should be equal to 635/1.05, or 604 degrees Absolute Fahrenheit (145 F).

To calculate the balloon volume -- The heated air is determined to be able to lift .15 ounces per cubic foot.  Since the balloon weighs 3/4 of an ounce, its volume is equal to .75/.15, or 5 cubic feet.

To calculate the weight of air, at 69 degrees ambient (529 Absolute) --.  The heated air temperature is 145 degrees Fahrenheit (605 Absolute).  The balloon lifts 3/4 of an ounce, or .15 ounces per cubic foot.  Hence the following calculation:

(Ambient Air Weight  -  .15) * 605  =  Ambient Air Weight * 529  (the air weight times the temperature times is constant)

(Ambient Weight  -  .15) / Ambient Weight   =   529 / 605  =  .875   (rounded up)...
...
.15 / Ambient Weight  =  1 - .875  =  .125

Ambient Weight   =  .15 / .125  =  1.20 ounces per cubic foot

To calculate the temperature of Absolute Zero -- Imagine a cubic foot tank of air, at 69 degrees Fahrenheit, with a pressure of exactly two atmospheres (59.84 inches of mercury).  The pressure is released, and the weight of the tank goes down by exactly 1.20 ounces.  The 5 cubic foot balloon is heated to buoyancy, as before.  Hence the following calculation:

1.20 * (69 + Negative Absolute Zero)   =   1.05 * (145 + Negative Absolute Zero)

69 + Neg. Absolute Zero   =   .875 *  (145 + Neg. Absolute Zero)   =   126.875  + .875 * Neg. Absolute Zero

.125 * Negative Absolute Zero = 57.875

Negative Absolute Zero  =  57.875 / .125   =   463, or pretty close to the actual value.

--Stay tuned for information on advanced experiments

By Thomas Taylor -- balloons @overflite.com 
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