The Physics of Hot‑Air Ballooning

(A course offered for the continuing education of high school physical science teachers)

Hot-Air Ballooning is a compelling, 'real life' example of the fundamental concepts commonly taught in high school physical science curriculums; hot-air ballooning exemplifies many of the physical mechanisms taught in the disciplines of classical physics and thermodynamics.  For the student-pilot, an understanding of the controlling dynamics behind the sport can improve the ability to deal with a dangerous situation by replacing chance with knowledge.  Below we present a classroom oriented Analytical Model of the Hot-Air Balloon.  Specifically, the mathematical model we develop below exemplifies the mechanisms associated with heat transfer by radiation, convection and conduction, through “thermal energy regulation”.    Pre-flight planning and "thermal energy regulation” ultimately 'controls' the flight path of a Hot-Air Balloon.  Mastering the relatively simple concepts presented here allows the student to mathematically model the parameters that describe the motion of a dynamical system.  Together with modern numerical methods (computers) these techniques and disciplines can be employed to empower the student towards success in any technical endeavor.

The various concepts and scientific disciplines on point are presented one by one, then the concepts are combined in a force equation that mathematically models the motion of a Hot-Air Balloon.  The latest in ultra-lightweight high-performance hot-air balloon design and construction technologies are presented in the classroom to exemplify the principles taught. 

Imagine a cube of air; (10 ft x 10 ft x 10 ft), one thousand cubic feet of air.  Did you know.. that cube of air weighs more than eighty pounds?  Hot-air will safely and efficiently perform as a lifting gas when lifting 12 to 18 pounds per thousand cubic feet of envelope volume: load it down more and your balloon flies hot and performs poorly; load the system lighter and it won't penetrate in flight.  A hot-air balloon in motion has momentum based on it's total system mass and velocity, and the total system mass includes the mass of the air in and around the envelope; many thousands of pounds of air.  Traditional hot-air balloon designs compensate for their heavy empty weight by increasing the envelope volume and/or the operating temperature, at the expense of reduced performance/efficiency and dramatically increased in-flight momentum.  Every pound taken aloft is important: When hot-air is used as a lifting gas, seven pounds of 'air' in and around the envelope contribute to your in flight momentum for every pound lifted.  For in-flight vertical motion, this factor is multiplied again by a velocity factor resulting in dramatically increased momentum that must be overcome to maneuver in flight.  Traditional hot-air balloon designs, all, entrain thousands of pounds of unnecessary excess airframe weight; and require extra fuel for this 'extra unnecessary weight', which makes traditional balloon systems even bigger and heavier. 

Did you know..  The latest in modern inexpensive high-performance ultra-lightweight 2-place sport hot-air ballooning equipment... the aircraft empty weight is less than 4 pounds per 1,000 cubic feet: one-half the weight of a "traditional hot-air balloon" system design.   A 2-place ultra-lightweight balloon design can operate efficiently with a smaller envelope volume; because these balloons are dramatically lighter than traditional designs; a smaller envelope volume penetrates better and has less momentum...  Most competitors gravitate towards smaller lighter designs.  This is the year 2010; did you know... interestingly, the world class ballooning competitors dream machine... also happens to be the best... safest equipment for student training.... These balloons inflate faster and easier with fewer people, they are easier to transport and retrieve; they out perform other balloons in flight because they are lighter, they have less momentum to overcome to maneuver with safety in flight.  Did you know..   A student balloon pilot is safest when operating a simple 2-place ultra-lightweight high-performance sport hot-air balloon, one-on-one with their instructor.

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Kinetic Theory provides a physical basis upon which the concept of Temperature can be understood.  There is an equivalence between the Kinetic Energy of molecular motion and the Internal Energy of a system.  These relationships can be used to develop a Force equation, including all the fundamental forces acting upon the Hot‑Air Balloon system.  We will start with the basic definitions:

 I) Take Newton’s' Laws 

    A) Definition of Equilibrium     Weigh‑Off

     1)  1^st Law        SF=0

     2)  Inertia   The pound is a unit of weight. ‑ A Slug is a unit of mass.

    B) 2^nd Law           F=ma  W=mg           p=mv  (momentum)

    C) 3^rd Law   

     1) Force is a Vector quantity, it has  a magnitude and a direction.

     2) Friction C_d = .4                 A = Balloon Cross Sectional Area

                 r_{a  =}  Ambient Density     V^{2  =} The Velocity Squared

                        Aerodynamic Drag=(C_d) (A) (r_a) (V²/2)

II) A bit of Fluid Mechanics 

    A) Density and Pressure

                         Density=r=mass/Volume=m/V

                           Pressure=P=P_a+rgh

           P = The Pressure at any depth (h) in a fluid

          P_a = atmospheric pressure or initial pressure

          g = acceleration due to gravity

    B) And Archimedes' principle

                        B=rVg

          V = the Volume displaced

          g = acceleration due to gravity

The inertial force of the Hot‑Air Balloon System includes a quantity of the surrounding air that is moving with the balloon (virtual air mass) and is approximately equal to one half the mass of the air inside the balloon.  This increases the total inertial effects to well over 5,000 Kg for a balloon of International Class AX‑8, or 105,000 cubic feet in volume. 

III) Add the definitions of Temperature & Thermodynamics

    A)  Molecular Interpretation Definition:

     Temperature is a direct measure of the average molecular kinetic energy.

                   Temperature=T=2/3k (½mv²)

     B)  CH₃CH₂CH₃ + O₂ = CO₂ + H₂O  Envelope Gas

          1) The counter‑intuitive example of Density & Relative Humidity

          2) Density Altitude as a performance consideration 

    C) 0^th Law ....0 pressure = ‑273.15 degrees Celsius

      1) The Ideal Gas Law     PV=nrT

     2) Definition of Heat

          Heat flow is an energy transfer that takes place as a consequence of a temperature difference only.

      3) Mechanical Equivalent of heat       1 calorie = 4.186 Joules

     4) Latent Heat  ...Propane Emphasis ‑ tank refrigeration effect

                              Q=mcDT + mL

IV) And Heat Transfer

    A) Temperature Gradient  and The Law of Heat Conduction

    B) Radiation and Convection

                         Stefans Law      R=sAeT⁴  

     C) Black Body 

A Black Body, by definition, absorbs all energy incident upon it.  This gives a definition for (e) in Stefan's Law (emissivity).  The emissivity of a ideal absorber, or Black Body is equal to 1.  In contrast, an object that reflects all energy incident upon it has an emissivity of 0, and is a perfect reflector.

     1) A Hot‑Air Balloon of Volume 200,000 cu. ft. (min) can sustain flight,  without need of a burner.

Hot-Air Balloon Burner Design Philosophy for Safety, and Ergonomic Factors:

(Example: A burner designed by Paolo Bonanno of Mondovi Italy)

An understanding of the engineering problems behind the modern Hot-Air Balloon Burner reveal a variety of competing design parameters that must come together in compromise for a good result.

The burner must be engineered in detail to consistently and reliably operate within a range of both ambient atmospheric pressures and LPG fuel line pressures. There is a specific area of the burner flame producing maximum temperature. The burner flame pattern must consistently envelop the vaporization coils with this high temperature plasma throughout the entire operating range. Stoichiometric combustion is achieved when the quantitative relationship between reactants and products during combustion are optimized. The combustion products, which can be easily smelled during deflation and seen in condensation on the envelope fabric, or on the burner coils; characterize the efficiency of the combustion process. Thermal shock or ‘hot-spotting’ of the envelope is a common problem for balloon burners. Propane (C3H8), if burned efficiently in free air burns at approximately 2950°F and produces approximately 21,600 Btu/lb.; however, the flame must not be too hot (explosion) or to cold (radiant). The flame pattern must not be too thin and long (hot-spotting) or too diffuse (deflection in wind). A bad burner design can quickly destroy an envelope; on the other hand, an excellent design could double the life of your envelope...


Hot-air balloon main burner efficiency within a ‘normal’ range of fuel pressures is obtained when the vaporization coil is designed to process approximately 3.5kg/min of LPG, providing approximately 10 Million Btu/hour output. 3.5kg/min throughput is a 'steady-state' design goal when operating within a ‘normal’ range of fuel pressures. Some balloon manufactures advertise their burner’s output at line pressures above 230 psi (120 degrees F), which would be considered on the high end of any ‘normal’ range; any burner marketing specifications advertising output above 160psi is exaggerated, unrealistic and therefore this data is comparatively useless. Instead, the intelligent pilot should be concerned about how the burner performs when the burner is operating below the normal range as this is commonly the cause of dramatically reduced power/output. Most hot-air balloon burners available today fail to provide sufficient power/output below 80 psi. Some burners make use of bimetal strip technology in their coandă ring orifice manifold. The bimetal strips regulate the orifice cross-sectional area to compensate for reduced fuel pressure, maintaining 3.5kg/min flow through the coil over a wider range of fuel pressures and good burner output down to 44 psi. In addition, a properly designed liquid burner can provide 80% of the main burner’s output. When the flame patterns from the two systems (main & liquid) are properly designed to be engaged simultaneously; the available power/output from a Dual-Single Burner is dramatically increased.

Simplicity and Attention to Detail
Burner output and/or safety can be affected by something as simple as the hose fittings, every detail is considered. For example: one of many details; the corners in square or triangular vaporization coils interrupt the momentum, kinetic energy and/or flow/output of the LPG in the vaporization process, and so burner coils should be circular/helical. If the cross-sectional area of the fuel manifold expands in the proper area of the burner's flame pattern; a properly designed burner doesn't need a slurper.

Coandă jet ring orifice design
Air is entrained through both the center and outside of the coandă orifice jet ring, so much air, you can feel and even hear the sucking action when placing your hand on the outer screen. The coandă effect, also known as "boundary layer attachment", is the tendency of a stream of fluid to stay attached to a convex surface, rather than follow a straight line in its original direction. This principle was named after Romanian inventor Henri Coandă, who was the first to understand the practical importance of the phenomenon for aircraft development. The Bonanno Burner coandă orifice jet ring is used to shape the burner flame pattern, optimizing ignition reliability, flame stability, and mixing ratios for proper reliable combustion. The coandă jet ring I describe here has several major improvements over “the older Sirocco technology”. There are three concentric rings of orifices that serve different purposes. Flow through the orifices on the inside of the coandă ring are controlled by bimetal strips which are only activated when fuel pressures are low. The orifices at the outside of the coandă jet ring produce a low velocity cylindrical flame pattern. This low velocity fuel/air mixture is easily and consistently ignitable, without the sound of explosion that is experienced with many balloon burner systems. The flame emanating from the orifices at the middle of the coandă jet ring produces the higher velocity flame for directional control and power. This more powerful inner flame is mixed with and muffled by the quieter low speed outer cylindrical flame, which of course mixes with the surrounding air with less turbulence and noise. Most importantly; this coandă jet ring design is very simple and easy to disassemble; it requires very little maintenance, even when the nasty oily LPG fuel sold in Europe these days is burned…

Burner noise, gas velocity, and the speed of flame propagation
Many balloon burners on the market today operate at or near detonation/explosion of the propane, and so they are very loud, inefficient in this application, and can actually be damaging to your envelope fabric over time. A term used to describe the process of subsonic combustion that propagates through thermal conductivity is deflagration. Propane-air combustion begins with spark ignition; a spark thermally decomposes the propane-air mixture to produce free radicals and other reactive species. Deflagration burning then continues by the propagation of the reactive species generated by the heat of combustion. ‘Deflagration’ is different from ‘Detonation’ which is supersonic and propagates through shock compression. Sound familiar?...’Detonation’ is associated with the noise of explosion. When flame velocities are low, the effect of a deflagration is the release of heat. At flame velocities near the speed of sound, the energy released is in the form of pressure and the results resemble a detonation or explosion. The pressure wave developing during an explosion and, therefore, the effect of an explosion, strongly depend on the velocity at which the flame propagates. Between these extremes both heat and pressure are released; and here is the opportunity for design compromise. Orifice design, the temperature of combustion and flame propagation velocity all conspire to instantiate the sound signature and power output of the burner. Bonanno Burners are designed in detail to be the quietest and most powerful burners available.

Thermal Shock - situations that marketing hype won't explain
Your balloon is in a terminal descent at gross load. You are operating at the fuel pressures allowed (something greater than 200psi), and your 40+ Million Btu almost transparent pencil thin long blue flame is used to arrest your terminal descent. If your burner is turned on and left on until the terminal descent has been arrested, the temperature in the top of the balloon, in places, can be as much as 100 degrees F hotter than the temperature necessary to arrest the terminal descent, and in places, much hotter than that. In fact; the pressure wave from a poorly designed burner flame pattern in detonation can create a rising central core column above your burner that is so hot; the nadir of your parachute shroud lines will melt and the valve line will fall, coiling up in the bottom of your basket. These undesirable heat concentrations take time to dissipate. In fact, the FAA Approved Flight Manual/Operating Limitations for most hot-air balloons specifies that their burner systems must be throttled on and off to minimize the dangerous effects of thermal shock. Burners with flame patterns like this are examples of poor burner designs that can and will shorten the life of your envelope. Your almost transparent pencil thin long blue flame is so long and so hot and so loud, it's actually composed of a series of detonations; a dangerous rising column of plasma that must somehow, subjectively, be controlled sparingly so as not to destroy your other primary flight control system (your valve line). I'm guessing that the reason for the long burn was because of some sort of emergency situation, and now, you have made things even worse...
Bonanno burner flame pattern propagation and the resultant envelope heat pattern has been designed to produce less than 10 degrees F Thermal Shock. When arresting a terminal descent at gross load, this technology delivers power safely and more effectively which will stop your descent faster. When in a terminal descent with a Bonanno Burner, you turn the burner on and you can leave it on, until the terminal descent has been completely arrested, without danger or damage to your envelope.
 

Notes on inflation fan design:

A typical traditional 77K envelope design requires an inflation fan that produces about 20 pounds of thrust.  If your goal is to design a lightweight fan, you will sooner or later come to understand that increased prop diameter means increased weight.  At the time of this writing, the best compromise for a gas powered fan is 16"; a 3.5HP motor and a three blade 4oz  16" prop; together will produce 22lbs of thrust @ 8500 RPM.   You could go smaller in diameter but you will need more power and RPM, (example: an 8.5HP LiPo powered 7-blade 4.6" Ducted Electric Fan @ 30,000RPM), this solution works well, but you don't save much weight and at this time, it's quite expensive.

Propellers loose efficiency at transonic speeds due to compressibility/energy losses so the ideal tip speed for a propeller is about 600 to 650 feet per second; this is true for all propellers.  The ideal propeller diameter for any RPM is described mathematically in the formula:

[Diameter in inches = 138,000 / RPM]

The tip speed for a 24" prop on a 5hp motor approaches efficiency at about 5,700 RPM and produces 38lbs of thrust; how do I know this?

Using the diameter of the propeller, calculate the propeller disk area in square feet.
Area is defined as:
A [ft^2] = Pi * r^2
Where r is the radius of the propeller disk and D is the diameter in feet, Pi is the number 3.14159

Power loading is calculated by:
PL [hp/ft^2] = power / A
Where "power" is the horsepower (hp) delivered to the propeller and A is the area, calculated above.

Thrust Loading (TL) is calculated using an empirically defined formula.
TL [lb/hp]= 8.6859 * PL^(-0.3107)
Thrust loading is a function of power and propeller disk area. The theoretical amount of thrust from a properly loaded propeller turning at idea tip speed: Thrust = TL * power >>>[lb] = [lb/hp] * [hp]

The ubiquitous traditional balloon inflation fan motor tops out at 3,600 RPM - A more appropriate propeller for a 5hp motor reaching it's maximum power at 3,600 RPM would be a 40" propeller.  Today, the market for 1/4 scale RC Airplanes and US Military UAV funding drives the research and development.  Inexpensive ultra-lightweight high-performance hot-air balloon inflation fans are easily constructed from the powerful, lightweight RC/UAV solutions that are now available.  The LiPo powered electric ducted fans available today might be a bit expensive for the typical balloonist in today's market; but clearly the ubiquitous traditional 70 pound hot-air balloon inflation fan is obsolete for the ultra-lightweight balloon owner/operator.  Bigger, heavier props are more dangerous, unnecessarily more dangerous.

The design and maintenance parameters associated with a crankshaft-prop dynamical system are engineering concepts that are very well documented, and have been around since the invention of the airplane. Traditional balloon inflation fan designs use big, heavy props that have unnecessarily high moments of inertia, and for this reason, they are quite dangerous; here are two things that a balloon pilot needs to know about the inflation fan.

First, the prop must be balanced and the crankshaft must be normal to the plane of rotation. This procedure is maintenance information that should be provided by the manufacturer of the product. What happens when your prop is damaged in the course of normal operation?

Second, always throttle back your fan before moving it. During the inflation of a "traditional" hot-air ballooning system, it is common for a crew person, or the pilot to have need to reposition the fan.  A common scary situation arises when a basket is forcefully and undesirably repositioned by a cross wind, pushing/knocking an unattended traditional free-standing heavy inflation fan over; producing clouds of smoke and a very dangerous situation.  More commonly, the handle of the fan frame is used to lever back or shift the direction of the air-stream, ignoring the considerable gyroscopic inertial effects of the heavy spinning prop; this person is putting a strain on the crankshaft. The more abrupt this levering action is, the more strain is put on the crankshaft. This effect can be demonstrated in the following way: Put a handle on the axle of a bicycle wheel, spin it as fast as you can, and try to change the direction that the axle is pointing. Now, imagine that your wrist is a weakened crankshaft being levered around by an excited exasperated crew person.

Restated more precisely in scientific terms:

A rotating prop has an angular momentum vector. It resists directional change. The magnitude of this angular momentum vector is proportional to the moment of inertia of the prop and the RPM at which it is turning. The crankshaft absorbs the angular impulse (time integral of the applied torque) as the rotating prop is 'torqued' into position.  The faster your fan is running, the more it resists movement. The longer and or heavier your prop is, (30" vs 24" vs 16") the more it resists movement at any given RPM. When moving your fan (while it is running), the crankshaft becomes part of a lever arm and is forced to dissipate energy in a way that could snap a weakened crankshaft.  The metallurgical composition, diameter, and bearing surface of the crankshaft, the distance between (the center of gravity of the prop-hub assembly) and the (bearing or crankcase face-plate), and the moment of inertia of your prop, all play a role in the amount of vibration energy that a crankshaft will absorb before failure.

V) Balloon Flight is a manifestation of The First Law of Thermodynamics   

    A) The first law of thermodynamics is a generalization of the conservation of energy that includes  possible changes in the internal energy of a system.

I) Development of  The Lift Equation and an Analytical Model

     A)  Archimedes Principle, The Ideal Gas Law, and Aerodynamic friction, make a force equation.....LIFT ‑ Mg ½+ Drag = Ma.

         Add to this the various amounts of heat gained or lost due to the thermodynamic considerations and you have a working model.

Thermodynamic heat transfer quantities include the following examples: 

  Heat lost from The Mouth            Heat added by the Burner

  Heat lost by Leaks or Holes              Heat added by Solar Radiation

  Heat lost by Convection                  Heat added by The Atmosphere

  Heat lost by Radiation                   Heat added by The Earth

A) The model requires an understanding of The concept of Work

     a) as energy 

     b) by gravity

     c) done by a gas

     d) Mechanical Equivalent of Heat

     e) Burner power calculation; To convert Btu/hr  into gallons of Propane/hr divide by 79,000

The flight path of a Hot‑Air Balloon is ultimately controlled by thermal energy regulation.  This regulation of the heat gives the pilot a precise control of the altitude of the balloon, which in turn provides for any lateral control that might be possible, considering the atmospheric conditions.  There are many parameters that affect this thermal energy regulation.  Heat is added by the burner until the (ambient‑envelope) temperature differential is in excess of 100 degrees Fahrenheit, but the burner is not the only source that adds to (or subtracts from) this temperature differential.  The radiative and convective heat transfer processes that contribute to the problem of thermal energy regulation are very important, and can significantly affect balloon performance.