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.   As is true for all adventurous endeavors; an understanding of the controlling parameters and the science behind them can improve the ability to deal with a dangerous situation by replacing chance with knowledge.  Hot-Air Ballooning models many of the physical mechanisms taught in the disciplines of Classical Physics and Thermodynamics.  This treatise presents a classroom oriented Analytical Model of the Hot-Air Balloon.  Specifically, this treatise demonstrates 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 these relatively simple concepts presented in this treatise 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 will be presented one by one, then combined in a force equation that mathematically describes the motion of a Hot-Air Balloon.

..................................................................................Jim Rogina

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.

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) power  ‑‑‑ example: 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.