Expert: Paul Soderman - 11/18/2002

Question
I'm developing a hot air balloon simulation program for my research project and thus, I need to calculate the climbing rate of the hot air balloon as it rises.  

I've managed to calculate the load, ambient density and ambient temperature but I'm stuck in how to calculate the climbing rate. I used the formula below to calculate the  Net Force,F.

 F = ((D-d)*V - Mass of Hot air balloon)*g

D - ambient density
d - density of the hot air in the balloon
V - Volume of the balloon

What I get is that there will be a net upward force until a certain height.  Since F=ma, there is also a positive acceleration upwards and this results in that the climbing rate of the balloon is increasing as it goes higher.  This result doesn't seems correct because from my understanding, the climb rate of the hot air balloon should be decreasing as it goes higher (am I correct in this assumption?).  I think I could have missed out some other forces in the formula.

Please advise on the function of getting the climbing rate. Thank you in advance, your help wil be very much appreaciated :)  

Answer
Wong
This is a tricky question depending on the detail you want to get into.  There are many variables such as the rate of heat going into the baloon, the vent rate, the baloon taughtness, the atmospheric conditions, etc.  For simplicity, I would assume a constant ascent rate.  If you assume accelerations, you have to integrate things on the way up.  So, for constant ascent rate, the sum of forces, F is zero.

F = L-Fd-Fw = mdU/dT = 0
where L = hydrostatic lift = gVp(1/Ta - 1/Tg)/R
Fd = aerodynamic drag = pi d^2 rho U^2 Cd/8
Fw = weight of baloon
g = gravity constant
V = baloon volume
p = atmospheric pressure
Ta = atmospheric temperature
Tg = baloon gas temperature
R = gas constant of air
rho = air density
U = ascent speed
Cd = drag coefficient of baloon (which depends on Reynolds number, which depends on atmospheric conditions)
d = baloon diameter

If I got everything, you can calculate ascent rate.  A good reference is Prandtl and Tietjens - Fundamentals of Hydro- and Aeromechanics, Dover pub, 1957.  Good luck and thanks to my colleague Blair for the equations.

Paul