A hot-air balloon achieves its buoyant lift by heating the airinside the balloon

Question

A hot-air balloon achieves its buoyant lift by heating the airinside the balloon, which makes it less dense than the air outside.Suppose the volume of a balloon is 1900 m3 and the required liftis 2500 N (rough estimate ofthe weight of the equipment and passenger). Calculate thetemperature of the air inside the balloon which will produce therequired lift. Assume that the outside air temperature is 0°Cand that air is an ideal gas under these conditions. What factors limit the maximumaltitude attainable by this method for a given load? (Neglectvariables like wind.) What factors limit the maximumaltitude attainable by this method for a given load? (Neglectvariables like wind.)

Explanation / Answer

   in the given problem there are three forcesto consider
   the upward bouytant force (FB)    the downward force which is the weight of the hotair balloon (Wh)    the downward force which is the weight of thepassenger and his equipment (Wp)    accoeding to equilibrium when the blloon is to beraised at constant speed the bouyant force    must equal to the two weigts    FB = Wh +Wp    if mh is the mass of the hot air balloon and gis acceoleration due to gravity then    FB = mh + 2500 N    let c be the density of the codand h is the density of the hot then    as we know that    density = mass m / volume V    according to newtons second law we canwrite    V c g = V h g +2500 N    the ideal gas law is given by    PV = nRT    if the gas law is written in terms of the gasdensity and the molecular mass M then the ideal gas    law changes to    PV = nRT          = (m /M) RT    (PM / R) = (m / V) T                   = T       here we must recollect that    the gas inside and outside the balloon is thesame so M will be same and moreover the pressure    is also same so the above equation reducesto    T = C (some constant)          =( T)c          = (T)h    so finally we get    V c g = Vh g + 2500N                      = V c (Tc / Th) g + 2500N    temperature of the air inside the balloon willbe    Th = V cTc g / V cg - 2500 N         =...........K    the factor that will be limiting is, thedensity of the air decreases and the temperature required    will be increasing as a result the airbecomes too hot and the balloon fabricbecomes burnt      so finally we get    V c g = Vh g + 2500N                      = V c (Tc / Th) g + 2500N    temperature of the air inside the balloon willbe    Th = V cTc g / V cg - 2500 N         =...........K    the factor that will be limiting is, thedensity of the air decreases and the temperature required    will be increasing as a result the airbecomes too hot and the balloon fabricbecomes burnt  

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