Problem 37P

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1800 m3 and the required lift is 2700 N (rough estimate of the weight of the equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the required lift. Assume that the outside air temperature is 0°C and that air is an ideal gas under these conditions. What factors limit the maximum altitude attainable by this method for a given load? (Neglect variables like wind.)

Solution 37PIntroductionThis question can be solved by using the Poiseuille’s equation.SolutionGiven that,Volume flow rate Q = 0.25 L/sQ = 0.25 × 10 3 m /s ( since 1 m 3 = 1000 L)Length l = 10.0 mDiameter = 2.5 cmRadius r = 1.25 cmRadiusr = 0.0125 m 0 3Viscosity of water at 20 C is = 10 Pa.sSubstituting these values in Poiseuille’s equation, 4Q = Pr 8l 3 3 3.14×P×(0.0125) m0.25 × 10 m /s = 3 6 8×10 Pa.s ×10.0 mP = 80×0.25×10 4 Pa 3.14×(0.0125) 20 ×106P = 3.14 × 2.44×10Pa 2P = 2.61 × 10 PaP = 261 PaTherefore, the approximate gauge pressure of the water where it enters the hose is 261 Pa.