Explanation Draw the free body diagram for the hot air balloon. Figure-(1) Write the expression for the buoyancy force. F B = ρ g V (I) Here, the density of the air is ρ , the acceleration due to gravity is g and the volume of the balloon is V . Write the expression for the density of the balloon. ρ b = p R T (II) Here, the gas constant is R , the atmospheric pressure is p and the temperature of the balloon is T . Write the expression for the specific weight of the balloon. γ b = ρ b g (III) Write the expression for the total weight. W = W a + γ b V (IV) Here, the weight of the passenger, fabrics and basket is W a . Write the expression for the radius ( R ) of the sphere using volume expression. R 3 = 3 4 π V (V) Write the expression for the drag force. D = 1 2 ρ U 2 C D π R 2 (VI) Here, the velocity of the balloon is U and the drag coefficient is C D . Write the expression for the buoyancy force using the condition of equilibrium. F B = W + D (VII) Write the expression for the Reynolds number. Re = U ( 2 R ) ν (VIII) Here, the kinematic viscosity of air is ν . Conclusion: Refer to table B.3 “Physical properties of air at standard atmospheric pressure” to obtain the density of air as 2.286 × 10 − 3 slug / ft 3 and kinematic viscosity of the air as 1.69 × 10 − 4 ft 2 / s at the temperature of 80 ° F . Substitute 2.286 × 10 − 3 slug / ft 3 for ρ , 32.2 ft / s 2 for g and 70000 ft 3 for V in Equation (I). F B = ( 2.286 × 10 − 3 slug / ft 3 ) ( 32.2 ft / s 2 ) ( 70000 ft 3 ) = 5152.64 slug ⋅ ft / s 2 ( 1 lb slug ⋅ ft / s 2 ) = 5152.64 lb Convert the temperature from degree Fahrenheit to degree Rankin. T ( ° R ) = T ( ° F ) + 460 = ( 165 ° F + 460 ) ° R = 625 ° R Substitute 14.7 psi for p , 53.33 ft ⋅ lbf / lb ⋅ ° R for R and 625 ° R for T in Equation (II). ρ b = ( 14.7 psi ) ( 53.33 ft ⋅ lbf / lb ⋅ ° R ) ( 625 ° R ) = ( 14.7 psi ( 1 lbf / in 2 1 psi × 144 in 2 1 ft 2 ) ) ( 53.33 ft ⋅ lbf / lb ⋅ ° R ) ( 625 ° R ) = 2116.8 33331.25 lb / ft 3 ( 1 slug 32.2 lb ) = 1.98 × 10 − 3 slug / ft 3 Substitute 1.98 × 10 − 3 slug / ft 3 for ρ b and 32.2 ft / s 2 for g in Equation (III). γ b = ( 1.98 × 10 − 3 slug / ft 3 ) ( 32.2 ft / s 2 ) = 0

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

8th Edition

ISBN: 9781119080701

Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Publisher: WILEY

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Chapter 9.3, Problem 58P

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