The Figure Below Shows The Forces Exerted On A Hot Air Balloon System Dy Fd Fg Figure Forces On A Hot Air Balloon Fb 1 (98.68 KiB) Viewed 46 times
The figure below shows the forces exerted on a hot air balloon system. dy FD FG Figure Forces on a hot air balloon: FB = buoyancy, FG = weight of gas, Fp = weight of payload (including the balloon envelope), and Fp = drag. Note that the direction of the drag is downward when the balloon is rising. = v2 Formulate the drag force as Fp = pav?ACa, where Pa = air density (kg/m?), v = velocity (m/s), A = projected frontal area (m²), and Ca = the dimensionless drag coefficient (0.47 for a sphere). Note also that the total mass of the balloon consists of two components: m = mg + mp, where mg = the mass of the gas inside the expanded balloon (kg), and mp = the mass of the payload (basket, passengers, and the unexpanded balloon = 265 kg). Assume that the ideal gas law holds (P = PRT), that the balloon is a perfect sphere with a diameter of 17.3 m, and that the heated air inside the envelope is at roughly the same pressure as the outside air. Other necessary parameters are: Normal atmospheric pressure: P = 101,300 Pa, The gas constant for dry air, R = 287 Joules/(kg K), The air inside the balloon is heated to an average temperature, T = 100 C, The normal (ambient) air density, p = 1.2 kg/m²
= (a) Use a force balance to develop the differential equation for dv/dt as a function of the model's fundamental parameters. (b) Use Euler's method to compute the velocity from t = 0 to 6 s with At = 2 s given the previous parameters along with the initial condition: v(0) = 0. Hint: The buoyancy force is defined by Fp = Pavg, where V is the volume of the balloon envelope and g = 9.8 m/s2 (standard gravity). (Please solve this problem by hand, not MATLAB)
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