Figure P1.28 shows the forces exerted on a hot air balloon system.Formulate the drag force...

Figure P1.28 shows the forces exerted on a hot air balloon system.

Formulate the drag force as

where ρ_a = air density (kg/m³), υ = velocity (m/s), A = projected frontal area (m²), and C_d = the dimensionless drag coefficient (≅ 0.47 for a sphere). Note also that the total mass of the balloon consists of two components:

m = m_G + m_P

where m_G = the mass of the gas inside the expanded balloon (kg), and m_P = the mass of the payload (basket, passengers, and the unexpanded balloon = 265 kg). Assume that the ideal gas law holds (P = ρRT), 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, ρ = 1.2 kg/m³.

(a) Use a force balance to develop the differential equation for dυ/dt as a function of the model’s fundamental parameters.

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(b) At steady-state, calculate the particle’s terminal velocity.

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(c) Use Euler’s method and Excel to compute the velocity from t = 0 to 60 s with Δt = 2 s given the previous parameters along with the initial condition: υ(0) = 0. Develop a plot of your results.

Figure P1.28: Forces on a hot air balloon: F_B = buoyancy, F_G = weight of gas, F_P = weight of payload (including the balloon envelope), and F_D = drag. Note that the direction of the drag is downward when the balloon is rising.