A balloon with a volume of 2500 m ^ 3 and a shell mass of 400 kg, heated from below by a burner, suspended from a weight of 200 kg. Ambient temperature 7 degrees C, air density 1.2 kg / m ^ 3. At what minimum temperature difference between the air outside and inside the balloon will take off with the load? Consider the shell of a sphere inextensible.
Task data: V (ball volume) = 2500 m3; mo (shell weight) = 400 kg; mg (weight of the suspended load) = 200 kg; Тwo (ambient temperature) = 280 К (7 ºС); ρvo (ambient air density) = 1.2 kg / m3.
1) Air mass in the ball: Fa = Fт; ρw * g * V = (mo + mg + mw) * g and mw = ρw * V – mo – mg.
2) Air pressure outside the ball: Pn = m * R * Tn / (Mw * V) = ρw * R * Tn / Mw.
3) Air pressure inside the ball: Rsh = mw * R * Tv / (Mw * V).
4) Air temperature inside a given balloon: mw * R * Tv / (Mw * V) = Rsh = Pn = ρw * R * Tn / Mw; mw * Tw / V = ρw * Tn, whence we express: Tw = ρw * Tn * V / mw = ρw * Tn * V / (ρw * V – mo – mg) = 1.2 * 280 * 2500 / (1, 2 * 2500 – 400 – 200) = 350 K = 77 ºС.
5) Minimum temperature difference: ΔТ = Tв – Тн = 77 – 7 = 70 ºС.
Answer: The balloon will start to take off at a minimum temperature difference of 70 ºС.
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