Fatalities have resulted from the careless handling of compressed-gas cylinders, both during transportation and during use. The objective of this problem is to illustrate the consequences of a large, improperly secured, inert-gas cylinder falling over in a lab and shearing off the pressure regulator at its top.

A standard 44-liter steel gas cylinder is 9 inches in diameter, 51 inches tall, and has a tare weight of 133 pounds, corresponding to a mass when empty of 60.4 kg. Suppose that such a cylinder is filled with argon [M = 40 kg/(kg-mol), γ = 1.67] at 2000 psig. In SI units, V = 0.044 m³, P = 1.38 × 107 Pa (absolute), and T = 293 K. The inside diameter of the regulator connection is 5 mm. Thus, breaking off the regulator at t = 0 creates a nozzle with that throat diameter.

(a) Assuming that the gas expands isentropically, show that the initial throat velocity v_t will be sonic and find its value. Explain why v_t will be constant during almost the entire discharge, if the gas in the cylinder remains at 293 K.

(b) Find the initial mass flow rate w(0).

(c) Let U(t) be the velocity of the cylinder relative to the lab. If after tipping over it makes little contact with the floor, derive an expression for dU/dt.

(d) If the initial dU/dt is sustained and no obstacles are encountered, show that U = 9.4 m/s at t = 3 s. That is more than enough to severely damage a masonry wall, not to mention a person! (Accounting for both the decreasing w and the decreasing dU/dt as the cylinder empties gives U = 8.4 m/s at 3 s.)