Consider air escaping from a tank under choked flow conditions.

a$)$ Plot P$/$P$\_0$ vs t$/\backslash$tau for both isothermal and adiabatic conditions on the same graph. Real tanks are rarely $100\%$ isothermal or $100\%$ adiabatic, and so the curve for a real tank would likely like in between the two you$$ve plotted.

b$)$ For both isothermal and adiabatic conditions, at what non$-$dimensional time t$/\backslash$tau will we expect pressure to have dropped by $50\%?$

c$)$ Consider a tank with an initial pressure of P$\_0=85\mathrm{.}3$ psig in a place where atmospheric pressure is $14\mathrm{.}7$ psi. A pressure relief valve in this tank is opened until absolute pressure has fallen by $50\%.$ Is it valid to assume choked flow conditions for the entirety of the time the valve is open? Demonstrate why or why not.

d$)$ The tank in part c has a volume of $40$ ft$^3.$ The valve opens a circular orifice with a diameter of $1$ in$,$ and its discharge coefficient under choked flow conditions has been measured as $0\mathrm{.}9.$ The temperature of the tank when the valve is opened is $70\backslash$deg F$.$ For both isothermal and adiabatic conditions, find the time required for absolute tank pressure to drop by $50\%.$