Fundamentals of Fluid Mechanics Assignment 7 — Energy Conservation II
 Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
 Question #1
A spaceship cabin may be considered to be a rigid pressurized vessel, which contains the atmosphere required to support the life of its occupants. A puncture in the cabin when it is in space must be detected quickly, so that the cabin occupants can seal the puncture. The time interval that elapses before a dangerously low pressure is reached is a critical quantity for designing the warning devices and countermeasure systems. Derive an equation giving an estimate for the pressure $P$ after a puncture as a function of the time $t$, the initial pressure $P_{\rm i}$, the initial temperature $T_{\rm i}$, the area $A$ of the puncture, and the volume $V$ of the cabin. Consider the atmosphere of the cabin to be a perfect gas. If the cabin atmosphere is air at an initial temperature of 300 K, determine the ratio $P/P_{\rm i}$ as a function of time with the ratio $(A/V)$ as a parameter. Assume that for air, $\gamma=1.4$ and $R=287.4$ J/kg K.
 10.11.19
 Question #2
 (a) Prove that the total enthalpy $H=h+\frac{1}{2}q^2+gy$ is conserved on a streamline. Outline all assumptions. (b) Prove that the total enthalpy $H=h+\frac{1}{2}q^2+gy$ is conserved on a particule path (in space and time) under certain conditions. Outline these conditions.
 10.12.19
 Question #3
Starting from the isentropic relationship $$\frac{P}{\rho^\gamma} = \textrm{constant}$$ and the momentum equation along a streamline, show that the stagnation pressure corresponds to: $$P_\circ = P \left( 1 + \frac{\gamma-1}{2}M^2 \right)^\frac{\gamma}{\gamma-1}$$ Outline all assumptions. Can the former be used in the presence of a gravitational field? Explain. Can the latter be used in the presence of a gravitational field? Explain.
 Question #4
Consider air flowing in a duct oriented along the $x$ axis. The air enters the duct with a pressure of 0.4 atm, a speed of 100 m/s, and a density of 1 kg/m$^3$. The duct cross-sectional area is of 1.3 m$^2$ at the entrance and of 1 m$^2$ at the exit. The duct external surface is touching air at a pressure of 1 atm. Do the following:
 (a) Calculate the pressure, speed, and density of the air exiting the duct. (b) Calculate the net force acting on the duct. (c) Calculate the net force acting on the duct if the air touching the external surface would be replaced by a vacuum.
 1. $\displaystyle \frac{P}{P_{\rm i}}=\left( 1-\frac{b A}{8 V} t\sqrt{\frac{8RT_{\rm i}}{\pi}}\right)^\frac{2-2b}{b}$ with $b\equiv \left(1+\frac{8}{32\pi}\right)(1-\gamma)$ 4. 0.911 kg/m$^3$, 142.7 m/s, 0.351 atm, 18821 N, -11569 N.
 $\pi$