Fundamentals of Fluid Mechanics A Assignment 6 — Energy Conservation  
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)  Starting from the 1st law of thermo (with $\delta W=0$): $$ {\rm d}(mh)-V {\rm d}P=\delta Q $$ and the $x$ and $y$ momentum equations $$ \rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y}=-\frac{\partial P}{\partial x} $$ $$ \rho \frac{\partial v}{\partial t} + \rho u \frac{\partial v}{\partial x} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y} $$ show that the total energy transport equation for a fluid corresponds to: $$ \frac{\partial \rho E}{\partial t} + \frac{\partial \rho u H}{\partial x} + \frac{\partial \rho v H}{\partial y} = \dot{Q} $$ with the total energy $E\equiv e+\frac{1}{2}q^2$, the total enthalpy $H\equiv h +\frac{1}{2}q^2$, and $q^2=u^2+v^2$. Outline all assumptions.
(b)  Starting from the conservation of energy in conservative form derived in (a), show that the energy conservation in control volume form corresponds to: $$ \frac{{\rm d}}{{\rm d}t} \int_V \rho E{\rm d} V + \int_S \rho (\vec{v}\cdot\vec{n}) H{\rm d}S=\int_V \dot{Q} dV$$
Note: you can not assume that the density is constant.
Question #2
Starting from the 1st law of thermo: $$ {\rm d}(mh)-V {\rm d}P=\delta Q - \delta W $$ and the $y$ momentum equation: $$ \rho \frac{\partial v}{\partial t} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y} - \rho g $$ show that the total energy transport equation for a fluid corresponds to: $$ \frac{\partial \rho E}{\partial t} + \frac{\partial \rho v H}{\partial y} = \dot{Q} $$ with the total energy $E\equiv e+\frac{1}{2}v^2 + g y$, the total enthalpy $H\equiv h +\frac{1}{2}v^2 + g y$ and $y$ the distance from the ground. Outline all assumptions. Note: you can not assume that $\delta W$ is zero or that the density is constant.
09.18.19
Question #3
A 4 m$^3$ storage tank (see schematic below) containing 2 m$^3$ of liquid is to be pressurized with air from a large, high-pressure reservoir through a valve at the top of the tank to permit rapid ejection of the liquid:
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The air in the reservoir is maintained at 100 bar and 300 K. The gas space above the liquid contains initially air at 1 bar and 280 K. When the pressure in the tank reaches 5 bar, the liquid transfer valve is opened and the liquid is ejected at the rate of 0.2 m$^3$/min while the tank pressure is maintained at 5 bar. What is the air temperature when the pressure reaches 5 bar and when the liquid has been drained completely? You must use the integral form of the energy conservation equation to solve this problem. If the type of integral form needed is not in the tables, it needs to be derived.
Question #4
(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.
10.20.20
Question #5
Starting from the energy equation excluding work done except through pressure forces: $$ \frac{\partial }{\partial t} \rho E + \frac{\partial}{\partial x} \rho u H + \frac{\partial}{\partial y} \rho v H + \frac{\partial}{\partial z} \rho w H = \dot{Q} $$ and the inviscid momentum equations, prove that the energy equation for a constant-density fluid excluding work done except through pressure becomes: $$ \rho \frac{\partial e}{\partial t} + \rho u \frac{\partial e}{\partial x} + \rho v \frac{\partial e}{\partial y} + \rho w \frac{\partial e}{\partial z} = \dot{Q} $$ Outline all assumptions.
10.22.20
Answers
3.  382 K, 336 K.
Due on Wednesday October 21st at 11:59pm. Do questions #2 and #3 only.
10.12.26
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