Thermodynamics Assignment 3 — Conservation of Mass  
Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
05.04.14
Question #1
Starting from the conservation of mass principle, show that the conservation of mass equation in differential form corresponds to: $$ \frac{\partial }{\partial t}\rho + \frac{\partial}{\partial x} \rho v_x + \frac{\partial}{\partial y} \rho v_y + \frac{\partial}{\partial z} \rho v_z =0 $$
Question #2
A pipe with a diameter $D_1=0.30$ m is first reduced to a diameter $D_2=0.15$ m and later expanded to a diameter $D_3=0.25$ m:
figure3.png  ./download/file.php?id=1411&sid=96367ee047c06c4eaa3880735f36b495  ./download/file.php?id=1411&t=1&sid=96367ee047c06c4eaa3880735f36b495
If the mean velocity is 4.5 m/s in the narrowest cross-section, what is the mean velocity in the other two sections? The fluid media is water. Note: you have to solve this problem starting from the control volume form of the mass conservation equation; indicate clearly where you locate the control volume.
Question #3
Consider a tank that is being filled with water:
figure4.png
Knowing that the flow rate of water going in is of $\dot{m}_{\rm win}=3$ kg/s, that the flow rate of water going out is of $\dot{m}_{\rm wout}=(2+0.5\times\xi_1)$ kg/s, that the volume of air within the tank is of 10 m$^3$ and that the volume of water within the tank is of 15 m$^3$, find $\dot{m}_{\rm aout}$, the mass flow rate of air going out of the air vent. Take the density of water as $\rho_{\rm w}=1000$ kg/m$^3$ and the density of air as $\rho_{\rm a}=1$ kg/m$^3$.
Question #4
Consider a tank that is being filled with water with the mass flow rate $\dot{m}_{\rm in}$ and from which the water escapes through a vent with the mass flow rate $\dot{m}_{\rm out}$:
figure5.png  ./download/file.php?id=1413&sid=96367ee047c06c4eaa3880735f36b495  ./download/file.php?id=1413&t=1&sid=96367ee047c06c4eaa3880735f36b495
Knowing that the flow rate of water going in is of $\dot{m}_{\rm in}=2$ g/s, that the flow rate of water going out is of $\dot{m}_{\rm out}=(2+\xi_1)$ g/s, that the mass of the water at time $t_1$ is 10 kg, determine the mass of the water $t_2=t_1+4$ min. Start from the control-volume form of the mass conservation equation and take the density of water as $\rho_{\rm w}=1000$ kg/m$^3$.
Question #5
Consider the following two-tank system:
figure7.png  ./download/file.php?id=1414&sid=96367ee047c06c4eaa3880735f36b495  ./download/file.php?id=1414&t=1&sid=96367ee047c06c4eaa3880735f36b495
Knowing that the mass flow rate of water going in is of $\dot{m}_{\rm win}=(2+\xi_1)$ kg/s, that the mass flow rate of air coming in is of $\dot{m}_{\rm air}=0.02$ kg/s, that the volume of the water in tank 2 increases at the rate of $\rm 0.005~m^3/s$, find the mass flow rate of the water coming out $\dot{m}_{\rm wout}$. Also, find the ratio between the mass flow rate of air entering tank 2 and the mass flow rate of air entering the system $(\dot{m}_{\rm air2}/\dot{m}_{\rm air})$. Start from the control-volume form of the mass conservation equation and outline clearly where the control volume(s) is(are) located. Take the density of water as $\rho_{\rm w}=1000$ $\rm kg/m^3$ and the density of air as $\rho_{\rm air}=1~{\rm kg/m^3}$.
Due on April 2 2014
This assignment is not too difficult but make sure you understand well the control volume and the definition of the unit normal vector $\vec{n}$.
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