Intermediate 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.
 05.03.14
 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: 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: 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 $\dot{m}_{\rm wout}=2$ 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 $\dot{m}_{\rm out}$: Knowing that the flow rate of water going in is of $\dot{m}_{\rm in}=3$ g/s, that the flow rate of water going out is of $\dot{m}_{\rm out}=2$ 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: Knowing that the mass flow rate of water going in is of $\dot{m}_{\rm win}=2$ 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 Wednesday April 3rd at 16:30. Do Questions #1, #4, and #5 only.
 03.28.19
There was a mistake in the formulation of Q4: $\dot{m}_{\rm in}$ should be 3 gr/s, not 2 gr/s. This has been fixed.
 04.01.19
 $\pi$