Fluid Dynamics - Mass Conservation, State Equation for an Ideal Gas

[WhiteWolf98]

Homework Statement

Air flows steadily in a long pipe. The static, absolute pressure and the temperature at the pipe inlet are $77~kPa$ and $264~K$ respectively. At the outlet the static, absolute pressure and the temperature are $44~kPa$ and $244~K$ respectively. Assuming that the average air velocity at the inlet is $V=202~ms^{-1}$, use the mass conservation principle and the state equation for an ideal gas to determine the average air velocity (in $ms^{-1}$) at the outlet.

Relevant Equations

${\dot m}_{in}= {\dot m}_{out}$, $PV=nRT?$

I understand that $\dot m=\rho Q$ and ${\dot m}_{in}= {\dot m}_{out}$ . So one can say that $\rho Q_1 = \rho Q_2$. But I'm not sure if that equation is correct. I don't know if the density remains constant, or the volume flow rate. And then how I'm also supposed to tie a state equation in it too... I've thought about the problem a lot, but I don't seem to be getting anywhere. Any help in the right direction would be appreciated; thanks!

 

[TSny]

You can use the ideal gas law to express the mass density $\rho$ in terms of temperature and pressure. To do this, express the number of moles $n$ in a sample of gas in terms of the mass $m$ of the sample and the molar mass $M$.

 

[WhiteWolf98]

I still don't see how this helps me work out the velocity. All the formulas are confusing me greatly.

$\dot m = \rho Q$ and $\rho =\frac m V$ || $Q=AV$

$PV=nRT, n=\frac m M, m=\rho V$

$PV=\frac m MRT$

$PV=\frac {\rho V} M RT$

$PM=\rho RT$

Am I to assume next that: $\frac {PM} {\rho RT} = constant$?

 

[TSny]

What do you get if you set up $\dot m_{out} = \dot m_{in}$ in terms of $\rho$ and $Q$? Use what you learned about $\rho$ from your manipulations of the ideal gas law.

 

[WhiteWolf98]

$\dot m_1 = \dot m_2$

$\rho_1 Q_1 = \rho_2 Q_2$

$\frac {P_1 M} {R T_1} A V_1 = \frac {P_2 M} {R T_2} A V_2$

$\frac {P_1 V_1} {T_1} = \frac {P_2 V_2} {T_2}$

$V_2 = \frac {T_2 P_1 V_1} {T_1 P_2}$

Thank you :3