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