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- Homework Statement
- An air reservoir at ##T=20\, \textrm{ºC}## and ##p_0=150\, \textrm{kPa}## is emptied by a pump. The volume of the reservoir is ##V=1,5\, \textrm{m}^3##, and the pump evacuates ##\dot{V}=90\, \textrm{l}/\textrm{min}## of air, independent of pressure. Assuming that air is an ideal gas and the process isothermal, determine the time it takes to reduce the pressure to ##p=50\, \textrm{kPa}##.

Solution: ##t=18,3\, \textrm{min}##

- Relevant Equations
- ##\dfrac{d}{dt}\int_{VC}\rho\, dV=\dot{m}##

TRANSITIONAL REGIME: ##\dot{V}=90\, \textrm{L}/\textrm{min}=0,0015\, \textrm{m}^3/\textrm{s}##

$$\dfrac{d}{dt}\int_{VC}\rho\, dV=\dot{m}\rightarrow \dfrac{d}{dt}\rho V=\dot{m}\rightarrow $$

$$\boxed{\rho_{\textrm{air}}}\rightarrow pV=R'T\rightarrow \rho =\dfrac{p}{R'T}\rightarrow 287$$

$$\rightarrow \dfrac{dp}{dt}=\dfrac{\dot{m}R'T}{V}=\dfrac{\rho_{\textrm{air}}\dot{V}R'T}{V}=103,011\, \textrm{Pa}/\textrm{s}$$

$$50000=150000-\dfrac{dp}{dt}t\rightarrow \boxed{t=16,67\, \textrm{min}}$$

$$\rho_{\textrm{average}}\rightarrow p_{\textrm{avg}}=100000\rightarrow \rho_{\textrm{avg}}=1,189\, \textrm{kg}/\textrm{m}^3$$

Would it be well done like this?

$$\dfrac{d}{dt}\int_{VC}\rho\, dV=\dot{m}\rightarrow \dfrac{d}{dt}\rho V=\dot{m}\rightarrow $$

$$\boxed{\rho_{\textrm{air}}}\rightarrow pV=R'T\rightarrow \rho =\dfrac{p}{R'T}\rightarrow 287$$

$$\rightarrow \dfrac{dp}{dt}=\dfrac{\dot{m}R'T}{V}=\dfrac{\rho_{\textrm{air}}\dot{V}R'T}{V}=103,011\, \textrm{Pa}/\textrm{s}$$

$$50000=150000-\dfrac{dp}{dt}t\rightarrow \boxed{t=16,67\, \textrm{min}}$$

$$\rho_{\textrm{average}}\rightarrow p_{\textrm{avg}}=100000\rightarrow \rho_{\textrm{avg}}=1,189\, \textrm{kg}/\textrm{m}^3$$

Would it be well done like this?