A 1-m^3-rigid tank initially contains compressed air at 15°C. Air is being removed from the tank through a 3-cm-diameter pipe, as shown in Fig. P4.64. If the velocity and density of air in the pipe are 200 m/s and 1.8 kg/m^3 , respectively, determine the initial rate at which the pressure inside

Question

A 1-[latex]m^3[/latex]-rigid tank initially contains compressed air at 15°C. Air is being removed from the tank through a 3-cm-diameter pipe, as shown in Fig. P4.64. If the velocity and density of air in the pipe are 200 m/s and 1.8 [latex]kg/m^3[/latex], respectively, determine the initial rate at which the pressure inside the tank will vary assuming a constant temperature.

Accepted Answer

Choose the control volume to consist of the air volume inside the tank. The conservation of mass equation is

[latex]0 =\frac{d}{dt}\int_{CV} {ρdV} +\int_{CS} ρ\pmb {V.n} dA[/latex]

Since the volume of the tank is constant, and for no flow into the tank, the equation is

[latex]0 = V \frac{dρ}{dt}+ρ_e V_e A_e[/latex]

Assuming air behaves as an ideal gas, [latex] \rho=\frac{p }{RT} [/latex] At the instant of interest,

[latex]\frac{d\rho }{dt}=\frac{1}{RT}\frac{dp}{dt}[/latex] . Substituting in the conservation of mass equation, we get

[latex]\frac{dp }{dt} =-\frac{ρ_e V_e A_e}{V} (RT)=-\frac{( 1.8 kg/m^3 )(200 m/s )\pi ( 0.015 m )^2 }{1.5 m^3}[0.287 \frac{KJ}{Kg.K}× 298 K ][/latex]

∴[latex]\frac{dp }{dt} = \underline{-14.5 kPa/s} [/latex]

Question 4.64: A 1-m^3-rigid tank initially contains compressed air at 15°C...

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