The pressure head in an air pipe shown in Fig. 2.20 is measured with a piezometer as 16 mm of water. A pitot probe at the same location indicates 24 mm of water. The temperature is 20°C. Calculate the Mach number and comment on compressibility of air.

Question

The pressure head in an air pipe shown in Fig. 2.20 is measured with a piezometer as 16 mm of water. A pitot probe at the same location indicates 24 mm of water. The temperature is [latex]20 ^{\circ}C[/latex]. Calculate the Mach number and comment on compressibility of air.

Accepted Answer

Applying Bernoullis equation between two points 1 and 2 on the streamline, the point 1 refers to static pressure and point 2 is stagnation pressure, we can write.

[latex]\frac{P_{1}}{\gamma } + \frac{V^{2}_{1}}{2g} = \frac{P_{0}}{\gamma } \because V_{1} = \sqrt{\frac{2\left(P_{0}-P_{1} ight) }{ ho } }[/latex]

As air is considered ideal gas, and applying characteristic equation

[latex]P= ho R T or ho = \frac{P}{RT}[/latex]

[latex]P = P_{g} + P_{atm} = 9800 imes 0.016 + 101000 = 101.156 k Pa[/latex]

[latex] ho = \frac{101.156 imes 10^{3}}{287 imes \left(273+20 ight)} = 1.2 {kg}/{m^{3}}[/latex]

[latex] herefore V=\sqrt{\frac{2}{ ho }\left(P_{0}-P ight) } = \sqrt{\frac{2\left(0.032-0.016 ight)9800 }{1.2} }[/latex]

[latex] =16.17 {m}/{s}[/latex]

Velocity of sound [latex]C=\sqrt{KRT} =\sqrt{1.4 imes 287 imes 293} = 343 {m}/{s}[/latex]

Mach number [latex]M=\frac{V}{C} =\frac{16.17}{343} = 0.047 [/latex]

The flow is incompressible as Mach number is less than 0.3.

Question 2.9: The pressure head in an air pipe shown in Fig. 2.20 is measu...

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