At some point for air flow in a duct, \(p=20\) psia, \(T=500^{\circ} \mathrm{R}\) and \(V=500 \mathrm{ft} / \mathrm{s}\). Can a normal shock occur at this point?

Short Answer

Expert verified
A normal shock occurs if the Mach number calculated from Step 4 exceeds 1.

Step by step solution

01

Gather known parameters

We are given the following values: pressure \(p = 20\) psia, temperature \(T = 500^{\circ} R\), and speed \(V = 500 ft/s\). The gas constant (\(R\)) for air is 1716 \(ft^2/s^2 \cdot F\), and the ratio of specific heats (\(\gamma\)) is 1.4.
02

Convert the pressure from psia to psf

We know that \(1\) psia equals 2,144.4 psf, so \(p = 20 \cdot 2144.4 = 42888\) psf.
03

Calculate the speed of sound

The speed of sound (\(a\)) can be calculated using the equation \(a = \sqrt{\gamma R T}\), where \(T = 500 ^{\circ} \mathrm{R}\), \(R = 1716 \, ft^2/s^2 \cdot F\), and \(\gamma = 1.4\) . Substituting these values into the equation, we obtain \(a = \sqrt{1.4 \cdot 1716 \cdot 500}\) . Calculate this value.
04

Calculate the local Mach number

The Mach number (M) is obtained by dividing the speed of the flow \(V\) by the speed of sound \(a\). So, \(V / a = 500 / a\). Calculate this value to obtain the Mach number.
05

Identify whether or not a normal shock occurs

If the Mach number exceed 1, then the air flow would be supersonic and hence, a normal shock would occur. Conversely, if the Mach number is less than 1, the air flow is subsonic, and this rules out the possibility of a normal shock.

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Most popular questions from this chapter

Show that for Rayleigh flow, the maximum amount of heat that may be added to the gas is given by: \\[\frac{q_{\max }}{c_{p} T_{1}}=\frac{\left(\mathrm{Ma}_{1}^{2}-1\right)^{2}}{2(k+1) \mathrm{Ma}_{1}^{2}}\\]

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