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Chapter 11: Problem 73

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).

Short Answer

Expert verified
The Mach number at the point of maximum temperature in Rayleigh flow is \(1 / \sqrt{k}\). The solution is derived by using the ideal gas assumption, necessarily using the Newton's second law of motion and the relation for velocity of gas under adiabatic conditions.

Step by step solution

01

Derive the velocity and temperature relation

We start by expressing the velocity \(V\) of the gas as a function of the temperature \(T\) for an ideal gas under adiabatic conditions, using Newton's second law of motion: \(V^2 = 2kRT\). Here, \(R\) is the gas constant.
02

Derive the Mach number

Next, we derive the Mach number \(M\), which is defined as the ratio of the velocity of the gas \(V\) to the speed of sound \(a\). The speed of sound can be expressed as a function of temperature and the gas constant: \(a = \sqrt{kRT}\). Therefore, the Mach number \(M\) is given by: \(M = V/a = \sqrt{V^2 / kRT} = \sqrt{2}\).
03

Find condition of maximum temperature

The condition of maximum temperature \(T_{max}\) occurs when the velocity \(V\) is minimum that is \(V_{min}\). Using the relations derived earlier, we substitute \(V_{min}\) and \(T_{max}\) into the Mach number \(M = \sqrt{2}\) to get: \(M_min = \sqrt{2 / kRT_{max}} = 1 / \sqrt{k}\) when \(V_{min} = \sqrt{2kRT_{max}}\).

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Most popular questions from this chapter

At a certain point in a pipe, air flows steadily with a velocity of \(150 \mathrm{m} / \mathrm{s}\) and has a static pressure of \(70 \mathrm{kPa}\) and a static temperature of \(4^{\circ} \mathrm{C}\). The flow is adiabatic and frictionless.

The flow blockage associated with the use of an intrusive probe can be important. Determine the percentage increase in section velocity corresponding to a \(0.5 \%\) reduction in flow area due to probe blockage for airflow if the section area is \(1.0 \mathrm{m}^{2}, T_{0}=\) \(20^{\circ} \mathrm{C},\) and the unblocked flow Mach numbers are (a) \(\mathrm{Ma}=0.2\) (b) \(\mathrm{Ma}=0.8\) (c) \(\mathrm{Ma}=1.5,(\mathrm{d}) \mathrm{Ma}=30\)

Steam \(\left(\mathrm{H}_{2} \mathrm{O} \text { vapor }\right)\) flows in a pipeline in a power station. The steam pressure is 150 psia, its temperature is \(500^{\circ} \mathrm{F}\), and it flows with velocity \(750 \mathrm{ft}\) /s. Calculate the stagnation pressure and stagnation temperature. If you are familiar with Steam Tables or steam property software, use these tools to make an "exact" calculation. If you are not familiar with these tools, model the steam as an ideal gas with molecular weight of 18 and \(k=1.3\).

Distinguish between flow of an ideal gas and inviscid flow of a fluid.

An ideal gas flows isentropically through a convergingdiverging nozzle. At a section in the converging portion of the nozzle. \(A_{1}=0.1 \mathrm{m}^{2}, p_{1}=600 \mathrm{kPa}(\mathrm{abs}), T_{1}=20^{\circ} \mathrm{C},\) and \(\mathrm{M} \varepsilon_{1}=0.6 .\) For section (2) in the diverging part of the nozzle, determine \(A_{2}, p_{2},\) and \(T_{2}\) if \(\mathrm{Ma}_{2}=3.0\) and the gas is air.
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