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Chapter 11: Problem 42

The static pressure to stagnation pressure ratio at a point in a gas flow field is measured with a Pitot-static probe as being equal to \(0.6 .\) The stagnation temperature of the gas is \(20^{\circ} \mathrm{C}\). Determine the flow speed in \(\mathrm{m} / \mathrm{s}\) and the Mach number if the gas is air. What error would be associated with assuming that the flow is incompressible?

Short Answer

Expert verified
The velocity and Mach number calculated reveal properties about the gas flow's speed and compressibility. The error associated with assuming the flow is incompressible can be calculated by comparing results obtained using the Bernoulli's equation for incompressible and compressible flow.

Step by step solution

01

Calculate velocity

The flow speed of a compressible flow can be determined using Bernoulli's equation, which allows the calculation of fluid flow speed at a point in terms of energy conservation. First, calculate the total pressure \(P_0\), which is the sum of the static pressure \(P\) and dynamic pressure for the system. Use the given ratio, \(P/P_0 = 0.6\), to find \(P_0 = P / 0.6\). Now, apply Bernoulli's equation to solve for velocity \(V\), \( V = \sqrt{(2 * (P_0 – P))/\rho}\), where \(\rho\) is the density of air, which can be approximated as 1.225\(kg/m^3\) at sea level and 20 degrees Celsius.
02

Calculate Mach number

The Mach number, which expresses the speed of an object moving through air, or any fluid substance, as a multiple of the speed of sound through that substance, can be calculated using the formula: \(Ma = V/a\), where \(V\) is the velocity of the flow (calculated in the previous step), and \(a\) is the speed of sound in the medium, which for air at 20 degrees Celsius, is approximately 343.2 m/s.
03

Calculate error based on incompressibility assumption

Assuming the flow is incompressible can introduce a source of error, particularly at high flow speeds where compressibility effects can be significant. To estimate this error, recalculate the velocity and Mach number using Bernoulli's equation for incompressible flow, and compare the results with those obtained using the compressible flow assumptions. The percentage difference between these values is the error associated with assuming the flow is incompressible.
04

Interpret results

The calculated flow speed and Mach number provide information about the properties of the gas flow, more specifically about its speed and its compressibility. Higher Mach numbers imply greater compressibility effects.

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Most popular questions from this chapter

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 \(k=1.3\).

Sound waves are very small-amplitude pressure pulses that travel at the "speed of sound." Do very large-amplitude waves such as a blast wave caused by an explosion (see Video \(\vee 11.8\) ) travel less than, equal to, or greater than the speed of sound? Explain.

The Pitot tube on a supersonic aircraft (see Video \(\mathbf{V} 3.8\) ) cruising at an altitude of 30,000 ft senses a stagnatior pressure of 12 psia. If the atmosphere is considered standard, determine the airspeed and Mach number of the aircraft. A shock wave is present just upstream of the probe impact hole.

Air flows in a constant-area, insulated duct. The air enters the duct at \(520^{\circ} \mathrm{R}, 50\) psia, and \(\mathrm{Ma}=0.45 .\) At a downstream location, the Mach number is one. Find: (a) The pressure and temperature at the downstream location (b) The change in specific entropy (c) The frictional force if the duct is circular and \(1 \mathrm{ft}\) in diameter.

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