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Chapter 11: Problem 46

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) \(\mathrm{Ma}=1.5,(\mathrm{d}) \mathrm{Ma}=30\)

Short Answer

Expert verified
The percentage increase of the section velocity due to a reduction in the flow area will vary depending on the Mach number

Step by step solution

01

Identify Known Variables and Equations

Begin by identifying the known variables .We have Mach numbers (\(M\)) of \(M = 0.2, M = 0.8, M = 1.5, M = 3.0\), temperature(\$T_{0}=20^{\circ C} = 293.15 K\$), initial area (\(A_{0}=1.0 m^{2}\)), and the percentage area reduction due to probe blockage (\(0.5\%\)). The percentage blockage can be calculated using the equation: \(\% Increase = \frac{Area_{2}- Area_{1}}{Area_{1}} * 100\). In addition, the equation for velocity using the Mach number will be used: \(V = M \cdot \sqrt{γ \cdot R \cdot T}\) where \(γ\) denotes the ratio of specific heats (for air, \(γ = 1.4\)) and \( R = 287 J/ (K \cdot kg) \) is the specific gas constant for air.
02

Calculating Section Velocity

The initial section velocity can be obtained by substituting the known values of \(M, γ, R,) and \(T\) into the equation \(V = M \cdot \sqrt{γ \cdot R \cdot T}\). Calculate this for all Mach numbers given.
03

Calculate New Area

The new area due to probe blockage will be 99.5% of the original area since there is a reduction of 0.5%. This can be calculated using the equation: \(A_{2} = A_{1} * \frac{(100 - Reduction \%)}{100}\). After calculating the new area, obtain the new velocities using same equation as above \(V = M \cdot \sqrt{\frac{γ \cdot R \cdot T}{A}}\) by replacing \(T\) with the new areas obtained.
04

Calculate Percentage Increase

Finally, the percentage increase in velocity due to the probe blockage can be calculated using the percentage increase equation: \(\% Increase = \frac{V_{2}- V_{1}}{V_{1}} * 100 \) where \(V_{1}\) is the initial velocity and \(V_{2}\) is the new velocity. Calculate this for all Mach numbers given.

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Most popular questions from this chapter

An aircraft cruises at a Mach number of 2.0 at an altitude of \(15 \mathrm{km} .\) Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is \(1.2 .\) For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of \(59^{\circ} \mathrm{F}\) and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in \(\mathrm{ft} / \mathrm{s}\) if the gas is air.

Air flows steadily and isentropically from standard atmospheric conditions to a receiver pipe through a converging duct. The cross-sectional area of the throat of the converging duct is \(0.05 \mathrm{ft}^{2}\). Determine the mass flowrate through the duct if the receiver pressure is (a) 10 psia, (b) 5 psia. Sketch temperature-entropy diagrams for situations (a) and (b). Verify results obtained with values from the appropriate graph in Appendix D with calculations involving ideal gas equations. Is condensation of water vapor a concern? Explain.

An ideal gas flows with velocity \(V\), pressure \(p\), temperature \(T,\) and density \(\rho .\) Determine a set of equations for stagnation properties, including entropy, if the stagnation process is defined to be isothermal \((T=\text { constant ) rather than isentropic }(s=\text { constant })\).

Air flows isentropically through a duct to a section where \(p_{1}=25 \mathrm{kPa}, T_{1}=300 \mathrm{K},\) and \(V_{1}=900 \mathrm{m} / \mathrm{s} .\) For these conditions: (a) Determine the stagnation conditions for the flow. (b) What is the Mach number at station \(1 ?\) Show a \(T-s\) diagram displaying stagnation and static conditions. (c) Is the flow choked? Is the throat behind or ahead of section \(1 ?\) Label this state on the \(T-s\) diagram.
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