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Chapter 8: Problem 11

In an axial-flow compressor test rig with no inlet guide vanes, a 1 -m diameter fan rotor blade spins with a sonic tip speed, i.e., \(U_{\text {tip }} / a_{1}=1.0 .\) If the speed of sound in the laboratory is \(a_{0}=300 \mathrm{~m} / \mathrm{s}\), and the axial velocity to the fan is \(C_{z 1}=150 \mathrm{~m} / \mathrm{s}\), calculate the fan rotational speed \(\omega\) in rpm. $$ R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \text { and } \gamma=1.4 $$

Short Answer

Expert verified
The fan's rotational speed \(\omega\) must be calculated in the steps above, first in rad/s and then converted to rpm.

Step by step solution

01

Calculate the Blade Tip Speed

First, we need to calculate the blade tip speed \(U_{\text {tip}}\), which according to the sonic tip speed condition is equal to \(a_1\), where \(a_1\) is the speed of sound in the fluid moving to the fan. In this case, we know that \(a_0 = 300 \, \mathrm{m/s}\) is the speed of sound in the stationary laboratory and \(C_{z1} = 150 \, \mathrm{m/s}\) is the axial velocity to the fan. The speed of sound \(a_1\) in the moving fluid can be calculated using the formula: \[a_1 = (a_0^2 + C_{z1}^2)^{0.5}\]
02

Calculate Rotational Speed in Radians per Second

Given that \(U_{\text {tip}} = a_1 \), we can calculate the rotational speed \(\omega\) in radians per second using the blade tip speed and the radius of the blade which is half of its diameter, thus is \(0.5 \, \mathrm{m}\): \[\omega = U_{\text{tip}}/r = a_1/r\]
03

Convert Rotational Speed to RPM

The final step is to convert the rotational speed from radians per second to rounds per minute (rpm). The conversion formula is: \[\omega [\text{rpm}] = \omega[\text{rad/s}] * \frac{60}{2\pi}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sonic Tip Speed
In the world of axial-flow compressors, the concept of sonic tip speed is pivotal. When an axial-flow compressor rotor blade achieves a sonic tip speed, its speed equals the speed of sound in the surrounding medium. This condition is depicted mathematically as \( U_{\text{tip}} / a_{1} = 1.0 \), meaning the blade tip speed \( U_{\text{tip}} \) is precisely equal to the speed of sound \( a_{1} \). Understanding sonic tip speed is crucial as it influences the compressor's ability to efficiently compress the air without reaching shock wave conditions. When shock waves are present, they can disrupt the smooth flow of air, potentially causing vibrations and losses in efficiency.
Speed of Sound
The speed of sound is a key factor in aerodynamic and thermodynamic calculations, especially in axial-flow compressors. It varies based on the medium through which it travels and is influenced by factors such as temperature and pressure. The exercise states the local laboratory speed of sound as \( a_{0} = 300 \, \text{m/s} \). Additionally, the axial velocity of air moving to the fan is \( C_{z1} = 150 \, \text{m/s} \). The effective speed of sound \( a_{1} \) in the moving fluid environment is calculated using the relationship: \[ a_{1} = \sqrt{a_{0}^2 + C_{z1}^2} \] This formula accounts for both the ambient speed of sound and the velocity component of air as it approaches the fan.
Rotational Speed
The rotational speed \( \omega \) of the fan is indicative of how fast the blades are spinning. It is measured in radians per second (rad/s) or revolutions per minute (rpm) and is a direct result of the blade tip speed and the radius of the fan blades. With the blade tip's sonic speed meaning that \( U_{\text{tip}} = a_{1} \), the rotational speed can be derived by the formula: \[ \omega = \frac{a_{1}}{r} \] Where \( r \) is the radius of the blade, which is half the diameter, in this case, \( 0.5 \text{ m} \). This step allows us to understand the mechanical motion behind the rotational dynamics.
Blade Tip Speed
Blade tip speed is the linear velocity of the very tip of the fan blade as it moves through space. It represents how fast the tip of the blade covers distance and is a critical factor in determining the aerodynamic responsiveness and stress levels on the blade. Achieving the desired blade tip speed, which in this context equals the speed of sound (\( a_1 \)), is essential for maintaining optimum functionality without transgressing into supersonic realms where control becomes complex. The calculation of blade tip speed forms an essential foundation for understanding the movement and forces at play in axial-flow compressors, ensuring smoother and more controlled operations.

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Most popular questions from this chapter

The flow at the entrance to an axial-flow compressor rotor has zero preswirl and an axial velocity of \(175 \mathrm{~m} / \mathrm{s}\). The shaft angular speed is \(5000 \mathrm{rpm}\). If at a radius of \(0.5 \mathrm{~m}\), the rotor exit flow has zero relative swirl, calculate at this radius (a) rotor specific work \(w_{c}\) in \(\mathrm{kJ} / \mathrm{kg}\) (b) degree of reaction \({ }^{\circ} R\)

An axial-flow compressor stage is downstream of an IGV that tums the flow \(15^{\circ}\) in the direction of the rotor rotation, as shown. The axial velocity component remains constant throughout the stage at \(C_{z}=150 \mathrm{~m} / \mathrm{s}\). The rotor rotational speed is \(\omega=3000 \mathrm{rpm}\) and the pitchline radius is \(r_{\mathrm{m}}=0.5 \mathrm{~m}\). The rotor relative exit flow angle is \(\beta_{2}=-15^{\circ}\). The static temperature and pressure of air upstream of the rotor are \(T_{1}=20^{\circ} \mathrm{C}\) and \(p_{1}=10^{5} \mathrm{~Pa}\), respectively. Assuming \(c_{p}=1.004 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\) and \(\gamma=1.4\), calculate (a) relative Mach number to the rotor, \(M_{\mathrm{tr}}\) (b) absolute total temperature \(T_{\mathrm{u}}\) (c) rotor specific work \(w_{c}\) in \(\mathrm{kJ} / \mathrm{kg}\) (d) total temperature downstream of the rotor, \(T_{12}\) (e) relative Mach number downstream of the rotor, \(M_{2}\) (f) stage total pressure ratio at the pitchline radius for \(e_{c}=0.92\) (g) stage degree of reaction \({ }^{\circ} R_{\mathrm{m}}\) at the pitchline

An axial-flow compressor rotor at the pitchline has a radius of \(r_{\mathrm{m}}=0.35 \mathrm{~m}\). The shaft rotational speed is \(\omega=\) \(5000 \mathrm{rpm}\). The inlet flow to the rotor has zero preswirl and the axial velocity is \(C_{z 1}=C_{z 2}=C_{z 3}=175 \mathrm{~m} / \mathrm{s}\). The rotor has a \(50 \%\) degree of reaction at the pitchline. The stage adiabatic efficiency is nearly equal to the polytropic efficiency \(\eta_{\mathrm{s}} \cong e_{\mathrm{c}}=\) \(0.92\). Assuming the inlet total temperature is \(T_{\mathrm{t} 1}=288 \mathrm{~K}\) and \(c_{p}=1.004 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) rotor specific work at \(r=r_{\mathrm{m}}\) (b) stage loading \(\psi\) at \(r=r_{\mathrm{m}}\) (c) flow coefficient at \(r=r_{\mathrm{m}}\) (d) rotor relative Mach number at the pitchline, \(M_{1 r, m}\) (e) stage total pressure ratio at the pitchline

An axial-flow compressor rotor has an angular velocity of \(\omega=5000 \mathrm{rpm}\). The flow entering the compressor rotor has zero preswirl and an axial velocity of \(C_{z 1}=150\) \(\mathrm{m} / \mathrm{s}\). Assuming the axial velocity is constant throughout the stage, and the rotor specific work at the radius \(r=0.5 \mathrm{~m}\) is \(w_{c}\) \(=62 \mathrm{~kJ} / \mathrm{kg}(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})\) calculate (a) stage degree of reaction, \({ }^{\circ} R\), at this radius (b) total pressure ratio across the rotor, \(p_{12} / p_{11}\), at this radius, assuming a polytropic efficiency of \(90 \%\) and \(T_{1}=20^{\circ} \mathrm{C}\).

Apply Euler turbine equation to a streamtube that enters a compressor rotor blade row at \(r=2.5 \mathrm{ft}\) with zero preswirl and exits the row at \(r=2.7 \mathrm{ft}\) and attains \(1000 \mathrm{ft} / \mathrm{s}\) of swirl velocity in the absolute frame. Assume that the shaft rotational speed is \(5000 \mathrm{rpm}\).
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