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Chapter 8: Problem 34

A compressor stage has 37 rotor blades and 41 stator blades. The shaft rotational speed is \(5000 \mathrm{rpm}\). Calculate (a) the rotor blade passing frequency as seen by the stator blades (b) the stator blade passing frequency as seen by the rotor blades

Short Answer

Expert verified
The rotor blade passing frequency as seen by the stator blades is approximately 3083.33 Hz. The stator blade passing frequency as seen by the rotor blades is approximately 3416.67 Hz.

Step by step solution

01

Understand and define blade passing frequency

Blade passing frequency is the frequency at which a rotor or stator blade passes a point in a given time. It is usually measured in Hertz (Hz) and is calculated as the product of the number of blades and the rotational speed of the shaft.
02

Calculate rotor blade passing frequency

Given, the shaft speed is 5000 rpm. Convert this to revolutions per second (rps) by dividing by 60 seconds in a minute. This gives \( \frac{5000}{60} \) rps which is approximately 83.33 rps. The number of rotor blades is 37. Hence the rotor blade passing frequency viewed by the stator blades is \( 37 \times 83.33 \) Hz, which is approximately 3083.33 Hz.
03

Calculate stator blade passing frequency

Similarly, using the shaft speed of 83.33 rps, and the number of stator blades being 41, the stator blade passing frequency viewed by rotor blades can be calculated as \( 41 \times 83.33 \) Hz, which is approximately 3416.67 Hz.

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Key Concepts

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

Compressor Stage
The heart of many mechanical systems is a compressor stage, a critical component often found in engines, particularly in the aeronautics industry. The compressor stage's role is to increase the pressure of the air flowing through it, preparing the air for combustion in jet engines, or other applications that require compressed air.

In encompassing the concept, imagine the compressor stage as a segment in a larger compressor assembly. It typically includes a set of rotor blades attached to a rotating shaft and a set of stator blades fixed in place. These blades work in tandem to accelerate and decelerate the air, thus increasing its pressure through the dynamic action of the rotor and the static action of the stator.

The frequency at which these blades pass a given point, known as the blade passing frequency, is essential in diagnosing and monitoring the health of the compressor. An understanding of this frequency helps in avoiding resonance conditions and potential mechanical failures that could stem from operational stresses.
Rotor Blades
Rotor blades are the moving components in a compressor stage. They are attached to the central shaft and spin at a considerable speed鈥攄etermined by the shaft's rotational speed. These blades have a twofold purpose: they must capture the incoming air and impart kinetic energy to it, thrusting the air radially outward.

The design of rotor blades is complex, taking into account aerodynamic efficiency, material strength, and resistance to wear and tear. They interact with the air more directly than stator blades, thus their shape and build are crucial for efficient compression.

Each time a rotor blade passes a stationary point, such as a stator blade, it contributes to the blade passing frequency. This interaction is consistent and can be calculated by multiplying the number of rotor blades by the rotational speed of the compressor's shaft鈥攁 critical part of the health monitoring process in mechanical systems.
Stator Blades
In contrast to rotor blades, the stator blades in a compressor stage are stationary. They do not move with the central shaft but instead serve a key role in managing the flow of air being compressed by the rotor blades. As the air emerges from the rotor with high kinetic energy and potentially turbulent flow characteristics, the stator blades help to convert this kinetic energy into pressure.

They are carefully shaped and positioned to ensure minimal air resistance while directing the flow into the next set of rotor blades seamlessly. This strategic arrangement maintains constant pressure and controlled airflow through the entire compressor. The design considerations for stator blades, thus, revolve around their ability to handle the dynamic pressure exerted on them and to channel the air effectively for consecutive compression stages.

Despite their static nature, the passing of rotor blades by each stator blade contributes to the determination of blade passing frequency, which is as significant for stator blades as it is for rotor blades in predicting performance and potential mechanical issues.
Rotational Speed
Rotational speed is the rate at which the compressor's shaft鈥攁nd consequently the rotor blades attached to it鈥攔otates. It is typically given in revolutions per minute (rpm) and is a key factor in calculating the blade passing frequency. Higher rotational speeds mean that rotor blades will pass a stationary point more frequently within a given time frame.

The rotational speed is a prime indicator of the operational status of a compressor. It affects not only the efficiency of compression but also the stress exerted on both rotor and stator blades due to the increased frequency of blade interactions. It's crucial to understand how rotational speed contributes to the blade passing frequencies for both rotor and stator blades.

Engineers and maintenance technicians closely monitor the rotational speed to ensure that the compressor operates within safe and efficient parameters. Anomalies in rotational speed could signal potential mechanical issues or a need for adjustment in the compression process.

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Most popular questions from this chapter

An axial-flow compressor stage has a pitchline radius of \(r_{\mathrm{m}}=0.6 \mathrm{~m}\). The rotational speed of the rotor at pitchline is \(U_{\mathrm{m}}=256 \mathrm{~m} / \mathrm{s}\). The absolute inlet flow to the rotor is described by \(C_{z \mathrm{~m}}=155 \mathrm{~m} / \mathrm{s}\) and \(C_{\theta 1 \mathrm{~m}}=28 \mathrm{~m} / \mathrm{s}\). Assuming that the stage degree of reaction at pitchline is \({ }^{\circ} R_{\mathrm{w}}=0.50, \alpha_{3}=\alpha_{1}\), and \(C_{z \mathrm{~m}}\) remains constant, calculate (a) rotor angular speed \(\omega\) in rpm (b) rotor exit swirl \(\mathrm{C}_{\theta 2 \mathrm{~m}}\) (c) rotor specific work at pitchline, \(w_{c m}\) (d) relative velocity vector at the rotor exit (e) rotor and stator torques per unit mass flow rate (f) stage loading parameter at pitchline, \(\psi_{\mathrm{m}}\) (g) flow coefficient \(\varphi_{\mathrm{m}}\)

A compressor stage at its pitchline radius \(\left(r_{\mathrm{m}}=0.25\right)\) has \({ }^{\circ} R_{\mathrm{m}}=0.73\). The axial velocity is constant at \(C_{2 m}=150\) \(\mathrm{m} / \mathrm{s}\). The flow entering the compressor is swirl free, i.e., \(\alpha_{1}=0\). The flow exiting the stage at pitchline is also swirl free, i.e., \(\alpha_{3}=0 .\) The compressor angular speed is \(\omega=7500 \mathrm{rpm}\). The speed of sound at the entrance to the stage is \(a_{1}=330 \mathrm{~m} / \mathrm{s}\) and rotor solidity at the pitchline radius is \(\sigma_{\mathrm{m}}=0.80\). Assuming the gas constant is \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and \(\gamma=1.4\), calculate: (a) absolute swirl downstream of the rotor, \(C_{\theta 2 \mathrm{~m}}\), in \(\mathrm{m} / \mathrm{s}\) (b) rotor specific work at pitchline radius, \(w_{c}\), in \(\mathrm{kJ} / \mathrm{kg}\) (c) total temperature downstream of the rotor, \(T_{12 m}\), in K (d) de Haller criterion for the rotor at pitchline (e) the rotor diffusion factor at the pitchline radius, \(D_{\mathrm{m}}\)

An axial-flow compressor stage is designed on the principle of constant through flow speed. The flow at the entrance to the rotor has \(100 \mathrm{~m} / \mathrm{s}\) of positive swirl and 180 \(\mathrm{m} / \mathrm{s}\) of axial velocity. Assuming we are at the pitchline radius \(r_{\mathrm{m}}=0.5 \mathrm{~m}\), where the rotor rotational speed is \(U_{\mathrm{m}}=230 \mathrm{~m} / \mathrm{s}\), the degree of reaction \({ }^{\circ} R_{\mathrm{m}}=0.5\), the radial shift in the streamtube is negligible, i.e., \(r_{1 \mathrm{~m}} \approx r_{2 \mathrm{~m}} \approx r_{3 \mathrm{~m}}\), and also assuming a repeated stage design principle is implemented, calculate (a) \(\alpha_{1 \mathrm{~m}}\) and \(\beta_{1 \mathrm{~m}}\) (b) \(\alpha_{2 m}\) and \(\beta_{2 m}\) (c) rotor specific work at the pitchline \(w_{\mathrm{cm}}\) in \(\mathrm{kJ} / \mathrm{kg}\) (d) stator torque at the pitchline per unit mass flow rate, \(\tau_{\mathrm{s}} / m\)

A compressor stage at the pitchline, \(r_{\mathrm{m}}=0.5 \mathrm{~m}\), is shown. The inlet flow to the rotor has a preswirl with \(\alpha_{1}=\) \(22^{\circ} .\) The axial velocity is \(C_{21}=C_{22}=C_{23}=170 \mathrm{~m} / \mathrm{s}\), i.e. constant throughout the stage. The stage is of repeated design, with \(\alpha_{3}=\alpha_{1} .\) Rotor and stator solidities at pitchline are 1.2 and \(1.0\) respectively. The rotor inlet relative velocity at pitchline is sonic, i.e., \(W_{1}=a_{1}\) and the rotor relative exit velocity, following de Haller criterion, is \(W_{2}=0.75 W_{1}\). Calculate: (a) shaft rotational speed, \(\omega\), in rpm (b) rotor exit (absolute) swirl, \(C_{\theta 2}\) in \(\mathrm{m} / \mathrm{s}\) (c) stage degree of reaction, \({ }^{\circ} R_{\mathrm{w}}\) (d) rotor D-factor, \(D_{\mathrm{r}}\) (e) stage total temperature ratio (f) stage total pressure ratio for \(e_{c}=0.9\)

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
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