91Ó°ÊÓ

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 \(\varphi_{\mathrm{m}}\)

Step by step solution

01

Calculate Rotor Angular Speed

Given rotational speed in m/s, it needs to be converted into RPM (revolutions per minute). This can be done using the conversion factor of \(60/(2\pi)\). So, \(\omega = U_{m}/r_{m} \cdot 60/(2\pi)\). Substitute the given values to calculate \(\omega\).

02

Calculate Rotor Exit Swirl

Rotor exit swirl \(C_{\theta 2 m}\) can be calculated using the formula \(C_{\theta 2 m} = U_{m} - C_{\theta 1 m}/R_{w}\). Substitute the given values to get the result.

03

Find Rotor Specific Work at Pitchline

To calculate the rotor specific work at pitchline, \(w_{c m}\), use the formula \(w_{c m} = U_{m}(C_{\theta 2 m} - C_{\theta 1 m})\). Fill in the values to calculate the answer.

04

Calculate Relative Velocity Vector at the Rotor Exit

Relative velocity vector at the rotor exit, \(W_{2 m}\), can be calculated using the Pythagorean theorem for velocities: \(W_{2 m} = sqrt(C_{z m}^2 + (U_{m} - C_{\theta 2 m})^2)\), where \(C_{z m}\) is the absolute inlet flow to the rotor, \(U_{m}\) is the rotational speed of the rotor, and \(C_{\theta 2 m}\) is the rotor exit swirl.

05

Find Rotor and Stator Torques per Unit Mass Flow Rate

Rotor and stator torques per mass flow rate, \(T = r_{m}(C_{\theta 2 m} - C_{\theta 1 m})/2\). Substitute the given values to find the rotor and stator torques.

06

Calculate Stage Loading Parameter at Pitchline

Stage loading parameter at pitchline, \(\psi_{m} = w_{c m}/U_{m}^2\). Substitute the values to get the result.

07

Calculate Flow Coefficient

The flow coefficient, \(\varphi_{m} = C_{z m}/U_{m}\). Substitute the provided values to find the flow coefficient.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Rotor Angular Speed

In an axial-flow compressor, understanding the rotor angular speed is crucial. This speed, denoted by \(\omega\), represents how fast the rotor spins around its axis. To convert the speed from meters per second (m/s) to revolutions per minute (RPM), you use a conversion factor:

• Start by determining the "linear" speed at the pitchline, \(U_{m}\), which is given as 256 m/s.
• The pitchline radius, \(r_{m}\), is 0.6 m.

The formula to find \(\omega\) is:\[\omega = \frac{U_{m}}{r_{m}} \times \frac{60}{2\pi}\]By substituting the given values, you get the rotor's angular speed in RPM. This conversion helps understand how fast the rotor is rotating in a more relatable unit of measurement.

Rotor Specific Work

The rotor specific work is a crucial element in the analysis of compressors. It denotes the work done by the rotor on the fluid per unit mass. In this context, it is symbolized by \(w_{c m}\).

• To find \(w_{c m}\), identify the rotor exit swirl \(C_{\theta 2 m}\) using the formula:\[C_{\theta 2 m} = U_{m} - \frac{C_{\theta 1 m}}{R_{w}}\]
• Next, compute \(w_{c m}\) with:\[w_{c m} = U_{m}(C_{\theta 2 m} - C_{\theta 1 m})\]

Substituting the known values will yield the specific work, which illustrates the energy transfer efficiency of the rotor.

Stage Loading Parameter

The stage loading parameter, \(\psi_{m}\), is pivotal for understanding compressor performance. It relates the rotor specific work to the square of the blade speed.

• Given by the formula:\[\psi_{m} = \frac{w_{c m}}{U_{m}^2}\]
• This parameter helps in assessing the energy transfer efficiency in terms of the rotor's capability to handle the fluid dynamics effectively.

The stage loading parameter is critical for optimizing design and operational efficiency of compressors.

Flow Coefficient

Understanding the flow coefficient, \(\varphi_{m}\), is essential for analyzing the mass flow rate relative to the rotor speed. It provides insights into how effectively the compressor can intake and process the fluid.

• Defined as:\[\varphi_{m} = \frac{C_{z m}}{U_{m}}\]
• Where \(C_{z m}\) is the inlet flow, and \(U_{m}\) is the rotor speed.

The flow coefficient is crucial for determining how well the compressor stage handles the incoming air, affecting both design and performance aspects of the machine.