91Ó°ÊÓ

An axial-flow compressor rotor at the pitchline has a radius of \(r_{\mathrm{m}}=0.5 \mathrm{~m}\). The shaft rotational speed is \(\omega=6000\) rpm. The inlet flow to the rotor has zero preswirl and the axial velocity is constant, with \(C_{2}=150 \mathrm{~m} / \mathrm{s}\). The stage has a \(80 \%\) degree-of-reaction at the pitchline where the solidity is \(\sigma_{\mathrm{m}}=\) 1.2. The stage adiabatic efficiency is equal to the polytropic efficiency, \(e_{\mathrm{c}}=0.92\). Assuming that the inlet total temperature is \(T_{\mathrm{t} 1}=288 \mathrm{~K}\), \(\gamma=1.4\) and \(\mathrm{c}_{\mathrm{p}}=1.004 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), calculate: (a) rotor specific work at \(r=r_{\mathrm{m}}\) in \(\mathrm{kJ} / \mathrm{kg}\) (b) stage loading, \(\psi\), at \(r=r_{\text {w }}\) (c) flow coefficient, \(\phi\), at \(r=r_{\mathrm{m}}\) (d) rotor relative Mach number at the pitchline, \(M_{1 r, \ldots}\) (e) stage total pressure ratio at the pitchline (f) rotor diffusion factor at the pitchline (g) is the de Haller criterion satisfied?

Step by step solution

01

Compute the rotational speed in rad/s

First convert the rotational speed from rpm to rad/s using the formula \( \omega = \text{rpm} * \frac{2\pi}{60} \). This is used for further calculations. Now, \( \omega = 6000 * \frac{2\pi}{60} \) rad/s.

02

Compute the tangential velocity

Now that we have the rotational speed in rad/s, calculate the tangential (rotational) velocity, \(U\), using the formula \(U=\omega*r_{\mathrm{m}}\). Thus, \(U= \omega * r_{\mathrm{m}} \) m/s

03

Calculate Specific work

Next calculate the specific work, \(W_{\mathrm{m}}\), using the formula \(W_{\mathrm{m}}= U^2 / 1000\). This is because the specific work is given by \(U^2\) and is converted to kJ/kg by dividing by 1000. Thus, \(W_{\mathrm{m}}= U^2 / 1000\) kJ/kg.

04

Stageloading calculation

The stageloading, \(\psi\), can be calculated using the formula \(\psi = W_{\mathrm{m}} / U^2\). Thus, \( \psi = W_{\mathrm{m}} / U^2 \).

05

Calculate the flow coefficient

The flow coefficient, \(\phi\), is computed from the formula \(\phi = C_2/U\). Thus, \(\phi = C_2 / U \).

06

Find rotor relative Mach number

The relative Mach number at the pitchline, \(M_{1 r, \ldots}\), is determined from \(M_{1 r, \ldots} = C_2 / a_{1,\ldots}\), where \(a_{1,\ldots}\) is computed from \(a_{1,\ldots} = \sqrt{\gamma R T_{1,\ldots}}\). Thus we first need to find \(T_{1,\ldots} = T_{\mathrm{t} 1} - C_2^2 / 2 C_p\), where \(R\) is the specific Gas constant given by \(R = C_p - C_v\) and \(C_v = R / (\gamma - 1) \).

07

Compute stage total pressure ratio

The stage total pressure ratio at the pitchline is given by \(\pi_{\mathrm{t}} = \left(1 + \frac{\gamma - 1}{2} M_{1 r, \ldots}^2\right)^{\gamma/(\gamma-1)}\). Thus, to find \(\pi_{\mathrm{t}}\), we firstly _(sub-step)_ find the value for \(M_{1 r, \ldots}\) from Step 6.

08

Calculate rotor diffusion factor

The rotor diffusion factor at the pitchline, \(D_f\), is given by the formula \(D_f = 1 - \sigma_m(1-\phi^2)\). Calculation gives \(D_f = 1 - \sigma_m(1-\phi^2)\).

09

Check De Haller criterion

Finally, check the de Haller criterion. If the diffusion factor, \(D_f\), is greater than 0.72, the de Haller criterion is satisfied. Thus, \(D_f > 0.72\).

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Key Concepts

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

Rotor Specific Work

In an axial-flow compressor, understanding the rotor specific work is crucial to comprehending the energy dynamics. Rotor specific work refers to the energy added to the airflow per unit mass by the rotor. The calculation of rotor specific work, denoted as \(W_{\mathrm{m}}\), begins with the tangential velocity \(U\), which is merely the product of angular velocity \(\omega\) and the rotor radius \(r_{\mathrm{m}}\). The formula for specific work is \(W_{\mathrm{m}} = \frac{U^2}{1000}\), with the output in kilojoules per kilogram (kJ/kg). This measurement is a direct indicator of the energy transfer efficiency from the rotor to the air, which affects the compressor's overall performance.

Stage Loading

Stage loading, symbolized by \(\psi\), encapsulates the idea of how much work is completed per stage for a given velocity of the flow. It is a dimensionless parameter obtained by the formula \(\psi = \frac{W_{\mathrm{m}}}{U^2}\). Calculation of stage loading helps in the design and analysis of the compressor's aerodynamic performance. It is indicative of the amount of energy added to the flow by the rotor blades, and it aids in assessing the aerodynamic force exerted on the compressor blades.

Flow Coefficient

The flow coefficient \(\phi\) is yet another dimensionless parameter that is crucial for the assessment of compressor performance. It is defined as the ratio of the axial velocity of the flow, \(C_2\), to the tangential velocity of the rotor, \(U\), and is calculated using the simple formula \(\phi = \frac{C_2}{U}\). This coefficient helps determine the axial flow velocity component relative to blade speed and is a critical design variable affecting the compressor's aerodynamic loading and efficiency.

Rotor Relative Mach Number

The rotor relative Mach number gives an insight into the compressibility effects of the airflow within the compressor. It refers to the Mach number of the air relative to the moving rotor blades at the pitchline. The relative Mach number, \(M_{1 r}\), is calculated by taking the axial velocity \(C_2\) and dividing it by the speed of sound in the relative frame, derived from the temperature and specific heat ratio (\(\gamma\)). This number is vital in predicting the occurrence of shock waves and their possible effects on compressor performance.

Stage Total Pressure Ratio

In axial-flow compressors, the stage total pressure ratio, notated as \(\pi_{\mathrm{t}}\), defines the increase in the total pressure as the air progresses through one compressor stage. It is directly tied to the work done by the compressor and is determined using the relationship \(\pi_{\mathrm{t}} = \left(1 + \frac{\gamma - 1}{2} M_{1 r}^2\right)^{\gamma/(\gamma-1)}\). The total pressure ratio is of paramount importance in understanding how effectively the compressor increases the pressure of the working fluid, a fundamental aspect of its operating efficiency.

Rotor Diffusion Factor

The rotor diffusion factor, often notated as \(D_f\), quantifies the deceleration of the airflow within the rotor blade passages. The formula for the diffusion factor is \(D_f = 1 - \sigma_m(1-\phi^2)\). A low diffusion factor could indicate potential flow separation and loss of efficiency within the rotor stage, which engineers strive to minimize for optimal compressor function.

De Haller Criterion

Lastly, the De Haller criterion offers a simple check to ensure the flow remains attached to the rotor blades. It is based on the diffusion factor \(D_f\), and the criterion dictates that if \(D_f\) is greater than 0.72, the flow is less likely to separate from the blade surfaces, satisfying the criterion. This threshold helps guide designers to ensure a stable flow within the compressor, which is central to maintaining efficiency and preventing flow instabilities that could lead to compressor stall or surge.