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Chapter 8: Problem 31

The flow coefficient to a rotor at pitchline is \(\varphi_{\mathrm{m}}=\) \(0.8\), its loading coefficient is \(C_{\mathrm{m}}=1.0\). The inlet flow to the rotor has zero swirl in the absolute frame of reference. Assuming axial velocity \(C_{z \mathrm{~m}}=\) constant across the rotor, calculate (a) the relative inlet flow angle \(\beta_{1 \mathrm{~m}}\) (b) the relative exit flow angle \(\beta_{2 m}\) (c) the degree of reaction \({ }^{\circ} R_{\mathrm{w}}\)

Short Answer

Expert verified
Once you have evaluated the arctangents appropriately in steps 1 and 2, use these to find the degree of reaction in step 3. The units for all the angles should be in degrees.

Step by step solution

01

Calculation of Relative Inlet Flow Angle 尾1

The relative inlet flow angle \(\beta_{1m}\) can be evaluated from the equation \(tan(\beta_{1m}) = \frac{\varphi_m}{C_m}\). Here, \(\varphi_m\) is the flow coefficient and \(C_m\) is the loading coefficient. From the given information, \(\varphi_m = 0.8\) and \(C_m = 1.0\). So, \(\beta_{1m} = atan(\frac{0.8}{1.0})\).
02

Calculation of Relative Exit Flow Angle 尾2

The relative exit flow angle \(\beta_{2m}\) can be obtained from the expression \(tan(\beta_{2m}) = - \frac{\varphi_m}{C_m}\). Given that \(\varphi_m = 0.8\) and \(C_m = 1.0\), then \(\beta_{2m} = atan(-\frac{0.8}{1.0})\).
03

Calculation of Degree of Reaction R

The degree of reaction, \(R_w\), in degrees can be calculated using the equation \({ }^{\circ} R_{\mathrm{w}} = 50(\beta_{2m} + \beta_{1m})\). Utilize the calculated values of \(\beta_{1m}\) and \(\beta_{2m}\) in this equation to find \({ }^{\circ} R_{\mathrm{w}}\).

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Key Concepts

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

Flow Coefficient
The flow coefficient, often represented by the symbol \( \varphi \), is a crucial parameter in analyzing fluid flow, especially in axial-flow turbines. It quantifies the relationship between the axial velocity of the fluid and the tangential velocity of the rotor.
  • This ratio provides insights into the efficiency of the energy conversion process within the turbine.
  • An understanding of the flow coefficient helps engineers design turbines that maintain optimal performance in varying operational conditions.
In the given exercise, the flow coefficient is denoted as \( \varphi_m = 0.8 \), which indicates a balanced flow where the axial and circumferential components of the velocity play an essential role in the rotor's performance.
Loading Coefficient
The loading coefficient, represented by \( C_m \), is another vital parameter in turbine analysis. It explains the amount of energy transferred from the rotor to the fluid per unit of angular momentum.
  • A higher loading coefficient suggests that the rotor imparts more energy into the fluid, which usually corresponds to higher efficiency.
  • This coefficient allows designers to predict the power output of the turbine blades based on specified operational conditions.
In the exercise, the loading coefficient is given as \( C_m = 1.0 \), indicating a standard operating condition where the energy transfer is directly proportional to the rotational dynamics of the rotor.
Inlet and Exit Flow Angles
In axial-flow turbines, the relative inlet and exit flow angles define how the fluid interacts with the moving blades. These angles are depicted by \( \beta_{1} \) and \( \beta_{2} \) respectively.
  • Determining these angles is essential for optimizing blade design to ensure the turbine operates efficiently and reduces losses.
  • The angles assist in understanding the direction of flow, which is crucial for mechanical stability and efficiency.
The exercise calculates these angles using trigonometric relations:
  • The inlet flow angle is calculated using \( \tan(\beta_{1m}) = \frac{\varphi_m}{C_m} \), leading to \( \beta_{1m} = \tan^{-1}(0.8) \).
  • Similarly, the exit flow angle is derived by \( \tan(\beta_{2m}) = -\frac{\varphi_m}{C_m} \), resulting in \( \beta_{2m} = \tan^{-1}(-0.8) \).
These results enable a precise understanding of how flow angles influence the rotor鈥檚 performance.
Degree of Reaction
In turbomachinery, the degree of reaction (often denoted by \( R \)) signifies the proportion of total pressure energy converted into kinetic energy across the rotor. This concept is important for characterizing the type of stage (reaction or impulse) in a turbine.
  • A degree of reaction closer to 1 or 100% indicates a high reaction turbine where most energy conversion occurs in the rotor.
  • Conversely, a value near 0 signifies an impulse turbine, where energy conversion primarily happens in the nozzle.
In the exercise, the degree of reaction \( R_w \) is computed using the formula \[^{\circ} R_{\mathrm{w}} = 50(\beta_{2m} + \beta_{1m})\]This expression helps underline how
  • relative flow angles directly correlate with energy distribution between the rotor and the surrounding fluid,
  • enabling engineers to adjust designs accordingly for desired efficiencies.
Understanding the degree of reaction is important for balancing energy conversion processes within axial-flow turbines.

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Most popular questions from this chapter

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 \(T_{1}=20^{\circ} \mathrm{C}\).

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

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 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?

An axial-flow compressor stage at its picthline radius \(\left(r_{\mathrm{m}}=0.40 \mathrm{~m}\right)\) has zero preswirl, an axial velocity of \(175 \mathrm{~m} / \mathrm{s}\), a degree of reaction of \({ }^{\circ} R_{\mathrm{m}}=0.65\), with the shaft angular speed of \(6450 \mathrm{rpm}\). Assuming constant throughflow speed, i.e., \(C_{z}=\) const., and a repeated-stage design at the pitchline, calculate: (a) relative Mach number at rotor inlet, \(M_{\mathrm{ir}}\) (b) rotor exit relative velocity, \(W_{2}(\mathrm{~m} / \mathrm{s})\) (c) static temperature downstream of the rotor, \(\mathrm{T}_{2}\) inK (d) absolute Mach number downstream of the rotor, \(M_{2}\) (e) absolute Mach number downstream of the stator, \(M_{3}\) (f) stage total temperature ratio, \(\tau_{\mathrm{s}}\) (g) stage total pressure ratio if \(\eta_{\mathrm{s}}=0.88\)
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