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Chapter 10: Problem 5

An axial-flow turbine nozzle turns the flow from an axial direction in the inlet to an exit flow angle of \(\alpha_{2}=70^{\circ}\). The rotor wheel speed is \(U=400 \mathrm{~m} / \mathrm{s}\) at the pitchline. The rotor is of impulse design and the exit flow from the rotor has zero swirl, i.e., \(\alpha_{3}=0\). Calculate (a) the rotor-specific work (b) the stage loading at the pitchline

Short Answer

Expert verified
To compute (a) the rotor-specific work and (b) the stage loading at the pitchline in the given turbine, it's vital to apply the Euler turbine equation to the conditions represented by the problem parameters. It implies taking into account the turbine's exit flow angle and rotor wheel speed, and assuming a zero swirl at output.

Step by step solution

01

Rotor Specific Work Calculation

The rotor-specific work \(W_{s}\) (assumed per unit mass) for an impulse turbine can be calculated using Euler Turbine Equation: \(W_{s} = U * C_{w2}\), where \(C_{w2} = U * tan(\alpha_{2})\). So, substituting the value of \(C_{w2}\) in the Euler Turbine Equation, we get \(W_{s} = U^2 * tan(\alpha_{2})\). Given that \(U = 400 \mathrm{~m} / \mathrm{s}\) and \(\alpha_{2} = 70^{\circ}\), substitute these values into the equation.
02

Stage Loading Calculation

The stage loading coefficient \(\psi\) is a non-dimensional parameter used to characterize stages in turbomachines. The stage loading coefficient is given by \(\psi = W_{s} / U^2\). We already have the value of \(W_{s}\) from step 1 and we know that \(U = 400 \mathrm{~m} / \mathrm{s}\). Substituting these values into the equation, we can calculate \(\psi\).

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Key Concepts

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

Axial-Flow Turbine
An axial-flow turbine is a type of turbine where the flow of the working fluid is in the direction parallel to the axis of rotation. This setup allows the turbine to handle large volumes of fluid efficiently.
The main components of an axial-flow turbine include:
  • Nozzle: Directs the flow of fluid onto the rotor at a specific angle.
  • Rotor: Converts the fluid's energy into mechanical work.
  • Stator: Steadies the flow, guiding it into the rotor efficiently.
Axial-flow turbines are popular in power generation and jet engines because they can smoothly handle high fluid velocities and pressures. Their design minimizes energy losses by ensuring that the fluid flow is less turbulent, making them ideal for processes requiring continuous and high energy conversion.
Rotor-Specific Work
Rotor-specific work is the amount of energy transferred from the fluid to the rotor in a turbine, per unit mass of fluid. It's crucial for understanding the efficiency and performance of a turbine stage. In impulse turbines, the change in fluid flow direction provides the necessary work to the rotor without a change in static pressure.
To calculate rotor-specific work:
  • Use the Euler Turbine Equation: \(W_{s} = U \cdot C_{w2}\).
  • For an impulse turbine, \(C_{w2} = U \cdot \tan(\alpha_{2})\), where \(\alpha_{2}\) is the nozzle exit flow angle.
  • Substituting yields \(W_{s} = U^2 \cdot \tan(\alpha_{2})\).
Rotor-specific work provides insight into potential energy conversion, helping design efficient turbine systems.
Stage Loading Coefficient
The stage loading coefficient (\(\psi\)) is a dimensionless value used to evaluate the performance of a turbine stage. It relates the rotor-specific work done by the turbine to the square of the rotor speed.
To determine the stage loading coefficient:
  • Use the formula: \(\psi = \frac{W_{s}}{U^2}\).
  • Here, \(W_{s}\) represents the rotor-specific work calculated previously, and \(U\) is the rotor wheel speed.
The stage loading coefficient allows engineers to compare different turbine designs and operational conditions. A higher value generally indicates a better performance, as it reflects more work being done by the rotor without a significant increase in rotor speed.
Euler Turbine Equation
The Euler Turbine Equation is a fundamental formula used in the design and analysis of turbines. It describes the relationship between the torque produced by a turbine and the change in angular momentum of the fluid. This relationship helps engineers determine the work potential of the turbine system.
For an impulse turbine, the equation is simplified to:
  • \(W_{s} = U \cdot C_{w2}\).
  • Since \(C_{w2} = U \cdot \tan(\alpha_{2})\), it becomes \(W_{s} = U^2 \cdot \tan(\alpha_{2})\).
The Euler Turbine Equation is crucial because it provides a direct link between the geometry of the turbine (like blade angles) and its mechanical effectiveness. By understanding this equation, one can predict turbine performance and efficiency accurately under different operating conditions.

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Most popular questions from this chapter

The leading edge on a turbine nozzle is to be internally cooled using an impingement cooling technique, as shown. The leading-edge diameter is \(8 \mathrm{~mm}\). Calculate the heat transfer coefficient \(h_{\mathrm{g}}\) at the leading edge, assuming an augmentation factor \(a=1.5\) due to a high-intensity turbulent flow in the turbine (use Equation \(10.81\) for a cylinder in cross- flow). $$ \begin{aligned} &M_{g}=0.5 \\ &\gamma_{g}=1.33 \\ &c_{p g}=1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ &k_{g}=0.082 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \\ &T_{\mathrm{g}}=1500 \mathrm{~K} \\ &p_{g}=1.0 \mathrm{MPa} \\ &\operatorname{Pr}_{g}=0.70 \\ &\mu_{g}=4.9 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s} \end{aligned} $$

The combustor discharge into a turbine nozzle has a total temperature of \(1850 \mathrm{~K}\) and inlet Mach number of \(0.50\), as shown. Assuming that the nozzle is uncooled, the axial velocity remains constant across the nozzle and the absolute flow angle at the nozzle exit is \(\alpha_{2}=65^{\circ}\), calculate (a) inlet velocity \(C_{1}\) in \(\mathrm{m} / \mathrm{s}\) (b) the exit absolute Mach number \(M_{2}\) and (c) nozzle torque per unit mass flow rate for \(r_{1} \approx r_{2}=\) \(0.40 \mathrm{~m}\)

Consider the flow of high-temperature, high Mach number gas over a flat wall, \(\left(M_{\mathrm{g}}=1.0, \gamma_{\mathrm{g}}=1.33, c_{p g}=\right.\) \(1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, T_{g}=1420 \mathrm{~K}, p_{g}=12 \mathrm{kPa}, \operatorname{Pr}_{g}=0.70\) ). We intend to internally cool the wall to achieve a wall temperature of \(T_{\mathrm{wg}}=1200 \mathrm{~K}\). Assuming the gas-side Stanton number is \(\mathrm{St}_{g}=0.005\), calculate (a) the gas stagnation temperature \(T_{\mathrm{tg}} \mathrm{K}\) (b) the adiabatic wall temperature \(T_{\mathrm{aw}} \mathrm{K}\) for a turbulent boundary layer (c) the gas-side film coefficient \(h_{\mathrm{g}} \mathrm{W} / \mathrm{m}^{2} \mathrm{~K}\) (d) the heat flux to the wall \(q_{\mathrm{w}} \mathrm{kW} / \mathrm{m}^{2}\) For a wall thickness of \(t_{w}=3 \mathrm{~mm}\), and a thermal conductivity, \(k_{\mathrm{w}}=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), calculate (e) the wall temperature on the coolant side \(T_{\mathrm{wc}} \mathrm{K}\)

A gas turbine is to provide a shaft power of \(50 \mathrm{MW}\) to a compressor with mass flow rate of \(100 \mathrm{~kg} / \mathrm{s}\). The compressor bleeds \(10 \mathrm{~kg} / \mathrm{s}\) at its exit for turbine internal cooling purposes. The combustor exit mass flow rate is \(93 \mathrm{~kg} / \mathrm{s}\), which accounts for \(3 \mathrm{~kg} / \mathrm{s}\) of fuel flow rate in the combustor. The turbine inlet temperature is \(T_{14}=1850 \mathrm{~K}\) and \(p_{14}=2.0 \mathrm{MPa}\) with \(\gamma_{\mathrm{t}}=1.30, c_{p \mathrm{t}}=1244 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) Assuming the coolant total temperature is \(T_{\mathrm{w}}=785 \mathrm{~K}\) and the gas properties are \(\gamma_{c}=1.40, c_{p c}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) the turbine exit total temperature \(T_{\mathrm{t} 5 \text {-cooled }}\) (b) the turbine exit total pressure \(p_{\text {s-cooled }}\). You may assume that the effect of cooling on turbine adiabatic efficiency is about \(\sim 2.8 \%\) loss (of efficiency) per \(1 \%\) cooling.

A turbine stage has an inlet total temperature \(T_{\mathrm{t} 1}\) of \(2000 \mathrm{~K}\). If the rotational speed of the rotor at the pitchline is \(200 \mathrm{~m} / \mathrm{s}\) and the axial velocity component is \(C_{2}=250 \mathrm{~m} / \mathrm{s}\), calculate (a) the stagnation temperature as seen by the rotor, \(T_{\mathrm{t} 2, \mathrm{r}}\) (b) the gas static temperature upstream of the rotor, \(T_{2}\) (c) the adiabatic wall temperature for the rotor, \(T_{\text {aw, }}\) assuming a turbulent boundary layer Assume \(\gamma=1.33, c_{p}=1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and \(\operatorname{Pr}=0.71\)
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