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Chapter 8: Problem 75

A Pelton wheel has a diameter of \(5 \mathrm{~m}\) and a bucket angle of \(165^{\circ} .\) The effective head at the nozzle is \(550 \mathrm{~m}\), and the nozzle is set such that the velocity coefficient is 0.94 and the diameter of the jet is \(130 \mathrm{~mm}\). Estimate the optimal flow rate, rotational speed of the wheel, and power output. Assume water at \(20^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Optimal flow rate: 1347 L/s, Rotational speed: 193.6 RPM, Power output: 6410 kW.

Step by step solution

01

Convert Jet Diameter to Radius

First, convert the diameter of the jet from millimeters to meters and obtain the radius of the jet. The diameter of the jet is given as 130 mm, which is equivalent to 0.13 m. Thus, the radius is \[ r = \frac{0.13}{2} = 0.065 \text{ m} \]
02

Calculate Jet Area

The area of the jet can be calculated using the formula for the area of a circle: \[ A = \pi r^2 = \pi (0.065)^2 \approx 0.0133 \text{ m}^2 \]
03

Determine the Velocity of the Jet

Using the effective head and the velocity coefficient, calculate the velocity of the jet with the formula \[ v = C_v \sqrt{2gh} \]where \( C_v = 0.94 \), \( g = 9.81 \thinspace \text{m/s}^2 \), and \( h = 550 \thinspace \text{m} \). Therefore, \[ v = 0.94 \times \sqrt{2 \times 9.81 \times 550} \approx 101.3 \text{ m/s} \]
04

Calculate the Optimal Flow Rate

Flow rate \( Q \) can be calculated by multiplying the velocity of the jet by the cross-sectional area:\[ Q = A \cdot v = 0.0133 \times 101.3 = 1.347 \text{ m}^3/\text{s} \approx 1347 \text{ liters/second} \]
05

Determine Rotational Speed

Using the relationship between the jet velocity \( v \), the wheel diameter \( D \), and the speed ratio \( u/v \approx 0.5 \) for maximum efficiency (common assumption for Pelton wheels), determine the rotational speed.\[ u = 0.5 \cdot v = 0.5 \cdot 101.3 = 50.65 \text{ m/s} \]Now, find the rotational speed \( N \) by solving \( u = \pi D N/60 \):\[ N = \frac{60 \cdot u}{\pi \cdot D} = \frac{60 \cdot 50.65}{\pi \cdot 5} \approx 193.6 \text{ revolutions/minute (RPM)} \]
06

Calculate Power Output

Finally, calculate the power output using the formula \[ P = \rho Q g h \eta \]where \( \rho = 1000 \text{ kg/m}^3 \) is the density of water, \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity, \( h = 550 \text{ m} \), and \( \eta = C_v^2 \) is the efficiency (assuming that the efficiency is approximately the square of the velocity coefficient):\[ P = 1000 \times 1.347 \times 9.81 \times 550 \times 0.94^2 \approx 6.41 \times 10^6 \text{ Watts} = 6410 \text{ kW} \]

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Key Concepts

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

Flow Rate Calculation
The flow rate in a Pelton wheel is essential to determine as it affects the turbine's efficiency and performance. To calculate the flow rate (Q), we need to understand the relationship between the velocity of the water jet and the cross-sectional area of the jet. The flow rate is given by the expression:
  • Jet Area (\(A\)): Calculated using the formula for the area of a circle \(A = \pi r^2\), where \(r\) is the radius of the jet.
  • Jet Velocity (\(v\)): Obtained by the equation \(v = C_v \sqrt{2gh}\), utilizing the effective head and velocity coefficient.
Combining these, the flow rate is: \(Q = A \cdot v\). In this particular problem, the values are used to compute \(Q \approx 1.347 \text{ m}^3/\text{s}\)or \(1347 \text{ liters/second}\). This calculation considers the efficient conversion of potential energy from the water head into kinetic energy in the form of a jet.
Jet Velocity
The velocity of the jet is pivotal to the analysis of the Pelton wheel, as it directly influences the momentum transfer to the wheel's buckets. The effective head, or height from which the water falls, provides the potential energy converted into kinetic energy, determining the jet's velocity. This velocity is calculated by:
  • Effective Head (\(h\)): The vertical distance that the water falls, given as \(550\) meters in this scenario.
  • Velocity Coefficient (\(C_v\)): This reflects the efficiency of the nozzle, valued at \(0.94\).
  • Gravity (\(g\)): Standardized at \(9.81\) m/s虏.
The equation used here is \(v = C_v \sqrt{2gh}\), resulting in \(v \approx 101.3\) m/s. Understanding jet velocity is crucial for calculating the force exerted on the turbine.
Rotational Speed
The rotational speed of the Pelton wheel indicates how fast the wheel is turning. It depends heavily on the velocity of the water jet and the wheel diameter. In scenarios like this, the optimal speed is typically achieved when the speed ratio (\(u/v\)) is about \(0.5\), meaning the wheel's velocity (\(u\)) is half the water jet velocity.
  • Wheel Diameter (\(D\)): Provided as \(5\) meters in this exercise.
  • Jet Velocity (\(v\)): Previously calculated as \(101.3\) m/s.
  • Optimal Speed Ratio (\(u/v\)): A typical assumption for maximum efficiency.
With \(u = 0.5 \cdot v = 50.65\) m/s, the wheel's rotational speed \(N\)is \(\frac{60 \cdot u}{\pi \cdot D}\), yielding approximately \(193.6\) RPM. Fast rotation ensures that the kinetic energy transferred from water is effectively used.
Power Output
The power output of a Pelton wheel turbine is what ultimately measures its effectiveness at converting the energy of falling water into mechanical energy. It is determined by several factors, such as flow rate, head, and efficiency.
  • Flow Rate (\(Q\)): Already found to be \(1.347\) m鲁/s.
  • Effective Head (\(h\)): The given \(550\) meters.
  • Density of Water (\(\rho\)): Approximated to \(1000\) kg/m鲁 at \(20\) 掳C.
  • Gravity (\(g\)): \(9.81\) m/s虏 as standard.
  • Efficiency (\(\eta\)): Calculated from the square of the velocity coefficient \((C_v^2)\).
Using the formula \(P = \rho Q g h \eta\), the calculated power output comes to around \(6410\) kW. This high power output is symbolic of an efficacious energy transformation process.
Hydraulic Efficiency
Hydraulic efficiency quantifies how well the Pelton wheel converts the hydraulic energy of the water into mechanical energy. It is an essential measure to ensure optimal performance and is closely related to the velocity coefficient.
  • Velocity Coefficient (\(C_v\)): Utilized here to derive efficiency, reflecting nozzle performance.
  • Efficiency Calculation: Hydraulic efficiency is assumed to be \(C_v^2\), an assumption made for simplicity and practical estimates.
For this specific situation, \(\eta = 0.94^2\), equating to approximately \(0.8836\) or \(88.36\)% efficiency. Such a figure shows the effectiveness of the conversion process and is a vital parameter for assessing the overall performance of the Pelton turbine system.

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Most popular questions from this chapter

A hydropower installation with reaction-type (Francis) turbines is to be located where the downstream water surface elevation is \(100 \mathrm{~m}\) below the water surface elevation in the upstream reservoir. The 2.0 -m-diameter concrete-lined penstock is \(500 \mathrm{~m}\) long and has an estimated roughness height of \(15 \mathrm{~mm}\). When the flow rate through the system is \(20 \mathrm{~m}^{3} / \mathrm{s}\), the combined head loss in the turbine and draft tube is \(5.0 \mathrm{~m}\), and the average velocity in the tailrace is \(0.80 \mathrm{~m} / \mathrm{s}\). Estimate the power that can be extracted from the system.

A Pelton wheel is to be designed to harness the available hydropower from a site where the effective head on the turbine will be \(160 \mathrm{~m}\) and the reliable flow rate through the turbine will be \(6 \mathrm{~m}^{3} / \mathrm{s}\). The wheel is to have a rotational speed of \(500 \mathrm{rpm}\), the bucket angle will be \(165^{\circ}\), and the nozzle is expected to have a velocity coefficient of 0.92 . (a) What diameter wheel would maximize the efficiency of the turbine? (b) What power can be expected from the turbine? (c) Assess whether a different type of turbine should be considered for this site. Assume water at \(20^{\circ} \mathrm{C}\).

A pump with a rotary speed of 1725 rpm delivers \(25 \mathrm{~L} / \mathrm{s}\) at its most efficient operating point. Under this condition, the inflow velocity is normal to the inflow surface of the impeller, the component of the velocity normal to the outflow surface of the impeller is \(4 \mathrm{~m} / \mathrm{s}\), and the efficiency of the pump is \(80 \%\). The width of the impeller at the outflow surface is \(15 \mathrm{~mm}\), and the blade angle at the outflow surface is \(50^{\circ}\). (a) Estimate the head added by the pump. (b) Use the affinity laws to estimate the head added and the flow rate delivered by the pump when the rotational speed is changed to \(1140 \mathrm{rpm}\).

At a Pelton wheel installation, the water surface elevation in the supply reservoir is \(85 \mathrm{~m}\) above the nozzle; the delivery line has a diameter of \(600 \mathrm{~mm}\), a length of \(300 \mathrm{~m}\), and a roughness height of \(8 \mathrm{~mm}\). The discharge nozzle has a diameter of \(50 \mathrm{~mm}\) and a head loss coefficient of \(0.8 .\) The bucket friction loss coefficient is \(0.5,\) the velocity of water relative to the bucket at the exit from the bucket is \(2 \mathrm{~m} / \mathrm{s},\) and the absolute velocity of the water leaving the bucket is \(6 \mathrm{~m} / \mathrm{s}\). Determine the power that could be derived from the system and the hydraulic efficiency of the turbine.

The performance of a turbine is being studied using a \(\frac{1}{5}\) -scale model. The prototype (full-scale) turbine operates at a design head of \(35 \mathrm{~m}\) when the flow rate through the turbine is \(64.1 \mathrm{~m}^{3} / \mathrm{s}\) and the angular speed of the runner is \(600 \mathrm{rpm}\). The model is to be tested at a head of \(12 \mathrm{~m}\). (a) What should be the angular speed and flow rate in the model to achieve similarity with the prototype? (b) If the shaft power generated in the model is measured as \(110 \mathrm{~kW}\) and the efficiency in the prototype is assumed to be \(5 \%\) better than the efficiency in the model, estimate the power that is generated in the prototype under design conditions. (c) What is the specific speed of the turbine, and what should be its type? Assume water at \(20^{\circ} \mathrm{C}\).
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