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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.

Step by step solution

01

Calculate the Velocity of Water at the Nozzle

The velocity of water coming out from the nozzle can be determined using the relation between water head and water velocity:\[ v = \sqrt{2gh} \]where \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity, and \( h = 85 \, \text{m} \) is the water head.Thus,\[ v = \sqrt{2 \times 9.81 \times 85} = 40.78 \, \text{m/s} \]

02

Calculate the Energy Losses

Calculate the hydraulic friction loss using the formula:\[ h_f = K \cdot \left( \frac{v^2}{2g} \right) \]The head loss coefficient \( K = 0.8 \).Thus,\[ h_f = 0.8 \times \left( \frac{40.78^2}{2 \times 9.81} \right) = 64.35 \, \text{m} \]

03

Calculate Power Available at the Nozzle

The power at the nozzle can be calculated using the formula:\[ P = \rho \cdot g \cdot Q \cdot (h - h_f) \]where \( \rho = 1000 \, \text{kg/m}^3 \) is the density of water, \( Q \) is the flow rate, and \( h = 85 \, \text{m} \) is the head.For a pipe diameter of 0.6 m, area \( A = \pi D^2 / 4 \), flow rate \( Q = A \times v \).\[ Q = \frac{\pi \times 0.6^2}{4} \times 40.78 \approx 11.47 \text{ m}^3/\text{s} \]Then,\[ P = 1000 \cdot 9.81 \cdot 11.47 \cdot (85 - 64.35) \approx 2.31 \times 10^6 \text{ W} \]

04

Calculate Bucket Friction Loss

The power lost due to bucket friction can be calculated using \[ P_{friction} = \kappa \cdot \frac{\rho \cdot Q \cdot v}{2} \], where the loss coefficient \( \kappa = 0.5 \) and \( v = 2 \, \text{m/s} \).Thus,\[ P_{friction} = 0.5 \cdot \frac{1000 \cdot 11.47 \cdot 2}{2} = 5.735 \text{ kW} \]

05

Calculate Net Power Extracted by Bucket

Subtract the bucket friction loss from the nozzle power:\[ P_{net} = P - P_{friction} = 2.31 \times 10^6 \text{ W} - 5.735 \times 10^3 \text{ W} = 2.304265 \times 10^6 \text{ W} \]

06

Determine Hydraulic Efficiency

The hydraulic efficiency \( \eta \) of the system can be calculated using:\[ \eta = \frac{P_{net}}{P} \times 100 \%= \frac{2.304265 \times 10^6}{2.31 \times 10^6} \times 100 \%\approx 99.8\% \]

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Key Concepts

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

Nozzle Velocity Calculation

In Pelton Wheel Turbine systems, nozzle velocity determines how fast water exits the nozzle, affecting the system's power. The velocity is derived from the relationship between the gravitational potential energy and kinetic energy of water. Using the formula \( v = \sqrt{2gh} \), where \( g = 9.81 \text{ m/s}^2 \) (gravity's acceleration) and \( h = 85 \text{ m} \) (head), we find that \( v = \sqrt{2 \times 9.81 \times 85} \), resulting in a velocity of \( 40.78 \text{ m/s} \). This calculation shows us the transition of energy forms, from potential to kinetic, as water travels through the turbine system.

Hydraulic Efficiency

Hydraulic efficiency in a turbine system indicates how effectively the system converts water's potential energy into mechanical power. It's calculated by comparing the net power output after accounting for losses to the gross power input. Using the efficiency formula:

• \( \eta = \frac{P_{\text{net}}}{P} \times 100 \% \)

Where \( P_{\text{net}} = 2.304265 \times 10^6 \text{ W} \) and \( P = 2.31 \times 10^6 \text{ W} \), resulting in an efficiency of approximately 99.8%. This high efficiency means that most of the water's energy is effectively captured to do useful work, minimizing wastage.

Energy Losses in Turbines

Understanding energy losses is critical in optimizing turbine systems. These losses occur due to factors such as friction and turbulence.
In our scenario, the hydraulic friction loss was calculated using the formula:

• \( h_f = K \cdot \frac{v^2}{2g} \)

Given that the head loss coefficient \( K = 0.8 \), the calculated hydraulic friction loss \( h_f \) is \( 64.35 \text{ m} \). Such losses often stem from pipe roughness or bends that disrupt the water flow, which engineers always aim to reduce for better efficiency.

Bucket Friction Loss

As water hits the Pelton bucket, some energy is lost due to friction between the water and bucket surface. This bucket friction loss affects the net energy output of the turbine.
To calculate this loss, we use:

• \( P_{\text{friction}} = \kappa \cdot \frac{\rho \cdot Q \cdot v}{2} \)

Where \( \kappa = 0.5 \), \( \rho = 1000 \text{ kg/m}^3 \), \( Q = 11.47 \text{ m}^3/\text{s} \), and \( v = 2 \text{ m/s} \). The result is \( 5.735 \text{ kW} \). Reducing these losses can significantly improve the turbine's efficiency.

Power Calculation in Fluid Systems

Power calculation is crucial in assessing the performance of fluid systems like the Pelton Wheel Turbine. It helps determine how much mechanical work can be derived from the system. Power available at the nozzle is calculated with the formula:

• \( P = \rho \cdot g \cdot Q \cdot (h - h_f) \)

Here, \( \rho = 1000 \text{ kg/m}^3 \) (water density), \( g = 9.81 \text{ m/s}^2 \), and \( h - h_f = 85 - 64.35 \text{ m} \) give \( P = 2.31 \times 10^6 \text{ W} \). This calculation clarifies how fluid dynamics principles are applied to derive energy from water flow, essential knowledge for engineers involved in hydraulic projects.