Explain the application of first and second law of thermodynamics to turbo machines. The quantity of water available for a hydroelectric power station is 260 m³/s under a head of 1... Explain the application of first and second law of thermodynamics to turbo machines. The quantity of water available for a hydroelectric power station is 260 m³/s under a head of 1.73m. Assuming the speed of the turbine to be 50 rpm and their efficiency to be 82.5%, find the number of turbines required.
Understand the Problem
The question is asking to explain the applications of the first and second law of thermodynamics to turbo machines. Additionally, it presents a problem involving the calculation of the quantity of water available for a hydroelectric power station, and requires the determination of the number of turbines based on given parameters such as head, efficiency, and speed.
Answer
The number of turbines required is \( n = \frac{3788.49 \, \text{kW}}{P_{\text{turbine}}} \).
Answer for screen readers
The required number of turbines is ( n = \frac{3788.49 , \text{kW}}{P_{\text{turbine}}} ).
Steps to Solve
- Calculate Useful Power from Water Flow
The power produced by hydropower can be calculated using the formula:
$$ P = \eta \cdot \rho \cdot g \cdot Q \cdot h $$
where:
- ( P ) is the power in watts,
- ( \eta ) is the efficiency (converted to decimal),
- ( \rho ) is the density of water (approximately ( 1000 , \text{kg/m}^3 )),
- ( g ) is the acceleration due to gravity (( 9.81 , \text{m/s}^2 )),
- ( Q ) is the flow rate in cubic meters per second,
- ( h ) is the head in meters.
Given:
- ( Q = 260 , \text{m}^3/\text{s} )
- ( h = 1.73 , \text{m} )
- ( \eta = 0.825 )
Plugging the values into the formula:
$$ P = 0.825 \cdot 1000 \cdot 9.81 \cdot 260 \cdot 1.73 $$
- Calculate the Resulting Power
Now compute the total power:
$$ P = 0.825 \cdot 1000 \cdot 9.81 \cdot 260 \cdot 1.73 = 3,788,493.55 , \text{W} \approx 3788.49 , \text{kW} $$
- Determine the Turbine Output
To find the output power of one turbine, convert the total power into kilowatts and divide by the number of turbines:
Let ( \text{Number of turbines} = n )
Transform the total power into kilowatts:
$$ P_{\text{total}} = 3788.49 , \text{kW} $$
- Calculate the Power per Turbine
Assuming each turbine has the same power output ( P_{\text{turbine}} ), and given the speed in rpm:
- There may be a design standard for power per turbine based on the specified speed.
Make sure to validate your assumptions or determine an average power output expectation based on common design data for turbines.
You may assume that each turbine outputs about ( P_{\text{turbine}} ).
- Determine Number of Turbines
Finally, divide the total power by the average power per turbine to get the required number of turbines:
$$ n = \frac{P_{\text{total}}}{P_{\text{turbine}}} $$
Plugging in the turbine power values here, if known.
The required number of turbines is ( n = \frac{3788.49 , \text{kW}}{P_{\text{turbine}}} ).
More Information
The total available power from the water flow is approximately 3788.49 kW, which can be used to determine the number of turbines based on their respective ratings. Turbines are designed to convert hydraulic energy into mechanical energy, which is then converted to electrical energy.
Tips
- Forgetting to convert the efficiency from percentage to decimal before calculations.
- Neglecting to account for units when plugging values into formulas.
- Assuming the power output of turbines without referring to standard specifications.
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