Answer:
a) Discharge for the turbine: 0.053 m³/s
b) Jet diameter: 5.53 meters
c) Nozzle tip diameter: 6.91 meters
d) Pitch circle diameter of the wheel: 11.48 meters
e) Specific speed: 26.76
f) Number of buckets on the wheel: 43
Step-by-step explanation:
To estimate the required values for the Koyna hydroelectric project, we'll use the given information and apply relevant formulas. Let's calculate each value step by step:
a) Discharge for the turbine:
The power output of the four units combined is given as 260 MW. Since the efficiency is 91%, we can calculate the actual power output:
Power output = Efficiency * Total power output
Power output = 0.91 * 260 MW
The power output is related to the discharge (Q) and gross head (H) by the following formula:
Power output = Q * H * ρ * g / 1000
Where:
ρ = Density of water = 1000 kg/m³
g = Acceleration due to gravity = 9.81 m/s²
From this equation, we can solve for Q:
Q = (Power output * 1000) / (H * ρ * g)
Substituting the given values:
Q = (260 MW * 1000) / (505 m * 1000 kg/m³ * 9.81 m/s²)
Q = 0.053 m³/s
Therefore, the discharge for the turbine is 0.053 m³/s.
b) Jet diameter:
The flow coefficient (φ) is given as 0.98. The jet diameter (D) can be calculated using the following formula:
φ = (π * D² * Q * √(2 * g * H)) / (4 * A * √(2 * g * H))
Where:
A = Number of jets
Rearranging the formula, we get:
D² = (4 * A * φ * A * √(2 * g * H)) / (π * Q)
Substituting the given values:
D² = (4 * 4 * 0.98 * 4 * √(2 * 9.81 m/s² * 505 m)) / (π * 0.053 m³/s)
D² ≈ 30.66
Taking the square root:
D ≈ √30.66
D ≈ 5.53 m
Therefore, the jet diameter is approximately 5.53 meters.
c) Nozzle tip diameter:
The nozzle tip diameter (d) is given to be 25% larger than the jet diameter (D):
d = D + 0.25 * D
d = 5.53 m + 0.25 * 5.53 m
d ≈ 6.91 m
Therefore, the nozzle tip diameter is approximately 6.91 meters.
d) Pitch circle diameter of the wheel:
The speed ratio (λ) is given as 0.48. The pitch circle diameter (D_p) is related to the jet diameter (D) by the following formula:
D_p = D / λ
Substituting the given values:
D_p = 5.53 m / 0.48
D_p ≈ 11.48 m
Therefore, the pitch circle diameter of the wheel is approximately 11.48 meters.
e) Specific speed:
The specific speed (N_s) can be calculated using the formula:
N_s = (n * √Q) / (√H^(3/4))
Where:
n = Rotational speed of the turbine (rpm)
The rotational speed (n) can be calculated using the formula:
n = (120 * f) / p
Where:
f = Frequency (Hz)
p = Number of poles
Substituting the given values:
n = (120 * 50 Hz) / 10
n = 600 rpm
Substituting the calculated values into the specific speed formula:
N_s = (600 rpm * √0.053 m³/s) / (√505 m)^(3/4)
N_s ≈ 26.76
Therefore, the specific speed is approximately 26.76.
f) Number of buckets on the wheel:
The number of buckets (B) on the wheel is related to the specific speed (N_s) by the formula:
N_s = (n * B) / (√H^(5/4))
Solving for B:
B = (N_s * √H^(5/4)) / n
Substituting the given values:
B = (26.76 * √505 m^(5/4)) / 600 rpm
B ≈ 43.09
Therefore, the number of buckets on the wheel is approximately 43.