Energy Output of a Turbine in a Pipeline System
Problem Statement
For the turbine shown in the figure, the inlet pressure is P₁ = 200 kPa and the outlet pressure is P₂ = -35 kPa, with a discharge Q = 0.3 m³/s. If the turbine efficiency is 80%, determine the energy output of the machine.
Given Data
| Diameter at Section 1 (d₁) | 400 mm = 0.4 m |
| Area at Section 1 (A₁) | A₁ = (π/4) × (0.4)² ≈ 0.1256 m² |
| Diameter at Section 2 (d₂) | 800 mm = 0.8 m |
| Area at Section 2 (A₂) | A₂ = (π/4) × (0.8)² ≈ 0.5026 m² |
| Discharge (Q) | 0.3 m³/s |
| Velocity at Section 1 (V₁) | V₁ = Q / A₁ ≈ 0.3 / 0.1256 ≈ 2.38 m/s |
| Velocity at Section 2 (V₂) | V₂ = Q / A₂ ≈ 0.3 / 0.5026 ≈ 0.6 m/s |
| Pressure at Section 1 (P₁) | 200 kPa = 200000 N/m² |
| Pressure at Section 2 (P₂) | -35 kPa = -35000 N/m² |
| Datum Head at Section 1 (Z₁) | 1.5 m |
| Datum Head at Section 2 (Z₂) | 0 m |
| Turbine Efficiency (η) | 0.8 (80%) |
| Acceleration due to Gravity (g) | 9.81 m/s² |
| Specific Weight of Water (γ) | 9810 N/m³ |
1. Calculating Velocities
The velocities in the two sections are computed as follows:
V₁ = Q / A₁ = 0.3 / 0.1256 ≈ 2.38 m/s
V₂ = Q / A₂ = 0.3 / 0.5026 ≈ 0.6 m/s
2. Applying Bernoulli’s Equation
With the datum taken at section 2 (Z₂ = 0 m) and Z₁ = 1.5 m, Bernoulli’s equation (including the head extracted by the turbine, hₜ) becomes:
P₁/γ + V₁²/(2g) + Z₁ – hₜ = P₂/γ + V₂²/(2g) + Z₂
Substituting the values:
200000/9810 + (2.38²)/(2×9.81) + 1.5 – hₜ = (-35000)/9810 + (0.6²)/(2×9.81) + 0
Solving for the turbine head, hₜ, gives:
hₜ ≈ 25.72 m
3. Calculating Energy Output
The energy output of the turbine is calculated by:
Energy Output = η × (γ × Q × hₜ)
Substituting the known values:
Energy Output = 0.8 × (9810 N/m³ × 0.3 m³/s × 25.72 m) ≈ 60555 W
Physical Interpretation
This turbine problem illustrates how the energy available from a fluid is extracted using pressure and velocity differences:
Pressure and Velocity Changes:
The high inlet pressure (200 kPa) combined with the low outlet pressure (-35 kPa) creates a significant driving head for the turbine.
Turbine Head (hₜ):
The calculated head of about 25.72 m represents the energy potential available from the pressure and velocity differences.
Efficiency:
The actual energy output is reduced by the turbine’s efficiency (80%), which accounts for real-world energy conversion losses.
Detailed Explanation for Students
Step 1: Velocity Calculations
Calculate the cross-sectional areas of the inlet and outlet pipes. Use the discharge (Q) to determine the velocities (V₁ and V₂) in each section.
Step 2: Bernoulli’s Equation
Apply Bernoulli’s equation between the two sections, including the head extracted by the turbine (hₜ). The datum is taken at the turbine outlet (Z₂ = 0 m), and the elevation difference (Z₁ = 1.5 m) is considered.
Step 3: Energy Calculation
The turbine’s energy output is computed as the product of the specific weight, flow rate, and turbine head, adjusted by the efficiency factor. This quantifies the net power produced by the turbine.
