Oil Flow Rate Calculation
Problem Statement
Oil with specific gravity 0.75 is flowing through a 15 cm diameter pipe under a pressure of 105 kN/m². If the total energy relative to a datum plane 2.5 m below the center of the pipe is 18 m, determine the flow rate of oil.
Given Data
- Specific gravity of oil (sg) = 0.75
- Pipe diameter (d) = 15 cm = 0.15 m
- Pressure (P) = 105 kN/m² = 105 kPa
- Datum head (z) = 2.5 m (below pipe center)
- Total energy (E) = 18 m
1. Calculate Cross-Sectional Area of Pipe
The cross-sectional area of the pipe is given by:
A = π × (d/2)² = π × (0.15/2)² = π × 0.075² = π × 0.005625
A = 0.01767 m²
2. Apply Bernoulli’s Equation
According to Bernoulli’s equation, the total energy at any point in a fluid flow is:
E = z + P/(ρg) + V²/(2g)
Where:
E = Total energy (head) in meters
z = Elevation head (datum head) in meters
P/(ρg) = Pressure head in meters
V²/(2g) = Velocity head in meters
For our problem:
E = 18 m
z = 2.5 m
P = 105 kPa
ρ = 0.75 × 1000 kg/m³ = 750 kg/m³ (since specific gravity is 0.75)
g = 9.81 m/s²
Rearranging the Bernoulli equation to find velocity (V):
V²/(2g) = E – z – P/(ρg)
V²/(2g) = 18 – 2.5 – 105000/(750×9.81)
V²/(2g) = 18 – 2.5 – 14.27
V²/(2g) = 1.23
V² = 1.23 × 2g = 1.23 × 2 × 9.81 = 24.13
V = √24.13 = 4.91 m/s
3. Calculate Flow Rate
The volumetric flow rate (Q) is the product of cross-sectional area and fluid velocity:
Q = A × V = 0.01767 m² × 4.91 m/s = 0.0867 m³/s
Verification
We can verify our solution by checking if the calculated velocity satisfies the Bernoulli equation:
E = z + P/(ρg) + V²/(2g)
E = 2.5 + 105000/(750×9.81) + 4.91²/(2×9.81)
E = 2.5 + 14.27 + 1.23
E = 18 m
This confirms our calculation is correct.
Additional Notes
- The flow rate can also be expressed as 86.7 liters per second (0.0867 × 1000)
- This calculation assumes steady, incompressible flow with negligible frictional losses
- The specific gravity of 0.75 means the oil is lighter than water, which has implications for energy calculations
- For engineering applications, this flow rate would need to be considered when selecting pumps and designing the piping system
