Oil Flow Analysis in a Pipe with Diameter Change
Problem Statement
A pipe line carrying oil of sp.gr. 0.8, changes in diameter from 300mm at a position A to 500mm at position B which is 5m at a higher level. If the pressures at A and B are 19.62 N/cm² and 14.91 N/cm² respectively, and the discharge is 150 lps, determine the loss of head and the direction of flow.
Given Data
| Diameter at position A (dₐ) | 300 mm = 0.3 m |
| Diameter at position B (dᵦ) | 500 mm = 0.5 m |
| Elevation difference (B above A) | 5 m |
| Pressure at position A (Pₐ) | 19.62 N/cm² = 196,200 N/m² |
| Pressure at position B (Pᵦ) | 14.91 N/cm² = 149,100 N/m² |
| Discharge (Q) | 150 lps = 0.15 m³/s |
| Specific gravity of oil | 0.8 |
| Density of oil (ρ) | 0.8 × 1000 = 800 kg/m³ |
| Acceleration due to gravity (g) | 9.81 m/s² |
1. Key Principles and Equations
We’ll use the Bernoulli equation to analyze this problem and determine both the head loss and the direction of flow.
P₁/ρg + V₁²/(2g) + Z₁ = P₂/ρg + V₂²/(2g) + Z₂ + hₗ
Where:
P = Pressure (N/m²)
ρ = Density of fluid (kg/m³)
g = Acceleration due to gravity (9.81 m/s²)
V = Velocity of fluid (m/s)
Z = Elevation from datum (m)
hₗ = Head loss (m)
Q = A₁ × V₁ = A₂ × V₂
Where:
Q = Discharge (m³/s)
A = Cross-sectional area (m²)
V = Velocity (m/s)
2. Calculating Cross-sectional Areas
Cross-sectional area at position A:
A₁ = π × (dₐ/2)² = π × (0.3/2)² = π × 0.0225 = 0.0707 m²
Cross-sectional area at position B:
A₂ = π × (dᵦ/2)² = π × (0.5/2)² = π × 0.0625 = 0.1963 m²
3. Calculating Velocities
Using the continuity equation Q = A × V:
Velocity at position A (V₁):
V₁ = Q/A₁ = 0.15/0.0707 = 2.12 m/s
Velocity at position B (V₂):
V₂ = Q/A₂ = 0.15/0.1963 = 0.764 m/s
4. Converting Pressure to Pressure Head
Converting pressure to equivalent head using P/ρg:
Pressure head at position A:
P₁/ρg = 196,200/(800×9.81) = 25.00 m
Pressure head at position B:
P₂/ρg = 149,100/(800×9.81) = 19.00 m
5. Setting Datum and Elevation Values
We’ll set our datum at position A:
Z₁ = 0 m (Position A is our datum)
Z₂ = 5 m (Position B is 5 m higher than A)
6. Calculating Velocity Head
The velocity head is calculated as V²/(2g):
Velocity head at position A:
V₁²/(2g) = (2.12)²/(2×9.81) = 0.229 m
Velocity head at position B:
V₂²/(2g) = (0.764)²/(2×9.81) = 0.0298 m
7. Calculating Total Energy Head at Each Position
Total energy head at position A (Eₐ):
Eₐ = P₁/ρg + V₁²/(2g) + Z₁
Eₐ = 25.00 + 0.229 + 0
Eₐ = 25.229 m
Total energy head at position B (Eᵦ):
Eᵦ = P₂/ρg + V₂²/(2g) + Z₂
Eᵦ = 19.00 + 0.0298 + 5
Eᵦ = 24.030 m
8. Determining Head Loss and Flow Direction
Based on the energy equation, head loss is the difference in total energy between the two positions. The flow direction is from the position with higher energy to the position with lower energy.
Head loss calculation:
Since Eₐ > Eᵦ, flow must be from position A to position B.
hₗ = Eₐ – Eᵦ = 25.229 – 24.030 = 1.199 m
Direction of Flow: From Position A to Position B
9. Visual Representation of the Flow System
10. Physical Interpretation
This oil flow problem reveals several important fluid dynamics principles:
- Energy Balance: The total energy at position A (25.229 m) is higher than at position B (24.030 m), which determines that the flow direction is from A to B.
- Pressure-Velocity Relationship: The pipe expands from position A to B, causing the velocity to decrease from 2.12 m/s to 0.764 m/s. According to Bernoulli’s principle, this velocity decrease should cause a pressure increase, but we observe a pressure decrease (from 19.62 N/cm² to 14.91 N/cm²).
- Effect of Elevation: The 5m elevation gain from A to B consumes energy and contributes to the pressure drop between the two positions.
- Energy Losses: The calculated head loss of 1.199 m represents energy dissipated due to factors like friction, turbulence, and flow separation, especially in the expanding section of the pipe.
- Reverse Analysis Confirmation: If we were to analyze the flow in the reverse direction (B to A), we would calculate a negative head loss, which is physically impossible and confirms our determination of flow direction from A to B.
11. Conclusion
We have successfully determined both the head loss and the direction of flow for the given pipe system:
- Head Loss: 1.199 m
- Direction of Flow: From Position A (smaller diameter, 300 mm) to Position B (larger diameter, 500 mm)
This analysis demonstrates the application of the energy equation to determine flow characteristics in a pipe with changing diameter and elevation. The pressure drop between positions A and B is influenced by three factors: the elevation increase (which requires energy), the pipe expansion (which decreases velocity and should recover some pressure), and frictional losses (which dissipate energy). The calculated head loss quantifies the energy dissipation in the system, which is crucial for practical engineering applications like pump sizing, system efficiency evaluation, and pressure regulation.
