The diameter of a pipe bend is 30cm at inlet and 15cm at outlet, and the flow is turned through 120° in a vertical plane. The axis at inlet is horizontal and the center of the outlet section is 1.5m below the center of the inlet section. Total volume of water in the bend is 0.9m³. Neglecting friction, calculate the magnitude and direction of the force exerted on the bend by water flowing through it at 250lps and when the inlet pressure is 0.15N/mm².

Fluid Mechanics Problem Solution

Problem Statement

Given Data

Diameter at inlet (d₁) 30 cm = 0.3 m

Diameter at outlet (d₂) 15 cm = 0.15 m

Flow turn angle 120°

Outlet below inlet 1.5 m

Volume of water in bend 0.9 m³

Flow rate 250 lps = 0.25 m³/s

Inlet pressure (P₁) 0.15 N/mm² = 0.15×10⁶ N/m²

Friction Neglected

Solution Approach

To determine the force exerted on the pipe bend by flowing water, we’ll apply momentum principles and Bernoulli’s equation. We’ll first calculate key parameters including areas, velocities, and the outlet pressure, then resolve forces in x and y directions.

Calculations

Basic Parameters Calculation

Step 1: Calculate the cross-sectional areas at inlet and outlet.

A₁ = π/4 × d₁² = π/4 × 0.3² = 0.07068 m²

A₂ = π/4 × d₂² = π/4 × 0.15² = 0.01767 m²

Step 2: Calculate velocities at inlet and outlet using the continuity equation (Q = AV).

V₁ = Q/A₁ = 0.25/0.07068 = 3.54 m/s

V₂ = Q/A₂ = 0.25/0.01767 = 14.15 m/s

Step 3: Calculate the weight of water within the control volume.

W = γ_water × Vol = 9810 × 0.9 = 8829 N

Step 4: Apply Bernoulli’s equation to determine the outlet pressure.

P₁/ρg + V₁²/2g + Z₁ = P₂/ρg + V₂²/2g + Z₂

With Z₁ = 1.5 m, Z₂ = 0 (taking outlet as reference), ρ = 1000 kg/m³, g = 9.81 m/s²:

(0.15×10⁶)/(1000×9.81) + 3.54²/(2×9.81) + 1.5 = P₂/(1000×9.81) + 14.15²/(2×9.81) + 0

P₂ = 70870 N/m²

Force Analysis

Step 5: Determine the angle for force resolution.

θ = 180° – 120° = 60°

Step 6: Calculate the force in the X direction (horizontal).

∑Forces in X direction = Rate of change of momentum in X direction

(P₁A₁ + P₂A₂Cosθ) – R_x = ρQ(V_2x – V_1x)

(P₁A₁ + P₂A₂Cosθ) – R_x = ρQ(-V₂Cosθ – V₁)

R_x = (P₁A₁ + P₂A₂Cosθ) + ρQ(V₁ + V₂Cosθ)

R_x = (0.15×10⁶×0.07068 + 70870×0.01767×Cos60°) + 1000×0.25(3.54 + 14.15×Cos60°)

R_x = 13882 N

Step 7: Calculate the force in the Y direction (vertical).

∑Forces in Y direction = Rate of change of momentum in Y direction

R_y – P₂Sinθ A₂ – W = ρQ(V_2y – V_1y)

R_y – P₂A₂Sinθ – W = ρQ(-V₂Sinθ – 0)

R_y = P₂A₂Sinθ – ρQV₂Sinθ + W

R_y = 70870×0.01767×Sin60° – 1000×0.25×14.15×Sin60° + 8829

R_y = 6850 N

Step 8: Calculate the resultant force and its direction.

R = √(R_x² + R_y²) = √(13882² + 6850²) = 15481 N

Direction θ_R = Tan^-1(R_y/R_x) = Tan^-1(6850/13882) = 26.3°

Resultant Force: 15481 N at 26.3° (to the right and downward)

Detailed Explanation

Principles Applied

This problem involves the application of two fundamental principles in fluid mechanics:

Conservation of Energy (Bernoulli’s Equation): Used to determine the pressure at the outlet based on the given inlet conditions, velocities, and elevation changes.
Conservation of Momentum: Applied to calculate the forces exerted by the flowing water on the pipe bend in both horizontal and vertical directions.

Significance of Each Force Component

The horizontal force component (R_x = 13882 N) is primarily due to:

Pressure forces at inlet and outlet
Change in momentum as water changes direction

The vertical force component (R_y = 6850 N) is influenced by:

Weight of water within the bend
Pressure force at the outlet in the vertical direction
Momentum change in the vertical direction

Engineering Implications

The resultant force of 15481 N (approximately 1.5 tons) is significant and has several practical implications for the design and installation of the pipe system:

Structural Support: The pipe bend must be adequately supported to withstand this force without deformation or failure.
Anchoring: Proper anchoring systems must be designed to counteract this force, especially considering its direction at 26.3° from horizontal.
Joint Design: Pipe joints and connections must be engineered to handle both the pressure and the mechanical forces.
Foundation Requirements: The supporting structure or foundation must be capable of handling the load transmission from the pipe bend.

Effects of Flow Rate Variations

If the flow rate were to increase:

Both velocities V₁ and V₂ would increase proportionally
The momentum change would increase as the square of the velocity
The resultant force would increase significantly

This highlights the importance of considering maximum flow conditions in the design of pipe bends, particularly in systems where flow rates may vary or where surge conditions might occur.

Practical Applications

Understanding the forces in pipe bends is crucial in many engineering applications:

Municipal water distribution networks
Industrial fluid transport systems
Hydropower installations
Irrigation systems
Fire protection systems

In all these applications, proper calculation and accounting for the forces exerted on bends and other components is essential for safe, reliable, and efficient operation.