A 45° pipe bend tapers from 600mm diameter at inlet to 300mm diameter at outlet. The pressure at inlet is 140 kN/m² and the rate of flow is 0.425 m³/s. At outlet the pressure is 123 kN/m² gauge. Neglecting friction, calculate the resultant force exerted by the water on the bend in magnitude and direction. The bend lies in a horizontal plane.

Pipe Bend Force Calculation – Fluid Mechanics Solution

Pipe Bend Force Calculation

Fluid Mechanics Problem Solution

Problem Statement

A 45° pipe bend tapers from 600mm diameter at inlet to 300mm diameter at outlet. The pressure at inlet is 140 kN/m² and the rate of flow is 0.425 m³/s. At outlet the pressure is 123 kN/m² gauge. Neglecting friction, calculate the resultant force exerted by the water on the bend in magnitude and direction. The bend lies in a horizontal plane.

Given Data

Pipe diameter at entrance (d₁) 600 mm = 0.6 m

Pipe diameter at exit (d₂) 300 mm = 0.3 m

Flow rate (Q) 0.425 m³/s

Fluid Water (density ρ = 1000 kg/m³)

Pressure at entrance (P₁) 140 kN/m² = 140,000 Pa

Pressure at exit (P₂) 123 kN/m² = 123,000 Pa

Bend angle (θ) 45°

Solution Approach

To find the resultant force exerted by the water on the bend, we need to:

Calculate the cross-sectional areas and velocities at entrance and exit
Apply the momentum equation to determine the forces in both X and Y directions
Calculate the resultant force and its direction

Preliminary Calculations

Step 1: Calculate the cross-sectional areas:

A₁ = π/4 × d₁² = π/4 × 0.6² = 0.2827 m²

A₂ = π/4 × d₂² = π/4 × 0.3² = 0.07068 m²

Step 2: Calculate the velocities:

V₁ = Q/A₁ = 0.425/0.2827 = 1.5 m/s

V₂ = Q/A₂ = 0.425/0.07068 = 6.01 m/s

Force in X-Direction

Step 1: Apply the momentum equation in the X-direction:

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

(P₁A₁ – P₂Cosθ A₂) – F_x = ρQ(V₂_x – V₁_x)

Step 2: At section 1, the velocity is entirely in the X-direction (V₁_x = V₁ = 1.5 m/s). At section 2, after the bend, the X-component of velocity is (V₂_x = V₂Cosθ = 6.01×Cos45° = 4.25 m/s).

(P₁A₁ – P₂A₂Cosθ) – F_x = ρQ(V₂Cosθ – V₁)

Step 3: Solve for F_x:

F_x = (P₁A₁ – P₂A₂Cosθ) + ρQ(V₁ – V₂Cosθ)

F_x = (140000 × 0.2827 – 123000 × 0.07068 × Cos45°) + 1000 × 0.425 × (1.5 – 6.01 × Cos45°)

F_x = (39578 – 6133) + 1000 × 0.425 × (1.5 – 4.25)

F_x = 33445 – 1183 = 32262 N

Force in Y-Direction

Step 1: Apply the momentum equation in the Y-direction:

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

F_y – P₂Sinθ A₂ = ρQ(V₂_y – V₁_y)

Step 2: At section 1, there is no Y-component of velocity (V₁_y = 0). At section 2, after the bend, the Y-component of velocity is (V₂_y = V₂Sinθ = 6.01×Sin45° = 4.25 m/s).

F_y – P₂A₂Sinθ = ρQ(V₂Sinθ – 0)

F_y – P₂A₂Sinθ = ρQV₂Sinθ

Step 3: Solve for F_y:

F_y = P₂A₂Sinθ + ρQV₂Sinθ

F_y = 123000 × 0.07068 × Sin45° + 1000 × 0.425 × 6.01 × Sin45°

F_y = 6133 + 1820 = 7953 N

Resultant Force Calculation

Step 1: Calculate the magnitude of the resultant force:

F_R = √(F_x² + F_y²)

F_R = √(32262² + 7953²)

F_R = √(1,040,837,444 + 63,250,209)

F_R = √1,104,087,653 = 33,228 N

Step 2: Calculate the direction of the resultant force:

θ = tan⁻¹(F_y/F_x) = tan⁻¹(7953/32262) = 13.8°

The resultant force exerted by the water on the bend is 33,228 N at an angle of 13.8° (to the right and downward).

Summary

The fluid velocities in the pipe were calculated:
- Entrance velocity (V₁) = 1.5 m/s
- Exit velocity (V₂) = 6.01 m/s
The cross-sectional areas were determined:
- Entrance area (A₁) = 0.2827 m²
- Exit area (A₂) = 0.07068 m²
The force components were calculated using the momentum equation:
- X-direction force: F_x = 32,262 N
- Y-direction force: F_y = 7,953 N
The resultant force on the bend:
- Magnitude: 33,228 N
- Direction: 13.8° from the X-axis (to the right and downward)

This problem demonstrates the application of the momentum equation in fluid mechanics to determine forces on pipe bends. The resultant force is significant due to both the pressure forces and the momentum change of the fluid as it changes direction through the 45° bend. The reduction in pipe diameter from 600mm to 300mm also contributes to the change in fluid velocity, affecting the overall force on the bend.