A steel pipeline carrying gas has an internal diameter of 120cm and an external diameter of 125cm. It is laid across the bed of a river, completely immersed in water and is anchored at intervals of 3m along its length. Calculate the buoyancy force per meter run and upward force on each anchorage. Take density of steel = 7900 kg/m3.

Buoyancy Force on Steel Pipeline

Problem Statement

A steel pipeline carrying gas has the following specifications:

Internal diameter: 120 cm
External diameter: 125 cm
Density of steel: \( 7900 \) kg/m³
The pipeline is completely immersed in water and anchored at intervals of 3m along its length.

Determine:

The buoyancy force per meter run.
The upward force on each anchorage.

Solution

1. Calculate Buoyant Force per Meter

\[ F_B = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ = 9810 \times \frac{\pi}{4} \times (1.25)^2 \times 1 \] \[ = 12039 \text{ N/m} \]

2. Calculate Buoyant Force for 3m

\[ F_{B3} = 12039 \times 3 \] \[ = 36117 \text{ N} \]

3. Calculate Weight of 3m Steel Pipe

\[ W_3 = 3 \times \gamma_{\text{steel}} V_{\text{steel}} \] \[ = 3 \times 7900 \times 9.81 \times \frac{\pi}{4} \times ((1.25)^2 – (1.20)^2) \] \[ = 22369 \text{ N} \]

4. Calculate Upward Force on Each Anchorage

\[ P = F_{B3} – W_3 \] \[ = 36117 – 22369 \] \[ = 13748 \text{ N} \]

Final Results:

Buoyancy force per meter: 12039 N/m
Upward force on each anchorage: 13748 N

Explanation

The problem involves the concept of buoyancy, which states that an object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced.

1. Buoyant Force:
The buoyancy force is determined by the volume of water displaced by the submerged pipeline. The volume of displaced water per meter is found using the outer diameter of the pipe.

2. Buoyancy Force for 3m Length:
Since the pipeline is anchored at intervals of 3m, we multiply the buoyancy force per meter by 3 to find the total buoyant force acting on a 3m segment.

3. Weight of the Pipeline:
The weight of the steel pipeline for a 3m section is determined by calculating the volume of steel using the difference between the outer and inner diameters.

4. Net Upward Force on the Anchorage:
The total buoyant force is counteracted by the weight of the pipeline. The net upward force on each anchorage is the difference between the buoyant force and the weight of the pipe.

Physical Meaning

1. Importance of Anchoring:
The pipeline must be anchored properly to prevent it from floating or shifting due to the buoyant force. If the buoyancy force exceeds the weight of the pipe, it will tend to rise.

2. Role of Density in Buoyancy:
The density of steel is significantly higher than that of water, meaning that the steel pipeline has sufficient weight to counteract most of the buoyancy force. However, additional anchoring is needed to stabilize it.

3. Application in Submerged Pipelines:
This principle is crucial for designing underwater pipelines, such as those used for transporting oil and gas. Engineers must ensure proper anchoring to prevent displacement due to buoyant forces.