I. Introduction Water pipelines refer to the system that transports water from the source to the purification plant or from the plant to the distribution network. Due to the extensive nature of the water supply system and the high pressure involved, ensuring the safe operation of these pipelines has always been a priority for the water industry and engineering departments. Common incidents include pipe bursts and explosions, often caused by factors such as thermal stress, pipe quality, construction issues, geological conditions, and water hammer effects. Although air pockets themselves do not directly cause water hammer, they can contribute to serious damage through this phenomenon. This article explores how to properly install air vents on water pipelines to prevent gas accumulation and the formation of dangerous air pockets. II. Examples and Analysis In areas with varied terrain, it is standard practice to install air vents at the highest points of the pipeline. However, in real-world scenarios, many pipe bursts occur not at the high points but at bend sections, even in low-pressure areas. A typical example is the pipe burst in Hegang, Heilongjiang. Hegang is a hilly region where the water treatment plant and pumping station are located 5 km apart. Purified water flows by gravity to the pumping station, with the treatment plant at an elevation of 210 meters and the pump station's clear well at 185 meters. The pipeline is made of DN800 cast iron pipes with an average flow rate of 1.0 m/s. The pumping station has a high point at 500 meters, with an elevation of 185 meters. An air vent was installed at the highest point, but despite this, pipe bursts occurred frequently. Later, additional air vents were installed near the burst locations, and since then, no further bursts have occurred between the two valves. Only one burst occurred 10 meters beyond the new vent. This indicates that the presence of air pockets is closely related to the occurrence of bursts, as no bursts occurred after the proper installation of vents. III. Formation and Stress Analysis of Air Pockets Under normal operating conditions, the water flow in the pipeline can be considered steady, with constant pressure, flow rate, and temperature. In this state, any air present in the water gradually rises due to buoyancy and forms small bubbles that move along the pipe walls. In uphill sections, the bubbles may move faster than the water due to their buoyancy. However, the roughness of the pipe walls prevents the bubbles from merging into large ones. Small bubbles travel along the wall until they reach the highest point, where the air vent allows some of them to escape. Others are pushed downstream by the water flow. Due to turbulence and flow characteristics, some bubbles pass through the vent and continue downstream. The movement direction of the bubbles is opposite to the component of the buoyancy force (P1). This resistance slows down the bubble’s movement, causing subsequent bubbles to collide and merge into larger ones. Larger bubbles generate greater buoyancy. The formula for buoyancy is: P1 = P * sinα Where: P = Buoyancy of the bubble (P = 1/6 πd³ * ρ) ρ = Water density d = Bubble diameter α = Angle of buoyancy force The thrust of the water flow is given by: P' = V² / (2g * S) Where: V = Flow velocity S = Maximum cross-sectional area of the bubble (S = 1/4 πd²) When P1 = P', the bubble remains stationary in the pipeline. From the equilibrium equation: d * sinα = C / (10.4 * p * g) This shows that under steady flow conditions, the bubble size is inversely proportional to the sine of the slope angle. When d * sinα > C, the bubble moves upward. IV. Shape of Air Bubbles in the Pipeline Assuming the bubble is spherical, this approximation works for micro-bubbles. However, when the bubble grows large enough to touch the pipe wall, surface tension causes it to take on a semi-elliptical shape. As it continues to grow, its shape becomes influenced by water flow, gravity, and the pipe geometry, elongating along the length and arching across the width. According to simulations, the relationship between the length (L) and height (h) of the air pocket is approximately L ≈ 15h. V. Analysis of Air Pocket Forces and Critical Points According to fluid mechanics principles, the force exerted on an air pocket in the pipeline is equal to the pressure acting perpendicular to the flow. This creates a "bow" shape with height h. The thrust on the air pocket is calculated using: P' = V² / (2g * S) Where S is the cross-sectional area of the bow. If the pressure exceeds the structural capacity of the pipe, especially during rapid valve operations or pump start-stop cycles, the compressed air can cause localized stress, leading to a pipe burst. Research suggests that when the height of the air pocket reaches one-quarter of the pipe diameter, it becomes a critical risk point. For a DN800 pipe with a flow rate of 1.0 m/s, the thrust can be calculated as: P' = (1.0²) / (0.2π * 40²) = 5.124 kg Using the buoyancy formula: P1 = 0.5π(0.4³) × 1000 × sinα = 100.5 × sinα Setting P' = P1 gives: sinα = 5.124 / 100.5 ≈ 0.051 → α ≈ 2.92° This matches the measured depression angle at the burst location. VI. Conclusion 1. Air vents must be added at the descending slopes of the pipeline, with their placement determined by the formula P' = P sinα and appropriate flow rates. 2. If the actual slope is less than the calculated value, the air vent should be placed at the intersection of the curved line and the straight line. 3. To ensure early gas release and prevent air pocket formation, air vents should be installed at the junctions of bends and straight sections. 4. The air vent cannot replace the high-point vent, which remains essential for effective air removal.

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