Wave Reflection and Transmission in Pipelines
In addition to the equations describing transient flow, it is important to know about the affect of boundaries - such as tanks, dead ends, and pipe branches - that modify the effects of hydraulic transient phenomena.
Hydraulic systems commonly have interconnected pipelines with differing characteristics, such as material and diameter. These pipeline segments and connection points (nodes) define a system's topology.
When a wave traveling in a pipe and defined by a head pulse H0 comes to a node, it is transmitted with a head value Hs to all other connected pipes and reflects back to the initial pipe with a head value Hr. The wave reflection occurring at a node changes the head and flow conditions in each of the pipes connected to the node.
If the distances between the pipe connections are small, the head at all connections can be assumed to be the same (that is, the head loss through the node is negligible), and the transmission factor (s) can be defined as
| Where: | s | = | transmission factor (dimensionless) |
| Hs | = | head of transmitted wave (ft, m) | |
| Ho | = | incident head pulse (ft, m) | |
| Ao | = | incoming pipe area (ft2, m2) | |
| ao | = | incoming wave speed (ft/sec., m/s) | |
| Ai | = | area f i-th pipe (ft2, m2) | |
| ai | = | wave speed of i-th pipe (ft/sec, m/s) | |
| n | = | number of outgoing pipes | |
| i | = | pipe number index |
In a closed system without friction to dampen transients, transients would persist indefinitely. However, viscous and friction effects typically cause transients to attenuate within seconds to minutes. Bentley HAMMER HAMMER CONNECT is an essential tool to keep track of the transient pressure-wave reflections and the friction and elastic effects during the simulation, as follows:
- Because friction does exist in an actual system, the potential head change calculated using the Joukowsky equation underestimates the actual head rise. This underestimation is due to packing—an additional increase in head occurring at the valve as the pressure wave travels upstream.
- The small velocity behind the wave front means that the velocity difference across the wave front is less than Vo, so the pressure change is progressively less than the potential surge as the wave travels upstream. This effect, which is concurrent with line packing, is called attenuation or reduction.
- Transient pressure waves are partially transmitted and simultaneously reflected back at every junction with other pipes, depending on their wave speed and diameter.
Although Bentley HAMMER HAMMER CONNECT calculates the proportion of an incoming transient energy pulse that is transmitted and reflected at each junction node, it is useful to consider how this phenomenon takes place in a typical hydraulic system using the relation for the reflection factor:
Several special cases can be considered, including:
- Pipe connected to a reservoir-In this case, n = 1, s = 0, and r = -1. In other words, a wave reaching a reservoir reflects with the opposite sign.
- Pipe connected to a dead-end or closed valve-In this case, n = 1, and, through the derivation of an equation for r similar to Equation , it can be shown that r = 1. In other words, a wave reflects at a closed extremity of a pipe with the same sign and, therefore, head amplification occurs at that extremity. If a flow-control operation causes a negative pressure wave that reaches a closed valve, the wave's reflection causes a further reduction in pressure. This transient flow condition can cause liquid column separation and, in low-head systems, potential pipeline collapse. At a dead end, the wave is reflected with twice the pressure head of the incident wave.
- Pipe diameter reduced (celerity increase)-In this case, A 1 < A 0 , and s > 1, so the head that is transmitted is amplified. For example, if A 1 = A 0 /4 (or D1 = D 0 /2), then s = 8/5=1.6 and r = s - 1 = 0.6, and the head transmitted to the smaller pipeline is 60 percent greater than the incoming head. The larger pipeline is also subjected to this head change after the wave partially reflects at the node. If the diameter is reduced to zero, the junction becomes a dead end.
- Pipe diameter increased (celerity decrease)-In this case, an attenuation of the incident head occurs at a pipeline diameter increase. The smaller pressure wave is transmitted to the larger pipeline and, after the reflection, the smaller pipeline is subjected to the lower final head. At an expansion, the reflected wave has the opposite sign of the incident wave. In the limit, as the diameter increases indefinitely, the reservoir case is obtained.