Pressure Transients in Liquid-Gas Lines - Part 2
This is the second in a series of articles on fluid transients in piping systems.
A trapped gas pocket in a liquid system can occur in several configurations, as illustrated in Figure 1. Consider the first case where the gas pocket is trapped in-line at a dead-end, at an initial pressure Pg0. As the pump starts, the gas pocket is compressed, causing an oscillatory pressure in the gas.
The gas pressure can be calculated by closed-form solution based on mass conservation and momentum principles, with gas state equations. A closed-form solution to the differential equations (DE) is plotted in Figure 2, labeled “DE Model”.
A more realistic pressure can be calculated by computational fluid dynamics, taking into consideration the compressibility of the liquid. Figure 2 shows a comparison of the computational fluid dynamics numerical solution with liquid compressibility (labeled “Numerical”) to the rigid column differential equation closed-form solution (labeled “DE Model”). The CFD method goes one step further than just modeling compressibility; it also models the propagation of pressure waves; this is why smaller precursor pressure spikes representing the pump start-up pressure can be found before the main pressure peak.
The second consideration that can be added to these equations is the pump startup characteristics, which may be represented by an exponential ramp up function, where the time constant is pump-specific:
P(t) = Pop exp [-1 / (t + 0.1)s]
The differential equations can then be solved by a numerical stepping method. Figure 3 shows a comparison of an instantaneous pressure increase (pump pressure as step function, labeled “Numerical”) to a pump ramp up without liquid compressibility (labeled “DE Model W/Startup”) and with liquid compressibility (labeled “Numerical Model W/Startup”). These solutions are still bounding as they do not account for the liquid velocity, which is limited by the pump capacity.