| The transient response associated with systems experiencing pump power failure having undulating profiles and containing air vacuum valves (AVVs) is systematically explored to characterize the critical high points where placing AVVs has the maximum influence on the resulting transient pressures. For this purpose, this work seeks to understand the principal function of AVVs and their relation to the geometry and hydraulic characteristic of the system. Semi-analytical formulas are developed for the time of air cavity growth and collapse, maximum air cavity growth, and the secondary transient pressures due to cavity collapse at AVV location. Such relations are developed considering an undulating pipeline with an exaggerated high point at an intermediate location with the assumption of sudden upstream flow curtailment. Through these semi-analytical formulas, it is shown how the presence of AVVs at particular high points influences the system's transient response. Key characteristics of the high point---namely, its elevation and distance from the upstream and downstream boundaries---are explored and those values that lead to the most severe transient response are identified.;In order to numerically investigate those parameters most influential to a system's transient response, the sensitivity of hypothetical pipeline systems to different input and design parameters is systematically explored. To this end, a numerical model is developed to solve the water hammer governing equations in conjunction with the AVV boundary equations by application of the method of characteristics. This numerical model is also coupled to a Monte Carlo simulation procedure. Monte Carlo Filtering is applied to help isolate and understand the most influential parameters. Also, a variance-based method is applied to explore the main and interaction effects between the influential parameters and to map these as a function of the terms defining the AVV properties and the high point location. |
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