When the Joukowsky Equation Does Not Predict Maximum Water Hammer Pressures
AFT Impulse™ Technical Paper
Authors: Trey Walters, P.E., Applied Flow Technology; Robert A. Leishear, Ph. D., P. E., Leishear Engineering, LLC
Presented at the 2018 ASME PVP Conference July 16, 2018
ASME Journal of Pressure Vessel Technology, December 2019, Vol. 141 / 060801-1
Abstract
The Joukowsky equation has been used as a first approximation for more than a century to estimate water hammer pressure surges. However, this practice may provide incorrect, non-conservative, pressure calculations under several conditions. These conditions are typically described throughout fluid transient text books, but a consolidation of these issues in a brief paper seems warranted to prevent calculation errors in practice and to also provide a brief understanding of the limits and complexities of water hammer equations.
To this end, various issues are discussed here that result in the calculation of pressures greater than those predicted by the Joukowsky equation. These conditions include reflected waves at tees, changes in piping diameter, and changes in pipe wall material, as well as frictional effects referred to as line pack, and the effects due to the collapse of vapor pockets. In short, the fundamental goal here is to alert practicing engineers of the cautions that should be applied when using the Joukowsky equation as a first approximation of fluid transient pressures.
CONCLUSION
The Joukowsky equation should be used judiciously in piping systems for several conditions:
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- Piping systems that contain tees
- Piping systems that contain changes in pipe diameter, pipe material, pipe wall thickness, or frictional coefficients
- Piping systems where increased pressures due to line pack may be an issue (examples are long pipelines and/or higher viscosity fluids)
- Systems where pressures drop to the vapor pressure of the liquid in the piping system
All in all, when these complex conditions are present in piping systems, numerical methods are preferred to the simplified Joukowsky equation to prevent a misunderstanding of system performance.
Significant mistakes can be made by using the simplified Joukowsky equation without a more complete awareness of its limitations.
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Introduction
The rigorous study of water hammer reaches back into the 19th century (Bergant et al. [1], Ghidaoui et al. [2]). Among the excellent text books on water hammer are those of Thorley [3], Wylie and Streeter [4], Swaffield and Boldy [5], Leishear [6], and Chaudhry [7].
The field of water hammer is well established in academia as well as in industry, where industry is often tasked with designing complicated piping systems sometimes many kilometers in length. Further, engineers in industry are typically under budgetary and schedule constraints and often need to make decisions with sometimes incomplete and imperfect information. As a result, in many cases engineers in industry rely on quick, handbook formulas to make decisions based on estimates.
One such powerful and important formula for water hammer is usually credited to Joukowsky [8] and is therefore often called the “Joukowsky equation”. Other names that one finds in industry for this equation are, in various forms, the “Basic Water Hammer Equation”, the “Instantaneous Water Hammer Equation” and the “Maximum Theoretical Water Hammer Equation”. Note that research in recent years showed that researchers prior to Joukowsky discovered this same equation, but Joukowsky’s name is most often associated with this equation. For more on the history of water hammer, consult Tijsseling and Anderson [9].
Just two decades ago water hammer was still largely considered by industry to be a niche specialty. As a result, water hammer studies were often outsourced to specialized and experienced consultants. In the last two decades, there has been a significant growth in the availability of user-oriented (i.e., graphically based and menu-driven) commercial software for water hammer simulation (Ghidaoui et al. [2]).
While there is certainly an overall appreciation for the water hammer phenomenon in industry, mistakes are easily made when using the simplified Joukowsky equation. This equation can be mistakenly misunderstood in the industry to be a worst-case, conservative equation. A clear understanding is demanded with respect to the situations where non-conservative pressure estimates are obtained when using this equation.
Academic papers acknowledge that the Joukowsky equation is not always conservative, but that knowledge does not always make its way into industrial applications. The purpose of this paper is to identify for the practicing engineer those situations where the Joukowsky equation does not provide worst-case, conservative pressure predictions. This knowledge should result in safer and more cost-effective piping systems.