One of the key assumptions in modern steam hammer load analysis is based on an incomplete understanding of steam wave behavior. Compression waves generated after a valve closure steepen much more quickly than previously thought. The resulting forces generated on pipe runs separated by elbows can be significantly higher than those predicted using current methods.

To be more specific, current methods may give unconservative pipe force estimates – potentially resulting in unsafe designs. Using a more complete understanding of gas wave speed, quantitative examples are given which show the larger forces which occur as a result of wave steepening. Suggestions on how to improve load estimation are discussed.

Author: Trey W. Walters, PE, Applied Flow Technology, USA; Presented at the ASME PVP Pressure Vessels & Piping Conference, July 2022

Traditional methods for evaluating transient piping loads (such as the Goodling Method [2-3]) do not reliably give conservative pipe loads. Wave steepening is not accounted for in such methods.

Systems designed using the Goodling Method are likely not as safe as previously believed. This includes nuclear power station piping in recent decades. 

The Goodling Method should not be used for steam piping load estimation except perhaps in very short pipe runs. Engineers should seek other methods that can yield more accurate estimates.

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1. INTRODUCTION

One of the key unstated assumptions in modern steam hammer load analysis is that wave steepening is not significant in typical lengths of pipe runs. Recently Walters and Lang [1] conclusively showed that this assumption is mistaken. For the last four decades the standard procedure for estimating steam hammer piping loads resulting from valve closures has been the Goodling Method [2-3]. There are several key assumptions made in the Goodling Method. One unstated assumption is that acoustic waves do not appreciably steepen.

Several papers have been published in recent years that used transient compressible flow numerical methods to evaluate wave steepening and the resultant generated forces. These papers used different numerical methods. Rovagnati and Gray [4-5] used an MOC (Method of Characteristics) tool, while Mayes, Gawande and Williams [6] and Mayes and Gawande [7] used a CFD tool. These four papers all observed wave steepening and larger forces than those given using Goodling or similar methods. However, no reasons for this behavior based on physics or theory were given by these authors.

Moody and Stakenborghs [8] recently discounted the claims in these papers and did so using physics and theory. However, Walters and Lang [1] showed that the reasoning in [8] had an oversight and neglected to consider how changes in bulk fluid velocity of the gas affected wave speed. Walters and Lang developed analytical relationships to show why and how fast wave steepening occurs. They also used a commercially available software tool [9] to show the same wave steepening in a more real-world system which included friction and real gas effects. They showed that pipe forces can be much higher than Goodling predicts.

It will be shown in this paper what this means for steam hammer load analysis. Current methods of predicting forces are based on a mistaken assumption and are not conservative. Existing power station steam piping designed using the Goodling Method are not as safe as believed.

ACOUSTIC WAVE BEHAVIOR IN STEAM FLOW

When a valve closes in a steam line, a compression wave is generated upstream of the valve. This wave travels at a speed that depends on the bulk fluid velocity and the acoustic velocity of the steam. More generally, when a valve closes over some finite time, a family of compression waves is generated. The wave speed at the front of the wave family is not the same as the wave speed at the back of the wave family. This is due in part to a difference in acoustic velocities at the front and back (the back is at a higher temperature after the compression event and thus has a higher acoustic velocity) [8]. But, more importantly, the wave speed is different because the bulk steam velocity is different at the front and back of the wave family. This is discussed in detail using analytical relationships and diagrams by Walters and Lang [1]. It is well known that gas wave velocity, a, is a function of gas acoustic velocity, c, and gas bulk velocity, V.

In that gas bulk velocity, V, is a vector, that makes gas wave velocity, a, also a vector. A compression wave like that discussed in this paper will follow Eq. 1 with the negative sign. A nomenclature distinction used here is that c represents gas acoustic velocity and a represents gas wave velocity. The terms gas wave velocity and gas wave speed are used interchangeably here with the understanding that both terms are vectors. Previous authors at times confused the concepts of gas acoustic velocity and gas wave velocity. This is part of the reason why wave steepening has been underestimated in the past. Walters and Lang [1] gave an analytical relationship for the speed of wave steepening (the speed at which the back of the wave family catches the front) for a perfect gas in frictionless, adiabatic flow where VSS is the steady-state velocity and γ is the isentropic expansion coefficient. They showed that Eq. 2 is constant over time for frictionless, adiabatic flow of a perfect gas. 

For adiabatic flow of real gases with friction the initial speed of wave steepening where cb is the sonic velocity at the back of the wave family and cSS is the sonic velocity at the front (equal to the steady-state value). The bulk gas velocity at the front of the wave family is also the same as the steady-state gas velocity, VSS. Eqs. 2 and 3 give the same result if the flow is frictionless and adiabatic and the gas is a perfect gas. A quantitative example will be given which demonstrates the validity of Eq. 3 and also shows how the speed of wave steepening changes over time and distance from the initiating transient event. In other words, Eq. 3 is not constant in the flow of real gases with pipe friction.

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