Solving the transient compressible flow equations is hard. To help engineers make everyday decisions, simplified methods have been developed. One such simplification is used in the estimation of transient forces in steam pipe systems. An incomplete understanding related to gas wave speed has resulted in a method which does not reliably give conservative pipe force estimates, potentially resulting in unsafe designs. This paper develops gas wave speed predictions from first principles for compression waves moving into a non-zero steady-state flow. It is shown how the length of a family of waves steepens more quickly than previously thought. Implications for transient pipe force estimation are discussed.

Authors:
Trey Walters, PE, Applied Flow Technology, 
Presented at ASME 2023 Pressure Vessels & Piping Conference  |  July 16-21, 2023, Atlanta, Georgia, USA

An improved understanding of gas wave speed for compression waves is offered and should be considered in future gas transient analyses. Simplified methods of estimating transient pipe forces do not consider wave steepening and may not be conservative – especially at elbow pairs further from the source of the transient. Improvements to the Goodling Method should be made by the engineering community and a review of existing steam systems designed and built using Goodling should be considered for safety reasons. 

The analysis in this paper clearly shows wave steepening, and it shows it can happen much faster and in shorter pipe lengths than previously thought – particularly in methods used in power station steam piping design. This raises questions about simplified methods like the Goodling Method. It appears this simplified method inadvertently overlooked something. What was it?

Reviewing a commentary on Goodling in Moody and Stakenborghs, 2018 (7) sheds some light on this. It appears that when considering how fast wave steepening could occur, the authors in (7) essentially used Eq. 15. But they had an oversight and neglected to consider steady state in Eq. 15 and only considered the difference in sonic velocities. This resulted in the wave speed estimated difference between the back of the wave to the front to be much lower than determined in our paper and lent credence to their conclusion that wave steepening was not important.

1.  Introduction

The topic of transient compressible flow has many important applications in industry. Among these are the prediction of:

• pipe forces in high pressure steam piping in nuclear and fossil power stations during shut down events
• the rate of pressure change in gas turbine supply conditions during system transients in order to avoid unplanned shut downs
• the disruption of flow conditions to air and gas compressors to avoid unplanned shut downs
• pipe forces during pressure relief events
• pipe flow during tank blowdown and charging events

The complications of accurately simulating such behavior have been noted by many authors over many decades (Safwat, 1978 (1), Thorley and Tiley, 1987 (2), Vardy and Pan, 2000 (3)). As a result, it is often the case that simplified calculation methods have been adopted to assist engineers in making practical design decisions. However, when evaluating finite magnitude and finite length waves, the authors note frequent misconceptions in published methodologies. Some of these misconceptions can result in significantly unconservative predictions used for design purposes.

The purpose of this paper is to untangle some of these misconceptions as they relate to wave speed in steam and gas piping. More specifically, it is typical in industrial systems that waves have a finite magnitude (they cannot be accurately treated with infinitesimal wave methodology) and, perhaps more importantly, they have a finite length (as part of a wave family, discussed below). In other words, they are not instantaneous. They have a starting time (e.g., when a valve begins to close) and an ending time (e.g., when the valve finally closes). As is commonly known, this finite process over time generates a family of waves. The family of waves results when, during each infinitesimal increment of time, the valve position changes slightly thereby generating a new incremental wave. Over the entire valve closure time numerous incremental waves are generated. These waves are referred to here as a wave family. The length of this family of waves can change with time. Why? Because the wave speed at the front of the wave family is not the same as the wave speed at the back.

Properly understanding why this length changes over time leads to a better understanding of how fast a family of compression waves can steepen (i.e., the back of the wave family catching up with the front). This has immensely important applications in predicting forces (e.g., on pipe sections bounded by direction changes such as elbows). If a wave can steepen more quickly, it can exert a greater imbalanced force when it passes through a given pipe section.

In order to better understand why wave steepening is important, we will first give a summary of calculating transient pipe forces in gas systems. Second, an analytic solution of compressible gas flow will be reviewed for the case of perfect gases in frictionless, adiabatic pipe flow. Fortunately, the analytical solution will help us clearly determine gas wave speed and see how and why wave steepening happens. Third, simulation results of compressible flow of real gases with friction included will be discussed. This will be compared and contrasted to analytic solutions to better understand the wave steepening effect in industrial systems.

Use the links above to view the full papers.