The Method of Characteristics is widely preferred for analyzing unsteady liquid networks,  in part due to the straightforward modeling of complex inline equipment. More general analyses, including unsteady gas flow, tend to favor Finite Volume formulations due to their conservative nature. These solvers are comparatively complex, making their implementation and use more challenging. Advantages and disadvantages of each method for the analysis of gas networks are discussed here. The Method of Characteristics has limitations, but it is shown here to be a technique worthy of consideration for typical engineering analyses.

Authors:
Scott Lang, PE, and Mark Dudley, Applied Flow Technology, USA
Presented at the 14th International Conference on Pressure Surges 
April 12-14, 2023, Eindoven, Netherlands

The MOC as a tool will never replace FVM, which excels in both accuracy and flexibility. However, industrial tools must consider practicality of both development of tools and their use. To that end, the authors believe the tests shown here demonstrate that the MOC is worthy of consideration for the analysis of unsteady compressible transients under some conditions and should be considered a valuable piece of the engineer’s toolkit.

The authors intend to investigate further the effects of real gasses, source terms, and complex boundaries in both solution methods. The present work was intended primarily to show that the MOC is a viable option for the analysis of compressible flow in piping networks, and therefore focused on relatively simple test cases. The authors believe the method shows promise, and that further research is warranted.

Like the FVM, future investigations could result in vast improvements in simulation quality. The simplest FVM formulations often have results that are very inaccurate, while complex schemes can make them some of the most accurate tools available. The authors believe it is reasonable to assume that with dedicated research effort, dramatic improvements to the MOC are possible, building on results that are already practically actionable in many engineering contexts.

1.  Introduction

1.1 Motivation

Accurate numerical modeling of large piping networks is often critical to ensuring their optimal and safe operation. Many such techniques are established for liquid systems [1] [2], but tools for compressible gas are both rarer and more complex [3] [4] [5].

Steady analysis of compressible systems is common, while analysis of acoustic transients in large compressible flow piping networks appears to be an uncommon practice. Problematic situations are often either unaddressed (putting the system at risk) or avoided (via expensive overdesign), which may be due to a lack of knowledge that unchecked compressible surge can pose serious risk to the integrity and safety of the system. Designers or operators may be unaware of practical tools to analyze gas surge and thus resort to simple empirical rules, unclear codes and standards, or overbuilt systems [6].

Consequently, it is desired to develop a tool that allows for practical analysis of unsteady compressible flow for piping networks. It is argued here that some amount of error in results is acceptable if the tool allows for both ease of use and straightforward development of complex devices, especially when the usual alternative is no analysis whatsoever.

1.2  Accuracy and practicality

In piping network analysis, an engineer rarely seeks an exact answer. Even if one was attainable, piping networks change regularly. Equipment is frequently added, replaced, or removed to mitigate component wear and degradation. Few system analysts will know with certainty the exact specifications of every component in the system. Even with exact numerical solutions, approximation and uncertainty will still creep their way in.

Safety factors or design margins can account for this uncertainty. Because it is impossible to eliminate this uncertainty, it is unproductive to demand perfection in a practical numerical model. Some uncertainty is acceptable, provided the model enables an engineer to make valid and actionable conclusions for the system’s design or operation. Given the choice between a very accurate but difficult to use tool, and a less accurate but more accessible tool, many engineers will sacrifice some accuracy for better ease of use and more immediate results.

1.3 Common approaches to simulation

Two general frameworks to solving the non-linear equations of gas flow are discussed here: the Method of Characteristics (MOC), and the Finite Volume Method (FVM). This paper is not intended to be a survey of all methods for unsteady gas analysis, nor is it intended to be a survey of all possible MOC or FVM formulations – only selected approaches for each method are discussed herein.

2 A brief review of fundamentals

2.1 First principles

Accurate compressible flow solutions must obey conservation of mass (), momentum (), and energy (). Also needed is an appropriate equation of state.

2.2  Appropriate assumptions and simplifications for piping networks

In a typical network, piping lengths are orders of magnitude larger than piping diameters, and any given pipe has constant diameter. These conditions make non-axial flow effects negligible in most cases, so a one-dimensional analysis is reasonable.

In one dimensional flow, three forces act on the fluid, barring more exotic situations:

• Upstream and downstream pressure forces ().
• Gravity ().
• Friction, which imparts a shear stress () that opposes flow along the outer surface of the control volume (. Determining analytically is extremely difficult – it is much more practical to use an empirical relation. Models of varying complexity exist – herein it is assumed the frictional term can be estimated from the conditions at the previous time level, referred to as a quasi-steady friction approximation [7]. The Darcy-Weisbach equation () is assumed to apply, meaning that the frictional force can be determined with. For later simplicity, we define the value.

Heat transfer (Q) is assumed to be known or to be a function only of the previous time level. The only work done by the system is assumed to be pressure work.

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