Pulsating Gas Flow In Pipes Part I -- Damping Coefficients
Authors: Converse A.O. , Pigford R.L. 
Published: December 2007

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This paper is to be presented at the 36th Annual Fall Meeting of the Society of Petroleum Engineers of AIME in Dallas October 8–11, 1961, and is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustrations, unless the paper is specifically released to the press by the Editor of JOURNAL OF PETROLEUM TECHNOLOGY or the Executive Secretary. Such abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in JOURNAL OF PETROLEUM TECHNOLOGY or SOCIETY OF PETROLEUM ENGINEERS JOURNAL is granted on request, providing proper credit is given that publication and the original presentation of the paper.
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Introduction
Pulsating flow of a gas in a piping system closely resembles sound transmission. As a result acoustic theory has been used in the design of such systems, e.g. Isakoff (1955), Chilton and Handby (1952), and Murphy (1945). There are, however, two principal differences:the amplitudes of the oscillations in a compressor piping system are likely to be much greater than those in acoustic systems, andthere is a time-average flow of gas through the piping system. This paper is concerned only with the effects resulting from the high (or finite) amplitude oscillations, however.
The presence of the finite-amplitude pulsations has two effects: first, the flow regime becomes at least partially turbulent rather than laminar; and, second, the dependent variables, such as pressure, density, velocity, and temperature, can not always be considered to be related to each other in a linear manner. Hence, the waveform of the resulting pulsation can be quite different from that of the impressed disturbance. This paper is thus divided into two parts. Part I deals with wave damping; the studies of waveform change are reported in Part II.

THEORY OF PULSATION DAMPING

Definition of the Damping Coefficient
As is shown below, the linearized solution to the differential equations which represent the gas oscillations, is of the form
(1)
where the symbol "Re" means the real part of what follows. (Other nomenclature used is listed at the end of the paper.) The damping coefficient, , is of particular interest because as the real part of the exponential dependence on distance, it sets the rate of decay of a wave as it progresses down the pipe; if wave reflection occurs at the open or closed end of a pipe, sets the amplitude of the standing wave. The principal aim of Part I of this paper is to find a way of predicting from fluid mechanical principles.

The Acoustic Damping Coefficient
Consider transmission of sound in a tube, the wave length being much greater than the tube's radius. The fluid motion is damped, primarily owing to heat and momentum exchange with the pipe wall. Kirchhoff (1868) developed the expression for the damping factor which results from the consideration of these effects in the limit of small amplitude. It was given by Rayleigh (1945) and by Weston (1953).
(2)
It is important to note that, according to the above expression, is dependent on the frequency, , but is independent of the wave amplitude.

Damping of Waves of Finite Amplitude
Lebmann (1934) and Fredericksen (1954) have previously measured damping coefficients for finite amplitude waves. They found to be much greater than indicated by Eq. (2).