Flow of Natural Gas

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Catalog No. L41022

Flow Of Natural Gas Thru High-Pressure Transmission Lines

Monograph 6

Prepared for theUnderground Storage Committee

of

Pipeline Research Council International, Inc.

Prepared by the following Research Agencies:

U.S. Department Of The Interior Bureau Of Mines

Authors:T.W. Johnson and W.B. Berwald

Publication Date:February , 1935



“This report is furnished to Pipeline Research Council International, Inc. (PRCI) under the terms of PRCI Report Of Investigations 6763, between PRCI and U.S. DepartmentOf The Interior Bureau Of Mines. The contents of this report are published as receivedfrom U.S. Department Of The Interior Bureau Of Mines. The opinions, findings, andconclusions expressed in the report are those of the authors and not necessarily thoseof PRCI, its member companies, or their representatives. Publication and disseminationof this report by PRCI should not be considered an endorsement by PRCI or U.S.Department Of The Interior Bureau Of Mines, or the accuracy or validity of any opinions,findings, or conclusions expressed herein.

In publishing this report, PRCI makes no warranty or representation, expressed or implied, with respect to the accuracy, completeness, usefulness, or fitness for purpose of the information contained herein, or that the use of any information, method, process, or apparatus disclosed in this report may not infringe on privately owned rights. PRCIassumes no liability with respect to the use of, or for damages resulting from the use of,any information, method, process, or apparatus disclosed in this report.

The text of this publication, or any part thereof, may not be reproduced or transmitted inany form by any means, electronic or mechanical, including photocopying, recording,storage in an information retrieval system, or otherwise, without the prior, writtenapproval of PRCI.”

Pipeline Research Council International Catalog No. L41022

Copyright, 1935 All Rights Reserved by Pipeline Research Council International, Inc.

PRCI Reports are Published by Technical Toolboxes, Inc.

3801 Kirby Drive, Suite 340Houston, Texas 77098Tel: 713-630-0505Fax: 713-630-0560Email: [email protected]



CONTENTS

PageIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgments

. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Development and review of formulas for flow of natural gas through pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

General consideration of formulas for flow of natural gas throughpipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Formulas for calculating flow of natural gas. . . . . . . . . . . . . . . . . . . . . . . 5Formulas where coefficient of friction is constant . . . . . . . . . . . . . . . . 6Formulas where coefficient of friction is expressed as some func-

tion of diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Formulas where coefficient of friction is expressed as a function

. . . . . . . . . . . . .

of Reynolds’ criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Flow tests of commercial pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Purpose of tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Method of making flow tests . . . . . . . . . . . . . . . . . . . . . . . . . . .Preparation for tests

. . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Apparatus and data. . . . . . . . . 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Method of calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Standards used in calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Selection of test periods for calculationCalculation of volumes by orifice measurement . . . . . . . . . . . . . . . . . . 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Description of pipe lines tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Data from flow tests of commercial pipe lines . . . . . . . . . . . . . . . . . . . . . . 17

Flow tests of experimental pipe line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Apparatus and experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Results of flow tests of commercial pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Variations from conditions for which general flow formula is developed 25Sources of error in test data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Comparison of test results and flow formulas . . . . . . . . . . . . . . . . . . . . . . 27Relationship between coefficient of friction and diameter of pipe . . . . . . . . . . . . .

31Relationship between coefficient of friction and Reynolds’ criterion . . . . . . . . . . . . . . 34Complex pipe-line systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Formulas for complex systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Examples of complex systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Tests of commercial complex systems 42Tests of experimental pipe line consisting of several diameters of pipe .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Special formulas for designing pipe-line systems consisting of parallellines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples showing use of formulas 47. . . . . . . . . . 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . .Design of parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Construction and operating conditions influencing delivery capacities of natural-gas pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Liquids in natural-gas pipe lines52

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Sources of liquids in natural-gas pipe lines . . . . . . . . . . . . . . . . . . . . 54Condensation of water vapor due to changes in temperature and

pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Removal of liquids from natural-gas pipe lines . . . . . . . . . . . . . . . . . 61Location of drips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Design of drips . . . . . . . . . . . . . . .

Rust scale in natural-gas pipe lines. . . . . . . . . . . . . . . . . . . . . . . . . . 63

65Stored gas in pipe-line systems . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

Differences in elevation at inlet and outlet ends of a pipe line. . . . . . . 72. . . . . . . . . . . . . . . . . . .

Types of construction and equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Physical and chemical properties of natural gas . . . . . . . . . . . . . . . . . . . . . . . . 76

Chemical composition of natural gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Compressibility of natural gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Viscosity of natural gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

II I



IV CONTENTS

PageSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Appendix 1. - Derivation of general formula for flow of natural gas

through pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Appendix 2. - Reynolds’ criterion of coefficient of friction . . . . . , . . . . . . . . . . 93 Appendix 3. - Formulas having coefficient of friction expressed as a func-

tion of Reynolds’ criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Appendix 4. - Description of pipe lines tested . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Appendix 5. - Derivation of special formulas for designing pipe-line sys-

Appendix 6. - Tables and formulastems consisting of parallel lines . . . . . . . . . . . . . . . . . . . . . . 108

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Fig.1.2.3.4.

5.

6.

7.

8.

9.

10.11.

12.13.

14.

15.16.

17.18.

19.20.21.22.

23.

24.25.

ILLUSTRATIONS

Typical installation of apparatus for flow test. . . . . . . . . . . . . . . . . . . . . . 12Record of pressure observations (line 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Plan of experimental pipe line for study of gas flow. . . . . . . . . . . . . . . . . 23Comparisons of several flow formulas with Weymouth’s formula for different pipe diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Relationship between coefficient of friction and diameter of pipe fromflow tests (actual values) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Relationship between coefficient of friction and diameter of pipe fromflow tests (average values) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Relationship between the coefficient of friction and Reynolds’ cri-terion-commercial pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Relationship between coefficient of friction and Reynolds’ criterion-experimental pipe lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Layout of parallel system for illustrating the calculation of equivalentdiameters and lengths and importance of cross connections in someparalleled systems (example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Design of pipe line 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Flow of gas through experimental pipe line consisting of several

different diameters of pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Flow of gas through the different diameter sections comprising theentire length of experimental pipe line . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Relationship between percentage increase in the deliver and per-

centage of the line paralleled for different values of d1/d . . . . . . . . . . 49Record of pressure observations showing drop in line pressure due to

blowing drips (line 11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Percent by volume of water vapor in saturated gas. . . . . . . . . . . . . . . . . . 56Gallons of water per million cubic feet of gas contained in gas sat-

urated with water vapor at the indicated temperatures and pressures 57Daily atmospheric temperature variations . . . . . . . . . . . . . . . . . . . . . . . . . . .Daily temperature variations of ground at different depths..........................

5960

Simple drip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Complex drip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Storage capacity of 1 mile of pipe of various diameters . . . . . . . . . . . . . 69Comparison between input and withdrawal rates and the corresponding

changes in line pressures for line 3 (table 4) . . . . . . . . . . . . . . . . . . . 71Relationship between viscosity and temperature for natural gas and

gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Relationships between Reynolds’ criterion and the coefficient of friction. 95Relationship between ratio of inlet to outlet pressure and percentage

of maximum flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117



FLOW OF NATURAL GAS THROUGH HIGH-PRESSURE

TRANSMISSION LINES1

A Joint Report

BY T. W. JOHNSON2 AND W.B. BERWALD3

INTRODUCTION

During the last 60 years the transportation of natural gas fromwells to city distribution plants has developed from a single low-

pressure line 25 miles long, made of short lengths of 8-inch-diameter wooden pipe, to one of the most important branches of the petro-leum and natural-gas industry. Thousands of miles of large-diameter steel pipe are in use carrying natural gas, as far as 1,000miles in some instances, between the sources of supply and pointsof consumption.

The engineering problems involved have multiplied and becomemore difficult as the length, size, and operating pressures of natural-gas transmission lines have increased. The solution of many of these problems has made possible the present long-distance trans-mission systems and the consequent large-scale utilization of anatural resource which otherwise often is wasted. Conservation,insofar as it prevents waste, and utilization of natural gas beganwith the first transmission line and have depended almost entirelyon the development of transmission systems.

The Bureau of. Mines and the Natural-Gas Department of the American Gas Association long have recognized the need for reli-able data pertaining to flow of gas suitable for commercial designand operation of high-pressure transmission systems. Severalyears ago these organizations cooperatively began a study of theflow of gas through commercial transmission lines. Numerous dataon gas flow through pipe lines under many conditions of operationwere obtained from carefully conducted tests, and these data,together with all other available information, were analyzed care-fully to obtain practical information for the use of operators indesigning pipe lines and determining the effects of different operat-ing conditions on pipe-line capacities. In other words, the inves-

t igation was primari ly an engineering study of representativenatural-gas transmission lines operating under numerous conditionsof flow.

During this study the existing data pertaining to the flow of natural gas through pipe lines were reviewed thoroughly, and itwas found that the several pipe-line flow formulas developed by



2

different investigators gave widely varying results when appliedto problems of pipe-line design. The factors that influence theflow of gas through transmission lines are numerous and varied.Roughly, they may be divided into two classes, those inherent to

pipe lines and flowing gases and those due to construction character-istics and operation of the lines.

The first group includes dimensions and properties of the pipeline (length, diameter, and internal wall roughness), the proper-ties of the flowing gas (density and viscosity), and velocity of thegas. The relationships between these factors are the basis for pipe-line flow formulas; appendixes to this report give in detailderivations of fundamental relationships and their application inthe development of practical flow formulas. Experimental workpertaining to such factors was done to verify some of the assump-tions necessary in developing a practical flow formula. The pur-pose of such work and of the discussion of the fundamental equa-tions of fluid flow was to emphasize the conditions for which flow

formulas have been developed and to obtain data on the accuracyof the formulas.The second group of factors influencing the flow of gas through

transmission lines includes any foreign material in the line (con-densates, rust, and dust) or any construction or operating featurethat influences the resistance to flow or decreases the effectivediameter of the pipe. These factors usually are peculiar to indi-vidual pipe lines and seldom can be considered in a flow formula.The effects of these incidental factors have been analyzed for cer-tain individual lines and expressed in terms of reduced efficiencyof the pipe line to deliver gas.

For the purpose of presentation the discussions in this bulletinhave been arranged generally to include, first, a listing of pipe-

line formulas used by different investigators; second, a discussionof flow tests made by the writers on commercial and experimentalpipe lines, including methods of testing and of analyzing flow data;and third, a discussion of the application of flow data to commer-cial pipe-line transmission problems. Detailed mathematical de-velopment of formulas is included in appendixes with appropriatereferences in the main text of the bulletin.

ACKNOWLEDGMENTS

This study of the flow of natural gas through high-pressuretransmission lines was made under the general supervision of H. C. Fowler, former acting chief engineer, and R. A. Cattell, chief engineer, Petroleum and Natural-Gas Division; N. A. C. Smith,

supervising engineer, Petroleum Experiment Station, Bartlesville,Okla.; and E. L. Rawlins, senior petroleum engineer, all of theBureau of Mines. Constructive suggestion& for outlining experi-mental work and correlating the data were made by members of the pipe-line flow subcommittee of the main technical and researchcommittee of the Natural-Gas Department of the American Gas Association and by others active in the technical work of the gasassociation. Special acknowledgments are made to H. C. Cooper,Hope Natural Gas Co., Pittsburgh, Pa.; H. D. Hancock, Henry L.



3

Doherty & Co., New York; G. H. Baird, Cities Service Gas Co.,Bartlesville, Okla.; E. F. Schmidt, Lone Star Gas Co., Dallas, Tex.;R. W. Hendee, Oklahoma Natural Gas Corporation, Tulsa, Okla.;T. R. Weymouth, Columbia Gas & Electric Corporation, New York;

F. M. Towl, Southern Pipe Line Co., New York; E. A. Clark,Columbia Gas & Electric Corporation, Pittsburgh, Pa.; and Dr.Edgar Buckingham, National Bureau of Standards, Washington,D. C.

Work on this problem was done in cooperation with the Natural-Gas Department of the American Gas Association. The splendidcooperation and willing assistance rendered by the operating com-panies and their representatives in permitting use of their pipelines and assisting in the tests are gratefully acknowledged.

The manuscript was reviewed constructively by G. H. Baird,Cities Service Gas Co., Bartlesville, Okla.; E. S. Burnett, Bureauof Mines, Amarillo, Tex.; R. A. Cattell, Bureau of Mines, Wash-ington, D. C.; Joseph Chalmers, Shell Petroleum Corporation;Houston, Tex.; H. D. Hancock, Henry L. Doherty & Co., New York;R. W. Hendee, Oklahoma Natural Gas Corporation, Tulsa, Okla.;B. E. Lindsly, Petroleum Administrative Board, Department of the In ter ior , Washington, D. C. ; Ben jamin Mi l le r , Henry L .Doherty & Co., New York; H. C. Miller, Bureau of Mines, SanFrancisco, Calif.; E. L. Rawlins, Bureau of Mines, Bartlesville,Okla.; H. P. Rue, Bureau of Mines, Laramie, Wyo.; E. F. Schmidt,Lone Star Gas Co., Dallas, Tex.; T. R. Weymouth, Columbia Gas& Electric Corporation, New York; F. M. Towl, Southern PipeLine Co., New York; and Gustav Wade, Bureau of Mines, Dallas,Tex. Special acknowledgment is made to Mabel E. Winslow,Bureau of Mines, for edit ing and indexing this report and to

J. M. Seward, Bureau of Mines, for preparation of the illustrations.The State of Oklahoma cooperated in the study.

DEVELOPMENT AND REVIEW OF FORMULAS FOR FLOW OFNATURAL GAS THROUGH PIPE LINES

The data and formulas for determining the flow of natural gasthrough pipe lines were reviewed carefully and comprehensively.Many formulas are given in the literature, and results using thedifferent formulas vary widely. To ascertain the theoretical dif-ferences between these formulas the literature pertaining to flowof fluids was studied thoroughly. This study brought out the factthat certain assumptions must necessarily be made in the deriva-

tion of the fundamental formula for the flow of compressible fluidsand that additional assumptions are required before the funda-mental formula can be transposed into a practical form for deter-mining the flow of natural gas through pipe lines.

The flow of fluids is a broad subject, and it is beyond the scopeof this report to enumerate the contributions of its many investi-gators, but it is well to bear in mind that the laws and expressionsfor fluid flow were not developed by one individual or within ashort period of time.



4

GENERAL CONSIDERATION OF FORMULAS FOR FLOW OF NATURAL GASTHROUGH PIPE LINES

The mathematical development of the general formula for theflow of natural gas through pipe lines is given in appendix 1.

The assumptions made in the derivation of the general equationas well as the conditions to which the formula applies are discussedin detail. Briefly, the assumptions and conditions relating to theflow through a length of pipe line are:

(1) No work is done upon the fluid by external means, for example, by apump.

(2) The flow is steady; that is, the same weight of gas passes each point inthe pipe line during an interval of time.

(3) The flow is-isothermal; that is, the temperature of the gas remainsunchanged throughout the length of the line.

(4) Natural gas behaves according to Boyle’s law, which states that atconstant temperature the volume occupied by a gas is inversely proportionalto the absolute pressure (P1V1=P2V2=K).

(5) There is no difference in elevation throughout the length of the line.

The general equation for the flow of natural gas through pipelines (see appendix 1) is:

(23)

whereQ = volume, cubic feet of gas per hour at pressure and temperature bases

of Po and To:To= temperature basis defining a cubic foot of gas, degrees F. absolute;Po=pressure basis defining a cubic foot of gas, pounds per square inch

absolute;K = constant, 1.6156;P2 = inlet pressure, pounds per square inch absolute;P2 = outlet pressure, pounds per square inch absolute;d = internal diameter of pipe, inches;G = specific gravity of gas (air=1.000);

T = temperature of flowing gas, degrees F. absolute;L = length of pipe, miles;f = coefficient of friction.

The general equation (equation 23) is expressed in several modi-fied forms depending upon the factors (such as the coefficient of friction and temperature and pressure bases) that are includedwith the constant K to give a different constant, K,. A commonform of the modified general equation is

(24)

The most important fundamental difference between the manyformulas for the flow of natural gas through pipe lines lies in the

evaluation of the coefficient of friction; therefore the followingdiscussion is limited to the coefficient of friction, the factors influ-encing it, and the methods that have ‘been used to express it.

In the development of many of the formulas for flow of gasthrough pipe lines little analytical consideration was given to thecoefficient of friction. The experimental values obtained for thecoefficient of friction often have not been correlated with conditionsproducing them, and important factors influencing the coefficientwere overlooked or given no consideration. As a result many of



5

the formulas that have been offered for calculating the flow of gasthrough pipe lines, although giving accurate results under certainlimited conditions, do not define these conditions exactly andusually are in error when applied under other conditions,

Formulas have been developed that express the coefficient of friction in terms of one or more of the variable factors that influ-ence the flow. The conditions that affect the coefficient of frictionare numerous and variable, and to predict the exact influence thatcertain combinations of conditions have on the coefficient of fric-tion is impracticable. For example, the roughness of the internalwalls of the pipe is a variable element which is exceedingly difficultto incorporate into a formula.

The methods of treating or expressing the coefficient of frictionwill be considered in three general classifications. Some investiga-tors assumed that the value obtained for the coefficient of friction,from experiments on one pipe of given material and roughness.would apply to all pipes and substituted this value in the generalformula (equation 23); some considered the coefficient as vary-ing with the diameter of the pipe; and others have expressed thecoefficient of friction as a function of the ratio of the diameter (D)of the pipe multiplied by the density (S) and velocity (U) of the

gas to the viscosity of the gas (Z) or (Reynolds’ cr iter ion)

A general discussion of the Reynolds criterion and its applicationto flow problems’ is given in appendix 2.

FORMULAS FOR CALCULATING PLOW OF NATURAL GAS

To classify the gas-flow formulas the three following subdivi-sions are made: (1) Formulas in which the coefficient of friction

is constant; (2) formulas in which the coefficient of friction isa function of the diameter; and (3) formulas in which the coeffi-cient of friction is a function of the Reynolds criterion.

As previously stated, equation (23) forms the basis of formulasfor computing the flow of natural gas through pipe l ines. Thesame temperature and pressure bases seldom were used in any twoformulas, and often it was impossible to find exactly what baseswere used. To present the formulas as clearly as possible eachformula is given in the form presented in the literature and thenreduced to the form of equation (23) or a modified form, such asequation (24), with Q in cubic feet per hour at 14.4 pounds per square inch absolute pressure and 60o F. temperature, flowingtemperature 60o F., and specific gravity 0.600 (air = 1.000). Unless

specifically explained all other symbols used in the listed formulashave the same significance as those in equation (23).The following groups of formulas by no means contain all that

have been written, but they are representative of the three typesof equations that have been used for calculating the flow of gasthrough pipe lines. The first two groups contain formulas writtenexclusively for the flow of gas. Their coefficients of friction usuallyhave been determined experimentally from the flow of gas or air.The third group contains formulas whose coefficient of friction



6

depends primarily on the value of the Reynolds criterion. Theydiffer from each other in the value of the coefficient of frictionas obtained from empirically determined relationships between thatcoefficient and the Reynolds criterion.

FORMULAS WHERE THE COEFFICIENT OF FRICTION IS CONSTANT

Formulas having a constant coefficient of friction are given asfollows.

The Cox formula4 is

where Q1 = cubic feet per hour of 0.65 specific gravity gas at assumed5 basesof 14.7 pounds per square inch absolute pressure and 60o F. temperature andI = length of pipe, feet.

In a standard form similar to equation (24) the units of which

have been given previously the Cox formula may be written

The Rix formulao is

where Q1 = cubic feet of gas per hour at assumed bases of 14.7 pounds per square inch absolute pressure and 60oF. temperature and l = length of pipe,feet.

In the standard form previously mentioned the Rix formula is

The Towl formula’ is

where Q1 = cubic feet of gas of 0.59 specific gravity per hour, pressure base of 14.65 pounds per square inch absolute, temperature base of 50o F., and flowingtemperature of 32o F.

In the standard form the Towl formula becomes

The Towl formula in its original form, as stated above, makesthe coefficient of the formula constant for all diameters of pipe andflow conditions. However, Forrest M. Towl has advised the writers

that he proposes a variable value for the flow coefficient of theformula, depending upon the size of. the pipe and the amount of gas flowing.



7

The Pittsburgh formula 8 is

where Q1=cubic feet of gas per hour of 0.60 specific gravity at assumed

bases of 14.7 pounds per square inch absolute pressure and 60º F. temperatureand l =length of pipe, feet.

In the standard form the Pittsburgh formula becomes

FORMULAS WHERE COEFFICIENT OF FRICTION IS EXPRESSED AS SOME

FUNCTION OF DIAMETER

Formulas having the coefficient of friction expressed as somefunction of the diameter are given as follows,

The Oliphant formula 9 is

In the standard form the Oliphant formula is

The Unwin formula 10 is

where Q1= cubic feet per second at pressure P1;I)= internal diameter of pipe, feet;

U 1= initial velocity of gas, feet per second;

g = acceleration due to gravityf

=32.17 feet= thermodynamic constant

per or air = 53.33.

second per second;

Unwin determined the following value for the coefficient of fric-tion from tests on the Paris compressed-air mains;

However, he considered the interior of these mains “excep-tionally smooth” and allowing some weight to other gas experi-ments he presented the following expression for the coefficient of

fr ict ion.



8

Converted into the standard form, the Unwin formulas usingthe two values of the coefficient of friction become respectively

and

Weymouth’s formula 11 is obtained from the general formula

(equation 23) by substitut ing for the value of the coefficient

of friction, giving

where Q1=cubic feet per hour at temperature and pressure bases of T o and P o .

Reduced to the standard form, the Weymouth formula becomes

The California formula 12 (modified Weymouth formula) is

where Q1=cubic feet per hour at a pressure base of 14.65 pounds per squareinch absolute and a temperature base of 60° F.

Reduced to standard form the California formula becomes

The Spitzglass formula 13 is

where Q1= cubic feet ‘per hour at a pressure base of 14.7 pounds per squareinch absolute and a temperature base of 60” F.

In the standard form the Spitzglass formula becomes

FORMULAS WHERE COEFFICIENT OF FRICTION IS EXPRESSED AS A

FUNCTION OF REYNOLDS’ CRITERION

The use of the Reynolds criterion, to determine the coeffi-

cient of friction in developing pipe-line flow formulas is discussed



9

in detail in appendix 2. Some of the formulas expressing the coef-ficient of friction in terms of empirically established relationshipsbetween that coefficient and the Reynolds criterion (or relations

of a similar type) are now reviewed briefly.The Fritzsche formula is 14

where A =numerical constant depending on units,d =internal diameter of pipe,S =density of air (or gas),U =velocity of flow.

Modified negligibly and expressed in English units 15 equation(26) becomes

Substituting the value of f from equation (28) in equation (22)(see appendix 1),

The Lees formula 16 is

where =an expression for the coefficient of friction= f (see appendix 2).

V =kinematic viscosity (absolute viscosity divided by the density).

In terms suitable for substitution into the general formula (equa-tion 23) and converting from kinematic to absolute viscosity, Lees’equation becomes,

The White formula 17 is

where F=R in equation (31).

In terms suitable for substitution in the general formula (equa-tion 23), equation (33) becomes

The McAdams and Sherwood formula 18 is

where z =absolute viscosity, centipoises;d =internal diameter, inches;s =specific gravity (water=1.000).



10

In terms suitable for substitution in the general formula (equa-tion 23) equation (35) becomes

FLOW TESTS OF COMMERCIAL PIPE LINES

PURPOSE OF TESTS

A review of the literature on the flow of gas through pipe linesshowed that data applicable to the pipe and operating conditionsnormally used in the high-pressure transportation of natural gaswere insufficient to establish the relative accuracy of the manyformulas that have been developed for calculating the flow of natural gas. Therefore additional experimental data and tests under conditions of actual operation were considered essential before anyrecommendations could be made relative to the value and limita-

tions of the several formulas. Numerous data were analyzed care-fully to determine fundamental relationships between variablesthat affected the flow of gas through pipe lines; however, the inves-tigation was primarily an engineering study of pipe lines used inthe long-distance transportation of natural gas and of the factorsthat influenced the flow through each line tested.

Tests of the flow of gas through pipe lines were desired under as many different operating conditions as possible; however, therewas some difficulty in finding lines operating under various condi-tions which were adaptable to a field study.

Most of the pipe lines tested were comparatively new; however,several old lines were selected so that comparisons of flow datacould be made on lines of different ages. Several pipe lines con-

taining condensates or rust scale were selected to show the effectsof these obstructions on the flow of gas. The operating and deliver)records of the various companies showed that the amount of leak-age from most of the lines selected was small and did not exceedthat to be expected from pipe lines in good condition. On a num-ber of pipe lines the amount of leakage had been determined by the" shut-in pressure-drop " method.” There were no regulators or other mechanical devices for reducing pressure in any of the sec-tions’ of pipe lines tested.

An attempt was made to obtain sect ions of pipe lines equippedwith orifice flanges at the two ends, so the volume of gas passingthrough the test section might be measured at both the inlet andoutlet. In many sections, however, only one orifice flange was avail-

able, and therefore the rate of flow could be measured at only oneend of the section.In each test it was desirable to select some periods during which

the conditions of flow were steady and other periods when both theinlet- and outlet-line pressures increased or decreased at the samerate. It was desired also that the chemical composition of thenatural gas in the pipe line should remain constant during a test.



11

Special precautions were taken to select pipe lines, or sections of pipe lines, without laterals so that no gas entered or left the linebetween the inlet and outlet.

METHOD OF MAKING FLOW TESTS

Considerable thought was given to developing a method of con-ducting pipe-line flow tests so that their results might be of themost practical value to the largest number of companies and indi-viduals concerned directly with the high-pressure transportationof natural gas. The tests did not interfere with the operation of the lines. This was important from a commercial standpoint andwas necessary in order that the results of the tests be as nearly aspossible true criteria of the flow in the lines.

PREPARATION FOR TESTS

A thorough inspection and survey of the pipe line preceded therecording of any flow observations. Special attention was givento the following :

(1) Topography of the country traversed by the line.(2) Elevation of the inlet and outlet.(3) Number of drips, type of drip construction, and the amount of con-

densate removed. (All drips were blown periodically during a test.)(4) Number of river crossings, with size and length of pipe used and type

of construction of each.(5) Condition of orifice plates. The orifice plates were inspected, and only

orifices with edges in good condition and of micrometered diameter were used. Wherever possible the internal diameter of the pipe atthe orifice was micrometered.

(6) Number and size of main-line valves.

APPARATUS AND DATA

The data required for a flow test of a transmission line consistedof the pressures at each end of the section being tested, rates of gas flow, and temperatures and specific gravities of the flowing gas.

The static pressures were measured with dead-weight gages inal l of the flow tests, as a study of the accuracy of spring gagesshowed that they were not reliable enough for work of this nature.The dead-weight gages were made from ordinary dead-weight gagetesters constructed to measure the pressures directly by balancingwith calibrated weights the pressure exerted against a small pis-ton. They are sensitive to within 0.1 pound per square inch. Thedifferential pressures at the orifices were measured with water manometers. This offered the best means of observing the differen-tial pressure, for any pressure change is indicated more quickly ona direct-connected manometer than on a mechanical recording in-strument. Figure 1 is a diagram of the dead-weight gage andmanometer used to observe the static and differential pressures.

Where orifices were installed at both ends of the line being testedthe differential and static pressure readings were observed simul-taneously at the two ends at intervals of 3 or 4 minutes. Whereonly one orifice was installed the static pressure was taken at bothends of the line, but the measurement of the rate of flow could be



12

obtained at only one end. Where telephones were available at eachend of the section hourly checks of carefully adjusted watches weremade so that the data could be observed at both ends as nearlysimultaneously as possible. Usually the observations were takenfor periods of 7 or 8 hours for 2 days. On some lines where a

greater variety of flow conditions was obtainable the tests extendedover a longer period.

Copper tubing and “compression-ring” fittings were used to con-nect the water manometer to the pressure taps. The location of these taps with respect to the orifice plate varies; that is, in onetype of installation the taps are 23 pipe diameters upstream and



13

8 pipe diameters downstream from the orifice; in the other type,the taps are in the flange holding the orifice plate in place. Where“21- and 8-diameter” connections were used the dead-weight gagewas connected to the upstream tap; where “flange” connections

were used the gage was connected to the downstream tap. Aneroid barome ters ca li brated against mercur ia l barome ters

were used to obtain the barometric pressures at the inlet and outletof the test section,

The temperatures of the flowing gas were determined with cali-brated mercurial thermometers inserted in steel thermometer wellsfilled with mercury or oil and extending into the flowing gas.

The specific gravities of the natural gas were determined hourlywith a modified Edwards gravity balance-a suspension-type bal-ance with the balance rod and float fulcrumed on knife edges.

Samples of gas were collected from each pipe line tested. Chemi-cal analyses and determinations of the viscosity and deviation fromBoyle’s law were made of each sample collected.

The length of a line or section of line tested was found fromthe survey data or maps made by the company owning the pipe line. All apparatus installed on a section of pipe line selected for test-

ing was subjected to line pressure and tested for leaks with a soapsolution, and all joints were made gas-tight before any data wererecorded.

METHOD OF CALCULATION

The calculation of the results of the pipe-line flow tests consistedchiefly of determining the volume of gas flowing through the lineby substitution of the observed data in the formulas for flow throughorifices and comparing this volume with those computed from thedifferent pipe-line flow formulas. The selection of a test period for calculation and the determination of average pressures for theperiod selected were considered carefully.

STANDARDS USED IN CALCULATIONS

For the purpose of this report 1 cubic foot of gas is defined asthe amount that will occupy a volume of 1 cubic foot at a pressureof 14.4 pounds per square inch absolute and a temperature of 60o F. All gases were assumed to conform with Boyle’s law. The averageof the temperatures of the gas at the inlet and outlet was used asthe flowing temperature.

Orifice coefficients used in the volume-measurement calculationswere determined according to the methods recommended by themanufacturers of orifice meters. Each coefficient was corrected to

the temperature and specific gravity of the gas observed duringthe interval chosen for calculation.

SELECTION OF TEST PERIODS FOR CALCULATION

Figure 2 i l lustrates the method used in plotting the pressureobservations taken during each test period. The observed static anddifferential pressures at the inlet and outlet were plotted againsttime. These four pressure curves facilitate choosing test intervals



14

when the static pressures are constant and the rate of flow is thesame at both inlet and outlet. In selecting an interval for calculat-ing care was taken also to choose one when the barometric pressure,the temperature, and the specific gravity of the gas were approxi-

mately constant. In figure 2 the interval from 1:00 to 2:00 p.m.

represents conditions of almost constant flow. The test records for this interval showed that from 1:12 to 1:56 p.m. the static pres-sure decreased 0.2 pound per square inch at both the inlet and theoutlet. Periods of time during which the static pressure remainsunchanged are the most desirable for calculations; however, periodsof appreciable length during which the pressures remained exactly



15

constant occurred in very few tests; and during a period of 44minutes, a change of 0.2 pound per square inch in the static pres-sure indicates a flow condition that is almost steady.

Determination of average values of test data. After a test inter-val for calculation was selected the next step was to determine thebest average value of each of the variables observed during thisinterval. The intervals selected varied in length on different linesfrom 15 minutes with only 3 or 4 readings to 3 hours with 45 or more readings. It was necessary first to calculate an average for each time interval between successive readings; these averagesthen were weighted by multiplying each average by the number of minutes in its respective time interval. The products obtained wereadded, and their sum was divided by the total elapsed time inminutes. The method of weighting individual readings is illustrated

in table 1. Actually, weighting was unnecessary in this exampleexcept to show the method, as all of the intervals between readingswere equal, and the value desired could have been obtained bydividing the sum of the average values for the individual timeintervals by the number of time intervals.

Before these weighted average values were used it was neces-sary to apply corrections for known errors in the test apparatus.

The static pressure readings were recorded as indicated by thenominal values of the weights used on the dead-weight gages. Theaverages of these approximate values were corrected to correspondwith the actual values of these weights as determined by carefulweighing on a laboratory balance.

In many of the tests the observed static pressures at the inletand outlet were corrected for the differential pressures across theorifice to obtain the true inlet and outlet pressures of the line.Where the static pressure was taken upstream from the inlet orifice



16

the differential pressure in pounds per square inch was subtractedfrom the static pressure; where the static pressure was taken down-stream from the outlet orifice the differential pressure was added.

CALCULATION OF VOLUMES BY ORIFICE MEASUREMENT

The quantity of gas was computed from the weighted averagevalues calculated for the static and differential pressures by sub-stitution in the orifice-meter formula,

where Q m =metered volume of gas, cubic feet per hour;h =differential pressure across orifice, inches of water;P =static pressure at orifice, pounds per square inch absolute;C =hourly orifice coefficient, applicable to existing temperature and

specific gravity of the gas and on a pressure base of 14.4per square inch absolute and temperature base of 60° F.

The values used for the orifice coefficients (C) were determined

according to the methods recommended by the manufacturers of orifice meters, from a knowledge of the diameter of the orifice, theinternal diameter of the pipe, and the specific gravity and tempera-ture of the gas.20

DESCRIPTION OF PIPE LINES TESTED

A complete description of the pipe lines included in this investi-gation would be more voluminous than the importance of such datawarrants. In the group of lines tested there was wide variation inthe age, amount of leakage, design, and operation, as well as in thespecific gravity and chemical composition of the natural gases.

Almost al l condit ions of internal pipe surfaces from that of newto that of badly corroded pipe were included. Many of the data

relating to the construction, history, and description of the pipelines, the physical condition of the pipe, and the operating condi-tions at the time of the tests affect the analyses of the results of theflow tests. Some of the more important of these data, applying onlyto the sections of the pipe lines used for the tests, are given intable 2. Where the leakage rate per mile had not been determinedexclusively for the section subjected to flow tests it was assumedto be the same as that determined for the entire line. Accurateleakage data were obtained for some of the lines from leakage testsmade by the owners. For many of the lines no record of the actualleakage was available; however, previous to the flow tests theselines were subjected to pressure tests and carefully surveyed for leaks; leaks found were repaired. For a number of those lines

where no actual leakage data were available the pressure to whichthe line had been subjected is given in table 2. In only one test(line 3) was it convenient for the operators, at the time of theflow test, to shut in the line and make a pressure-drop leakage test.

In table 2 the type of drip is designated by simple, complex, andsiphon. Detailed descriptions of these types of drips are givenunder “Removal of liquids from natural-gas pipe lines.”



17

Data of particular value in analyzing the results of flow testsother than those given in table 2 were observed on individual lines.It was impracticable to present all of this information in tabular form, and in appendix 4 individual lines for which such additional

data of value were obtained are described.Tables 3 and 4 contain, respectively, the orifice-meter measure-

ment data for the test period selected for calculation and the descrip-tions of the lines, operating pressures, and rates of flow.

TABLE 2 -Description of drips, valves, leakage, right of way, and age of pipe lines studied

DATA FROM FLOW TESTS OF COMMERCIAL PIPE LINES

Table 3 contains a brief description of the orifice-meter installa-tion on each line and the observed data for calculating the volumeof gas flowing during each test. Some explanation of the data intable 3 is necessary to define clearly the nature and use of thesedata.

The orifice-meter installations were of two general types. The“line” type consisted simply of a direct installation of the orificeflange in the pipe line without any change in the vertical positionof the pipe. These were of single, double, or triple “meter-run”construction. The "meter-run" in a single-line installation usu-ally was an integral part of the pipe line, with or without a by-pass



18

T ABLE 3. Orifice-meter measurement data



19

T ABLE 3. Orifice-meter measurement data -Continued

around the orifice flange. For the double or triple “meter-run”installations of this type, Y or “header” connections were used,lying in the same horizontal plane as the pipe line. The other gen-eral type of ori f ice-meter instal lat ion was the “riser” type, inwhich the “meter-runs,” either single or multiple, were above thehorizontal plane of the pipe line and connected to it at each endby one or more vertical risers. In the riser type the single “run”installations were usually but not always in the same vertical plane

as the pipe line.Some variations from the two general types of installation werefound on a few lines, Meter-runs sometimes were offset at rightangles from the pipe line, and the gas had to flow through two or more addit ional turns in these instal lat ions. Combinations of “line” and “riser” types of meter installations were in use ontwo lines (lines 21 and 22), where the contour of the ground madeit convenient to use only one riser at the meter installation. Al!meter-runs were in a horizontal plane with one exception (line 15)



TABLE 4. Test data on



no of gas through pipe lines



22

which was of the double-line type with the meter runs conformingto the general grade of the pipe line. As a result, the outlet endof the meter-run was about 12 inches higher than the inlet end.

Where meter installations of multiple-run construction were inuse it was considered desirable for test purposes to use only onemeter-run where the delivery of gas through the line would permitits use without incurring conditions unfavorable to accurate mea-surement. Meter-runs not in use during the flow tests were closedby inserting “blind plates” in the or i f ice f langes. Where i t wasnecessary to use two meter-runs at either end of the line orificeplates as nearly the same size as possible were used, and static anddifferential readings were taken on both orifices.

Table 4 contains, in addition to many of the observed data oneach line and test, the metered volumes during each test and thevolumes calculated by the Weymouth flow formula. The volumescalculated by Weymouth’s formula serve as a basis for some gen-

eral comparisons between the 79 flow tests listed in the table, Moredetailed comparisons between the volumes calculated by the severalflow formulas and the metered volumes for the various flow testsare discussed later in this monograph.

FLOW TESTS OF EXPERIMENTAL PIPE LINE

Because of the wide range of flow conditions and variations inconstruction in commercial pipe-line systems it was impossible to And conditions that were exactly comparable. Consequently, it isdifficult to determine from flow tests of commercial pipe lines theexact effects of individual factors on the flow of gas. Even withindividual pipe lines where it was desirable to obtain various rates

of flow the necessary control and regulation of pressures and vol-umes were impracticable. To supplement the data from commer-cial pipe lines a number of tests were made of a small experimentalpipe line.

The chief purposes of these tests were: (1) To find values of thecoefficient of friction at different rates of flow for several small-diameter pipes; (2) to obtain data on the relationship between thecoefficient of friction and the Reynolds criterion; and (3) to obtaincomparative tests showing the effect of complex line construction(pipe line constructed of pipe of several different diameters).

The experimental tests supplement the tests of the commerciallines, and for the purpose of presentation the results of these ex-periments are included with the results, comparisons, and analysesof the results obtained with commercial pipe lines. The experi-mental flow tests check many of the comparisons made from theresults obtained on the commercial lines, where control of the flowconditions was limited and often the effects of some of the indi-vidual factors were difficult to evaluate. They serve also to extendthe range of conditions observed on the commercial lines and par-t icularly to show the effects of certain complex constructionfeatures.



23

APPARATUS AND EXPERIMENTAL PROCEDURE

For the experimental gas-flow study a “dry” gas well of largecapacity having a shut-in pressure of approximately 400 poundsper square inch supplied the gas. The experimental pipe line con-sisted of 5 sections of 3 “joints” each, of 1.03-, 1.38-, 1.59-, 2.07-,

and 2.49-inch internal-diameter pipe screwed together; the proper size swages connected these sections so that the line increased indiameter progressively from 1.03 to 2.49 inches. Figure 3 is adiagram of the experimental pipe line. Connections to the gas wellwere so arranged that without disturbing the five sections of pipeeither end could be made the inlet or outlet.



24

The line was laid in a straight and horizontal position with aminimum of valves and fittings. A covering of dirt was placedover the pipe to decrease the fluctuations of the temperature; tem-perature observations were made at both the inlet and outlet. Each

of the five sections of pipe making up the line had “pressure” tapsspaced 60 feet apart. Static pressures were measured at the inletand outlet of the combined sections, and differential pressures weremeasured over each of the sections and over the entire length of the five sections for each rate of flow. These measurements weremade with dead-weight gages for the static pressures and withhigh-pressure manometers for the differential pressures after asteady rate of flow was obtained. Measurements of the volume of gas were made with a 2-inch “critical-flow meter" (“prover")installed at the outlet of the line.

RESULTS OF FLOW TESTS OF COMMERCIAL PIPE LINES

The purpose of the tests on the commercial pipe lines was todetermine the relative values and limitations of different pipe-lineflow formulas; also to develop a new formula, if none of the currentformulas agreed satisfactorily with the experimental data.

In the section, “Formulas for Calculating Flow of Natural Gas,”numerous formulas were quoted and as far as possible reduced tocommon bases of temperature and pressure as well as to generalterms that are comparable. Formulas where the coefficient of fric-tion either is constant or expressed as a function of the diameter of the pipe may be reduced to a standard form (modified form of equation 23) which permits comparison between the formulas with-out specifying a set of flow conditions, such as is required for for-mulas where the coefficient of friction depends, among other things,upon the volume of gas flowing.

This standard form of equation is

where Q =the rate of flow, cubic feet per hour;C =the coefficient for each flow formula, including the temperature-

and pressure-base conversions, and the specific gravity of the gas;ø(d) =a function of pipe diameter d, inches;

P 1 =inlet pressure, pounds per square inch absolute;P 2 =outlet pressure, pounds per square inch absolute;L =length of pipe, miles.

For the individual formulas which may be converted to the form

of the above equation, the values of both the coefficient (C) and thefunction of the diameter, ø (d), vary. The term com-

mon to all of them, and therefore the formulas of this classifica-tion may be compared by comparing the values of the term Cø (d)for the different formulas.

Table 5 includes a list of the formulas that permit comparisonsby this method, and the value for C+(d) for each formula. The



25

value of C is on an hourly basis and is applicable to a pressurebase of 14.4 pounds per square inch absolute, a temperature baseand flowing temperature of 60º F., and a specific gravity of 0.600.

T ABLE 5. -Formulas where coefficient of friction is either a constant or a function of diameter of pipe

Standard form:

Bases: Q =cubic feet per hour at pressure base of 14.4 pounds per squareinch absolute and temperature base of 60º F. The value o thecoefficient (C) is applicable to gas of 0.600 specific gravity and aflowing temperature of 60º F.

VARIATIONS FROM CONDITIONS FOR WHICH GENERAL FLOW FORMULA ISDEVELOPED

In the derivation of the general formula (equation 23, appen-dix 1) for the flow of natural gas through long pipe lines manyassumptions were made and conditions specified to which the for-mula is applicable. These conditions are not always met in actualoperation, and the variations therefrom often are difficult to con-sider in a formula.

The flow formulas are developed on the assumption that the rateof flow is steady, that is, that the same quantity of gas passes eachpoint of the pipe line during an interval of time. In the commer-cial operation of most pipe lines it is impracticable to maintain

steady pressures and rates of flow, since the delivery is subject tochanges in the demand. Variations in the rate of flow were indi-cated by change in the inlet or outlet static pressure over a periodof time or, where orifice meters were installed at both the inletand outlet, by the difference between the inlet and outlet volumes.The effect of unsteady rates of flow on comparisons betweenmetered volumes and volumes calculated by formulas for the flowof gas through pipe lines is illustrated by line 3, tests 2, 3, and 5(see table 4). In tests 2 and 3 the line pressures were decreasing,



26

indicating withdrawal of the stored gas in the line, and in bothtests the actual measured delivery out of the line was more thanthe input into the line. In test 5 the reverse was true, and gas wasbeing stored in the line; the delivery was 4.0 percent less than the

input into the line. Therefore error is possible in comparing thevolume calculated by a pipe-line flow formula with the meteredvolume when the rate of flow is not constant, especially where thevolume through the line is metered only at one point.

Several other tests where extreme changes in the pressure over the period of the test were observed, such as line 5, test 3, and line 20,test 2, were included in table 4 to show only the effect of changesin pressure on the value of comparisons between metered volumesand volumes calculated from pipe-line flow formulas.

Coefficients of friction ordinarily used in the flow formulas areapplicable for a wall roughness of new steel pipe; and no provisionis made for pipes having interior walls exceptionally rough or smooth. The resistance to flow (frictional resistance) is increasedgreatly in pipes whose interior walls have been made excessivelyrough by internal corrosion. The products of corrosion frequentlybecome dislodged from the walls of the pipe and are deposited inlow places or other favorable points of accumulation in the line.These deposits change the shape of the cross-section of the pipefrom that of circular, and retard the flow of gas. Lines 4, 15, and18 (table 4) are examples of reduced capacity caused by the pres-ence of corrosion products in the pipe.

Liquids in gas lines also increase resistance to flow; they shouldbe removed or, better still, prevented from entering the line. Theflow formulas are applicable only to lines free of liquids. Lines 11,14, and 17 (table 4) are examples of reduced capacity of linescaused by the presence of liquids.

In the derivation of the flow formulas it is assumed that thepipe is horizontal; that is, each point on the pipe line has the sameelevation. Ordinarily a formula for the flow of gas based on thisassumption is satisfactory for commercial design and operation.However, in several of the lines tested the differences in elevationbetween the inlet and outlet were appreciable, and their effect onthe flow could not be neglected, In table 4 notes under “Remarks”indicate the pipe lines where the effect of differences in elevationwas sufficient to be taken into account (lines 8, 25, 26, and 28).Results of flow tests of these lines are discussed in detail under "Differences in Elevation at Inlet and Outlet Ends of a Pipe Line.”

SOURCES OF ERROR IN TEST DATA

The pipe-line flow tests were made during the commercial opera-tion of the pipe lines; consequently it was impracticable to regu-late, for test purposes, either the pressures or rates of flow. Like-wise, on many lines there were undesirable features of constructionand operation which could not be changed to satisfy test require-ments completely. For these reasons data from many of the flowtests of commercial pipe lines were not used to determine theaccuracy of different pipe-line flow formulas.



27

The static pressures were determined with dead-weight gages,and a series of these pressure observations was used to determinethe average pressure during the period of time selected for calcula-tion (see “Method of Calculation“). An average static pressure

can be determined accurately in this way if the rates of change inthe pressure during the period selected for calculation are notappreciable; however, if this rate of change is large there is somequestion as to the accuracy of the calculated average pressure (see“Selection of Pipe Lines“). Therefore, where the change in pres-sure at either the inlet or outlet of the line was large compared tothe difference between the average inlet and outlet pressures calcu-lated for the period the data were not used when comparing pipe-line flow formulas. For this reason line 6 (tests 1 and 2), line 13A(test 2), and line 27A (test 2) were not included in the analysisof data for comparing pipe-line flow formulas. The irregularityof the results for line 6 indicates the importance of obtaining anappreciable drop in pressure between the inlet and outlet, whenchecking the efficiency of a pipe line, especially where the linepressure is either increasing or decreasing.

In several tests there was a surge in the differential pressureacross the orifice (table 3). The effect of this surge on the accuracyof the volume measurement was not known. In the test of line SD,the differential pressure across the orifice surged about 4 inches ina total of 9 inches; this surge was considered to be of such magni-tude as to preclude accurate measurements of rates of flow.

The amount of leakage from the lines studied was small exceptin line 25 (tables 2 and 4) ; this was the only line where the meteredvolume was corrected for leakage.

In the descriptions of individual pipe lines 9 and 27 mention ismade of the locations of the orifice meters with respect to the sec-

tions tested. It is preferable for the orifice meter to be at the inletor outlet of the section under test, rather than to have several milesof pipe between the test section and the meter used to measure thequantity of gas flowing through the line. In the sections of line Slisted as 9A, 9B, 9C, and 9D the metered volume of gas passingthrough each section is taken as that determined by the orificemeter at the inlet or outlet of that section. The location of theorifice meter with respect to the test sections was fairly satisfac-tory for line 27 and the individual sections, 27A and 27B, since theline was new and the leakage very small; however, the pressureconditions were not especially steady.

COMPARISON OF TEST RESULTS AND FLOW FORMULAS

Comparison of the metered volumes to volumes calculated byeach formula listed. in table 5 (as given in table 4 for Weymouth’sformula) would be unduly long. Comparisons between the meteredvolume and that calculated by any of the other formulas given intable 5 may be made easily by means of the relationship betweenthese formulas and the Weymouth formula, given in figure 4. For example, the metered inlet volume from line 1, test 1, may be com-pared with the volume calculated by Oliphant’s formula, instead of Weymouth’s, by referring to the percentage values given in figure 4.



28

From figure 4, for-a 21.4-inch internal-diameter pipe line the volumecalculated by Oliphant’s formula will be approximately 21.3 percentless than that calculated by Weymouth’s formula, therefore thevolume calculated by the Oliphant formula for line 1, test 1, is

1,345,000 cubic feet (table 4) less 21.3 percent or 1,059,000 cubicfeet, and the metered volume of 1,359,000 cubic feet (table 4) isapproximately 28.3 percent greater than the volume calculated bythe Oliphant formula.



29

Comparisons of the metered volumes to volumes calculated byany of the formulas in table 5 may be made as in the precedingexample. From figure 4 it is apparent that for pipe lines above 6inches in diameter the Weymouth formula gives volumes consis-tently larger than those calculated by any of the other formulas,with the exception of the Unwin (A) formula and that the amountof this difference increases with increase in the diameter of thepipe. Furthermore, analysis of the data given in table 4 for theflow tests made under favorable test conditions (tests are listed intable 6) shows that in the majority the metered volume is larger than the volume calculated from Weymouth’s formula. Conse-quently, the difference between the metered volumes and the vol-umes calculated by the other formulas (fig. 4) usually is greater than the difference between the metered volumes and these calcu-lated by the Weymouth formula.

A series of calculations similar to the example previously givenillustrating the Oliphant formula showed that, except for the Wey-

mouth and Unwin (A) formulas, the formulas compared in figure 4gave results so far from the actual, especially for the larger sizesof pipe, as to merit no further consideration in this study.

The development of pipe-line flow formulas showed that all of them are basically alike but vary in the evaluation of the coefficientof fr iction. The general formula (equation 23) as previouslygiven is,

(23)

Differences between the flow formulas lie in the determination of the value to be assigned to the coefficient of friction (f) ; accord-ingly, comparisons can be made readily by reference to thecoefficient of friction:

In the sections, “Variations from Conditions for Which Gen-eral Flow Formula Is Developed” and “Sources of Error in TestData” mention is made of a number of pipe lines and tests whichare excluded from an analysis of the data used to determine theaccuracy of different pipe-line flow formulas. Furthermore, thetests of the complex systems (systems consisting of parallel lines.or of two or more different diameters of pipe, etc.) were not usedin these comparisons between the test results and the flow formulas.However, the tests of line 9 in particular were so arranged as togive valuable data relative to complex systems, and the tests onthis line will be discussed under “Complex Pipe-Line Systems,”

Table 6 is based on the tests selected for determining the accuracyof the Weymouth, Unwin (A), Fritzsche, and Lees pipe-line flow

formulas. For each test the actual coefficient of friction was calcu-lated by equation 23, using the metered volume for the value of Q,together with the observed values (table 4) for the other variables,and comparisons were made to coefficients of friction calculated bythe Weymouth, Unwin (A), Lees, and Fritzsche equations. Thecoefficients of friction calculated from these four equations gavethe closest comparisons to the actual coefficients,

The value of Q in equation (23) varies inversely as the squareroot of the coefficient of friction, and consequently, the percentage



30

difference in volumes compared is much less than the percentagedifference in coefficients of friction. For example, a difference of 10 percent in the coefficients of friction is comparable to a differenceof approximately 5 percent in the volumes.

The data in table 6 show that for the 38 tests the coefficient of friction calculated by the Weymouth formula is closest to the actualcoefficient in 15 tests; the coefficient by the Unwin (A) formula isclosest in 10 tests; by the Fritzsche formula, in 5 tests; and by theLees formula, in 8 tests. In the 38 tests, the coefficient of friction

TABLE 6.-Comparisons of actual coefficientspipe-line flow formulas

of friction and those calculated by various

calculated by the Weymouth formula is within 10 percent of the

actual coefficient in 23 tests; the coefficient calculated by the Unwin(A) formula in within 10 percent in 20 tests; by the Fritzscheformula, in 12 tests; and by the Lees formula, in 13 tests. Con-sidering the 27 tests of the lines larger than 8 inches in diameter (table 6), the coefficient of friction calculated by the Weymouthformula is within 10 percent of the actual in 21 tests; the coefficientcalculated by the Unwin (A) formula in within 10 percent in 15tests, by the Fritzsche formula in 8 tests, and by the Lees formulain 9 tests.



31

The results of the tests made in this investigation show that for the larger-diameter lines, free from condensates and other foreignmaterials and operating under steady flow conditions, the Wey-mouth flow formula agrees with the actual data better than anyof the other formulas discussed in this report, and will give avolume within a few percent of the metered delivery.

RELATIONSHIP BETWEEN COEFFICIENT OF FRICTION AND DIAMETER OFPIPE

Both the Unwin (A) and Weymouth formulas consider the coeffi-cient of friction as a function of the diameter of the pipe.

The Unwin (A) formula for the coefficient of friction

was determined from tests on the flow of

air through compressed air mains.21 The Weymouth formula for

the coefficient of friction was determined from data

collected by Harris on the flow of air in pipes from 1 to 12 inchesin diameter.” The Weymouth formula, embodying the above ex-pression of the coefficient of friction in the general equation for flow of compressible fluids (equation 23) agrees closely with theresults of a test made by Towl23 in 1901 of an g-inch gas line near Buffalo, N. Y. The rate of flow was found to be 221,000 cubic feetof gas per hour, and substituting the observed values in Weymouth’sformula the calculated rate of flow is 221,400 cubic feet of gas per hour.

Figure 5 shows graphically the relationship between the coef-ficient of friction and the diameter of the pipe as expressed by the

Weymouth and Unwin (A) formulas and as determined from thetests of the commercial (table 6) and experimental pipe lines.For some pipe sizes several values for the coefficient of friction

were observed. It is thought that a large part of this variation isdue to the differences in roughness of the pipe walls. There was noevidence of condensate, rust scale, or other obstructions in the pipelines used for plotting figure 5; however, the actual conditions of the interior walls were unknown. It is logical to assume that thedegrees of roughness of these pipe lines were of the order of thatof commercial steel pipe. Figure 6 shows the same relationshipas figure 5 except that the average coefficient of friction for eachof the different sizes of pipe is plotted from the tests on the com-mercial pipe lines. These two figures show that for pipes larger than 6 inches in diameter the Weymouth formula for the coefficientof friction fits the data better than the Unwin (A) formula andthat the curve based on the Weymouth formula satisfies the testdata as well as any curve that could be drawn through the plottedpoints.



32

Values of the coefficients of friction obtained from tests of the6-inch commercial pipe lines and of the 1- and 2 1/2-inch experimentalpipe lines are considerably less than the values obtained from either the Weymouth or Unwin formulas. However, it should be empha-

sized that the observed values for the small-diameter pipe lines

are based entirely on data from two commercial 6-inch lines and thetests of the short 1- and 2 1/2-inch diameter experimental lines. Although these data indicate the relation between the coefficient of friction and diameter of the pipe on smaller-diameter pipe linesthis relationship on smaller-sized lines is not substantiated so wellas for pipe lines over 6 inches in diameter.



33



34

RELATIONSHIP BETWEEN COEFFICIENT OF FRICTION AND REYNOLDS’CRITERION

Figure 7 shows graphically the relationship between the coef-

ficient of friction and the Reynolds criterioncalculated from the 38 flow tests of commercial pipe lines listedin table 6 and discussed under “Relationship between coefficient of friction and diameter of pipe.” The data from which the value of

was calculated for each test are contained in table 7. Table 7

also contains values of the coefficient of friction for lated from the general flow formula (equation 23).

each test calcu-

FIGURE 7 - Relationship between coefficient of friction and Reynolds' cri-

terion: ----- mean curve through experimental data on commercial

pipe linesThe plotted points in figure 7 are too scattered to establish a

satisfactory relationship between the coefficient of friction and theReynolds criterion. A mean line (dashed curve, figure 7) has con-siderable slope for smaller values of the Reynolds criterion andgradually approaches a horizontal line for the higher values. Ingeneral, the mean line obtained from flow tests on commercial pipe

lines has characteristics similar to curves obtained by other investi-gators. Curves based on the Lees formula and the works of Darcy,Lander, and McAdams and Sherwood are included in figure 7. Theliterature references to these curves are given in appendix 2.

The curve representing the Lees formula which was based on theexperimental data of Stanton and Pannell is extended in figure 7to include much larger values of the Reynolds criterion than arecovered by Stanton and Pannell. For the smaller values of the Reyn-olds criterion, which in the tests (table 7) generally are for the



35

T ABLE 7.- Data used for studying the relationship between the coefficient of friction and theReynolds criterion on the commercial and experimental pipe-line flow tests



36

smaller-diameter pipes, the curve representing the Lees formulaagrees reasonably well with the curve representing the mean of theobserved friction coefficients for the commercial lines; for thelarger values of the Reynolds criterion there is considerable devia-

tion between the two curves. The curve based on the work of Darcyfor the range of the Reynolds criterion covered satisfies the ob-served data probably as well as the curve based on the Lees for-mula. Although the curve representing the work of Lander is basedon flows through steel tubes the diameters of these tubes weresmall, and it is reasonable to expect higher values for the coefficientsof friction than those observed on the larger-diameter commercialpipe lines. The curve published by McAdams and Sherwood, which

FIGURE 8. - Relationship between coefficient of friction and Reynolds’criterion; -------- mean curve through data on experimental pipe lines

obviously gives values of coefficients of friction considerably larger than the observed coefficients of the commercial pipe lines, is in-cluded in figure 7 primarily to show to what extent the curve repre-senting a relationship between the coefficient of friction and theReynolds criterion for rough pipe flattens at the higher values of the criterion.

Figure 8 presents graphically the relationship between the coef-ficient of friction and the Reynolds criterion calculated from theflow tests of the 1- and 2 1/2-inch-diameter sections of the experi-

mental pipe line. The data from which the values of were

calculated are contained in table 7. The flow conditions, under which the experimental data on the 1- and 2 1/2-inch pipes wereobtained, were subject to precise control and regulation. The less-



37

pronounced scattering of the points plotted in figure 8 comparedto those plotted in figure 7 indicates that better control of the vari-ables and a more satisfactory degree of similarity in conditions

were obtained in the tests of the experimental than of the com-mercial pipe lines. A mean line (dashed curve) drawn through the. experimental

points plotted in figure 8 satisfies the data from the 1- and 2 1/2-inchpipes equally well. Values of the Reynolds criterion in these testsdo not extend over as great a range as in the tests of the commer-cial pipe lines. Over the same range in values of the Reynoldscriterion the mean curves in figures 7 and 8 are comparable as toshape; and probably there is less variation in the coefficient of friction with changing values of the Reynolds criterion for thesmaller-diameter experimental pipes. The values of the coefficientof friction, however, are relatively higher for the smaller-diameter experimental pipes than for the larger-diameter commercial pipelines. Curves based on the Lees formula and the works of Darcy,Lander, and McAdams and Sherwood are included in figure 8.

Considering the data in figures 7 and 8, apparently no one rela-tionship between the Reynolds criterion and the coefficient of fric-tion is applicable to the flow of natural gas over the extremely widerange of conditions observed in practice. Probably the reason liesin the absence of sufficient similarity of conditions (see appendix 2).In figure 7 it appears that numerous and more or less parallelcurves, depending upon the extent to which individual sets of flowconditions are similar, could be drawn through the data from thecommercial tests. On individual pipe lines or on a group of linesunder certain l imited conditions some function of the Reynoldscriterion serves as a suitable means of determining the coefficientof friction or analyzing the flow data.

The apparent slope of the curves representing the relationshipbetween the coefficient of friction and the Reynolds criterion, if thedata plotted in figures 7 and 8 are sufficient criteria, leads to arather important general deduction pertaining to the applicationof the Reynolds criterion as a means of determining the coefficientof friction in the commercial transportation of gases. It is apparentfrom the data plotted in figures 7 and 8, and from the data of numerous experimenters, that the coefficient of friction, for largevalues of the Reynolds criterion and for pipes of a roughness com-parable to that of commercial steel pipes, changes very little withchanges in the Reynolds criterion, and this change apparentlybecomes negligible for extremely large values of the Reynoldscriterion. In the commercial transportation of natural gas the value

of the Reynolds criterion usually is large, and unless the pipe hasextremely smooth internal walls the coefficient of friction is affectedvery little by changes in the value of the Reynolds criterion. Toillustrate, the mean line in figure 7 shows that within the range of

from 500,000 to 5,000,000 the change in the value of f was

about 6 percent; for the curve suggested by McAdams and Sher-

wood (f ig. 7) this change over the same range of was



38

approximately 1 percent; and the curve representing the Lees for-mula, which is based on smooth brass tubes, when extended over

the same range of gives a change in f of approximately 26

percent.It is possible that an equation or a group of equations expressing

a relationship between the coefficient of friction and the Reynoldscriterion could be obtained from an extensive research includingappreciable lengths of pipes of several diameters and internal wallroughnesses and a wide range in rates of flow of several differentnatural gases; however, the application of such an equation, or group of equations, would be exceedingly complex mathematicallyand limited in practical use.

COMPLEX PIPE-LINE SYSTEMS

Most natural-gas pipe-line systems do not consist of a single

length of one-diameter pipe but rather of sections of different-diameter pipes or of sections paralleled or looped with other lines. As the demand for gas increases it is common practice to parallelsections of the system with pipe of the same or different diameter.Extensions of old systems often are constructed of a diameter of pipe different from the original. Ever-changing conditions of boththe source of supply and the demand have resulted in complex pipe-line systems, the design of which requires special handling of theflow formulas. The commoner types of complex systems, withexamples of calculation and formulas used, are discussed in thefollowing.

FORMULAS FOR COMPLEX SYSTEMS

The general principle involved in the solution of complex pipe-line problems is to convert the various lengths and diameters of pipe used in the complex system to equivalent lengths of a commondiameter or to equivalent diameters of a common length. Fromthese determinations the dimensions of a single line with a deliverycapacity equivalent to the complex system is obtained. Conversionformulas are derived from the pipe-line flow formulas on the basesthat for given inlet and outlet pressures, specific gravity, and tem-perature the volume of gas flowing depends upon a definite relation-ship between length and diameter of the pipe line. In other words,when the pressures, temperature, and specific gravity are constantthe same quantity of gas will be delivered for numerous combina-tions of length and diameter of pipe. The following derivations arebased on Weymouth’s pipe-line flow formula.

Mathematically, the flow of a given quantity of gas under thesame pressure and temperature conditions and of the same specificgravity through any two pipe lines with different diameters andlengths is expressed as

(37a)

and

(37b)



3 9

By definition, the values of Q, Po, To, P1, P2, G, and T are thesame in both equations, and by equating the right sides of the equa-tions and canceling the like factors the equality becomes,

or

(38)

where L1= the equivalent length of any pipe of length L1 and diameter d2 interms of diameter d1.

By rearranging equation (38),

(39)

where d1= the equivalent diameter of any pipe of given diameter d1 andlength L1 in terms of any other length L1.

The equivalent diameter for use in calculating the flow of gasthrough a complex system consisting of parallel lines is obtainedas follows. The volume flowing through the system is

(40)

where d1, d2, ...d11 are the diameters of the individual lines.

TABLE 8. - Equivalent lengths of various diameter pipes in terms of other diametersfor use in Weymouth's formula



40

The equivalentdelivery capacity

o r

diameter (do) of a single line that has the sameas that of the parallel lines is

( 4 1 )

This calculated value of the equivalent diameter (d,) may be sub-stituted directly into the Weymouth formula for computing theflow.

Table 8 (p. 39) gives the equivalent lengths of various diameter pipes in terms of other diameters and is convenient in calculatingproblems dealing with complex systems.

EXAMPLES OF COMPLEX SYSTEMS

The preceding formulas and their application to pipe-line cal-culations are illustrated by the following examples.

Example 1 (Series System)Given : A pipe line consisting of 10 miles of 15.5-inch internal-diameter

(I.D.) pipe, 15 miles of 17.5-inch I.D. pipe, and 25 miles of 19.4-inch I.D. pipeconnected in series.

And P1 = absolute inlet pressure, 400 pounds per square inch (gage pressureplus barometric pressure) ;

P2,=absolute outlet pressure, 150 pounds per square inch (gage pressureplus barometric pressure) ;

T = absolute flowing temperature, 530oF. (70o F. plus 460o F.);G =To=

specific gravity of the gas, 0.645 (air=1.000);absolute temperature basis, 520oF.;

Po = absolute pressure basis, 14.4 pounds per square inch.To determine: Q (rate of flow, cubic feet per hour).Problems dealing with this type of pipe-line system may be computed by

first determining the equivalent length of each section, using a commondiameter. The system is then equivalent to a continuous line of one diameter having a length equal to the sum of the equivalent lengths of the severalsections. Equivalent lengths of the several sections of common diameter arecalculated by equation (38).

Choosing 17.5 inches, the diameter of the 15-mile section, as the commondiameter, the 10-mile section of 15.5-inch pipe by equation (38) is equivalent to

L1=19.10 miles of 17.5-inch-diameter pipe, and the 25-mile section of 19.4inch I.D. pipe is equivalent to

L1=14.43 miles of 17.5-inch-diameter pipe.The entire system is equivalent to the sum of these lengths (19.10+14.43+

15) or 48.53 miles of 17.5-inch pipe.

The final step in the problem of computing the rate of flow is then accom-plished by applying Weymouth’s formula,

Substituting the values from the example,

Q=3,876,000 cubic feet of gas per hour.



41

Example 2 (Looped System)

Given: A pipe line consisting of 50 miles of 19.4-inch I.D. pipe, looped for adistance of 20 miles with 20 miles of 15.5-inch I.D. pipe and with the samepressures, temperatures, and specific gravity as example 1.

To determine: Rate of flow, cubic feet per hour.The 20-mile looped section is converted into a single line 20 miles long with adiameter equivalent to the two parallel lines of 19.4- and 15.5-inch pipe, asfollows:

By equation (41), for equal lengths of looped lines of different diameters,

= 22.86 inches, equivalent diameter of a singleline 20 miles long.

The system is now equivalent to two sections in series, one consisting of 20 miles of 22.86-inch pipe, and the other of 30 miles of 19.4-inch pipe. Either section can be computed to an equivalent length of pipe of the same diameter asthe other section. To convert the 20-mile section of 22.86-inch pipe to anequivalent length of 19.4-inch pipe equation (38) is used, and

L1=8.33 miles of 19.4-inch pipe, equivalentto 20 miles of 22.86-inch pipe.

The complete system, in terms of a single line of one diameter, consists of 30+8.33 or 38.33 miles of 19.4-inch pipe. Substituting these values, together with the given values of pressure, temperature, etc., in Weymouth’s formula,the rate of flow is

Q=5,741,000 cubic feet of gas per hour.

When the looped section consists of lines of unequal length the equivalentdiameter for one of the lines is computed for a length equal to the other lineby equation 39 before the section is converted into an equivalent single line of one diameter (equation 41).

Example 3 (Special Complex System)

The following example is given to illustrate not only the application of theformulas to complex pipe-line systems but also the importance of connectionsor “tie-overs” between certain looped systems.

The system to be considered consists of two parallel lines (A and B)connected at the inlet and outlet. Line A consists of 22 miles of 12.25-inchI.D. pipe and 16 miles of 15.50-inch I.D. pipe. Line B consists of 22 miles of 15.50-inch I.D. pipe and 16 miles of 12.25-inch I.D. pipe. The arrangement of the pipe is shown in figure 9.

Figure 9.- Layout of parallel system for illustrating the calculation of equivalentdiameters and lengths and also the importance of cross connections in some

paralleled systems (Example 3).

For line A the equivalent length in 15.50-inch pipe of the 22 miles of 12.25-inch is, by equation (38),

L1=77.16 miles of 15.50-inch pipe,



42

or line A is equivalent to 77.16 miles plus 16.00 miles or 93.16 miles of 15.50-inch pipe.

For line B, by equation (38), the equivalent length in 15.50-inch pipe of 16 miles of 12.25-inch pipe is

L1=56.12 miles of 15.50-inch pipe,

or line B is equivalent to 56.12 miles plus 22 miles or 78.12 miles of 15.50-inchpipe.

The entire system is now the equivalent of two looped lines of 15.50-inchdiameter, one 93.16 miles long and the other 78.12 miles long. The next step isto determine the equivalent diameter of one of the lines when it is made of equal length to the other; or the 78.12 miles of 15.50-inch pipe is the same as93.16 miles with an equivalent diameter determined by equation (39) of

d1=16.02 inches.

The equivalent of the system now consists of two looped lines each 93.16miles long, one with a diameter of 16.02 inches and the other 15.50 inches.The equivalent of this system is reduced still further to one line 93.16 mileslong, with a diameter determined by equation (41), or

The system as shown in figure 9 is now equivalent to a single line 93.16 mileslong with an internal diameter of 20.47 inches.

The capacity of this particular system may be increased by connecting thetwo lines together at point , figure 9, so that the system may be considered astwo looped lines, both 38 miles long, one 15.50 inches in diameter and the other 12.25 inches in diameter. By equation (41) the equivalent diameter of thetwo looped lines is,

In other words, the system constructed with a “tie-over” at point (figure 9)is equivalent to a single line 38 miles long with a diameter of 18.17 inches.For purposes of comparison to the system without the “tie-over” the 38 milesof 18.17-inch pipe, by equation (39), is equivalentpipe. The advantage in this example of

to 93.16 miles of 21.50-inchconnecting the system at point

(figure 9) is indicated, therefore, by the difference in delivery capacity betweena line 93.16 miles long with an internal diameter of 21.50 inches and a line of the same length with an internal diameter of 20.47 inches, all other factorsinfluencing the flow being the same in both lines.

TESTS OF COMMERCIAL COMPLEX SYSTEMS

Several pipe lines on which the writers made flow tests containedsections of different diameter pipe, and therefore it was necessaryto determine equivalent lengths before the volume flowing couldbe computed by the flow formula. On line 9 (table 4) data wereobtained that show the accuracy of calculations of equivalentlengths.

The general design of line 9 is shown in figure 10; pressure mea-surements were made at the inlet and outlet meters and at thepoints marked A and B. The section between the inlet meter andpoint B (line 9B) consisted of 7.707 miles of single 12.125-inchpipe and 5.882 miles of double 12.125-inch pipe. The equivalentlength of this section (line 9B) in terms of 15.25inch-diameter pipe



43

is 31.18 miles. The section from point B to the outlet meter (line 9C) consists of 31.39 miles of 15.25-inch pipe.

Since the calculated equivalent length of section 9B expressed interms of 15.25-inch pipe is almost equal to the actual length of the15.25-inch-diameter section (line 9C), the difference between thesquares of the inlet and outlet pressures (P 1

2-P 22) for line 9B

should be nearly the same as that for line 9C whenthe volume flowing in the two lines or sections isthe same. Two test periods were selected when theflow was steady, as indicated by almost the samequantity of gas passing through the inlet and out-let meters. In the first test, the value of P1

2-P22

for line 9B was 133.0 and for line 9C 131.8. Theother test gave a difference in the squares of the in-let and outlet pressures on line 9B of 187.4 andon line 9C of 183.3. The close agreement in the

values of the quantity (P1

2

-P2

2

) for lines 9B and9C, observed in the two tests, indicates that themethod of calculating equivalent lengths based onWeymouth’s pipe-line flow formula is satisfactoryfor practical purposes.

TESTS OF EXPERIMENTAL PIPE LINE CONSISTING OFSEVERAL DIAMETERS OF PIPE

A series of tests was made to determine the rela-tive capacities of a pipe line for different rates of flow when the gas was flowing in the directionof increased diameters of pipe and when the direc-

tion of flow was toward reduced diameters. Thesetests were conducted on a small experimental pipeline consisting of three “joints” each of 1.03-,1.38-, 1.57-, 2.07-, and 2.49-inch-diameter pipe con-nected together as shown in figure 3 and describedunder “Flow Tests of Experimental Pipe Line.”The drops in pressure over the entire system andover the 60-foot lengths of each of the five sectionswere determined for the different rates of flow andpressures. The inlet pressure ranged from 5.0 to225 pounds per square inch, and for each inletpressure the outlet pressure was varied to giveseveral different rates of flow. The linear velocityof the gas in the several tests ranged from 6 to140 feet per second. During the tests steady flowconditions were attained before any observations were recorded.

To show the relative capacities of the system for the two direc-tions of flow the differences between the square of the inlet pres-sure and the square of the outlet pressure, (P1

2-P22), are com-

pared to the corresponding rates of flow. The relationships betweenP1

2-P22 and the delivery from the line for the two directions of



44

flow are plotted on logarithmic cross-section paper in figure 11.The results indicate that under the conditions of the tests and for the same value of P1

2 -P22 a greater volume of gas per hour was

flowing through the line with the direction of flow toward increas-

ing pipe diameters than when the direction of flow was towarddecreasing pipe diameters. This difference probably can be attrib-uted to the effect of the swages, because when the direction of flow was toward increasing diameters some of the velocity headwas converted into static pressure at each swage and the staticpressure immediately beyond the swage was higher than the pres-sure on the upstream side. This condition was observed in all of

FIGURE 11.-Flow of gas through experimental pipe line consisting of several different diameters of pipe

the tests with the flow in the direction of increasing pipe diameters,and the magnitude of the pressure increase across the swages de-pended upon the static pressure, rate of flow, and diameter ratioat each swage. This would indicate that the gain in volume withthe flow in the direction of increasing pipe diameters depended

upon the number of swages and diameter ratio at each swage.In figure 12 the volume of gas flowing per hour is plotted against

the corresponding values of P12-P2

2 over each of the 60-foot sec-tions comprising the entire length of the experimental line and for both directions of flow. These data show that regardless of thedirection of flow through each of the 60-foot sections the volumeof gas flowing per hour was the same for the same value of P1

2-P2

2.



45

SPECIAL FORMULAS FOR DESIGNING PIPE-LINE SYSTEMS CONSISTING OFPARALLEL LINES

As a result of the large expansion of the natural-gas industry

almost all markets within economic distance of the developed gasfields have at least one pipe line supplying them with natural gas.

Small increases in the consumers’ demand for natural gas usuallycan be met by increasing operating pressures, but any markedincrease in demand necessitates paralleling sections of the existingsystems. Several formulas have been developed to aid in design-ing such systems. Curves and tables were prepared from theseformulas to simplify the calculations and to facilitate determina-



46

tion of the sizes of pipe to be used in paralleling various diameter lines to give the most economical design. The formulas are listedbelow, and the detailed derivations are given in appendix 6.Examples included here show the detail of application of the severalformulas to problems of design of parallel lines.

In the following discussion the pipe lines are referred to as theoriginal line and the parallel line. The parallel line may or maynot extend the entire length of the original line; however, theparallel line is connected to the original line at its two ends. Thesespecial formulas are derived from Weymouth’s formula and areapplicable under the conditions that after a section of the originalline (or all of it) has been paralleled the temperature and specificgravity of the gas and the pressures at the inlet and outlet endsof the original line are the same as they were before paralleling.

Definitions of symbols used are:

X=portion of length of original line paralleled, expressed decimally;

d=internal diameter of original line, inches;d1=internal diameter of parallel line, inches;Q=rate of flow through the system before paralleling, at base conditions of

pressure and temperature; andQ1=rate of flow through the system after paralleling, at base conditions of

pressure and temperature.

The following formula is used to determine the portion of theentire length of line of a given diameter that must be paralleledwith the same or a different diameter of pipe to increase the volumethe desired amount.

(40)

Equation (49) may be rearranged to give an equation for thevolume flowing through a system for any percentage of length of line paralleled with any diameter pipe.

(50)

Equation (49) also may be rearranged to give an expression for the diameter of the paralleling line that will be required to increasethe flow of gas a given amount for a given portion of line paralleled.

(51)

When the value of X is 1; in other words, when the entire lengthof the original line has been paralleled, the following relationshipmay be written,

(52)



47

When the diameters of the original and parallel lines are thesame equations (49) and (50) simplify to

and

(53)

(54)

EXAMPLES SHOWING USE OF FORMULAS

The following examples are given primarily to show applicationof equations (49), (50), and (51).

Problem 1

Given: A pipe line 75 miles long, internal diameter (I.D.) 19.4 inches,delivering 40,000,000 cubic feet of gas per day. It is desired to increase the

delivery to 50,000,000 cubic feet per day bwith 17.5-inch I.D. pipe while maintaining theparalleling a section of the linesame inlet and outlet pressureson the system. Find the length of the original line that must be paralleled.

Substituting in equation (49),

Thus it is necessary to parallel 53.2 percent (39.9 miles) of the original 75-mileline to increase the delivery from 40,000,000 to 50,000,000 cubic feet, using

17.5-inch pipe.Problem 2

Given: A pipe line 50 miles long, internal diameter (I.D.) 12.25 inches,deliverinparalleled

25,000,000 cubic feet of gaswith 15.5-inch I.D. pipe and th

per day. If 20 miles of the length is

maintained on the system, finde same inlet and outlet pressures are

is paralleled.the delivery from the system after the section


= 31,044,000 cubic feet of gas per day.

Thus the delivery is increased to 31,044,000 cubic feet per day, or there is again of 24.2 percent,

Problem 3

Given: A pipe line 70 miles long, internal diameter (I.D.) 19.5 inches,delivering 45,000,000 cubic feet of gas’ per day. It is desired to increase thedelivery to 63,000,000 cubic feet per day, or by 40 percent, by paralleling 42miles (60 percent of the length) while maintaining the same ‘inlet and outletpressures on the system. Find the required diameter of the parallel line.



48


from whichfrom whichd1= (19.5) (1.114) = 21.7 inches, internal diameter of the

Oparallel line.line.

In the actual design of the line a 22-inch outside-diameter (In the actual design of the line a 22-inch outside-diameter ( .D.).D.) pipe with apipe with awall thick enough to withstand the line pressure with a reasonable safety factor wall thick enough to withstand the line pressure with a reasonable safety factor would be used.would be used.

DESIGN OF PARALLEL LINES

The percentages of length of line that must be paralleled to givevarious percentage increases in gas delivery are shown in table 9

Table 9.- Percentage of length of original line that must be paralleled to givevarious percentage increases in delivery

for different ratios of the diameter of the parallel line to thediameter of the original line. The curves in figure 13 are plottedfrom these data and show the nature of the relations expressed byequations (49), (50), and (51). Some interesting factors andcomparisons pertaining to the design of parallel systems arebrought out by the equations and by the curves in figure 13.

The preceding formulas for solving problems in the design of parallel lines eliminate the numerous computations of the “trial-



49

and-error” methods commonly used. Furthermore, a table or aseries of curves similar to those in figure 13 (based on the equationspresented in this report) will serve the purpose almost as well ascalculations made by the formula , and will reduce materially the

time required to design parallel lines. For example, in problem 3the computed internal diameter of the parallel line is a size notregularly manufactured, and in practice a 22-inch outside-diameter (O.D.) pipe of sufficient wall thickness would be selected; another calculation must then be made, using equation (49) with the cor-rect internal diameter of the 22-inch pipe to determine the actual

FIGURE 13.- Relationship between percentage increase in the delivery and

percentage of the line paralleled for different values of

Note: d1 = diameter of parallel lined = diameter of original line

length of the parallel line. The number of calculations in problemsof this type would be reduced materially by the use of the curves infigure 13. Another advantage in the use of such a series of curvesis that for any immediate increase in gas delivery the length

of the parallel line necessary for different ratios is obtained

readily, and from the curves the additional length of the parallel

line necessary for estimated future increases in gas demand canbe determined with ease.

From equation (51) the minimum portion (expressed decimally)of the length of the original line that must be paralleled for any



50

desired increase in volume equals the arithmetical value of the

quantity and for this condition the diameter of the

parallel line is infinitely large. The column of figures under letter

M in table 9 and the curve in figure 13 for a ratio of infinitely

large give the minimum percentage of the length that must beparalleled to obtain specific increases in delivery; however, for this condition the diameter of the parallel line is infinitely large.In other words, for real conditions the percentage of the length of the original line that must be paralleled by another line for aspecific increase in delivery is greater than the value given under

M in table 9 or by the curve in figure 13 for a ratio infinitely

large.The figures at the bottom of table 9 following letter A are the

percentage increases in delivery by paralleling the entire lengthof a line for different ratios of internal diameter of parallel line

(d1) to internal diameter of original line (d). Obviously for a

ratio of 1 (parallel line same diameter pipe as original line) thedelivery is increased 100 percent when the entire length has been

paral leled. This same ratio requires parallel ing of 74.1 per-

cent of the length of the line to increase the delivery of gas byonly 50 percent.

A group of curves similar to those in figure 13 il lustrates theeconomic advantages of certain types of design in parallel pipe

lines. The following discussion and examples bring out some of the factors to be considered.

For the small values of ( ra tio of d iameter of paral le l line

to original l ine) an increase in value of reduces materially the

length of the parallel line required for any desired increase in

delivery. However, for increasing values of the amount of

change in the length of the parallel line, with change in the

ratio, decreases. For instance, to increase the delivery 10 percent

(see table 9) it is necessary to parallel a lo-inch internal-diameter (I.D.) line for 47.4 percent of its length using 6-inch I.D. pipe

29.7 percent of i ts length using 8-inch I.D. pipe

18.9 percent of its length using 14-inch I.D. pipe

and for 16-inch I.D. pipe the percentage of



51

line paralleled decreases to only 18.3; for still larger values of the

ratio the percentage of the original line required to be paralleled

decreases gradually to a minimum value of 17.35, and for thispercentage the diameter of the parallel line is infinitely large. Inother word:, for a lo-percent increase in delivery of gas a percent-age of the length of the original line greater than 17.35 must beparalleled, and if the ratio of the diameter of the parallel line tothe diameter of the original line is 2 it is necessary to parallel17.7 percent of the length.

The following examples illustrate a method analyzing the eco-nomics of various designs of parallel lines. The costs per mile of different-diameter lines employed in the examples are averagevalues and should be used with caution when applied to specificinstallations. Variations in cost of labor and materials and intopography of the right-of-way preclude any definite cost figures.However, the costs per mile used for different-diameter lines are

comparable and for the purpose of the following analysis are accu-rate enough to indicate certain general trends in the economics of the design of parallel lines.

The problem is to determine the cost of increasing-the deliver]of a pipe line 100 miles long of 15.432-inch I.D. pipe by 20 and50 percent, using different-diameter parallel lines. Table 10 showsthe length of the parallel line required and its cost, using 22-, 20-,

TABLE 10. - Costs of increasing the gas delivery by using different-diameter parallel lines

16-, 12.75- and 10.75-inch pipe. The general trend (table 10) isa decrease in cost as smaller-diameter lines (necessitating paral-leling more of the entire length of the original line) are used.However, for the 20-percent increase in gas delivery and for theparticular cost figures used, the cost of the 10.250-inch I.D. parallelline is slightly more than the cost of the project using 12.188- or 15.432-inch I.D. pipe. For the 20-percent increase in delivery the16-inch line probably would be the best size to use, since its cost isconsiderably less than that of the 22- or 20-inch lines and aboutthe same as that of the 12.75 or 10.75-inch lines. Furthermore,the 16-inch-diameter parallel line has several advantages over the12.75- and 10.75-inch lines, chiefly in that the shorter length of the16-inch line that would be required, compared to the Iength of



52

either of the two smaller sizes, allows more freedom in extendingthe parallel line to take care of future increases in gas demand.The shorter length of line also has the advantage of lower mainte-nance expense.

From table 10 the cost of the 50-percent increase in gas delivery,using the same diameter parallel lines, is proportionately muchless than the cost of the 20-percent increase in delivery. To illus-trate, using 22-inch pipe the average cost of each percent increase

for the 20-percent increase in delivery is = $50,340,

whereas the cost of each percent for the 50-percent increase in

delivery is = $36,600.

However, for any particular gas-transmission system other fac-tors than those considered may enter into determination of themost economical design of parallel lines. For instance, only alimited length of the pipe-line right-of-way may be easily accessibleand satisfactory for laying pipe lines, whereas the remainder of the pipe line may traverse swamps, mountains, or areas of corro-sive soil, where the cost of construction and maintenance is muchhigher, and paralleling this section may not be economical.

CONSTRUCTION AND OPERATING CONDITIONS INFLUENCING

DELIVERY CAPACITIES OF NATURAL-GAS PIPE LINES

The data obtained by the writers from flow tests of representa-tive pipe lines throughout the United States not only have beenhelpful for comparing accuracies of different pipe-line flow for-mulas but also have indicated the influence of different constructionand operating conditions on the delivery capacities of natural-

gas pipe lines. These construction and operating conditions in-clude liquids and rust scale in natural-gas pipe lines, stored gas inpipe-line systems, difference in elevation at inlet and outlet endsof a pipe line, and types of construction and equipment.

LIQUIDS IN NATURAL-GAS PIPE LINES

The presence of liquids in a natural-gas pipe line or pipe-linesystem is one of the most important factors to be considered whenabnormal delivery capacities of pipe lines are analyzed. Theseliquids usually consist of natural gasoline and water. During ayear’s operation few pipe l ines are entirely free from l iquidsthroughout their entire length. During the winter months the tem-perature of the pipe line decreases, and if large volumes of water

vapor condense “frozen lines” may result, which add to the diffi-culties of operation when the maximum capacity of the pipe lineusually is desired.

Instances of "frozen lines" at temperatures considerably abovethe freezing point of water have been called to the attention of thewriters. In the laboratory, several natural gases containing water vapor were subjected to pressure and a solid substance (gas hy-drates) resembling ice was produced at temperatures above 32o F.,



53

depending on the pressure and chemical composition of the gas.Enough study has not been made to determine the composition of the solid substance, the circumstances incident to its formation,and the relationship between the pressure and temperature of formation. A very interesting report on gas hydrates has beenpublished.24

Liquids in pipe l ines tend to conglomerate any other foreignmaterial (sand, dust, rust scale, etc.) present in the pipe line intoa solid mass which partly or entirely fills the pipe and obstructsthe flow of gas.

Flow tests were conducted on several lines known to containliquids to show the extent to which liquids affect the capacities of pipe lines to deliver gas. For the purpose of analyzing the effectof liquids on delivery capacities the metered-deliveries through thelines were compared with those calculated by the Weymouth pipe-line flow formula. On pipe lines known to contain liquids adequatedrip facil i t ies were not provided; consequently, l iquids were in

direct contact with the flowing gas. The following data and testsillustrate the extent to which liquids in these pipe lines have affectedtheir capacities to deliver gas.

In line 11, table 4, part of the gas entered the line directly fromgas wells that were producing water with the gas. Drip facilitiesat these wells were inadequate, and water entered the pipe line.The metered delivery from the line was 41.5 percent less than thevolume calculated by substituting the observed values of pressureand other terms in Weymouth’s formula. A large number of thegas wells connected to the system did not have enough pressure todeliver gas into this line, and it was necessary to compress the gasfrom these wells before it entered the line. Inlet pressures (P1)given in table 4 indicate the discharge pressure of the compressors,

since the inlet of line 11 was taken a short&distance downstreamfrom the compressors. During the period of test 1 the averageinlet pressure was 464.36 pounds per square inch absolute. If theline had been free from water the inlet pressure could have beenreduced to 438.00 pounds per square inch absolute and the samequantity of gas delivered at the same outlet pressure. In other words, with the line free of liquids the discharge pressure of thecompressors could have been reduced approximately 26 pounds per square inch and a saving in the cost of operation effected thereby.

Also, with the line pressure reduced 26 pounds per square inch thewells could produce directly into the line for a longer period.

An ef fort was made to remove the water from this line duringpart of the flow test. A drip 2,000 feet downstream from the inlet

was “blown” for 5 to 8 minutes at 2-hour intervals, and thevolume of gas released in blowing the drip was enough to causea drop in line pressure at the outlet 18 miles away. (See figure 14.) An Indication of the amount of gas wasted due to blowing dr ipshas been given by Rawlins and Wosk25 as follows:

A test of a drip that consisted of a 2-inch inlet from the pipe line, a water reservoir that was 8 inches in diameter and about 10 feet in length, and a



54

L-inch blow-off line showed that 3,194 cubic feet of as was lost in 1 minutewhen the line was blown off at a pressure of 250 pounds per square inch. If thedrip is blown 1 minute each day for 1 year, 1,116,000 cubic feet of gas is lost.

Line 17, table 4, began at an oil-absorption gasoline plant and

contained both water and absorption oil which had been carriedinto the line by the gas. Some of this oil was trapped in the drips,and when the drips were blown the oil came out as a froth resem-bling an emulsion. The metered deliveries from the line for thetwo tests were 11.5 and 39 percent less than the volumes calculatedfrom Weymouth’s formula.

Line 14, table 4, began at a charcoal-adsorption gasoline plant.The gas from the cooling coils entered a small “header” and carriedwith it water condensed during the cooling of the gas. A “blow-off” at the bottom of this header was partly open during operation

FIGURE 14.- Record of pressure observations showing drop In line pressuredue to blowing drips (line 11)

of the plant, and a continuous stream of water flowed from theheader. However, all of the water was not removed at the header,and some of it was carried with the gas into the main line. Water accumulated in many low places in the line not provided with dripsor traps. The metered volume of gas passing through this line was

48 percent less than the delivery, calculated by Weymouth’s for-mula. Although the internal diameter of this line was 12.125 inchesthe delivery of gas under the conditions of the test was equivalentto the delivery that could be obtained through a clean line of onlyabout 9.5 inches internal diameter.

SOURCES OF LIQUIDS IN NATURAL-GAS PIPE LINES

A study of the sources of liquids in the pipe lines tested indi-cated ways of improving operations to prevent liquids from enter-ing natural-gas pipe lines,



55

Gas wells often produce quantities of free water or gasolinealong with the gas. Water may come from gas-producing horizonsor from upper formations not properly “cased off.” Methods of excluding water from gas and oil wells are given in various publica-tions26 of the United States Bureau of Mines. Drips or traps usuallyare provided at each well to collect liquids produced with the gas,but often the size or design of the drip is inadequate and liquidsare carried into the pipe line.

Before natural gas enters a pipe line from a compressor stationor gasoline-extraction plant it should be cooled at least to the low-est temperature on the pipe line so that any condensation of water vapor can take place before the gas enters the pipe line. The cool-ing system should be designed not only to cool the gas sufficientlybut to remove from the gas flow all condensate resulting from thedecrease in temperature. Where the traps or drips are inadequateto collect the condensate it is carried into the pipe line by the gas.

In this investigation several installations were noted where the

gas was cooled insufficiently at the discharge of gasoline plantsbefore it entered the pipe lines, and quantities of water accumu-lated in the lines as the gas temperature decreased from the dis-charge temperature of the gasoline plants. Line 14 is an exampleof such an installation; it began at a charcoal-adsorption gasolineplant and the gas was not cooled sufficiently before it entered theline. The temperature of the gas entering the pipe line was 85 o F.,and large quantities of water collected in the drips near the inletas the temperature of the gas decreased. Line 17 began at an oil-absorption gasoline plant, and the gas was not cooled enough beforeit entered the line. The temperature of the gas in the line about1/2 mile from the discharge of the gasoline plant was between 85 o

and 90o F., whereas about 2 1/2 miles from the plant it had decreased

to 62o

F., which was approximately the temperature at the outletof the line 16 miles distant. Large quantities of water collectedin the drips in the first 3 miles of line from the gasoline plant, andit was considered that all of the water had been removed from theline. However, a low place in the line about 9 miles from the plantand not provided with a drip was tapped, and large quantities of water and absorption oil were found in the line.

CONDENSATION OF WATER VAPOR DUE TO CHANGES IN TEMPERATURE

AND PRESSURE

The maximum volume of water vapor that natural gas can hold(saturation) for a given temperature can be computed from Dal-ton’s law of partial pressures and the vapor pressure of water at

the given temperature. Dalton’s law of partial pressures statesthat the total pressure of a mixture of several gases equals thesum of the pressures each of the individual gases would exert if it were present alone in the space occupied by the mixture. Thepressure exerted by each component of the mixture (partial pres-



56

sure) indicates the portion of the total volume occupied by thatcomponent. At saturation, the partial pressure of the water vapor in a gaseous mixture equals the vapor pressure of water at thetemperature of the mixture. That is, in a gaseous mixture saturated

with water vapor at a given temperature, the partial pressure of the water vapor is independent of the total pressure on the mixtureof gases. Then, for any definite volume (V) of saturated gas, the

TABLE 11.- Percent by volume of waler vapor in saturated gas

FIGURE 15.- Percent by volume of water vapor in saturated gas

volume of the mixture that is water vapor (V w) is determinedfrom the ratio of the partial pressure of the water vapor (P w)tothe total pressure (P) as follows:

In the above equation it must be remembered that the volume of water vapor (V10) t the pressure and temperature bases of thevolume of saturated gas (V). Table 11 and figure 15 show the



57

volume of water vapor (Vw) expressed as a percentage of the totalvolume of saturated gas (V) for different total pressures (P) andtemperatures.

Considering the water-vapor content of natural gas on a weightbasis the weight of water vapor in a given saturated space dependsupon the temperature and is independent of pressure. For example,natural gas, in a container having a volume of 100 cubic feet, willhold the same weight of water vapor when the gas is saturated at

TABLE 12.- Gallons of water per million cubic feel of gasof 14.4 pounds per square inch and 60

oF.

(pressure and temperature basesrespectively contained in gas saturated with

water vapor at indicated pressures and temperatures

FIGURE 16. - Gallons of water per million cubic-feet of gas (pressure andtemperature bases of 14.4 lb. per sq. in. and 60

oF. respectively) contained

in gas saturated with water vapor at the indicated temperatures andpressures

a given temperature, whether the pressure is atmospheric or 500pounds per square inch. If, however, -the container is filled withsaturated natural gas at atmospheric pressure and this volume of gas is then compressed isothermally to 500 pounds per square inchsome of the water vapor will condense; the weight of water vapor that remains in the gas will depend upon the volume occupied bythe gas after compression. For these conditions the weight of water vapor per unit of space occupied by the gas remains the same,

The data in table 12, which are plotted in figure 16, give thenumber of gallons of water per million cubic feet of gas saturated



58

with water vapor (pressure and temperature bases of 14.4 poundsper square inch absolute and 60o F., respectively) when subjectedto the indicated absolute pressures and temperatures. The data intable 12 were determined from the volumes of water vapor ex-

pressed as a percentage of the total volume of saturated gas intable 11 and Avogadro’s law, from which 22.4 liters of water vapor at 760 millimeters of mercury pressure and 0

oC. weighs 18 grams.

As shown in figure 16, a given quantity of gas at saturation andconstant temperature releases one half of its water-vapor contentwhen the absolute pressure of the gas is doubled. Furthermore, for each increase in temperature of 20

oF., the capacity of a given

quantity of gas to hold water vapor is approximately doubled,The relationships between water vapor, pressure, and tempera-

ture are important in the compression and transmission of gas.Referring to figure 16, each million cubic feet of free gas (pressureand temperature bases of 14.4 pounds per square inch and 60o F.respectively) entering the compressors at 100 pounds per square

inch absolute pressure and 60o F. contains 14.3 gallons of water at saturation. If the absolute pressure is doubled the temperaturemust increase to about 80

oF. to prevent condensation of the water

vapor. However, if the temperature is maintained at 60o F., thenat 200 pounds per square inch absolute pressure the water capacitywill be only 7.2 gallons, and the difference between 14.3 and 7.2,or 7.1 gallons of water, will have condensed. Another interestingfact is that at low temperatures and high pressures a change ineither temperature or pressure produces a small change in thewater content, whereas at high temperatures and low pressures achange in either temperature or pressure produces a much greater change in the water content of the gas,

Condensation of water vapor in natural gas results in a decrease

in the total volume of gas. Under some conditions this decrease ingas volume due to condensation of water vapor is appreciable andmay account in part for the apparent loss in gas volume throughcompressor stations.

Comparison of ground temperature at different depths and tem-perature fluctuations of atmosphere.-The temperature of gas flow-ing through a buried pipe line depends to a large extent upon thetemperature of the ground adjacent to the pipe. The daily atmos-pheric temperature fluctuations at Bartlesville, Okla., were recordedfor 12 months, and a similar concurrent record of the temperaturevariations 1, 2, 3 and 4 feet below the surface of the ground wasmade as well.” The soil where the tests were made was a heavy clay,tightly packed. Since these temperature tests were conducted inone locality only the actual data are applicable strictly to the con-ditions under which the tests were made; however, the results inthese particular tests indicate what may be expected under other conditions.

Ground temperatures were obtained by placing the external bulbsof the recording thermometers in direct contact with the soil atthe different depths. Thermometers for recording the atmospheric



59

temperature were placed within a specially constructed louver house. All thermometers were of the external bulb type and hadbeen calibrated against standardized mercurial thermometers.

The results of the tests are shown in figures 17 and 18. Graphs 1and 2 in figure 17 show respectively the maximum and minimum

daily temperatures of the atmosphere. Graphs 1 and 2, 3 and 4,5 and 6, and 7 and 8 in figure 18 show the maximum and minimumdaily temperatures at depths of 1, 2, 3, and 4 feet respectively.

As shown in figure 17 there was a daily atmospheric temperaturevariation ranging from 2o to 42o F. By calculation, the average



61

mum temperature of one day with the minimum temperature of the day following shows a still greater variation. On the abovedates this temperature difference was 53o F.

Comparing atmospheric temperatures with underground tem-peratures indicated by the graphs in figure 18 it is seen that temper-ature fluctuations are reduced greatly beneath the surface of theground. Even at the shallow depth of 1 foot there were no exces-sively rapid fluctuations, as noted for atmospheric temperatures.Graphs 1 and 2 show that at a depth of 1 foot the daily fluctuationwas (never greater than 3o to 4o F., and that the total variationduring the entire period of the tests was 52o F.; graphs 3 and 4show that at a depth of 2 feet the total temperature variation was40o F.; graphs 5 and 6 show that at a depth of 3 feet this variationwas 34o F.; and graphs 7 and 8 show that at 4 feet this variationwas 22o F.

Graphs in figure 18 show also that the average temperature be-neath the surface of the ground for the entire test approximated60o F. The average daily temperature from June 1, 1926 to May 26,1927 at a depth of 1 foot was 60o F.; at a depth of 2 feet, 62o F.;at a depth of 3 feet, 65o F.; and at a depth of 4 feet, 66o F.

As the depth is increased the minimum temperature to which aburied pipe line will be subjected is raised. Graphs in figure 18show that the minimum temperature recorded for a depth of 1 footwas 36o F.; for 2 feet, 39o F.; for 3 feet, 48o F.; and for 4 feet,54o F.

Results of these tests show the amount of variation between thetemperature of the atmosphere and the temperature at differentdepths below the surface of the ground, and therefore the impor-tance of burying pipe lines. They also give some indication of theeconomic depths of cover required to offset undesirable temperature

fluctuations.

REMOVAL OF LlQUIDS FROM NATURAL-GAS PlPE LINES

The most satisfactory method of keeping natural-gas pipe linesfree from liquids is to remove these liquids before the natural gasenters the pipe line. One of the largest natural-gas transportationsystems includes a complete gas-conditioning plant at the inlet of its main transmission line. The gas is passed through high-pressureoil absorbers which remove the small amount of gasoline vaporspresent in the natural gas. The residue gas from these absorbersthen is passed through a refrigerating plant where it is cooledbelow any temperature anticipated in the transmission line. Thelow temperatures cause most of the water vapor in the gas to con-

dense, and it is removed before the gas enters the main transmis-sion line. As a result of this process of conditioning the gas beforeit is allowed to enter the main line an unusually clean, dry gas isobtained, which eliminates numerous operating difficulties in thetransportation of natural gas and the necessitytraps, and scrubbers in the main line.

of numerous drips,

A comparison of the operation of three of the pipe lines studiedby the writers illustrates the effect of cooling gas before it entersa pipe line. These lines are referred to in table 4 and in appendix 4



62

as lines 17, 22, and 28. Lines 17 and 22 both began at the dischargeof the same gasoline-extraction plant. Line 17 operated at the dis-charge pressure of the plant; the temperature of the gas enteringthe line was high, and water condensed in the line as the tempera-ture decreased. The inlet pressure on line 22 was normally about100 pounds per square inch less than on line 17, and this reductionof pressure produced sufficient cooling to condense enough of thewater vapor before the gas entered the line to prevent depositionof water in the line. Line 28 also began at the discharge of an oil-absorption gasoline plant. The discharge pressure of the gas fromthe plant was about 200 pounds per square inch and was reducedto about 100 pounds per square inch by a pressure regulator beforethe gas entered the line. The reduction in pressure decreased thetemperature of the gas, resulting in condensation of water vapor,and the header at the plant directly below the pressure regulator collected large quantities of water, whereas the drips in the mainline were almost dry.

Liquids usually are removed from natural-gas pipe lines by theproper location of traps, commonly known as drips, whose func-tions are to separate and remove liquids from the gas stream. Theseparation of liquids from gas often is difficult, depending uponthe amount of liquid with the gas and the state in which it ispresent, that is, whether it is in the form of a mist or is free fromthe gas and moving along the bottom of the pipe.

Many gas wells produce liquid hydrocarbons with the gas. If these condensed hydrocarbons are not removed from the drip reser-voir and the temperature increases some of them will vaporize,will enter the flowing gas, and may liquefy again at other pointsin the line where the temperatures are lower. This alternate vapori-zation and condensation may account for distribution of liquids in

pipe lines that is not easily explained otherwise. Free water inpipe-line drips also should be removed because if unsaturated gascomes in contact with water in the line it will tend to become satu-rated, and at some other point in the line this water vapor maycondense due to changes in temperature or pressure, or both.

LOCATION OF DRIPS

The location of drips for the most efficient removal of liquids isan important factor in the design of a pipe-line system. Everyeffort should be made to remove the liquids before the gas entersthe pipe line. Where gasoline or water is produced with the gasit is customary to install a drip or trap at the well to collect theliquid before the gas enters the pipe-line system. It is good prac-

tice to cool the gas at the discharge of a gasoline-extraction plantor compressor station to a temperature such that no further cool-ing takes place in the pipe line, and adequate drips should be pro-vided at the point of cooling to collect resulting condensates. How-ever, it is not always economically practicable to cool the gas atthe discharge of a gasoline-extraction plant or compressor stationenough to eliminate the possibility of condensation in the pipe line.If condensation occurs it usually takes place a relatively short dis-tance from the gasoline plant or compressor station, and in this



63

section of the pipe line special attention should be given the design,location, and number of drips.

Low places or sags in the pipe line, resulting from the topography

of the right-of-way, serve as natural locations for accumulation of liquids, and it is good practice to install the drips at these points.The drips are not always installed at the bottom of the low places or sags but rather a short distance beyond. The reason is that themomentum of the condensate may carry it past a drip installed atthe bottom of the sag, and for this reason the drip is placed a shortdistance beyond where the velocity of the condensate has decreased.

DESIGN OF DRIPS

It is beyond the scope of this. report to discuss in detail themany different designs of drips for removing liquids from natural-gas pipe lines and the relative efficiencies of the different designs.The use of drips has been given much attention by the natural-gas

industry, and although drip designs vary, the general principlesof decreased velocity of the gas and change in direction of gas flowusually are embodied. For the purpose of this report, drips havebeen divided into three general classes, namely, simple, complex,and siphon.

Simple drips.- Drips of this class consist primarily of a reser-voir connected to the main line at two separate points. The reser-voir is lower than the section of line to which it is connected; thetwo communicating passages enable the pressure to equalize inthe line and reservoir and permit the liquid to flow by gravity intothe reservoir. The short section of main line to which the reservoir is connected is sometimes of larger diameter than the pipe line,thereby reducing the velocity of the gas and facilitating separation

of the condensate. The reservoir usually is installed on a slopewith a blow-off at its lowest point so that all accumulated liquidcan be removed without unnecessary waste of gas. The size of thedrip reservoir depends on the amount of liquid to be removed andordinarily is sufficient to contain the liquid collected during severaldays. Simple drips offer no abnormal resistance to the flow of gas.

A drip usually is of welded construction, fabricated separately,and installed in the pipe line as a unit. This method of construction



64

allows much freedom in drip design and even the simplest dripsusually embody features that facil i tate the separation of l iquidfrom the gas. Figure 19 illustrates an all-welded simple drip of common design, The enlarged pipe diameter causes the velocityof gas passing through this drip to decrease and allows liquids todrop out of the gas. The liquids collect in the reservoir section of the drip and are removed through the blow-off.

Complex drips. - I n a complex drip the direction of flow of gaschanges several times; if the dr ipwill flow in a zigzag course.

is equipped with baffles the gasMost complex drips are made of

larger-diameter pipe than the main-line pipe; the velocity throughthe drip therefore is less than in the main line, which allows liquidsto drop out of the gas, as in a simple drip. In addition, the changesin direction of the gas flow in complex drips facilitate the separa-

tion of liquids from the gas stream. The large-diameter pipe in thedrip compensates for the increased resistance caused by changesin direction of the gas flow or by obstructions such as baffles.

Figure 20 is a diagram of a complex drip suitable for most pipe-line requirements.28

The main section of the drip is appreciablylarger in diameter than the pipe line. A “solid baffle” is weldedinto this section at a point about two thirds of the-distance fromthe inlet to the outlet of the drip. The baffle causes the gas to flowthrough the by-pass which often is of the same size as the pipeline. In some designs the by-pass is made of larger-diameter pipeto offset the increased resistance to the flow of gas caused by thefour right-angle turns in the drip. Upstream from the baffle twoconnections are made between the main section of the drip and alarge reservoir directly below. Also the main section of the dripand the reservoir are connected downstream from the baffle andopposite the outlet of the by-pass; this connection is swaged down



65

to include a small check valve. Any liquid accumulating down-stream from the baffle enters the reservoir through the check valve,which prevents flow of gas from the reservoir to the main line.The Connection downstream from the baffle is placed directly oppo-site the outlet of the by-pass so that any liquid separating fromthe gas in the by-pass will be removed from the path of flow; more-over, advantage is taken of the impact pressure of the gas to actuatethe check valve and allow liquid to drain into the reservoir moreeasily.

In some designs the main section of the drip and the reservoir are not connected downstream from the baffle. Instead, a blow-off is provided at the lower end of the swaged nipple to remove liquidsthat accumulate in the main section of the drip downstream fromthe baffle. If large quantities of liquid accumulate downstream fromthe baffle the connection to the reservoir is desirable to keep theliquid from the path of the flowing gas.

Complex drips of the design illustrated in figure 20 remove liquids

from a pipe line both upstream and downstream from the place atwhich they are installed. Often it is difficult to determine exactlythe correct location for a drip in a pipe line; if the drip is designedto accumulate liquids from both directions, more freedom is allowedin placing the drip.

Siphon drips. - Siphon-type drips consist of a small-diameter pipeinserted into the main line from the top and extending to within1 inch or less of the bottom. This type of drip is not used exten-sively for several reasons: (1) Liquid as it accumulates remainsin contact with the gas stream (the pipe line itself serves as areservoir) ; (2) the small siphon pipe extending through the mainline at right angles to the direction of flow obstructs the gas flow;and (3) removal of the accumulated liquid through a siphon of

this type often involves an undue waste of gas.

RUST SCALE IN NATURAL-GAS PIPE LINES

Corrosion of the internal walls of natural-gas pipe lines causesmany operating difficulties. Continued corrosion weakens the pipeand in some instances necessitates frequent- replacements. Theproducts of internal corrosion, commonly called rust scale, arcobjectionable in several ways. Particles of rust scale on the wallsof the pipe increase its roughness. Some of the particles of rustscale may become loosened from the walls of the pipe and be carriedalong by the moving gas. These particles rub together and againstthe walls of the pipe forming a fine powder, which is distributedthroughout the pipe-line system, accumulating in the bottom of

the pipe and at low places in the line, causing reduction of theeffective diameter of the pipe and increased resistance to flow.Where it is necessary to compress the gas small rust particlescarried by the gas have an abrasive action on piston rings andcylinder liners in compressors resulting in additional maintenanceexpense. The particles of rust also cause excessive wear of equipment on the line such as regulators and valves.

Several types of scrubbers have been developed to remove rustscale carried by natural gas. Generally the rust scale is removed



66

by passing the gas through a chamber or scrubber containing oilor hemp. In the oil-type scrubber the gas is in intimate contact’with oil and the particles of rust scale are retained in the scrubber by the oil. Baffles and screens prevent the oil in the scrubber from

being carried away by the gas. Scrubbers are connected directlyin the main line with suitable by-passes for the gas so that occa-sionally they can be inspected and the oil or hemp removed andcleaned. Scrubbers are placed usually at such points of vantage asthe inlets to compressor stations and where the main transmissionline connects with the distribution systems,

Three pipe lines that contained quantities of rust scale wereselected for study by the writers to indicate the effect of rust scaleon delivery capacity. These are listed as lines 4, 15, and 18 intable 4 and appendix 4. Lines 4 and 18 are two different lines of the same general system. Line 18 begins at an oil-absorption gaso-line plant and is one of several gathering lines delivering gas intothe main line of which line 4 is a section. Line 18 contained more

rust scale than line 4, as evidenced when the drips were blown andnew orifice plates installed in the lines. The metered gas deliveryof line 4 was approximately 19 percent less and that of line 18 wasabout 48 percent less than the respective volumes calculated fromWeymouth’s pipe-line flow formula. Using this method of analysisline 18, of 10-inch-pipe, had a delivery capacity about equal to thatof an 8-inch clean line under-the same operating pressures. Asample of the rust scale taken from this system and dried at 105o C.had the following composition :

Internal corrosion of the pipe on line 15, which started at thedischarge of an oil-absorption gasoline plant, had produced largequantities of rust scale, and the metered delivery from the line wasabout 52 percent less than the volume calculated from Weymouth’sformula. This line delivered gas to a compressor station; however,before the gas entered the compressors it passed through a scrubber filled with hemp for removing the rust scale. A sample of scaletaken from this scrubber and dried at 105 o C. had the followinganalysis :

The chemistry of corrosion of metals is complicated and a subject of much invest igation and study. It is beyond the scope of this report to discuss the chemical reactions and conditions inci-dent to corrosion; however, some of the more common conditionsfound to exist where the internal walls of natural-gas pipe lineswere becoming corroded are mentioned briefly.



67

Water and oxygen in natural gas result in corrosion of the inter-nal walls of natural-gas pipe lines.29 Burton30 points out the corro-sive action of sulphur-bearing gases, the sulphur existing in the

natural gas in the form of hydrogen sulphide. Devine, Wilhelm,and Schmidt31 draw the following conclusions from a Bureau of Mines study of conditions incident to corrosion of steel equipmentused in the production and transmission of natural gas containing"traces” of hydrogen sulphide. In their report, “ traces” wereconsidered to be 12 grains or less of hydrogen sulphide per 100cubic feet of gas.

(1) Traces of hydrogen sulphide in a gas may cause severe corrosion if oxygen and water are present.

(2) Two forms of corrosion have been identified :(a) A less serious form, ordinary gaseous corrosion, in which no

precipitated water is present on the surface of the metal, although thegas may be saturated with water vapor.

(b) A severe form, modified gaseous corrosion, in which precipitatedwater is present on the surface of the metal.

(3) Pressure accelerates the reaction in both forms of corrosion, but thiseffect is particularly noticeable with modified gaseous corrosion.(4) The application of pressure to a gas also may increase the corrosion

rate by causing precipitation of moisture on the surface of the metal, thustransforming ordinary gaseous corrosion to modified gaseous corrosion.

(5) The corrosion-threshold concentration of amount necessary for

hydrogenreducing severe corrosion, is lower for modified gaseous

sulphide, or the

corrosion than for ordinary gaseous corrosion.(6) Modified gaseous corrosion appears to be the result of an electrochemical

reactjon, while ordinary gaseous corrosion has the characteristics of a chemicalattack.

(7) The corrosion rate with modified gaseous corrosion increases with adecrease in the thickness of the moisture film on the surface of the metal: butwith continued reduction, the reaction finally loses its electrochemical character and becomes ordinary gaseous corrosion.

(8) Modified gaseous corrosion can occur in the absence of hydrogen sulphide,if oxygen is present; such corrosion, also, is accelerated by pressure.

(9) In natural-gas pipe-line systems, modified gaseous corrosion is the chief consideration in combating corrosion caused by gas containing traces of hy-drogen sulphide.

(10) Steps that may be taken to reduce corrosion losses in pipe-line systemsinclude :

(a) Removal of hydrogen sulphide from thesures up to 300 pounds per square inch it is believed that the corrosive

as; for operating pres-

effect of hydrogen sulphide is eliminated if its content in the gas isreduced to 0.10 grain per 100 cubic feet.

(b) Reduction of the oxygen content of the gas: the practical stepis to eliminate from the gathering system all unnecessary points of partial vacuum, and to repair all leaks at necessary points of partialvacuum.

(c) Control of the humidity conditions: this can best be accomplishedby reducing the relative humidity of the gas to such a degree that nowater-is precipitated within the line itself; where dehumidification of the

gas is not feasible, and precipitation of water within the line cannot beaverted, the alternative is to prevent the precipitated water from beingdispersed extensively in the line.



68

STORED GAS IN PIPE-LINE SYSTEMS

The stored gas in a pipe-line system may be defined as the volumeof gas measured under standard conditions of pressure and tem-

perature contained in the system. If the rate of gas withdrawal isthe same as-the rate of gas input (steady flow) the volume of storedgas in the system remains constant. When the rate of gas with-drawal is greater than the rate of input the mean line pressure andthe stored gas in the system are decreasing; conversely, when therate of withdrawal is less than the input rate, the mean line pres-sure and the stored gas are increasing. In the operation of mostpipe lines the pressure conditions generally are unsteady becauseof periodic variations in the demand for gas. That is, the averageline pressure usually is increasing or decreasing, resulting inchanges in the quantity of gas stored in the line. Where storagefacilities, such as gas holders, are provided to take care of varia-tions in the rate of demand the operating conditions and pressuresof the transportation system are less subject to fluctuation.

The quantity of stored gas in a pipe line depends upon the lengthand diameter of the line and upon the mean pressure and tempera-ture of the gas. The equation for the mean pressure in the line32 is

where P111 = the mean pressure, pounds per square inch absolute;P1 = inlet pressure, pounds per square inch absolute;P2 = outlet pressure, pounds per square inch absolute.

If a pipe line is operating at the maximum mean pressure thatcan be carried in it safely, the available stored gas is that portion

TABLE 13 - Volume of 1 mile of different-diameter pipes

of the total storage between this pressure and the mean pressurein the line under the lowest operating pressures permissible.Table 13 gives the volume in cubic feet of 1 mile of various diameter pipes commonly used in transporting gas. Figure 21 shows graphi-cally the total volume, in cubic feet, of gas stored in 1 mile of various diameter pipes at pressures ranging from zero to 1,000pounds per square inch. The following example of a pipe line



69

50 miles long and 12.188 inches internal diameter illustrates theapplication of figure 21 for determining the available stored gas. Assume that the mean line pressure, under the lowest operat ingpressures necessary to deliver a desired quantity at a minimumoutlet pressure, is 200 pounds per square inch absolute, and thatthe mean line pressure when the pipe is subjected to its maximumsafe working pressure is 450 pounds per square inch absolute.From figure 21, the total storage in 1 mile of 12.188-inch-diameter pipe at a pressure of 200 pounds per square inch is 60,000 cubicfeet of gas. In 50 miles the total storage would be 50 times thisamount, or 3,000,000 cubic feet. Similarly, the total storage in the

FIGURE 21. - Storage capacity of 1 mile of pipe of various diameters

50 miles of pipe at 450 pounds per square inch absolute is 6,750,000cubic feet of gas. The available stored gas between these two pres-sures is therefore the difference between 6,750,000 and 3,000,000or 3,750,000 cubic feet.

The curves in figure 21 are based on the assumption that natural

gas follows the pressure-volume relation expressed by Boyle’s law.Experiments show that natural gas is more compressible than isindicated by Boyle’s law,

33 and therefore more natural gas of standard pressure base is stored in a pipe line at a given pressurethan is calculated by Boyle’s law. Through the range of pressuresused in natural-gas transmission the deviation of natural gas fromBoyle’s law increases as the pressure increases, and therefore the



70

difference between the actual and calculated stored volumes of gasincreases with pressure. In the preceding example of calculatingavailable stored gas, assume that the. deviation from Boyle’s lawat 200 and 450 pounds per square inch absolute is 2.5 and 5.6 per-cent, respectively, The total quantity of stored gas at 200 poundsper square inch absolute, considering deviation from Boyle’s law,then is 3,000,000 times 1.025 or 3,075,000 cubic feet. Likewise, thevolume of stored gas at 450 pounds per square inch is 6,750,000times 1.056 or 7,128,000 cubic feet. The available stored gas, or thedifference between the quantities stored at 450 and 200 poundsper square inch, corrected for the deviation from Boyle’s law, is4,053,000 cubic feet.

The delivery of gas from pipe lines operating under unsteadypressures is not comparable to the delivery calculated by the pipe-line flow formulas which are based on steady flow conditions. Thetests of commercial pipe lines, particularly those provided withmeters at both the inlet and outlet of the line (see table 4) showthat the rate of gas withdrawal from a line is greater when themean line pressure is decreasing than when it is constant and thatthe rate of gas withdrawal is less when the mean line pressure isincreasing. The following examples of both the rate of drainageand rate of increase of stored gas in pipe lines are taken from datacollected by the writers on pipe lines listed in table 4.

In figure 22 the rates of gas input and withdrawal for line 3and the corresponding inlet and outlet line pressures are given for a period of 7 hours. The area between the gas input and with-drawal curves represents approximately the quantity of additionalstored gas or the quantity of gas drained from the stored gas inthe line. Changes in line pressures also indicate the variation inthe quantity of stored gas. The rate of gas withdrawal from the

line was greater than the input rate during the period from 9:00to approximately 12:45 o’clock, and gas was drained from thestored gas in the line, causing a decrease in the inlet- and outlet-line pressures; for example, the outlet pressure decreased from 318to 300 pounds per square inch absolute. In the period between10:00 and 11:00 o’clock the average input rate was 1,380,000cubic feet per hour and the average rate of withdrawal was1,482,000 cubic feet of gas per hour, or the withdrawal rate was7.4 percent greater than the input rate. The difference between theinput and withdrawal rates, 102,000 cubic feet per hour, is the rateat which gas was being drained from the stored gas in the line.

During the period between 12:45 and 4:00 o’clock the rate of gas withdrawal from the line was lower than the input rate, and

the stored gas in the line was increased, causing the line pressuresto increase; for example, the outlet pressure increased from 300to 317.5 pounds per square inch absolute. At 1:30 o’clock the inputrate was 1,224,000 cubic feet of gas per hour and the withdrawalrate 1,110,000 cubic feet per hour, or the input rate was 10.3 per-cent greater than the withdrawal rate.

On line 1, test 1, of table 4 the inlet and outlet pressures increased0.7 and 0.5 pound per square inch, respectively, during a period of 1 hour, and 0.4 percent more gas entered than left the line; in test



71

2 the pressure decreased 0.7 and 0.9 pound per square inch, respec-tively, and 5.1 percent more gas left than entered the line. In test 1the metered delivery rate at the outlet was 0.5 percent greater

than that calculated by Weymouth’s pipe-line flow formula; and intest 2, under decreasing line pressures indicating drainage of storedgas, the metered outlet rate of flow was 5.5 percent greater thanthat calculated by Weymouth’s formula.

The pressures were decreasing in line 2, tests 1 and 2, with thegreater decrease in test 2. During the tests, the metered outlet ratesof flow were 0.5 and 1.8 percent, respectively, greater than themetered inlet rates. The metered outlet rate of flow during test 1was 0.5 percent above that calculated by Weymouth’s formula;during test 2, because of the higher rate of drainage from the storedgas, as indicated by the greater decrease in line pressures, the out-



72

let metered rate of flow was 2 percent above that calculated by theWeymouth flow formula.

In line 5, test 2, both the inlet and outlet pressures increased1 pound per square inch, indicating that additional gas was being

stored in the line, while in test 3 the inlet and outlet pressuresdecreased 1.5 and 1.7 pounds per square inch, respectively, indicat-ing drainage from the stored gas in the line. The metered rate of withdrawal from the line during test 2 was 1.5 percent above thatcalculated by Weymouth’s formula; and in test 3, when gas wasbeing drained from the stored gas in the line, the rate of withdrawalwas 5 percent above the Weymouth rate of flow.

In the tests of line 10 measurements of the rate of flow wereobtained at the outlet only. The tests of this line, however, indicatethe rate at which the stored gas was drained from the line. Dur-ing test 1 the decrease in line pressure was 0.4 pound per squareinch in 52 minutes, and in test 2 the fate of decrease was about3.6 pounds per square inch in 56 minutes; or during test 2 the rate

of decrease in line pressure was approximately 9 times greater thanin test 1. The difference between the metered rate of withdrawaland the withdrawal calculated by Weymouth’s formula in test 1was negligible, whereas in test 2 the metered rate of withdrawalwas 2 percent above the Weymouth rate of flow.

Another example is that of line 24, which is comparable to line 10in that an orifice meter was provided at the outlet only, In test 2the line pressure increased 0.3 pound per square inch, indicatingadditional storage of gas, and the metered rate of delivery was6 percent less than the delivery rate calculated by Weymouth’sformula. The inlet and outlet pressures decreased 1.4 and 1.1pounds per square inch, respectively, during test 1, indicating adrainage from the stored gas, and the metered rate of delivery was

8 percent greater than that calculated by the Weymouth formula.DIFFERENCES IN ELEVATION AT INLET AND OUTLET ENDS OF PIPE LINE

The derivation of the general formula for the flow of naturalgas through pipe lines is given in appendix 1. The balance betweenthe input and output energies, upon which this derivation is based,includes an expression for the difference in elevation at the inletand outlet ends of a pipe line. It was assumed that the energyrepresented by the difference in elevation is relatively small com-pared to the energy represented by other terms in the energy bal-ance, and therefore the effect of elevation difference was neglected.The magnitude of the error introduced in the flow formulas byneglecting the effect of elevation difference at the ends of a pipe

line depends on the operating conditions as well as on the amountof the elevation difference. The density of a gas increases withpressure, therefore the weight of a column of gas of definite heightincreases with pressure. However, the necessity or importance inpipe-line design of considering the energy due to elevation differ-ence diminishes with increase in the difference between the inletand outlet pressures. Most gas pipe lines are operated with enoughdifference between the inlet and outlet pressures so that the effectin the flow formulas of the elevation difference is negligible,



73

In some pipe lines on which flow tests were made by the writersthe pressure drop was small, and there was a considerable differ-ence in elevation at the ends of the line. For these lines it was

necessary to consider the effect of elevation difference on the flowof gas before comparisons could be made with pipe lines that werealmost horizontal. The determination of the exact effect of eleva-tion difference on the flow of gas would require a more detailedstudy than was possible in this investigation; however, data col-lected on elevation differences and methods for taking them intoaccount are presented in this report for their general value. It isrealized that their scope does not warrant anything more thangeneral deductions.

In correcting for a difference in elevation the pressure due to theweight of a column of gas equal in height to the elevation differencewas subtracted from either the observed inlet or outlet pressurebefore these pressures were substituted in the flow formulas. When

the elevation at the outlet of the line was higher than the elevationat the inlet the observed inlet pressure included the pressure due tothe weight of the column of gas; when the inlet of the line washigher than the outlet the pressure due to the weight of the columnof gas was included in the observed outlet pressure. Therefore,when the direction of flow was upgrade, the pressure due to theweight of the gas was subtracted from the observed inlet pressure;when the flow was downgrade, the pressure due to the weight of the column of gas was subtracted from the observed outlet pressure.

The pressure due to the weight of the column of gas was deter-mined from the following formula:

Pc= (Pa-Pd)Gs,where Pc= pressure due to the weight of the column of gas, pounds per square

Pa = atmospheric pressure at the end with lowest elevation, pounds per square inch,

Pb = atmospheric pressure at the end with highest elevation, pounds per square inch,

G = specific gravity of the gas (air=1.000), and

s = density factor, or

Atmospheric pressures Pa and Pb were obtained with aneroidbarometers or calculated from elevations of the inlet and outletends by an equation expressing the relationship between elevationsand barometric pressures. A formula31 for computing the baro-metric pressure at various altitudes is,

where h = altitude, feet;Ph = barometric pressure at altitude h, pounds per square inch;Po = barometric pressure at sea level, 14.7 pounds per square inch,

Table 14 contains data on the pipe lines included in the inves-tigation that had appreciable differences in elevation at the inlet.and outlet ends. Comparisons were made between the metered



74

volumes and those calculated by the Weymouth pipe-line flow for-mula, using both observed pressures and pressures corrected for the difference in elevation. The results for lines 7, 10, 12, and 22show virtually the same variation between the metered volumes andthe volumes calculated by the Weymouth formula, whether or notthe effect of elevation difference is included in the calculations.These tests bear out the assumption made in the derivation of theflow formula that for most commercial pipe lines the effect of dif-ference in elevation at the inlet and outlet is negligible when thedifference between the inlet and outlet pressures is large. Of thefour remaining lines listed in table 14 the volumes by Weymouth’sformula for lines 8, 25, and 28 were closer to the metered volumeswhen the line pressures corrected for the pressure due to the columnof gas were used, whereas for line 26 the uncorrected pressures

TABLE 14.- Corrections for differences in elevation

gave a volume by Weymouth’s formula closer to the actual meteredvolume,

Lines 25 and 28 had very little pressure drop, and the effect onthe results of the difference in elevation was very noticeable. Inline 28, test 2, the actual observed outlet pressure was higher thanthe observed inlet pressure; however, after corrections for eleva-tion difference were applied there was a small difference betweenthe metered volume and the Weymouth volume.

In line 26 the first 16 miles of line traversed almost level coun-try, and the 3,150-foot difference in elevation occurred beyond thissection. Since the density of gas is a function of pressure the aver-age density of the gas over the mountainous area located towardthe outlet of the line was less than the average density over theentire line. The pressure due to the weight of the column of gasfor test 2 of line 26 based on the average density (assuming theslope was gradual from inlet to outlet) over the entire line was20.99 pounds per square inch, whereas, based on the average den-



75

sity over the mountainous area only, the pressure due to the weightof the column of gas was 18.95 pounds per square inch. The per-centage difference between the metered volume and Weymouth

volume was 13.5 percent, using the value for the pressure due tothe weight of the column of gas of 18.95 pounds per square inch,compared with a difference of 14.5 percent, using 20.99 pounds per square inch. As shown in this example, where there is a differencein elevation between the inlet and outlet of a pipe line the exactlocation of the change in elevation with respect to the inlet andoutlet has an effect on the calculations.

In another method of considering the effect of difference in ele-vation, suggested by E. S. Burnett, mechanical engineer, U. S.Bureau of Mines Helium Plant, Amarillo, Tex., the term P1

2-P22

in the flow formulas is written [P1+P2] [P1-P2] and the pressuredue to the weight of the column of gas (PC) is considered only inthe values used in the term P1 -P2.

The quantity P1

2

-P2

2

in the flow formula then is replaced whenthe flow is downgrade by

[P1+P2] [P1-(P2-P c)]

and when the flow is upgrade by

[P1+P 2] [(P1-P c)-P2].

The difference between the method used to compile table 14 andthat suggested by Burnett depends chiefly upon the pressure dueto the column of gas (PC). The latter method reduced the percent-ageline

difference between the metered and Weymouth volumes for

13.58, test 1, from 12.0 to 11.5; for line 26, test 1, from 16.0 to; and for line 28, test 1, from 7.0 to 6.5; these are representa-

tive comparisons of the two methods.

TYPES OF CONSTRUCTION AND EQUIPMENT

The pipe lines on which the writers conducted flow tests includedalmost all of the common types of construction and auxiliary equip-ment, such as valves, drips, and meter settings. Although thenature of the data collected does not permit definite comparisonsof resistance to flow between the various types of construction andof equipment some general deductions of practical value are possible.

The more common means of joining the sections of pipe are bywelding and by couplings. The data in table 4 show no apparentdifference in the resistance to flow between pipe lines of weldedand coupled construction. The length of the sections of pipe between

joints was approximately 30 feet in lines 1, 2, and 8 and about20 feet in the other pipe lines. The data obtained were insufficientto establish the relative effect of different lengths of pipe between

joints; however, the writers believe that pipe lines made up of longer sections of pipe have less frictional resistance, unless specialcare is exercised in making the joints.

In most pipe-line systems orifice meters are installed at variouslocations on the lines to obtain check measurements of the volumeof gas. The pressure drop across an orifice-meter setting dependsupon its construction and upon the amount of differential pressureacross the orifice. Drips, valves, bends, and other fittings also influ-



7 6

ence the resistance to flow by varying amounts. In the complex-type drip the diameter of the pipe usually is larger than that of the pipe in the main line, and this tends to offset some of the dropin pressure caused by the changes in direction of flow. For lines 7and 10 in table 4 the presence of a large number of drips, valves,and bends apparently had little effect on the delivery capacities of the lines. Line 7 traverses mountainous country, and in the 32-milesection included in the test there were many sharp vertical bends,14 ninetyo turns, 37 forty-fiveo turns, and many other turns of lesser degree. In addition to the numerous bends and turns thisline contained 1 complex drip, 41 simple drips, and 13 twelve-inchvalves connected into the 15-inch line by swaged nipples. Regard-less of all these bends, turns, drips, and valves on line 7 the meteredvolume was very close to that calculated by Weymouth’s formula.Line 10 also traverses mountainous country, and in the 50 milestested there were 106 simple drips, 1 complex drip, 15 valves of the same diameter as the pipe, and numerous bends and turns. The

drop in pressure over the section tested was almost the same as thatcalculated by Weymouth’s formula.

The pipe lines included in this investigation ranged in age fromseveral days to approximately 30 years. The results of flow testsof these lines show that the delivery capacity of a pipe line is notaffected by length of service as long as the pipe line is in goodoperating condition. For example, the metered flow through line 21,which was about 30 years old, agreed closely with that calculatedfrom the Weymouth flow formula.

Pipe-line river crossings are of two general types: (1) Over-head crossings, where the pipe is placed on a bridge ; and (2) sub-merged crossings, where the pipe is placed on or below the river bed. In the overhead crossing, which is the newer and less common

type of construction, the bends used to place the pipe on the bridgeare the only additional source of pressure loss. Where the sub-merged type of crossing is used several lines of smaller diameter than the main line usually are placed across the river. In mostinstallations the equivalent diameter of the lines laid across theriver at least equals the diameter of the main line, and thereforeno additional resistance to the flow of gas is introduced.

PHYSICAL AND CHEMICAL PROPERTIES OF NATURAL GAS

Natural gas usually is a mixture of several gases. Its physicaland chemical properties therefore depend upon those of its con-stituents, and upon properties peculiar to gaseous mixtures. Thisreport does not propose to discuss in detail the physical and chemi-

cal properties of individual gases and their theoretical behavior but to discuss the more common properties that affect the flow of natural gas through pipe lines under the conditions observed incommercial transmission.

The physical and chemical properties of natural gases are influ-enced by the relative quantities of their various hydrocarbons andother constituents, and these individual components differ widelyin their characteristics and properties. For this reason, no one setof definite values for the properties of natural gas would apply



77

strictly to all natural gases; instead, values must be determinedfor each particular natural gas. Since the physical properties of natural gas depend upon its chemical composition and the com-position may change with a change in pressure or temperature dueto liquefaction of some of the constituents, it is advisable to deter-mine the physical properties of the natural gas through the desiredrange of temperatures and pressures.

CHEMICAL COMPOSITION OF NATURAL GAS

The term “natural gas” is applied to mixtures of gases, as foundin nature, composed of hydrocarbon gases of the paraffin series(methane, ethane, propane, butane, etc.) and usually small amountsof nitrogen and carbon dioxide, and still smaller amounts of heliumand other inert gases, The wide variation in both the actual num-ber of components and the percentages of each that constitutenatural gas gives the term an indefinite chemical significance. Usu-

ally the greater parts of the heavier hydrocarbons (propane, butane,pentane, etc.) present in natural gas in varying amounts are re-moved as gasoline or “liquefied-gas products.” Some natural gasescontain varying amounts of hydrogen sulphide, and if considerablequantities are present the commercial use of the gas as fuel islimited, These several qualifications and processes tend to limitthe variation in the composition of natural gas as used for fuel.The chemical composition of natural gas as obtained from the wellsvaries in different localities and producing formations; this studyincluded pipe lines transporting gas from many of the more impor-tant natural-gas areas, in some instances before the gas had beenprocessed to remove its gasoline vapors.

Table 15 gives the chemical composition of the natural gases

from the 29 pipe lines used in this investigation (samples 1 to 29,inclusive) and of 8 additional natural gases (samples 30 to 37,inclusive). The chemical analyses of all but 2 of the gases intable 15 were made by the combustion method

35in which the hydro-

carbon constituents of the natural gas are determined by slowcombustion.

The usual method of reporting a combustion analysis of naturalgas is to divide the total hydrocarbon content into methane andethane (or into some other pair of hydrocarbons both heavier thanmethane if the average molecular weight of hydrocarbons in thegas is heavier than that of ethane), according to a mathematicalinterpretation of the results of the combustion. Natural gas usuallycontains several hydrocarbon gases besides methane and ethane;

for example, in "wet-casinghead" natural gas the percentage of hydrocarbons heavier than methane and ethane usually is large.To determine the percentage of each hydrocarbon in a natural gascontaining more than two hydrocarbons, it is necessary to use the"fractional"

36 method of analysis.



78

TABLE 15.- Chemical composition of natural gases

COMPRESSIBILITY OF NATURAL GAS

The volume occupied by a definite quantity of gas depends uponthe absolute pressure and temperature to which it is subjected. If the temperature is constant changes in volume may be consideredin terms of pressure only. The relationship, at constant tempera;ture, between the pressure and volume of a gas is known as the com-pressibility. A relationship between volume and pressure is ex-pressed by Boyle’s law, which states that at constant temperaturethe volume occupied by a gas is inversely proportional to the abso-lute pressure. The mathematical expression of this law is

PV=P2V1=K,where V is the volume at pressure P, and V1 is the volume at pres-sure P1; or, the pressure multiplied by the volume is equal to aconstant (K).

The writers have discussed the pressure-volume relationship for natural gas in detail in a previous publication37 which includes amethod for determining the deviation of a natural gas from Boyle’s



79

law, the method of applying deviation corrections to the pressure-volume relationships, an analysis of the results of deviation deter-minations for a number of representative natural gases, and a

discussion of deviation from Boyle’s law as applied to some gas-engineering problems. Table 16 shows the percentage deviationfrom Boyle’s law of a number of natural gases, the gas-samplenumbers corresponding to numbers used in table 15, Chemicalcomposition of natural gases.

The general equation (equation 23, appendix 1) for the flow of gas through pipe lines is based on the assumption that naturalgases behave according to Boyle’s law; furthermore, the meteredvolumes of gas given in table 4 of this bulletin are based on theassumption that natural gas does not deviate from Boyle’s law. In

TABLE 16.- Deviation of representative natural gases from Boyle’s law

comparing the metered volumes from clean lines having no unusualfeatures with volumes calculated from Weymouth’s pipe-line flowformula, it is interesting to note that the agreement is close enoughfor most practical purposes, although deviation of natural gas fromBoyle’s law was not considered in the calculation of any of the data.

The data obtained by the writers on the flow of natural gasthrough the commercial and experimental pipe lines were notenough to indicate the quantitative effect of deviation from Boyle’s

law in the flow of gas or to determine a method of applying thedeviation factors to the flow formulas. The operating conditionsfor commercial pipe lines were not flexible enough to establish thecorrections necessary to the flow formulas for deviations fromBoyle’s law. Possibly, the deviation can be taken into account bya method similar to that used when deviation factors are appliedin computing the flow of gas through orifices.38 However, if at anytime the need arises for considering the deviation from Boyle’s law



80

in the practical use of pipe-line flow formulas, a systematic seriesof tests should be made to establish definitely the effect of devia-tion from Boyle’s law and methods of correcting the formulas for

the flow of natural gas through pipe lines.

VISCOSITY OF NATURAL GAS

The absolute viscosity of a fluid is defined as the internal r&s-tance offered to the relative movement of its particles. Knowledgeof the magnitude of this resistance is important in calculating theflow of a fluid in the straight-line flow region. In the commercialtransportation of natural gas the flow. is turbulent, and among thevarious factors that influence the flow of gases through pipe linesthe variation in the viscosities of natural gases seldom is impor-tant; however, the viscosity of natural gas enters into the deter-mination of the Reynolds criterion of the coefficient of friction,and therefore data are given pertaining to the magnitude of natural-

gas viscosities and the extent to which this property of natural gasaffects its flow.

Table 7 of this bulletin contains values for the viscosity (Z) at60

oF. and at about atmospheric pressure of the natural gases

from the pipe lines tested. These values range from 0.00000646pound per second-foot for the natural gas from line 21 to 0.00000866pound per second-foot for the natural gas from line 27. The methodand apparatus used in determining the viscosity of natural gaseshave been described in a recent publication

39by the writers.

Figure 23 shows the relationship between temperature and vis-cosity for three natural gases over a temperature range of from40o to 80o F. and the same relationship for several pure gases andgas mixtures over a considerably larger temperature range. Vis-

cosity values for these three natural gases (see table 15 for chemicalcomposition) were determined by the writers. Those for puregases and gas mixtures were taken from the published data of Trautz and others40 and those for air from the published data of Harrington.

41

The relationships between viscosity and temperature for naturalgas as well as for other gas mixtures depends largely upon theviscosity and percentage of each component gas in the mixture.No reliable methods are available for calculating the viscosity-temperature relationship for a complex mixture of gases. Thecurves in figure 23 indicate that the viscosity of natural gas in-creases approximately 7 percent from 40o to 80o F. and that thisrate of increase in viscosity is approximately the same for theother hydrocarbon-gas mixtures over a considerably greater rangeof temperature. Since the temperature gradient on most natural-gas transmission pipe lines seldom is appreciable, the changes in



81

viscosity with temperature are negligible. Even seasonal changesin pipe-line temperatures seldom are large enough to cause as muchas a 25-percent change in viscosity, which, as will be shown, would

introduce no appreciable error in the determination of the coeffi-

cient of friction by the Reynolds criterion under the conditions of high-pressure gas transmission.

For the common conditions of temperature and pressure of natural-gas transmission and particularly in view of the relativemagnitudes of the numerous other factors influencing the coefficientof friction, the viscosity of the gas usually is a factor of minor



82

importance. The viscosity of natural gases is approximately con-stant over the range of pressures commonly used in transportingnatural gas.42

The following example offers some indication of theextent to which the viscosity may enter into an analysis of the

coefficient of friction. Appendix 2 shows that the coefficient of friction in fluid flows (both liquid and gas) may be considered as

some function of the Reynolds criterion . Since the ratio

includes the absolute viscosity of the fluid (Z), any change in (Z)produces a change in the ratio and consequently a change in thecoefficient of friction. By substituting the maximum and mini-mum observed viscosity values for natural gas (0.00000866 and0.00000646 pound per second-foot from table ‘7) into the ratio

and considering the mean curve through the experimental

data on the relationship between the coefficient of friction and theReynolds criterion in figure 7, some indication may be obtained of the effect of a change in viscosity of the gas on the coefficient of friction. For example, if the value of D is 1.0157 feet, U 3.5311feet per second, and S 1.4149 pounds per cubic foot, then the value of

is 586,000 when Z is 0.00000866 pound per second-foot. If

the same values of D, U, and S are used the value of i s

785,000 when the value of Z is 0.00000646 pound per second-foot.From the experimental curve in figure 7 for this change in theReynolds criterion, the coefficient of friction changes from approxi-mately 0.00354 to approximately’ 0.00348 or about 1.7 percent.Since the rate of flow is inversely proportional to the square root

of the coefficient of friction (equation 23), the rate of flow wouldbe changed, by approximately half of this percentage or by about0.85 percent.

The values of Z used in the above example are the highest andlowest observed for any of the gases tested; therefore, the effecton the rate of flow indicated for this change in viscosity is notan average, but is the maximum for the given values of D, U,and S. It is obvious from figure 7 that the change in the coefficientof friction with changes in the Reynolds criterion becomes less asthe value of the criterion increases, and therefore changes in theviscosity of the gas affect the coefficient of friction less as the valueof the product, DUS, increases. As indicated in the preceding ex-ample, for the flow of natural gas through pipe lines having a valueof the Reynolds criterion above 500,000, a change in viscosity of asmuch as 25 percent produces a negligible effect on the rate of flow.Since the value of the Reynolds criterion observed for large-diameter pipe lines transporting naturalfor long distances usually is well over 50

as under high pressure0,000 and the average

variation observed in the viscosities of the natural gases flowingin these pipe lines usually is much less than 25 percent, the relativeeffect of the viscosity of the natural gas on the rate of flow isnegligible under these conditions.



83

SUMMARY

In initiating the study discussed in this bulletin a review andcompilation were made of the published data and formulas for deter-mining the flow of natural gas through pipe lines. Many formulasare given in the literature, and the results using the different for-mulas vary widely. The basic differences between these formulasare explained in the derivation of the general flow formula and bythe methods of considering the coefficient of friction.

Because of the lack of sufficient data on flow of natural gas incommercial pipe l ines, further experiments under conditions of natural-gas transportation were essential before any recommenda-tions could be made relative to the accuracies and limitations of the various formulas. Flow tests have been made of 29 pipe lines,totaling 757 miles of pipe, in the principal natural-gas areas of thecountry. The diameters of the pipe lines tested range from 6 to22 inches and the operating pressures from 30 to 600 pounds per

square inch. The description, test data, and results for each lineare given in tables 2, 3, and 4, which are augmented by a moredetailed discussion of important factors on each line in appendix 4.

The results of flow tests of the commercial pipe l ines showthat for the larger-diameter lines free from condensates and other foreign materials and operating under steady flow conditions themetered rates of delivery agreed more closely with the rates calcu-lated from Weymouth’s formula than with those calculated fromany of the other formulas; and in almost every test, under theseconditions, the volume calculated from Weymouth’s formula waswithin a few percent of the metered delivery.

Comparisons were made of the relationship between the coeffi-cients of friction and the internal diameter of the pipe as expressed

by the pipe-line flow formulas and as *determined from the tests of the pipe lines used in the investigation. The coefficient of frictiontends to decrease slightly with increasing diameter of pipe. Thecurve based on the Weymouth formula for the relationship betweenthe coefficient of friction and diameter of the pipe when applied tothe lines larger than 6 inches in diameter agreed closely with theaverage of the experimental data and satisfied the test data as wellas any curve that could be drawn through the plotted points.

Further analyses’ of the data from the flow tests on commerciallines, together with the* results of laboratory investigations andtests on a small experimental pipe line, indicate small variationsin the coefficient of friction, depending upon the velocity, density,and viscosity of the gas and the diameter of the pipe. A mathe-

matical analysis of the variations observed in the calculated coeffi-cient of friction was made on the basis that the coefficient of friction for the flow of any fluid through a circular pipe is somefunction of the ratio of pipe diameter multiplied by the velocity anddensity of the fluid divided by the viscosity of the fluid (Reynolds’criterion). The use of the Reynolds criterion to determine thecoefficient of friction results in a flow formula more complicatedthan the Weymouth-type formula and difficult to apply to designand operating problems. Enough data are not available to estab-lish the relationship between the coefficient of friction and the



84

Reynolds criterion throughout the. range of flow conditions found incommercial practice. The tests of the small-diameter experimentallines indicate the nature of this relationship; however, before its

general use could be adopted it would be necessary to determinethe exact relationship for all sizes of pipe, varying the velocity anddensity of the gas and using several different natural gases andpipes of different roughness. Precise analyses of flows throughindividual lines, or geometrically similar pipe lines, may be madeby means of the relationship between the coefficient of friction andthe Reynolds criterion: however, the relationship is influenced byfactors peculiar to individual lines, and no average relationshipcan be recommended that would apply over all ranges of pipe sizesand flow conditions.

Since most natural-gas pipe-line systems do not consist of a singlelength of one-diameter pipe but rather of sections of different-diameter pipes or of sections paralleled with other lines, the designof pipe-line systems often requires special handling of flow for-mulas. In the solution of complex pipe-line problems the variouslengths and diameters of pipe used in the complex system are con-verted to equivalent lengths of a common diameter or to equivalentdiameters of a common length,

A special series of tests of a small experimental pipe line wasmade to determine the relative delivery capacities of a pipe linewhen the gas was flowing in the direction of increased diametersof pipe and when the direction of flow was toward reduced diame-ters. The results indicate that under the conditions of the tests,and for the same pressure conditions, the rate of flow with increas-ing pipe diameters was slightly greater than that toward decreas-ing pipe diameters. This difference probably can be attributed tothe effect of the wages in the experimental pipe line.

The paralleling of sections of existing pipe-line systems is oneimportant method of increasing the delivery capacities of the sys-tems. Several formulas have been developed in this bulletin to aidin the design of such systems, and curves and tables have been pre-pared from these formulas to simplify the calculations and to facili-tate the determination of sizes of pipe to be used in parallelingvarious diameter lines to give the most economical design.

Condensates or rust scale in a pipe line retard the flow of gas,thereby decreasing the delivery capacity of the pipe line. Severalpipe lines containing condensates or rust scale, or both, were in-cluded in the investigation to show the extent to which their pres-ence affected the delivery capacity of the lines. The actual deliveryfrom some of these lines was about 50 percent below the delivery

calculated from Weymouth’s flow formula. A detailed study madeof each of the pipe lines containing liquids to determine the sourcesof liquids in the lines indicated ways of improving operation toprevent liquids from entering the pipe lines. A study also was madeof methods of removing liquids from pipe lines, Rust scale causedby corrosion of the internal walls of the pipe results in many operat-ing difficulties, such as reduced capacity of the pipe line, and there-fore conditions incident to corrosion of the internal walls of gaspipe lines and methods of eliminating rust scale from the lines areimportant considerations.



85

In several pipe lines tested the rate of flow was unsteady, indi-cating additional storage of gas or drainage of stored gas fromthe lines. The flow formulas are not applicable to unsteady flow

conditions. As was to be expected, the tests of commercial pipelines showed that the rate of gas withdrawal from a line is greater when the mean line pressure is decreasing than when it is constantand that the rate of gas withdrawal is less when the mean linepressure is increasing.

The assumption was made in the derivation of the flow formulasthat the effect of differences in elevation along a pipe line on thedelivery capacity is negligible. Most gas pipe lines operate withenough difference between the inlet and outlet pressures so thatthe effect of a difference in elevation at the inlet and outlet neednot be considered. However, for several of the lines studied thepressure drop was small, and there was a considerable differencein elevation at the ends of the line. For these lines it was necessary

to consider the effect of elevation difference on the flow of gasbefore comparisons could be made with pipe lines that were almosthorizontal.

A study was made of the physical and chemical properties of natural gas that influence the flow through pipe lines, which in-cluded the determination of the chemical compositions, the devia-tions from Boyle’s law, and the absolute viscosities of the differentnatural gases in the pipe lines studied. Since the physical proper-ties of natural gas depend upon its chemical composition, whichmay change with a change in pressure or temperature due to lique-faction of some of the constituents, it is advisable to determine thephysical properties of the natural gas through the applicable rangeof temperatures and pressures. Deviation of natural gas fromBoyle’s law and viscosity of natural gas are of minor importancein pipe-line flow calculations compared with the other factors thatinfluence the flow of gas through pipe lines. Enough data werenot obtained to determine the quantitative effect of deviation fromBoyle’s law on the flow of gas; however, it was interesting to notethat the metered volumes from clean lines and those calculated fromthe Weymouth formula agreed closely enough for practical pur-poses, although deviation from Boyle’s law was not considered inthe calculation of any of the data. The variations in viscosity be-tween natural gases of different chemical composition and for indi-vidual natural gases over the range of pressures and temperaturesobserved in commercial transportation of natural gas are small;and the effect of these small variations in viscosity on the flow of natural gas under commercial conditions is negligible.

The value of applying many minor corrections in pipe-line flowcalculations, such as those due to small differences in elevation atthe inlet and outlet of the line, reduced diameter fittings, bends,right-angle turns, metering stations, drips, type of joints, the lengthof pipe between Joints, deviations of natural gas from Boyle’s law,and variations in the viscosity of natural gas, often is greatlydiminished because of conditions pertaining to individual lines,such as unsteady rates of flow, storage or drainage of gas, thepresence of condensates, rust, and foreign material in the line, andthe variations in roughness of the interior walls of different lines.



86

The effect of such conditions often is many times greater than thesum of all of the measurable corrections and, for practical pur-poses of design, minor corrections may be neglected in the flow

formulas.APPENDIX L-DERIVATION OF GENERAL FORMULA FOR FLOW

OF NATURAL GAS THROUGH PIPE LINES

The value of knowledge of the conditions for which the formulasused to compute the flow of natural gas through pipe lines areapplicable, as well as of the assumptions made in the derivationof these formulas, justifies a detailed analysis of the basic equations.From such an analysis, differences between the formulas for the flowof natural gas may be understood more easily. The mathematicalderivations include the fundamental formula for the flow of com-pressible fluids and the general formula for the flow of naturalgas through pipe lines. The theory of the flow of compressible

fluids and the derivation of the fundamental formuIas are containedin most textbooks relating to thermodynamics.The general formula for the flow of natural gas through pipe

lines may be derived in a number of ways; the following methodseemed the most direct : Consider a length of pipe line between anytwo cross-sections which are normal to the walls of the pipe. Twoconditions are specified relative to the flow between these cross-sections :

(1) No work is done upon the fluid by external means.(2) The flow is steady; that is, the same weight of gas passes each

cross-section in the pipe line during an interval of time.

Gases usually are measured in volumetric terms rather than byweight; however, energy relations used in deriving the funda-mental formula for the flow of compressible fluids are presentedmore easily when a given weight of the fluid is considered. Con-version factors from weight to volume are introduced late in thederivation.

In the following derivation of the fundamental equation for theflow of a compressible fluid through a pipe line the first step is toapply the law of conservation of energy, balancing only the me-chanical energy. With respect to the arbitrarily selected length of pipe line, the mechanical energy balance for a unit weight of theflowing fluid is:

( 1 )

where subscripts 1 and 2 designate conditions at the inlet and outlet cross-sections, respectively.

The notation for equation (1) may be in any consistent system of units.Using the English system (foot-pound-second) the unit weight of fluid is 1pound, and the notation of the equation is:

X=potential energy of the pound of fluid due to its position, measured byits height in feet above an assumed datum plane.

pv=mechanical work performed in forcing the pound of fluid across thecross-section, where p=absolute pressure of the flowing fluid in pounds per square foot and v=the specific volume of the fluid at pressure p, in cubic feetper pound.

=kinetic energy of the pound of fluid, where U=velocity of the fluid jn

feet per second and g=acceleration due to gravity in feet per second per second.



I87

We = mechanical work done by and received by the pound of fluid due to itsexpansion while passing from the inlet to the outlet. In the flow of a com-pressible fluid through a pipe line, each pound of fluid in expanding from apressure p1 and specific volume v1 to a pressure p2 and specific volume v2 does

the work p dv upon the surrounding fluid, and in a pipe line where the

flow is steady each pound of fluid receives this same amount of work fromthe rest of the fluid in the line; therefore, each pound of fluid may be con-sidered as doing this work upon itself. Therefore,

W1 = the mechanical work per pound of fluid done in overcoming frictionalresistance between the inlet and the outlet.

Formulas can be derived from the mechanical energy balance(equation 1) for numerous flow conditions. In the development of a general formula for the flow of natural gas through pipe linesonly conditions that pertain to the commercial transportation of gas are considered, In applying equation (1) to the flow of naturalgas through pipe lines some of the factors are of relatively littlemagnitude and may be neglected; also, several assumptions aremade that result in simplifications without materially affecting thevalue of the resulting equation. Three of these assumptions or conditions are as follows:

(1) That the flow takes place under isothermal conditions ; thatis, the temperature of the gas remains unchanged. The temperatureof the gas agrees closely with that of the containing pipe, and sincenatural-gas pipe lines usually are buried the temperature of theflowing gas is not affected appreciably by rapid changes of atmos-pheric temperature. Temperature changes of the gas usually areseasonal and for simultaneous observations the temperatures at

the inlet and outlet of a section of pipe line are almost the same.(2) That natural gas behaves according to Boyle’s law, which

states that at constant temperature the volume occupied by a gasis inversely proportional to the absolute pressure (p1,v1 = p2v2).Therefore, for the assumption of isothermal flow, the products of pressure and volume that appear on both sides of equation (1) arecanceled and the equation becomes

(2 )

It is well-known, however, that no real gas behaves strictlyaccording to Boyle’s law. The deviation of natural gas from Boyle’slaw is significant at higher pressures and depends on the chemicalcomposition of the natural gas as well as the pressure and tempera-ture conditions under which it exists. Deviations from Boyle’s lawhave been determined by the writers for a representative groupof natural gases, and some detailed data pertaining to their magni-tude and effect on calculating rates of flows are presented in thisbulletin.

(3) That the pipe line is horizontal. Elevation changes along apipe line seldom are very large, and their effect in calculating theflow of gas usually is negligible; for liquid flows, however, theweight of the liquid makes it impossible to neglect differences in



88

elevation in writing an energy balance. Density of natural gasunder ordinary pipe-line pressures is small compared to that of liquids, and under most conditions the differences in potential ener-

gies of the gas due to elevation differences have relatively littlesignificance. The rate of gas flow usually is high enough to givelarge values for other terms in equation (1) compared to the valueof the difference between the terms X1 and X2. Therefore X1 andX2 may be eliminated in equation (1). Data from flow tests per-taining to observed differences in elevation, and their effect on cal-culating the rate of flow are presented elsewhere in this bulletin.

In accordance with the three assumptions, isothermal flow, nodeviations from Boyle’s law, and no differences in elevation, equa-tion (1) becomes

o r

( 3 )

(3a)

In the flow of natural gas through long pipe lines there is usuallyconsiderable pressure drop between the inlet and outlet; there-fore the flow conditions relative to “large percentage pressured r o p ” 4 3

are considered. As the pressure along the pipe line de-creases and the temperature remains constant, the volume of thegas increases and since the same weight of gas passes each pointin the line during the same interval of time and the pipe is of con-stant cross-sectional area the velocity of flow increases. There-fore, the energy relations are considered for an infinitesimal or differential length (dl). For the differential length dl equation(3a) is

( 4 )o r

(4a)

Before evaluating the term dW1 it is necessary to define brieflythe nature of the flow of natural gas in commercial transporta-tion. In the flow of fluids the movement of the particles of fluidthrough the pipe is either straight-line or turbulent. As the nameimplies, in straight-l ine flow the movement of the particles isparallel to the walls of the pipe and there are no transverse cur-rents, whereas in turbulent flow transverse or eddy ‘currents exist.Straight-line flow is considered to take place usually at low veloci-

ties. The work of Reynolds, discussed in this report, deals withthe relationship between the type of flow and the diameter of thepipe and the velocity, density, and viscosity of the fluid. In com-mercial transportation of natural gas through pipe lines, the flowis of the turbulent type and it is for this type of flow that theenergy balance is expressed.

The work done in overcoming frictional resistance through dis-tance dl is equal to frictional resistance times the distance through



89

which the resistance is overcome. For turbulent flow the frictionalresistance is proportional to the surface in contact with the fluid.approximately proportional to the square of the velocity, and pro-

portional to the density of the fluid. If the resistance is propor-tional to the square of the velocity it is also proportional to the

first power of the kinetic energy of a pound of the fluid.

Expressed by symbols, the frictional resistance is proportional to:

where dl = length of pipe, feet;Per = perimeter of pipe, feet;

U = velocity, feet per second;S = density, pounds per cubic foot.

Or frictional resistance= (f) (dl) (Per) ( 5 )

The f-term then becomes a ratio or proportionality factor to satisfythe equality; it is commonly called the coefficient of friction.

Expressed in symbols, the work in overcoming frictional resis-tance in the length of pipe, dl, is

(5a)

The weight of fluid in the length of pipe, dl, is equal to the cross-sectional area multiplied by the length and by the density of thefluid, or in symbols (A) (dl) (S).

The work to Overcome frictional resistance in the length, dl,per pound of the fluid, is

( 6 )

For a circular pipe the perimeter divided by the area equals

where D is the internal diameter of the pipe in feet. Simplifying

equation (6) and substituting

( 7 )

The coefficient of friction in equation (7) often is made by other writers to include one or both of the numerical constants in theequation; that is, the coefficient of friction in equation (7) has

been defined by some writers as (4f), (2f) or . These defini-

tions lead to confusion and uncertainty in the numerical signifi-cance of the coefficient of friction. In this bulletin the term “coef-ficient of friction” refers to f, as that symbol is used in equation (7),except when the context indicates that reference is being made tocoefficients of friction as defined by other writers.

In equation (4) dWc may be evaluated as

( 8 )



90

Substituting the values of dW f and dWc, as given by equations (7)and (8), in equation (4a),

( 9 )To simpli fy the solution of equation (9) U, dU, and p are

expressed as follows :

and p = (for an ideal gas),

where N = rate of flow of gas, pounds per second;U = velocity, feet per second; A = cross-sectional area of pipe, square feet;T = absolute temperature, oF.;

b= , where B = 1544 (gas constant) and M = molecular weight.

Substituting , and for U, dU, and p, respec-

tively, in equation (9),

(10)

Dividing both sides of equation (10) by v2,

(11)

Integrating between the limits of 0 (zero) and I for the lengthand v1 and v2 for the volume,

(12)

Since p1

v1

=p2

v2

= b T, equation (12) may be written,

(13)

Multiplying both sides of equation (13) by

(14)

Solving for by rearranging and substituting

the right side of the equation,

for b on

Since

andTherefore

(15)

(16)

(17)

(18)

For commercial pipe lines the ratio of the length to the diameter



91

of the pipe is large compared with the ratio of the inlet to the out-let pressure; accordingly in equation (15) the value of the term

l o g , is negligible in comparison with the value of term

and for ordinary computations may be neglected.Equation (15) may now be written

(19)

Since p12-p2

2 = 1442(P12-P 2

2) and p12=1442P1

2, where P ex-presses pressure in pounds per square inch, and since the volumeof gas (q) at temperature T and pressure p1 flowing per secondis equal to the cross-sectional area (A) multiplied by the velocity(U1), equation (19) may be written in the form,

(20)

Equations (19) and (20) are fundamental formulas for the flowof compressible fluids through long pipe lines.Equation (20) is simplified further for use in the natural-gas

industry and is expressed in variables that are measured easily.The chief function of any pipe-line flow formula is in its applica-tion to the design of pipe-line systems. For this reason it is desiredto express the relationships of the various factors influencing theflow in their simplest form to facilitate computation of any oneof the factors when values are given for the others. It is customaryto express the quantity of gas that will flow through a pipe of givendimensions, under various pressure conditions, in terms of cubicfeet per hour at definite pressure and temperature bases. To con-vert volume (q) in equation (20) from cubic feet per second attemperature (T) and pressure (P

1) to cubic feet per hour (Q)

at a base temperature of (To) and base pressure of (Po) it isnecessary to multiply by

It is customary also to express the diameter of the pipe in inches,the length in miles, and the area in terms of diameter. Conversionsto these units are obtained by substituting in equation (20) thefol lowing,

l = 5280 L, where l is in feet and L in miles,

D= , where D is in feet and d in inches, and

A=

The specific gravity of natural gas usually is determined rela-tive to air; therefore, equation (20) is made applicable to air. Thisis accomplished by substituting the value of B and M, where B

(“gas constant” in pv = = b T) equals 1,544 and M is the

molecular weight (about 29 for air). The quantity equals b

(above equation), and the value of the constant for air (ba) is 53.33.



92

The value of b for any gas (bg) equals the value of b for air (ba) divided by the specific gravity (G) of the gas compared toair, or

(21)

Introducing the preceding conversions and taking the value of constant g (acceleration due to gravity) as 32.17 feet per second per second, equation (20) becomes,

Simplifying,

(22)

(23)

where Q = volume of gas, cubic feet per hour at pressure base of P o andtemperature base of Ta;

K = numerical constant, 1.6156;To = temperature base defining 1 cubic foot of gas, oF. absolute;Po = pressure base defining 1 cubic foot of gas, pounds per square inch

P1

absolute;= inlet pressure, pounds per square inch absolute;

P2 = outlet pressure, pounds per square inch absolute;d = internal diameter of pipe, inches;G = specific gravity of gas (air=1.000) ;T = temperature of flowing gas, oF. absolute;L = length of pipe, miles;f = coefficient of friction, no units.

Including the coefficient of friction (f), the temperature base (To),and pressure base (Po) in the value of K, a modified form of equa-tion (23) is

(24)

Equation (23) and its modified form, equation (24), representthe most common forms of expressing the relationship between thefactors influencing the flow of natural gas through pipe lines.These two equations form the bases for most of the pipe-line flowformulas that have been developed.

In equation (23) the numerical value of K based on the valuesof the constants listed in the report is 1.6156. The value of K1 inequation (24) is subject to the value given to the coefficient of friction (f) and to the temperature and pressure bases used todefine 1 cubic foot of gas. A number of formulas of the general

type of equation (24) have been developed ; some of them includethe specific gravity (G) and the absolute temperature of the flow-ing gas (T) in the value of K,. However, fundamentally, the mostimportant factor influencing the value of K, is the value assignedto the coefficient of friction. In several formulas where the coef-ficient of friction is expressed as a function of the diameter of thepipe, the exponent of the diameter (equations 23 or 24) is changedfrom 2.5 to include the relationship indicated by the coefficient of fr iction.



93

APPENDIX 2.- REYNOLDS' CRITERION OF COEFFICIENT

OF FRICTION

The experimental work44of Reynolds and of subsequent investi-

gators malting use of the Reynolds criterion, because of its contri-bution to the knowledge of the motion of fluids and the general lawsof resistance in fluid flows, deserves a brief review. From experi-ments on the flow of water through glass tubes Reynolds observed,by introducing coloring matter, a marked change in the nature of the flow, depending on the velocity when all other conditions affect-ing the flow were constant. At low velocities the flow proceeded ina steady straight-line manner, in which it appeared that all of theparticles moved parallel to the walls of the tube. The velocity wasincreased and straight-line flow continued until at a certain valueof the velocity (critical) a definite change in the character of theflow occurred. Small eddies or swirls were noticed, giving the flowa turbulent appearance in which the particles of colored matter no

longer moved parallel to the walls of the tube.The experimental data on flow of water at various temperaturesthrough different-diameter tubes indicated that the velocity (criti-cal) at which the character of the flow changed from straight-lineto turbulent depended upon the diameter of the tube and the densityand viscosity of the fluid. In other words, the character of the flowdepended upon the diameter of the tube and the density and vis-cosity of the fluid as well as on the velocity of flow. From theseobservations Reynolds arrived at a ratio involving these four vari-ables, the numerical value of which offered a means of predictingthe nature of the flow. This ratio or criterion of Reynolds consistsof the product of the diameter, density, and velocity divided by theviscosity, or, expressed in symbols:

where D = diameter, Reynolds’ criterion ,

U = velocity,S = density,Z = viscosity.

The numerical value of this ratio presented itself to Reynolds asa criterion not only of the character of the fluid flow but also of the resistance to flow, for which he developed an equation. Follow-ing the work of Reynolds, Lord Rayleigh pointed out that the Rey-nolds law of resistance was a particular case of a more general lawor relationship between the resistance and the dimensions of thetube, the velocity, density, and viscosity of the fluid.45 This rela-tionship of Rayleigh’s expressed mathematically is

(25)

where R = frictional resistance per unit of pipe surface in contact withflowing fluid (absolute units),

φ = function of,

= Reynolds’ criterion.



94

The relationship expressed by equation (25) offered a definitemathematical expression for a coefficient of friction in fluid motion,which was a considerable improvement over the older expressions,since they often lacked any mathematical significance by which theymight be correlated. A recently published study of fluid flow by

Beale and Docksey gives an excellent account of the use of

in expressing coefficient& of friction.46

Briefly, from their discus-sion, the values of the coefficients of friction used by various writers,expressed in their formulas by such symbols as F, f, B, C, K, etc.,

usually are equal either to and always equal to

some multiple of .

In appendix 1, Derivation of General Formula for the Flow of Natural Gas through Pipe Lines, the letter f with the significance

given it in equation (5) became the means of expressing the coef-ficient of friction in the general equation of flow (equation 23).Equation (5) states that:

Frictional resistance through a small length of pipe (dl) is

The frictional resistance per unit area of pipe surface exposed tothe fluid, or R1, is equal to the frictional resistance per unit of length divided by the product of the perimeter of the pipe and theunit length. Expressed in symbols,

or

The frictional resistance per unit of surface exposed to the fluidexpressed in absolute units (poundals) R, is

from which

Applying the general law expressed by equation (25), Stantonand Pannell47 found that it not only correlated their results for allconditions of flow for both water and air, but the same relation-

ship between existed for the flow of either air or

water through geometrically similar pipes. They point out thatgeometric similarity as applied to different pipes should includethose irregularities in the surface of the walls that constitute rough-ness. However, this condition could not be fulfilled exactly, and theexperiments were made with smooth-drawn brass pipes; the gen-eral agreement of the results for different pipes indicates that



95

slight variations in the roughness did not have a marked effect onthe resistance within the range of diameters used.

It is not necessary to describe in detail the experimental work

of Stanton and Pannell or of any of the numerous investigators of

the flow of fluids who have studied the application of the Reynoldscriterion to determination of the coefficient of friction. The experi-ments of these investigators cover a wide range of fluids and flowconditions. Figure 24 illustrates a method of plotting the results



96

and contains some of the experimental curves of Stanton and Pan-nell, Reynolds, Darcy, Lander, Beale and Docksey, and several other investigators of the frictional resistance in the flow of fluids.Table 17 contains the source and description of the data from which

the curves in figure 24 were plotted. The range of the Reynoldscriterion in figure 24 has been limited to the turbulent flow regionsince straight-line flow conditions seldom are found in commercialgas-pipe lines. The curves in figure 24 represent a mean or averagerelationship as found by each of the various investigators and in

TABLE 17.- Description of data in figure 24

any series of experiments where more than one pipe or tube wasused, the actual data if plotted would not fall on the same curvebut would form a group of curves parallel and close to the averagecurve. Many of the data on which figure 24 is based are of littlevalue for direct application to commercial gas pipe l ines. Thecurves indicate that different coefficients of friction should be usedfor pipes of different degrees of roughness or for other variationsin the similarity of the pipe dimensions and flow conditions.

The same relationship between the coefficient of friction and theReynolds criterion for different pipes could exist only if the require-



97

ments of the similarity theory, which is applicable to numerous” andvaried phenomena wherein a similarity of motions, dimensions, or other properties or characteristics is possible, can be fulfilled. True

similarity exists only when the values of all variables influencingthe particular phenomenon change proportionally.The theory of similarity as applied to the flow of fluids imposes

several important requirements upon the conditions under whichthe flow occurs. For true similarity to exist between the flows of any two fluids in any two pipes there must be a geometric similaritybetween all dimensions of the two pipes, and these dimensions mustinclude all physical characteristics of the pipe material that affectthe flow under any conditions. If the two pipes are of such lengththat joints are needed the dimensions should include welding,couplings, fittings, ratio of pipe diameter to length of pipe between

joints, and other construction features.Under this rigorous interpretation of the requirements imposed

by the theory of similarity in its application to the flow of fluids itis questionable whether true similarity is possible except under ideal or laboratory conditions. Certainly, under the conditions of the flow of fluids commonly observed in practice, it is doubtfulwhether true similarity between any two sets of conditions exists.The commercial conditions under which fluid flows occur are numer-ous and varied. Although many factors are common to all fluidflows, often there are particular factors difficult to evaluate in theconditions of each flow that tend to destroy similarity. Neverthe-less, as demonstrated by the experiments of Reynolds, Stanton andPannell, and others, a reasonable or satisfactory similarity of thefactors influencing fluid flows can be obtained for certain limitedflow conditions. However, to evaluate the experimental work of Reynolds, Stanton and Pannell, and others in respect to its appli-

cation to fluid flows in commercial practice it is necessary to under-stand the conditions under which most of these experimental datawere obtained. Many different conditions of flow and pipe charac-teristics were used in these experiments (table 17 and fig. 24), butlimits for these conditions had to be established to obtain a. particu-lar relationship between the coefficient of friction and the Reynoldscriterion. If no restrictions are placed on the conditions of flow(pipe characteristics, etc.) a large number of relationships betweenthe coefficient of friction and the Reynolds criterion seem possible.This is well-illustrated when relationships between the coefficientof friction and the Reynolds criterion obtained from flows throughtubes of different material or of different wall roughness are com-pared. For further illustrations of numerous relationships between

and reference is made to the work of Kemler,48 White,49

and Beale and Docksey50 on tubes of different internal-wall rough-

nesses, diameters, various degrees of linear curvature, and tubeshaving various types of joints and variable distances between joints.



98

It is bel ieved that much of the observed variat ion in the func-

tion of is due to di f fer ences in roughness and to the fact

that for dif ferent diameter tubes of the same material the effectof roughness is not the same. Pipe or tube materials (steel, i ron,brass, copper, lead, glass, rubber, etc.) have surfaces of defini tephysical or geometrical character, and while the character of thesurface may be uniformly constant for any one material i t is notthe same for all materials, and therefore pipes of the same diameter but of dif ferent materials have walls of dif ferent roughness. Withreference to the effect of roughness for dif ferent diameter tubesof the same material, McAdams51 states,

It has also been pointed out by several writers that the size of internalprojections at the wall might affect f. If this assumption is made, dimensional

analysis tells us that f should be a function not only of but also of

the ratiowhere a represents the size of the projection. If this be the

case, with pipes of the same material and having the same value of a, theeffect of diameter should not be equal to the effect of velocity as D wouldappear in two functions instead of one.

He states further that some observations made on the flow of water through clean pipes shows that this second function exists; how-

ever, the effect of the function is not nearly so marked as the

effect of the group , and unless large changes in diameter are

considered the value of f for a given kind and condition of pipe may

be taken as a function of alone. Therefore, pipes of the same

material become “relat ively smoother” as the diameter increases,and it is possible to have pipes of different diameters and materialswith the same “relat ive roughness.” For dif ferent-diameter pipes,the roughness of the walls should change i f the requirements of geometric s imilari ty are to be satisf ied.

In v iew of the preceding d iscuss ion re la t ive to d iameter androughness i t i s reasonab le to expec t , fo r the same va lue o f theReynolds criterion, an increase in the coefficient of friction for thesmaller-diameter pipe of the same material, an expectat ion well-s ubs tan t i a ted by ex pe r imen ta l da ta o f t he w r i t e rs ( s ee f i g s . 7and 8).

I f a means o f eva lua t ing wa l l roughness were ava i lab le re la -t ionships between the coeff ic ient. of fr ict ion and the Reynolds cri-

terion for pipes of dif ferent roughness might be correlated empiri-cal ly, and i t might be possible to develop an equation taking intoaccount the effect of various degrees of roughness. To be of themost value such a correlation or equation would have to be basedupon extensive experimental data. For tubes of a part icular ma-terial (surface roughness) i t may be possible that an exponent or

power of the diameter other than that used in the ratio could



99

be determined empirical ly that would make a constant function of

sa t is fac to ry fo r a l l -d iamete r p ipes o f the same mate r ia l .

Theoretical discussions of such factors as roughness and its effecton fluid flow, and the mathematical interpretation of these effects,could be extended to great length. Surface roughness of pipe hasbeen used only as an example of one of the factors that influences imi la r i ty ; the re a re many o ther fac to rs sub jec t to as much d is -cussion and analyses. This is true part icularly in the commercialt ranspor ta t ion o f gases where , in add i t ion to d i f fe rences in wa l lroughness , the re a re numerous and var ied types o f p ipe jo in ts ,variations in the ratio of distance between joints to the diameter of the pipe for lines of different diameters, tolerances in pipe diametersand cross-sectional shapes, and numerous construction features and“ f i t t ings ” pecu l ia r to ind iv idua l p ipe l ines . There fo re , the accu-racy with which any relationship between the coefficient of friction

and the Reynolds criterion can be used to determine the coefficientof fr ict ion over a wide range of al l condit ions is questionable. I tis apparent that for any one specific pipe line or tube the factor of s imilari ty of condit ions ordinari ly is a constant, and equation (25)becomes an excellent means of analyzing the flow data.

Briefly, the result of all the work that has been done on the rela-tion of the coefficient of friction to the Reynolds criterion has beento give an accurate and desirable method of analyzing the frictionalres is tance o f f lu ids f low ing th rough geometr ica l ly s imi la r p ipes .However, its application to commercial problems pertaining to gaspipe l ines is l imited, because factors often are present for indi-v idual pipe l ines that destroy or l imit the similari ty in dimensionsnecessary for any one relationship to exist for all pipe lines. These

factors are numerous, and often of uncertain nature, and at presentmeans of comparative measurements are lacking. If more of themcould be incorporated into the analysis of the coefficient of frictionit is conceivable that more exact flow formulas could be developedfor gases.

APPENDIX 3 .-FORMULAS HAVING THE COEFFICIENT OF

FRICTION EXPRESSED AS A FUNCTION OF

THE REYNOLDS CRITERION

Relationships between the coeff ic ient of fr ict ion f and the Rey-

no lds c r i te r ion have been de te rmined exper imenta l ly by a

number of investigators and used to evaluate the coefficient of fric-t i o n i n t h e g e n e r a l f l o w e q u a t i o n ( e q u a t i o n 2 3 ) . T h e r e f o r e , i npresenting formulas of this type, s ince they vary from each other only in the value of f, it is necessary only to give the several equa-tions by which f may be calculated for substi tut ion in the generalf low equation.

From a group of carefully conducted experiments on the flow of air in zinc pipes of approximately 1 and 1.5 inches internal diame-



100

ter and 50 feet long Fritzsche52 obtained the fol lowing expression

for the coefficient of fr iction.

(26)

where A = numerical constant depending on units,d = internal diameter of pipe,S = density of air, andU = velocity of flow.

It is apparent from equation (26) that Fritzsche did not give anydirect considerat ion to the viscosity of the air used in his experi-ments. Obviously equat ion (26) is not a relat ionship between the

coeff ic ient of f r ict ion f and the Reynolds cr i ter ion ; however,

i t may be considered to be of this general type of relat ionship. Atleast one published analysis 53

of Fritzsche’s data was made on thisbasis.

The Fr i tzsche f low formula has received considerable recogni -t ion and perhaps more use than any other formula of this general

type. To show how various relat ionships between f and , or

between f and some factor similar to , may be substituted in

the genera l equat ion fo r f lu id f low to g ive var ious f low fo rmulasof this type, the fol lowing derivat ion of Fr i tzsche’s f low formula isoffered as an example.

E q u a t i o n ( 2 6 ) m o d i f i e d 5 1

units may be wri t ten,neg l ig ib ly and expressed in Eng l i sh

(27)

where N = pounds of fluid flowing per second.

To express equat ion (27) in te rms o f cub ic fee t per hour (Q) o f a gas of specif ic gravi ty G (air= 1.000) at a pressure base of P o

pounds per square inch absolute and a temperature base of T ooF.

absolute the value of N becomes

By substitution of the above value of N equation (27) becomes

(28)

The general formula for the flow of gas (equations 22 and 23) mayb e w r i t t e n

(22a)



I101

Subst i tu t ing the va lue o f f f rom equat ion (28) in equat ion (22a)and s imp l i f y ing ,

(29)

Fr i t zsche ’s f low fo rmula was wr i t ten fo r the f low o f a i r and ex-

pressed in cub ic fee t per minu te a t 60 o F . t empera tu re and an

absolute pressure base of 14.7 pounds per square inch, through a

pipe l feet in length. By making the fol lowing subst i tut ions,

Qa = cubic feet per minute=Q/60,G=1.000 for air,

T, To=60° F. (520’ F. absolute),Po=14.7 pounds per square inch absolute,

equat ion (29) becomes(30)

which is the Fritzsche flow formula as it is commonly written. 55 Thecoefficient 35.08 obviously applies only to the flow of air expressedin cubic feet per minute at bases of 60 o F. temperature and 14.7pounds per square inch. absolute pressure. A series of coefficientsfor gases of different specific gravities (air= 1.000) usually is givenfor use with the formula.56 Fritzsche’s formula is perhaps the leastcomplicated of any of the formulas of i ts general type. Inspect ionof equat ion (29) shows that no addit ional terms have been addedto the general equation and that the main d i f ference is the powers

to which the terms are raised.Other equat ions in which the coeff ic ient of f r ict ion is a funct ion

of the Reynolds criterion are not incorporated into the general f lowformula as easiIy as the Fri tzsche equation. In fact, most of theempir ica l equat ions developed f rom the exper imental work of nu-

merous inves t iga tors o f the re la t ionsh ip be tween f and are in

themselves not simple and i f subst i tuted in the general f low for-mula in the way the Fritzsche equation has been substituted resultin an exceedingly complex formula rather unwieldly in appl icat ionto f low problems. To avoid compl icat ing the general f low formulato such an ex ten t tha t app l i ca t ion becomes d i f f i cu l t the mathe-mat ica l a l te rna t i ve o f ca lcu la t ing the coef f i c ien t o f f r i c t ion ( f rom

a g i v e n r e l a t i o n s h i p b e t w e e n f a n d a n d k n o w n o r a s s u m e d

values of D, U, S, and Z) and substituting this value in the generalf l o w f o r m u l a u s u a l l y i s p r e f e r a b l e . T h i s a l t e r n a t i v e m e t h o d o f calculat ion often involves the use of tables, curves, and approxi-mat ions which are l ike ly to become burdensome.



10 2

Lees’ equation57 for the relationship between the coefficient of

friction and the Reynolds criterion is

(31)

where = an expression for the coefficient of friction as defined by some

investigators = see appendix 2, Reynolds’ Criterion of the

Coefficient of Friction),V=kinematic viscosity (absolute viscosity divided by density).

In terms suitable for substitution into the general formula (equa-tion 23) and converting from kinematic to absolute viscosity, Lees’equation becomes

(32)

which may be substituted in equation (23) giving a complex flowformula, or may be used to determine the value of f directly when

is known or assumed. Lees’ equation was written to satisfy

the experimental data of Stanton and Pannell,58 and over the rangeof Reynolds’ numbers and conditions in these experiments it gives anexcellent mathematical expression of the relationship between thecoefficient of friction and the Reynolds criterion. The experimentsof Stanton and Pannell on smooth drawn brass extended to Reyn-olds’ numbers of about 450,000. However, the Lees equation basedon these experiments has been applied to much larger values of theReynolds criterion.

Another empirical equation somewhat less complex than that of Lees and based on the same experimental data of Stanton andPannell, together with the data of Saph and Schoder and those of Schiller, was published by White

59in the following form :

(33)

where F=R in equation (31).

In terms suitable for substitution in the general formula (equa-tion 23), equation (33) becomes

(34)

According to White, equations (33) and (34) are recommendedonly for Reynolds’ numbers between 2,300 and 100,000. For thisrange equation (33) gives results almost identical with Lees’equation.



103

McAdams and Sherwood60

present the following equation for the coefficient of friction (f).

(35)

where z=absolute viscosity, centipoises;d=internal diameter, inches;s=specific gravity (water=1.000).

Equation (35) is based on extensive experimental data publishedby numerous investigators of fluid flow. McAdams and Sherwood

recommend its use for values of greater than 1.5. Substitut-

ing consistent units (centimeters,’ grams, and seconds; or feet,

pounds, and seconds) for those used in the ratio equation (35)

may be written

(36)

Values of greater than about 11,500 correspond to the

values of greater than 1.5 for which the equation is recom-

mended by &Adams and Sherwood.Many other equations based on experimental relationships be-

tween the coefficient of friction and the Reynolds criterion couldbe, and very likely have been, written from the data in the litera-ture on the flow of fluids. The four equations that have been pre-sented are based on reliable data; although they do not all give thesame value for the coefficient of friction, individually they expressthe relationship between the coefficient of friction and the Reynoldscriterion for the conditions upon which they are based.

APPENDIX &-DESCRIPTION OF PIPE LINES TESTED

The general construction and operating features of pipe linestested are described as follows :

Lines 1 and 2.-These lines represent two different sections of a pipe-line system in the Gulf coast area. The pipe in both lineswas in 30-foot lengths. Orifice meters were installed at 35 to 40-mile intervals over the entire length of the system; distances be-tween the orifice meters determined the lengths of the test sections.The gas was relatively free of gasoline vapors, having passedthrough a gasoline-extraction plant at the inlet of the system. The

section known as line 1 crossed two small rivers, where the systemconsisted of five lo-inch lines with a lo-inch valve at each end of each line,

Line 3.-Line 3 represents a section between two orifice metersin a large transmission system. It is paralleled by an l&inch lineto which it is connected by three 12-inch "tie-overs" which wereclosed while the tests were in progress. There was an overhead river



104

crossing in the test section; midway across the river an expansionloop 40 feet high provided for l inear movement of the l ine result-ing from temperature changes. Except for this loop the l ine madeno right-angle or sharp bends. The only drip in the section testedwas a shor t d is tance ups t ream f rom th is expans ion loop, and i tcol lected very l i t t le condensate during the tests.

The gas had passed through a gasoline-extraction plant approxi-mately 75 miles upstream from the test section. A recording gravi-tometer, checked every 2 hours against a suspension-type gravi tybalance, indicated that the specific gravity of the gas did not changemore than 0.004 during any one day.

Line 4. - Line 4 contained large quantities of finely divided par-ticles of rust, caused by internal corrosion of the pipe, which mayhave been formed within the test section or carried by the gas fromother sect ions in the system. Rust part ic les that had accumulatedin the meter settings to depths of 1 or 2 inches were removed beforethe tests were begun. Although most of the gas had passed through

a gasoline extraction plant before entering this l ine, some of it wasfrom wel ls producing direct ly into the system. The dr ips at thesewel ls d id not remove a l l o f the gasol ine f rom the gas and conse-quently large quant i t ies of gasol ine had been carr ied into the testsection.

Line 5. - This l ine served as a loop to the 12-inch section of am a i n t r a n s m i s s i o n s y s t e m c o n s i s t i n g o f 1 2 - , 1 6 - , a n d H - i n c h -diameter pipe. Several large gathering lines were tied into the sys-tem near the inlet of l ine 5. Line 4 represents a section of one of these gathering l ines. Although l ine 4 contained large quanti t iesof rust, there was no evidence of internal corrosion or of any appre-ciable accumulation of rust in l ine 5. The orif ice plates at both theinlet and outlet ends of l ine 5 had not been removed for 2 months,

and a small amount of rust had settled on the inlet plates; however,the outlet plates were clean.

The tes t sec t ion c rossed two r i vers where i t cons is ted o f twoparal lel 12-inch l ines; these crossings were 4,598 feet and 1,579feet long.

L ine 7 . - Th is l i ne represents the f i r s t 32 mi les o f a sys temapproximately 60 miles long, starting at a gasoline-extraction plant.The test section included 17 miles of l ine over mountainous coun-try where the l ine made 14 ninety o turns, 37 forty-f ive o turns, andnumerous other turns of less than 45 o . The elevat ion of the testsect ion was about 200 feet at the inlet and about 900 feet at theoutlet; in the mountains the highest elevation was about 1,250 feet.Drips had been installed in many of the sags and low places in thel ine; however, almost al l of the condensate was removed by thef irst few drips after the gasol ine plant.

Between the inlet and outlet of the test section a volume of about2,500 cubic feet of gas daily was delivered to domestic consumersalong the r ight of way.

Recording thermometers were provided at each of the or i f ice-me te r se tt ings. S t ra i gh ten ing vanes o f S - i nch ga l vani zed t ubes30 inches long were installed upstream from the orifice plate.

L i n e 8. - Line 8 i s a sec t ion o f 14- inch p ipe approx imate ly 60miles long of a system consisting of 14-, 18-, and 20-inch-diameter



105

pipe; the diameter increasing in the direction of flow. The line con-sisted of 30-foot lengths of pipe connected by welded “bel l-and-s p igo t ” j o i n t s , w i th “ex pans ion j o in t s ” we lded i n to the l i ne a t

intervals of 11 pipe lengths. The r ight-of-way decreased graduallyin elevation from about 800 feet at the inlet to about 200 feet atthe outlet.

Line 9. - This l ine represents a complex system, consist ing of

7.707 miles of single 12-inch-, 5.882 miles of double 12-inch-, and31.390 miles of s ingle 16-inch-diameter pipe. Pressure observa-tions were made at both the inlet and outlet meters and at point Aduring part of the test and for the remaining t ime at point B (seef ig . 10 ) . In th is way a tes t was made o f the to ta l leng th and o f separate sections of the line at the same time. For one period datawere ob ta ined (1 ) on the en t i re leng th o f the l ine , (2 ) on sec -tion 9A, consisting of the single 16-inch and double 12-inch lines,and (3) on section 9D, consisting of the single 12-inch line. Obser-

vations then were taken at point B instead of at A, and during thisperiod data were obtained (1) on the total length of the line, (2) onsection 9B, consisting of the single 12-inch and double 12-inch lines,and (3) on section 9C, consisting of the 16-inch-diameter pipe.

The ma in - l ine va lves in the 16- inch-d iamete r sec t ion were 12inches in d iamete r and ins ta l led in pa i rs para l le l to each o ther .The two va lves were connected wi th 12- by 18- inch n ipp les to16-inch headers joined to the main l ine.

There was an appreciable difference between the volumes of gasmeasured at the inlet and outlet of the system. The volume measure-ments at the inlet meter, which was 3 miles from the discharge of a compressor stat ion, probably were in error because of pulsatingf low th rough the o r i f i ce . The ou t le t -o r i f i ce f lange was ins ta l led

d i rec t ly in the ma in l ine w i thou t a bypass , mak ing i t d i f f i cu l t tochange ori f ice plates. The demand on this system was subject torap id and p ronounced changes , requ i r ing ’ an o r i f i ce p la te la rgeenough to handle high rates of flow. The volume of gas that passedthrough any test section was assumed to be that measured by theorifice meter in that section in preference to an average of the inletand outlet metered volumes, because of leakage and of drainageor storage of gas in the other sections between the two meters.

L ine 10 . -The e leva t ion o f th is l ine was abou t 100 fee t a t theinlet and about 1,050 feet at the outlet. A compressor stat ion 2miles upstream from the inlet of the test section caused disturbancesin the f low that precluded satisfactory ori f ice-meter measurementat the inlet.

Lines 11, 12, and 13. - These l ines represent dif ferent parts of a large transmission system : line 12 was a section of the main line;line 11 connected into the main line about 45 miles beyond the out-let end of section 12; and l ine 13 del ivered gas to a compressor s ta t ion , wh ich supp l ied mos t o f the gas fo r l ine 11 . Some gasen te red l ine 11 d i rec t f rom h igh-p ressure gas we l ls ad jacen t tothe inlet.

Large amounts of water were present in l ine 11 at the t ime of the tests because the drips at the wells and in the line were inade-quate. The drips in l ine 12 were almost free of water, and there



106 0

were no unusual features in the design or operation of this l ine.Two dif ferent lengths were used for l ine 13; the longer length isknown as line 13 and the shorter length as line 13A. Line 13A was

included in line 13.Line 14. - This section of line began near a charcoal-adsorptiongasoline plant and ended at a main-line check meter. The gas leav-ing the gasoline plaint contained large quantities of water and hada r e l a t i v e l y h i g h t e m p e r a t u r e ( 8 5 o F . ) . Du r ing the tes ts l a rgequanti t ies of water were removed continual ly from the l ine by thedrips near the inlet of the section. Because of the hilly right-of-waythere were numerous sags and low places in the line, many of whichwere not provided with drips.

Line 15. - This section of line began near an oil-absorption gaso-l ine plant and terminated at a compressor stat ion. The l ine con-ta ined la rge quan t i t ies o f rus t sca le and some l iqu ids . A h igh-pressure scrubber filled with hemp was installed to remove the rustscale from the gas before it entered the compressors. About a year before the flow tests several low places in the line were tapped, andquanti t ies of condensate and rust scale were removed. Moreover,the line was subjected to a pressure test, and leaks were repaired(see table 2).

L ine 17 . - Two d i f fe ren t leng ths were used fo r th is l ine . Thefirst test was made of a section extending from a point about 1/2 miledowns t ream f rom a gaso l ine -ex t rac t ion p lan t near the in le t to amain-line orifice meter. The drips in the first 2 miles from the gaso-l ine plant col lected large volumes of water. For this reason, dur-ing the second test the inlet of the section was taken at a pointabout 2 1/2 miles downstream from the gasoline plant. The tempera-ture of the gas leaving the gasol ine plant was relat ively high; atthe inlet for the f irst test i t was 85 o to 90o F., and at the inlet of

the second test section it was about 62 o F. The temperature at theoutlet of both test sections was 58 o to 62o F.

It had been assumed that the drips in the f irst 2 or 3 miles of l ine from the gasol ine plant had col lected al l of the l iquids. How-ever, in repairing a leak 9 miles from the gasoline plant a few daysafter the tests were completed a low place in the line was tappedand found to contain large quanti t ies of absorption oi l and water.

Line 18. - This line, part of the same general system as lines 4and 5, extends from a gasoline plant to line 4. Due to internal cor-rosion of the pipe large amounts of rust scale were present in thel ine. The volume of gas f lowing through the l ine was determinedby an o r i f i ce mete r a t the ou t le t end . A l though an o r i f i ce mete r was installed about 1 1/4 miles from the gasoline plant and near the

point chosen as the inlet of the test section the differential pressureacross the orifice surged so much (about 15 inches of water) thatvolumetric observations were not taken at this meter.

Lines 19 and 20. - These two lines are separate parts of a systemoriginal ly intended for transport ing oi l . However, the system never had been used fo r th is purpose and was recond i t ioned to ca r rynatural gas. The section known as line 19 began near the gas fieldsupplying the system and ended at an oil-absorption gasoline plant;line 20 began at the discharge of this plant and ended at a main-l ine pressure-regulator stat ion. Oil pump stat ions had been bui l t



107

a long the l ine approx imate ly every 18 mi les , and th ree o f thesewere on line 20. Bypasses had been built around these stations sothat the gas passed through eight right-angle bends at each station.

The sys tem had been car ry ing na tu ra l gas abou t 6 months , andat the t ime the l ine was condit ioned for gas i t was tested to 500pounds hydraul ic pressure. I t was d i f f i c u l t t o d ra in wa te r f romthe l ine, and there was doubt whether al l of the water had beenremoved.

Line 21. - This l ine is of special interest because i t is the onlyline made of wrought-iron pipe included in the tests and had beenin operation longer than any other l ine tested. At the t ime of thetests the line had been reconditioned, subjected to a pressure test,and all leaks repaired (see table 2).

Line 22. - The gas in this line came from the same gasoline plantthat delivered gas into line 17. The inlet of line 22 also was takennear the gaso l ine p lan t . Very l i t t le condensa te was found in thedrips on line 22, indicating absence of water, whereas the drips inline 17 continually collected liquid. This difference may be due, inpart at least, to the fact that the pressure of the gas from the gaso-line plant was reduced before it entered line 22. This point is dis-cussed under Removal of Liquids from Gas Pipe Lines.

Line 23. - During part of the test on this line the inlet pressureswere taken near a junction point of several small gathering l inesfrom nearby compressor stations. The orifice meter at the inlet wasnot used for volume measurements because of surging of the differ-en t ia l p ressure caused by the compressors . Moreover , the s ta t icpressure of the gas at the inlet fluctuated as much as 3 or 4 pounds ina 4-minute interval. For the remaining period of the test the inlet

of the test section was selected about 1 mile beyond the junctionpoint of the several gathering l ines and the pressures were muchsteadier.

Line 25. - The elevation at the inlet of this l ine was 4,446 feetand at the outlet 3,715 feet, giving a difference of 731 feet in eleva-t ion, There was an ori f ice meter at the inlet of the l ine. Approxi-mately 30 feet downstream from this meter was an 8-inch pressureregulator with a I- inch bypass, which were opened completely dur-ing the tes ts . Under the ex is t ing ra te o f f low the p ressure d ropth rough the opened regu la to r and bypass was abou t 0 .1 inch o f

w a t e r .The leakage on line 25 was so great that it could not be neglected

when the metered volume of gas at the inlet was compared withthe volume calculated by pipe-l ine f low formulas. From companyrecords of the leakage from this line (table 2) a correction of 5,000cubic feet of gas per hour was determined to apply to the observedinlet volume; that is, the observed data in table 3 will give rates of f low abou t 5 ,000 cub ic fee t per hour g rea te r than the mete redvolumes listed in table 4.

Line 26. - The elevation at the outlet of this line was 4,000 feetand at the inlet 850 feet, a difference of 3,150 feet in elevation. Anorifice meter at the inlet was about 22 feet upstream from a regu-lator which was opened completely during the tests, and under the



108

existent rate of f low the pressure drop through the regulator wasabout 5 pounds per square inch . The ma in - l ine p ressure a t theinlet was taken below the regulator. The flow through the line was

fa i r l y s teady fo r shor t in te rva ls . The p r inc ipa l consumer was alarge cement plant, and the demand from the l ine was subject tothe operation of this plant. The middle section of this l ine, con-s i s t i ng o f app rox ima te l y 2 m i l es o f 6 .625 - inc h p ipe , was ov e r e x t r e m e l y m o u n t a i n o u s c o u n t r y a n d h a d n u m e r o u s r i g h t - a n g l eturns and bends of lesser angles.

L ine 27 . - Th is l ine cons is ted o f 6 - inch- and 8 - inch-d iamete r pipe; the 8-inch section was toward the outlet end. Both the 6-inch

and 8-inch sections were tested separately and designated as lines27B and 27A, respectively. The volume of gas was determined byan orifice meter at the discharge end of the 8-inch section. A regu-lator 0,419 mile upstream from the orifice meter necessitated takingthe main-line pressure upstream from the regulator, and that point

was selected as the outlet of the test section. The arrangement was‘no t en t i re ly sa t is fac to ry fo r the tes ts . The mete red vo lume maydiffer from the actual volume flowing through either section becauseof leakage and storage or drainage of gas in the sections betweenthe ori f ice meter and the test section. For example, between thesection of 6-inch pipe ( l ine 27B) and the ori f ice meter there wasabout 4 mi les o f 8 - inch l ine . Any leakage o r i r regu la r i ty o f f lowin this 8-inch section would be reflected in the volume through theorifice and thereby introduce an error when this volume is assumedto be that passing through the 6-inch section.

Line 28. - The elevation at the inlet of this l ine was 6,070 feetand at the outlet 4,960 feet, a difference of 1,110 feet in elevation.The inlet of the l ine was near the discharge of an oi l-absorptiongaso l ine p lan t . The p ressure o f the gas a t the d ischarge o f theplant was reduced from about 200 pounds per square inch to about100 pounds per square inch before the gas entered the l ine. Theheader at the plant, directly below the pressure regulator, collectedlarge quanti t ies of water, whereas the drips in the main l ine col-lected small amounts.

L ine 29. - Th is l ine is a sect ion of a 6- inch-d iameter systemextending from several gasol ine plants to a large industr ia l con-sumer . The sec t ion inc luded one r ive r c ross ing 1 ,760 fee t longcons is t ing o f two 6 - inch l ines . The par t icu la r po in t o f in te res t inthis test is that the gas contained about 88 percent ethane accord-ing to a combustion analysis.

APPENDIX 5. DERIVATION OF SPECIAL FORMULAS FORDESIGNING PIPE-LINE SYSTEMS CONSISTING

OF PARALLEL LINES

The der iva t ion o f spec ia l fo rmu las fo r des ign ing para l le l - l inesystems is based on Weymouth’s formula for the flow of gas throughpipe l ines. The expressions for equivalent lengths and diametersused in this derivation are given in the fol lowing discussion.



109

The flow of a given quantity of gas under the same pressureand temperature conditions and of the same specific gravitythrough any two pipe lines with different diameters and lengths

is expressed as (37a)

(37b)

By definition the value of Q, Po, To, P1, P2, G, and T are the samein both equations, and by equating the right sides of the equationsand canceling the like factors the equality becomes

or

(38)where L1= the equivalent length of any pipe of length L2 and diameter d2 interms of diameter d1.

By rearranging equation (38)

(39)

where d1=the equivalent diameter of any pipe of given diameter d2 and lengthL2 in terms of any other length (L1).

The equivalent diameter for use in calculating the flow of gasthrough complex systems consisting of parallel lines is obtained asfollows. The volume flowing through the system is

(40)

where d1, d2. . .dn are the diameters of the individual lines.

The equivalent diameter (do) of a single line that has the samedelivery capacity as that of the parallel lines is

or (41)

This calculated value of the equivalent diameter (d,) may be sub-stituted directly in the Weymouth formula for computing the flow.

In the following derivation of special formulas for designingcomplex systems the pipe lines are referred to as the original andthe parallel line. The parallel line may or may not extend the entirelength of the original line; however, the parallel line is connectedat its two ends to the original line. The formulas are applicableunder the conditions that after all or a section of the original linehas been paralleled the temperature and specific gravity of the gasand the pressures at the inlet and outlet ends of the original lineare the same as they were before paralleling.



110

The symbols used in the derivation of the formulas and their definitions are :

X = portion of length of original line paralleled, expressed decimally;d = internal diameter of original line, inches;

d1 = internal diameter of parallel line inches;Q = volume of gas flowing through the system before paralleling, cubic feetper hour at base conditions of pressure and temperature;

Q1 = volume of gas flowing throughper hour at base conditions of

L = length f

the system after paralleling, cubic feetpressure and temperature;

rom inlet to outlet of original line, miles;Ln = length of a single line of diameter d equivalent

paralleling a section, miles;to the entire system after

Lm = length of a single line of diameter d equivalent to 1 mile of the parallelsection, miles.

The diameter of a single line equivalent to the two lines makingup the parallel section from equation (41) is equal to (d8/3+d1

8/3)3/8.The equivalent length (L

m) of 1 mile of pipe with a diameter equal

to (d 8 / 3 +d 18 / 3 ) 3 / 8

in terms of diameter d is, from equation (38),

(42)

The total length of the system (L) minus the length of the paral-leled section plus the equivalent length of the paralleled section interms of the diameter of the original line (d) equals the length of a single line of diameter (d) equivalent to the entire system after a section has been paralleled, or

(43)

By Weymouth’s flow formula the quantity of gas per hour (Q)before paralleling is

For the same specific gravity (G), temperature (T), and inletand outlet pressures (P1 and P2) the quantity of gas (Q1) after paralleling is

From which

and

(44)

Substituting this value of Ln in equation (43),(45)

Simplifying and solving equationing equations are obtained;

(45) for X and Q1, he follow-

(46)



and

111

(47)

Equations (46) and (47) may be written in a slightly differentform, with the values of d and d1 expressed as a ratio.

Dividing the numerator and denominator of the term,

(48)

Substituting in equations (46) and (47), respectively, the abovevalue for the quantity;

(49)

and(50)

Equation (49) may be rearranged as follows to give an expression

for the calculation of the ratio .

Rearranging equation (49),

Dividing by X and transposing,

from which

and

Solving for ,

(51)



112

The maximum value of the term X (portion of line paralleled)is 1, which signifies that the entire length of the line has beenparalleled. When the value of X is 1 the following relationship maybe written. Substituting 1 for the value of X in equation (49),

or

and

(52)

By equation (52), when the entire length of the line is paralleledthe ratio of the volume after paralleling to the volume before paral-leling is equal to 1 plus the eight-thirds power of the ratio of thediameter of the parallel line to the diameter of the original line.

Equations (49) and (50) can be simplified for the conditionwhere the original line and parallel line are of the same diameter.

For this condition the value of the term is 1, and equation (49)

becomes,

or

and

(53)

(54)

APPENDIX 6.-TABLES AND FORMULAS

The Weymouth formula for the flow of natural gas through pipelines is

(55)

where Q=cubic feet of ture basis of To;

gas per hour, at a pressure basis of Po and a tempera-

Po=pressure basis, pounds per To=temperature basis, degrees square inch absolute;Fahrenheit absolute;P1=inlet pressure, pounds per square inch absolute;P2=outlet pressure, pounds per square inch absolute;d=internal diameter of the pipe, Inches;G=specific gravity of the gas (air=1.000);T=flowing temperature, degrees Fahrenheit absolute_;L=length of pipe, miles.

For gas of a specific gravity of 0.600, flowing temperature of 60

oF. (520

oF. absolute) and the volume (Q1) expressed in cubic



113

feet per hour at a pressure basis of 14.4 pounds per square inchabsolute and a temperature basis of 60o F. (520o F. absolute),equation (55) becomes

or (56)

(57)

The values of (d16/3 and d8/3 are given in table 18 for differentdiameters of pipe from 2 to 30 inches. Table 19 contains values of

for a number of lengths of pipe line ranging from 1 to 500 miles.

Equation (57) may be simplified further by combining the con-stant 36.926 and the values of d8/3 for the different diameters of pipe; that is, by letting 36.926 x d

8/3=K, equation (57) becomes

(58)

The values of K applicable to equation (58) for the differentdiameters of pipe are given in table 20.



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TABLE 19. - Values of

TABLE 20. - Value of K for use in equation (58)

TABLE 21. - Pressure-base multipliers

TABLE 22. - Temperature-base multipliers



115

Tables 21, 22, 23, and 24 contain factors expressed as multipliersfor equations (56), (57), and (58) when the conditions of pressureand temperature bases defining the volume (Q1) or the specific

gravity or temperature of the flowing gas are other than thosedefining equation (56).Equations (55), (56), (57), and (58) are readily applicable to

problems where the volume of gas flowing is the unknown quantityand all other factors are given. However, in many flow problemsthe quantity of gas to be delivered and the diameter of the pipeare known, and it is desired to determine (1) the necessary inlet

TABLE 23. - Specific-gravity multipliers

TABLE 24. - Flowing-temperature multipliers

pressure for gas delivery at a certain outlet pressure or (2) theoutlet pressure that will exist for gas delivery at a given inletpressure. In other problems, the diameter of the pipe is the un-known term, and values are given for the other terms in the equa-tions. To simplify calculations, equations (55) and (56) have beenrearranged to give direct expressions for the unknown quantities.

The equations for the inlet pressure based on equations (55) and(56), respectively, are,

(59 )

(60)



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The equations for the outlet pressure based on equations (55)and (56), respectively, are

(61)

(62)

The equations for the diameter based on equations (55) and (56),respectively, are

(63)

(64)

The delivery (Q) in Weymouth’s formula (equation 55) is ex-pressed at the temperature and pressure bases, To and Po; accord-ingly, the numerical value of Q depends upon the ‘values used for

the temperature and pressure bases. For a temperature basis of 60o F. (520o F. abs.) and a pressure basis of 14.4 pounds per square

inch absolute the value of 18.062 times is 652.24. The extent

to which changes in the base temperature and pressure from 60o F.and 14.4 pounds per square inch absolute affect the numerical valueof Q is indicated by the conversion factors in tables 21 and 22.

The delivery (Q) as expressed by Weymouth’s formula (equa-tion 55) is directly proportional to the square root of the differencebetween the squares of the inlet and outlet pressures and to thesquare root of the sixteen-thirds power of the diameter and isinversely proportional to the square roots of the specific gravityof the gas, the absolute temperature of the flowing gas and the

length of the line. With all factors constant except the diameters,the deliveries through any two pipe lines calculated by Weymouth’sformula vary directly as the eight-thirds power of the respectivediameters, or

Table 18 gives the values of d8/3 and indicates the change in thedelivery with different diameters when all other factors are con-stant. For gas of 0.600 specific gravity, 60o F. (520o F. abs.) flow-ing temperature, and volume (Q) expressed in cubic feet per hour at a pressure base of 14.4 pounds per square inch absolute andtemperature base of 60o F. (520o F. abs.) the value of the quantity,

becomes 36.926 (equation 56). For values of

the specific gravity, flowing temperature, and pressure and tem-perature bases other than those given above the change in the valueof constant 36.926 is indicated by the conversion factors in tables 21,22, 23 and 24. As mentioned above, the delivery (Q) also variesdirectly as the square root of the difference between the squares of the inlet and outlet pressures and the effect of changes in either the inlet or outlet pressure is indicated by the change in the value



117

of the quantity (P12-P2

2)1/ 2. F igure 25 shows the relat ionshipbetween the ratio of the inlet pressure (P1) to outlet pressure (P2)and the percentage of the total possible flow determined from theratio,

FIGURE 25. - Relationship between ratio of inlet to outlet pressure andpercentage of maximum flow

The value of the outlet pressure (Po in the denominator) for

maximum flow is taken as zero; therefore, the ratio reduces toand the value of P1 is the same in both numerator

and denominator. From figure 25, when the ratio of the inlet pres-sure to the outlet pressure is 2½, the rate of flow is 91.5 percent of the total possible amount. This figure indicates that at the higher values of the ratio between the inlet and outlet pressures an increasein that ratio causes relatively little gain in the rate of flow.



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