modeling of oil product.pdf

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Michael V. Lurie

Modeling of Oil Product andGas Pipeline Transportation

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Michael V. Lurie

Modeling of Oil Product andGas Pipeline Transportation

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The Author

Prof. Dr. Michael V. LurieRussian State Universityof Oil and GasMoscow, Russian Federation

Translation

Emmanuil G. SinaiskiLeipzig, Germany

Cover Picture

Trans-Alaska Pipeline

All books published by Wiley-VCH arecarefully produced. Nevertheless, authors,editors, and publisher do not warrant theinformation contained in these books,including this book, to be free of errors.Readers are advised to keep in mind thatstatements, data, illustrations, proceduraldetails or other items may inadvertently beinaccurate.

Library of Congress Card No.: applied forBritish Library Cataloguing-in-Publication DataA catalogue record for this book is availablefrom the British Library.

Bibliographic information published bythe Deutsche NationalbibliothekDie Deutsche Nationalbibliothek lists thispublication in the Deutsche National-bibliograe; detailed bibliographic data areavailable in the Internet at.

2008 WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

All rights reserved (including those oftranslation into other languages). No part ofthis book may be reproduced in any form byphotoprinting, microlm, or any othermeans nor transmitted or translated into amachine language without written permissionfrom the publishers. Registered names,trademarks, etc. used in this book, even whennot specically marked as such, are not to beconsidered unprotected by law.

Printed in the Federal Republic of GermanyPrinted on acid-free paper

Composition Laserwords, Chennai

Printing Strauss GmbH, Movlenbach

Bookbinding Litges & Dopf GmbH,Heppenheim

ISBN: 978-3-527-40833-7

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VIn memory of the Teacher academician Leonid I. Sedov

Modeling of Oil Product and Gas Pipeline Transportation. Michael V. LurieCopyright 2008 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40833-7

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VII

Foreword

This book is dedicated rst and foremost to holders of a masters degree andpostgraduate students of oil and gas institutes who have decided to specializein the eld of theoretical problems in the transportation of oil, oil products andgas. It contains methods of mathematical modeling of the processes takingplace in pipelines when transporting these media.

By the term mathematical model is understood a system of mathematicalequations in which framework a class of some processes could be studied. Thesolution of these equations provides values of parameters without carrying outmodel and, especially, full scale experiments.

Physical laws determining the dynamics of uids and gases in pipes arepresented. It is then shown how these laws are transformed into mathematicalequations that are at the heart of one or another mathematical model. Inthe framework of each model, are formulated problems with the aim ofinvestigating concrete situations. In doing so there are given methods of itssolution.

The book is self-sufcient for studying the subject but the text is outlined insuch a way that it impels the reader to address oneself to closer acquaintanceof considered problem containing in special technical literature.

Professor Michael V. LurieMoscow


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IX

Contents

Dedication Page V

Foreword VII

Preface XIII

List of Symbols XV

1 Fundamentals of Mathematical Modeling of One-DimensionalFlows of Fluid and Gas in Pipelines 1

1.1 Mathematical Models and Mathematical Modeling 11.1.1 Governing Factors 31.1.2 Schematization of One-Dimensional Flows of Fluids and Gases

in Pipelines 41.2 Integral Characteristics of Fluid Volume 51.3 The Law of Conservation of Transported Medium Mass.

The Continuity Equation 71.4 The Law of Change in Momentum. The Equation of Fluid Motion 91.5 The Equation of Mechanical Energy Balance 111.5.1 Bernoulli Equation 151.5.2 Input of External Energy 161.6 Equation of Change in Internal Motion Kinetic Energy 171.6.1 Hydraulic Losses (of Mechanical Energy) 181.6.2 Formulas for Calculation of the Factor (Re, ) 201.7 Total Energy Balance Equation 221.8 Complete System of Equations for Mathematical Modeling

of One-Dimensional Flows in Pipelines 29

2 Models of Transported Media 312.1 Model of a Fluid 312.2 Models of Ideal and Viscous Fluids 322.3 Model of an Incompressible Fluid 342.4 Model of Elastic (Slightly Compressible) Fluid 34


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X Contents

2.5 Model of a Fluid with Heat Expansion 342.6 Models of Non-Newtonian Fluids 362.7 Models of a Gaseous Continuum 382.7.1 Model of a Perfect Gas 392.7.2 Model of a Real Gas 392.8 Model of an Elastic Deformable Pipeline 42

3 Structure of Laminar and Turbulent Flows in a Circular Pipe 453.1 Laminar Flow of a Viscous Fluid in a Circular Pipe 453.2 Laminar Flow of a Non-Newtonian Power Fluid in a Circular Pipe 473.3 Laminar Flow of a Viscous-Plastic Fluid in a Circular Pipe 493.4 Transition of Laminar Flow of a Viscous Fluid to Turbulent Flow 513.5 Turbulent Fluid Flow in a Circular Pipe 523.6 A Method to Control Hydraulic Resistance by Injection of

Anti-Turbulent Additive into the Flow 623.7 Gravity Fluid Flow in a Pipe 65

4 Modeling and Calculation of Stationary Operating Regimes of Oiland Gas Pipelines 73

4.1 A System of Basic Equations for Stationary Flow of anIncompressible Fluid in a Pipeline 73

4.2 Boundary Conditions. Modeling of the Operation of Pumps andOil-Pumping Stations 75

4.2.1 Pumps 754.2.2 Oil-Pumping Station 784.3 Combined Operation of Linear Pipeline Section and Pumping

Station 814.4 Calculations on the Operation of a Pipeline with Intermediate

Oil-Pumping Stations 844.5 Calculations on Pipeline Stationary Operating Regimes in

Fluid Pumping with Heating 874.6 Modeling of Stationary Operating Regimes of Gas-Pipeline Sections 924.6.1 Distribution of Pressure in Stationary Gas Flow in a Gas-Pipeline 944.6.2 Pressure Distribution in a Gas-Pipeline with Great Difference

in Elevations 964.6.3 Calculation of Stationary Operating Regimes

of a Gas-Pipeline (General Case) 974.6.4 Investigation of Thermal Regimes of a Gas-Pipeline Section 984.7 Modeling of Blower Operation 100

5 Closed Mathematical Models of One-Dimensional Non-StationaryFlows of Fluid and Gas in a Pipeline 109

5.1 A Model of Non-Stationary Isothermal Flow of a Slightly CompressibleFluid in a Pipeline 109

5.2 A Model of Non-Stationary Gas Flow in a Pipeline 112

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Contents XI

5.3 Non-Stationary Flow of a Slightly Compressible Fluidin a Pipeline 113

5.3.1 Wave Equation 1135.3.2 Propagation of Waves in an Innite Pipeline 1155.3.3 Propagation of Waves in a Semi-Innite Pipeline 1175.3.4 Propagation of Waves in a Bounded Pipeline Section 1195.3.5 Method of Characteristics 1215.3.6 Initial, Boundary and Conjugation Conditions 1245.3.7 Hydraulic Shock in Pipes 1275.3.8 Accounting for Virtual Mass 1345.3.9 Hydraulic Shock in an Industrial Pipeline Caused by Instantaneous

Closing of the Gate Valve 1355.4 Non-Isothermal Gas Flow in Gas-Pipelines 1385.5 Gas Outow from a Pipeline in the Case of a Complete Break

of the Pipeline 1465.6 Mathematical Model of Non-Stationary Gravity Fluid Flow 1495.7 Non-Stationary Fluid Flow with Flow Discontinuities

in a Pipeline 152

6 Dimensional Theory 1576.1 Dimensional and Dimensionless Quantities 1576.2 Primary (Basic) and Secondary (Derived) Measurement Units 1586.3 Dimensionality of Quantities. Dimensional Formula 1596.4 Proof of Dimensional Formula 1616.5 Central Theorem of Dimensional Theory 1636.6 Dimensionally-Dependent and Dimensionally-Independent

Quantities 1646.7 Buckingham -Theorem 168

7 Physical Modeling of Phenomena 1737.1 Similarity of Phenomena and the Principle of Modeling 1737.2 Similarity Criteria 1747.3 Modeling of Viscous Fluid Flow in a Pipe 1757.4 Modeling Gravity Fluid Flow 1767.5 Modeling the Fluid Outow from a Tank 1787.6 Similarity Criteria for the Operation of Centrifugal Pumps 179

8 Dimensionality and Similarity in Mathematical Modelingof Processes 183

8.1 Origination of Similarity Criteria in the Equations of aMathematical Model 183

8.2 One-Dimensional Non-Stationary Flow of a Slightly CompressibleFluid in a Pipeline 184

8.3 Gravity Fluid Flow in a Pipeline 1868.4 Pipeline Transportation of Oil Products. Batching 187

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XII Contents

8.4.1 Principle of Oil Product Batching by Direct Contact 1888.4.2 Modeling of Mixture Formation in Oil Product Batching 1898.4.3 Equation of Longitudinal Mixing 1928.4.4 Self-Similar Solutions 194

References 199

Appendices 201

Author Index 205

Subject Index 207

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XIII

Preface

This book presents the fundamentals of the mathematical simulation ofprocesses of pipeline transportation of oil, oil products and gas. It is shownhow the basic laws of mechanics and thermodynamics governing the ow ofuids and gases in pipelines are transformed into mathematical equationswhich are the essence of a certain mathematical model and, in the frameworkof a given physical problem, appropriate mathematical problems are formulatedto analyze concrete situations.

The book is suitable for graduate and postgraduate students of universitieshaving departments concerned with oil and gas and to engineers and researchworkers specializing in pipeline transportation.

Beginners will nd in this book a consecutive description of the theory andmathematical simulation methods of stationary and non-stationary processesoccurring in pipelines. Engineers engaged in the design of and calculations onpipelines will nd a detailed theoretical and practical text-book on the subjectof their work. Graduate and postgraduate students and research workers willbecome acquainted with situations in the theory and methods in order togeneralize and develop them in the future.

The author of the book, Professor Dr. M. Lurie, is a great authority in Russiain the eld of the hydromechanics of oil and gas pipeline transportation.

Prof. Emmanuil SinaiskiLeipzig

. . . No human investigation could be referred to as true when it is not supportedby mathematical proof

Leonardo da Vinci


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XV

List of Symbols

Symbol Denitiona radius of the ow corea dimensionless constanta parameter of the (Q H) characteristicA proportionality factorA+ value of parameter A to the left of the discontinuity frontA value of parameter A to the right of the discontinuity front[A] = A+ A jump of parameter A at the discontinuity frontdAin elementary work of internal forcedAex elementary work of external forceb parameter of the (Q H) characteristicc velocity of wave propagation in a pipelinec sound velocity in gasC2 integration constantCf friction factorCp heat capacity at constant pressureCSh Chezy factorCv heat capacity at constant volumecP centipoise, 0.01 PcSt centistokes, 0.01 St = 106 m2 s1d pipeline internal diameterd diameter incrementd0 nominal internal diameter of pipeline; cylinder internal

diameterD pipeline external diameterD velocity of hydraulic shock wave propagation in a pipelineD velocity of discontinuity front propagation in the positive

direction of the x-axisDim diameter of impellerDp diameter of pump impellerD JouleThompson factorein internal energy density; specic internal energyekin kinetic energy density; specic kinetic energy


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XVI List of Symbols

E elastic modulus in extension and compression,Youngs modulus

Ein internal energyEkin kinetic energyEi(z) Euler functionf1 dimensionless factorf(Q) friction forceF restoring forcedFn elementary forceFr Froude numberg acceleration due to gravityg0, g1 dimensionless constantsh piezometric headh(S) depth of pipeline cross-section lling with uidhc head losses in station communicationshcr critical depthh. head before PLPhn normal depth of gravity ow in the pipeH headH differential headH = F(Q) head-discharge (Q H) characteristic of a pumpH1 hydraulic headH1 hydraulic headHe Hedstroem numberi hydraulic gradienti0 hydraulic gradientI momentumI Ilyushin numberI1, I2 Riemann invariantsJ gas enthalpyk factor of string elasticityk factor of power, Ostwald uidk parameter of non-Newtonian uid factor; heat-transfer factor; empirical factor 1/K Karman constantk dimensionless constantk kinematic consistencyK heat transfer factorK elastic modulus of uid, PaK factor of longitudinal mixing of oil productlc length of the mixture regionL length of a pipeline or a pipeline sectionM mass ow rateM0 initial mass ow rateOPS oil-pumping station

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List of Symbols XVII

n factor of power, Ostwald uidn exponent in Ostwald rheological lawn exponentn number of revolutions of centrifugal blower shaftn unit normal vectorn0 nominal number of revolutions of blower shaftnin specic power of internal friction forcesN power consumption, kWNmech power of external mechanical devicesNus useful power of mechanical force acting on gasN/e specic powerp pressurep difference between internal and external pressures, pressure

dropp0 nominal pressure, initial pressure, normal pressure, pressure

at the beginning of the pipeline sectionpen pressure of gas at the entrance of compressor station

and blowerpcr critical pressurepex external pressure; pressure at initial cross-section

of the pipeline sectionpin internal pressurepL pressure at the end of the pipeline sectionpl, pe, p pressure at the pressure line of pumps (PLP)pr reduced pressurepst standard pressure, pst = 101 325 Papu pressure before oil-pumping stationpu head before stationpv saturated vapor tension (pressure)[p] pressure jump[pinc] incident pressure wave amplitude[pre] reected pressure wave amplitude[ptrans] transmitted pressure wave amplitudeP poise, 0.1 kg m s1Ps wetted perimeterPa pascal (SI unit), kg m1 s2Pe Peclet numberqh specic heat uxqex heat inow (qex > 0) to gas; heat outow (qex < 0) from gasqM specic mass ow rateqn external heat uxQ volume ow rate; uid ow rateQe ow rate of gas at the entrance to the compressor stationQk commercial ow rate of gasQM mass ow rate

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XVIII List of Symbols

Qv volume ow rate of gas at pipeline cross-sectionr radial coordinater0 pipeline radiusR gas constant (R = R0/g)R0 universal gas constantRh hydraulic radiusRim radius of the impellerRr reduced gas constantRe Reynolds numberRecr critical Reynolds numberRe generalized Reynolds numberS area of a cross-section; area of pipeline cross-section part lled

with uidS0 area of pipeline cross-section; nominal (basic) areaSt stokes, 104 m2 s1t timeT absolute temperatureT0 nominal temperature; initial temperature; temperature of uid

at normal conditionTav average temperature over pipeline section lengthTcr critical temperatureTB temperature of gas at the entrance to the compressor stationTex temperature of external mediumTL temperature at the end of pipeline sectionTm mean temperatureTr reduced temperatureTst standard absolute temperatureu(y) velocity distribution over cross-sectionumax maximum value of velocityuw uid velocity at pipe wallu dynamic velocityv velocity averaged over cross-sectionv mean ow rate velocityvcr critical velocityV volume[v] uid velocity jumpw accelerationx coordinate along the pipeline axisx1 coordinate of gravity ow section beginningx2 coordinate of gravity ow section endy coordinate transverse to the pipeline axis; direction of a normal

to the elementary surface dz(x) elevation level of a pipeline cross-section x(z1 z2) geometrical height differences of sections 1 and 2Z over-compressibility factor

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List of Symbols XIX

Zav average over-compressibility factorZ = Zr reduced gas over-compressibility

angle of inclination of the pipeline axis to the horizontal, factorsv volume expansion factorT thermal expansion factor compressibility factor adiabatic index ratio between the hydraulic gradient of pipeline section

completely lled with uid and the absolute value of thegravity ow section with slope p to the horizontal

shear rate, s1 pipeline wall thickness absolute equivalent roughness; roughness of wall surface relative roughness; compression ratio; thickness ratio(t) local resistance factor dimensionless radius(%) efciency function of temperature; concentration; parameter of over-

compressibility factor; parameter of state equation of real gas;concentration of anti-turbulent additive

hydraulic resistance factoreff effective factor of hydraulic resistance dynamic viscosity factor kg m1 s1g molar mass of gas; molecular weightt turbulent dynamic viscosity apparent viscosity of power Ostwald uid kinematic viscosity factor m2 s10 kinematic viscosity factor at temperature T01 kinematic viscosity factor at temperature T1P Poisson ratiot turbulent kinematic viscosity factor of volumetric expansion, K1; self-similar coordinate;

dimensionless coordinate dimensionless parameter; similarity criterion(x) initial pressure distribution density, v, p, S values of parameters before hydraulic shock wave+, v+, p+, S+ values of parameters after hydraulic shock wave0 nominal density; uid density at p0; density of uid under

normal conditionsst gas density under standard conditions area of suction branch pipe cross-section; hoop stress; degree

of pipe lling; circumferential stress

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XX List of Symbols

d elementary surface area; surface element tangential (shear) stress

tangential friction stress0 critical (limit) shear stressw tangential (shear) stress at the pipeline internal surface specic volumecr critical specic volume angle of inclination of a straight line to the abscissa; central

angle(x) initial uid velocity distribution frequency of rotor rotation; angular velocity of impeller

rotation

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11Fundamentals of Mathematical Modeling ofOne-Dimensional Flows of Fluid and Gas in Pipelines

1.1Mathematical Models and Mathematical Modeling

Examination of phenomena is carried out with the help of models. Each modelrepresents a denite schematization of the phenomenon taking into accountnot all the characteristic factors but some of them governing the phenomenaand characterizing it from some area of interest to the researcher.

For example, to examine the motion of a body the material point model isoften used. In such a model the dimensions of the body are assumed to beequal to zero and the whole mass to be concentrated at a point. In other wordswe ignore a lot of factors associated with body size and shape, the materialfrom which the body is made and so on. The question is: to what extent wouldsuch a schematization be efcient in examining the phenomenon? As we allknow such a body does not exist in nature. Nevertheless, when examining themotion of planets around the sun or satellites around the earth, and in manyother cases, the material point model gives brilliant results in the calculationof the trajectories of a body under consideration.

In the examination of oscillations of a small load on an elastic spring wemeet with greater schematization of the phenomenon. First the load is takenas a point mass m, that is we use the material point model, ignoring bodysize and shape and the physical and chemical properties of the body material.Secondly, the elastic string is also schematized by replacing it by the so-calledrestoring force F = k x, where x(t) is the deviation of the material pointmodeling the load under consideration from the equilibrium position and k isthe factor characterizing the elasticity of the string. Here we do not take intoaccount the physical-chemical properties of the string, its construction andmaterial properties and so on. Further schematization could be done by takinginto account the drag arising from the air ow around the moving load andthe rubbing of the load during its motion along the guide.

The use of the differential equation

md2x

dt2= k x, (1.1)


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2 1 Mathematical Modeling of One-Dimensional Flows of Fluid and Gas in Pipelines

expressing the second Newtonian law is also a schematization of thephenomenon, since the motion is described in the framework of Euclidiangeometry which is the model of our space without taking into account therelativistic effects of the relativity theory.

The fact that the load motion can begin from an arbitrary position with anarbitrary initial velocity may be taken into account in the schematization byspecifying initial conditions at

t = 0 : x = x0; v =(

dx

dt

)0

= v0. (1.2)

Equation (1.1) represents the closed mathematical model of the consideredphenomenon and when the initial conditions are included (1.2) this is theconcrete mathematical model in the framework of this model. In the given casewe have the so-called initial value (Cauchy) problem allowing an exact solution.This solution permits us to predict the load motion at instants of time t > 0and by so doing to discover regularities of its motion that were not previouslyevident. The latest circumstance contains the whole meaning and purpose ofmathematical models.

It is also possible of course to produce another more general schematizationof the same phenomenon which takes into account a great number ofcharacteristic factors inherent to this phenomenon, that is, it is possible, inprinciple, to have anothermore general model of the considered phenomenon.

This raises the question, how can one tell about the correctness orincorrectness of the phenomenon schematization when, from the logicalpoint of view, both schematizations (models) are consistent? The answer is:only from results obtained in the framework of these models. For example,the above-outlined model of load oscillation around an equilibrium positionallows one to calculate the motion of the load as

x(t) = x0 cos(

k

m t)

+

m

kv0 sin

(k

m t)

having undamped periodic oscillations. How can one evaluate the obtainedresult? On the one hand there exists a time interval in the course of whichthe derived result accords well with the experimental data. Hence the modelis undoubtedly correct and efcient. On the other hand the same experimentshows that oscillations of the load are gradually damping in time and cometo a stop. This means that the model (1.1) and the problem (1.2) do not takeinto account some factors which could be of interest for us, and the acceptedschematization is inadequate.

Including in the number of forces acting on the load additional forces,namely the forces of dry f0 sign(x) and viscous f1 x friction (where thesymbol sign(x) denotes the function x sign equal to 1, at x > 0; equal to 1,at x < 0 and equal to 0, at x = 0), that is using the equation

md2x

dt2= k x f0 sign(x) f1 x (1.3)

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1.1 Mathematical Models and Mathematical Modeling 3

instead of Eq. (1.1), one makes the schematization (model) more complete.Therefore it adequately describes the phenomenon.

But even the new model describes only approximately the model underconsideration. In the case when the size and shape of the load strongly affectits motion, the motion itself is not one-dimensional, the forces acting on thebody have a more complex nature and so on. Thus it is necessary to use morecomplex schematizations or in another words to exploit more complexmodels.Correct schematization frequently represents a challenging task, requiringfrom the researcher great experience, intuition and deep insight into thephenomenon to be studied (Sedov, 1965).

Of special note is the continuum model, which occupies a highly importantplace in the following chapters. It is known that all media, includingliquids and gases, comprise a great collection of different atoms andmolecules in permanent heat motion and with complex interactions.By molecular interactions we mean such properties of real media ascompressibility, viscosity, heat conductivity, elasticity and others. Thecomplexity of these processes is very high and the governing forcesare not always known. Therefore such seemingly natural investigationof medium motion through a study of discrete molecules is absolutelyunacceptable.

One of the general schematization methods for uid, gas and otherdeformable media motion is based on the continuum model. Because eachmacroscopic volume of the medium under consideration contains a greatnumber of molecules the medium could be approximately considered as ifit lls the space continuously. Oil, oil products, gas, water or metals may beconsidered as a medium continuously lling one or another region of thespace. That is why a system of material points continuously lling a part of spaceis called a continuum.

Replacement of a real medium consisting of separate molecules by acontinuum represents of course a schematization. But such a schemati-zation has proved to be very convenient in the use of the mathemat-ical apparatus of continuous functions and, as was shown in practice,it is quite sufcient for studying the overwhelming majority of observedphenomena.

1.1.1Governing Factors

In the examination of different phenomena the researcher is always restrictedby a nite number of parameters called governing factors (parameters) withinthe limits of which the investigation is being studied. This brings up thequestion: How to reveal the system of governing parameters?

It could be done for example by formulating the problem mathematicallyor, in other words, by building a mathematical model of the considered

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phenomenon as was demonstrated in the above-mentioned example. In thisproblem the governing parameters are:

x, t,m, k, f0, f1, x0, v0.

But, in order to determine the system of governing parameters, there is noneed formathematical schematization of the process. It is enough to be guided,as has already been noted, by experience, intuition and understanding of themechanism of the phenomenon.

Let us investigate the decrease in a parachutists speed v in the air whenhis motion can be taken as steady. Being governed only by intuition itis an easy matter to assume the speed to be dependent on the mass ofthe parachutist m, acceleration due to g, the diameter of the parachutecanopy D, the length L of its shroud and the air density . The viscosity ofthe air owing around the parachute during its descent can be taken intoaccount or ignored since the force of viscous friction is small compared toparachute drag. Both cases represent only different schematizations of thephenomenon.

So the function sought could be assumed to have the following general formv = f (m, g,D, L, ). Then the governing parameters are:

m, g,D, L, .

Theuse of dimensional theory permits us to rewrite the formulated dependencein invariant form, that is, independent of the system of measurement units(see Chapters 6 and 7)

vgD

= f(

m

D3,L

D

), v = gD f ( m

D3,L

D

).

Thus, among ve governing parameters there are only two independentdimensionless combinations, m/D3 and L/D, dening the sought-fordependence.

1.1.2Schematization of One-Dimensional Flows of Fluids and Gases in Pipelines

In problems of oil and gas transportation most often schematization of theow process under the following conditions is used: oil, oil product and gas are considered as a continuum continuously llingthe whole cross-section of the pipeline or its part;

the ow is taken as one-dimensional, that is all governing parametersdepend only on one space coordinate x measured along the pipeline axisand, in the general case, on time t;

the governing parameters of the ow represent values of the correspondingphysical parameters averaged over the pipeline cross-section;

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1.2 Integral Characteristics of Fluid Volume 5

the prole of the pipeline is given by the dependence of the height of thepipeline axis above sea level on the linear coordinate z(x);

the area S of the pipeline cross-section depends, in the general case, on xand t. If the pipeline is assumed to be undeformable, then S = S(x). If thepipeline has a constant diameter, then S(x) = S0 = const.;

the most important parameters are:(x, t) density of medium to be transported, kg m3;v(x, t) velocity of the medium, m s1;p(x, t) pressure at the pipeline axis, Pa = N m2;T(x, t) temperature of the medium to be transported, degrees;(x, t) shear stress (friction force per unit area of the pipeline internalsurface), Pa = N m2;Q(x, t) = vS volume ow rate of the medium, m3 s1;M(x, t) = vS mass ow rate of the medium, kg s1 and other.

Mathematical models of uid and gas ows in the pipeline are basedon the fundamental laws of physics (mechanics and thermodynamics) of acontinuum, modeling a real uid and a real gas.

1.2Integral Characteristics of Fluid Volume

In what follows one needs the notion of movable uid volume of the continuumin the pipeline. Let, at some instant of time, an arbitrary volume of themedium be transported between cross-sections x1 and x2 of the pipeline(Figure 1.1).

If the continuum located between these two cross-sections is identiedwith a system of material points and track is kept of its displacement intime, the boundaries x1 and x2 become dependent on time and, togetherwith the pipeline surface, contain one and the same material points ofthe continuum. This volume of the transported medium is called themovable uid volume or individual volume. Its special feature is that italways consists of the same particles of the continuum under consideration.If, for example, the transported medium is incompressible and the pipelineis non-deformable, then S = S0 = const. and the difference between thedemarcation boundaries (x2 x1) dening the length of the uid volumeremains constant.

Figure 1.1 Movable uid volume of thecontinuum.

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Exploiting the notion of uid or individual volume of the transportedmedium in the pipeline one can introduce the following integral quantities:

M = x2(t)

x1(t)(x, t) S(x, t) dx mass of uid volume (kg);

I = x2(t)

x1(t)(x, t) v(x, t) S(x, t) dx momentum of uid volume

(kg m s1);

Ekin = x2(t)

x1(t)k

v2

2S(x, t) dx kinetic energy of the uid volume (J),

where k is the factor;

Ein = x2(t)

x1(t)(x, t) ein(x, t) S(x, t) dx internal energy of the uid

volume, where ein is the density of the internal energy (J kg1), that is theinternal energy per unit mass.

These quantities model themass, momentum and energy of amaterial pointsystem.

Since the main laws of physics are often formulated as connections betweenphysical quantities and the rate of their change in time, we ought to adducethe rule of integral quantity differentiation with respect to time. The symbolof differentiation d()/ dt denotes the total derivative with respect to time,associated with individual particles of a continuum whereas the symbol ()/tdenotes the local derivative with respect to time, that is the derivative of aow parameter with respect to time at a given space point, e.g. x = const. Thelocal derivative with respect to time gives the rate of ow parameter change ata given cross-section of the ow while, at two consecutive instances of time,different particles of the continuum are located in this cross-section.

The total derivative with respect to time is equal to

d

dt

x2(t)x1(t)

A(x, t) S(x, t) dx.

From mathematical analysis it is known how an integral containing aparameter, in the considered case it is t, is differentiated with respect tothis parameter, when the integrand and limits of integration depend on thisparameter. We have

d

dt

x2(t)x1(t)

A(x, t) S(x, t) dx = x2(t)

x1(t)

t[A(x, t) S(x, t)] dx

+ A(x, t) S(x, t)|x2(t) dx2dt

A(x, t) S(x, t)|x1(t) dx1dt

.

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1.3 The Law of Conservation of Transported Medium Mass. The Continuity Equation 7

First, at frozen upper and lower integration limits, the integrand isdifferentiated (the derivative being local) and then the integrand calculated atthe upper and lower integration limits is multiplied by the rates of change ofthese limits dx2/ dt and dx1/ dt, the rst term having been taken with a plussign and the second with a minus sign (see Appendix B).

For the case of the uid volume of the medium the quantities dx2/ dt anddx1/ dt are the corresponding velocities v2(t) and v1(t) of the medium in theleft and right cross-sections bounding the considered volume. Hence

d

dt

x2(t)x1(t)

A(x, t) S(x, t) dx = x2(t)

x1(t)

t[A(x, t) S(x, t)] dx

+ A(x, t) v(x, t) S(x, t)|x2(t) A(x, t) v(x, t) S(x, t)|x1(t).If, in addition, we take into account the well-known NewtonLeibniz formula,according to which

A(x, t) v(x, t) S(x, t)|x2(t) A(x, t) v(x, t) S(x, t)|x1(t)

= x2(t)

x1(t)

x[A(x, t) v(x, t) S(x, t)] dx,

we obtain

d

dt

x2(t)x1(t)

A(x, t) S(x, t) dx = x2(t)

x1(t)

(AS

t+ ASv

x

)dx. (1.4)

1.3The Law of Conservation of Transported Medium Mass. The Continuity Equation

The density (x, t), the velocity of the transported medium v(x, t) and thearea of the pipeline cross-section S(x, t) cannot be chosen arbitrarily sincetheir values dene the enhancement or reduction of the medium mass inone or another place of the pipeline. Therefore the rst equation would beobtained when the transported medium is governed by the mass conservationlaw

d

dt

x2(t)x1(t)

(x, t) S(x, t) dx = 0, (1.5)

This equation should be obeyed for any uid particle of the transportedmedium, that is for any values x1(t) and x2(t).

Applying to Eq. (1.4) the rule (1.5) of differentiation of integral quantity withregard to uid volume, we obtain x2(t)

x1(t)

(S

t+ vS

x

)dx = 0.

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Since the last relation holds for arbitrary integration limits we get the followingdifferential equation

S

t+ vS

x= 0, (1.6)

which is called continuity equation of the transported medium in the pipeline.If the ow is stationary, that is the local derivative with respect to time is

zero (()/t = 0), the last equation is simplied todvS

dx= 0 M = vS = const. (1.7)

This means that in stationary ow the mass ow rate M is constant along thepipeline.

If we ignore the pipeline deformation and take S(x) = S0 = const.,from Eq. (1.7) it follows that v = const. From this follow two importantconsequences:

1. In the case of a homogeneous incompressible uid (sometimes oil andoil product can be considered as such uids) = 0 = const. and theow velocity v(x) = const. Hence the ow velocity of a homogeneousincompressible uid in a pipeline of constant cross-section does not changealong the length of the pipeline.

Example. The volume ow rate of the oil transported by a pipeline withdiameter D = 820 mm and wall thickness = 8 mm is 2500 m3 h1. It isrequired to nd the velocity v of the ow.

Solution. The internal diameter d of the oil pipeline is equal to

d = D 2 = 0.82 2 0.008 = 0.804 m;v = 4Q/d2 = const.v = 4 2500/(3600 3.14 0.8042) = 1.37 m s1.

2. In the case of a compressible medium, e.g. a gas, the density (x)changes along the length of pipeline section under consideration. Sincethe density is as a rule connected with pressure, this change representsa monotonic function decreasing from the beginning of the section toits end. Then from the condition v = const. it follows that the velocityv(x) of the ow also increases monotonically from the beginning of thesection to its end. Hence the velocity of the gas ow in a pipeline withconstant diameter increases from the beginning of the section betweencompressor stations to its end.

Example. The mass ow rate of gas transported along the pipeline(D = 1020 mm, = 10 mm) is 180 kg s1. Find the velocity of the gas ow

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1.4 The Law of Change in Momentum. The Equation of Fluid Motion 9

v1 at the beginning and v2 at the end of the gas-pipeline section, if thedensity of the gas at the beginning of the section is 45 kg m3 and at theend is 25 kg m3.

Solution. v1 = M/(1 S) = 4 180/(45 3.14 12) = 5.1 m s1;v2 = M/(2 S) = 4 180/(25 3.14 12) = 9.2 m s1, that is the gas owvelocity is enhanced by a factor 1.8 towards the end as compared with thevelocity at the beginning.

1.4The Law of Change in Momentum. The Equation of Fluid Motion

The continuity equation (1.6) contains several unknown functions, hence theuse of only this equation is insufcient to nd each of them. To get additionalequations we can use, among others, the equation of the change inmomentumof the system of material points comprising the transported medium. This lawexpresses properly the second Newton law applied to an arbitrary uid volumeof transported medium

dI

dt= d

dt

x2(t)x1(t)

v S dx = (p1 S1 p2 S2) + x2(t)x1(t)

pS

xdx

x2(t)x1(t)

d w dx x2(t)x1(t)

g sin (x) S dx. (1.8)

On the left is the total derivative of the uid volume momentum of thetransported medium with respect to time and on the right the sum of allexternal forces acting on the considered volume.

The rst term on the right-hand side of the equation gives the differencein pressure forces acting at the ends of the single continuum volume.The second term represents the axial projection of the reaction forcefrom the lateral surface of the pipe (this force differs from zero whenS = const.). The third term denes the friction force at the lateral surfaceof the pipe (w is the shear stress at the pipe walls, that is the frictionforce per unit area of the pipeline internal surface, Pa). The fourth termgives the sliding component of the gravity force ((x) is the slope of thepipeline axis to the horizontal, > 0 for ascending sections of the pipeline; < 0 for descending sections of the pipeline; g is the acceleration due togravity).

Representing the pressure difference in the form of an integral over thelength of the considered volume

p1 S1 p2 S2 = x2(t)

x1(t)

pS

xdx

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and noting that

x2(t)

x1(t)

pS

xdx +

x2(t)x1(t)

pS

xdx =

x2(t)x1(t)

Sp

xdx,

we obtain the following equation

d

dt

x2(t)x1(t)

vS dx = x2(t)

x1(t)

(S p

x S 4

dw Sg sin (x)

)dx.

Now applying to the left-hand side of this equation the differentiation rule ofuid volume x2(t)

x1(t)

(vS

t+ v

2 S

x

)dx

= x2(t)

x1(t)

(S p

x S 4

dw Sg sin (x)

)dx.

As far as the limits of integration in the last relation are arbitrary one candiscard the integral sign and get the differential equation

vS

t+ v

2 S

x= S

( p

x 4

dw g sin (x)

). (1.9)

If we represent the left-hand side of this equation in the form

v

(S

t+ vS

x

)+ S

(v

t+ v v

x

)and take into account that in accordance with the continuity equation (1.6) theexpression in the rst brackets is equal to zero, the resulting equation may bewritten in a more simple form

(v

t+ v v

x

)= p

x 4

dw g sin (x). (1.10)

The expression in brackets on the left-hand side of Eq. (1.10) represents thetotal derivative with respect to time, that is the particle acceleration

w = dvdt

= vt

+ v vx

. (1.11)

Now the meaning of Eq. (1.10) becomes clearer: the product of unit volumemass of transported medium and its acceleration is equal to the sum of allforces acting on the medium, namely pressure, friction and gravity forces. SoEq. (1.10) expresses the Newtons Second Law and can therefore also be calledthe ow motion equation.

Remark. about the connection between total and partial derivatives with respectto time. The acceleration w = dv/ dt is a total derivative with respect to time(symbol d()/ dt), since we are dealing with the velocity differentiation of oneand the same xed particle of the transported medium moving from one

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1.5 The Equation of Mechanical Energy Balance 11

cross-section of the pipeline to another one, whereas the partial derivative withrespect to time (symbol ()/t) has the meaning of velocity differentiation ata given place in space, that is at a constant value of x. Thus such a derivativegives the change in velocity of different particles of the transported mediumentering a given cross-section of the pipeline.

Let a particle of the medium at the instant of time t be in the cross-sectionx of the pipeline and so have velocity v(x, t). In the next instant of time t + tthis particle will transfer to the cross-section x + x and will have velocityv(x + x, t + t). The acceleration w of this particle is dened as the limit

w = dvdt

= limt0

v(x + x, t + t) v(x, t)t

= vt

x

+ vx

t

dxdt

.

Since dx/ dt = v(x, t) is the velocity of the considered particle, from the lastequality it follows that

dv

dt= v

t+ v v

x. (1.12)

A similar relation between the total derivative ( d/ dt), or as it is also calledthe individual or Lagrangian derivative, and the partial derivative (/t), or asit is also called the local or Eulerian derivative, has the form (1.12) no matterwhether the case in point is velocity or any other parameter A(x, t)

dA(x, t)

dt= A(x, t)

t+ v A(x, t)

x.

1.5The Equation of Mechanical Energy Balance

Consider now what leads to the use of the mechanical energy change law asapplied to the system of material points representing a uid particle of thetransported medium. This law is written as:

dEkindt

= dAex

dt+ dA

in

dt(1.13)

that is the change in kinetic energy of a system of material points dEkin is equalto the sum of the work of the external dAex and internal dAin forces acting onthe points of this system.

We can calculate separately the terms of this equation but rst we shoulddene more exactly what meant by the kinetic energy Ekin. If the transportedmedium moves in the pipeline as a piston with equal velocity v(x, t) over thecross-section then the kinetic energy would be expressed as the integral

Ekin = x2(t)

x1(t)

v2

2S dx.

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But, in practice, such a schematization is too rough because, as experimentsshow, the velocity of the separate layers of the transported medium (uid orgas) varies over the pipe cross-section. At the center of the pipe it reaches thegreatest value, whereas as the internal surface of the pipe is approached thevelocity decreases and at the wall itself it is equal to zero. Furthermore, if ata small velocity of the uid the ow regime is laminar, with an increase invelocity the laminar ow changes into a turbulent one (pulsating and mixingow) and the velocities of the separate particles differ signicantly from theaverage velocity v of the ow. That is why models of the ow are, as a rule,constructed with regard to the difference in ow velocity from the averagevelocity over the cross-section.

The true velocity u of a particle of the transportedmedium is given as the sumu = v + u of the average velocity over the cross-section v(x, t) and the additiveone (deviation) u representing the difference between the true velocity andthe average one. The average value of this additive u is equal to zero, butthe root-mean-square (rms) value of the additive (u)2 is non-vanishing.The deviation characterizes the kinetic energy of the relative motion of thecontinuum particle in the pipeline cross-section. Then the kinetic energy of thetransported medium unit mass ekin may be presented as the sum of two terms

ekin = v2

2+ (u)

2

2

namely the kinetic energy of the center of mass of the considered point systemand the kinetic energy of the motion of these points relative to the center ofmass. If the average velocity v = 0, then

v2

2+ (u)

2

2= v

2

2(1 + (u)

2

v2

)= k v

2

2

where k = 1 + (u)2/v2 > 1. For laminar ow k = 4/3, while for turbulentow the value of k lies in the range 1.021.05.

Remark. It should be noted that in one-dimensional theory, as a rule, the casesv = 0 and (u)2 = 0 are not considered.

With regard to the introduced factor the kinetic energy of any movablevolume of transported medium may be represented as

Ekin = x2(t)

x1(t)k v

2

2 S dx.

Let us turn now to the calculation of the terms in the mechanical energyequation (1.13). Let us calculate rst the change in kinetic energy

dEkindt

= ddt

( x2(t)x1(t)

k v2

2S dx

).

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Employing the rule of integral quantity integration with reference to the uidvolume, that is an integral with variable integration limits, we get

dEkindt

= x2(t)x1(t)

[

t

(k v

2

2S

)+

x

(k v

2

2S v

)]dx.

The work of the external forces (in this case they are the forces of pressure andgravity), including also the work of external mechanical devices, e.g. pumps ifsuch are used, is equal to

dAex

dt= (p1Sv1 p2Sv2)

x2(t)x1(t)

g sin v S dx + Nmech

= x2(t)

x1(t)

x(pSv) dx

x2(t)x1(t)

g sin v S dx + Nmech.

The rst term on the right-hand side of the last expression gives the workperformed in unit time or, more precisely, the power of the pressure forceapplied to the initial and end cross-sections of the detached volume. Thesecond term gives the power of the gravity force and the third term Nmech thepower of the external mechanical devices acting on the transported mediumvolume under consideration.

The work of the internal forces (pressure and internal friction) executed inunit time is given by

dAin

dt= x2(t)

x1(t)p(Sv)

xdx +

x2(t)x1(t)

nin S dx.

The rst term on the right-hand side gives the work of the pressure force inunit time, that is the power, for compression of the particles of the medium,the factor (Sv)/x dx giving the rate of elementary volume change. Thesecond term represents the power of the internal friction forces, that is theforces of mutual friction between the internal layers of the medium, nin

denoting specic power, that is per unit mass of the transported medium.In what follows it will be shown that this quantity characterizes the amountof mechanical energy converting into heat per unit time caused by mutualinternal friction of the transported particles of the medium.

Gathering together all the terms of the mechanical energy equation we get x2(t)x1(t)

[

t

(k v

2

2S

)+

x

(k v

2

2vS

)]dx

= x2(t)

x1(t)Sv

[(1

p

x

)+ g sin

]dx +

x2(t)x1(t)

nin S dx + Nmech.

If the transported medium is barotropic, that is the pressure in it depends onlyon the density p = p(), one can introduce a function P() of the pressure

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such that dP = dp/, P() = dp/ and 1

px = P()x . If, moreover, we take

into account the equality sin (x) = z/x, where the function z(x) is referredto as the pipeline prole, the last equation could be rewritten in the simpleform x2(t)

x1(t)

[S

t

(kv2

2

)+ vS

x

(kv2

2+ P() + gz

)]dx

= x2(t)

x1(t)nin S dx + Nmech. (1.14)

If we assume that in the region [x1(t), x2(t)] external sources of mechanicalenergy are absent. Then Nmech = 0 and we can go from the integral equality(1.14) to a differential equation using, as before, the condition of arbitrarinessof integration limits x1(t) and x2(t) in Eq. (1.14). Then the sign of the integralcan be omitted and the corresponding differential equation is

S

t

(kv2

2

)+ vS

x

(kv2

2+ P() + gz

)= S nin (1.15)

or

t

(kv2

2

)+ v

x

(kv2

2+

dp

+ gz

)= nin. (1.16)

This is the sought differential equation expressing the lawofmechanical energychange. It should be emphasized that this equation is not a consequence of themotion equation (1.10). It represents an independent equation for modelingone-dimensional ows of a transported medium in the pipeline.

If we divide both parts of Eq. (1.16) by g we get

t

(kv2

2g

)+ v

x

(kv2

2g+

dp

g+ z

)= n

in

g.

The expression

H = kv2

2g+

dp

g+ z (1.17)

in the derivative on the left-hand side of the last equation has the dimension oflength and is called the total head. The total head at the pipeline cross-sectionx consists of the kinetic head (dynamic pressure) kv2/2g, the piezometric head

dp/g and the geometric head z. The concept of head is very important in thecalculation of processes occurring in pipelines.

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1.5.1Bernoulli Equation

In the case of stationary ow of a barotropic uid or gas in the pipeline thederivative ()/t = 0, hence the following ordinary differential equations apply

vd

dx

(kv2

2g+

dp

g+ z

)= n

in

g

or

d

dx

(kv2

2g+

dp

g+ z

)= n

in

gv= i, (1.18)

where i denotes the dimensional quantity nin/gv called the hydraulic gradient

i = dHdx

= nin

gv.

Thus the hydraulic gradient, dened as the pressure loss per unit length ofthe pipeline, is proportional to the dissipation of mechanical energy into heatthrough internal friction between the transported medium layers (i < 0).

In integral form, that is as applied to transported medium located betweentwo xed cross-sections x1 and x2, Eq. (1.18) takes the following form(

kv2

2g+

dp

g+ z

)1

(kv2

2g+

dp

g+ z

)2

= x2x1

i dx. (1.19)

This equation is called the Bernoulli equation. It is one of the fundamentalequations used to describe the stationary ow of a barotropic medium ina pipeline.

For an incompressible homogeneous uid, which under some conditions canbe water, oil and oil product, = const., dp/g = p/g + const. Thereforethe Bernoulli equation becomes(

kv2

2g+ p

g+ z

)1

(kv2

2g+ p

g+ z

)2

= x2

x1

i dx.

If in addition we take i = i0 = const. (i0 > 0), then(v2

2g+ p

g+ z

)1

(v2

2g+ p

g+ z

)2

= i0 l12 (1.20)

where l12 is the length of the pipeline between cross-sections 1 and 2.This last equationhas a simple geometric interpretation (see Figure 1.2). This

gure illustrates a pipeline prole (heavy broken line); the line H(x) denotingthe dependence of the total head H on the coordinate x directed along theaxis of the pipeline (straight line) with constant slope to the horizontal(i = dH/ dx = tg = const.) and three components of the total head at an

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Figure 1.2 Geometric interpretation of the Bernoulli equation.

arbitrary cross-section of the pipeline: geometric head z(x), piezometric headp(x)/g and kinetic head kv2(x)/2g.

The line H(x) representing the dependence of the total head H on thecoordinate x along the pipeline axis is called the line of hydraulic gradient.

It should be noted that if we neglect the dynamic pressure (in oil andoil product pipelines the value of the dynamic pressure does not exceed thepipeline diameter, e.g. at v 2 m s1, k 1.05 then v2/2g = 0.25 m), andthe length of the section between the pipeline prole and the line of hydraulicgradient multiplied by g gives the value of the pressure in the pipeline cross-section x. For example, when the length of the section AA (see Figure 1.2) is500 m and diesel fuel with density = 840 kg m3 is transported along thepipeline, then

p

840 9.81 = 500 p = 500 840 9.81 = 4 120 200 (Pa)

or 4.12 MPa (42 atm).

1.5.2Input of External Energy

In uid ow in the pipeline the mechanical energy is dissipated into heat andthe pressure decreases gradually. Devices providing pressure restoration orgeneration are called compressors.

Compressors installed separately or combined in a group form the pumpingplant destined to set the uid moving from the cross-section with lesserpressure to the cross-section with greater pressure. To do this it is required toexpend, or deliver from outside to the uid, energy whose power is denoted byNmech.

Let index 1 in the Bernoulli equation refer to parameters at the cross-sectionx1 of the pump entrance (suction line) and index 2 at the cross-section x2 of the

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1.6 Equation of Change in Internal Motion Kinetic Energy 17

pump exit (discharge line). Since vS = const., the Bernoulli equation (1.14)may be written as: x2

x1

d

dx

[vS

(kv2

2+ p

+ gz

)]dx =

x2x1

nin S dx + Nmech.

Ignoring the difference between the kinetic and geometric heads we get

vS p2 p1

x2

x1

nin S dx = Nmech.

Denoting by H = (p2 p1)/g the differential head produced by the pump orpumping plant and taking into account that vS = Q = const. and nin = gv i,we obtain

Nmech = gQ H x2

x1

gQ i dx = gQ H (1

x2x1

i

Hdx

).

The expression in parentheses characterizes the loss of mechanical energywithin the pump. Usually this factor is taken into account by insertion of thepump efciency

=(1

x2x1

i/H dx)1

< 1

so that

Nmech = gQ H(Q)

. (1.21)

The relation (1.21) is the main formula used to calculate the power of thepump generating head H in uid pumping with ow rate Q .

1.6Equation of Change in Internal Motion Kinetic Energy

At the beginning of the previous section it was noted that the totalkinetic energy of the transported medium consisted of two terms thekinetic energy of the center of mass of the particle and the kinetic energyof the internal motion of the center of mass, so that the total energyof a particle is equal to kv2/2, where k > 1. Now we can derive anequation for the second component of the kinetic energy, namely the kineticenergy of the internal or relative motion in the ow of the transportedmedium.

Multiplication of motion equation (1.10) by the product vS yields

Sd

dt

(v2

2

)= p

x vS 4

dw vS gvS sin (x).

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Subtracting this equation term-by-term from the Bernoulli equation (1.15),one obtains

Sd

dt

[(k 1)v

2

2

]= 4

dw vS + S nin.

Introduction of nin = gv i0 gives

Sd

dt

[( 1)v

2

2

]=(4

dw v

)S gvS i0. (1.22)

This is the sought equation of change in kinetic energy of internal motion ofone-dimensional ow of the transported medium. Its sense is obvious: thepower of the external friction forces (4w vS/d) in one-dimensional ow minus thepower gS(v i0) of internal friction forces between the particles causing transitionof mechanical energy into heat is equal to the rate of change of internal motionkinetic energy in the ow of the transported medium.

For stationary ow ( d/ dt = 0 + v /x) of the transported mediumEq. (1.22) gives

d

dx

[(k 1)v

2

2

]= 4

d

w

g i0. (1.23)

If v = const., which for the ow of an incompressible medium in a pipelinewith constant diameter is the exact condition, the left-hand part of the equationvanishes. This means that the tangential friction tension w at the pipelinewall and the hydraulic gradient i0 are connected by

w = gd4 i0. (1.24)

It must be emphasized that in the general case, including non-stationary ow,such a connection between w and i0 is absent (see Section 4.1).

1.6.1Hydraulic Losses (of Mechanical Energy)

The quantity nin entering into Eq. (1.16) denotes the specic power of theinternal friction force, that is per unit mass of transported medium. Thisquantity is very important since it characterizes the loss of mechanical energyconverted into heat owing to internal friction between layers of the medium.In order to derive this quantity theoretically one should know how the layersof transported medium move at each cross-section of the pipeline but this isnot always possible. In the next chapter it will be shown that in several cases,in particular for laminar, ow such motion can be calculated and the quantitynin can be found. In other cases, such as for turbulent ows of the transported

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medium, it is not possible to calculate the motion of the layers and othermethods of determining nin are needed.

The quantity of specic mechanical energy dissipation nin has the followingdimension (from now onwards dimension will be denoted by the symbol [ ])

[nin] = Wkg

= Js kg

= N ms kg

= kg m s2 m

s kg= m

2

s3=[v3

d

].

So the dimension of nin is the same as the dimension of the quantity v3/d,hence, without disturbance of generality, one can seek nin in the form

nin = 2

v3

d(1.25)

where is a dimensional factor ( > 0), the minus sign shows that nin < 0,that is themechanical energy decreases thanks to the forces of internal friction.The factor 1/2 is introduced for the sake of convenience.

The presented formula does not disturb the generality of the considerationbecause the unknown dependence of nin on the governing parameters of theow is accounted for by the factor . This dependence is valid for any mediumbe it uid, gas or other medium with complex specic properties, e.g. waxycrude oil, suspension or even pulp, that is a mixture of water with large rigidparticles.

For stationary uid or gas ow one can suppose the factor to be dependenton four main parameters: the ow velocity v (m s1), the kinematic viscosityof the ow (m2 s1), the internal diameter of the pipeline d (m) and themean height of the roughness of its internal surface (mm or m), so that = f (v, , d,). The density of the uid and the acceleration due to gravityg are not included here because intuition suggests that the friction betweenuid or gas layers will be dependent on neither their density nor the force ofgravity.

Note that the quantity is dimensionless, that is its numerical valueis independent of the system of measurement units, while the parametersv, , d, are dimensional quantities and their numerical values depend onsuch a choice. The apparent contradiction is resolved by the well-knownBuckingham I-theorem, in accordance with which any dimensionless quantitycan depend only on dimensionless combinations of parameters governing thisquantity (Lurie, 2001). In our case there are two such parameters

v d

= Re and d

= ,

the rst is called the Reynolds number and the second the relative roughness ofthe pipeline internal surface. Thus

= (Re, ).

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The formula (1.25) acquires the form

nin = (Re, ) 1d

v3

2. (1.26)

The factor in this formula is called the hydraulic resistance factor, oneof the most important parameters of hydraulics and pipeline transportation.Characteristic values of lie in the range 0.010.03. More detailed informationabout this factor and its dependence on the governing parameters will bepresented below.

Turning to the hydraulic gradient i0, one can write

i0 = nin

gv= 1

d v

2

2g. (1.27)

Characteristic values of the hydraulic slope are 0.000050.005.If we substitute Eq. (1.27) into the Bernoulli equation (1.20), we obtain(

kv2

2g+ p

g+ z

)1

(kv2

2g+ p

g+ z

)2

= (Re, ) l12d

v2

2g. (1.28)

The expression h = l12/d v2/2g on the right-hand side of this equationis called the loss of head in Darcy-Veisbach form.

Using Eq. (1.27) in the case of stationary ow of the transported mediumpermits us to get an expression for the tangential friction stress w at thepipeline wall. Substitution of Eq. (1.27) into Eq. (1.24), yields

w = gd4 i0 =gd

4(1

d

v2

2g

)=

4 v

2

2= Cf v

2

2, (1.29)

Cf (Re, ) = (Re, )4where the dimensional factor Cf is called the friction factor of the uid on theinternal surface of the pipeline or the Funning factor (Leibenson et al., 1934).

1.6.2Formulas for Calculation of the Factor (Re, )

Details of methods to nd and calculate the factor of hydraulic resistance inEqs. (1.26)(1.29) and one of the primary factors in hydraulics and pipelinetransportation will be given in Chapter 3. Here are shown several formulasexploiting the practice.

If the ow of uid or gas in the pipeline is laminar, that is jetwise or layerwise(the Reynolds number Re should be less than 2300), then to determine theStokes formula (see Section 3.1) is used

= 64Re

. (1.30)

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As the Reynolds number increases (Re > 2300) the ow in the pipelinegradually loses hydrodynamic stability and becomes turbulent, that is vortexow with mixing layers. The best known formula to calculate the factor inthis case is the Altshuler formula:

= 0.11 ( + 68

Re

)1/4(1.31)

valid over a wide range of Reynolds number from 104 up to 106 and higher.If 104 < Re < 27/1.143 and Re < 105, the Altshuler formula becomes the

Blasius formula:

= 0.31644

Re(1.32)

having the same peculiarity as the Stokes formula for laminar ow, whichdoes not consider the relative roughness of the pipeline internal surface .This means that for the considered range of Reynolds numbers the pipelinebehaves as a pipeline with a smooth surface. Therefore the uid ow in thisrange is ow in a hydraulic smooth pipe. In this case the friction tension w atthe pipe wall is expressed by formula

w = 4 v2

2= 0.0791

4

vd/ v

2

2 v1.75

signifying that friction resistance is proportional to uid mean velocity to thepower of 1.75.

If Re > 500/, the second term in parentheses in the Altshuler formula canbe neglected compared to the rst one. Whence it follows that at great uidvelocities the uid friction is caused chiey by the smoothness of the pipelineinternal surface, that is by the parameter . In such a case one can use thesimpler Shiphrinson formula = 0.11 0.25. Then

w = 4 v2

2= 0.11

1/4

4 v

2

2 v2.

From this it transpires that the friction resistance is proportional to the squareof the uid mean velocity and hence this type of ow is called square ow.

Finally, in the region of ow transition from laminar to turbulent, thatis in the range of Reynolds number from 2320 up to 104 one can use theapproximation formula

= 64Re

(1 ) + 0.31644Re , (1.33)

where = 1 e0.002(Re2320) is the intermittency factor (Ginsburg, 1957). Itis obvious that the form of the last formula assures continuous transfer fromthe Stokes formula for laminar ow to the Blasius formula for turbulent owin the zone of hydraulic smooth pipes.

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To calculate the hydraulic resistance factor of the gas ow in a gas main,where the Reynolds number Re is very large and this factor depends only onthe condition of the pipeline internal surface, Eq. (1.34) is often used.

= 0.067 (2

d

)0.2(1.34)

in which the absolute roughness is equal to 0.030.05 mm.

Exercise 1. The oil ( = 870 kg m3, = 15 s St) ows along the pipeline(D = 156 mm; = 5 mm; = 0.1 mm) with mean velocity v = 0.2 m s1.Determine through the Reynolds criterion the ow regime; calculate factors and Cf .

Answer. Laminar; 0.033; 0.0083.

Exercise 2. Benzene ( = 750 kg m3, = 0.7 s St) ows along the pipeline(D = 377 mm; = 7 mm; = 0.15 mm) with mean velocity v = 1.4 m s1.Determine through the Reynolds criterion the ow regime; calculate factors and Cf .

Answer. Turbulent; 0.017; 0.0041.

Exercise 3. Diesel fuel ( = 840 kg m3, = 6 s St) ows along the pipeline(D = 530 mm; = 8 mm; = 0.25 mm) with mean velocity v = 0.8 m s1.Determine the ow regime; calculate factors and Cf .

Answer. Turbulent; 0.022; 0.0054.

1.7Total Energy Balance Equation

Besides the law (1.13) of mechanical energy change of material points,applied to an arbitrary continuum volume in the pipeline there is one morefundamental physical law valid for any continuum the law of total energyconservation or, as it is also called, the rst law of thermodynamics. This lawasserts that the energy does not appear from anywhere and does not disappearto anywhere. It changes in total quantity from one form into another. Asapplied to our case this law may be written as follows

d(Ekin + Ein)dt

= dQex

dt+ dA

ex

dt(1.35)

that is the change in total energy (Ekin + Ein) of an arbitrary volume of thetransported medium happens only due to the exchange of energy withsurrounding bodies owing to external inow of heat dQex and the workof external forces dAex.

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1.7 Total Energy Balance Equation 23

In Eq. (1.35) Ein is the internal energy of the considered mass of transportedmedium, unrelated to the kinetic energy, that is the energy of heat motion,interaction between molecules and atoms and so on. In thermodynamicsreasons are given as to why the internal energy is a function of state, thatis at thermodynamic equilibrium of a body in some state the energy has awell-dened value regardless of the means (procedure) by which this statewas achieved. At the same time the quantities dQex/ dt and dAex/ dt arenot generally derivatives with respect to a certain function of state but onlyrepresent the ratio of elementary inows of heat energy (differential dQex) andexternal mechanical energy (differential dAex) to the time dt in which theseinows happened. It should be kept in mind that these quantities depend onthe process going on in the medium.

In addition to function Ein one more function ein is often introduced,representing the internal energy of a unit mass of the considered bodyein = Ein/m, where m is the mass of the body.

We can write Eq. (1.35) for a movable volume of transported mediumenclosed between cross-sections x1(t) and x2(t). The terms of this equation are

d(Ekin + Ein)dt

= ddt

[ x2(t)x1(t)

(k

v2

2+ ein

)S dx

],

dQex

dt= x2(t)

x1(t)d qn dx,

dAex

dt=

x2(t)x1(t)

x(pSv) dx

x2(t)x1(t)

g sin v S dx + Nmech

where qn is the heat ux going through the unit area of the pipeline surfaceper unit time (W m2); d dx is an element of pipeline surface area and d isthe pipeline diameter.

Gathering all terms, we obtain

d

dt

[ x2(t)x1(t)

(k v

2

2+ ein

)S dx

]= x2(t)x1(t)

d qn dx

x2(t)

x1(t)

x(pSv) dx

x2(t)x1(t)

g sin v S dx + Nmech.

Differentiation of the left-hand side of this equation gives x2(t)x1(t)

{

t

[(kv2

2+ ein

)S

]+

x

[(kv2

2+ ein

)vS

]}dx

= x2(t)

x1(t)d qn dx

x2(t)x1(t)

x

(p

vS

)dx

x2(t)x1(t)

vSgz

xdx + Nmech

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or x2(t)x1(t)

{

t

[(kv2

2+ ein

)S

]+

x

[(kv2

2+ ein + p

)vS

]}dx

= x2(t)

x1(t)d qn dx

x2(t)x1(t)

vSgz

xdx + Nmech. (1.36)

If we assume that inside the region [x1(t), x2(t)] the external sources of me-chanical energy are absent, that is Nmech = 0, then it is possible to pass fromintegral equality (1.36) to the corresponding differential equation using, asbefore, the condition that this equation should be true for any volume of thetransported medium, that is the limits of integration x1(t) and x2(t) in (1.36)are to be arbitrarily chosen. Then the sign of the integral can be omitted andthe differential equation is

t

[(kv2

2+ ein

)S

]+

x

[(kv2

2+ ein + p

)vS

]= d qn vSg z

x. (1.37)

Excluding from Eq. (1.37) the change in kinetic energy with the help of theBernoulli equation with term by term subtraction of Eq. (1.16) from Eq. (1.37)we get one more energy equation

S

t

(kv2

2

)+ vS

x

(kv2

2+

dp

+ gz

)= vSg i

called the equation of heat inow.This equation could be variously written. First, it may be written through

the internal energy ein:

t(ein S) +

x(ein vS) = d qn pvS

x vSg i

or

S

(eint

+ veinx

)= d qn p vS

x vSg i. (1.38)

This equation proved to be especially convenient for modeling ows ofincompressible or slightly compressible uids because the derivative (vS)/xexpressing the change in uid volume in the pipeline cross-section is extremelysmall as is the work p (vS)/x of the pressure forces. With this in mindEq. (1.38) may be written in a particularly simple form:

deindt

= 4d

qn vg i. (1.39)

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This means that the rate of internal energy change of the transported mediumis determined by the inow of external heat through the pipeline surface andheat extraction due to conversion of mechanical energy into heat produced byfriction between the continuum layers.

Second, the equation of heat inow can be written using the functionJ = ein + p/ representing one of the basic thermodynamic functions, enthalpyor heat content, of the transported medium

t(ein S) +

x

[(ein + p

)vS

]= d qn + vSg

(1

g

p

x i

)or

t(ein S) +

x[J vS] = d qn + vSg

(1

g

p

x i

). (1.40)

If we take into account (as will be shown later) that the expression inparentheses on the right-hand side of this equation is close to zero, since for arelatively light medium, e.g. gas, the hydraulic slope is expressed through thepressure gradient by the formula i = 1/g p/x, the equation of heat inowcan be reduced to a simpler form

S eint

+ vS Jx

= d qn (1.41)

in which the dissipation of mechanical energy appears to be absent.

Temperature Distribution in Stationary FlowThe equation of heat inow in the form (1.39) or (1.41) is convenient todetermine the temperature distribution along the pipeline length in stationaryow of the transported medium.1. For an incompressible or slightly compressible medium, e.g. dropping liquid:

water, oil and oil product, this equation has the form

v deindx

= 4d

qn vg i. (1.42)

The internal energy ein depends primarily on the temperature of the uidT , the derivative dein/ dT giving its specic heat Cv (J kg1 K1). If we takeCv = const. then ein = Cv T + const.

To model the heat ux qn the Newton formula is usually used

qn = (T Tex), (1.43)by which this ow is proportional to the difference between the temperaturesT and Tex in and outside the pipeline, with qn < 0 when T > Tex and qn > 0when T < Tex. The factor (W m2 K1) in this formula characterizes theoverall heat resistance of the materials through which the heat is transferred

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from the pipe to the surrounding medium (anticorrosive and heat insulation,ground, the boundary between ground and air and so on) or the reverse. Thisfactor is called the heat-transfer factor.

The hydraulic gradient i can sometimes be considered constant i = i0 const., if the dissipation of mechanical energy in the stationary uid ow inthe pipeline with constant diameter is identical at all cross-sections of thepipeline.

With due regard for all the aforesaid Eq. (1.42) is reduced to the followingordinary differential equation

Cvv dTdx = 4

d(T Tex) + vgi0 (1.44)

for temperature T = T(x). From this equation in particular it follows that theheat transfer through the pipeline wall (the rst term on the right-hand side)lowers the temperature of the transported medium when T(x) > Tex or raisesit when T(x) < Tex, whereas the dissipation of mechanical energy (the secondterm on the right-hand side) always implies an increase in the temperature ofthe transported medium.

The solution of the differential equation (1.44) with initial conditionT(0) = T0 yields

T(x) Tex TT0 Tex T = exp

( d

CvMx

). (1.45)

Where T = gi0M/d is a constant having the dimension of temperature;M = vS is the mass ow rate of the uid (M = const.). The formula thusobtained is called the Shuchov formula.

Figure 1.3 illustrates the distribution of temperature T(x) along the pipelinelength x in accordance with Eq. (1.45).

The gure shows that when the initial temperature T0 is greater than(Tex + T), the moving medium cools down, while when T0 is less than(Tex + T), the medium gradually heats up. In all cases with increase in thepipeline length the temperature T (Tex + T).

In particular from Eq. (1.44) it follows that if the heat insulation of thepipeline is chosen such that at the initial cross-section of the pipeline x = 0

Figure 1.3 Temperaturedistribution along the pipeline length.

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the condition of equality to zero of the right-hand side is obeyed

4d

(T0 Tex) + vgi0 = 0

that is the factor satises the condition

= vdgi04(T0 Tex) =

gM i0d (T0 Tex) .

And the temperature of the transported medium would remain constant andequal to its initial value over the whole pipeline section. In such a case theheat outgoing from the pipeline would be compensated by the heat extractedby internal friction between the layers. Such an effect is used, for example,in oil transportation along the Trans-Alaska oil pipeline (USA, see the coverpicture). Through good insulation of the pipeline the oil is pumped overwithout preheating despite the fact that in winter the temperature of theenvironment is very low.

From Eq. (1.45) follows the connection between the initial T0 and nal TLtemperatures of the transported medium. If in this formula we set x = L,where L is the length of the pipeline section, we obtain

TL Tex TT0 Tex T = exp

(dL

CvM

). (1.46)

Expressing now from (1.46) the argument under the exponent and substitutingthe result in Eq. (1.45), we get the expression for the temperature distributionthrough the initial and nal values

T(x) Tex TT0 Tex T =

(TL Tex TT0 Tex T

)x/L. (1.47)

Exercise 1. The initial temperature of crude oil ( = 870 kg m3, Cv =2000 J kg1 K1, Q = 2500 m3 h1), pumping over a pipeline section (d =800 mm,L = 120 km, i0 = 0.002) is 55 C.The temperature of the surroundingmedium is 8 C. The heat insulation of the pipeline is characterized by theheat-transfer factor = 2 W m2 K1. It is required to nd the temperature atthe end of the section.

Solution. Calculate rst the temperature T:

T = gi0Md

= 9.81 0.002 870 (2500/3600)3.14 0.8 2

= 2.36 K.

Using Eq. (1.46) we obtain

TL 8 2.3655 8 2.36 = exp

( 3.14 0.8 2 120 10

3

2000 870 (2500/3600))

,

from which follows TL = 37.5 C.

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Exercise 2. By how much would the temperature of the oil (Cv =1950 J kg1 K1) be raised due to the heat of internal friction when theoil is transported by an oil pipeline (L = 150 km, d = 500 mm, i0 = 0.004)provided with ideal heat insulation ( = 0)?Solution. In this case it is impossible to use at once Eq. (1.45) since = 0. Touse Eq. (1.47) one should go to the limit at 0, therefore it would be betterto use Eq. (1.44)

Cvv dTdx = vgi0 or Cv dT

dx= gi0,

from which T = gi0L/Cv = 9.81 0.004 150 103/1950 = 3 K.Exercise 3. It is required to obtain the temperature of oil pumping over thepipeline section of length 150 km in cross-sections x = 50, 100 and 125 km,if the temperature at the beginning of the pipeline T0 = 60 C, that at theend TL = 30 C, and that of the environment Tex = 8 C. The extracted heat ofinternal friction may be ignored.

Solution. Using Eq. (1.46), one gets

T(x) 860 8 =

(30 860 8

)x/Land T(x) = 8 + 52 (0.4231)x/150.

Substitution in this formula of successive x = 50, 100 and 125 givesT(50) = 47 C; T(100) = 37.3 C; T(125) = 33.4 C.2. For stationary ow of a compressible medium, e.g. gas, the equation of heat

inow (1.41) takes the form

vSdJ

dx= d qn.

In the general case, the gas enthalpy J is a function of pressure and temperatureJ = J(p,T), but for a perfect gas, that is a gas obeying theClapeyron law p = RT ,where R is the gas constant, the enthalpy is a function only of temperatureJ = Cp T + const., where Cp is the gas specic heat capacity at constantpressure (Cp > Cv; Cp Cv = R). Regarding Cp = const. and taking as beforeqn = (T Tex), we transform the last equation to

CpMdT

dx= d (T Tex)

or

dT

dx= d

CpM (T Tex).

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1.8 Mathematical Modeling of One-Dimensional Flows in Pipelines 29

The solution of this differential equation with initial condition T(0) = T0 givesT(x) TexT0 Tex = exp

( d

CpMx

), (1.48)

which is similar to the solution (1.45) for temperature distribution in anincompressible uid. The difference consists only in that instead of heatcapacity Cv in the solution (1.47) we use heat capacity Cp and the temperatureT taking into account the heat of internal friction is absent (for methaneCp = 2230 J kg1 K1; Cv = 1700 J kg1 K1).

The temperature TL of the gas at the end of the gas pipeline section is foundfrom

TL TexT0 Tex = exp

(dL

CpM

)(1.49)

with regard to which the distribution (1.47) takes the form

T(x) TexT0 Tex =

(TL TexT0 T

)x/L(1.50)

allowing us to express the temperature through the initial and naltemperatures.

Note that for a real gas the enthalpy J = J(p,T) of the medium depends notonly on temperature but also on pressure, so the equation of heat inow has amore complex form. By the dependence J(p,T) is explained, in particular, theJoule-Thomson effect.

1.8Complete System of Equations for Mathematical Modeling of One-DimensionalFlows in Pipelines

This system consists of the following equations.

1. Continuity equation (1.6)

S

t+ vS

x= 0;

2. Momentum (motion) equation (1.10)

(v

t+ v v

x

)= p

x 4

dw g sin (x);

3. Equation of mechanical energy balance (1.15)

t

(kv2

2

)+ v

x

(kv2

2+ P() + gz

)= vg i;

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4. Equation of total energy balance (1.37)

t

[(kv2

2+ ein

)S

]+

x

[(kv2

2+ J

)vS

]= d qn vgS dzdx .

The number of unknown functions in this equation is 10: , v, p, S, ein,T, w,i, qn, k, while the number of equations is 4. Therefore there are neededadditional relations to close the system of equations. As closing relations thefollowing relations are commonly used: equation of state p = p(,T), characterizing the properties of thetransported medium;

equation of pipeline state S = S(p,T) characterizing the deformation abilityof the pipeline;

calorimetric dependences ein = e(p,T) or J = J(p,T); dependence qn = (T Tex) or more complex dependencesrepresenting heat exchange between the transported medium and theenvironment;

hydraulic dependence w = w(, v, v, d, , . . .); dependences k = f (, v, , d, . . .), or i = f (w),characterizing internal structure of medium ow.

To obtain closing relations a more detailed analysis of ow processes isneeded. It is also necessary to consider mathematical relations describingproperties of the transported medium and the pipeline in which the mediumows.

The division of mechanics in which properties of a transported mediumsuch as viscosity, elasticity, plasticity and other more complex properties arestudied is called rheology.

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31

2Models of Transported Media

Algebraic relations connecting parameters of the transported medium such asdensity, pressure, temperature and so on are called equations of state. Each ofthese relations represents of course a certain schematization of the propertiesof the considered medium and is only a model of a given medium. Let usconsider some models.

2.1Model of a Fluid

By uid ismeant a continuum inwhich the interaction of the contacting interiorparts at rest is reduced only to the pressing force of pressure. If uid particlesinteract along the surface element d with the unit normal n (Figure 2.1), theforce dFn with which the uid particles on one side of the element act on theuid particles on the other side of the element is proportional to the area d,is directed along the normal n and has a pressing action on them. Then

dFn = pn d. (2.1)The magnitude p of this force does not depend on the surface elementorientation and is called pressure.

Thus, p = | dFn/ d|. The absence of tangential friction forces in the state ofrest models the fact that the uid takes the shape of the vessel it lls.

Further classication of uids is dependent on whether or not tangentialfriction forces are taken into account on exposure to uid ow. In accordance

Figure 2.1 A scheme of force interactions in a uid.


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32 2 Models of Transported Media

with this there are two models: the model of an ideal uid and the model of aviscous uid.

2.2Models of Ideal and Viscous Fluids

In the model of an ideal uid it is assumed that tangential friction forcesbetween uid particles separated by an elementary surface are absent, not onlyin the state of rest but also in the state of ow. Such a schematization(or model) of a uid appears to be very fruitful when the tangentialcomponents of interaction forces, that is friction forces, are far smallerthan their normal components, that is pressure forces. In other cases whenthe friction forces are comparable with or even exceed the pressure forces themodel of an ideal uid has proved to be inapplicable. Hence expression (2.1)for an ideal uid is true in the state of rest as well as in the state ofow.

In the model of a viscous uid tangential stresses resulting in uid ow aretaken into account. Let for example theuid layersmove as shown inFigure 2.2.

Here u(y) is the velocity distribution in the ow and y is the direction of anormal to the elementary surface d.

In the model of a viscous uid it is accepted that the tangential stress between the layers of the moving uid is proportional to the velocity differenceof these layers calculated per unit length of the distance between them, namelyto the velocity gradient du/ dy:

= dudy

. (2.2)

The tangential stress is dened as the friction force between the uid layersdivided by the area of the surface separating these layers. Then the dimensionof the stress is

[] = forcearea

= M L/T2

L2= M

L T2 .

In the SI system of units the stress is measured by Pa = kg m1 s2.

Figure 2.2 Illustration of the denition of the viscousfriction law.

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2.2 Models of Ideal and Viscous Fluids 33

The proportionality factor in the law of viscous friction (2.2) is called thefactor of dynamic viscosity or simply dynamic viscosity. Its dimension is

[] = [] T = ML T

In SI units is measured in kg m1s1 and is expressed through poise P,where 1 P = 0.1 kg m1 s1. For example the dynamic viscosity of water isequal to 0.01 P = 0.001 kg m1

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