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Liviu COSTIUC

Virgil-Barbu UNGUREANU

MECANICA FLUIDELOR

FLUID MECHANICS

2014 ISBN: 978-606-19-0492-1

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CUPRINS / TABLE OF CONTENTS PREFA ............................................................................................................................................................................................................................................. 7 FOREWORD ........................................................................................................................................................................................................................................ 8 1. NOIUNI INTRODUCTIVE / INTRODUCTION ................................................................................................................................................................... 9 1.1. OBIECTUL DE STUDIU / THE OBJECT OF STUDY ........................................................................................................................................................................... 9 1.2. NOIUNEA DE FLUID / THE NOTION OF FLUID ............................................................................................................................................................................ 10 2. MRIMI DE STARE I PROPRIETI FIZICE ALE FLUIDELOR / PARAMETERS AND PHYSYCAL PROPERTIES OF FLUIDS ............... 13 2.1. PROPRIETILE FIZICE FUNDAMENTALE COMUNE TUTUROR FLUIDELOR / BASIC PROPERTIES OF ALL FLUIDS .......................................................................... 13

2.1.1. Densitatea i greutatea specific / The density and specific weight ............................................................................................................................... 14 2.1.2. Presiunea / Pressure ...................................................................................................................................................................................................... 16 2.1.3. Viscozitatea fluidelor / Viscosity of fluids ...................................................................................................................................................................... 21

2.2. PROPRIETI SPECIFICE LICHIDELOR / PROPERTIES OF LIQUIDS ................................................................................................................................................ 26 2.2.1. Compresibilitatea izotermic i dilatarea izobar a lichidelor / The izothermal compressibility and isobaric expansion of liquids ........................... 26 2.2.2. Adeziunea fluidelor la suprafee solide / Fluid adhesion to solid surfaces .................................................................................................................... 31 2.2.3. Tensiunea superficial i capilaritatea lichidelor/ Surface tension and liquid capillary ............................................................................................... 32 2.2.4. Absorbia i degajarea gazelor, cavitaia / Absorbtion of gas, cavitation ..................................................................................................................... 39

2.3. PROPRIETI FIZICE SPECIFICE GAZELOR / PHYSICAL PROPERTIES OF GASES ............................................................................................................................ 41 2.3.1. Expansibilitatea / Expansibility ...................................................................................................................................................................................... 41 2.3.2. Proprieti termodinamice / Thermodynamic properties ............................................................................................................................................... 42

2.4. TESTE DE AUTOEVALUARE / SELF-ASSESSMENT TESTS ............................................................................................................................................................. 43 3. ECUAIILE ECHILIBRULUI STATIC AL FLUIDELOR/ STATIC BALANCE EQUATIONS OF FLUIDS .............................................................. 52 3.1. FORE CARACTERISTICE FLUIDELOR/ CHARACTERISTIC FLUID FORCES ................................................................................................................................... 52 3.2. ECUAIILE LUI EULER PENTRU ECHILIBRUL STATIC AL UNUI FLUID / EULER'S EQUATIONS FOR STATIC EQUILIBRIUM OF A FLUID ............................................ 54 3.3. ECHILIBRUL STATIC AL UNUI FLUID UOR/ STATIC EQUILIBRIUM OF A LIGHT FLUID ................................................................................................................. 56 3.4. ECHILIBRUL STATIC AL UNUI FLUID GREU I INCOMPRESIBIL/ STATIC EQUILIBRIUM OF A HEAVY INCOMPRESSIBLE FLUID ....................................................... 57

3.4.1. Ecuaia fundamental a hidrostaticii/ The basic equation of hydrostatics .................................................................................................................... 57 3.4.1.1. Deducerea ecuaiei / Equation Deduction ............................................................................................................................................................................................. 57 3.4.1.2. Interpretarea ecuaiei fundamentale a hidrostaticii/ Interpretation of the fundamental equation of hydrostatics .................................................................................. 58

3.4.2. Consecine deduse din ecuaia fundamental a hidrostaticii/ Consequences derived from the fundamental equation of hydrostatics ......................... 60 3.4.3. Aplicaii ale legii fundamentale a hidrostaticii/ Applications of the fundamental law of hydrostatics .......................................................................... 62

3.4.3.1. Aplicaii ale principiului vaselor comunicante/ Application of the principle of communicating vessels .............................................................................................. 62 3.4.3.2. Aplicaii ale principiului lui Pascal / Applications of Pascal's principle ............................................................................................................................................... 68

3.5. TESTE DE AUTOEVALUARE / SELF-ASSESSMENT TESTS ............................................................................................................................................................. 70 4. FORE DE ACIUNE ALE FLUIDELOR N REPAUS ASUPRA UNOR PEREI SOLIZI/ ACTING FORCES OF REST FLUIDS ON SOLID

WALLS 75 4.1. GENERALITI / GENERAL ISSUES ............................................................................................................................................................................................ 75 4.2. FORE DE PRESIUNE ALE FLUIDELOR N REPAUS ASUPRA UNOR SUPRAFEE PLANE .................................................................................................................. 76

4.2.1. Ecuaii generale / Basic equations ................................................................................................................................................................................. 76 4.2.1. Aciunea unui fluid uor n echilibru static pe o suprafa plan/ The action of a light fluid in static equilibrium on a hard flat surface ................... 77

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4.2.2. Aciunea unui fluid greu n echilibru static asupra unei suprafee plane/The action of a heavy fluid in static equilibrium on a hard flat surface ...... 78 4.3. FORE DE ACIUNE ALE FLUIDELOR N REPAUS ASUPRA UNOR SUPRAFEE CURBE DESCHISE / ACTING FORCES OF THE FLUID AT REST ON OPEN CURVED

SURFACES 85 4.3.1. Generaliti / General issues .......................................................................................................................................................................................... 85 4.3.2. Fore de aciune ale fluidelor uoare n repaus pe suprafee curbe deschise/ Acting forces of the light fluid at rest on open curved surface.............. 85 4.3.3. Fore de aciune ale fluidelor grele n repaus pe suprafee curbe deschise/ Acting forces of the heavy fluid at rest on open curved surface .............. 88

4.4. FORE HIDROSTATICE PE SUPRAFEE CURBE NCHISE / HYDROSTATIC FORCES ON CLOSED CURVED SURFACES ....................................................................... 90 4.5. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ........................................................................................................................................................... 96 5. CINEMATICA FLUIDELOR / FLUID KINEMATICS ........................................................................................................................................................ 98 5.1. CLASIFICAREA MICRII FLUIDELOR / CLASSIFICATION OF FLUID MOVEMENT ......................................................................................................................... 98 5.2. REPREZENTAREA STRII DE MICARE A FLUIDELOR / REPRESENTATION OF THE STATE OF FLUID MOTION ................................................................................ 99 5.3. DEFINIREA NOIUNILOR GENERALE DIN CINEMATICA FLUIDELOR / GENERAL DEFINITIONS IN KINEMATICS OF FLUIDS .......................................................... 101 5.4. DESCOMPUNEREA MICRII UNEI PARTICULE DE FLUID / DECOMPOSITION OF THE MOVEMENT OF THE FLUID PARTICLE ........................................................ 106 5.5. LEGEA CONTINUITII/ CONTINUITY LAW .............................................................................................................................................................................. 109

5.5.1. Legea continuitii pentru micri spaiale / Continuity law for spatial movements ................................................................................................... 109 5.5.2. Legea continuitii pentru un tub de curent / The law of continuity for the streamtube .............................................................................................. 112

5.6. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ......................................................................................................................................................... 114 6. ECUAIILE DE MICARE ALE FLUIDELOR PERFECTE / THE EQUATIONS OF MOTION OF IDEAL FLUIDS ........................................... 116 6.1. ECUAIILE EULER PENTRU ECHILIBRUL DINAMIC AL FLUIDELOR PERFECTE / EULERS EQUATIONS FOR DYNAMIC EQUILIBRIUM OF IDEAL FLUIDS ................ 116 6.2. INTEGRAREA ECUAIILOR ECHILIBRULUI DINAMIC AL FLUIDELOR PERFECTE / THE INTEGRATION OF DYNAMIC EQUILIBRIUM EQUATIONS OF IDEAL FLUIDS . 118 6.3. ECUAIA LUI BERNOULLI PE O LINIE DE CURENT PENTRU MICAREA PERMANENT I ABSOLUT A UNUI FLUID IDEAL N CMP GRAVITAIONAL / BERNOULLI'S

EQUATION ON A STREAMLINE FOR STEADY AND ABSOLUTE FLOW OF AN IDEAL FLUID IN THE GRAVITATIONAL FIELD .................................................................................... 125 6.4. ECUAIA LUI BERNOULLI PENTRU MICAREA PERMANENT I ABSOLUT A UNUI FLUID COMPRESIBIL N CMP GRAVITAIONAL / BERNOULLIS EQUATION FOR

A STEADY AND ABSOLUTE MOVEMENT OF A COMPRESSIBLE FLUID IN THE GRAVITATIONAL FIELD ................................................................................................................. 129 6.5. EXTINDEREA ECUAIEI LUI BERNOULLI LA CURENI DE SECIUNE FINIT N MICARE PERMANENT / EXTENSION OF THE BERNOULLIS EQUATION AT FOWS OF

FINITE SECTION IN STEADY FLOW ................................................................................................................................................................................................................... 132 6.6. TEOREMA IMPULSULUI / MOMENTUM THEOREM ..................................................................................................................................................................... 135

6.6.1. Enunul teoremei / Enunciation of the theorem ............................................................................................................................................................ 135 6.6.2. Teorema impulsului aplicat unui tub de curent / Momentum theorem applied to a streamtube ................................................................................ 138 6.6.3. Aciunea dinamic a unui jet de fluid asupra unei suprafee solide / Dynamic action of a fluid jet over a solid surface ............................................ 145

6.7. TESTE DE AUTOEVALUARE / SELF ASSESSEMENT TESTS .......................................................................................................................................................... 150 7. MICAREA LAMINAR A FLUIDELOR VSCOASE / THE LAMINAR FLOW OF VISCOUS FLUIDS .............................................................. 152 7.1. ECUAIILE FUNDAMENTALE ALE MICRII LAMINARE A FLUIDELOR VSCOASE / BASIC EQUATIONS OF VISCOUS FLUIDS FLOW ............................................ 152 7.2. ECUAIA LUI BERNOULLI PENTRU FLUIDE VSCOASE I INCOMPRESIBILE / BERNOULLIS EQUATION FOR VISCOUS AND INCOMPRESIBILE FLUIDS ................ 154 7.3. REZISTENE HIDRAULICE, COMPUNEREA PIERDERILOR DE SARCIN / HYDRAULIC RESISTANCES, HEAD LOSSES COMPOSITION .............................................. 157 8. ANALIZ DIMENSIONAL I SIMILITUDINE HIDRODINAMIC / DIMENSIONAL ANALYSIS, AND HYDRODYNAMIC SIMILARITY163 8.1. ANALIZA DIMENSIONAL / DIMENSIONAL ANALYSIS .............................................................................................................................................................. 163 8.2. SIMILITUDINEA / SIMILITUDE (SIMILARITY) ........................................................................................................................................................................... 166 8.3. CRITERIILE DE SIMILITUDINE MAI IMPORTANTE / MORE IMPORTANT CRITERIA OF SIMILARITY............................................................................................... 172

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8.4. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ......................................................................................................................................................... 179 9. CURGEREA FLUIDELOR VSCOASE N REGIM LAMINAR PRIN CONDUCTE FORATE / LAMINAR VISCOUS FLOW THROUGH

PRESSURIZED PIPES ............................................................................................................................................................................................................................... 182 9.1. LEGEA DE DISTRIBUIE A VITEZEI / DISTRIBUTION LAW OF VELOCITIES ................................................................................................................................. 182 9.2. LEGEA DE DISTRIBUIE A EFORTULUI DE FRECARE N SECIUNEA TRANSVERSAL A UNEI CONDUCTE CIRCULARE / DISTRIBUTION LAW FOR FRICTION STRESS IN

THE CROSS-SECTION OF A CIRCULAR PIPE ....................................................................................................................................................................................................... 186 9.3. CALCULUL DEBITULUI I VITEZEI MEDII / CALCULATION OF FLOW RATE AND AVERAGE VELOCITY ........................................................................................ 187 9.4. CALCULUL COEFICIENTULUI PIERDERILOR LINIARE DE SARCIN / CALCULATION OF LINEAR HEAD LOSSES COEFFICIENT (FRICTION FACTOR) ....................... 189 9.5. DETERMINAREA COEFICIENILOR CORIOLIS I BOUSSINESQ / DETERMINATION OF CORIOLIS AND BOUSSINESQ COEFFICIENTS ............................................. 191 10. MICAREA TURBULENT / TURBULENT FLOW .................................................................................................................................................... 193 10.1. CARACTERISTICILE MICRII TURBULENTE / CHARACTERISTICS OF TURBULENT FLOW .......................................................................................................... 193 10.2. STRATUL LAMINAR NTR-UN CURENT TURBULENT. CONTACTUL FLUIDULUI CU PEREII / BOUNDARY LAYER IN A TURBULENT STREAM / FLUID CONTACT WITH

WALLS 197 10.3. RELAIA LUI BERNOULLI PE O LINIE DE CURENT N MICAREA TURBULENT / BERNOULLI'S RELATIONSHIP ON A STREAMLINE IN THE TURBULENT FLOW .... 199 10.4. DISTRIBUIA VITEZELOR PE SECIUNEA TUBULUI DE CURENT / DISTRIBUTION OF VELOCITIES ON THE STREAM TUBE SECTION.............................................. 201 10.5. CALCULUL COEFICIENILOR CORIOLIS I BOUSSINESQ / CALCULATION OF CORIOLIS AND BOUSSINESQ COEFFICIENTS......................................................... 204 10.6. TEST DE AUTOEVALUARE / SELF-ASSESSEMENT TEST ............................................................................................................................................................. 206 11. PIERDERI DE SARCIN N INSTALAIILE HIDRAULICE / HEAD LOSS IN HYDRAULIC SYSTEMS ........................................................ 207 11.1. PIERDERI DE SARCIN LOCALE N INSTALAII HIDRAULICE / LOCAL PRESSURE LOSSES IN HYDRAULIC SYSTEMS ................................................................... 207 11.2. PIERDERI DE SARCIN LINIARE N INSTALAIILE HIDRAULICE ................................................................................................................................................. 217 11.3. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ......................................................................................................................................................... 228 12. MICAREA PERMANENT N CONDUCTE SUB PRESIUNE / STEADY FLOW IN PRESSURIZED PIPES................................................... 235 12.1. CARACTERISTICA UNEI CONDUCTE / CHARACTERISTIC OF A PIPELINE .................................................................................................................................... 235 12.2. CONDUCTE SCURTE SHORT PIPES ......................................................................................................................................................................................... 239 12.3. CONDUCTE LUNGI LONG PIPES ............................................................................................................................................................................................. 240 12.4. CONDUCTE LEGATE N SERIE / SERIES CONNECTED PIPES ........................................................................................................................................................ 240 12.5. CONDUCTE LEGATE N PARALEL / PARALLEL CONNECTED PIPES ............................................................................................................................................. 242 12.6. PROBLEME TIP I METODE DE REZOLVARE / STANDARDIZED PROBLEMS AND METHODS TO SOLVE ......................................................................................... 243

12.6.1. Probleme de exploatare/- Operational problems ......................................................................................................................................................... 243 12.6.2. Probleme de proiectare Design problems ................................................................................................................................................................. 246

12.7. TEST DE AUTOEVALUARE / SELF-ASSESSEMENT TEST ............................................................................................................................................................. 250 13. PERTURBAII N FLUIDE COMPESIBILE / DISTURBANCES IN COMPESSIBILE FLUIDS ........................................................................... 251 13.1. VITEZA DE PROPAGARE A SUNETULUI / VELOCITY OF SOND PROPAGATION ............................................................................................................................ 251 13.2. PERTURBAII N MEDII INFINITE - DISTURBANCES IN INFINITE MEDIA ..................................................................................................................................... 257 13.3. MICAREA VARIABIL N CONDUCTE SUB PRESIUNE - VARIABLE MOVEMENT IN PRESSURE PIPES .......................................................................................... 258

13.3.1. Analiza fenomenului Phenomenon analysis .............................................................................................................................................................. 258 13.3.2. Calculul suprapresiunii maxime Calculation of the maximum overpressure ............................................................................................................ 261 13.3.3. Calculul vitezei de propagare a loviturii de berbec / Calculation of the water hammer velocity of propagation ....................................................... 264 13.3.4. Metode de atenuare a loviturii de berbec / Attenuation methods for the water hammer ............................................................................................. 268

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13.4. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ......................................................................................................................................................... 270 14. APARATE PENTRU MSURAREA PARAMETRILOR HIDRODINAMICI BAZATE PE ECUAIA LUI BERNOULLI / APPARATUS FOR

MEASURING HYDRODYNAMIC PARAMETERS BASED ON BERNOULLI'S PRINCIPLE ...................................................................................................... 276 14.1. TUBUL PITT-PRANDTL / PITOT-PRANDTL TUBE .................................................................................................................................................................... 277

14.1.1. Parametrii frnai ai fluidelor / Stagnation parameters of fluids ................................................................................................................................ 277 14.1.2. Principiul msurrii vitezei cu ajutorul tubului Pitt-Prandtl / The principle of measuring the velocity using the Pitot-Prandtl tube ...................... 282 14.1.3. Determinarea debitului folosind sonda de vitez / Flow rate determining using the velocity probe ........................................................................... 285

14.2. DISPOZITIVE STANDARDIZATE PENTRU MSURAREA DEBITULUI PRIN RESTRICIA SECIUNII DE CURGERE / STANDARD FLOW MEASUREMENT DEVICES BY RESTRICTING THE FLOW PATH......................................................................................................................................................................................................................... 288

14.3. ALTE DISPOZITIVE PENTRU MSURAREA DEBITULUI PRIN RESTRICIA SECIUNII DE CURGERE / OTHER DEVICES FOR THE MEASUREMENT OF FLOW RATE USING THE FLOW RESTRICTION .................................................................................................................................................................................................................................. 293

14.4. TESTE DE AUTOEVALUARE / SELF-ASSESSEMENT TESTS ......................................................................................................................................................... 298 15. MICRI EFLUENTE / EFFLUENT FLOWS ................................................................................................................................................................ 303 15.1. CURGEREA FLUIDELOR INCOMPRESIBILE PRIN ORIFICII MICI N REGIM PERMANENT / INCOMPRESSIBLE FLUID FLOW THROUGH SMALL ORIFICES IN PERMANENT

REGIME 303 15.2. CURGEREA FLUIDELOR INCOMPRESIBILE PRIN ORIFICII MICI N REGIM VARIABIL / INCOMPRESSIBLE FLOW THROUGH SMALL ORIFICES IN A NON STEADY REGIME

311 15.3. CURGEREA FLUIDELOR INCOMPRESIBILE PRIN AJUTAJE / INCOMPRESSIBLE FLUID FLOW THROUGH NOZZLES ......................................................................... 315 15.4. CURGEREA FLUIDELOR INCOMPRESIBILE PRIN ORIFICII MARI I DEVERSOARE / INCOMPRESSIBLE FLOW THROUGH LARGE ORIFICES AND WEIRS .................... 319 15.5. CURGEREA FLUIDELOR EXPANSIBILE PRIN ORIFICII MICI / EXPANSIBLE FLUID FLOW THROUGH SMALL ORIFICES ................................................................... 325 15.6. CURGEREA FLUIDELOR EXPANSIBILE PRIN AJUTAJE / EXPANSIBLE FLUID FLOW THROUGH NOZZLES ...................................................................................... 330 15.7. TEST DE AUTOEVALUARE / SELF-ASSESSEMENT TEST ............................................................................................................................................................. 334 16. APARATE CU JET / JET APPARATUS .......................................................................................................................................................................... 335 16.1. INTRODUCERE / INTRODUCTION .............................................................................................................................................................................................. 335 16.2. PRINCIPII CONSTRUCTIVE I FUNCIONALE / STRUCTURAL AND FUNCTIONAL PRINCIPLES ...................................................................................................... 336 16.3. INDICI PENTRU APRECIEREA CALITII APARATELOR CU JET / INDICATORS FOR ASSESSING THE QUALITY OF JET DEVICES .................................................... 340 16.4. DOMENII DE UTILIZARE / AREAS OF USE ................................................................................................................................................................................. 343 17. MAINI HIDRAULICE / HYDRAULIC MACHINES ................................................................................................................................................... 345 17.1. DEFINIII / DEFINITIONS ......................................................................................................................................................................................................... 345 17.2. CLASIFICRI / CLASSIFICATIONS ............................................................................................................................................................................................ 348 17.3. PARAMETRII ENERGETICI PRINCIPALI / THE MAIN ENERGETIC PARAMETERS ........................................................................................................................... 351

17.3.1. Parametrii fundamentali / Basic parameters ............................................................................................................................................................... 351 17.3.2. Puteri i randamente / Powers and efficiencies ........................................................................................................................................................... 354 17.3.3. Test de autoevaluare / Self-assessement test ................................................................................................................................................................ 364

17.4. MAINI VOLUMICE / VOLUMETRIC MACHINES ........................................................................................................................................................................ 368 17.4.1. Principii de funcionare i tipuri constructive / Working principles and constructive types ........................................................................................ 368 17.4.2. Pompe cu piston / Piston pumps ................................................................................................................................................................................... 370

17.4.2.1. Caracteristicile principale ale pompelor cu piston / The main characteristics of piston pumps ........................................................................................................ 370 17.4.2.2. Randamente / Efficiencies ................................................................................................................................................................................................................ 372

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17.4.2.3. Recomandri privind utilizarea i funcionarea pompelor volumice alternative / Recommendations for utilization and exploitations of alternatives volumetric pumps 375

17.4.3. Motoare hidraulice liniare - Linear hydraulic motors ................................................................................................................................................. 377 17.4.3.1. Construcie i funcionare / Construction and operation ................................................................................................................................................................... 377 17.4.3.2. Calculul hidraulic pentru alegerea motoarelor hidraulice liniare / The hydraulic calculus for linear hydraulic motors choice ......................................................... 380

17.4.4. Pompe i motoare rotative / Rotating pumps and motors ............................................................................................................................................ 384 17.4.5. Acionri hidrostatice / Hydrostatic drives .................................................................................................................................................................. 388 17.4.6. Teste de autoevaluare / Self-assessement tests ............................................................................................................................................................. 390

17.5. POMPE HIDRODINAMICE / HYDRODYNAMIC PUMPS ................................................................................................................................................................. 405 17.5.1. Definiii / Definitions .................................................................................................................................................................................................... 405 17.5.2. Clasificarea pompelor hidrodinamice / Hydrodynamic pumps classification ............................................................................................................. 405 17.5.3. Puteri, pierderi energetice i randamente / Powers, energetic losses and efficiencies ................................................................................................ 411

17.5.3.1. Pierderi mecanice i randament mecanic / The mechanical losses and the mechanical efficiency ................................................................................................... 412 17.5.3.2. Pierderi volumice i randament volumic / The volumetric losses and volumetric efficienciency ..................................................................................................... 413 17.5.3.3. Pierderi hidraulice i randament hidraulic / Hydraulic losses and hydraulic efficiency .................................................................................................................... 414 17.5.3.4. Puterea consumat i randamentul global / The consumed power and the global efficiency ............................................................................................................ 415

17.5.4. Mrimi caracteristice pompelor hidrodinamice i instalaiilor de pompare / Characteristic magnitudes of hydrodynamic pumps and pumping installations 417

17.5.5. Curbele caracteristice ale pompelor centrifuge / The characteristic curves of centrifugal pumps .............................................................................. 421 17.5.5.1. Determinarea curbelor caracteristice / The determination of characteristic curves ........................................................................................................................... 421 17.5.5.2. Similitudinea pompelor centrifuge / The similarity of the centrifugal pumps ................................................................................................................................... 428

17.5.6. Cavitaia pompelor hidrodinamice / Hydrodynamic pumps cavitation ........................................................................................................................ 437 17.5.6.1. Prezentarea fenomenului / Presentation of the phenomenon ............................................................................................................................................................. 437 17.5.6.2. Sarcina geometric la aspiraie / Geometric height from suction ...................................................................................................................................................... 439 17.5.6.3. Evitarea cavitaiei / Avoiding cavitation ........................................................................................................................................................................................... 449

17.5.7. Funcionarea n comun a sistemului pomp-reea - Joint operation of pump - network system .................................................................................. 451 17.5.7.1. Curbele caracteristice de exploatare ale pompelor hidrodinamice - Operating characteristic curves of the hydrodynamic pumps .................................................. 451 17.5.7.2. Punct de funcionare Point of operation ......................................................................................................................................................................................... 451 17.5.7.3. Reglarea punctului de funcionare / Adjustment of the operating point ............................................................................................................................................ 455 17.5.7.4. Cuplarea pompelor centrifuge / Coupling of centrifugal pumps ....................................................................................................................................................... 462

17.5.7.4.1. Cuplarea pompelor n paralel / Parallel coupling of pumps ...................................................................................................................................................... 463 17.5.7.4.2. Cuplarea pompelor n serie / Series coupling of pumps ............................................................................................................................................................ 467

17.5.8. Test de autoevaluare / Self-assessement test ................................................................................................................................................................ 472 18. ANEX - MRIMI I UNITI DE MSUR / ANNEX - MAGNITUDES AND UNITS OF MEASURE ............................................................ 487 18.1. NOIUNI GENERALE / GENERALITIES ...................................................................................................................................................................................... 487 18.2. SISTEMUL INTERNAIONAL DE UNITI DE MSUR / THE INTERNATIONAL SYSTEM OF UNITS OF MEASUREMENT ................................................................ 488 18.3. UNITI DE MSUR CARE NU FAC PARTE DIN SI / UNITS OF MEASUREMENT WHICH ARE NOT PART OF THE SI ...................................................................... 491 18.4. TRANSFORMAREA RELAIILOR LA SCHIMBAREA UNITILOR DE MSUR / TRANSFORMING RELATIONSHIPS CHANGING UNITS OF MEASURE ....................... 495 19. BIBLIOGRAFIE / REFERENCES .................................................................................................................................................................................... 497

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Prefa

Studiul aplicaiilor mecanicii fluidelor precum i principiile constructive ale instalaiilor, aparatelor i mainilor hidraulice stau astzi la

baza a numeroase aplicaii n cele mai variate domenii ale activitii inginereti i de cercetare, iar cunoaterea lor este absolut necesar pentru formarea complet a unui inginer.

Lucrarea de fa i propune s prezinte printr-o tratare unitar noiunile teoretice i practice ale mecanicii fluidelor, cuprinznd capitolele de statica, cinematica i dinamica fluidelor, iar legate de acestea, curgerea permanent i nepermanent n conducte, msurri n instalaii hidraulice, micri efluente i aparate cu jet. Legile generale i gsesc aplicaii n construcia i funcionarea mainilor hidraulice, acestea fiind tratate ntr-un capitol separat.

Scopul lucrrii este dobndirea unor cunotine n domeniul fenomenelor hidraulice cu formarea unor deprinderi de calcul minimal al instalaiilor i mainilor hidraulice innd cont de condiii optimizatoare cuantificate prin eficien i randament. Astfel, la sfritul fiecrui capitol sunt prevzute cteva teste de autoevaluare. La problemele care pot prea mai dificile s-au prevzut indicaii mai detaliate, cuprinznd rezultatele pariale ale mrimilor care conduc la determinarea mrimilor finale cerute de problem.

Lucrarea se adreseaz n primul rnd studenilor de la cursurile de licen ale programelor de studii de inginerie mecanic, design, ingineria mediului, energii regenerabile i inginerie tehnologic dar, prin diversitatea aplicaiilor, poate fi utilizat la toate programele de studii de licen i master.

Braov, decembrie 2014 Autorii

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Foreword

Research applications of fluid mechanics and constructive principles of hydraulic installation, apparatus and machines are today to the

basis for many applications in various fields of engineering and research activity, and their knowledge is absolutely necessary for the

complete formation of an engineer.

This work aims to present in a treatment unit, theory and practice of the fluid mechanics, comprising chapters of statics, kinematics and

fluid dynamics and related to him, the permanent and non permanent flow in pipes, measuring in hydraulic installations, efluent flows and jet

apparatus. General laws find their applications in the construction and operation of hydraulic machines, which are treated in a separate

chapter.

The aim of this work is knowledge acquisition in hydraulic phenomena with minimal computing skills formation of the hydraulic plant

and machinery considering optimizing conditions measured by effectiveness and efficiency. Thus, at the end of each chapter are provided

several self-assessment tests. The problems may seem more difficult were provided detailed information, including the results of partial

quantities leading to the final determination of the required quantities of matter.

The work is addressed primarily at students from the undergraduate courses of study programs of mechanical engineering, design,

environmental engineering, renewable energy and technological engineering but, by the diversity of applications, can be used in all programs

of Bachelor and Master.

Braov, December 2014 The Authors

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1. NOIUNI INTRODUCTIVE / INTRODUCTION

1.1. Obiectul de studiu / The object of study

Mecanica teoretic, studiind cele mai simple forme de micare i cauzele care le produc, se folosete de noiunile de punct material sau sistem de puncte materiale. Un sis-

tem de puncte materiale ns poate fi discret sau continuu. Lichidele i gazele sunt medii continui fluide, deci au

proprietatea de curgere datorit coeziunii moleculare mult mai mici dect a corpurilor solide. n stare de repaus,

fluidele nu suport dect eforturi normale. Eforturile tangeniale ar determina ieirea din starea de repaus, deci

curgerea lor.

Mecanica are trei mari diviziuni:

mecanica general studiaz legile universale ale mic-rii punctului material, sistemelor de puncte materiale i a corpurilor solide rigide;

mecanica solidelor deformabile studiaz deformarea corpurilor solide sub aciunea forelor exterioare i n func-ie de natura materialului (elasticitate, plasticitate etc.); mecanica fluidelor este o ramur a mecanicii mediilor continui, desprins ca tiin de sine stttoare, care studi-az repausul i micarea fluidelor, precum i interaciunea lor mecanic cu corpurile cu care vin n contact.

The theoretic mechanics investigates the common

movements and that causes uses notions of material point

or system of material points. A system of material points

can be discrete or continuous. Liquids and gases are

continuous fluid mediums, therefore has the flow property

because of the smaller molecular cohesion than solid

bodies. In position of repose, fluids no support than

normal stress. Tangential thrusts can determine the leave

from the position of repose, therefore their flow.

Mechanics has three big divisions:

general mechanics investigates the universal laws of

the material point, systems of material points and rigid

solids;

mechanics of strain solids investigates the straining of

solid bodies under the external forces and function of the

material properties (elasticity, plasticity et al.);

fluid mechanics is a continuous mediums branch,

detached that an independent science, that investigates the

repose and movement of fluids, as well as their mechanical

interaction with bodies that has contact.

Sometimes, in order to study the movement of sys-

10

Uneori, pentru a studia micarea unor sisteme, se fac unele ipoteze simplificatoare cum ar fi, de exemplu, absen-

a frecrilor. n toate aceste cazuri, micarea apare ca un fenomen pur mecanic, legea fundamental fiind legea conservrii energiei mecanice.

Dar ipotezele simplificatoare nu sunt niciodat riguros realizate n natur, astfel nct fenomenele meca-nice sunt totdeauna nsoite de femomene termice. Numai n msura n care aceste fenomene termice au efecte negli-jabile putem spune c acceptarea ipotezelor simplificatoare reprezint o aproximaie acceptabil a realitii, [17].

tems to make some simplifying assumptions such as, for

example, the absence of friction. In all these cases, the mo-

vement appears as a purely mechanical phenomenon, the

fundamental law is the law of conservation of mechanical

energy.

But the simplifying assumptions made are never rigo-

rous in nature, so that mechanical phenomena are always

accompanied by thermal phenomenon that punctuate. Only

to the extent that these thermal phenomena have negligible

effects simplifying assumptions we can say that

acceptance is an acceptable approximation of reality [17].

1.2. Noiunea de fluid / The notion of fluid

Mecanica fluidelor opereaz cu noiunile particul de fluid i domeniu fluid.

Particula fluid este o poriune de fluid avnd dimensiuni cu mult mai mari dect dimensiunile molecu-

lelor, dar cu mult mai mici fa de dimensiunile corpuri-lor n raport cu care se studiaz echilibrul sau micarea fluidului.

Mecanica fluidelor se ocup de studiul repausului sau micrii fluidelor considerate ca medii continue, omogene i izotrope n care, n stare de repaus, pe supra-feele de contact ale diferitelor particule se exercit numai eforturi normale, iar sub aciunea unor fore care nu tind s-i modifice volumul se deformeaz cu uurin. Mobili-

The fluid mechanics uses notions: the particle of

fluid and the fluid domaine. The particle of fluid is a portion of fluid having

dimensions greater than the molecular dimensions, but

smaller than dimensions of bodies with that are referred

the equilibrium and movement of fluid.

Fluid mechanics investigate the study of the repose

and movement of fluids considered that continuum,

homogeneous and izotropic media in that, in the idle

state, on the contact surfaces of different particles there

are carried out only normal stress, but, under the action of

forces that no tend to modify his volume is easily

deformed. The mobility of fluid particles is due of the

11

tatea particulelor fluide este datorat slabei coeziuni a moleculelor.

Lichidele au volum propriu i iau forma vaselor n care sunt coninute. Gazele i vaporii, avnd o coeziune mult mai mic datorit spaiilor intermoleculare mari, nu au volum propriu, ci sunt expansibile, deci ocup tot spaiul disponibil. De asemenea, sunt cu mult mai compresibile i mai uoare dect lichidele.

Ipoteza general a continuitii unui fluid exprim faptul c n fiecare punct P(x,y,z) i la orice moment t se pot determina mrimi fizice (de exemplu: o

densitate, tzyx ,,, , o presiune tzyxpp ,,, , o vitez tzyxvv ,,, etc.) i c aceste funcii de coordonatele punctului i de timp sunt continue aproape peste tot, deci cu excepia unui numr finit de suprafee sau linii singulare, [1], [6], [9].

Ca exemple de suprafee de discontinuitate se pot cita suprafeele de contact: suprafaa de contact dintre un lichid i vasul care l conine, suprafaa care delimiteaz un jet de fluid, suprafaa liber a unui lichid.

Practic, n natur nu exist fluid omogen i izotrop, cu mici excepii: apa pur i unele lichide cu nalt puri-tate din industria chimic. De cele mai multe ori, apa se gsete n amestec cu particule solide sau bule de gaz, iar aerul este n amestec cu fum, alte gaze, picturi fine de lichid cea. Prezena fazelor diferite ntr-un mediu fluid modific proprietile acestuia. Fluidul polifazic real

small coezion forces of molecules.

The liquids have own volume and take over the

shape of vessel into that there are contained. Gases and

vapours, having small molecular cohesion because their

big espaces between molecules, have not own volume

and are expandable, therefore occupiyng all diponible

espace. Also, there are much compresible and light than

liquids.

The general hypothesis of the continuity of fluids

express the fact of, in each point P(x,y,z) and in each

moment t can be determined physical magnitudes (for

example: a density, a pressure

, a speed et al.) and these

are function of coordinates of point and time are

continuous nearby everywhere so, with exception of

finite number of surfaces or singular lignes, [1], [6], [9].

For examples, surfaces of discontinuity are contact

surfaces: the surface of contact between a liquid and the

vase that it is contained, the surface that delimit a jet

fluid, the open space of liquid.

Practically, in nature is not an homogeneous and

izotropic fluid, with some exceptions: pure water, some

chemical liquids whith high purity. For example, water is

in mixture with solid particles or gas bubbles and air is in

mixture with smoke, other gases, fine liquid drops mist. The presence of different phases in a fluid medium

changes its properties. The real poliphasic fluid is a

tzyx ,,, tzyxpp ,,, tzyxvv ,,,

12

este un mediu continuu, neomogen i neizotrop caracte-rizat prin proprietatea de fluiditate. Dinamica fluidelor

polifazice studiaz micarea, aciunea amestecurilor asupra corpurilor solide cu care vin n contact i transfor-mrile energetice care apar n cursul deplasrii mediului multifazic innd cont de forele care determin curgerea i de interaciunea dintre fazele constituente, [7].

continuum unhomogeneous and anizotropic medium

characterized by the property of fluidity. The dynamic of

poliphasic fluids investigates the movement, the action of

mixtures on the solid bodies whith that they are in contact

and energetic conversions that occur in the multiphasic

medium deplacement taking into account between

constitutive phases [7].

13

2. MRIMI DE STARE I PROPRIETI FIZICE ALE FLUIDELOR / PARAMETERS AND PHYSYCAL PROPERTIES OF

FLUIDS

2.1. Proprietile fizice fundamentale comune tuturor fluidelor / Basic properties of all fluids

n ecuaiile care descriu repaosul sau micarea fluide-lor intervin mrimile care definesc starea fluidului luat ca sistem termodinamic (temperatura, presiunea, volumul

specific sau densitatea etc.) sau mrimi care definesc unele proprieti (viscozitate, compresibilitate etc.). n acest capitol se prezint proprietile fizice mai importante ale fluidelor utilizate n cartea de fa, precum i unele apli-caii simple care pot fi rezolvate folosind definiiile acestor proprieti.

Proprietile fizice ale fluidelor sunt variabile cu temperatura i presiunea. Determinarea corect a valorii lor presupune cunoaterea funciilor care aproximeaz aceste variaii. Se vor avea n vedere aplicaii tehnice uzuale n care nu se cer precizii foarte ridicate.

In equations that describe the repos and movement of

fluids intervene quantities that define the fluid state

considered a thermodynamic system (temperature,

pressure, specific volume or density) or magnitudes that

define some physical properties of fluids used in the actual

book, as well as some simple applications that can be

solved by using definitions of these properties.

Physical properties of fluids are variable with

temperature and pressure. The accurate determination of

yours values supposes the knowledgement of functions

that approximate these variations. It shall to refer to usual

technical applications in that there are not requested high

precisions.

14

2.1.1. Densitatea i greutatea specific / The density and specific weight

Densitatea este masa unitii de volum, definit pentru un punct dintr-un fluid prin relaia matematic:

The density is the mass of volume unity, defined for a

point of a fluid by the mathematical relation:

.d

dlim

0 V

m

V

m

V

(2.1)

Dup cum s-a artat anterior, n fenomenele n care este admisibil ipoteza continuitii, densitatea este o funcie continu de punct i de timp, tzyx ,,, . Dimensional,

Because in fluids is admitted the continuity

hypothesis, the density is a continuum function of point

and time, . Dimensional,

3lm . (2.2)

n Sistemul Internaional de Uniti de Msur (notat pe scurt SI), unitatea de msur este kg/m3.

Densitatea variaz n funcie de temperatur i presiune. Densitatea lichidelor variaz foarte puin cu presiunea.

Densitatea fluidelor scade cu creterea temperaturii. Apa prezint o anomalie din acest punct de vedere, densi-tatea maxim fiind la temperatura de 3,98oC (1000 kg/m3).

Pentru calculul densitii lichidelor n funcie de temperatur i presiune se recomand ecuaii de tip polino-mial, suficient de precise n domeniul tehnic uzual. Tabelul

2.1 prezint densitatea unor lichide uzuale la temperatura de 20

oC.

In the International System of Units (abbreviated by

SI) the unity of measurement is kg/m3.

The density varies function of temperature and

pressure. The density of liquids has a little variation with

pressure.

The density of fluids diminishes with the temperature

increasing. Water presents an anomaly from this point of

vue, the maximum density being at 3.98 C (1000 kg/m3).

The density of liquids can be obtained function of

temperature and pressure by polynomial equations used

for the technical domain. Table 2.1. presents the density of

useful liquids at 20 C.

For gases there are recommended the equation of

tzyx ,,,

15

Pentru gaze se recomand ecuaia de stare care d rezultate foarte bune pentru domeniul n care acestea pot fi

asimilate unui gaz ideal:

state from that can be obtained very good results for the

domain in that these can be assimilated with an ideal gas:

,RT

p (2.3)

unde p reprezint presiunea absolut a gazului, T este temperatura lui absolut, iar R este constanta caracteristic a gazului. Pentru aer, constanta caracteristic este R = 287,04 J/kgK.

where p repersents the absolute pressure of gas, T absolute temperature, and R characteristic constant. For air, the characterisitic constant is R = 287.04 J/kgK.

Tab. 2.1. Densitatea unor lichide / Density of some liquids

Lichidul / Liquide [kg/m3] / 20 oC

Aceton /Acetone 790

Alcool etilic / Ethilic alcohol 789,5

Alcool metilic / Metilic alcohol 792

Ap de mare / Sea water 1010 - 1050

Benzin / Gasoline 710 - 740

Lapte / Milk 1020 - 1050

Mercur / Mercury 13545,7

Ulei de ungere / Luboil 871

Ulei de transformator / Electrical insulating oil 866

Greutatea specific este definit ca greutatea unitii de volum cu ajutorul ecuaiilor:

The specific weight (unit weight) is the weight per

unit volume of a fluid, being defined by equations:

16

.ggd

d

d

)(d

d

d

V

m

V

mg

V

G (2.4)

n Sistemul Internaional, unitatea de msur este N/m

3. Acceleraia medie a gravitaiei terestre este g = 9,80665 m/s

2.

In the International System of Units, the unity of

measurement is N/m3. The medium acceleration of gravity

in the terrestrial field gravity is g = 9,80665 m/s2.

2.1.2. Presiunea / Pressure

Prin definiie, presiunea este raportul dintre fora normal i aria suprafeei pe care se exercit aceast for. ntr-un punct dintr-un fluid n repaus, presiunea se definete ca fiind limita raportului dintre fora normal i aria suprafeei pe care se exercit aceast for, cnd aria tinde ctre zero, n jurul punctului respectiv:

By a general definition, pressure is the ratio of the

normal force and the area on that is exerted these force.

In a point from a fluid in repos is defined by the limit of the

ratio of normal force and surface area on which they are

exerted this force, when the area tends to zero, around the

point in question:

A

F

A

Fp

A d

dlim

0

. (2.5)

Dac fora elementar nu ar fi perpendicular pe suprafa, ar nsemna c admitem ipoteza existenei unor eforturi tangeniale n fluidul n repaus, ceea ce contrazice ipoteza dinainte.

Trebuie accentuat faptul c ntr-un fluid n echilibru, presiunea este funcie de punctul n care se determin.

Unitatea de msur n Sistemul Internaional este N/m

2 denumit i pascal:

If the elementary force is not orthogonal on the

surface, it can suppose the existence of tangential stress in

the fluid being in static equilibrium and contradicts the

above hypothesis.

It must to mention that in a fluid being in equilibrium,

pressure is function of the point in that it is determinated.

The unit of measurement in the International System

of Units is N/m2 named pascal too:

17

Pam

Np

2SI

. (2.6)

Deoarece aceasta este o unitate de msur foarte mic n comparaie cu presiunile uzuale ntlnite n instalaiile industriale, se folosesc multiplii: kilopascalul, kPa

(denumit i piez - prescurtat pz): 1kPa = 103Pa i megapascalul, MPa: 1 MPa = 10

6Pa.

n aplicaiile tehnice curente se folosete barul (prescurtat bar), o unitate care, dei nu aparine Sistemului Internaional este tolerat pe o perioad nedefinit datorit obinuinei utilizrii ei n diferite ri, printre care i ara noastr:

Because this is a small unit of measure comparatively

with usual pressure from industrial insatllations, there are

used multiple units: kilopascal, kPa (named piez too,

symbolized by pz): 1kPa = 103Pa and megapascal, MPa:

1 MPa = 106Pa.

In the ordinary technical applications it is used the

bar, an unit that is not in the International System of

Units, but accepted to be used for an indefinite period with

the SI units. The bar is legally recognized in countries of

the European Union:

pz10kPa10MPa0,1m

N10bar1 22

2

5 . (piez) (2.7)

n tehnic s-a mai utilizat i se mai ntlnete nc destul de frecvent o unitate de msur denumit atmosfer tehnic, prescurtat at i definit astfel:

In the techical domaine it was often used a measure

unit named technical atmosphere, at, defined by

kilogram-force per square centimeter:

Pa1019,8m

N1006659,8

m10

N06659,81

cm

kgf1at1 4

2

4

242

. (2.8)

Pentru definirea strii normale fizice se utilizeaz atmosfera normal, prescurtat atm sau At,:

In order to deffine the normal physical state it is used

the standard atmosphere, symbol atm:

2m

N101325atm1 . (2.9)

18

Deoarece pentru msurarea presiunilor n fluide se pot utiliza cu succes aparate bazate pe principiul presiunii

hidrostatice create de o coloan cu lichid (numite i piezometre), se definesc:

milimetrul coloan de ap:

Because for measuring pressure in fluids it can be

used aparatus based on the hydrostatic pressure created by

a liquid column (named piezometers), there are defined:

milimeter of water:

aP81,9m

N06659,8m10

s

m9,80665

m

kg10OmmH1

2

3

23

3

2

; (2.10)

metrul coloan de ap: meter of water:

kPa9,81Pa109,80665m1s

m9,80665

m

kg10mca1 3

23

3 ; (2.11)

milimetrul coloan de mercur cunoscut i sub denumirea de torr:

milimeter of mercury, named also Torr:

Pa133,32m10s

m9,80665

m

kg13595mmHg1torr1 3

23 . (2.12)

milimetrul coloan de alcool: milimeter of alcohol

Pa7,875m10s

m9,80665

m

kg803alc mm1 3

23 . (2.13)

n cazul utilizrii piezometrelor, pentru creterea preciziei msurrilor este necesar a se ine seama de varia-ia densitii lichidului piezometric cu temperatura. n ecuaia (2.13) s-a dat densitatea alcoolului la 20oC.

Dup nivelul fa de care se face msurarea presiunii, ntlnim dou noiuni: presiune absolut i presiune relativ

For piezometers, in order to increase the mea-

surement precision is needed to take into account the vari-

ation with temperature of the piezometric liquid density.

In the above equation the density of alcohol is at 20 C.

There are used two notions: absolute pressure and

gauge pressure (Fig. 2.1):

19

(fig. 2.1):

Fig. 2.1. Presiuni absolute i relative Fig. 2.1. Absolute and relative pressures

Presiunea absolut este presiunea care are ca nivel de referin vidul absolut. Presiunea atmosferic este presiunea absolut a atmosferei n punctul de msurare. Ea se msoar cu ajutorul barometrului i de aceea se mai numete i presiune barometric. Presiunea relativ este presiunea care are ca nivel de referin presiunea atmosferic a locului unde se efectueaz msurarea. Aceasta este mrimea care se determin n mod curent n practica msurrilor din instalaiile industriale. ntre aceste mrimi exist urmtoarea relaie:

Absolute pressure is zero-referenced against a perfect

vacuum.

Atmospheric pressure is the absolute pressure of the

atmosphere in the measuring point. It is measured with a

barometer and so is named barometric pressure.

Gauge pressure is zero-referenced against ambient air

pressure in the place where is accomplished the

measurement. This is the magnitude that is ususlly

measured in industrial installations.

There are the below relation between these

magnitudes:

relatabs ppp . (2.14)

Presiunea relativ poate fi: o suprapresiune denumit i presiune manometric;

The gauge pressure can be:

a pressure gauge or manometric gauge; it is positive

20

ea este pozitiv, iar presiunea absolut calculat cu ecuaia de mai sus are o valoare mai mare dect presiunea

atmosferic; o depresiune sau presiune vacuummetric; ea este negativ, iar presiunea absolut rezultat din ecuaia de mai sus are o valoare mai mic dect presiunea atmosferic.

n practica msurrilor industriale se ntlnesc diferite uniti de msur. Cnd se utilizeaz ecuaia de mai sus tre-buie ca cele dou presiuni s fie exprimate n aceleai uni-ti de msur. De cele mai multe ori, presiunea atmosfe-ric se determin n mmHg sau mbar. Ea prezint variaii n funcie de altitudine, (scade cu creterea altitudinii) dar chiar i variaii sptmnale sau diurne.

Totdeauna n problemele de termodinamic se utilizeaz presiunea absolut. Astfel, n ecuaia de stare a gazului perfect, n toate ecuaiile obinute cu ajutorul acesteia, sau n curbele care descriu tranziiile de faz (vaporizare-condensare, sau solidificare lichefiere) a unui agent termodinamic se folosete presiunea absolut.

n general, n problemele de mecanica fluidelor se

utilizeaz presiunea relativ deoarece forele care apar n instalaii sunt rezultatul diferenei dintre presiunea din interiorul instalaiei i presiunea ambiant. Excepie fac: problemele de cavitaie (fenomenul depinde de presiunea absolut de vaporizare a lichidului dependent la rndul ei de temperatur) i

and the absolute pressure obtained with the above

equation has a bigger value that the atmospheric pressure;

a vacuum pressure; it is negative and the absolute

pressure obtained with the above equation has a value

smaller that the atmospheric pressure.

In the industrial measuring processes there are used

different units of measurement. When it is used the above

equation, the pressures must to be expressed in the same

units of measure. Usually, the atmospheric pressure is

measured in mmHg or mbar. It presents variation function

of altitude (it diminishes with the altitude increasing) but

it has weekly or daily variations.

Always in thermodynamics it is used the absolute

pressure. Thus, in the ideal gas equation of state, in all

equations obtained from this equations, or in curves that

describes phase transitions (vaporization condensation or freezing liquefying) of a thermodynamic agent there are used the absolute pressure.

In common problems of fluid mechanics, it is used

the gauge pressure because forces that appear in

hydraulic installations are the results of the difference

between the internal pressure of the installation and the

ambient pressure, excepting:

problems of cavitation (the phenomenon depends of

the vapor absolute pressure of the liquid, that depends of

temperature) and

problems in that are studied the repos and

21

problemele n care se studiaz repausul sau micarea gazelor perfecte.

movement of ideal gases.

2.1.3. Viscozitatea fluidelor / Viscosity of fluids

Viscozitatea este proprietatea fluidelor de a se

opune deformaiilor atunci cnd sunt supuse la lunecare relativ a straturilor suprapuse.

Aceast proprietate reprezint mecanismul de transmitere a micrii ntr-un fluid. Constatarea a fost fcut de Newton n 1687, care a stabilit i expresia efortu-lui unitar tangenial de viscozitate n micarea laminar.

Se consider dou straturi de fluid cu aria infinit mic dA, situate la distana elementar dn msurat pe normal i aflate n micare relativ unul fa de cellalt astfel: stratul inferior are viteza v, iar stratul superior o vitez cu un infinit mic mai mare dect aceasta: v+dv (fig. 2.2).

Datorit frecrii, apare o for elementar dF care se opune acestei deplasri relative.

Newton a presupus c efortul unitar tangenial de frecare este proporional cu variaia vitezei pe direcia normal:

Viscozity is the fluids property to oppose to

deformations when there are supposed to relative slip

of adjacent layers.

This property reprsets the mechanism of transmition

of movement in a fluid. The phenomenon was observed

by Newton in 1687, which was established the expressi-

on of the tangential effort of viscozity in the laminar

flow.

There are considered two layers of fluid with the

infinitesimal area dA, emplaced at the elementary dis-

tance dn, measured on crosswise (normal) direction and

being in relative movement one to the other one, so: the

lower layer has the speed v and the upper layer with a

positive infinitesimal value: v+dv (Fig. 2.2). Because

the friction, it exists an elementary force dF that opposes

to these relative deplacement.

Newton supposed that the unit shear stress is

linearly proportional with the speed variation on normal

direction:

n

v

A

F

d

d

d

d . (2.15)

22

Acest efort tangenial are tendina de a egala vitezele straturilor, deci se opune micrii stratului cu viteza mai mare (semnul -).

Coeficientul de proporionalitate este coeficientul viscozitii dinamice sau, pe scurt, viscozitatea dinamic, deoarece ecuaia de definiie a sa conine o mrime dinamic (efortul unitar tangenial):

This shear stress has the tendency to equalize the

speed of layers, so it opposes to the movement of the layer

having the greater speed (sign -). The factor of proportionality is the dynamic

viscosity because its equation of definition include a

dynamic quantity (the unit shear stress):

n

v

d

d

, (2.16)

Fig. 2.2. Modelul pentru ecuaia lui Newton Fig. 2.2. The model for the Newtons equation

nlocuind unitile de msur corespunztoare din SI se obin succesiv egalitile:

By replacing the unities of measuring there are

obtained the succesive equalities:

23

sPam

sN

m

smm

N

2

2

. (2.17)

n sistemul de uniti CGS, unitatea de msur este numit poise de la numele savantului francez Poiseuille care a studiat curgerea laminar:

In the system CGS the unity of measurement is

named poise from the name of the French physicist

Poiseuille which studied the laminar flow:

2

1

24-

5

2 m

sN10

m10

sN101

cm

sdyn1P1

. (2.18)

Prin raportarea viscozitii dinamice la densitatea fluidului se obine o mrime cinematic numit viscozitate cinematic:

By relating the dynamic viscosity to the fluid

density it is obtained a kinematic quantity, the kinematic

viscosity:

. (2.19)

n SI unitatea de msur este m2/s, iar n sistemul CGS, unitatea de msur este cm2/s care poart denumirea de stokes:

In SI the unit of measurement is m2/s, and in CGS,

the unit of measurement is cm2/s, stokes:

.sm10s

m101

s

cm1St1 24

242

(2.20)

Se mai folosete centistokesul: It is more used the centistokes:

.sm10St10tcS1262 (2.21)

24

Fluidele al cror efort de vscozitate n micare laminar (curgere n straturi paralele) este dat de ecuaia (2.15) se numesc fluide newtoniene. n aceast categorie se nscriu destul de bine fluidele uzuale: aerul, apa i uleiurile aflate n micare laminar. Fluidele care nu respect legea lui Newton se numesc ne-newtoniene.

Viscozitatea dinamic crete foarte puin cu presiunea dar variaz foarte mult cu temperatura. La creterea temperaturii, viscozitatea lichidelor scade, pe cnd cea a gazelor crete.

Explicaia n cazul lichidelor const n faptul c prin creterea temperaturii, dilatarea conduce la scderea fore-lor de coeziune molecular. n cazul gazelor, agitaia molecular crescnd cu temperatura, are loc un transfer de particule materiale (la nivel molecular) ntre straturile de

fluid aflate n micare laminar, ceea ce conduce la o cretere a eforturilor de frecare.

Variaia viscozitii cinematice prezint aceleai caracteristici cu cea a viscozitii dinamice, cu excepia variaiei cu presiunea n cazul gazelor. Astfel, prin creterea presiunii, viscozitatea dinamic a gazelor crete, dar densitatea crete mai accentuat, astfel nct rezult o scdere a viscozitii cinematice.

Pentru variaia viscozitii dinamice cu temperatura, n cazul gazelor se recomand formula semiempiric dat de Southerland:

Fluids being in laminar flow (flow in parallel

layers) that comply with the Newtons equation (2.15) are named Newtonian fluids. This category includes the

usual fluids: air, water and oils, all in the laminar flow.

Fluids that no comply with the Newtons law are named non-Newtonian fluids.

The dynamic viscosity presents a little increasing

with the pressure but presents a great variation with

the temperature. For the temperature increasing, the

liquids viscosity decreases while the gases viscosity

increases.

In the case of liquids the explanation is the

dilatation with the temperature that pursues to the

decreasing of molecular cohesion forces. In the case of

gases, the thermal molecular agitation pursues to a

molecular transfer between layers being in laminar flow,

which determine an increasing in the shear stress.

The variation of the kinematic viscosity presents

the same characteristic with the dynamic viscosity,

excepting variation with pressure of gas. Thus, by the

pressure increasing, the dynamic viscosity of gas

increases, but the density increasing is greater, so it

results a decreasing of the kinematic viscosity.

It is recommended the below relation for the gas

dynamic viscosity variation with the temperature, (the

semi empirical relation of Southerland):

25

,2

3

TS

TS

T

T N

N

N

(2.22)

unde T este temperatura absolut, S este o constant a crei valoare este n funcie de gazul respectiv, iar N este

viscozitatea gazului la temperatura normal fizic TN. n cazul aerului, unul dintre cele mai ntlnite gaze n

natur i n aplicaiile industriale, constantele sunt: S =

111K, .s/mN 1,717.102-5 N

Pentru variaia viscozitii lichidelor cu tempera-tura se utilizeaz formule diferite. Astfel, pentru ap se recomand formula, [14]:

where T is the absolute temperature, S is a constant that

value is function of the gas, and N is the dynamic

viscosity at the temperature TN, corresponding to the

normal physical state.

For air, the constants are: S = 111K,

For the variation with temperature of liquids

viscosity there are different formulas. So for water it is

recommended the formula [14]:

sm00022,00337,01

1078,1 22

6

tt

, (2.23)

unde t este temperatura relativ [oC]. Pentru calculul viscozitii uleiurilor n funcie de

temperatur se poate utiliza formula [14]:

where t is the relative temperature [oC].

For the variation with temperature of oils viscosity

it can be used formula [14]:

K

00

78,17

78,17

t

t, (2.24)

unde 0 este viscozitatea cinematic la temperatura 0t , iar

exponentul 25,3...5,2K .

Bazat pe modelul creat de Newton se poate da o rezol-

vare destul de corect a unor probleme simple legate de

where 0 is the kinematic viscosity at the standard

temperature 0t , and the exponent 25,3...5,2K .

Based on the model proposed by Newton it can be

approximately solved some elementary problems of lubri-

.s/mN 1,717.10 2-5 N

26

lubrificaie, cum ar fi determinarea aproximativ a forelor de frecare vscoas i a puterii consumate prin frecare n cazul unor lagre avnd forme relativ simple. n astfel de probleme se poate presupune c pelicula de lubrifiant are o grosime foarte mic, considerndu-se c n ecuaia (7.1) se poate trece la diferene finite fr o eroare prea mare.

Este important ca n practic s se in seama de variaia viscozitii cu temperatura. Pornirea unei maini de la rece presupune fore de viscozitate mai mari dect n regimul de funcionare continu. n afar de aceasta, creterea temperaturii lubrifiantului conduce la scderea viscozitii, iar ndeprtarea defectuoas a cldurii de frecare vscoas generat n funcionarea unei maini determin scderea proprietilor de ungere, micorarea grosimii peliculei de lubrifiant i n final griparea lagrelor.

cation, for example the approximatively determination of

viscous friction forces in the cases of bearings having rela-

tive simple shapes. In these problems it can be supposed

that the lubricant layer has a very little thickness, bay con-

sidering that in the Newtons equation it can be replaced the infinitesimal quantities by the finite differences.

In practice it is important to considerate the

variation of the viscosity with temperature. The starting

a cold machine supposes viscous friction forces higher

than in the nominal working regime. In addition to this,

the lubricant temperature increasing determine the

viscosity decreasing, but the flawed viscous friction heat

removal determines the decreasing of lubrication

properties, the lubricant layer thickness decreasing and

finally the bearing sticking.

2.2. Proprieti specifice lichidelor / Properties of liquids

2.2.1. Compresibilitatea izotermic i dilatarea izobar a lichidelor / The izothermal compressibility and isobaric expansion of liquids

Compresibilitatea izotermic a lichidelor este proprietatea de variaie a densitii unui lichid datorit variaiei presiunii la temperatur constant.

Fie V0 volumul ocupat de un fluid la presiunea p0.

Dac presiunea are o variaie 0ppp , are loc o

The isothermal compressibility of liquids is the

property of variation of the density because the

pressure variation at a constant temperature.

It is supposed V0 the volume occupied by a liquid at

the pressure p0. If the pressure has a variation

0ppp , it results a relative variation of the volume

27

variaie relativ de volum 0VV proporional cu variaia

presiunii:

in a linearly proportionality with the pressure

variation:

,0

pV

V

(2.25)

unde 0VVV . Semnul minus arat c unei creteri de

presiune i corespunde o scdere de volum, iar factorul de proporionalitate este coeficientul (modulul) de compresi-bilitate cubic notat . Este similar coeficientului de com-

presibilitate izotermic definit pentru gazul perfect i obinut prin diferenierea ecuaiei de stare n forma general. Din ecuaia de mai sus rezult ecuaia de definiie a acestui coeficient:

where 0VVV . The negative sign indicate that for a

pressure increasing corresponds to a volume decreasing,

the proportionality factor being the cubic coefficient of

compressibility, noted with . It is similar to the

isothermal coefficient defined for the ideal gas and

obtained from the equation of state in a general shape by

differentiation. From the above equation it results the

equation of definition for this coefficient:

.0

p

V

V

(2.26)

Coeficientul de compresibilitate cubic a lichidelor scade puin cu creterea presiunii i temperaturii.

ntr-o alt variant, ecuaia (2.25) poate fi scris utiliznd coeficientul (modulul) de elasticitate cubic:

The cubic compressibility of liquids has a little

decreasing with the pressure and temperature increasing.

In another variant, equation (2.25) can be write using

the bulk modulus elasticity:

.Pam/N 1 2

(2.27)

Pentru ap la temperatura ambiant, = 2,11 109 N/m

2. Apa este deci de aproximativ 100 de ori mai

For water at the ambient temperature, = 2.11 109

N/m2. Water is of about 100 tan more than the steel.

0VV

28

compresibil dect oelul. Tabelul 2.2 prezint modulul de elasticitate al ctorva

lichide la temperaturi uzuale. Totui, n majoritatea feno-menelor studiate, lichidele se consider ca fluide incompresibile. Fac excepie fenomenele: ocul hidraulic (cunoscut i sub numele de lovitur de berbec) i sonicitatea (propagarea energiei n lichide prin

comprimri i dilatri succesive ale straturilor de lichid). Teoria sonicitii, cu numeroase aplicaii tehnice

(motorul sonic, pompa sonic, transmisia sonic, ciocanul sonic .a.) a fost fundamentat de savantul romn Grigore (Gogu) Constantinescu.

Gazele i vaporii sunt cu mult mai compresibile dect lichidele.

Table 2.2. presents the bulk modulus elasticity of

some liquids at usual temperatures. However, in the

majority of studied phenomena, liquids are considered

incompressible fluids. There are excepted phenomena:

hydraulic shock (known also that hammer) and

sonicity (study laws for transmitting the mechanical

power through oscillations that propagate in continuous

environments (liquid or solid) due to their elasticity.

The Romanian scientist Grigore (Gogu)

Constantinescu found the theory of the sonicity with

many technical applications (the sonic engine, sonic

pump, sonic transmition, sonic hammer et al.).

Gas and vapor are much compressible that liquids.

Tab. 2.2. Coeficientul de elasticitate al unor lichide uzuale Tab. 2.2. The bulk modulus elasticity of usual liquids

Lichidul / Liquid t [oC] [N/m

2]

Ap / Water 0 1,954.109

Ap / Water 20 2,11.109

Petrol / Petroleum 20 1,154.10

9

Ulei / Oil 20 1,443.10

9

Pentru variaii infinit mici ale presiunii i volumului, ecuaia (2.25) devine:

For infinitesimal variations of the pressure and volume,

equation (2.25) becomes:

29

pV

Vd

1d

. (2.28)

Dilataia termic izobar a lichidelor reprezint creterea volumului unui fluid datorit creterii temperaturii (la presiune constant).

Legea matematic se exprim sub forma:

The isobaric thermal expansion of liquids

represents the increasing of a fluid volume because the

temperature raising (at a constant temperature).

The mathematicl law is expressed by the relation:

TV

V

0

. (2.29)

Deci creterea relativ a volumului unui fluid este direct proporional cu creterea absolut a temperaturii.

Din aceast ecuaie rezult definiia coeficientului de dilatare izobar:

Therefore, the relative raising of the volume of the

liquid is in proportion to the absolute increasing of the

temperature.

From this equation results the definition of the isobaric

expansion coefficient:

,0T

V

V

(2.30)

din care se poate obine unitatea de msur: K-1. Este similar coeficientului de dilatare izobar definit pentru gazul perfect i obinut prin diferenierea ecuaiei de stare n forma general.

n cartea de fa prezint interes aceast proprietate a lichidelor. O aplicaie foarte important o reprezint termo-metrele din sticl cu lichid. Deoarece coeficientul de dilata-re izobar al lichidelor folosite n astfel de termometre are

from that it can be obtained the unity of measurement: K-1

.

It is similar to the isobaric expansion coefficient defined for

the ideal gas and obtained from the equation of state in a

general shape by differentiation.

In the present book it is of interest this property of

liquids. A significant application is the glass thermometers

with liquid. Because the isobaric expansion coefficient of

liquids used in these thermometers is constant in the usual

30

o valoare constant n domeniul uzual de utilizare, va rezulta o scar termometric liniar, avnd diviziunile echidistante.

Pentru ap la 20oC, = 1,5.10-4 K-1. Trebuie totui s reamintim faptul c apa prezint o anomalie fa de aceast lege deoarece n intervalul 0...4

oC volumul apei scade cu

creterea temperaturii, astfel nct la temperatura de 3,98oC apa are cea mai mare densitate. Astfel este posibil viaa florei i faunei acvatice n anotimpul rece deoarece gheaa i apa aflat la temperaturi ntre punctul de nghe i 3,98oC au densitate mai mic, deci se afl deasupra unui strat de ap cu densitatea maxim.

Tabelul 2.3 prezint coeficientul de dilatare izobar al unor lichide la temperatura de 20

oC (pentru pcur o medie n intervalul 0...100

oC).

domain, it shall result a thermometric linear scale, having

equidistant divisions.

For water at 20oC, = 1,5.10

-4 K

-1. However, it must

to remind that water presents an anomaly because in the

temperature interval 0...4oC the water volume decreases

with temperature increasing, so that at the temperature of

3,98oC water has the greater density. So it is possible the

aquatic flora and fauna life in the winter season because the

ice and water at temperatures between the freeze point and

3,98oC has a lower density, so they are upper from layer

water with the maximum density.

Table 2.3. presents the isobaric expansion coefficient

of liquids at the temperature of 20 oC (for mazout, is

presented an average value in the interval 0...100 oC).

Tab.2.3. Coeficientul de dilatare izobar Tab. 2.3. The coefficient of isobaric expansion

Lichidul / Liquid [10-6 . K-1] Lichidul / Liquid [10-6 . K-1]

Aceton / Acetone 1487 Benzin / Gasoline 1100

Alcool etilic / Ethanol 1100 Glicerin / Glycerine 505

Alcool metilic / Methanol 1220 Mercur / Hydrargyrum 181

Pcur, ulei / 600

Petrol / Petroleum 900

Pentru o variaie infinit mic a temperaturii i volumului, ecuaia (2.29) devine:

For an infinitesimal of the temperatuer and volume,

equation (2.29) becomes:

31

TV

Vd

d . (2.31)

Reunind ecuaiile (2.28) i (2.31) ntr-o singur relaie, se obine ecuaia general de transformare a lichidelor:

By assembling equations (2.28) and (2.31) in a

single equation, it is obtained the general equation of

liquids transformation:

.dd1d

TpV

V

(2.32)

Ecuaia se poate folosi cu rezultate bune i dac, pentru variaii relativ mici ale temperaturii i presiunii diferenialele se nlocuiesc prin diferene finite.

This equation can be used with good results and if,

in the case of relative small variations of temperature and

pressure there are replaced the differentials by finite

differences.

2.2.2. Adeziunea fluidelor la suprafee solide / Fluid adhesion to solid surfaces

Adeziunea este consecina atraciei dintre moleculele unui fluid i moleculele unui perete solid aflat n contact cu el.

Prin urmare, pe conturul corpului solid se formeaz un strat foarte subire de fluid care are viteza frontierei sale solide, astfel nct deplasarea relativ ntre peretele solid i pelicula fluid n zona de contact este nul.

Proprietatea de adeziune a fluidelor la contururile

solide are o prim importan practic n explicarea distribuiei vitezei pe seciunea unei conducte prin care curge un fluid. Astfel, de exemplu, la peretele solid al unei

The adhesion is the consequence of the attraction

between molecules of a fluid and molecules of the solid

in contact with him.

Therefore, on the solid body outline is formed a

thick layer of fluid that has the speed of their solid

boundary, so that the relative displacement between the

solid wall and fluid layer in the contact zone is null.

The property of fluid adhesion at the solid

boundaries has a first practical importance in the

explanation of the speed distribution on a pipe section

through that flows a fluid. Thus, for example, at the solid

32

conducte se poate pune condiia la limit c viteza fluidului este nul.

O aplicaie practic important a adeziunii lichidelor la pereii solizi este folosirea adezivilor i a straturilor de protecie a solidelor (vopseluri, lacuri, emailuri).

La gaze, adeziunea este neglijabil, dar legea de distribuie a vitezei pe seciunea unei conducte pornete de la aceei ipotez, confirmat experimental, c viteza este zero la contactul gaz-perete solid.

wall of a pipe it can be established the boundary

condition that the fluid velocity is null.

An important application of the liquids adhesion at

solid walls is the using of adhesives and of the protective

layers of solids (dyestuff, lacquer, enamel).

The gas adhesion is negligible, but the speed

distribution law on the pipe section becomes from the

same hypothesis, experimentally confirmed, that the

speed is null at the solid wall-gas contact.

2.2.3. Tensiunea superficial i capilaritatea lichidelor/ Surface tension and liquid capillary

Tensiunea superficial este consecina neechilibrrii ctre exterior a forelor de coeziune molecular exercitate de moleculele din interiorul unui lichid. Aa cum s-a artat mai sus, adeziunea gazelor este neglijabil. Pentru fiecare molecul aflat la suprafaa de contact ntre lichid i gazul aflat deasupra, apare o rezultant a forelor de coeziune ndreptat nspre lichid, aa cum se prezint n figura 2.3.

The surface tension is the consequence of the non-

equilibration to exterior of molecular cohesion forces

exerted of molecules from the interior of the liquid. So it

is above presented, the gas adhesion is negligible. For

each molecule of the liquid being near the upper situated

gas, it has a resultant of cohesion forces oriented through

liquid, so being presented in figure 2.3.

Fig. 2.3. Forele moleculare de legtur ntr-un lichid Fig. 2.3. The binding molecular forces in a liquid

33

Se consider o tietur rectilinie de lungime s n suprafaa liber a lichidului (fig. 2.4). Pentru a menine suprafaa n echilibru, trebuie s se aplice o for F i reaciunea ei F pe cele dou margini ale tieturii. Se definete tensiunea superficial prin relaia:

It is considered a rectilinear section of length s in

the liquid surface (fig. 2.4). In order to maintain the

surface in equilibrium, it must to apply a force F and its

reaction F on both margins of the section. It is defined

the surface tension coefficient with the relation:

m

N

d

dlim

0 s

F

s

F

s. (2.33)

Fig. 2.4 Modelul fizic pentru definirea tensiunii

superficiale

Fig. 2.4. The physical model for the surface tension

definition

Valoarea coeficientului de tensiune superficial depinde de natura fluidelor n contact i de temperatur.

De exemplu, pentru ap/aer la 20oC, mN0279,0 .

Se fac urmtoarele observaii: tensiunea superficial este definit ca o for raportat la lungime, putnd fi interpretat ca lucru mecanic pe unitatea de suprafa; tensiunea superficial modific presiunea n lichid.

Se propune determninarea unei relaii ntre diferena de presiune pe cele dou fee ale elementului de suprafa

The value of the surface tension coefficient depends

on the nature of fluids in contact and on temperature.

For example, for water/air at 20oC, mN0279,0 .

There are the below observations:

the surface tension is defined that a force related to

length, that can be interpreted that work on the surface

unity;

the surface tension modify the pressure in a liquid.

It is proposed to obtain a relation between the

differential pressure on both surfaces of the elementary

34

curb Sd i caracteristicile curburii (raza de curbur).

Suprafaa elementar are razele de curbur 1r i respectiv

2r pe dou direcii perpendiculare.

Observnd figura 2.5, modulul rezultantei forelor de tensiune superficial este:

curved surf