Chapter 1 Fluid Mechanics Introduction

8/13/2019 Chapter 1 Fluid Mechanics Introduction

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CFD Institute of Nuclear Engineering and Scienc

NPP CFD Modeling and Safety nalysis Lab

Chapter 1Introduction


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Institute of Nuclear Engineering and SciencNPP CFD Modeling and Safety nalysis LabCFD

Main Topics

1.1 Some Characteristics of Fluids1.2 Dimensions and Units1.3 Analysis of Fluid Behaviors1.4 Measures of Fluid Mass and Weight1.5 Ideal Gas Law1.6 Viscosity

1.7 Compressibility of Fluids1.8 Vapor Pressure1.9 Surface Tension1.10 A Brief Look Back in History


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1.1 Characteristicsof Fluids

Whats a Fluid ? Whats difference between a solid and a fluid ?


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Definition of Fluid

Fluids comprise the liquid and gas (or vapor) phaseof the physical forms.

A fluid is a substance that deforms continuouslyunder the application of a shear stress no matter howsmall the shear stress may be.

A shearing stress is created whenever a tangentialforce acts on a surface.


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Fluid and Solid _1

When a constant shear force is applied:Solid deforms or bends

Fluid continuously deforms.


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Fluid and Solid _2

Vague ideaFluid is soft and easily deformed.

Solid is hard and not easily deformed.Molecular structure

Solid has densely spaced molecules with

large intermolecular cohesive force allowedto maintain its shape.


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Fluid and Solid _3

Liquid has further apart spaced molecules, theintermolecular forces are smaller than for solids,and the molecules have more freedom ofmovement. At normal temperature and pressure,the spacing is on the order of 10 -6mm. Thenumber of molecules per cubic millimeter is on theorder of 10 18 .Gases have even greater molecular spacing andfreedom of motion with negligible cohesiveintermolecular forces and as a consequence areeasily deformed. At normal temperature andpressure, the spacing is on the order of 10 -7mm.The number of molecules per cubic millimeter ison the order of 10 21 .


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Fluid? Solid ?

Some materials, such as slurries, tar, putty,toothpaste, and so on, are not easily classified sincethey will behave as solid if the applied shearingstress is small, but if the stress exceeds some criticalvalue, the substance will flow. The study of suchmaterials is called rheology.


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1.2 Dimension and Units

Primary quantities also referred as basicdimensions

Such as Length,L, time, T, mass, M, andtemperature, . Used to provide a qualitative description of anyother secondary quantity.

Secondary quantitiesFor example, area L2,velocity Lt -1 ,density ML -3.


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System of Dimensions

Mass[M], Length[L], Time[T] MLT systemForce[F], Length[L], Time[T] FLT system


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Dimensions Associatedwith Common Physical Quantities
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DimensionallyHomogeneous

All theoretically derived equations aredimensionally homogeneous that is, thedimensions of the left side of the equation must bethe same as those on the right side, and all additiveseparate terms have the same dimensions.

General homogeneous equation: valid in anysystem of units.

Restricted homogeneous equation : restricted toa particular system of units.

2

2 gt d 24.90 t d

Valid only for the system ofunits using meter andseconds


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InternationalSystem (SI)

Length: mTime: second

Mass: KgTemperature: K oC+273.15Force: Newton: 1 N (1 Kg)(1 m / sec 2 )Work: Joule ( J ) ; 1 J 1 N m

Power: Watt (W) ;1 W 1 J / sec 1 N

m/secGravity: g = 9.807 m / sec 2 Weight: W (N) = m (Kg) g (m/ sec 2) : 1 kg-mass weights9.81N


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EnglishEngineering (EE) System

Mass: lb m Force: lb f

Length: ftTime: secondTemperature: oR (absolute temperature)

F ma / g c gc : the constant of proportionality1 lb f lbm 32.174 ft / sec

2 / g cgc lbm 32.174 ft / sec

2 / lb f


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Exam p le 1.2

A tank of water having a total mass of 36 kg rests on thesupport in the equipment bay of the Space Shuttle.Determine the forces (in newtons) that the tank exerts onthe support shortly after lift off when the shuttle isaccelerating upward as shown in Fig.E1.2a at 4.5 m/s 2.

2

22

/515

/5.4/81.936

)(

smkg

sm smkg

a g m F

maW F or ma F

f

f


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Analysis ofFluid Behaviors _1

Analysis of any problem in fluid mechanicsnecessarily includes statement of the basic lawsgoverning the fluid motion. The basic laws, which

applicable to any fluid, are:Conservation of massNewtons second law of motion The principle of angular momentum

The first law of thermodynamicsThe second law of thermodynamics


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Analysis ofFluid Behaviors _2

NOT all basic laws are required to solve any oneproblem . On the other hand, in many problems it isnecessary to bring into the analysis additionalrelations that describe the behavior of physicalproperties of fluids under given conditions.Many apparently simple problems in fluid mechanicsthat cannot be solved analytically . In such cases

we must resort to more co m pl ica ted num erica lso lu t ions and/or resu l t s o f exper imental t es t s .


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1.4 Measurement

of Fluid Mass and WeightDensitySpecific weightSpecific Gravity


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Density

The density of a fluid, designated by the Greek symbol (rho), is defined as its mass per unit volume.Density is used to characterize the mass of a fluid system.In the BG system has units of slug/ft 3 and in SI theunits are kg/m 3.The value of density can vary widely between differentfluids, but for liquids, variations in pressure andtemperature generally have only a small effect on the valueof density.The specific volume, , is the volume per unit mass that is, /1


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Specific Weight

The specific weight of a fluid, designated by theGreek symbol (gamma), is defined as its weight

per unit volume.

Under conditions of standard gravity (g= 9.807m/s2), water at 15C(288K) has a specific weight of9.80kN/m 3. The density of water is 999kg/m 3.

g


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Specific Gravity

The specific gravity of a fluid, designated as SG,is defined as the ratio of the density of the fluid to

the density of water at some specified temperature.

C4@OH2

SG

H 2O , 4 oC 1.94slug/ft 3 or 999kg/m 3.


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1.6 Viscosity

The properties of density and specific weight aremeasures of the heaviness of a fluid.

It is clear, however, that these properties are notsufficient to uniquely characterize how fluids behave since two fluids can have approximatelythe same value of density but behave quitedifferently when flowing.There is apparently some additional property thatis needed to describe the FLUIDITY of thefluid.


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Fluidity of Fluid _1

How to describe the fluidity of the fluid? The bottom plate is rigid fixed, but the upper plateis free to move.

If a solid, such as steel, were placed between thetwo plates and loaded with the force P , the topplate would be displaced through some smalldistance, a.The vertical line AB would be rotated through the

small angle, , to the new position AB .

P A


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What happens if the solid is replaced with a fluid suchas water?

When the force P is applied to the upper plate, it willmove continuously with a velocity U.The fluid sticks to the solid boundaries and is referredto as the NON-SLIP con ditions.

The fluid between the two

plates moves with velocityu=u(y) that would be assumedto vary linearly, u=Uy/b. Insuch case, the velocitygradient is du / dy U / b.
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The shearing stress is increased byP,

dydu

dydu P / The common fluids such as water, oil, gasoline, and air.The shearing stress and rate of shearing strain can berelated with a relationship


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Viscosity Definition

The constant of proportionality isdesignated by the Greek symbol (mu) and is called the

absolute viscosity, dynamicviscosity, or simply the

viscosity of the fluid.The viscosity depends on theparticular fluid, and for aparticular fluid the viscosity isalso dependent on temperature.

dydu


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Dimension and Unit of

The dimension of : Ft/L 2 or M/Lt.The unit of :

In B.G. : lb f -s/ft2 or slug/(ft-s)

In S.I. : kg/(m-s) or N-s/m 2 or Pa-sIn the Absolute Metric: poise=1g/(cm-s)


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Exam p le 1.4 Visc o s i tyand Dim ens ion less Quant i t ies

A dimensionless combination of variables that is importantin the study of viscous flow through pipes is called theReyno lds num ber , Re , defined as VD/ where is the

fluid density, V the mean velocity, D the pipe diameter,and the fluid viscosity. A newtonian fluid having aviscosity of 0.38 Ns/m 2 and a specific gravity of 0.91flows through a 25-mm-diameter pipe with a velocity of 2.6m/s. Determine the value of the Reynolds number using (a)SI units.

156//156...Re/910)/1000(91.0

2

334@2

N smkg VD

mkg mkg SG C O H


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Exam p le 1.5New ton ian Fluid Shear St ress

The velocity distribution for the flow of a Newtonianfluid between two sides, parallel plates is given by theequation

2

hy

12V3

u


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Solution

0hVy3

dydu

ft/lb4.14hVy3

dydu

hVy3

dydu

0y2midplane

2

hy2wall bot tom

2

where V is the mean velocity. The fluid has a velocity f0.04 lbs/ft2. When V=2 ft/s and h=0.2 in. determine: (a)the shearing stress acting on the bottom wall, and (b) theshearing stress acting on a plane parallel to the walls and

passing through the centerline (midplane).


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Viscosity vs.Temperature _1

For fluids, the viscositydecreases with an

increase in temperature.For gases, an increasein temperature causesan increase in viscosity.

WHY? molecularstructure .


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The liquid molecules are closely spaced, with strongcohesive forces between molecules, and theresistance to relative motion between adjacent layersis related to these intermolecular force.

As the temperature increases, these cohesive forceare reduced with a corresponding reduction inresistance to motion. Since viscosity is an index of

this resistance, it follows that v i scos i ty i s r educedby an inc rease in tem perature .

The Andrades equation De B/T


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In gases, the molecules are widely spaced andintermolecular force negligible.The resistance to relative motion mainly arises due tothe exchange of momentum of gas moleculesbetween adjacent layers.

As the temperature increases, t he r andomm olecular ac t iv i ty inc reases w i th a co r resp ond ingincrease in v i sc os i ty.The Sutherland equation CT 3/2 / (T+S)


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Newtonian andNon-Newtonian Fluid

Fluids for which the shearing stress is linearly relatedto the rate of shearing strain are designated asNewtonian fluids after I. Newton (1642-1727).

Most common fluids such as water, air, and gasolineare Newtonian fluid under normal conditions.Fluids for which the shearing stress is not linearlyrelated to the rate of shearing strain are designatedas non-Newtonian fluids.


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Non-Newtonian Fluids

Shear thinning fluids .Shear thickeningfluids.Bingham plastic
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Kinematic Viscosity

Defining kinematic viscosity = / [Ny]The dimensions of kinematic viscosity are

L2/T.The units of kinematic viscosity in SI systemare m 2/s.In the CGS system, the kinematic viscosityhas the units of cm 2 /s, is called a stoke,abbreviated St.


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1.7 Compressibilityof Fluids

Bulk modulus.Compression and expansion of gases.Speed of sound.


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Speed of Sound _1

Another important consequence of thecompressibility of fluids is that disturbancesintroduced at some point in the fluid propagate ata finite velocity .For example, if a fluid is flowing in a pipe and avalve at the outlet is suddenly closed, the effect ofthe valve closure is not felt instantaneously upstream.It takes a finite time for the increased pressurecreated by the valve closure to propagate to anupstream location.


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Speed of Sound _2

The velocity at which small disturbances propagate in a fluid is called the speed of sound .

The speed of sound is related to change in pressure and density of the fluid medium through

For isentropic process

For ideal gas

vEddp

c

kP c

kRT c


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1.9 Surface Tension _1

At the interface between a liquid and a gas, orbetween two immiscible liquids, forces develop inthe liquid surface which cause the surface tobehave as if it were a skin or membranestretched over the fluid mass.

Although such a skin is not actually present, thisconceptual analogy allows us to explain severalcommonly observed phenomena.


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Surface Tension _2

Surface tension: theintensity of the molecularattraction per unit length

along any line in the surfaceand is designated by theGreek symbol .

R

2 p p p

R pR 2

ei

2 Where p i is the internal pressure and p e is the external pressure

The force due tosurface tension

The force due topressure difference

=
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Surface Tension _3

A common phenomena associated with surfacetension is the rise or fall of a liquid in a capillary tube.

cosR 2hR 2

R cos2

h

is the angle of cont act between the fluid and tube.

Balance for equilibrium
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Surface Tension Effects

Surface tension effects play a role in many fluidmechanics problems including the movement ofliquids through soil and other porous media, flow ofthin film, formation of drops and bubbles, and thebreakup of liquid jets.Surface phenomena associated with liquid-gas,liquid-liquid or liquid-gas-solid interfaces are

exceedingly complex.


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Fig. 1.10


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Chapter 1 Fluid Mechanics Introduction

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