CHAPTER 2 PROPERTIES OF FLUIDS - Faculty of …ummi/SME1313/Chapter 2.pdf · SME 1313 Fluid...

SME 1313 Fluid Mechanics I

CHAPTER 2PROPERTIES OF FLUIDS

ByUmmikalsom Abidin

C24-316FKM, UTM


IntroductionProperty – any characteristic of a system

E.g pressure P, temperature T, volume V, and mass m

Intensive propertiesProperties that are independent of the mass of a system e.g temperature, pressure, density

Extensive propertiesProperties whose values depend on size-or-extent-of the system e.g mass, total volume, total momentum

Specific propertiesExtensive properties per unit mass e.g specific volume (v=V/m) and specific total energy (e=E/m)


Introduction

½m

½V

T

P

ρ

½m

½V

T

P

ρ

Extensive

propertiesmVTPρ Intensive

properties

Criteria to differentiate intensive and extensive properties


IntroductionState postulate – the state of a simple compressible system is completely specified by two independent, intensive properties


Continuum Concept of a FluidThe number of molecules involved is immense, and the separation between them is normally negligible by comparison with the distances involved in the practical situation being studiedAlthough the properties of a fluid arise from its molecular structure,engineering problem are usually concerned with the bulk behavior of fluidsUnder these conditions, it is usual to consider a fluid as a continuum - a hypothetical continuous substance – and the conditions at a point as the average of a very large number of molecules surrounding that point within a distance which is large compared with the mean intermolecular distance


DensityDensity is defined as mass per unit volume

ρ = m (kg/m3)V

Specific volume is defined as volume per unit massv = V = 1 (m3/kg)

m ρDensity of a substance, in general depends on temperature and pressure


Specific GravitySpecific gravity or relative density is defined as the ratio of the density of a substance to the density of some standard substance at a specified temperature (usually water at 4oC, for which ρH2O=1000 kg/m3)

SG = ρρH2O

Specific gravity is a dimensionless quantity


Specific Gravity

0.0013Air (at 1 atm)

0.92Ice

1.7-2.0Bones

19.2Gold

0.3-0.9Wood

13.6Mercury

0.79Ethyl Alcohol

0.7Gasoline

1.025Seawater

1.05Blood

1.0Water

SGSubstance

Specific gravities of some substances at 0oC


Specific WeightSpecific weight is the weight of a unit volume of a substance

γs= ρg (N/m3)

Where,

g = gravitational acceleration, 9.81 m/s2


Density of Ideal GasIdeal gas equation

Pv = RT or P = ρRT

Where,P = absolute pressurev = specific volumeT = thermodynamic (absolute temperature)

T(K) = T(oC) + 273.15ρ = gas densityR = gas constant


Vapor Pressure and CavitationSaturation temperature, Tsat – the temperature at which a pure substance changes phase at a given pressureSaturation pressure, Psat – the pressure at which a pure substance changes phase at a given temperatureVapor pressure, Pv of a pure substance – the pressure exerted by its vapor in phase equilibrium with its liquid at a given temperaturePartial pressure – the pressure of a gas or vapor in a mixture with other gases


Vapor Pressure and CavitationPossibility of the liquid pressure in liquid-flow system dropping below the vapor pressure at some location, and resulting unplanned vaporizationA fluid vaporizes when its pressure gets too low, or its temperature too high. All centrifugal pumps have a required head (pressure) at the suction side of the pump to prevent this vaporization. Suction cavitation occurs when the pump suction is under a low pressure/high vacuum condition where the liquid turns into a vapor at the eye of the pump impeller. Cavitation must be avoided in flow systems since it reduces performance, generates vibration and noise and causes damage to the equipment


Energy and Specific HeatEnergy – mechanical, thermal, kinetic, potential, electric, etc.Microscopic energy – the forms of energy related to the molecular structure of a system and the degree of the molecular activity

Internal energy, U – sum of all microscopic forms of energy

Macroscopic energy – energy related to motion and the influence of some external effects such as gravity, magnetism, electricity, surface tension

E.g potential energy, kinetic energy


Energy and Specific HeatEnthalpy, h

h = u + Pv = u + Pρ

Where,P = flow energy, energy per unit mass needed toρ move the fluid and maintain flow


Energy and Specific HeatFor ideal gas, the internal energy and enthalpy can be expressed in terms of specific heat∆u = cv,ave ∆T and ∆h = cp,ave ∆T

Where,cv,ave = average specific heat at constant volumecp,ave = average specific heat at constant pressure

For incompressible substance,∆h = ∆u + ∆ P/ρ ≅ cave ∆T + ∆ P/ρ


Coefficient of CompressibilityCoefficient of compressibility κ (also called the bulk modulus of compressibility or bulk modulus of elasticity) for fluids as

κ = - v ∂P = ρ ∂P (Pa)∂V T ∂ρ T

In term of finite changesκ ≅ - ∆P ≅ ∆P (T=constant)

∆v/v ∆ρ/ρLarge κ, a large change in pressure is needed to cause a small fractional change in volume (incompressible e.g liquids)


Coefficient of CompressibilityFor an ideal gas,P = ρRT and (∂P/ ∂ρ)T = RT = P/ρAnd thus,

κideal gas = P (Pa)Coefficient of compressibility of an ideal gas is equal to its absolute pressure and it increases with increasing pressureSubstituting κ = P into definition of the coefficient of compressibility and rearranging gives

∆ρ = ∆P (T=constant)ρ P

The percent increase of density of an ideal gas during isothermal compression is equal to the percent increase in pressure


Isothermal CompressibilityIsothermal compressibility is the inverse of the coefficient of compressibilityα = 1 = - 1 ∂V = 1 ∂ρ (1/Pa)

κ v ∂P T ρ ∂P TIsothermal compressibility of fluid represents the fractional change in volume or density corresponding to a unit change in pressure


Coefficient of Volume ExpansionCoefficient of volume expansion provide information on variation of density of a fluid with temperature at constant pressure

β = 1 ∂v = - 1 ∂ρ (1/K)v ∂T P ρ ∂T P

In term of finite changesβ ≅ ∆v/v ≅ - ∆ρ/ρ (P=constant)

∆T ∆TLarge β for a fluid means a large density change in density with temperature


Coefficient of Volume Expansion

(∂v/∂T)P

20oC100 kPa

1 kg

21oC100 kPa

1 kg

(∂v/∂T)P

20oC100 kPa

1 kg

21oC100 kPa

1 kg

(a) Substance with a large β (b) Substance with a small β

The coefficient of volume expansion is a measure of the change in volume of a substance with temperature at constant pressure


Coefficient of Volume ExpansionFor ideal gas, the volume expansion coefficient at a temperature T isβideal gas = 1 (1/K)

TWhere,

T=absolute temperature


Coefficient of Volume ExpansionThe combined effects of pressure and temperature changes on volume change of a fluid can be determined by taking the specific volume to be a function of T and PDifferentiating v=v(T,P) and using the definitions of compression and expansion coefficient α and β give

dv = ∂v dT + ∂v dP =(β dT – α dP) v∂T P ∂P P

The fractional change in volume (or density) due to changes in pressure and temperature can be expressed approximately as

∆v = - ∆ρ ≅ β∆T - α∆Pv ρ


Problem 2-32Problem 2-32

Water at 1 atm pressure is compressed to 800 atm pressure isothermally. Determine the increase in the density of water.Take the isothermal compressibility of water to be 4.80 x 10-5 atm-1


Example 2-3Example 2-3

Consider water initially at 20oC and 1 atm. Determine the final density of water (a) if it is heated to 50oC at a constant pressure of 1 atm, and (b) if it is compressed to 100-atm pressure at constant temperature of 20oC. Take the isothermal compressibility of water to be α=4.80x10-5 atm-1


ViscosityViscosity is a property that represents the internal resistance of a fluid to motion or the ‘fluidity’Consider a fluid layer between two very large parallel plates separated by distance l

x Velocity profile

u(y)=(y/l)V

Area A

Behavior of a fluid in a laminar flow between two parallel plates when the upper plate moves with a constant velocity

dβl

dAForce F

N N’ u=V

Velocity V

y

u=0M


ViscosityA constant parallel force F is applied to the upper plate, the lower plate is held fixedThe upper plate moves continuously at constant velocity under the influence of FThe shear stress τ acting on the fluid layer is

AF

=τ

Where,

A=contact area between the plate and the fluid


ViscosityFluid in contact with lower plate has zero velocity (no-slip condition)In steady laminar flow, the fluid velocity between the plates varies linearly between 0 and V, and thus the velocity profile and the velocity gradient are

Vyyul

=)(l

Vdydu

=and

Where,

y=vertical distance from lower plate


ViscosityThe angular displacement or deformation (or shear strain) can be expressed as

Rearranging, the rate of deformation

dtdydudtVdad ===≈

llββ tan

dydu

dtd

=β


ViscosityThe rate of deformation of a fluid element is equivalent to the velocity gradient du/dyFluids rate of deformation (and thus the velocity gradient) is directly to the shear stress τ

Newtonian fluids for fluids which its rate of deformation is proportional to the shear stress

dydu

∝τdtdβ

τ ∝ or


ViscosityIn 1-D shear flow of Newtonian fluids, shear stress can be expressed by linear relationship

dydu

µτ =

Where,

µ=coefficient of viscosity/dynamic(absolute) viscosity of the fluid, N.s/m2 (or Pa.s)

Poise=0.1 Pa.s


ViscosityThe shear force acting on a Newtonian fluid layer is

The force F required to move the upper plate at constant velocity of V while the lower plate remains stationary is

dyduAAF µτ == (N)

l

VAF µ= (N)


ViscosityKinematic viscosity is the ratio of dynamic viscosity to density

ρµ

υ = (m2/s)


Viscosity

Viscosity=slopeOil

Water

Air

a

b

ba

dydu==

τµ

Shear stress, τ

Rate of deformation, du/dy

The rate of deformation (velocity gradient) of a Newtonian fluid is proportional to shear stress, and the constant of proportionality is the viscous


ViscosityBingham plastic

PseudoplasticNewtonian

Dilatant

Shear stress, τ

Rate of deformation, du/dy

Variation of shear stress with the rate of deformation for Newtonian and non-Newtonian fluids (the slope of a curve at a point is the apparent viscosity of the

fluid at that point)


ViscosityViscosity is due to the internal frictional force that develops between different layers of fluids as they are forced to move relative to each otherGases

Viscosity is caused by molecular collisions Viscosity of gas increases with temperature

In a gas, molecular forces are negligible and the gas molecules at high temperatures move randomly at higher velocitiesThis results in more molecular collisions per unit volume per unit time, thus greater resistance to flow


Viscosity of GasViscosity of gases is expressed as function by the Sutherland correlation

Where, T=absolute temperature (K)a and b = experimentally determined constants

TbaT+

=1

2/1

µ


Viscosity of LiquidLiquids

Viscosity is due to the cohesive forces between the molecules At higher temperatures, molecules posses more energy, and they can oppose the large cohesive intermolecular forces more stronglyThe liquid viscosity is approximated as

)/(10 cTba −=µ

Where, T=absolute temperature (K)a, b and c = experimentally determined constants


Viscosity of Gas and LiquidViscosity

Gases

Liquids

TemperatureThe viscosity of liquids decreases and the viscosity of gases increases with

temperature


Example 2-4 Determining the Viscosity of a Fluid

The viscosity of a fluid is to be measured by viscometer constructed of two 40-cm-long concentric cylinders (Fig. 2-18).the outer diameter of the inner cylinder is 12 cm, and the gap between the two cylinders is 0.15 cm. The inner cylinder is rotated at 300 rpm, and the torque is measured to be 1.8 N.m. Determine the viscosity of the fluid

SOLUTION The torque and the rpm of a double cylinder viscometer are given. The viscosity of the fluid is to be determined

Assumptions 1) The inner cylinder is completely submerged in oil 2) The viscous effects on the two ends of the inner cylinder are negligible

Analysis The velocity profile is linear only when the curvature effects are negligible, and the profile can be approximated as being linear in this case since l/R<<1


Example 2-4 Determining the Viscosity of a Fluid

R

N=300 rpm

Shaft

Stationary cylinder

l

Fluid

Figure 2-18


Surface TensionLiquid droplets behave like small spherical balloons filled withthe liquid, and the surface of the liquid acts like a stretched elastic membrane under tension. The pulling forces that causes this tensions acts parallel to the surface and is due to the attractive forces between the molecules of the liquid.

A molecule on the surface

A molecule inside the liquid

Attractive forces acting on a liquid molecule at the surface and deep inside the liquid


Surface TensionThe magnitude of the force per unit length is called surface tension, σs (N/m)


Capillary EffectCapillary effect is the rise or fall of a liquid in a small-diameter tube inserted into liquidCapillaries is a narrow tubes or confined flow channelsThe curved free surface of a liquid in a capillary tube is called meniscus

Φ

Φ

a) Wetting fluid b) Nonwetting fluid


Capillary EffectContact (or wetting) angle Φ, is defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact

Φ<90o The liquid is to wet the surfaceΦ>90o The liquid is not to wet the surface

Cohesive forces is forces between like molecules (water-water), adhesive forces is forces between unlike molecules (water-glass). Water molecules are more strongly attracted to the glass molecules than they are to other water molecules, and thus water tends to rise along the glass surface

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