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PROPERTIES OF FLUIDS Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill 1 Chapter 2
PROPERTIES OF FLUIDS
Fluid Mechanics: Fundamentals and Applications, 2nd EditionYunus A. Cengel, John M. Cimbala
McGraw-Hill
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Chapter 2
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PROPERTIES OF FLUIDS
2–1 IntroductionContinuum
2–2 Density and Specific GravityDensity of Ideal Gases
2–3 Vapor Pressure and Cavitation
2–4 Energy and Specific Heats
2–5 Coefficient of CompressibilityCoefficient of Volume Expansion
2–6 Viscosity
2–7 Surface Tension and Capillary EffectCapillary Effect
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ObjectivesObjectives
• Have a working knowledge of the basic properties of fluids andunderstand the continuum approximation.
• Have a working knowledge of viscosity and the consequences ofthe frictional effects it causes in fluid flow.
• Calculate the capillary rise (or drop) in tubes due to the surfacetension effect.
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• Property: Any characteristic of a system.• Some familiar properties are pressure P,
temperature T, volume V, and mass m.• Properties are considered to be either intensive or
extensive.
• Intensive properties: Those that areindependent of the mass of a system, such astemperature, pressure, and density.
• Extensive properties: Those whose valuesdepend on the size -or extent - of the system.
• Specific properties: Extensive properties perunit mass.specific volume (v=V/m)specific energy (e=E/m)
• Property: Any characteristic of a system.• Some familiar properties are pressure P,
temperature T, volume V, and mass m.• Properties are considered to be either intensive or
extensive.
• Intensive properties: Those that areindependent of the mass of a system, such astemperature, pressure, and density.
• Extensive properties: Those whose valuesdepend on the size -or extent - of the system.
• Specific properties: Extensive properties perunit mass.specific volume (v=V/m)specific energy (e=E/m)
Criterion to differentiate intensiveand extensive properties.
Criterion to differentiate intensiveand extensive properties.
2–1 INTRODUCTION2–1 INTRODUCTION
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ContinuumContinuum • Atoms are widely spaced in the gas phase.• However, we can disregard the atomic nature
of a substance.• View it as a continuous, homogeneous matter
with no holes, that is, a continuum.
• This allows us to treat properties as smoothlyvarying quantities.
• Continuum is valid as long as size of the systemis large in comparison to distance betweenmolecules.
• Atoms are widely spaced in the gas phase.• However, we can disregard the atomic nature
of a substance.• View it as a continuous, homogeneous matter
with no holes, that is, a continuum.
• This allows us to treat properties as smoothlyvarying quantities.
• Continuum is valid as long as size of the systemis large in comparison to distance betweenmolecules.
Despite the relatively large gapsbetween molecules, a substance can betreated as a continuum because of thevery large number of molecules even inan extremely small volume.
Despite the relatively large gapsbetween molecules, a substance can betreated as a continuum because of thevery large number of molecules even inan extremely small volume.
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The diameter of the oxygen molecule is about3x10-10 m and its mass is 5.3x10-26 kg. Also,the mean free path of oxygen at 1 atm pressureand 20°C is 6.3x10-8 m.
At very high vacuums or very high elevations,the mean free path may become large (forexample, it is about 0.1 m for atmospheric air atan elevation of 100 km).
The length scale associated with most flows, such as seagulls in flight, is orders ofmagnitude larger than the mean free path of the air molecules. Therefore, here, and for allfluid flows considered in this book, the continuum idealization is appropriate.
The length scale associated with most flows, such as seagulls in flight, is orders ofmagnitude larger than the mean free path of the air molecules. Therefore, here, and for allfluid flows considered in this book, the continuum idealization is appropriate.
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2–2 DENSITY AND SPECIFIC GRAVITY2–2 DENSITY AND SPECIFIC GRAVITY
Density is mass per unit volume;Density is mass per unit volume;Specific gravity: The ratio of the density of asubstance to the density of some standardsubstance at a specified temperature (usuallywater at 4°C).
DensityDensity
Specific weight: Theweight of a unit volume ofa substance.
Specific volumeSpecific volume
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specific volume is volume per unit mass.
Density of Ideal GasesDensity of Ideal Gases
Equation of state: Any equation that relates the pressure,temperature, and density (or specific volume) of a substance.Ideal-gas equation of state: The simplest and best-known equationof state for substances in the gas phase.
Equation of state: Any equation that relates the pressure,temperature, and density (or specific volume) of a substance.Ideal-gas equation of state: The simplest and best-known equationof state for substances in the gas phase.
The thermodynamic temperature scale in the SI is the Kelvin scale.In the English system, it is the Rankine scale.The thermodynamic temperature scale in the SI is the Kelvin scale.In the English system, it is the Rankine scale.
where
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Air behaves as an ideal gas, even at veryhigh speeds. In this schlieren image, abullet traveling at about the speed of soundbursts through both sides of a balloon,forming two expanding shock waves. Theturbulent wake of the bullet is also visible.
Air behaves as an ideal gas, even at veryhigh speeds. In this schlieren image, abullet traveling at about the speed of soundbursts through both sides of a balloon,forming two expanding shock waves. Theturbulent wake of the bullet is also visible.
An ideal gas is a hypothetical substance thatobeys the relation Pv = RT.
At low pressures and high temperatures, thedensity of a gas decreases and the gasbehaves like an ideal gas.
In the range of practical interest, many familiargases such as;air, nitrogen, oxygen, hydrogen, helium, argon,neon, and kryptonand even heavier gases such as carbondioxide can be treated as ideal gases withnegligible error.
Dense gases such as water vapor should notbe treated as ideal gases since they usuallyexist at a state near saturation.
An ideal gas is a hypothetical substance thatobeys the relation Pv = RT.
At low pressures and high temperatures, thedensity of a gas decreases and the gasbehaves like an ideal gas.
In the range of practical interest, many familiargases such as;air, nitrogen, oxygen, hydrogen, helium, argon,neon, and kryptonand even heavier gases such as carbondioxide can be treated as ideal gases withnegligible error.
Dense gases such as water vapor should notbe treated as ideal gases since they usuallyexist at a state near saturation.
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EXAMPLE 2–1Determine the density, specific gravity, and mass of the air in a roomwhose dimensions are 4x5x6m at 100 kPa and 25°C.
Solution: The density, specific gravity, and mass of the air in aroom are to be determined.
Assumptions: At specified conditions, air can be treated as anideal gas.
Properties: The gas constant of air is R= 0.287 kPa.m3/kg.K.
Discussion Note that we converted the temperature to the unit K from °Cbefore using it in the ideal-gas relation.Discussion Note that we converted the temperature to the unit K from °Cbefore using it in the ideal-gas relation.
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2–3 VAPOR PRESSURE AND CAVITATION2–3 VAPOR PRESSURE AND CAVITATION
• Saturation temperature Tsat: The temperature at which a puresubstance changes phase at a given pressure.
• Saturation pressure Psat: The pressure at which a pure substancechanges phase at a given temperature.
• Vapor pressure (Pv): The pressure exerted by its vapor in phaseequilibrium with its liquid at a given temperature. It is identical to thesaturation pressure Psat of the liquid (Pv = Psat).
• Partial pressure: The pressure of a gas or vapor in a mixture with othergases. For example, atmospheric air is a mixture of dry air and watervapor, and atmospheric pressure is the sum of the partial pressure of dryair and the partial pressure of water vapor.
• Saturation temperature Tsat: The temperature at which a puresubstance changes phase at a given pressure.
• Saturation pressure Psat: The pressure at which a pure substancechanges phase at a given temperature.
• Vapor pressure (Pv): The pressure exerted by its vapor in phaseequilibrium with its liquid at a given temperature. It is identical to thesaturation pressure Psat of the liquid (Pv = Psat).
• Partial pressure: The pressure of a gas or vapor in a mixture with othergases. For example, atmospheric air is a mixture of dry air and watervapor, and atmospheric pressure is the sum of the partial pressure of dryair and the partial pressure of water vapor.
The partial pressure of a vapor must be less than or equal to the vaporpressure if there is no liquid present.
However, when both vapor and liquid are present and the system is in phaseequilibrium, the partial pressure of the vapor must equal the vapor pressure,and the system is said to be saturated.
The partial pressure of a vapor must be less than or equal to the vaporpressure if there is no liquid present.
However, when both vapor and liquid are present and the system is in phaseequilibrium, the partial pressure of the vapor must equal the vapor pressure,and the system is said to be saturated.
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The vapor pressure (saturation pressure) of apure substance (e.g., water) is the pressureexerted by its vapor molecules when the systemis in phase equilibrium with its liquid moleculesat a given temperature.
The vapor pressure (saturation pressure) of apure substance (e.g., water) is the pressureexerted by its vapor molecules when the systemis in phase equilibrium with its liquid moleculesat a given temperature.
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• There is a possibility of the liquid pressure in liquid-flow systems droppingbelow the vapor pressure at some locations, and the resulting unplannedvaporization.
• The vapor bubbles collapse as they are swept away from the low-pressureregions to extremely high-pressure waves.
• This phenomenon, which is a common cause for drop in performance and eventhe erosion of impeller blades, is called cavitation, and it is an importantconsideration in the design of hydraulic turbines and pumps.
Cavitation damage on a 16-mmby 23-mm aluminum sampletested at 60 m/s for 2.5 h. Thesample was located at thecavity collapse regiondownstream of a cavitygenerator specifically designedto produce high damagepotential.
Cavitation damage on a 16-mmby 23-mm aluminum sampletested at 60 m/s for 2.5 h. Thesample was located at thecavity collapse regiondownstream of a cavitygenerator specifically designedto produce high damagepotential.
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Cavitation
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EXAMPLE 2–2. In a water distribution system, the temperature of water is observed tobe as high as 30°C. Determine the minimum pressure allowed in the system to avoidcavitation.
Solution: The minimum pressure in a water distribution system to avoid cavitation isto be determined.
Properties: The vapor pressure of water at 30°C is 4.25 kPa.
Analysis: To avoid cavitation, the pressure anywhere in the flow should notbe allowed to drop below the vapor (or saturation) pressure at the giventemperature.
Discussion: Note that the vapor pressure increases with increasing temperature,and thus the risk of cavitation is greater at higher fluid temperatures.Discussion: Note that the vapor pressure increases with increasing temperature,and thus the risk of cavitation is greater at higher fluid temperatures.
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2–4 ENERGY AND SPECIFIC HEATS2–4 ENERGY AND SPECIFIC HEATS
• Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential,electric, magnetic, chemical, and nuclear, and their sum constitutes the totalenergy, E of a system.
• Thermodynamics deals only with the change of the total energy.• Macroscopic forms of energy: Those a system possesses as a whole with respect to
some outside reference frame, such as kinetic and potential energies.• Microscopic forms of energy: Those related to the molecular structure of a system
and the degree of the molecular activity.• Internal energy, U: The sum of all the microscopic forms of energy.
• Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential,electric, magnetic, chemical, and nuclear, and their sum constitutes the totalenergy, E of a system.
• Thermodynamics deals only with the change of the total energy.• Macroscopic forms of energy: Those a system possesses as a whole with respect to
some outside reference frame, such as kinetic and potential energies.• Microscopic forms of energy: Those related to the molecular structure of a system
and the degree of the molecular activity.• Internal energy, U: The sum of all the microscopic forms of energy.
The macroscopic energy of an objectchanges with velocity and elevation.The macroscopic energy of an objectchanges with velocity and elevation.
Kinetic energy, KE: The energy that asystem possesses as a result of itsmotion relative to some referenceframe.Potential energy, PE: The energy thata system possesses as a result of itselevation in a gravitational field.
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for a P=cnst. proc.
Energy of a flowing fluidEnthalpy
For a T = const. process
The internal energy u represents the microscopicenergy of a nonflowing fluid per unit mass, whereasenthalpy h represents the microscopic energy of aflowing fluid per unit mass.
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P/ is the flow energy, also called theflow work, which is the energy per unitmass needed to move the fluid andmaintain flow.
P/ is the flow energy, also called theflow work, which is the energy per unitmass needed to move the fluid andmaintain flow.
Specific HeatsSpecific Heats
Specific heat at constant volume, cv : The energy required to raise the temperature of theunit mass of a substance by one degree as the volume is maintained constant.Specific heat at constant pressure, cp: The energy required to raise the temperature of theunit mass of a substance by one degree as the pressure is maintained constant.
Specific heat at constant volume, cv : The energy required to raise the temperature of theunit mass of a substance by one degree as the volume is maintained constant.Specific heat at constant pressure, cp: The energy required to raise the temperature of theunit mass of a substance by one degree as the pressure is maintained constant.
Specific heat is the energy required toraise the temperature of a unit mass of asubstance by one degree in a specifiedway.
Specific heat is the energy required toraise the temperature of a unit mass of asubstance by one degree in a specifiedway.
Constant-volume and constant-pressure specific heats cv and cp
(values are for helium gas).
Constant-volume and constant-pressure specific heats cv and cp
(values are for helium gas).
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2–5 COMPRESSIBILITY AND SPEED OF SOUND2–5 COMPRESSIBILITY AND SPEED OF SOUND
Coefficient of Compressibility
Fluids, like solids, compresswhen the applied pressure is
increased from P1 to P2.
Fluids, like solids, compresswhen the applied pressure is
increased from P1 to P2.
We know from experience that the volume(or density) of a fluid changes with a changein its temperature or pressure.
But the amount of volume change is differentfor different fluids, and we need to defineproperties that relate volume changes to thechanges in pressure and temperature.
Two such properties are:the bulk modulus of elasticity
the coefficient of volume expansion .
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Coefficient of compressibility (alsocalled the bulk modulus ofcompressibility or bulk modulus ofelasticity) for fluids
Coefficient of compressibility (alsocalled the bulk modulus ofcompressibility or bulk modulus ofelasticity) for fluids
The coefficient of compressibility represents the change in pressurecorresponding to a fractional change in volume or density of the fluid while thetemperature remains constant.What is the coefficient of compressibility of a truly incompressiblesubstance (v = constant)?
A large value of indicates that a large change in pressure is needed to causea small fractional change in volume, and thus a fluid with a large isessentially incompressible.
This is typical for liquids, and explains why liquids are usually considered to beincompressible.
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Water hammer arrestors: (a) Alarge surge tower built to protectthe pipeline againstwater hammer damage.
(b) Much smaller arrestors usedfor supplying water to ahousehold washing machine.
Water hammer arrestors: (a) Alarge surge tower built to protectthe pipeline againstwater hammer damage.
(b) Much smaller arrestors usedfor supplying water to ahousehold washing machine.
Water hammer: Characterized by a sound that resembles the sound producedwhen a pipe is “hammered.” This occurs when a liquid in a piping networkencounters an abrupt flow restriction (such as a closing valve) and is locallycompressed.The acoustic waves that are produced strike the pipe surfaces, bends, and valvesas they propagate and reflect along the pipe, causing the pipe to vibrate andproduce the familiar sound.Water hammering can be quite destructive, leading to leaks or even structuraldamage. The effect can be suppressed with a water hammer arrestor.
Water hammer: Characterized by a sound that resembles the sound producedwhen a pipe is “hammered.” This occurs when a liquid in a piping networkencounters an abrupt flow restriction (such as a closing valve) and is locallycompressed.The acoustic waves that are produced strike the pipe surfaces, bends, and valvesas they propagate and reflect along the pipe, causing the pipe to vibrate andproduce the familiar sound.Water hammering can be quite destructive, leading to leaks or even structuraldamage. The effect can be suppressed with a water hammer arrestor.
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Water Hammer
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The coefficient of compressibility of an ideal gas is equal to its absolute pressure,and the coefficient of compressibility of the gas increases with increasingpressure.
The coefficient of compressibility of an ideal gas is equal to its absolute pressure,and the coefficient of compressibility of the gas increases with increasingpressure.
The percent increase of density of an ideal gas during isothermal compression isequal to the percent increase in pressure.
Isothermal compressibility: The inverse of the coefficient of compressibility.The isothermal compressibility of a fluid represents the fractional change involume or density corresponding to a unit change in pressure.
Isothermal compressibility: The inverse of the coefficient of compressibility.The isothermal compressibility of a fluid represents the fractional change involume or density corresponding to a unit change in pressure.
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Coefficient of Volume ExpansionCoefficient of Volume Expansion
Natural convection over awoman’s hand.
Natural convection over awoman’s hand.
The density of a fluid depends morestrongly on temperature than it does onpressure.
The variation of density with temperatureis responsible for numerous naturalphenomena such as winds, currents inoceans, rise of plumes in chimneys, theoperation of hot-air balloons, heattransfer by natural convection, and eventhe rise of hot air and thus the phrase“heat rises”.
To quantify these effects, we need aproperty that represents the variation ofthe density of a fluid withtemperature at constant pressure.
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The volume expansion coefficient of an ideal gas (P = RT ) at a absolutetemperature T is equivalent to the inverse of the temperature:The volume expansion coefficient of an ideal gas (P = RT ) at a absolutetemperature T is equivalent to the inverse of the temperature:
A large value of for a fluid means a large change indensity with temperature,and the product T represents the fraction of volume change ofa fluid that corresponds to a temperature change of T at constantpressure.
A large value of for a fluid means a large change indensity with temperature,and the product T represents the fraction of volume change ofa fluid that corresponds to a temperature change of T at constantpressure. The coefficient of volume
expansion is a measure ofthe change in volume of asubstance with temperatureat constant pressure.
The coefficient of volumeexpansion is a measure ofthe change in volume of asubstance with temperatureat constant pressure.
It can also be expressed approximately in terms of finitechanges as
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The coefficient of volume expansion (or volumeexpansivity): The variation of the density of a fluid withtemperature at constant pressure.
In the study of natural convection currents, the condition of the main fluid body thatsurrounds the finite hot or cold regions is indicated by the subscript “infinity” toserve as a reminder that this is the value at a distance where the presence of thehot or cold region is not felt. In such cases, the volume expansion coefficient canbe expressed approximately as
In the study of natural convection currents, the condition of the main fluid body thatsurrounds the finite hot or cold regions is indicated by the subscript “infinity” toserve as a reminder that this is the value at a distance where the presence of thehot or cold region is not felt. In such cases, the volume expansion coefficient canbe expressed approximately as
The combined effects of pressure and temperature changes on the volume change of afluid can be determined by taking the specific volume to be a function of T and P.The combined effects of pressure and temperature changes on the volume change of afluid can be determined by taking the specific volume to be a function of T and P.
The fractional change in volume (or density) due to changes in pressure and temperaturecan be expressed approximately asThe fractional change in volume (or density) due to changes in pressure and temperaturecan be expressed approximately as
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Solution: Water at a given temperature and pressure is considered. The densities ofwater after it is heated and after it is compressed are to be determined.
Assumptions 1 The coefficient of volume expansion and the isothermal compressibility ofwater are constant in the given temperature range.2 An approximate analysis is performed by replacing differential changes in quantities byfinite changes.
(a) The change in density due to the change of temperature from 20°C to 50°C atconstant pressure is;
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(b) The change in density due to a change ofpressure from 1 atm to 100 atm at constanttemperature is
(b) The change in density due to a change ofpressure from 1 atm to 100 atm at constanttemperature is
Discussion Note that the density of water decreases while being heated andincreases while being compressed, as expected.
This problem can be solved more accurately using differential analysis whenfunctional forms of properties are available
Discussion Note that the density of water decreases while being heated andincreases while being compressed, as expected.
This problem can be solved more accurately using differential analysis whenfunctional forms of properties are available
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Speed of sound (sonic speed): The speed at which an infinitesimally small pressurewave travels through a medium.
Prop
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ion
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ave
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duct
.Pr
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Control volume moving with the smallpressure wave along a duct.Control volume moving with the smallpressure wave along a duct.
For an ideal gasFor an ideal gasFor any fluidFor any fluid
Speed of Sound and Mach NumberSpeed of Sound and Mach Number
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The speed of sound changes withtemperature and varies with the fluid.The speed of sound changes withtemperature and varies with the fluid.
The Mach number can be different at differenttemperatures even if the flight speed is the same.The Mach number can be different at differenttemperatures even if the flight speed is the same.
Mach number Ma: The ratio of the actual speedof the fluid (or an object in still fluid) to thespeed of sound in the same fluid at the samestate.
Mach number Ma: The ratio of the actual speedof the fluid (or an object in still fluid) to thespeed of sound in the same fluid at the samestate.
The Mach number depends on the speed ofsound, which depends on the state of the fluid.The Mach number depends on the speed ofsound, which depends on the state of the fluid.
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2–6 VISCOSITY2–6 VISCOSITY
Viscosity: A property that represents the internalresistance of a fluid to motion or the “fluidity”.
Drag force: The force a flowing fluid exerts on a bodyin the flow direction. The magnitude of this forcedepends, in part, on viscosity
A fluid moving relative to abody exerts a drag force onthe body, partly because offriction caused by viscosity.
A fluid moving relative to abody exerts a drag force onthe body, partly because offriction caused by viscosity.
The viscosity of a fluid is a measure of its “resistance to deformation.”
Viscosity is due to the internal frictional force that develops between different layers offluids as they are forced to move relative to each other.
When two solid bodies in contact move relative to eachother, a friction force develops at the contact surface inthe direction opposite to motion.
When two solid bodies in contact move relative to eachother, a friction force develops at the contact surface inthe direction opposite to motion.
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ViscosityViscosity
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Shear stressShear stress
Shear forceShear force
coefficient of viscosity, Dynamic (absolute) viscosity (kg/ms or Ns/m2 or Pas ) 1 poise = 0.1 Pas
The behavior of a fluid in laminar flowbetween two parallel plates when the upperplate moves with a constant velocity.
The behavior of a fluid in laminar flowbetween two parallel plates when the upperplate moves with a constant velocity.
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Newtonian fluids: Fluids for which the rate of deformation is proportional to the shear stress.Newtonian fluids: Fluids for which the rate of deformation is proportional to the shear stress.
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the shear stress
velocity profile andthe velocity gradient
The force F required to move the upper plate at a constant velocity of Vwhile the lower plate remains stationary is;
The rate of deformation (velocity gradient) of aNewtonian fluid is proportional to shear stress,and the constant of proportionality is theviscosity.
The rate of deformation (velocity gradient) of aNewtonian fluid is proportional to shear stress,and the constant of proportionality is theviscosity.
Variation of shear stress with the rate ofdeformation for Newtonian and non-Newtonian fluids (the slope of a curve at apoint is the apparent viscosity of the fluid atthat point).
Variation of shear stress with the rate ofdeformation for Newtonian and non-Newtonian fluids (the slope of a curve at apoint is the apparent viscosity of the fluid atthat point).
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Non_Newtoinan Fluid
Fluids for which shearing stress is not linearly related to the rate of shearingstrain are designated as non-Newtonian fluids.Fluids for which shearing stress is not linearly related to the rate of shearingstrain are designated as non-Newtonian fluids.
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Dynamic viscosity, in general,does not depend on pressure,but kinematic viscosity does.
Dynamic viscosity, in general,does not depend on pressure,but kinematic viscosity does.
Kinematic viscosityKinematic viscosity
m2/s or stoke 1 stoke =1 cm2/sm2/s or stoke 1 stoke =1 cm2/s
For gases: For liquids
For liquids, both the dynamic and kinematic viscosities are practicallyindependent of pressure and any small variation with pressure is usuallydisregarded, except at extremely high pressures.For gases, this is also the case for dynamic viscosity (at low to moderatepressures), but not for kinematic viscosity since the density of a gas isproportional to its pressure.
For liquids, both the dynamic and kinematic viscosities are practicallyindependent of pressure and any small variation with pressure is usuallydisregarded, except at extremely high pressures.For gases, this is also the case for dynamic viscosity (at low to moderatepressures), but not for kinematic viscosity since the density of a gas isproportional to its pressure.
T is absolute temperature and a, b, and c are experimentally determined constants.
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The viscosity of liquids decreases andthe viscosity of gases increases withtemperature.
The viscosity of liquids decreases andthe viscosity of gases increases withtemperature.
Viscosity is caused by the cohesive forces between themolecules in liquids and by the molecular collisions ingases, and it varies greatly with temperature.In a liquid, the molecules possess more energy athigher temperatures, and they can oppose the largecohesive intermolecular forces more strongly. As aresult, the energized liquid molecules can move morefreely.
In a gas, the intermolecular forces are negligible, andthe gas molecules at high temperatures moverandomly at higher velocities. This results in moremolecular collisions per unit volume per unit time andtherefore in greater resistance to flow.
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40
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• How is viscosity measured? A rotating viscometer.– Two concentric cylinders with a fluid in the
small gap ℓ.– Inner cylinder is rotating, outer one is fixed.
• Use definition of shear force:
• If ℓ/R << 1, then cylinders can be modeledas flat plates.
• Torque T = FR, and tangential velocityV=wR
• Wetted surface area A=2pRL.
• Measure T and w to compute m
duF A A
dy
Most devices (called viscometers) used to determineviscosity do not measure it directly, but instead measuresome characteristic with a known relationship to viscosity.
The capillary tube viscometer involves the laminar flow of afixed volume of fluid through a capillary tube.
The time required for the fluid to pass through the tube is ameasure of the kinematic viscosity of the fluid.
As shown with the four tubes, the drain times can varydepending on the viscosity of the fluid and the diameter ofthe capillary tube.
Most devices (called viscometers) used to determineviscosity do not measure it directly, but instead measuresome characteristic with a known relationship to viscosity.
The capillary tube viscometer involves the laminar flow of afixed volume of fluid through a capillary tube.
The time required for the fluid to pass through the tube is ameasure of the kinematic viscosity of the fluid.
As shown with the four tubes, the drain times can varydepending on the viscosity of the fluid and the diameter ofthe capillary tube.
Capilar tube ViscometerCapilar tube Viscometer
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EXAMPLE 2–4The viscosity of a fluid is to be measured by a viscometer constructed of two 40-cm-long concentriccylinders. The outer diameter of the inner cylinder is 12 cm, and the gap between the two cylindersis 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 viscometerare 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 arenegligible.
Analysis The velocity profile is linear only when the curvature effectsare negligible, and the profile can be approximated as being linear inthis case since l/R<< 1. Solving for viscosity and substituting thegiven values, the viscosity of the fluid is determined to be;
Discussion Viscosity is a strong function of temperature, and a viscosity value without acorresponding temperature is of little value. Therefore, the temperature of the fluid should havealso been measured during this experiment, and reported with this calculation.
Discussion Viscosity is a strong function of temperature, and a viscosity value without acorresponding temperature is of little value. Therefore, the temperature of the fluid should havealso been measured during this experiment, and reported with this calculation.
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Example 2-47: The clutch system shown in Fig. P2–47 is used to transmit torque through a 3-mm-thick oil film with m 0.38 N s/m2 between two identical 30-cm-diameter disks. When the drivingshaft rotates at a speed of 1450 rpm, the driven shaft is observed to rotate at 1398 rpm. Assuming alinear velocity profile for the oil film, determine the transmitted torque. we can assume one of thedisks to be stationary and the other to be rotating at an angular speed of
Discussion: Note that thetorque ransmitted isproportional to thefourth power of diskdiameter, and is inverselyproportional to thethickness of the oil film.
Example 2–53: In regions far from the entrance, fluid flow through a circular pipe is one-dimensional, and the velocity profile for laminar flow is given by u(r) umax(1 r 2/R2), where R isthe radius of the pipe, r is the radial distance from the center of the pipe, and umax is themaximum flow velocity, which occurs at the center. Obtain (a) a relation for the drag forceapplied by the fluid on a section of the pipe of length L and (b) the value of the drag force forwater flow at 20°C with R 0.08 m, L 15 m, umax 3 m/s, and m 0.0010 kg/m s.
2–7 SURFACE TENSION AND CAPILLARY EFFECT2–7 SURFACE TENSION AND CAPILLARY EFFECT
Liquid droplets behave like small balloons filled with the liquid on a solid surface, and thesurface of the liquid acts like a stretched elastic membrane under tension.
The pulling force that causes this tension acts parallel to the surface and is due to theattractive forces between the molecules of the liquid.
The magnitude of this force per unit length is called surface tension (or coefficient ofsurface tension) and is usually expressed in the unit N/m.
This effect is also called surface energy [per unit area] and is expressed in the equivalentunit of N m/m2.
A drop of blood forms a hump on a horizontal glass
A drop of mercury forms a near-perfect sphere and can be rolled just like a steel ball over asmooth surface
Water droplets from rain or dew hang from branches or leaves of trees
A soap bubble released into the air forms a spherical water beads up into small drops on flowerpetals.
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Some consequences of surface tension.Some consequences of surface tension.
This can be observed by slightlyoverfilling a drinking glass; the water willstand above the rim without spilling.
This can be observed by slightlyoverfilling a drinking glass; the water willstand above the rim without spilling.
Surface tension depends on the natureof the liquid, the surroundingenvironment and emperature. Liquidswere molecules have large attractiveintermolecular forces will have a largesurface tension
Surface tension depends on the natureof the liquid, the surroundingenvironment and emperature. Liquidswere molecules have large attractiveintermolecular forces will have a largesurface tension
Some insects can land on water or even walk on waterand that small steel needles can float on water. Thesephenomena are again made possible by surfacetension that balances the weights of these objects.
Some insects can land on water or even walk on waterand that small steel needles can float on water. Thesephenomena are again made possible by surfacetension that balances the weights of these objects.
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A drop forms when liquid is forced out of a small tube. The shape of thedrop is determined by a balance of pressure, gravity, and surface tensionforces.
A drop forms when liquid is forced out of a small tube. The shape of thedrop is determined by a balance of pressure, gravity, and surface tensionforces.
Attractive forces acting on a liquidmolecule at the surface and deep insidethe liquid.
Attractive forces acting on a liquidmolecule at the surface and deep insidethe liquid.
Stretching a liquid film with a U-shaped wire,and the forces acting on the movable wire oflength b.
Stretching a liquid film with a U-shaped wire,and the forces acting on the movable wire oflength b.
Surface tension: The work doneper unit increase in the surfacearea of the liquid.
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The free-body diagram of half a droplet or air bubbleand half a soap bubble.The free-body diagram of half a droplet or air bubbleand half a soap bubble.
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Capillary EffectCapillary Effect
Capillary effect: The rise or fall of a liquid in a small-diameter tube inserted into theliquid.Capillaries: Such narrow tubes or confined flow channels. The capillary effect is partiallyresponsible for the rise of water to the top of tall trees.Meniscus: The curved free surface of a liquid in a capillary tube.
The contact angle for wetting and nonwettingfluids.The contact angle for wetting and nonwettingfluids.
The strength of the capillary effect is quantified by the contact (or wetting) angle,defined as the angle that the tangent to the liquid surface makes with the solid surfaceat the point of contact.
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The meniscus of colored water in a 4-mm-inner-diameter glass tube. Note thatthe edge of the meniscus meets the wallof the capillary tube at a very smallcontact angle.
The meniscus of colored water in a 4-mm-inner-diameter glass tube. Note thatthe edge of the meniscus meets the wallof the capillary tube at a very smallcontact angle.
The capillary rise of water and the capillaryfall of mercury in a small-diameter glasstube.
The capillary rise of water and the capillaryfall of mercury in a small-diameter glasstube.
The forces acting on a liquid column thathas risen in a tube due to the capillaryeffect.
The forces acting on a liquid column thathas risen in a tube due to the capillaryeffect.
Capillary rise is inversely proportional to the radius of the tube and density of the liquid.
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EXAMPLE 2–5A 0.6-mm-diameter glass tube is inserted into water at 20°C in a cup. Determinethe capillary rise of water in the tube.
Solution: The rise of water in a slender tube as a result of thecapillary effect is to be determined.
Assumptions 1 There are no impurities in the water and nocontamination on the surfaces of the glass tube.
2 The experiment is conducted in atmospheric air.Properties The surface tension of water at 20°C is 0.073 N/m. Thecontact angle of water with glass is 0°. We take the density of liquidwater to be 1000 kg/m3.
Discussion: Note that if the tube diameter were 1 cm, the capillary rise would be 0.3 mm,which is hardly noticeable to the eye. Actually, the capillary rise in a large-diameter tubeoccurs only at the rim. The center does not rise at all. Therefore, the capillary effect can beignored for large-diameter tubes.
Discussion: Note that if the tube diameter were 1 cm, the capillary rise would be 0.3 mm,which is hardly noticeable to the eye. Actually, the capillary rise in a large-diameter tubeoccurs only at the rim. The center does not rise at all. Therefore, the capillary effect can beignored for large-diameter tubes.
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SummarySummary
• Introduction– Continuum
• Density and Specific Gravity– Density of Ideal Gases
• Vapor Pressure and Cavitation• Energy and Specific Heats• Compressibility and Speed of Sound
– Coefficient of Compressibility– Coefficient of Volume Expansion– Speed of Sound and Mach Number
• Viscosity• Surface Tension and Capillary Effect
• Introduction– Continuum
• Density and Specific Gravity– Density of Ideal Gases
• Vapor Pressure and Cavitation• Energy and Specific Heats• Compressibility and Speed of Sound
– Coefficient of Compressibility– Coefficient of Volume Expansion– Speed of Sound and Mach Number
• Viscosity• Surface Tension and Capillary Effect
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