Chapter 1FLUID PROPERTIES

The engineering science of fluid mechanics fluid properties the application of the basic laws of mechanics and thermodynamics and orderly experimentation

The properties of density and viscosity play principal roles in open- and closed-channel flow and in flow around immersed objects

Surface-tension effects are important in the formation of droplets in the flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occur as well as in the formation of capillary waves

The property of vapor pressure which accounts for changes of phase from liquid to gas becomes important when reduced pressures are encountered

This chapter liquid is defined and the International System of Units (SI) of force mass length and time units are discussed before the discussion of properties and definition of terms is taken up

11 DEFINITION OF A FLUID

Fluid substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be

Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point

Fig 11

Figure 11 Deformation resulting from application of constant shear force A substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

Fig11 fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate

Experiments other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The engineering science of fluid mechanics fluid properties the application of the basic laws of mechanics and thermodynamics and orderly experimentation

The properties of density and viscosity play principal roles in open- and closed-channel flow and in flow around immersed objects

Surface-tension effects are important in the formation of droplets in the flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occur as well as in the formation of capillary waves

The property of vapor pressure which accounts for changes of phase from liquid to gas becomes important when reduced pressures are encountered

This chapter liquid is defined and the International System of Units (SI) of force mass length and time units are discussed before the discussion of properties and definition of terms is taken up

11 DEFINITION OF A FLUID

Fluid substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be

Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point

Fig 11

Figure 11 Deformation resulting from application of constant shear force A substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

Fig11 fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate

Experiments other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

11 DEFINITION OF A FLUID

Fluid substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be

Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point

Fig 11

Figure 11 Deformation resulting from application of constant shear force A substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

Fig11 fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate

Experiments other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 11 Deformation resulting from application of constant shear force A substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

Fig11 fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate

Experiments other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

Fig11 fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate

Experiments other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

If τ = FA for the shear stress

The ratio Ut angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)

The angular velocity may also be written dudy ndash more general The velocity gradient dudy may also be visualized as the rate a

t which one layer moves relative to an adjacent layer in differential form

(111)

- Newtons law of viscosity - proportionality factor μ viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Materials other than fluids cannot satisfy the definition of a fluid A plastic substance will deform a certain amount

proportional to the force but not continuously when the stress applied is below its yield shear stress

A complete vacuum between the plates would cause deformation at an ever-increasing rate

If sand were placed between the two plates Coulomb friction would require a finite force to cause a continuous motion

plastics and solids are excluded from the classification of fluids

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Fluids Newtonian non-Newtonian

Newtonian fluid linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] (Fig 12)

Non-Newtonian fluid nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation An ideal plastic has a definite yield stress and a constant linear rel

ation of τ to dudy A thixotropic substance such as printers ink has a viscosity that

is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest

Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 12 Rheological diagram

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

For purposes of analysis the assumption is frequently made that a fluid is nonviscous

With zero viscosity the shear stress is always zero regardless of the motion of the fluid

If the fluid is also considered to be incompressible it is then called an ideal fluid and plots as the ordinate in Fig 12

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

12 FORCE MASS LENGTH AND TIME UNITS

Force mass length and time consistent units greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particular c

onsistent system A system of mechanics units consistent when unit force causes

unit mass to undergo unit acceleration The International System (SI)

newton (N) as unit or force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion

(121)

The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g

(122) For example where g = 9876 ms2 a body with gravity force of 10 N

has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is

Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Table 11 Selected prefixes for powers of 10 in SI units

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

13 VISCOSITY

Viscosity requires the greatest consideration in the study of fluid flow

Viscosity is that property of a fluid by virtue of which it offers resistance to shear

Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity

Molasses and tar are examples of highly viscous liquids water and air have very small viscosities

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity The resistance of a fluid to shear depends upon its

cohesion and upon its rate of transfer of molecular momentum

A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion - predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise

A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Fig13 rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks

Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 13

Model illustrating transfer of momentum

Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13 The measure of the motion of one layer relative to an adjacent layer is dudy

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature

For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure

A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid fluid statistics - no shear forces considered and the only stresse

s remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Dimensions of viscosity from Newtons law of viscosity ndash solving for the viscosity μ

and inserting dimensions F L T for force length and time

shows that μ has the dimensions FL-2T With the force dimension expressed in terms of mass by use of

Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1

The SI unit of viscosity which is the pascal second (symbol Pas) has no name

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Kinematic Viscosity

μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)

(131)

- occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)

The dimensions of ν are L2T-1 SI unit 1 m2s has no name Viscosity is practically independent of pressure and depends u

pon temperature only The kinematic viscosity of liquids and of gases at a given p

ressure is substantially a function of temperature

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Example 11

A liquid has a viscosity or 0005 Pas and a density or 850 kgm3 Calculate the kinematic viscosity

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πw where w = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ we get

Figure 15 lists the program in which the variable RR represents the crank radius r

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 14 Notation for sleeve motion

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 15 BASIC program to determine loss in sleeve motion

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

14 CONTINUUM

In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum

Example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood

This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations

Rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow

The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow

In this text only the gas-dynamics regime is studied

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

15 DENSITY SPECIFIC VOLUME UNIT GRAVITY FORCE RELATIVE DENSITY PRESSURE

The density ρ its mass per unit volume Density at a point the mass Δm of fluid in a small volume ΔV su

rrounding the point

(151) For water at standard pressure (760 mm Hg) and 4oC ρ = 1000

kgm3

The specific volume vs the volume occupied by unit mass of fluid

(152)

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The unit gravity force γ the force of gravity per unit volume It changes with location depending upon gravity

(151) Water γ = 9806 Nm3 at 5oC at sea level

The relative density S of a substance the ratio of its mass to the mass of an equal volume of water at standard conditions (may also be expressed as a ratio or its density to that of water)

The average pressure the normal force pushing against a plane area divided by the area The pressure at a point is the ratio of normal force to area as the area

approaches a small value enclosing the point If a fluid exerts a pressure against the walls or a container the contain

er will exert a reaction on the fluid which will be compressive Liquids can sustain very high compressive pressures but are very weak

in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)

Units force per area which is newtons per square metre called pascals (Pa)

Absolute pressure P gage pressures p

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

16 PERFECT GAS

This treatment thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)

The perfect gas substance that satisfies the perfect-gas-law (161)

and that has constant specific heats P is the absolute pressure vs is the specific volume R is the gas constant T is the absolute temperature

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)

Eq(161) the equation of state for a perfect gas may be written

(162) The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepance increases and becomes serious near the critical point

The perfect-gas law encompasses both Charles law and Boyles law Charles law for constant pressure the volume of a given m

ass of gas varies as its absolute temperature Boyles law (isothermal law) for constant temperature the

density varies directly as the absolute pressure

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The volume v of m mass units of gas is mvs

(163) With being the volume per mole

(164) If n is the number of moles of the gas in volume ϑ

(165) The product MR called the universal gas constant has a value

depending only upon the units employed(166)

The gas constant R can then be determined from (167)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The specific heat cv of a gas number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant

The specific heat cp the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant

The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) the energy p

er unit mass due to molecular spacing and forces The enthalpy h important property of a gas given by h=u+Pρ cv and cp units joule per kilogram per kelvin (JkgK)

4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions

R is related to cv and cp by

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Example 12

A gas with relative molecular mass of 44 is at a pressure or 09 MPa and a temperature of 20oC Determine its density

From Eq(167)

Then from Eq(162)

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

17 BULK MODULUS OF ELASTICITY

For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important also when temperature changes are involved eg free convection

The compressibility of a liquid is expressed by its bulk modulus of elasticity

If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio dpdV is the bulk modulus of elasticity K

For any volume V (171)

Expressed in units of p For water at 20oC K = 22 GPa

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Example 13

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNn2 What is its bulk modulus of elasticity

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

18 VAPOR PRESSURE

Liquids evaporate because or molecules escaping from the liquid surface vapor molecules exert partial pressure in the space - vapor pressure

If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists

Depends upon temperature and increases with it Boiling when the pressure above a liquid equals the vapor pressure of the liquid

20oC water 2447 kPa mercury 0173 Pa When very low pressures are produced at certain locations in t

he system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

19 SURFACE TENSION

Capillarity

At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface

The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium

The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Table 12 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet

For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec 26)

For the cylindrical liquid jet of radius r the pipe-tension equation applies

Both equations the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid

A liquid that wets the solid has a greater adhesion than cohesion Surface tension in this case causes the liquid to rise within a small vertical tube that is partially immersed in it

For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus

Figure 14 the capillary rise for water and mercury in circular glass tubes in air

Figure 15 Capillarity in circular glass tubes

Figure 15 Capillarity in circular glass tubes