fundamentals fluid properties

7/30/2019 Fundamentals Fluid Properties

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A significant portion of these notes summarizes various sections of Massey, but additional materia

from other sources is also included. Note that the notes are incomplete; they will be completed

during the lectures, so please attend

Fluid mechanics may be used to answer the following interesting questions:

How can a rocket generate thrust (forward motion) without having air to push

against in outer space?

How can we design an airplane from the experiments performed on a miniature

model?

How much less work and gas may be used by improving the aerodynamic design of

vehicles?

How does a turbine convert fluid power into electricity?

Fundamental Concepts in Fluid Mechanics1. Definition of Fluid Mechanics

2. Fluids

3. Concept of a Continuum

4. Dimensions and Units used in Fluid Mechanics

5. Fluid Properties

Density and Specific Weight

Compressibility

Surface tension

Vapor Pressure

Viscosity

1. DEFINITION OF FLUID MECHANICS

Fluid mechanics is that branch of applied mechanics that is concerned with the statics

and dynamics of liquids and gases. The analysis of the behaviour of fluids is based upon

the fundamental laws of applied mechanics that relate to the conservation of mass,

energy and momentum. The subject branches out into sub-disciplines such as

aerodynamics, hydraulics, geophysical fluid dynamics and bio-fluid mechanics.

2. FLUIDS

A fluid is a substance that may flow. That is, the particles making up the fluid

continuously change their positions relative to one another. Fluids do not offer any

lasting resistance to the displacement of one layer over another when a shear force is

applied [Fig. 1,2 &3]. This means that if a fluid is at rest, then no shear forces can exist

in it, which is different from solids; solids can resist shear forces while at rest. To

summarize, if a shear force is applied to a fluid it will cause flow. Recall the example in


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class when a book was placed between my hands that were previously moving parallel to

one another, even in the presence of the fluid, air. The book was somewhat distorted by

the shear forces exerted on it by my hands, but eventually adopted a deformed position

that resisted the force.

A further difference between solids and fluids is that a solid has a fixed shape whereas

a fluid owes its shape at any particular time to that of the vessel containing it.

Fig. 1: Plate sits on the surface of afluid at rest. All particles in eachlayer are aligned.

Fig. 2: The plate ispushed horizontallycausing the surfacelayer to slide.Particles in eachlayer are caused toslide. They moverelative to the other

layers.

Fig. 3. The force onthe plate is removedand the plate comesto rest. Displacedparticles in each layerremain in newposition.

Figures 1,2 &3 demonstrate that when a force is applied to a fluid the layers becomepermanently displaced.


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3. CONTINUUM CONCEPT

The behaviour of individual molecules comprising a fluid determines the observed

properties of the fluid and for an absolutely complete analysis, the fluid should be

studied at the molecular scale. The behaviour of any one molecule is highly complexcontinuously varying and may indeed be very different from neighbouring molecules at any

instant of time. The problems normally encountered by engineers do not require

knowledge and prediction of behaviour at the molecular level but on the properties of the

fluid mass that may result. Thus the interest is more on the average rather than the

individual responses of the molecules comprising the fluid. At a microscopic level, a fluid

consists of molecules with a lot of space in between. For our analysis, we do not consider

the actual conglomeration of separate molecules, but instead assume that the fluid is a

continuum, that is a continuous distribution of matter with no empty space. The sketch

below illustrates this. Note that the fluid particle consists of an assembly of moleculeseach having properties such as pressure, temperature, density etc. However, we are

interested in the property of the fluid particle at P and therefore we regard P as being a

smear of matter (represented as a solid filled circle in the figure) with no space.

Recall the example of a crowd in a stadium given in class.

4. DIMENSIONS AND UNITS

Physical quantities require quantitative descriptions when solving engineering problems

Density, which is one such physical quantity, is a measure of the mass contained in unit

volume. Density, however, does not represent a fundamental magnitude. There are nine

quantities considered to be fundamental magnitudes, and they are: length, mass, timetemperature, amount of a substance, electric current, luminous intensity, plane angle, and

solid angle. The magnitudes of all the quantities can be expressed in terms of the

fundamental magnitudes.

To give the magnitude of a quantity a numerical value, a set of units must be selected.

Two primary systems of units are commonly used in Fluid Mechanics, namely, the Imperia

PIndividualmolecules

Macroscopicview of afluid particlePFluid

mass


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System (sometimes called the English units) an the International System, which is

referred to as SI (Systeme International) units.

The fundamental dimensions and their units and the factors for conversion from the

English unit system to the SI are shown in the two tables on the following pages.

All theoretically derived equations are dimensionally homogeneousthat is, the

dimensions of the left side of the equation must be the same as those on the right side

and all additive separate terms must have the same dimensions. We accept as a

fundamental premise that all equations describing physical phenomena must be

dimensionally homogeneous. If this were not true, we would be attempting to equate or

add unlike physical quantities, which would not make sense. For example, the equation for

the velocity, V, of a uniformly accelerated body is

V _ V0 _ at (1.1)

where is the initial velocity, a the acceleration, and t the time interval. In terms of

dimensions the equation is

LT_1 = LT_1 + LT_1

and thus Eq. 1.1 is dimensionally homogeneous.


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Table 1. Fundamental Dimensions and Their Units

Quantity Dimensions SI unit English unit

Length, l L metre m foot ft

Mass, m M kilogram kg slug slug

Time, t T second s second sec

Eelctric

current, iampere A ampere A

Temperature, T kelvin K Rankine R

Amount of

substanceM kg-mole kg-mol lb-mole lb-mol

Luminousintensity

candela rd candela cd

Plane angle radian rad radian rad

Solid angle steradian sr steradian sr


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Table 2. Derived Dimensions

Quantity Dimensions SI unit English unit

Area A L2 m2 ft2

Volume V L

3

M

3

; L (litre) ft

3

Veloctiy v LT-1 m/s ft/sec

Acceleration a LT-2 m/s2 ft/sec2

Angular velocity

T-1s-1 sec-1

Force FMLT-2

kg m/s2

N (newton)

slug-ft/sec2

lb (pound)

Density ML-3 kg/m3 slug/ft3

Specific weight ML-2T-2 N/m3 lb/ft3

Frequency f T-1 s-1 sec-1

Pressure pML-1T-2

Pa (pascal)N/m2

lb/ft2

Stress ML-1T-2 N/m2 lb/ft2

Surface tension

MT-2N/m lb/ft

Work W ML2T-2 J (joule) N m ft-lb

Energy E ML2T2 J (joule) N m ft-lb

Heat rate Q ML2T-3 J/s Btu/sec

Torque T ML2T-2 N m ft-lb

Power P ML2T-3 J/s W (watt) ft-lb/secViscosity ML-1T-1 N s/m2 lb-sec/ft2

Mass flux m MT-1 kg/s Slug/sec

Flow rate Q L3T-1 m3/s ft3/sec

Specific heat c L2T2-1 J/(kg K) Btu/slug-R

Conductivity K MLT-3-1 W/(m K) lb-sec-R

5. FLUID PROPERTIES

i. DensityDensity is the ratio of the mass of a given amount of the substance to the volume it

occupies.

Mean density is defined as the ratio of a given amount of a substance to the volume that

this amount occupies. The density is said to be uniform if the mean density in all parts of

the substance is the same.


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The value of density can vary widely between different fluids, but for liquids,variations in pressure and temperature generally have only a small effect on the

value of . The small change in the density of water with large variations intemperature is illustrated in Fig. 4.

Fig. 4. Variation of density of water with temperature

Density at a point is the limit to which the mean density tends as the volume considered

is indefinitely reduced. Expressed mathematically, it is:

V

mlimV

where is taken as the minimum volume of a fluid particle below which the continuum

assumption fails.

This is illustrated in the sketch below (as completed in class).

Specific Weight

The specific weight of a fluid, designated by the Greek symbol (gamma), is defined asits weight per unit volume. Thus, specific weight is related to density through the

equation

V


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= g (1.6)

where g is the local acceleration of gravity. Just as density is used to characterize the

mass of a fluid system, the specific weight is used to characterize the weight of the

system.

In SI system the units are N/m3 .Under conditions of standard gravity (g = 9.807 m/s2

water at 600F (15.556 0C) has a specific weight of 9.80 KN/m3.

Specific Gravity

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

the fluid to the density of water at some specified temperature. Usually the specified

temperature is taken as 40C (39.20F) and at this temperature the density of water is

1000 kg/m3. In equation form, specific gravity of a liquid of density x is expressed as

Cwater

xSG

04@

And since it is a ratio of densities, there are no units (dimensionless). SG for mercury is

13.55 or 13.6.

ii. Compressibility

An important question to answer when considering the behavior of a particular fluid is

how easily can the volume (and thus the density) of a given mass of the fluid be changed

when there is a change in pressure? That is, how compressible is the fluid?

Gases are highly compressible in comparison to liquids, with changes in gas density

directly related to changes in pressure and temperature through the equation

P = RT (1.8)

where p is the absolute pressure, the density, T the absolute temperature, and R is a

gas constant. Equation 1.8 is commonly termed the ideal or perfect gas law, or theequation of statefor an ideal gas. It is known to closely approximate the behavior of rea

gases under normal conditions when the gases are not approaching liquefaction.

Pressure in a fluid at rest is defined as the normal force per unit area exerted on a plane

surface (real or imaginary) immersed in a fluid and is created by the bombardment of the

surface with the fluid molecules. From the definition, pressure has the dimension of FL-2

and in British units is expressed as 1lb/ft2 (psf2) or (psi) and in SI units as N/m2. In SI, 1


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N/m2 is defined as a pascal, abbreviated as Pa, and pressures are commonly specified in

pascals. The pressure in the ideal gas law must be expressed as an absolute pressure,

which means that it is measured relative to absolute zero pressure (a pressure that would

only occur in a perfect vacuum). Standard sea-level atmospheric pressure (by

international agreement) is 14.696 psi (abs) or 101.33 kPa (abs). For most calculations

these pressures can be rounded to 14.7 psi and 101 kPa, respectively. In engineering it iscommon practice to measure pressure relative to the local atmospheric pressure, and

when measured in this fashion it is calledgaugepressure. Thus, the absolute pressure can

be obtained from the gauge pressure by adding the value of the atmospheric pressure

For example, a pressure of 30 psi (gauge) in a tire is equal to 44.7 psi (abs) at standard

atmospheric pressure. Pressure is a particularly important fluid characteristic and it wil

be discussed more fully later.

The gas constant, R, which appears in Eq. 1.8, depends on the particular gas and is related

to the molecular weight of the gas. Values of the gas constant for several common gasesare listed in Tables 1.7 and 1.8. Also in these tables the gas density and specific weight

are given for standard atmospheric pressure and gravity and for the temperature listed.

A property that is commonly used to characterize compressibility is thebulk modulus.The reciprocal of the bulk modulus is compressibility.

The degree of compressibility of a substance is characterized by the bulk modulus of

elasticity, K, defined as:

VV

pK

where p represents the small increased in pressure applied to the substance that causes

a decrease of the volume by V from its original volume of V.

Note the negative sign in the definition to ensure that the value of K is always positive.

K has the same dimensional formula as pressure, which is: [ML-1T-2]

K can also be expressed as a function of the accompanying change in density caused by

the pressure increase as in the above equation.

Note that the value of K depends on the relation between pressure and density under

which the compression occurs. The isothermal bulk modulus is the value when


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compression occurs while the temperature is held constant. The isentropic bulk modulus

is the value when compression occurs under adiabatic conditions.

For liquids, K is very high (2.05 GPa for water at moderate pressure) and so there is very

little change of density with pressure. For this reason, the density of liquids can be

assumed to be constant without any serious loss in accuracy. On the other hand, gases arevery compressible.

iii.Surface Tension

Surface tension is the surface force that develops at the interface between two

immiscible liquids or between liquid and gas or at the interface between a liquid and a

solid surface. Because of surface tension, small water droplets, gas bubbles and drops of

mercury tend to maintain spherical shapes.


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These various types of surface phenomena are due to the unbalanced cohesive forces

acting on the liquid molecules at the fluid surface. Molecules in the interior of the fluid

mass are surrounded by molecules that are attracted to each other equally. However

molecules along the surface are subjected to a net force toward the interior. The

apparent physical consequence of this unbalanced force along the surface is to create thehypothetical skin or membrane. A tensile force may be considered to be acting in the

plane of the surface along any line in the surface. The intensity of the molecular

attraction per unit length along any line in the surface is called the surface tension and is

designated by the Greek symbol (sigma).

The ultimate magnitude and direction of this tension force is determined not only by

what happens on either side of the interface, but by the way molecules of the two fluids

interact with each other. Surface tension, therefore, is specific to the participating

fluids. Surface tension forces are also sensitive to the physical and chemical condition of

the solid surface in contact, such as its roughness, cleanliness, or temperature.

If a line is imagined drawn in a liquid surface, then the liquid on one side of the line pulls

that on the other side. The magnitude of surface tension is defined as that of the

tensile force acting across and perpendicular to a short, straight element of the line

drawn in the surface divided by the length of that line.

Dimensional Formula: [MLT-2]/[L] = [MT-2]

The forces of attraction binding molecules to one another give rise to cohesion, the

tendency of the liquid to remain as one assemblage of particles rather than to behave as

a gas and fill the entire space within which it is confined. On the other hand, forces

between the molecules of a fluid and the molecules of a solid boundary give rise to

adhesion between the fluid and the boundary. It is the interplay of these two forces that

determine whether the liquid will wet the solid surface of the container. If the

adhesive forces are greater than the cohesive forces, then the liquid will wet the

surface; if vice versa, then the liquid will not. It is rare that the attraction between

molecules of the liquid exactly equals that between molecules of the liquid and moleculesof the solid and so the liquid surface near the boundary becomes curved.


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Fig.5 Forces acting on half of a liquid drop.

The pressure inside a drop of fluid can be calculated using the free-body diagram in Fig.

5. If the spherical drop is cut in half as shown the force developed around the edge due

tosurface tension is 2 R. This force must be balanced by the pressure difference, pbetween the internal pressure, pi, and the external pressure, pe, acting over the circular

area,R2

Thus,

2R = pR2

p = pi pe = 2/R

It is apparent from this result that the pressure inside the drop is greater than the

pressure surrounding the drop.

For a curved surface, the resultant surface tension forces is towards the concave side.

For equilibrium, the pressure on the concave side must be greater than that on the

convex side by an amount equal to

21 R

1

R

1

where, R1 and R2 are the surface radii of curvature in two perpendicular directions.

The capillarity phenomenon is due to the rise or depression of the meniscus of the liquiddue to the action of surface tension forces.

The water column in the sketch below rises to a height hsuch that the weight of the

column is balanced by the resultant surface tension forces acting at to the vertical at

the contact with the tube.


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And from equilibrium of forces,

gd

cos4h

iv.Vapour Pressure

At the surface of a liquid, molecules are leaving and re-entering the liquid mass. The

activity of the molecules at the surface creates a vapour pressure, which is a measure of

the rate at which the molecules leave the surface. When the vapour pressure of the

liquid is equal to the partial pressure of the molecules from the liquid which are in the gas

above the surface, the number of molecules leaving is equal to the number entering. At

this equilibrium condition, the vapour pressure is known as the saturation pressure.

The vapour pressure depends on the temperature, because molecular activity depends

upon heat content. As the temperature increases, the vapour pressure increases unti

boiling is reached for the particular ambient atmospheric pressure.

Dimensional Formula: [ML-1T-2]

An important reason for our interest in vapor pressure and boiling lies in the common

observation that in flowing fluids it is possible to develop very low pressure due to the

fluid motion, and if the pressure is lowered to the vapor pressure, boiling will occur. For

example, this phenomenon may occur in flow through the irregular, narrowed passages of

a valve or pump. When vapor bubbles are formed in a flowing fluid they are swept along

into regions of higher pressure where they suddenly collapse with sufficient intensity to

actually cause structural damage. The formation and subsequent collapse of vapor

bubbles in a flowing fluid, called cavitation, is an important fluid flow phenomenon to be

given further attention.

h

d


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v. Viscosity

Viscosity can be thought of as the internal stickiness of a fluid. It is one of the

properties that controls the rate at which fluids are transported in a pipeline, or along a

channel. It provides friction during flow with the walls of the container. It accounts for

the energy losses associated with the transport of fluids in ducts, channels and pipes.Further, viscosity plays an important role in the generation of turbulence. Needless to

say, viscosity is an extremely important fluid property in our study of fluid flows.

All real fluids resist any force tending to cause one layer to move over another, but the

resistance occurs only when the movement is taking place. On removal of the externa

force, flow subsides because of the resisting forces both between layers and between

the fluid and walls of the container. But unlike solids that may return to their origina

position, the fluid particles stay in the position they have reached and have no tendency

to return to their original positions. The resistance to the movement of one layer offluid over an adjoining one is due to the viscosity of the fluid.

Causes of Viscosity

To understand the causes of viscosity of a fluid, consider the observed effects of

temperature on the viscosity of a gas and a liquid. It has been noted that for gases

viscosity increases with increasing temperature and for liquids, viscosity decreases with

increasing temperature. The reason for this is that viscosity appears to depend on two

phenomena, namely the transfer of momentum between molecules and the intermolecular

(cohesive) forces between molecules of the fluid.

Consider a fluid consisting of two layers aa and bb as shown below, with the layer aa

moving more rapidly than bb. Some molecules in aa owing to their thermal agitation wil

migrate to bb and take with them the momentum they have as a result of the overal

velocity of aa. These molecules on colliding with molecules in the bb layer transfer their

momentum resulting in an overall increase in the velocity of bb. In turn, molecules from

bb, also owing to thermal agitation cross over to layer aa and collide with molecules there

The net effect of the crossings is that the relative motion between the two layers is

reduced: layer aa is slowed down because of the collision with the slower molecules; layerbb is accelerated because of collision with faster molecules.

Now use this to explain why it is observed that viscosity of a gas increases with

increasing temperature.

With a liquid, transfer of momentum between layers also occurs as molecules move

between the two layers. However, what is different from the gas is the strong


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intermolecular forces in the liquid. Relative movement of layers in a liquid modifies these

intermolecular forces, thereby causing a net shear force that resists the relative

movement. The effect of increasing the temperature is to reduce the cohesive forces

while simultaneously increasing the rate of molecular interchange. The net effect of

these two in liquids is a decrease in viscosity.

Quantitative Definition of ViscosityThe experiment described in Figs. 1,2 &3 for defining a fluid showed the deformation of

a fluid with constant pressure. See the first figure below. Here the top boundary was

moving with constant velocity and due to the NO SLIP condition, the velocity profile

shows zero (0) at the bottom and U at the top (equal to that of the plate). Other flow

conditions are given below.

Fig 4. Movement of a fluid between plates (top plate moving, bottom plate stationary).

Insert the velocity profile next to each figure.

(a)

y

c

a

b

d

b c

F

H u

U

y

b

a a

b

Faster

U m/s


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(b)

(c)

From experiments with various fluids, Sir Isaac Newton postulated that for the straight

and parallel motion of a given fluid, the tangential stress between two adjoining fluid

layers is proportional to the velocity gradient in a direction perpendicular to the layers

That is:

yu

(1)

where is a constant for a particular fluid at a particular temperature. The coefficient

of proportionality is the absolute viscosity (sometimes referred to as the coefficient of

viscosity). Note that is a scalar quantity, while the other terms are vector quantities.

Note also that the surface over which the stress acts is perpendicular to the velocity

gradient Fig 4. If the velocity u increases with y, then the velocity gradient is positive

and so also must be positive. So the positive sense of the shear stress is defined as

being the same as the positive sense of the velocity.

In accordance with Eq. 1, plots of versus du/dyshould be linear with the slope equal to

the viscosity as illustrated in Fig. 5.

Direction

of flowH

y

x

y

x

H

Direction

of flow

0 m/s


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.

Fig 5. Linear variation of shear stress with rate of shear strain.

Fluids for which the shearing stress is linearly relatedto the rate of shearing strain (also

referred to as rate of angular deformation) are designated as Newtonian fluids. Fluids

for which the shearing stress is not linearly relatedto the rate of shearing strain

are designated as non-Newtonian fluids.

From these definitions, the dimensional formula for viscosity is:

Dimensional Formula:

Kinematic Viscosity:

The kinematic viscosity, , is defined as the ratio of absolute viscosity to density:


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(2)

Dimensional formula: [L-2T-1]

The interest in expressing this ratio will become clearer in discussion on Reynolds numberand its use in turbulent and laminar flows where the ratio of viscous forces, (which is

proportional to ), to the inertial forces (which is proportional to ) is involved.

vi.Pressure

To define pressure, consider some imaginary surface of area A at an arbitrary part of a

fluid. This surface must experience forces, say of magnitude F, due to a very large

number of molecular collisions from the fluid adjoining it. Pressure, which is a scalar

quantity, is defined as the ratio of the force and the area, that is F/A.

Dimensional Formula is: [ML-1T-2]

The units are: the pascal (Pa) N/m2. Sometimes pressures of large magnitude are

expressed in atmospheres (atm). One atmosphere is taken as 1.03125 x 105 Pa. A

pressure of 105 is called a bar. For pressures less than that of the atmosphere, the

units are normally expressed as millimetres of mercury vacuum.

Pascals LawIt is important to realize that fora fluid having no shear forces, the direction of the

plane over which the force due to pressure acts has no effect on the magnitude of the

pressure at a point. This result is known as the Pascals Law and its derivation is given

below.

Point P in a fluid having

pressure p. The pressure has

the same magnitude regardless

of which plane the force due to

the pressure acts.

Imagine a small prism (element of fluid) with plane faces and triangular section

surrounding the point, P, in question experiencing external forces normal to the faces (F1

F2, F3). The rectangular face ABBA is assumed vertical and the rectangular face BBCC

is horizontal, and the face AACC slants at an arbitrarily defined angle to the horizontal

X

a

a

b

b c

c

d

d


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Assume the mean density to be . The most general case is for a fluid accelerating with

an acceleration component in the x and y direction being ax and ay respectively. Note the

fluid accelerates as a whole body with no relative motion between its layers. That is to

say that no shear forces are acting.

This means that the forces on the two end faces, ABC and ABC are acting onlyperpendicular to these faces.

Resolving in the horizontal direction, we get:

ABLppAcosACLpABLp 3131 (3)

because AC cosA = AB.

From Newtons Second Law, the net force is equal to the product of the mass of the fluidand the mean acceleration in the horizontal direction.

Therefore,

x31 aBCABL2

1ABLpp

(4)

That is,

x31 aBC21pp

(5)

If the size of the prism is reduced so that it converges on the then its dimensions

approach zero. Therefore, the right hand side of the above equation tends to zero.

So,

31 pp (6)

mg

F1

F3

F2

Pp1

A

A

BB C

C

p2

p3

L


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Now consider the forces acting on the prism in the y (vertical) direction, and they are

due to the weight and pressure. The resultant of these forces is the product of the

mass of the prism and its acceleration in the y direction. So,

y23 aBCABL21BCLpgBCABL

21CcosACLp

(7)

which becomes after rearrangement:

gaAB2

1pp y23 (8)

In the limit as the prism converges to the point P, the length AB approaches zero, and

hence the right side of the above equation approaches zero. So from this,

23 pp (9)

So from the above,

321 ppp (10)

Now recall that the direction of the sloping face, AACC was arbitrarily chosen.

Therefore, the results above will be valid for any value of the angle ACB.

Also, the plane ABBA may face any point of the compass and so we may conclude that:

The pressure is independent of the direction of the

surface used to define it.

6. EQUATION OF STATE

A perfect gas is one in which its molecules behave like tiny, perfectly elastic spheres in

random motion, and would influence each other only when collided.

The kinetic theory of gases, which is based on perfect gases, states that for equilibrium

conditions, the absolute pressure, p, the volume V occupied by mass m, and the absolute

temperature T would be related as follows:

mRTpV (11)

or


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RTp (12)

where is the density and R the gas constant whose value depends on the gas concerned.

Any equation relating p, and T is known as the equation of state. Note that the

equation of state is valid even when the gas is not in mechanical or thermal equilibrium.

The dimensional formula for R can be derived as follows:

M

FL

L

M

LF

:T

p

3

2

where [F] is the dimensional symbol for force and [] is the for temperature. The unitsare in J/kg K.

Universal gas constant The product of the relative molecular mass, M and the gas

constant R. This value is constant for all perfect gases.

Isothermal process: change of density of a gas occurring such that the temperature

remains constant.

Adiabatic process: change of density of a gas occurring with no heat transfer to or fromthe gas.

Isentropic process: If in addition to the adiabatic process, no heat is generated within

the gas, say, by friction, then the process is isentropic. The absolute pressure and

density of a perfect gas are related by the additional expression:

ttanconsp (13)

where = cp/cv, cp andcv being the specific heat capacities at constant pressure andconstant volume respectively.


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EXAMPLE PROBLEMS

Eg. #1

. A Newtonian fluid having a viscosity of and a specific gravity (SG) of 0.91 flows

through a 25-mm-diameter pipe with a velocity of 2.6m/s. Determine the value of the

Reynolds number using SI units.Reynolds number,Re, is defined as VD/ where is the

fluid density, V the mean fluid velocity, D the pipe diameter, and the fluid viscosity.

The fluid density is calculated from the specific gravity as

= SG * [H2O@4oC] = 0.91 *1000 kg/m

3 = 910 kg/m3

and from the definition of the Reynolds number

=156 (kg.m/s2)/N

However, since 1 N = 1 kg.m/s2 it follows that the Reynolds number is unitlessthat

is,

Re = 156

fundamentals fluid properties

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