Application of Environment Spatial Information System CHAPTER 1 FLUID PROPERTIES

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Title: Application of Environment Spatial Information System CHAPTER 1 FLUID PROPERTIES

1
Application of Environment Spatial Information
SystemCHAPTER 1FLUID PROPERTIES
Minkasheva AlenaThermal Fluid Engineering
Lab.Department of Mechanical EngineeringKangwon
National University
2

The engineering science of fluid mechanics has
been developed through an understanding of 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 (stream,
river, sewer line, pipe line, groundwater, sea
water, overland flow).
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 (phase,
water, air, solid phase).
The property of vapor pressure, which accounts
for changes of phase from liquid to gas, becomes
important when reduced pressures are encountered.
We will use International System of Units (SI) of
force, mass, length, and time units.

3
1.1 DEFINITION OF A FLUID

A fluid is a substance that deforms continuously
when subjected to a shear stress, no matter how
small that shear stress may be.
Shear stress Tangential (Shear )Force / Area
Normal Stress (pressure) Normal Force /Area
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.

4
In figure 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 F/A
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.
Figure 1.1 Deformation resulting from application
of constant shear force
5

The fluid in immediate contact with a solid
boundary has the same velocity as the boundary
(no slip at the boundary).
In the figure 1.1 the fluid in the area abcd
flows to the new position ab'c'd, 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 show that, other quantities being
held constant, F is directly proportional to A
and to U and is inversely proportional to
thickness t. In equation form
Where µ is the proportionality factor and
includes the effect of the particular fluid.

If t F/A for the shear stress,
The ratio U/t is 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 du/dy
is more general.
The velocity gradient du/dy may also be
visualized as the rate at which one layer moves
relative to an adjacent layer, in differential
form,
- Newton's law of viscosity
(1.1.1)
is the relation between shear stress and rate of
angular deformation for one-dimensional flow of a
fluid.
The proportionality factor µ is called 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. Hence, plastics and
solids are excluded from the classification of
fluids.

Fluids may be classified as
Newtonian
non-Newtonian.
In Newtonian fluid there is a linear relation
between the magnitude of applied shear stress and
the resulting rate of deformation µ constant in
Eq. (1.1.1), as shown in Fig. 1.2.
In non-Newtonian fluid there is 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 relation of t to du/dy.
A thixotropic substance, such as printer's 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.

9
Figure 1.2 Rheological diagram
10

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. 1.2.

11
1.2 FORCE, MASS, LENGTH, AND TIME UNITS

Consistent units of force, mass, length, and
time
greatly simplify problem solutions in mechanics
derivations may be carried out without reference
to any particular consistent system.
A system of mechanics units is said to be
consistent when unit force causes unit mass to
undergo unit acceleration.
The International System (SI)
newton (N) as unit of 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
Newton's second law of motion
(1.2.1)
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
(1.2.2)
For example, where g 9.876 m/s2, a body with
gravity force of 10 N has a mass m 10/9.806 kg.
At the location where g 9.7 m/s2, the force of
gravity is
Standard gravity is 9.806 m/s2. Fluid properties
are often quoted at standard conditions of 4oC
and 760 mm Hg.

13
Table 1.1 Selected prefixes for powers of 10 in
SI units
14
1.3 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.
Newton's law of viscosity Eq. (1.1.1) 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 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 appears to be the 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.

As a 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 (see Fig.
1.3).
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.

17
Figure 1.3 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. 1.3.
The measure of the motion of one layer relative
to an adjacent layer is du/dy.

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 du/dy is zero throughout the
fluid.
Hence, in the study of fluid statistics, no shear
forces considered, and the only stresses
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.

The dimensions of viscosity are determined from
Newton's law of viscosity Eq. (1.1.1). 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 Newton's 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 Pa s) or kilograms per
meter-second (kg/m s), has no name.

20
Kinematic Viscosity

µ - absolute viscosity or the dynamic viscosity
? - kinematic viscosity (the ratio of viscosity
to mass density)
(1.3.1)
? occurs in many applications (e.g., in the
dimensionless Reynolds number for motion of a
body through a fluid, Vl/?, in which V is the
body velocity and l is a representative linear
measure or the body size).
The dimensions of ? are L2T-1. The SI unit of
kinematic viscosity is 1 m2/s, and has no name.
Viscosity is practically independent of pressure
and depends upon temperature only. The kinematic
viscosity of liquids, and of gases at a given
pressure, is substantially a function of
temperature.

Example 1.1
A liquid has a viscosity or 0.005 Pa s and a
density of 850 kg/m3. Calculate the kinematic
viscosity

Example 1.2
In Fig. 1.4 the rod slides inside a concentric
sleeve with a reciprocating motion due to the
uniform motion of the crank. The clearance is d
and the viscosity µ. Write a program in BASIC to
determine the average energy loss per unit time
in the sleeve. D 0.8 in, L 8.0 in, d 0.001
in, R 2 ft, r 0.5 ft, µ 0.0001 lb s/ft2,
and the rotation speed is 1200 rpm.
Solution 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 2p/?,
where ? d?/dt. 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 f, we get
Figure 1.5 lists the program, in which the
variable RR represents the crank radius r.

23
Figure 1.4 Notation for sleeve motion
24
Figure 1.5 BASIC program to determine loss in
sleeve motion
25
1.4 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.
For 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-1/3 cm.

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

27
1.5 DENSITY, SPECIFIC VOLUME, UNIT GRAVITY FORCE,
RELATIVE DENSITY, PRESSURE

The density ? of a fluid is defined as its mass
per unit volume. To define density at a point,
the mass ?m of fluid in a small volume ?V
surrounding the point is divided by ?V and the
limit is taken as ?V becomes a value e3 in which
e is still large compared with the mean distance
between molecules,
(1.5.1)
For water at standard pressure (760 mm Hg) and
4oC, ? 1000 kg/m3.
(1.5.2)
The specific volume vs is the reciprocal of the
density ? that is, it is the volume occupied by
unit mass of fluid.

The unit gravity force, ? - the force of gravity
per unit volume. It changes with location
depending upon gravity.
(1.5.3)
Water ? 9806 N/m3 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.
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 container will exert a reaction
on the fluid which will be compressive.
Pressure has the units force per area, which is
newtons per square metre, called pascals (Pa).

29
1.6 PERFECT GAS

In this treatment, thermodynamic relations and
compressible-fluid-flow cases have been limited
generally to perfect gases.
The perfect gas is defined as substance that
satisfies the perfect-gas-law
(1.6.1)
and that has constant specific heats.
p is the absolute pressure vs is the specific
volume R is the gas constant and 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. (1.6.1).
Eq. (1.6.1) is the equation of state for a
perfect gas may be written
(1.6.2)
The units of R can be determined from the
equation when the other units are known.

Real gases below critical pressure and above the
critical temperature tend to obey the perfect-gas
law. As the pressure increases, the discrepancy
increases and becomes serious near the critical
point.
The perfect-gas law encompasses both Charles' law
and Boyle's law.
Charles' law states that for constant pressure
the volume of a given mass of gas varies as its
absolute temperature.
Boyle's law (isothermal law) states for constant
temperature the density varies directly as the
absolute pressure.

The volume V of m mass units of gas is mvs ?
(1.6.3)
With being the volume per mole, the
perfect-gas law becomes
(1.6.4)
If n is the number of moles of the gas in volume
V,
(1.6.5)
The product MR, called the universal gas
constant, has a value depending only upon the
units employed. It is
(1.6.6)
The gas constant R can then be determined from
(1.6.7)
? knowledge of relative molecular mass leads to
the value of R.

The specific heat cv of a gas is the 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 is 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 cp/cv.
The intrinsic energy u (dependent upon P, ? and
T) is the energy per unit mass due to molecular
spacing and forces.
The enthalpy h is important property of a gas
given by h u P/?.
cv and cp have the units joule per kilogram per
kelvin (J/kg?K). 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 1.3
A gas with relative molecular mass of 44 is at a
pressure of 0.9 MPa and a temperature of 20oC.
Determine its density.
Solution
From Eq. (1.6.7),
Then, from Eq. (1.6.2),

35
1.7 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. Liquid (and
gas) compressibility also becomes important when
temperature changes are involved, e.g., 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 -dp/dV is the bulk modulus of
elasticity K.
For any volume V of liquid,
K expressed in units of p. For water at 20oC K
2.2 GPa.

Example 1.4
A liquid compressed in a cylinder has a volume
of 1 liter (L 1000 cm3) at 1 MN/m2 and volume
of 995 cm3 at 2 MN/m2. What is its bulk modulus
of elasticity?

37
1.8 VAPOR PRESSURE

Liquids evaporate because or molecules escaping
from the liquid surface. The vapor molecules
exert partial pressure in the space, known as
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.
The vapor pressure of a given fluid depends upon
temperature and increases with it. When the
pressure above a liquid equals the vapor pressure
of the liquid, boiling occurs.
At 20oC water vapor pressure - 2.447 kPa
mercury vapor pressure - 0.173 Pa
When very low pressures are produced at certain
locations in the system, pressures may be equal
to or less than the vapor pressure ? the liquid
flashes into vapor. This is the phenomenon of
cavitation.

38
1.9 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
0.074 N/m at 20oC to 0.059 N/m at 100oC (Table
1.2)

39
Table 1.2 Approximate properties of common
liquids at 20oC and standard atmospheric pressure
40

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 s
calculated in terms of the forces which act on a
hemispherical free body,
For the cylindrical liquid jet of radius r, the
pipe-tension equation applies
Both equations show that 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
liquid and solid to cohesion of the liquid.
A liquid that wets the solid has a greater
adhesion than cohesion. The action of surface
tension in this case is to 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 1.4 shows the capillary rise for water and
mercury in circular glass tubes in air.