CHAPTER 2
PROPERTIES OF FLUIDS
By
Ummikalsom Abidin
C24-316
FKM, UTM
SME 1313 Fluid Mechanics I
Introduction
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Property – any characteristic of a system
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Intensive properties
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Properties that are independent of the mass of a system
e.g temperature, pressure, density
Extensive properties
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E.g pressure P, temperature T, volume V, and mass m
Properties whose values depend on size-or-extent-of the
system e.g mass, total volume, total momentum
Specific properties
„
Extensive properties per unit mass e.g specific volume
(v=V/m) and specific total energy (e=E/m)
SME 1313 Fluid Mechanics I
Introduction
m
V
T
P
ρ
½m
½m
½V
½V
T
T
P
P
ρ
ρ
Criteria to differentiate intensive and extensive properties
SME 1313 Fluid Mechanics I
Extensive
properties
Intensive
properties
Introduction
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State postulate – the state of a simple compressible
system is completely specified by two independent,
intensive properties
SME 1313 Fluid Mechanics I
Continuum Concept of a Fluid
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„
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The number of molecules involved is immense, and the
separation between them is normally negligible by comparison
with the distances involved in the practical situation being
studied
Although the properties of a fluid arise from its molecular
structure,engineering problem are usually concerned with the
bulk behavior of fluids
Under these conditions, it is usual to consider a fluid as a
continuum - a hypothetical continuous substance – and the
conditions at a point as the average of a very large number of
molecules surrounding that point within a distance which is
large compared with the mean intermolecular distance
SME 1313 Fluid Mechanics I
Density
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Density is defined as mass per unit volume
ρ=m
(kg/m3)
V
Specific volume is defined as volume per unit mass
v = V = 1 (m3/kg)
m ρ
Density of a substance, in general depends on
temperature and pressure
SME 1313 Fluid Mechanics I
Specific Gravity
„
Specific gravity or relative density is defined as the
ratio of the density of a substance to the density of
some standard substance at a specified temperature
(usually water at 4oC, for which ρH O=1000 kg/m3)
SG = ρ
ρH2O
2
„
Specific gravity is a dimensionless quantity
SME 1313 Fluid Mechanics I
Specific Gravity
Substance
SG
Water
1.0
Blood
1.05
Seawater
1.025
Gasoline
0.7
Ethyl Alcohol
0.79
Mercury
13.6
Wood
0.3-0.9
Gold
19.2
Bones
1.7-2.0
Ice
0.92
Air (at 1 atm)
0.0013
Specific gravities of some substances at 0oC
SME 1313 Fluid Mechanics I
Specific Weight
„
Specific weight is the weight of a unit volume of a
substance
γs= ρg
(N/m3)
Where,
g = gravitational acceleration, 9.81 m/s2
SME 1313 Fluid Mechanics I
Density of Ideal Gas
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Ideal gas equation
Pv = RT
or
P = ρRT
Where,
P = absolute pressure
v = specific volume
T = thermodynamic (absolute temperature)
T(K) = T(oC) + 273.15
ρ = gas density
R = gas constant
SME 1313 Fluid Mechanics I
Vapor Pressure and Cavitation
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„
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Saturation temperature, Tsat – the temperature at
which a pure substance changes phase at a given
pressure
Saturation pressure, Psat – the pressure at which a
pure substance changes phase at a given
temperature
Vapor pressure, Pv of a pure substance – the
pressure exerted by its vapor in phase equilibrium
with its liquid at a given temperature
Partial pressure – the pressure of a gas or vapor in a
mixture with other gases
SME 1313 Fluid Mechanics I
Vapor Pressure and Cavitation
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„
„
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Possibility of the liquid pressure in liquid-flow system dropping
below the vapor pressure at some location, and resulting
unplanned vaporization
A fluid vaporizes when its pressure gets too low, or its
temperature too high. All centrifugal pumps have a required
head (pressure) at the suction side of the pump to prevent this
vaporization.
Suction cavitation occurs when the pump suction is under a low
pressure/high vacuum condition where the liquid turns into a
vapor at the eye of the pump impeller.
Cavitation must be avoided in flow systems since it reduces
performance, generates vibration and noise and causes damage
to the equipment
SME 1313 Fluid Mechanics I
Energy and Specific Heat
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Energy – mechanical, thermal, kinetic, potential,
electric, etc.
Microscopic energy – the forms of energy related to
the molecular structure of a system and the degree
of the molecular activity
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Internal energy, U – sum of all microscopic forms of energy
Macroscopic energy – energy related to motion and
the influence of some external effects such as
gravity, magnetism, electricity, surface tension
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E.g potential energy, kinetic energy
SME 1313 Fluid Mechanics I
Energy and Specific Heat
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Enthalpy, h
h = u + Pv = u + P
ρ
Where,
P = flow energy, energy per unit mass needed to
ρ move the fluid and maintain flow
SME 1313 Fluid Mechanics I
Energy and Specific Heat
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For ideal gas, the internal energy and enthalpy can be
expressed in terms of specific heat
∆u = cv,ave ∆T and
∆h = cp,ave ∆T
Where,
cv,ave = average specific heat at constant volume
cp,ave = average specific heat at constant pressure
For incompressible substance,
∆h = ∆u + ∆ P/ρ ≅ cave ∆T + ∆ P/ρ
SME 1313 Fluid Mechanics I
Coefficient of Compressibility
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„
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Coefficient of compressibility κ (also called the bulk
modulus of compressibility or bulk modulus of
elasticity) for fluids as
(Pa)
κ = - v ∂P = ρ ∂P
∂ρ T
∂V T
In term of finite changes
κ ≅ - ∆P ≅ ∆P
(T=constant)
∆v/v
∆ρ/ρ
Large κ, a large change in pressure is needed to
cause a small fractional change in volume
(incompressible e.g liquids)
SME 1313 Fluid Mechanics I
Coefficient of Compressibility
„
„
„
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For an ideal gas,
P = ρRT and (∂P/ ∂ρ)T = RT = P/ρ
And thus,
(Pa)
κideal gas = P
Coefficient of compressibility of an ideal gas is equal to its
absolute pressure and it increases with increasing pressure
Substituting κ = P into definition of the coefficient of
compressibility and rearranging gives
∆ρ = ∆P
(T=constant)
ρ
P
The percent increase of density of an ideal gas during
isothermal compression is equal to the percent increase in
pressure
SME 1313 Fluid Mechanics I
Isothermal Compressibility
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Isothermal compressibility is the inverse of the
coefficient of compressibility
α = 1 = - 1 ∂V
= 1 ∂ρ
(1/Pa)
κ
v ∂P T
ρ ∂P T
Isothermal compressibility of fluid represents the
fractional change in volume or density corresponding
to a unit change in pressure
SME 1313 Fluid Mechanics I
Coefficient of Volume Expansion
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Coefficient of volume expansion provide information
on variation of density of a fluid with temperature at
constant pressure
= - 1 ∂ρ
(1/K)
β = 1 ∂v
ρ ∂T P
v ∂T P
In term of finite changes
β ≅ ∆v/v ≅ - ∆ρ/ρ
(P=constant)
∆T
∆T
Large β for a fluid means a large density change in
density with temperature
SME 1313 Fluid Mechanics I
Coefficient of Volume Expansion
(∂v/∂T)P
(∂v/∂T)P
20oC
100 kPa
1 kg
21oC
100 kPa
1 kg
(a) Substance with a large β
20oC
100 kPa
1 kg
21oC
100 kPa
1 kg
(b) Substance with a small β
The coefficient of volume expansion is a measure of the change in volume
of a substance with temperature at constant pressure
SME 1313 Fluid Mechanics I
Coefficient of Volume Expansion
„
For ideal gas, the volume expansion coefficient at a
temperature T is
βideal gas = 1
(1/K)
T
Where,
T=absolute temperature
SME 1313 Fluid Mechanics I
Coefficient of Volume Expansion
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The combined effects of pressure and temperature
changes on volume change of a fluid can be
determined by taking the specific volume to be a
function of T and P
Differentiating v=v(T,P) and using the definitions of
compression and expansion coefficient α and β give
dv = ∂v dT + ∂v dP =(β dT – α dP) v
∂T P
∂P P
The fractional change in volume (or density) due to
changes in pressure and temperature can be
expressed approximately as
∆v = - ∆ρ ≅ β∆T - α∆P
v
ρ SME 1313 Fluid Mechanics I
Problem 2-32
Problem 2-32
Water at 1 atm pressure is compressed to
800 atm pressure isothermally. Determine the
increase in the density of water.Take the
isothermal compressibility of water to be
4.80 x 10-5 atm-1
SME 1313 Fluid Mechanics I
Example 2-3
Example 2-3
Consider water initially at 20oC and 1 atm.
Determine the final density of water (a) if it is
heated to 50oC at a constant pressure of 1
atm, and (b) if it is compressed to 100-atm
pressure at constant temperature of 20oC.
Take the isothermal compressibility of water
to be α=4.80x10-5 atm-1
SME 1313 Fluid Mechanics I
Viscosity
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Viscosity is a property that represents the internal
resistance of a fluid to motion or the ‘fluidity’
Consider a fluid layer between two very large parallel
plates separated by distance l
dA
N
Area A
N’
Force F
u=V
Velocity V
dβ
l
y
Velocity profile
M
u=0
u(y)=(y/l)V
x
Behavior of a fluid in a laminar flow between two parallel plates when the upper plate moves with
a constant velocity
SME 1313 Fluid Mechanics I
Viscosity
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A constant parallel force F is applied to the upper
plate, the lower plate is held fixed
The upper plate moves continuously at constant
velocity under the influence of F
The shear stress τ acting on the fluid layer is
F
τ=
A
Where,
A=contact area between the plate and the fluid
SME 1313 Fluid Mechanics I
Viscosity
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Fluid in contact with lower plate has zero velocity
(no-slip condition)
In steady laminar flow, the fluid velocity between the
plates varies linearly between 0 and V, and thus the
velocity profile and the velocity gradient are
y
u( y) = V
l
and
du V
=
dy l
Where,
y=vertical distance from lower plate
SME 1313 Fluid Mechanics I
Viscosity
„
The angular displacement or deformation (or shear
strain) can be expressed as
dβ ≈ tan β =
„
da
dt du
dt
=V =
l
l dy
Rearranging, the rate of deformation
dβ du
=
dt dy
SME 1313 Fluid Mechanics I
Viscosity
„
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The rate of deformation of a fluid element is
equivalent to the velocity gradient du/dy
Fluids rate of deformation (and thus the velocity
gradient) is directly to the shear stress τ
dβ
τ∝
dt
„
or
du
τ∝
dy
Newtonian fluids for fluids which its rate of
deformation is proportional to the shear stress
SME 1313 Fluid Mechanics I
Viscosity
„
In 1-D shear flow of Newtonian fluids, shear stress
can be expressed by linear relationship
du
τ = µ
dy
Where,
µ=coefficient of viscosity/dynamic(absolute)
viscosity of the fluid, N.s/m2 (or Pa.s)
Poise=0.1 Pa.s
SME 1313 Fluid Mechanics I
Viscosity
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The shear force acting on a Newtonian fluid layer is
du
F = τA = µA
dy
„
(N)
The force F required to move the upper plate at
constant velocity of V while the lower plate remains
stationary is
F = µA
V
l
(N)
SME 1313 Fluid Mechanics I
Viscosity
„
Kinematic viscosity is the ratio of dynamic viscosity to
density
µ
(m2/s)
υ =
ρ
SME 1313 Fluid Mechanics I
Viscosity
Oil
Viscosity=slope
Shear stress, τ
µ=
a
b
τ
a
=
du dy b
Water
Air
Rate of deformation, du/dy
The rate of deformation (velocity gradient) of a Newtonian fluid is proportional to
shear stress, and the constant of proportionality is the viscous
SME 1313 Fluid Mechanics I
Viscosity
Bingham plastic
Shear stress, τ
Pseudoplastic
Newtonian
Dilatant
Rate of deformation, du/dy
Variation of shear stress with the rate of deformation for Newtonian and nonNewtonian fluids (the slope of a curve at a point is the apparent viscosity of the
fluid at that point)
SME 1313 Fluid Mechanics I
Viscosity
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Viscosity is due to the internal frictional force that
develops between different layers of fluids as they
are forced to move relative to each other
Gases
„
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Viscosity is caused by molecular collisions
Viscosity of gas increases with temperature
„
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In a gas, molecular forces are negligible and the gas
molecules at high temperatures move randomly at higher
velocities
This results in more molecular collisions per unit volume
per unit time, thus greater resistance to flow
SME 1313 Fluid Mechanics I
Viscosity of Gas
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Viscosity of gases is expressed as function by the
Sutherland correlation
aT 1 / 2
µ=
1+ b T
Where,
T=absolute temperature (K)
a and b = experimentally determined constants
SME 1313 Fluid Mechanics I
Viscosity of Liquid
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Liquids
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Viscosity is due to the cohesive forces between the
molecules
At higher temperatures, molecules posses more energy, and
they can oppose the large cohesive intermolecular forces
more strongly
The liquid viscosity is approximated as
µ = a10 b /(T −c )
Where,
T=absolute temperature (K)
a, b and c = experimentally determined
constants
SME 1313 Fluid Mechanics I
Viscosity of Gas and Liquid
Viscosity
Liquids
Gases
Temperature
The viscosity of liquids decreases and the viscosity of gases increases with
temperature
SME 1313 Fluid Mechanics I
Example 2-4
Determining the Viscosity of a Fluid
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The viscosity of a fluid is to be measured by viscometer
constructed of two 40-cm-long concentric cylinders (Fig. 218).the outer diameter of the inner cylinder is 12 cm, and the
gap between the two cylinders is 0.15 cm. The inner cylinder is
rotated at 300 rpm, and the torque is measured to be 1.8 N.m.
Determine the viscosity of the fluid
SOLUTION
The torque and the rpm of a double cylinder viscometer are
given. The viscosity of the fluid is to be determined
Assumptions
1) The inner cylinder is completely submerged in oil 2) The
viscous effects on the two ends of the inner cylinder are negligible
Analysis
The velocity profile is linear only when the curvature effects are
negligible, and the profile can be approximated as being linear in this case since
l/R<<1
SME 1313 Fluid Mechanics I
Example 2-4
Determining the Viscosity of a Fluid
Stationary cylinder
R
N=300 rpm
Shaft
l
Fluid
Figure 2-18
SME 1313 Fluid Mechanics I
Surface Tension
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Liquid droplets behave like small spherical balloons filled with
the liquid, and the surface of the liquid acts like a stretched
elastic membrane under tension.
The pulling forces that causes this tensions acts parallel to the
surface and is due to the attractive forces between the
molecules of the liquid.
A molecule on the surface
A molecule inside the liquid
Attractive forces acting on a liquid molecule at the surface and deep inside the liquid
SME 1313 Fluid Mechanics I
Surface Tension
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The magnitude of the force per unit length is called
surface tension, σs (N/m)
SME 1313 Fluid Mechanics I
Capillary Effect
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Capillary effect is the rise or fall of a liquid in a smalldiameter tube inserted into liquid
Capillaries is a narrow tubes or confined flow
channels
The curved free surface of a liquid in a capillary tube
is called meniscus
Φ
Φ
a) Wetting fluid
b) Nonwetting fluid
SME 1313 Fluid Mechanics I
Capillary Effect
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Contact (or wetting) angle Φ, is defined as the angle
that the tangent to the liquid surface makes with the
solid surface at the point of contact
„
„
„
„
Φ<90o
Φ>90o
The liquid is to wet the surface
The liquid is not to wet the surface
Cohesive forces is forces between like molecules
(water-water), adhesive forces is forces between
unlike molecules (water-glass).
Water molecules are more strongly attracted to the
glass molecules than they are to other water
molecules, and thus water tends to rise along the
glass surface
SME 1313 Fluid Mechanics I