Chapter 2 Notes Fluid Mechanics, 1 Edition Tom Barwell
Properties of Fluids
Intensive properties are those that are independent of the mass of a system i.e. temperature, pressure and density.
Extensive properties are those whose values depend on the size of the system. Generally uppercase letters are used
to denote extensive properties and lowercase letters are used for intensive properties. Extensive properties per unit
mass are called specific properties. The number of properties required to fix the state of a system is given by the
state postulate: the state of a simple compressible system is completely specified by two independent, intensive
A continuum is a continuous, homogeneous matter with no holes. The continuum idealization allows us to treat
properties as point functions and to assume the properties vary continually in space with no jump discontinuities.
This idealization is valid as long as the size of the system we deal with is large relative to the space between the
Density and specific gravity
Density is defined as mass per unit volume:
The reciprocal of density is the specific volume which is defined as volume per unit mass. The density of a substance
depends on temperature and pressure.
The density of most gases is proportional to pressure and inversely proportional to temperature.
Liquids and solids are essentially incompressible substances and the variation of their density with pressure is
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. That is
The specific gravity of a substance is a dimensionless quantity. However in SI units the numerical value of the specific
gravity of a substance is exactly equal to its density. Substances with specific gravities less than 1 are lighter than
water, and thus will float on water.
The weight of a unit volume of a substance is called specific weight and is expressed as:
(N/m ) where G is gravitational acceleration.
Recall that densities of liquids are essentially constant and thus they can often be approximated as being
incompressible substances during most processes.
Density of Ideal Gases
Any equation that relates the pressure, temperature and density (or specific volume) of a substance is called an
equation of state. The simplest is the ideal-gas equation of state:
Where P is absolute pressure, v is specific volume, T is thermodynamic temperature, is density and R is gas
The gas constant R is different for each gas and is determined from
Where R iU the universal gas constant which equals 8.314kJ/kmol K and M is the molar (molecular) mass.
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In SI the thermodynamic temperature scale is the Kelvin Scale (K) and the English system uses the Rankine scale (R)
Any gas that obeys the ideal-gas equation of state (or ideal-gas relation) is called an ideal gas. For an ideal gas of
Volume V, mass m and number of moles N =m/M the ideal gas equation of state can also be written as:
For a fixed mass m, the properties of an ideal gas at two different states are related to each other by:
An ideal gas is a hypothetical substance that obeys the ideal gas relation; however it has been experimentally
observed that it closely approximates the P-v-T behaviour of real gases at low densities. At low pressures and high
temperatures the density of a gas decreases and the gas behaves much like an ideal gas.
Some familiar gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon and krypton and even heavier
gases such as carbon dioxide can be treated as ideal gases with neglible error. Dense gases such as water vapour and
refrigerant vapour however should not be treated as ideal gases since they usually exist at a state near saturation.
Vapor Pressure and Cavitation
Temperature and pressure are dependent properties for pure substances during phase change processes, and there
is one-to-one correspondence between temperatures and pressures. At a given pressure the temperature at which a
pure substance changes phase is called the saturation temperature T . Similiarly the pressure at which a pure
substance changes phase is called the saturation pressure P . Thsatapor pressure P of a puve substance is defined
as the pressure exerted by its vapour in phase equilibrium with its liquid at a given temperature.
Vapor pressure is a property of the pure substance and turns out to be identical the to the saturation pressure of the
liquid (P =P ).
Partial pressure is defined as the pressure of a gas or vapour in a mixture with other gases. The partial pressure of a
vapour must be less than or equal to the vapour pressure if there is no liquid present. However, when both vapour
and liquid are present and the system is in phase equilibrium the partial pressure of the vapour must equal the
vapour pressure and the system is said to be saturated. For phase-change processes between the liquid and vapour
phases of a pure substance, the saturation pressure and the vapour pressure are equivalent since the vapour is pure.
Note that the pressure would be the same whether it is measured in the vapour or liquid phase, provided that it is
measured at a location close to the liquid-vapor interface to avoid hydrostatic effects). Vapor pressure increases with
The reason for our interest in vapour pressure is the possibility of the liquid pressure in liquid-flow systems dropping
below the vapour pressure at some locations, and the resulting unplanned vaporization. For example, water at 10°C
will flash into vapour and form bubbles at locations (such as the tip of impellers or suction sides of pumps) where the
pressure drops below 1.23kPa. The vapour bubbles (Cavitation bubbles since they create cavities in the liquid)
collapse as they are swept away from the low-pressure regions, generating highly destructive, extremely high
pressure waves. This phenomenon is called cavitation and it is an important consideration in the design of hydraulic
turbines and pumps. Cavitation must be avoided or at least minimized in flow systems since it reduces performance,
generates annoying vibrations and noise and causes damage to equipment. It can be characterised by a tumbling
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Energy and specific heats
Total Energy E The summation of all other forms of energy within a given system
Internal Energy U The sum of all microscopic forms of energy of a system
Kinetic Energy Ke The energy that a system possesses as a result of its motion relative to some reference
frame, when all parts of the system move with the same velocity Ke = V /2 2
Potential Energy Pe The energy that a system possesses as a result of its elevation in a gravitational field can
be expressed as pe = gz (g = Gravity, z = elevation above an arbitrary axis)
Thermal Energy The sensible and latent forms of internal energy (heat) simililarly known as thermal
The international unit of energy is the Joule or kilojoule.
In the analysis of systems that involve fluid flow we frequently encounter the combination of properties u and Pv.
For convenience, this combination is called Enthalpy, h.
where ( ). Which is the energy per unit mass needed to move the fluid and maintain
flow. In the energy analysis of flowing fluids, it is convenient to treat the flow energy as part of the energy of the fluid
and to represent the microscropic energy of a fluid stream by enthalpy. Enthalpy is a quantity per unit mass, and thus
it is a specific property.
In the absence of magnetic, electric and surface tension a system is called a simple compressible system. The total
energy of a simple compressible system consists of three parts: Internal, kinetic and potential energies. Then the
total energy of a flowing fluid on a unit-mass basis becomes.
(kJ/kg) where h = is the enthalpy.
V is velocity, z is elevation.
By using enthalpy instead of the internal energy to represent the energy of a flowing fluid, one does not need to be
concerned about the flow work. The energy associated with pushing the fluid is automatically taken care of by
The differential and finite changes in the internal energy and enthalpy of an ideal gas can be expressed in terms of
the specific heats as and
Where C avd C arp the constant-volume and constant pressure specific heats of the ideal gas. Using specific heat
values at the average temperature, the finite changes in internal energy and enthalpy can be expressed
approximately as and
For liquids and are identical therefore and the change in the internal energy of liquids can be
expressed as .
Enthalpy change can be written as
Therefore for constant pressure processes and for constant-temperature processes of
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Coefficient of compressibility
To determine the amount of volume change you need to define properties that relate volume changes to the
changes in pressure and temperature. Two such properties are the bulk modulus of elasticity κ and the coefficient
of volume expansion β. It is known that fluids act like solids with respect to pressure. Therefore the coefficient of
compressibility κ (also called the bulk modulus of compressibility or bulk modulus of elasticity) for fluids is:
( ) ( ) ( )
In terms of finite changes it can be expressed as
( ) ( ) ( )
( ) ( )
( ) ( )are dimensionless, must have the dimension of pressure. Also, the coefficient of compressibility
represents the change in pressure corresponding to a fractional change in volume or density of the fluid while the
temperature remains constant. Then it follows that the coefficient of compressibility of a truly incompressible
substance (v = constant) is infinity.
A large coefficient of compressibility ( ) indicates that a large change in pressure is needed to cause a small
fractional change in volume, and thus a fluid with a laris essentially incompressible.
Note that volume and pressure are inversely proportional.
Also differentiating pressure
that Is, the fractional changes in the specific volume and the d