alinik
۱۳ مهر ۱۳۸۸, ۱۷:۰۴
Gases are highly compressible in comparison to liquids, with changes in gas density directly
related to changes in pressure and temperature through the equation
(1.8)
where p is the absolute pressure, the density, T the absolute temperature,2 and R is a gas
constant. Equation 1.8 is commonly termed the ideal or perfect gas law, or the equation of
state for an ideal gas. It is known to closely approximate the behavior of real 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 1real or imaginary2 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
and in BG units is expressed as 1psf2 or 1psi2 and in SI units as In SI,
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 1a pressure that would only occur
in a perfect vacuum2. Standard sea-level atmospheric pressure 1by international agreement2
is 14.696 psi 1abs2 or 101.33 kPa 1abs2. For most calculations these pressures can be rounded
to 14.7 psi and 101 kPa, respectively. In engineering it is common practice to measure pressure
relative to the local atmospheric pressure, and when measured in this fashion it is called gage
pressure. Thus, the absolute pressure can be obtained from the gage pressure by adding the
value of the atmospheric pressure. For example, a pressure of 30 psi 1gage2 in a tire is equal
to 44.7 psi 1abs2 at standard atmospheric pressure. Pressure is a particularly important fluid
characteristic and it will be discussed more fully in the next chapter.
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
gases are 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. More
complete tables for air at standard atmospheric pressure can be found in Appendix B 1Tables
B.3 and B.42.The properties of density and specific weight are measures of the “heaviness” of a fluid. It
is clear, however, that these properties are not sufficient to uniquely characterize how fluids
behave since two fluids 1such as water and oil2 can have approximately the same value of
density but behave quite differently when flowing. There is apparently some additional property
that is needed to describe the “fluidity” of the fluid.
To determine this additional property, consider a hypothetical experiment in which a
material is placed between two very wide parallel plates as shown in Fig. 1.2a. The bottom
plate is rigidly fixed, but the upper plate is free to move. If a solid, such as steel, were placed
between the two plates and loaded with the force P as shown, the top plate would be displaced
through some small distance, 1assuming the solid was mechanically attached to the plates2.
The vertical line AB would be rotated through the small angle, to the new position
We note that to resist the applied force, P, a shearing stress, would be developed at the
plate-material interface, and for equilibrium to occur P tA where A is the effective upperplate area 1Fig. 1.2b2. It is well known that for elastic solids, such as steel, the small angular
displacement, 1called the shearing strain2, is proportional to the shearing stress, that is
developed in the material.
What happens if the solid is replaced with a fluid such as water? We would immediately
notice a major difference. When the force P is applied to the upper plate, it will move
continuously with a velocity, U 1after the initial transient motion has died out2 as illustrated
in Fig. 1.3. This behavior is consistent with the definition of a fluid—that is, if a shearing
stress is applied to a fluid it will deform continuously. A closer inspection of the fluid motion
between the two plates would reveal that the fluid in contact with the upper plate moves with
the plate velocity, U, and the fluid in contact with the bottom fixed plate has a zero velocity.
The fluid between the two plates moves with velocity that would be found to vary
linearly, as illustrated in Fig. 1.3. Thus, a velocity gradient, is developed
in the fluid between the plates. In this particular case the velocity gradient is a constant since
but in more complex flow situations this would not be true. The experimental
observation that the fluid “sticks” to the solid boundaries is a very important one in fluid
mechanics and is usually referred to as the no-slip condition. All fluids, both liquids and
gases, satisfy this condition.
In a small time increment, an imaginary vertical line AB in the fluid would rotate
through an angle, so that
Since it follows that
We note that in this case, is a function not only of the force P 1which governs U2 but also
of time. Thus, it is not reasonable to attempt to relate the shearing stress, to as is done
for solids. Rather, we consider the rate at which is changing and define the rate of shearing
strain, as
which in this instance is equal to
A continuation of this experiment would reveal that as the shearing stress, is increased
by increasing P 1recall that 2, the rate of shearing strain is increased in direct
proportion—that is,
t PA
t,
g #
U
b
du
dy
g #
lim
dtS0
db
dt
g #
,
db
t, db
db
db
U dt
b
da U dt
tan db db
da
b
db,
dt
related to changes in pressure and temperature through the equation
(1.8)
where p is the absolute pressure, the density, T the absolute temperature,2 and R is a gas
constant. Equation 1.8 is commonly termed the ideal or perfect gas law, or the equation of
state for an ideal gas. It is known to closely approximate the behavior of real 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 1real or imaginary2 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
and in BG units is expressed as 1psf2 or 1psi2 and in SI units as In SI,
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 1a pressure that would only occur
in a perfect vacuum2. Standard sea-level atmospheric pressure 1by international agreement2
is 14.696 psi 1abs2 or 101.33 kPa 1abs2. For most calculations these pressures can be rounded
to 14.7 psi and 101 kPa, respectively. In engineering it is common practice to measure pressure
relative to the local atmospheric pressure, and when measured in this fashion it is called gage
pressure. Thus, the absolute pressure can be obtained from the gage pressure by adding the
value of the atmospheric pressure. For example, a pressure of 30 psi 1gage2 in a tire is equal
to 44.7 psi 1abs2 at standard atmospheric pressure. Pressure is a particularly important fluid
characteristic and it will be discussed more fully in the next chapter.
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
gases are 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. More
complete tables for air at standard atmospheric pressure can be found in Appendix B 1Tables
B.3 and B.42.The properties of density and specific weight are measures of the “heaviness” of a fluid. It
is clear, however, that these properties are not sufficient to uniquely characterize how fluids
behave since two fluids 1such as water and oil2 can have approximately the same value of
density but behave quite differently when flowing. There is apparently some additional property
that is needed to describe the “fluidity” of the fluid.
To determine this additional property, consider a hypothetical experiment in which a
material is placed between two very wide parallel plates as shown in Fig. 1.2a. The bottom
plate is rigidly fixed, but the upper plate is free to move. If a solid, such as steel, were placed
between the two plates and loaded with the force P as shown, the top plate would be displaced
through some small distance, 1assuming the solid was mechanically attached to the plates2.
The vertical line AB would be rotated through the small angle, to the new position
We note that to resist the applied force, P, a shearing stress, would be developed at the
plate-material interface, and for equilibrium to occur P tA where A is the effective upperplate area 1Fig. 1.2b2. It is well known that for elastic solids, such as steel, the small angular
displacement, 1called the shearing strain2, is proportional to the shearing stress, that is
developed in the material.
What happens if the solid is replaced with a fluid such as water? We would immediately
notice a major difference. When the force P is applied to the upper plate, it will move
continuously with a velocity, U 1after the initial transient motion has died out2 as illustrated
in Fig. 1.3. This behavior is consistent with the definition of a fluid—that is, if a shearing
stress is applied to a fluid it will deform continuously. A closer inspection of the fluid motion
between the two plates would reveal that the fluid in contact with the upper plate moves with
the plate velocity, U, and the fluid in contact with the bottom fixed plate has a zero velocity.
The fluid between the two plates moves with velocity that would be found to vary
linearly, as illustrated in Fig. 1.3. Thus, a velocity gradient, is developed
in the fluid between the plates. In this particular case the velocity gradient is a constant since
but in more complex flow situations this would not be true. The experimental
observation that the fluid “sticks” to the solid boundaries is a very important one in fluid
mechanics and is usually referred to as the no-slip condition. All fluids, both liquids and
gases, satisfy this condition.
In a small time increment, an imaginary vertical line AB in the fluid would rotate
through an angle, so that
Since it follows that
We note that in this case, is a function not only of the force P 1which governs U2 but also
of time. Thus, it is not reasonable to attempt to relate the shearing stress, to as is done
for solids. Rather, we consider the rate at which is changing and define the rate of shearing
strain, as
which in this instance is equal to
A continuation of this experiment would reveal that as the shearing stress, is increased
by increasing P 1recall that 2, the rate of shearing strain is increased in direct
proportion—that is,
t PA
t,
g #
U
b
du
dy
g #
lim
dtS0
db
dt
g #
,
db
t, db
db
db
U dt
b
da U dt
tan db db
da
b
db,
dt