Transcript 幻灯片 1

Chapter 14
The Concepts of Pressure
I wanted to make an instrument which would show the
changes in the air, which is at times heavier and thicker, and at
times lighter and more rarefied.
Evangelista Torricelli (1644)
14.1 THE CONCEPTS
For a fluid at rest, pressure p can be defined as the force F
exerted perpendicularly by the fluid on an area A of any
bounding surface [1], that is,
dF
p 
dA
Thus pressure is seen to be basically a mechanical concept
(in the field of mechanics it is called “compressive stress”)
that can be fully described in terms of the primary dimensions
of mass, length, and time (Figure 14.1).
This definition and the following three observations
encompass the whole of pressure measurement.
1. It is a familiar fact that pressure (since it is a local property
of the fluid) is strongly influenced by position within a static
fluid, but at a given position, it is quite independent of
direction (Figure 14.2). Thus we note the expected variation in
fluid pressure with elevation dp   w  dh
FIGURE 14.1 Basic definition of fluid pressure.
By definition p=F/A, where F=Ma and A=bh. Typical units
of pressure are lbf /in2, lbf/ft2, and dyn/cm2 ,
Primary dimensions:
since F=MLT-2, and A=L2, p=MLT-2 In dimensional analysis
T is the symbol for the primary dimension of time.
FIGURE 14.2 Static fluid pressure and position.
(a) Qualitatively: fluid pressure varies with depth but is the
same in all directions a: a given depth.
(b) Quantitatively: variation in fluid pressure with elevation is
obtained by balancing forces on a static fluid element. The
following equations may be used:
F
n
0
F1  F2  W
( p  dp)dA  pdA wdAdh
dp  wdh
For a constant-density fluid
p1  p2  w(h2  h1 )
Pressure is independent of the size and shape of its
confining boundaries:
p1  p2  wh
where w is represents the specific weight of the fluid, and h
represents vertical height of the fluid. Equation (14.2)
accounts for the usefulness of manometry.
2. It also has been amply demonstrated that pressure is
unaffected by the shape of the confining boundaries. Thus a
great variety of fluid pressure transducers are available
(Figure 14.3).
3. Finally, it is well known that a pressure applied to a
confined fluid via a movable surface. Thus hydraulic lifts
and deadweight testers make their appearance (Figure 14.4).
14.2 HISTORICAL RESUME
These basic concepts, which today are generally taken for
granted, emerged only slowly over the years.
A short historical review concerning the development of the
pressure principle is given next. Much can be learned from it,
for it involves some of the great names in the physical
sciences [2]-[4].
Evangelista Torricelli (born October 15, 1608), briefly a
student of Galilee, induced his friend Viviani (another pupil of
Galilee, see Section 1.2) to experiment with mercury and
atmospheric pressure in 1643.
Inverting a glass tube, closed at one end and initially filled
with mercury, into a shallow dish also filled with mercury, the
mercury in the tube was found to sink to a level of about 30in
above the mercury level in the dish.
Torricelli realized that the atmosphere exerted a pressure on
the earth which maintained the mercury column in equilibrium.
He reasoned further that the height of mercury thus
supported varied form day to day, and he also concluded that
this height would decrease with altitude.
Incidentally, by this method of inverting the filled mercury
tube, Torricelli successfully pulled a vacuum (i.e., a region
essentially devoid of matter).
FIGURE 14.4 (a) Hydraulic lift uses force multiplication based on
undiminished pressure transmission in a confined fluid.
Basic principle of deadweight testing:
at balance of known weight the gauge pressure is p=W/A.
14.2 Historical Resume
In 1660, Robert Boyle discerned that “whatsoever is
performed in the material world, is really done by particular
bodies, acting according to the laws of motion.” He further
stated the now famous relation: “The product of the measures
of pressure and volume is constant for a given mass of air at
fixed temperature.”
And it was Boyle who first used the word barometer in print
[7]: “…consulting the barometer (if to avoid circumlocutions
I may so call the whole instrument wherein a mercurial
cylinder of 29 or 30 inches is kept suspended after the manner
of the Torricellian experiment) I found…”
Robert Hooke, at one time Boyle’s assistant, considered the
pressure of an enclosed gas as resulting from the continuous
impact of large numbers of hard, independent, fast-moving
particles on the container walls.
However, it remained for Daniel Bernoulli, in 1738, to
develop the impact theory of gas pressure to the point where
Boyle’s law could be deduced analytically.
Bernoulli also anticipated the Charles-Gay-Lussac law by
stating that pressure is increased by heating a gas at constant
volume.
In 1811, Amedeo Avogadro at Turin declared that equal
volumes of pure gases, whether elements or compounds,
contained equal numbers of molecules at equal temperatures
and pressures (see Section 2.4).
The number, later determined to be 2.69×1019
molecules/cm3at 0℃ and 1 atm, attests well to the insight of
Hooke and Bernoulli as to the very large numbers of
particles involved in a gas sample under usual conditions.
In rapid succession James Prescott Joule, Rudolf Clausius,
and James Clerk Maxwell, in the years 1847-1859, developed
the kinetic theory of gas pressure in which pressure is viewed
as a measure of the total kinetic energy of the molecules, that
is,
2 KE 1
p
  C 2  NRT
3 V
3
(14.3)
Since kinetic energies are additive, so are pressures, and this
leads to Dalton’s law (to the effect that the pressure of a
mixture is made up of the sum of the partial pressure exerted
separately by the constituents of the mixture).
If volume changes when kinetic energy (i.e., temperature)
is held constant, pressure is seen to vary inversely with
volume, which is Boyle’s law.
Alternatively, if pressure is held constant, Charles’s law
follows. Such were the immediate successes of the kinetic
theory of gas pressure.
Although we have confined our discussion so far to fluid
pressure, the concept of pressure as a result of impacts is not
so-restricted.

As Sir James Jeans [8] has pointed out, these laws are also
found to hold for
(1) osmotic pressure of weak solutions,
(2) pressure exerted by free electrons moving about in the
interstices of a conducting solid, and
(3) pressure .. by the atmosphere of electrons surrounding a
hot solid.
In this case electrons parameter N of equation (14.3)
represents the number of free electrons per unit volume.
Still another viewpoint of pressure is gained from macroscopic
thermodynamics [9].
The reversible work done by a closed system is given by
the product pdV,
which is a path function, that is, one strongly dependent on the
process joining the end states.
For the more realistic irreversible process, the internal heat
generated (i.e., the friction) is defined as
 F  pdV  Wclosed
(14.4)
where friction, like work and heat, is also inherently a path
function.
However, when equation (14.4 ) is rewritten as
dV 
W   F
(14.5)
p
We see that pressure can be viewed as an integrating factor
that transforms the “path” function W  F into the “point”
function dV. This relationship is analogous to the familiar
second law of thermodynamics
dS 
Q  F
T
where temperature serves as the integrating factor for heat
absorbed
14.3 BRIEF SUMMARY
In mechanics, pressure is force per unit area,
dF
p
(14.1)
dA
In kinetics, pressure is molecular kinetic energy per unit
volume,
2 KE
p
3 V
(14.2)
In thermodynamics, pressure is work per unit volume,
p
W   F
dV
(14.4)
All basic pressure measurements are made in accord with
equation (14.1) or equation (14.2).
Equations (14.3) and (14.4), however, serve to broaden our
understanding of the Pressure principle.