Chapter 14The 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,

dA

dFp

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 dhwdp

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:

0nF WFF 21

wdAdhpdAdAdpp )(wdhdp

For a constant-density fluid

)( 1221 hhwpp

Pressure is independent of the size and shape of its confining boundaries:

whpp 21

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,

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).

22 1

3 3

KEp C NRT

V (14.3)

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

closedF pdV W (14.4)

where friction, like work and heat, is also inherently a path function.

However, when equation (14.4 ) is rewritten as

We see that pressure can be viewed as an integrating factor that transforms the “path” function into the “point” function dV. This relationship is analogous to the familiar second law of thermodynamics

FW

where temperature serves as the integrating factor for heat absorbed

W FdV

p

(14.5)

Q FdS

T

14.3 BRIEF SUMMARY

In mechanics, pressure is force per unit area,

In kinetics, pressure is molecular kinetic energy per unit volume,

In thermodynamics, pressure is work per unit volume,

dFp

dA (14.1)

2

3

KEp

V (14.2)

W Fp

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.