5. Measurement of Pressure

5. MEASUREMENT OF PRESSURE* 5.1. Introduction

List of Symbols Cross sectional area; parameter Parameter Electrical capacitance; parameter Dilatational elastic modulus Component of electric field relative to axes of piezoelectric element emf Fringe count; wave front Gage factor for resistance strain gage Frequency response function Electrical current; light intensity; time dependent input I ( ( ) Exponential transform of input I ( t ) Zero order Bessel function of first kind of imaginary argument Zero order Bessel function of first kind with real argument Relative electrical permittivity; K = (w/c,h)”* in diaphragm theory Electrical inductance Elastic modulus unspecified Time dependent output Resonant period = 2 m / o q ; 9, longest resonant period Electrical charge Electrical resistance Unit step at time I = 0 Pressure sensitivity; surface tension Hold time of gage Time variable in addition to t

Time dependent response to unit step input S ( t ) Electrical voltage Shock wave velocity Volume Single crystal axes Young’s modulus of elasticity radius, or half-width, of elastic element Speed of electromagnetic radiation = 0.3 Gm . s-’ nominal value Velocity of strain propagation unspecified C; = Y / [ 1 2 p ( l - v’)] for linear bending diaphragm cg = Y / p strain wave speed in bar c%= E / p dilation wave speed for one-dimensional wave : C = p / p shear wave speed c: = u o / p for pretensioned diaphragm Diameter, or lateral dimension, of elastic element Stress related piezoelectric constant Frequency in Hertz (complete cycles per second) Weight per unit mass = 9.8 N . kg-’ nominal value Thickness of diaphragm; dimension of sensor in direction of strain propagation Dimension

* Part 5 is by R. I. Soloukhin, C. W. Curtis, and R. J. Emrich. 499 METHODS OF EXPERIMENTAL PHYSICS, VOL. 18B

Copyright 0 1981 by Academic Press. lnc All rights of reproduction in any form reserved

ISBN 0 12-475956-4

5 00 %n

I

P Po r

t U

UO

v

M’,W o

5.

MEASUREMENT OF PRESSURE

Unit vector, component of unit vector in ith direction i = 1,2,3 Pressure Amplitude of pressure step or of periodic pressure variation Spatial coordinate, usually radial, rectangular in case of slit diaphragm Time Longitudinal displacement for propagating strain; displacement in plane of diaphragm Maximum displacement in plane of diaphragm due to pretension Material velocity, usually in direction of propagation of strain wave Displacement perpendicular to plane of diaphragm, maximum displacement

Cartesian coordinates, usually z in direction of strain wave propagation Optical path length Sensing element Strain element Permittivity of vacuum Strain, strain components Wavelength = c/f Viscosity coefficient; shear modulus of elasticity Kinematic viscosity coefficient = p / p ; Poisson’s ratio Mass density; electrical resistivity Stress Stress component in one-, twoindex notation Response time of gage Radian frequency = 2nf

Measurement of pressure is often as simple as reading a pointer on a dial calibrated in pascals, bars, or millimeters of mercury or some other set of units in which pressure is expressed. Before proceeding to describe how such simple pressure gages operate and are used, however, it is worthwhile reviewing the inherent assumptions made in supposing such a variable as pressure has a meaning. There is more than the usual confusion among users in different professions regarding the meaning of the variable and the units in which it is measured. Certain kinds of “pressure” will not be discussed in the current part at all. 5.1.l.Mechanical Concept of Pressure

The existence of contact forces everywhere between contiguous parts of matter is one of the basic concepts of continuum mechanics. Pressure is an especially simple case of this concept. In its most general form, the concept is expressed by postulating the existence of astress tensor, which provides that nine numbers specified at a point in the matter and a specified set of rectangular coordinate axes can describe the contact force per unit area acting across uny small area at the point. Specifically the nine numbers, called components of the stress tensor (uI1,ulz, (TI37 azl, . . . , c ~ ~allow ) , calculation of the three components of the vector force per unit area P with the components (P1, Pz , P3) for an area whose orientation is described by a unit vector ii normal to it with components

5.1. INTRODUCTION

50 I

(nl , n 2 , n3). The calculation is carried out using the three formulas

C uonj, 3

Pi

=

i = 1, 2, 3 .

j= 1

For any of the infinite number of orientations fl may have, the force per unit area P pulling by the material on one side on the material on the opposite side is thus given by the nine components of the stress tensor. For all processes observed in nature, only six of the nine components are needed, because u21= u12,( ~ 3 2= ( ~ 2 3and (TI3 = ~ 3 1 . That is, the stress tensor is symmetric. The six components of the stress tensor provide the means of calculating the forces per unit area at a point. At neighboring points, the stress is in general different, and the full description of the internal contact forces in matter requires a stressjeld. This means that each component is a function of x, y , and z. In this chapter, we will not discuss general methods of measuring the stress field, Indeed, measurement is quite difficult and is accomplished only in very special cases. Two special cases will concern us. The term pressure applies to both, but it is a good idea to recognize that there are two and to be aware which one is under consideration when we speak of pressure. The term “pressure” is sometimes used in older literature to be synonymous with “negative stress,” particularly in cases of uniaxial stress (all other 8 components zero). “Isotropic pressure” or “hydrostatic pressure” was then used for the modern term. The first special case applies to a fluid at rest, for which case ull = ( T = ~ u33 ~ = - p and all other components of the stress tensor are zero. In fact, a fluid, as distinguished from a solid, is usually defined as a substance having this property. The existence of such substances as oils and greases, pitch and structural polymers such as rubber, nylon, and plexiglas, for which “at rest” may demand waiting for very long times, illustrates that the concept of a fluid at rest is only a limiting case. The single number p in this case is called “pressure.” Since the contact forces are expressible in terms of a single number times the unit tensor 1=(;

8 ;)

and the stress field is expressible as a sculurjeld times the unit tensor, pressure is often referred to as a scalar. This can be very misleading in understanding the physical meaning of the concept.

502

5.

MEASUREMENT OF PRESSURE

The second special case applies to a fluid in motion where the fluid has properties of isotropy and forgetfulness of its previous motion sufficient to allow the assumption of linear viscosity. This is the assumption that each stress component is a linear function of all rate of strain components and that a single material constant, called the viscosity coefficient, is sufficient to provide the interdependence of stress and rate-of-strain components. In this special case, although the six components of the stress are in general all different, it is useful to employ the differences of the diagonal stress components from their average and call that average the negative of the pressure: (5.1.1)

This procedure has the advantage that the resulting equations of motion for the fluid-called Navier-Stokes equations-reduce to the hydrostatic equations as the motion ceases. The kinetic theory of gases provides a different concept of pressure from the concept we have presented of a force per unit area between contiguous parts of matter. In the kinetic theory, pressure is thought of as net transport through an element of area of momentum component normal to the area, per unit of area and per unit of time. Since the kinetic theory does not deal with contiguous matter but only with separate molecules moving through empty space and colliding with other molecules, the momentum transport is wholly by the material molecules themselves. Fundamental theoretical problems still exist when momentum transport by long range action-at-a-distance forces is contemplated; the reader is referred to treatises on nonequilibrium statistical mechanics, kinetic theory, and particularly plasma theory. The pressure of gases at rest and equations of state such as the laws of Boyle, Gay-Lussac, and Charles and the ideal gas law provide an elementary range of experience from which much of our thinking about pressure emerges. The kinetic theory elucidates phenomena in low density gases very well, and extensions of the concept of pressure as momentum transport by molecules can provide modified equations of state such as the van der Waals equation, but there is no meaningful relation between molecular transport and contact forces when liquid densities are reached. The lower limit of the range of pressure measurements is in the region where the two concepts overlap, at about 0.01 Pa. Confusion between the mechanical concept of pressure and the kinetic theory concept of pressure is common, and it is helpful to keep in mind that the range of phenomena treated jointly is limited. In particular, the

5.1.

INTRODUCTION

503

equality of the concepts must be limited to surfaces across which there is no net transfer of matter. Thermodynamic equations of state of condensed matter, as well as of gases, employ pressure as one of the thermodynamic variables. Experiments to date have shown that, for a fluid in motion, the quantity defined by Eq. (5.1.1) serves for this purpose so long as the Navier-Stokes equations describe the mechanical properties of fluids. Extension of the concept of pressure to higher values than can be attained with rigid materials using the piston and cylinder gage (Section 5.2.3), by thermodynamics and fluid dynamics, employs shock wave relations in explosively driven metals. Data on the pressure dependence of ruby fluorescence wavelength shift based on shock wave experiments have been used to measure steady pressure achieved by piston and cylinder methods employing diamonds. Steady pressure of 170 GPa is reported to have been measured, representing the current top of the range of pressure measurements. Finally, we call attention to the tendency of scientists to use the word “pressure” to describe physical quantities which are not within the scope of the meaning of the word in this article, namely contact force per unit area between contiguous parts of matter. We list these as a warning to the reader that he needs to look elsewhere for methods of measuring these “pressures.” Category of action-at-a-distance forces on matter: (i) (ii) (iii) (iv)

gravitational pressure. radiation pressure. magnetic pressure. electrostriction pressure.

Category of analogous equations of state: (i) partial pressure. (ii) vapor pressure. (iii) osmotic pressure. Two other uses of the word pressure in the parlance of fluid dynamicists cause conceptual confusion and are discussed in Chapter 5.3. They are listed here along with other “pressures” which are not pressure with the aim of clarifying what the meaning of the word is in this chapter:

(i) impact pressure, also called total pressure, Bernoulli pressure, or stagnation pressure; (ii) dynamic pressure, which is merely two words designating Bpvz at the point within the fluid:

5.

5 04

MEASUREMENT OF PRESSURE

5.1.2. Contact with Gage Element Necessary

A contactless pressure measuring device cannot exist. However, if the thermodynamic equation of state is known and pressure is calculated from this equation and other measured variables, one often says that the pressure has been measured; for example, by measuring molecule number density n and temperature T of a gas, one can calculate the pressure from the equation p = nkT, where k is Boltzmann’s constant. ( k = 1.38 X 10-235

a

K-1).

The insertion of a gage is very likely to change the pressure at a point in a flow from the value that would exist there without the gage element. This classic problem will be dealt with in detail in Chapter 5.3, and one important feature, which is employed in velocity measurement, is dealt with in Section 1.2.2, Pitot Probe. At this point, we emphasize that measurement of pressure with a probe means finding the pressure that would be there in the absence of the probe. Most of this part deals with the gages inserted into fluids for pressure measurement. A wide range of gages is manufactured and sold by commercial companies for industrial and research use. 5.1.3. Calibration and Standards

The most accurate method of pressure measurement employs a piston fitting tightly in a cylinder but not touching. (See Section 5.2.3.) Combining this with the principle that pressure in a homogeneous fluid at rest is uniform throughout if there are no action-at-a-distance forces permits calibration and comparison of pressure gages. Accuracies to within uncertainties of 0.01 percent can be achieved with static fluid calibration. Fidelity of dynamic response of gages having undergone steady pressure calibration is inferred from an understanding of their behavior and from shock tube tests; reliability in the range of 1 percent is rarely achieved, however, as discussed in Chapter 5.8. Extraneous effects on pressure gage readings are numerous and difficult to avoid. Means of intercomparing measurements made by gages operating on different principles are much to be desired. Provision for frequent calibrations of gages under conditions where their behavior is well-understood can be helpful, both in routine monitoring and in research investigations. It is our intention to list and illustrate the types of gage designs that have been recommended or manufactured, especially to clarify the principles of action employed, and to provide recommendations for specific situations.

5.2.

MEASURING CONSTANT AND SLOWLY VARYING PRESSURES

505

5.2. Gages for Measuring Constant and Slowly Varying Pressures The most familiar and widely used pressure measuring devices are U-tube manometers and dial and digital gages. Manometers are easily constructed of equipment found in every laboratory and, for rough measurements, fairly insensitive to errors. Dial gages are cheap, sturdy, and easily connected. A manometer can provide absolute readings, with suitable precautions, but a dial or digital gage must always refer to another gage for calibration. One ordinarily takes it for granted that the reading of a manometer or a dial or digital gage can be carried out at one’s leisure. Both require a few seconds typically to respond to a changing pressure, and the assumption is made that they have had an indefinitely long time to come to mechanical equilibrium with the fluid whose pressure is measured. 5.2.1. Liquid Manometers The liquid manometer, typified by two columns of liquid partly filling a piece of glass tubing bent into a “U” shape with hoses connecting to two reservoirs, employs the hydrostatic law p - pgy = const

(5.2.1)

applicable to a homogeneous fluid at rest with no forces except pressure and weight. The liquid in the manometer typically has a density lo3 times the density of the gas which is connecting the manometer to the reservoirs; correction for the pressure difference associated with the weight of the gas in the connecting tubes can be made, but usually the correction is negligible in comparison with other corrections which can only be estimated. To the extent that Eq. (5.2.1) is valid, the pressure difference between the two reservoirs is measured by p z - p 1 = pg(yz - yl) = p g h , where y z and y1 are the vertical coordinates of the respective surfaces between liquid and gas on the two sides of the U-tube, p is the mass per unit volume of liquid, and g is weight per unit mass, nominally 9.8 N kg-’. One disadvantage of the U-tube manometer is the ease with which the liquid is blown out of the manometer when the pressure difference exceeds the range. A trap to catch the liquid in case of this accident is advisable. Chemical contamination of the reservoirs where the pressure is being measured on either side is avoided by using a liquid with low “vapor pressure” such as mercury or silicone oil. Mercury has the additional advantage, due to its high density, of measuring high pressure; a manometer to

5 06

5 . MEASUREMENT OF PRESSURE

FIG.I . Modification of U-tube manometer to provide increased sensitivity in reading difference in heights of two surfaces.

measure a pressure difference larger than that corresponding to about 1 meter of height difference is seldom used, however. In the direction of small pressures, a liquid of density lower than that of silicone oil (approximately 0.8 the density of water) is impractical. One arm of the manometer may be bent to an almost horizontal position as shown in Fig. 1 to “magnify” the position of the liquid-air interface to aid in the measurement. Elaborate techniques have been developed to aid in precision measurement of the height of the surfaces’; only a few will be mentioned. One commonly used technique is to mount a pointer internally which does not wet the liquid (ivory-mercury) and which is attached to an accurately readable micrometer scale. The pointer is adjusted until the observer does not see a depression in the mirrorlike liquid surface. Another method for mercury uses a steel float carrying a glass mast on which is engraved an accurate scale; the height of the mercury column is obtained in terms of the position of the glass scale read with a microscope (Betz manometer). An ultrasonic pinger and receiver at the base of a mercury column is used in a commercial instrument to detect the time of travel of an ultrasonic pulse from the base to the surface of the mercury. A fringe counting laser interferometer allows measurements of the light reflecting surface to a sensitivity of less than one micr~rneter.~.~ Other methods of pressure measurement are probably more practical than these, however, since so much trouble is required to operate the measuring equipment. Pressure differences smaller than measured by approximately 10 mm of oil (
C. R . Tilford, Rev. Sci. Instrum. 44, 180 (1973). S . J. Bennett, P. B. Clapham, J. E. Daborn, and D. I . Simpson, J . Phys. E 8 , s (1974).

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

507

Reference may be made to a vacuum technology handbook for descriptions of a thermocouple gage, a Pirani gage, a Phillips corona gage and an ionization gage. The first two of these “pressure” measuring methods employ the heat transfer properties of the particular gas and the others are measurements of density rather than of pressure. The most reliable method of measuring pressure in the range below 100 Pa is by means of diaphragm type gages, calibrated by a piston gage, as described below. 5.2.1.l. Sources of Inaccuracy in Liquid Manometers. Liquid manometers fail most frequently as absolute pressure measuring instruments on two accounts: (1) lack of homogeneity of the liquid; and (2) presence of other forces than weight. No recipe can be given for avoidance of error in either category. Many metals dissolve in mercury and change its density; it is similarly difficult to be certain that water or oils obtained for use in a manometer are homogeneous, pure and of known density. Temperature variations in the parts of the manometer will cause expansion of the scale for reading h , and one must also be aware that the density of the liquid depends on its temperature. For very precise measurement, departure of the local value of g from 9.80665 N kg-I must be accounted for. In avoiding the effects of extraneous forces, one must, of course, avoid magnetic forces and electrostatic forces and not use the manometer in an accelerating frame of reference. The most bothersome type of extraneous force is that of surface tension which may prevent the pressures on the two sides of the liquid gas interface being equal as is assumed when the heights of the two manometer columns are measured and formula (5.2.1) used to determine the pressure difference. If the tube is sufficiently small, so that the shape of the interface is approximately a sphere, an additional height of the manometer is present of about

2 Y cos a (5.2.2) (PL - Pc)& where Y is surface tension, a is the contact angle between the liquid-gas interface and the wall, pLand p c are the liquid and gas mass densities, and r is the radius of the tube. a = 0 for a water-air-glass interface, and a = 140” for a mercury-air-glass interface. If the glass is extremely clean and outgassed, the mercury-glass angle may become less than 900.4 If the tube is sufficiently large in diameter, the surface will not be a hemisphere and may become sufficiently plane at the center of the tube so that a correction does not have to be applied. The measure of the distance from the wall at which the curvature of the surface is noticeable is ( 2 Y / ~ g ) ” ~ , e.g., 3.9 mm for water and 2.7 mm for mercury. The tube radius must be y’

=

‘ N . K . Adam, “The Physics and Chemistry of Surfaces,” p. 185.

Dover, N e w York.

5.

SOX

MEASUREMENT OF PRESSURE

TABLE1. Pressure Units Name of unit

Valueb

1 kgf/cmz

98.07 0.1 100 101.3

kPa Pa kPa kPa 10 Pa (transducers) or 20 @Pa (hearing)

1 dyne/cm2 1 bar 1 atm 0 acoustic decibel (reference level)”

-~

[

~~

~~

Name of unit 1 “in Hg” 1 “in

HzO”

1 “mm Hg” (Torr) 1 “mm HzO” 1 Ib/ft2 1 Ib/inz

Value 3.38 kPa 249 Pa 133.3 Pa 9.81 Pa 47.9 Pa 6.895 kPa

~~

" Sound pressure level, in decibels, is 20 times the logarithm to the base

10 of the ratio of the sound pressure to the reference pressure. The reference pressure should be explicitly stated. Unless otherwise explicitly stated, it is understood that the sound pressure is the rms pressure change. The SI unit of pressure, pascal (symbol Pa), is 1 N . m-*.

of the order of 10 times this value to avoid measurable curvature in the most exacting devices. The effect of the capillary elevations in two arms-both vertical-of a manometer will cancel in determining the difference in heights if the two arms are of equal diameter tubing. Most stock laboratory glass tubing does not have uniform diameter, but uniform bore tubing can be obtained. Small amounts of foreign substances, e.g. grease, on the glass can destroy the compensation hoped for. The manometer is so widely used that a variety of units for pressure has grown up. Some are quite confusing. Table I includes some of the units which have been found in the literature and gives their value in terms of the SI unit pascal ( 1 Pa = 1 newton per square meter). 5.2.1.2. Response Time of Liquid Manometer. A manometer subjected to a time-varying pressure behaves as a damped oscillator. The frequency of oscillation is (5.2.3)

where L is the length of the liquid column, u is its radius and v is the kinematic viscosity of the liquid. Typically, manometers are heavily damped, but not overdamped; in this case, the damping time is of the order of magnitude u2/4v. This tells us, for example, that the time elapsing after a pressure change in a water (v = m2 s-l) manometer of 2-mm radius before we can read it is greater than 1 s. The detailed behavior is more c~mplicated.~

’ W. L. Holley and J . R. Banister, J . Ind. Arrodyn.

1, 139-165 (1975).

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

509

Equation (5.2.3) assumes that there is no lag in the equalization of pressure in the connecting tubing. Such a lag can be appreciable in a 1 mm bore tube many meters in length.6

5.2.1.3. McLeod Gage. Precise measurement of pressure in the range below 100 Pa is difficult to carry out with a U-tube manometer, even with the refinements described in preceding sections. Down to about 0.1 Pa, manometric methods can be extended by isothermally reducing a volume of a gas such as H z , He, or Nz at a low pressure and measuring the pressure by a manometer a t the higher pressure. Since the virial coefficients* of these gases are known with high accuracy, the pressure ratio can be accurately calculated from the volume ratio. The McLeod gage illustrates one method of making a known reduction in volume and of measuring the higher pressure. It has been used for many years but is cumbersome to use in routine pressure measurement. It does not measure pressure of a condensible vapor or of a mixture of gases and vapor. Figure 2 illustrates the gage. A glass bulb of known volume "Irl communicates with the vessel where the pressure is to be measured; at its top, a glass capillary with precision bore and a flat closed end is sealed. Mercury is caused to rise from the lower reservoir, trapping gas at the pressure to be determined p1 in the bulb and compresses it to a small "Irz. As the mercury rises in the bulb, it also rises in the nearby tubing, which includes a side arm capillary of the same size as the capillary sealed to the top of the bulb. If the trapped gas was in the correct range of pressure initially, it is contained entirely in the capillary when the mercury in the side arm reaches the level of the closed end. Then the pressure p z of the compressed gas is measured by the difference in vertical heights h of the two mercury columns in the capillaries. "Irz is given by this same height difference multiplied by the capillary cross section A . The two identical precision bore capillaries are used in the hope that the capillary depression will be the same and cancel in the calculation of the pressure p z . The volume "Ir, is determined before the lower parts of glass tubing are attached to the bulb during manufacture; Sr, is determined by weighing the bulb and capillary when completely filled with mercury, weighing the empty bulb, and knowing the density of mercury. The cross section of the capillary is likewise found during manufacture by moving a mercury drop of known weight (and volume) along inside and measuring its length.' A . L. Ducoffe, J . Appl. Phy.\. 24, 1343 (1953). Strong, "Procedures in Experimental Physics," p. 138. Prentice-Hall, Englewood Cliffs, New Jersey, 1938.

'J.

* See Section 6. I . 1 and Eq. (6.1.1)for further information on the virial equation of state.

5 10

5.

MEASUREMENT OF PRESSURE

be measured

1-l;

capillary

1 roughing Pump

FIG.2. McLeod gage. Pressure of gas trapped in volume V , is “magnified” when it is compressed by mercury rising in the bulb. Both pressure and volume are indicated by the dimension h in the compressed state.

Assuming p l y l = p z 7 f 2 ,i.e., neglecting higher-order terms in the virial equation of state,

where p is the density of mercury. The McLeod gage’s scale is nonlinear; the range is p g A 1 2 / V l , where 1 is the length of the capillary’s precision bore. The bore loses its precision near the joint with the bulb, s o l may not extend that far. 5.2.2. Deformation Gages with Mechanical Readout

A mechanical structure with a cavity connected to a vessel whose pressure is to be measured will deform. Three important structures are in use. The most convenient and most highly developed of these are the Bourdon gage, the capsule (also called sylphon or bellows) gage and the diaphragm gage. The diaphragm has unique properties and is discussed

5.2.

MEASURING CONSTANT AND SLOWLY V A R Y I N G PRESSURES

5I I

separately in Chapter 5.9. Here we discuss the Bourdon and capsule type gages briefly because of their wide use. For details and design information, see Andreeva.8 5.2.2.1. Bourdon Gage. A metal tube of elliptical cross section which is bent to form a nearly complete circle has one end fixed to a base. The elastic displacement of the other end as the circle tries to straighten out is nearly proportional to the pressure inside the tube. By means of a rack and pinion connection, the displaced end of the tube moves a needle whose position is read on a circular dial. This is the common gage one sees so frequentiy on machinery and control panels. Metallurgical skill in fabricating the elliptical cross section tube and watchmaker-type skill in the linkage magnifying the displacement of the tube end has produced gages capable of maintaining remarkable accuracy over many years. Ambient temperature changes, corrosive fluids and damage caused by overload affect the reading; therefore, most such gages are demountable so they may be taken to a calibrating station or replaced easily by a calibrated unit. Variations in the size, shape and material of the tube have been made in a large range of applications. Particularly noteworthy is use of fused quartz tubing, fabricated into a spiral, fixed at one end and suspending a mirror at the other. Rotation when the spiral unwinds is detected and converted into a pressure reading. The rotation may be linked to a mechanism with a digital readout as an operator maintains a null position of a light beam reflected from the mirror. Another readout employs an electromagnetic counter-torque automatically maintained by a photocell detector of the light beam deflection and a digital readout of the current required to balance the spiral torque change. The advantage of fused quartz is its long term mechanical stability, relatively small sensitivity to temperature, and ability to withstand somewhat higher overloads without loss of calibration. 5.2.2.2. Capsule Gage. The capsule, or bellows-type gage, is similar in construction to a diaphragm, but the elastic element is convoluted to magnify the motion of the diaphragm. By combining metallurgical art with mechanical art to translate the motion into the rotation of a needle on a dial without backlash, manufacturers have been able to supply gages with a wide variety of operating ranges, compensated for temperature L. E. Andreeva, "Uprugie elementy priborov," Mashgiz, Gosudarstvennoe NauchnoTekhnicheskoe Izdatel'stvo Mashinostroitel'noi Literatury, Moskva, 1962; English translation, A . Baruch and D. Alster, in "Elastic Elements of Instruments" (H. Schneider, ed.), pp. 194-360. Israel Program for Scientific Translations, Jerusalem, 1966.

512

5.

MEASUREMENT OF PRESSURE

POINTER

CAPSULE CALIBRATION ADJUSTMENT

PINION GEARED SECTOR BACKLASH ELIMINATOR REVOLUTION INDICATOR FLEXURE

FIG.3. Capsule-type dial gage. The dial on which the pointer is read is not shown. [Courtesy Wallace and Tiernan Division of Pennwalt Corp., Newark, New Jersey.]

change, and retaining precision of the order of 0.1 percent of the range. To achieve this order of reliability, each gage is individually calibrated at several places on its dial since nonlinearities arise in several parts of the mechanism. Such gages are easily damaged by surges and excessive pressures beyond about 125 percent of the intended range, and by mechanical shock such as is sustained by being dropped on the floor. Frequent intercomparison of such gages, particularly with gages which have been tenderly protected from hazards, is necessary to assure reliability. Figure 3 illustrates the construction of a capsule gage.

5.2.3.Piston and Cylinder Gage This gage is the basic standard for pressure measurements above about 100 kPa. Even within the range of usefulness of the U-tube manometer, the piston and cylinder instrument is used for absolute pressure measure-

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

5 I3

I

2

--

\ I

(a) (b) FIG.4. Piston and cylinder gage. (a) Arrangement for use. (b) Detail of clearance area. 1 : piston; 2: cylinder: 3: bleed hole; 4:jacket pressure inlet. The jacket pressure inlet allows the clearance between piston and cylinder to be maintained at an optimum value and the “effective area” to be accurately evaluated.

ment and calibration of other types of instruments. Also called a “deadweight gage” and a “gage tester,” it is seldom seen outside a standards laboratory because its operation requires skill, care, patience, and conditions of cleanliness. Figure 4 shows the principal parts of the gage. Known masses are stacked on the piston of known mass and cross sectional area. The piston fits with extremely small clearance in a cylinder resting on a firm base. A liquid (usually oil) or a gas (nitrogen or air) fills the cavity in the cylinder below the piston and communicates via tubing and fittings with a dial gage to be calibrated and a plunger pump allowing fine adjustment of the volume. By means of the pump, the pressure is increased until the piston rises and is supported. Fluid escapes through the small clearance region between the piston and cylinder, and the piston gradually settles. While the piston is settling, the pressure in the fluid at the level of the closely fitting portion of the cylinder is known to be the weight of the piston and supported masses divided by the effective area of the piston. The eflectiw area has been determined in principle by measurement with a metrologic microscope of the piston and cylinder diameters at the region of small clearance and is the average of the two area^.^ The temperature at which the areas were measured must be known; in use, the area depends on the temperature of the test together with the coefficient J . L. Cross, Reduction of data for piston gage pressure measurements. N u t / . Bur. Stund. ( U . S . ) ,Monogr. 65 (1970).

514

5.

MEASUREMENT OF PRESSURE

of thermal expansion of the piston and cylinder material. Changes in the areas are also caused by the pressure to which they are subjected. If the pressure on the cylinder is not increased by introducing pressure in the jacket (see Fig. 4) when the pressure being measured is large, the clearance becomes large, and the piston falls too rapidly. The changes in areas of piston and cylinder must be calculated from a knowledge of the elastic modulus of the material and the pressures beneath the piston and in the jacket. For the highest pressures, the piston may be made of sintered tungsten carbide. Piston and cylinder gages are obtainable from commercial manufacturers. The effecfive areas of gages are supplied by the manufacturer, having been obtained by comparing gages with gages built and maintained by national standards bureaus. Because the clearance between cylinder and piston is so small, especially when gases are used, cleanliness is mandatory. The gage must be frequently disassembled, cleaned and reassembled to dispose of fine particles which lodge in the clearance region. The presence of foreign particles is detected by spinning the piston and masses very slowly while they are settling. If the clearance region is unblocked, the supported masses decrease their slow rotation very evenly and gradually. With care, pressures can be measured with errors as small as 0.01 percent in ranges from 100 Pa to 100 MPa (1000 bars). In addition to corrections noted for temperature and pressure to determine the effective areu, corrections must be made for oil buoyancy of the immersed portion of the piston, air buoyancy of masses (0.15 percent), local value of g (up to 0.3 percent), fluid head when oil is used and pressure at a level other than the clearance region between piston and cylinder is sought, and surface tension of oil. The last two corrections mentioned are needed only when pressures in the kilopascal range are measured; in this range, mercury and oil manometers can be used for absolute measurement with about the same care as is required for piston and cylinder gages. 5.2.4. Calibration Procedures for Constant Pressure

Use of absolute instruments-manometers and piston gages-is possible in routine pressure measurement, but usually sufficient care is not taken to avoid errors that may exceed 1 percent. Dirty mercury, dissolved gases in oil, and scratches causing sticking of pistons easily occur to invalidate the readings. In practice, it is convenient to obtain dial gages, or deformation gages with digital readout, from a manufacturer who maintains a “standards laboratory” for calibrating his products. Intercomparison of older gages with newly acquired gages is relatively simple.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

515

Damage to calibrated gages in transport can always be suspected. A new differential pressure gage that does not read “zero” on being unpacked can be assumed to have suffered damage. Most large industries maintain “standards laboratories,” keeping secondary standards traceable to national standards laboratories. Inquiry in a large city will usually locate one or more where a pressure gage can be calibrated . The difficulties of establishing exact pressures in the range below 10 Pa have led some to propose that a prestressed diaphragm gage with capacitance bridge detection of the diaphragm deflection may be the best method of measuring such pressures.lO The readability of the tensioned diaphragm gage is the order of 1 part in lo6, and gages have been built which are linear (at higher pressures) to 1 part in lo4. So one is inclined to think that such a gage calibrated by a piston gage at 100 Pa will be able to measure 0.01 Pa to an accuracy of 0.1 percent. Attempts to calibrate diaphragm gages with McLeod gages have been troubled by mercury streaming in a cold trap, inaccuracies in mercury column readings due to variable capillary depression and unknown “thermal transpiration” (associated with maintaining the diaphragm gage at a different temperature) so that demonstrated accuracy in the 0.01-Pa range is not better than 0.5 percent .I1

5.3. Pressure Measurement in Moving Fluid A principle of mechanics is that forces are independent of the frame of reference for all inertial frames. Thus pressure is the same at a place and time in a fluid whether we imagine that we are at rest or in (constant velocity) motion with respect to the fluid. However, if a stationary probe is inserted in a steadily moving fluid the pressure over the surface of the probe is not constant and in general is different from the pressure in the fluid before the probe was inserted. The highest pressure on the probe is at the stagnation point which is usually the point facing the flow; this highest pressure is often called stagnation pressure, or impact pressure (or “total” pressure) and measurement of its value is one of the measurements required when the Pirot tube is used for fluid velocity measurement (see Section 1.2.2). At other points on the surface of the probe the pressure has values ranging from the stagnation pressure to below that of the fluid without the probe. The purpose being to measure the pressure in the undisturbed fluid, lo

I’

N . G . Utterback and T. Griffith, R e v . Sci. Instrum. 37, 866-870 (1966). J. P. Bromberg,J. V a c . Sci. Techno/. 6, 801-808 (1969).

5.

516

MEASUREMENT OF PRESSURE

probes have been developed to sense the pressure at the intermediate places on their surfaces where one has reason to believe that the pressure is the same as in the undisturbed fluid. Such probes are called stutic probes because they respond in the way any gage would if it were at rest relative to the fluid, i.e., moving with the fluid. The terminology is as confusing as the explanation. Confusion can be lessened if one remembers that “static pressure” is merely pressure. The term is used to emphasize that one is not referring to some of the many other kinds of ”pressure,” such as “stagnation pressure.”

5.3.1.Wall Taps A small hole in the wall of a pipe or duct, or in the surface of an im-

mersed object, connected by tubing to a pressure gage is often used to measure the pressure of a Newtonian fluid at the surface containing the hole. The wall tap method cannot be used for non-Newtonian fluids such as polymers, where stresses are not linearly related to rates of strain. Since even in supersonic flow the gas velocity approaches zero at all solid surfaces (except for very rarefied flows), the measurement of pressure at any surjucr is relatively easy. However, even when the flow is steady, the pressure in the connecting tubing is a bit larger than the pressure at the surface without a hole. When the flow is changing, or turbulent, the pressure in the connecting tubing may be considerably higher than the time average of the pressure at the surface. If there is a protuberance comparable in dimension with the hole diameter, the pressure in the connecting tubing may be either higher or lower than the pressure that would exist at the smooth surface; higher if the protuberance is on the downstream edge of the hole, and lower in the opposite case. The hole-error, assuming no

3E, -

-

c,

FIG.5 . Streamlines in the neighborhood of a hole in a wall or surface of an immersed body.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

!

5 I7

7 1-

/burr

3%d

bob o 6 -

\

Q

-4

5-

z

4 -

L

L I

0

100

zoo

300

Hole size Reynolds number, d(p t~~)'''/p FIG.6. Dimensionless hole-error versus Reynolds number for hole diameter d = 1.6 mm and varying sizes of burr. Wall stress IT,,= p ( a ~ / d y ) ~ - , , .p, are density and viscosity of fluid. u is mean speed of fluid at distance y from surface. Solid curves are dimensionless hole-errors for the indicated size of burr. Dashed curve is hole-error for a sharp edged "well-finished" hole. [From Shaw.'*]

protuberance, is smaller the smaller the hole, and it is assumed that the hole-error would extrapolate to zero for hole size zero; however, the difficulty in avoiding burrs and other mechanical irregularities at small hole sizes requires that the hole-error be calculated for a larger hole in practice. The computed inviscid steady flow pattern in the neighborhood of a wall tap is sketched in Fig. 5 . The streamlines curve at the hole; such curvature is associated with a pressure gradient. In a real flow, there is also an eddy or system of eddies set up in the fluid within the hole, caused by flow separation at the hole leading edge, and an increase in pressure at the stagnation point near the downstream edge. These three factors combine to give the net hole-error.*2 The hole-error in straight pipe flow of air has been measured by Shawl' and Franklin and W a l l a ~ e . ' ~Figure 6 displays results for various flow speeds (Reynolds numbers) and for the effects of burrs of various sizes. Shaw, on the basis of dimensional argu-

l3

R . Shaw, J . Fluid Mech. 7, 550-564 (1960). R. E . Franklin and J . M . Wallace, J . Fluid Mech. 42, 33-48 (1970).

518

5.

MEASUREMENT OF PRESSURE

ments, deduces that the hole-error can be determined most easily in terms of the "wall stress" wo = , U ( L ~ V / L ~ ~ ) , , = ~ as the mean flow speed in the pipe, or the free stream velocity outside the boundary layer, varies. In a long uniform steady pipe flow, wois readily found from the axial pressure drop dp/dx since d u o= (7#/4)(dp/dx) where D is pipe diameter. In a boundary layer, wo is found from an estimate or measurements of (av/ay),=o. It is evident from Fig. 6 that the effect of relatively small burrs is significant and the hole-error cannot be reduced safely merely by using small holes. As an example of the magnitude of hole-error, it may be noted that a hole size Reynolds number of 300 in air at 100 kPa (1 bar) with hole diameter d = 1.6 mm corresponds to a mean pipe flow speed of 66 m s-' in a pipe of 51 mm diameter, and for this case u0 = 9.6 Pa = 0.01 percent of the pressure. The pipe flow Reynolds number pv,,,,D/,~ for this case is 225,000. The dimensionless hole-error reaches its maximum value for hole size Reynolds number 800 and hole depth 1S d and does not increase beyond; it is then 3.8. Again considering a 51-mm diameter pipe, air at 100 kPa and hole diameter d = 1.6 mm, the hole-error of 3 . 8 is~ 260 ~ Pa, or 0.3 percent of the pressure; this case corresponds to a mean pipe flow speed of 200 m s-' and a pipe flow Reynolds number of 700,000.

-

5.3.2.Static Probe in Steady Flow In view of the situation described in the preceding section, one sees that if a very small object with a hole in its wall connecting to a pressure gage is inserted into a steady flow, it can measure the pressure there as disturbed by the presence of the probe. Figure 7 indicates how flow with straight streamlines is disturbed by introduction of a small bent tube.

s

_ I _

3

3

3

-

4

FIG.7. Streamlines about a bent tube immersed in a uniform flow. 1: side view. 2: nose alone. 3: bottom view. 4: stem alone.

5.3. PRESSURE

MEASUREMENT IN MOVING FLUID

519

stem4 effect

FIG.8. Pressure variation along wall of a bent tube due to streamline curvature. Separate effects due to the nose and due to the stem can combine to yield the undisturbed stream pressure at position of holes.

Imagine that this tube is small in comparison with the extent of the flow being probed; if the hole in the side wall of the tube is judiciously located, it may read a pressure which would be the same as the pressure would have been without the probe. The pressure on the wall of a closed round-headed tube aligned with a steady flow has been measured by a series of holes and increases with distance from the shoulder of the tube, but never reaches the pressure in the undisturbed flow. However, the stem of a bent tube causes the pressure in front of the stem to be larger; the effect is less at greater distances in front. For a given tube, the two effects cancel at some intermediate distance as indicated in Fig. 8, and this is the place chosen for the hole(s) in the tube wall. An 8-mm-diameter tube with nose 3d long and with holes 6d back of the shoulder, where d is tube diameter, has been accepted by workers in wind tunnel laboratories for use in subsonic and supersonic flows, up to Mach 2. Mach number corrections must be made above M = 0.7 which are ddpendent on the geometry of the tube.l4-Is A tube with a single hole in the wall making an angle a with the direction of the steady flow can have a higher or a lower pressure than the I‘ D. W. Bryer and R. C. Pankhurst, “Pressure Probe Methods for Determining Wind Speed and Flow Direction.” HM Stationery Office, London, 1971. Is A. N. Petunin, “Methody i tekhnika izmerenii parametrov gazovogo potoka.” Makhostroeniye, Moscow, 1972 (in Russian). S. H. Chue, Pressure probes for fluid measurement. f r o g . Aerosp. Sci. 16, 147-223 (1975).

5 20

5.

MEASUREMENT OF PRESSURE

Undisturbed stream velocity

-20"

FIG.

-10"

,

0"

loo

20°

9. Pressure inside "static probe" inclined at various angles to steady uniform flow.

undisturbed pressure, depending on whether the single hole is in front or in back. Usually multiple holes symmetrically placed around the tube are used. Then the measured pressure pm is always lower as indicated in Fig. 9. On the simple hypothesis that the deviation is proportional to pv:, where u,, = urn sin (I! and p is fluid density, pm - pm = B p ( v , sin The coefficient B is found experimentally to be about - 0.25. An airplane, or other vehicle, in motion through the air requires a knowledge of the pressure outside the vehicle. The pressure outside is needed when a barometer is used as an altimeter, and also when a Pitot tube (see Section 1.2.2) is used for measurement of the vehicle's air speed. General regions suitable for the location of pressure taps, called "static vents," on the sides of an airplane are found from model tests in wind tunnels. In most cases the taps are arranged in pairs, one on each side of the fuselage, either between the nose of the fuselage and the leading edge of the wing or between the trailing edge of the wing and the leading edge of the stabilizer. The pressures measured at these holes may be appreciably affected by the attitude of the airplane and the amount of thrust in the engines. Calibration of the readings of "static vents" is carried out in flight tests near a site where the pressure on the ground is known and radar measurements of height permit corrections for difference in pressure with height to be made." The need for empirical search for the places on the surface of an airplane where the undisturbed pressure may be found characterizes the problem of designing any probe to measure "static pressure," i.e., the pressure in a fluid were it not moving over the probe.

'' F. J . Bailey, Jr.. J . A . Zalovcik, W. H. Phillips, and W. B. Huston, Piloted aircraft testing. "High Speed Aerodynamics and Jet Propulsion," Vol. VIII, Article N,1, page 839. Princeton Univ. Press, Princeton, New Jersey, 1961.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

52 1

5.3.3. Static Probe in Unsteady Flow In an unsteady flow, such as a sinusoidally oscillating sound field, or a fluctuating random field characteristic of turbulent flow, the pressure is a function of time, and probes display a fluctuating output. Whether the indicated pressure is related to the true pressure that would have been there in the undisturbed flow is a question open to debate. An acoustic cavity, driven by oscillating one wall and with calibrated (at steady pressures) diaphragm type gages mounted in other walls is used for "dynamic calibration" of a probe either also mounted in the walls or immersed in the interior of the cavity. Nonlinear responses and generation of harmonics are familiar problems discussed in acoustics handbooks, textbooks and journals,'* and the problems associated with fluctuating flows which have a mean flow component are to some extent an extension of these. However, an additional feature of the problem with a mean flow is the accompanying fluctuation in both the direction and magnitude of the velocity of the fluid impinging on the probe. Elliott19 approached this problem by designing a probe with a shape found to give pressure readings relatively independent of pitch and yaw when mounted in a steady wind tunnel flow. The probe was then immersed in an acoustic "dynamic calibration" cavity and its fidelity tested at various frequencies. In the range from 0.01 Hz to 10 Hz the response was found to be within 20 percent in amplitude and 20" in phase of the pressure at the wall of the cavity. The probe consisted of a 2.5-mm thick circular disk, 45 mm in diameter, mounted on a sting of tubing 0.56 m long connecting to a differential tensioned diaphragm gage with capacitor sensor (see Fig. 22 and Section 5.9.3.) The disk was an ellipsoid of revolution but with the flat sides smoothly indented on both sides to give a thickness at the center of 1.8 mm; 0.5-mm ports in the center of each indentation connected, via extended 0.5-mm channels, to the supporting tube. When the disk is tilted at more than 10" to the main flow in the wind tunnel, the flow over the disk separates and the pressure reading changes drastically. This probe was designed to measure fluctuations in pressure in the turbulent boundary layer in the atmosphere, with minimum interference due to velocity fluctuations. Elliott concludes that it is usable within statistical accuracy of 10 percent in amplitude for wind range of 3 to 10 m s-' in stable atmospheric conditions. Under unstable conditions the angle between the velocity vector and the plane of the probe can attain L. L. Beranek, "Acoustics." McGraw-Hill, New York, 1954.

'13

J . A. Elliott, Instrumentation for measuring static pressure fluctuations within the atmo-

spheric boundary layer. Bnundary-Layer Metearol. 2, 476-495 (1972); see also J . Fluid Mech. 53, 351-383 (1972).

522

5.

MEASUREMENT OF PRESSURE

20-30" and the pressure readings would be seriously in error. Obviously this probe could not detect rapid fluctuations, and the extent to which its readings are affected by small scale rapid fluctuations is unknown. The diameter of the connecting tubing from the ports to the diaphragm gage is purposely small to delay, and time-average, rapid pressure fluctuations which, in turbulent flow, arise from small scale eddies carried across the probe. Another attack on the measurement of pressure within a fluctuating fluid flow has been made by Siddon.20 Following earlier workers, Siddon suggests treating steady (time averaged) and unsteady components of the flow variables and of the measured variables: P ( t ) = P + p ( t ) , V d t ) = P, + ul(t), V 2 ( f )= V2 + u2(t) and V3(t)= U3 + u3(t), where V , ( t ) is the component of velocity along the probe axis and V2(t)and V3(t)are components normal to the axis. The measured (indicated) pressure P,(t) = F,,,+ p,(t) is a function of the (true) pressure Pt(t)and of V , ( t ) ,V2(t),and V3(t). Assuming isotropy, and assuming that averages do not change with time, Siddon proposes that separate equations can be written: Steady error: Unsteady error:

p,,, - p , = A p ( P : + 2)+ B p ( p : + p m ( t ) - p t ( t ) = Ap(2v1v1 + u: - $1

z),(5.3.1) -

+ Bp(2V2p2 + 2U3V3 + V i - v:).

(5.3.2)

For convenience the notation 8: = + Vi and = u: + ui is used for net cross-stream components. The coefficients A and B are to be determined by a calibration procedure. The form of these equations is analogous to the corrections found to apply in steady flow and illustrated in Figs. 8 and 9. Coefficient A is largely a form factor relating to the imperfect cancellation of the streamline curvature effects of nose and stem; coefficient B represents the cross-stream effects. Eqs. (5.3.1) and (5.3.2) might be expected to apply to a turbulent flow with eddies large in comparison to the size of the probe, for it is assumed that the pressure gage recording p&) reaches a quasi equilibrium with the flow in the time such an eddy passes over the probe. Stated in the conventional language of turbulence researchers, this means that the unsteady flow must have a spatial scale larger than the probe if Eqs. (5.3.1) and (5.3.2) apply. The flow is then approximately uniform over the probe at all times. Siddon built a probe 3 mm in diameter whose construction is shown in Fig. 10 to test these hypotheses. The probe not only had a diaphragm u, T. E. Siddon, On the response of pressure measuring instrumentation in unsteady flow. Univ. Toronto Inst. Aerosp. Stud., Rep. UTIAS 136, Jan. 1969.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

CROSS SECTION

523

6 2

4

FIG.10. "Static probe" to measure p ( t ) and cross-stream velocity components v,(r) and u,(t). 1: epoxy mounting of nosepiece on I beam. 2: balsa nosepiece with epoxy surface. 3: four element piezoelectric I beam. 4: pressure sensing slit. 5: leads with u1 and q signals. 6: cotton plug. 7: pressure gage. [From Sidd~n.~']

pressure gage (see Fig. 23 and Chapter 5.9) but also had a piezoelectric crystal beam-supported nosepiece which recorded the fluctuating vz(t) and u3(t) components of the stream flow velocity. The pressure gage was calibrated by placing it in a cavity, one end of which was a piston, driven sinusoidally to produce a 31 Pa rms amplitude. The ug and us sensors were calibrated by placing the assembled probe in a rotating inclined channel flow; the frequency of rotation (5- 150 Hz), gas speed (35 to 65 m s-') and angle of inclination (2"-7.5") could be varied. The pressure gage calibration and the u2 and u3 sensors calibrations were each accurate to 5 percent. The exterior contour of the probe, Fig. 10, was chosen to make A < 0.001 in a steady wind tunnel flow. With A negligible, and the probe inserted into the rotating inclined channel flow with pt(t) = 0, Eq. (5.3.2) becomes p,(t)

=

2Bpvl, sin a sin Bo sin 21rfi,

(5.3.3)

where a is the probe inclination to the axis of rotation, 0, is the angle the channel makes with the axis of rotation and f is the frequency of the channel's rotation. Callingpk the rms value of the sinusoidally varying p m signal, the observedp; divided by q sin S o , whereq = Bpufm,is plotted against sin a in Fig. 11. Three values of q and three values of O0 give overlapping data points indicating the linear dependence, predicted by

5 24

5 . MEASUREMENT OF PRESSURE

0.4

rn

0.3

.-C

in U

- E

0 2

a

01

7 -----7 7 ~__1 _ I _ ~ .

0

01

0 2

0 3

04

sin a

FIG. I I . Pressure signal amplitude versus sine of angle of inclination of probe in rotating inclined channel flow. q = &pv:-; 6,-angle of inclination of rotating flow to axis of rotation, p6 is the rms value of measured pressure oscillation. (0)ul0, = 62 ms?: (0) ulo, = 50 ms-'; (0) ulm = 35 ms-I. [From S i d d ~ n . * ~ ]

Eq. (5.3.3), valid up to 0.24 (sin 14") on the abscissa. The value of B = -0.46 5 10 percent obtained by measurement of the slope of the graph of Fig. 11 was shown also to be independent of channel rotation frequency in its range of operation, namely 5 Hz to 150 Hz. The probe was used to explore the pressure variations in three different turbulent flows (channel, jet wake, and grid). Corrections employing Eq. (5.3.2) were applied, and it was found that the correction to rms fluctuation levels was small, generally amounting to less than 20 percent. This last result is surprising, since the corrections themselves were as much as 100 percent of the measured fluctuating pressures. The validity of Eq. (5.3.1) for correcting static probe readings for mean pressure in a turbulent flow can not be evaluated. Showing that Eqs. (5.3.1) and (5.3.2) apply in any given flow is a bootstrap operation. A probe of size smaller than Siddon's probe, i.e., smaller than 5 mm, would have to be used to determine whether the spatial scale of the flow is smaller than 5 mm. It is noteworthy, if Eq. (5.3.1) is valid, that there is a substantial correction due to 3 even if A is made small (by shaping the probe) and the probe is aligned with the mean direction of the flow field (so = 0.)

v:

5.4.

TIME-DEPENDENT PRESSURE MEASUREMENTS: PREVIEW

525

5.3.4. Fluctuating Wall Pressure near Turbulent Flow

Measurement at the wall underneath a fluctuating turbulent flow is possible with miniature diaphragm and stub gages (see Sections 5.9.3 and 5.10.3 where some gages built for this purpose are described). Willmarth and Yangzl describe measurements using 13 gages of 1.5-mm diameter by 0.5-mm-thick barium zirconate and barium titanate mounted flush in the surface of a 100-mm diameter cylinder concentric with a wind tunnel axis. A frequency response flat from 5 Hz to 50 kHz and a sensitivity of 13 p V Pa-', with simultaneous recording of 3 channels on magnetic tape gave both spatial and time data. Other studies with batteries of diaphragm gages in a wind tunnel wall, with optical recording, have also attempted to observe fluctuating pressure^.^^*^^ Despite the small size and close spacing of the pressure gages in these studies, the measurements still suffer from poor spatial resolution; still smaller gages are needed. In an attempt to improve the spatial resolution small pinhole microphones have been used. The pinholes and connecting channels to the microphones disturb the flow, however, and cause serious errors in fluctuating pressure meas~rements.~~ Although one thinks of a wind tunnel as a steady flow with superposed fluctuations, the pressure at the wall is essentially time dependent and the measurement of time dependent pressure is the subject of the remainder of this part.

5.4. Time-Dependent Pressure Measurements: Preview Nearly all of today's gages designed to measure pressure as a function of time consist basically of an element whose deformation varies with the magnitude of the applied pressure and a sensor which continuously converts some aspect of the deformation, such as displacement or strain, into an electrical output. An exception to gages with an electrical output, is one using laser interferometry and photographic recording (Section 5.6.3). Also purely mechanical gages are still being used but most can follow only extremely slow pressure changes and are therefore limited to essentially steady measurements (Chapter 5.2). Various gages differ from one another in the type of sensor employed W. W. Willmarth and C. S. Yang, J . HuiJ Mech. 41,47-80 (1970). Dinkelacker, M. Hessel, G. E. A . Meier, and G. Schewe, P h y s . Fluids 20, S216 (1977). 23 R . I . Soloukhin, Yu. A . Yakobi, and D . I Margulis, Z h . Prikla. M e k h . Tekh. Fiz. 1, 88-92 (1975). *' M. K . Bull and A. S. W. Thomas, P h y s . Nuids 19, 597 (1976). 21

** A .

526

5.

MEASUREMENT OF PRESSURE

and also in the mounting, the size and particularly the shape of the elastic element. A suitable choice depends on the amount and rate by which the pressure is expected to change. The main characteristics for judging the suitability of a gage are considered in the following chapter (Chapter 5.5). A variety of sensors, operating on quite different principles, can be used with different types of elastic element. Since there is little correlation between type of sensor and geometry of the elastic element, sensors are considered first in Chapter 5.6 with little reference to gage application. Following initial development of a signal by a sensor the signal is “conditioned” or “processed” (amplify, filter, digitize, etc.) for purposes of display and recording (Chapter 5.7). Dynamic calibration is considered in Chapter 5.8. With respect to the elastic element, gages for measuring time varying pressures belong to one of six basic types which differ from each other in the shape of the element, the support supplied by the mounting, and the way in which pressure is applied. The schematic diagrams of Fig. 12 illustrate these types which are designated diaphragm gage, stub gage, slab gage, probe gage, bar gage, and dilatational gage. As far as the shape of the elastic element is concerned, the probe and dilatational gage might have the same name, say block gage; they differ in that the strain in a probe gage is a volumetric change whereas during a limited time in which measurements are carried out with a dilatational gage the strain is strictly one-dimensional. The strain is also primarily one-dimensional in the slab gage; in this case it is because of the constraint provided by the mounting

P

t t t t t;t t 1 1 t

(a) Element Rigid Mount

TP 1

t t t t;t t t t

f e 1

=] l;f

FIG.12. Gages for measuring time varying pressures: (a) diaphragm; (b) stub; (c) bar; (d) slab; (e) dilatational; (f) probe. Diaphragm deforms by bending and stretching; other types deform by compression.

5.5.

GAGE CHARACTERIZATION

527

material. There is a major difference between the diaphragm gage and the other types; a diaphragm deforms by bending and stretching whereas the elastic elements of the other types deform primarily by compression. Theory and examples of diaphragm gages are considered in Chapter 5.9. The other types are grouped under the general heading Fast response gages in Chapter 5.10. This is not meant to imply that a moderately fast diaphragm gage cannot be made. Roughly, the time response of which a particular type of gage is capable decreases in the order diaphragm, stub or probe, bar,,slab and dilatational, with very little in favor of the stub or probe over the diaphragm.

5.5. Gage Characterization This chapter is concerned with methods for evaluating gage capabilities and with criteria for determining the suitability of a gage for a particular application. Definitions of terms are given.

5.5.1. Time-Dependent Response One way of indicating a gage’s ability to respond to changing pressure is to give its frequency response function, X(o);another way is to give U(t), its time varying response to a step function load. Other characteristics, such as response time, hold time and resonant period are less informative but useful. These are considered following a brief discussion of X ( w ) and

W t ).

The prevalent use of the term frequency r e ~ p o n s e ~ ~to- characterize ~’ the ability of a system to reproduce time dependent changes of the input reflects the mathematical power of Fourier analysis, or more generally, exponential transformation t h e ~ r y . ~By ~ . definition, ~~ a system’s frequency response function, X ( w ) , is the amplitude (and phase) of the output when the input is a steady-state sinusoidally varying function of time having unit amplitude, radian frequency w and zero phase. For a perfect W. Bleakney and A. B. Arons, Pressure measuring manometers and gauges. In “Physical Measurements in Gas Dynamics and Combustion” (R. Ladenburg, ed.), Artic. B2. Princeton Univ. Press, Princeton, New Jersey, 1954. O6 W. W. Willmarth, Unsteady force and pressure measurements. Annu. Rev. Fluid Mech. 3, 147 (1971). L. Bernstein, in “Measurement of Unsteady Fluid Dynamic Phenomena” (B.E. Richards, ed.), Chapter 3. McGraw-Hill, New York, 1977. I* I. N . Sneddon, “Fourier Transforms.” McGraw-Hill, New York, 1951. ** L. W. T. Thornson, “Laplace Transformation.” Prentice-Hall, Englewood Cliffs, New Jersey, 1950.

*’

528

5.

MEASUREMENT OF PRESSURE

system %(a) does not depend on o. In principle, when X ( w ) is known one can determine the output due to any physically realistic input by use of a Fourier series or inversion integral of the form

where 9 ( w ) is the Fourier component (i.e., transform) of the input and 00)is the output. Characterization by specification of X(o)is particularly useful for pressure sensitive systems, such as a microphone, which are designed to respond to a periodically varying pressure of small amplitude and low frequency. For gages whose function is to measure large amplitude, high frequency pressure oscillations such as occur in rocket motors, %(w) is useful but hard to determine.30 Gages planned for these purposes are usually complicated, making mathematical analysis difficult and often unrealistic. On the other hand, %(w) cannot be determined directly by experiment because no satisfactory sinusoidally varying pressure source of large amplitude and high frequency is available.31 Lack of a suitable periodic source for determining X ( w ) also presents a problem for gages whose purpose is to measure large, rapid, nonperiodic pressure changes, such as produced by explosions and projectile impacts. Furthermore, even if a theoretical expression for %(a) is accepted, evaluation of the inversion integral, Eq. (5.5.1) is difficult for any but the simplest types of nonperiodic input. An alternative to using X ( o ) is to specify the system’s response to a nonperiodic input which is simple enough to be mathematically tractable and realistic enough to permit experimental duplication. The input should also contain an abrupt change and be of long duration in order to test both the fast and slow response of the system. A step function meets these requirements. Analytical expressions for the response to unit step function load U ( t )are given for several simple types of gage in following sections. Experimental testing and calibration methods are considered in Chapter 5.8. If U(r) is known, a system’s response to any nonperiodic input can be obtained with the use of a relatively simple superposition (convolution) integral, one form of which is (5.5.2) R. Bowersox, ISA J . 5, 98 (1958). D. S . Bynum, R . L. Ledford, and W. E. Smotherman, “Wind Tunnel Pressure Measuring Techniques,” Advisory Group for Aerospace Research and Development, NATO, AGARD-AG-145-70. Available through NASA, Washington, D.C., 1970. 31

5.5.

GAGE CHARACTERIZATION

529

where O(r)is the response at time t to an input Z ( 8 applied at time F and U(t - .T)is the response at t to a unit step applied at F.32*33 Equation (5.5.2) is usually much easier to evaluate than the inversion integral, Eq. (5.5.1). Other types of input, such as a delta function or a ramp function (linear increase over a finite time to a constant value), are sometimes considered in mathematical analyses, but they are less suitable than a step for use as a standard, since corresponding pressure changes cannot be as satisfactorily produced by experiment. On the other hand, response to such inputs can be readily calculated with the use of Eq. (5.5.2), or an equivalent form, when U ( t ) is known. In finding an expression for U ( t )by mathematical analysis, using transformation theory, the first and easy step is the determination of X ( w ) . On substituting i/o for 9(w), the inversion integral of Eq. (5.5.1) then provides a formal representation of U ( t ) . Usually the most difficult part of the problem is the reduction of the inversion integral to a simple, readily useable form. Many reported analyses are not carried beyond a determination of X ( w ) . When the gage system is too complicated for a reliable mathematical analysis, or, more likely, to provide a check, a numerical representation of U ( t ) can be obtained directly from experimental records. As indicated previously, however, direct measurement of X(w) is often difficult, if not impossible. For cases in which U(t) alone is known, but for which X ( w )is more useful, it has been proposed that a numerical procedure on a computer be used to determine X ( w ) from U(t).30.31*34 One method of calculation is based on the following integral relationship:

(5.5.3) where S ( t ) is a step function. The denominator reduces immediately to i / w . If the transients of U ( t)decay so that U(t )can be considered constant after a finite time t , , the contribution from the integral in the numerator between the limits t, and 03 is given by ie'""/o. This leaves the finite integral from 0 to t, to be determined by numerical integration for specific values of o. In brief summary either X ( w ) or U(t)furnishes, at least in principle, 3p L. W. T . Thomson, "Laplace Transformation," p. 36. Prentice-Hall, Englewood Cliffs, New Jersey, 1950. 33 F. B. Hildebrand, "Advanced Calculus for Applications," p. 451. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 34 R. B. Bowersox and J. Carlson, Digital computer calculation of transducer frequency response from its response to a step function. Jet Propul. Lab., Prog. Rep. 20-331 (1957).

5.

530

MEASUREMENT OF PRESSURE

complete information for evaluating the time response cabability of a gage. Depending on the purpose of the gage, one or the other of these forms may be more appropriate, with the preference going to W t ) for gages measuring nonperiodic or large pressure changes. Lacking specification of %(a) or U ( t ) ,other characteristics, which can be described by a single number, can provide a partial guide. Among these are response time, hold (or dwell) time, and resonant period. The response time T may be described as the time needed to determine the steady state output of a gage following step function pressure loading and is a measure of the gage’s ability to follow rapid changes. Having reached a steady state value following step function loading the output of some gages begins to decay. The hold time T of a gage is the period during which the decay is acceptably small. To measure slow variations in pressure the hold time must be long and, strictly speaking, it should be infinite for steady measurements. A resonant period P, is the period of a gage’s free or characteristic vibration. The applicability of these characteristics and the exact criteria for assigning numbers depend on the type of response of a particular gage. The drawings of Fig. 13 illustrate typical types of response to step function loading. Drawing (a) resembles records from a dilatational gage (Section

output

U(t’

i^”-.I

:mTime - t

-

(b)

~=.+-‘-ii--output U(t)

I

Time

-

(C)

t

FIG. 13. Typical types of gage response to step function loading. T = response time; T = hold time; 9, = resonant period.

5.5.

GAGE CHARACTERIZATION

53 1

5.10.6); drawing (b), a bar gage (Section 5.10.5); drawing (c), an undamped diaphragm gage (Chapter 5.9). The feature of decaying oscillations, drawing (b), is also typical of records from damped diaphragm, stub (Section 5.10.3) and probe (Section 5.10.4) gages, although details of the oscillations may differ considerably from those shown: The terms “overshoot” and “ringing” are sometimes used to signify this type of behavior. For a response of type (a) the meaning of response time T is clear, but resonant period is not an applicable characteristic since, within the time of measurement, resonance has not yet developed. On the other hand, the record of type (c) consists entirely of resonant oscillations which begin immediately and, without damping, continue indefinitely. For this case response time is a vague concept with no generally accepted, exact definition. For the purpose of comparison with other types of gage, the response time may be considered to be simply some stated fraction or small multiple of the longest resonant period PI.When the spurious oscillations are small and decay with time as in drawing (b), the response time is a more useful characteristic and may be considered to be some specified large fraction of the rise time to the first maximum. The “acceptable decay” needed to determine the hold time T may be taken to be some stated small fraction of what is judged to be the steady state value; a precisely stated value of the fraction is relatively unimportant if an obviously spurious change in the response occurs suddenly as shown in drawings (a) and (b). Response time and resonant period provide good figures of merit for judging the effect of changes in the parameters of a particular type of gage. They also serve as a semiquantitative guide for comparing the capabilities of gages of different types, but for this purpose are much less informative than a knowledge of U ( t ) . Unfortunately, a graph or analytical expression for U(t) is seldom given in specifications for commercial gages.

5.5.2.Sensitivity and Range Disregarding distortions in time, the change in output of a gage due to a change in pressure from p o to p is given by ro

(5.5.4)

where Y ( p ) ,the rate of change of output with respect to pressure ( a O / d p ) , is the sensitivity, which in general is a function of p . The range is the difference between the maximum allowable pressure, pmaxand the lowest (zero), or usually simply pmax. Primarily to simplify data reduction by

532

5.

MEASUREMENT OF PRESSURE

eliminating the need for evaluation of a cumbersome integral, gages are usually designed so that 3 p ) will be independent of pressure over their useable range, thus providing a linear response. PO(p) = Y ( p - P o)

for 9’const.

(5.5.5)

Often changing a parameter to improve one of a gage’s characteristics automatically degrades another characteristic. Sensitivity and range are likely to be competing characteristics. For example, in the case of a diaphragm gage decreasing the thickness of the diaphragm keeping its other dimensions unchanged increases sensitivity but reduces the pressure at which the response becomes nonlinear (or, beyond this, the pressure at which the diaphragm deforms permanently or ruptures) thus reducing pmax. Other characteristics competing with sensitivity are response time or hold time. It is a rule, almost without exception, that cutting down the response time to improve the capability of measuring rapid changes will result in a decrease in sensitivity. Thus scaling down all dimensions of an elastic element and sensor will reduce the response time (or resonant period), but it will also decrease the sensitivity, although it will not change the range. Also gages having very short response times often have short hold times, so they cannot be used for measuring slowly varying or steady pressures and must be calibrated dynamically. Fortunately, in practice, if the requirement for one of a pair of conflicting characteristics is stringent the requirement for the other is often lenient. For example, except during severe storms, changes in atmospheric pressure are small and slow, so that an ordinary barometer must be relatively sensitive and have an essentially infinite hold time but its response time can be long-of the order of seconds and greater-and its range small. On the other hand pressure changes due to sonic booms, explosions, etc., are extremely rapid-occurring within microseconds or less-but they are usually large and do not continue over a long period, so that although a gage must have a very fast response time and large range its sensitivity can be low and its hold time relatively short.

5.5.3. Pressure as a Function of Position A single gage measures average pressure over a finite area. In order that this correspond as nearly as possible to the idealization of pressure at a point, the size of the sensitive element should obviously be as small as possible. A decrease in size of a particular type of gage thus increases its space resolution, as well as its time resolution, but at the expense of a decrease in sensitivity, which sets a practical lower limit on size.

5.5.

GAGE CHARACTERIZATION

533

Arrays, composed of a large number of gages, are used to measure pressure distribution over an extended region in space. A system which records continuously from each gage, and thus simultaneously for all gages, usually requires an inordinate amount of expensive equipment. Two schemes, each involving a compromise between time and space measurement, are used to reduce the equipment requirement. In one, called multiplexing, the electrical or mechanical responses from the individual gages are recorded in succession, each for a short period of time.31 In the other scheme, optical sensors are used to produce a photograph from which the response of all gages at a particular instant can be determined (Section 5.6.3). Either scheme is quite easily adapted to the study of steady state phenomena.

5.5.4.Environmental Effects A part of the problem of pressure gage design is eliminating or compensating for response to changes other than those of pressure. Among these are changes of temperature, of external electric and magnetic fields and vibrations of the base on which the gage is mounted. An unwanted change may occur either during a measurement and cause a spurious response directly, or between measurements and lead to a lack of repeatability (long term stability). For example, a piezoelectric sensor may also be pyroelectric so that a change in temperature accompanying a change in pressure will produce a spurious response during a measurement. But even if the sensor is not pyroelectric a change in ambient temperature between measurements may change the values of the piezoelectric constants and consequently the pressure sensitivity of the gage. Gage properties may also change over a long time due to ageing. Even repeated changes in pressure itself occurring during application can cause changes in proper tie^,^^ particularly mechanical deterioration leading to a short life. 5.5.5. Accuracy The problem of attaining and being assured of a given accuracy is in general the same for a pressure gage as for any measuring instrument. If a pressure gage is unique, it is because its operation is usually based on the deformation of a supposedly purely elastic element. Since no material is strictly elastic, to assume that it is can lead to systematic error. A material can be used only well below its yield o r fracture point and should not behave anelastically, ie strain should depend only on the magnitude of A . H . Meitzler, R e v . Sci. Insirurn. 27, 56 (1956).

534

5.

MEASUREMENT OF PRESSURE

the applied stress and not on the rate of application, otherwise hysteresis ~’ solids in particular are suspect and or relaxation will O C C U ~ . ~ ~ ,Polymer should be carefully chosen and tested. A s for all measuring instruments, the sensitivity must be large enough so that the smallest readable division corresponds to a pressure increment less than the required accuracy; also the sum of the random errors, including noise, should not be appreciably greater than this least count. Probably the best method of assuring that there is no large unsuspected systematic error is to compare readings with those of a gage of an entirely different type. Calibration should ultimately refer to values obtained with a simple, absolute gage, such as a liquid manometer or piston gage (Chapter 5.2). Most gages have an accuracy between 2 and 10 percent. An accuracy of 10 percent is usually relatively easy to obtain but extreme care, including frequent calibration, is needed for an absolute accuracy less than 2 percent.

5.5.6. Ease of Construction, Calibration, and Operation Although there is no quantitative measure of simplicity, its advantages are obvious. Complexity leads to high construction costs, both in time and in money, and is likely to increase the need for lengthy and frequent calibration. Ease of operation decreases the chance of mistakes.

5.6. Sensors This chapter emphasizes physical phenomena basic to sensor operation and does not purport to be an exhaustive compendium of all proposed devices. It includes a selection of commonly used or unique means of measuring the strain within an elastic element or the displacement or velocity of one of its surfaces. Most pressure gages use a sensor which responds directly to strain or displacement. There is at least one important case (Section 5.10.6.2), however, in which the velocity of a surface, rather than its displacement is proportional to the applied pressure and in this case a velocity sensor is the most suitable. There are also accelerometers which are sometimes used as part of a “force transducer” to measure the components of force due to the integrated pressure over the surface of a body such as an artilC . M. Zener, “Elasticity and Anelasticity of Metals.” Chicago Univ. Press, Chicago, Illinois, 1956. Wiley, New York, 1956. 37 J. C. Jaeger, “Elasticity, Fracture and Flow.”

5.6. SENSORS

535

lery shell or an air vehicle. But the sensors employed in an accelerometer measure the instantaneous displacement of an inertial mass working against an “elastic spring” of some type and so involve no new principles. Special problems associated with accelerometers and force transducers are considered in previous review^^^.^^ and are not treated here. Acceleration is important in another way, however. Most pressure gages are attached to a base which supposedly is perfectly rigid and stationary, but which in fact often vibrates. If the mounting does not isolate the gage from vibrations of the base and if the sensor responds to acceleration of the gage, spurious oscillations will appear in the recorded signal (Section 5.10.3). For descriptions of sensors not considered here and for more detailed discussions of those which are, refer to earlier reviews26*27.31 and their bibliographies. Some and a compendium of commercial gages,*O although older, may prove useful. Sensors can be classified as mechanical, electrical, or optical. 5.6.1. Mechanical Sensors

The liquid column (Section 5.2.1) and dead weight tester (Section 5.2.3) fall into this category. A more convenient type consists of a system of levers and gears which translates the displacement of a point on an elastic element (Bourdon tube, capsule) into movement of a pointer along a scale (Section 5.2.2). Construction of sensors of the lever-gear type might best be described as a watch maker’s art and, like the spring-driven watch, they are reliable and simple to operate. Also like the spring-driven watch, and probably for much the same reasons, many gages of this type are still in use and only gradually being replaced by those with electronic mechanisms. But, as previously noted, they are only useful for measuring steady or slowly varying pressures. The fact that large pressures produce plastic flow of soft metals with resulting permanent deformation has been used to measure peak values of nonperiodic pressure changes.25 For blast pressures produced by explosions, a number of holes of different diameters can be drilled through the side of a heavy steel box and covered with a sheet of soft aluminum to form a cluster of diaphragms. After an explosion near the box, the largest unbroken and the smallest broken diaphragm serve, with suitable calibration, to bracket the peak pressure. Or the peak pressure is revealed in 38 H. K . P. Neubert, “Instrument Transducers.” Oxford Univ. Press, London and New York, 1965. 39 K . S. Lion, “Instrumentation in Scientific Research.” McGraw-Hill, New York, 1959. G . F. Harvey, ed., “Transducer Compendium.” ISA/PIenum, New York, 1969.

536

5.

MEASUREMENT OF PRESSURE

a continuous fashion by the amount of permanent bulge of an unbroken diaphragm. For gun pressures, an undersized cylinder or sphere of soft copper can be placed in a cylindrical cavity in the wall of the gun barrel or shell chamber. A nonleaking piston of hard steel touching the copper piece closes the cavity and causes permanent deformation of the copper when pressure is applied. The amount of the permanent set is related to the peak value of the pressure. A purely mechanical method of measuring the amplitude and time variation of an unsteady pressure in the form of a pulse, such as produced by an explosion, was developed shortly after the turn of the century by Hopkinson. It is described in Section 5.10.5.1 because it is ingenious and unique. 5.6.2. Electrical Sensors

Electrical sensors may be characterized in several ways. They may be passive or active depending respectively on whether an external source is or is not required.*’ They may also be either intrinsic or extrinsic.26 An intrinsic sensor (sometimes called “molecular”) depends on an electric or magnetic change (resistive, piezoelectric, magnetostrictive, etc.) which takes place at the atomic-molecular level within a material; the associated property (resistivity, piezoelectric charge per unit volume, etc.) is independent of a body’s size. An extrinsic sensor (sometimes called parametric) operates because of the macroscopic movement or change in shape of a body; the related property (resistance, capacitance, inductance) is a “lumped” electrical parameter whose value is a function of a sensor’s overall dimensions. All sensors are extrinsic, but some operate primarily because of an intrinsic change. A third division is based on the quantity (displacement, strain etc.) to which a sensor directly responds and a fourth depends on principle of operation (Ohm’s law, Faraday’s law of induction, etc.). The following divisions are based roughly on principle of operation. 5.6.2.1. Resistance Sensors: Current-Voltage Dependence. Passive sensors of a large class depend on a change in their current-voltage characteristic. Most are resistance sensors for which current I and voltage V are related by

V = RI

(5.6.1)

and sensing is accomplished by finding the change in the total resistance

R.

Some old types of sensor consisted of a rheostat or potentiometer with a mechanically sliding ~ o n t a c t , but ~ ’ nearly all of today’s sensors respond

5.6. SENSORS

537

FIG. 14. Element of a resistance or capacitance sensor. Shaded areas are electrodes, i.e., surfaces of constant voltage.

to a change in geometry of the resistance element, which may or may not be accompanied by a change in the resistivity p of the material. A slab of homogeneous, isotropic material with plane parallel electrodes as shown in Fig. 14 has total resistance given by

R = pl/A,

(5.6.2)

where A is the effective cross-sectional area of the element and 1 is the distance between the electrodes. Sensors for medical applications have been made with a liquid as the resistive material.41 In this case I is made small compared to the dimensions of the electrodes so that the electric field is essentially confined to the region between the electrodes. Provided changes in 1 are not too rapid, the liquid merely moves in or out of the electric field, p and A can be considered constant and R is a linear function of 1 alone. Ordinarily resistance sensors use a solid for the resistive material and are called strain sensors. Particularly in gages designed to measure pressures in the gigapascal range, the sensing element and the mechanical element to which pressure is applied may be physically the same. The value of 1 is comparable to the dimensions of A . For this type it is convenient to define sensitivity 9, in terms of applied pressure instead of resulting strain

ARIR YP = AP ’

(5.6.3)

where Ap is the change in pressure. If the pressure is transmitted equally 41

J . R . Pappenheimer, Rev. Sci. fnsfrum. 25, 912 (1954).

538

5.

MEASUR.EMENT OF PRESSURE

to all faces of the element (hydrostatic stress) and if the element remains elastic,

Y*= c + (3B)-’,

(5.6.4)

where C is a piezoresistive constant (fractional change in resistivity per unit change in stress) and B is the bulk modulus of the material. See the last part of Section 5.10.4 for a method of calibrating resistance sensors at high pressures and for representative values of 9,for several materials. The most common type of resistance strain sensor has the form of a thin filament with electrical connections at the end^.^*-^^ The filament may be a fine wire, a thin foil, a deposited film or a region of impurity concentration within a semiconductor. The filament is attached to or made an integral part of an elastic body and ideally, without introducing constraint, undergoes the same strain as the body, or at least undergoes the same strain as the body along one or two directions. This ideal is almost exactly satisfied in case the body is a pure semiconductor in which the filament is formed by impurity doping and can be closely approximated by a film deposited directly on the body, or on a previously deposited insulating film. For some older type sensors where the filament is embedded in an insulating sheet (e.g., paper) glued to the body, the bond may not hold. Either the assumption of no slip may not be justified or the bond may break during use. Another drawback of stick-on sensors is that they are difficult to apply to extremely small areas. Conventionally, the sensitivity of resistance strain sensors is given in terms of the so-called gage factor

G=-ARIR E

’

(5.6.5)

where E is the strain sensed. Sometimes the expression for the change in resistance AR,is quite complicated. In the case of a single crystal semiconductor, AR depends on the orientation of the filament relative to the crystal axes, on changes in the various components of strain and on accompanying piezoresistive changes in the resistivity tensor. Although bothersome, this complexity can be an advantage to a designer, because of the large number of parameters at his disposal. In other cases, a sensor may be highly sensitive only to one component of strain in the

‘*

C. C. Perry and H. R . Lissner, “The Strain Gage Primer.” McGraw-Hill, New York, 1955. M. Dean and R . D. Douglas, “Semi-Conductor and Conventional Strain Gages.” Academic Press, New York, 1962. H. K. P. Neubert, “Strain Gauges.” Macmillan, New York, 1967.

5.6.

SENSORS

539

body, usually in a direction along the length of the filament, because of the relative strength of the filament and the way it is attached. Often A l / l , where 1 is the effective length of the filament, is used for E in Eq. ( 5 . 6 . 3 , with the understanding that AR depends on other factors (such as transverse constraint) as well. This can, however, be misleading if one or more of the other factors does not remain constant during a measurement or is not directly related to Al/l; A R / R may change even when All1 is zero. Wire strain sensors have a gage factor of about 2 which is mainly attributable to a change in geometry rather than to a change in resistivity. The wire is usually a metal alloy having a low temperature coefficient of resistance to minimize temperature dependence. On the other hand semiconductor resistance sensors (e.g., silicon doped with N- or P-type impurities) have gage factors in the range 50 to 150. This large sensitivity is due to the fact that the material is highly piezoresistive and so undergoes large changes in resistivity; the sensitivity to strain is a function of impurity concentration. The principal disadvantage of semiconductors compared to metals is their much greater temperature sensitivity. The temperature sensitivity of a semiconductor is also a function of impurity concentration; unfortunately it increases as the strain sensitivity increases. To reduce temperature dependence, some strain sensitivity is usually sacrificed and the circuit is designed to provide temperature compensation. The change in resistance is sensed either by providing a constant current (perhaps with a constant voltage source in series with a large ballast resistance) and by measuring the change in terminal voltage or by keeping the terminal voltage constant and measuring the change in current. A dc source is usually preferred for very fast response gages since dc provides continuous recording, but an ac source can be used if its period is considerably less than the response time of the gage. Excitation by ac has the advantage that unwanted pickup is less troublesome. A bridge network can be useful, particularly when two or four sensors are used to increase sensitivity or to provide temperature compensation. For example, consider four sensors attached to a diaphragm as shown in Fig. 15. As indicated by the arrow tips, the sensors are directionally sensitive. As the diaphragm distorts, the resistance of sensors 1 and 2 increases since they sense tension whereas that of sensors 3 and 4 decreases since they sense compression. When connected as shown in Fig. 15c all sensors contribute to a change in output due to pressure change but there is very little output due to equal temperature changes of the sensors. An unusual type of sensor which is not a resistance sensor, but which depends on a change in current-voltage characteristic is a so called piezo-

5 40

5.

;;;,

M E A S U R E M E N T OF PRESSURE

Source

FIG. 15. Disposition of resistance sensors on diaphragm to enhance pressure sensitivity and minimize temperature sensitivity. R , and RPincrease (tension) and RS and R4decrease (compression) as the diaphragm deflection increases. (a) section through diaphragm; (b) plan of diaphragm; (c) connections in bridge circuit.

junction.45 This is an N-P semiconductor junction whose characteristic is sensitive to applied stress. It is used as the emitter-base junction of an N-P-N transistor so that the applied stress controls the current from emitter to collector, thus exploiting the gain of the transistor. The dimensions of the active semiconductor region are only a few micrometers, so that the sensor can be small and the gage resonance frequency high. 5.6.2.2. Capacitance Sensors: Charge-Voltage Dependence. Equations for capacitance are formally the same as those for the conductance of resistance sensors. Thus Eq. (5.6.2) becomes the equation for the capacitance C of a parallel plate capacitor upon substitution of C for the conductance R-', and permittivity KeOfor the conductivity p-'. C = K&oA/I.

(5.6.6)

See Fig. 14. The circuit relation for the operation of the resistance sensor is V = RZ, Eq. (5.6.1): the analogous relation for the capacitance sensor is Q

=

CV,

(5.6.7)

where V is the voltage difference between the electrodes and + Q and - Q are the charges on the electrodes. Both the resistance and the capacitance sensor respond to relative displacement of the electrodes. An important difference is that a resistance sensor must have a material, with mass and usually some strength, between the electrodes, but a capacitance sensor need not. Another difference is that a resistance sensor is a heat generator, whereas a capacitance sensor is not. An air or vacuum spaced capacitor is commonly used to sense the deformation of a metal diaphragm, which serves as one of the electrodes. 45

W. Rindner and R. Nelson, Proc. IRE 50, 2106 (1962).

5.6.

54 I

SENSORS

Although the displacement, w = A I , varies across the face of a diaphragm, its maximum value wo can be used as a parameter related both to C and to the applied pressure. For small values of w o and change in capacitance AC AC = (dC/dl)wo.

(5.6.8)

In other applications, the capacitor has an elastic dielectric solid between the electrodes and is basically a strain sensor. Pressure may be applied either directly to the electrodes or to a larger elastic element in which the capacitor is embedded. Formally, AC is again given by Eq. (5.6.8) with the understanding that K and A may, but usually do not, depend on the strain A l / l . In analogy to the gage factor for a resistance strain sensor, Eq. (5.6.5), the sensitivity Y for a capacitance sensor is

y=- AC/C &

= -

A(l/.C)/(l/C) E

,

(5.6.9)

where E = w / l = Al/I. For a sensor in which the electrodes remain plane and K and A do not change, Y = 1. A capacitance sensor requires an external electrical source. For measuring rapid pressure changes a dc source in series with a large ballast resistor can be used to charge the capacitor. The charge remains essentially constant during rapid changes in capacitance which, according to Eq. (5.6.7),produce changes in voltage across the electrodes. These are amplified and measured. The main difficulty is that the capacitance is usually small and may be the same order of magnitude as the distributed capacitance of connecting cables and associated circuitry; electrostatic pickup and reduction in sensitivity often loom as major problems. To minimize these problems the capacitor plates are connected to a nearby miniature amplifier with a high impedance input which develops a voltage signal across a lower impedance for remote monitoring and recording. Because of charge leakage and low frequency disturbances, the constant charge method is unsuitable for sensing steady and very slowly varying pressures. At this extreme an ac source, usually feeding into a bridge containing one or two capacitance sensors, can be used. To determine capacitance changes, one can either monitor changes in bridge output or use null balancing. Fig. 22b displays the bridge circuit used widely for capacitance sensing of diaphragm deflection. Instead of deriving a voltage change in the constant charge mode, or a change in C when the capacitor is in an ac bridge, sometimes the capacitor is used with a fixed inductor L to determine the resonant frequency of an oscillator. The frequency o = (LC)1’2changes as C changes. The signal

542

5.

MEASUREMENT OF PRESSURE

is transmitted, or radiated by an antenna, to a remote receiver where the value of C is derived by frequency demodulation. The capacitance sensor is simple, can be made with low inertia and little inherent strength, and is not a heat source. 5.6.2.3. Piezoelectric Sensors: Voltage-Charge Generation. The basic element of a piezoelectric sensor is a dielectric lacking a center of symmetry and having the property that an applied stress produces an internal electric field terminating on positive and negative surface charges. Sheets of conducting material placed adjacent to appropriate areas of the element serve as capacitor electrodes whose open-circuit voltage is controlled by the piezoelectric field. No external electric source is needed.27.4s-48 There are three types of piezoelectric sensor. In one type the basic element is a properly cut single crystal of a piezoelectric material, such as quartz, tourmaline, etc. Single crystal plates or bars are naturally polarized in a direction determined by the crystal structure. In a second type, the basic element is a ceramic composed of tiny crystals of a ferroelectric material such as barium titanate, lead zirconate, etc. embedded in clay. The ferroelectric clay mixture is molded into shape and fired. It is then subjected to a strong electric field, its temperature is raised above the Curie point of the ferroelectric and it is allowed to cool slowly to a temperature below the Curie point while the electric field is maintained. When the electric field is then removed, the element is left in a polarized state, with the direction of polarization the same as that of the previously applied field, and it remains in this state unless its temperature is again raised above the Curie point. The third type comes in the form of a thin plastic film made from a polymer such as PVFz (polyvinylidene f l ~ o r i d e ) .Piezoelectric ~~ films of PVFzhave been suggested for practical use only recently and their preparation at present is as much an art as a science. In general there are three steps. First, following polymerization of the vinylidene fluoride monomer, which usually contains impurities whose role is still a subject for debate, the resin is pressed into sheet form. Next it is stretched, either uni- or biaxially, to form an oriented, semicrystalline matrix whose W. G . Cady, “Piezoelectricity.” McGraw-Hill, New York, 1946. J. F. Nye, “Physical Properties of Crystals.” Oxford Univ. Press, London and New York, 1957. W. P. Mason, “Piezoelectric Crystals and Their Applications to Ultrasonics.” Van Nostrand-Reinhold, New York, 1950. * A. L. Robinson, Science 200, 1371 (1978). 47

5.6.

SENSORS

543

crystallites are thought to be mainly in a particular phase, called the pphase. Finally, the film is subjected to a large electric field perpendicular to the plane of the film. If this has been done at an elevated temperature, which is not always the case, the film is allowed to cool and the field is removed. With removal of the field, the film is left with a large permanent electrical polarization across its thickness and will behave in the same way as a poled ceramic. For all three types, the electrical detection system may be a voltage or charge amplifier connected, usually through a network including a cable, to the electrodes of the piezoelectric sensor. In part because of charge leakage between the electrodes of the piezoelectric element in the case of a voltage amplifier or across the feedback capacitor of a charge amplifier, piezoelectric sensors are not suitable for measuring steady pressures. They can only be used to measure changes taking place in an interval much less than the effective time constant (RC) of the circuit, but in some cases this can be quite large (greater than 100 s). They are ideally suited, however, for use in fast response gages (milliseconds and less). The open-circuit voltage V and the short-circuited charge Q developed by a single crystal or electrically poled element, can be calculated by the following equation (the open-circuit voltage and the short-circuited charge are, respectively, the quantities measured by a voltage and a charge amplifier):

where & is the electric field produced by unit voltage applied to the electrodes of a mechanically free element, d& are the stress related piezoelectric constants, c { k are the stress components in an electrically free material and Co is the geometric capacitance of the element. The integral in Eq. (5.6.10) is to be taken over the volume 'V of the element and the summation convention is implied by the repeated subscripts. Because of symmetry not all of the piezoelectric constants are independent and many are zero. As Eq. (5.6.10) indicates, V and Q depend on the detailed distribution of stress, including shear as well as normal components, within the element. They also depend, through &, on the orientation of the conducting electrodes relative to the crystal or polarization axes. In most pressure applications, the piezoelectric element is a slab of thickness 1, between plane parallel conducting plates whose normal will be taken in the direction of the unit vector ii. Since the relative permittivity K of the element is usually large, the direction of E will to a fair approximation be along ii and its magnitude will be I-'. Using for ajkthe

544

5 . MEASUREMENT OF PRESSURE 0x1s of polarizai ion

axis of polarization

OXIS o f polarization

3- x

3- x

z I-Y

eio

I-Y

I-Y

(a) axis o f polarizo tion

3-x

directions of

P

I-Y

(e) FIG.16. Operating modes for piezoelectric elements. ( I , 2, 3) denote axes for a polarized ceramic or polymer. (A', Y , 2)denote axes for a single crystal.

six component notation of Mason,4B,*Eq. (5.6.10) becomes Q=- &ntA

1

v=

- -1 1

I,.

n<&Uj d7f

(j= 1, 2 , 3, 4, 5 , 6 ) (5.6.11)

(i = 1, 2 , 3),

where A is the area of one of the electrodes and the Ki are permittivity constants. Using linear stress-strain relations, Eq. (5.6.11) can be , strain components, and written in terms of strain by replacing ujby E ~ the dg by d t the piezoelectric constants related to strain. For relations between systems using different sets of dependent variables see Mason.5o Several idealized modes of operation are shown in Fig. 16 where the axes are those of a polarized ceramic or polymer (1, 2, 3) or of a single so

W. P. Mason, Bell S y s t . Tech. J . 26, 80 (1947).

* In this notation, stress is represented by a first rank tensor whose six components are related to the components of the more common second rank tensor in the following way: 0 1

=

UII,

US

us = um, =

031

=

0 3

UlJt

= u s s , u, = USBp = ups, = UBpI = ule

5.6.

545

SENSORS

TABLE11. Piezoelectric Components for Electrically Poled Materials” i

j

2

1

3

4

5

6

Entries in the table are values of the components du in the notation detailed in the text.

crystal (X, Y , 2). The axes of the ceramic and crystal are labeled differently because the components of the piezoelectric tensor are given in terms of these axes and by convention the polarization direction of a ceramic is taken along the 3-axis, but for a crystal it is along the X axis, or by number designation the 1-axis. Thus the d33component for a ceramic is the dll component for a crystal and there are similar confusions with other components. Unfortunately, these conventions are too entrenched to be disregarded. The charge sensitivity to stress, defined by 9’ = Q / A u , can be calculated for different modes with the use of Eq. (5.6.1 l), on the assumption that stress does not vary throughout the element. As previously mentioned, the problem is simplified by the fact that, due to symmetries of the material, many of the components of the piezoelectric tensor are zero and relations exist among the nonzero component^.^^,^^ All electrically poled ceramics and polymers have transverse isotropic symmetry and the piezoelectric components have the values shown in Table 11. Quartz, which is the most used of the single crystals, has trigonal rhombohedra1 (D3) symmetry and the piezoelectric components have the values shown in Table 111. The scalar integrand of Eq. (5.6.11) becomes, for a poled material,

YU

=

&d@j

=

nid15V5

+ nzdl@4 - n,(d31ul + daiUz - d33CT3),

(5.6.12)

+ dl4V4) - nz(d14V5 + dIlu6).

(5.6.13)

and for quartz, 9’U = QdUUj

=

nl(d11Vi

-

d11Uz

In pressure gage applications, the shear mode Fig. 16c, which for poled material has 9’= dI5(in this case ul = u2= u3 = u4 = u 6 = 0, u5# 0; n2 = n3 = 0 , nl = l), is seldom used. st W. P. Mason, “Electromechanical Transducers and Wave Filters.” Van NostrandReinhold, New York, 1948.

5.

546

MEASUREMENT OF PRESSURE

TABLE111. Piezoelectric Components for Quartz”

.i 1

1

2

1

2

3

4

5

6

4I

-dii

0 0 0

d14 0

0 -dl4 0

0 -dii 0

0 0

3 ‘I

0 0

0

Entries in the table are values of the components du in the notation detailed in the text.

The hydrostatic mode is used in probe gages which are mounted in the interior of a fluid rather than on a boundary. In this case, Fig. 16d, (ar = (+I = u 6 = 0; u1 = o2= u3= - p ) the pressure sensitivity for a 2 4 , - d,, but for quartz it is, poled material is, from Eq. (5.6.12), Y’= from Eq. (5.6.13), 9’= d,, - dil = 0. Thus, since in general d, # 2ds1, a poled material can be used in the hydrostatic mode but quartz cannot. Single crystal tourmaline, unlike quartz, is hydrostatically sensitive and has been used in probe gage applications. A bimorph, Fig. 16e, which responds to bending, consists of two bonded piezoelectric elements, each of which operates in a transverse mode, Fig. 16b. If the polarization vector of each element had the same sense, bending of the bimorph would produce no voltage between electrodes on its outer faces, since the effect of contraction of one element would exactly balance that due to extension of the other. A signal can be obtained either by bonding the elements so that their polarization vectors are oppositely directed, Fig. 16e, or, with the polarization vectors having the same sense, by using a third electrode in the bonding surface as one terminal and the outer electrodes connected together as the other terminal (parallel connection). For sensors having comparable dimensions, thin bimorphs have an advantage relative to those operating in a thickness or longitudinal mode of higher overall sensitivity, but they have the disadvantage of a lower natural frequency. Fast response gages designed to measure pressure at a wall have customarily used a sensor made from a poled ceramic or X-cut quartz operating in the mode in which the applied pressure produces a compressive strain.” X-cut quartz has the normal to one pair of faces parallel to the X-axis. The element is shown in Fig. 16a with the normals to the other pairs of faces along the Y- and Z-axis. With this orientation, the quartz * The name “thickness mode” usually implies that its thickness is much less than at least one of its lateral dimensions and that its major resonant frequency is determined by its thickness. The name “longitudinal mode” implies that the major resonant frequency is determined by the greater lateral dimension.

5.6.

s47

SENSORS

element will respond in the transverse mode (Fig. 16b) (0,= - p , u3 = 0 or u2 = 0, u3 = - p and u1 = u4= u5= uB= 0; n, = 1, n2 = n3 = 0), only to pressure applied to the faces with their normal along the Y-axis but not to pressure applied along the Z-axis. This is seen from Eq. (5.6.13), with Y = dll when u2 = - p . u3= 0 but Y = 0 when uz= 0, u3= - p . Some other crystals, such as Rochelle salt, when cut with faces having normals along theX-, Y-, Z-axes, will not respond to pressure on any pair of faces, and it is sometimes said that Rochelle salt is sensitive only to shear stresses. However, if a Rochelle salt crystal is cut so that the normal to the electrode surfaces is along the X-axis and the normals to the other two pairs of faces are at 45” with respect to the Y- and Z-axis, it will respond to pressure on either pair of these surfaces. For a discussion of crystal cuts see For the purpose of comparison of ceramic and quartz sensors, nominal values for the compressive sensitivity of two poled ceramics (us= - p , u1 = u2= u4 = u5= cre = 0; nl = n, = 0 , n3 = 1) and of X-cut quartz (u,= - p , u, = u3 = w4 = u5 = u6 = 0; n , = 1, n2 = n3 = 0) are52 barium titanate lead zirconate titanate (PZT-5) quartz

Y = d33

Y

=

d33

=

149 PC N-’

= 593 pC

Y = dll = 2.3

. N-’

pC * N-’

Although the sensitivities of the poled ceramics are much greater than those for quartz, quartz has the advantages that it will respond linearly over a greater pressure range, will withstand higher pressures and temperatures without damage, has a less temperature dependent sensitivity, and in general has a higher resistivity and better long term stability. The piezoelectric property of the polymer PVF, has recently become of interest, although the polymer has for some time been used as a protective coating for metals because, like Teflon, it is chemically inert and electrically i n ~ u l a t i n g .Interest ~~ in its piezoelectric property stems in large part from the fact that it can be formed into very thin (6-50 pn), nonbrittle, highly flexible sheets. For example, ceramic and crystal sensors have long been used in hydrophones designed to detect bodies emitting or reflecting underwater acoustic waves, but they are highly susceptible to breakage under conditions in which they are likely to be used; the nonbrittle characteristic of PVF, provides an advantage. Since thin sheets of PVF, are light, flexible and of small tensile strength, they easily conform to a body of any shape and when attached virtually lose their mechanical sz W. P. Mason, “Physical Acoustics and Properties of Solids.” Van NostrandReinhold, New York, 1958. Values for more recently developed ceramics are found in manufacturer’s literature.

548

5.

MEASUREMENT OF PRESSURE

identity. Unlike a crystal or ceramic they do not appreciably constrain the body or impose their own resonant characteristics. Electrically poled PVFz, like poled ceramics, is not only piezoelectric, but is pyroelectric (a change in temperature produces an electric field) as well and this is a disadvantage for many pressure gage applications. 5.6.2.4. Electromagnetic, Magnetostrictive and Inductance Sensors: Change in Magnetic Flux. Faraday’s law of induction provides the operating basis for a large variety of possible sensors. According to the law, an emf is induced in a coil of conducting material whenever there is a change in the total magnetic flux linking the coil, the magnitude of the emf being proportional to the time rate of change of flux. In some sensors a pick-up coil is placed in a magnetic field produced by a permanent magnet; in others, the field is produced by a dc electric source maintaining a current through an excitation coil. A change in flux through the pick-up coil can be due to a movement of the permanent magnet or of a separate excitation coil, of a piece of magnetic material in a magnetic circuit, or of the pick-up coil as a whole or in part, thus allowing many possible sensor arrangements. Such devices are referred to as electromagnetic In general, the response of this type of device to the movement of a mechanical element to which pressure is applied is proportional to the velocity of the element rather than its displacement, because the measured emf is proportional not to the change in flux but to its rate of change. Since the pressure usually produces a proportional displacement (or relative displacement), as in the case of a diaphragm, electromagnetic sensors, in their primary form, are unsuitable for most pressure gage applications. They cannot be used to measure a steady pressure difference and to measure a pressure change the record must be integrated over time. The magnetic analogue of the piezoelectric sensor is the magnetostrictive sensor. Ferromagnetic materials, particularly nickel and nickel-iron alloys, have the property, when magnetized, that struin produces a change in magnetization. This results in a change in magnetic flux which can be detected with a coil magnetically linked to the material. Although magnetostrictive have been developed to measure pressure changes, they are rarely used today. Compared to piezoelectric sensors, they suffer from low sensitivity, nonlinearity and long term instability. Also, if one senses the induced emf by measuring the open circuit voltage across the terminals of the pick-up coil, the response is proportional to rate of strain rather than to strain and, like an electromagnetic sensor, the device has the disadvantage of being basically a velocity sensor. 53

A. W. Smith and D. K . Weimer, Rev. Sci. fnsrrum. 18, 188 (1947).

5.6. SENSORS

549

Unlike electromagnetic and magnetostrictive sensors, an inductance sensor responds to displacement. Physically an inductance sensor employs Faraday's law as does an electromagnetic sensor; the distinction is that an ac source is used for an inductance sensor, but a dc source for an electromagnetic sensor. The essential aspects are displayed in a simple series circuit consisting of a coil linking a magnetic circuit and having an inductance L , a resistor R , and an electric source 8. Suppose the magnetic circuit linked by the coil contains a part whose position changes with pressure, as in the case of the diaphragm gage shown in Fig. 22a, and that this change in position w causes a change in the self-inductance L of the coil. The equation for the series circuit is dl L-+RZ=-v dt

(5.6.14)

where I is the current and v is the velocity of the moving part. For a dc source, 8 is constant and we have approximately for Z(t), the time varying component of the current, (5.6.15)

where Zo(= 8 / R ) is the steady state component of the current. The driving emf, given by the right-hand side of Eq. (5.6.13, depends on the velocity of the moving part. On the other hand, if 8 varies periodically and L is a slowly varying function of time (v is small), the term in Eq. (5.6.14) containing v can be neglected and we have

L

=+ dt

RZ(t) = 80).

(5.6.16)

The effect of a change in L is to alter the amplitude and phase of the current. In other words, the ac source provides a carrier signal which is modulated by a change in L ; L in turn depends on the displacement rather than the velocity of the moveable part. An inductance sensor can measure a static pressure difference and follow a smooth variation in pressure as a function of time. Its response time can be no smaller than the period of the ac source.

5.6.3.Optical Sensors Specular reflection, the doppler effect, the photoelastic effect and various types of interference phenomena, including holography, have been suggested as a basis for using light to sense displacement, strain, or velocity. Recording is accomplished by visual observation, by pho-

550

5.

MEASUREMENT OF PRESSURE

tography or by means of a photocell. Compared to purely electrical sensors, light sensors usually require more space, are subject to more severe problems of alignment and need more complicated equipment for time-dependent measurements. Because of these drawbacks, light sensors have shown promise of competing successfully with electrical sensors only for special situations, such as in a noisy electrical environment where pickup precludes reliable transmission of electrical signals or for special problems of measurement. One special problem is that of measuring pressure at a particular time as a function of position. For example, one may wish to know how pressure varies over an extended region of a flow about an airfoil. As mentioned in Section 5.5.3, this requires a large array of small sensors; to record electrical signals from all sensing elements simultaneously may involve the use of an inordinate amount of equipment. To reduce the amount of equipment needed, signals from the individual elements are often recorded in sequence (multiplexing), but this in itself complicates the circuitry and at best is merely a trade-off of time resolution for space resolution. If displacement of the individual elements is the quantity to be sensed, interferometric holography23 offers an alternative. The holographic method is described in Section 8.2.4 of Part 8 of this volume. Its basis is illustrated here by Fig. 17. Consider the three pairs of light sources indicated at the left of the figure to be coherent and equally intense. Because of interference, the intensity of the light I producing a particular image at F depends on the spacing A between the corre-

FIG.17. Schematic arrangement for reconstruction of image at three points ( 1 , 2 , 3 ) from double exposed hologram H. Hologram is illuminated by coherent light beam S which is the same as the reference beam used in making the hologram. Images are focused on photographic film F by lens L. Virtual objects are shown at left at equivalent large distances. See Fig. 28 and Section 5.9.3 for method of making hologram.

5.6.

55 I

SENSORS

sponding pair of sources. If the angle 0 between the rays producing the image and the axis of the optical system is small, and if the wavelength A is the same for the two interfering waves,

I = 210 cos2

(F+

$9).

(5.6.17)

where Zo is constant and $9 is a constant phase factor. The photographic densities of the images on a film placed at F thus provide a measure of A for each pair of sources. The sources shown in the figure are not real but are virtual sources formed by illuminating a composite hologram at H by light from S. The virtual sources indicated by full lines in the figure are images reconstructed from one hologram, and those indicated by dashed lines are due to a second hologram made at a different time. The two holograms are superposed on the same film by double exposure so that reconstruction from the composite contains both sets of virtual sources; one set provides reference positions from which to measure displacements of the other. Description of an arrangement for making the hologram, together with an example of a reconstruction, is contained in Section 5.9.3. By using a classical interferometer, described in Chapter 2.4, Section 2.4.1 of Part 2 of this volume, the intermediate step of making the hologram can be eliminated.22 But the classical interferometer has the disadvantage that it requires optical components, including the surfaces whose displacements are to be measured of extremely high precision (defects no greater than a small fraction of A). On the other hand, in the holographic method phase shifts due to distortions resulting from poor optics are practically the same for the two holograms and therefore cancel out on reconstruction from the composite. Another special requirement of some experiments is an extremely short response time, in some cases much shorter than a microsecond. Interferometric sensors both of displacement and of velocity have been developed with response times in this range. These employ photodetectors. For descriptions see Section 5.10.6.2 and a review article by Barker.54 These are called chrono-interferometers because, in contrast to the more common type on which the holographic interferometer is based, they measure movement at a particular place or small region in space mm2) as a function of time. A moving film rather than a photodetector can be used with some interferometric arrangements. The interferometer is adjusted to form straight fringes which are focused on and at right angles to a slit so that a change in optical path length of one of the interfering beams causes a shift in all the 54

L. M. Barker, Exp. Mech. 12, 209 (1972).

552

5 . MEASUREMENT OF PRESSURE

fringes along the slit. The shift is recorded on film traveling perpendicularly to the slit. Cinematographic methods are also available for measuring displacements of objects moving at speeds requiring nanosecond framing rates (See Section 8.2.2.3 in Chapter 8.2 of this volume). While quite elaborate, these can provide both spatial and temporal resolution over limited space-time regions.

5.7. Pressure -Ti me Recording In experimental work, when the manometer or elastic element and the sensor have been chosen, assembled and tested for determining pressure as a function of space and time, an additional important requirement is the reliable recording of results. Many ingenious methods have been devised; we will survey several but omit many details. Many other clever methods and modifications will be omitted. Ease of operation, reliability and cost are changing so rapidly with the advent of miniature solid state electronic devices that our survey cannot be up-to-date even at the time of writing. 5.7.1. Nonelectrical Recording

Mechanical coupling of an inked penpoint to a deformation gage allows it to record on moving paper driven by clockwork. The mechanical requirements are not much different from those required to move a pointer on a dial for visual display. Such gages, once used widely for recording atmospheric pressure and biological pressures, are superceded by more reliable electrical recording methods. Banks of liquid manometers connected by tubing to wall taps in the surface of a model in a wind tunnel have been photographed as a method of recording pressure as the model’s angle of attack, or the tunnel’s speed, changes. Sequences of photographs formed a convenient record of a great deal of data in wind tunnel installations in earlier years, but electrical recording has replaced this method also. Optical sensors described in Section 5.6.3 may be used with photographic recording. When diaphragm deflection is detected by forming interference fringes, the interferometer may be adjusted so that fringes are focused on and at right angles to a slit in front of the moving film, so that fringe-shift versus time is displayed graphically on the developed film. As described in Section 5.6.3, holographic interferometry records information on the deflections of diaphragms distributed over space,

5.7.

PRESSURE-TIME RECORDING

553

5.7.2. Electrical Signal Recording At the present stage of technical development, once the quantity of data recorded exceeds pencil and notebook feasibility, practically all recording is electrical, or electrically facilitated to the stage of obtaining magnetic tape, “hard copy” of an oscilloscope display by photography, or graphical display by xerography or computer plotter actuating pen motion on paper. Once the pressure transducer has delivered the electrical signal, there is nothing unique to pressure measurement in the techniques of signal processing, display and recording. Acquaintance with the terminology and familiarity with the processing and recording units is necessary for one to be able to buy or use the electronic equipment, but it would be out of place to describe the principles here. Manufacturers’ literature and instruction manuals constitute the most common source of education and information, but there are useful textbookP where the terminology and principles of instrumentation are presented. Some examples of useful applications are given in the remainder of this section.

5.7.2.1. Analog Display on Oscilloscope with Linear Sweep. Suitable amplifiers are often available in a bench oscilloscope to display the voltage generated by a piezoelectric sensor or a bridge output from a resistance, capacitance, or inductance sensor. Sweep triggering may be obtained from the signal itself, but an external trigger may be needed to establish a time reference. Oscilloscope cameras are usually offered for sale by the oscilloscope manufacturers. Attention needs to be given to spurious electrical pickup (it may be reduced by use of shielded cable), cable and input capacitance, noise generation in high gain amplifiers, time response of oscilloscope (frequency response), linearity, trace brightness for sweep rate used, and ease of time and deflection calibration. Probably the chief drawback is the limited time of display when sufficient time detail is present. A raster displuy can increase the duration by a factor of 10 or so with accompanying decrease in accuracy of measurement of signal amplitude and danger of overlapping displays. Usually an auxiliary unit must be built or purchased to provide the raster display for a bench oscilloscope. An oscilloscope screen storage mode is advantageous for saving film for the camera and set-up time when the single sweep mode is employed and preliminary adjustments need to be made. The digital memory oscilloscopes provide this along with other advantages, however (see below). Achievement of better than 5 percent accuracy is difficult with analog display on an oscilloscope screen. 55 A . J. Diefenderfer, “Principles of Electronic Instrumentation,” 2nd ed. Saunders, Philadelphia, Pennsylvania, 1979.

554

5.

MEASUREMENT OF PRESSURE

5.7.2.2. Photography of Oscilloscope Spot on Moving Film. The difficulty of limited length of display has been overcome in some laboratories by providing mechanically for sweep. The image is projected by a rotating mirror on a stationary circular loop of film, or directly on to moving film. The film may be on the inside or outside of a rotating drum, or it may be driven by sprockets from one rotating spool to another. By using both x- and y-deflections of the oscilloscope trace, a raster may be recorded on the moving film and time detail at 0.1 ps is displayed for intervals of tens or hundreds of milliseconds. A disadvantage is that the record is not available until the film has been developed, which is inconvenient when many trial adjustments are needed. 5.7.2.3. Digital Memory Bank. Integrated circuit and microprocessor chips promise to make oscilloscope and photographic recording obsolete. Improvements in miniaturization, accuracy, reliability, time resolution, and cost are achieved; long time storage is provided by magnetic tape and magnetic disks, but permanent storage on integrated circuit chips may become even more advantageous. As an example of the current state of the art, a digital memory oscilloscope-not much more costly than analog display instruments -will be described.56 The analog signal, full scale 0.1 V, is sampled every 0.5 ps, and the size measured and converted to a 12 bit binary word (4096 fineness) and the words for 4000 successive time intervals are stored in a digital memory. Until the oscilloscope is triggered, these data are displayed once and continuously replaced by the next 4000 intervals sampled. The trigger “freezes” the memory and the last data sample is preserved. If desired, one may choose to save some data preceding the trigger in time. The captured data can be displayed in up to 64 times the detail seen on the original unexpanded sweep. Since the sweep time is recorded in 4000 individual intervals, the signal can be displayed with a fineness of 1/4000 of the sweep interval; the accuracy of the displayed time, and of the voltage, is 1/400 of the full range. All the data, or the portion it is desired to keep, may be transferred to an analog x - y recorder or digitally to magnetic tape, or the oscilloscope screen may be photographed with the best choice of magnification as the data are repetitively displayed. Panel controls permit more liesurely sampling times (up to many seconds per point) and larger full scale voltage ranges may be chosen before recording; these provide longer total recording time and capacity for recording the largest voltage signal expected from the transducer. Manufacturers provide separate units-analog-to-digital converter (ADC), oscilloscope display, enlarged memory bank, disk and tape 56 “Nicolet Explorer Digital Oscilloscopes,” catalog of Nicolet Instrument Corporation, 5225 Verona Rd., Madison, Wisconsin 53711.

5.8.

DYNAMIC CALlBRATlON

555

recorders-which are compatible with general purpose minicomputers so that data acquisition can be programmed and the resulting data analyzed and correlated. ADC’s with 0.01 ps sampling interval are currently available; few if any transducers can make use of this time resolution. Also none can be calibrated so accurately as to justify the 1/4096 sensitivity of ADC’s, but this sensitivity is quite useful because often the maximum voltage output is not known before recording and a margin of safety can be indulged in without fear of recording too small a signal.

5.7.3. Multiplexing When many pressure (and other) transducers are sensing at the same time, for example many pressure taps in the surface of a wind tunnel model, use of many separate recording channels becomes expensive in space, money, and repair time. A recording system with more rapid response than needed to record single channels may be used to sample and display the multiple outputs in turn. The fast ADC’s mentioned in the preceding paragraph have been used to record pressures from multiple wall-taps by leading the many tubes from the taps via a mechanically rotating valve to a single pressure transduceF so that only one recording channel is needed.

5.8. Dynamic Calibration Calibration of gages for measuring steady and slowly varying pressures was considered in Chapter 5.2. In Section 5.5.1 it was pointed out that the ability of a gage to measure time varying pressures can be judged either from its frequency response function or from its response to a specified nonperiodic input. Correspondingly, there are two general types of dynamic calibration. In one type, carried out in the frequency domain, the amplitude and phase of a periodic output due to a “steady state,” ideally sinusoidal, input is measured as a function of frequency. In the other type, which provides results in the real time domain, the nonperiodic input used is nearly always taken to be a step function, Frequency domain calibrators commonly have a cavity or chamber, in whose walls are placed the gage to be calibrated and usually a monitoring gage, and a source for producing and maintaining pressure oscillations within the contained fluid. A large number of quite different arrangements is possible.31 In one type of arrangement the cavity is excited so that it operates in one of its resonant modes, usually the one of lowest frequency. The exciting driver-a siren, a piezoelectric stack, an electro-

556

5 . MEASUREMENT

OF PRESSURE

statically driven diaphragm, rotating jet-has its frequency tuned to the resonant frequency of the cavity. The frequency is changed by altering the dimensions of the cavity or by using a fluid with a different sound speed. Other arrangements operate in a cavity but at nonresonant frequencies. In some nonresonant calibrators a fixed mass of fluid is periodically compressed, in others, the flow of fluid through a cavity is modulated causing a periodic pressure change. The compression or flow modulation is carried out by the same kinds of periodically driven devices as used with resonant cavity calibrators. An advantage of the nonresonant over the resonant calibrator is that it is easier to change the frequency. The upper frequency limit for the nonresonant calibrator is determined by the onset of strong standing waves within the cavity. For both resonant and nonresonant types, smaller cavities will produce higher frequency limits as will an anechoic configuration. With few ex~ - ~certain ~~~~ ceptions, such as provided by a reciprocity ~ a l i b r a t o r ’ and electrostatic actuators and pistonphones, frequency domain calibrators require a reference gage, whose sensitivity has in turn been related to static measurements made by such basic gages as the manometer and piston and cylinder described in Chapter 5 . 2 , to determine the input pressure to the gage being calibrated. A major problem of frequency domain calibrators is the requirement that they produce essentially sinusoidal pressure variations. Because of the inherent nonlinearity of fluid dynamic phenomena, sinusoidal variations have been possible only when the peak-to-peak variation is no more than a few percent of the ambient pressure and the frequency is less than a few tens of kilohertz.31 Consequently, the use of frequency domain calibrators has been limited mainly to gages designed for relatively low pressure level applications in the audio and near audio frequency range. Real time domain calibrators are essential for fast response gages designed to measure transient pressure changes and, with few exceptions, the type of pressure change employed is a step function. Subjecting a gage to a step function pressure change provides a test condition which is either similar to or more severe than encountered in most transient pressure applications. Furthermore, as pointed out in Section 5.5.1, even in applications where the shape of the pressure-time event to be recorded is quite different from a step function-a short rectangular pulse at one extreme, or a gradually rising pressure at the other extreme-the response of a gage can be predicted from a knowledge of its response to a step function by using a superposition integral, such as given by Eq. (5.5.2) of Section 5.5.1. 57

L. L. Beranek, “Acoustic Measurements.” Wiley , New York, 1949.

5.8.

DYNAMIC CALIBRATION

557

A linear shock tube, suitably instrumented for measuring shock strengths, provides an excellent test and calibration facility. Equations pertinent to the design of shock tubes and to the prediction of values for the parameters (pressure, temperature, density, velocity, etc.) of flow induced by a one-dimensional shock wave are contained in numerous arThe gage to be tested and calibrated can be ticles and mounted either in the end plate from which the shock reflects or in the side wall of the channel, with the wall mount having the obvious disadvantage that it takes the shock front a finite time to cross the sensitive face of the gage. Further details are in Chapter 9.2 of this volume. Important gage characteristics were considered in Chapter 5.5. For a gage designed to measure transient pressures they are: (a) its useful pressure range and sensitivity; (b) its hold time T , during which measurements can be made; and (c) its response time T and its freedom from spurious oscillations. The time dependent characteristics (b) and (c) can be determined from records made with shocks of unknown strength, but obviously the pressure increase across the applied shock must be known in order to measure the gage’s sensitivity. The pressure increase across the applied shock is best measured directly with a gage which has been calibrated at steady pressures and is mounted in the wall of the channel close to the gage under test. Although the gage for measuring shock strength should be accurate, have long term stability and have a long enough hold time so that it can be calibrated with essentially steady pressures which in turn can be measured with a manometer or piston and cylinder gage (Chapter 5.2), its response time need not be extremely short, nor are rapidly damped spurious oscillations particularly deleterious, since the pressure behind the shock can usually be arranged to be essentially constant for several hundred microseconds. A diaphragm gage, Section 5.9.3, which has been specially designed for accuracy and stability and which has a relatively short response time, or a stub gage, Section 5.10.3, such as pictured in Fig. 33, are good gages for shock strength determination following calibration at essentially steady pressures. Lacking a suitable gage for measuring shock strength directly, a reasonable value can be calculated under the assumption of ideal gas behavior from a knowledge of the temperature and composition and thus 58 R. I. Soloukhin, “Shock Waves and Detonations in Gases.” Mono Book Corp., Baltimore, Maryland, 1966. 5@ A . R . Hartunian, in “Methods of Experimental Physics” (L. Marton, ed.), Vol. 7, Part B, Chapter 7. Academic Press, New York, 1968. 6o D. E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed., p. 2-275. McGraw-Hill, New York, 1972. W. Bleakney and R. J . Emrich, The shock tube. “High Speed Aerodynamics and Jet Propulsion,” Vol. 8, Artic. J. Princeton Univ. Press, Princeton, New Jersey, 1961.

558

5.

MEASUREMENT OF PRESSURE

2

-3

-~4 -!(a)

FIG. 18. Mechanical devices for calibrating transient pressure gages. (a) Rapidly operating valve. I: test gage; 2: gas seal acting as valve; 3: pressurized chamber; 4: falling weight which breaks seal. (b) Spring loading. 1 : test gage; 2: compressed spring; 3: support wire; 4: wire cutter or electrically initiated explosion; 5: known weight reduces force applied to gage by spring until wire is broken.

of the sound speed and the heat capacities of the gas and either measurements of the initial pressures in the driver and channel sections (Eq. (5.2.4), Ref. 59) or measurements of the shock velocity and initial pressure in the channel (Eq. (7.3.7), Ref. 59). Even if the shock strength is determined indirectly, the accumulated error in calibration of a test gage with a shock tube is usually less than 5 percent. If a higher pressure step than that produced in a shock tube is required, a fast acting mechanical device similar to one of those shown in Fig. 18 can be used for c a l i b r a t i ~ n . ~ However, ~ - ~ ~ . ~ ~such devices do not apply pressure to the gage as rapidly as a shock, so they are unsuitable for measuring microsecond response times and usually provide a poor test of high frequency vibrational characteristics of the gage. The apparatus shown in Fig. 18a employs a falling weight to break a seal allowing pressure on the gage to build up to a known value. The pressure applied to the gage increases to the known value in approximately 70 ps. In the arrangement of Fig. 18b, a force is applied to the gage by a compressed spring. This force is reduced by a suspended body of known weight. When the suspending wire is broken, the force on the gage increases by an amount equal to the weight of the body. This increase occurs over the finite time required to break the wire completely.

'* G . D. Salamandra,T. V . Bazhenova, S. G . Zaitsev, R. I. Soloukhin, I. M. Naboko, and I. K . Sevast'yanova, "Nekotorye Metody Issledovaniya Bystroprotekayushchikh Processov." USSR Acad. Sci. Press, Moscow, 1960 (in Russian). K . Haider, H . Holtbecker, and E. Jorzik, J . Phys. E 3,945 (1970).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDlNG A N D STRETCHING

559

5.9 Diaphragm Gages: Strain by Bending and Stretching Some gages, such as a Bourdon type or those employing a bellows, have elastic elements of quite complicated shape which respond by bending and stretching. These were described in Section 5.2.2 and will not be considered further. Their use is restricted to the measurement of essentially steady pressures. The theory of the steady deflection of elements having complicated shapes is well covered by Andreeva.8 The following considerations deal only with simple diaphragms which are initially flat. 5.9.1. Theory of Diaphragm Deflection

Figure 19 shows the deflection of a clamped diaphragm produced by the application of a uniform pressure. Even when the time dependence of the behavior is not considered, the general problem of predicting the deflection due to a known pressure is nonlinear and cannot be solved exactly. Andreeva8 provides an excellent review of the static problem including approximate equations which are adequate for the design of slow response gages. A comparable review of the dynamic problem is not known, but aspects of the problem are covered by Den Hartoge4 and Timoshenko. 85 5.9.1.l.Static Behavior. As Andreeva8 points out, a useful practical solution to the general problem of predicting the deflection of a simple diaphragm due to a known applied pressure can be obtained by adding the solutions for three idealized situations. With the diaphragm clamped in a perfectly rigid mount, as in Fig. 19, the idealized situations are: (1) the diaphragm is initially unstressed and the maximum deflection is much less than the thickness of the diaphragm (pure bending), (2) the diaphragm is initially unstressed and the maximum deflection is much greater than the thickness of the diaphragm (pure stretching), (3) the diaphragm is prestressed and held by clamping so that it is initially in tension and the initial tension is of sufficient amount so that it is not effectively increased by subsequent stretching (constant tension). (Situation (3) is often referred to as the “membrane” problem.) Equation (5.9.1) relates the maximum deflection w oat the center of the diaphragm to the applied pressure p ; the first, second, and third terms on the left-hand side represent, respectively, J. P. Den Hartog, “Mechanical Vibrations.” McGraw-Hill, New York, 1940. S. Timoshenko, “Vibration Problems in Engineering,” 3rd ed. Van NostrandReinhold, New York, 1955. B(

5 . MEASUREMENT OF

5 60

PRESSURE

Section Circular Diaphragm

m 1 r

+u-i

-a--

.i

X E h

Section Slit Diaphragm

FIG.19. Diaphragm clamped at the boundary where r

= u, w = u =

0, and dw/dr = 0.

the contributions due to pure bending, pure stretching, and initial tension.

.(y)+.(y)”+ c@)(?)

=(a>’$.

(5.9.1)

In this equation (J is the minimum half width of the diaphragm (half width of shorter side of rectangle, radius of circle), h is its thickness, Y is Young’s modulus and uo is the amount by which the diaphragm is stretched before clamping. The dimensionless coefficients A , B , and C depend only on Poisson’s ratio and on the shape of the diaphragm; values range between 1 and 8. Expressions for A , B , C, and uo/a are given in Table IV for the extreme shapes of a circle and an infinitely long rectangular slit. As Eq. (5.9.1) indicates, wo does not in general increase linearly with p . As p increases, wo/h increases, the second term on the left hand side of the equation becomes relatively more important and for sufficiently large values of p the second term dominates. Gages can be designed, however, to operate in an essentially linear range. For a prescribed maximum value of p and with no initial tension, a / h can be made small enough so that the first term on the left of Eq. (5.9.1) dominates and the problem reduces to the idealization of pure bending. TABLEIV. Diaphragm Deflection Coefficients” Diaphragm shape Circle Slit

B

C

(7 - v)

4 ( 1 - v)

A 16 3(1 - 9)

3(1

- V)

4 __3(1 - v’)

(1

2

-

uda

(1

- v ’UOy U O

4)

(1 - 3’-j7

v , Poisson’s ratio; Y , Young’s modulus; N o , prestretch; u,,, initial tensile stress (hue = tension per unit length applied at boundary).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING AND STRETCHING

561

We shall refer to a diaphragm operating in this idealized way as a “linear bending diaphragm.” The displacement w in the case of linear bending is given in terms of the distance from the center of the diaphragm r by

wh (L) A =

(574

h

($) (1 - ! J

(5.9.2)

In terms of the maximum displacement, the sensitivity, .Yo= dwo/dp, is 9 0

=

(3(;)3

(5.9.3)

For some gages operating in the linear bending range, strain is sensed rather than displacement. For a circular diaphragm, the strain components at a surface are E,

= 2

-g)

($)(I

2);();(

and

(5.9.4) Ee

=

a 2 p * (2;i(x) ) (T) (1

-

f).

where E , is the radial strain and E~ is orthogonal to E,. The plus and minus signs apply to opposite faces and indicate, for a particular position, that if one face is in tension, the other is in compression. For a slit diaphragm, there is no strain along the length of the slit, but the strain across the slit has the same dependence as E, of Eq. (5.9.4). The strain sensitivity for E , , .Ye = de,/dp, is, for either a circular or slit diaphragm, YE=

*

(&)(;y

(1

-

g).

(5.9.5)

For a bridge circuit (see Fig. 15), it is useful to have two sensors whose response is the same in magnitude but opposite in sign. As indicated by Eq. ( 5 . 9 . 9 , this can be accomplished for sensors responding to E, by placing them at the same position ( r = 0) but on opposite faces. This has the disadvantage that the sensor on the surface where the fluid pressure is measured is more subject to damage and to sudden temperature changes of the fluid. They can both be on the protected surface if they are small and one is positioned at Y = 0 and the other at Y = (2/3)%. Instead of operating in the linear bending range, many gages use a highly pretensioned diaphragm. The advantage of using prestress, in addition to making linear behavior easier to achieve, is that it reduces erratic

5.

562

MEASUREMENT OF PRESSURE

response due to intitial unevenness of the diaphragm and to residual stresses, which are difficult to eliminate in the manufacture of thin diaphragms. With sufficiently large prestress, the third term on the left-hand side of Eq. (5.9.1)dominates and the problem reduces to the idealization of constant tension, For this case, the displacement is

w

=

h 1 a z p c1 (6)(;)3(;) ( 1 - 5) c,(7;) ( a , ) ( l - f). (5.9.6) =

where Co has a value of 4 for a circle and 2 for a slit. The sensitivity in terms of maximum displacement is

(5.9.7) 5.9.1.2. Dynamic Behavior. Following are expressions for the frequency response of a diaphragm (%To),Section 5.5.1). Specifically, the expressions give the amplitude, w(w, r), of the oscillating displacement produced by a sinusoidally varying pressure, p = e’”‘, of unit amplitude and frequency w . For a linear bending circular diaphragm

where K = (w/coh)”’, c%= Y/[12p(l - v’)], Jo(x) and Jl(x) are first-kind Bessel functions of zeroth and first order and Zo(x) and Z,(x) are the corresponding Bessel functions with imaginary arguments. Other symbols have the same meaning as for static loading. For the case of a prestressed circular diaphragm having constant tension

(5.9.9) where ct = a o / p . These transform to equations for a slit diaphragm by replacement of the Bessel functions with analogous trigonometric functions. For a linear bending slit diaphragm sinh(Ka) cos(Kr) + sin(Ka) cosh(Kr) - I] sinh(Ka) cos(Ka) + sin(Ka) cosh(Ka)

(5.9.10)

and for a prestressed slit diaphragm - 11.

(5.9.11)

While a slit is not practical, the slit and the circle are extreme shapes. These expressions are standing wave representations of the dia-

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

563

phragm's contour. For small frequencies they are the same as the contours produced by static loading, i.e., for small o, Eqs. (5.9.8) and (5.9.lo), when multiplied by p , reduce to Eq. (5.9.2) and Eqs. (5.9.9)and (5.9.11), when multiplied by p , reduce to Eq. (5.9.6). As the frequency increases the contour distorts from the shape produced by a steady pressure and the amplitude increases until resonance occurs. At still higher frequencies other resonances appear and the diaphragm vibrates in progressively more complicated modes. For measuring periodic variations, the frequency of the applied pressure should be well below the lowest resonant frequency of the diaphragm, in which case Eq. (5.9.2) or Eq. (5.9.6) with p = poebt, will provide a good approximation for the displacement. For nonperiodic pressure changes, response to a step function provides a better guide to behavior ( U ( t ) ,Section 5.5.1). A formal expression for the response to a step can be obtained by use of the inversion integral, Eq. (5.5.1) of Section 5.5.1. Use i/o (transform of unit step) for 9(o)and w ( o , r) from Eqs. (5.9.8), (5.9.5), (5.9.lo), or (5.9.11) for %lo). The inversion integral can be reduced to a more useable form by use of the Cauchy residue theorem. The form of the resulting expression for the displacement, W ( Y , t), depends upon whether the integrand of the inversion integral is viewed as being composed of traveling waves or standing waves. The traveling wave representation is easier to visualize in cases, such as that of the prestressed diaphragm, where the phase velocity of the waves is independent of the frequency (no dispersion) but is more complicated when it varies with frequency (dispersion), as in the case of the linear bending diaphragm. For consistency and ease of comparison the standing wave representation is used below for the prestressed as well as the linear bending diaphragm. The displacement due to a unit step applied at time t = 0 is given by Eq. (5.9.12)for linear bending circular diaphragm, by Eq. (5.9.13)for a prestressed circular diaphragm, by Eq. (5.9.14) for a linear bending slit diaphragm and by Eq. (5.9.15)for a prestressed slit diaphragm.* * The expressions of Eqs. (5.9.12) through (5.9.15) as well as those of Eqs. (5.9.8) through (5.9.11) are solutions to the following basic differential equations. Lineur bending diuphragm:

where the meanings of the symbols are the same as in Eqs. (5.9.8), (5.9.10). (5.9.12) and (5.9.14).

5.

5 64

MEASUREMENT OF PRESSURE

(5.9.12a) where K, = ( w , / ~ , h )is~ given /~ by (5.9.12b)

J,(K,a)/Jo(K,u) = - z l ( ~ , a ) / ~ o ( ~ & ) ,

where wq is given by Jo(w,a/ct) = 0, or approximately, wq

(5.9.13b)

= (Y -

c

2 tan(K,a) cos(K,r) pc;h3 q=l (aK5,) [cos(K,a)

w ( r , 1) = -

-

cosh(Kqr)] [l cosh(K,a)

-

cos(w,t)], (5.9.14a)

where K,

given by

= (w,/coh)1/2is

tan(K,a) = - tanh(K,a) or approximately, K,a

I :

(5.9.14b)

(4q - 1 ) ~ / 4 ;

where w, is given by cos(w,a/cl) 0,

=

0 or

= (2q -

1)7rct/2a.

(5.9.15b)

In all cases q is a running positive integer. Other symbols are the same as in Eqs. (5.9.8), (5.9.9), (5.9.10) and (5.9.11). The effect of applying a step function load is to produce a periodic moPrestressed diaphragm:

where the meanings of the symbols are the same as in Eqs. (5.9.9), (.5.9.11),(5.9.13) and (5.9.15).

For all equations p ( t ) represents the applied time-varying pressure. For nonvarying pressures, p(t) is constant and azw/dtz is zero.

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

56.5

Time

Position, r Position

Time, t

FIG.20. Prestressed membrane shape dependence with time and time dependence of displacement. (a) Contour at 118 period intervals following step function loading. (b) Time dependence of displacement at particular positions following step function loading. Full lines are exact; dashed lines are contributions of standing wave of lowest freqeuncy. c: = u o / p ;9, = 4a/c,.

tion which is expressed in Eqs. (5.9.12), ( 5 . 9 . 1 3 ) , (5.9.14), or (5.9.15) as a Fourier series whose components are resonant oscillations. The time average of the motion, about which the oscillations are centered, is the displacement that would be produced by a constant pressure having the magnitude of the step.* A good approximation to the exact motion is ob~~

~~

* To obtain displacements due to load p

(5.9.13), (59.14). and (5.9.15) hypo.

= poS(t)multiply expressions of

Eqs. (5.9.12),

5 . MEASUREMENT

566

OF PRESSURE

tained by neglecting all terms of the series except the one having the lowest frequency. To illustrate these points, consider the effect of applying a step load to a prestressed slit diaphragm, Eq. (5.9. 15a), noting first that the characteristic frequencies, wq of Eq. (5.9.15b), are the same as the resonant frequencies at which the frequency function, Eq. (5.9.11) becomes infinite. Figure 20a shows the contour of the diaphragm at intervals of one-eighth the longest resonant period and Fig. 20b pictures the time variation of the displacement at several positions. Considering the full-line curves, which represent exact values for the displacement, the central contour of Fig. 20a at time a/ct and the time averages of the curves of Fig. 20b are the same as for static loading, Eq. (5.9.6). The period of motion, g1= 27r/wl = 4a/c,, corresponds to the lowest resonant frequency of Eq. (5.9.15b) and, for this simple case where there is no dispersion, is the time for a wave to travel back and forth across the diaphragm. The dashed curves of Fig. 20a and Fig. 20b were calculated using only the first term of Eq. (5.9.15a), i.e. the term with the lowest characteristic frequency. The small differences between the dashed and full-line curves indicate that the first term is dominant and suggest the following as a good approximation for Eq. (5.9.15a): w(r, t ) = Y p o [ l - c o s ( 2 ~ t / 9 ~ ) ] ,

(5.9.16)

where Yis sensitivity, i.e., displacement at position r due to unit constant pressure, po is the magnitude of the applied pressure step and 9,is the longest resonant period. With appropriate values for 9'and PI, Eq. (5.9.16) provides a good approximation for Eqs. (5.9.12), (5.9.13), and (5.9.14) as well as Eq. TABLEV . Parameters of Approximate Expression for Displacement of Diaphragm Produced by Step Function Loading" ~~

Conditions Linear bending, circular diaphragm Prestressed, circular diaphragm Linear bending, slit diaphragm Prestressed slit diaphragm a

Y

[3( 1; 1 ] [$1 [ Y')

[7][$J[1 (1 - 3)

9 1 ~~

~~

- f] '

-5j'

~~~

~~

az

0.615 -

coh

2.612

ct

1.123- a2 coh 4'

~~~

~~

Derived from exact Eq. No.

Cl

~~~

(5.9.12)

(5.9.13) (5.9.14) (5.9.15)

Y (sensitivity) and ?PI (longest resonant period) are parameters to use in Eq. (5.9.16).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

567

(5.9.15). Expressions for Y and Pl for the four cases treated here are given in Table V. For a linear bending diaphragm the expression for Y comes from Eq. (5.9.2) and for a prestressed diaphragm it comes from Eq. (5.9.6). Expressions for the longest resonant period Pl = 27r/ol come from Eqs. (5.9.12b), (5.9.13b), (5.9.14b), and (5.9.15b). 5.9.2. Response Characteristics of a Diaphragm Gage Knowing the response to a unit step function load, the time dependent behavior of a diaphragm for any type of variable loading can readily be determined by the superposition integral Eq. (5.5.2) of Section 5.5.1. A good approximation for U(t - Y), the response at time t to unit step applied at time 9, is given by Eq. (5.9.16) of Section 5.9.1.2. (5.9.17) To illustrate the response of a diaphragm to a less precipitous loading than given by a step function, suppose the applied pressure, of Eq. (5.5.2) of Section 5.5.1, increases as a sine function for a quarter of some period Po,and subsequently remains constant:

Z(n

zcn =

po sin(2.rrT/Po)

for 0 < 9 c P0/4 for P0/4 < T < m

.

(5.9.18)

Equations (5.9.17) and (5.9.18) have been used in Eq. (5.5.2) of Section 5.5.1 to develop the graphs of Fig. 21. Figure 21 shows the response of a diaphragm gage for three values of Po (four times the rise time of the load) relative to PI (the longest resonant period of the gage). The three values of Poare (1) Po= 0, i.e., the load is a step function, (2) Po= 4P1, i.e., the rise time of the load is equal to the longest resonant period of the gage and (3) Po= lop1, i.e., the rise time of the load is 2.5 times the longest resonant period of the gage. As the graphs indicate, the size of the spurious oscillations in the response becomes smaller as goincreases relative to Pl. For Pogreater than 10P1 the spurious oscillations are for practical purposes inconsequential. This behavior is consistent with a diaphragm's frequency response [Eqs. (5.9.8)-(5.9.1 l)]. For a sustained sinusoidally varying load with period Po,the response varies sinusoidally with period Pobut the amplitude is a function of YO/Pl. The amplitude is virtually infinite at resonance (90/P1= 1) and decreases as Po/Pl increases. For Pogreater than a few multiples of B1 the amplitude is essentially independent of Po/Plwith a value determined by the steady state sensitivity. These two cases demonstrate a general rule of thumb. Provided no im-

5.

568

MEASUREMENT OF PRESSURE

2

FIG.21. Decrease in amplitude of resonant oscillations with increase in risetime Poof apis largest resplied pressure. 1: step function load, 9 0 = 0; 2: Po= 4P1 ; 3: go= 109,. 9, onant period of gage.

portant pressure change occurs in a time interval less than a few resonant periods of the gage, the response at each instant is practically equal to the applied pressure multiplied by the gage's steady state sensitivity. Sensitivity and longest resonant period are clearly important characteristics of a diaphragm, particularly since they are parameters of Eqs. (5.9.16) and (5.9.17). However, since step function loading produces large oscillations, response time is a vague concept and its exact definition is arbitrary. For purposes of comparison with other types of gage, the response time of a diaphragm gage will be considered to be its longest resonant period. Except for limitations set by the sensor or recording circuitry, the hold time of all diaphragm gages is practically infinite. To simplify the interpretation of records, resonant oscillations are sometimes reduced by adding mechanical damping or by using an electrical filter. These modifications do not, however, improve a gage's ability to measure rapid pressure changes. Real improvement is obtained only by shortening the resonant period. Of all shapes, a circular diaphragm has, for the same value of the smallest dimension a , the shortest resonant period and in practice most diaphragms have been made with this shape. For all gages the resonant period is inversely proportional to ct or co and thus decreases as these increase. See Table V. In the case of the prestressed diaphragm, where ct = ( ~ ~ / p ) the l ' ~resonant , period decreases slowly as the prestress uois increased, with an ultimate limit set by the condition that cro must be less than the material's yield stress, which is nominally at least two orders of

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

569

magnitude less than Young’s modulus. In the case of a linear bending diaphragm, co = ( Y / [12( 1 - ~ ~ ) p ] ) lwhich ’ ~ , depends only on material properties. Both ct and co are several times smaller than the propagation velocity for compressional strain which is the type of distortion basic to the operation of most fast response gages. The remaining parameters are the dimensions, a and h. The resonant period decreases with a decrease in a and for an unprestressed diaphragm with an increase in h. Changing a parameter to decrease the resonant period always leads to a decrease in displacement sensitivity. See Table V. It is usually advantageous to sensitivity to make a / h as large as feasible since for a prestressed diaphragm an increase in this ratio increases the sensitivity but does not change the resonant period and for a linear bending diaphragm the sensitivity increases as the cube of the ratio whereas the resonant period increases only as the first power. For a given maximum pressure an upper limit on a / h is set by the fact that as the ratio is increased the gage response becomes nonlinear and eventually the diaphragm will stretch beyond yield. For similar diaphragms, having the same value of a / h , both the resonant period and the sensitivity decrease in direct proportion to a so that decreasing the resonant period by miniaturization is eventually limited by inadequate sensitivity. The following values for the resonant period are of the order of the smallest period obtainable with a diaphragm gage using present day technology. Miniature gages using a semiconductor sensing technique have been made with very small, unprestressed silicon diaphragms. Assuming Y = 110 GPa, p = 2400 kg m-“, Y = 0.3 and a circular shape with a = 1 mm, h = 25 pm, the longest resonant period is 12 ps, as calculated from 8, = 0.615 a2/c& [Table V]. A very small gage has also been made with a prestressed, stainless steel diaphragm. Assuming uo = (2/3) yield stress = 500 MPa, p = 7800 kg m-3 and a circular diaphragm with a = 1.25 mm, the expression 8,= 2.612 a/ct [Table V] predicts 13 ps for the longest resonant period. The theory on which these predictions depend presupposes no damping, a perfectly rigid stationary mount and direct contact between fluid and diaphragm at the position where the pressure is to be measured. Damping causes decay of the spurious oscillations, but also increases their period slightly. In practice, the assumption of a stationary mount is often not justified; there are often unwanted vibrations in the base to which the gage is attached. In this case either the diaphragm must in some way be isolated from the unwanted vibrations or a means of compensation must be employed. Also in practice, the diaphragm is sometimes mounted in a cavity which has a tube extending to the position of measurement (Bynum el p. 31). This certainly increases the

-

570

5.

MEASUREMENT OF PRESSURE

response time and may introduce spurious oscillations due to resonance of the fluid in the tube or cavity. Other characteristics, such as accuracy, temperature independence, etc. may be important in selecting a gage for a particular application but no recipe for an optimum choice can be given. Gages offered foresale by several manufacturers offer a variety of compromises. 5.9.3. Types of Diaphragm Gage

Treatises on high fidelity microphones and loudspeakers contain much that is useful in the design of gages for measuring pressure changes in the audio range.18*6s-67 (Figure 22a is a schematic of a diaphragm gage whose electromechanical system is similar to an induction microphone. Fig. 22b pictures a gage which is similar to a condenser microphone. Both types are shown with two sensors symmetrically placed on opposite sides of the diaphragm. The inductance type sensor is considered in Section 5.6.2.4. The particular form shown here consists of a wire coil of many turns linking a core of high permeability which, together with the diaphragm, forms a magnetic loop. Displacement of the diaphragm changes the reluctance of the magnetic circuit and therefore the self inductance of the c ~ i l . * In ~ *the ~~ other type of gage, the diaphragm serves as one plate of a capacitance sensor, such as considered in Section 5.6.2.2. When the diaphragm is displaced the capacitance of the sensor changes, due to change in plate ~ e p a r a t i o n . ’ ~Both * ~ ~ types of sensor are shown as parts of an alternating current bridge whose output is modulated by a change either in inductance or capacitance. With this arrangement measurement of constant as well as changing pressures is possible, the push-pull aspect of the bridge makes the response more linear than that of a single sensor and some temperature compensation is provided. Gages of best longterm stability must be built in a thermostated enclosure to assure freedom from ambient temperature effects. For gages of moderate size, the inductance type sensor can be made with relatively high electrical sensitivity and signal-to-noise ratio, but the capacitance type is simpler and can be made with a mechanically superior diaphragm since it is not restricted to materials of high permeability. In part because a duct or a tube leading to the cavity must be provided, gages of these types are ordinarily useful only for measuring pressure changes in the audio frequency range and lower. McGraw-Hill, BB G . P. Harnwell, “Principles of Electricity and Magnetism,” p. 464. New York, 1938. *‘D.E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed., Sect. 3. McGraw-Hill, New York, 1972.

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

571

A

I

(a I

I

I

(b)

FIG.22. Typical diaphragm gages with cavity and tubing connecting to region where pressure is to be measured. (a) Variable reluctance (inductance)type. (b) Variable capacitance type. Electrical connections to ac bridge provide output signal proportional to pressure change.

Gages with faster response are smaller and usually employ dc sensors and amplifying circuits. An inductance type sensor is not a viable candidate for use in a fast response gage because it is difficult to miniaturize and has unattractive features when operated with a direct current (Section 5.6.2.4). On the other hand, a moderately fast diaphragm gage has been made with a capacitance type sensor.2o A schematic diagram is shown in Fig. 23. The problem of scaling down this type of gage is not only that the sensitivity is decreased directly because the capacitance of the sensor becomes smaller but also, unless the parallel capacitance of the connecting cable is made smaller in proportion, because the fraction of the total capacitance which is independent of pressure becomes larger. Furthermore, as mentioned in Section 5.6.2.2, when the capacitance of the sensor becomes comparable to that of the cable, pick-up of electrical disturbances by the cable becomes important and adversely affects the signal-to-noise ratio. These difficulties associated with the cable were alleviated in the gage of Fig. 23 by placing a small electronic impedance matching unit next to the capacitor of the sensor. A highly prestressed membrane is built on 1

4

6

FIG.23. Miniature capacitance gage, overall diameter 2.5 mm. 1 : Pretensioned diaphragm of stainless steel 2.6 pm thick. 2: Metal backplate. 3: Metal collar for diaphragm support. 4: Conducting cement. 5: miniature coaxial cable connector. 6: to adjacent miniature preamplifier. [From S i d d ~ n . ~ ' ]

572

5.

‘‘

MEASUREMENT OF PRESSURE

DIAPHRAGM THICKNESS

pyjL

(not to %ale)

-

STRESS DISTRIBUTION

\-‘---

\

-&,

FIG.24. Distribution of radial component of strain in a circular linear bending diaphragm clamped at its perimeter. 1 : uniform thickness diaphragm. 2: optimally chosen profile to provide uniformly distributed strain. 3: positions of strain resistive elements.

the end of a coaxial connector directly coupled to a cathode follower. With a prestressed stainless steel diaphragm of 2.5-mm diameter and 0.0025-mm thickness, the gage provided an overall sensitivity of 10 V Pa-’. The calculated resonant period is 13 ps. As noted in Section 5.6.2.2, a capacitance sensor has a distinct advantage in that it does not generate heat. This is a necessity, for example, in most cryogenic experiments. In experiments with helium performed at temperatures near 0.013 K, pressures of 10 MPa have been measured with a detection capability of 20 Pa using a capacitance sensor in conjunction with Be-Cu and polymer diaphragms.‘* Another type of gage uses a linear bending diaphragm and one of a variety of resistance sensors which respond to strain rather than displacement [Eq. (5.9.5)l. In the simplest construction the resistance sensor consists of a single fine wire or thin film stick-on element. This is likely to present a bonding problem, particularly since the strain varies across a diaphragm of constant thickness. Semiconductor elements of single crystal silicon are useful strain elements because of their high sensitivity. They are however limited to strain levels of 0.001 and have correspondingly high temperature sensitivity. Figure 24 shows the distribution of strain over the face of a uniform-thickness diaphragm, and it is seen that the maximum strain is reached at the center while the strain elsewhere is smaller. In a special variable-thickness diaphragmse it is possible to obtain a uniform distribution of radial strain, and the strain produced in such a diaphragm is shown in Fig. 24 together with the positions of cemented-on strain elements. The strain in the central region is positive and near the border it is nega-

*

G . C. Straty and E. D. Adams, Rev. Sci. Instrum. 43, 394 (1972). E. A. Samoletov, “Aerofizicheskiye Issledovaniya,” Annu. Rep., Inst. Pure Appl. Mech., p. 38. Novosibirsk, 1972 (in Russian). dQ

5.9.

DIAPHRAGM GAGES: STRAIN B Y BENDING A N D STRETCHING

573

tive. Several pairs of sensors are cemented to the central and border regions to provide symmetry and averaging of residual nonuniformities in the gage construction. At strain levels of 0.001, electric signal magnitudes of AVIV, --- 0.1 ( V , is bridge supply voltage) are obtained with silicon crystal sensors each several millimeters in extent, and some temperature compensation for ambient changes is provided. In another method of construction better bonding of resistive elements is obtained by evaporating thin resistance films directly onto the diaphragm. The technique of placing pairs of resistive elements on various regions of uniform thickness diaphragms to increase sensitivity and to provide temperature compensation in the bridge connection is generally used; see Section 5.6.2.1 and Fig. 15. A third type of diaphragm gage using resistance sensors employs techniques developed for forming microcircuits on semiconductors. In one version a small area of a silicon chip 250 pm in thickness is etched from one side leaving on the other side a rectangular diaphragm 2.5 mm by 1.25 mm and only 25 pm thick. Working in a vacuum a second chip is bonded to the first so as to form behind the diaphragm a cavity in which the pressure is zero. A tiny resistance bridge is formed in the outer face of the diaphragm by diffusion of an N- or P-type material. The arrangement is shown schematically in Fig. 25. The bridge is centrally located and its resistance arms are identical. The rectangular diaphragm provides a net strain sensitivity, whereas the equal arm bridge compensates for equal temperature changes of the resistors. An independently connected minitransistor placed at the center of the diaphragm serves as a temperature sensor and can be used either as a thermometer or as a thermostat for further temperature compensation. The calculated value for the resonant period of a silicon diaphragm without prestress and having dimensions 2500 X 1200 X 25 pm3 is 17 ps. Gages of this type have been built with different diaphragm thicknesses for pressure ranges from 0- 100 kPa to 0-30 MPa. Using techniques developed for manufacturing integrated circuits, this gage can be mass produced relatively inexpensively .'O A gage which uses a thin polymer membrane both to supply a piezoelectric emf (Section 5.6.2.3) and to provide an array of small diaphragms, each acting as one plate of a variable capacitor, is shown in Fig. 26 (Bernstein," p. 46). The poled piezoelectric polymer membrane is metallized on one side and supported on the other side along a network of lines by a corrugated metal electrode. The electrical system thus consists of a parallel-connected array of small capacitors with an electrical source resulting from the piezoelectric property of the membrane. The electrodes National Semiconductor Corp., Santa Clara, California.

574

5.

MEASUREMENT OF PRESSURE

E!zk3 I - - - - - -

-7

FIG.26 FIG.25 FIG.25. Miniature diaphragm gage with resistance sensor made by semiconductor microcircuit technology. 1: silicon chips bonded together. 2: chip etched to provide diaphragm and evacuated cavity. 3: resistance bridge formed by diffusion of N or P type impurity. 4: diffused P-N junction transistor senses temperature. FIG.26. Diaphragm gage using piezoelectric source. 1: Metallized surface. 2: Poled piezoelectric polymer film. 3: Corrugated metal electrode and diaphragm support.

are connected to a field-effect transistor deposited on the back of the corrugated electrode. The effect of a change in pressure on the membrane is to change the input voltage to the transistor. (The voltage may change due to a change either in capacitance or in the piezoelectric effect. That is, in Eq. (5.6.10) of Section 5.6.2.3 there may be a change either in C, or in the integral.) The sensitivity of the transducer is about 50 pV Pa-' and the quoted band width 3-250 kHz (response time approximately 4 ps). Its attractive features are its fast response time and relative insensitivity to vibrations of the mount, due to low inertia of the membrane, as well as its small size, 5-mm diameter by 1-mm thick. It was designed for use on thin aerofoils and compressor blades. Figure 27 shows a gage which uses a silicon NPN planar transistor as a sensor. See last sensor described in Section 5.6.2.1. A diaphragm is mechanically coupled to a transistor by a small cylinder whose conical tip bears against the pressure sensitive emitter-base junction. Although this gage uses a diaphragm, it might be classified as a stub gage, which is considered in Section 5.10.3, since a change in pressure causes very little movement of the diaphragm. The dynamic behavior is probably determined as much by the size, shape and elastic properties of the coupling cylinder as by the diaphragm. A response time in the neighborhood of 10 ps has been reported for a gage of this type. In order to measure the pressure distribution in a flow about a model in a wind tunnel, use has been made of a large array of small diaphragm sensors whose displacements were simultaneously determined by the

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

575

I

FIG.27. Diaphragm gage using transistor as sensor. 1: Diaphragm. 2: Pressure pin. 3: Transistor with pressure sensitive emitter-base junction. [After "Pitran Pressure Transistors," Stow Laboratories Catdog, Hudson, Massachusetts.]

3

double exposure holographic method considered in Section 5.6.3 .23 The hologram used. for reconstruction of the diaphragm deflections, by the method illustrated in Fig. 17 on page 550, was made with the arrangement shown in Fig. 28. The flow and model Ware shown at the left of a rigid plate containing an array of chambers (1, 2, 3, 4,etc.) each 3 mm in diameter and connected to the flow through a hole having a diameter of 0.2 mm. A diaphragm was formed at the end (right) of each chamber by soldering a brass foil of thickness 0.05 mm to the rigid plate. To obtain a hologram, coherent light from a laser is divided into beams 1 and S by a beam splitter. A lens L1 causes beam I to diverge so that after reflection from a mirror it illuminates all diaphragms of the array. The diaphragms have been etched so that they diffusely reflect some of the light from I

1

I

1

Laser

-1

___

H

---

1 .

V...

.L.W

.WL.,LW..W"

V"..'..

Y

I V

y . Y U Y " W

u *."'"6.U

I...

,"W"

a x e .

I ,

1"s

..IWI.I"U

" 1

L I W V I I

structing picture of diaphragms to measure deflections.) W is a model of a wing in a wind tunnel flow.

576

5.

MEASUREMENT OF PRESSURE

FIG.29. Interferometric reconstruction from a double exposure hologram. The intensity variations over an array of diaphragms correspond to differences in deflections, mapping out pressure variations around an airfoil.

onto the film at H where it combines with the diverging light from the reference beam S to produce a hologram. For the reconstruction shown in Fig. 17 on page 550, the light beam S used for illumination must be the same relative to the hologram H as in the diverging reference beam S of Fig. 28. Actually, the virtual sources of Fig. 17, which correspond to diaphragm positions, occur in pairs and thus produce interference at F, because H is a double exposure hologram. The first exposure is made with no flow and the second during the flow under study. Fig. 29 is an interferogram obtained by reconstruction from a double exposure hologram made in a study of flow about an airfoil. Using light of wavelength 600 nm, the detection sensitivity is estimated as Ap,,,,,, = 500 Pa. This is ample for measuring pressure variations in this example, where the total pressure difference is 7000 Pa. Most gages described in this chapter are commercially available.40 They provide a variety of attractive features including the one that both constant and moderately rapid changes can be measured. Gages capable of measuring more rapid changes are considered in the following chapter.

5.10. Fast Response Gages: Compressional Strain The gages to be considered in this chapter differ from those of Chapter 5.9 primarily in that the elastic element deforms by compressional strain rather than by bending and stretching. Most of the gages were developed for pressure measurements in studies of detonation waves ,71-7* underW . Turetric, “Probleme der Detonation.” Berlin Akademie der Luftfahrtforschung, 1941. D. H. Edwards, J . Sci. Instrum. 35, 346 (1958). 73 D . H. Edwards, L. Davies, and T. R . Lawrence, J . Sri. Instrum. 41, 609 (1964). ” M. K . Mclntosh, J . Phys. E 4, 145 (1971).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

577

water blast waves," and gaseous shock waves,58~62~76-* as well as acoustic waves and boundary layer fluctuations.26*81*82 These phenomena often involve large pressures which last only a few milliseconds but the characteristic of main concern in this chapter is that pressure changes may occur within a range from a fraction of a microsecond to several microseconds. Fast response is required. 5.10.1. Basic Elements of Fast Response Gage

Although proposals for gages using compressional strain involve a wide variety of geometries, the gages all contain the same basic elements pictured in Fig. 30. The time dependent pressure" p ( t ) is applied to the surface of the strain element C. Usually X contains a strain sensor q ,although sometimes v' is omitted and instead the motion of the back surface (bc) of Z is recorded. In regions B and C there may or may not be materials which support the strain element. As the pressure p ( f ) changes, it produces strain which propagates (straight dot-dash line, F) with finite velocity into the strain element 8 . Either the strain or the motion of the back surface of 2 is translated into an electrical signal which is recorded as a function of time. Gage types having elastic elements of different shapes were designated by name in Chapter 5.4 and pictured in Fig. 12. In a dilatational gage the dimensions d and 1 in Fig. 30 are usually of the same order but more importantly the recorded strain or surface motion is considered only up to the time that the one-dimensional behavior due to the passage of the front F is complicated either by the arrival of relief strain (dashed curves) from the lateral boundary (ab-dc) between Z and C or by the arrival of a disturbing reflection from the surface (cb) between C and B or, when movement of the back surface (cd) is sensed, by the arrival of the first reflection from the front surface (ad). '5

R. H. Cole, "Underwater Explosions." Princeton Univ. Press, Princeton, New Jersey,

1948.

S. G. Zaitsev, Prib. Tekh. Exsp. 6, 97 (1958). W. W. Willmarth, Rev. Sci. Instrum. 29, 218 (1958). 'I8 R. I. Soloukhin, Prib. Tekh. Exsp. 3, 170 (1961). 7e K. W. Ragland and R. E. Cullen, Rev. Sci. Insfrum. 38, 740 (i967). M. I. Vorotnikova and R. I. Soloukhin, Zh. Prikl. Mekh. Tekh. Fiz. 5 , 138 (1964). *I A. L. Kistler, Phys. Fluids 2, 290 (1959). * A. L. Kistler and W. S. Chen, J . Nuid Mech. 16, 41 (1963). 7e

'I7

* In general, the applied pressure p(r, t ) is space as well as time dependent, but a single pressure gage measures only the average of the pressure on its sensitive surface. For simplicity in analysis the applied pressure is usually assumed to be uniform.

578

5 . MEASUREMENT OF PRESSURE

FIG. 30. Principal parts of fast response pressure gage. p ( r , 1 ) is applied pressure, which changes with time t and may or may not depend on position r across the surface of the gage. For simplicity of analysis it is assumed to be uniform. S is the strain element (region abcd). P is the sensor or strain-sensitive element. B and C are regions which usually contain materials different from that of P. F is front of propagating strain pulse.

For a bar gage I is either ten or more times larger than d or the properties of the material in region B are such that no reflection occurs at the surface (bc) between Z and B. Often the lateral boundary of the bar can be taken to be stress free because region C contains only gas which provides negligible constraint for the bar. A stub gage is essentially a short bar gage backed by a dense, rigid material in region B whose lateral dimension d is much greater than for the strain element Z. Region C contains a gas and is sealed to isolate it from the applied pressure by a very thin diaphragm. Sizeable reflections occur at the back and front surfaces (bc-ad) of the strain element and its lateral surface (ab-dc) is practically stress free. For a slab gage the shape of the elastic element is similar to that of a diaphragm, with the thickness dimension l being much smaller than d. If the material in region B has very large elastic constants so that for practical purposes it can be considered to be perfectly rigid, it will constrain the elastic element laterally. The strain will be basically one dimensional with the displacement in the thickness direction. In a probe gage dimensions 1 and d usually have approximately the same value. The gage is either mounted in the interior of a fluid whose pressure is to be measured or stress due to applied pressure rapidly propagates into regions C and B. In either case the distinguishing feature is that all surfaces of the strain element 8 are rapidly subjected to a normal stress equal to the applied pressure. The steady state strain is volumetric. For the diaphragm gage, which was discussed in the preceding chapter, the dimension 1 is many times smaller than d, there is only gas in region B and a dense, rigid material in region C provides a mount to which the

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

579

elastic diaphragm is clamped. It was tacitly assumed that reflections between the faces of the diaphragm are so rapid that within time intervals of practical interest they can be neglected. Bending and stretching remain as the types of strain to be taken into account. 5.10.2. Theory of Behavior of Elastic Element

Predictions of the strain produced in an elastic element due to a time varying pressure are usually carried out at one of three general levels of exactness. First is a prediction of practical use, but one that provides little more than a rough criterion for satisfactory performance. The prediction requires only a knowledge of the gage’s response time. An order of magnitude value for the response time can be taken as l / c , where 1 is an appropriate dimension of the elastic element or sensor and c is the effective velocity at which strain propagates. Although the best values to use for 1 and c differ with the shape of the elastic element and the boundary conditions, only a rudimentary knowledge of how strain propagates and reflects at discontinuities is needed to choose a reasonable value for 1 and a value for c can be taken as the square root of the ratio Young’s modulus to density with an error less than a factor of two. Finally, if there are no important pressure changes in a time interval less than several, say 10, multiples of the response time, one can assume for practical purposes that no spurious oscillations are excited and that the applied pressure at each instant immediately produces the steady state (static) strain. This might be called the instantaneous steady state approximation. At the next level of approximation the elastic system of the gage is sometimes replaced by a model consisting of one or a number of rigid massive bodies connected to the mounting and perhaps to each other by massless elastic springs. Dashpots can be used to introduce dissinative effects. General mathematical methods are available for handling such coupled systems of massive bodiess4 and predictions of their behavior provide insight into the nature of the dynamic behavior of gages.27 But, like the analogous lumped parameter models for electric circuits, the approximation they provide usually becomes invalid at extremely high rates of change. These models will not be considered here since this chapter is primarily concerned with very fast response. At the level nearest to actual behavior, strain variations within the elastic element are treated as being due to traveling or standing waves. For gages with response times of the order of microseconds analysis of strain waves is usually necessary. For all but a few simple cases, however, exact solutions, even where possible, are too complicated to be useful.

580

5.

MEASUREMENT OF PRESSURE

The simplest type of wave is a one-dimensional, plane wave whose representation is a solution of the following well known equation

(5.10.1) where E is a strain component, c is a constant propagation velocity, z is a Cartesian coordinate and t is time. Alternatively, the dependent variable might be displacement u , material velocity = & / a t , or stress component cr = (elastic const)e. Referring to Fig. 30 and assuming that the pressure p ( t ) is uniform and suddenly applied, the strain at T is initially due only to a one-dimensional plane wave indicated by the wave front F. Later, with dimension 1 less than d/4, the strain due to direct propagation from the stressed surface is in general augmented by reflection from the interface (bc) between X and B. Although this adds to the complexity, the effect of such a reflection can readily be taken into account since the reflected strain is still onedimensional and plane. Later still, if again 1 is less than d/4, the strain at T is affected by the presence of the lateral boundary (ab-cd) between Z and C. This complication ordinarily results from a relief wave which develops at the boundary following passage of the initial compression front F and propagates inward as a three-dimensional wave (dashed curves). These three-dimensional waves cannot be described by a simple analytical expression. However, for the case of a bar where 1 is much larger than d, for a stress free lateral surface and for certain types of end conditions, an exact solution for the strain at q can be formulated.83 In this solution the effect of the lateral surface on the strain at q is not pictured as being due to three-dimensional relief waves which undergo multiple reflections. Instead the solution is given in terms of a Fourier integral representing the superposition of plane sinusoidally varying phase waves propagating along the bar. In this representation, the effect of the free lateral surface is to produce a dependence of phase velocity on frequency. The result is that a strain pulse propagating away from the end of the bar disperses, i.e., changes shape during the travel. Unfortunately, this exact integral formulation is in general too complicated to be useful. But relatively simple, approximate expressions have been developed which are valid when q is more than a few diameters away from the stressed end of the bar. No truly satisfactory description of the behavior near the stressed end of the bar has been proposed for times after the first arrival of relief strain. 83 R. Folk, G . Fox, C. A. Shook, and C. W.Curtis,J. Acousr. SOC.A m . 30,552 (1958); G . Fox and C. W.Curtis, ibid. p. 559.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

58 1

For the dilatational gage, Section 5.10.6, the one-dimensional plane wave representation is exact. Also, this representation provides a first approximation for a bar gage, Section 5.10.5, with a correction as described in Section 5.10.5.1 possible for a remote sensor. For a bar gage with a contact sensor, Section 5.10.5.2, and for a stub gage, Section 5.10.3, the one-dimensional plane wave approximation is usually poor, but it furnishes insight into the nature of the actual behavior and provides semiquantitative predictions. In the case of a probe gage, Section 5.10.4, an assumption of a one-dimensional wave is probably useful for no more than an order of magnitude calculation. In the remainder of the current chapter only the one-dimensional plane wave approximation will be considered. Well known relationships are given for easy r e f e r e n ~ e . ~ ~ * ~ ~ - ~ ~

5.10.2.1. Propagation of a One-Dimensional Plane Wave. Using the stress-strain relation ~ ( z t,) = ME(z, t )

(5.10.2)

between longitudinal strain E(Z, t ) and longitudinal stress u(z, r ) together with the boundary conditions that the applied pressure p( t ) is uniform and equal to longitudinal stress when z = 0, we have for a solution to Eq. (5.10.1) u(z, t )

=

p

(I -

f ) = p(t’)

(5.10.3)

and (5.10.4)

where t’ = t - z/c. These equations represent a wave traveling in the positive z-direction with constant velocity c. c = (M/p)1’2,

(5.10.5)

where p is density and the value of the elastic modulus M depends on the type of strain being propagated. A . E. H. Love, “On the Mathematical Theory of Elasticity,” 4th ed. Dover, New York, 1944. Bs H. Kolsky, ”Stress Waves in Solids.” Oxford Univ. Press, London and New York, 1953. E. M . Ewing, W. S . Jardetsky, and F . Press, “Elastic Waves in Layered Media.” McGraw-Hill, New York, 1957.

5. MEASUREMENT OF PRESSURE

5 82

For dilatational strain, where there is no displacement perpendicular to M is the dilatational modulus E which is related to Young’s modulus Y and Poisson’s ratio v.

z,

M = E =

Y(l - v) ( 1 + v)(l - 2v)

and

Cd = (5)l”.

(5.10.6)

The velocity c d is called the dilatational velocity. For a bar with a f r e e lateral surface and with the assumptions that only the stress component directed axially (z-direction) is nonzero and that inertia due to lateral expansion can be neglected,*

M

=

Y

and

cg=

(--JY

‘2

.

(5.10.7)

The bar velocity cb is a few percent smaller than c d . In addition to a ( z , t) and ~ ( zt),, other dependent variables are the longitudinal displacement u and the material velocity u , both in the z-direction. u(z, t ) =

JZ 0

a” dz = az

loZ

1 e dz = PC

J

f’

0

p(t’) dt’

(5.10.8)

and (5.10.9)

as before t’ = t -z/c. 5.10.2.2. Time to Travel Through a Sensor. A relation such as given by Eq. (5.10.4) is basic to the operation of gages using a strain sensor; Eq. (5.10.9)plays a similar role for gages using a velocity sensor. The attractive feature of these equations is the prediction that the time variation of strain, or velocity, at any point on a surface for which z is constant is the same as the time variation of the applied pressure. This behavior is exactly realized in the case of a dilatational gage with a velocity sensor, such as described in Section 5.10.6.2, since, with careful alignment, the wave is strictly one-dimensional and velocity is measured at a surface with z constant. However, for a gage with a strain sensor the behavior is less simple, perhaps because the wave is only approximately onedimensional, but, in any case, because the sensor is necessarily finite in size. Consider the effect of measuring strain over a finite distance in the direction of propagation of a wave.

* In this approximation, the lateral displacement is equal to rve, where r is the polar distance from the z-axis.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

583

The general equation for the electrical signal V ( t )produced by a strain sensor having a length h in the propagation direction of a strain wave is V ( t ) = k / =*I ' c ( t - f ) d z = $ [ ' p ( f - ; ) d z .

(5.10.10)

The parameter k depends on the type of sensor, its dimensions, its sensitivity to various components of strain, the elastic properties of the medium through which the wave is traveling and the type of strain being measured. Expressions for k are given for a resistance sensor and for a piezoelectric sensor in Section 5.10.5.1 and for a capacitance sensor in Section 5.10.6.1. For a continuous wave the limits of the integral may be taken to be zi = - h / 2 and zf = h / 2 . But for a pulse with a beginning and an end the limits depend on whether the head, the tail, or neither are in the region covered by the sensor. Assuming that both the head and tail of the pulse are not between the ends of the sensor at the same time and that the pressure is applied initially at time t = 0 and drops to zero at time t = t o , the limits are

zi = 0 zi = 0 zi = ct

and and and

zf = ct zf = h zf = h V ( t )= 0

when 0 < t < h / c , when h / c < t < t o , (5.10.11) when to < t < to + h / c , when to + h / c < r .

For a continuous, sinusoidally varying pressure having a radian frequency w,

~ ( t=) M

-hl2

po sin [o(t

-

f ) ] dz =

Pokh

sin

wh (x) sin(wt)

(5.10.12) The amplitude of the response is a function of frequency with large, approximately constant values occurring only for radian frequencies w considerably less than 2 c / h or, stated differently, only if the wavelength A = 27rc/w is much greater than h . For w = 27rc/h << 2 c / h ,

(5.10.13) If values of w are less than one-tenth 2 c / h , the amplitude of V ( t )will differ from pokh/M by less than 1 percent and pokh/M is the magnitude of the signal that would be obtained by applying a constant pressure p o to the

5 84

5.

MEASUREMENT OF PRESSURE

Time, t

-

FIG. 31. Comparison of gage signals V&) and V&) with applied pressure p(r) for two sensor lengths hl and h z . h, is ten times hl leading to a response time T* = hl/c which is also ten times T~ = hl/c. T is time at which relief strain arrives at sensor from lateral surfaces.

elastic element. Decreasing h improves the frequency response of the gage but at the expense of a decrease in sensitivity, which is k h / M . The gage response, calculated using Eqs. (5.10.10) and (5.10.11), is shown in Fig. 3 1 for three types of nonperiodic pressure-time profile and for two lengths of sensor. For step-function loading (middle graph) the signal increases linearly over the interval T = h / c , which is the response time of the gage. As shown by the top and bottom graphs, if the applied pressure varies appreciably over an interval equal to the response time 7, the resulting output of the gage (V, curves) differs considerably from the input and the integral of Eq. (5.10.10) would have to be inverted to obtain a reasonable representation of p ( t ) from V ( t ) . The tedious process of inverting the integral of Eq. (5.10.10) can be avoided without appreciable loss of accuracy provided the response time T is much smaller, say 10 times smaller, than the time during which a significant pressure change

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

585

occurs. In this case, Eq. (5.10.10) with limits zi = 0 and zf = h can be approximated by the simpler relation V(r) =

kh p (I -

%)

(5.10.14)

since p(r - z/c) is assumed to be a slowly varying function which can be removed from the integral. Equation (5.10.14) also follows directly from the assumption that the sensor measures strain at a point. The approximation is equivalent to replacing the V ( t ) curves in Fig. 31 by the p ( t ) curves. For the Vl(r) curves, which apply to the gage with the shorter response time, the approximation introduces an error of only about 5 percent except during the time interval 0 to T~ when the change in pressure is significant. Eqs. (5.10.13) and (5.10.14) are examples of the “instantaneous steady state” approximation mentioned at the beginning of Section 5.10.2. 5.10.2.3. Reflection and Transmission of an Elastic Wave at an Interface between Two Media. In general, a plane dilatational (longitudinal) wave incident on an interface between two elastic media at an oblique angle will spontaneously generate reflected and transmitted shear (transverse) waves in addition to reflected and transmitted dilatational Only for normal incidence are the shear waves missing, but for present purposes this is the case of most interest and is the only one considered here. For normal incidence, expressions for the reflected strain E;’ and the transmitted strain E;” at the interface (z = 0) are

(5.10.15) where single, double and triple primed quantities are respectively associated with the incident, reflected and transmitted waves. (The boundary conditions are that stress and material velocity be continuous across the interface. Use of these conditions, together with the relations of Eq. (5.10.9) leads to Eq. (5.10.15) and also Eq. (5.10.16).) The incident and reflected waves travel in the medium identified by the subscript 1 and the transmitted wave travels in the medium identified by the subscript 2. It is assumed that the incident wave is produced by the applied pressure p ( t ) , which is related to the material velocity u ; , the strain E; and the stress (+I of the incident wave by the relations of Eq. (5.10.9). There are similar expressions for the material velocity associated with the reflected and transmitted waves:

5.

5 86

v;t

=

[1

MEASUREMENT OF PRESSURE

-

[1 +

””1

PlCl

p(f)

3

;

PlCl

VLtf =

2

[1 + E]

’(‘)

.

(5.10.16)

According to Eqs. (5.10.15) and (5.10.16)a reflected wave does not exist if the product of the density p and the wave velocity c , called the acoustic impedance, is the same for the two materials. This prediction has been used as a basis for extending the length of a bar gage and for reducing disturbances produced by a sensor embedded in an elastic element. See Sections 5.10.5.1and 5.10.5.2. In some experimental arrangements, such as considered in Section 5.10.6.2, pressure applied to the front face of an elastic element produces a strain wave which propagates through the element and subsequently reflects from a stress free back face. During reflection the velocity of the back face is measured. Since the reason the back face is stress free is that there is no supporting material in contact with it, the measured velocity is given by v;” of Eq. (5.10.16)when pz is set equal to zero. Then we have the following relation between measured velocity and applied pressure:

(5.10.17) Measurements are discontinued after the arrival at the point of observation of a doubly reflected wave or of a disturbance which has propagated from a lateral boundary. With gages such as the stub and probe types, which are considered in Sections 5.10.3 and 5.10.4, measurements are continued long after the time needed for multiple reflections to occur at the faces of the elastic element. The behavior is then similar to that of a diaphragm gage. Step function loading produces oscillations of the strain which are centered on steady state values. Both the period of the oscillations and the magnitude of the steady state strain depend on the boundary conditions. For a one-dimensional wave, the well-known result of the normal reflection of strain from a free surface is obtained from Eq. (5.10.15)by setting pz = 0: EI’

=

- p(t)/M.

(5.10.18)

For a fixed surface, setting p p equal to infinity, we have El’

= p(f)/hf.

(5.10.19)

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

587

0 y/////fl/fl&//////m

V

V

0

-

1

Time (a 1

0

-

1

Time

(b)

FIG. 32. Normal reflections of one-dimensional strain waves at two parallel surfaces. Waves are produced at left surface by step loading, so stress at this surface is constant after time zero. The right surface is fixed in case (a) and free in case (b). Full line contour and hatching represents actual strain; dashed contour and hatching represents time average.

Equations (5.10.18) and (5.10.19) have been used as a basis for constructing Fig. 32. Figure 32a illustrates the situation in which one end of a bar* is subjected to a step function load and the other end is fixed. For Fig. 32b, one end is again subjected to a step function load, but the other end is free. As illustrated, the period of oscillation for the bar with a free end is 21/c; the time average of the strain varies linearly with distance along the bar and is equal to po[1 - ( z / l ) ] / M ,where p o is the magnitude of * Although this picture is oversimplified for the case of a bar, it illustrates certain general features of the behavior correctly. See Section 5.10.5.1 for a better representation.

588

5 . MEASUREMENT OF PRESSURE

the step, 1 is the length of the bar and z is the distance from the stressed end. For the bar with the fixed end, the period is 41/c; the time average of the strain is uniform along the bar and equal to p o / M . Actually, because of damping and of dispersion due to the free lateral surface, the head of the pulse does not remain a step as shown, but gradually spreads out during travel with the result that oscillatory variations in the strain die out leaving the bar at each position with a steady state strain equal to the time average. 5.10.3. Stub and Slab Gages

There are two fundamentally different approaches to problems resulting from the reflection of strain at discontinuities within a gage. One approach is to locate the discontinuity, such as the back end of a pressure bar, Section 5.10.5, or the lateral surface of a dilatational gage, Section 5.10.6, far enough from the sensor to prevent the arrival of a disturbance from the discontinuity before the pressure measurement is complete. The other approach is to make the strain element very small so that reflections from opposite boundaries follow in rapid succession and tend to cancel each other. This leads to resonance oscillations of high frequency so they are not excited with appreciable amplitude except by an abrupt pressure change, i.e., a change which occurs within less than a few resonant periods. See Fig. 21. As noted in Section 5.5.1 these oscillations are often referred to as “ringing” of the gage. Depending on the damping properties of the strain element and supporting material, as well as the overall geometry, these oscillations may or may not attenuate rapidly. The gages described in this and the next section use the second of the above approaches, so shock wave loading usually produces large ringing oscillations. The appearance of ringing is often removed from the record by an electrical filter, but this is done at the expense of increasing the response time. Essential features of a reliable, fast response stub gage are shown in Fig. 33. The strain element is the central cylinder or stub, which resembles a very short pressure bar, such as described in Section 5.10.5, with nearly rigid back end support. Due to reflections at the front and back faces, either an abrupt application of pressure to the front of the stub, or a sudden change in velocity of the back end, produces periodic changes in the stress and strain. According to the one-dimensional theory considered in the preceding section, the periods of the oscillations are proportional to the distance between the reflecting surfaces and therefore decrease as the size of the stub decreases. The theory also indicates

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

589

FIG.33. Fast response stub gage. (a) Construction: 1: thin diaphragm and seal, 2: front of stub, 3: back of stub, 4: stub support, 5: piezoelectric quartz crystal stack, 6: piezoelectric quartz crystal compensator. (b) Response to spark induced shock wave with no electrical filter. (c) Response with filter.

that the time-average of the stress or strain at every position along the stub is the same as the steady state value at that position. Due to the back end support, the steady state stress produced by pressure applied to the front face is essentially uniform along the length of the stub. On the other hand, changing motion of the base causes the stub to accelerate* and, since the stress induced by acceleration is due solely to inertia, the steady state stress due to uniform acceleration decreases from a maximum at the base to zero at the front because the mass being accelerated by the stress decreases in this direction. This difference in stress patterns provides the operating basis for the unusual arrangement of op* Step function pressure loading of the back end of the stub, which was considered in the preceding section, is equivalent to an abrupt change in velocity of the end followed by subsequent changes of twice the initial amount at intervals equal to the time for a strain wave to make a round trip of the stub. The resultant time-average acceleration is uniform along the length of the stub. For the stress or strain produced see Fig. 32b.

590

5 . MEASUREMENT

OF PRESSURE,

posing sensors shown in Fig. 33a. The purpose of the arrangement is to balance out the effects of accelerations due to vibrations of the support and at the same time provide a measurable response to the strain produced by the applied pressure. This is accomplished, at least for steady state conditions, by making the quartz sensor near the front of the stub thicker, and therefore more sensitivie to stress, than the opposing sensor near the back. The combination is sensitive to the essentially equal stresses produced by the applied pressure but is insensitive to the unequal stresses resulting from acceleration. An alternative but less attractive method of acceleration compensation is to use two identical elastic elements with their sensors connected to oppose each other; both are mounted on the same base, but the pressure to be measured is applied p. 21). only to one, the other being shieided (Bynum et Figs. 33b and 33c are schematic drawings showing the response of a tiny stub gage to a pressure pulse consisting of a leading shock front fob lowed by a rarefaction. The direct response of the sensors, shown in Fig. 33b, exhibits prominent ringing oscillations having a period of approximately 2 ps. In the record of Fig. 33c the ringing oscillations have been virtually eliminated by an electrical filter, which has also increased the response time by a factor of two or three. An advantage of this gage over the bar and dilatational gages described later is that its hold time can, with the use of an electrostatic charge amplifier, be made very long-long enough, for example, to permit calibration with essentially constant pressures. Usable pressure ranges extend from 10 kPa (fraction of a bar) to nearly 1 GPa (few kilobars). Gages of this type are made commercially in the United States and Switzerland. For a slab, Fig. 12d, which deforms under pressure in roughly the same way as a stub, a capacitor provides a useful means of sensing strain. Shock pressures from 0.5 to 20 MPa have been measured using capacitors with liquid electrolytess7or dielectric filmse8between the plates. See Fig. 34. It has also been suggesteds0that a capacitor using a thin polymer film between its electrodes be cemented to an airfoil to measure pressure on its surface. A third possibilityoois to use a piezoelectric film between the V. N . Kochnev, Electrokhirnicheskiye datchiki dinamicheskikh davlenii. Absrr., All-Union Conf. Dyn. Pressure M e a s . , Ist, 1973, pp. 9-10. VNIIFTRI, Moscow, 1973 (in Russian). G . V. Stepanov, Izmerenie davlenii v udarnykh volnakh dielektricheskirn datchikom. Absrr., All-Union Conf. Dyn. Pressure Measu., Ist, 1973, pp. 13-14. VNIIFTRI, Moscow, 1973 (in Russian). 88 M. Chatanier, “Capteurs de pression pelliculaires.” Office National d’Etudes et de Recherches ACrospatiales, ONERA, France, 1975 (in French). 00 A. L . Robinson, Science 200, 1371 (1978).

5.10. FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

* AP

AP

&$&--AV=?

\,el:+,cnl;

,I

f

“

59 1

G IT R

“0

AC

!rn

’’ Film

-

FIG.34. Pressure sensors which utilize pressure-dielectric effects in a liquid ele~trolyte,~’ and in a solid dielectric filmsa capacitor.

electrodes: with the “plate spacing” only a small fraction of a millimeter, the response time of the unit will be appreciably less than 1 ps. Also, because of small mass, the unit is relatively insensitive to acceleration. 5.10.4. Probe Gage; Volumetric Strain

A variety of gages has been developed to measure pressure within the body of a fluid, rather than at a wall. These are mainly the result of field75,@1*9Z and l a b o r a t ~ r y ’ ~ *studies ~ ~ * ~of, ~open ~ air and underwater explosions. An example of such a gages1-e3is shown in Fig. 35. The piezoelectric element is an x-cut tourmaline crystal, quartz being unsuitable because, as pointed out in Section 5.6.2.3, it is insensitive to a volumetric change. The sensor is embedded in a hard epoxy resin casing which is attached to the end of a hypodermic needle. In a steady state condition, the stress within the elastic element and sensor is a uniform pressure and the fractional change in volume u is given by

AT/T = p/B

=

(5.10.20)

3p(l - 2v)/Y,

where AVis change in volume, p is applied pressure, B is the bulk modulus of the material, Y is Young’s modulus, and v is Poisson’s ratio. Fast response with a minimum of spurious oscillations results from the small size and the damping properties of the casing. As with most gages of this type an exact response time or resonance period is difficult to predict. To the extent that the elastic element can be considered to be a sphere to which pressure is suddenly applied, the resonant periods Pqare given by solutions to the following equation.

[I

-

- 2(1

’)

- 2v)

sin(rna) - (ma) cos(ma) = 0, (5.10.21)

where v is Poisson’s ratio, a is radius of sphere, m

OP

=

2?r/cdP = w/cd,

I. B. Sinani, Prib. Tekh. Exsp. 4, 85 (1957). M. 1. Vorotnikova, Zh. Prikl. Mekh. Tekh. Fiz. 2, 110 (1962). M. I. Vorotnikova, V. K. Kedrinskii, and R. 1. Soloukhin, Fiz. Goreniyu Vzryvu 1, 5

(1965).

5.

592

MEASUREMENT OF PRESSURE 2

1

3

1 cm

FIG.35. Tourmaline probe gage for underwater blast measurement of pressure. 1: tourmaline platelet; 2: epoxy resin; 3: hypodermic needle. Size of tourmaline sensing element is 1 x 1 x 0.2 mm3.

and c d is dilatational wave velocity. For a nominal value of 0.3 for v, the longest resonant period is

(5.10.22) The gage pictured in Fig. 35 has a response time of the order of 1 ps and has been used to measure pressures of approximately 100 MPa. Another type of fast response gage for measuring pressure in the interior of a fluid" uses a small tube in the end of which is a tiny pressure bar such as described in Section 5.10.5. The high pressure gage shown in Fig. 36 measures pressure at a wall,

"2 t

v t (a)

(b)

FIG.36. Resistive probe gage for pressure measurement. (a) Water column impact tube for testing and calibratinggage. 1 : sensing element, 2: impacting shock front generated by a water-water collision, 3: thin diaphragm separating colliding water columns, 4: piston, 5 : gas driver chamber. (b) Calibration curves for Ge(Si) and Ge(P-N) samples.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

593

but very shortly after the pressure is applied the sensing element experiences compressive loading on all faces. Its operation depends on a resistance change of a metal such as manganing4or a sernicondu~tor.~~ According to Eq. (5.6.8) of Section 5.6.2.1, a measure of a material’s sensitivity for resistance change is its value for Y,,, defined by

y

1 dR R dp’

=--

(5.10.23)

where R is the resistance and p is pressure. Values of 9,for the metal manganin and for the germanium semiconductors Ge(Si) and Ge(P-N) are, respectively, 0.024, 0.03, and 0.01 MPa-’. Figure 36 shows a resistance sensor mounted in an apparatus for producing shock waves in water. The sensor was a semiconductor, Ge(Si) or Ge(P-N), having dimensions 1 X 1 x 0.5 mm. It was mounted in an epoxy resin case and measured pressures of nearly 1 GPa (10 kbar). Fig. 36 also shows calibration curves for Ge(Si) and Ge(P-N). 5.10.5. Pressure Bar Gage

Use of a long bar for the strain element of fast response gages has been popular since the idea was introduced by Hopkinson in 1905.9s-9sIn these gages the time varying pressure to be measured is applied to one end of a bar and either the resulting strain at some position along the bar or the movement of the far end is sensed. Measurements are made during the first passage of the strain wave and are completed before distortions are introduced by the arrival of a wave reflected from one of the ends. 5.10.5.1. Remote Sensor: Stress-Free Surface. Figure 37 shows a long bar with several types of sensor placed at some distance from the stressed end. It is supposed that effects due to supports or any surrounding material, such as shown in regions B and C of Fig. 30, can be neglected. The oldest and simplest theory of strain propagation along a bar with a stress-free lateral surface is the one-dimensional theory considered in P. W. Bridgman, Proc. A m . Acud. Arts Sci. 43, 347 (1911). V. K. Kedrinskii, R . I. Soloukhin, and S.V. Stebnovskii, Zh. Prikl. Mekh. Tekh. Fiz. 4, 93 (1969). ge H. Kolsky, “Stress Waves in Solids,” p. 87. Oxford Univ. Press, London and New York, 1973. O7 B. Hopkinson, Proc. R. Soc. London, Ser. A 74, 498 (1905). B . Hopkinson, Philos. Trans. R. Soc. London, Ser. A 213, 437 (1914). R . M. Davies, Philos. Trans. R. Soc. London. Ser. A 240, 375 (1948).

5 94

5.

MEASUREMENT OF PRESSURE

-

h

+ . ic

4

A

B

h=o

1

C

FIG. 37. Pressure bar and types of sensors. A: Surface type, B: cross section type, C: end type.

Section 5.10.2.1. Although this theory is only approximate, it predicts correctly for step pressure loading of one end of a bar that the head of the resulting strain pulse travels with the bar velocity c b defined by Eq. (5.10.7) and that the steady state longitudinal strain following passage of the head of the pulse is equal to the applied pressure divided by Young’s modulus. The one-dimensional theory is assumed in the following discussion of sensor types. Modifications due to exact theory are considered later. Three general types of sensor are indicated in Fig. 37: surface sensor, cross section sensor, and end sensor. In all cases the hold time of the gage T is determined by the first arrival of a reflection from one of the ends. Thus T = 21/cb, where 1 is the distance from the sensor to the far end of the bar for the surface and cross section sensors and is the length of the bar for an end sensor. Capacitance, piezoelectric, and resistance sensors have all been used for surface measurements. The capacitor is cylindrical, with the metal bar serving as the grounded e l e ~ t r o d e . It ~ ~responds to radial displacement which is related to longitudinal strain through Poisson’s ratio. The piezoelectric sensor is cemented to the side of the bar. One type is a thin slab of polarized ceramic (e.g., barium titanate) with a metal foil on the outer face for one electrode and the grounded bar for the other. The electrodes act as capacitor plates on which charge or voltage builds up as the element is strained. With the polarization direction perpendicular to the surface of the bar, the sensor response is proportional to change in surface area which is equal to the sum of the longitudinal and radial strains. Stick-on resistance sensors have also been used. These have consisted of a fine metal grid whose resistance changes when the grid expands or contracts in a particular direction. Thus the grid responds to longitudinal or radial strain depending on whether its sensitive axis is parallel or perpendicular to the axis of the bar. Or a fine resistance wire with an insulating coating has been wrapped around the circumference of the bar, in which case it responds to radial strain. On the basis of one-dimensional theory, all of these sensors operate according to Eq. (5.10. lo), or a similar equation for a change in charge or current. The velocity c is the bar velocity and k and M depend on the particular type of sensor and arrangement. The contribution to the response time due to the size of the sensor is given by T = h / c b . As an example, for a stick-on resistance sensor di-

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

595

rected so as to respond only to longitudinal strain, M is Young’s modulus and k = GIoR/h, where G is the gage factor, I, is the constant current through the sensor and R is the sensor’s total resistance. Equation (5.10.10) becomes (5.10.24)

and, for small h, we have for Eq. (5.10.14), (5.10.25)

Heating limits the value of I,,. Since R is proportional to h , the signal V(r),as well as the response time r , increase linearly with h. Of the three sensors the polarized ceramic has the greatest sensitivity, but the sensitivity is more dependent on ambient temperature and less stable over a long period. Lack of long term stability of the polarized ceramic necessitates frequent calibration, which must be carried out dynamically. The need for precise electrode alignment is an annoyance with an air-spaced capacitor whereas bonding problems can be troublesome with cemented sensors. For cases in which pressure is a monotonically increasing function, bonding is much less of a ,problem with the wraparound wire gage than with commercial stick-on resistance gages. A problem peculiar to the cross-section sensor, type B in Fig. 37, is the distortion of strain in the bar due to reflections at the interfaces. According to Eq. (5.10.19, reflections of strain will not occur if the product of density and bar velocity, p c b , is the same for the bar and the sensor. On this basis the following pairs of metal and piezoelectric material provide reasonable matches: aluminum-quartz, zinc - barium titanate, tinlead metaniobate. Use of such corn bin at ion^^^ has been found to minimize reflections greatly, often to the point of no practical importance. A cross section sensor using a piezoelectric material operates as a capacitor with adjacent sections of the bar acting as electrodes. Assuming one-dimensional theory, the voltage developed between the electrodes is given by Eq. (5.10.10), with k of this equation determined from Eq. (5.6.1 1) and Eq. (5.6.12) or Eq. (5.6.13) of Section 5.6.2.3 for either a poled ceramic or x-cut quartz with the direction of polarization along the axis of the bar.

v(t)= Q(r)/co= l z f p( t hCo 5,

-

k)

dz,

(5.10.26)

where the sensor has a thickness h , a cross-sectional area A, which is the same as that of the bar, and a geometrical capacitance Co. The piezoelec-

5 96

5.

MEASUREMENT OF PRESSURE

tric constant d,, is d,, for x-cut quartz and dS3for a poled ceramic. If h is very small,

(5.10.27) Since C, is inversely proportional to h, V(r) increases linearly as h increases but Q(t) is independent of h. End sensors measure displacement or velocity rather than strain. Again relying on one-dimensional theory, the operating equation for an end sensor is Eq. (5.10.17) of Section 5.10.2.3, with c1 equal to the bar velocity c b . This relates the material velocity u ( t ) of the far end of a bar of length I to the pressure applied at the other end:

(5.10.28) Hopkinson’s original experiment^*^*^* were carried out with an end sensor in the form of a short bar placed in contact with the pressure bar. The compression pulse propagating from the stressed end of the pressure bar passes through the contact interface and reflects a; a tension pulse from the free end of the sensor. When the reflected pulse reduces the compression at the contact interface to zero, the sensing bar separates, trapping momentum in the process. See Fig. 38. The trapped momentum, which depends on the pressure-time profile and the length of the sensor, was measured with a ballistic pendulum. The pressure-time profile was determined roughly from several measurements using sensing bars of different lengths. Although this technique is cumbersome and not very accurate, it is unique in that it uses no electrical measurement. Also modifications of this technique have been used in recent experiments not involving a pressure bar. One type of electrical sensor uses the end of the pressure bar as one plate of a parallel plate capacitor. With the other plate stationary, the voltage across the capacitor varies with the displacement of the end of the bar. An advantage of this sensor is that its contribution to the response time 7 is negligible (in effect, h = 0). Its serious disadvantage is that it measures displacement and therefore the voltage-time record must be differentiated to obtain the velocity u of Eq. (5.10.28). In principle, both types of laser interferometer described in Section 5.10.6.2 could be used as end sensors, but €or most pressure bar applications only the displacement interferometer would be suitable since the velocity interferometer would be too insensitive. The principle ambiguity in these analyses of pressure bar gages results from the approximate forms of Eqs. (5.10.4) and (5.10.9) of Section

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

IP

597

v

Tension

I

4 IP

I

q

I

I 1

I------

FIG.38. Reflections at end of Hopkinson sensor bar. Graphs, at successive times, of pressure p and material velocity u as function of spatial coordinate. All parts of pulses are assumed to propagate with unchanged form at speed c. Bottom graphs show conditions after separation. Next to bottom graphs show conditions at instant of separation (p = 0 at joint .)

5.10.2.1, which relate the applied pressure to the measured strain or material velocity. According to Eq. (5.10.4) a step function change in pressure should produce a step function change in strain at any position along the bar. For comparison with this prediction, Fig. 39 shows an actual record of strain some distance from the end of a bar which has been subjected to reflection of an air shock. The slow rise and complex oscillations are due to the lateral free surface, which makes the problem of prediction three-dimensional. Exact solutions to problems of this type have been f o r m ~ l a t e d ,but ~ ~in, ~general ~ they are too complicated to be useful. G. P. DeVault and C. W. Curtis, J. Acousf. SOC.A m . 34, 421 (1962).

598

5.

MEASUREMENT OF PRESSURE

F

S

TIME

-

FIG.39. Strain observed at surface of cylindrical bar subjected to step function end loading. Bar is magnesium with diameter 38 mm. Distance of sensor from loaded end is 1.51 m. A superposition of very high frequency, low amplitude oscillations begins at the time denoted by S. At the sensitivity of this oscillogram there are no observable oscillations following time marked F. [From the reports of Ref. 83.1

They can, however, be replaced by simple asymptotic expressions which are valid for large distances of travel (>10-20 diameter). The expression* most useful for pressure gage design describes the slow rise and the large, low frequency oscillations near the beginning of the pulse; the strain is

EJB) =

9 [k + 1A@) d B ]

=

poU(B),

(5.10.29)

where p o is the magnitude of the pressure step, Ai(B) is the Airy function, and (5.10.30)

where d is the bar diameter, u is Poisson’s ratio, and z is distance of travel. The time t’ = t - z/cb is measured relative to the arrival time for strain traveling with the bar velocity c b . A graph of U(B)versus B is shown in Fig. 40. The beginning strain arrives in the form of a precursor which travels with the dilatational velocity c d but this decays rapidly during travel and is negligible after a few diameters (> 10-20 diameter) beyond which Eq. (5.10.29) becomes a reasonable representation. Since U(B)of Eq. (5.10.29) is the strain produced by unit step function load, p ( B ) = S(B), it can be used in the superposition integral of Eq. (5.5.2) of Section 5.5.1, to predict the strain when the applied pressure p ( B ) changes gradually rather than abruptly with time. Similar to the case of a diaphragm, for which results are shown in Fig. 21, the integra* Other expressions describe the complex behavior in the region between the arrows at S and F in Fig. 39, but this behavior is less important because the oscillations are of much smaller amplitude and higher frequency.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

599

tion of Eq. (5.5.2)has the effect of averaging out the oscillations of U(B). When no significant pressure variations occur within the interval 7B [shown in Fig. 401 the expression for E,(B) given by Eq. (5.5.2)reduces to p(B)/ Y, which is the value predicted by the one-dimensional theory. Thus the rough one-dimensional theory is reasonable provided there are no appreciable pressure changes within an interval 7B. Call T~ the dimensionless response time, which is in addition to the response time due to the length of the sensor. With the assignment of a value of 3.00 to 78 = B in Eq. (5.10.30), the actual response time T = t’ is given by 7 =

3.00 (

(5.10.31)

3 ) 2 / 3 ($;)1’3.

Values for the magnesium bar used to obtain the record of Fig. 39 were d = 3.81 cm, c b = 5.02 km s-’, Y = 0.300, z = 1.51 m, which result in a response time of T = 20 ps. This can be reduced somewhat by measuring the strain closer to the stressed end of the bar. Taking z / d = 20 gives 7 = 16 ps. As z / d becomes smaller than about 20 the oscillations become more complicated than indicated by Eq. (5.10.29) and, contrary to that equation, the amount of overshoot to the first maximum increases. However, the rise time to the first maximum continues to decrease as z ” ~ in accord with Eq. (5.10.31). Edwards et ~ 1 . suggest ‘ ~ that a value of z / d between 2 and 4 provides the best compromise between short rise time and large overshoot. Assuming z / d = 2, 7 is 7.4 ps. A more significant reduction in 7 results from a decrease in the diameter of the bar. If the diameter is decreased to 5 mm, keeping z / d , cb and v unchanged, 7 = 1 ps. Aluminum, steel, and most other practical materials

-

Dilatational Precursor

\

-2

. 2

4

6

8

10

Time, B

FIG.40. Strain versus time predicted by asymptotic theory for a bar subjected to step function end loading. The dimensionless quantity B is proportional to t’ which is time measured from the instant at which strain traveling at the bar velocity cb arrives at the sensor.

600

5.

MEASUREMENT OF PRESSURE

have approximately the same values for c b and v , so 1 ks is about the lower limit for the resolving time of pressure bars with a free surface and a remote sensor. However, a possible exception has been pointed out by Baganoff.101*102 Beryliium has a bar velocity approximately 2.5 times larger and a Poisson's ratio about 10 times smaller than the above values, which could result in a response time of approximately 0.1 ps. Joneslo3 has examined this possibility experimentally. He used a beryllium bar with a thin cross-section sensor made from a poled ceramic (PZT-4) and measured a response time several times less than predicted for a bar having the same dimensions but made from magnesium, aluminum, or steel. The measured time was slightly greater than anticipated, possibly because of a mismatch between the elastic properties of beryllium and the ceramic PZT-4. This mismatch was thought to cause small spurious radial oscillations of the sensing disc. Problems resulting from the mismatch might be reduced or eliminated with further development. Although records from a pressure bar with remote sensor, like those from a stub or probe gage, contain spurious oscillations, the amplitude of the oscillations can in principle be smaller and its time rate of decrease larger in the case of the bar. On the other hand, the hold time can be greater for the stub or probe gages. Unlike the dilatational gage, which is described in Section 5.10.6, the bar can be used to measure average pressure when the distribution over the sensitive surface is nonuniform. This is because the effect of an asymmetrical distribution of pressure over the end of a bar is to produce flexural oscillations which can be cancelled out by the sensor.1oo Thus, a bar gage can be used to measure pressure at a wall as well as at the end of a shock tube. A pressure bar with a remote sensor can be particularly useful in cases, such as occur in plasma experiments, where the pressure to be measured is in an environment of large electrical disturbances, since a remote sensor can be placed outside such a region. For constructional details and typical records of bar gages with remote sensors, see reports of Refs. 72-74. There are alternatives to the pressure bar which also have the attractive feature that the record is not complicated by reflections from the back of the strain element. The alternatives suggested so far, however, have other characteristics which make them impractical or unattractive for general purpose measurements. One alternative is the torsion bar, described by Davies and Owen.lo4 Unlike the longitudinal strain pulse in a pressure bar, a torsional pulse lol lo*

Io3 lo'

D. Baganoff, R e v . Sci. Instrum. 35, 288 (1964). R . K . Hanson and D. Baganoff, R e v . Sci. Instrum. 43, 396 (1972). I. R. Jones, R e v . Sci. Instrum. 37, 1059 (1966). R . M. Davies and J . D. Owen, Proc. R . Soc. London, Ser. A 240, 17 (1950).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

60 I

I------% ___L( FIG. 41. Expansion of thin tube during passage of confined shock wave. a = initial radius, h = wall thickness, w = lateral expansion of wall. Y is a sensor of fine enamelled resistance wire wrapped around the tube. T = response time, V , = shock wave velocity.

propagating in the fundamental mode does not spread out or develop spurious oscillations as it travels along a bar. The propagation velocity is c, = ( ~ / p ) ” where ~, p is the shear modulus. Davies and Owen confirmed this experimentally by observing a torsional pulse produced by the impact of a bullet fired into a notch on the side of a bar. Large pressure changes taking place in a fraction of a microsecond were recorded. The obvious difficulty with this technique for general pressure measurement is translating pressure into torque. Another alternative is to use the wall of a tube as a strain element. For example, the pressure jump across a one-dimensional shock propagating through a fluid contained in a shock tube could be determined by measuring the expansion of the wall. Usually the velocity with which a strain pulse propagates along a solid is determined by the elastic properties of the material, but in this case, the expansion pulse travels with the velocity of the shock as shown in Fig. 41. Both experiment10sand theorylWindicate that, like the head of a longitudinal pulse in a bar, the head of the lateral expansion pulse in a tube is spread out in space and time. An approximate expression for the consequent response time is (5.10.32)

where h is tube thickness, a is tube radius, and V , is shock velocity. This expression is reasonable if h << a and V, << cp, where cp is the plate velocity ( Y / p ( 1 - Y ~ ) ) ~ / * . Even assuming h / a = 0.1 and a = 1 cm, which are unrealistically small for a conventional shock tube, T would be about ‘‘I W . R. Smith, Shock Tube R e s . , Proc. I n t . Shock Tube S y m p . . 8th, London, July 1971, Paper 59. Chapman and Hall, London, 1971. ‘OB S. Tang, Proc. A m . SOC. Civ. Eng., J . Eng. M e c h . Div. 91, 97 (1965).

602

5.

MEASUREMENT OF PRESSURE

20 ps when V , = 750 m s?. This response time is an order of magnitude larger than the response time of a small pressure bar, so the tube is much less suitable for measuring rapid pressure changes. 5.10.5.2. Contact Sensor: Lateral Constraint. A versatile bar gage, with a sensor at the pressure end, has been developed by Turetic, Zaitsev, Soloukhin and, later, by ~ t h e r ~ . ~Details ~ , ~ are ~ J shown ~ J ~ in ~ Fig. 42. Pressure is applied directly to one surface of a small piezoelectric sensor which is backed by a relatively long, matching metal rod. Both sensor and rod are embedded in a pliable material which is contained by a mounting tube. The purpose of the backup rod is to prevent oscillations of the sensor due to reflections of strain at its front and back surfaces. For a onedimensional wave it follows from Eqs. (5.10.15) and (5.10.16) of Section 5.10.2.3 that reflections at the interface between the sensor and the rod can be effectively eliminated by using materials having the same acoustic impedance, pc. With a polarized ceramic such as barium titanate (lead zirconate-titanate, lead metaniobate, etc.) for the sensor, the backup rod is matched by making it of zinc (zinc, tin, etc.). To avoid depolarization of the sensor, which occurs at high temperatures, a low melting point solder, such as Wood’s metal, is used to attach the sensor to the rod; some of the recently developed polymer cements might also be satisfactory. The embedding material has a number of functions in addition to that of supporting the sensor and rod. First, it insulates the sensor from disturbances due to extraneous vibrations of the mounting tube. Second, together with the pressure pulse it sustains, it tends to suppress the lateral expansion of the sensor, thus reducing the inward propagating relief pulse. See schematic pulse profiles in regions ‘c and C of Fig. 30. Third, it causes attenuation of the strain pulse in the backup rod so that reflection of the pulse from the far end of the rod has little effect on the sensor. Beeswax and silicone rubber have been found to be effective embedding materials. Use of a poled ceramic for the piezoelectric sensor has the advantage of high intrinsic sensitivity, thus permitting construction of a very small gage. Its response is linear at low pressures5E~82*76~78~108 but becomes nonlinear above a few MPa (tens of bars).log As mentioned in Section lo’ Yu. E. Nesterikhin and R . I . Soloukhin, “Metody Skorostnykh Izrnerenii b Gazdinarniki i Fiziki Plazmy.” Nauka, Moscow, 1967; English translation available from Federal Scientific and Technical Information, Springfield, Virginia (Doc. AD 682067). Io8 J. P. Huni, R. Ardila, and B. Ahlborn, Rev. Sci. Instrum. 41, 1074 (1970). ‘00 E. K . Dobrer and K . N . Karmen, Zh. Tekh. Fiz. 2, 455 (1957).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

603

FIG.42. Bar gage with piezoelectric end sensor. Two gages are shown, one mounted in the shock tube side wall and one in the end plate. The output signals are shown near each gage and the x - t diagram indicates the shock arrivals at each gage. 1: piezoelectric sensor, 2: acoustic absorbing rod, 3: beeswax potting, 4: supporting and mounting tube. Electrode at pressure loaded surface is connected to grounded mounting tube with a fine wire.

5.6.2.3, the mechanical strength of poled ceramics is less than that of single piezoelectric crystals, such as quartz and tourmaline. As with all gages, miniaturization is the key to fast response. Good performance has been obtained with contact sensors having a bar diameter d in the range 1 to 5 mm and a sensor thickness h of comparable value, 0.5 < h / d < 1. Experiment has indicated that the best compromise for the thickness of the embedding material 6 is S/d 0.5. Tests have also shown that the best position for the sensor is at the very end of the bar. Typical response of this gage to step loading produced by a shock wave is shown by the records reproduced in Fig. 43.* The response is close to the ideal predicted on the assumption of one-dimensional strain propagation. See middle graph of Fig. 3 1. The rise to a constant value is approximately linear and there are no sizeable oscillations. Distortions due to relief at the lateral surface are either suppressed by the embedding material or smoothed out by averaging over the volume of the sensor. If the lateral constraint were perfect, the response time would be T = h / c d . Measured response times are somewhat greater than this, 7 1.5 h / c d , but a value of 1 ps is readily attainable. The hold time depends on the length of the backup rod and is nominally T = 21/cb but is actually greater than this because of attenuation due to the embedding material. Hold times of approximately 100 ps are reasonable.

-

-

A. H. Meitzler, IRE Nutl. Conv. Rec. Part 9, p. 55 (1956); J. Miklowitz and C. R. Nisewanger, J . Appl. Mech. (Trans. ASME, Ser. E ) 24, 240 (1957).

* Exact theory is too complicated to be helpful in predicting the spacetime dependence of strain within the sensor. Near the end of the bar Eq. (5.10.29)does not provide a reasonable description. On the other hand, a tractable solution based on the idea of an inward propagating relief pulse has not been developed quantitatively for a bar whose side is free or partially constrained. Prediction is simple only if it is assumed that the lateral constraint is perfect. For experimental studies of strain near the end of a bar see Ref. 110.

604

5 . MEASUREMENT OF PRESSURE

FIG.43. Response of pressure bar gage with contact sensor and lateral constraint. Oscillographic records of pressure at the side wall of shock tube as incident and reflected shocks pass gage. (a): incident shock M s = 3 and reflected shock in argon, timing trace period 10 ps. (b): incident shock on expanded time scale, timing trace period 1 ps.

The versatility of this piezoelectric gage is illustrated by some of its applications. It has been successfully used (i) to measure pressures behind detonation and shock waves in shock tube experiment^,^^ (ii) to study shock front configurations and to determine pressure distributions behind high Mach number shock waves generated in an electromagnetic shock tube,81,82(iii) to measure pressure profiles of one dimensional shock waves produced in water-to-water impact, as well as (iv) for auxiliary purposes such as triggering and shock velocity measurement. 5.10.6. Dilatational Gage 5.10.6.1. Contact Strain Sensor. Figure 44 shows a gage due to Baganoff"' which has an extremely short response time. It is essentially the same as the basic gage depicted in Fig. 30 except that the length of the strain element is greater than its diameter and the sensor, which in this case is a capacitor, is in contact with the stressed surface. The gage operates on the principle that the strain propagating into the strain ele-

FIG.44. Dilatational gage using an electrical capacitor as sensor. 1: sensor electrode, conducting epoxy. 2: outer electrode, silver paint. 3: polycarbonate.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

605

ment is purely dilatational, i.e., the material undergoes no lateral extension. This should be true provided the applied pressure is uniform and measurements are completed before strain propagates to the sensor from the lateral boundary or from the back surface after reflection. Under these conditions, the equation for strain propagation is strictly one-dimensional, and the relationships of Section 5.10.2 should be exact. If the capacitor is initially charged with a voltage V , and if there is no leakage of charge during measurement, the voltage signal V ( t )of Eq. (5.10.10) becomes

(5.10.33) where E and c d are, respectively, the dilatational modulus and propagation velocity, Eq. (5.10.6). When all important pressure changes take place in an interval which is large compared to the time for a strain wave to propagate through the capacitor Eq. (5.10.33) then reduces to

'

V(t)= - p E

(t - c'-, )

(5.10.34)

Tests of this gage, using air shocks of negligible thickness, have shown that it responds to step function loading as predicted by Eq. (5.10.33), and shown by the middle graphs of Fig. 31. The records show no overshoot or spurious oscillations. With the strain element made of polycarbonate polymer (General Electric's Lexan) and with a separation h = 0.13 mm for the capacitor plates, the response time T = h/cd is about 0.1 ps. The hold time, during which the measurement must be completed, is set by the arrival of relief strain from the lateral boundary. For a gage having the dimensions given in Fig. 44, the hold time is approximately 5 ps. The value of the modulus E calculated by Eq. (5.10.6) from the measured value of c d is reported to be several times larger than the listed static value. This is not unusual for a polymer experiencing such a high rate of strain. Baganoff does not discuss the effect of material relaxation; presumably it is too slow to be important. A unique aspect of this gage is that the underlying propagation theory is both simple and exact. This and its extremely short response time are very attractive features. One of its limitations is that the applied pressure must be uniform across the sensitive surface of the strain element. Thus it is suitable for measuring pressures at the end plate of a linear shock tube, but, unlike a bar gage, not along a side wall. Another limitation is that the hold time is small, so measurements cannot be carried out over a long period. But it is ideal for certain applications, such as the one for

606

5.

MEASUREMENT OF PRESSURE

which it was designed, namely to measure pressure profiles across shock fronts about 1 cm thick traveling in low density argon.

5.10.6.2. Free Surface Motion. Another gage, which has the fastest response of any proposed, also depends on the propagation of dilatational strain. Pressure is applied to one surface of a plate and the subsequent motion of the opposite surface, which is free, is measured with a chr~no-interferometer,~~J~~~~~~ such as referred to in Section 5.6.3. The motion is observed up to the time that regularity of the first reflection is disturbed by the arrival of a signal either from the periphery of the region over which the applied pressure is uniform or from the lateral boundary of the plate or from a second internal reflection. The hold time is given by the smallest of the values given approximately by r / c d , W / k d or 2 1 / c d , where r is the radius of the region of uniform pressure, w is the width of the plate and 1 is its thickness. The measured velocity u ( t ) of the back surface of the plate is related to the applied pressure p (t) by Eq. (5.10.17), (5.10.35)

where the zero of time is 1 / c d after pressure is applied to the front surface. In many high pressure applications, the plate is stressed beyond its elastic limit, where c d is a slowly varying function of the pressure, so it is sometimes necessary to determine c d by measuring the time for stress to propagate through the plate. But a determination of v ( t ) is the basic measurement, which is made with the chrono-interferometer. In a chrono-interferometer the output of a photodetector V ( t )is proportional to the light intensity of two interfering beams Z(t) = 21,

cos2[aF(t)

+ p],

(5.10.36)

where F(t) is referred to as the “fringe count” because the intensity passes through a maximum and a minimum and returns to its original value when F(r) changes by 1. See Chapter 2.4 of Part 2 for more description of interferometers. In general, F(t) is not an integer but it and the intensity are constant provided the optical path difference A for the interfering waves is constant and their wave length A is the same. Two basically different types of interferometer have been proposed. In one type,”’ illustrated by the Michelson arrangement shown in Fig. 45, one of the interfering beams is reflected from a fixed mirror and the other L. M. Barker and R. E. Hollenbach, Rev. Sci. Instrum. 36, 1617 (1%5). L. M.Barker and R. E. Hollenbach, J . Appl. Phys. 41,4208 (1970).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

suring FIG. 45. surface Chronointerferometer displacement Al(t). for meal:

moving surface. 2: stationary mirror. 3: beam splitter. 4: light source. 5: photodetector.

607

"." all

2

is reflected from a mirror whose movement is to be sensed. If v ( t ) is the velocity of the moving mirror, the frequency f of the reflected wave is Doppler shifted an amount Af given to a good approximation by

(5.10.37) where the velocity of light F is much greater than v ( t ) . The rate of change of the fringe count is the frequency difference, Af, of the interfering waves, i.e., the beat frequency. Thus if u(t) is zero when t is zero,

(5.10.38) where Al(r) is the displacement of the mirror and A(t) is the increase in optical path length. This leads to Eq. (5.6.17) of Section 5.6.3 when substituted into Eq. (5.10.36). A drawback of this type of interferometer is that it is basically a displacement sensor since fringe count is the measured quantity. To obtain the velocity v ( t ) the fringe count must be differentiated with an accompanying loss in accuracy. A second type of chrono-interferometer, which is of more recent origin,54*112 measures velocity directly. Such an interferometer is shown in Fig. 46. In this case light from the source is reflected from the surface whose movement is to be sensed before it is split into two beams, one of which travels a greater optical length A before it arrives at the photodetector where it is combined with the beam which has traveled the shorter path. The distance A is constant and can be made fairly large by using laser light with a long coherence length. The frequencies of the interfering waves are Doppler shifted and by different amounts if the velocity of the moving surface is changing since they would have been reflected at different times. If the wave traveling the shorter path was reflected at time t when the velocity was v(t) it would interfere with the other wave

608

5.

MEASUREMENT OF PRESSURE

FIG.46. Chronointerferometer for measuring surface velocity u ( t ) . 1: moving surface, 2: laser, 3: lens, 4: delay path, 5: beam splitters, 6: photomultiplier, 7: alternate position for photomultiplier.

which had been reflected at time t - T when the velocity was v(t - T ) , where T is A/?, the “delay time” for the wave traveling the longer path. Using Eq. (5.10.37) and the fact that the rate of change of the fringe count is the beat frequency of the interfering waves, we have F(t) =

A

[1;

u ( t ) dt

-

1:

u(t

- T ) dt

1.

(5.10.39)

If t, the time after the reflecting surface started to move, is less than T , the second integral of Eq. (5.10.39) is zero and the expression for F(t) is the same as given by Eq. (5.10.38). For t greater than T , Eq. (5.10.39) reduces to (5.10.40)

or in terms of uav(r), the velocity averaged over the interval from t

-

7

to

(7

(5.10.41) If v ( t ) varies monotonically, ua,(t) can usually be replaced to a good approximation by u(t - 7/2), the instantaneous velocity at the midtime of the interval. For a step change in velocity of amount u , the fringe count increases lin-

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

609

early with time over an interval 7 and then remains constant at a value determined by Eq. (5.10.41) with u substituted for vav(f). An upper limit on the size of a velocity step is set by the fact that if it is too large the time between successive fringes will be so small that individual fringes cannot be resolved; this holds for the velocity interferometer during the interval T and for the displacement interferometer at all times. For an early velocity interferometer having a single delay line as shown in Fig. 46, nonresolution of fringes over the interval T results in a loss of the integer but not the fractional count of fringes. To be able to determine the integer count unambiguously, later velocity interferometers have been built with two, simultaneously operating, delay lines having T ’ S which differ by a noninteger multiple.113 The response time of the velocity interferometer is T = A/i. and its sensitivity is Flu = 2T/h = 2 A / t h . With values of T in the range 1-10 ns feasible, a dilatational gage using a velocity interferometer can provide one of the shortest response times available. With the simple chrono-interferometers pictured in Figs. 45 and 46, the mirrors must be precisely ground to provide essentially pure specular reflection and the source must supply light having a long coherence length. These requirements set a practical upper limit on the length of the delay time that can be employed. Consequently, very low velocities cannot be measured with velocity interferometers of this type because impossibly long delay times are needed to provide several fringes which are necessary for accurate measurement. On the other hand, as the velocity to be measured with a displacement interferometer is increased it becomes progressively more difficult to distinguish and count individual fringes because of an increase in fringe frequency. In Barker’s 1972 review,s4 he suggested the advantage lies with the velocity interferometer for measuring velocities greater than 100 m . s-l, but shifts to the displacement interferometer for velocities below this value. Since this review, however, velocity interferometers have been developed which can operate with a diffuse reflecting surface and for which the requirement of coherent light is much less severe, thus permitting larger values of T to be used. These interferometer^'^^^^^^ operate on the principle that good fringes can ‘I3 R . A. Lederer, S. A. Sheffield, A. C. Schwarz, and D. B. Hayes, The use of a dualdelay-leg velocity interferometer with automatic data reduction in a high explosive facility. In “6th Symposium (International) on Detonation” (D.J . Edwards, ed.), ACR-221, p. 668. Office of Naval Research, Arlington, Virginia, 1976. *I4 L. M . Barker and R . E. Hollenbach, J . Appl. Phys. 43, 4669 (1972). 115 B. T. Amery, Wide range velocity interferometer. In “6th Symposium (International) on Detonation” (D. J. Edwards, ed.), ARC-221, p. 673. Office of Naval Research, Arlington, Virginia, 1976.

610

5.

MEASUREMENT OF PRESSURE

be obtained with practically incoherent light provided the images of the source produced by the two interfering beams appear to be coincident from the point of view of the photodetector. This has been accomplished, and at the same time a nonzero value of T obtained, by the use of lenses in one design115and by the use of a thick plate with a high index of refraction in a n ~ t h e r . ” ~A velocity interferometer of this type has provided 1-2 percent accuracy over a range of u from 8 m s-’ to 400 m s-l. The measured velocity can be related to the applied pressure by Eq. (5.10.35). For example, ifp = 8000 kg mP3and c d = 5 km s-l, the applied pressure is 0.16 GPa (or 1.6 kbar) for u = 8 m SO and 8 GPa (or 80 kbar) for u = 400 m s-’. Considerably higher velocities and pressures have been measured with similar accuracy in other experiments. Chrono-interferometers have been used particularly in shock wave experiments with colliding plates and contact explosives to study the dynamical behavior of solids stressed well beyond their elastic limit. An important result of the measurements made with shocks in metal plates is the establishment of properties of materials which can be used to calibrate “gages” for high pressure measurements. Each of the four metals Cu, Mo, Pd, and Ag has been mixed with ruby crystals and the mixture subjected to steady high pressures in a piston and cylinder press employing diamonds. Pressure-volume relations for these metals at pressures beyond 100 GPa, established by shock wave experiments, were used to “measure” the pressure by determining volumes by x-ray diffraction inside the high pressure cell. Simultaneous optical measurements then provided a pressure calibration of the wavelength shift of the ruby R, fluorescent radiation .”” Subsequently the ‘‘ruby gage” calibration from 6 to 100 GPa was extrapolated and used to measure a sustained pressure of 170 GPa in the piston and cylinder press employing diamonds. This is the highest known pressure measurement. The reading of the ruby R1pressure gage is performed by determining the wavelength shift AA between 100 kPa (1 bar) and the high pressure, and using p = 380.8[(AA/694.2

+ 1)5 - 11,

(5.10.42)

where Ah is in nanometers and p is in gigapascals. The calibration carried out by Mao and B e l P is estimated to have systematic uncertainty of k 10 percent at 100 GPa and below, and to be within + 20 and - 10 percent at the highest pressure of 170 GPa.

li6

H.K.Mao and P. M. Bell, Science 200, 1147 (1978).