I

1]

]

ij

i

I

I

NBSIR 75-698 1? )

STABLE PRESSURE TRANSDUCER

Jack H. Colwell

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

April 1975

Interim Report

Prepared for

CCG-Army/Navy

US Army Missile Command

US Army Metrology and Calibration Center

Redstone Arsenal , AL 35809

Naval Plant Representative

Metrology Engineering Center

1675 W. Mission Blvd.

Pomona, CA 91766

ABSTRACT

This report describes recent v/ork on the solid-dielectric

capacitive pressure transducer. Capacitance measurements on parallel

and perpendicular cut calcite crystals and CaF2 crystals have been

determined at several temperatures against pressures determined by a

piston gauge. The results are used in an extensive discussion of the

effectiveness of various possible devices. Design parameters for

minimizing compressive heating in the pressure vessel are presented.

Data on several new materials are reported as well as the progress on

the automatic capacitance bridge to be use with the transducer.

INTRODUCTION

Since it was first demonstrated that solid dilectric capacitors could be

used as highly stable pressure transducers, ways have been sought to reduce

the pronounced temperature dependence of these devices. The search for a

material that would have a small temperature dependence of capacitance and

still retain a sizable pressure dependence has not been abandoned, but more

effort is now being directed toward other ways of circumventing the

problem. The current program in this laboratory is an attempt to use two

capacitors inside the pressure vessel with one in each side of the bridge

measuring circuit so that the bridge balance is a measure of the ratio of

the two capacitors,

R = C1/C2 . (1)

The temperature dependence of the ratio measurement is

i dR ^ 1 dCi_ 1_ dC2 (9\

R dT Ci dT C2 dT ^

'

and similarly, the pressure dependence is

1 dR _ 1_ dCi_ 1_ dC2 /^x

R dP Ci dP C2 dP

*

By finding a pair of materials where the difference in Eq. 2 is sufficiently

small relative to the difference in Eq. 3 we can produce a pressure transducer

with tractable temperature effects. Note that it is the fractional change in

capacitance that is of concern in Eqs. 2 and 3, this is a property of the

material and does not depend on the magnitude of the individual capacitances.

I
'Or, 5-
To date, paral lei -and perpendicular-cut calcite (CaCOa) crystals are the

only pair tested that approach our requirements. In the present report we

begin by presenting recent data on calcite which is used to critically

evaluate the transducer performance that can be achieved with this material.

The possibility of using one of the calcite crystals in combination with

either KCl or KBr is explored. In this evaluation we also consider the

measurement method being tested at Redstone Arsenal which utilizes two

capacitors of the same material with one inside the pressure vessel and one

at ambient pressure.

Temperature and Pressure Dependence of Calcite and CaF2 Capacitors

Detailed capacitance determinations as a function of temperature and

pressure were carried out on a parallel and perpendicular cut pair of calcite

samples which were measured both separately and in opposition to each other.

Data were simultaneously collected on CaF2j the material which has been used

as a benchmark for comparisons throughout this work. Pressures were precisely

determined by a Ruska type 2415, 0-20,000 psi piston gauge, rather than the

Heise Gauge used for all previous measurements in this laboratory. The precise

pressure measurements permitted a detailed analysis of the temperature and

pressure dependence of these samples. Pressure determinations were carried out

at four different temperatures so the cross derivatives of the capacitance could

be determined. The data were assumed to have the usual functional form

C = C° (1 + AP + BP2) (4)

which rearranges to

f^o = A + BP . (5)

Accordingly, with the data plotted as AC/PC° versus P as shown in Figs. 1^.3

the quantities A and B are determined, respectively, by the intercept and

slope of the lines. From these data dA/dT can be derived which is required

to determine the temperature dependence of the pressure equation and also

determines how the temperature dependence of the capacitance varies with pressun

Throughout the determination of these data we were experiencing

difficulties with the reproducibility of the capacitance measurements.

There were random shifts in the values amounting to several parts-per-million.

After completion of the measurements it was discovered that shifts of the

same magnitude in the capacitance measurement could be obtained by shifting the

relay rack containing the capacitance bridge from side to side. The bridge

problem is apparently due to ground loops which have now been partially correctec

We are assuming that all our irreproducibil ity in the data was due to this

cause for if the shifts were in the capacitor themselves they would indicate

pressure uncertainties up to about 20,000 Pa (several psi). These arbitrary

shifts in the capacitance show up as scatter in the data of Figs. 1-4 which

is- much larger than the expected imprecision of the measurements. The data at

20*C were take in two separate sets some two weeks apart and are distinquished

by open and closed symbols. The systematic shift between these two sets of data

is obvious.

The data for CaF2 are plotted in Fig. 1. These results are in good

agreement with the earlier data take by Andeen, Fontanel la, and Schuele

(Rev. Sci. Instr. 42, 495 (1971)) at 35°C; the 35°C curve drawn in Fig. 1 is

the equation given in their paper. The value for dA/dT derived from the

results in Fig. 1 is given in Table I.

In Fig. 4 are plotted the measured ratios of the calcite pair which are

fitted to an equation of the same functional form as used for the individual

capacitances,

R = R°(l + ApP + BpP2) (6)
:T1
TABLE I Temperature and Pressure Characteristics of Capacitors

Material

I^^xlOVK l^§^=AxlOi2/Pa ^xl0i2/(PaK) ^gylK-M BxlO^O/Pa^

CaF2

KBr

CaCOsX

CaCOs 1

1

AS2S3

BN

262

337

331.5

335

65

-52

-37.8

-147.7

12.0

71.0

116

126

-0.025

0.107

0.0149

0.0169

0.05.

0.05

0.00066

0.00072

0.00124

0.00024

0.0004

0.0004

0.27

5.6

0.42

1.25

-1.36

-6.9

Pair Combinations

Materials R° dT

CaCOs Pair 3.6

CaCOall-KBr -2

l^f^=AxlOi2/Pa ^xlOi2/(PaK)

59.0

219

0.0020

0.124

IdA

A dT ^

0.000034

0.00058

Bxl020/Pa:

0.76

-1.10

The curves in Fig. 4, except for a small correction to the slopes, are the

differences between the curves for corresponding temperatures in Figs. 2

and 3. Neglecting terms higher than quadratic in pressure, the correction

is

R°P ~ CfF ~ CIP ~ ^^^^^ ~ ^

To a good approximation the temperature dependence as a function of

pressure for the individual capacitances can be written as IdC^ldC^.dAp

C dT C° dT dT

^

It follows using Eqs. 2 and 6 that

ldR^l_dC!_ 1_ dC2+/dAi dAA 1_ dR^ , ^ p /qx

R dT Cf dT q dT \dT " dT /

^ R° dT dT ^

'

Values for the different parameters for the calcite crystals are given in

Table I.

The data in Fig. 4 appear nonuniform when compared with those of Figs. 2

and 3, they are, however, consistant with all the above equations within the

experimental uncertainty. This apparent inconsistency probably arises because

we are looking at small differences between relatively large numbers (note

that the scale on the ordinate of Fig. 4 is twice that of the other figures).

The zero-pressure temperature dependence of the parallel-cut calcite

crystal increases linearly with temperature but that of the one cut

perpendicularly varies somewhat from a linear behavior. As a result the

temperature dependence of the ratio, in units of 10"^/K, increases from 1.5 to

5 in the range 5 to 45°C but in the region 10 to 30°C it is essentially constant

at 2.7. We have made measurements on a second calcite pair and again found the

temperature dependence very small but different from the first pair. The

temperature dependence of the perpendicular cut crystal from this pair was

nearly identical to the first. That of the parallel cut was smaller than the

perpendicular cut at low temperatures but increased more rapidly so that the

temperature dependence of the ratio actually passed through zero at about

35°C. It should be noted that the electrode gap on the low voltage side of the

second parallel -cut crystal was initially shorted and was opened by physically

scraping away some of the aluminum coating. The loss component for this sample

was about five times larger than that for the other samples; this and the other

deviate behavior observed with this sample may be due to the treatment it

received. This result does point out, however, that considerable variation may

be found in the ratio measurements of different pairs.

Use of a Calcite Crystal With KC1 or KBr .

The temperature dependence of the two calcite crystals are very close

to the values for KCl and KBr so the there is the possibility of using one of

these combinations in the pressure transducer. The pressure dependence of

the alkali halides is negative whereas that of calcite is positive so that in

a ratio measurement their magnitudes will add. As a result the pressure

sensitivity of one of these combinations would be 2 1/2 to 3 1/2 times larger

than that of the calcite pair.

Values of the temperature and pressure dependencies of the four materials

at 35°C are plotted in Fig. 5. The arrows extending from the points indicate

the change in the two dependencies as the pressure is increased from zero to

138 MPa (20000 psi). The calcite values are taken from the present work using

Eq. 8 and the pressure derivative of Eq. 4. The pressure dependencies and

zero-pressure temperature dependence of the two potassium salts are from the

work of Fontanella, Andeen, and Schuele (Phys. Rev. B6^, 582 (1972) but unfortunate

they did not determine dA/dT for their samples. Smith and Riehl (J. Phys. Chem.

Solids 35_, 1327 (1974)) have measured the pressure dependence of KBr at 211 K

and 295 K which, if we assume A is a linear function of T, gives us a value

for dA/dT (See Table 1). The magnitude of this value appears rather large but

the ratio (1/A) dA/dT is comparable to other values listed in Table 1 so there

is no reason to suspect it is in error. No data is available to determine dA/dT

for KCl but it is probably similar to the value for KBr. As can be seen from

Fig. 5, subtracting the temperature dependence of KBr from that of parallel -cut

calcite, the dependence of the combination will vary from -2 to + 15 x 10"^/K

as the pressure is increased from zero to 138 MPa. The possible use of this

combination is discussed in the next section. ,

Resolution and Temperature Effects in the Pressure Transducer

It is now reasonable to expect accuracies from a ratio-transformer

capacitance bridge of at least 1:10^. Using the data in Table 1 for

individual capacitors we find that for CaF2j 1:10"^ accuracy in the capacitance

corresponds to a pressure resolution of

' ^

= 2600 Pa = 0.4 psi

37.8 X 10-12

With AS2S3 or BN the resolution is improved to about 850 Pa = 0.12 psi.

For the calcite pair the pressure resolution would be about 1700 Pa = 0.25 psi

and for a CaCOs - KBr pair it would be 450 Pa = 0.07 psi. With these systems

we can expect to achieve pressure resolution in the region of 700 pa = 0.1 psi

what demand does this put on the temperature regulation of the system?

We will consider the two measurement methods that use the ratio of two

capacitors, the pressurized pair described above and the pressurized and

unpressurized combination of the same material. There are three ways that

temperature effects can cause errors in the capacitance ratio measurements.

The first, and most important, arises from the two capacitors being at a

different temperature. The second way is for the temperature to be different

during the reference determination, R°;and the pressure determination, R.

The third possible error can arise from the absolute temperature of the device

being different between calibration and use.

We will consider how large each of these temperature effects can be

and still have pressure resolution of 700 Pa when using the two types of ratio

measurements. The error arising from a temperature difference between the

two capacitors depends on the temperature dependence of the individual capacitors

relative to the pressure dependence of the combination. For CaF2 in a

pressurized-unpressurized combination, resolution of 700 Pa requires that the

two capacitors be within

With AS2S3 or BN the requirements are less stringent, thermal equilibration

of only 1.2 and 1.6 mK, respectively, is required. For the calcite pair

we require the two capacitors to be within

With the parallel -cut calcite-KBr combination the requirement drops to

0.46 mK.

It is difficult to assess the size of the temperature gradients one

may expect in a system. They are certainly less likely to exist in the case

whereboth samples are in close proximity within the pressure vessel than

where the two are separated by a pressure bearing wall. Nevertheless, even

if heat flows in the system were large enough to cause sizable temperature

gradients, they would have to change significantly from those that existed

during the gauge calibration before errors would result.

37.8 X IQ-^^/Pa X 700 Pa

262 X IO'VK

= 0.10 mK .

59.0 X lO-^yPa X 700 Pa

333 X lO-^K

= 0.12 mK.

Temperature differences will be generated within the system when it

is pressurized or depressurized and the time it takes to reestablish

equilibrium between the two capacitors can be a major consideration. Again,

having the two capacitors within the pressure vessel has a definite advantage,

for the two capacitors will probably be in thermal equilibrium with each

other long before equilibrium is established throughout the system. Whether

this is an advantage or not will depend on the combinations response to the

actual temperature changes which we now consider.

The effect of temperature changes on the two measurement methods using

capacitance ratios is different and we have summarized the contributions for

the two methods in Table II. In this table the quantities given are R/R°

where R° is the value of the ratio measurement at zero pressure and temperature,

T. Quadratic terms in the pressure dependence have been neglected.

For the pressurized-unpressurized combination in Table I la, the observed

ratio at P = 0 does not change with temperature since the two capacitors

are of the same material and are affected identically. As a result, in changing

the pressure from 0 to P, the same change will be observed when starting at a

temperature T and ending at T + aT as when the whole process occurred at the

temperature T + AT. The magnitude of this change depends on the total pressure,

and its fractional dependence, (1/A) dA/dT, is tabulated in Table I for the

materials of interest to us here. At the maximum pressure we are considering,

138 MPa (20,000 psi ), 700 Pa amounts to 5:10^(5ppm). The temperature change

that would be equivalent to this pressure resolution with a combination of two

CaFa capacitors would be

5 X 10"^ _ 7 r 0.00066/K

"

With AS2S3 or BN this equivalent temperature change would be 12 to 15 mK. These

temperature values will vary inversely as the total pressure for the fixed

TABLE II Capacitance Ratios as a Function of Temperature and Pressure

(a) Pressurized-Unpressurized Combination

0 P

T 1 1 + AP

T+AT 1 1 + AP + ^ P AT = 1 + AP (1 A dT

(b) Pressurized Pair

0 P

T

T+aT

1

AT

dT

1 + AP

1 .AP .(^f . dA p^ AT = 1 + AP

7l dR» ^2

dT

dA
4 A -r

r i
resolution of 700 Pa.

Unless the temperature dependence of the two different capacitors in the

pressurized pair arrangement are identical, the ratio measurement will not be

completely independent of temperature as indicated in the first column of

Table II b. Now when the system is pressurized there are two correction terms

and as initially written in the table they are both pressure dependent. In

rewriting the equation in the table, Eq. 9 is used to substitute for

(l/R)dR/dT so that we are left with one correction term which is dependent

on pressure and one which is not. The equation for the temperature equivalent

AT for a given pressure resolution aP is

is 11.5 mK while at the maximum pressure of 138 MPa it is 10.0 mK. These

temperature changes are the permitted limits when the temperature at pressure

is different than it was in the reference state of zero pressure. When the

temperature remains at T + AT, ie it is in error by AT, when the system is

pressurized the term (1/R°) dR°/dT drops out of the observed capacitance

change and the temperature change at P = 138 MPa could be as large as 75 mK

before it would be equivalent to 700 Pa.

The KBr - CaCOs || pair is an interesting case, (l/R") dR°/dT and dA/dT

are of opposite sign so that the denominator in Eq. 10 goes through zero at

about 8 MPa (1200 psi). The temperature equivalents of a 700 Pa pressure change

at various pressures are given as aT(1) in the following table.

dA

dT

P

= AT (10)

For the calcite pair the temperature equivalent of AP = 700 Pa at P = 0

Dr Al u j A 1 Uj (psiT (mK) (mK)

U 0 77 00 oo

1 or\r\ 00 7-7 77

OA oUUU

69 10000 10 8.7

138 20000 4.5 4.3

The column headed aT(2) are the temperature equivalents for the case of

a pressure change with the temperature remaining at T + AT.

From these calculations it is evident that the KBr - CaCOa pair has less

stringent temperature requirements at pressures below 69 MPa and is only a

factor of two worse at 138 MPa when compared with a calcite pair. We saw

earlier that the thermal equilibrium requirement was a factor of 4 less

and the pressure resolution was a factor of 3.5 greater so on this basis

the KBr - CaCOs pair is superior in nearly every way.

In comparing the two pressurized pair combinations with the performance

expected of either AS2S3 or BN capacitor used in the pressurized-unpressurized

pair combination there is not a great contrast. In most of the cases the values

for the latter two combinations fall between the values for the former two.

Unless a material is found which would greatly enhance the performance of

one of the techniques the choice of which to use will rest on other

considerations.

TherkiOSTATINg Considerations for the Pressure Transducer

The simplest way to thermostat a system is to isolate it as well

as possible from its surroundings and to minimize any disturbances that

would tend to change its temperature. Utilizing this approach with the

pressure transducer is complicated to a large extent by the fact that the

pressurizing and depressurizing of the device results in considerable heating

and cooling of its component parts. We are starting the development of

system in which we believe the resultant heating effects can be made nearly

zero. This possibility exists because the material inside the pressure

vessel heats as the vessel is pressurized, while the vessel itself cools. If

these two opposing heating effects can be equated, a totally isolated system

will return to its original temperature when thermal equilibrium is

re-established after a pressure change.

The heating of a material under isothermal compression is given by

(see M. W. Zemansky, Heat and Thermodynamics , 5th Ed., p. 288)

Q = - tJVBdP n

where 3 is the volume thermal expansion. This can be usually written as

Q = T V3(Pf - Pi) 12

where V and B are average values as they are usually relatively independent

of pressure.

The cooling of the pressure vessel upon pressurization results from

the net tensile stress on the vessel. In the Appendix we show that for

pressure vessels of simple geometries the amount of cooling is given by an

equation identical in form (with opposite sign) to Eq. 11 or 12 but the

volume, V, is the internal volume of the vessel. 3 is the thermal

expansion of the pressure vessel material. Thus, to minimize the heating

effects in the vessel, the amounts and thermal expansions of the materials

used inside the vessel must average to the same thermal expansion as that

of the vessel material.

The development is complicated by the requirement of using oil as

a pressurizing fluid since all oils appear to have very large thermal

expansion coefficients which are also very pressure dependent. The thermal

expansion of the steel in the pressure vessel is about

3(steel) = 45 X lO-^/K

while that of most oils is in the range

3(oil) = (700 - 1000) X 10-VK

The resolution of the problem appears to lie with the use of Invar, a high

nickel steel which has a very small thermal expansion,

3(Invar) = 3.6 x IO'VK.

By filling the internal volume of the pressure vessel primarily with invar

and keeping the quantity of oil small the proper net expansion coefficient

be achieved.

We present' here a calculated example using a commercial 20,000 psi

(138 MPa) pressure vessel. The vessel has an internal diameter of 3.8 cm

with a depth of 15 cm so that the internal volume is

V^. = 173 cm3

The vessel has a material volume of 1700 cm^, its mass is 14.8 kg and its

heat capacity is 6.68 kJ/K.
r-h.

: if!':"

t
The heat taken up by the vessel when pressurized to 34.5 MPa (5000 psi)

Q=TV3(Pf-Pi)

= 308 X 173 X 45 X lO'^ x 34.5

• = 82.7 MPa.cm^ = 82.7 J

If the vessel were pressurized adiabatically the cooling of the vessel

would be

We have calculated the heating with the following amounts of material

inside the pressure vessel.

Calcite samples 1 cm^

Invar sample holder 160 cm^

Paraffin Oil 12 cm^

The heat evolved for a 34.5 MPa (5000 psi) pressure increase on the

calcite and invar are:

Calcite

Q = - T V B A P .

= - 308 X 1 X 14 X 10-6 X 34.5

= - 0.14 J

Invar

Q = - 308 X 160 X 3,6 x 10"^ x 34.5

= - 6.12 J

Since the volume and thermal expansion of steel, invar, and calcite

are relatively independent of pressure the net heating of these three

components, 82.7 - 0.14 - 6.12 = 76.4 J, will be about the same for any

34.5 MPa pressure change. For the paraffin oil we have determined average

values of the expansion coefficient for four different 34.5 MPa intervals

from 0 to 138 MPa. The amount of heating of the oil on pressurizing gets

smaller as the pressure gets higher. This reduction is compensated to a small extent by the small increase in the amount of oil in the system due

to the expansion of the vessel and the compression of the internal components.
i :: . I

;>v..
The heating due to the compression of the oil and the net heating

and temperature change of the system for the four pressure ranges are

given in the following table:

Pressure Paraffin Oil Q(oil) Q(oil) Q(net) AT(mK)

Change, MPa 3 X 10^ J corrected System System

0 - 34.5 680 -86.7 -86.9 -10.5 +1.4

34.5 - 69 605 -77.1 -77.5 - 1.1 + .1

69 - 103.5 535 -68.2 -68.7 + 7.7 -1.0

103.5 - 138 480 -61.2 -61.8 +14.6 -2.0

The temperature change for each 34.5 MPa interval was determined assuming

a total heat capacity of 7.42 kJ/K.

For this example, increasing the pressure from O'to 69 MPa will increase

the temperature of the system by 1.5 mK but in continuing to increase the

pressure to 138 MPa the temperature will drop to 1.5 mK below its initial

value. In going from zero pressure to any value below 138 MPa the total

temperature change will be no greater than 1.5 mK. The worst case would be

to allow the system to stand for a long period at 69 MPa so as to equilibrate

with the thermostat, pressurizing to 138 MPa would then cool the system to

3 mK below the thermostat temperature. These temperature changes are well

below the temperature limits required by the calcite pair transducer. We are

proceeding to check out these ideas in the laboratory.
1
Automatic Capacitance Bridge

The self-balancing capacitance bridge to be used with the pressure

transducer should soon be operational. Initial testing turned up numerous

wiring errors which are now being corrected. Two of the component amplifiers

have malfunctioned and have had to be returned to the manufacturer which

may delay resumption of testing.

New Materials Tested

A new sample of AS2S3 has been obtained from Barnes Engineering and

was found to have nearly the same temperature and pressure dependencies as

found earlier for the Servo Corporation samples. The values are d £nC/dT x

106 = 66.2/K and dilnC/dP x 10^2 = 114.4/Pa.

Magnesium aluminum spinel (MgAl20it) was measured and found to have

dependencies very similar to those of parallel cut sapphire. The values

are d in C/dT x 10^ = 135. 3/K and d ilnC/dP x 10^2 = 12.2/Pa.

Measurements have been made on LiNbOs which is a pyroelectric material,

i.e. a polarized ferroelectric. The samples were poled, single-domain

crystals cut parallel and perpendicular to the axis of polarization. The

temperature dependence of both samples was rather large and the pressure

dependence small so this material is not of particular interest to us.

In addition both samples shows long term relaxation effects that amounted

to tens of ppm in the capacitance. For the parallel cut sample d £,nC/dT x

10^ = 734/K and d s,nC/dP x 10^^ = 7.0/Pa and for the perpendicular cut

d M/dJ x 10^ = 387/K and d anC/dP x 10^^ = 59/pa. The loss component,
l'/ to
although about an order of magnitude larger than most of our samples,

was still small tan (}> =0.0002.

FUTURE WORK

The idea of using a calcite capacitor in combination with a potassium

halide capacitor will be checked out in the laboratory. Additional calcite

crystals will be obtained and tested so that we will be in a position to

proceed with the development of the transducer whichever combination appears

to work best. An invar sample h'older will be fabricated for use with a

capacitor pair in a commercial pressure bomb. Assuming the temperature changes

in the pressure vessel are within our estimated requirements we will proceed

with the development of the actual thermostat for the "transducer. If these

various aspects of the project go well we should be in a position to begin

evaluation tests of the transducer and its capacitance bridge early in the

next fiscal year.
4i
AppErroix

Cooling of a Pressure Vessel upon Pressurization

The isothermal heating which occurs in a piece of material under

compressive stress can be calculated by writing the formula

dS = EdS-j is the sum of the three components of the stress acting

on the volume dV. a = 1/3 3 is the linear thermal expansion coefficient.

Shear stresses are neglected.

The individual stress components for uniformly shaped pressure vessels

under internal pressure, P, can be calculated using standard formulae as

given by R. J. Roark ( Formulas for Stress and Strain , 4th Ed., p. 308).

As a practical example we consider a cylindrical vessel with spherical

ends. The internal and external radii are a and b and the stresses at a

point r in the vessel wall are given by:

(Al)

in a more general form as

Q = - T JJa dS dV. (A2)

Cylindrical Portion -

longitudinal stress

dP,

hoop stress

dS2 = - a2(b2-;-r2)

r2(b2-a2)

dP,
1- ^t-V-; -,%.-;'r ^
radial stress

"•^3 ^ r2(b2-a2)

'

Spherical Ends -

hoop stresses

^ - ^ 2r3(b3-a3)

radial stress

_ + a3(b3-r3)

" ^ r3(b3.a3)

•

The contributions of the individual stress components can be found

by multiplying by the appropriate volume element, Zirrdr for cylindrical

segments and 47rr^dr for spherical segments, and integrating over r from

a to b.

The total contribution is, however, more easily found since the sum

of the three components of both segments are independent of r,

dS = - dP

dS = - 3(4^) dP sph b3-a3

The volume integrations of Eq. A2 can now be carried out separately

giving

and (A4)

= 4/3 Tr(b3-a3)

Multiplying the appropriate parts of Eqs. A3 and A4 and substituting in

Eq. A2 we have

Q = t/3 aiva^Z + 4/3TTa3) dP

The expression in parentheses is just the internal volume V-,- of the

vessel so we can write
.r;r>
Q = t/3 VidP

This is just the expression for the compression heating by the

pressure change dP of a volume V-,- with thermal expansion 3 as given

by Eq. lA but with the sign changed.

FIGURE CAPTIONS

1. Pressure dependence of the CaF2 capacitor as a function of temperature.

2. Pressure dependence of the parallel-cut calcite capacitor as a function

of temperature.

3. Pressure dependence of the perpendicular-cut calcite capacitor as a

function of temperature.

4. Pressure dependence of the calcite-pair capacitor combination as a

function of temperature.

5. The temperature and pressure dependence of the capacitance of KCl , KBr

and the two cuts of calcite. Circles indicate the zero-pressure values

at 35°C and arrows indicate how the values change as the pressure is

increased to 138 MPa (20,000 psi). Data are not available on the variation

with pressure for KCl.
; i ..;)