Nat onal Bureau of Standards

Library, N.W. Bldg

JUN 2 8 1963

NBS MONOGRAPH 65

Reduction of Data for

Piston Gage Pressure Measurements

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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Documents, Government Printing Office, Washington 25, D.C.
UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Reduction of Data for

Piston Gage Pressure Measurements

J. L. Cross

National Bureau of Standards Monograph 65

Issued June 17, 1963

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 15 cents
Library of Congress Catalog Card Number: 63-60036
Contents

Page

1. Introduction 1 2. Pressure 1 3. Force and fluid heads 2

3.1. Mass of weights 2

3.2. Mass of air 2

3.3. Fluid head 3

3.4. Air head 3

3.5. Fluid buoyancy 3

3.6. Mass of fluid 4

3.7. Keference level 5

4. Area 5

4.1. Temperature coefficient of area 5

4.2. Effective area at atmospheric pressure 5

4.3. Pressure coefficient of area 5

4.4. Change of area with jacket pressure 6

5. Piston gage pressure 6

6. Conclusions 6

7. References ^ 6

8. Appendix A. Computation of pressure 6

8.1. Calculation of correction factors 6

8.2. Machine computation 7

8.3. Slide rule computation 7

8.4. Correction table computation 8

8.5. Conclusions 8

9. Appendix B. Working equations 8

10. Appendix C. Examples of calculations 9

m

Reduction of Data for Piston Gage Pressure Measurements

J. L. Gross

Pressure measurements made with piston gages are affected by gravity, temperature,

pressure, and several other variables. For accurate determinations of pressure the cal- culations must take these variables into account. A general equation is developed and

simplified procedures for calculating pressure are illustrated.

1. Introduction

The dead weight piston gage is one of the few

instruments that can be used to measure pressure

in terms of the fundamental units, force, and area. Piston gages are known by several names such as "dead weight gage," "dead weight tester," "gage

tester," and "pressure balance." The name,

"piston manometer," is used in Germany, and,

since it aptly describes the instrument, might very

well be used more extensively in this country. In principle, it is a piston inserted into a close

fitting cylinder. Weights loaded on one end of

the piston are supported by fluid pressure applied

to the other end. Construction of piston gages

varies as to method of loading, methods of rotating

or oscillating the piston to reduce friction, and

design of the piston and cylinder. Three designs

of cylinders are commonly used; the simple

cylinder with atmospheric pressure on the outside;

the re-entrant cylinder with the test pressure

applied to the outside as well as the inside; and

the controlled clearance cylinder with an external

jacket in which hydraulic pressure can be applied

so as to vary the clearance between the piston

and cylinder at the will of the operator. In order

to use the piston gage for the measurement of pressure, one must take into account a number of

parameters of the instrument and its environment.

Error in measurement results from failure to account for the parameters or from the uncertainty

of the measured values of them. It is obvious

that error results from the uncertainty of the mass

of the loading weights and the measurement of

the effective area of the piston and cylinder.

Other sources of error may not be so easily

recognized. Such sources include the air buoyancy

on the weights, the fluid buoyancy on the piston,

the value of local gravity, the force on the piston

due to the surface tension of the fluid, the thermal

expansion and elastic deformation of the piston

and cylinder, and the fluid heads involved. These

effects can be evaluated and corrections applied

to reduce the magnitude of overall error of

measurement.

Air buoyancy corrections amount to about 0.015

percent of the load. The corrections for the

buoyancy of the pressure fluid on the piston have

been found to range from zero to nearly 0.5 psi,

and could be larger. A brief discussion of gravity

error is given by Johnson and Newhall [l], 1 and

detailed information can be found in the Smithsonian Meteorological Tables [2]. The values of local gravity differ by over 0.3 percent at different

places in the United States. The pressure correc- tion due to surface tension is usually negligible,

but may amount to more than 0.003 psi. Thermal

expansion of the effective piston area is usually

about 0.003 percent per °C and elastic distortion

may amount to 0.05 percent at 10,000 psi. Fluid

head amounts to about 0.03 psi per inch for

lubricating oils. Other errors resulting from corkscrewing (verti- cal force caused by a helical scratch on the piston,

cylinder, or guide bearing), vertical component of

the force used to rotate or oscillate the piston,

eccentric loading of the piston, and the force of air currents against the weights, are not readily

evaluated. They may be kept small by good

design and workmanship and by careful operating

technique, but usually no attempt is made to apply corrections for them.

2. Pressure

The pressure in any system may be defined as measured at the piston gage is F

P=A (1)

where F=force and A

area over which the

force is applied. When a piston gage is in equilibrium with a pressure system, the pressure, pp ,

Pp=A,

(2)

where Fe =the force due to the load on the piston,

.Ae =the effective area'of the piston.

1 Figures in brackets indicate the literature references on page 6.

1
The pressure pp , is not necessarily the quantity desired. It is the pressure measured by the

piston gage at its reference level, whereas, the

pressure that we may wish to measure may be

at another level within a system which is con- nected to the piston gage by a length of tubing

filled with a pressure transmitting fluid . The pres- sure exerted by the head of fluid in the connecting

line, from the level within the system to the

reference level of the piston gage, must be added

to the piston gage pressure. If the total (absolute) pressure is to be determined, the atmospheric

pressure at the reference level of the piston gage

must also be added to the piston gage pressure, so we have

P=Pp +H,p +Pa (3)

where P=total pressure at a particular level in

the system,

pp =pressure at the reference level of the

piston gage (piston gage pressure),

Hfv =pressure exerted by the head of pressure

transmitting fluid,

Pa =atmospheric pressure at the reference

level of the piston gage.

Usually the quantity to be measured is the differ- ence between the total internal pressure in the

system and the atmospheric pressure outside of

the system. If the pressure is to be measured

at a level in the system (which might be the refer- ence level of another gage) which is different

from the reference level of the piston gage, the

atmospheric pressure at these levels is different

and the equation for the difference between the

internal and external pressure or gage pressure, pg, of the system is

pg =Vp+HfP -Ha (4) where Hn =ihe difference in the atmospheric pres- sure between the reference level of the piston

gage and the level in the system at which the

pressure is to be measured.

3. Force and Fluid Heads

The force, F, due to the gravitational attrac- tion between the earth and a mass, M, is numerically

F=kMgL (5) where gL is the local acceleration due to gravity,

and k is a proportionality factor, the numerical

value of which depends upon the units of F, M,

and gL as follows

k=l, for F in dynes, M in grams, and gL in

cm/sec2

, k=l, for F in newtons, M in kilograms, and

gL in meter/sec2, k=l, for .Fin poundals, Min pounds mass, and

gL in feet/sec2, k=l, for F in pounds force, M in slugs, and gL

in feet/sec2

, i k=

^2

f°r F m pounds force, M in pounds

mass, and gL in feet/sec2

, k= 1 for F in pounds force, M in pounds

980.665 mass, and gL in cm/sec2

, k= 1 > for F in kilograms force (in some

980.665

countries, kiloponds, kp), M in kilograms,

and gL in cm/sec2

.

There are several quantities that must be used

in the determination of the force, Fe , (eq (2)) acting upon the effective area of the piston.

These include, the mass of the weights, Mm, the mass of the air displaced by the load, Ma ; the mass of the pressure fluid contributing to the load, Mr ; the local acceleration due to gravity, gL, and

the force due to the surface tension, y, of the pres- sure fluid acting upon the circumference of the

piston, C, where it emerges from the surface of

the fluid. The value for Fe can be determined

from the following equation,

Fe = (Mm +Mf-Ma)kgL +yC. (6)

3.1. Mass of Weights

The mass of the weights, including the piston

and all parts which contribute to the load on the

piston when in operation, is determined by comparison with the mass of accurately known standard weights. This is usually done by means of an equal arm balance.

3.2. Mass of Air

The mass of air displaced by the load on the

piston is Ma =pa Vrn, (7)

where pa =the mean density of the air displaced

by the load . The volume of the load

T

, Mm ,Mf V m= 1

> Pm Pf

(8)

where pm =the density of the weights, Mm, and

P/=the density of the pressure fluid, Mr

. Substituting eq (8) in eq (7) we get

Ma Pm Pf(9)

2
The value, pm, depends upon the way in which the

values for the loading weights are reported.

When they are reported as true mass, the actual

value for the density of the weights should be

used. If, however, the apparent mass is given,

as determined by comparison with brass standards

in air having: a density of 0.0012 g/cm3

, the density

of the weights should be assumed to have the same density as the brass standards (8.4 g/cm3 ). Apparent mass values are usually used when re- porting loading weight values.

By substituting eq (9) in eq (6) we obtain the

following expression for Fe , Ft =^Mn (l-fy+M, (l-^)] kgL+yC.

(10)

Piston gages operating with the piston assembly

partially submerged in a liquid are subjected to a

force resulting from the surface tension of the

liquid acting on the periphery of the piston where

it emerges from the surface of the liquid. This

force, yC, is added to the force due to the load on

the piston.

3.3. Fluid Head

The fluid head pressure, Hfp, eq (3) and (4), exerted by the column of pressure fluid between

the reference level of the piston gage and the

reference level of the pressure system to be

measured, may be calculated as follows

:

Hfp = — PfPhfpkgL (ID

where p/2,=the density of the pressure fluid at pressure P,

hfP =ihe height of the column of fluid of

density pfp, measured from the refer- ence level of the piston gage. Measurements up from the reference level are positive and down from the refer- ence level are negative.

3.4. Air Head

The air head pressure, Ha , eq (4), exerted by

the column of air between the reference level of

the piston gage and the reference level of the

pressure system to be measured, may be calculated as follows

:

Ha = — pahakgL (12)

where pa =the density of the air at the ambient

atmospheric pressure, and temperature,

ha —ihe height of the air column measured

from the reference level of the piston

gage. Measurements above the refer- ence level have a positive sign and

those below have a negative sign.

3.5. Fluid Buoyancy

In certain instances, the pressure fluid in which

the piston is immersed contributes to the load on

the piston. This effect may be accounted for in two ways. One method is to compute the mass

of the fluid, Mf, contributing to the load on the

piston and include it in the calculation of the force

as shown in eq (6). The other method is to shift

the reference level (the level at which the piston

gage pressure, pP, is measured) from the lower end

of the piston by an amount equal to the height of a column of oil that will compensate for the mass

of the fluid acting on the piston.

In order to determine the contribution of the

pressure fluid to the load on the piston, first consider the case of a piston of uniform cross section

exactly fitting the cylinder (fig. 1). If the pres1 w

Y/y///, §

W,

#1 (

Figube 1. Piston of uniform cross section

sure is measured at the level of the lower end of

the piston (level R), the fluid exerts no vertical

forces other than that against the area of the end

of the piston. In this case, therefore, there is no

buoyancy correction to be applied.

Next consider a piston with grooves or holes

machined in it as shown in figure 2. The weight

of the fluid contained in the grooves and holes, is part of the load on the piston. That is to say

that all of the material included within a cylinder

having a cross section equal to the effective area

of the piston is included in the load on the piston.

In practice, the effective area of a piston is very

nearly equal to the mean of the areas of the

piston and the cylinder as shown in figure 3 by

the dotted lines C and C. Any metal extending

beyond those bounds displaces a volume of fluid the mass of which must be subtracted from the

load on the piston and any fluid within those

bounds must be added to the load on the piston.

3
) ( P

H

) (

i ,,1

Figure 2. Piston of irregular cross section.

C c'

Figure 3. Piston of irregular cross section.

From these examples we see that the fluid buoyancy correction can be either positive or

negative. It can be calculated from the following

equation

Mf =(Aeyf-Vt ) Pf (13)

where ^4e =the effective area of the piston as

before,

yf =the length of the submerged part of

the piston,

Fs =the volume of the submerged part of

the piston.

To determine the pressure equivalent, pf, of the

fluid, Mj, the following equation may be used:

(14)

Shifting the reference level of the piston gage by

an amount, Ah, effectively consists of adding the

pressure exerted by a column of fluid of height,

Ah, and density, pfp , and subtracting the pressure

exerted by a column of air of height, Ah, and

density, pa . The resulting pressure change,

Ay = pfpAhkgL —

p

aAhkgL . (15)

By shifting the reference level an amount sufficient to make the resulting fluid head compensate for the pressure equivalent of the fluid buoyancy

[3, 4] Ap=pr we obtain (from eqs 13, 14, and 15),

Ah- (Aey,—Vs ) (p,— Pa) Ae (P/p —Pa) (16)

When the buoyancy correction is determined as shown in eq (13) the density of the fluid, pf, is included. For the portion of the piston between

the cylinder and the surface of the fluid (level L)

the value of pf will be the density of the fluid at atmospheric pressure, pfa, but for the portion of

the piston below the cylinder the value of pf will

be the density of the fluid, pjv, at pressure, P,

and may not be easily determined.

On the other hand, when the buoyancy correction for the lower end of the piston is applied by

shifting the reference level, pf in eq (16) is equal

to Pfp, and the ratio S^-——. is equal to 1, so that

(Pfp— Pa) the value for pfv need not be known.

3.6. Mass of Fluid

By applying the buoyancy correction for the

upper portion of the piston as a load correction,

eq (13) becomes

Mfa = (Aeyfa — Vfa) Pfa (17)

where yfa =ihe length of the submerged part of the

piston above the cylinder,

V7a=the volume of tbe submerged part of

the piston above the cylinder,

and eq (10) becomes

Fe =Mm (l-^-)+Mfa (l-^)kgL +yC. (18)

\ Pm/ \ Pfa/
3.7. Reference Level *h=y„-?j* (19) „ , • ,1 i <• ..i where yf!) =the length of the piston below the

By applying the buoyancy correction lor the a ' p cylinder

lower portion of the piston as a reference level y/p =the volume of the piston below the

change, eq (16) becomes cylinder.

4. Area

The effective area, Ae , of the piston, can be

expressed by the relationship

Ae =Aa [\ +a(f-ts)](l+bpp) [1 +d(Vz-Vi)] (20)

where -A0 =the effective area of the piston at temperature, ts , and atmospheric pressure, a=the fractional change in effective area per

unit change in temperature,

t=the temperature of the piston and cylinder,

ts =the reference temperature,

6=the fractional change in effective area per

unit change in pressure, c?=the fractional change in effective area per

unit change in jacket pressure of a controlled clearance piston gage,

p2 =the jacket pressure required to reduce the

piston clearance to zero at pressure pp, p} =the jacket pressure.

4.1. Temperature Coefficient of Area

The fractional change in effective area per unit

change in temperature can be determined as

follows

:

a =ak +ac (21)

where a^the temperature coefficient of linear

expansion of the piston, ae =the temperature coefficient of linear expansion of the cylinder.

The most convenient reference temperature, ts , is the average temperature of the room in which

the instrument is used. In many instances the

difference t—ts may be insignificant.

The temperature, t, of the piston and cylinder is usually assumed to be the same as the temperature

of the base of the instrument. The fact is, in

practice, the piston and cylinder are usually at a temperature higher than that of the rest of the

instrument, although, in a gas lubricated piston

gage they may be lower. So many factors affect the temperature that the order of magnitude cal- culation of the temperature rise may be unreliable.

Some of the more important factors are: speed of

rotation, clearance, pressure, viscosity, JouleThompson coefficient, and thermal conductivity

of the pressure fluid. One precaution that can be

taken to keep the uncertainty of temperature, t,

from being unnecessarily large, is to keep the

speed of rotation no greater than is required to maintain hydrodynamic lubrication.

4.2. Effective Area at Atmospheric Pressure

A0 is very nearly equal to the mean of the area

of the piston and the area of the cylinder [3,4] at the reference temperature and can be calculated

as follows

:

A=^^[l+a(t-ta)] (22)

where .4*= the area of the piston,

-A<;=the area of the cylinder,

tm =the temperature at which Ak and Ac are measured.

A -\-A The value of —^

? may be determined from

direct measurements of the diameters of the piston

and cylinder, or by comparison with a piston gage

of known area. In a controlled clearance piston

gage, the jacket pressure, applied to the outside of

the cylinder, is used to vary the diameter of the

cylinder as desired and Ac is assumed to be equal

to Ak .

4.3. Pressure Coefficient of Area

The fractional change in area with pressure is most readily obtained by /comparing the instrument against a controlled clearance piston gage for which the pressure coefficient, b, can be computed,

or by comparison with a piston gage of known

characteristic. To the first approximation, for a

controlled clearance piston gage,

bJ*=± (23)

where /i=Poisson's ratio for the piston,

Y=Young's modulus for the piston.

The product, bp v , is small and therefore an approximate value for pv is adequate. Deffet and

Trappeniers [4], Dadson [6], Bridgman [7], and

Ebert |8] give more detailed discussions of the

elastic distortion of piston gages.

5
4.4. Change of Area With Jacket Pressure

The factor [l+d(p2 —pJ)] is used only for controlled clearance piston gages. The fractional

change of area with jacket pressure, d, is best

obtained experimentally by varying the jacket

pressure, pu and measuring the change in pressure, pv

. Since the change in pressure will be small,

the measuring instrument must be sensitive.

The jacket pressure required to reduce the

clearance between the piston and cylinder to zero,

at a particular value of pp, may be determined by

two methods. One method is to observe the

torque required to turn the piston as the jacket

pressure is varied. An abrupt increase of torque

is observed when the clearance is reduced to zero. The other method is to measure the fall rate of

the piston at several jacket pressures. The cube

root of the fall rate is then plotted against jacket

pressure, and the curves are extrapolated to zero

fall rate to get values of pz

.

5. Piston Gage Pressure

The complete equation for pp can be obtained

from combining eqs (2), (18), and (20);

Mn

[l+a(t-ts )](l +bpp)[l+d(p-pj)]

(24)

where the reference level is determined as shown

in eq (19).

Equation (24) would be rather formidable if it were necessary to solve it for each pressure measurement. Fortunately the terms can be grouped

so that the amount of calculation can be reduced

to practical proportions for some instruments, and some terms can be ignored if the accuracy re- quirements are low. It should be noted that eq

(24) is not exact. Some second order terms have

been dropped and the coefficients are constant

only to a first approximation.

6. Conclusions

The accuracy of pressure measurements depends

not only on the performance of the piston gage,

but on the application of corrections derived from

parameters of the system. These depend upon

the construction of the instrument, composition

of the pressure fluid, environment, pressure, and

physical arrangement of the pressure system.

The accuracy to which the values of these parameters are known usually establishes the overall

accuracy of the measurements.

7. References

[1] The piston gage as a precise pressure measuring [5] instrument, D. P. Johnson and D. H. Newhall,

Trans. ASME 75, 3, 301 (April 1953).

[2] Smithsonian Meteorological Tables, published by the [6] Smithsonian Institution, Washington, D.C., Sixth Revised Edition" (1951).

[3] A multiple manometer and piston gages for precision [7] measurements, C. H. Meyers and R. S. Jessup, BS

J. Research 6, 1061 (June 1931).

[4] Les balance manometriques, L. Deffet and N. Trap- [8] peniers, Memorial de l'Artillerie Francaise, 4 fas- cicule (1954).

Elastic distortion error in the dead weight piston gage, D. P. Johnson, J. L. Cross, J. D. Hill, and H. A. Bowman, Ind. Eng. Chem. 49, 2046 (Dec. 1957).

A new method for the absolute measurement of high

pressure, R. S. Dadson, Nature 176, 188 (July 30, 1955).

The measurement of hydrostatic pressures up to 20,000

kilograms per square centimeter, P. W. Bridgman,

Proc. Am. Acad. Arts and Sci. 47, 321 (1912). Aufstellung einer Druckskale und deren experimentelle

Erprobung bis 20,000 at., H. Ebert, Z. Angew. Phys.

7, 331 (March 1949).

8. Appendix A. Computation of Pressure

8.1. Calculation of Correction Factors

To illustrate a method for simplifying the cal- culations, suppose that a particular piston gage is to be used in Room 131, MTL Bldg. at the National Bureau of Standards in Washington, D.C.,

to calibrate Bourdon gages. The local value of

gravity #£=980.10 gals, pa will be assumed to have

the value of the average density of air at this

location, so pa =0.00H7 g/cm3. The fluid being

used in the instrument is aviation instrument oil for which the density, p/a =0.862 g/cm3 or 0.0321

lb/in.3 and the surface tension, 7=0.00017 lbf/in., at atmospheric pressure and 25 °C.

The piston gage is not a controlled clearance

piston gage, therefore the factor l-\-d(pz —

p

}) will be omitted. From direct measurements on the

instrument, we find that 0= 1.964 in., yfa =2.5 in., Vfa = 1.525 in.3

, yfp = 1.625 in., and F/p =0.2778 in. 3

. From a previous calibration against a controlled

clearance piston gage, we have: A0 =0. 13024 in. 2 (at *s =25 °C) and 6 = 1.48X10-7 in. 2 /in. 2 psi. We

also know that the piston is steel with the temperature coefficient, ai=12X10-6

in./in. deg C at 25 °C and the cylinder is brass with the temperature coefficient ac =18.4X10-6

in./in. degree C at 25 °C.

6
The masses of the loading weights have been

determined and have been reported in tabular form

as shown in columns 1 and 2 of table 1. The value

given in column 2 is the apparent mass and therefore the density, pm, is assumed to be S.4 g/cm3

. All of the quantities thus determined can be

listed as illustrated in table 2 and from these

values the factors and terms of the numerator of

eq (24) and the reference level, Ah, can be computed as shown in table 2. The importance of the various factors and terms

of the numerator can now be evaluated. In the

first term —r- is the dominant factor, while the

factor kgL accounts for about 0.06 percent and the

factor (1——) accounts for about 0.014 percent.

V Pm/

The contribution of the second term is —0.295 psi

with the factor ( 1 —— ) having an effect amounting

\ Pfa/ to about 0.0004 psi. In this case the third term

amounts to 0.0027 psi. The importance of each

factor and term depends upon the design of the

instrument, the environment, and the application,

and none should be neglected without first evaluating it to determine its significance.

8.2. Machine Computation

The temperature and pressure factors, l-\-a(t—ts and (1-^-bpp), in the denominator of eq (24) can

be combined in a double entry table to give values

of

1 [l+a(t-ts)](l+bpP)

for values of t and pv as illustrated in table 3. To further facilitate computations the weight

table, table 1, has been extended to give values

of Mm times the factor 7.6726 (from table 2) for each weight as shown in column (3) and the value

of

pa \ kgL Mta [l0

j^+^=-0.292

Pja/ -^0 -"-0

is added to the value of 7.6726 Mm for the piston.

Accumulative totals, for frequently used combinations of weights, are given in column (4). Computation of pressure pp

is now simplified to the

process of multiplying the sum of values from

column (3) or a value from column (4) by the

appropriate value from table 3. Greater simplification to suit the requirements of specific

applications will be left to the ingenuity of the

user.

8.3. Slide Rule Computation

The method just described can be modified

slightly for use with a slide rule. Using the same

values as were used in the preceding illustration,

eq (24) for pv may be written in the form

Table 1. List of weights and appropriate values for weight

set and piston gage No. 1357 used at Washington, D.C..

with aviation inslru incnl oil

Piston gage No. 1357 with weight set No. 1357 Location—N 13 S, Washington, D.C. Fluid—Aviation instrument oil

(1) Weight No.

piston

12_ _

45

6

7

(2) Mass (Mm)

lb 1.3024

2. 6042

2. 6045

0. 5123

2(1. 0473

26. 0454

65. 1095

65. 1190

65. 1065

65. 1165

(3)

Af.iX7.6720

psi

9. 993-0. 292* =9. 701

19. 981

19. 983

49. 966

199.85

199. 84

499. 56

499. 63

499. 54

499. 61

(4) Accumulative

Total

psi 9.701

29. 682

49. 665

99. 631

299. 48

499. 32

998. 88

1498. 51 1998. 0

2497. 7

•Fluid buoyancy and surface tension correction,

v kgL . yC Wi-^)^+2£= _0 . 292psi . V p/a/At Aa Table 2. Tabulated values of parameters for piston gage No. 1357 used at Washington, D.C, with aviation in- strument oil

Values for Piston Gage No. 1357 ^0=0.13024 in.'. y/a =2.5 in. V>.=1.525 in3. C=1.964 in. a=a t +ac = (12+18.4) X10-6=30.4X10-6in2./in2

. °C

«.=25 °C

6 = 1.48X10-? in2./in2 .psi ?//P=1.625 in. V>p=0.2778 in3. Values for Weight Set No. 1367

p m =8.4 g/cm3 Values for Room 131, MTL Bldg., NBS, Washington, D.C.

p a =0.00117 g/cm3 or 0.0000423 lb/in3. 91=980.10 cm/sec2 Values for Aviation Instrument Oil P/a=0.0321 lb/in'. 7=0.00018 lbf/in. Computations kgL= 0.99942 ^=7.6737/in». AO(*-£)-=0.99986 Mm (l-—) ^=M„X7.6726 psi V pm/ Aa

M/«=(A.2,/a-V/.)p/a=-0.03851b

(i-^-) = -0.9987 ^4-p7JSr-°-295 psi

(17)

^=0.0027 psi

Pp = (Mm X7.6726 0.292) [l+a(< _( ;)](1+()pj))

(25)

A/i = !//P --^=-0.508 in. (19)

pp = (Mm7. 6726 -0.292) + (A/m7.6726-0.292)

X([l+a«-<t)](l+62)J ,) 0" (25)
Table 3. Temperature and pressure correction factors for piston gage No. 1857, for calculation of pressure, in psi

Values of

1 \+a(t-t,)}(l+bpp )

- for piston gage No. 1357

Temperature, t, °C

Pp

23 24 25 26 27

psi

•2500 0. 99909 0. 99906 0. 99963 0. 99960 0. 99957 2000 . 99970 . 99973 . 99970 . 99967 . 99904

1500 . 99984 . 99981 . 99978 . 99975 . 99972

1000 . 99991 . 99988 . 99985 . 99982 . 99979

500 . 99999 . 99990 . 99993 . 99990 . 99987

0 1.00000 1.00003 1.00000 . 99997 . 99994

Table 4. Temperature and pressure correction factors for

piston gage No. 13.57, for calculation of corrections, in psi

Values of [l+u(t-l,)](l+bpp ) -1 for piston gage No. 1357

Temperature, t, °C

Pressure, Pp

23 24 25 20 27

psi 2500 -0. 00031 -0. 00034 -0. 00037 -0. 00040 -0. 00043

2000 -.00024 -.00027 -.00030 -.00033 -.00036

1500 -.00016 -.00019 -.00022 -.00025 -.00028

1000 -.00009 -.00012 -.00015 -.00018 -.00021

500 -.00001 -.00004 -.00007 -.00010 -.00013

0 +. 00000 +. 00003 . 00000 -.00003 -.00006

A double entry table for values of

1 l+a(t-ts)](l +bpp) 1

for various values of t and pp can be prepared as

illustrated in table 4. A slide rule can be used to multiply the appropriate value from this table by

the sum of values from column (3) or a value from

Table 5. Temperature and pressure corrections in psi, for piston gage No. 1357

Values of pp [ [1+a{l _t ] }] -l] for piston gage No. 1357

Pressure, Pp

psi 2500 2000

1500

1000 500

0

Temperature, 1,'c

25 26 27

Corrections, psi

-0.8 -.5

-.24

-.09

.00

.00

-0.8 -0.9 -1.0 -.5 -.6 -.7

-.28 -.33 -.38

-.12 -. 15 -.18

-.02 -.04 -.05

.00 .00 .00

-1.

-.7

-.42

-.21 -.06

.00

column (4) of table. 1 to obtain a correction to be

added to the value from table 1 to give pressure. ,p p

. 8.4. Correction Table Computation

A correction table may be preferred in many

instances. Again using the same values as before to illustrate, a double entry table of values

of

Pp \[l+a(t-ts)](l+bPp) -1

)

for various values of pp and t is prepared as

illustrated by table 5. Appropriate corrections

from the table are added to values of Mm 7.6726 —

0.292 obtained from table 1 to obtain pressure pp

.

8.5. Conclusions

By construction of tables a procedure similar

to one of those illustrated can be established to

suit the particular needs and application of the

user. The computation of pressure from piston

gage data is thereby reduced to a simple, fast operation.

The equations that may be required for the computation of the absolute or the gage pressure in a system, from measurements made with a piston

gage, are listed below:

9. Appendix B. Working Equations

P=pp +Hfp +Pa

pg =pp +Hfp -Ha HfP =pfPhfpk(jL K=—PaKkgL A0 =^±^[l+a(t, tm )\

(3)

(4)

(ID

(12)

(22)

i 3m— 1 Mfa = {Aeyfa —Vfa ) Pfa

Ah = yfp -^

(21)

(23)

(17)

(19)

Mm

~[l+a(t-t,)Kl +bpp)[l+d(p-Pj)] H^+%('-£)*»+g

(24)

8
fv

Definitions of the Symbols Used in the Above

Equations

Cylinder area.. Effective area of piston.

Piston area. Effective area of the piston at atmospheric

pressure and temperature ts

. Circumference of the piston at the surface

of the pressure fluid. Pressure difference in the atmosphere between the reference level of the piston

gage and the reference level of the system

to be measured.

Pressure head of the column of pressure

transmitting fluid between the reference

level of the piston gage and the reference

level of the system to be measured.

Mass of the pressure fluid at atmospheric

pressure contributing to the load on the

piston.

Mass of the loading weights, including the

piston assembly.

Absolute (total) pressure. Atmospheric pressure at the reference level

of the piston gage. Volume of the submerged part of the piston

above the cylinder.

Volume of the part of the piston below the

cylinder.

Young's modulus.

Fractional change in effective area with unit

change in temperature.

Fractional change in effective area with unit

change in pressure. Fractional change in area with unit change

in jacket pressure. Local acceleration due to gravity.

Height of the air column measured from

the reference level of the piston gage to the reference level of the system. Meas-

£age

<7p

Ah

Vfp

Ve

ViVp

p3

t

t/n

t,

Oik

7

M

Pa

P/a

PfP Pm

the

the

urements up from the piston

reference level are positive.

Height of the column of pressure fluid measured from the reference level of the

piston gage to the reference level of the

system. Measurements up from the

piston gage reference level are positive.

Height of the reference level of the piston

gage with respect to the bottom of the

piston. Measurements up from the bottom of the piston are positive.

Proportionality factor relating force, mass

and gravity.

Gage pressure. Jacket pressure. Pressure measured by piston gage at reference level of the piston gage.

Jacket pressure required to reduce

piston-cylinder clearance to zero. Temperature of the piston gage.

Temperature at which piston and cylinder

are measured.

Reference temperature (usually the nominal room temperature).

Length of the submerged part of the piston

above the cylinder.

Length of the part of the piston below the

cylinder.

Temperature coefficient of linear expansion

of the cylinder.

Temperature coefficient of linear expansion

of the piston.

Surface tension of the pressure fluid. Poisson's ratio for the piston.

Mean density of the air displaced by the

load.

Density of the pressure fluid at atmospheric

pressure. Density of the pressure fluid at pressure P.

Density of the weights.

10. Appendix C. Examples of Calculations

Fluid—Aviation instrument oil Piston gage No. 1357, Washington, D.C.

Machine Calculation

:

a. Weights: Piston, 1, 2, 3, 4, 5, 6, 7, 8

Accumulative total: 1998.0 psi (from table

1, column (4)) Temperature: 26 °C

Correction factor: 0.99967 (from table 3)

pp = 1998.0X0-99967 = 1997.3 psi

b. Weight No.

(3)) 9.701]

19.981 1 99.631 accumulative total

column

Piston

12

36

7S

Mm X 7.6726 (from table 1.

19.983 from column (4) 49.966J

499.56

499.63

499.54

1598.361 psi

Temperature: 26 °C

Correction factor: 0.99960 (from table 3)

pp = 1598.36X0.99967 = 1597.8 psi

Slide Rule Calculation:

a. Weights: Piston, 1, 2, 3, 4, 5, 6, 7, 8 Accumulative total: 1998.0 psi (from table

1, column (4))

Temperature: 26 °C

Correction factor: -0.00033 (from table 4)

Correction = -0.00033 X 1998.0 = -0.7 psi

p p = 1998.0-0.7 = 1997^ psi

Correction Table Calculation :

a. Weights: Piston, 1, 2, 3, 4, 5, 6, 7, 8

Accumulative total: 1998.0 (from table 1, column (4))

Temperature: 26 °C

Correction =—0.7 psi (from table 5) pg = 1998.0-0.7 = 1997.3 psi

U.S. GOVERNMENT PRINTING OFFICE: 1963 O—679411

I
j

[

:

I

i

i

S

1
U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

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