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NBSIR 83

Calculation of Compressibility

Factor for Air Over the Ranges of

Pressure, Temperature, and

Relative Humidity of Interest in

Flowmeter Calibration

-2652

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Chemical Engineering

Chemical Process Metrology Division

Washington, DC 20234

March 1 983

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U.S. DEPARTMENT OF COMMERCE

national bureau of standards

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NBSIR 83-2652

CALCULATION OF COMPRESSIBILITY

FACTOR FOR AIR OVER THE RANGES OF

PRESSURE, TEMPERATURE, AND

RELATIVE HUMIDITY OF INTEREST IN

FLOWMETER CALIBRATION

Frank E. Jones

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Chemical Engineering

Chemical Process Metrology Division

Washington, DC 20234

March 1983

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Calculation of Compressibility Factor for Air

Over the Ranges of Pressure, Temperature, and Relative

Humidity of Interest in Flowmeter Calibration

Frank E. Jones

Fluid Engineering Division

Center for Chemical Engineering

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

A simple yet precise equation has been developed to enable

calculation (using programmable calculators) of the compressibility factor, Z, for air from measurements of pressure, temperature, and humidity. The compressibility factor, a factor which

accounts for the non-ideality of air in real-gas equations of

state, is conventionally computed using virial coefficients. In

the present paper, an equation is fitted to tabulated values of

Z. The deviation between calculated and tabulated values is of

the order of 0.01% or less; this does not imply, however, that

the accuracy of calculated values is of this order.

Key Words: Air; compressibility factor; extrapolation formulas;

flowmeter calibration; pressure; relative humidity;

temperature.

1. Introduction

In the calculation of the flowrate of air through various metering

devices (Venturi tubes, orifice plates, and nozzles, for example), the

density of air enters. Since air is not an "ideal" gas, the compressibility factor (Z) a measure of the nonideality is introduced in the

real gas equation for density (p). The compressibility factor is

conventionally computed using the virial equation of state of an airwater vapor mixture (1,2). In this paper a simple interpolation

formula is developed which enables the engineer to conveniently make

precise calculations of Z using small readily available programmable

calculators.

2. Real Gas Equation for Density of Air

The real gas equation for the density, p, of dry air is [1]

PM

a RTZ 5

1

p =

( 1 )
where P is the total pressure; M is the apparent molecular weight of

dry air; R is the molar gas constant; T is absolute temperature; and

Z is the compressibility factor. The magnitude of the non-ideality of

the gas is reflected in the departure of Z from 1. For moist air, a mixture of dry air and water vapor, the real gas equation for p is [1]

PM

p - [1 + (e ' 1) • (2)

where e is the ratio of the molecular weight of water vapor to the

apparent molecular weight of dry air, and e‘ is the effective vapor

pressure of water in moist air. The value of Z depends on P, T, and

relative humidity of the air.

3. Development of the Equations

To develop the equation for Z, unpublished data for dry air

provided by Wexler and Hyland [4] and data for moist air in table 2

of reference [1] were used. The equation was developed in two parts:

the first includes the dependence of Z on pressure and temperature;

the second part is a correction due to the effect of relative humidity

(RH).

At each of nine values of P (covering the range 1 to 40 atmospheres), values of (1-Z) were fitted by least squares to an equation

quadratic in temperature, t, in °C. At each value of pressure, 19

values of (1-Z) covering the temperature range 0°C to 54°C at intervals of three degrees were used. The resulting equations are:

at P = 1 atm, (1-Z) x 10

6

= 581.98 - 12.471 t +

at P = 5 atm, (1-Z) x 10

6 = 2860.3 - 62.056 t +

at P = 10 atm, (1-Z) x 10

6

= 5594.3 - 123.32 t

at P = 15 atm, (1-Z) x 10

6

= 8195.8 - 183.24 t

at P = 20 atm, (1-Z) x 10

6 = 10670.6 - 243.22 t

at P = 25 atm, (1-Z) x 10

6

= 13006.4 - 301.64 t

at P = 30 atm, (1-Z) x 10

6

= 15204.0 - 358.89 t

at P = 35 atm, (1-Z) x 10

6

= 17261.7 - 415.00 t

063156 t

2

, (3)

31 525 1.2

5 (4)

.62894 t

2

. (5)

.93145 t

2

. (6)

1.2517 t

2

. (7)

1.5587 t

2

. (8)

1.8613 t

2

, (9)

2.1607 t

2

. (10)

2.4534 t

2

. (ID

The general form of eqs (3) through (11) is

(1-Z) x 10

6

= a(P) + b(P) t + c(P)t2

, ( 12 )

2
where the dependence of Z on P is assigned to the coefficients, a, b,

and c. Each of these coefficients was fitted by least squares to an equation quadratic in P (in atm). The resulting equations are

a ( P ) = -10.864 + 588.26 P - 2.7106 P

2

, (13)

b(P) = 0.33297 - 12.585 P + 2.0659 x 10" 2

P

2

, (14)

c(P) = -2.4925 x 10" 3 + 6.3706 x 10" 2 P - 5.5619 x 10" 5

P 2 (15)

An equation expressing the dependence of Z for dry air on t and P is synthesized by substituting eqs (13) through (15) in eq (12):

(1-Z) x 10

6

= -10.864 + 588.26 P - 2.7106 P

2

+ 0.33297 t - 12.585 Pt + 2.0659 x 10" 2

P 2

t

- 2.4925 x 10“ 3 t

2 + 6.3706 x 10" 2 Pt2

- 5.5619 x 10" 5 P 2

t

2 (16)

To develop an expression for the correction for the effect of RH on Z, values of Z for temperatures 19°C to 26°C at 1-degree intervals

for RH of 0, 25, 50, 75, and 100 percent were used. The five mean values of the difference (one corresponding to each value of RH)

between Z for moist air and Z for dry air, a( 1-Z), were fitted by

least squares to an equation quadratic in RH. The intercept of the

equation was forced sufficiently close to 0 by weighting the 0 percent

RH point. The resulting equation is

(1-Z) x 10

6

= -0.35 (RH) - 5.0 x 10" 3 (RH)

2

, (17)

where RH is in percent.

By adding eq (17) to eq (16) and rearranging, the equation

expressing the dependence of Z on P, t, and RH results:

Z = 1.00001 - 5.8826 x 10" 4

P + 2.7106 x 10" 6

P 2

- 3.3297 x 10‘ 7 t + 1.2585 x 10' 5 Pt - 2.0659 x 10' 8

P 2 t

+ 2.4925 x 10“ 9

t

2 -6.3706 x 10' 8

Pt

2 + 5.5619 x 10" 11

P

2 t2

- 3.5 x 10“ 7 (RH) - 5.0 x 10" 9

(RH)

2

, (18)

where P is in atm, t is in °C, RH is in percent, and Z is a pure

number.

For P in PSI, t in °C, and RH in percent.

3
Z = 1.00001 - 4.0029 x 10" 5

P + 1.2551 x 10" 8P 2

+

3.3297 x 10" 7 t + 8.5636 x 10“ 7 Pt - 9.5827 x

2.4925 x 10' 9t

2

- 4.3349 x 10“ 9

Pt

2 + 2.5753

3.5 x 10" 7 (RH) - 5.0 x 10" 9

(RH)

2

10" 11 P 2

t

x 10' 13

P 2 t

2

(19)

For P in MPa, t in °C, and RH in percent,

Z = 1.00001 - 5.8057 x 10" 3

P + 2.6402 x 10" 4P 2

- 3.3297 x 10" 7 t + 1.2420 x 10“ 4

Pt - 2.0158 x 10“ 6

P 2

t

+ 2.4925 x 10“ 9 t

2

- 6.2873 x 10" 7 Pt

2 + 5.4174 x 10' 9

P

2 t

2

- 3.5 x 10“ 7 (RH) - 5.0 x 10' 9

(RH)

2 (20)

The values of Z for dry air calculated using eq (18) have been

compared with the data used to develop the equation. The mean of the

171 differences between calculated and data values is 6 x 10" , and

the estimate of standard deviation is 2.6 x 10" 3 which corresponds to

0.0026%. At a pressur

deviation is 2.1 x 10"

of 1 atmosphere the estimate of standard

which corresponds to 0.00021%. The estimate

of overall uncertainty for the data on which the calculation of Z is

based [4] ranges from 0.00093% to 0.0011% at a pressure of 0.987

atmosphere over the temperature range of interest; at a pressure of

49.3 atmospheres the estimate of overall uncertainty ranges from

0.054% to 0.075%. The estimate of standard deviation for the difference between the calculated RH correction (the last two terms in

eq (18)^ and that inferred from the data on which it is based is

1 x 10~ 3 which corresponds to 0.001% in Z. Since the RH correction

was fitted over the range 19°C to 26°C it will have a greater deviation outside this range. However at the extremes of the temperature

range of interest here, approximately 0°C to 50°C, the deviation is

only of the order of 0.01% in Z.

4. Conclusions

A simple yet precise equation has been developed to enable

calculation of the compressibility factor, Z, for air from measurements of pressure, temperature, and humidity. The deviation between

calculated and data values of Z is of the order of 0.01% or less; this

does not imply, however, that the accuracy of calculated values is of

this order.

5. Acknowledgment

The author gratefully acknowledges the assistance of R. W. Hyland

in providing access to the compressibility factor data, the computational assistance of C. P. Reeve, and the typing of the manuscript by

Susan Johnson.

4
6. References

1. Jones, F.E. The Air Density Equation and the Transfer of the

Mass Unit, J. Res. Nat. Bur. Stand. (U.S.). 83 (5) : 41 9-428 ; 1978

September-October.

2. Hyland, R.W., and Wexler, A. The Second Interaction (Cross) Virial

Coefficient for Moist Air, J. Res. Nat. Bur. Stand. (U.S.).

77A ( 1

) : 1 33-1 47 ; 1973 January-February

.

3. Hyland, R.W. Correlation for the Second Interaction Virial

Coefficients and Enhancement Factors for Moist Air, J. Res. Nat.

Bur. Stand. (U.S.). 79A (4):551-560; 1975 July-August.

4. Wexler, A., and Hyland, R.W. Formulations for the Thermodynamic

Properties of Dry Air from 173.15 to 473.15 K, and of Saturated Moist

Air from 173.15 to 372.15 K, at Pressures to 5 MPa, May 1980,

to be published by ASHRAE.

5

NBS-114A irev. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 83-2652

2. Performing Organ. Report No 3. Publication Date March 1983

4. TITLE AND SUBTITLE

Calculation of Compressibility Factor for Air Over the Ranges of Pressure, Temperature , and Relative Humidity of Interest in Flowmeter Calibration

5. AUTHOR(S)

Frank E. Jones

6. PERFORMING ORGANIZATION (If joint or other than N BS. see in struction s) 7. Contract/Grant No. NATIONAL bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234 Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10.

SUPPLEMENTARY NOTES

~| Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A simple yet precise equation has been developed to enable calculation (using

programmable calculators) of the compressibility factor, Z, for air from measurements of pressure, temperature, and humidity. The compressibility factor, a factor

which accounts for the non-ideality of air in real-gas equations of state, is

conventionally computed using virial coefficients. In the present paper, an

equation is fitted to tabulated values of Z. The deviation between calculated

and tabulated values is of the order of 0.01% or less; this does not imply,

however, that the accuracy of calculated values is of this order.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semi colon s) Air; compressibility factor; extrapolation formulas; flowmeter calibration;

pressure; relative humidity; temperature.

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