Variation of specific heats and of specific heat ratio in air with humidity

Variation of specific heats and of specific heat ratio in air with humidity

George S. K. Wong and Tony F. W. Embleton Division of Physics, National Research Council of Canada, Ottawa, Ontario KL4 OR6, Canada

(Received 14 September 1983; accepted for publication 16 March 1984)

This paper described a theoretical investigation into the variations of the specific heats of air Cp and Co, and their ratio y, with relative humidity. The investigation is based on published equations for Cp, and the predicted values are derived from theoretical and experimental thermodynamic data on the constituents of humid air. For acoustical applications, such as the determination of the variation of the speed of sound in humid air, the perturbation of the term y/ M with relative humidity is also presented.

PACS numbers: 43.28.Kt, 92.60.Jq, 92.60. Dj, 51.30. + i, 43.35. -- c

LIST OF SYMBOLS

At .4 function of t

specific heat at constant pressure specific heat at standard pressure Po mean value

specific heat of constituent (i} at standard pressure Po specific heat at constant volume speed of sound enhancement factor

relative humidity, x•/x•, dimensionless molar mass of constituent (i) sum of the molar mass contributions

P

Xi

Xsw

X w

atmospheric pressure standard pressure (101.325 kPa) saturated water vapor pressure in air universal gas constant • gas constant for air temperature in degrees Celsius absolute temperature in Kelvins thermodynamic temperature in Kelvins mole fraction of constituent (i} of air mole fraction of water vapor in saturated humid air

mole fraction of water vapor in humid air modified mole fraction, X i( 1 -- xw } ratio of specific heats, Cp/Co, dimensionless

INTRODUCTION

The specific heats of air Cp and Co, and their ratio y, are some of the often encountered but less well quantified physical parameters in applied thermodynamics. One of the rea- sons for this unfamiliarity is that their present known numerical values are for dry air, and there are no publications on the corresponding values for humid air. One of the earlier values of 1.403 cited for y, "for atmospheric air at 0 øC and 1 atmosphere," was provided by the international critical tables. • In a relatively recent publication, 2 a value of 1.403, and in national and international standards, 3'4 a value of 1.402 for y (for dry air at 25 øC, Ref. 3, p. 35; at 20 øC, Ref. 4, p. 9) are recommended for precise reciprocity calibration of condenser microphones. There is no information available on the variation of y in air with relative humidity h though the latter is influenced by the air temperature t.

It is the aim here to provide theoretical values for the variation of Cp, Co, and y in air with relative humidity h. The variation of the term y/M with relative humidity is also shown graphically in view of the potential application of the above data in other areas of acoustics, such as the determination ofthe variation ofthe speed of sound 5 in humid air based on the perfect gas state: c = (y RTo/J•[) 1/•. I. THEORY

The theoretical calculations are based on the constitu- tion of standard air. •-8 Table I lists the characteristics of

each constituent and these are the same as given in Ref. 6. For dry air, at the standard pressure Po, Touloukian et al. 9 provided an equation [Eq. (3), p. 293] for C• which was derived from various experimental and theoretical data obtained for the real gas state. The equation was found to fit the data with a maximum deviation of 0.01%. Similar equations and recommended numerical data (see Table III) provided by the above authors for C• of the constituents of standard air, and of water vapor, enable us to calculate C• for humid

• for the constituents at 20 ø C air. The calculated values of Cp are listed in Table I.

Because our application is to real gases, the real gas formulas for C• are used where possible. However, of the 12 constituents listed in Table I, the real gas equations for five minor constituents (Ne, Kr, He, N:O, and Xe), and that of water vapor over the temperature range of our interest, 0 øC- 30 øC, are not provided; and their corresponding ideal gas state values and formulas are used. The real gas state equation for CH4 was derived from the ideal gas state, with a correction applied (Ref. 9, p. 246)for gas imperfection. From Table I, one can see that the molar mass contributions of the above five constituents are relatively small; and even with 10% errors in the determination of the respective values of Cp due to the adoption of the ideal gas values instead of their real values, the total deviation of the calculated Cp for dry air is less than 2 pPm. For water vapor, the ideal gas state equation was used in our computation. However, the enhance-

555 J. Acoust. Soc. Am. 76 (2), August 1984 0001-4966/84/080555-05500.80 555

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TABLE I. Composition of standard dry air. C• were derived from equations and numerical data listed in Table III.

Constituent

Molar mass Contribution

Mi Mole fraction xi.Mi (10 -3 kg mol -l) xi (10 -3 kg mo1-1) (kJ kg -1K -1)

N2 O2 Ar

CO2 Ne

Kr

CH4 He

N20 Xe

CO

H20

28.013 4 0.780 84 21.873 983

31.998 8 0.209 476 6.702 981

39.948 0.009 34 0.373 114

44.009 95 0.000 314 0.013 819

20.183 18.18 X 10 -6 0.000 367 83.80 1.14 X 10 -6 0.000 096 16.043 03 2 X 10 -6 0.000 032 4.002 6 5.24 X10 -6 0.000021

44.012 8 0.27 X 10 -6 0.000 012 131.30 0.087 X 10 -6 0.000 011 28.01 0.19 X 10 -6 0.000006 2.015 94 0.5 X 10 -6 0.000 001

18.015 34 0 0

1.0404

0.9187

0.5216

0.8460

1.0299

0.2480

2.2193

5.1931

0.8721

0.1583

1.0420

14.3020

1.8624

ment factor correction,8 f, was applied to obtain a better agreement with known values. The computation procedure for C• of humid air is as follows.

The mole fraction xw of water vapor in humid air is

x•, = hfPsv/P, (1)

where h is the relative humidity, f is the enhancement factor to correct for the departure of behavior of humid air from that of a perfect gas, and Psv is the saturated water vapor pressure at pressure P.

The enhancement factorf is a function of P and t and is based on an approximate equation [Ref. 8, Eq. (23)]. Pso is a function of T, and is calculated with Eq. (4) given on p. 5.2 of Ref. 10.

With x• known, from Eq. (1), for a particular value of h, the mole fraction xi of each constituent shown in Table I is modified in proportion, such that the sum of the modified mole fractions x;, of the constituents of humid air, is equal to unity. The molar mass of humid air M is given by the sum of the molar mass contributions from the constituents, includ-

ing the contribution from water vapor

•lr= Ex;Mio (2)

The C• of the constituents is calculated with the corresponding equations given in Ref. 9 (see Table III). The fractional contributions of the constituents to the C• of humid air are added to give a mean value

C• • • - =

For real gas at the standard pressure (101.325 kPa) which is the condition assumed throughout this investigation, the value of C• -- C• is not greatly different from the gas constant R• for air. Substituting C• for C•, one can deduce C•

(4) C• =C• , where

and R is the universal gas constant whose numerical value is

8314.48 J kmo1-1 K -1 (Ref. 11). From Eqs. (3) and (4), the ratio of specific heat of humid air is

l/C• (6) y= C•, . In the following presentation of data, a multiplication

factor of4.184 has been used to convert specific heats in units of Cal g-1 K-l (units of the equations given in Ref. 9) to units of kJ kg- l K- i.

A. Data presentation

The variation of C•, C•, and y with h are shown in Table II for temperatures from 0 øC to 30 øC in steps of 10 øC. For humid air, the specific heats increase and their ratio decreases with relative humidity. The ratio y/M increases

TABLE II. The variations of specific heats and specific heat ratio of air with relative humidity. In all cases the pressure P - 101.325 kPa.

Relative C• Co humidity (kJ kg-1 K-1) {kJ kg- • K-1)

OøC 0 1.0051 0.7181 1.3998

0.05 53 182 7

0.10 54 183 7

0.15 56 185 7

0.20 58 186 7

0.25 59 187 6

0.30 61 188 6

0.35 63 190 6

0.40 64 191 6

0.45 66 192 5

0.50 67 193 5

0.55 69 195 5

0.60 71 196 5

0.65 72 197 4

0.70 74 199 4

0.75 75 200 4

0.80 77 201 4

0.85 79 202 3

0.90 80 204 3

0.95 82 205 3

1.00 83 206 3

556 J. Acoust. Sec. Am., Vol. 76, No. 2, August 1984 G.S.K. Wong and T. F. W. Embleton: Variation of specific heats 556


TABLE II. (Continued.)

Relative C• Co humidity (kJ kg-• K- •) (kJ kg- • K-

(b) t = •0'C

0 1.0052 0.7182

0.05 056 184

0.10 059 187

0.15 062 190

0.20 065 192

0.25 069 195

0.30 072 197

0.35 075 200

0.40 078 202

0.45 082 205

0.50 085 208

0.55 088 210

0.60 091 213

0.65 095 215

0.70 098 218

0.75 101 221

0.80 104 223

0.85 108 226

0.90 111 228

0.95 114 231

1.00 117 234

(c) t = 20 øC • 0 1.0055 0.7184 0.05 061 189 0.10 067 194 0.15 073 199 0.20 079 204 0.25 - 086 209 0.30 092 214 0.35 098 219 0.40 104 224 0.45 110 229 0.50 117 234 0.55 123 238 0.60 129 243 0.65 135 248 0.70 142 253

0.75 148 258 0.80 154 263 0.85 160 268 0.90 167 273 0.95 173 278 1.00 179 283

(d) t- 30 øC 0 1.0058 0.7187

0.05 069 196 0.10 081 205 0.15 092 214

0.20 103 223

0.25 114 232

0.30 126 241 0.35 137 250 0.40 149 260

0.45 160 269 0.50 171 278

0.55 183 287

0.60 194 296 O.65 2O6 3O5 0.70 217 314

0.75 229 323 0.80 240 333 0.85 252 342

0.90 263 351 0.95 275 360

1.00 287 370

1.3997

96

96

95

95

94

94

93

93

92

92

91

91

90

90

89

89

88

88

87

87

1.3996

95

94

93

92

91

90

89

88

87

86

85

84

83

82

81

80

79

78

77

76

1.3994

92

90

89

87

85

83

81

80

78

76

74

72

71

69

67

65

63

62

58

with both humidity and temperature. The variations of •' and ?'/M with h over the above temperature range in steps of 5 øC are shown graphically in Fig. 1 and Fig. 2, respectively.

It is interesting to comment on two of the numerical values obtained. First, the value of 1.0051 kJ kg- • K- •, obtained for C• at 0 øC for h = 0 [Table II{a)], is identical to the recommended value of 0.2402 Cal g-• K-• for dry air C• {Ref. 9, p. 294). Second, it can be seen from Table II(c), that •' has a value of 1.3983 at 20 øC and at a relative humidity of 0.65; since the sensitivities of condenser microphones are inversely proportional to •,•/2 [see Ref. 12, p. 1276, Eqs. (11) and ( 12)], the sensitivities are underestimated by 0.011 dB if a value of 1.402 were to be used for •'.

We are aware of the proposed modification of the mole fraction of argon (Ar), Ref. 8, to 0.00917. The effects of this small change in mole fraction on our calculations are insigni- ficant. Also, from the same reference, an approximate method is available for calculating Psv, and this leads to a relatively small departure of 0.006% at 20 øC from the corresponding value obtained with the method used here.

B. An approximate equation for 7/

Based on the numerical data for •' shown in Table II, one can deduce the following approximateequation:

•, = 1.39984 -- At(h -I- 0.125), (7) where

A t: 5.2 X 10 -4 -t-4 X 10-st + 7.5 X 10-7t 2 -t-4.5 X 10-st 3. (8)

From Eqs. (7) and (8), one is able to calculate y as a function of relative humidity h (dimensionless)and temperature t expressed in degrees Celsius. The difference between the value of •/obtained with Eq. (7) and that in Table II is within one least significant digit.

1.400[ i I I I I I I I I

1399

o

1.395 I I I I I I I I I 0 0.1 0.:• 03 0.4 0.5 0.6 0.? 0.8 0.9 1.0

RELATIVE HUMIDITY - h

FIG. 1. The variations of specific heat ratio of air with relative humidity at various temperatures.

557 J. Acoust. Soc. Am., Vol. 76, No. 2, August 1984 G.S.K. Wong and T. F. W. Embleton: Variation of specific heats 557


0.0490

0.0489

0.0488

T 0.0487

0.0486

0.0485 -

0.0484 --

0.0485 -

0.0482 0

I I I I I I I I I

Temperature t ø C 3O

25

15

IO

5

o

I I I I I I I I I o.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

RELATIVE HUMIDITY - h

1.0

FIG. 2. The variations of 7/M with relative humidity at various temperatures.

II. UNCERTAINTIES

It is difficult to provide a detailed error analysis for the theoretical results presented here. Our estimation of their uncertainties is based on information supplied by the authors of Ref. 9; their recommended mean and maximum deviations, shown in Table III, are dependent on their judgment of the original experimental and theoretical data.

From the molar mass contributions of each constituent

{Table I), one can see that the predominant constituents are nitrogen, oxygen, and argon; and it is reasonable to weight their mean and maximum deviations with their corresponding fractional molar mass contributions: xi Mi/•xi Mi. For dry air, the weighted mean and maximum deviations for these constituents are as follows:

Percentage deviations Mean Max.

nitrogen (N2) 0.03 0.19 oxygen {O2) 0.02 0.09 argon {Ar) < 0.002 < 0.002.

For humid air, the uncertainty of the water vapor contribution is relatively small. It can be shown that for a relative humidity of 0.95 at 20 øC, the weighted mean and maximum deviations contributed by water vapor are only 0.0004% and 0.001%, respectively. From the above, one may assume that effects of the mean and maximum devia-

tions of humid air on values of C•1, C 1 and y are less than 0.03% and 0.19%, respectively. The percentage deviations shown in Table III apply to a relatively wide temperature range. For example, the maximum deviation of 0.25% for N2 applies to temperatures between 250 K and 780 K. By examining the graphical data presented in Ref. 9, and in view of our interest within a relatively limited temperature range {0 øC-30 øC), it may be reasonable to assume that the overall uncertainty of our tabulated results is of the order of 0.04%.

III. CONCLUSIONS

Based on published experimental and theoretical data, this theoretical investigation has provided useful informa-

TABLE III. Numerical data and equations provided by Ref. 9. Mean and maximum deviations were derived from least-squares fit on the original experimental and theoretical data.

Deviations, % Comments Page

Constituent (C•, units in cal g-• K-•) numbers mean max.

N2 Eq. (3) 41 0.04 0.25

02 Eq. (3) 50 0.09 0.41

Ar Least-squares fit of C• values from -- 18 øC to 38 øC 3 0.1 b

CO2 Eq. (3) 145 0.24 1.0

Ne" C•, = 0.246 15 37 ......

Kr • Cp = 0.059 284 36 ...... CH 4 Eq. (3). Derived from

ideal gas state with corrections for real gas 246 ...... states.

He" C•, = 1.2412 23 ...... N20" F__q. (1) 94 0.04 0.16

Xe" Ce = 0.037 837 59 ...... CO Eq. (3) 154 0.03 0.17

H2 Eq. {3) 28 0.10 0.37

H20" Eq. (1). Enhancement factors used in the 105 0.03 0.07 computation of Psv

Dry air Eq. (3) 293 0.01 0.01

Ideal gas states. An estimated value based on graphical data presented in Ref. 9.

558 J. Acoust. Sac. Am., Vol. 76, No. 2, August 1984 G.S.K. Wang and T. F. W. Embleton: Variation of specific heats 558


tion on the variations with humidity of the two principal specific heats of air and their ratio. An immediate application is in the area of precision reciprocity pressure calibration of condenser microphones. An underestimate of micro- phone sensitivity of 0.01 dB due to uncertainty in the value of y is greater than the currently attainable {see Ref. 12) uncertainty in reciprocity calibrations.

•International Critical Tables of Numerical Data, Physics, Chemistry, and Technology, edited by E. W. Washburn (McGraw-Hill, New York, 1929), Vol. 5.

2.4roerican Institute of Physics Handbook (McGraw-Hill, New York, 1972), 3rd ed., p. 3-71.

3American National Standards Institute, "Method for the Calibration of Microphones," ANSI, S1.10-1966 {R1976){1976).

4International Electrotechnical Commission, "Precision Method for Pres- sure Calibration of One-Inch Standard Condenser Microphones by the Reciprocity Technique," Publication 327 {1971).

•A. D. Pierce, Acoustics: .4n Introduction to its Physical Principles and .4p- plications {McGraw-Hill, New York, 1981 }, p. 28; see also I.Malecki, Phys- ical Foundations of Technical.4coustics {Pergamon, New York, 1969}, p. 291; and T. J. Quinn, Temperature {Academic, New York, 1983}, p. 16.

6U.$. Standard .4tmosphere, 1976 {U.S.G.P.O., Washington, DC, 1976}, pp. 3 and 33.

7International Organization for Standardization, "Standard Atmo- sphere," ISO 2533-1975 {E} {1975}.

sp. Giaeomo, "Equation for the Determination of the Density of Moist Air {1981}, "Metrologia 18, 3340 {1982}.

•Y. S. Touloukian and T. Makita, "Specific Heat, Nonmetallic Liquids and Gases," Thermophysical Properties of Matter {Plenum, New York, 1970}, Vol. 6.

•ø.4$HIL4E Handbook, 1981 Fundamentals {Am. Sec. Heat. Refrig. Air- Condition. Engs., Atlanta, GA, 1982}.

•A. R. Colclough, T. J. Quinn, and T. R. D. Chandler, "An Acoustic Rede- termination of the Gas Constant," Proc. R. Sec. London, Ser. A 368, 125- 13911979 ) .

•G. S. K. Wong and T. F. W. Embleton, "Arrangement for Precision Reci- procity Calibration of Condenser Microphones," J. Acoust. Sec. Am. 66, 1275-1280 (1979).

559 J. Acoust. Sec. Am., Vol. 76, No. 2, August 1084 G.S.K. Wong and T. F. W. Embleton: Variation of specific heats 559


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Variation of specific heats and of specific heat ratio in air with humidity