# Non-ideal behaviour

The application of the ideal gas laws to mixtures of gases and vapours assumes the validity of Dalton’s Law. To quote Goff (1949): ‘By present-day standards, Dalton’s Law must be regarded as an inaccurate conjecture based upon an unwarranted faith in the ultimate simplicity of nature.’ The ideal laws and the kinetic theory of gases ignore the effects of inter-molecular forces. The more up-to-date approach of statistical mechanics, however, gives us a theory of thermodynamics which takes account of forces between like molecules, in terms of virial coefficients, and between unlike molecules in terms of interaction coefficients.

The methods of statistical mechanics, in conjunction with the best experimental results, yield accurate evaluations of the properties of dry air and water vapour, published in NBS Circular 564 (1955) and in the NEL Steam Tables (1964), respectively.

The association of dry air and water vapour in a mixture is expressed by the moisture content on a mass basis and so percentage saturation is particularly relevant, since it defines the fraction of saturated water vapour present in a mixture, for a particular temperature. It is typified by the equation:

8 = Iqo" (2.32)

And the moisture content at saturation, gss, is defined more precisely by the CIBSE (1986): than is possible with the ideal gas laws

_ 0.621 97fspss ^

P,-f, P, <233>

The numerical constant in equation (2.33) is the ratio of the molecular mass of water vapour and dry air with respective values of 18.016 and 28.966 kg mol-1. The molecular masses used by ASHRAE (1997) are slightly different and yield a value of 0.621 98, to replace the value used in equation (2.33).

The symbol fs is a dimensionless enhancement factor and, according to Hyland and Wexler (1983), is dependent on: Henry’s constant (relating to the solubility of air in water), saturation vapour pressure, temperature, barometric pressure, and other factors. The related equations are complicated. The value of fs is approximately 1.004 at a barometric pressure of 101.325 kPa and a temperature of 0°C.

The data in Table 2.1 are based on Goff (1949) and form the basis of the calculations leading to the CIBSE psychrometric tables (1986) and chart according to Jones (1970). Other research by Hyland and Wexler (1983) yields slightly different values for the enhancement factor, namely, 1.0039 at a barometric pressure of 100 kPa over the temperature range from 0°C to 90°C. For all practical purposes/s can be taken as 1.004.

At sufficiently low pressures the behaviour of a mixture of gases can be accurately described by

Pv = RT-p’L XiXkAik(T) — p2 I XiXjXkam(T) (2.34)

Ik ijk

The term Aik(T) is a function of temperature and represents the second virial coefficient if i = k, but an interaction coefficient if i * k, the molecules being considered two at a time.

Table 2.1 Enhancement factor, fs

 KPa 0°C 10°C 20°C 30°C 40°C 50°C 60°C 101.325 1.0044 1.0044 1.0044 1.0047 1.0051 1.0055 1.0059 100 1.0044 1.0043 1.0044 1.0047 1.0051 1.0055 1.0059 97.5 1.0043 1.0043 1.0043 1.0046 1.0050 1.0054 1.0058 95 1.0042 1.0042 1.0043 1.0045 1.0049 1.0053 1.0057 92.5 1.0041 1.0041 1.0042 1.0045 1.0049 1.0053 1.0056 90 1.0040 1.0040 1.0041 1.0044 1.0048 1.0052 1.0056 87.5 1.0039 1.0039 1.0040 1.0043 1.0047 1.0051 1.0055 85 1.0038 1.0038 1.0040 1.0042 1.0046 1.0050 1.0054 82.5 1.0037 1.0037 1.0039 1.0042 1.0045 1.0049 1.0053 80 1.0036 1.0036 1.0038 1.0041 1.0045 1.0049 1.0052

The function Aijk(T) then refers to the molecules considered three at a time. The terms in x are the mole fractions of the constituents, the mole fraction being defined by

Xa = —^— (2.35)

«a + «w

Where na is the number of moles of constituent a and nw is the number of moles of constituent w.

Also, since pressure is proportional to the mass of a gas (see equation (2.8)) it is possible to express the mole fraction in terms of the partial pressures of dry air (pa) and water vapour (pw):

** = 77T’4Tn~l (2’36)

P a P w)

If xa is the mole fraction of constituent a then (1 — xa) is the mole fraction of constituent w and equation (2.34) can be rephrased as

Pv — RT — [xa + *a(l — xa)2Aaw + (1 — x%) Aww]p — [(1 — xa) Awwwp (2.37)

Where the subscripts ‘a’ and ‘w’ refer to dry air and water vapour, respectively. The third

Virial coefficient, Awww, is a function of the reciprocal of absolute temperature and is

Insignificant for temperatures below 60°C. Other third order terms are ignored.

If equation (2.37) is to be used to determine the specific volume of dry air then pa and Ra replace p and R. It then becomes

R ’J’

V — —— [-^a^aa — -^a )2Aaw + (1 — Xa) Aww]

P a

= 82.0567 X 101.325 _____ T_____ x2A + x (1 — x )2A + fl — x )2A 1

28 966 (p — p ) a’ “ V1 Aa/ ‘‘wwJ

(2.38)

For water vapour mixed with dry air

= „ by equation (2.35)

Lla T rlvj

But

 N^M^ 0.621 97n„ NaMa rca Therefore *w 0.621 97 +g (239)

Table 2.2 Vidal and interaction coefficients for moist air

 F(°C) ^aa ^aa A A Naw 0 4.540 x 10“4 4.56 X 10-4 6.318 X 10“2 1.45 X 10“3 10 3.728 x 10-4 3.76 X 10“4 5.213 X 10~2 1.36 X 10~3 20 2.983 x 10^ 3.04 X 10-4 4.350 X 10~2 1.27 X Io-3 30 2.296 x 10-4 2.37 X 10“4 3.708 X Ur2 0.19 X 10“3 40 1.663 x 10-4 1.76 X IQ“4 3.190 X Ur2 0.12 X 10"3 All the Coefficients are in M3 kg“1

The coefficient Aaa in the first column is due to Hyland and Wexler (1983) but the coefficient Aaa in the second column, and the coefficients Aww and Aaw, are due to Goff (1949).

The enthalpy of dry air increases very slightly as the pressure falls at constant temperature but, for most practical purposes, it can be regarded as a constant value at any given temperature from one standard atmosphere down to 80 kPa. It follows that the most significant effect of a change in barometric pressure on the enthalpy of moist air lies in the alteration of the moisture content. A revised expression for the enthalpy of moist air is

 H = h„ +
 H„
 0.621 97 fsPs; Pat ~fsPss (2.40)

EXAMPLE 2.15

Calculate the enthalpy of moist air at 20°C and 82.5 kPa (a) when saturated, and (b) at 50 per cent saturation.

(a) From CIBSE tables, fta = 20.11 kJ kg-1 and pss = 2.337 kPa at 20°C. From NEL Steam Tables (1964) or from the CIBSE Guide (1986) by interpolation hg = 2537.54 kJ kg-1 at 20°C saturated. From Table 2.1 /s = 1.0039 at 20°C and 82.5 kPa. If it is assumed that the liquid from which the steam was evaporated into the atmosphere was under a pressure of

82.5kPa then an addition of 0.08 x 82.5/101.325 = 0.07 kJ kg-1 should be made to the NEL value for A giving a new figure of 2537.61 kJ kg-1. Then

(0.62197 X 1.0039 x 2.337 x 2537.61 h -20.11 + (82.5 — 1.0039 x 2.337)

= 20.11 +46.20 = 66.31 kJ kg’1

The CIBSE Guide (1986) gives a table of additive values to be applied to enthalpies read from the psychrometric tables for 101.325 kPa at any value of the temperature of adiabatic saturation. From this table the additive value at 20°C adiabatic saturation temperature and

82.5 kPa is 8.77 kJ kg-1. The CIBSE psychrometric tables for 101.325 kPa and 20°C saturated quote 57.55 kJ kg-1. Hence the value to be compared with our result is 66.32 kJ kg-1.

(b) By equation (2.33)

0. 62197 x 1.0039 x 2.337 8ss ~ (82.5 — 1.0039 x 2.337)

= 0.018 205 kJ kg“1

By equations (2.20) and (2.32):

H = 20.11 + 0.5 x 0.018 205 x 2537.61 = 43.21 kJ kg“1

Posted in Engineering Fifth Edition