Functional Equations for the Enhancement Factors for CO2-Free Moist Air

Equations are presented which explicitly express the enhancement of water vapor in CO2-free air from 0.1 to 2 MPa. The equations are approximations to the formulation of Hyland and provide the means of obtaining enhancement with very modest computational facilities. The agreement with Hyland’s enhancement values is well within his estimated uncertainty.


Introduction
The effec tiv e saturation press ure of wate r vapor in equilibrium with a plan e s urface ofliquid or solid water in the p rese nce of an admixed gas above its criti cal te mpe rature diffe rs from th e saturation vapor press ure of th e pure phase. It is ex pressed by xu;P

1=es
where 1= the enhan ce me nt factor (1) Xw = the mole frac tion of water vapor in the saturated mixture. P = total press ure above the s urface of the co nde nsed phase (liquid or solid) es = the pure phase saturation vapor pressure of water.
In the fi eld of humidity it is important to know both es and lin order to d etermin e the quantity of saturated wate r vapor in a gas. As an example, the mixing ratio of a gas saturated with water vapor is give n by r = Ele s (2) Ple s whe re r = the mixin g ratio in mass of water vapor pe r unit mass of associated dry gas. E = the ratio of the molecular weight of water to the molec ular weight of the associa ted dry gas. Whe reas es is and has bee n known over a fairly wide range of te mperature to a high degree of accuracy, accurate values of I have not been a vaila ble for any gas until Hyland [1] 1 rece ntly published values of I for wate r vapor in CO 2 -free air over a wide ran ge of 41 te mperatures and press ures. These values are based on expe rim ents from which the approximate th eoretical form for the second interacti on (cross) virial coe ffi cie nt for air-water vapor mixtures was obtained. Use of thi s coe fficie nt in a th er modynamically based eq uation permits de termination of the enha nce ment fac tor.
This equation , though impli cit in I, can be used in an iterative mode to calcula te lover fairly wide ranges of te mpera ture and pressure. The equation is co mplex and impli cit in I and th erefore req uires high speed computer capability for th e de termin ation of I a t any giv e n te mperature and pressure. An alte rn ate a nd simpler method of obtaining values of I for specific condition s would be by interpolation in tables first obtained by co mputer from Hyland 's equ ation. Since a two way (te mpera ture a nd pressure) interpolation of a fun ction nonlinear in two parameters would be required, s uc h interpolation would, at best, be awkward unless the table were close spaced.
By ge neralizing an equa tion, originally due to Goff and Gratch [2] and fitting it to values ob taine d from Hyland's relation, we have obtain ed simplified explicit equations for I which can be easily programed for a computer or ca n be calculated with the aid of a pocket calc ula tor.

Method
Goff and Gratc h [2] used the following form of equation for I whe re a and f3 were obtain ed from theore ti cal equa· tions and presented in a table of a and {3 for water and ice as a function of temperature. The implication was that a and {3 were independent of pressure, which is not precisely true, but they are fairly insensitive over modest pressure ranges.
We obtained values of a and {3 based on the new data given by Hyland. We calculated these values of a and {3 for water at temperatures from -50 to 100°C and pressures from 0.101325 MPa (1 atm) to 2.0265 MPa (20 atm) and for ice from -100 to 0 °C over the same pressure range.
a and {3 were obtained by aleast square fit of Hyland's values to equation   Table 1 gives the values of A i and B i for three conditions at 0.1 to 2 MPa. The agreement between these equations and values obtained with the Hyland formulation is shown in table 2.
It is of interest to compare deviations between these formulations and Hyland's formulation with his estimate of the total uncertainty. This comparison IS given an estimated uncertainty. The deviations nowhere exceed Hyland's estimated uncertainties and in general are more than an order of magnitude less than the uncertainties.
It will be noted that these formulations have been compared with Hyland's formulation in regions where Hyland has declined to assign uncertainties (below -50°C and above 90 °C as well as in regions where he has published no values (below -80°C, above 90 °C and for water below O°C). Despite the questionable validity of his formulation in these areas, namely the region of supercooled water and the region of ice below -50°C, there is no formulation for J in these regions that is more valid at this time. It is our suggestion therefore that the Hyland formulation be used for all water-air J factors up to the boiling point. It is for this reason that we have proposed our simplified equations for this wide region.

42
As can be seen from table 2, the agreement between these formulations is exceedingly good above O°C and quite good for water down to -50°C. Because there does not appear to be any need for f factors for supercooled water below -50°C, we have terminated our equation at that point.
There is a need for Jfactors for ice below -50°C and we have therefore extended our formulation to -100°C. As can be seen from table 2, the agreement with Hyland is considerably poorer for ice than for water. The agreement can be improved by using two sets of coefficients, one for 0 to -50°C and the other for -50 to -100°C, in the same form of the equation. The coefficients, are given in

Discussion
We have atte mpted to prese nt equations that will be useful to a wid e range of workers in the fi eld of humidity measurement. We have used a historical form of the equation which appears to give reasonably adequate results up to 2. MPa.
It should be pointed out that the deviations between the values obtained from these equations and the values obtained from the Hyland formulation are not random. These equations therefore are not an alternative expression of the Hyland formulation, but merely a convenience for obtaining approximate values of Hyland 's f factor.
Of course, there are those who have need for Hyland's results above 2 MPa, and those people we have not helped. It is regrettable that we required three equations to cover the desired range (some may feel that the four equations are necessary) and that each of the equations has eight coefficients. We do not s uggest that the equations presented here are the optimum equations available, but only that they appear adequate for most purposes and appropriate for adoption where uniform simple computations are desirable. It must also be pointed out that these formulations are based on the 1968 IPTS temperature scale, the Wexler [3] formulation for the vapor pressure of water and the Goff [4] formulation for the vapor 44 pressure of ice, which is completely consistent with the work of Hyland. Were new vapor pressure equations to be used which give values that differ markedly from the referenced equations, these formulations would no longer be acceptable. It is important to be aware that under the same circumstances, the values of f calculated with Hyland's formulation would differ from his published values.