**INTRODUCTION **

Long time readers may recall that one answer to the question of how humidity changes the thermal conductivity of air is “Nothing to worry about” [1]. Generally, the effect of humidity on air properties is small enough that it can be neglected. However, at high ambient temperatures and low pressure, humidity effects may need to be addressed in applications such as free air cooling or cooling of outdoor equipment. This article provides a method for estimating the properties of humid air, in a temperature range of 0-100˚C, and is based on a previous publication [2].

**CALCULATION METHOD **

Since humid air is a mixture of dry air and water vapor, it is logical that a good first step in determining its properties is to determine the properties of its constituents. These properties are dependent only on the fluid temperature and coefficients for polynomial curve fits for both fluids are summarized in *Table 1 *and *Table 2*. Terms in these tables are used to calculate properties with *Equation (1):*

where T is the absolute temperature in K (1)

For example, the specific heat of air at 26.85C (300K) would be calculated as:

Cp, air = 1.034090E+00*300^{0} – 2.848870E-04*300^{1 }+7.816818E- 07*300^{2 }– 4.970786E-10*300^{3 }+1.077024E-13* 300^{4 }+ 0*300^{5 }= 1.0064 kJ/kg K

Once the properties of the two constituents of humid air have been determined, they can be used to determine the properties of a mixture with a given ambient pressure, P_{o }and relative humidity, RH. This process requires a number of terms to be calculated: the enhancement factor (F), the compressibility factor (z), two interaction parameters (Φ_{av }and Φ_{va}) and the molar fraction of water vapor (x_{v}).

The interaction factor is determined using *Equation (2) *

The constants in *Equation (2) *are calculated using the same approach as in *Equation (1) *with the coefficients shown in *Table 3. *

For example, for a temperature of 26.85C (300K), the two coefficients would be calculated as:

At that temperature, the saturation pressure for water is calculated to be 3.56 kPa. If the atmospheric pressure is 101.35kPa, the pressure ratio P_{sv}/P_{o }is 0.0351 and the value of F would then be calculated as:

F = exp(0.0015*(1 – 0.0351) + 0.000098*(0.0351 – 1)) = 1.0014

Once the enhancement factor has been calculated, it can be used to determine the molar fraction of vapor for a given relative humidity using *Equation (3)*.

x_{v }= F * RH * P_{sv}/P_{o }(3)

If, in the case described above, the humidity is RH=50%, the molar fraction of vapor would then be

x_{v }= 1.0014 * 0.5 * 0.0351 = 0.0176

The effective molar mass of the mixture, M_{m}, can them be determined using *Equation (4)*.

M_{m }= (1 – x_{v}) * M_{a }+ x_{v }* M_{v }(4)

Where M_{a} is the molar mass of air (28.97 g/mol) and M_{v} is the molar mass of water (18 g/mol). For the previously determined conditions then, the mixture’s molar mass would be

M_{m }= (1 – 0.0176) * 28.97 + 0.0176 * 18 = 28.78 g/mol

The compressibility factor, z, is a function of the temperature and saturation pressure. It can be estimated using *Equation (5) *

The coefficients A and B are calculated using the coefficients in *Table 4*, where:

For T = 300K, A = -4.700E-07, B = -6.332E-13 and z = 0.99832.

Finally, the interaction parameters can be calculated using *Equations (6a) *and *(6b)*.

For a temperature of 300K, *Equations (6) *produce interaction parameters of Φ_{av }= 0.4973 and Φ_{va }= 2.4507.

Once all of these parameters have been calculated, the properties of humid air can be calculated using *Equations (7 – 10)*. These properties of the mixture of water and air include the density, ρ_{m}, the specific heat, c_{p,m}, the thermal conductivity, k_{m}, and the dynamic viscosity, μ_{m}.

For the conditions described previously, these equations calculate the properties as ρ_{m }= 1.171 kg/m^{3}, c_{p,m}, = 1.0162 kJ/kg K, k_{m }= 0.02598 W/mK, and μ_{m}. = 1.903E-05 kg/m s.

*Figure 1 *shows calculated properties for air with relative humidity ranging from 0 to 100%. Calculations are made for six combinations — three ambient temperatures are used (25, 50 and 75˚C) and two pressures (101.325 and 78.1 kPa). These correspond to sea level (s.l.) and ~2000m (2km) above sea level for a standard atmosphere.

These plots show that impact of humidity on fluid properties is quite small at lower temperatures, particularly for thermal conductivity. But since hot air can hold more moisture the effects do become more pronounced at higher temperatures and lower pressures, particularly for viscosity. This is clearly illustrated in *Figure 2*, which shows the change in each property at each of the six combinations when comparing dry air (0% RH) and air with 100% RH. Again, the effects of humidity are negligible at low temperatures, but can be significant at high temperatures, particularly if the ambient pressure is low.

**SUMMARY**

- Humidity generally has a small impact on the properties of air, but under some conditions the presence of humidity can affect properties to a sufficient degree that it needs to be accounted for.
- This article outlined a method for calculating the properties of humid air using a number of curve fits to determine the properties of air and water vapor as well as correction factors. These curve fits were developed for a specific temperature range of 0 – 100˚C. The equations in this article should not be used outside this temperature range.
- This analysis implicitly assumes that the saturation pressure of water at ambient temperature is not higher than the ambient pressure. If this is not the case, incorrect results will be calculated.
- A spreadsheet with the procedure described in this article is available. Send an email with the word ‘Humidity’ in the subject line to ElectronicsCoolingWilcoxon@gmail.com to receive a link for a spreadsheet that can be downloaded for editing.

1. Reference [2] reported coefficients for calculating the properties of water vapor using the temperature in C. To simplify the analysis for this article, values generated with those equations were used to generate curve fits for the absolute temperature, in K.

2. The coefficients for determining saturation pressure reported in [2] seemed to have a typo that led to somewhat larger errors than reported. The coefficients shown here were generated with by generating a curve fit from the a standard steam table [3].

**REFERENCES**

1. C. Lasance, “The Thermal Conductivity of Moist Air”, *Electronics Cooling Magazine, November 1, 2003*, https://www. electronics-cooling.com/2003/11/the-thermal-conductivity-of-moist-air/#

2. P. T. Tsilingiris, “Thermophysical and transport properties of humid air at temperature range between 0 and 100C”, *Energy Conversion and Management*, Vol 49, pp. 1098-1110, 2008

3. 1967 ASME Steam Tables, http://www.che.ksu.edu/docs/imported/SteamTable.pdf

ACKNOWLEDGMENTS

I would like to thank *Parizad Shojaee Nasirabadi *for her help in interpreting the equations in *Ref [2] *and for recognizing the occasional typo.