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Psychrometrics

Overview

Psychrometrics is the study of the thermodynamic properties of moist air. Accurate psychrometric calculations are essential for cooling tower analysis because both the driving force for heat transfer (enthalpy difference) and the driving force for mass transfer (humidity ratio difference) depend on the thermodynamic state of the air.

Psychrometric chart with cooling tower process
Figure 1. Simplified psychrometric chart showing constant relative humidity lines and the air process path through a cooling tower (from inlet at lower-left toward saturation at upper-right).

Saturation Vapor Pressure

The saturation vapor pressure of water, \( p_s(T) \), is calculated using the ASHRAE correlation (ASHRAE, 2017):

\[ \ln(p_s) = \frac{C_1}{T} + C_2 + C_3 T + C_4 T^2 + C_5 T^3 + C_6 \ln(T) \]

where \( T \) is in Kelvin and \( p_s \) in Pascals. The coefficients are:

Coefficient Value Applicable Range
\( C_1 \) -5.8002206 x 10³ 273.15 K to 473.15 K
\( C_2 \) 1.3914993
\( C_3 \) -4.8640239 x 10⁻²
\( C_4 \) 4.1764768 x 10⁻⁵
\( C_5 \) -1.4452093 x 10⁻⁸
\( C_6 \) 6.5459673

In this implementation, the Antoine-type correlation is used for computational efficiency:

\[ p_s(T) = \frac{\exp\left(77.3450 + 0.0057\,T - \dfrac{7235}{T}\right)}{T^{8.2}} \]

Temperature in Kelvin

All psychrometric equations in this implementation use absolute temperature (Kelvin). Ensure correct conversion from user inputs.

Humidity Ratio

The humidity ratio (or moisture content) \( w \) is the mass of water vapor per unit mass of dry air.

From Relative Humidity

Given the relative humidity \( \phi \) and dry-bulb temperature \( T_{db} \):

\[ p_v = \phi \cdot p_s(T_{db}) \]
\[ w = 0.62198 \times \frac{p_v}{P_{atm} - p_v} \]

where \( P_{atm} \) is the atmospheric pressure (Pa).

From Wet-Bulb Temperature

The humidity ratio can also be determined from the wet-bulb temperature using the psychrometric equation (ASHRAE, 2017):

\[ w = \frac{(2501 - 2.326\,T_{wb}) \cdot w_s(T_{wb}) - 1.006 \cdot (T_{db} - T_{wb})}{2501 + 1.86\,T_{db} - 4.186\,T_{wb}} \]

where:

  • \( T_{db} \), \( T_{wb} \) are in degrees Celsius
  • \( w_s(T_{wb}) \) is the saturation humidity ratio at the wet-bulb temperature

Moist Air Enthalpy

The enthalpy of moist air per unit mass of dry air is:

\[ h_a = c_{pa} \cdot T + w \cdot (h_{fg,0} + c_{pv} \cdot T) \]

In simplified form (temperatures in Celsius):

\[ h_a = 1.006\,T + w\,(2501 + 1.86\,T) \quad \text{[kJ/kg dry air]} \]

where:

  • \( c_{pa} = 1.006 \) kJ/(kg·K) — specific heat of dry air
  • \( c_{pv} = 1.86 \) kJ/(kg·K) — specific heat of water vapor
  • \( h_{fg,0} = 2501 \) kJ/kg — latent heat of vaporization at 0 °C

Saturated Air Enthalpy

The enthalpy of saturated air at a given temperature is obtained by setting \( w = w_s(T) \):

\[ h'_s(T) = 1.006\,T + w_s(T) \cdot (2501 + 1.86\,T) \]

This is the saturation enthalpy curve and plays a central role in both the Merkel and Poppe methods.

Latent Heat of Vaporization

The latent heat of vaporization decreases with increasing temperature:

\[ h_{fg}(T) = 2\,501\,000 - 2\,326 \cdot T_C \quad \text{[J/kg]} \]

where \( T_C \) is the temperature in degrees Celsius. At 40 °C, \( h_{fg} \approx 2{,}408 \) kJ/kg.

Lewis Factor

The Lewis factor (or Lewis number relation) relates the heat and mass transfer coefficients:

\[ Le_f = \frac{h_c}{h_D \cdot c_{pm}} \]

where:

  • \( h_c \) = convective heat transfer coefficient (W/m²·K)
  • \( h_D \) = mass transfer coefficient (kg/m²·s)
  • \( c_{pm} \) = specific heat of moist air (kJ/kg·K)

Merkel vs Poppe

The Merkel method assumes \( Le_f = 1 \) (Lewis analogy), which simplifies the governing equations by combining sensible and latent heat transfer into a single enthalpy driving force. The Poppe method accounts for \( Le_f \neq 1 \), providing more accurate results especially when the air approaches saturation.

The Bosnjakovic formula (Bosnjakovic, 1965) is commonly used:

\[ Le_f = 0.865^{0.667} \times \frac{\left(\frac{w_s + 0.622}{w + 0.622}\right) - 1}{\ln\left(\frac{w_s + 0.622}{w + 0.622}\right)} \]

Water Specific Heat

The specific heat of liquid water varies with temperature. The implementation uses a temperature-dependent correlation:

\[ c_{pw}(T) = 4.2174 - 3.7 \times 10^{-3}\,(T_C - 25) + 1.4 \times 10^{-4}\,(T_C - 25)^2 \quad \text{[kJ/(kg·K)]} \]

At the commonly used reference condition of 15.6 °C (60 °F): \( c_{pw} = 4.186 \) kJ/(kg·K).

Impact on Merkel Number

Using a constant \( c_{pw} = 4.186 \) kJ/(kg·K) instead of a temperature-dependent value introduces errors of approximately 0.5-1.5% in the calculated KaV/L, which is within engineering accuracy for most applications.

References

  1. ASHRAE. ASHRAE Handbook — Fundamentals, Chapter 6: Psychrometrics. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA, 2017. https://www.ashrae.org/

  2. Bosnjakovic, F. Technische Thermodynamik. Theodor Steinkopff, Dresden, 1965.

  3. Kroger, D.G. Air-Cooled Heat Exchangers and Cooling Towers: Thermal-Flow Performance Evaluation and Design, Vol. II. PennWell Books, Tulsa, OK, 2004. ISBN: 978-0878148967.

  4. Singh, K. and Das, R. "An experimental and multi-objective optimization study of a forced draft cooling tower with different fills." Energy Conversion and Management, Vol. 111, pp. 417-430, 2016. https://doi.org/10.1016/j.enconman.2015.12.080