Psychrometrics

Concept

Psychrometrics — Thermodynamics of Moist Air (First Principles + Engineering Correlations)

Scope: rigorous derivation and engineering implementation of moist-air thermodynamics. Includes both first-principles molecular reasoning and practical ASHRAE-style correlations. Covers ideal and non-ideal moist air, humidity and enthalpy relations, psychrometric chart derivation, adiabatic saturation, condensation, and evaporative processes.

1. Definition of Moist Air

Moist air is a binary mixture of dry air (a mixture of N₂, O₂, Ar, CO₂) and water vapor, often approximated as a mixture of two perfect gases.

At total pressure $P$ : $P = P_a + P_v$ where $P_a$ is the partial pressure of dry air, $P_v$ that of water vapor.

Mass basis:

Mass of dry air: $m_a$
Mass of water vapor: $m_v$
Humidity ratio (specific humidity): $w = \frac{m_v}{m_a} = 0.622 \frac{P_v}{P - P_v}$

This follows from the ideal gas law and molecular-weight ratio $M_v/M_a = 18.016/28.97 = 0.622$ .

2. Thermodynamic Foundation (Chemical Potential Equality)

At equilibrium between vapor and liquid water: $\mu_v(T,P_v) = \mu_l(T,P)$

Using differential $d\mu = -s\,dT + v\,dP$ : integrate at constant T to yield the Clausius–Clapeyron relation for saturation vapor pressure: $\frac{dP_{sat}}{dT} = \frac{h_{fg}}{T (v_g - v_f)}.$

Assuming $v_g \gg v_f$ and vapor ideal: $\frac{d\ln P_{sat}}{dT} = \frac{h_{fg}}{R_v T^2}.$ Integration gives the exponential form used in Antoine and related equations.

3. Saturation Pressure Correlations

3.1 Antoine Equation (Empirical)

$\log_{10} P_{sat} = A - \frac{B}{T + C}$ Typical constants (water, 1–100 °C): $A = 8.07131, B = 1730.63, C = 233.426$ .

3.2 Goff–Gratch (Internationally accepted for precise work):

$\log_{10}\left(\frac{P_{sat}}{P_c}\right) = -7.90298\left(\frac{T_c}{T}-1\right)+5.02808\log_{10}\left(\frac{T_c}{T}\right) -1.3816\times10^{-7}(10^{11.344(1-T/T_c)}-1)+8.1328\times10^{-3}(10^{-3.49149(T_c/T-1)}-1).$

3.3 Hyland–Wexler Formulation (ASHRAE standard)

Used in high-accuracy psychrometrics; polynomial-log fit for $P_{sat}(T)$ up to 200 °C.

4. Moist-Air Thermodynamic Properties

4.1 Specific Humidity and Relative Humidity

$w = 0.622 \frac{P_v}{P - P_v}, \qquad \phi = \frac{P_v}{P_{sat}(T)}.$

Given $T, P, \phi$ : $P_v = \phi P_{sat}(T)$ , hence w follows directly.

4.2 Dew-Point Temperature

Dew point $T_{dp}$ satisfies $P_v = P_{sat}(T_{dp})$ . Numerically, invert $P_{sat}(T)$ correlation.

Approximate closed form (Magnus–Tetens): $T_{dp} = \frac{b\,\gamma(T,\phi)}{a - \gamma(T,\phi)}, \qquad \gamma(T,\phi) = \frac{a T}{b+T} + \ln \phi, \; a=17.27, b=237.7~°C.$

4.3 Enthalpy of Moist Air

Neglecting non-idealities: $h = h_a + w h_v = c_{p,a} T + w (h_{g}(T) + c_{p,v} T) \approx (1.005 T + w (2501 + 1.88 T))~[\text{kJ/kg dry air}].$ Reference: liquid water at 0 °C has h = 0.

4.4 Entropy of Moist Air

Using mixture formulation: $s = s_a + w s_v = c_{p,a}\ln\frac{T}{T_0} - R_a \ln\frac{P_a}{P_{a,0}} + w\left[c_{p,v}\ln\frac{T}{T_0} - R_v\ln\frac{P_v}{P_{v,0}} + s_{fg,0}\right].$

4.5 Density of Moist Air

$\rho = \frac{P_a}{R_a T} + \frac{P_v}{R_v T} = \frac{P}{R_a T (1 + 1.607 w)}.$ This expression shows vapor makes moist air less dense.

5. Adiabatic Saturation and Wet-Bulb Temperature

Adiabatic saturation: steady-flow process where unsaturated air contacts liquid water adiabatically. $h_1 = h_2, \qquad w_2 = w_{sat}(T_2).$

Define wet-bulb temperature $T_w$ satisfying equality of enthalpy between given state and saturated state at $T_w$ : $h(T, w) = h(T_w, w_{sat}(T_w)).$ Numerical iteration gives $T_w$ . For engineering use, Stull (2011) correlation: $T_w = T \arctan(0.151977(\phi + 8.313659)^{1/2}) + \arctan(T + \phi) - \arctan(\phi - 1.676331) + 0.00391838 \phi^{3/2}\arctan(0.023101\phi) - 4.686035.$

6. Non-Ideal and High-Pressure Corrections

At elevated pressures (>200 kPa) or near condensation:

Replace ideal gas with virial correction: $Z = 1 + B(T)/v + \ldots$
Correct humidity ratio: $w = 0.622 \frac{\phi P_{sat}(T)\exp(V_l(P-P_{sat})/(R_v T))}{P - \phi P_{sat}(T)\exp(V_l(P-P_{sat})/(R_v T))}.$
Use fugacity coefficient from real-gas EOS for accurate condensation prediction.

7. Psychrometric Chart Equations

7.1 Axes and Primary Relations

Axes: dry-bulb temperature T (x-axis) vs. humidity ratio w (y-axis). For each T: $P_v = \phi P_{sat}(T), \quad w = 0.622 P_v/(P-P_v).$

7.2 Constant Enthalpy (Adiabatic Mixing) Lines

From $h = 1.005 T + w(2501 + 1.88 T)$ , constant h ⇒ linear relation: $w = \frac{h - 1.005 T}{2501 + 1.88 T}.$ Lines slope downward slightly with T.

7.3 Constant Relative Humidity Lines

For given $\phi$ , use saturation correlation to compute $w(T,\phi)$ . Plot by looping T and computing $w=0.622\phi P_{sat}(T)/(P-\phi P_{sat}(T))$ .

7.4 Constant Wet-Bulb Lines

From enthalpy constancy between $(T, w)$ and $(T_w, w_{sat}(T_w))$ , solve for $w(T, T_w)$ . These lines approach the saturation curve as $\phi \to 1$ .

7.5 Constant Specific Volume

Using $v = RT(1+1.607 w)/P$ . Constant-v lines are slightly curved on the chart.

8. Mixing, Heating, and Cooling Processes

8.1 Adiabatic Mixing of Two Air Streams

Mass and energy balance per kg dry air: $w_3 = \frac{\dot m_1 w_1 + \dot m_2 w_2}{\dot m_1 + \dot m_2}, \qquad h_3 = \frac{\dot m_1 h_1 + \dot m_2 h_2}{\dot m_1 + \dot m_2}.$ Intersection of lines on psychrometric chart gives state 3.

8.2 Sensible Heating/Cooling

Change T, constant w: $\Delta h = c_{p,ma} \Delta T, \; c_{p,ma} = 1.005 + 1.88 w.$

8.3 Evaporative Cooling (Adiabatic Humidification)

Adiabatic → constant h; water evaporates, T decreases, w increases until saturation or device limit.

8.4 Dehumidification and Cooling Below Dew Point

If $T < T_{dp}$ , condensation occurs. Latent heat removal $= h_{fg}\,\Delta w$ .

9. Condensation and Supersaturation (Thermodynamic Description)

At onset of condensation: $\mu_v = \mu_l$ . Supersaturated vapor has $\mu_v > \mu_l$ ; condensation rate $\propto \exp(-\Delta G^*/kT)$ , where $\Delta G^*$ is nucleation barrier depending on surface tension and supersaturation ratio $S = P_v/P_{sat}$ .

Kelvin equation (curvature effect): $\ln \frac{P_v}{P_{sat}} = \frac{2\sigma v_l}{R_v T r}.$ Small droplets require higher vapor pressure to be stable.

10. Computational Implementation (Engineering Use)

10.1 Input Variables

Given any two independent variables among { $T, \phi, w, h, T_w, T_{dp}$ }, the remaining can be computed using iterative evaluation of:

$P_{sat}(T)$ via correlation.
$P_v = \phi P_{sat}(T)$ .
$w = 0.622 P_v/(P - P_v)$ .
$h = 1.005 T + w (2501 + 1.88 T)$ .

10.2 Example Calculation

At T = 30 °C, $\phi = 0.6$ , P = 101.325 kPa:

$P_{sat}=4.247~kPa$
$P_v=2.548~kPa$
$w=0.622 \times 2.548/(101.325−2.548)=0.0159~kg/kg$
$h=1.005 \times 30+0.0159(2501+1.88 \times 30)=72.6~kJ/kg \text{ dry air}.$

11. Non-Equilibrium Moisture and Hygroscopic Materials

For porous solids, sorption isotherm relates equilibrium moisture content $X_{eq}$ to $\phi$ : $X_{eq} = X_s (a \phi)/(1 - (1 - a) \phi), \quad 0<a<1.$ Desorption/adsorption hysteresis arises from pore morphology. Moisture diffusion follows Fick’s law with $D(\phi,T)$ . Nonequilibrium surfaces: mass-transfer rate $\dot m = h_m A (\rho_{v,\infty} - \rho_{v,s})$ .

Coupling with energy balance gives simultaneous heat–mass transfer model used in drying and evaporative coolers.

12. Summary Equations

Property	Symbol	Equation (SI units)
Humidity ratio	w	$0.622 P_v/(P-P_v)$
Relative humidity	$\phi$	$P_v/P_{sat}(T)$
Enthalpy	h	$1.005 T + w(2501 + 1.88 T)$ kJ/kg dry air
Density	$\rho$	$P/[R_a T(1+1.607w)]$
Wet-bulb (enthalpy const.)	h(T,w) = h(T_w,w_{sat}(T_w))
Dew point	$P_{sat}(T_{dp})=P_v$
Adiabatic mixing	mass and energy balances for h and w
Saturation correlation	Antoine/Goff–Gratch/Hyland–Wexler

13. Cross-Links

See mixtures-phases.md for molecular mixture foundations.
See Fluid_Dynamics/07_Convective_Mass_Transfer.md for detailed derivations of evaporation and diffusion.
See HVAC_Applications/00_Psychrometric_Calculations.jl for code-level implementation in Julia.