Thermodynamic Properties of Humid Air Mixtures

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humid-air mixtures enthalpy entropy

Core Idea

Enthalpy of humid air per unit mass of dry air: h = h_da + ω*h_g, where ω is humidity ratio (kg water / kg dry air) and h_g is saturated vapor enthalpy. Entropy calculations account for the low partial pressure of water vapor. Psychrometric processes like adiabatic saturation and evaporative cooling relate wet-bulb temperature to mixture state.

Explainer

From psychrometric analysis, you already know the key state variables: humidity ratio ω (kg water vapor per kg dry air), relative humidity φ = p_v / p_sat(T), and how to locate states on the psychrometric chart. Now you need to compute actual thermodynamic properties — enthalpy h and entropy s — so that you can apply the first and second laws to HVAC processes and calculate real energy requirements.

The enthalpy of humid air is expressed per unit mass of *dry air*, which is the natural bookkeeping unit because dry air mass is conserved in all typical psychrometric processes (humidification, dehumidification, mixing, sensible heating) while water vapor mass changes. The formula h = h_da + ω × h_g decomposes into two contributions: h_da ≈ c_p,da × T = 1.006T (kJ/kg·K) is the sensible enthalpy of the dry air fraction, and ω × h_g is the enthalpy carried by the water vapor. The vapor enthalpy h_g is taken from saturated steam tables at the mixture temperature — this is valid because at the low partial pressures of water vapor in air (well below 0.1 atm typically), the vapor behaves nearly ideally and its enthalpy depends on temperature alone, not on partial pressure. A convenient approximation: h_g ≈ 2501 + 1.86T (kJ/kg) where T is in °C, combining the latent heat of vaporization at 0°C with the sensible heating of the vapor.

Entropy of the mixture requires more care. From gas-mixture-thermodynamics and Dalton's law, each component's entropy is evaluated at its own partial pressure, not the total mixture pressure. The dry air entropy is s_da(T, p_da) and the vapor entropy is ω × s_v(T, p_v), where p_v = φ × p_sat(T). Because water vapor in air is at a partial pressure much lower than its saturation pressure, it has *higher* specific entropy than saturated steam at the same temperature — lower pressure always increases entropy at fixed temperature. This has a practical consequence: humidification by evaporation always increases mixture entropy, consistent with the second law.

The adiabatic saturation process connects these properties to the wet-bulb temperature. In an adiabatic saturator, unsaturated inlet air contacts a large water surface, evaporating water until the exiting air is saturated at the adiabatic saturation temperature T_as. With no heat exchange, an energy balance gives: h_inlet + (ω_s − ω_1) × h_f(T_as) = h_outlet, where ω_s is the saturation humidity ratio at T_as and h_f is the liquid water enthalpy at T_as. Substituting the enthalpy expressions and solving yields ω_1 as a function of T_1 and T_as. This is the working equation for determining the inlet state from wet-bulb and dry-bulb thermometer readings. For air-water mixtures specifically (not general gas-vapor pairs), the wet-bulb temperature is very nearly equal to the adiabatic saturation temperature, which is why psychrometric charts label those slanted lines as both wet-bulb temperature and adiabatic saturation temperature lines.

In HVAC system analysis, these enthalpy calculations let you quantify the energy cost of every process on the psychrometric chart. Heating along a constant ω line costs Δh_da = c_p,da ΔT per kg dry air. Humidification at constant T costs Δh = Δω × h_g. Mixing two airstreams requires a mass-weighted enthalpy balance to find the mixed-state point. Every arrow on the psychrometric chart corresponds to a first-law calculation using h = h_da + ω h_g, making the enthalpy formula the central computational tool for psychrometric engineering.

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Prerequisite Chain

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIterated Integrals and Fubini's TheoremDouble Integrals in Cartesian CoordinatesDouble Integrals over Rectangular RegionsDouble Integrals in Polar CoordinatesDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsCenter of MassConservation of Linear MomentumElastic CollisionsInelastic CollisionsCoefficient of RestitutionCollision Analysis and Real-World ApplicationsTwo-Body Collisions in the Center-of-Mass FrameReduced Mass and Two-Body ProblemsKinematics in Two DimensionsProjectile MotionCircular Motion: KinematicsRotational KinematicsTorqueMoment of InertiaRotational Kinetic EnergyThe Work-Energy TheoremConservation of Mechanical EnergyFirst Law of ThermodynamicsThermodynamic Processes and the PV DiagramIsobaric and Isochoric ProcessesHeat EnginesThermal Efficiency of Heat EnginesRefrigerators and Heat PumpsSecond Law of ThermodynamicsEntropyMicrostates and MacrostatesEnsemble Theory FundamentalsCanonical Ensemble (NVT)Partition Function: Definition and PropertiesThe Canonical Partition Function and Thermodynamic DerivationFree Energy and Thermodynamic Relations from Partition FunctionsPhase Transitions and Equilibrium Phase DiagramsSpontaneous Symmetry BreakingOrder Parameters and Phase TransitionsMean Field Theory and Self-ConsistencyVan der Waals Equation from Statistical MechanicsCritical Point and Supercritical Fluid BehaviorReal Gas Thermodynamics and Equations of StateCompressibility Factor and Generalized CorrelationsIdeal and Real Gas BehaviorGas Mixture Thermodynamics and Dalton's LawPsychrometric Analysis and Humid Air PropertiesThermodynamic Properties of Humid Air Mixtures

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