Thermodynamic Relations, Mixtures, and Psychrometrics

Introduction

Thermodynamic relations provide a compact framework for connecting measurable properties such as pressure, temperature, volume, and enthalpy with energy functions that are not always directly measurable. They are especially valuable in heat-transfer analysis because thermal systems frequently involve phase change, gas mixtures, humid air, and property variations.

Mixture thermodynamics extends the analysis from pure substances to multicomponent systems. Psychrometrics is the specialized study of mixtures of dry air and water vapour, central to air-conditioning, drying, ventilation, cooling-tower design, and indoor environmental control. At postgraduate level, these topics should be treated not merely as collections of formulas, but as consequences of the first and second laws, equations of state, chemical-potential concepts, and carefully stated assumptions.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Derive and apply the fundamental thermodynamic relations and Maxwell relations.
2. Evaluate property changes using measurable variables such as pressure and temperature.
3. Explain fugacity, activity, partial properties, and chemical potential in mixtures.
4. Distinguish ideal-gas mixtures, real-gas mixtures, and ideal solutions.
5. Calculate humidity ratio, relative humidity, dew-point temperature, enthalpy, and volume of moist air.
6. Analyse mixing, heating, cooling, humidification, dehumidification, and adiabatic saturation processes.
7. Recognize the assumptions, limitations, and measurement uncertainties in psychrometric calculations.

1. Fundamental Thermodynamic Relations

For a simple compressible, closed system of fixed composition, the combined first and second laws give

where U is internal energy, T absolute temperature, S entropy, p pressure, and V volume. This equation applies to a reversible differential change, or more generally as a property identity when the system is in equilibrium at neighbouring states. It is not a statement that every real process is reversible.

Introducing enthalpy, H = U + pV, gives

The Helmholtz and Gibbs free energies are defined as

Their differentials are

The symbols A and G denote Helmholtz and Gibbs free energy, respectively. These relations assume a closed, fixed-composition, simple compressible system. For variable composition, additional terms involving chemical potential must be included.

1.1 Maxwell Relations

Because thermodynamic potentials are state functions, their mixed second derivatives are equal when the relevant properties are continuous. From the differential forms above:

These are Maxwell relations. They replace difficult-to-measure entropy derivatives with pressure–volume–temperature derivatives.

For example, using the first relation in the total differential of entropy,

Similarly,

where C_V = (∂U/∂T)_V and C_p = (∂H/∂T)_p. The units of C_p and C_V are J·kg⁻¹·K⁻¹ for specific properties or J·mol⁻¹·K⁻¹ for molar properties.

1.2 General Property Relations

For a substance described by an equation of state V = V(T,p), the enthalpy differential can be written as

The internal-energy differential is

These equations are exact for a simple compressible substance. They show that enthalpy is not generally a function of temperature alone, except under special conditions.

For an ideal gas, pV = nR_uT, or on a mass basis pv = RT, where n is amount in moles, R_u is the universal gas constant, v is specific volume, and R is the gas-specific constant. Since (∂V/∂T)_p = V/T,

and similarly dU = C_V dT. Thus, ideal-gas enthalpy and internal energy depend only on temperature, although C_p and C_V may vary with temperature.

For a real gas, the Joule–Thomson coefficient is

A positive μ_JT means that throttling causes cooling, while a negative value means heating. The coefficient is important in gas liquefaction and refrigeration. At an inversion condition, μ_JT = 0.

1.3 Clapeyron and Clausius–Clapeyron Equations

For two phases in equilibrium, equality of molar Gibbs free energy gives the Clapeyron equation:

Here p_sat is saturation pressure, Δh_phase is latent enthalpy per mole or unit mass, and Δv_phase is the corresponding change in specific or molar volume. This relation applies along a phase-equilibrium curve.

For liquid–vapour equilibrium when vapour volume is much larger than liquid volume and the vapour behaves ideally,

where h_fg is latent heat of vaporization per unit mass and R_v is the vapour gas constant. If h_fg is treated as constant, integration between states 1 and 2 gives

The constant-property approximation is useful over a limited temperature range; accurate psychrometric work should use a validated saturation-pressure correlation or steam tables.

2. Mixtures and Partial Properties

A mixture contains two or more chemical species. Extensive properties depend on the total amount of material, whereas intensive properties do not. For a mixture property M,

where nᵢ is the amount of species i and \bar{M}ᵢ is its partial molar property. The bar notation does not mean the property of pure species i; it means the change in total property caused by adding species i to the mixture at fixed temperature, pressure, and amounts of all other species:

Euler’s theorem for an extensive property gives the summation relation above. The Gibbs–Duhem equation imposes a consistency condition among partial properties:

At constant temperature and pressure,

2.1 Chemical Potential and Phase Equilibrium

The chemical potential of species i is its partial molar Gibbs energy:

At equilibrium between phases α and β, the chemical potential of every species is equal:

This is the rigorous criterion for phase equilibrium. Temperature and pressure equality are also required for ordinary thermal and mechanical equilibrium, but equality of chemical potentials determines species equilibrium.

For an ideal-gas mixture,

where yᵢ is mole fraction, p° is a standard pressure, and μᵢ° is the standard chemical potential. The partial pressure is pᵢ = yᵢp.

2.2 Ideal-Gas Mixtures

For a mixture of ideal gases,

and Dalton’s law gives pᵢ = yᵢp. Mole and mass fractions are

The mixture molar mass is

and the mixture gas constant is

The mixture equation of state is pV = nR_uT or pv = R_mixT.

For ideal gases, internal energy and enthalpy are additive:

The entropy includes a mixing contribution. On a molar basis,

The ideal entropy of mixing at fixed temperature and total pressure is

It is non-negative because 0 < yᵢ ≤ 1 and ln yᵢ ≤ 0. The corresponding Gibbs-energy change is negative, while the enthalpy of ideal mixing is zero.

2.3 Real Mixtures

Real mixtures depart from ideal behaviour because molecular interactions differ between like and unlike molecules. Fugacity replaces pressure in chemical-potential expressions:

The fugacity coefficient is φᵢ = fᵢ/pᵢ. For an ideal gas, φᵢ = 1. In liquid solutions, activity is commonly used:

where xᵢ is liquid mole fraction and γᵢ is the activity coefficient. Raoult’s law for an ideal liquid solution is

For nonideal solutions, a simple modified form is yᵢp = xᵢγᵢpᵢ^sat, subject to vapour-phase corrections when pressure is high. Activity coefficients must come from experimental data or a validated model such as NRTL, UNIQUAC, or Wilson; they should not be assumed to equal unity without justification.

3. Psychrometric Fundamentals

Psychrometrics concerns the thermodynamic properties of moist air, normally treated as a binary mixture of dry air and water vapour. In ordinary HVAC calculations, dry air and water vapour are approximated as ideal gases, while the liquid-water reference state and saturation properties are obtained from steam data.

Let p be total atmospheric pressure and p_v the water-vapour partial pressure. The dry-air partial pressure is p_da = p − p_v.

3.1 Humidity Measures

The humidity ratio, or specific humidity in many engineering texts, is the mass of water vapour per mass of dry air:

The factor 0.622 is approximately M_v/M_da, where M_v ≈ 18.015 kg·kmol⁻¹ and M_da ≈ 28.965 kg·kmol⁻¹. Humidity ratio has units kg water vapour per kg dry air.

Relative humidity is

where T_db is dry-bulb temperature and p_sat is saturation pressure at that temperature. Relative humidity is dimensionless and is usually reported as a percentage.

Combining these equations,

This equation is valid when the vapour behaves ideally and the saturation pressure is known accurately.

The dew-point temperature T_dp is defined by

It is the temperature at which condensation begins during cooling at essentially constant vapour partial pressure.

3.2 Enthalpy and Specific Volume of Moist Air

Using a common reference in which liquid water at 0 °C has zero enthalpy, the approximate enthalpy per kilogram of dry air is

where h is in kJ·kg⁻¹ dry air and T_db is in °C. The coefficients are approximate: 1.005 kJ·kg⁻¹·K⁻¹ is the specific heat of dry air, 1.88 kJ·kg⁻¹·K⁻¹ is the specific heat of water vapour, and 2500 kJ·kg⁻¹ approximates the latent enthalpy at 0 °C. For high-accuracy work, temperature-dependent property data should be used.

The specific volume per kilogram of dry air is approximately

where R_da = 0.287055 kPa·m³·kg⁻¹·K⁻¹, T_K is absolute temperature in kelvin, p is in kPa, and v is in m³·kg⁻¹ dry air. The 1.607 factor is approximately R_v/R_da − 1.

3.3 Wet-Bulb Temperature and Adiabatic Saturation

A wet-bulb thermometer measures a temperature influenced by evaporation from a wetted surface. Under idealized adiabatic saturation conditions, unsaturated air contacts liquid water until it approaches saturation. An energy balance per unit mass of dry air can be expressed as

where states 1 and 2 refer to inlet and outlet moist air, and h_f is liquid-water enthalpy. If the outlet is saturated and the apparatus is well insulated, T_2 approximates the thermodynamic wet-bulb temperature. Real wet-bulb measurements depend on air velocity, radiation, wick condition, and instrument design.

3.4 Psychrometric Processes

For steady-flow equipment, neglecting kinetic and potential-energy changes, the energy balance is

where ṁ is mass flow rate, Q̇ is heat transfer to the control volume, and Ẇ is shaft work. Moist-air balances must be made on a dry-air basis, with separate water-mass balances when condensation or injection occurs.

Sensible heating or cooling without moisture transfer: ω remains constant. Heating increases dry-bulb temperature and enthalpy; cooling decreases both until the dew point is reached.

Cooling and dehumidification: after reaching the apparatus dew point, further cooling causes condensation. The outlet humidity ratio decreases, and condensate rate is

Humidification by steam injection:

where ṁ_s and h_s are the steam mass flow rate and steam enthalpy.

Mixing of two moist-air streams: if no heat is exchanged with the surroundings,

and

with ṁ_da,3 = ṁ_da,1 + ṁ_da,2. These balances are more reliable than reading graphical intersections from a chart.

Figure: Conceptual psychrometric process path for cooling followed by dehumidification.

4. Worked Examples

Example 1: Humidity Ratio and Moist-Air Enthalpy

Air at 25 °C dry-bulb temperature and 60% relative humidity is at 101.325 kPa. Use p_sat(25 °C) = 3.169 kPa.

The vapour partial pressure is

p_v = 0.60 × 3.169 = 1.901 kPa.

Therefore,

ω = 0.622(1.901)/(101.325 − 1.901) = 0.01190 kg/kg dry air.

The approximate enthalpy is

h = 1.005(25) + 0.01190[2500 + 1.88(25)]

h ≈ 55.4 kJ/kg dry air.

The result is meaningful only with the stated atmospheric pressure and saturation-pressure value; changing altitude changes the humidity ratio even if temperature and relative humidity remain the same.

Example 2: Mixing Two Air Streams

Stream 1 has ṁ_da,1 = 2 kg/s, h₁ = 45 kJ/kg dry air, and ω₁ = 0.008. Stream 2 has ṁ_da,2 = 3 kg/s, h₂ = 70 kJ/kg dry air, and ω₂ = 0.014. The mixer is adiabatic with no moisture loss.

The outlet humidity ratio is

ω₃ = [2(0.008) + 3(0.014)]/(2 + 3) = 0.0116 kg/kg dry air.

The outlet enthalpy is

h₃ = [2(45) + 3(70)]/5 = 60 kJ/kg dry air.

Thus the mixed state must satisfy both h₃ = 60 kJ/kg dry air and ω₃ = 0.0116 kg/kg dry air. A psychrometric chart or property routine can then determine its temperature and relative humidity.

5. Illustrative Case: Data-Centre Cooling and Humidity Control

Background. A data centre requires supply air near 18 °C to protect electronic equipment and maintain acceptable human comfort in adjacent occupied areas. Outdoor air enters at 32 °C, 50% relative humidity, while return air is at 24 °C, 50% relative humidity.

Problem. The cooling coil must remove sensible heat and, during humid weather, latent heat. A design based only on dry-bulb temperature may underestimate condensate production and fan-coil capacity.

Analysis. For each stream, engineers calculate p_v = φp_sat(T), humidity ratio ω, enthalpy h, and dry-air mass flow. The mixed-air state is found from dry-air-weighted mass and energy balances. The coil outlet is checked against the saturation curve. If its apparatus dew point is below the required value, condensation occurs and the water-removal rate is ṁ_da(ω_in − ω_out). Coil capacity is approximately ṁ_da(h_in − h_out), with corrections for casing losses and fan heat.

Approach. A validated psychrometric software package is used, but the underlying balances are independently reproduced in a spreadsheet. Sensor calibration is checked because a 2% relative-humidity bias can materially alter latent-load estimates at high humidity. Control logic modulates chilled-water flow and, where necessary, reheating to avoid over-cooling occupied zones.

Outcome. The design identifies both refrigeration capacity and condensate-drain capacity. It also prevents operation below the dew point of insulation or structural surfaces, reducing mould and corrosion risk.

Learning points. Psychrometric design is a coupled heat-and-mass-transfer problem. Enthalpy alone is insufficient when moisture is added or removed; humidity-ratio balances are essential. Software should support, not replace, conservation-law checks and uncertainty analysis.

6. Limitations, Best Practices, and Contemporary Developments

Psychrometric charts are usually based on a specified barometric pressure, commonly standard atmospheric pressure. They are therefore approximate at altitude. Charts also have limited resolution, and interpolation can introduce errors. Accurate design should use pressure-corrected equations, reliable saturation correlations, and consistent reference states.

The ideal-gas approximation becomes less reliable at high pressure, near saturation at unusual temperatures, or when contaminants and noncondensable gases are significant. Refrigerant mixtures, combustion products, and humid gases containing chemicals may require equations of state, fugacity coefficients, or multicomponent mass-transfer models.

Best practice includes:

• use absolute temperature in thermodynamic equations;
• use kelvin in gas-law calculations and °C only where a correlation explicitly permits it;
• state whether mass flow is total moist-air flow or dry-air flow;
• use one consistent enthalpy reference state;
• check p_v ≤ p_sat and ω ≥ 0;
• verify energy and water balances independently;
• report pressure, uncertainty, and property-source information;
• calibrate temperature and humidity sensors and protect them from radiation and condensation.

Contemporary HVAC systems increasingly combine psychrometric models with building-energy simulation, model-predictive control, and machine-learning fault detection. AI can identify sensor drift, estimate unmeasured humidity, and optimize coil operation, but it cannot safely replace thermodynamic constraints. Training data may not cover extreme weather, sensor faults, or unusual occupancy. A physically constrained model, transparent validation, and human oversight are therefore required.

Chapter Summary

Thermodynamic potentials generate exact differential relations that connect energy, entropy, pressure, volume, and temperature. Maxwell relations enable entropy and other difficult properties to be evaluated from measurable derivatives. Clapeyron-type equations describe phase-equilibrium curves, while the Joule–Thomson coefficient characterizes throttling temperature changes.

Mixture thermodynamics introduces partial properties, chemical potential, Gibbs–Duhem consistency, fugacity, and activity. Ideal-gas mixtures obey Dalton’s law and have additive energy properties, but their entropy includes a positive entropy of mixing. Real mixtures require experimentally supported models for nonideal interactions.

Psychrometrics applies these principles to moist air. Humidity ratio, relative humidity, dew point, enthalpy, and specific volume provide the basis for analysing heating, cooling, humidification, dehumidification, and mixing. Correct calculations require a specified pressure, consistent property references, separate dry-air and water balances, and awareness of measurement and model limitations.

Key Terms

Thermodynamic potential: A state function formed by Legendre transformation, such as H, A, or G.

Maxwell relation: An equality of mixed derivatives obtained from a thermodynamic potential.

Chemical potential: Partial molar Gibbs free energy of a species.

Partial molar property: Change in a mixture property per infinitesimal addition of a species.

Gibbs–Duhem equation: Consistency relation among intensive properties and chemical potentials.

Fugacity: Effective pressure used to represent nonideal chemical-potential behaviour.

Activity: Effective concentration or composition measure in a nonideal phase.

Ideal-gas mixture: Mixture whose species and total behaviour follow ideal-gas relations.

Mole fraction: Amount of a species divided by total amount of mixture.

Mass fraction: Mass of a species divided by total mixture mass.

Humidity ratio: Mass of water vapour per mass of dry air.

Relative humidity: Ratio of vapour partial pressure to saturation pressure at the same temperature.

Dew point: Temperature at which water vapour first becomes saturated during cooling.

Dry-bulb temperature: Ordinary measured air temperature shielded from evaporative effects.

Wet-bulb temperature: Temperature indicated by a wetted thermometer under specified airflow conditions.

Psychrometrics: Study of thermodynamic properties and processes of moist air.

Sensible heat: Heat associated primarily with temperature change without phase change.

Latent heat: Energy associated with phase change at approximately constant temperature.

Apparatus dew point: Effective coil-surface temperature used to represent cooling and dehumidification.

Entropy of mixing: Entropy increase caused by mixing species, particularly in an ideal mixture.

Review Questions

Short Answer Questions

1. State the fundamental differential relations for U, H, A, and G.
2. What physical meaning is assigned to chemical potential?
3. Distinguish relative humidity from humidity ratio.
4. Why does the entropy of ideal-gas mixing increase?
5. State two assumptions commonly used in basic psychrometric analysis.

Long Answer Questions

1. Derive the Maxwell relations from the differentials of the four thermodynamic potentials.
2. Explain partial molar properties, the Gibbs–Duhem equation, and their role in mixture analysis.
3. Derive the humidity-ratio relation from the ideal-gas law and molecular masses.
4. Discuss ideal and real mixtures using chemical potential, fugacity, and activity.
5. Explain cooling and dehumidification in an air-conditioning coil using simultaneous energy and water balances.

Analytical / Application Questions

1. Air at 30 °C, 70% relative humidity, and 95 kPa is cooled at constant humidity ratio until condensation begins. Determine the initial humidity ratio and identify the equation required to find the dew-point temperature.
2. Two moist-air streams are mixed adiabatically. Explain why both enthalpy and humidity-ratio balances are required, and formulate the two outlet equations.
3. A humidity sensor in a high-altitude HVAC plant reports values inconsistent with the measured dew point. Diagnose possible thermodynamic, instrumentation, and data-processing causes, and propose a validation procedure.

Multiple Choice Questions with Answers

Question 1

Which differential is the fundamental relation for enthalpy in a simple compressible closed system of fixed composition?

A. dH = T dS + V dp

B. dH = −S dT − p dV

C. dH = C_V dT + p dV

D. dH = V dT + S dp

Correct Answer: A. dH = T dS + V dp

Question 2

For an ideal-gas mixture, the partial pressure of species i is

A. p_i = x_i/p

B. p_i = y_i p

C. p_i = p/y_i

D. p_i = y_i + p

Correct Answer: B. p_i = y_i p

Question 3

At phase equilibrium, the chemical-potential criterion for species i is

A. μ_i^α = 0

B. μ_i^α = μ_i^β

C. μ_i^α > μ_i^β

D. μ_i^α + μ_i^β = 1

Correct Answer: B. μ_i^α = μ_i^β

Question 4

At 25 °C and 101.325 kPa, air has 60% relative humidity and p_sat = 3.169 kPa. Its humidity ratio is approximately

A. 0.00119 kg/kg dry air

B. 0.00622 kg/kg dry air

C. 0.01190 kg/kg dry air

D. 0.1190 kg/kg dry air

Correct Answer: C. 0.01190 kg/kg dry air

Question 5

The dew-point temperature is defined by the condition

A. The dry-bulb temperature is zero

B. The vapour partial pressure equals saturation pressure

C. Relative humidity equals zero

D. The total pressure equals saturation pressure

Correct Answer: B. The vapour partial pressure equals saturation pressure

Question 6

For ideal gases, the enthalpy of a mixture is calculated by

A. Subtracting all component enthalpies

B. Using only the highest component temperature

C. A mass- or mole-weighted sum of component enthalpies on a consistent basis

D. Setting mixture enthalpy equal to zero

Correct Answer: C. A mass- or mole-weighted sum of component enthalpies on a consistent basis

Question 7

During sensible heating of moist air with no moisture transfer, which property remains constant?

A. Humidity ratio

B. Relative humidity

C. Dew-point pressure always increases

D. Water-vapour mass flow always increases

Correct Answer: A. Humidity ratio

Question 8

For adiabatic mixing of two moist-air streams, which pair of balances is essential?

A. Momentum and electrical charge only

B. Dry-air mass and water mass, or equivalently dry-air-weighted enthalpy and humidity-ratio balances

C. Entropy and volume only

D. Relative humidity and dry-bulb temperature only

Correct Answer: B. Dry-air mass and water mass, or equivalently dry-air-weighted enthalpy and humidity-ratio balances

Question 9

A positive Joule–Thomson coefficient indicates that throttling at the specified state generally causes

A. A temperature decrease

B. A temperature increase

C. No pressure change

D. Condensation in every gas

Correct Answer: A. A temperature decrease

Question 10

The entropy of mixing for an ideal mixture at fixed temperature and pressure is

A. Δs_mix = R_u∑ y_i ln y_i

B. Δs_mix = −R_u∑ y_i ln y_i

C. Always negative

D. Always zero

Correct Answer: B. Δs_mix = −R_u∑ y_i ln y_i