Gas Mixtures and Psychrometrics

Introduction and Context

Many engineering systems handle mixtures rather than pure substances. Combustion gases, refrigerant blends, natural gas, atmospheric air, and humid air are familiar examples. The thermodynamic analysis of a mixture requires both a description of its composition and methods for determining its pressure, volume, temperature, energy, and phase-equilibrium properties.

Psychrometrics is the study of mixtures of dry air and water vapour. It is central to heating, ventilation, and air-conditioning (HVAC), drying, cooling towers, refrigeration, combustion-air preparation, and indoor environmental control. Unlike an ordinary ideal-gas mixture, moist air may contain water vapour that condenses or evaporates. Consequently, psychrometric processes involve both sensible and latent energy transfer.

This chapter develops the Dalton and Amagat models for gas mixtures, introduces the principal properties of moist air, and applies mass and energy balances to common HVAC processes.

Learning Objectives

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

1. Distinguish mole fraction, mass fraction, partial pressure, and partial volume.
2. Apply Dalton’s law and Amagat’s law to ideal-gas mixtures.
3. Calculate mixture molar mass, gas constant, density, and specific heats.
4. Define humidity ratio, relative humidity, degree of saturation, dew-point temperature, and enthalpy of moist air.
5. Analyze sensible heating and cooling, humidification, dehumidification, mixing, and adiabatic saturation.
6. Use psychrometric charts or equations while recognizing their assumptions and limitations.
7. Evaluate an HVAC application using mass and energy balances.

1. Composition of Gas Mixtures

Consider a mixture containing components indexed by i. If nᵢ is the amount of component i and n is the total amount, the mole fraction is

The mole fractions satisfy ∑xᵢ = 1. For a mixture of ideal gases, mole fraction is directly related to pressure; for real gases this relationship is approximate and requires a suitable equation of state.

The mass fraction is

where mᵢ is the mass of component i, m is total mass, and Mᵢ is the molar mass. The mass fractions satisfy ∑yᵢ = 1.

The average molar mass of a mixture is

The mixture-specific gas constant is

where Rᵤ = 8.314 kJ/(kmol·K) when M is expressed in kg/kmol. Thus R_mix has units kJ/(kg·K), equivalent to kPa·m³/(kg·K).

An alternative useful relation is

These equations apply to any composition, although the ideal-gas pressure and volume relations introduced below require ideal-gas behavior.

Worked Example 1: Composition and Mixture Molar Mass

A gas contains 0.70 kmol nitrogen, 0.20 kmol oxygen, and 0.10 kmol carbon dioxide. Determine mole fractions, mass fractions, and average molar mass. Use M_N₂ = 28.013 kg/kmol, M_O₂ = 31.999 kg/kmol, and M_CO₂ = 44.01 kg/kmol.

The total amount is 1.00 kmol, so the mole fractions are 0.70, 0.20, and 0.10. The component masses are

m_N₂ = 0.70 × 28.013 = 19.609 kg

m_O₂ = 0.20 × 31.999 = 6.400 kg

m_CO₂ = 0.10 × 44.01 = 4.401 kg

The total mass is 30.410 kg, giving

y_N₂ = 0.645, y_O₂ = 0.210, y_CO₂ = 0.145

The average molar mass is

M_mix = 30.410 kg/kmol per kmol = 30.410 kg/kmol.

The mass fractions sum to unity within rounding error.

2. Pressure and Volume Models

Dalton’s Law of Partial Pressures

For a mixture of nonreacting ideal gases occupying a common volume V at temperature T, each component behaves as though it alone occupied V at T. The partial pressure of component i is

The total pressure is the sum of partial pressures:

Since pᵢ/p = nᵢ/n,

This is Dalton’s law. It is most accurate at low and moderate pressures, where intermolecular effects are small. For real mixtures, fugacity or an appropriate mixture equation of state may be required.

Amagat’s Law of Additive Volumes

At a specified mixture pressure p and temperature T, the partial volume Vᵢ is the volume that component i would occupy alone at p and T. For an ideal-gas mixture,

and

Therefore,

Dalton’s and Amagat’s laws are equivalent for ideal-gas mixtures but represent different conceptual paths: Dalton uses additive pressures at a common volume, whereas Amagat uses additive volumes at a common pressure.

Density and Specific Properties

The ideal-gas equation for a mixture is

Thus density is

For an ideal-gas mixture with negligible chemical reaction, an extensive property is obtained by summing component contributions. On a mass basis,

If each component has approximately constant specific heat,

and, for an ideal gas,

Specific heats may vary significantly with temperature, particularly for combustion products and high-temperature systems. In such cases, enthalpy tables or temperature-dependent specific-heat correlations should be used.

3. Moist Air and Psychrometric Properties

Moist air is treated as a binary mixture of dry air and water vapour. Dry air is itself a mixture, but it is normally treated as one component with molar mass approximately 28.97 kg/kmol. The total pressure is

where pₐ is the partial pressure of dry air and pᵥ is the partial pressure of water vapour.

Humidity Ratio

The humidity ratio, or specific humidity in HVAC practice, is the mass of water vapour per unit mass of dry air:

The value 0.622 is the ratio M_v/M_a, using M_v ≈ 18.015 kg/kmol and M_a ≈ 28.97 kg/kmol. The equation assumes ideal-gas behavior and uses consistent pressure units.

For unsaturated moist air, pᵥ is below the saturation pressure p_g(T). At saturation,

Relative Humidity

Relative humidity is

where p_g(T) is the saturation pressure of water at the mixture temperature. Relative humidity is commonly reported as a percentage, 100φ%. It is not the same as humidity ratio: relative humidity depends strongly on temperature, while humidity ratio is a mass ratio.

Degree of Saturation

The degree of saturation is

At ordinary atmospheric pressures, μ is numerically close to relative humidity but is not identical because the denominators in the two definitions differ.

Dew-Point Temperature

The dew point T_dp is the temperature at which water vapour in a moist-air sample becomes saturated during cooling at essentially constant pressure and constant humidity ratio. It satisfies

If a surface is below the dew point, condensation occurs on that surface. This explains moisture on cold pipes, windows, and evaporator coils.

Enthalpy of Moist Air

Using a reference temperature of 0 °C and neglecting the small effect of liquid-water reference choices, the enthalpy per kilogram of dry air is commonly approximated by

where h is in kJ/kg dry air, T is in °C, and ω is in kg water/kg dry air. The first term represents sensible enthalpy of dry air; the second represents water-vapour enthalpy, including latent energy.

The formula is suitable for ordinary HVAC temperatures and moderate pressures. At high temperatures, high pressures, or when high precision is required, property tables and more rigorous water-vapour models should be used.

The specific volume per kilogram of dry air is approximated by

where v is in m³/kg dry air, Rₐ = 0.287055 kPa·m³/(kg·K), T_K is absolute temperature in kelvin, and p is in kPa.

Figure: Schematic path for cooling moist air first to its dew point and then below the dew point with condensation.

4. Psychrometric Processes

Sensible Heating

In sensible heating, dry-bulb temperature increases while humidity ratio remains constant, provided no moisture is added or removed. Relative humidity decreases because saturation pressure increases with temperature. For a steady-flow heater with negligible kinetic and potential energy changes,

where q is heat transfer per kilogram of dry air. If no moisture is added, m_da,1 = m_da,2 and ω₂ = ω₁.

Sensible Cooling Above the Dew Point

Cooling above the dew point also leaves ω constant. Relative humidity increases as temperature falls. The process continues until the air reaches its dew point.

Cooling and Dehumidification

Below the dew point, water condenses. The outlet humidity ratio is lower than the inlet value. The condensate rate is

The cooling load, neglecting kinetic and potential energy changes, is

if condensate leaves the control volume as liquid. In many HVAC calculations the condensate enthalpy term is small relative to the air-stream enthalpy difference, but omitting it should be recognized as an approximation. A coil energy balance can also be written directly using inlet and outlet moist-air enthalpies and the liquid-water exit enthalpy.

Humidification

Humidification increases ω. In steam humidification, the injected steam adds both mass and enthalpy. In evaporative humidification, liquid water evaporates into the air and may cool the air as latent energy is supplied by sensible energy from the air.

For a steady humidifier with no liquid water leaving and injected steam at enthalpy h_s,

where ṁ_w = ṁ_da(ω₂ − ω₁). For an adiabatic device, Q̇ = 0. The exact process path depends on the equipment and water condition; therefore, “humidification” alone does not imply constant dry-bulb temperature or constant enthalpy.

Mixing of Two Moist-Air Streams

For adiabatic mixing of two streams at the same total pressure, conservation of dry air, water vapour, and energy gives

These equations are more reliable than attempting to estimate the mixed state by visual chart interpolation alone.

5. Worked Example 2: Humidity, Dew Point, and Enthalpy

Moist air at 25 °C and 101.325 kPa has relative humidity φ = 60%. At 25 °C, take the saturation pressure of water as p_g = 3.169 kPa. Determine pᵥ, humidity ratio, and moist-air enthalpy.

The vapour partial pressure is

p_v = φp_g = 0.60 × 3.169 = 1.901 kPa.

The humidity ratio is

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

Using the enthalpy approximation,

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

h = 25.125 + 0.01190(2547) ≈ 55.4 kJ/kg dry air.

The dew point is the temperature at which saturation pressure equals 1.901 kPa. From steam tables, this is approximately 16.7 °C. Any surface below approximately 16.7 °C can cause condensation under these conditions.

6. Worked Example 3: Cooling and Dehumidification Coil

An air-conditioning coil processes 2.0 kg/s of dry air. The entering air is at 30 °C and 50% relative humidity; the leaving air is at 14 °C and 90% relative humidity. The total pressure is 101.325 kPa. Use p_g(30 °C) = 4.247 kPa and p_g(14 °C) = 1.599 kPa.

Entering humidity ratio:

p_v1 = 0.50(4.247) = 2.124 kPa

ω₁ = 0.622(2.124)/(101.325 − 2.124) = 0.01345 kg/kg dry air.

Leaving humidity ratio:

p_v2 = 0.90(1.599) = 1.439 kPa

ω₂ = 0.622(1.439)/(101.325 − 1.439) = 0.00896 kg/kg dry air.

Condensate production is

ṁ_cond = 2.0(0.01345 − 0.00896) = 0.00898 kg/s.

The entering enthalpy is

h₁ = 1.005(30) + 0.01345[2500 + 1.88(30)] ≈ 64.4 kJ/kg dry air.

The leaving enthalpy is

h₂ = 1.005(14) + 0.00896[2500 + 1.88(14)] ≈ 36.7 kJ/kg dry air.

Neglecting the relatively small condensate enthalpy correction, the cooling capacity is

Q̇ ≈ 2.0(64.4 − 36.7) = 55.4 kW.

The condensate rate is approximately 32.3 kg/h. In a detailed coil calculation, the liquid condensate temperature and the enthalpy of condensate leaving the coil should be included.

7. Psychrometric Charts and Measurement

A psychrometric chart is normally constructed for a fixed total pressure, commonly standard atmospheric pressure. Its principal coordinates are dry-bulb temperature and humidity ratio. Curves or families of curves represent relative humidity, wet-bulb temperature, dew point, specific volume, and enthalpy.

Charts are valuable for rapidly visualizing process direction:

• horizontal movement commonly represents constant humidity ratio;
• movement upward represents humidification;
• movement downward represents dehumidification;
• movement toward higher temperature represents heating;
• movement toward lower temperature represents cooling.

Because charts are graphical approximations, numerical balances should be used for final equipment sizing. Pressure changes require a chart prepared for the new pressure or direct equations.

Wet-bulb temperature is measured using a wetted sensor exposed to moving air. Evaporation cools the sensor, and the reading depends on air velocity, pressure, and moisture content. A sling psychrometer estimates relative humidity from dry-bulb and wet-bulb temperatures, but modern HVAC practice increasingly uses calibrated electronic temperature and humidity sensors.

Sensor errors can be consequential. A small temperature error near saturation may produce a large error in calculated relative humidity or dew point. Instruments should be calibrated, protected from radiant heat, and installed where the sample represents the actual air stream.

8. Applied Case: Humidity Control in a University Computer Laboratory

Background

This illustrative case concerns a computer laboratory requiring a comfortable indoor condition of 22 °C and 50% relative humidity. Outdoor summer air enters at 32 °C and 60% relative humidity. The HVAC unit mixes outdoor air with return air, cools the mixture over a coil, and reheats it slightly before delivery.

Problem

The design team must prevent indoor condensation, maintain acceptable comfort, and avoid excessive energy use. Outdoor air introduces both sensible heat and moisture. If the coil only cools sensibly, indoor humidity will rise.

Analysis

The outdoor humidity ratio is calculated from

At 32 °C, p_g is approximately 4.76 kPa, giving pᵥ,o ≈ 2.86 kPa and ω_o ≈ 0.0184 kg/kg dry air. The indoor design humidity ratio at 22 °C, where p_g ≈ 2.64 kPa, is approximately

ω_i = 0.622(0.50)(2.64)/(101.325 − 0.50×2.64) ≈ 0.0082 kg/kg dry air.

Thus outdoor air contains substantially more moisture than the desired indoor condition. A cooling coil must reduce the humidity ratio, requiring a coil surface temperature below the incoming air dew point. Reheat may then be needed to achieve the desired dry-bulb temperature without raising humidity.

Approach and Solution

The engineering sequence is:

1. Determine outdoor and return-air states from measured or design data.
2. Apply dry-air, water-vapour, and energy balances to obtain the mixed-air state.
3. Select a coil apparatus dew-point condition low enough to remove the required moisture.
4. Calculate coil cooling capacity and condensate rate.
5. Evaluate whether reheat is necessary and compare reheat sources, such as heat recovery or condenser heat.
6. Verify that supply-air temperature, humidity, filtration, and ventilation meet applicable building and indoor-air-quality requirements.

Outcome

The resulting system controls both sensible and latent loads. A humidity sensor can modulate outdoor-air quantity or coil capacity, but control logic should include low-temperature and condensation safeguards. Heat recovery can reduce the energy penalty of reheat and ventilation.

Learning Points

The case demonstrates that relative humidity alone is insufficient for equipment sizing. Humidity ratio and enthalpy are needed for mass and energy balances. It also shows why ventilation, comfort, condensation prevention, and energy efficiency must be considered together.

9. Limitations, Safety, and Contemporary Practice

The ideal-gas model is highly effective for ordinary atmospheric air, but errors increase at high pressure, near saturation, and in mixtures with strong molecular interactions. Gas composition may also change through combustion, chemical reaction, adsorption, or leakage. In such cases, composition and energy balances must account for reaction and possibly nonequilibrium behavior.

Psychrometric formulas generally assume uniform, well-mixed air; negligible changes in atmospheric pressure over the device; and equilibrium between vapour and liquid when condensation occurs. These assumptions may fail in ducts with stratification, rapidly changing transient systems, desiccant wheels, spray chambers, and high-speed flows.

Moisture control has health and building-safety implications. Excessive humidity can promote microbial growth and damage materials, while excessively dry air can cause discomfort and electrostatic discharge. HVAC designs should comply with applicable ventilation, refrigerant, energy, fire, and occupational standards. Condensate drainage must prevent standing water and biological growth.

Building-management systems increasingly use sensors, model-based control, and machine-learning fault detection. AI can identify fouled coils, sensor drift, abnormal condensate rates, and inefficient simultaneous heating and cooling. However, AI predictions should not replace physical balances, calibrated sensors, engineering judgment, or safety interlocks. Data governance is also important when occupancy sensors or environmental monitoring could reveal personal information.

Best-Practice Calculation Procedure

1. State the basis: total mass, dry-air mass, kmol, or time rate.
2. Identify pressure and temperature units and use absolute temperature where required.
3. Determine composition before calculating mixture properties.
4. For moist air, use dry-air mass as the basis for ω and h.
5. Check whether water is saturated and whether condensation is possible.
6. Apply separate balances for dry air and water vapour.
7. Apply the energy balance using consistent enthalpy references.
8. Check that mass fractions or mole fractions sum to one.
9. Check physical plausibility: 0 ≤ φ ≤ 1 for unsaturated air, ω ≥ 0, and condensate rate ≥ 0.
10. Use property tables or validated software when conditions exceed ordinary HVAC ranges.

Chapter Summary

Gas-mixture analysis begins with composition. Mole fractions describe amounts on a molar basis, mass fractions on a mass basis, and the average molar mass connects the two. For ideal-gas mixtures, Dalton’s law states that total pressure equals the sum of component partial pressures, while Amagat’s law states that total volume equals the sum of component partial volumes. These laws lead to mixture gas constants, density, enthalpy, and specific-heat relations.

Moist air is a mixture of dry air and water vapour. Its important properties include humidity ratio, relative humidity, degree of saturation, dew-point temperature, enthalpy, and specific volume. Sensible heating or cooling changes temperature at constant humidity ratio, whereas cooling below the dew point produces condensation. Humidification adds water vapour, and adiabatic mixing requires separate dry-air, water-vapour, and energy balances.

Psychrometric charts provide useful visualization, but equations and conservation laws are required for accurate design. HVAC engineers must evaluate both sensible and latent loads, verify equipment and sensor performance, and consider condensation, indoor-air quality, energy efficiency, and applicable safety standards.

Key Terms

Mole fraction: Ratio of the amount of a component to the total amount of mixture.

Mass fraction: Ratio of component mass to total mixture mass.

Partial pressure: Pressure contributed by one component of an ideal-gas mixture.

Dalton’s law: Total pressure equals the sum of component partial pressures.

Amagat’s law: Total volume of an ideal-gas mixture equals the sum of component partial volumes.

Average molar mass: Total mixture mass per unit amount of mixture.

Mixture gas constant: Gas constant based on the average molar mass of a mixture.

Moist air: Mixture of dry air and water vapour.

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

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

Degree of saturation: Ratio of actual humidity ratio to saturated humidity ratio.

Dew point: Temperature at which water vapour begins to condense during cooling.

Saturation pressure: Equilibrium vapour pressure of water at a specified temperature.

Dry-bulb temperature: Ordinary air temperature measured without evaporative cooling.

Wet-bulb temperature: Temperature measured by a wetted sensor exposed to moving air.

Sensible heat: Heat transfer associated primarily with temperature change.

Latent heat: Energy associated with phase change or moisture transfer.

Psychrometric chart: Graphical representation of moist-air properties at a specified pressure.

Cooling and dehumidification: Process that lowers air temperature and removes water vapour.

Apparatus dew point: Effective coil-surface temperature used to approximate cooling-coil behavior.

Review Questions

Short Answer Questions

1. Define mole fraction and mass fraction, and state how they differ.
2. State Dalton’s law for an ideal-gas mixture.
3. What is the physical meaning of humidity ratio?
4. Define dew-point temperature.
5. Why does relative humidity decrease during sensible heating?

Long Answer Questions

1. Derive the relationship between partial pressure and mole fraction for an ideal-gas mixture.
2. Explain Dalton’s and Amagat’s laws and compare their assumptions and applications.
3. Develop the principal psychrometric properties of moist air and explain their engineering significance.
4. Describe cooling and dehumidification using dry-air and water-vapour balances.
5. Explain how a psychrometric chart can be used to analyze heating, cooling, humidification, and mixing processes.

Analytical / Application Questions

1. Air at 25 °C and 101.325 kPa has 70% relative humidity. Determine its humidity ratio and dew point using appropriate saturation-pressure data.
2. Two moist-air streams mix adiabatically. Stream 1 is 1 kg/s of dry air with ω₁ = 0.008 and h₁ = 42 kJ/kg dry air; stream 2 is 0.5 kg/s of dry air with ω₂ = 0.016 and h₂ = 65 kJ/kg dry air. Determine the mixed humidity ratio and enthalpy.
3. An air-conditioning coil processes 3 kg/s of dry air, reducing humidity ratio from 0.014 to 0.009 kg/kg dry air. Determine the condensate production rate and explain why the coil surface must be below the entering-air dew point.

Multiple Choice Questions with Answers

Question 1

For an ideal-gas mixture, the partial pressure of component i is given by p_i = x_i p. What does x_i represent?

A. Mass fraction of component i

B. Mole fraction of component i

C. Volume occupied by component i in cubic metres

D. Specific gas constant of component i

Correct Answer: B. Mole fraction of component i

Question 2

A gas mixture has mole fractions 0.75 nitrogen and 0.25 oxygen at a total pressure of 200 kPa. What is the partial pressure of oxygen?

A. 25 kPa

B. 50 kPa

C. 150 kPa

D. 200 kPa

Correct Answer: B. 50 kPa

Question 3

Which law states that the total pressure of an ideal-gas mixture equals the sum of its component partial pressures?

A. Fourier’s law

B. Amagat’s law

C. Dalton’s law

D. Fick’s law

Correct Answer: C. Dalton’s law

Question 4

Moist air at 25 °C has a water-vapour partial pressure of 1.901 kPa. If the saturation pressure at 25 °C is 3.169 kPa, what is the relative humidity?

A. 40%

B. 50%

C. 60%

D. 75%

Correct Answer: C. 60%

Question 5

During sensible heating of a closed moist-air stream with no moisture addition or removal, which property remains constant?

A. Relative humidity

B. Humidity ratio

C. Saturation pressure

D. Dew-point pressure only

Correct Answer: B. Humidity ratio

Question 6

The dew point of moist air is the temperature at which:

A. Dry air becomes an ideal gas

B. Relative humidity becomes zero

C. Water vapour begins to condense during cooling

D. The humidity ratio becomes equal to one

Correct Answer: C. Water vapour begins to condense during cooling

Question 7

For moist air at total pressure p, the humidity ratio is calculated from ω = 0.622p_v/(p − p_v). What does p_v denote?

A. Total atmospheric pressure

B. Dry-air partial pressure

C. Water-vapour partial pressure

D. Saturation temperature

Correct Answer: C. Water-vapour partial pressure

Question 8

A cooling coil processes 2 kg/s of dry air and reduces humidity ratio from 0.012 to 0.008 kg/kg dry air. What is the condensate rate?

A. 0.004 kg/s

B. 0.008 kg/s

C. 0.016 kg/s

D. 0.020 kg/s

Correct Answer: B. 0.008 kg/s

Question 9

In adiabatic mixing of two moist-air streams, which balances are required to determine the outlet state?

A. Only a total-volume balance

B. Dry-air, water-vapour, and energy balances

C. Only a momentum balance

D. Only a relative-humidity balance

Correct Answer: B. Dry-air, water-vapour, and energy balances

Question 10

Why does a cold surface below the dew-point temperature become wet?

A. Dry air changes into oxygen

B. Relative humidity becomes negative

C. Water vapour condenses on the surface

D. The total pressure becomes zero

Correct Answer: C. Water vapour condenses on the surface