Psychrometry and vehicle air conditioning

Why it matters

A car's air-conditioner does two jobs: it cools the cabin air and it removes moisture from it. In humid Indian conditions more than half of the evaporator's load can be moisture removal, and getting it wrong means fogged windscreens or an under-sized system. Psychrometry, the thermodynamics of moist air, is how that load is calculated and how the evaporator, blower and demist strategy are sized.

Key ideas

Moist air is treated as an ideal-gas mixture of dry air and water vapour at atmospheric pressure P = p_a + p_v (Dalton's law). Properties are expressed per kg of dry air, because the dry-air mass stays constant through most processes.

Humidity measures.

• Specific humidity (humidity ratio) ω: kg of water vapour per kg of dry air.
• Relative humidity φ: ratio of the actual vapour partial pressure to the saturation pressure at the same dry-bulb temperature, p_v/p_sat(T). It changes with temperature even when ω is fixed.
• Degree of saturation μ: ω/ω_s at the same temperature; close to φ but not equal.

Temperatures.

• Dry-bulb temperature (DBT): ordinary air temperature.
• Dew-point temperature (DPT): the temperature at which the vapour starts to condense when air is cooled at constant ω (and pressure); it is T_sat at p_v. A windscreen colder than the cabin dew point fogs up.
• Wet-bulb temperature (WBT): read by a thermometer with a wetted wick in moving air; close to the adiabatic saturation temperature for air–water. For unsaturated air, DPT < WBT < DBT; all three are equal at saturation.

Enthalpy of moist air per kg dry air is the sum of the dry-air and vapour enthalpies, measured from 0 °C.

Psychrometric chart. DBT on the horizontal axis, ω on the vertical axis, with curves of constant φ, inclined lines of constant WBT (nearly constant enthalpy) and constant specific volume. Any two independent properties locate the state.

Air-conditioning processes.

• Sensible heating or cooling: ω constant, horizontal line on the chart.
• Cooling and dehumidification: the evaporator surface is below the dew point, so moisture condenses. The air leaves close to the coil's apparatus dew point (ADP). The bypass factor BF is the fraction of air that effectively misses the coil: BF = (T₂ − T_ADP)/(T₁ − T_ADP).
• Heating and humidification, adiabatic (evaporative) cooling along a constant-WBT line, and adiabatic mixing of two streams (the mixture lies on the straight line joining them, divided in inverse ratio of the dry-air masses).
• Sensible heat factor SHF = sensible load / total load. Low SHF means a large latent (moisture) load.

Vehicle air-conditioning loads. Solar radiation through glass (often the largest), heat conducted through the body, passengers (sensible and latent), fresh-air ventilation, and engine and exhaust heat. Recirculation mode reduces the fresh-air load but lets CO₂ and humidity build up. Demisting uses the evaporator to dry the air and then reheats it with the heater core, so the air reaching the glass is dry and warm.

Formulas

φ = p_v / p_sat(T) — p_v: partial pressure of vapour, kPa; p_sat: saturation pressure at the DBT (steam tables).

ω = 0.622 p_v / (P − p_v) — kg vapour/kg dry air; P: total pressure, kPa.

μ = ω / ω_s = φ (P − p_sat)/(P − p_v) — degree of saturation.

DPT = T_sat(p_v) — from the saturation table.

h = 1.005 t + ω (2501 + 1.88 t) — kJ/kg dry air, t in °C.

ṁ_w = ṁ_a (ω₁ − ω₂) — condensate rate, kg/s.

Q̇ = ṁ_a [(h₁ − h₂) − (ω₁ − ω₂) h_f] — coil load for cooling and dehumidification, kW; h_f: enthalpy of condensate (often neglected).

BF = (T₂ − T_ADP)/(T₁ − T_ADP); contact factor = 1 − BF.

SHF = Q̇_s / (Q̇_s + Q̇_L)

ṁ_a = P_a V̇ / (R_a T) — R_a = 0.287 kJ/kg·K, P_a = P − p_v in kPa, T in K.

Worked examples

Example 1 (standard). Outside air is at 35 °C DBT and 60% RH at 101.325 kPa. Find ω, h and the dew point. Take p_sat(35 °C) = 5.629 kPa.

1. p_v = φ p_sat = 0.60 × 5.629 = 3.377 kPa.
2. ω = 0.622 p_v/(P − p_v) = 0.622 × 3.377/(101.325 − 3.377) = 0.0214 kg/kg dry air.
3. h = 1.005 t + ω (2501 + 1.88 t) = 1.005 × 35 + 0.02145 × (2501 + 65.8) = 35.2 + 55.1 = 90.2 kJ/kg dry air.
4. Dew point = T_sat at 3.377 kPa ≈ 26.1 °C (from the saturation table). A windscreen below about 26 °C would fog if this air touched it.

Example 2 (GATE level, car evaporator). The blower pushes 0.1 kg/s (dry-air basis) of the outside air from Example 1 through the evaporator, which delivers it saturated at 12 °C (p_sat = 1.403 kPa). Find the condensate rate, the coil load and the SHF. h_f at 12 °C = 50.4 kJ/kg.

1. Exit state: p_v = 1.403 kPa; ω₂ = 0.622 × 1.403/(101.325 − 1.403) = 0.00873 kg/kg.
2. h₂ = 1.005 × 12 + 0.00873 × (2501 + 22.6) = 12.06 + 22.03 = 34.1 kJ/kg.
3. Condensate: ṁ_w = ṁ_a (ω₁ − ω₂) = 0.1 × (0.02145 − 0.00873) = 0.00127 kg/s ≈ 4.6 kg/h (the water that drips under a parked car).
4. Coil load: Q̇ = ṁ_a [(h₁ − h₂) − (ω₁ − ω₂) h_f] = 0.1 × [(90.2 − 34.1) − 0.01272 × 50.4] = 0.1 × (56.1 − 0.64) = 5.55 kW (about 1.6 TR).
5. Sensible part: cool at constant ω₁ to 12 °C: h = 1.005 × 12 + 0.02145 × 2523.6 = 66.2 kJ/kg, so Q̇_s = 0.1 × (90.2 − 66.2) = 2.40 kW.
6. SHF = 2.40/5.61 ≈ 0.43 (using the load before the small condensate correction). More than half the load is moisture removal.

Common mistakes

• Using p_sat at the wet-bulb temperature to compute φ; φ always uses p_sat at the dry-bulb temperature.
• Forgetting that properties are per kg of dry air, not per kg of moist air.
• Treating RH as constant during sensible cooling; ω is constant and RH rises.
• Using P instead of P − p_v in ω, or mixing kPa and Pa.
• Ignoring the latent load and sizing an evaporator on ṁ c_p ΔT alone.
• Reading enthalpy lines as wet-bulb lines without checking the chart's convention; they are close but not identical.

For GATE ME

Expect numericals on ω, φ, dew point and enthalpy from given saturation pressures; mass of water condensed in a cooling coil; adiabatic mixing of two airstreams; bypass factor and SHF; and conceptual MCQs on the order of DBT, WBT and DPT and on process lines on the psychrometric chart. Practise sketching each process on the chart before calculating.

Quick check

1. If p_v = 1.2 kPa and p_sat = 2.0 kPa at the DBT, what is the relative humidity?
2. For saturated air, how do DBT, WBT and DPT compare?
3. During sensible cooling above the dew point, which stays constant: ω or φ?
4. Air at 30 °C has p_v = 2.123 kPa at 101.325 kPa. What is ω?

Answers: 1. 60%. 2. All three are equal. 3. ω stays constant; φ rises. 4. 0.622 × 2.123/99.20 = 0.0133 kg/kg dry air.