13.4 Psychrometric Properties, Processes & Air Conditioning Systems
Key Takeaways
- Moist air properties follow Dalton's law of partial pressures P = P_a + P_v, with specific humidity omega = 0.622*P_v/(P - P_v) and relative humidity phi = P_v/P_vs * 100%; at 100% saturation DBT = WBT = DPT.
- Moist air specific enthalpy is calculated as h = 1.005*DBT + omega*(2501 + 1.88*DBT) kJ/kg d.a., where 1.005 kJ/kg·K is dry air specific heat and 2501 kJ/kg is water latent heat of vaporization at 0°C.
- Fundamental psychrometric processes are mapped as directional vectors on the psychrometric chart: sensible heating/cooling (horizontal omega = const), humidification/dehumidification (vertical DBT = const), cooling & dehumidification (diagonal down-left), and adiabatic evaporative cooling along constant WBT/enthalpy lines.
- Cooling coil performance is governed by the Bypass Factor BPF = (T_exit - T_ADP)/(T_inlet - T_ADP) and Contact Factor eta_c = 1 - BPF, where Apparatus Dew Point (ADP) is the effective coil surface temperature.
- The Sensible Heat Factor SHF = RSH / (RSH + RLH) establishes the slope of the room condition line on the psychrometric chart, targeting industrial comfort design standards of 24°C DBT and 50-60% RH.
Psychrometric Properties, Processes & Air Conditioning Systems
Psychrometry is the branch of engineering science dedicated to the thermodynamic evaluation of moist air—a binary mixture of dry air and water vapor. In coal mining operations, precise psychrometric control is essential for maintaining human thermal comfort, mitigating heat stress in deep underground shafts (where auto-compression and geothermal gradients raise intake air temperatures), preventing spontaneous combustion in coal storage silos, and ensuring proper climate control in heavy equipment control cabins and computerized washery dispatch rooms.
1. Atmospheric Air Composition & Dalton's Law of Partial Pressures
Atmospheric moist air is treated as an ideal gas mixture obeying Dalton's Law of Partial Pressures. The total barometric pressure $P$ is the sum of the partial pressure of dry air ($P_a$) and the partial pressure of water vapor ($P_v$):
- Standard Atmospheric Pressure at Sea Level: $P = 101.325\text{ kPa} = 760\text{ mm Hg} = 1.01325\text{ bar}$.
- Characteristic Gas Constants: Dry air $R_a = 0.2870\text{ kJ/kg}\cdot\text{K}$, Water vapor $R_v = 0.4615\text{ kJ/kg}\cdot\text{K}$.
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| PSYCHROMETRIC TEMPERATURE RELATIONSHIPS |
| |
| UNSATURATED MOIST AIR (Relative Humidity < 100%): |
| |
| Dry Bulb Temp (DBT) > Wet Bulb Temp (WBT) > Dew Point Temp (DPT) |
| |
| - Dry Bulb Temp (DBT): True sensible kinetic temperature measured by bare thermometer.|
| - Wet Bulb Temp (WBT): Lowest adiabatic evaporation temp measured by wet-wick sensor. |
| - Dew Point Temp (DPT): Saturation temperature corresponding to partial pressure Pv. |
| |
| SATURATED MOIST AIR (Relative Humidity = 100%): |
| |
| Dry Bulb Temp (DBT) = Wet Bulb Temp (WBT) = Dew Point Temp (DPT) |
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2. Fundamental Psychrometric Properties & Governing Formulations
1. Specific Humidity / Humidity Ratio ($\omega$)
The mass of water vapor present per unit mass of dry air:
2. Relative Humidity ($\phi$ or $\text{RH}$)
The ratio of the actual partial pressure of water vapor $P_v$ to the saturation pressure of water vapor $P_{vs}$ at the same dry bulb temperature:
3. Degree of Saturation ($\mu$)
The ratio of actual specific humidity $\omega$ to the saturated specific humidity $\omega_s$ at the same dry bulb temperature:
[!NOTE] Because $P_v \le P_{vs}$, the term $\left[\frac{P - P_{vs}}{P - P_v}\right] \le 1$. Therefore, for unsaturated air, $\mu \le \phi$; at saturation ($\phi = 1.0$), $\mu = \phi = 1.0$.
4. Specific Enthalpy of Moist Air ($h$)
The total thermal energy per kilogram of dry air, referencing dry air and liquid water at $0^{\circ}\text{C}$ ($h_a = 0, h_w = 0$):
Taking $c_{pa} = 1.005\text{ kJ/kg}\cdot\text{K}$, $h_{fg,0} = 2501\text{ kJ/kg}$, and $c_{pv} = 1.88\text{ kJ/kg}\cdot\text{K}$:
5. Moist Air Specific Volume ($v$)
Volume occupied by the mixture per unit mass of dry air:
3. The Psychrometric Chart: Coordinate Geometry & Lines of State
The psychrometric chart visually maps thermodynamic states of moist air at standard atmospheric pressure ($101.325\text{ kPa}$).
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| PSYCHROMETRIC CHART TOPOLOGY & LINES |
| |
| Specific |
| Humidity (w) Saturation Curve (phi=100%) |
| ^ / |
| | / |
| | / / Constant Rel. Humidity |
| | / / |
| | / / |
| | Constant Enthalpy (h) / / |
| | & Constant WBT Lines / / |
| | \ / / |
| | \ / / Constant Specific |
| | \ / / Volume Lines (v) |
| | \ / / \ |
| | \ / / \ |
| | \ / / \ |
| | \ / / \ |
| +---------------------------+--+----------------------+--------------------> |
| Constant Dry Bulb Temp (DBT) Lines |
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+-----------------------------------------------------------------------------------------+
| ORIENTATION OF LINES ON THE CHART |
| |
| 1. Constant Dry Bulb Temp (DBT): Vertical, parallel straight lines. |
| 2. Constant Specific Humidity (w): Horizontal, parallel straight lines. |
| 3. Constant Dew Point Temp (DPT): Horizontal lines extending left to Sat. Curve. |
| 4. Constant Relative Humidity (phi): Curved lines bowing upward from bottom-left. |
| 5. Constant Enthalpy / WBT: Diagonal lines sloping downward to the right. |
| 6. Constant Specific Volume (v): Steeply inclined diagonal lines. |
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4. Fundamental Psychrometric Processes
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| VECTOR MAP OF PSYCHROMETRIC PROCESSES |
| |
| Humidification (w increases) |
| ^ |
| Heating & Humidification | Cooling & Humidification |
| (Winter A/C) | (Air Washer / Evaporative) |
| \ | / |
| \ | / |
| \ | / |
| Sensible Cooling <----------------------+----------------------> Sensible Heating |
| (w = const, DBT drops) | (w = const, DBT rises|
| / | \ |
| / | \ |
| / | \ |
| Cooling & Dehumidification| Chemical Dehumidification |
| (Summer A/C) v (Desiccant Dryer) |
| Dehumidification (w decreases) |
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Summary of Key Thermodynamic Air Conditioning Processes
| Process | DBT | WBT | DPT | Specific Humidity $\omega$ | Relative Humidity $\phi$ | Enthalpy $h$ |
|---|---|---|---|---|---|---|
| Sensible Heating | $\uparrow$ | $\uparrow$ | Constant | Constant | $\downarrow$ | $\uparrow$ |
| Sensible Cooling | $\downarrow$ | $\downarrow$ | Constant | Constant | $\uparrow$ | $\downarrow$ |
| Humidification | Constant | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| Dehumidification | Constant | $\downarrow$ | $\downarrow$ | $\downarrow$ | $\downarrow$ | $\downarrow$ |
| Cooling & Dehumidification | $\downarrow$ | $\downarrow$ | $\downarrow$ | $\downarrow$ | $\uparrow$ (usually) | $\downarrow$ |
| Heating & Humidification | $\uparrow$ | $\uparrow$ | $\uparrow$ | $\uparrow$ | Varies | $\uparrow$ |
| Evaporative Cooling | $\downarrow$ | Constant | $\uparrow$ | $\uparrow$ | $\uparrow$ | Constant |
| Chemical Dehumidification | $\uparrow$ | $\approx$ Const | $\downarrow$ | $\downarrow$ | $\downarrow$ | $\approx$ Const |
5. Adiabatic Mixing of Two Moist Air Streams
When two air streams 1 (flow rate $\dot{m}_{a1}$, state $\text{DBT}1, \omega_1, h_1$) and 2 (flow rate $\dot{m}{a2}$, state $\text{DBT}_2, \omega_2, h_2$) mix adiabatically under steady flow:
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| ADIABATIC MIXING & LEVER RULE |
| |
| Conservation of Dry Air: m_dot_a1 + m_dot_a2 = m_dot_a3 |
| Conservation of Water Vapor: m_dot_a1 * w1 + m_dot_a2 * w2 = m_dot_a3 * w3 |
| Conservation of Energy: m_dot_a1 * h1 + m_dot_a2 * h2 = m_dot_a3 * h3 |
| |
| Mixed State Point (3): |
| w3 = (m_dot_a1 * w1 + m_dot_a2 * w2) / (m_dot_a1 + m_dot_a2) |
| h3 = (m_dot_a1 * h1 + m_dot_a2 * h2) / (m_dot_a1 + m_dot_a2) |
| T3 approx (m_dot_a1 * T1 + m_dot_a2 * T2) / (m_dot_a1 + m_dot_a2) |
| |
| GEOMETRIC LEVER RULE: |
| State 3 lies on the straight line connecting State 1 and State 2 such that: |
| (Line Segment 1-3) / (Line Segment 3-2) = m_dot_a2 / m_dot_a1 |
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6. Cooling Coil Performance: Bypass Factor (BPF) & Apparatus Dew Point (ADP)
In an air conditioning cooling coil, a fraction of air contacts the coil tubes and fins directly, reaching the Apparatus Dew Point ($T_{\text{ADP}}$) at 100% saturation, while the remaining fraction bypasses the coil surface completely unaffected.
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| COIL BYPASS FACTOR & CONTACT FACTOR |
| |
| Inlet Air (State 1: T1, w1) |
| ========================> [ COOLING COIL (Surface Temp = T_ADP) ] ====> Exit Air (T2, |
| w2) |
| |
| Bypass Factor (BPF or X): |
| BPF = (T2 - T_ADP) / (T1 - T_ADP) = (w2 - w_ADP) / (w1 - w_ADP) |
| = (h2 - h_ADP) / (h1 - h_ADP) |
| |
| Contact Factor (eta_c): |
| eta_c = 1 - BPF = (T1 - T2) / (T1 - T_ADP) |
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Factors Governing Bypass Factor (BPF)
- Number of Coil Rows: Increasing tube rows increases air-surface contact $\implies \text{BPF}$ decreases.
- Fin Density (Fins per inch): Increasing fin density reduces open gap area $\implies \text{BPF}$ decreases.
- Air Face Velocity: Increasing velocity reduces contact dwell time $\implies \text{BPF}$ increases.
7. Sensible Heat Factor (SHF) & Space Air Conditioning Loads
Thermal loads in an air-conditioned space comprise sensible heat gain ($\text{RSH}$, raising temperature from people, lights, equipment) and latent heat gain ($\text{RLH}$, adding moisture from respiration, infiltration, steam):
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| ROOM CONDITION LINE ON PSYCHROMETRIC CHART |
| |
| Specific |
| Humidity (w) |
| ^ |
| | Saturation Curve |
| | / |
| | / Room State (R) |
| | / * |
| | / / |
| | / Coil Exit (2)/ <--- Slope of Line = SHF |
| | / * / (RSH / RTH) |
| | ADP *-------/-------/ |
| | / |
| +--------------+-------------------------------------------------------------> |
| T_ADP T2(Supply) T_Room DBT |
+-----------------------------------------------------------------------------------------+
8. Human Thermal Comfort & Mine Ventilation Standards
Human thermal equilibrium is governed by the body heat balance: $M \pm R \pm C - E = S = 0$, where $M$ is metabolic rate, $R$ is radiation, $C$ is convection, $E$ is evaporative sweating, and $S$ is rate of heat storage.
Industrial Design Guidelines (ASHRAE / Indian Standards IS 1391)
- Comfort Zone Dry Bulb Temperature: $23^{\circ}\text{C} - 25^{\circ}\text{C}$ (Standard Design Target: $24^{\circ}\text{C}$).
- Comfort Relative Humidity: $50% - 60%$ (Below 30% causes mucosal drying and electrostatic discharge; above 70% inhibits sweat evaporation, inducing heat strain).
- Air Velocity: $0.15 - 0.25\text{ m/s}$ in occupied zones.
- Deep Underground Mine Environments: Coal Mines Regulations (CMR) mandate effective temperature thresholds ($< 30.5^{\circ}\text{C}$ wet bulb) in working coal faces to prevent heat stroke and fatal miner fatigue.
9. Step-by-Step Worked Problems: Psychrometric Load & Coil Sizing
Worked Example 1: Psychrometric Properties Calculation
Atmospheric air is at $\text{DBT} = 35^{\circ}\text{C}$ and $P = 101.325\text{ kPa}$ with a relative humidity of $\phi = 60%$. Saturated vapor pressure at $35^{\circ}\text{C}$ is $P_{vs} = 5.628\text{ kPa}$. Calculate:
- Vapor partial pressure $P_v$
- Specific humidity $\omega$
- Specific enthalpy $h$
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| PSYCHROMETRIC PROPERTY CALCULATION |
| |
| STEP 1: Calculate Water Vapor Partial Pressure Pv |
| Pv = phi * Pvs = 0.60 * 5.628 kPa = 3.3768 kPa |
| |
| STEP 2: Calculate Specific Humidity (w) |
| w = 0.622 * Pv / (P - Pv) |
| w = 0.622 * (3.3768) / (101.325 - 3.3768) = 2.10037 / 97.9482 |
| w = 0.02144 kg w.v. / kg d.a. = 21.44 g/kg d.a. |
| |
| STEP 3: Calculate Specific Enthalpy (h) |
| h = 1.005 * DBT + w * (2501 + 1.88 * DBT) |
| h = 1.005 * (35) + 0.02144 * (2501 + 1.88 * 35) |
| h = 35.175 + 0.02144 * (2501 + 65.8) = 35.175 + 0.02144 * (2566.8) |
| h = 35.175 + 55.032 = 90.21 kJ/kg d.a. |
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Worked Example 2: Cooling Coil Sizing with Bypass Factor
An air handling unit (AHU) conditions $1.5\text{ kg/s}$ of outdoor air entering at $T_1 = 38^{\circ}\text{C}$ DBT and $\omega_1 = 0.020\text{ kg/kg d.a.}$ across a cooling coil with $\text{ADP} = 10^{\circ}\text{C}$ and Bypass Factor $\text{BPF} = 0.15$. Calculate the off-coil temperature $T_2$ and the total cooling coil load in kW (assuming $h_1 = 89.5\text{ kJ/kg}$ and $h_2 = 42.0\text{ kJ/kg}$).
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| COIL SIZING & BPF CALCULATION STEPS |
| |
| STEP 1: Calculate Off-Coil Dry Bulb Temperature T2 |
| BPF = (T2 - T_ADP) / (T1 - T_ADP) |
| 0.15 = (T2 - 10) / (38 - 10) |
| 0.15 * 28 = T2 - 10 ===> 4.20 = T2 - 10 ===> T2 = 14.2 deg C |
| |
| STEP 2: Calculate Coil Cooling Load |
| Q_coil = m_dot_a * (h1 - h2) |
| Q_coil = 1.5 kg/s * (89.5 - 42.0 kJ/kg) |
| Q_coil = 1.5 kg/s * 47.5 kJ/kg = 71.25 kW |
| Tonnage = 71.25 kW / 3.517 kW/TR = 20.26 TR |
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The barometric pressure in a mine ventilation shaft is 100 kPa. If the dry bulb temperature is 30°C and the partial pressure of water vapor is measured as 2.40 kPa, what is the specific humidity (humidity ratio ω) of the air?
An air stream at 40°C DBT passes through a cooling and dehumidifying coil maintained at an Apparatus Dew Point (ADP) of 8°C. If the coil has a Contact Factor (η_c) of 0.80, what is the dry bulb temperature of the air leaving the coil?
A space conditioning calculation for a coal dispatch control center establishes a Room Sensible Heat (RSH) load of 45 kW and a Room Latent Heat (RLH) load of 15 kW. What is the Sensible Heat Factor (SHF) of the room, and what does it represent on the psychrometric chart?