6.1 Psychrometric Principles & Air-Conditioning Processes

Psychrometric Principles & Air-Conditioning Processes

Psychrometrics is the branch of thermodynamics dedicated to atmospheric air combined with water vapor. It serves as the foundational core for heating, ventilating, air-conditioning, and refrigeration (HVAC&R) system design. On the NCEES PE Mechanical exam, psychrometric analysis is essential for evaluating air handling unit (AHU) performance, sizing direct expansion (DX) and chilled water cooling coils, determining required supply airflow rates, and analyzing mixing plenums and evaporative coolers.

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1. Thermodynamic Properties of Moist Air & Dalton's Law

Atmospheric moist air is treated as a binary mixture of dry air (a pseudo-pure gas with apparent molecular weight $M_a = 28.966\text{ lbm/lbmol} = 28.966\text{ kg/kmol}$) and water vapor ($M_w = 18.015\text{ lbm/lbmol} = 18.015\text{ kg/kmol}$). Under typical HVAC operating ranges (temperatures from $-40^\circ\text{F}$ to $140^\circ\text{F}$ and pressures near atmospheric), both dry air and water vapor behave in close accordance with the ideal gas law ($p V = m R T$).

+-----------------------------------------------------------------------------------------+
|                        MOIST AIR CONSTITUENTS & DALTON'S LAW                            |
|                                                                                         |
|   Total Barometric Pressure:   p = p_a + p_w                                            |
|                                                                                         |
|   - Dry Air Partial Pressure:   p_a = \frac{m_a R_a T}{V}                                |
|   - Water Vapor Partial Press:  p_w = \frac{m_w R_w T}{V}                                |
|   - Standard Atmospheric Press: p = 14.696 psia = 29.921 in. Hg = 101.325 kPa            |
+-----------------------------------------------------------------------------------------+

Primary Psychrometric Variables

1. Dry-Bulb Temperature ($T_{db}$ or $T$): The true thermodynamic temperature of the moist air mixture measured by an ordinary, unshielded thermometer unaffected by radiation or moisture.
2. Wet-Bulb Temperature ($T_{wb}$ or $T^*$): The equilibrium temperature registered by an accurate thermometer whose sensor is covered by a wetted wick and exposed to rapid airflow ($> 900\text{ FPM}$). At saturation (100% relative humidity), dry-bulb, wet-bulb, and dew-point temperatures are all identical: $T_{db} = T_{wb} = T_{dp}$.
3. Dew-Point Temperature ($T_{dp}$): The temperature at which water vapor in moist air begins to condense when the mixture is cooled at constant moisture content ($\omega = \text{const}$) and constant total pressure ($p = \text{const}$). $T_{dp}$ is strictly a function of the water vapor partial pressure $p_w$, satisfying $p_w = p_{ws}(T_{dp})$, where $p_{ws}$ is the saturation pressure of water.
4. Relative Humidity ($\phi$): The ratio of the actual water vapor partial pressure $p_w$ to the saturation pressure $p_{ws}$ at the same dry-bulb temperature: $$\phi = \left(\frac{p_w}{p_{ws}}\right)_{T_{db}} \times 100$$
5. Humidity Ratio / Specific Humidity ($\omega$ or $W$): The ratio of the mass of water vapor ($m_w$) to the mass of dry air ($m_{da}$): $$\omega = \frac{m_w}{m_{da}} = \frac{M_w p_w}{M_a p_a} = \frac{18.015}{28.966} \left(\frac{p_w}{p - p_w}\right) = 0.622 \frac{p_w}{p - p_w} \quad \left[\frac{\text{lbm}_w}{\text{lbm}_{da}} \text{ or } \frac{\text{kg}_w}{\text{kg}_{da}}\right]$$ In US Customary units, moisture content is frequently expressed in grains of moisture per pound of dry air ($1\text{ lbm} = 7,000\text{ grains}$): $$W_{gr} = 7,000 \times \omega \quad \left[\frac{\text{grains}_w}{\text{lbm}_{da}}\right]$$
6. Specific Volume ($v$): The total volume of the moist air mixture per unit mass of dry air: $$v = \frac{V}{m_{da}} = \frac{R_a T}{p_a} = \frac{R_a T}{p - p_w} = \frac{R_a T (1 + 1.608\omega)}{p} \quad \left[\frac{\text{ft}^3}{\text{lbm}_{da}} \text{ or } \frac{\text{m}^3}{\text{kg}_{da}}\right]$$ Where $R_a = 53.35\text{ ft}\cdot\text{lbf}/(\text{lbm}\cdot^\circ\text{R}) = 0.2870\text{ kJ}/(\text{kg}\cdot\text{K})$ and $T$ is absolute temperature ($^\circ\text{R}$ or $\text{K}$).
7. Specific Enthalpy ($h$): The total thermal energy of the moist air mixture per unit mass of dry air, using $0^\circ\text{F}$ (or $0^\circ\text{C}$) dry air and $32^\circ\text{F}$ liquid water as reference states:
   • US Customary Units: $$h = c_{pa} T_{db} + \omega h_g \approx 0.240 T_{db} + \omega (1061 + 0.444 T_{db}) \quad \left[\frac{\text{BTU}}{\text{lbm}_{da}}\right]$$
   • SI Units: $$h = c_{pa} T_{db} + \omega h_g \approx 1.006 T_{db} + \omega (2501 + 1.86 T_{db}) \quad \left[\frac{\text{kJ}}{\text{kg}_{da}}\right]$$

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2. Navigating the Standard Psychrometric Chart

The standard sea-level psychrometric chart ($p = 14.696\text{ psia} = 101.325\text{ kPa}$) plots thermodynamic properties on a 2D coordinate system.

+-----------------------------------------------------------------------------------------+
|                        PSYCHROMETRIC CHART GEOMETRY & AXES                              |
|                                                                                         |
|   Humidity Ratio (\omega) / Enthalpy (h)                                                  |
|   ^                                                                                     |
|   |                                                      / Constant Volume (v)          |
|   |                                        Saturation   /                               |
|   |                                        Curve       /                                |
|   |                                        (\phi=100%)/    Constant Wet-Bulb (T_wb)     |
|   |                                          ,-''''' /    /  & Specific Enthalpy (h)    |
|   |                                      _,-'       /   ,'                              |
|   |                                  _,-'          /  ,'                                |
|   |                              _,-'             / ,'                                  |
|   |                          _,-'                /,'    --- Constant Relative           |
|   |                      _,-'                   /'          Humidity (\phi = 50%)       |
|   |                  _,-'                      /                                        |
|   |              _,-'                         /                                         |
|   |          _,-'                            /                                          |
|   |      _,-'                               /                                           |
|   +----+-----------------------------------+-------------------------------------->     |
|       Dew Point (T_dp)                    Dry-Bulb Temperature (T_db)                   |
+-----------------------------------------------------------------------------------------+

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3. Core Psychrometric Air-Conditioning Processes

Every HVAC cycle decomposes into basic linear or multi-step psychrometric transformations on the chart.

+-----------------------------------------------------------------------------------------+
|                        SUMMARY OF CORE PSYCHROMETRIC PROCESSES                          |
|                                                                                         |
|                      [2] Humidification Only (\Delta T_db = 0, \omega \uparrow)           |
|                                     ^                                                   |
|   [6] Evaporative Cooling           |             [1] Sensible Heating                  |
|       (Adiabatic Saturation)        |                 (\omega = const, T_db \uparrow)   |
|         \                           |            /                                      |
|          \                          |           /                                       |
|           \                         |          /                                        |
|            <---------------------- [0] ---------------------->                          |
|           /                     State Point    \                                        |
|          /                          |           \                                       |
|         /                           |            \                                      |
|   [4] Cooling & Dehumidification    |             [5] Chemical Dehumidification         |
|       (T_db \downarrow, \omega \downarrow)   v                 (T_db \uparrow, \omega \downarrow)  |
|                      [3] Dehumidification Only (\Delta T_db = 0, \omega \downarrow)       |
+-----------------------------------------------------------------------------------------+

1. Sensible Heating and Cooling

Sensible processes occur when moist air passes over a heating or cooling coil whose surface temperature remains above the dew-point temperature of the entering air ($T_{\text{surface}} > T_{dp}$). Moisture is neither added nor removed ($\Delta \omega = 0$).

$$\dot{q}_s = \dot{m}_{da} c_{pa} \Delta T = (\rho \cdot CFM \cdot 60) c_{pa} \Delta T$$

Applying standard sea-level air properties ($\rho = 0.075\text{ lbm/ft}^3$, $c_{pa} = 0.240\text{ BTU/lbm}\cdot^\circ\text{F}$):

$$\dot{q}_s = (0.075)(60)(0.240) \times CFM \times \Delta T = 1.08 \times CFM \times \Delta T \quad [\text{BTU/hr}]$$

In SI units ($\rho = 1.204\text{ kg/m}^3$, $c_{pa} = 1.006\text{ kJ/kg}\cdot\text{K}$):

$$\dot{q}_s = (1.204)(1.006) \times \dot{V}_{L/s} \times \Delta T = 1.23 \times \dot{V}_{L/s} \times \Delta T \quad [\text{W}]$$

2. Cooling and Dehumidification

When air passes over a cooling coil whose effective surface temperature is below the entering air dew point ($T_{\text{coil}} < T_{dp}$), moisture condenses out of the air stream. The process line on the psychrometric chart curves downward and leftward toward the Apparatus Dew Point (ADP), which represents the effective coil surface temperature on the saturation curve ($\phi = 100\%$).

+-----------------------------------------------------------------------------------------+
|                        COIL BYPASS FACTOR & APPARATUS DEW POINT                         |
|                                                                                         |
|   Saturation Curve                                                                      |
|          ,-'                                                                            |
|        ,'                                                                               |
|      ,'   [ADP] (T_adp, \omega_adp)                                                     |
|     /       \                                                                           |
|    /         \______ [Leaving Air Point: 2] (T_db,2, \omega_2)                          |
|   /                 \                                                                   |
|  /                   \______ [Entering Air Point: 1] (T_db,1, \omega_1)                 |
| /                                                                                       |
+-----------------------------------------------------------------------------------------+

• Bypass Factor ($BF$): The fraction of total airflow that passes through the cooling coil completely unconditioned: $$BF = \frac{T_{db,2} - T_{ADP}}{T_{db,1} - T_{ADP}} = \frac{\omega_2 - \omega_{ADP}}{\omega_1 - \omega_{ADP}} = \frac{h_2 - h_{ADP}}{h_1 - h_{ADP}}$$
• Contact Factor ($CF$): The fraction of total air that comes into direct contact with the coil surface and achieves thermal equilibrium at the ADP: $$CF = 1 - BF$$

3. Latent Heat and Total Heat Equations

• Latent Heat Transfer Rate (Moisture Addition or Removal): $$\dot{q}_L = \dot{m}_{da} h_{fg} \Delta \omega = (0.075)(60)(1076) \times CFM \times \Delta \omega = 4840 \times CFM \times \Delta \omega \quad [\text{BTU/hr}]$$ When moisture is given in grains per pound of dry air ($W_{gr}$): $$\dot{q}_L = \frac{4840}{7000} \times CFM \times \Delta W_{gr} \approx 0.69 \times CFM \times \Delta W_{gr} \quad [\text{BTU/hr}]$$
• Total Heat Transfer Rate (Enthalpy Method): $$\dot{q}_t = \dot{q}_s + \dot{q}_L = \dot{m}_{da} \Delta h = (0.075)(60) \times CFM \times \Delta h = 4.5 \times CFM \times \Delta h \quad [\text{BTU/hr}]$$ In SI units: $$\dot{q}_t = 1.20 \times \dot{V}_{L/s} \times \Delta h \quad [\text{W}]$$

4. Sensible Heat Ratio (SHR)

The ratio of sensible heat removal rate to total heat removal rate:

$$SHR = \frac{\dot{q}_s}{\dot{q}_t} = \frac{\dot{q}_s}{\dot{q}_s + \dot{q}_L}$$

• Room Sensible Heat Ratio (RSHR): $\frac{\dot{q}_{s,\text{room}}}{\dot{q}_{t,\text{room}}}$, defining the slope of the condition line between the supply air state and the room design state.
• Grand Sensible Heat Ratio (GSHR): $\frac{\dot{q}_{s,\text{total}}}{\dot{q}_{t,\text{total}}}$, defining the slope of the process line across the central cooling coil.

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4. Adiabatic Mixing of Two Air Streams

In standard air handling systems, outdoor ventilation air (Stream 1) mixes adiabatically with recirculated return air (Stream 2) to produce mixed air (Stream 3) entering the conditioning coils.

+-----------------------------------------------------------------------------------------+
|                        ADIABATIC AIR STREAM MIXING SCHEMATIC                            |
|                                                                                         |
|   Outdoor Air (Stream 1)   ---> [  MIXING  ] ---> Mixed Air (Stream 3)                  |
|   (m_1, T_1, \omega_1, h_1)      [ PLENUM  ]      (m_3 = m_1 + m_2, T_3, \omega_3, h_3) |
|   Return Air (Stream 2)    ---> [          ]                                            |
|   (m_2, T_2, \omega_2, h_2)                                                             |
+-----------------------------------------------------------------------------------------+

Conservation Governing Equations:

1. Dry Air Mass Balance: $\dot{m}_{da,3} = \dot{m}_{da,1} + \dot{m}_{da,2}$
2. Energy Balance: $\dot{m}_{da,1} h_1 + \dot{m}_{da,2} h_2 = \dot{m}_{da,3} h_3 \implies h_3 = \frac{\dot{m}_{da,1} h_1 + \dot{m}_{da,2} h_2}{\dot{m}_{da,1} + \dot{m}_{da,2}}$
3. Moisture Balance: $\dot{m}_{da,1} \omega_1 + \dot{m}_{da,2} \omega_2 = \dot{m}_{da,3} \omega_3 \implies \omega_3 = \frac{\dot{m}_{da,1} \omega_1 + \dot{m}_{da,2} \omega_2}{\dot{m}_{da,1} + \dot{m}_{da,2}}$
4. Dry-Bulb Temperature Approximation (Volumetric Basis): $$T_{db,3} \approx \frac{CFM_1 T_{db,1} + CFM_2 T_{db,2}}{CFM_1 + CFM_2}$$

The Psychrometric Lever Rule

The mixed state point (Point 3) lies on the straight line connecting Point 1 and Point 2 on the psychrometric chart. The line segment lengths are inversely proportional to the mass flow rates:

$$\frac{\text{Distance }(1-3)}{\text{Distance }(1-2)} = \frac{\dot{m}_{da,2}}{\dot{m}_{da,1} + \dot{m}_{da,2}} = \frac{CFM_2}{CFM_3}$$ $$\frac{T_{db,3} - T_{db,1}}{T_{db,2} - T_{db,3}} = \frac{\dot{m}_{da,2}}{\dot{m}_{da,1}}$$

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5. Evaporative Cooling (Adiabatic Saturation)

In direct evaporative cooling (swamp coolers), unsaturated air passes through a wetted porous medium without external heat transfer ($\dot{q} = 0$).

• The sensible heat of the air stream provides the latent heat required to vaporize the liquid water.
• Thermodynamic Path: The process moves upward and leftward along a constant wet-bulb temperature / constant enthalpy line ($T_{wb} = \text{const}$, $h \approx \text{const}$).
• Outcome: Dry-bulb temperature decreases ($T_{db} \downarrow$), while humidity ratio increases ($\omega \uparrow$) and relative humidity increases ($\phi \uparrow$).
• Direct Evaporative Saturation Effectiveness ($\epsilon_{\text{evap}}$): $$\epsilon_{\text{evap}} = \frac{T_{db,\text{in}} - T_{db,\text{out}}}{T_{db,\text{in}} - T_{wb,\text{in}}}$$
• Indirect Evaporative Cooling: Primary air passes through dry channels of a heat exchanger while secondary air is cooled evaporatively in adjacent wet channels, providing sensible cooling at constant moisture content ($\omega = \text{const}$) without adding humidity to the supply air.

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6. Step-by-Step Worked Engineering Problem

Problem Statement

An air handling unit (AHU) serves a commercial office zone. The space design condition is $75.0^\circ\text{F } T_{db}$ and $50\%\text{ RH}$ ($h_{\text{room}} = 28.10\text{ BTU/lbm}_{da}, \omega_{\text{room}} = 0.00925\text{ lbm/lbm}_{da}$). The total supply airflow rate is $12,000\text{ CFM}$.

• Outdoor Air (OA): $3,000\text{ CFM}$ at $95.0^\circ\text{F } T_{db}$ and $76.0^\circ\text{F } T_{wb}$ ($h_{\text{OA}} = 39.40\text{ BTU/lbm}_{da}, \omega_{\text{OA}} = 0.01480\text{ lbm/lbm}_{da}$).
• Return Air (RA): $9,000\text{ CFM}$ at room conditions ($75.0^\circ\text{F } T_{db}, 50\%\text{ RH}$).
• The mixed air enters a chilled water cooling coil with an Apparatus Dew Point (ADP) of $52.0^\circ\text{F}$ ($h_{\text{ADP}} = 21.40\text{ BTU/lbm}_{da}, \omega_{\text{ADP}} = 0.00820\text{ lbm/lbm}_{da}$) and a Bypass Factor ($BF$) of $0.12$.

Calculate:

1. The mixed air dry-bulb temperature ($T_{db,m}$), enthalpy ($h_m$), and humidity ratio ($\omega_m$).
2. The leaving air dry-bulb temperature ($T_{db,L}$) and enthalpy ($h_L$).
3. The total cooling coil capacity required in Tons of Refrigeration (TR).
4. The coil Sensible Heat Ratio (SHR).

+-----------------------------------------------------------------------------------------+
|                          AHU COIL CALCULATION SCHEMATIC                                 |
|                                                                                         |
|   OA: 3,000 CFM, 95°F ----> [ MIXING ]                                                  |
|                             [ PLENUM ] ---> MIXED AIR ---> [ COOLING COIL ] ---> SUPPLY |
|   RA: 9,000 CFM, 75°F ----> [        ]     (T_m, h_m)      [ ADP = 52°F   ]     (T_L, h_L)|
|                                                            [ BF = 0.12    ]             |
+-----------------------------------------------------------------------------------------+

Step 1: Mixed Air Properties

Fraction of Outdoor Air: $x_{OA} = \frac{3,000}{12,000} = 0.25$, Fraction of Return Air: $x_{RA} = \frac{9,000}{12,000} = 0.75$.

$$T_{db,m} = x_{OA} T_{db,OA} + x_{RA} T_{db,RA} = (0.25)(95.0^\circ\text{F}) + (0.75)(75.0^\circ\text{F}) = 23.75 + 56.25 = 80.0^\circ\text{F}$$ $$h_m = x_{OA} h_{OA} + x_{RA} h_{RA} = (0.25)(39.40) + (0.75)(28.10) = 9.85 + 21.075 = 30.925\text{ BTU/lbm}_{da}$$ $$\omega_m = (0.25)(0.01480) + (0.75)(0.00925) = 0.00370 + 0.0069375 = 0.010638\text{ lbm/lbm}_{da}$$

Step 2: Leaving Air Conditions from Coil Bypass Factor

$$T_{db,L} = T_{ADP} + BF(T_{db,m} - T_{ADP}) = 52.0^\circ\text{F} + 0.12(80.0^\circ\text{F} - 52.0^\circ\text{F}) = 52.0 + 0.12(28.0) = 55.36^\circ\text{F}$$ $$h_L = h_{ADP} + BF(h_m - h_{ADP}) = 21.40 + 0.12(30.925 - 21.40) = 21.40 + 0.12(9.525) = 22.543\text{ BTU/lbm}_{da}$$ $$\omega_L = \omega_{ADP} + BF(\omega_m - \omega_{ADP}) = 0.00820 + 0.12(0.010638 - 0.00820) = 0.008493\text{ lbm/lbm}_{da}$$

Step 3: Total Cooling Coil Capacity

$$\dot{q}_t = 4.5 \times CFM \times (h_m - h_L) = 4.5 \times 12,000 \times (30.925 - 22.543) = 54,000 \times 8.382 = 452,628\text{ BTU/hr}$$ $$\text{Cooling Capacity in TR} = \frac{452,628\text{ BTU/hr}}{12,000\text{ BTU/hr}\cdot\text{TR}} = 37.72\text{ TR}$$

Step 4: Sensible Coil Capacity & SHR

$$\dot{q}_s = 1.08 \times CFM \times (T_{db,m} - T_{db,L}) = 1.08 \times 12,000 \times (80.0 - 55.36) = 12,960 \times 24.64 = 319,334\text{ BTU/hr}$$ $$SHR = \frac{\dot{q}_s}{\dot{q}_t} = \frac{319,334}{452,628} = 0.7055 \approx 0.71$$

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7. Common Exam Traps & PE Pro-Tips

• Trap 1 — Using Standard Air Multipliers at High Elevation: The constants $1.08$, $4840$, and $4.5$ are valid only at standard sea-level air density ($\rho = 0.075\text{ lbm/ft}^3$, $p = 14.696\text{ psia}$). At non-standard altitudes, you must calculate mass flow directly using $\dot{m} = \rho \dot{V} = \frac{p \dot{V}}{R_a T}$ and apply $\dot{q}_s = \dot{m} c_p \Delta T$ and $\dot{q}_t = \dot{m} \Delta h$.
• Trap 2 — Misinterpreting Apparatus Dew Point Location: The Apparatus Dew Point (ADP) is always located on the saturation curve ($\phi = 100\%$). It represents the effective coil surface temperature if the coil had a bypass factor of zero ($BF = 0$). Leaving air dry-bulb is always higher than the ADP whenever $BF > 0$.
• Trap 3 — Mixing Moisture Units: When computing latent heat with humidity ratio $\omega$ in $\text{lbm}_w/\text{lbm}_{da}$, use the $4840$ constant. When using moisture in grains per pound ($W_{gr}$), use the $0.69$ constant ($4840 / 7000 = 0.691$).