Formula Sheet: Psychrometrics

Fundamental Properties of Moist Air

Dry-Bulb Temperature

• Dry-bulb temperature (T_db or DBT): The actual temperature of air measured by a standard thermometer
• Units: °F or °C

Wet-Bulb Temperature

• Wet-bulb temperature (T_wb or WBT): The temperature indicated by a thermometer covered with a water-saturated wick over which air is passed
• Units: °F or °C
• Note: T_wb ≤ T_db for unsaturated air; T_wb = T_db at saturation (100% RH)

Dew Point Temperature

• Dew point temperature (T_dp or DPT): The temperature at which condensation begins when moist air is cooled at constant pressure
• Units: °F or °C
• Note: T_dp ≤ T_wb ≤ T_db for unsaturated air

Humidity Properties

Humidity Ratio (Absolute Humidity)

• Humidity ratio (W): Mass of water vapor per unit mass of dry air

\[W = \frac{m_w}{m_a}\]

• m_w = mass of water vapor (lb or kg)
• m_a = mass of dry air (lb or kg)
• Units: lb_w/lb_a or kg_w/kg_a (dimensionless ratio)

Humidity Ratio from Partial Pressures

\[W = 0.622 \frac{p_w}{p - p_w}\]

• p_w = partial pressure of water vapor (psia or kPa)
• p = total atmospheric pressure (psia or kPa)
• 0.622 = ratio of molecular weights (M_w/M_a = 18.015/28.965)
• Note: (p - p_w) = partial pressure of dry air

Relative Humidity

• Relative humidity (φ or RH): Ratio of actual water vapor pressure to saturation pressure at the same temperature

\[\phi = \frac{p_w}{p_{ws}} \times 100\%\]

• p_w = partial pressure of water vapor (psia or kPa)
• p_ws = saturation pressure of water vapor at dry-bulb temperature (psia or kPa)
• Units: % (expressed as percentage)
• Range: 0% (completely dry) to 100% (saturated)

Alternative Definition Using Humidity Ratios

\[\phi = \frac{W}{W_s} \times 100\%\]

• W = actual humidity ratio (lb_w/lb_a)
• W_s = humidity ratio at saturation at the same temperature (lb_w/lb_a)

Degree of Saturation

\[\mu = \frac{W}{W_s}\]

• μ = degree of saturation (dimensionless ratio)
• W = actual humidity ratio (lb_w/lb_a)
• W_s = humidity ratio at saturation (lb_w/lb_a)
• Note: Degree of saturation is similar to relative humidity but not identical

Saturation Humidity Ratio

\[W_s = 0.622 \frac{p_{ws}}{p - p_{ws}}\]

• W_s = saturation humidity ratio (lb_w/lb_a)
• p_ws = saturation pressure at the given temperature (psia or kPa)
• p = total atmospheric pressure (psia or kPa)

Specific Volume

Specific Volume of Moist Air

\[v = \frac{V}{m_a}\]

• v = specific volume (ft³/lb_a or m³/kg_a)
• V = total volume of moist air (ft³ or m³)
• m_a = mass of dry air (lb_a or kg_a)

Using Ideal Gas Law (US Customary Units)

\[v = \frac{R_a T}{p_a} = \frac{53.352 \, T}{p - p_w}\]

• R_a = gas constant for dry air = 53.352 ft·lbf/(lb_a·°R)
• T = absolute temperature (°R = °F + 459.67)
• p_a = partial pressure of dry air (psia)
• p = total pressure (psia)
• p_w = partial pressure of water vapor (psia)
• Units: ft³/lb_a

Using Ideal Gas Law (SI Units)

\[v = \frac{R_a T}{p_a} = \frac{287.055 \, T}{p - p_w}\]

• R_a = gas constant for dry air = 287.055 J/(kg·K)
• T = absolute temperature (K = °C + 273.15)
• p_a = partial pressure of dry air (kPa)
• Units: m³/kg_a

Corrected for Water Vapor Content

\[v = \frac{R_a T}{p} (1 + 1.6078 \, W)\]

• W = humidity ratio (lb_w/lb_a)
• 1.6078 = ratio R_w/R_a for water vapor to dry air

Enthalpy of Moist Air

Total Enthalpy

• Enthalpy (h): Total heat content of moist air per unit mass of dry air

\[h = h_a + W \, h_w\]

• h = total enthalpy of moist air (Btu/lb_a or kJ/kg_a)
• h_a = enthalpy of dry air (Btu/lb_a or kJ/kg_a)
• W = humidity ratio (lb_w/lb_a)
• h_w = enthalpy of water vapor (Btu/lb_w or kJ/kg_w)

Standard Approximation (US Customary)

\[h = 0.240 \, T_{db} + W (1061 + 0.444 \, T_{db})\]

• h = enthalpy (Btu/lb_a)
• T_db = dry-bulb temperature (°F)
• W = humidity ratio (lb_w/lb_a)
• 0.240 = specific heat of dry air (Btu/(lb_a·°F))
• 1061 = latent heat of vaporization at 0°F (Btu/lb_w)
• 0.444 = specific heat of water vapor (Btu/(lb_w·°F))

Standard Approximation (SI)

\[h = 1.006 \, T_{db} + W (2501 + 1.86 \, T_{db})\]

• h = enthalpy (kJ/kg_a)
• T_db = dry-bulb temperature (°C)
• W = humidity ratio (kg_w/kg_a)
• 1.006 = specific heat of dry air (kJ/(kg_a·K))
• 2501 = latent heat of vaporization at 0°C (kJ/kg_w)
• 1.86 = specific heat of water vapor (kJ/(kg_w·K))

Enthalpy at Saturation

\[h_s = c_{pa} T + W_s (h_{fg} + c_{pw} T)\]

• h_s = enthalpy at saturation (Btu/lb_a or kJ/kg_a)
• c_pa = specific heat of dry air
• W_s = saturation humidity ratio
• h_fg = latent heat of vaporization
• c_pw = specific heat of water vapor

Psychrometric Processes

Sensible Heating or Cooling

• Process: Temperature changes with no change in moisture content (W = constant)
• Horizontal line on psychrometric chart (left to right for heating, right to left for cooling)

\[q_s = m_a \, c_{pa} (T_2 - T_1)\]

• q_s = sensible heat transfer rate (Btu/hr or kW)
• m_a = mass flow rate of dry air (lb_a/hr or kg_a/s)
• c_pa = specific heat of dry air = 0.240 Btu/(lb_a·°F) or 1.006 kJ/(kg_a·K)
• T₂, T₁ = final and initial dry-bulb temperatures

Using Enthalpy Difference

\[q_s = m_a (h_2 - h_1) = 60 \, \rho \, Q (h_2 - h_1)\]

• ρ = density of air (lb_a/ft³ or kg_a/m³)
• Q = volumetric flow rate (CFM or m³/s)
• 60 = conversion factor for CFM to ft³/hr (US units only)

Humidification (Adding Moisture)

• Process: Adding water vapor to air

\[m_w = m_a (W_2 - W_1)\]

• m_w = mass flow rate of water added (lb_w/hr or kg_w/s)
• m_a = mass flow rate of dry air (lb_a/hr or kg_a/s)
• W₂, W₁ = final and initial humidity ratios

Energy Balance for Humidification

\[m_a h_1 + m_w h_w = m_a h_2\]

• h₁ = initial enthalpy of air (Btu/lb_a or kJ/kg_a)
• h₂ = final enthalpy of air (Btu/lb_a or kJ/kg_a)
• h_w = enthalpy of water added (Btu/lb_w or kJ/kg_w)

Dehumidification (Removing Moisture)

• Process: Cooling below dew point to condense and remove water vapor

\[m_w = m_a (W_1 - W_2)\]

• m_w = mass flow rate of condensate removed (lb_w/hr or kg_w/s)
• W₁, W₂ = initial and final humidity ratios

Cooling and Dehumidification Heat Transfer

\[q_t = m_a (h_1 - h_2)\]

• q_t = total heat removal rate (Btu/hr or kW)
• h₁ = initial enthalpy (Btu/lb_a or kJ/kg_a)
• h₂ = final enthalpy (Btu/lb_a or kJ/kg_a)

Adiabatic Mixing of Air Streams

• Process: Combining two air streams with no external heat transfer

Mass Balance for Dry Air

\[m_{a3} = m_{a1} + m_{a2}\]

• m_a1, m_a2 = mass flow rates of dry air in streams 1 and 2
• m_a3 = mass flow rate of dry air in mixed stream 3

Mass Balance for Water Vapor

\[m_{a3} W_3 = m_{a1} W_1 + m_{a2} W_2\] \[W_3 = \frac{m_{a1} W_1 + m_{a2} W_2}{m_{a1} + m_{a2}}\]

• W₁, W₂, W₃ = humidity ratios of streams 1, 2, and 3

Energy Balance

\[m_{a3} h_3 = m_{a1} h_1 + m_{a2} h_2\] \[h_3 = \frac{m_{a1} h_1 + m_{a2} h_2}{m_{a1} + m_{a2}}\]

• h₁, h₂, h₃ = enthalpies of streams 1, 2, and 3

Mixing Ratio

\[\frac{m_{a1}}{m_{a2}} = \frac{W_3 - W_2}{W_1 - W_3} = \frac{h_3 - h_2}{h_1 - h_3}\]

• Note: The mixed state point lies on a straight line connecting states 1 and 2 on the psychrometric chart

Adiabatic Saturation

• Process: Air passes over water in an insulated chamber until saturated
• Final temperature = wet-bulb temperature (for water)

\[h_1 + (W_s - W_1) h_w = h_s\]

• h₁ = initial enthalpy of air (Btu/lb_a or kJ/kg_a)
• W₁ = initial humidity ratio
• W_s = saturation humidity ratio at final temperature
• h_w = enthalpy of makeup water
• h_s = enthalpy at saturation (final state)

Evaporative Cooling

• Process: Adding water to unsaturated air in an adiabatic process
• Follows constant wet-bulb temperature line on psychrometric chart
• Dry-bulb temperature decreases, humidity ratio increases
• Enthalpy remains approximately constant

\[h_1 \approx h_2\]

• Note: Process follows lines of constant wet-bulb temperature sloping downward to the right

Sensible Heat Factor (SHF)

Definition

\[SHF = \frac{q_s}{q_t} = \frac{q_s}{q_s + q_l}\]

• SHF = sensible heat factor (dimensionless ratio)
• q_s = sensible heat load (Btu/hr or kW)
• q_t = total heat load (Btu/hr or kW)
• q_l = latent heat load (Btu/hr or kW)
• Range: 0 to 1 (0 = pure latent load, 1 = pure sensible load)

Using Enthalpy and Temperature

\[SHF = \frac{c_{pa} (T_2 - T_1)}{h_2 - h_1}\]

• T₂, T₁ = final and initial temperatures
• h₂, h₁ = final and initial enthalpies

Latent Heat Load

\[q_l = m_a (W_2 - W_1) h_{fg}\]

• q_l = latent heat load (Btu/hr or kW)
• W₂, W₁ = final and initial humidity ratios
• h_fg = latent heat of vaporization ≈ 1060 Btu/lb_w or 2501 kJ/kg_w

Simplified Form (US Customary)

\[q_l = 4840 \, Q (W_2 - W_1)\]

• Q = volumetric flow rate (CFM)
• 4840 = conversion factor (approximately 60 × 0.075 × 1076)
• Units: Btu/hr

Psychrometric Chart Applications

Chart Components

• Horizontal axis: Dry-bulb temperature
• Vertical axis (right): Humidity ratio
• Curved lines: Constant relative humidity
• Nearly vertical lines: Constant wet-bulb temperature
• Curved saturation line: 100% relative humidity (φ = 100%)
• Diagonal lines: Constant enthalpy (nearly parallel to wet-bulb lines)
• Oblique lines: Constant specific volume

Process Lines on Chart

• Sensible heating: Horizontal line to the right (W constant, T increases)
• Sensible cooling: Horizontal line to the left (W constant, T decreases)
• Humidification (steam injection): Vertical or nearly vertical upward (W increases)
• Cooling and dehumidification: Downward and to the left
• Evaporative cooling: Along constant wet-bulb line, downward to right
• Heating and humidification: Upward and to the right
• Adiabatic mixing: Straight line connecting two state points

Density and Volumetric Flow Relationships

Density of Moist Air

\[\rho = \frac{1}{v}\]

• ρ = density of moist air (lb_a/ft³ or kg_a/m³)
• v = specific volume (ft³/lb_a or m³/kg_a)

From Ideal Gas Law

\[\rho = \frac{p}{R_a T (1 + 1.6078 \, W)}\]

• p = atmospheric pressure (psia or kPa)
• R_a = gas constant for dry air
• T = absolute temperature (°R or K)
• W = humidity ratio

Mass Flow Rate from Volumetric Flow

\[m_a = \frac{Q}{v} = \rho \, Q\]

• m_a = mass flow rate of dry air (lb_a/min or kg_a/s)
• Q = volumetric flow rate (CFM or m³/s)
• v = specific volume (ft³/lb_a or m³/kg_a)
• ρ = density (lb_a/ft³ or kg_a/m³)

Conversion to Standard Flow Rate (US Customary)

\[m_a = \frac{Q \times 60}{v}\]

• Q = volumetric flow rate (CFM)
• 60 = conversion from minutes to hours
• m_a = mass flow rate (lb_a/hr)

Heat and Mass Transfer Rates

Total Heat Transfer Rate

\[q_t = m_a (h_2 - h_1)\]

• q_t = total heat transfer rate (Btu/hr or kW)
• m_a = mass flow rate of dry air (lb_a/hr or kg_a/s)
• h₂, h₁ = final and initial enthalpies (Btu/lb_a or kJ/kg_a)

Using Volumetric Flow (US Customary)

\[q_t = 60 \, \rho \, Q (h_2 - h_1) = 4.5 \, Q (h_2 - h_1)\]

• Q = volumetric flow rate (CFM)
• ρ = density ≈ 0.075 lb_a/ft³ (standard air)
• 4.5 = approximate conversion factor (60 × 0.075)
• Units: Btu/hr

Sensible Heat Rate

\[q_s = m_a \, c_{pa} (T_2 - T_1)\]

• c_pa = 0.240 Btu/(lb_a·°F) or 1.006 kJ/(kg_a·K)

Using Volumetric Flow (US Customary)

\[q_s = 1.08 \, Q (T_2 - T_1)\]

• Q = volumetric flow rate (CFM)
• T₂, T₁ = final and initial temperatures (°F)
• 1.08 = conversion factor (60 × 0.075 × 0.240)
• Units: Btu/hr

Using Volumetric Flow (SI)

\[q_s = 1.2 \, Q (T_2 - T_1)\]

• Q = volumetric flow rate (m³/s)
• T₂, T₁ = final and initial temperatures (°C)
• 1.2 = approximate conversion factor (ρ × c_pa ≈ 1.2 × 1.006)
• Units: kW

Latent Heat Rate

\[q_l = m_a \, h_{fg} (W_2 - W_1)\]

• h_fg = latent heat of vaporization ≈ 1060 Btu/lb_w or 2501 kJ/kg_w

Using Volumetric Flow (US Customary)

\[q_l = 4840 \, Q (W_2 - W_1)\]

• Q = volumetric flow rate (CFM)
• W₂, W₁ = final and initial humidity ratios (lb_w/lb_a)
• 4840 = conversion factor (60 × 0.075 × 1076)
• Units: Btu/hr

Cooling Coil Performance

Apparatus Dew Point (ADP)

• Apparatus dew point: The effective surface temperature of a cooling coil
• Represents the temperature at which all air would exit if the coil had 100% contact efficiency
• Located on saturation curve of psychrometric chart

Bypass Factor

\[BF = \frac{T_2 - T_{ADP}}{T_1 - T_{ADP}}\]

• BF = bypass factor (fraction of air that bypasses the coil without contact)
• T₁ = entering air dry-bulb temperature
• T₂ = leaving air dry-bulb temperature
• T_ADP = apparatus dew point temperature
• Range: 0 to 1 (0 = perfect contact, 1 = no contact)

Contact Factor

\[CF = 1 - BF = \frac{T_1 - T_2}{T_1 - T_{ADP}}\]

• CF = contact factor (fraction of air effectively contacting the coil)
• Range: 0 to 1 (1 = perfect contact)

Coil Heat Removal

\[q_{coil} = m_a (h_1 - h_2)\]

• q_coil = total heat removal by coil (Btu/hr or kW)
• h₁ = enthalpy of entering air
• h₂ = enthalpy of leaving air

Standard Air Conditions

Standard Air Properties (US Customary)

• Temperature: 70°F (529.67°R)
• Pressure: 14.696 psia (1 atm)
• Relative humidity: 0% (dry air)
• Density: 0.075 lb/ft³
• Specific volume: 13.33 ft³/lb

Standard Air Properties (SI)

• Temperature: 20°C (293.15 K)
• Pressure: 101.325 kPa (1 atm)
• Relative humidity: 0% (dry air)
• Density: 1.204 kg/m³
• Specific volume: 0.8305 m³/kg

Useful Conversion Factors

Temperature Conversions

• °F to °R: °R = °F + 459.67
• °C to K: K = °C + 273.15
• °F to °C: °C = (°F - 32) / 1.8
• °C to °F: °F = 1.8 × °C + 32

Pressure Conversions

• 1 atm = 14.696 psia = 101.325 kPa
• 1 psi = 6.895 kPa
• 1 in Hg = 0.4912 psi = 3.386 kPa

Common Constants

• Gas constant for dry air (R_a): 53.352 ft·lbf/(lb_a·°R) or 287.055 J/(kg_a·K)
• Gas constant for water vapor (R_w): 85.78 ft·lbf/(lb_w·°R) or 461.52 J/(kg_w·K)
• Molecular weight ratio (M_w/M_a): 0.622
• Specific heat of dry air (c_pa): 0.240 Btu/(lb_a·°F) or 1.006 kJ/(kg_a·K)
• Specific heat of water vapor (c_pw): 0.444 Btu/(lb_w·°F) or 1.86 kJ/(kg_w·K)
• Latent heat of vaporization (h_fg): ≈ 1060 Btu/lb_w at 70°F or ≈ 2501 kJ/kg_w at 0°C