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# Cooling and Heating Equations

## Latent and sensible cooling and heating equations - imperial units

### Sensible Heat

The sensible heat in a heating or cooling process of air (heating or cooling capacity) can be calculated in SI-units as

hs = cp ρ q dt                                              (1)

where

hs = sensible heat (kW)

cp = specific heat of air (1.006 kJ/kg oC)

ρ = density of air (1.202 kg/m3)

q = air volume flow (m3/s)

dt = temperature difference (oC)

Or in Imperial units as

hs = 1.08 q dt                                            (1b)

where

hs = sensible heat (Btu/hr)

q = air volume flow (cfm, cubic feet per minute)

dt = temperature difference (oF)

### Example - Heating Air, Sensible Heat

Metric Units

An air flow of 1 m3/s is heated from 0 to 20oC. Using (1) the sensible heat added to the air can be calculated as

hs = (1.006 kJ/kg oC) (1.202 kg/m3) (1 m3/s) ((20 oC) - (0 oC))

= 24.2 (kW)

Imperial Units

An air flow of 1 cfm is heated from 32 to 52oF. Using (1b) the sensible heat added to the air can be calculated as

hs = 1.08 (1 cfm) ((52 oF) - (32 oF))

= 21.6 (Btu/hr)

#### Sensible Heat Load and Required Air Volume Chart

Sensible heat load and required air volume to keep the temperature constant at various temperature differences between make up air and room air: ### Latent Heat

Latent heat due to the moisture in air can be calculated in SI-units as:

hl = ρ hwe q dwkg                                      (2)

where

hl = latent heat (kW)

ρ = density of air (1.202 kg/m3)

q = air volume flow (m3/s)

hwe = latent heat evaporization water (2454 kJ/kg - in air at atmospheric pressure and 20oC)

dwkg = humidity ratio difference (kg water/kg dry air)

Latent evaporation heat for water can be calculated as

hwe = 2494 - 2.2 t                  (2a)

where

t = evaporation temperature (oC)

Or for Imperial units:

hl = 0.68 q dwgr                                      (2b)

or

hl = 4840 q dwlb                                     (2c)

where

hl= latent heat (Btu/hr)

q = air volume flow (cfm, cubic feet per minute)

dwgr = humidity ratio difference (grains water/lb dry air)

dwlb = humidity ratio difference (lb water/lb dry air)

### Example - Cooling Air, Latent Heat

Metric Units

An air flow of 1 m3/s is cooled from 30 to 10oC. The relative humidity of the air is 70% at the start and 100% at the end of the cooling process.

From the Mollier diagram we estimate the water content in the hot air to be 0.0187 kg water/kg dry air, and the water content in the cold air to be 0.0075 kg water/kg dry air.

Using (2) the latent heat removed from the air can be calculated as

hl = (1.202 kg/m3(2454 kJ/kg) (1 m3/s) ((0.0187 kg water/kg dry air) - (0.0075 kg water/kg dry air))

= 34.3 (kW)

Imperial Units

An air flow of 1 cfm is cooled from 52 to 32oF. The relative humidity of the air is 70% at the start and 100% at the end of the cooling process.

From the psychrometric chart we estimate the water content in the hot air to be 45 grains water/lb dry air, and the water content in the cold air to be 27 grains water/lb dry air.

Using (2b) the latent heat removed from the air can be calculated as

hl = 0.68 (1 cfm) ((45 grains water/lb dry air) - (27 grains water/lb dry air))

= 12.2 (Btu/hr)

#### Latent Heat Load and Required Air Volume Chart

Latent heat load - humidifying and dehumidifying - and required air volume to keep temperature constant at various temperature differences between entering air and room air are indicated in the chart below: ### Total Heat - Latent and Sensible Heat

Total heat due to both temperature and moisture can be expressed in SI units as:

ht = ρ q dh                                            (3)

where

ht = total heat (kW)

q = air volume flow (m3/s)

ρ = density of air (1.202 kg/m3)

dh = enthalpy difference (kJ/kg)

Or - in imperial units:

ht = 4.5 q dh                                        (3b)

where

ht= total heat (Btu/hr)

q = air volume flow (cfm, cubic feet per minute)

dh = enthalpy difference (btu/lb dry air)

Total heat can also be expressed as:

ht = hs + hl

= 1.08 q dt + 0.68 q dwgr                                      (4)

### Example - Cooling or Heating Air, Total Heat

Metric Units

An air flow of 1 m3/s is cooled from 30 to 10oC. The relative humidity of the air is 70% at the start and 100% at the end of the cooling process.

From the Mollier diagram we estimate the water enthalpy in the hot air to be 77 kJ/kg dry air, and the enthalpy in the cold air to be 28 kJ/kg dry air.

Using (3) the total sensible and latent heat removed from the air can be calculated as

ht = (1.202 kg/m3) (1 m3/s) ((77 kJ/kg dry air) - (28 kJ/kg dry air))

= 58.9 (kW)

Imperial Units

An air flow of 1 cfm is cooled from 52 to 32oF. The relative humidity of the air is 70% at the start and 100% at the end of the cooling process.

From the psychrometric chart we estimate the water enthalpy in the hot air to be 19 Btu/lb dry air, and the enthalpy in the cold air to be 13.5 Btu/lb dry air.

Using (3b) the total sensible and latent heat removed from the air can be calculated as

ht = 4.5 (1 cfm) ((19 Btu/lb dry air) - (13.5 Btu/lb dry air))

= 24.8 (Btu/hr)

### SHR - Sensible Heat Ratio

The Sensible Heat Ratio can be expressed as

SHR = hs / ht                              (6)

where

SHR = Sensible Heat Ratio

hs = sensible heat

ht = total heat (sensible and latent)

## Related Topics

• Air Conditioning - Air Conditioning systems - heating, cooling and dehumidification of indoor air for thermal comfort

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## Citation

• Engineering ToolBox, (2004). Cooling and Heating Equations. [online] Available at: https://www.engineeringtoolbox.com/cooling-heating-equations-d_747.html [Accessed Day Mo. Year].

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