Latent and Sensible Heat Formulae and Derivations

Home , Sensible heat

Total Heat Formula: QT = 4.5 x cfm x Δh

Sensible Heat Formula: QS = 1.08 x cfm x ΔT

Latent Heat Formula: QL = 0.68 x cfm x ΔW

where

Δh = The change in enthalpy between return and supply air

ΔT = The difference between the dry bulb temperature of the return air and the dry bulb temperature of the supply air

ΔW = The difference between the moisture content (in grains per pound of dry air) of the return and supply air

CFM = Volume of air passing through the heat exchange coil, in cubic feet per minute

Derivation of the Total Heat Formula

The total heat content of air takes into account both latent and sensible heat aspects. The total heat also referred to as enthalpy. The enthalpy is provided in btu/lb and can be found by following the wet-bulb temperature lines on the psychrometric chart. Total heat is in the units of btu/hour, so the resulting calculations must also provide the correct units.

QT  (btu/hr) = Δh (btu/lb) x ? x ? x ?

Since the amount of heat transfer depends in part by the amount of air moving through the heat exchanger, we must take the cfm into account.

QT  (btu/hr) = Δh (btu/lb) x Airflow (cubic feet/minute) x ? x ?

Since our total heat is in the units of btu/hour and we presently have “minute” units, we must convert to hours by mulitplying by 60.

QT  (btu/hr) = Δh (btu/lb) x Airflow (cubic feet/minute) x 60 minutes/hour x ?

We must also get rid of the cubic feet in the numerator as well as the lb in the denominator to keep the units in order. The specific volume of the air is in the units of cubic feet per pound. Since the “cubic feet” is in the numerator of the above equation, it must be in the denominator to cancel out when multiplied. So, we use the density of the air, which is in the units of lb/cubic foot.

When dealing with return and supply air, the average density is used. We calculate this by adding the density of the return air to the density of the supply air and then divide by 2.

On average, the density of the air is about 0.075 lb/cubic foot.

3 3 QT  (btu/hr) = Δh (btu/lb) x Airflow (ft /minute) x 60 minutes/hour x 0.075 lb/ft

Since the “60” conversion between minutes and hours and the 0.075 lb/ft3 remain constant (more or less), we simplify the equation by multiplying these two items together to get 4.5.

3 QT  (btu/hr) = Δh (btu/lb) x Airflow (ft /minute) x 4.5

It is important to keep in mind that this calculation is only an estimate. To ensure a more accurate calculation, determine the actual densities of the return and supply air and average these two values.

Derivation of the Sensible Heat Formula

The sensible heat content of air takes into account only the sensible heat portion. Sensible heat transfers are measured with a thermometer and are represented on the psychrometric chart by changes in the dry bulb temperature. Sensible heat is also expressed in the units of btu/hour, so the resulting calculations must also provide the correct units.

QS  (btu/hr) = ΔT (°F) x ? x ? x ?

Since the amount of heat transfer depends in part by the amount of air moving through the heat exchanger, we must take the cfm into account.

QT  (btu/hr) = ΔT (°F) x Airflow (cubic feet/minute) x ? x ?

Since our sensible heat is in the units of btu/hour and we presently have “minute” units, we must convert to hours by mulitplying by 60.

QT  (btu/hr) = ΔT (°F) x Airflow (cubic feet/minute) x 60 minutes/hour x ?

We must also get rid of the “cubic feet” and “°F” in the numerator. The specific volume of the air is in the units of cubic feet per pound. Since the “cubic feet” is in the numerator of the above equation, it must be in the denominator to cancel out when multiplied. So, we use the density of the air, which is in the units of lb/cubic foot.

When dealing with return and supply air, the average density is used. We calculate this by adding the density of the return air to the density of the supply air and then divide by 2.

On average, the density of the air is about 0.075 lb/cubic foot.

3 3 QT  (btu/hr) = ΔT (°F) x Airflow (ft /minute) x 60 minutes/hour x 0.075 lb/ft

To complete the formula, we need a “btu” term on the right hand side, we need to get rid of the “°F” on the right side of the equation and we need to get rid of the “lb.” term on the right hand side.

Specific heat represents the amount of heat required to raise one pound of a substance 1 degree F and is in the units of btu/lb/°F. The specific heat of air is 0.24. This gives us

3 3 QT (btu/hr) = ΔT (°F) x Airflow (ft /minute) x 60 min/hour x 0.075 lb/ft x 0.24 btu/lb/°F

Since the 60, 0.075 and the 0.24 are constants (more or less) we can multiply them together to simplify the calculations.

60 x 0.075 x 0.24 = 1.08

So the formula becomes

QS = 1.08 x cfm x ΔT

Derivation of the Latent Heat Formula

The latent heat content of air takes into account only the heat transfers that resuilt in a change of state with no resulting change in temperature. Latent heat transfers are indicated on the right hand side of the psychrometric chart on the “grains per pound of dry air” and “pounds of moisture per pound of dry air” scales. Latent heat, as with the sensible and latent heat calculations, is also expressed in the units of btu/hour, so the resulting calculations must also provide the correct units.

QL  (btu/hr) = ΔW (grains/lb) x ? x ? x ?

Since the amount of heat transfer depends in part by the amount of air moving through the heat exchanger, we must take the cfm into account.

QL  (btu/hr) = ΔW (grains/lb) x Airflow (cubic feet/minute) x ? x ?

Since our latent heat value is in the units of btu/hour and we presently have “minute” units, we must convert to hours by mulitplying by 60.

QL  (btu/hr) = ΔW (grains/lb) x Airflow (cubic feet/minute) x 60 minutes/hour x ?

We must also get rid of the “cubic feet” and “grains” in the numerator and the “lb” in the denominator. The specific volume of the air is in the units of cubic feet per pound. Since the “cubic feet” is in the numerator of the above equation, it must be in the denominator to cancel out when multiplied. So, we use the density of the air, which is in the units of lb/cubic foot.

When dealing with return and supply air, the average density is used. We calculate this by adding the density of the return air to the density of the supply air and then divide by 2.

On average, the density of the air is about 0.075 lb/cubic foot.

3 3 QL (btu/hr) = ΔW (grains/lb) x Airflow (ft /minute) x 60 min/hour x 0.075 lb/ft

To complete the formula, we need a “btu” term on the right hand side, we need to get rid of the “grains” on the right side of the equation and we need to get rid of the “lb” term on the right hand side.

Since we have air entering the coil and are removing moisture, we must determine the latent heat activity within the coil. We do this by subtracting the enthalpy of the condensed vapor at 50°F from the enthalpy of the 75°F water vapor that is entering the coil. The enthalpy of 75°F water vapor is about 1094 btu/lb while the enthalpy of the 50°F condensed water vapor is about 18 btu/lb. This gives us a net latent heat transfer of about 1076 btu/lb (1094 btu/lb – 18 btu/lb)

3 3 QL (btu/hr) = ΔW(grains/lb) x Flow(ft /minute) x 60 min/hour x 0.075 lb/ft x 1076 btu/lb

It should be noted that the “lb” in “grains/lb” represents dry air, while the “lb” in “btu/lb” represents water

We now need to get rid of the “grains” term by using the conversion 7,000 grains/lb. This “lb” represents water and will cancel out the “lb” in “btu/lb”.

The formula now becomes:

3 3 QL = ΔW(gr/lb) x Flow(ft /min) x 60 min/hour x 0.075 lb/ft x 1076 btu/lb x 1 lb/7000gr

By simplifying the constant terms we can get:

(60 x 0.075 x 1076) ÷ 7000 = 0.68

So the formula becomes

QL = 0.68 x cfm x ΔW