# Humid Air - Heating

## Enthalpy change and temperature rise when heating humid air without adding moisture.

The process of **sensible heating** of air - heating without adding moisture - can be visualized in the Mollier diagram as:

Sensible heating of air changes the state of the air from A to B along the constant specific humidity - *x* - line. The supplied heat - *dH* - can be estimated as indicated in the diagram above.

The heating process can also be visualized in the psychrometric chart

Note! - when sensible heating of air - the specific moisture remains constant - the relative humidity is decreased.

### Calculating Enthalpy

The enthalpy of moist air can be calculated as:

h = c_{pa}t + x [c_{pw}t + h_{we}](1)

where

h= specific enthalpy of moist air (kJ/kg)

c_{pa}= 1.01 - specific heat capacity of air at constant pressure (kJ/kg^{o}C, kWs/kgK)

t= air temperature (^{o}C)

x= humidity ratio (kg/kg)

c_{pw}= 1.84 - specific heat capacity of water vapor at constant pressure (kJ/kg.^{o}C, kWs/kg.K)

h_{we}= 2502 - evaporation heat of water at 0^{o}C (kJ/kg)

(1) can be modified to:

h = 1.01 (kJ/kg.^{o}C) t + x [1.84 (kJ/kg.^{o}C) t + 2502 (kJ/kg)](1b)

### The Enthalpy Difference

The enthalpy difference when heating air without adding moisture can be calculated as:

dh_{A-B}= c_{pa}t_{B}+ x [c_{pw}t_{B}+ h_{we}] - c_{pa}t_{A}+ x [c_{pw}t_{A}+ h_{we}]

= c_{pa}(t_{B}- t_{A}) + x c_{pw}(t_{B}- t_{A})(2)

#### Example - Enthalpy Change when Heating Air

The specific humidity of air at 25^{o}C and relative moisture *50%* is *0.0115 kg/kg* - check the Mollier diagram. The change in enthalpy when heating the air to *35 ^{o}C* can be calculated as:

dh_{A-B}=(1.01 kJ/kg^{o}C)(35^{o}C - 25^{o}C) + (0.0115 kg/kg) (1.84 kJ/kg^{o}C) (35^{o}C - 25^{o}C)

= (10.1 kJ/kg) + (0.2 kJ/kg)

= 10.3 (kJ/kg)

**Note!** - the contribution from the water vapor is relatively small and can for practical purposes often be neglected. (2) can then be modified to:

dh_{A-B}= c_{pa}( t_{B}- t_{A})(2b)

### Increase in Temperature when Heating Air

If heat is added to humid air the increase in air temperature can be calculated by modifying *(2b)* to:

t_{B}- t_{A}= dh_{A-B}/ c_{pa}(2c)

#### Example - Heating Air and Temperature Rise

If *10.1 kJ* is added to *1 kg* air the temperature rise can be calculated as:

t_{B}- t_{A}=(10.1 kJ/kg) / (1.01 kJ/kg^{o}C)

= 10^{ }(^{o}C)

### Heat Flow in a Heating Coil

The total heat flow rate through a heating coil can be calculated as:

q = m (h_{B}- h_{A})(3)

where

q= heat flow rate (kJ/s, kW)

m= mass flow rate of air (kg/s)

The total heat flow can also be expressed as:

q_{s}= L ρ (h_{B}- h_{A})(3a)

where

L= air flow rate (m^{3}/s)

ρ= density of air (kg/m^{3})

**Note!** The density of air varies with temperature. At *0 ^{o}C* the density is

*1.293 kg/m*. At

^{3}*80*the density is

^{o}C*1.0 kg/m*.

^{3}It's common to express sensible heat flow rate as:

q = m c_{pa}(t_{B}- t_{A})(3b)

or alternatively:

q = L ρ c_{pa}(t_{B}- t_{A})(3c)

### Heating Coil Effectiveness

For a limited heating coil surface the average surface temperature will always be higher than the outlet air temperature. The effectiveness of a heating coil can be expressed as:

μ =(t_{B}- t_{A}) /(t_{HC}- t_{A})(4)

where

μ= heating coil effectiveness

t_{HC}= mean surface temperature of the heating coil (^{o}C)

### Example - Heating Air

*1 m ^{3}/s* of air at

*15*and relative humidity

^{o}C*60%*(A) is heated to

*30*(B). The surface temperature of the heating coil is

^{o}C*80*. The density of air at

^{o}C*20*is

^{o}C*1.205 kg/m*.

^{3}From the Mollier diagram the enthalpy in (A) is *31 kJ/kg* and in (B) *46 kJ/kg*.

The heating coil effectiveness can be calculated as:

μ =(30^{o}C - 15^{o}C) / (80^{o}C - 15^{o}C)

= 0.23

The heat flow can be calculated as:

q =(1 m^{3}/s) (1.205 kg/m^{3}) ((46 kJ/kg) - (31 kJ/kg))

= 18 (kJ/s, kW)

As an alternative, as one of the most common methods:

q =(1 m^{3}/s) (1.205 kg/m^{3}) (1.01 kJ/kg.^{o}C) (30^{o}C - 15^{o}C)

= 18.3 (kJ/s, kW)

**Note!** Due to the inaccuracy when working with diagrams there is a small difference between the total heat flow and the sum of the latent and sensible heat. In general - this inaccuracy is within acceptable limits.