Let’s assume a simple steady state air system where moisture air flows in one direction, and an enveloping duct is perfectly insulated. And Let’s say 1 refers to inlet status, 2 outlet status, \(\dot{Q}\) heat added (or extracted if negative), \(\dot{m}_{da}\) dry air mass rate, \(\dot{m}_{w}\) mass rate of water added (extracted if negative), W humidity ratio, h moisture air enthalpy in the state specified by a subscript, \(h_{f}\) liquid water enthalpy. Then the following laws of conservation of mass and energy apply:

- Mass balance of dry air

\(\dot{m}_{da,1} = \dot{m}_{da,2}\) - Mass balance of water vapor

\(\dot{m}_{da,1}W_{1} + \dot{m}_{w} = \dot{m}_{da,2}W_{2}\) - Heat balance of moist air

\(\dot{m}_{da,1} + \dot{m}_{w}h_{w} + \dot{Q} = \dot{m}_{da,2}\)

These laws of conservation are the basis of all HVAC process applications that follow. The same symbols hold true unless noted otherwise.

- energy balance

\(\dot{m}_{da,1}h_{1} + \dot{Q} = \dot{m}_{da,2}h_{2}\)

- mass balance of dry air

\(\dot{m}_{da,1} + \dot{m}_{da,2} = \dot{m}_{da,3}\) - mass balance of water

\(\dot{m}_{da,1}W_{1} + \dot{m}_{da,2}W_{2} = \dot{m}_{da,3}W_{3}\) - energy balance

\(\dot{m}_{da,1}h_{1} + \dot{m}_{da,2}h_{2} = \dot{m}_{da,3}h_{3}\)

where 1 inlet 1 state, 2 inlet 2 state, 3 outlet state

- application: cooling coil
- Dehumidification occurs only when the cooling coil surface temperature is lower than the dew point of incoming air.
- The condition of air leaving the cooling coil is not on the saturation curve because a portion of air bypasses or is not in intimate contact with the cooling coil.
- lower bypass rate is not necessarily good because it increases pressure drop
- mass balance of water and condensation rate

\(\dot{m}_{da}W_{1} = \dot{m}_{w} + \dot{m}_{da}W_{2}\)

\(\dot{m}_{w} = \dot{m}_{da}(W_{1} - W_{2})\) - energy balance and cooling coil load

\(\dot{m}_{da}h_{1} = q_{cc} + \dot{m}_{da}h_{2} + \dot{m}_{w}h_{f}\)

\(\dot{q}_{cc} = \dot{m}_{da}(h_{1} - h_{2}) - \dot{m}_{w}h_{f} \approx \dot{m}_{a}(h_{1} - h_{2})\)

where \(h_{f}\) liquid water enthalpy at the cooling coil leaving temperature condition, \(q_{cc}\) cooling coil load.

\(\dot{m}_{w}h_{f}\) is usually negligible compared to \(\dot{m}_{da}(h_{1} - h_{2})\)

- application: space sensible and latent cooling load, heater and water spray
- mass balance of water and evaporation rate

\(\dot{m}_{da}W_{1} + \dot{m}_{w} = \dot{m}_{da}W_{2}\)

\(\dot{m}_{w} = \dot{m}_{da}(W_{2} - W_{1})\) - energy balance

\(\dot{m}_{da}h_{1} + q_{heat} + \dot{m}_{w}h_{f} = \dot{m}_{da}h_{2}\) - cooling sensible and latent load

\(\dot{q}_{sen} + \dot{q}_{lat} = \dot{m}_{da}(h_{2} - h_{1})\) - heating and humidification

\(\dot{q}_{hc} + \dot{m}_{w}h_{f} = \dot{m}_{da}(h_{2} - h_{1})\)

where \(\dot{q}_{hc}\) is heating coil load

- application: humidifier water spray, cooling tower
- when unsaturated moist air passes through water, a certain amount of water evaporate into the air. The latent heat is drawn from the air. This results in temperature drop and humidity ratio rise of the air.
- Although the process occurs along a constant wet-bulb temperature
line, it is not strictly isenthalpic because of a latent heat term.
- mass balance of water

\(\dot{m}_{da}W_{1} + \dot{m}_{w} = \dot{m}_{da}W_{2}\)

\(\dot{m}_{w} = \dot{m}_{da}(W_{2} - W_{1})\) - energy balance

\(\dot{m}_{da}h_{1} + \dot{m}_{w}h_{w} = \dot{m}_{da}h_{2}\) - overall energy balance on a cooling tower

\(\dot{m}_{da}(h_{air,out} - h_{air,in}) = \dot{m}_{w}(h_{water,in} - h_{water,out}) + \dot{m}_{da}(W_{out} - W_{in})h_{f}\)

The evaporative cooling term, \(\dot{m}_{da}(W_{out} - W_{in})h_{f}\), is smaller compared to the other two major terms.

- T.A. Reddy, et al (2016) Heating and cooling of buildings. Principles and practice of energy efficiency deisgn. 3rd edition