Energy Balance of basic HVAC processes

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.

1. Sensible heating and cooling

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

2. Adiabatic mixing

• 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

3. Cooling and dehumidification

• 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 \(\dot{m}_{da}(h_{1} - h_{2})\)

4. Heating and humidification

• 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

5. Evaporative cooling

• 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.

References

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

Back