3.4 Mixed Air Calculations & Complex Air Conditioning Cycles (Preheat, Reheat & Bypass)

3.4 Mixed Air Calculations & Complex Air Conditioning Cycles (Preheat, Reheat & Bypass)

Practical HVAC systems rarely operate with a single heating or cooling coil treating 100% recirculated air. Modern air-handling units (AHUs) blend outdoor ventilation air with recirculated return air, navigate freeze-protection constraints, compensate for supply fan thermal energy dissipation, and incorporate reheat or face-and-bypass dampers for precise zone humidity and temperature regulation.

1. Adiabatic Mixing of Two Airstreams

When outdoor air ($OA$) and return air ($RA$) mix adiabatically in an AHU mixing box, the resulting mixed air ($MA$) state point is governed by the conservation of dry air mass, water vapor mass, and thermal energy:

1. Dry Air Mass Balance: $\dot{m}_{ma} = \dot{m}_{oa} + \dot{m}_{ra}$

2. Water Vapor Mass Balance: $\dot{m}_{ma} W_{ma} = \dot{m}_{oa} W_{oa} + \dot{m}_{ra} W_{ra} \implies W_{ma} = \left( \frac{\dot{m}_{oa}}{\dot{m}_{ma}} \right) W_{oa} + \left( \frac{\dot{m}_{ra}}{\dot{m}_{ma}} \right) W_{ra}$

3. Enthalpy (Energy) Balance: $\dot{m}_{ma} h_{ma} = \dot{m}_{oa} h_{oa} + \dot{m}_{ra} h_{ra} \implies h_{ma} = \left( \frac{\dot{m}_{oa}}{\dot{m}_{ma}} \right) h_{oa} + \left( \frac{\dot{m}_{ra}}{\dot{m}_{ma}} \right) h_{ra}$

4. Dry-Bulb Temperature (Sensible Approximation): $T_{ma} = \left( \frac{\dot{m}_{oa}}{\dot{m}_{ma}} \right) T_{oa} + \left( \frac{\dot{m}_{ra}}{\dot{m}_{ma}} \right) T_{ra} = \%OA \cdot T_{oa} + (1 - \%OA) \cdot T_{ra}$

Where: $\%OA = \frac{\dot{m}_{oa}}{\dot{m}_{ma}} \approx \frac{\text{CFM}_{oa}}{\text{CFM}_{ma}}$

Geometric Lever Rule on Psychrometric Chart: The mixed air state point $MA$ lies directly on the straight line connecting state $OA$ and state $RA$. The distance from $RA$ to $MA$ is proportional to the fraction of outdoor air: $\frac{\overline{RA-MA}}{\overline{RA-OA}} = \frac{\dot{m}{oa}}{\dot{m}{ma}}$.

2. Advanced Cycle Components and Phenomena

1. Preheat Coil & Freeze Protection

In cold climates when outdoor air design temperatures fall below $35^\circ\text{F}$, cold outdoor air entering an AHU mixing box risks freezing chilled water or domestic water coils downstream due to air stratification. A preheat coil is located in the 100% outdoor air stream to elevate $T_{oa}$ to a safe mixing temperature (typically $40^\circ\text{F}$ to $45^\circ\text{F}$):

$\dot{q}_{preheat} = 1.08 \times \text{CFM}_{oa} \times (T_{oa,leaving} - T_{oa,entering}) \quad [\text{Btu/hr}]$

2. Supply and Return Fan Heat Gain

Fans convert electrical and mechanical shaft power into kinetic and thermal energy. All aerodynamic losses and internal flow friction dissipate directly into the airstream as heat, raising the dry-bulb temperature:

$\Delta T_{fan} = \frac{\Delta P_{total} \text{ [in. wg]}}{6356 \times \eta_{total} \times \rho \times c_p} \approx \frac{\Delta P_{total} \text{ [in. wg]}}{2.69 \times \eta_{total}} \quad [^\circ\text{F}]$

Where:

• $\Delta P_{total} = \text{fan total pressure rise } (\text{in. w.g.})$
• $\eta_{total} = \text{combined fan and drive efficiency (decimal, typically } 0.60 - 0.75)$
• $\rho = 0.075\text{ lb/ft}^3$, $c_p = 0.24\text{ Btu/lb}\cdot^\circ\text{F}$

For a typical draw-through system with $\Delta P_t = 3.5\text{ in. wg}$ and $\eta = 0.65$: $\Delta T_{fan} = \frac{3.5}{2.69 \times 0.65} = 2.00^\circ\text{F}$

If the fan motor is located inside the airstream (common in packaged units), motor inefficiency heat is also added: $\eta_{total} = \eta_{fan} \times \eta_{motor}$.

3. Dehumidification with Subcooling and Reheat

When an occupied space has high internal latent loads (e.g., hospital operating rooms, natatoriums, cleanrooms) or during humid part-load conditions, simply cooling air to the required room sensible supply temperature will not remove enough water vapor.

The Subcool/Reheat Process:

1. Cool air down to a low apparatus dew point (e.g., $48^\circ\text{F}-52^\circ\text{F}$) across the cooling coil to wring out excess moisture.
2. Reheat the dry air sensibly (via hot water, electric, or condenser heat recovery) to the design supply temperature ($60^\circ\text{F}-65^\circ\text{F}$) to prevent overcooling the space.

$\dot{q}_{reheat} = 1.08 \times \text{CFM}_{sa} \times (T_{supply} - T_{leaving\_coil}) \quad [\text{Btu/hr}]$

4. Face and Bypass Damper Systems

In face-and-bypass systems, a portion of the mixed air is directed through the cooling coil (face damper) while the remainder bypasses around the coil (bypass damper). The two streams remix downstream. This provides precise space temperature control without varying the total airflow or modulating coil water temperature.

3. Step-by-Step Worked Example: Comprehensive Complex AHU Cycle Analysis

Problem Statement

A commercial AHU serves an office suite with the following design specifications at standard sea-level pressure ($14.696\text{ psia}$):

• Total Supply Airflow: $\text{CFM}_{sa} = 12,000\text{ CFM}$
• Outdoor Ventilation Air: $\text{CFM}{oa} = 3,000\text{ CFM}$ ($25%\text{ OA}$) at $92.0^\circ\text{F}$ DB, $76.0^\circ\text{F}$ WB ($h{oa} = 39.6\text{ Btu/lb}$, $W_{oa} = 0.0158\text{ lb/lb}$)
• Recirculated Return Air: $\text{CFM}{ra} = 9,000\text{ CFM}$ ($75%\text{ RA}$) at $76.0^\circ\text{F}$ DB, $50%\text{ RH}$ ($h{ra} = 28.3\text{ Btu/lb}$, $W_{ra} = 0.0094\text{ lb/lb}$)
• Cooling Coil: Cools and dehumidifies mixed air to a leaving condition of $52.0^\circ\text{F}$ DB, $51.0^\circ\text{F}$ WB ($h_{cc} = 20.8\text{ Btu/lb}$, $W_{cc} = 0.0078\text{ lb/lb}$)
• Supply Fan: Draw-through configuration with a measured temperature rise of $\Delta T_{fan} = 1.5^\circ\text{F}$
• Reheat Coil: Heats the air from the fan discharge state to a final supply air temperature of $58.0^\circ\text{F}$ DB

Calculate:

1. Mixed air properties ($T_{ma}, W_{ma}, h_{ma}$)
2. Total cooling coil load in tons of refrigeration
3. Supply fan heat gain in Btu/hr
4. Reheat coil thermal duty in Btu/hr
5. Net sensible cooling delivered to the conditioned space (room)

Solution Steps

Step 1: Calculate Mixed Air State Points $\%OA = \frac{3,000}{12,000} = 0.25 \quad \text{and} \quad \%RA = \frac{9,000}{12,000} = 0.75$ $T_{ma} = 0.25(92.0^\circ\text{F}) + 0.75(76.0^\circ\text{F}) = 23.0 + 57.0 = 80.0^\circ\text{F}$ $W_{ma} = 0.25(0.0158) + 0.75(0.0094) = 0.00395 + 0.00705 = 0.0110\text{ lb}_w/\text{lb}_{da}$ $h_{ma} = 0.25(39.6) + 0.75(28.3) = 9.90 + 21.225 = 31.13\text{ Btu}/\text{lb}_{da}$

Step 2: Calculate Total Cooling Coil Load $\dot{q}_{coil} = 4.5 \times \text{CFM}_{sa} \times (h_{ma} - h_{cc})$ $\dot{q}_{coil} = 4.5 \times 12,000 \times (31.13 - 20.80) = 54,000 \times 10.33 = 557,820\text{ Btu/hr}$ $\text{Coil Capacity (Tons)} = \frac{557,820}{12,000} = 46.49\text{ tons}$

Step 3: Calculate Supply Fan Heat Addition $\dot{q}_{fan} = 1.08 \times \text{CFM}_{sa} \times \Delta T_{fan} = 1.08 \times 12,000 \times 1.5^\circ\text{F} = 19,440\text{ Btu/hr}$ $T_{fan,out} = T_{cc,out} + \Delta T_{fan} = 52.0^\circ\text{F} + 1.5^\circ\text{F} = 53.5^\circ\text{F}$

Step 4: Calculate Reheat Coil Thermal Duty $\Delta T_{reheat} = T_{supply} - T_{fan,out} = 58.0^\circ\text{F} - 53.5^\circ\text{F} = 4.5^\circ\text{F}$ $\dot{q}_{reheat} = 1.08 \times \text{CFM}_{sa} \times \Delta T_{reheat} = 1.08 \times 12,000 \times 4.5^\circ\text{F} = 58,320\text{ Btu/hr}$

Step 5: Calculate Net Space Sensible Cooling Capacity $\dot{q}_{s,space} = 1.08 \times \text{CFM}_{sa} \times (T_{room} - T_{supply})$ $\dot{q}_{s,space} = 1.08 \times 12,000 \times (76.0^\circ\text{F} - 58.0^\circ\text{F}) = 12,960 \times 18.0 = 233,280\text{ Btu/hr} = 19.44\text{ tons}$ (Notice how the total refrigeration capacity of 46.5 tons satisfies outdoor ventilation load + coil latent load + fan heat + reheat load to deliver 19.4 tons of net space cooling).