DIY Forced Air Cooling System🌡️

Our rental apartment came without air conditioning, so we installed a unit in the bedroom and explored ways to distribute cooling to other rooms. This article compares two approaches.

1. System Definition

This model employs the Lumped Parameter Method to simulate a dual-room thermal system. The system simplifies the space into two thermal capacitance nodes with uniform temperature:

• Node 1: Bedroom, equipped with an active cooling source (AC).
• Node 2: Living Room, equipped with temperature sensor, serving as the target control zone.

Symbol Definitions

• $T_1(t)$: Bedroom temperature [${}^\circ C$]
• $T_2(t)$: Living Room temperature (Sensor) [${}^\circ C$]
• $T_{amb}$: Ambient temperature [${}^\circ C$]
• $C_1, C_2$: Room thermal capacitance (air mass $\times$ specific heat) [$J/K$]
• $R_{w1}, R_{w2}$: Wall thermal resistance to outdoor [$K/W$]
• $Q_{AC}(t)$: Air conditioner cooling power [$W$]
• $P_{fan}$: Circulation fan waste heat [$W$]

---

2. Energy Conservation Differential Equations

According to the First Law of Thermodynamics, the rate of temperature change is determined by the net heat flow in and out.

2.1 Bedroom (Node 1)

The bedroom is the cooling source generation point, while simultaneously receiving heat exchange from the living room, fan waste heat, and wall heat leakage.

\[C_1 \frac{dT_1}{dt} = \dot{Q}_{wall,1} + \dot{Q}_{coupling} + P_{fan} - Q_{AC}(t)\]

2.2 Living Room (Node 2)

The living room is the passive cooling zone, with heat source from the environment and cooling source from convection with the bedroom.

\[C_2 \frac{dT_2}{dt} = \dot{Q}_{wall,2} - \dot{Q}_{coupling}\]

---

3. Heat Transfer Mechanisms

3.1 Wall Conduction

Follows the simplified form of Fourier’s law of heat conduction:

\[\dot{Q}_{wall, i} = \frac{T_{amb} - T_i}{R_{wi}} \quad, \text{where } i=1,2\]

3.2 Convective Coupling

The heat flow between the two rooms $\dot{Q}_{coupling}$ is defined as:

\[\dot{Q}_{coupling} = U_{eff} \cdot (T_2 - T_1)\]

where the effective heat transfer coefficient $U_{eff}$ is determined by the operating mode:

Case 1: Natural Convection (Door Open, No Fan)

With the door open, air circulates naturally between rooms through buoyancy-driven convection. However, without forced airflow, the heat exchange efficiency remains extremely low.

\[U_{eff} = U_{natural} = 35 \text{ W/K}\]

Case 2: Forced Convection (Door Open + Duct Fan)

A duct fan actively circulates air between rooms, establishing a mass flow rate $\dot{m}$. Heat exchange efficiency is significantly enhanced through forced fluid enthalpy transport:

\[U_{eff} = U_{forced} = \dot{m} \cdot c_p = 68.4 \text{ W/K}\]

Mass flow rate calculation based on airflow setting:

\[\dot{m} = \frac{\text{Flow}_{CMH}}{3600} \cdot \rho_{air}\]

Key Difference: Both cases have the door open, but Case 2 uses a 40W fan (200 CMH airflow) to force air circulation, achieving nearly 2× higher heat transfer efficiency compared to natural convection alone.

---

4. Control Logic

The system uses P-Control to simulate an inverter AC with saturation limits and low-frequency protection. While most AC systems employ PI control in practice, a simple proportional controller is used here for demonstration purposes.

\[Q_{AC}(t) = \begin{cases} 0 & \text{if } T_1 \le T_{set} \\ \min\left( P_{max}, \ P_{max} \cdot K_p \cdot (T_1 - T_{set}) \right) & \text{if } T_1 > T_{set} \end{cases}\]

• If the calculated $Q_{AC}$ is less than the minimum operating power (e.g., 800W), it is forced to 800W (simulating low-frequency compressor operation).

---

5. Numerical Simulation Algorithm

The Forward Euler Method is adopted for discrete-time simulation. With time step $\Delta t = 1.0s$, the state update formula at step $k$ is:

\[T_1[k+1] = T_1[k] + \frac{\Delta t}{C_1} \left( \frac{T_{amb}-T_1[k]}{R_{w1}} + U_{eff}(T_2[k]-T_1[k]) + P_{fan} - Q_{AC}[k] \right)\] \[T_2[k+1] = T_2[k] + \frac{\Delta t}{C_2} \left( \frac{T_{amb}-T_2[k]}{R_{w2}} - U_{eff}(T_2[k]-T_1[k]) \right)\]

---

6. Model Parameter Table (Simulation Parameters)

Parameter settings based on the provided Python code (Fri Jan 30 2026 version):

Parameter Category	Variable Name	Value	Unit	Notes
Environment	T_amb	30.0	${}^\circ C$	Outdoor temperature
	T_init	28.0	${}^\circ C$	Initial room temperature
Space	vol_1	35.0	$m^3$	Bedroom volume
	vol_2	70.0	$m^3$	Living room volume
Thermal Resistance	R_wall1	0.020	$K/W$	Bedroom insulation
	R_wall2	0.011	$K/W$	Living room insulation (fast heat loss)
Equipment	AC_power_max	5200.0	$W$	AC capacity
	AC_setpoint	16.0	${}^\circ C$	Extremely low setpoint
	Fan_power_heat	40.0	$W$	Fan waste heat
Convection	U_natural	35.0	$W/K$	Natural convection (door open, no fan)
	U_forced	68.4	$W/K$	Forced convection (door open + fan)
	flow_rate_cmh	200.0	$CMH$	Duct airflow (Case 2 only)
Physical	rho_air	1.225	$kg/m^3$	Air density
	cp_air	1005.0	$J/kgK$	Air specific heat