## Motivation Bob and Alice are concerned about the air quality in their living room. They've heard about Air Changes per Hour (ACH) as an important metric for indoor air quality and want to calculate if their room meets recommended ventilation standards. They don't have a carbon dioxide monitor to do the tracer-gas approach so they are trying to bootstrap a calculation together with physics! Here are the generally accepted ACH values for living rooms: | ACH Value | Classification | Description | |-----------|----------------|-------------| | < 0.5 | Poor | Insufficient ventilation, potential health risks | | 0.5 - 2 | Minimum | Basic ventilation, acceptable for short periods | | 2 - 4 | Good | Recommended for living spaces | | 4 - 6 | Excellent | Ideal for high-occupancy rooms | | > 6 | High | May lead to energy inefficiency | Bob and Alice want to calculate their room's ACH to ensure it meets the "Good" classification (2-4 ACH). Let's help them solve this problem. ## Problem Statement A rectangular room has dimensions 4m × 3m × 2.5m (length × width × height). The room has a single window of area 1.2m² located on one wall. The window is positioned 1.5m from the floor. The room contains a heat source that generates 500W of thermal energy. Outside, there is a steady wind blowing at 5 m/s at an angle of 30° to the window plane. The outside temperature is 20°C, and the room temperature is initially 25°C. Calculate: 1. The pressure difference across the window due to wind 2. The natural ventilation rate through the window 3. The Air Changes per Hour (ACH) in the room 4. The time it takes for the room temperature to reach 22°C ## Relevant Equations ### 1. Wind Pressure The pressure difference due to wind can be calculated using: $\Delta P_w = \frac{1}{2} \rho v^2 C_p \cos^2 \theta$ where: - $\rho$ is air density (typically 1.225 kg/m³) - $v$ is wind velocity - $C_p$ is pressure coefficient (typically 0.7 for windows) - $\theta$ is the angle between wind direction and window normal [1] ### 2. Stack Effect The pressure difference due to temperature difference (stack effect): $\Delta P_s = \rho g h \left( \frac{T_i - T_o}{T_o} \right)$ where: - $g$ is gravitational acceleration (9.81 m/s²) - $h$ is height difference - $T_i$ is inside temperature - $T_o$ is outside temperature [2] ### 3. Ventilation Rate The total ventilation rate can be calculated using: $Q = C_d A \sqrt{\frac{2|\Delta P|}{\rho}}$ where: - $C_d$ is discharge coefficient (typically 0.6 for windows) - $A$ is window area - $\Delta P$ is total pressure difference ($\Delta P_w + \Delta P_s$) [3] ### 4. Air Changes per Hour (ACH) The ACH can be calculated using: $ACH = \frac{Q \times 3600}{V}$ where: - $Q$ is the ventilation rate in m³/s - $V$ is the room volume in m³ - 3600 is the conversion factor from seconds to hours [4] ### 5. Temperature Change The rate of temperature change in the room: $\frac{dT}{dt} = \frac{Q_{in} - Q_{out}}{m c_p}$ where: - $Q_{in}$ is heat input (500W) - $Q_{out}$ is heat loss through ventilation - $m$ is mass of air in room - $c_p$ is specific heat capacity of air (1005 J/kg·K) [5] ## Solution Approach 1. Calculate wind pressure difference using the wind pressure equation 2. Calculate stack effect pressure difference 3. Determine total pressure difference 4. Calculate ventilation rate 5. Calculate Air Changes per Hour (ACH) using the room volume 6. Use the temperature change equation to find the time required to reach 22°C ## Additional Notes - Assume the room is well-sealed except for the window - Neglect heat transfer through walls - Assume air properties remain constant - Consider both wind-driven and buoyancy-driven ventilation - The problem combines principles of: - Fluid dynamics - Thermodynamics - Heat transfer - Building physics This problem demonstrates how natural ventilation is influenced by both wind pressure and temperature differences, which is crucial for understanding building ventilation and indoor air quality. The ACH calculation helps Bob and Alice determine if their room meets recommended ventilation standards. ## Acknowledgements Thank you to George Smith, Paul Jules Micolet, Cyrus Hafezparast, Angus Fotherby and Natasha Tilokani for feedback and ideas. ## References [1] ASHRAE Handbook of Fundamentals (2021), Chapter 16: Ventilation and Infiltration, Section 16.2.2 [2] CIBSE Guide A: Environmental Design (2015), Chapter 4: Ventilation, Section 4.3.2 [3] Awbi, H.B. (2003) "Ventilation of Buildings", 2nd Edition, Chapter 5: Natural Ventilation, Section 5.2.3 [4] ASHRAE Standard 62.1-2019, Section 6.2.1: Ventilation Rate Procedure [5] Incropera, F.P. and DeWitt, D.P. (2002) "Fundamentals of Heat and Mass Transfer", 5th Edition, Chapter 1: Introduction, Section 1.3.2 [6] ASTM E741-11(2017) Standard Test Method for Determining Air Change in a Single Zone by Means of a Tracer Gas Dilution