Estimation and prediction with CO₂

Aaron Nielsen

Warning

This presentation is designed to access a local flask app web server pulling data from an Aranet 4 CO2 sensor. If you are viewing it on the web, portions of the presentation will not display properly.

Estimation

  • What’s the temperature of a room?
  • You have a noisy thermometer.

Estimation

  • What’s the temperature of a room?
  • You have a noisy thermometer.
  • Assume constant temperature
  • Take the average and standard deviation

Estimation

  • What’s the temperature of a room?
  • You have a noisy thermometer.
  • Assume constant temperature
  • Take the average and standard deviation

Averaging

\[ T_{avg} = \frac{1}{N} \sum_{i=1}^N T_i \]

  • Requires all the measurements
  • But, I’m impatient
  • Average the points I have up to now

Averaging

\[ T_{avg} = \frac{1}{N} \sum_{i=1}^N T_i \]

  • Requires all the measurements
  • But, I’m impatient
  • Average the points I have up to now

Recursive estimation

  • I need to remember all the measurements: \[ T_{avg,N} = \frac{1}{N} \sum_{i=1}^N T_i \]
  • I don’t want to remember, so instead: \[ \color{red}{\underbrace{T_{avg,N}}_{\text{New average}}} = \color{green}{\underbrace{\frac{T_N}{N}}_{\text{New value}}} + \color{blue}{\frac{N-1}{N} \underbrace{\frac{1}{N-1}\sum_{i=1}^{N-1} T_i}_{\text{Old average}}} = \color{green}{\frac{T_N}{N}} + \color{blue}{\frac{N-1}{N} T_{avg,N-1}} \]

Recursive estimation: Kalman Filter

Atmospheric CO2

Indoor CO2

  • people inhale O2 and exhale CO2
  • outside the exhaled breath dilutes into the atmosphere
  • indoors CO2 builds up unless vented without fresh air
  • CO2 concentration is a good proxy measurement for fresh-air ventillation in an occupied space

Why monitor indoor CO2?

  • Elevated CO2 negatively impact people
    • OSHA limit 5000 ppm (permanent poor health outcomes)
    • Cognitive impairment >1000 ppm
  • Poor ventillation allows bad things to accumulate in the air
    • toxins (e.g. VOCs)
    • airborne illnesses

Mass balance equation

\[ \color{red}{{V \frac{dC}{dt}}}= \color{green}{E} - \color{blue}{\left[ Q C - Q C_R \right]} \]

CO2 change = Incoming CO2 - Outgoing CO2
  • \(C\): CO2 concentration [mg/m3]
  • \(V\): volume of room [m3]
  • \(C_R\): replacement air CO2 level [mg/m3]
  • \(E\): CO2 emission rate [mg/hr]
  • \(Q\): air flow rate [m3/hr]
  • \(N\): number of occupants
  • \(E \propto N\)

CO2 sensors

  • Hand-held Aranet 4
  • Non-Dispersive InfraRed (NDIR)

  • Logs data up to 1 minute intervals
  • Batteries last several months
  • Bluetooth connectivity

Example measurement

Kalman filter solution

  • Estimate \(C\), \(E\)
  • \(Q\), \(V\) constant
  • Adult \(E \approx 15\) L/hour
  • CO2 1.96 PPM \(\rightarrow\) 1 mg/m3

Results

References

Allen, Joseph G., Piers MacNaughton, Jose Guillermo Cedeno-Laurent, Xiaodong Cao, Skye Flanigan, Jose Vallarino, Francisco Rueda, Deborah Donnelly-McLay, and John D. Spengler. 2018. “Airplane Pilot Flight Performance on 21 Maneuvers in a Flight Simulator Under Varying Carbon Dioxide Concentrations.” Journal of Exposure Science &Amp\(\mathsemicolon\) Environmental Epidemiology 29 (4): 457–68. https://doi.org/10.1038/s41370-018-0055-8.
Batterman, Stuart. 2017. “Review and Extension of CO 2 -Based Methods toDetermine Ventilation Rates with Application toSchool Classrooms.” Int J. Of Env. Res. And Pub. Health. https://doi.org/doi:10.3390/ijerph14020145.
Haverinen-Shaughenssy, U., D. J. Moschandreas, and R. J. Shaughnessy. 2010. “Association Between Substandard Classroom Ventilation Rates andstudentsÕ Academic Achievement.” Indoor Air. https://doi.org/10.1007/978-3-322-89521-9_13.
Sakamoto, Mitsuharu, Mengze Li, Kazuki Kuga, Kazuhide Ito, Gabriel Bekö, Jonathan Williams, and Pawel Wargocki. 2022. “CO2 Emission Rates from Sedentary Subjects Under Controlled Laboratory Conditions.” Building and Environment 211 (March): 108735. https://doi.org/10.1016/j.buildenv.2021.108735.