Motivation

Initially I always thought of ventilation as something bad, a terrible way to lose valuable hot air for which you have paid money at the expel of cold air. Later I came to realise that proper ventilation is key to drive out pollutants from buildings. These pollutants often arise from indoor activities such as the presence of humans, by cooking, smoking or burning a candle. Such processes typically involve the emission of moisture, CO₂ and other metabolites. Less known, maybe, is the accumulation of radioactive radon and thoron-gas in poorly ventilated buildings, which results from the breakdown of uranium in soil, rock, water and building materials.^{1, 2}

Very recently we have seen attention for ventilation picking up in the context of “the spread of a novel airborne virus”. The airborne spread of germs is not a new phenomenon and prevalent in poorly ventilated spaces. The concentration of airborne particles (that probably contain virus) emitted after a sneeze, or by simply breathing remains much higher in a poorly ventilated building than in a well ventilated spot, such as the outdoors.

But also a bit less nuanced and maybe not harmful to your physical health but sometimes inflicting your mental well-being: the odour of a big fart is mitigated much more rapidly in the outdoor due to rapid mixing of the surrounding air, thereby bringing down the concentration of these odour generating molecules, compared to the indoors. It is therefore much more recommended spending beer-drinking activities with your friends in the outdoors than the indoors, unless bothering each other is your main activity, which it is for most groups.

Without ventilation pollutants accumulate within a building. It is like that the garbage man does not pick up your garbage any longer; soon you will sit on a stockpile of trash. So sometimes you have to pay the garbage man to empty your “dirty” air and replace it with “fresh” air. Obviously, you don’t want the garbage man to become rich and therefore need to apply a balancing act; you want sufficient ventilation, not too much, not too less.

On a more serious note, higher concentration of pollutants have been linked to less and more severe health issues. To touch upon a few examples: 1) there is ongoing discussion that elevated CO₂ levels impair cognitive function and decision making 2) “Radon is estimated to cause tens of thousands of lung cancer deaths each year. In fact, the Surgeon General has warned that radon is the second leading cause of lung cancer in the United States. Only smoking causes more lung cancer deaths. “ 3) an increased change on the development of allergic disease in poorly ventilated buildings 4) high moisture levels facilitate mould growth which is harmful and damages buildings ref,ref 5) and the consequences of airborne pathogens speaks for itself.

To finalize this too long introduction, yes, ventilation is very important for many reasons, and I probably missed out on a lot of them. On the contrary, with our recent heavy focus on the reduction of the energy consumption, ventilation results in a heat loss. Thus, something like ventilation which appears rather trivial, actually touches upon two of the world most urgent problems: 1) health and 2) the reduction of energy consumption.

For the reasons discussed I have developed an interest in quantifying, at least to make an attempt to do so, of the ventilation rate of my house. In parallel, when exploring the topic of ventilation I found affiliation by Jan Sundell’s plea on “VENTILATION: WHY does no one take it seriously?”.

Measurement approach

So, I needed a measure of the ventilation rate. A simple but elegant method is by the injection of a tracer-component. One measures the concentration of this tracer-component in time and can thereby gets an idea of the airflow. If the concentration of this tracer-component in a room drops rapidly, it means that there is appropriate ventilation. On the contrary, if its concentration remains high, the ventilation is poor. A few options for the tracer-component came quickly to my mind, moisture, CO₂ and volatile organic components (VOC). Importantly, sensors that can detected these components are nowadays pretty affordable for mere mortals. A stable outdoor concentration eases comparison which rules out humidity, as humidity fluctuates and is strongly temperature depended. Outdoor CO₂ levels are rather stable, well apparently there is some debate about that, but you get the point and therefore much more suitable. So, I went for the CO₂-sensor, but VOC would work probably fine as well and would allow to trace a “pulse” of tracer gas, for example by the spray of deodorant (maybe a future project…).

I bought the MH-Z16 which is a NDIR (non-dispersive infrared) sensor for some ~30E. The working principle of this sensor is the selective absorption by CO₂ of IR light with a certain wavelength. The degree of absorption is a measure for the CO₂ concentration. I created a tutorial to get this sensor running with an Arduino microcontroller if you want to attempt to reproduce my experiments

I placed the sensor in the living room and started logging CO₂ data (every 5 min I saved a reading). Prior to any attempt to calculate the ventilation rate I had a look at the CO₂ profiles to get a grasp of what is happening. An example of a CO₂ profile for a day is shown in Figure 1. This is just a random day and note that I do not recall exact details of that day, but I just want to illustrate a typical day. Initially the CO₂ concentration is stable at ~390 ppm (5:00 till 8:00). This concentration should be 414 ppm nowadays, but hey, limitations of the sensor (improper baseline adjustment). We see a rapid increase in CO₂ levels when person(s) are entering the room, which makes sense as humans emit CO₂. The increase is rather fast, which suggest that there is proper airflow/air mixing in the room. On two occasions we see that the CO₂ concentration is rather stable, which means that CO₂ produced in the room = CO₂ removed by ventilation. The system is in equilibrium. Around 22:30 we leave the living room and probably went to sleep as every ordinary person would do. Consequently, the CO₂ level drops as the CO₂ “source” is removed until a new equilibrium is achieved, i.e. the indoor CO₂ concentration = outdoor CO₂ concentration, when sufficient time is provided.

Figure 1: Example of a CO₂ profile throughout a day

At this point, I do not have a proper explanation for the “sawtooth” curves, but my hypothesis is that this has to do with non-ideal mixing effects in combination with movement of the CO₂ source(s) (i.e. persons). A support for this argument is that the “sawtooths” are not there when there are no persons in the room or during rapid increase or decline of CO₂ levels. Moreover, I noticed that opening/closing the door to the room (relatively close to the sensor) typically results in a big dip followed by a rapid increase of CO₂ concentration, for example seen at ~13:00 in Figure 1.

This CO₂ profile learns us that I am quite an ordinary person as I go to bed at 22:30, maybe I wake up a bit late, but hey, I need my sleep. On a more serious note, something simple as CO₂ sensing reveals privacy sensitive information: the time that I get up, the time that I go to bed, when I am home, when I have guests and probably an estimate of the number of guests. So be warned when some company that claims “don’t be evil” or the likes, is selling you sensors and owns the data.

Mathematical description

Back to the objective; estimate the ventilation rate. The CO₂ concentration can be mathematically modelled, in which the ventilation rate would be one of the fit parameters. If the model is based on valid assumptions and fits the experimental data well, we could find at least an approximate number of the ventilation rate. We consider our room as a vessel with an inlet and outlet, see Figure 2.

Figure 2: Schematic representation of a room

“Fresh” air from the outside with a CO₂ concentration of ~414 ppm is entering the room and at the same time “Dirty” air is expelled from the room. It is fair to assume that, in a relatively open system, the flow (Φ_v,in) of “Fresh” air in = the flow (Φ_v,out) of “Dirty” air out, as any disbalance would result in a pressure difference, which automatically corrects itself, i.e. equilibrate with the outdoor environment. Moreover, I assumed the room is ideally mixed, i.e. a package of fresh air that enters the room is immediately dispersed in the room. The concentration at every position in the room is considered equal, C = C_out. That is why I drew a stirrer (grey thing) in the Figure 2, I apologize for my drawing skills but use your imagination, consider yourself privileged with it. This hypothesis could be relatively easily checked by measurement of the CO₂ concentration at different positions in the room at the same time. I have studied this hypothesis and it seems pretty reasonable. Nevertheless, the rapid increase in CO₂ concentration followed by a plateau, as was displayed twice in Figure 1, suggests that air in the room is properly mixed. Note that in case of perfect mixing we would expect a step change in the CO₂ level upon the injection of tracer-gas. However, we do not inject tracer-gas but we add a source (person(s)) that continuously emits tracer-gas (CO₂), and the “step-change” is therefore more gradual.

To summarize we assume the following:

The flow (Φ_v,in) of “Fresh” air in = the flow (Φ_v,out) of “Dirty” air out.
The room is perfectly mixed, C_CO₂ is constant throughout the room. Thus C = C_Out

We can then set up the following molar balance for CO₂:

Accumulation = in – out + production

To make our life easy, we only look at the CO₂ profile during the night. Typically, there is no person present in the room during the night, i.e. is no “production” term in our equation. Moreover, there are no “external” disturbances of the airflow during the night; no opening/closing doors, no people wandering around, etc. The airflow should just be the result of “passive ventilation”.

This reduces the equation:

\[Accumulation = in – out\]
\[\frac{dCV}{dt} = \phi_{in} C_{in} – \phi_{out} C_{out}\]

We then implement the previously mentioned assumptions, namely C = C_Out, Φ_v,in = Φ_v,out and V is constant, which simplifies our the equation to:

\[V \frac{dC}{dt} = \phi \cdot (C_{in} – C)\]

We introduce the residence time, \(\tau = V / \phi\):

\[\tau \frac{dC}{dt} = C_{in} – C\]
\[\frac{dC}{(C_{in} – C)} = \frac{dt}{\tau}\]

We integrate and define integration borders from t = 0 and C = C₀ till t = t and C = C.

\[-\ln (C_{in} – C) \Biggr|_{C_0}^{C} = \frac{1}{ \tau} \cdot t\Biggr|_{0}^{t}\]
\[\ln (C_{in} – C) – \ln (C_{in}-C_{0}) = - \frac{t}{\tau}\]
\[\ln \left( \frac{ C_{in} – C} { C_{in}-C_{0} } \right)= -\frac{t }{ \tau }\]
\[\frac{ C_{in} – C }{ C_{in} – C_{0}} = e ^{ ( -t / \tau)}\]

The final equation is:

\[C = C_{in} – (C_{in} – C_{0}) e ^{(-t / \tau) }\]

In which C₀ is set to the start value and 1/ \(\tau\) is the fit parameter.

Model vs Measurement

For a certain day we observe a “stable” CO₂ concentration during the evening, denoted as “equilibrium” in Figure 3. When we leave the living room the CO₂ concentration exponentially decays until it approaches the outdoor CO₂ level of ~414 ppm (at the time of writing) and shoots up again when we someone enters the living room in the morning. Note that a different day(s) is shown in Figure 3 than in Figure 1, but that the patterns are similar.

As discussed before, the absence of persons during the night is exploited to estimate the ventilation rate. The derived equation is fitted to measurement data with the initial value (C₀) set at the measured concentration when we leave the living room (Red dot -> start model in Figure 3) and 1/τ is adjusted as such that that measurement and model give the best agreement.

Figure 3: CO₂ measurement and model comparison

It is evident from Figure 3 that model and measurement agree well, which suggests that the assumptions made are reasonably valid. Yes, I did a bit of cherry picking by showing this nice example, but my general experience is that the model fits the measurement data well. I have investigated the model deviation in more detail, and found that it is more accurate than I had initially hoped for (press this link) Let me put a disclaimer here that for you it might not work out so nicely. The characteristics and conditions of your room/building might be too different that the assumptions underlying the model are not valid (i.e. no perfect-mixing).

Ventilation rate

So what is the ventilation rate? The fit constant 1/\(\tau\) = Air changes per hour (ACH) is 0.36 for this particular example, which is close to the 0.27 ACH that Jan Sundell typically measured for naturally ventilated homes. In other words, it takes 2.8 hours to replenish the air in the room. Now is this number good, or bad? Limited robust correlation between health and ventilation rate exists but an ACH greater than 0.5 is suggested . Still my gut feeling says it is bad, it is way too low. But I believe I am one of many that lives in a poorly ventilated house…