Motivation

I previously estimated the ventilation rate based on the decline CO2 profile when no person is present in a room. I here want to take this one step further and incorporate the number of people present in a room and mathematically derive this effect on the CO₂ concentration.

Mathematical description

If you do not want to be bothered with mathematics or its derivation, just skip to the final equation or if you are a real lazy ass, just look at the pictures. Let me remind you that you came here for fun, not for mathematics.

\[Accumulation = in – out + Production\]

We introduce here the production term P which represents the amount of CO₂ emitted by one individual in mg per second, and n the number of people. Note here that P could have different values depending on the metabolism and activity of the person. I will come back to that later.

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

The volume of the room (V) is constant and $\phi_{in} = \phi_{out}$ as the pressure in the room does not change. We assume that the air in the room is ideally mixed, as such C = C_Out.

\[V \frac{dC}{dt} = \phi (C_{in} – C ) + nP\]
\[\frac{dC}{\phi (C_{in} - C) + nP} = \frac{dt}{V}\]
\[\int \frac{dC}{\phi (C_{in} - C) + nP} = \frac{1}{V} \int dt\]

The integral on the left is “complicated”, we therefore apply the substitution rule. We introduce the variable u and define it as:

\[u = \phi (C_{in} - C) + nP\]

The differential of u is:

\[du = -\phi dC \ \ \ \ \ \ \ \ \ \ \rightarrow \ \ \ \ \ \ \ \ \ \ dC = -\frac{du}{\phi}\]

Now we make the subsitution:

\[-\frac{1}{\phi} \int \frac{ du}{u} = \frac{1}{V} \int_{0}^{t} dt\]

The integral of 1/u is “simple”, which is ln(u). Thus:

\[-\frac{1}{\phi} \ln (u) \Biggr| = \frac{1}{V} t\Biggr|_{0}^{t}\]

Now that the integration is performed, we (re)-substitute u again.

\[\ln \left( \phi (C_{in} - C) + nP \right) \Biggr|_{C_0}^{C} = -\frac{\phi}{V} t\Biggr|_{0}^{t}\]

Now implement the integration borders.

\[\ln \left( \phi (C_{in} - C) + nP \right) - \ln \left( \phi (C_{in} - C_0) + nP \right) = -\frac{\phi}{V} (t-0)\]
\[\ln \left( \frac{ \phi (C_{in} - C) + nP } { \phi (C_{in} - C_0) + nP } \right) = -\frac{\phi}{V} t\]

We are interested in C, let’s therefore rearrange the equation.

\[\frac{ \phi (C_{in} - C) + nP } { \phi (C_{in} - C_0) + nP } = e ^{-\frac{\phi}{V} t }\]
\[\phi (C_{in} - C) + nP = \left[ \phi (C_{in} - C_0) + nP \right] e ^{-\frac{\phi}{V} t }\]
\[\phi (C_{in} - C) = \left[ \phi (C_{in} - C_0) + nP \right] e ^{-\frac{\phi}{V} t } - nP\]
\[C_{in} - C = \frac{1}{\phi} \left( \left[ \phi (C_{in} - C_0) + nP \right] e ^{-\frac{\phi}{V} t } - nP \right)\]
\[C = C_{in} - \frac{1}{\phi} \left( \left[ \phi (C_{in} - C_0) + nP \right] e ^{-\frac{\phi}{V} t } - nP \right)\]
\[C = C_{in} - \left( \left[ C_{in} - C_0 + \frac{nP}{\phi} \right] e ^{-\frac{\phi}{V} t } - \frac{nP}{\phi} \right)\]

The final equation is:

\[C = C_{in} + \frac{nP}{\phi} - \left( \left[ C_{in} - C_0 + \frac{nP}{\phi} \right] e ^{-\frac{\phi}{V} t } \right)\]

No person present

Now we can define a couple of “edge” cases. The first important one is the scenario for which no person is present in the room. In that case, n is 0 and the equation is reduced to:

\[C = C_{in} - \left( C_{in} - C_0 \right) e ^{-\frac{\phi}{V} t }\]

This is exactly the same equation as I have previously derived in another post, which describes the change in CO₂ concentratin in a room in case no person is present, under the assumption that the air in the room is ideally mixed.

After a long time

Obviously, I am now more interested in the scenario for which person(s) are present in the room. An interesting scenario is when sufficiently long time (i.e. infinitely) is provided.

\[e ^{-\frac{\phi}{V} t } \ \ \rightarrow \ \ e ^{-\frac{\phi}{V} \infty} = e ^{ -\infty } =0\]

This reduces the equation to:

\[C = C_{in} + \frac{nP}{\phi}\]

The production term P, the amount of CO₂ produced per second, can be estimated based on the recommended average energy intake, which is around 2500 kcal. This is equal to a power consumption of 10 460 000 / (24 x 3600) = 121 J / s = 121 W. To estimate the amount of CO₂ emitted, we consider a diet fully composed of sugar, as most people are addicted to such diet. One gram of carbohydrate provides roughly 17 kJ of energy. Hence, on a daily basis a person with this great diet would need to consume 10 460 000 / 17 000 = 615 g of carbohydrate to meet the recommended caloric intake. The combustion of one mole of sugar results in 6 mole of CO₂, thus, 615 / 180 x 6 x 44 = 902 g of CO₂ per day. This estimation is pretty close to what is reported on the first reference that popped up on the magical place that we call the internet: ~ 1 kg CO₂ per day, which translates to 12 mg per second.

I plotted the result for equation 22 when varying the air flow rate and the number of people, see Figure 2. Initially raising the air flow rate is very effective, but for higher flows it approaches the assymptote of ~400 ppm (C_in). Interestingly, we can calculate the air flow ($\phi$)when the number of persons in a room are known and sufficient time is applied (i.e. CO₂ concentration is constant). This was not possibly with my previous approach based on equation 20 for which I would need the volume of the room to calculate the air flow ($\phi$).

However, it is important to realise that slope is pretty small for higher flows and makes such approach error sensitive. For example when one person is present in the room and we consider raising the air flow ($\phi$) from 15 L / s to 50 L / s the CO₂ concentration only drops from 800 to 550 ppm. An error in the sensors CO₂ reading then translate in a big error in the air flow estimate.

Figure 2: CO₂ concentration as function of the flow rate (air) for 1, 2, 3 or 4 persons in a room. When sufficient time (e.g. t = $\infty$) is provided and ideal mixing is assumed

Variables: Y = CO2 concentration, X = Flow, N = persons

Obviously, we can improve the air quality by increasing the air flow. However, there is a penalty to pay, which is energy consumption. We can calculate the power consumption via the following equation:

\[P = \phi_{v} \cdot \rho \cdot C_{p} \cdot \Delta T\]

If we consider the scenario where there is a delta T of 10 K and a air flow rate of 10 L / s we would need 120W:

\[P = \phi_{v} \cdot \rho \cdot C_{p} \cdot \Delta T = 10\ \frac{L}{s} \cdot 1.2 \cdot 10^{-3}\ \frac{kg}{L} \cdot 1000 \ \frac{J}{kg\ K} \cdot 10\ K = 120 \frac{J}{s} = 120 W\]

I have displayed the result of this equation for a varied flow rate for various $\Delta$T, see Figure 3. It is evident that the power consumption is a linear function of the ventilation rate.

Figure 3: Power consumption as function of air flow rate and $\Delta$T

A person entering the room

Another interesting edge case is when a person enters a room which was completly replenished with outside air (CO₂ concentration = 400 ppm). In mathematical terms, C₀ = C_in, which will reduce the equation to:

\[C = C_{in} + \frac{nP}{\phi} \left( 1- e ^{-\frac{\phi}{V} t } \right)\]

The term $\phi/V$ can be defined as the number of Air Changes per Hour (ACH). In a previous example I found an ACH of 0.36. But my so far not well supported observation is that the ACH of my living room is varying between 0.25 and 1. The plot below shows the CO₂ profile as function of time for the previously mentioned ACH’s.

Figure 4: CO₂ profile when a person enters and stays in a room

We can see that ultimately the profile converges to a stable value. For t = $\infty$ the right term in the equation vanishes and we end up with equation 22. Obviously, the profile converges faster for higher ACH numbers as e$^{-ACH \ \cdot \ t}$ approaches zero faster. Nevertheless, it easily takes a few hours to approach a stable CO₂ concentration. Interestingly, in the first (half) hour the slope is very similar for the various ACH.

Having a party?

Now let us consider the scenario when you invite some friends, and to make it easy; no one was in the room as such that the starting CO₂ concentration was 400 ppm (C₀ = C_in). If we rearrange equation 25 a bit we can see that number of persons in a room just amplifies the CO₂ profile. Contrary to changing the ACH (air changer per hour) the slope does increase. We can use this to our advantage to differentiate between air flow and the number of people in a room. Alternatively reasoned, if you are too lazy to count the number of people in your room or simply too wasted, the CO₂ profile would help you to do so.

\[C = C_{in} + n \cdot \left[ \frac{P}{\phi} \left( 1- e ^{-\frac{\phi}{V} t } \right) \right]\]

Figure 5: CO₂ profile for a number (n) of people entering a room (ACH=1)

To conclude

Yes, first of all it would be very nice if I would add some measurement data to support the model findings, I am working on it. But trust me, on a first eye the model readings do make sense (I typically check the CO₂ log data daily and developed a feeling for it). To summarize a bit the findings:

With a few people in the room it is pretty hard to keep the CO₂ concentration below 1000 ppm.
The airflow and consequently energy loss can be determined when a known number of people are present in the room and a stable CO₂ concentration is reached
It can easily take a couple of hours to reach a stable CO₂ concentration (ACH <=1)
An initial increase in the slope means more people are entering the room