2.1 Model development

The model space is a seminar room with a total volume of 159 m³. The room is located on the third floor of the engineering building at the Kookmin University. One side of the room’s wall is exposed to the outdoor environment with four operable windows that were kept closed during the measurement. There is no mechanical ventilation system for the room. Adjacent rooms are present on both sides and the hallway is connected to the fourth side of the room, as illustrated in Figure 1.

Figure 1: Space layout.

2.2 Bayesian Markov-chain Monte Carlo (MCMC) approach

The Bayesian MCMC approach is a stochastic simulation technique for computing inferential quantities. According to the Bayesian method, the posterior probability is computed on the basis of the prior probability and the likelihood function is derived from a probability model for the data to be observed. The posterior-distribution relation is given by a combination of the prior distribution and the likelihood as

π( N|C )= π( N )f( C|N ) ∫π( N )f( C|N )dN (1)

where π(N) is the prior probability of proposed number of occupants and f(C|N) is the likelihood of observing CO₂ given number of occupants. π(N|C) is the posterior probability of the number of occupants. Carbon concentration in the room is generated by the CO₂ mass-balance equation as follows:

V dC( t ) dt = m ˙ N+Q( C out − C t−1 ) (2)

where V is the volume of the room, m is the CO₂-generation rate per person, N represents the number of occupants, Q is ventilationrate, and C_out is outdoor CO₂ concentration. The dynamic model as a solution of Eq. (2), which is used to generate the likelihood function, can be expressed as

C t =α· C t−Δt +( 1−α )· C ∞ (3)

The concentration at the current time step, C_t, is determined by the concentration at the previous time step, C_t-Δt and the steady concentration, C ∞ =( C SA + m ˙ N i Q i ) . The weighting factor, α=( e − Q i V i Δt ) , depends upon the air-exchange rate, i, and the time step, Δt. We used the informative Bayesian prior of N with a calculation time step of 3 min.

We input the prior information based on the most likely occurrences in the observed system, which should approximate the true values as closely as possible. In this study, the prior of N was uniformly distributed with minimum and maximum levels of 0 and 25 persons, respectively. The remaining prior information was assumed to have a Gaussian distribution. The mean of the prior probability, m, was assumed to be 0.553 g/min, equivalent to sitting with a standard deviation of 30%. We set the mean and the standard deviation of the prior Q according to the measured steady-state-ventilation rate in section 2.3. The means of the prior concentrations of CO₂ outdoors and CO₂ background is 480 PPM at 5% standard deviation. The MH-MCMC algorithm for each time step is shown in Figure 2. We performed 10,000 iterations to collect 5,000 posterior samples after a burn-in period comprising the first 5,000 iterations.

Figure 2: The MH-MCMC algorithm.

2.3 Quantification of the ventilation rate

The prior information used to calculate the Bayesian model should be identified with high specificity in order to avoid over- and underestimation. However, some parameters cannot be measured accurately due to uncontrollable characteristics. For instance, there could be difficulties measuring and controlling the ventilation rate in a space without a ducting system. Infiltration takes place through cracks in the walls and windows depending on the outdoor-wind conditions. In this study, the ventilation rate is determined using two methods: the concentration decay method and the sum-up method. Both the methods evaluate CO₂ levels with respect to time. CO₂ concentration was recorded minutely for seven days during typical weekdays in the observed room. The alteration of CO₂ concentration is illustrated in Figure 3.

Figure 3: CO₂ concentration measured in the space over seven days.

The concentration decay method is commonly used to measure the ventilation rate by applying a tracer gas technique [6]. We utilize the CO₂ naturally generated by humans as a tracer gas. The ventilation rate is estimated by analyzing the concentration decay rate at the end of each day.

The space concentration decays down to the background concentration after unoccupied time due to dilution of outdoor concentration. Assuming that the space concentration is mixed uniformly and the ventilation rate is constant, the exponential decay from Figure 4 can be observed. The decay concentration is normalized from the observed period as

C n = C ( t ) − C out C 0 − C out , (4)

Figure 4: Log-normalized concentration of selected periods.

where C_O is the initial concentration at the start of the decay observation period. The log-normalized concentration during the selected period is illustrated in Figure 4. The slopes of linear fitting in the figure represent the air-exchange rates summarized in Table 1. The steady-state-ventilation rate can be obtained by averaging the seven observed air-exchange rates and then multiplying the result by the given space volume. The calculated ventilation rate was defined to have 0.60 ACH (1.6 ±0.74 m³/min) with a standard deviation of 0.74 m³/min or 46% of the mean.

Table 1: Specific airflow rates calculated from various decay periods.

The sum-up method integrating total CO₂ generation rate and number of occupant over the concentration deference between indoor and outdoor as formulated in Eq. 5. A summary of the appraised ventilation rates obtained by the sum-up method is shown in Table 2. The average of ventilation rate is 1.52 m³/min and the standard deviation is 0.30 m³/min or 20% from the mean:

Q ¯ = ∫ t=i t=f ( m ˙ N (t) )dt ∫ t=i t=f ( C (t) − C out )dt (5)

Table 2: Calculation summary of daily ventilation rate by the sum-up method.