We first considered air change rate measurements in buildings of n fully mixed zones, where n is a natural number. The ventilation situations in buildings with 1–3 zones are illustrated in Figure 1. Measurements aim to determine the volume of air uptake from the outside into the n zones (air change rate). The uptake volume of air, F_uptake, can be represented by the following equation:

where F_0i represents the air uptake volume to each zone. A single tracer gas is emitted at a known rate, E_i, in zone i. The emitted tracer gas is collected by the passive-type air sampler and the average tracer gas concentration in zone i, C_i, is measured. The volume of each zone, V_i, is considered to be a known parameter as it can be measured. On the other hand, the volume of airflow from zone i to zone j, F_ij, cannot be measured. All F_ij have non-negative values. Generally, a gas that does not exist naturally outdoors is selected as the tracer gas, which means its concentration outside (zone 0), C₀, is zero. Usually, the measurement time for the tracer gas method is sufficient that the tracer gas concentration is considered to be constant. In each zone, the air volume is also considered to be constant:

This equation reflects the conservation of mass (air) in each zone.

If all fully mixed zones are in contact with each other and the air can freely pass between any two zones, then the number of unknown parameters F_ij is n(n + 1). It should be noted that this provides the maximum number, and the actual number of unknown parameters is reduced compared with this when F_ij = 0 before measurement. If zone i and j is not physically connected, F_ij and F_ji can be assumed as 0. A total of 2n balanced equations can be established, which includes tracer mass balance equations and air mass balance equations for the n fully mixed zones. Therefore, when calculating air change rate the information deficit is equal to n(n − 1) (= n(n + 1) − 2n). This is only equal to 0 when n = 1, and in all other cases we cannot obtain an accurate air change rate. Table 1 shows the maximum information deficit for n fully mixed zones.

To avoid this deficit, we have to supplement n(n − 1) data. For example, preparing and measuring the steady state concentrations of n types of tracer gases for n zones allows calculation of an accurate ventilation rate. Otherwise, with one tracer gas, n ventilation measurement trials are required to calculate the steady state concentration of the tracer gas in n zones. In this case, for each ventilation measurement, the location of the tracer gas doser is varied in n ways. However, these methods require either many tracer gases or many measurements, which means they cannot be practically applied. In addition, the time and money available for air change rate measurements are limited. Consequently, we evaluated the estimation of air change rate from a single measurement using one tracer gas.

As already mentioned, F_uptake for a single fully mixed zone [Figure 1(a)] can be calculated accurately even from only a single measurement with one tracer gas. The tracer mass balance is given by:

The indoor air mass balance is given by:

where E₁ is a known value and C₁ is obtained by a tracer gas concentration measurement. F_uptake can then be accurately calculated as follows:

If building consists of two fully mixed zones [Figure 1(b)], it is more difficult to measure the air uptake F_uptake than for n = 1. The tracer mass balance for the two fully mixed zones is as follows:

where C₁ and C₂ are considered to be constant after stabilization of tracer gas concentration. Therefore, we can transform the above equations to:

The air balance is represented by:

There are now six unknown parameters, F₀₁, F₀₂, F₁₀, F₁₂, F₂₀, and F₂₁ in the above four balanced equations, which means F_uptake, which is defined by the following equation, cannot be calculated accurately:

A situation such as this usually occurs with n ≥ 2, and the maximum information deficit is equal to n(n − 1). However, practically it is difficult to increase the number of tracer gases used or the number of measurements, which both consequently increase the measurement time. Therefore, it is necessary to estimate the ventilation volume for these zones.

For air change rate estimation of two fully mixed zones from limited information, we can utilize information about the volume of each zone, and tracer gas emission rates in each zone. Information about tracer gas concentrations in each zone can be obtained from passive air sampler measurements. As discussed in Section 2.3., four equations can be established for tracer gas mass balance and air balance for the two zones. There are six unknown parameters, F₀₁, F₀₂, F₁₀, F₁₂, F₂₀, and F₂₁, and F_uptake cannot be accurately estimated. However, even in this case, we can estimate the air change rate with a practical level of accuracy using the following estimation method. This is due to the symmetry of Equations (8)–(12) and non-negative restrictions for the six unknown parameters.

Here, we use estimated air intake, F_e, as an indicator of F_uptake. F_e is given by the sum of emission rates divided by the volume-weighted average tracer gas concentration, C̄:

where C̄ is given by:

One of the purposes of the study is to show that F_e is suitably accurate for practical applications and has a small range of error. The error between F_e and F_uptake can be indicated by the normalized parameter y:

With non-negative restrictions on both parameters, the range of y is at least:

Because F_e and F_uptake are not independent, the range of y is even more limited. A further parameter, x, can be introduced as an indicator of concentration differences between the two zones:

With non-negative restrictions on C₁ and C₂, the range for x is:

If the steady state concentration of tracer gas in each zone is equal, then the two zones can be treated as one fully mixed zone. So, when x = 0, y is 0, and the ventilation rate can then be accurately calculated. The range for y is mathematically more limited.

The restriction condition is non-negative restriction of six airflow volumes. There are six unknown parameters in the four balanced equations (Equations (8)–(11)), and we can express y as a function of two arbitrary unknown parameters. If F₁₀ and F₂₀ are selected as these two parameters, then y can be expressed as:

If we select F₀₁ and F₁₂, then y can be expressed as:

and for F₀₂ and F₂₁:

It should be noted that only the symmetry of parameters was considered when selecting these three pairs, and other expressions would also give the same results.

In Equation (19), from non-negative restriction of F₁₀ and F₂₀, the range of y as a function of x can be calculated as follows:

In Equation (20), from non-negative restrictions of F₀₁ and F₁₂, we can obtain the range of y as a function of x as follows:

In Equation (21), from non-negative restrictions of F₀₂ and F₂₁, we can obtain the range of y as a function of x as follows:

If non-negative restrictions for all six unknown parameters are considered, then combination of Equations (22)–(27) can accurately express the ranges for y (Appendices I–III). If f₁(x) − f₄ (x) are determined as follows:

the ranges for y can be expressed as follows:

The estimation method can then be illustrated in application to a specific situation. It should be noted that the kind of tracer gas does not affect the results. This method is based on steady state. Thus, that requires constant airflow rates over a sufficient period of time for the concentrations to stabilize. Here, we attempt to simply measure the air change rate of two fully mixed zones. V₁ and V₂ are assumed to be 10 m³ and 30 m³, respectively. E₁ and E₂ are set as 100 μg/h and 200 μg/h, respectively. Figure 2 illustrates this situation. In the case, f₁(x) − f₄(x) were calculated as:

Thus, the range of y is limited as shown in Figure 3. If C₁ and C₂ are measured to be 30 μg/m³ and 10 μg/m³, respectively, air intake F_e can be estimated at 20 m³/h using Equations (13) and (14). In this case, x is calculated to be 0.5, so, f₁ (0.5) = −0.33, f₂ (0.5) = 1, and f₃ (0.5) = 0. Thus, from Equation (32):

So, the estimated F_e (20 m³/h) is accurate within 33%, and is an underestimate. From Equation 15:

It should be noted that F_e can be appropriately normalized in accordance with the aim of measurement. If we want to minimize the error rate, we can choose 25 ± 5 m³/h as F_e (error range is ±20%). To minimize the error rate, E₁ and E₂ should be set at the same value. In this case, the maximum error range of air intake F_e estimated by the methods is always <33%. This error value is equivalent to that reported by Riffat with single tracer gas measurement [14]. Miller et al. [12] reported that airflow rates in two-zone building could be estimated with an accuracy of 8% by two tracer gas decay experiments with the nonlinear least-squares minimization method in controlled conditions. One feature of this method is its simplicity in calculation operations. As described above, the proposed method can estimate air uptake volume with a practical accuracy only by using four arithmetic operations. If this estimation method can be combined with real time monitoring of a tracer gas concentration, we can extemporarily obtain the estimated air uptake without complex calculations. Furthermore, by using this method, overestimation of air change rate can be easily avoided. When aiming to avoiding overestimation of air change rate, the estimation method can give a minimum air change rate (in this case, F_e,min. = 20 m³/h). Figure 4 summarizes the estimation flow chart for air intake and its error range.