#### 3.1. Estimation Method for Air Change Rate of Two Fully Mixed Zones from a Single Measurement with One Tracer Gas

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,

F_{01},

F_{02},

F_{10},

F_{12},

F_{20}, and

F_{21}, 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 C_{1} and C_{2}, 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

F_{10} and

F_{20} are selected as these two parameters, then

y can be expressed as:

If we select F_{01} and F_{12}, then y can be expressed as:

and for F_{02} and F_{21}:

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

F_{10} and

F_{20}, the range of

y as a function of

x can be calculated as follows:

In

Equation (20), from non-negative restrictions of

F_{01} and

F_{12}, we can obtain the range of

y as a function of

x as follows:

In

Equation (21), from non-negative restrictions of

F_{02} and

F_{21}, 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

f_{1}(

x) −

f_{4} (

x) are determined as follows:

the ranges for y can be expressed as follows:

#### 3.2. Application of the Estimation Method

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.

V_{1} and

V_{2} are assumed to be 10 m

^{3} and 30 m

^{3}, respectively.

E_{1} and

E_{2} are set as 100 μg/h and 200 μg/h, respectively.

Figure 2 illustrates this situation. In the case,

f_{1}(

x) −

f_{4}(

x) were calculated as:

Thus, the range of

y is limited as shown in

Figure 3. If

C_{1} and

C_{2} are measured to be 30 μg/m

^{3} and 10 μg/m

^{3}, respectively, air intake

F_{e} can be estimated at 20 m

^{3}/h using

Equations (13) and

(14). In this case,

x is calculated to be 0.5, so,

f_{1} (0.5) = −0.33,

f_{2} (0.5) = 1, and

f_{3} (0.5) = 0. Thus, from

Equation (32):

So, the estimated

F_{e} (20 m

^{3}/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 m

^{3}/h as

F_{e} (error range is ±20%). To minimize the error rate,

E_{1} and

E_{2} 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 m

^{3}/h).

Figure 4 summarizes the estimation flow chart for air intake and its error range.