A multiple PTF method using the steady-state assumption was initially developed by D'Ottavio and Dietz (1985). In the current research, a similar approach was applied: the number of different tracer gases used must be equal to the number of measured zones/floors/dwelling units (henceforth referred to as “compartments”), with each compartment containing exactly one tracer gas. In the equations that follow, it is assumed that tracer k is placed in compartment i . Additionally, the following assumptions must be met: 1) the air in each measured compartment must be well-mixed, and the tracer should be emitted uniformly in the compartment. 2) All the measured compartments are at steady-state over the measurement period, which should usually be in the order of weeks. 3) The tracers used should be emitted into each compartment with a known and constant entry rate. 4) The tracers used should not have a background in the outdoor air and should mix well with the air. 5) The outdoor concentration of the tracers used should be negligible.

The theoretical basis of the model is the continuity equation, which expresses the conservation of a tracer gas in the measured volume of the building (compartment). In a general multi-zone environment of N compartments, where a different type of constant tracer source k ∈ 1 , 2 , … , N is used for each measured compartment i ∈ 1 , 2 , … , N , the continuity equation can then be rewritten as follows: V i d C k i t d t = m k i + ∑ j = 1 , j ≠ i N R j i C k j t − C k i t ∑ j = 1 , j ≠ i N R i j − R e i C k i t , (1) where:

N denotes a number of investigated compartments;

The continuity equation expresses the mass balances of the tracer gases used, and therefore their concentrations are expressed as mass per unit volume to ensure accuracy, even if the air density differs between zones (e.g. if the zones have different temperatures).

The left-hand side of Equation 1 expresses the change of total mass V i C k i t of tracer k in compartment i at time t . The first term on the right-hand side of the equation represents the known emission rate of tracer k in compartment i , the second term represents the unknown inflows R j i of the measured mass of tracer k from compartment j ( C k j ) to compartment i   i ≠ j , the third term represents the unknown outflows R i j of the measured mass of tracer k from compartment i ( C k i ) to the other compartments j. The last term represents the unknown air exfiltration rate R e i of the measured mass concentration of tracer k from compartment i ( C k i ) to the outside environment.

The steady-state multi-tracer constant-emission model for N compartments (and N different tracers) of average AER in the compartment i can then be written as m k i + ∑ j = 1 , j ≠ i N R j i C k j − C k i ∑ j = 1 , j ≠ i N R i j − R e i C k i = 0 . (2)

Time variable t is removed from the equation since, in the steady state, the concentrations C i j are constant.

The measured average mass concentration of tracer k in compartment i ( C k i ) in (ng m⁻³) during sampling time t in Equations 1,2, can be calculated as follows: C k i = K R d k i   M k   p i U k   t   R   T i , (3) where:

K denotes a conversion coefficient from g ppm m⁻³ to ng m⁻³ (K = 1,000);

The average AER of the multi-compartment building n (h⁻¹) can be calculated thus n = 1 V ∑ i = 1 N R e i , (4) where:

V denotes total the internal volume of a studied dwelling (m³), = ∑ i = 1 N V i .

The most common use of the multi-tracer gas steady-state ventilation model is measuring the average AER in multi-story family houses. The airflow schematics for a house that contains three investigated compartments is illustrated in Figure 1 considering exfiltration, R e i , infiltration, R I i , and average airflows between each compartment. Assuming the model approach indicated in the beginning of this section, the average airflows during measurement time from each compartment to the outdoor environment can be described. The derivation of the equations necessary for the AER calculation coming from the measured steady-state mass concentrations of the tracer gases used, C k i , and the assumed number of measurement zones N = 1 − 3 can be found in Supplementary Appendix S1. The investigated average air exchange rate n between the building as a whole and the outdoor environment is eventually calculated using Equation 4 and expressions from the equations in Supplementary Appendix S1.

An analysis of the errors in calculated airflows related to the use of PFT tracer gas techniques can be found in D'Ottavio and Dietz (1985) and Sherman (1989). The usual approach for determining the average AER uses constant emission sources of the tracers injected into a building for a sufficiently long period compared to the time rate of change in the tracer concentration. The steady-state tracer gas concentration is then assumed to be reached. Additionally, if the exposure time is sufficiently long, the mixing of tracers used with indoor air will not be a significant factor in estimating random error. The key errors will be determined only by the precision of the source strength and the measured tracer gas concentrations. Both Monte Carlo and matrix inversion (Lefebvre, et al., 2000) approaches are used to calculate the average AER and its uncertainty. The error propagation law using a Taylor series with only first derivatives is considered. All input variables are calculated as independent, and therefore the error estimates are overestimated. Because the matrix inversions to calculate the AER are used, the experimental setup of each measurement is designed to ensure a positive and non-zero discriminant. Table 1 lists the sources of partial relative uncertainties of types A and B (JCGM, 2008) that are used to calculate the AER (see general Equations 1–3). Table 1 illustrates typical results of the estimation of airflows and their total uncertainties for a coverage factor of k = 1 from occupied multi-story buildings. The results are sorted by the type of tracer gases used. The conventional true value of the measurement quantity lies in the dedicated range of values with the probability of approximately 68%.

To avoid multiple testing problems, multivariate multiple regression is used first. The statistical significance of independent variables (exposure time, temperature, absolute humidity, pressure, and their nonlinear components represented as quadratic terms in the model) was tested using the Pillai trace test (Pillai, 1955). In addition, the interaction, represented as a product between absolute humidity and temperature, was included in the model. The significance between a full model and a sub-model (nested model) without all the terms that include tested quantity was assessed. Follow-up univariate tests were performed if testing in multivariate multiple regression was significant, and only significant independent variables from the first step were used in the model. F-tests of sub-models (Chambers and Hastie, 1992) were used and per-dependent variable p-values adjustments were performed to adjust the p-values using the Holm method (Holm, 1979).

This study has thus far outlined the theoretical and statistical background of AER determination. The current section covers the material description and methodology of the detection system. The method’s basic principle is that the measurement of the average AER in buildings is composed of a two-component system: tracers containing VOCs and sorption tubes containing specific sorbent with significant affinity to the VOC. The principle of the AER measurement is the utilization of the tracer source container (TSC)–sorption tube (ST) system in single or multiple compartments of a building for a specific exposure time that can vary from 1 week to several months. The tracer contains compounds based on either fluorinated cycloalkanes‐PFT–or trichloro- and tetrachloro-ethylene (TCE and PCE, respectively). Using multiple VOCs during the same campaign offers the possibility of calculating the airflow between several compartments in one building. The sorption tube contains grains of sorbent based on activated carbon with specific grain size and porosity. After the exposure of tracer and sorption tubes in a dwelling, the tubes are analyzed using a thermal desorption–gas chromatography system to determine the amount of specific tracers adsorbed onto a tube (Leaderer, et al., 1985; Center for Environmental Research Information, 1999). Using the weight difference of TSC before and after the campaign, the known temperature in a dwelling throughout the campaign, the known volume of the rooms comprised in a dwelling, and the total weight of each VOC on a sorption tube used in a campaign, the AER can eventually be calculated.

The source container with an internal volume of approximately 4.5 mL was composed of borosilicate glass with a capillary of well-defined diameter and length. Approximately half of the container volume comprised fiberglass to capture the VOC and prevent the content from spilling. The container was filled with a specific amount of liquid depending on the boiling point of the VOC. The emission rate of the container was calculated by mass difference over a time interval (usually before the start and after the end of a campaign). The physical characteristics of the VOC tested during method development are shown in Table 2, and the layout of the TSC is illustrated in Figure 2.

TCE and PCE were used early in the research due to their wide application, good detection sensitivity, and suitable system tracer–sorbent. However, with the subsequent testing of several PFTs and owing to their additional suitable physicochemical properties such as near-zero background concentration (Dietz, et al., 1986; Dietz, 1991), chemical stability, good resolution and identification on a chromatogram (Lagomarsino, 1996), and no known adverse effects related to human toxicity or ecological toxicity, the chloroethenes became dispensable and redundant as tracer analytes.

For all the measurements and assessments of the AER, stainless steel tubes filled with a sorbent based on activated carbon were used. The sorbent was fixed in a tube by a set of two gauzes and a spring. During the tubes’ transport to and from dwellings, brass caps were used on both ends; during exposure, a diffusion cap on one end was used. The tubes were filled with sorbent based on either graphitized carbon black (Carbopack B™) or porous polymer adsorbent matrix (Tenax TA™ and Chromosorb®). Selected physical properties of these sorbents are detailed in Table 3.

Tenax TA™ first used as a chlorinated VOC was commonly applied in most measurements and campaigns. However, with the expansion of PFT usage, this sorbent was omitted from the succeeding AER measurements and served only as a backup option. Chromosorb 102™ was used in the earlier stages of PFT introduction, but it was soon replaced entirely by Carbopack B™ due to high artifacts and lower sorbent maximum temperature (Table 3).

A thermal desorption system (Markes International Ltd. model TD100-xr) and gas chromatography (Agilent Technologies. Inc., model 7890B) were used to pre-concentrate and analyze the type and amount of the VOC adsorbed onto a sorption tube. A DB-VRX column with a length of 30 m, inner diameter of 0.25 mm, and film thickness of 1.40 µm is usually used. Along with the EC detector, this arrangement is a suitable technique for the sorption tube analysis of chloro- and fluorocarbons. The advantages of this TD-GC system are a low detection limit (order of magnitude 10⁻¹ ng of PFT per tube), simultaneous analysis of up to six compounds, semi-automatic loader with a capacity of 100 tubes working on a 24/7 regime, and a favorable S/N ratio. Fast sample processing was achieved by efficient electrical cooling of the focusing trap. The digital mass flow controllers gave double-split ratios of up to 1:125 000, which was desirable for the more concentrated samples analysis or for the samples of unpredictable amounts of VOCs on the sorption tubes. The most used double-split ratios in the experiments were 1:25, 1:55, and 1:155. For every split and tracer, a calibration procedure was created consisting of a preparation of a liquid multi-standard solution containing five tracers (MCH, MDC, PCH, TCE, and PCE) dissolved in methanol. The concentration of these solutions ranged from 10 to 1,000 ng μL⁻¹ for MCH, MDC and PCH, from 10 to 800 ng μL⁻¹ for TCE, and from 10 to 400 ng μL⁻¹ for PCE. They were prepared according to ISO 16017-2 (ISO 16017-2:2003, 2003) by the injection of 1 µl of multi-standard solution onto a tube using a calibration solution loading rig. At least five different concentrations were used to prepare a calibration curve.

The quality assurance of AER measurements was primarily based on the control of a ST for every sorbent and tracer used by spraying known concentrations of standard tracer gases into a tube and their subsequent evaluation on the TD-GC unit. The validation process of the VOC analysis covered internal control background measurement, control sample measurement, and the measurement of two identical STs installed at a same location/site with the same sampling time, where available. Additionally, the laboratory took part in regular proficiency testing for tubes spiked with trichloroethene/tetrachloroethene solution, with satisfactory results. An interlaboratory comparison with the University of Chemistry and Technology, Prague, regarding the amount of MDC adsorbed on the Carbopack™ tubes in the range of 52–184 ng was performed. The calibration curves from the two laboratories were analyzed, and it was determined that the difference between them was statistically insignificant, with p = 18.3% on a 5% confidence level.

Furthermore, the uptake rate of the TD tubes ( U k ) was controlled on a statistically significant sample for each tracer gas used in a large NRPI radon chamber until the manufacturer’s maximum recommended number of the TD cycles was achieved. For U k determination, only tubes filled with 400 mg of Carbopack B™ with 60–80 mesh size were used. The uptake rate, U k , was calculated using the relationship in Equation 3. According to this equation, the TD tube responses, R d k i , were measured directly using a TD-GC calibrated to the appropriate gas standard. The tracer mass concentration, C k i , was calculated as the ratio of the constant tracer gas entry rate into the chamber and air exchange rate between the inner chamber volume and its outdoor air. The tracer gas entry rate was calculated as the mass time difference TSC. A defined and stable air exchange rate in the chamber was measured and controlled by the chamber ventilation system, which allowed the known volumetric airflow rates to be applied in the chamber. The one-time injection of CO gas into the chamber and its continuous measurement with a Polytron CO gas flow current sensor (Dräger, DE) allowed an independent estimation of the air exchange rate. Temperature, barometric pressure, and relative humidity sensors in the chamber allowed estimation of the required molar volumes of tracer gas to be tested. A known, defined, and stable air exchange rate set up in the chamber allowed estimation of the time at which a steady-state concentration of tracer gas was reached. This was considered the initial time for the deployment of the ST into the defined exposure atmosphere.

The deployment of emission sources used and TD tubes in houses was then standardized in the instructions, depending on the layout and volume of individual rooms in the measured building. The instructions to the owners of the measured objects and the measurement time were written to meet the main requirements and assumptions of the method. The correct progress of the approach in the deployment and instructions to the owners was verified by proficiency testing with the National Brookhaven Laboratory in New York, USA, which indicated a mutual agreement of the measured values of the average AER from 15 tested objects (family houses and apartments), with a relative difference between the results of less than 10%. A standard operating procedure (SOP), “Determination of the amount of the tracer gases adsorbed onto a sorption tube using gas chromatography system with electron capture detector,” was created as a part of the accreditation process for the VOC amount determination. The laboratory underwent the assessment of conformity of the laboratory’s quality, and along with the SOP, the NRPI was granted an accreditation certificate according to international standard ISO/IEC 17025:2018 (ISO/IEC 17025, 2017). This procedure describes the preparation of calibration standards and the analysis of the ST and highlights the validation and verification process by regularly checking the desorption cycles, limits of quantification, retention times of specific VOCs and stability of background pulses, and also by measuring a control sample. However, the sampling could not be a part of the accreditation since the laboratory is neither responsible for the information provided by the indweller during the campaign nor can be liable for the potential misuse of either the TSC or ST. Two tubes are usually added to a package, with the ST and TSC being en route to a designated building and back to the laboratory, in order to track the sorption of the unwanted VOC during transport (called “blank tubes”).

The first step needing evaluation before the in situ usage of a TSC–ST system in dwellings was the determination of the emission rates of different tracer source containers filled with the specific VOC analytes. As the emission rate of every VOC depends on the ambient temperature, it had to be measured for every temperature expected in a dwelling in the Czech environment—in the range 12 °C–33 °C. Such a low minimum temperature was chosen to also cover conditions in basements and during transport. For the purpose of laboratory testing, a batch of TSC was put in a dry block heater. Emission ranges of PFT analytes in relation to the ambient temperature are depicted in Figure 3, and emission ranges of chlorinated VOCs are outlined in Figure 4. Each point in the figure represents an average value of the emission rates of ten source containers with very similar capillary length. The capillary length of container sets filled with each tracer varied. The data points can be fitted using linear function with coefficient of determination well above 0.90, ranging between 0.91 (for PCH) and 0.97 (for TCE). The fraction of unexplained variability is therefore low.

To estimate the time dependency of the R d k i , several one-month-long campaigns in a large radon chamber were performed under controlled ambient conditions (temperature, humidity, and air exchange rate). Figures 5–7 represent VOC amounts adsorbed onto a tube relative to the exposure time at 20, 22, and 25 °C during the campaign, respectively. The points represent the average value of an adsorbed amount of analytes on an ST, with the number of STs withdrawn each time ranging from 2 to 17 tubes. It is important to note that in each steady-temperature campaign, the TSC with different emission rates were used. The exposure time was chosen to represent the possible timespan for in situ measurements from 7 days to 1 month (30 days). Since the fraction of unexplained variability is low in this case too, linear fitting of the data points yields the average value of the coefficient of determination for a tracer gas of 0.99.

When the temperature, relative humidity, air exchange rate, and emission rates of TSC are known and under control, it is possible to measure the average R d k i relative to their exposure time and to the expected temperatures in a dwelling, as shown in Figures 5–7 for 20, 22, and 25 °C, respectively. Thus, it is possible to calculate the expected amount of the VOC onto an ST during a campaign with a specific duration using the proper split ratio and corresponding calibration. For a 1-week long exposure, a 1:25 ratio was applied, and for all the other exposure experiments, a 1:155 ratio was used in order to avoid saturation of the ECD detector.

In principle, to estimate U k for a tracer under study, Equations 2,3, are used. The initial conditions were one compartment ( i = 1) and no background, where R i j = R j i = 0. The uptake rate can be dependent on five physical quantities measured in this experiment: exposure time (t), temperature (T), pressure (p), relative humidity (RH), and absolute humidity (AH). The dataset contains 20 measurements overall retrieved from the radon chamber experiments. Due to their interconnection, these quantities are affected by multicollinearity; RH, AH, T, and the potential nonlinearity and interaction terms in statistical models exhibit this issue. For AH, RH, and T, any two variables contain similar statistical information as the third variable. To mitigate multicollinearity, RH was excluded from the model assessment. The effect of the exposure time on the set of uptake rates is altogether significant on statistical level 0.05 (p = 0.047). Any nonlinearity, represented as quadratic component, is not observed (p = 0.54) in this case. The effect of temperature is significant (p = 0.0014) and is nonlinear (p = 0.0057). The effect of absolute humidity is also significant (p = 0.011) and can be considered nonlinear (p = 0.0016). Additionally, the interaction between absolute humidity and temperature is significant (p = 0.0019). However, for a given dataset, this effect is almost impossible to distinguish from the quadratic term of temperature and of absolute humidity. The effect of absolute humidity remains unclear. The effect of the pressure is statistically insignificant (p = 0.267).

More detailed analysis and quantification of the effects of independent variables (t, T, AH) per each analyte is problematic due to the multicollinearity and small sample size (lack of statistical power). Table 4 summarizes the significance of variables for each analyte.

The dataset provides the clearest results for pressure (insignificant) and exposure time. For the latter, the effect is approximately linear, and the coefficients are −2.1E-5 min⁻¹ (MDC) and −2.6E-5 min⁻¹ (TCE). The results for temperature and absolute humidity that are represented in the model with linear, quadratic, and interaction terms are fuzzy, with no possibility of precisely quantifying their effect. For temperature, the uptake rates decrease with possible nonlinearity where the uptake rate decreases with higher temperatures and with relatively constant rate at 20 °C–25 °C (MDC, TCE, and PCE). For MCH and PCH, the effect is approximately linear with coefficients −0.28 and −0.19 °C⁻¹, respectively. The effect of absolute humidity is significant for PCE. Its interaction with temperature can be significant (depending on selection of the model used for testing). The effect of interaction is almost impossible to distinguish from the quadratic component of temperature and humidity. A similar issue is seen in the difference between the quadratic component of temperature and quadratic component of humidity.

In some field measurements of AER in multi-zone buildings, Cl- and PFT-based tracer gases were used simultaneously for comparison. Tables 5,6 illustrate the results of the airflows investigated and their typical total measurement uncertainties for a coverage factor of k = 1   J C G M , 2008 in buildings with two different volumes (300 and 500 m³, respectively) obtained from the three-week-long measurements. In the case of AER measurements using Cl-based tracer gases, TENAX™ TA was used as the sorbent in the TD tubes, and Carbopack B was used for the PFT tracer gases—detailed in the preceding text of Table 3.

Here,

t exp denotes exposure duration;

The meanings of all the symbols in Table 6 are the same as in Table 5.

Descriptive statistics of measured weekly AER averages obtained from measurements in single and multi-story detached houses during the heating and non-heating season are presented in Table 7. The measured houses were occupied without any mechanical ventilation system to control the AER and were different ages, as were their owners. The aim of the measurements was to evaluate the seasonal effect and the influence of building envelope tightness on average air change values.

Here,

Q₁ and Q₃ denote first and third quartile, respectively;