#### 2.1. Approaches to Estimating Ventilation Rates

Methods using CO

_{2} as a tracer gas are based on a fully mixed mass balance model:

where V = room (or zone) volume (m

^{3}); C = CO

_{2} concentration in the room (mg·m

^{−3}); C

_{R} = CO

_{2} concentration in outdoor air or replacement air (mg·m

^{−3}); Q = flow rate of outdoor or replacement air (m

^{3}·h

^{−1}), and E = CO

_{2} emission rate of indoor sources (mg·h

^{−1}). Generally, E is calculated as n G

_{P}, where n = number of persons in the space, and G

_{P} = CO

_{2} generation rate per person (L·h

^{−1}), which is age- and activity level-specific (as described later). The air change rate, A (h

^{−1}), is Q/V. Concentrations throughout the zone are assumed to be equal; this should be confirmed using measurements at multiple locations [

24]. This assumption can be violated by the uneven distribution of CO

_{2} sources and by limited air distribution effectiveness. The CO

_{2} methods discussed in this paper apply only to a single and fully mixed zone. Also, VRs derived using CO

_{2}-methods will include outdoor air delivered via both the ventilation system and infiltration. Finally, most applications of tracer gas methods assume that the VR is constant over a specific time window, i.e., the period over which the CO

_{2} concentration trend or peak level is measured and analyzed.

The outdoor air flow rate per person, V

_{0} (L·s

^{−1}·person

^{−1}), is obtained from air flow rate Q (m

^{3}·h

^{−1}) or A (h

^{−1}):

where the constant accounts for the volume and time conversions. If the number of occupants varies over the time when the VR is determined, then the choice of n is critical. In schools, students and staff often leave for lunch, recess and for other reasons, thus the average occupancy over the school day can be much lower than the maximum occupancy. In consequence, V

_{0} determined using the average occupancy is higher, and often considerably (by about 50%) than that based on the maximum occupancy. To assess the adequacy of ventilation by reference to VR guidelines expressed as L·s

^{−1}·person

^{−1}, the use of the maximum occupancy appears most consistent. To understand contaminant exposures, the use of the average occupancy is preferable. This difference does not appear to have received attention in the literature, possibly because prior studies have not had continuous occupancy information, however, this issue appears important based on our recent experience in school classrooms [

36].

VRs in both naturally- and mechanically-ventilated buildings can be affected by time-varying factors including internal heating and cooling loads, outdoor temperature and the indoor-outdoor temperature difference, and wind speed and direction [

37]. In buildings using variable air volume (VAV) systems, air flow and the VR will depend on thermal load. If the VR varies in the study time window used to estimate the VR, then the assumption of a constant VR required by most of the CO

_{2}-based methods will be violated, although the VR estimate may be useful if the variation is small.

#### 2.2. Steady-State Methods

Steady-state or equilibrium methods apply after CO

_{2} levels have reached a steady-state concentration. The method is described by guidance and standards [

16,

24,

33]. The steady-state air change rate, A

_{S} (h

^{−1}), is calculated as:

where n = number of persons in the space; G

_{P} = average CO

_{2} generation rate per person (L·min

^{−1}·person

^{−1}); V = volume of the room or space (m

^{3}); C

_{S} = steady-state indoor CO

_{2} concentration (ppm); and C

_{R} = CO

_{2} concentration in replacement or outdoor air (ppm) (Given the widespread practice, the remainder of this paper uses concentration units of ppm). Steady-state methods assume that the CO

_{2} generation rate (i.e., the number and physical activity of occupants) over the study time window is constant for a sufficiently long period to reach the indoor equilibrium concentration C

_{S}. (If C

_{R} changes over the time window, then the difference between C

_{S} and C

_{R} should approach a steady-state level.) ASTM [

24] suggests that the measured C

_{S} should reflect at least 95% of the equilibrium value (i.e., as attained after three complete air changes), and provides comprehensive guidance for this method including methods to estimate uncertainty. As noted, steady-state methods assume the VR is constant over the study time window, and most assume that the outdoor CO

_{2} concentration is constant.

In practice, the average age and average activity level of occupants are used to estimate G

_{p}, the replacement air concentration C

_{R} is preferentially measured [

24] or assumed to be 400 ppm [

38], and C

_{S} is determined as the maximum 5- to 20-min average concentration over the study time window [

1]. Some spaces may not reach steady-state conditions over the workday or study period. For this reason, the steady-state method is not recommended in schools if classes last 45 min or less and the air change is below 4 h

^{−1} [

21,

25]. Many schools have much lower VRs, thus, the number of occupants must be constant for at least several hours to approach steady-state levels. If occupancy varies around the time of the peak concentration, then the method’s assumptions are not met. In classrooms, averaging the number of occupants n and the generation rate G

_{P} just prior to the observed CO

_{2} peak may avoid anomalies if the occupancy fluctuates widely.

Steady-state methods may be used to estimate the VR per person using Equation (2) or as:

where V

_{0,S} = outdoor air flow rate per person (L·min

^{−1}·person

^{−1}), and the constant converts the generation rate from hours to seconds and concentrations C

_{S} and C

_{R} from ppm to a mixing ratio. For example, using a generation rate for a moderately active adult (1.7 MET; G

_{P} = 0.46 L·min

^{−1}·person

^{−1} (

Section 2.5), C

_{S} = 1479 ppm, and C

_{R} = 400 ppm, Equation (4) gives V

_{0,S} = 7.1 L·s

^{−1}·person

^{−1}, the minimum VR recommendation for many indoor spaces, including classrooms [

12].

#### 2.3. Decay Methods

Decay or step-down methods can be used when a space is vacated after occupancy, or if there is a stepwise decrease in occupancy [

16,

24]. The decay air change rate, A

_{D} (h

^{−1}), for a single (and well-mixed) zone can be estimated using two CO

_{2} measurements:

where Δt = period between measurements (h); C

_{0} and C

_{1} = measured CO

_{2} concentrations over the decay period (ppm); and C

_{R} = CO

_{2} concentration (ppm) in replacement air or the steady-state concentration at the lower occupancy in the case of stepwise decrease in occupancy. The stability of this 2-point estimate can be checked using an upper bound estimate of A

_{D}, obtained by selecting C

_{1} and C

_{0} as the maximum and minimum concentrations, respectively, occurring near the nominal times specified, e.g., within ±1 h.

Alternatively, a sequence of CO

_{2} concentrations over a portion of the decay period, C

_{t}, may be used to fit a solution to Equation (1) using regression or other means:

where t = time of the measurement (h). Equation (6) can be linearized:

where C

_{S} = the steady-state CO

_{2} concentration. The estimated decay air change rate A

_{D} is the slope of the regression of ln(C

_{1} − C

_{R}) against time t. The regression intercept is ln(C

_{S} − C

_{R}), thus C

_{S} is equal to exp(intercept) + C

_{R}.

Decay methods have been used to estimate VRs in schools [

21,

32,

39,

40,

41]. The decay method is simple, and the regression approach does not require knowledge of C

_{R}, C

_{S}, G

_{P},

n or even V. However, there are important caveats. First, the appropriate time window for analysis can require careful selection [

39,

42]. The concentration change over the period must be large relative to the variation in C

_{R} and CO

_{2} measurement error; typically, changes of 100 ppm or more may be sufficient, but at least several hundred ppm are desirable given the performance of typical instrumentation. (Many CO

_{2} instruments report accuracies of ±50 ppm plus 1% to 2% of readings.). Second, while CO

_{2} decay curves often display near ideal behavior in many spaces, in rooms that utilize natural ventilation, opening windows during class breaks can overestimate VRs during classes [

32]. Conversely, if windows are opened during occupancy but closed afterwards, VRs will be underestimated. Opened windows also can lead to significant variation in concentrations in the space, thereby violating the well-mixed assumption. Third, in mechanically-ventilated school buildings, while windows are rarely opened (and sometimes not openable), HVAC systems typically are shut down at the end of the school day. Because of the shut-down, decay air change rates will not apply to the occupied portion of the day. This applies to most U.S. schools where shut-down may occur immediately following the last class, e.g., as early as 14:20 (2:20 p.m.). (24-h time notation is used throughout this paper.) Fourth, VAV systems will provide less ventilation air if the thermal load diminishes after the space becomes unoccupied, which would have the effect of lowering the VR during the decay period even if the HVAC system is not shutdown. This especially applies to densely occupied spaces, including some classrooms. Fifth, in both naturally- and mechanically-ventilated buildings, VRs and/or infiltration rates will depend on the indoor-outdoor temperature difference, wind speed, heating and cooling load, and other factors that may change over time. For these reasons, the decay-based VR may not reflect conditions during the (occupied) school day, although it may provide information regarding the infiltration rate. Finally, while both the two-point and multipoint methods provide identical results under idealized circumstances, the former is more sensitive to measurement error and thus may be less accurate.

#### 2.4. Build-Up Methods

Build-up methods use the increase in CO

_{2} concentrations following occupancy to determine VRs with the assumptions that the CO

_{2} generation rate and VR are constant over the study time window, and the zone is fully mixed (The assumption of a constant CO

_{2} generation rate in classrooms is discussed in

Section 2.5). Build-up methods can estimate the VR just after a building is occupied, or after a step-wise increase in occupancy. The method may be preferable to both the steady-state and decay methods since the derived VR applies to the occupied period, and since steady-state conditions are not required.

There are many approaches to solving the single zone mass balance for the build-up period. One approach calculates the build-up air change rate A

_{B} (h

^{−1}) using two sequential CO

_{2} measurements:

where Δt = period between C

_{0} and C

_{1} measurements (h), C

_{S} = steady-state concentration (ppm), and C

_{0} and C

_{1} = CO

_{2} concentrations measured at start and end of the observation time window, respectively (ppm). C

_{S} may be derived in several ways. A 3-point method uses a third concentration, C

_{M}, measured at the midpoint in time between the C

_{0} and C

_{1} measurements [

25]. If the build-up curve follows the expected approach for the method’s assumptions (i.e., smooth and initially rapid increase that gradually plateaus), and if the method’s assumptions are valid, then Ĉ

_{S} may be estimated as:

Appropriate times for measuring C_{0}, C_{1} and C_{M} in classrooms and other applications will depend on occupancy patterns and building system operation. Because build-up curves may not follow the expected approach due to time-varying occupancy or other reasons, Equation (9) can be sensitive to the time period selected (e.g., the estimated C_{S} can be negative and meaningless if C_{M} is less than the average of C_{0} and C_{1}).

A second solution technique for the build-up method is proposed that circumvents some of the issues associated with the 3-point method. This uses an estimate of C

_{S} obtained by simultaneously solving Equations (3) and (8) so that A

_{S} = A

_{B}. Using the average number of occupant n (persons), age-adjusted CO

_{2} generation rate G

_{P} (L·min

^{−1}·person

^{−1}), zone volume V (m

^{3}), replacement air concentration C

_{R} (ppm), and an initial air change rate estimate denoted as Â

_{B} (h

^{−1}), an initial estimate of the steady-state concentration Ĉ

_{S} follows from Equation (3):

By substituting Equation (10) into Equation (8) and simplifying, Â

_{B} can be solved for as the root of the following equation:

While no analytical solution exists, Equation (11) can be solved by numerical root-finding methods. Because Equation (11) has local minima and inflection points, a robust algorithm and an appropriate starting solution should be used. Since C

_{S} must exceed C

_{1}, Â

_{B} is bounded and an upper bound estimate (that can be reduced slightly to use as a starting estimate) is:

Equation (11) was solved using a modified Newton-Raphson method (described in

Supplementary Materials). This implicit solution to the build-up method will yield results identical to the 3-point or other (exact) build-up methods under ideal circumstances. However, it may be more stable and less sensitive to the time window selected. It somewhat resembles a steady-state method encompassing a “correction factor” to account for measurements taken prior to reaching steady-state conditions [

26].

A third solution method can be adopted from the ASTM E471 Standard, which includes a build-up method for a continuous injection of a tracer gas [

16]. This is adopted for CO

_{2} by subtracting C

_{R} from the series of CO

_{2} observations measured over the study period, C

_{t} at times t = 1 … T, expressing concentrations in ppm, and calculating the air change rate (rather than the air flow):

where T = number of measurements in the summation, i.e., the expression T

^{−1} Σ

_{t} (C

_{t} − C

_{R})

^{−1} is the average of the inverse of the CO

_{2} measurements after subtracting C

_{R}. The first term in Equation (13) is equivalent to the steady-state solution shown as Equation (3), but with the use of multiple measurements; the second term provides a “correction” given that concentrations C

_{0} and C

_{1}, taken at the beginning and ending of the study time window, respectively, are not at steady-state. This solution to the non-steady-state problem can be sensitive to the time window selected, e.g., using the early part of the build-up curve can inflate the summation term due to relatively large values of inverse C

_{t}, which leads to an overestimate of A

_{B}.

A fourth solution technique for the build-up VR also uses the series of concentration measurements C

_{t} over the build-up period with a solution to the fully mixed model

which may be linearized:

allowing A

_{B} to be obtained as the (negative) slope of ln(C

_{S} − C

_{t}) versus time t. The regression intercept is equal to ln(C

_{S} − C

_{R}), thus, C

_{R} is estimated as C

_{S} – exp(intercept). This approach requires an estimate of C

_{S}, along with the sequence of CO

_{2} measurements. A fifth and related solution method might use a non-linear solver to simultaneously estimate A

_{B} and C

_{S} in Equation (15), e.g., by minimizing the squared residuals.

Only a few applications of the build-up method in schools were identified [

43,

44]; these appear to have used the 3-point method (Equations (8) and (9)) to estimate C

_{S}. The build-up method may be used with relatively short occupied classroom periods and with low air change rates [

32].

Build-up methods can be sensitive to the time window selected, as noted. In U.S. schools, the school day in primary and secondary schools generally starts about 07:30 to 08:00 and lasts till about 14:30 or 15:00, however, there can be considerable variation and many classrooms have substantial drops in occupancy during midday (e.g., due to lunch), in the afternoon, and at other times. In addition to sensitivity to the time window, the build-up air change rate may be more affected by incomplete mixing than the decay method [

32]. The method assumes a constant CO

_{2} generation rate, which requires that the number of occupants in the space and level of physical activity are unchanged over the time window, assumptions that are examined in

Section 2.7. Finally, while the various solution techniques for the build-up method can provide identical results under idealized circumstances, the three point method requires that concentrations closely follow the expected trend; this method as well as the implicit method can be sensitive to measurement error; the implicit method also depends on the accuracy of the occupancy and emission estimates; and the ASTM method requires near-steady-state conditions to minimize errors. In contrast, the multipoint solution method using Equations (14) and (15) is potentially more robust. These issues are explored in

Section 3.

#### 2.5. Transient Mass Balance Methods

Transient mass balance methods to determine the VR use single or multiple occupancy phases, time-resolved occupancy and CO

_{2} observations, and a numerical solution to the mass balance model, Equation (1). An approximate but useful solution is:

where C

_{t} = observed CO

_{2} concentration at time t (ppm); n

_{t} = sequence of occupancy rate observations at time t (persons); G

_{P} = average CO

_{2} emission rate (L·min

^{−1}·person

^{−1}); Q = replacement air flow rate (m

^{3}·h

^{−1}), V = volume of the space (m

^{3}); C

_{R} = replacement air CO

_{2} concentration (ppm); and Δt = time interval for CO

_{2} and occupancy observations (h). The estimated air change rate, A

_{TMB} (h

^{−1}), is Q/V. The CO

_{2} build-up from emission sources present in the space during the period Δt is expressed as the first exponential term, and the CO

_{2} decay as the second term. If flow rate Q is constant, then the expression exp(−A/V Δt) in both terms is constant. Equation (16) is exact for step-wise changes in occupancy at Δt intervals (as obtained by occupant survey data). The key unknown, Q, is determined as the value that meets an error criterion, e.g., a numerical solver may be used to fit Q by minimizing the sum of squares between predicted and observed concentrations. We also recommend fitting the CO

_{2} concentration at the beginning of the study time window using constraints, e.g., minimum of 375 ppm. Estimating this concentration, rather than selecting the CO

_{2} measurement, will increase model fit and may avoid issues associated with an anomalous measurement, minor sensor errors, and in cases, CO

_{2} levels above C

_{R} that remain elevated from the previous day. The term exp(−Q/V Δt), which does not change with time, may be precomputed to increase the optimization speed.

The feasibility of using transient mass balance methods with CO

_{2} to estimate VRs was shown for a university library in England in 1980 [

45]; more recently, the method was used to determine outdoor air flow rates per person (V

_{0}) in 16 classrooms in nine schools (seven naturally ventilated) in England [

46]. Applications of the method using tracer gases other than CO

_{2} may be more common, e.g., the method was used with SF

_{6} to determine VRs in 62 naturally ventilated classrooms in 27 schools in Greece; results showed strong correlation between VRs and CO

_{2} concentrations [

47]. CO

_{2} simulations have been used to evaluate VRs in a few mechanically- and naturally-ventilated school rooms in the U.S. [

22]. Smith [

23] presents additional analyses, including extensions to multizone applications. Methods to estimate uncertainty using transient methods have been discussed elsewhere [

23,

45].

The transient mass balance method is very flexible. It does not require steady-state conditions, and it can be used with arbitrary (but known) occupancy patterns. It can estimate the VR for different portions of the day and different occupancy phases, e.g., morning, afternoon and evening periods, and include occupied and unoccupied periods, either separately or combined. (However, periods that might have different VRs should be analyzed separately.). The method has less sensitivity to the time window selected, any time interval Δt can be used, and the replacement air concentration C

_{R} can vary in time (if known). In addition, other unknown or uncertain parameters can be estimated, e.g., the replacement air concentration C

_{R} and the metabolic activity level used to determine G

_{P} (as well as the initial CO

_{2} concentration mentioned). Such applications, however, should constrain estimated variables to ensure plausible values, aid convergence, and limit the sensitivity of results. While application-specific parameters should be used, for example, C

_{R} if estimated might be constrained to the range of 350 to 500 ppm, reflecting measurement uncertainty of ±50 ppm and an urban increment over average global levels of 50 ppm. Potentially the method can be used to account for the post-exercise recovery period that may temporally increase G

_{P} (see

Section 2.5). Finally, fitting criteria can be calculated as a quality check, e.g., a minimum fraction of variance explained (R

^{2}) might be required, thus ensuring acceptable agreement between predicted and observed CO

_{2} concentrations.

#### 2.6. CO_{2} Generation Rates

The CO

_{2}-based VR methods, other than the decay method and some forms of the build-up method, require estimates of the CO

_{2} generation rate G

_{P}, and all VR methods assume that G

_{P} is constant (or that its variation is known) over the time window used for analysis. This section uses a method documented elsewhere [

24] to calculate school grade-level specific rates suitable for classroom applications. In addition, we include updated physical activity estimates and consider the post-exercise recovery period that can temporarily increase G

_{P}.

Estimates of G

_{P} can be derived using relationships established between occupant weight, height (or age and gender) and metabolic activity. The Dubois equation estimates an individual’s skin area:

where S = skin area (m

^{2}); H = height (m); and W = weight (kg) of an individual. The CO

_{2} generation rate per person, G

_{P} (L·min

^{−1}·person

^{−1}), is the O

_{2} consumption rate multiplied by the respiratory coefficient (0.83):

where MET = metabolic equivalent, a measure of an individual’s energy expenditure of physical activities.

Height and weight data of children for Equation (17) come from U.S. representative growth charts that list sex-specific data at the monthly level (

Table 1) [

48]. We took the 12-month average by grade level (assuming 1st grade children are between 6 and 7 years of age, 2nd grade between 7 and 8 years of age, and so on). For teachers, height and weight data for adults aged 20 to 70 years use representative U.S. statistics derived from NHANES 1999–2006 (

Table 1) [

49]. The relative variation in these data were calculated as the interquartile range (75th − 50th percentile) divided by the median (in percent). For height, this relative variation was 6% for both boys and girls, and 5% for adults. For weight, the relative variation was 22% and 30% for boys and girls, respectively, and 26% and 32% for men and women, respectively.

Physical activity levels depend on task and intensity, and standard MET-based definitions have long been used to classify physical activity levels into sedentary (<1 to 1.5 MET), light (1.5 to <3 MET), moderate (3 to <6 MET) and vigorous (>6 MET) classifications. For youth (8–18 years of age), recent analyses suggest that resting energy expenditures exceed adult METS by about 33%, thus increasing the ranges stated; consequently, sedentary behaviors of children may range of up to 2.0 MET [

50]. The recent compendium of physical activity for children most commonly lists 1.4 MET for children sitting quietly, studying, taking notes and writing, and having class discussion [

51]. This value exceeds the “typical” MET levels stated in the guidance for using CO

_{2} for ventilation evaluations, i.e., 1.0 to 1.2 MET for reading, writing and typing while seated [

24]. For adults, the most recent compendium lists 1.5 MET for sitting tasks with light effort, e.g., office work, reading, computer work and talking; standing tasks with light effort, e.g., directing traffic, filing, talking and teaching physical education, range from 2.0 to 3.0 MET [

52]. Again, these exceed values in the guidance noted earlier (1.0 to 1.2 MET for sitting activities noted earlier, and 1.4 and 2.0 MET for filing while standing and walking, respectively). Teachers and other adults in classrooms are expected to engage in a combination of sitting, standing and walking activities, thus a blend of sedentary and light activity levels is appropriate, e.g., 1.7 MET may be appropriate for a teacher who occasionally stands and walks about the classroom.

Using a physical activity value of 1.4 MET and averaging across boys and girls, the CO

_{2} generation rate G

_{P} in classrooms ranges from 0.147 L·min

^{−1}·person

^{−1} for pre-kindergarten children to 0.343 L·min

^{−1}·person

^{−1} for 12th graders (

Table 1). Several studies have used a single generation rate for 5th grade children, equivalent to 0.258 L·min

^{−1}·person

^{−1} [

1,

6,

8]; the strong dependence on age is important to recognize. For women, using a physical activity value of 1.7 MET and averaging across ages, G

_{P} is 0.442 L·min

^{−1}·person

^{−1}. G

_{P} estimates for adults have little sensitivity to age (<3% variation), unlike for children. The assumption of female teachers, used since most teachers (88%) in our school surveys have been women, is not significant since the total CO

_{2} generation rate in classrooms is usually dominated by students. The total classroom emission rate can be calculated as the sum of emissions from children (grade level-specific G

_{P} multiplied by the number of students present) and adults (adult G

_{P} multiplied by the number of adults present).

The variability of the population’s height and weight can affect G

_{P}, as demonstrated using a limited sensitivity analysis. Using the relative variability described earlier and examining 6th grade children, a 6% increase in height over the nominal case (

Table 1) would increase G

_{P} by 4% (average for boys and girls); 22% and 30% increases in weights of boys and girls, respectively, would increase G

_{P} by 10%, and increasing both height and weight by these amounts would increase G

_{P} by 15%. In most classrooms with a mix of children and imperfect correlation between height and weight, uncertainties due to height and weight variation are expected to be smaller. Uncertainties related to physical activity and metabolism levels, discussed below, are likely much more significant.

In most applications, the physical activity level of building occupants is unknown, and the actual value might considerably exceed the assumed level. Also, if the activity level changes, the assumption of a single and constant activity level might not adequately represent the CO

_{2} generation rate G

_{P}. In addition, energy expenditures of occupants depend on activity levels prior to entering the space. In schools, for example, students may enter a classroom following more vigorous activity, e.g., walking/running to class, playing at recess, etc. Factors that affect the post-exercise recovery time needed to reach resting (or sedentary) metabolism levels include the duration and intensity of the preceding exercise and the individual’s condition and age, e.g., recovery periods for children are about half that of adults [

53]. Information relevant to previous activity levels and the corresponding G

_{P} pertaining to students entering a classroom in the recovery period is not directly available, but can be inferred from several sources. For example, guidance for measuring the resting metabolic rate in adults in clinical settings suggests a minimum rest period of 10 to 20 min, and an abstention period of 2 h following moderate aerobic or anaerobic exercise [

54]. The duration of excess post-exercise oxygen consumption (EPOC) for short and low intensity exercise across a number of studies examining adults listed in a comprehensive review of EPOC studies ranged between 6 and 30 min [

55]. Comparable EPOC studies for healthy children engaged in low intensity exercise relevant for schools were not identified.

To evaluate effects of prior physical activity, we analyzed several scenarios that considered children exercising just prior to entering a classroom. For exercise, we assumed walking with light to hard effort, equivalent to 2.9 to 4.6 MET [

51], and a nominal value of 3.5 MET. For recovery periods, we assumed a recovery time t

_{REC} (h) from 0.1 to 0.5 h, and a nominal period of 0.25 h (only adult values were available). The literature often represents recovery trajectories as a first-order (exponential decay) trend, but recovery durations have been reported using different methods. We assumed that the reported recovery time reached the resting metabolism rate within 5% of the pre-exercise value, and thus the recovery time rate constant K

_{REC} (h

^{−1}) = ln(0.05)/t

_{REC} (Larger percentages, 10% and 15%, were are also tested). With these assumptions, an individual’s instantaneous activity level after entering a classroom, MET

_{t}, will exponentially decline to the sedentary level, MET

_{SED}:

where MET

_{EXER} = physical activity level during exercise (MET); and t = time elapsed since entering the classroom and the cessation of exercise (h). To gauge the importance of the recovery effect over the CO

_{2} monitoring time window, MET

_{t} is averaged over the time period the student is in the classroom and compared to MET

_{SED} (assumed to be 1.4 MET), and expressed as a relative bias (%). This represents the amount of additional activity resulting from the post-exercise recovery period, and also the extent to which CO

_{2} emissions will be underestimated if the post-exercise recovery effect is ignored.

#### 2.7. Application

Four classrooms in different primary schools were selected to demonstrate the VR methods, several of which have not been applied previous in schools, and to show a range of ventilation and occupancy conditions. The selected schools are a subset of those in a larger (Environmental Quality and Learning in Schools or EQUALS) study. These schools were located in the U.S. Midwestern states of Michigan, Illinois, Ohio and Indiana; all are mechanically-ventilated, and all were constructed or fully renovated in the past 15 years. Schools were visited and monitored in winter during the heating season (November 2015 to February 2016) when window or door opening was at a minimum. Our technicians conducted walk-through surveys of each classroom, ventilation system, school building, and grounds. Classroom dimensions were obtained using a laser tape measure. HVAC information, including system type, configuration, design and minimum air flow rates, were obtained from HVAC drawings, schedules and photos, as well as manufacturer’s specifications; this information was used to calculate VRs for comparison with those determined using the CO

_{2}-based methods. The studied spaces are summarized in

Table 2.

CO

_{2} concentrations in each classroom were measured at a central location away from windows and doors using NDIR sensors (C7632A, Honeywell Corp., Morristown, NJ, USA); similar instrumentation on the school grounds or rooftop measured outdoor CO

_{2} concentrations. CO

_{2} data were reduced to 15 min averages. CO

_{2} sensors were calibrated quarterly using zero air and a certified standard. Drift per season was typically below 25 ppm. Teachers in the selected classrooms were asked to maintain an occupancy log by recording the number of adults and children present every 15 min. This information was used to estimate CO

_{2} generation rates on a 15 min basis using metabolic activity levels of 1.4 and 1.7 MET for children and adults, respectively, and the generation rates in

Table 1. The effect of post-exercise recovery was not modeled, although a sensitivity analysis is performed in

Section 3.3. One day with full data in each school was selected for analysis. To calculate build-up, steady-state and decay air change rates, study days were separated into several periods (see below) in which the VR was assumed to be constant. The 06:00 to 08:00 and 15:00 to 18:00 periods were excluded since the ventilation systems were turned on and off during these periods, which would change the VR. In addition, during these periods, occupancy rates changed dramatically, and the collected occupancy data may have been incomplete.

Steady-state VRs were calculated using Equation (3), the maximum 15-min CO_{2} concentration over the school day (08:00 to 15:00), the measured room volume, the average generation rate over the 2 h prior to the time of the maximum CO_{2} concentration (determined from the 15-min average data), and 400 ppm for the measured outdoor CO_{2} concentration, which was confirmed using the outdoor measurements. V_{0,S} was calculated using Equation (4), the average per person CO_{2} generation rate in the classroom, and the number of persons in the classroom in the 2 h period just prior to the peak CO_{2} concentration. We determined the sensitivity to the time period and outdoor air concentration.

Decay air change rates were estimated using Equation (5), 15-min average CO_{2} measurements measured at 18:00 and 24:00 for the “evening” period, and concentrations at 24:00 and 06:00 for the “early morning” period. We searched for the maximum and minimum concentrations occurring within ±1 h of the nominal period, which were used as C_{0} and C_{1} in Equation (5). Comparable decay air change rates were found using regression to fit Equation (7) (results not shown). In addition, transient mass balance air change rates were estimated for the same periods using the sequence of 15 min average CO_{2} measurements.

Build-up air change rates were determined using the 2-point method in Equation (8) with an implicit estimate of C

_{S} determined using Equations (10) and (11), C

_{0} and C

_{1} measured as 15-min averages, a starting estimate given as 0.9999 × Â

_{B,0} from Equation (12), and the Newton-Raphson search method, described earlier (also see

Supplemental Information). Given the occupancy patterns noted, we selected a nominal period from 08:00 to 12:00, but allowed actual start and end times to vary by ±1 h (as described earlier). The average CO

_{2} generation rate over the selected time window was used to estimate C

_{S}. The build-up method was also implemented using the 3-point and ASTM methods (Equations (8), (9) and (13), respectively), the same selected times, and the 15-min concentration at the midpoint time (or the average of two consecutive 15-min concentrations given an even number of observations between C

_{0} and C

_{1}).

Transient mass balance air change rates were determined using Equation (16) for three periods: the school day (8:00 to 15:00 p.m.), an evening period (18:00 to 24:00), and an early morning period 24:00 to 06:00). We used an interval of 15 min (Δt = 15 min) for observed and simulated CO_{2} concentrations, and the CO_{2} generation rate (using teacher-reported occupancy data). In addition to A_{TMB}, we estimated the replacement air concentration (C_{R}) with constraints (375 ppm < C_{R} < 450 ppm), the children’s metabolic equivalent level within constraints (1.2 < MET < 1.6), and the CO_{2} concentration at the beginning of the simulation period with a constraint (C > 375 ppm). A generalized reduced gradient solver using central derivatives estimated these unknown parameters by minimizing the sum of squares between observed and simulated CO_{2} concentrations. VRs were estimated separately for each period. Each problem used the same initial solution. We also tested the sensitivity of results to the start and stop times and other parameters. Lastly, V_{0} is reported using both the average and the maximum number of occupants, and the estimated VR.

Outdoor temperatures and wind speeds measured at the airport nearest each school are shown in

Table 2. During the school day (08:00–15:00), average temperatures varied considerably (−11 to 13 °C), depending on the day studied. Several days had moderately high wind speeds (up to 9 m/s for the school day average).