Methodology for Objective 1

Two SOA formation model frameworks, called SOA-M1 and SOA-M2, were developed to account for the ozonolysis of a single terpenoid in a dynamic environment. First, elements com-mon to both models are introduced, then elements unique to each model are developed, and then finally the model validation efforts are discussed.

In an indoor setting, it is a common practice to assume the building envelope as the control volume of a reactor, and then write constant mixed flow reactor (CMFR) mass balance equations for the reactants and products, which assume that air is well-mixed so the concentration of a com-ponent is spatially identical over the entire reactor volume but can change with time. Assuming that reactions are limited to that of monoterpenes with ozone, the volume-normalized mass balances for ozone and the terpenoid are:

d𝐶𝐶_O3

d𝐶𝐶terp con-centrations of ozone and the terpene, respectively; λ^j (h^-1) is an air exchange rate, which can be due to infiltration or mechanical or natural ventilation; Eterp (μg/h) is indoor emission rate a terpene;

and β^O3 (h^-1) is the surface loss rate of ozone. These mass balances account for sources due to air exchange and emissions of a terpene inside the building, and losses due to air exchange, surface deposition of ozone, as well as homogenous gas phase reactions.

3.2.1 Modeling SOA formation with SOA-M1

The first SOA formation model, SOA-M1, uses a standard volume-normalized mass bal-ance for CSOA:

where CSOA (μg/m³) is the SOA concentration being determined; βSOA (h^-1) is the deposition rate of SOA, which can account for losses to surfaces or mechanical filters; and the AMF term changes with the organic aerosol concentration according to the two-product model formulation:

AMF =∆𝐶𝐶_SOA

∆ROG = 𝐶𝐶^SOA� 𝛼𝛼₁𝐾𝐾₁

1 + (𝐶𝐶SOA+ 𝐶𝐶Mb)𝐾𝐾1+ 𝛼𝛼₂𝐾𝐾₂

1 + (𝐶𝐶SOA+ 𝐶𝐶Mb)𝐾𝐾2� Eq.(3-5)

where CMb (μg/m³) is a term that accounts for any type of background organic aerosol present besides formed SOA. The rightmost side of Eq.(3-5) is the two-product form of the incremental increase in SOA mass per some decrease in the terpene mass. SOA-M1 directly substitutes that term for the AMF in Eq.(3-4), and then Eq.(3-2) to Eq.(3-4), as well as one for CMb (not shown), may be solved simultaneously using a Runga-Kutta order 4 (RK4) numerical solution.

3.2.2 Modeling SOA formation with SOA-M2

SOA-M2 takes a different approach than SOA-M1; rather than tracking CSOA with time, SOA-M2 tracks the change in the concentration of products owing to terpenoid ozonolysis indoors.

It assumes that ozone and terpenoid react to form a hypothetical product, CdROG (μg/m³), which has the same molecular weight as the parent terpenoid. CdROG physically represents a lumped concen-tration of all products of the oxidized terpenoid that remains indoors at a particular instant in time, and this CdROG is used to predict the CSOA. That is, some of the CdROG will exist in the particle phase as CSOA and some of it will exist the gas-phase as volatile products, CV(μg/m³), at fractions that depend on the amount of CdROG indoors. In batch reactor systems with negligible losses owing to air exchange and deposition, CdROG = ΔROG. However, there are losses indoors because of air exchange and deposition, so CdROG < ΔROG.

To determine CSOA as a function of CdROG, one recognizes that Eq.(3-5) holds not only for incremental changes in products and reactants but also for instantaneous equilibrium partitioning of products (Hoffmann et al., 1997). Therefore, by assuming instantaneous equilibrium and that there is not any background organic aerosol present (i.e., CMb= 0 μg/m³) the AMF equation can be recast as:

𝐶𝐶SOA

𝐶𝐶_dROG= 𝐶𝐶_SOA� 𝛼𝛼1𝐾𝐾1

1 + 𝐶𝐶_SOA𝐾𝐾₁+ 𝛼𝛼2𝐾𝐾2

1 + 𝐶𝐶_SOA𝐾𝐾₂� Eq.(3-6)

Thus, Eq.(3-6) can be solved to yield algebraic solutions of CSOA as a function of CdROG. Kroll and Seinfeld (2005) solved Eq.(3-5) for one- and two-product model solutions of CSOA as a function of ΔROG to parameterize batch reactor formation, but their solutions are applicable for Eq.(3-6) as well. The one-product solution is a linear equation, as in Eq.(3-7):

𝐶𝐶_SOA= 𝛼𝛼₁𝐶𝐶_dROG− 1

𝐾𝐾₁ Eq.(3-7)

The two-product model solution is quadratic and more complex, and it is Eq.(3-8):

𝐶𝐶SOA=1

2 �𝐶𝐶^dROG(α1+ α2) − 1 𝐾𝐾₁− 1

𝐾𝐾₂�

+

�4𝐾𝐾₁𝐾𝐾₂(𝐾𝐾₁α₁𝐶𝐶_dROG+ 𝐾𝐾₂α₂𝐶𝐶_dROG− 1) +�𝐾𝐾1+ 𝐾𝐾₂− 𝐾𝐾₁𝐾𝐾₂𝐶𝐶_dROG(α1+ α₂)�²�

12

2𝐾𝐾₁𝐾𝐾₂

Eq.(3-8)

Kroll and Seinfeld (2005) also solved Eq.(3-5) assuming that there is background organic aerosol, but those solutions were not used in this work.

To use this method, a differential equation for CdROG was derived by substituting the term (AMF·CdROG) for CSOA in Eq.(3-4) and computing the derivative, d(AMF·CdROG)/dt, which is:

d𝐶𝐶_dROG

d𝑡𝑡 = 𝑘𝑘_O3/terp𝐶𝐶_O3𝐶𝐶_terp𝛤𝛤_terp− �� 𝜆𝜆_𝑗𝑗

𝑛𝑛 𝑗𝑗=1

+ 𝛽𝛽_SOA+ 1 AMF

d(AMF)

d𝑡𝑡 � 𝐶𝐶^dROG Eq.(3-9)

Eq.(3-9) implies that all products of terpenoid oxidation, not just SOA phase products, are subject to particle loss mechanisms. Particle losses applying to volatile products may seem counterintui-tive; however, volatile products are affected by particle-specific loss mechanisms because parti-tioning is an equilibrium process. As CSOA is lost by deposition mechanisms to filters or surfaces, there is a driving force for some of the volatile products to further partition to the SOA phase, at a fraction dependent on the total amount of CdROG currently indoors.

For SOA-M2, Eqs.(3-2), (3-3) and (3-9) were solved numerically using Runge-Kutta order 4 (RK4) method, then CdROG was found algebraically with Eq.(3-8). One important point regarding Objective 1 is that the RK4 transient solution for CdROG with Eq.(3-9) used values for AMF and d(AMF)/dt that were determined from the AMF values of the solution from SOA-M1. This choice was made to verify that SOA-M1 and SOA-M2 yielded identical solutions.

3.2.3 Validating the SOA formation models

To validate the SOA-M1 and SOA-M2 formation models, steady state and transient exper-iments in the literature for limonene ozonolysis were simulated using kO3/terp = 0.0183 ppb^-1 h^-1 at 25 °C (Atkinson et al., 1990), including those (i) steady state cases from Fadeyi et al. (2009), Cole-man et al. (2008) and Li et al. (2002), which were performed in large and small chambers as well as an actual office, and (ii) a transient case from Singer et al. (2006a), who measured SOA transient formation due to ozonolysis of a limonene containing orange-oil degreaser. The two-product AMF parameters for limonene were synthesized from the literature and were: α¹ = 0.082, α² = 0.86, K1

= 1 m³/μg, and K² = 0.0055 m³/μg. To assess the model performance, the American Society for Testing and Materials (ASTM) method D5157-91 (ASTM 1991) was used.