Model development

Limonene ozonolysis experiments from Objective 2 were used to tune and then evaluate the model. The experimental procedure was explained fully in Chapter 3, but I briefly describe it here: in a 1-m³ stainless steel chamber system operated as a well-mixed, continuous flow mixed reactor (CMFR) initially free from ROGs and aerosols, ozone was generated (Absolute Ozone NANO) until it reached a desired concentration. Then, a limonene solution was injected with a syringe pump for one minute at a certain rate that would yield its desired concentration in the chamber. The size distributions of the produced SOA were measured in one-minute intervals by

using a Fast Mobility Particle Sizer (FMPS, TSI 3091), while ozone (1 min averages, 2B Technol-ogies 205) and terpene concentrations (every ~20 min, GC-FID, SRI Instruments) were also meas-ured.

The SOA formation in our transient experiments followed the trends discussed above in re-lation to Figure 6-1. Though nucleation occurred at some rate over most of the reaction process, it is a very strong mechanism near the reaction commencement, and after the appearance of nuclei, other processes such as partitioning and coagulation dominated. The purpose of this Objective 4 was to develop a model of the evolution of the SOA size distribution, with a strong focus on im-proving partitioning predictions. It did not have the goal of advancing nucleation theory, which is a separate and complex topic. As such, the starting point of the model is after the primary nucleation has produced the peak SOA number with a size distribution of certain GM and GSD (i.e., at the end of Stage 1).

When considering the mechanisms of coagulation, partitioning, air exchange, and surface deposition, the governing equation is a form of the continuous general dynamic equation, as in:

𝜕𝜕𝑛𝑛(𝑣𝑣, 𝑡𝑡)

𝜕𝜕𝑡𝑡 =1

2 �^{𝑣𝑣−𝑣𝑣0}𝐾𝐾(𝑣𝑣 − 𝑞𝑞, 𝑞𝑞) 𝑛𝑛(𝑣𝑣 − 𝑞𝑞, 𝑡𝑡)𝑛𝑛(𝑞𝑞, 𝑡𝑡)𝑑𝑑𝑞𝑞

𝑣𝑣0

− 𝑛𝑛(𝑣𝑣, 𝑡𝑡) � 𝐾𝐾(𝑣𝑣, 𝑞𝑞)𝑛𝑛(𝑞𝑞, 𝑡𝑡)𝑑𝑑𝑞𝑞^∞

𝑣𝑣₀ − 𝜕𝜕

𝜕𝜕𝑣𝑣[𝐼𝐼𝑣𝑣(𝑣𝑣, 𝑡𝑡) 𝑛𝑛(𝑣𝑣, 𝑡𝑡)]

− 𝜆𝜆 𝑛𝑛(𝑣𝑣, 𝑡𝑡) − 𝛽𝛽_SOA(𝑣𝑣) 𝑛𝑛(𝑣𝑣, 𝑡𝑡)

Eq.(6-1)

where n(v,t) (m^-3) is the number of particles with volume (m³) within v and (v + dv) at time t (s);

v0 and (v – v0 ) are volumes of two colliding particles and v is the volume of the resulting particle;

K is the coagulation kernel, and it depends on sizes of the two colliding particles; Iv (v,t) (m³/s) is the condensation rate, and it depends on the particle size and also gaseous compound characteristics

such as diffusivity and molecular weight; λ (s^-1) is the air exchange rate; and β^SOA(v) (s^-1) is the particle deposition to surfaces and it depends on the size of the particle, as well as surface and airflow characteristics.

The left hand side of Eq.(6-1) is the time rate of the change in the number of particles of volume v. On the right hand side, the first and the second terms correspond to coagulation gains and losses, respectively. The coefficients 1 and ½ in the coagulation gain and loss terms are only valid for a monodisperse distribution, which will be explained in detail later. The third term is partitioning due to net condensation; the fourth term is air exchange loss; and the last term is to particle loss due to surface deposition. As explained qualitatively above, Eq.(6-1) will be solved with the following initial condition: n(v, 0) = nmax(v); where nmax(v) is the peak SOA number con-dition and is represented by a measured distribution with a particular GM and GSD.

To solve the integro-differential equation, it was discretized. As mentioned earlier, the in-tegro-differential equation is valid for monodisperse distribution, where the volume in bin k is the volume in the smallest bin multiplied by k. Such a distribution requires a large number of bins, which is computationally intensive; for our model range of 5.6 to 560 nm, it would require 10⁶ bins.

So instead, we use a volume-ratio distribution, in which the volume in the next bin is defined by multiplying the volume in the previous bin by a constant “volume ratio”, or VR (Jacobson, 2005).

Smaller ratios lead to more bins, which increases accuracy. In a monodisperse size distribution, when a particle grows either due to coagulation or partitioning, the resulting particle will have a certain volume corresponding to one bin size. However, in a volume ratio size distribution, the resulting particle may have a volume that can be distributed between the lower and upper bins. I discretized Equation 6-1 and modified it to represent change in the volume of particles, v, rather than the particle number, as follows:

𝑉𝑉𝑘𝑘,𝑡𝑡− 𝑉𝑉𝑘𝑘,𝑡𝑡−∆𝑡𝑡 coeffi-cient between two colliding particles from bins i and j; λ (s^-1) is the air exchange rate; and β^{SOA, k} (s^-1) is the particle deposition rate of bin k.

On the right-hand side of Eq.(6-2), the first two terms represent gain and loss due to coag-ulation, respectively. fi,j,k is a coefficient that can be between 0 and 1 and describes the ratio of particles that generated from particles in bins i and j and end up in bin k. The second two terms represent gain and loss due to partitioning, respectively. The gain in the particle volume in bin k due to partitioning is the sum of particles from smaller bins (bin 1 to up to bin k) that grow due to partitioning and enter bin k. The total volume entering from a smaller bin j is the product of the particle number in that bin at time t, n j,t, and the volume of a single particle in bin k, vk. The particle loss rate from bin k due to partitioning is the sum of particles in bin k that grow and leave bin k and will move to bins bigger than bin k. The total volume of particles leaving bin k and going to bin j is the product of number of particles in bin k at time t, nk,t , and volume of a single particle in bin k. The gj,k,t is a coefficient that describes the ratio of particles from bin j that will go to bin kdue to partitioning at time t. The last term represents the losses due to air exchange and surface deposition, respectively.

Since I used a volume-ratio distribution in this model, when particles grow either due to coagulation or partitioning, the volume of the resulting particle may not correspond precisely to one unique bin. Therefore, this resulting volume can be distributed between upper and lower bins;

f i,j,k and gj,k,t, are volume fractions for coagulation and partitioning, respectively, and are between

0 and 1. The coagulation volume fraction, fi,j,k, for bin k is:

𝑅𝑅_{𝑖𝑖,𝑗𝑗,𝑘𝑘}=

where NB is the total number of bins. For partitioning, assuming that dvj,t is the volume that parti-tions to a particle in bin j at time span dt, the resulting particle will have volume of Vj,t = vj + dvj,t., and the partitioning volume fraction, gi,j,k, for bin k is:

𝑔𝑔_{𝑗𝑗,𝑘𝑘,𝑡𝑡}=

Unlike the coagulation volume fractions, fi,j,k, which are constant during the experiment, the parti-tioning volume fractions, gj,k,t, change as the rate of SOA formation (i.e., dv/dt) changes.

For the partitioning element and more specifically to estimate gj,k,t, it is necessary to deter-mine the condensation rate, dvj,t/dt, which is the rate of the transfer of condensable material from the gas to aerosol phase. The common practice uses Fick’s first law of diffusion for individual SOA precursors, but it requires detailed knowledge of the complex chemistry that produces those pre-cursors, as well as their individual vapor pressures, diffusivities, and molecular weights. It is shown in Eq.(6-5).

d𝑣𝑣_𝑗𝑗 d𝑡𝑡 =

2𝜋𝜋𝑑𝑑p,𝑗𝑗𝐷𝐷𝑀𝑀

𝜌𝜌𝑅𝑅𝑅𝑅 𝑅𝑅(𝐾𝐾𝑛𝑛, 𝛼𝛼)(𝑃𝑃 − 𝑃𝑃_eq) Eq.(6-5)

where dvj/dt (m³/s) is the volume transfer rate of a SOA precursor from the gas to particle phase for a particle with diameter dp,j; D (m²/s) is the diffusion coefficient of the precursor in air; M (kg/kmol) and ρ (kg/m³) are its molecular weight and density; and f (Kn, α) is the correction due to non-continuum effects and imperfect surface accommodation. The difference between P (Pa), the vapor pressure of the SOA precursor far from the particle, and Peq (Pa), the equilibrium vapor pressure is the driving force (P – Peq) for the transport of the SOA precursor from the gas to particle phase.

To simplify the need for information of gas phase chemistry and precursor transfer param-eters, I instead use the well-constrained aerosol mass fraction (AMF) model framework to deter-mine the total volume of SOA produced at time t. As discussed in Objectives 1-3, AMF is the ratio of the amount of SOA produced, to that of the reactive organic gas reacted, (Grosjean and Seinfeld, 1989, Pandis et al., 1992, Pandis et al., 1993). AMF is not constant and it increases as more organic aerosol is formed. It indicates the strength of SOA formation for some amount of oxidized organic compound and is determined experimentally. By using experimental AMFs determined in Objec-tives 2 and 3, I can find the total amount of SOA produced as follows:

d𝑉𝑉 d𝑡𝑡 =

AMF ∙ 𝑘𝑘_O3∙terp𝐶𝐶_O3,𝑡𝑡𝐶𝐶_{terp,𝑡𝑡}𝛤𝛤

𝜌𝜌SOA Eq.(6-6)

where kO3-terp (ppb^-1 s^-1) is the reaction rate constant of ozone and d-limonene; CO3,t and Cterp,t (ppb) are the ozone and d-limonene concentrations at time t; ρ^SOA is the SOA density (kg/m³), taken to be a unit density in the AMF determinations in Objectives 2 and 3; and Γ is the temperature-dependent conversion factor to change from ppb to kg/m³.

Equating the total amount of partitioning dV to the summed amount of SOA that partition on particles in each bin j, over a time span, dt, is as follows:

d𝑉𝑉 of SOA that partition over particles in bin j at time t.

Combining Eqs.(6-5) to (6-7) gives the volume of condensable material that partitions to a particle with diameter dp , i as follows:

d𝑣𝑣_{𝑗𝑗,𝑡𝑡} d𝑡𝑡 =

𝑑𝑑_p,𝑗𝑗∙ 𝑅𝑅(𝐾𝐾𝑛𝑛, 𝛼𝛼)

∑^{𝑁𝑁𝑁𝑁}_𝑗𝑗=1𝑛𝑛_{𝑗𝑗,𝑡𝑡}.∙ 𝑑𝑑_p,𝑗𝑗∙ 𝑅𝑅(𝐾𝐾𝑛𝑛, 𝛼𝛼) ∙ AMF ∙ 𝑘𝑘_O3∙terp𝐶𝐶_O3,𝑡𝑡𝐶𝐶_{terp,𝑡𝑡}𝛤𝛤 Eq.(6-8)

Unlike Eq.(6-5), which is for individual precursors that may be unknown and require knowledge of variables such as D, M, and Peq, all parameters in Eq.(6-8), including relevant parti-cle diameters, ozone and terpene concentrations, AMF values, and f (Kn, α), are known, either by

theoretical or experimental means. To find f (Kn, α), I used the equation suggested by Fuchs, as follows:

𝑅𝑅(𝐾𝐾𝑛𝑛, 𝛼𝛼) = 0.75 𝛼𝛼(1 + 𝐾𝐾𝑛𝑛 )

0.75 𝛼𝛼 + 𝐾𝐾𝑛𝑛 + 𝐾𝐾𝑛𝑛² Eq.(6-9)

where Kn is the Knudsen number and α is the accommodation coefficient. The Knudsen number is the ratio of the mean free path of the particle, which is the length that an entity travels before it collides with an air molecule, and the particle diameter. The mean free path for a particle is:

Mean free path = 3𝐷𝐷p

𝐶𝐶̅𝑆𝑆 Eq.(6-10)

where Dp is the diffusivity of a particle with diameter dp and CA is its mean speed.

The f (Kn, α) is shown as a function of particle diameter in Figure 6-3. As mentioned, f (Kn, α) is the correction due to non-continuum effects and surface imperfections, and it depends on interplay of the particle’s diameter and its mean free path. When the diameter of a diffusing particle is bigger than the mean free path of the air, the air may be treated as a continuum; in other words, the diffusing particle is surrounded completely by air molecules and in every movement the particle makes, it hits air molecule. The accommodation factor, α, can have a value between 0 and 1, and it is assumed as α = 1, which is realistic according to reported values (Julin et al., 2014). A value of 1 indicates that when a condensable molecule from gas phase encounters the aerosol surface, it sticks to the surface with the probability of sticking equal to unity.

Furthermore, according to Eq.(6-8), the distribution of gas phase compounds to the aerosol phase proceeds as a function to the particles’ diameters. In order to improve the model performance, I tested it at different values for a power term, x, applied to the particle diameter in Fick’s law, i.e.,

as in dpx

. Though this alteration is predominately a tuning parameter, it is nevertheless grounded in a physical concept of the changing strength of the condensation mechanism as being dependent on particle size (i.e., x = 1), surface area (x = 2), or volume (x = 3).

Figure 6-3. Kn number and f (Kn, α) as a function of the particle diameter for the range of the for-mation experiments herein (5.6 to 560 nm).

For coagulation, I do not propose any new framework, but instead, I apply well-established theories to calculate these coefficients in the model, which I explain in more detail in the next sub-section. To predict the particle deposition loss rate to surfaces. I used the experimental measure-ments, performed in each experiment, as discussed in Objective 2. To solve Eq.(6-2), I used an explicit solution where I replace nj,t and vj,t on the right side of the equation by nj,t-Δt and vj,t-Δt

𝑉𝑉𝑘𝑘,𝑡𝑡− 𝑉𝑉𝑘𝑘,𝑡𝑡−∆𝑡𝑡

Furthermore, this numerical solution also requires finding the volume-ratio (VR) for the particle size bins. Smaller VRs lead to more bins, increasing accuracy. However, smaller VRs and more bins require more computational power. To find a reasonable VR for the model, I ran the model with various values of VR and compared the results with experimental values.

6.2.1. Coagulation coefficient calculation

The coagulation coefficient, which depends on the colliding particles’ diameters, can be affected by several forces: Brownian motion, convective Brownian motion enhancement, gravita-tional collection, turbulent inertial motion, turbulent shear, and van der Waals forces. The coagu-lation coefficient is the sum of coefficient regarding each of the above forces (Jacobson, 2005). For the most general case, which is coagulation due to the Brownian motion, the coagulation coefficient is:

𝐾𝐾₁₂= 2𝜋𝜋 �𝑑𝑑p1+ 𝑑𝑑_p2��𝐷𝐷_p1+ 𝐷𝐷_p2�𝛽𝛽 Eq.(6-12)

where dp1 and dp2 are the diameters of the two colliding particles and Dp1 and Dp2 are diffusivities of the two particles in the air and β is a coefficient to account the for the fluid regime (e.g., in the continuum regime it is equal 1). Particle diffusivity can be estimated by the following equation:

𝐷𝐷p𝑗𝑗= 𝐾𝐾_𝑁𝑁𝑅𝑅

3𝜋𝜋𝜋𝜋𝑑𝑑p,𝑗𝑗�5 + 4𝐾𝐾𝑛𝑛𝑗𝑗+ 6𝐾𝐾𝑛𝑛_𝑗𝑗²+ 18𝐾𝐾𝑛𝑛_𝑗𝑗³

5 − 𝐾𝐾𝑛𝑛_𝑗𝑗+ (8 + 𝜋𝜋)𝐾𝐾𝑛𝑛_𝑗𝑗² � Eq.(6-13)

where KB is the Boltzmann constant; Knj is the Knudsen number for a particle with diameter dp,j; and μ is the air viscosity. The coagulation coefficient for particles up to 10 µm is shown in Figure 6-4. When particles have similar diameters, they have the smallest coagulation coefficient. Figure 6-5 shows a contourplot of the coagulation kernel for the experimental range of particles (5 to 560 nm).

Figure 6-4. Brownian coagulation coefficient for particles with dp1 and dp2 diameters. To inter-pret this plot, find the smaller particle diameter on the abscissa and then locate the line that corre-sponds to the larger particle diameter.

1.E-16

Figure 6-5. Brownian coagulation kernel for experimental range of SOA diameters (5.6 to 560 nm).