# Estimating Inhalation Exposure Resulting from Evaporation of Volatile Multicomponent Mixtures Using Different Modelling Approaches

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Estimating Evaporation Rates of Liquid Mixtures

_{i,room}, which in this context is often referred to as “backpressure”. Raoult’s law relates the partial vapour pressure of compound i to its saturation vapour pressure p

_{i}

^{*}and its molar fraction x

_{i}, which is calculated by

_{i}. The activity coefficient γ

_{i}${,}_{\mathrm{liq}}$ is used to take non-ideality into account (vide infra).

_{i}is a function of the diffusivity D

_{i}of the substance in the air and of the air flow v

_{air}over the product surface. Parameter notation and corresponding symbols and units are listed in Table 1.

_{i}and substance vapour pressure are available for many substances in chemical property reference manuals. For chemicals for which the molecular diffusion data and vapour pressure are not available, there are some estimation methods, e.g., Equation (10) in McCready and Fontaine [11].

_{i,liq}, which is concentration dependent. The importance of taking deviations from Raoult’s law into account even for moderately non-ideal mixtures has been demonstrated by various authors (see, e.g., [12]). Activity coefficients for a wide range of mixtures can be estimated using the group-contribution concept of the UNIFAC method that was introduced by Fredenslund, Jones, and Prausnitz [13] and was further investigated by Gmehling [14]. The group-contribution concept is based on the idea that physicochemical properties of organic molecules can be represented reasonably well by segmenting a molecule into different “functional” groups and considering specific properties of these groups. This leads to the description of a liquid mixture as a “solution of groups” (instead of a solution of molecules). This concept has been proven useful; it is often a good approximation for the real behaviour of organic mixtures and aqueous–organic mixtures. However, there are also limitations to the group-contribution approach, especially when applied to inorganic–organic mixtures. In order to overcome these limitations, Zuend et al. [15,16] have developed the thermodynamic model AIOMFAC that is designed for the calculation of activity coefficients of different chemical species in inorganic–organic mixtures. The AIOMFAC model has been developed and tested against hundreds of experimental datasets and the model has been shown to be useful, reasonably accurate, and practical for the calculation of activity coefficients in complex multicomponent mixtures [17,18].

_{i}of a compound i with their mole fractions x

_{i}in the liquid phase by a separate function, i.e., γ

_{i}= f(x

_{i}). Basically, they can be used to estimate activity coefficients for all mixture compositions. In practice, however, there is often a lack of the needed model parameters, which prevents the models from being used consistently. Therefore, different activity coefficient models were used in each case in this study (s. chapter results), depending on the particular composition of the mixture and the data available on model parameters.

_{room,i}, denoted in mass per volume, is of interest. For setting up and discussing the mass balance equations, however, it is easiest to observe the molar amounts, for example the molar amount of substance i in the room air, n

_{room,i}. Considering that the transition between concentrations and molar amounts using the relation $C=\frac{n\xb7M}{V}$ is rather simple, the subsequent presentation of the computational methods will focus on the molar amounts.

#### 2.2. Well-Mixed Room Model for Instantaneous Application (WMR_inst)

_{init,A,i}of the substance i (as a component of the liquid mixture) is applied initially to a certain surface area A. Then, evaporation starts, and owing to differences in volatility of individual components, usually the composition, and therefore the evaporation rate of the liquid mixture, changes over time. During evaporation, the concentration of the evaporated substances is highest near the source and forms a concentration gradient throughout the room. However, for simplification, often an even distribution is assumed. This approach, which is known as a well-mixed room model [24], is reasonable when room volumes are small and when the room is well mixed due to either natural or induced air currents, resulting in nearly equal concentration levels throughout the room. Assuming ideal conditions, the molar amount of compound i in the room air, n

_{room,i}, is a function of the evaporation rate $\frac{d{n}_{\mathrm{evap},\mathrm{i}}}{dt}$ (s. Equation (7)) and the loss caused by the ventilation rate Q

_{vent}expressed in (m

^{3}/s). These relationships can be expressed as follows:

_{vent,i}describes the concentration of compound i in the supply air. As mentioned above, the desired air concentration at time t can be easily derived from this equation using the ideal gas law. This results in:

_{liq,i}in Equation (5). As a consequence, the evaporation of all (major) volatile compounds from the liquid layer has to be taken into account. This leads to the question of whether there is a closed-form analytical expression for the integration describing the time varying amounts n

_{room,i}of each substance in the room air. Unfortunately, this finding process, using symbolic mathematics programs such as Mathematica [25], was not successful. Hence, numerical solutions of the simultaneous differential equations for instantaneous application were obtained using a fourth-order Runge–Kutta method [26] (s. chapter results). For the rather simple Runge–Kutta method, the number of iteration steps must be chosen. During our work, a number in the range of 10,000 steps has proven to be a good compromise between computing speed and accuracy.

#### 2.3. Near-Field/Far-Field Model for Instantaneous Application (NF/FF_inst)

_{init,A,i}of substance i (as a component of the liquid mixture) is instantaneously applied to a certain surface area A which is completely located in the near field. In addition to the approach discussed above, this model comprises terms for the volumes of the near field (V

_{nf}) and the far field (V

_{ff}); the quantity of the interzonal air flowing from the near to the far field and vice versa is given by Q

_{nf/ff}, which is assumed to be due to natural convection. The distribution of the substance i in the near field (n

_{nf,i}) and the far field (n

_{ff,i}) is assumed to be homogenous within the respective volumes. The evaporation term changes slightly compared to Equation (5) in that now the backpressure inside the near field, p

_{nf,i}has to be considered:

_{vent}. It has to be noted that for simplification, this mass flow is assumed to be zero in all numerical calculations discussed in this paper (which is probably not the case in practice). The third equation quantifies the changes within the liquid layer.

^{3}cube centred on the workers head. That is, one side face of the cube is 4 m

^{2}and its volume is 8 m

^{3}. Unlike Cherrie and Schneider, we assume that the near field is centred on the workers body and thus includes the entire worker and the emission sources within reach of the worker (this is typically the case when liquid mixtures are manually applied to surfaces). In other words, we view the near field as encompassing the worker and an emission source area of 4 m

^{2}at maximum (s. Figure 1). With this approach it is also possible to let the worker move in and out of the emission source area. Here, it is assumed that the total applied area A is a horizontal rectangle with arbitrary width and a height h

_{A}that can be 2 m at maximum (s. Figure 1), but the same approach would be applicable to treatment of the floor for, e.g., mopping with a disinfectant.

_{nf/ff}. Following the approach of Cherrie [28] and Kulmala [29] to estimate Q

_{nf/ff}, it is assumed that there is a minimum convective airflow arising from the person’s body heat. Higher airflows are possible when air is moved through the near field from cross drafts, etc. For that reason, Cherrie has suggested interzonal airflows for three conditions: minimal likely convective airflow (3 m

^{3}/min), maximal convective airflow plus bulk air movement through the near field at 0.1 m/s (30 m

^{3}/min), and an intermediate value (10 m

^{3}/min). For the model calculation presented in this study, the intermediate value of 10 m

^{3}/min has been used (s. chapter results). Solutions for scenarios with instantaneous application (Equations (11)–(13)) were obtained numerically (s. chapter results).

#### 2.4. Well-Mixed Room Model for Continuous Application (WMR_cont)

_{ΔA}small area elements ΔA, which are numbered consecutively using the index l. We assume that fractions of the product are applied instantaneously, element by element, where the applied amount reflects the work rate of the worker. However, in principle, this approach can also take arbitrary values for the applied amount, or even zero to reflect periods where the worker stops the application (e.g., breaks, post application phase).

_{ΔA}determines the spatial resolution of the proposed approach. We then define the temporal resolution that is needed to describe the time-dependent evaporation from each area element appropriately. This is achieved by dividing the overall exposure time t

_{expo}into N

_{t}time steps so that Δt = t

_{expo}/N

_{t}. Basically, the time Δt

_{Δ}

_{A}needed to apply the product to one area element ΔA can be the same as Δt, so that each time step involves the complete application to one area element. However, for diminishing calculation time, it may also be useful to reduce the spatial resolution by allowing Δt

_{Δ}

_{A}to adopt integer multiples IM of Δt, that is Δt

_{Δ}

_{A}= Δt·IM, ΔA = (A/N

_{t})·IM and N

_{t}= N

_{ΔA}·IM. For simplification, however, in the further discussion we will use IM = 1 until mentioned otherwise. How the temporal and spatial resolution determined by the size of N

_{t}and IM, respectively, influences the numerical precision of this approach is addressed to some extent in the results chapter by some example calculations.

_{liq,l,i,init}of component i to an area element l and once Δt

_{ΔA}= Δt·IM seconds have passed, we begin application to the next element l + 1. While this is crude for a small number of time steps, the approximation becomes much more reasonable when the time interval Δt and the size of each area element ΔA decreases accordingly, approaching a quasi-continuous application. This discretisation approach is illustrated in Figure 2 for IM = 1 (that is Δt

_{ΔA}= Δt). For each time step k and each area element l the molar amount of component i at the beginning of time step k is represented by ${n}_{1,t}\left({t}_{\mathrm{k}}\right)$. For instance, ${n}_{2,1}\left({t}_{3}\right)$ means the molar amount of component 1 on area element 2 at the beginning of time step 3. The diagonal elements (that is, k = l) of the matrix in Figure 2 represent the initial molar amount ${n}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t},1,t}\left({t}_{\mathrm{k}}\right)$ of component i applied to area element l at the actual time t

_{k=l}= Δt ∙ (k − 1). The temporal iteration process starts with time step k = 1 and proceeds successively for each area element l to the maximum number of time steps N

_{t}. If IM > 1, the area size ΔA and the application time Δt

_{ΔA}of an area element needs to be adopted accordingly. Once some product is applied, the further course of the amount is determined by evaporation of the components i, which is again described by the Gmehling–Weidlich Equation (14), but now separately for each area element l and time step k using the results ${n}_{\mathrm{l}\mathrm{i}\mathrm{q},1,t}\left({t}_{\mathrm{k}-1}\right)$ of the previous time step k − 1 as initial conditions.

^{4}) than in liquids. This means that in this approach for each compound and each area element, individual evaporation rates will be computed, but only one rate in a common air space will be considered for a given point in time.

_{ΔA,k}= ceil(k/IM) ≤ N

_{ΔA}at time step k, where ceil(x) denotes the ceiling function, which gives the smallest integer greater than or equal to x). We can further assume that during sufficiently small time intervals Δt, the evaporation rate remains almost constant.

_{k}by:

^{2}of component i for an application cycle m and r the coverage velocity in m

^{2}/s. The initial surface coverage $S{C}_{\mathrm{i},\mathrm{m}}$ can take arbitrary positive values and can be calculated from the coverage of the product $S{C}_{\mathrm{P},\mathrm{m}}$ over the mass fraction w

_{i}of component i ($S{C}_{\mathrm{i},\mathrm{m}}=S{C}_{\mathrm{P},\mathrm{m}}\xb7{w}_{\mathrm{i}})$. The term t

_{a,m}is the starting time and t

_{b,m}the end time of an application cycle m and N

_{AC}denotes the total number of application cycles. Please note that in the case of intermittent application, the initial surface coverage can be zero as well, which means that no product is applied to the overall surface area A. This is, for instance, the case when the worker stops wiping, painting, etc., without leaving the room. A corresponding example for intermittent application is given in the results chapter.

#### 2.5. Near-Field/Far-Field Models for Continuous Application (NF/FF_cont and NF/FF_mov)

_{nf}in the near field is here a key parameter as in our example, we assume a near field with dimensions of 2 × 2 × 2 m

^{3}, assuming a rectangular horizontal (e.g., floor) or vertical (e.g., wall) surface just fitting into this near field, i.e., 2 × 2 m

^{2}, which seems reasonable.

_{ΔA,nf,k}= ceil(k/IM) ≤ N

_{ΔA,nf}and in the far field N

_{ΔA,ff,k}= ceil(k/IM) ≤ N

_{ΔA,ff}, respectively

_{t}; $\mathrm{l}\in \mathbb{N},\mathrm{l}=1,2..{N}_{\Delta \mathrm{A}}$, the iterative formulas are written as:

^{2}, which is the side face of the near-field cube. Hence, no product is applied from the worker located in the stationary near field to the far field. With Equation (20), for intermittent applications, the user needs to specify the surface coverage $S{C}_{\mathrm{i},\mathrm{m}}^{}$ and start/end times t

_{a,m}and t

_{b,m}of component i for each application cycle m, the total exposure time t

_{expo}and the number of iteration steps N

_{t}and IM to define the initially applied microscopic molar amounts. It should be noted that in the case of a stationary near field, continuous product application to the far field (by another worker) can be simulated as well. For reasons of brevity, however, we refrain from exemplifying.

_{A}, which can be 2 m at maximum (s. Figure 1). In the first phase, the product is applied by the worker continuously in the near-field area as long as the width of the applied area is less than 2 m. In this phase, the near field is assumed as stationary, and no product is applied to the far field. In the subsequent phase, the worker (and with him the near field) starts moving and continues applying the product until the entire rectangular area A is covered. That is, the worker and the corresponding near field can move out of the already applied area. This moves more and more applied near-field areas into the far field. Taking into account the height h

_{A}of the applied area, the coverage velocity r, the width of the near field of 2 m and the exposure time t

_{expo}, the number of area elements in the moving near field ${N}_{\Delta \mathrm{A},\mathrm{nf}}$ can be calculated using the following expression:

## 3. Results

_{2}O

_{2}or glutaraldehyde are applied as a biocidal product to surfaces by wiping, mopping, etc. In addition to the substance-specific saturation vapour pressures (p*H

_{2}O, p*H

_{2}O

_{2}, p*

_{glutarald}.), which have been looked up from the literature [33,34], the activity coefficients as a function of the mole fraction are required for the model calculations. While the activity coefficients for H

_{2}O

_{2}and H

_{2}O have been estimated using the equation of Schumb et al. [35], which is based on measured data, the UNIFAC method was used to estimate the activity coefficient of glutaraldehyde in aqueous systems (s. Figure 3 and Figure 4). Figure 4 reveals that UNIFAC reflects the nonlinear behaviour of glutaraldehyde in aqueous mixtures fairly, although the estimated partial vapour pressures deviate from measured values to some extent. The initial mass fraction w

_{i}of these example substances is assumed to be 1% w/w, the initial surface coverage of the product SC

_{P}= 0.1 kg/m

^{2}and the application velocity r = 1 m

^{2}/min. Further substance-specific parameters are the mass transfer coefficients β

_{H2O}, β

_{H2O2}, β

_{glutarald}. that were calculated according to Equation (6). Values for the diffusion coefficient of the substance in air D

_{air}and the kinematic viscosity of air ν

_{air}were looked up from the literature [32] or estimated using the method presented by McCready and Fontaine [11], which is based on the work originally presented in Perry and Chilton [36]. The air velocity v and the interzonal ventilation rate Q

_{NF/FF}as well as the air exchange ACPH of the room are assumed to be in a medium range. All numerical values of these input parameters are kept constant for all scenarios (s. Table 2).

^{2}is kept constant in all scenarios. In Table 4, the concentration time diagrams are depicted for the various scenarios for the systems H

_{2}O/H

_{2}O

_{2}and H

_{2}O/glutaraldehyde. Please note that the left-hand ordinate uses different scales then the right-hand ordinate.

_{2}O

_{2}and glutaraldehyde significantly. Basically, there is a time shift to be expected in the release of both substances because both substances are less volatile than water, which causes water to evaporate first. However, as the comparison between № 1 and № 2 shows, it should be highlighted that this delay is enhanced if there are larger surfaces and amounts involved, potentially leading to higher evaporation rates and consequently more pronounced backpressure, which in turn delays evaporation and decreases the peak concentration. While the backpressure has a significant influence on the shape of the time course, the influence of the activity coefficients is not so pronounced in our examples, but it should be noted that for other mixtures, activity coefficients can deviate substantially from 1, making the influence of non-ideality much more pronounced. The concentration vs. time diagrams related to scenario 3 also reveal that the activity coefficients provoke different effects for H

_{2}O

_{2}and glutaraldehyde. While the release of H

_{2}O

_{2}is delayed at the beginning, glutaraldehyde starts to evaporate earlier to some extent. This is due to the fact that activity coefficients of H

_{2}O

_{2}are below 1 (s. Figure 3) while activity coefficients for glutaraldehyde are significantly greater than 1 (s. Figure 4) for low concentrations. Thus, in the mixture, the effective vapour pressure of glutaraldehyde is greater than that of H

_{2}O

_{2}, although the saturation vapour pressure of pure glutaraldehyde is lower than that of H

_{2}O

_{2}. This effect can be observed if products are applied instantaneously (scenario № 3) and continuously (scenario № 4) as well. However, in contrast to instantaneous application, the concentration time curves are significantly broader with lower peak concentrations if products are applied continuously.

^{2}), quite sharp peaks occur in the near-field concentration curves, there is a broadening effect if large areas are applied continuously and a moving worker is assumed. As in the well-mixed room case, the release of H

_{2}O

_{2}is delayed at the beginning, while glutaraldehyde starts to evaporate earlier to some extent. It has to be noted that scenarios № 4 and 6 are quite similar with regard to the shape of the concentration time curves. This is because the worker, and with him the near field, moves and hence a significant portion of the applied area then contributes to the emission into the far field, where the air concentrations are rising accordingly.

_{2}O

_{2}and glutaraldehyde suggests that the worker should leave the room at least after the application phase to minimize exposure.

_{t}= 10,000 iteration steps (Δt = 1.8 s), both for Runge–Kutta and Euler, which has proven to be a good compromise between computing speed and accuracy.

_{t}and IM, respectively, influences the numerical precision of this approach. In Figure 5 and Figure 6, the results of some example calculations for scenario 4 are depicted. The concentration curves of H

_{2}O and H

_{2}O

_{2}reveal that IM values of 10 do not significantly change the shape of the curves. At the same time; however, the computing time, which is in the range of minutes for IM = 1, is reduced by a factor of about 1/10 (1/IM). Even with IM values of 100, the calculation accuracy may be acceptable for a quick rough estimate, with calculation times in the range of seconds.

## 4. Discussion

_{2}O

_{2}with biofilms). Although these effects can in principle be considered in the mass balance to some extent, the corresponding kinetics are regarded as complex and hence as a challenge for future research. Finally, it has to be noted that the proposed “well-mixed room” one- and two-box models are currently limited to scenarios where technical control measures such as local exhaust ventilation or containments are not used. However, as Ganser and Hewett [46,47] demonstrated, one- and two-box models can be extended to situations where various forms of a local control with exhaust are involved. This can be a topic for future work.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Pseudocode for the WMR_cont Scenario Using the Extended Euler Method

_{a,m}and end time t

_{b,m}of an application cycle m according to the function q

_{i}(for abbreviation), defined by Equation (20). The initial values for the airborne concentrations are ${C}_{1}\left(0\right)$, ${C}_{2}\left(0\right)$ (that is, t

_{k}= 0) and must be defined as well.

_{i}(t) as a discrete function time t is obtain by transposing the t

_{k}and C

_{i}(t

_{k}) lists.

#### Appendix A.2. Pseudocode for Continuous Application of a Binary Mixture in a Moving Near-Field Scenario Using Euler’s Method

_{i}. ${n}_{\mathrm{liq},\mathrm{ff},\mathrm{i},\mathrm{init}}$ is zero because no product i is initially applied to the far field. However, when the worker starts moving, this turns more and more applied near-field areas into applied far-field areas that can act as a far-field emission source. Whether a near-field area element starts emitting to the far field is decided by the criteria ${N}_{\Delta \mathrm{A},\mathrm{nf}}=\mathrm{k}-\mathrm{l}$. The initial value for the airborne concentration of component i is ${C}_{\mathrm{nf},\mathrm{i}}$ and ${C}_{\mathrm{ff},\mathrm{i}}$ (that, is k = 1), which are generally assumed to be zero. The functions fN

_{i}, fF

_{i}, hN

_{i,}hF

_{i,}and the sums gN

_{i}, gF

_{i}are defined according to the Equations (20), (21), (27) and (28).

## References

- Delmaar, J.E.; Park, M.V.D.Z.; van Engelen, J.G.M. ConsExpo 4.0 Consumer Exposure and Uptake Models Program Manual, RIVM Report 320104004/2005; RIVM: De Bilt, The Netherlands, 2006.
- Advanced REACH Tool. ART. 2013. Available online: https://www.advancedREACHtool.com/ (accessed on 17 August 2021).
- European Parliament and the Council [EP]. Regulation (EU) No 528/2012 of the European Parliament and of the Council of 22 May 2012 Concerning the Making Available on the Market and Use of Biocidal Products; European Parliament and the Council: Strasbourg, France, 2012. [Google Scholar]
- European Parliament and the Council [EP]. Regulation (EU) No 1907/2006 of the European Parliament and of the Council of 18 December 2006 Concerning the Registration, Evaluation, Authorisation and Restriction of Chemicals (REACH); European Parliament and the Council: Strasbourg, France, 2006. [Google Scholar]
- European Chemicals Agency. Recommendation No. 16, BPC Ad hoc Working Group on Human Exposure, Applicability of ConsExpo for Water Based Disinfectants; ECHA: Helsinki, Finnland, 2019.
- Abattan, S.F.; Lavoue, J.; Halle, S.; Bahloul Ali Drolet, D. Modeling occupational exposure to solvent vapors using the Two-Zone (near-field/far-field) model: A literature review. J. Occup. Environ. Hyg.
**2021**, 18, 51–64. [Google Scholar] [CrossRef] [PubMed] - Arnold, S.F.; Shao, Y.; Ramachandran, G. Evaluation of the well mixed room and near-field far-field models in occupational settings. J. Occup. Environ. Hyg.
**2017**, 9, 694–702. [Google Scholar] [CrossRef] [PubMed] - Lennert, A.; Nielsen, F.; Breum, N.O. Evaluation of Evaporation and Concentration Distribution Models in a Test Chamber Study. Ann. Occup. Hyg.
**1997**, 41, 625–641. [Google Scholar] [PubMed] - Gmehling, J.; Weidlich, U.; Lehmann, E.; Fröhlich, N. Verfahren zur Berechnung von Luftkonzentrationen bei Freisetzung von Stoffen aus flüssigen Produktgemischen (Teil 1). Staub-Reinhalt. Der Luft
**1989**, 49, 227–230. [Google Scholar] - Weidlich, U.; Gmehling, J. Expositionsabschätzung–Ein Methodenvergleich Mit Hinweisen für Die Praktische Anwendung; Schriftenreihe der Bundesanstalt für Arbeitsschutz, Fb 488: Dortmund, Germany, 1986. [Google Scholar]
- McCready, D.; Fontaine, D. Refining ConsExpo Evaporation and Human Exposure Calculations for REACH. Hum. Ecol. Risk Assess.
**2010**, 16, 783–800. [Google Scholar] [CrossRef] - Nielsen, F.; Olsen, E.; Fredenslund, F. Prediction of isothermal evaporation rates of pure volatile organic compounds in occupational environments—A theoretical approach based on laminar boundary layer theory. Ann. Occup. Hyg.
**1995**, 39, 497–511. [Google Scholar] - Fredenslund, A.; Jones, R.L.; Prausnitz, J.M. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J.
**1975**, 21, 1086–1099. [Google Scholar] [CrossRef] - Gmehling, J. Group contribution methods for the estimation of activity coefficients. Fluid Phase Equilibria
**1986**, 30, 119–134. [Google Scholar] [CrossRef] - Zuend, A. Modelling the Thermodynamics of Mixed Organic-Inorganic Aerosols to Predict Water Activities and Phase Separations. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2007. Available online: http://e-collection.ethbib.ethz.ch/view/eth:30457 (accessed on 17 August 2021). [CrossRef]
- Zuend, A.; Marcolli, C.; Booth, A.M.; Lienhard, D.M.; Soonsin, V.; Krieger, U.K.; Topping, D.O.; McFiggans, G.; Peter, T.; Seinfeld, J.H. New and extended parameterization of the thermodynamic model AIOMFAC: Calculation of activity coefficients for organic-inorganic mixtures containing carboxyl, hydroxyl, carbonyl, ether, ester, alkenyl, alkyl, and aromatic functional groups. Atmos. Chem. Phys.
**2011**, 11, 9155–9206. [Google Scholar] - Zuend, A.; Marcolli, C.; Luo, B.P.; Peter, T. A thermodynamic model of mixed organic-inorganic aerosols to predict activity coefficients. Atmos. Chem. Phys.
**2008**, 8, 4559–4593. [Google Scholar] [CrossRef] - Zuend, A.; Marcolli, C.; Peter, T.; Seinfeld, J.H. Computation of liquid-liquid equilibria and phase stabilities: Implications for RH-dependent gas/particle partitioning of organic-inorganic aerosols. Atmos. Chem. Phys.
**2010**, 10, 7795–7820. [Google Scholar] [CrossRef] - Margules, M. Über die Zusammensetzung der gesättigten Dämpfe von Misschungen. In Sitzungsberichte der Kaiserliche Akadamie der Wissenschaften Wien Mathematisch-Naturwissenschaftliche Klasse I; Kaiserliche Akadamie der Wissenschaften: Vienna, Austria, 1895; Volume 104, pp. 1243–1278. [Google Scholar]
- van Laar, J.J. Über Dampfspannungen von binären Gemischen (The vapor pressure of binary mixtures). Z. Physik. Chem.
**1910**, 72, 723–751. [Google Scholar] [CrossRef] - Wilson, G.M. Vapor-liquid equilibrium. XI. A new expression for the excess free energy of mixing. J. Am. Chem. Soc.
**1964**, 86, 127–130. [Google Scholar] [CrossRef] - Renon, H.; Prausnitz, J.M. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J.
**1968**, 14, 135–144. [Google Scholar] [CrossRef] - Abrams, D.S.; Prausnitz, J.M. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J.
**1975**, 21, 116–128. [Google Scholar] [CrossRef] - Heinsohn, R.J. General Ventilation Well-Mixed Mode (Chapter 5). In Industrial Ventilation Engineering Principles; John Wiley & Sons, Inc.: New York, NY, USA, 1991. [Google Scholar]
- Wolfram Research, Inc. Mathematica, Version 12.1; Wolfram Research: Champaign, IL, USA, 2020. [Google Scholar]
- Dahmen, W.; Reusken, A. Numerik für Ingenieure und Naturwissenschaftler; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Cherrie, J.W.; Schneider, T. Validation of a new method for structured subjective assessment of past concentrations. Ann. Occup. Hyg.
**1999**, 43, 235–245. [Google Scholar] [CrossRef] - Cherrie, J.W. The effect of room size and general ventilation on the relationship between near- and far-field concentrations. Appl. Occup. Environ. Hyg.
**1999**, 14, 539–546. [Google Scholar] [CrossRef] - Kulmala, I. Calculation of Vertical Buoyant Plumes. In I: Proceedings of Ventilation ‘97 Global Developments in Industrial Ventilation; Goodfellow, H., Tahti, E., Eds.; Canadian Environment Industry Association: Ottawa, ON, Canada, 1997; pp. 393–398. [Google Scholar]
- Evans, W.C. Development of continuous-application source terms and analytical solutions for one- and two-compartment systems. In Characterising Sources of Indoor Air Pollution and Related Sink Effects; ASTM STP 1287; Tichenor, B.A., Ed.; American Society for Testing and Materials: Philadelphia, PA, USA, 1996; pp. 279–293. [Google Scholar]
- Zemansky, M.W.; Dittman, R.H. Heat and Thermodynamics; McGraw Hill: New York, NY, USA, 1984. [Google Scholar]
- Thibodeaux, L.J. Exchange rates between air and water. In Chemodynamics; Wiley-Interscience: New York, NY, USA, 1979. [Google Scholar]
- Handbook of Chemistry and Physics, 59th ed.; CRC Press: Boca Raton, FL, USA, 1979.
- Olson, J.D. The vapor pressure of pure and aqueous glutaraldehyde. Fluid Phase Equilibria
**1998**, 150, 713–720. [Google Scholar] [CrossRef] - Schumb, W.C.; Satterfield, C.N.; Wentworth, R.L. Report No. 43, Hydrogen Peroxide Part Two; Massachusetts Institute of Technology, Division of Industrial Cooperation Project 6552: Cambridge, MA, USA, 1953. [Google Scholar]
- Perry, R.H.; Chilton, C.H. (Eds.) Chemical Engineer’s Handbook; McGraw-Hill: New York, NY, USA, 1973; pp. 233–234. [Google Scholar]
- Matthies, H.G. Quantifying uncertainty: Modern computational representation of probability and applications. In Extreme Man-made and Natural Hazards in Dynamics of Structures; Springer: Berlin/Heidelberg, Germany, 2007; pp. 105–135. [Google Scholar]
- WHO. Uncertainty and Data Quality in Exposure Assessment [Part 1: Guidance Document on Characterizing and Communicating Uncertainty in Exposure Assessment. Part 2: Hallmarks of Data Quality in Chemical Exposure Assessment]; WHO: Geneva, Switzerland, 2008. [Google Scholar]
- Hofstetter, E.; Spencer, J.W.; Hiteshew, K.; Coutu, M.; Nealley, M. Evaluation of recommended REACH exposure modeling tools and near-field, far-field model in assessing occupational exposure to toluene from spray paint. Ann. Occup. Hyg.
**2013**, 57, 210–220. [Google Scholar] - Jayjock, M.A.; Armstrong, T.; Taylor, M. The Daubert standard as applied to exposure assessment modeling using the two-zone (NF/FF) model estimation of indoor air breathing zone concentration as an example. J. Occup. Environ. Hyg.
**2011**, 8, D114–D122. [Google Scholar] [CrossRef] - Keil, C.B.; Nicas, M. Predicting room vapor concentrations due to spills of organic solvents. AIHA J.
**2003**, 64, 445–454. [Google Scholar] [CrossRef] - Spencer, J.W.; Plisko, M.J.; Occup, J. A Comparison Study Using a Mathematical Model and Actual Exposure Monitoring for Estimating Solvent Exposures During the Disassembly of Metal Parts. Environ. Hyg.
**2007**, 4, 253–259. [Google Scholar] [CrossRef] - Nicas, M.; Neuhaus, J. Predicting Benzene Vapor Concentrations with a Near Field/Far Field Model. J. Occup. Environ. Hyg.
**2008**, 5, 599–608. [Google Scholar] [CrossRef] - Zhang, Y.; Banerjee, S.; Yang, R.; Lungu, C.; Ramachandran, G. Bayesian Modeling of Exposure and Airflow Using Two-Zone Models. Ann. Occup. Hyg.
**2009**, 53, 409–424. [Google Scholar] [PubMed] - Keil, C.B.; Simmons, C.E.; Anthony, T. Mathematical Models for Estimating Occupational Exposure to Chemicals, 2nd ed.; American Industrial Hygiene Association: Fairfax, VA, USA, 2009. [Google Scholar]
- Ganser, G.H.; Hewett, P. Models for nearly every occasion: Part II—Two box models. J. Occup. Environ. Hyg.
**2017**, 14, 58–71. [Google Scholar] [CrossRef] - Hewett, P.; Ganser, G.H. Models for nearly every occasion: Part I—One box models. J. Occup. Environ. Hyg.
**2017**, 14, 49–57. [Google Scholar] [CrossRef] [PubMed]

Indices: | |
---|---|

_{i} | counting index for particular substances |

_{l} | counting index for particular area elements |

_{k} | counting index for particular time steps |

_{m} | counting index for the application cycle |

_{room} | parameters referring to the air space in the room in case of well-mixed-room models |

_{nf} | parameters referring to the air space in the near field in case of two-box models |

_{ff} | parameters referring to the air space in the far field in case of two-box models |

_{nf/ff} | used to indicate an air exchange rate between near and far field |

_{vent} | parameters referring the air exchanged by ventilation |

_{liq} | parameters referring to the liquid layer |

_{air} | parameters referring to air |

_{evap} | parameters indicating an evaporating fraction |

_{A} | parameters referring to the entire treated surface |

_{ΔA} | parameters referring to a fraction of the treated area (“area element”) |

_{app} | refers to the application duration |

_{expo} | refers to the total simulated exposure duration |

_{init} | initial value (for product amounts applied or air concentrations) |

_{a} | refers to starting time of application cycle |

_{b} | refers to end time of application cycle |

_{P} | refers to the entire product |

Parameters and units: | |

p: | vapour pressure [Pa] |

p^{*}: | vapour pressure of a pure substance [Pa] |

x: | molar fraction |

γ | activity coefficient |

$\mathrm{A}:$ | surface area of the applied product [m^{2}] |

$\mathit{\beta}:$ | mass transfer coefficient [m/s] |

$\mathrm{R}:$ | ideal gas constant (8.3145 Pa m^{3} K^{−1} mol^{−1}) |

$T:$ | temperature [K] |

$V:$ | volume of an air space [m^{3}] |

${v}_{}:$ | velocity [m/s] |

$D:$ | molecular diffusion coefficient in air [m^{2}/s] |

${\nu}^{}:$ | kinematic viscosity [m^{2}/s] |

$M$: | molecular weight [kg/mol] |

n: | molar amount [mol] |

w: | weight fraction |

SC: | the initial surface coverage with the product or a compound [kg/m^{2}] |

C: | concentration [kg/m^{3}] |

t: | time [s] |

Q: | air exchange rate [m^{3}/s] |

ACPH: | number of air changes [1/h] |

r: | work rate (rate at which the surface is covered with product) [m^{2}/s] |

N_{t}: | number of time steps |

N_{ΔA}: | number of area elements |

IM: | integer multiples of Δt |

N_{AC}: | number of application cycles |

h_{A}: | height of the area to which the product is applied |

T = 298.15 K | ν_{air} = 1.53∙10^{−5} m^{2}/s | V_{room} = 200 m^{3} |

p*_{H2O} = 3130 Pa | β_{H2O} = 2.4∙10^{−3} m/s | V_{nf} = 8 m^{3} |

p*_{H2O2} = 257 Pa | β_{H2O2} = 2.2∙10^{−3} m/s | Q_{nf/ff} = 600 m^{3}/h |

p*_{glutarald}. = 62 Pa | β_{glutarald}. = 1.9∙10^{−3} m/s | ACPH = 2 /h |

D_{H2O} = 2.4∙10^{−5} m^{2}/s | M_{H2O} = 18∙10^{−3} kg/mol | r = 1 m^{2}/min |

D_{H2O2} = 1.8∙10^{−5} m^{2}/s | M_{H2O2} = 34∙10^{−3} kg/mol | ν_{air} = 0.5 m/s |

D_{glutarald}. = 0.73∙10^{−5} m^{2}/s | M_{glutarald}. = 100∙10^{−3} kg/mol | SC_{P} = 0.1 kg/m^{2} |

η = 1.82∙10^{−5} kg/(m∙s) | w_{i} = 0.01 | ${C}_{\mathrm{i},\mathrm{init}}$ = 0 mg/m^{3} |

№ | Model/ Algorithm | Area A [m^{2}] | Application Time/Pattern | Back-Pressure | Activity Coeff. |
---|---|---|---|---|---|

1 | WMR_inst/ Runge-Kutta | 4 | instantaneous | with and without | with |

2 | WMR_inst/ Runge-Kutta | 40 | instantaneous | with and without | with |

3 | WMR_inst Runge-Kutta | 40 | instantaneous | with | with and without |

4 | WMR_kont/ Euler | 40 | continuous over 0.67 h (40 min) | with | with and without |

5 | NF/FF_inst/ Runge-Kutta | 4 | instantaneous | with | with |

6 | NF/FF_mov/ Euler | 40 | continuous over 0.67 h (40 min) | with | with |

7 | NF/FF_mov_int/ Euler | 40 | intermittent: t _{a,1} = 0 h, t_{b,1} = 0.25 h,t _{a,2} = 0.42 h, t_{b,2} = 0.67 h | with | with |

№ | H_{2}O/H_{2}O_{2} | H_{2}O/Glutaraldehyde |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tischer, M.; Roitzsch, M. Estimating Inhalation Exposure Resulting from Evaporation of Volatile Multicomponent Mixtures Using Different Modelling Approaches. *Int. J. Environ. Res. Public Health* **2022**, *19*, 1957.
https://doi.org/10.3390/ijerph19041957

**AMA Style**

Tischer M, Roitzsch M. Estimating Inhalation Exposure Resulting from Evaporation of Volatile Multicomponent Mixtures Using Different Modelling Approaches. *International Journal of Environmental Research and Public Health*. 2022; 19(4):1957.
https://doi.org/10.3390/ijerph19041957

**Chicago/Turabian Style**

Tischer, Martin, and Michael Roitzsch. 2022. "Estimating Inhalation Exposure Resulting from Evaporation of Volatile Multicomponent Mixtures Using Different Modelling Approaches" *International Journal of Environmental Research and Public Health* 19, no. 4: 1957.
https://doi.org/10.3390/ijerph19041957