2.1. Estimating Evaporation Rates of Liquid Mixtures
A number of models have been proposed for the evaporation of substances from open surfaces, each of which makes different assumptions and simplifications. The simplest approach regards a pure substance as applied instantaneously and then evaporated from a surface area that remains the same size over time. It is clear that instantaneous application is an ideal that will rarely occur in practice. However, the assumption “instantaneous” is regarded as reasonable if the application time is small in comparison to the whole evaporation time.
More importantly, exposure to vapours at workplaces often originates from liquid mixtures. Mixtures and especially the influence of liquid phase non-ideality on evaporation have gained relatively little recognition, however. One model that performed well under a variety of conditions [
8] and that is able to predict evaporation rates of liquid mixtures was developed by Gmehling and Weidlich [
9,
10], s. Equation (2). According to the two-film theory, in this model it is assumed that evaporation is driven by the difference between the partial vapour pressure of an individual component i, which can be derived from the equilibrium vapour pressure by Raoult’s law, and its vapour pressure in the room air
pi,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
pi* and its molar fraction
xi, which is calculated by
The evaporation rate of substance i,
, is proportional to this pressure difference and depends on the surface area A of the product and the mass transfer coefficient
ßi. The activity coefficient
γi is used to take non-ideality into account (vide infra).
The vapour pressure in the room air can be related to its concentration using the ideal gas law.
If the vapour pressure in the room air is higher than the partial vapour pressure, Equation (2) will result in a negative evaporation rate. This can be suppressed by introducing the Heaviside operator
H.
Finally, the modified Gmehling–Weidlich equation describing the evaporation rate can then be written as:
The mass transfer coefficient
ßi is a function of the diffusivity
Di of the substance in the air and of the air flow
vair over the product surface. Parameter notation and corresponding symbols and units are listed in
Table 1.
The values of the molecular diffusion coefficient
Di 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].
The parameter exponents in Equations (2) and (6) were fitted based on experiments carried out by Gmehling and Weidlich [
9,
10] for different solvents released from mixtures at air velocities from 0.2 to about 0.7 m/s and low air concentrations. Due to the low air velocities, it can be assumed that these equations best represent the laminar flow conditions. Slow, laminar flows have a different influence on mass transfer than fast turbulent air flows, as the latter additionally leads to turbulent back mixing above the evaporation surface. Hence, for fast turbulent airflows, other models may be more appropriate [
12]. It has to be noted that the model of Gmehling and Weidlich assumes isothermal evaporation at a constant room temperature
T, which means that cooling effects are neglected. The more volatile the substances are, cooling effects may play a significant role, leading to a reduction in vapour pressure and consequently in air concentrations. From this, it follows that Equation (2) may tend to overestimate exposure slightly. However, overestimation can often be accepted since conservative estimates are more justifiable from an occupational safety and health view.
As mentioned above, in the Gmehling–Weidlich equation, liquid phase non-ideality is corrected for by use of the activity coefficient
γi,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].
Another useful approach for predicting activity coefficients involves the use of an equation of state to represent the behaviour of the gas phase and an excess Gibbs energy model to represent the behaviour of the liquid phase. In particular, the Margules equation [
19], the van Laar equation [
20], the Wilson equation [
21], the NRTL equation [
22], and the UNIQUAC equation [
23] have found widespread usage. From a computational point of view, these methods correlate the activity coefficients
γi of a compound i with their mole fractions
xi in the liquid phase by a separate function, i.e., γ
i =
f(
xi). 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.
In some cases, the non-ideal effect is less pronounced, and it may be argued that non-ideal solution calculations are not necessary for certain mixtures. In practice, however, only people with in-depth thermodynamic knowledge can predict when mixtures do not deviate significantly from Raoult’s law. Hence, it is recommended to consider the relevance of non-ideal behaviour prior to any assessment.
For exposure assessments, usually the concentration of a substance i in the room air, Croom,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, nroom,i. Considering that the transition between concentrations and molar amounts using the relation 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)
In the instantaneous application scenario, a certain molar amount
ninit,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,
nroom,i, is a function of the evaporation rate
(s. Equation (7)) and the loss caused by the ventilation rate
Qvent expressed in (m
3/s). These relationships can be expressed as follows:
The term
Cvent,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:
The loss of compound i occurring from the liquid layer due to evaporation is considered by Equation (9), which is here just the negative of the evaporation rate (see Equation (5)).
Based on these assumptions, the complete mass balance of this system can be described by a set of two time-varying differential equations for each of the involved compounds. It is important to note that all differential equations describing the various components in the liquid are coupled via the molar fraction
xliq,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
nroom,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)
The obvious drawback to the well-mixed room model is that concentration gradients between the source and the rest of the room are ignored. This may result in underestimation of the exposure, because in most scenarios, the exposed person is located next to the source. One way to account for the concentration gradients that occur in a room is to divide the airspace into two or more “zones”. This has the advantage of accounting for the “positional variability” in concentration by using relatively simple mathematical approaches.
For the application pattern “instantaneous”, a two-compartment box model with an emission source in the near field was used. Here, it is assumed that an initial molar amount
ninit,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 (
Vnf) and the far field (
Vff); the quantity of the interzonal air flowing from the near to the far field and vice versa is given by
Qnf/ff, which is assumed to be due to natural convection. The distribution of the substance i in the near field (
nnf,i) and the far field (
nff,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,
pnf,i has to be considered:
The mass balance then comprises three simultaneous first-order differential equations for each component. The first expresses the concentration changes in the near field resulting from evaporation from the liquid into the near field and the air exchange between near and far field. The second equation describes the concentration changes in the far field due to exchange with the near field and the ventilation of the room (which is expected to occur inside the far field). As before, the last term on the right-hand side of Equation (12) represents the mass flow of component i by the incoming airflow
Qvent. 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.
Regarding the most appropriate volume/location or geometry for the near field, so far, no consensus has emerged. We use an approach that is slightly modified in comparison to Cherrie and Schneider [
27], who viewed the near field as a (2 × 2 × 2) m
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
hA 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.
The only model parameter which is essentially unknown is the interzonal airflow
Qnf/ff. Following the approach of Cherrie [
28] and Kulmala [
29] to estimate
Qnf/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)
Although continuous application of volatile mixtures is quite common in practice, little attention has been given to this situation so far from a modelling perspective. Evans [
30] developed continuous source terms for one- and two-compartment systems that break down the application into a sequence of differential area elements. Each area element has an emission profile assuming constant or exponentially decreasing emission rates. Hence, temporal and spatial differences in the emission rates of the area element can be represented. As time progresses during the application, the ensemble effect of the differential areas produces a “macroscopic” emission rate, which, in the absence of interaction, will be the sum of the microscopic rates. It must be emphasized that the approach of Evans [
30] assumes that there is no feedback or “backpressure” effect from the room air, which would tend to suppress the microscopic emission rates as the room concentration increases. This kind of effect is certainly possible and relevant for large evaporating sources indoors and therefore regarded as a limitation.
Another approach that is able to model continuous application of volatile products is integrated into the evaporation model of ConsExpo [
1] when choosing the release area mode “Increasing”. This model is based on a mass balance for the well mixed room. The mass transfer coefficient is estimated using alternatively Langmuir’s [
31] or Thibodeaux’s expression [
32]; or, in more recent versions, an empirically derived default value of 10 m/h is recommended. The model can take into account the increase in amount and surface area due to application of the product, thereby assuming a constant concentration of the mixture components and a constant film thickness over the surface. However, this assumption should be regarded as a limitation, because usually the composition and in consequence the evaporation of a liquid mixture in the surface layer changes over time
and space, owing to differences in volatility of individual components.
To overcome these limitations, we suggest an approach that takes into account the backpressure effect and spatial and temporal differences of the evaporation rate. It has to be noted that the mathematical formalism using the Heaviside operator, as proposed by Evans [
30], was not successful when backpressure plays a relevant role. Initially following Evans, we break down a certain area
A into a sequence of
NΔA 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).
The size of ΔA = A/NΔA 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 texpo into Nt time steps so that Δt = texpo/Nt. Basically, the time ΔtΔA 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 ΔtΔA to adopt integer multiples IM of Δt, that is ΔtΔA = Δt·IM, ΔA = (A/Nt)·IM and Nt = NΔA·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 Nt and IM, respectively, influences the numerical precision of this approach is addressed to some extent in the results chapter by some example calculations.
Within the proposed procedure, we apply the initial molar amount n
liq,l,i,init of component i to an area element l and once Δ
tΔA = Δ
t·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 Δ
t 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 Δ
tΔA = Δ
t). 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
. For instance,
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
of component i applied to area element l at the actual time
tk=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
Nt. If
IM > 1, the area size ΔA and the application time Δ
tΔA 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
of the previous time step k − 1 as initial conditions.
It is assumed that there is no mass flow between neighbouring surface elements, but there is in the gas phase, which can be justified because the diffusion in gases is much faster (≈104) 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.
Hence, the average evaporation rate over all area elements during the interval t at time step k is approximated by the sum over all “active” area elements NΔA,k = ceil(k/IM) ≤ NΔA 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.
This allows us to switch from differential terms to differences, i.e.,
, and consequently:
Since the molar fraction as well as the activity coefficients are concentration dependent, they have to be calculated individually for each area element and each time increment. Assuming the well-mixed room conditions, the temporal change of the air concentration
of component i at time step k is then approximated by Equation (16).
It is obvious that this approach will easily result in a massive number of differential equations when using a large number of area elements, which slows down the numerical solution with the Runge–Kutta method (or other methods such as DoPri-5). Since a very high numerical precision is not required in occupational exposure modelling, we therefore use the simplest method for numerical integration of differential equations, the Euler method [
26]. Starting with the initial value
y(
t = 0), the Euler method calculates the absolute change of the target variable during this first interval Δ
y(
t = 0) by multiplying the slope
dy(
t)/
dt at this point with the length Δ
t of this interval. It then adds this value to the initial value of
y(
t = 0) to estimate the start value of the next interval and iterates this process until the last interval, i.e.,
y(t + Δ
t) =
y(
t) +
dy/
dt · Δ
t.
Although not required by the Euler method, for simplification we here use a constant time interval Δ
t for the iteration steps. For an area element l and time step k, the iteration formulas of the Euler method for Equations (14)–(16) are then written:
Since all time increments k have the same duration Δ
t, they can simply be converted into an actual time
tk by:
Despite its flexibility, this extended Euler method can be programmed straight forward, essentially using two nested next loops. In
Appendix A.1, a piece of pseudocode is given that is intended to illustrate how the iteration algorithm works for a binary mixture assuming continuous application under well-mixed room conditions.
The iteration process over time starts with initial values
for each new area element added. In practical applications, the initial value can be constant or can change over time. For intermittent application, which is quite common in practice, using the Heaviside operator
H we propose the following expression for
:
is the initial surface coverage in kg/m2 of component i for an application cycle m and r the coverage velocity in m2/s. The initial surface coverage can take arbitrary positive values and can be calculated from the coverage of the product over the mass fraction wi of component i (. The term ta,m is the starting time and tb,m the end time of an application cycle m and NAC 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)
The mass balance model described above can provide reasonable estimates when room volumes are small and the air is indeed well mixed. For large rooms, however, gradients of the airborne concentration can be quite high, especially due to temporal and spatial changes in the composition of the applied liquid mixture and due to incomplete air mixing. In addition, workers often move in large rooms, e.g., when manually applying disinfectants, paints, etc., to surfaces, leading to additional positional variability and in consequence possibly to misleading estimates of the airborne concentration. We therefore propose a near-field/far-field model that is able to address these scenarios to some extent. As before, this method employs a direct numerical solution based on the Euler method. As described in the chapter on near-field/far-field models for instantaneous application, we assume the near-field to be centred on the workers body and thus including the entire worker and the emission sources within reach of the worker. In this way, we can model situations in which the worker and the corresponding near field are stationary (that is, the worker can only move within the near-field limits), as well as situations in which the worker and the corresponding near field are moving. In this setup, evaporation occurs into the near field from area elements located close to the worker, but at the same time more distant area elements may evaporate directly into the far field. Thus, we now need two different equations (see Equations (21) and (22)) to describe the evaporation rate of component i for area element l, depending on whether the element is located in the near or far field. The question then arises of which area element l evaporates into the near field and which into the far field during a time increment k. The size of the surface area
Anf 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.
In contrast to the well-mixed room model the mass balance now can take emission sources in the near and the far field into account. Assuming a stationary near field, the average evaporation rate over the interval
t at time-step k is approximated by the sum of all active area elements in the near field
NΔA,nf,k =
ceil(k/IM) ≤ NΔA,nf and in the far field
NΔA,ff,k =
ceil(k/IM) ≤
NΔA,ff, respectively
The changing amounts of component i in the air of a stationary near field around the worker and in the far field at time step k are then approximated by Equations (25) and (26), respectively:
As the search for an analytical solution to this sequence of differential equations was not successful, we propose a numerical approximation using the Euler method again. With
k = 1, 2…
Nt; , the iterative formulas are written as:
This iterative approach is quite flexible and allows us to model a variety of scenarios by choosing different initial conditions. We propose two scenarios that may be of practical relevance. In the case of a stationary near field, the product is applied continuously only within the reach of the worker. That is, the entire applied area must not exceed 4 m2, 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 and start/end times ta,m and tb,m of component i for each application cycle m, the total exposure time texpo and the number of iteration steps Nt 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.
The second scenario allows a moving near field (NF/FF_mov) and hence may be indicated if the worker needs to move when treating large surfaces/rooms. Here, it is assumed that the total applied area
A is a horizontal rectangle with height
hA, 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
hA of the applied area, the coverage velocity
r, the width of the near field of 2 m and the exposure time
texpo, the number of area elements in the moving near field
can be calculated using the following expression:
Whether a near-field area element starts emitting to the far field is decided by the criteria
. In
Appendix A.2, a piece of pseudocode is given that illustrates in more detail how the iteration algorithm works. It should be noted that the moving near-field algorithm assumes
IM = 1 to keep spatial resolution at maximum.