Intermittent Rainfall in Dynamic Multimedia Fate Modeling

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Environ. Sci. Technol. 2001, 35, 936-940

Intermittent Rainfall in Dynamic Multimedia Fate Modeling EDGAR G. HERTWICH* Institute for Product Design and Industrial Ecology Program, Norwegian University of Science and Technology, Kolbjørn Hejes vei 2b, 7491 Trondheim, Norway

It has been shown that steady-state multimedia models (level III fugacity models) lead to a substantial underestimate of air concentrations for chemicals with a low Henry’s law constant (H < 0.01 Pa m-3 mol-1) because they assume a steady rain. This can lead to substantial errors, especially when multimedia models are used to estimate the spatial range or inhalation exposure. A dynamic model of pollutant fate is developed for conditions of intermittent rainfall to calculate the time profile of pollutant concentrations in different environmental compartments. The model utilizes a new, mathematically efficient approach to dynamic multimedia fate modeling that is based on the convolution of solutions to the initial conditions problem. For the first time, this approach is applied to intermittent conditions. The investigation indicates that the time-averaged pollutant concentrations under intermittent rainfall can be approximated by the appropriately weighted average of steady-state concentrations under conditions with and without rainfall.

Introduction This paper introduces a new approach to the modeling of intermittent conditions in multimedia environmental fate models. This approach is based on a superposition of a small number of independent solutions to the mass balance equations and is simpler than the standard approach to dynamic multimedia modeling, which is based on numerical integration. The superposition approach is applied to intermittent rainfall in order to investigate whether dynamic modeling is necessary or a combination of different steady states is sufficient to determine the average concentrations in different compartments. Multimedia environmental fate models (also called fugacity models or Mackay-type models) are mass balance models that focus on the partitioning of chemicals among environmental compartments such as air, surface water, sediments, and soil (1). The models can be formulated in terms of the fugacity (2), concentration (3), or inventory (4) of the compartments. These formulations are equivalent, and the mathematical properties explored in this paper are independent of the formulation. Multimedia models are used today to set cleanup standards (5); to assess the relative importance of chemical emissions (6, 7); to evaluate the partitioning (2), persistence, and long-range transport (8) of organic pollutants; and to set priorities in pollution prevention. The most frequently used multimedia models are not equilibrium models but so-called level III or level IV models that describe the steady state and the dynamic response to pollutant input, respectively. * Telephone: +47-7355 0634; fax: [email protected]. 936

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The present investigation was prompted by the findings of a systematic uncertainty analysis (9) of the CalTOX multimedia fate and exposure model (4). This analysis has shown that, for air emissions of pollutants with a low Henry’s law constant, the pollutant inventory in the air compartment was significantly lower under conditions of average rainfall than if there was no rain (Figure 1). As Figures S2 and S4 in the Supporting Information illustrate, significant differences can also occur for emissions to surface water and surface soil. The uncertainty analysis demonstrated that, for 25% of the investigated 336 chemicals, the contributions of wet deposition and wash-out result in significant differences in the potential human exposure between the rain and no-rain situations (9). CalTOX takes into account diffusive pollutant transfer between air and surface water, surface soil, and plants as well as wash-out of gas-phase pollutants and dry as well as wet deposition of particle-bound pollutants. The mass transfer processes are described in the Supporting Information. This investigation focuses on pollutants with H < 1 Pa m-3 mol-1, the group affected by wash-out (Figure 1). The distinction between wet and dry periods has previously been demonstrated to be important for the long-range transport of particulate matter (10) and sulfur (11). Smith demonstrated that the long-range transport of sulfur was dominated by moving dry regions. The highest sulfur deposition resulted from the rain-out of a previously dry air mass (11). For both sulfur transport and air concentration predicted by a multimedia model (9), sensitivity analyses show that the results depend on the occurrence of rainfall but not on the quantity of rainfall. To account for the intermittent character of rainfall in multimedia models, Hertwich et al. (12) suggest combining the results of steadystate calculations with and without rainfall. In this paper, I investigate whether this combination of steady-state results is sufficient to determine the average compartment inventories or whether dynamic modeling is required (Figure 2). The type of dynamic modeling investigated here corresponds to the Mackay level IV models and is based on linear transport equations with constant transfer and decay coefficients (1). Dynamic multimedia models are less common than steady-state models because they are mathematically more difficult. The models that are currently operational are based on a numerical integration of the mass balance equations (13-16). In this paper, I utilize a new approach to dynamic multimedia modeling that is based on the convolution of dynamic system responses to an emissions pulse. This approach is mathematically simpler than the standard approach to dynamic multimedia modeling. Through the simplicity of explicit solutions, it offers interesting insights. For the first time, it is shown that this approach can also be applied to problems in which the transfer coefficients between different environmental compartments change with time.

Methods A level IV fugacity model is described by a set of coupled, linear, first-order differential equations that can be written in the matrix form as in eq 1:

dN h ) FN h + Sh dt

(1)

+47-7359 0110; e-mail:

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 35, NO. 5, 2001

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 2001 American Chemical Society Published on Web 01/18/2001

Li ) -ki -

∑T

ij

i*j

FIGURE 1. Ratio of steady-state inventories for air emissions of 306 volatile and semi-volatile pollutants in different compartments without and with rainfall. s, soil; w, water. Note that the ratios for root-zone soil and vadose-zone soil as well as for surface water and sediments are often the same.

In multimedia modeling, the standard approach to dynamic modeling is to solve eq 1 through numerical integration. This approach is computationally demanding because the time scales that occur in the problem vary widely. Two alternative approaches that take advantage of the linear nature of eq 1 may be simpler: a transformation of the problem into an algebraic form using Laplace functions or solving the homogeneous equation and including sources through the convolution integral. Both of these approaches are frequently used in dynamic modeling, for example, in control engineering and chemical reactor design (17, 18). They have previously been applied in environmental modeling (19, 20). The convolution approach is based on first finding the solutions to a system of homogeneous differential equations. Equation 1 can be converted to an initial conditions problem by describing the source as a series of pulse releases, represented by δ functions (ref 17, p 54):

∫

∞

-∞

Sh (τ)δ(t - τ) dτ ) Sh (t)

(2)

We can now find the system response to the δ function and then integrate the solutions across time. This approach is possible because the superposition principle allows for the addition of solutions to systems of linear differential equations (18). The system response to an emissions profile can thus be obtained through the convolution of the response to an initial condition problem ()pulse release) and the emissions profile Sh (t):

N h s(t) )

FIGURE 2. Comparison of the steady-state inventories (horizontal lines) of cyromazine (CAS No. 66215-27-8) with the dynamic profile resulting from continuous air emissions during a period that consists of 1.5 d without rain followed by 0.5 d with rain. The inventories in the air (broken lines) and surface soil (solid lines) are normalized by the time-weighted average of steady-state inventories under rain and no rain conditions. The figure also indicates the inventories NR(TR) and NN(TN) at the beginning of the rain-free (t ) 0) and rainy (t ) 1.5) periods (eq 7).

() () (

N1 . N h ) , Sh ) . Nn

S1 . ,F) . Sn

L1 T12 . .

T21 L2 . .

. . . .

. . . Ln

)

Ni represents the inventory of the pollutant in compartment i (measured in mol, molecules, or kg), Si is the emissions rate, n represents the number of compartments in the model, and F is the fate matrix. The elements of F are assumed to be time and space invariant. The positive off-diagonal elements, Tij, represent the transfer coefficients from compartment i to compartment j. The diagonal elements Li are the loss terms and are always negative for a primary pollutant. Li is the negative sum of the transformation rate ki and the transfer coefficients Tij:

∫

t

N(t - τ)Sh (τ) dτ

(3)

-∞

The column vectors of N(t) represent n independent solutions to the homogeneous mass balance equation (eq 1 with S ) 0). Each column vector represents the pollutant inventory resulting from initial conditions of 1 mol in one compartment (or, equivalently, a unit pulse release to one of the compartments at t ) 0). N(t) is obtained from a diagonalization of the fate matrix (eq 4) and the subsequent integration of the resulting n independent differential equations (18, p 187f):

F ) V Diag(λi)V-1

(4)

where V is the matrix of eigenvectors of F and λi is the corresponding eigenvalues. In this paper, the symbol Diag() always refers to a diagonal matrix as defined by

{

0 Diag(ai)ij ) a i

if i * j if i ) j

Expressing the time-dependent variable (eq 1) as V-1N h results in a set of n uncoupled, homogeneous, first-order differential equations. If these equations are solved and transformed back, eq 5 results:

N(t) ) VDiag(exp(λit))V-1

(5)

Equations 3 and 5 thus define the possible solutions to any multimedia problem with time-varying sources but constant transfer and transformation constants F. Intermittent Conditions. Intermittent conditions imply that the fate matrix F is not constant through time. While a continuously changing fate matrix cannot be modeled with the present approach, one that is piece-wise constant can be modeled. As noted in the Introduction, the air inventory VOL. 35, NO. 5, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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depends on whether it rains or not but not on the precipitation rate. For this investigation, we assume that the system flips back and forth between the rain and no rain conditions with a period of 2 d (Figure S1). Each rain event lasts for 0.5 d (TR) and follows a rain free period of 1.5 d (TN). We have two fate matrixes FN and FR (for no rain and rain, respectively), and the corresponding eigenvectors (VN, VR) and eigenvalues (λiN, λiR). We furthermore assume a constant emissions rate Sh , reflecting the potential dose calculations (6, 7) that prompted this investigation. The pollutant inventory at each point in time is the sum of two contributions: the emissions during the present period and the pollutants remaining from the point at which the conditions have changed (eq 6). For notational simplicity, we set t ) 0 each time the conditions change:

N h R(t) )

N h N(t) )

∫ V Diag(exp(λ t

R

0

∫

iR(t

- τ)))V-1 h dτ + R S

VRDiag(exp(λiRt))V-1 h N(TN) (6a) R N

t

0

VNDiag(exp(λiN(t - τ)))V-1 h dτ + N S VNDiag(exp(λiNt))V-1 h R(TR) (6b) N N

To fully define these equations, we need to evaluate eq 6a at t ) TR and eq 6b at t ) TN. This leaves us with two equations and two unknowns. Solutions are presented in eq 7:

N h N(TN) ) A(VNDiag((exp(λiNTN) - 1)/λiN)V-1 h+ N S

-1 VNDiag(exp(λiNTN))V-1 h ) (7a) N VRDiag(exp(λiRTR))VR S

-1 VRDiag(exp(λiRTR))V-1 h ) (7b) R VNDiag(exp(λiNTN))VN S

-1 -1 (I - VNDiag(exp(λiNTN))V-1 N VRDiag(exp(λiRTR))VR ) (7c)

where I is the identity matrix of dimension F. Equations 6 and 7 now provide us with a fully defined time profile of pollutant inventories in periodically changing intermittent conditions with a constant emissions rate. It is the purpose of this investigation to compare the timeaveraged pollutant inventories in the dynamic model (eq 8) with those obtained through the appropriate weighting of steady-state solutions (eq 9):

∫

N h dyn ) (

TN

0

N h N(τ) dτ +

∫

TR

0

N h R(τ) dτ)/(TN + TR) (8)

h SS h SS N h SS ) (N N TN + N R TR)/(TN + TR)

(9)

where N h SS h SS R and N N represent the steady-state inventories of the rain and no rain conditions, respectively. The timeintegrated pollutant inventories in eq 8 are defined by eq 10 (and a similar eq for N h R):

∫

TN

0

N h N(τ) dτ ) VNDiag((exp(λiNTN) - 1 -

2 )V-1 λiNTN)/λiN h + VNDiag((exp(λiNt) - 1)/λiN)V-1 h R(TR) N S N N (10)

Implementation. The fate matrixes for 306 pollutants were calculated using the CalTOX multimedia fate and exposure model. CalTOX assesses the pollutant inventories in air, surface water, plants, sediments, surface soil, root-zone soil, and vadose-zone soil resulting from contaminants released to air, water, and soil surface and for contaminant incorporated to some depth in soil (4, 6). The model is described in the Supporting Information. The model settings, the data, 938

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and the data sources used here are documented in Hertwich et al. (12). We used the CalTOX transfer factors and loss terms to assemble a 7-by-7 fate matrix. The matrix was exported from Excel using a Visual Basic macro. The matrix manipulations described above were conducted in Matlab.

Results

N h R(TR) ) A(VRDiag((exp(λiRTR) - 1)/λiR)V-1 h+ R S A)

FIGURE 3. Time profile of glyphosate (CAS No. 1071-83-6) inventories in the different compartments before and during a period of rain. Note that almost all of the pollutants is in the root-zone soil, so the lines for the root-zones soil and the total inventory overlap. The lines for surface water and vadose-zone soil also overlap.

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 35, NO. 5, 2001

A time profile of compartment inventories resulting from glyphosate emissions to air is displayed in Figure 3. As expected, the main effect of rainfall is on the concentration in the air. A similar investigation for cyromazine is presented in Figure 2. In Figure 2, the pollutant inventories have been normalized by the average obtained from eq 9, so that all inventories can be displayed on a linear scale. This case shows what the logarithmic display in Figure 3 hides: that the inventory in other compartments can change in response to changing rainfall. Table 1 presents the corresponding average inventories and the time constants. In the case of cyromazine, the surface soil compartment loses pollutants through evaporation and/or degradation during the rain-free period. With the beginning of rainfall, most of the air inventory is added to surface soil, and surface soil continues to receive input from the emissions to the air compartment. The surface water inventory is much larger than the air inventory (Table 1), so that it shows little change in response to the instantaneous input at the beginning of the rain event. The comparison of average dynamic and steady-state inventories for air emissions of 306 organic pollutants (as defined by eqs 8 and 9) is displayed in Figure 4. Similar figures for emissions to surface water and surface soil are included in the Supporting Information. Table 2 shows a number of statistical properties of the subset of pollutants with a Henry’s law constant H < 1 Pa m-3 mol-1. Table 2 also displays the statistical properties for emissions to surface water and surface soil. The steady state and average dynamic air inventories and the Henry’s law constants of all pollutants are listed in Table S1 in the Supporting Information.

Discussion Both Figures 2 and 3 seem to indicate that the systems approach a steady state within a few hours after conditions change. This suggests that the intermittent nature of the rain event does not have a large influence on the average concentrations. Indeed, the average inventories under intermittent conditions are within 50% of the appropriately

TABLE 1. Steady-State and Average Dynamic Compartment Inventories and Eigenvalues for Air Emissions of Cyromazine (CAS No. 66215-27-8) steady-state inventory (mol)

dynamic inventory

eigenvalues (1/d)

compartments

no rain

rain

time-averaged

time-averaged

no rain

rain

air plants surface soil root-zone soil vadose-zone soil surface water sediments

0.44 128 2.1 49 1.4 138 0.24

2.9 E-5 1.9 2.5 23 0.6 177 0.30

0.33 97 2.2 43 1.2 148 0.25

0.24 70 2.3 37 1.0 156 0.27

-2.3 -0.39 -8.7 E-3 -8.2 E-3 -6.1 E-3 -4.7 E-3 -1.3 E-2

-3.5 E+4 -0.39 -8.7 E-3 -6.1 E-3 -8.2 E-3 -4.7 E-3 -1.3 E-2

TABLE 2. Statistical Properties of Inventory Ratios for Chemicals with H < 1 Pa m-3 mol-1 (n ) 156) air

plants

surface soil

root-zone soil

vadose-zone soil

Relationship between Steady-State Results under No Rain and Rain Conditions emissions to air correlation log(Nnorain/Nrain), log H -0.81 -0.78 -0.10 -0.40 -0.40 geometric mean (Nnorain/Nrain) 1.87 1.50 -0.71 -0.18 -0.18 geometric SD (Nnorain/Nrain) 2.81 2.15 0.72 0.87 0.87 emissions to surface water correlation log(Nnorain/Nrain), log H -0.81 -0.78 -0.10 -0.39 -0.38 geometric mean (Nnorain/Nrain) 1.87 1.50 -0.71 -0.18 -0.19 geometric SD (Nnorain/Nrain) 2.81 2.15 0.72 0.88 0.85 emissions to surface soil correlation log(Nnorain/Nrain), log H -0.81 -0.15 -0.19 -0.19 -0.19 geometric mean (Nnorain/Nrain) 1.87 0.38 -0.02 -0.01 -0.01 geometric SD (Nnorain/Nrain) 2.81 0.67 0.10 0.07 0.07 Ratio of Time-Averaged Dynamic and Steady State Pollutant Inventories emissions to air correlation (Ndyn/Nss) with log H 0.63 0.61 -0.23 0.42 0.43 mean (Ndyn/Nss) 0.91 0.90 1.04 1.00 1.00 SD (Ndyn/Nss) 0.11 0.12 0.06 0.10 0.10 emissions to surface water correlation (Ndyn/Nss) with log H 0.13 0.65 -0.09 0.33 0.31 mean (Ndyn/Nss) 0.97 0.90 1.00 0.99 1.00 SD (Ndyn/Nss) 0.07 0.12 0.08 0.10 0.10 emissions to surface soil correlation (Ndyn/Nss) with log H 0.22 0.13 -0.10 0.05 0.06 mean (Ndyn/Nss) 0.97 0.95 1.00 1.00 1.00 0.07 0.08 0.01 0.01 0.01 SD (Ndyn/Nss)

surface water

sediments

0.29 -0.69 0.76

0.29 -0.69 0.76

-0.16 -0.002 0.01

-0.16 -0.002 0.01

-0.28 -0.01 0.02

-0.28 -0.01 0.02

-0.30 1.07 0.09

-0.27 1.08 0.10

0.14 1.00 0.0003

0.16 1.00 0.0003

0.10 1.00 0.002

0.15 1.00 0.003

between the rain and no rain conditions displayed in Figure 1. It is surprising that this is true for all pollutants because the time constants, represented by the eigenvalues in Table 1, indeed would permit a different result. For cyromazine, it takes >1000 d until the proper steady state is reached. The air inventories in Figures 2 and 3 appear to reach steady state because (i) the fastest time constant is the most important for the air inventory and (ii) the slower time constants would not result in a significant change in inventory levels during the two days displayed in the figures.

FIGURE 4. Ratio of the time-average dynamic inventories to the time-weighted average of steady-state inventories under “rain” and “no rain” conditions vs the Henry’s law constant. The graph shows the ratio of compartment inventories of 306 volatile and semi-volatile pollutants released to air. weighted steady-state inventories for all pollutants, as Figure 4 shows. The differences between dynamic and steady-state results in Figure 4 are small as compared to the differences

If it rains more frequently, the average compartment inventories in the dynamic situation approach the compartment inventories in the steady-state rain situation, as Figure 5 illustrates. In an intermediate situation, with 10 rainfall events per day, the air and plant concentrations are different from both the steady-state rain and the weighted average concentrations of the rain and no rain steady-state scenarios. In this situation, a dynamic modeling seems indeed necessary. For hydrophilic chemicals and the 2-day period, the main effect of the intermittent rain is the removal of pollutants from the air at the beginning of the rainfall event (Figure 2). This removal results in a lower average air and plant concentration. According to Figure 4, the time-averaged inventories in air and plants are tightly coupled. So are rootzone soil and vadose-zone soil as well as surface water and sediments. The reduced air concentration leads to an increase VOL. 35, NO. 5, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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the uncertainty in the input parameters (21). This conclusion could not have been reached, however, without first evaluating the intermittent conditions.

Acknowledgments I would like to thank Tom McKone for providing me with the CalTOX model and the description in the Supporting Information. This research was funded by the P2005 program of the Norwegian Research Council.

Supporting Information Available Description of the CalTOX multimedia fate model, figures illustrating the effect of rainfall on the fate of pollutants emitted to surface water and soil, and a table with the data underlying Figures 1 and 4 (18 pages). This material is available free of charge via the Internet at http://pubs.acs.org.

FIGURE 5. Dependency of the time-average dynamic compartment inventories on the length of the period between the onset of rainfall events. A period consists of 75% “no rain” and 25% rainfall. The markers on the left indicate the inventories under steady-state rain conditions. All inventories were normalized by the time-weighted average of steady-state inventories under rain and no rain conditions. in the inventory in surface water and surface soil. This is probably due to an increased pollutant influx at the beginning of the rain event and possibly also due to an increase of overall persistence, because degradation rates in the air are often higher than those in surface water or soil. The correlation of the ratio of the average dynamic and steady-state inventories and the logarithm of the Henry’s law constant (Table 2) indicates that for air emission there is indeed a relationship and that a lower H leads to an overestimate of the air and plant inventories from a combination of steady-state results. For root-zone soil and vadose-zone soil, a low H leads to a weak underestimate, whereas 1e - 2 < H < 1 tends to produce an overestimate. For surface water and sediments, there is a weak tendency toward an overestimate that is higher with lower H. For emissions to surface water and surface soil, the weighted average of the steady-state concentrations very closely reproduces the results of the dynamic calculations (Table 2 and figures in the Supporting Information). This is interesting because the no rain conditions lead to a significantly higher air concentration than the continuous rain. Note, however, that the air inventory is not very high for emissions to surface water and surface soil of compounds with low H. These results are therefore only important if the air inventory is of primary interest, as it may be for longrange transport. This paper has demonstrated a new approach to the modeling of intermittent conditions in dynamic multimedia models. This approach is mathematically efficient and fairly easy to implement. It can also be used to evaluate other intermittent conditions, such as the variations in temperature, snow cover, or photodegradation. For the case of intermittent rain, it is essential to take into account periods without rain, during which the air concentration is substantially higher than during periods with rain at any precipitation rate. This is especially important when multimedia models are used to assess inhalation exposure or the spatial range of pollutants. The exact modeling of rainfall events and the transition between rain and no rain conditions may not always be necessary. The differences between the average pollutant inventories in the dynamic model of intermittent rainfall and appropriately weighted steady-state inventories, displayed in Figure 4, are smaller than the typical uncertainty in multimedia models due to 940

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Received for review February 25, 2000. Revised manuscript received December 8, 2000. Accepted December 8, 2000. ES000041V