Correspondence Comment on “Intermittent Rainfall in Dynamic Multimedia Fate Modeling” In ref 1, Hertwich presents a modeling approach with which the effect of intermittent and continuous rain on the concentrations obtained with multimedia fate models can be quantified. His analysis shows that the unrealistic assumption of continuous rainfall, which is made in many existing multimedia models, significantly influences the model results for water-soluble chemicals. However, in his alternative approach presented in ref 1, we have found significant errors in the solution of the mass balance equations that make it impossible to reproduce the results. In this paper, we provide the correct expressions. The notation used is exactly as in ref 1. The errors occur in eq 7 of ref 1. Equation 7 provides the two vectors N B N(TN) and N B R(TR) that contain the masses of a chemical in all model compartments at times TN and TR; TN and TR denote the end of the alternating no-rain and rain periods assumed in the model. The mass in each compartment assumes equal values at some time t and at time t + TN + TR. Generally, the step from eq 6 to eq 7 constitutes the solution of 2n linear equations with 2n variables (n being the number of dimensions of the model matrix F, i.e., the number of compartments), so there are unique values of N B N(TN) and N B R(TR) for the system being in a periodical or cyclostationary state. The cyclostationary state is uniquely defined and independent of the initial condition; it is attained after sufficiently long times (t) after start of the continuous emission at t ) 0. This makes the concept of the cyclostationary state equivalent to the concept of the steady state for models with constant rainfall. The correct version of eq 7a in ref 1 is
The correct version of eq 7b in ref 1 is
Here, the same error in the diagonal elements occurs (denoted by E3). In addition,in the first line on the right-hand side, the matrix A defined in Hertwich’s eq 7c cannot be used (error E2). Instead, an expression with “R” and “N” in reversed order has to be used, see eq 2 above. The other equations in ref 1 are not affected by the errors clarified here and are correct except for a typing error in eq 10, where the “t” must be replaced by “TN”. In the actual computations performed in ref 1, only error E2 was made (E1 and E3 were just typographical errors in the manuscript of ref 1); the numerical error introduced by E2 is small. However, because of the typographical errors E1 and E3, the numerical results presented in ref 1 cannot be reproduced with the equations given in ref 1. To use the approach in other models, the expressions provided in eqs 1 and 2 above are necessary. Most affected by error E2 in ref 1 are the values in the column for the compartment air in the bottom half of Table 2. For emissions to surface water and emissions to surface soil, the correct mean of Ndyn/Nss for air is somewhat smaller, and the corresponding standard deviation is somewhat greater than in ref 1. In addition, the correlation of Ndyn/Nss with log H (for air) is 0.62 for all three emissions scenarios (air, surface water, and surface soil).
Acknowledgments We thank H. Held for a review of the mathematical expressions.
Literature Cited (1) Hertwich, E. G. Environ. Sci. Technol. 2001, 35, 936-940.
Maximilian Stroebe and Martin Scheringer* Institute for Chemical and Bioengineering Swiss Federal Institute of Technology Zu ¨ rich CH-8093 Zu ¨ rich, Switzerland
Edgar G. Hertwich The expression in the first line on the right-hand side of eq 1 is the matrix A defined in eq 7c in ref 1. E1 in eq 1 above denotes where the first error in ref 1 occurs; Hertwich has different diagonal elements at this point.
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ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 20, 2004
Department of Energy and Process Engineering Norwegian University of Science and Technology NO-7491 Trondheim, Norway ES0404668
10.1021/es0404668 CCC: $27.50
2004 American Chemical Society Published on Web 09/18/2004
