Correspondence/Rebuttal pubs.acs.org/est
Modeling Transfer of Dialkyl Organotins from PVC Pipe into Water: Comments on “Predicting the Migration Rate of Dialkyl Organotins from PVC Pipe into Water”
T
he recent paper by W. A. Adams et al.1 reports an impressive attempt to explore migration of alkyl groups dimethyl tin (DMT) and dibutyl tin (DBT) from polyvinyl chloride (PVC) pipe into drinking water in experimental and theoretical ways. In this correspondence, I would like to make a few comments on the mass transfer model used in their work. These remarks may contribute to a great simplification of the modeling and for a more thorough interpretation of the experimental data presented in.1 For clarity, their mathematical model is repeated below with the same nomenclature (see Figure 2 of ref 1):
∂C ∂ 2C =D 2 ∂t ∂x x = 0,
(1a)
∂C =0 ∂x
x = L, − D
y(t ) ≃
(1b)
⎛C ⎞ ∂C = h⎜ − y ⎟ ⎝ ⎠ ∂x K
t = 0, C = C0 x = L,
More importantly, the scale analysis implies that OT diffusion in PVC pipes can be treated as one occurred in finite media during very small period of time. Unfortunately, the authors1 neglect this specific time scale, an approach that leads to them having a complicated solution of an infinite series requiring roots of a transcendental equation; and an explicit expression for OT concentration in drinking water, y, is still not available. Actually, these difficulties and complexities are unnecessary. Solutions for small values of time are very useful for modeling such diffusion processes. Such a solution to the problem defined by eqs 1a−1g can be derived as in the Appendix, and the final result is
−
(1c)
⎤⎫ 1 exp(β2Dt )erfc(β Dt )⎥⎬ ⎦⎭ β
(3)
As shown in Figure 1, eq 3 gives the same prediction of the original model. Compared with the original model, eq 3 has
(1d)
V dy ∂C = −D A dt ∂x
⎡1 C0 ⎧ Ah 2 ⎨1 − ⎢ exp(α Dt )erfc(α Dt ) K ⎩ VD(β − α) ⎣ α
(1f)
t = 0, y = 0
(1g)
Adams et al. solved this mathematical model in an analytical way, and the final result consists of two parts; the first part is the solution to eq 1a subjected to eqs 1b−-1d) (i.e., eqs 6 or 7 in ref 1, and the second is eq 1f; these two parts are coupled with each other in an implicit way. Since PVC pipes are of circular cross-area, the first issue that needs to be clarified is whether it is reasonable and enough precise to use Cartesian coordinate systems and ignore the influence of curvature of pipes on mass transfer of OT. Although the answer is positive, interpretation has not been provided in the original work. It can be confirmed by the following scale analysis. For a given time scale t, the space range involved by mass diffusion processes can be estimated by
Δr ∼
Dt
(2)
Figure 1. Comparison of model eq 3 with experimental data. Equation 3 is applied with the parameters values given in Table 1 of ref 1.
Where D is material-phase mass diffusivity, Δr is the involved space dimension in polar coordinate systems. Equation 2 can provide estimated results within a factor of order one.2 The order-of-magnitude of D at indoor temperatures approximately is 10−18 ∼10−17 m2/s as mentioned in ref 1; based on eq 2, a time scale of 10 years gives a influenced space range Δr ∼ 10−5 m, which is very small compared to the thickness (L ∼ 10−3 m) and the inner radius (ri > 10−2 m) of PVC pipes. This extremely small region, in which variation of OT concentration in pipes occurs, implies that curvature of pipes can be ignored and Cartesian coordinate systems are applicable. © 2012 American Chemical Society
several advantages. First, it provides an explicit expression for y, which is the most interesting quantity in real applications. Second, eq 3 requires roots of a quadratic equation not those of a transcendental equation, so it is simple to implement and fast to compute. Last but not least, eq 3 is derived by the fact that diffusivity of OT in PVC pipes is extremely small, so this Published: February 21, 2012 4249
dx.doi.org/10.1021/es204281r | Environ. Sci. Technol. 2012, 46, 4249−4251
Environmental Science & Technology
Correspondence/Rebuttal
Using inversion tables, y as a function of time from eq A7 reads
methodology is also applicable for other situations where substances, or chemical pollutants, are of very small mass diffusivities, including, for example, transportation of formaldehyde, VOC and sVOC from indoor materials into air.3−5
Min Li*
■
Engineering College, Guangdong Ocean University, East of Hu Guang Yan, Zhanjiang, 524088, Guangdong, China
(A1)
c w = C0 − y
(A2)
⎪
C0hcosh qx (p + Ah/V )KDqsinh qL + phcosh qL
⎪
(A8)
Equation 2 shows that in common cases L/(Dt) ≫ 1, so we have erfc(L/(Dt)1/2) ≃ 0. Additionally, for large z, 1/2
erfc(z) ∼
(A3)
C0 AhC0 sinh qL cw ̅ = p − Vp [(p + Ah/V )K sinh qL + hqcosh qL]
