Comment on ‘‘Taylor Dispersion with Absorbing Boundaries: A Stochastic Approach’’ In a recent Letter [1], Biswas and Sen (BS) presented a stochastic approach to solve the problem of Taylor disper- sion with absorbing boundaries. The purpose of this Comment is to bring attention to prior work in which this problem has been treated more exhaustively with deeper issues such as the validity of Taylor dispersion coefficient concept, convergence of reduced order models, nonlinear sources and parabolic versus hyperbolic description were also discussed. First, we note that the basic idea behind Taylor disper- sion is to reduce or eliminate the local degrees of freedom and provide a low-dimensional description (LD) of the transport process. As shown by Taylor, for a nonreactive solute, the combined effect of transverse velocity gradients and molecular diffusion is equivalent to adding an effective axial dispersion coefficient in the transport equation that describes any cross-sectionally averaged concentration hCi (x, t). In this case, the mean axial velocity remains un- affected and the total amount of solute (or the zeroth mo- ment) remains constant. However, for the case of absorption or reaction at the wall, it was already known that any LD must include at least three effective transport coefficients, i.e., the mean axial velocity, the Taylor dis- persion coefficient and the effective rate or decay constant [2]. In an even more general treatment of this problem using the center or invariant manifold approach, it was shown that a LD for hCi (x, t) to all orders is given by [3] @hCi @t  u e @hCi @x  X 1 n2 D n;e @ n hCi @x n  k e hCi; (1) where the effective velocity, volumetric decay (rate) coef- ficient and all the effective dispersion coefficients were expressed in closed form in terms of the eigenvalues and matrix elements of the transverse diffusion operator. Further, u e , k e and the first 100 coefficients D n;e were computed for the pipe, Poiseuille and Couette flows as a function of the wall Damkohler number (), and the con- vergence of the infinite series in Eq. (1) was examined to determine the conditions under which a LD does not exist. For example, for the case of flow in a pipe with a fast nonlinear wall reaction, the characteristic reaction length scale can become much smaller than the transverse length scale and lead to transverse patterns [4]. BS [1] state that their work differs from the earlier literature in that they provide closed form expressions for the cumulants. In our view, it is not the cumulants but a LD and the corresponding effective transport coefficients that are important. Once the effective transport coefficients are known, the calculation of the spatial moments or cumu- lants is a trivial exercise and can be done using Eq. (1). In fact, an exact solution of Eq. (1) can be obtained using the Fourier transform method [3] and all the cumulants can be determined explicitly. For example, for the case of reflect- ing boundaries (k e  0), we get  n  1 n n!D n;e t (n  2), indicating that all cumulants increase linearly with time. This result is no longer true for the case of absorption at the wall since the zeroth moment decays exponentially with time (and all the effective transport coefficients change with ). We note that if all the moments are normalized by the zeroth moment (which is equivalent to normalizing the particle density at each cross-section with the number of surviving particles as was done by BS), then the cumulants are once again given by  n  1 n n!D n;e t. However, this is clearly incorrect as it is equivalent to ignoring the sink term in Eq. (1). It should also be pointed out that the moments approach presented by BS is only applicable for the restricted case of linear absorption whereas the recent methods [3,5] have treated linear as well as nonlinear sources or sinks either in the bulk or at the boundaries. Finally, it was also shown that multimode hyperbolic models containing transfer coefficients (be- tween different scales) lead to a better physical description of the Taylor dispersion phenomena compared to the tradi- tional parabolic models using a single transverse mode and the Taylor dispersion coefficient concept [4,5]. This work was supported by a grant from the Robert A. Welch Foundation. Vemuri Balakotaiah * Department of Chemical and Biomolecular Engineering University of Houston Houston, Texas, 77204, USA Received 24 April 2007; published 17 January 2008 DOI: 10.1103/PhysRevLett.100.029402 PACS numbers: 47.27.eb, 05.40.a, 47.55.dr, 47.57.eb *bala@uh.edu [1] R.R. Biswas and P.N. Sen, Phys. Rev. Lett. 98, 164501 (2007). [2] H. Brenner and D. A. Edwards, Macrotransport Processes (Butterworth-Heinemann, London, 1993). [3] V. Balakotaiah and H.-C. Chang, Phil. Trans. R. Soc. A 351, 39 (1995). [4] S. Chakraborty and V. Balakotaiah, Advances in Chemical Engineering (Elsevier, New York, 2005), Vol. 30, p. 206. [5] V. Balakotaiah and H.-C. Chang, SIAM J. Appl. Math. 63, 1231 (2003). PRL 100, 029402 (2008) PHYSICAL REVIEW LETTERS week ending 18 JANUARY 2008 0031-9007= 08=100(2)=029402(1) 029402-1 © 2008 The American Physical Society