Exact averaging of laminar dispersion

Abstract

We use the Liapunov–Schmidt (LS) technique of bifurcation theory to derive a low-dimensional model for laminar dispersion of a nonreactive solute in a tube. The LS formalism leads to an exact averaged model, consisting of the governing equation for the cross-section averaged concentration, along with the initial and inlet conditions, to all orders in the transverse diffusion time. We use the averaged model to analyze the temporal evolution of the spatial moments of the solute and show that they do not have the centroid displacement or variance deficit predicted by the coarse-grained models derived by other methods. We also present a detailed analysis of the first three spatial moments for short and long times as a function of the radial Peclet number and identify three clearly defined time intervals for the evolution of the solute concentration profile. By examining the skewness in some detail, we show that the skewness increases initially, attains a maximum for time scales of the order of transverse diffusion time, and the solute concentration profile never attains the Gaussian shape at any finite time. Finally, we reason that there is a fundamental physical inconsistency in representing laminar (Taylor) dispersion phenomena using truncated averaged models in terms of a single cross-section averaged concentration and its large scale gradient. Our approach evaluates the dispersion flux using a local gradient between the dominant diffusive and convective modes. We present and analyze a truncated regularized hyperbolic model in terms of the cup-mixing concentration for the classical Taylor–Aris dispersion that has a larger domain of validity compared to the traditional parabolic model. By analyzing the temporal moments, we show that the hyperbolic model has no physical inconsistencies that are associated with the parabolic model and can describe the dispersion process to first order accuracy in the transverse diffusion time.

Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe

“…In the study by series expansion and through illustration in Chatwin (1970), the longitudinal normality of the mean concentration has been well established at a time scale of 1.0R 2 /D, as verified by other studies (Houseworth 1984;Stokes & Barton 1990;Phillips & Kaye 1996). A generalized dispersion model was presented in Gill (1967), Gill & Sankaras (1970), Gill & Sankarasubramanian (1971), and strides towards a certain exact averaging model were made recently by applying the Lyapunov-Schmidt technique of bifurcation theory (Ratnakar & Balakotaiah 2011;Ratnakar, Bhattacharya & Balakotaiah 2012). Asymptotic analyses for the dispersion (Phillips & Kaye 1996, 1997 and numerical simulations of the transport process (Houseworth 1984;Stokes & Barton 1990;Shankar & Lenhoff 1991) have also been carried out.…”

Low-Dimensional Modeling of Transport and Reactions in Two-Phase Stratified Flow

“…This corresponds to the case of perfect transverse mixing. (20) Note that we have assumed Da to be O(1), i.e., the time scale of reaction is comparable to the time scale of convection. We can further simplify eqs 19 and 20 by using the relation between dφ/dx and c ̅ 2 , obtained by substituting eqs 9 into the averaged eq 14:…”

Spectral properties and low‐dimensional description of loop and recycle reactors

“…As the averaging of continuous and discrete models using the LS procedure is explained in detail elsewhere, we outline only some intermediate steps here. We expand the concentration vector as and take the inner product of Eq.…”

“…The second mixing coefficient depends on both the auxiliary flow rates and the inlet reactant concentrations and, hence, can be a function of time. As explained elsewhere, it is convenient to regularize the local Eq. by replacing in the third term on the r.h.s.…”