Short communicationAn analytical solution of the convection–dispersion–reaction equation for a finite region with a pulse boundary condition
Introduction
Particle transport in water flowing through porous media is of interest to many fields, from Earth sciences to various techniques for particle separation, e.g. filtration, membrane processes, chromatography, etc. One of the most convenient ways to examine transport and fate of particles is by tracer studies [1], [2], [3], [4], [5]. This approach allows obtaining abundant information on porous media [6], [7] without spending considerable efforts on preparation of the particles. Significant attenuation in concentration of particles at the outlet from a porous media column requires either high concentrations of the particles at the inlet, or major scale-down of the experimental system. Therefore, the experiments are often performed with a spike (pulse) introduction of the tracer at the inlet and collection of probes at the outlet [1], [2], [3], [4], [5]. Thus, the outlet tracer concentration plays a major role in understanding and characterizing the processes which take place inside the media. This concentration is profiled as a function of time to provide the residence time distribution (RTD), or breakthrough, curve [8], [9], [10].
The tracer concentration at the exit depends on such processes as convection, dispersion, reaction and accumulation. In terms of theory, it is described by the convection–dispersion (diffusion) equation and its extension which includes also the reaction term. This equation has been extensively studied in the past, both in general terms [11] and in specific applications related to chemical and environmental engineering [1], [2], [12].
In order to reflect the experimental conditions properly, this equation is to be solved for a finite region, with a pulse inlet boundary condition and a realistic outlet boundary condition. Since such a solution has not been found yet, in many cases the analytical solutions for semi-infinite regions are adopted for an analysis of finite regions [2], [3], [4], [5], e.g. filtration columns. It is obvious that this approach has an inherent problem due to the fact that the solutions for finite and infinite (or semi-infinite) regions differ in their form. As discussed in Ref. [2], when the transport equation is solved for an infinite region, the analytical solution is fitted to the observed experimental particle breakthrough curves using standard algorithms for the solution of nonlinear least-squares problems. A similar approach, which includes application of statistical analysis tools, e.g. regression, to adjust the predictions, is used also in Refs. [3], [4], [5]. It should be noted that although these works employ the solutions for the pulse inlet boundary condition, they are unable to yield an RTD curve in the form of a skewed Gaussian characteristic to finite systems.
As for the finite regions, some rather elaborate solutions exist [11], [13]. Certain boundary conditions, attempted to describe the processes, are rather problematic when the short columns are considered [14]. In particular, van Genuchten and Parker [14] conclude that only the solution of Brenner [15] for the flux-type inlet boundary condition conserves mass in the system, whereas the concentration-type condition at the inlet boundary incorrectly predicts the volume-averaged or resident concentration inside both semi-infinite and finite systems. This problem results, in particular, in an inaccurate prediction of effluent curves from finite columns. Van Genuchten and Parker [14] also find that applicability of certain boundary condition type may depend on the column Péclet number.
The so-called Danckwerts boundary condition [16] postulates that the concentration gradient at the exit of the column is zero, ∂C/∂x = 0. According to Brenner [15], it was imposed “in order to avoid the unacceptable conclusion that the solute concentration passes through a maximum (or minimum) in the interior of the medium.” We note that this problem does not exist when the tracer is introduced as a pulse. Still, Lee et al. [17] note that the Danckwerts condition is commonly used in the theoretical and numerical analysis of finite regions “despite the fact that artificial suppression of the exit concentration gradient to zero may be physically unrealistic.” The literature reports several approaches aimed at modifying this condition, e.g. based on the Péclet number, the high values of which result in a physically meaningless concentration profile when the concentration gradient at the exit is suppressed to zero [18].
To the best of the authors’ knowledge, the above-mentioned skewed breakthrough curve has not been reported yet analytically for a finite column. An exception is the work [12] where it is obtained for a pulse introduced inside the column.
In the present paper, an attempt is made to bring a theoretical analysis closer to practical situations. In the next section, an analytical solution of the convection–dispersion reaction equation for a finite column is suggested. The pulse boundary condition is applied at the inlet. As for the outlet, the Danckwerts boundary condition is used. The problem is solved using the Laplace transform. The inverse transform is performed based on the complex functions and residue theory. Limitations of the solution are discussed in Section 3, where it is rendered dimensionless. A parametric study is given in Section 4. Finally, a comparison with a broad variety of experimental results is presented and discussed.
Section snippets
Analytical solution
The equation in question is, in a general form:where a, b and c are constants. The initial and boundary conditions are:Thus, x = 0 denotes the exit, where the Danckwerts boundary condition [17] is applied. This is done to facilitate the solution by having a homogeneous boundary condition at x = 0. The form of the boundary condition at the entrance reflects pulse injection [2], [3], [4], [5].
The Laplace transform of Eq. (1) is:
Dimensional analysis
We now rewrite the problem in an appropriate form, where ω = C is the concentration, ω0 = m/Q is the ratio of the injected mass to the flowrate [2], [3], [4], [5], a = D is the dispersion coefficient, −b = U is the approach velocity, −c = K is the attachment coefficient, and l = L is the column length. Accordingly the equation and its solution are:The dimensionless
Results and discussion
In this section, we first describe the overall characteristics of the solution. Then, a parametric investigation is presented, in which the effects of D, U, and K are illustrated. The investigation was based on realistic values of these parameters. Finally, a comparison with the experiments, both performed by the authors and found in the literature, is presented and discussed.
Overall form of the solution. A typical example of the results is presented in Fig. 1. The space- and time-dependent
Conclusions
An analytical solution of the convection–dispersion–reaction equation has been obtained for a finite one-dimensional region, with a pulse boundary condition at the inlet and Danckwerts’ boundary condition at the outlet. The problem was solved using Laplace transform, with the inverse transform based on the complex formulation and residue theory.
The effect of various parameters, including the dispersion coefficient, approach velocity and attachment coefficient, on the breakthrough curve has been
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