Advances in Mathematical Physics

Volume 2015, Article ID 307312, 6 pages

http://dx.doi.org/10.1155/2015/307312

## Role of Time Relaxation in a One-Dimensional Diffusion-Advection Model of Water and Salt Transport

^{1}Department of Materials Engineering and Chemistry, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 16629 Prague, Czech Republic^{2}Department of Physics, Constantine the Philosopher University, 94974 Nitra, Slovakia

Received 6 September 2015; Revised 2 November 2015; Accepted 8 November 2015

Academic Editor: Yao-Zhong Zhang

Copyright © 2015 Igor Medved’ and Robert Černý. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The transport of salt, necessarily coupled with the transport of water, through porous building materials may heavily limit their durability due to possible deterioration and structural damage. Usually, the binding of salt to the pore walls is assumed to occur instantly, as soon as the salt is transported by water to a given position. We consider the advection-diffusion model of the transport and generalize it to include possible delays in the binding. Applying the Boltzmann-Matano method, we calculate the diffusion coefficient of the salt in dependence on the salt concentration and show that it increases with the rate of binding. We apply our results to an example of the chloride transport in a lime plaster.

#### 1. Introduction

One of the key problems in real situations where the mass transport by diffusion occurs is the fact that the corresponding diffusion coefficients may strongly depend on the concentration of the transported species. Without a detailed knowledge of this dependence, the analysis of the transport could be rather inaccurate and yield misleading results. If the transport is simple enough to be described as one-dimensional, then the Boltzmann-Matano method [1, 2] is often applied as a convenient technique to derive a varying diffusion coefficient. To do so, the method uses a known concentration profile (usually from experimental data), that is, the concentration as a function of the Boltzmann variable , where and are the position and time, respectively.

The calculation of the diffusion coefficient by the Boltzmann-Matano method is most frequently performed numerically. This has several drawbacks: the resulting diffusion coefficient is given in a discrete format and may require a final smoothing to remove illogical jumps or spikes, the reliability of the results should be verified by back-calculation of the original profile, and the sensitivity of the results to various model parameters is rather time consuming to examine [3]. In addition, the whole numerical determination procedure must be performed afresh for each studied situation. It is possible to avoid all of these drawbacks by employing an analytical approach. For such a case, the concentration profile is approximated by an analytical curve from which a formula for the varying diffusion coefficient is obtained [4–8].

The analytical approach can be a very useful tool for practical engineering calculations. We applied the approach in [9] to the diffusion-advection model of Bear and Bachmat [10] that has been used to study a coupled water and salt transport in porous building materials. In this case, proper understanding of the transport is crucial for the assessment of durability of the materials, because moisture and salt can cause serious deterioration and structural damage. In [9] it was implicitly assumed that the binding of salt to the pore walls occurs instantly whenever the salt is transported by water to a given position. Here we wish to generalize our analytical results to include the fact that the binding of salt is somewhat delayed and to study the effect of this phenomenon on the diffusion coefficient of salt. Such an effect has been lately studied in connection with the chloride ion penetration into concrete, using a transport model based on the concrete electrical resistivity [11]. They accounted for a slower chloride advance simply by adding a phenomenological multiplication factor in the apparent diffusion coefficient. In this study our approach will be more advanced: we implement the relaxation into the salt concentration profile and examine how this affects the diffusion coefficient.

#### 2. Theoretical Background

##### 2.1. The Diffusion-Advection Model

The Bear-Bachmat diffusion-advection model for the transport of a salt solution together with the water mass balance is given as [10]where is the volumetric moisture content, is the concentration of a free salt in water, is the concentration of a bound salt in the whole porous body, is the salt diffusion coefficient, and is the Darcy velocity. Since can be expressed via the moisture content as , where is the moisture diffusivity, model (1) is a system of two coupled partial differential equations with two material parameters, and , and three input quantities, , , and . Measurable quantities are the moisture content and the total concentration, , given asNevertheless, all three input quantities are known, as soon as the binding isotherm, that is, the dependenceis determined for the studied material from an experiment.

In the one-dimensional case, model (1) may be rewritten asHere is the solution of the equation (see (2) and (3)). In formulation (4) of the model there are two quantities, and , to be found.

In a typical experiment, a completely dry prismatic sample is exposed by one of its faces to a penetrating salt solution (water) with a concentration (with a moisture content ). At the same time, the opposite face is kept dry, while the lateral sides are insulated to make the solution transport one-dimensional. The initial and boundary conditions corresponding to such an experiment with an idealized infinitely long sample are given as

##### 2.2. Relaxation in Binding of Salt

In the binding isotherm (3) the salt is assumed to become bound at a given position, ,* immediately* at the time when the salt is first transported by water flow to . Note that this time depends on position . In reality, however, the binding is always somewhat* delayed*. We may take this fact into account by replacing isotherm (3) by the relationwhere characterizes the delay of the salt binding. It should be vanishing before the time, , when the salt is first transported by water to a position . In addition, it should be increasing after , approaching as . We will consider a simple form of given as follows (see Figure 1(a)):in which the onset of the salt binding (which begins at ) is described by the exponential with a relaxation time (see Figure 1). As approaches zero, turns into a step function, which corresponds to an immediate binding of salt. Thus, in the limit , the modified isotherm (6) yields the original isotherm (3) (note that at any position for ). On the other hand, the limit yields a vanishing , which corresponds to no binding of salt.