Analytic solution for the non-linear drying problem
Introduction
Drying of porous media is a current topic of research in many areas, e.g., chemical engineering, civil engineering and soil science. A thorough understanding of drying will have many technical and economic consequences. In engineering applications one prefers a simple, but adequate description, which simulates the drying of a material correctly. It is usual for the moisture transport to be described by a non-linear diffusion model (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17])In this equation θ is the moisture content and D(θ) is the moisture diffusivity. In this `lumped' model all mechanisms for moisture transport, i.e., liquid flow and vapour diffusion, are combined into a single moisture diffusivity, which is dependent on the actual moisture content. Whitaker [2] and Bear and Bachmat [3] provide a more fundamental basis for this diffusion model using volume-averaging techniques.
For the one-dimensional drying problem considered here the boundary condition at the left drying surface is given bywhere β is the mass transfer coefficient, ha is the relative humidity of the exterior air and hm(θ) is the relative humidity of the material at the surface which is determined by the desorption isotherm. At the vapour-tight right-hand side there is a no-flux condition.
In general two stages can be identified during the drying process. Initially the drying is determined by the external conditions, i.e., the moisture transport in the material is faster than the mass transfer out of the material by the air flow. As a result, the moisture profiles in the material will be rather flat and uniform. However, as soon as the air-blown surface becomes dry a drying front will enter the material. Now it is the internal moisture transport that limits the drying rate of the material. Both the drying rate and the heat extracted from the sample are strongly decreased. In general from this time onwards the drying experiment can be considered as isothermal (see e.g., [4], [5]). During this second stage the moisture profiles, which exhibit a moving drying front, are determined by the material properties such as moisture diffusivity.
This moisture diffusivity D(θ) must be determined experimentally for the porous material of interest. By measuring the transient moisture profiles during drying, the diffusion coefficient can be determined directly (see e.g., [4], [5]). The resulting moisture diffusivity has a deep minimum for many building materials, as shown for fired-clay brick [4], [5], [6], [7], [8], clay [9], [10], sand–lime brick [5], [6], gypsum [5], [11], cellular concrete [7], [12], mortar [8] and soil [13], [14]. In Fig. 1 the general form of the moisture diffusivity is indicated by the dashed line.
From the shape of the moisture diffusivity curve in Fig. 1, two regimes can be identified that relate to certain ranges of the moisture content. At high moisture contents the moisture transport is dominated by liquid transport. With decreasing moisture content the large pores will be drained and will therefore no longer contribute to liquid transport. Subsequently the moisture diffusivity will decrease. Below a so-called critical moisture content, the water in the sample no longer forms a continuous phase or the transport between remaining water clusters has become very low. Hence the moisture has to be transported by vapour and this transport will therefore be governed by the vapour pressure. For low moisture contents the moisture diffusivity begins to increase again. The minimum in the moisture diffusivity therefore indicates the transition from moisture transport dominated by liquid transport to moisture transport dominated by vapour transport. This transition from liquid to vapour transport corresponds to the drying front in the moisture profiles.
Using such measured moisture diffusivity relations, the drying process can be simulated under various conditions and geometries. Until now, numerical simulations have been the only way to solve the non-linear diffusion equation. However, numerical simulations provide no basic understanding of the drying process. Recently Landman et al. [18] proposed an analytic solution based on quasi-steady states which broadens our understanding of the drying process. It is beyond the scope of this article to go into the full details of the mathematical derivation of the solution. In this paper the discussion will be limited to the second drying stage during which the drying is internally limited and the moisture profiles exhibit a front. Here the analytic model will be compared with numerical simulations. Furthermore, a comparison is made between analytic solutions and the measured moisture profiles during drying of fired clay brick, sand–lime brick and gypsum. Finally, also a comparison is made between the moisture diffusivity determined from the measured moisture profiles and those determined by fitting the analytic model to the measured moisture profiles.
Section snippets
The analytic solution
The drying problem can be restated in the following dimensionless form:where the dimensionless spatial position is given in terms of L, the length of the sampleFor our materials, there is a clear and deep minimum in the moisture diffusivity, occurring at a moisture content denoted by θm. Here we have chosen to scale the moisture diffusivity relative to this minimum in the moisture diffusivity asand to scale the time variable by:In the second drying
Comparisons between the analytic model and numerical simulations
The constant speed of the drying front gives a good basis for benchmarking the analytic solution with a numerical simulation. Moisture profiles during drying were computed using standard procedures from the NAG-library for various combinations of b1,b2 and θm. From these simulated profiles the speed of the drying front was calculated from a linear fit of the position of the front between 0.1<ξ<0.8. In Fig. 4 the numerically determined speed from a simulation is plotted against the calculated
Examples of drying behaviour and comparison with the analytic model predictions
Finally a comparison was made between the moisture profiles measured during the drying process and those predicted by the analytic model. The experimental profiles were measured using nuclear magnetic resonance. One-dimensional drying experiments were performed on various samples, i.e. fired-clay brick, gypsum and sand–lime brick. In these experiments the samples were dried using air with a relative humidity of 45%. An extensive description of these experiments can be found in [5], [6].
In Fig. 5
Conclusions and discussion
An analytic solution has been given for the non-linear drying problem. It is shown that the minimum in the moisture diffusivity is related to the drying front. This drying front enters the material with a constant speed determined by the type of material, consistent with experimental results. The speed of the drying front is not only determined by the vapour transport but is also determined by the liquid moisture transport.
The analytic model has been shown to give an accurate solution when
Acknowledgements
Part of this project was supported by the Dutch Technology Foundation (STW).
References (19)
- et al.
Determination of moisture diffusivity in porous media using scanning neutron radiography
Int. J. Heat Mass Transfer
(1993) - et al.
Determination of moisture diffusivity in porous media using moisture concentration profiles
Int. J. Heat Mass Transfer
(1996) - et al.
Etude fondamentale des transfert couples chaleur-masse en milieux poreux
Int. J. Heat Mass Transfer
(1981) - et al.
Moisture movement in porous materials under temperature gradients
Trans. Am. Geophys. Un.
(1957) Simultaneous heat, mass and momentum transfer in porous media. A theory of drying porous media
Adv. Heat Transfer
(1977)- et al.(1990)
- L. Pel, Moisture transport in porous building materials, Ph.D. Thesis, Eindhoven University of Technology, The...
Die wissenschaftlichen Grundlagen der Trocknungstechnik
(1978)- H. Garrecht, Porenstrukturmodelle für den Feuchtehaushalt von Buastoffen mit und ohne Saltzbefrachtung und rechnerische...
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