Kristian Krabbenhoft;yand Lars Damkildez Department of Civil Engineering
Technical University of Denmark, Lyngby, Denmark e–mail:ykk@byg.dtu.dk,zld@byg.dtu.dk
ABSTRACT
Summary The simultaneous transfer of heat and mass in capillary-porous materials can be described by three coupled differential equations with saturation, gas pressure, and temperature as variables. In this abstract it is demonstrated how, under certain circumstances, it is crucial to include these couplings for a satisfactory description of the overall behaviour of the system.
An accurate description of simultaneous heat and mass transfer in porous media is essential to a wide class of problems. These include industrial processes such as the drying of building ma-terials, food stuffs, and other agricultural products. Other examples include concrete hardening, multiphase flow in soils, and pressure increases in concrete subjected to extreme conditions such as fire. Whereas heat and mass transfer as independent uncoupled phenomena can be described relatively easily, the complexity of the models grows significantly when the full coupling between moisture content, gas pressure, and temperature is taken into consideration.
In this work we use the model put forward by Whitaker [1]. Here a representative average vol-ume of the porous medium in question is considered, see Figure 1. Starting from the microscopic balance equations these are then averaged over the representative volume. This yields a set of macroscopic conservation equations for the total water content (liquid and vapour), dry air, and enthalpy. These can be written as
Total water: intrinsic phase average densities,v the velocities, and" are the volume fractions with"w
+"
g
=
'where'is the porosity. As for the enthalpy conservation equation is the average density,Cp the average specific heat,cp the specific heat of the different species,T the temperature,Keffthe effective conductivity, andhvapthe heat of evaporation. The liquid water and gas velocities are determined by Darcy’s law
v
Figure 1: Averaging volume.
where the effective permeability matrices KKr are the product of the intrinsic permeability K, which depends only on the structure of the porous medium, and the relative permeabilities Kr
which depend on the saturation. BothKandKr are diagonal matrices. The viscosities are given by . The motion of the liquid water is due to gradients in the total liquid pressure, i.e. gradients in the difference between total gas pressurepand capillary pressurepc, whereas gradients in total gas pressure alone results in motion of the gas species. Furthermore, following Fick’s law for a binary mixture of gases (air and vapour) the relative average velocities of the these species are related to the average total gas species velocity as
The system is closed by the thermodynamic relations
p
and the equations of state
p
wheres="w='is the saturation. The capillary pressure-saturation curves are usually determined experimentally, but a good first guess can also be made on the basis of the structure of the porous medium. An example of this is wood which has a rather regular cell structure and where there is a good correlation between analytically and experimentally determined capillary pressure-saturation relations.
The three coupled conservation equations can be discretized in space by finite elements such that the following system of ordinary differential equations is obtained
2
Here the subscripts l,e, andarefer to the physical quantities of liquid, enthalpy, and air respec-tively. As state variables we have used saturation, temperature, and total gas pressure, and the C and K matrices depend nonlinearly upon these state variables. For the time discretization Lee’s three-level time stepping scheme is applied
x
(a) (b)
Figure 2: Drying configurations (a) and (b).
Next, an example concerning the drying of a 20 cm one-dimensional slab of softwood is given.
Wood is a highly hygroscopic material and as such a complete analysis also needs to account for the transfer of bound water in the cell wall. In this case, however, we consider only the transfer of free water above the fiber saturation point. Two different configurations as shown in Figure 2 are considered. In (a) a temperature of60ÆC is maintained at the left end whereas the temperature at the right end is15ÆC, and in (b) these boundary conditions are then reversed. The initial saturation iss=0:44corresponding to a moisture content of MC=90%. For the material values used in the simulations we refer to[2]. To illustrate the effect of the temperature gradients both a fully coupled and an uncoupled computation is performed for each configuration. In the uncoupled computation the transfer equations are still nonlinear ins,T, andp, but the changes in the individual physical quantities occur only as a result of gradients in these quantities. This means that e.g. flow of gas due to temperature gradients does not occur which under certain circumstances is a seriously flawed assumption.
The results of the computations for the two configurations are shown in Figure 3 (a) and (b).
Several observations can be made. Whereas the uncoupled simulation for configuration (a) shows a more rapid drying than is the case with the use of the coupled transfer equations, the opposite is the case for configuration (b). Also, for the uncoupled simulation configuration (a) seems to be the most favourable, whereas the opposite is clearly the case for the coupled simulation. The explanation for these differences lies primarily in the equation of motion for the dry air species.
0 50 100 150 200 250 300
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44
Time (h)
Average saturation
Uncoupled Coupled
(a)
0 50 100 150 200 250 300
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44
Time (h)
Average saturation
Coupled Uncoupled
(b)
Figure 3: Results of drying simulation for configurations (a) and (b).
0 5 10 15 20
Distance from left end (cm) Pressure (p/p atm)
Distance from left end (cm) Pressure (p/patm)
50 hours 100 hours 200 hours
(b)
Figure 4: Drying configurations (a) and (b).
Rewriting (5) in terms of pressures, we have
Here we have used the fact that@pv=@s '0. Equation (10) expresses the fact that flow of air is owed to both pressure and temperature gradients, and in such a way the transfer will occur towards decreasing pressures and increasing temperatures. For configuration (a) the temperature gradients will cause a transport of air out of the slab and the resulting under-pressure shown in Figure 4 (a) is obviously quite unfavourable. For configuration (b) the situation is the opposite. Here an over-pressure develops at the right end, see Figure 4 (b), which has a positive effect both regarding the transport of water and gas. As for the uncoupled problems, the flow of gas is independent of the temperature distribution and the large under-pressure in case (a) never occurs. Similarly, the pressures in case (b) are also moderate. The result is two quite similar drying histories with case (a) being slightly more rapid since heat is here supplied at the left end of the slab where the bulk of the evaporation takes place.
The conclusions are as follows. In drying processes with considerable temperature variations the coupling between gas pressure and temperature can influence the results significantly, and as shown through an example it is not possible to make any general statements of whether this influence will increase or reduce the computed drying times.
REFERENCES
[1] S. Whitaker. Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying. Advances in Heat Transfer, 13, 119–203, (1977).
[2] P. Perr´e and I. W. Turner. TransPore: A generic heat and mass transfer computational model for understanding and visualising the drying of porous media. Drying Technology, 17(7&8), 1273–1289, (1999).