F. Kallel Effects of Moisture on Temperature During Drying of Consolidated Porous Materials

Introduction Porous materials attract and hold water molecules in quantities that depend directly on the ambient relative humidity. Even at low relative humidities a thin film of liquid is deposited on the internal surface of the material. At higher relative humidities, some of the pores can be entirely filled with liquid. The moisture, in either liquid or vapor form, migrates through the porous material as a result of molecular diffusion, gravity, capillary action, pressure, and thermal gradients. This movement contributes to other heat transfer mechanisms while eventual phase changes occurring within the material act as heat sources or sinks. Thus heat and mass transfer in porous bodies are highly coupled. The development of an integrated heat and mass transfer model began at the end of the 1930s when Henry (1939) studied the diffusion of vapor through air within the pores of a solid (cotton), which may absorb (or desorb) and immobilize some of the diffusing substances. Later, In this paper we present a transient one-dimensional model for undeformable porous materials based on the continuum mechanics approach. It is assumed that temperatures are above freezing so that water is only present in the liquid and vapor phases. The pressure of the gas phase is assumed to be uniform and constant throughout the material. Mass fluxes caused primarily by capillary forces, in the case of the liquid phase, and by diffusion, in the case of the gas, are expressed in terms of moisture content and temperature gradients. The equations resulting from the conservation principles are solved numerically for symmetrical and unsymmetrical ambient conditions using experimentally determined suction isotherms and transport properties for brick and mortar. The results are validated by comparison with measured temperature and moisture content values and the model is used to study the effect of initial moisture content and convection transfer coefficients on the temperature of such porous slabs during drying. Modeling Heat and Mass Transfer The situation under study is one-dimensional horizontal heat and mass transfer in a homogeneous porous slab of thickness L The formulation of the heat and mass transfer phenomena within the porous slab is based on the assumption that the spatial variation of all dependent variables is continuous. It also incorporates the assumption of local thermal equilibrium between the three coexisting phases under consideration: the solid matrix, the pure liquid (no dissolved gases), and the gaseous mixture of vapor and dry air. Furthermore, the densities of the solid matrix and liquid phases as well as the total pressure of the gas phase are assumed to be fixed and uniform. The conservation equations are derived by performing mass and energy balances on an elementary control volume treated as a continuum comprising all four components (solid matrix, liquid, vapor, and dry air). The resulting expressions are similar to those derived by A positive value of the source term indicates that vapor is condensing. Therefore, the plus sign appears in the equation for liquid, the minus in the equation for vapor, while for the dry air the source term is zero. In the case of the energy conservation, the following assumptions have been used: 8 viscous dissipation is neglected 9 the pressure work term for the gaseous components is also neglected, and • the internal energy of the liquid and solid components is identical to their enthalpy, which is a linear function of temperature. The energy equation is then written as: The summations in this expression include the four components, i.e., the solid, dry air, water vapor, and liquid. The conduction term comprises heat transfer through all four components by an appropriate definition of an equivalent thermal conductivity The mass fluxes appearing in the conservation equations can be caused by the following driving forces: 1 Total pressure gradient causing movement of both liquid and gas. In this work, the total pressure is considered to be constant and uniform throughout the porous material, so that the corresponding mass fluxes are nonexistent. This simplification has been used by 2 Capillary forces resulting in liquid movement. The corresponding flux is expressed using Darcy's equation, assuming the flow in the porous material is quasi-steady as do Since the capillary pressure P c is a function of the total moisture content W (by virtue of the experimentally determined sorption isotherm), and since this function depends on the temperature T, the flux due to capillary forces can rewritten as 3 Concentration gradients. The corresponding diffusion fluxes for liquid, vapor, and air are expressed using a modified version of Fick's law, which takes into account the resistance to diffusion inside the porous body: Philip and DeVries (1957) indicate that the value jj, which according to For the case of the dry air, its concentration is related to its own mass content W" and to the temperature, so that: 4 Gravity. In this paper, the effect of gravity is neglected since we are considering one-dimensional horizontal problems. By adding Eqs. This formulation is identical to that used by By replacing the expressions for the mass fluxes (Eqs. (3c) and (3d)) in the equation of continuity for each of the three mobile components we obtain: The effect of the porous material on the transport of the three mobile components is quantified by the values of the mass transfer coefficients D mJ , D Tj which, according to the previous discussion, depend on several properties of the porous material, such as its permeability to the liquid water, its porosity (which determines the maximum value of ^ in Eqs. (3b) and (3c)), the form of its suction characteristic, etc. For saturated materials, as well as for hygroscopic materials and high ambient relative humidities, W; is much larger than w v or, equivalently W = w t . Summing up the relations for vapor and liquid under these conditions results in: A comparison of Eqs. (4b) and (5a) gives the following expression for the rate of vapor condensation under this condition: which, by comparison with Eq. (4a), indicates that the conditions w/ > > vf", is equivalent to the condition of negligible vapor accumulation. Substituting in the energy relation, Eq. (2), the gradient of the mass fluxes from Eq. (1) and neglecting the term E Jj (dHj/ dx) (for an estimate of the relative importance of this term, see Appendix A) results in the following relation: where c =c 0 + w,c,+ w"c tt (6a) (6b) In the last expression, the term w a c pa , which is small compared with the specific heat of the solid matrix, has been neglected. As a result of this elimination, the temperature field, the moisture content, and the phase change rate can be calculated from Eqs. (5a), (5b) and (6o), provided the various material properties are known and appropriate boundary conditions are specified. It is therefore not necessary to know the dry air transport coefficients appearing in Eq. (4c) in order to calculate T and w/. This formulation is analogous to the one by DeVries (1958) except that in his case mass fluxes and mass conservation are expressed in terms of the volumetric moisture concentration. Here, mass concentration is preferred, since this variable can be determined directly by simply weighing the dry and moist material. Furthermore, the present formulation is complete, while the one proposed by Material Properties, Boundary Conditions and Solution Procedure The materials of interest in this study are brick and mortar. Their suction characteristics as well as their heat and mass transport properties have been determined and published previously by • K" 1 for W = 1 percent and ~ 2.0 W • m" 1 • KT 1 for W = 13 percent; the value of the mass transfer coefficient D"," for mortar decreases monotonically from about 10~9to7»10~ m 2 s ~' as ^increases from 1 percent to 6 percent. Furthermore, the experimentally determined dependence of the mass transport coefficients on W reflects, as expected, the relative magnitude of the different mass transfer mechanisms at different stages of the drying. Thus, for example, for mortar at values of moisture content higher than 3 percent, D ml is two order of magnitudes larger than D,"", which is consistent with the fact that capillary movement of the liquid dominates the early stages of drying. On the other hand, for W < 1 percent, D m " is at least an order of magnitude larger than D mh which is consistent with the fact that vapor diffusion is more important in the late stages of drying. Thus, it is not necessary to have • a different model for each drying stage, as proposed by For the present study, the effect of W on the heat and mass transfer coefficients appearing in Eqs. (5a), (5b), and (6a) has been taken into account using best fit polynomials to express their dependence on W while the effect of T on these coefficients has been neglected. On the other hand, the effect of 726 / Vol. 115, AUGUST 1993 Transactions of the ASME temperature on the specific heat capacities and the enthalpy of evaporation has been accounted for. The solution can only be obtained numerically because Eqs. (5a), (5b), and (6a) are highly coupled and nonlinear. For this purpose, the solid slab is discretized using n nodes. The first node is on the surface at x = 0, the last on the surface at x = L. Thus Ax = L/(n -1). Calculations have been performed with L = 0.20 m for both n = 9 and n = 17. For each one of the interior nodes, the following comments apply: • the appropriate expression for M 0 in Eqs. (5b) and The first expresses conservation of the water in both vapor and liquid phases, while the second expresses energy conservation. The coefficients a p , a p , . . . and the constants g p , g p depend on the heat and mass transport coefficients (i.e., on the hygrothermal state at the beginning of the timestep), on the timestep At, and on the distance Ax between the nodes. The implicit approach used in this part does not impose any stability condition on the numerical solution. The boundary conditions, i.e., the corresponding equations for nodes 1 (at x = 0) and n (at x = L) are obtained by taking into account the mass and energy accumulation term, which for these nodes is associated with the volume A • A x/2 (see 9 Conservation of the vapor component states that the difference between the mass flux from the ambient air to the surface, AJ" = Ah m (p vm -p v \), minus the vapor flux from node 1 to node 2 Ax Ax is equal to the rate of phase change at the surface node (this quantity is representative of conditions in the volume A • A x/2 which in the discretization process gets replaced by conditions at node 1). This principle leads to the following equation: The mass transport coefficients in this equation are evaluated at the average conditions between nodes 1 and 2. Conservation of H 2 0 (vapor and liquid phases) states that the difference between the mass flux from the ambient air to the surface, AJ" = Ah m (p"" -p v] ), minus the water flux from node 1 to node 2 Ax is equal to the rate of water accumulation in the volume associated with the surface node. This accumulation is: The mass transport coefficients in this equation are again evaluated at the average conditions between nodes 1 and 2. • Conservation of energy states that convective heat addition from the ambient air to the surface, expressed by h c A (T" -Tt), minus conductive heat transfer from node 1 to node 2, plus the heat generated by evaporation at node 1, is equal to the rate of energy accumulation within the volume replaced by node 1. In accordance with previous simplifications, this accumulation term is Ax *3T ,-r '-r 2 2 dt Ax where m/A is given by Eq. (8a) and the effective conductivity is evaluated at the average conditions between nodes 1 and 2. For a simple heat transfer problem (i.e., for m = 0) Eq. (8c) reduces to the relation found in many textbooks 9 the convective coefficient of heat transfer, which can be calculated from appropriate correlations, and 9 the convective mass transfer coefficient, which must be determined experimentally or estimated from the heat and mass transfer analogy; for natural convection on a vertical surface, the latter gives h," ~ 10" 3 h c (see Appendix B). The formulation is now complete. The system of algebraic Eqs. (la), (lb) can be solved from any fixed initial conditions, w, (x,0), T(x,0),m(x,0) for given ambient conditions T K (t), 0 M U) using the boundary conditions expressed by Eqs. Time Model Validation and Analysis of Results The numerical solution has been obtained for several different conditions for brick and mortar slabs of thickness L = 20 cm. In the results presented here, the slab is considered to be initially in hydrothermal equilibrium and at time t = 0 a sudden change of one or both of the ambient properties T m , 0a, is imposed on one or both surfaces of the slab. The results presented include comparisons with measured values and illustrate the influence of mass transfer and phase changes on the temperature profiles, and their evolution, within the slab. Calculations have also been performed using Luikov's model by setting the semi-empirical coefficient for the phase change rate equal to D mtt /(D ml + D mu ); in all cases, the results calculated with the two models were essentially identical. T" and $". Perrin (1985) has measured the temperature and moisture content evolution for slabs of uniform initial conditions T-, = 20°C, W-, = 0.07 which were suddenly put in contact with cold humid air at T m = 5°C, <j>" = 80 percent on one side (x = 0) and with warm, dry air at T" = 23°C, </ >«, = 45 percent on the other side (x = L). Calculations were performed for this case to validate the model. The average convective coefficients corresponding to the experimental conditions Unsymmetrical Conditions for Both By comparing the results of Figs. 2 and 3, it is noted that the moisture content difference between the two faces of the slab is, for the present conditions, quite small. Mass fluxes due to the moisture content gradient are therefore expected to be small. Furthermore, since D T is several orders of magnitudes smaller than D m for both materials under consideration, the total moisture flux is expected to be small (cf. The results of Figs. 2 and 3 further indicate that, as expected, higher values of the convective coefficients result in a quicker drying of the material. This is achieved by an important increase in the evaporation rate on the warm side: Indeed, as shown in It is interesting to note in Discussion of Drying Process With T" = T t . The minimum temperature of approximately 13.2 °C attained during the evaporative cooling shown in Another point of interest regarding this evaporative cooling process can be deduced from the above description of the phenomenon. It regards the influence of the initial excess moisture content (i.e., the difference between W,-and the equilibrium value corresponding to the ambient conditions) on the duration of the constant uniform temperature phase. For fixed convective transfer coefficients and fixed ambient conditions, a decrease of W, will shorten the time necessary for the surface to attain the equilibrium moisture content. Since from then on the evaporation rate becomes negligible and the surface temperature starts increasing, the duration of the first two phases of the temperature transient must decrease under these conditions. In fact, we can anticipate that if W-, is quite close to the equilibrium moisture content corresponding to the fixed ambient conditions, the constant uniform temperature phase 730/Vol. 115, AUGUST 1993 Transactions of the ASME may not occur at all. In other words, the equilibrium moisture content may be reached at the surface before the temperature reaches its minimum value. Conclusion The simple, two-variable model for coupled heat and moisture transfer in consolidated porous materials is a fairly accurate tool for predicting transient temperature and water content profiles. The agreement between calculated and measured results provides an indirect validation of the heat and mass transfer coefficients for brick and mortar published earlier Acknowledgments This study was carried out with the financial assistance of the Natural Sciences and Engineering Research Council of Canada

Dalle chauffante sur terre plein. Etude experimentale et numerique du comportement du sol et des ponts thermiques en regime varie

CNRS RP 440 (100) / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc

Commission "Architecture, urbanistique, société". "États des lieux, thèmes émergents": I. Construire la ville. II. Fonction et gestion urbaines. III. Milieux et réseaux urbains. IV. Fondements historiques et sociaux de la ville

Les textes rassemblés ici ont été rédigés en 1984 et 1985 pour répondre à deux demandes: celle d'un état de la recherche architecturale et urbaine, pour le Comité national du Centre National de la Recherche Scientifique et celle d'un repérage de thèmes émergents, pour le Comité national d'orientation du Programme urbanisme et technologies de l'habitat