Heat and Moisture Transfer in Hygroscopic Porous Media: Two Contrasting Analyses

The heat and mass transfer phenomena that occur in beds of ventilated hygroscopic porous media are coupled. This results in two transfer waves traversing bulks of porous medium that initially have a uniform temperature and moisture content, and that are subsequently ventilated with air that is not in thermodynamic equilibrium with the solids. The more rapidly travelling wave is known as a temperature wave and the slower wave is known as the moisture wave. Between the two waves the bed of porous medium is at its dwell state. In an Eulerian formulation of the differential equations that govern heat and mass transfer in beds of hygroscopic porous media a solution is obtained using computational fluid dynamics software. This approach has the advantage of enabling a wide range of geometries and operating conditions to be investigated. A second analysis relies on the observation that thermodynamic states of transfer waves lie on straight lines in ln r ln s p space. The Lagrangian analysis is used to investigate the effect on heat and mass transfer of the differential of the integral heat of wetting of the solids with respect to temperature. The numerical and analytical solutions are compared, and it is shown that the numerical solution captures dispersive effects arising from the finite intra-particle resistance to moisture transfer and thermal conductivity. It is observed that accounting for the differential of the integral heat of wetting of the solids with respect to temperature in the analysis has a significant effect on the predicted velocity of the temperature wave. However, this may be an artifice arising from the fact that the sorption isotherm was obtained by extrapolation to low moisture contents. It may be that sorption isotherms need to be constrained to ensure that the ratio of the differential heat of sorption to the latent heat of vaporisation of water is independent of temperature. It is shown that such isotherms do indeed lessen the effects of the integral heat of wetting on the velocity of the temperature wave. However, the velocity of the moisture transfer wave is found to be strongly dependent on the form of the isotherm. NOMENCALTURE ACP [1/oC] Constant in the Chung-Pfost isotherm equation Ai [..] Velocity ratio BCP [..] Constant in the Chung-Pfost isotherm equation CCP [oC] Constant in the Chung-Pfost isotherm equation A,..,D [..] Constants in Hunter’s isotherm equation ca [J/(kg.oC)] Specific heat of dry air. cs [J/(kg.oC)] Specific heat of dry grain. cw [J/(kg.oC)] Specific heat liquid water. Deff [m/s] The effective diffusion coefficient of water vapour in air. dma [kg] Mass of air that enters a region of a travelling wave. dma [kg] Mass of solids in an incremental length of a travelling wave. dx [m] Incremental distance along the bed of porous medium. h [J/kg] Specific enthalpy of moist air. ho [ J/kg] Specific enthalpy of dry air. o a h [J/kg] Specific enthalpy of dry air at o T °C. hs [J/kg] Differential heat of wetting of solids. o s h [J/kg] Specific enthalpy of dry solid at o T °C, hw [J/kg] Differential heat of wetting. o w h [J/kg] Specific enthalpy of liquid water at o T °C hv [J/kg] Latent heat of vaporisation of water. o v h [J/kg] Latent heat of vaporisation of water at o T °C. H [J/kg] Specific enthalpy of moist solids. HW [J/kg] Integral heat of wetting of solids, J/kg. i [..] Refers to the temperature wave (i =1) or moisture wave (i = 2) k [1/s] Drying constant. keff [W/(m.oC)] Effective thermal conductivity of a porous medium, p [Pa ] Vapour pressure of water. pd [Pa ] Saturation vapour pressure of water at the dwell state. ps [Pa ] Saturation vapour pressure of water. r [..] Relative humidity S1 [..] Slope of the temperature transfer wave in ln r ln s p space S2 [..] Slope of the moisture transfer wave in ln r ln s p space. rd [..] Relativive humidity at the dwell state R [Pam/(kgK)]Gas constant. t [s] Time. T [oC ] Temperature. va [m/s] Component of velocity of air through the bed of porous medium. va [m/s] Velocity of air through the bed of porous medium. vf,i [m/s] Component of velocity of the i drying wave that traverses the bed. vsi [m/s] Component of velocity of a point on the wave that traverses the bed. w [kg/kg] Humidity of air. W [kg/kg] Moisture content of solid phase. We [kg/kg] Equilibrium moisture content. Wo [kg/kg] An empirical moisture content in Hunter’s isotherm. x [m] Distance measured from the air inlet. Greek symbols ε [..] Void fraction of the bed of grains g ε [..] Emissivity of grains a ρ [kg/ m ] Density of dry air. s ρ [kg/ m ] Density of solid phase INTRODUCTION Hygroscopic porous media are used in a wide range of industries. There is increasing interest in using beds of silica gel, for example, to remove moisture from air in airconditioning systems such as those developed by Dai et al. [1] and Thorpe and Chen [2]. Other solid desiccants that comprise bentonite, calcium chloride and vermiculite have been used to dry air that is subsequently used to dry horticultural produce (Shanmugan and Natarajan, [3]). Stored grains, such as wheat and rice, also comprise hygroscopic media that benefit from being cooled and dried if their properties are to be preserved, hence a good understanding of the heat and mass transfer processes that occur in bulks of ventilated grain are extremely useful. When beds of non-hygroscopic media that have an initially uniform temperature distribution are ventilated with air of a different, but constant, temperature one temperature wave traverses the bed. After the bed has been ventilated for a while a temperature wave forms, indicated by line ‘a’ in Figure1. The solids downstream of the leading edge of the temperature wave remains unaffected by the conditions of the inlet air the porous medium remains at its initial temperature. The region upstream of the wave approaches thermal equilibrium with the ventilating air entering the bed. There is a continuous temperature wave through the bed of porous medium between the leading and trailing edges of the temperature wave. When beds of hygroscopic porous media are ventilated with moist air they usually adsorb and desorb moisture which greatly affects their thermodynamic behaviour. For example, consider a bed of hygroscopic porous medium that has been dried so that the relative humidity of the intergranular or interstitial air is low. The bed is then ventilated with air that has a high relative humidity and a lower temperature than the solids. Two waves form in this situation, as indicated by line ‘b’ in Figure 1. The wave with the higher velocity is known as a temperature wave because there is often a relatively large temperature difference across the wave, and the slower moving velocity is known as a moisture wave because there is typically a significant Figure 1. The effects of hygroscopy on the passage of transfer waves through beds of ventilated porous media. Curve ‘a’ depicts the passage of a temperature wave through a non-hygroscopic medium, and curve ‘b’ shows the formation of plateau or dwell state upstream of a temperature wave. A moisture transfer wave is forming upstream of the dwell state. difference in the solids moisture content across the wave. The region between the two waves is known as the dwell state. The velocity of the temperature wave is typically two orders of magnitude greater than that of the temperature wave; in turn the speed of the temperature wave is about 3 orders of magnitude less than the face velocity of the air through the bed of hygroscopic porous medium (Thorpe, 2001). TWO APPROACHES TO ANALYSING HEAT AND MOISTURE TRANSFER IN HYGROSCOPIC POROUS MEDIA The thermodynamic phenomena that occur in beds of ventilated hygroscopic porous media are analysed by formulating the partial differential equations that govern simultaneous heat and moisture transfer between the solid and fluid phases. These are prescribed by the physics of the system, and to that extent they are immutable in so far as the laws of physics are immutable. However, certain approximations can be made to their formulations that render their solutions analytically tractable. In this paper we consider two formulations and solutions of the governing equations, namely: 1) The governing equations are formulated to account for intra-particle resistance to mass transfer and thermal dispersion, and they are solved numerically. 2) Thermodynamic equilibrium is assumed between the hygroscopic porous solid and the interstitial air. The resulting equations have closed form solutions. The two methods are quite distinct, and both are useful under different circumstances. The capability of solving the equations numerically enables designers to analyse the performance of beds of hygroscopic materials operating with arbitrary geometries and with arbitrary initial and boundary conditions. In this work we shall demonstrate how the analysis can be incorporated as a user-defined function into commercial computational fluid dynamics software that increases its flexibility even further. Numerical solutions also permit more physical phenomena to be incorporated into the governing equations. For example, Thorpe and Whitaker [4],[5] have shown that thermal equilibrium is closely approached in beds of hygroscopic porous media, but moisture equilibrium is less likely to be approached, especially when it is desired to dry, rather than simply cool a bed of porous media. The effects of intra-particle resistance to mass transfer are included in the numerical model presented in this work. Analytical solutions to the governing equations remain extremely valuable because they enable the effects of changes to operating conditions and physical properties on the behaviour of ventilated beds of hygroscopic media to be obtained explicitly. In this work we consider a formulation of the equations that assume thermodynamic equilibrium exists between the solid and the interstitial air, and we use Hunter’s [6] method to obtain their solution. The solution of the governing equations provides useful information on the speeds of temperature and moisture fronts through beds of hygroscopic porous media, and the dwell temperature. The sorption isotherm that relates the moisture content of the interstitial air with the temperature and moisture content of the solids also plays a key rôle in the analysis. When Hunter [6] formulated the governing equations he assumed that the integral heat of wetting of porous hygroscopic porous media is not a function of temperature. An order of magnitude suggests that this may be approximately correct, particularly if one can assume that the ratio of the differential heat of sorption to latent heat of vaporisation of water is independent of temperature. This happens to be approximately the case for silica gel. If this is also true for hygroscopic media in general it must be reflected in the form of the sorption isotherm. In this work we investigate the effects of the apparent dependence of the integral heat of wetting on temperature on the behaviour of ventilated hygroscopic porous media. FORMULATION OF THE GOVERNING EQUATIONS – AN EULERIAN APPROACH In our numerical solution of the governing equation we adopt an Eulerian approach, that is we perform enthalpy and mass balances on a differential volume of the porous medium fixed in space. Moisture balance A moisture balance on a differential volume is expressed as ( ) ( ) ( ) 0 v s s a a a a a W w w t t ρ ε ρ ε ρ ε ∂ ∂ + +∇ ⋅ = ∂ ∂ (1) in which s ρ and s ε are respectively the density (dry matter basis) and the volume fraction of the solid matter in the hygroscopic porous medium, W is the mean moisture content (fractional dry basis) of the solids, a ρ is the density of dry air in the interstitial air, w is the absolute humidity of the air and a ε is the void fraction of the bed of grain ( 1 s ε = − ). The mean velocity of dry air through the interstices of the grain bed is represented by va . In some biological systems, such as bulk stored grains, the solid substrate can be consumed by fungi hence s s ρ ε and a ε are not necessarily constant, but in this study we assume that these biological activities are negligible. A moisture balance on the solid phase can be expressed in the form ( ) e W k W W t ∂ = − − ∂ (2) in which k is the ubiquitous drying constant and e W is the moisture content of the solid phase in thermodynamic equilibrium with the interstitial air. Enthalpy balance It has been noted above that the air and the grains are in thermal equilibrium and the enthalpy balance on the air and solids is written as ( ) ( ) ( ) 2 v s s a a a a a eff H h h k T t t ρ ε ρ ε ρ ε ∂ ∂ + +∇ ⋅ ⋅ = ∇ ∂ ∂ (3) in which H is the specific enthalpy of the moist solids phase and h is the specific enthalpy of the moist interstitial air. The thermal conductivity, keff, of the porous medium is taken to be isotropic. The specific enthalpy, H, of the moist solids is defined by ( ) ( ) ( ) s s W w w H h c T T H W h c T T = + − + + + − o o o o (4) in which the subscripts ‘s’ and ‘w’ refer to the solid substrate and moisture respectively. s h o and w h o are standard enthalpies determined at some standard temperature T o , and these do not have to be calculated because when enthalpy conservation equations are balanced they do not figure in the result. This helps to serve as a check on the algebra. HW is the integral heat of wetting of the solid phase expressed in J/kg of dry solid. The integral heat of wetting becomes increasingly negative as the moisture content increases and this implies that the internal energy of the solid decreases as it becomes wetter. The specific heats of dry solid and liquid water are cs and cw respectively. The enthalpy of water vapour, hw, is given by ( ) w w w v h h c T T h = + − + o o (5) where hv is the latent heat of vaporisation of liquid water at temperature T. Expanding equation 3 results in ( ) ( ) 2 v a a s s a a a eff h H h k T t t ρ ε ρ ε ρ ε ∂ ∂ + +∇ ⋅ ⋅ = ∇ ∂ ∂ (6) Equation 6 can be expressed in terms of physical properties by noting the following relationships H H W H T t W t T t ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂ (7) ( ) W w w H H h c T T W W ∂ ∂ = + + − ∂ ∂ o o (8) W s w H H c c W T T ∂ ∂ = + + ∂ ∂ (9) w v w h h T T c t t T t ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ (10) v w w h h c T T T ∂ ∇ = ∇ + ∇ ∂ (11) t T c t h a a ∂ ∂ = ∂ ∂ (12) a a h c T ∇ = ∇ (13) We also note that the enthalpy of the interstitial air is the sum of the enthalpies of dry air and water vapour, hence ( ) ( ) ( ) { } o o o o a a a a a a w w v h h c T T w h c T T h ρ ε ρ ε = + − + + − + (14) Making use of equations 7 to 14 and inserting them into equation 6 enables one to write