Modelling of hysteresis influence on mass transfer in building ...

ARTICLE IN PRESS638J. Kwiatkowski et al. / Building and Environment 44 (2009) 633–642A similar chart was obtained for the relative humidity pr<strong>of</strong>ile atdepth 25.0 mm. The changes <strong>of</strong> the vapour permeability <strong>influence</strong>also the vapour transfer and its accumulation in the material.Hereafter, in Table 6 the relative deviation for the highest value<strong>of</strong> the RH pr<strong>of</strong>ile (after 24 h) at depths <strong>of</strong> 12.5 and 25.0 mm ispresented.The same analysis has been done for the thicker material, andthe deviations <strong>of</strong> relative humidity at depths <strong>of</strong> 50.0 and100.0 mm are presented in Table 7.Again, the errors are bigger and deeper in the thinner material,but this time, for the thicker sample the deviations are lower anddeeper in the material. For both thicknesses <strong>of</strong> the material in thesimulation with lower values <strong>of</strong> water vapour permeability thanin the reference case, the problem with divergence occur. It mightbe connected with a too long time step and too thin layers used inthe numerical simulations.Some uncertainty is always associated with the experimentalcharacterization <strong>of</strong> materials. The variations used in the sensitivitystudy above seem reasonable approximation <strong>of</strong> experimentaluncertainty. Therefore it must be remembered that the calculatedvalues are also within a limit <strong>of</strong> several percent; here about 6% asshown in the results in the table above.5. <strong>Modelling</strong> <strong>hysteresis</strong> <strong>of</strong> the sorption isothermSorption <strong>hysteresis</strong> <strong>influence</strong>s the water vapour transport inthe material. To describe the effect <strong>of</strong> <strong>hysteresis</strong>, the empiricalmodel proposed by Pedersen [11] in MATCH s<strong>of</strong>tware was used inHumi-mur. Eq. (4) is used for desorption and Eq. (5) is used in thecase <strong>of</strong> adsorptionx d;h ¼ ðu u aÞ A x d þ Bðu u d Þ A x aðu d u a Þ A (4)x a;h ¼ Bðu u aÞ A x d þðu u d Þ A x a(5)ðu d u a Þ Awhere u a and u d are moisture contents in kg/kg, x a and x d are themoisture capacities (kg/kg) <strong>of</strong>, respectively, adsorption anddesorption isotherms for a given relative humidity, and u is theactual moisture content in kg/kg.Computed moisture capacity x d,h or x a,h is used in Eq. (3). Thefunction (4) is used for desorption if in the previous two timesteps the water content was decreasing, the function (5) is usedfor adsorption if in the previous two time steps the water contentwas increasing. Pedersen proposed the values <strong>of</strong> A and Bcoefficients as follows: A ¼ 2 and B ¼ 0.1. Although, like the otherresearchers showed [21,25], it is better to fit those coefficients foreach material. For fitting the experimental data, primary sorptionand desorption isotherm and secondary sorption or desorptionisotherm are needed.5.1. Setting A and B coefficientThe coefficients A and B from Eqs. (4) and (5) were chosenusing experimental data <strong>of</strong> primary adsorption and desorptionisotherm and secondary desorption isotherm, as plotted in Fig. 9.Finally, coefficient A was found to be equal to 1.6 and B to 0.68.Those values were used in Eqs. (4) and (5), implemented in Humimur,for simulations <strong>of</strong> mass flow in the material with the effect <strong>of</strong><strong>hysteresis</strong>.Even if the values <strong>of</strong> A and B coefficients fit well theexperimental data, the fully empirical procedure <strong>of</strong> selection isnot satisfactory from the theoretical point <strong>of</strong> view. It is still apresent challenge to enhance understanding and modelling <strong>of</strong>Water content [kg/kg]0.0180desorption 79.5-33%0.0160adsorption0.0140desorption0.01200.01000.00800.00600.00400.00200.00000.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80<strong>hysteresis</strong>. It seems interesting to exploit micro-structural properties,such as pore-size distribution.5.2. Simulations and resultsIn this part <strong>of</strong> the paper the results <strong>of</strong> mass flow calculation forthe model with the <strong>hysteresis</strong> effect are presented. The simulationswere done for gypsum board from Test 1 (Table 1), withthickness <strong>of</strong> the sample l ¼ 12.5 mm, number <strong>of</strong> layers z ¼ 5, timestep ¼ 60 s, and period <strong>of</strong> calculation ¼ 360 h. The boundarycondition, ambient relative humidity, was described by asinusoidal function with a 24-h period, oscillating between 30%and 50%. The initial relative humidity in the material was equal to40%. In Fig. 10, water content in the middle layer <strong>of</strong> the material isplotted.The darker line describes the process <strong>of</strong> change in watercontent when at the beginning the desorption process appears.The lighter line depicts the process starting with adsorption. Alarge difference can be seen during several initial cycles betweenthe two situations. However, after several cycles (approximately12 here), the material water content is predictably at the samelevel in both situations.Additional simulations <strong>of</strong> the mass flow in the material wereperformed with different levels <strong>of</strong> relative humidity: for the first 7days it was varying sinusoidaly between 55% and 45% and for thenext week between 50% and 40%. Fig. 11 shows that moisturecontent in the material, as expected, is situated in between thecurves <strong>of</strong> adsorption and desorption, and the shape <strong>of</strong> the curve issimilar to previous results from the literature [12].6. Impact <strong>of</strong> <strong>hysteresis</strong>Relative humidity [-]0.901.00Fig. 9. Curves <strong>of</strong> primary sorption and desorption isotherms, and seconddesorption isotherm from [24].To demonstrate the <strong>influence</strong> <strong>of</strong> sorption <strong>hysteresis</strong> effect onwater vapour transport in building materials, series <strong>of</strong> simulations<strong>of</strong> the 12.5-mm-thick gypsum board were performed. The samematerial and same boundary conditions were used with twomodels: with and without <strong>hysteresis</strong> effect. The convection masstransfer coefficient b and temperature were the same as thoseused in Test 1 (Tables 1 and 2), but the relative humidity in the airwas changing sinusoidally between 45% and 55%. The calculationswere made for four variants (see Fig. 12): Hysteresis: The model with <strong>hysteresis</strong> effect. Adsorption: Model without the <strong>hysteresis</strong> effect. The dependencybetween the relative humidity and the water content isdescribed by the adsorption isotherm.