Model for hysteretic moisture behaviour of wood Dominique Derome, Ph.D., Empa, Swiss Federal Laboratories for Materials Testing and Research, Wood Laboratory, Überlandstrasse 129, CH-8600 Dübendorf Hannelore Derluyn, Ph.D. student Katholieke Universiteit Leuven, Laboratorium Bouwfysica, Kasteelpark Van Arenberg 41, B-3001 Heverlee,
[email protected] Wolfgang Zillig, Ph.D. student Katholieke Universiteit Leuven, Laboratorium Bouwfysica, Kasteelpark Van Arenberg 41, B-3001 Heverlee,
[email protected] Jan Carmeliet, Ph.D., Chair of Building Physics, Swiss Federal Institute of Technology ETHZ, Zürich, ETH-Hönggerberg, CH-8093 Zürich, Empa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Building Technologies, Überlandstrasse 129, CH-8600 Dübendorf KEYWORDS: hysteresis, sorption isotherm, wood. SUMMARY: Sorption hysteresis of wood is modelled using a modification of the independent domain approach by taking into account capillary condensation as a hysteretic process and film adsorption as a non-hysteretic process. A change in moisture content in the porous material is the result of the behaviour of an assemblage of different sorption domains, described by a distribution function. A new description of the integrated distribution function is introduced, which can be identified from measured main adsorption and primary desorption curves. The model is validated by simulating the sorption moisture behaviour of wood in dynamic sorption tests for the three orthotropic directions.
1. Introduction Sorption hysteresis of wood is known already for more than 100 years. Kollmann (1968) refers to van Bemmelen (1896), who published work in 1896. A simple elegant framework for modelling hysteresis is the Preisach model (Preisach, 1935). The Preisach model has been extensively used by Everett (1967) to model scanning curves and subloops in capillary condensation hysteresis. In the independent domain theory by Everett, each pore domain behaves as an independent isolated system that is in direct contact with the external environment. Mualem (1974) presented an independent domain model for hysteresis based on the similarity hypothesis, that requires only the boundary isotherms to predict the scanning curves. These models were explored by Peralta (1995, 1998a, 1998b) for describing sorption hysteresis of wood. Coasne et al (2005) proposed a modification to the original domain theory by taking into account the presence of a film of absorbed water on the non-wetted pores. More sophisticated models have been developed to describe the interaction of individually non-hysteretic units, where rules for the evolution of fluid configurations in pore networks are included (Guyer and McCall 1996, Carmeliet et al. 1999). Recent work on sorption hysteresis was performed by Derome et al. (2003) and Time (2002) using scanning curves for sorption in wood. Carmeliet et al. (2005) formulated a hysteresis model based on the work of Mualem for oak and compared the results to the phenomenological model of Pedersen (1990) and Rode and Clorius (2004). In Zillig et al. (2007) and in Derluyn et al. (2007), the hysteresis model was applied respectively to spruce and paper, a wood-derived material. It was shown that water vapour permeability in a hysteretic model is dependent on the moisture content and not on relative humidity. In this paper, we first present a hysteresis model, based on the work of Coasne et al (2005), taking into account the presence of a film forming process in not-filled sorption sites. Then, the hysteresis model is validated by comparing experimental results obtained in a dynamic sorption test.
2. Hysteresis model 2.1 Independent domain theory and PM model Following the independent domain approach, the porous material contains a number of domains or sorption sites, which behave independently. Figure 1 describes the typical behaviour of a sorption site. With increasing relative humidity, first monolayer adsorption occurs, followed by multilayer adsorption forming a liquid film of increasing thickness. The film forming process is described by the function mf(φ). At a critical relative humidity φa, the sorption site becomes totally filled by capillary condensation, resulting in a jump mc(φa ,φd) in moisture content from mf(φa) to a moisture content mT(φa, φd). When the relative humidity decreases again, the site evaporates at a relative humidity φd , resulting in a jump back to a moisture content mf(φd). Since φd ≤ φa hysteresis occurs between adsorption and desorption. Remark that the jump mc in adsorption does not equal the jump in desorption, which leads to a hysteresis effect due to the difference in film thickness in adsorption and desorption.
mT (φa , φd ) mc (φa , φd ) m f (φa ) m f (φd )
φd
φa
Figure 1 Characteristic behaviour of a hysteretic sorption site. A porous material consists of a number of sorption sites. Changes in moisture content in a porous material are the outcome of the behaviour of an assemblage of independent sorption sites. An efficient way to represent sorption sites is to map a site characterized by (φa, φd), on the half φa-φd space, also called the PM space (referring to Preisach 1935 and Mayergoyz 1985). The PM space is represented in Figure 2b. On the PM space, a frequency density distribution ρ(φa, φd) is generated, called the PM distribution, representing the number of sorption sites in (φa+dφa,φd+dφd). The PM space is triangular, since φd ≤ φa. A typical snapshot of the PM space after a RH loading history is given in Figure 2a. The region of sites filled by capillary condensation is denoted Ω. The region of sites, where only film forming occurred is denoted Γ. Performing integration over the respective domains gives the moisture content due to respectively film forming and capillary condensation: M f (φ ) = ∫∫ ρ (φa , φd ) m f (φ ) dφa dφd Γ
, M c (φ ) = ∫∫ ρ (φa , φd ) mT (φ ) dφa dφd
(1)
Ω
with the total moisture content M given by M (φ ) = M f (φ ) + M c (φ )
(2)
For determining the integrals, the PM density ρ(φa, φd), and the functions mf(φ) and mT(φa, φd) have to be known. However, a direct determination of these functions is impossible. To simplify the identification process, we do not determine these specific functions directly, but only determine their integrals Mc and Mf.
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ρ(φa,φd) H(φa,φd)
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Figure 2 (a) A sequence of typical snapshot of the PM space after a RH loading history, where φa is the relative humidity in adsorption, φd the relative humidity in desorption, Ω is the region of domains completely filled by capillary condensation, Γ is the region containing the domains where only film forming occurred. (b) Example of a PM distribution function ρ(φa, φd), commonly attaining maximal values on the diagonal and decaying away from the diagonal. (c) Representation of integral function H(φa, φd) in the IPM space. The IPM function is zero on the diagonal and attains maximal values at the boundaries. Capillary condensation is considered to be hysteretic and described by the PM model. Capillary condensation leads to the total filling of a sorption site accompanied by the disappearance of the adsorbed film. This means that when capillary condensation takes place in more and more sites, the moisture content due to film forming Mf will decrease. At RH = 100%, no water film will be present, or Mf(1)=0. The moisture content due to capillary condensation Mc attains the maximum moisture content, or Mc(1)= Mmax. The influence of capillary condensation on the moisture content due to film forming can be described as ⎛ M (φ ) ⎞ M f (φ ) = G (φ ) ⎜ 1 − c ⎟ M max ⎠ ⎝
(3)
where G(φ) is a function which describes the film forming process, when no capillary condensation would occur. Equation 3 shows that, although we assume that the film forming process in the pore scale is non hysteretic, the resulting moisture content Mf becomes hysteretic, since the moisture content Mc in equation 3 is hysteretic. To determine the moisture content due to capillary condensation by the PM approach, we use the integrated PM (IPM) approach. The function H in the IPM space is defined as H (φa , φd ) =
φa xa
∫ ∫ ρ(x , x a
d
) mT ( x ) d xa d xd
(4)
φd φd
where H equals the integral over the triangle (φd1, φd1), (φa1, φd1) (φa1, φa1). By the introduction of the IPM function H, the integral Mc in equation 1 can be calculated as
M c (φ ) = H (φa1 , φd 1 ) − H (φa1 , φd 2 ) + H (φa 2 , φd 2 ) − H (φa 2 , φd 3 ) + ..... n
(5)
= ∑ H (φai , φd i ) − H (φai , φd i +1 ) i =1
with n the number of vertical boundary segments. Keeping track of the boundary between the regions Ω and Γ, the moisture content Mc can be determined summing and subtracting values of H. The IPM function H covers the complete PM space, attains its maximum at the boundary axes, decreases towards the diagonal and becomes zero on the diagonal (Figure 2c). Therefore, we describe the IPM function as H (φa , φd ) = H (1, φd ) * F ( y ) with
y=
φa − φd 1 − φd
(6)
where H(1,φd) is the boundary curve on the desorption axis. The function F(y) describes the decay from this boundary curve towards the diagonal. The functions H(1,φd), F(y) and G(φ) are described by exponential functions introducing five parameters. These parameters including Mmax are determined from measurement data.
2.2 Adsorption and desorption scanning isotherms The wood studied is spruce (Picea abies) from Bavaria, where all specimens were cut from the same log. The average dry density is 402 kg/m³. The sorption isotherm was measured by conditioning samples in a dessicator over saturated salt solutions at a controlled temperature of 23°C. The samples were initially dried at 50°C and 10 % RH. Intermediate desorption curves were obtained by a stepwise reduction of the relative humidity from the adsorption isotherm. The measurement data are given in Figure 3. The solid curves give the fitted adsorption and scanning desorption curves obtained from the hysteretic model.
2.3 Film forming and capillary condensation Figure 4b gives the moisture content curves due to film forming and capillary condensation for the main adsorption, main and intermediate desorption curves. The moisture content variation due to capillary condensation shows a typical hysteresis behaviour as described by the PM model. As foreseeable from equation 3, we observe that, although the film forming process itself is considered to be physically non hysteretic at the pore scale, the moisture content variations due to film forming become hysteretic. (b)
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Figure 3. (a) Measured adsorption isotherm (○), scanning desorption isotherms (▲) and hysteresis model predictions (solid lines).(b)Hysteretic isotherms due to capillary condensation and film forming
3. Experimental validation 3.1 Experimental procedures The samples were initially conditioned at 23°C and 54 % RH. Then the specimens were exposed to a step change in RH to 79.5 % during 14 days. Afterwards, the RH was again stepwise reduced to 54 % for another 14 days. Finally the RH was varied following a sinusoidal daily variation between 54 and 79 % RH. The test was performed for the three directions of wood (longitudinal, radial and tangential). The measurements are given in Figures 4. We observe that the response of wood in longitudinal direction is very fast compared to wood in tangential and radial directions. The different behaviour can be explained by the differences of the water vapour permeability for longitudinal, radial and tangential directions. The difference between tangential and radial directions is small. All specimens show hysteretic behaviour. Small differences in moisture content were observed between the two samples for each direction.
3.2 Water vapour transport model Water vapour transport is described by combining Fick’s law and the conservation of mass equation
ρd
∂u = ∇δ ∇pv ∂t
, u = M (φ )
(7)
with ρd the dry density of the material, u the moisture content, pv the water vapour pressure and δ the water vapour transport coefficient, depending on the vapour pressure. The boundary condition for vapour transport is given by q = β ( pve − pvs ) with β the water vapour surface coefficient, pve the water vapour pressure of the environment, pvs the water vapour pressure at the surface of the material. The water vapour transport coefficient is measured in adsorption for different RH ranges using the dry/wet cup method. The nonlinear adsorption vapour transport coefficient is described by
δ ad (φ ) =
δa a + b exp(c φ )
(8)
The increase of the water vapour permeability with relative humidity is commonly attributed to the enhanced microscopic liquid water transport in water filled pores due to capillary condensation. The water vapour transport coefficient for hysteretic materials thus depends on the moisture content. Using the main adsorption isotherm Mad
δ (u ) = δ ( M ad−1 (u ) )
(9)
The material properties for the different directions are taken from Zillig et al. (2007). The surface transfer coefficient is fitted to the measurement data and assumed to be a constant.
3.3 Results Figures 4 compare the simulated and measured response for the longitudinal, radial and tangential directions. A good agreement between simulations and measurements is obtained. Only the amplitude of the cyclic behaviour of spruce in longitudinal direction is underpredicted. In figure 4d, two additional variants for the tangential direction are examined. In a first case, the hysteretic effect is disregarded by using only the adsorption characteristics. In the second case, the water vapour permeability is assumed to depend on the relative humidity (as per eq. 8) and not on the moisture content. The results show clearly that in the first case (no hysteresis), the simulation overpredicts the moisture loss of the specimen in desorption. For the second case, the moisture loss in desorption is too slow when the water vapour permeability factor is assumed to be dependent on relative humidity. This can be explained by the fact that the vapour permeability dependent on RH is lower than the vapour permeability dependent on the moisture content.
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Figure 4
4. Conclusions Sorption hysteresis of wood (spruce) is modelled using a modified Preisach-Mayergoyz (PM) approach taking into account independently water film adsorption and capillary condensation. A change in moisture content in the porous material is the result of the behaviour of an assemblage of different sorption sites, which show a non-hysteretic film forming and a hysteretic capillary condensation process. It is found that, although film adsorption on the pore scale is non hysteretic, the moisture content due to film forming becomes hysteretic due to the hysteretic behaviour of the capillary condensation process. The model is validated by analysing the moisture content variations of specimens of wood exposed to different relative humidity conditions in a dynamic test. It is shown that the water vapour permeability in a hysteretic material is dependent on the moisture content and not on the relative humidity. The model demonstrates a good agreement with experimental data for the longitudinal, radial and tangential directions.
5. References Carmeliet, J., Descamps, F., Houvenaghel, G., (1999) Multiscale network model for simulating liquid water and water vapour transfer properties of porous materials, Transport in Porous Media, 35, 67-88. Carmeliet, J., de Wit, M., Janssen, H. (2005). Hysteresis and moisture buffering of wood, Symposium of Building Physics in the Nordic Countries, June 13-15 2005, Reykjavik, Iceland, 55-62. Coasne, B., K.E. Gubbins, R.J.-M. Pellenq (2005) Domain theory for capillary condensation hysteresis, Phys.Rev. B, Vol. 72, 024304 (9). Derluyn, H., Janssen, H. Diepens, J., Derome, D., Carmeliet, J. (2007). Hygroscopic behavior of paper and books, Journal of Building Physics, 31 (9): 9-34. Derome, D., Fortin, Y., Fazio, P. (2003) “Modeling of the moisture behavior of wood planks in non-vented flat roofs”, ASCE Journal of Architectural Engineering, American Society of Civil Engineers, New York, March, pp. 26-40. Guyer R.A., McCall, K.R. (1996) Capillary condensation, invasion percolation, hysteresis, and discrete memory, Physical Review B, Volume 54, number 1, pp. 18-21. Kollmann, F., Côté, W. A. (1968). Principles of wood science and technology. Part 1: Solid wood. Springer-Verlag. Mayergoyz, J.D. (1985) , Hysteresis models from the mathematical and control theory points of view, J. Appl. Phys., 57, 3803-3805 Mualem,Y. (1974). A conceptual model of hystersis, Water Resources research, Vol. 10 No. 3. Pedersen, C. Rode. (1990). Transient Calculations of Moisture Migration Using a Simplified Description of Hysteresis in the Sorption Isotherm. Proceedings of the 2nd symposion on Building Physics in the Nordic Countries. Technical University of Norway, Trondheim, Norway. Peralta, P. N. and A. Bangi (1998) Modeling wood moisture sorption hysteresis based on similarity hypothesis. Part 1. Direct approach. Wood and Fiber Science, Vol. 30, No. 1, pp. 48-55. Peralta, P. N. and A. Bangi (1998) Modeling wood moisture sorption hysteresis based on similarity hypothesis. Part II. Capillary-radii approach. Wood and Fiber Science, Vol. 30, No. 2, pp. 148-154. Peralta, P. N. (1995) Modeling wood moisture sorption hysteresis using the independent-domain theory. Wood and Fiber Science, Vol. 27, No. 3, pp. 250-257. Preisach, F. (1935) Über die magnetische Nachwirkung. Z. Phys. 94, 277. Rode C and Clorius C.O. (2004). Modelling of Moisture Transport in Wood with Hysteresis and Temperature Dependence Sorption Characteristics, Proceedings of the Conference Performance of Exterior Envelopes of Whole Buildings IX, Sheraton Sand Key Resort, Clearwater Beach, Florida, December 5-10, 2004. Time, B. (2002). Studies on hygroscopic moisture transport in Norway spruce (Picea abies). Part 2: Modelling of transient moisture transport and hysteresis in wood Holz als Roh- und Werkstoff, 60, 405-410. Van Bemmelen, J. M. (1896). Z. Anorg. Allgem. Chem. 23: 233. Zillig, W., Derome, D., Diepens, J., Carmeliet, J. (2007) “Modelling hysteresis of wood”, Proceedings of 12th Symposium for Building Physics, Technische Universität Dresden, Dresden, March 29-31, Vol. 1, pp. 406-413.
