Accuracy of modeling of bound water diffusion in wood induced by variation of ambient air parameters

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Action FP0802 Accuracy of modeling of bound water diffusion in wood induced by variation of ambient air parameters Wiesław OLEK and Jerzy WERES

1 Action FP0802 Accuracy of modeling of bound water diffusion in wood induced by variation of ambient air parameters Wiesław OLEK and Jerzy WERES Poznań University of Life Sciences, Poland Joint Focussed Meeting "Modeling mechanical behavior of wooden cultural objects" Kraków,, April 12-13, 13, 2010

2 Introduction Wooden objects are in permanent contact with moist air which is characterized by daily and seasonal variations of its parameters. The parameter variation induces changes in bound water content of wood. The water gain or loss is responsible for dimensional changes which lead to stress development and therefore to the risk of cracks and deformations.

3 Introduction The accurate modeling of bound water transport is critical for assessing environmental risks to wooden cultural objects. The quality of the modeling depends on transition between air parameters and wood equilibrium moisture content (EMC), structure of the diffusion model, values of coefficients of the model.

4 Introduction The objective of the present study is to identify and assess factors influencing accuracy of modeling of the bound water diffusion induced by variations in the air parameters.

5 Transition between air parameters and wood EMC

6 Indoor RH variation Indoor RH variation Santa Maria Maddalena church in Rocca Pietore Taken from Camuffo D.,, Pagan E., Rissanen S., Bratasz Ł., Kozłowski owski R., Camuffo M., della Valle A. (2010): Journal of Cultural Heritage, 11:

7 FPL Madison

8 FPL Madison

9 FPL Madison Parameterized by Hailwood-Horrobin Horrobin model (two hydrates model) M = 1800 W 1 Kh + Kh K 1 + K Kh + 1 Kh + K 1 2K 1 K 1 2 K K 2 2 K 2 h 2 2 h M (%) Equilibrium Moisture Content (EMC) h ( )) air relative humidity (RH) W, K, K 1, K 2 functions of temperature

10 FPL Madison The data do not distinguish between adsorption and desorption. Simpson W.T. (1973): Wood and Fiber,, 5(1): The data are based largely in Sitka spruce (Picea sitchensis) Simpson W.T., Rosen H.N. (1981): Wood and Fiber,, 13(3): The set of EMC values comes from different experiments originally reported in the 1930s.

11 FPL Madison t = 21.1 C EMC (%) RH (%)

12 Sorption hysteresis

13 Mature lime wood EMC (%) st desorption 2nd desorption 1st adsorption EMC differences 1st ads and 2nd des: for RH = 30% --- EMC = 0.7% for RH = 70% --- EMC = 2.4% RH (%)

14 Wood species

15 Adsorption lime & beech Lime Beech EMC (%) EMC differences: for RH = 30% --- EMC = 0.5% for RH = 70% --- EMC = 1.5% RH (%)

16 Structure of sorption models

17 3 parameters models One hydrate Hailwood-Horrobin Horrobin model M = 1800 W 1 Kh + Kh K 1Kh 1 + K Kh 1 GAB model M = m 0 (1 ckh kh)(1 kh + ckh)

18 3 parameters models The models can be reduced to the following form h M = A + 2 Bh Ch Nakano (2006) showed that h /M = f(h) ) is not symmetric for the FPL data.

19 8 3 parameters models 6 RH / EMC RH (%) Lime adsorption Lime desorption

20 Boundary conditions

21 Boundary conditions The 1st kind b. b condition = instant hygroscopic equilibrium at wood surface. ( x, t ) M, ( x, t ) Γ [ 0 ] M, = t F The 3rd kind b. condition = transport resistance between air and wood surface. M D = σ, x [ M ( x, t ) M ], ( x, t ) Γ [ 0 ] t F

22 Boundary conditions Modified 3rd kind b. condition = transport resistance between air and wood surface + exponential approach to EMC at wood surface. M D = σ, x [ M ( x, t ) M ], ( x, t ) Γ [ 0 ] [ 1 exp( t c )] M = a + b / t F

23 Methods for diffusion coefficient determination

24 Analytical methods Initial sorption D = π 4 / l 2 E t 2 Final sorption 2 4l 8 D = ln 2 2 π t π ( 1 E ) Assumptions: the wood surface obtains instant hygroscopic equilibrium with moist air, the diffusion coefficient is a constant (it does not depend on water content).

25 Analytical methods Separation of the diffusion coefficient and the surface emission factor Choong & Skaar (1969) (Liu 1989) Dt l l 2 1 Dt = = Assumption: D σ l D σ l 1 2 D = dt t de the diffusion coefficient is a constant (it does not depend on water content). l 2 σ = 0.701D t D l l

26 Analytical methods The analytical methods require information on EMC: from sorption isotherms, or from direct measurements during sorption experiments.

27 Inverse method Model of bound water diffusion = partial differential equation + initial and boundary conditions. Results from sorption experiments = bound water content in time. Software for finding model coefficients and equilibrium water content = solving diffusion model and optimizing objective function.

28 Inverse method Diffusion coefficient values dependent on bound water content and parameterized by functions. The boundary conditions of any kind. Equilibrium water content is not measured but estimated.

29 Results of modeling

30 Initial sorption method E (-) Experiment Initial sorption T ( )

31 Liu method 1.0 E (-) Estimated by the inverse method Experiment Liu method; EMC = 12.48% T ( )

32 Liu method E (-) Experiment Liu method; EMC = 12.48% Liu method; EMC = 14.3% T Estimated by the inverse method ( ) FPL sorption isotherm

33 Liu method 80 Local relative error (%) Liu method; EMC = 12.48% Liu method; EMC = 14.3% T ( )

34 Inverse method E (-) Experiment D = const & 3rd k.b.c. D = const & modified 3rd k.b.c T ( )

35 Concluding remarks 1. The application a of the sorption isotherms should account at least for wood species, hysteresis, structure of sorption models. 2. The first kind boundary condition is not acceptable for f r the diffusion modeling in wood. 3. Accuracy of the diffusion modeling is mostly affected not only by the third kind boundary condition but also by its modified form.

PREDICTING EQUILIBRIUM MOISTURE CONTENT OF WOOD BY MATHEMATICAL MODELS

PREDICTING EQUILIBRIUM MOISTURE CONTENT OF WOOD BY MATHEMATICAL MODELS PURCHASED BY THE FOREST PRODUCTS LABORATORY U.S. DEPARTMENT OF AGRICULTURE, FOR OFFIClAL USE WILLIAM T. SIMPSON Made in the United