Evaluation of Wood Sorption Models and Creation of Precision Diagrams for the Equilibrium Moisture Content

Precision has been evaluated of the 10 most often used wood sorption models, available in literature, for the calculation of the equilibrium moisture content of wood given a change in temperature within the range from 0 °С to 200 °С and in the relative humidity of the surrounding air environment from 0% to 100%. Based on the results of the critical analysis, an argumentative selection has been done of the models that can be purposefully used for the computer determination of wood equilibrium moisture content in contemporary systems for model-based or model predictive automatic control of different processes of hydrothermal treatment of wood and wood materials. With the help of these models, diagrams have been created for precise determination of wood equilibrium moisture content. The established high precision of both the Simpson and Ray et al. models and the Garsía model, which we have refi ned, makes them user friendly for model-based or predictive automatic systems and other engineering applications in the respective temperature ranges specifi ed in the paper.


INTRODUCTION 1. UVOD
In practice the most widely used method for the control of various processes of hydrothermal treatment of wood materials is based on ensuring scientifi cally based change during the time of wood equilibrium moisture content in a function of temperature t and relative humidity φ of the processing medium.In any combination of values t and φ after a certain time the wood reaches a state of stability and thus its moisture content is constant and it either receives, or emits moisture, i.e. it is in equilibrium with the surrounding environment.This moisture is defi ned as wood equilibrium moisture content U EMC .
The instructions for exploitation of the systems for automatic control of the processes of convective drying and other kinds of hydrothermal treatment of wood materials, as a rule consist of empirical tables and/or diagrams for the dependency of U EMC on t and φ.The implementation of modern control of technological processes with the help of programmable controllers or computers allows for the determination of U EMC with the help of software.For this purpose, it is necessary to have a precise mathematical description of U EMC depending on t and φ.
The effective control of U EMC can ensure a signifi cant reduction of the duration and specifi c energy expenses of the processes of hydro-thermal treatment of wood, as well as deviations in the fi nal moisture content in the separated materials, subjected to such treatment.
The aim of the present paper is to analyze the precision of wood sorption models, found in reference literature and most often used, and to provide an argumentative choice of those most suitable for use as mathematical description of U EMC in the systems of model-based and model predictive automatic control of various processes of hydrothermal treatment of wood and wood-composite materials (Deliiski, 2009b;Shubin, 1990;Trebula and Klement, 2002;Videlov, 2003).

MATERIJAL I METODE
In the reference literature we found 10 most often used mathematical models of the sorption behavior of wood, with the help of which U EMC can be calculated depending on temperature and relative humidity of the air.Chronologically, these models are published in the following way: Model 1:  Vidal and Cloutier (2005) make a relative assessment of the precision of the models 1, 2, 3, 4, 5, 6 and 9 in relation to experimental data published in the literature for U EMC at fi ve different values of relative humidity: 40, 52, 65, 75 and 85% for the temperature ranging between 0 o С and 160 o С.The authors determine, and our calculations also prove, that the models 1, 2 and 5 give the least accuracy.Because of this fact, these models are not evaluated below, and the calculations in the present paper at 0% ≤ φ ≤ 100% are limited to assessing the precision of the models 3, 4, 6, 7, 8, 9 and 10 within the range 0 o С ≤ t ≤ 100 o С and of the model 3, 4, 6, 7 and 9 within the range 100 o С ≤ t ≤ 200 o С.The models 8 and 10 represent regression equations, which indicate the change in U EMC depending on t and φ only within the range from 0 o С to 100 o С and are not applicable in the range100 o С ≤ t ≤ 200 o С (Deliiski et al, 2009a).
In the equations of the models below φ is labeled as the relative vapor pressure, which is the result of division by 100 so as to get the relative air humidity expressed in percentages.In the models 3, 4 and 6 thermo-dynamical temperature Т (in K) is taken into account and in the models 7, 8, 9 and 10 -the temperature t (in o С).
The calculated values of wood equilibrium moisture content U EMC of the models 3, 6, 7, 8 and 10 are expressed with dimension percentages, and in the models 5 (Malmquist, 1958) and 9 (Garsía, 2002) U EMC is expressed in kg•kg -1 and in order to express this in percentage terms, the obtained results have to be multiplied by 100.
The models 3, 4, 6, 7, 8, 9 and 10 are presented through the following equations: Model 3: Hailwood and Horrobin (1946) -two hydrate model where 1800 is the molecular weight of water x 100, g•mol -1 ; M p -molecular weight of a polymer unit that forms a hydrate, g•mol -1 .As a result of the parameterization procedure Vidal and Cloutier (2005) deduct the following equations for the calculation of the coeffi cients on the right side of equation (1): (2) where, according to the author K = 2.2885 -0.0016742T + 2.0637.10 - T 2 .( 7) K 1 = 0.40221 -0.00009736T -5.8964.10 - T 2 , ( 8) Model 6: Day and Nelson (1965 where K 1 , K 2 , K 3 , K 4 are constants, for which Avramidis (1989) has determined the following values: Model 8: Simpson (1991) where, according to the author: K 1 = 0.805 + 7.36.10 -4 t -2.73.10 -6 t 2 , ( 14) Model 9: Garsía (2002) where, according to the author: In 2007 Ray et al. (2007) published their results of research on the precision of the calculation of U EMC with the help of the Simpson model, i.e. using the equations ( 12) ÷ (16).They came to the conclusion that the average square error, when determining U EMC by these equations in the range 0 o С ≤ t ≤ 44 o С, is equal to ±0.5%, but in the range 44 o С < t ≤ 100 o С it increases to ±1.5%.
This inspired the authors to propose their regression equation for the computation of U EMC depending on φ and t in the range 44 o С < t ≤ 100 o С. Ray (19) where: The authors (Ray et al, 2007) prove that the error of the results obtained from the equations ( 19) ÷ (24), for the ranges 0% ≤ φ ≤ 100% and 44 o С < t ≤ 100 o С, is smaller by 44% than the error of the results obtained from the equations ( 12) ÷ (16).

REZULTATI I RASPRAVA
For the solution of the models 3, 4, 6, 7, 8, 9 and 10, which include the equations (1) ÷ (24), we created a program in the computing environment of VISUAL FORTRAN PROFESSIONAL supported by Windows.
With the help of the program, we calculated the values of U EMC when t changes from 0 o С to 200 o С in steps of 0.1 o С and φ from 0% to 100% in steps of 0.1%.The results are compared with the corresponding experimental data from the literature (FPL 1999) and (Kubojima et al, 2003) related to the change of U EMC depending on t and φ (Table 1, 2 and 3).
During the analysis of the results obtained, we did not take into consideration the sorption hysteresis, possible variations of the sorption isotherms due to different species of wood used for the determination of the sorption isotherms taken from the literature, and variations due to internal structures such as heartwood and sapwood (Ball et al, 2001;Pervan, 2000).
As the experimental data in FPL (1999) are temperatures expressed in degrees Fahrenheit, in the very fi rst column of both Table 1 and 2, the values of t are given in degrees Celsius, which were assigned to t during the experiments.Since the experiments are conducted using different values of φ, not always corresponding to φ values in Table 1, 2 and 3, in the very last column of the tables the exact experimental values of U EMC (when the experimental and computed values of φ coincide) are not marked with an asterix sign (*), while the interpolated experimental values of U EMC (when the experimental and computed values of φ do not coincide) are shown with an asterix sign (*).The computed values of U EMC according to the models 3, 4, 6, 7, 8, 9 and 10 and their corresponding experimental values from FPL (1999) are given in Table 1 (for the range 0% ≤ φ ≤ 50%) and in Table 2 (for the range 60% ≤ φ ≤ 94%).
The analysis of the data from Table 1 and 2, and also of the others not given in these tables, shows that the experimentally established change in U EMC depending on φ is described most accurately by the Simpson model within the range 0 o С ≤ t ≤ 50 o С and by the model of Ray et al. within the range 50 o С < t ≤ 100 o С.The absolute error of U EMC , which is obtained from these models in the given temperature ranges, is within the limits of ±0.4% at 0% ≤ φ ≤ 50% and ±0.7% at 60% ≤ φ φ ≤ 94%. Figure 1 shows the isotherms of U EMC derived using these two models when t = 0, 20, 40, 60, 80 and 100 o С with the change of φ from 0 to 100%.
The next most accurate model is that of Garsía with the absolute error within the limits of ± 0.7% at 0% ≤ φ ≤ 50% and 0.9% at 60% ≤ φ ≤ 94%.The models of Hailwood and Horrobin-2 and of Malmquist give very close results with the absolute error within the limits of ±0.7% at 0% ≤ φ ≤ 50% and ±1.2% at 60% ≤ φ ≤ 94%.The biggest inaccuracy is observed in the models of Day & Nelson and of Kaplan -they give the results of U EMC higher than the experimental ones within the limits of 3.3%.
It should be noted that all of the examined models refl ect very well the complicated character of change in U EMC depending on t and φ as shown in Fig. 1.The indicated limits of change of the absolute error in determining U EMC based on all the models refer to the relatively high values of φ.With smaller values of φ, the absolute error, as a rule, signifi cantly decreases.Only for Kaplan's model the opposite dependency can be observed -the more the value of φ decreases, the more the absolute error increases and reaches +3.3% at t = 0 o С and φ = 0%.The values of U EMC calculated using the models 3, 4, 6, 7 and 9 and their corresponding experimental values from FPL (1999) and Kubojima et al. (2003) are given in Table 3 within the range 100 o С ≤ t ≤ 150 o С and 10% ≤ φ ≤ 85%.
The comparison of the calculated and experimental results, partly presented in Table 3, shows that the experimentally established change of U EMC depending on t and φ is most accurately described by the Garsía model.Fig. 3 shows the isotherms of U EMC built using this model when t = 100, 120, 140, 160, 180 and 200 o С with the change of φ ranging from 0% to 100%.In the contemporary model-based and model predictive systems for automatic control of high temperature processes for wood hydrothermal treatment (e.g.veneer drying), it is required to compute continuously the set values of U EMC in the temperature range from 0 o С to 200 o С.For ensuring this requirement, the evaluation of the validity of the models has been extrapolated to 200 o С in Fig. 3.
The comparison of the results calculated based on the Garsía model at t = 100 o С as shown in Table 3, with the precisely analogous results in Fig. 2    on the Garsía model are higher than their corresponding values of U EMC in all the examined values of t ≥ 100 o С.
In order to increase the precision of the Garsía model and for a better qualitative and quantitative coordination of the calculated values of U EMC when t ≥ 100 o С with the values of U EMC based on the Ray et al. model within the range 50 o С < t ≤ 100 o С, we suggest adding a power coeffi cient of 1.33 to the denominator on the right side of the equation (17).Then the equation (17) becomes: .......Deliiski: Evaluation of Wood Sorption Models and Creation of Precision Diagrams...The column before the last one in Table 3 presents the obtained results of the change of U EMC based on the so-called Garsía-Deliiski model, which consists of both equations (25) and (18).The comparison of these results with the experimental data, presented on the right in Table 3, shows a signifi cant reduction of the absolute errors of U EMC when U EMC is calculated according to the absolute errors obtained by Garsía model (17) and (18).

ZAKLJUČCI
The present paper describes the evaluation of the precision of the 10 most often used wood sorption models, available in the literature, for the calculation of wood equilibrium moisture content U EMC given a change in temperature within the range from 0 o С to 200 o С and in relative humidity φ of the surrounding air environment from 0% to 100%.The calculated results were compared to corresponding precise experimental data from the literature.(Hadjiyski, 2003) of different processes of hydrothermal treatment of wood and wood materials.This way, for example, we have input the Sim pson model into the software of the microprocessor programmable controller in order to control the temperature conditioning process of dried lumber (Fig. 6).
The long use of the implemented automated installation in the conditioning storage house (Deliiski, 2009b) confi rmed completely the validity of the calculating and controlling algorithm used in the controller.It proved its high energy effi ciency, reliable functioning and suitability to assure the temperature-humidity parameters of the air, corresponding completely to the U EMC of the wood, required by the user.

Figure 2
Figure 2 shows a precise diagram of the change of U EMC depending on φ and t within the range 0% ≤ φ ≤ 100% and 0 o С < t ≤ 100 o С.The diagram curves are built based on the results obtained by the Simpson model at 0 o С ≤ t ≤ 50 o С, and with the help of Ray et al. model when 50 o С < t ≤ 100 o С.This diagram shows the experimentally established relations of U EMC depending on t and φ with a higher precision in comparison with analogous diagrams, usually referred to in the literature(Shubin, 1990;Trebula and Klement, 2002;Videlov, 2003).

6 Figure 3
Figure 3 Change in U EMC depending on φ and t, calculated based on Garsía model Slika 3. Promjene ravnotežnog sadržaja vode u ovisnosti o temperaturi i vlažnosti zraka, izračunane na temelju Garsíjeva modela za temperature 0 o С ≤ t ≤ 50 o С having the same temperature obtained based on the Ray et al. model, show that the Garsía model gives higher values of U EMC within the whole range of change of φ.

Fig. 4
shows isotherms of the change of U EMC , with full lines, obtained based on the Garsía model, and their analogues, calculated based on the Garsía-Deliiski model, shown with dotted lines.In the present paper the proposed Garsía-Deliiski model, which con-sists of equation (25) and (18), precisely refl ects qualitatively and quantitatively the relation of U EMC depending on φ and t within the range from 100 o С to 200 o С.A future clarifi cation of this model should be made when having extensive experimental data for the change in U EMC depending on t and φ within this temperature range.Fig. 5 shows for the fi rst time the summarized diagram of the change in U EMC depending on φ and t within the range 0% ≤ φ ≤ 100% and 0 o С < t ≤ 200 o С.The curves are built based on the results obtained by the Simpson model at temperatures 0 o С ≤ t ≤ 50 o С, by the Ray et al. model at 50 o С < t ≤ 100 o С and by the Garsía-Deliiski model at 100 o С < t ≤ 200 o С.

Figure 4 Figure 5 5 .
Figure 4 Change in U EMC depending on both φ and t, calculated using the Garsía model -full lines, and using the Garsía-Deliiski model -dotted lines Slika 4. Promjene ravnotežnog sadržaja vode u ovisnosti o temperaturi i vlažnosti zraka, izračunane na temelju Garsíjeva modela (pune linije) i Garsía-Deliiskijeva modela (iscrtkane linije) et al (2007) present the equation for the determination of U EMC depending on φ and on temperature t expressed in degrees Fahrenheit.After substituting t in the known relation from degrees Fahrenheit to degrees Celsius, namely: t[°F] = 1,8t [°C]+32 , we obtained the following equation for determining U EMC for the range 0% ≤ φ ≤ 100% and 44 o С < t ≤ 100 o С:

Table 3
Change of the calculated values and their corresponding experimentally established values of U EMC depending on t and φ within the range 100 o С ≤ t ≤ 150 o С and 0% ≤ φ ≤ 85% Tablica 3. Razlike između izračunane i odgovarajuće eksperimentalno dobivene vrijednosti ravnotežnog sadržaja vode u ovisnosti o temperaturi i vlažnosti zraka u rasponu 100 o С ≤ t ≤ 150 o С i 0% ≤ φ ≤ 85% The obtained results show that the Simpson model gave the best fi t to experimental data for the range 0 o С ≤ t ≤ 50 o С.The best precision within the range 50 o С < t ≤ 100 o С was provided by the Ray et al. model and within the range 100 o С < t ≤ 150 o С by the Garsía model.In order to increase the precision of the Garsía model and for better qualitative and quantitative coordination of the calculated values, with the help of U EMC at t ≥ 100 o С, with values of U EMC based on the Ray et al. model within the range 50 o С < t ≤ 100 o С, we suggest the clarifi cation of the Garsía model.The clarifi cation means introduction of a power coeffi cient, equaling 1.33, to the denominator on the right side of the equation of the Garsía model.With the results calculated from the Simpson and Ray et al. models, a diagram has been built for the change in U EMC depending on φ and t within the ranges 0% ≤ φ ≤ 100% and 0 o С ≤ t ≤ 100 o С.This diagram refl ects the experimentally established dependency of U EMC on φ and t with better precision in comparison to analogous diagrams, usually found in the literature.Using the results obtained by both the Simpson and Ray et al. models and the Garsía model, which we have refi ned, a summary diagram of the change in U EMC , depending on φ and t within the range 0% ≤ φ ≤ 100% and 0 o С ≤ t ≤ 200 o С, has been created for the fi rst time.This diagram can be used for the precise determination of U EMC when having different temperature-humidity impacts on the wood.The established high precision of both the Simpson and Ray et al. models and the Garsía model, which we have refi ned, makes them user friendly for contemporary systems for model-based and model predictive automatic control 5. LITERATURA 1. Avramidis, S., 1989: Evaluation of "three-variable" models for the prediction of equilibrium moisture content wood.Wood Sci.Technol.23: 251-258, http://dx.doi.org/10.1007/BF00367738 2. Ball, R.D.; Simpson, G.; Pang, S., 2001: Measurement, modelling and prediction of equilibrium moisture content in Pinus radiata heartwood and sapwood.Holz als Rohund Werkstoff 59(6): 457-462, http://dx.doi.org/10.1007/s001070100242 EMCravnotežni sadržaj vode U EMC , %