The rate at which wood dries can be expressed by two factors:(1) the diffusion coefficient, as an indication of internal resistance to moisture removal, and (2) the surface emission coefficient, as an indication of external resistance to moisture removal. An optimization technique was applied to calculate these two coefficients. Unlike existing methods where these coefficients are evaluated at only one point, usually at one-half of the evaporable water remaining in wood, this optimization technique used the entire drying data. The technique searches for the optimum pair of coefficients based on the least squares principle. The performance of this method was demonstrated by reanalyzing published data using a digital computer. The results indicate that a more accurate prediction of the drying process can be achieved by the optimization method than by existing methods. The dependence of the diffusion coefficient on moisture content and sample thickness is discussed

Chen, Yong

Choong, Elvin T.

Wetzel, David M.

English

Wood and Fiber Science (E-Journal)

EVALUATION OF DIFFUSION COEFFICIENT AND SURFACE EMISSION COEFFICIENT BY AN OPTIMIZATION TECHNIQUE ' Yong Chen Research Assistant Elvin T. Choong Professor Louisiana Forest Products Laboratory Louisiana State University Agricultural Center Baton Rouge, LA 70803 and David M. Wetzel Associate Professor Department of Chemical Engineering Louisiana State University Baton Rouge, LA 70803 (Received February 1994) ABSTRACT The rate at which wood dries can be expressed by two factors: (1) the diffusion coefficient, as an indication of internal resistance to moisture removal, and (2) the surface emission coefficient, as an indication of external resistance to moisture removal. An optimization technique was applied to calculate these two coefficients. Unlike existing methods where these coefficients are evaluated at only one point, usually at one-half of the evaporable water remaining in wood, this optimization technique used the entire drying data. The technique searches for the optimum pair of coefficients based on the least squares principle. The performance of this method was demonstrated by reanalyzing published data using a digtal computer. The results indicate that a more accurate prediction of the drying process can be achieved by the optimization method than by existing methods. The dependence of the diffusion coefficient on moisture content and sample thickness is discussed. Keywords: Diffusion coefficient, surface emission coefficient, optimization, conjugate gradient. INTRODUCTION The diffusion coefficient in wood determines the rate of internal moisture movement, while the surface emission coefficient describes the rate at which moisture is emitted from the surface of drying wood. Methods to evaluate I This paper (No. 93-22-7087) is published with the approval of the Director of the Louisiana Agricultural Ex- periment Station. The research was supported in part by the McIntire-Stennis Cooperative Forest Research Pro- gram. these two coefficients in wood drying practice have been studied extensively since Skaar (1 954) introduced Newman's general solution (Newman 193 1) ofthe unsteady-state diffusion equation into the field of wood drying. Based on a graphical solution of Newman's equation, choong and Skaar (1969) suggested an ap- proximate method to evaluate the diffusion coefficient, D, and the surface emission coef- ficient, S. According to this method, D and S can be calculated if the drying times for two samples with different thicknesses are known Wood and Fiber Sclence, 27(2), 1995, pp. 178-182 O 1995 by the Society of Wood Science and Technology Chen el al. -DIFFUSION AND SURFACE EMISSION COEFFICIENTS 179 for reduction of the fractional moisture con- tent (MC) E in wood to 0.5. Later, Choong and Skaar (1972) developed a more general form to enable D and S to be calculated from si- multaneous drying experiments on samples of several different thicknesses. Using the same analytical procedure, Rosen (1 978) proved that air velocity had a significant influence on the surface emission coefficient related to the ex- ternal resistance in the sorption process. He found that the relative importance of external resistance to moisture movement in wood in- creased as specimen thickness and/or air ve- locity decreased. Liu (1 989) described a mod- ified method based on Choong and Skaar's (1969) approach to estimate D and S from a single lumber-drying curve. His method re- quires knowledge of both the drying time and the slope of the fractional moisture content- drying time curve at E = 0.5. There are two common characteristics for the above approximate methods. First, the cal- culations are carried out at one point, = 0.5. Second, the resulting diffusion coefficient and surface emission coefficient are assumed to be constant, independent of E. If the final aim is to predict accurately the experimental data over the entire drying period, the pair of coefficients evaluated at E = 0.5 may not be the ideal ones since the diffusion coefficient is generally not constant but a function of wood MC (Com- stock 1963; Stamm 1964; Choong 1965). There were no published data available to describe how well the pair evaluated at E = 0.5 com- pared with those evaluated at other values of E. The objectives of this study were: 1) to search the best pair of diffusion coefficients and sur- face emission coefficients by using an opti- mization technique based on the least squares principle and Newman's general theoretical solution, 2) to evaluate how the variation of diffusion coefficients with MC affects the pre- dictability of the drying curve. BASIC EQUATIONS AND OPTIMIZATION PROCEDURE The general partial differential equation for unsteady-state moisture diffusion in the direc- tion perpendicular to the surface can be ex- pressed by Fick's second law: where C is moisture concentration (or MC), t time, X distance from the centerline, and D the diffusion coefficient, which is a function of moisture concentration. Initial and boundary conditions for mois- ture evaporation at the surface are given by C = C,, -a < X < a, t = 0 (2) and where C, is the moisture concentration in equi- librium with the water vapor pressure in the surrounding air, C, the actual moisture con- centration in the surface at any time, a the half- thickness of the specimen, and S the surface emission coefficient. If D is a constant, a theoretical solution for Eq. (I), as given by Newman (1 93 l), is: where is the average value of C across the specimen thickness, Pn are the positive roots of where and Since the diffusion coefficient in wood during drying is generally a function of the moisture concentration, the D calculated from Eq. (4) represents some average value over the con- centration range. In the present work, we take the alternate approach of searching for an optimum pair of constants, D and S, which minimize the sum 180 WOOD AND FIBER SCIENCE, APRIL 1995, V. 27(2) 0 5 0 100 150 DRYING TIME (Hrs )  FIG. 1 .  Drying curves from the optimization technique and the approximate method. (Data from Choong and Skaar 1969.) of squares of the differences between experi- mental data and Eq. (4). To fulfill this task, an ascent optimization algorithm was applied. The search can begin from any reasonable pair of D and S values. After each iteration, a new D and S pair with a smaller sum of squares is obtained. Finally, the search stops after the reduction in the sum of squares from one it- eration to the next is less than the given closure criterion. A FORTRAN program was written to accommodate the calculation. In this pro- gram, a bisection technique is employed to guarantee accurately finding all positive roots in Eq. (5) for any L. A conjugate gradient tech- nique is used to increase the climbing speed toward the minimum absolute value of Neg- ative Mean Sum of Squares (NMSS), which is defined as -2 (AE)~/N, where AE is the dif- ference in the fractional moisture content be- tween the fitted curve and the experimental data and N is the number of data points. When the optimum pair of D and S has been found, the corresponding drying curve can be com- puted from Eq. (4). NUMERICAL EXAMPLE AND DISCUSSION Choong and Skaar (1969) evaluated the D and S at E = 0.5 using their approximate meth- od for yellow-poplar sapwood and heartwood specimens, dried in both the tangential and the radial directions. Two samples of different thicknesses were dried for each combination in an environmental chamber at 32 k 0.2 C and 40% relative humidity at an air speed of 3.3 m/sec. Their results for sapwood in the tangential direction were D = 7.5 x 10-I0 m2/ sec and S = 1.52 x mlsec. Similar results were obtained by Liu (1 989) using one sample and the slope technique. The data for sapwood in the tangential direction were chosen for the evaluation of D and S in the present study. The predicted drying curves (dashed lines) from the approximate D and S above are shown in Fig. 1. Since they are based on data at = 0.5, the approximate techniques fit well at that point, but poorly elsewhere. The calculated drying curves seriously underpredict the dry- ing time necessary to reach lower E values. The application of the optimization technique sig- nificantly enhanced the predictability of the experimental data. The solid E curves (Fig. 1) are those that correspond to the optimum S and D using the entire drying data in accor- dance with minimum NMSS criterion. The ab- solute values of the NMSS by the two approx- imate methods shown in Table 1 (Choong and Skaar 1969; Liu 1989) were more than 3.4- 3.6 times larger than those resulting from the optimization technique for the thick and thin samples, respectively. The relationship of the optimum E curves to the experimental data (Fig. 1) indicates that the diffusion coefficient was a function of MC. For both thick and thin samples, the experi- mental E data are located below the optimum E curve in the higher MC range and above it in the lower MC range, suggesting that the dif- fusion coefficient decreased with a reduction in MC. This is consistent with similar findings by Comstock (1963) and Choong (1965). To confirm this point more quantitatively, an ex- tension of the optimization procedure was car- ried out by calculating a series of D and S pairs over reduced moisture ranges, using the same final MC. As shown in Table 1, the optimum diffusion coefficient tended to decrease grad- ually with a decrease in average MC. For the surface emission coefficient, the thick sample Chen et al. -DIFFUSION AND SURFACE EMISSION COEFFICIENTS 181 TABLE 1. Comparison of Value9 of S, D, and NMSS from the optimization technique with thosefrorn two approximate methods over the reduced moisture ranges for the thick and thin samples. MC range from S D NMSS Method E = l t o l  fmlsec) (m2/sec) data Approximate 1 Approximate 2C Optimization 0.89 0.86 0.82 0.79 0.75 0.73 0.68 0.63 0.60 0.46 0.40 0.37 0.28 0.22 0.18 0.16 0.13 0.10 0.07 Approximate 1 Approximate 2C Optimization Thick sample (half-thickness = 14.31 mm) 1.52N6 7.50N10 1.48N6 7.35Nl0 1.43N6 8.95N10 1.74N6 7.65N10 2.21N6 6.81N10 2.67N6 6.39N10 2.39N6 6.60N10 2.30N6 6.67N10 2.07N6 6.87N10 2.02N6 6.91N10 2.01N6 6.92N10 2.32N6 6.72N10 3.08N6 6.43N10 5.13N6 6.12N10 2.52N5 5.79N10 >0.40 5.62N10 >0.40 5.51N10 >0.40 5.41N10 >0.40 5.33N10 >0.40 5.28N10 >0.40 5.25N10 Thin sample (half-thickness = 4.79 mm) 1.52N6 7.50N10 1.66N6 7.14N10 >0.40 4.61N10 >0.40 4.61N10 >0.40 4.60N10 1.12N4 4.66N10 1.05N4 4.67N10 >0.40 4.57N10 >0.40 4.50N10 >0.40 4.43N10 >0.40 4.34N10 >0.40 4.29N10 >0.40 4.28N10 >0.40 4.28N10 a The notatlon xNy  means x x 10 ' b Values of S, D are from Choong and Skaar (1969). Values of S ,  D are from Llu (1989). and the thin sample seemed to follow some- what different trends. In the thick sample, the definite values of S could be searched out over the relatively high moisture ranges; but they became very large (>0.40 m/sec) when the rel- atively low moisture ranges were reached. In the thin sample, the definite values of S could be searched out only at the medium moisture ranges; and the values of S were very large (<0.40 m/sec) at rather high and low moisture ranges; thus the sample thickness could also be an affecting variable. The diffusion coefficient may depend on the sample thickness. The optimum D for the thin sample is 18% smaller than for the thick sam- ple (Table 1). A similar result was reported by Choong and Skaar (1969, 1972), who ex- plained that the discrepancy in the "apparent" 182 WOOD AND FIBER SCIENCE, APRIL 1995, V. 27(2) diffusion coefficients was due to a change in - , AND C. SKAAR. 1969. Separating internal and - the external resistances of samples of different thicknesses. However, their air speed was 3.3 m/sec, which is greater than the threshold val- ue of 3.0 m/sec (Rosen 1978) above which the external resistance is negligible. Therefore, other factors such as a time-dependent defor- mation of the cellulose-lignin matrix (Hart 1964) and drying stress could contribute to this discrepancy. CONCLUSIONS The optimum values of the diffusion coef- ficient and surface emission coefficient can be determined from a single drying curve using an optimization technique. This technique gives a more accurate prediction of experi- mental data than existing approximate meth- ods. In a typical drying case, the diffusion co- efficient was found to be a function of moisture content as well as sample thickness. REFERENCES -- -external resistance to moisture removal in wood drying. Wood Sci. 1 (4):200-202. , AND - . 1972. Diffusivity and surface emis- sivity in wood drying. Wood Fiber 4(2):80-86. COMSTOCK, G. L. 1963. Moisture diffusion coefficients in wood as calculated from absorption, desorption, and steady-state data. Forest. Prod. J. 13(3):97-103. HART, C. A. 1964. Principles of moisture movement in wood. Forest. Prod. J. 14(5):207-2 14. LIU, J. Y. 1989. A new method for separating diffusion coefficient and surface emission coefficient. Wood Fiber Sci. 21(2):133-141. NEWMAN, A. B. 193 1. The drying of porous solids: Dif- fusion calculations. Trans. Am. Inst. Chem. Eng. 27: 203-216, 310-330. ROSEN, H. N. 1978. The influence of external resistance on moisture adsorption rates in wood. Wood Fiber 1 O(3): 2 18-228. SKAAR, C. 1954. Analysis of methods for determining the coefficient of moisture diffusion in wood. Forest. Prod. J. 4(6):403-4 10. STAMM, A. J. 1964. Wood and cellulose science. Ronald Press, New York, NY. CHOONG, E. T 1965. Diffusion coefficients of softwoods by steady-state and theoretical methods. Forest. Prod. J. 15(11):21-27. 

Evaluation of Diffusion Coefficient and Surface Emission Coefficient by an Optimization Technique

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Evaluation of Diffusion Coefficient and Surface Emission Coefficient by an Optimization Technique

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