Ind. Eng. Chem. Res. 1991,30,1833-1836
1833
MATERIALS AND INTERFACES Moisture Desorption in Cellulosic Materials Sheng H.Lin Department of Chemical Engineering, Yuan Ze Institute of Technology, Neili, Taoyuan 320, Taiwan, Republic of China
A nonisothermal diffusion model is given in this paper for describing the moisture desorption in cellulosic materials. The physical model represents an extension of the isothermal one that simulates moisture absorption of cellulosic materials from exposure to moist air. A heat-transfer equation is needed in the moisture desorption model for the determination of the temperature change in the cellulosic materials during the drying period. The simultaneous heat- and mass-transfer process is simulated numerically for various operating conditions. The model predictions have been found to agree well with the observed data for the drying of Kraft paper.
Introduction In a previous paper (Lin,1990), an isothermal diffusion model was presented for describing moisture absorption in cellulosic materials. The nonlinear moisture-dependent model was found to predict well the experimental data of moisture absorption in Kraft paper. In the present paper, the isothermal diffusion model is extended to the case of moisture desorption in which the moisture moves in the opposite direction of the previous process. The moisture desorption, more generally known as the drying process, differs from the previous one in that the temperature in the cellulosic materials is no longer constant, but a function of time and distance from the surface. The temporal and spatial variations of the medium temperature strongly affect the diffusion coefficient and the moisture diffusion rate. Hence the temperature history during the drying period is required in order to adequately model the nonisothermal process. A heat-transfer model is developed in the present paper along with the nonlinear moisture diffusion equation for describing the simultaneous heatand mass-transfer phenomena. Moisture Diffusion Coefficient The isothermal diffusion coefficient was given previously as an exponential function of moisture concentration: D(c) = Do exp(kc) (1) where Do is the base diffusion coefficient and k the constant exponential parameter (0.5 for moisture diffusion in Kraft paper). For the nonisothermal case, the Arrhenius type of temperature dependence needs to be incorporated into the above equation: D(c,T) = Do exp(kc) exp
I);
[E-d( -io -
(2)
where E, represents the activation energy, R the gas constant, and Tothe base temperature (298 K). The parameters Do and k remain the same as those in eq 1 and have been given in the previous paper (Lin, 1990). Figure 1 shows the Arrhenius plots of D vs 1/T for the oil-free and oil-soaked Kraft papers for moisture content up to 15% dry weight (Ast, 1966). The slopes of the two straight lines
(i.e,, the activation energies) in this figure are in fact slightly different. They are 16 100 and 16OOO cal/g-mol, respectively, for the oil-free and oil-soaked Kraft papers. However, for practical purposes, an average value of 16050 cal/g-mol would be adequate and is adopted in the present study.
Nonisothermal Moisture Diffusion Model The moisture diffusion equation remains the same as the one given in the previous paper (Lin, 1990): - = a D(c,T) ac (3) at ax except that, for the present case, the diffusion coefficient D is a function of both the moisture concentration (c) and the temperature (T), as given by eq 2. In many practical circumstances, the cellulosic materials do not have an uniform initial moisture concentration at the beginning of the drying process, but assume a certain moisture profile. Usually, this moisture concentration profile is established by the absorption process. In other words, the drying process starts where the absorption process ends. To take this into consideration, the initial and boundary conditions for eq 3 can be represented by t = 0; c = co(x) (44 x=o; c=c, (4b) x = L; ac/ax = o (44 Equation 4a represents the condition of initial moisture concentration profile established at the end of the absorption process. Hence the moisture absorption model given in the previous paper (Lin, 1990) needs to be integrated first for a given operating condition and absorption time to establish this initial moisture concentration profile. It is noted that the surface moisture concentration (c,) in eq 4b is represented by the following equation (Lin, 1990) c, = (8.086 X 10-7)p0.6722 exp(4212/T) (5a)
-[
E]
p = 6.36 X lo8 exp(-5090/T)(%RH/100)
(5b) It is further noted that the assumption of an equilibrium moisture concentration at the surface in eq 4b does not
0888-5885/91/ 263O-1833$O2.5O/O 0 1991 American Chemical Society
1834 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
I
\
IO-'} Oil-Soaked
er
10-
27
,
,
,
29
31
33 VT,
' 35
!lo-.
],o-lo
37
I O - ~ O K - '
Figure 1. Arrhenius plot of oil-free and oil-soaked Kraft papers.
necessarily exclude the existence of the external mass transfer resistance outside the cellulosic materials. It simply implies that the external mass transfer resistance is significantlysmaller than the internal moisture diffusion resistance and hence an equilibrium moisture concentration is established. In fact, under the experimental conditions of Ast (1966) and Ewart (1966),the external mass transfer resistance was expected to be small in the light of relatively well mixed air exterior to the Kraft papers undergoing absorption or desorption. To predict the temperature progression in the cellulosic material, a heat-transfer model is required. A one-dimensional partial differential equation similar to eq 3 could be written for the temperature prediction. Instead, the following simplified macroscopic version of energy balance equation is adopted here:
subject to t=0, T=T, (64 where p and Md are the density and dry weight of cellulosic material, C, is the heat capacity, h is the external heat transfer coefficient, T, is the constant air temperature, h, is the heat of vaporization, A is the exposure area, and To is the initial temperature. The main reason for adopting eq 6 is due to the fact that the thermal response of cellulosic material is much faster than the moisture diffusion for the majority of practical cases (Keey, 1972). Furthermore, the moisture penetration distance in the cellulosic materials is relatively short, as noted previously (Lin, 1990), leading to relatively small temperature variation within the cellulosic layer of interest. This prompts other researchers (Mujumdar, 1982) to adopt a similar heattransfer model for the temperature prediction in foods during air dehydration and in other drying processes. An important parameter that characterizes the thermal response of the cellulosic materials is the time constant, t,, of the drying system. This constnat, as represented by (pLC ) / h and obtained from eq 6, is defined as the time for tke cellulosic material to reach 63% of the new steady-state temperature from its initial temperature. This parameter varies significantly from one drying system to the other and depends strongly on the size and type of cellulosic materials to be dried. It can be as short as a few
minutes and as long as several hours. However, one characteristic common to most drying systems is that the time constant, regardless of how long or short it is, represents only a small fraction of the total drying time. For instnace, in the drying of cellulosic materials in the large transformers, the time constant is approximately 4 h in comparison with several days or over a week of totaldrying time. In the food-drying processes, the time constant is usually a few minutes in comparison with hour(s) of total drying time. For the drying of Kraft paper of a few centimeters or less in thickness (Ewart, 1966),as the example illustrated in this paper, the time constant is no more than 10 min. This is another reason that the simplified heattransfer model is deemed adequate for the temperature prediction of cellulosic material during the drying period. The present nonisothermal model thus consists of the moisture diffusion equation, eq 3, subject to eqs 4a-c and the heat-transfer equation, eq 6, subject to eq 7. The isothermal moisture absorption problem as described in the previous paper (Lin, 1990) is integrated first to establish the initial moisture concentration profile. Equations 3 and 6 are then integrated simultaneously by the orthogonal collocation method (Finlayson, 1980; Lin, 1990) to establish the temperature and the moisture concentration profiles. To evaluate how much error was introduced by neglecting the temperature gradient in the cellulosic materials in eq 6, a complete conduction heat transfer model is needed. This model is represented by aT a2T pC -=k-+hv-ax, p at
Md ac AL at
(7)
subject to t=0; T=To
(74
x = 0;
in which k is the thermal conductivity of the Kraft paper, h the heat-transfer coefficient, and Tdthe constant drying temperature. For wet papers, the thermal conductivitywas estimated to be approximately 0.003 cal/(cmwK) and the heat-transfer coefficient was estimated to be 0.0047 cal/ (cm2-s-K)(Kreith, 1973). With the use of those constants and other known properties of the Kraft papers (Ast, 1966; Ewart, 1966), eq 7 was numerically solved by the implicit Crank-Nicolson finite difference method for the timedependent temperature distributions inside the paper which were then employed for calculating the average paper temperatures. Figure 2 shows the computed average paper temperature vs time. It is apparent that the time for the average paper temperature to approach the drying temperature is indeed relatively short in comparison with several days of total drying time, amply justifying the adoption of eq 6 for representing the heat-transfer process. Kreith (1973) also has shown that if the Biot number, h L / k , is equal to 0.1 or less, the simplified equation is adequate. For the present case, the Biot number is found to be 0.4, which is deemed sufficientlyclose to the criterion of Kreith. Discussion of Results Two sets of experimental data of Ewart (1966) are shown in Figures 3 and 4 along with the model predictions. Both experimental measurements were conducted for drying of oil-soaked Kraft paper under different operating absorp-
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1835
6 L 'f
32% RH 28OC Air Temperature
.-
i 05
I
4 . 5 day exposure
i
2ol 0
Observed (€wart) Predicted
0
-
1.0
20
1.5
Time, hr
Figure 2. Average paper temperature vs drying time.
Distance from Surface, 0.01 cm
Figure 4. Comparison of predicted absorption and adsorption moisture concentration profiles in oil-soaked Kraft paper with observed data of Ewart (1966).
6v 50% RH 22OC Air Temperature 80% Drying Temperature
3 7 % RH 26OC Air Temperature
-
8 day exposure
e,
a
-
2.5 day drying
0
5
IO
15
20
25
-
30
Distance from Surface, 0.01cm
Figure 3. Comparison of predicted absorption and desorption moisture concentration profiles in oil-soaked Kraft paper with observed data of Ewart (1966).
tion and desorption conditions. The Kraft papers were first exposed to moist air at a controlled relative humiduty for a given period. Those papers were then removed to the drying chamber, which was maintained at a constant temperature (80 "C). Small paper samples were taken at various drying times and the water contents determined. It is apparent that the agreement between the model predictions and the observed data is very good, considering the complexity of the drying process and the numerous assumptions involved in the physical model. This attesta the adequacy of the present nonisothermal diffusion model for describing the drying process. Figure 5 shows the results of a typical run of the simulation model. In this case, the oil-soaked Kraft paper is exposed to the air at 22 OC air temperature and 50% relative humidity for 10 days before the drying process starts. The drying is carried out at a constant external air temperature of 80 O C . The moisture concentration profiles at various drying times are shown to illustrate the moisture penetration during the drying period. It is of interest to note that while the peak moisture concentration in the Kraft paper decreases as the drying progresses, the moisture penetrates deeper into the paper. This is not surprising because of the existence of an internal moisture gradient. Hence, during the drying period, the moisture diffusion actually goes both ways.
I
0
I
2
3
4
5
6
Dl8tance from Surface, cm
Figure 5. Moisture migration in oil-soaked Kraft paper during drying period.
One important industrial application of the drying model is the prediction of drying time required to dry the cellulosic material. There are generally two criteria on which such a prediction is based. The first is that the maximum moisture concentration inside the cellulosic material is below a certain level. The second is that the drying rate is sufficiently low. Although both criteria are practiced in industries, the first one is deemed to be more reliable because of the difficulty in gathering accurate drying rate data. Hence the first criterion is adopted here in the determination of the drying time, as shown in Figure 6. For the drying example of oil-soaked Kraft paper in large transformers illustrated here, the acceptable maximum moisture concentration is 1.5%, which is chosen to ensure the electrical integrity of the transformers in high-voltage operation. The drying model can be integrated to determine the time when the maximum moisture concentration starts to fall below this level. Figure 6 shows the drying time vs the drying temperature for various exposure times of initial moisture absorption. It is of interest to note that, for a drying temperature above 100 O C , the initial absorption time does not appear to be a factor any more in determining the drying time. However, at a lower drying temperature, the initial exposure time becomes highly
1836 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1
601
30
50
70
Drying Temperature,
90
110
OC
Figure 6. Drying time of oil-soaked Kraft paper aa a function of drying temperature and initial exposure duration.
critical. For a drying temperature lower than 60 or 50 "C, there appears to be an exponential increase in the drying time, implying the low efficiency of drying at low air temperature. With the use of the drying model, the effects of other factors, like the initial absorption temperature and relative humidity, on the drying time can be explored also. It should be noted that the one-dimensional drying model presented in this paper is valid only if the dimension of cellulosic materials in the moisture diffusion direction is significantly smaller than those in two other directions, like the drying of thin Kraft papers considered here. Otherwise, a two or even three-dimensional physical model may have to be adopted. However, the basic model will be similar to the one presented here except that extra differential terms need to be added in the moisture diffusion equation. The resulting differential equations, although slightly more complicated, can be tackled in a similar fashion by the orthogonal collocation method. Conclusions A nonisothermal moisture desorption model is given in the present paper. The physical model consists of a moisture diffusion equation and a simplified heat-transfer equation. The model is employed to simulate the drying process starting with a known moisture profile in the cellulosic material established by the initial absorption process. The model predictions agree reasonably well with the experimental observations for the drying of oil-soaked
Kraft paper. The physical model can be employed for predicting the drying time of cellulosic material under various operating conditions. Nomenclature A = drying exposure area c = moisture concentration c, = equilibrium moisture concentration at the surface c,, = initial moisture concentration in the cellulosic material C, = specific heat capacity D = diffusion coefficient Do = base diffusion coefficient E , = activation energy h = external heat transfer coefficient h, = heat of vaporization k = constant parameter in the moisture-dependent function of diffusion coefficient L = thickness of cellulosic material Md = dry weight of cellulosic material p = vapor pressure R = gas constant RH = relative humidity t = time T = temperature T , = constant air temperature Td = drying temperature x = rectangular coordinate Greek Letters density of cellulosic material
p =
Registry No. Cellulose, 9004-34-6; water, 7732-18-5.
Literature Cited Ast, P. F. Test Report HV-ER-66-41; General Electric Company, Pittsfield, MA, 1966. Ewart, D. E. Test Report 60PT44; General Electric Company, Pittsfield, MA, 1966. Finlayson, B. A. Nonlinear Analyses in Chemical Engineering; McGraw-Hill: New York, 1980. Keey, R. B. Drying Principles and Practice; Pergamon Press: Oxford, England, i972. Kreith, F. Principles of Heat Transfer; International Textbook: Scranton, PA, 1973. Lin, S. H. Moisture Absorption in Cellulosic Materials. Int. J. Eng. Sci. 1990, 28, 1151. Mujumdar, A. S. Aduances in Drying; Hemisphere: Washington, DC, 1982.
Receioed for review January 17, 1991 Accepted April 15, 1991
