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Everett's domain and Mualem's capillary-radii similarity models of hysteresis ... and loop trajectories inside the main sorption hysteretic loop with ...
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188

Ind. Eng. Chem. Res. 2001, 40, 188-194

Comparison of Domain and Similarity Models for Characterizing Moisture Sorption Equilibria of Paper Siddharth G. Chatterjee* Empire State Paper Research Institute, Faculty of Paper Science and Engineering, SUNY College of Environmental Science and Forestry, 1 Forestry Drive, Syracuse, New York 13210

Everett’s domain and Mualem’s capillary-radii similarity models of hysteresis were used to characterize the moisture sorption equilibria of a bleached kraft paperboard of mean basis weight 230 g/m2. The models predicted the experimental equilibrium moisture content (EMC) of the paperboard for the case of desorption scanning (similarity model only), adsorption scanning, spiral, and loop trajectories inside the main sorption hysteretic loop with good precision. Over an RH range of 15-90%, the root-mean-square deviation between the predicted and experimental EMCs was 0.11-0.83% (domain model) and 0.21-0.64% (similarity model). However, the similarity model is easier to use and requires data of only the two boundary isotherms that envelop the main hysteretic loop. The similarity method obviates the need of taking additional labor-intensive and time-consuming scanning curve data that is necessary to develop the moisture distribution function of the domain model and predicts higher-order EMC trajectories that lie inside the main hysteretic loop in a consistent fashion. Introduction The moisture content of paper is a key parameter that affects its mechanical properties. There is a loss in strength of paper products when they are subjected to a changing relative humidity (RH) of the surrounding environment. Recent research suggests that the phenomenon of accelerated creep of paper under cyclic humidity conditions can be understood as resulting from the interaction of tensile loading and moisture gradients in the paper.1 Such transient moisture gradients are believed to result in localized load cycling, leading to accelerated creep. To describe moisture gradients in paper or paperboard under transient humidity conditions, it is necessary to develop a moisture transport model that has two aspects: (1) moisture sorption equilibria and (2) moisture transport characteristics of the paper under consideration.2-4 It is the objective of this work to address the first aspect, specifically, the mathematical characterization of moisture sorption equilibria, which, in the case of paper, is complicated by the phenomenon of hysteresis. The theoretical description of moisture sorption equilibria of paper will allow a more precise prediction of its equilibrium moisture content (EMC) as a function of its RH history and, thus, will also benefit personnel in paper testing laboratories in developing more informed conditioning procedures for their paper samples. In a previous publication, we reported a detailed experimental investigation of the moisture sorption equilibria of a machine-made bleached kraft paperboard (free of fillers and additives) with special attention focused on the interior of the main sorption hysteretic loop.5 Utilizing the domain model of hysteresis developed by Everett,6 a moisture distribution function (MDF) was constructed from experimental boundary adsorption, boundary desorption, and desorption scanning isotherms. The MDF completely characterizes the * Phone: (315) 470-6517. Fax: (315) 470-6945. E-mail: [email protected].

interior of the main hysteretic loop at a given temperature and can, in principle, predict the EMC of a paper sample (based on its oven-dry weight) given its past arbitrary RH history. Predictions of two adsorption scanning isotherms and two spiral EMC-RH trajectories inside the main hysteretic loop were made by using the MDF and were found to be in good agreement with experimental observations. The experimental determination of the MDF of the domain model is time-consuming and laborious, as it requires data from the boundary adsorption and desorption isotherms and a family of either desorption scanning or adsorption scanning isotherms. Under the hypothesis that the domain or moisture distribution function was mathematically self-similar, Mualem7,8 developed two similarity methods (direct and capillaryradii) to describe the hysteresis of the wetting and drying processes in porous media. These methods require information only from the experimental boundary isotherms that envelop the main hysteretic loop and provide analytical expressions for all paths within the main hysteretic loop in a consistent fashion. Recently, Peralta and Bangi9,10 used these methods to model the EMC-RH hysteresis of yellow poplar wood at 30 °C. They found good agreement of the theoretical EMC predictions with the experimental desorption scanning isotherms of yellow poplar wood that Peralta had reported earlier.11 The objectives of the present work, which is a continuation of our earlier work,5 are the following: (a) to present a geometrical interpretation of Mualem’s capillary-radii similarity model; (b) to compare the capillaryradii similarity model of hysteresis,8,10 which is mathematically simpler and easier to use than the directsimilarity approach,7,9 with Everett’s domain model of hysteresis; and (c) to test the models against a variety of experimental EMC-RH trajectories (desorption scanning, adsorption scanning, spiral, and loop) obtained with the bleached kraft paperboard whose sorption equilibria were reported earlier.5

10.1021/ie000437f CCC: $20.00 © 2001 American Chemical Society Published on Web 12/07/2000

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 189

Figure 1. Geometry of Mualem’s capillary-radii similarity model of hysteresis. The hysteretic factor f equals YZ/WZ for path 1 (varies from point to point along path) and QR/KR for path 2 (constant along path).

Capillary-Radii Similarity Model of Hysteresiss Geometrical Interpretation The capillary-radii similarity model of hysteresis developed by Mualem8 considers independent domains in the adsorbent material (conceptually thought of as capillaries) that adsorb and desorb moisture both irreversibly and reversibly. We will present a geometrical interpretation of this model that is easier to visualize than the theoretical development of Mualem8 or Peralta and Bangi.10 Figure 1 is a schematic diagram of the EMC-RH hysteresis of a cellulosic material like paper. The boundary adsorption and desorption curves are denoted by Ma(H) and Md(H), respectively, where M and H stand for the EMC and RH, respectively. C denotes the upper closure point of the main hysteretic loop where the RH is HC (taken as 100% in this work) and the EMC is MC. The lower closure point of the hysteretic loop is O, where the EMC of the paper sample is taken as 0 at 0% RH. Consider a paper sample at point O whose surrounding RH is raised in a quasi-static fashion along the boundary adsorption curve up to point U, where the RH is HU. The EMC of the sample at the reversal point U is MU, which can be obtained from the boundary adsorption isotherm, Ma(H). Subsequently, maintaining equilibrium at all times, the RH is then decreased along path 1 down to point L. The question that is of interest here is whether a simple method can be devised for calculating the moisture content MX at any point X along path 1 given MU, the two humidity levels HU and HX, and the boundary curves Ma(H) and Md(H). The capillary-radii similarity model,8,10 referred to in this work as simply the similarity model, provides one way of doing this. After performing an analysis, it can be shown that the entire theoretical development of Mualem8or Peralta and Bangi10 consists of the following fundamental equation:

MX ) MR - [Ma(HR) - Ma(HX)]

{

}

MC - Md(HM) MC - Ma(HM)

(1)

where MR and HR are the EMC and RH, respectively, at the reversal point R. In the case of path 1, MR ) MU and HR ) HU, i.e., the reversal point R is the point U. HM is the RH of either the reversal point R or the unknown point X, whichever is lower. Thus, for any

desorption (descending) path like path 1, HM ) HX, i.e., point M corresponds to the unknown point X. For any adsorption (ascending) trajectory, HM ) HR. The last bracketed term in eq 1 is called the hysteretic factor, f. Note that f is always less than 1 because of sorption hysteresis. In the absence of hysteresis, f would equal 1. From the definition of HM, for any descending trajectory (e.g., path 1), the hysteretic factor f varies from point to point, whereas for any ascending trajectory (e.g., path 2, to be discussed below), f is constant. It is clear from the above that, given MR (from the boundary adsorption isotherm in the case of path 1), the only data necessary to calculate MX are the boundary curves Ma(H) and Md(H), the humidity levels HR and HX, and the limiting EMC value, MC. Also note that, although eq 1 was presented in the context of Figure 1 where the point U lies on the boundary adsorption curve, it is actually a general equation that relates the EMC of any reversal point R (which can lie anywhere inside the main hysteretic loop) with that of an unknown point X as long as R and X are connected by a path. Next, the paper sample at point L is again brought back to point U by increasing the RH along path 2 (readsorption, see Figure 1). The point L now becomes the new reversal point whose EMC, ML, is already known as discussed above, i.e., by applying eq 1 with X equal to L. The EMC of the paper sample at any unknown point along path 2 can then be calculated by eq 1 once again with MR ) ML. Note that HM in this case is equal to HR () HL). The paper sample at point U is now brought to point P by raising the RH to HP. Because between points U and P, the sample traverses along the boundary adsorption isotherm Ma(H), the reversal point now shifts to the origin O. That is, for all ascending points along the boundary adsorption isotherm, the reversal point has MR ) 0 at HR ) HM ) 0. Substituting these values into eq 1, it is seen that MP ) Ma(HP). Analogously, for desorption along the boundary desorption isotherm Md(H), the reversal point is point C with MR ) MC at HR ) HC in eq 1 and, thus, MX ) Md(HX). Finally, by using eq 1 in conjunction with the rules mentioned above, it is possible to track the EMC of a paper sample given its past arbitrary RH history. Experimental Procedures We will present a variety of experimental EMC-RH trajectories inside the main hysteretic loop of the bleached kraft paperboard of mean basis weight 230 g/m2 whose sorption equilibria was reported earlier.5 The experimental results will be compared against the predictions of Everett’s domain model of hysteresis and Mualem’s similarity model. Because a detailed description of the experimental setup, experimental procedures, data analysis, and experimental errors was already given earlier,5 we offer only brief comments here. The experimental apparatus consisted of a humidity chamber inside which a paperboard sample (10 cm × 10 cm and approximately 0.03 cm thick) can be suspended from a balance connected to a computer. A stream of air of a definite humidity, obtained by mixing wet and dry air in specific proportions, continuously flowed over the sample. A Vaisala probe constantly monitored the RH and temperature of the chamber and sent its signal to the computer that maintained the RH in the chamber at any set level by controlling the wet

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Table 1. Boundary Adsorption and Desorption Isotherms of Bleached Kraft Paperboarda EMC (%) RH (%)

bound. ads.

bound. des.

hysteretic factor, f

15 30 45 60 75 80 90 100

4.84 6.23 7.74 9.67 12.00 12.90 17.23 29.51b

5.04 6.89 8.79 11.20 14.80 16.25 25.11 29.51b

0.992 0.972 0.952 0.923 0.840 0.798 0.358

a 23.8 °C; experimental data.5 text.

b

Estimated as explained in the

and dry airflows. Two sample replicates were used for each experimental EMC trajectory, and except for runs begun from the wet state (boundary desorption isotherm in Table 1 and Tables 3 and 4), all paper samples were preconditioned by holding them successively at RH levels of 50, 90, 15, 90, and 15% for 5 h at each RH level. The equilibrium experiments were begun after this preconditioning step was completed. For the boundary adsorption isotherm (Table 1), desorption scanning isotherms (Table 2), and the spiral trajectory from the boundary adsorption isotherm (Table 5 ), a sample hold time of 5 h was used at each RH level, whereas a sample hold time of 6 h at each RH level was used for the loop trajectory in Table 6. For the boundary desorption isotherm (Table 1), adsorption scanning isotherms (Table 3), and the spiral trajectory from the boundary desorption isotherm (Table 4), desorption was begun from a completely wet paperboard sample. During the main desorption branch of these runs, sample hold times of 15-17 and 15 h were used at RH levels of 90 and 80%, respectively. This was done to ensure that the sample weight attained a constant value (i.e., equilibrium was attained) as the drying was from an initially wet state. For all other RH levels in these trajectories, a hold time of 7 h at each RH level was used. The average temperature of all of the runs reported in this paper was 23.7 °C, with an average standard deviation of 0.3 °C. For any particular trajectory, the average of the standard deviations of the RH from its mean value was generally maintained within 1% RH (in the range of accuracy of the humidity probe). The oven-dry weight of the paperboard sample was determined by drying the sample in a Mark II moisture analyzer (Denver Instrument Company, Arvada, CO) at 105 °C until a constant weight was attained. The EMC of the sample at a particular RH was calculated by subtracting the ovendry weight from the equilibrium moist sample weight (at that RH level) and dividing this difference by the oven-dry weight. All sample weights were measured correct to a milligram. The boundary adsorption and desorption isotherm data of the bleached kraft paperboard are shown in Table 1. Because we were unable to reach 100% RH in our humidity chamber, the limiting EMC value at 100% RH, MC, was estimated by extrapolating the boundary adsorption and desorption data (fitted with the GAB isotherm12) to an RH of 100% and taking the average of the two EMC values obtained. For the data in Table 1, the value of MC was found to be 29.51%. The influence of the value of MC on calculations with the similarity model will be discussed later in the paper. Note that

Figure 2. Different EMC trajectories inside the main hysteretic loop.

the lowest RH level at which data were taken was 15%, where the hysteretic effect is minimal (f ) 0.992 in Table 1). The quality of the fit of the domain and similarity models to experiment will be judged from the root-meansquare (RMS) deviation defined as

RMS deviation )

x

n

[(EMCexpt avg)i - (EMCmodel)i]2 ∑ i)1 n

(2)

where n is the number of data points in any particular trajectory. Desorption Scanning Isotherms Figure 2 shows the schematic of a desorption scanning isotherm and Table 2 compares such experimental isotherms starting at 90, 80, and 75% RH on the adsorption boundary isotherm with predictions of the similarity model. To illustrate the similarity method, consider the desorption scanning curve that begins at an RH of 90% on the boundary adsorption curve. For all points on this trajectory, HR ) 90% and MR ) M(HR ) 90) ) Ma(90) ) 17.23% from Table 1. Suppose we wish to calculate the EMC of the paper sample at HX ) 80% by the similarity model, i.e., eq 1 (for this desorption trajectory, HM ) HX). From Table 1, Ma(HX) ) Ma(80) )12.90%, and Md(HX) ) Md(80) )16.25%. Because the limiting moisture content MC ) 29.51% as discussed earlier, the hysteretic factor f ) (29.51-16.25)/(29.5112.90) ) 0.798. From eq 1, we then obtain MX ) M(HX ) 80%) ) 17.23 - {17.23-12.90} × 0.798 ) 13.77%, as entered in Table 2. Similarly, at HX ) 75%, Ma(HX) ) Ma(75) ) 12.00%, and Md(HX) ) Md(75) ) 14.80% (Table 1), f ) (29.51-14.80)/(29.51-12.00) ) 0.840, and MX ) M(HX ) 75%) ) 17.23 - {17.23-12.00} × 0.840 ) 12.84% (eq 1). The EMC predicted by the similarity model at HX values of 60, 45, 30, and 15% can be calculated in a similar manner and they are shown in Table 2. Table 2 shows that the similarity method generally underpredicts the experimental EMC, which was also observed by Peralta and Bangi.10 However, the agreement between the values is quite good, with the RMS deviation in the predicted EMC decreasing from 0.29% for the 90% RH scanning curve to 0.23% for the 75% RH scanning curve. Predictions of the domain model are not given in Table 2, as the desorption scanning data

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 191 Table 2. Comparison of Similarity Model with Experimental Desorption Scanning Isothermsa EMC (%) 90% des. scan.

80% des. scan.

RH (%)

expt (23.8 ( 0.3 °C)

similarity

15 30 45 60 75b 80b 90b RMS dev.

4.97 6.82 8.58 10.63 13.19 14.10 17.23 0.00

4.94 6.54 8.20 10.25 12.84 13.77 17.23b 0.29

a

23.7 °C; experimental

data.5 b

75% des. scan.

expt (23.8 ( 0.3 °C)

expt (23.6 ( 0.3 °C)

similarity

similarity

4.90 6.65 8.40 10.34 12.35 12.90

4.91 6.42 7.99 9.92 12.14 12.90b

4.87 6.58 8.28 10.20 12.00

4.90 6.39 7.95 9.85 12.00b

0.00

0.27

0.00

0.23

Reversal point in the similarity model.

Table 3. Comparison of Domain and Similarity Models with Experimental Adsorption Scanning Isothermsa EMC (%) 45% ads. scan. RH (%) 100b 45b 60b 75 80 90 RMS dev. a

expt (23.8 ( 0.3 °C) 8.72 10.37 12.75 13.74 18.80 0.00

23.9 °C; experimental data.5

b

60% ads. scan.

domain5

similarity

8.79 10.21 12.51 13.29 17.44 0.65

8.79b 10.63 12.84 13.70 17.82 0.46

expt (24.1 ( 0.2 °C)

domain5

similarity

11.13 13.28 14.21 19.25 0.00

11.20 13.00 13.76 17.80 0.77

11.20b 13.35 14.18 18.18 0.54

Reversal point in the similarity model.

were themselves used to construct the moisture distribution function of that model from which all subsequent EMC predictions were made. Adsorption Scanning Isotherms An adsorption scanning isotherm is shown schematically in Figure 2. Table 3 compares such experimental isotherms (begun with a wet paper sample) starting at 45% and 60% RH on the boundary desorption isotherm with the domain and similarity models. Because calculations for the domain model were explained by us earlier,5 here we focus on the similarity model. Consider the adsorption scanning isotherm that starts at an RH of 45% on the boundary desorption curve. For all points on this trajectory, HR ) 45%, and MR ) M(HR ) 45) ) Md(45) ) 8.79% from Table 1. We can calculate the EMC of the paper sample at HX ) 60% by the similarity model, i.e., eq 1 (for this adsorption trajectory, HM ) HR) via the following steps. From Table 1, Ma(HR) ) Ma(45) ) 7.74%, and Ma(HX) ) Ma(60) ) 9.67%. Because the limiting moisture content MC ) 29.51%, the hysteretic factor f ) (29.51-8.79)/(29.51-7.74) ) 0.952. Note that f remains constant for all points in this ascending trajectory, unlike the case of the desorption scanning isotherms (descending trajectories). From eq 1, we then obtain MX ) M(HX ) 60) ) 8.79 - {7.749.67} × 0.952 ) 10.63%, as entered in Table 3. Similarly, at HX ) 75%, Ma(HX) ) Ma(75) )12.00% (Table 1), f ) 0.952, and MX ) M(HX ) 75) ) 8.79 - {7.7412.00} × 0.952 ) 12.84% (eq 1). The EMCs predicted by the similarity model at HX values of 80 and 90% can be calculated in a similar fashion, and they are given in Table 3. Table 3 shows that the domain model consistently underpredicts the experimental EMC, whereas the similarity model overpredicts it at low RH values while underpredicting it at high RH values. The maximum deviation between the models and experiment occurs

Table 4. Comparison of Domain and Similarity Models with Experimental Spiral Trajectory from Boundary Desorption Isotherma EMC (%) experimental RH (%) 100b 90 45b 90b 60b 75 80b 75 RMS dev.

repl. #1 repl. #2 (23.4 ( 0.2 °C) (23.3 ( 0.3 °C) 25.49 8.73 19.40 10.97 13.00 13.97 13.29 0.03

25.50 8.71 19.45 10.87 13.03 14.00 13.29 0.02

a 23.3 °C; experimental data.5 model.

b

theoretical avg 25.50 8.72 19.43 10.92 13.02 13.99 13.29 0.00

domain5 similarity 25.11 8.79 17.44 10.84 12.64 13.40 12.85 0.83

25.11 8.79b 17.82b 10.85b 13.00 13.83b 13.07 0.64

Reversal point in the similarity

at the 90% RH termination point of the trajectories, where the differences between experimental and predicted EMCs are 1.36% and 1.45% (domain) and 0.98% and 1.07% (similarity) for the 45 and 60% RH scanning curves, respectively. The similarity model performs better than the more complex domain model, with RMS deviations ranging from 0.46 to 0.54% compared to 0.65-0.77% of the domain model. Spiral Trajectories Tables 4 and 5 compare the predictions of the domain and similarity models with experimental spiral trajectories begun from the boundary desorption and adsorption curves, respectively (see Figure 2). The similarity model calculations for the EMC are similar to those described above for desorption and adsorption scanning isotherms, with the reversal point shifting appropriately when the direction of the RH change reverses. In the spiral trajectory begun from the boundary desorption isotherm shown in Table 4, an initially wet paper sample was subjected to an RH oscillation of

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Table 5. Comparison of Domain and Similarity Models with Experimental Spiral Trajectory from Boundary Adsorption Isotherma

Table 6. Comparison of Domain and Similarity Models with Experimental Loop Trajectorya EMC (%)

EMC (%)

experimental RH (%)

experimental RH (%) 0b 15 90b 30b 75b 45b 60 RMS dev.

repl. #1 repl. #2 (23.2 ( 0.2 °C) (23.2 ( 0.2 °C) 4.74 17.68 6.54 12.17 8.26 9.84 0.04

4.80 17.70 6.60 12.25 8.38 9.96 0.04

a 23.2 °C; experimental data.5 model.

b

theoretical avg 4.77 17.69 6.57 12.21 8.32 9.90 0.00

domain5 similarity 4.84 17.23 6.82 12.24 8.52 9.94 0.23

4.84 17.23b 6.54b 12.15b 8.09b 9.93 0.21

Reversal point in the similarity

gradually decreasing amplitude. There is very good agreement between the corresponding experimental EMCs of the two replicates with RMS deviations (from the experimental average EMC) of 0.02-0.03% and a maximum difference of 0.1% between their EMCs at 60% RH. The average EMC of the two replicates is compared with the predictions of the domain and similarity models. In general, agreement between experiment and models is quite good, except at the turning point of 90% RH where both models underpredict the experimental EMC. As discussed in our prior publication,5 the probable cause for this discrepancy is the artificial closure of the main hysteretic loop at its upper end (RH ) 100%), as mentioned earlier. In the 90-100% RH range, where we were unable to acquire experimental data, the EMC changes rapidly with RH. Because this spiral trajectory was begun from a completely wet paperboard sample, its adsorption scanning curves are likely to be higher in the 90-100% RH range than the predictions of the models, which are based on averaging of the (extrapolated) boundary adsorption and desorption isotherms at 100% RH. The RMS deviations of the domain and similarity models are 0.83% and 0.64%, respectively. Table 5 shows a spiral trajectory begun from the boundary adsorption isotherm in which the paper sample was subjected to an RH oscillation of gradually decreasing amplitude. Both experimental replicates compare very well with each other, with RMS deviations of 0.04% and a maximum difference of 0.12% between their EMCs at 45 and 60% RH. The agreement between the average experimental EMC and that predicted by the models is better than for the spiral trajectory from the boundary desorption isotherm, as reflected in lower RMS deviations of 0.23% and 0.21% for the domain and similarity models, respectively. Loop Trajectory This trajectory, shown in Figure 1 and Table 6, was carried out to test the validity of Theorems 4 and 5 of Everett and Smith13 for systems conforming to the theory of independent domains. It is best to understand this trajectory by looking at the theoretical EMC (domain or similarity model) variation with RH. During the initial part of this trajectory, we move up along the boundary adsorption isotherm along RH values of 15-30-45-60-75%, where the 75% RH corresponds to point U in Figure 1. Thereafter, we descend along the secondary desorption curve, starting at U, along RH values of 60 and 45% and stop at an

0b 15 30 45 60 75b 60 45 30b 45 60 75b 60 45 30b 45 60 75 80d 90d RMS dev.

repl. #1 repl. #2 (23.2 ( 0.2 °C) (23.4 ( 0.2 °C) 4.84 6.10 7.64 9.49 11.93 10.17 8.36 6.56 7.91 9.67 11.75 10.08 8.27 6.56 7.91 9.58 11.66 12.57 17.31 0.11

4.79 6.27 7.88 9.85 12.17 10.41 8.50 6.72 8.10 9.84 11.95 10.38 8.50 6.74 8.11 9.86 11.86 12.81 17.52 0.11

theoretical avg 4.82 6.19 7.76 9.67 12.05 10.29 8.43 6.64 8.01 9.76 11.85 10.23 8.39 6.65 8.01 9.72 11.76 12.69 17.42 0.00

domainc similarity 4.84 6.23 7.74 9.67 12.00 10.20 8.28 6.58 7.99 9.70 12.00 10.20 8.28 6.58 7.99 9.70 12.00 12.90 17.23 0.11

4.84 6.23 7.74 9.67 12.00b 9.85 7.95 6.39b 7.86 9.74 12.00b 9.85 7.95 6.39b 7.86 9.74 12.00 12.90d 17.23d 0.24

a 23.3 °C. b Reversal point in similarity model. c Calculated from the moisture distribution function.5 d In the similarity method, the EMC of this point was on the boundary adsorption curve as the reversal point was taken to be 0% EMC at 0% RH.

RH of 30%, corresponding to point L (path 1 in Figure 1). From L, we reverse direction and ascend along path 2 in Figure 1, passing through RH values of 45 and 60% and stop when we reach point U, which concludes one complete circuit between U and L. Note that the theoretical EMC of 12% at U, which corresponds to an RH of 75%, is the same at the beginning and end of this circuit. One more circuit between U and L along paths 1 and 2 is performed after which we ascend along the boundary adsorption isotherm passing 80% RH and terminate the trajectory at 90% RH. The agreement between the experimental EMCs of the two replicates along the entire trajectory is quite good, although not reaching the level of that of the spiral trajectories. This is reflected in higher RMS deviations of 0.11% and a maximum difference of 0.36% between the EMCs at the first 60% RH of the trajectory. The average experimental EMC is also in good agreement with predictions of the domain and similarity models, which have RMS deviations of 0.11% and 0.24%, respectively. However, there is an interesting feature in the experimental data shown in Table 6. The average experimental EMC at 75% RH appears to continuously decrease with RH cycling, i.e., 12.05, 11.85, and 11.76%. The EMCs of the individual replicates show this same behavior, so this effect is due to either a systematic error or some other physical cause. Recently, in our laboratory, experiments were conducted in which freshly made handsheets and a machine-made paper were subjected to repeated humidity cycling between 10 and 90% RH with a hold time of 6 h at both RH levels.14 The EMC of the sheets was measured as a function of the number of humidity cycles. The EMC at 10% RH was not affected by RH cycling, whereas that at 90% RH exhibited a continuous decrease with the number of RH cycles and appeared to reach a constant value after a certain number of cycles were completed. Over the course of eight humidity cycles between 10 and 90% RH,

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the EMCs at 90% RH of the various types of paper investigated decreased by about 6-14% of their initial values. Qualitatively, this behavior was independent of the methods of preparation of the handsheetssdrying (restrained and unrestrained) and aging in a vacuum oven at 60 °C. Quantitatively, the effect was smaller for the machine-made and (artificially) aged paper. Freshly made paper undergoes irreversible structural changes upon repeated humidification and drying. It is known that drying causes an irreversible reduction in the swelling ability of cellulose, the closing of large pores (2.5-50 nm), and cell-wall collapse.15 Recent research has also revealed that, when the moisture content of a cellulose fiber is reduced by drying or pressing, the pores in the fiber decrease in size, and the pore-size distribution becomes narrower.16 These irreversible changes in structure are manifested as an irreversible loss of hygroscopicity of freshly made paper that is subjected to humidity cycling. After the paper has attained a stable structure, a steady EMC value is obtained (at a particular RH), and reversible EMC-RH hysteretic behavior is seen. We have observed this experimentally for the case of a kraft handsheet that exhibited reversible (repeatable or time-independent) hysteretic EMC behavior after 40 humidity cycles between 10 and 90% RH were completed. It is this reversible hysteresis that is strictly amenable to analytical treatment. Although in practice it might not be feasible to conduct such a large number of RH cycles, it is nevertheless suggested that paper physicists and personnel in paper testing laboratories pay attention to adequate preconditioning of their samples in order to obtain consistent and reproducible results. With regard to the concrete case under discussion here (i.e., the experimental trajectory in Table 6), if a systematic experimental error can be discounted, then it can be conjectured that the preconditioning procedure used here and previously5 for the machine-made naturally aged bleached kraft paperboard used in this work might not have been perfect. Limiting Moisture Content, MC For all calculations so far with the similarity model, the value of the limiting moisture content MC was taken to be 29.51%, which is the average of the EMCs predicted by the fitted boundary adsorption and desorption isotherms extrapolated to 100% RH. As explained in our previous work,5 this method of obtaining MC is an artificial procedure that was necessitated by the fact that we were unable to reach 100% RH in our humidity chamber. However, MC physically represents the moisture content at the upper closure point of the main hysteretic loop, which, in general, will depend on experimental conditions. For example, in some of the trajectories that began from the 15% RH point in this work, the upper end point at which the trajectory was reversed was 90% RH. Thus, an MC value of 17.23%, which is the EMC corresponding to 90% RH on the boundary adsorption isotherm (Table 1), can be chosen as the limiting moisture content for these trajectories and for others that lie within them. That is, for all such trajectories, the main hysteretic loop can now be considered to be enclosed by the boundary adsorption curve (Table 1) and the experimental desorption scanning curve starting at 90% RH (Table 2). Using MC ) 17.23%, the boundary adsorption isotherm (Table 1), and the experimental 90% RH desorp-

tion scanning isotherm (Table 2), calculations using the similarity model were repeated for the EMC trajectories shown in Tables 2, 5, and 6. For the 80 and 75% RH desorption scanning isotherms in Table 2, the RMS deviations between the similarity model and experiment were 0.15 and 0.13%, respectively. For the trajectories in Tables 5 and 6, the RMS deviations were 0.24 and 0.17%, respectively. As expected intuitively, these errors are, in general, lower than those obtained when a value of MC ) 29.51% was used in the similarity model. Conclusions This work used Everett’s domain and Mualem’s capillary-radii similarity models of hysteresis to characterize the complex moisture sorption equilibria of paper. For a variety of EMC-RH trajectories (desorption scanning, adsorption scanning, spiral and loop) inside the main sorption hysteretic loop, the models were able to predict the experimental EMC with good precision. Over an RH range of 15-90%, the RMS deviation between the predicted and experimental EMCs was 0.11-0.83% (domain model) and 0.21-0.64% (similarity model). However, Mualem’s model, as conceptualized by eq 1, is easier to use and requires data from only the two boundary isotherms that envelop the main hysteretic loop. It also predicts higher-order EMC trajectories that lie inside the main hysteretic loop in a consistent fashion. The similarity approach thus obviates the need of taking additional labor-intensive and time-consuming scanning curve data that is necessary to develop the moisture distribution function of the domain model. Nomenclature f ) hysteretic factor, [MC - Md(HM)]/[MC - Ma(HM)] H ) relative humidity (RH) HC ) RH of upper closure point C of the main hysteretic loop HL ) RH of lower point L in Figure 1 HM ) RH of either the reversal point R or the unknown point X, whichever is lower. HM ) HR for any ascending path, whereas HM ) HX for any descending path HP ) RH of point P in Figure 1 HR ) RH of reversal point R HU ) RH of upper point U in Figure 1 HX ) RH of unknown point X Ma(H) ) equilibrium moisture content (EMC) at the RH of H on the boundary adsorption isotherm Ma(HM) ) EMC at the RH of HM on boundary adsorption isotherm Ma(HR) ) EMC at the RH of HR on boundary adsorption isotherm Ma(HX) ) EMC at the RH of HX on boundary adsorption isotherm MC ) EMC of upper closure point C of the main hysteretic loop Md(H) ) EMC at the RH of H on the boundary desorption isotherm Md(HM) ) EMC at the RH of HM on the boundary desorption isotherm ML ) EMC of lower point L in Figure 1 MP ) EMC of point P in Figure 1 MR ) EMC of reversal point R

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MU ) EMC of upper point U in Figure 1 MX ) EMC of unknown point X

(8) Mualem, Y. A Conceptual Model of Hysteresis. Water Resour. Res. 1974, 10 (3), 514.

Acknowledgment

(9) Peralta, P. N.; Bangi, A. P. Modeling Wood Moisture Sorption Hysteresis Based on Similarity Hypothesis. Part 1. Direct Approach. Wood Fiber Sci. 1998, 30 (1), 48.

This work was supported by Award 97-35103-4795 of the USDA NRI Competitive Grants Program. Mr. Harikrishnan Radhakrishnan recorded the experimental data reported in Table 6. Literature Cited (1) Habeger, C. C.; Coffin, D. W. The Role of Stress Concentrations in Accelerated Creep and Sorption-Induced Physical Aging. J. Pulp Pap. Sci. 2000, 26 (4), 145. (2) Ramarao, B. V.; Chatterjee, S. G. Moisture sorption by paper materials under varying humidity conditions. In The Fundamentals of Papermaking Materials: Transactions of the Eleventh Fundamental Research Symposium; PIRA International: Leatherhead, Surrey, U.K., 1997; Vol. 2, p 703. (3) Foss, W. R.; Bronkhorst, C. A.; Bennett, K. A.; Riedemann, J. R. Transient moisture transport in paper in the hygroscopic range and its role in the mechano-sorptive effect. In Proceedings of the Third International Symposium: Moisture and Creep Effects on Paper, Board and Containers; Chalmers, I. R., Ed.; PAPRO: Rotorua, New Zealand, 1997; p 221. (4) Radhakrishnan, H.; Chatterjee, S. G.; Ramarao, B. V. Steady-State Moisture Transport in a Bleached-Kraft Paperboard Stack. J. Pulp Pap. Sci. 2000, 26 (4), 140. (5) Chatterjee, S. G.; Ramarao, B. V.; Tien, C. Water-Vapor Sorption Equilibria of a Bleached-Kraft PaperboardsA Study of the Hysteresis Region. J. Pulp Pap. Sci. 1997, 23 (8), 366. (6) Everett, D. H. Adsorption Hysteresis. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel-Dekker: New York, 1967; Vol. 2, p 1055. (7) Mualem, Y. Modified Approach to Capillary Hysteresis Based on a Similarity Hypothesis. Water Resour. Res. 1973, 9(5), 1324.

(10) Peralta, P. N.; Bangi, A. P. Modeling Wood Moisture Sorption Hysteresis Based on Similarity Hypothesis. Part II. Capillary-Radii Approach. Wood Fiber Sci. 1998, 30 (2), 148. (11) Peralta, P. N. Sorption of Moisture by Wood within a Limited Range of Relative Humidities. Wood Fiber Sci. 1995, 27 (1), 13. (12) Bizot, H. Using the GAB Model to Construct Sorption Isotherms. In Physical Properties of Foods; Jowitt, R., Esher, F., Hallstro¨m, B., Meffert, H. T. Th., Spiess, W. E. L., Vos, G., Eds.; Applied Science Publishers: Essex, U.K., 1983; Chapter 4. (13) Everett, D. H.; Smith, F. W. A General Approach to Hysteresis. Part 2: Development of the Domain Theory. Trans. Faraday Soc. 1953, 50, 187. (14) Beck, K.; Radhakrishnan, H.; Chatterjee, S. G.; Ramarao, B. V.; Makkonen, H. P. Experiments on Moisture Sorption and Transport in Paper Materials; ESPRA Research Report No. 108; Empire State Paper Research Institute: Syracuse, NY, 1998; p 35. (15) Lindstro¨m, T. The Porous Lamellar Structure of the Cell Wall. In PAPER: Structure and Properties; Bristow, J. A., Kolseth, P., Eds.; Marcel-Dekker: New York, 1986; p 104. (16) Ha¨ggkvist, M.; Li, T. Q.; O ¨ dberg, L. Effects of Drying and Pressing on the Pore Structure in the Cellulose Fibre Wall Studied by 1H and 2H NMR Relaxation. Cellulose 1998, 5, 33.

Received for review April 27, 2000 Revised manuscript received September 26, 2000 Accepted October 4, 2000 IE000437F