Ind. Eng. Chem. Res. 2000, 39, 219-226
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Moisture Sorption Response of Paper Subjected to Ramp Humidity Changes: Modeling and Experiments A. Bandyopadhyay, H. Radhakrishnan, B. V. Ramarao,* and S. G. Chatterjee Empire State Paper Research Institute, SUNY College of Environmental Science and Forestry, Syracuse, New York 13210
Moisture has a profound effect on the physical properties of paper and thus determines the performance characteristics of paper products including the so-called mechanosorptive effect where the creep behavior of paper is accentuated by cyclic humidity variations. Moisture transport in a paper sheet exposed to ramp changes in the external humidity has been studied. Our transport model considers a paper sheet as a composite medium of fibers and void spaces with moisture transported by diffusion through each one. Diffusion in the fibers is approximated by a lumped linear relaxation process. The important dimensionless parameters consist of the Biot, fiber, and void diffusion numbers. A comparison of the model predictions with experimental transient moisture uptake of a bleached kraft paper board sample showed that a single value of the fiber relaxation parameter, ki, was sufficient to give good estimates of transient moisture sorption data. Introduction Moisture has a significant effect on the physical properties of paper materials leading to a degradation in performance at high moisture contents. For instance, the strength of paper has been shown to decrease by 5-10% for each unit percentage increase in the moisture content.1 Under high humidity conditions, paper has long been known to show lower elastic moduli, yield stresses, and tensile strengths.2 Under compressive loads, containers made of paper board fail substantially earlier when subjected to cyclic humidity conditions than when exposed to a constant yet high level of humidity.3-5 This shortened life span of boxes is a concern in paperboard manufacturing. The unsteady behavior of the moisture content profiles is referred to as moisture transients. Although the impact of moisture transients has been extensively investigated in the past,5-8 attention has been focused on the mechanical performance but not on moisture transients. To understand the variations in the mechanical properties of paper under changing humidity conditions, knowledge of the moisture transients is essential. There are two aspects to the interaction of moisture with paper under changing humidity conditions. The first is the equilibrium sorption capacity of a paper sheet described by the moisture sorption isotherm. This obviously determines the ultimate moisture content of the sheet as well as its limiting response to a changing humidity when transport resistances are negligible. The second aspect is the nature of transport resistances of the paper sheet. Transport resistance can be characterized by the transmittance of moisture through the paper sheet under steady-state conditions. The interaction between these two aspects gives the transient response of paper sheets under changing humidity conditions. The equilibrium moisture sorption behavior of paper is similar to that of other cellulosic materials. The GAB equation has been shown to represent moisture sorption isotherms in paper quite well (see eq 11).9,10 The theory of independent domain complexions has been applied
to describe sorption hysteresis and the evolution of the equilibrium moisture content under arbitrary equilibrium cycles in humidity.11-13 Analysis of moisture sorption under changing humidity conditions has been reported.10,12,14-16 Following conventional transport models for transient moisture sorption in wood, Lin14 modeled the transport of moisture in paper as adsorption on the surface followed by bound water diffusion through the sheets. The diffusivity of bound water was assumed to be an unknown but exponentially increasing function with moisture content. By comparison of predicted transient moisture content profiles with experimental data (reported by Steele17), Lin14 determined the parameters for the moisture diffusivity. Lin’s approach of extending models for transient sorption in wood is unsatisfactory for two reasons. First, paper is a porous medium with substantial pore volume in addition to cellulosic fibers. Thus, transport through paper is expected to occur simultaneously through both the pore and the fiber phases. Second, in this procedure, the moisture dependence of the diffusivity is not determined independently but by fitting to experimental transient sorption data. Lescanne et al.15 furnished a more elaborate model of moisture transport which incorporates the effects of fiber and pore diffusion. In their model, fiber diffusion is idealized as a relaxation process in addition to the diffusion in the pore space. The steady-state diffusivity of paper was measured at low humidities in independent experiments. At low humidities, the diffusion was assumed to occur only through the pore space. Experimental transient sorption data were then analyzed, and when the relaxation model is fitted to the experimental data, an estimate of the diffusivity of bound water through fibers was obtained as being on the order of 10-12 m2/s. Chatterjee et al.10 analyzed the phenomenon of transient moisture sorption under sinusoidal variation in humidity. This model assumed that the transient sorption process could be idealized as diffusion within the vapor space of the sheet and that the fibers were in equilibrium with the humidity in the local pore space. Foss et al.16 and Ramarao and Chatterjee12 presented
10.1021/ie990279w CCC: $19.00 © 2000 American Chemical Society Published on Web 12/08/1999
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port including fiber uptake is written as follows:
p
∂c ∂2c ) Dp 2 - Fpki(qsat - q) ∂t ∂z
[
(1)
]
∂ ∂q ∂q ) D (q) + ki(qsat - q) ∂t ∂z q ∂z
(2)
In the second equation, we have allowed for moisture diffusion in the z dimension within the fibers. The second term on the right-hand side of both of these equations represents the uptake of moisture by the individual fibers at a specific z location. We consider a ramped change in humidity given as
Hb(t) ) H0 + Figure 1. Representation of paper sheets for analyzing transient moisture sorption showing interfiber (i.e., void space) diffusion of water vapor, intrafiber diffusion of bound water, and external boundary layers. Half thickness is shown (z ) 0-L).
closely related analyses of the transient sorption of paper sheets subjected to ramp changes in humidity. Ramarao and Chatterjee12 idealized transient sorption as consisting of two simultaneous processes:diffusion of water vapor through the void space of the sheets and diffusion of bound water through the fiber space. Diffusion through the void space was represented by means of the unsteady-state diffusion equation, with the fibers representing a sink. Diffusion through fibers was represented by a linear driving force approximation similar to the one used in the analyses of adsorption.18 An analytical solution to the model was obtained for the special case when the sorption isotherm is linear and was compared with a limited set of experimental transient sorption data. In this paper, we present an analysis of the transient moisture sorption process in a paper sheet exposed to ramped external humidity. A model incorporating diffusion through the fiber space and the pore space of paper sheets is then developed. The intrafiber moisture transport is represented by the linear driving force approximation. The model equations are solved numerically using a finite volume method. A comparison of the model predictions with experimental data of the transient response of a sample bleached kraft paper board is provided. A single value of the fiber relaxation parameter within the model can provide good predictions for a multitude of humidity conditions even when the humidity is cycled. This analysis is expected to improve our understanding of how transient sorption impacts the mechanical behavior of paper materials. Mathematical Model of Transport Because paper is often substantially porous, it can be idealized as a composite medium of fiber layers: each layer consists of fibers oriented predominantly in the x-y plane. The sheet’s transverse dimension is denoted as z in accordance with the accepted nomenclature for paper. The transport process is shown schematically in Figure 1. We identify two concentration fields (averaged in the x-y plane), one for the moisture concentration within the void space, c(z,t), and one for the moisture content within the fiber matrix, q(z,t). The uptake of moisture by the fibers at any z location is idealized as a linear relaxation process. Thus, the model for trans-
(H1 - H0) t tR
(3)
for t < tR and
Hb(t) ) H1
(4)
for t g tR. The initial and boundary conditions are given as
c(t)0,z) ) c0
(5)
q(t)0,z) ) qsat(c0) ) q0
(6)
∂c (t,z)L) ) 0 ∂z
(7)
∂q (t,z)L) ) 0 ∂z
(8)
Equations 7 and 8 represent symmetry in the profiles at the center of the sheet. For the conditions at the sheet boundary, z ) 0, the following equations are assumed to represent the boundary conditions:
-Dp
∂ (t,z)0) ) kf[cb(t) - c(t,z)0)] ∂z q(t,z)0) ) qsat[c(t,z)0)]
(9) (10)
The complete mathematical model consists of eq 1 and 2 and the conditions described by eqs 5-10. Before a numerical solution of the model can be developed, we must specify the moisture sorption isotherm. Issues of hysteresis and reversibility do not directly impact the transient sorption response to ramped changes in humidity as they do for the response to cyclic humidity changes.9 Thus, the sorption isotherm can be represented by means of the GAB isotherm given by
q(c) ) qsat(c) )
M0KGABCGABc (1 - KGABc)(1 - KGABc + KGABCGABc) (11)
The constants M0, KGAB, and CGAB are determined by fitting eq 11 to experimental sorption equilibrium data.9 The average moisture content of the sheet is defined over the sheet’s thickness as
qav )
1 L
∫0L[q(t,z) + pc(t,z)/Fp] dz
(12)
In this equation the second term of the integrand is
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usually negligible compared to the first. The model equations are expressed in dimensionless variables (defined in the nomenclature section) as
p
∂C 1 ∂2C ) - Rβ[Qsat - Q] ∂τ P ∂ζ2
1 ∂ ∂Q ∂Q ) D (Q) + β[Qsat - Q] ∂τ P′ ∂ζ Q ∂ζ
[
]
(13) (14)
where DQ(Q) is a nondimensionalized version of the intrafiber moisture diffusivity, Dq(q). The initial and boundary conditions in dimensionless form are
C(t)0,ζ) ) Q(τ)0,ζ) ) 0
(15)
∂C ∂Q (τ,ζ)1) ) (τ,ζ)1) ) 0 ∂ζ ∂ζ
(16)
and
-
∂C (τ,ζ)0) ) γP[Cb(τ) - C] ∂ζ
(17)
The sorption equilibrium can be represented by the following relationship in dimensionless form:
Q(τ,ζ)0) ) Qsat(C) )
qsat(c) - qsat(c0) qsat(c1) - qsat(c0)
(18)
Table 1. Parameters for Experiments on Sorption Dynamics bound water diffusivitya
Dq, m2 s-1
water vapor diffusivityb sheet density porosity sheet thickness fiber radius external convective mass-transfer coefficientc internal diffusion parameterd temperature saturation vapor pressure slope of isotherm external humidity change ramp time GAB parameters MGAB KGAB CGAB
Dp, m2 s-1 Fp, kg m-3 p 2L, m R, m kf, ms-1
varies from 10-14 at low RH (below 75%) to 10-6 at higher values 6.3 × 10-7 839 0.5 0.325 × 10-3 10 × 10-6 0.0025
ki, s-1 T, C psat, kPa Kiso, m3 kg-1 H1 - H0 tR, s adsorption 0.05228 0.7719 35.241
0.0035 23.7 2.93 4.52 45-15% 3600 desorption 0.0661 0.6966 17.4927
a Estimated from steady-state water flux through sheets at high RH (>75% avg). b Calculated values from steady-state water vapor flux through sheets at low RH ( 1. A numerical solution of eqs 22-27 by using the finite volume method was developed. The baseline values of the parameters for this study were obtained for a sample sheet of a bleached kraft linerboard which was used in our previous studies.9-12 The parameter values and the corresponding dimensionless groups are given in Tables 1 and 2. In this paper, we confine our attention to sorption cases where moisture diffusion in the z direction through the fibers is negligible as compared to diffusion within the void spaces and local uptake of moisture by the fibers. For the water diffusivity Dq to be significant, the moisture content of the fibers should
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to transport in the z dimension is negligible although there still may be significant resistance in the radial dimension of the fibers. Integrating eq 22 with respect to ζ and applying the boundary condition leads to
p
dC dQ +R ) γ[Cb - C] dτ dτ dQ ) β[Qsat - Q] dτ
(28) (29)
The initial conditions are given at τ ) 0 as
C)Q)0
(30)
For the case of a nonlinear isotherm, a numerical solution of the above equations is necessary. However, when the isotherm is linear, an analytical solution is possible. Simplified Model(2): All Sheet Resistances Are Negligible. When β . 1, eq 29 indicates that the moisture content within the sheet is at equilibrium with the local concentration. Thus,
Q(τ) ) Qsat(C)
(31)
Therefore, eqs 28 and 29 can be combined and simplified as
dC ) γ[Cb(τ) - C] dτ
(32)
where
γ
λ)
p + R
dQsat dC
(33)
The initial condition is given by eq 30. Experimental Results Figure 2. (a) Moisture content profiles: Comparison of numerical and analytical solutions for RH ramps of two different magnitudes (15-45% and 15-75%). All other parameters are given in Table 2. (b) Average moisture content of paper sheets as a function of dimensionless time, τ. Comparison of numerical and analytical solutions for ramps as in part a.
be substantially high (greater than 13%).19 This corresponds to a relative humidity (RH) of above 75%. For the case of a linear sorption isotherm, an analytical solution to the previous equations has been obtained using the integral transform technique.12 Figure 2a shows a comparison of the moisture content profiles by the numerical method employing the complete nonlinear isotherm and the analytical solution using the linear isotherm. As can be expected, when the humidity change is small such that linearization of the isotherm is valid, the analytical solution describes the moisture profiles quite well. In the higher humidity ranges, however, significant deviation between the analytical and numerical solutions is found, indicating that the linearization is not valid anymore. Figure 2b shows a similar comparison of the average moisture content of the sheet. Simplified Model(1): Negligible Diffusion Resistance in the Sheet Thickness Direction. When RP , 1 and C and Q are independent of ζ, the resistance
To verify the mathematical model, transient sorption experiments were conducted in a controlled temperature and humidity chamber. Figure 3 shows a schematic of the experimental setup. In brief, the experimental setup consists of a humidity chamber in which a paper sheet sample can be suspended from a balance. The humidity chamber is continuously swept by air of controlled humidity obtained by mixing saturated and dry air in proportions. The humidity inside the chamber is monitored by a Vaisala temperature and humidity probe which sends its signal to a computer. A feedback controller is used to control the humidity of the entering air. A detailed description of the experimental apparatus was given by Chatterjee et al.11 Bleached kraft paper board of a basis weight of 240 gsm was used for the experiments. The diffusivity of water vapor through this board and the external masstransfer coefficient were determined by independent experiments and are reported in Table 3. The moisture sorption isotherm was obtained for this paper board and was fitted by the GAB isotherm (cf. eq 11), the parameters of which are shown in Table 1. Figure 4 shows the experimentally determined sorption isotherm for this board sample. Square samples measuring 10 × 10 cm were hung from the balance in this chamber. The samples were preconditioned by cycling the humidity between 90%
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Figure 3. Schematic of experimental setup for equilibrium and transient sorption experiments. The same setup was used also for determining the external mass-transfer coefficient and the steady-state water vapor diffusivity of paper sheets. Table 3. Mass-Transfer Coefficient and Steady-State Water Vapor Diffusivity (Radhakrishnan19) Steady-State Water Vapor Diffusivity average of RH on two sides of the sheet
Dp, cm2/s
52.5 57.6 62.7 67.6 72.4 77.9
0.0049 0.0053 0.0061 0.0068 0.0083 0.0110
Mass-Transfer Coefficient RH in the chamber
kf, cm/s
30 35 40 45 50 avg
0.26 0.24 0.24 0.25 0.30 0.26 Figure 5. Comparison of model predictions against experimental average moisture content evolution data for ramp changes in RH [15-45%]. Complete model (fiber diffusion in thickness direction negligible) predictions with (1) ki ) 0.001 s-1, (2) ki ) 0.0025 s-1, (3) ki ) 0.0035 s-1, (4) experimental data, (5) ki ) 0.005 s-1, (6) ki f ∞, and (7) qsat.
Figure 4. Equilibrium sorption isotherms for a sample bleached kraft paper board sheet. The upper curve is the desorption isotherm, and the lower curve is the adsorption isotherm. Points are experimentally measured data, and the dotted line is the fitted GAB isotherm with parameters as shown in Table 1.
and 10% as described by Chatterjee et al.11 After preconditioning, the samples were exposed to ramp changes in RH starting from a 15% level and increased to 45%, 60%, and 90% levels. The ramp times were
varied from 15 to 150 min to obtain different ramp slopes. The sample weight was monitored for a period of approximately 5 h, after which it appeared to reach a steady value. The sample was then taken out, and the oven dry weight of the sheet was determined. The following comparisons between the predictions of the various models described previously and experimental data will be made in the dimensional variables. Figure 5 shows a plot of the average moisture content of the sheet as a function of time for an increase in RH of up to 45% from a 15% level in 30 min (ramp slope of 1% RH/min). Also shown in this figure are the predictions of the model using various values of ki. Note that all other parameters in the model, namely, Dp, kf, and the sorption isotherm, have been determined by independent experiments. We observe that a value of ki of 0.0035 s-1 seems to give the best predictions which match with the experimental data. Further, we note that values of ki higher than 0.03 s-1 do not change the transient sorption curve significantly, indicating that at these levels sorption by the fibers is rapid and will
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Figure 6. Comparison between experimental data and model predictions for ramp humidity change 15-45%, simplified model (1): (1) ki ) 0.002 s-1, (2) ki ) 0.003 s-1, (3) experimental data, (4) ki ) 0.005 s-1, (5) qsat.
Figure 8. Comparison of model predictions with experimental data. Ramp changes in humidity 15-45% at 0.5% and 1% RH/ min. Complete model (Dp - kf - ki) with ki ) 0.0035 s-1.
Figure 9. Comparison of experimental data with model predictions. Case 1, with ki ) 0.0035 s-1. Ramp changes in humidity between 15 and 90% at two different ramp rates.
Figure 7. Comparison of model predictions (simplified model 2) with experimental data: (1) kf ) 0.09, (2) kf ) 0.11, (3) kf ) 0.2, (4) kf ) 0.25 (all in cm s-1), (5) qsat. Experimental data are shown as individual points.
not control the moisture profile. Local equilibrium with the moisture in the void space will be achieved in such situations. Figure 6 shows a plot of the same experimental transient sorption data along with predictions from the simplified model(1) (incorporating the parameters kf and ki and represented by eqs 28-30) for different values of ki (i.e., β in nondimensional terms). The mass-transfer coefficient kf used in this case was the experimentally measured value of 0.25 × 10-2 m/s. Note that the bestfitting value of the parameter ki is 0.003 s-1. Figure 7 shows the predictions of the simplified model(2) with the external surface resistance alone as the contributing factor (kf-1) [eq 32] compared to the same transient experimental data. The best-fitting value of kf is seen to be about 0.11 × 10-2 m/s. From Figures 6 and 7, we note that any of the solutions to the simplified models may be used to represent the transient sorption response of these paper sheets to RH ramps. However, in the latter situation, the best-fitting mass-transfer coefficient is 0.11 × 10-2 cm/s, which is
substantially smaller than the average experimentally measured value of 0.25 × 10-2 m/s (cf. Table 3). This indicates the presence of an additional resistance(s) within the sheet hindering moisture uptake. Figure 6 indicates that a ki value of 0.003 can describe the dynamics when the intrapore vapor-phase diffusion process is ignored. Figure 5 indicates that when the intrapore diffusion is considered (using a pore diffusivity determined from steady-state experiments), the bestfitting ki value is 0.0035 s-1. Because the best fitting values of ki are not substantially different from each other for either of the two cases, we conclude that the effect of pore diffusional resistance in this case is small. This, however, is not universal because an increase of the thickness of the paper will result in a more significant pore diffusion resistance in the transient sorption process. Figure 8 shows experimental moisture content data as a function of time for two ramps in RH ranging from 15% to 45% at slopes of 0.5% and 1% RH/min. The predictions of the complete sorption model (fiber diffusion in the z direction is negligible, incorporating the parameters Dp, ki, and kf, given by eqs 22-27) are shown to compare well with the experimental data for both RH ramps. Figure 9 shows experimental moisture content data for ramped changes taking place between 15% and 90% RH levels. Again, the model predictions compare well with the experimental data. Even though intrafiber
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Figure 10. Comparison of model predictions with experimental data. RH ramps 15-90% and 90-15% of different ramp rates: (1) 1% RH/min, upward, (2) 0.5% RH/min, upward, (3) 1% RH/min, downward, (4) 0.5% RH/min, downward. Experimental data are shown individually and merge with model predictions. Corresponding equilibrium (qsat) curves also are shown.
diffusion (Dq) may not be negligible at the upper limit of the RH, the model predictions seem to agree with the experimental data. This may be due to the fact that, by the time the RH increases to high values (i.e., significantly more than 75%), the effect of pore diffusion has become negligible during the transient. Figure 10 presents data of two experiments each consisting of a ramp from 15% to 90% RH followed by a ramp down from 90% to 15% RH levels. Note that after each upward ramp the samples were held at the higher humidity levels for sufficiently long periods for equilibrium to be attained within the sheets. Thus, for all of the RH changes, the sheets were initially in equilibrium. For the desorption step predictions, the appropriate desorption isotherm was used in place of the adsorption isotherm (cf. Figure 4). The predictions of the model can be seen to compare very well with the experimental data. In these calculations, we used values of Dp ) 0.0063 cm2/s, kf ) 0.25 cm/s, and ki ) 0.0035 s-1. The experimental data can also be fitted by the simplified model(1) (negligible diffusional resistance in the thickness direction) including the effects of kf and ki but ignoring the effect of pore diffusion as shown in Figure 6. However, note that the model represented by eqs 28 and 29 assumes that the parameter RP f 0; i.e., it is the leading order approximation to eqs 22-26. The predictions are coincident with those of eqs 22 and 23 only when Dp > 0.03 cm2/s for the conditions of these experiments. In conclusion, the mathematical model incorporating diffusion and convective surface resistances can be used to describe transient sorption dynamics in paper sheets. The model provides predictions of the evolution of the average moisture content as well as the moisture content profiles in the sheets. The analysis described in this paper can be used to make judgments and estimates of the mechanical behavior of paper materials under transient humidity conditions. This could also be of use in isolating diffusion effects and their contributions to phenomena such as mechanosorptive creep effects. Nomenclature Bi ) Biot number ) γP c ) water vapor concentration within the void space of the paper sheet (Hpsat/100RgT), gmol/cm3
C ) (c - c0)/(c1 - c0) CGAB ) parameter in the GAB sorption isotherm c0 ) initial concentration of water vapor within the sheet, gmol/cm3 cb ) concentration of water vapor in the bulk environment, gmol/cm3 Dq0 ) reference bound water diffusivity within the fiber matrix DQ ) dimensionless moisture diffusivity function DQ(Q) ) Dq(q)/Dq0 Dp ) diffusion coefficient of water vapor within the sheet’s void space, cm2/s Dq ) bound water diffusivity within the fiber matrix, cm2/s G ) geometric parameter H ) relative humidity, % H0 ) initial RH of the environment H1 ) final RH Hb ) RH of the environment undergoing change kR ) slope of the ramp change in RH KGAB ) parameter in the GAB sorption isotherm Kiso ) slope of the linearized isotherm kf ) mass-transfer coefficient corresponding to the external surface boundary layer, cm/s ki ) intrafiber mass-transfer coefficient, s-1 L ) half-thickness of the sheet, cm M0 ) parameter in the GAB sorption isotherm (defined in eq 11) psat ) saturation vapor pressure of water P ) dimensionless parameter, P ) L2/(tRDp) P′ ) dimensionless parameter, P′ ) L2/(tRDq0) q ) bound water concentration within the fiber matrix (gmol of water/g of dry fiber) q0 ) initial concentration of sorbed water within the sheet, gmol/g of dry fiber qsat ) equilibrium sorbed water concentration within fibers qav ) average moisture content of the sheet, described by eq 12 Q ) (q - q0)/(q1 - q0) Qsat ) dimensionless equilibrium moisture content Rg ) universal gas constant t ) time, s tR ) ramp time for the RH change, s z ) coordinate along the sheet thickness Greek Symbols p ) void fraction of the sheet R ) dimensionless parameter ) Fp(q1 - q0)/(c1 - c0) β ) kitR γ ) kftR/L ζ ) dimensionless position in the sheet ) z/L τ ) dimensionless time ) t/tR Fp ) fiber density, g/cm3
Literature Cited (1) Markstrom, H. The elastic properties of papersTest methods and measurement instruments; Lorentzen and Wettre: Stockholm, Sweden, 1991. (2) Benson, R. E. Effects of relative humidity and temperature on tensile stress-strain properties of kraft linerboard. Tappi J. 1971, 54 (4), 699. (3) Byrd, V. L. Effect of relative humidity changes during creep on handsheet paper properties. Tappi J. 1972, 55 (2), 247. (4) Back, E. L.; Salmen, L.; Richardson, G. Transient effect of moisture sorption on the strength properties of paper and woodbased materials. Sven. Papperstidn. 1983, 6, 61. (5) Gunderson, D. E.; Tobey, W. E. Tensile creep of paperboards effect of humidity change rates. Mater. Res. Soc. Symp. Proc. 1990, 197, 213. (6) Fellers, C., Laufenberg, T., Eds. Moisture-Induced Creep Behavior of Paper and Board, 2nd International Proceedings; STFI: Stockholm, Sweden, 1997.
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(7) Chalmers, I. Proceedings of the third international symposium on Moisture-induced creep on paper, board and containers; PAPRO, Forest Research Institute: Rotorua, New Zealand, 1997. (8) Gunderson, D. E. A method for compressive creep testing of paperboard. Tappi J. 1981, 64 (11), 67. (9) Ramarao, B. V.; Chatterjee, S. G.; Eusufzai, A. R. K.; Tien, C. Moisture transport and sorption by paper under cyclic variations in humidity. Mechanics of Cellulosic Materials; ASME/ AMD: Los Angeles, CA, 1995; Vols. 209/MD p 195. (10) Chatterjee, S. G.; Ramarao, B. V.; Eusufzai, A. R. K. Transient moisture transport in papersA preliminary analysis of diffusion and adsorption. Empire State Paper Research Institute Report 102; Empire State Paper Research Institute: Syracuse, NY, 1995; pp 143-172. (11) Chatterjee, S. G.; Ramarao, B. V.; Tien, C. Water vapor sorption characteristics of a bleached kraft paperboard: A study of the hysteresis region. J. Pulp Pap. Sci. 1997, 23 (8), 366. (12) Ramarao, B. V.; Chatterjee, S. G. Moisture sorption by paper materials under varying humidity conditions. The Fundamentals of Papermaking Materials: Transactions of the Eleventh Fundamental Research Symposium, Cambridge; PIRA Intl.: Leatherhead, Surrey, U.K. 1997, Vol. 2, p 703. (13) Chatterjee, S. G.; Ramarao, B. V. Hysteresis in water vapor sorption equilibria of bleached kraft paper board. Proceedings of the third international symposium on Moisture-induced creep on paper, board and containers; PAPRO, Forest Research Institute: Rotorua, New Zealand, 1997.
(14) Lin, S. H. Moisture absorption in cellulosic materials. Int. J. Eng. Sci. 1990, 28, (11), 1151. (15) Lescanne, Y.; Moyne, C.; Perre, P. Diffusion mechanisms in a sheet of paper. Drying ′92 1992, 1017. (16) Foss, W.; Bronkhorst, C.; Bennett, K. A.; Riedemann, J. R. Transient moisture transport in paper in the hygroscopic range and its role in the mechano-sorptive effect. Proceedings of the third international symposium on Moisture-induced creep on paper, board and containers; PAPRO, Forest Research Institute: Rotorua, New Zealand, 1997. (17) Steele, E. K. GE Rep. 1971, 71MATL205. (18) Tien, C. Adsorption calculations and modeling; Butterworth-Heinemann: Newton, MA, 1994. (19) Radhakrishnan, H. Transient and Equilibrium Moisture Sorption by Paper Sheets. M.S. Thesis, SUNY College of Environmental Science and Forestry, Syracuse, NY, 1999. (20) Chatterjee, S. G.; Tien, C. Adsorption in continuous flowwell mixed tanks: The effect of residence time distribution of adsorbents. Sep. Technol. 1990, 1, 79.
Received for review April 20, 1999 Revised manuscript received October 7, 1999 Accepted October 19, 1999 IE990279W
