Two-stage sorption

Recently, Long and Richman (1960) have shown that both sigmoid and two-stage sorption curves can be derived from Eq. (1) (with D dependent on Cj only) if, in place of Eq. (15), one assumes for Cj a somewhat more general equation of the form ... 

The two-stage sorption behavior is not a unique characteristic of the cellulosic systems investigated by Long et al. Subsequent studies, by... 

Joshi S and Astarita G. Diffusion-relaxation coupling in polymers which show two-stage sorption phenomena. Polymer 1979 20 455 58. 

Considering the two-stage sorption process of wool, David and Nordon [65] proposed an exponential relationship to describe the rate of water content change in the fibers, as shown in equations (3) and (4) ... 

Watt, I.C., Kinetic study of the wool-water system, Part I the mechanisms of two-stage sorption, Textile Res. J. 58, 581-592, 1988. 

T. goesingense. The required input parameters were partly measured for the experimental conditions of the rhizobox experiment and partly obtained from literature data. The curves that matched best measured labile Ni were those obtained by the most simple initial model, the model including release from a fixed Ni phase and the two-stage sorption model. The simulation results for each of the three models were quite similar, indicating that the influence of the additionally included processes was relatively small under the assumptions of equilibrium between fixed phase and solution in the bulk soil. The models including both root hairs and exudation overestimated depletion of labile Ni close to... 

Model 5 model 1 including a two-stage sorption model... 

Note Oj is the rate constant of the two-stage sorption process. 

The values for the exudation rate F, interaction coefficient (A), buffer power of exudate in soil b and the decomposition rate constant for the exudate k were adopted from Kirk (1999). The value of the forward rate constant was estimated from Scheckel and Sparks (2001), who evaluated kinetic adsorption data of Ni to different minerals where ranged from 2.5 x 10 to 9.78 X 10 s For the simulation, an average value of 5.00 x 10 was used. This value also coincides with the values that Kirk and Staunton (1989) suggested for the kinetic adsorption of Q to soil, where the values ranged from lO" to 10 2 s f This same value was assumed for the rate constant for the two-stage sorption model, a2- The fraction of type 1 sites (F ) was assumed to be 0.3. Table 7 summarizes all input parameter values. 

Fig. 2. Simulated curves of (a) labile and (b) dissolved Ni (NiJ vs. measurements. Model 1 initial model Model 2 plus root hairs Model 3 plus root exudation Model 4 plus slow reacting Ni Model 5 plus two-stage sorption Ni, initial concentration.

Fig. 7. Effect of varying the values of on (a) labile Ni and (b) dissolved Ni (Ni ) calculated by Model 4 including slow reacting Ni and a on (c) labile Ni and (d) dissolved Ni (Ni ) calculated by Model 5 including two-stage sorption.

In another work, Nordon and David [44] have modified their analysis to take account of the two-stage sorption behavior of a textile material and moistness dependence of the transfer rate. Use of this elaboration does not alter the essential conclusions of Henry s linearized analysis. A wad of textile material exposed to a sudden rise in ambient relative humidity experiences the passage of the fast temperature front with a rise in temperature, and a slow front with a drop in temperature of equal magnitude. Of the two corresponding humidity fronts, only the slow one is obvious as the fast one is very small and superimposed on the major one. 

When fresh sorbents are used at each stage and the pseudo-second order rate expression is used to describe equilibrium in the two-stage sorption system, then the mass balance equation can be obtained by combining Eqs. (15.19) and (15.20) ... 

In another work, Nordon and David [44] have modified their analysis to take account of the two-stage sorption behavior of a textile material and moistness dependence of the transfer rate. Use of... 

De Wilde WP, Shopov PJ (1994) A simple model for moisture sorption in epoxies with sigmoidal and two-stage sorption effects. Compos Struct 27(3) 243-252 Derrien K, Gilormini P (2007) The effect of applied stresses on the equilibrium moisture content in polymers. Scr Mater 56(4) 297-299... 

In these equations, both and X are functions of the concentration of water absorbed by the fibers. Most textile fibers have very small diameters and very large surface/volume ratios. The assumption in the second equation of instantaneous thermal equilibrium between the fibers and the gas in the inter-fiber space does not therefore lead to appreciable error. The two equations in the model are not linear and contain three unknown, i.e., Cj-, T, and. A third equation should be established appropriately in order to solve the model. Researchers derived a third equation to obtain an analytical solution by assuming thatC is linearly dependent on T and C , and that fibers reach equilibrium with adjacent air instantaneously. Considering the two-stage sorption process of wool. Scientists proposed an... 

The assumptions in his model do not allow for the complexity of the moisture sorption isotherm and the sorption kinetics of fibers. Scientists presented two mathematical models to simulate the interaction between moisture sorption by fiber and moisture flux through the air spaces of a fabric. In the first model, they considered diffusion within the fiber to be so rapid that the fiber moisture content is always in equilibrium with the adjacent air. In the second model, they assumed that the sorption kinetics of the fiber follows Fickian diffusion. Their model neglected the effect of heat of sorption behavior of the fiber. Scientists developed a new sorption equation that takes into account the two-stage sorption kinetics of wool fibers, and incorporated this with more realistic boundary conditions to simulate the sorption behavior of wool fabrics. They assumed that water vapor uptake rate of fiber consists of a two components associated with the two stages of sorption identified by researchers. 

Li, Y. and B. Holcombe, A two-stage sorption model of the coupled diffusion of moisture and heat in wool fabrics. Textile Research Journal, 1992, 62(4), 211-217. 

In the case of hydrophilic fiber assemblies, vapor diffusion is governed by a non-Fickian, anomalous diffusion [30]. This is a two stage diffusion process. The two stage sorption has been first expressed by Nordon and David [31], The first stage corresponds to a Fickian diffusion with a constant diffusion coefficient but the second stage is non-Fickian diffusion associated with viscoelastic relaxation of the fiber stmcture [32],... 

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