Abstract
In this work a mathematical model for homogeneous diffusion controlled batch adsorption that take nonequilibrium distribution between bulk liquid and solid average concentration into account is developed for the nonlinear adsorption isotherm case. We modified the conventional batch adsorption analysis by introducing an equation that enables the prediction of solid and liquid concentrations at the adsorbent particleliquid interphase. During the experimental phase of this study batch uptake experiments were carried out with αamylase, BSA and IgG, with molecular masses ranging from 51 000 to 160 000, onto a strong anion exchanger Q Sepharose XL. For these solutes and the adsorbent media, single component homogeneous diffusion coefficients were experimentally determined with the nonequilibrium methodology, introduced in the present study.
The simple homogeneous diffusion model appears to predict the experimentally observed trends with respect to protein concentration and boundary layer mass transfer effects for the gelcomposite adsorbents (Weaver and Carta, 1996; Hunter and Carta, 2002). Many authors for example, Skidmore et al., (1990); Yoshida et al., (1994); Tonga et al., (1994); Weaver and Carta, (1996); Chang and Lenhoff, (1998); Hunter and Carta, (2000); Chen et al., (2002); Tscheliessnig et al., (2005); Gao et al., (2006) have used batchadsorption rate measurements to determine the diffusivities of proteins in commercial media using other diffusivity models as well.
Theoretical
The solute mass balance around the reservoir shown in Fig. 1 gives:
_{ }
where c(t) is the time dependent bulk liquid concentration (mg/cm^{3}), m is the adsorbent weight within the reservoir (g), V is the liquid volume in batch adsorber (cm^{3}), _{ }
_{ }
where _{ }
_{ }
For the early stages of batch uptake from dilute solutions, and for favorable isotherms, _{ }
_{ }
Figure 2. Visualization of liquid and solid concentrations in homogeneous adsorption model
Eq. (4) indicates that k_{f} value can easily be determined for short times and low concentration batch experiment uptake data as long as the adsorption isotherm is favorable. Once k_{f} value is obtained from a low concentration experimental data, then homogeneous diffusivity, D_{s} can be calculated from high concentration experiments, by keeping all other experimental parameters the same with that of the low concentration experimentation, since the already calculated k_{f} value is needed in the D_{s} determination.
Equilibrium dispersive model (Guiochon et al, 1994) is employed by many authors, where instant equilibrium is assumed between the solid and the liquid phases In this work, the equilibrium dispersive between the solid average concentration, _{ }
Another feature of the nonequilibrium model presented in this study is that it obviates the usually employed solution of coupled ordinary and partial differential equation systems (McCue et al., 2003) and makes it possible to use the well known RungeKutta algorithms during simulation studies. Özdural et al., (2004) showed that for a parabolic solid concentration profile within an adsorbent particle, interphase liquid concentration can be given by Eq. (5).
_{ }
(5)
_{ }
It is assumed that Langmuir adsorption isotherm, as shown in Eq. (7), holds at the interphase.
_{ }
Eq. (3) can now be solved without assuming _{ }
_{ }
For finding homogeneous diffusivity, model predicted reservoir concentration versus time data for different D_{s} values were numerically generated and compared with the experimentally gathered reservoir concentration versus time data until the two profiles agree. D_{s} value which gives the best fit, evaluated through the rootmeansquare (rms) analysis, was taken as the homogeneous solid diffusion coefficient of the system of interest.
Results
The equilibrium uptake isotherms for IgG, BSA and aamylase on Q Sepharose XL were nonlinear and Langmuir adsorption isotherm parameters, q_{m} and b are summarized in Table 1. For the present experimental conditions k_{f} values, calculated through Eq. (4), were 0.78 x 10^{3} cm/s, 2.92 x !0^{3} cm/s and 0.19 x 10^{3} cm/s for αamylase, BSA and IgG respectively. It was found that the homogeneous diffusion model successfully predicts the observed uptake trends. The experimentally determined homogeneous diffusion coefficient values (given in Table 1) via nonequilibrium model indicate that D_{s} values decreases as the protein M.W. increases.
Table 1. Equilibrium parameters and homogeneous diffusivities
Protein
 M.W. (kDa)
 q_{m} (mg/cm^{3})
 b (cm^{3}/mg)
 D_{s} x 10^{9} (cm^{2}/s)

αamylase
 51
 161.3
 12.4
 1.38

BSA
 66
 153.9
 32.5
 1.05

IgG
 160
 170.1
 11.3
 0.92

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