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A New Methodology for the Determination of Homogeneous Diffusion Coefficients of Biomolecules in Adsorbents Via Non-Equilibrium Modeling

Gönül Akkaya and Ahmet R. Özdural. Chemical Engineering Department, Hacettepe University, Beytepe, Ankara, Turkey


In this work a mathematical model for homogeneous diffusion controlled batch adsorption that take non-equilibrium 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 particle-liquid interphase. During the experimental phase of this study batch uptake experiments were carried out with α-amylase, 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 non-equilibrium 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 gel-composite 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 batch-adsorption rate measurements to determine the diffusivities of proteins in commercial media using other diffusivity models as well.


The solute mass balance around the reservoir shown in Fig. 1 gives:


where c(t) is the time dependent bulk liquid concentration (mg/cm3), m is the adsorbent weight within the reservoir (g), V is the liquid volume in batch adsorber (cm3), is the time dependent solid average concentration (mg/cm3solid), ρp is the adsorbent particle density (g/cm3) and t is the time (s). With referring to Fig. 2, the solid uptake rate for spherical adsorbent particles can be expressed as following (Radcliffe et al., 1982; Cooney, 1991).


where is the time dependent interphase liquid concentration (mg/cm3), rp is the particle radius (cm) and kf  is the liquid film mass transfer coefficient (cm/s).  Let's substitute Eq. (2) into Eq. (1):


For the early stages of batch uptake from dilute solutions, and for favorable isotherms, , (Mc Cabe et al., 1985). Under these conditions integrating Eq. (3) between the limits of t = 0, c = c0; t = t, c = c(t) gives:  


Figure 2. Visualization of liquid and solid concentrations in homogeneous adsorption model

Eq. (4) indicates that kf value can easily be determined for short times and low concentration batch experiment uptake data as long as the adsorption isotherm is favorable. Once kf  value is obtained from a low concentration experimental data, then homogeneous diffusivity, Ds 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 kf value is needed in the Ds 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,  and the bulk liquid concentration,  is ruled out since under dynamic conditions with realistic mass transfer resistances this assumption is not valid (Skoog et al., 2003). The assessment presented above illustrates that in order to reach a factual batch uptake adsorption dynamics portrait the inclusion of non-equilibrium constraints in the modeling studies, is essential at least for the case of larger, slowly diffusing molecules. However still it is reasonable to assume that equilibrium is attained at the interface, i.e. resin surface concentration,  is considered to be in equilibrium with interphase liquid concentration, , since transfer of solute at the interphase to adsorbed state is generally very fast.

Another feature of the non-equilibrium 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 Runge-Kutta 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).




It is assumed that Langmuir adsorption isotherm, as shown in Eq. (7), holds at the interphase.


Eq. (3) can now be solved without assuming  since Eq. (5) gives value. During the numerical solution procedure, the    value at j time panel is evaluated from the c(t)  and   values at j-1 time panel through the mass balance methodology of  Özdural et al. (2004).


For finding homogeneous diffusivity, model predicted reservoir concentration versus time data for different Ds values were numerically generated and compared with the experimentally gathered reservoir concentration versus time data until the two profiles agree. Ds value which gives the best fit, evaluated through the root-mean-square (rms) analysis, was taken as the homogeneous solid diffusion coefficient of the system of interest.


The equilibrium uptake isotherms for IgG, BSA and a-amylase on Q Sepharose XL were nonlinear and Langmuir adsorption isotherm parameters, qm and b are summarized in Table 1. For the present experimental conditions kf 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 α-amylase, 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 non-equilibrium model indicate that Ds values decreases as the protein M.W.  increases.

Table 1.  Equilibrium parameters and homogeneous diffusivities


M.W. (kDa)

qm (mg/cm3)

b (cm3/mg)

Ds x 109 (cm2/s)

















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