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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.20 no.3 São Paulo July/Sept. 2003

#### https://doi.org/10.1590/S0104-66322003000300012

**Adsorption of aromatic amino acids in a fixed bed column**

**M.A.Cremasco ^{I}; R.Guirardello^{I}; N.-H.Linda Wang^{II} **

^{I}School of Chemical Engineering, State University of Campinas, P. O. Box 6066, 13083-970, Campinas - SP, Brazil, E-mail: cremasco@feq.unicamp.br

^{II}Dept. of Chemical Engineering, Purdue University West Lafayette, IN 47907-1283, U.S.A., E-mail: wangn@ecn.purdue.edu

**ABSTRACT**

Phenylalanine (Phe) and tyrosine (Tyr) are two of the twenty amino acids in proteins; they are classified as aromatic amino acids, because both have a benzene ring in their structures. These amino acids are important in the synthesis of several biologically active amines, such as b-endorphin, a neurotransmitter. Amino acids can be separated by ion-exchange chromatography. In this case, it is important that fixed-bed adsorber design adequately predict the breakthrough curve. This work presents a mathematical model for both fluid and porous phases. In the solution proposed for this model the liquid-phase concentration inside the particles is solved analytically and is related to the liquid-phase concentration in the bed using Duhamel's theorem. The solution for liquid-phase concentration in the bed is then solved numerically instead of analytically. The basic mass transfer parameters are from the literature. The results from the model are compared with those obtained experimentally using Phe and Tyr diluted in aqueous solutions in a fixed bed of PVP (poly-4-vinylpyridine) resin.

**Keywords:** adsorption, model, amino acids.

**INTRODUCTION**

Around twenty amino acids are found in proteins. Some amino acids are essential, while others are nonessential. Essential amino acids are those that cannot be produced by the body so must be obtained from the diet. Nonessential amino acids can be produced by the body from essential amino acids. Phenylalanine (Phe) is a ringed essential amino acid that is readily available in most food sources, particularly meats and milk products, with lower levels found in oats and wheat germ. It is essential for many bodily functions and is one of the few amino acids that can cross the blood-brain barrier and thus directly affect brain chemistry. Phenylalanine is the precursor of the amino acid tyrosine (Tyr), which cannot be reconverted, so Phe is essential in the diet. Tyrosine is important to the structure of almost all proteins in the body. Tyr and Phe are also the precursors of several neurotransmitters (Bohinski, 1987), including L-dopa, dopamine, norepinephrine, and epinephrine. Through its effect on neurotransmitters, L-tyrosine may have an effect on several health conditions, including Parkinson's disease, depression, and other mood disorders. Studies have suggested that tyrosine may help people with depression (Gelenberg et al., 1982). Preliminary findings indicate the beneficial effect of Tyr, along with other amino acids, on people affected by dementia, including Alzheimer's disease (Meyer et al., 1977). Because of its role as a precursor to norepinephrine and epinephrine (two of the body's main stress-related hormones) tyrosine may also ease the adverse effects of environmental, psychosocial, and physical stress (Banderet and Lieberman, 1989; Salter, 1989; Neri et al., 1995; Deijen et al., 1999; Shurtleff et al., 1994; Deijen and Orlebeke, 1994; Dollins et al., 1995; Koch, 1996; Chiarone et al., 1990; Alvestrand et al., 1983). Amino acids can be separated experimentally by ion-exchange chromatography, batch ion-exchange chromatography (Dechow, 1990), or a simulated moving bed system (SMB) (Hashimoto et al., 1989; Wu et al., 1998; Cremasco et al., 2000).

An important goal in the design of an adsorber is to predict the sorbate breakthrough curve. Several mathematical models were developed for that curve and analytical solutions can be found in the literature for systems with linear equilibrium (Ruthven, 1984; Carta, 1988). The analytical solution for the complete problem results in an integral expression that is quite difficult to compute numerically (Rasmuson, 1981, 1985) due to the oscillatory nature of the integrand. However, the complex model can be solved numerically using, for example, orthogonal collocation. On the other hand, when the numerical method is used, knowledge of physical adsorption might be lost.

In this paper, a hybrid method is utilized to describe a breakthrough curve for adsorption in a system containing a single adsorbate at a low concentration. The adsorbate is an aromatic amino acid (Phe or Tyr) diluted in water. The liquid-phase concentration inside the particles is found analytically and it is related to the liquid-phase concentration in the bed using Duhamel's theorem (Rosen, 1952). However, the solution for the liquid-phase concentration in the bed is found numerically instead of analytically.

**THEORY**

** Model Hypothesis**

In order to obtain the breakthrough curve for the fluid phase, the following assumptions were made:

a) the liquid mass flow rate is constant and the cross section is uniform;

b) the liquid concentration profile varies very little with radial position in the bed;

c) axial dispersion with a constant dispersion coefficient is taken into account;

d) resistance of the external mass transfer from the bulk liquid-phase to the resin is assumed to be very small (Bi ® ¥);

e) the solid phase (resin) is composed of small spheres of uniform radii;

f) the mobile phase is a dilute solution, so Henry´s law can be used to describe sorbate uptake;

g) intraparticle diffusion is described by pore diffusion. For a linear isotherm system, the flux due to surface diffusion, if important, can be lumped together with the pore diffusion flux (Ma et al., 1996);

h) no chemical reaction occurs;

i) the initial concentration inside the column (liquid and resin) is zero.

j) Applicability of the model will depend on the validity of these hypotheses. In many cases, the experimental conditions can be adequately described by the previous hypotheses, even when they are only approximations.

** Mathematical Model **

Based on these hypotheses, the following equations are obtained:

solute mass balance for the solid particles (resin):

where c is the solute concentration in the liquid phase inside the particle pores (volume fraction e_{p}) and q is the solute concentration in the resin (volume fraction 1- e_{p} );

solute mass balance for the mobile phase:

where C is the solute concentration in the liquid phase inside the void portions of the fixed bed (volume fraction e_{b}) and R_{a} is the rate of adsorption/desorption of solute by the resin by unit volume of reactor, given by

It is important to observe that C is a function of z and t, while c and q are functions of z, r and t, since each particle will be subject to different external concentrations along the length of the column. Using equation (2), equations (1) and (4) can also be written as

**Initial and Boundary Conditions **

The initial and boundary conditions that describe a step injection into a chromatographic column are given as follows:

initial condition:

boundary condition:

Since in this work we are assuming that Bi ® ¥, equation (11) reduces to

**Correlations for the Model Parameters**

The parameters used in the model were calculated using correlations from the literature. The axial dispersion coefficient can be calculated by (Koch and Brady, 1985)

with the molecular mass Peclet number for the particle defined by

and the diffusion coefficient estimated by Wilke and Chang's equation (1955), for the diluted system

with f =2.6 for water, where is calculated from Le Bas´ volume method (Cremasco, 1998).

The effective diffusion coefficient is calculated from Mackie and Meares (1955)

**Strategy for Numerical Resolution**

The method used in this paper is similar to the one from Rosen (1952), except that the solution for the PDE for the bed is found numerically instead of analytically. In this work, the liquid-phase concentration inside the particles is found analytically and is related to the liquid-phase concentration in the bed using Duhamel's theorem. However, the solution for liquid-phase concentration in the bed is found numerically instead of analytically. The procedure is described as follows:

The solute concentration in the liquid phase inside the particle pores, c, can be related to the solute concentration in the liquid phase inside the void portions of the fixed bed, C, using Duhamel´s theorem:

with

where u is the solution for the problem given by equation (5), initial condition (8), and boundary conditions (12) and (13) with C = 1. However, in order to apply equation (18) into equation (6), it is more interesting to work with the average concentration, :

The analytical solution for u can be found using the method of separation of variables, which results in a Fourier series (Cremasco, 1998). This solution is substituted into equation (20), resulting in

where

The series given by equation (21) has fast convergence for large values of t, but it has slow convergence for t approaching zero. Therefore, for very small values of t an asymptotic expression for , obtained using the Laplace transform method (Guy et al, 1982), was used:

which gives accurate results for 0 __<__ q __<__ 0.05.

Using Duhamel´s theorem and the average concentration , an expression for R_{a} as a function of C is obtained:

This expression was used in equation (3), which was solved numerically using a first-order discretization method that avoids both numerical diffusion and numerical instability, based on a combination of implicit and explicit finite differences with a fixed step for Dz and a calculated step for Dt as a function of the column parameters and Dz. The initial and boundary conditions are given by equations (7), (9), and (10). The resulting set of equations is a linear tri-diagonal system in the variables C(z_{i}, t_{j=1}), which was solved with Thomas´s algorithm at each time step, calculating each z profile C(z_{i}, t_{j=1}) from z profiles at previous times.

The main advantage of this approach is that it is easier to compute than the entirely analytical solution (Rosen, 1952). The disadvantage of the analytical solution is that it is quite difficult to calculate, since it is expressed as an integral of an oscillatory function that is very difficult to compute numerically (Rasmuson, 1981, 1985). This difficulty can be avoided with the hybrid approach used here.

The main limitation of the approach used is the fact that Duhamel´s theorem is only applicable to linear PDE with linear initial and boundary conditions. However, the same limitation holds for the entirely analytical solution, since it is based on the same model (Rosen, 1952). ** **

** EXPERIMENTAL**

The experimental setup is illustrated in Figure 1. The system consists of two Pharmacia P-500 low-pressure pumps, a Pharmacia LCC-500 controller, and a Pharmacia MV-7 injection valve. A Waters 990 photodiode array detector was used for data acquisition, and the data were processed using the Waters 900 software. A Pharmacia glass column was packed with Reillex^{TM}.-HP resin, which is poly-4-vinylpyridine cross-linked with divinylbenzene (PVP), purchased from Reilly Industries, Inc., Indianapolis, USA. The characteristics of the column and resin are shown in Table 1. The amino acids studied were Phe and Tyr, with purities of 98% and 99%, respectively. The partition coefficients were k_{p}=1.947 for Phe and k_{p}=3.229 for Tyr (Cremasco et al., 2001).

Breakthrough curves were obtained using injection of aqueous solutions of 0.5 g/l of Phe and of 0.04 g/l of Tyr. All experiments were performed at 1 atm and 25ºC. The feed flow rates were 1.5 ml/min, 2.0 ml/min, and 2.5 ml/min. The effluent was monitored within the range of ultraviolet wavelengths: 260 nm for Phe and 300 nm for Tyr.

**RESULTS AND DISCUSSION**

To solve the model described above, it is necessary to estimate the free diffusion coefficient value, equation (16). By Le Bas´s volume method, it is possible to estimate the molar volume at boiling point of Phe and Tyr. For Phe, = 193.4 cm^{3}/gmol, while for Tyr, this value is = 200.8 cm^{3}/gmol. If equation (16) is used at 25° C and infinite dilution is assumed, D_{AB} = 4.25 ´10^{–4} cm^{2}/min for Phe, which is close to the experimental value from Paduano et al. (1990) (D_{AB} = 4.25 ´10^{–4} cm^{2}/min). For Tyr, the value from equation (16) isD_{AB} = 4.25 ´10^{–4} cm^{2}/min.

The effective diffusion coefficients for Phe and Tyr are calculated by equation (17), using the values of D_{AB} from equation (16). In this case, D_{P} = 1.10´10^{–4} cm ^{2}/min for Phe and 1.07´10^{–4} cm ^{2}/min for Tyr, which are close to the values in from Cremasco et al. (2001) D_{P} = 1.02´10^{–4} cm^{2}/min for Phe and 1.01´10^{–4} cm ^{2}/min for Tyr).

The theoretical values for the diffusion coefficient and effective diffusion coefficient were used to describe the breakthrough curves. Other values used in the modeling are shown in Table 1. The resulting theoretical curves for 1.5ml/min, 2.0ml/min, and 2.5ml/min are compared with those of the experimental method. Figures 2 and 3 show a comparison between experimental adsorption curves and those from the model proposed in this paper.

It can be observed that the calculated values agree well with the experimental ones. This is even more interesting considering that mass transfer parameters were estimated from correlations but not fitted, i.e., no fitting method was used to adjust the parameters in this work. Therefore, the method has a good predictive capability since it required only previously available data to obtain good results.

However, there is a difference between the calculated and the experimental values, particularly in Figure 3, that needs to be explained. The area above the breakthrough curve given by

is related to the adsorption capacity of the column. In Figure 3, for example, it is clear that the areas for the experimental and calculated curves are not the same. It can be seen from equation (25) that this area does not depend on the mass transfer parameters (E_{b}, D_{p}, D_{AB}). Also, numerical tests and comparison with the analytical value given by the right hand side of equation (25) indicated that the difference between calculated and experimental values in the breakthrough curves was not due to the numerical method used. Therefore, this difference is very likely due to some of the other parameter values used in the simulations (V,Q,e_{b}, e_{p}, k_{p}), which may have been incorrectly determined in the experiments.

All calculations were performed in a Pentium 150 MHz with 16 Mbytes RAM, using Fortran 77. The number of intervals in the axial direction was 300, while the number of time intervals is shown in Table 2. The computational time was less than 10 seconds for each case. The precision of the numerical approach can be improved with more intervals, but that requires longer computational times. For the examples tested, it was not necessary to use a more refined grid of points.

**CONCLUSIONS**

A hybrid approach for solving the model for fixed-bed biosorption was proposed. The hybrid approach has the advantage of being easier to apply than the purely analytical solution. Results indicate that the method was efficient and easy to apply, solving the problem in a very short computational time. The numerical results described well the experimental data in the breakthrough curves for two amino acids and three different flow conditions, using only estimated parameters from the literature.

The performance of the model with the proposed method for solving it, using the parameters from correlations, indicates that it can be a useful tool for predicting breakthrough curves. Although the mathematical model has some limitations, it can be applied to a variety of situations, as long as the model hypotheses are satisfied.

**NOMENCLATURE**

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Received: June 10, 2002

Accepted: February 28, 2003