상세 동적 모사를 통한 회분 및 고정층 흡착탑의 비교

이혜진, 고대호, 문일

연세대학교 화학공학과

Comparison of Batch and Fixed Bed Adsorption using Rigorous Dynamic Simulation

Hyejin Lee, Daeho Ko and Il Moon

Department of Chemical Engineering, Yonsei University

1. Introduction

In this study, a rigorous mathematical model was developed to analyze the dynamics of the adsorption systems. The model includes film and pore diffusion mass transfer resistances, as well as the rate of interaction between solute and ligand. In addition, the efficiency of the protein adsorption was compared in the batch and the fixed bed modes by the rigorous dynamic simulation. The performance of this separation depends on several factors including solute concentration, surface reaction kinetics, ligand loading, and particle size. This paper also considers the variation of the radial concentration of the particle.

2. Model Formulation - Adsorption

In the adsorption step, solute P of interest gets adsorbed onto the matrix on which the ligand L is immobilized. The affinity interaction between the protein and the vacant immobilized ligand is assumed to be of the form:

P   +   L    =    PL

(protein)  (ligand)   (complex)

The mass balance equation for the solute adsorbed on solid phase is written as:

[eqn: partialq_i over partialt = k_i C_i ( Q_m -q_i ) -k_2 q_i~ ,~~~~~~~~ FORALLz IN [0,L]]	(1)

Eq. (1) is simplified to Langmuir's isotherm at equilibrium.

[eqn: q_i = {Q_m C_i} over {(K_D +C_i )}]	(2)

The following basic assumptions are made for the general rate models.

① The effective diffusivity does not depend on concentration.

② The film model governs the mass transfer to the surface of the adsorbent.

③ The immobilized ligand is distributed uniformly throughout the interior of the particle.

·Mass balance in the batch bed

[eqn: partialC over partialt =- 3V_s over RV_L k_f (C-C_i vert _r=R ), ~~~~ FORALLz IN (0,L )]	(3)

·Mass balance in the fixed bed

[eqn: u_i partialC over partialZ + partialC over partialt + 3V_s over RV_L k_f (C-C_i vert _r=R ) = E_z {partial^2 C} over partialZ^2 ,~~~~~ FORALLz IN (0,L)]	(4)

·Mass balance in the particle

[eqn: D_i LEFT ({partial^2 C_i} over partialr^2 + 2 over r partialC_i over partialr RIGHT ) - partialC_i over partialt - partialq_i over partialt = 0, ~~~~~~ FORALLz IN [0,L]]	(5)

·Initial condition (batch bed)

[eqn: C=C-_0 , ~~~~~~~~~~~FORALLz IN (0,L)]	(6)

[eqn: C_i =0,~~q_i = 0,~~~~~~ FORALLz IN [0,L]]	(7)∼(8)

·Initial condition (fixed bed)

[eqn: C=0 , ~~~~~~~~~~~~~~FORALLz IN (0,L)]	(9)

[eqn: C_i =0,~~q_i = 0,~~~~~~ FORALLz IN [0,L]]	(10)∼(11)

·Boundary condition

[eqn: u_i C_0 = u_i C ~ RIGHT |_z=0 -E_z partialC over partialZ RIGHT | _z=0 ~, ~~~~~~partialC over partialZ RIGHT | _z=L =0]	(12)∼(13)   [eqn: partialC over partialr RIGHT |_ r=0 =0~,~~~~~~k_f~ (C-C_i vert_r=R ) =D_i partialC_i over partialr RIGHT | _r=R]	(14)∼(15)

3. Numerical Solution Strategy

A mathematical model including mass balance is formulated and described by a set of partial differential and algebraic equations (PDAEs). In this study, method of lines (MOL) and finite difference method (FDM) are adopted for the conversion of PDAEs system to differential algebraic equations (DAEs) system. For dynamic simulation, integration for time is achieved by using DAE integrator based on backward differentiation formula (BDF). Main advantage of this method is to minimize an integral error efficiently by this DAE integration technique. The dynamic simulation is performed by the dynamic simulator, gPROMS.

4. Result and Discussion

For the comparison purpose the effect of process variables such as solute concentration, ligand loading, reaction kinetics, and particle size has been evaluated in terms of solute recovery and adsorbent utilization efficiencies for the both batch and column modes.

The corresponding efficiencies are defined as:

Solute recovery efficiency = [eqn: 1-C over C_0]	(16)

Adsorbent utilization efficiency =[eqn: q over Q_m]	(17)

Fig. 1 displays the effect of the solute concentration on the both batch and fixed bed modes. A lean feed has better solute recovery efficiency and less adsorbent utilization efficiency than rich feeds. Fig. 2 shows the effect of reaction kinetics on the breakthrough behavior of the solute. For both the cases, higher forward rate constant lead to better efficiencies with regard to both solute recovery and adsorbent utilization. Fig. 3 depicts the effect of ligand concentration on both batch and fixed bed modes. For the batch case, the higher the ligand concentration, the less the adsorbent utilization and better the solute recovery. In case of the fixed bed, the two efficiencies increase according to the rise of the ligand content. Fig. 4 is the comparison of breakthrough curves for three particle sizes. A change in particle size affects the film mass transfer coefficient. The film mass transfer coefficient is inversely proportional to the particle size for the batch adsorption. Fig. 5 and Fig. 6 are the concentration profile for the radial direction of particle in both modes when operating time is 1000 second.

5. Conclusion

This study showed the dynamic changes of the solute recovery and adsorbent utilization efficiencies for batch and fixed bed modes as the following parameters vary: solute concentration, surface reaction kinetics, ligand loading, and particle size. By analyzing the relationship among the variables, we formulated design strategies. As a result of rigorous dynamic simulation, we found that the fixed bed adsorption is more efficient than the batch adsorption for the rich feed. When the feed is lean, the batch mode is preferable. This paper also analyzed the adsorption behavior of the bulk as well as the particle and considered the concentration varies with the radial direction of the particle.

References

1. P. Sridhar, Chem. Eng. Technol. 19, pp.357-363, 1996.

2. D. J. Hwang, M. Oh, and Il Moon, Journal of the Korean Institute of         Chemical Engineering, 36, No. 2, p151-158, 1998.

3. D. H. Ko and Il Moon, '99 Korea-China Joint Workshop on Process          Systems Engineering, Taejon, Korea, pp.111-114, 1999.

4. W. E. Schiesser, The Numerical Method of Lines, Academic Press, New

York, 1991.

5. R. B. Jarvis, Robust Dynamic Simulation of Chemical Engineering            Processes, PhD Thesis, Imperial college of Science, Technology and           Medicine, 1993.

Fig 2. Effect of reaction kinetics

Fig 1. Effect of solute concentration

Fig 4. Effect of particle size

Fig 3. Effect of ligand concentration

Fig 5. Concentration of radial direction in the batch bed

Fig 6. Concentration of radial direction in the fixed bed