활성탄 흡착에 있어서 입자내유효확산계수 및 흡착속도상수의 입도 의존성

송 훈, 한 능 원

전남대학교 화학공학과

The Dependence of Intraparticle Effective Diffusivity and Adsorption Rate Constant on Particle Size in Activated Carbon Adsorption

Hoon Song, Neung-Won Han

Dep. of Chem. Eng. Chonnam National University

Introduction

Complex transfer mechanisms contribute to the transportation of adsorbent into activated carbon particles in liquid phase adsorption. There are several models presented, such as pore diffusion model, surface diffusion model, macropore diffusion model, and micropore diffusion model. These model equations, while realistic, contribute little to the design of adsorber with complex calculation processes (Ruthven, 1984). The linear driving force model (LDF) is, on the other hand, very simple and practical in use. However, LDF rate equation needs to be tried in many situations because of its emprical nature.

This study concerned the validity of LDF rate equation for different size particle and mixed size particles.

Theoretical

Unsteady state diffusion equation for overall diffusion flux based on adsorption phase concentration gradient is

[eqn: rm partialq over partialt = D_q ({partial^2 q}over partialr^2 + 2 over r partialq over partialr )]                                       (1)

This equation assumed pseudo-homogeneous medium and constant effective diffusivity.

The solution of Eq 1. for finite bach adsorption and linear isotherm was given by Crank(1975). Dryden and Kay(1954) derived a similar solution after Peterson(1941). Their parameter, a relates to the linear slope of isotherm between initial and final concentration, as

[eqn: rm a= V over M left({q_o - q_inf} over {c_o - c_inf} right)]                                             (2)

The linear isotherm approximation between co and c∞ is more acceptable than the linear isotherm for entire concentration range to co. The solution is,

[eqn: rm E=1-(2/3a)sum from n=1 to inf {exp(-p_n ^2 theta)} over {1+p_n /9a(1+a)}]                             (3)

and

[eqn: rm tan p_n = 3ap_n over 3a+p_n ^2]                                               (4)

LDF rate equation is

[eqn: rm {d overlineq} over dt = it k rm (q_s - overlineq )]                                              (5)

The LDF rate constant, k is generally related to Dq as

[eqn: k = {PHI D_q} over R^2]                                                      (6)

Glueckauf(1955) approximated □=15 at large time. The same result was also given by Rice(1982) for a parabolic concentration particle in a particle.

For mixed size particle LDF rate equation can be written as,

[eqn: rm {d overlineq}_m over dt = it k_m rm(q_s - {overline q}_m )]                                         (7)

and for constant Dq and □, km is with  mean radius

[eqn: k_m = rm {PHI D_q} over R_m ^2]                                                    (8)

otherwise

[eqn: k_m = rm summ_i PHI_i D_qi over R_m ^2 = sum m_i it k_i rm left(R_i over R_m right) ^2]                         (9)

Experimental

Experimental apparatus for batch adsorption kinetics is shown in Fig 1. Concentration of bath solution(phenol) was traced by circulating the solution to UV flow cell. Pre-wetted carbon particles (Calgon F400 4 sizes and mixed size) was agitated in stirred solution to minimized the effect of stirring speed. A cylinderical wire screen barrier was set around magnetic bar to prevent crushing of particles.

Initial concentration was 120 to 130 ppm and M/V was fixed to 0.5 g/l

Results and Discussion

One set of experimental results was shown in Fig 2. Eq. 3 & 4 were incorporated with the results and obtained t vs. □, which was plotted in Fig 3. Dq was evaluated form the slope of Fig 3. The slopes were nonlinear for large time, usually when E exceeded 0.75 and Dq was determined from the slope between 0.3 and 0.7 of E. Short time region(below 0.3 of E)was not included to meet Glueckauf's approximation. Dq values, as presented in Table 1., decreased slightly with larger particle sizes.

Eq. 5 was plotted in Fig 4. to evaluate k(Miller, 1970). Since correlation were nonlinear, k value were determined from linear approximation between 0.3 and 0.7 of E, as the case for Dq. The rate constant values for different particle sizes were shown in Fig 5. The constant is proportional to 1/R instead of 1/R2 as Eq. 6.

Rate constant for mixed size particle agree well to Eq. 9, when mass mean radius was employed.

Conclusion

In liquid phase adsorption of phenol into activated carbon of different particle sizes, Glueckauf's coefficient did not hold 15 and varied by particle size. The LDF rate constant was then inversely propertional to particle radius.

Notation

a : Dryden-Kay parameter                   t : time

c : concentration                            V : volume of bath solution

Dq : intraparticle effective diffusivity         □ : dimensionless time, Dqt/R2

E  : uptake fraction

k, km, ki : LDF rate constant

M : mass of adsorbent

mi : mass fraction

q : adsorption at a location

[eqn: rm overlineq ,{overlineq}_m] : average adsorption of particles

qs : adsorption at surface

R, Ri, Rm : particle radius

r : radial distance from surface

References

1. Crank, J. : " The Mathematics of Diffusion" , 2nd ed., Clarendon Press. Oxford (1975).

2. Dryden,  C. E. and Kay, W. B. : Ind. Eng. Chem., 46, 2294(1954).

3. Glueckauf, E. : Trans. Faraday Soc., 51, 1540(1955).

4. Miller, C. O. M. and Clump, C. W. : AIChE J., 16, 169(1970).

5. Peterson, S. : Proc. Phys. Soc. (London), 59, 50(1947).

6. Rice, R. G. : Chem. Eng. Sci., 37, 83(1982).

7. Ruthven, D. M. : "Principles of Adsorption and Adsorption processes", John Wiley & Sons. Inc. (1984).