A one-compartment single-pore model for extracorporeal hemoperfusion

0010-4825189 $3.‘JO+.C+l e 1989 Pergamon Press plc

Compur. 8101. Med Vol. 19. No. 2. pp. 83-94. 1989 Printed in Great Bn~a\n

A ONE-COMPARTMENT SINGLE-PORE MODEL EXTRACORPOREAL HEMOPERFUSION

FOR

CHAU-JEN LEE*, SHIH-TONG Hsu and SU-CHIEH Hu Department

of Chemical

Engineering,

National

(Received6 October 1987; in rerisedform

Tsing Hua University,

6 May 1988; receioedfor

Hsinchu. publicatron

Taiwan,

R.O.C.

19 May 1988)

Abstract-A one-compartment single-pore model is proposed for the dynamic analysis and performance evaluation of a hemoperfusion column under operation. Experimental data obtained from binary systems containing creatinine and uric acid may be well predicted by the theoretically computed curves. This theory offers a means for the preclinical evaluation of any prototype hembperfusion column.

Adsorption

Hemoperfusion

Activated

1.

carbon

Mass transfer

INTRODUCTION

Hemoperfusion which utilizes the adsorption principles by passing the patient’s blood directly through a fixed bed of activated carbon particles (specially treated), is clinically accepted for cleansing the blood from unwanted metabolic wastes or intoxicated materials. A typical system flow diagram of clinical hemoperfusion is shown in Fig. 1. Yatsidis in 1964 [l] undertook an in uiuo hemoperfusion experiment with animal and demonstrated that a fixed-bed cartridge containing activated carbon particles could

venous pmu, monitor

Arterial pr.rrur* monitor Heporin-cd&d physdopicol saline solution

Fig. 1. Typical *To whom correspondence

should

ifow diagram

be addressed. 83

of clinical hemoperfusion.

CHAU-JENLEE er al.

84

successfully remove creatinine, uric acid, phenol and barbiturates from the animal blood. Since then, much research work on hemoperfusion has been reported in the literature but mostly on the results of animal experiment or clinical experiences. There has been few literature which studied the hemoperfusion from its basic adsorption kinetics and problems related to the engineering design of the hemoperfusion columns [2-41. The adsorption of an adsorbate onto activated carbon depends on both external and internal mass transfer and is mechanistically complicated [S-7]. In the literature, most studies on liquid phase adsorption kinetics were reported on systems related to industrial waste-water decontamination [S], ion-exchange processes [9] and catalytic reactors [lo]. In this paper, a one-compartment single-pore model is proposed for the dynamic analysis and performance evaluation of a hemoperfusion column under extracorporeal operations. Langmuir isotherm, axial dispersion, film and pore diffusion mechanisms are assumed to formulate a set of system equations. The equations were then solved numerically with Crank-Nicolson and implicit finite backward difference method [l 11. Experimental data obtained from binary systems containing creatinine and uric acid respectively, fit the theoretically computed curves fairly well. This simple model thus enables one to predict the dynamic performance characteristics of a hemoperfusion column. 2. MODEL

DEVELOPMENT

Figure 2 depicts a single compartment model for the hemoperfusion system concerned, where the body was assumed as a well-mixed tank. The solute to be adsorbed in the fixed particle bed diffuses through the bulk liquid phase towards the solid adsorbent. The mass transport occurs in four steps: (a) film diffusion; (b) pore diffusion; (c) pore-surface diffusion; (d) adsorption at interior sites of particles. With those transport mechanisms, the following differential equations may be formulated for different mass transfer conditions (i) around a single particle; (ii) in the fixed bed, and (iii) in the well-mixed tank, respectively. (i) Mass transfer around a single particle,

-

aq

(1)

PP&l

the initial and boundary conditions are

c, = 0,

aC Sr

--‘=

0

’

(t = 0)

(2)

(r = 0)

(3)

D,,,!&,=k,(C, - C,lr=d.

(4)

(ii) Mass transfer in the fixed bed,

acb

&bat

a2cb =

DzEb

az2

~~

-

A,k,(Cb

-

Crlr=R)

Well

mlxed tank Fixed bed

/ i

Fig. 2. A single-compartment

model for hemoperfusion

system.

(5)

Extracorporealhemoperfusion

85

the initial and boundary conditions are Cb = 0,

(r = 0)

(6)

(r = 0).

(10)

(iii) Mass transfer in the well-mixed tank,

c, = co,

Thus, the one-compartment model is able to predict the time-history of solute concentration change in the body during the hemoperfusion, and that the efficacy of solute removal can be determined. Nevertheless, equations (l)-( 10) can not be solved analytically because they are nonlinear partial differential equations. They have to be solved numerically with Crank-Nicolson and implicit finite backward difference method [ll].

3. PARAMETER

EVALUATION

Several parameters and coefficients have to be evaluated or determined numerically before model equations may be computed. (i) The adsorption capacity for each species was obtained experimentally and correlated with Langmuir or linear form of isotherm equations,

q*c = f (C,) =

Langmuir isotherm Linear isotherm

_-EL 1 + aC,

qAc=KCr

(111

(12)

where a, /I are Langmuir constants, and K represents the Henry constant. At 37°C the experimentally determined isotherms for creatinine (mol.wt = 113.1) and uric acid (mol.wt = 168.1) are respectively (Figs 5 and 6), 305 c, qAC= 1 + 4750 C,’ qAC

=

250c,,

for creatinine

for uric acid.

(13) (14)

(ii) The axial dispersion coefficient D, in the fixed bed was calculated using the empirical equations given by Ebach et al. [ 121, i.e. y=

l,92(Q$J’06,for(Q$)<100

(15)

(iii) The external mass transfer coefficient, kf, was calculated using the equation presented by Wilson et af. [13]. i.e.

for Eb = 0.35 h 0.75 Re = 0.0016 5 55.

CHAU-JEN LEE et al

86

(iv) The film mass transfer coefficient around a particle in a stirred batch system, k,, was calculated using the equation derived by Hixson et al. [14] i.e. ShT = 0.16Re T ‘.u SC’.~’

(17)

for ReT > 67,000 and 486 < SC < 2.56 x lo6 where,

Sh =k,T T 012 Re, =NTZp,JC P

=-!!PDIZ’

(v) The effective diffusivity, Den, which characterizes the adsorption steps (b) and (c), was obtained from the experimental data in batch system operation by equations (l)-(3), and the following equations (18)-(20).

(18) (19)

c, = co,

(t = 0).

(20)

The effective diffusion paramater can be determined simply by using the standard error of regression (SER) technique, and acquired when the value of SER is minimum, where SER is defined as

SER =/q.

4. FINITE

(21)

MASS TRANSFER

RESISTANCE

4.1. Fixed-bed adsorption systems Applying van der Laan’s theorem [ 15 a,b] and generalized Glueckauf approximation [16], we can suggest a relationship for fixed-bed adsorption systems in which more than one mass transfer resistance is significant, e.g. axial dispersion, liquid film resistance, and intraparticle pore diffusion resistance. R,, = R, -I- R, + R,

(22)

or (23)

However, this approximation c171. 4.2.Batch

adsorption

is only applicable to linear and nearly linear isotherm systems

systems

For the linear driving force (LDF) approximation and a finite fluid phase of batch system, in which adsorption follows a linear iostherm, Hills suggested a factor F, = ~‘(1 + A)/(1 + A - x2/15) to be suitable for a film-surface diffusion model [ 183. Analogously, we can define a parameter K, to be suitable for a film-pore diffusion model, i.e. R, = R, + RI

(24)

Extracorporeal hemoperfusion

87

or, R2

1

R

(25)

K,=F,D,,,+3

where, F

=

a

A=

n2(l+ N 1 4 A - x2/15 volume of solution vb K(P,)K x volume of solid = K(p,),,V,.

5. EXPERIMENTAL The experimental set-up and detailed procedures were reported elsewhere [19]. 5.1.Batch system experiment The experimental data for adsorption isotherms were taken in an agitated vessel with suspended carbon particles at 37°C and at 620 rev./min stirring rate as shown in Fig. 3. Before introducing carbon particles in the vessel, they were weighed and then wetted to avoid the adherence of air bubbles on the surface of particles. The absorbent particles used were self-made spherical activated carbon (SAC) [21] and the particle size, in the range of 20-40 mesh. Pertinent properties of the adsorbent are tabulated in Table 1. The test fluids were prepared by dissolving either creatinine or uric acid in a Na,HPO,-NaH2P04 buffer solution. The solutions were titrated to pH = 7.4 before they were diluted to give the desirable initial test concentrations. The pseudo-adsorption iostherms were obtained for creatinine and uric acid respectively and with various levels of initial concentrations.

(I) (2) (3 ) (4) (5) (61

Water bath Batch reactor Stirrer Motor pH meter Taking sample side

Fig. 3. Batch

system

experimental set-up.

CHAU-JEN LEE et al.

Table 1. The pertinent properties of SAC particles [21] Property

Unit

SAC

P. PP

(g/cm ? (g/cm’)

Gelaf;n/AC SF.’ V,(r, < 20 A) s/r, > 20 A)

(g/g)

1.663 0.4907 0.705 1,/3 159 0.075 0.18

(cm3/g particle) (cm’/g particle)

* SF. = swelling factor =

5.2. Fixed-bed column experiment

The flow sheet of the fixed-bed experiment is as shown in Fig. 4. The reservoir which contains approximately 2000 ml of creatinine- or uric acid-buffered solution represents the body. The solution is well stirred and temperature controlled at 37°C and is pumped with a LAB pump (Model RP-D, Fluid Metering CO, U.S.A.) and circulated through a double lumen adsorbent cartridge (an acrylic pipe of 16 mm ID and with 400 mm SAC packed height). In addition, glass beads of 16-20 mesh are packed in both entrance and exit ends (about 6 cm height) to help flow distribution and bubble elimination. Small samples were drawn intermittently from the inlet and outlet of the fixed bed column (Fig. 4) and analyzed to give the time-varying concentrations. 6. RESULTS

AND

DISCUSSION

6.1. Pseudo-adsorption isotherms Metabolic wastes such as creatinine and uric acid may decompose within two or three days during the isotherm test using the bottle point method. The reaction behaviour caused by decomposed ingredient should not be neglected, especially in the dilute concentration range. As Peel et al. [6] have observed that the behaviour in long period batch kinetic experiment is not consistent with the single intraparticle diffusion rate model in the phenolic adsorption. It is also true for the metabolic wastes batch kinetic experiment. Fortunately, the period of clinical hemoperfusion treatment is usually shorter than 3 h. The adverse effect caused by prolonged experimental time is thus eliminated. If the activated carbon adsorption is considered as the sole mechanism of the process, then pseudo-equilibrium isotherms and relatively short-term batch kinetic data should be compatible. Figures 5 6 I Varmble-speed 2 3 4

Pump Surge Control

5 6

Flow meter Pressure gouge

valve

motor

7 8 3. IO

Thermometer Inlet of heating water Outlet of heating water Taking sample side I I pH meter 12. Reservoir

Fig. 4. Flow sheet for the fixed-bed experiment.

89

Extracorporeal hemoperfusion

and 6 show pseudo-adsorption isotherms of creatinine and uric acid in Na2HP04NaH,PO, buffer solution onto SAC particles at 37°C respectively. 6.2. Batch contact-time studies The experimental data obtained in the batch system operation are presented in Fig 7 and 8 for adsorption of creatinine and uric acid, respectively. The solid curves represent the calculated values obtained by substituting the best-fitted value of Den into equations (l)-(3), (13)-( 14), and (18)-(20). The effective diffusion parameters of creatinine and uric acid for SAC particles from different initial concentrations can be determined using SER technique (Figs 9 and 10). The average values of Deff tabulated in Table 2 were used for numerical calculation in the flow system. 6.3. Fixed-bed column studies The experiment results of operation in flow system are shown in Figs 11 and 12. The one-compartment single pore model seems to fit the experimental data reasonably well for both creatinine-SAC and uric acid-SAC systems for initial solution concentrations of Co = 20 mg/dl and Co = 12 mg/dl. Table 3 gives the calculated values for both creatinine-SAC and uric acid-SAC flow systems. 100

r

8 C,

Fig. 5. Pseudo-adsorption

12 (mg/dl)

isotherm of creatinine in Na,HPO,-NaH,PO, SAC particles at 37°C.

buffer solution onto

IO,3 -

83

.

::

/

,” F -

6040-

/ i

u 0

. /

0

5

1 IO C,

Fig. 6. Pseudo-adsorption

I

I

I

I

15

20

25

30

(mg/dl)

isotherm of uric acid in Na,HPO,-NaH,PO, particles at 37°C.

buffer solution onto SAC

CHAU-JENLEE et al.

90

Equilibrium represented by (D,,,),, = 0 156 D,e

0.6 t

0”

\

5

-

Longmuir Isotherm

(Cb)theo=12 mg/dl

04 .

Co = 25 mg/di

,

0

Co=12

mg/dl

0

Co =20

I

v

Co =8

mg/dl

0.2

mg/dl

t 0

I

I

02

04

I

I

I

0.6

I .o

06

Time

I

/

I

12

14

I .6

I

I

I .8

2.0

(h)

Fig. 7. Calculated and experimental concentrations of single solute creatinine on SAC particles in batch system.

---+----a

IO

0.0

r----q

i

Equilibrium represented by Linear isotherm =0.246 D,, (D,ttk

0.6 u” \ ;

04

(Colthe = 20 mg/dl .

Co =40

mg/dL

0

C, = 20 mg/dl

0

C, =30

mg/di

v

C, = 12 mg/di

I 02

t 0

I

I

I

I

I

I

I

06

IO

12

14

I .6

I .8

2.0

I

I

I

0.2

04

0.6

Time

(h)

Fig. 8. Calculated and experimental concentrations of single solute uric acid on SAC particles in batch system. 0 07

0 06

0.05 k (I) 0.04

0 03

0 02’

0

I 0.2

I 04

I 0.6

I 06

I

I .o

‘Ltr (* De) Fig. 9. Determination of D,,, for creatinine on SAC particles using the SER technique: Cq = 12 mg/dl, D,,,J(SER),,, = 0.162 D r2.

Extracorporeal hemoperfusion

0 02

Fig. 10. Determination

I

1 0

I

02

91

I

/

I

04

0.6

06

I I .o

of D,,, for uric acid on SAC particles using the SER technique: Co = 12 mg/dl. D,r,J(SER),,, = 0.256 D, 2.

Table 2. Values of V,, & pH, N, Co, C,, qac,D,rrI(SER),,,, (SER),,,, (De&, Solute (mol.wt)

V, W (ml) g

and in each run of batch process

pH

N Co C, qac D,rrI(SER),i, (SERLS (rev./min) (mnidl) (mg/dl) WAC) 1900 5 7.40 * 0.03 620 25 18.86 0.0311 0.112 D,,* 0.0205

(113.1) Creatinine

1900 5 7.40 * 0.03

620

20 12

13.14 8.47

0.0178 0.0245

0.162 D12 D,,

;:;;;;

7.40 * 0.03

620

0.0132

0.187 D,,

0.0309

7.40 * 0.03 7.40 k 0.03 7.40 f 0.03

620 620 620

8 40 30 20

5.39

Uric acid (168.1)

1900 5 1900 3 1900 3 1900 3

31.10 23.03 15.79

0.0752 0.0589 0.0425

0.216 D,,t 0.229 D,, 0.283 D,,

;:;;;

1900 3 7.40 + 0.03

620

8.77

0.0228

0.256 D,,

0.0306

(D,rr),v§

r

0.156 D,, 4.52

0.0142 0.246 D,2 2.87

*(D,,) creatinine = 1.185 x lo-> cm’/sec, estimated by reference [19]. t (D,,) uric acid = 1.097 x 10-s cm*/sec. also estimated by reference [19].

$ SER = standard error of regression = sPDtz _ _ 0.705D,, T

6.4. Finite mass transfer resistance

Tables 4 and 5 show the various mass transfer resistance and their relative importance for batch and flow systems. It is obvious that the intraparticle pore diffusion resistance plays a major role in both systems. Furthermore, in the initial stage (t < 0.2 h) of operation in every flow system, as shown in Figs 9 and 10, the experimental data are always slightly larger than those of numerical solution. The main reason is that the film transfer resistance dominates the resistivity of the system in the initial stage. But in the fixed-bed operation, as tabulated in Table 5, the intraparticle pore diffusion is in fact the controlling mechanism.

7. CONCLUSIONS A one-compartment single-pore model is postulated for the dynamic analysis and performance evaluation of a hemoperfusion column under operation. Experimental data obtained from binary system containing creatinine and uric acid fit the thoretically computed curves fairly well. Also, we suggest two relationships for fixed-bed adsorption and batch systems in which more than one mass transfer resistance is significant.

CHAU-JENLEE et ~1.

92

(0) Co ~20

tb)

0

02

04

06

Co

mg/dl

=I2

mg/dl

I.0

12

08

Tune

1.4

16

I8

2.0

lh)

Fig. 11. Comparison between experimental data and model calculations for creatinine adsorption on SAC particles in flow system, with equilibrium represented by Langmuir isotherm. (a) Co = 20 mg/dl; (b) Co = 12 mg/dl.

IO OB 06

. d ’

04

02

(a) Co =20

mg/dt

tb)

mg/dt

0 Y

O/IO

u’ 08

06

04

02

I 0

0 2

C,=12

I 0.4

I 0.6

I

I

I

I

0 8

I 0

1.2

14

Time

I I6

I I-3

J 2.0

(h)

Fig. 12. Comparison between experimental data and model calculations for uric acid adsorption on SAC particles in flow system, with equilibrium represented by linear isotherm. (a) Co = 20 mg/dl; (b) Co = 12 mg/dl. Table 3. The calculated values of kr, D,, Q,, (SER)t,, and (SER),,, for both creatinine-SAC flow systems Solute (mol.wt) Creatinine (113.1) Uric Acid (168.1) l

Vb Isotherm a (ml) (ml/g) W~ACJ ZOlOLangmuir 4750 305 2010Langmuir 4750 305 2020 Linear 0 250’ 2020Linear 0 25O*

K = 250 in q*c = KC,

fz

5 5 6 10

CO

k

hw/dl)

(cm;h)

D (cmljh)

20 12 20 12

25.91 25.91 23.76 24.70

1950 1950 1879 1958

and uric acid-SAC

~b

(SERlin

(SERhut

0.531 0.53 1 0.550 0.529

0.0229 0.0290 0.0424 0.0271

0.0218 0.0305 0.0372 0.0485

Extracorporeal Table 4. The liquid film, intraparticle K (ml/aAC) /-

Solute Creatinine (relative importance

179.6

Uric acid

250

A 2.82

hemoperfusion

pore diffusion, overall resistance, in each run of batch process Fa

k, (cm/h) .,,

and their relative

29.94

0.666 x 1O-2

28.80

11.92

0.973 x lo-* or 2.70 X 1o-6 (cm’s)

1relative tmportancel

RI

11.62

0.00041

0.0170

0.01741

(2.4%)

(97.6%)

(100%)

0.00043

0.01194

0.01237

(3.4%)

(96.6%)

(100%)

Table 5. The axial dispersion, liquid film, intraparticle pore diffusion, and their relative importance for flow systems Solute

Co (mg/dJ)

importance

(h)

1.85”: 10-h (cm’/s) 3.38

93

RZ (h)

Rf (h)

R, (h)

4.835 x 10-s

4.728 x 1O-4

0.0135

(0.3%)

(3.4%)

20/12

Uric acid (relative Importance)

20

4.316 x lO-5 5.156 x lo-“

12

(94.3%) (0.496) (5.3%) (100%) 4.894 x IO-’ 4.960 x 10m4 9.254 x 10m3 9.799 x 10-s

Acknowledgvmenr ~ The financial support (Project No. NSC 73-0402-EOO7-05).

(5.1%)

of National

Science

0.01404

(96.3%)

(100%)

9.254 x lo-’

9.813 x 1O-3

(94.4%)

Council,

80.8

overall resistance,

Creatinine (relative Importance)

(0.5%)

57.4

71.2

101.9

102.1

(100%)

R.O.C.

is gratefully

acknowledged,

REFERENCES H. Yatzidis, Proc. Eur. Dial. Transpl. Ass. 1, 83-84 (1964). E. H. Dunlop and P. G. Langley, Artificial Liver Support. pp. 310-318. Pitman Medical, London (1975). R. L. Dedrick and R. B. Beckman, Chem. Engng Prog. Symp. Ser. 63, 68 (1967). D. 0. Cooney and D. F. Shieh, A.1.Ch.E.J. 18, 245, (1972). J. W Eagle and J. W. Scott, Ind. Engng Chem. 42, 1287. (1950). R. G. Peel. A. Benedek and C. M. Crowe. A.I.Ch.E.J. 27, 26-32 (1981). R. D. Fleck Jr, D. J. Kirman and K. R. Hall, Ind. Engng Chem. Fundam. 12. 95-99 (1973). G. Mckay, Chem. Engng J. 28. 955104 (1984). K. Kawazoe. Kagaku Kogaku 32, 175-181 (1968). C. N. Satterfield. Mass Transfer in Heterogeneous Catalysis, pp.78-207. M.I.T. Press, Cambridge, MA (1970). B. Carnahan. H. A. Luther and J. 0. Wilkes, Applied Numerical Methods, pp. 440-451. John Wiley. New York (1969). 12. E. A. Ebach and R. R. White, A.1.Ch.E.J. 4, 161 (1958). 13. E. J. Wilson and J. R. Geankoplis. Ind. Engng Chem. Fundam. 5. 9-14 (1966). 14. A. W. Hixson and S. J. Baum, Ind. Engng Chem. 36, 528 (1944). 15a. E. Th. Van der Laan. Chem. Engng Sci. 7, 187 (1958). 15b. H. W. Haynes Jr and N. S. Phanindra. A.1.Ch.E.J. 19, 1043-1046 (1973). 16. E. Glueckauf. Trans. Faraday Sot. 51, 1540-1551 (1955). 17. D. M. Ruthven, Principles of Adsorption and Adsorption Processes. pp. 235-244. John Wiley, New York (1984). 18. J. H. Hills, Chem. Engng Sci. 41, 277992785 (1986). 19. S. C. Hu, Studies on adsorption kinetics of encapsulated activated carbons. M.S. Thesis, National Tsing Hua LJniversity. Hsinchu. Taiwan, R.O.C. (1984). 20. C. R. Wilke and P. Chang, A.1.Ch.E.J. I, 264 (1955). 21. C. J. Lee, S. T. Hsu, J. Biomed. Mat Res. (submitted).

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

About the Author - CHALJ-JEN LEE received the B.S. degree from National Taiwan University in 1957 and the Ph.D. degree in Chemical Engineering from Oklahoma State University in 1965. From 1964 to 1972, Dr Lee worked as a senior research engineer at the Research Centers of Phillips Petroleum Co. and subsequently of Air Products & Chemical Co. He then joined the Faculty of Engineering at the National Tsing Hua University (Hsinchu, Taiwan), and has been teaching there ever since. During his 15 years teaching career, he has experienced several administrative positions, i.e. Department Chairman, Dean of Engineering College, and Director of Bioengineering Center. His current research activities focus on transport phenomena in various biomedical engineering systems.

CHALJ-JEN LEE et al.

94

Dr Lee is a member of CIChE and the Japanese Society of the Artificial Organs. He is also among the directors of the Chinese Institute of Chemical Engineering and the Chinese Society of Biomedical Engineering. About the Author - SHIH-TONG Hsu was born in Tainan, Taiwan, on 10 January 1953. He received the BS and MS degrees in Chemical Engineering from Chung Yuan Christian University, Chung Li, Taiwan in 1979 and 1981, respectively. He is currently completing doctorial dissertation on ‘Hemoperfusion and its adsorption characteristics’ at National Tsing Hua University.

About the Author - SU-CHIEH HU received the MS degree in Chemical Engineering from National Tsing Hua University in 1984. He is currently enrolled as a Ph. D. student at Pennsylvania State University, College Park. PA, U.S.A.

APPENDIX:

NOMENCLATURE

A,: adsorbent external surface area per unit volume of fixed bed (cm’ cmm3), AC: activated carbon, C,: bulk fluid-phase concentration in the fixed bed (g ml-‘), C,: bulk fluid-phase initial concentration in the well-mixed tank or batch system (g ml-‘, C,: fluid-phase concentration within adsorbent pores at radius r(g ml-‘), C,: bulk fluid-phase concentration at time t(g ml-‘). C,: equilibrium fluid-phase concentration (g ml-‘, mg dl-‘), d,: diameter of carbon particle (cm), D 1?: molecular dithrsivity in solution (cm2 s _ ’ ), D,,,: effective digusivity within the adsorbent (cm’ h- I, cm’ s-l). D,: axial dispersion diffusivity (cm2 h- ‘), F,: n2(1 + A)/( 1 + A - n/15), factor in equation (25). k,: external liquid-film mass-transfer coefficient in the fixed bed (cm h- I), k,: external liquid-film mass-transfer coefhcent for batch process (cm h- ‘), K: Henry constant (ml/g of activated carbon), K,: overall mass-transfer coefficient (h- ‘), L: bed length (cm), LDF: linear driving force, mol.wt: molecular weight (g mole- I), n: number of data, IV: rotation speed of rmpeller (rev./mink 4: solid-phase concentration (g g-‘). q,,c: adsorption capacity per unit g of activated carbon (g g- ’ of activated carbon), Q: average flow rate (ml h I), r: radial distance from center of carbon particles (cm), rp: radius of pore (A). R: radius of carbon particle (cm), Re: dpup/p, Reynolds number, Re,: NT’p/p(, Reynolds number referred to the vessel, R,: liquid film resistance (h), R,: overall mass-transfer resistance (h), R,: intraparticle pore diffusion resistance (h), R,: axial dispersion resistance (h), S,: total external surface area of adsorbent within the batch reactor (cm’), SAC: self-made spherical activated carbon, SC: p/(pD12), Schmidt number, SER: standard error of regression, S.F.: (Rwe,/Rd,J3. Swelling factor, Sh,: k,TJD,,. Sherwood number referred to the vessel, t: time (h), T: vessel diameter (cm), u: fluid velocity in bed based on total cross-sectional area (cm h-l), V,: the volume of batch system or the well-mixed tank (ml), V,: total pore volume per unit g of adsorbent (cm3 g-l), V,: volume of wet solid (cm% W adsorbent weight (g). 2: longitudinal coordinate (cm), 1: constant of Langmuir isotherm (ml g-‘), fi: constant of Langmuir isotherm (ml g-r of activated carbon), sb: fixed-bed void fraction, sp: adsorbent wet-particle porosity, p: dynamic viscosity (g cm _ ’ h _ I). r: tortuosity factor for pore volume diffusion. p: liquid density (g cme3), pp: adsorbent wet-particle density (g cm-a), pE: true density of solid phase (g cme3), A: WK(P,)*,K

mg dl-‘),