Kinetics and equilibrium of dissolved oxygen adsorption on activated carbon

Kinetics and equilibrium of dissolved oxygen adsorption on activated carbon

Chemical Engineering Science 63 (2008) 609 – 621 www.elsevier.com/locate/ces Kinetics and equilibrium of dissolved oxygen adsorption on activated car...

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Chemical Engineering Science 63 (2008) 609 – 621 www.elsevier.com/locate/ces

Kinetics and equilibrium of dissolved oxygen adsorption on activated carbon Vassileios M. Matsis, Helen P. Grigoropoulou ∗ Laboratory of Chemical Process Engineering, School of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou, 15780, Athens, Greece Received 12 December 2006; received in revised form 25 September 2007; accepted 4 October 2007 Available online 10 October 2007

Abstract The macroscopic adsorption behavior of dissolved oxygen on a coconut shell-derived granular activated carbon has been studied in batch mode at 301 and 313 K for initial dissolved oxygen concentrations of 10–30 mg/l and oxygen/carbon ratios of 2–180 mg/g. BET (Brunauer, Emmett, and Teller) surface area, micropore volume, and pore size distribution were determined from N2 isotherm data for fresh and used samples of carbon. The surface groups were characterized using Boehm titrations, potentiometric titrations, and FTIR study. The material is characterized by its high specific surface area (1307 m2 /g), microporocity (micropore volume 0.54 cm3 /g), its basic character (0.57 meq/g total basic groups) and its high iron content (15,480 ppm Fe). BET n-layer isotherm describes adsorption equilibrium suggesting cooperative adsorption and important adsorbate–adsorbate interactions. Kinetic data suggest a process dependent on surface coverage. At low coverage a Fickian, intraparticle diffusion rate model assuming a local equilibrium isotherm (oxygen dissociation reaction) adequately describes the process. The calculated diffusion coefficients (D) vary between (4.7.8.2) × 10−9 m2 / min and (3.5.5.3) × 10−9 m2 / min for initial oxygen concentration of 10 and 20 mg/l, respectively. Sensitivity analysis shows that the oxygen dissociation equilibrium constant determines the equilibrium concentration, whereas the diffusion coefficient controls the kinetic rate of the adsorption process having no effect at the final equilibrium concentration. A combined kinetic mass transfer model with concentration-dependent diffusion (parabolic form) has been developed and successfully applied on the dissolved oxygen adsorption system at high surface coverage. For equilibrium uptake of 0.08 mg/m2 the estimated mean mass transfer coefficient and adsorption rate constant are 3.38 × 10−5 m/ min and 1.0 × 10−2 l/(m2 min), respectively. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Activated carbon; Adsorption isotherm; Kinetics; Porous media; Dissolved oxygen; Mass transfer

1. Introduction Activated carbon is a microporous adsorbent that can be produced from a variety of carbonaceous materials, including wood, coal, lignin, coconut shells, and sugar. Its unique adsorption properties result from its high surface area, high adsorption capacity, micropores, and broad range of surface functional groups. It is an excellent and versatile adsorbent, widely applied in the adsorptive removal of color, odor, taste, and other undesirable organic and inorganic pollutants from drinking water; the treatment of industrial wastewater; solvent recovery; air purification and purification of many chemical, food, and pharmaceutical products. It is increasingly being used in the field of hydrometallurgy and in medicine and health ∗ Corresponding author. Tel.: +30 2107723222; fax: +30 2107723155.

E-mail address: [email protected] (H.P. Grigoropoulou). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.10.005

applications to combat certain types of bacterial ailments and for the adsorptive removal of certain toxins and poisons, and for the purification of blood (Levine and LaCourse, 1967). Currently granular activated carbon (GAC) adsorption is proved to be an attractive and effective process in wastewater and water treatment for removing color, heavy metals, and organic compounds. However this process is expensive due to both high cost of GAC and regeneration. On the other hand, problems such as competition between adsorbates and fouling arise during the operation (Karimi and Farooq, 2000). Adsorptive properties of activated carbon are defined by retention capacity (adsorption equilibrium which is a function of the type, concentration and distribution of adsorption sites, adsorbent pore structure, adsorptive and experimental conditions) and by adsorption kinetics. Both kinetics and thermodynamics are of critical importance in assessing the performance of activated carbon for the adsorption of pollutant species. Two

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types of surface sites are involved in adsorption on activated carbon, these being hydrophobic sites, consisting of grapheme basal plane layers, and hydrophilic functional group sites. The adsorption potential of a non-polar adsorbate is determined by both sites, such an adsorbate not distinguishing between them. A polar adsorbate will be preferentially adsorbed, initially, on the polar sites. Water itself is a very poor adsorbate on graphitic surfaces, nevertheless, upon activation, these same surfaces appear to be hydrophilic, sometimes blocking out other more compatible substances. Oxide groups increase the polarity of activated carbon and contribute to improve its wetability. Thus, adsorption of non-polar species should be impaired as a result of increasing surface oxides, but polar species may be adsorbed to a greater or lesser extent, depending on their specific oxide–adsorbate interactions. Although adsorption onto activated carbon is mainly of diffusive interaction type, surface chemistry plays an important role when specific interactions are considered. Generally, the adsorption from solution depends upon the properties of the organic and inorganic solutes such as their molecular mass, molecular size and geometry, their polarity and solubility, and the structure and nature of the carbon surface. On the other hand, the experimental conditions such as the pH, the ionic strength of the solution, and the temperature of adsorption determine the adsorption capacity and the mechanism of the adsorption process. Various adsorption isotherms have been developed, the most widely adopted in monolayer surface adsorption on an ideal surface being the Langmuir equation. Brunauer, Emmett, and Teller (BET) were the first to develop a theory to account for multilayer adsorption. The time dependence of the adsorption process leading to the equilibrium state is actually driven by two processes that can have different characteristic times according to the molecular structure of the adsorbate: adsorption onto the solid and diffusion, usually Fick-type with the effective diffusivity being a function of flow mechanism around and within the particle. Various models are available to determine the rate constant corresponding to different mechanisms associated with aqueous solute transfer (Duong, 1998). However, no single relationship is available combining the various mechanisms involved causing a major problem in adsorber design. Previous analyses on the single and double resistance mass transfer models (Mckay, 1984) show that such models agree with experimental results but only over very limited regions of the adsorption period indicating that either another internal mass transfer resistance or a variable solid diffusivity combined with changes on the surface becomes rate controlling over long time periods. Studies of the adsorption of gases and water vapor on carbon have shown that the adsorption kinetics follow a linear driving force (LDF) model (Chagger et al., 1995; O’Koye et al., 1997; Reid et al., 1998; Reid and Thomas, 1999, 2001), a mass transfer model (Duong, 1998; Aksu and Kabasacal, 2003), a combined barrier resistance/diffusion model (Karger and Caro, 1977; Loughlin et al., 1993), or Fickian diffusion, depending on the experimental conditions, adsorbate, and adsorbent. The presence of dissolved oxygen in adsorptive separations from aqueous solutions complicates sorption leading to a synergistic

effect. It has been demonstrated (Vidic and Suidan, 1991; Karanfil et al., 1996) that the presence of molecular oxygen has a significant influence on the adsorption capacity of activated carbon for several organic compounds and natural organic matter. Oxygen is believed to induce polymerization reactions that contribute to the removal of organic compounds from the liquid phase. Furthermore, biological activity in a carbon contactor or extensive bed depth can cause significant depletion of oxygen which will facilitate anoxic conditions. Therefore, it is very important to evaluate adsorptive and kinetic properties of activated carbon under different environmental conditions. Despite this fact, few data are available on the adsorption kinetics of dissolved oxygen. Prober et al. (1975) studied the adsorption of dissolved oxygen on activated carbon and concluded that the rate data generally conform √to a model of adsorption limited by intraparticle diffusion ( t variation). The present study aims to investigate the equilibrium and kinetics of dissolved oxygen adsorption onto GAC in aqueous solution. Adsorption dynamics are investigated and global kinetic and mass transfer models are proposed and validated through experimental data. In order to interpret equilibrium/kinetic behavior the physicochemical properties of carbon are determined. The effect of parameters such as initial concentration of dissolved oxygen, amount of catalyst, temperature and mixing rate on the rate, and extent of dissolved oxygen adsorption is studied in a completely mixed batch reactor. The surface groups are characterized using various techniques such as Boehm titrations, potentiometric titrations, and FTIR study. 2. Experimental section 2.1. Materials The adsorbent used in this study was a coconut shell-derived GAC commercially formed via treatment with concentrated phosphoric acid, supplied by Merck (Pr. no. 102518) having a density of 1.95 mg/cm3 . Carbon activated with phosphoric acid exhibits a decrease in aliphatic groups, increased aromaticity, and increased stability of carbonyl functions, whereas coconut shell carbons are particularly efficient in adsorbing small molecular species (Toles et al., 1996; Braymer et al., 1994). The solid was ground and sieved to different fractions of which geometric mean particle diameters of 2.18 × 10−3 m and 9.2 × 10−4 m were used in the study. It was thoroughly washed with deionized water, dried at 393 K for 24 h to remove fine dust and adsorbed impurities from the adsorbent surface and finally stored in a desiccator until use. Ultrapure water (specific resistance > 18 M cm), reagent grade sodium hydroxide, sodium nitrate, nitric acid, sodium carbonate, sodium hydrogen carbonate, and hydrochloric acid were used. 2.2. Functional group characterization BET surface areas, micropore volume, and pore size distribution were determined from N2 isotherm data collected at 77 K (Quant Chrome Nova 2000) for fresh and used samples of carbon. The t-method (Duong, 1998) was applied to determine the volume of micropores, external surface area, and specific surface area.

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

solution (1.1×10−3 m3 ) was poured and oxygenated with pure oxygen gas up to the desired level of dissolved oxygen concentration. The stirring speed was 550 rpm. A 0.2–6.0 g of activated carbon was added at an initial dissolved oxygen concentration of 2–30 mg/l and a pH-meter pH538 and an oxygen meter Oxi 340i from WTW were used for on-line measurements. Isothermal conditions (301–313 K) were assured using a thermostatic bath from Julabo. Certain experiments were duplicated under identical conditions to estimate the reproducibility of the experimental setup which was better than 5%.

The titration of the chemical surface groups was carried out following the method proposed by Boehm (1994). One gram of carbon samples was placed in 5 × 10−5 m3 of the following 0.05 N solutions: sodium hydroxide, sodium carbonate, sodium bicarbonate, and hydrochloric acid. The vials were sealed and shaken for 24 h 301 K and the excess of base and acid was titrated with HCl and NaOH, respectively, in 5×10−6 m3 of the filtrate. The numbers of acidic sites of various types were calculated under the assumption that NaOH neutralizes carboxyl, phenolic, and lactonic groups; Na2 CO3 neutralizes carboxyl and lactonic groups; and NaHCO3 neutralizes only carboxyl groups. The number of surface basic sites was calculated from the amount of hydrochloric acid that reacted with the carbon. Fourier transform infrared (FTIR) spectra were acquired using a Perkin Elmer Spectrum 2000 FTIR, by averaging 256 scans in the 4000.400 cm−1 spectral range at 4 cm−1 resolution. Pressed KBr pellets at a sample/KBr weight ratio of 1:300–1:1300 were scanned. Potentiometric titration measurements were performed with a Titrando 808 automatic titrator (Metrohm) (Jagiello et al., 1995). Subsamples of the activated carbon at amounts of about 0.3 g in 5 × 10−5 m3 of 0.1 N NaNO3 were placed in a container thermostated at 25 ◦ C and equilibrated overnight with the electrolyte solution. The carbon suspensions were stirred throughout the measurements. Volumetric standard 0.1 N HCl was used as the titrant. The experiments were conducted in the pH range of 3–10. The point of zero charge (PZC) of fresh carbon was determined using a simple method proposed by Noh and Schwarz (1989). Slurries of 10 wt% were prepared by mixing particles of carbon with deionized water in a plastic bottle sealed under N2 . The pH of the slurry was measured after shaking for 24 h. Some deionized water was subsequently added and the previous procedure was repeated for slurries of 6, 2, 1, and 0.5 wt%. Hydrogen peroxide content of the slurry was determined by a simple colorimetric method for determining micromolar quantities: H2 O2 was reacted with potassium titanium oxalate (5 × 10−2 kg/m3 ) in acid solution to form the yellow pertitanic acid complex, which was measured spectrophotometrically at 400 nm.

3. Results and discussion 3.1. Adsorbent characterization In Table 1 the surface characteristics of the fresh and used activated carbon is reported. The results indicate that the material has a high micropore volume and some mesopores for efficient transport. The pore size distribution does not show sig˚ Macro/mesoporocity nificant number of pores wider than 50 A. is low, thus reducing particle density and hence volumetric capacity. Efficient transport of dissolved oxygen is expected to be reduced due to insufficient number of macro- and mesopores, ˚ (Bondi, 1968). given its molecular diameter of 3 A As it can be seen in Table 1, the results from the samples which adsorbed various amounts of dissolved oxygen (qe ) reveal that oxygen adsorption at 301 and 313 K did not produce any significant changes in the physical (textural) properties of the carbon leaving it in essentially its original condition. The amphoteric character and functional groups present in carbon were assessed and it was found that the quantities of total basic and acidic functional groups present were 0.57 ± 0.04 and 0.11 ± 0.01 meq/g while the total amount of polar carbon–oxygen surface groups was 0.31 sites/nm2 . The concentrations of phenolic/carbonyl, lactone/lactol and carboxylic functional groups were 0.01, 0.00, and 0.10 meq/g, respectively. Therefore, the basic groups have the highest contribution (about 84%) to the oxygen functional groups. Basic groups, detected by multibase titrations, exhibit a slight increase and then they are significantly reduced after oxygen sorption as shown in Fig. 1. On the other hand, probable changes in acid groups are not detected by the methods employed. A multipart mechanism can be adopted to explain the basic character under the prism of the experimental findings:

2.3. Adsorption studies Adsorption experiments were conducted in a routine manner by the batch technique. The aqueous phase was ultrapure water and equilibrium and kinetic experiments were performed in a Pyrex glass vessel of 0.1 m i.d. and 0.135 m high. An aqueous .

• A chromene type model (benzpyran-type structures) accounts for deprotonation/adsorption process (Leon et al., 1992) as it is described by the following model reactions:

H

O

R

611

.

+

H

+

+

1/ 2 O 2

+

R

+

.

H 2O 2

. . .

O

.

CH.

+

H

+

+

1/ 2 O 2

.

O

+

CH

+

1/ 2 H2O 2

612

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Table 1 Surface characteristics of the fresh and treated activated carbon Method

Specific surface area (m2 /g)

External surface area (m2 /g)

Micropore volume (cm3 /g)

1307 ± 105a 1346 ± 188b

178 ± 19.6b

0.54 ± 0.06b

Sample 1 (qe = 3.79 mg/g) BET 1306 ± 105a t-method 1343 ± 188b

177 ± 19.6b

0.54 ± 0.06b

Sample 2 (qe = 0.82 mg/g) BET 1276 ± 102a t-method 1356 ± 190b

170 ± 18.7b

0.56 ± 0.06b

Fresh sample BET t-method

a Expanded b Expanded

Average pore ˚ diameter (A)

0.76 ± 0.08a

23.4 ± 2.3a

0.75 ± 0.08a

23.0 ± 2.3a

0.75 ± 0.08a

23.6 ± 2.3a

uncertainties have been calculated according to error propagation theory based on repeatability and weighing uncertainty. uncertainties have been calculated according to error propagation theory based on repeatability, weighing uncertainty and calculation uncertainty.

“graphitized carbon surface plate having maximum itinerant  electrons”. These delocalized electrons were envisioned to give rise to Lewis basicity. However, after oxygen adsorption they were localized weakening such type of basicity.

0.7 0.6 Basic functional groups, meq/g carbon

Total pore volume (cm3 /g)

0.5 0.4 0.3 0.2 0.1 0 0

2

4 6 Oxygen sorbed, mg/g carbon

8

10

Fig. 1. Results of basic functional groups (meq/g) and adsorbed oxygen (mg O2 /g carbon) (: C0 =20 mg/l, w=3.0 g, : C0 =10.30 mg/l, w=1.0.6.0 g).

In the present study small amounts of H2 O2 were detected (up to 2 × 10−4 kg/m3 ) in the liquid after oxygen adsorption (1 mol H2 O2 /100 mol oxygen adsorbed) revealing the possible contribution of chromene (benzpyran)-type structures to the deprotonation/adsorption process. However, the amount of peroxide liberated was considerably less than the amount predicted by the model reactions mentioned above. This could be attributed to the known ability of the carbon to catalyse the decomposition of H2 O2 . However, the ratio of oxygen uptake to H+ consumption (100 mol O2 /1 mol H+ ) (as calculated from uptake and pH measurements) compared to reaction stoichiometry (1 mol O2 /1 mol H+ ) denotes their small contribution to the process. In addition, H2 O2 oxidizes Fe2+ (15480±364 ppm iron content of the carbon material) producing the very active species · OH according to the Fenton reaction: Fe2+ +H2 O2 → Fe3+ + · OH. • A “Lewis-base concept” introduces Lewis bases (electron pair donors) as a source of basicity which was claimed (Puri, 1970) to be weakened by the presence of adsorbed oxygen. A generalized expression to account for Lewis basicity is: C + 2H2 O ⇔ C H3 O+ + OH− , where C was defined as

The calculated loss in net basicity could also be attributed to the following: the basic sites in the neighborhood of non-basic surface groups are likely to be shielded from reaction with HCl by internal neutralization, electronic rearrangement, and/or chelation (Van Krevelen, 1981). According to the potentiometric titration studies two types of basic centers with pKb1 = 5.3 and pKb2 = 8.4, respectively, are identified on fresh carbon samples. At high oxygen uptake (greater than 100 mg O2 /g carbon) the weak basic center of pKb2 =8.4 is not detected. Carbon’s strong basic character relies on the strong basic character of active sites with pKb1 = 5.3 which is detected in fresh and all used carbon samples. Finally, the pHzpc of the fresh carbon is 7.6 so that a positive surface charge of the adsorbent is acquired when pHsolution < 7.6 and a negative one at greater values. FTIR analysis revealed large P&O and C&O peaks at 1190 and 1097 cm−1 , respectively. The activated carbon in this study was an H-type. Although a thorough rinsing procedure was followed, a considerable pH elevation was observed during the experiments. Its hydrophobic nature adds positive charge by absorbing H+ when immersed in deionized water. This adsorption tends to point of zero charge where the surface capacity has been reached. 3.2. Evaluation of adsorption data 3.2.1. Adsorption equilibria The adsorption isotherms for dissolved oxygen at 301 and 313 K are shown in Fig. 2. Negligible uptake is acquired by carbon at low relative adsorbate concentrations, and not until the bulk concentration is near the bulk saturation point (Csat = 7.81 and 6.41 mg/l at 301 and 313 K, respectively) does significant uptake occur. This behavior is a consequence of the weak carbon-oxygen interaction relative to the strong oxygen-oxygen interaction (cooperative nature of adsorption).

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

613

Table 2 Multilayer isotherms

Reduced uptake

1.2 1.0 0.8

BET-2 layers LGD Sircar exp. data Huttig

0.6 0.4

n-layer BET

n-BET (x) =

Hüttig

Hutt (x) =

q qmax

Sircar

Sirc (x) =

q qmax

LGD

LGD (x) =

q qmax

q qmax

=

1 1−x



Kx[1−(+1)x  +x +1 ] 1+(K−1)x−Kx +1



Kx = (1 + x) 1+Kx   Kx/(1−x) ∞ −1)x = 1−(C1−x 1+Kx/(1−x)

= 21 [n-BET (x) + Hutt (x)]

0.2 0 0.98

0.985

0.99

0.995

1

Reduced concentration

2.0

Reduced uptake

1.8

exp. data

1.6

Huttig

1.4

Sircar

1.2

BET-2 layers

1.0

LGD

0.8 0.6 0.4 0.2 0 0.95

0.96

0.97 0.98 Reduced concentration

0.99

1

Fig. 2. Experimental and calculated adsorption isotherm data at 301 and 313 K ((a) and (b), respectively).

Given the hydrophobic nature of carbon a type III isotherm in the Brunauer classification, which is characteristic of weak nonspecific interaction between adsorbate and solid, is expected, since the non-polar oxygen molecules interact through dispersion forces only on sites other from the functional groups. The “hydrophobic effect” of dissolved oxygen in water is primarily a consequence of changes in the clustering in the surrounding water rather than water–solute interactions. The liquid water lying against extended hydrophobic surfaces has low density, encouraging non-polar dissolved oxygen gas accumulation. The oxygen atoms from the oxygen molecules will form multiple van der Waals interactions with water molecules. The nearly horizontal plateau seen in the isotherms for reduced concentration values up to 0.99, is a macroscopic consequence of the formation of the first monolayer; the steep vertical rise which follows it indicates a sharp transition to multilayer adsorption. The increased uptake at high relative equilibrium concentrations may represent adsorption onto preadsorbed sites. The convexity of the isotherms is suggestive of cooperative adsorption, which means that the already adsorbed molecules tend to enhance the adsorption of other molecules. In other words, it implies that the adsorbate–adsorbent interactions are of less importance than the adsorbate–adsorbate interaction. The weak adsorbate–adsorbent interactions result in small adsorption at lower relative concentrations. But once a

molecule has been adsorbed, the adsorbate–adsorbate interactions will tend to promote the adsorption of more molecules so that the isotherm becomes convex to the concentration axis. The limited adsorption at the lower part of the isotherm is due to the fact that the activated carbon used in this study possesses appreciable amounts of polar carbon–oxygen surface groups that result in increasing the interaction of the carbon surface with water molecules, due to the formation of hydrogen bonding. Strong hydrogen bonds are formed between activated sites and water molecules, and these adsorbed molecules become nucleation sites for other water molecules to adhere to; once a water molecule is adsorbed in this way, it can form hydrogen bonds with other molecules, forming three-dimensional clusters on the surface. This aggregation of water molecules puts a surface barrier to non-polar oxygen adsorption although the average pore diameter is 7–8 oxygen molecular diameters. When the surface is covered with a layer of adsorbed water, the adsorbent–adsorbate interaction energy is virtually reduced to the weak dispersion energy between water and oxygen molecules so that a Type III isotherm is obtained. A series of multilayer isotherm expressions (Table 2) is used to model the experimental data and the results are plotted in Fig. 2. The isotherm equations examined were: a 2-parameter n-layer BET equation, a 1-parameter Hüttig equation, a 2-parameter Sircar equation and a 3-parameter Lopez–Gonzalez and Dietz (LGD) equation (Fergusson and Barber, 1950; Duong, 1998; Sircar, 1984). The criterion for best fit was the minimization of sum of square errors (SSE) value where SSE =

k 

(qi,exp − qi,est )2 .

(1)

i=1

As it can be seen from Fig. 2 the n-layer BET model shows the best fit and the parameters are reported in Table 3. The low value of parameter K indicates that forces between adsorbate and adsorbent are much smaller than that between adsorbate molecules in the liquid state. The total dissolved oxygen uptake values for run times up to 48 h measured in this study are of the order of 1–60 mg/g, whereas in a past study Prober et al. (1975), using both manometric (batch operation) and packed column apparatus, found total adsorptions on the order of 10–40 mg/g (296–313 K). 3.2.2. Analysis of adsorption kinetics 3.2.2.1. Adsorption characteristics The typical variation of bulk concentration (C) and reduced uptake (q/qe ) vs. time for different initial oxygen concentrations is shown in Fig. 3.

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V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

Table 3 n-Layer BET model coefficients and sum of square errors (SSE) values n-layer BET model parameters Temperature (K)

K

n

301 313

0.0002 0.0005

2 3

Fitting criterion Isotherm

SSE values

n-layer BET Hüttig Sircar LGD

301(K)

313(K)

0.028 1.333 0.184 12.060

0.050 1.195 0.098 71.483

30 Co = 10 mg/l

Co = 20 mg/l

Co = 30 mg/l

25

C, mg/l

20 15 10

3.2.2.1.1. Diffusion limited process investigation In order to confirm whether internal mass transfer was limiting adsorption the following method was followed: (i) The initial stage of adsorption is considered and its behavior is investigated. The fractional uptake at short times during the initial stage of adsorption and for linear local isotherm is given by the following equation:  2(s + 1) Dapp √ M t − M0 ≈ √ t, s = 2 (2) Me − M 0 rp  for spherical particles (Duong, 1998). The apparent diffusion coefficient is estimated from the gradient of a(Mt /M0 −M0 /− M0 ) vs. t 1/2 plot, which is considered as a fingerprint for a diffusion limited process, where Mt the amount adsorbed up to time t. The analysis yielded apparent diffusion coefficients that varied between 6 × 10−10 and 6 × 10−9 m2 / min without exhibiting a general trend. These estimates are one or two orders of magnitude below the value of 1.5 × 10−7 m2 / min for dissolved oxygen diffusion in water (Perry and Green, 1984). (ii) To confirm further kinetic limitation the characteristic half-time t0.5 (the time for the system to reach half of its equilibrium capacity) of a diffusion controlled adsorption process for a spherical particle, assuming a linear local adsorption, was estimated: t0.5 = 0.03055

rp2 Dapp

.

(3)

5 0 0

200

400

600 Time, min

800

1000

1200

Fig. 3. Variation of dissolved oxygen bulk concentration vs. time at different initial concentrations (: w=3.00 g, ♦: w=2.00 g, ∗: w=6.00 g, T =301 K).

A two-step kinetic may be observed: a very rapid initial adsorption over a few minutes, followed by a long period of much slower uptake. As the concentration increases, the surface coverage increases leading to mono- and multilayer formation in the larger pores. The effective pore width is reduced and overlap of the potential energy surfaces is enhanced leading to pore filling. The decrease in effective pore width and length hinders oxygen diffusion and leads in variable diffusivity. Adsorbate–adsorbate interactions involved are cluster formation around active sites and subsequent merging of clusters in oxygen adsorption to give diffusion barriers. Penetration into the micropore can be viewed as controlled by the surface resistance of the pore mouth, equilibrium being rapidly established between the gas phase and the adsorbed phase at the pore mouth and the equilibrated adsorbed molecule having to overcome the surface barrier at the pore mouth to enter the micropore interior. The complexity of the analysis of the examined adsorption system is due to both the type of global non-linear adsorption isotherm involved (Type III) and the variable degree of masstransfer resistance encountered.

Results of this calculation are provided in Table 4. From Table 4 it can be seen that at low initial bulk concentrations (10 mg/l) the diffusion times are comparable with the observed time constants. However, at higher initial bulk concentrations (> 20 mg/l) the adsorption of dissolved oxygen is considerably faster than would be expected under internal diffusion control assuming a linear local adsorption isotherm. To determine the relative importance of the external and internal mass transfer resistances, experiments were performed with two different particle sizes, 2.18 and 0.92 mm at different agitation rates. Above 550 rpm the results, for the whole range of initial oxygen concentration were identical, within experimental uncertainty, for the two particle sizes implying that the external film resistance can be neglected under the experimental conditions. 3.2.2.1.2. Rate law of the adsorption system Additionally, analyzing the kinetic data using the pseudo-second order kinetic model (Supporting material) showed that at early time (up to 100 min) the second-order equation was the more appropriate among the pseudo-first order and saturation-type kinetic models (Ho and McKay, 1999). If it is assumed that the adsorption capacity is proportional to the number of active sites occupied on the adsorbent, then it may be concluded that a dissociation of oxygen is dissociated on the solid surface. The LDF model for adsorption (Braymer, 1994) is described by the equation: qt = 1 − e−kLDF t . qe

(4)

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

615

Table 4 Characteristic diffusion time at different bulk dissolved oxygen concentrations calculated by using Eq. (3) Initial dissolved oxygen concentration (mg/l)

Carbon load (g)

Calculated diffusion time constant (min) (t0.5 )

Experimental value (min)

T = 301 K 10 10 10 20 20 20 30

1 2 3 1 2 4 6

20.9 11.7 28.1 96.8 95.6 99.2 62.5

25.4 9.1 30.3 66.7 57.2 60.3 50.8

T = 313 K 10

3

52.7

39.0

0

ln(1-qt/qe)

-0.1

0

5

10

15

20

25

-0.2 -0.3 -0.4 -0.5

R2 = 0.9924

A general trend was depicted, with the rate constant increasing with increasing initial concentration and amount of carbon. The results indicate that a LDF mass transfer coefficient cannot adequately describe the short time behavior of dissolved oxygen adsorption onto activated carbon at all initial concentrations and surface coverage, even though the oxygen uptake in the form of ln(1−qt /qe ) vs. time is linear (r 2 > 0.99) in every single experiment. At high surface coverage the calculated lower rate constants imply that another mechanism (“cooperative”, “bridging” phenomena) becomes important.

-0.6

3.2.2.2. Global kinetics

t, min Fig. 4. The variation of oxygen uptake ln(1 − qt /qe ) against time (C0 = 10 mg/l, w = 2 g, T = 301 K, regressed slope = −0.0446).

Rate constant, min-1

0.06 Co = 10 mg/l

0.05

Co = 30 mg/l

0.04

3.2.2.2.1. Diffusion rate model The rate limiting step in adsorption dynamics is considered to be intraparticle diffusion (Duong, 1998). The diffusion of adsorbate within particle can be described by  jq jq D j = 2 r2 (5) jt jr r jr and the volume average adsorbate concentration and its time derivative are

rp

3 rp 1 q(r, t) dV = 3 q(r, t)r 2 dr (6) q¯ = Vp 0 rp 0

0.03 0.02 0.01 0 0

1

2

3

4

5

6

7

w, g Fig. 5. Rate constants vs. mass of carbon at two levels of initial oxygen concentration (T = 301 K).

and 3 jq¯ = 3 jt rp



rp 0

jq 2 3D jq , r dr = jt rp jr r=rp

(7)

respectively. Assuming a hyperbolic profile, An effective LDF coefficient was obtained for each run by fitting the uptake data to Eq. (4) and a typical plot is reported in Fig. 4, with the LDF coefficient being calculated from the initial slope. The behavior exhibited in Fig. 4 is consistent with a LDF approximation in the solid phase only for the short time region (0–10 min) and consequently for low surface coverage (qt /qe = 0.30). The coefficients obtained varied between 1.54 × 10−2 and 5.12 × 10−2 min−1 , depending systematically on initial bulk concentration and carbon loading (Fig. 5).

q(t, r) = Y (t) + Z(t)∗ r n ,

(8)

Eq. (7) is reduced into: 15D dq¯ ¯ = 2 (q ∗ − q). dt rp

(9)

For a particle subjected to a step change, the solution of Eq. (9) subjected to the boundary conditions t = 0, q = 0;

616

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

t > 0, q(t, rp ) = q ∗ is given by the familiar expression: q¯ = q ∗ − 6q ∗

∞  n=1

1 (n)

2

exp(−n)2

Dt . rp2

(10)

Using the linear relationship obtained between oxygen dissociation equilibrium constant Keq and equilibrium uptake qe , Eq. (15) is transformed to  (17) q ∗ (t) = Aq e C(t),

If the quantity of adsorbate adsorbed is not negligible compared to the quantity introduced from the ambient fluid phase, the adsorbate concentration in the fluid will not remain constant after the initial step, giving rise to a time-dependent boundary condition at the surface of the adsorbent particle. The solution for the uptake curve becomes

where A is the slope of the relationship referred above. Combining Eqs. (9), (13), and (17) the following equation is obtained:  15D wAq e √ dx =− 2 x + x − 1 , where x = C/C0 . √ dt rp V C0 (18)

∞  exp(−Dp 2n t/rp2 ) q =1−6 , qe 9/(1 − ) + (1 − )pn2

(11)

At the equilibrium dx/dt = 0 and Eq. (18) gives   V C0 . qe = √ Aw 1−

(12)

A complex dependence of equilibrium uptake is demonstrated with the above equation, since the absolute amount of oxygen adsorbed is derived as V V Ce . (20) wq e = √ C0 − A A Ce

n=1

where pn is given by the non-zero roots of tan pn =

3pn 3 + (1/ − 1)pn2

and  = (C0 − Ce )/C0 , the fraction of adsorbate ultimately adsorbed by the adsorbent. Considering a batch adsorber that contains w mass of adsorbent we can write the mass balance for the system: qt = (C0 − Ct )V /w,

(13.1)

while qt dp = q¯ .

(13.2)

Eqs. (9) and (13.1) can be solved numerically together with the isotherm equation q ∗ = f (C).

(14)

Local equilibrium was assumed and an equilibrium reaction for the dissociation of oxygen on the solid surface was used. A reaction for the dissociation of oxygen on the solid surface is 1 O2(gas) ⇔ O(solid) 2 with Keq =

[O] [O2 ]1/2

=

q∗ , C 1/2

dq = k1 q ∗ − k−1 C 1/2 , dt

(15)

(16)

where rate constants k1 and k−1 are considered surface concentration dependent. The gas–solid system is at a pseudo-steady state (steady state on a local but not on a global scale). This local equilibrium is feasible only when at any point within the particle the local adsorption kinetics is much faster than the diffusion process into the particle. As time approaches infinity, the local adsorption isotherm will become the global adsorption isotherm (true equilibrium) as there is no gradient in concentration either in the pore space or on the surface phase at t =∞.

(19)

Figs. 6a–d show the variation of calculated values of equilibrium dissociation constant and diffusion coefficient vs. mass of carbon and equilibrium uptake, whereas Fig. 7a represents diffusion rate model fitting to the experimental data. It can be seen that oxygen dissociation equilibrium constant tends to an asymptotic value of 0.25 and 0.66 (mg/cm3 )1/2 for initial bulk concentration of 10 and 20 mg/l, respectively. However, at low carbon loading (< 1 g) a considerably increased value of Keq is observed in the case of high initial bulk concentration, a finding associated with the nature of the global equilibrium isotherm where the adsorbate–adsorbate interaction will tend to promote adsorption. The higher calculated values of Keq at higher initial dissolved oxygen concentrations are expected since the local equilibrium attained on the solid/liquid interphase is concentration-driven. When equilibrium dissociation is assessed for a wide range of oxygen uptake where the adsorption and desorption rate constants vary with uptake, a wide range of values is calculated. This variation is diminished at low surface coverage (below 0.004 mg/m2 equilibrium uptake). For constant liquid volume and given equilibrium concentration, Eq. (20) indicates a linear relationship with respect to the initial bulk concentration, which is consistent with the experimental data (Supporting material). The diffusion coefficient, for initial concentration of 20 mg/l, increases with carbon mass from 3.5 × 10−9 to 5.0 × 10−9 m2 / min while oxygen dissociation equilibrium constant decreases from 12.1 to 0.66 (l/mg)1/2 /g for the same range of carbon mass. The small increase in diffusion coefficient compared to the large decrease of dissociation constant reveals the higher contribution of cooperative character of adsorbate–adsorbate interaction with respect to the barrier resistance. As the barrier to diffusion decreases at lower surface coverage, the diffusion coefficient increases resulting to dissociation constant rate controlling and diffusion is having less influence on the overall kinetics of the process.

10mg/l

12

20mg/l

12 Keq, (l mg)1/2/g

10 8 6 4

10mg/l

20mg/l

10 8 6 4 2

2 0

0 0

2 4 mass of carbon, g

0

6

0.02

0.04

0.06

Adsorbed amount per unit surface, mg/m2

9.00E-09

9.00E-09 Diffusion coefficient, m2/min

Diffusion coefficient, m2/min

617

14

14

(l mg)1/2/g

Dissociation equilibrium constant,

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

8.00E-09 7.00E-09 6.00E-09 5.00E-09 4.00E-09 3.00E-09 2.00E-09 1.00E-09

10mg/l

20mg/l

8.00E-09

10mg/l

20mg/l

7.00E-09 6.00E-09 5.00E-09 4.00E-09 3.00E-09 2.00E-09 1.00E-09 0.00E+00

0.00E+00 0

1

2 3 4 mass of carbon, g

5

0 0.02 0.04 0.06 Adsorbed amount per unit surface, mg/m2

6

Fig. 6. (a) Oxygen dissociation equilibrium constant Keq vs. carbon loading at two levels of initial oxygen concentration (T = 301 K). (b) Oxygen dissociation equilibrium constant Keq vs. equilibrium uptake at two levels of initial oxygen concentration at T = 301 K (regressed slope 248.8, R 2 = 0.9997). (c) Diffusion coefficient D vs. mass of carbon at two levels of initial oxygen concentration (T = 301 K). (d) Diffusion coefficient D vs. equilibrium uptake at two levels of initial oxygen concentration (T = 301 K).

At low surface coverage, adsorption occurs on high energy sites where adsorbed molecules are less mobile. With the increase of carbon loading, oxygen molecules would diffuse at a rate faster than that for the case of low available surface area, assuming a net surface diffusion mechanism. This is not the case in the present study where oxygen molecules exhibit a cooperative behavior and compete with water molecules which put a surface barrier to oxygen adsorption, a key factor for the adsorption process. For a given number of available binding sites at high surface coverage area of adsorption, considerable adsorbate–adsorbate interactions are present (merging of clusters of adsorbed oxygen creates diffusion barrier) and this results in lower diffusion coefficients. The adsorption of dissolved oxygen at the beginning of the adsorption process (first adsorbed layer) is always faster than that at high surface coverage. This indicates that the interaction between carbon surface and oxygen molecules is weaker than that between oxygen molecules, leading to a lower mobility of the adsorbed oxygen at higher surface coverage than that at low surface coverage. In conclusion at high surface coverage a combined cooperative (local equilibrium)/barrier (diffusivity resistance) phenomenon occurs consistent with the experimental data, whereas at low surface coverage the cooperative character is diminished and adsorbate–adsorbate interactions are less important

producing a lower transport resistance. This behavior explains the pseudo-steady state condition of the system and is consistent with the failure of diffusion rate model coupled with a global adsorption isotherm instead of a local one. 3.2.2.2.2. Sensitivity analysis Sensitivity analysis regarding Keq and D was conducted and it was shown that oxygen dissociation equilibrium constant determines the equilibrium concentration, since it is a measure of the carbon capacity, and through this the whole oxygen uptake profile, whereas diffusion coefficient controls the kinetic rate of the adsorption process having no effect at the final equilibrium concentration. An important requirement in parameter estimation is that the sensitivity coefficients should not be of small magnitude (Beck et al., 1985). The scaled sensitivity coefficients, defined as Xs (t) =

jC(t) s , jt

s = 1, 2, . . . I ,

(21)

where s are the parameters to be estimated. (D, Keq ) are calculated and the results are reported in Fig. 7b for C0 = 10 mg/l. The scaled sensitivity coefficients have all the same units of oxygen concentration, and a direct comparison is then possible. It can be seen that their effect on the model response is similar during the first 50 min and it is impossible therefore to

618

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

30 1.2

25

25 1.0

0.6 10 0.4 5

0.2

0 0

0 1500

500 1000 Time, min

0 0

200

400

C, mg/l

C, mg/l

0.8 15

20 q/qe, dimensionless

20

15 10 5 0 0

200

400

600 t,min

800

1000

1200

Fig. 8. Failure of diffusion rate model at high initial bulk concentration (linear equilibrium isotherm: dotted line, oxygen dissociation equilibrium reaction, solid line, : experimental data). Experimental conditions: w = 6 g, C0 = 30 mg/l, T = 301 K.

600

Xs, mg/l

-0.5 -1.0

D K

-1.5 -2.0 -2.5

A simple adsorption model for high surface coverage, comprising two processes: adsorption and desorption, is proposed. Oxygen molecules are considered as “reactants” while adsorbed molecules are considered as “products”. The model used here incorporates the rate of diffusion through an “unstirred” layer of constant thickness () by solving for the Fick’s law, under the following assumptions:

Time, min

Fig. 7. (a) Diffusion rate model fitting (w = 1.00 g, C0 = 20 mg/l, T = 301 K) and (b) Sensitivity coefficients for C0 = 10 mg/l.

tell them apart. After that time the sensitivity coefficient for D decreases so that this parameter cannot be accurately estimated after this time period. On the contrary the sensitivity coefficient for Keq increases (in an absolute value) which is related to better chances of obtaining a good estimate. 3.2.2.2.3. Combined kinetic mass transfer model The experimental data derived from low and intermediate initial dissolved oxygen concentration were successfully simulated using the diffusion rate model. The above model, however, fails to account for the dependence of the adsorption rate on the concentration when the initial adsorbate concentration is high (30 mg/l) and it also fails to exhibit the rapid initial uptake behavior at the beginning as it can be seen in Fig. 8. The adequacy of the Fickian model is expected to decrease with increasing initial oxygen concentration since diffusion becomes less dominant and another process such as cooperative adsorption or bridging of isolated sites begins to have an increasing effect on transport. A combined kinetic mass transfer model is proposed to explain the variation of oxygen transport from classical concentration gradient driven diffusion. At high surface coverage, the pore mouth is very restricted and therefore the rates of adsorption and desorption at the micropore mouth are expected to be comparable to the diffusion rate and under severe constriction they become the rate-limiting step.

(i) The adsorbent is a porous material into which the solute must diffuse from a sub-interfacial region to an interfacial monolayer, described in terms of diffusivity D and adsorption/desorption rate constants ka and kd , respectively. (ii) Dissolved oxygen molecules are distributed evenly. (iii) Adsorption is isothermal. The rate of adsorption on the surface, d/dt, filled with identical, independent binding sites can be expressed as d(t) (t) 2 = ka 1 − C(0, t) − kd (t), dt max

(22)

where ka and kd are determined by the rate at which molecules strike the interface. The fraction of these molecules “activated” for adsorption,  = (t)/max is the fractional interfacial coverage, C(0, t) is the instantaneous concentration next to the adsorbing surface, and the exponent is the number of sites required to hold an oxygen molecule or its dissociation product. In Eq. (22) adsorption is considered as an elementary reaction between the oxygen molecules next to the adsorbing surface (first order in C(0, t)) and the free sites for adsorption at the surface (second order in 1 − /max ), whereas the desorption is considered as a first order reaction of the oxygen molecules at the surface (). The resulting concentration gradient between the subsurface and the bulk solution drives a diffusive flux of oxygen molecules from the bulk solution towards the surface. The transport of molecules in the bulk solution can be modeled as Brownian diffusion obeying Fick’s law. The Fick’s diffusion equation

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

expressed as

is solved with the following boundary conditions and assumptions: (i) The rate of adsorption is linearly related to the concentration gradient on the surface j d C(0, t) = . jt dt

(24)

(ii) The bulk concentration at a distance  from the surface is equal to the bulk concentration: C(, t) = Cbulk .

(25)

(iii) A linear concentration gradient in the “unstirred” layer is formed. d(t) j Cbulk − C(0, t) = D C(0, t) = D . dt jy 

(26.1)

Solving with respect to C(0, t) we acquire C(0, t) = Cbulk −

 d(t) . D dt

(26.2)

(iv) A generalized parabolic concentration-dependent diffusivity is assumed:  2 C D = D(0) . (27) C0 The mass balance for bulk concentration can be expressed as V C 0 = V C bulk + SBET w(t).

(28)

Substituting Eq. (26.2) into Eq. (22) we obtain d app app = k (1 − (/max ))2 Cbulk − kd , dt app

(29)

app

where k and kd are the apparent adsorption and desorption rate constant defined, respectively, as app

ka

=

ka

(30.1)

1 + ka (1 − (/max ))2 D

and app

kd

=

kd 1 + ka (1 − (/max ))2 D

.

(30.2)

¯ The mean diffusion coefficient D(C) at the adsorbed concentration C can be determined by the following equation:

C0 1 ¯ 0 , Ce ) = D(C) dC D(C C0 − Ce Ce

 2  Ce Ce 2 = D(0)C0 1 + (31) + C0 C0

max,

(23)

1.0

Reduced uptake / dimensionless

1.2

j2 jC(y, t) = D 2 C(y, t) jt jy

D

619

0.8 0.6 0.4 0.2 0 0

200

400

600 Time, min

800

1000

1200

Fig. 9. Comparison of mixed kinetic-mass transfer model prediction based on constant diffusivity (dotted line) and variable diffusivity (solid line) with experimental data (). Experimental conditions w = 6 g, C0 = 30 mg/l, T = 301 K.

while the mean mass transfer coefficient can be defined as kL,mean =

D¯ D(0) 2 = C0 (3 − 3 + 2 ).  

(32)

An important feature of the proposed model is the hypothesis of variable surface coverage leading to a concentration-dependent diffusivity obtaining better fitting results as presented in Fig. 9, where the uptake experimental data and predicted values at high initial adsorbate concentration are compared. In the case of low surface coverage an average diffusivity was used in the diffusion rate model and the effect of the local surface loading on surface diffusivity at any given time was ignored since its contribution in the process was not of major importance. The estimated parameters of the proposed model are the kinetic parameters ka , kd , and the transport parameter kL,mean . The transport parameter is a measure of diffusivity since the thickness of the “unstirred” layer can be considered constant due to the experimental hydrodynamic conditions being identical. In Fig. 10 values of mean mass transfer coefficient and adsorption rate constant determined from combined kinetic mass transfer model vs. adsorbate uptake are reported. According to the proposed model, the rate of adsorption will be limited by the rate of oxygen appearance in the interfacial region. No kinetic barrier will be observed (high adsorption rate constant) when the subsurface contains a sufficient amount of oxygen (high mass transfer coefficient at high equilibrium concentrations). This is evident in Fig. 10a, where mean mass transfer coefficient (kL,mean ) exhibits a linear increase with oxygen amount adsorbed per unit surface at equilibrium. At higher surface coverage the movement of oxygen into the interfacial region will be energetically favorable causing an increase in the adsorption rate constant (k ) maybe due to cooperation. However, at very high equilibrium concentrations qe , this behavior is diminished leading to an almost constant adsorption rate constant, as can be seen in Fig. 10b.

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

4.0E-05 3.5E-05 3.0E-05 2.5E-05 2.0E-05 1.5E-05 1.0E-05 5.0E-06

1.2E-02 Adsorption rate constant, l/m2/min

Mean mass-transfer coefficient, m/min

620

0.0E+00

1.0E-02 8.0E-03 6.0E-03 4.0E-03 2.0E-03 0.0E+00

0 0.02 0.04 0.06 0.08 0.1 Adsorbed amount per unit surface, mg/m2

0 0.02 0.04 0.06 0.08 0.1 Adsorbed amount per unit surface, mg/m2

Fig. 10. (a) Variation in mean mass transfer coefficient as determined from mixed kinetic mass transfer model (301 K). (b) Variation in adsorption rate constant as determined from mixed kinetic mass transfer model (301 K).

4. Conclusion The macroscopic adsorption behavior of dissolved oxygen on GAC has been studied. The material is characterized by its high specific surface area (1307 m2 /g), microporocity (micropore volume 0.54 cm3 /g), its basic character (0.57 meq/g total basic groups) and its high iron content (15,480 ppm Fe). BET n-layer adsorption takes place suggestive of cooperative adsorption and important adsorbate–adsorbate interactions. Adsorption kinetics is dependent on surface coverage: at low surface coverage the transport mechanism can be explained by an intraparticle diffusion rate model, equipped with an oxygen dissociation equilibrium reaction (local equilibrium isotherm), while at high surface coverage a combined kinetic mass transfer model with concentration-dependent diffusion is proposed and validated through experimental data.

ka kd kLDF kL,mean K Keq M n q

radial distance, m specific surface area, m2 /g time, min volume, l mass of carbon, g reduced concentration (C/Csat ), dimensionless function of time, mg/cm3 function of time, mg/cm3+n

Greek letters   

adsorbed amount per unit surface, mg/m2 fractional surface coverage, dimensionless fraction of adsorbate adsorbed by the adsorbent, dimensionless

Subscripts

Notation A C d D D(0) k1 , k−1

r SBET t V w x Y Z

slope between Keq and qe , l1/2 /mg1/2 concentration, mg/l density, g/cm3 diffusion coefficient, m2 / min diffusion coefficient at zero loading, m2 / min reaction rate constants, min−1 and mg1/2 /m1/2 min adsorption rate constant, l/m2 min desorption rate constant, min−1 linear driving force mass transfer coefficient, min−1 mean liquid phase mass transfer coefficient, m/min constant at multilayer isotherms, dimensionless dissociation equilibrium constant, (l/mg)1/2 /g amount adsorbed, g exponent in Eq. (8), dimensionless oxygen uptake, mg/g

0.5 app b e est exp max 0 p s sat t zpc

half-time apparent bulk equilibrium estimated experimental maximum initial particle surface saturation at time t zero point of charge

Superscripts * –

local equilibrium between water and solid phase volume average

V.M. Matsis, H.P. Grigoropoulou / Chemical Engineering Science 63 (2008) 609 – 621

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