ISSN 0104-6632 Printed in Brazil

www.abeq.org.br/bjche

Vol. 33, No. 02, pp. 373 - 382, April - June, 2016 dx.doi.org/10.1590/0104-6632.20160332s20140077

*To whom correspondence should be addressed

Brazilian Journal of Chemical Engineering

MATHEMATICAL MODELING OF BATCH ADSORPTION OF MANGANESE ONTO

BONE CHAR

M. E. Maria and M. B. Mansur*

Departamento de Engenharia Metalúrgica e de Materiais, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Escola de Engenharia, Bloco 2, Sala 3634, Pampulha,

CEP: 31270-900, Belo Horizonte - MG, Brazil. Phone: + (55) (31) 3409-1811, Fax: + (55) (31) 3409-1815

E-mail: marcelo.mansur@demet.ufmg.br

(Submitted: September 25, 2014 ; Revised: January 7, 2015 ; Accepted: March 20, 2015)

Abstract - The present study investigated the dynamics of batch adsorption of manganese onto bone char by using two distinct mathematical formulations: the diffusion model and the shrinking core model. Both models assumed spherical particles and adequately described the transient behavior of metal adsorption under changing operating conditions. Comparatively, the diffusion model described the manganese adsorption better at distinct particle sizes even when small particles were used (dp ≤ 0.147 mm); the shrinking core model proved to be more reliable when larger adsorbent particles were used (dp > 0.147 mm), and it described experimental data better at changing solid-liquid ratios. Manganese adsorption was favored when: (i) smaller adsorbing particles were used due to the increase in the contact area and easier access to reacting sites of the char; however, such an effect proved to be limited to dp ≤ 0.147 mm, and (ii) higher solid-liquid ratios were used due to the increase in the available reacting sites. External and intraparticle mass transfer dependences on particle size and solid-liquid ratio were also investigated, and results corroborated with prior investigations found in the literature. Keywords: Adsorption; Manganese; Bone char; Diffusion model; Shrinking core model.

INTRODUCTION

The treatment of industrial effluents containing dissolved metals is commonly done using chemical precipitation, which is a relatively simple operation and an efficient method to attend to legal require-ments. The method, however, is not advantageous in the treatment of sulfuric aqueous solutions contain-ing manganese, as is the case of acid rock drainage (ARD) effluents generated in the southeastern region of Brazil, because manganese presents high solu-bility within a wide range of pH (Bamforth et al., 2006). In fact, the precipitation of manganese may occur at high pH values of approximately 10-11.

After the manganese has been removed, the pH of the effluent must be neutralized in order to discharge the treated water; therefore, the consumption of rea-gents is considerable. In addition, the quantity and the toxicity of the sludge will depend on the type and concentration of the dissolved metals. Therefore, it may result in a costly operation.

The removal and/or recovery of manganese from a typical nickel laterite waste solution was evaluated by Zhang et al. (2010) using distinct precipitation agents, such as hydroxides, carbonate, and SO2/air oxidative mixture. The best result in terms of effi-ciency and economics was obtained by applying a combined method. Hydroxide or carbonate precipita-

374 M. E. Maria and M. B. Mansur

Brazilian Journal of Chemical Engineering

tion was adequate for the removal/recovery of a large proportion of Mn(II), followed by oxidative precipi-tation applied to reduce manganese concentrations to a very low level. A literature review covering various separation and recovery methods, including solvent extraction, ion exchange, as well as hydroxide, car-bonate, sulfide, and oxidative precipitations, is pre-sented by Zhang and Cheng (2007). These methods are briefly compared and assessed in terms of their selectivity, efficiency, reagent costs, and product quality.

The adsorption of metals onto low cost sorbents, such as bone char, has also been evaluated as an alternative method to treat effluents (Choy and McKay, 2005; Giraldo and Moreno-Piraján, 2008; Kumar et al., 2010; Moreno et al., 2010; Martins et al., 2014; Vieira et al., 2014; Sicupira et al., 2014). Manganese, as well as cadmium, copper, iron, nickel and zinc, could be efficiently removed from sulfuric solutions at nearly neutral pH. As bone char consists basically of hydroxyapatite (Ca10(PO4)6(OH)2) and calcite (CaCO3), it also raises the pH of the aqueous solution due to the dissolution of the calcite, thus contributing to a reduction in the lime consumed in the current treatment of ARD solutions containing manganese by means of chemical precipitation. In fact, the efficiency of metal adsorption is not favored in acidic medium due to proton competition at pH < 5.

Experimental results obtained by Moreno et al. (2010) and Sicupira et al. (2014) using ARD efflu-ents revealed that manganese adsorption onto bone char followed satisfactorily the Langmuir equilib-rium isotherm, with a value of qm between 22 and 30 mg g-1. This proved to be a chemisorption process, which is strongly influenced by operating variables, such as the pH of the aqueous phase and the solid/ liquid ratio. Regarding its dynamics, manganese adsorption follows a pseudo-second order kinetic model (Ho, 2006). In addition, data fitting to simpli-fied intraparticle diffusion models, such as those of Weber and Morris (1963), revealed that manganese diffusion within particles is the main rate-limiting step, whereas external mass transfer and intraparticle diffusion phenomena may also affect the removal of manganese when particles of smaller sizes are used.

In an attempt to evaluate the dynamics of manga-nese adsorption onto bone char and to identify the main phenomena that affect the process as a whole, two distinct models were developed in the present paper. In the diffusion model, the transient concen-tration of manganese is assessed by differential mass balances in the external aqueous phase and inside particles (Costodes et al., 2003). According to this model, both intraparticle and chemical reaction rates

may occur simultaneously. Hence, metal concentra-tion inside the particles depends on time and space. The diffusion model consists of a system of partial differential equations to be solved numerically. In the shrinking core model, two distinct regions (named reacted, or ash region, and non-reacted region) may exist inside particles, and the reaction front moves towards the center of the particle, while adsorption proceeds. In this case, metal concentration depends solely on time. The model consists of a system of ordinary differential equations whose numerical so-lution is much simpler to be obtained. However, the diffusion model is classical and it is normally applied to describe adsorption processes. In this context, the aim of the present work is evaluate how adequate the shrinking core model could describe manganese adsorption. A diffusion model was also developed for comparison purposes.

DEVELOPMENT OF BATCH ADSORPTION MODELS

To model the manganese adsorption onto bone

char in a batch operation, the following assumptions were formulated: (i) the aqueous phase is an isother-mal and incompressible fluid containing manganese at a known initial concentration and pH; (ii) the po-rous adsorbent particles of bone char are perfectly spherical, containing reacting sites that are homoge-neously distributed within them; (iii) the reaction of manganese in the bone char particles is governed by a chemisorption mechanism (Sicupira et al., 2014); (iv) equilibrium is described by the Langmuir ad-sorption isotherm (Sicupira et al., 2014); (v) the sys-tem is perfectly mixed, so the external mass transfer process occurs solely in a thin boundary layer sur-rounding the particles, and (vi) the pH of the external phase is constant with time due to the buffer effect of calcite dissolution from the bone char. Diffusion Model (DM)

The mass-balance equation of manganese in the bulk solution is given as follows:

pe S i r R

dCk A C C

dt

(1)

The mass balance of manganese inside the spheri-

cal porous particles of bone char considers that: (i) mass transfer of metal is described mathematically by Fick’s law, assuming an effective intraparticle diffusion coefficient that is constant and independent

Mathematical Modeling of Batch Adsorption of Manganese Onto Bone Char 375

Brazilian Journal of Chemical Engineering Vol. 33, No. 02, pp. 373 - 382, April - June, 2016

of concentration, and (ii) the rate of adsorption of manganese is due to the chemical reaction that occurs at reacting sites that are homogeneously distributed within the particle (Crank, 1975).

22efi i

a

DC C qr

t r r tr

(2)

The equilibrium relation between q and Ci is

given by the Langmuir adsorption isotherm (Sicupira et al., 2014):

1m i

i

aq Cq

aC

(3)

The reaction by which the immobilized manga-

nese is formed proceeds quite rapidly when com-pared with the diffusion process, and for this reason the kinetics are assumed to be instantaneous and not involved in the rate-controlling process (Teixeira et al., 2001; Lee and McKay, 2004; Jena et al., 2004); therefore, local equilibrium can be assumed to exist between the free Ci(r,t) and immobilized q(r,t) com-ponents of the diffusing manganese species. Hence, deriving Eq. (3) and substituting into Eq. (2):

2(1 )i m i

i i

C aq Cq q

t C t taC

(4)

2

2 21

efa m i i

i

Daq C Cr

t r rraC

(5)

Equations (1) and (5) are subjected to the follow-

ing initial and boundary conditions:

00C C ,0 0iC r (6)

0

0i

r

C

r

(7)

pp

ief e i r R

r R

CD k C C

r

(8)

Shrinking Core Model (SCM)

According to this model, the adsorbing particle is surrounded by an external film and consists of two distinct regions schematically shown in Figure 1: the

permeable product layer, which is loaded in manga-nese (Rp ≤ r < rc), and the unreacted core, which shrinks uniformly as the reaction progresses (rc ≤ r ≤ 0).

Figure 1: Spherical particle representation using the shrinking core model.

The reaction between Mn2+ ions and the reactant

solid B of the char is assumed to be irreversible and occurs solely at the product-reactant interface (r = rc) according to the following stoichiometric reaction:

) (2

( ) ( ) aq s immobilizedMn b B Mn B

(9)

The model assumes that a pseudo steady-state ap-

proximation is valid and that the driving force in both external and particle mass transfer is linear, while the driving force in the reaction core incorpo-rates the Langmuir isotherm. In the adsorbing parti-cle, the relative effect of each resistance may change as the reaction proceeds because the length of the permeable product layer increases with time. If all mechanisms occur simultaneously, the system should be handled accordingly and the following rate ex-pressions for each stage (film diffusion, product layer diffusion, and chemical reaction, respectively) can be formulated (Levenspiel, 1999; Han, 2002):

2

2

( ) 4

4

1 1

4

p

p c

c

p e i r R

ef i ir R r r

c p

c r i r rm

N t R k C C

D C C

r R

qr k C

a q q

(10)

Bulk solution

C(t)

rc

Rp

Film diffusion layer

Permeable product layer

Core shrinks as reaction progresses

376 M. E. Maria and M. B. Mansur

Brazilian Journal of Chemical Engineering

The adsorption rate N(t) is related to the differen-tial mass balance over the system by equating the decrease in manganese concentration in the solution with the accumulation of the adsorbate in the bone char and the mass balance on a spherical element of adsorbate particle as given by, respectively:

24( ) c m cr q drdC dq

N t V Wdt dt b dt

(11)

Depending on which step of Equation (10) is slow-

est, that step is limiting for the overall adsorption process. Three limiting steps may occur: (i) diffusion through the film boundary layer (in this case,

pi r R

C = 0), (ii) diffusion through the porous product

layer (in this case, p

i r RC

= C and c

i r rC = 0), and

(iii) chemical reaction at the reactant-product inter-face (in this case,

ci r r

C = C). For each case, an

equation giving the time required for a reaction to proceed from particle radius Rp to rc can be obtained by integrating drc/dt with time from 0 to t (Han, 2002). If all mechanisms occur simultaneously, the mass-balance equation for manganese in the bulk solution is:

2

2

2

4

1

p

m

p c p p

e c ef c r

R qC

V a q qdC

dt R r R R

k r D r k

(12)

the average loading of manganese in the char is:

2

2

2

4

1

p

m

p c p p

e c ef c r

R qC

W a q qdq

dt R r R R

k r D r k

(13)

and the temporal variation of the reacting-core radius is given by:

2

2

1

m mc

p c cc

p ef rp e

bC qC

q a q qdr

dt R r rr

R D kR k

(14)

Equations (12)-(14) were subjected to the follow-ing initial conditions:

00C C 0q 0 c pr R (15)

According to this model, the adsorption rate of

manganese onto bone char is controlled by the re-spective resistances to film diffusion, product layer diffusion, and chemical reaction. If an instantaneous reaction within spherical adsorbent particles prevails, the third resistance of Eqs. (12)-(14) is null. Numerical Solution of Models and Estimation of Parameters

Both models were solved numerically using MATLAB software (Version 7.0). The diffusion model, Eqs. (1), (5)-(8), contains a partial differential equation, which was discretized using the implicit finite difference method. Other numerical methods, such as Crank-Nicholson’s implicit finite difference method; a semi-analytical solution method; and the Cartesian collocation method were compared else-where in terms of accuracy, stability, convergence, and computation in CPU time (Lee and McKay, 2004; McKay, 2001). The solution of the shrinking core model, Eqs. (12)-(15), was obtained using the fourth-order Runge-Kutta method. The same numeri-cal solution was used by Sarkar and Bandyopadhyay (2011) and Jena et al. (2004). Properties such as accuracy, stability, convergence and computational time of the numerical methods used in the present work are available elsewhere (Hoffman, 1992; Pinto and Lage, 2001).

Experimental data obtained by Sicupira et al. (2014) using laboratory solutions containing manga-nese were fitted to both models in an attempt to as-sess the values of mass transfer and chemical reac-tion rate parameters under changing operating condi-tions. The estimation of parameters was done nu-merically by minimizing an objective function (F) using an optimization routine found in MATLAB software based on the Nelder-Mead simplex direct search method (Lagarias et al., 1998). The objective function adopted in this work is the sum of the squared deviation between estimated (Cest) and ex-perimental (Cexp) concentrations of manganese in the external aqueous phase:

exp

2

, exp,1

N

est j jj

F C C

(16)

cuti

r

wfirechcocoTacTthmitth

The relativulated to meions are to th

elative error

RE

Physical cwas evaluatedilm diffusioneaction resihanging the oefficient (koefficient (D

The simulatioccording to

Table 1. Typhe external

manganese cot reached a che operating

Brazi

ve error in teeasure how

he experimen

exp

exp 1

100N

jN

ESULTS AND

consistency od by simulatin, product lastances. Camagnitude o

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the operatinical transienaqueous ph

oncentration condition of conditions.

Mathematical M

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Manganese Onto

o. 02, pp. 373 - 3

gure 2: Senodel: (a) Filmd kr = 10-5

stance (ke = 1hemical reacef = 10-10 m2 s = 100 mg L

able 1: Operdsorption of al., 2014).

ParamInitial concentrmanganese in tVolume of aquSurface area ofStoichiometric Particle porositReal density of

Intraparticlerption procesass transfer rons simulated

shown in Fiom or to solquid are affecrticle diametller, clearanc. The smallel resistance rticles; constained. Jadhrvey of the

ons and the och mass traass transfer lue of Def deffusion as shfect of chemveals that faon. This effecan those obs

Bone Char

82, April - June,

nsitivity analm diffusion rem s-1), (b) P10-5 m s-1 anction resistas-1) (S/L = 2/4L-1; pHi = 5.7

rating condimanganese

meter ration of the external phaueous phase f bone char constant

ty f particle

e resistance sses; howeveresistance isd herein, maigure 2(a). Elid particles cted by a numter, viscosityce, agitationer the value for mangane

sequently, a hav and Pang

literature, iobserved effeansfer coeff

resistance ecreases, slowhown in Figmical reactioster kinetics ct, however,served for m

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ysis of the sesistance (Def

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itions of simonto bone c

Symbol

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V

ap mb

g

normally goer, the effect significant

ainly when kExternal mass

suspended mber of vari

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lower adsogarkar (1991including vaects of typicaficients. Thealso increas

wing down tgure 2(b). Anon shown in favor mang, is much les

mass transfer

3

shrinking coef = 10-10 m2 r diffusion rm s-1), and (0-5 m s-1 an

dp = 0.833 mm

mulated batcchar (Sicupir

Unity Valuemg L-1 100

L 0.4m2 g-1 93

- 1- 0.7975

g cm-3 2.9

overns the aof the externfor the cond

ke < 10-5 m ss transfer ratin an agitateiables, such id, type of imssel geometrgher the extethe adsorbe

orption rate 1) presented arious correlal variables oe intraparticses when ththe manganend finally, thn Figure 2(

ganese adsorss pronounceeffects in th

377

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imulated conhe effect of he diffusion p

The effect f manganesencluding simDM) and theation coefficre displayed esulted in a inetics is qui

mainly if smao a larger conng sites of tht reaches equdsorption kiffect at longavior shownimited when p

Table 2: Fitthrinking coS/L = 2/400

Diameter of particle (mm)

Dec

< 0.053 0.104-0.147 0.417-0.833

Figure 3: Effmanganese on

00 mg L-1; p

By compat can be seenxperimental nd relative eiameters weonditions, a

nditions. As einstantaneouprocess. t of particle se onto bone mulations ue shrinking cocients and rel

in Table 2. Dshorter timeite rapid at thaller particlentact area an

he char, and uilibrium. Tnetics at shoer times. Ac

n in Figure 3particles with

ting parameore models fg mL-1; C0 =

Diffusion moetermination coefficient

(R2)

R

0.9701 0.9719 0.9731

fect of particnto bone cha

pHi = 5.76).

aring the simn that the diftransient behrror < 15%),ere used (dp

sharp decre

evidenced byus reaction is

size on the bchar is show

using the diore model (Slative errors fDecreasing te to reach eqhe beginnings are used, w

nd easier conslows down

The particle orter times, ccording to th3, however, sh dp ≤ 0.147

eters of the for different= 100 mg L-1

del Shrin

Relative error (%)

Determcoef

(R14.2 0.913.7 0.913.9 0.9

cle size on thar (S/L = 2/40

mulations fromffusion modehavior quite , even when

dp ≤ 0.147 ease in the c

M. E.

Brazilian Jou

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batch adsorptwn in Figureiffusion moSCM). Determfor both modthe particle squilibrium. Tg of the procewhich is relantact with rea

over time usize affects but it has lihe transient such effects mm were use

diffusion at particle si1; pHi = 5.76

nking core modmination fficient R2)

Relaterro(%

9667 18.9626 19.9972 2.9

he adsorption00 g mL-1; C

m both model described well (R2 > 0smaller partimm). In thoncentration

Maria and M. B.

urnal of Chemica

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al Engineering

anganese octerwards. Thch behavior ror < 20%).orer than th

hen small pae unreacted aell defined irinking corehen larger pa0.417 mm, w

Regarding sults portrayrcentage of me increase icrease in the onger time we increase in netics of ma

first contacre model des99 and relatodel (R2 > 0.cause larger ese runs (Tab

able 3: Fittirinking cortios (dp = 0.476).

gure 4: Effeon of manganm; C0 = 100

olid-liquid ratio

(g mL-1) Dete

co

1/400 2/400 3/400

ccurred at she shrinking

satisfactoril However, hat obtainedarticles were and reacted in smaller de model proarticles of th

with R2 > 0.99the effect

yed in Figurmanganese rin the solidquantity of r

was required the solid-liq

anganese adsct. In these sscribed data tive error < 95 and relatiparticles of t

ble 3).

ng parametre models f417-0.833 mm

ect of solid-lnese onto bomg L-1; pHi

Diffusion modermination oefficient

(R2)

Reer(

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shorter timecore model ly (R2 > 0.9fitting was

d by the difused, most regions migh

dimensions. Toved to be he adsorbent 9 and relativof solid-liqure 4 reveal removal was d-liquid ratioreacting charto load the a

quid ratio, desorption idensimulations, comparative10%) than

ive error < 3the adsorben

ters of the dfor differenm; C0 = 100

liquid ratio oone char (dp = 5.76).

del Shrinkelative rror (%)

Determicoeffic

(R2

2.8 0.9913.9 0.9932.8 0.99

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ntified at timthe shrinkin

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The transpata fitting uus reaction tanese onto nvestigated bn Figure 5. Tarameters wting conditiohe same resuameters. Powate data of thhe investigatome cases thre commonlyno preferredtructures sucDo and Nguiffusion modetter, while ffect of the he estimatesncluded in F

model with a epresented bther is repres

The averagally obtained(a). Such a rnvestigations

Asai et al., adhav and P

mass transfer phere to an in(Sherwood

with moleculDMn,water = 6.8menico and S

(a). The poss attributed ively higher

models were sed (dp ≤ 0.ant decreasef manganeseents. Concern the extern

Figure 5(b), iwith an averaonsidering erating the e1988).

An opposiwas identifiemodel (Figur

Brazi

port coefficiesing both mto describe thbone char iby Sicupira The effect ofas not prono

on, the solutiult with diffwer law equahe transport cted operatinhe R2-valuesy used to de

d length scalch as fractauyen, 1988). del describedthe shrinkingsolid-liquid

obtained byFigure 5 forhigher deter

by a wide cosented by a nge dependend for both mresult corrobos (Harriot, 11988; Arme

Pangarkar, 19coefficients

nfinite stagnnumber is g

lar diffusivit8x10-10 m2 s-

Schwartz, 199sitive deviati

to the convdeviations b

observed w.147 mm), th in the exter

e when microrning the effnal mass trait was found

age value of kstimates fromexperimenta

ite behavior wed by data re 5(c)). Ex

Mathematical M

ilian Journal of C

ents ke and Dmodels assum

he batch adsoin the operaet al. (2014f correlationunced, sinceion practicallferent initial ations were ccoefficients a

ng condition were poor, escribe self-sle), as verifials and adso

As previoud the effect g core mode

d ratio bettery using both

r comparisonrmination coontinuous cunarrow dashence of ke on dmodels, as shorates a num1962; Kuboenante and 991). Values of mangane

nant fluid, whgiven by Sh =ty of manga-1 at 25 oC, ac98) is also shon from thisvective contetween estimwhen small hus evidenc

rnal mass traoparticles arefect of the soansfer coefficd to be of litke = (13±4)xm both model findings o

with dependfitting usingxternal and

Modeling of Batc

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Def assessedming instantaorption of m

ating conditio4) are displayn between fite, for each oply convergedguesses of

chosen to coras a function

ns. Althoughsuch equatio

similar systefied in ramiforbent particusly shown, of particle s

el described r. Nevertheleh models wn purposes; oefficient (R2

urve, while ed curve. dp

-0.5 was prahown in Fig

mber of previoi et al., 19Kirwan, 19of the exter

ese to or fromhich yields S= kedp/DMn,w

anese in waccording to Dhown in Figs limiting vatribution. Remates from b

particles wing the sign

ansfer resistane used as adsolid-liquid racient shownttle importan

x10-6 m s-1 whels, thus corrof Asai et

dence Def dg the diffus

internal m

ch Adsorption of M

ering Vol. 33, No

by ane-

man-ons yed tted per-d to pa-rre-n of h in ons ems fied cles the

size the

ess, were

the 2) is the

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974; 989; rnal m a

Sh = water, ater Do-gure alue ela-

both were nifi-nce sor-atio n in nce, hen ob-al.

dp0.6

sion mass

trato parlonbentaiferliqshointratdu

Manganese Onto

o. 02, pp. 373 - 3

ansfer resistathe boundarrticle resistnger diffusiont particles ned may be rent class sizquid ratio onown in Figuternal resistantios were use

ue to the incre

Bone Char

82, April - June,

ances are clory condition, ance was e

on path is expare used. Tattributed to

zes. As regarn the intraparure 5(d), a snce was foun

ed (dependenease in the am

2016

osely connecEq. (8), so

expected. Inpected whenTherefore, tho diffusion inrds the effecrticle diffusisignificant innd when highnce Def (S/Lmount of poro

3

cted accordina higher intrn addition,

n bigger adsohe results on pores of dict of the soliion coefficiencrease in thher solid-liquL)-1.5), which ous particles.

379

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.

38

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Figure 5: Tranata fitting wore model asperating con2014).

The Biot nalogous to ter) is definedransfer resistaransfer resistropera et al.ore diffusionting conditio

Therefore, thot uniform th

The batchhar was mroaches assu

model and thwere compareowing conclu Both m

nt behavior ooncentrationshed monotoibrium, whicitions; The di

article size bescribed the

Manganese addsorbing parhe contact arhe char, but o dp ≤ 0.147

nsport coeffiwith the diffussuming instanditions inve

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)

Mathematical Modeling of Batch Adsorption of Manganese Onto Bone Char 381

Brazilian Journal of Chemical Engineering Vol. 33, No. 02, pp. 373 - 382, April - June, 2016

Sh Sherwood number (-) S/L solid-liquid ratio (g mL-1) t time (s) V volume of aqueous phase (L) W weight of the adsorbent (g) Greek Symbols particle porosity (-) density of adsorbent particle (g m-3) a apparent density of adsorbent particle

(g m-3)

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