Colloids and Surfaces A: Physicochem. Eng. Aspects 335 (2009) 181–188
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Colloids and Surfaces A: Physicochemical andEngineering Aspects
journal homepage: www.elsevier.com/locate/colsurfa
Biosorption of copper(II) ions from aqua solutions using dried yeast biomass
Corneliu Cojocaru
a
, Mariana Diaconu
a
, Igor Cretescu
a
,
∗∗
, Jasmina Savi´c
b
,
∗
, Vesna Vasi´c
b
a
Department of Environmental Engineering and Management, Technical University of Iasi, Bvd. D. Mangeron 67, 700050 Iasi, Romania
b
Department of Physical Chemistry, Vinˇ ca Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia
a r t i c l e i n f o
Article history:
Received 18 April 2008Received in revised form 5 October 2008Accepted 7 November 2008Available online 14 November 2008
Keywords:
BiosorptionAdsorption isothermHeavy metalsResponse surface methodology (RSM)
a b s t r a c t
Theabilityofdriedyeast
Saccharomyces
biomasstoremoveCu(II)ionsfromaqueoussolutionswasinvestigated by using of batch techniques. The inﬂuence of different parameters on copper uptake by driedyeast, such as initial Cu(II) concentration, initial pH of solution and temperature, was studied. The Freundlich,Langmuir,Redlich–PetersonandSipsisothermswereappliedtotheobtainedexperimentaldata.According to Langmuir isotherm the maximum adsorption capacity of investigated nonliving biomasswas found to be 2.59mg/g. The thermodynamic parameters (e.g. free energy and enthalpy) were calculated and discussed. The adsorption of Cu(II) onto the dried cells of
Saccharomyces cerevisiae
is anendothermic process and become more favorable with the increasing of temperature in pH range from 3to 4. Optimization studies by means of response surface methodology were carried out, which resultedin improvement of the efﬁciency of sorption removal by using of biomass. The removal efﬁciency of real wastewater srcinating from electroplating industry which contains Sn(II) ions was determined andcompared with synthetic wastewater obtained in laboratory.© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Wastewater contamined with heavy metals is a serious environmental problem because they do not undergo biodegradationand are accumulated into the organism entering into the foodchains.Duringrecentyears,theintensiveindustrialactivities,suchas electroplating, microelectronics, battery manufacture, dyestuff,chemical, metallurgical, pharmaceutical and other, greatly contribute to the increase of heavy metals in the environment [1].The stringent limits of different pollutant concentrations inindustrial and municipal wastewaters, imposed by the environmental legislation, make the treatment to be imperative.Conventional methods for removing of metal ions from aqueoussolutions, like chemical precipitation, ion exchange, electrochemical treatment, and adsorption on activated carbon, have signiﬁcantdisadvantages. Chemical precipitation and electrochemical treatmentbecomeineffectiveparticularlywhenmetalionconcentrationin the solution is low (in the range from 10 to 100mg/L), becausethey produce large quantity of sludge to be treated and whichrequiredisposal.Ionexchangeandactivatedcarbonadsorptionareextremely expensive processes, especially for the treatment of alarge amount of wastewater containing low heavy metal concentrations [2,3].
∗
Corresponding author. Tel.: +381 11 2453 967; fax: +381 11 2447 207.
∗∗
Corresponding author. Tel.: +40 741 914342; fax: +40 232 271311.
Email addresses:
icre@ch.tuiasi.ro (I. Cretescu), jasnas@vinca.rs (J. Savi´c).
Thesedisadvantagesofconventionalmethodstogetherwiththeneed of more effective and lowcost methods for the metal ionsremovalfromwastewaterresultedinthedevelopmentofnewseparationtechnologies.Thebiosorptionhasattractedtheattentionasa lowcost treatment technology for the removal of heavy metalsfrom wastewaters [1–15]. The biosorbents are prepared from naturally abundant materials and from byproducts or waste biomassfrom other industries [16]. Among biosorbents, those of microbiological srcin (e.g. bacteria, fungi, yeast and algae biomass) are of especial interest and were studied as potential heavy metal sorbents in various environments [4,6,7,11,12,14,17–23].Although many effects on biosorption have been widely studied, such as initial metal concentration, pH, temperature, andsorbent dose, the mechanism of metal biosorption is not completely understood because of its complexity. The metal uptake bybiosorption can occur due to physicochemical interactions suchas complexation, coordination, chelation, ion exchange, physicaladsorption or microprecipitation. The biosorption (passive sorption) by biomass occurs through interactions between metal ionsand functional groups at the cell surface like amino, phosphoryl,carboxyl,sulphydrilandhydroxylgroups[3].Inspiteofgreatnumber of papers reported for biosorption, this process is still at thestage of laboratoryscale study [3]. Therefore complex research isnecessary in order to select the best sorbent for certain pollutant[12].This work is focused on Cu(II) ions removal from aqueoussolutions using the dried nonliving yeast biomass (
Saccharomycescerevisiae
) as biosorbent. An effort to discuss the equilibrium
09277757/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.colsurfa.2008.11.003
182
C. Cojocaru et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 335 (2009) 181–188
adsorption isotherms on basis of Freundlich, Langmuir,Redlich–Peterson (R–P) and Sips models as well as to optimize the sorption process using response surface methodology(RSM) is also included in this work. In order to check the RSMoptimization of sorption process, the real sample obtained fromelectroplating industry was compared with synthetic wastewaterattained in laboratory conditions.
2. Experimental
2.1. Reagents
Commercial yeast biomass
S. cerevisiae
, available from localcommercial company, was prepared as nonliving biomass by drying in a hot air oven at 105
◦
C for 24h. Copper stock solution of 1000mg/L was prepared using CuSO
4
·
5H
2
O of analytical reagentgrade. This salt was chosen since sulphate ions exist in mostwastewater and mine drainage and thus simulating the potentialapplication of biosorption for copper ions removal. All solutionsandtheirdilutionswerepreparedbyusingofbidistilledwater.Theacidity of the solutions was adjusted by addition of 0.1N sulphuricacid or 0.1N sodium hydroxide solutions.
2.2. Instrumentation
The concentrations of copper solutions were determinedin aqueous solutions spectrophotometrically by measuring theabsorbance of complex formed between rubeanic acid and Cu(II)at 390nm. The absorbance was recorded using the direct reading spectrophotometer (HELIOS, USA). For pH measurements a pHmeter BOECO PT370, with a combined glass electrode, was used.The batch experiments were carried out using an orbital shaker.The temperature adjustment, in the range from 20 to 50
◦
C, wasperformed by using of BMT Ecocell incubator, with the precision of
±
0.5
◦
C.
2.3. Sorption studies
In order to obtain the adsorption isotherms, batch experimentswere conducted with copper aqueous solutions of initial concentration varying from 25 to 200mg/L and in pH range from 3 to5. Samples were mixed and left for 24h to reach the equilibrium.The resulting ﬁltrate was analyzed for copper. The biosorbent dose(BSD)intheseexperimentswasmaintainedconstant,i.e.1.5%(w/v)thatcorrespondsto1.5gofbiosorbentaddedto100mLofaqueoussolution. The adsorption isotherms were determined as the function of pH and temperature. The equilibrium adsorption capacitywas determined by the following expression:
Q
e
=
(
C
o
−
C
e
)
V m
1000 (1)where
Q
e
is the equilibrium adsorption capacity, i.e. the amount of copper ions adsorbed onto the biosorbent (in mg/g);
C
o
and
C
e
arethe initial and ﬁnal metal concentrations in the solution, respectively (in mg/L);
V
is the volume of the solution (in mL) and
m
represents the weight of the dried biosorbent (in g).In addition, another set of the experiments was performed inorder to optimize the process efﬁciency. The experiments concerning optimization were conducted in batch mode. In theseexperiments,theinitialcopperconcentrationwasvariedfrom20upto80mg/L.Afterreachingtheequilibriumtheresultingﬁltratewasanalyzed for ﬁnal copper concentration and the removal efﬁciencywas determined.In this respect the experimental design was applied and theremoval efﬁciency of copper ions from aqueous solution was chosenastheresponseforoptimization.Theremovalefﬁciency
Y
%was
Table 1
pH dependence of the isotherm parameters for copper biosorption onto
Saccharomyces
from aqueous solutions at
T
=293K.Type of isotherm pH 3 pH 4 pH 5Freundlich
n
=2.234
n
=2.226
n
=2.814
K
F
=0.260
K
F
=0.252
K
F
=0.287
R
2
=0.623
R
2
=0.673
R
2
=0.686
ARE
=24.230
ARE
=21.889
ARE
=16.653Langmuir
Q
max
=2.533
Q
max
=2.489
Q
max
=1.794
K
L
=4.241
×
10
−
2
K
L
=3.980
×
10
−
2
K
L
=4.872
×
10
−
2
R
2
=0.958
R
2
=0.940
R
2
=0.975
ARE
=18.832
ARE
=16.256
ARE
=11.422Redlich–Peterson
A
=5.858
×
10
−
2
A
=6.551
×
10
−
2
A
=4.698
×
10
−
2
B
=9.407
×
10
−
4
B
=2.313
×
10
−
3
B
=1.583
×
10
−
3
g
=1.637
g
=1.482
g
=1.550
ARE
=10.883
ARE
=10.423
ARE
=5.342Sips
Q
SM
=2.060
Q
SM
=2.019
Q
SM
=1.524
b
=1.274
×
10
−
5
b
=4.114
×
10
−
5
b
=1.977
×
10
−
4
n
S
=0.271
n
S
=0.303
n
S
=0.348
ARE
=2.743
ARE
=1.823
ARE
=1.852
determined by means of subsequent equation:
Y
=
1
−
C
e
C
0
100 (2)
3. Results and discussion
3.1. Adsorption isotherms
In the design of sorption systems, the equilibrium sorptionisotherms are very important from fundamental point of view. Theequation parameters and the underlying thermodynamic assumptions of these equilibrium models often provide some insight intoboth the sorption mechanism and the surface properties and afﬁnity of the sorbent [1]. In order to investigate the sorption isothermFreundlich,Langmuir,Redlich–PetersonandSipsequilibriummodels were applied.The wellknown expressions of the Freundlich and Langmuirmodels with two parameters are given by the following Eqs. (3)and (4).TheFreundlichequationisanempiricalrelationshipestablishedupon sorption onto a heterogeneous surface supposing that different sites with several adsorption energies are involved [5].
Q
e
=
K
F
C
1
/ne
(3)where
K
F
denotes the relative adsorption capacity and
n
the intensity of adsorption.The Langmuir isotherm:
Q
e
=
Q
max
K
L
C
e
1
+
K
L
C
e
(4)where
Q
max
and
K
L
are Langmuir isotherm constants. Thus,
Q
max
denotes the maximum adsorption capacity (mg/g) while
K
L
is theequilibriumconstantconnectedtotheenergyofsorptionthatquantitatively reﬂects the afﬁnity between the biosorbent and sorbate[5].The simplest method to determine isotherm constants for twoparameter isotherms is to transform the corresponding equationintothelinearformandthentoapplylinearregression[1].Thus,theapplicability of Freundlich and Langmuir sorption isotherms havebeen checked by plotting log(
Q
e
) versus log(
C
e
) and (
C
e
/
Q
e
) versus(
C
e
),respectively.ThevaluesofparametersinFreundlich(
K
F
and
n
)and Langmuir (
Q
max
and
K
L
) were determined and are reported inTables 1 and 2.The Redlich–Peterson and Sips isotherm equation with threeparameters are given by Eqs. (5) and (6).
C. Cojocaru et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 335 (2009) 181–188
183
Table 2
Temperature dependence of the isotherm parameters for copper biosorption onto
Saccharomyces
from aqueous solutions at pH 4.Type of isotherm
T
=308K
T
=323KFreundlich
n
=2.444
n
=3.176
K
F
=0.323
K
F
=0.546
R
2
=0.707
R
2
=0.890
ARE
=19.075
ARE
=8.409Langmuir
Q
max
=2.402
Q
max
=2.595
K
L
=5.619
×
10
−
2
K
L
=7.466
×
10
−
2
R
2
=0.982
R
2
=0.994
ARE
=13.652
ARE
=5.524Redlich–Peterson
A
=6.326
×
10
−
2
A
=0.118
B
=1.021
×
10
−
3
B
=2.007
×
10
−
2
g
=1.630
g
=1.153
ARE
=6.345
ARE
=4.073Sips
Q
SM
=2.154
Q
SM
=2.592
b
=3.16518
×
10
−
4
b
=4.110
×
10
−
2
n
S
=0.369
n
S
=0.861
ARE
=2.649
ARE
=4.498
The Redlich–Peterson isotherm:
Q
e
=
AC
e
1
+
BC
g e
(5)where
Q
e
and
C
e
have the same deﬁnitions as in Eqs. (3) and (4),
A
(L/g) and
B
(Lmmol
−
1
)
g
are the Redlich–Peterson isotherm constants, while
g
is the Redlich–Peterson isotherm exponent whichhas values between 0 and 1. For
g
=1 the Redlich–Peterson modelconverts to the Langmuir model.The Sips isotherm:
Q
e
=
Q
SM
bC
1
/n
S
e
1
+
bC
1
/n
S
e
(6)where
Q
SM
is the Sips maximum adsorption capacity (mg/g) and
b
is the Sips constant related to afﬁnity between solute andsorbent.It is noticeable that, at low solute concentrations the Sips equation is reduced to a Freundlich isotherm, while at high soluteconcentration it predicts a monolayer adsorption capacity characteristic of Langmuir equilibrium equation [24].The parameters of Redlich–Peterson (
A
,
B
,
g
) and Sips (
Q
SM
,
b
,
n
S
) isotherms, listed in Tables 1 and 2, were determined by meansofnonlinearregressionmethodusingaGauss–Newtontechnique.The nonlinear regression method provides a mathematically rigorous method for determining isotherm parameters by using of the srcinal form of the isotherm equation. This is an advantagecomparing to linear regression approach where the deviation biasappears as a result of linearization.Inordertorevealtheagreementoftheequilibriummodelswithexperimentalresults,theaveragerelativeerror(
ARE
)wascomputedas follows [20]:
ARE
(%)
=
100
z
z
i
=
1

Q
exp
−
Q
calc

Q
exp
i
(7)where
z
is the number of data points;
Q
exp
and
Q
calc
are theexperimentalsorptioncapacityandcalculatedsorptioncapacitybytheoretical models, respectively.
Fig. 1.
Adsorption isotherms obtained at 293K for pH 4 (a) and pH 5 (b).
Fig. 2.
Adsorption isotherms obtained at pH 4 for
T
=308K (a) and
T
=323K (b).
184
C. Cojocaru et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 335 (2009) 181–188
The average relative errors, represented in Tables 1 and 2,approve the effectiveness of the applied models in ﬁtting of theexperimental data.According to results represented in Tables 1 and 2, the copper uptake was in the range from 2.40 to 2.59mg/g at pH 4.0 andthe values of the Langmuir isotherm parameters indicate that themonolayer saturation capacity of Cu(II) ions,
Q
max
, is 2.59mg/g for
S. cerevisiae
dried cells. This value is comparable with the copper ions uptake by yeast biomass reported in the literature [2,25].As the illustration, the copper uptake by
S. cerevisiae
was rangedfrom 2.04 to 9.05mg/g for adapted and growing cells, while forcommercially available dry cells it was in the range from 2.98to 12.03mg/g [2,25,26]. The Sips model provided slightly lowervalues of maximum adsorption capacity than those achieved byLangmuirequation.AccordingtoSipsmodel,themaximumadsorption capacity for dry yeast cell was in the range from 2.02 to2.59mg/g for pH 4.0 and different temperature conditions. TheFreundlich isotherm constant
n
indicates that the intensity of adsorption is increasing with the increment of temperature. TheRedlich–Petersonisothermconstantsindicatethatthismodeltendstoward a Langmuir isotherm.Usingtheisothermparametersandmassbalanceequations,thepredicted amount of copper adsorbed per gram of sorbent wasdetermined and compared with experimental data (Figs. 1 and 2).As can be seen from Figs. 1 and 2 and
ARE
values listed inTables 1 and 2, the Sips isotherm give the best ﬁt with experimentalresults.Theothermodels,inorderofdecreasingagreementwithexperimentally obtained results, are as follows: Redlich–Peterson,Langmuir and Freundlich. Thus, the equilibrium models with threeparameters (Sips and R–P) ﬁt better to the experimental resultscomparing to the two parameters models (Langmuir and Freundlich).Likewise,Fig.1showstheeffectofpHontheadsorptioncapacity.IncreasingpHfrom4to5leadtoadecreaseofadsorptioncapacity.This could be attributed to the strong effect of initial concentration of solute which is very high comparing with the relatively lowsorptioncapacityofbiosorbent.Also,parallelprecipitationofCu(II)hydroxideispossiblewhichmeansthattheinitialconcentrationof Cu(II) ion is lower than it should be. Besides, increase in densityof the negative charge on the cell surface, causing proton removalfromthesolution,therebydecreasingbiosorptioncapacityofCu(II)because of competitively adsorption of protons.The adsorption capacities, reported for different temperatures,arerepresentedinFig.2.Ascanbeseen,aslightincreaseofadsorption capacity was induced with temperature growth. According tothe obtained results, the isotherm obtained by using of Langmuirmodel at 323K ﬁt the experimental data better than the isothermsreportedforlowertemperaturesandalsobetterthanthreeparameters models.On the basis of additional analysis of the Langmuir equation,the dimensionless parameter of the equilibrium (
R
L
—known alsoas separation factor) may be calculated as [27]
R
L
=
11
+
K
L
C
o
(8)where
C
o
(in mg/L) is the initial concentration of solute. Accordingto the value of separation factor
R
L
, following types of adsorption exists [27]: (1) favorable adsorption 0<
R
L
<1; (2) unfavorableadsorption
R
L
>1; (3) linear adsorption
R
L
=1; (4) irreversibleadsorption
R
L
=0. In our experiments, the initial solute concentration was in the range of 25–200mg/L and corresponding valuesof the separation factor
R
L
has been found to be in the range of 0.063–0.501. Hence, the results underlines that the adsorption of Cu(II) onto the dried cells of biomass is favorable under all conditions considered in this work.
Fig. 3.
Dependence of ln(
K
L
) vs. 1/
T
for Cu(II) adsorption on dried cells of
Saccharomyces cerevisiae
.
3.2. Thermodynamics of adsorption
The thermodynamics of sorption of the copper onto the driedcells of
S. cerevisiae
biosorbent was evaluated using the followingequations:
G
= −
R
g
T
ln
K
L
(9)ln
K
L
= −
H R
g
T
+
const
(10)
G
=
H
−
T S
(11)where
H
,
S
,
G
and
T
are the enthalpy, entropy, Gibbs freeenergy, and absolute temperature, respectively;
R
g
is the gas constant and
K
L
the equilibrium constant (L/mol). The computationof thermodynamic parameters gives the following numerical values of
G
=(
−
20.98
±
1.72)kJ/mol,
H
=(14.33
±
0.51)kJ/mol and
S
=(114.67
±
0.11)J/molK. The negative value of
G
indicates thespontaneous nature of adsorption process. Positive value of
H
(computedfromtheslopeoflineardependenceofln(
K
L
)versus1/
T
(
R
2
=0.997) as shown in Fig. 3) indicates the endothermic enthalpyof adsorption, favored by increased temperatures, while the positive value of
S
shows the afﬁnity of the adsorption for copperions.As we noticed, according to data shown inFig. 3, the adsorptionof Cu(II) onto the dried cells of
S. cerevisiae
is an endothermic processwhichbecomesmorefavorablewiththeincreasingofsolutiontemperature.
3.3. Response surface modeling
The response surface methodology deals with experimentaldesign strategy, statistical modeling and process optimization. TheRSM applications concerns the particular situations where severalinput variables (
factors
), which can be set by experimenter, inﬂuence the result of measurement (
response
). Also, RSM includes theempirical statistical modeling in order to develop an appropriateapproximatingrelationshipbetweentheresponseandfactorstobeused for process optimization. In this work the response surfacemodeling was applied to develop an appropriate regression modelto ﬁnd out the optimal values of experimental factors in order toenhanceremovalefﬁciency.Themostsigniﬁcantfactorsthataffectsorptionremovalefﬁciency(response)aretheinitialconcentrationofcopperionsinaqueoussolutions
C
o
(mg/L),initialpHofsolutionand the biosorbent dose (%, w/v). The real values of independent
C. Cojocaru et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 335 (2009) 181–188
185
Table 3
Central composite orthogonal design and experimental response.Run number (
N
) Factors (input variables) ResponseInitial concentration of Cu(II) pH of solution Biosorbent dose (%, w/v) Removal efﬁciency,
b
Y
%
C
o
(mg/L) Level
a
x
1
pH Level
a
x
2
BSD Level
a
x
3
1 75 1 5 1 2.25 1 32.732 25
−
1 5 1 2.25 1 37.963 75 1 3
−
1 2.25 1 36.984 25
−
1 3
−
1 2.25 1 41.225 75 1 5 1 1.50
−
1 29.476 25
−
1 5 1 1.50
−
1 34.697 75 1 3
−
1 1.50
−
1 34.208 25
−
1 3
−
1 1.50
−
1 38.949 80
˛
4 0 1.88 0 20.6410 20
−
˛
4 0 1.88 0 37.4011 50 0 5.2
˛
1.88 0 33.9612 50 0 2.8
−
˛
1.88 0 32.9813 50 0 4 0 2.33
˛
39.9214 50 0 4 0 1.42
−
˛
35.0215 50 0 4 0 1.88 0 36.3316 50 0 4 0 1.88 0 36.06
a
1=low value, 0=center value, +1=high value,
±
˛
=star point value.
b
The removal efﬁciency
Y
% was determined experimentally by using Eq. (2).
variables as well as their coded limits are listed in the central compositional experimental design given in Table 3.As can be seen from Table 3, the coefﬁcients of the empiricalmodelhavebeencalculatedbymeansofmultiplelinearregression(MLR) method [28] and according to MLR empirical model (withcoded variables) may be written as follows:ˆ
Y
=
33
.
87
−
3
.
633
x
1
−
1
.
397
x
2
+
1
.
602
x
3
−
2
.
205
x
21
+
3
.
523
x
23
subjected to :
−
˛
≤
x
j
≤ +
˛
;
j
=
1
,
2
,
3; (12)where
x
1
,
x
2
and
x
3
are the coded levels of factors and
˛
=1.215 isthe star point in experimental design that gives the limits of thevalid region (region of experimentation).TheregressioncoefﬁcientsweretestedforsigniﬁcancebymeansofStudent’s
t
test[28]retaininginEq.(12)onlythesigniﬁcantones.
For the graphical representation and analysis of the factors’ inﬂuence upon response it is worth converting the response surfacemodel in terms of coded variables to an empirical model in termsofactualvariables.Forthispurposethesubstitutiontechniquewasapplied and the empirical coefﬁcients were computed. Thus, interms of actual variables the empirical model may be written asˆ
Y
=
117
.
969
+
0
.
207
C
0
−
1
.
397pH
−
89
.
675BSD
−
3
.
528
×
10
−
3
C
20
+
25
.
052 BSD
2
subjected to : 20
≤
C
0
≤
80 (mg/L); 2
.
8
≤
pH
≤
5
.
2;1
.
42
≤
BSD
≤
2
.
33(%
,
w/v) (13)The goodnessofﬁt between empirical model and experimental data was veriﬁed using
F
C
ratio test [28] for a conﬁdence level
p
=0.05 and degrees of freedom
f
1
=9 and
f
2
=1. The
F
C
ratio wascomputed as the ratio between the variance of residual and thevarianceofexperimentalerror(replication).The
F
C
ratiowasfoundto be
F
C
=234.1 that is smaller than tabulated value
F
tab
(
p
,
f
1
,
f
2
)=240.5. This means that the mathematical model is validatedfrom statistical standpoint. Fig. 4 compares the experimental andpredicted data of removal efﬁciency. Likewise, the model adequacy has been investigated by the examination of the residuals(
e
=
Y
exp
−
ˆ
Y
predic
).TheresidualanalysisisalsoshowninFig.4outlining a good concordance between experimental and predictedresponses.As one can see the response model shows a goodnessofﬁt toexperimental data (Fig. 3). Therefore, the model has been considered adequate for the prediction and optimization. Having a validmodel the graphical representations of the response surface were
Fig. 4.
Removal efﬁciency, experimental data plotted against the predicted ones and residual analysis of empirical model.