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Biochemical Engineering Journal
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ybrid modeling of counterion mass transfer in a membrane-supportediofilm reactor
na R. Ricardo, Rui Oliveira, Svetlozar Velizarov, Maria A.M. Reis, João G. Crespo ∗
EQUIMTE/CQFB, Department of Chemistry, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, P-2829-516 Caparica, Portugal
r t i c l e i n f o
rticle history:eceived 15 June 2011eceived in revised form1 November 2011ccepted 20 December 2011vailable online 14 January 2012
eywords:
a b s t r a c t
This paper presents a hybrid mechanistic/statistical model for predicting counterion fluxes across an ion-exchange membrane in a membrane-supported biofilm reactor. The model was calibrated with operatingdata for the removal of nitrate and perchlorate from a simulated contaminated drinking water stream.Two different modeling strategies were tested: a cooperative parallel hybrid model and a competitivemixture-of-experts (MOE) structure both joining a mechanistic Donnan-dialytic transport model and amultivariate projection to latent structures (PLS) model. The MOE structure proved to be a better pre-dictive tool since it combines the two hybrid model elements in a mediated network. The PLS model
was used to identify the process variables that are responsible for the mechanistic model inaccuracy.The results showed that biocompartment physicochemical data need to be considered in the modelingof the transport of counterions across the membrane, especially in situations in which the target counte-rion (e.g., perchlorate or nitrate) is metabolically reduced in the biocompartment. By using this strategy,the complex biofilm contribution to the transport was accounted for, without the need of developingmechanistic models built on simplified and/or inaccurate assumptions.
. Introduction
Membrane-attached biofilms (MABs) have been widely used inater and wastewater treatment [1–5]. Since the membrane allows
or the physical separation of the microbial culture from a pol-uted water stream, the biological treatment of toxic organics from
astewater was proved possible using the extractive membraneioreactor concept [1–3]. MABs have been also successfully com-ined with gas delivery systems: O2 to aerobic biofilms and H2 tonaerobic biofilms, used as electron acceptor or donor, respectively4,5]. Additionally, biological purification of air by transferringaseous pollutants through a membrane into a biofilm has beeneported [6].
If an electron donor and an electron acceptor diffuse intohe membrane-attached biofilm from opposite sides (counter-iffusion), the process modeling differs significantly from theodeling of conventional biofilm reactors with substrates enter-
ng into the biofilm from one and the same side (co-diffusion)1]. In the case of counter-diffusion, the reaction zone could be
ocated in different regions within the biofilm depending on theocal concentrations of substrates required for a given reaction [7].
number of mathematical models applied to membrane-attached
biofilms have been developed so far [7–12]. Nicolella et al. [9]presented a reaction-diffusion model to predict substrate concen-tration profiles and the biofilm thickness evolution over time inan extractive membrane bioreactor. Several assumptions, such asconstant biofilm density, constant diffusivity and uniform biofilmthickness were considered. Even with these simplifications, thebiofilm dynamic model developed required numerical solutions ofpartial differential and integral equations.
The ion exchange membrane bioreactor (IEMB) is a processthat combines the transport of target counterions (e.g., nitrate,perchlorate) from contaminated water streams through an anion-exchange membrane to an anoxic membrane-attached biofilm [13].The transport between the two compartments is governed by theDonnan dialysis principles, thus enhancing the transport of tar-get counterions from the water to the biological compartment byadding an excess of suitable “driving” counterions (e.g., chloride)to this compartment. After transport through the anion-exchangemembrane, the ionic pollutants are reduced to innocuous species(such as nitrogen and chloride) by a mixed microbial culture.
The mechanism of transport of ionic pollutants in the IEMB wasextensively studied [14–16] and a mechanistic counterion trans-port model was previously developed [17].
This model was shown to predict accurately the fluxes of coun-terions across the membrane on the basis of physicochemical andhydrodynamic data, in situations of complete bioreduction of thetarget counterions in the IEMB biocompartment [14,17].
The model was developed assuming that the process is mass-ransfer limited. The “resistances-in-series” approach was followedonsidering the sum of the membrane resistance and the diffusionesistances of the two liquid films adjacent to the membrane in theater and biocompartment, respectively. Thereby, the model pre-
upposes non-limiting biological conditions, and neglects possibleass transfer limitations due to the biofilm itself [18]. However,
n the initial phase of the process when the biofilm is still noteveloped, or in situations of nutrient limitation, the rates of theiological reactions may become limiting.
In a previous work, a projection to latent structures (PLS)odel was developed to predict the counterion transport during
n IEMB operation [19]. This model was shown to have a goodrediction potential under both mass-transfer and reaction rate
imiting conditions. The model was developed using a projection toatent structures technique that captured the underlying relations,ssuming that the experimental data contained all the informa-ion needed for an adequate process description. Therefore, these of simplifying assumptions considered in previously developedABs mechanistic models was avoided. However, the use of purely
tatistical models ignores any mechanistic knowledge, requires aelatively large amount of experimental data and has a limitedxtrapolation potential [20].
This study aims at developing a hybrid model, for a membrane-ttached biofilm reactor by using statistically based modelingechniques for capturing underlying information from processperating conditions. The modeling strategy evaluated in theresent study was a combined use of mechanistic and statisti-al models. The working hypothesis was that such a combinationight gather the best of both approaches, thus allowing for a cer-
ain level of process mechanistic interpretation and, at the sameime, for the inclusion of the relevant physicochemical phenom-na, which are usually simplified by mechanistic modeling. In thiselation, a statistical fraction might account for effects not con-idered by the mechanistic model, while the mechanistic modelould potentially manage the extrapolation of the hybrid model toegions lacking calibration data [21].
The statistical analysis is also important to identify the operationariables that are limiting the mass-transfer of a target counte-ion across the membrane. This identification is crucial for processptimization and practical implementation. Eventually, the mech-nistic formulations might be also improved with the incorporationf the key variables.
. Materials and methods
.1. Experimental installation and procedure
The IEMB layout is illustrated in Fig. 1. The membrane modulesed had two identical rectangular channels in a flat parallel-plate
configuration. A Neosepta ACS membrane (Tokuyama Soda, Japan)with 34.5 cm2 separated water polluted with nitrate and per-chlorate from the biomedium, thus organizing two differentcompartments, a water compartment and a biological compart-ment, respectively. Polluted water was prepared by supplementingtap water from the Lisbon public distribution network with dif-ferent concentrations of nitrate and perchlorate, according to thedesign of experiments described in Table 1. The polluted water wascontinuously fed to the water compartment at 0.18 ml/min to main-tain a hydraulic retention time (HTR) of 8 h in this compartment.For maintaining good hydrodynamic conditions, the water wasre-circulated at a volumetric flow rate of 1620 ml/min (Reynoldsnumber of 3000).
A mixed microbial culture was used to reduce nitrate and per-chlorate in the biocompartment under anoxic conditions. Thisculture was enriched from a primary inoculum taken from a munic-ipal wastewater treatment plant [14]. Two different biomediumre-circulation flow rates were used (82 ml/min and 1620 ml/min),corresponding to Reynolds numbers of 150 and 3000, respectively.The total biocompartment volume, including the membrane mod-ule channel, the re-circulation loop and the anoxic vessel was550 ml. The composition of the nutrient media for the referenceexperiments was: 1 g/l of K2HPO4, 0.592 g/l of KH2PO4, 0.5 g/l ofNaH2PO4, 0.233 g/l of NH4Cl, 0.1 g/l of MgSO4.7H2O, 5.84 g/L ofNaCl and 0.56 g/l of ethanol. In the remaining experiments, thebiomedium composition, except for ethanol used as the carbonsource and electron donor, was changed according to the exper-imental design (Table 1). The biomedium feeding rate and therecirculation flow rate in the biocompartment were introducedas variables in the experimental design in order to investi-gate their effects on the fluxes of target counterions across themembrane.
All experiments were run at 23 ± 1 ◦C with periodic sampling foroff-line analyses of conductivity and pH and for analytical determi-nation of the concentrations of anions, ammonium and ethanol.
2.2. Analytical methods
Nitrate, phosphate, sulfate and chloride concentrations weredetermined by an ion exchange chromatography system (Dionex,USA), constituted by an Ionpac AG9 guard and analytic AS9 columns(4 mm), an Anion Supressor-ULTRA (4 mm) and an ED50 electro-chemical detector. The analysis was performed at 23 ◦C using a9 mM Na2CO3 aqueous solution as the mobile phase at a flow rateof 1 ml/min. Perchlorate concentration was measured at 30 ◦C with50 mM NaOH aqueous solution at a flow rate of 1 ml/min in the
same system using AG16 and AS16 columns. The ClO4
− detectionlimit was 1 ppb with the injection of 1 ml of sample. In the biocom-partment samples analysis, due to interference of the Cl− peak withthe ClO4
− peak, 500 �l of sample was injected and, consequently,
Fig. 1. Schematic diagram of the experimental set-up and counterio
he limit of detection increased to 2 ppb of ClO4−. For the bicar-
onate analysis, the Dionex system was operated with AG11 andS11 columns with a 25 mM NaOH aqueous solution at a flow ratef 1 ml/min at 30 ◦C.
Phosphate and bicarbonate were determined in the form ofO4
3− and CO32−, respectively, due to the high eluent pH. The con-
ersion to phosphate and bicarbonate species actually present inhe samples was based on the sample pH according to the pKa
alues of each acid–base pair [22,23].Ethanol in the biocompartment and biofeed was determined by
PLC using a mobile phase of 0.01 N H2SO4 aqueous solution at aow rate of 0.5 ml/min at 30 ◦C with an Aminex HPX-87H columnBioRad, USA) and a differential refractometric detector RI-71. Thethanol detection limit was 1 ppm.
Ammonium was determined with a gas-sensitive electroderion 95-12 (Thermo, USA) with a detection limit of 1 ppm.
.3. Experimental design
The experiments were designed with a screeninglackett–Burman design [24] in order to obtain data allowingor statistical interpretation through a reduced number of exper-ments. In the design, seven factors, previously identified as mostontributing to the transport of counterions across the membrane19], were tested at 2 levels of variation. The investigated factorsere: phosphate, ammonia, sulfate and chloride concentrations in
he biomedium, the hydraulic retention time (HRT) in the biocom-artment, the hydrodynamic conditions in the biocompartmentecirculation loop, characterized by the Reynolds number (Re), andhe concentrations of nitrate and perchlorate in the water com-artment. Each factor was tested at two levels of variation basedn previously operating conditions [14] as described in Table 1.n experiments in which the biocompartment HRT was increased,he concentrations of the nutrients in the biofeed (Table 1) were
ncreased correspondingly in order to assure the same mass loadso the biocompartment.
Moreover, two experiments (experiment 1 and 8) were per-ormed according to previously reported conditions [14] under
sport mechanism in the ion-exchange membrane bioreactor (IEMB).
which the mechanistic model was found to accurately predict thefluxes of nitrate and perchlorate across the membrane.
3. Mathematical modeling
3.1. Mechanistic transport model
The steady-state transport of a trace counterion, i, from com-partment 1 (water compartment) across the membrane intocompartment 2 (biocompartment) can be approximated by a bi-ionic equation (Eq. (1)) if a bulk counterion, a, is present in excess inboth compartments [17]. This equation combines the driving forceand the resistance (the “resistances-in-series” in the denominator)to mass transfer of a target trace counterion from the water to thebiocompartment. In Eq. (1), L is the membrane thickness (m); Pm
is the membrane permeability to the target counterion (m2/h); Qis the membrane ion exchange capacity (mmol/m3 of wet mem-brane); ı1 and ı2 are the thickness (m) of the liquid boundary layerscontacting the membrane surface in the water and biocompart-ment, respectively, and Di,w (m2/h) is the diffusion coefficient ofthe target counterion i in water.
Ji =Ci,1
CZbulk,1
− Ci,2
CZbulk,2
L
Pm,i(Q/a)Z + ı1Di,wCZ
bulk,1
+ ı2Di,wCZ
bulk,2
(1)
Since the transport in the IEMB process is governed by the Don-nan dialysis principles, the flux of a target counterion i across themembrane is proportional to the difference between the targetcounterion (Ci) to driving counterion (Cbulk) molar concentrationratios in the two compartments. The parameter Z represents theratio between the valence of the target counterion and the valence,a, of the major bulk counterion (Cl− in the case of this study). Thethicknesses of the liquid boundary layers at the membrane surfacesin the two channels of the module were estimated using Sherwood
number correlations for flat channels [25]. The membrane-relatedparameters and the Pm values for the studied counterions in themechanistic transport model were assessed in a previous study[19,23].
The transport flux of a target counterion across the membraneas calculated using the steady-state mass balance for the water
ompartment:
i = F
A(C in
i − Couti ) (2)
In Eq. (2), F is the inlet polluted water flow rate to theater compartment (1.08 × 10−5 m3/h), A is the membrane area
3.45 × 10−3 m2), and Ci is the target counterion concentration inhe polluted water (in) and treated water (out), respectively.
.2. Hybrid models
In the present study, two different approaches were tested: – a hybrid parallel structure (Fig. 2A) and 2 – a mixture-of-xperts structure (MOE) (Fig. 2B) [26–28]. Both strategies are basedn the previously described mechanistic model complementedith a PLS model. However, the PLS contribution, in the parallel
tructure, can be interpreted as cooperative since the PLS modelomponent forecasts the corrections that are needed to be incor-orated in the mechanistic model. On the other hand, in the MOE
tructure a competitive contribution between the mechanistic andhe PLS model is observed since both models individually predicthe target counterion flux and their respective contributions arealanced.
able 2nitial inputs used in PLS model calibration.
Input number Abbreviation Stream/Compartmen
1 NO3− ,W Polluted water
composition(mmol/m3)
2 ClO4− , W
3 SO42− , W
4 Cl− ,W
5 HCO3− ,W
6 Feed rate PO43− ,F Biofeed mass flow
rate (mmol/h)7 Feed rate H2PO4− ,F
8 Feed rate HPO42− ,F
9 Feed rate SO42− ,F
10 Feed rate Cl− ,F
11 Feed rate NH4+,F
12 NO3− ,B Biocompartment
mediumcomposition(mmol/m3)
13 ClO4− , B
14 PO43− ,B
15 H2PO4− ,B
16 HPO42− ,B
17 SO42− , B
18 Cl− ,B
19 NH4+,B
20 EtOH,B
21 Re,B Operatingconditions22 HRT,B
23 pH,W
24 pH,B
rix of inputs; y: vector of output; J: target counterion flux across the membrane; g:
In the hybrid parallel structure, the estimation of a target coun-terion flux combines the prediction of the previous developedmechanistic model and the PLS model in a single output (Fig. 2A).The procedure for parallel hybrid calibration followed two steps.First, a target counterion flux was estimated using the mechanis-tic transport model and the residuals were calculated. Then, a PLSregression analysis was used to estimate these residuals as thetarget output. It has been demonstrated that by using such a struc-ture, a better interpolation and range extrapolation proprieties canbe achieved in comparison with mechanistic or statistical modelsalone [20].
Fig. 2B illustrates the MOE approach, where the final model is aresult of the mediated contribution of the mechanistic and the PLSmodels. In this structure, the two models predict the counterionmass-transfer across the membrane and the contribution of eachmodel is mediated by a gating system that depends on the operat-ing conditions. The gating system applied was based on ‘softmax’functions (Eq. (3)) in which the subscript 1 refers to the mechanis-tic model contribution [29,30]. The ‘softmax’ output is considered aprobability choice since its value varies from 0 to 1 [31]. Therefore, ifg = 1 the counterion flux prediction is only described by the mecha-
1nistic model while when g1 = 0 the output is determined by the PLSmodel. Consequently, the contribution of the PLS model is quan-tified by the ‘softmax’ function g2 = 1 − g1. Since the MOE network
t Compound/Condition
NO3−
ClO4−
SO42−
Cl−
HCO3−
PO43−
H2PO4−
HPO42−
SO42−
Cl−
NH4+
NO3−
ClO4−
PO43−
H2PO4−
HPO42−
SO42−
Cl−
NH4+
Ethanol
Reynolds number in the biocompartment recirculation loop (-)Hydraulic retention time in the biocompartment (days)pH of polluted water (-)pH of the medium in the biocompartment (-)
present in both compartments. In previous studies, the biomediumcomposition was formulated with excess of phosphate in order toprevent nutrient limitation by phosphorous. Therefore, in experi-ment 1, transport of phosphate from the biological compartment
0 2 4 6 8 10
NO
3
- co
nc
en
tra
tio
n (
pp
m)
ClO
4
- co
nc
en
tra
tio
n (
pp
b)
0
10
20
30
40
50
60
70
80
90
100
110
NO3- limit
ClO4- limit
NO3-
ClO4-
6 A.R. Ricardo et al. / Biochemica
omprises two expert systems, the ‘softmax’ equation denominators composed by two terms in the denominator (Eq. (3)).
1 = e
∑n
j=1wjXj
e
∑n
j=1wjXj + e
∑n
k=1wkXk
(3)
Using this equation, the degree of contribution of each expertodel is defined according to the experimental conditions, since
he ‘softmax’ is a non-linear relation between the inputs (X) andheirs weights (w) for all the n variables of the PLS model. The samerocess information is used to feed both the PLS model and the
softmax’ equation. The w determination for the optimization ofhis combinatory problem was based on minimizing the MOE fluxesrediction error, using a modified Levenberg–Marquardt methodor estimation.
The PLS model, for both hybrid models, was implemented onatlab 2006b [32] with the toolbox N-Way [33] using randomly
5% of the initial data set for calibration. The remaining 25% weresed for predicting the model goodness of fitting (validation set).he number of samples was standardized to have the same contri-ution to the model calibration, and the data obtained was scaled byubtracting from each of the variables their averages and dividingy their respective standard deviations (auto-scaling).
The PLS model calibration started by computing a X-score matrix = XW for an appropriate weight matrix W by using the originalimensional input data (X). These weights are estimated so thatach of them maximizes simultaneously the covariance between Xnd T and between Y and T by minimizing the error (noise) termn each equation (E, F): X = TP′ + E and Y = TQ′ + F, where T and Qre the matrix of X-loadings and Y-loadings, respectively. Once theoadings are computed, the above equations can be combined tobtain a multiple regression model: Y = TQ′ + F = XWQ′ + F = XB + F,here the PLS regression coefficients, B, equals to WQ. In PLS,imension of X is reduced since the data set is transformed intonother coordinate system, T, which is composed by orthogo-al variables [29]. The incorporation of new vector on T occurss long as it is predictively significant. Therefore, PLS algorithms specifically designed to deal with noisy, collinear and numer-us variables and to eliminate redundant information [29]. Dueo this property, it is beneficial to use all available process infor-
ation to calibrate the model. Therefore, the PLS model wased with physicochemical data from the two compartments: theolluted water composition, the biofeed mass flow rate, the bio-ompartment medium composition, the hydraulic retention timend the Reynolds number in the biocompartment and the pHalues in the two compartments (giving a total of 24 input vari-bles, presented in Table 2). The remaining operating conditions:embrane area, Reynolds number in the water compartment,
nd hydraulic retention time in the water compartment wereaintained constant in all experiments. The contribution of these
actors to the anions transport was previously investigated andorrectly taken into account by the mechanistic transport model17].
The selection on the number of latent variables to use was basedn cross-validation [34]. The PLS model was calibrated accordingo a flowchart described in [19]. Briefly, the initial PLS model wasubjected to several procedures in order to distinguish the variableshat contribute to predict the output from uninformative predictorshat only introduce noise. The selection of useful model descrip-ors was done using 8 different techniques: forward selection [30],ackward selection [30], stepwise selection [30], iterative stepwiselimination (ISE) [35], iterative predictors weighting (IPW) [36],
ninformative variables elimination (UVE) [37] and Martens uncer-ainty test [38] with regression coefficients confidence intervalstimated with Jackknife [39] and Bootstrapping [40] resamplingechniques.
eering Journal 62 (2012) 22– 33
The validation set was compared with the predicted valuesusing different criteria: the correlation coefficient (R2) and theroot-mean-square-error-of-prediction (RMSEP). Both methods arebased on the model residuals and quantify the prediction capacityof the model [28].
The model obtained was also evaluated for its lack-of-fit inorder to compare its prediction capacity with the experimentalvariance of the errors (�2
exp) by the �2 statistic [41,42]. Since theerror term is supposedly independent and normally distributed, theweighted residuals should approximate a chi-square distributionfor n–p degrees of freedom (n: number of observation; p: numberof model parameters). Therefore, since the experimental varianceof the measurements error is known, the obtained model can beconsidered statistically fitted if the value of �2 is below the criticalvalues of tabulated �2 for the model degrees of freedom.
�2 =∑n
i=1˙(ymodel − y)2
�2exp
(4)
4. Results and discussion
4.1. IEMB performance and mechanistic model predictions
Before carrying out the design of experiments, the IEMB perfor-mance was evaluated for tap water supplemented with nitrate andperchlorate (experiment 1 in Table 1) and operated under previ-ously defined conditions [14]. These conditions were demonstratedto guarantee effective bioreduction of nitrate and perchlorate in thebiocompartment, since all nutrients were in excess and the processlimiting step was the counterion mass transport through the mem-brane. The results obtained in this work were in agreement with thepreviously obtained data [14]. Under these conditions, the concen-trations of nitrate and perchlorate in the treated water (Fig. 3) werebelow the drinking water quality guidelines: 45 ppm for NO3
− [43]and 15 ppb for ClO4
− [44].The experimental data obtained were used to validate the mech-
anistic model (Eq. (1)) predictions for transport flux of nitrate,perchlorate, phosphate, sulfate and bicarbonate. These anionsare the major species transported through the membrane. Thefirst two anions were initially present in the polluted water, thesecond two mainly in the biomedium, while bicarbonate was
Time (days)
Fig. 3. Time course of nitrate and perchlorate concentrations in the treated waterfor a typical IEMB process (experiment 1 in Table 1) fed with water polluted with60 ppm of NO3
Table 3Comparison of different models for prediction of the flux of target anions for the validation set (RMSEP: root-mean-square-error-of-prediction; TNP: total number ofparameters).
a The MOE structure has 53 estimated parameters (1 from mechanistic model, 14 from the PLS model and 38 from the gating system), but only parameters from themechanistic and from the PLS model are involved in the input–output regression.
Expe riment I D
Mo
de
l re
sid
ua
ls (
g/m
2h
)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Model residua ls
Exper ime ntal flux erro rC
Exp1 Exp2 E xp3 E xp4 E xp5 Exp6 Exp7 Exp8 E xp9 Exp10
Experiment ID
Model re
sid
uals
(g/m
2h)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Model residualsExperi men tal flux erro r
Exp1 E xp2 Exp3 E xp4 Exp5 Exp6 E xp7 Exp8 Exp9 Exp10
Fig. 4. Mechanistic model prediction residuals and experimental standard deviation (ı) of the flux error for nitrate (A), perchlorate (B) and sulfate (C) flux across themembrane.
Fig. 5. Mechanistic model contribution for the MOE structure (g1) and me
o the water compartment was documented. In order to preventnd/or minimize changes in the ionic composition of the treatedater, the control of the fluxes of all counterions presents (and not
nly of nitrate and perchlorate) is mandatory.The accuracy of the mechanistic transport model predic-
ions for experiment 1, performed under standard conditions,as confirmed with a predicted nitrate flux across the mem-
rane of 0.21 g/(m2h) and an experimentally determined value of.22 ± 0.01 g/(m2h). For perchlorate, the mechanistic model predic-ion was also accurate since the predicted value of 0.30 mg/(m2h)orresponded exactly to the experimentally determined value of.30 ± 0.05 mg/(m2h).
The mechanistic model accuracy was afterwards evaluated forll experiments performed. Whenever the residuals rise above thexperimental error standard deviation, the model prediction isonsidered to be not consistent with the values obtained experi-entally. As can be seen from the data presented in Fig. 4A and B, in
of the 10 experiments performed the mechanistic model predic-ions for the fluxes of nitrate and perchlorate across the membraneere inconsistent. As expected, model inconsistencies coincidedainly with experimental conditions that do not fulfill the build-
ng assumptions of the mechanistic transport model (experiments, 5, 7 and 10). A relatively low ratio of driving (chloride) to targetounterion was used in experiments 5 and 7 and an incompleteerchlorate reduction, due to ammonia nutrient limitation, was
bserved in the biocompartment in experiment 2. However, inxperiment 10, the lack of accuracy of the mechanistic modelannot be attributed to these factors, which is an indication thathe simplified mechanistic model does not account for all possible
stic model residuals for NO3− (A), ClO4
− (B) and SO42− flux prediction (C).
parameters and, therefore, its prediction may become unreliable incomplex systems.
The mechanistic model was not capable of predicting accuratelythe phosphate, sulfate and bicarbonate fluxes across the mem-brane (Table 3). For the cases of phosphate and bicarbonate, thismay probably relate to the pH-dependent character of their spe-ciation in water and/or the co-ion (proton) exclusion from theanion-exchange membrane, resulting in a higher pH inside themembrane compared to the bulk solution pH, thus makingimpossible a correct estimation of the actual transport driving forcedefined by Eq. (1) for these anions [19].
In the case of SO42−, the mechanistic model inaccuracy cannot
be attributed to a particular set of conditions (Fig. 4C). The SO42−
mechanistic model prediction mismatch is more significant in theinitial phase of each experiment since the mechanistic transportmodel is applicable under steady-state conditions. This behaviour isnot so evident in the NO3
− and ClO4− flux prediction, most probably
since the membrane permeability to these anions are about 4 ordersof magnitude higher than that of SO4
2− and, consequently, steady-state transport conditions for these anions are much more rapidlyestablished.
4.2. Parallel hybrid model development and assessment
A parallel hybrid model was developed as illustrated in Fig. 2A.
With this structure, the mechanistic and the PLS models outputswere combined in a cooperative way allowing the PLS model torectify the mechanistic model prediction. Therefore, the deviationsof the mechanistic model prediction from the experimental flux
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ata were subjected to a PLS regression analysis in order to establishinear correlations between process informative variables (inputs)nd model residuals (target output). In this parallel structure, theLS model was used to extract information from the mechanisticodel residuals.Table 3 compiles the results obtained in terms of goodness of fit
or this parallel structure. As can be observed, a clear improvementas obtained with the PLS model, compensating the mechanisticodel inaccuracies. Considering that the PLS regression analysis
rganizes the data into latent variables, a dimensional reductionas observed from initially a 24-dimensional co-ordinate system
o a 7-dimensional one (in the case of the HCO3− flux modeling).
oreover, since a calibration selection of the informative predictorsas performed in the model, the number of model inputs (latent
ariables) decreased. The selected inputs are indicated in Section.5.
The root-mean-square-error-of-prediction (RMSEP) is accept-ble compared with the experimental standard deviation, exceptor the case of phosphate (� = 0.001), for which the hybrid modelhows a clear deterioration of its prediction capacity. In this par-icular case, since the hybrid model is more inaccurate than the
echanistic model alone, the residuals used for the PLS calibrationbviously included not only process information but also vari-nce introduced by the mechanistic model error. Consequently, theesulting PLS was not able to find a correct correlation between theEMB operating data and the mechanistic model residuals and therediction results became unsatisfactory.
On the other hand, the value of �2 obtained for the hybrid models a clear indication of the improvement of the flux prediction com-ared to the mechanistic model. The value indicated in Table 3orresponds to the sum of �2 for all observations and hence forhe model to be considered statistically significant the value shoulde below the tabulated one at 95% confidence. The �2 calculationllowed for concluding that the hybrid model can be consideredtatistically accurate, since it compares its sum of squares withhe experimental standard deviation. Considering the number ofbservations for the validation set, and the number of parameters,he tabulated value varies between 251 and 266, for n–p degreesf freedom [45], where n is the number of observations and p ishe number of parameters. The total number of parameters of thisybrid model corresponds to the sum of the parameters from thewo models, one from the mechanistic model (the Pm parameter)nd the number of latent variables selected by cross-validation inhe PLS model calibration.
Although, the developed hybrid model improved the predic-ion capacity, it is rather sensitive to a possible mechanistic modeliscrepancy. When the mechanistic model is inappropriate for pre-iction (as in the case of phosphate) the parallel hybrid model islso inappropriate. In such situations, the PLS model captures notnly the underlying mechanisms from the process data but alsohe variance introduced by the mechanistic model. In such cases, aifferent type of hybrid structure needs to be developed.
.3. “Mixture of experts” development and characterization
The mixture-of-experts structure is a distinct modelingpproach since, contrary to the previous parallel hybrid approach,he mechanistic and the PLS model have a competitive contribu-ion. In this hybrid approach, the PLS model is weighted. A differentontribution of each expert (the mechanistic or the PLS model) isonsidered in each observation. In the MOE, the PLS model wasalibrated to fit target counterion flux values across the mem-
rane and not their residuals, as in the case of the parallel hybridodel. Therefore, two independent models for flux prediction are
vailable: a mechanistic and a PLS model and a gating system medi-tes and weights the contribution of these two experts.
eering Journal 62 (2012) 22– 33 29
This hierarchical structure combines the best of each model inorder to allocate the degree of contribution to capture from eachmodel. It combines linear functions for non-linear regression prob-lems using a gating system, referred to as a “generalized linear”network [28]. The gating system used in the present study wasbased on a softmax function (Eq. (3)) that can be inferred as provid-ing a “soft” division of the input space. This function was calibrated(with the training set of data) in order to maximize the MOE finalprediction. Therefore, the analysis of each input weight can be aclear indication of its contribution to each expert. For instance,when modeling the ClO4
− flux across the membrane, perchlorateconcentration in the polluted water is the input with the highestcontribution to the gating equation (g1).
Fig. 5 illustrates the value of g1 for the estimation of the fluxes ofNO3
−, ClO4− and SO4
2− in all experiments performed. When g1 isequal to 1, the target counterion flux prediction is totally (or 100%)obtained by the mechanistic model. In Fig. 5, the mechanistic modelresiduals are also represented in order to facilitate the identifica-tion of the experimental conditions, under which the mechanisticmodel failed (as discussed in Section 4.1).
In the NO3− flux prediction, the mechanistic model contribu-
tion is insignificant in almost all experiments (Fig. 5A). Except forthe case of experiment 7, the flux prediction is mainly obtained bythe PLS model alone (g1 = 0). This is a clear indication of the bet-ter capacity of the PLS model to predict the NO3
− flux across themembrane. Furthermore, despite the fact that in experiment 7 theNO3
− flux was mainly described by the mechanistic model, the MOEstructure improved this counterion flux prediction when comparedwith the parallel hybrid model and also with the PLS model alone(RMSEP = 0.012) [19].
In respect to the ClO4− flux, the experiments that were not
captured by the mechanistic model were mainly described by thePLS model. The 100% mechanistic model contribution to the ClO4
−
flux prediction in the remaining experiments improves the finalMOE modeling prediction since the mechanistic model deviationsin these experiments are minimal (see Fig. 4 for comparison). Thisis not observed for the NO3
− flux prediction since a considerablemechanistic model residual is detected in all experiments. Sincethe mechanistic model was developed for transport of trace coun-terions, it is not surprising that ClO4
−, which was present in theppb concentration range, is the anion that most closely obeys tothis condition. The other target counterions were all present in theppm concentration range and, therefore, higher deviations of theirfluxes from the mechanistic model predictions were observed.
In the case of SO42− flux, it was not possible to identify spe-
cific conditions under which the mechanistic transport model fluxprediction is not accurate. As can be observed in Fig. 5C, the mecha-nistic model contribution to the global output prediction was at themaximum 20%. Therefore, the PLS model had the major contribu-tion in the final model since it was able to describe more accuratelythe SO4
2− flux (R2 of 0.83) [19]. Thereby, the PLS model capturedthe underlying process behaviour directly from process operationaldata and the balanced combination with the mechanistic modelallowed for obtaining a more accurate SO4
2− flux prediction sinceall available sources of knowledge were incorporated.
4.4. Selection of an appropriate hybrid model structure
Different criteria were used for the analysis of the predictionpower in order to compare the models (Table 3). The MOE provedto be the best model since R2, RMSEP and �2 were improved incomparison with either mechanistic or parallel hybrid models.
Fig. 6 compares the predicted fluxes plotted against the exper-imental flux values for all counterions studied. The dashed linerepresents an ideal model prediction with a 100% correlationbetween the experimental and the estimated flux value. As can be
ig. 6. Predicted versus experimental flux values for the mechanistic transport moFor interpretation of the references to color in this figure legend, the reader is refe
een, significant improvement is obtained through the MOE whenomparing with the mechanistic and the parallel hybrid modelrediction. Besides an improved R2 value, a more accurate model isbtained since the slope of the straight line between experimentalnd simulated models is closer to 1 for the MOE.
However, even in the MOE, the H2PO4− and HPO4
2− flux pre-ictions were not implemented accurately (�2 not significant athe 95% confidence level), indicating that the variables consid-red were not sufficient to describe phosphate transport acrosshe membrane. As already discussed, the phosphate flux predictionan be more complex due to the pH-dependent speciation of thehosphorus-containing species. Due to co-ion (cation in this case)xclusion by the anion exchange membrane, the measured bulkH differs from the pH inside the membrane that is not measur-ble experimentally and, therefore, cannot be used as an operatingariable. Nevertheless, the model obtained is a clear improvementompared to the mechanistic model.
For the SO42− flux prediction, an increase in the model residu-
ls was detected for values higher than 0.1 g/(m2h). These valuesorrespond to the initial phase of experiments 1 and 7. Despitehe improvement of the prediction obtained with the MOE for this
articular situation, a deviation from the experimental SO4
2− fluxas found (see Fig. 6). Most probably, the PLS model was not able
o predict these experimental results since they corresponded tonly 6% of all data used for calibration. Under these circumstances,
een triangles), parallel hybrid model (yellow circles) and MOE model (red circles). the web version of the article.)
the PLS model did not have enough variance to describe thesesituations. In any case, an improvement in prediction was obtainedwhen using the MOE structure for fluxes above 0.10 g/(m2h).
4.5. Identification of critical process variables not included in themechanistic model
The use of hybrid models can be helpful not only to increasethe model predictive power but also to determine key parame-ters of a given process. This information can be useful for planningfuture experiments and for improving mechanistic formulationsby identifying the main process variables. Since the PLS model inthe parallel hybrid structure was calibrated with the mechanistictransport model residuals, the analysis of the most contributingpredictors to that PLS model is an indication of the main variablesnot accounted for in the mechanistic model.
Since the PLS is a linear model, the regression coefficients canbe quantified as the contribution of each predictor to the model.The results, illustrated in Fig. 7, represent the normalized regres-sion coefficients of the predictors and their respective degree ofuncertainty. The regression coefficients are constants in the PLS
linear equation that represent the change in the predicted variable(y) as a function of a change in the predictor (x) value. A predictoris considered to be important if its absolute regression coefficientis relatively high and its uncertainty value is as small as possible.
Fig. 7. PLS regression coefficients in the hybrid model
he confidence intervals were obtained by the Jackknife resamplingechnique [39].
The results obtained suggest that the main missing contributionomes from the biocompartment related parameters. This appearsogical since the mechanistic transport model cannot predict sit-ations in which the biological reactions kinetics are the limitingtep.
In the NO3− flux prediction, the major contribution comes from
he chloride concentration in the biocompartment. For nitrate, theain deviations of the mechanistic transport model predictions
re related with a low chloride concentration in the biocompart-ent, thus violating the model assumption of chloride as the major
ulk counterion present in this compartment. Additionally, theydraulic retention time in the biocompartment was also foundo have a considerable contribution. This parameter inherentlyontains the Cl− effect since its concentration in the biofeed was
olluted water stream; F: biofeed; B: biocompartment).
adjusted to the different HRT in order to maintain the same massload. Therefore, in the experiments performed at a HRT equal to 10days, two times higher concentration of Cl− was used in the biofeed.The contributions of the remaining biocompartment-relatedparameters are less important for the nitrate transport model.
On contrary, the most important descriptors not accounted forby the mechanistic model in the case of ClO4
− flux prediction wereparameters related with the biological conditions, especially theammonia concentration (used as nitrogen source) which, in someexperiments, became limiting. This limitation was not observed inthe NO3
− bioreduction since NO3− can be used as the N source
[46]. In the ClO4− anion flux prediction, the Cl− concentration con-
tribution is not as evident as in the NO3− model, since ClO4
− isa trace counterion and the driving force for its transport to thebiocompartment is much less sensitive to the Cl− concentrationin this compartment. The H2PO4
− content in the biocompartment
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2 A.R. Ricardo et al. / Biochemica
nd SO42− concentration in the polluted water stream were also
ound to have an important contribution to the PLS model for pre-icting the ClO4
− flux across the membrane. Their contributionsre not taken into account in the mechanistic model since theriving force term (the nominator of Eq. (1)) is calculated consid-ring chloride as the only driving counterion. The opposite signs of2PO4
− and SO42− regression coefficients is linked to their oppo-
ite transport directions through the membrane. Since H2PO4− was
nitially present only in the biocompartment, it was transported tohe water compartment under some of the conditions tested.
The PLS regression coefficient values in the hybrid model for pre-icting H2PO4
− and HPO42− membrane fluxes seem to be affected
nly by the respective concentrations. The mechanistic model wasnable to describe correctly the membrane fluxes of phosphatepecies since their transport was affected by the difference betweenhe pH inside the membrane and the pH values of the two contact-ng bulk solutions. Given the fact that the PLS model in a parallelybrid structure was used to correct the mechanistic model residu-ls, the regression coefficients obtained indicate that the PLS modelas also not able to extract valid information from the process data
f using such type of a hybrid structure.In the case of SO4
2−, a considerable amount of information fromhe process needs to be taken into account by the mechanisticransport model, especially the phosphate species concentrationn the biological compartment. These species can be used as driv-ng counterions, especially when the chloride concentration in theiocompartment is low. For instance, in experiment 7, in order toaintain the same sulfate flux to the biocompartment, the total
hosphorus flux increased 10 times in order to compensate theecrease in the Cl− flux to the water compartment. This effect wasaptured by the PLS model and the contribution of the phosphatepecies to the sulfate flux was emphasized (see Fig. 7).
The HCO3− flux prediction by the mechanistic model requires
he incorporation of biocompartment-related inputs as well asnformation about concentrations of other anions in the polluted
ater stream. The contribution of biocompartment-related param-ters concerned mainly anions that can potentially be used asriving counterions for its transport (e.g. Cl−, H2PO4
− and HPO42−).
he pH in the biocompartment has also an important contribu-ion to the PLS model because it influences the speciation of thisnion (HCO3
− versus CO32−) for the pH range used in the present
tudy.
. Conclusions
The use of hybrid mechanistic-statistical modeling to describehe counterion transport in an ion-exchange membrane-supportediofilm reactor proved to be adequate for expanding the mecha-istic model to situations beyond its building assumptions, such asiological reaction limiting situations.
The statistical analysis of the mechanistic model residuals sug-ested that the main missing mechanistic information came fromiocompartment-related parameters.
The parallel hybrid model allowed for covering a number ofrocess situations outside the domain of applicability of the mech-nistic model. In these cases, the PLS model was able to capture thenformation missing in the mechanistic model from the processperating data. However, for some counterions (e.g., phosphate)he prediction was unsatisfactory, since the mechanistic modelrror variance was included in the PLS calibration. On the otherand, in the mixture-of-experts structure, this limitation was
voided since both models predict the flux of a target counterionn a competitive way. Thereby, this modeling strategy proved toe a better choice, which could be successfully used as a processredictive and optimization tool.
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eering Journal 62 (2012) 22– 33
The results obtained can be also applied for a straightforwardbiomedium design. In that way, the transport of certain undesirablecounterions to the water compartment can be either avoided orminimized.
Acknowledgments
A.R. Ricardo acknowledges Fundac ão para a Ciência e a Tec-nologia, Portugal for the PhD scholarship SFRH/BD/25275/2005.This work has been supported by the FCT through grant no. PEst-C/EQB/LA0006/2011.
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