Noureddine Lebaz

Arnaud Cockx

Mathieu Spérandio

Alain Liné

Jérôme Morchain
email: jerome.morchain@insa-toulouse.fr

Application of the Direct Quadrature Method of Moments for the modelling of the enzymatic hydrolysis of cellulose: I. Case of soluble substrate

Keywords: Enzymatic hydrolysis, population balance, cellulose, Endoglucanase, Exoglucanase

The modelling of the enzymatic hydrolysis of cellulosic polymers is investigated through a population balance approach. Both Endoglucanase (EG) and Exoglucanase (CBH) activities are taken into account. EG achieves random attacks along cellulosic chains and cleaves the β-glycosidic bonds whereas CBH produces cellobiose molecules by chain-end scission mechanism. The EG activity is modelled as a pure breakage while the CBH activity is assimilated to an erosion process with a specific product (cellobiose). In the two

cases, the inhibition of the cellulases activity by the end-product is incorporated. The population balance equation (PBE) accounting for breakage processes is solved using the Direct Quadrature Method of Moments (DQMOM) coupled to a distribution reconstruction technique based on the Maximum Entropy (ME) principle in order to track the time evolution of the chain length distribution (CLD) during the hydrolysis reaction. The β-glucosidase activity transforming the produced cellobiose into glucose is modelled as a Michaelis-Menten type kinetic with a competitive inhibition effect and solved simultaneously with the PBE. The numerical results show the time-evolution of the CLD during the hydrolysis reaction as well as the rate of conversion of the substrate into simple sugars. These results are in concordance with those predicted analytically. The synergistic action of the EG and CBH is highlighted and discussed and the inhibition effect is investigated.

The approach is promising by its accuracy and fastness for the analysis of dynamic experimental data of the

Introduction

The growing demand for energy, the depletion of fossil fuels and the global warming have sparked a growth in research for renewable energy sources [START_REF] Sun | Hydrolysis of lignocellulosic materials for ethanol production: a review[END_REF]. One of the promising alternatives is the bioconversion of biomass materials to bioethanol [START_REF] Jorgensen | Enzymatic conversion of lignocellulose into fermentable sugars: challenges and opportunities[END_REF]. The biochemical conversion of lignocellulosic biomass to ethanol involves several processing steps [START_REF] Gregg | Bioconversion of lignocellulosic residue to ethanol: Process flowsheet development[END_REF]. The reduction of cellulose, which is the major component of plant material, into fermentable sugars is carried out by means of enzymatic hydrolysis.

Cellulose is a homopolymer of β-1,4 linked D-glucose units [START_REF] Mittal | Effects of alkaline or liquid-ammonia treatment on crystalline cellulose: changes in crystalline structure and effects on enzymatic digestibility[END_REF]. Enzymes cocktails capable of degrading this polymer to sugars are a mixture of at least three different activities. The endoglucanase activity (EG) cleaves randomly the β-1,4-glycosidic linkages, the cellobiohydrolase activity (CBH) attacks the chain-ends and releases cellobiose (dimer of glucose) and the β-glucosidase activity hydrolyzes cellobiose into D-glucose [START_REF] Andersen | Hydrolysis of cellulose using mono-component enzymes shows synergy during hydrolysis of phosphoric acid swollen cellulose (PASC), but competition on Avicel[END_REF]. A typical kinetic of the enzymatic hydrolysis of cellulose is characterised by a rapid initial rate followed by a gradual slowdown caused by a multitude of factors affecting the effectiveness of the biocatalysts, leading, finally, to a partial conversion of the substrate [START_REF] Yang | Changes in the enzymatic hydrolysis rate of Avicel cellulose with conversion[END_REF]. These factors are related to the substrate structure (nature, pretreatments), the operating conditions (pH, temperature and mixing) and the enzymes (nature, deactivation during the reaction, inhibition by the end-products) [START_REF] Van Dyk | A review of lignocellulose bioconversion using enzymatic hydrolysis and synergistic cooperation between enzymesFactors affecting enzymes, conversion and synergy[END_REF].

Understanding the complex interactions between the enzymes elementary mechanisms and the dynamic evolution of the substrate features during the hydrolysis reaction is of critical importance for the modelling, the design and the optimization of the transformation process. Michaelis-Menten based models are widely used for the modelling of this dynamic process however, since they have been developed initially for homoge-neous reaction conditions, the parameters fitting the experimental data are only apparent and constants are sometimes lacking in physical or biochemical sense [START_REF] Bansal | Modeling cellulase kinetics on lignocellulosic substrates[END_REF]. Mechanistic models incorporating one or several rate limiting factors such as the substrate accessibility, the cellulases inhibition/deactivation and the change in the substrate reactivity can be found in literature [START_REF] Fan | Kinetic studies of enzymatic hydrolysis of insoluble cellulose: Derivation of a mechanistic kinetic model[END_REF][START_REF] South | Modeling simultaneous saccharification and fermentation of lignocellulose to ethanol in batch and continuous reactors[END_REF][START_REF] Xu | A new kinetic model for heterogeneous (or spatially confined) enzymatic catalysis: Contributions from the fractal and jamming (overcrowding) effects[END_REF]. A convincing critical analysis of the identifiability of these models has been published by [START_REF] Sin | Assessing reliability of cellulose hydrolysis models to support biofuel process designIdentifiability and uncertainty analysis[END_REF].

Recently, models based on the population balance approach have been proposed in order to better describe the heterogeneity of the substrate during the dynamic enzyme catalysis. Hosseini & Shah (2011a,b) assimilated the cellulose hydrolysis to soluble polymer degradation. The endoglucanase activity (EG) has been modelled as a pure random breakage of the cellulose chains and the exoglucanase activity (CBH) as a chain-end scission [START_REF] Kostoglou | Mathematical analysis of polymer degradation with chain-end scission[END_REF]. To be more realistic, Griggs et al. (2012a,b) developed a population balance model based on the depolymerisation mechanisms of EG and CBH assuming a population of monodisperse cylindrical cellulose particles since the reaction is heterogeneous. To solve the system of rate equations, Hosseini & Shah (2011a,b) considered all the possible chains in the system when Griggs et al. (2012a,b) reduced the number of equations by mapping the continuous particle size distribution (PSD) to a fixed discrete grid points. [START_REF] Lebaz | Population balance approach for the modelling of enzymatic hydrolysis of cellulose[END_REF] used the discretization method developed by Kumar & Ramkrishna (1996a) for the resolution of the population balance equation (PBE) with a fixed pivot technique for the EG activity and a moving pivot technique [START_REF] Kumar | On the solution of population balance equations by discretiza-tionII. A moving pivot technique[END_REF] for the CBH activity since it was assimilated to a continuous dissolution of the cellulosic chains. [START_REF] Ho | Modeling chain-end scission using the Fixed Pivot technique[END_REF], meanwhile, proposed a modified fixed pivot technique for the modelling of chain-end scission by CBH. All of these modelling approaches are computationaly expensive.

Cellulose particles can be characterised by their Degree of Polymerization (DP) which is the number of monomers (D-glucose) constituting each particle. If we note N the maximum DP in a given particulate system, one needs to solve N equations to describe the time-evolution of all the possible sizes undergoing breakage processes. The discretization method (Kumar & Ramkrishna, 1996a) reduces considerably the number of equations by distributing the particles into a set of classes M < N. Unfortunately, the accuracy of this method depends on the number of classes M . Increasing M implies more equations to solve and, thereby, substantial computational time. In this study, we develop a new approach for solving the PBE in the case of cellulose depolymerisation by EG and CBH. For the sake of accuracy and computational efficiency, the Direct Quadrature Method of Moments (DQMOM) is used as solution method [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF]. Since DQMOM provides information only on the time evolution of the moments of the particle size distribution (PSD), the recovering of the full distribution is needed. For this, the PSD is reconstructed from its first moments using the Maximum Entropy based method [START_REF] Tagliani | Hausdorff moment problem and maximum entropy: A unified approach[END_REF]. This reconstruction method is preferred to other techniques (e.g. Spline based method, Beta Density Function based technique) for its fastness, accuracy and the small number of moments required [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process[END_REF]. This is of critical importance when the PSD reconstruction and the PBE resolution are conducted simultaneously. Note that more recent moment methods are proposed in literature to solve the PBE, maintly the Conditional Quadrature Method of Moments (CQMOM) developed by [START_REF] Yuan | Conditional quadrature method of moments for kinetic equations[END_REF] which is more suitable for multivariate problems and the Extended Quadrature Method of Moments (EQMOM) [START_REF] Yuan | An extended quadrature method of moments for population balance equations[END_REF] allowing the reconstruction of the distribution all along the resolution using kernel density functions.

For sake of clarity, this contribution is organized in two parts. In this first part, the focus will be put on soluble substrates. The general theoretical framework including the DQMOM approach for solving the homogeneous monovariate PBE and the PSD reconstruction by the Maximum Entropy based method is developed and validated against analytical solutions. We show how the cellulolytic activities are modelled in the case of depolymerisation of soluble cellulose chains before testing and discussing the robustness of the model in the results section. The second part deals with the case of particulate systems.

In both cases, the β-glucosidase activity which converts cellobiose into D-glucose is modelled as a Michaelis-Menten type kinetic with competitive inhibition and resolved simultaneously with the PBE.

Theoretical framework

The PBE describes the evolution of a density function of a given population of chains/particles undergoing different transformation processes. In the case of enzymatic hydrolysis of cellulose chains, the substrate chains are degraded during the reaction by the synergistic action of an enzyme mixture. Thus, considering the chain length L as the internal coordinate, the PBE expressing the depolymerisation process in a spatially homogeneous system is written in its continuous form as:

∂n(L, t) ∂t = ∞ L β(L, λ)Γ(λ)n(λ, t)dλ -Γ(L)n(L, t) (1)
where n(L, t) is the length-based number density function, Γ(L) the breakage frequency for a chain of length L which is related to the enzyme activity, β(L, λ) is the daughter distribution function giving the probability of obtaining a chain of length L from the breakup of a chain λ. The first term on the right hand side of equation (1) accounts for the formation (birth) of chains with length L resulting from the breakage of larger chains . The last term is the death term due to the loss of chains of length L because of their depolymerisation. Note that, in the particular case of soluble substrates, the internal coordinate L refers to the degree of polymerization (DP) of the cellulose chains.

Except for some cases where Γ(L) and β(L, λ) have simple mathematical expressions [START_REF] Ziff | The kinetics of cluster fragmentation and depolymerisation[END_REF], equation (1) has no analytical solution. Numerical methods are used to solve it. We adopt in this contribution the Direct Quadrature Method of Moments (DQMOM). A brief description of this method is given in Appendix A. For further details, one can refer to the original paper by [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF].

The k th order moment of the chain length distribution (CLD) is defined as:

m k (t) = ∞ 0 L k n(L, t)dL (2)
The moments provide the main properties of the CLD. The zeroth order moment, m 0 (t), represents the molar concentration of the cellulose chains since n(L, t)dL is the number of cellulose chains per volume having lengths between L and L + dL. The first order moment, m 1 (t), gives the total concentration of monomers present in cellulose chains. Average properties of the CLD can be derived from the moments such as the number-averaged chain length given by the ratio of the two first moments m 1 (t)/m 0 (t) and the mass-averaged chain length m 2 (t)/m 1 (t).

By applying the moment transformation of equation ( 1), the final moment transport equation is given as:

(1 -k) N i=1 L k i a i + k N i=1 L k-1 i b i = N i=1 b (k) i Γ i w i - N i=1 L k i Γ i w i (3)
where

                 a i = ∂w i ∂t b i = ∂c i ∂t c i = w i L i b (k) i = ∞ 0 L k β(L, L i )dL (4)
(L i (t), w i (t)) are the abscissa and weights of the N Gaussian quadrature nodes. (L i (0), w i (0)) can be computed using the Product-Difference (PD) algorithm [START_REF] Gordon | Error Bounds in Equilibrium Statistical Mechanics[END_REF]. The different steps of the transformation are explicited in Appendix A .

Although the intrinsic activities of EG and CBH are different, they lead to the depolymerisation of the substrate chains during the reaction. Thus, the two activities are modelled as breakage processes (equation 1), the distinction between them will appear in the formulation of the daughter distribution functions.

The validation of the DQMOM implementation in the case of breakage processes is given in Appendix B. This numerical validation is based on the analytical solution of equation (1) [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process[END_REF]. Moreover, the CLD reconstruction from the moments using the maximum entropy technique is embedded with the DQMOM algorithm in order to access the CLD evolution simultaneously. The maximum entropy technique is briefly described in Appendix C and its numerical validation is fully discussed in [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process[END_REF].

Daughter distribution functions formulation

The depolymerisation mechanism depends on the nature of the cellulases. In our case, two main activities (EG and CBH) are considered. This leads to the resolution of the global PBE below:

∂n(L, t) ∂t = S EG L (L, t) + S CBH L (L, t) (5)
with

     S EG L (L, t) = ∞ L β EG (L, λ)Γ EG (λ)n(λ, t)dλ -Γ EG (L)n(L, t) S CBH L (L, t) = ∞ L β CBH (L, λ)Γ CBH (λ)n(λ, t)dλ -Γ CBH (L)n(L, t) (6)
where β EG and β CBH are the daughter distribution functions reflecting the mode of action of the enzymes.

Γ EG and Γ CBH are the breakage frequencies directly related to the enzymatic activities [START_REF] Lebaz | Population balance approach for the modelling of enzymatic hydrolysis of cellulose[END_REF].

The expression of the daughter distribution functions for both EG and CBH depends on the nature of the substrate : soluble polymer chains or particulate system. In the following paragraph, the formulation will be restricted to the case of polymer chains. The case of particulate substrates is treated in the second part of the contribution.

EG activity: random breakage

The EG activity cleaves randomly the β-1,4-glycosidic bonds along the cellulosic chains. Thus, in the case of soluble substrate, each EG attack leads to the scission of the initial chain λ producing thereby two smaller chains (λ -L) and L:

P (λ) -→ P (λ -L) + P (L) (7)
The daughter distribution function β(λ, L) defines the probability that a chain of length λ leads to a chain of size L when it breaks up. Under the assumption of a binary scission, two particles are always produced and the factor 2 takes into account that either of these particles can be of size L < λ. The time-evolution of the chain length number density function, n(L, t) is given by the PBE :

S EG L (L, t) = ∂n(L, t) ∂t EG = 2 ∞ L 1 λ Γ EG (λ)n(λ, t)dλ -Γ EG (L)n(L, t) (8)

CBH activity: chain-end scission

The CBH activity has not a random aspect as the EG one, each effective attack produces one soluble cellobiose molecule and reduces, thereby, the DP of the attacked chain by 2. In the case of a soluble substrate, the CBH mechanism is described by:

P (L) -→ P (L -L C ) + P (L C ) (9)
where L C is the length of the cellobiose molecule.

Since the CBH activity leads to a specific product, the daughter distribution function is given by a Dirac delta function [START_REF] Mccoy | Continuous-mixture fragmentation kinetics: particle size reduction and molecular cracking[END_REF][START_REF] Wang | Continuous kinetics for thermal degradation of polymer in solution[END_REF] and the corresponding source term can be expressed as :

S CBH L (L, t) = ∂n(L, t) ∂t CBH = ∞ L Γ CBH (λ)δ (L -(λ -L C )) n(λ, t)dλ -Γ CBH (L)n(L, t) (10)
Additionally the cellobiose concentration C C (t) evolution during the reaction is given by :

∂C C (t) ∂t = ∞ L Γ CBH (λ)δ (L -L C ) n(λ, t)dλ + S EG L C (L, t) (11)
Where S EG L C (L, t) is the contribution of EG activity to cellobiose release which is negligible thus, the right hand side of equation ( 11) is reduced to the first term.

135

Moment transformation using the DQMOM formalism

The moment transformation of equations ( 8) and ( 10) is given by equation ( 3). We explicit hereafter the moment transformation of the equation ( 11) accounting for the cellobiose release after each CBH effective attack. Equation ( 11) can be written in a compact form as:

∂C C (t) ∂t = Q L (L, t) (12)
where Q L (L, t) is the source term due to the erosion process.

140

One has to precise that C C (t) is a delta function centered on the cellobiose length L C with a time increasing weight. Thus, C C (t) can be expressed as:

C C (t) = w C (t)δ(L -L C ) (13)
By introducing equation ( 13) in ( 12), we obtain:

δ(L -L C ) ∂w C ∂t -δ (L -L C )w C ∂L C ∂t = Q L (L, t) (14)
Since L C is constant and by applying the moment transformation, equation ( 14) is reduced to:

(1 -k)L k C ∂w C ∂t = ∞ 0 L k Q L (L, t)dL (15)
In the right hand side, the source term is transformed as follows :

Q (N ) k = ∞ 0 L k Q L (L, t)dL = ∞ 0 ∞ L L k Γ(λ)δ(L -L C )n(λ, t)dLdλ = N i=1 b (k) i Γ i w i (16)
In equation ( 16), w i refers to the weight of the node i of the density n(L, t). In the specific case of erosion process :

b (k) i = ∞ 0 L k δ(L -L C )dL = L k C (17)
Finally, equation 17 is tranformed to :

(1 -k)L k C ∂w C ∂t = N i=1 L k C Γ i w i (18)
Since C C (t) is fully defined by the concentration of cellobiose molecules produced at the time t, one needs only the zeroth order moment (k = 0), thus:

∂w C ∂t = N i=1 Γ i w i (19)
Thus, equation ( 19) gives the time evolution of the cellobiose concentration.

Breakage frequencies formulation

The breakage frequencies for the two enzymatic activities give the rate of the depolymerisation process, in other words, the number of effective attacks per time unit. They are directly related to the cellulases concentration, more specifically to the active adsorbed enzymes concentration.

Different mathematical expressions can be used for the breakage frequencies. One can, for example, assume that Γ EG is proportional to the chain length L since longer chains are more likely to be attacked which is not necessarily the case for Γ CBH because of the nature of the exoglucanase attacks. Different expressions for these frequencies are to be tested and confronted to the trends observed in experimental data sets (expressed as CLD at various times during hydrolysis, simple sugars concentrations, . . . ). The general form Γ = αL p is often used [START_REF] Nopens | Modelling the activated sludge flocculation process combining laser light diffraction particle sizing and population balance modelling(PBM)[END_REF][START_REF] Ding | Population balance modelling of activated sludge ical engineering science[END_REF]. This form is adopted and discussed in our case with different parameters (α and p).

The EG and CBH inhibition effect

The end-products inhibition effect is known as one of the most important factors affecting the hydrolysis rate [START_REF] Xiao | Effects of sugar inhibition on cellulases and $\beta$glucosidase during enzymatic hydrolysis of softwood substrates[END_REF][START_REF] Gruno | Inhibition of the Trichoderma reesei cellulases by cellobiose is strongly dependent on the nature of the substrate[END_REF]) causing a progressive decrease of the cellulases activity as the endproduct concentration increases with substrate conversion. To incorporate this critical effect in the model, we express the breakage frequencies of EG and CBH as functions of the end-product concentration which is time-dependent as follows (equations 20 and 21):

Γ EG = Γ EG0 • K EG C C (t) + K EG (20) Γ CBH = Γ CBH0 • K CBH C C (t) + K CBH (21)
where C C (t) is the cellobiose molar concentration, K EG and K CBH are the inhibition constants for EG and CBH respectively, Γ EG0 and Γ CBH0 are the intrinsic EG and CBH activities respectively.

The β-glucosidase activity

Since the β-glucosidase activity takes place in the liquid phase and turns cellobiose into D-glucose, Michaelis-Menten type kinetic is used and competitive inhibition by the accumulation of glucose is incorpo-rated as shown in equation ( 22).

dC G dt = 2V m C C K m 1 + C G K P + C C (22)
where C G is the glucose molar concentration, K m and V m the Michaelis-Menten parameters, K P the inhibition constant for the β-glucosidase activity. The factor 2 is due to the fact that one mole of cellobiose gives two moles of glucose.

Summary

Starting from a given chain length distribution (CLD), we solve simultaneously a set of equations (equation ( 23)) including a PBE accounting for a random breakage process for the EG activity, a PBE for an erosion process refering to the CBH activity and a Michaelis-Menten equation for the β-glucosidase activity.

                         ∂n(L, t) ∂t = 2 ∞ L 1 λ Γ EG (λ)n(λ, t)dλ -Γ EG (L)n(L, t) + ∞ L Γ CBH (λ)δ (L -(λ -L C )) n(λ, t)dλ -Γ CBH (L)n(L, t) ∂C C (t) ∂t = ∞ L Γ CBH (λ)δ (L -L C ) n(λ, t)dλ - 1 2 dC G (t) dt dC G (t) dt = 2V m C C K m 1 + C G K P + C C (23)
The integration of this set of equations using the DQMOM approach coupled with the reconstruction technique gives the time evolution of the CLD and the kinetic of the substrate conversion into simple sugars (cellobiose and glucose).

The initialisation of the resolution procedure requires the initial chain length distribution n(L, 0) and the model parameters. These parameters are the breakage frequencies for both EG and CBH activities (Γ EG and Γ CBH ), the Michaelis-Menten parameters for the β-glucosidase activity (V m and K m ) and the inhibition constants K EG , K CBH and K P . One has to point out the low number of parameters required compared to the common kinetic-based models such as the model described by [START_REF] Kadam | Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass[END_REF] in which 18 parameters are to be identified. Thus, the low number of parameters in our case, clearly associated to the physics of the modelled phenomena, enhances their reliable identification from experimental data.

Numerically, the ode45 integrator of MATLAB is used to solve the set of equations ( 23) under a workstation comprising of an Intel R Core TM i7-3740QM CPU with a clock speed of 2.7 GHz and 16 GB of installed memory (RAM).

Results and discussion

For all the results presented in this article, the initial CLD is assimilated to a Gamma distribution given by the equation below:

n(L, 0) = L κ-1 e -L/θ Γ(κ)θ κ (24)
where κ and θ are the Gamma distribution parameters and Γ the classical Gamma function. The mode of the Gamma distribution is given by (κ -1)θ and its variance by κθ 2 . In our case, the Gamma distribution parameters are κ = 60 and θ = 10.

The two main activities EG and CBH are treated separately before showing the combination and the inhibition effect. The breakage frequencies are set proportional to the polymer chain length which means that longer chains are more likely to be attacked (Γ = αL p ). This is the case especially for the EG random activity. Other breakage frequency expressions can be tested easily. The daughter distribution functions and frequencies for both EG and CBH are summerized in table 1. The system of ordinary differential equations (equation 23) is integrated over the time interval [0,10] hours.

For the erosion process, the formation of cellobiose is not integrated in the full CLD because it leads to a strong peak at the length L C = 2 which constitutes a singularity and disturbs by the way the reconstruction procedure. Thus, the daughter distribution function is reduced to b(k

) i = (L i -L C ) k . This is consistent with Mechanism β(L, λ) b (k) i Frequency (Γ) EG: Uniform breakage 2/λ L k i 2 k+1 αL p CBH: Erosion process              1 if L = L C 1 if L = λ -L C 0 otherwise L k C + (L i -L C ) k αL p
Table 1: The daughter distribution functions and breakage frequencies for EG and CBH activities in the case of depolymerisation of soluble chains the idea that cellobiose is no longer a chain per se but rather an elementary molecule. As a result, the total mass balance refers to the mass of polymer chains only. The cellobiose concentration is accessible via the mass loss in the system as well as by equation ( 11).

CBH + β-glucosidase activities

CBH attacks the chain-ends and releases cellobiose molecules as a product. The loss of two monomers by the attacked polymers is modelled as an erosion process leading to a progressive shortening of the chains length until their total transformation. The breakage frequency is set to Γ CBH = α.L (p = 1). The Figure 1 gives the time evolution of the first four normalised moments of the CLD undergoing CBH attacks. The parameter α is set equal to 4.10 -2 for practical considerations.

The total molar concentration (zeroth order moment) remains constant within the process which means that the number of polymer chains is constant. In the meantime, the total mass of the polymer chains (the first order moment) decreases due to the loss of cellobiose molecules. In other words, the initial number of chains is unchanged but the chains length is decreasing. This is consistent with the fact that the initial numberaveraged chain length is far from the origin (much larger than the dimer of cellobiose). In order to observe the total transformation of the substrate (and the simultaneous diminution of the total molar concentration of soluble polymer) it would be necessary to increase the hydrolysis time and/or erosion frequency. Note that the set of equations ( 23) being conservative, the loss of mass due to the enzymatic action on the polymer with a frequency proportional to the polymer length chains (Figure 1) is identical to the accumulation of cellobiose and glucose. The interpretation of higher order moment is not direct. Indeed, in the present case, the mass is directly proportional to the chain length.

However, the ratio ( m2 m1 ) represents the mass averaged-chain length and is analogous to d 43 for spherical particles. The results shows that this mass averaged-chain length decreases.

Figure 2 gives the evolution of the initial CLD reconstructed by the ME based method. The initial distribution is shrinking as it moves to the left since the erosion frequency is not equally distributed all along the CLD: longer chains are more frequently attacked than smaller ones.

These numerical simulations are validated against the analytical solution developed in Appendix D. The zeroth order moment is constant while the first order moment decreases exponentially with a time constant

235 equal to τ 1 = 1 α.L C .
Introducing non dimensional parameters will now help in analyzing the trends when different erosion laws are considered.

From the analytical solution, one can build a generic behaviour of the chains population undergoing erosion and the adimensional time is t τ2 .

Figure 3 shows the evolution of the first normalised moment ( m1(t) m1(0) ) reflecting the total mass in the system (without simple sugars: cellobiose and glucose). Two cases with length dependent/independent erosion frequencies, with the adimensional times t τ1 and t τ2 respectively are considered. The evolutions are independent from the absolute time and the numerical value of the factor α. They are caracteristic of the erosion frequency form.

The relative evolution of the first moment of the CLD considering the CBH attacks shown in Figure 3 is of critical importance to determine the CBH elementary mechanism by fitting experimental data. A linear fit of the mass versus time curve reveals a length independent frequency while an exponential fit is the footprint of a length dependent frequency.

EG activity

EG attacks randomly the cellulose chains and breaks up the glucosidic bonds leading to two daughter smaller chains after each attack. The fragmentation process increases the total molar concentration of the cellulose chains in the system as shown by the evolution of the zeroth order normalised moment in Figure 4.

255

In this case, the breakage frequency is length-dependent Γ EG = αL and the factor α is set equal to 10 -4 .

The total number of chains increases while the total mass represented by the first order moment remains constant (batch reactor). The number-averaged chain length given by the ratio of the first two moments decreases from a DP of 600 to 375 at the end of the process.

Figure 5 gives the time-evolution of the CLD undergoing fragmentation process. The molar concentration 260 of the longest chains decreases while small chains appear in the system. The initial CLD shifts toward the smaller lengths with a consequent shape transformation (bimodal distributions appear). The CLDs are reconstructed over the interval [0, +∞[. The action of EG on small chains with L ∈ [0, 2] is therefore thus ensure low computational cost for the PBE resolution. This is conceptually erroneous but there is no 265 consequence from a quantitative point of view as mentionned before (equation ( 11)). The probability to produce cellobiose when cutting a chain of length L is very small, the breakage frequency goes to zero when L → 2. As a result, the amount of cellobiose produced by EG is several order of magnitude smaller than that resulting from CBH activity. This supports the neglection of the term S EG L C in equation ( 11).

The analytical results relative to the random breakage given in Appendix D are in concordance with the 270 numerical ones. The total mass remains constant while the molar concentration of the polymers (zeroth order moment) increases linearly with a slope equal to αm 1 (0). In this case the adimensional time is t τ1 with τ 1 = m0(0) αm1(0) .

In the case where the breakage frequency is constant (Γ EG = α), the zeroth order moment increases exponentially with a time-constant equal to τ 2 = 1 α . Figure 6 shows the relative evolution of the normalised 275 zeroth order moment of the CLD when pure breakage process is considered (EG activity) with lengthdependent/independent frequencies. As mentioned previously in the case of CBH activity, the relative evolution of the zeroth order moment when considering EG activity (pure breakage process) shown in Figure 6 is time and parameter independent.

It depends only on the mathematical form of the breakage frequency and here again the fitting of experimental data (by either linear or exponential functions) can be used to identify the breakage law.

EG-CBH + β-glucosidase combined activity

The combination of the three different enzymatic activities results typically in a CLD evolution presented in Figure 7. In this example, the breakage frequencies those used in the previous cases (Γ = αL). The initial CLD in Figure 7 shifts toward small sizes and new small chains are continuously formed by the breakage of the longest ones because of the EG activity. This is reflected by the increase of the zeroth order moment and the exponential decrease of the first order moment as predicted via the analytical solution.

Clearly the trends depends on the relative values of α 1 and α 2 i.e. the composition of the enzymatic cocktail.

In order to get access to the general behaviour of the first two moments of the CLD depending on the form of the breakage/erosion frequencies, three different combination forms are tested. The first one is a length-dependent frequency (Γ EG = α 1 .L and Γ CBH = α 2 .L), the second one is a length-independent frequency (Γ EG = α 1 and Γ CBH = α 2 ) and the last one combines a length-dependent breakage frequency and a length-independent erosion frequency (Γ EG = α 1 .L and Γ CBH = α 2 ). The time constants for these three cases are respectively

τ 1 = 1 α2L , τ 2 = 1 α1 and τ 3 = 1 √ α1α2L C
. The results are shown in Figure 8. Figure 8 shows that the relative evolution of each moment depends on the form of the breakage/erosion frequencies. Thus, we have three generic trends for each moment. The combination of these informations (relative to the total molar concentration of the substrate and its total mass in the system) allows the determination of the breakage/erosion frequencies forms without any need of separation of the different activities at the experimental scale, step which can be critical.

Influence of the frequency formulation on the prediction of simple sugars 300

The synergistic action of the different activities increases the rate of degradation of the substrate. Additionally this can be quantified by comparing the time-evolution of the number-averaged chain length when using the enzymes separately and in cocktail as shown in Figure 9 (the right subplot). This chosen criterion is the ratio of the first two moments ( m1(t) m0(t) ). The total amount of substrate converted into soluble sugars is reflected by the diminution of the first moment (left subplot in Figure 9). The lower the curve the faster the conversion.

In the case presented in Figure 9 where Γ EG = α 1 .L and Γ CBH = α 2 , the number-averaged chain length decreases from an initial degree of polymerization (DP) of 600 to 520 through the sole action of CBH. The decrease reaches a DP of 375 when one considers the EG activity only. The combination of the two activities affects substantially the number-averaged chain length. At the end of the time-process, the mean DP is only of 320. The repercussion of the combination of the two different activities on the production of simple sugars is given in the first subplot (Figure 9). As shown, the conversion of the substrate is boosted when the two enzymes are combined. This Endo-Exo synergistic effect is explained by the appearance of new chains in the system due to the EG activity and, since all chains are equiprobably attacked by CBH, the conversion is accelerated. In the case presented in Figure 10 (Γ EG = α 1 .L and Γ EG = α 2 .L), the combined action of the two different activities has no incidence on the substrate conversion curves (in terms of simple sugars). The fractions of glucose and cellobiose produced by EG and CBH independently and in cocktail are compared and there is no noticeable difference in the two cases. This is confirmed by the analytical solution given in Appendix D in which the evolution of the first order moment of the CLD is similar when considering CBH activity only and both EG and CBH activities. This is not the case for the number-averaged chain length (right subplot in Figure 10). So one would conclude that there is no synergistic effect.

These two numerical examples ([Γ

EG = α 1 .L, Γ CBH = α 2 ] and [Γ EG = α 1 .L, Γ CBH = α 2 .L]) reveal that
the erosion frequency can not be length dependent. In this case, no synergistic effect is possible observed in terms of simple sugars production. This is in contradiction with experimental results that reveal an increase of the conversion rate (and hence sugar production) when the two types of enzymes are combined [START_REF] Sun | Hydrolysis of lignocellulosic materials for ethanol production: a review[END_REF][START_REF] Van Dyk | A review of lignocellulose bioconversion using enzymatic hydrolysis and synergistic cooperation between enzymesFactors affecting enzymes, conversion and synergy[END_REF]). It appears that a size independent erosion process is necessary to observe synergism. From a physical point of view it is consistent to assume that the CBH activity does not depend on the chain length since such enzymes operate at both ends of the chain irrespective of its length. We considered synergism as a phenomenon leading to a faster conversion of the substrate. With this definition, a size dependent erosion process does not lead to a synergetic effect (left subplot figure 10). Yet there is actually a consequence of the combined action of the two types of enzymes on the chain length. This could also be regarded as a form of synergism (but the information is rarely accessed experimentally). Comparing the right plots in figure 9 and 10 we can claim that there is always a synergetic effect (in terms of average chain length) and that the contribution of CBH to the average chain length reduction is more pronounced in the case of a linear dependency.

EG-CBH + β-glucosidase inhibition effect

In a batch reactor, the disintegration of the polymer chains by the cooperative action of the different cellulolytic activities leads to the accumulation of the end-products in the system. This causes the so-called inhibition effect since the enzymes are sensitive to their own products, inactive enzyme-product complexes are formed. As a consequence, the enzymes activity slows down as reported in Figure 11 where no inhibition and inhibition resultas are compared in terms of cellobiose and glucose fractions (last two subplots) and the first two normalised moments ot the CLD (first two subplots).

The introduction of the inhibition effect in the population balance model leads to the slowdown of the conversion rate of the substrate as shown in Figure 11. The total molar concentration (m 0 (t)) as well as the overall conversion rate (m 1 (t), cellobiose and glucose fractions) are affected. The parameters K EG , K CBH and K P corresponding to the three main activities control the significance of the inhibition effect.

By introducing the inhibition effect, the formulation of the model is more complete. The model inhibition parameters can be reached experimentally by fitting typical saccharification data in the specific case of depolymerisation process for example. The model is built considering the elementary mechanisms of the three main cellulase activities on soluble polymers.

Conclusion

In this first part, a dynamic population balance model has been developed in the case of cellulose polymers degradation by the synergistic action of cellulases. The three main cellulolytic actions are taken into account since they act differently with specific mechanisms. First, the general theoretical framework is developed in the case of enzymatic hydrolysis process. The random action of the endoglucanase activity (EG) is assimilated to a pure breakage process when the cellobiohydrolase activity (CBH) is modelled as an erosion phenomenon since it proceeds with a chain-end scission and produces cellobiose molecules. We give hereafter a succinct description of the Direct Quadrature Method of Moments (DQMOM) applied for the population balance equation accounting for breakage processes, for more detail one can refer to the original work by [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF].

The homogeneous monovariate PBE for breakage processes can be written in a compact form as:

∂n(L, t) ∂t = S L (L, t) (A.1)
where S L (L, t) is the source term due to the breakage, given in its explicit form in equation ( 1).

In the monovariate DQMOM approach, the continuous distribution function n(L, t) is expressed as a discrete summation of Dirac delta functions [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF]:

n(L, t) = N i=1 w i (t)δ [L(t) -L i (t)] (A.2)
where N is the number of delta functions (nodes i), L i is the property of the node and w i its weight. By 395 substituting n(L, t) by its discrete decomposition (equation A.2 in equation A.1), one obtains:

N i=1 δ (L -L i ) ∂w i ∂t - N i=1 δ (L -L i ) w i ∂L i ∂t = S L (L, t) (A.3)
where δ (L -L i ) is the first derivative of the delta function δ(L -L i ). By introducing the weighted abscissa c i instead of the abscissa L i (equation A.4), .4) and by setting :

c i = w i L i (A
∂w i ∂t = a i ∂c i ∂t = b i (A.5) Marchisio & Fox (2005) reformulate the equation A.3 as : 400 N i=1 [δ (L -L i ) + δ (L -L i )L i ] a i - N i=1 δ (L -L i ) b i = S L (L, t) (A.6)
The unknowns a i and b i are only time-dependent and are reachable by applying moment transformation. [START_REF] Hulburt | Some problems in particle technology: A statistical mechanical formulation[END_REF] defined the k th integer moment of the distribution function n(L, t) as:

m k (t) = ∞ 0 L k n(L, t)dL = N i=1 w i L k i (A.7)
The moment transformation (multiplying by L k and integrating over L) is applied to equation A.6. The final equation is given as:

(

1 -k) N i=1 L k i a i + k N i=1 L k-1 i b i = ∞ 0 L k S L (L, t)dL (A.8)
Considering the definition of S L given in equation ( 1), the right hand side term of equation (A.8) now 405 writes:

L k S L (L, t)dL = ∞ 0 ∞ 0 L k β(L, λ)Γ(λ)n(λ, t)dLdλ - ∞ 0 L k Γ(L)n(L, t)dL (A.9)
Thus, equation (A.8) becomes :

(

1 -k) N i=1 L k i a i + k N i=1 L k-1 i b i = N i=1 b (k) i Γ i w i - N i=1 L k i Γ i w i (A.10) where b (k) i = ∞ 0 L k β(L, L i )dL (A.11)
The system in equation A.10 can be written in a matrix form as :

Ax = d (A.12)
In order to assess the error induced by DQMOM, we define the global error function as:

E(k) = |m k -m * k | m * k (B.1)
where m k is the k th order moment estimated via DQMOM and m * k the k th order moment calculated via the analytical solution.

As we can see in figure B.12 , there is a good agreement between the moments calculated from the analytical solution and those estimated via DQMOM. The first order moment refers to the total chains length and remains constant within the time since the polymers are broken up but their total mass correlated with the total length is constant.

The figure B.13 represents the time evolution of the global error function E(k) for the four first moments when using a quadrature with four nodes. approach [START_REF] Tagliani | Hausdorff moment problem and maximum entropy: A unified approach[END_REF] is used in this work. We give hereafter a synthetic description of this technique.

We note by f (x) the probability density function (PDF) and by f M (x) its approximation. The ME method is based on the maximization of the Shannon entropy H[f ] given by:

H[f ] = - ∞ 0 f (x)lnf (x)dx (C.1)
Under the moments constraints:

m k = ∞ 0 x k f (x)dx k = 0, 1, . . . (C.2)
The explicit representation of the ME approximate is given by:

440 f M (x) = exp   - 2N -1 j=0 ξ j x j   (C.3)
To be supplemented by the 2N constraints:

m i = ∞ 0 x i f M (x)dx i = 0, 1, . . . , 2N -1 (C.4)
The (2N ) Lagrange's multipliers ξ 0 , . . . , ξ 2N -1 are obtained by solving the following set of 2N nonlinear equations:

∞ 0 x i exp   - 2N -1 j=0 ξ j x j   dx = m i i = 0, 1, . . . , 2N -1 (C.5)
Numerically, iterative methods are used to solve equation (C.5) [START_REF] Mead | Maximum entropy in the problem of moments[END_REF]. In our case, we use the standard Newton method starting from an initial choice of the Lagrange's multipliers as ξ = (-ln(m 0 )/(x max -x min ), 0, . . . , 0) with an accuracy of 10 -6 . The set of moments are normalised before such as m 0 = 1. For more details as well as for the validation, one can refer to [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process[END_REF].

Appendix D: Analytical solution of the PBE in the case of breakage processes

The evolution of the CLD moments when considering breakage processes (EG+CBH) can be written in a discret form as :

∂m k (t) ∂t = N i=1 b (k) i Γ i w i - N i=1 L k i Γ i w i EG + N i=1 b (k) i Γ i w i - N i=1 L k i Γ i w i CBH (D.1)
By substituting the breakage and erosion daughter distribution functions and frequencies in equation (D.1) as reported in table 1, we obtain the general equation below :

∂m k (t) ∂t = α 1 2 k + 1 -1 m k+p1 + α 2 i (L i -L C ) k L p2 i w i -m k+p2 (D.2)
with (α 1 , p 1 ) the parameters of the breakage frequency of EG and (α 2 , p 2 ) the parameters of the erosion frequency of CBH.

We explicit hereafter the expected evolution of the two first moments considering length-dependent (p 1 = p 2 = 1) and length-independent (p 1 = p 2 = 0) frequencies.

• Case 1 : p 1 = p 2 = 1 -Random breakge :

In this case, equation (D.2) is reduced to :

∂m k (t) ∂t = 2 k + 1 -1 α 1 m k+1 (D.
3)

The evolution of the two first moments is given by :

∂m k (t) ∂t = α 2 i (L i -L C ) k L i w i -m k+1 (D.6)
The evolution of the two first moments is : The combination of the two processes is traduced by the equation below :

∂m k (t) ∂t = α 1 2 k + 1 -1 m k+1 + α 2 i (L i -L C ) k L i w i -m k+1 (D.9)
The evolution of the two first moments is :

   ∂m0(t) ∂t = α 1 m 1 (t) ∂m1(t) ∂t = -α 2 L C m 1 (t) (D.10)
The two first moments in this case are :

     m 0 (t) = α 1 m 1 (0) α 2 L C 1 -e -α2L C t + m 0 (0)
m 1 (t) = m 1 (0).e -α2L C t (D.11)

• Case 2 : p 1 = p 2 = 0 470 -Random breakge :

In this case, equation (D.2) is reduced to :

∂m k (t) ∂t = 2 k + 1 -1 α 1 m k (D.
12)

The evolution of the two first moments is given by : The combination of the two processes is traduced by the equation below :

∂m k (t) ∂t = α 1 2 k + 1 -1 m k + α 2 i (L i -L C ) k w i -m k (D.18)
The evolution of the two first moments is :

   ∂m0(t) ∂t = α 1 m 0 (t) ∂m1(t) ∂t = -α 2 L C m 0 (t) (D.19)
The two first moments in this case are : We explore the combination of a random breakage with a length-dependent frequency and an erosion process with length-independent frequency. Equation (D.2) is written as :

∂m k (t) ∂t = α 1 2 k + 1 -1 m k+1 + α 2 i (L i -L C ) k w i -m k (D.21)
The evolution of the two first moments is :

   ∂m0(t) ∂t = α 1 m 1 (t) ∂m1(t) ∂t = -α 2 L C m 0 (t) (D.22)
The analytical resolution of the set of equations in (D.22) is given below :

   m 0 (t) = m 0 (0) cos √ α 1 α 2 L C .t + α1 α2L C .m 1 (0) sin √ α 1 α 2 L C .t m 1 (t) = m 1 (0) cos √ α 1 α 2 L C .t -α2L C α1 .m 0 (0) sin √ α 1 α 2 L C .t (D.23)
Appendix E: Supplementary data

Figure 1 :

 1 Figure 1: The time evolution of the first four normalised moments of the chain length distribution undergoing CBH attacks

Figure 2 :

 2 Figure 2: The time evolution of the CLD undergoing CBH attacks with a frequency proportional to the polymer length

Figure 3 :

 3 Figure 3: Evolution of the first normalised moment of the chain length distribution undergoing erosion process with a lengthdependent (continuous line) and a length-independent (dashed line) frequencies

Figure 4 :

 4 Figure 4: The time evolution of the first four normalised moments of the chain lenght distribution undergoing EG attacks with a frequency proportional to the polymer length

Figure 6 :

 6 Figure 6: Evolution of the zeroth normalised moment of the chain length distribution undergoing pure breakage process with a length-dependent (continuous line) and a length-independent (dashed line) frequencies

Figure 7 :

 7 Figure 7: The time evolution of the CLD undergoing EG and CBH attacks with a frequency proportional to the polymer length for the two activities

Figure 8 :

 8 Figure 8: Relative evolution of the two first normalised moments when the two activities EG and CBH are combined with different breakage/erosion frequency forms: Γ EG = α 1 .L and Γ CBH = α 2 .L by the continuous line, Γ EG = α 1 and Γ CBH = α 2 represented by the dashed line, Γ EG = α 1 .L and Γ CBH = α 2 by the circles.

Figure 9 :

 9 Figure 9: Evolution of the first normalised moment (left subplot) and the number-averaged chain length (right subplot) for the three different cases: EG (dashed line), CBH (continuous line) and EG-CBH (circles). The breakage/erosion frequencies are Γ EG = α 1 .L and Γ CBH = α 2 .

Figure 10 :

 10 Figure 10: Evolution of the first normalised moment (left subplot) and the number-averaged chain length (right subplot) for the three different cases: EG (dashed line), CBH (continuous line) and EG-CBH (circles). The breakage/erosion frequencies are Γ EG = α 1 .L and Γ CBH = α 2 .L.

Figure 11 :

 11 Figure 11: Comprison between the evolution of the two first normalised moments of the CLD and the cellobiose and glucose fractions when considering EG-CBH combined action without inhibition effect (continuous line) and with the inhibition effect (dashed line)

Γ

  Since the resolution of the population balance equation is computationaly intensive, the Direct Quadrature Method of Moments (DQMOM) is adopted. This resolution method based on the time tracking of a finite number of moments of the chain length distribution (CLD) has been validated against an analytical solution in the case of a specific breakage process. With a view to future comparison with transient experimental data, the Maximum Entropy based method is coupled to DQMOM for a simultaneous reconstruction of the full CLDs during the PBE resolution. The β-glucosidase activity is modelled as a Michaelis-Menten type kinetic and incorporated to the PBE model. The actions of EG and CBH activities are tested independently and in combination. Different breakage/erosion frequency laws are used. The numerical results are consistent with the analytical predictions. Global trends of the two first CLD moments (related to the molar concentration and to the mass of the substrate) are shown in the different cases. This guides the experimental determination of the breakage/erosion frequency laws. The EG/CBH synergistic action is discussed. This confirms the discriminatory character of the model since it proves numerically the fact that the erosion frequency can not be length dependent. The incorporation of the inhibition effect is investigated in the last section since the accumulation of cellobiose and glucose affects substantially the enzymes activity. The use of DQMOM resolution method coupled with the ME-based recontruction technique reduces the computational time drastically. It doesn't exceed few seconds in all the considered cases offering by the fact the possibility to perform a global optimization loop for the determination of the model parameters based on experimental data. Aknowledgments The authors would like to thank Toulouse White Biotechnology and Région Midi-Pyrénées for their financial support. Nomenclature A Square matrix of the linear system in DQMOM approach a i First time derivative of the weights of the quadrature nodes b i First time derivative of the product of the weighted abscissas β EG (L, λ), β CBH (L, λ) Daughter distribution function of EG an CBH respectively Monovariate Direct Quadrature Method of Moments (DQMOM)

  430

Figure B. 13 :

 13 Figure B.13: The time evolution of the global error function E(k) using a four nodes quadrature (eight first moments)

  the two differential equations in (D.4) leads to :

  t) = m 0 (0) m 1 (t) = m 1 (0).e -α2L C t (D.8)-Random breakage + Erosion processes :

  the two differential equations in (D.13) leads to :   m 0 (t) = m 0 (0).e α1t m 1 (t) = m 1 (0) (D.14) -Erosion process : Equation (D.2) is written as : ∂m k (t) ∂t = α 2 i (L i -L C ) k w i -m k (D.15)The evolution of the two first moments is : t) = m 0 (0)m 1 (t) = -α 2 L C m 0 (0)t + m 1 (0) (D.17)-Random breakage + Erosion processes :

•

  Case 3 : p 1 = 1 and p 2 = 0

Figures E. 14

 14 Figures E.14, E.15 and E.16 summarize the numerical results compared to those predicted analytically

Figure E. 14 :

 14 Figure E.14: Comparison between the analytical solution (continuous line) and the numerical results (circles) for the three different cases with Γ EG = α 1 L and Γ CBH = α 2 L.

Figure E. 15 :

 15 Figure E.15: Comparison between the analytical solution (continuous line) and the numerical results (circles) in the case of EG+CBH with Γ EG = α 1 and Γ CBH = α 2 .

Figure

  Figure E.16: Comparison between the analytical solution (continuous line) and the numerical results (circles) in the case of EG+CBH with Γ EG = α 1 L and Γ CBH = α 2 .

flocculation: Investigating the size dependence of aggregation, breakage and collision efficiency. Chem-

where A is a square matrix (2N, 2N ), x is the unknowns vector (2N : a 1 , . . . , a N , b 1 , . . . , b N ) and d is the term source vector (2N ). Thus, 2N moments (k = 0, . . . , 2N -1) are needed to solve the system. Generally, DQMOM requires at least a three nodes quadrature (N=3) for an accurate tracking of the time evolution of the moments [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF]). The initial abscissas L i (0) and weights w i (0) are obtained using the Product-Difference algorithm [START_REF] Gordon | Error Bounds in Equilibrium Statistical Mechanics[END_REF].

Appendix B: DQMOM validation in the case of breakage process

The DQMOM implementation was validated considering a normal law as initial CLD in the case where the PBE has analytical solution [START_REF] Ziff | The kinetics of cluster fragmentation and depolymerisation[END_REF]. In one hand, the time evolution of the CLD is known (see the analytical solution of the PBE for pure breakage process given in [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process[END_REF]), thus the moments m k (t) of the distribution can be easily derived (equation 2). In the other hand, DQMOM gives directly the time evolution of the CLD moments. The comparison between the two methods is given in the figure bellow:

Appendix C: Number density distribution reconstruction from the moments

The recovery of a distribution knowing only a finite number of its moments is known as the moment problem in mathematical analysis and arises in many scientific applications [START_REF] Gavriliadis | Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation[END_REF]. Different numerical methods have been developed for the reconstruction of distributions from the moments reviewed by [START_REF] John | Techniques for the reconstruction of a distribution from a finite number of its moments[END_REF]. For its simplicity and accuracy, the Maximum Entropy (ME)


Taous Aissani
email: taous.aissani@outlook.fr.tel

Idris Yahiaoui

Farouk Boudrahem

Lamia Yahia Cherif

Florence Fourcad

Abdeltif Amrane

Farida Aissani-Benissad

Sulfamethazine degradation by heterogeneous photocatalysis with ZnO immobilized on glass plate using heat attachment method and its impact on the biodegradability

Keywords: Photocatalytic process, Sulfamethazine, immobilized ZnO, Central composite design

Sulfamethazine degradation by heterogeneous photocatalysis with ZnO immobilized on a glass plate using the heat attachment method and its impact on the biodegradability

Taous Aissani, Idris Yahiaoui, Farouk Boudrahem, Lamia Yahia Cherif, Florence Fourcade, Abdeltif Amrane,

Introduction

Pharmaceuticals are being continuously released into the aquatic environment through different anthropogenic activities [START_REF] Babić | Photolytic and thin TiO 2 film assisted photocatalytic degradation of sulfamethazine in aqueous solution[END_REF]. In particular, antibiotics can cause ecotoxicological problems, because of their potential persistence in the environment [START_REF] Cesarino | Electrochemical oxidation of sulfamethazine on a glassy carbon electrode modified with graphene and gold nano particles[END_REF], since they are not readily biodegradable. Consequently, they are detected in surface water and groundwater; several pharmaceuticals have been detected in the μg L -1 range in wastewaters [START_REF] Mansour | Biodegradability Improvement of Sulfamethazine Solutions by Means of an electro-Fenton Process[END_REF]. For sulfamethazine (SMT), Concentrations in the range 15-328 ng.l -1 were detected in tropical waters in the Mekong Delta, Vietnam [START_REF] Managaki | Distribution of macrolines, sulfonamides, and trimethoprim in tropical waters: ubiquitous occurrence of veterinary antibiotics in the Mekong Delta[END_REF]. In China, comparable concentrations of SMT were observed in natural waters, e.g, 0.53-89.1 ng L -1 in surface water of the Yangtze estuary [START_REF] Yan | Antibiotics in the surface water of the Yangtze Estuary: occurrence, distribution and risk assessment[END_REF] and 2.05-623.27 ng L -1 in the Huangpu River, Shanghai [START_REF] Jiang | Occurrence, distribution and seasonal variation of antibiotics in the Huangpu River Shanghai, China[END_REF]. Such diffuse pollution is difficult to treat, and hence one solution would be to treat the considered pollution on site, to get low volumes highly concentrated [START_REF] Ledjeri | Combination of the Electro/Fe 3+/ peroxydisulfate (PDS) process with activated sludge culture for the degradation of sulfamethazine[END_REF]. For this purpose, several processes to degrade/remove antibiotics have been studied, including conventional techniques (biological processes, filtration, coagulation, flocculation, and sedimentation), advanced oxidation processes (AOPs), adsorption, membrane processes, and combined methods [START_REF] Dinga | Simultaneous removal and degradation characteristics of sulfonamide, tetracycline, and quinolone antibiotics by laccase-mediated oxidation coupled with soil adsorption[END_REF]. The advanced oxidation processes (AOP S ) appears as interesting tools in comparison with other technique [START_REF] Elmolla | Photocatalytic degradation of amoxicillin, ampicillin and cloxacillin antibiotics in aqueous solution using UV/TiO 2 and UV/H 2 O 2 /TiO 2 photocatalysis[END_REF]. They are based on the generation of very reactive species such as hydroxyl radicals, that have been proposed to oxidize quickly and not selectively a broad range of organic pollutants [START_REF] Sano | Degradation of C 2 H 2 with modified-TiO 2 photocatalysts under visible light irradiation[END_REF][START_REF] Kitano | Recent developments in titanium oxide-based photocatalysts[END_REF]. Among AOPs, photocatalysis using viz., TiO 2 and ZnO, appears effective for the pharmaceutical wastewater treatment. Photocatalytic conversion using ZnO showed better degradation efficiency than TiO 2 for removing organic compounds from water [START_REF] Movahedi | Photodegradation of congo red in aqueous solution on ZnO as an alternative Catalyst to TiO 2[END_REF][START_REF] Sapkal | Photoelectrocatalytic decolorization and degradation of textile effluent using ZnO thin films[END_REF][START_REF] Waghmode | Sequential photocatalysis and biological treatment for the enhanced degradation of the persistent azo dye methyl red[END_REF].

The technique is more effective as compared to other AOPs because semiconductors are inexpensive and can easily mineralize various organic compounds [START_REF] Gao | Preparation of porous TiO 2 /Ag heterostructure films with enhanced photocatalytic activity[END_REF]. Among different conventional catalysts, ZnO nanoparticles have been widely used to treat wastewater because of wide band gap, large volume-area ratio, low cost, large initial rate of activities and UV A c c e p t e d M a n u s c r i p t adsorption potential [START_REF] Soltani | Photocatalytic process by immobilized carbon black/ZnO nanocomposite for dye removal from aqueous medium: Optimization by response surface methodology[END_REF]. However, the elimination of ZnO suspended in the treated water causes running costs and induces secondary pollution, which greatly restricts its practical applications. Consequently, several methods have been implemented to immobilize ZnO nanoparticles on various substrates, such as activated carbon [START_REF] Aber | Application of immobilized ZnO nanoparticles for the photocatalytic regeneration of ultrasound pretreated-granular activated carbon[END_REF], carbon black (CB)/ZnO nanocomposite film [START_REF] Soltani | Photocatalytic process by immobilized carbon black/ZnO nanocomposite for dye removal from aqueous medium: Optimization by response surface methodology[END_REF], polystyrene pellets [START_REF] Vaiano | Visible light driven oxidation of arsenite to arsenate in aqueous solution using Cu-doped ZnO supported on polystyrene pellets[END_REF], glass substrates [START_REF] Boughelout | Photocatalysis of rhodamine B and methyl orange degradation under solar light on ZnO and Cu 2 O thin films[END_REF], L-Cysteine capped zinc oxide nanoparticles were immobilized onto a biodegradable poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) [START_REF] Selvin | Photocatalytic degradation of rhodamine B using cysteine capped ZnO/P(3HB-co-3HHx) fiber under UV and visible light irradiation[END_REF]. Maučec et al. immobilized ZnO nanoparticles in mesoporous silica supports [START_REF] Maučec | Titania versus zinc oxide nanoparticles on mesoporous silica supports as photocatalysts for removal of dyes from wastewater at neutral pH[END_REF]. These techniques must meet the following conditions: i) Strong adherence between the catalysts and the substrate, ii) The Crystal structure of the catalysts must be kept during the preparation and the immobilization process, iii) low cost of the substrate and the used method [START_REF] Han | The effect of solvent in the sol-gel synthesis of visible light-activated, sulfur-doped TiO 2 nanostructured porous films for water treatment[END_REF][START_REF] Tseng | A review of photocatalysts Prepared by Sol-Gel Method for VOCs Removal[END_REF][START_REF] Kumar | Sol-gel TiO 2 in self-organization process: growth, ripening and sintering[END_REF][START_REF] Li | Synergetic effect between adsorption and photodegradation on nanostructured TiO 2 /activated carbon fiber felt porous composites for toluene removal[END_REF][START_REF] Belver | Zr-doped TiO 2 supported on delaminated clay materials for solar photocatalytic treatment of emerging pollutants[END_REF][START_REF] Vicent | Preparation of high solids content nano-titania suspensions to obtain spray-dried nanostructured powders for atmospheric plasma spraying[END_REF].

Physicochemical methods such as flocculation-coagulation, adsorption, have been widely investigated. However, The removal of dyes and recalcitrant pollutants from wastewater using these methods was not as effective as expected; in addition, most of these methods are nondestructive, since they only consist in a transfer of organic compounds from water to another phase, thus causing secondary pollution [START_REF] Cherif | Heat Attachment Method for the Immobilization of TiO 2 on Glass Plates: Application to Photodegradation of Basic Yellow Dye and Optimization of Operating Parameters, Using Response Surface Methodology[END_REF][START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF][START_REF] Yahiaoui | The feasibility of combining an electrochemical treatment on a carbon felt electrode and a biological treatment for the degradation of tetracycline and tylosin-application of the experimental design methodology[END_REF][START_REF] Madi | Basic red dye removal by coupling electrocoagulation process with biological treatment[END_REF][START_REF] Dibene | Central composite design applied to paracetamol degradation by heat-activated peroxydisulfate oxidation process and its relevance as a pretreatment prior to a biological treatment[END_REF]. For the degradation of persistent organic compounds contained in wastewater, several studies recommend integrated processes, especially the coupling of AOPs and biological treatment [START_REF] Cherif | Heat Attachment Method for the Immobilization of TiO 2 on Glass Plates: Application to Photodegradation of Basic Yellow Dye and Optimization of Operating Parameters, Using Response Surface Methodology[END_REF][START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF][START_REF] Yahiaoui | The feasibility of combining an electrochemical treatment on a carbon felt electrode and a biological treatment for the degradation of tetracycline and tylosin-application of the experimental design methodology[END_REF][START_REF] Madi | Basic red dye removal by coupling electrocoagulation process with biological treatment[END_REF][START_REF] Dibene | Central composite design applied to paracetamol degradation by heat-activated peroxydisulfate oxidation process and its relevance as a pretreatment prior to a biological treatment[END_REF][START_REF] Yahiaoui | Removal of a mixture tetracycline-tylosin from water based on anodic oxidation on a glassy carbon electrode coupled to activated sludge[END_REF][START_REF] Ferrag-Siagh | Tetracycline degradation and mineralization by the coupling of an electro-Fenton pretreatment and a biological process[END_REF]. The main objectives of this study are: i) to use a heat attachment method to immobilize ZnO on glass plate (9 cm × 35 cm). This material was of soda-lime type, soda-lime glass is inexpensive and chemically stable, such immobilization method remains scarcely investigated in this research area; ii) to use the experimental design methodology to investigate the influence of the main experimental parameters (flow rate (Q v ), the pH and the initial SMT concentration) on the efficiency of the photocatalytic degradation of SMT with ZnO immobilized on glass plate in batch reactor, iii) to examine the feasibility of combining UV/ZnO process with biological A c c e p t e d M a n u s c r i p t treatment to eliminate the SMT from aqueous solutions, such combination has not been extensively examined in the available literature.

Materials and methods

Chemicals and photocatalytic system

The SMT (C 12 H 14 N 4 O 2 , 99% purity) frοm Alfa Aesar, sulfuric acid (96 % purity) and sοdium hydrοxide (97 % purity) bοth frοm Biοchem Chemοpharma. Zinc οxide (99 % purity) frοm Sigma Aldrich and hydrofluoric acid (HF) (40%) were purchased from prolabo (Paris, France). The Experimental setup is described in the supplementary material (See Fig. 1). It is composed of a Vessel (1) fitted with a Pyrex glass jacket (2) connected to a thermostated bath (3) in order to maintain the solution at a temperature of 25°C and a stirrer in stainless steel [START_REF] Managaki | Distribution of macrolines, sulfonamides, and trimethoprim in tropical waters: ubiquitous occurrence of veterinary antibiotics in the Mekong Delta[END_REF] was used for the homogenization. A glass plate loaded with ZnO (5), one UV lamp (6) (30 W,UVA, λ max = 360 nm, manufactured by Philips, Netherlands), a centrifugal pump [START_REF] Ledjeri | Combination of the Electro/Fe 3+/ peroxydisulfate (PDS) process with activated sludge culture for the degradation of sulfamethazine[END_REF] which allows to circulate the solution through the plate, and an air pump [START_REF] Dinga | Simultaneous removal and degradation characteristics of sulfonamide, tetracycline, and quinolone antibiotics by laccase-mediated oxidation coupled with soil adsorption[END_REF] was used to provide the solution with dissolved oxygen required for the experiment. 1 L of SMT solution was placed into the reactor; its pH was adjusted by adding concentrated solutions of H 2 SO 4 or NaOH.

Experiments were monitored for 5 h at a temperature of 25°C. Samples of 5 mL were withdrawn at regular time intervals for the measurement of the SMT residual concentrations.

Immobilization of ZnO on glass plates

ZnO nanoparticles were fixed on glass plate (9 cm×35 cm) by heat attachment method; the glass surface was pretreated with HF solution (0.1 M) and then washed in a solution of NaOH (0.01M) in order to increase the number of OH groups and ensure a good fixing of the ZnO on the glass plate. Twenty millilitres (20 mL) of suspension containing 4 g/L ZnO were prepared using ethanol as solvent and the suspension was stirred magnetically for 2 h times. The suspension was poured into a glass plate and then placed in a drying oven at 30°C. After A c c e p t e d M a n u s c r i p t drying, the plates were calcined at 500°C for 1h and the surface was then washed with distilled water for the removal of weakly attached ZnO [START_REF] Cherif | Heat Attachment Method for the Immobilization of TiO 2 on Glass Plates: Application to Photodegradation of Basic Yellow Dye and Optimization of Operating Parameters, Using Response Surface Methodology[END_REF]. Under the effect of heat the particles became welded together, forming the cohesion [START_REF] Cherif | Heat Attachment Method for the Immobilization of TiO 2 on Glass Plates: Application to Photodegradation of Basic Yellow Dye and Optimization of Operating Parameters, Using Response Surface Methodology[END_REF].

Characterization

The X-ray diffraction (XRD) patterns of ZnO nanoparticles powder, glass plate and ZnO immobilized on glass plate were obtained using a diffractometer panalytical type MPD/system vertical θ/θ, using radiation source (CuKα = 0.15406 nm) operating at 40 kV and 30 mA. The XRD diffraction patterns were collected over a 2θ range located between 10° and 80° using an incremental step size of 0.02° with 6 s of acquisition time per step. In addition, the band gap was estimated by means of UV-vis diffuse reflectance spectra (DRS) and by UV-Vis Spectroscopy (Shimadzu, UV-2600). Fourier transform infrared (FTIR) transmission spectra of ZnO nanoparticles powder and ZnO immobilized on glass plate were obtained on an IR Affinity-1 Shimadzu FTIR spectrometer. FTIR analysis was done by scraping the ZnO deposit on the glass plate then KBr Pellet method was applied to form the FTIR spectra. All FTIR spectra were obtained using an FTIR spectrometer over the wavenumber range of 400 and 4000 cm -1 .

Sulfamethazine analysis

SMT was analyzed by High Performance Liquid Chromatography (HPLC ACC 3000 HPLC).

The HPLC was equipped with a standard degasser, an autosampler and a pump (Model LPG 3400 SD), as well as a detector with visible ultraviolet ray (UV/ Vis detector VWD 3400 RS).

The separation was achieved on a Thermo Fisher scientific (Germany) C18 (5 mm; 4.6 ×150 mm) reversed-phase column. The mobile phase consisted of acetonitrile/ultrapure water (35/65 v/v) with a flow rate of 0.5 mL min -1 and the detection of sulfamethazine was carried out at 268 nm.

COD and BOD 5 measurements

All COD and BOD 5 measurements were duplicated. Chemical oxygen demand (COD) was measured by means of Kits Nanocolor® 15-160 mg/L. COD according to DIN ISO 15705 at A c c e p t e d M a n u s c r i p t 148°C. The amount of oxygen required for the oxidation of the organic and mineral matter at 148 °C for 2 h was quantified after oxidation with K 2 Cr 2 O 7 at acidic pH and under heating.

The BOD 5 measurements were carried out in Oxitop IS6 (WTW, Alès, France). Activated sludge from a wastewater treatment plant (Sidi Ali Labhar, Béjaia-Algeria) was used to inoculate the flasks; the initial microbial concentration was 0.5 g L - 1 . More details about BOD 5 measurement with Oxitop IS6 (WTW, Alès, France) can be found in a previous study [START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF].

Zn 2+ measurements

The concentrations of Zn 2+ ions was determined by using SCHIMADZU AA6500 atomic absorption spectrophotometer (SAA) connected to the microcomputer; the analysis was conducted with an oxidizing air-acetylene at 213.8 nm [START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF].

Response Surface Methodology (RSM)

Statistically designed experiments based on a RSM were conducted to study the combined effects and the optimization of the flow rate (Q V ), the pH, and the initial SMT concentration.

The combinations of the model performance was described using a second-order polynomial equation (Eq. 1) based on a central composite design (CCD). More details about the construction of the CCD model can be found in previous papers [START_REF] Yahiaoui | Optimization of silver cementation yield in fixed bed reactor using factorial design and central composite design[END_REF][START_REF] Yahiaoui | Response surface methodology for the optimization of the electrochemical degradation of phenol on Pb/PbO 2 electrode[END_REF][START_REF] Yahiaoui | Full factorial design applied to a biosorption of lead (II) ions from aqueous solution using a brewer's yeast (Saccharomyces cerevisiae)[END_REF].

ŷ = 𝑏𝑏 0 + � 𝑏𝑏 𝑖𝑖 𝑥𝑥 𝑖𝑖 3 𝑖𝑖=1 + � 𝑏𝑏 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑖𝑖 2 3 𝑖𝑖=1 + � � 𝑏𝑏 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑖𝑖 𝑥𝑥 𝑖𝑖 3 𝑖𝑖=𝑖𝑖+1 3 𝑖𝑖=1 (1)
Where:

• Predicted yield (ŷ) • the coefficients of the response model (b 1 , b 2 , ...,b n )
• linear terms, corresponding to the variables (x 1 , x 2 , ..., x n )

• squared terms, corresponding to the variables (x 2 1 , x 2 2 , ...,x 2 n )

A c c e p t e d M a n u s c r i p t

• first-order interaction terms for each paired combination (x 1 x 2 , x 1 x 3 ,...,x n-1 x n )

The response was expressed as the percent of yield abatement of SMT (Eq.2):

𝑦𝑦(%) = [𝑆𝑆𝑆𝑆𝑆𝑆] 0-[𝑆𝑆𝑆𝑆𝑆𝑆] 𝑡𝑡 [𝑆𝑆𝑆𝑆𝑆𝑆] 0 100 (2)
Where [SMT] 0 and [SMT] t were SMT values of sample at the initial time 0 and at a given time t (mg L -1 ), respectively.

The original values of each factor and their corresponding levels are shown in the supplementary material (Table 1).

Results and discussion

Characterization studies

• ZnO immobilized on glass plate was characterized by X-ray diffraction (XRD), acccording to the figure 2 given in supplementary material the following conclusion can be drawn: The results shown in Fig. 2 indicate that the glass plate without ZnO nanoparticles presented some amorphous peaks in the XRD curve. The XRD patterns of ZnO nanoparticles powder shows that no diffraction peaks of other impurities were detected.

The ZnO nanoparticles diffraction peak appeared in the XRD pattern of ZnO immobilized on glass plate; indicating that the process of dispersion in glass plate did not have an influence on the crystal form of the commercial ZnO nanoparticles powder corresponding to a wurtzite phase, characteristic of the commercial ZnO nanoparticles powder [START_REF] Kosera | Triclosan degradation by heterogeneous photocatalysis using ZnO immobilized in biopolymer as catalyst[END_REF]. The displayed in the figure 4 given in supplementary material. In the spectrum (a), the band at 500 cm -1 was assigned to the Zn-O. In addition, a weak band can also be perceived in the range of 3550-3450 cm -1 , attributed to the stretching of the O-H, which could be localized on the surface of the catalyst. It should be observed that the characteristic bands of ZnO were maintained in the spectra of ZnO immobilized in the glass plate.

.

Effect of the amount of catalyst

The effect of the amount of ZnO immobilized on glass plate was investigated for 50 and 100 mg/L of SMT. The amount of ZnO was varied in the range from 0.08 to 0.24 g, corresponding to one to three ZnO deposits. As can be seen in figure 1, the increase of the amount of ZnO immobilized on glass plate from 0.08 to 0.16g promoted the removal efficiency of SMT from 76 to 83 % and from 42 to 65 % after 300 min of reaction time for 50 and 100mg/L of SMT respectively. While, increasing the amount of ZnO from 0.16 to 0.24g did not increase significantly the removal efficiency of SMT, namely 2% regardless of the initial concentration of SMT. Consequently, two layers of ZnO (0.16 g) was selected as the optimal value of the amount of catalyst.

Regeneration of the photocatalyst

To study the stability of ZnO immobilized on the glass plate, at the end of a cycle, the glass plate was washed with distilled water in order to remove the SMT and by-products formed during the previous experiment and adsorbed on the catalyst. The degradation efficiency is shown in figure 2, showing a decrease in the removal efficiency of SMT from 84.5 to only 70 % after six cycles. It was probably due to the accumulation of by-products on the surface of catalyst.

Fig. 2 3.5. Modeling and optimization

Elaboration and analysis of the model

The results of the second-order models are presented in the supplementary material (Table 2).

The coefficients of polynomial model were calculated by a standard least-square regression method using "Excel" software. Three tests were required to evaluate the adequacy of the model: the Student's t test, which was used to determine the significant coefficients, discarding the insignificant ones, the R 2 test and the Fisher test. All these statistical tests were previously described in details [START_REF] Yahiaoui | Optimization of silver cementation yield in fixed bed reactor using factorial design and central composite design[END_REF][START_REF] Yahiaoui | Response surface methodology for the optimization of the electrochemical degradation of phenol on Pb/PbO 2 electrode[END_REF][START_REF] Yahiaoui | Full factorial design applied to a biosorption of lead (II) ions from aqueous solution using a brewer's yeast (Saccharomyces cerevisiae)[END_REF].

According to these results, the coefficients of the polynomial model can be expressed by the following equation: (3)

According to the regression equation (Eq.3), the initial concentration of SMT (x 2 ) had the strongest effect on the response (-11.6) and the negative sign of the b 2 coefficient suggests that the degradation of SMT decreased for increasing initial SMT concentration. Kinetic study confirmed this tendency as can be seen in figure 3. According to the figure 3, The results obtained indicate that the degradation efficiency of SMT was inversely proportional to its initial concentration; this may be attributed to a competitive consumption of the hydroxyl radicals between the SMT molecules and the generated by-products, or the decrease of generated h + holes and/or • OH radicals at the catalyst surface since the active sites are occupied by the pollutant, or by a decrease of the intensity of the radiation absorbed by the catalyst due to the molecules of the pollutant [START_REF] Mansour | Biodegradability Improvement of Sulfamethazine Solutions by Means of an electro-Fenton Process[END_REF][START_REF] Dinga | Simultaneous removal and degradation characteristics of sulfonamide, tetracycline, and quinolone antibiotics by laccase-mediated oxidation coupled with soil adsorption[END_REF][START_REF] Khenniche | Photocatalytic degradation of bezacryl yellow in batch reactors-feasibility of the combination of photocatalysis and a biological treatment[END_REF][START_REF] Andronic | Photocatalytic activity of cadmium doped TiO 2 films for photocatalytic degradation of dyes[END_REF]. The same tendency was previously reported by Kheniche et al. [START_REF] Khenniche | Photocatalytic degradation of bezacryl yellow in batch reactors-feasibility of the combination of photocatalysis and a biological treatment[END_REF] and Andronic et al. [START_REF] Andronic | Photocatalytic activity of cadmium doped TiO 2 films for photocatalytic degradation of dyes[END_REF].

Fig. 3

The second significant factor was the flow rate (x 3 ) with positive effect on the yield of SMT degradation (+ 2.06). In order to confirm this effect, kinetic experiments were conducted (Figure 4) at different flow rates (0.15 -0.68 L/min). The results showed that when the flow rates was increased from 0.15 to 0.52 L/min, the SMT degradation over 300 min of reaction time increased from 74 % to 85%. Contrarily, beyond 0.52 L/min the yield of the SMT degradation decreased from 85% to 81% for 0.68 L/min flow rate. To better explain the relationship between the degradation rate and the flow rate, it was considered necessary to A c c e p t e d M a n u s c r i p t determine the value of the film thickness of the liquid which can be calculated by the following equation [START_REF] Lou | Study of a photocatalytic process for removal of antibiotics from wastewater in a falling film photoreactor: Scavenger study and process intensification feasibility[END_REF]:

𝑇𝑇ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑖𝑖𝑓𝑓𝑓𝑓 = � Γ 2 𝑔𝑔 � 1/3 (4)
Where: Γ = Q 𝐿𝐿 is the flow rate per unit length (m 3 /s m), L is the length of the glass plate, Q is the flow rate (m 3 /s) and g the gravity (m/s²).

According to the table 1, the thickness of the liquid film increased with increasing of the flow rate, The decrease in the rate of degradation beyond 0.52 L/min flow rate may be attributed to the limitation of the UV-light penetration on account of the rise of the liquid thickness, in agreement with other reports, Yahia cherif et al [START_REF] Cherif | Heat Attachment Method for the Immobilization of TiO 2 on Glass Plates: Application to Photodegradation of Basic Yellow Dye and Optimization of Operating Parameters, Using Response Surface Methodology[END_REF] and Zayani et al [START_REF] Zayani | Solar photocatalytic degradation of commercial textile azo dyes: Performance of pilot plant scale thin film fixed-bed reactor[END_REF], and to the decrease of the residence time in the reactor. The decrease in the rate of degradation of SMT for decreasing flow rates (Q v = 0.15) can be consequently attributed to the fact that the residence time rose, favoring the surface reactions; this may be also due to the degradation of byproducts adsorbed on the surface of the catalyst.

Fig. 4

Table 1 The solution pH did not have significant effect on the yield of SMT degradation, which was confirmed by the time-courses (Figure 5a). On the other hand, pH time-courses were also displayed in figure 5b. The pH increased until being close to pH 5 for acidic initial pH = 3 and decreased from 9 to 7 in the alkaline medium. This effect was attributed to the ZnO dissolution and the formation of Zn 2+ [START_REF] Daneshvar | Photocatalytic degradation of azo dye acid red 14 in water on ZnO as an alternative catalyst to TiO 2[END_REF][START_REF] Zhang | Photocorrosion suppression of ZnO nanoparticles via hybridization with graphite-like carbon and enhanced photocatalytic activity[END_REF], which can be attributed to the oxidation of ZnO by h + VB according to the equation 5. Daneshvar et al [START_REF] Daneshvar | Photocatalytic degradation of azo dye acid red 14 in water on ZnO as an alternative catalyst to TiO 2[END_REF] reported that the photocorrosion of ZnO is complete at pH lower than 4, while no photocorrosion of ZnO takes A c c e p t e d M a n u s c r i p t place at pH higher than 10. The ZnO photocorrosion in an acidic medium and alkaline medium were given by Eqs. ( 6) and [START_REF] Ledjeri | Combination of the Electro/Fe 3+/ peroxydisulfate (PDS) process with activated sludge culture for the degradation of sulfamethazine[END_REF]. In this study, the concentration of Zn 2+ ions were 17.9, 6.5, and 4.1 mg/ L for pH solution values of 3, 6, and 9 respectively.

𝑍𝑍𝑖𝑖𝑍𝑍 + 2ℎ 𝐵𝐵 𝑉𝑉 + → 𝑍𝑍𝑖𝑖 2+ + 𝑍𝑍 • (5) 𝑍𝑍𝑖𝑖𝑍𝑍 + 2𝐻𝐻 + → 𝑍𝑍𝑖𝑖 2+ + 𝐻𝐻 2 𝑍𝑍 (6) 𝑍𝑍𝑖𝑖𝑍𝑍 + 𝐻𝐻 2 𝑍𝑍 + 2𝑍𝑍𝐻𝐻 -→ 𝑍𝑍𝑖𝑖 2+ + 𝐻𝐻 2 𝑍𝑍 (7)
Fig. 5 The model also underlined that the strongest interaction was between the solution pH (x 1 ) and the flow rate (x 3 ); this interaction is illustrated in the supplementary material (see the figure 6).

Kinetic studies

In this study, the First-order, Second-order and Langmuir-Hinshelwood (L-H) (see table 3 in in the supplementary material) were tested in order to determine the kinetic model of SMT degradation. A nonlinear method was used in order to optimize the different constants of the models (see supplementary materiel) by minimizing the error function using the solver add-in with Microsoft's spreadsheet, Microsoft Excel. The error function employed (Eq. ( 8)) was as follows (Aissani et al. [START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF] and Boudrahem et al. [START_REF] Boudrahem | Removal of basic yellow dye from aqueous solutions by sorption onto reed as an adsorbent[END_REF]).

𝐷𝐷(%) = �(� 𝐶𝐶 𝐶𝐶 0 � 𝐸𝐸𝐸𝐸𝐸𝐸 - 𝐸𝐸 𝑖𝑖=1 � 𝐶𝐶 𝐶𝐶 0 � 𝐶𝐶𝐶𝐶𝐶𝐶 ) 2 (8)
Where � The results given in the figure 6 and the table 2 show that for low concentrations (10 and 50 mg/L), the kinetic was described by a pseudo-first order kinetic model. However, for 100 mg/L of SMT the kinetic was described by a pseudo-second order kinetic model. The absence of adsorption in dark of SMT onto ZnO immobilized on glass plate and the kinetic degradation of SMT was non described by Langmuir-Hinshelwood model confirm that the photocatalytic degradation of SMT does not take place on the surface of the catalyst but in the close vicinity of the catalyst surface [START_REF] Aissani | The combination of photocatalysis process (UV/ TiO 2 (P25) and UV/ZnO) with activated sludge culture for the degradation of sulfamethazine[END_REF][START_REF] Sakthivel | Solar photocatalytic degradation of azo dye: comparison of photocatalytic efficiency of ZnO and TiO 2[END_REF].

Optimization of the operating variables

The model equation (Eq. 3) obtained by the CCD was used to find the optimal values of the operating parameters in order to obtain the highest degradation rate of SMT. Response surface and contour plot were drawn using STATISTICA software (Figure 7 in supplementary material), which allowed to deduce the following optimal conditions for the yield of SMT degradation, namely pH=6, Q v =0.56 L/min and [SMT] 0 = 11 mg/L. Under these optimal conditions, the SMT degradation yield experimentally obtained was y exp = 98%, which should be related to that given by the model ŷ = 96 %. The relative error between y exp and ŷ was below 2.8 %, validating the model, and hence the developed model describes accurately experimental data.

Biodegradability test

The biodegradability test was realized on solutions irradiated under the optimal values obtained by the CCD: pH=6, Q v =0.56 L/min and [SMT] 0 = 11 mg/L. The biodegradability test was estimated by means of the BOD 5 /COD ratio of the solutions irradiated during 5h and 8h. According to the table 3, it can be drawn that:

A c c e p t e d M a n u s c r i p t

• The removal efficiency of SMT increased from 0 to 98 and 100 % when the irradiation time was increased from 0 to 5and 8h respectively.

• The COD abatement increased from 0 to 75% after 5h irradiation time; while an increase of the irradiation time from 5h to 8h did not show a significant impact on the COD abatement.

• The BOD 5 /COD ratio increased from 0 initially to 0.20 after 5h of irradiation, showing that the biodegradability was enhanced even if the limit of biodegradability (0.4) was not achieved. However, further increase of the irradiation time, from 5 to 8h, induced a decrease of the BOD 5 /COD ratio from 0.20 to 0, showing that the by-products formed after 8h of irradiation are not biodegradable. This decrease of the BOD 5 /COD ratio may be also attributed to the adsorption of Zn 2+ ions by activated sludge, which was reported to be the main mechanism of toxicity [START_REF] Liu | Effect of ZnO particles on activated sludge: Role of particle dissolution[END_REF], while Wang et al [START_REF] Wang | Modeling heavy metal uptake by sludge particulates in the presence of dissolved organic matter[END_REF] reported that Zn 2+ ion is highly adsorbable on activated sludge. Indeed, the concentrations of Zn 2+ ions after 8h of reaction time was estimated to 6 mg/L, while Lui et al [START_REF] Liu | Effect of ZnO particles on activated sludge: Role of particle dissolution[END_REF] reported that the IC 50 values of Zn 2+ ions on biodegradation was 1.3 mg-Zn/ L.

Table 3 4. Conclusion

A response surface methodology based on CCD was successfully used for the study and the optimization of the removal efficiency of SMT by using UV/ZnO immobilized on glass plate.

The application of the CCD with a second-order polynomial regression model satisfactorily interpreted the experimental data. It was shown that increasing the initial SMT concentration had negative effect on the removal efficiency. In addition, when the flow rates was increased A c c e p t e d M a n u s c r i p t from 0.15 to 0.52 L/min, the SMT degradation increased from 74 % to 85%; while beyond 0.52 L/min flow rate, the yield of SMT degradation decreased from 85% to 81% for 0.68 L/min. The solution pH did not show significant impact on the removal efficiency of SMT in the studied interval. The use of CCD allowed the identification of the most significant interaction between parameters, shown to be between the pH and the flow rate. Analysis of the response surfaces and contour plots clearly indicated that the optimal values of solution pH, flow rate and initial SMT concentration were: 6, 0.56 L/min and 11 mg/L respectively.

Under these optimal conditions, the SMT removal efficiency was 96 % and the BOD 5 /COD ratio was increased from 0 to 0.20 after 5h of irradiation time.

figure 3

 3 figure 3 in supplementary material shows the band gap οf the ZnΟ pοwder (a) and ZnΟ

A c c e

  p t e d M a n u s c r i p t • FTIR spectra of ZnO powder (a) and ZnO immobilized on glass plate (b) are

Fig. 1 3. 3 .

 13 Fig.1

A c c e p t e d M a n u

  s c r i p t 𝑦𝑦 = 74.59 -11.6𝑥𝑥 2 -2.73𝑥𝑥 1 𝑥𝑥 3 + 2.06𝑥𝑥 3 2

  and calculated data, respectively, and P is the number of experimental data.

A c c e p t e d M a n u s c r i p t

Fig. 6

 6 Fig.6

Figure 1 :Figure 2 :

 12 Figure captions

Figure 3 :

 3 Figure 3: Effect of initial concentration of sulfamethazine (SMT) on the photocatalytic degradation process. Conditions: 0.16g of ZnO, pH = 6, Q v =0.52 L/min, and T = 25°C.

Figure 4 :

 4 Figure 4: Effect of flow rate (Q v ) on photodegradation efficiency of SMT. Conditions: [SMT] 0 = 50 mg/L, T=25°C, time = 300 min and 0.16g of ZnO immobilized on glass plate .

Figure 5 :

 5 Figure 5: Effect of pH solution on photodegradation efficiency of SMT. Conditions: [SMT] 0 = 50 mg/L, Q v =0.52 L/min, T = 25°C and 0.16 g of ZnO immobilized on glass plate.

Figure 6 :

 6 Figure 6: Experimental data, first-order kinetic model, second-order kinetic model, and Langmuire-Hinshelwood model for photocatalytic degradation of SMT by ZnO Immobilized on glass plate. Experimental conditions: 0.16g of ZnO, pH = 6, Q V = 0.52L/min, and T = 25°C A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t Table 3 .

 3 Figures

  of SMT (%) Number of deposed layers of ZnO [SMT]=50 mg/L [SMT]=100 mg/L A c c e p t e d M a n u s c r i p t

Fig. 2 84

 2 Fig. 2

  A c c e p t e d M a n u s c r i p t

  Fig.3

  Fig.5

Fig. 6 0

 6 Fig.6

Table 1 .

 1 The values of film thickness of the liquid at the different flow rates tested

	Tables
	Q v (L/min)	0.15	0.371	0.521	0.681
	Film thickness(mm)	0.19	0.34	0.43	0.51

Table 2 .

 2 Kinetic constants predicted by non-linear method for the photocatalytic degradation of SMT by ZnO immobilized on glass plate.

		First order model		Second-order model	L-H Kinetic model
	[SMT] 0 (mg/l)	k 1 (min -1 )	D (%)	k 2 (l/mg min)	D (%)	k(min -1 )	K	D (%)
	10	0.012 ±1.8410 -3	0.085	0.002 ±3.41 10 -4	5.2	0.34 ±8.35 10 -2	0.067 ±3.05 10 -2	0.023
	50	0.006 ±5.910 -4	0.11	0.0001 ±1.36 10 -5	2.5	0.26 ±8.9 10 -4	0.64 ± 7.9 10 -3	0.34
	100	0.005 ±1.110 -3	0.2	5.7.10 -5 ±1.03 10 -5	0.3	6.18 ±1.4 10 -2	0.00085 ± 2.68 10 -5	0.56