Guillaume Silhol

GOVERNING CATHOLIC RELIGIOUS EDUCATION IN ITALIAN STATE SCHOOLS: BETWEEN THE REVISION OF THE CONCORDAT AND SOCIAL MOVEMENTS, 1974-1984 SUMMARY

Keywords:

This article focuses on the redefinition of Catholic Religious Education in Italian State schools, from compulsory religious instruction into a non-compulsory discipline of "religious culture", by analyzing how the issue is framed and negotiated by political, religious and educational actors between 1974 and 1984. The negotiations between governmental and Church representatives in the revision of the Concordat led to attempts at a compromise on Religious Education, its regime and its guarantees for students' choices. However, social movements and school reforms forced various actors and institutions to reframe it in nonconfessional, pedagogical and professional terms in public arenas. "Religious culture", as a category promoted by teachers and intellectuals, became both a social problem and the main justification for the ownership of the Catholic Church over the problem.

supreme constitutional principle: in particular two sentences of the Italian Constitutional Court respectively on the non-compulsory character of the alternative hour to IRC and on the possibility for opting-out students to leave the school 3 . An important part of the research focuses on the legal consequences of the revision of the Concordat, in particular the guarantees on religious freedom and the use of school premises for electives and other types of religious instruction [START_REF] Fiorita | Scuola pubblica e religioni[END_REF] . A few studies have been conducted in disciplines other than ecclesiastical law, in particular in political history, centered on the production of the Agreement or on the bilateral commission [START_REF] Giovagnoli | Guido Gonella tra Chiesa e Stato (1968-1982)[END_REF] . In the case of IRC in State schools, most research is centered on students, their skills and the effects of pedagogy [START_REF] Malizia | L'evoluzione dell'insegnamento religioso in Italia alla luce delle ricerche sul campo[END_REF] . Some attention has been granted by historians of education to the reforms and the curricular changes of IRC [START_REF] Butturini | La religione a scuola[END_REF] . Yet, these studies have mostly been conducted separately from the Concordat as an international legal and political process, and from the politicization of IRC. This article proposes a complementary perspective with a socio-historical analysis of the requalification of IRC, in the research field of religion in public policies [START_REF] Reeh | Danish State Policy on the Teaching of Religion from 1900 to 2007[END_REF] . Neither a mere semantic change nor a onesided secularization, the conflictual redefinition of the former Religious Education (insegnamento religioso, hereafter IR), from a compulsory weekly hour of pastoral lessons to a non-compulsory subject of "religious culture", with the explicit "Catholic" adjective in art. 9.2 of the 1984 Agreement on the revision of the Concordat, is but the most visible aspect of a case study of the governance of religion [START_REF] Pazzaglia | L'insegnamento della religione nei dibattiti culturali e pedagogicali dall'ultimo governo Moro alla revisione concordataria[END_REF] .

From the perspective of political sociology, this research has been conducted on archives from political parties and teachers' unions over the period 1974-1991, completed with a survey of main Italian newspaper articles on these topics from digital databases and with 25 semi-structured interviews with actors involved in "religious culture" social movements. Social movements are defined as formal or informal networks of actors that use various types of resources (material, intellectual, political…) and frames of perception to achieve specific goals. The aim of the article is not to draw a history of Church-State relations, but to analyze the social construction and stabilization of definitions of "religious culture" as a category of public policy. It follows the hypothesis that these competing movements produce constraints and opportunities for Catholic institutions and government officials to endorse "religious culture" as a paradoxically consensual, functionally ambiguous category, to justify the preservation of IR in State schools with its own group of teachers. This diachronic and critical perspective also uses concepts from the study of social problems. A social problem is defined as a situation publicly and collectively deemed detrimental, requiring a political solution. It embodies one peculiar definition of a situation amongst others, including competing definitions as a "non-problematic" case in public arenas and behind closed doors, as in the example of the historical redefinition of drunk driving, from an erratic individual behavior to a public threat according to physicians and lawyers in postwar America, studied by Joseph Gusfield [START_REF] Gusfield | The Culture of Public Problems: Drinking-driving and the Symbolic Order[END_REF] . In this case, "religious culture" in Italy has acquired rival meanings: a necessary subject and professional skill that should be secularized for some groups of Catholic teachers and intellectuals; an ambiguous pedagogical expression that should not be left to religious organizations for dissident Catholics and secular activists; and a justification for leaders of the Christian Democratic party (hereafter DC) and the Catholic Church hierarchy to preserve an IR controlled by dioceses. Conflicts over the definition of a problem are made visible by public rhetoric and competing frames on religious issues [START_REF] Ozzano | European Culture Wars and the Italian Case. Which Side Are You On?[END_REF] .

Indeed, governing a problem in a policy sector requires symbolic control and the capacity from groups and institutions to claim its ownership [START_REF] Gilbert | La définition des problèmes publics : entre publicité et discrétion[END_REF] . While the revision of the Concordat consisted in collective writing and producing a political compromise, it interfered with administrative aspects of the governance of religion in schools beyond curricular changes: this rewriting occurred after 1986, once IRC had already been redefined as default "religious culture" [START_REF] Mayrl | Administering Secularization: Religious Education in New South Wales Since 1960[END_REF] . The redefinition of IR into IRC was inseparable from expert talks behind closed doors as well as from protests in public arenas (education, Parliament, journals…), on rival definitions of what should be taught in State schools. Three phases can be distinguished in this analysis: a first phase of confidential Church-State mediation (1974-1976), a second phase of attempts at making a political compromise with the drafts of article 9.2 of the Agreement on IRC (1976-1981), and a third phase of politicization and school reforms which favored the reframing of "religious culture" as both a social problem and an official justification (1981-1984). CONCORDAT, 1974-1976 In the aftermath of Unification and conflict with the Papacy, Religious Education was legally excluded from official curricula in the national education system with the 1877

RELIGION IN SCHOOLS AND THE

Coppino Act. This regulation changed with the fascist regime: in the 1923 Gentile Act and in the framework of the 1929 Lateran Treaties (art. 36), IR was defined as "the foundation and coronation of the public instruction system", compulsory for pupils from primary to Butturini,op. cit., Franco Bolgiani, "La catéchèse en Italie", Les quatre fleuves, 11 (1980), 95-124. 16 Francesco Margiotta Broglio, "Lo stato degli insegnanti di religione nell'ordinamento statuale", Rivista giuridica della scuola, 2-6 (1963), 770-797. (1946-1951) and Catholic intellectual close to Andreotti's right-wing faction, was then appointed as President of the governmental delegation in the bilateral commission. The two other scholars in the delegation were Arturo Carlo Jemolo, a jurist and specialist of Church-State relations with a "liberal Catholic" stance, and Roberto Ago, another international law jurist, closer to secular sensibilities 20 . As a public intellectual distant from the DC, Jemolo was known for his stance on the "obsolescence" of parts of the Concordat and for his criticism 17 Luciano Caimi, "L'insegnamento della religione nel quadro della pastorale della Chiesa. Dalla fine della guerra alla revisione del Concordato (1945-1984)", in: eds Luciano Caimi andGiovanni Vian, op. cit., 239-245. 18 Gian Franco Pompei, Un ambasciatore in Vaticano. Diario 1969-1977 (Bologna: Il Mulino, 1994), 456-458. 19 Gian Franco Pompei, Letter to Giulio Andreotti, 06-08-1976, ASILS, f. Andreotti, s. Vaticano, sottos. 10-4, b. 163, fasc. 10 20 Nomination Decree, 15-10-1976, ASILS, f. Gonella, s. 3.2.5.3, b. 88, fasc. 11 of the "ignorance in religious matters" among Italians 21 . While Gonella used his social capital in both the DC and the Catholic Church, the two other members were appointed as "technical" experts, reputedly distant from politics. Conversely, the Vatican delegation was made of two prelates and diplomats from the Council of Church Affairs, Agostino Casaroli (president) and Achille Silvestrini, and a Jesuit Canon law jurist, Salvatore Lener. As Gonella and Jemolo were both members of a previous commission on the Concordat in 1969, they worked on the basis of Jemolo's proposal. In the case of IR, the 1969 draft on art. 36 of the Concordat stated that "the principles of Christian religion [were] part of the spiritual heritage and Italian historical tradition" to justify the teaching of IR in State schools 22 . After a few meetings between October and November 1976, the commission handed over a first reduced draft of 14 articles, in which art. 9-2 tackled Religious Education with the explicit "Catholic" adjective (IRC) 23 . The draft was read entirely by President Andreotti in the Chamber of Deputies and discussed, including art. 9-2:

The State, recognizing the value of religious culture, and considering the affiliation of the vast majority of Italian people to the Catholic Church, ensures Catholic Religious Education in all public schools, infancy, elementary, medium and medium-secondary, in accordance with particular agreements as regards members of other confessions.

Current dispositions for infancy and elementary schools remaining valid, at the act of inscription in medium and medium-secondary school, pupils who have the required age, or their parents or tutors, state whether they intend or not benefit from such school subject 24 .

The two justifications based upon "religious culture" and the rhetoric of numbers were criticized by left-wing deputies, in particular by socialist Deputy Gaetano Arfè, who stated that the PSI would support a revision of the whole Concordat and the end of the regime of 21 Arturo Carlo Jemolo, Chiesa e Stato in Italia, dalla Unificazione a Giovanni XXIII (Turin: Einaudi, 1965), 330.

22 "Riassunto Commissione Gonella (1969)", 9, ASILS, f. Andreotti, s. Vaticano, sottos. 10-4, b. 162, u.a. 1 23 Apart from an intermediary version in 1977, all successive drafts followed this order: preamble, art. 1 on mutual independence between Church and State, art. 2 on the mission of the Church, art. 3 on Church organization, art. 4 on religious personnel, art. 5 on places of worship, art. 6 on religious holidays, art. 7 on Church institutions, art. 8 on matrimonial norms, art. 9 on Catholic interests in the education sector, art. 10 on Catholic higher education, art. 11 on chaplaincies, art. 12 on artistic heritage and catacombs, art. 13 on procedures of ratification and art. 14 on bilateral commissions for implementation of the agreement. 24 Giuseppe Dalla Torre, La riforma della legislazione ecclesiastica. Testi e documenti per una ricostruzione storica (Bologna: Pàtron, 1985), 362-365.

State religion only if the "facultative" character of IRC was explicit 25 . Most DC deputies approved the draft, while the PCI endorsed a non-separatist and pro-revision position.

However, as the case of IR remains one of the most discussed, along with matrimonial and property issues, the technical debates were translated in the sphere of political mediation, from the second of the seven drafts between 1976 and 1984. The members of the governmental delegation had to produce a broader political compromise, by alternating between argumentation in public arenas and bargaining with Church diplomats behind closed doors 26 . Between 1977 and 1981, "religious culture" was both a secondary legitimization in collective writing, and the object of marginal professional claims from teachers and Catholic activists. From the second draft, the new regime of IRC in art. 9-2 was justified by "the value of religious culture", by the "development of the character of youth" and by the spiritual and historical heritage of Italians. In his speech to heads of Senate groups in June 1977, President Andreotti openly quoted "experts in pedagogy" as an authoritative source. He described the respect of subject choice as a guarantee for religious freedom and a didactic matter, rather than a "referendum for or against religion" 27 . The mediation for the second and third drafts of the Agreement in the parliamentary political arena showed the conflictual character of IR among other issues. Subsequently, Gonella took part in sixteen external working reunions with all the political parties represented in Senate in October and November 1977. He had to mediate with proposals from experts and officials of left-wing parties, such as Senator Paolo Bufalini for the PCI, Senator Giovanni Spadolini for the Republican Party, and Francesco Margiotta Broglio, a jurist specialized on Church-State relations and advisor in the PSI, who criticized the "pedagogical" justifications and the vague modalities of choice 28 . However, Gonella was aware of adamant criticism voiced by Catholic Church diplomats about the "risks of dilution" of religious transmission, if "religious culture" prevailed over doctrine and if an excessive freedom of choice was granted to students 29 . Meanwhile, two types of social movements contributed to the construction of "religious culture" as a social problem, contesting IR publicly and separately from the Concordat. The first mobilization of expert criticism came from jurists of the Union of Italian Jewish Communities and, mostly, from officials of Protestant Churches. Indeed, in November 25 Notes on the first draft on the revision of the Concordat, FSSFT, f. Arfè, s. IV, b. 66, fasc. 13 26 Claude Gilbert and Emmanuel Henry, art. cit., p. 47-56 27 "Dichiarazioni del Presidente Andreotti ai capi-gruppo del Senato", 22-06-1977, 16-18, ASILS, f. Andreotti, s. Vaticano, sottos. 10-4, b. 163, fasc. 5 28 "Partito Socialista Italiano. Incontro del 19-11-1977", 10-12, ASILS, f. Andreotti, s. Varicano, sottos. 10-4, b. 162, u.a. 4 29 "Testo con Silvestrini, Giacchi, Fumagalli, Ciprotti, Dalla Torre, Lajolo", 27-09-1977, 1-2, ASILS, f. Gonella, s. Furthermore, initially marginal social actors began protesting publicly on the issue of "religious culture" by claiming the matter should not be governed by the Concordat, but guided by pedagogical and professional principles. Indeed, the political configuration of the late 1970s, with a dominant left-wing faction in the DC, favored actors opposed to the existing system of IR in teachers' unions, in universities and in left-wing Catholic associations. These moral entrepreneurs can be analytically distinguished in four categories according to their positions related to the Catholic Church and to the public instruction sector: intellectual, educational and dissident Catholic activists, and secular activists 32 . Most of them belonged to the first two categories: IR teachers or intellectuals linked to the professional journal Religione e Scuola, and activists from teachers ' unions. For example, between 1978 and 1984, a group of IR teachers from the section of the SISM-CISL in Turin actively asked for a regular professional profile, and for a "historicized", secular IR in secondary schools 33 .

Other proposals included cultural revisions of a Catholic IR, or the institutionalization of a dual system (doppio binario) of Catholic and non-confessional IR from Catholic historians Pietro Scoppola, elected in Senate in 1983, and Luciano Pazzaglia. The latter argued that "religious culture" was a legitimate pedagogical object for the full development of character, 30 Draft of the Agreement between the Italian Republic and the Waldensian andMethodist Churches, 04-02-1978, 11-12, ASILS, f. Andreotti, s. Vaticano, sottos. 10-4, b. 163, fasc. 6 31 Arturo Carlo Jemolo, Letter to Guido Gonella, 02-03-1981, ACS, f. Jemolo, b. 27, fasc. 52, sottof. 1 32 Howard S. Becker, Outsiders. Studies in the Sociology of Deviance (New York: The Free Press, 1963), 147-153. 33 "Sintesi della prima assemblea degli insegnanti di religione della CISL di TORINO", FIGT, f. SNS CGIL Segreteria Regionale, b. 92, fasc. 263 and for the necessity of "religious literacy" in society [START_REF] Pazzaglia | Come cambierà l'ora di religione[END_REF] . Other activists in secular teachers' unions such as CGIL-Scuola aimed at replacing IR by critical notions of history of religions [START_REF] Maria | Religione e Scuola[END_REF] .

As these debates gained visibility in newspapers, they shaped an informal consensus, from secular actors to left-wing Catholics, on IR as the "weak spot" of the Concordat. AND JUSTIFICATION, 1981-1984 From the third draft of the Agreement on the revision of the Concordat in 1978 to the final, seventh in 1984, the two justifications of "the value of religious culture" and the inclusion of the principles of Catholicism in the historical heritage of the Italian people remained stable elements of art. 9-2 36 . However, the extent of precision on students' rights and on control of teachers by the Catholic Church varied in the following paragraphs of the drafts, and the destabilization of the political majority made the revision of the Concordat unlikely in the early 1980s. The fifth draft written in 1980 was not discussed either in Parliament or in the Presidency of the Council, and the outcome of the 1981 referendum on abortion was deemed provocative by the Church hierarchy. The definitive failure of the "historic compromise" between the DC and the PCI, as well as financial and political scandals such as the P2 and the Banco Ambrosiano, favored precarious five-party governing coalitions (Pentapartito) from 1981 to 1992, involving the DC, the PSI, the Republican Party, the Social-Democratic Party and the Liberal Party [START_REF] Ginsborg | L'Italia del tempo presente. Famiglia, società civile[END_REF] . The first cabinet led by a non-DC President since 1948, Republican Giovanni Spadolini, summoned a technical commission, presided by Judge Vincenzo Caianiello, with Francesco Margiotta Broglio, Antonio Malintoppi, an international law jurist close to the Republican Party, and Catholic lawyer Pio Ciprotti. Their alternative draft mentioned the only justification of "religious heritage" in art. 9-2 on IRC, while insisting on guarantees of choice and freedom of conscience. After Jemolo's death in 1981, Gonella and Ago criticized it informally as humiliating and as unreasonable to negotiate with the Vatican [START_REF]Colloquio del Sen. Gonella con il Presidente Spadolini a Palazzo Chigi[END_REF] .

"RELIGIOUS CULTURE", SOCIAL PROBLEM

Simultaneously, two attempted reforms of educational policy publicized "religious culture" in the governmental agenda separately from the Concordat, and they provided opportunities for moral entrepreneurs to promote a redefinition of IR against its ownership by Catholic institutions. Indeed, a commission of fifty-nine experts including Luciano Pazzaglia, presided by Liberal Senator Giuseppe Fassino and Catholic pedagogy scholar Mauro Laeng, was put in charge of writing new curricula for primary schools in May 1981. The final report, was handed over to the Minister of Public Instruction Franca Falcucci (DC) in November 1983: it promoted the withdrawal of IR, replaced by "the study of religious facts and phenomena" 39 . Meanwhile, in June and July 1982, the parliamentary debates on the reform of secondary schools led to the isolation of DC deputies on the defense of the inclusion of "Religious Education" among the basic knowledge and skills of all students. Conversely, PSI and PCI deputies disagreed over the explicit lack of a "facultative" character for IR, while secular activists in left-wing unions used petitions to reinforce the teaching of history of religions in high schools. In addition, in November 1982, the Catholic journal Religione e Scuola organized a congress in Rome, with representatives of major political parties, to promote a cultural reform of IR separately from the Concordat 40 .

In front of these open contestations, the Catholic hierarchy adopted a defensive strategy, from the Vatican to the CEI. The pedagogical proposal of Catholic intellectuals, which entailed the distinction between IR in primary schools and "religious culture" in secondary schools, was publicly denounced in the pontifical rhetoric of "complementarity and distinction" between catechesis and IR 41 . The proposal of a curricular reform in primary schools was not implemented by the Minister of Public Instruction, as the revision of the Concordat found relevance again after the 1983 general elections. After the formation of a coalition government led by Socialist Secretary Bettino Craxi, the opportunity of new relations with the Catholic world, independently from the DC, gave strategic value to the revision, which was restored to the governmental agenda. This move was also likely to show a "united front" between the PSI and the PCI on the Concordat, while the two major left-wing parties fought over the deindexation of basic wages on inflation. Two new members of the governmental delegation were appointed to replace Jemolo and Gonella, deceased in 1982:

Catholic jurist Pietro Gismondi, and Paolo Rossi, a former social-democrat deputy and judge. 39 Francesco Bevilacqua, Verso la scuola elementare del duemila. Testi programmatici a confronto. Invito ad una lettura critica dei programmi per la scuola elementare nel testo elaborato dalla Commissione "Fassino", con il riscontro delle indicazioni contenute nei programmi Ermini del 1955 (Soveria Mannelli: Rubbettino, 1984), 250-255. 40 Lino Prenna and Flavio Pajer, Società civile, scuola laica e insegnamento della religione. Atti del Convegno Nazionale di "Religione e Scuola", 17-19 novembre 1982 (Brescia: Queriniana, 1983). 41 Pope John Paul II, "Insegnamento della religione e catechesi, ministeri distinti e complementari", L'Osservatore romano, 07-03-1981. Consequently, the "Concordat-framework" was endorsed by PSI officials and advisors to conclude talks, leading up to a final version which stated in art. 9-2:

The Italian Republic, recognizing the value of religious culture and considering that the principles of Catholicism are part of the historical heritage of the Italian people, will continue to ensure, in the framework of school aims, Catholic Religious Education in public schools of all orders and degrees. In the respect of freedom of conscience and parents' educative responsibility, the right to benefit or not from this discipline is guaranteed to everyone. Upon enrollment students or their parents will exert such a right on request of school authority, without entailing any form of discrimination related to their choice. The hypothesis of causal relations between "religious culture" as a category of public policy in the Italian education sector, and the negotiations and justifications of interests, is confirmed by the analysis of the socio-historical redefinition of Catholic IR between 1974 and 1984. Rather than a direct outcome of secularization on the religious socialization of youth, the reduction of possible alternative policy paths played a central role in the requalification of IR into a default discipline of "religious culture" for students. This category became a "cause without opponent" because its ambiguity was the product of protests and conflicting uses, as a social problem as well as a justification to preserve the interests of the dominant religious institution in Italy. Despite the external consensus of the hierarchy, this situation was blurred in the Catholic Church in early 1984. Some liberal bishops had to dissociate themselves from promoters of the doppio binario, while more conservative bishops, in conformity with the Vatican, anticipated and tried to limit some outcomes of an interpretation as a non-compulsory status: the marginalization of IRC in schools and a massive opting-out of parents and students 42 . Although the implementation did not produce any significant statistical drop in the following years, this redefinition brought about changes in the groups previously involved , 26-04-1984, ASILS, f. Andreotti, s. Vaticano, sottos. 10-4, b. 163, fasc. 10 43 Andrea Rota, La religion à l'école. Négociations autour de la présence publique des communautés religieuses (Geneva: Seismo, 2017).

This version was presented in

  secondary schools and initially justified by Public Instruction Minister Gentile as a "minor philosophy" for the youth. The legal provisions on teachers and Church control became part of the policy of Conciliation between the Italian State and the Holy See, through the recognition of the State of the Vatican, despite conflicts on Catholic Action movements 14 . After the inclusion of the Lateran Treaties in the 1947 Constitution of the Italian Republic (art. 7), the implementation remained largely unchanged apart from revised primary school curricula in 1959. IR was part of school routines, with the possibility of exemption used by parents of students from religious and secular minorities. Although Catholic IR in schools remained formally distinct from catechism in parishes, the main professional training initiatives and organizations relied on Catholic catechetical associations and diocesan structures until the 1970s 15 . The first attempts at integrating IR teachers into regular Public Instruction services, favored by Christian-Democrat representatives in the wake of the democratization of the first-degree secondary school (scuola media), began as they were still priests for the most part in secondary schools, whereas IR was taught by schoolteachers in primary schools 16 . Yet, most debates occurred in the Catholic Church following the Second Vatican Council: the creation of a National Office of Catechism under the Presidency of the Italian Bishops Conference (hereafter CEI) in 1961 and the gradual disconnection between the DC and Catholic associations favored new frames and claims of professional pedagogical 14 Emilio

  skills. A 1970 document by the CEI, "The Renewal of the Catechesis", argued for new contents and methods for IR, compatible with the aims of the schools, and a few groups of IR teachers in secondary schools adhered to Catholic teachers' unions such as the SISM-CISL 17 . Consequently, the first attempts at redefining IR as a pedagogical problem preceded the abrogative referendum on divorce in 1974, which marked a defeat for the DC and the Catholic Church. The negative result forced the DC to relaunch a process of revision of norms of the Concordat, including IR. After May 1974, the President of the Council of Ministers, Aldo Moro, asked the Italian Ambassador to the Holy See, Gian Franco Pompei, to negotiate on the matter with the Catholic hierarchy. The first talks on the revision of the Concordat in 1975 between the ambassador, and the Secretary of the CEI, Archbishop Enrico Bartoletti, revealed a gap between political parties and the Church. The first proposal from the CEI consisted in a merely symbolic withdrawal of the formula "foundation and coronation of the Public Instruction system" to retain a compulsory IR. Instead, a significant part of the leftwing of the DC, the Italian Communist Party (PCI) and the Italian Socialist Party (PSI) seemed to agree to make all IR non-compulsory 18 . Pompei presented a draft of eleven articles in 1976, with the abolition of the compulsory regime of IR and its organization on request of parents and students. However, the newly formed government led by President of the Council, Giulio Andreotti (DC), in June 1976, sustained by a "non-defiance vote" from the PCI, disqualified the ambassador's proposal, in favor of negotiations with Vatican officials 19 . 2. NORMALIZING OR REFORMING CATHOLIC IR, 1976-1981 Senator Guido Gonella, cofounder of the DC, former Minister of Public Instruction

  3.2.5, b. 78, fasc. 90 1976, the same governmental delegation led by Gonella was put in charge of talks with three scholars representing the Union of Waldensian and Methodist Churches of Italy (Tavola Valdese), President Giorgio Peyrot, Sergio Bianconi and Giorgio Spini, two jurists and a historian. The negotiations aimed at producing an internal law Agreement (Intesa) of recognition of the Tavola Valdese by the State, based on art. 8 of the Constitution for non-Catholic religions. Their proposal on religion in schools, written in art. 9 and 10 of the first draft in 1978, defined religious transmission as a duty of the families, not of State schools. It asked for the respect of freedom of conscience of non-Catholic students in general and it required limiting the organization of IR to the first or the last hour of schooldays. It also endorsed the possibility of creating alternative lessons on "religious phenomenon and its implications" 30 . Later negotiations reached the point of open conflict with the governmental delegation in 1981, leading to Jemolo's confidential proposal to Gonella to abruptly cease these talks because of their counterpart's stance on IR 31 .

  Senate in January 1984 by President Craxi, who stressed the justifications based on "religious culture" and guarantees of religious freedom, while Pietro Scoppola's stance was isolated in the DC. The Agreement on the revision of the Concordat was approved by the majority and the PCI, then signed by President Craxi and Cardinal Casaroli on February, 18 th 1984. The "technical aspects" in the 5 th Additional Protocol to the Agreement, such as the time slot of IRC, professional training, textbooks and curricular changes, remain undecided. The signing of the Intesa with the Tavola Valdese three days later confirmed the transition to a new system of governance of religion in Italy, opening a phase of conflictual implementation on IRC until the 1990s.

  FIGT: Fondazione dell'Istituto Gramsci, Turin FSSFT: Fondazione degli Studi Storici Filippo Turati, Florence


Ruben Teixeira Franco
email: ruben.teixeirafranco@insa-lyon.fr

Rémy Bayard
email: remy.bayard@insa-lyon.fr

Pierre Buffière
email: pierre.buffiere@insa-lyon.fr

Mathematical modelling of the ensiling process before biogas production: strengthening the links between biomass storage and anaerobic digestion

Keywords: Biomass, Ensiling, Anaerobic digestion, Mathematical modelling, Kinetics, Stoechio-kinetic model Highlights

teaching and research institutions in France or abroad, or from public or private research centers.   Mathematical modelling of the ensiling process before biogas production: strengthening the links between biomass storage and anaerobic digestion

Introduction

Society is now confronted with the energy transition, which is considered as an integral part of the solution for the sustainable development of future generations. Anaerobic digestion (AD) is one of the renewable energy production sectors that has attracted increasing interest in recent years. In this process, the bioconversion of complex organic materials is carried out, under anaerobic conditions and by successive reactions. The biogas produced, mainly composed of CH4 and CO2, can be then upgraded and injected into the gas network, converted into electricity or used as transportation fuel. Today, there are more than 17 000 biogas plants operating in Europe, including around 12 500 agricultural AD plants [START_REF]European Biogas Association, 6th Edition of the Statistical Report of the European Biogas Association[END_REF].

One of the major challenges for the optimization of the AD sector concerns the management of feedstocks before biogas production. Indeed, with the need for a continuous supply of AD plants throughout the year, some seasonal production inputs have to be stored in most cases for extended periods (several months). Since these feedstocks are readily biodegradable, long-term storage can result in significant energy losses and greenhouse gas emissions.

Ensiling is usually pointed out as the logical choice to preserve the energy potential of wet bio-products in the long term [START_REF] Franco | Optimizing storage of a catch crop before biogas production: impact of ensiling and wilting under unsuitable weather conditions[END_REF]. It relies on the anaerobic conservation through the stabilization of organic matter at acidic pH. Ensiling can be described as follows [START_REF] Franco | Ensiling for biogas production: Critical parameters. A review[END_REF][START_REF] Mcdonald | The Biochemistry of Silage, Second Edi[END_REF][START_REF] Egg | Feedstock storage, handling and processing[END_REF][START_REF] Weinberg | Engineering aspects of ensiling[END_REF][START_REF] Herrmann | Effects of ensiling, silage additives and storage period on methane formation of biogas crops[END_REF][START_REF] Zimmer | Efficient silage systems[END_REF].

First, aerobic respiration occurs during several hours, converting water-soluble carbohydrates (WSC) into carbon dioxide, water and releasing energy. Once oxygen is depleted, microorganisms capable of anaerobic growth will compete for the substrate.

Under suitable conditions, lactic acid bacteria (LAB), naturally present in wet raw materials, will produce lactate from available carbohydrates, decreasing and stabilizing the pH to around 4.0. This prevents further microbial activity (lactate degraders: clostridia, acetogens, etc.) and enables the long term preservation of the organic matter.

These chemical pathways are illustrated in Table 1. Ensiling is particularly interesting for AD as it can lead to full preservation of the biomethane potential (BMP) of feedstocks for prolonged durations (that can reach up to 1 year) [START_REF] Franco | Optimizing storage of a catch crop before biogas production: impact of ensiling and wilting under unsuitable weather conditions[END_REF][START_REF] Herrmann | Effects of ensiling, silage additives and storage period on methane formation of biogas crops[END_REF][START_REF] Liu | Effect of urea addition on giant reed ensilage and subsequent methane production by anaerobic digestion[END_REF][START_REF] Liu | Comparison between ensilage and fungal pretreatment for storage of giant reed and subsequent methane production[END_REF][START_REF] Kreuger | Ensiling of crops for biogas production: effects on methane yield and total solids determination[END_REF][START_REF] Menardo | Effect of Conservation Time and the Addition of Lactic Acid Bacteria on the Biogas and Methane Production of Corn Stalk Silage[END_REF][START_REF] Lehtomäki | Biogas Production from Energy Crops and Crop Residues[END_REF][START_REF] Franco | Optimizing agricultural wastes storage before anaerobic digestion : impact of ensiling on methane potential of lignocellulosic biomass[END_REF][START_REF] Franco | Co-ensiling of cattle manure before biogas production: effects of fermentation stimulants and inhibitors on biomass and methane preservation[END_REF][START_REF] Franco | Cattle manure for biogas production. Does ensiling and wheat straw addition enhance preservation of biomass and methane potential?[END_REF][START_REF] Lou Hillion | Co-ensiling as a new technique for long-term storage of agro-industrial waste with low sugar content prior to anaerobic digestion[END_REF][START_REF] Larsen | Co-ensiling of straw with sugar beet leaves increases the methane yield from straw[END_REF]. However, ensiling performance is extremely dependent on the feedstock and its biochemical properties, notably the total solids (TS) level, the amount of accessible carbohydrates, the buffering capacity and the nature of the initial endogenous microflora [START_REF] Franco | Ensiling for biogas production: Critical parameters. A review[END_REF][START_REF] Mcdonald | The Biochemistry of Silage, Second Edi[END_REF]. For instance, even though cattle manure is a major feedstock for agricultural AD it does not naturally lead to an optimal ensiling conservation. In fact, for this feedstock the low acidification degree during confined storage cannot prevent further bacterial activity and spoiling. Therefore, secondary (clostridial) fermentation, acetogenesis and methanogenesis will occur, resulting in vast energy losses. This is especially due to lack of easily fermentable sugars, high moisture content and high methanogenic activity found on this type of substrate [START_REF] Franco | Co-ensiling of cattle manure before biogas production: effects of fermentation stimulants and inhibitors on biomass and methane preservation[END_REF][START_REF] Franco | Cattle manure for biogas production. Does ensiling and wheat straw addition enhance preservation of biomass and methane potential?[END_REF]. Co-ensiling with other substrates containing easily accessible carbohydrates and/or low moisture content has been proposed as an effective technique to enhance energy conservation of cattle manure during storage [START_REF] Franco | Co-ensiling of cattle manure before biogas production: effects of fermentation stimulants and inhibitors on biomass and methane preservation[END_REF][START_REF] Franco | Cattle manure for biogas production. Does ensiling and wheat straw addition enhance preservation of biomass and methane potential?[END_REF].

The understanding of the biochemical mechanisms that occur during ensiling has been addressed by few authors in modeling approaches [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Pitt | A quantitative model of the ensilage process in lactate silages[END_REF]. These studies are mainly based on the lactic and clostridial fermentations phenomena, as well as hydrolysis reactions of the particulate compounds. Nevertheless, these reported modelling works do not include the reactions that take place during inefficient storage (such as acetogenesis and methanogenesis). Also, they are still very far from AD models (e.g. [START_REF] Fezzani | Implementation of IWA anaerobic digestion model No. 1 (ADM1) for simulating the thermophilic anaerobic co-digestion of olive mill wastewater with olive mill solid waste in a semi-continuous tubular digester[END_REF][START_REF] Rivas-García | Numerical study of microbial population dynamics in anaerobic digestion through the Anaerobic Digestion Model No[END_REF][START_REF] Shi | Modeling of the methane production and pH value during the anaerobic co-digestion of dairy manure and spent mushroom substrate[END_REF][START_REF] Batstone | Anaerobic Digestion Model No[END_REF]) and the logic of conservation of energy potential for biogas production. In order to fill these knowledge gaps, structured models of biomass storage should be developed. They should be applicable for the ensiling of the main feedstocks used in agricultural AD (manure, etc.). Furthermore, they should provide hints about the best storage practices before anaerobic digestion for a given raw material.

In this work, a mathematical model of the ensiling process with a focus on its anaerobic phase was developed. A set of biological reactions and chemical equilibria was used to simulate the temporal variation of the main compounds during storage, including pH. Additionally, BMP preservation was estimated through CH4 and H2 production during ensiling. The model design is based on the mentioned ensiling models [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Pitt | A quantitative model of the ensilage process in lactate silages[END_REF], but with an implementation approach similar to that of Anaerobic Digestion Model No.1 (ADM1) [START_REF] Batstone | Anaerobic Digestion Model No[END_REF]. Degradable chemical oxygen demand (COD) was used as unit. A particular attention has been paid to the fractionation of organic matter (from measurements to state variables). The model implementation was carried out with two different types of feedstock widely used in agricultural AD: catch crops and cattle manure. This should validate the applicability of this model framework for a large range of raw materials with different biochemical properties.

Finally, this model is intended to develop a better understanding of methane potential preservation by:  Describing the main phenomena involved during the ensiling process;  Investigating both efficient (e.g. energy crops) and inefficient (e.g. manure) feedstocks for ensiling;  Establishing a mathematical link between biomass storage and anaerobic digestion. This is accomplished by using degradable COD as unit for organic carbon assessment, and estimating the pH evolution and energy conservation during ensiling.

Experimental trials

Feedstock and co-substrates

Samples of catch crops and fresh cattle manure were collected from an agricultural site in the Auvergne-Rhône-Alpes region of France (Gaec Béreyziat, Les Teppes, 01340 Béréziat, France). Catch crop was composed by a mixture of sunflower, sorghum, peas, Vicia sp. and Trifolium alexandrinum. In order to assess the impact of TS content, both fresh and dried catch crop were ensiled. Catch crop drying was performed during 96 h: wilting, i.e., natural drying of organic matter, for 72 h; and then 24 h on air-ventilated stove at 35 °C. Before storage, fresh cattle manure (10% of TS content) was mixed with wheat straw (0.10 m maximum length) in order to reach a final TS content of 19% (ensilable TS level [START_REF] Franco | Cattle manure for biogas production. Does ensiling and wheat straw addition enhance preservation of biomass and methane potential?[END_REF]). Cattle manure was ensiled with and without co-substrate addition. Anhydrous glucose (D-(+)-glucose for analysis, Chimie-Plus, France) was used as co-substrate and represented around 40% of volatile solids (VS) content.

Conditions of ensiling experiments are summarized in Table 2.

Ensiling experiments

Laboratory trials were performed in 3.5 L airtight round plastic storage drums. In order to enable biogas output and at the same time minimizing headspace, silos were filled up to 2.55 L with raw material at packing density of 700 kg.m -3 , the remaining volume being filled with gravel, using a geotextile membrane to separate it from the raw material. Proper plastic lid and rubber ring were used for silo sealing and its airtightness was reinforced with silicone sealant. Then, silos were weighed and placed in a controlled-temperature room at 25±2 °C. Experiments were conducted during 7, 15, 30 and 120 days for cattle manure and 7, 14, 28 and 98 days for catch crop.

Analytical methods

For each sample time, one silo (per tested condition) was sacrificed. It was opened and weighed. Then, biomass was homogenized and two samples were taken. The first sample was used for direct analyses on the crude material and the other one was mixed with water in order to get two fractions: a water-soluble phase and a particulate phase. This leaching test was performed with a 10:1 water/TS ratio during 2 h under constant bottle rotation. Phase separation was achieved by centrifugation (5000 G; 10 min) followed by 0.7 µm particle size filtration. Finally, the particulate phase was dried at 70 °C until constant weight and ground at 2 mm theoretical length. Crude material/watersoluble and particulate samples were stored at 4 °C and -20 °C, respectively, until analysis.

Crude material was analyzed for its TS/VS content and BMP. For the water-soluble phase, besides TS/VS content, pH, WSC, volatile fatty acids (VFA), COD, total Kjeldahl nitrogen (TKN) and ammonia nitrogen (NH3-N) fractions were determined.

Particulate solid was analyzed for its TS/VS, COD and TKN.

Full description of experimental procedure, including phase separation, methods for chemical analysis and BMP measurement can be found elsewhere [START_REF] Franco | Optimizing storage of a catch crop before biogas production: impact of ensiling and wilting under unsuitable weather conditions[END_REF][START_REF] Franco | Co-ensiling of cattle manure before biogas production: effects of fermentation stimulants and inhibitors on biomass and methane preservation[END_REF]. It should be mentioned that BMP trials followed the recommendations of the international task group on the standardization of BMP tests [START_REF] Holliger | Towards a standardization of biomethane potential tests[END_REF].

Mathematical model

Units

Degradable COD (kgCOD.m -3 ) basis was used for the concentration of chemical compounds because of its flexibility in the study of anaerobic processes, its use in ADM1 [START_REF] Batstone | Anaerobic Digestion Model No[END_REF] and its experimental accessibility to liquid and particulate samples. For inorganic carbon (CO2 and HCO3 -) and inorganic nitrogen (NH4 + and NH3) kmol.m -3 was used as unit.

Biochemical processes

Aerobic phase

When the silo is sealed, aerobic respiration of organic matter occurs until the oxygen is completely consumed. This is described by the following reaction [START_REF] Mcdonald | The Biochemistry of Silage, Second Edi[END_REF][START_REF] Pitt | A quantitative model of the ensilage process in lactate silages[END_REF]:

C 6 H 12 O 6 + 6O 2 → 6CO 2 + 6H 2 O (1)
Since this phase only lasts for a few hours, its impact on the evolution of ensiling is negligible while compared to the anaerobic phase. For instance, sugars consumed during respiration were estimated to be less than 2% of the amount found in ensilable raw materials. Therefore, the aerobic phase was not accounted for in the dynamic model. Nevertheless, its effects on some initial conditions (CO2 concentration and pH) used to model the anaerobic phase were considered. In fact, since respiration leads to CO2 production and solubilization, the pH will slightly drop during aerobic phase. In this model it was considered an initial CO2 concentration in the gas phase of 0.038% (V/V) and 21% (V/V) at the end of the aerobic phase. The last concentration was calculated considering the exclusive conversion of O2 to CO2 under constant gas pressure (which is the case in our experiments). Then the inorganic carbon dissolved in the liquid phase (𝑆 𝐼𝐶 ) was calculated according to Henry's law:

𝑆 𝐼𝐶 = 𝐻 𝐼𝐶 × 𝑃 𝐼𝐶 (2)
Where the Henry constant for CO2 (𝐻 𝐼𝐶 ) is 0.035 kmol.m -3 .bar -1 (25 °C) [START_REF] Batstone | Anaerobic Digestion Model No[END_REF] and 𝑃 𝐼𝐶 , the partial pressure of CO2, is equal to the total gas pressure (assumed equal to the Patm = 1.013 bar) multiplied by the volume fraction of CO2 in the gas phase.

Anaerobic phase

The anaerobic phase of ensiling comprises a large number of reactions that occur either sequentially or in parallel, due to the complexity of the endogenous microflora. In the present model we did not represent all the conceivable biochemical reactions. We rather described the confined storage from the main observable phenomena. The anaerobic model is schematized in Figure 1 and includes the following biochemical reactions:

1. Hydrolysis of particulate carbohydrates into water-soluble sugars;

2. Lactic fermentation of soluble sugars;

3. Clostridial fermentation of lactic acid;

4. Hydrolysis of proteins;

5. Fermentation of amino acids; 6. Methanogenesis.

These reactions can be separated into extracellular and intracellular mechanisms. The enzymatic/chemical hydrolysis reactions (1 and 4) are extracellular and are presumed to follow a first order kinetics, which is a simplification based on empiricism [START_REF] Batstone | Anaerobic Digestion Model No[END_REF]. The reaction rate (𝜌 𝑗 ) of these two reactions is described through Eq. ( 3:

𝜌 𝑗 = 𝑘 𝑗 × 𝑋 𝑖 (3)
Where 𝑘 𝑗 is the rate coefficient of the process j and 𝑋 𝑖 is the concentration of the particulate compound i.

Fermentation (2, 3 and 5) and methanogenesis [START_REF] Weinberg | Engineering aspects of ensiling[END_REF] are intracellular biochemical reactions that result in both cell growth and decay. Decay was not considered in the present work, since its impact was found negligible (results not shown). Thus, the biological kinetics are described by the consumption of the substrate and growth of the biomass. These biological reactions are presented according to a Monod kinetics based on the consumption of the substrate:

𝜌 𝑗 = 𝑋 𝑖 × 𝜇 max_𝑖 𝑌 𝑖 × 𝑆 𝑖 𝐾 𝑆_𝑖 + 𝑆 𝑖 × 𝐼 𝑝𝐻_𝑖 (4)
Where, 𝑋 𝑖 is the concentration of biomass that consume the substrate i, 𝜇 𝑚𝑎𝑥_𝑖 the maximum specific growth rate of biomass, 𝑌 𝑖 the biomass yield on the substrate i, 𝑆 𝑖 the concentration of substrate i and, 𝐾 𝑆_𝑖 the half-saturation value of biomass that consumes the substrate i. The term 𝐼 𝑝𝐻_𝑖 reflects an inhibition mechanism based on the pH of the medium. The use of such biological inhibition is mandatory in this type of kinetic model since biomass growth and substrate consumption only occur when the pH of the silage is above a critical value. This function is represented by the following equation [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF]:

𝐼 𝑝𝐻_𝑖 = (𝑝𝐻 𝑝 𝑀_𝑖 ⁄ ) 𝑞 𝑖 1 + (𝑝𝐻 𝑝 𝑀_𝑖 ⁄ ) 𝑞 𝑖 (5)
Where, 𝑝 𝑀_𝑖 describes the pH for which the activity is half the maximum and 𝑞 𝑖 is the sensitivity of the activity around 𝑝 𝑀_𝑖 . The sensitivity of the inhibition function to these two parameters is presented in Figure 2.

The rate equations for the biochemical processes of the anaerobic phase model are detailed in Table 3. This is presented as a Güjer matrix, which has an implicit COD balance. In this table, the stoichiometry of the reactions is presented as product yield on the substrate (𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡,𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ). In addition, the acid/base pairs are presented as the sum of the concentrations of the acid and base forms (e.g. SIC = SCO2 + SHCO3-). Since neither inorganic carbon nor inorganic nitrogen has COD, these compounds have been implemented with a molar base conversion. For example, in the case of clostridial fermentation, the conversion of organic carbon was carried out knowing that the biodegradation of 1 gCOD of lactate produces 1/64 moles of inorganic carbon (not counting the bacterial yield). For inorganic nitrogen, a conversion based on the nitrogen molar content of the amino acids (NAA) was applied.

This representation includes several assumptions made in order to simplify the system and its determination from experimental measurements:

 The fraction of easily accessible carbohydrates includes not only soluble monosaccharides, but also a portion of non-structural particulate compounds such as starch. The hydrolysis reactions of the latter are not considered to be a limiting factor and, subsequently, were not added to the model. For the fraction of hardly accessible sugars, mainly the structural compounds (hemicellulose and cellulose) were considered.

 Acetic acid is used as a model molecule for other minor VFA formed during storage (propionic, valeric, caproic and formic acids). This may lead to differences in the by-products formed and in the pH evolution.

 Each fermentation process is represented by a general reaction, catalyzed by a single bacterial trophic group. For instance, in the case of lactic fermentation, a homofermentative reaction was used for the production of both lactic acid and ethanol (by applying stoichiometric coefficients). However, since the production of ethanol is theoretically carried out by hetero-fermentative bacteria, this leads to some inaccuracy with respect to the production of other by-products such as CO2. In the same way, for the consumption of lactic acid, a single trophic group (clostridial bacteria) was supposed to carry out the production not only of butyric acid, but also of acetic acid, so as to consider both other minority VFA and a possible acetogenesis reaction. This assumption underestimates the energy losses due to H2 production and may lead to deviations in the pH value.

 Biodegradation reactions of ethanol and butyric acid were not taken into account, which can lead to imprecisions for inefficient silages.

 The inorganic nitrogen consumed for bacterial growth was assumed negligible and was not included in the nitrogen balance.

 Since the silos were regularly degassed, the closed system equilibrium model was not compatible with the experiments. Therefore, this equilibrium was not incorporated in the anaerobic model. However, this leads to the total solubilization of CO2 produced. This issue is discussed in further detail in section 3.4. Model implementation.

Physicochemical processes

The prediction of the pH is a crucial point for the ensiling process. The pH is a result of acid-base equilibria of the different chemical species in the system. In this model, it were considered the equilibrium reactions of the important compounds in ensiling with pKa values (Table 4) close to the operating pH of this process. This included the following acid-base pairs: HLA / LA -; HBA / BA -; HAC / AC -; NH4 + / NH3; CO2 / HCO3 -and; H2O / (OH -+ H + ). These reactions constitute a set of algebraic equations which were integrated in the model by a charge balance.

So that the acid-base phenomena can be modeled as implicit algebraic equations, for each reaction the sum of the acid-base pair concentrations was considered as a dynamic state variable, as shown in the following equations:

𝑆 𝐿𝐴 -𝑆 𝐿𝐴--𝑆 𝐻𝐿𝐴 = 0 (6)
𝑆 𝐵𝐴 -𝑆 𝐵𝐴--𝑆 𝐻𝐵𝐴 = 0

𝑆 𝐴𝐶 -𝑆 𝐴𝐶--𝑆 𝐻𝐴𝐶 = 0 (8)

𝑆 𝐼𝐶 -𝑆 𝐻𝐶𝑂3--𝑆 𝐶𝑂2 = 0 (9) 𝑆 𝐼𝑁 -𝑆 𝑁𝐻3 -𝑆 𝑁𝐻4+ = 0 (10)
Where, for example in equation 6, 𝑆 𝐿𝐴 is the acid-base pair combination, 𝑆 𝐻𝐿𝐴 is the associated form and 𝑆 𝐿𝐴-is the dissociated form. Thus, these algebraic equations can be formulated considering the acid-base equilibrium and the equilibrium constants (Table 4) respectively:

𝑆 𝐿𝐴-- 𝐾 𝑎_ 𝐿𝐴 𝑆 𝐿𝐴 𝐾 𝑎_ 𝐿𝐴 + 𝑆 𝐻+ = 0 (11) 𝑆 𝐵𝐴-- 𝐾 𝑎_𝐵𝐴 𝑆 𝐵𝐴 𝐾 𝑎_𝐵𝐴 + 𝑆 𝐻+ = 0 (12)
𝑆 𝐴𝐶--𝐾 𝑎_𝐴𝐶 𝑆 𝐴𝐶 𝐾 𝑎_𝐴𝐶 + 𝑆 𝐻+ = 0 (13)

𝑆 𝐻𝐶𝑂3-- 𝐾 𝑎_ 𝐶𝑂2 𝑆 𝐼𝐶 𝐾 𝑎_𝐶𝑂2 + 𝑆 𝐻+ = 0 (14) 𝑆 𝑁𝐻4+ - 𝑆 𝐻+ 𝑆 𝐼𝑁 𝐾 𝑎_ 𝑁𝐻4+ + 𝑆 𝐻+ = 0 (15)
The ionic equilibrium of water is also described by the following relation:

𝑆 𝑂𝐻-- 𝐾 𝑎_𝐻 2 0 𝑆 𝐻+ = 0 (16)
Then, the concentration of the H + ions and the pH were calculated using a charge balance with the main species present during ensiling:

∑ 𝑆 𝐶 + -∑ 𝑆 𝐴 -= 0 (17)
Where ∑ 𝑆 𝐶 + and ∑ 𝑆 𝐴 -correspond to the total concentrations of cationic and anionic equivalents, respectively. The equivalent concentration of each ion is its valence multiplied by the molar concentration [START_REF] Batstone | Anaerobic Digestion Model No[END_REF]. Since organic acids were expressed on a COD basis, a conversion of its content in grams of COD per charge was applied in the denominators of the equation. The charge balance with all the ions used in the model is represented by the following equation:

𝑆 𝐵𝐶+ + 𝑆 𝑁𝐻4+ + 𝑆 𝐻+ -𝑆 𝐻𝐶𝑂3-- 𝑆 𝐿𝐴- 96 - 𝑆 𝐵𝐴- 160 - 𝑆 𝐴𝐶- 64 -𝑆 𝑂𝐻--𝑆 𝐵𝐶-= 0 (18)
In the above equation 𝑆 𝐵𝐶+ and 𝑆 𝐵𝐶-are the acid and base concentrations, respectively, of the buffer compounds of the system. These compounds represent acids and bases present in the raw material that do not have terms of consumption or production and, thereafter, are assumed to be constant throughout storage. These terms may also include other unmeasured organic acids or proteins and derived products, assuming that they are not degraded during ensiling.

For the calculation of the basic and acidic buffer concentrations, the Neal and Thornley [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] approach was used. This hypothesis considers that association and dissociation are mutually exclusive properties of the buffer. The difference 𝑆 𝐵𝐶--𝑆 𝐵𝐶+ is given by the relation below as described by Neal and Thornley [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF]:

𝑆 𝐵𝐶--𝑆 𝐵𝐶+ = 𝑆 𝐵𝐶 (𝐾 𝑎_𝐵𝐶 𝐾 𝑏_𝐵𝐶 -𝑆 𝐻+ 2 ) (𝑆 𝐻+ + 𝐾 𝑎_𝐵𝐶 )(𝑆 𝐻+ + 𝐾 𝑏_𝐵𝐶 ) -𝑆 𝐻+ 𝐾 𝑎_𝐵𝐶 (19)
Where 𝑆 𝐵𝐶 refers to the initial molar concentration of the buffer compounds, 𝐾 𝑎_𝐵𝐶 is the dissociation constant of the acid compounds of the buffer and 𝐾 𝑏_𝐵𝐶 is the association constant of the basic compounds of the buffer.

Model implementation

Sequential modelling

In order to be able to solve the model of the anaerobic phase with the charge balance, two preliminary steps have been carried out, namely:

 Estimation of properties of the buffer compounds and;

 Determination of pH after the aerobic phase.

According to Eq. ( 19), the buffer compounds (𝑆 𝐵𝐶-and 𝑆 𝐵𝐶+ ) are defined by their initial concentration (𝑆 𝐵𝐶 ) and its dissociation/association constants (𝐾 𝑎_𝐵𝐶 and 𝐾 𝑏_𝐵𝐶 ).

To reduce the number of variables to be estimated, Neal and Thornley's hypothesis [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] of consider 𝐾 𝑎_𝐵𝐶 = 𝐾 𝑏_𝐵𝐶 was performed. Then, the difference 𝑆 𝐵𝐶--𝑆 𝐵𝐶+ was calculated with the charge balance at t=0. Values of 𝑆 𝐵𝐶 and 𝐾 𝑎_𝐵𝐶 /𝐾 𝑏_𝐵𝐶 were finally refined with the charge balance by comparing the simulations results to the experimental decrease of pH during initial lactic acid production (7 days). This approach can also be considered as an indirect measure of silage buffering capacity, which is often estimated as the amount of base needed to, e.g., vary the pH from 4 to 6.

The pH after aerobic respiration was determined by applying the charge balance with the previously estimated buffer properties. The only change made was to consider the increase in the CO2 content in the gaseous phase from 0.038% (V/V) (SIC = 1.35 x 10 -5 kmol.m -3 ) initially up to 21% (V/V) (SIC = 7.45 x 10 -3 kmol.m -3 ) after respiration.

After these preliminary calculations, the anaerobic phase of the ensilage was simulated through the implementation of the dynamic model on MATLAB™. Differential and Algebraic Equation (DAE) implementation was used so that physicochemical processes can be integrated algebraically. Since ensiling is a batch process, the model was implemented in the following configuration:

 16 differential equations to represent dynamic state variables, according to the Güjer matrix (Table 3) and exemplified below for both hardly accessible carbohydrates (Eq. ( 20)) and inorganic carbon (Eq. ( 21)).

𝑑𝑋 𝐶𝐻 𝑑𝑡 = -𝑘 1 𝑋 𝐶𝐻 ( 20
)
𝑑𝑆 𝐼𝐶 𝑑𝑡 = ( 1
Since this anaerobic model does not take into account the liquid-gas equilibrium, 𝑆 𝐼𝐶 balance (Eq. ( 21)) considers that all the inorganic carbon produced remains solubilized.

To avoid significant impacts on the pH, it was assumed that the partial pressure of CO2 in the gas phase cannot exceed 1 bar, which corresponds to a maximum 𝑆 𝐼𝐶 of 3.50 x 10 -2 kmol.m -3 (25 °C). This has been integrated into the model by the following condition:

𝑑𝑆 𝐼𝐶 𝑑𝑡 = { Eq. 21, 𝑆 𝐼𝐶 < 0.035 𝑘𝑚𝑜𝑙. 𝑚 -3 0, 𝑆 𝐼𝐶 ≥ 0.035 𝑘𝑚𝑜𝑙. 𝑚 -3 (23)
Even so, the accumulation rate of inorganic carbon in the liquid phase will be greater in the model than in reality. This difference was neglected since silos normally have small headspace and consequently, the CO2 concentration in the gas phase rapidly increases and stabilizes in the first weeks of ensiling.

Finally, it is important to notice that this model was developed to work at constant temperature. Simulations in this study were performed for a standard temperature of 25 °C (the same used in experimental trials).

From measurements to state variables

In order to setup the anaerobic model, values of state variables were estimated as follows (initial values are presented in Table 5):

 XCH and XPR: The COD of the particulate phase was measured, as well as its biodegradable fraction (from the BMP tests). However, these analysis do not make it possible to distinguish each biodegradable fraction from the particulate COD. As a result, it was assumed that all the COD of the particulate phase corresponded to biodegradable proteins and sugars. This is an assumption that has no practical impact: during ensiling, the bio-accessible particulate fraction is never completely metabolized. Thus, XPR was estimated from the TKNparticulate measurement (using the NAA parameter) and the XCH was determined by the difference between the CODparticulate and the XPR. Since in the majority of the trials there was a solubilization of the readily available particulate at the beginning of storage, the initial XCH and XPR were determined by extrapolation of the hydrolysis trend for the experimental points from 7 days of silage. However, this approach should not be used for lipidrich feedstocks, which are poorly soluble in water. Under these conditions, an experimental determination of the lipid compounds must be carried out.

Nevertheless, the feedstocks studied here have been supposed to have very low fat content.

 SCH: The experimental determination of this variable was not totally possible, since for this parameter only monosaccharides were analyzed by HPLC. For this reason, the initial SCH was estimated as the required amount of sugars for VFA formation in the first week of ensiling. For all other points, it was assumed that only the soluble sugars analyzed remained in the SCH fraction.

 SLA, SET, SBA, SAC: Calculated from HPLC data and gas chromatography after conversion to COD. It is important to note that SAC includes the COD of acetic, propionic, valeric, caproic and formic acids.

 SAA: This parameter cannot be reliably compared to the results of soluble organic nitrogen (NTKsoluble -NH3-N), since it was experimentally observed that some of the amino acids present are not degradable under ensiling conditions. Therefore, the evolution of this variable was not compared to the experimental results. Its initial value was estimated as the amount needed to simulate the fast production of inorganic nitrogen (SIN) during the first few weeks of ensiling.

 SIN: Calculated from ammonia (NH3-N) analysis in the liquid phase.

 SIC, SH2 and SCH4: Initially, SIC was calculated from the CO2 gas content after respiration, while for SH2 and SCH4 it was assumed a zero concentration at t=0. On the other hand, the model outputs were not compared with the experimental results,

given the insufficient measurements of gas composition and the poor gas production.

 XSU, XLA, XAA, XAC: None of the measurements performed allowed a quantitative determination of the concentration of active biomass. As a result, these variables were estimated from bibliographic data. The value of 0.1 kgbiomass.m -3 estimated by Neal and Thornley [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] for lactic acid bacteria was used for all the bacterial populations in the fresh material of each feedstock tested in this model. This value was converted to COD using the empirical formula for biomass C5H7O2N proposed by the ADM1 model [START_REF] Batstone | Anaerobic Digestion Model No[END_REF] (M = 113 kg.kmol -1 , COD = 160 kg.kmol -1 ).

 SH+ (pH): For the starting point, the pH after aerobic respiration was determined and used as initial value for the simulation.

Kinetic parameters

For stoichiometric and kinetic parameters, when possible, values from bibliography [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Batstone | Anaerobic Digestion Model No[END_REF] were used and then optimized during the simulations. This was the case for the yields of the product on the substrate, the parameters pM_i and qi for the inhibition functions and the maximum specific growth rates of Monod reactions (𝜇 max_𝑖 ). For the nitrogen content of proteins and amino acids (NAA), the value of 0.007 kmol.kgCOD -1 suggested by the ADM1 model [START_REF] Batstone | Anaerobic Digestion Model No[END_REF] was used. Similarly, biomass yields were estimated from literature data ( [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] for YSU, YLA and YAA, [START_REF] Batstone | Anaerobic Digestion Model No[END_REF] for YAC). In contrast, since the kinetic constants of hydrolysis (kj) (storage conditions) cannot be compared with those used to simulate anaerobic digestion (biogas production conditions), their values were estimated from the biodegradation rates of experimental XCH and XPR particulate fractions during storage. Finally, according to the literature [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Bouguettoucha | Unstructured Models for Lactic Acid Fermentation -A Review[END_REF], the half-saturation values of Monod kinetics are very low and, therefore, have negligible impacts on the simulation results. In the present case, a value of 0.1 kgCOD.m -3 was used for all the reactions where this parameter is involved. Kinetic and stoichiometric parameters set according to literature data are summarized in Table 6. Optimized parameters during simulations are further described in section 5. Suggested model parameters and discussion.

Simulations and comparison with experimental results

The simulation were performed for the input variables of different feedstocks in order to calibrate the stoichiometric and kinetic parameters of the model. This included different moisture contents, soluble carbohydrates and initial pH (Table 2). The simulations were run until the end of ensiling experiments, i.e. 98 days for catch crop and 120 days for cattle manure (at 25 °C).

Hardly accessible fractions

The results of simulations for hardly accessible fractions (XCH and XPR) are shown in Figure 3. The estimated data followed a slow first order degradation during silage and as a result the model was properly fitted to the variables XCH and XPR.

Easily accessible fractions

Experimental trends for readily accessible COD (SCH, SLA, SBA, SAC and SET) during ensiling (Figure 4) are, in their majority, well described by the simulation after calibration of kinetic and stoichiometric parameters (detailed in section 5. Suggested model parameters and discussion).

Still, for some variables there is a gap between the evolution of the model predictions and experimental results. Indeed, the simulation for CM-Glucose condition forecasts an accumulation of easily accessible sugars, due to the hydrolysis reaction. However, this was not observed experimentally. Since the SCH measured during ensiling includes only soluble sugars, this difference indicates that the sugars produced during hydrolysis would remain in the accessible particulate fraction. In addition, ethanol produced during the storage of catch crop is degraded in the long term. This was not accounted for in the simulations, as this reaction was not considered in the model design. Finally, for the worst ensilable feedstock -CM-Freshit was observed experimentally a total loss of easily degradable compounds. This is the case, e.g., of butyric acid, which does not accumulate during storage since it is rapidly consumed. However, this phenomenon cannot be described by the simulations, since the model does not include any direct acetogenesis reactions.

Inorganic nitrogen

In most storage conditions the model allows the representation of inorganic nitrogen evolution for a given content of accessible amino acids (Figure 5). On the one hand, the modelling results suggest that there is a large amount of SAA in catch crop. This compounds are degraded in SIN when there is not a strong acidity. On the other hand, cattle manure does not have a significant amount of accessible amino acids. Therefore, it is the slow mineralization of the proteins that leads to the slight increase of inorganic nitrogen content. In addition, experimental SIN content of CM-Fresh decreased between 30 and 120 days of ensiling, while the model predicted a slight increase for this variable. This discrepancy might be explained by a partial volatilization and loss of the inorganic nitrogen (given the high pH), which has not been taken into account in the model. Finally, the evolution of the experimental inorganic content for CC-Wilted was not correctly modeled. In fact there was a huge increase in the SIN content in the first 7 days of this experiment, which can not only be attributed to the phenomena embedded in the model.

pH

Generally speaking, since many chemical species affect the charge balance and consequently the H + ion concentration, pH is one of the most difficult parameters to predict. Nevertheless, the modelling of this algebraic variable was a great success in most of the cases tested (Figure 6). This was partly due to the use of the buffer capacity of the system. Moreover, the efficiency of these simulations was strictly related to the stability of the organic matter. In fact, for silages without clostridial development -CC-Wilted and CM-Glucose -pH variations are perfectly described by the model.

Conversely, for fresh cattle manure (CM-Fresh) significant differences between measured pH and simulations were observed, especially in the first weeks of ensiling.

This inconsistency can be explained by the difficulties in simulating the VFA evolution under acetogenic and methanogenic conditions.

BMP conservation

Since COD was chosen as unit of the model and degradation reactions take place under anaerobic conditions, COD loss only occurs through the gas phase. Therefore, simulated BMP conservation for a given storage duration (%𝐵𝑀𝑃 𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑒𝑑,𝑡 ) was calculated from the cumulative production of hydrogen and methane, according to the following equation:

%𝐵𝑀𝑃 𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑒𝑑,𝑡 = 1 - 𝑆 𝐶𝐻4,𝑡 + 𝑆 𝐻2,𝑡 𝐶𝑂𝐷 𝑑e𝑔𝑟𝑎𝑑𝑎𝑏𝑙𝑒,𝑖 (24)
Where 𝑆 𝐶𝐻4,𝑡 and 𝑆 𝐻2,𝑡 designate, respectively, the cumulative productions of methane and hydrogen at time t (kgCOD.m -3 ) and 𝐶𝑂𝐷 𝑑𝑒𝑔𝑟𝑎𝑑𝑎𝑏𝑙𝑒,𝑖 represents the degradable COD of raw material under anaerobic digestion conditions, obtained from BMP and COD analysis.

The simulations for the conservation of BMP are qualitatively consistent with the experimental results (Figure 7). This indicates that this model can provides hints on the best storage practices to preserve the energy potential of different feedstocks before anaerobic digestion. However, the results show that in some cases it is very difficult to obtain a reliable quantitative estimation of BMP preservation from COD losses by H2 and CH4 production. For certain conditions, this should be due to the fact that during ensiling some gains in bioavailability may occur (due to further hydrolysis of the inaccessible fractions [START_REF] Franco | Ensiling for biogas production: Critical parameters. A review[END_REF][START_REF] Franco | Cattle manure for biogas production. Does ensiling and wheat straw addition enhance preservation of biomass and methane potential?[END_REF][START_REF] Larsen | Co-ensiling of straw with sugar beet leaves increases the methane yield from straw[END_REF]), which has not been integrated into the model. In addition, when the ensiling is very inefficient, namely for CM-Fresh, the preservation of the BMP is overestimated by the simulations. This can partly be explained by the omission of the biodegradation reactions of butyric acid in the model.

Suggested model parameters and discussion

The results of the simulations were obtained by varying only a limited number of parameters (Table 7), i.e. kinetic constants and stoichiometry, which illustrates the robustness of the model.

Regarding the stoichiometric coefficients, only one part varied according to the type of feedstock used. Homo-fermentative LAB populations dominated during ensiling, but this was more evident for cattle manure, for which no ethanol was produced during storage. In addition, during clostridial fermentation the yield of butyric acid was lower with cattle manure. This may suggest that the activity of acetogenic populations is greater in cattle manure. In contrast, identical stoichiometric yields of proteolytic fermentation were found for both raw materials.

Hydrolysis reactions had very low kinetic constants, which expresses the fact that ensiling is above all a method of organic matter preservation. Kinetic constant of protein hydrolysis was higher during storage of catch crop and with low TS levels. This may be related not only to the greater enzyme activity with higher moisture content, but also to the degradation of proteolytic enzymes during drying. Nevertheless, this moisture effect was not observed for trials with cattle manure. This may be due to the fact that a limited range of TS content was used. Concerning the degradation of structural carbohydrates, kinetic coefficient varied for all storage conditions and it was not possible to establish any correlation (either with respect to the feedstock type, TS content or level of accessible carbohydrates). On the one hand, this can be linked to inaccuracies in the estimation of the real fraction of carbohydrates, which are difficult to access experimentally. On the other hand, it may not be conceivable to obtain a trend by using a simple first order reaction, given the variety of anaerobic mechanisms that can lead to the hydrolysis of complex sugars. Indeed, the conversion of these polymers under ensiling conditions can be, according to Dewar et al. [START_REF] Dewar | The hydrolysis of grass hemicelluloses during ensilage[END_REF] and McDonald et al. [START_REF] Mcdonald | The Biochemistry of Silage, Second Edi[END_REF] catalyzed either enzymatically or chemically (depending, e.g., on the pH level). Maximum specific growth rates of microorganisms in Monod equations varied among the tested conditions. For lactic acid bacteria, it decreased slightly with the TS increase in both raw materials. For instance, the increase of TS content of catch crop from 10% to 39% resulted in a decrease of about 10% in 𝜇 𝑚𝑎𝑥 . This same TS variation led to a reduction of almost 30% in the maximum specific growth rate of clostridial bacteria.

These results show that the growth of these two bacterial populations is dependent on the TS content. Also, the sensitivity to water availability was much higher for clostridial bacteria, which is why it is preferable to use higher TS content in silage to achieve low pH stability. This is in line with data found in literature suggesting that different levels of tolerance to dryness are noticed among the silage microorganisms [START_REF] Franco | Ensiling for biogas production: Critical parameters. A review[END_REF][START_REF] Mcdonald | The Biochemistry of Silage, Second Edi[END_REF][START_REF] Borreani | Fermentative profiles of field pea (Pisum sativum), faba bean (Vicia faba) and white lupin (Lupinus albus) silages as affected by wilting and inoculation[END_REF], which is also the case for other anaerobic environments [START_REF] Hyaric | Influence of moisture content on the specific methanogenic activity of dry mesophilic municipal solid waste digestate[END_REF][START_REF] Hyaric | Influence of substrate concentration and moisture content on the specific methanogenic activity of dry mesophilic municipal solid waste digestate spiked with propionate[END_REF]. For cattle manure, the sensitivity of clostridial bacteria to the water content could not be verified for the conditions tested.

In fact, for Fresh CM simulations the accurate value of 𝜇 𝑚𝑎𝑥 of clostridial bacteria could not be calculated because the reaction took place before the first sampling. As with previous populations, when methanogenesis reactions occurred during silage, it was possible to identify an increase in μ max of methanogenic bacteria at lower TS levels. On the other hand, it was not possible to establish a general correlation for the μ max of proteolytic bacteria. Indeed, for catch crop there was an increase in proteolytic activity at higher TS concentrations, whereas for cattle manure no trend was observed among the TS contents tested. Nevertheless, according to the literature [START_REF] Borreani | Fermentative profiles of field pea (Pisum sativum), faba bean (Vicia faba) and white lupin (Lupinus albus) silages as affected by wilting and inoculation[END_REF][START_REF] Mahmoud | Factors affecting the microbial and chemical composition of silage: IV. Effect of wilting on maize silage[END_REF][START_REF] Wilkinson | A laboratory evaluation of comfrey (Symphytum officinale L.) as a forage crop for ensilage[END_REF], a decrease in proteolytic bacterial activity is also expected for lower moisture levels. This deviation from the literature may be due to the fact that the available amino acid content was estimated from inorganic nitrogen production during silage, certainly having a significant associated error. Finally, it is also important to note that no comparison between the values of μ max of catch crop and cattle manure was done. In fact, they depend on the amount of initial active biomass, which was arbitrarily set for each feedstock from literature data. Among the remaining parameters for which identical values were used for all storage conditions, the pH inhibition function 𝐼 𝑝𝐻 showed interesting results (Figure 8). Indeed, the model suggests that LAB were not inhibited over most of the pH range tested, because of their low half-inhibition pH (pM,SU = 4.3) and the high sensitivity of their activity around this value (qSU = 35). This sensitivity also implies that, for pH values below 4.3, the lactic activity is very reduced, which explains the partial conservation of easily accessible sugars during CM-Glucose experiment. For the other biological reactions, the half-inhibition pH values used were 5.5 for clostridial and proteolytic fermentations and 7.0 for methanogenesis. Since for these three reactions the sensitivity around these pH values is also high, this inhibition function mathematically explains the delay in the growth of these bacterial populations for properly acidified silages.

Moreover, this evidences that methanogens are more sensitive to low pH conditions than other fermentative microorganisms [START_REF] Kamali | Anaerobic digestion of pulp and paper mill wastes -An overview of the developments and improvement opportunities[END_REF].

The results of the simulations also showed that this model can describe in a very correct way silages where lactic fermentation proliferates and dominates. However, when one advances on the chain of fermentation reactions (acidogenesis, methanogenesis), the simulations are less efficient. In fact, the model is simplified at the level of acetogenesis and methanogenesis and, consequently, these phenomena cannot be described quantitatively. This means that, in its current form, the model can be used to expose the evolution of ensiling of other types of feedstocks that have, at least, a high accessible sugar content, such as energy crops, forages, or sugar-rich agro-industrial waste. On the contrary, for inefficient inputs such as single-handedly cattle manure, this model may only be used to indicate that silage will be ineffective, that there will be substantial energy losses and that it will be necessary to intervene on the initial characteristics of these feedstocks to improve the performance of BMP preservation (for instance, by adding easily biodegradable sugars [START_REF] Franco | Co-ensiling of cattle manure before biogas production: effects of fermentation stimulants and inhibitors on biomass and methane preservation[END_REF]).

In addition, the model allows a qualitative estimation of the suitable conditions for preserving the energy potential of feedstocks. Indeed, it is shown that the conservation of methane potential is positively correlated with the content of easily accessible carbohydrates and total solids in the raw material. Nevertheless, quantitatively there are still differences between some simulations and the experimental results. For very inefficient storages, this can again be due to the simplification of the model in the final stages of anaerobic digestion. At the same time, for some conditions there are hydrolysis phenomena that may lead to bio-accessibility gains, which are not taken into account in the model.

Conclusions

A mathematical link was established between biomass storage and anaerobic digestion.

This soechio-kinetic model explained the main biochemical phenomena that occur during ensiling, including the evolution of pH and methane potential. The model framework is applicable for the main feedstocks used in agricultural anaerobic digestion (manure, crops). Therefore, this model can be a crucial support tool for the management of resources before anaerobic digestion. Since storage practices have an important impact on the conservation of the energy content of biomass, this work should contribute to optimize the performance of agricultural biogas plants. [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] YLA 0.265 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] YAA 0.265 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] YAC 0.055 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF] KS_SU kgCOD.m -3 0.1 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Bouguettoucha | Unstructured Models for Lactic Acid Fermentation -A Review[END_REF] KS_LA 0.1 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Bouguettoucha | Unstructured Models for Lactic Acid Fermentation -A Review[END_REF] KS_AA 0.1 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Bouguettoucha | Unstructured Models for Lactic Acid Fermentation -A Review[END_REF] KS_AC 0.1 [START_REF] Neal | A model of the anaerobic phase of ensiling[END_REF][START_REF] Bouguettoucha | Unstructured Models for Lactic Acid Fermentation -A Review[END_REF] NAA kmol.kgCOD -1 0.007 [START_REF] Batstone | Anaerobic Digestion Model No[END_REF]

Figure 1 -

 1 Figure 1 -Structure of the ensiling model -anaerobic phase. (1) Hydrolysis of

Figure 2 -Figure 7 -Figure 8 -

 278 Figure 2 -Variation of the pH inhibition function (𝐼 𝑝𝐻 = (𝑝𝐻 𝑝 𝑀 ⁄ ) 𝑞 1+(𝑝𝐻 𝑝 𝑀 ⁄ ) 𝑞 ) with the

Table 1 -

 1 Examples of chemical pathway for LAB and clostridial fermentation

	Reaction	Possible pathway	Consequence
	Homofermentative lactic acid fermentation	C 6 H 12 O 6 → 2C 3 H 6 O 3	pH decrease; conservation of organic matter
	Saccharolytic clostridial fermentation	2C 3 H 6 O 3 → C 4 H 8 O 2 + 2CO 2 + 2H 2	pH increase; increase of biomass activity/degradation

Table 3 -

 3 Rate coefficients and kinetic rate equations involved in biochemical reactions.

		Component (i) 	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
																		Rate, 𝝆 𝒋 (kgCOD.m -3 j -1 )
	(j) Process 	XCH	SCH	SLA	SET	XSU	SBA	SAC	SH2	SIC	XLA	XPR	SAA SIN	XAA	SCH4	XAC
	1	Hydrolysis of carbohydrates	-1	1												𝑘 1 𝑋 𝐶𝐻
	2	Lactic fermentation			-1	(1 -𝑌 𝑆𝑈 ) × 𝑓 𝐿𝐴,𝐶𝐻	(1 -𝑌 𝑆𝑈 ) × 𝑓 𝐸𝑇,𝐶𝐻	𝑌 𝑆𝑈								𝑋 𝑆𝑈 𝜇 max _𝑆𝑈 𝑌 𝑆𝑈	𝑆 𝐶𝐻 𝐾 𝑆_𝑆𝑈 + 𝑆 𝐶𝐻	(𝑝𝐻 𝑝 𝑀_𝑆𝑈 ⁄ 1 + (𝑝𝐻 𝑝 𝑀_𝑆𝑈 ) 𝑞 𝑆𝑈 ⁄ ) 𝑞 𝑆𝑈
	3	Clostridial fermentation					-1			(1 -𝑌 𝐿𝐴 ) × 𝑓 𝐵𝐴,𝐿𝐴	(1 -𝑌 𝐿𝐴 ) × 𝑓 𝐴𝐶,𝐿𝐴	(1 -𝑌 𝐿𝐴 ) × 𝑓 𝐻2,𝐿𝐴	1 -𝑌 𝐿𝐴 96	𝑌 𝐿𝐴			𝑋 𝐿𝐴 𝜇 𝑚𝑎_𝐿𝐴 𝑌 𝐿𝐴	𝑆 𝐿𝐴 𝐾 𝑆_𝐿𝐴 + 𝑆 𝐿𝐴	(𝑝𝐻 𝑝 𝑀_𝐿𝐴 ⁄ 1 + (𝑝𝐻 𝑝 𝑀_𝐿𝐴 ) 𝑞 𝐿𝐴 ⁄ ) 𝑞 𝐿𝐴
	4	Hydrolysis of proteins														-1	1	𝑘 4 𝑋 𝑃𝑅
	5	Fermentation of amino acids									(1 -𝑌 𝐴𝐴 ) × 𝑓 𝐵𝐴,𝐴𝐴	(1 -𝑌 𝐴𝐴 ) × 𝑓 𝐴𝐶,𝐴𝐴	(1 -𝑌 𝐴𝐴 ) × 𝑓 𝐻2,𝐴𝐴				-1	𝑁 𝐴𝐴	𝑌 𝐴𝐴	𝑋 𝐴𝐴 𝜇 𝑚𝑎𝑥_𝐴𝐴 𝑌 𝐴𝐴	𝑆 𝐴𝐴 𝐾 𝑆_𝐴𝐴 + 𝑆 𝐴𝐴	(𝑝𝐻 𝑝 𝑀_𝐴𝐴 ⁄ 1 + (𝑝𝐻 𝑝 𝑀_𝐴𝐴 ) 𝑞 𝐴𝐴 ⁄ ) 𝑞 𝐴𝐴
	6 Methanogenesis										-1		1 -𝑌 𝐴𝐶 64				1 -𝑌 𝐴𝐶	𝑌 𝐴𝐶	𝑋 𝐴𝐶 𝜇 𝑚𝑎𝑥_𝐴𝐶 𝑌 𝐴𝐶	𝑆 𝐴𝐶 𝐾 𝑆_𝐴𝐶 + 𝑆 𝐴𝐶	(𝑝𝐻 𝑝 𝑀_𝐴𝐶 ⁄ 1 + (𝑝𝐻 𝑝 𝑀_𝐴𝐶 ) 𝑞 𝐴𝐶 ⁄ ) 𝑞 𝐴𝐶
			Hardly accessible carbohydrates	(kgCOD.m -3 )	Easily accessible carbohydrates	(kgCOD.m -3 )	Lactic acid	(kgCOD.m

Table 4 -

 4 Acid-base equilibrium constants (25 °C)[START_REF] Lide | CRC Handbook of Chemistry and Physics, Internet Version[END_REF].

	Acid/base pair	pKa	Ka
	CO2 / HCO3 -	6.35	4.47 x 10 -7
	H2O / (OH -+ H + )	13.995	1.01 x 10 -14
	HBA / BA -	4.83	1.48 x 10 -5
	HAC / AC -	4.756	1.75 x 10 -5
	HLA / LA -	3.86	1.38 x 10 -4
	NH4 + / NH3	9.25	5.62 x 10 -10

Table 5 -

 5 Initial state variables for the tested raw materials. Concentrations are expressed in kgCOD.m -3 except for inorganic carbon/nitrogen (kmol.m -3 ).

			Condition
	State variable	Catch crop	Cattle manure
		CC-Fresh	CC-Wilted	CM-Fresh	CM-Glucose
	XCH	30.0	235	205	227
	SCH	44.0	74.0	9.00	170
	SLA	0.0	0.0	0.0	0.0
	SET	0.0	0.254	0.0	0.0
	XSU	0.142	0.208	0.142	0.155
	SBA	0.0	0.0	0.0	0.0
	SAC	0.541	0.848	13.8	7.13
	SH2	0.0	0.0	0.0	0.0
	SIC	0.00735	0.00735	0.00735	0.00735
	XLA	0.142	0.208	0.142	0.155
	XPR	28.0	160	24.3	27.6
	SAA	5.00	13.0	3.50	9.97
	SIN	0.00912	0.101	0.0547	0.0586
	XAA	0.142	0.208	0.142	0.155
	SCH4	0.0	0.0	0.0	0.0
	XAC	0.142	0.208	0.142	0.155

Table 6 -

 6 Fixed kinetic and stoichiometric parameters based on literature data.

	Parameter	Unit	Value	Reference
	YSU		0.265
		kgCOD.kgCOD -1

Table 7 -

 7 Suggested values for the kinetic and stoichiometric parameters optimized during simulations.

				Condition
			Catch crop	Cattle manure
	Parameter	Unit
			CC-Fresh	CC-Wilted	CM-Fresh	CM-Glucose
	k1 k4	d -1	0.002 0.012	0.0033 0.0017	0.004 0.0005	0.002 0.0005
	𝝁 𝒎𝒂𝒙_𝑺𝑼		0.72	0.65	>0.85 1	0.71
	𝝁 𝒎𝒂𝒙_𝑳𝑨 𝝁 𝒎𝒂𝒙_𝑨𝑨	d -1	0.84 0.56	0.60 0.58	>0.65 1 0.23	0.70 0.23
	𝝁 𝒎𝒂𝒙_𝑨𝑪		0.54	<0.46 2	0.394	<0.22 2
	pM_SU		4.3	4.3	4.3	4.3
	pM_LA pM_AA	-	5.5 5.5	5.5 5.5	5.5 5.5	5.5 5.5
	pM_AC		7.0	7.0	7.0	7.0
	qSU		35	35	35	35
		-
	fLA,CH		0.92	0.92	1.00	1.00
	fET,CH		0.08	0.08	0.00	0.00
	fBA,LA		0.54	0.54	0.33	0.33
	fAC,LA	kgCOD.	0.29	0.29	0.50	0.50
	fH2,LA	kgCOD -1	0.17	0.17	0.17	0.17
	fAC,AA		0.88	0.88	0.88	0.88
	fBA,AA		0.06	0.06	0.06	0.06
	fH2,AA		0.06	0.06	0.06	0.06

1,[START_REF] Franco | Optimizing storage of a catch crop before biogas production: impact of ensiling and wilting under unsuitable weather conditions[END_REF] The exact values were not determined with the experimental results since reactions either occurred before 7 days of storage 1 or were not significant until the end of ensiling 2 .

Acknowledgements

Ruben Teixeira Franco held a doctoral fellowship from the Rhône-Alpes region. This work has been undertaken within the SAM project (Stockage Avant Méthanisation -Storage Before AD) funded by ADEME (# 1506C0038). The authors thank the DEEP laboratory team, including David Lebouil, Hervé Perier-Camby, Nathalie Dumont and Richard Poncet for the given support during the tests. We are grateful to Franck Barra for his permanent availability for discussion and raw material supply. Mathilde Hardier and SUEZ are also acknowledged for the inoculum provided for the BMP tests.