Eloi Bernier

Vincent Kaufmann

Florian Masse

Juliana Gonzalez

Analysing the potential for modal shiftbased on the logic of modal choice

This article explores how the logic underlying modal practices tends to modulate the population's responsiveness to improvements in global transport supply. Based on a quantitative survey conducted in 2018-2019 among the working population of the cantons of Bern, Geneva and Vaud, it presents the construction of eight types, each of which corresponds to a specific combination of action logic and allows segmentation of travel behaviour. The exploration of the links between this typology and three metropolitan areas shows that the weight of the types within the active population is associated with the quality of the transport offered, especially at the workplace, and that the use of the different means of transport for commuting within the types stays consistent with the opinions of the respondents, as confirmed by an ordinal regression. All the results presented in this article show that a modal shift objective in daily mobility requires a strategy that is segmented by these types.

Introduction

The objectives set by the Paris Agreement regarding the decarbonisation of mobility require a policy of modal shift from cars to public transport and active modes of transport (cycling, walking). In this context, the modal shift is now the stated objective of a large majority of urban and transport policies in Switzerland. This ambitious objective implies significant investments and, therefore, goes hand in hand with major political challenges. All policies and measures in this area thus require a detailed understanding of demand, its evolution and its sensitivity to changes in transport supply. Without a good understanding of these elements, a certain number of investments and measures may not be very effective -or may even be counterproductive -in relation to a modal shift objective.

With regard to the use of means of transport in everyday life, recent studies have agreed that the comparison of travel times and prices is not the only factor explaining the use of means of transport. Indeed, modal practices are based on a much wider range of reasons, which have tended to diversify over the two last decades [START_REF] Jindra | The Dilemma of Equality and Diversity[END_REF][START_REF] Lufkin | Linking spatial characteristics to residential lifestyles. A framework for analysing the hospitality potential of urban and architectural designs[END_REF][START_REF] Spread | The Emerging Visions for Future Sustainable Lifestyles, Part 2. D 3.1. Visual material presenting emerging best practices and emerging visions on sustainable lifestyles[END_REF]. In particular, with the development of remote communication technologies, travel time has ceased to be an interstitial gap with a duration that should be minimised, provided that conditions of comfort allow time to be used. These same technologies have also contributed to the development of new modes of transport, such as carsharing, carpooling and Uber taxis, which allow users to limit the cost of their journeys.

Following these observations, and given the diversity of the population's attitudes towards means of transport, this article explores how the logic underlying modal practices is likely to modulate the population's reactivity to various improvements of the transport supply and thus constitutes an instrument for the strategic management of modal shift policies.

State of the art

With the growing awareness of the nuisances associated with automobile traffic, the modal shift from motorised individual transport (MIT) to means of transport that generate fewer negative externalities such as public transportation (PT) is an objective of urban transport policies throughout Europe [START_REF] Banister | Unsustainable Transport[END_REF][START_REF] Flamm | Comprendre le choix modal: les déterminants des pratiques modales et des représentations individuelles des moyens de transport[END_REF]). For more than 20 years, urban policies have attempted to achieve this type of objective in many European cities by acting on land-use planning and transport supply [START_REF] Apel | Transport concepts in European cities[END_REF]. This has led to proactive spatial planning policies aimed at reducing urban sprawl, concentrating urban growth near railway stations and in areas that are well connected by public transport. In Switzerland, this has been the case through the federal government's agglomeration policy, which invites city centres and adjacent municipalities to work together to implement urban-transport coordination projects in order to obtain a share of federal funding for major urban projects. Significant investments have been made in providing public transport, including tram networks, metro systems and regional rail services such as the Eloi Bernier is a Ph.D. student at the Urban Sociology Laboratory of the Ecole Polytechnique de Lausanne (EFPL), with a degree in civil engineering and urban mobility (EIVP -Paris). His work focuses on the logic of modal choice at the individual level and on the quantification of mobility potential as a critical factor for socio-professional integration.

Vincent Kaufman is an associate professor of urban sociology and mobility at Ecole Polytechnique Fédérale de Lausanne (EPFL). He has also been a scientific director of the Mobile Lives Forum in Paris since 2011. After a Master's degree in sociology (University of Geneva) he did his Ph.D. at EPFL on rationalities underlying transport modal practices. He has been an invited lecturer at Lancaster University (2000)(2001) S-Bahn. However, these policies have generally been hampered by behavioural inertia and resistance to change among motorists [START_REF] Canzler | Möglichkeitsräume -Grundrisse einer modernen Mobilitäts-und Verkehrspolitik[END_REF][START_REF] Fouille | L'attachement automobile mis à l'épreuve. Etude des dispositifs de détachement et de recomposition des mobilités[END_REF][START_REF] Buhler | Eléments pour la prise en compte de l'habitude dans les pratiques de déplacements urbains: le cas des résistances aux injonctions au changement de mode de déplacement dans l'agglomération lyonnaise[END_REF][START_REF] Buhler | L'habitude automobile face à la menace de congestion[END_REF].

De [START_REF] De Witte | Linking modal choice to motility: a comprehensive review[END_REF] highlighted four types of factors influencing mode choice for daily mobility, including spatial environment, travel characteristics, travellers' socio-demographic characteristics and socio-psychological factors. The study revealed that, among the determining factors, car availability, income, age, household characteristics, and density are often found to be significant in the literature that authors reviewed for their study. However, income effects tend to fade away in the wealthiest countries as most households own at least one car [START_REF] Lipps | Measuring and Explaining the Increase of Travel Distance: A Multi-Level Analysis Using Repeated Cross-Sectional Travel Surveys[END_REF]. This means that other sociodemographic characteristics may be more relevant to explain the modal choices. As an example, the effects of car use seem to be differentiated by gender, since men are more likely to use a car than women, or age, as older people over 60 are more likely to travel by public transport for long-distance commuting [START_REF] Ko | Exploring factors associated with commute mode choice: An application of city-level general social survey data[END_REF] or even walk or bike [START_REF] Buehler | Determinants of transport mode choice: a comparison of Germany and the USA[END_REF]. However, the oldest as well as the youngest people make fewer and shorter trips compared to employed males (Giuliano, Dargay 2006).

Nevertheless, socio-economic attributes appear to be insufficient to make complete predictions about travel behaviour. Consequently, studies are increasingly incorporating latent variables representing attitudes, perceptions and preferences into choice models (Rieser-Schüssler, Axhausen 2012; [START_REF] Atasoy | Attitudes towards mode choice in Switzerland[END_REF]. For approximately the last 20 years, the use of socio-psychological approaches has made it possible to observe behaviour in detail through social representations, which introduce elements relating to perceptions and attitudes to explain certain forms of dissociation between opinions and actual behaviour (Gärling et al. 1998;[START_REF] Van Acker | When Transport Geography Meets Social Psychology: Toward a Conceptual Model of Travel Behaviour[END_REF][START_REF] Wang | Attitudes, Mode Switching Behavior, and the Built Environment: A Longitudinal Study in the Puget Sound Region[END_REF][START_REF] Ma | Mode Choice with latent preference heterogeneity: a case study for employees of the EU institutions in Luxembourg[END_REF]. A close examination of the literature indicates that the difficulty in bringing about transfers from car use to other means of transport in daily life also depends on two orders of factors [START_REF] Brög | Changer de comportement c'est d'abord changer d'état d'esprit, Marketing et qualité de service dans les transports en commun[END_REF][START_REF] Scheiner | Interrelations between travel mode choice and trip distance: trends in Germany 1976-2002[END_REF]): people's mobility skills (in particular, their access to networks and their knowledge of transport offerings) and the attitudes and experiences they have developed towards different means of transport [START_REF] Canzler | Möglichkeitsräume -Grundrisse einer modernen Mobilitäts-und Verkehrspolitik[END_REF][START_REF] Ettema | How in-vehicle activities affect work commuters' satisfaction with public transport[END_REF]De Vos et al. 2015;[START_REF] Higgins | All minutes are not equal: Travel time and the effects of congestion on commute satisfaction in Canadian cities[END_REF][START_REF] Rérat | Cycling to Work: Meanings and Experiences of a Sustainable Practice[END_REF]. A large body of social sci-ence research has shown that the mobility skills that determine the use of transport in everyday life are not reducible to the minimisation of travel time and price comparison [START_REF] Dupuy | À La recherche du temps gagné[END_REF][START_REF] Fichelet | Éléments pour une compréhension des pratiques de déplacement automobile[END_REF]Duhemet al. 1995;[START_REF] Kaufmann | Pratiques modales des déplacements de personnes en milieu urbain: des rationalités d'usage à la cohérence de l'action publique[END_REF][START_REF] Jain | The gift of travel time[END_REF][START_REF] Metz | The Myth of Travel Time Saving[END_REF]. Factors relating to value systems and habits play a central role in these practices, as do the perceptions of possible alternatives.

The reception potential of these practices in the territory -and especially the set of factorsrefers to what George Amar (1993) calls urban adhesion; that is, the adequacy of transport modes with respect to the spaces crossed. Thus, the form and diversity of the built environment are central in accounting for the use of different transport modes [START_REF] Cervero | Walking, bicycling, and urban landscapes: evidence from the San Francisco bay area[END_REF][START_REF] Handy | Correlation or causality between the built environment and travel behavior? Evidence from Northern California[END_REF][START_REF] Kaufmann | Les paradoxes de la mobilité -Bouger, s'enraciner[END_REF]). The literature on car dependency reflects this problem [START_REF] Newman | Cities and automobile dependence[END_REF][START_REF] Dupuy | La dépendance automobile: symptômes, analyses, diagnostic, traitements[END_REF][START_REF] Ewing | Travel and the built environment: A meta-analysis[END_REF][START_REF] Buehler | Reducing car dependence in the heart of Europe: lessons from Germany, Austria, and Switzerland[END_REF][START_REF] Li | Exploring car ownership and car use in neighborhoods near metro stations in Beijing: Does the neighborhood built environment matter?[END_REF], pointing out that universes of coherence make territories dependent on the automobile system and lead to the intensive use of the car as a driver [START_REF] Buhler | Eléments pour la prise en compte de l'habitude dans les pratiques de déplacements urbains: le cas des résistances aux injonctions au changement de mode de déplacement dans l'agglomération lyonnaise[END_REF][START_REF] Munafò | Typologie et évolution des logiques de choix modal chez les actifs motorisés urbains -Étude comparée des agglomérations de Geneva[END_REF].

This position leads us to consider the use of means of transport as the activation of people's dispositions to use it in a given supply context. Dispositions to use reflect more or less positive attitudes towards different modes of travel, which are measured and analysed, in particular, through the image that people have of the different modes of transport. The context of supply is understood through the concept of affordances [START_REF] Gibson | The ecological approach to visual perception[END_REF]Wilhoit 2017). According to Gibson, urban context and transport offerings constitute as many holds for people, offering various possibilities of action in the urban environment. Thus, the use of a means of transport, that is, the use that is actually made of it, results from the match between the qualities offered by a means of transport and the population's willingness to use it.

Taking Max [START_REF] Weber | Wirtschaft und Gesellschaft[END_REF] three logics of action as a starting point, the willingness to use can be conceptualised in the form of three principles that can be combined based on the potential for reception that these logics encounter in the territory.

Instrumental logic implies that users choose their modes of transport according to objectified values of time and money. It is the efficiency of the car in terms of marginal cost -but, above all, of speed -which is the mechanism behind its preferential use. According to this rationale, cars are the means of transport that Juliana Gonzalez holds an engineering degree in industrial engineering (UIS -University, focusing on individual travel behaviour regarding leisure mobility in coastal metropolitan areas. His work focuses on understanding the underlying determinants of travel behaviour. conducted in the shortest time and at the lowest cost. The second logic of action does not refer to the comparison of alternatives based on their respective performances but on value systems. The predominant use of the car would then result from a 'differential desire' to use the private car compared to other means of transport. Car use would a priori be preferred to that of public transport because of the intrinsic characteristics of the mobility offered by it. Driving one's own car and moving around in the private space of the car's cabin encourage individualisation and individuation and make the car a powerful symbol of freedom. The third logic of action relates to habits and routines. Having to make choices implies effort to seek information about the alternatives available and effort of comparative experimentation. It can, therefore, be rational to make a choice because the automatism of habits is a comfort. This logic refers directly to lifestyle. It follows that the use of one means of transport cannot be substituted for another without calling into question certain aspects of social integration. This anchoring of modal habits in lifestyle would then be a major obstacle to change in modal practices.

Data collection and methods

Our study is based on data from the 2018 modal choice survey [START_REF] Kaufmann | Analyse des logiques de choix modal auprès de la population active du Grand Geneve (Cahier du LaSUR, 33A)[END_REF]), produced at the request of several public partners in the cantons of Bern, Geneva and Vaud (see acknowledgements) in order to collect updated data to identify the potential for modal shift among the working population, which can be compared with those collected in two similar surveys conducted in 1994 and 2011 [START_REF] Kaufmann | Sociologie de la mobilité urbaine: la question du report modal[END_REF]Munafo et al. 2012). The questionnaire investigates factors that influence mobility behaviours such as: 1) the equipment (in subscriptions, vehicles and connected tools); 2) the habits of use of the different means of transportation and their evolution over time; 3) the image and opinion about their quality; 4) the use of travel time.

Among the available data from the survey (also including several cities and periurban sectors in the three cantons), this article focuses on the metropolitan areas surveyed (Geneva, Lausanne, and Bern as a reference for the benchmark), each consisting of the central city with suburban municipalities (Figure 1).

The data were collected in the autumn of 2018 (Geneva and Lausanne) and the summer of 2019 (Bern). The data collection protocol was based on a telephone survey (RBS landline method with a CATI treatment) conducted by a polling company with a randomly selected sample (among quota and survey sectors defined with the partners), which was representative of the target population in terms of sex, age and geographical location. All individuals in the target population met the following two criteria: "Be active (full or part-time)" and "Live within 500 m walking distance of a public transport stop (or 800 m from a railway station) with a relatively good service (main public transport network)".

According to the survey institute and common standards (AAPOR 2015), our response rate is close to 30%, (with no prior recruitment, incentives nor information letters). The number of pages or questions, the complexity of the questions posed, and the saliency of the survey are also determinants. Schmidt and Axhausen (2019) reviewed these determinants and proposed an objective way of rating the response burden with a scoring approach. With 48 questions, eight transitions, and answering actions of multiple types (from simple yes/no actions to more complex tables to complete), the response burden estimate reaches a score of 404.5 points for the choix modal 2018 questionnaire. This score is average compared to the 65 reported surveys employed in the operationalisation of the response rate estimate (ibid.: 7). The average completion time for our questionnaire was approximately 30 minutes.

Finally, the 2598 workers studied in this article (601 in Lausanne; 1739 in Geneva; 258 in Bern) were distributed as follows within the sample (Table 1) 1 . In terms of methods, the first step in the analysis of this dataset for this study consisted in developing a typology of the logics of action that makes it possible to link dispositions with regard to means of transportation. The typology enables accounting for the combination of the three logics of action that may be involved in mobility behaviour: the instrumental logic of action, broken down into functional, sensitive or social orientation; the preferences for use and the environmental values attributed to the various means of transport (by integrating active modes); and the anchoring of mono-or multi-modal habits in daily life 2 .

In order to identify the willingness of the respondents to use the different means of transport, they were asked in the survey to spontaneously name three adjectives to describe the car, public transport and the bicycle. This corpus of adjectives made it possible to create several variables: (1) positive or negative attitude variables with regard to the various modes of transport (one per mode -for the car, public transport and the bicycle), (2) an identification variable for the registers of action contained in the corpus of adjectives, i.e. the search for efficiency, the search for sensorial comfort and the search for sobriety. The variables thus created were used to construct the typology.

Methodologically, the construction of the typology is based on the results of a discriminant cluster analysis. It is an update of the typology of logics of action developed in the 1990s, which was based on a cluster analysis of the variables associated with the three parameters presented above (for a scheme of its construction see [START_REF] Kaufmann | Sociologie de la mobilité urbaine: la question du report modal[END_REF][START_REF] Munafo | Dispositions et usages de l'automobile et des transports publics entre 1994 et 2011: Analyse des cas de Bern[END_REF]or Kaufmann et al. 2019: 93). Eight types, each corresponding to a specific disposition towards the use of different means of transport, were thus obtained (see section results Table 4 for a description of the types).

The last step of this work was to test the influence of the dispositions underlying modal practices on the frequency of MIT use for home-work trips. To assess the factors that may influence modal practices, these dispositions identified in the typology were tested in conjunction with control variables related to public transport accessibility, metropolitan area of residence and other socio-demographic control variables (Table 2). Two variables related to the The method used for the modelling was an ordinal logistic regression that compared the probability of choice between several ordered modalities of the variable to be explained ( McFadden 1975). This method is based on the postulates of transparency of the environment, that is, the knowledge by individuals of all avail-able alternatives as well as the rationality of their choices. This approach has historically been based on journey-related characteristicscost, comfort, connection, etc. -to which differences related to socio-demographic characteristics -age, education, household composition, etc. -are then added (Souche 2010). The characteristics of the places used are also considered, namely the origin and destination of trips.

Results

Starting from the first descriptive results on the modal practices of the respondents, we present the results of the analytical typology of modal choice logics. Finally, we analyse the association of the typology, among other variables, on the MIT use for commuting with our interpretation of the ordinal regression.

Equipment, practices and modal habits of respondents

The analysis of the average number of trip-days (per week and per respondent) highlights the comparative intensity of transport mode use. Work-related trips appear to be less frequent than non-work-related trips, accounting for less than 30% of total trips (Figure 2).

The results confirm the choice of Bern as a reference, where public transport and cycling are used more than in Geneva and Lausanne, where MIT trips remain predominant, even though Bern has the highest total number of trip-days. A detailed study of modes shows that the MIT category primarily corresponds to working people who drive their car, while public transport is more oriented towards urban public transport than trains, particularly in Geneva (22% of trip-days). Regarding bicycles, electric models are still marginal, including in Lausanne, where the declivity already explains the low use of conventional bicycles. Generally speaking, the differences in modal share between the three conurbations can be explained by the accessibility of the different means of transport (i.e., MIT, PT, active modes, etc.). This is particularly true for the car in Bern, where a large proportion of jobs are located in the city centre and near railway stations, and where parking facilities are limited within the city and the agglomeration railway network is highly developed.

From the point of view of reducing polluting emissions, it is important to emphasise the predominant role of daily use ("every day or almost every day") in car traffic: approximately 75% of car journeys (all motives) correspond to those of daily motorists who, nevertheless, represent less than 50% of the respondents.

Non-motorised households are over-represented in Bern (31%, compared with 18% in Geneva and 16% in Lausanne), while the most frequent situation in the three metropolises is of households with a car (Geneva and Lausanne 58%, Bern 45%). As the vehicle may be shared within the household or linked to a mobility service, around one-third of our respondents have a vehicle every day, and another third has one two or three times a week.

Parking remains a key variable in terms of car accessibility, which greatly determines the constraints linked to the use of this mode in an urban environment. Between 8% and 20% of workers employed in the central areas use reserved parking spots at their place of work; this proportion exceeds 20% for workers coming to work in suburban areas, and reaches more than 30% in Geneva and Lausanne if non-reserved parking spots at the place of work are also considered. Working people have relatively safe parking.

As far as public transport season tickets are concerned, the territorial differences within our sample of working people are striking (Table 3). Bern stands out from the rest with a particularly high subscription rate: only 7% of Bern respondents do not have any of the subscriptions offered, compared to 29% in Lausanne and 38% in Geneva. As the average price of local subscriptions is slightly superior in Bern than in French-speaking cities 4 , we suspect that this prevalence of PT subscriptions in Bern is mainly due to the strong performance of public transportation, particularly for the trains, thanks to the central position of this city in Switzerland, which is much better than the location of Geneva, for example (where the Geneva-Lausanne railway is the only alternative to access the rest of the country by train). This hypothesis stands in line with the fact that the subscriptions in Bern are primarily due to a large proportion of SBB passes.

By providing car parking spaces at the workplace, often at advantageous rates, the employer participates directly in financing car use to go to work (depending on the sector, between 50% and 75% of workers going to work by car could be affected). For users who go to work by public transport and have a season ticket, the employer contributes in about a third of cases. Full coverage represents about 15% of the workers considered in the three metropolises, while partial coverage is more variable.

Generally speaking, connections are associated with the use of public transport: the more the number of connections increases, the more Residents of Bern have a high proportion of two-wheelers (90% of respondents personally own at least one, compared to 68% in Geneva and 53% in Lausanne), particularly conventional bicycles (83%, 52% and 42%, respectively).

Typology of logics underlying modal practices

The dispositions towards means of transport are synthesised in the analytical typology constructed from Max Weber's three logics of action [START_REF] Munafo | Dispositions et usages de l'automobile et des transports publics entre 1994 et 2011: Analyse des cas de Bern[END_REF] based on the methodology presented in Section 3. These different variables combined enable identification of eight logics or types associated with a specific disposition towards the use of different means of transport (Table 4).

The distribution of the eight types identified in Geneva, Lausanne and Bern reveals several general trends (Table 5). On the one hand, 'exclusive motorists' are marginal in the two French-speaking cities and have now completely disappeared in Bern. This seems counterintuitive in relation to the connotations of the adjectives cited to describe the modes of transport. Indeed, the image of the car is more negative in Lausanne (43%) and Geneva (35%) than in Bern (26%). In the typology, we find more MIT-predisposed people, especially in Lausanne (9%), while the 'comfort comparators' and 'environmentalist' reach a threshold for all the areas (around 10% and 7% of respondents, respectively).

With regard to the differences in the logics of action based on the geographical context, the 'exclusive motorists', 'MIT-predisposed', 'efficiency comparators' and 'predisposed to individual modes' types are systematically over-represented in the suburban areas compared to the central cities. This seems logical since these types are favourable to the car, the use of which is more important in suburban municipalities, even if the adjectives used to describe the car in the two types of territory remain similar. For the 'predisposed to alternative modes' and 'ac-Tab. 4: Description of the eight modal choice logics in the typology used since 2018.

Types of modal choice logic Description

Exclusive motorists

They only use the car in their daily lives; their activity programmes are structured around the accessibility offered by this means of transport.

Predisposed to MIT

They have a strong preference for the use of cars and motorised two-wheelers for the freedom of space and time that these means of transport allow. These people are attached to the rapid and individual crossing of space.

Comparing efficiency

They are primarily responsive to comparisons of the efficiency of transport modes. They will therefore favour the fastest and most cost-effective means of transport.

Comparing comfort

They are, above all, reactive to comparisons of travel comfort. The search for comfort and ergonomics in travel time is motivated, in particular, by the desire to use travel time as free time or, on the contrary, as working time.

Predisposed to individual means of transportation

This group is characterised by an attachment to autonomous travel. As much as possible, these people avoid being confronted with the constraints of public transport systems (i.e. lines, schedules and shared travel).

Predisposed to sustainable means of transportation

They do not like driving and prefer to use other means of transport. It should be noted that their motivation for not using the car is not related to ecological considerations but rather to the stress of driving (traffic jams, accidents etc).

Predisposed to active means of transportation

They favour the use of active means of transport (bicycle, electric bicycle and walking) and avoid using motorised means of transport as much as possible in their daily life.

Environmentalists

They favour the use of environmentally-friendly means of transport in line with their convictions. Their image of the different modes of transport is marked by environmental considerations. The distribution of types by education level indicates that the 'environmentalist' type is correlated with longer education, while respondents with upper-secondary education are under-represented among the 'alternative mode inclined'. Those with an apprenticeship (or less) are numerous among the 'efficiency comparators'.

This segmentation by logic of action allows an understanding of the levers, enabling a modal shift towards sustainable modes of transport: each type is characterised by a spe-Tab. 5: Distribution by modal choice logic by residential context (metropolitan areas), in % of respondents. timodal transport supply and accessibility. If we look closely at the use of MIT through the prism of modal choice logics, we notice that the types favourable to MIT indeed make more intensive use of it 5 , and the opposite is true for the types unfavourable to MIT (Table 6). The 'comparators' show intermediate use.

Exclusive

However, the types that use MIT most for individual travel are not necessarily those that contribute most to car traffic. The intensity of use must be qualified based on the number of users of the type in question. It is, therefore, interesting to translate these proportions of users into shares of total car traffic to obtain a vision of the impact of each type (Table 7).

Ordinal regression analysis: the influence of disposition on MIT commuting habits

To assess in the presence of control variables the association of the typology with the actual use of MIT, we ran an ordinal regression (link: logit) on 2230 individuals, where the dependent variable is the frequency of commuting days with motorised individual transport (MIT), as a function of socio-demographic characteristics, context and concurrent PT accessibility, associated with individual modal dispositions. This variable is coded with the five following levels: Never / One day per month or fewer / Two to four days per month / Two to three days per week / Four days per week or more.

The prediction model indicates goodness of fit to the observed data (p<.001). The Nagelkerke R-squared (0.269) and the non-significant results of Deviance and Pearson chisquare tests also show that the model fits the data well [START_REF] Petrucci | A primer for social worker researchers on how to conduct a multinomial logistic regression[END_REF]. The assumption has been satisfied because the test of parallel lines is not significant (p>.01). We also checked the absence of multicollinearity and the low correlation between the variables.

Firstly, the results of this analysis seem globally consistent with the expected outputs. If we study the disposition to MIT use based on the logics of modal choice, our results confirm the significant impact of each type on MIT use, all other things being equal. Apart from exclusive motorists and those predisposed to MIT (as references), those predisposed to individual modes still have a higher probability of using MIT compared to other efficiency or comfort comparators. Individuals who are disposed to alternative, active or environmentally proactive modes consistently choose other modes of transport over MIT. Nevertheless, we see that the environmentalists are not totally able to become independent of the use of MIT compared to the respondents predisposed to use active or alternative modes of transport. This fact could reflect a cognitive dissonance among the adjectives stated during the survey.

All other things being equal, the populations of Lausanne and Geneva, to a lesser extent, are more inclined to use MIT. This result is in line with figure 2, due to the fact the three Swiss cities considered here have had different development regarding public transportation and land use, and have different topographical and spatial configurations [START_REF] Kaufmann | The Coordination of Local Policies for Urban Development and Public Transportation in Four Swiss Cities[END_REF].

Regarding the number of commuting days, only a small number of commuting days per week (2 or fewer) have a significant influence on reducing MIT use, probably due to lifestyles associated with part-time jobs (low income compared to expenses for the use of a car, and a flexible schedule can encourage the use of PT services even with low coverage) while a distant workplace doesn't significantly reduce MIT use. These results are interesting for teleworkers and their potential trade-off between frequency and distance for commuting (Ravalet, Rerat 2019). We can see that good or very good PT service coverage at the workplace significantly reduces the use of MIT for commuting, whereas only a very good level at home will have a significant but weaker impact on it. Furthermore, the number of connections in the journey by PT encourages the use of MIT as soon as there is one connection to be made. This result is probably reflecting the fact that commuters are sensitive to the optimisation of their daily trips, which are often during peak hours (even for flexitime workers: Wöhner 2022) and sometimes used as part of their working time.

Regarding sociodemographic variables, women seem to be less willing to use MIT (this result is in line with most research on this topic for regular trips in the daytime, see McCright, Xiao 2014) as well as people between 36 and 49 years old and younger workers even if the coefficient is not significant for this generation (which was more difficult to survey with landlines). Education seems not to be significant among our sample of working people, as we can assume that all respondents have a minimum level of education. Surprisingly, household size is not significant, again we can understand that commuting trips do not necessarily imply transportation of children, which can attenuate the expected effect of this variable (Mc-Carthy et al. 2021 The dispositions highlighted by the typology reveal the potential for modal shift, particularly among individuals sensitive to the instrumental logic of time and money. This is the case for individuals willing to compare the efficiency of transport modes and those predisposed to individual transport, who are more involved in the global share of MIT use in the three metropolitan areas. Finally, the issue of connections seems to be a determining factor in the attractiveness of public transport, particularly for working people who compare the comfort provided during the whole trip. Indeed, the presence of a chain of journeys between the home and the workplace significantly favours a shift towards car use.

In view of these results, two types of strategies can be adopted to encourage modal shift. The first concerns the improvement of transport services, especially relating to the sensitivity of the service at the place of work for our sample of residents in urban areas. For example, in commercial and industrial areas where accessibility by public transport is often neglected, they are often located close to major roads, offering a real potential to improve travel conditions, which could lead to a signifi-cant modal shift if an integrated approach was adopted by all the players concerned (including workers and employers).

The second strategy focuses on the influence of dispositions regarding modes of transport, based on the fact that environmentalists and efficiency comparators seem to be the most sensitive types to modal shift in the case of a variation in the quality of service at the workplace. As we suggest improving public transport in working areas, we also propose to focus on the benefits to exploit these topics in communication or pricing campaigns. The objective of such a proposal is to influence individuals' dispositions to shift from an insensitive form of disposition, to environmental or comparative efficiency sensitivities. What we mean here is that we can't push people to switch from exclusive motorists to environmentalists in one step. However, the intermediate status of comparators is an achievable milestone that can lead to progressive modal shifts thanks to the improvement of sustainable transport accessibility around work. With this aim, both the focus on improving PT accessibility and attracting through communication and pricing is necessary for such a paradigm to engage.

As future research directions, a longitudinal study following the same people going over several years 7 will bring us findings about what determinants influence shifting between predispositions in order to identify the process to support populations into evolving progressively from MIT use to more sustainable solutions. We are also interested in producing detailed results in order to understand the differences revealed by our results between German-and French-speaking respondents, for example, in a mutual urban context such as the bilingual city of Biel/Bienne, which is also available in our database.

Notes

1 The under-representation of respondents aged 35 and under (who use landlines less) was not adjusted for due to the sampling of sectors based on isochrones (6 min, i.e. about 500 m travelled) around public transport stops and the low impact on the results obtained for the active population. 2 For more details on the corpus of data used to construct the typology, see the working paper 'Analyse des logiques de choix modal auprès de la population active du Grand Genève' [START_REF] Kaufmann | Analyse des logiques de choix modal auprès de la population active du Grand Geneve (Cahier du LaSUR, 33A)[END_REF]). 3 Source: https://www.are.admin.ch/are/fr/home/ mobilite/bases-et-donnees/desserte-en-suisse. html 4 According to this comparator: https://www.combien-coute.net/abo-bus/suisse/ 5 The number of trip-days per week can exceed 7 as the MIT category results from the additional use of several means of transportation. 6 A value less than 1 means that the modality negatively influences the probability of strongly using MIT, while a value greater than 1 means that the modality positively influences the probability of strongly using MIT, compared to the modality of reference for each qualitative variable and all other things being equal. Modalities with confidence intervals that intersect the value 1 are not significant (threshold: 0.05). 7 Lake Geneva Sustainability Monitoring Panel (Ongoing research)

Fig. 1 :

 1 Fig. 1: Metropolitan areas ( Geneva, Lausanne and Bern) and sectors for surveying.

Fig. 2 :

 2 Fig. 2: Average weekly frequency of mode use by purpose and residential context (metropolitan areas), in trip-days (for calculation method: Kaufmann et al. 2020).

  disP 231 • 58.4 (4/2022) 65 bility Department), City of Nyon (Mobility Department), City of Biel (Urban Planning Department), Federal Institute of Technology of Lausanne (EPFL): Laboratory of Urban Sociology (LaSUR).

  ).

	The reference is the last modality of each factor 64 disP 231 • 58.4 (4/2022) 5 Discussion and conclusion Odds	Standard	95% confidence interval	disP 231 • 58.4 (4/2022) 63
	ratio The logic of action underlying modal practices, error * p < 0.05, ** p < 0.01, *** p < 0.001	Lower	Upper
		Environmentalists as measured by the typology based on adjec-0.184*** 0.227 tives for each mode and consistently associated	0.118	0.287
		Predisposed to active means of transportation with modal practices (e.g., car use for com-0.137*** 0.215 muting as confirmed by the ordinal regression	0.090	0.208
		Predisposed to sustainable means analysis), highlights opportunities and obsta-
		of transportation cles to a modal shift from cars to other means 0.125*** 0.188	0.087	0.181
	Disposition	Predisposed to individual means of transport. In light of the analyses presented,
		of transportation we should focus on two main types of factors: 0.454*** 0.197	0.309	0.668
		Comparing comfort those related to the quality of PT accessibility, 0.257*** 0.199 and those associated with the phenomena of	0.174	0.380
		Comparing efficiency cognitive dissonance (Masse et al. 2020). We 0.235*** 0.184	0.164	0.336
		Exclusive motorists / Predisposed observe this with the bicycle in Geneva, which
		to MIT	1 has a good image but is used only marginally.
	Metropolitan area	Geneva Lausanne Some contexts also seem to favour an ambig-2.117*** 0.172 uous attitude towards cars, without having any 2.414*** 0.183 effect on usage. This is the case in Bern, where Bern positive opinions about cars are the highest, 1 yet usage is the most virtuous. Thus, the sym-	1.511 1.687	2.966 3.453
		1 or fewer bolic dimension of the car (Rubens et al. 2011) 0.355*** 0.142	0.269	0.469
		2	0.648* seems to persist without altering the possibility 0.170	0.464	0.904
	# days outside for work	3	0.778 of a fall in modal share, in this case, minimis-0.137 ing the influence of values and preferences in	0.595	1.017
		4	0.809 favour of other instrumental logics. Conversely, 0.119	0.640	1.022
		5 or more we observe the opposite fact in Lausanne and 1 Geneva, where negative opinions towards cars
	PT service coverage at	Very good (A-level) tend to suggest a lack of performant alterna-0.309*** 0.148 Good (B-level) 0.449*** 0.158 tives for commuting.	0.232 0.329	0.413 0.611
	workplace	Bad (C-level or less)	1
		Very good (A-level)	0.677*	0.179	0.477	0.961
	PT service cov-erage at home	Good (B-level)	0.828	0.184	0.578	1.187
		Bad (C-level or less)	1
	Home to work
	distance	(Continuous, unit: 10 km)	0.972	0.031	0.915	1.033
		0		0.235***	0.132	0.181	0.304
	Home to work connections	1		0.392***	0.146	0.294	0.521
		2 or more	1
	Sex	Male		1.808***	0.093	1.506	2.171
		Female		1
		35 and under	0.805	0.146	0.605	1.071
	Age	36-49		0.744**	0.098	0.615	0.901
		50 and over	1
		Apprenticeship or less	1.186	0.105	0.964	1.458
	Education	Secondary	1.061	0.116	0.845	1.333
		University or equiv.	1		1.000	1.000
	Household	No children	0.927	0.092	0.774	1.110	Tab. 8: Ordinal logistic regression 6 analysis (dependent
	composition	With children	1				variable: ordinal frequency of MIT use for commuting).

Acknowledgements

This work was supported by the following partners: Agglomeration of Yverdon-les-Bains (Agglo Y), BLS SA, State of Geneva (Department of Infrastructure and Department of Territory), State of Vaud (General Directorate of Mobility and Roads), Greater Geneva, Lémanis SA, Swiss League for the Rational Organisation of Traffic (LITRA), Region of Nyon (Territory, Environment and Mobility Pole), Public Transport of the Lausanne region (TL), Unireso, Union des Transports Publics, Union des Villes Genevoises (Carouge, Geneva, Lancy, Meyrin, Onex, Vernier), City of Lausanne (Mobility Division), City of Morges (Urban Planning, Construction and Mo-


domain: spi chim

text:
Jean-François Cornet

Jérémi Dauchet

Thomas Vourc'h
email: thomas.vourc-h@sigma-clermont.fr

Carolina Arnau

David Garcia-Gragera

Francesc Gòdia

Fabrice Gros

Enrique Peiro

A Simple and Reliable Method to Determine Mean Incident Light Flux Densities on Cylindrical Photoreactors and Photobioreactors from a Unique Fluence Rate Measurement

come

INTRODUCTION

The use of solar or artificially-illuminated photoreactors and photobioreactors (P(B)R) is increasing with the growing interest in photo-reactive processes. They are used for the development of microalgae with various applications [START_REF] Borowitzka | high-value products from microalgae -their development and commercialization[END_REF], the depollution of liquid [START_REF] Serrano | Photocatalytic degradation of water organic pollutants. Kinetic and energy efficiency[END_REF][START_REF] Kabra | Treatment of hazardous organic and inorganic compounds through aqueous phase photocatalysis: a review[END_REF][START_REF] Janin | Solar photocatalytic mineralization of 2,4-dichlorophenol and mixtures of pesticides: kinetic model of mineralization[END_REF] and gaseous effluents [START_REF] Ku | Photocatalytic degradation of gaseous benzene in air streams by using an optical fiber photoreactor[END_REF][START_REF] Da | Strategies for the intensification of photocatalytic oxidation processes towards air stream decontamination: a review[END_REF], the production of solar fuels [START_REF] Xing | On the engineering part of solar hydrogen production from water splitting: photoreactor design[END_REF][START_REF] Nguyen | Recent developments in the design of photoreactors for solar energy conversion from water splitting and CO 2 reduction[END_REF], the synthesis of molecules of interest for chemistry and fine chemistry [START_REF] Braun | Photochemical technology[END_REF], the synthesis of platform molecules and high-value products, etc. Different geometries co-exist [START_REF] Braham | Review of major design and scale-up considerations for solar photocatalytic reactors[END_REF][START_REF] Legrand | A review on photobioreactor design and modelling for microalgae production[END_REF], but for many processes, the cylindrical geometry predominates as it presents a large specific illuminated area and is suitable for controlling reactor mixing. Regardless of geometry, radiative transfer always controls the kinetic and energetic performances of a P(B)R [START_REF] Dauchet | Photobioreactor modeling and radiative transfer analysis for engineering purposes[END_REF] and the first quantity that determines the maximum performances of a P(B)R is the mean hemispherical incident light flux density on the illuminated surface. This is the reason why it is of considerable importance to know accurately its value [START_REF] Dauchet | Photobioreactor modeling and radiative transfer analysis for engineering purposes[END_REF][START_REF] Cassano | Photoreactor analysis and design: fundamentals and applications[END_REF][START_REF] Rochatte | Radiative transfer approach using Monte Carlo method for actinometry in complex geometry and its application to Reinecke salt photodissociation within innovative pilot-scale photo(bio)reactors[END_REF]. Several methods exist for this, each with their strengths and weaknesses.

Mapping from a flat hemispherical incident flux density sensor is often used because it is quite easy to implement, but suffers from a significant lack of accuracy because it is difficult to take into account the heterogeneities often existing at the level of light sources or to be adapted to cylindrical geometries [START_REF] Dahi | A novel experimental bench dedicated to the accurate radiative analysis of photoreactors: the case study of CdS catalyzed hydrogen production from sacrificial donors[END_REF]. Actinometry may be preferred; it perfectly averages the surface heterogeneities, but appears much more tedious to implement, especially in cylindrical or complex geometries [START_REF] Rochatte | Radiative transfer approach using Monte Carlo method for actinometry in complex geometry and its application to Reinecke salt photodissociation within innovative pilot-scale photo(bio)reactors[END_REF][START_REF] Zalazar | The extended use of actinometry in the interpretation of photochemical reaction engineering data[END_REF][START_REF] Roibu | An accessible visible-light actinometer for the determination of photon flux and optical pathlength in flow photo microreactors[END_REF].

Many cylindrical P(B)R are radially illuminated with a surrounding light source device, since this design is more favorable in terms of specific illuminated area compared to an annular reactor with a central light source. They can be either artificially-illuminated reactors or solar reactors with cylindrical parabolic compounds. In this particular case, there is another method based on the central measurement of fluence rate with a spherical sensor. This method is very simple to implement and can be extremely accurate if the theory for deducing the average hemispherical incident light flux density is properly established. The purpose of this article is to present in detail this accurate, simple and reliable method with the help of all the necessary theoretical developments. The proposed approach will then be confronted with experimental results to validate the method in two different and complementary situations, using different lamps, angular distributions of incident radiation as well as different geometries and scale factors.

MATERIALS AND METHODS

This section presents the two experimental devices of photo(bio)reactors used in this article to validate the methodology together with important considerations regarding fluence rate measurement and conversion units.

Lighting system and measurements for the photo(bio)reactor with halogen lamps BAB 36-38°

The first experimental device used in this study is a photo(bio)reactor illuminated by halogen lamps. In more details, the lighting system is composed of 55 halogen lamps BAB 36-38° radially arranged with a fixed internal radius for emission R C = 12 cm. The internal radius of the P(B)R is R P(B)R = 8 cm and the semi-height of the vessel is H = L/2 = 14 cm (see Figure 1).

The working volume of this reactor is 5 L. For a beam angle of 36-38° given by the manufacturer for this kind of lamps (Osram, Sylvania,....), i.e. a viewing angle of 18.5°, equation [START_REF] Zalazar | The extended use of actinometry in the interpretation of photochemical reaction engineering data[END_REF] gives the degree of collimation n = 13 (see Figure 2 and theoretical section). Lighting system and measurements for the photo(bio)reactor with LEDs + optics For this second experimental device, the lighting system is composed of 80 white LEDs + optics by panel × 8 panels arranged in two octagonal identical layouts around the two cylindrical parts of the reactor (for a total of 640 LEDs). The radius of the P(B)R is R P(B)R = 7.5 cm and the semi-height of the vessel is H = L/2 = 75 cm (see Figure 3). The working volume of this reactor is roughly 85 L. For a beam angle of 24° given by the manufacturer of the LEDs (Luxeon, Sunplus 20),

R S R P(B)R 2H R C (a) (b)
i.e. a viewing angle of 12°, equation [START_REF] Zalazar | The extended use of actinometry in the interpretation of photochemical reaction engineering data[END_REF] gives the degree of collimation n = 31 (see Figure 4 and theoretical section). 9.2 cm -13.9 cm -18.9 cm. For these three situations, we have calculated the view factor C S F → according to Eqs. ( 20) and ( 23) hereafter with our Monte Carlo algorithm (see Supporting Information). The results are summarized in Table 2, in the Results & Discussion section of the paper.

Fluence rate measurement

It is very important here to point out that, whatever the external lighting device used as emitting source and its surface heterogeneity, the measurement of the fluence rate by a spherical sensor placed in the center of a cylindrical tank and which is moved from bottom to top has a remarkable stability. The measurement obtained is very accurate (better than +/-5%) and perfectly averages the fluctuations observed at the emitting wall, even with the very small diameter sensor (3 mm). The two sensors used are quantum-meters delivering the fluence rate in µmol hν .m -2 .s -1 in the visible light spectrum (more precisely in the photosynthetically active radiation PAR [400 -700 nm]), but the results for incident radiant flux densities q R are given hereafter in W.m -2 (in the same range of wavelengths). The conversion factors for each kind of illuminating source have been obtained by the following relation from the knowledge of the lamp energy distribution spectrum E λ (pdf) given by the suppliers [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène (in french)[END_REF]:

1 W.m -2 = 1.1974 × 10 mol . m -2 .s -1 = , 1.1974 × 10 mol . m -2 .s -1
where the term 1.1974 × 10 is obtained from the Avogadro number, the Planck constant h and the light celerity in vacuum c 0 from: 1.1974 × 10 = 6.02 × 10 " × ℎ × $ .

This leads to the following conversion factors in the PAR range of wavelengths:

-for the white halogen lamps BAB: 1 W.m -2 = 5.1 mol . m -2 .s -1

-for the white LEDs used: 1 W.m -2 = 4.6 mol . m -2 .s -1

THEORETICAL CONSIDERATIONS

The aim of this theoretical section is to establish rigorously the relations between the fluence rate measurement ψ by a spherical sensor (radius R S ) located at the centre of a cylindrical and radially illuminated device (cylinder of radius R C , see Figure 5) and the hemispherical mean incident photon (or radiative) flux density q 0 (PFD) at the wall of this device. By "mean" it is assumed here that this incident PFD is homogeneous on the emitting surface; even if practically it is not the case (it is generally not the case), we will observe in the experimental result section that this assumption gives excellent (accurate) results with much less effort than using actinometry for example. In this work, we will also assume a cosine power law for the angular distribution of the emission directions; this law allows to describe a full range of situations from collimated to Lambertian sources.

Once the hemispherical flux density at the emitting wall q 0 is known, it is then easy, using the conservation law for radiant energy, to calculate the incident PFD at any concentric cylindrical wall located inside the emitting cylinder (with a radius R P(B)R ), i.e. the incident PFD q R for any photo(bio)reactor. In such conditions, this incident PFD q R corresponds to the radiative source at the boundary (rear of the vessel wall). This PFD along with q R homogeneity assumption and cosine power law for emission directions fully provide the boundary condition for the photoreactive process. The general system of coordinates, the main considered variables and the angular distribution of light at the boundaries are summarized on Figure 5. Any spherical sensor is calibrated to measure a photon (or radiant) fluence rate defined by the double integral [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène (in french)[END_REF][START_REF] Naught | Compendium of Chemical Terminology[END_REF]:

2 2 2 2 2 ( ) ( ) 1 1 ( , ) ( ) ( , ) cos S S S S S S dr q r n r dr d L r n r dr d L r R R R π π ψ ω ω ω ω ω ξ π π π + + ∩ ⋅ = = ⋅ =      (1)
in which the directional quantity ( , ) L r ω is the luminance. The luminance is also referred in textbooks of radiative heat transfer as specific intensity (I) [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF]. This definition ensures that for R S → 0, ψ tends to the spherical irradiance

( ) ( , ) G r L r d

π ω ω

=  [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène (in french)[END_REF][START_REF] Naught | Compendium of Chemical Terminology[END_REF]. Alternatively, Eq. ( 1) may be also rewritten in a simple way from the total radiative flux received by the sensor Q S :

2 S S Q R ψ π = (2)
Relations for the total radiative flux Q S at the sensor surface

In this context, the establishment of a relation between the measured fluence rate ψ and the unknown hemispherical incident flux density q 0 becomes then a "simple" problem of radiation exchange between surfaces. This approach is extremely well documented in the literature [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF] and requires knowing the radiative view factor of the given studied geometry:

- S C
F → is the view factor of the sensor (S) toward the surrounding cylinder (C);

- C S
F → is the view factor of the surrounding cylinder (C) toward the sensor (S).

The radiative balance in the system is then straightforward (since the surrounding cylinder of radius R C and height 2H is the unique radiative source):

C S C S 0 C S F 2 2 F C C S Q Q q R H Q π → → → = = = (3)
Using Eq. (2-3), we then obtain the following useful relationship:

C S 0 C S 2 2 2 F 4 F S C C S S S Q Q q R H R R R ψ π π → → = = = (4)
This equation demonstrates that it is possible to find the relation between the fluence rate ψ and the incident PFD q 0 (at the sources) from the knowledge of only the view factor C S F → .

Moreover, once q 0 is known, it will be easy to determine the incident PFD q R on any concentric cylindrical surface with same height (the reactor vessel surface A P(B)R ) located between the sources and the spherical sensor of fluence rate (R S < R P(B)R < R C , see Figures 1 &3) by applying the conservation of the radial radiative flux Q r :

0 ( ) ( ) cte 2 2 2 r C R P B R P B R Q q R H q R L π π = = = (5)
i.e., since 2

PBR H L ≃ : 0 ( ) C R P B R R q q R = (6)
The practical relations if the incident luminance (intensity) is isotropic in any direction ω (Lambertian boundary condition) from classical approach of radiative thermodynamic equilibrium (black body theory)

In this case, at the boundary of the cylinder (the source), the incident luminance (intensity) is independent of the direction ω and has the same value L 0 in any incoming direction (see Figure 5). Consequently, the hemispherical incident flux density (the unknown of the problem) is given by (see Figure 5): 0

0 q L π = .
Additionally, this situation is described in the literature providing the view factor S C F → [START_REF] Feingold | New analytical approach to the evaluation of configuration factors in radiation from spheres and infinitely long cylinders[END_REF] and using the reciprocity relation that is available in the case of equilibrium -Lambertian -boundary conditions

2 C C S C S F F 2 2 F 4 F S S C C S S C A A R H R π π → → → → = = = (7)
we obtain the view factor C S F → :

2 2 C 2 2 2 F 1 S S S C C C C R R R H R R R H H → = = +   +     (8)
Finally, substituting this equation into the general expression (4) gives the expression of the incident PFD:

2 0 1 4 C R H q ψ   +     = (9)
Attention must be paid here, however, to the fact that this definition imposes to put the spherical sensor at the middle of the total height L = 2H (see Figure 5) of the cylinder [START_REF] Feingold | New analytical approach to the evaluation of configuration factors in radiation from spheres and infinitely long cylinders[END_REF].

The practical relations if the incident luminance is perfectly collimated (with normal incidence)

In this case, at the boundary of the cylinder, the incident luminance (intensity) is non-zero only in the direction

( ) C R L r q r = .
Especially, for the integration on the fluence rate sensor area, and restricting the application conditions to H > R S (see Figure 5), we have (

0 0 sin C C S S dQ q R dz d dQ L R dz d φ θ φ = = = ): 0 0 ( ) sin C S R L r q R θ = (10)
The definition of the fluence rate (Eq. 1) may then be rewritten as follows:

( )

2 2 0 2 0 0 1 sin ( ) ( ) S x S d d R L r e n R π π ψ φ θ θ θ π = -⋅   (11)
Because the normal ( ) n θ is clearly a function of the angle θ in the integration (see Figure 5) [ ]

with
2 0 0 0 2 0 0 2 sin 2 cos 4 S C C C S S S S R R R R q d q q R R R R π π π ψ θ θ θ π = = - =  (12)
Finally, Eq. ( 12) leads to the practical relationship giving the hemispherical incident quasicollimated flux density q 0 from the experimental measurement of the fluence rate ψ from:

0 4 S C R q R ψ = (13)
It must be noticed that using Eq. ( 4) with Eq. ( 12) enables to demonstrate that the view factor is:

C S C S F if H F 1 if H S S S R R H R → → = > = < (14)
which is of course, a very intuitive value for a normal collimated incidence (see Figure 5).

The practical relations if the incident luminance is neither collimated nor Lambertian but diffuse with any degree of collimation 0 < n < ∞ -The general case.

In this general situation, any point source of light emits a radiation in a cone of beam angle θ B (twice the viewing angle θ v ) with azimuth symmetry over φ = 2π. This situation is here approached by a directional distribution of the luminance 0 ( ) L Ω assumed to obey the cosine power law:

n 0 0 0 0 0 ( , ) cos ( ) L L θ φ θ = ( 15
)
where n is called the degree of collimation of the radiation field (a real value, not necessarily an integer). This law is often used to simply model the radiation pattern of sources [START_REF] Wu | A precise model of LED lighting and its application in uniform illumination system[END_REF] as the simplest one, among more complicated approaches [START_REF] Moreno | Modeling the radiation pattern of LEDs[END_REF].

From this definition, any radiation field may be represented if θ v (or beam angle θ B ) is known and n determined in agreement with this information generally given by the manufacturer. For this, one needs to identify the correct value of n fitting with the definition of the viewing angle (see above Figures 2 &4) for which the luminance is 50% of the direct forward luminance (θ = 0), i.e.:

1 2 cos ( )

n V θ = or 1 2 ln( ) ln(cos ) V n θ = (16)
Eq. ( 15) then enables to treat any incident characteristic of the source between the collimated case ( n = ∞ ) and the Lambertian case (n = 0) discussed above.

In this general case, the incident flux density at the source is always given from Eq. ( 4) by:

2 0 C S 4 F S C R q R H ψ → = (17)
and the incident flux density at the rear PBR wall from Eq. ( 6):

2 ( ) C S 4 F S R P B R R q R H ψ → = (18)
In the two simple cases already discussed, the view factor C S F → has an analytical form:

-For diffuse -Lambertian incident luminance, n = 0 and the view factor is given by eq. ( 8),

2 C 2 F 1 S S C C R R R H H → =   +    
-For collimated incident luminance, n = ∞ , and the view factor is given by eq. ( 14),

C S F S R H → = .
But for any other value of 0 n < < ∞ generally encountered, the view factor C S F → must be defined and evaluated by a numerical procedure. The general definition of the view factor C S F → is given by: ( )

0 0 C 0 0 0 0 2 2 2 1 0 0 0 0 0 0 0 ( , ) 1 F cos ( ; ; ) cos sin ( 2) ( ) 2 2 2 S S C C H n H L C dC d C R A q d dy d d n H π π π ω ω θ ω φ φ θ θ θ π π → + - = ∆ = + ∆             H H ( 19
)
where 19) becomes:

2 1 1 0 C 0 0 0 0 F ( 2) ( ) 2 2 H n S H d dy d n H π φ µ µ π + → - = + ∆        H ( 20
)
From the notations defined on Figure 5 and the symmetry already discussed, we can easily define the ( , , )

C S S S R R a y R b R c σ θ θ φ σ θ φ θ σ θ φ θ φ - = + = = (21)
Note that for convenience in this demonstration, because of a symmetry around the x axis, we have used y < 0 in order to have 0 0; 2

π θ   ∈    
. Taking the square of each term and summing, we get:

( ) ( ) ( ) 2 2 2 2 0 0 0 0 0 cos sin cos sin sin C S R y R σ θ σ θ φ σ θ φ - + + + = (22)
Rearranging the polynomial finally leads to:

2 2 2 2 0 0 0 2 ( sin cos cos ) 0 C C S y R R y R σ θ φ θ σ + - + + - =
the discriminant of which being:

2 2 2 2 0 0 0 4 ( sin cos cos ) 4 ( ) C C S y R R y R θ φ θ ∆ = - - + - (23)
The sign of this discriminant concludes the demonstration since it allows to evaluate the Heaviside function H (∆) in Eq. ( 20) according to whether it is positive or negative. The view factor C S F → given by the triple integral Eq. ( 20) can be easily interpreted as an expectation in the statistical point of view and evaluated by a numerical procedure using a Monte Carlo algorithm and a Python code summarized in Supporting Information. It enables to achieve the calculation of the view factor C S F → for any degree of collimation n by solving Eqs ( 20) and ( 23) with a 95% confidence interval. A web application where the interested reader can implement this calculation is also made available at http://gspr.ip.uca.fr/fluence_rate_view_factor.

As already mentioned in the Materials & Methods section, two actual situations are presented in this article to illustrate the procedure in the general case: a)-validation of the approach with a 5 L cylindrical photoreactor using white halogen lamps BAB 36-38° at constant R C , but with two diameters of fluence rate sensors R S ;

b)-validation of the approach on a 85 L cylindrical photobioreactor using a scalable illuminating system with white LEDs 24° with a constant radius of sensor R S but with three radii R C for the distance of the sources.

Hence, for these two kinds of P(B)R, the view factor C S F → has been evaluated by the Monte Carlo algorithm (see Supporting Information) from Eqs. ( 20) and ( 23) after specifying the degree of collimation n and the geometry for each case (radius sensor R S , radius for the illuminating device R C , radius for the P(B)R, R P(B)R and total height 2H).

Correction of interfaces from geometrical optics.

Experimentally and theoretically speaking, the final quantity that we try to evaluate q R is the hemispherical incident photon flux density at the rear of the wall (glass, quartz, etc.) of the photo(bio)reactor (input of the reaction medium) in the case of radial illumination. If the fluence rate measurement is performed with the photo(bio)reactor filled with water or any solvent (using underwater sensors), so no correction is necessary in the incident PFD calculation q R by the preceding relations (Eq. 18 for instance). At the opposite, if the fluence rate measurement is performed with an empty P(B)R (in air) or without P(B)R, it is possible to increase the accuracy of the q R calibration from the knowledge of the reflectivity ρ of an air-material (glass) interface correction. In case of a measurement in air (no vessel between the fluence rate sensor and the illuminating device), the calculated value of q R must be finally roughly multiplied by a factor (1 -ρ) to take into consideration the air-glass interface (the second interface with water or any liquid has a negligible reflectivity). If the measurement is done with an empty cylindrical vessel (the photo(bio)reactor), then, the calculated value of q R must be divided by (1 -ρ) because using finally aqueous solution inside the reactor (or any other solvent) almost completely extinguishes the reflectivity at that second interface.

In the general case (i.e. for any degree of collimation considering the incident radiation at the wall), the directional reflectivity ρ(θ) at an interface is a function of the angular direction θ and can be obtained, for unpolarized radiation, by the algebraic average, knowing the relative refractive index n r between the glass and the fluid (air, water, etc.) [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF]:

& ∥ ()) = + , - . cos

&()) =

7 ∥ (8)97 : [START_REF] Nguyen | Recent developments in the design of photoreactors for solar energy conversion from water splitting and CO 2 reduction[END_REF] .

(

) 24
The mean normalized reflectivity ρ is then obtained by the following integrals for any degree of collimation n:

& = 2;< &()) cos ()) cos()) sin ()) = > ) 2;< cos 9" ())sin ()) = > ) = ? &()) cos 9" ())(n + 2)sin ()) = > ) (25)
It is well known that for the two extreme situations (collimated and Lambertian incidence) already discussed, the integral (Eq. 25) is analytical [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF]:

-Normal collimated radiation field (, = ∞): & = A B " B 9" C .
-Lambertian radiation field (n = 0):

& = " . + (D B 9")( B ") E( B 9") > + B > F B > "G > F B > 9"G H ln A B " B 9" C -2 B H F B > 9. B "G F B > 9"GF B I "G + 8 B I F B I 9"G F B > 9"GF B I "G > ln (, -)
For any other situation with 0 < n < ∞ , the integral (Eq. 25) must be solved numerically.

As a case study, we can invoke here the calculation of the reflectivity for an air-glass interface, assuming a relative refractive index of 1.55 for glass. In this case, the analytical relation for a collimated radiation field (, = ∞) gives ρ = 0.046 and the analytical relation for a Lambertian radiation field (n = 0) leads to ρ = 0.099. This demonstrates that the proposed correction in this case for the incident PFD may vary from 5 to 10% and is not always negligible. The behaviour of this same reflectivity with the degree of collimation n may be observed on Figure 6 that displays values given by Eqs. (24-25). One can observe that the quasi-collimated hypothesis for reflectivity calculation (the limit dotted line on Figure 6) is accurate for n higher than 20 (with less than 1% deviation).

RESULTS AND DISCUSSION

In this section, we present and discuss all the experimental results obtained for the two technical setups studied: i) a first setup with constant radial distance for the sources (radius R C ) but varying the radius R S of the spherical sensor, and ii) a second setup with a unique sensor radius but varying the distance for the sources. For each situation, the light source also is different and representative of the diversity of experimental illuminating systems with degree of collimation varying between n = 13 to 31. For each illuminating system, the value of the mean incident PFD q R on the fixed P(B)R surface is of course independent of the fluence rate sensor size and the diameter of the illuminating device. Therefore, we expect to obtain the same q R value for each of our case study. Indeed, modifications of sensor size R S or illuminating device radius R C in Eq. ( 18) is compensated by the corresponding modification of the view factor C S F → .

Thus, obtaining a constant value of the PFD q R for any of the following experimental situation guarantees first that the view factor C S F → is properly determined and second that the theoretical relation between fluence rate and incident PFD is correct. It is particularly important to mention at this stage that except from edge effects, it is possible, for each experimental device, to move the fluence rate sensor vertically upwards or downwards at the position R (or x) = 0 (center of the photo(bio)reactor, see Figure 5) during a measurement. In this case, a remarkable stability of the signal is observed, even for sources having high degrees of collimation with a great heterogeneity of the sources on the emission surface. This result shows that the proposed method has a strong ability to average these source heterogeneities and at the same time, it ensures a very high accuracy for the average value of the PFD thus determined (estimated experimentally at 5%).

First case study with fixed diameter of the illuminating system, halogen lamps BAB 36-38° (n = 13) and two sensor diameters For this first situation depicted in Material & Methods section (see Figure 1), the results of fluence rate measurements with two sensors and the calculation of the incident PFD from Eq. ( 18) for three values of the view factor C S F → (Eqs. 20 & 23) are summarized in Table 1. It clearly appears that the collimated or Lambertian classical assumptions for sources with a beam angle of 36-38° are not adapted to the calculation of the view factor C S F → and then to the incident PFD at the rear of the PBR wall. Nevertheless, with a general treatment using a degree of collimation n = 13 for the calculation of the exact view factor C S F → gives exactly the same value q R = 450 W.m -2 = 2300 µmol hν .m -2 .s -1 with the two sensors of diameters 3 and 0.15 cm respectively.

This emphasizes the importance to have a complete angular model for the incident radiation generated by the sources, with an exact degree of collimation. This is required to properly calculate the view factor C S F → according to Eq. 20 & 23, and then obtain an accurate (uncertainty 5%) estimation of the mean incident PFD from the fluence rate measurement at a position R (or x) = 0 at the center of the photo(bio)reactor (See Figure 5). 1: Results for the incident hemispherical radiative flux density (or PFD) q R in the first case study: P(B)R with radius R P(B)R = 8 cm and semi-height H= 14 cm illuminating system composed of halogen lamps (36-38°) at a fixed emission position radius (R C = 12 cm). Results obtained with 2 different radii of fluence rate sensor R S . The PFD can be obtained from the radiative flux (last column) with the conversion factor 1 W.m -2 = 5.1 mol . m -2 .s -1 .

Second case study with fixed sensor diameter, three variable diameters for the illuminating system with LEDs and optics at a beam angle of 24° (n = 31)

For this second situation depicted in Material and Methods section (see Figure 3), the results of fluence rate measurements with three different diameters for the illuminating octagonal system and the calculation of the incident PFD from Eq. ( 18) for three values of the view factor C S F → (Eq. 20 & 23) are summarized in Table 2. Once again, it clearly appears that the collimated or Lambertian classical assumptions for sources with a beam angle of 24° are not adapted to the calculation of the view factor C S F → and then to the incident PFD at the wall. Nevertheless, with a general treatment using a degree of collimation n = 31 and calculating the exact view factor (Eq. 20 & 23) gives exactly the same value q R = 360 W/m 2 = 1660 µmol hν .m -2 .s -1 for the three distances of the sources (R C of 9.2 cm, 13.9 cm and 18.9 cm respectively) as it should be.

Finally, because all the measurements have been performed in air with the presence of the photo(bio)reactor between the sensor and the sources (two air-glass the q R values obtained in Tables 1 and2 must be corrected by dividing them by (1 -ρ), with ρ = 0.047 for n = 13 (first case study) and ρ = 0.046 for n = 31 (second case study). This leads respectively to q R = 470 ± 20 W/m 2 or 2400 ± 100 µmol hν .m -2 .s -1 for the first case and q R = 380 ± 20 W/m 2 or 1750 ± 90 µmol hν .m -2 .s -1 for the second case. Table 2: Results for the incident hemispherical radiative flux density (or PFD) q R in the second case study: P(B)R with radius R P(B)R = 7.5 cm and semi-height H=75 cm illuminating system composed of white LEDs (24°) with fixed fluence rate sensor radius (R S =3 cm). Results obtained for 3 different positions of the sources at distance R C . The PFD can be obtained from the radiative flux (last column) with the conversion factor 1 W.m -2 = 4.6 mol . m -2 .s -1 .

photon flux density, which is an experimental characteristic essential to the description of the kinetic and thermodynamic performances of these photo(bio)reactors. In this case, it appears to be simpler, faster and more accurate than actinometry, which remains the reference method in complex geometry [START_REF] Rochatte | Radiative transfer approach using Monte Carlo method for actinometry in complex geometry and its application to Reinecke salt photodissociation within innovative pilot-scale photo(bio)reactors[END_REF]. Additionally, this article clarifies and extends the domain of validity of the relations of the literature for the calculation of reflectivities of the interfaces to the general case of a source with any degree of collimation. This is indeed an important correction step to increase the accuracy in the incident PFD determination. • An angle S is uniformly sampled between 0 and 2;

• A value = cos () ) is sampled according to the probability density function T( ) = (, + 2) 9" . To do so, we uniformly sample a random number L between 0 and 1. The cumulative density function (CDF) corresponding to T( ) is $( ) = U T( U ) = 9.

V , thus the value of that we retain by inverting the CDF is = L W XY> . The pair (S , ) defines a direction in the 3D space.

• To assess whether there is an intersection, we compute the discriminant Δ according to Eq [START_REF] Moreno | Modeling the radiation pattern of LEDs[END_REF]:

o If Δ ≥ 0, there exists an intersection and we count \ = 1 o If Δ < 0, there is no intersection and we retain \ = 0 Finally, the above sampling procedure is repeated N times and the view factor is estimated by the average of the N \ samples. The 95% confidence interval is provided from the standard deviation of the \ set .

A Python code has been developed to implement this algorithm and is also proposed hereafter for the interested reader (the input data correspond here to the first case study of the paper with halogen lamps BAB 36-38° and the collimation factor n = 13):

Figure 1 :

 1 Figure 1: Representation of the P(B)R with the illuminating device using 55 BAB 36-38° halogen lamps. (a) Photograph of the P(B)R and (b) Scheme for the main variables and sensor location under experimental fluence rate measurement procedure.

Figure 2 :

 2 Figure 2: Polar representation of the normalized luminance (L(θ)/L max (0), thick line) with a degree of collimation n = 13. Case of halogen lamps BAB 36-38° with a mean beam angle of 37° (thin line). The dotted line represents the location of iso-luminances independent of the θ angle for half of the maximum normalized luminance value (L(θ)/L max = ½), corresponding to the definition of the beam angle.

  Normalized polar angle θ (+/-2θ/π, adim.) Normalized luminance (adim.) beam angle, θ b = 37°d egree of collimation, n = 13

Figure 3 :

 3 Figure 3: Representation of the P(B)R with the illuminating device composed of 80 white LEDs + optics by panel × 8 panels arranged in two octagonal identical layouts around the two cylindrical parts of the reactor. In this case, the beam angle is of 24°. (a) Photograph of the P(B)R, (b) Scheme for the main variables and sensor location under experimental fluence rate measurement procedure and (c) Top view for different measurement positions of the sources at distance R C during fluence rate (ψ) measurement (distance units are in mm).

Figure 4 :

 4 Figure 4: Polar representation of the normalized luminance (L(θ)/L max (0), thick line) with a degree of collimation n = 31. Case for white LEDs + optics with a mean beam angle of 24° (thin line). The dotted line represents the location of iso-luminances independent of the θ angle for half of the maximum normalized luminance value (L(θ)/L max = ½), corresponding to the definition of the beam angle.

Figure 5 :

 5 Figure5: Notations for the academic case establishing relations between the fluence rate ψ (measured by a spherical sensor of radius R S at the centre of an illuminating cylindrical device of radius R C and height L = 2H) and the radial hemispherical incident photon flux density q 0 (at the sources) for i) a normal collimated incidence, and ii) a Lambertian (diffuse) incidence.

  δ (x-a) is the Dirac distribution centred in %. The view factor of the problem is not published in the literature, but in this simple situation, we can envisage to evaluate directly the fluence rate from its integral formula (Eq. 1). In cylindrical coordinates indeed, as already mentioned, we must verify the radial radiative flux conservation (Q r = cte), i.e. a concentration effect in the form: 0

  Eqs. (10) and (11):

∈

  A C is the cylinder C area and H (∆) is the Heaviside generalized function which is 0 if ∆ is negative and 1 if ∆ is positive. ∆ is a discriminant defining the condition that the half-line 0 in direction ω at the cylinder boundary intercepts the spherical sensor. Because of the symmetry around the y axis, the integral on φ vanishes and introducing the notation µ = cos(θ), Eq. (

5 .& 6 (

 56 ()) -1, - . -sin . ()) , -. cos()) + 1, -. -sin . ()) )) = + 1, -. -sin . ()) -cos()) 1, - . -sin . ()) + cos()) 5.

Figure 6 :

 6 Figure 6: reflectivity of an air-glass interface (n r = 1.55) as a function of the degree of collimation n (Eq. 25) for the radiation field. One can observe that the two analytical limiting cases for Lambertian radiation (n = 0; ρ = 0.099) and collimated radiation (, = ∞; ρ = 0.046) are perfectly given by the general Eq. (25).

ACKNOWLEDGMENTψ∥

  ABBREVIATIONSPFD, mean hemispherical incident photon flux density; P(B)R photo(bio)reactor.

Table

CONCLUSIONS

In this paper, we rigorously established a theoretical relationship between fluence rate measurement using a spherical sensor placed at the center of a cylindrical illumination device and the average incident photon flux density emitted by the sources (PFD). This theoretical approach not only makes it possible to deal with the classic cases of a collimated or Lambertian incident radiation field, but above all, it is valid for any kind of source as soon as its degree of collimation n (properly defined and introduced in the paper from the manufacturers' data) is known. It is then possible to numerically calculate the view factor C S F → in all generality as established in the article, for example using the web application or the Python code provided along with this article (http://gspr.ip.uca.fr/fluence_rate_view_factor). Knowing this view factor is the preliminary step to calculate the incident PFD in all situations, from a simple measurement of the fluence rate using a spherical sensor put at the centre of the cylindrical device. The approach was then validated on two experimental devices by a redundant approach. In the first case, the diameter of the illuminating system is fixed (constant R C ), but the diameter of the fluence rate sensors varies (variable R S ); in the second case, it is the diameter of the sensor which is fixed (constant R S ) and the diameter of the illuminating system which varies (variable R C ). In all the tested situations, whatever the degree of collimation, the method proved to be very accurate both for the stability of the measurement of the fluence rate (which makes it possible to average the heterogeneities of the emitting wall in a remarkable way) and for the theoretical relations established in this article.

Thus, the method proposed here in the case of radially illuminated photo(bio)reactors proves to be extremely reliable and very simple to implement for accurately calibrating the mean incident

SUPPORTING INFORMATION

Algorithm and Python code for the Monte Carlo assessment of the view factor C S F → in the general case with any degree of collimation n for incident radiant emission at the boundary.

This section summarizes the way to evaluate the view factor C S F → from eqs (20):

and [START_REF] Moreno | Modeling the radiation pattern of LEDs[END_REF] in the paper:

As explained in the text, the triple integral Eq. ( 20) can be easily interpreted as an expectation in the statistical point of view and evaluated by a numerical procedure using the Monte Carlo algorithm as follows:

Except for the inputs/outputs, this Python code is exactly the code executed when using the web interface for the C S F → view factor calculation at http://gspr.ip.uca.fr/fluence_rate_view_factor.