M Dali-Youcef
email: manel.dali-youcef@inrae.fr

J Harmand
email: jerome.harmand@inrae.fr

A Rapaport
email: alain.rapaport@inrae.fr

T Sari

Some non-intuitive properties of serial chemostats with and without mortality

Keywords: chemostat, gradostat, biogas production, mortality

This paper discusses a model of two interconnected chemostats in series, characterized by biomass mortality. A comparison is established with a single chemostat of the same total volume in two different cases, that are with or without mortality rate. The outlet substrate concentration and the biogas flow rate are the main criteria for comparison. According to conditions depending on the operating parameters and the distribution of the total volume, our results show which structure, the series of the two chemostats or the single chemostat, performs better in terms of minimizing the outlet substrate concentration or maximizing the biogas flow rate, and this with or without account of mortality. Moreover, the differences and similarities in the results corresponding to the case with mortality and the one without mortality, are highlighted.

INTRODUCTION

'Process design' means determining which type of ideal reactor (e.g. CSTR; continuous stirred-tank reactor, PFR; plug flow reactor) -or interconnection of reactors -is best suited for a given 'task', the degrees of freedom being whether or not to multi-feed, the volumes of the tanks, the different interconnection points, etc... The 'task' in question here is the transformation of some matter -the 'substrate' -into a 'product' through a bioreaction autocatalyzed by a microbial ecosystem, hereafter called the 'biomass'. This -very general -problem is of a crucial practical importance when dealing with large industries where the number of tanks, their volumes, the technology on which they rely on or the way they are connected to others directly impact not only the capital but also the operating costs. Being the first industry in terms of the volume of matter processed, the study of wastewater treatment plants (WWTP) have attracted a lot of attention for quite a long time, cf. for instance [START_REF] Scuras | Optimization of activated sludge reactor configuration : kinetic considerations[END_REF]. Because of their interest in producing energy from waste, the optimal design of anaerobic processes (AD) has also been extensively studied, cf. [START_REF] Schievano | Can two-stage instead of one-stage anaerobic digestion really increase energy recovery from biomass[END_REF]. Such studies -among many others mainly conducted in the field of chemical and/or biochemical engineering -have These results are part of the JPI project Control4reuse (http://control4reuse.net) financed by the French Research National Agency under the contract ANR-18-IC4W-0002.". The authors thank the Algerian program "Bourses d'Excellence" for the funding of the PhD grant of Manel Dali Youcef.

in common to consider steady state characteristics of the bioprocess. Indeed, in these industries it is the rule rather than the exception to process continuously very large quantities of matter instead of stocking them, notably because the matter in question (liquid or solid waste in the examples here above) are produced continuously: the price to pay to stock them would be very high. They are however different in the objectives pursued: it is expected from a classical WWTP that the output concentration of pollutant (the output substrate concentration) be minimal while in the case of the AD, the objective is to maximize biogas production, the final product of the bioreaction. And looking at the results of the different available studies, it appears the best designs are not the same. In other words, given a bio-transformation process, different objectives may lead to different 'optimal' configurations. This is precisely summarized as a preamble by Gooijer and his coauthors in their 1996 survey on the design procedures of bioprocesses in series, citing what Herbert claimed in 1964 at a conference of bioengineering: 'If one fermenter gives good results, two fermenters will give better results and three fermenters better still. This is sometimes true, but often false. ', cf. de Gooijer et al. (1996). One difficulty when working with the design of bioprocesses comes from the fact that the quantities assumed to be known and the degress of freedom are not the same depending on the specific problem to be solved. For instance, when dealing with water treatment, it is the rule rather than the exception to minimize the total volume of the system given input-output constraints specified by normative values.

But, if the system considered is already existing, it could also be interested to find conditions such that the output pollution concentration be minimized, for instance in playing with operating conditions or with the interconnection of reactors. In addition, excluding a very few number of studies (cf. for instance [START_REF] Harmand | Optimal design of interconnected bioreactors : new results[END_REF]; Dali-Youcef (2022) who worked with kinetics defined by qualitative properties), most available literature solve the problem in fixing kinetics once and for all, limiting the genericity of the results. Thus, it is particularly difficult to draw general conclusions and get a global idea about the advantages and drawbacks of a given configuration. In the present paper, we revisit the properties of a specific configuration -the two tanks in series -from a generic viewpoint. To do so, we adopt an approach in which we systematically compare the performances of a single CSTR with those obtained using two tanks in series in which both r, the fraction of the total volume used to calculate the volume of each tank as rV and (1 -r)V , respectively, and the input substrate concentration S in , are free. It should be noticed that it is easy to compare both configurations since the CSTR can be seen as a special case of the two tanks in series model when r = 0 or r = 1. The paper is organized as follows: first, we recall the model of the systems. It should be noticed that we consider here the presence of a mortality term which, as we will see, may play an important role in the design results. Then, considering the biomass growth rate follows a 'Monod-like' kinetics we establish the properties of the CSTR and the two tanks in series configuration for two different objectives pursued that are the search for the best design when we want to minimize the output substrate concentration and when we want to maximize biogas production. The best design is given in the following two cases; when in all chemostats mortality is neglected, then when in all chemostats mortality is positive and identical. Finally, some conclusions are drawn.

PRESENTATION OF THE MODEL AND PRELIMINARIES

We consider two serial interconnected chemostats of total volume V . The first tank is of volume V 1 := rV and the second tank is of volume V 2 := (1 -r)V where r ∈ (0, 1). The substrate and the biomass concentrations in the tank i are respectively designated S i and x i , i = 1, 2. At the input, the first tank is fed by the substrate concentration denoted S in . The dilution rate of the whole structure denoted D is defined by D := Q/V where Q designating the flow rate is constant. The dilution rates of the tanks i are different and are defined by

D i := Q/V i , i = 1, 2.
The mortality of the biomass is denoted a such that a ≥ 0. The mathematical model is given by the following equations:

Ṡ1 = D r (S in -S 1 ) -f (S 1 )x 1 ẋ1 = - D r x 1 + f (S 1 )x 1 -ax 1 Ṡ2 = D 1 -r (S 1 -S 2 ) -f (S 2 )x 2 ẋ2 = D 1 -r (x 1 -x 2 ) + f (S 2 )x 2 -ax 2 (1)
Notice that for r = 0 or r = 1, the configuration is reduced to a single reactor, which corresponds to the single chemostat model given by the system

Ṡ = D(S in -S) -f (S)x ẋ = -Dx + f (S)x -ax. (2)
For sake of completeness, the complete analysis of systems (1) and ( 2) are given in [START_REF] Dali-Youcef | Performances study of two serial interconnected chemostats with mortality[END_REF]. The considered growth function f is assumed to be of Monod type i.e. it verifies the following assumption. Hypothesis 1. The function f belongs to C 1 (R + , R + ) and satisfies f (0) = 0, f (S) > 0 for all S > 0.

As f is increasing, then the break-even concentration is well defined by λ(D) = f -1 (D) for 0 ≤ D < m with m := sup S>0 f (S), (that may be +∞). System (1) can admits at most three steady states: the washout steady state

E 0 = (S in , 0, S in , 0), the steady state E 1 = (S in , 0, S 2 , x 2 )
of washout in the first reactor but not in the second one and the steady state

E 2 = (S * 1 , x * 1 , S * 2 , x *
2 ) of persistence of the biomass in both reactors. Expressions of equilibria E 1 and E 2 are given in Table 1 where E 2 requires the definition of the function h as

h(S 2 ) = D+(1-r)a 1-r S * 1 -S2 DS in +raS * 1 D+ra -S2 , S * 1 = λ D r + a . (3)
In order to simplify the notations, we posit δ := f (S in )-a.

Table 1. The steady states of (1) and their conditions of existence and stability.

E 1 = (S in , 0, S 2 , x 2 ) S 2 = λ D 1-r + a and x 2 = D D+(1-r)a (S in -S 2 ). E 2 = (S * 1 , x * 1 , S * 2 , x * 2 ) S * 1 = λ D r + a , x * 1 = D D+ar (S in -S * 1 ), S * 2 is the unique solution of equation h(S 2 ) = f (S 2 ) and x * 2 = D D+(1-r)a (x * 1 + S * 1 -S * 2 ).

Existence condition Stability condition E 0

Always exists

D > rδ and D > (1 -r)δ E 1 D < (1 -r)δ D > rδ E 2 D < rδ Stable if it exists
Proposition 2. Let a ≥ 0, D ≥ 0 and r ∈ (0, 1) be fixed. For all S in > λ(D/r+a), the function

S in → S * 2 (S in , D, r) is decreasing.
The proof of Proposition 2 is given in [START_REF] Dali-Youcef | Study of performance criteria of serial configuration of two chemostats[END_REF][START_REF] Dali-Youcef | Performances study of two serial interconnected chemostats with mortality[END_REF]. This Proposition is illustrated in Figure 1 (a,b) which shows that for r = 0.4 (resp. r = 0.6), from λ(D/r + a) ≈ 17.22 (resp. λ(D/r + a) ≈ 6.54), increasing S in , decreases the output substrate concentration defined by S * 2 (S in , D, r). In addition, one remarks that for r = 0.4 and S in ∈ [6.54, 17.22], increasing S in has no impact on the output substrate concentration that is constant and defined by λ(D/(1 -r) + a). This remains true for every r ∈ (0, 1/2) and S in ∈ (λ(D/(1 -r) + a), λ(D/r + a)). Furthermore, in (a) and (b), and in general for any r ∈ (0, 1), one remarks that for S in ∈ [0, λ(D/(1r) + a), increasing S in , increases the output substrate concentration which is defined by S in .

OUTPUT SUBSTRATE CONCENTRATION

We consider the map r → S out r (S in , D) defined by ( 4), which represents the output substrate concentration at steady state of the serial configuration.

S out r (S in , D) =              S in if D ≥ rδ and D ≥ (1 -r)δ, λ D 1 -r + a if D ≥ rδ and D ≤ (1 -r)δ, S * 2 (S in , D, r) if D < rδ.
(4) The following results draw comparisons of S out r with the output substrate concentration S out of the single chemostat at steady state given by

S out (S in , D) = S in if D ≥ δ, λ(D + a) if D < δ.
(5)

For a fixed r ∈ (0, 1), let the function g r : [0, r(m-a)) → R be defined by

g r (D) := λ D r + a + r(D+ar) (1-r)(D+a) λ D r + a -λ(D + a) .
(6) We introduce the following hypothesis that is satisfied by any concave growth function (and hence by the Monod growth function), and also by some non concave functions such as Hill functions. For more details, see Sections 5 of [START_REF] Dali-Youcef | Study of performance criteria of serial configuration of two chemostats[END_REF][START_REF] Dali-Youcef | Performances study of two serial interconnected chemostats with mortality[END_REF]. Hypothesis 3. For any a ≥ 0 and

D ∈ [0, m -a), the function r ∈ (D/(m -a), 1) → g r (D) ∈ R is decreasing. Let the function D ∈ [0, m -a) → g(D) ∈ R be defined by g(D) := λ(D + a) + Dλ (D + a) (7
) whose expression is given in Table 2 for the Monod function.

Table 2. Analytical expressions obtained for the Monod function f (S) = mS K+S .

gr(D)

K(D+ar)((D+a) 2 -m(D(1+r)+a)) (m-a-D)(mr-ar-D)(D+a) g(D) K(m(2D+a)-(D+a) 2 ) (m-a-D) 2
Proposition 4. Assume that Hypothesis 1 and 3 are satisfied, and take a ≥ 0,

• If S in ≤ g(D) then for any r ∈ (0, 1), S out r (S in , D) > S out (S in , D).

• If S in > g(D) then S out r (S in , D) < S out (S in , D) if and only if r 1 (S in , D) < r < 1 where r 1 (S in , D) is the unique solution of S in = g r (D).

In addition, S out r (S in , D) = S out (S in , D) for r = 0, r = r 1 (S in , D) and r = 1.

Proposition 4 ensures that for given input substrate concentration S in and dilution rate D, the serial configuration can be more efficient than a single chemostat in terms of minimizing the output substrate concentration, if practitioner can choose arbitrarily the volumes distribution among the two tanks. A proof of this proposition is available in Dali-Youcef et al. ( 2020) for a = 0 and in Dali-Youcef et al. ( 2021) for any a > 0.

In the Monod case, S in = g r (D) is a second degree algebraic equation in r that gives two solutions, one corresponds to r 1 (S in , D) considered in the proposition and the second one is not considered as it does not belong to (0, 1). Proposition 4 is illustrated in Figure 2 (resp. S in = 18 > g(1)), it exists r = r 1 (6.4, 1) (resp. r = r 1 (18, 1)) solution of 6.4 = g r (1) (resp. 18 = g r (1)), such that for any r > r 1 (6.4, 1) (resp. r > r 1 (18, 1)), the output substrate concentration of the serial configuration is smaller than the output substrate concentration of the single chemostat. In addition, as shown in Figure 2, there exists a unique value r min of the parameter r which gives the smallest output substrate concentration of the series configuration, defined as

r min := argmin 0<r<1 S * 2 (S in , D, r). ( 8
)
According to Proposition 2, notice that for a fixed dilution rate D, the lowest output substrate concentration S out rmin (S in , D) = S * 2 (S in , D, r min ) gets smaller by increasing S in . This was numerically highlighted in [START_REF] Zambrano | Optimizing zone volumes in bioreactors described by monod and contois growth kinetics[END_REF] for Monod and Contois growth functions.

BIOGAS FLOW RATE

Recall that the biogas production rate of a single chemostat is proportional to the microbial activity f (S)x and to its volume V (see [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]; [START_REF] Polihronakis | Parameter adaptive control techniques for anaerobic digestersreal-life experiments[END_REF]). For the two series interconnected chemostats, the total biogas flow rates G 1 and G 2 corresponding respectively to the steady states E 1 and E 2 , are thus given by G 1 (S in , D, r

) := V 2 x 2 f (S 2 ), G 2 (S in , D, r) := V 1 x * 1 f (S * 1 ) + V 2 x * 2 f (S * 2 ). (9)
According to Table 1 and( The steady-state biogas flow rate of the single chemostat is defined by

G chem (S in , D) := 0 if D ≥ δ, V D(S in -λ(D + a)) if D < δ.
(12) Using ( 10), ( 11) and ( 12), we have the following result: Proposition 5. Assume that Hypothesis 1 and 3 are satisfied. For all 0 ≤ D < δ, r ∈ (0, 1) and a ≥ 0,

• When rδ ≤ D and D < (1 -r)δ, one has G 1 (S in , D, r) < G chem (S in , D). • When D < rδ, if S in ≤ g(D) then one has G 2 (S in , D, r) < G chem (S in , D), and if S in > g(D)
then one has G 2 (S in , D, r) < G chem (S in , D) if and only if r 1 (S in , D) < r < 1, where r 1 (S in , D) is the unique solution of S in = g r (D).

In addition, G 1 (S in , D, r) = G chem (S in , D) for r = 0, and In addition, one remarks in Figure 3 that the biogas flow rate of the serial configuration without mortality never exceed the maximal biogas flow rate of the single chemostat. This observation is indeed always true and is given in Proposition 8 below.

G 2 (S in , D, r) = G chem (S in , D) for r = r 1 (S in
For S in > 0 and a ≥ 0, let G max be defined by G max (S in ) := max 13), G = G(r, S in ) defined by ( 14), S in = 2, f (S) = 4S/(5 + S) and a = 0.6.

D∈(0,δ) G chem (S in , D). ( 13
) D Biogas Drrδ (1 -r)δ δ (a) r = 0.3 D Biogas Dr rδ (1 -r)δ δ G G max (b) r = 0.75
Proposition 8. Let S in be fixed and a = 0. For any D > 0 and r ∈ (0, 1), one has G 1 (S in , D, r) < G max (S in ) and

G 2 (S in , D, r) < G max (S in ).
The proof of Proposition 8 is given in [START_REF] Dali-Youcef | Study of performance criteria of serial configuration of two chemostats[END_REF]. Thus, for a null mortality rate, when S in and r are fixed, if the dilution rate D can be chosen then there is no interest to consider a series configuration. Therefore, for optimizing the biogas flow rate production, the practitioner should consider a single chemostat of same total volume. However, in Figure 4, one notices that the maximal biogas flow rate of the series configuration with mortality, at the positive steady state E 2 (that is max D≥0 G 2 (S in , D, r), the maximum of the red curve), can exceed the maximal biogas flow rate of the single chemostat (which is G max (S in ), the maximum of the black curve). This phenomenon can happen only when the mortality rate is non null. Proposition 9 asserts that without mortality, for a fixed r close enough to 1, the maximal biogas flow rate of the series configuration with mortality is certainly higher than the one of the single chemostat.

For S in > 0 fixed, we assume that the maximum of the function D → G 2 (S in , D, r) is unique, and we denote by G(r, S in ) the maximal value of G 2 (S in , D, r):

G(r, S in ) = max D∈[0,rδ] G 2 (S in , D, r). ( 14
)
Note that G max (S in ) = G(1, S in ). The following result ensues from Proposition 8 of [START_REF] Dali-Youcef | Performances study of two serial interconnected chemostats with mortality[END_REF]. Proposition 9. Assume that f is C 2 . If a > 0, then there exists r * ∈ (0, 1) such that for any r ∈ (r * , 1) we have G(r, S in ) > G max (S in ).

Proposition 9 is illustrated with Figure 5, where one observes that the tangent of G at r = 1 is horizontal i.e G (1, 2) = 0 and that G (1, 2) is positive and remains pos- itive in a neighborhood V 1 of r = 1. In real ecosystems, a biomass mortality often occurs, but is sometimes neglected when it is very small. Indeed, this last result shows that the occurrence of a biomass mortality, even small, may be advantageous: considering two interconnected reactors in series where the volume of the first reactor rV is close to the total volume V (but not equal to, so that the second reactor has a small but non null volume (1 -r)V ) gives the possibility to choosing a dilution rate sufficiently close or equal to D(r) := argmax D∈[0,rδ] G 2 (S in , D, r), which optimizes the production of the biogas flow rate and gives a better and more efficient functioning than a single tank configuration.

In Figure 6 of functions is non monotonic and admits a maximum which is not reached for r = 1 (see the blue curve). Moreover, all the curves are locally decreasing about r = 1, excepted the one for D that realizes the maximum of D → G chem (S in , D) (the green one), which is increasing with an horizontal slope at r = 1. Indeed, all the other curves reach 0 (wash-out) at smaller values of r and thus intersect the green one. We propose a heuristic to better understand this non-intuitive phenomenon. Let us consider a non-zero but very small volume of the second reactor. The dilution term Q/V 2 is thus large compared to the mortality term a. Thus, the dynamic equations of the second reactor are well approximated by those without mortality term (which is not the case for the first reactor). Therefore, we find ourselves in a situation virtually close to the one where the mortality would act only in the first reactor, which is less penalizing than having a mortality which applies uniformly on all the total volume of a single reactor.

CONCLUSION

In this paper, we present different results following the study of a mathematical model of two interconnected chemostats in series. The particularity of our study is the consideration or not of mortality. On the one hand, we show that for fixed input substrate concentration S in and dilution rate D, whatever the mortality is; positive or null, for a volume distribution such that r is greater than the threshold r 1 (S in , D), there exists a series configuration of two chemostats that gives a smaller output substrate concentration than a single chemostat. In addition, with respect to the parameter r, we can even define the best performing series configuration which corresponds to r = r min defined by (8). This result was numerically noticed by [START_REF] Hill | Minimum tank volumes for cfst bioreactors in series[END_REF] who built on the work done by [START_REF] Grieves | Piston-flow reactor model for continuous industrial fermentations[END_REF] and [START_REF] Bischoff | Optimal continuous fermentation reactor design[END_REF]. Indeed, in the case with null mortality, using a growth function of type Monod, [START_REF] Hill | Minimum tank volumes for cfst bioreactors in series[END_REF] found that two or three reactors in series optimally designed (e.g. the first reactor has the biggest volume), could provide substrate conversions similar to a PFR. According to the experimental study done by [START_REF] Grieves | Piston-flow reactor model for continuous industrial fermentations[END_REF], where when using a Monod growth function, it is deduced that a PFR is always more efficient than a single chemostat, [START_REF] Hill | Minimum tank volumes for cfst bioreactors in series[END_REF] deduced that under precise conditions, a configuration of two or three reactors in series more efficient than a single chemostat can exist. On the other hand, for a fixed inlet substrate concentration S in and a previously defined volume distribution into the two tanks i.e. r, if the practitioner can choose the dilution rate D then, for null mortality, we prove that the single chemostat is always more efficient and allows to have a higher biogas flow rate than the serial device. Knowing that biogas flow rate is proportional to the microbial activity and as shown in [START_REF] Dali-Youcef | Study of performance criteria of serial configuration of two chemostats[END_REF], the biogas flow rate and the productivity of the biomass of the two series interconnected chemostats configuration are defined by the same steady-state equations, then all the results with no mortality, quoted so far, are valid for three performances criteria: output substrate concentration, biogas flow rate and biomass productivity. Now, in the case where mortality is positive, this last result is no longer verified and changes significantly. Indeed, when mortality is positive, for a fixed S in and r, if the practitioner can choose the dilution rate D, then the series configuration becomes the structure that should be considered. Therefore, for the right chosen dilution rate D, the series configuration gives a higher biogas flow rate than the single chemostat. This can only happen with presence of mortality and is a definitely non intuitive result. Consequently, considering that mortality is positive in a biological experiment corresponding to the hypotheses made in our results, may represent a solution to improve the outcome of the experiment.

  Fig. 1. The curve of the output substrate concentration with respect to S in , with f (S) = 4S/(5 + S), D = 1 and a = 0.6. In (a): λ(D/(1 -r) + a) ≈ 6.54 and λ(D/r + a) ≈ 17.22. In (b): λ(D/r + a) ≈ 6.54.

  for f (S) = 4S/(5 + S), a = 0.3 and D = 1. For these parameter values, one has g(1) ≈ 5.1509. As shown in panel (a) (resp. panel (b)) of the figure, for S in = 6.4 > g(1)

Fig. 2 .

 2 Fig. 2. The output substrate concentrations of the serial configuration in red and the single chemostat in blue with r 0 = D/δ, f (S) = 4S/(5 + S), a = 0.3, D = 1 and r min is defined by (8). (a): S in = 6.4, r 0 (6.4, 1) ≈ 0.51, r 1 (6.4, 1) ≈ 0.75 and r min ≈ 0.67. (b): S in = 18, r 0 (18, 1) ≈ 0.35, r 1 (18, 1) ≈ 0.39 and r min ≈ 0.86.

  Fig. 3. The curves of the functions D → G 1 (S in , D, r) (in blue), D → G 2 (S in , D, r) (in red) and D → G chem (S in , D) (in black), with D r = D r (S in ), S in = 7, f (S) = 4S/(5 + S) and a = 0. The proof of Proposition 6 is given in Dali-Youcef et al. (2020, 2021). This proposition is illustrated in Figures 3 and 4 which have been established for fixed S in and r (i.e. both volumes of the two reactors are fixed). The difference between the figures is due to the mortality rate which is null in Figure 3 and positive in Figure 4. On the one hand, observe that for any D ∈ (0, δ), the biogas flow rate of the serial configuration for the steady state E 1 with or without mortality (curves in blue) is always smaller than the biogas flow rate of the single chemostat (curves in black). That is the first item of the proposition. On the other hand, notices that for any D ∈ (0, D r (S in )), the biogas flow rate of the serial configuration for the steady state E 2 , with or without mortality (curves in red) is greater than the biogas flow rate of the single chemostat (curves in black). Therefore, for any D ∈ (0, D r (S in )) the serial configuration is more efficient than the single chemostat. That is the second item of the proposition. Notice that for fixed S in , a and r, the value D = D r (S in ) is solution of the equation S in = g r (D) where g r is defined by (6). This value exists and is unique according to the following hypothesis which is satisfied by any concave growth function such as the Monod function, but also by the Hill functions. For more details one can see Sections 5 of Dali-Youcef et al. (2020, 2021). Hypothesis 7. For any a ≥ 0 and r ∈ (0, 1), the function D ∈ [0, r(m -a)) → g r (D) ∈ R is increasing.

Fig. 4 .

 4 Fig. 4. The curves of the functions D → G 1 (S in , D, r) (in blue), D → G 2 (S in , D, r) (in red), and D → G chem (S in , D) (in black), with G max = G max (S in ) defined by (13), G = G(r, S in ) defined by (14), S in = 2, f (S) = 4S/(5 + S) and a = 0.6.

Fig. 5 .

 5 Fig. 5. The map r → G(r, S in ) with G defined by (14), f (S) = 4S/(5 + S), a = 0.6 and S in = 2.

Fig. 6 .

 6 Fig. 6. Curves r → G 2 (S in , D, r) for different values of D,with f (S) = 4S/(5 + S), a = 0.6 and S in = 2. In blue, the one for D 0.213 such that max r G(r, S in ) = max r G 2 (S in , D, r). In green, the one for D 0.283 such that G max (S in ) = G chem (S in , D).

ACKNOWLEDGEMENTS

These results are part of the JPI project Control4reuse (cf. http://control4reuse.net) financed by the French Research National Agency under the contract ANR-18-IC4W-0002. The authors thank also the support of the Euro-Mediterranean network TREASURE.