Henri Loubergé
email: henri.louberge@unige.ch

Yannick Malevergne
email: yannick.malevergne@univ-paris1.fr

Béatrice Rey

New Results for Additive and Multiplicative Risk Apportionment

Keywords: Additive risks, Constant relative risk aversion, Multiplicative risks, Preserved preference ranking, Risk apportionment JEL classification: D81 Additive risks, Constant relative risk aversion, Multiplicative risks, Preserved preference ranking, Risk apportionment

We start by pointing out a simple property of risk apportionment with additive risks in the general stochastic dominance context defined by Eeckhoudt et al. (2009b). Quite generally, an observed preference for risk apportionment with additive risks in a specific risk environment is preserved when the decision-maker is confronted to other risk situations, so long as the total order of stochastic dominance relationships among pairs of risks remains the same. Our objective is to check whether this simple property also holds for multiplicative risk environments. We show that this is not the case, in general, but that the property holds and more strongly for the case of CRRA utility functions. This is due to a particular feature of CRRA functions that we unveil.

Introduction

The work of [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] provided a refreshing perspective for the study of decisions under risk. Until the publication of their seminal article, and since [START_REF] Friedman | The utility analysis of choices involving risk[END_REF] at least, specific properties of the von Neumann-Morgenstern utility function were associated to specific traits of attitudes towards risk. A negative second derivative of the utility function reflected risk aversion [START_REF] Friedman | The utility analysis of choices involving risk[END_REF][START_REF] Pratt | Risk aversion in the small and in the large[END_REF]. A positive third derivative reflected prudence [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF] and was associated to precautionary behavior. A negative fourth derivative was defined as temperance and invoked to explain the demand for risky assets in the presence of background risks [START_REF] Kimball | Precautionary motives for holding assets[END_REF]. In the same vein, a positive sign of the fifth derivative was more recently associated to edginess to explain the effects of background risk on precautionary saving [START_REF] Lajeri-Chaherli | Proper prudence, standard prudence and precautionary vulnerability[END_REF]. Similarly, a S-shaped utility function or a state-dependent utility function was used to rationalize simultaneous purchasing of insurance and lottery tickets, etc.

Instead of focusing ab initio on properties of the utility function, the two authors started from individual choices in simple equal-probability lotteries in order to define risk aversion, prudence, temperance, and so on, and they coined the term "risk apportionment" to describe how these behavioral traits were reflected in individual choices. For example, temperance reflects the preference for not associating an additional zeromean risk ε 2 to a situation where the decision maker is already exposed to a prevailing zero-mean risk ε 1 . The 50-50 lottery [ε 1 , ε 2 ] is preferred to the 50-50 lottery [0, ε 1 + ε 2 ]. Preference for risk apportionment means preference for disaggregation of harms, given risk aversion. The beauty of this more primitive approach to attitudes towards risk is that it does not need familiarity with utility theory to be understood, although it is perfectly consistent with the traditional approach based on specific properties of the utility function. Preference for risk apportionment (or harms disaggregation) in successive increasingly complex lotteries translates into alternating signs of successive derivatives of the utility function (mixed risk aversion, as defined by [START_REF] Caballé | Mixed risk aversion[END_REF].

The approach proposed by [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] using zero-mean risks received support in experimental work [START_REF] Deck | Exploring higher-order risk effects[END_REF][START_REF] Deck | Consistency of higher order risk preferences[END_REF][START_REF] Deck | On the robustness of higher-order risk preferences[END_REF][START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF]. It was also generalized by Eeckhoudt et al. (2009b) to any couple of risks linked together by properties of increases in risk or stochastic dominance at any order. 1 Indeed, considering four mutually independent risks X 1 , Y 1 , X 2 and Y 2 such that Y i dominates X i by s th i -order stochastic dominance for i = 1, 2, an expected utility maximizer with a mixed risk averse (MRA) utility function up to order s = s 1 + s 2 1 Increases in risk were introduced by [START_REF] Rothshild | Increasing risk: 1. A definition[END_REF] and generalized to any order by [START_REF] Ekern | Increasing n th degree risk[END_REF]. Stochastic dominance was introduced by [START_REF] Hadar | Rules for ordering uncertain prospects[END_REF] and [START_REF] Hanoch | The efficiency analysis of choices involving risk[END_REF] and extended to any order by [START_REF] Ingersoll | Theory of Financial Decision Making[END_REF]. The generalization has shown that there is a correspondence between n th degree stochastic dominance and the preference for a non dominated risk by an expected utility maximizer with a mixed risk averse utility function up to order n. An equivalent but less stronger relationship holds for increases in risk of order n (see [START_REF] Ekern | Increasing n th degree risk[END_REF].

prefers the 50-50 lottery [X 1 + Y 2 ; Y 1 + X 2 ] to the 50-50 lottery [X 1 + X 2 ; Y 1 + Y 2 ]. In the former lottery risk apportionment holds. There is disaggregation of harms. In the latter lottery this is not the case. Instead of combining "good with bad" in the two possible lottery outcomes, the lottery yields "bad with bad" in the first outcome and "good with good" in the second outcome. [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF], as well as Eeckhoudt et al. (2009b), consider additive risks. In the above lotteries, the final outcomes are either X 1 + Y 2 and Y 1 + X 2 on one hand, or X 1 + X 2 and Y 1 + Y 2 on the other hand. Subsequent research addressed risk apportionment for multiplicative risks. The analysis is thus restricted to non-negative random variables. Multiplicative risks are observed in various circumstances in economic and social life. For example, investing in an asset denominated in foreign currency exposes the domestic investor to two multiplicative risks, the risk of the asset itself and the risk of variations in the domestic currency value of the foreign currency. Similarly, taking a job with a variable income in a firm exposed to bankruptcy results for the wage earner in a range of final outcomes where the two risks interact multiplicatively. [START_REF] Wang | Multiplicative risk apportionment[END_REF] addressed risk apportionment with multiplicative risks specifically. Building on results obtained by Eeckhoudt et al. (2009a) in a related context -see also [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF] -they reach the conclusion that there is a direct link between multiplicative risk apportionment at order n + 1 and the value of n th degree relative risk aversion.2 A similar result was obtained by [START_REF] Chiu | On relative and partial risk attitudes: Theory and implications[END_REF] using a model combining two additive risks with a multiplicative effect on the first risk, a n-degree shift of stochastic dominance on this risk, and a first-degree shift of stochastic dominance on the second risk. Again, the value of n th degree relative risk aversion is critical to conclude whether risk apportionment in the sense of [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] is obtained or not.

This literature was soon superseded by work addressing risk apportionment in a bivariate context, a context where the decision-maker's preferences are driven by the joint effects of two independent risks, for instance risks affecting wealth and health. This work is based on a paper by [START_REF] Denuit | A class of bivariate stochastic orderings with applications in actuarial sciences[END_REF] defining mixed risk aversion for bivariate functions. In particular, [START_REF] Jokung | Risk apportionment via bivariate stochastic dominance[END_REF] and [START_REF] Denuit | Another look at risk apportionment[END_REF] provided a new look on risk apportionment with additive or multiplicative risks by analyzing these cases as specific cases of risk apportionment in a bivariate context. Indeed, when the two attributes defined in the bivariate context are of the same nature (financial risks, for instance), they can be combined (additively or multiplicatively) to yield a single attribute of utility. This allows the application to the univariate context of the more general results derived in a bivariate context.

In this paper, we start with additive risks and we obtain a simple result for risk apportionment preferences of one decision-maker (DM) in different situations. When the DM displays a preference for risk apportionment when risk X 1 is dominated by risk Y 1 at order s 1 and risk X 2 dominated by risk Y 2 at order s 2 , then the same DM necessarily displays a preference for risk apportionment when two other couples of risks (X 1 , Y 1 ) and (X 2 , Y 2 ) are related by stochastic dominance orderings at orders s 1 and s 2 ,

s i = s i for i = 1, 2, if s 1 + s 2 = s 1 + s 2
. This is our Proposition 1. Its contribution to the literature is to emphasize a correspondence between risky choices by one DM in two different circumstances involving risk combinations, as long as the sum of stochastic dominance orders linking the risk combinations in each circumstance is the same. Our main motivation is then to check whether this result holds when the two risks combine multiplicatively, instead of additively. We find that the answer is negative in general and we explain why. The answer is positive only in the case where one of the two couples of risks is related by a first-degree stochastic dominance (FSD) relationship, and provided relative risk aversion at order n is larger than n.

However, by turning to a specific case, the case where the DM's preferences are reflected in a Constant Relative Risk Aversion (CRRA) utility function -the function most commonly used in the literature -we obtain that our Proposition 1 is valid for multiplicative risks (Proposition 3). In addition, prior to obtaining this result, we also obtain in Proposition 2 that risk apportionment holds with multiplicative risks and a CRRA utility functions for all such functions where relative risk aversion exceeds one, independently of the stochastic dominance orders relating the two couples of variables X and Y . This much stronger result is driven by a property of CRRA utility functions that we also unveil. If relative risk aversion (at order 1) is larger than one, then relative risk aversion at order n is larger than n, and conversely (The result applies mutatis mutandis if relative risk aversion is less than one). This means that the above-mentioned results [START_REF] Wang | Multiplicative risk apportionment[END_REF][START_REF] Chiu | On relative and partial risk attitudes: Theory and implications[END_REF], dealing with multiplicative risk apportionment in a specific context, and pointing to the role of relative risk aversion at order n, are down-graded to the role of relative risk aversion at order 1, even in a general stochastic dominance context, if CRRA utility is considered.

Our paper is organized as follows. Section 2 presents the Eeckhoudt et al. (2009b) result on risk apportionment in a univariate additive risks context and derives our result of equivalence between risk apportionment at order s 1 + s 2 and risk apportionment at order s 1 + s 2 (s i = s i for i = 1, 2) when s 1 + s 2 = s 1 + s 2 . Section 3 addresses risk apportionment in a bivariate and univariate multiplicative risks contexts and explains why our result from section 2 does not hold in the latter context, except in the specific cases s 2 = 1 and s 1 = 1, or s 1 = 1 and s 2 = 1, and provided relative risk aversion at order n is larger than n. Section 4 turns to the specific case of CRRA utility functions and shows that our result from Section 2 holds generally in a univariate multiplicative risks context, even when s 1 + s 2 = s 1 + s 2 , provided relative risk aversion is larger than 1. This result derives from a property of CRRA utility functions so far only known for orders 2 and 3 and that we generalize to order n: if relative risk aversion (order 1) is larger/less than 1, relative risk aversion at order n is larger/less than n. Section 5 concludes briefly.

2 Additive risks and preserved preference ranking Eeckhoudt et al. (2009b) show that mixed risk aversion (MRA), characterized by a utility function u such that (-1) (1+k) u (k) ≥ 0 ∀k, means preference for harms disaggregation or -equivalently -preference to combine good with bad and bad with good rather than good with good and bad with bad where bad is defined as stochastically dominated. More formally, let us consider four mutually independent risks, Eeckhoudt et al. (2009b) show that a MRA DM from 1 to s 1 + s 2 (i.e. a DM with a utility funtion u such that (-1) (1+k) u (k) ≥ 0 ∀k = 1, . . . , s 1 + s 2 ) will allocate the state-contingent risks in such a way as not to gather the two bad risks in the same state. Such an individual prefers the 50

X 1 , X 2 , Y 1 , Y 2 such that Y i dominates X i by s th i order stochastic dominance for i = 1, 2 (X i s i -SD Y i for i = 1, 2).
-50 lottery [X 1 + Y 2 , Y 1 + X 2 ] to the 50 -50 lottery [X 1 + X 2 , Y 1 + Y 2 ]: 1 2 E [u(X 1 + Y 2 )] + 1 2 E [u(Y 1 + X 2 )] ≥ 1 2 E [u(X 1 + X 2 )] + 1 2 E [u(Y 1 + Y 2 )] . (1)
Equation ( 1) means that the DM prefers to disaggregate bad outcomes rather than aggregate them. In what follows, we denote this disaggregation preference relation (Eq. 1) by PD

({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ).
Let us consider a given DM with a MRA utility function from 1 to 4. We consider a first set of four mutually independent risks X 1 , X 2 , Y 1 , Y 2 such that X 1 2-SD Y 1 and X 2 2-SD Y 2 , i.e. a first set of four mutually independent risks with a total order equal to 4 (s 1 + s 2 = 2 + 2 = 4). The EST (2009b) theorem states that PD({(X 1 , Y 1 ); (X 2 , Y 2 )} (2,2) ) holds for the given DM. Let us now consider a second set of four mutually independent risks X 1 , X 2 , Y 1 , Y 2 such that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 with s 1 = 1 and s 2 = 3, i.e. such that the total order is also equal to 4. For the same given DM and using Eeckhoudt et al. (2009b), we obtain that PD({(X 1 , Y 1 ); (X 2 , Y 2 )} (1,3) ) holds since s 1 + s 2 = 4 and since u (s 1 +s 2 ) = u (s 1 +s 2 ) for all s 1 , s 2 , s 1 , s 2 such that s 1 + s 2 = s 1 + s 2 with s 1 = s 1 and s 2 = s 2 .

We can then derive the following result. Preference ranking is preserved when we replace the first set of risks by the second one and vice-versa. We understand that this result can be easily extended to the more general case s 1 + s 2 = s 1 + s 2 with s 1 = s 1 and s 2 = s 2 . This is our first proposition.

Proposition 1. Let X 1 , X 2 , Y 1 , Y 2 be
a first set of four mutually independent risks such that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 , and let X 1 , X 2 , Y 1 , Y 2 be a second set of four mutually independent risks such that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 with s 1 + s 2 = s 1 + s 2 = s, s 1 = s 1 and s 2 = s 2 . For a given mixed risk-averse DM from 1 to s, the disaggregation preference relation is preserved when the first set of risk is replaced by the second set of risks and conversely.

More formally, PD({(X

1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) ⇐⇒ PD({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ).
This result deals with additive risks. Our question is then the following: does the result still hold when risks are multiplicative? More generally, let us denote by PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) the disaggregation preference relation when risks interact multiplicatively:

1 2 E [u(X 1 • Y 2 )] + 1 2 E [u(Y 1 • X 2 )] ≥ 1 2 E [u(X 1 • X 2 )] + 1 2 E [u(Y 1 • Y 2 )] , (2)
i.e. the preference to disaggregate bad outcomes rather than to aggregate them when outcomes combine in a multiplicative form. Assume that s 1 +s 2 = s 1 +s 2 = s with s 1 = s 1 and s 2 = s 2 . Our question is then the following. Is preference ranking preserved when we replace

{(X 1 , Y 1 ); (X 2 , Y 2 )} by {(X 1 , Y 1 ); (X 2 , Y 2 )} and conversely? More formally, is PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) equivalent to PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) )?
In section 3, we show that the answer is no in the general case of a MRA DM from 1 to s and we explain why. In section 4, we restrict our attention on CRRA utility functions. In this specific case, we show that the result holds and we explain why. We also show that with CRRA utility functions, we obtain a stronger result than the general one obtained in the additive case.

Additive and multiplicative contexts: two particular cases of the bivariate context

Consider the preferences for disaggregation displayed in equations ( 1) and (2) -respectively PD({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) in the additive context and PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) in the multiplicative context. Introducing a bivariate utility function V such that V (x 1 , x 2 ) = u(x 1 + x 2 ) in the additive case and V (x 1 , x 2 ) = u(x 1 • x 2 ) in the multiplicative case, the two relations (1) and ( 2) can be summarized by

1 2 E [V (X 1 , Y 2 )] + 1 2 E [V (Y 1 , X 2 )] ≥ 1 2 E [V (X 1 , X 2 )] + 1 2 E [V (Y 1 , Y 2 )] . (3)
Considering bivariate utility functions quite generally, Denuit and Rey (2013) -see also [START_REF] Jokung | Risk apportionment via bivariate stochastic dominance[END_REF] -obtain the following result. The inequality (3) holds for all bivariate utility function V such that (-1) (k 1 +k 2 +1) V (k 1 ,k 2 ) ≥ 0 ∀k 1 = 1, . . . , s 1 and ∀k 2 = 1, . . . , s 2 . Eq. ( 3) states the condition for observing risk apportionment as well for general bivariate functions, as for univariate functions where u reflects u(x 1 + x 2 ) in the additive case and u(x 1 • x 2 ) in the multiplicative case. Note that cross-derivatives play a key role in this condition.

Let us make several important remarks.

Remark 1. When V takes the additive form V (x 1 , x 2 ) = u(x 1 + x 2 ), the inequality (3) holds provided that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 and the single-attribute utility function u exhibits MRA from 1 to s 1 + s 2 . This is the result obtained by Eeckhoudt et al. (2009b). In the additive case, the analysis is simple because all the cross-derivatives of V for a given total order are captured by the expression of a unique derivative of u. Indeed V (s 2 ,s 1 ) = u (s 1 +s 2 ) = u (s 1 +s 2 ) = V (s 2 ,s 1 ) for all s 1 + s 2 = s 1 + s 2 and that is why we obtain Proposition 1, i.e. the result of preserved preference ranking replacing

{(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) by {(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 )
and conversely.

Remark 2. When the bivariate utility function V reads u(

x 1 + x 2 ) or u(x 1 • x 2 ), for all k 1 and k 2 , sgn V (k 1 ,k 2 ) = sgn V (k 2 ,k 1 )
since the utility function is symmetric in its arguments x 1 and x 2 . Indeed, the combination of x 1 and x 2 yields one single variable x, and u = u(x). For instance, in the additive case, we have

V (3,2) (x 1 , x 2 ) = u (5) (x 1 +x 2 ) = V (2,3) (x 1 , x 2 ).
In the multiplicative case, the equality does not hold, but the signs remain equal, given that x 1 and x 2 are both assumed positive. For instance,

V (3,2) (x 1 , x 2 ) = x 2 • Ω(x) and V (2,3) (x 1 , x 2 ) = x 1 • Ω(x), where Ω(x) = x 2 u (5) (x) + 6xu (4) (x) + 6u (3) (x).
Remark 3. However, when V takes the multiplicative form, the signs of the crossderivatives are not necessarily equal if k 1 + k 2 > 3. Consider the above example again. With V (3,2) and V (2,3) , k 1 + k 2 = 5. A total derivative order of 5 can also be obtained by 1+4 and 4+1 in the case of cross-derivatives V (1,4) and V (4,1) . These two derivatives are respectively equal to (x 1 ) 3 Ψ(x) and to (x 2 ) 3 Ψ(x) where Ψ(x) = xu (5) (x)+4u (4) (x). They have same signs. But since Ω = Ψ, sgn V (2,3) and sgn V (1,4) may differ. Proposition 1 does not apply. Of course, the possibilities of divergence increase when the sum k 1 + k 2 increases. For instance, when k 1 + k 2 = 9, four different pairs of cross-derivatives need to be considered.

Remark 4. Several authors remarked that relative risk aversion plays a role when risks are multiplicative. See [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF], Eeckhoudt et al. (2009a), [START_REF] Wang | Multiplicative risk apportionment[END_REF], [START_REF] Chiu | On relative and partial risk attitudes: Theory and implications[END_REF], [START_REF] Denuit | Another look at risk apportionment[END_REF]. More specifically, starting from [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF], and using lotteries with multiplicative risks defined by Eeckhoudt et al (2009a), [START_REF] Wang | Multiplicative risk apportionment[END_REF] generalize the latter contribution by showing that risk apportionment of order n + 1 occurs if and only if the n th degree relative risk aversion r

(n) u exceeds n: r (n) u = - xu (n+1) (x) u (n) (x) ≥ n . (4)
We observe then the important following point. A MRA DM from 1 to s, who prefers to disaggregate risks X 1 , X 2 , Y 1 , Y 2 with X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 and s 1 + s 2 = s, when risks are additive does not necessarily disaggregate them when they are multiplicative. Indeed, a MRA utility function does not necessarily imply r [START_REF] Denuit | Another look at risk apportionment[END_REF] show that they can adapt their general result above (Eq. 3) to the multiplicative case if they set s 1 = 1 (and s 2 ≥ 1). Indeed, they show that the condition (-1) (1+k+1) V (1,k) ≥ 0 for all k (which is equivalent to (-1) (k+1+1) V (k,1) ≥ 0 following remark 2) reads equivalently r (k) u ≥ k for all k when risks are multiplicative. Specifically, they consider the degenerate lotteries X 1 = a and Y 1 = b where a and b are two positive constants such that a < b (then X 1 1-SD Y 1 obviously holds). They obtain the following result (see also Eeckhoudt et al. (2009a) and [START_REF] Chiu | On relative and partial risk attitudes: Theory and implications[END_REF]: the inequality

(n) u ≥ n.
1 2 E [u(aY 2 )] + 1 2 E [u(bX 2 )] ≥ 1 2 E [u(aX 2 )] + 1 2 E [u(bY 2 )] , (5)
holds for all utility function u that satisfies r

(k) u ≥ k for all k = 1, . . . , s 2 .
These authors explain that they cannot extend their result to higher values of s 1 (or s 2 ), because for a given total order s (s 1 +s 2 = s), the signs of the higher cross-derivatives of V are not necessarily the same, i.e. (-1) (1+s 2 +1) V (1,s 2 ) ≥ 0 is not equivalent to (-1) (2+(s 2 -1)+1) V (2,s 2 -1) ≥ 0 for all s 2 ≥ 3 (see remark 3).

Let us now return to our question of Section 2. If the total order s 1 + s 2 = s 1 + s 2 ≤ 3 then we can extend Proposition 1 to multiplicative risks. Indeed sgn V (1,2) = sgn V (2,1) (see remarks 2 and 3). The result still holds when the total order is equal to 1 + n with s 1 = 1 and s 2 = n on the one hand, and s 1 = n and s 2 = 1 on the other hand. But when the total order is greater than or equal to four (s ≥ 4) with s 1 > 1 and s 2 > 1, then we cannot extend the result.

4 Multiplicative risks and CRRA utility functions CRRA utility functions u verify

d dx -xu (x) u (x) = 0 ∀x , (6)
and they read

u(x) = x 1-γ 1-γ , γ > 0, γ = 1 , ln x, γ = 1 . (7)
In this case, the relative risk aversion index, r

u , is equal to the constant γ:

-xu (x) u (x) = γ . ( 8
)
We can state the following (see proof in appendix).

Lemma 1. Consider multiplicative risks and a CRRA utility function with γ > 0.

(a) For all n ≥ 2, the relation -p,p) and V (m,0) , with m > p for all p ≥ 1, have the same (opposite) sign if and only if γ > 1 (γ < 1).

sgn V (1,n) = sgn V (2,n-1) = sgn V (3,n-2) = • • • = sgn V (n,1) , holds (b) Additionally, V (m
As V (m,0) is negative when m is even and positive when m is odd, we obtain from item (b) of Lemma 1:

(-1) (1+(m-p)+p) V (m-p,p) > 0 ⇔ r (1) u = γ > 1. ( 9
)
Using Denuit and Rey (2013) -see equation ( 3) above -this means that risk apportionment holds, by item (b) of Lemma 1, for all p ≥ 1 and m > p, if and only if γ > 1. The restrictions spelt out in equation ( 4) about the value of s 1 (or s 2 ) and about the value of relative risk aversion at orders greater than 1 do not apply any more. For instance, if s 2 = p = 10, and s 1 = m -p = 5, multiplicative risk apportionment holds assuming a CCRA utility function with X 1 5-SD Y 1 and X 2 10-SD Y 2 provided γ > 1. This result is expressed in the following proposition.

Proposition 2. Let us assume that (ı) Y i dominates X i in the sense of the s th i order stochastic dominance for i = 1, 2 (X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 ) and (ıı) X 1 , X 2 , Y 1 , Y 2 are mutually independent risks. The inequality

1 2 E [u(X 1 • Y 2 )] + 1 2 E [u(Y 1 • X 2 )] ≥ (≤) 1 2 E [u(X 1 • X 2 )] + 1 2 E [u(Y 1 • Y 2 )] ,
holds for all CRRA utility function u verifying r

(1) u > (<)1.

Turning now to part (a) of Lemma 1, we see that, if risk apportionment holds for a total order s = 1 + n, with s 1 = 1 and s 2 = n, it also holds for the same total order with s 1 = 2 and s 2 = n -1, or s 1 = 3 and s 2 = n -2, and so on. We can then reproduce, for multiplicative risks and CCRA utility, the same result as in Proposition 1 dealing with additive risks. This is our Proposition 3. Proposition 3. Given a decision maker with a CRRA utility function u such that r

(1) u > 1. Given a first set of four mutually independent risks X 1 , X 2 , Y 1 , Y 2 such that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 and given a second set of four mutually independent risks X 1 , X 2 , Y 1 , Y 2 such that X 1 s 1 -SD Y 1 and X 2 s 2 -SD Y 2 for all s 1 , s 2 , s 1 , s 2 with s 1 = s 1 and s 2 = s 2 . When risks are multiplicative, the disaggregation preference relation is preserved when the first set of risk is replaced by the second set of risks and conversely.

More formally, PD

M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) ⇐⇒ PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ).
Remark 5. Note that the inequality with ≥ (≤) of proposition 2 holds for all CRRA utility function u verifying r

(1) u > (<)1 whatever the total derivative order s 1 + s 2 = s. Consequently, the inequality holds for s 1 + s 2 = s, it holds for s 1 + s 2 = s with s 1 = s 1 , s 2 = s 2 , but it also holds for s 1 +s 2 = s, i.e. for s 1 +s 2 < s 1 +s 2 and for s 1 +s 2 > s 1 +s 2 .

to our knowledge and is a direct implication of Property 1. Note also that these two properties are in the background of our Proposition 2, that holds with the simple condition r

(1) u > (<)1, given CRRA utility.

Conclusion

In this paper, we extend the Eeckhoudt et al. (2009b) results on risk apportionment with two additive risks by showing that, for one DM, risk apportionment at total order s = s 1 + s 2 implies risk apportionment at total order s = s 1 + s 2 , with s 1 = s 1 and s 2 = s 2 , provided s = s . Our main motivation is then to check whether this simple property holds in a multiplicative risk context. We show that this is not the case, in general. We also show that the property is however recovered, when we restrict the analysis to CRRA utility functions. In this case, the property is even stronger, as it holds also, and more generally, for all s = s , and as well for multiplicative risks as for additive risks, provided relative risk aversion is larger than 1. We are able to link these last results to a particular feature of constant relative risk aversion functions ignored so far: relative risk aversion at order n + 1 is equal to relative risk aversion at order n, plus one.

Hence

-x • u (n+1) (x) u (n) (x) = -x • (-1) n+2 n-1 j=0 (γ + j) • x -γ-n (-1) n+1 n-2 j=0 (γ + j) • x -γ-n+1 , (15)
= γ + n -1 .

So

-x • u (n+2) (x) u (n+1) (x) = γ + n -1 + 1 , (17)
= -x • u (n+1) (x) u (n) (x) + 1 , (18)
that is

r (n+1) u = r (n) u + 1 , (19)
which concludes the proof.

Proof of Lemma 1

We consider a CRRA utility function u with γ = 1 (if γ = 1, all the cross-derivatives of V (x 1 , x 2 ) = ln(x 1 • x 2 ) vanish). Let us express the cross-derivatives of V .

We start with the first order derivatives:

V (1,0) (x 1 , x 2 ) = ∂ ∂x 1 u(x 1 x 2 ) , (20)
= x 2 • u (x 1 x 2 ) , (21)
= x -γ 1 • x 1-γ 2 , (22)
and

V (0,1) (x 1 , x 2 ) = x 1 • u (x 1 x 2 ) . (23)
It shows that V (1,0) (x 1 , x 2 ) and V (0,1) (x 1 , x 2 ) have both the same sign and have the sign of u , i.e. are positive.

For the second order derivatives, we get

V (2,0) (x 1 , x 2 ) = x 2 2 • u (2) (x 1 x 2 ) , (24)
V (1,1) (x 1 , x 2 ) = - 1 -γ γ x 1 x 2 • u (2) (x 1 x 2 ) , (25)
V (0,2) (x 1 , x 2 ) = x 2 1 • u (2) (x 1 x 2 ) , (26)
meaning that V (2,0) and V (0,2) always have the same signs (are negative as u (2) ) but have the same sign as V (1,1) if and only if γ > 1.

and, with p = n -k, k ≤ n, we finally get

V (k,n-k+1) (x 1 , x 2 ) = γ + n -k -1 γ + k -1 • x 1 x 2 • V (k+1,n-k) (x 1 , x 2 ) , ∀n ≥ k . (45)
Hence V (k,n-k+1) and V (k+1,n-k) have the same sign if and only if γ +n-k-1 > 0, that is, if and only if n-k > 1-γ. When γ > 1, this condition holds for all n ≥ k since n and k are integers. When γ ≤ 1, the condition still holds for all n > k but is not met when n = k. It is however sufficient for our purpose since statement (a) in lemma 1 does not involve derivatives of the form V (0,n) nor V (n,0) .

Eventually irrespective of γ ≷ 1, we conclude that sgn V (k,n-k+1) = sgn V (k+1,n-k) while, with k+1 instead of k, sgn V (k+1,n-k) = sgn V (k+2,n-k-1) so that sgn V (k,n-k+1) = sgn V (k+1,n-k) = sgn V (k+2,n-k-1) . Repeating this substitution for all k ranging from 1 to n -1, we obtain the set of equalities that establishes statement (a) in lemma 1.

Their definition of relative risk aversion at order n + 1 corresponds to what is generally defined now as relative risk aversion at order n (see Section

below).

Thus, the disaggregation preference relation is preserved when the first set of risk is replaced by the second set of risks and vice-versa: PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) ⇐⇒ PD M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) for all total order level i.e. such that s 1 +s 2 = s 1 +s 2 or s 1 + s 2 = s 1 + s 2 . For this last reason, with CRRA utility functions, the result obtained when risks are multiplicative (Proposition 3) is stronger than the one obtained with additive risks (Proposition 1). Indeed, in the additive case, the condition (-1) (1+s) u (s) ≥ 0, that holds for a s-order MRA DM, does not necessarily hold at order s + 1, if the DM is only MRA up to order s. Note, however, that CRRA utility yields MRA at any order. In this sense, even in the additive case, a preference for harm disaggregations at total order s implies a preference for disaggregation at order s+n, whatever n, provided the DM is endowed with a CRRA utility function. This remark underlines the power of CRRA utility functions.

Let us denote by PA M ({(X 1 , Y 1 ); (X 2 , Y 2 )} (s 1 ,s 2 ) ) the aggregation preference relation when risks are multiplicative:

We can derive the following result.

Proposition (3'). Given a decision maker with a CRRA utility function u such that r

(1)

When risks are multiplicative, the aggregation preference relation is preserved when the first set of risk is replaced by the second set of risks and conversely.

More formally, PA

Finally, having in mind that the condition (-1) (1+k+1) V (1,k) ≥ 0 for all k (which is equivalent to (-1) (k+1+1) V (k,1) ≥ 0 following remark 2) reads equivalently r (k) u ≥ k for all k when risks are multiplicative (see remark 4), we derive two noteworthy properties of CRRA utility functions.

Property 1 of CRRA utility function u, with γ > 0.

(See proof in the appendix). From Property 1, one easily derives Property 2.

Property 2 of CRRA utility function u, with γ > 0.

Note that the equivalence r

> 2 already appears in the literature (see, for instance, [START_REF] Eeckhoudt | Economic and Financial Decisions under Risk[END_REF], but the generalization to order n is new

Appendix

Proof of Property 1 of CRRA utility function.

Let us show that, for any CRRA utility function u(•)

Let us recall that

For all n ≥ 0, with u given by eq. ( 7)

Let us now consider the n th order derivatives. A straightforward recursion shows that

from which we obtain, with n ≥ 2,

so that V (n,0) and V (n-1,1) have same signs if and only if γ-1 γ+n-2 > 0 that is γ > 1 whenever n ≥ 2.

We then get, with n ≥ 3,

• V (n,0) and V (n-2,2) have the same sign if and only if

• V (n-2,2) and V (n-1,1) always have the same sign (irrespective of γ ≶ 1), whenever n ≥ 3.

We can proceed one step further, with n ≥ 4,

so that

• V (n,0) and V (n-3,3) have the same sign if and only if

• V (n-3,3) and V (n-2,2) always have the same sign (irrespective of γ ≶ 1), whenever n ≥ 4.

The recursion is now straightforward and we derive the following results.

• By generalization of equation ( 40), we obtain the relation between V (n-p,p) and V (n,0) for all n > p ≥ 1:

It shows that V (n-p,p) and V (n,0) have the same sign if and only if p-1 k=0 γ+k-1 γ+n-k-2 > 0. Since γ > 0, all the terms of the product are necessarily positive excepted the term corresponding to k = 0, that is the term γ-1 γ+n-2 , which is positive if and only if γ > 1. This result establishes statement (b) of lemma 1.

• By generalization of equation ( 41), we have

Replacing n by n + 1, we obtain

x 2 • V (n+1-p,p) (x 1 , x 2 ) , ∀n ≥ p , (44)


Gaëlle Lebrun

Annabelle Couvert

Eric Dumont
email: eric.dumont@imt-atlantique.fr

Éric Dumont

H2S removal using cellular concrete waste as filtering material Reaction identification and performance assessment

Keywords: Cellular concrete, Packing material, H2S, CaCO3, Iron oxide

come

Introduction

Hydrogen sulfide (H2S) is a toxic, flammable and hazardous pollutant. Its rotten eggs smell is noticeable at very low concentrations, between 0.02 and 0.1 ppm depending on the individual. At higher concentrations, between 10 and 500 ppm, it may cause disorders in respiratory, nervous, cardiovascular and hematological systems, and immediate death above 500 ppm [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF].

Sources of H2S can be anthropogenic since it is discharged by many industries using chemical reaction with sulfur such as paper manufacturing, wastewater treatment and petroleum. It also occurs naturally in carbon and volcanic gases. Nowadays, gaseous H2S in air or biogas can be treated in several ways such as absorption or adsorption processes, and biological processes.

The interest for biotechniques is increasing because of their high removal efficiency and low running cost [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF]. Incidentally, recent study has highlighted that cellular concrete waste, which can be used as packing material for biofiltration [START_REF] Ben Jaber | Hydrogen sulfide removal from a biogas mimic by biofiltration under anoxic conditions[END_REF][START_REF] Ben Jaber | Biofiltration of H2S in air-Experimental comparisons of original packing materials and modeling[END_REF][START_REF] Ben Jaber | Biofiltration of high concentration of H2S in waste air under extreme acidic conditions[END_REF], could also be an effective medium for the treatment of H2S in air without any microbial population [START_REF] Ben Jaber | Removal of hydrogen sulfide in air using cellular concrete waste: Biotic and abiotic filtrations[END_REF] (i.e in abiotic conditions). For an H2S concentration of 100 ppmv, a removal efficiency around 70% was reported. Therefore, such a waste material could appear as a cheap and eco friendly solution for H2S treatment in air or biogas at industrial scale. Cellular concrete is a building material mainly made up of calcium oxide (CaO), iron oxide (Fe2O3), alumina (Al2O3), silicon dioxide (SiO2) and sulfur trioxide (SO3). However, interactions between cellular concrete components and H2S are not yet fully understood, nor are the operating conditions to reach the optimum recovery. The aim of this study is therefore to determine the physical reactions that occur between H2S and the main components of cellular concrete. For this purpose, the removal of H2S in air was experimentally study in three abiotic filters filled with: (i) waste of cellular concrete, (ii) expanded schist, and (iii) a mixture of expanded schist and cellular concrete waste. Thereby, the difference in composition between cellular concrete and expanded schist could allow the identification of major components involved in H2S removal.

Characterization and composition of materials

The cellular concrete waste used in this study is a mineral material provided by the Florentaise company based in Nantes, France, and sold as "copolight" (https://www.florentaise.com/fr), whereas expanded schist is an inorganic material provided by the Granulex company based in Mayenne, France (https://www.granulex.fr/). Properties of both materials are reported in Table 1. Analytical materials used to determine these properties are described by Ben Jaber et al. [START_REF] Ben Jaber | Removal of hydrogen sulfide in air using cellular concrete waste: Biotic and abiotic filtrations[END_REF]. Table 2 shows that both materials exhibit close chemical compositions determined by using an Energy Dispersive X-ray Fluorescence Spectrometer (EDX-800HS, Shimadzu Company). As a result, comparing performances of filters filled with cellular concrete waste and expanded schist, respectively, could ultimately highlight which compounds are mainly involved in H2S removal in abiotic conditions.

A C C E P T E D

M A N U S C R I P T

Possible mechanisms involved in H2S removal

Literature data are used to investigate possible mechanisms that occur between H2S and the components of materials. Findings are displayed in Figure 1. Sorption mechanisms and chemical reactions between H2S and the cellular concrete components can be considered.

Cellular concrete may be used as adsorbent for heavy metals or halogens [START_REF] Grandclerc | Impact of cement composition on the adsorption of hydrogen sulphide and its subsequent oxidation onto cementitious material surfaces[END_REF][START_REF] Martemianov | Cellular concrete-supported cost-effective adsorbents for aqueous arsenic and heavy metals abatement[END_REF][START_REF] Oguz | Adsorption of fluoride on gas concrete materials[END_REF]. However, due to its specific surface area (44 m 2 g -1 ), adsorption tests carried out at laboratory scale indicated that cellular concrete waste used in this study is not a good adsorbent for H2S removal [START_REF] Ben Jaber | Removal of hydrogen sulfide in air using cellular concrete waste: Biotic and abiotic filtrations[END_REF].

Chemical reactions between H2S and the cellular concrete components are consequently more probable. Nonetheless, there are no data suggesting a single-step reaction between H2S and cellular concrete. Considering the composition of cellular concrete waste (Table 2), the

A C C E P T E D M A N U S C R I P T
thermodynamically favorable reactions for H2S removal involve iron oxides, calcium oxides, and potassium oxides (Table 3). However, it has to be noted that the role of potassium oxides in H2S removal in gases is not specifically described in the literature. FeO + H2S → FeS + H2O rG° = -67.4 kJ mol -1 It is known that iron oxides are currently used to remove H2S in sour gas at industrial scale. In such a case, gas is contacted with a liquid containing soluble ferric ions Fe 3+ . H2S is then oxidized to elemental sulfur and Fe 3+ is reduced to ferrous ions Fe 2+ according to the reaction:

H2S + 2Fe 3+ + 2OH -→ S 0 + 2Fe 2+ + 2H2O (1)
The liquid can be regenerated. Thus, Fe 2+ can subsequently be reconverted to Fe 3+ by oxidation with air (e.g. the Shell Iron Redox process SulFerox®) or via a biological means using Thiobacillus ferrooxidans [START_REF] Dumont | H2S removal from biogas using bioreactors: a review[END_REF].

Additionally, cheap materials containing iron oxides, like laterite [START_REF] Thanakunpaisit | Removal of Hydrogen Sulfide from Biogas using Laterite Materials as an Adsorbent[END_REF], waste pyrolysis chars [START_REF] Hervy | H2S removal from syngas using wastes pyrolysis chars[END_REF], sewage-sludge [START_REF] Ortiz | Biogas desulfurization by adsorption on thermally treated sewage-sludge[END_REF], red mud [START_REF] Sahu | Removal of hydrogen sulfide using red mud at ambient conditions[END_REF], municipal solid waste incineration bottom ash [START_REF] Obis | The effect of the origin of MSWI bottom ash on the H2S elimination from landfill biogas[END_REF] or steel slags [START_REF] Sarperi | The use of an industrial by-product as a sorbent to remove CO2 and H2S from biogas[END_REF], have been recently identified as potentially interesting media for H2S removal. Moreover, synthetic biofiltration media containing iron was also developed to speed up the biological oxidation of H2S [START_REF] Shareefdeen | Development of a biofilter media for removal of hydrogen sulphide[END_REF]. Reactions between H2S and iron oxide then lead to the formation of FeS and FeS2:

Fe2O3 + 3H2S → 2FeS + S + 3H2O (2)
FeS + H2S → FeS2 + H2 (3)

A C C E P T E D M A N U S C
CaCO3 + H2S → CaS + H2O + CO2 (5)

CaS + 2CO2 + 2H2O → CaSO4 2H2O + 2C (6)
The gypsum formed can then react with aluminates to form ettringite (Ca6Al2(SO4)3(OH)12•26H2O) known to be an expansive material causing the disintegration of concrete [START_REF] Zhang | Characterization of pozzolan-amended wellbore cement exposed to CO2 and H2S gas mixtures under geologic carbon storage conditions[END_REF]. Note that Ca-based sorbents (Ca(OH)2 and CaCO3) are applied to remove H2S

from fuel gases at high temperatures [START_REF] Agnihotri | Mechanism of CaO reaction with H2S: Diffusion through CaS product layer[END_REF] but it seems that there are only few available data at ambient temperature. For instance, the reactivity of (CaCO3)-based solid wastes for the removal of H2S from simulated biogas matrix was investigated using triphasic gas/liquid/solid process at room temperature and atmospheric pressure [START_REF] Awe | Laboratory-scale investigation of the removal of hydrogen sulfide from biogas and air using industrial waste-based sorbents[END_REF]. In the absence of oxygen in the biogas, this study suggested that dissolved H2S is neutralized by CO3 2-released by dissolution of CaCO3. However, as CaCO3 is only slightly soluble in water, H2S removal would not be efficient in this case. Calcium oxide and calcium carbonate have been also used for the development of synthetic media enabling the removal of H2S by biofiltration. A cylindricalshaped extrudate, called UP20, containing mainly calcium carbonate, an organic binder, and nutrients, has thus been specifically developed for the treatment of H2S. Several studies have highlighted that UP20 can offer a real advantage for H2S treatment, either alone, or in association with others packing materials [START_REF] Dumont | H2S biofiltration using expanded schist as packing material: performance evaluation and packed-bed tortuosity assessment[END_REF][START_REF] Dumont | Evaluation of a new packing material for H2S removed by biofiltration[END_REF][START_REF] Dumont | Evaluation of innovative packing materials for the biodegradation of H2S: a comparative study[END_REF][START_REF] Romero Hernandez | Steady-and transient-state H2S biofiltration using expanded schist as packing material[END_REF]. However, the ability of UP20 to improve H2S removal has been attributed rather to a slow-release of nutrients and to a buffering effect than to possible reactions between H2S and CaCO3. Another composite packing material, called CM-5, mainly consisting of compost (17%w) with functional microorganisms, calcium

A C C E P T E D M A N U S C R I P T carbonate (13%w)
, perlite (18%w), cement (18%w), plant fiber (7%w), and inorganic bender (27%w) was developed for H2S biofiltration [START_REF] Zhu | Performance evaluation of a slow-release packing material-embedded functional microorganisms for biofiltration[END_REF][START_REF] Zhu | Comparison of biological H2S removal characteristics between a composite packing material with and without functional microorganisms[END_REF]. A comparison between two identical biofilters filled with original CM-5 (i.e. medium embedded with microorganisms) and sterilized CM-5 (i.e. medium without microorganisms) was carried out. Based on the maximum elimination capacity (ECmax) determined from the Michaelis-Menten model, it was highlighted that performance of the biofilter filled with sterilized CM-5 was eight times lower than that of the biofilter filled with original CM-5 (8 and 62 g m -3 h -1 , respectively).

Consequently, the ability of CM-5 to remove H2S is mainly due to the presence of functional microorganisms rather than to possible reactions with the material. Nonetheless, it has to be noted that performance of the sterilized CM-5 is not insignificant. This could be due to reactions between H2S and several components of compost, calcium carbonate, perlite, cement, plant fiber, or due to synergic interactions between all of them. However, no information about the role of calcium carbonate was reported in these studies.

The comparison between experimental results obtained for the 3 abiotic filters could then be used to identify which components, either iron oxides or calcium oxides, are mainly involved in H2S removal.

A C C E P T E D M A N U S C R I P T

Materials and methods

Experimental set up

The experimental set up is shown in Figure 2. It consisted of 3 PVC cylindrical columns (internal diameter 100 mm) filled with a volume of 7.8 L of material (1 m height). Filter "BF1" was filled with a mixture of cellular concrete waste and expanded schist (54%/46% weight, corresponding respectively to 65%/35% in volume). Filters "BF2" and "BF3" were filled with 100% expanded schist and 100% cellular concrete waste, respectively (Table 4).

Materials were sieved and fractions exceeding 10 mm and less than 5 mm were removed.

Thermocouples (K type) were installed on each column to measure temperatures. For each

A C C E P T E D M A N U S C R I P T
column, the air flow was generated using a regulated fan. If necessary, it bubbled through a humidification system (internal diameter of 200 mm and 30 cm height) filled with tap water.

A stream of H2S (99.7% purity), controlled by a mass flow meter (5850S, Brooks Instruments, Hatfield, USA) was diluted in the air after the humidification system and before entering the column. The H2S concentration was measured by an Onyx 5220 analyser (Cosma, France, measurement accuracy ± 1%) all along the column, which was equipped with 9 sampling ports located at 0, 5, 20, 40, 50, 60, 80, 90, 95 and 100 cm from the bottom.

These sampling ports could also be used to measure the pressure drops (using a pressure sensor Setra, Setra Systems, Inc, Boxborough, USA; 0-700 Pa).

Operating conditions

In order to identify the possible reactions occurring between the cellular concrete components and H2S, the polluted gas was firstly flowed through the packing filters in dry conditions, i.e without the humidification system, at a constant airflow (0.5 m 3 h -1 corresponding to an EBRT of 56 s) and inlet H2S concentration (50 ppmv). Experiments were carried out over five days from 8:00 AM to 18:00 PM each day. H2S filtration was afterwards investigated in wet conditions for various operating conditions over 4 months (Table 5). For these conditions, the three packing materials were initially sprayed one time. The moisture of the bed materials were then only maintained by the humidity of the gas flow entering the columns. Experiments in wet conditions were usually carried out from 8:00 AM to 18:00 PM each day (experiments were not carried out after 18:00 PM during the week nor during the weekend for security reasons). Nonetheless, some experiments were carried out continuously for 80 h in order to determine the variation in measurements and to check the "steady state"

A C C E P T E D M A N U S C R I P T condition.
Parameters used to describe the operating conditions and to determine the removal performances are defined in Table 6.

Loading Rate LR (g m -3 h -1 ) Q V C G in C G in : Inlet concentration (g m -3 ) Elimination Capacity EC (g m -3 h -1 ) Q V (C G in -C G out ) C G out : Outlet concentration (g m -3 ) Empty Bed Residence Time EBRT (s) V Q Q: Gas flow rate (m 3 s -1 )
Removal Efficiency RE (%) 100

C G in -C G out C G in V: Packing bed volume (m 3 )
5 Results and discussion

H 2 S treatment in dry conditions vs wet conditions

In dry conditions, H2S removal efficiency was relatively low. At EBRT = 56 s, RE = 18% was obtained for filters using cellular concrete waste whereas no removal was recorded for expanded schist (Figure 3). In wet conditions, H2S removal efficiency was significantly

A C C E P T E D M A N U S C R I P T
achieved for cellular concrete waste whereas a slight H2S removal was measured for expanded schist (Figure 4). For each filter, results given in this figure correspond to 17 measurements carried out regularly between 2 and 79 h after the beginning of the experiment, which explains why results are relatively dispersed. On average, H2S removal efficiency was 47% and 4% for filters BF3 and BF2, respectively, whereas 28% was obtained for the filter BF1 (mixture). Results obtained for cellular concrete waste alone were nonetheless lower than those reported in Ben Jaber et al. [START_REF] Ben Jaber | Removal of hydrogen sulfide in air using cellular concrete waste: Biotic and abiotic filtrations[END_REF], i.e. RE = 70% for an H2S concentration of 100 ppm at EBRT = 63 s. In terms of sulphur proportion (in % weight), analysis using X-ray Fluorescence spectrometer reveals that H2S reacted with the components of the materials (Table 7). Results obtained are in agreement with removal efficiencies. For cellular concrete waste used alone (BF3), sulphur proportion increased from 6.8% at the beginning to reach 20.9% at the end of the experiment in wet conditions. It has to be noted that the pH of each material measured after H2S exposition was unchanged in regards to initial value. Overall, it was calculated that 130 mg of H2S was captured per g of cellular concrete waste during the 4 months of experiment. For expanded schist used alone (BF2), the increase was less noticeable, in agreement with the low RE recorded, but demonstrated nonetheless that expanded schist reacted with H2S. Regarding both materials used as a mixture in filter BF1, the increase in H2S proportion in the cellular concrete waste is similar to that recorded for filter BF3, albeit in a lesser extent (19.7% and 20.9%, respectively). Conversely, the H2S proportion in the expanded schist in BF1 was double that in BF2 (14.2% vs 7.1%) showing probable interactions between both materials used in the mixture. Differences between these results will be discussed hereafter. According to Table 4, CaO and SO3, might be involved. SO3 can be excluded since it is not identified as a possible reactant with H2S (Figure 1). Moreover, it can be assumed that components like Fe2O3 and Al2O3, largely present in expanded schist and marginally present in cellular concrete (Table 1), are therefore only slightly involved in H2S removal.

Considering that materials containing iron oxides are usually identified as potential interesting media for H2S removal, this last finding may seem surprising at first. Besides, as better removal efficiencies were recorded in wet conditions, results can be explained by the appearance of CaCO3, formed according to reactions that occur naturally in concrete surface in wet conditions.

CaO + H2O → Ca(OH)2 ( 7)

Ca(OH)2 + CO2 → CaCO3 + H2O (8)
As a result, cellular concrete waste used in this study should not act as an iron sponge with iron regeneration. Chemical reactions between H2S and calcium oxides leading to the formation of gypsum according to equations 4-6 are mostly probable. The formation of the

A C C E P T E D M A N U S C R I P T
components Ca(OH)2 and CaCO3, although slightly soluble in water, could explain the double proportion of sulphur measured in the expanded schist in the BF1filter in regard to BF2. It could be assumed that during the formation of calcium compounds, a part of these compounds moved by lixiviation from the cellular concrete waste to the expanded schist. In regards to literature data, reactions that occur between CaCO3 and H2S could also explain the ability of synthetic media developed specifically for biofiltration, and containing CaCO3 such as UP20 [START_REF] Dumont | Evaluation of a new packing material for H2S removed by biofiltration[END_REF][START_REF] Corre | Efficiency of Biological Activator Formulated Material (BAFM) for volatile organic compounds removalpreliminary batch culture tests with activated sludge[END_REF] and CM-5 [START_REF] Zhu | Performance evaluation of a slow-release packing material-embedded functional microorganisms for biofiltration[END_REF][START_REF] Zhu | Comparison of biological H2S removal characteristics between a composite packing material with and without functional microorganisms[END_REF], to improve biofilter performances. Moreover, in the case of UP20, these physical reactions could explain the ability of biofilters to withstand sudden high load H2S changes [START_REF] Romero Hernandez | Steady-and transient-state H2S biofiltration using expanded schist as packing material[END_REF]. As previously indicated, the use of Ca-based sorbents in the removal of H2S from fuel gases was extensively studied at high temperatures, i.e. 650-1300°C [START_REF] Agnihotri | Mechanism of CaO reaction with H2S: Diffusion through CaS product layer[END_REF], but the use of carbonaceous materials for gas desulfurization at ambient temperature is only now starting to be studied [START_REF] Hervy | H2S removal from syngas using wastes pyrolysis chars[END_REF]. To date, and from the current state of our knowledge, reported results have been obtained in the presence of water. For instance, crushed oyster shells, mainly composed of CaCO3, were used to remove H2S in water [START_REF] Asaoka | Removal of hydrogen sulfide using crushed oyster shell from pore water to remediate organically enriched coastal marine sediments[END_REF]. Mechanisms of adsorption followed by an oxidation to obtain S°, SO3 2-and S2O3 -were suggested. Therefore, based on the study of Asaoka and co-workers [START_REF] Asaoka | Removal of hydrogen sulfide using crushed oyster shell from pore water to remediate organically enriched coastal marine sediments[END_REF], an attempt to remove H2S using a filter filled with crushed oyster shells was carried out. Filter BF2 was used for this purpose (Figure 2), expanded schist being replaced by crushed oyster shells (particle size distribution: 3-6 mm). For the operating conditions EBRT = 56 s and CG in = 50 ppm, removal efficiencies were 7.5% and 21.5% in dry and wet conditions, respectively, which is lower than results obtained for cellular concrete waste (Figure 5). However, it has to be noted that the oyster shell used in this study is a smooth material. Consequently, from these findings, it can be argued that: (i) the porosity of the material is a parameter that also has to be considered; (ii) the ability of the cellular concrete to react with H2S in wet conditions is mainly due to the presence of calcium

A C C E P T E D M A N U S C R I P T
oxides, but reaction could be catalyzed by the presence of other trace metal elements not present in the oyster shell. Both arguments can obviously be conjugated.

Impact of experimental conditions

The influence of inlet H2S concentration was studied in wet conditions at a constant EBRT of 56 s. Results are reported in Figure 5. For both filters filled with cellular concrete waste, the removal efficiency logically decreased with the increase in CG in , the shapes of the curves being quite similar. The drop in RE were significant for inlet concentrations up to 200 ppm and moderate for higher concentrations. For a filter filled with cellular concrete only, RE was around 40-45% for an H2S inlet concentration of 50 ppm. This result confirms the potential advantage of using cellular concrete waste as a first stage in treatment of gaseous H2S effluents. For expanded schist, RE ranged from 1.5% to 5.8% irrespective of the H2S inlet concentration.

The influence of EBRT change in the removal efficiency is clearly highlighted in Figure 6 for filters filled with cellular concrete waste. For BF1 and BF3, RE increased significantly with EBRT. As the change in EBRT was obtained by varying the gas flow-rate, the maximum EBRT value was 56 s, corresponding to a minimum flow-rate of 0.5 m 3 h -1 . In this case, RE was around 40-45% for the filter filled with cellular concrete waste. Considering that measured pressure drops were low for this material (around 12 Pa m -1 ), it will be interesting to investigate the impact of higher EBRT obtained by increasing the height of the packing material. Furthermore, it could be argued that H2S filtration through a long horizontally packed filter (several meters leading to high EBRT values) could be very efficient. For BF1

(mixture), it can be considered that H2S removal is mainly due the presence of cellular concrete waste. Since the contact time between the gas phase and this material in BF1 is approximately half that in BF3 (due to the presence of expanded schist in BF1), it can be

A C C E P T E D M A N U S C R I P T
considered that the performances of both filters were similar (on the basis of the amounts of cellular concrete waste in filters) as it will be shown hereafter from the determination of parameter lump using the Ottengraf model.

Figure 5: Removal efficiency vs H2S concentration at the inlet of filters in wet conditions at EBRT = 56 s (blue diamond: cellular concrete waste; green triangle: mixture of cellular concrete waste and expanded schist; red square: expanded schist).

Performances of packing materials

In order to assess the performances of the packing materials filling the filters in wet conditions, the Ottengraf and Michaelis-Menten models have been applied, respectively.

Experimental results were fitted to determine the parameters lump, ECmax and Ks. The choice to use these models to determine the filter performances in abiotic conditions may seem surprising at first but can easily be explained. Firstly, it should be noted that the equation of the Michaelis-Menten model describes physically the rate of a reaction by relating the reaction rate to the concentration of the substrate. This equation describes the biochemical reactions involving a single substrate but may also be formally applied to other physical reactions. In the present case, the form of the equation of the Michaelis-Menten model is:

With:

C ln = C G in -C G out ln( C G in C G out ) (10)
It has to be noted that equations (9-10) have been successfully applied to describe the performances of different packing materials used in H2S biofiltration [START_REF] Ben Jaber | Biofiltration of H2S in air-Experimental comparisons of original packing materials and modeling[END_REF][START_REF] Dumont | H2S biofiltration using expanded schist as packing material: performance evaluation and packed-bed tortuosity assessment[END_REF][START_REF] Dumont | Evaluation of a new packing material for H2S removed by biofiltration[END_REF][START_REF] Courtois | H2S biofiltration using expanded schist as packing material: influence of packed bed configurations at constant EBRT: H2S biofiltration using expanded schist as packing material[END_REF][START_REF] Dumont | Biofiltration using peat and a nutritional synthetic packing material: influence of the packing configuration on H2S removal[END_REF].

Secondly, it has to be remembered that the Ottengraf model, developed by Ottengraf and Van

den Oever [START_REF] Ottengraf | Kinetics of organic compound removal from waste gases with a biological filter[END_REF] to describe the mechanisms of transfer and biodegradation in a biofilter, is based on the Michaelis-Menten equation. Thus, one of the main assumptions of this model is that the reaction rate constant of the substrate elimination in the biofilm is of zero-order in the pollutant concentration, which assumes a very low value of the Michaelis-Menten constant Ks. However, the same assumption of a zero-order reaction rate could be made in the absence of biofilm provided that a chemical reaction occurs between H2S and the components of the material. Usually, zero-order kinetics are encountered at high concentrations of H2S, which is generally the case in laboratory experiments. Two situations were then considered by the Ottengraf model: zero-order kinetics with diffusional limitation and zero-order kinetics with biological limitation in the biofilm. Considering that H2S reaction limitation or H2S diffusion limitation can also occur in the present case due to a physical reaction between H2S and the components of the material in place of a biological degradation in a biofilm, it therefore appeared interesting to apply this model to determine the parameter lump. This parameter is interesting because it was demonstrated that it is a suitable tool to determine the performance of a biofilter as a whole whatever its composition (mixture or layers of different packing materials) and whatever the EBRT [START_REF] Ben Jaber | Biofiltration of H2S in air-Experimental comparisons of original packing materials and modeling[END_REF]. Thus, assuming that the reaction rate is diffusion limited, the relation between the loading rate and the removal capacity of the filter is given by the relation (the way to obtain this relation is extensively described in [START_REF] Ben Jaber | Biofiltration of H2S in air-Experimental comparisons of original packing materials and modeling[END_REF]):

A C C E P T E D M A N U S C R I P T EC = LR [1 -(1 -α lump √ EBRT LR ) 2 ] (11)
From Equation [START_REF] Hervy | H2S removal from syngas using wastes pyrolysis chars[END_REF], the parameter lump is obtained by drawing the curve (1-EC/LR) 1/2 versus (EBRT/LR) 1/2 . A linear curve with an intercept of 1, and a slope corresponding to -lump has to be obtained. Using the definitions of LR and EC given in Table 6 (i.e. g m -3 h -1 ) and EBRT in h, the unit of lump is then in g 1/2 m -3/2 h -1/2 .

Experimental data and modeled curves provided by the Michaelis-Menten and Ottengraf models are displayed in Figures 7 and8, respectively. From Figure 7, it is shown that data obtained for filters containing cellular concrete waste (i.e. BF1 and BF3) can be satisfactorily described by the model of Michaelis-Menten (Microsoft ® Excel solver was used to determine

ECmax and Ks values). For BF2, data cannot be modeled (calculated ECmax being an unrealistic value) which is not surprising since RE was close to zero whatever the operating conditions.

As indicated in Table 8, ECmax value determined for cellular concrete waste was in the same order of magnitude as the value obtained for sterilized CM-5, a synthetic material specifically developed for H2S treatment, and significantly lower than the value obtained using laboratoryscale biofilters. Nonetheless, this result reinforces the idea that cellular concrete waste could be used as a first stage in H2S treatment. Figure 8 shows that the experimental data for cellular concrete waste can also be modelled using the Ottengraf model. Consequently, it can be considered that H2S reaction with cellular concrete is of zero-order kinetic with diffusional limitation. As the parameter lump allows to characterize the performance of a filter as a whole whatever its composition (mixture of different packing materials for instance), it seems reasonable that lump value for BF1 was lower than for BF3 (2.70 vs 4.26 g 1/2 m -3/2 h -1 , respectively). The ratio between lump values corresponds well to the ratio of the amounts of cellular concrete waste filling each filter (65% and 100% in volume, respectively). The comparison between lump values determined for biofilters (Table 9) and cellular concrete waste in abiotic conditions are in agreement with the findings deduced from the analysis of ECmax values. Even if the lump value determined for cellular concrete waste is lower than the values reported in the literature in the case of a

Conclusion

The ability of an abiotic filter filled with cellular concrete waste as packing material to treat H2S in air was investigated. It was demonstrated that wet conditions are required to obtained H2S removal. Removal efficiency around 40-45% was obtained at an EBRT of 56 s and for a H2S concentration of 50 ppm. This result was mainly due to reactions occurring between H2S and calcium carbonate leading to gypsum formation. A maximum elimination capacity of 7.8 g m -3 h -1 was calculated using the Michaelis-Menten model. Therefore, H2S filtration using cellular concrete waste could be carried out beneficially as the humidification step of biofiltration systems. Additionally, this mode of filtration could be used as a first step treatment to remove H2S from biogas. Considering that pressure drops were low for this material (around 12 Pa m -1 ), it can be assumed that a H2S filtration through a long packed filter would be very efficient (a horizontal mode would be more appropriate than a very high column). However, the fact that the gypsum formed can potentially react with aluminates to form ettringite causing the disintegration of concrete has to be investigated in order to determine the lifetime of the material for such applications.

Figure 1 :

 1 Figure 1: Possible interactions between H2S and cellular concrete components.

Figure 2 :

 2 Figure 2: Flowsheet of the filters used for H2S removal.

Figure 3 :

 3 Figure 3: H2S relative concentration along the three filters in dry conditions (blue diamond: cellular concrete waste; green triangle: mixture of cellular concrete waste and expanded schist; red square: expanded schist). CG in = 50 ppm; EBRT = 56 s.

Figure 4 :

 4 Figure 4: H2S relative concentration along the three filters in wet conditions (CG in = 50 ppm; EBRT = 56 s).

Figure 6 :

 6 Figure 6: Removal efficiency vs EBRT in wet conditions at Cg in = 50 ppm (blue diamond: cellular concrete waste; green triangle: mixture of cellular concrete waste and expanded schist; red square: expanded schist).

Figure 7 :

 7 Figure 7: Elimination capacity vs H2S concentration. Experimental points (blue diamond: cellular concrete waste; green triangle: mixture of cellular concrete waste and expanded schist; red square: expanded schist) and the Michaelis-Menten model (continuous lines).

  biofilter filled with inoculated materials, the potential advantages of using this waste material for H2S treatment is nonetheless proved.

Figure 8 :

 8 Figure 8: lump determination from the Ottengraf model (blue diamond: cellular concrete waste; green triangle: mixture of cellular concrete waste and expanded schist; red square: expanded schist).

Table 9 :

 9 αlump values of different packing materials used in biofiltration[START_REF] Ben Jaber | Biofiltration of H2S in air-Experimental comparisons of original packing materials and modeling[END_REF][START_REF] Dumont | Evaluation of innovative packing materials for the biodegradation of H2S: a comparative study[END_REF].

Table 1 :

 1 Physico-chemical properties of material used in the study

	Properties	Cellular concrete	Expanded schist
	Density (kg m -3 )	547 ± 5	1248 ± 12
	Specific area BET (m 2 g -1 )	44 ± 0.8	-
	Porosity (%)	64	47
	pH of surface	9.0 ± 0.1	7.0 ± 0.1
	Water retention capacity (%)	56 ± 2	-

Table 2 :

 2 Composition of cellular concrete and expanded schist (weight)

	Elemental composition		Components
	Element	Cellular concrete (%)	Expanded schist (%)	Oxide	Cellular concrete (%)	Expanded schist (%)
	Si	25	40	SiO2	50.5	56.4
	Fe	5	25	Fe2O3	1.3	12.4
	Al	(nd)	15	Al2O3	2.2	20.5
	Ca	51	2	CaO	24.6	0.9
	K	0.6	10	K2O	0.2	5
	S	7	2	SO3	19.7	1.6
	Ti	(nd)	2	TiO2	(nd)	(nd)
	P	12	3	P2O5	1.4	12.4
	nd: not determined

Table 3 :

 3 Possible reactions between H2S and cellular concrete components (rG° is the

	Gibbs free energy of reaction):
	SiO2 + 2H2S → SiS2 + 2H2O	rG° = + 262.7 kJ mol -1
	CaO + H2S → CaS + H2O	rG° = -61.8 kJ mol -1
	Al2O3 + 3H2S → Al2S3 + 3H2O	rG° = + 284 kJ mol -1
	K2O + H2S → K2S + H2O	rG° = -237.1 kJ mol -1

Table 4 :

 4 Composition of filters

	Filter	Material (weight)	Main Components (weight)
			SiO2 (53%)
	BF1	54% Cellular concrete 46% Expanded schist	CaO (14%) SO3 (11%) Al2O3 (10%)
			Fe2O3 (6%)
			P2O5 (6%)
			SiO2 (56%)
	BF2	100% Expanded schist	Al2O3 (20%) Fe2O3 (12%)
			P2O5 (12%)
			SiO2 (50%)
			CaO (25%)
	BF3	100% Cellular concrete	SO3 (20%)
			Al2O3 (2%)
			Fe2O3 (1%)

Table 5 :

 5 Operating conditions

	Conditions	Air flowrate (m 3 h -1 )	Inlet concentration (ppm) H2S	LR (g m -3 h -1 )	EBRT (s)
	Dry	0.5	50	4.5	56
			100	9.0	56
			200	17.9	56
		0.5	300	26.9	56
			400	35.9	56
	Wet	1.0	500	44.9 9.0	56 28
		1.5		13.5	19
		2.0	50	18.0	14
		3.0		26.9	9
		4.0		35.9	7

Table 6 :

 6 Parameters used in this paper

	Parameter	Definition	Nomenclature

Table 7 :

 7 Change in sulphur proportion (% weight) in packing materials during filtration experiment.

	Filter	Material	Before exposure	After exposure dry conditions	After exposure wet conditions
	BF3	Cellular concrete	6.8	9.9	20.9
	BF1	Cellular concrete Expanded schist	6.8 2.3	7.6 4.5	19.7 14.2
	BF2	Expanded schist	2.3	1.8	7.1
	All these findings suggest that one or several compounds, mainly present in cellular
	concrete and present at a negligible concentration in expanded schist, are reacting with H2S.

Table 8 :

 8 Parameters ECmax et Ks reported in the literature.

	EBRT	KS
	(s)	(g m

-3 ) ECmax (g m -3 h -1 ) Reference

	Cellular concrete waste (abiotic)	56	0.21	7.8	Present study
	Expanded schist + microorganisms	63	0.070	28.6	[4]
	Expanded schist + microorganisms	35	0.050	32.8	[23]
	Peat + microorganisms		0.113	53.4
	Sapwood + microorganisms	57	0.017	8.3	[22]
	Pine bark + microorganisms		0.036	16.5
	Sterilized CM-5 (abiotic) CM-5 + microorganisms	48	0.030 0.080	8.4 62.2	[25]