Carmen Camacho Cho

Carmen Camacho

Pjse Umr8545

Hye-Jin Cho

Sciencespo

Sorting in Credit Rationing: An Elementary Survey

Keywords: D31, D43, D82, D86 single crossing property, screening, credit rationing, performance incentives single crossing property, screening, credit rationing, performance incentives D31, D43, D82, D86

This paper reviews the literature in sorting providing with some of the key analytic elements to understand its causes. Among others, sorting has been applied to mechanism design, games and growth theory, allowing for the analysis of strategic behaviors in principal-agent problems. In some applications, an optimal solution can be obtained by introducing a one-dimensional screening device. In practice, screening devices are interest rates, high performance incentives or non-convex technologies that can yield for instance non convex salaries as functions of human capital.

Introduction

Any type of market imperfection allows to generate profits. Firms, governments, or in a general sense, principal agents, will try to sort individuals, i.e., to strategically provide accurate incentives leading agents to reveal their true profile, minimize agent's profits and maximize theirs. In more general and formal terms, sorting is a methodology which ranks individuals and facilitates obtaining a problem's equilibria. Sorting is specially useful in problems with moral hazard, adverse selection or games with asymmetric information. Due to the complexity of these problems, conclussions are usually built on comparative statics, that is, on how outcomes change with a key parameter. After presenting with some detail a first benchmark model of sorting or screening, this survey provides varied applications of sorting in credit rationing in the fields of mechanism design, games with asymmetric information, development and growth theory.

Credit rationing is a well suited problem in which banks need to sort individuals. Roughly speaking, credit rationing arises when risk-averse banks cannot observe the type of potential borrowers. The easiest example of credit rationing is when banks are risk-averse and they cannot observe the type of the potential borrower. Without full disclosure of information, banks will appeal to screening devices to sort individuals. Screening devices have consequences increases, safe-type borrowers drop out from the credit market. This is called adverse selection and it diminishes the number of loan applicants. Second, a higher interest rate may induce borrowers to take riskier projects, which in the end, can cause the problem of debt overhang.

For instance, a highly indebted farmer has very little stake of a good harvest. Large repayment implies that the borrower obtains a lower return than late-due returns after the harvest. This survey is organized as follows. Next section presents the simple strategic models with one-dimensional screening in Freixas and Laffont (1990) and [START_REF] Guesnerie | A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm[END_REF].

We also take the opportunity to introduce the single crossing property as a sorting device in higher dimension. Section 3 applies sorting mechanisms to different economic fields, while Section 4 summarizes some results in [START_REF] Hellwig | Liquidity Provision, Banking, and the Allocation of Interest Rate Risk[END_REF], namely how individuals are sorted and choose either short or long-term investments. Conclussions are presented in Section 5.

A Model of Screening

Freixas and Laffont (1990) underline that "a weakness of most contribution is to carry out their analysis with standard debt contracts which have no theoretical justification." Incentive theory has been used since then to enrich (among others) the theoretical framework of debt contracts. Usually, a loan policy providing agent-specific incentives will take into account the distribution of agents or firms and divide them in groups according to their ability to reimburse the loan. Let us illustrate this issue by describing first the strategic behaviour between a screening principal and the agent in Freixas and Laffont (1990). Then, we present the results of [START_REF] Guesnerie | A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm[END_REF]. Laffont (1990) Suppose that agents can be either efficient, with productivity θ, or inefficient, characterized by a lower productivity θ. All agents would like to obtain a loan from the principal, who ignores agents' types at the time of signing a contract. Regardless of information asymmetry, an interest rate is agreed on the contract.

The strategic behaviour in Freixas and

Strategic interaction among players arises because the principal needs to design incentives to reveal the type of each agent. On the one hand, the principal maximizes her expected payoff knowing that the probability of finding a high type is ν. On the other hand, the agent tries to extract the maximum rent of a loan k, up to t, and may agree a risk-free loan at the risk-free interest rate R. Hence in this case, the lender's utility is V = t -kR and the borrower's utility is U = θ • f (k) -t, where f (k) is the net return from the loan, after repayment. θ may be regarded as a productivity shock. It is assumed that a productivity shock has a lesser effect on efficient agents because the high type has more skills to solve problems and pay back loans successfully. Summing up, the principal offers a menu of contracts ( t, k) to the high type and (t, k) to the low type, in order to maximize her expected returns max

( t, k),(t,k) ν • t -R k + (1 -ν) • (t -Rk)
The information rent of the high-type agent is Ū = θ • f ( k) -t and the rent of the low-type

agent is U = θ • f (k) -t.
If types are revealed truthfully, then the incentive constraints for the high and the low type are respectively

Ū = θ • f ( k) -t ≥ θ • f (k) -t and U = θ • f (k) -t ≥ θ • f ( k) -t.
Agents participate in the market as long as Ū , U ≥ 0. Therefore, the principal problem can be written as

max k,k ν • θ • f ( k) -R k + (1 -ν) • (θ • f (k) -Rk) -ν • Ū + (1 -ν) • U .
Since the low type agent has no incentive to mis-report, the principal sets U = 0, and she does not offer the efficient type more than required so that Ū

= U + ∆θ • f ( k). The first-best solution provides θ • f ( k * ) = R and θ • f (k * ) = R, so that k * ≤ k * .
That is, at equilibrium, the return on capital equals to the risk-free interest rate:

R = θ - ν 1 -ν • ∆θ .
2.2 One Dimensional Screening à la [START_REF] Guesnerie | A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm[END_REF] In the principal-agent model of [START_REF] Guesnerie | A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm[END_REF], adverse selection arises since there is a one-dimensional parameter θ, which is known to the agent but unobservable to the principal. There are two types of variables linking the principal and the agent. The first type are multi-dimensional action variables, which both the agent and the principal can observe.

The second type of variable is a one-dimensional variable, t(θ), which is a money transfer from the principal to the agent and which depends on the type θ.

Consider a principal who wants to delegate to an agent the production of x units of a good.

The agent can be either efficient or inefficient, measured by a parameter θ in the set Θ, with respective probabilities v and 1 -v. The principal aims at establishing the first-best contract (x, θ) implying a transfer t from the principal to the agent, which fulfills both the utilities of the principal, V , and of each of the agents, U . It implies that higher (efficient) types get more goods.

The optimal necessary conditions describe (1) the marginal rate of substitution between x and θ, (2) the monotonicity of x, and (3) truth-telling incentive compatibility (IC) constraints.

The revelation mechanism gives the key constraints under which a contract is incentive feasible: Definition 1. An action profile x : θ ∈ Θ → x(θ) is implementable via a compensatory transfer t, if there exists a function of transfer, x(θ) such that the revelation mechanism (x(θ), t(θ))

induces truthful revelation, i.e. U (x(θ), t(θ), θ) ≥ U (x(θ ), t(θ ), θ), ∀θ, θ ∈ Θ 2 .
The following assumption is a necessary condition for implementation.

Assumption 1. U is strictly increasing in t, continuously differentiable of class C 2 .
Then, Assumption 1 implies that agents' utility is increasing in actions and transfers for almost all types; and the indifference curves for agent θ and another agent θ +dθ have constant sign.

Theorem 1. Under Assumption 1, if a piecewise C 1 action profile x is implementable via compensatory transfers t, then necessarily

∂ ∂θ ∂ x U ∂ t U (x,t,θ) dx dθ (θ) ≥ 0 (1)
or equivalently:

Σ i ∂ ∂θ ∂ x U ∂ t U (x,t,θ) dx i dθ (θ) ≥ 0 (2)
for any x, t, θ such that x = x(θ), t = t(θ) and x is differentiable at θ.

Theorem 1 shows that if the marginal rate of substitution between x and t increases with the agent's type, then any implementable contract is also necessarily increasing in types. Hence, the higher the type θ, the higher x(θ).

The following theorem further requires the marginal rate of substitution between x and t to be bounded, meaning that the transfer t cannot tend to infinity as a reaction to an infinitesimal change of x.

Theorem 2. Assume that the agent's utility function satisfies Assumption 1 and that U is bounded in the action-transfer domain in the following sense

| ∂xU ∂tU |≤ K | t | for K > 0.
Then, any piecewise C 1 action profile is implementable via compensatory transfers such that its derivative is non-negative if and only if

dx dθ ≥ 0. (3)
Note that the boundedness assumption | ∂xU ∂tU |≤ K | t | for K > 0 implies that the transfer t is large enough to fulfill implementability so that | dt dθ |< K, with K > 0.

Theorem 2 ensures that under certain conditions, there exists at least one transfer function such that (x(θ), t(θ)) is implementable, and this for any action profile x that optimizes the principal's utility. In this case, both the principal and the agents will all be satisfied, for all θ ∈ Θ.

Unfortunately, it's not simple to extend this result to multi-dimensional actions. As Guesnerie and Laffont underline, if x(θ) has a finite number of increasing portions, then any constant piece in x(θ) must join two different increasing pieces.

Multidimensional Sorting with the Single Crossing Property

Sorting conditions simplify allocative distortions. In multi-dimensional problems where the action variable has dinension higher than

A utility function U : R 2 → v which is C 1 , U (x 1 , x 2 , θ) is said to satisfy the (strict) Spence Mirrlees condition ∂ 2 U ∂θ∂x (x, θ) > 0 (4) if U x 2 U x 1
is (increasing) non-decreasing in θ with U x 1 = 0 and it has the same sign for every (x 1 , x 2 , θ). This is equivalent to saying that the ratio ∂U ∂x 1 / ∂U ∂x 2 is increasing in θ, and

∂U ∂x 2 is never 0.
This property is exploited in numerous applications. Despite the importance of the Spence-Mirless condition in multidimensional problems, we cannot develop them in this short survey, choosing instead to overview only one-dimensional problems.

Economic Applications of Sorting

This section presents some applications of sorting in the literature, underlining the mechanisms, the intuitions and consequences depending on each context. We start with [START_REF] Stiglitz | Credit Rationing in Markets with Imperfect Information[END_REF], which uses the interest rate as an incentive mechanism.

Credit Rationing à la Stiglitz and Weiss (1981)

In [START_REF] Stiglitz | Credit Rationing in Markets with Imperfect Information[END_REF] the principal is a bank which receives agents' demands for loans.

The principal ignores the agents' types, which is here the riskiness of the project that the agent would like to undertake. The agents' type is measured by a parameter θ ∈ R + . Here, the principal uses the interest rate, r, as screening device. In face of some unknown default risk at the individual level, the principal maximizes its expected return. As a result, the interest rate cannot be too low. But, as the interest rate increases, a part of potential borrowers drops out, and the risk of projects undertaken increases, so that the mix of borrowers changes adversely.

If an agent borrows an amount B, and the ceiling interest rate is r, then we say that the agent defaults on her loan if the return R plus the collateral C is insufficient to pay back the promised amount.

Definition 2. The value of θ for which expected profits are zero satisfies

Π(r, θ) ≡ ∞ 0 max [R -(r + 1)B; -C] dF (R, θ) = 0 ( 5
)
Theorem 3. For a given interest rate, r, there is a critial value θ such that an agent borrows from the bank if and only if θ > θ.

Stiglitz and Weiss also show that there could exist an equilibrium with two interest rates with an excess demand for credit for the lower one. This situation can arise if the bank's expected return has several modes as a function of the interest rate and if the Walrasian interest rate at which credit demand equals credit supply lies between two of these modes.

High performance incentives of Holmstrom and Milgrom (1994)

Let x(θ) denote the optimal action of a type θ with incentives t. The objective in higher dimensional screening is to determine how to design types θ of agents (sorting individuals) in a principal-agent relationship. The first step is to characterize how action choices x vary with types θ. Holmstrom and Milgrom (1994) use the following theorem from Topkis (1978) on supermodularity and complementarity:

Theorem 4. Let f (x, θ) be a continuous supermodular function and X a compact sublattice of R n . Let X(θ) = argmax{f (x, θ)|x ∈ X} be the set of maximizers with least upper bound

x * (θ) = supX(θ) and greatest lower bound x * = inf X(θ). Then both x * (θ) and x * (θ) belong to X(θ) and are nondecreasing functions (from R m to R n ).

When θ is a multidimensional parameter, then sorting depends on the distribution of θ as for instance in the distributional comparative statics in Jensen (2016). Note that in most examples you may think of, not all parameters characterizing types of agents are complementary with all the incentive instruments. For this reason, it is absolutely not straightforward to obtain a general sorting mechanism nor an optimal solution.

In the context of multitask agents Holmstrom and [START_REF] Milgrom | Monotone Comparative Statics[END_REF] introduce the idea of associated random variables. A vector θ of random variables is associated if for all real-valued non decreasing functions f and g, Cov(f (θ), g(θ)) ≥ 0, that is, if

E [f (θ)g(θ)] ≥ E [f (θ)] E [g(θ)] (6)
Theorem 5. Let θ = (θ 1 , ..., θ m ) be a vector of associated random variables. Then:

1. any subvector of θ is associated; in particular, any single random variable is associated;

2. Cov(θ) ≥ 0;

3. If γ is a random variable, independent of θ, then (θ, γ) is associated; in particular, any vector of independent random variables is associated;

4. if x i : R m → R, i = 1, ..., n
, is a collection of nondecreasing functions, then (x(θ), θ) is associated; and in particular, x(θ) is associated.

Then, if a subvector θ of the incentive vector θ is associated, then any policy adopted by the principal that increases one of the elements of θ will increase all the elements of θ .

Note that we can apply Theorem 5 using the law of iterated expectations in a context of lowpowered incentives: if x : R n → R is an arbitrary function and f : R → R a non-decreasing function, then, f (x(θ)) and x(θ) are associated irrespective of how the vector θ is distributed and whether x is monotone in θ. Using these results, Holmstrom and [START_REF] Milgrom | Monotone Comparative Statics[END_REF] conclude that in their model of industrial selling, low-powered incentives can be important to inspire cooperation and coordination, and may be enhanced by placing constraints on employees' freedom to act.

Sorting, Development and Growth

We present next two papers which evaluate the effect of sorting in inequality and its evolution in time. A common feature is the existence of different types of investments, which end up by sorting individuals. Indeed, an individual's initial wealth determines the type of investment one is entitled to. Although the unicity of investment types and of the associated returns is usually assumed in general equilibrium models, it is debatable whether this assumption is realistic and whether in doing so the modeler does not only lose an opportunity to seize market imperfections but also to better understand access to credit and the ensuing inequality. In Banerjee and Newman (1993) there are three distinct types of investment which determine individuals' careers, and the focus in put on the evolution of the distribution of wealth. Using a more standard approach, [START_REF] Piketty | The Dynamics of the Wealth Distribution and the Interest Rate with Credit Rationing[END_REF] analyzes the dynamics of the distribution of wealth in a classic Solow model.

In [START_REF] Banerjee | Occupational Choice and the Process of Development[END_REF] there is a continuum of risk neutral individuals who only differ at birth in their bequeathed wealth. There are three ways to invest, which determine the individuals' capacity to accumulate more or less wealth for future generations. Nevertheless, agents are not free to choose their type of investment since they are constrained by their initial wealth.

First, there exist a divisible, safe asset which does not require any labor effort from the agent and which yields a fixed gross return. Second, there exists a risky and indivisible project, which does not require any specific skill but an initial investment, I, and it generates a random return. The third project is a monitoring technology which will allow entrepreneurs to set firms and monitor µ > 1 workers. This project requires an initial investment of I > I units, and it also yields a random return. The second and the third project have the same expected rate of return, the difference between them being that the second allows an individual to become self-employed and the third generates µ jobs.

The loan market is based on Kehoe and Levine (1993), and through a monetary punishment threat, all loans will be honored. The agent's wealth determines the project they can undertake, since the lender will only accept loan requests with a positive return. In this setup, agents are credit constrained if their initial wealth is not enough as collateral to balance possible losses. At equilibrium, access to credit divides the population in three. The poorer agents are credit constrained, they become workers and can only invest in the safe project. Agents with an intermediate level of wealth become self-employed, and the upper tier of the distribution becomes entrepreneur.

From a dynamic perspective, the limited access to credit can generate a dual society with two types of agents: prosperous entrepreneurs and socially stagnating workers. Note that if all agents have a small wealth, then the whole economy stagnates.

Piketty (1997) analyzes a classic Solow growth model with a continuum of heterogeneous agents. Here, there is moral hazard, uncertainty about physical capital and additionally, interest rate(s) is endogeneously found as the rate which brings equilibrium to the capital market.

With respect to the standard Solow model, Piketty shows that under credit constraints, the initial distribution of wealth does have a preeminent role in the long-term distribution of wealth and the economic outcomes.

Since agents can privately choose whether to exert effort or not, lenders need to find contracts that will encourage individuals to exert maximum effort, taking into account that projects can fail. A key founding assumption is that all agents desire to invest an identical amount of capital, the first best optimal, which depends uniquely on the prevailing interest rate. Individual wealth plays a major role here since the poorer an agent, the more she will have to borrow, making risky investments, which are less profitable for the agent. As a result, there exists a critical level of wealth below which the agent will not make any effort. These agents will not receive any loan, they will be credit constrained. ii) there is some credit rationing. Some agents are credit-rationed and can only make the low investment. Other agents can attain sufficient credit to make the optimum investment.

iii) all agents are rationed, so that everybody chooses the low investment.

Short and Long term positions

While the previous section focused on situations in which a multiplicity of projects was available at a given date, we present next [START_REF] Hellwig | Liquidity Provision, Banking, and the Allocation of Interest Rate Risk[END_REF], where agents may have access to investment goods with different maturities, risks and returns.

The economy goes through 2 periods. All decisions and contracts are undertaken at t = 0, and consumption can be made at times 1 or 2. Actually, there is a probability p that an agent will need to consume early, at t = 1, and invest short-term accordingly; and a probability 1 -p that she will consume late, at t = 2. If banks can distinguish among agents who invest either short or longterm, then unregulated (Bertrand) competition will implement the firstbest allocation. Otherwise, if banks cannot distinguish them, then competition forces banks to offer contracts that will implement a second-best allocation.

Let us denote by k 01 and k 02 the allocation of the initial capital to short-term and longterm investment, with returns θ 1 and η, respectively. If agents are not truthful and they do not consume all their investment revenue in the first period, that is, c 1 -θ 1 k 01 > 0, then they can re-invest it to consume in the second period (with un uncertain return θ 2 ). Incentive compatibility will require that c 2 ≥ θ 2 c 1 , that is, consumption in the second period is larger if the agent is truthful than if she cheats, investing short and then re-investing from the first to the second period. If θ 2 is sufficiently low, then in the first best solution c 1 = θ 1 k 01 p and c 2 = ηk 02 1-p .

Agents can liquidate prematurely part of their long-term investment, getting a return .

Although there are no premature liquidations in the first best solution, they do arise when the second best is implemented. Then, if the unknown return θ 2 is large enough, premature liquidations can overwhelm the bank, who will not accept all liquidation demands. Forced by competition forces, banks then need to offer the best implementable contracts to the different households so that they will implement a second best allocation, while revealing their type, avoiding a bank-run. First, to avoid bank-runs, banks will suspend payments at date 1 when the fraction of individuals claiming their investment reaches p, the fraction of early consumers.

Second, Hellwig shows that if is sufficiently low, then there exists a value for θ 2 , R 2 , such that the economy reaches an equilibrium where

c 1 = θ 1 k 01 + η k 02 R 2 ,
if consumption takes place in the first period, and c 2 = R 2 θ 1 k 01 + ηk 02 if consumption takes place in the second period. R 2 is actually the equilibrium relative price of date 1 consumption versus date 2 consumption.

We can summarize the results of [START_REF] Hellwig | Liquidity Provision, Banking, and the Allocation of Interest Rate Risk[END_REF] presented here by saying that there exists a contract which implements a second best allocation in which the bank provides liquidity, and even allows early consumers to liquidate earlier their long term investment. However, in this allocation individuals bear all interest rate risk.

Conclusion

After reading this brief review, one may wonder to what extent sorting controls individuals' actions. We have seen that screening devices help achieve expected profits, optimum investment and the second-best welfare. Among all, we have explored various sorting mechanisms like for instance monetary punishment and low powered incentives. These devices sort agents who reveal their type by accepting contracts with monetary transfers (incentives). Hence sorting modifies somehow agents' behavior while helping all parties in a game to reach a second best solution.

  Proposition 1. In the long-term, the economy could reach three types of steady state types depending on the probability of failure and the initial wealth distribution. We could observe a steady state in which

	Piketty analyzes the long-term of the economy which may result in a multiplicity of in-
	terest rates depending on the initial distribution of wealth, among other factors. This single
	interest rate assumption is convenient, among many different issues, to understand business
	cycles (Jaffee-Modigliani, 1969, Stiglitz-Weiss, 1981); nevertheless, it is inconsistent with Nash
	equilibria in which the first best may be either no credit rationing (low interest rate) or low
	investment (high interest rate). The following proposition summarizes the most interesting
	long-term properties of credit rationing in Piketty (1997):

i) there is no credit rationing. The interest rate is sufficiently low because net returns become sufficiently high to give proper incentives to agents with no collateral.

We believe that the literature would benefit from a nobel comprehensive interpretation of the economy, before and after sorting takes place, that would represent problems in a sufficiently abstract manner so as to exploit the structural similarities among all problems. More precisely, we are thinking about exploiting the most important of these similarities: the fact that the agents' iso-utility curves only cross once.


Ahmed Hassan
email: a.hassan@itv.rwth-aachen.de

Taraneh Sayadi

Martin Schiemann

Viktor Scherer

Jean Le Rond

Adjoint-based sensitivity analysis of char combustion surface reaction kinetics

Keywords: Char burnout, sensitivity analysis, adjoint-based methods

. Adjoint-based methods are then employed to extract sensitivities of various quantities of interests, including the burning rate and total heat release, with respect to surface model parameters, such as, the activation energy, and the pre-exponential factor. The variation of these sensitivities are then assessed as the particle size and free stream composition are varied.

Introduction

Simulations of reactive flows are performed by accounting for both transport and chemistry and their interaction. Due to the prohibitive costs of reactive flow computations, arising from the large range of scales, and the equations governing the evolution of all the species, models are employed to help approximate the underlying chemistry. In the case of solid particle combustion, due to the presence of reactions on the surface of the particle, the underlying dynamics are even more complex, and the accuracy of the simulation results are highly dependent on reliable surface reaction models employed.

The most conventional models used to simulate coal particle combustion are the single film [START_REF] Nusselt | Der verbrennungsvorgang in der kohlenstaubfeuerung[END_REF][START_REF] Essenhigh | Predicted burning times of solid particles in an idealized dust flame[END_REF], double film [START_REF] Spalding | Combustion of fuel particles[END_REF], and continuous film [START_REF] Hecht | Analysis of the errors associated with typical pulverized coal char combustion modeling assumptions for oxy-fuel combustion[END_REF] models. These models are mostly developed in onedimensional settings and in the steady state limit. The fidelity of the model can be increased by considering ever more complex gas and particle-phase chemistry [START_REF] Mcconnell | The effect of model fidelity on prediction of char burnout for single-particle coal combustion[END_REF]. Regardless of the fidelity, however, these models generally rely on many parameters in order to approximate the physical problem. These parameters are then determined using experimental conditions resembling typical char combustion environment. Careful selection of experimental conditions, favoring the char reaction process by de-emphasizing heat and mass transfer, has allowed minimising uncertainties in species transport properties [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF]. Nevertheless, a certain level of uncertainty still remains in such experimental measurements. Therefore, it is of great importance to determine the sensitivity of the output (final solution or quantity of interest) to the existing parameters in the model.

Most common methods in extracting the gradient information are analytical or use finite differences. The subtleties associated with the numerical approach, such as the meshing process, discretization, and complex governing equations, especially when chemical reactions and regions with steep gradients (flames) are present, make extracting the analytical expression close to impossible. In the case of finite differences, the numerous parameters involved in modeling make this approach prohibitively expensive for large problems and can also be easily overwhelmed by numerical noise. Adjoint-based methodology overcomes these two shortcomings, by allowing the determination of the gradient at a cost comparable to a single function evaluation [START_REF] Giles | An introduction to the adjoint approach to design[END_REF]. The backward solution (the solution of the adjoint equation) in effect provides gradient or sensitivity information. Given an objective function (quantity of interest), as well as a set of governing equations and side conditions, using the adjoint solution, a gradient of the quantity of interest with respect to specified input variables (model parameters) can be determined, while the governing equations are satisfied. This gradient is computed in the form of algebraic expressions based on the problem's Lagrange multipliers or adjoint variables, which in turn is used in standard optimization algorithms that rely on Jacobian information (such as the conjugate-gradient family). This approach, although more involved, is gaining traction especially in the field of acoustics and thermoacoustics [START_REF] Juniper | Triggering in the horizontal rijke tube: non-normality, transient growth and bypass transition[END_REF][START_REF] Lemke | Adjoint-based analysis of thermoacoustic coupling[END_REF][START_REF] Freund | Adjoint-based optimization for understanding and suppressing jet noise[END_REF]. Recently, nonlinear problems have also been tackled, within the context of optimal control of separation on a realistic high-lift airfoil or wing, enhancement of mixing efficiency, minimal turbulence seeds and shape optimization [START_REF] Nemili | Discrete adjoint based optimal active control of separation on a realistic high-lift configuration[END_REF][START_REF] Schmidt | Three-dimensional large-scale aerodynamic shape optimization based on shape calculus[END_REF][START_REF] Rabin | Designing a more nonlinearly stable laminar flow via boundary manipulation[END_REF][START_REF] Foures | Optimal mixing in two-dimensional plane poiseuille flow at finite péclet number[END_REF]. Adjoint-based methods have also been recently used to perform sensitivity analysis with respect to model parameters in reactive flow regimes [START_REF] Braman | Adjoint-based sensitivity analysis of flames[END_REF][START_REF] Sashittal | Adjoint based sensitivity analysis for a reacting jet in crossflow[END_REF][START_REF] Lemke | Adjoint-based sensitivity analysis of quantities of interest of complex combustion models[END_REF][START_REF] Hassan | Adjoint-based sensitivity analysis of steady char burnout[END_REF].

Previous studies have demonstrated the importance of extracting in situ particle temperature and size data, in order to extract the desired kinetic models, which outperforms the more common approach, relying on bulk measurements. In this study, therefore, we rely on a surface model, extracted using this approach, in order to capture surface reactions of char particle combustion [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF], where particle temperature and size data are gathered, and combined with other experimental information, to produce the best fit parameter of the chosen char combustion model. Since heat and mass transfer is described reasonably well with established models, the kinetics parameters of the surface reaction mechanism are those fitted through gathered experimental data. Adjoint-based methods are then employed to extract the sensitivity of the described surface reaction model, with respect to various quantities of interest, to provide a detailed assessment of the existing sensitivities in the model parameters.

The study is structured as follows: The governing equations describing the evolution of species and temperature on the particle surface as well as in the gas phase, together with appropriate boundary conditions are presented in section 2. The numerical approach is briefly described in section 3. The methodology for developing the adjoint framework is then presented in section 4. The results of the extracted sensitivities are reported in section 5. Finally, summary and conclusion of this work are discussed in section 6.

Governing equations

In this section, the governing equations describing flow/chemistry, in the gas phase, and on the particle surface, along with the accompanying boundary conditions, are presented. Throughout the study, the superscript "g" denotes variables defined in the gas phase, and superscript "s" refers to the variables defined on the surface of the particle. Superscript "*" signifies non-dimensional quantities, whenever applicable.

Equations describing the gas phase

Assuming a coal particle, the following homogeneous reaction is considered for the oxidation of carbon monoxide in the gas phase,

CO + 1 2 O 2 → CO 2 . ( 1
)
The evolution of the gaseous composition is described by the species mass fraction equations,

∇ • (ρY i (v + V i )) = ω i , (2)
where i = 1, . . . , 4 refer to fuel (CO), oxidizer (O 2 ), product (CO 2 ), and inert gas (H 2 O), respectively. ρ, denotes the mixture density, v, the mass-averaged velocity, and V i the diffusion velocity for species i. The term on the right hand side of the above equation reads

ω i = W i (ν i -ν i )ω,
where ν" i and ν i are the stoichiometric coefficients of the product and reactants in the homogeneous reaction, and W i denotes the molecular weight of species i. The homogeneous reaction rate obeys Arrhenius law [START_REF] Howard | Kinetics of carbon monoxide oxidation in postflame gases[END_REF][START_REF] Matalon | Complete burning and extinction of a carbon particle in an oxidizing atmosphere[END_REF],

ω = k [CO] [O 2 ] 1 2 [H 2 O] 1 2 exp - E g a RT (3)
= k

W CO W 1 2 O 2 W 1 2 H 2 O ρ 2 Y 1 Y 1 2 2 Y 1 2 4 exp - E g a RT , (4)
where, T , E g a , and R denote the gas temperature, the activation energy for the gas reaction, and the universal gas constant, respectively. Using Fick's law, the diffusion velocity for species i can be closed as,

Y i V i = -D∇Y i , (5)
where the diffusion coefficient is assumed identical for all species. Substituting Eq. 5 in Eq. 2 leads to the following transport equation for the species,

∇ • (ρvY i -ρD∇Y i ) = ω i . (6)
Similarly, assuming identical and constant specific heats C p for all species, the energy equation yields the following relation for the temperature in the gas phase,

∇ • (ρC p vT -λ∇T ) = - 4 i=1 h 0 i ω i , (7)
where h 0 i is the enthalpy of formation of species i, and λ is the coefficient of heat conduction. For low speed and steady flow, viscous effects are often negligible and the momentum equation reduces to the pressure being approximately constant. Finally, the continuity equation holds,

∇ • (ρv) = 0. ( 8
)
For the purpose of this study, ρD is assumed constant and the Lewis number, Le = λ/ρDC p is set to unity. The variables are made dimensionless by introducing the following characteristic values, as proposed by Matalon [START_REF] Matalon | Complete burning and extinction of a carbon particle in an oxidizing atmosphere[END_REF]: (i) the particle radius a for length, (ii) Q/C p for temperature, where Q is the heat of the homogeneous reaction per unit mass of CO 2 , namely,

Q = - 4 i=1 h 0 i W i ν i -ν i W CO 2 ν CO 2 -ν CO 2 , (9)
(iii) ρ 0 = p 0 C P /Q R, where R is the specific gas constant based on average molecular weight, and p 0 is the uniform pressure, finally (iv) λ/ρ 0 C p a, for velocity. This non-dimensionalisation yields the following expression for the Damköhler number of the homogeneous reaction

Da g = k ν CO 2 -ν CO 2 C 3 p λQ 2 R2 W CO 2 W CO W 1 2 O 2 W 1 2 H 2 O a 2 p 2 0 . (10)
In non-dimensional form, the equation of state (ideal gas) reads ρT = 1. Finally, the non-dimensional equations are:

∀i ∈ {1, 4} , L (Y i ) = Ωα i , L (T ) = Ωα 5 ,
where, (α i ) i∈{1,5} = (α -1, -α, 1, 0, 1) and

α = W O 2 /2W CO 2 , L (ϕ) = ∇ • (ρvϕ -∇ϕ) ,
where, ϕ = {Y i , T * }, and

Ω = Da g Y 1 Y 1 2 2 Y 1 2 4 T * 2 exp - E g * a T * .
If axi-symmetry is assumed, the governing equations reduce to a set of coupled second order ordinary differential equations (ODE). In the case of a single spherical particle in particular, the continuity equation (in spherical coordinates) yields

r 2 ρv r = Ṁ = constant, ( 11
)
where r is the radial coordinate, v r the radial velocity component and Ṁ denotes the "burning rate". The non-dimensional linear operator L then reduces to

L spherical (ϕ) = Ṁ * r * 2 dϕ dr * - 1 r * 2 d dr * r * 2 dϕ dr * ,
where, Ṁ * = λ C p a Ṁ , and r * = r a . In this study, heat capacity is C p = 29.1 J moleK , and the gas conductivity is λ = 39.4 × 10 -3 J mole.K.m.s .

Equations at the particle surface

The particle is assumed isothermal (T = T s for r < 1), and the temperature profile is also assumed continuous, which yields the following boundary condition

T * (r = 1) = T s . ( 12
)
The remaining conditions at the particle surface are extracted assuming the following heterogeneous reaction on the surface of the particle [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF]:

C [s] + 1 + ψ 2 O 2 → (1 -ψ) CO + ψ CO 2 . ( 13
)
The stoichiometric ratio of oxygen depends on the value of ψ, which is determined from the relation given by Tognotti [START_REF] Tognotti | The products of the high temperature oxidation of a single char particle in an electrodynamic balance[END_REF]. In non-dimensional form the relation is given by,

ψ 1 -ψ = 0.02P * s (O 2 ) 0.21 exp( T a C p QT s ), (14)
where, T a = 3070 K, P * = P P 0 is the non-dimensional pressure, and Q is the heat of the homogeneous reaction per unit mass of CO 2 , defined by equation 9. The oxygen partial pressure P s (O 2 ) is computed from the following expression,

P s (O 2 ) = P 0 Y s 2 W W O 2 . ( 15
)
Due to large solid to gas density ratio, the quasi-steady assumption holds. Conservation statements for each species i, which match the differences between the net fluxes of each species to the rates of consumption/production per unit area due to the heterogeneous reaction rates, results in the following mass balance:

ρv • nY s i -ρD ∇Y i | s • n = ω i , (16)
where, n denotes the outward pointing normal to the particle surface, and ω i the reaction rate of species i due to the heterogeneous reaction. The non-dimensional and axi-symmetric form of the above equation reads,

Ṁ * Y s i - dY i dr * s = ω i C p a λ .
The global reaction rate for surface reaction obeys Arrhenius law, and is given as [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF],

ω s = A s P s (O 2 )exp -E s a RT p , (17)
where, T p is particle temperature measured in Kelvin. A s = 21 * 10 3 mole m 2 .s.atm is the pre-exponential factor, and E s a = 80 kJ/mole is the activation energy, as specified by the experiments [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF]. Using Eq. 17, the reaction rate of each species at the surface is given as,

ω i = β i Da s Y s 2 W W O 2 , (18)
where, W is the average molecular weight, (

β i ) i∈{1,4} = [1 -ψ]W CO , -1+ψ 2 W O 2 , ψW CO 2 , 0
, and the surface Damköhler number Da s is,

Da s = A s C p a λ exp -E s * a T s * P 0 . (19)
Non-dimensionalisation has been performed as described in section 2.1. Finally, the solution satisfies the species conservation equation at the particle surface,

n=4 i=1 Y s i = 1 (20)

Remaining boundary conditions

Away from the particle, composition (Y ∞ CO , Y ∞ O 2 , Y ∞ CO 2 and Y ∞ H 2 O
) and temperature (T ∞ ) are set.

Numerical method

In its first-order form, the two-point boundary value problem, described in section 2, may be solved numerically using well-documented techniques, such as collocation or shooting methods. The latter was used in this work: it consists in solving the original problem as an initial value problem using readily available ODE integrators. The integration is performed outward from the particle surface but, since initial data such as mixture composition are missing, it is estimated and iteratively updated until the far field conditions are matched. The iterative solver employed in this work is the Levenberg-Marquardt algorithm, which proved able to handle all the cases presented in the results section (for more information regarding the numerical procedure refer to [START_REF] Hassan | Adjoint-based sensitivity analysis of steady char burnout[END_REF]).

Adjoint-based methodology

Traditionally, adjoint equations are derived from the continuous equations by applying a variational principle to the unconstrained optimisation problem and setting the first variations with respect to all involved dependent variables to zero. This results in governing equations for the direct (original) and for the adjoint variables, together with appropriate boundary conditions, initial conditions, and optimality expressions, which subsequently have to be discretized and implemented [START_REF] Jameson | Aerodynamic shape optimization using the adjoint method[END_REF][START_REF] Bewley | Dns-based predictive control of turbulence: an optimal benchmark for feedback algorithms[END_REF]. This approach is most commonly referred to as differentiate-then-discretize or continuous adjoint approach, and in particular, for complex governing equations and/or optimisation objectives, can become very cumbersome and error-prone. Alternatively, the spatially discretized equations (e.g. resulting from the application of the method of lines) can be used and processed by automaticdifferentiation (AD) software to produce the associated adjoint code. This approach, on the other hand, is otherwise referred to as discretize-then-differentiate or discrete adjoint formalism. Due to the superior performance of the discrete adjoint methodology [START_REF] Hassan | Adjoint-based sensitivity analysis of steady char burnout[END_REF], when applied to system of equations similar to those described in section 2, it has been selected to extract the sensitivities in this study.

Discrete adjoint formulation

Following the approach of discretize-then-differentiate, the discrete adjoint equations are extracted directly from the discretized form of the primal problem, where linearisation is performed using complex step differentiation. It should be noted that since the discrete adjoint equations are extracted from the primal problem after the discretization is performed, the adjoint and the primal grids are identical. This is one of the main differences between the continuous and the discrete approaches [START_REF] Hassan | Adjoint-based sensitivity analysis of steady char burnout[END_REF], applied to this problem.

Results

In this section, sensitivities of multiple quantities of interests with respect to existing model/physical parameters are extracted. This study mainly focuses on the sensitivities with respect to the heterogeneous model parameters governing the reaction on the surface of the particle, as described in section 2. The variation of model sensitivities are then examined as particle size, particle temperature, and free stream composition are varied as specified in the experiments [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF].

As particle size and shape are crucial parameters for reaction kinetics determination, two measurement techniques were employed to reduce uncertainty [START_REF] Vorobiev | Comprehensive data set of single particle combustion under oxy-fuel conditions, part i: Measurement technique[END_REF]: (i) Stereoscopic Camera system for Optical Thermography (SCOT), which is based on two-color pyrometry and (ii) light microscopy (Zeiss Axiophot, image scale 1.375 µm/pixel), which allows a much easier sample handling and a higher accuracy due to the higher resolution and the stationary fixation of the particles on a flat surface. Particle surface temperature was also measured using two-color pyrometry at λ 1 = 785 and λ 2 = 650 micrometers, where similar measures were taken to reduce the uncertainty of measurements (≈ 5%). Finally gas flow controllers provide accurate measurement of the reaction atmosphere.

Due to access to detailed and accurate measurement data, the adjoint-based analysis can be used to numerically provide sensitivities with respect to each measured model parameter, at each measurement location. Using the simulations to study the evolution of these sensitivities as free stream and/or particle conditions change, a trend can be extracted for the variation of sensitivities of the experimentally measured model parameters. For this propose, eight different experimental conditions are considered and analysed as shown in table 1. The symbols are used to identify the measured data in all the figures of this section. Note that the particle temperature, T p and free stream temperature (gas temperature in the experiment), T ∞ , are reported in non-dimensional form in the table. Before analysing the evolution of model sensitivities, the effect of the choice of the quantity of interest (QoI) on the sensitivities, at a given measurement point (case 1 of table 1), is determined. For this purpose, sensitivities of representative model/physical parameters are considered. By comparing the sensitivities of the heterogeneous model parameters to the sensitivities in free stream compositions and particle temperature, one can estimate the impact of the accuracy of measurements on the prediction of the quantity of interest. Due to large variation in their values, absolute values of the extracted sensitivities are plotted in logarithmic scale. The selected quantities of interest are: (i) the total heat release (integrated in the spatial domain), (ii) burning rate, and (iii) ψ, given by equation 14. Figure 1 shows the distribution of the extracted sensitivities for the identified QoIs. Comparing the sensitivities of the parameters determining the value of the heterogeneous reaction: A s , E s a , and T a , suggests that the activation energy E s a is generally the most sensitive parameter, as opposed to the pre-exponential factor, A s , having the lowest sensitivities. T a dominates the extracted sensitivities for ψ, which is also expected, as it appears directly in the expression defining this quantity (equation 14). While, the activation energy shows high sensitivities, depending on the QoI, other physical parameters, such as the temperature at the particle surface, T p , can contribute to comparable and sometimes larger sensitivities, suggesting that particle temperature needs to be measured extremely accurately, for the sensitivities with respect to model parameters to be meaningful.

Case Y ∞ O 2 Y ∞ CO 2 Y ∞ H 2 O T p T ∞ 2 × a [
As far as experimental observations are concerned, several studies have reported on the uncertainties in particle temperature measurements for burning char particles. Murphy & Shaddix [START_REF] Murphy | Combustion kinetics of coal chars in oxygen-enriched environments[END_REF] use a ratio-pyrometer based on photo-multiplier tubes (PMT). The reported uncertainty in a single particle temperature measurement is reported to be better than 1% (≈ 20K) for typical char combustion conditions. Bejarano & Levendis used another type of PMT-based three-color pyrometer and investigated classy carbon spheres, which were supposed to be of relatively constant quality [START_REF] Bejarano | Combustion of Coal Chars in Oxygen-Enriched Atmospheres[END_REF]. They observed a standard deviation of 70K and some particle-to-particle variations, such that the systematic error in the temperature measurement is certainly smaller than this value. The technique adopted for experimental measurements of this study, is based on high-speed intensified CCD cameras (a very detailed analysis of the temperature uncertainty of single particle measurements, using this method, is given in [START_REF] Vorobiev | Comprehensive data set of single particle combustion under oxy-fuel conditions, part i: Measurement technique[END_REF]). Although this work reports an average uncertainty of up to 4.7% for all particles which are accepted for further evaluation, recording of several hundred particles at one residence time, or conversion level, reduces the error in the mean temperature. This leads to comparably low uncertainties in the average temperature, which is used for further calculations, e.g. the determination of reaction rate parameters (a study of the influence of the measurement uncertainty is given in [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF]). Figure 1 also shows that the extracted sensitivities are directly dependent on the choice of QoI. While one QoI might lead to high sensitivities with respect to a model parameter or physical compositions of the problem, another QoI might show no sensitivities at all. Therefore, if the QoI is not chosen correctly, it might have misleading consequences in the interpretation of the extracted sensitivities. In case no sensitivities are reported with respect to a certain QoI for a certain parameter, it should not be concluded that the overall system is independent on the parameter, just the prediction of the selected QoI.

A s E s T T a T p Y C O 2 Y H 2 O Y O 2
Since the main objective of this study is determining the sensitivities in kinetic model parameters, the evolution of the extracted sensitivities with respect to the model parameters are examined, as the conditions specified in the free stream and at the particle surface change according to the values reported by the experimental measurements. The evolution is defined such that, the parameters of each experimental configuration is fixed as specified by table 1, except for the particle diameter, which is allowed to vary, as shown in figure 2. The symbols show the corresponding measurement location of each experiment. This figure, also compares the evolution of sensitivities extracted for all heterogeneous model parameters: A s , E a , and T a , although the activation energy E a remains the most sensitive throughout. Considering the experimental conditions (symbols in the figures), independent of the model coefficient, case 8 results in the highest sensitivities, and case 1 leads to the lowest, as far as the prediction of the burning rate is concerned. The burning rate, using the heterogeneous model presented in section 2, is directly related to the global reaction (a) A s rate, described by equation 17, which in turn depends on the partial the surface, P s (O 2 ). Increasing the oxygen concentration in the free stream, results in higher concentrations at the surface as well, strengthening the effect of the surface reaction, which leads to higher sensitivities in prediction of the burning rate. On the other hand, experiments have shown [START_REF] Vorobiev | Comprehensive data set of single particle combustion under oxy-fuel conditions, part i: Measurement technique[END_REF], that higher oxygen concentrations lead to higher particle temperatures, which in turn reduce uncertainties in measurements. More accurate measurements of surface temperature, may compensate the higher sensitivities reported here. Confirmation of this fact requires further research, beyond the scope of this work. When studying the evolution of sensitivities with respect to particle size (lines in figure 2), experimental condition imposed through case 8, still produces the highest sensitivities of the burning rate. This figure also suggests that particle size has a weak impact on the sensitivities of the pre-exponential factor (figure 2(a)), with smaller particles having slightly higher sensitivities. However, the effect of particle size is negligible as far as activation energy and T a are concerned, figures 2(b) and 2(c), respectively. Since these two parameters show higher sensitivities in prediction of the burning rate than the pre-exponential factor, A s , it can be concluded that, the particle size has a minor effect on the extracted sensitivities of the most dominant model parameters.

Figure 1 shows that, as far as the burning rate is concerned, the dominant parameter regarding the prediction of its value, could indeed be the particle temperature or the free stream composition (Y CO 2 is selected here as a representative, other quantities such as oxygen mass fraction follow the same trend). As a result, figure 3 shows the evolution of sensitivities extracted with respect to these two parameters. Comparison of figure 3(a) and 2(b) shows that the sensitivities with respect to the particle temperature follow the same trend as that of the activation energy E s a . On the other hand, the sensitivities of Y CO 2 follow the opposite trend, with highest sensitivities reported for case 1. A closer inspection of this case shows higher concentrations of Y CO 2 in the experimental condition as well, which could explain the impact on the extracted sensitivities. The sensitivities with respect to particle temperature is higher for all the considered cases compared to the other existing model/physical parameters, with the sensitivities with respect to the activation energy of the heterogeneous reaction following closely. This results confirm the conclusion, that for all experimental conditions considered here, particle temperature should be measured accurately to allow sensitivity analysis of the model parameters.

Here, the impact of the choice of QoI on the evolution of sensitivities is examined, and instead of the burning rate, the total heat release is selected as the desired output. The evolution of sensitivities, extracted for this QoI, is shown in figure 4. Figure 1 showed that, while the activation

& •, case 1; • • • & •, case 2; ---& , case 3; -& , case 4; • • • & •, case 5; ---& •, case 6; -& , case 7; • • • & , case 8
. Lines: simulations, where the free stream and particle conditions are fixed similar as specified by the experiment, and particle size is allowed to vary; Symbols: experimental conditions.

(a) A s energy E s a remains the most sensitive model parameter for both quantities of interest, burning rate results in higher sensitivities compared to total heat release. This proves to be the case for all the experimental conditions that are considered here, as comparison of figures 4(b) and 2(b) suggests. Similar to the case of the burning rate, the highest sensitivities are calculated for case 8 and the lowest for case 1. The higher oxygen concentration together with higher particle temperature in cases 7 and 8, result in higher temperatures throughout the domain, increasing the sensitivities of prediction as far as the heat release is concerned. In contrast to figure 2, however, the particle size has a larger effect on the extracted sensitivities, with larger particle sizes leading to higher sensitivities. For cases with highest sensitivities, cases 7 and 8, the effect of particle size is also more substantial. The pronounced dependence on particle size is due to the particular definition of Da g (equation 10), used in this model, which results in quadratic dependence on the particle (a) A s radius. Note that the total heat release is calculated by integrating the reaction rate in the whole spatial domain, containing the gas phase. As a result, although the sensitivities are extracted with respect to the surface parameters, the behaviour is dependant on the dynamics, prescribed by both heterogeneous and homogeneous reactions. As described in detail by Schiemann et al. [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF], ψ is an important parameter in the calculation of rate parameters A s and E s a . As a result, figure 5 shows the evolution of the sensitivities with respect to ψ, replacing particle size. This figure includes only the evolution of the sensitivity with respect to the most dominant parameter E s a , and for the experimental setup of case 8 and 7, which were the most sensitive (figure 4(b)). Comparing figures 4(b) and 5 shows that, plotting versus ψ rather than particle radius changes the trend in the evolution of sensitivities. Lower values of ψ result in higher sensitivities, suggesting an inverse relationship between the value of ψ and the particle size. This inverse proportionality can be explained by considering the definition of the Da s in equation 19. This equation shows the direct relation between the particle size and Da s . Increasing particle size increases Da s , resulting in faster reactions, and higher carbon dioxide production, hence reducing the oxygen mass fraction on the surface, and in turn, reducing the value of ψ, defined by equation 14. Therefore, all the plots reported for particle size can be interpreted similarly for ψ, considering this inverse proportionality.

Figure 6 shows the evolution of sensitivities extracted for ψ as the quantity of interest. As demonstrated by equation 14, this variable is one of the necessary parameters in determining the reaction rate of the heterogeneous reaction, and is also the solution of the equations governing the steady char combustion, explained in section 2. In contrast to the other quantities of interest, T a is the most sensitive parameter as far as the prediction of ψ is concerned. Another difference to the other QoIs is that, higher sensitivities are reported for case 1 as opposed to case 8. This is expected, since T a appears directly in the definition of ψ (equation 14). Considering the form of the equation, it is obvious that the sensitivity of ψ is adversely related to the partial pressure of oxygen, favoring case 1, where lower free stream oxygen concentrations are imposed by the experimental conditions. Similar to the burning rate, however, there seems to be a week dependence on the particle size, and this could be attributed to the dominance of the heterogeneous reaction in determining the value of this parameter.

As a final remark, it should be noted that, while extracted sensitivities are important in assessing and interpreting model/flow dynamics, they are, in effect, a linear local interpretation of the response of the underlying system to slight variations in model or flow parameters. For a nonlinear alternative, uncertainty quantification should be used. In experimental studies, the results are commonly reported with error bars indicating the level of uncertainties in the measurements. In order to provide similar information from simulations, the extracted sensitivities can be used to approximate such uncertainty information. Such analysis will be the subject of future work for this study.

It should be also noted that, the extracted sensitivities are directly dependent on the choice of the kinetic model used to perform the simulations. Despite well-known weaknesses of the onestep heterogeneous model analysed here, it is commonly used in literature to give a quantitative description of different experimental or fuel-dependent parameters on the resulting reactivity, motivating its use in the experimental and numerical investigations here. However, a variety of other char combustion models exist, many of them consider more physical and chemical effects than the one-step kinetics model, which has been applied here (for example, the CBK model family [START_REF] Holland | Comprehensive model of single particle pulverized coal combustion extended to oxy-coal conditions[END_REF][START_REF] Niksa | Coal conversion submodels for design applications at elevated pressures. Part I. Devolatilization and char oxidation[END_REF], and Senneca's model [START_REF] Senneca | A semi-detailed kinetic model of char combustion with consideration of thermal annealing[END_REF], to name a few). Although different in their overall structures, these models all consider multiple chemical reactions for char conversion, coupled serially and/or in parallel. Nevertheless, the model calibration to a fuel relies on a single or a few parameters of specific reactions, and the remaining parameters are defined by fixed ratios in these models. Due to the different structure of these models, the sensitivity of the model performance might not be equal to the presented case, but similar tendencies are expected. These models are far more complicated in their mathematical structure, therefore a detailed analysis of the sensitivities of the involved parameters is left to later work.

Summary and conclusions

This study focuses on a heterogeneous model, commonly implemented in numerical simulations, in order to predict the reaction process on the surface of a solid particle (in this case char particle), during the combustion process. The particular model of interest to this project is extracted through detailed measurement of Schiemann et al. [START_REF] Schiemann | Determination of char combustion kinetics parameters: Comparison of point detector and imaging-based particle-sizing pyrometry[END_REF], where model parameters are reported for various, particle sizes, free stream compositions, and particle temperatures. Using adjoint-based methodology, sensitivities of the model parameters are extracted for various quantities of interests, including the burning rate and the total heat release. Finally, the variation of the sensitivities are reported as the particle size, and free stream compositions are varied, as specified by the experimental measurements. The analysis shows the activation energy to be the most sensitive parameter, as far as, the burning rate and total heat release are concerned. It should be noted, however, that particle temperature can also result in comparable sensitivities, when approximating the burning rate. The resulting parameter sensitivities vary slightly with particle diameter for burning rate as QoI. On the other hand, particle size shows a much higher impact in the extracted sensitivities when considering the total heat release. This is attributed to the fact that the total heat release is an integrated quantity, evaluated within the entire domain, whereas the burning rate is a local quantity determined mainly by the heterogeneous reaction rate on the surface of the particle. The analysis shows that the free stream composition has a large impact in the extracted sensitivities. Free stream conditions with higher oxygen concentrations, generally result in higher sensitivities in the model coefficients. Higher oxygen concentration leads to stronger reactions on the surface, increasing the burning rate, leading to higher sensitivities in the model parameters as well.

Fig. 1 :

 1 Fig. 1: The sensitivity of different quantities of interest with respect to existing model/physical parameters. (green), total heat release; (red), ψ; (blue) burning rate.

Fig. 2 :

 2 Fig.2: Evolution of model sensitivities with respect to particle size (radius), for burning rate as QoI:-& •, case 1; • • • & •, case 2; ---& , case 3; -& , case 4; • • • & •, case 5; ---& •, case 6; -& , case 7; • • • & , case[START_REF] Hassan | Adjoint-based sensitivity analysis of steady char burnout[END_REF]. Lines: simulations, where the free stream and particle conditions are fixed similar as specified by the experiment, and particle size is allowed to vary; Symbols: experimental conditions.

CO 2 Fig. 3 :

 23 Fig.3: Evolution of sensitivities for T p and Y CO 2 with respect to particle size (radius), for burning rate as QoI:-& •, case 1; • • • & •, case 2; ---& , case 3; -& , case 4; • • • & •, case 5; ---& •, case 6; -& , case 7; • • • & , case 8. Lines: simulations, where the free stream and particle conditions are fixed similar as specified by the experiment, and particle size is allowed to vary; Symbols: experimental conditions.

Fig. 4 :

 4 Fig.4: Evolution of model sensitivities with respect to particle size (radius), for total heat release as QoI:-& •, case 1; • • • & •, case 2; ---& , case 3; -& , case 4; • • • & •, case 5; ---& •, case 6; -& , case 7; • • • & ,case 8. Lines: simulations, where the free stream and particle conditions are fixed similar as specified by the experiment, and particle size is allowed to vary; Symbols: experimental conditions.

Fig. 5 :

 5 Fig. 5: Evolution of sensitivities in E s a , with respect to ψ, for heat release as QoI:-& , case 7; • • • & , case 8

Fig. 6 :

 6 Fig. 6: Evolution of sensitivities with respect to particle size (radius), for ψ as QoI: -& •, case 1; • • • & •, case 2; ---& , case 3; -& , case 4; • • • & •, case 5; ---& •, case 6; -& , case 7; • • • & , case 8. Lines: simulations, where the free stream and particle conditions are fixed similar as specified by the experiment, and particle size is allowed to vary; Symbols: experimental conditions.

Table 1 :

 1 Experimental conditions, considered here for analysis.

	µm] Da g Symbol

Acknowledgements The authors kindly acknowledge financial support through Deutsche Forschungsgemeinschaft (DFG)-Projektnummer 215035359 -TRR 129.


Méo Pauline Cavagna
email: pauline.cavagna@aphp.fr

Stéphane Ikama

Euloge Kouadio

Jean Laurent Kramoh

Ibrahima Bara Takombe

Ibrahim Ali Diop

Dadhi M Toure

Anastase Balde

Beatriz Dzudie

Martin Ferreira

Houenassi

MD Kouadio Euloge Kramoh

MD Jean Laurent Takombe

MD Ibrahima Bara Diop

MD Ibrahim Ali Toure

MD Dadhi M Balde

MD Anastase Dzudie

MD Beatriz Ferreira

MD Martin Dèdonougbo Houenassi

MD Murielle Hounkponou

MD Adama Kane

MD Suzy Gisèle Kimbally-Kaki

MD Samuel Kingue

MD Charles Kouam Kouam

MD Emmanuel Limbole

MD Liliane Mfeukeu Kuate

Jean Bruno Mipinda

MD Roland N'guetta D

MD Carol Nhavoto

MD Jean Marie Damorou

MD Abdallahi Sidy Ali

MD Bamba Gaye

PhD Gabriel S Tajeu

PhD Diane Macquart De Terline

Pharm D Marie Cécile Perier

Michel Mch

Azizi

PhD Xavier Jouven

PhD Marie Antignac

Antihypertensive strategies and hypertension control in Sub-Saharan Africa

Keywords: Hypertension, cardiology, antihypertensive medications, developing countries

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

High blood pressure(BP) is the main contributor to the global burden of chronic disease reaching 10 million deaths each year [START_REF] Gbd | Risk Factor Collaborators. Global, regional, and national comparative risk assessment of 84 behavioural, environmental and occupational, and metabolic risks or clusters of risks for 195 countries and territories, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF]. In Africa, 130.2 million people suffer from hypertension and this figure is expected to reach 216.8 million by 2030 [START_REF] Adeloye | Estimating the prevalence and awareness rates of hypertension in Africa: a systematic analysis[END_REF]. A combination of lifestyle modification and blood pressurelowering medications are the cornerstone of hypertension control. Even though antihypertensive medications reduce hypertension-related complications, large-scale data on their use in Sub-Saharan Africa(SSA) are scarce [START_REF] Attaei | Availability and affordability of blood pressure-lowering medicines and the effect on blood pressure control in high-income, middle-income, and low-income countries: an analysis of the PURE study data[END_REF]. Furthermore, most studies in SSA are limited to single countries or centers [START_REF] Mutua | Level of blood pressure control among hypertensive patients on follow-up in a regional referral hospital in Central Kenya[END_REF].

We aimed to describe BP control and antihypertensive drug strategies using a large multinational study conducted in 12 SSA countries: the EIGHT Study(Evaluation of Hypertension in Sub-Saharan Africa) [START_REF] Antignac | Socioeconomic Status and Hypertension Control in Sub-Saharan Africa: The Multination EIGHT Study (Evaluation of Hypertension in Sub-Saharan Africa)[END_REF].

We conducted an observational cross-sectional study using data collected during outpatient consultations for hypertension in cardiology departments of 29 hospitals from 17 cities across 12 SSA countries. The EIGHT study was conceived by a multidisciplinary collaborative team of epidemiologists, cardiologists and pharmacists from Africa and France.

The study was approved by the Ile-de-France III ethics committee(Number_2014-A00710-47) and was exclusively supported by a public grant.

Consecutive men and women ≥18 years of age with hypertension were enrolled at any visit during outpatient consultations in the cardiology departments of the participating hospitals. Hypertension diagnosis was defined according to a BP level≥140/90 mmHg. Seated office BP was measured twice by physicians, at least 15 minutes apart; participants were instructed to avoid caffeine and smoking within 30 minutes prior to BP measurement. Uncontrolled hypertension was defined by a systolic BP(SBP) ≥ 140 mm Hg and/or a diastolic BP(DBP) of ≥90 mmHg of the measured office BP values in the clinic [START_REF]Practice guidelines for the management of arterial hypertension of the European Society of Hypertension (ESH) and the European Society of Cardiology (ESC): ESH/ESC Task Force for the Management of Arterial Hypertension[END_REF]. Severity of hypertension was defined according to European Society of Cardiology guidelines(6): SBP:140-159 mmHg and/or DBP:90-99 mmHg, SBP:160-179 mmHg and/or DBP:100-109 mmHg and SBP≥180 mmHg and/or DBP ≥110 mmHg. Antihypertensive drugs classes were reported by the physician during the consultation.

Continuous and categorical variables were expressed as mean(standard deviation) and numbers(percentage) where appropriate. Categorical variables were compared using Chi-square tests.

A two tailed p value of <0.05 was considered significant. Analyses were performed with R software(version 3.5.1(2018-07-02)).

A total of 2198 hypertensive patients (58.4±11.8years; 39.9% male) were included from six low-income countries(Benin, Democratic Republic of the Congo, Guinea, Mozambique, Niger and Togo) and six middle-income countries(Cameroon, Congo(Brazzaville), Gabon, Côte d'Ivoire, Mauritania, Senegal).

Among whom 2123(96.6%) had at least one antihypertensive drug, ranged from 88.8% in Niger and Senegal to 100% in Mauritania and Mozambic(Table 1).

The use of a monotherapy ranged from 13.5% in Togo to 48.2% in Democratic Republic of the Congo.

The use of two-drug strategies ranged from 31.8% in Mozambic to 52.4% in Guinea. Three-and more drug strategies varied from 2.9% in Guinea to 44.2% in Togo. The proportion of drugs strategies differed significantly according to countries(p<0.001). Furthermore, we observed a significant difference of strategies between low-and middle-income countries(55.3% and 44.7% of monotherapy, respectively; p<0.001). In addition, in low income countries monotherapy was prescribed more than two or three-and more drug strategies(p<0.001) whereas in middle income countries, three-and more drug strategies were more frequently prescribed than monotherapy or two-drug strategies p<0.001).

Among patients prescribed antihypertensive medication, mean systolic BP was 148.9±23.4mmHg and mean of diastolic BP measures was 88.2±14.2mmHg(Table1). Indeed, 1630(76.7%) had uncontrolled BP(range: 57.1%(140/245) in Benin to 100%(89/89) in Gabon, Figure 1). Among patients taking antihypertensive medication with uncontrolled hypertension (n=1630), 625(38.3%), 543(33.3%) and 462(28.3%) patients had hypertension 140/90mmHg, 160/100mmHg and 180/110mmHg respectively(Table 1). The percentage of patients with hypertension 180/110mmHg varied from 16.2%(20/123) in Guinea to 58.7%(47/80) in Gabon. Among patients with hypertension 180/110mmHg, 127(27.5%) were treated with a monotherapy. This proportion varied from 5.9% in Niger to 53% in Democratic Republic of the Congo.

In discussion, antihypertensive drug strategies differed largely according to country. The proportion of monotherapy in Democratic Republic of the Congo was three times higher than in Togo where the proportion of three-and more drug strategies was fifteen times higher than that of Guinea. In contrast, Guinea was the country where the proportion of two-drug strategies is the highest. Furthermore, hypertension is poorly controlled in this region even among those who are taking medications. A high proportion(28.3%) of patients with uncontrolled BP had hypertension 180/110mmHg, of whom 27.5% were still treated with a monotherapy. Our results are in keeping with the results collected in different regions of the world. Primary prevention efforts are poorly developed in people with high cardiovascular risk [START_REF] Kotseva | Lifestyle and impact on cardiovascular risk factor control in coronary patients across 27 countries: Results from the European Society of Cardiology ESC-EORP EUROASPIRE V registry[END_REF][START_REF] Kotseva | Primary prevention efforts are poorly developed in people at high cardiovascular risk: A report from the European Society of Cardiology EURObservational Research Programme EUROASPIRE V survey in 16 European countries[END_REF]. According to international guideline(6), these patients should be uptitrated with double, triple or more combination therapy [START_REF] Ojji | Comparison of Dual Therapies for Lowering Blood Pressure in Black Africans[END_REF]. The high proportion of patients with hypertension 180/110mmHg who were only treated with a monotherapy may be due not only to physicians' lack of knowledge of guidelines but, may be also due to clinical inertia, patient socioeconomic factors and behavior, or availability and affordability of antihypertensive medications. However, even though combination therapy is known to improve BP control [START_REF] Ojji | Comparison of Dual Therapies for Lowering Blood Pressure in Black Africans[END_REF], the proportion of uncontrolled hypertension remained very high even in patients who were prescribed combination therapy. We cannot exclude that the pharmaceutical quality of drugs did not contribute to treatment failure. Indeed, we have previously shown in the SEVEN study that 16% of cardiovascular drugs have produced in Asia are sold in street markets [START_REF] Antignac | Fighting fake medicines: First quality evaluation of cardiac drugs in Africa[END_REF].

We acknowledged the following limitations. The sampling framework in each country was not nationally representative, and therefore caution is needed in extrapolating the information in the current study to other populations.

Our study had many strengths including its multisite design, with over 2000 patients from 29 medical centers in 17 cities from 12 countries. Our analysis provided the opportunity to extend knowledge on effectiveness of antihypertensive strategies in the African continent.

In conclusion, our study described antihypertensive drug strategies prescribed in 12 Sub-Saharan countries. Even among patients recruited from tertiary cardiology centers in urban areas in Sub-Saharan countries and prescribed antihypertensive medications for almost all of them, BP control remains very poor and there is a large room for improvement in medical treatment. * Uncontrolled hypertension was defined by a systolic BP (SBP) ≥ 140 mm Hg and/or a diastolic BP (DBP) of ≥90 mmHg on either of office BP measures in the clinic.

Figure 1 :

 1 Figure 1: Proportion of patients with uncontrolled hypertension

Table 1 : Treated patients characteristics by country

 1

		GLOBAL	Niger	Togo	Benin	Guinea	Mozambic	Dem. Rep of	Congo	Senegal	Côte	Cameroon	Mauritani	Gabon
								the Congo			d'Ivoire		a
	N, (%)	2198	80 (3.6)	106 (4.8)	250 (11.4)	172 (7.8)	148 (6.7)	261 (11.9)	178 (8.1)	160 (7.3)	295 (13.5)	375 (17.1)	84 (3.8)	89 (4)
	Age (year), mean (sd)	58.3 (11.8)	56.6 (11.9)	55.7 (10.6)	56 (12.6)	57.8 (8)	53.4 (11.2)	62.5 (11.1)	55.9 (11.9)	58.9 (12.5)	60.1 (11.5)	60.9 (12.1)	57.1 (9.5)	53.7 (11.9)
	Male, N (%)	874 (39.8)	34 (42.5)	44 (41.5)	91 (36.4)	114 (66.3)	71 (48.0)	89 (34.1)	63 (35.4)	50 (31.2)	135 (45.8)	126 (33.6)	32 (38.1)	25 (28.1)
	Patients on antihypertensive medication, N (%)	2123 (96.6)	71 (88.8)	104 (98.1)	246 (98.4)	170 (98.8)	148 (100)	255 (97.7)	177 (99.4)	142 (88.8)	292 (99)	354 (94.4)	84 (100)	80 (89.9)
	Antihypertensive drug class, N
	(%)
	Calcium channel blocker 1219 (57.4) 38 (53.5) 67 (64.4)	176 (71.5)	54 (31.8)	114 (77.0)	146 (57.3)	93 (52.5)	74 (52.1)	159 (54.5)	208 (58.8)	51 (60.7)	39 (48.8)
	Diuretic	1167 (55.0) 54 (76.1)	71 (68.3)	97 (39.4)	85 (50.0)	70 (47.3)	145 (56.9)	125 (70.6)	80 (56.3)	130 (44.5)	225 (63.6)	30 (35.7)	55 (68.8)
	RAS Blocker : Angiotensin-converting-enzyme inhibitor	981 (46.2)	50 (70.4)	69 (66.3)	95 (38.6)	91 (53.5)	14 (9.5)	100 (39.2)	109 (61.6)	77 (54.2)	141 (48.3)	191 (54.0)	13 (15.5)	31 (38.8)
	RAS Blocker :
	Angiotensin II receptor	321 (15.1)	2 (2.8)	17 (16.3)	60 (24.4)	2 (1.2)	37 (25.0)	4 (1.6)	37 (20.9)	15 (10.6)	70 (24.0)	20 (5.6)	47 (56.0)	10 (12.5)
	antagonist
	Beta-blocker	466 (22.0)	12 (16.9)	36 (34.6)	45 (18.3)	37 (21.8)	30 (20.3)	25 (9.8)	51 (28.8)	41 (28.9)	86 (29.5)	70 (19.8)	31 (36.9)	2 (2.5)
	Centrally active drug	79 (3.7)	2 (2.8)	4 (3.8)	14 (5.7)	0 (0.0)	11 (7.4)	15 (5.9)	1 (0.6)	5 (3.5)	11 (3.8)	10 (2.8)	0 (0.0)	6 (7.5)
	Vasodilator	33 (1.6)	7 (9.9)	0 (0.0)	1 (0.4)	0 (0.0)	3 (2.0)	0 (0.0)	1 (0.6)	4 (2.8)	12 (4.1)	1 (0.3)	0 (0.0)	4 (5.0)

Office Blood pressure, mean (sd)

	Systolic blood pressure, mmHg	148.9 (23.4)	153.9 (25.1)	142.7 (20.2)	139.3 (22.5)	139.9 (17.5)	144.1 (19.8)	150.01 (22.5)	148.5 (20.5)	146.9 (22.5)	155.8 (25.7)	151.4 (23.4)	149.4(22.2)	175.6 (21.2)
	Diastolic blood pressure, mmHg	88.2 (14.2)	91.8 (13.8)	86.9 (14.6)	84.3 (13.3)	87.6 (10.2)	91.4 (13.8)	91.0 (15.2)	87.1 (12.3)	90.4 (13.4)	84.7 (15.1)	87.1 (14.1)	87.8 (11.8)	104.5 (13.4)
	Blood pressure control* (Uncontrolled vs controlled), N (%)	1630 (76.7) 65 (84.4)	77 (72.6)	142 (57)	125 (73.5)	119 (80.4)	232 (89.2)	134 (75.3)	128 (80.5)	228 (77.8)	285 (76.6)	68 (81)	89 (100)
	NA	10	3	0	1	2	0	1	0	1	2	3	0	0
	Hypertension severity, N (%)
	Grade 1	625 (38.3)	15 (23.1)	40 (51.9)	51 (35.9)	70 (56.0)	31 (26.1)	86 (37.1)	70 (52.2)	50 (39.1)	77 (33.8)	115 (40.4)	35 (51.5)	8 (9.0)
	Grade 2	543 (33.3)	31 (47.7)	22 (28.6)	59 (41.5)	35 (28.0)	47 (39.5)	69 (29.7)	38 (28.4)	49 (38.3)	87 (38.2)	77 (27.0)	16 (23.5)	28 (31.5)
	Grade 3	462 (28.3)	19 (29.2)	15 (19.5)	32 (22.5)	20 (16.0)	41 (34.5)	77 (33.2)	26 (19.4)	29 (22.7)	64 (28.1)	93 (32.6)	17 (25.0)	53 (59.6)

Acknowledgements: -Funding:

The study was exclusively supported by a public grant:

-INSERM (Institut national de la Santé et de la Recherche Médicale), -AP-HP (Assistance Publique -Hôpitaux de Paris), Paris Descartes University Disclosures: none declared. No author has any competing interests Author's contribution: All authors have substantial contributions.

M. Antignac and X. Jouven had full access to the whole data in the study and take responsibility for integrity of the data and accuracy of data analysis. They take responsibility for all aspects of the reliability and freedom from bias of presented data and their discussed interpretation. Study concept and design: M. Antignac, IB Diop, X. Jouven. Responsible of data collection : Dr. DIOP, Dr. KRAMOH, Dr. ALI TOURE, Dr. BALDÉ, Dr. FERREIRA, Dr. HOUENASSI, Dr. IKAMA, Dr. KANE, Dr. KIMBALLY KAKI, Dr. KINGUE, Dr. KOUAM KOUAM, Dr. KUATE MEKFEU, Dr. NHAVOTO, Dr. LIMBOLE, Dr. TAKOMBE, Dr. MIPINDA, Dr. SIDY ALI, Dr. N'GUETTA, Dr. DZUDIE, Dr. DAMOROU. Interpretation of data: all authors. Statistical analysis: P. Cavagna, M. Antignac, MC Perrier, X. Jouven Drafting of the manuscript: P. Cavagna, M. Antignac, M. Azizi, X. Jouven Critical revision of the manuscript for important intellectual content: M. Antignac, M. Azizi, G. S. Tajeu, X. Jouven. Final approval of the version to be published: all authors

No author has any competing interests. EIGHT study was supported by a public grant Disclosure: none declared.