Nicolas Blet
email: nicolas.blet@univ-lorraine.fr

Denis Maillet

Analytical steady-state model based on Fourier integral transforms for cylindrical heat pipes under axisymetric conditions

Keywords: Cylindrical heat pipe, Steady-state analytical model, Thermal quadrupoles, Thermohydraulic coupling

A thermohydraulic analytical model of a capillary cylindrical heat pipe in steady-state is proposed in this article. It is based on an original representation by thermal quadrupoles to describe heat transfer in the wall and in the porous wick, via the use of Fourier integral transforms. Thanks to a validation from literature results, this model provides two-dimensional axisymetric thermal fields and one-dimensional pressure and velocity profiles of both liquid and vapour flows. Different developments and solutions are introduced according to the kind of boundary conditions at evaporator and at condenser, and with a more or less strong thermohydraulic coupling at the liquid/vapour interface. For the simple case with imposed uniform heat fluxes, intrinsic properties of the heat pipe are originally defined. The introduced model offers a generalisation of analytical models of standard heat pipe as a design or optimisation tool. Wider developments of analytical models for more complex three-dimensional geometries of heat pipe and in transient regime can be expected.

Introduction

Many modelling developments and analyses of heat pipes have been carried out and can be found nowadays, thanks to the increase of application opportunities, either to improve heat transfer (electronic cooling, ...) or to reduce thermal gradients [START_REF] Blet | Heats pipes for temperature homogenization : a literature review[END_REF]. While some modelling improvements still remain for loop heat pipes and pulsating heat pipes, much work has already been done to develop sizing or analysis tools for capillary heat pipes.

Today, numerical studies are often selected to study precise thermohydraulic effects in the vapour flow or in the liquid flow in the porous media, but some analytical developments can be found, even in recent years. While sizing models were firstly focused on hydraulic considerations, that is evaluating a capillary limit from liquid and vapour pressure drops [START_REF] Faghri | Heat Pipe Science and Technology[END_REF], more precise thermal models have been introduced now.

Zuo and Faghri [START_REF] Zuo | A network thermodynamic analysis of the heat pipe[END_REF] propose a transient thermal network to describe the behaviour of a standard cylindrical heat pipe through simple radial or axial thermal resistances. Solving the ordinary differential equations system is done by numerical standard Runge-Kutta method. From boundary conditions, the operating temperature of the heat pipe can be evaluated in particular. With Darcian pressure drops for the vapour and liquid flows and from the Clausius-Clapeyron equation, a dimensionless criterion of "thermal compatibility" is proposed between the geometric parameters of the heat pipe, the thermal conductivity of the material and the fluid properties.

Zhu and Vafai [START_REF] Zhu | Analysis of cylindrical heat pipes incorporating the effects of liquid-vapor coupling and non-darcian transport -a closed form solution[END_REF] focus on an analytic development of velocity and pressure profiles of the vapour and liquid flows in a cylindrical heat pipe under axisymetric boundary conditions in steady-state. In particular, non-Darcian transport and interfacial effects are taken into account. The method of matched asymptotic expansions is used to obtain a closed-form solution. Assuming that the vapour phase is saturated with a uniform temperature along the heat pipe, the external wall temperature profile is obtained by a simple radial heat conduction model without heat losses, from the heat input at evaporator and the coolant temperature at condenser.

In the case where the capillary structure is merely grooves in the heat pipe wall, some specific modelling have been also developed. For example, Kim and Seo [START_REF] Kim | Analytical and experimental investigation on the operational characteristics and the thermal optimization of a miniature heat pipe with a grooved wick structure[END_REF] propose a mathematical model to evaluate the maximum heat transport capability of a miniature heat pipe with a special shape grooved wick structure. From continuity and momentum balances of the vapour and liquid flows, the evaluation of the axial evolution of the interface curvature and the liquid block length at condenser are assessed first. A mean thermal resistance of the heat pipe is then calculated from mean resistances specific to heat transfer at evaporator, to the evolution of the saturated vapour temperature and to heat transfer at condenser, according to the various liquid levels in the grooves at both evaporator and condenser.

In 2011, Aghvami and Faghri [START_REF] Aghvami | Analysis of flat heat pipes with various heating and cooling configurations[END_REF] introduce a two-dimensional analytical thermohydraulic model for flat heat pipes with various sollicitation configurations in steady state. Temperature field in the wall of the heat pipe are evaluated under the form of infinite Fourier cosine series, with coefficients under exponential form. A constant saturation temperature is imposed at the liquid/vapour interface and a linear temperature profile across the wick structure is assumed, through an equivalent thermal conductivity. The velocity and pressure distributions for the vapour flow and for the liquid flow are obtained thanks to the assumption of a parabolic velocity profile. The overheating of the vapour is then evaluated from the Clapeyron equation and the effective capillary pressure is deduced. In a parallel work, Shabgard and Faghri [START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF] propose an equivalent analytical model for capillary cylindrical heat pipes. With the same assumptions, two-dimensional wall temperature field, and velocity and pressure profiles for vapour and liquid flows are written under the form of infinite series of products of cosine and modified Bessel functions. All the analytical results are compared with full numerical simulations. From a parametric analysis, Shabgard and Faghri propose an improvement of the previous equivalent thermal networks by introducing some additional thermal resistances for a better consideration of axial heat transfer in the wall, which is not necessarily negligible in certain cases, especially in the calculation of the pressure drops in the heat pipe. In 2014, Lips and Lefèvre [START_REF] Lips | A general analytical model for the design of conventional heat pipes[END_REF] generalise this kind of approach by introducing an analytical design model for flat and cylindrical capillary heat pipes. Three-dimensional calculations are here made with double Fourier cosine and sine series. The thermal problem of the cylindrical case is solved in rectangular coordinates as well as in the flat case, by assuming a thin thickness for the wall with respect to the perimeter of the heat pipe. An imposed uniform saturation temperature and imposed heat fluxes at evaporator and at condenser are taken as boundary conditions.

In the last years, full numerical analyses have been achieved. Three-dimensional thermohydraulic solving is performed by the finite volume method [START_REF] Pooyoo | Numerical simulation of cylindrical heat pipe considering non-darcian transport for liquid flow inside wick and mass flow rate at liquid-vapor interface[END_REF][START_REF] Brahim | Effect of the heat pipe adiabatic region[END_REF], by finite elements [START_REF] Naemsai | Numerical model of heat transfer characteristics for sinteredgrooved wick heat pipes under non-uniform heat loads[END_REF] or by the more recent Lattice-Boltzmann method [START_REF] Grissa | Performance of a cylindrical wicked heat pipe used in solar collectors: Numerical approach with lattice boltzmann method[END_REF]. The main disadvantage of these numerical modelling is naturally the long duration of the calculation compared to previous analytical models. Simulation time of several hours are needed against few seconds for the calculation of analytical Fourier series (with 200 terms) [START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF]. So, even if some additional assumptions are required with some small inaccuracies, the analytical models are always interesting in order to develop a fast and quite accurate design tool for a capillary heat pipe. The present article introduces thus a generalisation of the analytical approach by Fourier series [START_REF] Aghvami | Analysis of flat heat pipes with various heating and cooling configurations[END_REF][START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF][START_REF] Lips | A general analytical model for the design of conventional heat pipes[END_REF] for a capillary cylindric heat pipe via an original mathematical representation by thermal quadrupoles.

Problem description

The developed model is focused in this article on the standard axisymetric problem of a capillary cylindrical heat pipe in steady state. The heat source and the heat sink can be either located at each extremity (Fig. 1) for usual applications or anywhere along the axis. The problem of multi-sources positioning of the x axis can even be considered. The only constraint is to consider these sources as uniform around the axis of the heat pipe. Furthermore, the studied heat pipe is composed of a hollow tube with an inner porous layer, but the model can be extended to a heat pipe with several porous layers.

The considered problem relies on the following assumptions, mainly posed for an achievable analytical solution:

• the vapour and liquid flows are considered unidirectional along the x axis ;

• the vapour flow is supposed at saturation with a parabolic axial velocity profile ; • the thermodynamic properties of the fluid and of the solid parts are supposed constant ; properties of the vapour and of the liquid are assumed to be evaluated at the mean saturation temperature at the liquid/vapour interface along the heat pipe ; due to a low velocity liquid flow, an equivalent thermal conductivity λ eq is assumed for the capillary structure full of liquid. In the literrature, different formulations can be found to determine it, according to the fluid-material pair and the kind of capillary structure [START_REF] Atabaki | Effective thermal conductivity of water-saturated sintered powder-metal plates[END_REF]. For example, for a wrapped screen [START_REF] Chi | Heat Pipe Theory and Practice[END_REF], λ eq can be evaluated by:

λ eq = λ l [λ l + λ w -(1 -w )(λ l -λ w )] λ l + λ w + (1 -w )(λ l + λ w ) (1)
where w is the porosity of the wick ;

• the heat pipe operation is supposed to be regular : the liquid/vapour interface is located at the inner diameter, corresponding to the vapour domain, and no thermal effect of the interface curvature or of a possible liquid slug is considered.

In the rest of the document, the solid geometry of the heat pipe will be defined by the axial abscissa x from 0 to the length of the heat pipe L, and by the radius abscissa r varying from the inner radius R i of the vapour domain to the outer radius R o of the solid wall. The radial location of the contact between the wall and the porous wick is noted R w .

The following boundary conditions for the global problem are considered (they correspond to the most standard conditions met in the literature):

• the heat pipe is insulated at each extremity, x = 0 and x = L ;

• a heat flux q E is imposed at the evaporator ; according to axisymetric assumptions, q E can only depend on

x ;

• a condition of convection is considered at the condenser, with a reference temperature T C and a heat transfer coefficient h C ;

• heat losses with ambience can be taken into account at a temperature T ∞ through a heat transfer coefficient h ∞ .

Some of these conditions could be simplified according to cases considered in the next sections.

With these external conditions, different models are developed depending on the coupling between thermal model and hydraulic model (with or without the assumption of a uniform saturation temperature for the evaluation of heat transfer) and on the considered vapour pressure drop (with or without non-linear dynamic pressure drop).

Model development

Thermal model

Heat equations in the two layers (solid and porous wick) are simply defined in steady state by the Laplace equation, written in cylindrical coordinates:

∂ 2 T ∂x 2 + 1 r ∂ ∂r r ∂T ∂r = 0 (2)
To solve more easily this Laplace equation, it is interesting to work with temperature variation around a reference temperature, rather than directly with absolute temperature. In our case, a good idea is to take the reference temperature as the mean fluid saturation temperature in the axis direction and at r = R i , noted T sat . So we pose thereafter θ = T -T sat . The heat equation is then written:

∇ 2 θ = 0 (3)
The adiabatic boundary conditions at x = 0 and x = L are written:

∂θ ∂x x=0 = ∂θ ∂x x=L = 0 (4)
Due to these conditions, the Fourier integral transform of θ, noted θn , relative to the space variable x ∈ [0; L], is defined as: θn (r )

= L 0 θ(x, r )cos(α n x)d x ( 5
)
where the eigenvalues α n are defined for any natural number n by α n = nπ/L. The associated inverse transform for returning to the original domain is:

θ(x, r ) = θ0 (r ) L + 2 L ∞ n=1 θn (r )cos(α n x) (6)
The integral tranforms of the heat equation (3), taking into account boundary conditions (4), become:

• for the fundamental eigenvalue (n = 0): 1

r d d r r d θ0 d r = 0 (7)
• for the harmonic eigenvalues (n > 0): 1

r d d r r d θn d r -α 2 n θn = 0 (8)
Introduction of the integral transform of the rate of heat flux through a cylindrical surface, per unit length in the axial direction, φ = -2πr λ∂θ/∂r , makes a corresponding quadrupole form possible for a layer with constant thermal conductivity located between two radii R 1 and R 2 :

θn φn r =R 2 = A n B n C n D n θn φn r =R 1 (9) Notation Expression A 0 1 B 0 l n (R 1 /R 2 ) /(2πλ) C 0 0 D 0 1 A n α n R 1 [I 0 (α n R 2 )K 1 (α n R 1 ) + I 1 (α n R 1 )K 0 (α n R 2 )] B n [I 0 (α n R 1 )K 0 (α n R 2 ) -I 0 (α n R 2 )K 0 (α n R 1 )]/(2πλ) C n 2πλα 2 n R 1 R 2 [I 1 (α n R 1 )K 1 (α n R 2 ) -I 1 (α n R 2 )K 1 (α n R 1 )] D n α n R 2 [I 0 (α n R 1 )K 1 (α n R 2 ) + I 1 (α n R 2 )K 0 (α n R 1 )
] I 0 , I 1 and K 0 , K 1 are the modified Bessel functions of zero and first order, respectively, of first and second kind [START_REF] Maillet | Thermal Quadrupoles : solving the heat equation through integral transforms[END_REF]. It has to be noted that the coefficient B 0 corresponds to the radial conductive resistance, and so the two fundamental scalar equations above correspond to a one-dimensional model.

The two-layers conductive problem can also be written under the quadrupole form, linking both interfaces

r = R i and r = R o : θn φn r =R o = A n B n C n D n θn φn r =R i ( 10
)
where the quadrupole matrix is the product of the two matrices related to each of the two layers.

Thermodynamic considerations at the liquid/vapour interface

The boundary condition at r = R i corresponds to the liquid/vapour interface.

Thermal conditions

By definition of the mean vapour temperature T sat , the fundamental θ0 r =R i is null. Furthermore, a heat balance written for the whole vapour domain implies :

L 0 φ r =R i = 0 ⇒ φ0 r =R i = 0 (11)

Thermodynamic condition of saturation

As a first approximation, the saturation temperature variation along the heat pipe can be neglected for the estimation of the heat transfer, typically when the vapour pressure drop is small compared to the variation of the saturation pressure, for example with a large vapour diameter. In that case, it implies in addition :

θn r =R i = 0 (12)
However, in order to have a stronger coupling between heat transfer and hydraulics, for example for heat pipes with smaller vapour space, it can be interesting to take into account the axial variation of the saturation temperature.

The idea of the model is to linearise the saturation equation near the mean temperature T sat :

d P v d T v = K sat ( 13
)
where K sat is a constant depending on T sat only. This linearisation implies the relationship :

P v -P sat (T sat ) = K sat θ| r =R i ( 14
)
and so the mean vapour pressure is equal to the saturation pressure at T sat .

Hydrodynamic model 3.3.1. Vapour flow Velocity profile.

The axial velocity profile of the vapour flow is assumed to be parabolic. With symetry at r = 0 and slip condition at the interface (r = R i ) [START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF], it is expressed as:

u x (x, r ) = 2u v (x) 1 - r 2 R 2 i ( 15
)
where u v (x) is the mean vapour velocity defined by:

u v (x) = 2 R 2 i R i 0 r u x (x, r )d r (16)
The mass balance is written:

r ∂u x ∂x + ∂(r u r ) ∂r = 0 (17)
Integrating Eq. ( 17) between 0 and r provides the expression of the radial velocity:

u r (x, r ) = - d u v d x r 1 - r 2 2R 2 i ( 18
)
At r = R i , assuming that conductive effects are negligible compared to phase change, the balance at the interface gives the relation between the mean velocity and heat flux:

d u v d x = -φ r =R i πρ v h l v R 2 i ( 19
)
Integrating Eq. ( 19) between 0 and x, and developing the Fourier series of φ r =R i , the mean velocity of vapour is finally expressed as:

u v (x) = -2 πρ v h l v R 2 i L ∞ n=1 φn r =R i α n si n(α n x) (20)
Pressure profile.

The momentum balance of the vapour flow is written assuming that pressure variations are unidirectional following x:

∂P v ∂x = µ v 1 r ∂ ∂r r ∂u x ∂r + ∂ 2 u x ∂x 2 -ρ v u r ∂u x ∂r + u x ∂u x ∂x (21)
Thanks to expressions of axial and radial velocities (Eqs ( 15) and ( 18)), integrating Eq. (21) between 0 and R i gives:

d P v d x = -8µ v R 2 i u v + µ v d 2 u v d x 2 - 4 3 ρ v d u 2 v d x ( 22
)
Integrating Eq. ( 22) between 0 and x and introducing the mean vapour pressure result in a vapour pressure profile that can be expressed by:

P v (x) = P sat (T sat ) + 8µ v R 2 i 1 L L 0 x 0 u v (x )d x d x - x 0 u v (x )d x + µ v d u v d x + 4 3 ρ v 1 L L 0 u 2 v (x)d x -u 2 v (x) (23)
From the expression of the mean vapour velocity (20), Eq. ( 23) finally gives:

P v (x) = P sat (T sat )- 16µ v πρ v h l v R 4 i L ∞ n=1 1 α 2 n + R 2 i 8 φn r =R i cos(α n x)+ 16 3π 2 ρ v h 2 l v R 4 i L 2   1 2 ∞ n=1 φ2 n r =R i α 2 n - ∞ n=1 φn r =R i α n si n(α n x)
Simplifying assumption.

If the dynamic pressure drop (last term in Eq. ( 24)) is considered negligible, the vapour pressure profile becomes linear and can therefore be written as a Fourier serie where the pressure harmonics are linked to the heat flux harmonics at r = R i by:

Pv,n = -8µ v πρ v h l v R 4 i 1 α 2 n + R 2 i 8 φn r =R i (25)
Thanks to the linearisation of the saturation curve [START_REF] Atabaki | Effective thermal conductivity of water-saturated sintered powder-metal plates[END_REF], heat flux and temperature harmonics at r = R i are then linked by:

φn r =R i = -K sat 8µ v πρ v h l v R 4 i 1 α 2 n + R 2 i 8 θn r =R i = -K n θn r =R i (26)

Liquid flow Velocity profile.

In the same way as the vapour flow, the global mass balance of the liquid flow can be written:

d u l d x = φ r =R i ρ l A w h l v ( 27
)
with

A w = π(R 2 w -R 2 i
) the liquid section, and u l the mean liquid velocity defined by:

u l (x) = 2 R 2 w -R 2 i R w R i r u x d r (28)
Integrating Eq. ( 27) between 0 and x gives:

u l (x) = 2 ρ l A w h l v L ∞ n=1 φn r =R i α n si n(α n x) (29)
Pressure profile. Liquid pressure drop in the porous wick is supposed to follow the Darcy law. Neglecting the dynamic pressure, the global momentum balance of the liquid flow is:

d P l d x = - µ l k w u l + ρ l g si n(γ) ( 30
)
Integrating Eq. (30) between 0 and x, and then assuming that the minimum capillary pressure (corresponding to the minimum of the difference ∆P v (x) -∆P l (x)) is null (flat interface) and located at x = x 0 , the expression of the liquid pressure can be written as:

P l (x) = P v (x 0 ) + 2µ l ρ l k w A w h l v L ∞ n=1 φn r =R i α 2 n (cos(α n x) -cos(α n x 0 )) + ρ l g si n(γ)(x -x 0 ) (31)
where P v (x 0 ) is evaluated directly from Eq. (24).

Capillary pressure

In the general case, the capillary pressure is just determined from the general expressions of the vapour and liquid pressures (Eqs (24) and (31)):

P c ap (x) = P v (x) -P l (x) (32)
If the dynamic vapour pressure and the effects of the variation of the mean vapour velocity are considered negligible, with the evaporator and the condenser at each side of the heat pipe (standard configuration), the minimum capillary pressure is located at x 0 = L and the maximum capillary pressure at x = 0. In this case, this maximum capillary pressure can be simply written:

P cap,max = ρ l g si n(γ)L - 4 L 8µ v πρ v h l v R 4 i + µ l ρ l k w A w h l v ∞ m=0 φ2m+1 r =R i α 2 2m+1 (33)

Calculation method 3.4.1. Quadrupole model used according to the model assumptions

From the previous development, the thermal quadrupole model can be more or less simplified according to the assumptions made on the effects of the temperature axial variation on heat transfer, and of taking into account the dynamic vapour pressure drop.

Whatever the assumptions, the fundamental (n = 0) of the quadrupole model gives:

θ0 r =R o = 0 (34a) φ0 r =R o = 0 (34b)
The first equation corresponds to the equality between the two mean temperatures at r = R i and r = R o . The second equation explicits a null global heat balance for the heat pipe.

With regard to the thermal harmonics, the different assumptions lead to more or less simple models:

• if the saturation temperature is considered uniform to estimate heat transfer, the quadrupole model is simplified by: θn

φn r =R o = A n B n C n D n 0 φn r =R i ( 35
)
This model is then decoupled from the hydrodynamic model. Both models can so be used independently of each other. From this quadrupole model, the harmonics of temperature and heat flux at r = R o can be directly linked:

φn r =R o = D n B n θn r =R o (36)
• if the thermohydraulic coupling is taken into account at the liquid/vapour interface :

in the case that dynamic vapour pressure drop is neglected, the thermodynamic coupling is linear (thanks to the linearisation of the saturation curve). The thermal quadrupole model can be written under the form:

θn φn r =R o = A n B n C n D n θn 0 r =R i ( 37
)
where the modified quadrupole coefficients are defined by:

A n B n C n D n = A n B n C n D n 1 0 -K n 1 ( 38
)
with K n defined in Eq. ( 26).

From this modified quadrupole model, we can write then the direct relationship:

φn r =R o = C n A n θn r =R o ( 39
)
if no vapour pressure drops are neglected, the initial quadrupole model ( 10) is used.

Estimation of temperature and heat flux fields

Solving the considered thermal quadrupole model from the external conditions of the heat pipe (see next section) provides the expressions (or calculations) of the integral transforms of θ and φ at the outer wall of the heat pipe (r = R o ). The integral transforms at r = R i are easily deduced from the global quadrupole model. From there, the integral transforms can be evaluated for any radius r thanks to the thermal quadrupole of each layer (Eq. 9), with a reference : for the wall layer the results at r = R o , and for the porous layer the results at r = R i . Finally, the temperature and heat flux fields can be calculated for all x and r by the inverse transform (Eq. 6), knowing that the fundamental transforms (n = 0) are null according to the developed model.

Model solution following external conditions

Solution with two heat flux sources

Problem description

The external boundary conditions can be simplified (Fig. 2) at first, with some assumptions:

• heat fluxes at both boundaries (evaporator and condenser) are imposed, with respective values q E and q C ; this assumption is perfectly correct for the evaporator but it can be questionable for the condenser, for example if its size is not small compared to the size of the heat pipe ; both heat fluxes can be considered non-uniform on their respective length ; finally, the two heat fluxes can be considered as a single heat source, noted q, defined on the entire length of the heat pipe and equal to -q E at the evaporator, equal to q C at the condenser and equal to zero along the adiabatic zone ;

• the heat losses with ambience are considered over the entire length of the heat pipe.

x r 0 The integral tranforms of the heat flux at r = R o can be directly evaluated here:

R o L x E x C q E (x) q C (x) h ∞ , T ∞ q(x) R i saturation R w
φ0 r =R o = q0 + 2πR o h ∞ θ0 r =R o + 2πR o h ∞ L(T sat -T ∞ ) (40a) φn r =R o = qn + 2πR o h ∞ θn r =R o (40b)

Solution of the quadrupole model

Eqs. (34a) and (34b), coupled with the expression (40a), provide the expression of T sat :

T sat = T ∞ - q0 2πR o h ∞ L (41)
From this expression, it can be noticed that this mean saturation temperature can be only calculated if heat losses are defined (h ∞ = 0). If it is not the case, the value of T sat must be provided with this model.

Direct analytical solution achievable.

If the saturation temperature is considered uniform in the estimation of heat transfer, Eqs. ( 36) and (40b) provide the expressions of the harmonics at r = R o :

θn r =R o = B n D n -2πR o h ∞ B n qn (42a) φn r =R o = D n D n -2πR o h ∞ B n qn (42b)
If the thermohydraulic coupling is considered with linear vapour pressure drop, from Eqs (39) and (40b), these equations turn into:

θn r =R o = A n C n -2πR o h ∞ A n qn (43a) φn r =R o = C n C n -2πR o h ∞ A n qn (43b)
With both above first assumptions, if in addition the heat fluxes at evaporator and at the condenseur are uniform, the integral transforms of the heat source q can be easily evaluated. The temperature and the heat flux fields can be explicitly evaluated from the two total sources Q E and Q C , under the form:

θ φ x,r = Z E Z C W E W C x,r Q E -Q C ( 44
)
The impedances Z and the transmittances W are independent of the values of the imposed heat fluxes. So they are intrinsic properties of the heat pipe, depending on the fluid and material properties and on the location of the boundary conditions. Their expressions, with a thermohydraulic coupling, are for J ∈ {E ,C } (in parentheses, the references R 1 and R 2 for the coefficients of Table 1):

• for the porous wick layer (R 1 = R i and R 2 = r , where R i ≤ r ≤ R w ):

Z J x,r = 2 L ∞ n=1 A n -B n K n 2πR o h ∞ A n -C n χJ,n χJ,0 cos(α n x) (45a) W J x,r = 2 L ∞ n=1 C n -D n K n 2πR o h ∞ A n -C n χJ,n χJ,0 cos(α n x) (45b)
• for the wall layer (R 1 = R o and R 2 = r , where R w ≤ r ≤ R o ):

Z J x,r = 2 L ∞ n=1 A n A n + B n C n 2πR o h ∞ A n -C n χJ,n χJ,0 cos(α n x) (46a) W J x,r = 2 L ∞ n=1 C n A n + D n C n 2πR o h ∞ A n -C n χJ,n χJ,0 cos(α n x) (46b)
The χ J (x) functions are the indicator functions, equal to 1 if x is in the considered domain (referenced by subscript J ) and equal to 0 everywhere else.

Finally, the reference temperature T sat is, from Eq. (41):

T sat = T ∞ + Q E -Q C 2πR o h ∞ L (47)
From this last equation, the temperature and heat flux fields can be rather expressed as a function of the arithmetic average of the source at evaporator and the source at condenser, (

Q E + Q C )/2, and of the temperature dif- ference T sat -T ∞ : θ φ x,r = Z W sat W Y sat x,r Q E +Q C 2 T sat -T ∞ ( 48
)
with the relationships:

Z = Z E -Z C , W = W E -W C , W sat = πR o h ∞ L(Z E + Z C ) and Y sat = πR o h ∞ L(W E + W C ).
If no heat loss is taken into account (h ∞ = 0), no more link exits between saturation temperature and heat sources. The mean saturation temperature has to be imposed and the last relation can not be written. However, in that case, the heat fluxes at evaporator and condenser are necessarily equal: 44) is then simply written as:

Q E = Q C = Q. Eq. (
θ φ x,r = Z W x,r Q (49)
The previously defined transmittance W sat and the admittance Y sat characterise thus the effects of heat losses with ambience for a given temperature difference T sat -T ∞ .

With regard to the capillary pressure, in the case of uniform heat fluxes at evaporator and at condenser, located at each extremity of the heat pipe, and with Darcy-Weisbach vapour pressure drop only, the following explicit expression of the maximum capillary pressure is obtained from Eq. ( 33), all calculations done (with a thermohydraulic coupling):

P c ap,max = ρ l g si n(γ)L - 4L π 3 L eff 8µ v L eff πρ v h l v R 4 i + µ l L eff ρ l A w h l v k w J ∈E ,C Q J L J L ∞ m=0 -K 2m+1 2πR o h ∞ A 2m+1 -C 2m+1 si n(α 2m+1 L J ) (2m + 1) 3 (50) with L eff = L -0.5(L E + L C
), the effective length of the heat pipe.

Moreover, if the heat pipe is perfectly insulated, it can be assumed that h ∞ = 0 and the heat power Q is the same at evaporator and at condenser. The following expression of the heat pipe capillary limit can be deduced:

Q cap,max = 2σ R eff -ρ l g si n(γ)L 8µ v L eff πρ v h l v R 4 i + µ l L eff ρ l k w A w h l v π 3 L eff 4L J ∈{E ,C } ∞ m=0 L L J si n (2m+1)πL J L (2m+1) 3 -K 2m+1 C 2m+1 (51)
with R eff the effective pore radius of the wick. This expression is similar to the analytical expression given by Lips and Lefèvre [START_REF] Lips | A general analytical model for the design of conventional heat pipes[END_REF]. The first factor in the product corresponds to the well-known analytical expression of the capillary limit when the heat conduction in the wall and in the porous wick is neglected [START_REF] Faghri | Heat Pipe Science and Technology[END_REF]. The second term is a correction factor taking into account this phenomenon. This term is always greater than 1. It has to be noticed that, when the solid structure is neglected, coefficients C n tend to -K n , and so the correction factor is well equal to 1:

π 3 L eff 4L J ∈{E ,C } ∞ m=0 L L J si n (2m+1)πL J L (2m+1) 3 = 1 (52)
Finally, still in the case of the simplified model with uniform heat fluxes at evaporator and at condenser, intrinsic relationships can be developed between each hydrodynamic variables (pressure or velocity) and the heat flux inputs, as well as for the thermal variables (with impedances and transmittances).

Iterative analytical solving required.

If no assumption is made (thermohydraulic coupling and nonlinear vapour pressure drop), a direct analytical solution is not possible. An iterative procedure is thus necessary.

On the one hand, from the quadrupole model [START_REF] Brahim | Effect of the heat pipe adiabatic region[END_REF] and Eq. (40b), the following linear relation can be written:

φn r =R i = qn -(C n -2πR o h ∞ A n ) θn r =R i D n -2πR o h ∞ B n (53)
On the other hand, the linearisation of the saturation curve ( 14) and the expression of the vapour pressure (24) provides the expression for θ| r =R i as a function of the harmonics of φ r =R i . From the definition of the integral transform (5), the following equation is finally obtained:

θn r =R i = - 8µ v πρ v h l v R 4 i K sat 1 α 2 n + R 2 i 8 φn r =R i - 4 3π 2 ρ v h 2 l v R 4 i LK sat 2 ∞ m=1 φm r =R i φm+n r =R i α m α m+n - n-1 m=1 φm r =R i φn-m r =R i α m α n-m (54)
The very fast iterative procedure between the two above equations provides the harmonics of temperature and heat flux at r = R i . The quadrupole model allows to go back to the values of the harmonics at r = R o .

Solving with one heat flux source and one temperature source

Under real conditions, described in section 2, the problem can not be solved directly by thermal quadrupoles anymore, because the external conditions are localised at specific lengths of the heat pipe (Fig. 3). The idea is here to mesh the external wall following the direction x, with points x i ,1≤i ≤N ∈ [0; L] and to find the unknown variables (temperature and/or heat fluxes) on this side of the heat pipe. To simplify the notation, the mentioned variables in the original domain (depending on x and r ) are at r = R o in the rest of this paragraph.

x r 0 R o L x E x C q E (x) h ∞ , T ∞ R i saturation h C , T C R w
On the one hand, the heat flux at r = R o can be written:

φ(x) = -χ E (x)q E (x) + χ A (x)2πR o h ∞ (T (x) -T ∞ ) + χ C (x)2πR o h C (T (x) -T C ) (55)
On the other hand, from the quadrupole model ( 10) and for any case, this heat flux can be written:

φ(x) = 2 L ∞ n=1 D n B n θn r =R o cos(α n x) - 2 L ∞ n=1 1 B n θn r =R i cos(α n x) (56)
If uniform saturation temperature is supposed in the thermal model (with any kind of vapour pressure drop) or if a thermohydraulic coupling is considered with linear vapour pressure drop, Eq (56) can be simplified with only the harmonics θn r =R o (Appendix A). In any case, the expressions of these harmonics are developed as function of the external temperatures from Eq (5) by a standard Newton-Cotes method.

Equality of Eqs. ( 55) and ( 56) is then written for all x i of the mesh. For both cases of uniform T sat for the thermal model and of thermohydraulic coupling with linear ∆P v , it leads to a linear system under the form AX = BU (Appendix A), linking the unknown variables X , namely the temperatures T (x i ), and the inputs U , namely q E (x i ), T ∞ and T C . An iterative procedure is nevertheless necessary to solve this system by introducing a closure equation to evaluate T sat on which matrix A depends (Fig. A.1).

For the case of a thermohydraulic coupling with nonlinear vapour pressure drop, a more complex iterative procedure is necessary (Fig. A.2), involving in particular the nonlinear expression of θn r =R i (Eq. ( 54)). It leads to a system under the form AX = BU +C , with C is a vector taking into account the nonlinearities at the liquid/vapour interface (Appendix A).

Once the system is solved, the integral transforms of temperature at r = R o are deduced from the discretisation:

θn r =R o = N j =1 w j (T (x j ) -T sat )cos(α n x j ) ( 57
)
with w j the coefficients of the Newton-Cotes method.

Then, the quadrupole model will provide temperature and heat flux fields (see section 3.4.2).

Summary

The different ways to solve both models (according to the external thermal conditions) for the three assumptions introduced at the liquid/vapour interface are summarised in Table 2 as a synthesis of the introduced mathematical developments. It can be noticed that iterative procedures are very fast with respect to other numerical methods, even in the worst case (less than five minutes with a standard computer for the case of a thermohydraulic coupling with nonlinear vapour pressure drop).

External

Comparison between model and literature results

To confirm the results of the developed model, they are compared with two different cases from Shabgard and Faghri studies [START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF] on copper/water heat pipes. The first case, named case A, corresponds to a standard heat pipe, with evaporator and condenser at each extremity, without heat losses and with a convective cooling boundary condition at the condenser. The second case, named case B, is a heat pipe subjected to four evaporators and one condenser with conditions of imposed heat fluxes, still without heat losses. The main specifications of both cases are given in Table 3.

While case A is compared directly to the model with one heat flux and one temperature sources (section 4.2), case B is compared to the model with imposed heat fluxes (section 4.1.2). The different conditions at the liquid/vapour interface are studied for both cases.

Single uniform heat source and convective cooling

For case A, the mean saturation temperature calculated by the model is in particular evaluated from boundary conditions. The axial temperatures are plotted in Fig. 4a. The axial capillary pressure and vapour velocity are compared in Fig. 4b. It appears that the influence either of the thermohydraulic coupling or of the vapour pressure drop on these variables is not very significant. The presented profiles are thus quite similar regardless of these conditions.

External wall temperatures fit almost perfectly. It can be noticed that the mean saturation temperature matches to the temperature in the adiabatic zone very well, even if the thermal gradient in this zone is the strongest.

The capillary pressure and vapour velocity profiles are likewise well reproduced. It can be noticed in that case, that the capillary pressure mainly balances the liquid pressure drop. The evolution of the mean vapour velocity is standard with an increase along the evaporator, a maximum in the adiabatic zone and a decrease along the condenser.

To complete this study, it is interesting to analyse the evolution of the vapour pressure according to the modelling assumptions (Fig. 5). Even if the vapour pressure drop is of the order of a dozen pascals, the nonlinear

C = 1800 W m -2 K -1 Q C = 200 W T C = 26°C T sat = 71.4°C

Comparative model

Heat flux source + Two heat Temperature source flux sources Table 3: Main specifications of the two heat pipes from Shabgard and Faghri [START_REF] Shabgard | Performance characteristics of cylindrical heat pipes with multiple heat sources[END_REF] dynamic pressure drop is important to consider and are of the same order of magnitude as viscous pressure drop.

Considering these latter, it can be noticed that the term depending on the acceleration of the vapour velocity (second term in Eq. ( 22)) is negligible compared to Darcy-Weisbach pressure drop. Taken all the pressure drops into account, the well-known phenomena of injection at the evaporator (due to the increase of the vapour velocity) and of suction at the condenser (due to the decrease of the velocity) are highlighted.

Finally, regardless of the losses considered, a shifting of the absolute value of the vapour pressure is noted if the thermohydraulic coupling at the liquid/vapour interface is considered. This difference comes simply from the evaluation of the mean saturation temperature corresponding to the mean external temperatures. From a zoom on the wall temperatures at the evaporator and the condenser (Fig. 6), a slight difference on the profiles is noted. It comes directly from the slight axial variation of the saturation temperature. With any vapour pressure drop, the thermohydraulic coupling implies a small change of T sat and so of the reference vapour pressure.

Multiple uniform heat sources and uniform cooling

For case B, equal multi-sources are directly applied at four different evaporators and the cooling is also considered uniform. However, no heat losses are considered, which implies that no reference temperature exists and the mean saturation temperature has to be provided for the model as previously mentioned. As for case A, axial temperatures, and axial capillary pressure and vapour velocity, are plotted, respectively, in Fig. 7a and7b.

Results of literature and the simulations match perfectly. The increases and decreases of temperature at and between evaporators are well determined. Here the mean saturation temperature corresponds to the temperature in the main adiabatic zone (between evaporators and condenser). While capillary pressure evolution is similar to case A with one heat source, the impact of the multi-sources is well highlighted by the distribution of the vapour velocity, with a more or less high increase following the evaporators and the adiabatic zones. The impact of the dynamic pressure drop on the vapour pressure is the same as for case A, but the thermohydraulic coupling has no 48), is highlighted in Fig. 9. This transmittance is positive in evaporators and condenser, and negative in the adiabatic zones. It means that, for a fixed ambience temperature and at fixed mean heat flux, if the saturation temperature T sat increases, the difference between T sat and the temperatures at evaporators and at condenser increases, while the temperatures in adiabatic zones get closer to T sat . It implies an increase of the axial thermal gradients. It can be noticed that this impact affects both temperature fields in the wall and in the porous wick, even if it is attenuated in the latter. Finally, it is verified that this transmittance W sat is near zero at the liquid/vapour interface, as well as the impedance Z , which indicates a quasi uniformity of the saturation temperature along the axial direction.

Significance of the thermohydraulic coupling

After the previous validation of the model, an analysis is finally proposed to study a case where the influence of the non-uniformity of the saturation temperature along the length of the heat pipe affects substantially its thermohydraulic behaviour. The fixed geometrical specifications and boundary conditions of the studied case are gathered in Table 4. The standard problem of a heat pipe under realistic boundary conditions (model with heat source and one temperature source), with evaporator and condenser at each extremety is considered. The chosen heat pipe is especially small in diameter and the heat flux imposed at evaporator is relatively low (small mean saturation temperature). Due to the effects of the thermohydraulic coupling on the evaluation of T sat (see section 5.1), the imposed heat flux at evaporator is adapted to have nearly the same T sat in both cases (Q E = 4.7 W with thermohydraulic coupling and Q E = 5.2 W without). Table 4: Specifications of the heat pipe for the analysis the saturation temperature around 20°C. It involves an increase of 1.2°C of the maximal temperature at evaporator and a decrease of 1.15°C of the minimal temperature at condenser, compared to the case without thermohydraulic coupling. It implies an increase of the lengthwise thermal gradient and a non-uniformity of temperatures in the adiabatic zone, equal to that of saturation. The consideration of the uniformity of the saturation temperature for the thermal model implies also a decrease of both liquid and vapour pressure drops, slightly more substantial for the vapour flow. It involves a 10% decrease of the maximal capillary pressure in that case. Compared to a standard heat pipe, it can be noticed that liquid and vapour pressure drops are of the same order of magnitude.

Finally, we can look in more details at the difference of temperature fields inside the wall and the porous of the heat pipe ( Fig 11). With the thermohydraulic coupling, the lengthwise thermal gradient is conserved in the wall thickness while the thermal gradient in the porous media is more two-dimensional. With a considered uniform saturation temperature, the thermal gradient is mainly radial in the porous wick, while relatively constant temperatures are observed in each part of the wall of the heat pipe. To conclude this section, it is to highlight that, in the studied conditions, the model with two heat flux sources with a uniform imposed heat flux at condenser gives similar results, whatever the coupling is taken into account. The only drawback of this simple model is that the variation of the mean saturation temperature with respect to the transferred heat flux has to be known if no heat losses with ambience are considered.

Conclusion

A generalisation of steady state analytical models of cylindrical heat pipe by Fourier integral transforms has been proposed in this article. By assuming axisymetric boundary conditions, two types of models have been developed via an original mathematical representation by thermal quadrupoles for the two-dimensional conductive heat transfer. A first simplified model with both imposed heat fluxes at evaporator and at condenser enables a direct analytical solution for heat transfer in the wall and in the porous wick and for the pressure drops in liquid and vapour flows, under specific thermodynamic conditions at the liquid/vapour interface. A second model with more realistic boundary conditions, especially a convective cooling at condenser, requires a mesh over the outer wall of the heat pipe and a fast iterative procedure. For either models, three kind of inner thermohydraulic conditions have been taken into account: a decoupling of thermal and hydraulic models, or a stronger coupling between heat transfer at the interface and the vapour pressure drop, considered either linear or nonlinear. It has been shown that the operating temperature of the heat pipe can be evaluated from boundary conditions in the case where some heat losses with ambience are taken into account. If no reference temperature exists in the problem definition, this operating temperature has to be provided additionally to solve analytically the heat transfer equations.

The model results have been validated by comparison with literature results in both cases with uniform or convective cooling at condenser, as well through temperature, capillary pressure and velocity distributions. An example with multiple sources at evaporator has been easily treated with this methodology.

In the case of uniform heat fluxes at evaporator and at condenser, an original determination of temperature and radial heat flux fields for imposed boundary conditions have been introduced, with the definition of thermal impedances, transmittances or admittances of the heat pipe. These latter are intrinsic properties of the heat pipe subject to any thermal sollicitations. They depend directly on fluid and material properties and on the location of the boundary conditions.

The introduced analyses have shown that the uniformity of saturation temperature can be considered to model a large diameter heat pipe, but not a micro heat pipe where strong vapour pressure drop and two-dimensional thermal gradients in the wick are present.

As well as previous analytical models proposed in literature, this model have some limits regarding some specific phenomena, as for example no thermal effect on the interface curvature or no possible inclusion of liquid slug at the condenser. However, as the previous models, the introduced model can be used to design a capillary cylindrical heat pipe or to be included more easily in an optimisation process than a complete numerical tool. Furthermore, the original quadrupoles representation suggests simpler developments of more complex models in three-dimensional or in transient regime, with an additional Laplace time integral transform. The development of analytical models for flat heat pipes can also be easily considered under the same formalism. If a uniform saturation temperature is supposed for the thermal model (with any vapour pressure drop) or if a thermohydraulic coupling is considered with linear vapour pressure drop, Eq. ( 56) can be simply written as: From the equality of Eqs. ( 55) and (56), for all x i of the mesh, a relationship of the following form is derived:

Nomenclature

2πR o [h ∞ χ A (x i ) + h C χ C (x i )]T (x i ) + N j =1 ω i j T (x j ) = χ E (x i )q E (x i ) + χ A (x i )2πR o h ∞ T ∞ + χ C (x i )2πR o h C T C (A.3)
with the coefficients ω i j defined as:

ω i j = - 2 L w j ∞ n=1
Y n cos(α n x i )cos(α n x j ) (A.4)

Eq. (A.3) can be written under matrix form, for 1 ≤ i ≤ N and 1 ≤ j ≤ N :

  ω i j +   0 N E 0 N E ,N A 0 N E ,N C 0 N A ,N E 2πR o h ∞ I N E 0 N A ,N C 0 N C ,N E 0 N C ,N A 2πR o h C I N C     T (x j ) =   I N E 0 N E ,1 0 N E ,1 0 N A ,N E 2πR o h ∞ 1 N A ,1 0 N A ,1 0 N C ,N E 0 N C ,1 2πR o h C 1 N C ,1     q E (x i ) T ∞ T C
  (A.5) which can be written under the form AX = BU , with X the unknown external temperatures and U the heat and temperature sources.

Figure 1 :

 1 Figure 1: Heat pipe geometry

Figure 2 :

 2 Figure 2: Simplified model of the cylindrical heat pipe with two heat flux sources

Figure 3 :

 3 Figure 3: Model of the cylindrical heat pipe with one heat flux source and one temperature source

Figure 6 :

 6 Figure 6: Case A -Zoom on wall temperatures at evaporator and condenser depending on model assumptions

Figure 9 :-9 m 2

 92 Figure 9: Case B with additional heat losses (h ∞ = 10 W.m -2 .K -1 ) -Thermal transmittance field (W sat from Eq. (48)) of θ from the temperature difference T sat -T ∞ Working fluid Water Wall material Copper Wick material Copper Total length 1 m Location of evaporator [0, 0.3] m Location of condenser [0.7, 1] m Inner radius 1.5 mm Wall thickness 1 mm Wick thickness 0.5 mm Wick porosity 0.7 Wick permeability 1.5 10 -9 m 2 Effective thermal conductivity of the wick 1 W m -1 K -1 Mean saturation temperature 20°C Convective cooling at condenser h C = 100 W m -2 K -1 T C = 10°C

Figure 10 :

 10 Figure 10: Micro heat pipe with an operating temperature of 20°C

Figure 11 :

 11 Figure 11: Temperature fields (°C) for micro heat pipe with an operating temperature of 20°C

Y 0 T

 0 n θn r =R o cos(α n x) (A.[START_REF] Blet | Heats pipes for temperature homogenization : a literature review[END_REF] with the admittance Y n = D n /B n in the case where a uniform saturation temperature is assumed and Y n = C n /A n with thermohydraulic coupling and linear vapour pressure drop.In this equation, the integral transforms of temperature can be evaluated by a standard Newton-Cotes method, as trapezoidal rule, under the form:θn r =R o = L (x )cos(α n x )d x = N j =1w j T (x j )cos(α n x j ) (A.2)

Table 1 :

 1 Quadrupole coefficients in cylindrical coordinateswhere the matrix coefficients A n , B n , C n and D n are defined in Table1

  in the case of imposed heat flux sources and with no ambient losses, the value of T sat is required

		Uniform T v	Thermohydraulic coupling	Thermohydraulic coupling
	thermal	for thermal model	at the interface	at the interface
	conditions	Any ∆P v	Linear ∆P v	Nonlinear ∆P v
	Two heat	Direct analytical expressions	Iterative solving
	flux sources*	of thermal harmonics	for inner thermal harmonics
	Heat flux source	Iterative solving for (x,r ) depending	Iterative solving for (x,r ) depending
	+	external thermal variables	external thermal variables
	Temperature source			and for inner thermal harmonics

*

Table 2 :

 2 Solving summary of the model according to external conditions and saturation thermohydraulic coupling

Setting up the system for the model with one heat flux source and one temperature source and iterative procedure

	Subscripts and superscripts
	A	adiabatic zone
	C		condenser
	E		evaporator
	eff		effective
	eq		equivalent
	i		inner radius
	l		liquid
	o		outer radius
	r		radial
	sat		saturation
	Latin letters v	vapour
	A w		porous wick	section	m 2
	g x		axial	gravitational acceleration	m.s -2
	h ∞		ambience	heat transfer coefficient	W.m -2 .K -1
	h l v				latent heat	J.kg -1
	k w K sat linearisation coefficient of the saturation curve permeability of porous wick Appendix A.	m 2 Pa.K -1
	L				heat pipe length	m
	N				number of mesh points
	P				pressure	Pa
	q				heat flux	W.m -1
	Q				heat power	W
	r				radius coordinate	m
	R				radius	m
	T				temperature	K or	°C
	u				velocity	m.s -1
	W				transmittance
	w j				quadrature coefficients	m
	x				length coordinate	m
	Y				admittance	W.K -1
	Z				impedance	K.W -1
	Greek letters
	α n	eigenvalue of order n following x	m -1
	γ	angle of heat pipe with horizontal
				porosity
	θ		temperature difference	K
	λ		thermal conductivity	W.m -1 .K -1
	µ		dynamic viscosity	Pa.s
	ρ				density	kg.m -3
	σ		surface tension	N.m -1
	φ		radial heat flux	W.m -1
	χ		indicator function

Appendix A.1. Linear system

  (24)

more effects because T sat is imposed.

As mentioned in section 4.1.2, the behaviour of the heat pipe in this example can be intrinsically characterised by impedances and transmittances. For example, the temperature field can be deduced from the transferred heat flux and the impedance of the system, illustrated in Fig. 8, by T | x,r = T sat + Z | x,r Q (from Eq. ( 49)). This thermal impedance is completely independent of the total imposed heat flux Q. In Fig. 8, it can be observed that this impedance is strongly linked to the locations of the evaporators and the condenser, especially in the wall of the heat pipe. Its value becomes more and more homogeneous along the axial length within the porous, until a zero value at the liquid/vapour interface. It means that, for an imposed saturation temperature, when the heat flux increases, the axial temperature gradient increases more in the wall than in the porous wick. Finally, in the main adiabatic zone, its value is also close to zero over most of the area, both in the porous wick and in the wall.

For the same case with heat losses, with for example h ∞ = 10 W.m -2 .K -1 , the thermal impedance field is quite similar to the previous one in Fig. 8 with a maximal variation of 2.5 10 -4 K.W -1 . The transmittance W sat field, The temperature and pressure profiles are illustrated in Fig. 10. A variation of more than 2.5°C is present for Through the expressions of the quadrupole coefficients in the terms of ω i j coefficients, matrix A depends on the unknown temperature T sat . So, an iterative procedure is necessary. The iterative closure equation to evaluate the new value of T sat stems from Eq. (34a) by using the mesh: If the assumption of a uniform saturation temperature is considered and if the liquid conductivity can be considered as independent of T sat , the same is true for matrix A. No more iterative procedure is necessary and the linear system (A.5) can be solved directly.

Appendix A.2. Nonlinear system

In the case of a thermohydraulic coupling with nonlinear vapour pressure drop, Eq. (56) can not be simplified any more. With a similar development as in the previous linear case, the equality of Eqs (55) and (56) leads to a system under the form AX = BU +C , with Y n = D n /B n in matrix A while vector C is expressed as:

In that case, each harmonic θn r =R i can be expressed from Eq. (54) as nonlinear function of the harmonics φn r =R i . Furthermore, from the quadrupole model and the Newton-Cotes method, these latter harmonics can be expressed as:

where the heat fluxes φ(x j ) are evaluated from Eq (55) as function of the T (x j ) variables (deduced by solving the matrix system).

The iterative procedure involves finally the closure equation for the evaluation of the new value of T sat (Eq. (A.6)).

Appendix A.3. Comments

It can be noticed that some numerical precautions must be implemented for a good convergence of iterations. The main one is imposing the equality between the number of subintervals for the Newton-Cotes method (N -1) and the number of harmonics for the calculation of the infinite sum. The purpose is to numerically preserve the orthogonality of the cosine functions.

The previous developments can be done for any kind of boundary conditions, especially at the condenser.


Brigitte Lamy

Sandrine Baron

Olivier Barraud
email: olivier.barraud@unilim.fr

Aeromonas: the multifaceted middleman in the One Health world

publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Aeromonas in the One Health world: a ubiquitous "Jack of all trades"

The genus Aeromonas belongs to the Aeromonadaceae family, which itself is part of the Aeromonadales order and Gammaproteobacteria class [START_REF] Fernandez-Bravo | An update on the genus Aeromonas: taxonomy, epidemiology, and pathogenicity[END_REF]. This genus, which currently comprises 36 species (https://www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cgi), is organized in a complex of species, i.e., heterogeneous groups of closely-related species with unclear boundaries [START_REF] Roger | Multilocus genetics to reconstruct aeromonad evolution[END_REF], as observed for other Gram-negative bacteria like Vibrio, Acinetobacter, Pseudomonas, Burkholderia, and several Enterobacterales [START_REF] Talagrand-Reboul | Delineation of taxonomic species within complex of species: Aeromonas media and related species as a test case[END_REF]. From this complex of species, some more specialized clones may emerge [START_REF] Roger | Ribosomal multi-operon diversity: an original perspective on the genus Aeromonas[END_REF]. Of these, the complex A. salmonicida, which is responsible for fish diseases, is to date the only species that has successfully become specialized to cause a specific disease [START_REF] Reith | The genome of Aeromonas salmonicida subsp. salmonicida A449: insights into the evolution of a fish pathogen[END_REF]. While the primary reservoir of Aeromonas is aquatic environments and soils where it is almost always present, it is also capable of rapidly colonizing a wide range of niches and hosts [START_REF] Fernandez-Bravo | An update on the genus Aeromonas: taxonomy, epidemiology, and pathogenicity[END_REF]. Aeromonas has a notable ability to colonize the animal gut, including the guts of leeches, fish, cows and humans [START_REF] Gowda | Isolation and seroprevalence of Aeromonas spp. among common food animals slaughtered in Nagpur, Central India[END_REF][START_REF] Igbinosa | Antibiogram characterization and putative virulence genes in Aeromonas species isolated from pig fecal samples[END_REF] as well as many other host organisms.

Data from Pubmed, using close to 10,000 descriptions since 1944, (https://pubmed.ncbi.nlm.nih.gov/?term=aeromonas&sort=date&size=20) provides a biased view of this epidemiology, with >50% of Aeromonas publications linked to humans or animals including livestock, companions and wildlife. The IMNGS 16S rRNA-based metagenomic website provides a more accurate and realistic picture of Aeromonas epidemiology [START_REF] Lagkouvardos | IMNGS: A comprehensive open resource of processed 16S rRNA microbial profiles for ecology and diversity studies[END_REF]. Aeromonas is mostly detected from water and soil samples, and is present in 20% of rhizosphere and aquatic metagenomes (Figure 1). Detection from animal samples is infrequent, except from fish gut metagenomes and, to a lesser extent, from bovine gut metagenomes [START_REF] Igbinosa | Detection of antibiotic resistance, virulence gene determinants and biofilm formation in Aeromonas species isolated from cattle[END_REF][START_REF] Ruzauskas | Microbial and antimicrobial resistance profiles of microbiota in common carps (Cyprinus carpio) from aquacultured and wild fish populations[END_REF]. In humans, detection remains exceptionally rare. Relative abundance data from positive metagenomes also highlights the importance of Aeromonas in hydrotelluric environments with anthropic pollution. The highest relative abundance is found in wastewater metagenomes (Figure 1), as a partial consequence of gut-related waste. Relative abundance of Aeromonadaceae in animals is very low, below 0.1% (Figure 1). One exception is leech gut where relative abundance of Aeromonas seems very important, estimated at 36% [START_REF] Maltz | Metagenomic analysis of the medicinal leech gut microbiota[END_REF].

The environmental reservoir of Aeromonas allows it to be at the interface of all the One Health components (Figure 2). With its ability to produce biofilm and its large metabolic capacity that facilitates its rapid adaptation to environmental changes [START_REF] Seshadri | Genome sequence of Aeromonas hydrophila ATCC 7966T: jack of all trades[END_REF][START_REF] Zhong | Comprehensive analysis reveals the evolution and pathogenicity of Aeromonas, viewed from both single isolated species and microbial communities[END_REF], Aeromonas is viable in a large range of conditions. These extended capabilities that allow Aeromonas to persist and rapidly propagate in continuously changing environments [START_REF] Seshadri | Genome sequence of Aeromonas hydrophila ATCC 7966T: jack of all trades[END_REF][START_REF] Zhong | Comprehensive analysis reveals the evolution and pathogenicity of Aeromonas, viewed from both single isolated species and microbial communities[END_REF][14][15][16][17] has earned this genus the nickname of a "Jack-of-all-trades" [START_REF] Seshadri | Genome sequence of Aeromonas hydrophila ATCC 7966T: jack of all trades[END_REF].

Aeromonas in close contact with human activities

Aeromonas is primarily able to develop long-term symbiotic relationships (e.g. with leeches), or to transiently colonize other animals and humans, where it can trigger cutaneous or digestive infections [16,[START_REF] Lorenz | Bacterial gene transfer by natural genetic transformation in the environment[END_REF]. Aeromonas spp. are responsible for economic burdens of dozens of millions of dollars in aquaculture due to mortality, lost feeding days and costs associated with chemical and antibiotic therapy [19][20][21][22]. Among fish pathogens, A. salmonicida subsp. salmonicida (referred to as "typical") is responsible for furunculosis disease in salmonids

[23], while some hypervirulent clones have emerged (e.g., A. hydrophila ST251) in the catfish farming and shrimp industry [24,25].

Another Aeromonas-impacted area concerns ornamental fish that are traded worldwide (2 billion/year). These exchanges occur mainly between Asian countries (the primary producer) and USA/Europe. In this industry, as observed for several years in fish farms [26,27], the misuse or overuse of antibiotics to fight bacterial diseases has selected for drug-resistant pathogens [28][29][30]. Recent studies underline the potential role of ornamental fish in the dissemination of antibiotic-resistant bacteria, with possible consequences for private aquariums and human public health in importing countries [28].

Aeromonas spp. have also been isolated from a wide range of foods of both plant and animal origin [31], including minimally processed and ready-to-eat seafood products that are gaining popularity [32]. The prevalence of Aeromonas in ready-to-eat products has been estimated to be between 8 and 52%, with resistance to clinically-relevant antibiotics also emerging [START_REF] Lee | Aeromonas spp. isolated from ready-to-eat seafood on the Norwegian market: prevalence, putative virulence factors and antimicrobial resistance[END_REF][START_REF] Hoel | The significance of mesophilic Aeromonas spp. in minimally processed ready-to-eat seafood[END_REF][START_REF] Umutoni | Occurrence, diversity and temperature-dependent growth kinetics of Aeromonas spp. in lettuce[END_REF]. This raises concerns of the potential for food spoilage and foodborne infections.

Aeromonas is associated with human activities such as animal husbandry, animal slaughter [START_REF] Fontes | Phylogenetic identification of Aeromonas simiae from a pig, first isolate since species description[END_REF][START_REF] Fontes | Phylogenetic identification of Aeromonas from pigs slaughtered for consumption in slaughterhouses at the North of Portugal[END_REF][START_REF] Li | Antimicrobial resistance and resistance genes in aerobic bacteria isolated from pork at slaughter[END_REF], wastewater treatment [START_REF] Harnisz | The prevalence of multidrug-resistant Aeromonas spp. in the municipal wastewater system and their dissemination in the environment[END_REF], reclaiming irrigation water [START_REF] Latif-Eugenin | Chlorinated and ultraviolet radiation -treated reclaimed irrigation water is the source of Aeromonas found in vegetables used for human consumption[END_REF][START_REF] Solaiman | Longitudinal assessment of the dynamics of Escherichia coli, total coliforms, Enterococcus spp., and Aeromonas spp. in alternative irrigation water sources: a CONSERVE study[END_REF], recreational or occupational activities that are related to water exposure and those that exploit microbiological metabolic capabilities in biotechnological applications (Figure 2). These include the treatment of sewage and industrial wastewater with activated sludge that contains aeromonads at up to 2% of the total biomass [14,[START_REF] Kampfer | Characterization of bacterial communities from activated sludge: culture-dependent numerical identification versus in situ identification using group-and genus-specific rRNA-targeted oligonucleotide probes[END_REF]. Here, aeromonads are organized in aggregates and can resist the extreme conditions of pH, salinity, heavy metal toxicity, and temperature [14]. These aggregates are inserted in mixed species biofilms that are able to remove organic pollutants and improve both denitrification and phosphorus removal [START_REF] Andersson | Influence of microbial interactions and EPS/polysaccharide composition on nutrient removal activity in biofilms formed by strains found in wastewater treatment systems[END_REF][START_REF] Chong | The presence and role of bacterial quorum sensing in activated sludge[END_REF][START_REF] Chen | Impact resistance of different factors on ammonia removal by heterotrophic nitrification-aerobic denitrification bacterium Aeromonas sp. HN-02[END_REF].

Human infections: by accident or by Trojan horse

In humans, Aeromonas infections are primarily associated with contact with soil and/or water e.g., during recreational or occupational activities, or following road accidents [16,[START_REF] Lamy | Prospective nationwide study of Aeromonas infections in France[END_REF]. Cutaneous, eye or digestive infections are the most frequent, although almost any type of infection is possible. Aeromonads are weak opportunistic pathogens in humans, and most infections are associated with a barrier disruption. Aeromonads can also spread from a colonized niche, typically the gut, and cause biliary or bloodstream infections notably in immunocompromised patients [16,[START_REF] Parker | Aeromonas spp. clinical microbiology and disease[END_REF]. While human infections are widely reported through hundreds of published case reports [START_REF] Bhowmick | Bacteriological, clinical and virulence aspects of Aeromonasassociated diseases in humans[END_REF], they are poorly documented compared to other bacteria, with the notable exception of natural disaster situations, when thousands of wounded individuals are exposed to contaminated waters with poor access to healthcare facilities [16].

No global study exists that compares prevalence among countries. The prevalence of bloodstream infections seems higher in Asia compared with European countries [START_REF] Lamy | Prospective nationwide study of Aeromonas infections in France[END_REF]49], and some species are more prevalent in tropical countries (e.g., A. dhakensis) [START_REF] Fernandez-Bravo | An update on the genus Aeromonas: taxonomy, epidemiology, and pathogenicity[END_REF]. One can expect that use of gastrointestinal syndromic tests targeting aeromonads will increase documentation and will provide a more exhaustive picture, as it was observed in this recent German study with a prevalence of 2.9% of Aeromonas sp. from 5032 stool samples [START_REF] Zimmermann | Workflow optimization for syndromic diarrhea diagnosis using the molecular Seegene Allplex GI-Bacteria(I) assay[END_REF].

The particular case of healthcare-associated infections caused by ciprofloxacin-resistant aeromonads following leech therapy is an example of the One Health challenges posed by these bacteria, with resistant strains selected from the environment and spread to patients (Figure 3). Medicinal leeches may be administered to patients after orthopedic surgery or reconstructive surgery to treat venous congestion that threatens flap outcome through necrosis. This salvage treatment actively removes blood and improves outcomes, at the expense of a risk of serious wound infection leading to flap loss, amputation and sepsis. Indeed, the leech regurgitates its symbiont during bloodlettings. While the most abundant species among leech symbionts is by far A. veronii, A. hydrophila is the most frequently reported species associated with infection following leech therapy (88%) [START_REF] Whitaker | The efficacy of medicinal leeches in plastic and reconstructive surgery: a systematic review of 277 reported clinical cases[END_REF]. This underlines that the bacterial load during leech regurgitation is not the only factor to contribute to infection and that some yet unknown virulence factors in A. hydrophila are likely involved [START_REF] Beka | role of low-level ciprofloxacin in the leech, and as corollary in the leech environment, and consequences on the leech microbiota[END_REF]. This risk of infection is controlled with ciprofloxacin-based prophylactic treatment [START_REF] Whitaker | The efficacy of medicinal leeches in plastic and reconstructive surgery: a systematic review of 277 reported clinical cases[END_REF][START_REF] Herlin | Leech therapy in flap salvage: systematic review and practical recommendations[END_REF]. However, since the early 2010s, several cases of infections following leech therapy were suddenly reported worldwide by aeromonads resistant to ciprofloxacin [START_REF] Patel | Ciprofloxacin-resistant Aeromonas hydrophila infection following leech therapy: a case report and review of the literature[END_REF][START_REF] Barraud | It takes two to tango: two Aeromonas isolates combine virulence and multidrug resistance in flap infection following leech therapy[END_REF]. As detailed in Figure 3, investigations showed that these events resulted from complex relationships between the antibiotic that disseminates in the environment, the leech microbiota where there is competition between fluoroquinolone (FQ)-resistant and FQ-susceptible bacteria, a leech trade based on a limited number of suppliers that disseminate leech with closely related FQ-resistant strains, and immunocompromised patients [START_REF] Beka | role of low-level ciprofloxacin in the leech, and as corollary in the leech environment, and consequences on the leech microbiota[END_REF].

The use of antimicrobials in a natural setting facilitates the spread of resistance markers in Aeromonas, even when used at very low-levels [START_REF] Moraru | Changes in concentrations of fluoroquinolones and of ciprofloxacin-resistant Enterobacteriaceae in chicken feces and manure stored in a heap[END_REF][START_REF] Rutgersson | Fluoroquinolones and qnr genes in sediment, water, soil, and human fecal flora in an environment polluted by manufacturing discharges[END_REF], and this spread may occur between distant locations through a vector and Trojan horse such as the leech. This example highlights the consequences of the selective pressure in the leech environment with subsidiary consequences to human health. Such resistance spread is not limited to FQ, with some reports recently identifying Aeromonas that harboured resistance to 3 rd generation cephalosporins or cotrimoxazole, which exemplifies the capability of aeromonads to acquire diverse antibiotic resistance genes (ARGs) [START_REF] Barraud | It takes two to tango: two Aeromonas isolates combine virulence and multidrug resistance in flap infection following leech therapy[END_REF][START_REF] Floug | A case of leech-associated infection involving an extended-spectrum beta-lactamase-producing and extensively drug-resistant Aeromonas hydrophila[END_REF].

How to uptake DNA: the Swiss Army knife

Bacteria of the Aeromonas genus have a 4-5 Mbp genome, which is smaller than other Gramnegative bacteria like Enterobacterales or Pseudomonas. Aeromonas compensates for this relatively small genome size by its tremendous capacity to acquire several mobile genetic elements (MGE), including plasmids, transposons, genomic islands or integron gene cassettes.

The core genome of these bacteria is represented by only 16% of the genes in the whole genome, with the remaining (84%) consisting of variably represented genes. This highlights the exceptional level of genome plasticity in Aeromonas due to horizontal gene transfers (HGT) both from close relatives in the genus and from several other families [START_REF] Zhong | Comprehensive analysis reveals the evolution and pathogenicity of Aeromonas, viewed from both single isolated species and microbial communities[END_REF][START_REF] Kloub | Systematic detection of largescale multigene horizontal transfer in prokaryotes[END_REF]]. An exciting yet unanswered question is to know whether the environment shapes aeromonad plasticity and whether their accessory genome correlates with their origin. This mobilome [START_REF] Piotrowska | Insight into the mobilome of Aeromonas strains[END_REF] allows for the acquisition of hundreds of antibiotic or heavy metal resistance genes, genes involved in molecular function or biological process, as well as genes encoding metabolic enzymes, transporters and bacterial defense mechanisms [START_REF] Kloub | Systematic detection of largescale multigene horizontal transfer in prokaryotes[END_REF][START_REF] Rasmussen-Ivey | Virulence factors of Aeromonas hydrophila: in the wake of reclassification[END_REF]. These genes enable the survival of Aeromonas in hostile environments. All known mechanisms of HGT have been described within the Aeromonas genus [START_REF] Bello-Lopez | Horizontal gene transfer and its association with antibiotic resistance in the genus Aeromonas spp[END_REF]: transformation (as most Aeromonas species are naturally competent [63,[START_REF] Huddleston | Natural transformation as a mechanism of horizontal gene transfer among environmental Aeromonas species[END_REF]), transduction and conjugation. A wide variety of lysogenic phages has been described especially amongst the species A. salmonicida and A. hydrophila [START_REF] Yuan | Characterization and genomic analyses of Aeromonas hydrophila phages AhSzq-1 and AhSzw-1, isolates representing new species within the T5virus genus[END_REF][START_REF] Park | Recent insights into Aeromonas salmonicida and its bacteriophages in aquaculture: A comprehensive review[END_REF]. Conjugation remains the most efficient mechanism of HGT, with hundreds of conjugative plasmids identified in aeromonads. This plasmidome remains poorly characterised, with the exception of that of A. salmonicida [START_REF] Vincent | The Aeromonas salmonicida plasmidome: a model of modular evolution and genetic diversity[END_REF].

Aeromonads are diversely equipped with the type II, III, IV and VI secretion systems (TSS) [START_REF] Zhong | Comprehensive analysis reveals the evolution and pathogenicity of Aeromonas, viewed from both single isolated species and microbial communities[END_REF]. Whilst the study of these nanomachines in Aeromonas has mostly focused on their virulence properties, the T4SS and T6SS are also likely to be implicated in gene transfer, as they are typically involved in Gram-negative bacterial conjugation and transformation respectively [START_REF] Veening | Interbacterial predation as a strategy for DNA acquisition in naturally competent bacteria[END_REF][START_REF] Jaboulay | Regulatory networks of the T4SS control: from host cell sensing to the biogenesis and the activity during the infection[END_REF]. For instance, competence-induced T6SS-mediated bacterial killing has been demonstrated in V. cholerae, and occurs through a complex system that involves chitincomposed exoskeleton-associated-biofilms (e.g. in copepods), quorum sensing and the metabolic activity of chitinase [START_REF] Veening | Interbacterial predation as a strategy for DNA acquisition in naturally competent bacteria[END_REF][START_REF] Chourashi | Chitin-induced T6SS in Vibrio cholerae is dependent on ChiS activation[END_REF]. Because Aeromonas shares aforementioned features with the closely-related Vibrio spp., it is conceivable that the T6SS plays a role, albeit still unravelled, in Aeromonas DNA uptake. Whatever the mechanisms involved, the lifestyle of aeromonads is associated with biofilm in sheltering organisms (e.g., copepods) that are hotspots for genetic exchanges [START_REF] Aujoulat | From environment to man: genome evolution and adaptation of human opportunistic bacterial pathogens[END_REF].

Given its importance in human and animal health, the acquisition of ARGs is the most described event of this genetic exchange capability of Aeromonas. The genus can be considered as a "sponge" because it can easily absorb and redistribute genes [START_REF] Vincent | The Aeromonas salmonicida plasmidome: a model of modular evolution and genetic diversity[END_REF]. A recent in silico analysis on plasmids retrieved from Aeromonas species revealed that more than one third of plasmids carry ARGs [START_REF] Nwaiwu | An in silico analysis of acquired antimicrobial resistance genes in Aeromonas plasmids[END_REF]. These ARGs can affect all antibiotic families, including βlactams, with extended-spectrum β-lactamases (CTX-M, GES, VEB) and carbapenemases (VIM, KPC, NDM-like), but also aminoglycosides, cyclins, trimethoprim, sulfonamides, FQ (with notably the qnr and qepA genes [START_REF] Ayobola | Occurrence of plasmid mediated fluoroquinolone resistance genes amongst enteric bacteria isolated from human and animal sources in Delta State, Nigeria[END_REF]) and even colistin, with the recent description of the mcr genes carried on Aeromonas plasmids [START_REF] Barraud | It takes two to tango: two Aeromonas isolates combine virulence and multidrug resistance in flap infection following leech therapy[END_REF][START_REF] Floug | A case of leech-associated infection involving an extended-spectrum beta-lactamase-producing and extensively drug-resistant Aeromonas hydrophila[END_REF][START_REF] Piotrowska | Occurrence and variety of betalactamase genes among Aeromonas spp. isolated from urban wastewater treatment plant[END_REF][START_REF] Bosch | A blaVIM-1 positive Aeromonas hydrophila strain in a near-drowning patient: evidence for interspecies plasmid transfer within the patient[END_REF][START_REF] Hu | Emergence and characterization of a novel IncP-6 plasmid harboring blaKPC-2 and qnrS2 genes in Aeromonas taiwanensis isolates[END_REF][START_REF] Lin | Molecular and functional characterization of a novel plasmid-borne blaNDM-like gene, blaAFM-1, in a clinical strain of Aeromonas hydrophila[END_REF][START_REF] Tuo | The prevalence of colistin resistant strains and antibiotic resistance gene profiles in Funan River, China[END_REF].

Integrons in Aeromonas: witnesses of horizontal gene transfers?

Due to the environmental location of Aeromonas and its capability to acquire multiple MGE under selective pressure, it can be expected that detection of virulence or ARGs in aeromonads could constitute potential biomarkers of anthropic pollution. Unfortunately, these genes are too diverse for this approach to be feasible. One alternative attractive possibility in the search for a unique biomarker of anthropic pollution would be to detect integrons from Aeromonas isolates. Combining a relevant microorganism capable of acquiring hundreds of genes under selective pressure (Aeromonas) and a marker of antimicrobial resistance frequently hosted by Aeromonas isolates and easy to detect (integrons) could constitute a potential interesting tool. In fact, class 1 integrons can be easily detected [START_REF] Barraud | Quantitative multiplex real-time PCR for detecting class 1, 2 and 3 integrons[END_REF] and are now considered as a good proxy for anthropic pollution [START_REF] Gillings | Using the class 1 integronintegrase gene as a proxy for anthropogenic pollution[END_REF][START_REF] Stalder | Quantitative and qualitative impact of hospital effluent on dissemination of the integron pool[END_REF]. Integrons are non-mobile genetic elements carried by plasmids or transposons [START_REF] Mazel | Integrons: agents of bacterial evolution[END_REF][START_REF] Cambray | Integrons[END_REF] that can carry hundreds of mobile gene cassettes, most of them involved in antimicrobial resistance. The prevalence of class 1 integrons within the Aeromonas genus is difficult to determine and can greatly vary (from 7 to 72%) depending on the origins of the samples [START_REF] Deng | Multi-drug resistance mediated by class 1 integrons in Aeromonas isolated from farmed freshwater animals[END_REF][START_REF] Lee | Molecular diversity of class 1 integrons in human isolates of Aeromonas spp. from Southern Taiwan[END_REF][START_REF] Carnelli | Characterization of genetic determinants involved in antibiotic resistance in Aeromonas spp. and fecal coliforms isolated from different aquatic environments[END_REF]. Irrespective, their description among the Aeromonas species is frequent, as is their association with multidrug resistance, as previously demonstrated for Enterobacterales [START_REF] Barraud | Value of integron detection for predicting antibiotic resistance in patients with Gram-negative septicaemia[END_REF][START_REF] Mendes Moreira | Proteae: a reservoir of class 2 integrons?[END_REF]. Whilst gene cassettes are relatively diverse in Aeromonas [START_REF] Moura | INTEGRALL: a database and search engine for integrons, integrases and gene cassettes[END_REF] (Table S1), the presence of these integrons demonstrates that aeromonads readily acquire MGEs following selective pressure.

Conclusions

Although aeromonads are often described as possessing a significant pathogenic potential, these opportunistic pathogens seldom cause infections, and when they do, it is usually in the context of a disrupted barrier or a compromised immune system. One intriguing question is why a microorganism that bears many so-called virulence factors rarely causes infection. Understanding pathogenicity in Aeromonas remains elusive [15], with virtually all studies failing to successfully predict pathogenicity in aeromonads from the expression of so-called virulence factors. One assumption to reconcile this paradox is that the genes that we, from a very anthropocentric analysis, consider to encode virulence factors may indeed have other primary functions. Several arguments suggest that exaptation, the process by which features acquire functions for which they were not originally adapted or selected [START_REF] Pievani | Exaptation in human evolution: how to test adaptive vs exaptive evolutionary hypotheses[END_REF], may have occurred in Aeromonas. For example, aeromonad hydrolytic enzymes are now thought to be primarily involved in essential functions of general cell metabolism [15].

Exaptation may have then changed the function of these genes during the course of evolutionary succession. Other examples of possible exaptation are mcr-3 or qnr genes, which evolved long before the synthesis and clinical use of colistin and quinolones, respectively [START_REF] Cabello | Aquaculture, exaptation, and the origin of mcr-positive colistin resistance[END_REF][START_REF] Sanchez | Differential epigenetic compatibility of qnr antibiotic resistance determinants with the chromosome of Escherichia coli[END_REF]. A complementary hypothesis is that the multiple varied environments and organisms that aeromonads colonize act as a nursery to pre-adapt to pathogenicity, because such relationships need to overcome the innate defense of their hosts [START_REF] Aujoulat | From environment to man: genome evolution and adaptation of human opportunistic bacterial pathogens[END_REF].

Finally, some questions also arise regarding the role of integron gene cassettes in Aeromonas.

While mostly associated with ARGs, they existed even prior to the antibiotic era, and they operate as a general gene capture system. As such, they are likely to be an important factor in the more general process of HGT during the evolution of bacterial genomes and in bacterial adaptation [START_REF] Da | Integrons: natural tools for bacterial genome evolution[END_REF], and they may contribute to shaping the plasticity, versatility and flexibility of the Aeromonas genus [START_REF] Vincent | The Aeromonas salmonicida plasmidome: a model of modular evolution and genetic diversity[END_REF].

Irrespective of the original functions in the genus and its evolutionary changes, Aeromonas to date carries all the characteristics of a potential pathogen with resistance to multiple antibiotics. Taking into account its weaponry and its lifestyles that encompass all the One Health components, aeromonads have the potential to one day emerge as more prevalent pathogens than they are currently, and to wreak havoc in the future. Therefore, it is essential that we maintain a close watch on these fascinating bacteria.

** This study based on Whole Genome Sequencing of 29 Aeromonas isolates from different species revealed a limited core genome and a high level of genome plasticity. More than four fifth of the pangenome was variable with hundreds of gene expansions and contractions, horizontally transferred genes, and mobile genetic elements, which emphasized the heterogeneity of the genus. This is one of the first study that deeply compared genome contents; this bioinformatics analysis highlighted differences between acquired genes and abundance of some genes that may reflect environmental selection. * This article underlines the diversity of the species A. salmonicida, recognized as fish pathogen, described as psychrophilic bacteria. A. salmonicida able to growth at 37°C and to colonize/ infect mammals and birds is described here and linked to the genetic plasticity of the species. Figure 1A presents for the 3 compartments of the One Health world the percentage of metagenomes in which Aeromonadaceae are detected. This ecological distribution is based on 16S rRNA studies: grey, human metagenomes; yellow, fish and other animal metagenomes; blue, water metagenomes; green, soil and plant metagenomes. These data were obtained using the IMNGS database (http://imngs.org) [START_REF] Lagkouvardos | IMNGS: A comprehensive open resource of processed 16S rRNA microbial profiles for ecology and diversity studies[END_REF] queried in May 2021 using reads matching the Aeromonadaceae taxonomy. Figure 1B indicates the relative abundance of Aeromonadaceae retrieved for each type of positive metagenome, presented with box plots. A, aeromonads are ubiquitous bacteria present primarily in waters. They usually exhibit very high rate of susceptibility to ciprofloxacin (green circles) [START_REF] Lamy | Prospective nationwide study of Aeromonas infections in France[END_REF]. B, Human activities with either misuse or overuse of antimicrobials, including quinolones, lead to antimicrobial selective pressure on aeromonads that can become resistant to ciprofloxacin (red circles). C, during leech colonization by Aeromonas, its main symbiont, ciprofloxacin resistant strains outcompete the susceptible strains when some trace of quinolones are present in the leech environment [START_REF] Beka | role of low-level ciprofloxacin in the leech, and as corollary in the leech environment, and consequences on the leech microbiota[END_REF]. D, the leech trade is a global trade with diverse distant supplying sources, which allows the dissemination of bacteria from distanced parts of the world. E, patients with cancer are managed in hospital. F, during care pathway, cancer patients may benefit from reconstructive surgery, which may include flap surgery and may be associated with leech therapy to circumvent flap venous congestion.

There is a risk of aeromonosis when leech regurgitates bacteria into the flap during bloodletting. Risk of infection is controlled with an antimicrobial prophylactic treatment, mainly ciprofloxacin [START_REF] Herlin | Leech therapy in flap salvage: systematic review and practical recommendations[END_REF]. G, because leeches are now frequently colonized with ciprofloxacin-resistant aeromonads, infection occurs despite prophylactic treatment and results in serious skin and soft tissue infection with flap loss and sepsis. Such event are posterior to 2010 and are regularly reported [START_REF] Barraud | It takes two to tango: two Aeromonas isolates combine virulence and multidrug resistance in flap infection following leech therapy[END_REF]. Some parts of the figure were created in biorender.com. Bibliometry histogram in G was obtained from Pubmed with keywords Aeromonas, leech therapy, infection and antimicrobial resistance. R: resistant. AMR: antimicrobial resistant. AM: antimicrobial.

  14. Talagrand-Reboul E, Jumas-Bilak E, Lamy B: The social life of Aeromonas through biofilm and quorum sensing systems. Front Microbiol 2017, 8:37. 15. Talagrand-Reboul E, Colston SM, Graf J, Lamy B, Jumas-Bilak E: Comparative and evolutionary genomics of isolates provide insight into the pathoadaptation of Aeromonas. Genome Biol Evol 2020, 12:535-552. * The authors analysed 64 genome from strains representative of the genus Aeromonas to investigate the correlation between genome-based phylogeny and virulence associated factors to provide insight into pathoadaptation through the evolutionary process of virulence associated genes. 16. Janda JM, Abbott SL: The genus Aeromonas: taxonomy, pathogenicity, and infection. Clin Microbiol Rev 2010, 23:35-73. 17. Awan F, Dong Y, Wang N, Liu J, Ma K, Liu Y: The fight for invincibility: environmental stress response mechanisms and Aeromonas hydrophila. Microb Pathog 2018, 116:135-145. * This paper presents an extensive review on the environmental stress response mechanisms of A. hydrophila and links it with its ability to survive and propagate under environmental various conditions 18. Nelson MC, Graf J: Bacterial symbioses of the medicinal leech Hirudo verbana. Gut Microbes 2012, 3:322-331. 19. Bandeira Junior G, Baldisserotto B: Fish infections associated with the genus Aeromonas: a review of the effects on oxidative status. J Appl Microbiol 2020. 20. Menanteau-Ledouble S, Kumar G, Saleh M, El-Matbouli M: Aeromonas salmonicida: updates on an old acquaintance. Dis Aquat Organ 2016, 120:49-68. 21. Wallace IS, McKay P, Murray AG: A historical review of the key bacterial and viral pathogens of Scottish wild fish. J Fish Dis 2017, 40:1741-1756. 22. Zhang D, Xu DH, Shoemaker C, Hemstreet W: Virulent Aeromonas hydrophila in channel catfish. Edited by: Global Aquaculture Alliance; 2016. 23. Charette SJ: Microbe profile: Aeromonas salmonicida: an opportunistic pathogen with multiple personalities. Microbiology (Reading) 2021, 167.

AFigure 1 :

 1 Figure 1: Prevalence of Aeromonadaceae among human, animal and environmental metagenomes.

Figure 2 :Figure 3 :

 23 Figure 2: Aeromonas in the One Health world, a complex network of relationships.The figure presents the 3 parts of the One Health world where aeromonads are present: environment, mainly waters (blue), animals (green), human beings and related activities (light orange) and the compartments to which they are connected are shown with red dotted connectors. The intricate and complex network in which aeromonads interface the different compartments, mostly through waters, and in which they play a role of courier is shown with blue dotted lines. Flashes highlight the compartments where aeromonads contribute to disseminate antimicrobial resistance, dark orange specific host-bacteria interactions; beneficial relationships with host (e.g., human being) are highlighted in purple while detrimental relationships are yellow highlighted. WWTP, Wastewater treatment plant. Some parts of the figure were made with biorender (https://biorender.com).

Acknowledgments

The authors are indebted to Akshay Sabnis who reviewed English language with great elegance. Authors would like also to thank Thomas Jové who provided with integron gene cassette arrays from Aeromonas.

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Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.