Pierre-Yves Fravallo

Marc Prat
email: mprat@imft.fr

Vincent Platel

Numerical study of flow and heat transfer in the vapour grooves of a loop heat pipe evaporator

Keywords: Two-phase cooling systems, Loop Heat Pipe, Numerical simulations, Convective heat transfer coefficient, Porous media, 𝐶𝐶 Compensation chamber inj. Injection 𝑠𝑎𝑡 Saturation 𝑣 Vapour

A three-dimensional numerical study is developed to investigate the heat and mass transfer in the vapour removal channels of a Loop Heat Pipe (LHP). The heat transfer governing equations and the Navier-Stokes equations are solved under steady-state conditions for two vapour removal channel configurations: when the channels are in the wick and when the channels are in the casing. The boundary conditions for both configurations are extracted from numerical simulations using a model of the evaporator in which the modelization of the transfers in the wick is based on a pore network model. The results show that assuming that the vapour temperature in the channels is close to the saturation temperature is acceptable when the channels are within the wick but questionable when the vapour removal channels are within the casing. The simulations also indicate significant variations of the wall temperature between the various channel limiting surfaces. The heat transfer coefficient is different on the various surfaces and cannot be predicted using standard correlations for convective heat transfer in pipes.

Nomenclature

AR

Aspect Ratio With the miniaturization of electronic components and the increasing required power, the thermal control of electronic devices is a crucial issue. In this context, capillary pumped loops (CPLs) and loop heat pipes (LHPs) are very efficient systems to dissipate high heat fluxes over a large distance. First developed for spatial applications, this type of cooling device is now also considered for terrestrial applications such as automobile or railway transport [START_REF] Ayel | Romestant Experimental Analysis of a Capillary Pumped Loop for Terrestrial Application[END_REF][START_REF] Lachassagne | Experimental study of capillary pumped loop for integrated power in gravity field[END_REF][START_REF] Lachassagne | Steady-state modeling of Capillary Pumped Loop in gravity field[END_REF][START_REF] Accorinti | Steady-state analysis of a Capillary Pumped Loop for Terrestrial Application with methanol and ethanol as working fluids[END_REF]. CPLs and LHPs are diphasic thermal control devices based on a vaporizationcondensation process of a working fluid to transfer the heat. One can refer for example to [START_REF] Nikitkin | CPL and LHP Technologies: What are the Differences, What are the Similarities?[END_REF] for a comparative discussion of both systems. In what follows, a LHP is more specifically considered but the results presented later in the paper apply to both systems. As sketched in Fig. 1, a LHP is composed of an evaporator connected to the condenser through a vapour line, a liquid line between the condenser and the reservoir which is itself connected to the evaporator. The evaporator is made of a metallic casing in contact with a porous wick sucking the fluid by capillarity. The vapour forming in the evaporator is collected in vapour removal channels, referred to as vapour grooves in what follows, and exits the evaporator in the vapour line. The grooves are typically straight channels of rectangular cross-section machined either in the casing or the porous wick. Fig. 2 shows the schematic cross-section of a flat evaporator for both groove configurations. LHPs have been the subject of many experimental and/or numerical studies and is still a very active research topic, e.g. [START_REF] Nakamura | Study on a loop heat pipe for a long-distance heat transport under anti-gravity condition[END_REF][START_REF] Nishikawara | Numerical simulation of capillary evaporator with microgap in a loop heat pipe[END_REF][START_REF] Boubaker | A numerical comparative study of the effect of working fluids and wick properties on the performance of capillary pumped loop with a flat evaporator[END_REF][START_REF] Nishikawara | Numerical study on heat-transfer characteristics of loop heat pipe evaporator using three-dimensional pore network model[END_REF][START_REF] Giraudon | Design, manufacturing, and characterization of copper capillary structures for loop heat pipes[END_REF][START_REF] Ar Anand | Experimental studies on a miniature loop heat pipe with flat evaporator with various working fluids[END_REF][START_REF] Boubaker | Experimental study of the liquid/vapour phase change in a porous media of twophase heat transfer devices[END_REF][START_REF] Nishikawara | Static and dynamic liquid-vapour phase distribution in the capillary evaporator of a loop heat pipe[END_REF][START_REF] Odagiri | Investigation on liquid-vapour interface behavior in capillary evaporator for high heat flux loop heat pipe[END_REF][START_REF] Ar Anand | Investigations on vapour blanket formation inside capillary wick of loop heat pipe[END_REF] to mention only a series of recent works. One can refer to [START_REF] Launay | Parametric analysis of loop heat pipe operation: a literature review[END_REF][START_REF] Maydanik | Review: loop heat pipes with flat evaporators[END_REF] for literature reviews. As discussed in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF], one can distinguish two main investigation scales. The first one is the whole loop. At this scale, modelling of transfers in each LHP component is typically performed via a simplified 0D or 1D approach, e.g. [START_REF] Ku | Operating Characteristics of Loop Heat Pipes[END_REF][START_REF] Launay | Transient Modeling of Loop Heat Pipes for the Oscillating Behavior Study[END_REF][START_REF] Adachi | Numerical study of temperature oscillation in loop heat pipe[END_REF]. The second category of models aims at developing detailed modelling of the transfers at the scale of a component or a sub-region of the component. The 3D simulations in an evaporator element presented in [18; 22-24] are representative of this type of approach. At this scale, the heat and mass transfer problem in the evaporator can be seen as a conjugate heat and mass transfer problem between the various components of the evaporator, namely the casing, the porous wick, the grooves, the liquid line connecting the evaporator to the reservoir (also referred to as the compensation chamber (CC)). A detailed modelling approach then requires to compute the flow in the grooves and the liquid line in the evaporator by solving the Navier-Stokes equations and the heat transfer equation coupled with a model describing the heat and mass transfer in the porous wick and the heat transfer in the casing. To the best of our knowledge, such a complete 3D numerical approach has not yet been developed. In [START_REF] Cao | Conjugate analysis of a flat-plate type evaporator for capillary pumped loops with threedimensional vapour flow in the groove[END_REF], the 3D flow in the groove was computed but only the regime where the wick is fully saturated was considered. A somewhat similar approach was considered in [23] again considering the wick as fully saturated. A difference between these two studies lies in the position of the groove: in the casing in [START_REF] Cao | Conjugate analysis of a flat-plate type evaporator for capillary pumped loops with threedimensional vapour flow in the groove[END_REF] and in the wick in [23]. As discussed in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF], the regime where the wick is fully saturated corresponds to low heat loads and therefore is not representative of the LHP nominal operating conditions. According to [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF], the wick is partially invaded by the vapour under the nominal conditions. This corresponds to a situation where the wick at the wick-groove interface is partially saturated with coexistence of pores filled by the liquid and of pores filled by the vapour. The detailed simulations of the liquid-vapour distribution in the wick reported in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] were based on a pore network approach in which the pore space is represented as a network of pore bodies connected by narrower conduits referred to as throats. However, the modelling of the conjugate transfers between the pore network model (PNM) and the grooves and between the casing and the grooves was simplified assuming that the temperature in the grooves was the saturation temperature and using the correlation proposed in [START_REF] Sleicher | A convenient correlation for heat transfer to constant and variable property fluids in turbulent pipe flow[END_REF] for the computation of the convective heat transfer in the grooves. This correlation was also used in [START_REF] Kaya | Numerical analysis of heat and mass transfer in the capillary structure of a loop heat pipe[END_REF] in a numerical study of heat and mass transfer in a LHP evaporator. Thus, the heat transfer and momentum equations were not explicitly solved in the vapour grooves.

Also, the temperature and pressure in the grooves were assumed to be constant along the grooves. In this context, the objective of the present study is to explore whether these simplifications are acceptable. In other words, the objective is to make a step forward toward more comprehensive and accurate simulations through detailed computations of the heat and mass transfer in the grooves for both cases, i.e. when the grooves are in the wick and when the grooves are machined in the casing. The literature is scarce on this topic. To the best of our knowledge, the most detailed investigation is the study presented in the Ph.D thesis by Jesuthasan [START_REF] Jesuthasan | Modeling of Thermofluid Phenomena in Segmented Network Simulations of Loop Heat Pipes[END_REF].

However, only the case of the grooves in the casing was considered and the temperature of the groove-casing limiting surface was assumed to be spatially uniform and identical on each wall. Also, the vapour injection velocity at the wick-groove interface was assumed spatially uniform. The simulations presented in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] indicate that these features are questionable simplifications. In order to consider more representative conditions, the boundary conditions for the computation of the heat and mass transfer in the groove are obtained in the present study from numerical simulations over an evaporator unit cell using the model presented in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF].

In addition to a contribution to the CPL / LHP study, the work presented in this paper can be also seen as an extension of the works devoted to the analysis of the effects of blowing (injection) from the wall on heat transfer and pressure losses in laminar channel flows, e.g. [START_REF] Hwang | Developing laminar flow and heat transfer in a square duct with one-walled injection and suction[END_REF][START_REF] Yuan | Simulation of fully developed laminar heat and mass transfer in fuel cell ducts with different cross-sections[END_REF].

The paper is organized as follows. The model of heat and mass transfer in the groove for both cases is presented in section 2. The method to specify the boundary conditions from simulations using the model presented in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] is presented in section 3 together with the input data required for the groove heat and mass transfer simulations. The main results are presented in section 4. A short discussion is presented in section 5.

Conclusions are given in section 6.

Modelling of fluid flow and heat and mass transfer in vapour groove

The computational domains considered in the present study are displayed in Fig. 3 for the case of the grooves in the wick and in Fig. 4 for the case of the grooves is in the casing. Because of the symmetry about the lateral planes (surfaces 3 and 10 in Fig. 3, surfaces 3 and 9 in Fig. 4), only a half groove and a half casing (in the xdirection) are considered. The corresponding geometrical dimensions are reported in Table 1.

Table 1 -Geometrical dimensions of computational domains shown in Fig. 3 and4. 1, the groove length is 10 cm, which is representative of grooves in LHP evaporators.

Groove in casing and

Since grooves are symmetrical about their longitudinal vertical middle plane, computations are performed over a half groove. Note also that the computational domain include the dead-end limiting surface of the groove (groove end surface in Figs. 3 and4). As can be seen from Figs. 3 and4, the groove is closed at this end by the casing. Note also that the heat leak between the groove and the compensation chamber through the casing is taken into account (heat leak through the casing wall closing the groove at one end).

Governing equations

Influence of gravity is neglected as well as the impact of viscous dissipation on heat transfer. All thermophysical properties are considered constant except for the vapour density which is computed using the ideal gas law. The working fluid is ammonia. Steady states are considered. The mathematical formulation of the considered problem is now presented.

Heat conduction in the casing

The heat transfer governing equation in the casing reads

𝛻. (𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 𝛻𝑇) = 0 ( 1
)
where T is the temperature and 𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 is the casing thermal conductivity. The casing is assumed to be in

stainless steel (𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 = 16.3 W m -1 K -1 )
Heat Transfer in the groove Because of the fluid flow inside the groove, the heat transfer occurs through conduction and forced convection

(𝜌 𝑣 𝐶 𝑝𝑣 𝑈). 𝛻𝑇 = 𝛻. (𝜆 𝑣 𝛻𝑇) (2)
where 𝜌 𝑣 is the vapour density computed using the ideal gas law, 𝜌 𝑣 = 𝑝 𝑀𝑅𝑇 , where 𝑝 is the vapour pressure, M is the vapour molecular weight, R is the universal gas constant; 𝐶 𝑝𝑣 is the vapour heat capacity (𝐶 𝑝𝑣 =2841.3 J K -1 kg -1 ), 𝜆 𝑣 is the vapour thermal conductivity 0.024 W m -1 K -1 ) and U is the vapour velocity vector.

Flow in the groove

The pressure and velocity fields in the groove are computed solving the continuity and Navier-Stokes 𝜇 𝑣 (𝛻. 𝑈)𝐼)

where I is the identity matrix and 𝜇 𝑣 is the vapour dynamic viscosity (𝜇 𝑣 =9.63 x 10 -6 Pa s).

Boundary conditions

The boundary conditions are specified considering separately the two cases: groove in the casing and groove in the wick. The numbers refer to the numbering of boundary surfaces in Fig. 3 (groove in the wick) and Fig. 4 (groove in the casing).

Groove in the wick

The boundary surface numbering is given in Fig. 3.

-Top of the Casing (boundary surface #1)

𝑇 = 𝑇 𝑐𝑎𝑠𝑖𝑛𝑔 (5)
As detailed in section 3, 𝑇 𝑐𝑎𝑠𝑖𝑛𝑔 is obtained from PNM simulations.

-Casing limiting surface in contact with Compensation Chamber (boundary surface #2),

𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 𝛻𝑇. 𝑛 = 𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 (𝑇 𝑠𝑎𝑡 𝐶𝐶 -𝑇) 𝐿 𝑧𝑐 (6)
Where 𝑛 is a unit vector normal to the considered surface, 𝐿 𝑧𝑐 is the half thickness of the casing region between the groove dead-end surface and the compensation chamber (𝐿 𝑧𝑐 = 0.5 mm), and 𝑇 𝑠𝑎𝑡 𝐶𝐶 is the temperature in the compensation chamber (𝑇 𝑠𝑎𝑡 𝐶𝐶 = 283.15 K). In this approach, only one half of the casing region in contact with the compensation chamber is taken into account in the 3D simulation whereas the second half is taken into account through the BC Eq.( 6). Actually, although this reduces a bit the computational domain, it would have been simpler to consider the whole region and impose the temperature in the CC as boundary condition. However, since the thermal conductivity of the casing is high, the latter option should lead to results very similar to the ones presented in the article.

-Groove and casing symmetry surface at x = 0 (boundary surface #3):

𝛻𝑇. 𝑛 = 0 (7)
In addition, the vapour velocity component normal to this groove limiting surface is imposed to zero,

𝑈. 𝑛 = 0 (8)
Symmetry conditions are imposed as regards the two other components of the velocity vector.

-Casing bottom limiting surface at the beginning of evaporator (boundary surface #4):

This temperature cannot be obtained from the PNM simulations. The temperature at this interface is assumed to be close to the saturation temperature since this surface is between the compensation chamber and the wickgroove bottom interface, which are both to temperatures close to the saturation temperature.

𝑇 = 𝑇 𝑠𝑎𝑡 𝐶𝐶 (9)
-Wick-groove bottom interface (boundary surface # 5):

The temperature at this interface is assumed to be close to the saturation temperature since the PNM simulations reported in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] show that this interface remains highly liquid saturated.

𝑇 = 𝑇 𝑠𝑎𝑡 𝐶𝐶

𝑈. 𝑛 = 𝑢 𝑖𝑛𝑗-𝑏𝑜𝑡𝑡𝑜𝑚 (𝑥) [START_REF] Ar Anand | Experimental studies on a miniature loop heat pipe with flat evaporator with various working fluids[END_REF] where 𝑢 𝑖𝑛𝑗 is the injection velocity resulting from the vaporization of the liquid within the wick or at its surface. As detailed in section 3, 𝑢 𝑖𝑛𝑗 is obtained from PNM simulations. A no-slip condition is used as regards the velocity components parallel to the considered interface.

-Wick-groove side interface (boundary surface #6)

The temperature at this interface is assumed to be close to the saturation temperature since the PNM simulations reported in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] show that this interface remains highly liquid saturated.

𝑇 = 𝑇 𝑠𝑎𝑡 𝐶𝐶 (12)
The velocity normal component is specified as

𝑈. 𝑛 = 𝑢 𝑖𝑛𝑗-𝑠𝑖𝑑𝑒 (𝑦) (13)
-Groove outlet (boundary surface # 7): An outflow condition is applied,

𝜆 𝑣 𝛻𝑇. 𝑛 = 0 (14)
whereas a no stress condition is chosen for the flow:

(-𝑝𝐼 + 𝜇 𝑣 (𝛻𝑈 + (𝛻𝑈) 𝑇 ) - 2 3 𝜇 𝑣 (𝛻. 𝑈)𝐼) . 𝑛 = 0 (15)
-Wick-casing interface (boundary surface #8):

As detailed in Section 3, the temperature distribution at this interface is obtained from PNM simulations and is therefore an input data for the heat and mass transfer simulations in the groove,

𝑇 = 𝑇 𝑤𝑖𝑐𝑘𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥) (16)
-Casing end surface #2 (boundary surface # 9)

The following condition is applied on this surface

𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 𝛻𝑇. 𝑛 = 0 (17)
-Casing second lateral face (boundary surface # 10):

𝛻𝑇. 𝑛 = 0 (18)
-Casing end-wick vertical limiting surface (boundary surface #11, not visible in Fig. 3):

As explained in the next section, the conditions at this interface cannot be imported from the simulations performed to obtain the boundary condition missing information. For simplicity, a no heat flux condition is imposed

𝛻𝑇. 𝑛 = 0 (19)

Groove in the casing

Although many boundary conditions are identical for both cases, the corresponding boundary conditions are given again for clarity.

-Top of the Casing (boundary surface #1):

𝑇 = 𝑇 𝑐𝑎𝑠𝑖𝑛𝑔 (20)
-Casing limiting surface in contact with Compensation Chamber (boundary surface #2),

𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 𝛻𝑇. 𝑛 = 𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 (𝑇 𝑠𝑎𝑡 𝐶𝐶 -𝑇) 𝐿 𝑧𝑐 (21)
-Groove and casing symmetry plane at x = 0 (boundary surface #3):

𝛻𝑇. 𝑛 = 0 (22)
𝑈. 𝑛 = 0

-Casing bottom limiting surface at the beginning of evaporator (boundary surface #4):

𝑇 = 𝑇 𝑠𝑎𝑡 𝐶𝐶 (24)
-Wick-groove bottom interface (boundary surface #5):

𝑇 = 𝑇 𝑠𝑎𝑡 𝐶𝐶 (25)
𝑈. 𝑛 = 𝑢 𝑖𝑛𝑗-𝑏𝑜𝑡𝑡𝑜𝑚

-Groove outlet (boundary surface # 6):

𝜆 𝑣 𝛻𝑇. 𝑛 = 0 (27)
(-𝑝𝐼 + 𝜇 𝑣 (𝛻𝑈 + (𝛻𝑈) 𝑇 ) -

-Wick-casing interface (boundary surface # 7):

𝑇 = 𝑇 𝑤𝑖𝑐𝑘𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥) (29)
-Casing end surface adjacent to the vapor groove outlet (boundary surface #8):

𝛻𝑇. 𝑛 = 0 (30)
-Casing second end surface (boundary surface # 8):

𝛻𝑇. 𝑛 = 0 (31)
For the internal boundaries between casing and groove (groove end surface in Fig. 4 and groove top surfacecasing interface and groove side surface -casing interface) temperature and heat flux continuity and velocity wall conditions are applied.

Specification of boundary condition input data

To solve the above problems, several boundary conditions must be quantitatively specified, namely 𝑇 𝑐𝑎𝑠𝑖𝑛𝑔 , 𝑇 𝑤𝑖𝑐𝑘-𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥), 𝑢 𝑖𝑛𝑗-𝑠𝑖𝑑𝑒 (𝑦), 𝑢 𝑖𝑛𝑗-𝑏𝑜𝑡𝑡𝑜𝑚 (𝑥). As mentioned in the introduction, the heat and mass transfer problem in the evaporator grooves is actually a conjugate heat and mass transfer problem in which the heat and mass transfer in the grooves is coupled with the transfers in the casing and the wick (and actually in the other components of the loop). In other words, one cannot specify𝑇 𝑐𝑎𝑠𝑖𝑛𝑔 , 𝑇 𝑤𝑖𝑐𝑘𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥), 𝑢 𝑖𝑛𝑗-𝑠𝑖𝑑𝑒 (𝑦) and 𝑢 𝑖𝑛𝑗-𝑏𝑜𝑡𝑡𝑜𝑚 (𝑥) independently from each other. They all depend on the applied heat flux on top of the casing.

In order to take into account this feature, the needed data were computed from simulations using the evaporator unit cell model presented in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF]. This model is referred to as the PNM (Pore Network Model) because a pore network model is used to compute the heat and mass transfer in the wick. The method with inputs from PNM simulations to compute the heat and mass transfer in the groove is summarized in Fig. 5.

Fig. 5 -Sketch of the one-way coupling method used to specify the boundary conditions needed to solve the heat and mass transfer problem in the groove.

Pore network modelling of evaporator unit cell

The PN Model and the associated numerical PNM code used in this study are the same ones as in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF]. Only the main features of direct interest for the present study are presented. Figure 6 shows a sketch of the PNM computational domain, which is referred to as a 3D unit cell, for both cases, i.e. when the grooves are in the wick and when the grooves are in the casing. The unit cell can be divided into 3 areas: the vapour grooves, the casing and the wick. Heat and mass transfers in both the liquid and vapour phases take place in the wick whereas heat conduction is considered in the casing. The lattice spacing, i.e. the distance between two pores, is 61 µm. This corresponds to the mesh size used in the PNM computations. For the considered computational domains (Fig. 6), which include the wick and the casing, the computational grid has about 50000 nodes (see [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] for the details). The conditions imposed in the vapour grooves are spatially uniform homogeneous conditions and are specified as follows.

a) grooves in wick b) grooves in casing Fig. 6 -Sketch of computational domains considered for the PNM simulations (figures from [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF])

Conditions in the grooves

The groove temperature is computed using the Clausius-Clapeyron relationship from the saturation pressure (𝑃 𝑠𝑎𝑡 𝐶𝐶 ) and temperature ( 𝑇 𝑠𝑎𝑡 𝐶𝐶 ) in the compensation chamber (CC), which are input data for the model, as

𝑇 𝑠𝑎𝑡-𝑔𝑟𝑜𝑜𝑣𝑒𝑠 = (( -𝑅 ℎ 𝑙𝑣 𝑀 (𝑙𝑛(𝑃 𝑔𝑟𝑜𝑜𝑣𝑒𝑠 ) -𝑙𝑛(𝑃 𝑠𝑎𝑡 𝐶𝐶 ))) + 1 𝑇 𝑠𝑎𝑡 𝐶𝐶 ) -1 (32)
where hlv is the enthalpy of vaporization. The pressure 𝑃 𝑔𝑟𝑜𝑜𝑣𝑒𝑠 in the grooves are the addition of pressure losses in grooves, pressure losses in the loop and pressure in the CC,

𝑃 𝑔𝑟𝑜𝑜𝑣𝑒𝑠 = 𝑃 𝑠𝑎𝑡 𝐶𝐶 + ∆𝑃 𝐿𝑜𝑜𝑝 + ∆𝑃 𝑔𝑟𝑜𝑜𝑣𝑒𝑠 (33)
As explained in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF], the pressure losses∆𝑃 𝐿𝑜𝑜𝑝 in the elements of the loop elements (liquid line, condenser and vapour line) are computed using Poiseuille law. The pressure losses in the grooves is computed as in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] using Eq. (33),

∆𝑃

Wick-groove interface

In order to determine the groove heat transfer coefficient from Eq.( 35) as well as the saturation pressure and temperature in the groove, the average vapour velocity in the groove must be determined. This velocity is obtained from the total vaporization mass flow rate produced in the 3D unit cell. It is further assumed that this vaporization rate is about the same all along the groove. This leads to express the vapour average velocity at the groove exit as,

𝑈 𝑧 = 𝑚 ˙𝑡𝑜𝑡 𝜌 𝑣 2𝐿 𝑥𝑔 𝐿 𝑦𝑔 𝐿 𝑒𝑣𝑎𝑝𝑜𝑟𝑎𝑡𝑜𝑟 𝐿 𝑧 (36)
where 𝑚 ˙𝑡𝑜𝑡 is the total mass flow rate, i.e. the sum of the mass flow rate produced at the wick/groove interface from the pores in the liquid state and the mass flow rate produced at liquid-vapour interface within the wick.

As indicated in Fig. 6; 𝐿 𝑧 is the length of the unit cell (Fig. 6 and Table 2), 𝐿 𝑒𝑣𝑎𝑝𝑜𝑟𝑎𝑡𝑜𝑟 is the length of the evaporator (Table 2).

The method to compute the vapour-fluid flow and thermal transfer in the wick is detailed in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF]. The point of interest here is the thermal coupling between the wick and the groove which can be separated in two cases.

When the pore at the wick-groove interface is in the liquid state, i.e. occupied by the liquid, the interface temperature is imposed equal to the saturation temperature in the groove 𝑇 𝑠𝑎𝑡-𝑔𝑟𝑜𝑜𝑣𝑒𝑠 (Eq. [START_REF] Tl Bergman | Fundamentals of heat and mass transfer[END_REF]. When the pore is in the vapour state, a convective heat transfer boundary condition is applied as

𝜆 𝑣𝑎𝑝𝑜𝑢𝑟-𝑤𝑖𝑐𝑘 𝛻𝑇. 𝑛 = ℎ 𝑐 (𝑇 -𝑇 𝑠𝑎𝑡-𝑔𝑟𝑜𝑜𝑣𝑒𝑠 ) (37)
where the heat transfer coefficient ℎ 𝑐 is computed using Eq.(35).

Casing -groove interface

As sketched in Fig. 6, the heat flux is imposed over the casing top surface, spatially periodic boundary conditions are imposed on the lateral sides of the casing and a no flux condition is imposed on the side faces at z = 0 and z = Lz. At the casing-groove interface, the convective heat transfer condition is imposed,

𝜆 𝑐𝑎𝑠𝑖𝑛𝑔 𝛻𝑇. 𝑛 = ℎ 𝑐 (𝑇 -𝑇 𝑠𝑎𝑡-𝑔𝑟𝑜𝑜𝑣𝑒𝑠 ) (38)

PNM simulation results

The PNM simulations were performed for the parameter values indicated in Table 2 for grooves in the casing or in the wick. The PNM is a discrete model in which the temperature and vapour velocity are computed in each pore of the wick. In contrast, the groove heat and mass transfer problem presented in section 2 is set according to a continuum framework. The commonly used procedure to produce continuous field from discrete field is volume averaging, e.g. [START_REF] Moghaddam | Kinematics in a slowly drying porous medium: Reconciliation of pore network simulations and continuum modeling[END_REF] for an example. A somewhat similar procedure is used here by smoothing out the PNM data by performing spatial averages (as expressed by Eqs. (39-41) below). The maximal temperature on top of casing (Fig. 7) and the average temperature at the wick/casing interface (Fig. 8) are extracted from the pore network model simulations for each applied heat load for both cases. As can be seen from Fig. 8, the wick-casing temperature increases with the distance from the groove. The temperature is maximum in the middle of the rib formed either by the casing (groove in casing case) or the wick (groove in wick case). At a given location, 𝑇 𝑤𝑖𝑐𝑘-𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥) increases with the applied heat flux. As can be seen the variation of 𝑇 𝑤𝑖𝑐𝑘-𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥) with the heat flux is non-linear. The variation is more important for the higher heat fluxes and one can also see a jump in the temperature increase (between the heat fluxes 5.6 W/cm 2 and 6.6 W/cm 2 in Fig. 8 right and between 10.6 W/cm 2 and 12.6 W/cm 2 in Fig. 8 left). As discussed in [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] and sketched in Fig. 9, three main evaporator operating regimes can be distinguished depending on the applied heat flux. For a sufficiently low heat flux, the wick is saturated and the vaporization takes place only at the wick surface. When the heat flux is sufficiently high, a two-phase zone takes place within the wick. The vapour and liquid phase coexists within a region of the wick and the vaporization takes place both with the wick and at its surface. When the heat flux is further increased, a dry zone eventually forms within the wick.

The afore-mentioned temperature jumps correspond to the transition between regime II and regime III (sketched in Fig. 9) when a vapour blanket starts forming in the wick The formation of the vapour blanket induces a higher resistance to heat transfer within the wick and consequently a comparatively higher increase in the casing temperature with an increasing heat flux.

with y = -Ly (Figs. 3 and4).

with x = Lxg (Fig. 3). As can be seen from Figs.10 and 11, the injection velocity varies significantly over the groove limiting surfaces. As regards the groove bottom surface, the injection velocity increases significantly in the region close to the casing (case of the grooves in the casing in Fig. 10). This is because all the vapour forming in the region of the wick under the casing exits the wick in the region of the groove bottom surface close to the casingwick -groove triple contact line. A somewhat similar situation can be observed when the groove is in the wick (Fig. 10) with the greater velocities in the region close to the vertical groove-wick interfacial surface. However, the velocity magnitude is significantly lower when the grooves are in the wick compared to the case of the grooves within the casing. This is because the most active vaporization region inside the wick is under the casing. This is illustrated in Fig. 11 for the groove side surface. The velocity increases in the region of the groove side surface close to the casing and the velocity magnitude is comparable to the one of the bottom injection velocity for the case of the grooves in the casing in Fig. 10. In summary, these figures well illustrate the fact that the injection velocity is not uniform over the groove-wick interfacial surfaces.

Results

In this section, the results of the 3D simulations of the vapour flow inside the vapour removal channel of a Loop Heat Pipe are presented and discussed. The first subsection is dedicated to the temperature behaviour, the second one to the pressure and then, in the last part, the convective heat transfer coefficient is discussed.

To obtain these results, the simulations were performed with finite elements of variable sizes, smaller in the vicinity of the interfaces or boundaries. The mesh was adjusted through a series of test using a COMSOL relative tolerance criterion of 10 -10 . For instance, for the case of the groove in the casing, this led to a mesh containing 2 824 778 elements. The size of the elements varied but was on average of the same order as the mesh size in the PNM simulations in the groove transverse directions, i.e. on the order of 60 µm. This means that there were about 12 elements over the width of the groove. Convergence was not possible or judged too long to achieve with a more refined grid. This is a consequence of the relatively high groove aspect ratio (length: 100 mm for a 0.7 mm × 0.7 mm cross-section), which implies a much greater number of elements along the longitudinal direction than in the transverse directions. .

Vapour Temperature

A common simplification in the evaporator models, i.e. [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF] and references therein, is to consider that the vapour temperature in the groove is not significantly different from the saturation temperature. In order to assess the deviation from this assumption, a representative vapour temperature in the groove is extracted from the 3D field. This is performed considering the mixing temperature,

𝑇 𝑚𝑖𝑥𝑖𝑛𝑔 (𝑧) = ∫ ∫ 𝑇‖𝑈‖𝑑𝑥𝑑𝑦 -(𝐿 𝑦 -𝐿 𝑦𝑔 ) -𝐿𝑦 𝐿𝑥𝑔 0 𝑈 𝑚𝑖𝑥𝑖𝑛𝑔(𝑧) (42)
where

𝑈 𝑚𝑖𝑥𝑖𝑛𝑔 (𝑧) = ∫ ∫ ‖𝑈‖𝑑𝑥𝑑𝑦 -(𝐿 𝑦 -𝐿 𝑦𝑔 ) -𝐿𝑦 𝐿𝑥𝑔 0
, ‖𝑈‖is the velocity norm. The axial variation of the mixing temperature is displayed in Fig. 12. As can be seen from Fig. 12, the mixing temperature increases rapidly at the beginning of the groove and then varies little. The mixing temperature reaches 90 % of its maximum value (located at the groove outlet) after only 2 mm when the grooves are in the casing, consistently with the results reported in [START_REF] Jesuthasan | Modeling of Thermofluid Phenomena in Segmented Network Simulations of Loop Heat Pipes[END_REF]. This short distance is significantly less (~0.25 mm) when the grooves are in the wick. When the grooves are in the wick, the mixing temperature remains relatively close to the saturation temperature imposed at the wick-groove interfacial surfaces. As illustrated in Fig. 12 (left), the difference between the mixing temperature and the (wall

)
saturation temperature is less than 2K. The commonly used assumption that the temperature in the groove can be approximated by the saturation temperature is therefore reasonable in this case. By contrast, the mixing temperature is significantly greater than the saturation temperature when the grooves are in the casing. The difference is as high as about 15 K for the highest considered heat flux in Fig. 12 and is significant even for a relatively low heat flux (~4-5 K for example for the heat flux of 5.6 W/cm 2 ). In this case, the commonly used procedure consisting in neglecting the vapour overheating in the groove is therefore questionable. The temperature variations in Figs.12 and 13 reflect the various regimes sketched in Fig. 10 (for the case of the grooves in the casing, the transition between regimes I and II occurs at about 2 W/cm 2 and the transition between regime II and regime III occurs at about 6 W/cm 2 ). As shown in Fig. 13, this difference between both cases is notably due to the temperature over the groove dead-end surface, which is significantly higher when the grooves are in the casing, i.e. closer to the maximum temperature located on top of the casing. The results show in Fig. 14 further illustrate the fact that the temperature distribution is much more homogeneous in the groove when the groove is in the wick. By contrast, the temperature on the top surface of the groove becomes significantly higher than the groove mixing temperature when the groove is in the casing.

In other words, there is a noticeable convective heat transfer between the casing and the groove due to the vapour flow in the wick when the grooves are in the casing since the temperature difference between the mixing temperature and the casing -groove top interface temperature is as high as 17 K for the highest considered heat flux. As depicted in Fig. 15, somewhat similar results are obtained as regards the difference between the groove side surface temperature and the mixing temperature. As can be seen, the mixing temperature is actually slightly greater than the wick surface temperature with the difference in temperature on the order 1-2 K when the grooves are in the wick. Although less than for the groove top surface, the temperature difference between the groove side surface temperature and the mixing temperature is significantly higher when the grooves are in the casing. For the highest considered flux, the mixing temperature is about 7-8 K lower than the casing side surface temperature.

Also two groups of curves can be distinguished in Fig. 15, as well as in Figs. 13 and14, when the grooves are in the casing. There is a significant increase in the temperature difference when the heat flux is increased from 5.6 W/cm 2 to 6.6 W/cm 2 . As for the temperature profiles depicted in Fig. 8, the jump in the temperature difference can be related to the three main evaporator operating regimes sketched in Fig. 10. The temperature jump between the fluxes 5.6 W/cm 2 and 6.6 W/cm 2 in Figs. 13-15 corresponds to the transition between regime II and regime III when a vapour blanket starts forming in the wick.

Pressure drop

The axial velocity is continually increasing within the groove as a result of the uniform injection velocity at the wall. This is illustrated in Fig. 16 where the variation of the Reynolds Numberℜ = 𝑈 𝑣𝑧 𝜌 𝑣 𝐷 𝐻 𝜇 𝑣 along the groove is plotted for both cases. For the range of considered heat fluxes, the Reynolds number is always lower than 2500 which is the transition value between the laminar and turbulent regimes in a pipe. Although, the shape of the groove (different from a circular pipe) and the presence of the wall injection velocity should modify the value of the critical Reynolds number, assuming a laminar flow in the groove seems a quite reasonable assumption based on the results shown in Fig. 16. The pressure in the groove is analysed through the consideration of the axial variation of the pressure averaged over the groove cross-section,

𝑝(𝑧) = ∫ ∫ 𝑝𝑑𝑥𝑑𝑦 -(𝐿 𝑦 -𝐿 𝑦𝑔 ) -𝐿𝑦 𝐿𝑥𝑔 0 𝐿 𝑥𝑔 𝐿 𝑦𝑔 (43)
As shown in Fig. 17, the pressure varies non-linearly along the groove (note that the figure actually shows the difference 𝑝(𝑧) -𝑝(𝑧 𝑒𝑥𝑖𝑡 ) where 𝑝(𝑧 𝑒𝑥𝑖𝑡 ) is the pressure at the groove exit). The parabolic shapes in Fig. 17 are typical of a laminar flow with uniform injection velocity at the wall [START_REF] Hwang | Developing laminar flow and heat transfer in a square duct with one-walled injection and suction[END_REF][START_REF] Terrill | On laminar flow through a uniformly porous pipe[END_REF].

The greater pressure drop for the case of the grooves in the wick compared to case of the grooves in the casing for the same heat flux is consistent with the comparatively higher velocities in Fig. 16 when the grooves are in the wick.

Also, it can be noticed that such pressure variations correspond to a quite small saturation temperature variations (0.02 K for the highest considered flux) according the Clausius-Clapeyron relationship. In this respect, neglecting the impact of the pressure variation within the groove on the saturation temperature is a quite reasonable assumption.

Heat transfer coefficient

As illustrated in Fig. 16, the axial velocity starts from a null value and increases continually along the groove.

Thus, the heat transfer is by conduction at the beginning of the groove. Then when the velocity increases the importance of heat transfer by forced convection becomes important. This can be characterized by the Peclet number 𝑃𝑒 = ℜ𝑃𝑟, where Pr is the Prandtl number comparing the significance of convective and conductive heat transfers. Since the Prandtl number for ammonia is closed to 1 (Pr = 1.12), the variations of the Peclet number are actually very similar to the ones of the Reynolds number depicted in Fig. 16. As can be seen from Fig. 16, the convective heat transfer becomes very rapidly dominant. In other words, the extension of the zone where the heat conduction is on the same order as the convective heat transfer is quite small for both cases (< 0.25 mm).

The heat transfer within the groove is characterized from the computation of heat transfer coefficients defined For the case of the grooves in the wick, the side surface (surface #6 in Fig. 3) is a wick-groove interfacial surface. Therefore, the heat transfer at the wall also depends on the blowing effect associated with the injection velocity resulting from the vaporization. However, the average injection velocity over this surface is much smaller than the axial velocity, which leads to a small wall Peclet number. For this reason, the convective heat transfer associated with the injection velocity was computed using the same convention as for the casing solid surface, that is from equations similar to Eqs.(46-47). For the same reason, the heat transfer coefficient at the groove bottom surface was computed as

ℎ 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑧) = | 𝜑 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑧)
𝑇 𝑚𝑖𝑥𝑖𝑛𝑔 (𝑧)-𝑇 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑧) | (50)

𝜑 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑧) = 1 𝐿 𝑥𝑔 ∫ (-𝜆 𝑣 𝛻𝑇. 𝑛)𝑑𝑥 𝐿 𝑥𝑔 0 (51) 𝑇 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑧) = 1 𝐿 𝑥𝑔 ∫ 𝑇𝑑𝑥 𝐿 𝑥𝑔 0 (52)
The variations of the above defined heat transfer coefficients along the groove are depicted in Figs. 18-20 for both cases. As can be seen, the curves are noisy with noticeable longitudinal fluctuations (dotted lines in Figs. [START_REF] Mottet | Three-dimensional liquid and vapour distribution in the wick of capillary evaporators[END_REF][START_REF] Ku | Operating Characteristics of Loop Heat Pipes[END_REF][START_REF] Launay | Transient Modeling of Loop Heat Pipes for the Oscillating Behavior Study[END_REF]. As the convective coefficient is computed from the division of the mean heat flux by the average temperature difference, a small variation in the temperatures can lead to a noticeable variation of the coefficient. Note for instance that, as shown in Fig. 17, the temperature difference for the case of the grooves in the wick is quite small. Thus a small inaccuracy in the temperature differences between two successive mesh points along the groove can cause fluctuations. Also as depicted in Figs.10-11, the injection velocities at the wick-groove interfaces vary significantly. This can be a source of fluctuations with a too coarse grid. The fluctuations were reduced by increasing the mesh fineness. However, when the mesh fineness was too large, the computation could not converge. The results depicted in Figs.18-20 represent the best trade-off we could obtain between mesh-fineness and convergence. In other words, the fluctuations can be attributed to a lack of mesh refinement. In order to obtain a clearer view of the heat transfer coefficient behaviour, the fluctuations were smoothed out by using a running average procedure. This led to the solid lines shown in Figs 18-20. As can be seen, these lines show clear and consistent trends. Although performing the simulations over a more refined mesh remains desirable, it is our belief that the results should be very similar to the ones depicted in Fig. 181920. As can be seen, various behaviours can be observed depending on the considered case and the considered surface. For a given flux, the heat transfer coefficient can be almost constant outside the region at the beginning of the groove for some surfaces. Otherwise, the trend is a slight increase along the groove.An exception is the coefficient for the groove bottom limiting surface for the case of the grooves in the wick. As illustrated in Fig. 20 left, this heat transfer coefficient decreases along the groove in qualitative agreement with the behaviour reported in [START_REF] Hwang | Developing laminar flow and heat transfer in a square duct with one-walled injection and suction[END_REF] for the simpler case of a duct with one wall subjected to uniform injection and constant heat flux For the sake of comparison, Fig. 21 shows the variation of the heat transfer coefficient computed using Eq.(35), i.e. a classical correlation [START_REF] Sleicher | A convenient correlation for heat transfer to constant and variable property fluids in turbulent pipe flow[END_REF]. Contrary to our results, this correlation predicts a noticeable increase in the coefficient along the groove. In addition to a different spatial variation, the coefficient values are also quite different. The correlation significantly overestimates the coefficients compared to our simulation results. It can be argued that this correlation has been developed for turbulent flows whereas our simulations indicate a laminar flow. However, all the commonly used correlations of the literature for the convective heat transfer in pipes, e.g. [START_REF] Tl Bergman | Fundamentals of heat and mass transfer[END_REF], predict an increase of the heat transfer coefficient with the velocity. An important feature illustrated in Figs.14 and 15 lies in the fact that the temperature varies from one limiting surface to the other. In other words, the assumption of a similar wall temperature on the two casing-groove interfacial surfaces in the case of the grooves in the casing, as considered for instance in [START_REF] Jesuthasan | Modeling of Thermofluid Phenomena in Segmented Network Simulations of Loop Heat Pipes[END_REF], is not correct. This variability between the groove limiting surfaces can also be seen with the heat transfer coefficients and is further illustrated in Fig. 22 showing the variation of the mean heat transfer coefficient for the various surfaces for both cases.

Discussion

The results presented in Section 4 indicate that the coupling between the transfer in the grooves and the transfer in the other elements of the evaporator (casing, wick, etc) must be improved since most models of the literature are based on questionable assumptions and make use of questionable heat transfer coefficient correlations.

However, it can be noticed that the PNM simulations themselves, which were used to generate the boundary conditions for the groove COMSOL simulations (as sketched in Fig. 5) imply to take into account the heat and mass transfer in the groove. As explained in section 3, this was performed using a simplified approach. In that sense, the coupling between the COMSOL simulations and the PNM simulations used in the present study can be seen as a one-way coupling. In other words, the results of the COMSOL simulations were not taken into account to possibly improve the PNM simulations. Thus, still more accurate results can be expected through an iterative procedure between the evaporator PNM simulations and the groove COMSOL simulations (for instance by using the convective heat transfer coefficient determined from the COMSOL simulations in the PNM simulations). This is left for a future work. It can be noticed, however, that such a two-way coupling approach would be quite computationally demanding. Also, contrary to the COMSOL simulations, the PNM simulations did not take into account the groove dead-end region where the groove is closed by the casing.

The PNM computational domain corresponds to a region of the evaporator away from this region. In spite of these shortcomings, we believe that the one way coupling method used in the present study and the associated results are a step forward bringing interesting results on the heat and mass transfer in the evaporator grooves.

Conclusion

Three-dimensional simulations of the convective heat transfer inside the vapour removal channels (grooves) of a LHP evaporator were performed for two channel configurations: when the channels are in the wick and when the channels are in the casing. The boundary conditions were imported from simulations based on a discrete pore network representation of the wick referred to as a pore network model (PNM).

The results show significant differences between the two cases and also with commonly made assumptions.

When the groove is in the wick, the commonly used assumption that the vapour temperature is close to the saturation temperature is acceptable. This assumption is not valid anymore when the groove is in the casing.

A significant vapour overheating is obtained (up to 10 K). However, the assumption of a constant vapour temperature within the groove is acceptable for both cases except in a short region at the very beginning of the groove.

The study of the pressure drop along the groove confirms that the assumption of a constant saturation temperature within the groove is acceptable since the pressure variation along the groove remains small compared to the mean pressure in the groove.

The temperature varies from one limiting surface of the groove to the other for a given heat flux. This is also the case as regards the heat transfer coefficient. Also, this coefficient was found to be almost constant or varying only slightly along the groove over several of the groove limiting surfaces for a given heat flux. This is contrast with the commonly used correlations of heat transfer in pipes in the absence of injection which all predicts an increase of the Nusselt number with the Reynolds number. Contrary to the classical situation where the velocity is constant within the pipe, the fluid velocity increases continuously in an evaporator groove as the result of the vaporisation along the evaporator. This has a great impact on the heat transfer coefficient behaviour, which is thus markedly different from the behaviour in pipes without wall injection.

In summary, the present study indicates that the coupling between the transfer in the grooves and the transfer in the other elements of the evaporator (casing, wick, etc) must be improved since most models are based on questionable assumptions and make use of questionable heat transfer coefficient correlations.

Fig. 1 -

 1 Fig.1-Sketch of a loop heat pipe (figure from [18])

Fig. 2 -

 2 Fig.2 -Schematic of a flat evaporator cross-section with the vapour grooves in the wick (left) or in the casing (right).

Fig. 3 -Fig. 4 -

 34 Fig.3 -Sketch of computational domain for the case of the vapour grooves in the wick.

  𝜌 𝑣 (𝑈. 𝛻𝑈) = 𝛻. (-𝑝𝐼 + 𝜇 𝑣 (𝛻𝑈 + (𝛻𝑈) 𝑇 ) -2 3

2 3

 2 𝜇 𝑣 (𝛻. 𝑈)𝐼) . 𝑛 = 0

Fig. 7 -

 7 Fig. 7 -PNM computation of temperature on top of the casing as a function of heat flux for both cases

Fig. 8 -

 8 Fig. 8 -PNM computation of 𝑇 𝑤𝑖𝑐𝑘-𝑐𝑎𝑠𝑖𝑛𝑔 (𝑥) (temperature at wick-casing interface) for the case of the grooves in the wick and the case of the grooves in the casing; x = 0 corresponds to the triple contact line wick-casing-groove interface.

Fig. 9 -

 9 Fig.9 -Schematic of the evaporator main operating regimes (illustrated for the groove in casing case).

Fig. 10 -

 10 Fig.10 -PNM computation of𝑢 𝑖𝑛𝑗-𝑏𝑜𝑡𝑡𝑜𝑚 (𝑥): vapour injection velocity at wick-groove interface for the case of the grooves in the wick and the case of the grooves in the casing for several heat fluxes; x = 0 corresponds to the middle of the groove.

Fig. 11 -

 11 Fig.11 -PNM computation of𝑢 𝑖𝑛𝑗-𝑠𝑖𝑑𝑒 (𝑦): vapour injection velocity at wick-groove lateral interface for groove in wick for several heat fluxes. In this figure, y = 0 corresponds to the intersection between the horizontal and vertical groove limiting surfaces as sketched in the figure.

Fig. 12 -

 12 Fig. 12 -Variation of the mixing temperature along the groove for case of the grooves in the wick (left) and the case of the grooves in the casing case (right) for various heat fluxes.

Fig. 13 -

 13 Fig. 13 -Average temperature on the groove end surface (see Figs.3 and 4) as a function of heat flux for the case of the groove in the wick and the case of the grooves in the casing.

Fig. 14 -Fig. 15 -

 1415 Fig.14 -Difference between the temperature on the groove top surface and the mixing temperature along the groove for various heat fluxes for the case of the grooves in the wick and the case of the grooves in the casing.

Fig. 16 -

 16 Fig.16 -Axial Reynolds number for the case of the grooves in the wick and the case of the grooves in the casing.

Fig. 17 -

 17 Fig.17 -Variation of groove cross-section averaged pressure in the groove for the cases of the grooves in the wick and in the casing.

  using a classical Newton's law formulation. The heat transfer coefficient along the groove top surface is computed as ℎ 𝑡𝑜𝑝 (𝑧) = | 𝜑 𝑡𝑜𝑝 (𝑧) 𝑇 𝑚𝑖𝑥𝑖𝑛𝑔 (𝑧)-𝑇 𝑡𝑜𝑝 (𝑧) | (44) where 𝜑 𝑡𝑜𝑝 (𝑧) is the average heat flux along the x direction at the z position over the groove top surface (groove-casing top interface), 𝜑 𝑡𝑜𝑝 (𝑧) = 1 𝐿 𝑥𝑔 ∫ (-𝜆 𝑣 𝛻𝑇. 𝑛)𝑑𝑥 𝐿 𝑥𝑔 0 (45) and 𝑇 𝑡𝑜𝑝 (𝑧) is the average temperature along the x direction at the z position over the groove top surface 𝑇 𝑡𝑜𝑝 (𝑧) heat transfer coefficient at the casing-groove vertical interfacial surface (for the case of the grooves in the casing) is defined as ℎ 𝑠𝑖𝑑𝑒 (𝑧) = | 𝜑 𝑠𝑖𝑑𝑒 (𝑧) 𝑇 𝑚𝑖𝑥𝑖𝑛𝑔 (𝑧)-𝑇 𝑠𝑖𝑑𝑒 (𝑧) |

Fig. 18 .Fig. 19 -

 1819 Fig.18. Groove top limiting surface heat transfer coefficient for the cases of the grooves in the wick and in the casing. The solid lines represent running averages over 50 successive points along the groove.

Fig. 20 -

 20 Fig.20 -Groove bottom limiting surface heat transfer coefficient for the cases of the grooves in the wick and in the casing. The solid lines represent running averages over 50 successive points along the groove.

Fig. 21 -Fig. 22 -

 2122 Fig.21 -Heat transfer coefficient computed using the classical correlation Eq.(35) for the cases of the grooves in the wick and in the casing .

  The heat transfer convective coefficient ℎ 𝑐 in the grooves is determined using the correlation proposed in[START_REF] Sleicher | A convenient correlation for heat transfer to constant and variable property fluids in turbulent pipe flow[END_REF],

	where𝐷 𝐻 is the groove hydraulic diameter: 𝐷 𝐻 =	4𝑆 𝑃𝑒𝑟. =	8𝐿 𝑥𝑔 𝐿 𝑦𝑔 4𝐿 𝑥𝑔 +2𝐿 𝑦𝑔	, 𝛾 = 4.7 + 19.64 ( (𝐴𝑅+1) 2 ) where 𝐴𝑅 = 𝐴𝑅 2 +1
	𝐿 𝑦𝑔 2𝐿 𝑥𝑔	.
			𝑁𝑢 =	ℎ 𝑐 𝐷 𝐻 𝜆 𝑣	= 5 + 0.015ℜ 𝑎 𝑃𝑟 𝑏	(35)
	Whereℜ =	𝑈 𝑧 𝜌 𝑣 𝐷 𝐻 𝜇 𝑣	,𝑃𝑟	𝐶𝑝 𝑣 𝜇 𝑣 𝜆 𝑣	,𝑎 = 0.88 -	0.24 4 + 𝑃𝑟 and 𝑏 =	1 3 + 0.5𝑒 -0.6𝑃𝑟 , 𝑈 𝑧 is the groove cross-section
	averaged axial vapour velocity.
				𝑔𝑟𝑜𝑜𝑣𝑒𝑠 =	2𝜌 𝑣 𝜈 𝑣 𝛾𝑈 𝑧 𝐿 𝑧 𝐷 𝐻 2	(34)

Table 2 -PNM simulation parameters (notations refer to Fig.6)

 2

	Parameter
	Heat flux [W/m²]	1000 to 101000 by step of 5000
	Lx [mm]	1.9
	Lxg [mm]	0.35
	Ly [mm]	3.4
	Lyw [mm]	1.5
	Lyg [mm]	0.7
	Lz [mm]	2
	Levaporator (mm)	100
	Fluid	Ammonia
	𝑃 𝑠𝑎𝑡 𝐶𝐶 [Pa]	615050
	𝑇 𝑠𝑎𝑡 𝐶𝐶 [K]	283.15
	Wick	Nickel
	Casing	Stainless Steal

Acknowledgments

The financial supports from CNES and Airbus Defence and Space are gratefully acknowledged.


M Saidi

K Khalfaoui

Ahcène Chaouchi
email: ahchaouchi@yahoo.fr

Mohamed Sophie D'astorg

Christian Rguiti

Courtois

S D'astorg

M Rguiti

C Courtois

piezoelectric, and Impedance spectroscopy characterizations of CuO doped (Na0.52K0.44)(Nb0.9 Sb0.06)O3-0.04LiTaO3 lead free ceramics

Keywords: Na 0, 52 K 0, 44 )(Nb 0, 9 Sb 0

Introduction

The perovskite oxides were found to be useful for some solid state electronic devices such as random access memory devices, high dielectric constant capacitor, pyro-electric detectors, imaging devices, electro-optic devices, modulators etc. [START_REF] Uchino | Ferroelectric Devices[END_REF][START_REF] Kulawik | Investigations of properties of ceramic materials with perovskite structure in chosen electronic applications[END_REF]. Among the studied ferroelectric oxides, some lead based ferroelectric oxides such as lead titanate, lead zirconium titanate etc. are found to be most suitable for fabrication of various solid state devices. Unfortunately, because of the toxic nature they produce environmental pollutions [START_REF] Maeder | Lead free piezoelectric materials[END_REF]. Therefore, attempts are now being made to search lead free new ferroelectric materials which can be replacement of lead based ferroelectrics without losing much of their physical properties required for devices.

For this purpose, (Na 0.5 K 0.5 )NbO 3 (NKN) based lead-free piezoelectric ceramics, based on their good dielectric, piezoelectric and electromechanical properties, and relatively high Curie temperatures. Particularly, the (Li, Ta, Sb) substituted NKN ceramics exhibit significantly improved piezoelectric properties such that they have been expected as a typical candidate to replace conventional lead-based piezoelectric ceramics [4][5][6][7][8][9].

However, the fabrication of these ceramics is very difficult because: firstly, the high hygroscopicity potassium carbonate. Secondly, the temperature stability of these ceramics is so bad because it have lower orthorhombic-tetragonal transition temperature (T o-t ). Thirdly, it is very difficult to obtain fully densified ceramics due to the evaporation of potassium during the sintering process [10][11][12][13][14]. Therefore, a number of studies have been carried out to improve the sinterability and then the piezoelectric properties of (Na,K)NbO 3 ceramics, maintaining higher orthorhombic-tetragonal transition temperature for good temperature stability of the electrical properties [15]; these include the formation of solid solutions of NKN-SrTiO 3 and NKN-LiTaO 3 , and the use of sintering aids, e.g., K 4 CuNb 8 O 23 ,MnO 2 , CuO, and Bi 2 O 3 [16,15].

Among them, it is well known that the CuO is good sintering aid for low temperature sintering of (Na,K)NbO 3 -based ceramics as well as shifting the tetragonal-orthorhombic phase transition toward higher temperature. In addition, it is also a helpful sintering aid to greatly promote the densification of NKN based on the formation of liquid phases during sintering [17]. Although the addition of a small amount of CuO greatly enhances Q m and improves densification of NKN (ρ ≃ 98% theoretical density), the d 33 value remains low (mostly <100 pC/N).

Furthermore, many compositions that possess different Curie temperatures, a broad range of electrical properties and have their own morphotropic phase boundaries (MPB), can be produced by changing the ratio of Li to Ta or Sb within the NKN systems. Particularly, the textured (Li, Ta, Sb)-modified NKN ceramics exhibit piezoelectric properties comparable to those of a hard PZT [18].

It is well known that the interior defects such as A-site vacancies, space charge electrons or oxygen vacancies have great influence on ferroelectric fatigue or ionic conductivity of the material [19][20][21]. Considering that the solid defects play a decisive role in all of these applications, we find that it is very important to gain a fundamental understanding of their conductive mechanism. Various kinds of defects are always suggested as being responsible for the dielectric relaxations at high temperature range. The ac impedance analysis is a powerful means to separate out the grain boundary and grain-electrode effects, which usually are the sites of trap for defects. It is also useful in establishing its relaxation mechanism by appropriately assigning different values of resistance and capacitance to the grain and grain boundary effects.

In this work, CuO was used to reduce the sintering temperature of (Na 0.52 K 0.44 )(Nb 0.9 Sb 0.06 )O 3 -0.04LiTaO 3 piezoelectric ceramics, and a detailed analysis of ac impedance spectroscopy has been carried out to characterize the dielectric, electric and conductivity properties in order to gain insight into the relaxation mechanism and defects relation.

Experimental

The (Na 0.52 K 0.44 )(Nb 0.9 Sb 0.06 )O 3 -0.04LiTaO 3 (NKNSLT) compound was prepared by solid state reaction using reagent grades powders of Na 2 CO 3 ,S b 2 O 3 ,K 2 CO 3 ,L i 2 CO 3 , Nb 2 O 5 , and Ta 2 O 5 (purity >99%). These compounds were stoichiometrically mixed using isopropanol and zircon balls in a Teflon jar for 2 hours. The slurry was subsequently dried and the powder was manually reground and heat treated at 850 • C for 2 hours in air. The powder was finally reground using the same process than before in isopropanol for 2 hours.

The mixtures NKNSLT +xwt.%CuO (with x = 0.6, and 1) were prepared by mixing the powders manually in porcelain mortar. To manufacture pellets, an organic binder (Polyvinyl alcohol, 5 vol%) was manually added to the powder and disks (7 and 13 mm in diameter, 1.5 mm and 1mm thickness, respectively) were shaped by uni-axial pressing under 100MPa. The green samples were finally sintered in air at 1080 • C for 2 hours, with heating and cooling ratesof150 • C/h. The crystallised phase composition has been identified by X-ray diffraction (XRD) technique using the Cukα X-ray radiation (Philips X' Pert) and the microstructures were observed using a scanning electron microscope (SEM Philips XL '30). The specimens were polished and electroded with a silver paste. The dielectric and electric properties was determined using HP4284A meter versus temperature (from 20 • C to 500 • C), and the frequency range from 100 Hz to 1 MHz. For piezoelectric measurements, the specimens were polled in a silicone oil bath under different poling. The piezoelectric coefficient (d 33 ) was measured using a piezoelectric d 33 meter (Piezotest PM 200) at a frequency of 100 Hz. P-E hysteresis loops were obtained using Radiant Precision Workstation ferroelectric testing system.

Results and Discussions

Phase Analysis and Microstructure

Fig. 1 shows the XRD results of CuO doped NKNSLT ceramics. All the XRD patterns of the ceramics are single phase. The diffraction peaks shift toward a higher angle with the increase of the copper addition, which indicates the decrease of lattice constant. Their decrease is due to the incorporation of Cu 2+ into the B-site [22].

Fig. 2 shows the SEM micrographs of the NKNSLT +xw.%CuO (with x = 0.6, and 1) ceramics. Even though the sintering temperature is as low as 950 • C, the ceramics can be well sintered, as shown in Fig. 2(a) and (b). All the ceramics show the formation of large grains is attributed to presence of the liquid phase, which is consistent with the other reports that the formation of large grains is ascribed to the existence of the liquid phase in the KNN-based ceramics [START_REF] Zhen | Normal sintering of (K, Na)NbO 3 -based ceramics: influence of sintering temperature on densification, microstructure, and electrical properties[END_REF].

Dielectric and Piezoelectric Studies

Fig. 3 shows the temperature dependences of dielectric constant and dielectrics loss for the NKNSLT +xw.%CuO ceramics at different frequency (1kHz, 10 kHz, and 100 kHz). As shown in Fig. 3, there are two phase transition temperatures. For the higher phase transition temperatures, corresponding to tetragonal to cubic phase (Tc), all the ceramics present similar temperatures around 360 • C, which suggests that the excess CuO added would not obviously affect the Tc of NKNSLT ceramics [START_REF] Pang | Effects of Sb content on electrical properties of lead-free piezoelectric (K 0.4425 Na 0.52 Li 0.0375[END_REF]. For the lower phase transition temperatures, corresponding to orthorhombic to tetragonal phase (T o-t ).

At room temperature, the dielectric loss (tanδ) was lower, and the dielectric constant (ε r ) varied from 800 to 1400. The dielectric loss increases rapidly at higher temperature due to the rapid increase of conductivity. The temperature T m corresponding to the maximum value of the dielectric constant decreases with increasing frequency. The broadening of the ε r peaks and the obvious frequency dispersion indicate that NKNSLT +xw.%CuO ceramics present relaxor ferroelectric phenomena. A modified empirical expression proposed by Uchino and Nomura [25] is employed in this work to further characterize the dielectric dispersion and diffuseness of the phase transition:

(1/ε r ) -(1/ε m ) = A(T -T m ) γ
where ε m is the peak value of dielectric constant, and T m is the temperature at which ε r value reaches the maximum. The terms γ and A are assumed to be constant, γ value is between 1 and 2. The limiting values γ = 1 and 2 reduce the expression to the Curie-Weiss law, which is valid for the case of normal ferroelectric, and to the quadratic dependence, which is valid for an ideal relaxor ferroelectric [26], respectively. The logarithmic plots related to this equation for NKNSLT +xw.%CuO compositions are shown in Fig. 4. A linear relationship is observed for two samples. By linear fitting the experimental data to the modified Curie-Weiss law, γ is determined to be 1.27 for NKNSLT +1%CuO, and γ = 1.76 for NKNSLT +0.6%CuO compositions revealing relaxors ferroelectrics of these ceramics. These results are in accordance with the results of Figs. 3(a) and 3(b). Fig. 5 shows the P-E hysteresis loops of the NKNSLT +xw.%CuO (with x = 0.6, and 1) ceramics, all the ceramics exhibit a well-saturated P-E hysteresis loop, suggesting that the ceramics have good ferroelectric properties. From the loops Fig. 5, the remnant polarization P r and the coercive field E C (indicating the electric field required to zero the polarization) are determined to be near to 7 µC/Cm 2 , and 1000V/cm for these compositions, respectivity.

The piezoelectric constant d 33 and planar electromechanical coefficient k p also show interesting values. For NKNSLT +1.%CuO compositions, the d 33 and k p reach the maximum values, which are 91 pC/N and 23%, respectively. Theses high values reveal the promotion of piezoelectric properties at NKNSLT +1w.%CuO is expected owing to the coexistence region where both the orthorhombic and tetragonal phases exist and thus more possible polarization states. Another possible reason is partly attributed to dense microstructure with high bulk density which can lower the leakage current and enhance the poling process [27].

Complex Impedance Spectrum Analysis

Impedance spectroscopy (IS) is a relatively new and powerful method of characterizing the electrical properties of materials and their interface with electrodes. It may be used to investigate the dynamics of bound or mobile charge in the bulk or interfacial regions of any kind of solid including ionic or mixed electronic-ionic materials and even insulators [START_REF] Macdonald | Impedance Spectroscopy: Emphasizing Solid State Material and Systems[END_REF].The resultant response (when plotted in a complex plane) appears in the form of a succession of semicircles representing contributions due to the bulk, grain boundaries and interfaces. A polycrystalline material usually shows both grain and grain boundary effects with different time constants (relaxation times) leading to two successive semicircles [START_REF] Behera | [END_REF]. Electrical properties of a material are often represented in terms of some complex parameters like complex impedance (Z * ), complex admittance (Y * ), complex modulus (M * ), complex permittivity (ε * ) and dielectric loss (tanδ). These frequency dependent parameters are related to each other by the following relations [30]:

ε * =ε ′ + jε ′′ (1)
Z * = Z ′ + jZ ′′ = (1/jC 0 ε * ω)( 2 ) Y * = Y ′ + jY ′′ = jωC 0 ε * (3) M * = M ′ + jM ′′ = (1/ε * ) = jω C 0 Z * (4) tan δ = (ε ′′ /ε ′ ) = (M ′ /M ′′ ) = (Y ′ /Y ′′ ) = (Z ′ /Z ′′ )( 5 )
where ω = 2 ¶,f is the angular frequency, C 0 is the free geometrical capacitance, and j 2 =-1. These relations offer wide scope for a graphical analysis of the various parameters under different conditions of temperature or frequency. The useful separation of intergranular phenomena depends ultimately on the choice of an appropriate equivalent circuit to represent the sample properties. Fig. 6 shows the variation of the real part of impedance (Z ′ ) with frequency at various temperatures for both NKNSLT +0.6%CuO and NKNSLT +1%CuO specimens. It is observed that the magnitude of Z ′ decreases with increase in both frequency and temperature, indicating an increase in ac conductivity with rise in temperature and frequency. The Z ′ values for all temperatures merge at high frequency. This may be attributed to the increase of ac conductivity with temperature and frequency in the higher frequency region due to the removal of space charge as a result of reduction in barrier height [31]. Further, at low frequency, Z ′ values decrease with an increase in temperature and show negative temperature coefficient of resistance (NTCR) type behavior similar to that of semiconductors. Fig. 7 shows the variation of imaginary part of impedance (Z ′′ ) with frequency. The plots show that Z ′′ values attain a peak (Z ′′ max ) at all measuring temperatures.

The magnitude of Z ′′ max decreases with temperature indicating decrease in the resistive property of the sample. The value of Z ′′ max shifts to the higher frequencies with increasing temperature, this indicates the occurrence of relaxation in the system [32]. The peak (Z ′′ max ) broadening with increase in temperature suggests the presence of temperature dependent electrical relaxation phenomenon in the material. This relaxation process is due to the presence of immobile species at lower temperatures and defects /vacancies at higher temperatures [33]. Fig. 8 shows the complex impedance plot of Z ′′ vs. Z ′ (Nyquists plot) for NKNSLT +xw.%CuO at different temperatures. From Fig. 8 it is clear that the semicircular arc shifts towards origin indicating the increase in conductivity of the sample as the temperature increases. All the curves start at the origin, and hence there is no series resistance of the sample. The semicircles have their centers located slightly away from the real axis, indicating the presence of relaxation species with distribution of relaxation times in the sample. Furthermore, the shape of the curve suggests that the electrical response is completely dominated by the bulk properties of the material.

Thus, the electrical properties of NKNSLT + xw.%CuO can be represented by a parallel combination of RC circuits (inset of Fig. 8(a)). The value of R g (dc resistance) of the materials given in Table 1, was obtained from the intercept of the semicircular an arc on the real axis (Z ′ ).The capacitance values of the grains (C g ) have been calculated by this equation: C g = 1/R g .2π f max, where f max is the relaxation frequency of Z ′′ vs frequency (the value of frequency at Z ′′ max obtained at Fig. 7). The decrease in grain boundary resistance with rise in temperature may be due to the lowering of barrier favouring the increase of mobility of charge carriers that adds to the conduction. The values of grain resistance (R g ) reveal the high value in the case of NKNSLT + 1%CuO. This suggests that the conductivity in the perovskites can be described as an ordered diffusion of oxygen vacancies [START_REF] Macdonald | Impedance Spectroscopy: Emphasizing Solid Materials and Systems[END_REF]. These quantities of oxygen vacancies can be decreased by addition of donors; since the donor oxide contains more oxygen per cation than the host oxide it replaces [START_REF] Selvasekarapandian | [END_REF]. In ours work, the CuO is a donor, and can replace the evaporation of K or Na, it may be expressed by Kroger-Vink notation as: 2V

2V ′ Na + V •• o Cuo --→ Cu • Na + O x O + e ′
′ k + V •• O Cuo --→ Cu • k + O x O + e ′
where, Na + or K + -site vacancy in the structure (noted V In the consideration of the above reactions, we can note that if the CuO quantity increases, the oxygen vacancy decreases. In this case, the increase substitution of CuO in the solid solutions may decrease the source of oxygen vacancies (V o •• ). Furthermore, the increased activation energy of solid solutions with the increase the CuO content (results cited above), which may be evidence of decreased concentration of oxygen vacancies in the ceramics. As a result, the high value resistivity of NKNSLT + 1%CuO ceramics.

Electrical Modulus Spectrum Analysis

Complex electric modulus formalism/ analysis gives information on the nature of polycrystalline (homogeneous or inhomogeneous) samples, which can be resolved into bulk and grain boundary effects. It indicates also the electrical phenomenon with the smallest capacitance occurring in a dielectric system [36]. The complex electric modulus (M * )was calculated using The equations: M ′ = ωCoZ ′′ and M ′′ = ωC o Z ′ ,C o = ε o A/t, (where ε o = permittivity of free space, A = area of the electrode surface and t = thickness of the sample). The frequency dependence of the imaginary part of the electric modulus (M ′′ ) at different temperatures is shown in Fig. 9. The frequency region below peak maximum (M ′′ ) represents the range in which the charge carriers are mobile at long distances. At frequencies above peak maximum, the carriers are confined to potential wells and are mobile at short distances [32].

It may be mentioned here that as the samples NKNSLT have been synthesized at high temperature, a slight amount of oxygen loss can occur and it may be expressed by the Kroger-Vink notation [START_REF] Selvasekarapandian | [END_REF]:

O x O = V •• O + 1 / 2 O 2 + 2e ′
The formation of oxygen vacancies facilitate the appearance of dipoles formed with an adjacent host ion and enlarge the rattling space available for dipole vibration, which as a consequence leads to the short-range hopping of the ions and gives rise to relaxation.

Electrical Conductivity Analysis

Fig. 10 shows the variation of ac conductivity (σ ac ) with frequency at higher temperatures for NKNSLT +0.6%CuO ceramics. The σ ac was calculated from dielectric data using the following relation: σ ac = d/A.Z ′ , where d is the thickness and A is the surface area of the specimen. The Jonscher power law is used to fit the ac conductivity of the material as follows: σ ac = σ dc + Aω n where σ dc is frequency independent conductivity that related to dc conductivity, A is the temperature dependent pre-exponential factor and n is the frequency exponent. The value of n varies between 0 and 1 [START_REF] Jonscher | Dielectric relaxation in solids[END_REF].

Two regions are observed in the conductivity spectrum. The long-range translational hoping conduction in low frequency region gives rise to dc conductivity (σ dc ). The frequency dependant short-range translational and localized hoping conduction at high frequency is assigning by Aω n term. The frequency at which a change in the slop takes place is known as the "hopping frequency ω p ". The cross over from the frequency independent region to the frequency dependent region shows the onset of the conductivity relaxation, which further indicates the transition from long range hopping to the short-range ionic motion. The onset of conductivity relaxation shifts to higher frequency side with rise in temperature. At low frequency, the σ ac increases with temperature confirming the negative temperature coefficient of resistance (NTCR) behaviour that is a typical characteristic of a semiconductor.

The impedance (Z ′′ ) vs frequency spectrums were used to evaluate the relaxation time (τ ) of the electrical phenomenon occurring in the material at different temperature using the relation: τ = 1/ω = 1/2πf max , where f max is the relaxation frequency (the value of frequency at Z ′′ max ). Fig. 11 shows a typical variation of the relaxation time with inverse of absolute temperature (10 3 /T). The activation energy (E a ) was evaluated from the two different slop of the graph using the Arrhenius expression:

τ = τ 0 exp(-E a /kT)
where τ 0 is the pre-exponential factor, k is Boltzmann constant and T is the absolute temperature.

The obtained value of activation energy are 0,87 and 0,96 for NKNSLT + 0.6%CuO and NKNSLT +1%CuO compositions respectively, indicates that the conduction in the materials may be due to the oxygen vacancies. At high temperature range, the activation energy for conduction clearly suggested a possibility of formation of vacancies. In perovskite ferroelectric materials, oxygen vacancies are considered as one of the mobile charge carriers and mostly in titanates, niobates and tantalates, ionization of oxygen vacancies creates conduction of electrons, a process which is defined by the Kroger-Vink notation as:

O x O → 1/2O 2 (g) + V •• O + 2e ′
Thus excess electrons and oxygen vacancies are formed in the above reduction reaction. The volatility of alkaline element (K and Na) at high temperature during sintering [39], conduct also to the formation of oxygen vacancy, which also noted by the Kroger-Vink notation as: These electrons ions or oxygen vacancies can be thermally activated, thus enhancing the conduction process. Doubly charged oxygen vacancies are considered to be the most mobile charges in perovskite ferroelectrics and play an important role in conduction [38].

Conclusion

Polycrystalline (Na 0.52 K 0.44 )(Nb 0.9 Sb 0.06 )O 3 -0.04LiTaO 3 +xw%CuO ceramics sintered at low temperature and has been studied by a complex impedance spectroscopy technique. Structural and electrical properties of the material have been estimated from impedance data. These results have been expressed in terms of complex impedance spectrum, complex permittivity spectrum, complex modulus spectrum and conductivity spectrum. The compound shows a diffuse type of ferroelectric-paraelectric phase transition having relaxor behaviour. Complex impedance analysis suggests dielectric relaxation to be of polydispersive non-Debye type with a temperature dependent behaviour. Modulus analysis has established the possibility of hopping mechanism for electrical transport process in this material. The activation energy for conduction (E a ) of the material is found to be near 1 eV, which is attributed to the energy required for the motion of oxygen vacancies confirming the conductivity is due to the movements of oxygen vacancies in the material.

Figure 1 .

 1 Figure 1. X-ray diffraction patterns of NKNSLT +xwt.% CuO sintered ceramics sintered at 950 • C.

Figure 2 .

 2 Figure 2. Scanning electron micrographs of fracture for the sintered ceramic.

Figure 3 .

 3 Figure 3. Temperature dependence of permittivity and dielectric loss at different frequency of NKNSLT +xwt.% CuO compositions: (a) x = 0.6 and (b) x = 1.

Figure 4 .

 4 Figure 4. Plots of ln((1/ε r )-(1/ε m )) vs ln(T -Tm) at 1kHz for of sintered ceramics.

Figure 5 .

 5 Figure 5. P-E hysteresis loops: (a) NKNSLT +0.6% CuO and (b) NKNSLT +1% CuO.

Figure 6 .

 6 Figure 6. Variation of real part of impedance (Z ′ ) of NKNSLT +xwt.% CuO with frequency at different temperature.

Figure 7 .

 7 Figure 7. Variation of imaginary part (Z ′′ ) of impedance of NKNSLT +xwt.% CuO with frequency at different temperature.

Figure 8 .

 8 Figure 8. Complex impedance plots (Z ′′ vs. Z ′ ) of (a) NKNSLT +0.6%CuO and (b) NKNSLT +1%CuO at different temperature.

′

  Na and V ′ k , respectively), V o •• is the presence of oxygen-ion vacancy, e ′ is the electron released or captured, and Cu 2+ can replaces the Na + or K + -site vacancy (designated by Cu Na • and Cu k • , respectively).

Figure 9 .

 9 Figure 9. Variation of imaginary part of modulus (M ′′ ) with frequency.

Figure 10 .

 10 Figure 10. Frequency dependence of ac conductivity vs frequency at different temperatures.

Figure 11 .

 11 Figure 11. Arrhenius diagram of relaxation times, Ln(τ Z ′′ ) as a function of reciprocal temperature.

Table 1

 1 Grain resistance and capacitance obtained from Nyquist plot (Z ′′ vs Z ′ )

		NKNSLT +0.6%CuO	NKNSLT +1%CuO
	Temperature	Rg ( )C g ( F )R g ( )C g ( F )
	400 • C	54000	3,37E-09	80000	5,00E-09
	425 • C	30000	2,56E-09	50000	4,44E-09
	475 • C	16000	3,13E-09	24000	3,21E-09
	500 • C	12500	2,67E-09	16000	3,13E-09
	525 • C	10000	1,67E-09	12000	2,38E-09
	550 • C	8000	1,56E-09	9500	2,11E-09