(First received \1arch 6th, 1990; revised manuscript receivcd June 26th, 1990) (E)-Urucanic acid, a major metabolite of histidine [1,2] and a naturally occur ing sunscrccn, accumulates in the skin and is eventually excreted in sweat [3,4]. This compound, which undergoes E ;;= L photoisomerization, is one of the major UV light absorbers in the epidermis [5,6]. Recent studies, however, have revealed additional photobiological propertics, i.e., pholochemical binding to DNA and photoimmuno suppressive activity for the Z isomer [7][8][9][10]. Thcse observations rnle againsl the use of urocanic acid as a sunscreen and led us to undertake a study of urocanic acid long chain esters.

--

The separation of the E and 'L isomers of urocanic esters has a twofold interest: photochemical. as it is necessary for the study of E � Z isomerization and of the possible photocycloadditions; and biological, for the study of the inlluenoe of the configuration on the immunosupprcssivc propcrtics.

J\lthough separations of urocanic acid [11][12][13][14] and methyl urocanate isomers have bcen rcportcd [151, thosc of for long-chain esters have not. Wc thcreforc sel out to separate the E and Z isomers of n-dodecyl urocanate and to determine the propor tions of mixtures of these compounds by chromatography.

EXPERIMENTAL

Instrumentation and procedure

Photoisomerization reactions were carried out in a Rayonet-type rcactor (New England Ultraviolet) at 254 nm using 10-ml quartz tubes on a rotating rack.

Thin-layer chromatography (TLC) was done on 2.5 x 7.5 cm silica gel 60 À plates (250-µm layer) (Whatman) with UV detection al 254 nm. The solvent was chloroform methanol (95:5, v /v).

A 20 x 2 cm I .D. wlumn of silica gel (250--400 mesh) (Fluka) was used for the preparativc scparation of the L isomer from a 160-mg mixture of Z and E isomers (ZE � 75:25) using chloroform-methanol (95:5, v/v) as clucnt. The elution was followed by TLC. The first isomcr appcarcd aftcr 75 ml of eluent and was pure in the Lwelve following 3.5 ml-fractions collected.

High-performance liquid chromatography (HPLC) wa;; performed using a sys tem consisting of a 'Vlillipore Waters unit (Modcl 510) with an automatic gradient controller, a Waters Assoc. 990 photodiode array detector and a Waters Assoc. pPo rasil (10 µm) column (30 cm x 4 mm I.D.). The dectection wavelength was set at 310 nm. where hoth isomers absorb. The eluenl was chlorofurm-ethanol (HPLC grade) (85:15. v/v) aflcr having bccn fîltcrcd ovcr Millipore membranes (0.22 µm). The flow ratc was of O. 7 ml/min.

UV spectra were recorded on a Hewlett-Packard HP 8451 A speclropholom eter. lR spectra were recorded on an FT-IR Perkin-Elmcr 1760 X spcctrophotomcter. NMR spectra werc recorded in deutcrochloroform on a Bruker AC 80 apparatus.

Chemica/s

(E)-Dodecyl urocanatc was obtaincd by esterifi cation [! 6] of urocanic acid (Al drich). lts purity was checked by TLC ( R ,. = 0.21 ).

Photoisomerization was achieved by a 2-h irradiation, at 254 nm of a 6.5 • 10-3 mol 1 -1 solution of 160 mg of (E)-dodccyl urocanatc in 80 ml of butanol at 35 °C.

After having evaporated the solvent, the separation of the Z isomer was per formed by colum chromatography using the procedure described before. The twelve fractions containing the isomer cluted first wcrc evaporatcd to give l 10 mg of a solid of m.p. 61 °C (uncorrected). This compound obtained inca. 90% yield, was identified as the Z isomer by TLC [R,. (E) = 0.21: R F (Z) = 0.44] and 1 H NMR spectroscopy: -CH=CH-COO(CH 2 ) 11 CH 3 : E isomer, JH, = 6.73 ppm, JHp = 7.58 ppm, J(H,.Hp) = 16 Hz; Z isomer, bH, = 5.60 ppm, bH p = 6.83 ppm, J(H,,Hp) = 13 Hz.

The UV spectra were recorded in hutanol: Eisomer, Â m ., = 292 nm, c = 19 200 1 mol-1 cm -1 ; Z isomer, i m ,, = 302 nm, e = 17 100 l mol-1 cm -1 .

The IR spcctra wcre rccordcd in carbon tetrachloride: E isomer, v NH (free) = 3470 cm-1 (sharp band), v,rn(bonded) = 3200-3425 cm-1 (intermolecular); Z iso mer, v ,.,H (bonded) = 3258 cm 1 (intramolecular hydrogen bond).

RESUL TS AND DISCUSSION

The HPLC separation of an artificial mixture of (Z)-and (E)-dodecyl urocanate is shown in Fig. 1. Under the conditions described above, the elution limes were 15.25 min and 6.75 min for the Z and Eisomers, respectively. The weaker retention of the Z isomcr is probably related to the strong intramolecular hydrogen bond in this mole cule, which is readily seen in infrared spectroscopy. This bond, already reported for methyl urocanate [17], is not disrupted in the long-chain ester.

fhe pcaks obtained in HPLC were assigned by analysing the previously identi fied pure isomers separately. Quantitative HPLC was thcn undcrtaken. Calibration was pcrformcd by plotting known concentrations of each isomer against peak area. Unlike urocanic acid [18], the isomers did not show a deviation from the Beer Lambert law in the concentration range tested (seven concentrations from 4 • 10-5 to 5 -10-3 mol 1 -1 ). This HPLC method was tcsted using two artificial mixtures of Z and E isomcrs, whose proportions were measured by weighing and chosen in the range cxpected in photochemical experiments (45% ,( Z ,( 75%). The results ob tained, given in Table J, illustrate the precision of the method. Having confirmed the precision of the HPLC analysis for these compounds, we set out to analyse the mixture obtained after a 2-h irradiation of a 6.5 • 1 o- We lherefore decided to use HPLC to determine the photostationnary state of the photoisomerization of (E)-dodccyl urocanate in bulanol (6.5 • 10-3 mol l-1 ), and found that it was reached after I h of irridiation and corresponded to the ratio Z:E = 2.6 ± 0.02.

CONCLUSION

The preparative method for the separation of (Z)-and (E)-dodecyl urocanate dcscribcd has provcd to be a readily applicable technique that allows the rapid pro duction of large amounts of these two isorners. This will ena ble us to continue the sludy of the biological activities of these compounds, for instance the irnmunosup pression phenomenon. The quantitative analysis by HPLC of E and Z isomer mix tures greatly facilitates the study of the photoisomerization of dodccyl urocanatc in a wide range of solvents and concentrations.

7 minFig. 1 .

 71 Fig. 1. JJPLC separation of an artificial mixture of (Z)-and (li)-dodecyl urocanate isomers. Column, 11Porasil (JO /Lm) (30 cm x 4 mm l.D.); detection, absorption al 310 nm; mobile phase, chloroform-ethanol (85:15); fl.ow�rate, 0.7 ml/min� tempcrnture = 20 °C; injection: 15 µ1 of an approximalcly 10 3 mol l 1 solution. Values on the y-axis arc absurhancc units.

  (E) AND (Z)-DODECYL UROCANATE Mixture Z:E (weighing) Z:E (HPLC)" 1 (Z:t.' "° 3) 3.33 ± 0.03 3.35 + 0.02 2 (Z:F. "° 1) 0.92 _l_ 0.01 0.91 ± 0.01 " Values ± S.D. (n = 4).

Separation and determination of Z and E isomers of dodecyl urocanate

	:\!ARJE CARMEN MONJE. ARMAND LATTES and MONIQUE RIVIÈRE*

luhora!UÎrr! IA1 RC P, URA CNRS 470, Univt'rsilé Puul Sahalier. 118 route d{' lVarhonne, F-31062 Toulouse Cedex (France)

TABLE 1 TEST

 1 OF HPLC ASSAY OF

ACKI\OWI.EDGEMENT

The authurs lhanks J. C. Ader for helpful technical collahoration with the HPLC experiments.


N Baudin
email: baudin.na@gmail.com

C Colin

P Ruyer

J Sebilleau

Turbulent flow and transient convection in a semi-annular duct

Keywords:

Turbulent flow and heat transfer in an annular geometry have been previously studied experimentally or numerically. Velocity and temperature profiles have been measured and correlations for the wall shear stress and heat transfer have been derived. However there exists no study in turbulent flow for a semi annular geometry. This work aims to study steady and transient convection in a semi annular test section for a wide range of Reynolds numbers from 10,000 to 60,000, the inner cylinder being heated by Joule effect. The velocity profile in the symmetry plane is measured by Particle Image Velocimetry and the temperature of the inner heated cylinder is measured by infrared thermography. The experimental re sults are complemented by numerical simulations which give also access to the velocity and temperature profiles in the whole test section.

These results are compared to those obtained in an annular geometry for the same inner and outer cylinders radii and an equivalent flow rate. The velocity and temperature profiles and the wall shear stress are the same as in an annular section in an angular sector of p/2 around the symmetry plane. Both velocity and temperature profiles follow a logarithmic law. In steady convection, the local heat transfer has been characterized in several azimuthal positions. The local Nusselt number can be expressed versus a Reynolds number based on the local friction velocity. Characteristic thermal boundary layer thicknesses are also defined. Finally, transient convection tests are performed with a square power generation. The wall heat transfer and the evolution of the liquid temperature near the wall have the same self similar evolution, with a characteristic time scale, which only depends on the flow Reynolds number.

Introduction

The core of a Pressurised Water Reactor (PWR) contains thou sands of cylindrical fuel rods. Between them, some control rods allow controlling the nuclear reaction. All these rods are immersed in high pressure water that acts as coolant and moderator fluid. If a control rod is accidentally ejected, a local and sudden reactivity increase happens leading to a reactivity initiated accident (RIA). It leads to an increase of the rod temperature with first convective heat transfer and then boiling heat transfer. There is still a lack of knowledge for the rod's clad to coolant heat transfer in those transient conditions. The present study takes part of the research program defined by IRSN (Institut de Radioprotection et de Sûret e Nucl eaire) on this topic.

For a RIA, the wall to fluid heat transfer till the onset of nucleate boiling occurs over a very short time scale. Therefore, only the very near wall flow and heat transfer has a relevant impact on the fuel rod behavior. The transient is thus studied only around a single rod surrounded by a vertical flow. The latter is idealized as a flow in an annular cross section with equivalent hydraulic diameter of the typical sub channel defined between neighboring rods (in yellow in Fig. 1). To conduct experiments in a simple geometry but similar to this sub channel, an experimental set up was designed by Visentini et al. [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]. To measure the wall temperature with a high temporal and spatial precision, the infrared thermography is used and so an optical access to the wall is needed. Therefore a half annular cross section with the heated inner half cylinder has been preferred to the annular one. Before the boiling incipience, heat is transferred from the wall to the liquid by single phase convection. It is thus necessary to characterize the velocity and temperature fields and heat transfer in a semi annular geometry. Depending on the power generation in the heated wall, the con vection is either steady or transient. In the literature, the studies concerning heat transfer in annular sector ducts mainly concern heat exchangers. The cross section (Fig. 2) is characterized by the inner and outer wall radii r i and r o and the apex angle g. Sparrow et al. [START_REF] Sparrow | Laminar flow and pressure drop in internally finned annular ducts[END_REF], Soliman et al. [START_REF] Soliman | Laminar heat transfer in annular sector ducts[END_REF] and Ben Ali et al. [START_REF] Ben-Ali | Further results for laminar heat transfer in annular sector and circular sector ducts[END_REF] studied heat transfers in sector annulus with apex angles from 5 to 350 . However, these studies are all in laminar flow. Several studies were also performed in turbulent flows with different inner and outer radii, apex angle, Reynolds number Re, fluids, heated walls (Table 1). Tao et al. [START_REF] Tao | Experimental study on developing and fully developed fluid flow and heat transfer in annular-sector ducts[END_REF] and Li et al. [START_REF] Li | Numerical simulation of fully developed turbulent flow and heat transfer in annular-sector ducts[END_REF] studied turbulent flow and heat transfer in sector annulus with apex angles from 18 to 40 . The boundary conditions for the heat transfer are different from ours because they heat the outer wall. Li et al. found that the friction factor increases with the apex angle and varies from 0.03 to 0.02 when the Reynolds number Re ranges from 10,000 to 50,000 for a 40 apex angle. Another geometry of interest is the annular geometry, for which there are a lot of studies. Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF] conducted experiments for isothermal and heated turbulent upward flow in a vertical annular channel. The inner part is heated with a steady heat flux and the fluid is a refrigerant R 113. They measured liquid velocity profiles with laser Doppler velocimetry and the temperature field with a cold wire. Using the same test section, Hasan et al. [START_REF] Hasan | Heat transfer measurements in turbulent liquid flow through a vertical annular channel[END_REF] deduced a correlation for the Nusselt number Nu versus the Reynolds number Re and the Prandtl number Pr: Nu 0:0106Re 0:88 Pr 0: 4 (1)

It is known that the correlations in convection depend on the

T þ Dimensionless temperature, T þ TwÀT l T f t c
Characteristic time to establish stationary convection, (s)

T f
Friction temperature, T f

f w rCpu à i , (K) u Axial velocity, (m s À1 ) u',v',w' Turbulent velocities, (m s À1 ) u * Friction velocity u à tw r q , (m s À1 ) u þ Dimensionless axial velocity, u þ u u à u m Mean velocity u m Q/S, (m s À1 ) y
Radial position from the inner wall, y r r i (m) working liquid: water, liquid metal or refrigerant. Since we also use a refrigerant in our experiments and a similar heating of the inner cylinder, eq. ( 1) will be used for comparison with our results. Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF] developed a model for a turbulent flow and convection in an annulus, validated by many numerical results and the experimental results of Rehme [START_REF] Rehme | Turbulent flow in smooth concentric annuli with small radius ratios[END_REF]. The mean velocity u m is defined with the flowrate Q by u m Q/S, S being the surface of the cross section of the duct. The Reynolds number of the duct writes Re rumD h m where r is the fluid density, m the dynamic viscosity and D h the hydraulic diameter. The local velocity and distance to the wall y are scaled by the friction velocity u * :

y þ Dimensionless position, y þ ryu à m Subscripts q At a
u þ u u à u r t w r (2)
y þ ryu à m rðr r i Þu à m (3)
r being the radial position. The wall shear stresses t w can be calculated from the mean Fanning friction factor f F (Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF]):

f F t w;m 1 2 ru 2 m (4) t w;m r i t w;i þ r o t w;o r i þ r o (5) t w;i t w;o r o r i r 2 0 r 2 i r 2 o r 2 0 ! (6)
r 0 being the position where the total shear stress is equal to zero. It is calculated according to Rehme [START_REF] Rehme | Turbulent flow in smooth concentric annuli with small radius ratios[END_REF]:

r 0 r i r o r 0 r i r o 0:386 (7)
For the Fanning friction factor, Kaneda et al. propose the following equation:

2 f F s 1:61 þ 1 0:436 lnðReÞ 550 Re fF 2 q 1 0:436 ln 2 f F s ! (8)
With eq. ( 8), we calculated that the friction factor in an annular duct is much lower than the one in a sector annular duct, where there is a strong influence of the side walls. For the dimensionless velocity, Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF] propose the following log law:

u þ A þ 1 k ln y þ (9)
where they suggest to take A 6.13 and to set the Von Karman constant k 0.436.

To our knowledge, there is no study of turbulent convection in a semi annulus while the annular geometry is well documented. The objective of this work is to characterize flow velocity and heat transfer in a semi annular geometry and to derive scaling laws for the wall friction and heat transfer in different azimuthal positions for a wide range of Reynolds numbers. The velocity field in the symmetry plane is measured by PIV for a wide range of Reynolds number Re. The temperature of the heated wall is measured by Infrared Thermography in the symmetry plane. 3D numerical simulations with the software STAR CCM þ are also performed and allow to quantify the velocity and temperature profiles in the whole cross section. They also give access to the wall shear stress and wall temperature at all azimuthal positions.

Firstly in Section 2, we describe the experimental set up and the measurement techniques: PIV measurements and infrared ther mography. Because of technical constraints, we can only measure quantities in the symmetry plane of the semi annulus section. To investigate the whole section, we performed numerical simulations with the software STARCCMþ. Thus in Section 3, we present the computation method used and the validation of the mesh by comparison with results of the literature.

In Section 4, the characteristics of the turbulent flow in the semi annulus are presented: mean velocity profiles, turbulent ki netic energy and turbulent shear stress profiles, friction velocity. They are compared to results obtained in an annular cross section and to correlations of the literature. The flow is also characterized at different azimuthal positions. In Section 5, the steady convection is studied. The temperature profiles in the liquid are determined and characteristic thermal boundary layer thicknesses are defined. The heat transfer coefficient is evaluated and compared to the one obtained in annular flow. Then a local heat transfer coefficient depending on the azimuthal position is also determined. Finally the transient convection is studied and a similarity law for the evolu tion of the wall temperature and the development of the thermal boundary layer thickness is proposed.

Experimental set-up and measurements techniques

In this section, we introduce the main features of the facility. More details can be found in [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]. The test section has a semi annular section. The inner half cylinder consists of a 50 mm thick stainless foil heated by Joule effect. Its diameter is 8.4 mm and its length 200 mm. The outer part consists of a 34 mm internal diameter glass half cylinder (Fig. 3). The semi annular section is filled with a coolant 1 methoxyheptafluoropropane (C 3 F 7 OCH 3 ), which will be referred as HFE7000 (3 M). This fluid has been chosen because of its low saturation temperature (35 C at atmospheric pressure), its low latent heat of vaporisation (ten times smaller than the water one), and the smaller critical heat flux value compared to water. These characteristics have been chosen to perform highly transient boiling experiments with a lower power than required for water. The main physical properties of HFE7000 at saturation temperature and atmospheric pressure are given in Table 2, where r, C p , l and n are the density, heat capacity, thermal conductivity and kinematic viscosity, respectively.

The metal half cylinder is glued to two lateral quartz glass plates (thickness: 3 mm, width: 42 mm, length: 200 mm). The plates, together with the foil, are placed in an aluminum cell. In this cell, a visualization box filled with HFE7000 is used to reduce optical distortions (Fig. 3). The experiments are carried out in flow boiling conditions by inserting the test cell in a two phase flow loop con sisting of a gear pump, a Coriolis flow meter, a preheater, a 1 m long channel upstream the test section to establish the flow and a condenser.

The electrical power is provided by a power supply SORENSEN SGA. It covers the 0e40 V voltage range and the 0e250 A current range. The power supply is driven by an arbitrary generator where all kind of signals can be programmed. The power supply can provide a current increase up to 45 A s À1 . A voltmeter and an ammeter measure the tension U and current I to calculate the generated power in the metal foil P UI . Particle Image Velocimetry measurements are carried out in the symmetry plane of the test section (Fig. 4). The laser used is a "Nd:YAG" QUANTEL TWIN ULTRA with a wavelength of 532 nm and an energy of 30 mJ for two pulses of 7 ns at 15 Hz. The time interval between the 2 pulses of the laser is varied between 80 and 140 micro seconds depending on the flow velocity. Images are recorded by a Sensicam PCO camera with an optical MACRO NIKON lens of 105 mm. The resolution of the CCD sensor is 1376 Â 1040 px 2 . Methyl Metacrylate spheres with encapsulated Rhodamine B are used for seeding the flow. They have a mean diameter of 10 mm. The Stokes number associated to these particles in the flow has been evaluated. It is equal to the ratio of the response time of the seeding particles of density r p : t p r p Ãd 2 p =m 0.1 ms over the response time of the flow t f le/u * 0.3 s, where le is the integral length scale. The Stokes number is below 0.05 for our experiments.

The mean velocities and RMS values are calculated with the software Davis. They are calculated with cross correlations in three passes, first on 64 Â 64 px 2 windows to finish with 16 Â 16 px 2 windows, with an overlap of 50%. The convergence of mean and RMS values of the velocity is reached with about 1000 images. The optical distortions are negligible near the inner cylinder but are important near the outer one. The experimental velocity distribu tion is thus not studied in this area.

The metal foil is painted in black in order to increase its emis sivity (the emissivity of the paint is 0.94). Thermal measurements are performed with an infra red camera looking at the foil from backward (see Fig. 3). A CEDIP JADE III camera with a sensitivity range between 3.

C p (J kg 1 K 1 ) l (W m 1 K 1 )
n (m 2 s 1 ) 1385.8 1300 0.075 3.2 10 7

Fig. 4. Schema of PIV measurements.

Due to the very low heat transfers between air and the back of the foil, this surface is assumed adiabatic. The thermal gradient across the foil thickness is expected to be negligible, since the Biot number Bi characterizing the thermal resistance of the foil by comparison to the thermal resistance of the flow is very small. For a high heat transfer coefficient characteristic of convection h 1000 W m À2 K À1 , the Biot number is equal to Bi h.e w / l w 0.003, e w being the foil thickness and l w its thermal conduc tivity. Due to the small value of Bi, in the following the wall tem perature measured by the infra red camera at the backside of the metal foil will be considered equal to the temperature of the wall in contact with the fluid. Moreover, the diffusion time across the foil thickness, e 2 w =a w 0.6 ms, is actually small enough to assume that there is no time lag between the thermal evolution of both sides of the metal foil. e w is the metal foil thickness and a w 4.08$0.10 À6 m 2 s À1 its thermal diffusivity. Thus, it is possible to measure the temperature of the wall that is in contact with air, in order to know the temperature of the wall that is wetted by the refrigerant.

The heat flux transmitted to the liquid is calculated from the power generated in the metal foil by Joule effect f gen and from the energy stored in the foil itself. A calculation of the 2D transient heat conduction in the metal foil and the quartz plates has been per formed using COMSOL Multiphysics and showed that there are limited losses toward the quartz planes (Visentini [11]). It is possible to estimate the heat flux f w transferred to the fluid by a simple thermal balance:

f w f gen r w $C pw $ dT w dt $e w (10)
where r w 7930 kg m À3 , C pw 500 J kg À1 K À1 and dTw dt are the density, the heat capacity of the foil and the wall variation of temperature, respectively. In this article, the temperature and the heat flux measurements are averaged on a 1 cm height by 2 mm width area.

A National Instruments box is used to acquire the thermocou ples signals. Three K type thermocouples measure the HFE7000, air, and infrared camera temperatures.

The uncertainties of the electronic devices (ammeter, National Instrument box) are very low and of the order of magnitude of 0.1% (Visentini [START_REF] Visentini | Etude exp erimentale des transferts thermiques en ebullition transitoire[END_REF]). The power generated by Joule effect per surface unit f gen is equal to U :I /L.l where L the length of the foil and l its width. The main uncertainty is on the length of the part where the voltage is measured (about ±0.2 cm on 8.5 cm i.e. about 2%). Thus the total uncertainty on f gen is 2.5%. The heat transfer from the wall to the fluid is calculated with eq. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF]. Assuming that r w Cp w e w 198.3 J m À2 has no uncertainty, the absolute uncertainty on the heat flux is:

Df w f w Df gen f gen $ f gen f w þ r w Cp w e w DðdT=dtÞ f w ( 11
)
The ratio f gen /f w never exceeds 2 in the experiments. In steady state, the uncertainty on the heat flux is below 5%. In transient convection, the heat flux is generally above 10 kW m À2 and the uncertainty on dT/dt is less than 2 K s À1 , so, the absolute uncertainty on the heat flux is below 10%.

Numerical simulations: details and validation

Simulations are conducted with the software STAR CCM þ to investigate the velocity and temperature fields in the whole test section. First the numerical model and the mesh grid are presented. The validation results are provided in the corresponding result sections of the present paper. The numerical simulations are first validated in a 2D axisymmetrical configuration by comparison with experimental and theoretical results in annular flow. Finally 3D numerical simulations in a semi annular geometry are performed in adiabatic conditions, in steady and transient convection.

The flow is modeled in two regions. The turbulent flow is computed by using a k ε model far from the walls and a low Reynolds model near the walls. The space scheme used is an hybrid Gauss LSQ gradient method. The simulations of transients are computed with a 2nd order implicit scheme in time, with a time step of 1 ms. To reproduce the experiment, the metal foil is modeled and the Joule power generation is simulated by a volu metric heat source. The time step was validated by transient sim ulations of conduction with comparison with the analytical result from Carslaw and Jaeger [START_REF] Carslaw | Conduction of heat in solids[END_REF]. The numerical simulations are per formed with a wall heated by Joule effect corresponding to heat flux of 20,000 W m À2 on one side and an adiabatic wall at a distance of 13 mm.

A polyhedral mesh is used in the cross section with cells size about 0.1 mm in the center region. This mesh is plotted in Fig. 5 for the semi annular geometry. Due to symmetry consideration, only one half of the cross section is meshed. Face a is a symmetry axis, faces b, c and d are wall boundary conditions. Close to the walls (b, c and d) for y þ < 60 a refined mesh consists of n cells with a stretching of 1.1 (according to the user guide), n being between 20 and 60 (Fig. 6). For 2D axisymmetrical simulations a slide of the computation domain between two symmetry planes is used. The refined mesh in the two wall regions is also used. This mesh has been validated by comparison of numerical simulations with experimental results of Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF]. In the flow direction, the cross sectional mesh is extruded into 100 layers on 1 m. This length has also been extended to 2 m depending on the establishment length of the flow. A convergence study has been performed. For a cell size of 0.1 mm in the center region, the results are sensitive to the number n of cells in the near wall region. For n > 20, the results are converged.

Concerning the flow computation, the mass, momentum, en ergy balance equations can be solved in decoupled or coupled manner. The complete formulation can be described as using a co located variable arrangement (as opposed to staggered) and a Rhie and Chow type pressureevelocity coupling combined with a SIMPLE type algorithm. k ε turbulence model is used in the most part of the flow except close to the wall (y þ < 60) where a low Reynolds model is used. The time step is taken equal to 0.001 s with Fig. 5. Mesh of the cross section.

10 iterations per time step. The convergence criteria is a critical value of the residus smaller that 0.01 on mass, energy and dissi pation rate of the TKE and 0.001 on the other variables. Computa tion are performed on a PC with 8 cores Xeon(R) E5 1650 0 @ 3.20 GHz. The computation time ranges between a few hours (for 2D axisymmetrical simulations) to a few days (3D simulations).

Numerical simulations are first validated by comparison with the experimental results of Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF] with R113 in an annular geometry with inner and outer radii r i 7.9 mm and r o 19 mm. Fig. 7 shows a comparison of the velocity profiles for experiment 4 with a Reynolds number of Re 31,500 and two adiabatic walls. The mesh previously described is used for the simulations. There is a good agreement between the simulation, the experiment and the log law proposed by Kaneda (eq. ( 9) with u * calculated from eqs. ( 6)e( 8)). In the central region (0.003 < r r i < 0.007), we observe a slight difference between the model and both the experimental and simulated data. In the theory, this region is simply the junction of two wall laws that could explain this difference. In the near wall region, we can observe a slight difference between experimental results and both theory and simulations. The origin of this differ ence may reside in the difficulty to make accurate measurements near the wall.

For the convection simulations, the turbulent Prandtl number is set to 0.9 as deduced from Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF] with experimental data in an annulus. To estimate the importance of thermal convection compared to the forced convection, we calculate the Richardson number, which is ratio of the Grashof number Gr gbDTD 3 h n 2 over the Reynolds number squared. g is the gravity acceleration, b the isobar expansion coefficient and DT a characteristic difference temperature between the wall and the liquid. In the experiments, the ratio Gr/Re 2 is between 0.001 and 0.1. Thus there is no signifi cant buoyancy effect. Simulations with Boussinesq's approximation also show that buoyancy effect is negligible.

The outlet fields calculated in adiabatic flow simulations are used as inlet and initial conditions for the simulations with wall heat transfer. In these calculations, only the 20 cm heated length is simulated, and is discretized in 500 layers in the axial direction. The wall to fluid heat flux is set to f w 20,000 W m À2 and the inlet temperature is set to 300 K. These values are averages from the experimental values.

To validate the convective simulations, we use again the results from Kang et al., in particular the experiment 5. The wall to fluid heat flux f w 9000 W m À2 is the boundary condition and the Reynolds number is Re 22,800. The inlet temperature is equal to 316 K and the initial temperature of the fluid and the wall is also equal 316 K. Fig. 8 shows temperature profiles from experiment 5 in Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF] and simulation. As for velocity fields, the agreement between the numerical simulations and the experimental results is good. Additional numerical simulations were also performed in a 3D geometry with a mesh similar to Fig. 5, between 2 symmetry planes a and c. The results are in very good agreement with those of the 2D axisymmetrical simulations. The mesh is well adapted to the computation for both velocity and temperature fields. The same method of meshing is adopted to simulate the semi annular flow.

Turbulent flow in a semi-annulus

To analyse the wall to fluid heat transfer, the study of the ve locity distribution is needed. Experimental and numerical results in the symmetry plane are first compared. Then we study the shear stress and velocity field in the whole cross section.

We want to compare our results in a semi annular geometry with the results in the annular geometry from Kaneda et al.'s theoretical. To have an equivalence, we choose to keep the same inner and outer radii r i and r o and so, to vary the flowrate Q. The equivalent flowrate Q eq for the annular geometry is calculated to have the same maximum velocity u max as in the symmetry plane of the semi annular geometry. The parameters corresponding to different experimental runs are gathered in Table 3. The value of the ratio Qeq Q is constant (around 2.2) for Reynolds number values larger than 15,000. Simulations of the flow were computed with HFE7000 properties and it seems that the ratio Qeq Q is independent from the fluid used. It may be only a geometry characteristics which can be explained as follows: the annulus area is twice the semi annulus (r i 7.9 mm, r o 19 mm, Re 22,800, f w 9000 W m 2 , T inlet 316 K), experiment 5

from Kang et al. [START_REF] Kang | Isothermal and heated turbulent upflow in a vertical annular channel -part I. Experimental measurements[END_REF].

area so, to have the same maximum of velocity, the flow rate must be at least twice too. Moreover, out of the symmetry plane in the semi annular geometry, and due to the friction against the lateral walls, the velocity magnitude is always less than the maximum in the symmetry plane. Since in an annular geometry the velocity profile is the same everywhere, the equivalent flow rate must be larger than the semi annular flow rate. The ratio Qeq Q is not far from 2 so we can guess that a great part of the semi annulus has a velocity profile very similar to the symmetry plane profile.

The mean axial velocity u and the axial and radial fluctuating velocities u 0 and v 0 are measured in the symmetry plane of the semi annular geometry by PIV for different Reynolds numbers, as previously described. The numerical simulations are performed by STAR CCM þ using the mesh displayed in Figs. 5 and6. In the streamwise direction the computation domain is 1.4 m long with a mesh length of 1 cm. Fig. 9 shows the axial velocity profiles for the Reynolds numbers 8700, 17,400 and 26,500. The experimental, theoretical (eq. ( 9)) and simulated profiles are similar, with absolute errors less than 5%. There is the same asymmetry of the profiles in annular and semi annular geometries with a shift of the radial position of the maximum axial velocity toward the inner wall. There is only a slight difference because the semi annulus profile is a little less shifted. The profiles of the simulated and theoretical total shear stress (Fig. 10) are similar too. The inner and outer wall shear stresses in the equivalent annular geometry well model the ones in semi annular geometry. The location of the zero in the total shear stress is also shifted toward the inner wall with the same difference between annular and semi annular geometry. The difference be tween the inner and the outer wall shear stresses explains the asymmetry of the velocity and total shear stress profiles.

Fig. 11 shows experimental and simulated profiles of turbulent kinetic energy k e 0.5〈u 02 þ v 02 þ w 02 〉, with u 0 , v 0 and w 0 the axial, radial and azimuthal turbulent velocities, and <> is the symbol for time average. We did not measure the azimuthal component of the turbulent velocity so, to calculate k e we assume orthotropy be tween radial and azimuthal velocity fluctuation and therefore w 0 v 0 .

The wall shear stress can be deduced from the velocity profiles. Due to a slight uncertainty on the inner wall position, a log law is assumed with the same parameters as in Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF] (eq. ( 9)) and then an optimization to have the log law in a range of y þ be tween 30 and 100 is conducted. The values obtained are given in Table 4. The theoretical friction velocities at the inner wall u à i and outer wall u à o are calculated with eq. ( 6). The experimental, theo retical and simulated data are in good agreement. The simulated wall shear stress is given by the software STAR CCMþ and not calculated with the theoretical parameters: the good agreement is an other way to validate the simulations. For the both sides, the equivalent annular flow has the same shear stress as the semi annular flow. The simulations and theory show that the inner wall shear stress is more important than the outer one. We remind that u * depends on the fluid characteristics and the Reynolds number, instead of the ratio u * /u m which only depends on the Reynolds number. Fig. 12 shows the dimensionless velocity u/u * . The simulated and experimental profiles are well superimposed with the log law in the y þ range [30, 300]. The equivalent annular friction velocities are similar to the semi annular ones, for the inner and the outer walls. u à i;exp is determined by fitting the experimental velocity profile with the log law (eq. ( 9)) for A 6.13 and k 0.436, as the theory, and so the data are superimposed. It is important to see that the log laws for the simulations are obtained independently, and the agreement with the other curves is also good. Fig. 13 shows isocontours and radial profiles of the axial velocity at different azimuthal angles q. We choose the value of the angle q equal to 0 for the wall (x axis) and p/2 for the symmetry plane (y axis). The velocity profile does not vary a lot in ±p/4 sector around the symmetry plane. For angles larger than p/4, the profiles tends to symmetrize, i.e. the radial position of the maximum velocity shifts toward the center and the inner and outer wall shear stresses tend to be equal. When the angle tends to q 0, i.e. near the walls, the velocity decreases due to the friction. There is the same trend with the inner friction velocity (Fig. 14): the friction velocity does not vary in the quarter near the symmetry plane and decreases near the wall. Since the fluid velocity is higher in the center of the duct, the shear stress is more important in this area.

Integrating the wall shear stress in the cross section, we calcu late the pressure gradient along the duct. The Darcy friction factor f D can be evaluated with:

f D dp dz D h 0:5ru 2 m ( 12
)
The Darcy friction factor f D is four times the Fanning friction factor f F and these two definitions are used in function of the context. On the left hand side of Fig. 15 the friction factors f D are plotted versus the Reynolds number for different geometries and compared to Blasius correlation for a cylinder f D 0.316Re À0.25 . The friction factor in an annulus and a cylinder are equal, as shown by Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF]. The friction factor is also similar between the semi annular and annular sector geometries. However, there is a difference between the sector annular geometries (g 40 ) and the annular or cylindrical geometries. The additional walls (in com parison with the cylindrical or the annular geometry), induce a different friction even with the same Re number value based on the flow velocity. Using the equivalent Reynold number (based no the maximal velocity as defined in Table 3), the different friction factors obtained for the different geometries agree better. These walls slow down the flow, and it is thus relevant to define an equivalence to compare annular and semi annular geometries. That is why we plot the friction factor versus the equivalent Reynolds number Re eq (Table 3) for annular and cylindrical geometries on the right hand side of Fig. 15. In this figure, all curves better superimpose. As a partial conclusion, the flow in our test section at a given Reynolds number value is similar, in an angular sector of ±p/4 around the symmetry plane, to a flow of quasi twice the Reynolds number value in an annular test section of same inner and outer radii. Being an advantage to model this part, it is also a validation of the simplification from the nuclear core geometry. Moreover, the results from the wall shear stress study will help the analysis of the wall heat transfer.

Steady convection

In this part we study the steady convection in a semi annulus with an imposed heat flux at the inner wall in a test section of 20 cm long, with an imposed heat flux of 20,000 W m À2 . This heated section is placed downstream an adiabatic section of 1.4 m long. The flow is steady and established, so we can use the results from the previous part as inlet condition in the heated test section. We study the temperature and heat transfer far from the inlet of the test section, where thermal and dynamic boundary layers are established. It corresponds to a length of about 13 cm from the bottom of the heated part, i.e. z 7D h .

We are first interested in the temperature distribution in the symmetry plane of the test section. Fig. 16 shows the radial tem perature profiles in the liquid for different values of the Reynolds number. Most part of the liquid is at the bulk temperature T b and there is an heated boundary layer of only 1 mm thick. The bulk temperature does not vary along the duct and is equal to the fluid inlet temperature.

We define the friction temperature T f , by analogy with the friction velocity:

T f f w rC p u à i (13)
The temperature difference between the wall and the liquid scaled by this friction temperature is plotted in Fig. 17 versus y þ and follows a log law (eq. ( 15)).

T þ T w T l T f ðT w T l ÞrC p u à i f w (14)
. 17. T þ (y þ ). T þ A 0 þ 1 k T ln y þ (15)
with A' a constant equal to 43.8 and k T , the Von Karman constant for the temperature equal to about 0.43. We found that the ratio of the friction temperature T f over the temperature difference T w T b is a constant and equal to 0.0176. The Nusselt number Nu is therefore scaled by the friction Reynolds numbers Re à u à i D h =n as:

Nu T f T w T b Pr Re à 0:0176PrRe à (16)
To study the radial distribution of liquid temperature, we define two different boundary layer thicknesses: the first one is the con ventional thermal boundary layer thickness d 99% defined as:

T w T l ðy d 99% Þ T w T b 0:99 (17)
with T w and T l the wall and liquid temperatures respectively. This thickness is scaled by the viscous lengthscale n=u à i and is plotted in the left hand side of Fig. 18. Its evolution can be fitted by the law d þ 99% 0:183Re 0:7 and we see that the values correspond to the upper limit of the log laws validity (for both the velocity and the temperature profiles). It also corresponds to the end of the boundary layer, where the radial heat flux is near zero (for u à i 0.09 m s À1 , d þ 150 corresponds to d 0.5 mm).

If we consider the bulk temperature as reference, the enthalpy flux F h in the symmetry plane is defined as:

F h r l Cp l Z ro r i uðT l T b Þdr (18)
We define the second boundary layer d h as the thickness of liquid carrying the same enthalpy flux while flowing at the mean velocity u m,q 0 and being at T w (the rest of the liquid being sup posed at T b ):

F h r l Cp l ðT w T b Þu m;q 0 d h (19)
With equations ( 18) and (19) we calculate d h :

d h Z ro r i u u m;q 0 1 T w T l T w T b dr (20)
This thickness scaled by the viscous lengthscale n=u à i is plotted in the right hand side of Fig. 18. Its evolution can be fitted by the law d þ h 0:014Re 0:64 and we see that the values correspond to the viscous and buffer layers i.e. d þ < 20. The thickness d h corresponds to the near wall region where the heat flux and the liquid tem perature variations are the largest.

The evolution of the Nusselt number with the Reynolds number is plotted in Fig. 19. The experimental and simulated results are similar, and the variation deducted from the log law (eq. ( 16)) fits the simulation data. We compare these results with two different correlations of the literature: the Dittus Boelter correlation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] for a cylinder and the Hasan et al. correlation for an annulus. Both of them are calculated with the equivalent Reynolds number. Though these correlations overestimate the Nusselt number, they give a good approximation.

Fig. 20 shows the evolution of the Nusselt number with the azimuthal angle q. The heat transfer well estimated by eq. ( 16)

where u à i is considered as being a function of q. It fits the experi mental data on a large angular sector. The heat transfer is the largest in the mid plane of the cross section where the wall shear stress is the largest, and decreases when q goes to 0 or p.

The temperature profiles in steady convection can be modeled by two layers. A layer near the heated wall where the radial heat flux is the largest and whose layer is close to the enthalpy layer thickness d h (which is about 0.1 mm). The second part corresponds to the log law in the velocity profiles. The radial heat flux is smaller and the thickness of this layer corresponds to the more classical boundary layer thickness d 99% which is about 1 mm thick. The temperature in this last part also follows a log law from which we can deduce a relation between the Nusselt number and the wall shear stress. This result is valid for any angle. The simulated results well fit the experimental results and can be also approximated by difference with the experiments. The boundary layer based on the enthalpy is characteristic of the heat transfer at the wall and es tablishes in the same time as the characteristic time of wall tem perature variation.

Conclusions

Velocity and temperature profiles have been characterized in a semi annular geometry in steady and transient convection. In an angular sector of p/2 around the symmetry plane, the velocity and temperature profiles are the same as in an annular geometry. The semi annular geometry is therefore relevant to simulate the flow around a fuel rod in a nuclear core.

To compare the velocity profile in the semi annulus symmetry plane with the results in an annulus, we defined an equivalent flow rate for the 2 geometries with identical inner and outer radii. Since side walls induce additional friction, the flow rate in an equivalent annulus is 2.2 times that in the semi annulus. With this equiva lence, the velocity profile and the wall shear stress are in good agreement with the log law proposed by Kaneda et al. [START_REF] Kaneda | The characteristics of turbulent flow and convection in concentric circular annuli. Part I: flow[END_REF]. In an angular sector of p/2 around the symmetry plane, the velocity profile and wall shear stress remain the same as in the symmetry plane. Near the side walls, the mean velocity and wall shear stress rapidly decrease.

In steady convection, the bulk temperature is constant along the 20 cm of the heated test section and the heat transfer is established after 10 cm from the inlet. The liquid temperature scaled by the friction temperature is well modeled by a log law. The relation established between the Nusselt number Nu and the friction ve locity u à i is valid in the whole cross section. Two characteristic thermal boundary layer thicknesses are defined: d 99% corresponds to the upper limit of validity of the log law, where the liquid tem perature reaches the bulk temperature and where the radial heat transfer is close to zero. d h is a thickness based on the enthalpy flux and it is limited to the viscous and buffer layers near the wall where most of the heat transfer takes place. Transient convection tests were performed with a square power generation. The characteristic time of the transient decreases when the Nusselt number increases. The results of the numerical simu lations are well fitted by an analytical solution. The wall heat transfer and the evolution of the liquid temperature near the wall have the same self similar evolution, which can be easily modeled. This results is of great interest for the study of the onset of nucleate boiling on a wall with a transient heating. It gives the time evolu tions of the wall temperature and thermal boundary layer thickness which are required to determine the onset of nucleate boiling.

l 2 h

 2 Thermal conductivity, (W m À1 K À1 ) m Dynamic viscosity, (Pa s) V Cinematic viscosity, (m 2 s À1 ) f Heat flux, (W m À2 ) F h Enthalpy flux, (W m À1 ) r Density, (kg m À3 ) t Total shear stress, (m 2 s À2 ) Pressure gradient, (Pa m À1 ) A,A 0 Constants Bi Biot number, Bi h ew lw C p Specific heat capacity, (J kg À1 K À1 ) Heat transfer coefficient, h f w TwÀT l , (W m À2 K À1 ) k e Turbulent kinetic energy, k e 0.5〈u 02 þ v' 2 þ w' 2 〉, (m 2 s À2 ) Nu Nusselt number, Nu f w D h lðTwÀT b Þ Pr Prandtl number, Pr n a Q Flowrate (m 3 s À1 ) r Radial position (m) r * Dimensionless radial position, r à rÀri roÀri r 0 Radial position of the zero of total shear stress (m) Re Reynolds number, Re rumD h

Fig. 1 .

 1 Fig. 1. Scheme of fuel rods and sub-channel.

Fig. 2 .

 2 Fig. 2. Scheme of annular sector cross section.

  5 and 5.1 mm is used. It has a focal plane array type and a Stirling cooled MCT detector of 240 Â 320 px 2 . For this study, the acquisition frequency is 150 fps and the height of the metal foil investigated is about 7 cm. The infra red camera is calibrated thanks to a DCN 1000 N4 black body. The uncertainty on the measurement temperature is about ±0.6 C at 20 C and ±0.2 C at 70 C. It takes into account the non uniformity of the black body temperature surface and the noise on the camera sensor.

Fig. 3 .

 3 Fig. 3. Schema of the test section, Visentini et al. [1].

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Refined mesh close to the wall.

Fig. 9 .Fig. 10 .Fig. 11 .

 91011 Fig. 9. Mea, axial velocity profiles: experiments and simulations in a semi-annulus, theory for an annular geometry -Re 8700 in blue, 17,400 in red and 26,500 in green. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Comparison of experimental, simulated and theoretical values for u þ (y þ ): Re 8700 in blue, 17,400 in red and 26,500 in green (a) inner wall region -(b) outer wall region. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14 .

 14 Fig. 14. Normalized friction velocity along the azimuth angle (Re 17,400).

Fig. 15 .

 15 Fig. 15. Darcy friction factor versus the Reynolds number on the left hand side and versus the equivalent Reynolds number on the right hand side.

Fig. 16 .

 16 Fig. 16. Liquid temperature profiles.

Fig

  Fig. 17. T þ (y þ ).

Fig. 18 .

 18 Fig. 18. Thermal boundary layer thicknesses: simulated values and power law fit.

Fig. 19 .

 19 Fig. 19. Comparison between experimental, theoretical and simulated Nusselt numbers versus Reynolds numbers.

Table 1

 1 Studies of flow and heat transfer in sector-annulus.

	Authors	Fluids	g( o )	r i (mm)	r i /r o	10 3 Re	f w (kW m 2 )
	Tao et al. [5]	Air	18 40	9 16	0.25	5 50	Outer wall
	Li et al. [6]	Air	18 40		0.2 0.5	10 50	Outer wall
	Kang et al. [7]	R113	360	7.9	0.415	22 46.4	0, 9, 16 (in. wall)
	Hasan et al. [8]	R113	360	7.9	0.415	22 46.4	0, 9, 16 (in. wall)
	Rehme [9]	Air	360	2 10	0.02 0.1	20 200
	Kaneda et al. [10]	Air	360		0.001 0.999	20 200

Table 2

 2 HFE7000 properties at saturation temperature and at atmospheric pressure.

	r (kg m 3 )

Table 3

 3 Flow parameters.

	Q (L s 1 )	Re	u max (m s 1 )	Q eq (L s 1 )	Re eq
	0.20	8700	0.630	0.44	13,200
	0.31	13,400	0.944	0.67	20,200
	0.40	17,400	1.238	0.894	26,800
	0.51	22,100	1.542	1.122	33,700
	0.61	26,500	1.858	1.360	40,800

Table 4

 4 Friction velocities for all Reynolds numbers.

	Re	u à i;exp (m s 1 )	u à i;th (m s 1 )	u à i;simul (m s 1 )	u à o;th (m s 1 )	u à o;simul (m s 1 )
	8700	0.034	0.034	0.036	0.031	0.032
	13,400	0.048	0.049	0.051	0.045	0.046
	17,400	0.061	0.062	0.062	0.057	0.056
	22,100	0.075	0.076	0.077	0.070	0.070
	26,500	0.088	0.090	0.090	0.083	0.081

Appendix

The wall to fluid heat flux can be deduced from the wall energy balance:

The heat flux and the wall temperature can be expressed in function of T l , which leads to:

Writing the difference between the steady and transient tem perature distributions DT(y,t) T l,stat (y) T l (y,t), eq. ( 23) becomes:

In steady state condition, the wall to fluid heat flux reduces to:

Then, using eqs. ( 24) and (25), the equation with DT becomes:

Assuming that DT is the product of a space and a time function, i.e. DT f(y)g(t), allows to write:

It leads to gðtÞ e À t tc with:

r w C pw e w f ð0Þ lf 0 ð0Þ

(28)

With the initial condition at t 0 on the temperature and the heat flux, and assuming a constant bulk temperature T b :

we finally find the expression for t c :

Actually, DT should be a sum of space and time functions product i.e. DT P f n ðyÞe À t tc;n . However, the evolution of DT in the simulation is only one exponential evolution, so we keep this form for the analytical model of the time constant.