Jorge A Rueda
email: jorge.rueda@icra.it

Y Aimuratov

U Barres De Almeida

L Becerra

C L Bianco

C Cherubini

S Filippi

M Karlica

M Kovacevic

J D Melon

J D Melon Fuksman

R Moradi

M Muccino

A V Penacchioni

G B Pisani

D Primorac

R Ruffini

N Sahakyan ¶ ¶

S Shakeri

Y Wang

The binary systems associated with

Keywords: Gamma-ray bursts, neutron stars, black holes PACS Number(s): 04.30.Tv, 04.40.Dg, 97.60.Bw, 97.60.Jd, 97.60.Lf, 98.62.Mw

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Introduction

There has been a traditional phenomenological classification of GRBs based on the observed prompt duration, T 90 : long GRBs for T 90 > 2 s and short GRBs for T 90 < 2 s. [1][START_REF] Klebesadel | The durations of gamma-ray bursts[END_REF][START_REF] Dezalay | [END_REF][4][5] In this paper, we shall review the recent progress reached in the understanding of the nature of long and short GRBs that has led to a physical GRB classification, proposed in Refs. 6-8. Such a classification, as we will see below, is based on the possible outcomes in the final stages of the evolution of the progenitor systems.

Long GRBs

The induced gravitational collapse (IGC) scenario introduces, as the progenitor of the long GRBs associated with SNe Ib/c, binaries composed of a carbon-oxygen core (CO core ) on the verge of supernova with a NS companion. [START_REF] Ruffini | Black hole physics and astrophysics: The GRB-Supernova connection and URCA-1-URCA-2[END_REF][START_REF] Izzo | [END_REF][11][12][13][14][15] The explosion of the CO core as SN, forming at its center a newly-born NS called hereafter νNS, triggers an accretion process onto the NS binary companion. Depending on the parameters of the in-state, i.e. of CO core -NS binary, two sub-classes of long GRBs with corresponding out-states are envisaged 6 :

• X-ray flashes (XRFs). Long bursts with E iso 10 52 erg are produced by CO core -NS binaries with relatively large binary separations (a 10 11 cm). The accretion rate of the SN ejecta onto the NS in these systems is not high enough to bring the NS mass to the critical value M crit , hence no BH is formed. The out-state of this GRB sub-class can be either a νNS-NS binary if the system keeps bound after the SN explosion, or two runaway NSs if the binary system is disrupted. • Binary driven hypernovae (BdHNe). Long bursts with E iso 10 52 erg are instead produced by more compact CO core -NS binaries (a 10 11 cm, see e.g. Refs. 13 and 15). In this case, the SN triggers a larger accretion rate onto the NS companion, e.g. 10 -2 -10 -1 M s -1 , bringing the NS to its critical mass M crit , [11][12][13] namely to the point of gravitational collapse with consequent formation of a BH. Remarkably, in Ref. 14, it was recently shown that the large majority of BdHNe leads naturally to NS-BH binaries owing to the high compactness of the binary that avoids the disruption of it even in cases of very high mass loss exceeding 50% of the total mass of the initial CO core -NS binary.

In addition, it exists the possibility of BH-SNe. 6 Long burst with E iso 10 54 erg occurring in close CO core -BH binaries in which the hypercritical accretion produces, as out-states, a more massive BH and a νNS. These systems have been considered in Ref. 6 as a subset of the BdHNe but no specific example have been yet observationally identified.

Short GRBs

There is the consensus within the GRB community that the progenitors of short GRBs are mergers of NS-NS and/or NS-BH binaries (see, e.g. Refs. 16-20 for a recent review). Similarly to the case of long GRBs, in Ref. 6 short GRBs have been split into different sub-classes:

• Short gamma-ray flashes (S-GRFs). Short bursts with energies E iso 10 52 erg, produced when the post-merger core do not surpass the NS critical mass M crit , hence there is no BH formation. Thus, these systems left as byproduct a massive NS and possibly, due to the energy and angular momentum conservation, orbiting material in a disk-like structure or a low-mass binary companion.

• Authentic short gamma-ray bursts (S-GRBs). Short bursts with E iso 10 52 erg, produced when the post-merger core reaches or overcome M crit , hence forming a Kerr or Kerr-Newman BH, 8 and also in this case possibly orbiting material. • Ultra-short GRBs (U-GRBs). A new sub-class of short bursts originating from νNS-BH merging binaries. They can originate from BdHNe (see Ref. 14) or from BH-SNe.

In addition, it exists the possibility of gamma-ray flashes (GRFs). These are bursts with hybrid properties between short and long, they have 10 51 E iso 10 52 erg. This sub-class of sources originates in NS-WD mergers. 6 Table 1 summarized some observational aspects of the GRB sub-classes including the occurrence rate calculated in Ref. 6.

Table 1. Some observational aspects of the GRB sub-classes. In the first three columns, we indicate the GRB sub-class and their corresponding in-states and the out-states. In column 4, we list the E iso (rest-frame 1-10 4 keV), columns 5-6 list, for each GRB sub-class, the maximum observed redshift and the local occurrence rate computed in Ref. 6.

GRB sub-class

In-state Out-state We focus here on the physical properties of the above progenitors, as well as on the main properties of NSs that play a relevant role in the dynamics of these systems and that lead to the above different GRB sub-classes. We shall discuss as well recent estimates of the rates of occurrence on all the above subclasses based on X and gamma-ray observations, and also elaborate on the possibility of detecting the gravitational wave (GW) emission originated in these systems.

E iso zmax ρ GRB (erg) (Gpc -3 yr -1

IGC, Hypercritical Accretion, and Long GRBs

We turn now to the details of the accretion process within the IGC scenario. Realistic simulations of the IGC process were performed in Ref. 12, including: (1) detailed SN explosions of the CO core ; (2) the hydrodynamic details of the hypercritical accretion process; (3) the evolution of the SN ejecta material entering the Bondi-Hoyle region all the way up to its incorporation into the NS. Here, the concept of hypercritical accretion refers to the fact the accretion rates are highly super-Eddington. The accretion process in the IGC scenario is allowed to exceed the Eddington limit mainly for two reasons: (i) the photons are trapped within the infalling material impeding them to transfer momentum; (ii) the accreting material creates a very hot NS atmosphere (T ∼ 10 10 K) that triggers a very efficient neutrino emission which become the main energy sink of these systems unlike photons.

The hypercritical accretion process in the above simulations was computed within a spherically symmetric approximation. A further step was given in Ref. 13 by estimating the angular momentum that the SN ejecta carries and transfer to the NS via accretion, and how it affects the evolution and fate of the system. The calculations are as follows: first the accretion rate onto the NS is computed adopting an homologous expansion of the SN ejecta and introducing the pre-SN density profile of the CO core envelope from numerical simulations. Then, it is estimated the angular momentum that the SN material might transfer to the NS: it comes out that the ejecta have enough angular momentum to circularize for a short time and form a disc around the NS. Finally, the evolution of the NS central density and rotation angular velocity (spin-up) is followed computing the equilibrium configurations from the numerical solution of the axisymmetric Einstein equations in full rotation, until the critical point of collapse of the NS to a BH taking into due account the equilibrium limits given by mass-shedding and the secular axisymmetric instability. Now we enter into the details of each of the above steps. The accretion rate of the SN ejecta onto the NS can be estimated via the Bondi-Hoyle-Lyttleton accretion formula:

ṀB (t) = πρ ej R 2 cap v 2 rel + c 2 s,ej , R cap (t) = 2GM NS (t) v 2 rel + c 2 s,ej , ( 1
)
where G is the gravitational constant, ρ ej and c s,ej are the density and sound speed of the SN ejecta, R cap is the NS gravitational capture radius (Bondi-Hoyle radius), M NS , the NS mass, and v rel the ejecta velocity relative to the NS: The material shocks and decelerates as it piles over the NS surface. At the neutrino emission zone, neutrinos take away most of the infalling matter gravitational energy gain. The neutrinos are emitted above the NS surface in a region of thickness ∆rν about half the NS radius that allow the material to reduce its entropy to be finally incorporated to the NS. The image is not to scale. For further details and numerical simulations of the above process, see Refs. 12-15.

v rel = v orb -v ej ,
with |v orb | = G(M core + M NS )/a, the module of the NS orbital velocity around the CO core , and v ej the velocity of the supernova ejecta (see Fig. 1).

Extrapolating the results for the accretion process from stellar wind accretion in binary systems, the angular momentum per unit time that crosses the NS capture region can be approximated by: Lcap = (π/2)( ρ /2 -3 ν )ρ ej (a, t)v 2 rel (a, t)R 4 cap (a, t), where ρ and ν are parameters measuring the inhomogeneity of the flow (see Ref. 13 for details).

In order to simulate the hypercritical accretion, it is adopted an homologous expansion of the SN ejecta, i.e. the ejecta velocity evolves as v ej (r, t) = nr/t, where r is the position of every ejecta layer from the SN center and n is called expansion parameter. The ejecta density is given by ρ ej (r, t) = ρ 0 ej (r/R star (t), t 0 ) Menv(t) Menv(0) ( Rstar(0) Rstar(t) ) [START_REF] Dezalay | [END_REF] , where M env (t) the mass of the CO core envelope, namely the mass of the ejected material in the SN explosion and available to be accreted by the NS, R star (t) is the position of the outermost layer of the ejected material, and ρ 0 ej is the pre-SN density profile. The latter can be approximated with a power law: ρ ej (r, t 0 ) = ρ core (R core /r) m , where ρ core , R core and m are the profile parameters which are fixed by fitting the pre-SN profiles obtained from numerical simulations.

For the typical parameters of pre-SN CO core and assuming a velocity of the outermost SN layer v sn (R star , t 0 ) ∼ 10 9 cm s -1 and a free expansion n = 1 (for details of typical initial conditions of the binary system see Refs. 12 and 13), Eq. (1) gives accretion rates around the order of 10 -4 -10 -2 M s -1 , and an angular momentum per unit time crossing the capture region Lcap ∼ 10 46 -10 49 g cm 2 s -2 .

We consider the NS companion of the CO core initially as nonrotating, thus at the beginning, the NS exterior spacetime is described by the Schwarzschild metric. The SN ejecta approach the NS with specific angular momentum, l acc = Lcap / ṀB , thus they will circularize at a radius r st if they have enough angular momentum. What does the word "enough" means here? The last stable circular orbit (LSO) around a nonrotating NS is located at a distance r lso = 6GM NS /c 2 and has an angular momentum per unit mass l lso = 2 √ 3GM NS /c. The radius r lso is larger than the NS radius for masses larger than 1.67 M , 1.71 M , and 1.78 M for the GM1, TM1, and NL3 nuclear equation-of-state (EOS). 13 If l acc ≥ l lso , the material circularizes around the NS at locations r st ≥ r lso . For the values of the IGC systems under discussion here, r st /r lso ∼ 10 -10 3 , thus the SN ejecta have enough angular momentum to form a sort of disc around the NS. Even in this case, the viscous forces and other angular momentum losses that act on the disk will allow the matter in the disk to reach the inner boundary at r in ∼ r lso , to then be accreted by the NS.

Within this picture, the NS accretes the material from r in and the NS mass and angular momentum evolve as:

ṀNS = ∂M NS ∂M b JNS Ṁb + ∂M NS ∂J NS M b JNS , JNS = ξl(r in ) ṀB , (2)
where M b is the NS baryonic mass, l(r in ) is the specific angular momentum of the accreted material at r in , which corresponds to the angular momentum of the LSO, and ξ ≤ 1 is a parameter that measures the efficiency of angular momentum transfer. We assume in our simulations Ṁb = ṀB . In order to integrate Eqs. ( 1) and ( 2), we have to supply the two above partial derivatives which are obtained from the relation of the NS gravitational mass with M b and J NS , namely from the NS binding energy. The general relativistic calculations of rotating NSs in Ref. 21 show that, independent on the nuclear EOS, this relation is well approximated by the formula

M b M = M NS M + 13 200
M NS M 2 1 - 1 137 j 1.7 NS , (3)
where j NS ≡ cJ NS /(GM 2 ). In addition, since the NS will spin up with accretion, we need information of the dependence of the specific angular momentum of the LSO as a function of both the NS mass and angular momentum. For corotating orbits, the following relation is valid for all the aforementioned EOS 13 :

l lso = GM NS c   2 √ 3 -0.37   j NS M NS M   0.85   . (4)
The NS accretes mass until it reaches a region of instability. There are two main instability limits for rotating NSs: mass-shedding or Keplerian limit and the secular Table 2. Critical NS mass in the nonrotating case and constants k and p needed to compute the NS critical mass in the nonrotating case given by Eq. ( 5). The values are given for the NL3, GM1 and TM1 EOS. axisymmetric instability. The critical NS mass along the secular instability line is approximately given by 21 :

EOS M J =0 crit (M ) p k NL3 2
M crit NS = M J=0 NS (1 + kj p NS ), (5)
where the parameters k and p depends of the nuclear EOS (see Table 2). These formulas fit the numerical results with a maximum error of 0.45%. Along the mass-shedding sequence, the NS has the maximum possible angular momentum 21 : J NS,max ≈ 0.7GM 2 NS /c. Figure 2 shows the evolution of the NS dimensionless angular momentum, cJ NS /(GM 2 NS ), as a function of the NS mass for ξ = 0.5 and for selected values of the initial NS mass. The NS fate depends of the NS initial mass and the efficiency parameter ξ. The less massive initial configurations reach the mass-shedding limit with a maximum dimensionless angular momentum value while the initially more massive configurations reach the secular axisymmetric instability. It is interesting to note that the total angular momentum of the SN ejecta entering the Bondi-Hoyle region, L cap , is much larger than the maximum angular momentum that a uniformly rotating NS can support, J NS,max . The numerical simulations in Ref. 13 indicate L cap ∼ 10J NS,max . Thus, part of this angular momentum must be lost or redistributed before the material can reach the NS surface. This result leads to a clear prediction: the BHs produced through the IGC mechanism, namely those formed in BdHNe, have initial dimensionless spin ∼0.7 and the excess of angular momentum could lead to a jetted emission with possible high-energy signatures and/or to the presence of a disk-like structure first around the NS as shown above and possibly also around the BH originated from the gravitational collapse of the NS.

Most recent simulations of the IGC process

Additional details and improvements of the hypercritical accretion process leading to XRFs and BdHNe have been recently presented in Ref. 15. In particular:

(1) It was there improved the accretion rate estimate including the density profile finite size/thickness and additional CO core progenitors leading to different SN ejecta masses were also considered. (2) It was shown in Ref. 13, the existence of a maximum orbital period, P max , over which the accretion onto NS companion is not high enough to bring it to the critical mass for gravitational collapse to a BH. Therefore, CO core -NS binaries with P > P max lead to XRFs while the ones with P P max lead to BdHNe. In Ref. 15, the determination of P max was extended to all the possible initial values of the mass of the NS companion and the angular momentum transfer efficiency parameter was also allowed to vary.

(3) It was computed the expected luminosity during the hypercritical accretion process for a wide range of binary periods covering XRFs and BdHNe. (4) It was there shown that the presence of the NS companion originates large asymmetries (see, e.g. simulation in Fig. 3) in the SN ejecta leading to observable signatures in the X-rays.

Figure 3 shows a simulation of an IGC process presented in Ref. 15. We considered the effects of the gravitational field of the NS on the SN ejecta including the orbital motion as well as the changes in the NS gravitational mass owing to the accretion process via the Bondi formalism. The supernova matter was described as formed by point-like particles whose trajectory was computed by solving the Newtonian equation of motion. The initial conditions of the SN ejecta are computed assuming an homologous velocity distribution in free expansion. The initial powerlaw density profile of the CO envelope is simulated by populating the inner layers with more particles. For the M ZAMS = 30 M progenitor which gives a CO core with envelope profile ρ 0 ej ≈ 3.1 × 10 8 (8.3 × 10 7 /r) 2.8 g cm -3 , we adopt for the simulation a total number of N = 10 6 particles. We assume that particles crossing the Bondi-Hoyle radius are captured and accreted by the NS so we removed them from the system as they reach that region. We removed these particles according to the results obtained from the numerical integration explained above. Figure 3 shows the orbital plane of an IGC binary at selected times of its evolution. The NS has an initial mass of 2.0 M ; the CO core leads to a total ejecta mass 7.94 M and a νNS of 1.5 M . The orbital period of the binary is P ≈ 5 min, i.e. a binary separation a ≈ 1.5 × 10 10 cm. For these parameters, the NS reaches the critical mass and collapses to form a BH.

Hydrodynamics and neutrino inside the accretion region

We turn now to give some details on the properties of the system inside the Bondi-Hoyle accretion region. We have seen that the accretion rate onto the NS can be as high as ∼10 -2 -10 -1 M s -1 . For these accretion rates:

(1) We can neglect the effect of the NS magnetic field since the magnetic pressure remains much smaller than the random pressure of the infalling material. 11,22 (2) The photons are trapped in the accretion flow. The trapping radius, defined at which the photons emitted diffuse outward at a slower velocity than the one of the infalling material, is 23 : r trapping = min{ ṀB κ/(4πc), R cap , where κ is the opacity. For the CO core , in Ref. 12, a Rosseland mean opacity roughly 5 × 10 3 cm 2 g -1 was estimated. For the range of accretion rates, we obtain that ṀB κ/(4πc) ∼ 10 13 -10 19 cm, a radius much bigger than the NS capture radius which is in our simulations at most 1/3 of the binary separation. Thus, in our systems, the trapping radius extends all the way to the Bondi-Hoyle region, hence the Eddington limit does not apply and hypercritical accretion onto the NS occurs. (3) Under these conditions, the gain of gravitational energy of the accreted material is mainly radiated via neutrino emission (see below). 11,12,22,24,25

Convective instabilities

As the material piles onto the NS and the atmosphere radius, the accretion shock moves outward. The post-shock entropy is a decreasing function of the shock radius position which creates an atmosphere unstable to Rayleigh-Taylor convection during the initial phase of the accretion process. These instabilities can accelerate above the escape velocity driving outflows from the accreting NS with final velocities approaching the speed of light. 26,27 Assuming that radiation dominates, the entropy of the material at the base of the atmosphere is 22 : S bubble ≈ 16(1.4 M /M NS ) -7/8 (M s -1 / ṀB ) 1/4 (10 6 cm/r) 3/8 , in units of k B per nucleon. This material will rise and expand, cooling adiabatically, i.e. T 3 /ρ = constant, for radiation dominated gas. If we assume a spherically symmetric expansion, then ρ ∝ 1/r 3 and we obtain k B T bubble = 195 S -1 bubble (10 6 cm/r) MeV. However, it is more likely that the bubbles expand in the lateral but not in the radial direction, 27 thus we have ρ ∝ 1/r 2 , i.e. T bubble = T 0 (S bubble )(r 0 /r) 2/3 , where T 0 (S bubble ) is given by the above equation evaluated at r = r 0 ≈ R NS . This temperature implies a bolometric blackbody flux at the source from the bubbles

F bubble ≈ 2 × 10 40 M NS 1.4 M -7/2 ṀB M s -1 R NS 10 6 cm 3/2 × r 0 r 8/3 erg s -1 cm -2 , ( 6
)
where σ is the Stefan-Boltzmann constant.

In Ref. 12, it was shown that the above thermal emission from the rising bubbles produced during the hypercritical accretion process can explain the early (t 50 s) thermal X-ray emission observed in GRB 090618. [START_REF] Izzo | [END_REF]28 In that case, T bubble drops from 50 keV to 15 keV expanding from r ≈ 10 9 cm to 6 × 10 9 cm, for an accretion rate 10 -2 M s -1 .

It is interesting that also r-process nucleosynthesis can occur in these outflows. 26 This implies that long GRBs can be also r-process sites with specific signatures from the decay of the produced heavy elements, possibly similar as in the case of the kilonova emission in short GRBs. 29 The signatures of this phenomenon in XRFs and BdHNe, and its comparison with kilonovae, deserves to be explored.

Neutrino emission

Most of the energy from the accretion is lost through neutrino emission. For the accretion rate conditions characteristic of our models ∼10 -4 -10 -2 M s -1 , e + e - pair annihilation dominates the neutrino emission and electron neutrinos remove the bulk of the energy. The temperature of these neutrinos can be roughly approximated by assuming that the inflowing material generally flows near to the NS surface before shocking and emitting neutrinos. For accretion rates ∼10 -4 -10 -2 M s -1 , neutrino energies ∼5-20 MeV are obtained. 15 A detailed study of the neutrino emission will be the presented elsewhere.

For the developed temperatures (say k B T ∼ 1-10 MeV) near the NS surface, the dominant neutrino emission process is the e + e -annihilation leading to ν ν. This process produces a neutrino emissivity proportional to the ninth power of the temperature. The accretion atmosphere near the NS surface is characterized by a temperature gradient with a typical scale height ∆r ν ≈ 0.7 R NS . 15 Owing to the aforementioned strong dependence of the neutrino emission on temperature, most of the neutrino emission occurs in the region ∆r ν above the NS surface.

These conditions lead to the neutrinos to be efficient in balancing the gravitational potential energy gain allowing the hypercritical accretion rates. The effective accretion onto the NS can be estimated as 22 : Ṁeff ≈ ∆M ν (L ν /E ν ), where ∆M ν and L ν are the mass and neutrino luminosity in the emission region (i.e. ∆r ν ). E ν is half the gravitational potential energy gained by the material falling from infinity to the R NS + ∆r ν . Since L ν ≈ 2πR 2 NS ∆r ν e -e + with e -e + the e + e -pair annihilation process emissivity, and E ν = (1/2)GM NS ∆M ν /(R NS + ∆r ν ), it can be checked that for M NS = 1.4 M this accretion rate leads to values Ṁeff ≈ 10 -9 -10 -1 M s -1 for temperatures k B T = 1-10 MeV.

Accretion luminosity

The gain of gravitational potential energy in the accretion process is the total one available to be released, e.g. by neutrinos and photons. The total energy released in the star in a time-interval dt during the accretion of an amount of mass dM b with angular momentum l Ṁb , is given by 13,30

L acc = ( Ṁb -ṀNS )c 2 = Ṁb c 2 1 - ∂M NS ∂J NS M b l - ∂M NS ∂M b JNS . ( 7
)
This upper limit to the energy released is just the amount of gravitational energy gained by the accreted matter by falling to the NS surface and which is not spent in changing the gravitational binding energy of the NS. The total energy releasable during the accretion process, say ∆E acc ≡ L acc dt, is given by the difference in binding energies of the initial and final NS configurations. The typical luminosity will be L acc ≈ ∆E acc /∆t acc where ∆t acc is the duration of the accretion process.

The duration of the accretion process is given approximately by the flow time of the slowest layers of the supernova ejecta to the NS. If the velocity of these layers is v inner , then ∆t acc ∼ a/v inner , where a is the binary separation. For a ∼ 10 11 cm and v inner ∼ 10 8 cm s -1 , we obtain ∆t acc ∼ 10 3 s, while for shorter binary separation, e.g. a ∼ 10 10 cm (P ∼ 5 min), ∆t acc ∼ 10 2 s, as validated by the results of our numerical integrations.

For instance, the NS in the system with P = 5 min accretes ≈ 1 M in ∆t acc ≈ 100 s. With the aid of Eq. ( 3), we estimate a difference in binding energies between a 2 M and a 3 M NS, i.e. ∆E acc ≈ 13/200(3 2 -2 2 ) M c 2 ≈ 0.32 M c 2 leading to a maximum luminosity L acc ≈ 3 × 10 -3 M c 2 ≈ 0.1 Ṁb c 2 . This accretion power, which could be as high as L acc ∼ 0.1 Ṁb c 2 ∼ 10 47 -10 51 erg s -1 for accretion rates in the range Ṁb ∼ 10 -6 -10 -2 M s -1 , necessarily leads to signatures observable in long GRBs (see, e.g. Refs. 10 and 12).

Post-explosion orbits and formation of NS-BH binaries

We turn now to discuss the out-states of the IGC process. The SN explosion of the CO core leaves as a central remnant, the νNS, while the IGC process triggered by the hypercritical accretion of the SN ejecta onto the NS companion leads to the formation of a BH. Thus, the question arises if BdHNe are natural sites for the formation of NS-BH binaries or if these binaries become disrupted during the SN explosion and the consequent IGC process. The answer to this question was recently given in Ref. 14, where it was shown that indeed most of BdHN form NS-BH binaries since the high compactness of the orbit avoids the unbinding of the orbit.

In typical systems, most of the binaries become unbound during the SN explosion because of the ejected mass and momentum imparted (kick) on the newly formed compact object in the explosion of the massive star. Under the instantaneous explosion assumption, if half of the binary system's mass is lost in the SN explosion, the system is disrupted. In general, the fraction of massive binaries that can produce double compact object binaries is thought to be low: ∼0.001-1%. [31][32][33] The mass ejected during the SN alters the binary orbit, causing it to become wider and more eccentric. Assuming that the mass is ejected instantaneously, the post-explosion semi-major axis is a/a 0 = (M 0 -∆M )/(M 0 -2a 0 ∆M/r), where a 0 and a are the initial and final semi-major axes respectively, M 0 is the total initial mass of the binary system, ∆M is the change of mass (equal to the amount of mass ejected in the SN), and r is the orbital separation at the time of explosion. 34 For circular orbits, the system is unbound if it loses half of its mass. However, for very tight binaries as the one proposed in the IGC scenario, a number of additional effects can alter the fate of the binary.

The time it takes for the ejecta to flow past a companion in a SN is roughly 10-1000 s. Although the shock front is moving above 10 4 km s -1 , the denser, lowervelocity ejecta can be moving at 10 3 km s -1 . 12 The broad range of times arises because the SN ejecta velocities varies from 10 2 -10 4 km s -1 . The accretion peaks as the slow-moving (inner) ejecta flows past the NS companion. For normal (wide) binaries, this time is a small fraction of the orbital period and the "instantaneous" mass-loss assumption is perfectly valid. However, in the compact binary systems considered in the IGC scenario, the orbital period ranges from only 100-1000 s, and the mass loss from the SN explosion can no longer be assumed to be instantaneous.

We have seen how in BdHNe, the accretion process can lead to BH formation in a time-interval as short as the orbital period. We here deepen this analysis to study the effect of the SN explosion in such a scenario with a specific example of Ref. 14. Figure 4 shows as the ejecta timescale becomes just a fraction of the orbital timescale, the fate of the post-explosion binary is altered. For these models, we assumed very close binaries with an initial orbital separation of 7 × 10 9 cm in circular orbits. With CO core radii of 1-4 × 10 9 cm, such a separation is small, but achievable. We assume the binary consists of a CO core and a 2.0 M NS companion. When the CO core collapses, it forms a 1.5 M NS, ejecting the rest of the core. We then vary the ejecta mass and time required for most of the ejected matter to move out of the binary. Note that even if 70% of the mass is lost from the system (the 8 M ejecta case), the system remains bound as long as the explosion time is just above the orbital time (T orbit = 180 s) with semi-major axes of less than 10 11 cm.

The short orbits (on ejecta timescales) are not the only feature of these binaries that alters the post-explosion orbit. The NS companion accretes both matter and momentum from the SN ejecta, reducing the mass lost from the system with respect to typical binaries with larger orbital separations and much less accretion. In addition, as with common envelope scenarios, the bow shock produced by the accreting NS transfers orbital energy into the SN ejecta. Figure 4 shows the final orbital separation of our same three binaries, including the effects of mass accretion (we assume 0.5 M is accreted with the momentum of the SN material) and orbit coupling (30% of the orbital velocity is lost per orbit). With these effects, not only do the systems remain bound even for explosion times greater than 1/2 the orbital period but, if the explosion time is long, the final semi-major axis can be on par with the initial orbital separation. Fig. 4. Left panel: semi-major axis versus explosion time for three binary systems including mass accretion and momentum effects. Including these effects, all systems with explosion times above 0.7 times, the orbital time are bound and the final separations are on par with the initial separations. Right panel: merger time due to GW emission as a function of explosion time. Beyond a critical explosion time (0.1-0.6 T orbit depending on the system), the merger time is less than roughly 10,000 yr. For most of our systems, the explosion time is above this limit and we expect most of these systems to merge quickly.

The tight compact binaries produced in these explosions will emit GW emission, ultimately causing the system to merge. For typical massive star binaries, the merger time is many Myr. For BdHNe, the merger time is typically 10,000 yr, or less, as shown in the right panel of Fig. 4. Since the merger should occur within the radius swept clean by the BdHN, we expect a small baryonic contamination around the merger site which might lead to a new family of events which we term ultrashort GRBs, U-GRBs. to this new family of events.

NS-NS/NS-BH Mergers and Short GRBs

Let us turn to short GRBs. We have mentioned that the most viable progenitors of short GRBs appear to be mergers of NS-NS and/or NS-BH binaries. Specifically, in the case of NS-NS mergers, the value of the critical mass of the NS, which crucially depends on the nuclear EOS, has been also found to be a most relevant parameter since it defines the fate of the post-merger object. 8 In this section, we discuss the conditions that determine the fate of the NS-NS binary merger by estimating the mass and angular momentum of the post-merger object. Once we know these values, we can compare the mass of the merged core with the value of the NS critical mass obtained for uniformly rotating NSs. Based on this, we can asses whether a massive NS or a BH is formed from the merger.

We proceed to estimate the mass and the angular momentum of the post-merger core via baryonic mass and angular momentum conservation of the system. We adopt for simplicity that nonrotating binary components. We first compute the total baryonic mass of the NS-NS binary M b = M b1 + M b2 using the relation between the gravitational mass M i and the baryonic mass M bi (i = 1, 2) recently obtained in Ref. 21 and given in Eq. (3) assuming j NS = cJ NS /(GM 2 ) = 0. The post-merger core will have approximately the entire baryonic mass of the initial binary, i.e. M b,core ≈ M b , since little mass is expected to be ejected during the coalescence process. However, the gravitational mass of the post-merger core cannot be estimated using again the above formula since, even assuming nonrotating binary components, the post-merger core will necessarily acquire a fraction η ≤ 1 of the binary angular momentum at the merger point. One expects a value of η smaller than unity since, during the coalesce, angular momentum is lost, e.g. by gravitational wave emission and it can be also redistributed, e.g. into a surrounding disk.

To obtain the gravitational mass of the post-merger core, we can use again Eq. ( 3) relating the baryonic mass M b,NS and the gravitational mass M NS in this case with j NS = 0. The mass and angular momentum of the post-merger core, respectively M core and J core , are therefore obtained from baryon mass and angular momentum conservation, i.e.

M core = M NS , M b,core = M b,NS = M b1 + M b2 , J core = J NS = ηJ merger , (8)
where J merger is the system angular momentum at the merger point. The value of J merger is approximately given by J merger = µr [START_REF] Klebesadel | The durations of gamma-ray bursts[END_REF] merger Ω merger , where µ = M 1 M 2 /M is the binary reduced mass, M = M 1 + M 2 is the total binary mass, and r merger and Ω merger are the binary separation and angular velocity at the merger point. If we adopt the merger point where the two stars enter into contact we have r merger = R 1 + R 2 , where R i is the radius (which depend on the EOS) of the i-component of the binary.

Given the parameters of the merging binary, the above equations lead to the merged core properties M core and J core (or j core ). These values can be therefore confronted with the values of uniformly rotating, stable NSs to check if such a merger will lead either to a new massive NS or to an unstable merged core collapsing to a BH.

For the sake of exemplifying, let us assume a mass-symmetric binary, M 1 = M 2 = M/2. In this case, Eq. ( 8) together with the above estimate of J merger lead to the angular momentum of the merged core

J core = (η/4)(GM 2 /c)C -1/2 , where C ≡ GM 1 /(c 2 R 1 ) = GM 2 /(c 2 R 2 )
is the compactness of the merging binary components. Therefore, if we adopt M 1 = 1.4 M and C = 0.15 the above equations imply a merged core mass M core = (2.61, 2.65) M for η = (0, 1), i.e. for a dimensionless angular momentum of the merged core j core = (0, 5.06). Whether or not these pairs (M core , j core ) correspond to stable NSs depend on the nuclear EOS. A similar analysis can be done for any other pair of binary masses.

Detectability of GWs Produced by the GRB Progenitors

Having established the nature of the progenitors of each GRB sub-class, we turn now to briefly discuss the detectability of their associated GW emission. The minimum GW frequency detectable by the broadband aLIGO interferometer is f aLIGO min ≈ 10 Hz. 35 Since during the binary inspiral, the GW frequency is twice the orbital one, this implies that a binary enters the aLIGO band for orbital periods P orb 0.2 s. Thus, CO core -NS binaries, in-states of XRFs and BdHNe, and CO core -BH binaries, in-states of BH-SN, are not detectable by aLIGO since they have orbital periods P orb 5 min 0.2 s. Concerning their out-states after the corresponding hypercritical accretion processes, namely νNS-NS, out-states of XRFs, and νNS-BH, out-states of BdHNe and BH-SNe, they are not detectable by aLIGO at their birth but only when approaching the merger. Clearly, the analysis of the νNS-NS mergers is included in the analysis of the S-GRFs and S-GRBs and, likewise, the merger of νNS-BH binaries is included in the analysis of U-GRBs. In the case of NS-WD binaries, the WD is tidally disrupted by the NS making their GW emission hard to be detected (see, e.g. Ref. 36).

A coalescing binary evolves first through the inspiral regime to then pass over a merger regime, the latter composed by the plunge leading to the merger itself and by the ringdown (oscillations) of the newly formed object. During the inspiral regime, the system evolves through quasi-circular orbits and is well described by the traditional point-like quadrupole approximation. [37][38][START_REF] Rees | Black Holes, Gravitational Waves and Cosmology[END_REF] The GW frequency is twice the orbital frequency (f s = 2f orb ) and grows monotonically. The energy spectrum during the inspiral regime is:

dE/df s = (1/3)(πG) 2/3 M 5/3 c f -1/3 s
, where M c = µ 3/5 M 2/5 = ν 3/5 M is the so-called chirp mass and ν ≡ µ/M is the symmetric mass-ratio parameter. A symmetric binary (m 1 = m 2 ) corresponds to ν = 1/4 and the test-particle limit is ν → 0. The GW spectrum of the merger regime is characterized by a GW burst. [START_REF] Davis | [END_REF] Thus, one can estimate the contribution of this regime to the signal-to-noise ratio with the knowledge of the location of the GW burst in the frequency domain and of the energy content. The frequency range spanned by the GW burst is ∆f = f qnmf merger , where f merger is the frequency at which the merger starts and f qnm is the frequency of the ringing modes of the newly formed object after the merger, and the energy emitted is ∆E merger . With these quantities defined, one can estimate the typical value of the merger regime spectrum as: dE/df s ≈ ∆E merger /∆f . Unfortunately, the frequencies and energy content of the merger regime of the above merging binaries are such that it is undetectable by LIGO. 41 Since the GW signal is deep inside the detector noise, the signal-to-noise ratio (ρ) is usually estimated using the matched filter technique. 42 The exact position of the binary relative to the detector and the orientation of the binary rotation plane are usually unknown, thus it is a common practice to average over all the possible locations and orientations, i.e. 42 :

ρ 2 = 4 ∞ 0 | h(f )| 2 /S n (f )df = 4 ∞ 0 h 2 c (f )/[f 2 S n (f )]
df , where f is the GW frequency in the detector frame, h(f ) is the Fourier transform of h(t), and S n (f ) is the one-sided amplitude spectral density of the detector noise, and h c (f ) is the characteristic strain, h c = (1+z)/(πd l ) (1/10)(G/c 3 )(dE/df s ). We recall that in the detector frame, the GW frequency is redshifted by a factor 1 + z with respect to the one in the source frame, f s , i.e. f = f s /(1 + z) and d l is the luminosity distance to the source. We adopt a ΛCDM cosmology with H 0 = 71 km s -1 Mpc -1 , Ω M = 0.27 and Ω Λ = 0.73. 43 A threshold ρ 0 = 8 in a single detector is adopted by LIGO. 44 This minimum ρ 0 defines a maximum detection distance or GW horizon distance, say d GW , that corresponds to the most optimistic case when the binary is just above the detector and the binary plane is parallel to the detector plane. In order to give an estimate, the annual number of merging binaries associated with the above GRB subclasses detectable by aLIGO, we can use the lower and upper values of the aLIGO search volume defined by V s = V GW max T , where V GW max = (4π/3)R 3 , where T is the observing time and R is the so-called detector range defined by R = Fd GW , with F -1 = 2.2627 (see, Refs. 44 and 45, for details). For a (1.4 + 1.4) M NS binary and the three following different observational campaigns we have 44 : 2015/2016 (O1; T = 3 months) V S = (0.5-4) × 10 5 Mpc 3 yr, 2017/2018 (O3; T = 9 months) V S = (3-10) × 10 6 Mpc 3 yr, and the entire network including LIGO-India at design sensitivity (2022+; T = 1 yr) V S = 2 × 10 7 Mpc 3 yr. The maximum possible sensitivity reachable in 2022+ leads to d GW ≈ 0.2 Gpc, hence V GW max ≈ 0.033 Gpc 3 , for such a binary. One can use this information for other binaries with different masses taking advantage of the fact that d GW scales with the binary chirp mass as M 5/6 c . The expected GW detection rate by aLIGO can be thus estimated as: ṄGW ≡ ρ GRB V GRB max , where ρ GRB is the inferred occurrence rate of GRBs shown in Table 1 computed in Ref. 6. Bearing the above in mind, it is easy to check that there is a low probability for aLIGO to detect the GW signals associated with the GRB binary progenitors: indeed in the best case of the 2022+ observing rung one obtains, respectively, ∼1 detection every 3 and 5 yr for U-GRBs and S-GRFs.

Conclusions

There is accumulated evidence on the binary nature of long and short GRBs. Such binaries are composed of CO cores , NSs, BHs and WDs in different combinations. We have here focused on the salient aspects of the NS physics relevant for the understanding of these binaries and their implications in GRB astrophysics, including their associated GW emission. We have discussed the crucial role of the NS critical mass in discriminating the GRB sub-classes. Therefore, we expect that the increasing amount of GRB high-quality data will help in constraining the NS critical mass with high accuracy with the most welcome result of constraining the NS matter content and the corresponding nuclear EOS.

Fig. 1 .

 1 Fig. 1. Scheme of the IGC scenario: the COcore undergoes SN explosion, the NS accretes part of the SN ejecta and then reaches the critical mass for gravitational collapse to a BH, with consequent emission of a GRB. The SN ejecta reach the NS Bondi-Hoyle radius and fall toward the NS surface.The material shocks and decelerates as it piles over the NS surface. At the neutrino emission zone, neutrinos take away most of the infalling matter gravitational energy gain. The neutrinos are emitted above the NS surface in a region of thickness ∆rν about half the NS radius that allow the material to reduce its entropy to be finally incorporated to the NS. The image is not to scale. For further details and numerical simulations of the above process, see Refs. 12-15.

Fig. 2 .

 2 Fig. 2. Evolution of NSs of different initial masses M NS = 2.0, 2.25 and 2.5 M during the hypercritical accretion in a BdHN. 13 It is shown the dimensionless angular momentum as a function of the NS mass. The binary parameters are: COcore of a M ZAMS = 30 M progenitor star (m = 2.801, Menv = 7.94 M , ρcore = 3.08 × 10 8 g cm -3 and R 0star = 7.65 × 10 9 cm), a free expansion (n = 1) and a SN outermost ejecta velocity v 0star = 2 × 10 9 cm s -1 . The orbital period is of approximately 5 min.

Fig. 3 .

 3 Fig.3. Hypercritical accretion process in the IGC binary system at selected evolution times. In this example, the COcore has a total mass of 9.44 M divided in an ejecta mass of 7.94 M and a νNS of 1.5 M formed by the collapsed high density core. The supernova ejecta evolve homologously with outermost layer velocity v 0,star = 2 × 10 9 cm s -1 . The NS binary companion has an initial mass of 2.0 M . The binary period is P ≈ 5 min, which corresponds to a binary separation a ≈ 1.5 × 10 10 cm. The system of coordinates is centered on the νNS represented by the white-filled circle at (0, 0). The NS binary companion, represented by the gray-filled circle, orbits counterclockwise following the thin-dashed circular trajectory. The colorbar indicates values of ejecta density in logarithmic scale. Left upper panel : initial time of the process. The supernova ejecta expand radially outward and the NS binary companion is at (a, 0). Right upper panel: the accretion process starts when the first supernova layers reach the Bondi-Hoyle region. This happens at t = t acc,0 ≈ a/v 0,star ≈ 7.7 s. Left lower panel : the NS binary companion reaches the critical mass by accreting matter from the SN with consequent collapse to a BH. This happens at t = t coll ≈ 254 s ≈ 0.85 P . The newly-formed BH of mass M BH = M crit ≈ 3 M is represented by the black-filled circle. It is here evident the asymmetry of the supernova ejecta induced by the presence of the accreting NS companion at close distance. Right lower panel: t = t coll + 100 s = 354 s ≈ 1.2 P , namely 100 s after the BH formation. It appears here the new binary system composed of the νNS and the newly-formed BH.

  of the Ministry of Education and Science of the Republic of Kazakhstan. C. C. and S. F. acknowledge INdAM-GNFM for support.

Acknowledgments M. Karlica, M. Kovacevic and Y. A. are supported by the Erasmus Mundus Joint Doctorate Program Grant Nos. 2013-1471, 2013-1471 and 2014-0707, respectively, from EACEA of the European Commission. M. M. and J. A. R. acknowledge the partial support of the Project No. 3101/GF4 IPC-11, and the target program F.0679


Frederic Poggi
email: frederic.poggi@cemagref..fr.

Helene Macchi-Tejeda

Alain Marechal

Denis Leducq

Experimental study of single and two-phase adiabatic flow distribution in compact heat exchangers

Keywords: Distribution, compact beat exchanger, small channel, pressure drop 1

Small-channel heal exchangers are widely used for car air-conditioning. However information about theh" behaviour remains scarce. An experime111al device was built to characterize the distribu1ion o. f single and two-phase adiabatic flows in a small channel heal exchanger, and to measure ils singular and regular pressure losses. A transparent polycarbonate test section comprised a 16 mm dia1neter header connected to 8fl at tubes with 7 small channels (D11 = 0.889 mm). Both were studied with horizontal flow and vertical down-flow through the mani f old. The mass flux range was between I 5 and 650 kgl(m2.s) in the manifold and between 100 and 3 000 kgl(m2.�) in the small channels. The range of vapour quality values at the test section inlet was b. etween 0.1and0.9. The two-phase refrigerant was condensed at the outlet of each flat tube in order to measure each flow rate separately.

Resume:

Les echangeurs de chaleur compacts a mini-canaux sont largement repandus dans la climatisalion automobile. Cependant /es informations sur leur comportement demeurent rares. Un dispositif experimental est realise pour caracteriser la dislribution des ecoulements adiabatiques mono-et diphasiques dans un echangeur compact a mini-canaux. Les pertes de pres sion singulieres et regulieres sonl mesurees. Une section d'es sai en polycarbonate transparente comportant un distributeur de 16 millimetres de diametre relie a 8 barrettes de 7 mini-canaux (D1, = 0.889 mm) est realisee. Deux configurations sont etudiees : ecou/ements horizontal et vertical descendant dans le distributeur. La gamme de densite de flux massique varie de 15 a 650 kgl(m2.s) dans le distributeur et de 100 a 3 000 kgl(m2.s) dans Jes mini-canaux. Les valeurs de titre en vapeur en entree de section d'essai sont comprises entre 0,1 et 0,9. le refrigerant diphasique est condense en sortie de chaque barrette a . fin de mesurer chaque debit separement.

Introduction

The study described in this paper answers a triple objective: -Environmental objective: reduction of the refrigerant charge of refrigerating systems thanks to the implementation of small-channel heat exchangers (protocols of Montreal [1987] and Kyoto [I 997]), -Industrial objective: the studied exchanger is similar to automobile air-conditioners or plate heat exchangers for cooling fuel cells, l -Scientific objective: to folftl the lack of data and models concerning distribution and singular pressure drops on millimen• e scale n1bes.

Thus, this work experimentally snidies a p011ion of small-channel beat exchanger with liquid single-phase flow and two-phase flow. The selected geometry is that of the automobile's equipments (FIG. 1). A manifold distributes the refrigerant in the small channels of 8 successive extmded flat tubes in two positions (holizontal flow and vertical down-fl ow in the manifold).

This geometry can be industrially manufacmred by fining together the fl at tube inlets inside the manifold by a protrnsion. The tube protrnsion depth into the manifold will also affect the distribution. [START_REF] Webb | Two-phase flow distribution in tubes of parallel flow heat exchangers[END_REF][START_REF] Hmjak | Flow distribution issues the parallel flow heat exchangers[END_REF]Hrnjak (2004) provided recent reviews on this subject. In this paper, we smdied the refii. gerant distribution; regular and singular pressure drops in the small-channels and along the distributor were also studied by Poggi and al. (2007), but are not presented in this paper.

Test section

The tes t section was built in polished polycarbonate in order to visualise the flows in the manifold and in the small ch annels. The manifold diameter is 16 mm with a 90 mm length; 8 multi-port flat tubes are inserted inside the manifold. Each fiat tube con tains 7 rec tan gular small channels (Dh = O.S89 nun) (FIG. 1,3). The r efrigerant HFE-7100 w as used in order to cany ou t experiments dose to atmospheric pressure and ambient temperature. / • / • 0,14 ";;0 This quite good distribution in both distributor positions is in good agreement with the rule given by Peny & Green (1997). These authors indicated that a homogeneous distribution is obtained wi th a ratio of 10 between the pressure drop in the manifold and that in the small channels (for vertical down-flow. the maximal pressure drop in the manifold was to 300 Pa and 17 000 Pa in the small channels; for horizontal fl ow, the maximal pressure drop in the manifold was to 340 Pa and 20 000 Pa in small-channels). Nevertheless, the "accumulative pressure" forces the refrigerant towards the last multi-port flat tubes.

I • • • • 0 12 • • • .

Two-phase flow distribution

The fl ow rate distribution in the small channels was measured for different vapour qualities at the test section inlet. The mass flux was in the range of 35 to 120 kg/(ni2.s) and the flow inside the manifold was recorded for each flow rate and vapour quality.

.2.1. Visualisation of the flow patterns

FIG. 5 shows the flow pattern in the manifold for a 55 kglh mass flow rate. Ill the case of horizontal flow, at low vapour qualities \x � 0. 15), an intermittent fl ow was observed \slug flow + stratified fl ow), with the whole manifold fl ooded. For medium vapour qualities (0. 15 < x < 0.60), a wavy flow was obse1ved, as already seen by Watanabe and al. (1995). For x > 0.60, tlle flow was semi-annular with stratification and droplet wrenching at the impact with the first flat tube. For tbe higber vapour quality (x = 0.85), an annular flow was obse1ved before tbe impact of tbe fluid witb the fir st fl. at tube; tben the droplet wrenching led to fog formation. In tbe case of ve11ical down-fl ow, for all vapour qualities, we obse1ved an annular flow with liquid accumulation at bottom. All the bigber then the inlet vapour quality was low. On the contra1y, tbe thickness of tbe annular liquid fihn decreased as tbe qu ality increase. Ftllthennore the gravity effect amplified botb tbe thinning of the liquid film in annular down-fl ow before the impact of tbe fluid on tbe first fl at tube and tbe liquid accumulation in tbe rear of the manifold.

Droplet wrenching was also observed wben the fluid impacted tbe fir st flat tube; wben the vapour quality increased, tbe droplet wrencbing led to fog formation. Flow distribution in small channels of compact heat exchanger has been experimentally studied. In single-phase flows, the distribution did not depend on the inlet mass flow rate and it was rather homogeneous in both configurations. However it was less homogeneous in the horizontal flow than in the vertical one because of gravity. In two-phase flows, for the horizontal position, the flow patterns depended on the inlet vapour quality. On the contrary, in the vertical position, the flow at the manifold inlet was always annular, whatever the inlet vapour quality. As for the distribution, it was significantly affected by the manifold orientation and the inlet vapour quality. When the inlet vapour quality increased, the phase distribution was more homogeneous.

M

  FIG.1-Distributor for small-channel heat exchangers used in industrial applications.

FIG. 3 -

 3 FIG. 3 -Representation and photographs of the test section. The instrnme ntation allows the measurement of the flow rate and the vapom quality at the outlet of each multi-p011 flat tube. The distribution of the fl ow rate in the manifold can be determined as a fhnct ion of the inlet vapour quality and total tl ow rate. A Danfoss Mass 2100 mass flowmeter (/!! .+/. 0.033%) was i nstalled at the inlet of the test section. The temperature was measured with a p l at i num resistance sensor (il+/-0.3%) and the pressure was measured with an absolute pressure sensor Rosem ount 0/2 bars (A+/-0.2%). At the outlet of the test section, 8 plate condensers were placed for each fl at tube (link between 200 and 300 loops). Both the inlet

  FIG. 4 -Repartition of the mass fl ow rates at the outlet of the multi-poll fl at tubes. (a) Horizontal flow through the manifold, (b) Vertical down-flow through the manifold.

Flow

  FIG. 5 -Photogrnph of the fl ow pat terns in the manifold for a 55 kg/b mass fl ow rate and various inlet vapour qualities for both positions (a) horizontal flow (b) vertical down-flow through the manifold. 3.2.2. Influence of the inlet vapour quality on tbe distribution FIG. 6 shows tbe liquid and vapour dimensionless mass flow rates in tbe 8 multi-port flat tubes for both hmizontal flow and vertical down-flow througb the manifold in adiabatic condition for a 55 kg/h mass fl ow rate. In the case of borizontal fl ow, the liquid dimensionless mass flow rate bad a quite high value for the first channel because of gravity. Then, tbe liquid dimensionless mass flow rate grndually decreas ed as tbe cbannel number inc reased, exceptedfor the last flat tube. Indeed the "accumulative pressure" forces the liquid towards the last flat n1bes. The vapour dimensionless mass fl ow rate showed an opposite trend, especially for the low vapour qualities. Wben tbe inlet vapour quality increased, the liquid dimensionless mass fl ow rate decreased strongly in the first flat tube, but increased ligbtly in tbe last flat tube, so tbat the distribution tends to be more bomogeneous. In tbe ve11ical down-flow case, tbe liquid dimensionless mass flow rate also decreased as the cbannel number increased, fmther on the liquid dimensionl ess mass fl ow rate highly increased under the "accumulative pressure" forcing the liquid towards the last flat tube, effect amplified by the gravity. Tbe vapour dimensionless mass flow rate increased lightly as the channel nurriber increased, excepted the two last flat tube because of the liquid accumulation at tbe manifold bottom. Otberwise, for both manifold pos.itions, the va1iation of vapour disUibution was mucb less significant tban that oftbe liquid.

  FIG. 6 -Di mensionless vapour quality for each flat tube for various inlet vapour qualities and an inlet mass flow rate of M 0 = 55 kg/h for both positions (a) horizontal (b) vertical down-flow.