Henri Boutin
email: henri.boutin@ircam.fr

John Smith
email: john.smith@unsw.edu.au

Joe Wolfe
email: j.wolfe@unsw.edu.au

THE ACOUSTICS OF THE TROMBONE : HOW DO PLAYERS LIP UP AND DOWN?

Keywords: brass instruments, lip motion, pitch

Four advanced trombonists and three beginners played the note B ♭2 (the lowest normal note played with the slide retracted), then 'lipped up and down'. The normal playing frequency lies above that of the bore impedance peak, so the bore is a compliant load. However, the range reached while lipping is approximately centered on this peak. To investigate how the lip oscillation is regenerated with inertive or compliant acoustic loads, acoustic pressure and flow waveforms were determined both up-and downstream from the lips, while playing. The lip opening area and the flow component due to the lips' sweeping motion were also estimated using a transparent mouthpiece and high-speed video. The lips move forward before separating, then backward before closing. Consequently, the acoustic flow into the mouthpiece becomes positive before the lips open. Further, the volume V of air swept by the lips in the direction of the flow is positive around one cycle. A model in which the lips execute out-of-phase simple harmonic motions in the forwards and vertical directions is consistent with the results. The pressure difference Δ𝑃 across the lips does nett positive Δ𝑃𝑑𝑉 work on the lips, providing sufficient energy for regeneration of the oscillation. As the note is lipped down to inertive load, the phase of the mouthpiece pressure moves ahead of that of the flow, and Δ𝑃 moves further ahead of the volume of air swept in the horizontal direction. The latter effect decreases Δ𝑃𝑑𝑉 work, so, for very inertive loads, the oscillation stops. Lipping upwards is limited partly because negative mouthpiece pressure must always overlap largely with the lips being closed, and the limited magnitude of sweeping flow limits the extent to which flow can lead the lip aperture. Consequently, further increases in lip stiffness produce jumps to the next impedance peak.

Introduction

In brass and other lip-valve instruments, the player's lips usually vibrate at a frequency slightly above a peak in the acoustic impedance spectrum 𝑍 &'() (𝑓) of the instrument bore, measured at the mouthpiece. Nevertheless, by changing playing parameters, they are able to 'lip up and down' so as to play over a range of frequencies from above this peak (a compliant load) to below (inertive). To understand lip 2 vibration and the effect of up-and downstream acoustic loads, it is helpful to know the phase and magnitude of 𝑍 &'() (𝑓) and 𝑍 -'./0 (𝑓), of the acoustic pressures 𝑝 &'() (𝑡) and 𝑝 -'./0 (𝑡), the airflow 𝑈 4 (𝑡) between the lips and that due to the sweeping motion of the lips 𝑈 56 (𝑡). The motion of the lips, the total upstream pressure 𝑃 -'./0 and the total flow 𝑈 are also important. Here we report measurements of all of these quantities for lipping up and down. Seven players were observed and their general results are discussed but, in this preliminary study, the results for only one of the players are analysed in details.

Previous studies of lip-valve instruments have focused on lip motion, e.g. [START_REF] Martin | Lip vibrations in a cornet mouthpiece[END_REF][START_REF] Copley | A stroboscopic study of lip vibrations in trombone[END_REF][START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF][START_REF] Newton | Mechanical response measurements of real and artificial brass players lips[END_REF][START_REF] Bromage | Open areas of vibrating lips in trombone playing[END_REF][START_REF] Chen | Do trumpet players tune resonances of the vocal tract?[END_REF][START_REF] Fréour | Acoustical interaction between vibrating lips, downstream air column, and upstream airways in trombone performance[END_REF]. We have previously studied lip motion and related it to the up-and downstream AC and DC pressure and flow components [START_REF] Boutin | Relationships between pressure, flow, lip motion and upstream and downstream impedances for the trombone[END_REF], but that study considered only normal playing pitch. Here we apply these techniques to notes lipped up (where the load is even more compliant) and down (inertive). The ways in which an auto-oscillating valve adapts to acoustic loads with large phase differences can be expected to give insights into its basic mechanics.

Materials and Methods

The instrument and mouthpiece used in the previous study (a Yamaha trombone YBL 321, [START_REF] Boutin | Relationships between pressure, flow, lip motion and upstream and downstream impedances for the trombone[END_REF]) are used with the slide in the contracted position. Players played B ♭2, whose nominal frequency at A440 is 117 Hz, but 'lipped' it over the range 100 Hz to 125 Hz, which is approximately four semitones.

𝑍 -'./0 was measured during playing as reported previously [START_REF] Chen | Do trumpet players tune resonances of the vocal tract?[END_REF]. The impedance head comprised two small cylindrical ducts glued together. One duct connected to a pressure transducer (8507C-2, Endevco, CA) and preamplifier connected to a FireWire audio interface (MOTU 828, Cambridge, MA) (see Fig. 1). The second delivered a sound current source comprising a sum of sine waves from 50 to 1000 Hz with spacing 0.67 Hz. The impedance head was calibrated with an acoustically infinite duct [START_REF] Dickens | Improved precision in acoustic impedance measurements by using calibration loads without resonances[END_REF]. The pressure transducer signal was modulated to preserve its DC component and later demodulated. With no injected sound signal, this transducer measured the mouth pressure. Mouthpiece pressure was measured with a similar transducer. In separate experiments, measurements of 𝑍 &'() were made as a function of time 3 s after players had been playing a cool dry instrument for 10 s (to simulate conditions in the experiment) and extrapolated by linear regression back to the time at which playing stopped.

3

A high-speed camera (X-stream VISION TM XS-4), synchronized with the interface, recorded (x, z) images directly through the window from the side of the lips and (y, z) images opposite the lips via a mirror parallel to 𝑧 and at -45 ° to 𝑥 (Figs 1 &2). Seven players were measured: four advanced and three beginners. The advanced player analysed in this study initially spent several minutes becoming familiar with playing the instrument, positioning the impedance head and triggering measurements. For measurements, he began by playing a note sharp and then lipping it up as far as he could without 'jumping' to the next resonance. After a few seconds and when satisfied with the stability of the new pitch, the player pushed a switch, which started recording of images by the camera. Later, the player started with a note played flat and lipped down as far as possible. The instrument was dried with compressed air between each measurement.

Following [START_REF] Boutin | Relationships between pressure, flow, lip motion and upstream and downstream impedances for the trombone[END_REF], the total acoustic flow is calculated by dividing the acoustic pressure measured up-and downstream by the mouth and bore impedances respectively (all quantities complex, of course). The total flow swept by the motion of the lips is calculated from the shapes of the lips in each camera frame; this is discussed further below. The difference between the total flow and the sweeping flow is the flow through the lip aperture. When the lips are closed, the aperture flow is zero, so the total flow is equal to the sweeping flow. This identifies the DC flow; a constant value to add to the total acoustic flow, to minimize its difference from the sweeping flow while the lip aperture is closed.

Results and discussion

Figure 2 shows still images for one cycle for each of lipping down, normal and lipping up of B ♭2. In the side view (x, z), x is the longitudinal displacement and z the vertical; the lower series shows the front (y, z) view through the main mouthpiece lens, including the aperture between the lips.

The images show that the x motion leads the z motion in phase: the first few (x, z) images show that the lips move forward into the mouthpiece while closed; later images show that most of the opening of the lip aperture occurs while they are displaced forward and the last several in the series show that most of the closing occurs either during the end of the lips' retraction or while retracted against the teeth. Similarly, the volume 𝑉 >?@ of lip tissue in the mouthpiece is ahead in phase of the aperture between the lips. Qualitatively, this is similar to reports of brass players' lips made (for normal playing) by Copley and Strong [START_REF] Copley | A stroboscopic study of lip vibrations in trombone[END_REF] and Yoshikawa and Muto [START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF].

Fig. 3 plots 𝑥 A , 𝑥 B and 𝑧 A over the cycle, where the 𝑥 are the maximum longitudinal displacements of the upper and lower lips and 𝑧 A the maximum value of the aperture below the top lip. The phasor diagrams of Fig. 4 show magnitudes and phases of the fundamental component of eight waveforms: mouthpiece pressure 𝑃 &'() , ∆𝑃 across the lips, total acoustic flow 𝑈 4D , longitudinal sweeping flow 𝑈 E , opening area, and the 𝑥-and 𝑧-displacements of the lips. The length of each line corresponds to the magnitude of the fundamental; its angle relative to the horizontal axis shows the phase difference with the mouthpiece pressure, chosen as a reference. (For lipping down, 𝑈 E > 𝑈 4D at the fundamental frequency, but the positive average flow means that aperture flow is unidirectional.)

Δ𝑃 is almost exactly at π with respect to 𝑃 &'() ; this is because mouth impedance is small, so the acoustic pressure in the mouth is much smaller than that in the bore. The total acoustic flow is 58° behind 𝑃 &'() for the inertive load when lipping down, but ahead for the compliant load; 62° when lipping up and 59° for normal playing. The aperture is approximately in phase with 𝑃 &'() for lipping up (0.2° ahead) and normal pitch (5° ahead) and 45° behind for down. The forwards position x of the leading lip is ahead of the aperture for lipping up and normal pitch, respectively 42° and 33°, and further ahead, 55°, for lipping down: as remarked above, the lips move forward before opening. The aperture flow is a large fraction of total acoustic flow, so this limits how far its phase can differ from that of the lip aperture A (Figs 234). Because of the small upstream impedance, the pressure difference across the lips Δ𝑃 is almost in antiphase with 𝑃 &'() . Δ𝑃 can easily be large when the lips are closed (and it accelerates them forwards). When the lips are open, Δ𝑃 can include a Bernoulli term, i.e. a pressure difference ~H I 𝜌𝑣 I required to accelerate the air, plus small viscous losses. (At the low pitch considered here, the inertance of the air between the lips has little effect.) However, a large 𝑣 with even a modest lip aperture requires a very large flow. So the fundamental of Δ𝑃 is roughly in antiphase with A. These two effects limit how far the phase of 𝑃 &'() can be from that of U, Nevertheless, U is not sinusoidal and its maximum may occur early during the lip-open period, especially if the sweeping flow is large, or it may occur late. The large, short, negative pulse in 𝑃 &'() must occur while the lips are closed, as observed in [START_REF] Boutin | Relationships between pressure, flow, lip motion and upstream and downstream impedances for the trombone[END_REF], but can occur early in that period (for lipping down) or late (for lipping up). These effects might constrain the range of lipping up or down. So might energy effects, which we now discuss.

The images, displacement curves and phasors in Figs 234show that the x motion of the lips leads the z motion: the lips come forwards while closed and return towards the teeth while open. This phase difference is exaggerated a little in the simple diagram in Fig. 5, which follows a model of Strong and Dudley [START_REF] Strong | Simulation of a player-trumpet system[END_REF] and Adachi and Sato [START_REF] Adachi | Trumpet sound simulation using a two-dimensional lip vibration model[END_REF], a model that treats the lips as swinging elastic plates. For an approximate calculation, we assume that the pressures 𝑃 -'./0 and 𝑃 &'() acting on the inside and outside lip surfaces respectively are uniform. The flow is assumed to be laminar in the channel between the lips but turbulent when it leaves the channel, so the pressure in the channel is 𝑃 &'() , which is lower than 𝑃 -'./0 . Because the channel pressure is 𝑃 &'() , there is no pressure-volume work associated with the changes represented by the dotted areas in Fig. 5.

As we argued previously [START_REF] Boutin | Lipping down on the trombone: phases of lip motion and pressures[END_REF], the pressure-volume work produced around a cycle by 𝑃 -'./0 -𝑃 &'() = Δ𝑃 can be positive. First, the DC component of Δ𝑃 does positive work around a cycle because the flow swept in the x direction 𝑈 M = 𝑑𝑉 M 𝑑𝑡 ⁄ is positive around a cycle, as shown by the difference between the light gray and the dark gray areas in Fig. 5 ii and iv. Second, in cases where the phase difference between the AC components of Δ𝑃 and 𝑈 M lies between -𝜋/2 and +𝜋/2, this also gives positive work. Thus, it is necessary to distinguish the flow swept by the lips in the x and z directions.

The outside outline of the lips (𝑥(𝑧)) and opening aperture (in the (y, z) plane) are available from the video. For each measurement, the effective sweeping width 𝐿 is estimated as the value which minimizes the difference between total flow and total sweeping flow, when the lips are closed. (This gives values between 13 and 16 mm, in a mouthpiece width of 23 mm, which is about what one would calculate if the lip x motion had an envelope with shape between sine and sin I (𝜋𝑦/23 mm) across the mouthpiece). From the upper and lower limits of the aperture, effective upper and lower aperture limits, 𝑧 /'@ (𝑡) and 𝑧 &'//'-(𝑡), were determined. The volume component 𝑑𝑉 M (𝑡) swept by the lips in the x direction is:

𝑑𝑉 M (𝑡) = 𝐿 YZ [𝑥(𝑧, 𝑡) -𝑥(𝑧, 𝑡 -𝑑𝑡)]𝑑𝑧 ^_``ab cdedf ^fg`( h) + Z [𝑥(𝑧, 𝑡) -𝑥(𝑧, 𝑡 -𝑑𝑡)]𝑑𝑧 ^igffge (h) ^cgjab cdedf k (1)
From this, the longitudinal sweeping flow 𝑈 M = 𝑑𝑉 M /𝑑𝑡 is calculated, and the longitudinal sweeping work is 𝑑𝑊 = ∆𝑃𝑑𝑉 M (𝑡). As shown in Figs 2 to 5, the lips are closed when they move forwards and open when they retract. Hence the quantity ∮ 𝑑𝑉 M round a cycle is positive: +37 mm o while lipping down, +7 mm o at normal pitch and +9 mm o while lipping up. In addition, the lips sweep air in the x direction before opening. Consequently, the longitudinal sweeping flow leads the total acoustic flow: by 147°, 52° and 51° for the three pitches. Therefore, this component of flow contributes to increase the phase of the total acoustic flow, relative to the mouthpiece pressure. We also note that 𝑈 M is about 90° ahead of the average 𝑥 displacement of the upper and lower lips: 83° for lipping down, 96° for normal pitch and 93° for lipping up. This is because 𝑈 M = 𝑑𝑉 M /𝑑𝑡 is proportional to the time derivative of 𝑥, as defined in (1). 1 gives the sweeping power for lipping down, normal and up, expressed in energy per cycle for comparison with rough upper estimates of the maximum kinetic energy of the lips: 𝐾 q = 𝑚𝑣 Mstt I cos I (𝛼/2), where 𝑚 = 𝜌 𝜋𝑅 I 𝑇 >?@ = 2 g is the mass of the lips, taking 𝜌 ≈ 1000 kg. m €o , 𝜋𝑅 I = 4.2 cm I the area inside the mouthpiece rim, 𝑇 >?@ the lip thickness ~ 5 mm, and 𝛼, respectively equal to 55°, 34° and 42° for lipping down, normal playing and lipping up, is the phase difference between 𝑥 and 𝑧. The peak effective 𝑥-velocity of the lip motion, is taken as the peak in the derivative of 𝑉 >?@ , divided by 𝜋𝑅 I .

These rough estimates of maximum kinetic energy of the lips during a cycle (Table 1) are lower than or comparable with the sweeping energy gained by the lips during each cycle. Thus, even if the lip-teeth and lip-lip collisions were completely or largely inelastic, i.e. all or nearly all the lips' kinetic energy were lost in that collision, the sweeping power delivered by Δ𝑃 would be large enough to replace it for the next cycle. (The collisions are expected to lose more energy than internal losses.)

Dividing ∆𝑃 into its steady and oscillatory components, ∆𝑃 = ∆𝑃 •••• + ∆𝑃 ' , we write the sweeping work, 𝑊 = ∮ ∆𝑃𝑈 M 𝑑𝑡 = ∆𝑃 •••• ∮ 𝑈 M 𝑑𝑡 + ∮ ∆𝑃 ' 𝑈 M 𝑑𝑡. The first term (first line of Table 1) is positive because the steady component of Δ𝑃 and the sweeping flow ∮ 𝑈 M 𝑑𝑡 are both positive. The second term (second line of Table 1) depends on the phase difference between 𝑈 M and the acoustic pressure difference across the lips, ∆𝑃 ' . This phase difference increases as f decreases: 72° at 119.5 Hz, 73° at 118.4 Hz and 90° at 107.9 Hz; in the last case, it thus contributes negligible sweeping work (as in Table 1). At the limit of lipping down, this player increases the x motion and increases the phase difference between x and z motion. Thus, even with the mouth pressure at a similar value, the steady component of 𝑊 increases. This keeps the total 𝑊 comparable with the maximum kinetic energy of the lips, and hence maintains their oscillation.

Conclusions and perspectives

Players normally played a little above the impedance peak, and so the bore is a slightly compliant load. However, the player in this study could lip down to have an inertive load, with 𝑃 &'() leading U by up to 58° and with 𝑍 &'() at 57% of maximum, or lip up to have U lead 𝑃 &'() by up to 62° and 𝑍 &'() at 35% of maximum.

For this player lipping up and at normal pitch, the steady and oscillatory components of Δ𝑃 provide positive work 𝑊 on the non-zero sweeping flow ∮ 𝑈 M 𝑑𝑡. While lipping down, ∆𝑃 ' is π/2 out of phase with the sweeping flow 𝑈 E , and only Δ𝑃 •••• contributes work. In all cases, the sweeping work exceeds or is comparable with estimates for the maximum kinetic energy of the lips. Thus, the simple model in Fig 5 explains the source of energy for regeneration. Let us model it as a simple one degree-of-freedom harmonic oscillator, whose 𝑥 motion is driven by Δ𝑃. To lip up, players increase the lip stiffness and hence the lip resonance frequency 𝑓 >?@ . This helps reduce the phase lag of 𝑥 and hence 𝑈 E behind Δ𝑃, and the oscillatory component of 𝑊 rises. As players relax their lips, the resulting decrease of oscillatory sweeping work is compensated by the increase of its steady component, because Δ𝑃 is maximum around the peak of 𝑍 &'() . This allows players to lip As the lip stiffness further decrease, 𝑊 eventually falls below the collision and internal losses, so the oscillation stops, or (for higher bore resonances) the playing frequency jumps to the next resonance below. Analysis of the results for all of the subjects, figures for the time variation of all variables and a further discussion of lip mechanics will be included in a paper with longer page limit.

Figure 1 :

 1 Figure 1: Schematic of the experiment (not to scale).

Figure 2 .

 2 Figure 2. Twenty four equally spaced stills of side views (x,z) and six equally spaced front views (y,z) from one cycle of the videos of lipping down (top), normal playing (middle) and lipping up (bottom). The numbers above indicate the side views that are simultaneous with each of the front views.

Figure 3 .

 3 Figure 3. Plots of 𝑥 (the maximum longitudinal displacement of the lips) and 𝑧 A (the maximum vertical displacement of the top lip relative to the aperture reference, where the lips separate) around two cycles for lipping down (a), normal playing (b) and lipping up (c). The circled numbers correspond to the front views of Fig.2.

Figure 4 .

 4 Figure 4. Phasor diagrams of the fundamental components for lipping down (a), normal pitch (b) and lipping up (c): mouthpiece pressure, acoustic pressure difference across the lips, total acoustic flow and longitudinal sweeping flow in the mouthpiece, longitudinal and vertical components of the lips' motion.

Figure 5 .

 5 Figure 5. A simple model of the motion of the upper lip: the lower lip would be roughly symmetric below the dashed line. From (i) to (ii) the lip moves forwards (x direction) while closed. From (ii) to (iii), it contracts in the z direction. From (iii) to (iv) it retracts while open. From (iv) to (i) it closes while against the teeth.

Table 1 :

 1 algebraic sweeping power and work done on the sweeping motion in one cycle, compared with an estimate of the lips' kinetic energy, for lipping down, normal pitch and lipping up.

			Lipping down (107.9 Hz)	Normal pitch (118.4 Hz)	Lipping up (119.5 Hz)
	Longitudinal	DC component	+42 µJ/cycle (+4.6 mW)	+12 µJ/cycle (+1.4 mW)	+9 µJ/cycle (+1.0 mW)
	sweeping power	AC component	-3 µJ/cycle (-0.3 mW)	+67 µJ/cycle (+7.9 mW) +31 µJ/cycle (+3.7 mW)
	Estimate of maximum kinetic energy of the lips	37 µJ	25 µJ	14 µJ
	Table


Dr Bruno Agostini
email: bruno.agostini@epfl.ch

André É Bontemps

Bernard Thonon

Effects of Geometrical and Thermophysical Parameters on Heat Transfer Measurements in Small-Diameter Channels

published or not. The documents may come

Effects of geometrical and thermophysical parameters on heat transfer measurements in small diameter channels

INTRODUCTION

In recent years, an interest in heat transfer at small scales has led to an increase in their reference in publications. In particular, mini-channels are of high interest to the industry due to the possibility of manufacturing efficient compact heat exchangers. Indeed, mini-channel heat exchangers with a hydraulic diameter of about 1 mm achieve higher heat transfer coefficients and thermal efficiency and a lower fluid charge, all of which allow for space savings and lower costs.

Table 1 summarizes the experimental conditions of previous studies on mini-channels. In the sixth column, the critical macro-tubes, while in a number of cases, new physical processes or phenomena are supposed to be preponderant at small scales to explain significant deviations. However, phenomena like the Electrical Double Layer, the viscous dissipation, or the breaking of the continuous medium approximation for gases should not occur over about ten µm. It is also reported that vapor trapped in the wall bumps or the effect of tube roughness creating turbulent wakes may have an increased influence in mini-channels. This study presents experimental friction factors and heat transfer coefficients obtained in four mini-channels of varied size and geometry, using refrigerant R134a liquid flow. Experimental results in mini-channels agree with those obtained in macro-tubes, provided that experimental conditions are tightly controlled, that measurements are carried out with increased care, and that the usual hypotheses in the data analysis are still valid. Three error sources of importance at small scales were identified: large uncertainties on the channel dimensions, singular pressure losses in manifolds, and longitudinal heat conduction in the tube walls in the laminar regime.

EXPERIMENTAL DEVICE

The test loop (shown in Figure 2) includes a liquid pump (10-100 l/h) and a glycol-water mixture circuit for heat evacuation. The test section, shown in Figure 3, consists of industrial flat tubes made of extruded aluminium and composed of numerous parallel rectangular channels or of independent parallel tubes for the circular geometry. Table 2 gathers the characteristics of all the tested geometries. The inlet and outlet manifolds are 10 mm-diameter tubes at 90 degrees. The manifold diameter is between five and thirteen times greater than the channel hydraulic diameter to limit the fluid distribution effects. The whole test section is thermally insulated with 40 mm-thick wrapping foam. For heat transfer measurements, the length L j of the tube was heated using Joule effect by passing an electric current (up to 2800 A) from two brased electrodes through the tube wall. Experimental conditions are summarized in Table 3.

Figure 3 shows the test sections and instrumentation. It is worth noting that in the case of the circular channels, the test section was built with independent mini-tubes, and the electrodes were manufactured with the manifold on the same aluminium piece. Wall temperatures T w,k (0 < k < 9) on the tube external surface were measured with 0.5 mm-diameter calibrated type E thermocouples. Fluid inlet and outlet mixed mean temperatures (respectively, T f,in and T f,out ) were measured with 1 mmdiameter calibrated type K thermocouples in a T junction to ensure mixing. Calibration was carried out every five K between 268 and 333 K with a Rosemount 162-CE platinum thermome- ter. The thermocouples used for wall temperature measurements were equally spaced and fixed on the tube surface. The fluid inlet pressure was measured with calibrated Rosemount type II absolute pressure sensors. Three Kent-Deltapi K differential pressure sensors calibrated from 0 to 7.6, 40.5, and 182.2 kPa measured the pressure loss through the test section. A Rosemount Micromotion coriolis flowmeter was used to measure the mass flux of R134a downstream of the pump. The heating voltage V and current I (via a shunt) were measured directly via a HP 3421A multiplexer.

The heat flux was varied for every mass flow rate in order to keep the fluid temperature rise between 20 and 40 K. Steady state values were monitored using the Hewlett Packard 3421A with a 30-minute time lapse between each mass flow rate or heat flux change. Averaging occurred after every twenty values, and the standard deviation was used to calculate the uncertainties according to the Kline & McClintock [4] and Moffat [START_REF] Moffat | Contributions to the Theory of Single-sample Uncertainty Analysis[END_REF][START_REF] Moffat | Using Uncertainty Analysis in the Planning of an Experiment[END_REF] methods. The uncertainty on temperature measurements was estimated to be ±0.1 K. The thermocouples' locations were measured with a calliper rule, and the corresponding uncertainty, ±0.5 mm, was taken into account in the Kline & McClintock method. The other uncertainties are reported in Tables 2 and3. The total electrical power dissipated in the test section was calculated as the product of voltage and current. The variation of R134a thermophysical properties with temperature was calculated with the REFPROP 6.01 software from the NIST Standard Reference Database 23.

Non-uniform flow distribution can occur in multi-channels systems. In order to avoid this effect, the engineering rule that the manifolds diameter should be at least five times higher than the channel hydraulic diameter to equalize the fluid distribution was used. This does not ensure that non-uniform distribution is totally suppressed, but it should reduce it to within acceptable limits. However, even if non-uniform distribution occurs, it will not affect the inlet and outlet measurements, which are performed out of the manifolds. The local temperature measurements should also not be affected because of the averaging of wall temperatures across the N channels due to the aluminium's very high thermal conductivity.

THE DRAMATIC ROLE OF THE GEOMETRICAL UNCERTAINTIES

A classical uncertainty analysis in the case of a nearly square channel, taking only into account errors on dimensions, immediately demonstrates the influence of small geometrical uncertainties on 4f and Nu. If l 1 is the length of a channel side, it is known that

4 f = p f • D h L • 2ρ l • S 2 f Ṁ2 , ( 1
)
yielding 4 f = p f L • 2l 1 • l 2 l 1 + l 2 • 2ρ l (l 1 • l 2 ) 2 • N 2 Ṁ2 , ( 2
)
so that 4 f 4 f = 3 l 1 l 1 + 3 l 2 l 2 + (l 1 + l 2 ) l 1 + l 2 . ( 3
)
If

l 1 = l 2 , then 4 f 4 f ≈ 7 l 1 l 1 . ( 4
)
The same reasonning leads to

Nu Nu = V • I k f • L • (T w -T f ) l 1 • l 2 (l 1 + l 2 ) 2 ≈ 4 l 1 l 1 . ( 5
)
As a consequence, an error as low as ±2% on the channels' dimensions leads to uncertainties of ±14% on 4f and ±8% on Nu, for the geometrical part of the uncertainty only. Upon adding the other uncertainties onto pressure or temperature measurements, for instance, the total uncertainty can then very easily reach ±30% in the laminar regime. This obvious remark should be an incentive to determine the dimensions of mini-channels, in particular rectangular channels, with great care.

In order to reach the required degree of accuracy, the determination of the channel dimensions was carried out using scanning electron microscopy on several channels on different sections of tubes, allowing measurements to be done with precision. It is evident from Figure 4, which shows an electron microscopy photograph of the mini-channels, that the knowledge of l 1 and l 2 dimensions is not sufficient to calculate the hydraulic diameter. Indeed the circular corners and the rounded side channels must be taken into account to obtain an accurate value of D h . Accordingly, the hydraulic diameter was calculated by

D h = 4A f P f . ( 6
)
A f and P f are the total wetted area and perimeter, including the N channels. Both A f and P f were determined from several electron microscopy pictures. Several values were averaged and twice the standard deviation (95% confidence level) was taken for the uncertainties. This method ensured that all the geometric features were included. Table 4 shows the difference between the actual hydraulic diameter (Eq. 6) and that calculated from the manufacturer's blueprints, or the dimensions l 1 and l 2 . These differences can be important and justify the use of Eq. ( 6) to calculate D h . Furthermore, roughness measurements were carried out on tubes 1, 3, and 4, leading to R a /D h = 3 × 10 -4 , so that these tubes can be treated as smooth tubes.

FRICTION FACTOR AND SINGULAR PRESSURE LOSSES

Pressure loss measurements were performed under adiabatic conditions. The total pressure loss through the test section is:

p = p f r + p sin g (7)
with the frictional pressure loss

p f r = G 2 2ρ l • 4 f • L D h ( 8
)
and the singular pressure loss

p sin g = G 2 2ρ l • ξ (9)
Since pressure loss measurements were performed with a horizontal configuration, there is no gravitational pressure loss. In parenthesis are the differences with column 2. 3.

Thus, experimental pressure losses can be expressed under the dimensionless form

2ρ l • D h G 2 • L • p = 4 f + ξ • D h L . ( 10
)
This dimensionless pressure loss is plotted in Figure 5 as a function of the Reynolds number and is compared with the friction factors for rectangular, circular, and square ducts. In the laminar regime, these friction factors are given by the generic relation 4 f • Re = constant, the constant depending on the geometry; in the turbulent regime, they are given by the Blasius relation. It is apparent from Figure 5 that friction pressure losses cannot account for the experimental ones. The following argument demonstrates that this is due to singular pressure losses.

The frictional pressure loss in the straight part between the pressure sensor and the manifold is about 0.3% of the total pressure loss. This effect is then negligible. In order to assess the singular pressure loss coefficient caused by the T junctions in the manifolds, Eq. ( 10) multiplied by Re was plotted against the Reynolds number for 140 < Re < 1 700 (see Figure 6). In the laminar regime, the friction factor should not depend on the tube roughness so that the inlet and outlet effects can be isolated in this region. The linearity of the points on Figure 6 indicates that Eq. ( 10) is equivalent to

4 f • Re + ξ • D h L × Re = b + a • Re, (11)
with a and b being numerical constants. The singular pressure loss coefficient is a function of the manifolds geometry only so that it is independent of Re. Thus, it is possible to identify the right and left hand terms of Eq. [START_REF] Ghajar | Flow Regime Map for a Horizontal Pipe with Uniform Wall Heat Flux and Three Inlet Configurations[END_REF]. Since ξ • D h /L is constant, this identification necessarily gives

4 f • Re = b and ξ • D h L = a. ( 12
)
The singular pressure loss coefficient taking into account inlet and outlet effects can be calculated with this method, and the 4 f • Re product is found to be constant as expected. Results are 3.

gathered in Table 5. Uncertainties are calculated by evaluating Eq. ( 11) with lower and higher error marks. The 4 f factors evaluated with Eq. ( 10) can now be plotted as functions of Re (see Figure 7), together with the Shah and London [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] correlation for a laminar flow in rectangular channels,

4 f • Re = 96(1 -1.3553 • γ + 1.9467 • γ 2 -1.7012 • γ 3 +0.9564 • γ 4 -0.2537 • γ 5 ), (13)
the well known Poiseuille law for circular channels,

4 f • Re = 64 ( 14
)
and the Blasius [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF] correlation for turbulent flows,

4 f = 0.316 • Re -0.25 . ( 15
)
The laminar-turbulent transition is clearly visible for 1400 < Re < 2000, very close to the accepted value for macrotubes. As highlighted by Tam and Ghajar [START_REF] Tam | Effect of Inlet Geometry and Heating on the Fully Developed Friction Factor in the Transition Region of a Horizontal Tube[END_REF], who studied the effect of the inlet conditions on the laminar-turbulent transition, severe inlet conditions like re-entrant inlets should cause the transition to happen at lower Reynolds numbers than smooth inlets. This effect is indeed observed in the present results. In the laminar regime, the friction factor is in good agreement, 3.

within uncertainties, with Eqs. ( 13) and ( 14). In the developing turbulent regime, the friction factor is underpredicted by Eq. ( 15) for tubes 1 and 2 and within uncertainties for tubes 3 and 4. Table 3 indicates uncertainties on 4 f and Re. Finally, results in both the laminar and turbulent regime are in quite good agreement with the results in macro-tubes. The observed maximum deviation is +37% with the Blasius correlation for tube 1, which has a behavior slightly different from the others. This might be due to the relative importance of the side channels because of the reduced number of mini-channels (11) in tube 1.

In the last column of Table 5, the ratio between the manifolds and channels flow areas are reported. Compared to the ξ column, it appears that the ratio A /A f is characteristic of the singular pressure loss coefficient, which decreases with increasing A /A f . From the data on tube 4, for which uncertainties alone can explain the small deviations from known results, it can be assumed that for A /A f > 10, ξ is negligible. This may be taken as a rule of thumb for manifold design.

HEAT TRANSFER COEFFICIENT AND LONGITUDINAL CONDUCTION

The variation of Pr 1/3 due to the fluid temperature rise is about 2% so that it cannot explain the observed variations of the Nusselt number. The implicit assumption of one-dimensional conduction, i.e. between the external and internal tube wall only, is the starting point to calculate Nusselt numbers from experimental heat transfer data. Our goal is to show from experimental data analysis that this assumption is frequently wrong in the case of mini-channels for which the wall thickness can be of the same order of magnitude as the hydraulic diameter (e is 30% of D h for tube 4).

Heating by means of the Joule effect ensures that the volumetric power density is uniform within the aluminium and that the heat flux does not depend on z if heat losses are neglected.

However, it cannot be inferred that the peripheral heat flux around a channel is uniform. Some numerical simulations [START_REF] Agostini | Étude Expérimentale de l' Ébullition en Convection Forcée de Fluide Réfrigérant dans des Mini-Canaux[END_REF] with the FLUENT software proved that as a consequence of the high thermal conductivity of aluminium and the low heat transfer coefficients in the laminar regime, the internal wall temperature is uniform at the periphery of a channel. On the contrary, the heat flux varies and is nearly nil in the corners. If A x , qy (x), h(x) and A y , qx (y), h(y) are the heat transfer areas, heat fluxes, heat transfer coefficients parallel to the ox and oy axis, respectively, the average heat transfer coefficients on each wall can be calculated as follows (see Figure 3 for x,y coordinates):

h(x) = qy (x) T w -T f and h(x) = qx (y) T w -T f , ( 16
)
and the average peripheral heat transfer coefficient is

h = h(x) • A x + h(y) • A y A j = q y (x) • A x + qx (y) • A y A j • (T w -T f ) (17) = q T w -T f with q • A j = q y (x) • A x + qx (y) • A y , (18)
as the temperature difference T w -T f is constant over the periphery of a channel. As a consequence, the average heat flux, subsequently noted q,

q = V • I A j = V • I P f • L j . ( 19
)
can be used to calculate the average peripheral heat transfer coefficient, which is the relevant parameter.

Let us now assume that only one-dimensional phenomena occur, and let us conduct the classical reasoning to extract Nusselt numbers from measurements. Due to the high thermal conductivity of the aluminium (≈200 W/m K) and the low thickness of the tube walls, the measured temperature (T w,ext ) is equal to the wall temperature in contact with the fluid (T w,int ). This can be shown by using classical one-dimensional conduction (see Eq. [START_REF] Peng | The Effect of Thermofluid and Geometrical Parameters on Convection of Liquids through Rectangular Microchannels[END_REF]). Using this equation, the temperature difference is estimated to be less than 0.01 K,

T w,ext -T w,int = ω • e 2 k al + qleak • e k al , ( 20
)
where ω is the volumetric power density and qleak is the heat leaks density. The energy balance was checked, and thermal leaks, defined by

Qleak = Ṁ • c p,l • (T f l,out -T f l,in ) -V • I. ( 21
)
were found to be less than 15%, 10%, and 5% of the injected power for 100%, 87%, and 52% of the data points, respectively. Therefore the fluid temperature at z position was calculated via is the asymptotic theoretical prediction from [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] when Re → 0; uncertainties are in Table 3.

an energy balance ( Qleak = 0), giving

T f (z) = T f,in + V • I Ṁ • c p,l • L j • z ( 22
)
Two different heat transfer coefficients, a local and an averaged one, were determined. The local heat transfer coefficient and Nusselt numbers are defined as

h(z) = q T w (z) -T f (z) and Nu(z) = h(z) • D h k l (z) . ( 23
)
The average Nusselt number is defined as

Nu = 1 10 9 k=0 Nu k . ( 24
)
Accordingly, the local and average Reynolds numbers are

Re(z) = G • D h µ l (z) and Re = 1 10 9 k=0 Re k . ( 25
)
It was verified that mixed convection could not occur. Indeed, according to Ghajar and Tam [START_REF] Ghajar | Flow Regime Map for a Horizontal Pipe with Uniform Wall Heat Flux and Three Inlet Configurations[END_REF], over the present Reynolds number range with a uniform heat flux condition, the mixed convection should occur for Gr • Pr •D h /L j > 330, while the value for the present experiment was approximately 20. Thus, only forced convection can take place.

In Figure 8, the averaged Nusselt numbers are plotted against the Reynolds number for each tube. In every case, as already observed from the friction factor, the transition from the laminar to the turbulent regime occurs for Re ≈ 2000. In the turbulent regime, the global Nusselt number agrees with the Gnielinski [START_REF] Gnielinski | New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow[END_REF] correlation, modified to take into account the thermal entry length effect with f calculated using Eq. ( 15). This correlation is valid for 2300 < Re < 10 6 and 0.6 < Pr < 10 5 .

Nu = ( f /2) • (Re -1000) • Re 1 + 12.7 • √ f /2 • (Pr 2/3 -1) • 1 + D h L j 2/3 , ( 26
)
In Figure 9 the Nusselt numbers obtained with the 0.77mmhydraulic diameter tube are compared with some recent heat transfer correlations. The Ghajar and Tam [START_REF] Ghajar | Heat Transfer Measurements and Correlations in the Transition Region for a Circular Tube with Three Different Inlet Configurations[END_REF] correlation was designed to take into account entry effects in macroscale tubes. Adams et al. [START_REF] Adams | An Experimental Investigation of Single-Phase Forced Convection in Microchannels[END_REF], Choi and Barron [START_REF] Choi | Fluid Flow and Heat Transfer in Micro Tubes[END_REF], and Garimella et al. [START_REF] Garimella | The Effect of Simultaneously Developing Flow on Heat Transfer in Rectangular Tubes[END_REF] correlations were obtained with heat transfer data in minichannels, as reported in Table 1. The correlation proposed by Ghajar and Tam [START_REF] Ghajar | Heat Transfer Measurements and Correlations in the Transition Region for a Circular Tube with Three Different Inlet Configurations[END_REF] for a re-entrant inlet, similar to the present case, is compared to the present results. The prediction is the best of the four tested correlations, in particular in the laminar regime, but the slope of the Nu(Re) curve is still wrong. However, in the turbulent regime, the presented experimental results are still best predicted by the Gnielinski [START_REF] Gnielinski | New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow[END_REF] correlation.

If the laminar regime is fully developed, the Nusselt numbers should be constant. Figure 8 shows that this is not the case, and an increase of Nu with Re is observed. As a consequence, the effect of the entry length is first studied. The thermal entry length for constant heat flux heating is given by L th /D h = 0.0431 • Re • Pr [START_REF] Taine | Transfers Thermiques[END_REF]. In the present experiment, for 300 < Re < 2000 and tube 1, L j /D h = 343 and 50 < L th /D h < 320; thus, the heated and entry lengths are of the same order of magnitude. It is then possible that the flow is not fully developed. As a result, Nu slightly increases with Re in the laminar region. The Shah and London [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] tabulated values for a thermally developing flow are reported in Figure 8. It follows that the entrance effect explains the slight increase of Nu for tubes 1 and 3 when 500 < Re < 2000 but does not account for the strong drop of the Nusselt number at low Reynolds numbers.

In order to explain this behavior, the wall and fluid temperatures are plotted against the z coordinate in both the laminar and turbulent regimes. In Figure 10, the wall and fluid temperatures of tube 4 are reported for Re = 380 in the laminar regime, and in Figure 11, they are reported for Re = 4004 in the turbulent regime. In addition, in Figure 12, the local Nusselt 3.

number is plotted against the local Reynolds number. As expected for constant heat flux heating, the two profiles are parallel for Re = 4004 (Figure 11) so that the ten local Nusselt numbers are constant along the test section (Figure 12). On the contrary, for Re = 380, the two profiles are divergent (Figure 10), and the ten local Nusselt numbers regularly decrease along the test section (Figure 12). The theoretical wall temperature profile (i.e., calculated from the heat transfer coefficient predicted by Shah and London [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF]) is also reported in Figure 10, showing that the experimental slope is about twice the theoretical one.

Such an effect can be attributed to longitudinal heat conduction within the channels walls in the oz direction, which modifies the wall temperature profile. The Biot number defined as Bi = h • L j k al [START_REF] Celata | Experimental Investigation of Hydraulic and Single-phase Heat Transfer in 0.130 mm Capillary Tube[END_REF] may be taken as the dimensionless number representative of longitudinal conduction. For example, in the case of tube 4, Bi is 3.

Figure 12

Local Nusselt number versus local Reynolds number (tube 4); uncertainties are in Table 3.

about 1.6 in the laminar regime, so that conductive and convective effects compete to determine the wall temperature profile. On the contrary, Bi is about 14 in the turbulent regime, so that only convective heat transfer drives the wall temperature profile. Furthermore, despite the high temperature gradient in the z direction (up to 67 K/m), as the aluminium cross-section of the flat tube is small, the longitudinal heat transfer rate

Qz = k al • T w9 -T w,0 z 9 -z 0 • A al ( 28
)
is very small. In other words, a small imposed longitudinal heat transfer rate (e.g., heat leaks via external connections) will induce a high temperature gradient because of this small crosssection. This observation leads to important consequences, as shown in Table 6, where two data sets are examined and experimental values are compared to predictions from Shah and London [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF]. For tube 4, with as little as a 0.1 W change in the longitudinal heat transfer rate (that is, 2% of thermal leaks or 0.4% of the injected power), the average Nusselt number is overpredicted by a factor of 4.1. The conclusion is not less surprising in the case of tube 1, for which a 0.04 W change in the longitudinal heat transfer rate (that is, 4% of thermal leaks or 0.06% of the injected power) causes the average Nusselt number to be overpredicted by a factor of 1.5. As a consequence, even with a negligible longitudinal heat transfer rate, one can find average Nusselt numbers increasing with the Reynolds number in the laminar regime because of longitudinal conduction. It appears that in small-scale experiments, the data analysis must be revised. Equation ( 20) is no longer valid because it only takes into account the transversal heat conduction flux, and the internal wall temperature can no longer be assumed to be equal to the measured external wall temperature. However, because the Nusselt numbers for tubes 1 and 3 when 500 < Re < 2000 agree with Shah and London [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] predictions, it is felt that heat transfer is not so different in macro-and minichannels in the laminar regime, and that Nusselt numbers independent of Re should be obtained in mini-channels in the absence of any longitudinal conduction. Furthermore, a closer look at Table 2 shows that tubes 1,3 and tubes 2,4, of which the Nusselt numbers follow the same trend, have almost equal aluminium cross-section areas. Equation [START_REF] Agostini | Experimental Study of Single-Phase Flow Friction Factor and Heat Transfer Coefficient in Mini-Channels[END_REF] shows that this is not a coincidence. For a given Qz , T w,9 -T w,0 is inversely proportional to A al , all other parameters being equal. Thus, increasing A al for a given wall thickness reduces longitudinal conduction influence, but in doing so, the heat transfer area is increased so that one must also increase the injected power in order to keep the heat flux constant. Not doing so would reduce the wall to fluid temperature difference to the point that uncertainties would become unacceptable. This is not a trivial problem, as a higher injected power means an increased outlet temperature and a higher working pressure to keep the fluid subcooled at the outlet.

THE LAMINAR-TURBULENT TRANSITION

The transition between the two regimes was found to be conveniently characterized by plotting the uncertainties on the wall temperatures against the Reynolds number (see Figure 13). The uncertainty on T w is

T w = T 2 cal + (2σ) 2 , ( 29
)
T cal being the calibration error and σ the standard deviation calculated from each group of twenty measurements so that 2σ is a statistical uncertainty. Figure 13 exhibits an important peak of uncertainty for 2000 < Re < 4000, and T w is multiplied by ten at its maximum for Re ≈ 2700. The following interpretation is proposed. When the transition toward the turbulent regime begins at Re > 2000, at a fixed position along the tube, the flow will periodically become turbulent, and then fall back to the laminar regime. This causes a periodic shift of the heat transfer coefficient and consequently of the wall temperature, which finally increases the statistical uncertainty on the wall temperature. At mid transition T w is maximum (the flow is supposed to be turbulent half of the time) before decreasing as the turbulence prevails. Then, when the flow is finally always turbulent for Re > 4000, the same value as in the laminar regime is reached with no more heat transfer coefficient fluctuations. 3.

CONCLUSIONS

Frictional pressure loss and heat transfer in mini-channels were experimentally investigated. It was emphasized that three phenomena were likely to explain differences reported between mini-channels and macro-tubes. First, one must note that very precise dimensions measurements (i.e., better than ±1%) are essential when dealing with rectangular mini-channels because of the high sensitivity of 4 f and Nu on geometrical uncertainties.

Second, singular pressure losses were shown to affect the determination of the friction factor in a significant way. Fortunately, these are easy to take into account, and it was demonstrated that with a simple correction, good agreement was found most of the time with friction factors in macro-tubes.

Third, accurate dimension measurements were sufficient to obtain very good agreement between the Gnielinski [START_REF] Gnielinski | New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow[END_REF] correlation and our experimental heat transfer results in the turbulent regime. However no accurate prediction was found in the laminar regime. The Ghajar and Tam [START_REF] Ghajar | Heat Transfer Measurements and Correlations in the Transition Region for a Circular Tube with Three Different Inlet Configurations[END_REF] correlation gave the best prediction but was still unsatisfactory. Thanks to local temperature measurements, it was shown that this discrepancy is probably due to longitudinal conduction. The transversal one-dimensional conduction assumption seems to be no longer valid in minichannels because these have thick walls compared to their hydraulic diameter. In particular, the convective effects between the internal wall and the fluid, and the conductive effects in the mass of the aluminium, become uncoupled at low Reynolds numbers. Then, extremely small longitudinal heat transfer rates (that is, rates that are negligible compared to the heat flux toward the fluid) can modify the external wall temperature profile so that the Nusselt number calculated with the one-dimensional assumption is greatly overpredicted by classical correlations in the laminar regime. Unfortunately, the very smallness of this longitudinal heat transfer rate prevents any correction on the temperature measurements, and close examination of Figure 1 shows that experimentally determined laminar Nusselt numbers in minichannels are still far from macro-tubes results. Thus, new experimental methods should be invented to obtain expected constant Nusselt numbers in the laminar regime for mini-channels.

NOMENCLATURE

Figure 1

 1 Figure 1 Convergence toward macro-tubes correlations.

Figure 2

 2 Figure 2 Test loop.

Figure 3

 3 Figure 3 Test section, for rectangular and circular channels, dimensions in mm.

Figure 4

 4 Figure 4 Electron microscopy picture of the 0.77 mm hydraulic diameter mini-channel.

Figure 5 4

 5 Figure 5 4 f + ξ • D h /L versus Re; uncertainties are in Table3.

Figure 6

 6 Figure 6 Determination of the singular pressure loss coefficient; uncertainties are in Table3.

Figure 7

 7 Figure 7 Friction factor versus Reynolds number; uncertainties are in Table3.

Figure 8

 8 Figure 8 Averaged Nusselt number versus averaged Reynolds number. Nu(0)is the asymptotic theoretical prediction from[START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] when Re → 0; uncertainties are in Table3.

Figure 9

 9 Figure 9 Comparison of averaged Nusselt number versus averaged Reynolds number with some recent correlations (tube 4).

Figure 10

 10 Figure 10 Wall and fluid temperature profile, Re = 380 (tube 4); uncertainties are in Table3.

Figure 11

 11 Figure 11 Wall and fluid temperature profile, Re = 4004 (tube 4); uncertainties are in Table3.

Figure 13

 13 Figure[START_REF] Ghajar | Heat Transfer Measurements and Correlations in the Transition Region for a Circular Tube with Three Different Inlet Configurations[END_REF] Wall temperatures uncertainties versus Re (tube 3); uncertainties are in Table3.

Table 1

 1 Summary of experimental conditions of previous studies

	Year	Author	Fluid	Geometry	D h (mm)	Re	Re cr
	1991	Choi and Barron [15]	Nitrogen	Rectangular	0.003-0.081	30-3000	≈ 2000
	1994	Peng et al. [18]	Water	Rectangular	0.133-0.367	50-4000	200-700
	1994	Wang and Peng [19]	Water	Rectangular	0.311-0.747	80-3000	1000-1500
	1995	Peng and Peterson [20]	Water	Rectangular	0.311-0.646	200-2500	400-1000
	1996	Peng and Peterson [21]	Water	Rectangular	0.133-0.367	450-4000	70-400
	1996	Yang and Webb [22]	R12	Rectangular	1.56-3.25	2500-23000	-
	1998	Adams et al. [14]	Water	Circular	0.76-1.09	2600-23000	-
	1999	Harms et al. [23]	Water	Rectangular	0.4-1.92	173-12900	≈500
	1999	Mala and Li [24]	Water	Rectangular	0.05-0.254	100-2000	≈500
	1999	Yan and Lin [25]	R134a	Circular	2.00	400-10000	≈2000
	2001	Garimella et al. [16]	Glycol	Rectangular	1.74-3.02	118-10671	800-2000
	2001	Hegab et al. [26]	R134a	Rectangular	0.112-0.210	1200-13200	2000-4000
	2002	Ohwaib and Palm [2]	R134a	Circular	0.8-1.7	1000-16000	-
	2002	Celata et al. [27]	R114	Circular	0.130	100-8000	1880-2480
	2002	Agostini et al. [28]	R134a	Rectangular	0.77-1.17	320-12691	≈2000
	2003	Agostini et al. [1]	R134a	Rectangular	2.01	242-7356	≈2000

Table 2

 2 Tested geometries

	Tube	1	2	3	4
	D h (mm)	2.01 ± 3 %	1.44 ± 1 %	1.17 ± 4 %	0.77 ± 7 %
	Shape	Rectangular	Circular	Square	Square
	l 1 (mm)	3.28 ± 0.02	-	1.11 ± 0.02	0.73 ± 0.01
	l 2 (mm)	1.47 ± 0.02	-	1.22 ± 0.01	0.72 ± 0.01
	e (mm)	0.35	0.29	0.3	0.25
	N	11	24, 12	31	18
	L j (mm)	690 ± 2.5	1060 ± 0.4	688 ± 5	695 ± 2.5
	A al (mm 2 )	51± 2.5	19 ± 0.4	52 ± 1	1 3± 0.5

Table 3

 3 Operating conditions

		Friction Factor
		Value	Error
	Re	142-8,093	±3-8%
	T w , T f (K)	273-281	±0.1K
	T sub,in (K)	≈ 40	-
	p in (kPa)	≈ 1700	-
	p (kPa)	0.3-172.2	±0.3-25%
	4 f	0.04-0.45	±7-31%
	Flow	Horizontal
		Heat transfer coefficient
	Re	277-13,335	±3-8%
	q (kW/m 2 )	0.84-22	±3%
	T w , T f (K)	270-332	±0.1-1.4 K
	T sub,in (K)	≈ 70	-
	p in (kPa)	≈ 2000	-
	Pr	3.2-4	0.05-0.70%
	Nu	0.9-91	±3-28%
	Flow	Ascendant

Table 4

 4 Comparison of different hydraulic diameters

	mm	4A f /P f	D h (blueprint)	2l 1 • l 2 /(l 1 + l 2 )
	Tube 1	2.01	2.06 (+2%)	2.03 (+1%)
	Tube 2	1.44	1.40 (-3%)	-
	Tube 3	1.17	1.29 (+10%)	1.16 (-1%)
	Tube 4	0.77	0.70 (+9%)	0.72 (-6%)

Table 5

 5 Comparison of pressure drop parameters for channels of different

	hydraulic diameters
		4 f •Re	4 f • Re 0.25
	D 2.01	61 ± 8.5 63.8	0.388 ± 0.04 0.316	21 ± 0.5 1.5
	1.44	74 ± 12	64	0.434 ± 0.03 0.316	11 ± 3 2.0
	1.17	63 ± 13	57	0.329 ± 0.03 0.316	12 ± 6 1.9
	0.77	57 ± 11	57	0.347 ± 0.04 0.316	-	7.4

h (mm) Re < 1800 Eq. (13, 14) Re > 2000 Eq. (15) ξ A /A f

Table 6

 6 The longitudinal conduction: comparison of tubes 1 and 4

		Tube 1, Re = 280	Tube 4, Re = 380
	V • I (W)	6 4		2 6
	Qleak (W)	1		4
	Nu Shah /Nu exp	1.5		4.1
		Measured	Theory	Measured	Theory
	Nu 0 /Nu 9	1.2	1	3.8	1
	T w,9 -T w,0 (K)	33	30	40	18
	Qz (W)	-0.47	-0.43	-0.2	-0.1

B. Agostini received support from ADEME (French Agency for Environment and Energy Management) and CEA (French Atomic Energy Commission) for this project.


B Thonon

A Bontemps

Condensation of Pure and Mixture of Hydrocarbons in a Compact Heat Exchanger: Experiments and Modelling

This article presents a study on heat transfer in condensation of pure and mixtures of hydrocarbons in a compact welded plate heat exchanger. Three pure uids (pentane, butane, and propane) and two mixtures (butane C propane ) have been used. The operating pressure ranges from 1.5 to 18 bar. For pure uids, two heat transfer mechanisms have been identi ed. For low Reynolds numbers, the condensation occurs almost lmwise and the heat transfer coef cient decreases with increasing Reynolds number. For higher values of the Reynolds number, the heat transfer coef cient increases gently. The transition between the two regimes is between Re D 100 and 1,000 and depends on the operating conditions. For mixtures, the behavior is different. For low Reynolds numbers, mass transfer affects heat transfer and reduces the heat transfer coef cient by a factor of up to 4. Correlations for lmwise and in-tube condensation do not predict the results accurately, and a speci c correlation is proposed for pure uid condensation. For mixtures, the condensation curve method does not allow mass transfer effects to be taken into account, and more work is required to establish an accurate predictive model.

Condensation occurs in many industrial processes, but rarely with pure uids. The uids encountered are mixtures and often noncondensabl e gases are present, and this makes the condensation process very complex. This project, JOE3-CT97-0062, has been partially supported by the EC, within the frame of the Non-nuclear Energy Programme. The authors gratefully acknowledg e Alfa-Laval Vicarb for supporting this study and coordinating the European project.

Address correspondenc e to Dr. B. Thonon, CEA-Grenoble, GRETh, 17 rue des martyrs, 38054 Grenoble, France. E-mail: thonon@cea.f r In the case of mixtures or in the presence of a noncondensable gas, the vapor condensing must diffuse through the gas to the interface. This requires a partial pressure gradient toward the interface.

The partial pressure of the vapor falls from a constant value at a rather large distance from the interface to a lower value at the interface. Correspondingl y, the accompanying saturation temperature also falls toward the interface. Therefore, during condensation the condensing vapor arrives by diffusion at the condensate surface, and it is the thermal resistance in the vapor that limits the process. Hence, in order to improve the heat transfer, one must reduce the thermal resistance on the vapor side.

Several factors can enhance the condensation process by reducing the vapor-side resistance. During condensation of mixtures or of vapors that contain noncondensables, the heat transfer on the vapor side can be improved by raising the vapor velocity. It has been shown that the heat transfer coef cient can be improved by approximately 30% by increasing the vapor velocity. The use of nely undulated surface can also achieve signi cant augmentations in heat transfer during condensation. It has been shown that corrugation can promote turbulent equilibrium between the phases and thus contribute to the increase in heat transfer.

Compact heat exchangers are characterized by small hydraulic diameters (1-10 mm), and there is no reliable design method to estimate heat transfer coef cients during condensation in such small passages. In the open literature, condensation of mixtures and of vapor in the presence of noncondensable s has been studied, but essentially for conventional geometries (plain tubes), and only a few results have been published with uids representative of actual process conditions (hydrocarbons).

HEAT TRANSFER IN CONDENSATION

Gravity-Controlle d Regime

For the gravity-controlle d regime, the Nusselt theory for a plain wall allows the local heat transfer coef cient to be calculated. The average heat transfer coef cient is given by [START_REF] Rohsenow | Condensation: Part 1[END_REF] a Nu D Co 1:47 Re where Co is a physical property number:

Co D k L µ l 2 L q L (q L ¡ q G )g ¶¡1=3 (2)
The lm Reynolds number is calculated by

Re L D 4 Ç m L D h l L P (3)
where P denotes the wetted perimeter (for at channels twice the plate width). The Nusselt theory is valid only if the liquid lm is smooth, but for lm Reynolds numbers over 30, waves appear at the interface and increase the heat transfer coef cient. Kutatelatze has proposed a modi ed Nusselt correlation. The average heat transfer coefcient is given by [START_REF] Rohsenow | Condensation: Part 1[END_REF]

a Ku D Co Re L 1:08 Re 1:22 L ¡ 5:2 (4)
For Reynolds numbers above 1,600, the liquid lm becomes turbulent and the Labuntsov correlation can be applied. The average value is given by [START_REF] Rohsenow | Condensation: Part 1[END_REF]

a La D Co Re L 8570 C 58 Pr ¡0:5 L ¡ Re 0:75 L ¡ 253 ¢ (5)
A plot of these three models is given in Figure 1. For practical use, asymptotic models are often introduced in order to take into account laminar, wavy lm, and turbulent lm condensation simultaneousl y (Figure 2):

a LW D ¡ a 2 Nu C a 2 Ku ¢0:5 (6) a WT D ¡ a 2 Ku C a 2 La ¢0:5 (7)
Figure 2 Asymptotic models for lmwise condensation.

Forced-Convection Regime

In the shear-controlled regime, several theoretical or semiempirical correlations exist for predicting the heat transfer coef cient. The most common ones are the Akers correlation modi ed by Cavallini and Zecchin (as presented by Cavallini et al. [START_REF] Cavallini | Condensation Heat Transfer and Pressure Drop of Refrigerants in Tubes of Finned Tube Heat Exchangers[END_REF]), the Boyko-Kruzhilin [START_REF] Boyko | Heat Transfer and Hydraulic Resistance During Condensation of Steam in an Horizontal Tube and in a Bundle of Tubes[END_REF], and the Shah [START_REF] Shah | A General Correlation for Heat Transfer During Film Condensation in Pipes[END_REF] correlations. Akers or Cavallini-Zecchin:

a cv D a Re b eq Pr 0:33 L k L D h ( 8
)
where Re eq is an equivalent Reynolds number for the liquid C vapor mixture:

Re eq D Ç m[(1 ¡ x ) C x(q L =q G ) 0:5 ]D h l L (9)
Boyko and Kruzhilin or Shah:

a cv D a LO F ( 10
)
where the heat transfer coef cient of the liquid phase (a LO ) is deduced from speci c correlations for forced convection and F is the enhancement factor.

CONDENSATION IN PLATE HEAT EXCHANGERS Steam Condensation

Tovazhnyanski and Kapustenko [START_REF] Tovazhnyanski | Intensi cation of Heat and Mass Transfer in Channels of Plate Condensers[END_REF] have performed some tests on a plate heat exchanger with a 60Ê corrugation angle. Their tests covered a large range of Reynolds numbers. For Reynolds numbers above 300, a shear-controlled regime dominates and the heat transfer coefcients are compared to a modi ed Boyko and Kruzhilin correlation, the Boyko-Kruzhilin enhancement factor being

F D µ 1 C x q L q G ¡ 1
´ ¶0:5

The use of this correlation means that the enhancement of the heat transfer coef cients in condensation is comparable to the enhancement in single-phase ows. For example, for a 60 ± corrugation angle, the heat transfer coef cient is six times larger than for a comparable plain channel (Thonon et al. [START_REF] Thonon | Recent Research and Developments in Plate Heat Exchangers[END_REF]). For lower Reynolds numbers (Re L < 300), condensation seems to be gravitycontrolled. However, the Nusselt law underestimates the heat transfer coef cient. The herringbone pattern of the corrugation tends to drain the condensate to the periphery of the channel and keeps the condensate lm thin. Di-an et al. [START_REF] Di-An | Steam Condensation in a Vertical Corrugated Duct[END_REF] have studied steam condensation in two different geometries: a plain rectangular channel and a corrugated channel (45 ± corrugation angle). For the plain rectangular channel with Reynolds numbers between 70 and 500, the Nusselt theory predicts the heat transfer coef cients with good accuracy. For the corrugated channels, their results clearly outline the two regimes of condensation . For Reynolds numbers below 350 a gravity-controlle d regime exists, and for Reynolds numbers above 350 a shear-controlled regime dominates. The change of regime at a Reynolds number of 350 is explained by the change in the ow pattern. In corrugated channels, the transition from laminar to turbulent regime occurs at a lower Reynolds number than for a plain tube (about Re L D 400). The intensi cation compared to the plain channel is 60% at a Reynolds number of 250 and of 220% for a Reynolds number of 900.

Wang and Zao [START_REF] Wang | Analysis of Performance of Steam Condensation Heat Transfer and Pressure Drop in Plate Condensers[END_REF] studied steam condensation in a plate heat exchanger with a 45 ± corrugation angle. Their results clearly indicate that the heat transfer coef cient is shear-controlled : the heat transfer coef cient increases with the liquid lm Reynolds number. The intensication of the heat transfer coef cient compared to the Nusselt theory varies from a factor of 2 to 3 depending on the ow conditions.

Wang et al. [START_REF] Wang | Analysis of Steam Condensation Heat Transfer in Plate Heat Exchangers[END_REF][START_REF] Wang | Calculation Procedure for Steam Condensation in Plate Heat Exchangers[END_REF] have studied steam condensation in one plate and frame and two brazed heat exchangers. The effect of mass ux and pressure has been studied. The results have been compared to the Shah and Boyko-Kruzhilin correlations and a large discrepancy has been observed. The density exponent of the Boyko-Kruzhilin correlation has been reduced to 0.45 and allows better predictive accuracy.

Condensation of Refrigerants

Compact heat exchangers are often used in refrigerant systems. A review has been recently presented by Palm and Thonon [START_REF] Palm | Thermal and Hydraulic Performances of Compact Heat Exchangers for Refrigeration Systems[END_REF].

For ammonia applications , Panchal and Rabas [START_REF] Panchal | Thermal Performance of Advanced Heat Exchanger for Ammonia Refrigeration Systems[END_REF] have carried out two series of tests with high and low corrugation angle (60 ± and 30 ± ). The results indicate that the plates with low corrugation angle give higher heat transfer coef cients than the plates with a 60 ± corrugation angle. This is probably caused by an increase of the liquid hold-up in high-corrugation-angl e plates.

Kumar [START_REF] Kumar | The Design of Plate Heat Exchangers for Refrigerants[END_REF] has reported results for condensation of R22 and ammonia in various types of plate heat exchangers. The results indicate that the condensation is gravity-controlle d for low Reynolds numbers, and shear-controlled for higher Reynolds number. The measured heat transfer coef cients are 1.5 to 4 times larger than those expected in a comparable plain tube. The same type of results was also obtained by Navarro and Bailly [START_REF] Navarro | Compact Brazed Plate Heat Exchangers[END_REF], for condensation of refrigerants in compact brazed heat exchangers.

Chopard et al. [START_REF] Chopard | Assessment of Heat Transfer Performance of Rectangular Channel Geometries: Implication on Refrigerant Evaporator and Condenser Design[END_REF] have performed tests for condensation of R22 at high Reynolds numbers in welded plate heat exchangers. Three different geometries were tested (plain rectangular, in-line, or staggered studs). The results are presented in terms of an intensi cation factor of the single-phase heat transfer coef cient. The heat transfer coef cients of in-line or staggered studs are significantly higher than those of the plain channel. Furthermore, even for the plain channel, the Shah correlation underestimates the heat transfer coef cients. Local measurements of the void fraction have shown that the liquid is drained to the periphery and that a thin lm of liquid remains in the central part of the channel (Figure 3). Arman and Rabas [START_REF] Arman | Condensation Analysis for Plate and Frame Heat Exchangers[END_REF] made a serious effort to develop a general computer program for heat transfer and pressure drop of single-component and binary mixtures during condensation in plate heat exchangers. Starting with a review of the sparse literature in the area, they conclude that the condensation heat transfer is shearcontrolled. The computation scheme for heat transfer was based on a correlation by Tovazhnyanski and Kapustenko [START_REF] Tovazhnyanski | Intensi cation of Heat and Mass Transfer in Channels of Plate Condensers[END_REF] and for pressure drop on the Lockhart-Martinelli method. The condensation channel is divided into a number of steps, and the local heat transfer coef cient and pressure drop of each step is calculated iteratively. A number of constants in the correlations were determined by comparison with experimental data from the literature. Results from new experiments with pure saturated ammonia were then compared to the predictions of the computer program, and the deviations were shown to be small for the case of single-componen t uids.

More recently, Yan et al. [START_REF] Yan | Condensation Heat Transfer and Pressure Drop of Refrigerant R-134a in a Plate Heat Exchanger[END_REF] have presented results on condensation of R134a in a compact brazed heat exchanger having a 60 ± corrugation angle. The effects of mass ux, average heat ux, local vapor quality, and operating pressure have been studied. The results indicate that the heat transfer coef cient increases with mass ux, indicating forced-convection heat transfer. To correlate the results a modi ed Akers correlation is proposed:

Nu D a Re b eq Pr 1 / 3 L ( 12
)
As outlined by Srinivasan and Shah [START_REF] Srinavasan | Condensation in Compact Heat Exchangers[END_REF], the heat transfer performances are closely linked to the ow pattern.

Most of the conventional in-tube correlations fail in predicting the heat transfer coef cient in compact heat exchangers, as the ow patterns are radically different from in-tube ows. Furthermore, for very small channels, the surface tension has to be taken into account.

EXPERIMENTAL APPARATUS General Description

A test rig devoted to hydrocarbon applications has been used for the experiments. Fluids such as butane, propane, or mixtures can be used. The heat exchangers tested can be either evaporators or condensers. The pressure can be varied up to 20 bar with temperature up to 80 ± C. The hydrocarbon ow rate is up to 600 kg/h with a maximum heat duty of 70 kW.

The test rig has three independent circuits (Figure 4):

The hot water circuit (100) The cold water circuit (200) The hydrocarbon circuit (300)

The condenser is installed after the evaporator and is fed by saturated vapor. The saturation conditions are checked by measurement of the inlet pressure and temperature and by comparison with the saturation curve of the uid [T D f ( p sat )]. At the heat exchanger outlet, the liquid ows directly to a storage tank. Consequently, the uid temperature is assumed to be close to the saturation temperature, therefore neglecting possible subcooling of the liquid.

Heat Exchanger Geometry

The plate arrangement of the tested heat exchanger is given in Figure 5 heated to avoid the presence of liquid in the connecting pipes, and the accuracy is §1%.

RESULTS

Data Reduction

Experimental differences have been observed between the heat duties calculated on the water and hydrocarbon sides. These differences are mainly due to uctuations on the hydrocarbon mass ow rate. The hydrocarbon mass ow meter is located between two tanks, and at low ow rates (or low heat duties) instabilities are measured by the ow meter. For heat duties greater than 20 kW, the energy balance is within §10%, which is acceptable for such two-phase ow experiments.

Therefore, the actual heat duty is assumed to be the one measured on the water side:

Ç Q D Ç Q water D Ç V 200 q 200 c p;200 (T out;200 ¡ T in;200 ) (13)
Then, from this value and assuming an exact heat balance, the hydrocarbon mass ow rate is recalculated from

Ç M 300 D Ç Q water (h in ¡ h out ) 300 (14)
The heat transfer coef cient on the condensation side is deduced from the overall heat transfer coef cient, where the heat duty is estimated from the water side:

U D Ç Q DT ln : A (15)
The average heat transfer coef cient on the condensation side is deduced from

1 U D 1 a 1 C R w C 1 a 2 ( 16
)
where a 1 and a 2 are the heat transfer coef cients on the hydrocarbon and water sides. The latter is calculated with the correlations presented by Thonon et al. [START_REF] Thonon | Recent Research and Developments in Plate Heat Exchangers[END_REF] obtained for several plate heat exchangers. For the geometry studied (corrugation angle of 45 ± ), the correlation is Nu D 0:347 Re 0:653 Pr 0:33 [START_REF] Yan | Condensation Heat Transfer and Pressure Drop of Refrigerant R-134a in a Plate Heat Exchanger[END_REF] where the Reynolds and Prandtl numbers are calculated for the single-phase liquid ow on the cooling side.

The reference heat transfer surface ( A) is based on the projected area. Therefore, the surface extension (less than 20%) is included directly in the heat transfer correlation.

The results concerning the heat transfer coef cients are presented using the liquid Reynolds number assuming all the ow is liquid (condenser outlet):

Re LO D Ç m D h l L ( 18
)
where Ç m is the mass velocity of the hydrocarbon ow. The estimated uncertainty for the heat transfer coefcient on the condensation side ranges between 15% at high Reynolds number and 30% at low Reynolds numbers.

Tests Performed

Five sets of tests have been performed: three with pure uids and two with mixtures. For each uid, the absolute pressure and the mass ow rate on the condensation side were varied. The range covered is presented in Figure 6 for pentane, butane, propane, butane-propane (49%-51% by mass), and butane-propane (28%-72% by mass).

The pure uids used are of high quality and contain less than 0.1% of other components. The physical properties have been calculated using the Prophy software developed by PROSIM. For the two butane C propane mixtures, the mass concentration has been estimated by measuring the dew and bubble temperatures at different pressures and by comparing with the temperature predicted by the physical property software.

Pure Fluid Condensation

Three series of tests have been performed using pure hydrocarbons (pentane, butane, and propane) and the overall thermal and hydraulic performances have been recorded and are presented in Figures 789. The operating pressure ranges from 1.5 up to 18 bar, with Reynolds numbers between 100 and 2,000.

The results clearly indicate that for low Reynolds numbers the heat transfer coef cient in condensation is similar to that of a laminar falling lm and for higher Reynolds numbers turbulent effects tend to increase the heat transfer coef cient. The transition between these two mechanisms occurs for Reynolds numbers between 100 and 1,000, depending on the pressure.

The operating pressure has two effects. First, in the laminar zone, the heat transfer coef cients increase with pressure, which is essentially an effect of the physical properties. Second, the higher pressures give higher transition Reynolds numbers, and this might be linked to an effect of the condensate Prandtl number.

Mixture Condensation

For the mixtures (Figures 10 and11), the behavior is quite different. In the rst zone, the heat transfer coef cient increases with the Reynolds number, and then in the second zone the behavior is similar to the one of pure uids. These observations outline the effect of mass transfer on the overall heat transfer coef cient. For low Reynolds numbers (laminar lm) and for pure uids the heat transfer resistance is mainly in the liquid lm, and this resistance increases with the Reynolds number (Nusselt theory). For higher Reynolds numbers, the liquid lm becomes turbulent and with increasing Reynolds numbers the heat transfer coef cient increases.

For pure uids there is no mass transfer effect during condensation . On the contrary, with a mixture, there is a mass transfer resistance due to molecular diffusion in the gas phase. For low Reynolds numbers, the velocity is low and there is almost no mixing in the gas phase, consequently mass transfer effects will reduce the heat transfer coef cient. For higher Reynolds numbers, mixing occurs in the gas phase and enhances mass transfer; consequently the heat transfer resistance is mainly in the liquid lm and the mixture behaves as a pure uid.

DATA ANALYSIS

Pure Fluids

To correlate the results, two approaches can be adopted. We can consider either falling-lm or forcedconvection correlations. For falling-lm models, the heat transfer coef cients are plotted introducing the physical property number Co. For forced convection, the enhancement factor F is used [Eq. [START_REF] Wang | Calculation Procedure for Steam Condensation in Plate Heat Exchangers[END_REF]]. The data have been compared to several correlations for lmwise (Figure 12) and in-tube forced convection (Figure 13) condensation , but all these correlations fail in predicting the heat transfer performances. In the laminar regime, condensate drainage may occur in the furrow of the corrugation, and this will reduce the mean lm thickness and consequently increase heat transfer. The vapor velocity at the heat exchanger inlet is relatively low and the shear effect should be negligible.

A new correlation is proposed and adjusted based on the measurements. This correlation is based on lm condensation and introduces a corrective term taking into account the geometry of the corrugated channel. The effect of the geometry is introduced by the heat transfer coef cient of the liquid phase (a LO ). The use of the equivalent Reynolds number (Re eq ) rather than the liquid Reynolds number (Re L ) gives better results (Figure 14). F D 1564 Re ¡0:76 eq D a a LO [START_REF] Silver | Gas Cooling with Aqueous Condensation[END_REF] This new correlation gives satisfactory results for the three pure uids tested (Figure 15): 80% of the data are predicted within §20%. The three pure uids used (pentane, butane, and propane) have physical properties that are quite similar, and it is not surprising that a single correlation, based on a nondimensiona l approach, gives good results. To validate and generalize this correlation, other operating conditions (lower pressures) and more viscous uids should be used. Only one geometry has been tested, but the correlation has been built with nondimensiona l numbers. The general form of this correlation (enhancement factor) is similar to the one proposed by Tovazhnyanski and Kapustenko [START_REF] Tovazhnyanski | Intensi cation of Heat and Mass Transfer in Channels of Plate Condensers[END_REF].

Mixture

The experimental data (Figure 16) suggests two heat transfer mechanisms: (1) a laminar regime in which the mass transfer resistance is the dominant resistance; (2) a turbulent regime in which heat transfer is controlled by the liquid lm.

The condensation curve method has been applied to our results. The mass transfer resistance is assumed to be proportional to the heat transfer resistance in the vapor phase (1=a G ). A correction factor (z) is introduced to take into account the concentration pro le between the bulk of the ow and the liquid lm interface. The 13 resulting heat transfer coef cient is given by Silver [START_REF] Silver | Gas Cooling with Aqueous Condensation[END_REF] or Bell and Ghaly [START_REF] Bell | An Approximate Generalized Design Method for Multicomponent /Partial Condensation[END_REF]:

a D 1 (1=a pure uid condensatio n ) C (z=a G ) (20)
where the vapor heat transfer coef cient is estimated from single-phase heat transfer correlation, calculating the Reynolds number with the vapor fraction; z is given by

z D xc pG dT dh ( 21
)
where dT=dh is the inverse of the slope of the condensation curve h D h(T ). The calculation is carried out using an average vapor quality of x D 0.5. The physical properties are estimated at the dew temperature for the vapor phase and at the bubble temperature for the liquid phase.

The results are presented in Figure 17. The equivalent mass transfer resistance decreases with the Reynolds number; on the contrary, the condensate heat transfer resistance increases. The experimental trends are well predicted by this model. For the turbulent regime (Re > 1,500) there is very good agreement, but for the laminar regime the heat transfer coef cient is overpredicted (Figure 18).

CONCLUSION

A small industrial compact condenser has been installed and instrumented on a hydrocarbon test rig at GRETh. Five sets of tests have been performed using hydrocarbons: three with pure uids and two with mixtures. The pressure was varied between 1.5 and 18 bar for a range of Reynolds numbers between 100 and 2,000. These conditions are representative for industrial cases.

For pure uids (pentane, butane, and propane), two regimes have been identi ed. For low Reynolds numbers, the heat transfer coef cient in condensation decreases, indicating a laminar regime. For higher Reynolds numbers, the heat transfer coef cient increases gently, indicating a transition to turbulent ow.

For mixtures (butane C propane ), the behavior is different. At low Reynolds numbers, the heat transfer coefcient is much lower than with pure uids and increases with the Reynolds number. For higher Reynolds numbers, it remains almost constant or increases slightly, and the values are close to those obtained with pure uids. Furthermore, a signi cant pressure effect is observed for the laminar regime: low pressures give higher heat transfer coef cients. These observations indicate that mass transfer effects are signi cant and affect heat transfer during condensation.

A model for pure uid condensation has been established and gives satisfactory results within the range of tested uids. For mixtures, the condensation curve method does not allow estimating the heat transfer coefcient in the laminar regime. But for turbulent regimes, as the mass transfer effect is negligible , the pure uid model allows predicting the condensation heat transfer coef cient of mixtures.

In order to validate the measurements and the correlations, obtained on small-scale units, industrial measurements have been performed. A prototype compact heat exchanger has been designed and manufactured by Alfa-Laval Vicarb and installed in an industrial plant producing chloroform.

NOMENCLATURE

Figure 1

 1 Figure 1 Filmwise condensation models.

Figure 3

 3 Figure 3 Flow structure in a rectangular channel.

Figure 4

 4 Figure 4 Description of the hydrocarbon test rig. T, temperature measurement; P, pressure measurement; DP, drop pressure measurement; F, ow meter.

Figure 5

 5 Figure 5 Detail of the compact heat exchanger. (courtesy of Alfa-Laval Vicarb.)

Figure 6

 6 Figure 6 Experimental conditions (all data).

Figure 7

 7 Figure 7 Condensation of pure pentane.

Figure 8

 8 Figure 8 Condensation of pure butane.

Figure 9

 9 Figure 9 Condensation of pure propane.

Figure 10

 10 Figure 10 Condensation of a 49%-51% butane-propane mixture.

Figure 11

 11 Figure[START_REF] Palm | Thermal and Hydraulic Performances of Compact Heat Exchangers for Refrigeration Systems[END_REF] Condensation of a 28%-72% butane-propane mixture.

Figure 12

 12 Figure[START_REF] Panchal | Thermal Performance of Advanced Heat Exchanger for Ammonia Refrigeration Systems[END_REF] Comparison of measured heat transfer coef cient with lmwise condensation models.

Figure 13

 13 Figure[START_REF] Kumar | The Design of Plate Heat Exchangers for Refrigerants[END_REF] Comparison of condensation enhancement factors using butane with condensation correlations (F B-K , Boyko-Kruzhilin[START_REF] Boyko | Heat Transfer and Hydraulic Resistance During Condensation of Steam in an Horizontal Tube and in a Bundle of Tubes[END_REF]; F Yan , Yan et al.[START_REF] Yan | Condensation Heat Transfer and Pressure Drop of Refrigerant R-134a in a Plate Heat Exchanger[END_REF]).

Figure 14

 14 Figure 14 Condensation of pure uids.

Figure 15

 15 Figure 15 Comparison of predicted and measured heat transfer coef cients.

Figure 16

 16 Figure[START_REF] Arman | Condensation Analysis for Plate and Frame Heat Exchangers[END_REF] Enhancement factor for pure uids and mixtures.

Figure 17

 17 Figure 17 Evaluation of mass and heat transfer resistances.

Figure 18

 18 Figure 18 Condensation of mixtures.