A review of heat transfer between concentric rotating cylinders with or without axial flow

International Journal of Thermal Sciences 50 (2011) 1138e1155

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Review

A review of heat transfer between concentric rotating cylinders with or without axial flow M. Fénot*, Y. Bertin, E. Dorignac, G. Lalizel Institut P, Cnrs, ENSMAeUniversité de Poitiers, UPR 3346, Département fluides, thermique, combustion, 1 Avenue Clément Ader, BP 40109, 86961 Futuroscope Chasseneuil Cedex, France

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 July 2010 Received in revised form 11 February 2011 Accepted 12 February 2011 Available online 29 March 2011

Heat transfer in flow between concentric rotating cylinders, also known as TayloreCouette flows, constitutes a long-existing academic and industrial subject (in particular for electric motors cooling). Heat transfer characteristics of those flows are reviewed. Investigations of previous works for different gap thickness, axial and radial ratio, rotational velocity are compared. Configurations with axial flow and/or with slots on the cylinders are also considered. For each case, different correlations are presented. Finally, unresolved issues are mentioned for further research. Ó 2011 Elsevier Masson SAS. All rights reserved.

Keywords: Rotating cylinders Heat transfer TayloreCouette TayloreCouetteePoiseuille Slotted gap

1. Introduction Flow dynamics between two concentric rotating cylinders constitutes an old academic subject since Couette [1] and Taylor [2], whose names are recalled in the term TayloreCouette flow, which has become a reference in stability studies due to the gradual destabilizing of a flow lending itself to a rigorous mathematical approach. Moreover, this kind of flow has many industrial applications, particularly in the fields of mechanical or chemical mixing equipment. It has consequently been the subject of several bibliographic reviews by Di Prima and Swinney [3], Cognet [4], and Maron and Cohen [5]. The heat transfer in this flow and the impact of flow structures on heat transfer were more recently studied (Gazley [6]); there already existed numerous industrial applications of the rotating elements (rotation, outer wall of the rotating heat pipes, cooling of the lower extremities of the turbojet turbine.), especially in electric motors. Indeed, different studies [7,8] on the heat transfer of electric motors have demonstrated the importance of convective heat transfer within the cylindrical gap (area separating the rotor from the stator). In fact, the rotor is the locus for large-scale dissipations of electromagnetic origin, and its cooling is ensured principally by the air flow of the cylindrical gap. Two main families of rotating electric machines may

* Corresponding author. E-mail address: [email protected] (M. Fénot). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.02.013

be distinguished; closed machines (rotor rotation without axial air flow: TayloreCouette flow), and open machines (axial flow combined with rotor rotation: TayloreCouetteePoiseuille flow). Moreover, as regards some motor technologies, the existence of grooves on the rotor (where copper threads, for example, may be coiled) is liable to significantly modify the dynamic and the thermal flow behavior. Excepted a short part of the bibliographic review by Maron and Cohen [5], no very detailed analysis of heat transfer in a rotating annular gap has ever been carried out. This bibliographic review is focused on the heat transfer of TayloreCouette flow patterns. The kind of thermal behavior to be studied is obviously linked to the dynamics of these kinds of flow. We shall present in detail the different forms of flow already encountered, but our survey is not exhaustive. 2. The fundamentals of TayloreCouette flow Let us first look at the different parameters of influence. We shall consider a basic system composed of two concentric cylinders (Fig. 1). Its geometry is characterized by two radii, the outer radius of the inner cylinder R1 and the inner radius of the outer cylinder R2, as well as their length L. The flow is then characterized by the following geometric parameters: hydraulic diameter: Dh ¼ ð4Sp =Pm Þ ¼ ð2½pðR22  R21 Þ=pðR2 þ R1 ÞÞ, annular gap thickness (also known as cylindrical gap): e ¼ R2  R1, radial ratio h ¼ R1/R2, and axial ratio: G ¼ L/R2  R1.

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

S Sp T Tm

Nomenclature

Indexes 1 2 a c e eff eq m w o t turb fm wm

rotor stator axial critical input/entrance effective equivalent mean wall output/exit tangential turbulent mean in the fluid mean on the wall

Tmean V

4 l q n r u f

Dimensional numbers hydraulic diameter (m) Dh e cylindrical gap thickness or rib height (m) h convective heat transfer coefficient (W/m2 K) l slot width or rib width (m) p slot depth or relative space between two ribs (m) t time (s) specific heat at constant pressure (J/kg K) cp L actual cylindrical gap length or canal length (m) N rotation velocity (rpm) P static pressure (Pa) W canal width (m) H canal height (m) wetted perimeter (m) Pm volume flow rate (m3/s) Qv R cylinder radius (m)

The main dynamic parameter characterizing cylindrical gap flow is the rotor rotation speed u (rad/s). One may consequently define the tangential Reynolds number Ret ¼ uR1Dh/n. The Taylor number is often preferred to the Reynolds number, for it may be interpreted as the ratio between centrifugal force and viscous force [9]. It may be expressed in the form Ta ¼ ðu2 R1 ðR2  R1 Þ3 =n2 Þ

L Va

R1

R2

Fig. 1. TayloreCouette flow geometry.

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reference heat transfer area (m2) cross sectional area (m2) RRR temperature ( C) Va $T dS  ( C) mean temperature Tm ¼ RRR Va dS Te þ T0  ( C) mean temperature Tmean ¼ 2 velocity (m/s) flux density (W/m2) thermal conductivity (W/m K) open slot angle or chamfer (45 ) rib angle ( ) kinematic viscosity (m2/s) density (kg/m3) rotation velocity (rad/s) heat flux (W)

Dimensionless numbers n number of slots pressure coefficient Cp geometric factor Fg Gr Grashof number Nu Nusselt number Pr Prandtl number Re Reynolds number Sh Sherwood number St Stanton number Ta Taylor number a weighted rotation coefficient at effective speed g slot aspect ratio h radial cylindrical gap aspect ratio or canal performance coefficient s ratio between the Taylor number and its critical value G axial cylindrical gap aspect ratio g heat transfer function

ð1=Fg Þ with Fg as a geometrical factor that differs from one author to the next and may take into account the aspect ratio of the smooth annular gap. This ratio approaches 1 once the cylindrical gap has become narrow; this is the number used by most authors. Gardiner and Sabersky [10] and Becker and Kaye [11] propose: Fg ¼ ðp4 ððR1  R2 Þ=ð2R1 ÞÞ=1697ð0:0571ð1  0:652ðe=R1 ÞÞÞ þ 0:00056 ð1  0:652ðe=R1 ÞÞ1 Þ. Some authors (Ref. [12] for example) prefer the formulation Ta ¼ ðu2 R1 ðDh =2Þ3 =n2 Þ which may take into account possible geometrical variations by hydraulic diameter. It should be noted that several authors, particularly in the more ancient studies, define the Taylor number as the square root of the classical form. In addition, heat transfer is defined by the Nusselt number, whose definition may likewise vary from one author to the next. Today’s most widely used definition is:Nu ¼ ðhDh =lÞ ¼ ðl grad Tjp =Tw  Tref ÞðDh =lÞ ¼ ðF=SðTw  Tref ÞÞðDh =lÞ, where the reference temperature is often postulated as equal to the fluid temperature or else, more generally, to the temperature of the second wall. The Nusselt number thereby deduced serves as a translation of the efficiency of heat transfer from one wall to the other by means of the fluid. As for the heat transfer surface S, it is often defined as the surface of the heating wall (S ¼ 2$pLR1 or S ¼ 2$pLR2 [13]) Sometimes this surface is considered as intermediate with regard to the walls, and its exact definition differs from one author to the next (generally S ¼ 2$pLðe=lnð1=hÞÞ [14]). The more narrow the cylindrical gap, the smaller the differences between the definitions of S.

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Fig. 2. Regimes observed in flow between two rotating cylinders [16].

Some authors do not use the hydraulic diameter as the characteristic length, but rather prefer the thickness of the annular gap: e ¼ R2  R1. In the following pages, we will use the most widely accepted definitions: Ta ¼ ðu2 R1 ðDh =2Þ3 =n2 Þ and Nu ¼ ðhDh =lÞ ¼ ðl grad Tjp =Tw  Tref ÞðDh =lÞ. 3. Smooth cylindrical gap, TayloreCouette flow In this part, we shall focus on flow and convective heat transfer in a smooth annular gap subjected to inner cylinder rotation alone. A description of the flow structure due to rotation alone (TayloreCouette flow) will be given prior to a presentation of the results concerning heat transfer. 3.1. Flow structure Flow in a smooth and closed annular gap has been the subject of numerous theoretical and experimental studies. The first experiments were conducted by Couette [1] and Mallock [15] in 1887 and 1888. Most studies involve a single rotating cylinder (generally the inner one). The rotation of the second cylinder makes more complex the flow as recorded by Andereck et al. [16] (Fig. 2). It is important to note that in this study a strict procedure has been followed to reduce the problem range (first, the outer cylinder is slowly accelerated, and then the inner cylinder is slowly accelerated as well). The following flow description will involve only one rotating cylinder configurations since no heat transfer studies have been conducted on two rotating cylinders. 3.1.1. Couette flow Couette observed that the torque required for rotation of the central cylinder rises linearly to a critical speed, and then the torque rises still more rapidly. In 1923, Taylor [2] highlighted the existence of a critical rotation speed uC. Under this rotational speed, the flow, referred to as Couette flow, is steady and laminar. It is driven by viscous drag force acting on the fluid. The streamlines are annular and centered on the rotation axis. Natural convection induced by gravity can greatly affect this flow. It is generally recognized that for a Rayleigh number Ra ¼ ðPr$b$DT$e3 g=n2 Þ lower than 104, the effects of natural convection are negligible. For a higher Rayleigh number, flow depends on the ratio between the Grashof number and the tangential Reynolds number Ret ¼ uR1Dh/n (Guo and Zhang [17]).

So, the effects of natural convection occasioned by gravity are preponderant when rotation speed is particularly low. Along with an increase in rotation speed (and in Ret), the effects of natural convection seem to disappear and are replaced by classical cylindrical gap flow. When none of the effects are dominant, structures with two or three cells are obtained [18]. These effects are also dependant on the way the cylinder is oriented, that is to say vertically [18,19] or horizontally [20,21]. 3.1.2. Taylor vortex flow Once the critical rotation speed uC noted by Taylor is exceeded, the flow presents instabilities structured in a toric (O ring) form and known as “Taylor vortices”. They are counter-rotative and associated by pairs, as is indicated in Fig. 3. From both an experimental and a theoretical standpoint, Taylor [2] determined the critical value of rotor rotation speed for an infinitely long and pronouncedly narrow annular gap Tac ¼ 1708 z 1700. As soon as they appeared, Taylor vortices are arranged periodically by pairs, which rendered it possible to define their axial wavelength as the axial space taken up by a doublet of vortices. The axial

Fig. 3. Taylor vortices [6].

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

wavelength of a pair of vortices for the critical Taylor number (or critical wavelength) is generally slightly less than the theoretical critical wavelength for an infinitely long and pronouncedly narrow cylindrical gap. Kirchner and Chen [22] showed that Taylor vortices originate, at the level of the inner cylinder, in the form of axially and radially propagated discs, which are transformed into classical vortices. Ghayoub et al. [23] provided a number-based demonstration of the appearance of small and non-stationary Gortler-type vortices in the zone where Taylor vortices leave the stator. In cases where the rotation speed is still rising and the Taylor number exceeds a second critical threshold, (Ta/Tac ¼ 1.2), Coles [24] (for a narrow cylindrical gap: h ¼ 0.95) underscored the presence of an azimuth wave regime (wavy mode); the Taylor cell boundaries are no longer perpendicular to the cylindrical axis, but rather present undulations or waves (Fig. 4). The flow consequently becomes doubly periodic. As speed was gradually heightened, Cognet [4] (once again with regard to a narrow cylindrical gap: h ¼ 0.908) observed a rise in the number of azimuth waves up to a maximum number that remains constant for 4.5  Ta/Tac  25, and subsequently decreases until the azimuth waves disappear (Ta/ Tac z 96). The movement then becomes virtually periodic. Moreover, experiments by Coles [24] entailed the observation of up to 26 different states (number of vortices.) for the same Taylor number. Each state depends not only on rotation speed, but also on previous flow history, which involves hysteresis effects marked by transitions that differ with regard to increasing and then decreasing rotation speeds. Each of these states corresponds to a given number of pairs of vortices (number of axial waves) and of azimuth waves (number of azimuth wave periods). Cylindrical gap geometry likewise assumes an important role with regard to the transitions. As concerns annular space thickness, Sparow et al. [25] and Roberts [26] have used linear stability theory when showing that the wider the cylindrical gap (small h), the more frequent the appearance of vortices corresponding to high Taylor numbers. Cognet [4] provided experimental confirmation of these results (Fig. 5). In addition, Snyder and Lambert [27] showed that in large annular gaps (small h), Taylor cells are less rapidly affected by azimuth waves. Another parameter liable to exert influence on flow transitions is the axial length of the cylindrical gap. The results gathered by Cole [28] facilitate comprehension of the role assumed by annular gap length. In Fig. 6 and in accordance with axial cylindrical gap length, we may note: the rotation speed starting at which Taylor

Fig. 4. Azimuth wave regime (wavy mode).

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Fig. 5. Influence on the transition of annular gap thickness [4].

vortices appear (curve i), the rotation speed starting at which their waves appear (curve ii), and the corresponding theoretical numbers ( and *). Fig. 6 shows that regardless of cylinder length, vortices show up at essentially the same rotation speed. On the other hand, cylindrical gap length has a far greater impact on wave appearance. When cylinder length goes up, rotation speed goes down exponentially and is stabilized close to its theoretical number once the cylindrical gap has lengthened (G > 40). Lastly, the presence of a temperature gradient can likewise affect flow stability. This is due to variable fluid properties (density and viscosity), to the natural convection occasioned by centrifugal forces, and to the natural convection occasioned by gravity, which is generally considered separately. Becker and Kaye [11] as well as Walowit [29] have shown that while a negative radial gradient stabilizes flow (the critical Taylor number increases), a positive gradient has the opposite effect. According to Walowit [29], the effects of the convection occasioned by centrifugal forces are negligible and the variations in fluid properties take on the main role. As regards the effects of natural convection occasioned by gravity, Ali and Weidman [30] have shown that the flow depends on the ratio between the Grashof numbers and the tangential Reynolds numbers Ret and that, as regards the Couette flow, the effects are negligible for a Rayleigh number Ra lower than 104.Gardiner and Sabersky [10] and Aoki et al. [31] have observed, in their strictly thermal studies, an unusually high critical Taylor number (104 instead of 1700), which they attribute to the influence of natural convection in their experiments. Their conclusions lead

Fig. 6. Transitions and cylinder length [28].

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one to believe that natural convection modifies e and stabilizes e cylindrical gap flow. 3.1.3. Turbulent flow Finally, as the rotation speeds continue to rise after the disappearance of Azimuthal waves, random fluctuations come to progressively dominate the flow. In the end (above Ta/Tac z 1300), flow becomes turbulent. The transition toward turbulence has been confirmed experimentally by Gollub and Swinney [32] and numerically by Alziary de Rocquefort and Grillaud [33]. The Taylor numbers nonetheless differ, probably as a result of dissimilar geometries. In fact, Cognet [4] noticed that an infinite sequence of fluctuation appears progressively inserting a new flow frequency for each one of them until the flow becomes turbulent. 3.2. Convective heat transfer

that heat transfer first diminishes along with rotation speed (Fig. 7) on account of the disturbance provoked by centrifugal forces with regard to natural convection. The higher the Rayleigh number, the more pronounced and belated the diminution. It may be noted that in Fig. 6, the threshold value of the Nusselt number in the case of heightened rotation speed is 1, which is tantamount to strictly conductive transfer. 3.2.2. Taylor flow and turbulent flow heat transfer Taylor flow and turbulent flow heat transfer are studies jointly here as transition from one to the other seems to have no effect at all on the heat transfer rate. None of the authors have noticed heat transfer change with turbulent transition (Ta z 106). This is probably due to the progressive appearance of fluctuations bringing the flow to turbulent (see paragraph 2.1.3 Turbulent flow): as transition to turbulence is progressive, heat transfer variation is continuous.

The initial approaches to convective heat transfer were carried out using an analogy between heat transfer and momentum. Couette [1] and Taylor [2] both measured the resistant torque due to the fluid’s being viscously rubbed between the two cylinders. This type of measurement was likewise employed more recently by Yamada [34]. On the basis of these studies, it is possible to deduce friction factors or coefficients that can be linked to heat transfer by means of Reynolds analogy. In 1958, Gazley [6] was the first author to show a sustained interest in thermal study of TayloreCouette flow. Flow dynamics were found to create three different heat transfer situations as speed and consequently the Taylor number goes up. The first corresponds to the laminar regime below the critical threshold, the second to the laminar regime with superimposed Taylor vortices occupying the annular gap, and the third to the turbulent regime with regard to the highest Taylor numbers presenting more or less fully formed structures. One should recall that the Nusselt number is calculated by these authors on the basis of the difference of temperature from one wall to the other. It should also be noted that all of these studies deal only with average Nusselt numbers for the cylindrical gap taken as a whole. 3.2.1. Couette flow heat transfer Below the critical Taylor number, most of the authors have concurrently found a constant value equal to 1; this is particularly the case for Tachibana et al. [35]. The value may vary in accordance with Nusselt; for instance, Becker and Kaye [11] found a value of 2 by using the definition of a Nusselt number: Nu ¼ hDh/l (Nu ¼ h2e/ l) as regards a smooth cylindrical gap. This is the definition we have chosen in order to ensure that as thereby postulated, the Nusselt number remains identical, whatever be the configuration (smooth with or without clipping flow, slotted.). In every case, this Nusselt number corresponds to a conductive heat transfer between the two walls: only parallel to these walls does the fluid move, and the heat transfer is consequently carried out perpendicularly to the flow, which may be considered as conductive. Two phenomena may nonetheless arise and modify the above value. First, a radiative heat transfer involving the two cylinders probably explains the high Nusselt numbers observed by Gazley [6], who does not seem to have dealt specifically and at length with this kind of transfer. Second, there is the natural convection of which Gardiner and Sabersky [10], on the other hand, have taken particular note; since their results are in close agreement with those reported by Aoki et al. [31], it would appear that the same phenomenon of natural convection comes into play. Finally, Yoo [19] focused on the air driven into rotation between two horizontal cylinders for Rayleigh numbers ranging from 1000 to 50000 and a relatively large cylindrical gap (aspect ratio h ¼ 0.5), and observed

Fig. 7. Evolution of the Nusselt number along with the tangential Reynolds number for different Rayleigh numbers and an aspect ratio h ¼ 0.5 with Re ¼ ðuR2 e=nÞ.

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

So, above the critical Taylor number, Nu rises appreciably in conjunction with the latter. And above this critical threshold, the transportation of matter from one wall to the other of the annular gap favors heat transfer. Becker and Kaye [36] observed a second, less pronounced transition (simple modification of slope) for Ta ¼ 104. The results were confirmed in the respective works of Nijaguna and Mathiprakasam [37] as well as Bouafia et al. [12,14]. There would appear to exist another dynamic transition or differing flow behavior altering the evolution of convective heat transfer. It bears mentioning that thermal transition coincides for Peres [38] with the Taylor number corresponding to the shortest axial wavelength. In fact, wavelength diminishes, according to Peres [38] reaching a minimum corresponding to the second transition, and then it increases. Consequently, the transition seems to correspond to a situation in which the vortex number is maximum. Several reasons can explain the lack of this second transition for several authors. First of all, as flow state depends on previous flow history (Coles [24]), so evolution of axial wavelength can be different depending on the experimental protocol (rarely described) and so, there may be no shortest wavelength. Moreover, Refs. [31,39,40] have large Taylor ranges and their measurements are less numerous. So, it is possible that the authors have failed notice the first region (before the transition) which is relatively small. A third possible transition is shown by Tachibana et al. [35] and Tachibana and Fukui [41]. Comparison between their two studies seems to show a transition for Ta ¼ 108. They are the only authors working at such a high Taylor number. The results obtained by Taylor [2] lead to the conclusion that following the appearance of vortices: Nu z Ta1/4. Most authors have formulated the Nusselt number through experimental correlations of the aspect Nu ¼ ATan with A and n of the constants depending a priori on the aspect ratio of the cylindrical gap. Several precise correlations have been put forward by different authors, among whom we wish to cite Becker and Kaye [36], Tachibana and Fukui [35] and Bouafia et al. [12,14]. These correlations and the conditions of use are reported in Table 1. The curves in Fig. 8 show sensitivity identical to the Nusselt number of variations in the Taylor number recorded above Ta ¼ 104 by researchers as different as Becker and Kaye [36] and Bjorklund and Kays [39], Aoki et al. [31] and Tachibana et al. [35]; the exponents n presented by these authors are all situated within the same range; while the multiplying factor may vary, it nevertheless remains close to 0.2. It should also be noted that the geometric studies are far from identical. As seen in Table 1, h, e/R1 and G vary from one author to the next. Once the Taylor number exceeds 108,Tachibana and Fukui [41] have recorded an exponent n

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of 1/3 rather than ¼, The deviation from their prior numbers (and from other authors) is explained by the non-overlaping of the measuring ranges utilized in the two studies: the studies of Tachibana and Fukui [41] is conducted for greater value of Taylor numbers. It likewise bears mentioning that Tzeng [42] observed much higher heat transfer than the other authors without any marked differences with regard to their respective configurations. The results reported by Tzeng [42] were associated with Taylor numbers higher than those of the many other authors. But, the more significant difference between Tzeng’s experiment and other studies resides in the presence of high natural convection which seems to greatly affect heat transfer (see below). Aside from this particular case, the different results concerning to heat transfer are in close correspondence, with scattering occurring once the rotation speed goes up. Several parameters may exert some influence on the heat transfer depicted. Natural convection may continue to affect heat transfer e even after the appearance of Taylor vortices. Tzeng [42] has postulated a correlation taking into account the natural convection effects based on the tangential Reynolds number Ret ¼ uR1Dh/n for a horizontal configuration with h ¼ 0.895, e/ R1 ¼ 0.12 and G ¼ 17. This correlation provides the ratio between the Nusselt number with rotation NuU and the Nusselt number without rotation Nu0:ðNuU =Nu0 Þ ¼ 0:375ðGr$ðuR1 Dh =nÞ$1011 Þ0:328 with Gr ¼ b$DT$e3 g=n2 . The correlation should not be used unreservedly, for there are differences between the thermal results of Ref. [42] and those of the other authors (Fig. 8). The effect of cylindrical gap size on heat transfer has been studied by par Ball et al. [40]. They observed that the more pronounced the rise of h, the more pronounced the rise of the radial Reynolds number, and consequently of the Taylor number. This observation would appear logical, since a wide cylindrical gap tends to stabilize the flow and thereby delay the development of Taylor cells and diminish their effects on heat transfer. Most of the studies presented above involve global heat transfer, and only the numerical studies record local results. Thus, Fig. 9 presents the axial evolution of the Nusselt number numerically recorded by Ghayoub et al. [23] with a cooled stator and a heated rotor. The reference temperature chosen was the mixing temperature, of which the evolution is likewise depicted in Fig. 8; the choice renders it possible to distinguish the Nusselt number Nu2 of the stator from that Nu1 of the rotor. The sinusoidal evolution corresponds to the alternation of the pairs of vortices presented in Fig. 10. To be precise, when two vortices meet at the level of the

Table 1 Recapitulation of the different correlations for convective heat transfer in a smooth and open cylindrical gap. Note: The definition of the Nusselt number has been modified so as to have it coincide with the other authors, and it corresponds to the definition used in the present text.

h

e/R1

G

Ta

Tac

C.L. thermal

Correlations

Becker and Kaye [36]

0.807

0.238

172

0 à 3,3  105

1994

Cooled stator Heated rotor

Tachibana and Fukui [41]

0.75 / 0.938

0.07 / 0.33

2.25 / 11.25

108 / 5  1012

e

Tachibana et al. [34]

0.522 / 0.971

0.13 / 0.92

220 / 7000

0 / 108

1730e3000

Bjorklund and Kays [35] Aoki et al. [31] Ball et al. [40]

0.8 / 0.948

0.054 / 0.246

32 / 147

8000 / 4.106

1770e1994

e 0.437 / 0.656

e 0.26 / 0.64

e 31.4 / 77.2

5000 / 2  105 4000 / 4  105

e e

Tzeng [42]

0.895

0.12

17

7962 / 2  108

e

Cooled stator, Heated rotor or Heated stator, Cooled rotor Cooled stator Heated rotor Cooled stator Heated rotor e Cooled stator Heated rotor Heated rotor

Ta < Tac: Nu ¼ 2 Tac < Ta < 104: Nu ¼ 0,128 Ta0,367 104 < Ta < 3.3  105: Nu ¼ 0.409Ta0.241 Nu ¼ 0.092(TaPr)1/3

Nu ¼ 0.42(TaPr)0.25 Nu ¼ 0.35(Ta)0.25 Nu ¼ 0.44(Ta)0.25(Pr)0.3 Nu ¼ 0.069$h2.9084$ (R1ue/n)0.4614ln(3.3361h) Nu ¼ 8.854 (R1ue/n)0.262(Pr)0.4

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1000

Nu

100

10

1 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 Ta Tachibana et Fukui [41] Aoki et al.[31] Tzeng [42]

Tachibana et al. [35] BECKER et Kaye [36]

Bjorklund et Kays[39] Ball et al.[40]

Fig. 8. Comparison of the different correlations Note: the definition of the Nusselt number has been modified so as to have it coincide with the other authors, and it corresponds to the definition used in the present text.

rotor (A), the hot flow is evacuated in the direction of the stator, thereby provoking maximal Nu1 and increased mixing temperature. Conversely, when two vortices meet at the level of stator (B), Nu2 reaches a maximum and the mixing temperature a minimum. One may also note that as regards Ta ¼ 2  106, the secondary Nu1 maxima present at the bottom (B) might be in correspondence with the presence of the Gortler vortices. Moreover, the Nu peak grows more pronounced as the Taylor number goes up. This finding is most likely explained by a heightening of centrifugal forces and consequently of the speed of the vortices when they meet (in (A) or (B)). The impact of the Gortler vortices on total heat transfer remains relatively insignificant.

phenomena are combined, we may discern the four types of flow described by Kaye and Elgar [43] (Fig. 11): -

Laminar flow Laminar flow with Taylor vortices Turbulent flow Turbulent flow with Taylor vortices

Nevertheless, most of the authors have focused their studies on the two main transitions (Laminar to turbulent transition and Taylor vortex transition) rather than on the characterization of the flow encountered. So, we will analyze separately the two major transitions.

4. Smooth cylindrical gap, TayloreCouetteePoiseuille flow In the configuration of the smooth and open cylindrical gap, axial flow is superimposed on the rotation effect provided by the rotor. As in the preceding part of this paper, a brief description of flow structure will be given prior to a section devoted to heat transfer. In the case of TayloreCouetteePoiseuille flow, a new dynamic parameter is to be used in addition to rotation velocity u: average axial velocity of the fluid Va (m/s). The axial Reynolds number may now be defined as Rea ¼ VaDh/n. The most widely accepted definition of the Nusselt number remains: Nu ¼ ðhDh =lÞ ¼ ðl gradTjp =Tw  Tref ÞðDh =lÞ ¼ ðF=SðTw  Tref ÞÞðDh =lÞ, where the reference temperature is usually taken to be equal to a fluid temperature (mean temperature Tmean ¼ (Te þ T0)/2), even though, as is the case with TayloreCouette flow, some authors take it as equal to the temperature of the second wall. 4.1. Flow structure Flow results from the superimposing of two discrete mechanisms, one of them linked to the centrifugal effects of rotational flow, the other one axially driven. Two main transitions may be distinguished: 1) a transition from laminar flow to turbulent flow when axial velocity rises; 2) the appearance of «Taylor vortex» structures above critical rotation velocity. When these two

4.1.1. Influence of axial flow: turbulent flow transition Few studies have focused on the transition from laminar to turbulent flow, while most of them have concentrated on the influence of axial flow on vortex appearance. It is generally considered that, before the appearance of Taylor vortices, these flows are similar to those encountered in duct flow with a radial velocity as well. Some authors have analyzed the influence of parameters on turbulence transition. As is the case in TayloreCouette flow, influence parameters are both geometrical and thermal. Cylindrical gap geometry assumes a prominent role in transitions. Kaye and Elgar [43] were the first researchers to experimentally determine the transitions with regard to two different numerical degrees of relative cylindrical gap thickness h for a wide range of Ta and Rea (see Fig. 12). The transition to turbulence is more rapidly reached with a more narrow cylindrical gap for Rea < 1500, Reynolds number above which the effects are likewise reversed. Concerning thermal effects on flow, Becker and Kaye [36] studied the influence of existing temperature gradient and heat transfer on flow structure in a wide range of axial flow rates and rotation velocities (see Fig. 13). According to these authors: e For Ta ¼ 0, turbulence appears in both cases at Rea ¼ 1800 e Transition from laminar flow to turbulent flow appears to be independent of the axial Reynolds number when heat transfer

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

1145

Fig. 11. Nature of the flow in accordance with axial Re and Ta [43].

Given the unique character of Becker and Kaye [36] study, it would be dangerous to extrapolate and generalize on the basis of the reported results. The absent influence of Rea on the transition from laminar to turbulent in a diabatic setting appears rather astonishing. 4.1.2. Influence of rotation: Taylor vortex flow transition Studies about the Taylor vortex transition have been much more numerous. So, Fig. 11 clearly shows the stabilizing effect of axial velocity with regard to Taylor vortices: The initial instabilities (the vortices) appear for higher Taylor numbers in the presence of axial flow. It mention that the flow’s nature has already been established by the different authors cited [34,43e45] if: 0  Rea  104 and 0  Ta  106. The study by Jakoby et al. [46] bears special mention on account of a particularly high axial Reynolds number: 2  104 < Re < 3  104; this entails equally high critical Taylor numbers, which may reach 4  107 and even 2  109. In order to determine the critical Taylor number, Chandrasekhar [44] employed linear stability theory so as to calculate this number for a narrow cylindrical gap (h ¼ 1; 0.95; 0.90). He showed that with regard to narrow annular space and particularly low Reynolds numbers (the authors suppose that Rea approaches zero), we may 26.5Rea2. The Taylor number is defined write: Tac(Rea) ¼ Tac(0)pþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 2 u n as: Tac ¼ ð R1 e = Þ ð2h=1 þ hÞ. This analysis would appear to Fig. 9. Axial evolution of the mixing temperature and the Nusselt number [23].

takes place (as for adiabatic flow, Kaye and Elgar [43] have shown highly pronounced dependence with regard to axial flow rate and rotation velocity). e The flow is stabilized by the temperature gradient, as transition to turbulence occurs for greater axial Reynolds number for the diabatic configuration.

Fig. 10. An example of flow in the presence of Taylor vortices [23]. Velocity field in a radially profiled cylindrical gap.

Fig. 12. Nature of the flow in accordance with Reynolds and Taylor numbers: Influence of cylindrical gap thickness, according to Ref. [43].

1146

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

Fig. 13. Nature of the flow in accordance with the Taylor and Reynolds numbers for an adiabatic and a non-adiabatic flow [36].

indicate that zones with and without vortices are distinctly separated. On the other hand, the results of experiments by Yamada [34] show that the transition from turbulent flow to turbulent flow with vortices is punctuated by a critical zone and not by an analytically defined separation. Jakoby et al. [46] have likewise confirmed the existence of a transition area rather than a pronounced boundary between flows with as opposed to flows without vortices; one may consult their frequency study for high axial Reynolds and Taylor numbers (2  104 < Re < 3  104 et 105 < Ta < 2  108). Most of the publications cited involve relatively low Taylor numbers, that is to say approximately 0 < Ta/Tac < 100. The studies are consequently focused on the appearance of vortices and their deformation under the influence of heightened rotation velocity. Some studies have also described the different flow and velocity profiles involved. For instance, Simmers and Coney [47] analyzed flow with velocity profile measurements for Re ¼ 500; Ta was made to vary with regard to relatively narrow cylindrical gaps (h ¼ 0.955 and 0.8). Following the appearance of vortices, a constant velocity zone develops; at once tangential and axial, it is located halfway between the two walls. This zone would seem to correspond to relocation of the vortices in an overall movement at a speed equal to axial speed. This phenomenon has similarly been observed by Astill [48] and by Gu and Fahidy [49]. Lueptow et al. [50] carried out a highly meticulous experimental study pertaining to types of flows for low Taylor numbers (Ta < 3000) and very low Reynolds numbers (Rea < 540) in the framework of a narrow cylindrical gap: h ¼ 0.885, e/R1 ¼ 0.083 and G ¼ 41. They observed a strikingly high number of flow regimes ranging from laminar flow to turbulent flow along with

Taylor vortices. Limits and small description of these flows are presented in Fig. 14. The results reported by Coney [47,51e54] show that when the rotation velocity of the inner cylinder rises along with the Taylor number, vortex wavelength likewise goes up. This phenomenon, which also exists without axial flowing and is then contingent upon the way in which the nominal value of the Taylor number is reached, may be due to the merging of adjacent cells. Finally several authors have focused on influence parameters on transition between flows with and without vortices. Cylindrical gap geometry assumes a prominent role in this transition. Kaye and Elgar [43] have determined the transitions for two different relative cylindrical gap thicknesses h (see Fig. 12). As regards low axial Reynolds numbers (Rea < 100), annular gap thickness does not seem to exert a pronounced influence on the transitions. As for higher Reynolds numbers, Taylor vortices would seem to appear for lower Taylor numbers in the case of a narrow cylindrical gap up to Rea z 1500, Reynolds number above which the effect is reversed. Similar results have been observed by Jakoby et al. [46] as concerns transition between flows, with or without vortices. On the contrary, Hasoon and Martin [55] employed the Galerkin method by using mean axial velocity to analytically analyze the effect of relative cylindrical gap thickness h on the transition from laminar flow to laminar flow with vortices, and they show (Fig. 15) that the heightening of h stabilizes the flow (the appearance of vortices occurs with higher Taylor numbers). The influence of h is consequently identical to the influence of TayloreCouette flow. The influence of h seems to diminish as the axial Reynolds number rises. These results were confirmed by Di Prima and Pridor [56], who studied the problem of theoretically linear stability while nonetheless using a developed and thereby parabolic axial velocity profile, rather than a uniform velocity profile. This difference no doubt explains the deviations between the two types of curves (Fig. 15), particularly the difference in slope for the highest Reynolds axial numbers. These results are equally confirmed by those of Wan and Coney [51], whose work has been focused on a Ta and Rea range close to that studied by Kaye and Elgar [43]. These contradictory findings involving on the one hand Kaye and Elgar [43] and on the other hand Hasoon and Martin [55], Di Prima and Pridor [56] and Wan and Coney [51] can be explained by another parameter of influence, the axial aspect ratio G. Indeed, it is hard to achieve variation of h without variation of G, which may vary in a proportion of 1e2 for Kaye and Elgar [43], and in a proportion of 1e5 for Wan and Coney [51]. Fig. 13 provides some relevant information on the influence of the axial aspect ratio G. Becker and Kaye [36] therein show the results of three authors ([43,57,71]) who started with highly different geometries, with the flow always remaining adiabatic. An initial reading of this figure shows that a longer cylindrical gap allows vortices to

Fig. 14. Nature of the flow for Rea < 40 and Ta < 3000 [50].

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

Fig. 15. Critical Taylor number, influence of cylindrical gap thickness with N ¼ 1  h [56].

appear for lower Taylor numbers (comparison of the results reported by Kaye and Elgar [43] with those of FAGE [57], for example). Moreover, the transition situation proposed by Cornish [58] appears to prove that the role of length is dominant in comparison with the role of annular gap thickness. The pronounced influence of G may be explained by the input effects and the development of axial velocity profiles. Indeed, Astill [48] has taken note of a precise distance, short of which Taylor cells either do not or only very partially appear. This distance rises along with Rea and falls along with rotation velocity. It functions with an axial velocity profile fully developed at the cylindrical gap entry. Molki et al. [59] likewise observed, with an adequately high Taylor number (higher than Tac) the appearance of vortices at a distance from the cylindrical gap entrance proportionally as sizable as Rea. Hasoon and Martin [55] have likewise shown that the critical Taylor number decreases in accordance with distance from the cylindrical gap entrance. It would consequently seem that only with pronouncedly higher rotation velocities do Taylor vortices appear in the so-called zones of development. Finally and in addition to the geometrical parameters, Becker and Kaye [36] studied the influence of existing heat transfer on vortex structures (see Fig. 13). Transition from laminar flow to laminar flow with vortices proceeds similarly within each of the two configurations. This finding is confirmed through the experimental measurements carried out by Sorour and Coney [53], the critical Taylor number being independent of thermal limit conditions in the absence of axial flow. On the contrary, transition from turbulent flow to turbulent flow with vortices seems to occur for smaller Taylor numbers for the diabatic configuration. As for transition to turbulence (3.1.1), the unique character of the Becker and Kaye [36] should be borne in mind when considering the results they have reported. Notwithstanding their limits, these results show that the structure of combined flow in annular space hinges not only on the operating point (axial Reynolds and Taylor numbers), but also e and strongly e on geometry and, to a lesser degree, on parietal thermal conditions.

1147

the choice of a reference temperature may be particularly difficult. As regards closed cylindrical gap space, the temperature difference used when calculating the heat transfer coefficient is consequently (T1  T2). With regard to axial flow rate, on the other hand, a third temperature that of entering fluid must also be taken into account and it is that much harder to establish a reference temperature. Becker and Kaye [36] decided to maintain the distinction between the two walls, but they could compare their results only to the temperature at entrance of a given fluid; they could not dissociate the heat transfer on each one of the walls. The mean temperature calculated from the enthalpy assessment utilized by Bouafia et al. [14] in particular seems to be the most physically accurate, but also the most difficult to obtain experimentally. An intermediate solution occasionally adopted (by Grosgeorge [45] for instance) involves calculation of the mean value between entrance and exit temperature. As a rule, the main results available in the bibliography are preliminarily limited to the field: 0  Rea  5  104, 0  Ta  107, with the exception of a few publications such as those of Hirai et al. [60] (Even though they do not put forward any experimental correlations), Grosgeorge [45] and, especially, Childs and Turner [61]. As is the case with closed smooth cylindrical gaps, heat transfer in an open annular space is integrally linked to the different types of flow commonly encountered (Table 2). The influence of the two main parameters, axial and radial velocity, has been differently quantified by the authors. So, before studying the effects and contributions of each of these velocities on heat transfer, we need to observe how heat transfer has been modeled and connected with axial and radial velocities. Then, contrary to fluid flow, two heat transfer regions have to be considered: the stator and the rotor. 4.2.1. Modelizations and correlations Regardless of the techniques put to use by specific authors, estimation of the Nusselt number is frequently provided in the form of experimental correlations established under particular conditions. The most widely proposed expressions take on the following forms:

Axial flow rate þ rotation/Nu ¼ AReaa Tab Prg or Nu ¼ Reaeff Prg

The number Reeff is based on an effective velocity defined by:

Veff ¼



Va2 þ aðuR1 Þ2

1=2

0Reeff ¼



Veff Dh =n



with A, a, b and g as constants depending on experimental conditions these expression depends on whether they describe the influence of axial flow rate, rotation velocity or a combination of the two by means of the effective Reynolds number Reeff, based on effective velocity, a combination of the axial and radial contributions. Table 3 includes some of the experimental correlations that best represent the smooth cylindrical gap of a rotating electric machine as well as experimental conditions and their field of application.

Table 2 Influence of rotation speed and axial flow rate on nature of flow and heat transfer [36]. Ta < Tac(Rea)

Ta > Tac(Rea)

4.2. Convective heat transfer

Rea < ReaC(Ta)

Nu constant with Ta and Rea

In this type of «open» configuration, defining a convective heat transfer coefficient is more complex than in a «closed» configuration;

Rea > ReaC(Ta)

Nu constant with Ta and increasing with Rea

Nu increasing with Ta and decreasing with Rea Nu increasing with Ta and decreasing with Rea

1148 Table 3 Recapitulation of the main correlations for convective heat transfer in a smooth open cylindrical gap. Note: the Nusselt number definition has been modified so as to make it coincide with those of the other authors and correspond to the definition given in the present article.

Tachibana and Fukui [41]

h

e/R1

G

Rea

Ta

C.L. thermique

DT

Correlation

0.75 / 0.937

0.03 / 0.17

2.25 / 11.25

380 / 4200

71 / 3400

Variables

Rotation: T1  T2 Axial flow: T1  Tmean

Rotation: Nu ¼ 0.092(TaPr)1/3 Axial flow: Nu ¼ ARea0.8Pr1/3 and A ¼ 0.015(1 þ 2,3(Dh/L))h0.45 Global: f ¼ ft þ fa

Kuzay and Scott [62]

0.571

0.75

12

1.5  104 / 6.5  105

4.87  109 / 8.65  109

Heated stator Insulated rotor

T 1  T2

0 / 3.6  105

Insulated stator Heated rotor

T 1  T2

Axial flow: Nu0 ¼ 0.022Rea0.8Pr0,5 Global: Nu ¼ Nu0(1 þ b2)0.87

b ¼ (1/p)(Dh/R1)(uR1/Va) 0.75

0.165

195

293 / 1995

Global: s ¼ Ta/Tac 1 < s < 4.817: Nu ¼ 4.294s0.4845

s > 4.817: Nu ¼ 5.08s0.3507 Kosterin and Finat’ev [67]

0.78

0.0271

77.5

4

3.10 / 3  10

5

0 / 8  10

5

Insulated stator

T1  Tmean

Reeff ¼ (Rea2 þ 0.6Rerot2)1/2

Heated rotor 5

Grosgeorges [45]

0.98

0.02

200

9900 / 26,850

1.4.10 / 4.9  10

Childs and Turner [61]

0.869

0.15

13.3

1.7.105 / 13.7  105

6.107 / 12  1010

6

Insulated stator Heated rotor

T1  Tmean

Rotor: Global: Nu ¼ 0.023j(Rea)Pr1/3Reeff0.8 with j(Rea) ¼ 0.16Rea0.175 and Reeff ¼ (Rea2 þ 0.8Rerot2)1/2

Insulated stator

T 1  T2

Axial flow: Nu ¼ 0.023Rea0.8Pr0.5 Global:((Nu  Nuz)/Nuz) ¼ 0.068(uR1/Va)2

Heated rotor Simmers and Coney [54]

0.955 and 0.8

0.024 / 0.124

65 / 288

400 / 1200

4

10 / 2  10

6

Global: rotor: Nu ¼ 0.018 Reeff0.8

Heated stator Insulated rotor

T 2  Tm

Stator: Nu ¼

0:3675 4$Pr$Re0:5 a Ta

BðA= Þ1=2 $ðh= Þ1=4 Ta0:6175 c 1h 1h 2 with B ¼ Pr$f1 þ Pr$½expð ðh= Þ1=4 3 1h

ðAh= Þ1=2 Re0:5 Ta0:1325 $Ta0:1175  1Þ  1g a c ð1  hÞ2 1 þ h2 þ ð1  h2 Þ= lnðhÞ 2 þ ð1  h2 Þ= lnðhÞ Global: Rotor: Nu ¼ 0.025 Reeff0.8 and A ¼

Bouafia et al. [14]

0.956

0.045

98.4

1.1.104 / 3.1  104

1800 / 4  106

Cooled stator Heated rotor

T 1  Tm T2  Tm

Reeff ¼ (Rea2 þ 0.5Rerot2)1/2 Au stator: Nu ¼ 0.046 Reeff0.7 Reeff ¼ (Rea2 þ 0.25Rerot2)1/2

Gilchrist et al. [64]

0.833

0.2

56.9

950 / 2080

6

10 / 5  10

7

Heated fluid

DTLM

Hanagida and Kawasaki [65]

0.99

0.0094

283

900 / 10

4

4

2.3.10 / 2  10

5

Heated stator Heated rotor

Rotor: Nu ¼ 0.65Ta0.226Pr0.333

Cooled rotor T 1  Tm T2  Tm

Stator or rotor: St ¼ 0.218X1/2Pr2/3 quand X  5000 St ¼ 0.0072X0.1 Pr2/3 quand X  5000 Avec X ¼ Rea2/Rerot

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

Nijaguna and Mathiprakasam [37]

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

As regards combined flow, the authors mentioned in Table 3 propose several different ways of quantifying the respective roles of the Taylor and axial Reynolds numbers. Tachibana and Fukui [41] multiply their results by a coefficient that takes into account the axial lengthening of annular space (Table 3). Their approach consists in directly adding the heat flux transferred by simple rotation to the heat flux transferred through simple axial flow. The temperature difference used for the contribution of rotation alone is (T1  T2), while for axial flow, it is (T1  (To þ Te)/2). Tachibana and Fukui [41] consequently cannot take into account any possible combining of axial and radial flow. Their approach has remained less influential than those of the other authors mentioned in Table 3. Kuzay and Scott [62] propose a correlation based upon a characteristic flow parameter. 1/b represents the axial path of the flow in terms of hydraulic diameter during half a revolution of the inner cylinder: b ¼ ð1=pÞðDh =R1 ÞðuR1 =Va Þ. Through their use of b, Kuzay and Scott [62] modify the axial contribution to heat transfer so as to take into account the role assumed by rotation. Since there exists a relation between the axial Reynolds number and the critical Taylor number, Nijaguna and Mathiprakasam [37] utilized a different approach, which consisted in correlating the Nusselt number only to the ratio: s ¼ Ta/Tac. This approach may be explained by the low degree and amplitude of Rea variation. A supplementary difficulty consists in accurately assessing the critical Taylor number. Nijaguna and Mathiprakasam [37] partially extricated themselves from this problem as follows: when the axial Reynolds number is sufficiently low (Re < 2000 in this case), the transition is highly pronounced, and the crucial Taylor number is that much easier to measure. Under a turbulent regime, on the other hand, the critical Taylor number is considerably more difficult to estimate. Moreover, Nijaguna and Mathiprakasam [37] provide no explanation as to the significance of the critical s value assumed at 4.817. This value may perhaps be linked to the «transition» observed by Becker and Kaye [36] and by Bouafia et al. [12,14] for Ta ¼ 104 in a case without axial flow (Fig. 8). We may also cite Simmers and Coney [47], who correlated the Nusselt number to the product ReanTam. Gazley [6], Luke [63] and Bouafia et al. [12,14] correlated their thermal results with an effective Reynolds number characteristic of helicoid flow

Reeff ¼



Veff Dh =n



¼



Va2 þ aðuR1 Þ2

1=2

 Dh

n

where a is a weighted coefficient potentially taking into account the rotation of the inner cylinder at the level of heat transfer. The different studies by these authors allow for characterization of convective heat transfer with regard to relatively high Taylor and Reynolds numbers, and their correlations are worthy of consideration. Gilchrist et al. [64] are among the few authors to have worked with a fluid different from air, namely water. Moreover, their measurements are based on heated cylindrical gap fluid and a cooled rotor. Their correlations do not take into account the axial Reynolds number, which does not appear to have affected heat transfer during their experiments e a finding that is largely explained by the limited Rea area. Hanagida and Kawasaki [65] worked in a more traditional manner, with air as the fluid being studied and while using electric heating systems to create pronounced variations in temperature. And yet, contrarily to some other authors, they determined the heat transfer on both the rotor and the stator and went so far as to observe a lack of influence on fluid/rotor heat transfer in stator heating. One may also note their utilization of an original parameter

1149

of influence: X ¼ Rea2/Rerot. Based on the ratio rather than the sum total of two Reynolds numbers, this parameter is derived from studies of the friction coefficient. Childs and Turner [61] studied convective heat transfer in a relatively short and wide annular area; the Taylor and Reynolds numbers were quite high in comparison with most currently available references. Without rotation, they found a usual expression of the Nusselt number, which varies in conjunction with the axial Reynolds number at the power of 0.8. With rotation, the researchers modified the correlation through a coefficient in accordance with the ratio of the two velocities characteristic of the flow. This ratio may be likened to the b utilized by par Kuzay and Scott [62] and also to the X utilized by Hanagida and Kawasaki [65]. Finally, Jakoby et al. [46] are the only authors to have considered local heat transfer at the entrance of the cylindrical gap, taking into account the entrance region. They have provided a correlation of the overall Nusselt number as integrated on a cylindrical gap length of L (coefficients are presented in Table 4):

0 Nu ¼ C @

Va2 þ ðuR1 Þ2

y

1=2 1N L A with C

¼

a0 $etðGbÞ þ au $etðGþbÞ and N etðGbÞ þ etðGþbÞ

¼

a0 $etðGbÞ þ au $etðGþbÞ etðGbÞ þ etðGþbÞ

These observations once again confirm the specific character of each study. The extrapolation of the results to other particular cases is necessarily problematic. 4.2.2. Rotor convective heat transfer In the first part, we will analyze heat transfer on the rotor, which has been the focus of most of the studies. Differences between rotor and stator heat transfer and specific evolution on the stator will be presented in the paragraph 3.2.3. 4.2.2.1. Influence of axial flow. According to paragraph 3.2.1, most of the authors qualified the axial speed influence using an axial Reynolds number Rea. This axial Reynolds number does not have the same effect on convective heat transfer in a laminar as in a turbulent regime when the structures are present. As regards laminar flow (see Fig. 16 lower part), most of the authors [14,36] agree that radial velocity exerts little if any influence: Nusselt number is constant until a critical Taylor number, corresponding to vortex appearance. The higher the Reynolds number, the greater the critical Taylor number (and the larger the “low influence” area). In fact, a stronger axial flow rate drives away the Taylor vortices toward the exit. As for turbulent flow, the diminution in heat transfer caused by the number of structures may be compensated for by increased convective heat transfer due to turbulence. That is why, in one example, for Ta ¼ 5  104, the Nusselt number declines along with the axial number up to 1592, and rises once that figure is exceeded.

Table 4 Correlations coefficients. Steady flow

a0 au t b

Periodic flow

N

C

N

C

0.8 0.5 0.27 12.1

0.04 0.6 0.32 8.85

0.8 0.625 0.27 15

0.04 0.136 0.3 12.9

1150

M. Fénot et al. / International Journal of Thermal Sciences 50 (2011) 1138e1155

that their flow changes from laminar to turbulent with regard to the dimensions of a cylindrical gap. They went on to observe a rise in the overall Nusselt number when cylindrical gap length has been reduced, but they essentially attribute this variation to the appearance of a turbulent flow zone at the end of the cylindrical gap with regard to the shortest lengths of the latter.

Fig. 16. Evolution of the Nusselt number in accordance with Taylor numbers given at Rea by Becker and Kaye [36].

After the transition to turbulence and even if annular spaces, experimental conditions and the axial Taylor and Reynolds numbers are quite different. Study of Table 3 shows that the influence of the axial Reynolds number is expressed approximately by a “classical” Rea0.8 (encountered in many duct correlations such as Dittus and Boelter [66]). Previous results pertain to developed axial flow. But in many TayloreCouetteePoiseuille flow applications, the cylindrical gap is too short to permit a developed flow. Given the importance of the entrance area in classical axial duct flow, numerous recent studies have dealt with the development of an axial regime and its influence on heat transfer (and of mass, as well). One of the less recent studies of this type was carried out by Hirai et al. [60], who worked on a water flow. They observed that the Nusselt number declines from the onset of the cylindrical gap and then tends to reach a constant number. This finding perfectly corresponds to thermal development of flow in a pipe or duct. The higher the rotation speed, the more rapidly Nu tends to reach its numerical limit. This observation is apparently confirmed by the results reported by Molki et al. [59], who worked on mass transfer (naphthalene) with an air flow of which the input had been controlled in such a way as to reduce the dimensions of the zone preceding the cylindrical gap. It should nonetheless be noted that the cylindrical gap is proportionately much shorter than the one presented by Hirai et al. [60] and consequently does not quite allow for developed flow to be reached. On the other hand, heightened spatial accuracy allows for a rise of the Sherwood number (and thus of the Nusselt number) for high Taylor numbers from a given length upwards; the higher the axial Reynolds number, the greater the length. Smoke-based visualizations have enabled researchers to confirm that the heightening corresponds to the appearance of Taylor vortices. A heightened Nusselt number was likewise observed by Bouafia et al. [14] in spite of the fact that their flow was already partially developed prior to cylindrical gap entrance. This fact also explains the highly limited decrease of the Nusselt number at the onset of the cylindrical gap. In one particular case [60], as in another, [14], the local heat transfer appears on an overall basis to have risen slightly when Ta rose, except in the zone where Taylor vortices appeared; in this zone the transfer would seem to have risen more sharply. The last study of this kind was carried out by Jakoby et al. [46]. It involves an air flow for several cylindrical gap sizes. Total cylindrical gap length is not modified, and G consequently varies together with e. Researchers have also noted the pronounced influence of the entrance zone and of regime development; their finding is underscored by the fact

4.2.2.2. Influence of rotation. Heat transfer is subsequently favored by the appearance of vortices once the Taylor number rises (see Fig. 16). In accordance with the flow regime, these structures come about in different ways. In laminar flow, the structures’ appearance is manifested through an abrupt change in the slope of the Nusselt number variation in accordance with the Taylor number. In a turbulent regime, on the other hand, this change of direction is identified not with a threshold but rather in terms of a critical field. Considering the correlations of Table 3, the influence of the Taylor number after the appearance of vortices, is expressed at approximately Ta1/3 by nearly all the authors. Small differences between authors can be attributed to differences in annular spaces, experimental conditions and Taylor and axial Reynolds numbers areas. But, concerning the influence of rotation compared to axial flow, opinions tend to differ greatly. Thus, comparison of results of the authors using effective Reynolds number ðReeff ¼ ððVa2 þ aðuR1 Þ2 Þ1=2 Dh =nÞÞ shows great differences. Unfortunately, axial and radial velocity profiles are quite different from one author to another and so, it is difficult to compare studies. For example, Hanagida and Kawasaki [65], who have carried out measurements in laminar as well as turbulent flow, have observed only a weak influence of rotation in turbulent regime, and a pronouncedly greater influence for laminar regime. Nevertheless, many authors have noted a pronounced preponderance of the influence of axial velocity over that of radial velocity; they include Bouafia et al. [14] (a ¼ 0.5 for the rotor), and Kosterin and Finat’ev [67] (a ¼ 0.6). Gazley [6] indicated in his study a greater influence (a ¼ 0.25) but axial flow is clearly turbulent and not developed before entering the cylindrical gap. So, heat transfer due to axial flow is high at the entrance length, a factor that may help to explain the weak influence of rotation. Grosgeorge [45] seems to report strong dependence of the Nusselt number on rotation as regards high axial Reynolds numbers (rotation influence coefficient a ¼ 0.8). But, as we can see in Table 3, axial Reynolds number appears twice in his correlation. In the final analysis, axial velocity is also quite influent (even if it is difficult to compare with other authors). Jacoby et al. [46] observe that the contribution of Taylor vortices to heat transfer is relatively weak. According to these authors, radial velocity exerts as much influence on heat transfer as axial velocity (a ¼ 1). So, radial velocity has a greater influence in comparison with other authors. This is probably due to the principally laminar axial flow in their experiment since influence of rotation is greater in laminar regime (Hanagida and Kawasaki [65]). Finally, Gilchrist et al. [64] go so far as to observe that as regards the configuration they have studied, rotation alone exerts influence on heat transfer. It is true that in their experiment, a particularly small Rea area (950 / 2080) as compared with the Ta area (106 / 5  107) may explain the absent observable variation of the Nusselt number. As concerns the influence of thermal conditions applied to each heated, cooled or insulated wall, Hanagida and Kawasaki [65] studied the case of one wall (rotor or stator) heated alone and of two walls heated simultaneously, but we have not detected any pronounced differences in their respective configurations. 4.2.3. Stator convective heat transfer As may be observed in Table 3, few authors have studied stator heat transfer. This is largely due to their simple definition of the

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heat transfer coefficient (heat transfer between rotor and stator: h ¼ ðl gradTjw =Trotor  Tstator Þ) [37,41,61,62] and to the fact that their experiments allow for measurement of only one of the two walls, which is generally the rotor. Kuzay and Scott [62] studied the thermal behavior of the fixed stator wall. Through their use of b (see paragraph 3.2.1), they modified the axial contribution to heat transfer so as to take into account the role assumed by rotation. They note a limited role of rotation and estimate a lower Nusselt number than those estimated on the rotor by other authors. Simmers and Coney [47], who correlated the Nusselt number at the stator to the product ReanTam have also studied stator heat transfer. The Rea and Ta areas are unfortunately so far from those discussed by Kuzay and Scott [62] that it is all but impossible to make a comparison. In both cases, the limited influence of rotation is mentioned. We may go on to note that two studies have been focused on comparison of heat transfer to the stator and the rotor: Hanagida and Kawasaki [65], and Bouafia et al. [14]. Hanagida and Kawasaki [65] detected no noteworthy difference between the two walls, but their Taylor number range was relatively limited. The study conducted by Bouafia et al. [14] was more wide-ranging; the researchers observed rather similar stator and rotor developments with regard to axial flow and pronounced differences with regard to radial flow (rotation coefficient influence at coefficient a ¼ 0.5 for the rotor and at a ¼ 0.25 for the stator). The weighted coefficient for the stator is lower than with regard to the rotor. This appreciable difference stems from the dynamic conditions imposed on the walls. The influence exerted by rotation on convective heat transfer on the stator is likely to be attenuated to the degree that the cylindrical gap is widened. 5. Slotted cylindrical gap The studies involving non-smooth cylindrical gaps may be classified in two categories. Some of them attempt to reproduce the cylindrical gaps of electric machines, the slots being carved out in the packs of sheet metal comprising the stator and the rotor that allow for the passage of various coils. Other studies are aimed at improving heat transfer within the cylindrical gap. The objective of this part is to consider the thermo-aerodynamic behavior of the slotted cylindrical gap. Only a few articles contain references to an axially grooved cylindrical gap. What is more, most of the reported results are of experimental origin and remain general (or global) [6,10,41]. Only the analyses carried out by Hayase et al. [13] and Hirai et al. [60] put forward, by means of numerical simulation, a more fully nuanced description of flow and heat transfer in a closed and slotted cylindrical gap. Moreover, highly diversified geometry renders it exceedingly difficult to compare, much less generalize. Among the possible geometric parameters, one may cite the following: existence of slots on the rotor, on the stator, or on both walls simultaneously, the width 1 and the depth (p) of the slots, which may vary from 0.05e to 10e, their number, their layout, and possible geometric peculiarities (Jeng et al. [68] have studied a rotor presenting longitudinal slots themselves equipped with internal slots). In addition, there remain the dynamic parameters: presence or absence of axial flow, axial Reynolds number, and Taylor number. As regards the last two parameters (Rea and Ta), it should be noted that while some authors take into account any geometrical differences between cases with and without slots, others apply the same definition to both cases. And finally, rigor demands that when slots are to be found, the following, above-mentioned parameters must be added: number of slots n, their width l, their depth p, which enters into the slot aspect ratio: The characteristic physical quantities have got to be modified (Fig. 17).

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l

p

R1

L

R2

Fig. 17. Geometry of a slotted TayloreCouette flow.

The hydraulic diameter is expressed as:Dh ¼ ð2½pðR22  R21 Þþ nlp=pðR2 þ R1 Þ þ npÞ. One may define an equivalent smooth cylindrical gap with the same hydraulic diameter and the same rotor diameter with an equivalent width for the annular eeq and an equivalent inner radius for the stator R2eq:Dh ¼ 2ðR2eq  R1 Þ ¼ 2eeq , which gives eeq ¼ ðpðR22  R21 Þ þ nlp=pðR2 þ R1 Þ þ npÞ. 5.1. Flow structure Most of the studies pertaining to slotted cylindrical gaps are thermal studies; they may nonetheless provide useful information on flow dynamics. 5.1.1. TayloreCouette flow As regards flow without axial flow, Tachibana and Fukui [41] have compared the critical Taylor number of smooth cylindrical gaps with the critical Taylor number at the rotor for slotted cylindrical gaps. The cylindrical gaps they studied are not described in their entirety; the number of slots on the rotor and the diameter of the rotor and the stator have not been specified. The cylindrical gap would appear to be wide (up to 20.5 mm), and the slots are likewise wide (10 mm for 3 mm of depth). The rotor is heated. The authors show that when slots are present, the transition toward flow with Taylor vortices is delayed: Tac ¼ 1680 if the rotor is smooth, and TaC ¼ 6400 if the rotor is slotted. It should be noted that the authors take the slot’s existence into account by modifying the size of the cylindrical gap: eslotted ¼ esmooth þ p/2. Their results were reached on the basis of several cylindrical gap geometries, and it is difficult to dissociate the role directly assumed by the slots from that attributable to annular space thickness. Tachibana and Fukui [41] attribute the variation in Tac to the variation in cylindrical gap size, and not to the slots. In total contradiction with the preceding result, the study authored by Pecheux et al. [69] observe no difference as regards the value of Tac between the smooth and the slotted cases (Tac ¼ 1790 for the slotted case). The study proceeds by visualization in water as regards a slotted cylindrical gap: R1 ¼14 cm, R2 ¼ 14.5 cm, L ¼ 64 cm, 48 slots on the stator: p ¼ 1.5 cm and l ¼ 0.83 cm and Ta ¼ ðu2 R1 e3 =n2 Þ. The apparent contradiction of

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these results with regard to those reported by the other authors may be partially explained by the difference in geometry; Pécheux et al. [69] worked on a cylindrical gap equipped with particularly large slots (p equals 3 times the cylindrical gap) located only in the stationary part. Moreover, no smooth geometry was studied, and the smooth/slotted comparison was carried out with regard to the results reported by other authors. Finally, it should be noted that when defining the Taylor number, the researchers did not take the geometry into account. And use of the hydraulic diameter would yield: Tac w 3900. The above remark appears particularly plausible in light of the fact that a geometric study quite similar to that carried out by Pécheux et al. [69] was performed with regard to air by Bouafia et al. [14], who indeed observed thermal transition neighboring Tac ¼ 3900. Considering all those remarks, it seems that transition to Taylor flow is delayed if a correct definition of the Taylor number is used. Above and beyond the transition, only a few experimental studies have been conducted as regards the dynamics of TayloreCouette flow in a slotted cylindrical gap, and most of them have been based on numerical studies. Pécheux et al. [69] noted that the wavelength of the Taylor vortices is about twice the length of the vortices present in the smooth cylindrical gap. Moreover, they observed that the structures fluctuate in time, and they attributed this fact to interaction with the internal structure of the slots. Numerically, Hayase et al. [13] and Bouafia et al. [12] have worked with slotted configurations (especially on a large-slotted stator), and they took note of interaction between cylindrical gap flow and slot flow (Fig. 18). The higher the rotation velocity, the more pronounced the interaction. 5.1.2. TayloreCouetteePoiseuille flow In the presence of axial flow, Gardiner and Sabersky [10] underscored the paradoxical behavior of the critical Taylor number, which at first goes up, along with the Reynolds number, before it goes down: at first it is 104 without flow, then it is 6  104 for a Reynolds number of 800, finally it is 1.5  104 for 2700. As for Lee and Minkowycz [70], they reported that with slots and with regard to different cylindrical gap geometries, the length of dynamic flow development is small and even of negligible extension for high Taylor numbers (Ta > 105). They also observed that given the respective values of rotation velocity and axial flow (103 < Ta < 2.107 and 52 < Rea < 1000), a heightened axial Reynolds number has little effect on flow and heat transfer. In their panoramic study and by means of a more local approach to flow in a slotted cylindrical gap, Bouafia et al. [14] carried out an analysis devoted to the respective roles of rotation and axial flow. The results reported by these authors are applicable with regard to 4400  Rea  1:7104 ; 300  Ret  6:4  104 , (corresponding to 103  Ta  4:8  107 ) for deep slots located at the stator. The authors

Fig. 18. Velocity fields in a schema (r, z) crossing the channel (according to Ref. [12]).

noted that the higher the rotation velocity, the greater the interaction of cylindrical gap flow with slot flow, which drives some of the air from the gap toward the channels when rotation velocity remains high. 5.2. Convective heat transfer 5.2.1. TayloreCouette flow In the absence of axial flow, and below the critical threshold, Gardiner and Sabersky [10] reported that the presence or absence of the slot has no effect on heat transfer, which rather tends to be driven, given their location, by natural convection. On the contrary, Bouafia et al. [14] observed slight evolution of Nu in the slotted case, but not in the smooth case (Fig. 19); this is due to heightened interaction with rotation speed between cylindrical gap flow and slot flow. Bouafia et al. [12,14] have shown that above the critical threshold, the presence of channels at the level of the stator favors heat transfer, especially for high Taylor numbers. They go on to put forward a correlation: 6000 < Ta < 1.4  106: Nu ¼ 0.132(Ta)0.3 4  106 < Ta < 2  107: Nu ¼ 0.029(Ta)0.4 The convective heat transfer is equally heightened from 40 to 50%, above the threshold, when the rotor is slotted (Gardiner and Sabersky [10]). According to these authors, it is once again difficult to define the exact impact of natural convection as regards these results. In addition, Hayase et al. [13] have shown in their numerical study that when vortices are present, the Nusselt number is higher with slots on the rotor than with slots on the stationary wall. As for the evolution of local Nu, it generally corresponds to that reported by Bouafia et al. [12,14], in spite of the fact that in their study, these researchers dealt with slots on the stator. Axial distribution of the averaged rotor-located Nusselt number employed along the internal cylinder and based on the temperature difference between rotor and stator has the same overall composition for two-dimensional and three-dimensional calculations (Fig. 20). One may note that the heat transfer is weak at the slots and preponderant in the annular part. The more globally oriented study by Gazley [6] presents results contradicting those reported by Gardiner and Sabersky [10] and by Hayase et al. [13] and by Bouafia et al. [12,14]. According to Gazley [6], in the case of laminar flow the presence of slots entails an

Fig. 19. Distribution of mean Nusselt number for smooth and slotted cylinders (according to Ref. [14]).

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Gazley’s [6] and other authors’ results). Moreover, according to these authors it is not possible to correlate Nu according to (Rea þ aRerot) with regard to a slotted stator. Several authors such as Gardiner and Sabersky [10], Gilchrist et al. [64], and Hanagida and Kawasaki [65] have observed heightened heat transfer, but have not observed any noteworthy difference between slotted and smooth cases in the evolution of the heat transfer with regard to characteristic quantities. On the contrary, Jeng et al. [68] have noted different forms of evolution at the level of the rotor. As for Bouafia et al. [14], they are convinced that if there is no actual difference at the rotor; the Nusselt number at the stator depends essentially on the ratio Vt/Va. In terms of correlations, let us mention those reported by Hanagida and Kawasaki [65] at the stator and the rotor with regard to a slotted stator with 60 axial channels at h ¼ 0.99, e/R1 ¼ 0.0094, G ¼ 283, 4 mm < p < 22.5 mm and l ¼ 12.5 mm, and 4400 < Rea < 1.7  104, 500 < Ret < 6.4  104: St ¼ 0.297X1/2 Pr2/3 when X  4000 St ¼ 0.0026X1/5 Pr2/3 when X  4000 with X ¼ Rea2/Rerot And those of Bouafia et al. [14] with regard to a slotted stator with 48 axial channels, h ¼ 0.956, e/R1 ¼ 0.045, G ¼ 98.4, p ¼ 15 mm and l ¼ 8.3 mm and 4400 < Rea < 1.7  104, 300 < Ret < 6.4  104: Fig. 20. Distribution of the Nusselt number for the slotted cylinder e three-dimensional calculation, q ¼ 10 , Ta ¼ 218 (according to Ref. [13]).

overall diminution of the rotor-located Nusselt numbers (from 10% when only the rotor is slotted, to 20% when the two walls are channeled). In the case of turbulent flow, he observed a slight rise of the Nusselt number when the slots were present. It is nonetheless worth noting that Gazley [6] based his study on thermally insulated slots. The efficient surface offered for heat transfer is consequently quite small, and contrarily to a laminar case, it can be only partially compensated by supplementary flow disturbance. As for Tachibana and Fukui [41], they have shown that above approximately Ta ¼ 4  104, slot presence has no effect on the heat transfer coefficient. 5.2.2. TayloreCouetteePoiseuille flow In the presence of axial flow, several authors have recently devoted attention to the problem, but taken as a whole, their results remains just as self-contradictory as the previous ones. A majority of the authors nevertheless observes heightened heat transfer when slots are present. For example, Gardiner and Sabersky [10] take note, with a low Reynolds number (800), of heightened convective heat transfer, even prior to the transition. Gilchrist et al. [64] and Jeng et al. [68] (who work on configurations equipped with small « teeth » at the rotor) have calculated an overall improvement in Nu ranging from 5 to 40%. It should be noted that in determination of the characteristic quantities (Reynolds number.), slot geometry is not taken into account, and that this omission may modify the results (variation as regards the heat transfer surface). According to Gazley [6], on the other hand, the presence of slots at the stator or rotor has no effect on the rotor-located Nusselt number. But slots are thermally insulated in Gazley [6] study. Moreover, heat transfer coefficient is calculated between rotor and stator whereas previous authors have calculated heat transfer coefficient between rotor and fluid. Yet, Bouafia et al. [14] (once again in the case of a slotted stator) have distinguished stator from rotor-based heat transfer. They have observed that convective heat transfer is heightened with the presence of channels at the stator, but they have also noted a degradation of rotor-based heat transfer with regard to the smooth cylindrical gap (which can explain the contradiction between

At the rotor: Nu ¼ 0.021 Reeff0.8 with Reeff ¼ (Rea2 þ 0.5Rerot2)1/2 At the stator: (Nu  Nu0)/Nu0 with Nu0 ¼ 0.021 Rea0.8 6. Conclusion In light of the results summarized above, we may conclude that notwithstanding the sizable quantity of studies focused on the subject of flows in rotating annular space, the similarly large number of influence and impact factors still leaves many question marks pertaining to the dynamics and, more specifically, to the heat transfer in the flow. As regards a smooth closed cylindrical gap, that is to say a TayloreCouette flow, its dynamics are now well-documented. Heat transfer has been rather comprehensively studied, at least from a global standpoint; only a few configurations, such as very large cylindrical gaps (h < 0.8) and a rotating external cylinder, remain virtually unexplored. Moreover, a high degree of congruence between numerical and experimental results appears to open up the possibility of studying local heat transfer, which up until now have remained somewhat neglected, by means of numerical simulations. Lastly, the complex effects of natural convection on flow and heat transfer call for more in-depth studies, even though such influence appears negligible when rotation velocity grows high (Ta > Tac), as is the case in the majority of industrial applications. On the other hand, the multiplication of influential factors renders open cylindrical gap (TayloreCouetteePoiseuille) far less widely understood, and it leads to contradictory conclusions among the authors. This state of affairs would seem to be essentially due to the entrance or input conditions for axial flow, whether it be dynamic or thermal. Indeed, the impact of velocity profile and development length on cylindrical gap flow, and particularly on the Taylor structures, appear to be thermally confirmed by the influence of cylinder length on heat transfer. It should be added that important data such as entry velocity profile and turbulence rate are generally not given consideration by the authors, and this omission no doubt partially explains to striking disparities in their reported results. What is more, there exists a multitude of definitions pertaining to heat transfer and, more particularly, to the reference temperature. One may suppose that only more local studies, in fluid as well as on the wall surface, are likely to eliminate the many question marks

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with regard to flow in general and TayloreCouetteePoiseuille heat transfer flow in particular. Though many years have elapsed since publication of the initial studies, the works on slotted cylindrical gaps have not allowed researchers to derive general analyses from the reported results; in any event, the large number of possible geometries renders any analysis particularly complex. A wide range of investigation remains open for exploration. To conclude, the published results hinge to a great extent on the configurations that have been studied, and do not yet offer an adequate basis for large-scale extrapolation.

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