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Convective heat transfer on a rotating finned cylinder with transverse airflow
International Journal of Heat and Mass Transfer 54 (2011) 4710–4718
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Convective heat transfer on a rotating finned cylinder with transverse airflow Benjamin Latour a,b,c, Pascale Bouvier a,b,c, Souad Harmand b,c,⇑ a
EEA Department, HEI, 13 rue de Toul, 59046 Lille, France Université Lille Nord de France, F-59000 Lille, France c UVHC, LME, F-59313 Valenciennes, France b
a r t i c l e
i n f o
Article history: Received 12 July 2010 Received in revised form 17 May 2011 Accepted 30 May 2011 Available online 25 June 2011 Keywords: Rotating fins Air crossflow Forced convection Mean heat transfer Infrared thermography
a b s t r a c t The present experimental investigation relates to the convective heat transfer determination around annular fins mounted on a rotating cylinder with air crossflow. The mean convective heat transfer coefficient can be identified by solving the inverse conduction heat transfer problem during the fin cooling process. We used an inverse method, based on the mean squared error, to develop a model of mean convective heat transfer, taking lateral conduction into account. Tests were carried out for rotational Reynolds numbers Rex between 2150 and 17,200, air crossflow Reynolds numbers ReU between 0 and 39,600, and fin spacings u in the range 10 mm to 1, u = 1 corresponding to the single disk case. For each fin spacing, the relative influences of the rotational and airflow forced convections on the heat transfer were analyzed and correlations of the mean Nusselt number on the fin, relative to both Reynolds numbers, are proposed. Moreover, an efficiency definition, that allows optimal geometrical configurations of the finned cylinder to be identified for the given operating conditions, is proposed. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Convective heat exchanges from a finned cylinder subjected to an airflow parallel to the fins find an application in the standard heat exchangers. The heat transfer study of the rotating finned cylinder with no superposed airflow is relevant to magnetic disc storage systems, which are pertinent to the computer industry, as well as to the cooling of electrical machinery. Indeed, rotors are often provided with fins, on their surface, which improve cooling by increasing the surface area available for heat dissipation. Moreover, in the thermal behavior study of the heat-pipe brake disc during the braking process [1] or of the cooling of electrical devices provided with one or more fans, the presence of a superposed airflow must be considered. Nevertheless, in spite of the large application field of rotating devices, relatively few studies of convective phenomena on rotating finned cylinder in air crossflow are available. To predict finned systems performance, the smooth cylinder case could be taken in reference (Fig. 1a). To study the fins effect on heat dissipation around a cylinder, the local heat transfer and flow structure around a smooth cylinder must be known. From the literature [2,3], in the case of a rotating cylinder in still air, the flow and heat transfer are governed by centrifugal effects. The Couette flow is observed for Reynolds numbers lower to a critical value of 900 and the flow remains laminar until Rex = 14,500. ⇑ Corresponding author at: UVHC, LME, F-59313 Valenciennes, France. E-mail address: [email protected] (S. Harmand). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.06.006
For a stationary cylinder in airflow, Labraga and Berkah [4] show that the local mass transfer increases from the angle value h = 0° (stagnation point) to a critical angle, depending on the airflow Reynolds number, at which it reaches its maximum. Then, its decrease is observed and the lowest transfer, corresponding to the flow separation, is detected at about ±80°. Beyond this separation point, heat transfer increases to a value similar to the stagnation point one. The combined effect of the rotation and air crossflow on heat exchanges around a cylinder has been studied by different authors [2,4], who highlighted three domains with regard to the ratio x = Rex/ReU. For low x values, a domain of crossflow predominance is observed since heat transfer results are in accordance with those corresponding to the non-rotating cylinder case. On the other hand, for high x values, a uniformity of heat transfer around the cylinder, which is characteristic of the rotating cylinder in the still air case, is observed. As for intermediate x values, local heat transfer greatly depends on the ratio x. Kays and Bjorklund [2] has shown that the combined effects of rotation, crossflow and free convection on mean convective heat exchanges, for 2000 < Rex < 45,000, can be correlated by:
Nu ¼ 0:135 ½ð0:5 Re2x þ Re2U þ GrÞ Pr1=3
ð1Þ
The case of coaxial disks mounted on a rotating cylinder in a fixed cylindrical enclosure (Fig. 1b), filled with air, has been studied by Abramhamson et al. [5], Chang et al. [6], Humphrey et al. [7] and Herrero and Giralt [8]. In this configuration, the disks make the flow non-uniform, setting up secondary vortices. The symmetrical vortex pair is stable up to a critical value of the
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Nomenclature C Di e e0 Fab H h I J L Nure ReU ReU;re Rex Rex;re r ri re t T u u0 U
specific heat, J kg1 K1 cylinder outer diameter, m fin thickness, m (=e/Di) dimensionless fin thickness view factor between a and b (=(re ri)) disk height, m convective coefficient, W m2 K1 thermal level radiosity, W m-2 cylinder length, m e ka ): Nusselt number (¼ hr (=U Di/ma): airflow Reynolds number (=U re/ma): airflow Reynolds number (=x ri Di/ma): rotational Reynolds number ð¼ x r e 2=ma Þ: rotational Reynolds number radial coordinate, m (=Di/2) inner radius of the disk, m outer radius of the disk, m time, s temperature, K fin spacing, m (=u/Di) dimensionless fin spacing airflow velocity, m s1
Reynolds number ReH ¼ xR2ext =ma , at which point the 2D (axisymmetrical) flow becomes unsteady and 3D (asymmetrical with respect to the interdisk plane). Using an (ReH, SH) map, where SH is the ratio that links the interdisk spacing to the disk radius, Herrero [8] obtained a 3D unsteady flow for a high aspect ratio SH coupled to a high Reynolds number ReH and a 2D steady flow for low ReH coupled to low to moderate SH. Concerning the 3D flow, Abrahamson [5] observed that the flow field consisted of a 2D core sandwiched between thin 3D boundary layers on each disk. This 2D core can be divided into three distinct regions: the inner region, the outer region and the shroud boundary layer. In a plane parallel to the fins, the boundary of the inner region is polygonal and tends to be oval when the separation distance between the edge of the fins and the shroud approaches infinity [9]. In this region, the fluid is in solid body rotation characterized by a low air circulation. The outer region, dominated by large vortical structures whose vorticity is counter-aligned with the spin axis of the fins, is between the inner region and the edge of the disk. The absolute rotational velocity of the vortices series distributed around a circle is equal to about 75% of the fin velocity. The outer region is thus more actively turbulent than the inner region, but a regular fluid exchange occurs between these two regions. The shroud boundary layer is the third region. To the best of our knowledge, no study is dealing with flow mechanics around annular fins of a finned tube rotating in free
Greek symbols e emissivity u heat flux density, W m2 k heat conductivity, W m1 K1 m kinematic viscosity, m2 s1 q density, kg m3 h angular coordinate, rad r Stefan Boltzmann constant, W m2 K4 s transmittivity x rotational velocity, rad s1 Subscripts al aluminum amb ambiance cal computed value conv convective heat transfer cyl cylinder fin fin meas measured value n black paint rad radiative heat transfer 1 outside the boundary layer A average A
flow without a shroud. Concerning heat transfer, Sparrow [10] and Watel [11] have studied the mean convective exchanges in an unenclosed air-cooled rotating finned cylinder. The former identified heat transfer coefficients through analogy with mass transfers, using the naphthalene sublimation technique, for 700 < Rex < 6000 and 0.089 < u0 < 1. The latter used an experimental thermal transient method that uses temperature variations over time, measured by infrared thermography, for 400 < Rex < 30,000 and 0.034 < u0 < 0.69. Her study was concerned with fins of 21-mm height and 1-mm wide mounted on a 58-mm diameter cylinder. In fact, both Watel and Sparrow showed that the Nusselt number increases with the rotational Reynolds number and fin spacing. Moreover, the fin spacing effect is shown to be more important at low rotational velocities. From the same experimental methodology as Watel, the authors developed a new model allowing them to identify the local convective heat transfer coefficient on the central fin of a rotating finned cylinder in still air. This coefficient has been identified from solving the inverse conduction heat transfer problem during the fin cooling process. With this new model, they carried out a study at the laboratory [12] on the height (0.69 < H0 = H/Di < 1.38), fin spacing (0.10 < u0 < 0.69) and rotational velocity (4300 < Rex < 17200) influence on local convective exchanges around annular 2-mm wide fins. It was shown that the local Nusselt number is constant in the region near the cylinder and increases with r from a critical radius depending on the
Fig. 1. Geometrical configurations presented in the introduction.
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operating conditions. Concerning the mean heat transfers, it was shown that the fin height has no influence, whereas the relative influences of u0 and Rex observed are in accordance with the Sparrow and Watel results. Finally, the authors identify the optimal geometric configuration at u0 = 0.17, whatever the fin height and rotational velocity. It corresponds to the best compromise between the increase in the exchange surface and the decrease in the average Nusselt number, as the fin spacing decreases. The case of a fixed finned tube in transverse airflow has been extensively studied [13–21]. In this configuration, the presence of the cylinder generates flow perturbations, at the disk/cylinder junction. From experimental observations, a boundary layer development, from the leading edge of the disk associated to a reduction of velocity due to the adverse pressure gradient in the stagnation zone upstream of the cylinder, are observed. This causes the flow to separate and to form a horseshoe vortex system, consisting of counter-rotating vortices swept around the cylinder base. Zones of lower convective heat transfers correspond to the wake and the flow separation located at h = ±90° from the front stagnation point. On the other hand, zones of higher heat transfers are located at +110° < h < +150° and 150° < h < 110°, where the legs of horseshoe vortex system appears. The air velocity and the fin spacing influence on the flow structure and convective heat transfers, in the space between fins, has been studied by different authors [15–18,21]. They showed that the boundary layers interaction, the velocity reduction, the vortices characteristics, like number or size, building up in front of the cylinder and the wake zone depend greatly on the air velocity and the fin spacing. Furthermore, characteristics of the flow structure allow the same authors to link the convective heat transfer variations at the fin surface to the air velocity and geometrical parameters. Concerning mean convective exchanges around a rotating finned cylinder in transverse airflow (Fig. 1c), a previous study has been made at the laboratory [22–24]. The authors showed that the convective heat transfer is controlled by the air crossflow for Rex/ReU < x0, with x0 between 0.5 and 1 and by the rotation for Rex/ReU > x1, with x1 between 5 and 12, according to the u0 value. Furthermore, for a fixed airflow Reynolds number, the reduction in fin spacing leads to the decrease in the Nusselt number, due to the boundary layers interaction formed on two adjacent fins and the subsequent velocity reduction between fins. On the other hand, for a fixed fin spacing, as the airflow Reynolds number increases, the convective coefficient on the fin tends to reach the characteristic value of the single disk case. Indeed, the boundary layers interaction and the proportion of kinetic energy lost between the fins decreases with an increase in the airflow Reynolds number. For their tests carried out with rotating fins without airflow, an identical analysis can also be used to explain the variation of the Nusselt number as a function of the rotational Reynolds number and fin spacing. In this paper, a transient method for identifying heat transfers, using infrared thermography, is proposed. In this method, the infrared camera measures the time evolution of the fin temperature in the radial direction, at eight angular locations. Solving the inverse conduction heat transfer problem allowed us to identify the local and mean convective heat transfer from the fin surface, while taking into account both conductive and radiative fluxes [25]. Tests are carried out within the dimensionless fin spacing range 0.10 < u0 < 0.69 and for a single fin mounted on the cylinder (corresponding to u0 = 1), within the Reynolds number ranges 0 < ReU < 39,600 and 2150 < Rex < 17,200. For all of these tests, the influence of natural convection on the convective heat transfer remains negligible [12]. Thus, the Nusselt number on the fin surface is controlled by dimensionless numbers Rex and ReU expressing rotational and airflow forced convections, respectively. By comparing the total convective heat flow dissipated by the finned
cylinder surface to that dissipated by the smooth cylinder surface, a finned cylinder efficiency is evaluated in order to determine the optimal geometrical configurations leading to a better cooling.
2. Experimental setup Mean convective heat exchanges from the surface of a rotating fin in air crossflow are obtained by measuring its spatio-temporal temperature variations. The temperature is measured on the central fin a3 of a cylinder which has five annular fins (Fig. 2a and b). The 2-mm wide and 60-mm high fins were made of aluminum (kal = 200 W m1 K1, qal = 2700 kg m3, Cal = 0.96 kJ kg1 K1) and mounted on a 58-mm diameter cylinder (Fig. 2). The fin spacing u (10, 14, 24 and 40 mm) was obtained using bakelite rings of various thicknesses. In addition to the characteristics given in Fig. 2b and c, we also studied the case in which u = 1 (one fin on the cylinder) in order to compare it with the single disk case. Fins and rings were covered with a thin coat of black paint, whose relatively high emissivity allowed the radiative heat flux emitted at a given point of the fin a3 to be more accurately determined, thus improving the accuracy of the relationship between the camera exit signal and the disk temperature. The finned cylinder was driven by an electric motor coupled to a frequency variator, which allowed us to vary the rotational velocity from 250 to 2000 rpm. Experiments are performed in a 2.4 m long wind-tunnel in which the test finned cylinder is positioned at 2-m distance from the fan in order to obtain a straightened airflow upstream from the fins. A diaphragm allows us to vary airflow velocity from 0 to 14 m s1. The turbulence level of the incoming flow has been estimated in a previous work [23] by using a hot wire anemometry. It is equal to 5% for U = 12 m s1 and 10% for U = 4 m s1. In order to measure fin temperatures by thermography, the front wall of the wind tunnel includes a porthole made up of an infrared transparent film. The calibration law with the film presence has been determined with an extended black body at the laboratory. A radiant panel emitting short infrared waves was placed horizontally above the finned cylinder, heating them uniformly to temperatures of about 120 °C. Once the steady thermal state was reached, the heat source was shut off. The fins then cooled through radiative and convective heat transfers, depending on the rotational and air crossflow velocities. The surface temperatures during the cooling of the fins were recorded using an infrared (IR) camera (JADE 3 from CEDIP INFRARED SYSTEMS). Fins a1 and a2 had a rectangular slit along their entire radius to allow the central fin temperature to be measured. The presence of these slits was assumed to have no effect on the convective heat transfer in the central fin a3. A photoelectric sensor was used to detect the eight reflective bands stuck on the cylinder. Once the sensor had detected the bands, it triggered the IR camera, which then scanned the whole fin. In our study, the camera was placed 2.50 m in front of the central fin, with an acquisition frequency of 600 Hz and a window of 160 120 pixels, where each pixel corresponds to almost 2.5 mm2 of the fin surface. The eight angular locations taken at each turn of the finned cylinder are represented in Fig. 2b. The camera exit signals obtained during the cooling were collected and processed with the software MATLAB to determine the radial temperature profiles of the fin. They are expressed in thermal levels. Their processing is explained in a previous article, as well as the radiative heat flux computation [12]. An infrared pyrometer was placed perpendicular to the cylinder’s rotational axis in order to measure the cylinder’s temperature evolution over time. The air temperature measured by a K-thermocouple placed in the middle of the wind tunnel section at 0.5 m upstream of the finned cylinder, during fin cooling, was used as the reference temperature.
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Fig. 2. Representation of the test facility.
3. Convective heat transfer identification method Unlike the fin temperature and the radiative heat flux, the convective heat flux cannot be determined directly from measurements because convective phenomena depend on unknown flow characteristics. This type of problem is an inverse heat conduction problem, which has been described by Beck et al. [26]. In such problems, temperature measurements can be exploited to identify surface conditions, such as convective heat flux, using the direct model and inverse method described below.
to characterize mean convective heat transfers, the convective heat flux density uconv ðr; tÞ ¼ hðT cal ðr; tÞ T 1 Þ is characterized by a uniform on the fin surtime-averaged heat transfer coefficient h, face. To solve the direct model, the mean heat transfer coefficient is assumed to be known. The equations in the system (Eq. (2)) h are discretized to be solved by finite differences, with an implicit scheme using right differences for first-order terms and central differences for second-order terms. A sensitivity study allowed us to determine the adequate fixed step Dr = 2.5 mm (C = 19 nodes). 3.2. Inverse method
3.1. Direct model The direct model involves solving partial differential equations related to the cooling of the fin. Since in all of our experimental conditions the Biot number Bi ¼ h e=kal ¼ h 105 1 8h, the conductive heat transfer in the fin thickness direction (z) can be neglected. So, we use an unidimensional direct model, allowing to compute the spatio-temporal evolution of the fin temperature Tcal(r, t), described by the following equations:
qal C al @T cal ðr; tÞ
kal
@t
uconv ðr; tÞ þ uray ðtÞ
F¼
K X C h X
i2 T kcal ðiÞ T kfin ðiÞ
ð3Þ
k¼1 i¼1
e kal
2
@ T cal ðr; tÞ 1 @T cal ðr; tÞ ¼ þ @r2 r @r T cal ðr; t ¼ 0Þ ¼ T fin ðr; t ¼ 0Þ
The inverse method allows the mean heat transfer coefficient h to be determined by comparing the computed and measured temperature values. It involves a function specification method, which by minimizing the difference, in the least consists in determining h square sense, between calculated and measured temperatures [27]. In this way, the following function, based on the sum of the mean squared deviations, is used:
ð2Þ
where K is the time steps number, given by the acquisition frequency, and C the radial steps number (C = 19). This method allows the mean convective heat transfer coefficient to be determined while taking radial conduction into account [25].
T cal ðri ; tÞ ¼ T fin ðri ; tÞ T cal ðre ; tÞ ¼ T fin ðre ; tÞ
3.3. Convective heat transfer uncertainty
In this model, the temperature distribution at the initial time and the boundary conditions on the inner and outer radii (ri and re) of the fin are obtained from IR camera thermal levels. In order
The uncertainty associated with the convective heat transfer coefficient is determined by the confidence intervals Dh determined for each case. They are obtained from the sensitivity
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Table 1 Uncertainties for different typical cases. Case
u0 = 0.41 ReU = 11,350 Rex = 4300
u0 = 0.41 ReU = 11,350 Rex = 12,900
u0 = 0.41 ReU = 33,950 Rex = 4300
u0 = 0.41 ReU = 33,950 Rex = 17,200
(W/m2 K) h DT (°C)
27
40.8
66.8
71.9
Dh (W/m2 K)
0.3
0.38
8.0
9.6
coefficient Q k ¼ @T kcal =@h and the mean deviation DT between calculated and measured temperatures. For some typical cases, we obtained the results given in Table 1.
4. Results and discussion Radial profiles of the measured temperature, taken at different times and relative to the typical case characterized by u0 = 0.41, Rex = 4300 and ReU = 33,950, are presented in Fig. 3. The observed weak temperature gradients with r allow the hypothesis relative to the heat transfer coefficient uniformity to be validated at the fin surface. Moreover, Fig. 4 presents radiative and convective heat flux density evolutions corresponding to Fig. 3 case for h = 180°. The radiative heat flux density represents only 3% of convective
0.18 11.8
0.16 11.5
heat flux density. Moreover, as for all of our considered cases, the ratio between radiative and convective heat flux densities does not exceed 10%, the global approach used in the radiative heat flux determination is justified. The results reported in this section are expressed in terms of mean Nusselt number Nu, which is characteristic of convective heat transfers, and in terms of rotational and air crossflow Reynolds numbers (Rex and ReU), based on the diameter of the cylinder, which are respectively characteristic of rotation and air crossflow. Physical properties were evaluated at the air film temperature. To study the influence of rotation and air crossflow on convective heat transfers over the finned cylinder, we performed tests using the following conditions: Rex between 2150 and 17,200 and ReU between 0 and 39,600. The geometry of the finned cylinder was characterized by the dimensionless fin spacing u0 = u/ Di, varying from 0.17 to 0.69. 4.1. Rotating finned cylinder in still air The experimental and correlated results for the mean convective heat transfer [12] are presented as a double logarithmic plot of the mean Nusselt number against the rotational Reynolds number in Fig. 5. The correlated relationship of the mean Nusselt with the rotational Reynolds number and dimensionless fin spacing, for our tests, is determined by Watel’s method [22] and expressed as:
NuU¼0 ¼ 2:221 X Re0:5 x 0 0:5 0:5 e K1 a1 1 with X ¼ 1 Re x u0 þ 1 u0 b1
ð4Þ
The values of the different coefficients operating in Eq. (4) are K1 = 1.022, b1 = 0.031, a1 = 0.989 and e0 = e/Di = 0.034. 4.2. Rotating finned cylinder in air crossflow Fig. 3. Radial temperature evolutions at different times for u0 = 0.41, Rex = 4300, ReU = 33,950 and h = 180°.
Fig. 4. Radiative and convective flux density evolutions for u0 = 0.41, Rex = 4300, ReU = 33,950 and h = 180°.
As the thermal behavior of the central fin is strongly dependent on the couple (Rex, ReU) [22–24], the ratio Nu=Nux¼0;u0¼1 is intro-
Fig. 5. Mean heat transfer from a rotating finned cylinder in still air.
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duced to highlight the influence zones of rotation, airflow and coupling rotation/airflow, by comparison with the case of a single stationary fin with air crossflow (x = 0, u0 = 1). For this last case, our results obtained for 5700 < ReU < 39,600 [28], are correlated by:
Nux¼0;u0 ¼1 ¼ 0:036 Re0:8 U
ð5Þ
The mean Nusselt numbers ratio evolution Nu=Nux¼0;u0 ¼1 against the Reynolds numbers ratio Rex/ReU is plotted Figs. 6–9, for different values of u0 . For each fin spacing, different influence
Fig. 9. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.17.
Fig. 6. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.69.
Fig. 7. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.41.
Fig. 8. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.24.
zones are highlighted. For Rex/ReU < 0.2, our experimental results fluctuate around the mean value Nu=Nux¼0;u0¼1 ¼ A for all the fin spacings studied (A varying from 1 for u0 = 1, to 0.73 for u0 = 0.17, see Table 2). These results show that, in this zone, convective exchanges do not depend on rotation, and are therefore exclusively controlled by the airflow. Furthermore, the comparison of Figs. 6–9 highlights the mean Nusselt number decrease with fin spacing, due to the interaction of the boundary layers between the fins which increases with the decrease of fin spacing, leading to bad air circulation [15–18,21–23]. From Rex/ReU > 0.2, convective exchanges are governed (i) by rotation flow superposed to air flow and (ii) by fin spacing. Indeed, for u0 = 0.69 and 1 (Fig. 6) both characteristic zones are highlighted. In the first one corresponding to 0.2 < Rex/ReU < 0.8, rotation influence becomes significant, leading to a convective exchanges increase of 1:3Nux¼0;u0¼1 for u0 = 1 and 1:13Nux¼0;u0¼1 for u0 = 0.69. In the second one, corresponding to Rex/ReU > 0.8, rotating effects generates more and more flow perturbations, leading to convective exchanges intensification compared with the first zone. In the studied Rex/ReU range, the difference between the ratio Nu=Nux¼0;u0¼1 calculated for u0 = 1 and u0 = 0.69 is about 13%, for both zones. A change in physical phenomena occurs as fin spacing decreases (0.24 < u0 < 0.41, Figs. 7 and 8) since a single zone of rotation influence is highlighted. Indeed, the increase of boundary layers interactions between co-rotating fins [10–12], combined to flow perturbations, generated by the superposition of rotating and airflow effects, leads to identical behavior of convective exchanges for Rex/ReU > 0.2 and 0.24 < u0 < 0.41 (small fin spacings). By comparison with the case 0.69 < u0 < 1 (important fin spacings), in the range 0.2 < Rex/ReU < 0.8, we observe a higher increase in the Nusselt number ratio for small fin spacings than for large ones, whereas for Rex/ReU > 0.8 it is the contrary. Furthermore, concerning small fin spacings (0.24 < u0 < 0.41) in the considered Rex/ReU range, the mean difference between the ratio Nu=Nux¼0;u0¼1 calculated (i) for u0 = 1 and u0 = 0.41 is about 15% and (ii) for u0 = 1 and u0 = 0.24 is about 20%. Concerning the smallest fin spacing u0 = 0.17, the mean thermal behavior of the fin is different (Fig. 9), even if two zones are highlighted as for higher fin spacings. In the first one corresponding to 0.2 < Rex/ReU < 0.4, a Nusselt number ratio increase similar to one of the other cases presented Figs. 6–8, is observed. Nevertheless, this zone size is shorter than the others probably because of an effect of boundary layer interactions more intense from a lower Reynolds numbers ratio (Rex/ReU = 0.4 instead of 0.8). Furthermore, this phenomenon seems to be accentuated for Rex/ReU > 0.4 since the Nusselt number ratio increase is a lot smaller than the other cases.
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Table 2 Eq. (6) coefficient values for different fin spacings. u0
Rex/ReU < 0.2
0.2 < Rex/ReU < x
x < Rex/ReU
1 0.69 0.41 0.24 0.17
A = 1; B, C, D = 0 A = 0.87; B, C, D = 0 A = 0.85; B, C, D = 0 A = 0.79; B, C, D = 0 A = 0.73; B, C, D = 0
x = 0.8; A = 1; B = 0.45; C = 0.2; D = 0.77 x = 0.8; A = 0.87; B = 0.40; C = 0.2; D = 0.78 A = 0.85; B = 0.52; C = 0.2; D = 0.79 A = 0.79; B = 0.44; C = 0.2; D = 0.6 x = 0.4; A = 0.73; B = 0.44, C = 0.2; D = 0.66
x = 0.8; A = 1.3; B = 0.45; C = 0.8; D = 0.82 x = 0.8; A = 1.13; B = 0.40; C = 0.8; D = 0.81
x = 0.4; A = 0.89; B = 0.14; C = 0.4; D = 0.47
The complexity of the physical phenomena engendered by the simultaneous effects of u0 , Rex and ReU brings us to correlate our data for a fixed value of u0 . So, correlations between Nu, Rex and ReU are determined for different value of u0 , in term of the ratio Nu=Nux¼0;u0¼1 against the ratio Rex/ReU, as in Eq. (6):
" Nu ¼ Nux¼0;u0 ¼1 A þ B
D # Rex C ReU
ð6Þ
The different values of the coefficients A, B, C and D occurring in Eq. (6), depending on Rex/ReU and u0 , are presented in Table 2. The computed values (Eq. (6)) are equal to the experimental ones with a mean error of about 12%. The fin spacing influence in the airflow predominance zone is characterized by the coefficient A. So, we verify that the convective exchanges coefficient is all the more important as the fin spacing is high. For rotation/airflow coupling, the fin spacing influence is mainly characterized by the coefficient D. For u0 P 0.41, fin spacing influence remains constant since D 0.8 whereas for u0 = 0.17 and 0.24, the reduction in fin spacing leads to a convective transfer decrease characterized by D 0.6. Concerning the C coefficient, it is directly linked to the different influence zone boundaries.
Fig. 10. Correlated data from NuU=0 for u0 = 0.69.
4.3. A general correlation In spite of all the differences of behavior, according to the fin spacing, highlighted in the previous sections, it is possible to predict convective exchanges on a rotating finned cylinder in air crossflow by a correlation including the three parameters u0 , Rex and ReU from RexReU > 0.2. The following equation is proposed:
Nu ¼ NuU¼0
a K Rex u0b ReU
ð7Þ
With K = 1.280, b = 0.160 and a = 0.576. Figs. 10–13 present experimental and correlated mean Nusselt numbers (Eq. (7)) against Rex varying from 2150 to 17,200, ReU, from 5700 to 39,600 and u0 from 0.17 to 0.69. The computed values (Eq. (7)) are equal to the experimental ones with a mean error of about 15%. In conclusion, the correlation (Eq. (7)) allows us to predict the global thermal behavior of a rotating finned cylinder in an air crossflow, by taking into account the combined effects of rotation and airflow, for different fin spacings. It could be used for practical applications to determine convective exchanges on the rotating finned cylinder in airflow. 5. Evolution of the mean convective exchanges on a finned cylinder in air crossflow by comparison with the smooth cylinder case For a given cylinder length L, reducing the fin spacing allows to increase the cylinder-air exchange surface by increasing the number of fins. However, reducing the spacing also causes a decrease in the convective exchange coefficient. In order to find a compromise between these two contradictory effects, we evaluated the ‘‘efficiency’’ n of the finned cylinder relatively to the smooth cylinder [12]. The ratio variation of the total convective heat flow dissipated
Fig. 11. Correlated data from NuU=0 for u0 = 0.41.
by the finned cylinder surface and the one dissipated by the smooth cylinder can be expressed by the following equation:
n¼
h2p ðr 2e r 2i Þ þ hbf pDi u hcyl pDi ðu þ eÞ
ð8Þ
where hcyl is the heat transfer coefficient of the smooth cylinder surface determined from Eq. (1). Concerning the heat transfer coefficient at the cylinder surface comprised between fins hbf, taking hbf ¼ hcyl or hbf ¼ h in the numerator of Eq. (8) does not affect the n value significantly. Therefore, we chose to use hbf ¼ h to express the convective heat flux dissipated by the finned cylinder, in order to take into account the convective transfer decrease at the cylinder wall due to fin presence. Efficiency evolution n according to the
B. Latour et al. / International Journal of Heat and Mass Transfer 54 (2011) 4710–4718
Fig. 12. Correlated data from NuU=0 for u0 = 0.24.
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In a previous study on a rotating finned cylinder in still air [12], we identified the optimal fin spacing for u0 = 0.17. For u0 > 0.17, the increase of the mean Nusselt number does not offset the decrease in efficiency due to the exchange surface drop, engendered by larger fin spacings. On the other hand, for u0 < 0.17, it is the contrary, since the fall in efficiency is directly linked to the mean Nusselt number decrease, whereas the heat exchange surface is increased. Indeed, with u0 = 0.10, the exchange surface increase did not offset the fall of heat exchange coefficient due to the interaction of the boundary layers between two adjacent fins. In the case of a rotating finned cylinder in air crossflow, like in the previous case, efficiency greatly depends on the exchange surface extent, as well as boundary layer interactions. We have seen in the previous sections that these interactions are directly dependent on the ratio Rex/ReU in the presence of an airflow. Fig. 14 presents five evolutions relative to the different characteristic zones highlighted previously (Rex/ReU < 0.2; 0.2 < Rex/ReU < 0.4; 0.2 < Rex/ReU < 0.8; Rex/ReU > 0.8 and ReU = 0) which are representative of all the experimental cases carried out. Indeed, for Rex/ ReU < 0.8, efficiency continually increases with the decrease of u0 until u0 = 0.17, whereas for Rex/ReU > 0.8, the optimal fin spacing is equal to 0.24. This result can be linked to the drop in convective heat transfers observed Fig. 9 for the higher ratios (the third zone relative to u0 = 0.17).
6. Conclusion
Fig. 13. Correlated data from NuU=0 for u0 = 0.17.
Fig. 14. Efficiency of the finned cylinder according to u0 for different Rex/ReU.
dimensionless fin spacing u0 and the ratio Rex/ReU are presented in Fig. 14. For comparison, we added results corresponding to the case of a rotating finned cylinder in still air for 0.10 < u0 < 0.69 [12].
The experimental setup presented in this paper allowed us to evaluate the mean Nusselt number on a rotational finned tube submitted to airflow parallel to the fins. In each test, the radial evolution of the ‘‘thermal levels’’ at eight angular positions of the central fin was recorded by the IR camera while the system was cooling. The thermal levels were converted to temperature values using the calibration law and the radiative balance of the surface scanned. An inverse method based on the mean squared error allowed us to determine the mean convective heat transfer coefficient by taking both the radial conduction and radiative fluxes into account. Results presented in this paper are valuable in the operating conditions domain defined by 0.17 < u0 < , 2150 < Rex < 17,200 and 0 < ReU < 39,600. Concerning the rotating finned cylinder in still air, a general correlation taking into account the fin spacing and the rotation is proposed. Concerning the rotating finned cylinder in air crossflow, the fin spacing and rotation effects are presented by comparison with the single stationary fin in air crossflow. Several influence zones, function of the ratio Rex/ReU, are highlighted. For any fin spacings, in the first one, characterized by Rex/ReU < 0.2, convective heat transfers are exclusively controlled by the airflow. Concerning the other zones, the effects of an airflow superposed to the rotation induce different mean Nusselt number evolutions according to the fin spacing. Indeed, on one hand, precise correlations of mean Nusselt numbers, function of Rex/ReU, are proposed for each zone and fin spacing with a mean error of about 12%. On the other hand, in order to characterize the convective heat transfer in terms of all the parameters studied, a more general correlation is proposed from Rex/ReU = 0.2 with a mean error of about 15%. The maximal value of the finned tube efficiency in the presence of an airflow depends on the ratio Rex/ReU. The efficiency, based on the comparison with the smooth cylinder, allowed us to detect the optimal fin spacing at u0 = 0.17 for the rotating finned cylinder in still air and at u0 = 0.24 for Rex/ ReU > 0.8, corresponding to the best compromise between the increase of exchange surface and the decrease in Nusselt number, engendered by the fin spacing reduction. Moreover, for Rex/ ReU < 0.8, efficiency continually increases with the decrease of u0 until u0 = 0.17.
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References [1] B. Watel, S. Harmand, B. Desmet, Etude expérimentale de faisabilité du refroidissement d’un disque de frein équipé d’un caloduc, Entropie 191 (1995) 25–33. [2] W.M. Kays, I.S. Bjorklund, Heat transfer from a rotating cylinder with and without cross flow, Trans. ASME 80 (1958) 70–78. [3] B. Ozerdem, Measurement of convective heat transfer coefficient for an horizontal cylinder rotating in quiescent air, Int. Commun. Heat Mass Transfer 27 (3) (2000) 389–395. [4] L. Labraga, T. Berkah, Mass transfer from a rotating cylinder with and without cross flow, Int. J. Heat Mass Transfer 47 (2004) 2493–2499. [5] S.D. Abrahamson, J.K. Eaton, D.J. Koga, The flow between shrouded corotating disks, Phys. Fluids A 1 (2) (1989) 241–251. [6] C.J. Chang, C.A. Schuler, J.A.C. Humphrey, R. Greig, Flow and heat transfer in the space between two corotating disks in an axisymmetric enclosure, J. Heat Transfer 111 (1989) 625–632. [7] J.A.C. Humphrey, C.A. Schuler, I. Iglesias, Unsteady laminar flow between a pair of disks co-rotating in a fixed cylindrical enclosure, Phys. Fluids 7 (1995) 1125– 1240. [8] J. Herrero, F. Giralt, Influence of the geometry on the structure of the flow between a pair of corotating disks, Phys. Fluids 11 (1) (1999) 88–96. [9] E. Lenneman, Aerodynamic aspects of disks files, IBM J. Res. Develop. 18 (6) (1974) 480–488. [10] E.M. Sparrow, C.S. Preston, Heat transfer from rotating annular fins, Int. J. Heat Mass Transfer 29 (1986) 831–839. [11] B. Watel, S. Harmand, B. Desmet, Influence of fin spacing and rotational speed on the convective heat exchanges from a rotating finned tube, Int. J. Heat Fluid Flow 21 (2000) 221–227. [12] B. Latour, P. Bouvier, S. Harmand, Experimental study of convective heat transfer on a finned rotating cylinder, Int. J. Thermal Sci. 49 (9) (2010) 1742– 1751. [13] V.M. Legkiy, V.P. Pavlenko, A.S. Makarov, YA.S. Zheludov, Investigation of local heat transfer in a tube with annular fins in transverse air flow, Heat Transfer – Soviet Res. 6 (6) (1974) 101–107. [14] R.J. Goldstein, J. Karni, The effect of a wall boundary layer on local mass transfer from a cylinder in crossflow, Trans. ASME 106 (1984) 260–267.
[15] H.J. Sung, J.S. Yang, T.S. Park, Local convective mass transfer on circular cylinder with transverse annular fins in crossflow, Int. J. Heat Mass Transfer 39 (5) (1996) 1093–1101. [16] G. Schüz, V. Kottke, Visualization of flow heat and mass transfer on finned tubes in cross flow, in: Proceedings of the IV International Symposium on Flow Visualization, Hemisphere Washington, Paris, 1986, pp. 637–642. [17] G. Schüz, V. Kottke, Local heat transfer and heat flux distributions in finned tube heat exchangers, Chem. Eng. Technol. 15 (1992) 417–424. [18] X. Hu, A.M. Jacobi, Local heat transfer behaviour and its impact on a singlerow, annularly finned tube heat exchanger, J. Heat Transfer 115 (1993) 66–74. [19] E.M. Sparrow, S.R. Chastain, Effect of angle of attack on the heat transfer coefficient for an annular fin, Int. J. Heat Mass Transfer 29 (8) (1986) 1185– 1191. [20] E.M. Fisher, P.A. Eibeck, The influence of a horseshoe vortex on local convective heat transfer, J. Heat Transfer 112 (1990) 329–335. [21] R. Romero-Méndez, M. Sen, K.T. Yang, R. McClain, Effect of fin spacing on convection in a plate fin and tube heat exchanger, Int. J. Heat Mass Transfer 43 (2000) 39–51. [22] B. Watel, S. Harmand, B. Desmet, Experimental study of convective heat transfer from a rotating finned tube in transverse air flow, Exp. Fluids 29 (2000) 79–90. [23] B. Watel, Etude des échanges convectifs sur un cylindre aileté en rotation soumis à un courant d’air parallèle aux ailetés, Ph.D. Thesis, Université de Valenciennes et du Hainaut Cambresis, 1997. [24] B. Watel, S. Harmand, B. Desmet, Etude des échanges convectifs sur un arbre aileté tournant, soumis à un courant d’air parallèle aux ailetés, Int. J. Heat Mass Transfer 41 (1998) 3741–3757. [25] D. Bougeard, Infrared thermography investigation of local heat transfer in a plate fin and two-tube rows assembly, Int. J. Heat Fluid Flow 28 (2007) 988– 1002. [26] J.V. Beck, B. Blackwell, C.R. St Clair, Inverse Heat Conduction – Ill-posed Problems, John Wiley Interscience, New York, 1985. [27] G. Blanc, M. Raynaud, T.H. Chau, A guide for the use of the function specification method for 2D inverse heat conduction problems, Revue Générale de Thermique 37 (1998) 17–30. [28] B. Latour, P. Bouvier, S. Harmand, Convective heat transfer on a rotating disk with transverse air crossflow, J. Heat transfer 133 (2) (2011) 021702.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Convective heat transfer on a rotating finned cylinder with transverse airflow Benjamin Latour a,b,c, Pascale Bouvier a,b,c, Souad Harmand b,c,⇑ a
EEA Department, HEI, 13 rue de Toul, 59046 Lille, France Université Lille Nord de France, F-59000 Lille, France c UVHC, LME, F-59313 Valenciennes, France b
a r t i c l e
i n f o
Article history: Received 12 July 2010 Received in revised form 17 May 2011 Accepted 30 May 2011 Available online 25 June 2011 Keywords: Rotating fins Air crossflow Forced convection Mean heat transfer Infrared thermography
a b s t r a c t The present experimental investigation relates to the convective heat transfer determination around annular fins mounted on a rotating cylinder with air crossflow. The mean convective heat transfer coefficient can be identified by solving the inverse conduction heat transfer problem during the fin cooling process. We used an inverse method, based on the mean squared error, to develop a model of mean convective heat transfer, taking lateral conduction into account. Tests were carried out for rotational Reynolds numbers Rex between 2150 and 17,200, air crossflow Reynolds numbers ReU between 0 and 39,600, and fin spacings u in the range 10 mm to 1, u = 1 corresponding to the single disk case. For each fin spacing, the relative influences of the rotational and airflow forced convections on the heat transfer were analyzed and correlations of the mean Nusselt number on the fin, relative to both Reynolds numbers, are proposed. Moreover, an efficiency definition, that allows optimal geometrical configurations of the finned cylinder to be identified for the given operating conditions, is proposed. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Convective heat exchanges from a finned cylinder subjected to an airflow parallel to the fins find an application in the standard heat exchangers. The heat transfer study of the rotating finned cylinder with no superposed airflow is relevant to magnetic disc storage systems, which are pertinent to the computer industry, as well as to the cooling of electrical machinery. Indeed, rotors are often provided with fins, on their surface, which improve cooling by increasing the surface area available for heat dissipation. Moreover, in the thermal behavior study of the heat-pipe brake disc during the braking process [1] or of the cooling of electrical devices provided with one or more fans, the presence of a superposed airflow must be considered. Nevertheless, in spite of the large application field of rotating devices, relatively few studies of convective phenomena on rotating finned cylinder in air crossflow are available. To predict finned systems performance, the smooth cylinder case could be taken in reference (Fig. 1a). To study the fins effect on heat dissipation around a cylinder, the local heat transfer and flow structure around a smooth cylinder must be known. From the literature [2,3], in the case of a rotating cylinder in still air, the flow and heat transfer are governed by centrifugal effects. The Couette flow is observed for Reynolds numbers lower to a critical value of 900 and the flow remains laminar until Rex = 14,500. ⇑ Corresponding author at: UVHC, LME, F-59313 Valenciennes, France. E-mail address: [email protected] (S. Harmand). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.06.006
For a stationary cylinder in airflow, Labraga and Berkah [4] show that the local mass transfer increases from the angle value h = 0° (stagnation point) to a critical angle, depending on the airflow Reynolds number, at which it reaches its maximum. Then, its decrease is observed and the lowest transfer, corresponding to the flow separation, is detected at about ±80°. Beyond this separation point, heat transfer increases to a value similar to the stagnation point one. The combined effect of the rotation and air crossflow on heat exchanges around a cylinder has been studied by different authors [2,4], who highlighted three domains with regard to the ratio x = Rex/ReU. For low x values, a domain of crossflow predominance is observed since heat transfer results are in accordance with those corresponding to the non-rotating cylinder case. On the other hand, for high x values, a uniformity of heat transfer around the cylinder, which is characteristic of the rotating cylinder in the still air case, is observed. As for intermediate x values, local heat transfer greatly depends on the ratio x. Kays and Bjorklund [2] has shown that the combined effects of rotation, crossflow and free convection on mean convective heat exchanges, for 2000 < Rex < 45,000, can be correlated by:
Nu ¼ 0:135 ½ð0:5 Re2x þ Re2U þ GrÞ Pr1=3
ð1Þ
The case of coaxial disks mounted on a rotating cylinder in a fixed cylindrical enclosure (Fig. 1b), filled with air, has been studied by Abramhamson et al. [5], Chang et al. [6], Humphrey et al. [7] and Herrero and Giralt [8]. In this configuration, the disks make the flow non-uniform, setting up secondary vortices. The symmetrical vortex pair is stable up to a critical value of the
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Nomenclature C Di e e0 Fab H h I J L Nure ReU ReU;re Rex Rex;re r ri re t T u u0 U
specific heat, J kg1 K1 cylinder outer diameter, m fin thickness, m (=e/Di) dimensionless fin thickness view factor between a and b (=(re ri)) disk height, m convective coefficient, W m2 K1 thermal level radiosity, W m-2 cylinder length, m e ka ): Nusselt number (¼ hr (=U Di/ma): airflow Reynolds number (=U re/ma): airflow Reynolds number (=x ri Di/ma): rotational Reynolds number ð¼ x r e 2=ma Þ: rotational Reynolds number radial coordinate, m (=Di/2) inner radius of the disk, m outer radius of the disk, m time, s temperature, K fin spacing, m (=u/Di) dimensionless fin spacing airflow velocity, m s1
Reynolds number ReH ¼ xR2ext =ma , at which point the 2D (axisymmetrical) flow becomes unsteady and 3D (asymmetrical with respect to the interdisk plane). Using an (ReH, SH) map, where SH is the ratio that links the interdisk spacing to the disk radius, Herrero [8] obtained a 3D unsteady flow for a high aspect ratio SH coupled to a high Reynolds number ReH and a 2D steady flow for low ReH coupled to low to moderate SH. Concerning the 3D flow, Abrahamson [5] observed that the flow field consisted of a 2D core sandwiched between thin 3D boundary layers on each disk. This 2D core can be divided into three distinct regions: the inner region, the outer region and the shroud boundary layer. In a plane parallel to the fins, the boundary of the inner region is polygonal and tends to be oval when the separation distance between the edge of the fins and the shroud approaches infinity [9]. In this region, the fluid is in solid body rotation characterized by a low air circulation. The outer region, dominated by large vortical structures whose vorticity is counter-aligned with the spin axis of the fins, is between the inner region and the edge of the disk. The absolute rotational velocity of the vortices series distributed around a circle is equal to about 75% of the fin velocity. The outer region is thus more actively turbulent than the inner region, but a regular fluid exchange occurs between these two regions. The shroud boundary layer is the third region. To the best of our knowledge, no study is dealing with flow mechanics around annular fins of a finned tube rotating in free
Greek symbols e emissivity u heat flux density, W m2 k heat conductivity, W m1 K1 m kinematic viscosity, m2 s1 q density, kg m3 h angular coordinate, rad r Stefan Boltzmann constant, W m2 K4 s transmittivity x rotational velocity, rad s1 Subscripts al aluminum amb ambiance cal computed value conv convective heat transfer cyl cylinder fin fin meas measured value n black paint rad radiative heat transfer 1 outside the boundary layer A average A
flow without a shroud. Concerning heat transfer, Sparrow [10] and Watel [11] have studied the mean convective exchanges in an unenclosed air-cooled rotating finned cylinder. The former identified heat transfer coefficients through analogy with mass transfers, using the naphthalene sublimation technique, for 700 < Rex < 6000 and 0.089 < u0 < 1. The latter used an experimental thermal transient method that uses temperature variations over time, measured by infrared thermography, for 400 < Rex < 30,000 and 0.034 < u0 < 0.69. Her study was concerned with fins of 21-mm height and 1-mm wide mounted on a 58-mm diameter cylinder. In fact, both Watel and Sparrow showed that the Nusselt number increases with the rotational Reynolds number and fin spacing. Moreover, the fin spacing effect is shown to be more important at low rotational velocities. From the same experimental methodology as Watel, the authors developed a new model allowing them to identify the local convective heat transfer coefficient on the central fin of a rotating finned cylinder in still air. This coefficient has been identified from solving the inverse conduction heat transfer problem during the fin cooling process. With this new model, they carried out a study at the laboratory [12] on the height (0.69 < H0 = H/Di < 1.38), fin spacing (0.10 < u0 < 0.69) and rotational velocity (4300 < Rex < 17200) influence on local convective exchanges around annular 2-mm wide fins. It was shown that the local Nusselt number is constant in the region near the cylinder and increases with r from a critical radius depending on the
Fig. 1. Geometrical configurations presented in the introduction.
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operating conditions. Concerning the mean heat transfers, it was shown that the fin height has no influence, whereas the relative influences of u0 and Rex observed are in accordance with the Sparrow and Watel results. Finally, the authors identify the optimal geometric configuration at u0 = 0.17, whatever the fin height and rotational velocity. It corresponds to the best compromise between the increase in the exchange surface and the decrease in the average Nusselt number, as the fin spacing decreases. The case of a fixed finned tube in transverse airflow has been extensively studied [13–21]. In this configuration, the presence of the cylinder generates flow perturbations, at the disk/cylinder junction. From experimental observations, a boundary layer development, from the leading edge of the disk associated to a reduction of velocity due to the adverse pressure gradient in the stagnation zone upstream of the cylinder, are observed. This causes the flow to separate and to form a horseshoe vortex system, consisting of counter-rotating vortices swept around the cylinder base. Zones of lower convective heat transfers correspond to the wake and the flow separation located at h = ±90° from the front stagnation point. On the other hand, zones of higher heat transfers are located at +110° < h < +150° and 150° < h < 110°, where the legs of horseshoe vortex system appears. The air velocity and the fin spacing influence on the flow structure and convective heat transfers, in the space between fins, has been studied by different authors [15–18,21]. They showed that the boundary layers interaction, the velocity reduction, the vortices characteristics, like number or size, building up in front of the cylinder and the wake zone depend greatly on the air velocity and the fin spacing. Furthermore, characteristics of the flow structure allow the same authors to link the convective heat transfer variations at the fin surface to the air velocity and geometrical parameters. Concerning mean convective exchanges around a rotating finned cylinder in transverse airflow (Fig. 1c), a previous study has been made at the laboratory [22–24]. The authors showed that the convective heat transfer is controlled by the air crossflow for Rex/ReU < x0, with x0 between 0.5 and 1 and by the rotation for Rex/ReU > x1, with x1 between 5 and 12, according to the u0 value. Furthermore, for a fixed airflow Reynolds number, the reduction in fin spacing leads to the decrease in the Nusselt number, due to the boundary layers interaction formed on two adjacent fins and the subsequent velocity reduction between fins. On the other hand, for a fixed fin spacing, as the airflow Reynolds number increases, the convective coefficient on the fin tends to reach the characteristic value of the single disk case. Indeed, the boundary layers interaction and the proportion of kinetic energy lost between the fins decreases with an increase in the airflow Reynolds number. For their tests carried out with rotating fins without airflow, an identical analysis can also be used to explain the variation of the Nusselt number as a function of the rotational Reynolds number and fin spacing. In this paper, a transient method for identifying heat transfers, using infrared thermography, is proposed. In this method, the infrared camera measures the time evolution of the fin temperature in the radial direction, at eight angular locations. Solving the inverse conduction heat transfer problem allowed us to identify the local and mean convective heat transfer from the fin surface, while taking into account both conductive and radiative fluxes [25]. Tests are carried out within the dimensionless fin spacing range 0.10 < u0 < 0.69 and for a single fin mounted on the cylinder (corresponding to u0 = 1), within the Reynolds number ranges 0 < ReU < 39,600 and 2150 < Rex < 17,200. For all of these tests, the influence of natural convection on the convective heat transfer remains negligible [12]. Thus, the Nusselt number on the fin surface is controlled by dimensionless numbers Rex and ReU expressing rotational and airflow forced convections, respectively. By comparing the total convective heat flow dissipated by the finned
cylinder surface to that dissipated by the smooth cylinder surface, a finned cylinder efficiency is evaluated in order to determine the optimal geometrical configurations leading to a better cooling.
2. Experimental setup Mean convective heat exchanges from the surface of a rotating fin in air crossflow are obtained by measuring its spatio-temporal temperature variations. The temperature is measured on the central fin a3 of a cylinder which has five annular fins (Fig. 2a and b). The 2-mm wide and 60-mm high fins were made of aluminum (kal = 200 W m1 K1, qal = 2700 kg m3, Cal = 0.96 kJ kg1 K1) and mounted on a 58-mm diameter cylinder (Fig. 2). The fin spacing u (10, 14, 24 and 40 mm) was obtained using bakelite rings of various thicknesses. In addition to the characteristics given in Fig. 2b and c, we also studied the case in which u = 1 (one fin on the cylinder) in order to compare it with the single disk case. Fins and rings were covered with a thin coat of black paint, whose relatively high emissivity allowed the radiative heat flux emitted at a given point of the fin a3 to be more accurately determined, thus improving the accuracy of the relationship between the camera exit signal and the disk temperature. The finned cylinder was driven by an electric motor coupled to a frequency variator, which allowed us to vary the rotational velocity from 250 to 2000 rpm. Experiments are performed in a 2.4 m long wind-tunnel in which the test finned cylinder is positioned at 2-m distance from the fan in order to obtain a straightened airflow upstream from the fins. A diaphragm allows us to vary airflow velocity from 0 to 14 m s1. The turbulence level of the incoming flow has been estimated in a previous work [23] by using a hot wire anemometry. It is equal to 5% for U = 12 m s1 and 10% for U = 4 m s1. In order to measure fin temperatures by thermography, the front wall of the wind tunnel includes a porthole made up of an infrared transparent film. The calibration law with the film presence has been determined with an extended black body at the laboratory. A radiant panel emitting short infrared waves was placed horizontally above the finned cylinder, heating them uniformly to temperatures of about 120 °C. Once the steady thermal state was reached, the heat source was shut off. The fins then cooled through radiative and convective heat transfers, depending on the rotational and air crossflow velocities. The surface temperatures during the cooling of the fins were recorded using an infrared (IR) camera (JADE 3 from CEDIP INFRARED SYSTEMS). Fins a1 and a2 had a rectangular slit along their entire radius to allow the central fin temperature to be measured. The presence of these slits was assumed to have no effect on the convective heat transfer in the central fin a3. A photoelectric sensor was used to detect the eight reflective bands stuck on the cylinder. Once the sensor had detected the bands, it triggered the IR camera, which then scanned the whole fin. In our study, the camera was placed 2.50 m in front of the central fin, with an acquisition frequency of 600 Hz and a window of 160 120 pixels, where each pixel corresponds to almost 2.5 mm2 of the fin surface. The eight angular locations taken at each turn of the finned cylinder are represented in Fig. 2b. The camera exit signals obtained during the cooling were collected and processed with the software MATLAB to determine the radial temperature profiles of the fin. They are expressed in thermal levels. Their processing is explained in a previous article, as well as the radiative heat flux computation [12]. An infrared pyrometer was placed perpendicular to the cylinder’s rotational axis in order to measure the cylinder’s temperature evolution over time. The air temperature measured by a K-thermocouple placed in the middle of the wind tunnel section at 0.5 m upstream of the finned cylinder, during fin cooling, was used as the reference temperature.
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Fig. 2. Representation of the test facility.
3. Convective heat transfer identification method Unlike the fin temperature and the radiative heat flux, the convective heat flux cannot be determined directly from measurements because convective phenomena depend on unknown flow characteristics. This type of problem is an inverse heat conduction problem, which has been described by Beck et al. [26]. In such problems, temperature measurements can be exploited to identify surface conditions, such as convective heat flux, using the direct model and inverse method described below.
to characterize mean convective heat transfers, the convective heat flux density uconv ðr; tÞ ¼ hðT cal ðr; tÞ T 1 Þ is characterized by a uniform on the fin surtime-averaged heat transfer coefficient h, face. To solve the direct model, the mean heat transfer coefficient is assumed to be known. The equations in the system (Eq. (2)) h are discretized to be solved by finite differences, with an implicit scheme using right differences for first-order terms and central differences for second-order terms. A sensitivity study allowed us to determine the adequate fixed step Dr = 2.5 mm (C = 19 nodes). 3.2. Inverse method
3.1. Direct model The direct model involves solving partial differential equations related to the cooling of the fin. Since in all of our experimental conditions the Biot number Bi ¼ h e=kal ¼ h 105 1 8h, the conductive heat transfer in the fin thickness direction (z) can be neglected. So, we use an unidimensional direct model, allowing to compute the spatio-temporal evolution of the fin temperature Tcal(r, t), described by the following equations:
qal C al @T cal ðr; tÞ
kal
@t
uconv ðr; tÞ þ uray ðtÞ
F¼
K X C h X
i2 T kcal ðiÞ T kfin ðiÞ
ð3Þ
k¼1 i¼1
e kal
2
@ T cal ðr; tÞ 1 @T cal ðr; tÞ ¼ þ @r2 r @r T cal ðr; t ¼ 0Þ ¼ T fin ðr; t ¼ 0Þ
The inverse method allows the mean heat transfer coefficient h to be determined by comparing the computed and measured temperature values. It involves a function specification method, which by minimizing the difference, in the least consists in determining h square sense, between calculated and measured temperatures [27]. In this way, the following function, based on the sum of the mean squared deviations, is used:
ð2Þ
where K is the time steps number, given by the acquisition frequency, and C the radial steps number (C = 19). This method allows the mean convective heat transfer coefficient to be determined while taking radial conduction into account [25].
T cal ðri ; tÞ ¼ T fin ðri ; tÞ T cal ðre ; tÞ ¼ T fin ðre ; tÞ
3.3. Convective heat transfer uncertainty
In this model, the temperature distribution at the initial time and the boundary conditions on the inner and outer radii (ri and re) of the fin are obtained from IR camera thermal levels. In order
The uncertainty associated with the convective heat transfer coefficient is determined by the confidence intervals Dh determined for each case. They are obtained from the sensitivity
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Table 1 Uncertainties for different typical cases. Case
u0 = 0.41 ReU = 11,350 Rex = 4300
u0 = 0.41 ReU = 11,350 Rex = 12,900
u0 = 0.41 ReU = 33,950 Rex = 4300
u0 = 0.41 ReU = 33,950 Rex = 17,200
(W/m2 K) h DT (°C)
27
40.8
66.8
71.9
Dh (W/m2 K)
0.3
0.38
8.0
9.6
coefficient Q k ¼ @T kcal =@h and the mean deviation DT between calculated and measured temperatures. For some typical cases, we obtained the results given in Table 1.
4. Results and discussion Radial profiles of the measured temperature, taken at different times and relative to the typical case characterized by u0 = 0.41, Rex = 4300 and ReU = 33,950, are presented in Fig. 3. The observed weak temperature gradients with r allow the hypothesis relative to the heat transfer coefficient uniformity to be validated at the fin surface. Moreover, Fig. 4 presents radiative and convective heat flux density evolutions corresponding to Fig. 3 case for h = 180°. The radiative heat flux density represents only 3% of convective
0.18 11.8
0.16 11.5
heat flux density. Moreover, as for all of our considered cases, the ratio between radiative and convective heat flux densities does not exceed 10%, the global approach used in the radiative heat flux determination is justified. The results reported in this section are expressed in terms of mean Nusselt number Nu, which is characteristic of convective heat transfers, and in terms of rotational and air crossflow Reynolds numbers (Rex and ReU), based on the diameter of the cylinder, which are respectively characteristic of rotation and air crossflow. Physical properties were evaluated at the air film temperature. To study the influence of rotation and air crossflow on convective heat transfers over the finned cylinder, we performed tests using the following conditions: Rex between 2150 and 17,200 and ReU between 0 and 39,600. The geometry of the finned cylinder was characterized by the dimensionless fin spacing u0 = u/ Di, varying from 0.17 to 0.69. 4.1. Rotating finned cylinder in still air The experimental and correlated results for the mean convective heat transfer [12] are presented as a double logarithmic plot of the mean Nusselt number against the rotational Reynolds number in Fig. 5. The correlated relationship of the mean Nusselt with the rotational Reynolds number and dimensionless fin spacing, for our tests, is determined by Watel’s method [22] and expressed as:
NuU¼0 ¼ 2:221 X Re0:5 x 0 0:5 0:5 e K1 a1 1 with X ¼ 1 Re x u0 þ 1 u0 b1
ð4Þ
The values of the different coefficients operating in Eq. (4) are K1 = 1.022, b1 = 0.031, a1 = 0.989 and e0 = e/Di = 0.034. 4.2. Rotating finned cylinder in air crossflow Fig. 3. Radial temperature evolutions at different times for u0 = 0.41, Rex = 4300, ReU = 33,950 and h = 180°.
Fig. 4. Radiative and convective flux density evolutions for u0 = 0.41, Rex = 4300, ReU = 33,950 and h = 180°.
As the thermal behavior of the central fin is strongly dependent on the couple (Rex, ReU) [22–24], the ratio Nu=Nux¼0;u0¼1 is intro-
Fig. 5. Mean heat transfer from a rotating finned cylinder in still air.
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duced to highlight the influence zones of rotation, airflow and coupling rotation/airflow, by comparison with the case of a single stationary fin with air crossflow (x = 0, u0 = 1). For this last case, our results obtained for 5700 < ReU < 39,600 [28], are correlated by:
Nux¼0;u0 ¼1 ¼ 0:036 Re0:8 U
ð5Þ
The mean Nusselt numbers ratio evolution Nu=Nux¼0;u0 ¼1 against the Reynolds numbers ratio Rex/ReU is plotted Figs. 6–9, for different values of u0 . For each fin spacing, different influence
Fig. 9. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.17.
Fig. 6. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.69.
Fig. 7. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.41.
Fig. 8. Correlated data from Nux¼0;u0 ¼1 for u0 = 0.24.
zones are highlighted. For Rex/ReU < 0.2, our experimental results fluctuate around the mean value Nu=Nux¼0;u0¼1 ¼ A for all the fin spacings studied (A varying from 1 for u0 = 1, to 0.73 for u0 = 0.17, see Table 2). These results show that, in this zone, convective exchanges do not depend on rotation, and are therefore exclusively controlled by the airflow. Furthermore, the comparison of Figs. 6–9 highlights the mean Nusselt number decrease with fin spacing, due to the interaction of the boundary layers between the fins which increases with the decrease of fin spacing, leading to bad air circulation [15–18,21–23]. From Rex/ReU > 0.2, convective exchanges are governed (i) by rotation flow superposed to air flow and (ii) by fin spacing. Indeed, for u0 = 0.69 and 1 (Fig. 6) both characteristic zones are highlighted. In the first one corresponding to 0.2 < Rex/ReU < 0.8, rotation influence becomes significant, leading to a convective exchanges increase of 1:3Nux¼0;u0¼1 for u0 = 1 and 1:13Nux¼0;u0¼1 for u0 = 0.69. In the second one, corresponding to Rex/ReU > 0.8, rotating effects generates more and more flow perturbations, leading to convective exchanges intensification compared with the first zone. In the studied Rex/ReU range, the difference between the ratio Nu=Nux¼0;u0¼1 calculated for u0 = 1 and u0 = 0.69 is about 13%, for both zones. A change in physical phenomena occurs as fin spacing decreases (0.24 < u0 < 0.41, Figs. 7 and 8) since a single zone of rotation influence is highlighted. Indeed, the increase of boundary layers interactions between co-rotating fins [10–12], combined to flow perturbations, generated by the superposition of rotating and airflow effects, leads to identical behavior of convective exchanges for Rex/ReU > 0.2 and 0.24 < u0 < 0.41 (small fin spacings). By comparison with the case 0.69 < u0 < 1 (important fin spacings), in the range 0.2 < Rex/ReU < 0.8, we observe a higher increase in the Nusselt number ratio for small fin spacings than for large ones, whereas for Rex/ReU > 0.8 it is the contrary. Furthermore, concerning small fin spacings (0.24 < u0 < 0.41) in the considered Rex/ReU range, the mean difference between the ratio Nu=Nux¼0;u0¼1 calculated (i) for u0 = 1 and u0 = 0.41 is about 15% and (ii) for u0 = 1 and u0 = 0.24 is about 20%. Concerning the smallest fin spacing u0 = 0.17, the mean thermal behavior of the fin is different (Fig. 9), even if two zones are highlighted as for higher fin spacings. In the first one corresponding to 0.2 < Rex/ReU < 0.4, a Nusselt number ratio increase similar to one of the other cases presented Figs. 6–8, is observed. Nevertheless, this zone size is shorter than the others probably because of an effect of boundary layer interactions more intense from a lower Reynolds numbers ratio (Rex/ReU = 0.4 instead of 0.8). Furthermore, this phenomenon seems to be accentuated for Rex/ReU > 0.4 since the Nusselt number ratio increase is a lot smaller than the other cases.
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Table 2 Eq. (6) coefficient values for different fin spacings. u0
Rex/ReU < 0.2
0.2 < Rex/ReU < x
x < Rex/ReU
1 0.69 0.41 0.24 0.17
A = 1; B, C, D = 0 A = 0.87; B, C, D = 0 A = 0.85; B, C, D = 0 A = 0.79; B, C, D = 0 A = 0.73; B, C, D = 0
x = 0.8; A = 1; B = 0.45; C = 0.2; D = 0.77 x = 0.8; A = 0.87; B = 0.40; C = 0.2; D = 0.78 A = 0.85; B = 0.52; C = 0.2; D = 0.79 A = 0.79; B = 0.44; C = 0.2; D = 0.6 x = 0.4; A = 0.73; B = 0.44, C = 0.2; D = 0.66
x = 0.8; A = 1.3; B = 0.45; C = 0.8; D = 0.82 x = 0.8; A = 1.13; B = 0.40; C = 0.8; D = 0.81
x = 0.4; A = 0.89; B = 0.14; C = 0.4; D = 0.47
The complexity of the physical phenomena engendered by the simultaneous effects of u0 , Rex and ReU brings us to correlate our data for a fixed value of u0 . So, correlations between Nu, Rex and ReU are determined for different value of u0 , in term of the ratio Nu=Nux¼0;u0¼1 against the ratio Rex/ReU, as in Eq. (6):
" Nu ¼ Nux¼0;u0 ¼1 A þ B
D # Rex C ReU
ð6Þ
The different values of the coefficients A, B, C and D occurring in Eq. (6), depending on Rex/ReU and u0 , are presented in Table 2. The computed values (Eq. (6)) are equal to the experimental ones with a mean error of about 12%. The fin spacing influence in the airflow predominance zone is characterized by the coefficient A. So, we verify that the convective exchanges coefficient is all the more important as the fin spacing is high. For rotation/airflow coupling, the fin spacing influence is mainly characterized by the coefficient D. For u0 P 0.41, fin spacing influence remains constant since D 0.8 whereas for u0 = 0.17 and 0.24, the reduction in fin spacing leads to a convective transfer decrease characterized by D 0.6. Concerning the C coefficient, it is directly linked to the different influence zone boundaries.
Fig. 10. Correlated data from NuU=0 for u0 = 0.69.
4.3. A general correlation In spite of all the differences of behavior, according to the fin spacing, highlighted in the previous sections, it is possible to predict convective exchanges on a rotating finned cylinder in air crossflow by a correlation including the three parameters u0 , Rex and ReU from RexReU > 0.2. The following equation is proposed:
Nu ¼ NuU¼0
a K Rex u0b ReU
ð7Þ
With K = 1.280, b = 0.160 and a = 0.576. Figs. 10–13 present experimental and correlated mean Nusselt numbers (Eq. (7)) against Rex varying from 2150 to 17,200, ReU, from 5700 to 39,600 and u0 from 0.17 to 0.69. The computed values (Eq. (7)) are equal to the experimental ones with a mean error of about 15%. In conclusion, the correlation (Eq. (7)) allows us to predict the global thermal behavior of a rotating finned cylinder in an air crossflow, by taking into account the combined effects of rotation and airflow, for different fin spacings. It could be used for practical applications to determine convective exchanges on the rotating finned cylinder in airflow. 5. Evolution of the mean convective exchanges on a finned cylinder in air crossflow by comparison with the smooth cylinder case For a given cylinder length L, reducing the fin spacing allows to increase the cylinder-air exchange surface by increasing the number of fins. However, reducing the spacing also causes a decrease in the convective exchange coefficient. In order to find a compromise between these two contradictory effects, we evaluated the ‘‘efficiency’’ n of the finned cylinder relatively to the smooth cylinder [12]. The ratio variation of the total convective heat flow dissipated
Fig. 11. Correlated data from NuU=0 for u0 = 0.41.
by the finned cylinder surface and the one dissipated by the smooth cylinder can be expressed by the following equation:
n¼
h2p ðr 2e r 2i Þ þ hbf pDi u hcyl pDi ðu þ eÞ
ð8Þ
where hcyl is the heat transfer coefficient of the smooth cylinder surface determined from Eq. (1). Concerning the heat transfer coefficient at the cylinder surface comprised between fins hbf, taking hbf ¼ hcyl or hbf ¼ h in the numerator of Eq. (8) does not affect the n value significantly. Therefore, we chose to use hbf ¼ h to express the convective heat flux dissipated by the finned cylinder, in order to take into account the convective transfer decrease at the cylinder wall due to fin presence. Efficiency evolution n according to the
B. Latour et al. / International Journal of Heat and Mass Transfer 54 (2011) 4710–4718
Fig. 12. Correlated data from NuU=0 for u0 = 0.24.
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In a previous study on a rotating finned cylinder in still air [12], we identified the optimal fin spacing for u0 = 0.17. For u0 > 0.17, the increase of the mean Nusselt number does not offset the decrease in efficiency due to the exchange surface drop, engendered by larger fin spacings. On the other hand, for u0 < 0.17, it is the contrary, since the fall in efficiency is directly linked to the mean Nusselt number decrease, whereas the heat exchange surface is increased. Indeed, with u0 = 0.10, the exchange surface increase did not offset the fall of heat exchange coefficient due to the interaction of the boundary layers between two adjacent fins. In the case of a rotating finned cylinder in air crossflow, like in the previous case, efficiency greatly depends on the exchange surface extent, as well as boundary layer interactions. We have seen in the previous sections that these interactions are directly dependent on the ratio Rex/ReU in the presence of an airflow. Fig. 14 presents five evolutions relative to the different characteristic zones highlighted previously (Rex/ReU < 0.2; 0.2 < Rex/ReU < 0.4; 0.2 < Rex/ReU < 0.8; Rex/ReU > 0.8 and ReU = 0) which are representative of all the experimental cases carried out. Indeed, for Rex/ ReU < 0.8, efficiency continually increases with the decrease of u0 until u0 = 0.17, whereas for Rex/ReU > 0.8, the optimal fin spacing is equal to 0.24. This result can be linked to the drop in convective heat transfers observed Fig. 9 for the higher ratios (the third zone relative to u0 = 0.17).
6. Conclusion
Fig. 13. Correlated data from NuU=0 for u0 = 0.17.
Fig. 14. Efficiency of the finned cylinder according to u0 for different Rex/ReU.
dimensionless fin spacing u0 and the ratio Rex/ReU are presented in Fig. 14. For comparison, we added results corresponding to the case of a rotating finned cylinder in still air for 0.10 < u0 < 0.69 [12].
The experimental setup presented in this paper allowed us to evaluate the mean Nusselt number on a rotational finned tube submitted to airflow parallel to the fins. In each test, the radial evolution of the ‘‘thermal levels’’ at eight angular positions of the central fin was recorded by the IR camera while the system was cooling. The thermal levels were converted to temperature values using the calibration law and the radiative balance of the surface scanned. An inverse method based on the mean squared error allowed us to determine the mean convective heat transfer coefficient by taking both the radial conduction and radiative fluxes into account. Results presented in this paper are valuable in the operating conditions domain defined by 0.17 < u0 < , 2150 < Rex < 17,200 and 0 < ReU < 39,600. Concerning the rotating finned cylinder in still air, a general correlation taking into account the fin spacing and the rotation is proposed. Concerning the rotating finned cylinder in air crossflow, the fin spacing and rotation effects are presented by comparison with the single stationary fin in air crossflow. Several influence zones, function of the ratio Rex/ReU, are highlighted. For any fin spacings, in the first one, characterized by Rex/ReU < 0.2, convective heat transfers are exclusively controlled by the airflow. Concerning the other zones, the effects of an airflow superposed to the rotation induce different mean Nusselt number evolutions according to the fin spacing. Indeed, on one hand, precise correlations of mean Nusselt numbers, function of Rex/ReU, are proposed for each zone and fin spacing with a mean error of about 12%. On the other hand, in order to characterize the convective heat transfer in terms of all the parameters studied, a more general correlation is proposed from Rex/ReU = 0.2 with a mean error of about 15%. The maximal value of the finned tube efficiency in the presence of an airflow depends on the ratio Rex/ReU. The efficiency, based on the comparison with the smooth cylinder, allowed us to detect the optimal fin spacing at u0 = 0.17 for the rotating finned cylinder in still air and at u0 = 0.24 for Rex/ ReU > 0.8, corresponding to the best compromise between the increase of exchange surface and the decrease in Nusselt number, engendered by the fin spacing reduction. Moreover, for Rex/ ReU < 0.8, efficiency continually increases with the decrease of u0 until u0 = 0.17.
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