APP LIC ATION OF GROOVED FINS TO ENHANCE FORCED CONVECTION TO TRANSVER SE FLOW

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1 Numerical Heat Transfer, Part A, 58:491^512, 2000 Copyright # 2000 Taylor & Francis /00 $ APP LIC ATION OF GROOVED FINS TO ENHANCE FORCED CONVECTION TO TRANSVER SE FLOW Sang-Jong Lee and Tae-Ho Song Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-gu, Taejon, , Korea A numerical study is performed to investigate the heat transfer characteristics of a two-dimensional forced convection over a plate with protruded transverse groove ns. The study is made for four geometries, ve different Reynolds numbers, and for more groove ns with smaller size. Numerical analysis is carried out to investigate the ow patterns, isotherms, heat transfer rates, and the effectiveness of the increased grooves. Local Nusselt number and total heat transfer rate are calculated and the average Nusselt number is correlated as a function of Reynolds and Prandtl numbers. Protruded grooves yield a much larger heat transfer rate than a at plate owing to ow penetration into the groove. Grooved ns with protruded mounting are suitable for heat transfer enhancement, and the total length is recommended not to exceed the reattachment length signi cantly. The derived correlations and physical considerations may be used when designing grooved plates to enhance heat transfer. INTROD UCTION Extended surfaces are frequently employed to enhance heat transfer. Current enhancement efforts for gases are directed toward extended surfaces that provide a higher heat transfer coef cient than a conventional n, and a grooved n is widely used for this purpose. Extended surfaces used for gases typically have greater n heights than those used for liquids because gases exhibit lower heat transfer coef cients than liquids. There are both externally nned and internally nned tubes. Various grooved n shapes are employed. Offset strip ns, louvered ns, and segmented ns are used for external application, and helical ns and coiled ns are used internally [1]. The best alignment of the grooved ns depends heavily on the ow situation, and frequently they are arranged in transverse direction to the ow. Recently, grooved ns with forced convection have been commonly applied to electronic components cooling to handle the high chip power density over a small area. Cooling methods for various geometrical con gurations and ow conditions are summarized by Incropera [2] and Peterson and Ortega [3] including natural con- Received 24 June 1999; accepted 12 April This work was supported by a grant from the Korea Science and Engineering Foundation, project , and also by a grant from the Critical Technology 21 Project of the Ministry of Science and Technology of Korea. Address correspondence to Professor Tae-Ho Song, Korea Advanced Institute of Science and Technology, Department of Mechanical Engineering, Kuson-dong, Yusong-gu, Science Town, Taejon , South Korea. 491

2 492 S.-J. LEE AND T.-H. SONG NOM ENCLATUR E C p speci c heat at constant pressure H depth of the groove h average heat transfer coef cient k thermal conductivity L total length of numerical models L r reattachment length Nu L average Nusselt number based on total length L ˆ q=k T W T 1 ] Nu W local Nusselt number based on total length W P pressure Pr Prandtl number q total heat transfer rate per unit depth Re L Reynolds number based on total length L Re W Reynolds number based on characteristic length W T temperature u x-direction velocity u y-direction velocity W 1 width of the protruded top surface W 2 width of the groove W groove pitch ( ˆ W 1 W 2 ) a thermal diffusivity ( ˆ k/rc p n kinematic viscosity y dimensionless temperature ˆ T T 1 = T W T 1 Š r density of the uid Subscripts w wall 1 surrounding Superscript * dimensionless quantity vection, forced convection, mixed convection, and heat transfer with phase change, some of which are better than others. Also, Garimella [4] has summarized enhancement techniques of electronic equipment by air cooling using various means: controlling geometric layout, extended surfaces, ribs and elements, vortex generators, ow modulation, and impinging jets. Chu and Simons [5] report that forced air cooling has been and still is the most widely used cooling technology, and the principal advantage of cooling with air is its ready availability and easy application. Typically, electronic components are cooled either as they protrude from a substrate or as ush-mounted to the substrate. The grooved plate is one of the most popular geometries in such applications. Moffat et al. [6] have carried out an experimental study of forced convection in an air-cooled array of electronic components. Davalath and Bayazitoglu [7] have computed the heat transfer over an array of rectangular blocks to nd the optimum spacing between heat sources for a xed heat input with an allowable maximum temperature at the heat source. Jubran and Al-Salaymeh [8] report an experimental investigation varying the size and shape of arrayed ribs. The maximum Nusselt number is attained at the rst row of modules, which is attributed to the impingement of the ow there. Sparrow et al. [9, 10] have performed experiments using the naphthalene sublimation technique for arrayed modules. The results indicate that heat transfer is enhanced when the module is either taller or shorter than the modules of the reference array. Combined effects of rib alignment and channel aspect ratio in short square and rectangular channels are determined by Hong and Hsieh [11]. The Nusselt number appears higher in staggered alignment than in in-line alignment. McEntire and Webb [12] have focused on the local forced convective heat transfer from protruded and ush-mounted discrete heat sources varying channel spacing. It is found experimentally that protrusion of the heat sources results in enhanced heat transfer particularly for the second and subsequent heaters because of ow separation, whereas the

3 APPLICATION OF GROOVED FINS 493 ush-mounted heat sources show heat transfer enhancement from heater to heater by interruption of the thermal boundary layer. Heat transfer performance in small-scale ns and channels has been investigated by Mizunuma et al. [13]. Heat transfer enhancement to the channel height and the coolant velocity has been experimentally and numerically studied. They nd that the sensitivity of heat transfer performance to the channel height variation depends on the type of heat transfer surface. Another investigation has been made to enhance heat transfer by ow modulation techniques. Greiner [14] has performed experiments to study hydrodynamic resonance and its effect on heat transfer in laminar ows through ducts with periodically spaced transverse grooves cut into one wall. It has been found that oscillatory perturbation of the ow rate is actively modulated at an appropriate frequency to excite ow instabilities and thus to enhance the channel mixing, and it effectively more than doubles the heat transfer coef cient. Active, passive, and supercritical ow destabilization techniques have been compared based on an equal pumping power by Amon [15] via numerical simulations. The ow destabilization mechanism promotes lateral, large-scale convective mixing and increases heat transport in the direction normal to the heat transfer surface. Although passive ow modulation is the best enhancement system based on minimum power dissipation at low Nusselt numbers, supercritical ow destabilization becomes competitive at a higher Nusselt number. Also, depending on the shape and the material of grooved ns, grooved ns may function as an insulator. Bejan [16] has studied the insulating effect of hair strands. Insulation effect appears when the hair strands are suf ciently dense so that they trap a blanket of air in the tight spaces created between them. Similar insulating behavior is expected to occur in some extreme cases of grooved ns where the air ow is highly immobile in the groove. The literature on enhancement of forced convection is abundant. Regarding heat transfer from grooved surface to blowing air, there are extensive studies of the effect of rib alignment and the channel aspect ratio, the average heat transfer from multiple heat sources, and so on. However, it is necessary to examine how the local heat transfer from each surface may be enhanced by properly employing the interaction of ow pattern and thermal wake. Also, the effect of various groove shapes is not fully revealed either. In addition, little attention has been paid to how many serial grooves are desirable to enhance heat transfer. Thus we perform this study to investigate forced convection from transverse rectangular grooves in protruded mounting and to examine how many grooves are optimal. Since the ow is mostly laminar, we limit the study to two-dimensional laminar steady convection. The results can be utilized in designing groove ns to enhance heat transfer. M ATHEM ATICAL M ODEL AND NUM ERICAL ANALYSIS Numerical analysis is performed to investigate the ow patterns, isotherms, and heat transfer rates for the protruded grooves of four geometrical con gurations. The representative geometrical con guration is shown in Figure 1. Nondimensionalized geometrical con gurations of the models are summarized in Table 1. The four geometries are named square, shallow, deep, and wide grooves,

4 494 S.-J. LEE AND T.-H. SONG Figure 1. The transverse rectangular grooves with pitch numbers (1 to 5) and surface numbers; I, upper surface; II, outer surface; III, bottom surface; IV, inner surface. respectively. The square groove consists of square protrusions and grooves as shown in Figure 1. The shallow groove is made by reducing the groove height to half of the square groove. Thus H is 0.25 and other dimensions are the same as in the square groove. The deep groove is taken by increasing the groove height, and the wide groove has a larger width. Five protrusions are shown in a series and four grooves are placed between them. They are at uniform temperature T W. The lengths are as shown in the gure. One pitch, W, is de ned as the characteristic length. The external ow enters from the top with uniform velocity, V 1, and temperature, T 1, and leaves at the bottom. Thus ve pitches are expressed as 1 to 5 from the inlet and each pitch has four faces: upper surface I, outer surface II, bottom surface III, and inner surface IV.

5 APPLICATION OF GROOVED FINS 495 The two-dimensional Navier^Stokes and energy equations for steady state, laminar, forced and incompressible ow are given ˆ 0 @y ˆ @y ˆ @y ˆ where u is kinematic viscosity, and a is thermal diffusivity expressed as a ˆ k=rc p. The employed dimensionless quantities to nondimensionalize Eqs. (1)^(4) are de ned as x ˆ x W y ˆ y W y ˆ T T 1 T W T 1 u ˆ u u 1 Pr ˆ n a u ˆ u u 1 P ˆ P ru 2 1 Re W ˆ u1w n 5 Substituting Eq. (5) into Eqs. (1)^(4), we get Eqs. (6)^(9) ˆ 0 @y ˆ 1 Re @y ˆ 1 Re @y ˆ 1 Pr Re The upper numerical boundary is given at the top of the grooves. The right boundary is located at a far distance, 12 times the depth of the groove away from the n. The outlet boundary is also located far down, two times the total vertical length of the geometrical con guration, which is far enough to assume that there

6 496 S.-J. LEE AND T.-H. SONG is no upward ow. The boundary conditions are given in Eqs. (10)^(14) as follows: u ˆ 0 u ˆ 1 at the inlet and ˆ ˆ at the right boundary 11 y ˆ 0 at the inlet 12 y ˆ 1 at the surface of the ˆ 0 at the right boundary and the outlet 14 The SIMPLER algorithm [17] is utilized to solve Eqs. (6)^(9). The numerical calculation is made by taking ve air velocities of 2, 3, 5, 7, and 10 m/s. The corresponding Reynolds numbers based on the characteristic length W are 1880, 2820, 4700, 6580, and 9400, respectively. A constant Prandtl number 0.71, of air, is taken. For other uids with Pr close to 1, the widely accepted correlation that the Nusselt number is proportional to Pr 1=3 may be safely used. It is uncertain whether transition to turbulence may occur at the higher Reynolds numbers considered above because the transition in a practical situation is strongly dependent on the array con guration, the ow rate, and location in the array. Wirtz and Chen [18] and Garimella and Eibeck [19] have performed experiments to investigate the onset of transition in the ow over an array of protrusions mounted on rectangular air and water channels, respectively. They have found that increase the streamwise spacing between protrusions and decrease of channel height causes transition to occur at relatively low Reynolds numbers. From Wirtz and Chen s results, the onset of transition begins at a critical Reynolds number Re C of 3000, based on the local mass ux and the approximate hydraulic diameter, for a channel height of 3.0 protrusion heights and at Re C < 2000 for a channel height of 2.0 protrusion heights. Garimella and Eibeck have found that transition occurs in the fully developed region of the array at a channel-height-based Reynolds number of 700 for a channel height of 1.2 protrusion heights, increasing to 1900 for a channel height of 3.6 protrusion heights. Also, McEntire and Webb [12] have found in their experiment that ow instability leads to an eventual downstream transition to turbulent ow above Re C &2000, based on hydraulic diameter, for a channel height of 3.0 protrusion heights. Nonetheless it is dif cult to directly apply their results to this paper: the geometry of this paper may cause the onset of transition to occur at a higher Reynolds number because streamwise spacing between protrusions in Figure 1 is similar to the above experiments and the channel height is much larger. The Reynolds number corresponding to Re W ˆ 9400 on the at plate is about 1/10 of the critical Reynolds number Thus the assumption that ow is laminar in the range of the Reynolds number considered in this paper is not very deviated from a practical situation. In the process of validation of the numerical results, grid dependence is rst examined to minimize the numerical uncertainty. Nonuniform grid structure is applied by distributing dense grids near the surfaces of protrusions and sparse grids at locations far from the grooves. The total length of computation domain is 6W in the x-direction and 11.5W in the y-direction when the protrusion height H of the square groove is, for instance, 0.5. Length L of numerical model is 5.5W

7 APPLICATION OF GROOVED FINS 497 in the y-direction. This computational domain is veri ed to be appropriate by checking that further increased length alters the results by less than 1%. The heat transfer rate is calculated for two different grid structures in the range of Reynolds numbers 1880 to The computational results show less than a 4.5% difference in the total heat transfer rate between the employed 6700 cells and the increased 13,832 cells. Compromising between the computational cost and the accuracy, we nd that the employed cell is considered appropriate. To verify the result of the numerical study, we compute also the average Nusselt number for a at plate and compare it with that of the analytical calculation [20]. A discrepancy of less than 3% between them is found for the ve Reynolds numbers mentioned earlier. Another validation is made to con rm the prediction capability of the code. Computation is made for the geometry with three protrusions in the channel that is similar to the con guration investigated by Davalath and Bayazitoglu [7] as shown in Figure 2. They have suggested the correlation of average Nusselt number as a function of the channel-height-based Reynolds number for each protrusion. The results are compared at each protrusion in Figure 2 for the Reynolds numbers 750 and 1000, which are the same as those in [7]. Differences for the protrusions 1, 2, and 3 at the Reynolds number of 750 are 0.2%, 7.5% and 3.2%, respectively, and those at the Reynolds number of 1000 are 0.3% 6.9% and 2.5%, respectively. Thus the results show that the code predicts well so that its simulation is valid. Although not shown here, a separate experimental study using a Mach^Zehnder interferometer shows good agreement with the numerical result for the square groove [21]. Indeed, it is shown that the numerical results are more accurate than the interferometric ones. R ESULTS AND D ISCUSSIONS Five Protrusions of D iff erent Geometries Streamlines and isotherms for ve protrusions of the four geometries of Table 1 are shown in Figures 3a and 3b when Re W is Indeed, velocities in the groove are very small and thus the streamlines in the groove are drawn with a very ne interval. Flow separation occurs above the rst protrusions. It generates main recirculation ow above the leading grooves. It then reattaches downstream. Flow separation occurs when an adverse pressure gradient exists, that > 0. It means that curvature effect exists at the leading edge and the angle of incident is larger than zero. Reattachment length L r is studied by Hsieh and Huang [22] and Goldstein [23] in laminar for the protrusion on the at plate. They suggest that L r =H is 4, 6, and 7.5 for Re H of 100, 200, and 250, respectively. Moss and Baker [24] have found the reattachment length L r =H to be 5 for turbulent ow in a contraction channel. A detailed review of this length has been made carefully by Eaton and Johnson [25] for turbulent ow in the backward-facing step with various aspect ratios and Reynolds numbers from 22 studies. They summarize that L r =H is 5*8.5. Thus L r =H does not signi cantly change as it comes in 4*8.5. From Figure 3a, L r =H for the square groove is 6. It is 9 for the shallow groove, 4.5 for the deep groove, and 6.5 for the wide groove.

8 498 S.-J. LEE AND T.-H. SONG Figure 2. Comparison of simulation results with the air-cooling study of Davalath and Bayazitoglu [7] for the code validation. Penetration of the main recirculation ow occurs at the reattachment point. It depends heavily on the geometrical structures as the streamlines of Figure 3a show. For the square groove, the ow is reattached at the edge of the fourth protrusion. The extent to which the isotherms penetrate into the groove is similar to that of the streamline (compare Figures 3a and 3b). Generally, the outer parts of the protrusion have dense isotherms and those in the groove are sparse. Also, isotherms in the grooves are dense when the main recirculation ow penetrates deeply into the groove. Therefore, the wall heat ux in the third groove is the highest for the square

9 APPLICATION OF GROOVED FINS 499 Table 1. Nondimensionalized geometrical con guration of numerical models Total heat W Total transfer area per Con guration H ( ˆ W 1 W 2 ) H/W W 2 =W length/w unit depth Square groove W Shallow groove W Deep groove W Wide groove /14 7/15 4/ W Figure 3. (a) Streamlines and (b) isotherms for the four geometries in protruded mounting with Re W ˆ 1880.

10 500 S.-J. LEE AND T.-H. SONG geometry. Isotherms in the shallow groove penetrate deeply into the second groove as shown in Figures 3a and 3b. For the deep groove, the isotherms penetrate deeply into the fourth groove because the main recirculation is very large because of the large H/W. Thus the wall heat ux in the fourth groove is the largest. Similarly, isotherms in the third groove penetrate more deeply than other ones for the wide groove. The local Nusselt number is calculated at each surface of each pitch for the four geometries. Local Nu W distribution for the square groove when Re W is 1880 is shown in Figure 4. The Nusselt number at the upper surface (I) of the rst pitch is the greatest because the ow directly impinges on this surface. Heat transfer at the fourth pitch is greater than the rest because of reattachment of the above ow. The Nusselt number at the outer surface (II) of the third pitch is larger than that at any other outer surface. Heat transfer performance of each surface is known Figure 4. Local Nusselt number at each surface (square groove, protruded mounting, Re W ˆ 1880).

11 APPLICATION OF GROOVED FINS 501 to depend heavily on ow pattern caused by the geometrical con guration. McEntire and Webb s protruded heaters show the highest forced convection coef cient at the second and subsequent heaters [12], whereas the mixed convection for the protruded heat sources shows the most heat transfer enhancement at the rst and the subsequent heaters [26]. Here, the appearance of the maximum heat transfer rate at the third and the subsequent outer surfaces is thought to be because of ow reattachment of the main ow there (see Figure 3a). The heat transfer rates from the upper (I) and the bottom (III) surfaces are greater at the outer edge than at the inner corner. The innermost (IV) surfaces show a very small heat transfer rate. The behaviors of these three surfaces (I, III, and IV) are very similar to the case of natural convection from vertical plates with horizontal grooves as described in detail by Kwak and Song [27]. The most distinguishable difference from the natural convection is the existence of the recirculation ow over the leading grooves, which totally alters the heat transfer pattern. Contribution to the total heat transfer rate is the greatest at the outer surfaces because they are directly contacting the ambient ow and it is the smallest at the inner surfaces. Total heat transfer rate is computed as follows. The total heat transfer rate q from the entire n per unit depth is q ˆ hl T W T 1 15 Nondimensionalizing Eq. (15) by dividing by k T W T 1 we nd, that it reduces to Eq. (16) Nu L ˆ q k T W T 1 It is the very average Nusselt number Nu L hl=k when we calculate q by Eq. (15). It is summarized for the square groove when Re W ˆ 1880 in Table 2 with a contribution from each pitch and surface. The heat transfer rate at the upper surface (I) of the rst pitch is the largest because the ow directly impinges on this surface. 16 Table 2. Dimensionless heat transfer rate per unit depth [q=k T W T 1)] from each surface at each pitch in the square groove with Re W ˆ 1880 Upper Outer Bottom Inner Contribution to surface surface surface surface total heat I II III IV Total transfer rate, % Pitch Pitch Pitch Pitch Pitch Total Contribution to total heat transfer rate, % ö

12 502 S.-J. LEE AND T.-H. SONG The next largest occurs in pitch 4 where the main recirculation ow penetrates and it is the smallest in the second pitch. The second pitch is covered above by the recirculation blanket. The heat transfer rate on the outer surface is the largest at the third pitch. Comparing with other surfaces, the total heat transfer rate on the outer surface is the largest. For the bottom surfaces, the largest contribution occurs at the second pitch. The heat transfer rate at the inner surfaces is smaller than that at any other surface. The difference of heat transfer rate from each pitch is not so large, however. The heat transfer rates for the other three con gurations depend heavily on the main recirculation ow pattern. They are summarized for each surface and pitch in Table 3. In the shallow groove, the difference of heat transfer rates between pitches is small as in the square groove. In the deep groove, besides the rst pitch, the heat transfer rate is the maximum at the last pitch, because the attachment length is large and the main recirculation ow penetrates into the fourth groove. The inner surface has the minimal contribution to the total heat transfer rate because the ow is almost stagnant near this surface. For the wide groove, dimensionless heat transfer from the inner surface is as large as 15.6, however. In general, heat transfer from the rst upper surface occupies the greatest portion, and the next largest contribution is made at the reattachment point. In a ush-mounted at plate, the greatest heat transfer rate takes place at the leading edge and gradually deceases downstream. However, the sequence of the heat transfer rate in the protruded grooves is not the same as in the at plate because of ow separation and reattachment. Although not shown here to save space, the isotherms for Re W ˆ 2820, 4700, 6580, and 9400 for the four geometries are also Table 3. Dimensionless heat transfer rate [q=k T W T 1 ] from each surface and pitch for four con gurations with Re W ˆ 1880 Square Shallow Deep Wide groove groove groove groove Heat transfer rate from each surface Upper surface I Outer surface II Bottom surface III Inner surface IV Total Heat transfer rate from each pitch Pitch Pitch Pitch Pitch Pitch Total

13 APPLICATION OF GROOVED FINS 503 compared to examine the effect of the Reynolds number. They are qualitatively the same. The higher Re W is, the more deeply the isotherms and streamlines penetrate into the groove so that the isotherms at each surface become denser. Thus the heat transfer increases at all surfaces as Re W increases. The total heat transfer rate calculated using Eq. (15) is summarized in Table 4 as a function of Re W for all protruded groove shapes (called ``original ) together with the cases of a at plate and protruded smaller grooves, which will be discussed in the next section. The total heat transfer rate from the grooved ns is always greater than that of a at plate. For Re W ˆ 1880, the total heat transfer rate is the largest for the deep groove, and it is about 2.7 times that of a at plate. It is smaller in the sequence: wide groove (1.9 times), square groove (1.8 times), and shallow groove (1.3 times). The higher Re W is, the higher heat transfer rate results for all cases, and the ranking of heat transfer rate among the four geometries is not changed. The total heat transfer rate from the wide groove is greater than that from the square groove because of the greater contribution from the upper, bottom, and inner surfaces by the active ow motion in the wide grooves. Remember the heat transfer areas of the two cases are almost the same. The total heat transfer rate is the greatest for the deep groove because of the largest heat transfer area, and it is the smallest for the shallow groove for the same reason. The correlations of Nusselt number versus Reynolds number for the four geometries are made using the above results. The length scale of the Reynolds number is changed to L to be consistent with Nu L. Note that Re L ˆ 5.5 Re W since Table 4. Dimensionless heat transfer rate per unit depth [q=k T W T 1 ] for each geometry with original and protruded smaller grooves varying Re W Re W Flat platea Square groove Original Smaller Shallow groove Original Smaller Deep groove Original Smaller Wide groove Original Smaller a Calculated analytically from the exact solution [14].

14 504 S.-J. LEE AND T.-H. SONG L ˆ 5.5W. Nu L ˆ 0:22Re 0:67 L for square groove 17 Nu L ˆ 0:12Re 0:70 L for shallow groove 18 Nu L ˆ 0:36Re 0:66 L for deep groove 19 Nu L ˆ 0:27Re 0:65 L for wide groove 20 Note that, for a at plate, Nu L ˆ 0:59Re 0:5 L when Pr ˆ The above Eqs. (17)^(20) are valid in the range Re L The error of tting is less than 1% in the given range. Roughly speaking, the overall error, including numerical uncertainty, is less than 5%. Protruded Smaller Grooves Heat transfer characteristics have been investigated for ve protrusions in the previous section. In this section, they are examined for a greater number of smaller grooves. Computed ow patterns and isotherm distributions are presented for Reynolds number of 1880 in Figures 5a and 5b. Flow separation and reattachment are similar to the 5-protrusion case of Figure 3. The separation of ow occurs above the rst protrusion. Behind the recirculation cell, it penetrates into the different grooves depending on the geometrical con guration, that is, into the fth groove for the square groove, the third groove for the shallow groove, and the eighth groove for the deep groove. For the wide groove, it penetrates into the fth and the sixth grooves. Flow patterns in other grooves are similar to the lid-driven recirculation ows in a cavity. Flow velocities are the fastest in the grooves near the reattachment point. Reattachment length L r =H is about 5.5 for the square groove from the leading edge of the protrusion. It is 8 for the shallow groove, 4.5 for the deep groove, and 6 for the wide groove, respectively. It is basically the same as in the previous section. For the square groove, isotherms are the densest in the fth groove as penetrated by the main recirculation ow. Isotherms in the grooves between this and the rst grooves are sparse as covered by the recirculation blanket above. Note that the height of protrusion is different for the geometries, and the magnitude of the recirculation cell is dominated by the protrusion height. The heat transfer rate calculated using Eq. (15) is summarized for the square groove in Table 5 with a contribution from each pitch and surface to the total heat transfer rate. The heat transfer rate on the upper surface (I) is the largest as 21.8 in the rst pitch since ow directly impinges on this surface. It dramatically decreases and then gradually increases to 3.1 until it reaches pitch (6), where the main recirculation ow penetrates and decreases again to 0.9 in the last pitch. The total heat transfer rate slightly increases by about 5% compared with the 5 protrusions. The heat transfer contribution by the outer surface is the largest, whereas the inner surface contributes the least. The total heat transfer rate from each pitch is the largest at the impinging pitch and the reattachment pitch. It then sequentially decreases after the reattachment pitch.

15 APPLICATION OF GROOVED FINS 505 Figure 5. (a) Streamlines and (b) isotherms for the four geometries with smaller grooves in protruded mounting with Re W ˆ The heat transfer rates for four geometries are summarized in Table 6. In all the geometries, the following characteristics are pertinent although the numerical value depends more or less: The heat transfer from the grooves near the reattachment point is the most active together with that on the impinging surface, whereas below the recirculation cell and downstream from the reattachment point the heat transfer becomes inactive. Total heat transfer rates are summarized as a function of the Reynolds number in Table 4. Total heat transfer rates with smaller grooves are always larger than that from a at plate. However, they are smaller for all geometries compared with the original 5 protrusions. De ning the ef ciency as the ratio of the total heat transfer

16 506 S.-J. LEE AND T.-H. SONG Table 5. Dimensionless heat transfer rate per unit depth [q=k T W T 1 ] from each surface for the square con guration with smaller grooves with Re W ˆ 1880 Upper Outer Bottom Inner Contribution to surface surface surface surface total heat (I) (II) (III) (IV) Total transfer rate, % Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Total Contribution to total heat transfer rate, % ö Table 6. Dimensionless heat transfer rate per unit depth [q=k T W T 1 ] from each surface and pitch for four con gurations with smaller grooves with Re W ˆ 1880 Square groove Shallow groove Deep groove Wide groove Heat transfer rate from each surface Upper surface I Outer surface II Bottom surface III Inner surface IV Total Heat transfer rate from each pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch Pitch ö Total

17 APPLICATION OF GROOVED FINS 507 rates to the total heat transfer areas, we see that it is worse for the square and the shallow grooves; however, it is better for the deep groove. And, for the wide groove, it remains nearly the same. In summary, the total heat transfer rate decreases slightly by taking a larger number of smaller grooves. However, it may be a reasonable mounting method to enhance heat transfer when the problem of manufacturing and surface treatment is considered. M ore than 5 Protrusions of the Regular Size In the previous section, the protrusion height was greater than H by taking smaller grooves. A natural question that follows is, What if we employ more grooves with the same protrusion height as in the original con guration? To answer this question, we perform calculations for the grooves with more pitches added to the original con guration; grooves with 5, 6, and 7 pitches are considered. Streamlines for the four geometries with increased grooves are presented in Figure 6, and the isotherms are shown in Figure 7. Streamlines of the square groove remain almost the same, although the number of pitches is increased to 7 pitches. The point of reattachment is 6H from the edge of the rst protrusion. It is clearly seen that the upstream ow pattern is rarely affected by the added grooves downstream. Examination of the other geometries in Figure 6 con rms this conclusion more clearly. Only a slight variation of reattachment point is discernible. Isotherms also are presented in Figure 7. These characteristics are closely related to the ow patterns depicted in Figure 6. In Figure 7a, the isotherms for the square groove remain the same although the number of pitches is increased to 7. There are no large differences in isotherm distribution upstream of the newly added grooves. It is again con rmed for different geometries, too. In summary, ow pattern, isotherm distribution, and thus heat transfer are not greatly changed up to the fth groove by adding more pitches since the main recirculation ow is already fully developed within the leading 5 pitches. They may be changed when the protrusion height H is greater. The heat transfer rate per pitch and the total heat transfer rate are computed for the square groove, and the results show that the heat transfer rate per pitch is not changed much by increasing the number of pitches. In addition, the total heat transfer rates with 5, 6, and 7 grooves do not increase as much as the increase of heat transfer area. It is because the heat transfer is not increased much far downstream of the main recirculation. This trend is similar for the shallow and the wide groove con gurations. The effect of Reynolds number is investigated (see Table 7). In contrast to the change of heat transfer rate per pitch, the total heat transfer rates of the three cases are not much augmented. Even when the reattachment length is increased by increasing the protrusion height, the heat transfer rates from pitches below the recirculation blanket may be decreased. Thus the total heat transfer rates are not much augmented, if at all. Thus the number of pitches in protruded mounting is most effective when the total length is a little greater than the reattachment length. Finally, correlations are suggested to combine all these results into one. Correlation is formulated by using the concept of roughness on a at plate. Correlation of average Nusselt number as a function of Reynolds and Prandtl numbers is given

18 508 S.-J. LEE AND T.-H. SONG Figure 6. Streamlines for the four geometries with increased pitches (5, 6, and 7 pitches) in protruded grooves with Re W ˆ for a at plate as follows: Nu L;Flat ˆ 0:664Re 1=2 L Pr1=3 21 when we consider the in uence of roughness, Eq. (21) can be transformed into Nu L ˆ 0:664 1 e Re 1=2 L Pr1=3 22 where e is the roughness by protrusions. It becomes zero when the protrusions do not exist. Therefore, e has the following characteristics: e! 0 when H W! 0; W L! 0 W 2 W! 0 e! finite when H W! 1 23

19 APPLICATION OF GROOVED FINS 509 Figure 7. Isotherms for the four geometries with increased pitches (5, 6, and 7 pitches) in protruded grooves with Re W ˆ Thus we de ne e using the characteristics of Eq. (23) as follows: e ˆ a H=W b W =L c W 2 =W d 1 H=W 24 where, a, b, c, and d are constants. When we use the data in Table 7, we see that they are optimized as a ˆ 13:3 b ˆ 1:41 c ˆ 0:52 d ˆ 0:34 25

20 510 S.-J. LEE AND T.-H. SONG Table 7. Dimensionless heat transfer rate per unit depth [q=k T W T 1 ] from each pitch in increased protruded grooves varying Re W Re W ˆ 1880 Re W ˆ 2820 Re W ˆ 4700 Re W ˆ 6580 Re W ˆ 9400 Square groove 5 Pitches Pitches Pitches Shallow groove 5 pitches Pitches Pitches Deep groove 5 Pitches Pitches Pitches Wide groove 5 Pitches Pitches Pitches When we introduce Eqs. (24) and (25) to Eq. (22), Eq. (22) reduces to Nu L ˆ 0: :3 H=W 1:41 W =L 0:52 W 2 =W 0:34 1 H=W Re 1=2 L Pr1=3 26 Error of tting is less than 10% for the most part; however, it is as high as 19% in some cases. Equation (26) can be used to investigate the heat transfer characteristics or to design the groove con guration in the protruded mounting. This equation is applicable to the range Re L and Pr%1. C ONCLUSIONS Numerical studies on forced convection are carried out for four aspect ratios of 5 protrusions with transverse air ow. When using protrusion, the heat transfer rate is always larger than that from a at plate. This enhancement of heat transfer is caused by the main ow recirculation, reattachment, and penetration into the grooves. For more grooves with smaller size, the total heat transfer rate decreases a little compared with the original 5 protrusions. Test with more grooves for the same protruded mounting gives a recommendation to make the total length slightly

21 APPLICATION OF GROOVED FINS 511 longer than the reattachment length to enhance the heat transfer. The heat transfer characteristics for all the tested cases are summarized in an equation and it can be used as a design equation. REFERENCES 1. R. L. Webb, Principles of Enhanced Heat Transfer, pp. 1^32, Wiley, New York, F. P. Incropera, Convection Heat Transfer in Electronic Equipment Cooling, J. Heat Transfer, vol. 110, pp. 1097^1111, G. P. Peterson and A. Ortega, Thermal Control of Electronic Equipment and Devices, in J. P. Hartnett and T. F. Irvine, Jr. (eds.), Advances in Heat Transfer, vol. 20, pp. 181^314, Academic Press, San Diego, S. V. Garimella, Enhanced Air Cooling of Electronic Equipment, in S. J. Kim and S. W. Lee (eds.), Air Cooling Technology for Electronic Equipment, Chap. 5, CRC Press, Boca Raton, FL, R. C. Chu and R. E. Simons, Recent Development of Computer Cooling Technology, Proc. 6th Int. Symp. Transport Phenomena in Thermal Eng., pp. 17^25, Seoul, Korea, R. J. Moffat, D. E. Arvizu, and A. Ortega, Cooling Electronic Components: Forced Convection Experiments with an Air-Cooled Array, Proc. 23rd National Heat Transfer Conf., pp. 17^27, Denver, CO, J. Davalath and Y. Bayazitoglu, Forced Convection Cooling across Rectangular Blocks, J. Heat Transfer, vol. 109, pp. 321^328, B. A. Jubran and A. S. Al-Salaymeh, Heat Transfer Enhancement in Electronic Modules Using Ribs and ``Film-Cooling-Like Techniques, Int. J. Heat Fluid Flow, vol. 17, pp. 148^154, E. M. Sparrow, J. E. Niethammer, and A. Chaboki, Heat Transfer and Pressure Drop Characteristics of Arrays of Rectangular Modules Encountered in Electronic Equipment, Int. J. Heat Mass Transfer, vol. 25, pp. 961^973, E. M. Sparrow, A. A. Yanezmoreno, and D. R. Otis, Jr., Convective Heat Transfer Response to Height Differences in an Array of Block-Like Electronic Components, Int. J. Heat Mass Transfer, vol. 27, pp. 469^473, Y. J. Hong and S. S. Hsieh, Heat Transfer and Fiction Factor Measurements in Ducts with Staggered and In-Line Ribs, J. Heat Transfer, vol. 115, pp. 58^65, A. B. McEntire and B. W. Webb, Local Forced Convective Heat Transfer from Protruding and Flush-Mounted Two-Dimensional Discrete Heat Sources, Int. J. Heat Mass Transfer, vol. 33, pp. 1521^1533, H. Mizunuma, M. Behnia, and W. Nakayama, Heat Transfer from Micro-Finned Surfaces to Flow of Fluorinert Coolant in Reduced-Size Channels, IEEE Trans. Components, Packaging, Manufacturing Technology-A, vol. 20, pp. 138^145, M. Greiner, An Experimental Investigation of Resonant Heat Transfer Enhancement in Grooved Channels, Int. J. Heat Mass Transfer, vol. 34, pp. 1383^1391, C. H. Amon, Heat Transfer Enhancement by Flow Destabilization in Electronic Chip Con gurations, J. Electronic Packaging, vol. 114, pp. 35^40, A. Bejan, Theory of Heat Transfer from a Surface Covered with Hair, J. Heat Transfer, vol. 112, pp. 662^667, S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, R. A. Wirtz and Weiming Chen, Laminar-Transitional Convection from Repeated Ribs in a Channel, J. Electronic Packaging, vol. 114, pp. 29^34, 1992.

22 512 S.-J. LEE AND T.-H. SONG 19. S. V. Garimella and P. A. Eibeck, Onset of Transition in the Flow over a Three-Dimensional Array of Rectangular Obstacles, J. Electronic Packaging, vol. 114, pp. 251^255, F. P. Incropera and D. P. De Witt, Introduction to Heat Transfer, 2nd ed., pp. 357^409, Wiley, New York, S. J. Lee, Experimental and Numerical Study on Forced Convection from Plates with Transverse Rectangular Grooves, Ph.D. thesis, Korea Advanced Institute of Science and Technology, Taejon, S. S. Hsieh and D. Y. Huang, Flow Characteristics of Laminar Separation on Surface-Mounted Ribs, AIAA J., vol. 25, pp. 819^823, R. J. Goldstein, Laminar Separation, Reattachment and Transition of Flow over a Downstream Facing Step, J. Basic Eng., vol. 92, pp. 732^741, W. D. Moss and S. Baker, Re-circulating Flows Associated with 2-D Steps, Aero, Quart., pp. 151^172, J. K. Eaton and J. P. Johnson, A Review of Research on Subsonic Turbulent Flow Reattachment, AIAA J., vol. 19, pp. 1093^1100, S. Y. Kim, H. J. Sung, and J. M. Hyun, Mixed Convection from Multiple-Layered Boards with Cross-Steamwise Periodic Boundary Conditions, Int. J. Heat Mass Transfer, vol. 35, pp. 2941^2952, C. E. Kwak and T. H. Song, Experimental and Numerical Study on Natural Convection from Vertical Plates with Horizontal Rectangular Grooves, Int. J. Heat Mass Transfer, vol. 41, pp. 2517^2528, 1998.