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Performance analysis of rectangular ducts with staggered square pin fins
Energy Conversion and Management 44 (2003) 1787–1803 www.elsevier.com/locate/enconman
Performance analysis of rectangular ducts with staggered square pin fins O.N. S ß ara
*
Faculty of Engineering, Department of Chemical Engineering, Atat€ urk University, 25240 Erzurum, Turkey Received 26 April 2002; accepted 3 August 2002
Abstract This paper presents the heat transfer and friction characteristics and performance analysis of convective heat transfer through a rectangular channel with square cross-section pin fins attached over a flat surface. The pin fins were arranged in a staggered manner. Various clearance ratios (C=H ) and interfin distance ratios ðSx =DÞ were used. The performance analysis was made under a constant pumping power constraint. The experimental results showed that the use of square cross-section pin fins may lead to an advantage on the basis of heat transfer enhancement. For higher thermal performance, lower interfin distance ratio and clearance ratio and comparatively lower Reynolds numbers should be preferred for the staggered arrangement. The results of the staggered configurations were also compared with the results of the inline arrangement. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Pin fin; Heat transfer enhancement; Channel flow; Forced convection; Heat exchangers; Performance analysis
1. Introduction Extended surfaces (fins) are frequently used in heat exchanging devices for the purpose of increasing the heat transfer between a primary surface and the surrounding fluid. Various types of heat exchanger fins, ranging from relatively simple shapes, such as rectangular, cylindrical, annular, tapered or pin fins, to a combination of different geometry, have been used. These fins may protrude from either a rectangular or cylindrical base.
*
Tel.: +90-442-231-4556; fax: +90-442-236-0957. E-mail address: [email protected] (O.N. S ß ara).
0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 1 8 5 - 1
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Nomenclature A A C D De f H h kair L Lt Nf Nu NuD Pr Q_ Re ReD Sx T U Umax W DP q m
cross-section area of duct, m2 heat transfer area, m2 clearance between upper surface of fin and ceiling of duct, m thickness of fins, m hydraulic diameter of channel, m friction factor height of fins, m average convective heat transfer coefficient, W/m2 K thermal conductivity of air, W/m K length of test surface, m length of test section, m number of pin fins average duct Nusselt number average pin Nusselt number Prandtl number heat transfer rate, W duct Reynolds number pin Reynolds number distance between two fins in x direction, m mean temperature, K mean bulk flow velocity, m/s maximum flow velocity, m/s width of base plate, m pressure drop, N/m2 air density, kg/m3 kinematic viscosity of air, m2 /s
Subscripts in conditions at inlet out conditions at outlet s smooth w conditions at surface
One of the commonly used heat exchanger fins is the pin fin. A pin fin is a cylinder or other shaped element attached perpendicular to a wall, with the transfer fluid passing in crossflow over the element. There are various parameters that characterize the pin fins, such as shape, height, diameter, height to diameter ratio etc. In addition, the pin fins may be positioned in arrays that are either staggered or inline with respect to the flow direction. The heat transfer and friction characteristics of pin fin array systems have been the subject of extensive investigation because of its importance in a wide variety of engineering applications,
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such as compact heat exchangers and the cooling of advanced gas turbine blades and electronic devices. Pin fins having a pin height to diameter ratio, H =D, between 0.5 and 4 are accepted as short pin fins, whereas long pin fins have pin height to diameter ratio, H =D, greater than 4. The large height to diameter ratio is of particular interest in heat exchanger applications in which the attainment of very high heat transfer coefficients is of major concern. Arrays of pin fins with low/ intermediate height to diameter ratio (from 0.5 to 4), short pin fins, are commonly used in many industrial applications, especially in the trailing edges of gas turbine blades, in some modern electronic systems and in the aerospace industry [1]. There have been many investigations of the heat transfer and pressure drop of channels with pin fins, which are restricted to pin fins with circular cross-section [2–14]. Sparrow and coworkers [2,3] were among the first to investigate the heat transfer performance of inline and staggered wall attached arrays of cylindrical fins. Metzger et al. [4] investigated the heat transfer characteristics of staggered arrays of cylindrical pin fins. Simoneau and Vanfossen [5] also studied the heat transfer from a staggered array of cylindrical pin fins. Matsumoto et al. [13] studied the end wall heat transfer in the presence of inline and staggered adiabatic circular pin fins. A review of staggered array pin fin heat transfer for turbine cooling applications was presented by Armstrong and Winstanley [7]. While the studies regarding circular pin fin arrays are abundant, the research on pin fins with other cross-sections is relatively sparse. Grannis and Sparrow [15,16] investigated the heat transfer and pressure drop characteristics and numerically simulated the fluid flow through an array of diamond shaped pin fins. Tanda [1] performed an investigation of the heat transfer and pressure drop for a rectangular channel equipped with arrays of diamond shaped elements. Both inline and staggered fin arrays were considered in the thermal performance analysis under constant mass flow rate and constant pumping power constraints. The heat transfer performance of arrays of cubic and diamond shaped fins inside a rectangular channel was reported by Chyu et al. [17]. AlJamal and Khashashneh [18] performed a study on convective heat transfer of staggered pin fin and triangular fin arrays. Goldstein et al. [19] conducted an investigation to determine the effect of fin shape on the mass transfer and pressure loss of a staggered short pin fin array in a rectangular duct. S ß ara et al. [21] investigated the heat transfer and friction characteristics of rectangular channels with inline square pin fins. The literature survey on the investigations of pin fin array systems indicates that these studies have examined the heat transfer and friction characteristics and various parameters, such as interfin spacing in both streamwise direction and spanwise direction, gap clearance ratio, height to diameter ratios etc. on the basis of maximum heat transfer rate per unit base area. It is well known that the pin fin arrays produce higher heat transfer than plain channels without fins. However, the increase in heat transfer is always accompanied by a substantial increase in pressure loss. Therefore, in most applications of pin fins, both the heat transfer and pressure loss characteristics must be considered. Although there are some pin fin investigations in which the performance analysis is made by using a performance evaluation criterion (PEC) [1,3,19,21 etc.], in general, in these investigations, the heat transfer and friction characteristics have been obtained and the optimal parameters generally determined on the basis of maximum heat transfer rate or maximum heat transfer per unit base area [11,20 etc.]. Therefore, it is necessary to perform a performance analysis by the PEC and to state performance in terms of at least four interrelated characteristics: heat transfer, fluid pumping power, size and shape. On the other hand, the pin fins with various
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cross-sections have different heat transfer and flow resistance characteristics, and it is by no means clear that circular pin fins are the best for heat transfer, friction and performance. Therefore, it is essential to investigate various pin fins with different cross-sections in order to enhance the heat transfer and decrease the flow resistance. It is the aim of this study to investigate the heat transfer, pressure and performance characteristics for the staggered square pin fins array attached on a flat surface in a rectangular duct and to compare with those for the inline arrangement. 2. Experimental set-up Fig. 1(a) shows the channel flow experimental rig, which consisted of a closed rectangular channel with a removable test section (7), two fans (6 and 9), a data acquisition system (11), an inclined manometer (12), an anemometer/thermometer (2) and a number of thermocouples (8). The channel, constructed of wood of 20 mm thickness, had an internal cross-section of 160 mm width and 80 mm height and a total length of 2000 mm. A flow straightener (1) and a flow mixer (3) were fitted immediately after the inlet of the channel and 100 mm downstream of the heated surface prior to the mean outlet temperature measurement (4), respectively. The test section (7), located 900 mm downstream of the flow straightener and made of smooth aluminum plate of 2 mm thickness, 140 mm width, and 320 mm length, was heated using an electrical heater plate attached immediately underneath the plate. To minimize the heat loss, the outer surface of the heater was insulated by a combination of a 12 mm thick isolator layer and a 20 mm thick wood layer. Eight thermocouples of K-type, equally spaced along the base plate between the fins, were fitted on the central axis line of the plate, flush with its outer surface, for measurement of the local surface temperatures. Other thermocouples were used to measure the temperatures of the inlet and outlet of the air and of the outer surface of the bottom wall of the test section. The pressure drop across the test section was determined using pressure taps and an inclined glass tube kerosene manometer (12). The mean air velocity over the test surface (bulk mean velocity, U ) was determined by averaging the local measurements across the channel cross-section using an electronic thermal anemometer (2). The pin fins, made of the same aluminum material as the base plate, had a square cross-section of 10 mm by 10 mm and were attached on the upper surface of the base plate as shown in Fig. 1(b). Square pin fins with different lengths, corresponding to C=H values of 0, 0.6 and 1, were used. The pin fins were fixed uniformly on the base plate with a constant spacing between the spanwise directions of 2.25 mm, with different spacing between the pin fins in the streamwise direction. The spacings between the pin fins in the streamwise direction Sx were 1.58, 4.17 and 9.33 mm, giving different numbers of the pin fins (Fig. 1b). The Reynolds number range used in this experiment was 10,000–34,000, which was based on the hydraulic diameter of the channel over the test section ðDh Þ and the average velocity (U ). The experimental conditions used in the work are given in Table 1 collectively. For more detailed information about the experimental rig, see Refs. [23,24]. 3. Data reduction The convective heat transfer rate from the electrically heated test surface Q_ conv is calculated by using:
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Fig. 1. Experimental set-up.
Q_ conv ¼ Q_ elect Q_ cond Q_ rad
ð1Þ
where Q_ indicates the heat transfer rate and subscripts conv, elect, cond and rad denote convection, electrical, conduction and radiation, respectively. The electrical heat input is calculated
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Table 1 Experimental conditions used in the present investigation Parameter
Values
Pin fin spacing ratio in streamwise direction, Sx =D Streamwise fin number, Nfx Pin fin spacing ratio in spanwise direction, Sz =D Spanwise fin number, Nfz Clearance ratio, C=H Reynolds number, Re
1.58, 4.17, 9.33 13, 7, 4 2.25 5–4 0.0, 0.6, 1.0 10,000–34,000
from the electrical potential and current supplied to the surface. In similar studies, Naik et al. [25] and Hwang and Liou [26] reported that the total radiative heat losses from a similar test surface would be about 0.5% of the total electrical heat input. Therefore, the radiative heat loss can be neglected. The conductive heat losses from the sidewalls can be neglected in comparison to that from the bottom surface of the test section. Moreover, as mentioned above, the bottom surface of the test plate, which is not exposed to the flow, has been insulated by a combination of a 12 mm thick isolator layer and a 20 mm thick wood layer. Thus, it is thought that the conductive heat loss from the heated plate will be mainly from its back surface into the environment through the bottom wall of the channel. This was evaluated at steady state conditions by using the experimentally measured mean temperature of the outer surface of the bottom wall of the test section and a well established correlation equation suggested for natural convection for geometrically similar systems [27]. The average convective heat transfer coefficient is calculated by using [25]: h¼
Q_ conv A½T w ððT out þ T in Þ=2Þ
ð2Þ
where Q_ conv is the steady state convection heat transfer rate; T w , the average surface temperature; T out and T in , the mean temperatures of the flow at the outlet and the inlet, respectively; and A, the surface area. Q_ conv is calculated using Eq. (1). Either the projected area ðAP Þ or the total area ðAT Þ of the test surface can be taken as the heat transfer surface area in the calculations. The total area is equal to the sum of the projected area and the surface area contribution from the pin fins. Therefore, the projected and the total area can be related to each other by: AT ¼ WL þ Nf ½4ðHDÞ þ D2
ð3Þ
Two types of Reynolds number were used to characterize the flow condition. One is a Reynolds number based on the mean velocity (U ) in the smooth duct and the hydraulic diameter of the duct ðDe Þ and is expressed as Re ¼
qDe U l
ð4Þ
The other one is based on the maximum velocity through the pin fins and the thickness of the pin fins, i.e.
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qDUmax l where Umax is the maximum velocity through the pin fins and is given by A Umax ¼ U A Afront ReD ¼
1793
ð5Þ
ð6Þ
where A is the cross-sectional area of the duct and Afront , the frontal area of the fins. In this experimental setup, the values of A=ðA Afront Þ are found to be 1.1852, 1.2427 and 1.4545 for the values of C=H ¼ 1, 0.6 and 0, respectively. ReD has been widely used in many pin fin heat transfer studies [19 etc.], and it depends on the spanwise pin number and the height of the pin fins. In the following discussion, Re and ReD will be referred to as duct Reynolds number and pin Reynolds number, respectively. Similar to the Reynolds number definition, the average Nusselt number is represented by duct Nusselt number and pin Nusselt number, respectively, as: Nu ¼ hDe =k
ð7Þ
NuD ¼ hD=k
ð8Þ
where k is the thermal conductivity of air and h, the average convective heat transfer coefficient. While calculating Nu and NuD , h is based on the projected area and the total heat transfer area, respectively. An uncertainty analysis was performed using the method described by Holman [30]; the experimental uncertainty in Nusselt number was calculated to be 9.5%. For more detailed information about the data reduction, see Refs. [23,24]. 4. Results and discussion 4.1. Heat transfer The average Nusselt number for the staggered arrangement was correlated as a function of Reynolds number, Prandtl number, interfin spacing ratio and clearance ratio, and the following expression was obtained: Nu ¼ 2:8358 Re0:58 ð1 þ C=H Þ0:848 ðSx =DÞ0:251 Pr1=3
ð9Þ
which is valid for 10; 000 6 Re 6 34; 000, 1:58 6 Sx =D 6 9:33, 0 6 C=H 6 1 with correlation coefficient of r ¼ 0:9671. Fig. 2 shows the duct Nusselt number, Nu, as a function of the duct Reynolds number, Re, for the different pin heights, namely C=H ¼ 1, 0.6 and 0 at Sx =D ¼ 1:58. Using the same experimental system, the average Nusselt number for the smooth surface (without pin fins) ðNus Þ was found by S ß ara et al. as follows [21]: Nus ¼ 0:0919 Re0:706 Pr1=3
ð10Þ
The plot of the results of the smooth rectangular duct predicted from Eq. (10) is also included in Fig. 2. It can be easily seen that the use of pin fins in the duct causes a substantial increase in the Nu. It also increases with increasing Re and decreasing C=H . As Nu is based on the projected area, it reflects both the variation in the surface area and the disturbances in the flow due to the pin fins.
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Fig. 2. Duct Nusselt number as a function of the duct Reynolds number and clearance ratio at Sx =D ¼ 1:58.
Decreasing C=H means that the height of the fins increases. Therefore, decreasing the height of the fins leads to a decrease in Nu. This is because the surface area decreases with decreasing height of the fins, resulting in a decrease in Nu. Longer fins can also increase the turbulence of the flow in the channel, resulting in an increase in heat transfer. Fig. 3 shows the behaviour of the Nu as a function of the duct Reynolds number and interfin distance ratios for a constant clearance ratio of 1.0. As seen in this figure, Nu increases with decreasing Sx =D, which can be attributed to increasing the fin number. Since the number of fins increases with decreasing Sx =D, which also means an increase in the total heat transfer area, the heat transfer rate increases. Similar results were found for inline pin configurations [21]. Namely, Nu was found to increase with decreasing C=H and decreasing Sx =D. A comparison of Nu for the staggered and inline pin fin arrangements with the same number of pins is given in Fig. 4. It is evident from this figure that Nu is higher for the staggered pin fins arrangement than that for an inline arrangement for all values of C=H, especially at lower Reynolds number. An examination of Fig. 5, which is a plot of Nu as a function of the interfin distance ratio, reveals that similar results were found for the duct Nusselt number. A similar observation is reported in the literature [17]. As mentioned previously, the pin Nusselt number NuD as calculated by Eq. (8) has been widely used in the literature. NuD was correlated with the pin Reynolds number, ReD , in the form of the following equation: NuD ¼ m Ren
ð11Þ
The values of m and n change with Sx =D and C=H, and the changes, with correlation coefficient r, are summarized in Table 2. The values of m for the short pin fins are lower than those of the long pin fins. Brigham and VanFossen [28] studied the effect of length to diameter ratio (H=D) on short
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Fig. 3. Duct Nusselt number as a function of the duct Reynolds number and interfin spacing ratio at C=H ¼ 1:0.
Fig. 4. Comparison of staggered and inline arrangements.
pin fin heat transfer and reported that H =D was the dominant factor in pin fin heat transfer for H =D > 2, and for values of H=D < 2, the Nusselt number was independent of H =D. Armstrong and Winstanley [7] reported that the Reynolds exponent for the short pin fins ðH =D < 4Þ varied between 0.59 and 0.65 when correlated row by row, while for long pin fins ðH =D ¼ 7:72Þ, it was approximately 0.5.
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Fig. 5. Comparison of staggered and inline arrangements. Table 2 Constants of Eq. (11) C=H
Sx =D
m
n
r
1.0
1.58 4.17 9.33
0.201 0.364 0.168
0.547 0.473 0.569
0.994 0.987 0.998
0.6
1.58 4.17 9.33
0.723 0.642 0.182
0.415 0.410 0.575
0.956 0.921 0.987
0.0
1.58 4.17 9.33
0.524 0.782 0.4137
0.404 0.372 0.449
0.968 0.949 0.941
Fig. 6 shows the total heat transfer area-averaged pin Nusselt number, NuD , as a function of the pin Reynolds number, ReD . The results of the present work lie quite close to those of some works from the literature. Some previous correlations for array-averaged Nusselt number in the pin fin duct are also plotted in this figure and listed in Table 3, which also includes geometrical and flow conditions, for comparison. 4.2. Friction factor The experimental pressure drops over the test section in the duct were measured under heated flow conditions and were arranged in nondimensional form by using the following equation: f ¼
DP ðLt =De ÞqU 2 =2
ð12Þ
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Fig. 6. Pin Nusselt number as a function of pin Reynolds number and comparison with the results of other investigators.
Table 3 The previous correlations for pin fins No.
Equation 0:728
1
NuD ¼ 0:069 Re
2
NuD ¼ 0:414 Re0:511
3
NuD ¼ 0:145 Re0:646
4
NuD ¼ 7:04 103 Re0:953 ðSx =Wb Þ0:091 ðSy =LÞ0:053
5
NuD ¼ 0:3313ðRe PrÞ0:646
6
NuD ¼ 0:344ðRe PrÞ0:555
Conditions
Reference
Circular staggered pin fin, Sx =D ¼ 2:5, Sz =D ¼ 2:5, 1000 < Re < 2 105 , H =D ¼ 1, C=H ¼ 0, Rectangular duct Circular staggered pin fin, Sx =D ¼ Sz =D ¼ 2:5, 5000 < Re < 3 104 , H =D ¼ 1, C=H ¼ 0, Rectangular duct Circular staggered pin fin, Sx =D ¼ 2:5, Sz =D ¼ 2:5, 104 < Re < 5 104 , H =D ¼ 1, C=H ¼ 0, Wedge duct Circular staggered pin fin, 0:004 < Sx =Wb < 0:332, 0:033 < Sy =L < 0:152, 3:138 103 < Re < 4:98 103 , Rectangular duct Circular staggered pin fin, Sx =D ¼ 2:33, Sz =D ¼ 3, 100 < Re < 1100, H =D ¼ 5:5, C=H ¼ 1, Rectangular duct, triangular fin
[4]
[9]
[29]
[14]
[18]
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An uncertainty analysis, using the same method as in the case of Nusselt number, indicated that the uncertainty in experimental f measurements could be up to 18.8%. Using the experimental results, f was correlated as a function of the duct Reynolds number, Re, and geometrical parameters, and the resulting equation is: f ¼ 11:944 Re0:101 ð1 þ C=H Þ2:05 ðSx =DÞ0:582
ð13Þ
which is valid for 10; 000 6 Re 6 34; 000, 1:58 6 Sx =D 6 9:33, 0 6 C=H 6 1 with a correlation coefficient of r ¼ 0:9504. Fig. 7 depicts the friction factor for the staggered pin fins ducts as a function of the duct Reynolds number, Re, for Sx =D ¼ 1:58 and 4.17 at C=H ¼ 1. The results for the inline pin fin arrangement [21] and the smooth duct from the Blasisus equation are also included for comparison. It can be seen from Fig. 7 that the friction factor for the duct with staggered pin fins arrays is higher than that for inline pin fin arrays and also substantially higher than that for the smooth duct. This can be attributed to the fact that the staggered arrangement avoids the working fluid easily flowing between the fins as it flows in the inline arrangement, blockading the channel like passage of the fluid between the fins. The other notable result is that the effect of the streamwise pin row number, namely Sx =D, on the friction factor is higher for the staggered pin fins arrays than that for the inline arrangements. On the other hand, the pressure drop over the pin fins, generally, has been correlated in the following manner [4,7]: f ¼
DP 2 N 2qUmax
ð14Þ
where N is the number of rows of pins in the streamwise direction. Limited attention has been focused on the problem of hydraulic resistance in a pin fin bank. A literature review, which in-
Fig. 7. Duct friction factor as a function of Reynolds number and comparison with the results of inline arrangement.
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cludes some friction correlations, was reported by Armstrong and Winstanley [7]. In order to compare the friction factor with these correlations, the values of f , which are calculated from Eq. (12), were transferred to the fD by using the definition of Eq. (14). The results were plotted as a function of the pin Reynolds number, ReD , in Fig. 8. Some previous correlations for the pin friction factor were also plotted in this figure and listed in Table 4.
4.3. Performance analysis The appraisal of the possible benefits of using augmented heat transfer surfaces depends on the goal(s) to be fulfilled and the constraints being imposed, and several number of PEC have been
Fig. 8. Pin friction factor as a function of pin Reynolds number and comparison with the results of other investigators.
Table 4 The previous correlation of friction factor for pin fins No.
Equations
1 2 3 4
fD fD fD fD
¼ ½0:25 þ ð0:1175Þ=ðSz =D 1Þ1:08 Re0:16 D ¼ ½2:06ðSz =DÞ1:1 Re0:16 D ¼ 0:317 Re0:132 , 103 < ReD < 104 D 0:318 ¼ 1:76 ReD , 104 < ReD < 105
Reference Armstrong and Winstanley [7] Metzger et al. (1982) [4] Metzger et al. (1982) [4]
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developed. These PEC are classified into two general groups [22]: those based on the first law analysis and those based on the second law analysis. One of the PEC is to compare the heat transfer coefficient for constant pumping power for the channel with finned surface with that for the smooth surface. The pumping power is proportional to f Re3 , and the relationship between the finned and smooth duct for the same pumping power is expressed by [26]: f Re3 ¼ fs Re3s
ð15Þ
where f and Re are the values for the finned duct, and fs and Res are those for the smooth duct. It has been widely accepted in the literature that the friction factors for both the smooth and finned ducts are related to the Reynolds number by the following equation: f ¼ a Reb
ð16Þ
According to the constant pumping power constraint, the mass flow rates (or Reynolds number) passing through the finned and reference smooth channel cannot be the same, since the mass flow rate for the smooth channel must be increased to keep the fluid pumping power constant. Using Eqs. (15) and (16), the Reynolds number for the smooth duct, Res , can be expressed as: 1=ðbs þ3Þ a Reð3þbÞ=ð3þbs Þ ð17Þ Res ¼ as Using Eq. (13) and the following Blasius equation fs ¼ 0:316 Re0:25
ð18Þ
the constants a, as , b, and bs were found to be, respectively: a ¼ 11:944ð1 þ C=H Þ2:05 ðSx =DÞ0:582
as ¼ 0:316 bs ¼ 0:25
b ¼ 0:101
Inserting these constants into Eq. (17) yields the Reynolds number for the smooth channel at which the pumping power is the same as that occurring in the finned channel as: 0:746
Res ¼ 3:751ð1 þ C=H Þ
0:212
ðSx =DÞ
Re1:054
ð19Þ
In the present study, the following PEC, called the heat transfer enhancement factor, Nu , was used to appraise the possible benefits of using pin fins: ! Nu
ð20Þ Nu ¼
Nus equal pumping power
where Nus is the average Nusselt number for the smooth channel with Res at which the pumping
power is the same as that occurring in the finned channel. Nus was calculated from Eq. (10) by using the values of Res , which was calculated by using Eq. (19). In graphical representation, the
ratios of Nu=Nus are plotted against Res . Nu was plotted as a function of equivalent Reynolds number, Res , in Figs. (9) and (10) for the staggered configurations. Fig. 9 shows the effect of the interfin distance ratio on Nu , whereas Fig. 10 shows the effect of the clearance ratio on Nu . These
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Fig. 9. Heat transfer enhancement factor as a function of equivalent Reynolds number and interfin spacing ratio.
Fig. 10. Heat transfer enhancement factor as a function of equivalent Reynolds number and clearance ratio.
figures show that the heat transfer enhancement factor, Nu , decreases with increasing Res , Sx =D and C=H . In other words: (1) The heat transfer enhancement factors, Nu , are higher than unity for all investigated conditions. This means that the use of pin fins leads to an advantage on the basis of heat transfer enhancement.
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(2) At lower Reynolds numbers, the channels with pin fin arrays give higher performance than those at higher Reynolds numbers. (3) For higher thermal performance, a lower interfin distance ratio and a lower clearance ratio should be preferred. In other words, the higher number of pin fins and longer pin fins have better performance.
5. Conclusion The enhancement of the heat transfer from a flat surface in a rectangular channel flow by the attachment of staggered square cross-sectional pin fins was investigated experimentally. The effects of the flow and geometrical parameters on the heat transfer and friction characteristics were determined, and the constant pumping power criterion was used to evaluate the performance of the staggered pin fin array systems. The conclusions are summarized as: (1) The average Nusselt number increased with decreasing clearance ratio and interfin distance ratio. (2) The friction factor increased with decreasing clearance ratio and interfin distance ratio. (3) Nu increased with decreasing C=H , which means that longer pin fins have better performance. (4) Nu increased with decreasing Sx =D, which means that higher number of pin fins have better performance. (5) At lower Reynolds numbers, the channel with pin fin arrays gives higher performance than those at higher Reynolds numbers. (6) The heat transfer enhancement factors, Nu , are higher than unity for all investigated conditions. This means that the use of pin fins leads to an advantage on the basis of heat transfer enhancement.
References [1] Tanda G. Heat transfer and pressure drop in a rectangular channel with diamond-shaped elements. Int J Heat Mass Transfer 2001;44:3529–41. [2] Sparrow EM, Ramsey JW. Heat transfer and pressure drop for a staggered wall-attached array of cylinders with tip clearance. Int J Heat Mass Transfer 1978;21:1369–77. [3] Sparrow EM, Ramsey JW, Altemani CAC. Experiments on in-line pin fin arrays and performance comparison with staggered arrays. ASME J Heat Transfer 1980;102:44–50. [4] Metzger DE, Berry RA, Bronson JP. Developing heat transfer in rectangular ducts with staggered arrays of short pin fins. ASME J Heat Transfer 1982;104:700–6. [5] Simoneau RJ, Vanfossen GJ. Effect of location in an array on heat transfer to a short cylinder in crossflow. ASME J Heat Transfer 1984;106:42–8. [6] Sparrow EM, Samie F. Heat transfer and pressure drop results for one-and two-row arrays of finned tubes. Int J Heat Mass Transfer 1985;28:2247–59. [7] Armstrong J, Winstanley D. A review of staggered array pin fin heat transfer for turbine cooling applications. ASME J Turbomach 1988;110:94–103. [8] Lau SC, Han JC, Kim YS. Turbulent heat transfer and friction in pin fin channels with lateral flow ejection. ASME J Heat Transfer 1998;111:51–8.
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[9] Chyu MK. Heat transfer and pressure drop for short pin-fin arrays with pin-endwall fillet. ASME J Heat Transfer 1990;112:926–32. [10] Kumaran TK, Han JC, Lau SC. Augmented heat transfer in a pin fin channel with short or long ejection holes. Int J Heat Mass Transfer 1991;34:2617–28. [11] Jubran BA, Hamdan MA, Abdullah RM. Enhanced heat transfer, missing pin, and optimization for cylindrical pin fin arrays. ASME J Heat Transfer 1993;115:576–83. [12] BabusÕhaq RF, Akintunde K, Probert SD. Thermal performance of a pin-fin assembly. Int J Heat Fluid Flow 1995;16:50–5. [13] Matsumoto R, Kikkawa S, Senda M. Effect of pin fin arrangement on endwall heat transfer. JSME Int J 1997;40(Series B):142–51. [14] Tahat M, Kodah ZH, Jarrah BA, Probert SD. Heat transfer from pin-fin arrays experiencing forced convection. Appl Energy 2000;67(4):419–42. [15] Sparrow EM, Grannis VB. Pressure drop characteristics of heat exchangers consisting of arrays of diamond-shaped pin fins. Int J Heat Mass Transfer 1991;34:589–600. [16] Grannis VB, Sparrow EM. Numerical simulation of fluid flow through an array of diamond-shaped pin fins. Numer Heat Transfer 1991;19(Part A):381–403. [17] Chyu MK, Hsing YC, Natarajan V. Convective heat transfer of cubic fin arrays in a narrow channel. ASME J Turbomach 1998;120:362–7. [18] Al-Jamal K, Khashashneh H. Experimental investigation in heat transfer of triangular and pin fin arrays. Heat Mass transfer 1998;34:159–62. [19] Goldstein RJ, Jabbari MY, Chen SB. Convective mass transfer and pressure loss characteristics of staggered short pin-fin arrays. Int J Heat Mass Transfer 1994;37(Suppl 1):149–60. [20] Li Q, Chen Z, Fletchner U, Warnecke HJ. Heat transfer and pressure drop characteristics in rectangular channels with elliptic pin fins. Int J Heat Fluid Flow 1998;19:245–50. [21] S ß ara ON, Yapıcı S, Yılmaz M, Pekdemir T. Second law analysis of rectangular channels with square pin-fins. Int J Commun Heat Mass Transfer 2001;28(5):617–30. [22] Yilmaz M, S ß ara ON, Karsli S. Performance evaluation criteria for heat exchangers based on second law analysis. Exergy Int J 2001;1(4):278–94. [23] S ß ara ON, Pekdemir T, Yapıcı S, Yılmaz M. Enhancement of heat transfer from a flat surface in a channel flow by attachment of rectangular blocks. Int J Energy Res 2001;25(7):563–76. [24] S ß ara ON, Pekdemir T, Yapici S, Yilmaz M. Heat transfer enhancement in a channel flow with perforated rectangular blocks. Int J Heat Fluid Flow 2001;22:509–18. [25] Naik S, Probert SD, Shilston MJ. Forced convective steady-state heat transfers from shrouded vertical fin arrays aligned parallel to an undisturbed air-stream. Appl Energy 1987;26:137–58. [26] Hwang JJ, Liou TM. Heat transfer and friction in a low-aspect-ratio rectangular channel with staggered perforated ribs on two opposite walls. ASME J Heat Transfer 1995;117(11):843–50. [27] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. New York: John Wiley & Sons; 1996. p. 496. [28] Brigham BA, VanFossen GJ. Length-to-diameter ratio and row number effects in short pin fin heat transfer. J Eng Gas Turbines Power 1984;106:241–5. [29] Hwang JJ, Lui CC. Measurement of endwall heat transfer and pressure drop in a pin-fin wedge duct. Int J Heat Mass Transfer 2002;45:877–89. [30] Holman JP. Experimental methods for engineers. 5th ed. New York: McGraw-Hill; 1989. p. 41–9.
Performance analysis of rectangular ducts with staggered square pin fins O.N. S ß ara
*
Faculty of Engineering, Department of Chemical Engineering, Atat€ urk University, 25240 Erzurum, Turkey Received 26 April 2002; accepted 3 August 2002
Abstract This paper presents the heat transfer and friction characteristics and performance analysis of convective heat transfer through a rectangular channel with square cross-section pin fins attached over a flat surface. The pin fins were arranged in a staggered manner. Various clearance ratios (C=H ) and interfin distance ratios ðSx =DÞ were used. The performance analysis was made under a constant pumping power constraint. The experimental results showed that the use of square cross-section pin fins may lead to an advantage on the basis of heat transfer enhancement. For higher thermal performance, lower interfin distance ratio and clearance ratio and comparatively lower Reynolds numbers should be preferred for the staggered arrangement. The results of the staggered configurations were also compared with the results of the inline arrangement. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Pin fin; Heat transfer enhancement; Channel flow; Forced convection; Heat exchangers; Performance analysis
1. Introduction Extended surfaces (fins) are frequently used in heat exchanging devices for the purpose of increasing the heat transfer between a primary surface and the surrounding fluid. Various types of heat exchanger fins, ranging from relatively simple shapes, such as rectangular, cylindrical, annular, tapered or pin fins, to a combination of different geometry, have been used. These fins may protrude from either a rectangular or cylindrical base.
*
Tel.: +90-442-231-4556; fax: +90-442-236-0957. E-mail address: [email protected] (O.N. S ß ara).
0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 1 8 5 - 1
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Nomenclature A A C D De f H h kair L Lt Nf Nu NuD Pr Q_ Re ReD Sx T U Umax W DP q m
cross-section area of duct, m2 heat transfer area, m2 clearance between upper surface of fin and ceiling of duct, m thickness of fins, m hydraulic diameter of channel, m friction factor height of fins, m average convective heat transfer coefficient, W/m2 K thermal conductivity of air, W/m K length of test surface, m length of test section, m number of pin fins average duct Nusselt number average pin Nusselt number Prandtl number heat transfer rate, W duct Reynolds number pin Reynolds number distance between two fins in x direction, m mean temperature, K mean bulk flow velocity, m/s maximum flow velocity, m/s width of base plate, m pressure drop, N/m2 air density, kg/m3 kinematic viscosity of air, m2 /s
Subscripts in conditions at inlet out conditions at outlet s smooth w conditions at surface
One of the commonly used heat exchanger fins is the pin fin. A pin fin is a cylinder or other shaped element attached perpendicular to a wall, with the transfer fluid passing in crossflow over the element. There are various parameters that characterize the pin fins, such as shape, height, diameter, height to diameter ratio etc. In addition, the pin fins may be positioned in arrays that are either staggered or inline with respect to the flow direction. The heat transfer and friction characteristics of pin fin array systems have been the subject of extensive investigation because of its importance in a wide variety of engineering applications,
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such as compact heat exchangers and the cooling of advanced gas turbine blades and electronic devices. Pin fins having a pin height to diameter ratio, H =D, between 0.5 and 4 are accepted as short pin fins, whereas long pin fins have pin height to diameter ratio, H =D, greater than 4. The large height to diameter ratio is of particular interest in heat exchanger applications in which the attainment of very high heat transfer coefficients is of major concern. Arrays of pin fins with low/ intermediate height to diameter ratio (from 0.5 to 4), short pin fins, are commonly used in many industrial applications, especially in the trailing edges of gas turbine blades, in some modern electronic systems and in the aerospace industry [1]. There have been many investigations of the heat transfer and pressure drop of channels with pin fins, which are restricted to pin fins with circular cross-section [2–14]. Sparrow and coworkers [2,3] were among the first to investigate the heat transfer performance of inline and staggered wall attached arrays of cylindrical fins. Metzger et al. [4] investigated the heat transfer characteristics of staggered arrays of cylindrical pin fins. Simoneau and Vanfossen [5] also studied the heat transfer from a staggered array of cylindrical pin fins. Matsumoto et al. [13] studied the end wall heat transfer in the presence of inline and staggered adiabatic circular pin fins. A review of staggered array pin fin heat transfer for turbine cooling applications was presented by Armstrong and Winstanley [7]. While the studies regarding circular pin fin arrays are abundant, the research on pin fins with other cross-sections is relatively sparse. Grannis and Sparrow [15,16] investigated the heat transfer and pressure drop characteristics and numerically simulated the fluid flow through an array of diamond shaped pin fins. Tanda [1] performed an investigation of the heat transfer and pressure drop for a rectangular channel equipped with arrays of diamond shaped elements. Both inline and staggered fin arrays were considered in the thermal performance analysis under constant mass flow rate and constant pumping power constraints. The heat transfer performance of arrays of cubic and diamond shaped fins inside a rectangular channel was reported by Chyu et al. [17]. AlJamal and Khashashneh [18] performed a study on convective heat transfer of staggered pin fin and triangular fin arrays. Goldstein et al. [19] conducted an investigation to determine the effect of fin shape on the mass transfer and pressure loss of a staggered short pin fin array in a rectangular duct. S ß ara et al. [21] investigated the heat transfer and friction characteristics of rectangular channels with inline square pin fins. The literature survey on the investigations of pin fin array systems indicates that these studies have examined the heat transfer and friction characteristics and various parameters, such as interfin spacing in both streamwise direction and spanwise direction, gap clearance ratio, height to diameter ratios etc. on the basis of maximum heat transfer rate per unit base area. It is well known that the pin fin arrays produce higher heat transfer than plain channels without fins. However, the increase in heat transfer is always accompanied by a substantial increase in pressure loss. Therefore, in most applications of pin fins, both the heat transfer and pressure loss characteristics must be considered. Although there are some pin fin investigations in which the performance analysis is made by using a performance evaluation criterion (PEC) [1,3,19,21 etc.], in general, in these investigations, the heat transfer and friction characteristics have been obtained and the optimal parameters generally determined on the basis of maximum heat transfer rate or maximum heat transfer per unit base area [11,20 etc.]. Therefore, it is necessary to perform a performance analysis by the PEC and to state performance in terms of at least four interrelated characteristics: heat transfer, fluid pumping power, size and shape. On the other hand, the pin fins with various
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cross-sections have different heat transfer and flow resistance characteristics, and it is by no means clear that circular pin fins are the best for heat transfer, friction and performance. Therefore, it is essential to investigate various pin fins with different cross-sections in order to enhance the heat transfer and decrease the flow resistance. It is the aim of this study to investigate the heat transfer, pressure and performance characteristics for the staggered square pin fins array attached on a flat surface in a rectangular duct and to compare with those for the inline arrangement. 2. Experimental set-up Fig. 1(a) shows the channel flow experimental rig, which consisted of a closed rectangular channel with a removable test section (7), two fans (6 and 9), a data acquisition system (11), an inclined manometer (12), an anemometer/thermometer (2) and a number of thermocouples (8). The channel, constructed of wood of 20 mm thickness, had an internal cross-section of 160 mm width and 80 mm height and a total length of 2000 mm. A flow straightener (1) and a flow mixer (3) were fitted immediately after the inlet of the channel and 100 mm downstream of the heated surface prior to the mean outlet temperature measurement (4), respectively. The test section (7), located 900 mm downstream of the flow straightener and made of smooth aluminum plate of 2 mm thickness, 140 mm width, and 320 mm length, was heated using an electrical heater plate attached immediately underneath the plate. To minimize the heat loss, the outer surface of the heater was insulated by a combination of a 12 mm thick isolator layer and a 20 mm thick wood layer. Eight thermocouples of K-type, equally spaced along the base plate between the fins, were fitted on the central axis line of the plate, flush with its outer surface, for measurement of the local surface temperatures. Other thermocouples were used to measure the temperatures of the inlet and outlet of the air and of the outer surface of the bottom wall of the test section. The pressure drop across the test section was determined using pressure taps and an inclined glass tube kerosene manometer (12). The mean air velocity over the test surface (bulk mean velocity, U ) was determined by averaging the local measurements across the channel cross-section using an electronic thermal anemometer (2). The pin fins, made of the same aluminum material as the base plate, had a square cross-section of 10 mm by 10 mm and were attached on the upper surface of the base plate as shown in Fig. 1(b). Square pin fins with different lengths, corresponding to C=H values of 0, 0.6 and 1, were used. The pin fins were fixed uniformly on the base plate with a constant spacing between the spanwise directions of 2.25 mm, with different spacing between the pin fins in the streamwise direction. The spacings between the pin fins in the streamwise direction Sx were 1.58, 4.17 and 9.33 mm, giving different numbers of the pin fins (Fig. 1b). The Reynolds number range used in this experiment was 10,000–34,000, which was based on the hydraulic diameter of the channel over the test section ðDh Þ and the average velocity (U ). The experimental conditions used in the work are given in Table 1 collectively. For more detailed information about the experimental rig, see Refs. [23,24]. 3. Data reduction The convective heat transfer rate from the electrically heated test surface Q_ conv is calculated by using:
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Fig. 1. Experimental set-up.
Q_ conv ¼ Q_ elect Q_ cond Q_ rad
ð1Þ
where Q_ indicates the heat transfer rate and subscripts conv, elect, cond and rad denote convection, electrical, conduction and radiation, respectively. The electrical heat input is calculated
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Table 1 Experimental conditions used in the present investigation Parameter
Values
Pin fin spacing ratio in streamwise direction, Sx =D Streamwise fin number, Nfx Pin fin spacing ratio in spanwise direction, Sz =D Spanwise fin number, Nfz Clearance ratio, C=H Reynolds number, Re
1.58, 4.17, 9.33 13, 7, 4 2.25 5–4 0.0, 0.6, 1.0 10,000–34,000
from the electrical potential and current supplied to the surface. In similar studies, Naik et al. [25] and Hwang and Liou [26] reported that the total radiative heat losses from a similar test surface would be about 0.5% of the total electrical heat input. Therefore, the radiative heat loss can be neglected. The conductive heat losses from the sidewalls can be neglected in comparison to that from the bottom surface of the test section. Moreover, as mentioned above, the bottom surface of the test plate, which is not exposed to the flow, has been insulated by a combination of a 12 mm thick isolator layer and a 20 mm thick wood layer. Thus, it is thought that the conductive heat loss from the heated plate will be mainly from its back surface into the environment through the bottom wall of the channel. This was evaluated at steady state conditions by using the experimentally measured mean temperature of the outer surface of the bottom wall of the test section and a well established correlation equation suggested for natural convection for geometrically similar systems [27]. The average convective heat transfer coefficient is calculated by using [25]: h¼
Q_ conv A½T w ððT out þ T in Þ=2Þ
ð2Þ
where Q_ conv is the steady state convection heat transfer rate; T w , the average surface temperature; T out and T in , the mean temperatures of the flow at the outlet and the inlet, respectively; and A, the surface area. Q_ conv is calculated using Eq. (1). Either the projected area ðAP Þ or the total area ðAT Þ of the test surface can be taken as the heat transfer surface area in the calculations. The total area is equal to the sum of the projected area and the surface area contribution from the pin fins. Therefore, the projected and the total area can be related to each other by: AT ¼ WL þ Nf ½4ðHDÞ þ D2
ð3Þ
Two types of Reynolds number were used to characterize the flow condition. One is a Reynolds number based on the mean velocity (U ) in the smooth duct and the hydraulic diameter of the duct ðDe Þ and is expressed as Re ¼
qDe U l
ð4Þ
The other one is based on the maximum velocity through the pin fins and the thickness of the pin fins, i.e.
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qDUmax l where Umax is the maximum velocity through the pin fins and is given by A Umax ¼ U A Afront ReD ¼
1793
ð5Þ
ð6Þ
where A is the cross-sectional area of the duct and Afront , the frontal area of the fins. In this experimental setup, the values of A=ðA Afront Þ are found to be 1.1852, 1.2427 and 1.4545 for the values of C=H ¼ 1, 0.6 and 0, respectively. ReD has been widely used in many pin fin heat transfer studies [19 etc.], and it depends on the spanwise pin number and the height of the pin fins. In the following discussion, Re and ReD will be referred to as duct Reynolds number and pin Reynolds number, respectively. Similar to the Reynolds number definition, the average Nusselt number is represented by duct Nusselt number and pin Nusselt number, respectively, as: Nu ¼ hDe =k
ð7Þ
NuD ¼ hD=k
ð8Þ
where k is the thermal conductivity of air and h, the average convective heat transfer coefficient. While calculating Nu and NuD , h is based on the projected area and the total heat transfer area, respectively. An uncertainty analysis was performed using the method described by Holman [30]; the experimental uncertainty in Nusselt number was calculated to be 9.5%. For more detailed information about the data reduction, see Refs. [23,24]. 4. Results and discussion 4.1. Heat transfer The average Nusselt number for the staggered arrangement was correlated as a function of Reynolds number, Prandtl number, interfin spacing ratio and clearance ratio, and the following expression was obtained: Nu ¼ 2:8358 Re0:58 ð1 þ C=H Þ0:848 ðSx =DÞ0:251 Pr1=3
ð9Þ
which is valid for 10; 000 6 Re 6 34; 000, 1:58 6 Sx =D 6 9:33, 0 6 C=H 6 1 with correlation coefficient of r ¼ 0:9671. Fig. 2 shows the duct Nusselt number, Nu, as a function of the duct Reynolds number, Re, for the different pin heights, namely C=H ¼ 1, 0.6 and 0 at Sx =D ¼ 1:58. Using the same experimental system, the average Nusselt number for the smooth surface (without pin fins) ðNus Þ was found by S ß ara et al. as follows [21]: Nus ¼ 0:0919 Re0:706 Pr1=3
ð10Þ
The plot of the results of the smooth rectangular duct predicted from Eq. (10) is also included in Fig. 2. It can be easily seen that the use of pin fins in the duct causes a substantial increase in the Nu. It also increases with increasing Re and decreasing C=H . As Nu is based on the projected area, it reflects both the variation in the surface area and the disturbances in the flow due to the pin fins.
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Fig. 2. Duct Nusselt number as a function of the duct Reynolds number and clearance ratio at Sx =D ¼ 1:58.
Decreasing C=H means that the height of the fins increases. Therefore, decreasing the height of the fins leads to a decrease in Nu. This is because the surface area decreases with decreasing height of the fins, resulting in a decrease in Nu. Longer fins can also increase the turbulence of the flow in the channel, resulting in an increase in heat transfer. Fig. 3 shows the behaviour of the Nu as a function of the duct Reynolds number and interfin distance ratios for a constant clearance ratio of 1.0. As seen in this figure, Nu increases with decreasing Sx =D, which can be attributed to increasing the fin number. Since the number of fins increases with decreasing Sx =D, which also means an increase in the total heat transfer area, the heat transfer rate increases. Similar results were found for inline pin configurations [21]. Namely, Nu was found to increase with decreasing C=H and decreasing Sx =D. A comparison of Nu for the staggered and inline pin fin arrangements with the same number of pins is given in Fig. 4. It is evident from this figure that Nu is higher for the staggered pin fins arrangement than that for an inline arrangement for all values of C=H, especially at lower Reynolds number. An examination of Fig. 5, which is a plot of Nu as a function of the interfin distance ratio, reveals that similar results were found for the duct Nusselt number. A similar observation is reported in the literature [17]. As mentioned previously, the pin Nusselt number NuD as calculated by Eq. (8) has been widely used in the literature. NuD was correlated with the pin Reynolds number, ReD , in the form of the following equation: NuD ¼ m Ren
ð11Þ
The values of m and n change with Sx =D and C=H, and the changes, with correlation coefficient r, are summarized in Table 2. The values of m for the short pin fins are lower than those of the long pin fins. Brigham and VanFossen [28] studied the effect of length to diameter ratio (H=D) on short
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Fig. 3. Duct Nusselt number as a function of the duct Reynolds number and interfin spacing ratio at C=H ¼ 1:0.
Fig. 4. Comparison of staggered and inline arrangements.
pin fin heat transfer and reported that H =D was the dominant factor in pin fin heat transfer for H =D > 2, and for values of H=D < 2, the Nusselt number was independent of H =D. Armstrong and Winstanley [7] reported that the Reynolds exponent for the short pin fins ðH =D < 4Þ varied between 0.59 and 0.65 when correlated row by row, while for long pin fins ðH =D ¼ 7:72Þ, it was approximately 0.5.
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Fig. 5. Comparison of staggered and inline arrangements. Table 2 Constants of Eq. (11) C=H
Sx =D
m
n
r
1.0
1.58 4.17 9.33
0.201 0.364 0.168
0.547 0.473 0.569
0.994 0.987 0.998
0.6
1.58 4.17 9.33
0.723 0.642 0.182
0.415 0.410 0.575
0.956 0.921 0.987
0.0
1.58 4.17 9.33
0.524 0.782 0.4137
0.404 0.372 0.449
0.968 0.949 0.941
Fig. 6 shows the total heat transfer area-averaged pin Nusselt number, NuD , as a function of the pin Reynolds number, ReD . The results of the present work lie quite close to those of some works from the literature. Some previous correlations for array-averaged Nusselt number in the pin fin duct are also plotted in this figure and listed in Table 3, which also includes geometrical and flow conditions, for comparison. 4.2. Friction factor The experimental pressure drops over the test section in the duct were measured under heated flow conditions and were arranged in nondimensional form by using the following equation: f ¼
DP ðLt =De ÞqU 2 =2
ð12Þ
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Fig. 6. Pin Nusselt number as a function of pin Reynolds number and comparison with the results of other investigators.
Table 3 The previous correlations for pin fins No.
Equation 0:728
1
NuD ¼ 0:069 Re
2
NuD ¼ 0:414 Re0:511
3
NuD ¼ 0:145 Re0:646
4
NuD ¼ 7:04 103 Re0:953 ðSx =Wb Þ0:091 ðSy =LÞ0:053
5
NuD ¼ 0:3313ðRe PrÞ0:646
6
NuD ¼ 0:344ðRe PrÞ0:555
Conditions
Reference
Circular staggered pin fin, Sx =D ¼ 2:5, Sz =D ¼ 2:5, 1000 < Re < 2 105 , H =D ¼ 1, C=H ¼ 0, Rectangular duct Circular staggered pin fin, Sx =D ¼ Sz =D ¼ 2:5, 5000 < Re < 3 104 , H =D ¼ 1, C=H ¼ 0, Rectangular duct Circular staggered pin fin, Sx =D ¼ 2:5, Sz =D ¼ 2:5, 104 < Re < 5 104 , H =D ¼ 1, C=H ¼ 0, Wedge duct Circular staggered pin fin, 0:004 < Sx =Wb < 0:332, 0:033 < Sy =L < 0:152, 3:138 103 < Re < 4:98 103 , Rectangular duct Circular staggered pin fin, Sx =D ¼ 2:33, Sz =D ¼ 3, 100 < Re < 1100, H =D ¼ 5:5, C=H ¼ 1, Rectangular duct, triangular fin
[4]
[9]
[29]
[14]
[18]
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An uncertainty analysis, using the same method as in the case of Nusselt number, indicated that the uncertainty in experimental f measurements could be up to 18.8%. Using the experimental results, f was correlated as a function of the duct Reynolds number, Re, and geometrical parameters, and the resulting equation is: f ¼ 11:944 Re0:101 ð1 þ C=H Þ2:05 ðSx =DÞ0:582
ð13Þ
which is valid for 10; 000 6 Re 6 34; 000, 1:58 6 Sx =D 6 9:33, 0 6 C=H 6 1 with a correlation coefficient of r ¼ 0:9504. Fig. 7 depicts the friction factor for the staggered pin fins ducts as a function of the duct Reynolds number, Re, for Sx =D ¼ 1:58 and 4.17 at C=H ¼ 1. The results for the inline pin fin arrangement [21] and the smooth duct from the Blasisus equation are also included for comparison. It can be seen from Fig. 7 that the friction factor for the duct with staggered pin fins arrays is higher than that for inline pin fin arrays and also substantially higher than that for the smooth duct. This can be attributed to the fact that the staggered arrangement avoids the working fluid easily flowing between the fins as it flows in the inline arrangement, blockading the channel like passage of the fluid between the fins. The other notable result is that the effect of the streamwise pin row number, namely Sx =D, on the friction factor is higher for the staggered pin fins arrays than that for the inline arrangements. On the other hand, the pressure drop over the pin fins, generally, has been correlated in the following manner [4,7]: f ¼
DP 2 N 2qUmax
ð14Þ
where N is the number of rows of pins in the streamwise direction. Limited attention has been focused on the problem of hydraulic resistance in a pin fin bank. A literature review, which in-
Fig. 7. Duct friction factor as a function of Reynolds number and comparison with the results of inline arrangement.
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cludes some friction correlations, was reported by Armstrong and Winstanley [7]. In order to compare the friction factor with these correlations, the values of f , which are calculated from Eq. (12), were transferred to the fD by using the definition of Eq. (14). The results were plotted as a function of the pin Reynolds number, ReD , in Fig. 8. Some previous correlations for the pin friction factor were also plotted in this figure and listed in Table 4.
4.3. Performance analysis The appraisal of the possible benefits of using augmented heat transfer surfaces depends on the goal(s) to be fulfilled and the constraints being imposed, and several number of PEC have been
Fig. 8. Pin friction factor as a function of pin Reynolds number and comparison with the results of other investigators.
Table 4 The previous correlation of friction factor for pin fins No.
Equations
1 2 3 4
fD fD fD fD
¼ ½0:25 þ ð0:1175Þ=ðSz =D 1Þ1:08 Re0:16 D ¼ ½2:06ðSz =DÞ1:1 Re0:16 D ¼ 0:317 Re0:132 , 103 < ReD < 104 D 0:318 ¼ 1:76 ReD , 104 < ReD < 105
Reference Armstrong and Winstanley [7] Metzger et al. (1982) [4] Metzger et al. (1982) [4]
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developed. These PEC are classified into two general groups [22]: those based on the first law analysis and those based on the second law analysis. One of the PEC is to compare the heat transfer coefficient for constant pumping power for the channel with finned surface with that for the smooth surface. The pumping power is proportional to f Re3 , and the relationship between the finned and smooth duct for the same pumping power is expressed by [26]: f Re3 ¼ fs Re3s
ð15Þ
where f and Re are the values for the finned duct, and fs and Res are those for the smooth duct. It has been widely accepted in the literature that the friction factors for both the smooth and finned ducts are related to the Reynolds number by the following equation: f ¼ a Reb
ð16Þ
According to the constant pumping power constraint, the mass flow rates (or Reynolds number) passing through the finned and reference smooth channel cannot be the same, since the mass flow rate for the smooth channel must be increased to keep the fluid pumping power constant. Using Eqs. (15) and (16), the Reynolds number for the smooth duct, Res , can be expressed as: 1=ðbs þ3Þ a Reð3þbÞ=ð3þbs Þ ð17Þ Res ¼ as Using Eq. (13) and the following Blasius equation fs ¼ 0:316 Re0:25
ð18Þ
the constants a, as , b, and bs were found to be, respectively: a ¼ 11:944ð1 þ C=H Þ2:05 ðSx =DÞ0:582
as ¼ 0:316 bs ¼ 0:25
b ¼ 0:101
Inserting these constants into Eq. (17) yields the Reynolds number for the smooth channel at which the pumping power is the same as that occurring in the finned channel as: 0:746
Res ¼ 3:751ð1 þ C=H Þ
0:212
ðSx =DÞ
Re1:054
ð19Þ
In the present study, the following PEC, called the heat transfer enhancement factor, Nu , was used to appraise the possible benefits of using pin fins: ! Nu
ð20Þ Nu ¼
Nus equal pumping power
where Nus is the average Nusselt number for the smooth channel with Res at which the pumping
power is the same as that occurring in the finned channel. Nus was calculated from Eq. (10) by using the values of Res , which was calculated by using Eq. (19). In graphical representation, the
ratios of Nu=Nus are plotted against Res . Nu was plotted as a function of equivalent Reynolds number, Res , in Figs. (9) and (10) for the staggered configurations. Fig. 9 shows the effect of the interfin distance ratio on Nu , whereas Fig. 10 shows the effect of the clearance ratio on Nu . These
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Fig. 9. Heat transfer enhancement factor as a function of equivalent Reynolds number and interfin spacing ratio.
Fig. 10. Heat transfer enhancement factor as a function of equivalent Reynolds number and clearance ratio.
figures show that the heat transfer enhancement factor, Nu , decreases with increasing Res , Sx =D and C=H . In other words: (1) The heat transfer enhancement factors, Nu , are higher than unity for all investigated conditions. This means that the use of pin fins leads to an advantage on the basis of heat transfer enhancement.
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(2) At lower Reynolds numbers, the channels with pin fin arrays give higher performance than those at higher Reynolds numbers. (3) For higher thermal performance, a lower interfin distance ratio and a lower clearance ratio should be preferred. In other words, the higher number of pin fins and longer pin fins have better performance.
5. Conclusion The enhancement of the heat transfer from a flat surface in a rectangular channel flow by the attachment of staggered square cross-sectional pin fins was investigated experimentally. The effects of the flow and geometrical parameters on the heat transfer and friction characteristics were determined, and the constant pumping power criterion was used to evaluate the performance of the staggered pin fin array systems. The conclusions are summarized as: (1) The average Nusselt number increased with decreasing clearance ratio and interfin distance ratio. (2) The friction factor increased with decreasing clearance ratio and interfin distance ratio. (3) Nu increased with decreasing C=H , which means that longer pin fins have better performance. (4) Nu increased with decreasing Sx =D, which means that higher number of pin fins have better performance. (5) At lower Reynolds numbers, the channel with pin fin arrays gives higher performance than those at higher Reynolds numbers. (6) The heat transfer enhancement factors, Nu , are higher than unity for all investigated conditions. This means that the use of pin fins leads to an advantage on the basis of heat transfer enhancement.
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