- Email: [email protected]

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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