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Thermal performance of in-line diamond-shaped pin fins in a rectangular duct ☆ Tzer-Ming Jeng Department of Mechanical Engineering, Air Force Institute of Technology, GangShan 820, Taiwan, ROC Available online 12 July 2006

Abstract This work experimentally studied the pressure drop and heat transfer of an in-line diamond-shaped pin-fin array in a rectangular duct by using the transient single-blow technique. The variable parameters are the relative longitudinal pitch (XL = 1.060, 1.414, 1.979) and the relative transverse pitch (XT = 1.060, 1.414, 1.979). The empirical formula for the heat transfer is suggested. Besides, the optimal inter-fin pitches, XT = 1.414 and XL = 1.060, are provided based on the largest heat dissipation under the same pumping power. © 2006 Elsevier Ltd. All rights reserved. Keywords: Pressure drop; Heat transfer; Pin-fin array; Optimal inter-fin pitch

1. Introduction Pin-fins protruded from a heating surface can increase the surface area of dissipation and cause turbulent mixing of flow, subsequently enhancing the heat dissipation performance and protecting the reliability and life of devices. The affecting factors on the thermal performance of pin-fins include the velocity of fluid flow, the thermal properties of the fluid and the pin-fins, the relative fin height, the cross-sectional shape of the pin-fins, the relative inter-fin pitch, the arrangement of the pin-fins and the shroud clearance (bypass effect). Many researchers have considered the effects of the aforementioned parameters on heat transfer of pin-fins. Most popular cross-sectional shape of the pin-fins is circular [1–6]. Square pin-fin array is also more and more considered [7–15]. However, the study of diamond-shaped pin-fins is few [16–21]. Some of them are summarized below. Igarashi [16] and Yoo et al. [17] experimentally investigated the effect of angle of attack on heat and mass transfer from a square cylinder. They found that the average heat or mass transfer has a minimum value at the angle of attack α = 12°–13° and a maximum one at α = 20°–25°. Sparrow et al. [18] reviewed the correlations of average heat transfer coefficients for various cross-sectional shaped cylinders and for spheres in cross-flow. The above-mentioned studies focused on the single cylinder. About the diamond-shaped pin-fin array, Sparrow and Grannis [19], and Grannis and Sparrow [20] used the typical experiments to verify the accuracy of the numerical simulation of fluid flow through a diamond-shaped pin-fin array. They provided the correlation between the friction factor and the Reynolds number ☆

Communicated by W.J. Minkowycz. E-mail addresses: [email protected], [email protected]

0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.06.001

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Nomenclature afs Abase Atotal d fd hfs H k L Nud Nud ⁎ Red SL ST T Tfinal U W x XL XT ΔP

Heat transfer surface area per unit bulk volume, [m2/m3] Bottom area of the test channel, [m2] Total heat dissipation area of pin-fins, [m2] Transverse width of pin-fins, [m] DP d Friction factor, fd ¼ 0:5qu 2 L Interstitial heat transfer coefficients between fluid and solid, [W/m2 °C] Height of the channel or pin-fins, [m] Conductivity, [W/m°C] Length of the channel, [m] fin Nusselt number, Nud ¼ hkfsf d Global Nusselt number, Nu⁎d ¼ Nud AA Reynolds number based on d, Red ¼ qf lUd Center-to-center longitudinal distance between the adjacent fins, [m] Center-to-center transverse distance between the adjacent fins, [m] Volumetric average temperature, °C Final temperature of preheating air at the channel inlet, °C Volumetric average fluid velocity, [m/s] Width of the heat sink, [m] Streamwise Cartesian coordinate, [m] Relative longitudinal pitch, SL/d Relative transverse pitch, ST/d Pressure drop, [Pa] total

base

Greek symbols ε Porosity of pin-fin array μ Viscosity of fluid, [kg/m/s] ρ Density, [kg/m3] Subscripts 0 Initial f Fluid s Solid

based on the results of numerical calculations. Tanda [21] employed liquid crystal thermography to determine the heat transfer coefficients on the surface of the channel (endwall) on which the diamond-shaped pin-fins were mounted in inline and staggered arrangements. Big relative inter-fin pitches were adopted in his work (the ratio of the inter-fin spacing to the diamond side was larger than 4). Following the cited literature, this work experimentally explores the pressure drop and the heat transfer in diamondshaped pin-fins with small and independently variable inter-fin pitches, to improve our understanding of pin-fin arrays. An empirical formula for the heat transfer is suggested and the optimal inter-fin pitch is provided. Accordingly, the results can help cooling engineers design more efficient cooling devices based on pin-fin arrays. 2. Experimental apparatus Fig. 1 presents an experimental setup for measuring the flow drag and the heat transfer of pin-fin arrays. An openlooped suction-type wind tunnel was used herein. Firstly, a one-HP blower drove air into the wind tunnel through a 200 × 200 mm2 bell mouth with a honeycomb straightener. Then, the air passed through a contraction section to reduce the turbulence. Finally, the air entered the test section after it flowed through a specially designed plate heater made of

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Fig. 1. Experimental setup.

stainless steel foil. An inverter was employed to control the rate of rotation of the blower's motor to yield a specific flow rate. The test section was a rectangular channel that contained a diamond-shaped pin-fin array (see Fig. 2). The channel section was 81.5 mm wide, 76.5 mm high and 300 mm long. The channel walls were made of 20 mm thick Bakelite to reduce heat loss. The diamond-shaped pin-fin with a cross section of 9.6 × 9.6 mm2 (d = 13.58 mm) and a height of 76.5 mm was made from 6061 aluminum alloy. The variable parameters of the pin-fin array were the relative transverse pitch (XT = 1.060, 1.414, 1.979) and the relative longitudinal pitch (XL = 1.060, 1.414, 1.979). The Pitot tube and the micro-manometer measured the air velocity across the test channel. The digital pressure transmitter measured pressure drops across the pin-fin arrays. The time histories of the inlet and outlet air temperatures were obtained using

Fig. 2. Test section.

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T-type thermocouples. Eighteen thermocouples were installed at the cross-sections of the channel inlet and outlet. The transient temperature signals were transferred and subsequently recorded using a real-time hybrid recorder (Agilent 34970A) at a sample rate of 2 Hz. 3. Determining hfs and error analysis In the transient single-blow experiments [22,23], a heated fluid flow supplies thermal energy to aluminum pin-fins. The volumetric average fluid velocity in the test channel is assumed to be uniform. The temperatures of the solid and the fluid in the test channel depended on the distance along the flow direction and on the time. Therefore, this problem can be solved using a one-dimensional transient heat transfer model. The volume-averaged conservation equations of energy for the solid and fluid phases are [23] ð1−eÞðqCp Þs

ATs ¼ hfs afs ðTf −Ts Þ At

ATf ATf þU ðqCp Þf e ¼ hfs afs ðTs −Tf Þ At Ax

ð1Þ

ð2Þ

In these equations, ε denotes the porosity of the pin-fin array; U is the volumetric average fluid velocity; Ts and Tf represent the volumetric average temperature of the solid and fluid phases, respectively; hfs is the interstitial heat transfer coefficient between the fluid stream and the solid matrix, and afs is the surface area of heat transfer per unit bulk volume in the test channel. Besides, the initial and boundary conditions are Ts ð0; xÞ ¼ Tf ð0; xÞ ¼ 0

ð3Þ

Tf ðt; 0Þ ¼ measured data

ð4Þ

The control-volume-based finite-difference technique [24] is employed to solve the above equations associated with the initial and boundary conditions. The transient finite-difference form of the energy equations is obtained explicitly by line-by-line iteration. By the sensitive examinations of the outlet air temperature to the number of nodes and the time increment, 61 nodes and Δt = 0.1 s are employed to ensure the stability and accuracy of the numerical scheme.

Fig. 3. The measured nondimensional inlet and outlet temperatures for typical cases.

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The test channels with various XT and XL have various ε and afs. Eqs. (1)–(4) determine the total temperature field. The values of Tf(t,0), U, ε and afs can be measured. Substituting the measured data and the guessed hfs into the governing equations yields the predicted outlet air temperature numerically. If the predicted outlet air temperature matches the experimental value, then the guessed hfs is the interstitial heat transfer coefficient of the test channel with the specified XT and XL. Fig. 3 illustrates the predicted and measured outlet air temperature of diamond-shaped pin-fins with in-line arrangements and various inter-fin pitches and flow rates. According to our results, the predicted outlet air temperature correlates well with the experimental value because the estimate of hfs was accurate. The measured volumetric average fluid velocities (U ), pressure drops (ΔP) and interstitial heat transfer coefficients (hfs) were used to calculate the Reynolds numbers, the friction factors and the average fin Nusselt numbers: qf Ud ; l

ð5Þ

DP d ; 0:5qf U 2 L

ð6Þ

Red ¼

fd ¼

Nud ¼

hfs d kf

ð7Þ

where d and L are the transverse width of the pin-fin and the length of the test channel, respectively. Data supplied by the manufacturer of the instruments stated that the measurements of flow velocity, length scale and pressure drop have an error of ± 1%. The uncertainty in the measured temperature was ± 0.2 °C. The inlet air was typically heated from 20 °C to 60 °C. Each experiment took around 180 s. In the typical case with XT = XL = 2.0 and Red = 3361, the measurement errors in Tf(t,0), Tf(t,L), U, ε and afs yielded uncertainties of ± 4.2%, ± 4.6%, ± 3.0%, ± 3.2% and ± 2.2% in hfs. The standard single-sample uncertainty analysis, recommended by Kline and McClintock [25] and Moffat [26], yielded uncertainties in the Reynolds number, friction factor and average fin Nusselt number of ± 2.0%, ± 2.4% and ± 8.1%, respectively. 4. Results and discussion Fig. 4 shows the friction factor ( fd) against Reynolds number (Red) studied herein. The results indicate that the friction factor ( fd) increases as the relative transverse pitch (XT) decreases, because the case with a smaller XT has a larger velocity between the adjacent

Fig. 4. Friction factor as a function of Reynolds number.

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Fig. 5. Nusselt number as a function of Reynolds number.

fins. Besides, the friction factor ( fd) decreases when the relative longitudinal pitch (XL) declines. The reason is that the fluid flow is concentrated in lanes between the fin columns with small XL, reducing the turbulent wakes of the upstream pin-fins. Therefore, the pressure drop falls. Moreover, the correlation between fd and Red provided by Sparrow and Grannis [19] is also plotted in Fig. 4. The

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Table 1 Corresponding factors in the correlation of Nusselt number For and XL = 1.060–1.979 XT

m

n

1.060 1.414 1.979

0.164 0.164 0.143

0.727 0.727 0.689

data herein are generally consistent with the predictions of Sparrow and Grannis [19], revealing the validity of the tests performed herein. Fig. 5 depicts the relationship between the average fin Nusselt number (Nud) and Reynolds number (Red). Both correlations of Zukauskas and Ulinskas [4] for in-line circular pin-fins and Sparrow et al. [18] for single diamond cylinder are also plotted in Fig. 5. The comparison results indicate that the present data are within a reasonable range. The present Nud values also generally decrease as the relative transverse pitch (XT) increases, because the maximum velocity between fins decreases as XT increases, while this trend is similar with that of the in-line circular pin-fins [4]. On the other hands, the effect of the longitudinal pitch (XL) on the Nud seems negligible, while this find is also consistent with that of the in-line circular pin-fins [4]. Based on all experimental data, Table 1 gives the empirical equation of Nud for in-line diamond-shaped pin-fins. The average error between the values predicted using the empirical equation in Table 1 and experimental values is less than 11.7%. The convective heat dissipation of the pin-fin array in the channel is positively related to the amount of airflow that flow through it, but at high airflow, more pumping power is required to overcome the flow resistance. Therefore, in determining the optimal interfin pitches, the largest Nusselt number is reasonably used for a given pumping power. This section introduces fdRe3d as the nondimensional pumping power. Also, the fin Nusselt number (Nud) represents the average heat transfer capacity of the pin-fin surface, and the total heat dissipation area must be considered in determining the global heat transfer of the pin-fin array. Hence, this work proposes that Nud⁎ be used to represent the global Nusselt number, and the relationship between Nud⁎ and Nud is

Atotal Nu*d ¼ Nud Abase

ð8Þ

where Atotal is the total area of heat dissipation of pin-fins, and Abase represents the bottom area of the test channel (L × W). Fig. 6 represents the global Nusselt number as function of the nondimensional pumping power. The results demonstrate that, given a particular pumping power, decreasing XL will increase the Nud⁎, and the cases with moderate XT (XT = 1.414) yield the highest Nud⁎ among the present various XT. Therefore, the inter-fin pitches of diamond-shaped pin-fins in in-line arrangements are optimal at XT = 1.414 and XL = 1.060 herein.

Fig. 6. Global Nusselt number as a function of nondimensional pumping power.

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5. Conclusions The investigation of the pressure drop and heat transfer of an in-line diamond-shaped pin-fin array with various inter-fin pitches has been experimentally completed. The empirical formula of average fin Nusselt number (Nud) is provided based on all experimental data. Finally, the optimal inter-fin pitches are proposed (the moderate XT, XT = 1.414 and the smallest XL, XL = 1.060) herein based on the largest heat dissipation under the same pumping power. Acknowledgment The author would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 92-2212-E-344-005. References [1] G.J. Vanfossen, Heat-transfer coefficients for staggered arrays of short pin fins, ASME Journal of Engineering for Power 104 (1982) 268–274. [2] B.A. Brigham, G.J. Vanfossen, Length to diameter ratio and row number effects in short pin fin heat transfer, ASME Journal of Engineering for Gas Turbines and Power 106 (1984) 241–244. [3] D.E. Metzger, C.S. Fan, S.W. Haley, Effects of pin shape and array orientation on heat transfer and pressure loss in pin fin arrays, Journal of Engineering for Gas Turbines and Power 106 (1984) 252–257. [4] A. Zukauskas, R. Ulinskas, Efficiency parameters for heat transfer in tube banks, Heat Transfer Engineering 6 (1985) 19–25. [5] J. Armstrong, D. Winstanley, A review of staggered array pin fin heat transfer for turbine cooling applications, ASME Journal of Turbomachinery 110 (1988) 94–103. [6] B.A. Jubran, M.A. Hamdan, R.M. Abdualh, Enhanced heat transfer, missing pin, and optimization for cylindrical pin fin arrays, ASME Journal of Heat Transfer 115 (1993) 576–583. [7] M.A. Tahat, R.F. Babus'Haq, S.D. Probert, Forced steady-state convections from pin-fin arrays, Applied Energy 48 (1994) 335–351. [8] M. Tahat, Z.H. Kodah, B.A. Jarrah, S.D. Probert, Heat transfer from pin-fin arrays experiencing forced convection, Applied Energy 67 (2000) 419–442. [9] R.F. Babus'Haq, K. Akintunde, S.D. Probert, Thermal performance of a pin-fin assembly, International Journal of Heat and Fluid Flow 16 (1995) 50–55. [10] H.I. You, C.H. Chang, Determination of flow properties in non-Darcian flow, ASME Journal of Heat Transfer 119 (1997) 190–192. [11] H.I. You, C.H. Chang, Numerical prediction of heat transfer coefficient for a pin-fin channel flow, ASME Journal of Heat Transfer 119 (1997) 840–843. [12] D. Kim, S.J. Kim, A. Ortega, Compact modeling of fluid flow and heat transfer in pin fin heat sinks, ASME Journal of Electronic Packaging 126 (2004) 342–350. [13] O.N. Sara, S. Yapici, M. Yilmaz, T. Pekdemir, Second law analysis of rectangular channels with square pin-fins, International Communications in Heat and Mass Transfer 28 (2001) 617–630. [14] O.N. Sara, Performance analysis of rectangular ducts with staggered square pin fins, Energy Conversion and Management 44 (2003) 1787–1803. [15] T.M. Jeng, S.C. Tzeng, A semi-empirical model for estimating permeability and inertial coefficient of pin-fin heat sinks, International Communications in Heat and Mass Transfer 48 (2005) 3140–3150. [16] T. Igarashi, Heat transfer from a square prism to an air stream, International Communications in Heat and Mass Transfer 28 (1985) 175–181. [17] S.Y. Yoo, R.J. Goldstein, M.K. Chung, Effects of angle of attack on mass transfer from a square cylinder and its base plate, International Communications in Heat and Mass Transfer 36 (1993) 371–381. [18] E.M. Sparrow, J.P. Abraham, J.C.K. Tong, Archival correlations for average heat transfer coefficients for non-circular and circular cylinders and for spheres in cross-flow, International Communications in Heat and Mass Transfer 47 (2004) 5285–5296. [19] E.M. Sparrow, V.B. Grannis, Pressure drop characteristics of heat exchangers consisting of arrays of diamond-shaped pin fins, International Communications in Heat and Mass Transfer 34 (1991) 589–600. [20] V.B. Grannis, E.M. Sparrow, Numerical simulation of fluid flow through an array of diamond-shaped pin fins, Numerical Heat Transfer. Part A, Applications 19 (1991) 381–403. [21] G. Tanda, Heat transfer and pressure drop in a rectangular channel with diamond-shaped elements, International Communications in Heat and Mass Transfer 44 (2001) 3529–3541. [22] C.Y. Liang, W.J. Yang, Modified single-blow technique for performance evaluation on heat transfer surfaces, ASME Journal of Heat Transfer 96 (1975) 16–21. [23] C.C. Wu, G.J. Hwang, Flow and heat transfer characteristics inside packed and fluidized beds, ASME Journal of Heat Transfer 120 (1998) 667–673. [24] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere, New York. [25] S.J. Kline, F.A. Mcclintock, Describing uncertainties in single-sample experiments, Mechanical Engineering (1953) 3–8. [26] R.J. Moffat, Contributions to the theory of single-sample uncertainty analysis, ASME Journal of Fluids Engineering 104 (1986) 250–260.