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Forced Convection Heat Transfer Coefficient and Pressure Drop of Diamond-Shaped Fin-Array

Forced convection cooling of fins on a high-temperature wall has been used to cool high-power electronic devices. We numerically calculated and experimentally measured the forced convection heat transfer coefficient and pressure drop of a diamond-shaped fin-array with water flow in this study, which had been reported to have a self-induced flip-flop flow phenomenon. Although the flip-flop flow phenomenon occurred in calculations, it was not observed in experiments. The heat transfer and pressure drop of the diamond-shaped fin-array could be estimated with equations for turbulent flow in tubes.


Introduction
The heat density of high-load electronics, such as high-performance servers and the inverters used in vehicles, has been increasing as a result of higher processing speeds and higher power outputs. The method used to cool electronic devices depends on the level of heat generation and the operating environment, whereas available methods include natural convection cooling [1], forced convection cooling, and heat pipe cooling [2]. Forced convection cooling is used most often because of its practical advantages in cost and performance. For these reasons, we have been developing forced convection cooling technologies to cool high-power electronic devices [3]. Pin-fin arrays have been studied to enhance forced convection heat transfer in a wide variety of applications, such as heat exchangers and electrical devices. Pin-fin arrays usually consist of a number of cylinders attached S. Hirasawa et al. 79 perpendicularly to a high-temperature wall with a coolant fluid passing a cross flow over the cylinders. Tsuzuki et al. [4] studied an S-shaped fin-array and reported a method of calculating heat transfer in the fin-array. Tanda [5] studied heat transfer and pressure drop in a diamond-shaped fin-array with a leading angle of 90˚. Umeda et al. [6] [7] reported experimental results obtained from a self-induced flip-flop flow phenomenon in diamondshaped cylinder bundles with a leading angle of 30˚. The flip-flop flow oscillation promoted fluid mixing and had the potential to enhance convection heat transfer. However, they did not present experimental results for convection heat transfer in a diamond-shaped fin-array with flip-flop flow oscillation.
In this work, we numerically calculated and experimentally measured the forced convection heat transfer coefficient and pressure drop of a diamond-shaped fin-array with water passing a cross flow over the fin-array.

Numerical Calculation Method
We used a 3-dimensional thermo-fluid computation code called "STAR-CCM+" [8] in this work to calculate flow velocity and temperature distributions in a water flow around a diamond-shaped fin-array. We used the finite volume method for the numerical calculations. The conditions of incompressible fluid, unsteady turbulent flow with the large eddy simulation (LES) model, constant properties, and discretization with polyhedralmesh were used in the calculations. Figure 1 outlines the calculation conditions. The length was 66.3 mm in the x direction, the width of the flow region was 20.7 mm in the y direction, and the thickness was 2 mm in the z direction. The diamond-shaped fins were in a staggered array with six rows in the x direction and five rows in the y direction. The leading angle of the diamond-shaped fin was 30˚, their width was 2.07 mm in the y direction, and their length was 7.73 mm in the x direction. The width of the grooves between the fins was w = 2 mm in the flow direction of 15˚ was different from that in the x direction. We only calculated the temperature distribution in the fluid between the spaces created by fins and did not calculate the temperature distribution in the fins. The main reason for this was to understand the convection heat transfer coefficient for basic boundary conditions with the constant wall temperature of fins. The surface temperature of the fins was T w = 40˚C. The flow boundary condition at the surface of the fins had no-slip wall. The boundary condition for the outside wall of the flow region was symmetrical except for the inlet and outlet regions. The fluid was water with an inlet temperature of   the inlet velocity of u in = 3 m/s and Re = 26,900. The fluid flow caused separation behind the diamond-shaped fins and the flow pattern was not symmetrical in the y direction especially behind the fin-array near the outlet. Figure 3 shows the fluid flow velocity and temperature distributions at time (t = t 0 − 0.02 s), which is 0.02 s before from that for Figure 2. The separation and fluid flow behind the diamond-shaped fins are different from those in Figure 2 and there is a flip-flop flow phenomenon as was reported by Umeda et al. [6] [7].          Here, Pr is the Prandtl number of fluid (Pr = 6.0). The line in Figure 10 indicates the relation with Equation (1). Our calculated results agree with the relation in Equation (1). Figure 11 plots the relation between Reynolds number Re and the pressure drop through fins ∆P for our calculated results with the triangles "∆". Pressure drop ∆P for turbulent flow in a tube with the entrance and exit pressure loss can be calculated with Equation (2)

Calculation Results
Here, k is the friction coefficient with the function of the Reynolds number (k = 0.031, 0.024, and 0.013 at Re = 8980, 26,960, and 89,800 [9]), K c is the sum of entrance and exit loss coefficients (K c = 1), ρ is fluid density, and u is the flow velocity. The line in Figure 11 plots the relation with Equation (2). Our calculated results agree with the relation in Equation (2). Therefore, we found that heat transfer and pressure drop in a diamondshaped fin-array could be estimated with equations for turbulent flow in tubes though they had the flip-flop flow phenomenon. Figure 12 outlines the experimental apparatus. The test section was placed in an acrylic case (170 mm × 120

Re
Eq. (2) Calculation results mm × 30 mm) and the excess spaces in the case were filled with dummy acrylic blocks. Water at a temperature of 15˚C was supplied to the test section from a chiller after the flow rate was measured. Figure 13 shows the test fin-array. The diamond-shaped fin-array was made out of a copper block (70 mm × 70 mm × 15 mm) by cutting crossed grooves w = 2 mm in width, 2 mm in depth, and 4 mm in pitch. The leading angle of the diamondshaped fins was 30˚. The copper block was heated with three cartridge heaters (5 mm in diameter and a total heating rate of 200 W). The temperatures of the copper block and inlet water were measured with T-type thermocouples. The flow velocities u = 3.3 and 5 m/s in the grooves, which corresponded to the Reynolds number of Re = (ρud/µ) = 7410 and 11,200 using the hydraulic diameter d = w of rectangular grooves between the fins.

Experimental Results
The water flow pattern between the fins could be observed with tiny particles in water. The flip-flop flow phenomenon was not observed in our experiment under any experimental conditions. The reason for this is that the fluid space behind the fin-array was not enough to generate oscillation. Figure 14 plots the relation between Reynolds number Re and Nusselt number Nu in our experimental results with the circles "ο". The line in Figure  14 indicates the relation with Equation (1) for turbulent flow in a tube. Our experimental results agreed with the relation in Equation (1) which was similar to the calculated results in Figure 10.

Conclusions
We numerically calculated and experimentally measured the forced convection heat transfer coefficient and pressure drop of a diamond-shaped fin-array with water flow. The following results were obtained.  2) The heat transfer coefficient and pressure drop of the diamond-shaped fin-array could be estimated with equations for turbulent flow in tubes even though they had the flip-flop flow phenomenon.