Numerical study of turbulent fluid flow and heat transfer in lateral perforated extended surfaces

Energy 64 (2014) 632e639

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Numerical study of turbulent fluid flow and heat transfer in lateral perforated extended surfaces Md. Farhad Ismail a, *, Muhammad Noman Hasan b, Suvash C. Saha c a

School of Mechatronic Systems Engineering, Simon Fraser University, Surrey, B.C., Canada V3T 0A3 Department of Mechanical Engineering, The University of Akron, Akron, OH 44325, USA c School of Chemistry, Physics & Mechanical Engineering, Queensland University of Technology, Brisbane, QLD 4001, Australia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 May 2013 Received in revised form 18 October 2013 Accepted 25 October 2013 Available online 6 December 2013

Numerical study has been performed in this study to investigate the turbulent convection heat transfer on a rectangular plate mounted over a flat surface. Thermal and fluid dynamic performances of extended surfaces having various types of lateral perforations with square, circular, triangular and hexagonal cross sections are investigated. RANS (Reynolds averaged NaviereStokes) based modified keu turbulence model is used to calculate the fluid flow and heat transfer parameters. Numerical results are compared with the results of previously published experimental data and obtained results are in reasonable agreement. Flow and heat transfer parameters are presented for Reynolds numbers from 2000 to 5000 based on the fin thickness. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Turbulent convection heat transfer Perforation shape Fin effectiveness Friction drag coefficient

1. Introduction The power consumption rate of electronic systems is constantly increasing as the components size and mass shrink. For this reason, it is important to maintain the temperature of the systems within desired level. For this purpose, various cooling technologies i.e. cooling with heat pipe are introduced. Also different types of coolants are used to enhance the performance of heat transfer. But air cooling technology is still very popular for its cheap and easy thermal management issue. In general, electronic packages are attached at the bottom of the heat exchangers. It is important to maintain the electronic package temperature at a desired level. Various numerical and experimental investigations have been performed previously to improve the thermal performance of micro heat sinks. Diani et al. [1] conducted both experimental and numerical studies to find the effects of geometric parameters on the thermal performance of the heat sink. They also designed optimized fin configurations for a specific cooling application. Dorignac et al. [2] experimentally determined the Nusselt number correlation for perforated plates. El-Sayed et al. [3] determined the optimized fin array position to maximize heat transfer performance at high Reynolds number. Fluid flow parameters were measured for

* Corresponding author. Tel.: þ1 604 961 5856. E-mail address: [email protected] (Md.F. Ismail). 0360-5442/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.10.079

different flow orientations i.e. parallel, impinging and reverse impinging flow. Thermal performance of different types of extended surfaces has been studied numerically and/or experimentally by Ismail et al. [4e6], Kadle and Sparrow [7], Lau and Mahajan [8], Wirtz et al. [9], Suabsakul [10], Sata et al. [11] and Molki and Hashemi-Esfahanian [12]. Jonsson et al. [13,14] developed empirical correlation for the bypass flow under turbulent flow condition. The correlation predicts the Nusselt number and pressure drop along different types of fin geometry. Mahmood et al. [15] investigated the influence of temperature ratio and flow structures for a dimpled channel at different Reynolds number ranging from 600 to 11,000 with inlet temperature ratio of 0.78e0.94. Park et al. [16] numerically studied thermal performance of a channel with dimpled surfaces having dimple to channel diameter ratio of 0.3 using the Realizable ke3 model for Reynolds number from 2700 to 41,000. Sarkar et al. [17] improve thermal performance of a channel using perforated star shape insert. For various range of perforations, they proposed empirical relations of convective heat transfer co-efficient at various Reynolds number. Sahin and Demir [18] concluded that either increasing the inlet air velocity or increasing the surface area, heat transfer performance of extended surfaces can be enhanced. Sparrow et al. [19] studied the effects of bypass flow over the tip of pin fin heat sinks. Sparrow and Carranco Ortiz [20] also performed an experimental study to determine the Nusselt number value of a perforated surface which was

Md.F. Ismail et al. / Energy 64 (2014) 632e639

faced upstream. The perforations were placed on equilateral triangular centers. Two parameters e Reynolds number and pitch to diameter ratio had been used in their study to compute the results. Wu et al. [21] developed a practical heat sink model to predict the thermal performance of plate fin heat sinks. They proposed correlations to predict the value of friction factor and Nusselt number for laminar, transition and turbulent flow at various Reynolds number less than 5000. Sara et al. [22] compared the thermal hydraulic performances of perforated rectangular blocks with the solid ones. They concluded that performances of both solid and perforated blocks decrease at the higher values of Reynolds number. Yu et al. [23] performed both experimental and numerical studies to compare thermal performances between plate fin and plate-pin fin heat sinks. They found that thermal resistance of plate fins was about 30% higher than that of a plate-pin fins. Yaghoubi and Velayati [24] developed new correlations to predict the Nusselt number and fin efficiency for turbulent flow over an array of cubes. Heat transfer performance can be enhanced introducing perforations or other kinds of turbulence promoters. But these kinds of arrangements cause greater frictional loss and thus require higher cooling power. Velayati and Yaghoubi [25] numerically investigated turbulent heat transfer performance of rectangular plate fin heat sink. They also proposed a correlation of Nusselt number and fin efficiency for plate fins. Shaeri and Yaghoubi [26e28] compared thermal hydraulic performance of both lateral and longitudinal perforated fins having rectangular perforations with the solid ones. They considered both laminar and turbulent flow condition in their study. In the study of Shaeri et al. [28] only square/rectangular perforated fins are analyzed. Though they presented their results for different number of perforations they did not compare their results for different shaped perforations having a fixed fin volume. For comparison of thermal performance, it is also important to keep the fin volume fixed. To the best of the author’s knowledge, no studies have been performed previously to investigate the effects of lateral perforation geometry on the thermal performance of plate fin. In the present study four types of perforated fins i.e. square, circular, hexagonal and triangular lateral perforated fins having same fin volume are used and the thermo-fluid performances are compared with the solid ones. These types of shapes have been selected as these are the common types of perforations used in different types of heat exchanger equipments to enhance heat transfer. The main objective of the present study is to investigate the effects of lateral perforation shapes on the thermal and hydraulic performance of the heat sink. Heat transfer and fluid flow parameters are presented for two and three perforations. 2. Problem description and computational set-up Domain for the computation and the dimensions of square lateral perforated fins used for the present numerical study are shown in Fig. 1. Thermal and fluid dynamic boundary conditions for the numerical simulation are set considering the study of Velayati and Yaghoubi [25] and Shaeri et al. [28]. Fins with three perforations having square, circular, hexagonal and triangular cross sections are considered in this paper. These perforations are along the thickness of the fins which are in transverse direction of the flow. Air is considered incompressible in this simulation. Steady state thermal and fluid flow conditions and no slip condition at the wall interface between the solid and fluid domain are considered. Finite element based commercial software COMSOL Multiphysics is used to calculate the heat transfer and fluid flow parameters. The modified keu turbulence model [29] of the conjugate heat transfer module is used to predict the fluid flow and heat transfer performances. Three dimensional NaviereStokes

633

Fig. 1. Computational domain (wall function) starts with a distance ‘y’ from the solid wall [30].

equations have been solved with the continuity equation and the energy equation. Fin material is assumed as solid domain (Aluminum) having a constant thermal conductivity of 202 W m1 K1 and the fluid domain is considered with constant thermal properties i.e. density (rf), thermal conductivity (lf), viscosity (mf) and specific heat capacity. 3. Numerical modeling and procedures 3.1. Geometric configuration In this paper, fins are considered to have the following dimensions: length (L) of 24 mm, height (H) of 12 mm and thickness (D) of 4 mm. For square perforated fins (two perforation), each square perforation has height (HP) and width (WP) equal to 4 mm and their thickness is equal to fin’s thickness (Fig. 2). The other types of perforated fin shapes are calculated by keeping the fin volume fixed. The fin’s thickness, D is used as the characteristic length to compute the results. All the results are presented in this study for Reynolds number (ReD) in the range of 2  103 to 5  103 based on the fin thickness (D). The corresponding Reynolds number based on the fin length is in the range of 1.25  104 to 3  104. 3.2. Mathematical model In this study, the flow is assumed to be incompressible, steady state and turbulent. The modified keu method [29] has been chosen for this study of turbulent forced convection heat transfer. The reason for choosing this turbulence model is that it behaves considerably well for flows over flat plate with adverse pressure gradients [30e33]. Three dimensional governing equations i.e. continuity, momentum and energy can be described as follows: Continuity equation:

V$U ¼ 0

(1)

Reynolds averaged NaviereStokes (RANS) equation:




rU$VU þ V$ ru0 5u0



  ¼ VP þ V$m VU þ ðVUÞT

(2)

Eddy viscosity closure:




ru0 5u0

 ¼

  2 rk  mt VU þ ðVUÞT 3

(3)

Transport equations for keu model:

rU$Vk ¼ V$½ðm þ mT sk ÞVk þ mT PðUÞ  bk rku

(4)

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The wall function used for the current turbulent flow simulation is such that the computational domain is assumed to start a distance y from the wall as shown in Fig. 1. The distance y is computed iteratively by solving the following Eq. (7)

yþ ¼

rus y m

(7)

where the friction velocity is assumed such that

1 juj ¼ log yþ þ B k us Here k is the von Kӑrman constant whose value is 0.41 and B is an empirical constant equals to 5.2. For the computation the friction velocity us is assumed to be *1=4 equivalent to b0 Ok which becomes 11.06 and the boundary conditionsition for u can be defined as rk=ðkyþ mÞ [30]. This corresponds to the distance from the wall where the logarithmic layer meets the viscous sublayer. The solutions are always checked in such that the yþ value is 11.06 on all the walls of the heat sink. 3.3. Boundary conditions The domain has an inlet and an outlet plane that are abcd and ijkl, respectively (Fig. 2). These planes should be far enough from the fin surfaces to avoid the effects of boundary positions. In this study, the computational domain is assumed as 15D upstream and 30D downstream in X direction, 7H in Y direction and 5D in Z direction. Free stream temperature is assumed 298 K and constant temperature of 343 [K] is applied at the fin base plane efgh. Zero gradients of variables normal to the outlet plane is imposed as the outlet is sufficiently far from the plate. Symmetry conditions are applied at plane abji and dckl. Other planes are considered adiabatic and no slip condition. Fig. 2(b) shows the dimension of the heat sink and Fig. 2(c) shows a sample mesh view for the case of square perforated fin. The geometric configuration for circular, hexagonal, square and triangular perforated fins with 3 lateral perforations having same fin volume is shown in Fig. 3. Several grid configurations are studied to make the results grid independent. 3.4. Solver settings

Fig. 2. (a) Computational domain used in this study; (b) square perforated fins having two perforations with showing the dimensions in mm; (c) sample mesh view of square perforated fin.

rU$Vu ¼ V$½ðm þ mT su ÞVu þ aumT P

U  bru2 k

(5)

where U is the averaged velocity field, P(U) ¼ U:(VU þ (VU)T), mT ¼ rk=u, bk ¼ 0.09, b ¼ 0.072, a ¼ 0.52, sk ¼ 0.5, su ¼ 0.5. Energy Equation:

rCp uVT ¼ V$ðlVTÞ

Finite element based software COMSOL Multiphysics [30] is used to analyze the turbulent convection heat transfer in different types of perforated fins. To solve the governing equations, segregated solvers are used to calculate the fluid flow (velocity and pressure) and heat transfer (temperature) parameters. Generalized minimal residual (GMRES) iterative method solver is used to calculate the parameters with a tolerance of 103 and factor in error estimate of 20. As a preconditioner, Geometric Multigrid solver is used with PARDISO (Parallel Sparse Direct Linear Solver) coarse solver.

(6)

where u is the velocity vector, r is density, Cp is specific heat capacity at constant pressure, l is thermal conductivity of fluid.

3.5. Simulation validation Results of the numerical simulation are compared with the numerical study of Shaeri et al. [28] for the validation of the present simulation. Results of the two perforated fins are considered here with the same boundary conditions and governing equations assumed by Shaeri et al. [28]. For this purpose, results are compared for various mesh elements e coarser to finer mesh elements (Fig. 4). Comparison between present and previous numerical

Md.F. Ismail et al. / Energy 64 (2014) 632e639

635

Fig. 3. Different types of lateral perforated fins having same fin volume (a) circular, (b) hexagonal, (c) square, and (d) triangular perforations having three perforations.

results is also illustrated in Fig. 4(a)e(b). From the figures it can be stated that simulation results show good agreement with the numerical study of Shaeri et al. [28]. Thus, this confirms the reliability of the predicted results of the current study. Several parameters are used in the present study to compare the thermal performances of the fins such as Nusselt number, heat transfer performance enhancement, fin effectiveness and drag coefficient. For the comparison of thermal hydraulic performances of the fins, two parameters such as fin effectiveness [34] and heat transfer performance enhancement (HTPE) can be considered in this study. Fin effectiveness and HTPE values should be as high as possible (>1). Fin effectiveness is the ratio of heat transfer from fin to heat transfer from fin base without the fin as: 3f

¼

q hAb ðTb  TN Þ

(8)

where h is average convective heat transfer coefficient for heat sink base, Tb is the fin base temperature without the fin and q is heat transfer rate (W) of fin which can be obtained by the following equation:

Z$ q ¼

hðTs  TN ÞdAs

(9)

Af

where Af is the total fin surface area, Ts is the fin surface temperature of the fin, TN is the free stream fluid temperature. Perforated fin effectiveness, PFE (3 PF) can be obtained using the equation stated below

3 PF

qPF  qSF  100 qSF

¼

(10)

To compare both the frictional effects and heat transfer performance, another parameter Heat Transfer Performance Enhancement (HTPE) can be used [6] and it can be expressed in Eq. (5) as:

HTPE ¼ 

QPF =QSF 1=3 Cf;PF =Cf;SF

(11)

where QPF and QSF are the heat transfer rate of the perforated and solid fins, respectively and Cf,PF and Cf,SF are the skin friction coefficient for the perforated and solid fins respectively. The skin friction coefficient can be obtained using the following equation. Here uN is the free stream velocity (m/s), sw is the average wall shear stress over the faces of the fin including perforation (Pa) and r is the fluid density

Cf ¼

sw

0:5ru2N

(12)

The Nusselt number value can be obtained using the following equations:

Nu ¼ where

hD

l

(13)

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Average Friction Co-efficient ( Cf )

0.01 0.009 Shaeri et al. [28]

0.008 0.007

Present Simulation

0.006 0.005 0.004 0.003 0.002 1500

2000

2500

3000

3500

4000

4500

5000

Reynolds Number

(a) 30

Nusselt Number

25

20

15

10 Shaeri et al. [28] 5 Present Simulation 0 1500

2000

2500

3000 3500 Reynolds Number

4000

4500

5000

(b) Fig. 4. Comparison of the present numerical simulation with previously published study [28]: variation of (a) average friction coefficient and (b) Nusselt number.

h ¼

1 X hi DAi AT

and, AT is the total surface area including perforations. 4. Results and discussion This section describes the results of thermal hydraulic performance of various types of heat sinks which are investigated for Reynolds number: ReD ¼ 2  103 to 5  103. Flow path lines around solid and hexagonal lateral perforated fins are shown in Fig. 5(a)e(f). From Fig. 5 it is shown that with increase in height the length of recirculation becomes larger because of the viscous effect of the base plate. It is also seen that with the increase of height, the wake behind the fin surfaces reduces. Perforated fins have different Nusselt number values at different Reynolds number. Fig. 6(a)e(f) shows the temperature contours near the fin surfaces. From this it is clearly seen that temperature inside the perforations is higher than other external surfaces of the fins. Due to perforations some flow recirculate inside the perforations. It is because of the reduction of velocity inside the perforations . This flow is also confined inside the

perforations and separated from the main flow. This is also another reason of the higher temperature inside the perforations. Fig. 7 illustrates the Nusselt number of solid fins and different types of perforated fins at various Reynolds number. Fig. 7(a) illustrates the Nusselt number value for two perforations and Fig. 7(b) illustrates the value for three perforations. It can be noted from the figure that the solid fins have the highest and triangular perforated fins show the lowest Nusselt number values at all Reynolds numbers in the range considered here. It is also observed that all the perforated fins have lower Nusselt number value. It is because due to perforation the air velocity decreases inside the perforation and some flows are also confined inside the perforation. For two and three perforations the trend of the graphs is similar. It is also seen that the value of Nusselt number is little higher for the fins having two perforations in comparison with the fins having three perforations. Fig. 8 shows the values of average skin friction coefficient at different Reynolds number for different perforation configurations. Fig. 8(a) illustrates the value for two perforations and Fig. 8(b) illustrates the value for three perforations. Friction coefficient is calculated over the fin surfaces including perforations. It is seen that the friction coefficient values decrease with increase of Reynolds number. Solid fins have the largest friction coefficient value. Hexagonal perforated fins show almost same values as the square perforated ones. Total drag force has two components, one is due to surface shear stress which is called friction drag and another is called form or pressure drag. Due to lateral perforation, flow velocity decreases and flow circulates inside the perforations. For this reason the skin friction coefficient is smaller for fins with perforations. But in the case of solid fin, fluid interacts with larger surface area than the perforated ones. From Figs. 7 and 8 it can be stated that performance of perforated fins shows similar trend for two and three perforations. For this reason, further results (Figs. 9 and 10) will be presented only for the perforated fins having two perforations. Fig. 8 illustrates perforated fin effectiveness, PFE (3 PF) for different types of lateral perforated heat sinks. It can be said that fin effectiveness decreases with the increase of Reynolds number. From the figure it can also be predicted that perforated fins have higher fin effectiveness than that of solid fins (3 PF > 1) because of exchanging greater amount of heat to the ambient air. Though the Nusselt number value for solid fin is higher but due to greater surface area, perforated fins can perform better. From the figure it is also predicted that fins having triangular lateral perforations have the higher value of fin effectiveness. The main reason to use perforated fin is that it reduces the weight of the heat sink. For evaluating the thermal and fluid dynamic performances of the new types of fins, another term HTPE can be defined (Eq. (11)). From Fig. 9, it can be seen that HTPE values increase almost linearly with the increase of Reynolds number. From this analysis it can be predicted that hexagonal perforated fins have the largest value and the triangular perforated fins have the lowest Heat Transfer Performance Enhancement (HTPE) value. From the present analysis, it is seen that the thermal and fluid dynamic performances of the perforated fins can be increased by changing the perforation shapes. From this analysis it is also noted that triangular perforated fins have the lowest value of friction drag coefficient and thus the pumping power is the lowest. But its HTPE and fin effectiveness values are the lowest. From all the figures it is also observed that the HTPE value and PFE value are larger than one for the perforated fins. It means that all types of perforated fins show better thermal and hydraulic performance than the solid ones. The advantage of using perforated fin is that it reduces the weight of the heat sink as well as the cost.

Md.F. Ismail et al. / Energy 64 (2014) 632e639

Fig. 5. Flow path lines around solid and hexagonal perforated fins at different heights from the base (XZ plane), ReD ¼ 5000.

Fig. 6. Temperature distribution near the solid and hexagonal perforated fin surfaces at different heights from the base (XZ plane), ReD ¼ 5000.

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Md.F. Ismail et al. / Energy 64 (2014) 632e639

30

0.011

25

0.009

20

15

Solid Fins Square Perforated Circular Perforated Triangular Perforated Hexagonal Perforated

10

Average Friction coefficient

Nusselt Number

638

0.007

0.005 Solid Fins Square Perforated Circular Perforated Triangular Perforated Hexagonal Perforated

0.003

5

0.001

1500

2000

2500

3000

3500

4000

4500

5000

1000

1500

2000

3000

3500

4000

4500

5000

5500

Reynolds Number

Reynolds Number

(a)

(a) 0.011

Average Friction coefficient

30

25 Nusselt Number

2500

20

15 Solid Fins Square Perforated Circular Perforated Triangular Perforated Hexagonal Perforated

10

0.009

0.007

0.005

Solid Fins Square Perforated Circular Perforated Triangular Perforated Hexagonal Perforated

0.003

0.001 1000

2000

5

3000 4000 Reynolds Number

5000

6000

(b) 1500

2000

2500

3000 3500 4000 Reynolds Number

4500

5000

(b)

Fig. 8. Variation of skin friction coefficient at different Reynolds number: (a) fins having two perforations and (b) fins having three perforations.

Fig. 7. Nusselt number variation at different Reynolds number: (a) fins having two perforations and (b) fins having three perforations.

12

Square Perforated fins Hexagonal Perforated fin

10

5. Conclusions In this study three dimensional numerical computations are performed around different types of lateral perforated heat sinks. By introducing different shapes of perforations (having equal fin volume), better fin effectiveness and higher HTPE values for turbulent flow condition has been observed in this study. The purpose of this study is to investigate the effects of lateral perforation shapes on the thermal performance of heat sinks in comparison with the regular solid fins under turbulent flow condition. From this analysis, it is also observed that for the current range of Reynolds number (ReD ¼ 2000e5000), the modified keu turbulence model can successfully capture the flow recirculation and wake formation around the fin surfaces. The following conclusions can be drawn in brief from the present study:

Triangular Perforated fins Perforated fin effectiveness

Thus while designing perforated heat sink; designer should carefully select the perforation shape to get the desirable performance.

Circular Perforated fin

8

6

4

2

0 1500

2500

3500 Reynolds Number

4500

5500

Fig. 9. Perforated fin effectiveness (PFE) at different Reynolds number.

(1) Shape of lateral perforation has significant effects on the thermal performance of heat sinks under turbulent flow conditions. (2) Triangular perforated fins have the lowest and solid fins have the highest Nusselt number values.

Md.F. Ismail et al. / Energy 64 (2014) 632e639

Square Perforated Circular Perforated 1.2

Triangular Perforated Hexagonal Perforated

1.15

HTPE

1.1

1.05

1

0.95 1500

2500

3500

4500

5500

Reynolds Number Fig. 10. Heat transfer performance enhancement for different types of perforations.

(3) Hexagonal perforated fins show the highest fin effectiveness and the highest HTPE values. (4) Perforation shape is also an important factor for the better fluid dynamic performance. From this study, it is found that triangular perforated fins have lowest skin friction coefficient value than the other types of fins considered here.

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Nomenclature A: area AT: total fin area including perforation Cf,P: skin friction coefficient for the perforated fins Cf,S: skin friction coefficient for the solid fins D: fin thickness H: fin height h: convection heat transfer co-efficient L: fin length Nu: average Nusselt number, (hD)/l N: number of perforation Q: heat transfer rate ReD: Reynolds number (rUD)/m V: fin volume (L  H  D) 3 f: fin effectiveness l: fluid thermal conductivity m: fluid kinematics viscosity r: fluid density q: temperature s: shear stress N: free stream