1

Manuscript for Energy conversion and management 1

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Improved correlations of the thermal-hydraulic performance of large 3

size multi-louvered fin arrays for condensers of high power electronic 4

component cooling by numerical simulation 5

6

Jie Deng a, b,* 7

a School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, 8

Scotland, EH14 4AS, UK 9

b Southwest Jiaotong University, Chengdu 610031, China 10

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* Corresponding author: 12

School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, 13

Scotland, EH14 4AS, UK 14

Tel.: +44 (0) 1314514734 15

E-mail address: deng-jie2@163.com; j.deng@hw.ac.uk 16

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Improved correlations of the thermal-hydraulic performance of large 20

size multi-louvered fin arrays for condensers of high power electronic 21

component cooling by numerical simulation 22

23

Abstract 24

In view of a good thermal performance of the multi-louvered fin arrays, it is intended 25

to introduce the special fin structure into the design of condensers for high power 26

electricity converter cooling on the CRH (China Railway High-speed) trains in order 27

to improve the security of normal running for the high-speed trains. The geometrical 28

size of the multi-louvered fin arrays needed for the high power condensers is 29

relatively larger than that of the conventional louvered fin arrays and the flow state 30

within the large size multi-louvered fin arrays is different as well, with the Reynolds 31

numbers based on the louver pitch over the range of 2,850 to 11,000 in the present 32

study. Flow and heat transfer characteristics of the large size multi-louvered fin arrays 33

with various structural parameters are numerically simulated using Large Eddy 34

Simulation (LES). More reasonable parameters correlated with the thermal-hydraulic 35

performance of the large size multi-louvered fin arrays are identified compared to the 36

conventional characteristic parameters employed in the published work. Finally, 37

improved correlations for the Fanning friction f factor and the Colburn-j factor 38

representing the flow and heat transfer characteristics, respectively, are put forward in 39

terms of the identified parameters with special interval bounds for the condenser 40

design of high power electronic cooling. 41

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42

Keywords: Multi-louvered fin arrays; Condenser; High power electronic component 43

cooling; Thermal-hydraulic performance; Large Eddy Simulation (LES); Correlation 44

45

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List of symbols 46

47

Nomenclature

oA air-side total heat transfer area of the fin arrays, m2

cA minimum cross section area in the louver gap flow, m2

frA frontal surface area of the fin arrays, m2

pc isobaric specific heat capacity, J/(kg K)

vc volumetric specific heat capacity, J/(kg K)

hD equivalent hydraulic diameter of pipe flow, mm

f Fanning friction factor, –

dF flow length (or flow depth), mm

hF fin height, mm

pF fin pitch, mm

thF fin thickness, mm

oh air-side average heat transfer coefficient, W/(m2 °C)

j heat transfer Colburn factor, –

aL louver angle, °

flL non-louvered flat-landing region width, mm

hL louver height, mm

pL louver pitch, mm

trL louvered transitional region width, mm

LFA a characteristic parameter correlating louver pitch, fin pitch and louver

angle (Equation (22)), –

p pressure, Pa

Pr Prandtl number, –

q heat flux, W

5

DhRe Reynolds number based on the equivalent hydraulic diameter, –

LPRe Reynolds number based on the louver pitch, –

ijS resolved rate-of-strain tensor, –

T temperature, K

pT flat tube pitch, mm

t time, s

Nu Nusselt number, –

fru frontal velocity, m/s

iu filtered velocity in LES, m/s

maxV characteristic velocity in the louver gap flow, m/s

ix coordinate components, mm

x dimensionless distance of inlet effect region, –

Greek symbols

density, kg/m3

o kinematic viscosity of air, m2/s

dynamic or eddy viscosity, kg/(s m)

ij shear-stress tensor, –

thermal conductivity, W/(m K)

δ the effective spacing of the louver gap flow, mm

T temperature difference, K

totp total pressure drop, Pa

Subscript

air flowing air

in Inlet

out Outlet

48

49

50

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1 Introduction 51

Compact heat exchangers with multi-louvered fin arrays (also called parallel flow 52

heat exchangers) are widely used for automotive and air-conditioning cooling due to 53

their good thermal performances [1–5]. Investigations show that the high performance 54

of the multi-louvered fin arrays compared to smooth fins are attributed to the 55

continuous interruption of flow boundary layer by the extended louvered fins [2, 6]. 56

In view of a good performance of the louvered fin arrays, it is intended to introduce 57

the special fin structure into the design of condensers for high power electricity 58

converter cooling on the CRH (China Railway High-speed) trains. The high power 59

electricity converter as a power electronic component on the CRH trains, needs to be 60

cooled down during operation in order to sustain normal running of the high-speed 61

trains. Once the monitored core temperature in the converter achieves an upper limit 62

temperature (usually 80-85 °C), the converter will be compulsorily stopped by an 63

intelligent controller to protect the reliability and durability of the power electronic 64

component. In this situation, a good design of the condenser for the high power 65

electricity converter cooling is of significance to ensure the security of normal 66

running of the high-speed trains. 67

68

Figure 1 shows the photo of a power electronic converter and its condenser for the 69

component cooling on a CRH train. The air-side terminal heat radiator of the 70

prototype condenser made up of alloy aluminum is constructed by straight fin arrays 71

for forced air cooling. As the air-side thermal resistance of the condenser usually 72

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accounts for 70–90% of the whole condenser thermal resistance [1–3], it does make 73

sense to replace the original straight fins with multi-louvered fins to improve heat 74

dissipation effect. However, the air-side geometrical size of the condenser for the 75

power converter cooling shown in Figure 1 is relatively larger than that of the 76

compact heat exchangers with multi-louvered fin arrays for automotive and 77

air-conditioning cooling. Usually, most structural parameters of the multi-louvered fin 78

arrays for the compact heat exchangers are small, with fin pitch and louver pitch on 79

the magnitude order of several millimeters, as well as flow length in the range of 80

16–42 mm [1–5]. While the fin pitch and the louver pitch for the high power 81

condenser are around fivefold to tenfold as those for the small size multi-louvered fins. 82

Thus, it is necessary to carry out a further investigation to ascertain the 83

thermal-hydraulic performance of the large-size multi-louvered fin arrays used for 84

high power electronic cooling. 85

86

Figure 1 Photo of a high power electronic converter and its condenser for cooling on a 87

CRH train 88

89

With regard to the state of the art of investigations on the multi-louvered fin arrays, 90

the related subject matters mainly involve qualitative influence analysis of 91

geometrical parameters [1, 2, 7–12], effect mechanism of flow and heat transfer [6, 92

13–21], generalized correlations of thermal-hydraulic performance [22–32], frost 93

behaviors [33-37] and so on. Several types of typical multi-louvered fin arrays are 94

8

summarized in [29]. Specifically, the geometrical parameters comprise louver pitch 95

( pL ), fin pitch ( pF ), flow length ( dF ), louver angle ( aL ), fin thickness ( thF ), fin 96

height ( hF ), louver height ( hL ), non-louver region width ( flL ), louvered transitional 97

region width ( trL ), etc. In the aspect of a single geometrical parameter, it is reckoned 98

that the pressure drop and the average heat transfer coefficient of the louvered fin 99

arrays decrease as the increase of louver pitch [1, 2]. Tian [2] argued that an optimal 100

louver pitch to fin pitch ratio existed with a ratio range of 0.9–1.6. As for the flow 101

length, most of the sizes in the publications are in the range of 16–42 mm [1–5] and 102

the heat transfer coefficient decreases meanwhile pressure drop increases as the flow 103

length increases [9]. An optimal louver angle is also found in [1, 9]. The optimal angle 104

is 27° for the case of a flow length of 24 mm. Kim and Bullard [9] declared that the 105

flow length is one of the main influence parameters of pressure drop and the effect of 106

fin pitch on heat transfer is small. While Qi et al. [11] argued that the flow length, the 107

ratio of fin pitch to fin thickness and the number of louvers are the primary influence 108

parameters, using Taguchi method to evaluate the influence extent of five factors on 109

different control levels. 110

111

When it comes to the flow and heat transfer mechanism, it involves flow state, 112

thermal wake effect, flow transition and unsteadiness, etc. The flow state represents 113

the predominant flow direction, which can be delineated by flow efficiency [13, 14]. 114

The flow state is generally divided into two categories, duct flow and louver gap flow 115

[15–18]. Zhang and Tafti [19] classified two types of thermal wake interferences that 116

9

occurred in multi-louvered fins, inter-fin interference occurring between adjacent 117

rows of louvers and intra-fin interference appearing on subsequent louvers of the 118

same row or fin, respectively dominated by louver gap flow at a high flow efficiency 119

and by duct flow at a lower flow efficiency. It was verified by them that the thermal 120

wake effects could be expressed as functions of the flow efficiency and the fin pitch to 121

louver pitch ratio. Besides, the flow transition analysis is aimed at disclosing the flow 122

mechanism of multi-louvered fin arrays from steady flow to unsteady flow in terms of 123

flow state [6, 21]. It is argued in [6] that most of the louvers exhibit unsteadiness by a 124

Reynolds number of 1300 except for the entrance louver and the first two louvers 125

following it. 126

127

Apart from the qualitative analysis aforementioned, the correlations of 128

thermal-hydraulic performance respectively for the flow and heat transfer 129

characteristics are highly concerned as it can help to provide design rules for real 130

engineering. Most of the generalized correlations are fitted in terms of the 131

dimensionless Fanning friction f and Colburn-j factor, except that a few ones adopted 132

Stanton or Nusselt number for heat transfer characterization [2, 4, 9, 16, 22–32]. 133

Amongst all the correlations, it was worth mentioning that the generalized 134

correlations presented by Chang et al. [26, 27] was reckoned as the most widely 135

applicable formulas before 2005 because they collected the experimental data of 5 136

different types of louvered fin structures totally including 91 heat exchanger samples 137

from the literature. In 2006 they presented the amendment of their correlations with a 138

10

reduced error [29]. In 2009 Park and Jacobi [30] proposed more accurate, reliable and 139

updated predictive correlations using a comprehensive experimental database 140

consisting of 1030 heat-transfer and 1270 pressure-drop measurements from 9 141

independent laboratories for 126 sample heat exchangers. In recent years, relevant 142

studies tend to focus on the subject matter of frost behaviors of louvered fins [33–37] 143

and no obvious progress is observed for correlations. 144

145

Generally, the previous research work mainly considered the flow state with the 146

Reynolds numbers (based on the louver pitch) lower than 2,000 for the small size 147

multi-louvered fin arrays. The present study considered relatively large size 148

multi-louvered fin arrays with the Reynolds numbers (based on the louver pitch) over 149

the range of 2,850–11,000. To accurately capture the flow and temperature field 150

information in the multi-louvered fin arrays with higher Reynolds numbers, Large 151

Eddy Simulation (LES) was adopted to solve the 3-D transient flow and heat transfer 152

characteristics using commercial software package FLUENT. Furthermore, it was 153

noticed that the correlations [2, 4, 9, 16, 22–32] presented in the past usually utilized 154

the ratios pd LF , pp LF , ph LF , pth LF , etc. as the correlated parameters. For the 155

louver angle, usually 90aL was taken as corresponding correlated parameter. 156

General mathematical form of the conventional correlations is shown in Equations (1) 157

and (2). As a matter of fact, the dimensionless Fanning friction f and Colburn-j factor 158

are not a monotonic function (or not a rigorous exponential function) of each 159

correlated characteristic parameter, indicating that more reasonable parameters 160

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correlated with the thermal-hydraulic performance of the large size multi-louvered fin 161

arrays need to be identified for correlations. The present study will try to identify 162

more reasonable characteristic parameters and present improved correlations for the 163

multi-louvered fin arrays for the design of high power condenser unit. 164

165

65432

1

90Re1

b

p

l

b

p

d

b

p

h

b

p

p

b

ab

LPL

L

L

F

L

F

L

FLaf

(1) 166

167

3 4 5 6 72

1

2 Re90

c c c c cc

pc a h d l thLP

p p p p p

FL F F L Fj a

L L L L L

(2) 168

where ia , ib , and ic (i - integer) are fitting constants. 169

170

2 Numerical model 171

2.1 Geometrical model 172

The structure of typical multi-louvered fin arrays with flat tube is illustrated in Figure 173

2 [38]. In view of the complexity of the multi-louvered fin arrays in a heat exchanger 174

or condenser, it is efficient to make some assumptions for CFD (Computational Fluid 175

Dynamics) calculations to save the computational efforts. The louvered fin surface is 176

assumed to be flat. The heat source side at the flat tube wall is treated as equivalent 177

heat source boundary conditions. Single computational unit for the multi-louvered fin 178

arrays is considered for CFD calculations according to the symmetry and periodic 179

feature, as shown in Figure 3. Extended regions before the inlet and after the outlet 180

are considered to ensure the computational accuracy [6]. The inlet extended length is 181

12

chosen as 5 pF as it is found that the inlet effect is not significant, while an extended 182

length of 10–20 pF is considered for the outlet extended region since the wake flow 183

from the outlet has an evident effect on the flow characteristic. For a high frontal 184

velocity condition, a longer outlet extended length is needed. 185

186

Figure 2 Schematic sketch of the flat tube and multi-louvered fins 187

Figure 3 Computational domain of the flat tube and multi-louvered fins as well as 188

boundary conditions 189

190

2.2 Mathematical model 191

As the Reynolds numbers (based on the louver pitch) in the present study is over the 192

range of 2,850 to 11,000 presumably with flow unsteadiness, the Large Eddy 193

Simulation (LES) is adopted to solve the flow and heat transfer characteristics using 194

FLUENT software package. The top-hat or box filter is used in finite volume 195

implementation of LES in FLUENT [39]. Assuming an incompressible flow for the 196

problem studied, the space-filtering equations of continuity, momentum and energy 197

for the LES method are given in Equations (3–5) under Boussinesq hypothesis [40]. 198

199

0

i

i

ux

(3) 200

j

ij

j

ij

i

ji

j

ixxx

puu

xu

t

(4) 201

13

j

j

j

i

ij

i

j

j

j

vvx

q

x

u

x

upTu

xC

t

TC

(5) 202

203

where the shear-stress tensor ij is defined as 204

kkijijij SST

3

22)( (6) 205

The effects of the unresolved subgrid scales of motion on the resolved flow variables 206

are accounted for by the subgrid scale (SGS) terms in Equations (7) and (8). 207

i j i j i ju u u u (7) 208

(8) 209

210

The Smagorinsky-Lilly SGS model is used for the calculations and the local SGS 211

stresses are taken to be proportional to the local rate of strain of the resolved flow ijS , 212

with the SGS viscosity evaluated as [39, 40]. 213

214

ijijSGSSGS SSC 2)( 2 (9) 215

where SGSC is a constant; is the length scale; the resolved rate-of-strain tensor 216

ijS : 217

i

j

j

i

ijx

u

x

uS

2

1 (10) 218

219

In analogy to the SGS stresses, the subgird heat flux jq can be related to the filtered 220

large scale temperature gradient by an eddy diffusivity ( t ) as 221

j

tjx

Tq

(11) 222

14

For more information about the LES method, please refer to [39, 40]. 223

224

2.3 Boundary conditions and solving method 225

As shown in Figure 3, in the computational domain only half of the louver height is 226

considered due to symmetry and a symmetrical plane is justified at the half height 227

cross section of the louvered fins. The upper and the lower surface of the domain is 228

reckoned to be periodic boundary conditions due to the fact that there are 229

multi-louvers in the vertical direction with a regular distribution. The inlet is set as the 230

frontal face of the inlet extended region with an inlet temperature of 30 °C and a fixed 231

frontal velocity dependent upon the specific cases (2–15 m/s for the large size), while 232

the outflow condition is considered on the right side of the outlet extended region. 233

The inlet turbulent intensity and the turbulent length scale are determined by an 234

empirical formula referring to [40]. The wall surface of the side flat tube is set to be 235

the convective heat transfer boundary conditions as done in [41]. The thermal 236

conductivity of the alloy aluminum louvered fin is 110 W/(m K). Thermo-physical 237

properties of the flowing air are considered as piece-wise linear [39]. 238

239

With regard to the solving method based on the finite volume method, the 240

second-order interpolation scheme is used for the pressure terms and the bounded 241

central differencing scheme [39] is employed for the convective terms of the 242

momentum equations. The second-order upwind differencing scheme is used for the 243

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convective terms of the energy equations. To ensure the computational convergence, 244

calculation results of the laminar flow are adopted as the initial fields of the LES 245

calculations. The time step of computation is 0.03–0.05 s and the total calculating 246

time interval is 15–20 s. The residual control accuracies of the continuity, momentum 247

and energy equations for the convergent solution are 10-4, 10-4, 10-9, respectively. For 248

some special cases with a high frontal velocity, it is hard to attain a convergent 249

solution at some time steps and then the solution is reckoned to be convergent until 250

the variations of the outlet temperature and the pressure drop remain to be within 251

0.5%. 252

253

2.4 Parameter definitions 254

The maximum velocity through the louver gap flow is taken as the characteristic 255

velocity, as defined in Equation (12) [27, 38]. 256

max( cos )

fr p p

h p a th

u T FV

F F L F

(12) 257

258

The Reynolds numbers based on the louver pitch is defined in equation (13). 259

max cosRe P a

LP

o

V L L

(13) 260

261

Total pressure drop between the inlet and outlet 262

tot in outp p p (14) 263

264

16

The air-side average heat transfer coefficient ( oh ) is calculated by 265

TA

TTAuc

TA

qh

o

inoutfrfrairpair

o

air

o

)(, (15) 266

267

Please note that the numerical heat transfer temperature difference ( T ) in Equation 268

(15) is calculated by the temperature difference between the volume-averaged fin 269

temperature and the volume-averaged air temperature obtained by the 3-D CFD 270

calculations. 271

272

For the dimensionless processing of the pressure drop and heat transfer coefficient, 273

the numerical Fanning friction factor f is calculated by [22] 274

275

o

c

a i r

t o t

A

A

V

pf

2

m a x21

(16) 276

And the dimensionless heat transfer Colburn-j factor is given as [22, 26–30]. 277

3/1

m a x,

3/1Pr

PrRe Vc

hNuj

airpair

o

(17) 278

279

where the Nusselt number is 280

a i r

po LhNu

(18) 281

282

3 Model validation 283

The experimental data from Kim and Bullard [9] is chosen to carry out the CFD 284

model validation. The specific structural parameters of the multi-louvered fin are as 285

17

follows: dF = 20 mm, pF = 1.4 mm, pL = 1.7 mm, aL = 23°, pT = 10.15 mm, hF = 286

8.15 mm, hL = 6.4 mm. The structure of the corresponding flat tube can be found in 287

[38]. To begin with, it is necessary to conduct a grid independence testing to identify 288

the effective baseline cell size from the view point of saving computational efforts. 289

The condition with a frontal velocity of 1 m/s is chosen to verify the grid 290

independence, utilizing cell sizes of 0.1 mm, 0.09 mm, 0.85 mm, 0.08 mm, 0.075 mm, 291

0.07 mm for the fluid part (flowing air) and those of 0.08 mm, 0.07 mm, 0.06 mm, 292

0.05 mm, 0.05 mm, 0.05 mm for the solid part (alloy aluminum). Generated meshes 293

for the cases are comprised of 339, 458 cells, 519,544 cells, 629,419 cells, 821,744 294

cells, 893,234 cells, 985,590 cells, respectively. Figure 4 gives the calculated results 295

of the pressure drop and the average heat transfer coefficient, indicating that a 296

baseline cell size of 0.08 mm is suitable for the calculations. For the large size 297

multi-louvered fin arrays, similar grid independence is also carried out and a baseline 298

cell size of 0.4–0.5 mm is taken for the calculations. 299

300

Figure 4 Testing of different grid systems 301

302

Furthermore, three types of geometrical model, simplified full louver height model [3, 303

42], louvered fins model with flat-landing region [43], louvered fins model with both 304

flat-landing region and transitional region [44, 45] are compared for the model 305

validation. Note that the flow height ( hF = 8.15 mm) and the louver height ( hL = 6.4 306

mm) is aimed to the louvered fins model with flat-landing region. Concerning the 307

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simplified full louver height model, there is no flat-landing region and the louver 308

height should be treated the same as the fin height. As for the louvered fins model 309

with both flat-landing and transitional regions, there is not specific size reference and 310

thus the transitional region width is chosen as 0.5 mm, 0.8mm, 1.0 mm while the 311

flat-landing region width is set as 0.475 mm, 0.675mm, 0.875mm for different cases, 312

respectively. 313

314

Figures 5 and 6 give the calculation results of the total pressure drop and the average 315

heat transfer coefficient versus the frontal velocity for different calculation models, 316

respectively. It should be mentioned that the logarithmic mean temperature difference 317

is used here to calculate the average heat transfer coefficient [9] in order to make the 318

comparison on the same benchmark. Comparing with the experimental data from Kim 319

and Bullard [9], it suggests that the simplified full louver height model evidently 320

overestimates the pressure drop and the heat transfer coefficient, as the flat-landing 321

region is replaced by the louvered fin structure. Using mean deviation to evaluate the 322

relative error, the absolute mean deviation of the simplified full louver height model is 323

11.0%, while that of the louvered fins model with flat-landing region is 1.39%. 324

Regarding the louvered fins model with both flat-landing and transitional regions, it is 325

found that the absolute mean deviation for the cases of transitional region widths 0.5 326

mm, 0.8 mm, 1.0 mm are 2.32%, 4.61% and 6.08%, respectively. When it comes to 327

the heat transfer coefficient, a maximum absolute mean deviation of 23.1% is 328

observed for the simplified full louver height model. The absolute mean deviation of 329

19

the heat transfer coefficient for the louvered fins model with flat-landing region is 330

11.8%, while those for the louvered fins model with both flat-landing and transitional 331

regions with transitional region widths 0.5 mm, 0.8mm, 1.0 mm are 14.1%, 10.8% 332

and 8.8%, respectively. Combining the calculation errors of the pressure drop and the 333

heat transfer coefficient, it is assumed the louvered fins model with both flat-landing 334

and transitional regions can make a more accurate prediction, which will be used in 335

the following calculations. For the large size louvered fin arrays, the transitional 336

region width is chosen as 2.0 mm. 337

338

339

Figure 5 Calculated total pressure drop versus the frontal velocity for different 340

calculation models compared to the experimental data; FLR model – Louvered fins 341

model with the flat-landing region; FL&TR model – Louvered fins model with both 342

flat-landing and transitional regions 343

344

Figure 6 Calculated heat transfer coefficient versus the frontal velocity for different 345

calculation models compared to the experimental data; FLR model – Louvered fins 346

model with the flat-landing region; FL&TR model – Louvered fins model with both 347

flat-landing and transitional regions 348

349

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4 Results and discussion 350

4.1 Effect of the louver angle 351

A wide range of louver angles (15–30°) are chosen to explore the effect of the louver 352

angle on the flow and heat transfer characteristics of the large size multi-louvered fin 353

arrays, with specific geometrical parameters of thF = 0.5 mm, dF = 240 mm , 160 354

mm, pL = 8 mm, pF = 4.75 mm, hF = 25 mm, flL = 1.0 mm, trL = 2.0 mm. Figure 7 355

shows the total pressure drop and the average heat transfer coefficient versus various 356

louver angles with a flow length of 160 mm. It suggests that the total pressure drop 357

and the average heat transfer coefficient firstly decrease, followed with an increase 358

region till arriving at a maximum value after which a decrease appears again, as the 359

louver angle ( aL ) increases from 15° to 30° for a fixed frontal velocity ( fru ). The 360

minimum and maximum pressure drops arise at aL = 17° and aL = 27°, 361

respectively. The minimum heat transfer effect appears at aL = 17°, while the 362

maximum heat transfer effect occurs around aL = 25°–27°, implying that there is an 363

optimal louver angle for a good thermal performance of the multi-louvered fin arrays. 364

The optimal louver angle is close to the case of 24 mm flow depth in Ref. [9]. More 365

than that, the whole variation trend indicates that the pressure drop or the heat transfer 366

coefficient is not a monotonic function of the louver angle. It is therefore not accurate 367

to correlate the thermal-hydraulic performance with a monotonic mathematical 368

relationship in the form of equations (1) and (2) as employed in the conventional 369

correlations [2, 4, 9, 26–32]. 370

21

371

Figure 7 The pressure drop and the heat transfer coefficient versus various louver 372

angles ( dF = 160 mm) 373

374

With a careful examination on the geometrical relationship of the louvered fin arrays, 375

the effective spacing of the louver gap flow (δ) is defined in Equation (19) to 376

delineate the characteristic of the louver gap flow, as it is reckoned that the flow state 377

of the louver gap flow is effective with a high flow efficiency [15–19]. In order to get 378

a dimensionless effective spacing, the ratio of the effective spacing (δ) to the louver 379

spacing perpendicular to adjacent louvers ( )cos( ap LF ) is considered in Equation (20) 380

or (21). It is shown that )tan( app LFL is a characteristic parameter correlating 381

the louver pitch ( pL ), the fin pitch ( pF ) and the louver angle ( aL ). It suggests that the 382

three parameters pL , pF , aL take an interplay for the louvered gap flow. A notation 383

‘LFA’ is taken to represent the correlated parameter, as shown in Equation (22). 384

385

cos sinp a p aF L L L (19) 386

/ cosp am F L (20) 387

1 / tanp p am L F L (21) 388

/ tanp p aLFA L F L (22) 389

390

Figures 8 and 9 shows the pressure drop and the average heat transfer coefficient 391

versus LFA, respectively, for different frontal velocities ( fru ) with a flow length of 392

22

160 mm. The flow and heat transfer characteristics should be categorized by different 393

interval bounds of LFA. When 0.5 ≤ LFA ≤ 0.858, the pressure drop is an exponential 394

function of the LFA. The maximum pressure drop appears at δ ≈ Fth. In the ranges of 395

0< LFA ≤ 0.5 and 0.858 < LFA ≤ 1, the pressure drop decreases as the louver angle 396

increases, so does the average heat transfer coefficient. The average heat transfer 397

coefficient increases with the increase of LFA in the interval of 0.5 ≤ LFA ≤ 0.75 and 398

the optimal heat transfer effect appears in the interval of 0.75 ≤ LFA ≤ 0.858. The 399

maximum heat transfer tends to be where δ ≈ 2 thF . 400

401

Figure 8 The pressure drop versus LFA ( dF = 160 mm) 402

Figure 9 The average heat transfer coefficient versus LFA ( dF = 160 mm) 403

404

4.2 Effect of louver pitch to fin pitch ratio 405

The visualization experiment done by Howard [46] characterized the flow structure in 406

the louvered fin arrays as ‘effective flow’ and ‘low efficiency flow’ and declared that 407

the transitional region from a low efficiency flow to an effective flow happened at 408

pp FL = 0.7–0.8 with a louver angle of 20° in their study. When pp FL was higher 409

than 0.8, the flow structure was reckoned as effective, while a value of pp FL less 410

than 0.7 was assumed to be ineffective. A series of fin pitch values from 4.75 mm to 411

8.0 mm were chosen to ascertain the effect of pp FL over the range of 1.0–1.684. 412

Figures 10 and 11 give the pressure drop and the average heat transfer coefficient 413

23

versus louver pitch to fin pitch ratio, respectively. It shows that the pressure drop or 414

the average heat transfer coefficient is an exponential function of pp FL when 415

pp FL >1.3. While the mathematical relationship between the pressure drop (or the 416

heat transfer coefficient) and pp FL isn’t a monotonic function when pp FL < 1.3. 417

It indicates that the correlations for the thermal-hydraulic thermal performance of the 418

multi-louvered fin arrays should be fitted in different intervals of pp FL according 419

to the variation rules. UsingUse the characteristic parameter LFA identified in section 420

4.1 to show the variations of the Fanning friction factor f and the heat transfer 421

Colburn-j factor, as delineated in Figures 12 and 13, respectively. It is inferred that an 422

effective flow occurs when LFA > 0.5, corresponding to pp FL 1.3 and 423

23tanpp FL 0.56. Thus, for a good thermal performance of the multi-louvered 424

fin arrays, app LFL tan is recommended to be larger than 0.56. 425

426

Figure 10 The pressure drop versus the louver pitch to fin pitch ratio ( dF = 240 mm) 427

Figure 11 The average heat transfer coefficient versus the louver pitch to fin pitch 428

ratio ( dF = 240 mm) 429

Figure 12 The Fanning friction factor f versus LFA ( dF = 240 mm) 430

Figure 13 The heat transfer Colburn-j factor versus the louver pitch to fin pitch ratio 431

( dF = 240 mm) 432

433

4.3 Effect of the fin thickness 434

To figure out the effect of fin thickness on the thermal-hydraulic performance of the 435

24

multi-louvered fin arrays, various fin thicknesses (0.15mm, 0.25mm, 0.3mm, 0.4mm, 436

0.5mm, 0.6mm, 0.75mm, 0.9mm and 1.0mm) are considered for a comparison with 437

specific geometrical parameters dF = 240 mm and aL = 25°. Figure 14 gives the 438

pressure drop versus fin thickness for different frontal velocities ( fru ). It is further 439

found that the pressure drop is a piecewise exponential function of the fin thickness 440

( thF ) in the ranges of 0.15 mm ≤ thF ≤ 0.3 mm and 0.4 mm ≤ thF ≤ 1.0 mm, as 441

Figure 14 can be drawn in a semi-log plot given by Figure 15. Regarding the average 442

heat transfer coefficient versus fin thickness shown in Figure 16, no similar rule is 443

found as that of the pressure drop even using a semi-log plot or a double-log plot. To 444

further explore the rule of the average heat transfer coefficient versus fin thickness, 445

the ratio thF is identified as the characteristic parameter, inspiring by the 446

phenomenon that the louver gap flow is affected by the leading edge thickness of the 447

louvers. Table 1 lists the ratio thF for different fin thicknesses. Figure 17 shows 448

the variation trend of the average heat transfer coefficient in terms of the ratio thF . 449

It is found that the average heat transfer coefficient is a piecewise exponential 450

function of the fin thickness ( thF ) in the ranges of 0.924 ≤ thF ≤ 1.232 and 1.54 ≤ 451

thF ≤ 6.16. The line of demarcation tends to be δ = 1.5 thF . The rate of change of 452

the average heat transfer coefficient in the interval of 0.924 ≤ thF ≤ 1.232 is 453

slower than that in the interval of 1.54 ≤ thF ≤ 6.16. It seems that the parameter 454

thF is more suitable for characterizing the thermal-hydraulic performance 455

considering a specific interval. 456

457

25

Table 1. The ratio of the effective spacing of the louver gap flow to the fin thickness 458

459

Figure 14 The pressure drop versus the fin thickness ( dF = 240 mm) 460

Figure 15 The pressure drop versus the fin thickness in a semi-log plot 461

Figure 16 The average heat transfer coefficient versus the fin thickness ( dF = 240 462

mm) 463

Figure 17 The average heat transfer coefficient versus the ratio of thF 464

465

4.4 Effect of the flow length 466

As argued by [10, 11] that the flow length ( dF ) is one of the primary influence 467

parameters for pressure drop. Especially, for a longer flow length, the flow path 468

within the multi-louvered fin arrays will be very long at a high flow efficiency, 469

contributing to a higher pressure drop. Different flow lengths ( dF = 240 mm, 224 470

mm, 208 mm, 192 mm, 176 mm, 160 mm) are chosen by successively reducing 2 471

louvers (2 pL = 16 mm) for each case followed to determine the effect of flow length. 472

Figure 18 (a) and (b) give the pressure drop and the average heat transfer coefficient 473

versus flow length. It suggests that the pressure drop is an exponential function of 474

flow length ( dF ), while the heat transfer coefficient tends to be a linear variation as 475

the increase of flow length. 476

477

Figure 18 The pressure drop and the heat transfer coefficient versus the flow length 478

26

479

To effectively correlate the flow length with the thermal-hydraulic performance, the 480

dimensionless distance x is introduced in Equation (24) for the multi-louvered fin 481

arrays, in analogy to the definition of the inlet region dimensionless length (reciprocal 482

of the Graetz number) [47] in Equation (23). This is mainly because the flow 483

boundary layers along the louver gap flow are frequently distributed by the leading 484

edge of the louvers, representing a characteristic of the inlet length effect. 485

486

PrReDh

hDLx

(23) 487

PrRe LP

pd LFx

(24) 488

489

Figure 19 shows the Fanning friction factor f versus the dimensionless distance x 490

for different flow lengths. The absolute mean deviation of the scattered data points is 491

about 1.1% when a logarithmic relationship between the factor f and x is assumed. 492

It suggests that it is more reasonable to take the dimensionless distance x as the 493

correlated parameter in relation to the flow length ( dF ). Figure 20 provides the heat 494

transfer Colburn-j factor versus the dimensionless distance x . A logarithmic 495

relationship between the factor j and x is also observed, verifying the suitability of 496

the identified parameter in relation to the flow length ( dF ) as well as the Reynolds 497

number ( LPRe ) based on the presumption of inlet region effect. 498

499

27

Figure 19 The Fanning friction factor f versus the dimensionless distance x 500

Figure 20 The heat transfer Colburn-j factor versus the dimensionless distance x 501

502

4.5 Effect of louver height to fin height ratio 503

Different fin heights ( hF = 25 mm, 23 mm, 21.2 mm, 19.7 mm, 18.3 mm, 17.1 mm) 504

are chosen to get various ratios of louver height to fin height ( hh FL = 0.760, 0.7391, 505

0.7170, 0.6954, 0.6721, 0.6491), considering an identical condenser size with a 506

change of the number of tube rows in the transversal direction. Figure 21 shows the 507

pressure drop and the heat transfer coefficient versus hh FL in log-log plots, 508

indicating there is an exponential relationship between the pressure drop (or the heat 509

transfer coefficient) and hh FL . It is therefore reasonable to take hh FL as the 510

correlated parameter with regard to the louver height and fin height. 511

512

Figure 21 The pressure drop and the heat transfer coefficient versus hh FL 513

514

4.6 Correlations of the Fanning fraction f and the Colburn-j factor 515

According to the reasonable characteristic parameters identified for the 516

thermal-hydraulic performance of the large size multi-louvered fin arrays, the 517

correlations of the Fanning fraction f and the Colburn-j factor can be obtained through 518

multiple regression method [2, 26–30]. In view that the mathematical relationships of 519

the characteristic factors j, f versus the louver angle are not monotonic, the 520

28

correlations in terms of the Fanning fraction f and the Colburn-j factor are proposed in 521

the interval of 0.75 ≤ LFA ≤ 0.858, in which cases the heat transfer performance of the 522

multi-louvered fins is relatively better than in the other intervals. Additionally, the 523

ratio of louver pitch to fin pitch is limited to the ‘effective flow’ region, meeting 524

pp FL > 1)(tan56.0 aL . And the fin thickness is subject to the bound condition of δ 525

≥ 1.5 thF . Through multiple linear regression, the correlations of the Fanning fraction f 526

and Colburn-j factor are attained in Equations (25) and (26), respectively. 527

528

2.0217 2.5343 0.78620.3211

2.3501/0.07667 tan

Re Pr

d P hPa

LP p th h

F L LLf L

F F F

529

(25) 530

531

0.4825 0.433 1.1902 0.20740.6317

/ tan0.65842

Re Pr

d P d a hP

LP p p th h

F L F L LLj

F F F F

532

(26) 533

where the specific interval bounds are 0.75 ≤ LFA ≤ 0.858; pp FL > 1)(tan56.0 aL ; 534

δ ≥ 1.5 thF 535

536

Figure 22 shows the error limits of the fitting correlations for the Fanning friction f 537

and the Colburn-j factor. It indicates that 95% of the fitting data points are within the 538

±5% confidence interval limits. The mean deviation errors for the factors f, j are 539

2.39%, 1.76% respectively, which are very small compared to those in the literature 540

[26–30]. It is assumed that more reasonable characteristic parameters identified and 541

interval bounds give rise to a smaller error of the correlations, considering the specific 542

29

intervals of the correlated parameters for obtaining a good thermal-hydraulic 543

performance of the multi-louvered fin arrays. 544

545

Figure 22 Error limits of the fitting correlations for the Fanning friction f and the 546

Colburn-j factor 547

548

5 Conclusions 549

In order to enhance the heat dissipation effect of the condenser units for high power 550

electricity converter cooling on the CRH trains, the flow and heat transfer 551

characteristics of the relatively large size multi-louvered fin arrays are analyzed by 552

large eddy simulation for the condenser units. It is verified that the correlated 553

characteristic parameters of the thermal-hydraulic performance of the multi-louvered 554

fin arrays adopted in the previous studies, such as pd LF , 90aL , 555

pp LF , ph LF , pth LF , etc., are not optimum due to the fact that the dimensionless 556

Fanning friction f and Colburn-j factor are not a monotonic function (or not a rigorous 557

exponential function) of each correlated parameter. Thus, more reasonable parameters 558

correlated with the thermal-hydraulic performance of the large size multi-louvered fin 559

arrays are identified compared to the conventional characteristic parameters employed. 560

The correlated parameters are identified as app LFL tan , pp FL , thF , 561

PrReLPpd LFx and hh FL . 562

563

Moreover, specific interval bounds of the correlated parameters are ascertained for a 564

30

good thermal performance of the multi-louvered fin arrays. Improved correlations for 565

the Fanning friction f factor and the Colburn-j factor representing the flow and heat 566

transfer characteristics, respectively, are presented in terms of the identified 567

characteristic parameters with the specific interval bounds, with a view to obtaining a 568

good thermal-hydraulic performance of the multi-louvered fin arrays for high power 569

condenser design. Comparing with the error ranges of the correlations in the 570

published work, a smaller error of the correlations is obtained with 95% of the fitting 571

data points within the ±5% confidence interval limits. 572

573

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706

37

707

Table Captions: 708

709

Table 1. The ratio of the effective spacing of the louver gap flow to the fin thickness 710

711

712

Figure Captions: 713

Figure 1 Photo of a high power electronic converter and its condenser for cooling on a 714

CRH train 715

Figure 2 Schematic sketch of the flat tube and multi-louvered fins 716

Figure 3 Computational domain of the flat tube and multi-louvered fins as well as 717

boundary conditions 718

Figure 4 Testing of different grid systems 719

Figure 5 Calculated total pressure drop versus the frontal velocity for different 720

calculation models compared to the experimental data; FLR model – Louvered fins 721

model with the flat-landing region; FL&TR model – Louvered fins model with both 722

flat-landing and transitional regions 723

Figure 6 Calculated heat transfer coefficient versus the frontal velocity for different 724

calculation models compared to the experimental data; FLR model – Louvered fins 725

model with the flat-landing region; FL&TR model – Louvered fins model with both 726

flat-landing and transitional regions 727

Figure 7 The pressure drop and the heat transfer coefficient versus various louver 728

angles ( dF = 160 mm) 729

38

Figure 8 The pressure drop versus LFA ( dF = 160 mm) 730

Figure 9 The average heat transfer coefficient versus LFA ( dF = 160 mm) 731

Figure 10 The pressure drop versus the louver pitch to fin pitch ratio ( dF = 240 mm) 732

Figure 11 The average heat transfer coefficient versus the louver pitch to fin pitch 733

ratio ( dF = 240 mm) 734

Figure 12 The Fanning friction factor f versus LFA ( dF = 240 mm) 735

Figure 13 The heat transfer Colburn-j factor versus the louver pitch to fin pitch ratio 736

( dF = 240 mm) 737

Figure 14 The pressure drop versus the fin thickness ( dF = 240 mm) 738

Figure 15 The pressure drop versus the fin thickness in a semi-log plot 739

Figure 16 The average heat transfer coefficient versus the fin thickness ( dF = 240 740

mm) 741

Figure 17 The average heat transfer coefficient versus the ratio of thF 742

Figure 18 The pressure drop and the heat transfer coefficient versus the flow length 743

Figure 19 The Fanning friction factor f versus the dimensionless distance x 744

Figure 20 The heat transfer Colburn-j factor versus the dimensionless distance x 745

Figure 21 The pressure drop and the heat transfer coefficient versus hh FL 746

Figure 22 Error limits of the fitting correlations for the Fanning friction f and the 747

Colburn-j factor 748

Figures:

Figure 1 Photo of a high power electronic converter and its condenser for cooling on a

CRH train

Figure 2 Schematic sketch of the flat tube and multi-louvered fins

Figure 3 Computational domain of the flat tube and multi-louvered fins as well as

boundary conditions

Figure 4 Testing of different grid systems

Figure 5 Calculated total pressure drop versus the frontal velocity for different

calculation models compared to the experimental data; FLR model – Louvered fins

model with the flat-landing region; FL&TR model – Louvered fins model with both

flat-landing and transitional regions

Figure 6 Calculated heat transfer coefficient versus the frontal velocity for different

calculation models compared to the experimental data; FLR model – Louvered fins

model with the flat-landing region; FL&TR model – Louvered fins model with both

flat-landing and transitional regions

Figure 7 The pressure drop and the heat transfer coefficient versus various louver

angles ( dF = 160 mm)

Figure 8 The pressure drop versus LFA ( dF = 160 mm)

Figure 9 The average heat transfer coefficient versus LFA ( dF = 160 mm)

Figure 10 The pressure drop versus the louver pitch to fin pitch ratio ( dF = 240 mm)

Figure 11 The average heat transfer coefficient versus the louver pitch to fin pitch

ratio ( dF = 240 mm)

Figure 12 The Fanning friction factor f versus LFA ( dF = 240 mm)

Figure 13 The heat transfer Colburn-j factor versus the louver pitch to fin pitch ratio

( dF = 240 mm)

Figure 14 The pressure drop versus the fin thickness ( dF = 240 mm)

Figure 15 The pressure drop versus the fin thickness in a semi-log plot

Figure 16 The average heat transfer coefficient versus the fin thickness ( dF = 240

mm)

Figure 17 The average heat transfer coefficient versus the ratio of thF

Figure 18 The pressure drop and the heat transfer coefficient versus the flow length

Figure 19 The Fanning friction factor f versus the dimensionless distance x

Figure 20 The heat transfer Colburn-j factor versus the dimensionless distance x

Figure 21 The pressure drop and the heat transfer coefficient versus hh FL

Figure 22 Error limits of the fitting correlations for the Fanning friction f and the

Colburn-j factor

Tables:

Table 1. The ratio of the effective spacing of the louver gap flow to the fin thickness

apap LLLF sincos = 0.924 mm

thF (mm) 0.15 0.25 0.30 0.40 0.50 0.60 0.75 0.90 1.00

thF 6.160 3.696 3.080 2.310 1.848 1.540 1.232 1.0267 0.924