• No results found

Local Variation of Heat and Mass Transfer for Flow Over a Cavity and on a Flat Plate

N/A
N/A
Protected

Academic year: 2021

Share "Local Variation of Heat and Mass Transfer for Flow Over a Cavity and on a Flat Plate"

Copied!
369
0
0

Loading.... (view fulltext now)

Full text

(1)

Local Variation of Heat and Mass Transfer for

Flow Over a Cavity and on a Flat Plate

A DISSERTATION

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA

BY

Matthew E. Taliaferro

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Professor Richard J. Goldstein, Adviser

(2)
(3)

Acknowledgments

Choosing a PhD student requires a large risk on the part of the advisor, and I am grateful that Professor Goldstein took the chance and hired me. Working for him has been a great honor and pleasure. I can never thank you enough.

The OpenFOAM numerical simulations in section 2.1 were mostly set up and run by Dr. Matteo Angelino, and his continued help on otherOpenFOAM simulations are appreciated. Dr. John Gorman and Christopher Smith for their help on ANSYS CFX. What would have been three to four months of time spent on the turbulent numerical simulations was instead only a month because of their help.

Professors Terrance Simon and Alptekin Aksan always had generous open door policy. Professor Simon was always willing to patiently answer my numerous ques-tions on experimental heat transfer, while Professor Aksan was very generous with professional advice.

The machine shop was always extremely helpful to students that might’ve had a theoretical understanding of engineering, but little practical experience. Many thanks to Pam Wilson, her efforts helping me during the construction on the old ME building ensured the experiments were finished. Not sure how I could have completed this thesis without her. Many thanks to John Gardner who patiently answers the same questions from multiple graduate students everyday. The questions are new to me and your guidance is invaluable.

(4)

Finally, Susan Johnson. Your proof reading of this thesis was immensely helpful. Any errors, grammatical or otherwise, are solely my responsibility.

(5)

Dedication

To my wife, this dissertation is as much yours as it is mine. We have both dreamed of finishing this thesis, and have both dreamed of what might come after:

Had I the heavens’ embroidered cloths, Enwrought with golden and silver light, The blue and the dim and the dark cloths

Of night and light and the half light, I would spread the cloths under your feet:

But I, being poor, have only my dreams; I have spread my dreams under your feet; Tread softly because you tread on my dreams.

— W. B. Yeats

(6)

Abstract

Boundary layer theory for flat plates is fundamental to our understanding of fluid flow and heat transfer. However, most of the experimental and analytical work for thermal boundary layers focus on streamwise effects. Lateral changes of heat and mass transfer near a lateral singularity in the surface boundary conditions have not been as extensively studied. Lateral heat transfer is studied using OpenFOAM to run numerical simulations for heated strips of varying width, fluids with varying thermal properties, separation lengths, and unheated starting lengths. Turbulent mass transfer is studied using the naphthalene sublimation technique for heated strips of varying depths, widths, and freestream velocities. The lateral edge effect is found to scale with the conduction thickness for both turbulent and laminar boundary layer flows. For laminar boundary layer flow the lateral edge effect extends approximately three conduction thicknesses into the flow, while for turbulent boundary layer flow it extends approximately ten conduction thicknesses into the flow. The results are useful for modeling heat transfer from discrete electronic components. In addition, the results should serve as useful benchmarks for numerical fluid models and computations where lateral transport is important.

(7)

Table of Contents

Acknowledgments i Dedication iii Abstract iii Table of Contents v List of Tables x

List of Figures xii

Nomenclature xxxvi

1 Introduction 1

1.1 Outline of Present Work . . . 1

1.2 Theoretical Background . . . 2

1.2.1 General Equations . . . 2

1.2.2 Turbulent Boundary Layer Equations . . . 3

1.2.3 Convection Equations . . . 4

1.3 Literature Review . . . 5

(8)

1.3.2 Electronic Cooling . . . 6

1.3.3 Evaporation from Droplets . . . 10

2 Laminar Flow Over a Flat Plate 12 2.1 Semi-infinite Domain . . . 13 2.1.1 Similarity Solution . . . 13 2.1.2 Conduction Model . . . 18 2.1.3 Convection . . . 27 2.2 Finite Domain . . . 38 2.2.1 Conduction . . . 38 2.2.2 Convection . . . 42 2.3 Summary . . . 74

2.3.1 Semi-Infinite Lateral Extent . . . 74

2.3.2 Finite Lateral Extent . . . 75

3 Experimental Apparatus and Procedures 78 3.1 Instrumentation . . . 78

3.1.1 Voltage . . . 78

3.1.2 Temperature . . . 78

3.1.3 Velocity . . . 79

3.1.4 Data Acquisition and Control . . . 79

3.2 Apparatus . . . 80

3.2.1 Wind Tunnel . . . 80

3.2.2 Mass Transfer Mold . . . 81

3.2.3 Naphthalene Profile Measurement . . . 83

3.3 Experimental Procedures . . . 85

(9)

3.3.2 Flow Measurement . . . 94

3.3.3 Mass Transfer Plate . . . 95

3.3.4 Naphthalene Sublimation Technique . . . 97

3.4 Properties . . . 99 3.4.1 Air Properties . . . 99 3.4.2 Water Properties . . . 104 3.4.3 Naphthalene Properties . . . 106 3.5 Data Analysis . . . 108 3.5.1 Data Reduction . . . 108 3.5.2 Data Averaging . . . 115

3.5.3 Mass Stanton Number Calculation . . . 120

4 Turbulent Flow Over a Flat Plate 122 4.1 Flow Conditions . . . 123

4.1.1 Measurement Process . . . 123

4.1.2 Boundary Layer Profile . . . 124

4.1.3 Boundary Layer Parameters . . . 127

4.2 Mass Transfer Results . . . 132

4.2.1 Data Averaging . . . 132

4.2.2 Results . . . 138

4.2.3 Centerline Experimental Data . . . 139

4.2.4 General Correlation . . . 146

4.2.5 Semi-Infinite Correlation . . . 158

4.2.6 Lateral Extent of Edge Effect . . . 161

(10)

5 Turbulent Flow Over Cavities 170

5.1 Flow Conditions . . . 171

5.1.1 Numerical Simulations . . . 171

5.1.2 Grid Independence Study . . . 173

5.1.3 Flow Results for the Cavity . . . 175

5.2 Mass Transfer . . . 188 5.2.1 Data Averaging . . . 188 5.2.2 Results . . . 191 5.2.3 Lateral St Profile . . . 196 5.3 Summary . . . 226 6 Conclusion 228 6.1 Summary . . . 228

6.1.1 Laminar Flat Plate . . . 228

6.1.2 Turbulent Flat Plate . . . 229

6.1.3 Cavity . . . 230

6.2 Further Work . . . 231

Reference List 233 A Example Calculations 240 A.1 Example Calculations of Laminar Heat Transfer Quantities . . . 240

A.1.1 Example Calculation for Nu . . . 240

B Regression Analysis 245 B.1 Linear Least Squares with Errors in the Variables . . . 245

B.2 Polynomial Regression with Errors in the Variables . . . 247

(11)

C Uncertainty Calculations 249

C.1 Platinum Resistance Thermometer . . . 249

C.2 Thermocouples . . . 251

C.3 Uncertainty of the Experimental Stanton Number . . . 254

C.3.1 Uncertainty in the Naphthalene Vapor Density . . . 255

C.3.2 Uncertainty in the Freestream Velocity . . . 257

C.3.3 Uncertainty in the Sublimed Naphthalene Depth . . . 261

C.4 Uncertainty of the Experimental Normalized Stanton Number . . . . 261

D Code Examples 263 D.1 Data Acquisition . . . 263

E Selected data 270 E.1 Laminar . . . 270

E.1.1 Laminar Finite . . . 270

E.2 Turbulent . . . 292

E.2.1 Streamwise Strips . . . 292

E.3 Turbulent Cavities . . . 297

E.3.1 Centerline St . . . 297

E.3.2 St Contours . . . 305

(12)

List of Tables

2.1 Boundary conditions for laminar modeling . . . 15

2.2 Spanwise extent of lateral edge effect . . . 38

2.3 Grid information for finite laminar simulations . . . 43

2.4 Permutations of widths for numerical modeling . . . 43

3.1 Air properties at 1 bar . . . 99

3.2 List of naphthalene properties at 1.013×105Pa . . . 107

3.3 Example generalized ESD calculation . . . 109

3.4 Intersection Location . . . 115

4.1 Value and location of u0+ from various sources . . . 126

4.2 Boundary layer fitted parameters . . . 129

4.3 Boundary layer profile parameters . . . 131

4.4 Estimated random uncertainty of centerlineSt from turbulent experi-ments . . . 133

4.5 Estimated random uncertainty for normalized St . . . 138

4.6 Total random uncertainty for the St from turbulent experiments . . . 139

4.7 Increase in centerline St . . . 154

5.1 Property values for numerical simulations . . . 172

(13)

5.3 Estimated error of numerical simulations . . . 175

5.4 Various normalized reattachment lengths from numerical modeling . . 176

5.5 Estimates of vortex dissipation . . . 179

5.6 Estimated random uncertainty of centerline St from turbulent cavity experiments . . . 190

5.7 Estimated random uncertainty symmetrySt from turbulent cavity ex-periments . . . 191

5.8 Total estimated random uncertainty of St from turbulent cavity ex-periments . . . 192

5.9 . . . 193

5.10 Flow parameters . . . 194

5.11 Normalized location of experimental maximum St . . . 195

A.1 Values for Laminar Example Calculation . . . 241

C.1 Uncertainty of constants used for PRT calibration . . . 250

(14)

List of Figures

1.1 Temperature contour for laminar flow . . . 8

1.2 Raw data from Fassio (2009) . . . 9

1.2a U∞ = 3.93 m s−1 . . . 9

1.2b U∞ = 13.6 m s−1 . . . 9

1.3 Dimensionless fit of lateral sublimed naphthalene depth profile . . . . 10

2.1 Geometry for semi-infinite domain . . . 13

2.2 Nondimensional temperature contours for Pr = 1 . . . 16

2.2a Case 1 . . . 16

2.2b Case 2 . . . 16

2.2c Case 3 . . . 16

2.2d Case 4 . . . 16

2.3 Nondimensional temperature contours for Pr = 10. . . 17

2.3a Case 1 . . . 17

2.3b Case 2 . . . 17

2.3c Case 3 . . . 17

2.3d Case 4 . . . 17

2.4 Problem domain for laminar conduction . . . 20

2.4a Modified from fig. 2.1 showing conduction modeling domain. . 20

(15)

2.4c Case 4 from table 2.1 . . . 20

2.5 Comparison of estimated and least square fit of coefficients . . . 25

2.5a Case 1, eq. (2.15a) . . . 25

2.5b Case 2, eq. (2.31a) . . . 25

2.5c Case 3, eq. (2.31b) . . . 25

2.5d Case 4, eq. (2.25a) . . . 25

2.6 Conduction thickness . . . 28 2.7 Comparison of expected Nuz→∞, Reξ =5×103 . . . 32 2.7a Case 1 . . . 32 2.7b Case 2 . . . 32 2.7c Case 3 . . . 32 2.7d Case 4 . . . 32 2.8 Comparison of expected Nuz→∞, Reξ =5×104 . . . 33 2.8a Case 1 . . . 33 2.8b Case 2 . . . 33 2.8c Case 3 . . . 33 2.8d Case 4 . . . 33

2.9 Laminar grid independence study . . . 34

2.10 Convection results for case 1 . . . 35

2.11 Convection results for case 2 . . . 35

2.12 Convection results for case 3 . . . 36

2.13 Convection results for case 4 . . . 36

2.14 ζz∗ for case 1 . . . 37

2.15 Geometry for finite domain . . . 39

2.16 Modified from fig. 2.15 showing conduction modeling domain. . . 40

(16)

2.17a Case 1 from table 2.1 . . . 40

2.17b Case 4 from table 2.1 . . . 40

2.18 Estimate of parameter a from eq. (2.58) for case 2 . . . 45

2.19 Estimate of parameter pfrom eq. (2.58) for case 2 . . . 46

2.20 Estimate of parameter b from eq. (2.58) for case 2 . . . 46

2.21 Estimate of parameter a from eq. (2.58) for case 3 . . . 47

2.22 Estimate of parameter pfrom eq. (2.58) for case 3 . . . 47

2.23 Estimate of parameter b from eq. (2.58) for case 3 . . . 48

2.24 Effect of finite widths on centerline normalized Nu for case 1 . . . 49

2.25 Effect of finite widths on centerline normalized Nu for case 2 . . . 50

2.26 Effect of finite widths on centerline normalized Nu for case 3 . . . 50

2.27 Effect of finite widths on centerline normalized Nu for case 4 . . . 51

2.28 Center Nu for case 1,w1 = 4 mm,w2 = 4 mm . . . 52

2.29 Center Nu for case 1,w1 = 0.25 mm, w2 = 4 mm . . . 52

2.30 Center Nu for case 2,w1 = 0.25 mm, w2 = 50 mm . . . 53

2.31 Center Nu for case 2,w1 = 4 mm,w2 = 0.25 mm . . . 53

2.32 Center Nu for case 3,w1 = 4 mm,w2 = 4 mm . . . 53

2.33 Center Nu for case 3,w1 = 0.25 mm, w2 = 4 mm . . . 54

2.34 Center Nu for case 4,w1 = 0.25 mm, w2 = 0.25 mm . . . 54

2.35 Center Nu for case 4,w1 = 1 mm,w2 = 1 mm . . . 54

2.36 Lateral Nu for case 1,w1 = 0.25 mm,w2 = 50 mm . . . 56

2.37 Lateral Nu for case 1,w1 = 4 mm, w2 = 0.25 mm . . . 56

2.38 Lateral Nu for case 4,w1 = 0.25 mm,w2 = 50 mm . . . 57

2.39 Lateral Nu for case 4,w1 = 4 mm, w2 = 0.25 mm . . . 57

2.40 Lateral Nu for case 2,w1 = 0.25 mm,w2 = 50 mm . . . 59

(17)

2.42 Contour of θw2/θw1 for case 2 . . . 61

2.43 Lateral Nu for case 3,w1 = 0.25 mm,w2 = 50 mm . . . 61

2.44 Lateral Nu for case 3,w1 = 4 mm, w2 = 0.25 mm . . . 62

2.45 Contour of ζ1.1 for case 1 . . . 63

2.46 Contour of ζ1.1 for case 2 . . . 63

2.47 Contour of ζ1.1 for case 3 . . . 64

2.48 Contour of ζ1.1 for case 4 . . . 64

2.49 Contour of ζ1.1/w∗1 for case 1 . . . 65

2.50 Contour of ζ1.1/w∗1 for case 2 . . . 66

2.51 Contour of ζ1.1/w∗1 for case 3 . . . 66

2.52 Contour of ζ1.1/w∗1 for case 4 . . . 67

2.53 Spanwise average Nu and maximum nondimensional temperature for case 4 . . . 69

2.54 Spanwise average Nu/Nuz→∞ for heated section, case 4 . . . 70

2.55 Spanwise average Nu/Nuz→∞, case 2 . . . 71

2.56 Spanwise average Nu/Nuz→∞ for heated section, case 2 . . . 72

2.57 Spanwise average Nu/Nuz→∞ for heated section, case 3 . . . 73

2.58 Maximum nondimensional temperature,θw1/θz→∞, case 3 . . . 73

2.59 Spanwise heat transfer, shown in eq. (2.70), for case 3 . . . 74

3.1 Photograph of author with some of the test equipment . . . 80

3.2 Sketch of the wind tunnel . . . 81

3.3 Sketch of the wind tunnel test section . . . 81

3.4 Mass transfer base plate . . . 82

3.5 Rectangular active surface . . . 83

(18)

3.7 Phase diagram for sodium sulfate and water . . . 87

3.8 Fit for thermocouple temperature . . . 88

3.9 Goodness of thermocouple fit . . . 89

3.9a Mean square error for polynomial fitting . . . 89

3.9b Residuals for thermocouple fit . . . 89

3.10 Linear fit for LVDT . . . 90

3.11 Goodness of LVDT fit . . . 91

3.11a Mean square error for LVDT polynomial fitting . . . 91

3.11b Residuals for LVDT linear fit . . . 91

3.12 Histogram of random uncertainty of the LVDT depth measurements . 92 3.13 Fit for hotwire anemometer on 2015/08/07 . . . 93

3.13a Fit for hotwire anemometer . . . 93

3.13b Velocity residuals for hotwire anemometer fit . . . 93

3.14 Best fit for dynamic viscosity of air . . . 101

3.15 Residuals and MSE for dynamic viscosity of air . . . 102

3.15a MSE for dynamic viscosity of air . . . 102

3.15b Normalized residuals for dynamic viscosity of air . . . 102

3.16 Best fit of thermal conductivity of air . . . 102

3.17 Residuals and MSE for thermal conductivity of air . . . 103

3.17a MSE for thermal conductivity of air . . . 103

3.17b Normalized residuals for thermal conductivity of air . . . 103

3.18 Best fit of specific heat of air . . . 103

3.19 Residuals and MSE for specific heat of air . . . 104

3.19a MSE for specific heat of air . . . 104

3.19b Normalized residuals for specific heat of air . . . 104

(19)

3.20a Ideal gas law for air compared to table 3.1 . . . 105

3.20b Normalized residuals for density of air . . . 105

3.21 Histograms of generalized ESD applied to rim corrections . . . 110

3.21a Histogram with original slopes . . . 110

3.21b Histogram with corrected rim slopes . . . 110

3.22 Edge fit for data . . . 112

3.22a Edge fit for data closest to left side of data . . . 112

3.22b Edge fit for data closest to right side of data . . . 112

3.23 LVDT probe near the rim . . . 113

3.24 Polynomial intersection fit for data . . . 114

3.24a Polynomial estimate of intersection for left side of data . . . . 114

3.24b Polynomial estimate of intersection for right side of data . . . 114

3.25 Estimate of intersection using derivatives . . . 115

3.25a Estimate of intersection using derivatives for left side of data . 115 3.25b Estimate of intersection using derivatives of intersection for right side of data . . . 115

3.26 Estimate of intersection using derivatives . . . 116

3.26a Estimate of intersection using derivatives of intersection for left side of data . . . 116

3.26b Estimate of intersection using derivatives of intersection for right side of data . . . 116

3.27 Cumulative effect of successive corrections on raw sublimed naphtha-lene data . . . 116

4.1 Estimated boundary layer velocity profile . . . 125

(20)

4.3 Velocity defect profile . . . 128

4.4 Turbulent virtual origin calculation for U∞ = 23.13 m s−1 . . . 130

4.5 Skin friction coefficient . . . 131

4.6 Centerline St for 5 m s−1 freestream velocity and2.48 mm width strip 133 4.7 Centerline St for 5 m s−1 freestream velocity and19.77 mmwidth strip 134 4.8 Centerline St for 20 m s−1 freestream velocity and2.48 mmwidth strip 134 4.9 Centerline St for 20 m s−1 freestream velocity and19.77 mm width strip 135 4.10 Lateral averaging for 5 m s−1 freestream velocity and 2.48 mm width strip . . . 136

4.11 Lateral averaging for 5 m s−1 freestream velocity and 19.77 mm width strip . . . 136

4.12 Lateral averaging for 20 m s−1 freestream velocity and 2.48 mm width strip . . . 137

4.13 Lateral averaging for20 m s−1 freestream velocity and19.77 mmwidth strip . . . 137

4.14 Averaged St for U∞= 5.24 m s−1 for various widths . . . 140

4.14a Averaged St for W = 2.48 mm . . . 140

4.14b Averaged St for W = 5.00 mm . . . 140

4.14c Averaged St for W = 9.77 mm . . . 140

4.14d Averaged St for W = 19.8 mm . . . 140

4.15 Averaged St for U∞= 9.86 m s−1 for various widths . . . 141

4.15a Averaged St for W = 2.48 mm . . . 141

4.15b Averaged St for W = 5.00 mm . . . 141

4.15c Averaged St for W = 9.77 mm . . . 141

4.15d Averaged St for W = 19.8 mm . . . 141

(21)

4.16a Averaged St for W = 2.48 mm . . . 142

4.16b Averaged St for W = 5.00 mm . . . 142

4.16c Averaged St for W = 9.77 mm . . . 142

4.16d Averaged St for W = 19.8 mm . . . 142

4.17 Diffusion thickness . . . 143

4.18 Centerline St for U∞ = 5.24 m s−1, ξ= 0.53 m, n= 5.17 . . . 144

4.19 Centerline St for U∞ = 9.86 m s−1, ξ= 0.60 m, n= 5.95 . . . 145

4.20 Centerline St for U∞ = 20.72 m s−1, ξ= 0.65 m,n = 6.64 . . . 145

4.21 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 5.24 m s−1,W = 2.48 mm . . . 147

4.22 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 5.24 m s−1,W = 5.00 mm . . . 147

4.23 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 5.24 m s−1,W = 9.77 mm . . . 148

4.24 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 5.24 m s−1,W = 19.77 mm . . . 148

4.25 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 9.86 m s−1,W = 2.48 mm . . . 149

4.26 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 9.86 m s−1,W = 5.00 mm . . . 149

4.27 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 9.86 m s−1,W = 9.77 mm . . . 150

4.28 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 9.86 m s−1,W = 19.77 mm . . . 150

4.29 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 20.72 m s−1, W = 2.48 mm . . . 151

(22)

4.30 Comparison of normalized experimental data with fit from eq. (4.18) for U∞= 20.72 m s−1, W = 5.00 mm . . . 151 4.31 Comparison of normalized experimental data with fit from eq. (4.18)

for U∞= 20.72 m s−1, W = 9.77 mm . . . 152 4.32 Comparison of normalized experimental data with fit from eq. (4.18)

for U∞= 20.72 m s−1, W = 19.77 mm . . . 152 4.33 Centerline St for U∞ = 5.24 m s−1 and various widths . . . 154

4.34 Centerline St for U∞ = 9.86 m s−1 and various widths . . . 154

4.35 Centerline St for U∞ = 20.72 m s−1 and various widths . . . 155 4.36 Comparison of eq. (4.19a) with data from U∞ = 5.24 m s−1 . . . 156

4.37 Comparison of eq. (4.19a) with data from U∞ = 9.86 m s−1 . . . 156

4.38 Comparison of eq. (4.19a) with data from U∞ = 20.72 m s−1 . . . 157

4.39 Various curves for W∗ ≈10 . . . 158

4.40 Various curves for W∗ ≈17.8 . . . 159

4.41 Various curves for W∗ ≈31.6 . . . 159

4.42 Various curves for W∗ ≈56.2 . . . 160

4.43 Fit for normalized St for a strip of semi-infinite width. . . 164

4.43a Comparison of data with W∗ >40with eqs. (4.20) and (4.21) 164

4.43b Root mean square of the residuals (with a window of 31) be-tween eq. (4.20) and experimental results. . . 164 4.44 Dimensional and nondimensional lateral extent for a freestream

veloc-ity of 5.24 m s−1 . . . 165

4.44a Lateral extent for a freestream velocity of 5.24 m s−1 . . . 165

4.44b Lateral extent for a freestream velocity of 5.24 m s−1 . . . 165

4.45 Dimensional and nondimensional lateral extent for a freestream veloc-ity of 9.86 m s−1 . . . 166

(23)

4.45a Lateral extent for a freestream velocity of 9.86 m s−1 . . . 166

4.45b Lateral extent for a freestream velocity of 9.86 m s−1 . . . 166

4.46 Dimensional and nondimensional lateral extent for a freestream veloc-ity of 20.72 m s−1 . . . 167

4.46a Lateral extent for a freestream velocity of 20.72 m s−1 . . . 167

4.46b Lateral extent for a freestream velocity of 20.72 m s−1 . . . 167

5.1 Sketch of cavity geometry . . . 171 5.2 Wall shear stress for grid independence study . . . 179 5.2a Coarse grid . . . 179 5.2b Medium grid . . . 179 5.2c Fine grid . . . 179 5.3 Numerical skin friction coefficient . . . 180 5.4 Streamlines near the backward facing step . . . 181 5.4a 0.254 mm deep cavity . . . 181

5.4b 0.597 mm deep cavity . . . 181

5.4c 2.29 mm deep cavity . . . 181

5.4d 8.84 mm deep cavity . . . 181

5.5 Three dimensional streamlines . . . 182 5.6 Flow information in the z-yplane near the backward facing step, x=

2.96×10−3m, x/d= 0.335 . . . 183

5.6a In-plane streamlines and velocity vectors . . . 183 5.6b Filled non-dimensional temperature contour . . . 183 5.6c Filled contour of the normalized turbulent kinetic energy . . . 183 5.7 Flow information in thez-y plane in the primary recirculation region,

(24)

5.7a In-plane streamlines and velocity vectors . . . 184 5.7b Filled non-dimensional temperature contour . . . 184 5.7c Filled contour of the normalized turbulent kinetic energy . . . 184 5.8 Flow information in the z-y plane near the reattachment point, x =

0.0504 m,x/d= 5.70 . . . 185

5.8a In-plane streamlines and velocity vectors . . . 185 5.8b Filled non-dimensional temperature contour . . . 185 5.8c Filled contour of the normalized turbulent kinetic energy . . . 185 5.9 Flow information in thez-yplane after flow reattachment,x= 0.075 m,

x/d= 8.48 . . . 186

5.9a In-plane streamlines and velocity vectors . . . 186 5.9b Filled non-dimensional temperature contour . . . 186 5.9c Filled contour of the normalized turbulent kinetic energy . . . 186 5.10 Flow information in thez-yplane after flow reattachment,x= 0.12 m,

x/d= 13.6 . . . 187

5.10a In-plane streamlines and velocity vectors . . . 187 5.10b Filled non-dimensional temperature contour . . . 187 5.10c Filled contour of the normalized turbulent kinetic energy . . . 187 5.11 Centerline mass transfer between experimental results and numerical

simulation for U∞ ∼20 m s−1 and d= 0.254 mm . . . 188

5.12 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼20 m s−1 and d= 8.84 mm . . . 189

5.13 Lateral averaging for two streamwise locations and depths for10 m s−1 200

5.13a Lateral averaging for d= 0.254 mmat x= 4.59 mm . . . 200

5.13b Lateral averaging for d= 0.254 mmat x= 97.1 mm . . . 200

(25)

5.13d Lateral averaging for d= 8.84 mm atx= 101 mm . . . 200

5.14 Lateral averaging for two streamwise locations and depths for20 m s−1 201

5.14a Lateral averaging for d= 0.254 mmat x= 4.83 mm . . . 201

5.14b Lateral averaging for d= 0.254 mmat x= 97.3 mm . . . 201

5.14c Lateral averaging for d= 8.84 mm atx= 42.9 mm . . . 201

5.14d Lateral averaging for d= 8.84 mm atx= 101 mm . . . 201

5.15 Contour of experimentalSt forU∞ ∼10 m s−1 andd= 0.597 mmfrom

data taken on 2016/08/07-11 . . . 202 5.16 Contour of experimentalSt forU∞ ∼10 m s−1 andd= 0.597 mmfrom

data taken on 2016/08/08-22 . . . 202 5.17 Contour of experimentalSt forU∞ ∼20 m s−1 andd= 0.597 mmfrom

data taken on 2016/08/07-18 . . . 203 5.18 Contour of experimentalSt forU∞ ∼20 m s−1 andd= 0.597 mmfrom

data taken on 2016/08/08-11 . . . 203 5.19 Lateral profiles of theSt vsz at various streamwise locations forU∞∼

10 m s−1 and d= 0.254 mm. . . 204

5.20 Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 10 m s−1 and d= 8.84 mm . . . 204

5.21 Comparison of centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d = 0.254 mm . . . . 205

5.22 Comparison of centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d = 0.597 mm . . . . 205

5.23 Comparison of centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d = 2.29 mm . . . . 206

5.24 Comparison of centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d = 8.84 mm . . . . 206

(26)

5.25 Comparison of normalized centerline mass transfer between experimen-tal results and numerical simulation forU∞ ∼10 m s−1 andd= 0.254 mm207 5.26 Comparison of normalized centerline mass transfer between

experimen-tal results and numerical simulation forU∞ ∼10 m s−1 andd= 0.597 mm207

5.27 Comparison of normalized centerline mass transfer between experimen-tal results and numerical simulation for U∞∼10 m s−1 and d= 2.29 mm207

5.28 Comparison of normalized centerline mass transfer between experimen-tal results and numerical simulation for U∞∼10 m s−1 and d= 8.84 mm208

5.29 Location of global and centerline maximumSt, lines for experimental

data hold no significance beyond aiding readability. . . 208 5.30 Normalized location of maximum St . . . 209

5.31 Location of global and centerline maximumShd compared with lami-nar data of Kondoh et al. (1993). . . 209 5.32 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 0.0935 mm,x/d= 0.368 . . . 210

5.32a Filled non-dimensional temperature contour . . . 210 5.32b Filled contour of the normalized turbulent kinetic energy . . . 210 5.33 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 0.284 mm, x/d= 1.12 . . . 210

5.33a Filled non-dimensional temperature contour . . . 210 5.33b Filled contour of the normalized turbulent kinetic energy . . . 210 5.34 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 0.981 mm, x/d= 3.86 . . . 211

5.34a Filled non-dimensional temperature contour . . . 211 5.34b Filled contour of the normalized turbulent kinetic energy . . . 211

(27)

5.35 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 1.22 mm, x/d= 4.80 . . . 211

5.35a Filled non-dimensional temperature contour . . . 211 5.35b Filled contour of the normalized turbulent kinetic energy . . . 211 5.36 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 3.08 mm, x/d= 12.1 . . . 212

5.36a Filled non-dimensional temperature contour . . . 212 5.36b Filled contour of the normalized turbulent kinetic energy . . . 212 5.37 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 23.8 mm, x/d= 93.7 . . . 212

5.37a Filled non-dimensional temperature contour . . . 212 5.37b Filled contour of the normalized turbulent kinetic energy . . . 212 5.38 Flow information in thez-yplane after flow reattachment,d= 0.254 mm,

x= 121 mm, x/d= 477. . . 213

5.38a Filled non-dimensional temperature contour . . . 213 5.38b Filled contour of the normalized turbulent kinetic energy . . . 213 5.39 Comparison of the experimental St and numerical shear stress and

turbulent kinetic energy for d= 0.254 mm . . . 214

5.39a Filled contour of the skin friction coefficient from numerical simulation . . . 214 5.39b Contour of numerical turbulent kinetic energy at y+ = 17.6

above the cavity surface . . . 214 5.39c Contour of experimentalSt forU∞ ∼10 m s−1 from data taken

on 2016/08/15-17 . . . 214 5.40 Comparison of the experimental St and numerical shear stress and

(28)

5.40a Filled contour of the skin friction coefficient from numerical simulation . . . 215 5.40b Contour of numerical turbulent kinetic energy at y+ = 17.5

above the cavity surface . . . 215 5.40c Contour of experimentalSt forU∞ ∼10 m s−1 from data taken

on 2016/08/07-11 . . . 215 5.41 Comparison of the experimental St and numerical shear stress and

turbulent kinetic energy for d= 2.29 mm . . . 216

5.41a Filled contour of the skin friction coefficient from numerical simulation . . . 216 5.41b Contour of numerical turbulent kinetic energy at y+ = 31.4

above the cavity surface . . . 216 5.41c Contour of experimentalSt forU∞ ∼10 m s−1 from data taken

on 2016/07/28-20 . . . 216 5.42 Comparison of the experimental St and numerical shear stress and

turbulent kinetic energy for d= 8.84 mm . . . 217

5.42a Filled contour of the skin friction coefficient from numerical simulation . . . 217 5.42b Contour of numerical turbulent kinetic energy at y+ = 31.2

above the cavity surface . . . 217 5.42c Contour of experimentalSt forU∞ ∼10 m s−1 from data taken

on 2016/07/18-22 . . . 217 5.43 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations

for U∞∼10 m s−1 and d= 2.29 mm . . . 218

5.44 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼10 m s−1 and d= 4.32 mm . . . 218

(29)

5.45 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼10 m s−1 and d= 8.84 mm . . . 219 5.46 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations

for U∞∼20 m s−1 and d= 2.29 mm . . . 220

5.47 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼20 m s−1 and d= 4.32 mm . . . 220

5.48 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼20 m s−1 and d= 8.84 mm . . . 221

5.49 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼10 m s−1 and d= 0.254 mm . . . 222

5.50 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼10 m s−1 and d= 0.401 mm . . . 222

5.51 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations forU∞∼10 m s−1 and d= 0.597 mm. Note based on figs. 5.15 to 5.18,

the mass transfer data should be viewed with some skepticism. . . 223 5.52 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations

for U∞∼10 m s−1 and d= 0.960 mm . . . 223

5.53 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼20 m s−1 and d= 0.254 mm . . . 224

5.54 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations for U∞∼20 m s−1 and d= 0.401 mm . . . 224

5.55 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations forU∞∼20 m s−1 and d= 0.597 mm. Note based on figs. 5.15 to 5.18, the mass transfer data should be viewed with some skepticism. . . 225 5.56 Lateral profile of normalizedSt vs ζm∗ at various streamwise locations

(30)

C.1 95 % Uncertainty for thermocouple calibration . . . 253

E.1 Center Nu for case 1,w1 = 4 mm,w2 = 50 mm . . . 270

E.2 Center Nu for case 1,w1 = 1 mm,w2 = 50 mm . . . 271

E.3 Center Nu for case 1,w1 = 0.25 mm, w2 = 50 mm . . . 271

E.4 Center Nu for case 1,w1 = 4 mm,w2 = 0.25 mm . . . 271

E.5 Center Nu for case 1,w1 = 0.25 mm, w2 = 0.25 mm . . . 272

E.6 Center Nu for case 1,w1 = 1 mm,w2 = 1 mm . . . 272

E.7 Center Nu for case 2,w1 = 0.25 mm, w2 = 0.25 mm . . . 272

E.8 Center Nu for case 3,w1 = 1 mm,w2 = 1 mm . . . 273

E.9 Center Nu for case 2,w1 = 4 mm,w2 = 4 mm . . . 273

E.10 Center Nu for case 2,w1 = 0.25 mm, w2 = 4 mm . . . 273

E.11 Center Nu for case 2,w1 = 4 mm,w2 = 50 mm . . . 274

E.12 Center Nu for case 2,w1 = 1 mm,w2 = 50 mm . . . 274

E.13 Center Nu for case 3,w1 = 4 mm,w2 = 50 mm . . . 274

E.14 Center Nu for case 3,w1 = 1 mm,w2 = 50 mm . . . 275

E.15 Center Nu for case 3,w1 = 0.25 mm, w2 = 50 mm . . . 275

E.16 Center Nu for case 3,w1 = 4 mm,w2 = 0.25 mm . . . 275

E.17 Center Nu for case 3,w1 = 0.25 mm, w2 = 0.25 mm . . . 276

E.18 Center Nu for case 3,w1 = 1 mm,w2 = 1 mm . . . 276

E.19 Center Nu for case 4,w1 = 4 mm,w2 = 4 mm . . . 276

E.20 Center Nu for case 4,w1 = 0.25 mm, w2 = 4 mm . . . 277

E.21 Center Nu for case 4,w1 = 4 mm,w2 = 50 mm . . . 277

E.22 Center Nu for case 4,w1 = 1 mm,w2 = 50 mm . . . 277

E.23 Center Nu for case 4,w1 = 0.25 mm, w2 = 50 mm . . . 278

(31)

E.25 Lateral Nu for case 1,w1 = 4 mm, w2 = 50 mm . . . 279

E.26 Lateral Nu for case 1,w1 = 1 mm, w2 = 50 mm . . . 280

E.27 Lateral Nu for case 1,w1 = 0.25 mm,w2 = 0.25 mm . . . 280

E.28 Lateral Nu for case 1,w1 = 1 mm, w2 = 1 mm . . . 281

E.29 Lateral Nu for case 1,w1 = 4 mm, w2 = 4 mm . . . 281

E.30 Lateral Nu for case 1,w1 = 0.25 mm,w2 = 4 mm . . . 282

E.31 Lateral Nu for case 2,w1 = 4 mm, w2 = 50 mm . . . 282

E.32 Lateral Nu for case 2,w1 = 1 mm, w2 = 50 mm . . . 283

E.33 Lateral Nu for case 2,w1 = 0.25 mm,w2 = 0.25 mm . . . 283

E.34 Lateral Nu for case 2,w1 = 1 mm, w2 = 1 mm . . . 284

E.35 Lateral Nu for case 2,w1 = 4 mm, w2 = 4 mm . . . 284

E.36 Lateral Nu for case 2,w1 = 0.25 mm,w2 = 4 mm . . . 285

E.37 Lateral Nu for case 3,w1 = 0.25 mm,w2 = 0.25 mm . . . 285

E.38 Lateral Nu for case 3,w1 = 1 mm, w2 = 1 mm . . . 286

E.39 Lateral Nu for case 3,w1 = 4 mm, w2 = 4 mm . . . 286

E.40 Lateral Nu for case 3,w1 = 0.25 mm,w2 = 4 mm . . . 287

E.41 Lateral Nu for case 3,w1 = 4 mm, w2 = 50 mm . . . 287

E.42 Lateral Nu for case 3,w1 = 1 mm, w2 = 50 mm . . . 288

E.43 Lateral Nu for case 4,w1 = 0.25 mm,w2 = 0.25 mm . . . 288

E.44 Lateral Nu for case 4,w1 = 1 mm, w2 = 1 mm . . . 289

E.45 Lateral Nu for case 4,w1 = 4 mm, w2 = 4 mm . . . 289

E.46 Lateral Nu for case 4,w1 = 0.25 mm,w2 = 4 mm . . . 290

E.47 Lateral Nu for case 4,w1 = 4 mm, w2 = 50 mm . . . 290

E.48 Lateral Nu for case 4,w1 = 1 mm, w2 = 50 mm . . . 291

E.49 Centerline St for 5 m s−1 freestream velocity and5.00 mm width strip 292

(32)

E.51 Centerline St for 10 m s−1 freestream velocity and2.48 mmwidth strip 293

E.52 Centerline St for 10 m s−1 freestream velocity and5.00 mmwidth strip 294

E.53 Centerline St for 10 m s−1 freestream velocity and9.77 mmwidth strip 294

E.54 Centerline St for 10 m s−1 freestream velocity and19.77 mm width strip 295

E.55 Centerline St for 20 m s−1 freestream velocity and5.00 mmwidth strip 295

E.56 Centerline St for 20 m s−1 freestream velocity and9.77 mmwidth strip 296

E.57 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼10 m s−1 and d= 0.401 mm . . . 297

E.58 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼10 m s−1 and d= 0.960 mm . . . 298

E.59 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼10 m s−1 and d= 4.32 mm . . . 298

E.60 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼20 m s−1 and d= 0.401 mm . . . 299

E.61 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼20 m s−1 and d= 0.597 mm . . . 299 E.62 Centerline mass transfer between experimental results and numerical

simulation for U∞ ∼20 m s−1 and d= 0.960 mm . . . 300

E.63 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼20 m s−1 and d= 2.29 mm . . . 300

E.64 Centerline mass transfer between experimental results and numerical simulation for U∞ ∼20 m s−1 and d= 4.32 mm . . . 301

E.65 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d= 0.401 mm . . . 301

E.66 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d= 0.960 mm . . . 301

(33)

E.67 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼10 m s−1 and d= 4.32 mm . . . 302 E.68 Normalized centerline mass transfer between experimental results and

numerical simulation for U∞∼20 m s−1 and d= 0.254 mm . . . 302

E.69 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 0.401 mm . . . 302

E.70 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 0.597 mm . . . 303

E.71 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 0.960 mm . . . 303

E.72 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 2.29 mm . . . 303

E.73 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 4.32 mm . . . 304

E.74 Normalized centerline mass transfer between experimental results and numerical simulation for U∞∼20 m s−1 and d= 8.84 mm . . . 304 E.75 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 8.84 mm for

data taken on 2016/07/20-13 . . . 305 E.76 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 4.32 mm for

data taken on 2016/07/24-14 . . . 305 E.77 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 2.29 mm for

data taken on 2016/07/27-12 . . . 306 E.78 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 0.96 mm for

data taken on 2016/07/29-13 . . . 306 E.79 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 0.96 mm for

(34)

E.80 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 0.597 mm for data taken on 2016/08/08-22. Note based on figs. 5.15 to 5.18, the mass transfer data should be viewed with some skepticism. . . 307 E.81 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 0.401 mm for

data taken on 2016/08/09-13 . . . 307 E.82 Contour of experimental St for U∞ ∼ 10 m s−1 and d = 0.401 mm for

data taken on 2016/08/14-16 . . . 307 E.83 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 8.84 mm for

data taken on 2016/07/18-13 . . . 308 E.84 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 4.32 mm for

data taken on 2016/07/24-22 . . . 308 E.85 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 4.32 mm for

data taken on 2016/07/25-14 . . . 308 E.86 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 2.29 mm for

data taken on 2016/07/27-20 . . . 309 E.87 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 2.29 mm for

data taken on 2016/07/28-12 . . . 309 E.88 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.96 mm for

data taken on 2016/08/01-13 . . . 309 E.89 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.96 mm for

data taken on 2016/08/01-22 . . . 310 E.90 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.597 mm for

data taken on 2016/08/07-18 . . . 310 E.91 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.597 mm for

(35)

E.92 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.401 mm for data taken on 2016/08/13-10 . . . 311 E.93 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.401 mm for

data taken on 2016/08/13-22 . . . 311 E.94 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.254 mm for

data taken on 2016/08/17-23 . . . 312 E.95 Contour of experimental St for U∞ ∼ 20 m s−1 and d = 0.254 mm for

data taken on 2016/08/18-20 . . . 312 E.96 Lateral profiles of theSt vsz at various streamwise locations forU∞∼

10 m s−1 and d= 0.401 mm. . . 313

E.97 Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 10 m s−1 andd= 0.597 mm. Note based on figs. 5.15 to 5.18, the mass

transfer data should be viewed with some skepticism. . . 314 E.98 Lateral profiles of theSt vsz at various streamwise locations forU∞∼

10 m s−1 and d= 0.960 mm. . . 314

E.99 Lateral profiles of theSt vsz at various streamwise locations forU∞∼

10 m s−1 and d= 2.29 mm . . . 315

E.100Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 10 m s−1 and d= 4.32 mm . . . 315

E.101Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 and d= 0.254 mm. . . 316

E.102Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 and d= 0.401 mm. . . 316

E.103Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 andd= 0.597 mm. Note based on figs. 5.15 to 5.18, the mass

(36)

E.104Lateral profiles of theSt vsz at various streamwise locations forU∞∼

20 m s−1 and d= 0.960 mm. . . 317

E.105Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 and d= 2.29 mm . . . 318

E.106Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 and d= 4.32 mm . . . 318

E.107Lateral profiles of theSt vsz at various streamwise locations forU∞∼ 20 m s−1 and d= 8.84 mm . . . 319

E.108Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼10 m s−1 and d= 0.254 mm . . . 320

E.109Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼10 m s−1 and d= 0.401 mm . . . 320

E.110Lateral profile of normalized St vsz/dat various streamwise locations forU∞∼10 m s−1 and d= 0.597 mm. Note based on figs. 5.15 to 5.18,

the mass transfer data should be viewed with some skepticism. . . 321 E.111Lateral profile of normalized St vsz/dat various streamwise locations

for U∞∼10 m s−1 and d= 0.960 mm . . . 321

E.112Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼10 m s−1 and d= 2.29 mm . . . 322

E.113Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼10 m s−1 and d= 4.32 mm . . . 322

E.114Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼10 m s−1 and d= 8.84 mm . . . 323 E.115Lateral profile of normalized St vsz/dat various streamwise locations

(37)

E.116Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼20 m s−1 and d= 0.401 mm . . . 324 E.117Lateral profile of normalized St vsz/dat various streamwise locations

forU∞∼20 m s−1 and d= 0.597 mm. Note based on figs. 5.15 to 5.18,

the mass transfer data should be viewed with some skepticism. . . 325 E.118Lateral profile of normalized St vsz/dat various streamwise locations

for U∞∼20 m s−1 and d= 0.960 mm . . . 325

E.119Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼20 m s−1 and d= 2.29 mm . . . 326 E.120Lateral profile of normalized St vsz/dat various streamwise locations

for U∞∼20 m s−1 and d= 4.32 mm . . . 326

E.121Lateral profile of normalized St vsz/dat various streamwise locations for U∞∼20 m s−1 and d= 8.84 mm . . . 326

(38)

Nomenclature

Variables

English symbols

Aλ, Bλ Eigencoefficients

cf Skin friction coefficient, 1 τ 2ρU∞2

cp Specific heat at constant pressure (J kg−1K−1) C Mass concentration, kgsolute/kgmixture

d Cavity depth, m

D Diffusion coefficient (m2s−1)

E Voltage reading (V)

f Stream function from Blasius solution

h Heat transfer coefficient, q˙00/(Tw−T∞)(W m−2K−1) hm Mass transfer coefficient, m˙00/(Cw−C∞) (kg s−1m−2) H Profile shape factor, (n+ 2)/n

k Thermal conductivity (W m−1K−1)

K Acceleration parameter, ν U2∞

dU∞ dx

L Length of mass transfer plate (m)

(39)

˙

m Mass mass transfer (kg s−1) M Molar mass (kg kmol−1)

Nul Nusselt number with length scalel, hlk

p Exponent forp-norm, where thep-norm is (φp1+φp2)1/p

P Pressure (Pa) Pr Prandtl number cpµk

˙

q Heat transfer (W)

R Ideal gas constant,8314.4598 J kmol−1K−1 Rel Reynolds number with length scalel, U∞lν

Sc Schmidt number, ν D

Shl Sherwood number with length scale l, hmlρD

St Stanton number h

ρcpU∞

Stm Mass Stanton number, ρU∞gm¯ T Temperature (K)

Tu Turbulence intensity, uU0

U∞ Freestream velocity (m s−1)

U∞ Average freestream velocity (m s−1) ~

V Velocity vector with components u,v,w (m s−1)

u Velocity in thex direction (m s−1)

uτ Friction velocity at the wall,

p

τ /ρ (m s−1)

v Velocity in they direction (m s−1)

w Velocity in thez direction (m s−1)

w1 Half width of heated strip (m)

w∗1 Nondimensional half width of heated strip, w1/∆c w2 Half width of “cooled” strip (m)

(40)

W Width of subliming strip (m)

x Streamwise distance from the start of mass transfer plate or the backward facing step of a cavity (m)

x0 Streamwise distance from the virtual origin of the turbulent boundary layer (m)

y Distance normal to the mass transfer plate (m)

z Lateral distance from edge of heated surface, positive towards the center of the heated surface (m)

Greek symbols

α Thermal diffusivity,k/ρcp (m s−2) δ Depth or height (m) δ1 Displacement thickness, R∞ 0 1− u U∞ dy (m) δ2 Momentum thickness, R∞ 0 u U∞ 1− u U∞ dy (m)

δ99 Velocity boundary layer location whereu= 0.99U∞ (m)

∆c Conduction thickness,k/h orx/Nux (m)

∆d Mass diffusion thickness,ρD/hm or x/Shx (m) ζ z similarity variable for laminar flow, z

q

U∞ νx ζ∗ Non-dimensional distance from lateral edge,z/∆c ζm∗ Non-dimensional distance from lateral edge,z/∆d η y similarity variable for laminar flow,y

q

U∞ νx θ Temperature relative to freestream,T −T∞ (K)

κ von Kármán constant, 0.41 λ Eigenvalue (m)

(41)

µ Dynamic viscosity (Pa·s)

ν Kinematic viscosity (m2s−1)

ξ Unheated/inactive starting length (m)

π Ratio between a circle’s circumference and diameter ρ Density (kg m−3)

σ2 Variance

τ Shear stress,µ∂u∂y τt Elapsed time (s)

Math symbols

∇φ Gradient ofφ ∇ ·φ~ Divergence ofφ~ ∇ ×φ~ Curl of ~φ ∂φ

∂x Partial derivative ofφ with respect tox

exp (φ) Raising Euler’s number to power φ

sin (φ) Sine of angleφ

cos (φ) Cosine of angleφ

Rb

a φdx Integral ofφ from a tob with respect to x

P∞

n=1φi Sum of φi across index n from one to infinity I(φ) Imaginary component ofφ

log (φ) Natural logarithmφ

tan−1(φ) Arctangent ofφ

Li2(φ) Dilogarithm of φ

(42)

Subscripts

a Air

c Centerline

css Cubic smoothing spline

g Gas

hw Hotwire

∞ Evaluated in the free stream

j Component of mixture

n Naphthalene

N C Natural convection sat Saturated

tc Thermocouple tp Triple point of water

v Vortex

V D Van Driest

w Evaluated at the wall

z → ∞ Evaluated on the wall far from the lateral edge

Superscripts

φ Averaged quantity 0 Turbulent fluctuation

(43)

00 Per unit area

+ Wall coordinates

ˆ

(44)

Chapter 1

Introduction

1.1 Outline of Present Work

Thermal management of electronic systems is a multiscaled and increasingly impor-tant problem as miniaturization of electronic devices and circuits continues (Garimella et al. 2008). Particularly, heat transfer on the scale of the individual electronic com-ponents is an important design consideration for digital systems. The present study characterizes heat and mass transfer near the lateral edge of rectangular heat sources and subliming surfaces.

For laminar flow the present study characterizes the effects of varying the Prandtl number, unheated starting length, width of the heated elements, spacing between the heated elements, and surface boundary conditions on the heat transfer from a flat plate. The temperature field is solved numerically with OpenFOAM and the Blasius solution is used for velocity boundary layer. For turbulent flow the present study characterizes the effects of the free stream velocity, width of the subliming surface, and cavity depth on the mass transfer coefficient. To accomplish this, the present study will employ a naphthalene sublimation technique in a suction type wind tunnel at the Heat Transfer Laboratory.

(45)

The extent of the region affected by the lateral edge and general correlations for a the local heat and mass flux for a flat plate are presented. As conjugate heat transfer is expensive to fully model, the results can be used as boundary conditions for conduction modeling, eliminating the need to model the momentum and energy transport in the fluid. In addition, the results can serve as a benchmark for numerical simulations where lateral transport is important.

1.2 Theoretical Background

1.2.1 General Equations

The following two equations are the fundamental transport equations for energy (eq. (1.1)) and mass components (eq. (1.2)):

ρcp ∂T ∂t + ~ V · ∇T =∇ ·k∇T , (1.1) ρ ∂Cj ∂t +V~ · ∇Cj =∇ ·ρDj∇Cj. (1.2) The mass diffusion equation assumes no species creation through chemical reac-tions and the Soret and Dufour effects are neglected, as the temperature and concen-tration differences are relatively small. The thermal energy equation assumes there is only one species present, negligible viscous dissipation, and a constant specific heat.

Equations (1.1) and (1.2) are completely analogous, with the concentration field mapping to the temperature field when ρ is equivalent to ρcp, and ρDj is equivalent to k. If the flow has constant properties, then ρcpk is equivalent to Dj. This analogy is referred to as the heat-mass transfer analogy, and allows solutions with equivalent boundary conditions from one domain to be mapped to the other domain. For the

(46)

tur-bulent mass transfer experiments, a constant mass concentration boundary condition is realized, which is equivalent to a constant temperature boundary condition.

1.2.2 Turbulent Boundary Layer Equations

The mass concentration boundary layer is three dimensional near the edges of the mass transfer plate, so the spanwise gradients cannot be ignored. The following assumptions can be applied to the mass diffusion equation (eq. (1.2)) near the wall to derive the constant density mass diffusion boundary layer equation,

uv, w, w= 0, ∂Cj ∂y , ∂Cj ∂z ∂Cj ∂x , ∂2C j ∂y2 , ∂2C j ∂z2 ∂2C j ∂x2 .

Applying the listed assumptions yields,

∂Cj ∂t +u ∂Cj ∂x +v ∂Cj ∂y = ∂ ∂y Dj ∂Cj ∂y + ∂ ∂z Dj ∂Cj ∂z . (1.3)

Using Reynolds’ decomposition on eq. (1.3) and time averaging the equation, the turbulent mass diffusion boundary layer equation is

∂Cj ∂t + ¯u ∂Cj ∂x + ¯v ∂Cj ∂y = ∂ ∂y Dj ∂Cj ∂y −v 0C0 j + ∂ ∂z Dj ∂Cj ∂z −w 0C0 j . (1.4)

The relative strength of the transport by lateral diffusion, represented by the ∂ ∂z Dj ∂Cj ∂z

term, and the transport by turbulent eddies, represented by the term −∂ ∂z w

0C0 j

, is explored and commented on in chapter 4.

(47)

1.2.3 Convection Equations

The constitutive relations for the heat and mass fluxes are

˙

q00 =−k∇T (1.5)

˙

m00j =−ρDj∇Cj. (1.6)

Equations (1.5) and (1.6) are applicable for diffusive processes, and are used in eqs. (1.1) and (1.2) to model the diffusive fluxes. Note that this form of eq. (1.6) assumes an ideal mixture.

It is sometimes helpful to recast eqs. (1.5) and (1.6) as follows

˙

q00 =h(Tw−T∞) (1.7)

˙

m00 =hm(Cw−C∞). (1.8)

These equations are more practical than eqs. (1.5) and (1.6) for calculating the fluxes; as gradients can be difficult to measure with the required precision, or difficult to compute for complex fluid flow. In addition, for many situations the fluxes scale with temperature or mass concentration differences, and the heat and mass transfer coefficients are relatively independent of temperature and concentration in forced convection situations.

(48)

1.3 Literature Review

1.3.1 Naphthalene Sublimation Technique

A common way of conducting mass transfer experiments is to use naphthalene as the subliming substance. Naphthalene is often used because it sublimes at room temperature, and is easy to cast and machine (Souza Mendes 1991).

There are several advantages to using the naphthalene sublimation method as opposed to conducting a heat transfer experiment. Boundary conditions are easier to apply to a mass transfer experiment than to a heat transfer experiment because conduction and radiation losses are not present in a mass transfer study. Also, local measurements of the flux values can be taken nonintrusively, as depth measurements are made before and after the forced convection takes place (Goldstein and Cho 1995). It is worth mentioning several limitations of using the naphthalene sublimation method. First, the wall normal velocity can affect the boundary layer flow if the transpiration is large enough, invalidating the heat-mass transfer analogy for a no blowing condition. Second, the temperature of the mass transfer plate has to be fairly uniform, because of naphthalene properties’ sensitivity to temperature changes. Third, if the velocity is too high, viscous dissipation effects can invalidate the heat-mass transfer analogy. Fourth, external high shear flows cannot be studied, as this will cause some of the naphthalene to erode from the surface instead of subliming. Fifth, it is difficult to get transient data as all the sublimed naphthalene depth values are time averaged. Last, the naphthalene surface profile can deviate from the original surface profile by too much due to sublimation from a long experiment (Goldstein and Cho 1995). All of these concerns will be addressed in section 3.3.4.

(49)

1.3.2 Electronic Cooling

Many techniques are described in published research for increasing and managing heat transfer from heat sinks and sources, such as flow through porous metal foam (Yang and Garimella 2010; Mancin et al. 2013), pin fins (Khan et al. 2006; Siu-Ho et al. 2007), increased surface roughness (Honda and Wei 2004; Ventola et al. 2014), and microchannels (Kandlikar 2012). Despite the attention electronic cooling has received, there is little understanding of three dimensional thermal boundary layer effects on heat transfer from flush discrete heat sources.

When three dimensional effects are studied, often the local heat transfer is not presented, and rarely are lateral changes explicitly discussed. Rather, heat transfer characteristics from discrete sources are most often presented in terms of averaged Nusselt numbers. Examples of heat transfer characteristics of discrete heat sources in the open literature are Baker (1972), Incropera et al. (1986), and Tso et al. (1999). Baker (1972) noticed that for small, discrete sources, the familiar two-dimensional correlations give heat fluxes one-third smaller than experimentally found heat fluxes. Baker (1972) speculated that the increase was due to conduction and edge effects of the smaller heat sources embedded in the substrate. In addition to Incropera et al. (1986) noting that the two-dimensional turbulent correlations are inadequate for predicting the heat transfer from discrete sources; both Incropera et al. (1986) and Tso et al. (1999) reported similar results for the effect of wakes from upstream heat sources. Their experimental results showed large decreases between the first, second, and third heat sources, but almost no changes between the third and fourth heat sources. Both studies also concluded that the thermal characteristics of heat transfer are more representative of external flow; as the channel height did not appreciably affect the Nusselt number.

(50)

Finite Width Plates

In addition to looking at both single and multiple discrete heating elements with conjugate heat transfer, Ortega and Ramanathan (2003a) report the effects of finite size on local heat transfer from discrete sources. Ortega and Ramanathan (2003a) analytically describe wakes of arbitrary arrays of heated elements and local Nusselt numbers from arbitrary shaped, heated elements by using point source solutions for the energy equation, assuming bulk flow with constant surface heat flux, and then superposing the point source solution to form a general equation for the convection losses from rectangles and line sources. Using the methods developed in Ortega and Ramanathan (2003a), Ortega and Ramanathan (2003b) present numerical results for the average Nusselt number for heated elements with a variety of aspect ratios and substrate conductivities. Instead of assuming constant heat flux, as Ortega and Ramanathan (2003a) do, Yovanovich and Teertstra (2001) report the average Nu for

an isothermal plate by averaging the solutions for the diffusive limit and convective limit (i.e. for low and high Re).

Hauptmann and Rotem (1967) and Robbins (1969) looked at a three dimensional thermal boundary layer embedded in two dimensional boundary layer flow. Haupt-mann and Rotem (1967) propose a similarity transformation to map the three spacial dimensions of the energy equation to two dimensions based on a spanwise and wall normal similarity variables. Robbins (1969) found analytical and numerical solu-tions for the thermal boundary layer, and compares it to his own experimental work for laminar flow. To numerically solve for the thermal profile, he used the Blasius solution for the velocity field, and numerically solved the similarity equations from Hauptmann and Rotem (1967). To analytically solve the thermal boundary layer equations, Robbins used a Lèvêque approximation and employed operational calculus

(51)

and Fourier transforms. However, because of the Lèvêque approximation, this solu-tion is formally valid only as the Prandtl number approaches infinity, but Robbins (1969) reported accurate results for Prandtl numbers as low as one.

Figure 1.1: Nondimensional temperature contours for a constant temperature bound-ary condition, η =y q U∞ νx and ξ =z q U∞

νx, Pr = 1,taken from Robbins (1969). Note Robbins (1969) denotes the Prandlt number usingσ.

Fassio (2009) studied the lateral thermal boundary layer profile for turbulent flow. However, he only looked at the consequences of changing the flow speed in the wind tunnel, and did not look at other effects. Figure 1.2 shows the lateral profiles for the sublimed naphthalene depths at different streamwise locations at two different wind tunnel velocities. The dimensions for the subliming portion of Fassio’s mass transfer plate are121 mm×197 mm(corresponding to thez andxdirections). Overall, Fassio (2009) found that a complementary error function described both sets of data well, as shown below in fig. 1.3. In fig. 1.3, the abscissa is the distance from the edge of the plate normalized by the edge effect thickness.

Recently, Taliaferro et al. (2016) and Taliaferro et al. (2018) investigated heat transfer near a lateral edge of a flat plate in laminar and turbulent boundary layer flow. Using a conduction model, Taliaferro et al. (2016) describe the local variation

(52)

0 5 10 15 20 25 30 30 40 50 60 70 80 90 z(mm) Sublimation depth ( µ m) (a)U∞= 3.93 m s−1 0 5 10 15 20 25 30 60 80 100 120 140 160 z(mm) Sublimation depth ( µ m) (b) U∞= 13.6 m s−1

Figure 1.2: Change in surface height for the two free stream velocities at different stream wise locations. Symbols: • x = 2 mm, × x = 31.3 mm, ◦ x = 89.9 mm,

(53)

0 0.5 1 1.5 2 2.5 3 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 zs Normalized submilation depth x0 = 31.3 mm x0 = 89.9 mm x0 = 128.97 mm erfc (1.499zs) x0 = 31.3 mm x0 = 89.9 mm x0 = 128.97 mm erfc (1.401zs)

Figure 1.3: Dimensionless fit of lateral normalized sublimed naphthalene depth profile

δs−δc,s

δs,max−δc,s

vs. the normalized lateral distance

zs = δez

, taken from Fassio (2009). The edge effect thickness is denoted as δe.

of heat transfer for fluids of a variety of Prandtl numbers, unheated starting lengths, and boundary conditions for laminar flow. Taliaferro et al. (2018) used the numerical and analytical results from Taliaferro et al. (2016) with experimental results from turbulent boundary layer flow to construct a correlation describing the local mass transfer near a lateral edge. The results of Taliaferro et al. (2016) are presented in section 2.1 and some of the discussion from Taliaferro et al. (2018) is presented in section 4.2.5.

1.3.3 Evaporation from Droplets

Evaporation from droplets is closely related to heat transfer from discrete sources, and useful for printing and coating processes. Deegan et al. (1997) describe how mass fluxes for a droplet with a pinned contact line causes a “coffee-ring” effect. As the

(54)

contact line is pinned, mass must flow towards the edges to maintain the droplet footprint on the surface, depositing suspended particles near the pinned contact line. Measurements by Deegan et al. (1997) and later measurements and numerical simu-lations by Hu and Larson (2002), describe mass transfer with a power law that tends towards infinity near the pinned edge, with exponent values of -0.5 for no contact angle and zero at a90◦ contact angle. Deegan et al. (1997) used results from Jackson

(1999) to estimate the local fluxes while Hu and Larson (2002) used results from finite element method simulations.

(55)

Chapter 2

Laminar Flow Over a Flat Plate

Lateral variation of the wall heat transfer in laminar boundary layer flow is inves-tigated. For a geometry where both the heated and cooled/adiabatic surfaces are semi-infinite in lateral extent, both a similarity solution and a conduction model are compared to three dimensional numerical simulations for a variety of Prandtl num-bers, boundary conditions, and unheated starting lengths. For a geometry where either (or both) heated and cooled/adiabatic surfaces are finite in lateral extent, no similarity solutions exist, so only the conduction model is compared to the three dimensional numerical simulations.

It is found that the conduction model works remarkably well at predicting the wall heat transfer and the most relevant length scale for describing the problem is the conduction thickness (see fig. 2.6). Knowing that the conduction thickness is the main length scale for describing the lateral change in heat transfer allows for the definition of a key dimensionless spanwise variable,z/∆c. Furthermore, the spanwise

extent of the region affected by the lateral edge is of the same order of magnitude as the conduction thickness. Finally, the accuracy and computational simplicity of the conduction model allows for impressive savings in numerical modeling of conjugate heat transfer where lateral variation in heat transfer is important.

(56)

2.1 Semi-infinite Domain

The basic geometry for the semi-infinite heated plate is shown in fig. 2.1. The hot and “cold” plates are semi-infinite in lateral extent and meet at z = 0. The velocity

is assumed to conform to the Blasius solution for laminar boundary layers.

ξ y x z δ δT U∞ Hot plate “Cold” plate

Figure 2.1: Sketch of the problem domain domain showing the unheated starting length, ξ, momentum boundary layer, δ, and thermal boundary layer, δT. The hot and “cold” plate are coincident at z = 0 and are of semi-infinite width.

The results discussed in section 2.1 appear in briefer form in Taliaferro et al. (2016).

2.1.1 Similarity Solution

Similarity Equation

If the streamwise conduction and lateral velocity are assumed negligible, then eq. (1.1) can be simplified for steady, constant property, laminar flow as shown in eq. (2.1).

u∂T ∂x +v ∂T ∂y =α ∂2T ∂y2 + ∂2T ∂z2 (2.1)

(57)

Following the development described in Hauptmann and Rotem (1967), eq. (2.1) can be further simplified using the following similarity variable substitutions,

θ=T −T∞, η=y r U∞ νx, ζ =z r U∞ νx. (2.2)

The new temperature derivatives are:

∂T ∂x = ∂θ ∂η ∂η ∂x + ∂θ ∂ζ ∂ζ ∂x =−∂θ ∂η y 2x r U∞ νx − ∂θ ∂ζ z 2x r U∞ νx =− 1 2x η∂θ ∂η +ζ ∂θ ∂ζ (2.3) ∂T ∂y = ∂θ ∂η ∂η ∂y + ∂θ ∂ζ ∂ζ ∂y = r U∞ νx ∂θ ∂η (2.4) ∂2T ∂y2 = ∂ ∂y ∂θ ∂y = ∂ ∂y ∂θ ∂η ∂η ∂y = ∂ 2θ ∂η2 ∂η ∂y 2 + ∂θ ∂η ∂2η ∂y2 = U∞ νx ∂2θ ∂η2 (2.5) similarly, ∂2T ∂z2 = U∞ νx ∂2θ ∂ζ2. (2.6)

(58)

The usual substitution for the velocities are made (Lienhard IV and Lienhard V 2012, pg. 285, eq. 6.19) u=U∞f0, v = 1 2 r νU∞ x (ηf 0 f) (2.7)

where f is the stream function for a two dimensional momentum boundary layer. Combining eq. (2.1) with eqs. (2.2) to (2.7) results in the partial differential equation shown in eq. (2.8). − Pr 2 ζf0∂θ ∂ζ +f ∂θ ∂η = ∂ 2θ ∂η2 + ∂2θ ∂ζ2 (2.8)

With the similarity variable choices in eq. (2.2), the three spacial dimensions in eq. (2.1) have been collapsed down to two similarity spacial dimensions in eq. (2.8). The similarity equation as presented has dimensions of temperature, but can easily be made nondimensional by dividing the whole equation by some appropriate refer-ence temperature or temperature differrefer-ence. The similarity solution for the velocity boundary layer,f, is solved for separately and supplied in tabular for to the numerical solver when integrating eq. (2.8).

Similarity Solution

Equation (2.8) is numerically solved for four different boundary conditions and two different P r (1 and 10). Table 2.1 lists the boundary conditions used for the numer-ical modeling. All the laminar numernumer-ical simulations use the boundary conditions specified in table 2.1.

Table 2.1: Boundary conditions for laminar modeling Case Heated Surface Cooled Surface

1 Isothermal Isothermal 2 Isothermal Adiabatic

3 Isoflux Isothermal

(59)

References

Related documents

Taking an alternate view, Dimitriadis and Koritos (2014) argue that effect of personal relationships become non-significant once functional benefits are considered.

In the same way that the distribution of times to failures for a single mode will be a characteristic of that mode, so the more modes there are contributing to the failure data,

Specifically, traumatic and respiratory arrests will exhibit more severely impaired cognition when compared to cardiac etiology of arrest, and variables including

In practice, it appears that the aggregate stock of physical capital is mea- sured using a ‘perpetual inventory method.’ The way this is done is to take the investment flow (measured

The first chapter introduced research background, motivation and research methods; The second chapter analyzed financial leasing industry’s environment and competitive situation

Stuart, Megan R., "Domenico Scarlatti: The Beginnings and Creation of the Classical Piano Sonata" (2019).. The

leading private sector players involved in the scam included Fairgrowth financial services and Champaklal investments and finance (CIFCO)... The registration norms with

Although there was no significant difference (p>0.05), there existed slight changes in students’ opinions on the usefulness of e-models and real manipulations with e-models