A multi-phase supersonic jet impingement facility for thermal management


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A multi-phase supersonic jet impingement facility for thermal management
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Parker, Richard Raphael
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@ 2012 Richard Raphael Parker


Throughout my time in graduate school I have been helped by many people,

although I will try my best to include everyone, inevitably I will leave some people out.

For that I am sorry.

First and foremost I would like to thank my parents as without them this would not

be possible. Dad, you taught me what it meant to be a man, and I can't express my

gratitude enough.

There have been a multitude of people in my lab that have helped me immensely,

Dr. Patrick Garrity, Dr. Ayyoub Mehdizadeh, Dr. Jameel Khan, and Dr. Fadi Alnaimat

(my current roommate) who have all graduated and are going places in the world, you

have provided help and motivation. Bradley Bon, Fotouh AI-Ragom (the lab mother), and

Cheng-Kang (Ken) Guan who are currently preparing for their dissertation defense, we

have all helped motive each other. Ben Greek, Prasanna Venvanalingam, Kyle Allen,

Like Li, and Nima Rahmatian, who are currently completing their degree requirements,

have helped me develop some of my presentation skills. I hope you have learned as

much from me as I have from you.

My friends in the Interdisciplinary Microsystems Group (IMG) who have provided

many opportunities to escape the pressures of graduate studies. I would specifically like

to thank my old roommate, Dr. Drew Wetzel, who let me bounce ideas off of him, Dr.

Matt Williams, who always ran the group football competitions and helped me reminisce

about my roots in South Carolina, and Brandon Bertolucci our official social chair. Thank

you all and good luck.

I would also like to thank my professors at the University of South Carolina. Dr.

Jamil Khan, whose heat transfer classes helped shape the foundation of my knowledge

in the thermal and fluid sciences, Dr. David Rocheleau who provided me with contacts

at the University of Florida, Dr. Phillip Voglewede who encouraged me to attend the

University of Florida, and finally Dr. Abdel Bayoumi, my undergraduate research advisor

who saw potential in me and provided many opportunities for me to grow.

Last, but not least, I would like to thank my committee. Dr. Orazem has helped

me learn things outside of my field and my comfort zone and for that I am grateful. Dr.

Hahn, your rigorous heat conduction class helped build the foundation for much of my

studies. Dr. Mei, you have helped me realize my potential for numerical studies and how

to know when a solution is good enough or when perfection is essential. Lastly I would

like to thank my chair, Dr. James F Klausner. Dr. Klausner, you have kept me around,

even as I struggled, because you saw the potential in me. You've help keep some levity

in the lab with football talk and going to some excellent concerts. Most importantly,

you've helped me grow both professionally and as a person. For this you have my most

sincere gratitude.


ACKNOWLEDGMENTS ................... ............... 3

LIST O FTABLES ..................... ................. 8

LIST OF FIGURES .................... ................. 9

NOMENCLATURE .................... ................. 13

ABSTRACT .................... ................... .. 17



1.1 Literature Review ................... ............ 20
1.1.1 Single-Phase Jet Impingement .................. 20
1.1.2 Mist and Spray Cooling ....................... 24
1.1.3 Supersonic Jet Impingement ................ ... 25
1.2 Summary .................... ................ 34

2 JET IMPINGEMENT FACILITY ........................... 36

2.1 Impinging Jet Facility Systems ................ ........ 36
2.1.1 Air Storage System ........................... 36
2.1.2 Water Storage and Flow Control System ..... 36
2.1.3 Air Pressure Control System .. .. 39
2.1.4 Air Mass Flow Measuring System ..... 39
2.1.5 Temperature and Pressure Measurements .... 42
2.1.6 Converging-Diverging Nozzle . ... 42
2.1.7 Data Acquisition System ..... .... 43
2.2 Analysis of Impingement Facility ... 44
2.2.1 Temperature and Pressure Upstream of the Nozzle ... 44
2.2.2 Nozzle Exit Pressure Considerations ..... 46
2.2.3 Oblique Shock Waves at Nozzle Exit ... 48
2.2.4 Complete Shock Structure of an Overexpanded Jet ... 50
2.3 Sum m ary ... . .. .. 51

3 STEADY STATE EXPERIMENTS ........................ 53

3.1 Heater Construction .. .. .. .. .. ... 56
3.1.1 Physical Description .. .. .. .. .. .. 56
3.1.2 Theoretical Concerns ......................... 57
3.2 Experimental Procedure .. ............... ....... ..59
3.2.1 Two-Phase Experiments .... .. .. .. .. ..59
3.2.2 Single-Phase Experiments ..... .... 62

3.3 Experimental Results .. .............
3.3.1 Uncertainty Analysis .. ..........
3.3.2 Single-Phase Results .. .........
3.3.3 Two-Phase Results .. ...........
3.3.4 Evaporation Effects .. ...........
3.4 Comparison between Single and Two-Phase Jets
3.5 Discussion . .
3.6 Sum m ary . .

HEAT TRANSFER ANALYSIS ............................

4.1 Inverse Problems ........
4.2 Introduction to Inverse Problem
Method with Adjoint Problem .

Solution Using the Conjugate Gradient

4.2.1 The Direct Problem .
4.2.2 The Measurement Equation .
4.2.3 The Indirect Problem .
4.2.4 The Adjoint Problem .
4.2.5 Gradient Equation .
4.2.6 Sensitivity Equation .
4.2.7 The Conjugate Gradient Method .
4.3 Factors Influencing Inverse Heat Transfer Problems .
4.3.1 Boundary Condition Formulation Effects .
4.3.2 Sensor Location Effects .
4.3.3 Thermocouple Insertion Effects .
4.4 Inverse Heat Transfer Problem Formulation .
4.4.1 Direct Problem .................
4.4.2 Measurement Equation .
4.4.3 Indirect Problem ................
4.4.4 Adjoint Problem ..............
4.4.5 Gradient Equation ...............
4.4.6 Sensitivity Problem .
4.4.7 Conjugate Gradient Method .
4.4.8 Stopping Criteria .
4.4.9 Algorithm . .
4.5 Numerical Method and Limitations .
4.5.1 Alternating Direction Implicit Method .
4.5.2 Grid Stretching in the Z-Direction .
4.5.3 Time step size complications .
4.6 Deconvolution for Thermocouple Impulse Response
4.6.1 Direct Problem .................
4.6.2 Indirect Problem .. .. .. .. .
4.6.3 Adjoint Problem . .

. .
. .
. .
. .
. .
.. ..
. .
. .
. .
. .
. .
. .
. .
.. ..

4.6.4 Gradient Equation .. ..................


4.6.5 Sensitivity Problem .. .. .. .. .. .. 110
4.6.6 Conjugate Gradient Method .... 110
4.6.7 Stopping C riteria . 111
4.6.8 A lgorithm . . 111
4.6.9 TestCase ................... .......... .112
4.7 Sum m ary . . 114

TRANSFER ALGORITHM ................ ............... 116

5.1 Thermocouple Measurement Dynamics . ... 116
5.1.1 Low Biot Number Thermocouple Models ... 116
5.1.2 High Biot Number Thermocouple Models ... 121
5.1.3 Design of Experiment ......................... 122
5.1.4 Experimental Results ..........................126
5.1.5 Comparison to Established Models ... 127
5.2 Inverse Heat Transfer Algorithm Verification ... 131
5.2.1 Inverse Quenching Parametric Study Setup ... 132
5.2.2 Error Assessment Methods . 136
5.2.3 Parametric Study Results ... 137
5.2.4 Heat Loss/Gain Effects ......................... 141
5.2.5 Effectiveness of the Inverse Heat Transfer Algorithm ... 142
5.3 Sum m ary . . 144

6 CO NCLUSIO NS .. .. .. .. .. .. .. .. 147



CONTOUR PLOTS .................................. 166


REFERENC ES . . .. 175

BIOGRAPHICAL SKETCH ................................ 184


Table page

2-1 Water Mass Flowrate and Average Water Velocity for Different Regulator Pressures
and O rifice Sizes. . .. 40

2-2 Area, temperature, and pressure ratios at various points in the jet impingement
facility.... .. .. .. . ... 46

2-3 Nozzle exit pressure for various regulator pressures. ... 47

3-1 Reynolds number and corresponding heat fluxes. ... 62

5-1 Curve Fitting Constants for Rabin and Rittel's thermocouple impulse response
m odel, from [114] . . 121

5-2 RMS errors for L = 10 mm, and TDAQ = 0.2 C. ... 138

5-3 RMS errors for L = 10 mm, oDAQ = 1 OC, and M = 8. ... 139

5-4 RMS errors for L = 5 mm and 0DAQ = 0.2 C. . ... 140

5-5 RMS error for L = 10 mm,-DAQ = 0.2 oC, M = 8, and various values of a for
the Biot number distribution. .. ......... .. 140

5-6 RMS error for L = 10 mm and M = 8 with different actual and simulated time
co nstants . . 14 1





























Solution of stagnation point flow ....................

An illustration of the shock structure in the wall jet region. .

Grease streak photograph ......................

Numerical results of a flow field with a plate shock. .

Jet centerline pressure fluctuations with and without moisture. .

Adiabatic and heated temperature variation with z/D. .

Illustration of the jet impingement facility. .

Cross section of the mixing chamber .

Thread details of the mixing chamber cross section .

Orifice cross section ................... ........

Theoretical vs measured mir. ... .... ................

Cross section of nozzle .........................

Simplified view of the air flow path in the facility. .

Illustration of the limiting cases for shock waves in the nozzle. .

Illustration of an oblique shock wave at the nozzle exit. .

Variation of flow properties downstream of an oblique shock wave..

Structure of oblique shock waves ..................

Stainless steel heater assembly. . .

Copper heater assembly. ........................

Illustration of heater assembly used for steady state experiments. ..

Ice formation at adiabatic conditions. . .

Measured single-phase NuD spatial variation at different heat fluxes.

Spatial heater temperature variation at different applied heat fluxes.

Spatial variation of NuD at different ReD, a) unsealed and b) scaled.

Single-phase NuD at various nozzle height to diameter ratios. .


. 2 1

. 26

. 28

. 3 1

. 32

. 34

. 37

. 38

. 38

. 39

. 4 1

. 43

. 45

. 48

. 49

. 50

. 52

. 54

. 55

. 57

. 60

. 64

. 65

. 66

. 67

3-9 Measured two-phase NuD spatial variation at different heat fluxes without ice
formation ........... ... .. ....................... 68

3-10 Measured two-phase NuD spatial variation at different heat fluxes with and
w without ice form ation. . .. 69

3-11 Spatial heater temperature variation, two-phase jet results. ... 70

3-12 Two-phase NuD at various liquid mass fractions. ... 71

3-13 Two-phase NuD number at various nozzle height to diameter ratios. 72

3-14 Heat transfer enhancement ratio at various liquid mass fractions. ... 74

3-15 Heat transfer enhancement ratio at various nozzle height to diameter ratios. 76

4-1 1-dimensional solid for the sensitivity problem. ... 89

4-2 Relative step sensivity coefficients at x* = 0.1 as a function of time. ...... 92

4-3 Relative step sensivity coefficient for a heat flux input. ... 93

4-4 Relative step sensivity coefficient for a temperature input. ... 93

4-5 Illustration of the heat transfer physics of the inverse problem formulation. 96

4-6 Comparison of true temperature vs inverse results. ... 104

4-7 Effects of grid stretching. a) real domain and b) computational domain. 107

4-8 Time step results. .................................. 109

4-9 True and estimated impulse response function. ... 113

4-10 Convergence history.. .. .. .. .. .. .. .. .. 113

4-11 O utput com prison. . . 114

5-1 Illustration of the first order slab. ... .. 118

5-2 Illustration of the second order slab. ... .... .. 119

5-3 Example of a first and second order impulse response function. ... 120

5-4 Impulse response functions using the model of Rabin and Rittel, adapted from
[114]. .. .. .. .. ....................... .... 122

5-5 Diagram of the copper disc assembly. ... 123

5-6 Illustration of the experimental setup. .. .... ... 124

5-7 Back wall temperatures with non-ideal insulations. ... 125













Results of three separate impulse response experiments. .

Inverse method deconvolution results . .

First order response function comparison. ... ...

Best fit results using a first order impulse response function .

Comparison of model to deconvolution results . .

Best fit results using the model of [114] response function. ..

Comparison of the 2 exponential model to the deconvolution algorithm results.

Best fit results using the 2 exponential model. . .

Biot number distribution showing the effects of Bimax.. .

Biot number distribution showing the effects of . .

Truncation of Biot number ................... ..........

Error Contours for L = 10 mm, 7DAQ = 0.2 o C, and 8 measurement points. .

. 126

. 127

. 128

. 129

. 129

. 130


. 132

. 134

. 135



5-20 Comparison of the effects of heat gain. .... 143

5-21 Centerline NuD results from the inverse heat transfer algorithm. ... 144

5-22 Spatial NuD results from the inverse heat transfer algorithm. ... 145

A-1 Two-phase NuD results for various Z/D, nominal ReD = 4.46 x 10s ...... .152

A-2 Two-phase NuD results for various Z/D, nominal ReD 7.27 x 10 ... 153

A-3 Two-phase NuD results for various Z/D, nominal ReD 1.01 x 106 ... 154

A-4 Two-phase NuD results for various liquid mass fractions and Z/D = 2.0. ...... 155

A-5 Two-phase NuD results for various liquid mass fractions and Z/D = 4.0. ...... 156

A-6 Two-phase NuD results for various liquid mass fractions and Z/D = 6.0. ...... 157

A-7 Two-phase NuD results for various liquid mass fractions and Z/D = 8.0. ...... 158

A-8 Two-phase 0 results for various Z/D, nominal ReD = 4.46 x 10 ... 159

A-9 Two-phase 0 results for various Z/D, nominal ReD 7.27 x 10 ... 160

A-10 Two-phase 0 results for various Z/D, nominal ReD 1.01 x 106 ... 161

A-11 Two-phase 0

A-12 Two-phase d



for various liquid

for various liquid

mass fractions and Z/D = 2.0.

mass fractions and Z/D = 4.0.

. 162

. 163

A-13 Two-phase 0 results for various liquid mass fractions and Z/D = 6.0. 164

A-14 Two-phase 0 results for various liquid mass fractions and Z/D = 8.0. 165

B-1 Error Contours for L = 5 mm, ODAQ = 0.2 o C, and 8 measurement points 167

B-2 Error Contours for L = 5 mm, ODAQ = 0.2 o C, and 16 measurement points 168

B-3 Error Contours for L = 10 mm, ODAQ = 0.2 o C, and 8 measurement points 169

B-4 Error Contours for L = 10 mm, (DAQ = 0.2 o C, and 16 measurement points 170

B-5 Error Contours for L = 10 mm, (DAQ = 1 o C, and 8 measurement points 171

C-1 The 0.33 m m orifice . ... .. 172

C-2 The 0.37 m m orifice .. .. .. .. .. .. .. .. 173

C-3 The 0.41 m m orifice . . 173

C-4 The 0.51 m m orifice . . 174
























Area [m2]

Biot number = hL/k

Measurement equation operator

Diameter [m]


Mach number, Chapter 2

Total number of measurements

Nusselt number = hL/k

Pressure [Pa]

Gas constant for dry air [J/kg K, Chapter 2

Residual of modeling equation

Reynolds number = 4m/7r D

Least squares value

Temperature [oC or K]

Volume [m3]

Dimensionless relative sensitivity coefficient

Specific heat capacity at constant pressure [J/kg K]

General function

Heat transfer coefficient [W/m2 K], Chapter 3

Impulse response function [s]

Thermal conductivity [W/m K]













Greek letters










Mass flowrate [kg/s]

Internal heat generation rate [W/m3]

Radial coordinate [m]

time [s]

Velocity [m/s]

Dummy variable for partial differential equation, Chapter 4

Dummy variable for partial differential equation, Chapter 4

Liquid mass fraction

Length coordinate [m]

Width coordinate [m]

Generalize output variable, Chapter 4

Height coordinate [m]

Thermal diffusivity [m2/s]

Step size for Conjugate Gradient Method

Grid stretching parameter

Ratio of specific heats, Chapter 2

Conjugation coefficient for Conjugate Gradient Method, Chapter 4

Deflection angle in radians, Chapter 2

Thickness [m], Chapter 3

Dirac delta function, Chapter 4

Stopping criteria value

Transformed z coordinate





















Oblique Shock wave angle in radians, Chapter 2

Dimensionless temperature

Lagrange multiplier, Chapter 4

Effective time constant, Chapter 5

viscosity [Pa-s]

General parameter to be estimated, Chapter 4

Dummy integration variable, Chapter 5

Density [kg/m3]

Dummy variable of integration

Time constant [s]

Heat transfer enhancement ratio, Chapter 3

Dimensionless step response, Chapter 4

Humidity ratio

Solid domain

Thermocouple measured quantity

Adiabatic quantity

Air quantity

Experimental measurement

Fluid quantity

Liquid quantity

Measurement quantity

Mixture quantity

Modified quantity











Stagnation quantity, Chapter 2

Initial quantity, Chapter 4

Root-mean-square value

Surface quantity

Value at saturation conditions

Simulated measurement

Quantity at the speed of sound

Vapor quantity

Wall quantity

Critical quantity, Chapter 2

Dimensionless quantity

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Richard Raphael Parker

May 2012

Chair: James F Klausner
Cochair: Renwei Mei
Major: Mechanical Engineering

This study investigates the heat transfer characteristics of a multi-phase supersonic

jet impingement heat transfer facility. In this facility water droplets are injected upstream

of a converging-diverging nozzle designed for Mach 3.26 air flow. The nozzle is operated

in an overexpanded mode. Upon exiting the nozzle, the high speed air/water mixture

impinges onto a heated surface and provides cooling. Steady state heat transfer

measurements have been performed with peak heat transfer coefficients exceeding

200,000 W/m2. These heat transfer coefficients are on the same order as some of the

highest heat transfer coefficients ever recorded in the literature. Remarkably these heat

transfer coefficients are obtained using liquid flowrates ranging from 0.2 to 0.7 g/s, in

contrast to the several kg/s flowrates seen in studies that produce similarly high heat

transfer coefficients.

During steady state operation it is noted that no evidence of phase change was

experimentally observed. Preliminary investigations indicate that it may not be possible

to obtain evaporative heat transfer in the current facility. In order to investigate this

possibility higher surface temperatures are needed. However, designing a steady state

experiment to achieve high temperature operation is rife with difficulties and is likely to

be prohibitively expensive.

In order to overcome these challenges a transient inverse heat transfer (IHT)

method has been developed. One of the important issues revealed during this

investigation is that sensor dynamics will impact the measurements, thus diminishing

the measurement reliability. To alleviate this issue, a method of incorporating sensor

dynamics into the IHT method was developed. This type of method is not explicitly found

in the literature to the author's knowledge. A method for accurately determining the

impulse response function of the thermocouples used in the transient IHT experiments

yields good experimental results. Heat loss is discovered to be a critical factor in the IHT

method, and a difference in temperature of 3 C between that measured and the ideal

case renders the IHT results unusable.

A parametric study was performed to determine the effects of: disc height, impulse

response function, magnitude and shape of the heat transfer coefficient distribution,

the number of temperature sensors used, and the magnitude of the error in the data

acquisition system. It was discovered that the method was insensitive to noise levels

found in laboratory conditions and the accuracy increases for a decreasing disc height.

The relative slowness of the impulse response functions did affect the accuracy of the

IHT method as long as the time constant of the functions is accurately known.


Jet impingement produces high heat transfer coefficients, up to approximately
105 W/m2 -K. Liquid impinging jets have supported some of the highest recorded

surface heat fluxes, ranging from 100 to 400 MW/m2, [1]. Oh et al. [2] and Lienhard

and Hadaeler [3] have studied liquid jets and arrays that can produce heat transfer

coefficients of 200 kW/m2-K. These high heat transfer rates are accompanied by high

liquid flowrates of up to several kg/s of water, and such high water consumption may be

undesirable in some industrial settings. The current study proposes to use a supersonic

multiphase jet impingement facility designed after an experiment by Klausner et al. [4],

which uses the addition of liquid droplets to the impinging air-stream to enhance the

heat removal rate of the supersonic jet. The liquid flowrate will be orders of magnitude

lower than that used by the studies mentioned above, with less than one g/s, which may

be very desirable in applications where minimal water consumption is a concern.

Supersonic two-phase jet heat transfer is a field that has not been previously

studied. The contribution of the current study will largely consist of characterizing the

heat transfer capabilities of such a system including the effects of air and liquid mass

flowrates and nozzle spacing. Additionally, evaporative heat transfer capabilities of the

jet will be studied; in this scenario the latent heat of vaporization could potentially greatly

enhance the heat removal capabilities of the facility. However, it is not known whether

or not liquid evaporation can be achieved due to the high stagnation pressure. Due to

high impact pressures near the jet centerline, phase change is not likely in this region;

however, the conditions far removed from the impingement point may allow phase

change to occur.

1.1 Literature Review

Jet impingement heat transfer is a very diverse field and consists of single-phase

heat transfer and evaporative heat transfer, spray/mist cooling, and supersonic jet

impingement heat transfer. A brief review of jet impingement heat transfer is provided.

1.1.1 Single-Phase Jet Impingement

The analytical study of stagnation point flows largely begins with Hiemenz [5] who

studied the flow field of a laminar impinging jet by modifying the Blausis boundary layer

solution. Homann [6] extended this analysis to axisymmetric flows. These flows are part

of the Falkner-Skan boundary layer equations, which take the general form

f"' + of"f 3 (1 f'2) = 0

where (1-1)

f(O) = f'(O) =0 and f(oo) = 1.

The velocities u and v, and the similarity variable, Tl, are defined as

u = axf' (r)

v = -poVf avf )

where a is a proportionality constant and v is the kinematic velocity of the fluid. Note

that in the radial case the variable x is the radial distance from the origin. The particular

values of 3 and 3o are 1 and 1 for Hiemenz flow and 1 and 2 for Homann flow. The

variable f" is proportional to the shear stress, f' is the non-dimensional velocity u/U,,

and f is the stream function. Equation (1-1) represents a non-linear ordinary differential

equation (ODE) which must be solved numerically. The shooting type method is

generally used as the value of f" is unknown at the origin. The values of f" at the origin

Figure 1-1. Solution of a) Hiemenz stagnation point flow and b) Homann stagnation
point flow.

obtained numerically are 1.2325 for Hiemenz flow and 1.3120 for Homann flow, as found

in [7]. The solution for Hiemenz and Homann flow are shown in Figure 1-1. It is evident

that the the flow fields behave very similarly; however, the free stream velocity and

shear stress are reached for smaller values of the similarity variable for axisymmetric

stagnation point flow. A full derivation of these and other stagnation point flows using

a similarity type approach as that above can be found in the book by Schlichting and

Gersten [8].

While the above analysis is sufficient for completely laminar impinging jets, jets

which find industrial application must deal with the boundary layer approaching the free

surface of the jet far removed form the centerline as well as the transition to turbulence.

Due to the different flow regimes the jet analysis is typically broken up into several

different regions and analyzed through the use of a von Karmam momentum integral

analysis; for an analysis of stagnation region see [9].

The analysis of the temperature field within the boundary layer for these types of

flows is complicated due to the behavior of the thermal boundary layer that develops

on the surface. It is further complicated by the nature of the fluid itself as flows with

larger Prandlt (Pr) number behave very differently than flows with small ones. As the

boundary layer moves away from the centerline, the hydrodynamic boundary layer

reaches the free surface before the thermal boundary layer for Pr < 1 while the converse

is true for Pr > 1. Liu et al. have analyzed the flow in each of these regions for single

phase jets with constant surface temperature and heat fluxes, mostly through the use

of the von Karman-Pohlhausen integral solution [10, 11]. They were able to model the

transition to turbulence and the subsequent turbulent flow as well. Their solutions agree

exceptionally well with experimental results. In general the solutions have the form of

The analysis of the above flow field is not limited to integral solutions or to constant

boundary conditions. Wang et al. [12, 13] studied the effects of a spatially varying

surface temperature and heat flux on the solution using a perturbation method. They

found that the direction of increasing temperature affects the Nusselt number of the

flows, notably that increasing the wall temperature or heat flux with radial distance

from the origin will decrease the Nusselt number in the stagnation zone. Conversely

it increases the Nusselt number in the boundary layer region. Wang et al. additionally

studied the conjugate heat transfer problem where the temperature field is determined in

the liquid and solid simultaneously, [14]. They found that thickness of the heater can be

a contributing factor for the heat removal capability of the jet.

There are several additional phenomena that impact the cooling rates of impinging

jets. These include the effects of the jet nozzle diameter [15], hydraulic jump [16], and

the splattering of liquid from the resulting free surface [17]. In cases with sufficiently high

surface temperature, phase change can be observed under impinging jets including the

regions of nucleate boiling, departure from nucleate boiling, and transition boiling [18].

While most studies of liquid jet impingement find no appreciable effect on the nozzle

height above the heat surface, Jambunathan et al. [19] noted that some studies do show

an effect most notably at higher Reynolds numbers. An empirical correlation based

on heat transfer data available in the literature was proposed however, it provides no

physical insight of the flow field and heat transfer taking place.

Liquid jet impingement can support exceptionally high surface heat fluxes. Liu and

Lienhard [1] used a liquid jet with velocities exceeding 100 m/s, liquid supply pressures

of up to approximately 9 MPa, and flowrates of approximately 300 g/s to remove heat

fluxes of at least 100 MW/m2. These experiments were novel in the fact that they used

a plasma torch as a heat source. Surface temperatures were determined by coating

the top surface with a material of known melting temperature and completing several

experimental runs until the surface temperature could be isolated to lie within a range

of temperatures. Melting of the heated surface occurred due to the use of the torch and

the back wall temperature was assumed to be essentially the melting temperature of

the solid. The heat flux was determined by using the minimum thickness of the solid

where it had melted and then assuming a linear temperature profile. Because of the

coarse nature of the measurements, uncertainty is hard to quantify and heat transfer

coefficients were not reported. However, the heat fluxes measured are the highest

steady state valuesrecorded in the literature. To further enhance the study of high heat

flux removal Michels, Hadeler, and Lienhard [20] and Lienhard and Napolitano [21]

designed thin film heaters using vacuum plasma spraying and high velocity oxygen

fuel spraying. These heaters are supplied with dc electrical power of up to 3,000 A and

24 V, producing heat fluxes of up to 17 MW/m2. Lienhard and Hadeler [3] were able

to construct an array of liquid jets with liquid supply pressures of approximately 2 MPa

and flowrates of approximately 4 kg/s. These impinging jet arrays were able to support

heat fluxes of 17 MW/m2 with an average heat transfer coefficient of 200 kW/m2 with

uncertainties of 20%. Similar results were found in a study by Oh et al. [2]. These

studies help illustrate the high heat removal capabilities of jet impingement technology.

For an extensive review of the subject of liquid jet impingement the author

recommends the review articles of Liendhard [22], Webb and Ma [23], and Martin

[24]. These articles offer a complete review of the subject and include many effects not

discussed in this brief literature survey.

1.1.2 Mist and Spray Cooling

Mist/spray cooling of a heated surface is largely different from jet impingement due

to the fact that the liquid impinging on the surface is in the form of disperse droplets.

These droplets are usually generated by forcing liquid through very small orifices

within a nozzle which atomizes the liquid. The primary benefit of using this technique

is that the disperse droplets generally allow for evaporative heat transfer to dominate.

Mist/spray cooling produces heat transfer coefficients on the order of those found during

pool boiling. However, the critical heat flux can be several times higher [25].

The droplet size is an important parameter in mist/spray cooling, Estes and

Mudawar [26] correlated the critical heat flux (CHF) with the Sauter mean diameter

of the sprays; they also found that the apparent density of the spray can be an important

factor in mist/spray cooling as denser sprays are less effective. Mist/spray cooling

can also be applied to surfaces lower than the boiling point of the liquid. The reduced

evaporation can lead to a buildup of a liquid film on the heated surface which can have a

thickness less than the size of the droplets. Graham and Ramadhyani [27] performed an

experimental study which shows that increasing the amount of droplets on the surface

can lead to thicker films which may increase the thermal resistance at the surface;

however, this thick film may be able to convect the heat away better due to an increased

velocity. They were able to develop a simple model of the thin film dynamics and the

resulting heat transfer which had approximately 4% error for heat flux predictions with an

air/methanol mixture but, only provided qualitative agreement when used with air/water

data. It is noted here that because of the evaporation taking place in mist/spray cooling,

the heat transfer coefficient does not vary appreciably with the radial distance, a feature

quite different than that found in jet impingement heat transfer. Readers desiring a

comprehensive review of mist spray cooling are encouraged to consult the review article

of Bolle and Moureau [28].

1.1.3 Supersonic Jet Impingement

The flow field of a supersonic jet exhibits very complex phenomena. The nature

of the flow can change dramatically as the nozzle exit to ambient pressure ratio

changes, sometimes quantified by the stagnation to ambient pressure ratio. When

the nozzle exit pressure is lower than the ambient pressure oblique shock waves form

at the edge of the nozzle in order to compress the flow. These shock waves become

Prandtl-Meyer expansion fans when they meet at the jet centerline. This process leads

to a series of reflected shock waves and expansion fans forming in the flow field at the

exit of the nozzle, including the formation of normal shocks in the flow known as Mach

disks. When the jet exit pressure is larger than the ambient pressure, Prandtl-Meyer

expansion fans form at the exit of the nozzle and a similar series of events take place.

The flow field of the jet changes dramatically in the axial direction. Zapryagaev et al.

[29] noted that for overexpanded jets the radial pressure distribution upstream of the

first shock cell contains several local maxima with very sharp discontinuities present.

These discontinuities disappear downstream of the first shock cell and generally a non

centerline maximum appears in the pressure distribution. These features are present

in underexpanded jets as well, [30]. While these features are complex they can be

Figure 1-2. An illustration of the shock structure in the wall jet region. [Reprinted with
permission from Carling, J. C. and Hunt, B. L., The Near Wall Jet of a
Normally Impinging, Uniform, Axisymmetric, Supersonic Jet, Journal of Fluid
Mechanics 66 (1974) 159-176 (Page 174 Figure 9(b)), Cambridge University

modeled somewhat accurately by a method of characteristics approach as noted by

Pack, [31, 32] and Chu [33], among others.

Underexpanded impinging jets have been extensively studied in the literature

as they pertain to the launching of rockets and spacecraft, whereas overexpanded

impinging jets are relatively uncommon in industrial settings. When a jet impinges

upon a flat plate, a complex shock structure is formed. This shock structure forms

several complex features including a triple shock structure, where three shock waves

intersect near the impingement surface, a bow shock, also known as a plate shock in

the literature, as the flow must come to rest in the stagnation point on the surface, and

the shock waves which radiate from the triple shock point and slow the flow along the

plate to subsonic speeds. These features appear in all types of supersonic impinging

jets including underexpanded, ideally expanded, and overexpanded. The bow shock, if

curved, can form a recirculating stagnation region in the area of the center of the jet to

the edge of the nozzle. An illustration of the complex shock structure of the flow at larger

radial distances along the impingement plate is shown in Figure 1-2.

The stagnation region is very complex due to the formation of the above mentioned

bow shock and stagnation bubble. Donaldson and Snedeker [30] studied underexpanded

jets from a converging nozzle and performed many different measurements to help

characterize some of the important features of the flow including impingement angle

and nozzle pressure ratio. They were able to observe stagnation bubbles forming, but

noted that this phenomenon did not occur in every experiment. Schlieren photographs

were taken as well as total pressure measurements along the jet centerline, and it was

observed that the velocity and pressure vary greatly in the axial direction. The velocity

in the radial wall jet region was measured via the use of pitot-static pressure tube

measurements along the impingement surface, the effects of surface curvature were

also characterized. Gummer and Hunt [34] also studied the flow of uniform axisymmetric

ideally expanded supersonic jets with low nozzle to height spacing and noted the

presence of the bow shock and complex shock structure in the wall jet region. They

attempted to use a polynomial and integral relation method to model the bow shock

height and the pressure distribution under the nozzle. Some success was seen for high

Mach numbers but not in the region of the triple shock. Low Mach number calculations

contained as much as 60% error. Carling and Hunt [35] performed a theoretical and

experimental investigation using the nozzles of Gummer and Hunt. Their study mostly

comprised of the region just outside of the nozzle along the impingement plate. They

were able to note the presence of the stagnation bubble for some of their experiments,

but not all. The presence of the stagnation bubble can severely influence the pressure

distribution on the plate and an annular maximum is possible for some jet spacings.

Attempts were made to model the shock structure in the wall jet region using the method

of characteristics. Qualitative features of the flow were able to be reproduced. However,

there appears to be some error in the region near the triple shock region. The pressure

variation along the plate was measured which showed several regions of unfavorable

pressure gradient. Carling and Hunt were able to investigate these regions by coating

the impingement plate with a type of grease. When the jet is impinged upon the plate

these unfavorable pressure gradients cause the grease to be removed due to local

separation of the boundary layer. A photograph of one of these experiments is shown

Figure 1-3. Grease streak type photograph from [35], the dark areas contain no grease
and are areas of high wall shear stress. [Reprinted with permission from
Carling, J. C. and Hunt, B. L., The Near Wall Jet of a Normally Impinging,
Uniform, Axisymmetric, Supersonic Jet, Journal of Fluid Mechanics 66
(1974) 159-176 (Plate 3 Figure 6(d)), Cambridge University Press]

in Figure 1-3. The dark regions represent where no grease is present; note the very

dark region near a radial distance of 2 nozzle diameters from the center where evidence

of separation is clearly evident. The separation phenomenon was noted by several

investigators including Donaldson and Snedeker, [30]. Kalghatgi and Hunt provide a

qualitative analysis experimental study of overexpanded jets which concentrated on the

triple shock problem near the edge of the bow shock. Their analysis suggests that flat

bow shocks are a possibility and schlieren photographs of overexpanded impinging

jets with Mach numbers ranging from approximately 1.5 to 2.8 largely confirmed

their qualitative analysis. They also note that the formation of a flat bow shock is a

phenomenon that is hard to predict. Lamont and Hunt performed a comprehensive

experimental study on underexpanded jets oriented normally and obliquely to a flat plate

which includes pressure measurements and schlieren photographs. The stagnation

bubble phenomenon was noted as well as some unsteadiness in the jet. Velocity and

pressure profiles were seen to vary greatly with the nozzle to plate distance, and it was

noted that the local shock structure has a strong influence on the flow field.

Unsteadiness of the impinging jet is caused by a feedback phenomenon which

has been extensively studied due to its importance in air vehicle takeoff, including the

launching of rockets and short/vertical take off and landing vehicles, such as the Joint

Strike Fighter. This mechanism was successfully modeled by Powell, [36, 37]. The

mechanism is caused by acoustic phenomena occurring at the edge of the nozzle.

These acoustic waves cause vortical structures to be generated in the shear layer

of the jet and are convected towards the impingement point. Upon encountering the

region near the plate, these structures interact with the shock waves near the plate

generating strong acoustic waves, which travel upstream towards the nozzle where were

they interact with the nozzle edge generating more acoustic waves which then repeat

the process [38]. Krothapalli [39] was able to predict the frequencies generated by a

supersonic impinging rectangular jet using Powell's model, thus validating the theory.

The effects of the unsteadiness on the flow field will be detailed below.

Due to the complex shock structure and unsteady phenomena in impinging jets,

numerical simulations are often used to help enhance the knowledge in this area. Alvi et

al. [40] modeled the impingement of moderately underexpanded jets and used Particle

Image Velocimetry (PIV) to help verify their results. Their method had reduced temporal

accuracy, but was able to reproduce major flow features including the stagnation bubble

and wall jet region, although the region of the triple shock point had some disparity

between the numerical and experimental results. Klinkov et al. [41] compared numerical

results of the velocity, pressure, and density fields to experimental results in the form

of schlieren photographs and surface pressure measurements. Their study focused

on overexpanded jets with Mach numbers in the range of 2.6 to 2.8 at approximately

ambient stagnation temperatures. They found that the location of the bow shock can

change significantly with nozzle to plate spacing, with several regions of a near constant

shock height followed by an almost discontinuous change to another height. Regions

of high shock height represent a convex bow shock and regions of low shock height

represent a flat bow shock with unsteadiness noted as the shock transforms to from

a convex shock to a flat shock. They also noted that a stagnation bubble region is

typical of a convex bow shock and that regular stagnation flow accompanies a flat

bow shock. An illustration is shown in Figure 1-4. The behavior of the bow shock is

significantly affected by the unsteady feedback mechanism as it is seen to oscillate back

and forth along the axis of the jet. This causes correspondingly large fluctuations in the

surface pressure on the impingement plate. Kawai et al. performed a computational

aeroacoustic study which was 2nd order accurate in time and 7th order accurate in

space. This study was done to determine the effects of the presence or absence of

a hole in a launch pad configuration and primarily focused on large nozzle to plate

spacings and the effect of Reynolds number on the unsteady phenomena. It was seen

that high Reynolds numbers can significantly increase the sound power levels of the

jet and the magnitude of its oscillations. Their numerical code produced results which

agreed well with historical sound power level data maintained by NASA. This study is

useful in illustrating the complexity of the problem under study and how very complex

numerical simulations are needed to accurately reproduce the features of the flow.

The addition of moisture in the form of water vapor to the air supply of an impinging

jet can have a noticeable impact on the flow field. This was observed experimentally by

Baek and Kwon [42] who performed studies of air with varying degrees of supersaturation

of water vapor for a supersonic jet issuing into quiescent air. They found that the

location of the Mach disk was located further upstream in the flow for moist air jets

and its size was reduced. Empirical correlations for the location of quantities such

as the size and location of the Mach disk and the location of the jet boundary were

proposed, although little mechanistic insight to the flow was gained. Numerical studies

by Alam et al. [43, 44] and Otobe et al. [45] were performed for air with various values

of supersaturation of water vapor for a supersonic jet impinging on a flat plate. They

attempted to model the non-equilibrium condensation taking place in the region after the

Figure 1-4.

Numerical results of a flow field with a flat plate shock (left) and curved flat
plate shock (right). [Reprinted with permission from Klinkov K. et al,
Behavior of Supersonic Overexpanded Jets on Plats, in: H.-J. Rath, C.
Holze, H.-J. Heinemann, R. Henke, H. Hnlinger (Eds.), New Results in
Numerical Fluid Mechanics V, volume 92 of Notes on Numerical Fluid
Mechanics and Multidisciplinary Design, Springer, 2006, pp. 168-175 (Page
173 Figure 3]

first Mach disk in the flow. Their model assumes no slip between the liquid droplets that
condense and that these droplets do not influence the pressure of the flow downstream.
The flow field displays some noticeable differences than that of dry air. The authors
propose that this is due to the addition of the latent heat of condensation to the air by the
condensing water vapor. Unsteady behavior due to the acoustic feedback mechanism
by Powell was seen in the simulations. This unsteadiness was not present upstream
of the first Mach disk, but was seen down stream of it. The presence of condensate
particles combined with the addition of the latent heat reduces the magnitude of the
pressure fluctuations seen in the downstream portion of the flow which is illustrated in
Figure 1-5. The authors attempted to verify their simulations with experimental data,


Phic -I Lmtc

S: rTA4 Pc Static pressure
I ---:: ppT
S 3T4 T14
0.9 -.. ,-: 0.9 ---- :t= 2

S. : t=/ 3T-,4
0.6 A
Jw-, ,,0.6 1 "
0 II

0 1 2 3 4 o
D 1 2 3 x 1 2 3 4
L/ID=4.0 xiA)e
a b

Figure 1-5. Jet centerline pressure fluctuations with a) no moisture and b) 40%
supersaturation of water vapor. [Reprinted with permission from Ashraful
Alam, M. M. et al., Effect of Non-Equilibrium Homogeneous Condensation
on the Self-Induced Flow Oscillation of Supersonic Impinging Jets,
International Journal of Thermal Sciences 49 (2010) 2078-2092 (Page 2086
Figure 10(b) and Page 2088 Figure 13(c)), Elsevier]

mostly consisting of comparing the shock structure as seen in schlieren photographs like

those in the study by Baek and Kwon, along with noise tones for dry air generated by the

acoustic feedback mechanism. This proposed validation is weak because there is a lack

of experimental data of which to compare to in the literature.

The study of supersonic impinging jet heat transfer jets has been studied extensively

in the literature. Unfortunately most of these studies have focused on the heat transfer

from a rocket exhaust to a launch pad facility. Donaldson et al. [46] performed an

experimental study of impinging sonic jets and their turbulent structure. The authors

were able to develop a correlation for Nusselt number based on applying a turbulent

correction factor to laminar impinging jet theory near the stagnation point and further

away in the wall jet region. While good agreement was found for their correlation it

is for sonic or just slightly supersonic impinging jets and does not apply to the highly

supersonic jets previously mentioned. The unsteady acoustic feedback phenomenon

previously discussed causes interactions between acoustic waves and the shock

structure of the impingement region. This results in local cooling to occur in the

region of the jet edge and is very noticeable in the measurement of the adiabatic wall

temperature. This phenomenon is termed cooling by shock-vortex interaction by Fox and

Kurosaka [47] who investigated this phenomenon. Kim et al. [48] studied the surface

pressure and adiabatic wall temperature of an underexpanded supersonic impinging

jet. They noted that the acoustic vortical structure interaction significantly affects the

adiabatic wall temperature and surface pressure which also varies greatly with nozzle

height. The presence of a stagnation bubble, which enhances the cooling directly below

the nozzle, was noted as well. Rahimi et al. studied the heat transfer of underexpanded

impinging jets onto a heated surface. The temperature of the impingement surface with

uniform applied heat flux is noted to change dramatically with radial distance as well

as with nozzle spacing as shown in Figure 1-6. They note that Nusselt number scales

approximately with the square root of Reynolds number and that high heat transfer rates

are encountered in the stagnation zone when a stagnation bubble is present. Due to the

complexity of the problem, they note that a general correlation of Nusselt number should

be a function of not only Reynolds number and Prandtl number, as is common, but also

a function of Mach number and nozzle spacing. Yu et al. performed a similar study and

noticed similar trends; their measured Nusselt numbers exceed 1,500.

Studies of the heat transfer characteristics of supersonic moist impinging jets

are not found in the literature. They are likely to show very complex phenomena

as evidenced by the differences in the shock structure and general behavior of the

relevant flow quantities in the jet and along the impingement plate. The current study

uses discrete liquid droplets that are injected into the air upstream of the nozzle.

This will likely result in an air stream supersaturated with water vapor which is further

complicated by the behavior of the liquid droplets and their effects on the flow. As

Figure 1-6.

80 90
"o P,,jj- 91 1 I 1I 1 I 1 |
70 -0 P,, P =5.08 ~~~ P, P. ,- 5 0
60 70 -
50- 60 -0
40 -- J50
30 40 .. o
20 3-0 -- -- -- -0

-10 0
0 1 2 3 4 5 6 7 8 9 1011 0 1 2 3 4 5 6 7 8 9 1011

a b

Adiabatic (circles) and heated (diamonds) wall temperature for a nozzle
spacing of a) z/D = 3.0 and b) z/D = 6.0. [Reprinted with permission from
Rahimi M. et al, Impingement Heat Transfer in an Under-Expanded
Axisymmetric Air Jet, International Journal of Heat and Mass Transfer 46
(2003) 263-272 (Page 267 Figures 6(a) and 6(b)), Elsevier]

elucidated by the literature survey the flow structure associated with this technology is

very complex, and essentially no analytical solutions are available for the flow field and

heat transfer. The available empirical correlations do not cover two-phase supersonic

impinging jets. Numerical studies may provide some qualitative insight, but in most

instances they do not adequately capture all of the physics taking place in the flow field.

1.2 Summary

In this Chapter an introduction to the study was made and the relevance of

multiphase supersonic impinging jets was introduced. The contributions of this study

were also described, mainly that this is a technology that has not been studied until now.

A brief literature review of the different types of impingement heat transfer was

presented. Liquid and single-phase heat transfer was introduced starting with the

classic work of Hiemenz and Homann. The development of accurate Nusselt number

correlations based on von Karman-Pohlhausen integral method were detailed. The

agreement between theory and experiments is exceptional for these correlations.

Other aspects such as the flow hydrodynamics, transition to turbulence, nozzle height,

and non-uniform boundary conditions effects were discussed. High heat flux removal

technologies that are capable of heat transfer coefficients as high as 200 kW/m2 were

detailed as well.

The study of supersonic underexpanded and overexpanded impinging jets was

described as well. This field is complicated by the complex flow structure generated by

shock waves which form when a nozzle is operated away from its designed pressure

ratio. The details of these shock waves including the effects of the curvature of the

bow shock just above the impingement plate were discussed. Stagnation bubbles

formed just below the bow shock were discussed and their impact on the flow field

was detailed as well. Shock waves near the impingement region cause an unsteady

feedback phenomenon caused by the interaction of acoustic waves with the edge of

the nozzle. The effects of this feedback phenomenon and the unsteadiness it causes

and relevant changes in the local flow field were detailed. Moisture in the air stream and

how it changes the relevant flow field was briefly explored as it is a relatively new area

of study in the literature. The temperature profile on the impingement plate and how it

changes with the presence of the stagnation bubble and acoustic feedback mechanism

were discussed. Numerous experimental studies in the literature which include pressure

and temperature measurements, particle image velocimetry, and schlieren photographs

along with relevant numerical studies in the literature that discovered and confirmed

these phenomena were discussed where relevant.

Lastly the complexity of the current study was discussed. It is noted that an

analytical solution to the problem will not be attainable and a predictive numerical study

is not feasible as well. The contributions of this study will be in the form of developing

an understanding of the mechanisms taking place as the multiphase supersonic jet

removes heat from a surface.


The supersonic multi-phase jet facility should possess several traits in order to be

useful for an experimental apparatus. It should have sufficient air storage capacity so

that experiments can be run at steady state. The stored air should be pressurized to

such an extent that the desired Mach number can be achieved. Lastly it should contain

sufficient water, and a means to control the flow, so that the impinging jet will remain in

multiphase operation during experiments.

The design for the current setup is based on a similar experiment by Klausner et

al. [4]. The impinging jet consists of the following systems to be described below: air

storage system, water storage and flow control system, air pressure control system,

air mass flowrate measuring system, temperature and pressure measurement system,

converging-diverging nozzle, and data acquisition system (DAQ). A schematic diagram

illustrating the configuration of the jet impingement facility is shown in Figure 2-1.

2.1 Impinging Jet Facility Systems

2.1.1 Air Storage System

The air storage system consists of 9 'K' sized bottles which give a total volume of

0.45 m3 and are kept at a pressure of 14 MPa. The air storage system is filled with air

from a model UE-3 compressor from Bauer Corporation. The compressor is powered by

a 3-phase 240 V power supply and is capable of supplying 0.1 m3/min of approximately

moisture free air to the air cylinders, thus the air storage facility can be charged to

capacity in approximately 4.5 hrs.

2.1.2 Water Storage and Flow Control System

Water for the facility is contained in a stainless steel vessel with a capacity of 2

L and a pressure rating of 12.4 MPa. Water is forced into the mixing chamber by the

difference in pressure between the top of the water vessel (which is acted on the the full

force of the air supply pressure) and the pressure inside the mixing chamber, which is

From Air 14/2.8 MPa Pressure
Compressor Air Reducer Transducer T lowmeter


Storage Thermocouple (T)
i -1Mixing
Drain Orifice Chamber
Connection T



Figure 2-1. Illustration of the jet impingement facility.

lower due to a change in area and because of the friction acting in the system tubing. A

drawing the of the mixing chamber is found in Figures 2-2 and 2-3. The flowrate of the

water is controlled by means of an orifice between the water vessel and mixing chamber

and the air pressure in the system. Orifice diameters of 0.33, 0.37, 0.41, and 0.51 mm

are used during experiments, and a drawing of the orifice design is found in Figure 2-4.

While there is some variance in flowrate between each experiment for a given orifice

size this effect is eliminated in the analysis due to the fact the the flowrate of water

into the mixing chamber is measured during each experiment. This is accomplished

by recording the elapsed time of each experiment and measuring the difference in the

mass of liquid in the water vessel. Table 2-1 shows the nominal flowrate of liquid and the



Figure 2-2. Cross section of the mixing chamber.

9/16 18 THREAD
10.16 mm DEEP

-- 3/8 NPT
12.7 mm DEEP
1/2 20 THREAD
7.62 mm DEEP

' /

0 6.35

Figure 2-3. Thread details of the mixing chamber cross section.

average liquid velocity for the pressures and orifice sizes used during the experiments. It

is noted that the flowrate of the 0.37 mm orifice is less than that of the 0.33 mm orifice,

this is due to the fact that the 0.37 mm orifice is not perfectly circular. This condition is

also seen in the 0.51 mm orifice, but it is not as severe as that in the 0.37 mm orifice.

Pictures of each orifice taken with an optical microscope are shown in Appendix C.




Orifice Diameter


0 f 1 1

c o


2.1.3 Air Pressure Control System

The air pressure control system consists of an air regulator located between

the air storage tanks and the inlet to the air mass flowmeter and the top of the water

storage tank. The regulator is capable of reducing air pressure from 14 MPa down

to a maximum of 2.8 MPa. Air pressures of 1.0, 1.7, and 2.4 MPa are used during

experiments. As discussed later, these air pressures result in the converging-diverging

nozzle operating in an overexpanded manner.

2.1.4 Air Mass Flow Measuring System

Measurement of the air mass flowrate is accomplished using an Annubar Diamond

II model DNT-10 mass flowmeter located downstream of the regulator and before the

mixing chamber. The diamond cross section of the flowmeter is such that it has a fixed

separation point and also reduces pressure loss. The flowmeter senses differential

pressure which is measured with a differential pressure (DP) transducer, which is

Table 2-1. Water Mass Flowrate and Average Water Velocity for Different Regulator
Pressures and Orifice Sizes.
Orifice Regulator Water Mass Average Water
Size Pressure Flowrate Velocity
(mm) (MPa) (kg/s) (m/s)
1.0 3.08 x 10-4 3.60
0.33 1.7 3.99 x 10-4 4.67
2.4 4.73 x 10-4 5.53
1.0 2.67 x 10-4 2.50
0.37 1.7 3.31 x 10-4 3.11
2.4 3.97 x 10-4 3.73
1.0 4.61 x 10-4 3.55
0.41 1.7 5.72 x 10-4 4.41
2.4 6.79 x 10-4 5.24
1.0 4.85 x 10-4 2.39
0.51 1.7 6.90 x 10-4 3.00
2.4 7.20 x 10-4 3.55

calibrated to measure pressure differences of

equation for the flow meter is

up to 2.21 x 10-3 MPa. The calibration

m = 58.283KD2 APf


where A P is the measured differential pressure in kPa, D is the diameter of the

flowmeter in mm, in this case 15.80 mm, K is a gage factor of 0.6, and pf is the density

of the flowing air, in kg/m3 calculated via

pf = 539.5 (2-2)
where Pf is the pressure of the flowing air in kPa, as measured by the pressure

transducer upstream of the mass flowmeter and Tf is the temperature of the flowing

air in Kelvin as measured by the the thermocouple upstream of the mass flowmeter.

-- theoretical
0.03 --- +20%
20% I


0.02 -

0.015 -

0.01 -

0.01 0.015 0.02 0.025
Figure 2-5. A comparison of the theoretical and experimentally measured air mass

The theoretical mass flowrate through the nozzle for 1-D isentropic flow is
calculated using mass flow, m = pAV, the Mach number, M = a/V (the speed of sound for
a perfect gas is a = vYRT), and the ideal gas law, P = pRT,

m= D2MP (2-3)
4 RT
Knowing that at the throat of the nozzle the Mach number is one and using
Temperature and Pressure stagnation ratios of T/To = 0.8333 and P/Po = 0.5282
Equation (2-3) reduces to

m = 0.4545D2P /o (2-4)

Equation (2-4) neglects the effects of friction and heat transfer, which affect the flowrate

of air through the nozzle. A comparison of the air mass flowrate measured during the

course of experiments with the theoretical air mass flowrate is shown in Figure 2-5. The

agreement between theory and experiment is within 20%.

2.1.5 Temperature and Pressure Measurements

The temperature and pressure of the jet impingement system is monitored during

system operation for calculating various quantities of interest. The temperature

measurements are accomplished by the use of E type thermocouple probes which

are inserted into 'T' junction compression fittings at a depth such that the tip of the

probe is at the centerline of the fitting. The thermocouple probes used are grounded

and sheathed in stainless steel and have a nominal diameter of 1.59 mm. Temperature

measurements are taken at the following points: the outlet of the pressure regulator, the

outlet of the water reservoir, and at the outlet of the mixing chamber.

Pressure is measured at the outlet of the pressure regulator just before the location

of the air mass flowmeter. The pressure measurement is made using a strain gage

type pressure transducer, which has a range of 0 2.8 MPa. The output signal of the
pressure transducer is a current which varies between 4 20 mA; because the DAQ

system used in the experiments only senses voltages a resistor of 520 Q is used to

convert this current into a voltage in the 0 10 V range needed.

2.1.6 Converging-Diverging Nozzle

The converging-diverging nozzle is where the mixture of liquid and air are expanded

to supersonic speeds. The nozzle is constructed of stainless steel with a throat

diameter of 2.38 mm and an exit diameter of 5.56 mm, giving an exit Mach number

of 3.26. The nozzle is attached to a size 10 DN (1/2" NPS) stainless steel pipe with an

internal diameter of 13.51 mm, which is connected to a braided stainless steel hose

approximately 9 m long with an inner diameter of 9.53 mm and is connected to the

outlet of the mixing chamber. Although the hose adds some small amount of pressure



Figure 2-6. Cross sectional view of the converging-diverging nozzle used in

loss, it allows the nozzle to be located away from the air storage cylinders and near

the impingement heat transfer targets. A cross sectional view of the nozzle is shown in

Figure 2-6.

2.1.7 Data Acquisition System

The DAQ used during the course of steady state heat transfer experiments is a

DAS 1601 data acquisition PCI card and a CIOEXP32 analog to digital converter

board, both made by Measurement and Computing Inc. This DAQ consists of 32 16-bit

double ended channels and channel gains of 1, 10, 100, 200, and 500 are selectable.

The system has a maximum reliable sampling rate of 50 Hz and the software Softwire,

produced by Measurement and Computing Inc is used for programming data collection.

For transient measurements on heated targets during inverse heat transfer experiments,

the DAQ system is supplemented with a National Instruments (NI), NI USB-6210 system

which has 8 double ended 16-bit channels and has a maximum aggregate sampling rate

of 250 kHz. This system uses Labview software produced by NI which and is also able

to interface with the Measurement and Computing DAQ via the use of an NI supplied .dll


2.2 Analysis of Impingement Facility

Some analysis of the jet impingement facilities are warranted. The behavior of the

system upstream of the nozzle is examined to determine if there are any corrections that

need to be applied to the thermocouple or pressure transducer readings. Additionally,

the following is examined: the pressure required to operate the nozzle in a perfectly

expanded manner, the minimum and maximum pressure that cause a shock wave

to form inside the nozzle, and the nozzle exit pressure when operating at various

regulator pressures. Lastly the shock wave angles forming at the nozzle exit for various

operating pressure are calculated as well. One-dimensional gas dynamic relations are

used to investigate the quantities of interest. Here it is noted that the analysis used

has limitations, the one dimension gas dynamic relations are isentropic in nature, with

the exception of shock wave calculations. The jet impingement facility experiences

friction and heat transfer during operation, thus the isentropic assumption is not met.

Additionally after the mixing chamber the flow will contain water droplets which are not

compressible. The quantities calculated below will have some inherent error however,

they do provide a reasonable approximation of the physics taking place in the facility.

2.2.1 Temperature and Pressure Upstream of the Nozzle

Calculating the temperature and pressure at various points upstream of the nozzle

is a simple matter; the cross-sectional area of the points in the system are required for

this analysis; Figure 2-7 provides an illustration of the jet impingement facility and the

diameters of the points of interest. Using the commonly known one-dimensional gas

dynamics relationships found in various compressible flow textbooks, such as Liepmann

and Roshko [49] or John and Keith [50], the pressure and temperature ratios as well

as the Mach number of the flow in these areas can be determined. In the following

SSection 4
D = 6.22 mm

S Section 3
D = 15.80mm

Section 2
SD = 6.22 mm

-- Section 1
D = 13.51 mm

-- hD = 2.38 mm
SNozzle Exit
D = 5.56 mm

Figure 2-7. Simplified view of the air flow path in the facility.

equations 7 is the ratio of specific heats and is a constant equal to 1.4. To determine the

flow Mach number the following relationship is used

A 1 2 9 11 2(-1)
A* M 1 2 ) M2)\ (2-5)

where the superscript denotes the critical area where Mach = 1. Note that Equation (2-5)

is a quadratic equation in M and has two solutions thus careful attention must be paid

in selecting the proper Mach number given an area ratio, in the present case all Mach

numbers upstream of the throat of the converging-diverging nozzle are subsonic. Once

the Mach number of the given section is determined the pressure and temperature ratios

can be determined from the following

S=1+ 1M2 (2-6)
T 2

P = l1Y M2) (2-7)

where the subscript o is the stagnation property, which is simply the particular property

with zero velocity. The analysis results using Equations (2-5) to (2-7) are shown in

Table 2-2. The results show that the temperature and pressure upstream of the nozzle

throat differ from their stagnation point properties by less than 1%; no correction due the

the velocity of the flow is needed.

Table 2-2. Area, temperature, and pressure ratios at various points in the iet


impingement facility.
A/A, T/To P/Po M
5.44 0.3193 0.01840 3.26
1.00 0.5283 0.8333 1.00
32.20 0.9999 0.9998 0.018
6.83 0.9986 0.9950 0.085
44.02 0.9999 1.0000 0.013
6.83 0.9986 0.9950 0.085

2.2.2 Nozzle Exit Pressure Considerations

There are a few theoretical considerations that need to be explored at the nozzle

exit. First, the necessary regulator pressure in order to obtain a perfect expansion at the

nozzle exit is needed; then the actual exit pressures based on the regulator pressure

during experiments are determined. The results from the previous calculations listed in

Table 2-2 show the stagnation pressure ratio at the exit of the nozzle, simply carrying

the requisite algebra and assuming a back pressure of 101.4 kPa yields the nozzle

exit pressure, see the results of these calculations in Table 2-3. From these results it

is observed that the pressure necessary for ideal expansion is approximately twice the

pressure the regulator of the system can provide, and thus during normal operation of

the jet impingement facility the nozzle operates in an overexpanded manner.

Table 2-3. Nozzle exit pressure for various regulator pressures.
Regulator Nozzle Exit Nozzle
Pressure Pressure Operation
(MPa) (MPa)
5.5 0.1014 perfectly expanded
2.8 0.0508 overexpanded
2.4 0.0443 overexpanded
1.7 0.0317 overexpanded
1.0 0.0190 overexpanded

Due to the fact that the nozzle exit pressure is below the ambient pressure some

concern about a shock wave forming in the nozzle will be addressed. There are two

limiting pressures for this case, one is the pressure at which at shock wave forms at the

throat of the nozzle, the other is the pressure that a shock wave forms at the nozzle exit;

Figure 2-8 shows an illustration for both of these two cases. In the limiting case, a shock

wave occurring at the throat (where the Mach number is equal to unity), the stagnation

pressure ratio is 0.992. Assuming that the back pressure is atmospheric pressure, the

stagnation pressure that will cause a shock to be located at the throat is 0.102 MPa. To

calculate the limiting case of a shock wave occurring at the nozzle exit is just slightly

more complicated. When it is assumed that a shock wave is located at the exit plane of

the nozzle, the stagnation pressure ratio and Mach number just before the exit plane can

be found from Table 2-2. The normal shock wave relations for Mach number and static

pressure ratio across a shock (Equations (2-8) and (2-9)) can then be applied,

/M2(y7- 1) + 2
M2 = M 1) (2-8)

P2 2 M 7- 1
P2 2 1 (2-9)
P1 7<+1 7+1'

Nozzle Exit


Po I Patm

Figure 2-8. Illustration of the limiting cases for shock waves in the nozzle.

Using Equations (2-8) and (2-9) the Mach number just past the shock wave is

found to be 0.461 and the static pressure ratio is 12.268. Performing the requisite

algebra yields a back pressure of 0.449 MPa. Thus the nozzle will have a shock wave

located inside for a stagnation (regulator) pressure in the range of 0.102 to 0.449 MPa.

Since the minimum regulator pressure used during experiments is 1.0 MPa there is little

concern that a shock wave will form inside the nozzle.

2.2.3 Oblique Shock Waves at Nozzle Exit

It is known that the converging diverging nozzle operates in an overexpanded

manner; the exit conditions of the nozzle should be considered. When a nozzle

is overexpanded oblique shock waves form at the outlet of the nozzle, see [49] for

instance. These shock waves compress the air such that it is then equal to the

nozzle back pressure; the first oblique shock wave coming out of the nozzle can be

modeled using the standard one dimensional gas dynamic relations parameters such

as the shock angle, deflection angle, temperature ratio, stagnation pressure ratio, and

down stream Mach number. Figure 2-9 shows an illustration of the shock wave at the

nozzle exit. To determine these exit quantities, first the nozzle exit pressure should be

determined using the regulator pressure and Equation (2-7). The shock wave angle can

SOblique Shock Wave


Figure 2-9. Illustration of an oblique shock wave at the nozzle exit.

then be calculated using the following equation since the Mach number at the nozzle exit

is known from the design conditions, and the static pressure ratio can be calculated,

P2 2y M2 sin2 1
U1 (2-10)

The deflection angle, 6 can be determined from the following

M2sin20 1
tan6 = 2cotO (2-11)
M2 (y + cos20) + 2
The downstream Mach number, stagnation pressure ratio, and static temperature ratio

are easily calculated via the following equations:

1 1+ M2 sin20
M2 = 1 (2-12)
sin (0 6) M sin2 (2-12)

Po1 2 M sin2 1
Y (2-1 3)
Po2 [1+ 2M2 sin2O L 27 M2 sin2O (2-13)

TPo2 1 M2 sin20 ) ( M2 sin20 )_
T1 (7+1)'] M2 sin20
h -- )

40- 22
35 18
30 14
25 10
1 1.5 2 2.5 1 1.5 2 2.5
Pol (MPa) Po (MPa)

a b

1 2

0.95 1.9
0.9 1.8
0.85 1.7
0.8- 1.6

0.75- / 1.5
0.7 1.4
0.65 k 1.3

1 1.5 2 2.5 1 1.5 2 2.5
Pol (MPa) Pol (MPa)

c d

Figure 2-10. The variation of a) shock angle, b) deflection angle, c) stagnation pressure
ratio and d) static temperature ratio as a function of upstream stagnation
pressure, downstream of an oblique shock wave.

It is noted that the stagnation temperature across the shock is constant. Results of

these calculations are shown in Figure 2-10. It is briefly mentioned that the stagnation

pressure ratio reflects a loss of momentum going across a shock wave and that this loss

of energy is lessened at higher upstream pressure ratios.

2.2.4 Complete Shock Structure of an Overexpanded Jet

The first shock wave at the nozzle exit is easily modeled as shown above; however,

the subsequent behavior of those shock waves is quite complex. The oblique shock

waves can intersect at a point downstream or can merge and form a normal shock area

known as a Mach disk. Mach disks typically form after relatively strong shocks which

are typical for nozzles operating far removed from the ideal pressure ratio. These shock

waves compress the flow causing the formation of Prandtl-Meyer expansion fans which

turn the flow and lower the pressure. When expansion fans intersect the shear layer

which is formed at the boundary of the jet, they are reflected back as oblique shock

waves. This series of events causes the formation of "shock diamonds" in the flow and

is repeated until the combination of viscous effects and the influx of low momentum

fluid cause the jet to become subsonic, or when the jet interacts with an obstacle.

Figure 2-11 provides and illustration of the shock structure typically seen at the exit of an

overexpanded jet.

The pressure, temperature, and velocity of the flow in the down stream of the

nozzle changes very rapidly and is difficult to model analytically. Zapryagaev et al.

[29] performed experiments with an overexpanded nozzle and performed schlieren
photography as well. Their results show that the pressure in the flow upstream of

the first Mach disk varies greatly in the radial and axial direction with several sharp

discontinuities present. Downstream of the first Mach disk the variation in pressure is

still present. However, the discontinuities are no longer present. Many authors have

extensively studied underexanded jets [30, 34, 35, 46, 51, 52] and similar trends as the

above are observed.

2.3 Summary

In this Chapter the construction of the jet impingement facility has been explored.

The facility systems include the air storage, water storage and flow control, air pressure

control, air mass flow measuring, converging-diverging nozzle, and data acquisition.

A comparison between the mass flow rate measured during experiments and the

theoretical mass flowrate based on one-dimensional isentropic gas dynamic relations

was performed and the results differed by less than 20%. The pressure, temperature,

and Mach number various sections of the jet impingement facility were determined and

Oblique Shock Wave Viscous Shear Layer

I -----------------------

------- ---------------------

Mach Disk Expansion Fan
Nozzle Exit

Figure 2-11. Illustration of the structure of oblique shock waves at the exit of an
overexpanded nozzle.

are shown in Table 2-2. The nozzle exit pressure was calculated for each regulator
pressure used during the experiments, and it was found that the nozzle operates in an
overexpanded manner for the entire pressure range. No shock wave is expected inside
the nozzle. Finally, the shock structure at the exit of the nozzle has been described.


Jet impingement facilities are capable of very high heat flux removal, and an

experimental procedure to determine the heat transfer coefficient for the facility in

Chapter 2 is developed. Several different experiments were initially tested without

success before a successful experimental configuration was developed to measure the

steady-state heat transfer coefficient over a range of operating conditions.

Initially it was believed that phase change heat transfer would occur during

operation of the jet impingement facility to such an extent that no radial variation of

heat transfer coefficient would be observed. As such an experiment was designed in

which a thin sheet of stainless steel was machined into a metal blank and heated via

Joule heating and insulated at the bottom where the temperature was measured via

a thermocouple. The area of the strip was quite small (approximately 15 mm2) and

as such the heat fluxes produced during the experiment were high. The problem

experienced with this setup is that the back wall temperature measured by the

thermocouple is not very sensitive to changes in the heat transfer coefficient with the

applied heat fluxes and the metal thickness used. An illustration of the heater assembly

used is shown in Figure 3-1.

In order to alleviate this problem an experiment was conducted where thermocouples

were embedded inside of a copper cylinder which was heated from the bottom and

insulated along the side. The jet was allowed to impinge on the top of the cylinder

and the temperatures inside the copper piece were measured. One-dimensional heat

conduction in the axial direction was assumed and due to the absence of internal heat

generation, a linear fit of the measured temperature was then used to extrapolate the

temperature to the surface and allowed the determination of heat flux. Upon analyzing

the data gained from these experiments it was observed that the heat transfer coefficient

Thickness 0.3mm .

x *

- Area 3 x 5 mm

Figure 3-1. Stainless steel heater assembly.

for the jet impingement facility varies significantly in the radial direction. An illustration of

the copper heater assembly used is shown in Figure 3-2.

In order to gain some insight into the radial variation of the heat transfer coefficient

the heater in Figure 3-2 was modified to include temperature sensitive paint on the

top surface of the copper test piece. Steady state temperature distributions at the top

surface where then used as input to an inverse heat transfer algorithm to determine

the radially varying heat transfer coefficient. There were several drawbacks to this

study. First the temperature sensitive paint is very brittle and had to be protected from

the impinging jet via the use of thick clear coat applications to the surface of the paint

or via the use of transparent tape. These protective layers could not be neglected in

the inverse heat transfer analysis and complicated the algorithm. Lastly, and most

importantly, the impinging jet partially obscures visual observation of the temperature


Copper Test Piece


Cartridge Heater


Figure 3-2. Copper heater assembly.

sensitive paint. These complications render reliable experimental results difficult to


The third experimental configuration tested consists of a thin sheet of nichrome

which is heated by Joule heating. It is insulated at the bottom and 9 thermocouples

are used to measured the spatial variation of the back wall temperature. The area

of the nichrome strip is large and the heat fluxes produced are considerably smaller

than those applied to the stainless steel setup described earlier. This experimental

configuration proved to give reliable measurements of heat transfer coefficient, and the

abundance of thermocouples allows the spatial variation of heat transfer coefficient to be

determined. This experiment is now be described in detail. Later Sections will describe

the experimental procedure and explore the heat transfer results.

3.1 Heater Construction

3.1.1 Physical Description

The heater design used during the course of the following experiments is inspired

by the work of Rahimi et al. [51]. It is constructed using a thin nichrome strip which is

0.127 mm thick with an exposed area of 50.8 x 25.4 mm. The heater has 9 total E type

thermocouples attached to the backside of the strip. One thermocouple is attached

at the center of the strip and 7 additional thermocouples are attached every 3.79 mm

towards one side of the strip. Additionally one thermocouple is attached at 12.7 mm

from the center on the opposite side. This thermocouple is used to ensure the jet is

centered over the heater by verifying symmetry.

The nichrome strip is then epoxied on top of a Garolite slab that is 140 x 140 x

6.35 mm. Small holes are drilled in the slab so that the thermocouples penetrate it and

avoid deforming the flat surface of the nichrome strip. The slab of Garolite has a thermal

conductivity of 0.27 W/m-K compared with 13 W/m-K for the nichrome and thus acts to

insulate the backside of the nichrome strip.

Electrical power is supplied to the nichrome via two copper bus bars with dimensions

of 43 x 25 x 2 mm attached to the top of the strip. During operation of the two-phase jet,

liquid flows towards the bus bars and accumulates at the edge. This liquid film of

accumulation could affect the heat transfer physics; to lessen this effect the edges of

the bus bars are filed to an angle of approximately 300. An illustration of the heater

assembly is shown in Figure 3-3.

Power is supplied to the nichrome strip via a high current, low voltage dc power

supply. The power supply is capable of supplying 4.5 kW of power through a voltage

range of 0-30V and a current limit of 125 A. The maximum voltage seen during

experiments is approximately 5 V at 125 A.

Copper Bus Bar
I Garolite Base

N nnhr, : rr,- > Healc-r

z 6 1 \300

Power Supply Thermocouples

Figure 3-3. Illustration of heater assembly used for steady state experiments.

3.1.2 Theoretical Concerns

The thermocouples used for the experiment measure the temperature on the back

wall of the Nichrome strip. To determine the heat transfer coefficient for the impinging jet

the surface temperature of the nichrome is needed. Typical heat transfer coefficients for

impinging jets will yield Biot numbers (Bi = h6/k) much greater than unity thus a lumped

system assumption is not valid for the current experimental configuration.

The following analysis provides a method for evaluating the spatially resolved

surface temperature. First the steady state heat equation in Cartesian coordinates with

internal heat generation is examined.

2 T T2 T2 q'"
X2 2 + Z2 +- 0 (3-1)
ax2 ay2 az2 k

When Equation (3-1) is non-dimensionalized the following results:

()6 22 () 6 \2 2 9 z*22
0 y2 + l = 0, (3-2)


STk x y z
0= x = y= z =
q'562 D D' 6

Here D is the nozzle diameter, and 6 is the thickness of the strip. The coefficients in the

first 2 terms of Equation (3-2) are quite small, and three dimensional effects can be

neglected. As such the governing equation and boundary conditions are

d = 0 (3-3)
dz* z*_
dO kD
-- kDNUD (r*) (0 0, (r*))
dz* kuDr k,6

where r* = r/D is the non-dimensional radius with the origin at the centerline of the jet,

NuD (r*) = h(r*) D/ k, is the Nusselt number, k, is the thermal conductivity of water,

and 0o(r*) is the reference temperature. Due to the extreme difficulty in measuring the

expanding jet fluid temperature, the adiabatic wall temperature is commonly used as

a reference for purposes of computing a heat transfer coefficient [53] associated with

impinging jets. Note that the effects of the radial variation of heat transfer coefficient

come from the boundary conditions only. The solution of this ordinary differential

equation is:

S(r*, z*) = 1- z*2) +D w(r*) (3-4)
2 kD NUD(r*)
or in dimensional form

T (r, z) = q (62 2)- h + T. (r), (3-5)
2k h (r)
where Ta,w is the adiabatic wall temperature. It is observed that the difference in

temperature between the top and bottom surfaces is the quantity q"'62/2k, in dimensionless

form it is equal to 1/2. The maximum volumetric heat generation in the experiments is

3.66 x 109 W/m3. Using a value of 13 W/m-K for the thermal conductivity of nichrome

the maximum temperature difference between the upper and lower surface observed in

experiments is on the order of 2 C and cannot be neglected. However, the solution for

the heat transfer coefficient and thus the Nusselt number can be calculated knowing the

internal heat generation rate, back wall temperature, and the adiabatic wall temperature:

NUD(r*) NNUD,w(r*)
1 k D NuD,(r*)
kw6 2
where (3-6)
NUDw(r*) = D[O(r*, 0) e,(r*)]

3.2 Experimental Procedure

During the course of the experiments, the Measurement and Computing data

acquisition system is used Also note that the heat flux removed from the top of the

heater assembly is the quantity q"'6, since the heat generated internally can only be

removed from the top surface.

3.2.1 Two-Phase Experiments

During the course of experiments it was observed that running the impinging jet

would cause ice to form on the surface of the heater at low heat fluxes. At adiabatic

conditions the ice would begin to form into a cone shape, eventually this cone would

be broken off and a new cone would form in its place in a periodic manner, as shown in

Figure 3-4. At low but, non-zero heat fluxes, a thin sheet of ice would form that would

exhibit similar periodic behavior as the ice cones. This ice formation affects heat transfer

since this layer of ice is stationary and acts as an insulator. To avoid this condition a

minimum heat flux of 300 kW/m2 was chosen such that ice formation is not visually

observed during operation of the impinging jet.

Figure 3-4. Ice formation at adiabatic conditions. Note the conical shape of the ice

To run a complete experiment, the nozzle is aligned near the center of the heater

surface at a given height and the impinging jet is initiated with the pressure regulator

set at a desired pressure. The power supply to the heater is turned on and a heat flux

of approximately 470 kW/m2 is supplied to the heater. To ensure the impinging jet is

centered over the heater, the position of the heater is moved such that the temperatures


measured by the single thermocouple on one side matches the temperature of the

corresponding thermocouple on the opposite side to within approximately 0.5 OC.

Once this condition is achieved the heater is allowed to reach steady state, which is

determined by observing a graph of the heater temperatures vs. time. Upon reaching

steady state operation, the heater thermocouples are logged at a sampling rate of 50

Hz for approximately 2 minutes. Afterwards this same procedure is performed for heat

fluxes of approximately 430, 390, 350, and 315 kW/m2. These heat fluxes are chosen to

be as high as reasonably achievable with the given equipment for two reasons. First to

help prevent the formation of ice on the surface of the heater and such that the heater

wall temperature difference is as high as possible to minimize the uncertainty in the heat

transfer coefficient measurement.

The data reported for this study are averaged values from 100 samples taken

over the course of 2 minutes. While it is possible for high frequency oscillations in the

measured temperatures to exist due to the multitude of droplets impinging onto the

surface, this is not likely to be observed for several reasons. First, the data are averaged

which will lessen any transient effects. Second, the heater, although thin, does have a

finite thickness. So it will tend to act like a low pass filter and dampen fluctuations. Lastly

it is believed that a thin liquid layer exists on the surface of the heater. Any liquid drops

that impinge onto the heater will tend to coalesce with this liquid film, thus minimizing

transients of the local heat transfer coefficient.

The measurement of the adiabatic wall temperature is accomplished by a similar

procedure as above except after ensuring the jet is centered, the power supply to the

heater is turned off. The heater temperatures are allowed to reach steady state and then

measurement of the adiabatic wall temperature is commenced. It could take several

minutes (on the order of 5 minutes) for the heater to reach steady state. Because

of the absence of heat transfer, the measured temperature at the back wall is equal

to the surface adiabatic wall temperature of the jet. It is noted that the formation of

ice was observed on the surface of the heater during the adiabatic wall temperature

measurements, but because of the reasons listed above for neglecting fluctuations due

to multiple droplets impinging onto the surface, it is believed that this will have little to no

effect on the measurement of adiabatic wall temperature.

In computing the jet Reynolds number. the viscosity is based on the nozzle exit air

temperature and pressure as calculated by one-dimensional gas dynamic relations. In

computing the Nusselt number for the single-phase jet, thermal conductivity is evaluated

based on air and the adiabatic wall temperature. During two-phase jet impingement, a

thin liquid film exists on the heater surface, and thin liquid film dynamics dominate the

heat transfer physics. Thus the water thermal conductivity based on the adiabatic wall

temperature is used for Nusselt number calculations of the two-phase jet. Typical values

for air viscosity, water thermal conductivity, and air thermal conductivity respectively are

6.45 x 10-6 Pa-s with less than 1% variation, 0.588 W/m-K with a 3% variation, and

0.0243 W/m-K with a 5% variation.

3.2.2 Single-Phase Experiments

To provide a comparison for reference to the two-phase jet results single-phase

experiments are carried out with the jet impingement facility using only air expanding

through the nozzle, water flow is cut off. The experimental procedure is essentially

identical to that for the two-phase jet expect the applied heat flux is reduced. The heat

fluxes used during experiments are lower than those in the two-phase experiment to

ensure that the heater does not over-heat and de-laminate from the Garolite base.

Table 3-1 displays the corresponding heat fluxes for a given Reynolds number.

3.3 Experimental Results

3.3.1 Uncertainty Analysis

Uncertainty analysis for the experiments is done using the method of Kline and

McClintock [54]. The uncertainty of the calculated Nusselt number ranges from 2.0 to

4.0% for the single-phase jet and 2.5 to 18% for the two-phase jet near the centerline

Table 3-1. Reynolds number and corresponding heat fluxes.
Nominal lowest mid highest
ReD Heat Flux Heat Flux Heat Flux
(kW/m2) (kW/m2) (kW/m2)
4.5 x105 35 50 65
7.3 x105 60 80 100
1.0 x106 80 100 120

and 0.3 to 1.0% at the outer extents of the domain. The Reynolds number uncertainty

ranges from 2 to 3% and does not vary appreciable between the single and two-phase

experiments. The uncertainty in the temperature measurements are 0.2 OC.

To help ascertain the error in the experiments by assuming that the three dimensional

effects were neglected, a numerical analysis was conducted. This analysis uses a

second order accurate finite difference scheme to solve Equation (3-2) on a three

dimensional grid of 2" x 2" x 2" where n = 4, 5, 6, and 7. The top boundary is modeled

as having a Nusselt number distribution found in the experiments and the remaining

sides are modeled as being insulated. After completion of each simulation the Nusselt

number is calculated using Equation (3-6). To characterize the error in using a one

dimensional assumption the root-mean-square error is calculated via the following


f f (Nuexp Nsim)2 dxdy
errorrms 0 0 L W (3-7)
ff (Nuexp)2 dxdy
o o
the use of the different grid sizes allows the extrapolation of the error using Richardson's

extrapolation method [55, 56]. Using the highest Nusselt number distribution found

during the experiments, results in an rms error of 1.21% with a peak error of 3% located

near the origin, while using the lowest Nusselt number distribution results in an rms error

of 0.83 % with a peak error of 3.25% located near the edge of the domain. These errors

2500 q' = 65.3 kW/m2
0 q' = 49.3 kW/m2

2000 0 q' = 33.4 kW/m2

z/D = 2.0
1500 I ReD = 4.45 x 105

1000 -


0 I-- I I I I I I I
0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 3-5. Measured single-phase NuD spatial variation at different heat fluxes.

are less than the uncertainty contained in the experimental measurements and thus the

one-dimensional treatment for evaluating the Nusselt number is deemed satisfactory.

3.3.2 Single-Phase Results

During the course of the experiments it was observed that the heat transfer

coefficient is independent of heat flux, as expected. Results for a typical experiment

are shown in Figure 3-5. To illustrate the amount of uncertainty in the data error bars

have been included in this figure. However, to facilitate ease of viewing they are not

shown in the rest of this Section.

Figure 3-6 shows the measured thermocouple temperature profiles at various heat

fluxes for a nozzle spacing of 4 nozzle diameters for a single-phase experiment. Note

that the adiabatic wall temperature corresponds to that measured with zero applied

heat flux. Figure 3-7 shows that Nusselt number scales with ReD as reported by

1 1.5 2

2.5 3 3.5

Figure 3-6. Spatial heater temperature variation at different applied heat fluxes.

Donaldson et al. [46] and Rahim et al. [51], among others. As a matter of reference,

a single-phase Nusselt number on the order of 2,500 corresponds to a heat transfer

coefficient on the order of 11,000 W/m2-K in the present study. A single nozzle was used

in the experiments and thus the over-expansion pressure ratio and Reynolds number

are not independent of each other and the effects of over-expansion ratio could not be


Figure 3-8 compares the local Nusselt number for nozzle heights and Reynolds

numbers used during the experiments. For r/D > 0.5 and H/D > 2 the Nusselt number

distribution is not strongly dependent on the nozzle height. However, for r/D < 0.5 there

is a small variation in Nusselt number. This behavior results from the complex shock

structure at the nozzle exit and its interaction with the heater surface. For H/D < 2

Nusselt number is slightly elevated, but this effect appears to lessen at higher Reynolds


* q"= 102 kW/m2
0 q"= 79 kW/m2
0 q"= 62 kW/m2
S q"= 0 kW/m2
Z/D = 4.0
ReD = 7.50 x 105


i i i ii i*

0 0.5

ReD =4 57x105
4000 A Re = 7.355X 10s -
O ReD 1.05x10'
3500 z/D 6.0

3000 O
z A
2000 o
A 2


0 0.5 1 1.5 2 2.5 3 3.5 4


ReD =4.57x10
4 A ReD =755x10n
O ReD 1 05106
3.5 z/D = 6 0

Q 0


0 0.5 1 15 2 25 3 35 4


Figure 3-7. Spatial variation of NuD at different ReD, a) unsealed and b) scaled.

numbers. A possible explanation for this behavior is that due to the low temperature of

the flowing air, the nozzle becomes cooled. This will cause moisture in the surrounding

air to condense on the nozzle which can become entrained in the jet. This entrained

moisture will increase the amount of heat removed from the surface of the heater and

thus elevate the measured Nusselt number. To combat this issue the nozzle is insulated

to the best extent possible. With the added insulation, it is believed that the moisture

condensation effects are minimal but, non-zero.

0 0.5 1 1.5 2


2.5 3 3.5 4

0 05 1 15

25 3 35 4

0 0.5 1 1.5 2

2.5 3 3.5 4

Figure 3-8. Single-phase NUD at various nozzle height to diameter ratios, a) ReD = 4.57

x 105, b) ReD = 7.55 x 105, and c) ReD = 1.05 x 106.

3.3.3 Two-Phase Results

The two-phase jet experiments are performed in the same manner as the

single-phase jet with the exception of water being added to the air-stream. In order

to quantify the effect of the water on the heat transfer properties, the mass fraction of

water in the jet is calculated as

m + mar


H/D = 2 0
A H/D 4
SH/D = 6.0
+ H/D = b.
ReD = 4.37 x103

? *

H/D 2.0
A H/D =4.0
0 H/D = 6.0
+ H/D 8.0
* HeD = 7.55 x10'

6 *

A 1*

H/D 2.0
A H/D = 4.
0 H/D = .0
+ H/D = 3.0
RePD 1.05 x10"


P *
? *

q" = 478 kW/m2
0 q" = 440 kW/m2
0 q" = 397kW/m2
A q" -355 kW/m2
0 q" = 317 kW/m"
800 Z/D 2.0
w 0.0236
eD 4.50 x 105



0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 3-9. Measured two-phase NuD spatial variation at different heat fluxes without ice

In general, the two-phase heat transfer coefficient is found to be independent of

heat flux. However, as previously mentioned, when the heat flux at the heater surface

is too low, ice formation affects the heat transfer measurements. In order to combat

ice formation a minimum heat flux of 315 kW/m2 is used. Nevertheless, there are a

few cases where icing is observed in heat fluxes up 350 kW/m2. To identify and help

mitigate these effects, the mean and standard deviation of the heat transfer coefficient

as a function of space is taken. When the standard deviation of the experimental values

exceeded 20%, then heat fluxes of 470, 430, and 390 kW/m2 are used in the averaging

calculations. These heat fluxes are selected because the higher heat fluxes will result

in higher surface temperatures and inhibit ice formation. Also the higher temperatures

will result in a larger AT and less uncertainty in the computed heat transfer coefficient.

Approximately 20% of the measurements taken require these corrective measures,

and in all cases the resulting standard deviation is less than 20% of the mean. See

Figure 3-9 for an example of an experiment where the heat transfer coefficient is clearly

independent of heat flux and Figure 3-10 where a reduction in the heat flux used was


q' = 474 kW/m2
Sq" = 439 kW/m
A 0 q 395 kW/m2
2000 A q- 360 kW/m2
o q = 313 kW/m2
Z/D 8.0
1500 w 0.02
1500 ReD 1.02 x 10"



0, *
0 0.5 1 1.5 2 2.5 3 3.5 4


Sq = 474 kW/m'
1800 0 q" =439 kW/m
0 q = 395 kW/m
1600 Z/D = 8.0
w = 0.0255
1400 ReD =1.02 x 10"



200 *

0 0.5 1 15 2 25 3 35 4


Figure 3-10. Measured two-phase NuD spatial variation at different heat fluxes, a) ice

effects present and b) after removal of lowest heat fluxes.

Figure 3-11 shows the radial variation of measured thermocouple temperature for

various heat fluxes for a two-phase experiment. Note that the zero heat flux condition

represents the adiabatic wall temperature. Figure 3-12 shows the radial variation of

Nusselt number for the two-phase jet at different water mass fractions and constant

Reynolds number and nozzle height, Figure 3-13 shows the variation with nozzle height

with a constant Reynolds number with a nominally constant mass fraction of liquid. Note

that it is not possible in the current study to vary Reynolds number and the liquid mass

45 q"= 477 kW/m2
0 q"= 433 kW/m2
40- O q"= 391 kW/m2 O
+ q"= 354 kW/m2 O
35- q"= 315 kW/m2
q"= 0 kW/m2 O
30- w = 0.0255 0 +
ReD = 1.01 x 106 O 0
S25 O
o 0 0
20- 0
15 +



0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 3-11. Spatial heater temperature variation at different applied heat fluxes,
two-phase jet results.

fraction independently of each other; thus it is not possible to show how the Nusselt

number scales with Reynolds number.

Nusselt number generally increases with increasing Reynolds number and

increasing water mass fraction near the interior of the jet. For r/D > 1.5 there does

not appear to be a noticeable dependence of Nusselt number on the nozzle height.

There is some variation of Nusselt number with nozzle height in the jet interior, but a

definite trend is not apparent. For reference purposes, a two-phase Nusselt number

on the order of 2,000 corresponds to a heat transfer coefficient on the order of 200,000

W/m2 -K. More experimental results than those presented in this Chapter are presented

in Appendix A.

,',',',: + ',1,1.i
S0 .. .::::: 1400 ..
1000 Z/D = 2 0 Z/D =60
ReD =1. 42 x 105 1200 R,,eD = 4.45 x 105

800 .

600 o 800 +

A 600 0
+ 400
200* 200

JC 20
0 0.5 1 1.5 2 2.5 3 3.5 4 0 05 1 15 2 25 3 3.5 4
r/D r/D

a b

w= .01fi
A = 00189
Sw = 0.0255
+ + w -0.0301
2000 Z/D = 6.0
ReD = 7.24 x 105

1500 +



0 0.5 1 1.5 2 2.5 3 3.5 4


Figure 3-12. Two-phase NUD at various liquid mass fractions. a) Z/D = 2.0, ReD = 4.42 x

105, b) Z/D = 6.0, ReD = 4.45 x 105, and c) Z/D = 6.0, ReD = 7.24 x 105.

Heat transfer coefficients exceeding 400,000 W/m2-K are observed in Figure 3-13c,

which are on the same order as the highest liquid jet heat transfer coefficients, see [2],

to the author's knowledge. While there is more experimental uncertainty at these high

heat transfer rates (8 to 18%), the efficacy of the two-phase jet for high heat transfer

applications is clearly demonstrated.

It is briefly noted that the orifice for the 0.37 mm orifice had a defect and hence is

not perfectly circular; the liquid flowrate delivered was less than that for the 0.33 mm

orifice. The Nusselt number results for the 0.37 mm orifice are noticeably smaller that

1600-6 Z/D=20
A Z/D = 4,0
1400 0 Z/D = 6.0
+ Z/D = 8.0
w = 0.0375
1200 ReD 442x105

800 0

400- +

0 05 1 1 2 2 3

0 0.5 1 1.5 2 2.5 3










3.5 4










0 05 1 1.5

SZ/D = 2.0
-o A Z/D = 4.0
SZ/D = 6.0
+ Z/D = 8.0
w 0.0248
RoD = 1.01x10'6


-t 4


0 0.5 1 1.5 2 2.5 3 3.5 4


Figure 3-13. Two-phase NUD number at

0.0375, ReD = 4.42 x 105,

0.0248, ReD = 1.01 x 106.

Z/D =2.0
A Z/D = 4.0
0 Z/D = 6.0
0 + Z/D = 8.0
w= 0.0273
ReD 7.23x105




2 25 3


3.5 4

various nozzle height to diameter ratios. a) w =

b) w = 0.0273, ReD = 7.23 x 105, and c) w =

that of the 0.33 mm orifice and do not follow the expected trend. This is believed to be

due to the eccentricity of the orifice causing different behavior in the mixing chamber

and effecting the resulting droplet size/distribution at the nozzle exit. While the 0.51

mm orifice does have some eccentricity, it is not as severe as that found in the 0.37

mm orifice, and it does not seem to have a noticeable effect on the Nusselt number

measurements. Pictures of each orifice are shown in Appendix C.


3.3.4 Evaporation Effects

To help quantify the effect of evaporation on the heat transfer coefficient the

saturated humidity ratio at the impingement site (r = 0) and at the edge of the measurement

location (r = 50.8 mm) is carried out. The saturated humidity ratio is calculated from

Wsat = 0.622 PV,sat (3-9)
P Pv,sat
The vapor saturation pressure, Pv,sat is calculated from [57]

Pv,a = exp 64796 [-7.85951783v + 1.84408259v15 11.78664977v3

22.6807411v35 15.9618719v4 1.80122502v7. (3-10)
/ (3-10)
v= 1-
T has units of Kelvin, P has units of Pascals, and v is non-dimensional. At the jet

impingement zone the temperature is on the order of 10 C and the pressure is

approximately the stagnation pressure. Results for the saturated humidity ratio for

the three separate stagnation pressures used are all on the order of 10-4; thus any

effects due to evaporation near the centerline are considered negligible.

The pressure at the edge of the heater as well as the temperature at the surface

of the liquid film are unknown and a similar analysis cannot be performed. However,

no visual observation of phase change at the highest temperatures seen during

experiments is seen. It is observed in Figures 3-12 and 3-13, that Nusselt number

remains essentially constant near the edge of the heater. Evaporation would further

enhance heat transfer resulting in an increase in Nusselt number in this region thus it is

believed that evaporation is likely negligible in this area as well.

3.4 Comparison between Single and Two-Phase Jets

To gain an appreciation of the two-phase jet heat transfer enhancement, the

measured heat transfer coefficient is compared to the that for the single-phase case.

The heat transfer enhancement factor is defined here as

S= mx (3-11)

A F w 0 0211

9- Z/D = 2.0
ReD = 4 42 x 105


0 0.5 1 1.5 2


25 3 3.5 4

0 05 1 15

25 3 35 4



0.5 1 1.5 2

Z/D = 2.0
ReD = 1.01 x 106

2.5 3 3.5 4


Figure 3-14. Heat transfer enhancement ratio at various liquid mass fractions. a) Z/D =

2.0, ReD = 4.42 x 105, b) Z/D = 4.0, ReD = 7.35 x 105, and c) Z/D = 2.0,

ReD= 1.01 x 106.

Sw 0152
A 0 w 0.0189
16 a w0.0277
+ w 0.0277
SZ/D 4.0
14 IteD 7.35 x 10 -

12 +

100 0
10 O



Figure 3-14 show the heat transfer enhancement with the variation in water mass

fraction and Reynolds number. It is observed that the enhancement increases with

increasing mass fraction; the increase is diminished with increasing Reynolds number.

It is observed that in Figure 3-14 that for r/D < 0.5 there is a marked increase in the

measured heat transfer enhancement. At higher Reynolds numbers the maximum

enhancement occurs near the edge of the jet (r/D = 0.5).

It is observed in Figure 3-15 that the variation of the heat transfer enhancement with

nozzle height is similar to that for water mass fraction, it shows an increasing trend at

lower Reynolds numbers but the effect is damped at higher Reynolds numbers.

3.5 Discussion

One of the features apparent in all experiments is that at radial distances of 1.5 to 2

nozzle diameters the Nusselt number and heat transfer enhancement ratios approach a

near constant value which is indicative of film flow heat transfer. Various studies [34, 40]

have noted that there is a large adverse pressure gradient in this region which likely

causes boundary layer separation [35]. During the course of the present experiments,

two "rings" indicative of a separation region are observed on the heater surface. One of

which corresponds to the edge of the nozzle where there is a shock wave present, and

the other is located approximately 1.5 nozzle diameters from the jet centerline. Inside of

these region the heat transfer coefficient is affected by nozzle height and the water mass

fraction indicating that jet impingement is the dominating heat transfer mechanism for

r/D < 1.5.

All of the experiments reported are carried out at relatively low surface temperatures;

the highest temperature is on the order of 70 OC. Phase change due to mass transfer of

the liquid into the impinging air-stream is considered negligible for reasons discussed

in Section 3.3.4. Additionally the work of Benardin and Mudawar [58] explore the

Leidenfrost model for impinging drops and sprays. Their model predicts the pressure in

droplets using one-dimensional elastic impact theory [59], and a correction factor due to

1' '1

SZ/D =2 0 20A Z/D = 2.0
O Z/D =4.0 0 Z/D = 4.0
14 A A Z/D =6.0 18 A Z/D =6.0
+ Z/ 8.0 A + Z/D = 8.0
12 w = 0.0205 16-+ w 0.0290
ReD 4.54 x 10 UeD 7 24 x 10"
120 O
8O 10

6 8
0 A + 6
4 A I A A A6
0 0 0 4

0 0.5 1 1.5 2 2.5 3 3.5 4 0 05 1 15 2 25 3 3.5 4
r/D r/D

a b

F Z/D = 2.0
O ZiD = 4.0
+ A ZiD = 0.0
25 + /D = 8.0
w 0.0233
A RcD = 1.02 x 10"


15 A

10- o

0 0.5 1 1.5 2 2.5 3 3.5 4


Figure 3-15. Heat transfer enhancement ratio at various nozzle height to diameter ratios.

a) w = 0.0205, ReD = 4.54 x 105, b) w = 0.0290, ReD = 7.24 x 105, and c)

w = 0.0233, ReD = 1.02 x 106.

Engel [60, 61] gives good results. The pressure rise at the impingement surface can be

modeled as

AP = 0.20piuoUsnd (3-12)

where Uo is the droplet velocity and Usnd is the speed of sound in the liquid. Using

this equation it is seen that for any droplet velocity above approximately 7% of the

speed of the air in the impinging jet (approximately Mach 3) will yield surface pressures


above that of the critical pressure for water. Thus even if phase change occurs at the

impingement point, the latent heat of vaporization is zero and no enhancement of heat

transfer will occur. Because of the complex shock structures occurring when the jet

impinges onto the surface, making similar arguments for regions far removed from the

impingement zone are not reliable and thus are not attempted. However, it is noted that

most of the liquid droplets impinging onto the surface will occur near the centerline; thus

the pressure far removed from the jet centerline will be lower and evaporation may still

be possible at elevated surface temperatures and heat fluxes.

One of the current limitations of the current results is they lack information on the

liquid drop size distribution. Such measurements are not available at the current time

and future work is planned to address this deficiency.

The current heat transfer measurements are compared to the single-phase liquid jet

heat transfer measurements of Oh et al. [2], and the measured heat transfer coefficients

are on the same order of magnitude. The liquid flow rate in the current experiments is

very small when compared to those experiments, up to 0.7 g/s (542.5 g/m2 s, referenced

to heated area) compared with 4.3 kg/s (2.55 x 106 g/m2 s, referenced to heated

area), a feature which has significant industrial advantages. In the study performed by

Oh et al., liquid to vapor phase change is not observed, and those experiments were

performed at much higher heat fluxes (up to 30 times the heat fluxes reported in the

current study). Future investigations will explore higher heat flux regimes.

3.6 Summary

In this Chapter heat transfer enhancement measurements using two-phase

overexpanded supersonic impinging jets were presented for a wide range of Reynolds

numbers. These jets two-phase jets are generated by the addition of water droplets

upstream of a converging-diverging nozzle. Heat transfer measurements using a

single-phase jet is used for comparison. It is observed that the addition of water droplets

into the air flow significantly enhances the heat transfer rate. Enhancement is significant

near the jet centerline the enhancement factor exceeds 10 in most cases. The mass

fraction of water added to the jet is observed to by an important parameter for heat

transfer, generally increasing Nusselt follows increasing water mass fraction. However,

its influence diminishes at higher Reynolds numbers. Nozzle height appears to have a

small impact on the observed heat transfer rates.


As was seen in Chapter 3 the use of steady state measurement techniques yields

low surface temperatures which are not suitable for evaporating the liquid film and the

low AT between the wall and liquid film which creates uncertainty in evaluating the heat

transfer coefficient. In order to alleviate some of these effects a transient approach

involving an inverse heat transfer quenching problem is developed.

4.1 Inverse Problems

There are two basic paradigms in heat transfer. The most well known paradigm

is the solution of the temperature field within a medium subject to constraints such

as a given thermal conductivity, thermal diffusivity, and known boundary conditions.

If all of the constraints are known, then the resulting temperature field within the

medium of interest can be solved; in many cases an analytical solution can be

determined. However, if these conditions are not known then the problem is not unique,

is under-specified, and no solution can be determined.

The second paradigm in heat transfer, named inverse heat transfer, is when

the temperature at specific points inside of a medium are known and a constraint

needs to be determined e.g. contact resistance between two surfaces or a boundary

condition. Because of unavoidable measurement errors in the temperature field this

problem is ill-posed and can be difficult to solve. The difficulties of this problem can be

circumvented in very special circumstances, for example if one desires to determine

the heat flux applied at a boundary of a one-dimensional bar at steady state one can

ensemble average temperature measurements at a few locations along the length of the

bar and determine the temperature gradient via linear regression. With a known thermal

conductivity, Fourier's law can be used to determine the applied heat flux and the results

can be quite accurate. Unfortunately these simple problems do not come about in

practice often. For example if the heat flux varies in time, then the above method would

not be applicable and a different method would be needed.

Inverse Problems, in general, fall into one of two categories: parameter estimation,

in which one or more desired parameters are determined using experimental data

(e.g. thermal conductivity of a solid and a applied heat flux) and function estimation, in

which a desired function is to be estimated using experimental data (e.g. a boundary

condition which varies in space and time). It should be noted that many function

estimation problems can be formulated in terms of a parameter estimation problem if the

functional form of the of desired function is known, for instance if thermal conductivity

is a quadratic function of temperature the problem can be reduced to determining the

coefficients of the governing equation. This approach can yield good results, see Flach

and Oziik [62] for example, however, if the form of the equation is not known a priori

then this approach may not be useful.

Inverse Problems and Inverse Heat Transfer (IHT) problems have been studied

extensively in the literature and have been in use since at least the 1950's. Tikhonov,

[63-65] among others, was one of the first to tackle the challenge of inverse problems

and take into account measurement errors. His technique titled Tikhonov's regularization

minimized the least square error by adding a regularization term that penalizes

unwanted oscillations in the estimated function. Tikhonov's method can be related

to damped least squares methods, most notably the method due to Levenberg [66]

and Marquardt [67], known as the Levenberg-Marquardt method. These methods are

only suitable for parameter estimation. Stoltz [68] used a function estimation technique

based on Duhamel's principle and two simultaneous thermocouple measurements to

determine the surface heat flux in a one-dimensional problem. This process is known

as exact matching and does not take into account any measurement errors. Beck

[69-71] used a method similar to that by Stotlz; however, temperatures at future times

are used to provide regularization and reduce instabilities in the method. This method

can be used for parameter or function estimation but, can become unstable for small

time steps and thus highly transient phenomena cannot be accurately reproduced.

The Monte Carlo method can be used to estimate a parameter or function as was

demonstrated by Haji-Sheikh and Buckingham [72]; a good review of the technique can

be found in [73]. A method that is suitable for small time steps and performs parameter

or function estimation is Alifanvo's Iterative Regularization Method [74]. This method is

also known as parameter/function estimation with the adjoint problem and conjugate

gradient method, and is the method used for the present study. This method will be

able to determine a time and space varying heat transfer coefficient produced by a

multiphase supersonic impinging jet, as well as any temperature dependence due to any

evaporation of the liquid film.

4.2 Introduction to Inverse Problem Solution Using the Conjugate Gradient
Method with Adjoint Problem

Dealing with inverse problems, which by their nature are ill-posed, usually involves

some type of regularization technique or an optimization technique which inherently

regularizes the solution. The technique used in the current study is an optimization

technique known as function estimation using the conjugate gradient method with

adjoint problem. As the name implies this method uses the conjugate gradient method

to minimize the error in the least squares sense between the estimated output of an

equation/system of equations and the measured output which has been corrupted with

noise. The method will be described below in its general form to familiarize the reader.

Many references exist for functional estimation with the adjoint problem and conjugate

gradient method including Oziik [75], Oziik and Orlande [76], Alifanov [74], and the

Chapter by Jarny [77]. Much of the following analysis follows that of Jarny as the author

found that particular reference to be mathematically rigorous, thorough, general in

nature, and easy to follow. Specific implementations of this method will be discussed

where needed.

4.2.1 The Direct Problem

The direct problem is the model equation(s) for the system of interest. It can be

an algebraic, integral, ordinary differential, or partial differential equation or system of

equations or some combination therein.

y(x, t) = f(x, t, ) (4-1)

where y is the output of the system, is taken to be a parameters) or function to be

estimated and x and t are the independent variables. Note that although both space

and time are independent variables in this example it is not necessary for the output to

depend on both of them.

4.2.2 The Measurement Equation

The measurement equation exists due to the discrete nature of a sampling process

and due to changes brought about in data due to sensor dynamics. Although in modern

data acquisition systems it is possible to measure quantities at a near continuous rate

taking measurements still is an inherently discrete process. Bendat and Piersol [78]

have written a good reference on data measurement and analysis which includes

sensor/system dynamics.

Sensor dynamics can greatly effect the measurements of a system and their effects

can be quite significant. This process can be simplified if the sensor dynamics can be

approximated by a linear time invariant (LTI) system, which most sensors fall under. In

an LTI system the output of a sensor is the result of a convolution of its input with the

sensor's impulse response function. The impulse response function is the response of

the sensor, initially at rest (or zero), to an impulse input. The measurement equation can

be mathematically expressed as

Ym= h(t- T)y(T)dT (4-2)
( -

where Y, is the measured output, h is the impulse response function, and y is the true

output of the system. If the sensor is perfect and the goal is to simply denote its discrete

nature the impulse response function would simply be a delta function. For ease of

viewing the measurement equation can also be discussed in an operator form such that

Equation (4-2) is equal to

Ym = Cy (4-3)

4.2.3 The Indirect Problem

The indirect problem is actually the statement of the least squares criteria. When

solving an inverse problem with the current method the parameter or function sought is

the one which minimizes the least squares criteria. Simply stated the indirect problem is

M t
5(o) = m.,i- Ciyi(t,)]2 dt (4-4)
i=1 0
where S is the integrated squares (note that in the case of discrete data this would

be the sum of squares), i is the sensor number, and M is the total number of sensors.

The spatial dependence of y is left out of Equation (4-4) because it is assumed that

the sensors are placed at varying distances in space, thus the measurement operator,

C, would only operate on measurements at a location, xi. It can be useful to think of

the least squares criteria in the form of a norm operator, Ilull or sometimes(u, v), for

instance Equation (4-4) is equal to

S(O = II Ym Cy(t, 0)11 (4-5)

4.2.4 The Adjoint Problem

Formulating the adjoint problem correctly is a crucial step in the solution process.

Essentially this is where the optimization portion of the problem comes into play. To do

this the indirect problem is considered the modeling equation and the direct problem is

considered as a constraint such that the following equation holds,

R(y,) = y- f (x, t, )

These are then joined together through the use of a Lagrange multiplier.

L(y, A) = II|| Y Cy(t, ) I| (A, R(y, )) (4-6)

where L is the Lagrangian variable and A is the adjoint variable (also known as the

Lagrange multiplier) which, in general, can be a function of space and time.

When the correct parameter/function is inserted into Equation (4-6) the resulting

Lagrangian is zero for perfect measurements. Real world measurements will be

corrupted with noise and the resulting Lagrangian will be the minimum least squares

To determine the proper the Lagrangian must be minimized. If the adjoint variable

is treated as fixed the the differential of the Lagrangian is

dL = (VyS, Ay) (A, VyR(y, )Ay) (A, V-R(y, )A) (4-7)

or expressed in a more convenient form

dL = (VyS A [VyR(y, )], Ay) (A [V-R(y, A)], A ) (4-8)

Because the choice of the adjoint variable is not constrained it is chosen to be the

solution of

VyS A [VyR(y,)]= 0 (4-9)

Equation (4-9) is known as the adjoint equation. Note that this is implicit in A,

through mathematical operation (usually involving integration by parts) it can be

expressed as an explicit function of the adjoint variable.

4.2.5 Gradient Equation

Note that the adjoint equation renders the first term in Equation (4-8) to be zero.

At the solution point where is equal to the true value, the Lagrangian is equal to the

minimum of the least squares criteria

dL = dS = (VS((), A), (4-10)

and comparing the remainder of Equation (4-8) to Equation (4-10) the gradient

equation results

VS= -A [V R(y, )]. (4-11)

The gradient equation is used in the conjugate gradient minimization algorithm to

determine a step size and descent direction in order to minimize the least squares


4.2.6 Sensitivity Equation

As mentioned one of the parameters needed to find the minimum of the indirect

problem is the step size. This parameter can take a few different forms depending

on whether the inverse problem is linear or non-linear. In the interest of presenting a

general method, the form for non-linear problems are presented.

The step size to be determined is a perturbation in the parameter/function _, which

is to be estimated. To derive this quantity we simply perturb the direct problem

y +Ay= f(x, t, + A); (4-12)

generally the right hand side of Equation (4-12) is linearized such that


When Equation (4-1) is subtracted from the above equation the sensitivity equation is

the result

Ay = f(x, t, t, A) (4-14)

Note that the second term in Equation (4-13) and the right hand side of Equation (4-14)

contain both and Ai. Inverse problems in which the sensitivity equation contains both

parameters/functions and Ai are non-linear. Not all inverse problems are non-linear in

nature, and this form is used here for the sake of generality. The sensitivity equation

simply states that a perturbation in the parameter to be estimated will result in a

perturbation of the computed output.

4.2.7 The Conjugate Gradient Method

The conjugate gradient method is an optimization problem for solving linear or

non-linear equations. There are several references which detail the mathematics behind

this tool, for instance the books by Rao [79] and Fletcher [80], among others. As such

readers interested in a rigorous derivation of the method are encouraged to consult

these references.

The essential steps of the method are that a guess for is chosen, the above

equations are solved and a search direction, d< which is C-conjugate to the previous

direction is calculated using a conjugation coefficient, 7. The search direction is then

multiplied by the step size, 3 and is added to the previous guess for _. This iterative

process continues until the error between the measured output and calculated output

reaches a predetermined tolerance.

There are several different forms of the conjugation coefficient, 7 in the literature

such as the Hestenes-Stiefel [81], Polak-Ribiere [82], and Fletcher-Reeves [83], among

others. All of the mentioned forms are equivalent for linear equations; however it is

y + ay = f (x, t, +) f (x,t, t, a)O.

discussed in the literature [84, 85] that the Polak-Ribiere form of the equation has better

convergence properties for non-linear equations and as such will be used in the present

study unless otherwise noted. The Polak-Ribiere form of the conjugation coefficient is

Z(VSk, VSk VSk-1)
7k = i=- for k = 1,2,3...

Yk = O for k = 0

The search direction is then calculated by the following

A(k = VSk + kAk-1. (4-16)

The step size is defined by the following

M tf
/ = arg min [S( /3A)] = arg min [Yi Cyi(t, /A)]2 dt. (4-17)
i 10

The output of the direct problem is expanded in a Taylor series as

y(t, pAZ) y(t, ) POY(t) A y(t, ) 3Ay(t, A), (4-18)

and substituting the above into Equation (4-17) yields

M tf
= arg min [Y,i- C,yi(t, ) +/CAy(t, A)]2 dt. (4-19)
i=1 0
Performing the differentiation with respect to /, setting the result equal to zero and

solving for 6 yields the final form of the equation for the step size

M tf
E [Cy,(x, t, k) m,i(t)] CAy(t, Ak)dt
Mk = i-10 (4-20)
M tr
E J [Cayi(t, ak)]2 dt
i=1 0
'i 1
It is noted that the above equation can be simplified for linear problems when the

least squares criteria (the indirect problem) is cast in quadratic form. However, for the

sake of generality, the above equation will be used throughout the current Chapter.

4.3 Factors Influencing Inverse Heat Transfer Problems

There are several factors which can influence the solution of an inverse problem.

Some of these factors are discussed below.

4.3.1 Boundary Condition Formulation Effects

To perform a function estimation inverse problem to determine a spatially and

temporally varying heat transfer coefficient, the choice of the boundary condition

formulation is very important. The boundary condition can be formulated as a Dirichlet

(specified temperature) boundary condition where the surface temperature is determined

and the resulting heat flux is calculated in order to determine the heat transfer

coefficient, as a Neumann (specified heat flux) boundary condition where the heat

flux is determined and the resulting surface temperature is calculated, or as a Robin

(convection) type boundary condition where the heat transfer coefficient is directly

determined. At first glance, the Robin type boundary condition seems to be the best

choice as the underlying physics taking place are convective in nature. Upon further

analysis this is actually the worst choice. In order to demonstrate this an example

using a one-dimensional heat transfer problem with a time varying heat transfer

coefficient will be used because of the ease of calculation. The same concepts apply

to a two-dimensional problem with a spatially and temporally varying heat transfer


The following analysis can be found in [86] but is reproduced here for clarity and to

correct some errors contained therein. Suppose a time varying boundary condition is

30 X

Specified: : :::
T,q",orhTr.: : :

Figure 4-1. 1-dimensional solid for the sensitivity problem.

applied to the x = 0 surface of a one dimensional solid with an insulated boundary at x =

L, see Figure 4-1. The solution of this problem can be determined by using Duhamel's

principle for each type of boundary condition previously discussed. Essentially the time

varying boundary is convolved with the impulse response function of the slab. The

impulse response function is determined by solving the heat equation for the solid with a

boundary condition of unity (a step response function) and then taking the derivative of

that function with respect to time. For instance the solution for a time varying heat flux in

dimensionless form is

.t* 8 (x*, t* -)
0 (x*, t*) = 0 q* (7) (t* d

where Oq in dimensionless form is: (4-21)

1 x X* 2 1 2
,q (x*, t*) = t* + x* 1 2 exp (-n72t*) cos (nrx*)
0 \ ^- / / n- I

In order to analyze which type of boundary condition should be used in the inverse

problem formulation, a sensitivity analysis should be performed. The sensitivity

analysis is accomplished through the use of relative step sensitivity coefficients.

First the governing equations and boundary conditions are non-dimensionalized

and their solution obtained. The derivative of this solution is taken with respect to

the non-dimensional input parameter for the boundary condition (non-dimensional

temperature, heat flux, or Biot number). The result of these operations is the step

sensitivity coefficient, although the magnitude of the coefficient for the convection case

varies depending on the magnitude of the input Biot number. To allow direct comparison

of these sensitivity coefficients they are multiplied by their boundary condition inputs

transforming them to relative step sensitivity coefficients, denoted as Xipt. With

non-dimensionalization of the problem, the net effect is only seen in the convection case.

Xq (x*, t*) = q (*, t*) (4-22a)

) 21 12] (4-22b)1 1 2
X (x*, sin n -7x* exp n t* (4-22b)
n=- 1 22-
n-1 n 2 2

XB (x*, t*)= BIB-

=Bi exp (-At*) cos (An(1 x*))-
Cn [2Ant* cos (An(1 x*)) + (1 x*) sin (A(1 x*)]

tAn 1 (4-22c)
8Bi tanAn + AnSec2An
aCn 9An f 4cos(An) 8sin(An) [1+ cos(2An)]
OBi OBi 2An + sin(2An) [2An + sin(2An)]2

Antan(An) = Bi

Note that Equation (4-22c) depends on the input parameter (Bi) and thus the inverse

problem is non-linear in nature and can be difficult to solve. Also note that this equation

is different than that found in [86]; the equation in that reference contains x* as opposed

to 1 x*.

A plot of the sensitivity coefficients at x* = 0.1 is presented in Figure 4-2; it should

be mentioned that the magnitudes of the sensitivity coefficients are plotted and the

coefficients for Bi are actually negative and this is not clarified for the plot in Reference

[86]. A few points of interests should be pointed out. First, note that the coefficients

for Bi are lower than all of the others and that as Bi number increases its sensitivity

coefficient decreases. The coefficients for a temperature and heat flux input are much

larger than those for Bi number with the coefficients for temperature being larger than

those for heat flux until a value of t* z 0.75. Clearly an inverse problem formulated in

terms of an unknown convection coefficient is not a good choice.

4.3.2 Sensor Location Effects

The sensitivity coefficients also depend on position. The sensitivity coefficients

for a heat flux input are plotted in Figures 4-3 and 4-4. It is easily seen that the closer

1.8 Xe
-- Xq
1.6- X
--XBi (Bi = 1)
---- XBi (Bi = 10)
1.2- XBi (Bi = 100)

` ------- -


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4-2. Relative step sensivity coefficients at x* = 0.1 as a function of time.

a temperature sensor is placed to the boundary of interest the more sensitive it is

to changes of that boundary condition, as one would expect from simple physical


Comparing Figures 4-3 and 4-4 one can see that the magnitude of the relative

step sensitivity coefficient is larger at the back wall for an unknown surface temperature

formulation than for an unknown heat flux formulation. This characteristic will be

exploited in this study in order to minimize the effects of thermocouple insertion on the

inverse problem.

4.3.3 Thermocouple Insertion Effects

In order to perform temperature measurements, solid thermocouples are commonly

used as they are a robust, inexpensive method to the measurement. As was demonstrated

in Section 4.3.2, the closer to the surface of interest a sensor is placed, better

sensitivities to changes in the boundary condition are achieved. This can be accomplished

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4-3. Relative step sensivity coefficient for a heat flux input.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4-4. Relative step sensivity coefficient for a temperature input.

by drilling holes in the solid and inserting thermocouples inside the solid. Placing holes

in the solid can have adverse effects on the heat transfer dynamics.

t* 0
t* 0.1
--t* = 0.5
t- = 0.8


Several researchers have studied the problem of thermocouples inserted into a

solid and how they distort the thermal field as well as how this affects inverse problems.

Chen and Li [87] studied the problem numerically and found the the error produced by

the thermocouple insertion is proportional to the hole size and that the magnitude of the

error decreases in time. Chen and Danh [88] expounded upon the research in [87] by

performing experiments which confirmed some of the predicted results from numerical

simulations, these studied focused on thermocouples inserted parallel to the direction

of heat flow. Beck [89] used Duhamel's theorem to determine a correction kernel for

thermocouples inserted normal to a low thermal conductivity surface to compensate

for the insertion effects. Woodbury and Gupta [90] used numerical methods to study

thermocouple insertion and the effects on inverse heat transfer problems. Woodbury

and Gupta [91] also developed a simple one-dimensional sensor model to numerically

correct the effects for the thermocouple holes; this study also included the fin effect from

the thermocouple wires and is applicable to a thermocouple of any orientation to the

surface. Attia et al. [92] performed a very comprehensive numerical and experimental

study which helped quantify the error that the thermocouple insertion produces on

measurements including wire effects, filler material effects, and non-ideal contact

situations in which the thermocouple is inserted at an angle in the hole. Li and Wells

[93] performed numerical and experimental work studying the different factors affecting
the error due to thermocouple insertion. Interestingly they found that for a thermocouple

inserted perpendicular to the direction of heat flow (i.e. parallel to the surface of interest)

there would be no effect on the temperature measurements but, thermocouples oriented

parallel to the direction of heat flow would have noticeable effects on measurements

or on an inverse heat transfer analysis. Caron, Wells, and Li [94] continued this study

and found a correction model called the equivalent depth. This model implies that the

temperatures measured from an inserted thermocouple can be put into an inverse

analysis as measurements taken from a different position; this new position is the one

which would would experience the temperature transients recorded if there were no

thermocouples inserted. The correction model was only able to accurately reproduce

surface heat flux histories. Franco, Caron, and Wells [95] continued the work and

developed correction models which accurately reproduce surface temperatures.

At first inspection the work of Li and Wells [93] appears to give an ideal result,

that orienting the thermocouple in a specified direction will cause it to have no

noticeable impact. The author attempted to use this information and designed an

inverse experiment with sheathed thermocouples inserted 2 mm below the top surface

of a copper cylinder at a angular spacing every 45. After many trials to determine

the impulse response functions of these embedded thermocouples it was concluded

that thermocouples significantly impacted the heat flow and temperature field. This

experimental finding is contrary to the study by Li and Wells, but it could be due to

differing factors such as different types of thermocouples used, different solid material

(copper for the author's experiment, aluminum for Li and Wells), and the fact there were
many thermocouples inserted versus one for Li and Wells.

One commonality for the above work cited is that the correction models can be quite

complicated and they only assess the effects of a single thermocouple being inserted

into the solid. Because of these difficulties it was decided to use simple welded-bead

type thermocouples and silver solder them to the back of the solid (no insertion). This

configuration eliminated all insertion effects because there are no holes drilled. This will

affect the nature of the inverse problem because measurements performed at the back

surface will cause the sensitivity of the inverse method to decrease. This limitation can

be overcome by formulating the problem as an unknown surface temperature instead of

an unknown surface heat flux or convection coefficient.

4.4 Inverse Heat Transfer Problem Formulation

As was demonstrated in Section 4.3 the best choice for formulating an inverse

problem for determining the heat transfer coefficient is a specified temperature

Ts (rt0

. .

. . ..:* :*: *: :*: *:* *:* :*: :*: *:*
. .

. . ..^:^ :
. ..:: : :: ::: ..* '* *: : '* *: : *
:. ^ :^ *:*:.. .

Figure 4-5. Illustration of the heat transfer physics of the inverse problem formulation.

formulation with thermocouples measuring temperature at the back wall. Once the

temperature at the impingement surface is known the heat flux at the surface and heat

transfer coefficient can be determined. A schematic diagram illustrating the problem

formulation is shown in Figure 4-5. The inverse problem equations from Section 4.2

will now be cast into the proper from for an IHT problem for a cylinder at an initial

temperature that is exposed to a time and space varying surface temperature.

4.4.1 Direct Problem

The Direct problem in non-dimensional form is formulated as,

ai 1 ai0 0 20\
^-- r*+( (4-23a)
at* r* ar* r*) az*2 (4-23a)

0(r*,z* = 0, t*) = s(r*, t*) (4-23b)

S = 0 (4-23c)
9Z* *=1

= 0 (4-23d)
Or* r*=0

=0 0 (4-23e)
Or* r*=1

0 (r*,z*, t* = 0)= 0 (4-23f)

where the following non-dimensionalization is used

r z at
r*- z*= t*-
R' L' R2

To T r To Ts(r*, t*)
TO- To

4.4.2 Measurement Equation

The measurement equation is, in its rigorous form

t* 1i
m,i= JI I hi(t -T)O(r*,z*, )6(r* r)6(z* z) r* dr* dz* d. (4-24)
T=O r*=0z =0

Note that the delta functions in Equation (4-24) merely take into account the discrete

nature of the measurements. Noting this point, the measurement equation can now be

cast as

m,=i = hi(t- T)0j(r)dT (4-25)

The subscript i in the equation denotes the measurement location, of which there

are 7 total measurement points. Also note that each thermocouple can have its own

impulse response function and hence the subscript. This equation can take on an

operator form similar to Equation (4-3).

4.4.3 Indirect Problem

The corresponding indirect problem is

S (Os) = [Zm (t*) C, (r*, z*, t*, s)]2 dt*. (4-26)
Note that the operator from of the measurement equation is used.

4.4.4 Adjoint Problem

The development of the adjoint problem is quite involved mathematically. Because

of the sensor dynamics involved in the measurement equation the form of the adjoint

equation will look different than many of those found in the literature such as [96-100]

for example. To the author's knowledge there are no references in the literature that

explicitly take into account the sensor dynamics, except [86], which merely discusses the

convolution of the delta function to account for the discrete nature of the measurements.

Also Marquardt's analysis[101] which accounts for sensor dynamics, but uses a state

and disturbance observers model, which is different than using the adjoint problem such

as used for the current analysis.

To begin the derivation of the adjoint problem, the necessary substitutions are

carried out for Equation (4-6)

t M
L(O, O0, A) = [Ym,i CiO(r*, z*, t*, 0)]2
0 il (4-27)
-A r 1 a ( *o\ 2 dt.
at* r* r* r -(r*) z*2 dt

Here the norm in the second term of Equation (4-27) is equal to

1 R
(u, v) = I Iu v r* dr* dz* (4-28)
r*-0 z*-0
Next the second term of Equation (4-27) is integrated by parts. This allows for an
explicit function of the adjoint variable to appear. After using the boundary and initial
conditions of the direct problem, Equation (4-23), the result is the following


L(, 0s, A) = /{ [Ym CiO(r*, z*,t*Os)]2

X. O Id \ O 2X \
S0 t* r* Or* O-r*) )z*2 (4-29)
-0A +0OA -0OA ,+0sA

z*o+ t*- t

Next the derivative of Equation (4-29) is taken with respect to 0 and 0s.

dL(O, s,A) = (-2(Ym,i CO(r*, z*, t*, s)), CiAO)
Mi {=1

AO r*
a0 t* r* Or* r*j 9z*2 / (4-30)
_AO ,9 + AO ,_AO OA + 9
Or r*=1 r r*0 Oz z*= Az z =0
A + AAO I dt.
az *=o t=t
The goal is to specify the adjoint equation as the solution to the terms in Equation (4-30)
involving AO. However, the first term involves the measurement operator and AO. To
rectify this the adjoint of the measurement equation is sought such that

(ei(t*), CiA) = (Cjei(t*),AO)


ei(t*) = -2 [Ym(t*) m(t*)].


The operator Cq is known as the adjoint operator of C,. To solve for this operator
examination of the left hand side of Equation (4-31) gives

(ei(t*), CiA0)

tT t*
Sei(t*) Jhi(t* -T)AO(T) dTdt
t*=0 T=0


= A0(r) hi(t* T)e,(t*) dt dr.
T=0 t*=0
Comparing Equations (4-31) and (4-32) it is observed that

C*e,(t*)= h,(t* T)ei(t*)dt*.
Taking into account causality, it is known that



hi(t* T) = 0.

Therefore the equation for the operator C* is

Cje,(t*) = hi(t*- )ei(t*)dt*


Now knowing the adjoint operator of the measurement equation, the adjoint problem
is selected to be the solution of

A OA r* / ,a\
at* r* ar* \ r*




for T > t*,

C [Ym (t*)- -m (t*)]

Full Text




c 2012RichardRaphaelParker 2


ACKNOWLEDGMENTS ThroughoutmytimeingraduateschoolIhavebeenhelpedbymanypeople, althoughIwilltrymybesttoincludeeveryone,inevitablyIwillleavesomepeopleout. ForthatIamsorry. FirstandforemostIwouldliketothankmyparentsaswithoutthemthiswouldnot bepossible.Dad,youtaughtmewhatitmeanttobeaman,andIcan'texpressmy gratitudeenough. Therehavebeenamultitudeofpeopleinmylabthathavehelpedmeimmensely, Dr.PatrickGarrity,Dr.AyyoubMehdizadeh,Dr.JameelKhan,andDr.FadiAlnaimat mycurrentroommatewhohaveallgraduatedandaregoingplacesintheworld,you haveprovidedhelpandmotivation.BradleyBon,FotouhAl-Ragomthelabmother,and Cheng-KangKenGuanwhoarecurrentlypreparingfortheirdissertationdefense,we haveallhelpedmotiveeachother.BenGreek,PrasannaVenvanalingam,KyleAllen, LikeLi,andNimaRahmatian,whoarecurrentlycompletingtheirdegreerequirements, havehelpedmedevelopsomeofmypresentationskills.Ihopeyouhavelearnedas muchfrommeasIhavefromyou. MyfriendsintheInterdisciplinaryMicrosystemsGroupIMGwhohaveprovided manyopportunitiestoescapethepressuresofgraduatestudies.Iwouldspecicallylike tothankmyoldroommate,Dr.DrewWetzel,wholetmebounceideasoffofhim,Dr. MattWilliams,whoalwaysranthegroupfootballcompetitionsandhelpedmereminisce aboutmyrootsinSouthCarolina,andBrandonBertolucciourofcialsocialchair.Thank youallandgoodluck. IwouldalsoliketothankmyprofessorsattheUniversityofSouthCarolina.Dr. JamilKhan,whoseheattransferclasseshelpedshapethefoundationofmyknowledge inthethermalanduidsciences,Dr.DavidRocheleauwhoprovidedmewithcontacts attheUniversityofFlorida,Dr.PhillipVoglewedewhoencouragedmetoattendthe 3


UniversityofFlorida,andnallyDr.AbdelBayoumi,myundergraduateresearchadvisor whosawpotentialinmeandprovidedmanyopportunitiesformetogrow. Last,butnotleast,Iwouldliketothankmycommittee.Dr.Orazemhashelped melearnthingsoutsideofmyeldandmycomfortzoneandforthatIamgrateful.Dr. Hahn,yourrigorousheatconductionclasshelpedbuildthefoundationformuchofmy studies.Dr.Mei,youhavehelpedmerealizemypotentialfornumericalstudiesandhow toknowwhenasolutionisgoodenoughorwhenperfectionisessential.LastlyIwould liketothankmychair,Dr.JamesF.Klausner.Dr.Klausner,youhavekeptmearound, evenasIstruggled,becauseyousawthepotentialinme.You'vehelpkeepsomelevity inthelabwithfootballtalkandgoingtosomeexcellentconcerts.Mostimportantly, you'vehelpedmegrowbothprofessionallyandasaperson.Forthisyouhavemymost sinceregratitude. 4


TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................3 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 NOMENCLATURE.....................................13 ABSTRACT.........................................17 CHAPTER 1INTRODUCTIONANDLITERATUREREVIEW..................19 1.1LiteratureReview................................20 1.1.1Single-PhaseJetImpingement....................20 1.1.2MistandSprayCooling........................24 1.1.3SupersonicJetImpingement.....................25 1.2Summary....................................34 2JETIMPINGEMENTFACILITY...........................36 2.1ImpingingJetFacilitySystems........................36 2.1.1AirStorageSystem...........................36 2.1.2WaterStorageandFlowControlSystem...............36 2.1.3AirPressureControlSystem......................39 2.1.4AirMassFlowMeasuringSystem...................39 2.1.5TemperatureandPressureMeasurements..............42 2.1.6Converging-DivergingNozzle.....................42 2.1.7DataAcquisitionSystem........................43 2.2AnalysisofImpingementFacility.......................44 2.2.1TemperatureandPressureUpstreamoftheNozzle.........44 2.2.2NozzleExitPressureConsiderations.................46 2.2.3ObliqueShockWavesatNozzleExit.................48 2.2.4CompleteShockStructureofanOverexpandedJet.........50 2.3Summary....................................51 3STEADYSTATEEXPERIMENTS..........................53 3.1HeaterConstruction..............................56 3.1.1PhysicalDescription..........................56 3.1.2TheoreticalConcerns.........................57 3.2ExperimentalProcedure............................59 3.2.1Two-PhaseExperiments........................59 3.2.2Single-PhaseExperiments.......................62 5


3.3ExperimentalResults.............................63 3.3.1UncertaintyAnalysis..........................63 3.3.2Single-PhaseResults.........................64 3.3.3Two-PhaseResults...........................67 3.3.4EvaporationEffects...........................73 3.4ComparisonbetweenSingleandTwo-PhaseJets..............74 3.5Discussion...................................75 3.6Summary....................................77 4DETERMINATIONOFHEATTRANSFERCOEFFICIENTUSINGANINVERSE HEATTRANSFERANALYSIS............................79 4.1InverseProblems................................79 4.2IntroductiontoInverseProblemSolutionUsingtheConjugateGradient MethodwithAdjointProblem.........................81 4.2.1TheDirectProblem...........................82 4.2.2TheMeasurementEquation......................82 4.2.3TheIndirectProblem..........................83 4.2.4TheAdjointProblem..........................83 4.2.5GradientEquation...........................85 4.2.6SensitivityEquation...........................85 4.2.7TheConjugateGradientMethod...................86 4.3FactorsInuencingInverseHeatTransferProblems.............88 4.3.1BoundaryConditionFormulationEffects...............88 4.3.2SensorLocationEffects........................91 4.3.3ThermocoupleInsertionEffects....................92 4.4InverseHeatTransferProblemFormulation.................95 4.4.1DirectProblem.............................96 4.4.2MeasurementEquation........................97 4.4.3IndirectProblem............................98 4.4.4AdjointProblem.............................98 4.4.5GradientEquation...........................101 4.4.6SensitivityProblem...........................101 4.4.7ConjugateGradientMethod......................102 4.4.8StoppingCriteria............................103 4.4.9Algorithm................................104 4.5NumericalMethodandLimitations......................105 4.5.1AlternatingDirectionImplicitMethod.................105 4.5.2GridStretchingintheZ-Direction...................106 4.5.3Timestepsizecomplications.....................108 4.6DeconvolutionforThermocoupleImpulseResponseFunctions......108 4.6.1DirectProblem.............................109 4.6.2IndirectProblem............................110 4.6.3AdjointProblem.............................110 4.6.4GradientEquation...........................110 6


4.6.5SensitivityProblem...........................110 4.6.6ConjugateGradientMethod......................110 4.6.7StoppingCriteria............................111 4.6.8Algorithm................................111 4.6.9TestCase................................112 4.7Summary....................................114 5SENSORDYNAMICSANDTHEEFFECTIVENESSOFTHEINVERSEHEAT TRANSFERALGORITHM..............................116 5.1ThermocoupleMeasurementDynamics...................116 5.1.1LowBiotNumberThermocoupleModels...............116 5.1.2HighBiotNumberThermocoupleModels..............121 5.1.3DesignofExperiment.........................122 5.1.4ExperimentalResults..........................126 5.1.5ComparisontoEstablishedModels..................127 5.2InverseHeatTransferAlgorithmVerication.................131 5.2.1InverseQuenchingParametricStudySetup.............132 5.2.2ErrorAssessmentMethods......................136 5.2.3ParametricStudyResults.......................137 5.2.4HeatLoss/GainEffects.........................141 5.2.5EffectivenessoftheInverseHeatTransferAlgorithm........142 5.3Summary....................................144 6CONCLUSIONS...................................147 APPENDIX ACOMPLETESTEADYSTATETWO-PHASEHEATTRANSFERJETRESULTS151 BCOMPLETEINVERSEHEATTRANSFERALGORITHMERRORASSESSMENT CONTOURPLOTS..................................166 CIMAGESOFORIFICESUSEDDURINGEXPERIMENTS............172 REFERENCES.......................................175 BIOGRAPHICALSKETCH................................184 7


LISTOFTABLES Table page 2-1WaterMassFlowrateandAverageWaterVelocityforDifferentRegulatorPressures andOriceSizes...................................40 2-2Area,temperature,andpressureratiosatvariouspointsinthejetimpingement facility..........................................46 2-3Nozzleexitpressureforvariousregulatorpressures................47 3-1Reynoldsnumberandcorrespondingheatuxes..................62 5-1CurveFittingConstantsforRabinandRittel'sthermocoupleimpulseresponse model,from[114]...................................121 5-2RMSerrorsforL=10mm,and DAQ =0.2 C...................138 5-3RMSerrorsforL=10mm, DAQ =1 C,andM=8................139 5-4RMSerrorsforL=5mmand DAQ =0.2 C....................140 5-5RMSerrorforL=10mm, DAQ =0.2 C,M=8,andvariousvaluesof for theBiotnumberdistribution..............................140 5-6RMSerrorforL=10mmandM=8withdifferentactualandsimulatedtime constants........................................141 8


LISTOFFIGURES Figure page 1-1Solutionofstagnationpointow..........................21 1-2Anillustrationoftheshockstructureinthewalljetregion.............26 1-3Greasestreakphotograph..............................28 1-4Numericalresultsofaoweldwithaplateshock.................31 1-5Jetcenterlinepressureuctuationswithandwithoutmoisture..........32 1-6Adiabaticandheatedtemperaturevariationwithz/D................34 2-1Illustrationofthejetimpingementfacility.......................37 2-2Crosssectionofthemixingchamber........................38 2-3Threaddetailsofthemixingchambercrosssection................38 2-4Oricecrosssection..................................39 2-5Theoreticalvsmeasured m air .............................41 2-6Crosssectionofnozzle................................43 2-7Simpliedviewoftheairowpathinthefacility...................45 2-8Illustrationofthelimitingcasesforshockwavesinthenozzle...........48 2-9Illustrationofanobliqueshockwaveatthenozzleexit...............49 2-10Variationofowpropertiesdownstreamofanobliqueshockwave........50 2-11Structureofobliqueshockwaves..........................52 3-1Stainlesssteelheaterassembly...........................54 3-2Copperheaterassembly...............................55 3-3Illustrationofheaterassemblyusedforsteadystateexperiments.........57 3-4Iceformationatadiabaticconditions.........................60 3-5Measuredsingle-phaseNu D spatialvariationatdifferentheatuxes.......64 3-6Spatialheatertemperaturevariationatdifferentappliedheatuxes.......65 3-7SpatialvariationofNu D atdifferentRe D ,aunscaledandbscaled.......66 3-8Single-phaseNu D atvariousnozzleheighttodiameterratios...........67 9


3-9Measuredtwo-phaseNu D spatialvariationatdifferentheatuxeswithoutice formation........................................68 3-10Measuredtwo-phaseNu D spatialvariationatdifferentheatuxeswithand withouticeformation.................................69 3-11Spatialheatertemperaturevariation,two-phasejetresults............70 3-12Two-phaseNu D atvariousliquidmassfractions..................71 3-13Two-phaseNu D numberatvariousnozzleheighttodiameterratios.......72 3-14Heattransferenhancementratioatvariousliquidmassfractions.........74 3-15Heattransferenhancementratioatvariousnozzleheighttodiameterratios...76 4-11-dimensionalsolidforthesensitivityproblem...................89 4-2Relativestepsensivitycoefcientsatx =0.1asafunctionoftime........92 4-3Relativestepsensivitycoefcientforaheatuxinput...............93 4-4Relativestepsensivitycoefcientforatemperatureinput.............93 4-5Illustrationoftheheattransferphysicsoftheinverseproblemformulation....96 4-6Comparisonoftruetemperaturevsinverseresults.................104 4-7Effectsofgridstretching.arealdomainandbcomputationaldomain.....107 4-8Timestepresults...................................109 4-9Trueandestimatedimpulseresponsefunction...................113 4-10Convergencehistory..................................113 4-11Outputcomparison..................................114 5-1Illustrationoftherstorderslab...........................118 5-2Illustrationofthesecondorderslab.........................119 5-3Exampleofarstandsecondorderimpulseresponsefunction..........120 5-4ImpulseresponsefunctionsusingthemodelofRabinandRittel,adaptedfrom [114]..........................................122 5-5Diagramofthecopperdiscassembly........................123 5-6Illustrationoftheexperimentalsetup.........................124 5-7Backwalltemperatureswithnon-idealinsulations.................125 10


5-8Resultsofthreeseparateimpulseresponseexperiments.............126 5-9Inversemethoddeconvolutionresults........................127 5-10Firstorderresponsefunctioncomparison......................128 5-11Besttresultsusingarstorderimpulseresponsefunction............129 5-12Comparisonofmodeltodeconvolutionresults...................129 5-13Besttresultsusingthemodelof[114]responsefunction.............130 5-14Comparisonofthe2exponentialmodeltothedeconvolutionalgorithmresults.131 5-15Besttresultsusingthe2exponentialmodel....................132 5-16BiotnumberdistributionshowingtheeffectsofBi max ................134 5-17Biotnumberdistributionshowingtheeffectsof ..................135 5-18TruncationofBiotnumber...............................137 5-19ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints....139 5-20Comparisonoftheeffectsofheatgain.......................143 5-21CenterlineNu D resultsfromtheinverseheattransferalgorithm..........144 5-22SpatialNu D resultsfromtheinverseheattransferalgorithm............145 A-1Two-phaseNu D resultsforvariousZ/D,nominalRe D =4.46 10 5 .......152 A-2Two-phaseNu D resultsforvariousZ/D,nominalRe D 7.27 10 5 ........153 A-3Two-phaseNu D resultsforvariousZ/D,nominalRe D 1.01 10 6 ........154 A-4Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=2.0......155 A-5Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0......156 A-6Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=6.0......157 A-7Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=8.0......158 A-8Two-phase resultsforvariousZ/D,nominalRe D =4.46 10 5 .........159 A-9Two-phase resultsforvariousZ/D,nominalRe D 7.27 10 5 ..........160 A-10Two-phase resultsforvariousZ/D,nominalRe D 1.01 10 6 ..........161 A-11Two-phase resultsforvariousliquidmassfractionsandZ/D=2.0.......162 A-12Two-phase resultsforvariousliquidmassfractionsandZ/D=4.0.......163 11


A-13Two-phase resultsforvariousliquidmassfractionsandZ/D=6.0.......164 A-14Two-phase resultsforvariousliquidmassfractionsandZ/D=8.0.......165 B-1ErrorContoursforL=5mm, DAQ =0.2 C,and8measurementpoints....167 B-2ErrorContoursforL=5mm, DAQ =0.2 C,and16measurementpoints...168 B-3ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints...169 B-4ErrorContoursforL=10mm, DAQ =0.2 C,and16measurementpoints..170 B-5ErrorContoursforL=10mm, DAQ =1 C,and8measurementpoints....171 C-1The0.33mmorice..................................172 C-2The0.37mmorice..................................173 C-3The0.41mmorice..................................173 C-4The0.51mmorice..................................174 12


NOMENCLATURE Variables AArea[m 2 ] BiBiotnumber=hL/k CMeasurementequationoperator DDiameter[m] LLagrangian MMachnumber,Chapter2 MTotalnumberofmeasurements NuNusseltnumber=hL/k PPressure[Pa] RGasconstantfordryair[J/kgK,Chapter2 RResidualofmodelingequation ReReynoldsnumber=4 m / D SLeastsquaresvalue TTemperature[ CorK] VVolume[m 3 ] XDimensionlessrelativesensitivitycoefcient c p Specicheatcapacityatconstantpressure[J/kgK] fGeneralfunction hHeattransfercoefcient[W/m 2 K],Chapter3 hImpulseresponsefunction[s] kThermalconductivity[W/mK] 13


_ m Massowrate[kg/s] q 000 Internalheatgenerationrate[W/m 3 ] rRadialcoordinate[m] ttime[s] uVelocity[m/s] uDummyvariableforpartialdifferentialequation,Chapter4 vDummyvariableforpartialdifferentialequation,Chapter4 wLiquidmassfraction xLengthcoordinate[m] yWidthcoordinate[m] yGeneralizeoutputvariable,Chapter4 zHeightcoordinate[m] Greekletters Thermaldiffusivity[m 2 /s] StepsizeforConjugateGradientMethod Gridstretchingparameter Ratioofspecicheats,Chapter2 ConjugationcoefcientforConjugateGradientMethod,Chapter4 Deectionangleinradians,Chapter2 Thickness[m],Chapter3 Diracdeltafunction,Chapter4 Stoppingcriteriavalue Transformedzcorrdinate 14


ObliqueShockwaveangleinradians,Chapter2 Dimensionlesstemperature Lagrangemultiplier,Chapter4 Effectivetimeconstant,Chapter5 viscosity[Pa-s] Generalparametertobeestimated,Chapter4 Dummyintegrationvariable,Chapter5 Density[kg/m 3 ] Dummyvariableofintegration Timeconstant[s] Heattransferenhancementratio,Chapter3 Dimensionlessstepresponse,Chapter4 Humidityratio Subscripts DSoliddomain TThermocouplemeasuredquantity aAdiabaticquantity airAirquantity expExperimentalmeasurement fFluidquantity lLiquidquantity mMeasurementquantity mixMixturequantity modModiedquantity 15


oStagnationquantity,Chapter2 oInitialquantity,Chapter4 rmsRoot-mean-squarevalue sSurfacequantity satValueatsaturationconditions simSimulatedmeasurement sndQuantityatthespeedofsound vVaporquantity wWallquantity Superscripts *Criticalquantity,Chapter2 *Dimensionlessquantity 16


AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy AMULTI-PHASESUPERSONICJETIMPINGEMENTFACILITYFORTHERMAL MANAGEMENT By RichardRaphaelParker May2012 Chair:JamesF.Klausner Cochair:RenweiMei Major:MechanicalEngineering Thisstudyinvestigatestheheattransfercharacteristicsofamulti-phasesupersonic jetimpingementheattransferfacility.Inthisfacilitywaterdropletsareinjectedupstream ofaconverging-divergingnozzledesignedforMach3.26airow.Thenozzleisoperated inanoverexpandedmode.Uponexitingthenozzle,thehighspeedair/watermixture impingesontoaheatedsurfaceandprovidescooling.Steadystateheattransfer measurementshavebeenperformedwithpeakheattransfercoefcientsexceeding 200,000W/m 2 .Theseheattransfercoefcientsareonthesameorderassomeofthe highestheattransfercoefcientseverrecordedintheliterature.Remarkablytheseheat transfercoefcientsareobtainedusingliquidowratesrangingfrom0.2to0.7g/s,in contrasttotheseveralkg/sowratesseeninstudiesthatproducesimilarlyhighheat transfercoefcients. Duringsteadystateoperationitisnotedthatnoevidenceofphasechangewas experimentallyobserved.Preliminaryinvestigationsindicatethatitmaynotbepossible toobtainevaporativeheattransferinthecurrentfacility.Inordertoinvestigatethis possibilityhighersurfacetemperaturesareneeded.However,designingasteadystate experimenttoachievehightemperatureoperationisrifewithdifcultiesandislikelyto beprohibitivelyexpensive. 17


InordertoovercomethesechallengesatransientinverseheattransferIHT methodhasbeendeveloped.Oneoftheimportantissuesrevealedduringthis investigationisthatsensordynamicswillimpactthemeasurements,thusdiminishing themeasurementreliability.Toalleviatethisissue,amethodofincorporatingsensor dynamicsintotheIHTmethodwasdeveloped.Thistypeofmethodisnotexplicitlyfound intheliteraturetotheauthor'sknowledge.Amethodforaccuratelydeterminingthe impulseresponsefunctionofthethermocouplesusedinthetransientIHTexperiments yieldsgoodexperimentalresults.HeatlossisdiscoveredtobeacriticalfactorintheIHT method,andadifferenceintemperatureof3 Cbetweenthatmeasuredandtheideal caserenderstheIHTresultsunusable. Aparametricstudywasperformedtodeterminetheeffectsof:discheight,impulse responsefunction,magnitudeandshapeoftheheattransfercoefcientdistribution, thenumberoftemperaturesensorsused,andthemagnitudeoftheerrorinthedata acquisitionsystem.Itwasdiscoveredthatthemethodwasinsensitivetonoiselevels foundinlaboratoryconditionsandtheaccuracyincreasesforadecreasingdischeight. Therelativeslownessoftheimpulseresponsefunctionsdidaffecttheaccuracyofthe IHTmethodaslongasthetimeconstantofthefunctionsisaccuratelyknown. 18


CHAPTER1 INTRODUCTIONANDLITERATUREREVIEW Jetimpingementproduceshighheattransfercoefcients,uptoapproximately 10 5 W/m 2 -K.Liquidimpingingjetshavesupportedsomeofthehighestrecorded surfaceheatuxes,rangingfrom100to400MW/m 2 ,[1].Ohetal.[2]andLienhard andHadaeler[3]havestudiedliquidjetsandarraysthatcanproduceheattransfer coefcientsof200kW/m 2 -K.Thesehighheattransferratesareaccompaniedbyhigh liquidowratesofuptoseveralkg/sofwater,andsuchhighwaterconsumptionmaybe undesirableinsomeindustrialsettings.Thecurrentstudyproposestouseasupersonic multiphasejetimpingementfacilitydesignedafteranexperimentbyKlausneretal.[4], whichusestheadditionofliquiddropletstotheimpingingair-streamtoenhancethe heatremovalrateofthesupersonicjet.Theliquidowratewillbeordersofmagnitude lowerthanthatusedbythestudiesmentionedabove,withlessthanoneg/s,whichmay beverydesirableinapplicationswhereminimalwaterconsumptionisaconcern. Supersonictwo-phasejetheattransferisaeldthathasnotbeenpreviously studied.Thecontributionofthecurrentstudywilllargelyconsistofcharacterizingthe heattransfercapabilitiesofsuchasystemincludingtheeffectsofairandliquidmass owratesandnozzlespacing.Additionally,evaporativeheattransfercapabilitiesofthe jetwillbestudied;inthisscenariothelatentheatofvaporizationcouldpotentiallygreatly enhancetheheatremovalcapabilitiesofthefacility.However,itisnotknownwhether ornotliquidevaporationcanbeachievedduetothehighstagnationpressure.Dueto highimpactpressuresnearthejetcenterline,phasechangeisnotlikelyinthisregion; however,theconditionsfarremovedfromtheimpingementpointmayallowphase changetooccur. 19


1.1LiteratureReview Jetimpingementheattransferisaverydiverseeldandconsistsofsingle-phase heattransferandevaporativeheattransfer,spray/mistcooling,andsupersonicjet impingementheattransfer.Abriefreviewofjetimpingementheattransferisprovided. 1.1.1Single-PhaseJetImpingement TheanalyticalstudyofstagnationpointowslargelybeginswithHiemenz[5]who studiedtheoweldofalaminarimpingingjetbymodifyingtheBlausisboundarylayer solution.Homann[6]extendedthisanalysistoaxisymmetricows.Theseowsarepart oftheFalkner-Skanboundarylayerequations,whichtakethegeneralform f 000 + o f 00 f )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(f 0 2 =0 where f = f 0 =0 and f 1 =1. Thevelocitiesuandv,andthesimilarityvariable, ,aredenedas u = axf 0 v = )]TJ/F25 11.9552 Tf 9.298 0 Td [( o p a f = y r a whereaisaproportionalityconstantand isthekinematicvelocityoftheuid.Note thatintheradialcasethevariablexistheradialdistancefromtheorigin.Theparticular valuesof and o are1and1forHiemenzowand1and2forHomannow.The variablef 00 isproportionaltotheshearstress,f 0 isthenon-dimensionalvelocityu/U 1 andfisthestreamfunction.Equation1representsanon-linearordinarydifferential equationODEwhichmustbesolvednumerically.Theshootingtypemethodis generallyusedasthevalueoff 00 isunknownattheorigin.Thevaluesoff 00 attheorigin 20


a b Figure1-1.SolutionofaHiemenzstagnationpointowandbHomannstagnation pointow. obtainednumericallyare1.2325forHiemenzowand1.3120forHomannow,asfound in[7].ThesolutionforHiemenzandHomannowareshowninFigure1-1.Itisevident thatthetheoweldsbehaveverysimilarly;however,thefreestreamvelocityand shearstressarereachedforsmallervaluesofthesimilarityvariableforaxisymmetric stagnationpointow.Afullderivationoftheseandotherstagnationpointowsusing asimilaritytypeapproachasthatabovecanbefoundinthebookbySchlichtingand Gersten[8]. 21


Whiletheaboveanalysisissufcientforcompletelylaminarimpingingjets,jets whichndindustrialapplicationmustdealwiththeboundarylayerapproachingthefree surfaceofthejetfarremovedformthecenterlineaswellasthetransitiontoturbulence. Duetothedifferentowregimesthejetanalysisistypicallybrokenupintoseveral differentregionsandanalyzedthroughtheuseofavonK arm ammomentumintegral analysis;forananalysisofstagnationregionsee[9]. Theanalysisofthetemperatureeldwithintheboundarylayerforthesetypesof owsiscomplicatedduetothebehaviorofthethermalboundarylayerthatdevelops onthesurface.Itisfurthercomplicatedbythenatureoftheuiditselfasowswith largerPrandltPrnumberbehaveverydifferentlythanowswithsmallones.Asthe boundarylayermovesawayfromthecenterline,thehydrodynamicboundarylayer reachesthefreesurfacebeforethethermalboundarylayerforPr < 1whiletheconverse istrueforPr > 1 .Liuetal.haveanalyzedtheowineachoftheseregionsforsingle phasejetswithconstantsurfacetemperatureandheatuxes,mostlythroughtheuse ofthevonK arm an-Pohlhausenintegralsolution[10,11].Theywereabletomodelthe transitiontoturbulenceandthesubsequentturbulentowaswell.Theirsolutionsagree exceptionallywellwithexperimentalresults.Ingeneralthesolutionshavetheformof Theanalysisoftheaboveoweldisnotlimitedtointegralsolutionsortoconstant boundaryconditions.Wangetal.[12,13]studiedtheeffectsofaspatiallyvarying surfacetemperatureandheatuxonthesolutionusingaperturbationmethod.They foundthatthedirectionofincreasingtemperatureaffectstheNusseltnumberofthe ows,notablythatincreasingthewalltemperatureorheatuxwithradialdistance fromtheoriginwilldecreasetheNusseltnumberinthestagnationzone.Conversely itincreasestheNusseltnumberintheboundarylayerregion.Wangetal.additionally studiedtheconjugateheattransferproblemwherethetemperatureeldisdeterminedin theliquidandsolidsimultaneously,[14].Theyfoundthatthicknessoftheheatercanbe acontributingfactorfortheheatremovalcapabilityofthejet. 22


Thereareseveraladditionalphenomenathatimpactthecoolingratesofimpinging jets.Theseincludetheeffectsofthejetnozzlediameter[15],hydraulicjump[16],and thesplatteringofliquidfromtheresultingfreesurface[17].Incaseswithsufcientlyhigh surfacetemperature,phasechangecanbeobservedunderimpingingjetsincludingthe regionsofnucleateboiling,departurefromnucleateboiling,andtransitionboiling[18]. Whilemoststudiesofliquidjetimpingementndnoappreciableeffectonthenozzle heightabovetheheatsurface,Jambunathanetal.[19]notedthatsomestudiesdoshow aneffectmostnotablyathigherReynoldsnumbers.Anempiricalcorrelationbased onheattransferdataavailableintheliteraturewasproposedhowever,itprovidesno physicalinsightoftheoweldandheattransfertakingplace. Liquidjetimpingementcansupportexceptionallyhighsurfaceheatuxes.Liuand Lienhard[1]usedaliquidjetwithvelocitiesexceeding100m/s,liquidsupplypressures ofuptoapproximately9MPa,andowratesofapproximately300g/storemoveheat uxesofatleast100MW/m 2 .Theseexperimentswerenovelinthefactthattheyused aplasmatorchasaheatsource.Surfacetemperaturesweredeterminedbycoating thetopsurfacewithamaterialofknownmeltingtemperatureandcompletingseveral experimentalrunsuntilthesurfacetemperaturecouldbeisolatedtoliewithinarange oftemperatures.Meltingoftheheatedsurfaceoccurredduetotheuseofthetorchand thebackwalltemperaturewasassumedtobeessentiallythemeltingtemperatureof thesolid.Theheatuxwasdeterminedbyusingtheminimumthicknessofthesolid whereithadmeltedandthenassumingalineartemperatureprole.Becauseofthe coarsenatureofthemeasurements,uncertaintyishardtoquantifyandheattransfer coefcientswerenotreported.However,theheatuxesmeasuredarethehighest steadystatevaluesrecordedintheliterature.Tofurtherenhancethestudyofhighheat uxremovalMichels,Hadeler,andLienhard[20]andLienhardandNapolitano[21] designedthinlmheatersusingvacuumplasmasprayingandhighvelocityoxygen fuelspraying.Theseheatersaresuppliedwithdcelectricalpowerofupto3,000Aand 23


24V,producingheatuxesofupto17MW/m 2 .LienhardandHadeler[3]wereable toconstructanarrayofliquidjetswithliquidsupplypressuresofapproximately2MPa andowratesofapproximately4kg/s.Theseimpingingjetarrayswereabletosupport heatuxesof17MW/m 2 withanaverageheattransfercoefcientof200kW/m 2 with uncertaintiesof 20%.SimilarresultswerefoundinastudybyOhetal.[2].These studieshelpillustratethehighheatremovalcapabilitiesofjetimpingementtechnology. Foranextensivereviewofthesubjectofliquidjetimpingementtheauthor recommendsthereviewarticlesofLiendhard[22],WebbandMa[23],andMartin [24].Thesearticlesofferacompletereviewofthesubjectandincludemanyeffectsnot discussedinthisbriefliteraturesurvey. 1.1.2MistandSprayCooling Mist/spraycoolingofaheatedsurfaceislargelydifferentfromjetimpingementdue tothefactthattheliquidimpingingonthesurfaceisintheformofdispersedroplets. Thesedropletsareusuallygeneratedbyforcingliquidthroughverysmallorices withinanozzlewhichatomizestheliquid.Theprimarybenetofusingthistechnique isthatthedispersedropletsgenerallyallowforevaporativeheattransfertodominate. Mist/spraycoolingproducesheattransfercoefcientsontheorderofthosefoundduring poolboiling.However,thecriticalheatuxcanbeseveraltimeshigher[25]. Thedropletsizeisanimportantparameterinmist/spraycooling,Estesand Mudawar[26]correlatedthecriticalheatuxCHFwiththeSautermeandiameter ofthesprays;theyalsofoundthattheapparentdensityofthespraycanbeanimportant factorinmist/spraycoolingasdenserspraysarelesseffective.Mist/spraycooling canalsobeappliedtosurfaceslowerthantheboilingpointoftheliquid.Thereduced evaporationcanleadtoabuildupofaliquidlmontheheatedsurfacewhichcanhavea thicknesslessthanthesizeofthedroplets.GrahamandRamadhyani[27]performedan experimentalstudywhichshowsthatincreasingtheamountofdropletsonthesurface canleadtothickerlmswhichmayincreasethethermalresistanceatthesurface; 24


however,thisthicklmmaybeabletoconvecttheheatawaybetterduetoanincreased velocity.Theywereabletodevelopasimplemodelofthethinlmdynamicsandthe resultingheattransferwhichhadapproximately4%errorforheatuxpredictionswithan air/methanolmixturebut,onlyprovidedqualitativeagreementwhenusedwithair/water data.Itisnotedherethatbecauseoftheevaporationtakingplaceinmist/spraycooling, theheattransfercoefcientdoesnotvaryappreciablywiththeradialdistance,afeature quitedifferentthanthatfoundinjetimpingementheattransfer.Readersdesiringa comprehensivereviewofmistspraycoolingareencouragedtoconsultthereviewarticle ofBolleandMoureau[28]. 1.1.3SupersonicJetImpingement Theoweldofasupersonicjetexhibitsverycomplexphenomena.Thenature oftheowcanchangedramaticallyasthenozzleexittoambientpressureratio changes,sometimesquantiedbythestagnationtoambientpressureratio.When thenozzleexitpressureislowerthantheambientpressureobliqueshockwavesform attheedgeofthenozzleinordertocompresstheow.Theseshockwavesbecome Prandtl-Meyerexpansionfanswhentheymeetatthejetcenterline.Thisprocessleads toaseriesofreectedshockwavesandexpansionfansformingintheoweldatthe exitofthenozzle,includingtheformationofnormalshocksintheowknownasMach disks.Whenthejetexitpressureislargerthantheambientpressure,Prandtl-Meyer expansionfansformattheexitofthenozzleandasimilarseriesofeventstakeplace. Theoweldofthejetchangesdramaticallyintheaxialdirection.Zapryagaevetal. [29]notedthatforoverexpandedjetstheradialpressuredistributionupstreamofthe rstshockcellcontainsseverallocalmaximawithverysharpdiscontinuitiespresent. Thesediscontinuitiesdisappeardownstreamoftherstshockcellandgenerallyanon centerlinemaximumappearsinthepressuredistribution.Thesefeaturesarepresent inunderexpandedjetsaswell,[30].Whilethesefeaturesarecomplextheycanbe 25


Figure1-2.Anillustrationoftheshockstructureinthewalljetregion.[Reprintedwith permissionfromCarling,J.C.andHunt,B.L.,TheNearWallJetofa NormallyImpinging,Uniform,Axisymmetric,SupersonicJet,JournalofFluid Mechanics66159Page174Figure9b,CambridgeUniversity Press] modeledsomewhataccuratelybyamethodofcharacteristicsapproachasnotedby Pack,[31,32]andChu[33],amongothers. Underexpandedimpingingjetshavebeenextensivelystudiedintheliterature astheypertaintothelaunchingofrocketsandspacecraft,whereasoverexpanded impingingjetsarerelativelyuncommoninindustrialsettings.Whenajetimpinges uponaatplate,acomplexshockstructureisformed.Thisshockstructureforms severalcomplexfeaturesincludingatripleshockstructure,wherethreeshockwaves intersectneartheimpingementsurface,abowshock,alsoknownasaplateshockin theliterature,astheowmustcometorestinthestagnationpointonthesurface,and theshockwaveswhichradiatefromthetripleshockpointandslowtheowalongthe platetosubsonicspeeds.Thesefeaturesappearinalltypesofsupersonicimpinging jetsincludingunderexpanded,ideallyexpanded,andoverexpanded.Thebowshock,if curved,canformarecirculatingstagnationregionintheareaofthecenterofthejetto theedgeofthenozzle.Anillustrationofthecomplexshockstructureoftheowatlarger radialdistancesalongtheimpingementplateisshowninFigure1-2. Thestagnationregionisverycomplexduetotheformationoftheabovementioned bowshockandstagnationbubble.DonaldsonandSnedeker[30]studiedunderexpanded jetsfromaconvergingnozzleandperformedmanydifferentmeasurementstohelp 26


characterizesomeoftheimportantfeaturesoftheowincludingimpingementangle andnozzlepressureratio.Theywereabletoobservestagnationbubblesforming,but notedthatthisphenomenondidnotoccurineveryexperiment.Schlierenphotographs weretakenaswellastotalpressuremeasurementsalongthejetcenterline,anditwas observedthatthevelocityandpressurevarygreatlyintheaxialdirection.Thevelocity intheradialwalljetregionwasmeasuredviatheuseofpitot-staticpressuretube measurementsalongtheimpingementsurface,theeffectsofsurfacecurvaturewere alsocharacterized.GummerandHunt[34]alsostudiedtheowofuniformaxisymmetric ideallyexpandedsupersonicjetswithlownozzletoheightspacingandnotedthe presenceofthebowshockandcomplexshockstructureinthewalljetregion.They attemptedtouseapolynomialandintegralrelationmethodtomodelthebowshock heightandthepressuredistributionunderthenozzle.Somesuccesswasseenforhigh Machnumbersbutnotintheregionofthetripleshock.LowMachnumbercalculations containedasmuchas60%error.CarlingandHunt[35]performedatheoreticaland experimentalinvestigationusingthenozzlesofGummerandHunt.Theirstudymostly comprisedoftheregionjustoutsideofthenozzlealongtheimpingementplate.They wereabletonotethepresenceofthestagnationbubbleforsomeoftheirexperiments, butnotall.Thepresenceofthestagnationbubblecanseverelyinuencethepressure distributionontheplateandanannularmaximumispossibleforsomejetspacings. Attemptsweremadetomodeltheshockstructureinthewalljetregionusingthemethod ofcharacteristics.Qualitativefeaturesoftheowwereabletobereproduced.However, thereappearstobesomeerrorintheregionnearthetripleshockregion.Thepressure variationalongtheplatewasmeasuredwhichshowedseveralregionsofunfavorable pressuregradient.CarlingandHuntwereabletoinvestigatetheseregionsbycoating theimpingementplatewithatypeofgrease.Whenthejetisimpingedupontheplate theseunfavorablepressuregradientscausethegreasetoberemovedduetolocal separationoftheboundarylayer.Aphotographofoneoftheseexperimentsisshown 27


Figure1-3.Greasestreaktypephotographfrom[35],thedarkareascontainnogrease andareareasofhighwallshearstress.[Reprintedwithpermissionfrom Carling,J.C.andHunt,B.L.,TheNearWallJetofaNormallyImpinging, Uniform,Axisymmetric,SupersonicJet,JournalofFluidMechanics66 159Plate3Figure6d,CambridgeUniversityPress] inFigure1-3.Thedarkregionsrepresentwherenogreaseispresent;notethevery darkregionneararadialdistanceof2nozzlediametersfromthecenterwhereevidence ofseparationisclearlyevident.Theseparationphenomenonwasnotedbyseveral investigatorsincludingDonaldsonandSnedeker,[30].KalghatgiandHuntprovidea qualitativeanalysisexperimentalstudyofoverexpandedjetswhichconcentratedonthe tripleshockproblemneartheedgeofthebowshock.Theiranalysissuggeststhatat bowshocksareapossibilityandschlierenphotographsofoverexpandedimpinging jetswithMachnumbersrangingfromapproximately1.5to2.8largelyconrmed theirqualitativeanalysis.Theyalsonotethattheformationofaatbowshockisa phenomenonthatishardtopredict.LamontandHuntperformedacomprehensive experimentalstudyonunderexpandedjetsorientednormallyandobliquelytoaatplate whichincludespressuremeasurementsandschlierenphotographs.Thestagnation bubblephenomenonwasnotedaswellassomeunsteadinessinthejet.Velocityand pressureproleswereseentovarygreatlywiththenozzletoplatedistance,anditwas notedthatthelocalshockstructurehasastronginuenceontheoweld. 28


Unsteadinessoftheimpingingjetiscausedbyafeedbackphenomenonwhich hasbeenextensivelystudiedduetoitsimportanceinairvehicletakeoff,includingthe launchingofrocketsandshort/verticaltakeoffandlandingvehicles,suchastheJoint StrikeFighter.ThismechanismwassuccessfullymodeledbyPowell,[36,37].The mechanismiscausedbyacousticphenomenaoccurringattheedgeofthenozzle. Theseacousticwavescausevorticalstructurestobegeneratedintheshearlayer ofthejetandareconvectedtowardstheimpingementpoint.Uponencounteringthe regionneartheplate,thesestructuresinteractwiththeshockwavesneartheplate generatingstrongacousticwaves,whichtravelupstreamtowardsthenozzlewherewere theyinteractwiththenozzleedgegeneratingmoreacousticwaveswhichthenrepeat theprocess[38].Krothapalli[39]wasabletopredictthefrequenciesgeneratedbya supersonicimpingingrectangularjetusingPowell'smodel,thusvalidatingthetheory. Theeffectsoftheunsteadinessontheoweldwillbedetailedbelow. Duetothecomplexshockstructureandunsteadyphenomenainimpingingjets, numericalsimulationsareoftenusedtohelpenhancetheknowledgeinthisarea.Alviet al.[40]modeledtheimpingementofmoderatelyunderexpandedjetsandusedParticle ImageVelocimetryPIVtohelpverifytheirresults.Theirmethodhadreducedtemporal accuracy,butwasabletoreproducemajorowfeaturesincludingthestagnationbubble andwalljetregion,althoughtheregionofthetripleshockpointhadsomedisparity betweenthenumericalandexperimentalresults.Klinkovetal.[41]comparednumerical resultsofthevelocity,pressure,anddensityeldstoexperimentalresultsintheform ofschlierenphotographsandsurfacepressuremeasurements.Theirstudyfocused onoverexpandedjetswithMachnumbersintherangeof2.6to2.8atapproximately ambientstagnationtemperatures.Theyfoundthatthelocationofthebowshockcan changesignicantlywithnozzletoplatespacing,withseveralregionsofanearconstant shockheightfollowedbyanalmostdiscontinuouschangetoanotherheight.Regions ofhighshockheightrepresentaconvexbowshockandregionsoflowshockheight 29


representaatbowshockwithunsteadinessnotedastheshocktransformstofrom aconvexshocktoaatshock.Theyalsonotedthatastagnationbubbleregionis typicalofaconvexbowshockandthatregularstagnationowaccompaniesaat bowshock.AnillustrationisshowninFigure1-4.Thebehaviorofthebowshockis signicantlyaffectedbytheunsteadyfeedbackmechanismasitisseentooscillateback andforthalongtheaxisofthejet.Thiscausescorrespondinglylargeuctuationsinthe surfacepressureontheimpingementplate.Kawaietal.performedacomputational aeroacousticstudywhichwas2 nd orderaccurateintimeand7 th orderaccuratein space.Thisstudywasdonetodeterminetheeffectsofthepresenceorabsenceof aholeinalaunchpadcongurationandprimarilyfocusedonlargenozzletoplate spacingsandtheeffectofReynoldsnumberontheunsteadyphenomena.Itwasseen thathighReynoldsnumberscansignicantlyincreasethesoundpowerlevelsofthe jetandthemagnitudeofitsoscillations.Theirnumericalcodeproducedresultswhich agreedwellwithhistoricalsoundpowerleveldatamaintainedbyNASA.Thisstudyis usefulinillustratingthecomplexityoftheproblemunderstudyandhowverycomplex numericalsimulationsareneededtoaccuratelyreproducethefeaturesoftheow. Theadditionofmoistureintheformofwatervaportotheairsupplyofanimpinging jetcanhaveanoticeableimpactontheoweld.Thiswasobservedexperimentallyby BaekandKwon[42]whoperformedstudiesofairwithvaryingdegreesofsupersaturation ofwatervaporforasupersonicjetissuingintoquiescentair.Theyfoundthatthe locationoftheMachdiskwaslocatedfurtherupstreamintheowformoistairjets anditssizewasreduced.Empiricalcorrelationsforthelocationofquantitiessuch asthesizeandlocationoftheMachdiskandthelocationofthejetboundarywere proposed,althoughlittlemechanisticinsighttotheowwasgained.Numericalstudies byAlametal.[43,44]andOtobeetal.[45]wereperformedforairwithvariousvalues ofsupersaturationofwatervaporforasupersonicjetimpingingonaatplate.They attemptedtomodelthenon-equilibriumcondensationtakingplaceintheregionafterthe 30


Figure1-4.Numericalresultsofaoweldwithaatplateshockleftandcurvedat plateshockright.[ReprintedwithpermissionfromKlinkovK.etal, BehaviorofSupersonicOverexpandedJetsonPlats,in:H.-J.Rath,C. Holze,H.-J.Heinemann,R.Henke,H.HnlingerEds.,NewResultsin NumericalFluidMechanicsV,volume92of NotesonNumericalFluid MechanicsandMultidisciplinaryDesign ,Springer,2006,pp.168Page 173Figure3] rstMachdiskintheow.Theirmodelassumesnoslipbetweentheliquiddropletsthat condenseandthatthesedropletsdonotinuencethepressureoftheowdownstream. Theowelddisplayssomenoticeabledifferencesthanthatofdryair.Theauthors proposethatthisisduetotheadditionofthelatentheatofcondensationtotheairbythe condensingwatervapor.Unsteadybehaviorduetotheacousticfeedbackmechanism byPowellwasseeninthesimulations.Thisunsteadinesswasnotpresentupstream oftherstMachdisk,butwasseendownstreamofit.Thepresenceofcondensate particlescombinedwiththeadditionofthelatentheatreducesthemagnitudeofthe pressureuctuationsseeninthedownstreamportionoftheowwhichisillustratedin Figure1-5.Theauthorsattemptedtoverifytheirsimulationswithexperimentaldata, 31


a b Figure1-5.Jetcenterlinepressureuctuationswithanomoistureandb40% supersaturationofwatervapor.[ReprintedwithpermissionfromAshraful Alam,M.M.etal.,EffectofNon-EquilibriumHomogeneousCondensation ontheSelf-InducedFlowOscillationofSupersonicImpingingJets, InternationalJournalofThermalSciences492078Page2086 Figure10bandPage2088Figure13c,Elsevier] mostlyconsistingofcomparingtheshockstructureasseeninschlierenphotographslike thoseinthestudybyBaekandKwon,alongwithnoisetonesfordryairgeneratedbythe acousticfeedbackmechanism.Thisproposedvalidationisweakbecausethereisalack ofexperimentaldataofwhichtocomparetointheliterature. Thestudyofsupersonicimpingingjetheattransferjetshasbeenstudiedextensively intheliterature.Unfortunatelymostofthesestudieshavefocusedontheheattransfer fromarocketexhausttoalaunchpadfacility.Donaldsonetal.[46]performedan experimentalstudyofimpingingsonicjetsandtheirturbulentstructure.Theauthors wereabletodevelopacorrelationforNusseltnumberbasedonapplyingaturbulent correctionfactortolaminarimpingingjettheorynearthestagnationpointandfurther awayinthewalljetregion.Whilegoodagreementwasfoundfortheircorrelationit isforsonicorjustslightlysupersonicimpingingjetsanddoesnotapplytothehighly 32


supersonicjetspreviouslymentioned.Theunsteadyacousticfeedbackphenomenon previouslydiscussedcausesinteractionsbetweenacousticwavesandtheshock structureoftheimpingementregion.Thisresultsinlocalcoolingtooccurinthe regionofthejetedgeandisverynoticeableinthemeasurementoftheadiabaticwall temperature.Thisphenomenonistermedcoolingbyshock-vortexinteractionbyFoxand Kurosaka[47]whoinvestigatedthisphenomenon.Kimetal.[48]studiedthesurface pressureandadiabaticwalltemperatureofanunderexpandedsupersonicimpinging jet.Theynotedthattheacousticvorticalstructureinteractionsignicantlyaffectsthe adiabaticwalltemperatureandsurfacepressurewhichalsovariesgreatlywithnozzle height.Thepresenceofastagnationbubble,whichenhancesthecoolingdirectlybelow thenozzle,wasnotedaswell.Rahimietal.studiedtheheattransferofunderexpanded impingingjetsontoaheatedsurface.Thetemperatureoftheimpingementsurfacewith uniformappliedheatuxisnotedtochangedramaticallywithradialdistanceaswell aswithnozzlespacingasshowninFigure1-6.TheynotethatNusseltnumberscales approximatelywiththesquarerootofReynoldsnumberandthathighheattransferrates areencounteredinthestagnationzonewhenastagnationbubbleispresent.Duetothe complexityoftheproblem,theynotethatageneralcorrelationofNusseltnumbershould beafunctionofnotonlyReynoldsnumberandPrandtlnumber,asiscommon,butalso afunctionofMachnumberandnozzlespacing.Yuetal.performedasimilarstudyand noticedsimilartrends;theirmeasuredNusseltnumbersexceed1,500. Studiesoftheheattransfercharacteristicsofsupersonicmoistimpingingjets arenotfoundintheliterature.Theyarelikelytoshowverycomplexphenomena asevidencedbythedifferencesintheshockstructureandgeneralbehaviorofthe relevantowquantitiesinthejetandalongtheimpingementplate.Thecurrentstudy usesdiscreteliquiddropletsthatareinjectedintotheairupstreamofthenozzle. Thiswilllikelyresultinanairstreamsupersaturatedwithwatervaporwhichisfurther complicatedbythebehavioroftheliquiddropletsandtheireffectsontheow.As 33


a b Figure1-6.Adiabaticcirclesandheateddiamondswalltemperatureforanozzle spacingofaz/D=3.0andbz/D=6.0.[Reprintedwithpermissionfrom RahimiM.etal,ImpingementHeatTransferinanUnder-Expanded AxisymmetricAirJet,InternationalJournalofHeatandMassTransfer46 263Page267Figures6aand6b,Elsevier] elucidatedbytheliteraturesurveytheowstructureassociatedwiththistechnologyis verycomplex,andessentiallynoanalyticalsolutionsareavailablefortheoweldand heattransfer.Theavailableempiricalcorrelationsdonotcovertwo-phasesupersonic impingingjets.Numericalstudiesmayprovidesomequalitativeinsight,butinmost instancestheydonotadequatelycaptureallofthephysicstakingplaceintheoweld. 1.2Summary InthisChapteranintroductiontothestudywasmadeandtherelevanceof multiphasesupersonicimpingingjetswasintroduced.Thecontributionsofthisstudy werealsodescribed,mainlythatthisisatechnologythathasnotbeenstudieduntilnow. Abriefliteraturereviewofthedifferenttypesofimpingementheattransferwas presented.Liquidandsingle-phaseheattransferwasintroducedstartingwiththe classicworkofHiemenzandHomann.ThedevelopmentofaccurateNusseltnumber correlationsbasedonvonK arm an-Pohlhausenintegralmethodweredetailed.The agreementbetweentheoryandexperimentsisexceptionalforthesecorrelations. Otheraspectssuchastheowhydrodynamics,transitiontoturbulence,nozzleheight, 34


andnon-uniformboundaryconditionseffectswerediscussed.Highheatuxremoval technologiesthatarecapableofheattransfercoefcientsashighas200kW/m 2 were detailedaswell. Thestudyofsupersonicunderexpandedandoverexpandedimpingingjetswas describedaswell.Thiseldiscomplicatedbythecomplexowstructuregeneratedby shockwaveswhichformwhenanozzleisoperatedawayfromitsdesignedpressure ratio.Thedetailsoftheseshockwavesincludingtheeffectsofthecurvatureofthe bowshockjustabovetheimpingementplatewerediscussed.Stagnationbubbles formedjustbelowthebowshockwerediscussedandtheirimpactontheoweld wasdetailedaswell.Shockwavesneartheimpingementregioncauseanunsteady feedbackphenomenoncausedbytheinteractionofacousticwaveswiththeedgeof thenozzle.Theeffectsofthisfeedbackphenomenonandtheunsteadinessitcauses andrelevantchangesinthelocaloweldweredetailed.Moistureintheairstreamand howitchangestherelevantoweldwasbrieyexploredasitisarelativelynewarea ofstudyintheliterature.Thetemperatureproleontheimpingementplateandhowit changeswiththepresenceofthestagnationbubbleandacousticfeedbackmechanism werediscussed.Numerousexperimentalstudiesintheliteraturewhichincludepressure andtemperaturemeasurements,particleimagevelocimetry,andschlierenphotographs alongwithrelevantnumericalstudiesintheliteraturethatdiscoveredandconrmed thesephenomenawerediscussedwhererelevant. Lastlythecomplexityofthecurrentstudywasdiscussed.Itisnotedthatan analyticalsolutiontotheproblemwillnotbeattainableandapredictivenumericalstudy isnotfeasibleaswell.Thecontributionsofthisstudywillbeintheformofdeveloping anunderstandingofthemechanismstakingplaceasthemultiphasesupersonicjet removesheatfromasurface. 35


CHAPTER2 JETIMPINGEMENTFACILITY Thesupersonicmulti-phasejetfacilityshouldpossessseveraltraitsinordertobe usefulforanexperimentalapparatus.Itshouldhavesufcientairstoragecapacityso thatexperimentscanberunatsteadystate.Thestoredairshouldbepressurizedto suchanextentthatthedesiredMachnumbercanbeachieved.Lastlyitshouldcontain sufcientwater,andameanstocontroltheow,sothattheimpingingjetwillremainin multiphaseoperationduringexperiments. ThedesignforthecurrentsetupisbasedonasimilarexperimentbyKlausneret al.[4].Theimpingingjetconsistsofthefollowingsystemstobedescribedbelow:air storagesystem,waterstorageandowcontrolsystem,airpressurecontrolsystem, airmassowratemeasuringsystem,temperatureandpressuremeasurementsystem, converging-divergingnozzle,anddataacquisitionsystemDAQ.Aschematicdiagram illustratingthecongurationofthejetimpingementfacilityisshowninFigure2-1. 2.1ImpingingJetFacilitySystems 2.1.1AirStorageSystem Theairstoragesystemconsistsof9`K'sizedbottleswhichgiveatotalvolumeof 0.45m 3 andarekeptatapressureof14MPa.Theairstoragesystemislledwithair fromamodelUE-3compressorfromBauerCorporation.Thecompressorispoweredby a3-phase240Vpowersupplyandiscapableofsupplying0.1m 3 /minofapproximately moisturefreeairtotheaircylinders,thustheairstoragefacilitycanbechargedto capacityinapproximately4.5hrs. 2.1.2WaterStorageandFlowControlSystem Waterforthefacilityiscontainedinastainlesssteelvesselwithacapacityof2 Landapressureratingof12.4MPa.Waterisforcedintothemixingchamberbythe differenceinpressurebetweenthetopofthewatervesselwhichisactedonthethefull forceoftheairsupplypressureandthepressureinsidethemixingchamber,whichis 36


Figure2-1.Illustrationofthejetimpingementfacility. lowerduetoachangeinareaandbecauseofthefrictionactinginthesystemtubing.A drawingtheofthemixingchamberisfoundinFigures2-2and2-3.Theowrateofthe wateriscontrolledbymeansofanoricebetweenthewatervesselandmixingchamber andtheairpressureinthesystem.Oricediametersof0.33,0.37,0.41,and0.51mm areusedduringexperiments,andadrawingoftheoricedesignisfoundinFigure2-4. Whilethereissomevarianceinowratebetweeneachexperimentforagivenorice sizethiseffectiseliminatedintheanalysisduetothefactthetheowrateofwater intothemixingchamberismeasuredduringeachexperiment.Thisisaccomplished byrecordingtheelapsedtimeofeachexperimentandmeasuringthedifferenceinthe massofliquidinthewatervessel.Table2-1showsthenominalowrateofliquidandthe 37


Figure2-2.Crosssectionofthemixingchamber. Figure2-3.Threaddetailsofthemixingchambercrosssection. averageliquidvelocityforthepressuresandoricesizesusedduringtheexperiments.It isnotedthattheowrateofthe0.37mmoriceislessthanthatofthe0.33mmorice, thisisduetothefactthatthe0.37mmoriceisnotperfectlycircular.Thisconditionis alsoseeninthe0.51mmorice,butitisnotassevereasthatinthe0.37mmorice. PicturesofeachoricetakenwithanopticalmicroscopeareshowninAppendixC. 38


Figure2-4.Oricecrosssection. 2.1.3AirPressureControlSystem Theairpressurecontrolsystemconsistsofanairregulatorlocatedbetween theairstoragetanksandtheinlettotheairmassowmeterandthetopofthewater storagetank.Theregulatoriscapableofreducingairpressurefrom14MPadown toamaximumof2.8MPa.Airpressuresof1.0,1.7,and2.4MPaareusedduring experiments.Asdiscussedlater,theseairpressuresresultintheconverging-diverging nozzleoperatinginanoverexpandedmanner. 2.1.4AirMassFlowMeasuringSystem MeasurementoftheairmassowrateisaccomplishedusinganAnnubarDiamond IImodelDNT-10massowmeterlocateddownstreamoftheregulatorandbeforethe mixingchamber.Thediamondcrosssectionoftheowmeterissuchthatithasaxed separationpointandalsoreducespressureloss.Theowmetersensesdifferential pressurewhichismeasuredwithadifferentialpressureDPtransducer,whichis 39


Table2-1.WaterMassFlowrateandAverageWaterVelocityforDifferentRegulator PressuresandOriceSizes. OriceRegulatorWaterMassAverageWater SizePressureFlowrateVelocity mmMPakg/sm/s 1.03.08x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.60 0.331.73.99x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 4.67 2.44.73x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 5.53 1.02.67x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 2.50 0.371.73.31x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.11 2.43.97x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.73 1.04.61x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.55 0.411.75.72x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 4.41 2.46.79x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 5.24 1.04.85x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 2.39 0.511.76.90x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.00 2.47.20x 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 3.55 calibratedtomeasurepressuredifferencesofupto2.21 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(3 MPa.Thecalibration equationfortheowmeteris m =58.283 KD 2 p P f where PisthemeasureddifferentialpressureinkPa,Disthediameterofthe owmeterinmm,inthiscase15.80mm,Kisagagefactorof0.6,and f isthedensity oftheowingair,inkg/m 3 calculatedvia f =539.5 P f T f whereP f isthepressureoftheowingairinkPa,asmeasuredbythepressure transducerupstreamofthemassowmeterandT f isthetemperatureoftheowing airinKelvinasmeasuredbythethethermocoupleupstreamofthemassowmeter. 40


Figure2-5.Acomparisonofthetheoreticalandexperimentallymeasuredairmass owrate. Thetheoreticalmassowratethroughthenozzlefor1-Disentropicowis calculatedusingmassow, m = AV,theMachnumber,M=a/Vthespeedofsoundfor aperfectgasisa= p RT ,andtheidealgaslaw,P= RT, m = 4 D 2 MP r RT KnowingthatatthethroatofthenozzletheMachnumberisoneandusing TemperatureandPressurestagnationratiosof T = T o =0.8333 and P = P o =0.5282 Equation2reducesto m =0.4545 D 2 P o r RT o 41


Equation2neglectstheeffectsoffrictionandheattransfer,whichaffecttheowrate ofairthroughthenozzle.Acomparisonoftheairmassowratemeasuredduringthe courseofexperimentswiththetheoreticalairmassowrateisshowninFigure2-5.The agreementbetweentheoryandexperimentiswithin20%. 2.1.5TemperatureandPressureMeasurements Thetemperatureandpressureofthejetimpingementsystemismonitoredduring systemoperationforcalculatingvariousquantitiesofinterest.Thetemperature measurementsareaccomplishedbytheuseofEtypethermocoupleprobeswhich areinsertedinto`T'junctioncompressionttingsatadepthsuchthatthetipofthe probeisatthecenterlineofthetting.Thethermocoupleprobesusedaregrounded andsheathedinstainlesssteelandhaveanominaldiameterof1.59mm.Temperature measurementsaretakenatthefollowingpoints:theoutletofthepressureregulator,the outletofthewaterreservoir,andattheoutletofthemixingchamber. Pressureismeasuredattheoutletofthepressureregulatorjustbeforethelocation oftheairmassowmeter.Thepressuremeasurementismadeusingastraingage typepressuretransducer,whichhasarangeof0-2.8MPa.Theoutputsignalofthe pressuretransducerisacurrentwhichvariesbetween4-20mA;becausetheDAQ systemusedintheexperimentsonlysensesvoltagesaresistorof520 isusedto convertthiscurrentintoavoltageinthe0-10Vrangeneeded. 2.1.6Converging-DivergingNozzle Theconverging-divergingnozzleiswherethemixtureofliquidandairareexpanded tosupersonicspeeds.Thenozzleisconstructedofstainlesssteelwithathroat diameterof2.38mmandanexitdiameterof5.56mm,givinganexitMachnumber of3.26.Thenozzleisattachedtoasize10DN 1 / 2 NPSstainlesssteelpipewithan internaldiameterof13.51mm,whichisconnectedtoabraidedstainlesssteelhose approximately9mlongwithaninnerdiameterof9.53mmandisconnectedtothe outletofthemixingchamber.Althoughthehoseaddssomesmallamountofpressure 42


Figure2-6.Crosssectionalviewoftheconverging-divergingnozzleusedin experiments. loss,itallowsthenozzletobelocatedawayfromtheairstoragecylindersandnear theimpingementheattransfertargets.Acrosssectionalviewofthenozzleisshownin Figure2-6. 2.1.7DataAcquisitionSystem TheDAQusedduringthecourseofsteadystateheattransferexperimentsisa DAS-1601dataacquisitionPCIcardandaCIOEXP32analogtodigitalconverter board,bothmadebyMeasurementandComputingInc.ThisDAQconsistsof3216-bit doubleendedchannelsandchannelgainsof1,10,100,200,and500areselectable. Thesystemhasamaximumreliablesamplingrateof50HzandthesoftwareSoftwire, producedbyMeasurementandComputingIncisusedforprogrammingdatacollection. Fortransientmeasurementsonheatedtargetsduringinverseheattransferexperiments, theDAQsystemissupplementedwithaNationalInstrumentsNI,NIUSB-6210system whichhas8doubleended16-bitchannelsandhasamaximumaggregatesamplingrate of250kHz.ThissystemusesLabviewsoftwareproducedbyNIwhichandisalsoable 43


tointerfacewiththeMeasurementandComputingDAQviatheuseofanNIsupplied.dll library. 2.2AnalysisofImpingementFacility Someanalysisofthejetimpingementfacilitiesarewarranted.Thebehaviorofthe systemupstreamofthenozzleisexaminedtodetermineifthereareanycorrectionsthat needtobeappliedtothethermocoupleorpressuretransducerreadings.Additionally, thefollowingisexamined:thepressurerequiredtooperatethenozzleinaperfectly expandedmanner,theminimumandmaximumpressurethatcauseashockwave toforminsidethenozzle,andthenozzleexitpressurewhenoperatingatvarious regulatorpressures.Lastlytheshockwaveanglesformingatthenozzleexitforvarious operatingpressurearecalculatedaswell.One-dimensionalgasdynamicrelationsare usedtoinvestigatethequantitiesofinterest.Hereitisnotedthattheanalysisused haslimitations,theonedimensiongasdynamicrelationsareisentropicinnature,with theexceptionofshockwavecalculations.Thejetimpingementfacilityexperiences frictionandheattransferduringoperation,thustheisentropicassumptionisnotmet. Additionallyafterthemixingchambertheowwillcontainwaterdropletswhicharenot compressible.Thequantitiescalculatedbelowwillhavesomeinherenterrorhowever, theydoprovideareasonableapproximationofthephysicstakingplaceinthefacility. 2.2.1TemperatureandPressureUpstreamoftheNozzle Calculatingthetemperatureandpressureatvariouspointsupstreamofthenozzle isasimplematter;thecross-sectionalareaofthepointsinthesystemarerequiredfor thisanalysis;Figure2-7providesanillustrationofthejetimpingementfacilityandthe diametersofthepointsofinterest.Usingthecommonlyknownone-dimensionalgas dynamicsrelationshipsfoundinvariouscompressibleowtextbooks,suchasLiepmann andRoshko[49]orJohnandKeith[50],thepressureandtemperatureratiosaswell astheMachnumberoftheowintheseareascanbedetermined.Inthefollowing 44


Figure2-7.Simpliedviewoftheairowpathinthefacility. equations istheratioofspecicheatsandisaconstantequalto1.4.Todeterminethe owMachnumberthefollowingrelationshipisused A A = 1 M 2 +1 1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 +1 2 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 wherethe*superscriptdenotesthecriticalareawhereMach=1.NotethatEquation2 isaquadraticequationinMandhastwosolutionsthuscarefulattentionmustbepaid inselectingtheproperMachnumbergivenanarearatio,inthepresentcaseallMach numbersupstreamofthethroatoftheconverging-divergingnozzlearesubsonic.Once 45


theMachnumberofthegivensectionisdeterminedthepressureandtemperatureratios canbedeterminedfromthefollowing T o T =1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 P o P = 1+ )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 2 M 2 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 wherethesubscriptoisthestagnationproperty,whichissimplytheparticularproperty withzerovelocity.TheanalysisresultsusingEquations2to2areshownin Table2-2.Theresultsshowthatthetemperatureandpressureupstreamofthenozzle throatdifferfromtheirstagnationpointpropertiesbylessthan1%;nocorrectionduethe thevelocityoftheowisneeded. Table2-2.Area,temperature,andpressureratiosatvariouspointsinthejet impingementfacility. PointA/A T/T o P/P o M Exit5.440.31930.018403.26 Throat1.000.52830.83331.00 132.200.99990.99980.018 26.830.99860.99500.085 344.020.99991.00000.013 46.830.99860.99500.085 2.2.2NozzleExitPressureConsiderations Thereareafewtheoreticalconsiderationsthatneedtobeexploredatthenozzle exit.First,thenecessaryregulatorpressureinordertoobtainaperfectexpansionatthe nozzleexitisneeded;thentheactualexitpressuresbasedontheregulatorpressure duringexperimentsaredetermined.Theresultsfromthepreviouscalculationslistedin Table2-2showthestagnationpressureratioattheexitofthenozzle,simplycarrying therequisitealgebraandassumingabackpressureof101.4kPayieldsthenozzle exitpressure,seetheresultsofthesecalculationsinTable2-3.Fromtheseresultsit isobservedthatthepressurenecessaryforidealexpansionisapproximatelytwicethe 46


pressuretheregulatorofthesystemcanprovide,andthusduringnormaloperationof thejetimpingementfacilitythenozzleoperatesinanoverexpandedmanner. Table2-3.Nozzleexitpressureforvariousregulatorpressures. RegulatorNozzleExitNozzle PressurePressureOperation MPaMPa 5.50.1014perfectlyexpanded 2.80.0508overexpanded 2.40.0443overexpanded 1.70.0317overexpanded 1.00.0190overexpanded Duetothefactthatthenozzleexitpressureisbelowtheambientpressuresome concernaboutashockwaveforminginthenozzlewillbeaddressed.Therearetwo limitingpressuresforthiscase,oneisthepressureatwhichatshockwaveformsatthe throatofthenozzle,theotheristhepressurethatashockwaveformsatthenozzleexit; Figure2-8showsanillustrationforbothofthesetwocases.Inthelimitingcase,ashock waveoccurringatthethroatwheretheMachnumberisequaltounity,thestagnation pressureratiois0.992.Assumingthatthebackpressureisatmosphericpressure,the stagnationpressurethatwillcauseashocktobelocatedatthethroatis0.102MPa.To calculatethelimitingcaseofashockwaveoccurringatthenozzleexitisjustslightly morecomplicated.Whenitisassumedthatashockwaveislocatedattheexitplaneof thenozzle,thestagnationpressureratioandMachnumberjustbeforetheexitplanecan befoundfromTable2-2.ThenormalshockwaverelationsforMachnumberandstatic pressureratioacrossashockEquations2and2canthenbeapplied, M 2 = s M 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1+2 2 M 2 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 P 2 P 1 = 2 M 2 1 +1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +1 47


Figure2-8.Illustrationofthelimitingcasesforshockwavesinthenozzle. UsingEquations2and2theMachnumberjustpasttheshockwaveis foundtobe0.461andthestaticpressureratiois12.268.Performingtherequisite algebrayieldsabackpressureof0.449MPa.Thusthenozzlewillhaveashockwave locatedinsideforastagnationregulatorpressureintherangeof0.102to0.449MPa. Sincetheminimumregulatorpressureusedduringexperimentsis1.0MPathereislittle concernthatashockwavewillforminsidethenozzle. 2.2.3ObliqueShockWavesatNozzleExit Itisknownthattheconvergingdivergingnozzleoperatesinanoverexpanded manner;theexitconditionsofthenozzleshouldbeconsidered.Whenanozzle isoverexpandedobliqueshockwavesformattheoutletofthenozzle,see[49]for instance.Theseshockwavescompresstheairsuchthatitisthenequaltothe nozzlebackpressure;therstobliqueshockwavecomingoutofthenozzlecanbe modeledusingthestandardonedimensionalgasdynamicrelationsparameterssuch astheshockangle,deectionangle,temperatureratio,stagnationpressureratio,and downstreamMachnumber.Figure2-9showsanillustrationoftheshockwaveatthe nozzleexit.Todeterminetheseexitquantities,rstthenozzleexitpressureshouldbe determinedusingtheregulatorpressureandEquation2.Theshockwaveanglecan 48


Figure2-9.Illustrationofanobliqueshockwaveatthenozzleexit. thenbecalculatedusingthefollowingequationsincetheMachnumberatthenozzleexit isknownfromthedesignconditions,andthestaticpressureratiocanbecalculated, P 2 P 1 = 2 M 2 1 sin 2 +1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 +1 Thedeectionangle, canbedeterminedfromthefollowing tan =2 cot M 2 1 sin 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 M 2 1 + cos 2 +2 ThedownstreamMachnumber,stagnationpressureratio,andstatictemperatureratio areeasilycalculatedviathefollowingequations: M 2 = 1 sin )]TJ/F25 11.9552 Tf 11.955 0 Td [( s 1+ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 M 2 1 sin 2 M 2 1 sin 2 )]TJ/F26 7.9701 Tf 13.15 5.256 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 P o 1 P o 2 = +1 2 M 2 1 sin 2 1+ )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 M 2 1 sin 2 # )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 1 2 +1 M 2 1 sin 2 )]TJ/F26 7.9701 Tf 13.151 5.256 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 +1 # 1 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 T 2 T 1 = )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(1+ )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 M 2 1 sin 2 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 M 2 1 sin 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 h +1 2 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i M 2 1 sin 2 49


a b c d Figure2-10.Thevariationofashockangle,bdeectionangle,cstagnationpressure ratioanddstatictemperatureratioasafunctionofupstreamstagnation pressure,downstreamofanobliqueshockwave. Itisnotedthatthestagnationtemperatureacrosstheshockisconstant.Resultsof thesecalculationsareshowninFigure2-10.Itisbrieymentionedthatthestagnation pressureratioreectsalossofmomentumgoingacrossashockwaveandthatthisloss ofenergyislessenedathigherupstreampressureratios. 2.2.4CompleteShockStructureofanOverexpandedJet Therstshockwaveatthenozzleexitiseasilymodeledasshownabove;however, thesubsequentbehaviorofthoseshockwavesisquitecomplex.Theobliqueshock wavescanintersectatapointdownstreamorcanmergeandformanormalshockarea 50


knownasaMachdisk.Machdiskstypicallyformafterrelativelystrongshockswhich aretypicalfornozzlesoperatingfarremovedfromtheidealpressureratio.Theseshock wavescompresstheowcausingtheformationofPrandtl-Meyerexpansionfanswhich turntheowandlowerthepressure.Whenexpansionfansintersecttheshearlayer whichisformedattheboundaryofthejet,theyarereectedbackasobliqueshock waves.Thisseriesofeventscausestheformationofshockdiamondsintheowand isrepeateduntilthecombinationofviscouseffectsandtheinuxoflowmomentum uidcausethejettobecomesubsonic,orwhenthejetinteractswithanobstacle. Figure2-11providesandillustrationoftheshockstructuretypicallyseenattheexitofan overexpandedjet. Thepressure,temperature,andvelocityoftheowinthedownstreamofthe nozzlechangesveryrapidlyandisdifculttomodelanalytically.Zapryagaevetal. [29]performedexperimentswithanoverexpandednozzleandperformedschlieren photographyaswell.Theirresultsshowthatthepressureintheowupstreamof therstMachdiskvariesgreatlyintheradialandaxialdirectionwithseveralsharp discontinuitiespresent.DownstreamoftherstMachdiskthevariationinpressureis stillpresent.However,thediscontinuitiesarenolongerpresent.Manyauthorshave extensivelystudiedunderexandedjets[30,34,35,46,51,52]andsimilartrendsasthe aboveareobserved. 2.3Summary InthisChaptertheconstructionofthejetimpingementfacilityhasbeenexplored. Thefacilitysystemsincludetheairstorage,waterstorageandowcontrol,airpressure control,airmassowmeasuring,converging-divergingnozzle,anddataacquisition. Acomparisonbetweenthemassowratemeasuredduringexperimentsandthe theoreticalmassowratebasedonone-dimensionalisentropicgasdynamicrelations wasperformedandtheresultsdifferedbylessthan20%.Thepressure,temperature, andMachnumbervarioussectionsofthejetimpingementfacilityweredeterminedand 51


Figure2-11.Illustrationofthestructureofobliqueshockwavesattheexitofan overexpandednozzle. areshowninTable2-2.Thenozzleexitpressurewascalculatedforeachregulator pressureusedduringtheexperiments,anditwasfoundthatthenozzleoperatesinan overexpandedmannerfortheentirepressurerange.Noshockwaveisexpectedinside thenozzle.Finally,theshockstructureattheexitofthenozzlehasbeendescribed. 52


CHAPTER3 STEADYSTATEEXPERIMENTS Jetimpingementfacilitiesarecapableofveryhighheatuxremoval,andan experimentalproceduretodeterminetheheattransfercoefcientforthefacilityin Chapter2isdeveloped.Severaldifferentexperimentswereinitiallytestedwithout successbeforeasuccessfulexperimentalcongurationwasdevelopedtomeasurethe steady-stateheattransfercoefcientoverarangeofoperatingconditions. Initiallyitwasbelievedthatphasechangeheattransferwouldoccurduring operationofthejetimpingementfacilitytosuchanextentthatnoradialvariationof heattransfercoefcientwouldbeobserved.Assuchanexperimentwasdesignedin whichathinsheetofstainlesssteelwasmachinedintoametalblankandheatedvia Jouleheatingandinsulatedatthebottomwherethetemperaturewasmeasuredvia athermocouple.Theareaofthestripwasquitesmallapproximately15mm 2 and assuchtheheatuxesproducedduringtheexperimentwerehigh.Theproblem experiencedwiththissetupisthatthebackwalltemperaturemeasuredbythe thermocoupleisnotverysensitivetochangesintheheattransfercoefcientwiththe appliedheatuxesandthemetalthicknessused.Anillustrationoftheheaterassembly usedisshowninFigure3-1. Inordertoalleviatethisproblemanexperimentwasconductedwherethermocouples wereembeddedinsideofacoppercylinderwhichwasheatedfromthebottomand insulatedalongtheside.Thejetwasallowedtoimpingeonthetopofthecylinder andthetemperaturesinsidethecopperpieceweremeasured.One-dimensionalheat conductionintheaxialdirectionwasassumedandduetotheabsenceofinternalheat generation,alineartofthemeasuredtemperaturewasthenusedtoextrapolatethe temperaturetothesurfaceandallowedthedeterminationofheatux.Uponanalyzing thedatagainedfromtheseexperimentsitwasobservedthattheheattransfercoefcient 53


Figure3-1.Stainlesssteelheaterassembly. forthejetimpingementfacilityvariessignicantlyintheradialdirection.Anillustrationof thecopperheaterassemblyusedisshowninFigure3-2. Inordertogainsomeinsightintotheradialvariationoftheheattransfercoefcient theheaterinFigure3-2wasmodiedtoincludetemperaturesensitivepaintonthe topsurfaceofthecoppertestpiece.Steadystatetemperaturedistributionsatthetop surfacewherethenusedasinputtoaninverseheattransferalgorithmtodetermine theradiallyvaryingheattransfercoefcient.Therewereseveraldrawbackstothis study.Firstthetemperaturesensitivepaintisverybrittleandhadtobeprotectedfrom theimpingingjetviatheuseofthickclearcoatapplicationstothesurfaceofthepaint orviatheuseoftransparenttape.Theseprotectivelayerscouldnotbeneglectedin theinverseheattransferanalysisandcomplicatedthealgorithm.Lastly,andmost importantly,theimpingingjetpartiallyobscuresvisualobservationofthetemperature 54


Figure3-2.Copperheaterassembly. sensitivepaint.Thesecomplicationsrenderreliableexperimentalresultsdifcultto obtain. Thethirdexperimentalcongurationtestedconsistsofathinsheetofnichrome whichisheatedbyJouleheating.Itisinsulatedatthebottomand9thermocouples areusedtomeasuredthespatialvariationofthebackwalltemperature.Thearea ofthenichromestripislargeandtheheatuxesproducedareconsiderablysmaller thanthoseappliedtothestainlesssteelsetupdescribedearlier.Thisexperimental congurationprovedtogivereliablemeasurementsofheattransfercoefcient,andthe abundanceofthermocouplesallowsthespatialvariationofheattransfercoefcienttobe determined.Thisexperimentisnowbedescribedindetail.LaterSectionswilldescribe theexperimentalprocedureandexploretheheattransferresults. 55


3.1HeaterConstruction 3.1.1PhysicalDescription Theheaterdesignusedduringthecourseofthefollowingexperimentsisinspired bytheworkofRahimietal.[51].Itisconstructedusingathinnichromestripwhichis 0.127mmthickwithanexposedareaof50.8x25.4mm.Theheaterhas9totalEtype thermocouplesattachedtothebacksideofthestrip.Onethermocoupleisattached atthecenterofthestripand7additionalthermocouplesareattachedevery3.79mm towardsonesideofthestrip.Additionallyonethermocoupleisattachedat12.7mm fromthecenterontheoppositeside.Thisthermocoupleisusedtoensurethejetis centeredovertheheaterbyverifyingsymmetry. ThenichromestripisthenepoxiedontopofaGaroliteslabthatis140x140x 6.35mm.Smallholesaredrilledintheslabsothatthethermocouplespenetrateitand avoiddeformingtheatsurfaceofthenichromestrip.TheslabofGarolitehasathermal conductivityof0.27W/m-Kcomparedwith13W/m-Kforthenichromeandthusactsto insulatethebacksideofthenichromestrip. Electricalpowerissuppliedtothenichromeviatwocopperbusbarswithdimensions of43x25x2mmattachedtothetopofthestrip.Duringoperationofthetwo-phasejet, liquidowstowardsthebusbarsandaccumulatesattheedge.Thisliquidlmof accumulationcouldaffecttheheattransferphysics;tolessenthiseffecttheedgesof thebusbarsareledtoanangleofapproximately30 .Anillustrationoftheheater assemblyisshowninFigure3-3. Powerissuppliedtothenichromestripviaahighcurrent,lowvoltagedcpower supply.Thepowersupplyiscapableofsupplying4.5kWofpowerthroughavoltage rangeof0-30Vandacurrentlimitof125A.Themaximumvoltageseenduring experimentsisapproximately5Vat125A. 56


Figure3-3.Illustrationofheaterassemblyusedforsteadystateexperiments. 3.1.2TheoreticalConcerns Thethermocouplesusedfortheexperimentmeasurethetemperatureontheback walloftheNichromestrip.Todeterminetheheattransfercoefcientfortheimpingingjet thesurfacetemperatureofthenichromeisneeded.Typicalheattransfercoefcientsfor impingingjetswillyieldBiotnumbersBi=h /kmuchgreaterthanunitythusalumped systemassumptionisnotvalidforthecurrentexperimentalconguration. Thefollowinganalysisprovidesamethodforevaluatingthespatiallyresolved surfacetemperature.FirstthesteadystateheatequationinCartesiancoordinateswith internalheatgenerationisexamined. @ 2 T @ x 2 + @ T 2 @ y 2 + @ T 2 @ z 2 + q 000 k =0 WhenEquation3isnon-dimensionalizedthefollowingresults: D 2 @ 2 @ x 2 + D 2 @ 2 @ y 2 + @ 2 @ z 2 +1=0, where 57


= Tk q 000 2 x = x D y = y D z = z HereDisthenozzlediameter,and isthethicknessofthestrip.Thecoefcientsinthe rst2termsofEquation3arequitesmall,andthreedimensionaleffectscanbe neglected.Assuchthegoverningequationandboundaryconditionsare d 2 dz 2 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 d dz z =0 =0 )]TJ/F39 11.9552 Tf 12.68 8.088 Td [(d dz z =1 = kD k w Nu D r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 r wherer =r/Disthenon-dimensionalradiuswiththeoriginatthecenterlineofthejet, Nu D r =hr D/k w istheNusseltnumber,k w isthethermalconductivityofwater, and 1 r isthereferencetemperature.Duetotheextremedifcultyinmeasuringthe expandingjetuidtemperature,theadiabaticwalltemperatureiscommonlyusedas areferenceforpurposesofcomputingaheattransfercoefcient[53]associatedwith impingingjets.Notethattheeffectsoftheradialvariationofheattransfercoefcient comefromtheboundaryconditionsonly.Thesolutionofthisordinarydifferential equationis: r z = 1 2 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(z 2 + 1 kD k w Nu D r + a w r orindimensionalform T r z = q 000 2 k )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(z 2 + q 000 h r + T a w r whereT a,w istheadiabaticwalltemperature.Itisobservedthatthedifferencein temperaturebetweenthetopandbottomsurfacesisthequantity q 000 2 = 2 k ,indimensionless formitisequalto 1 / 2 .Themaximumvolumetricheatgenerationintheexperimentsis 58


3.66 10 9 W/m 3 .Usingavalueof13W/m-Kforthethermalconductivityofnichrome themaximumtemperaturedifferencebetweentheupperandlowersurfaceobservedin experimentsisontheorderof2 Candcannotbeneglected.However,thesolutionfor theheattransfercoefcientandthustheNusseltnumbercanbecalculatedknowingthe internalheatgenerationrate,backwalltemperature,andtheadiabaticwalltemperature: Nu D r = Nu D w r 1 )]TJ/F40 7.9701 Tf 13.459 4.707 Td [(kD k w Nu D w r 2 where Nu D w r = 1 kD k w [ r ,0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( a w r ] 3.2ExperimentalProcedure Duringthecourseoftheexperiments,theMeasurementandComputingdata acquisitionsystemisused.Alsonotethattheheatuxremovedfromthetopofthe heaterassemblyisthequantityq 000 ,sincetheheatgeneratedinternallycanonlybe removedfromthetopsurface. 3.2.1Two-PhaseExperiments Duringthecourseofexperimentsitwasobservedthatrunningtheimpingingjet wouldcauseicetoformonthesurfaceoftheheateratlowheatuxes.Atadiabatic conditionstheicewouldbegintoformintoaconeshape,eventuallythisconewould bebrokenoffandanewconewouldforminitsplaceinaperiodicmanner,asshownin Figure3-4.Atlowbut,non-zeroheatuxes,athinsheetoficewouldformthatwould exhibitsimilarperiodicbehaviorastheicecones.Thisiceformationaffectsheattransfer sincethislayeroficeisstationaryandactsasaninsulator.Toavoidthisconditiona minimumheatuxof300kW/m 2 waschosensuchthaticeformationisnotvisually observedduringoperationoftheimpingingjet. 59


Figure3-4.Iceformationatadiabaticconditions.Notetheconicalshapeoftheice structure. Torunacompleteexperiment,thenozzleisalignednearthecenteroftheheater surfaceatagivenheightandtheimpingingjetisinitiatedwiththepressureregulator setatadesiredpressure.Thepowersupplytotheheateristurnedonandaheatux ofapproximately470kW/m 2 issuppliedtotheheater.Toensuretheimpingingjetis centeredovertheheater,thepositionoftheheaterismovedsuchthatthetemperatures 60


measuredbythesinglethermocoupleononesidematchesthetemperatureofthe correspondingthermocoupleontheoppositesidetowithinapproximately 0.5 C. Oncethisconditionisachievedtheheaterisallowedtoreachsteadystate,whichis determinedbyobservingagraphoftheheatertemperaturesvs.time.Uponreaching steadystateoperation,theheaterthermocouplesareloggedatasamplingrateof50 Hzforapproximately2minutes.Afterwardsthissameprocedureisperformedforheat uxesofapproximately430,390,350,and315kW/m 2 .Theseheatuxesarechosento beashighasreasonablyachievablewiththegivenequipmentfortworeasons.Firstto helppreventtheformationoficeonthesurfaceoftheheaterandsuchthattheheater walltemperaturedifferenceisashighaspossibletominimizetheuncertaintyintheheat transfercoefcientmeasurement. Thedatareportedforthisstudyareaveragedvaluesfrom100samplestaken overthecourseof2minutes.Whileitispossibleforhighfrequencyoscillationsinthe measuredtemperaturestoexistduetothemultitudeofdropletsimpingingontothe surface,thisisnotlikelytobeobservedforseveralreasons.First,thedataareaveraged whichwilllessenanytransienteffects.Second,theheater,althoughthin,doeshavea nitethickness.Soitwilltendtoactlikealowpasslteranddampenuctuations.Lastly itisbelievedthatathinliquidlayerexistsonthesurfaceoftheheater.Anyliquiddrops thatimpingeontotheheaterwilltendtocoalescewiththisliquidlm,thusminimizing transientsofthelocalheattransfercoefcient. Themeasurementoftheadiabaticwalltemperatureisaccomplishedbyasimilar procedureasaboveexceptafterensuringthejetiscentered,thepowersupplytothe heateristurnedoff.Theheatertemperaturesareallowedtoreachsteadystateandthen measurementoftheadiabaticwalltemperatureiscommenced.Itcouldtakeseveral minutesontheorderof5minutesfortheheatertoreachsteadystate.Because oftheabsenceofheattransfer,themeasuredtemperatureatthebackwallisequal tothesurfaceadiabaticwalltemperatureofthejet.Itisnotedthattheformationof 61


icewasobservedonthesurfaceoftheheaterduringtheadiabaticwalltemperature measurements,butbecauseofthereasonslistedaboveforneglectinguctuationsdue tomultipledropletsimpingingontothesurface,itisbelievedthatthiswillhavelittletono effectonthemeasurementofadiabaticwalltemperature. IncomputingthejetReynoldsnumber.theviscosityisbasedonthenozzleexitair temperatureandpressureascalculatedbyone-dimensionalgasdynamicrelations.In computingtheNusseltnumberforthesingle-phasejet,thermalconductivityisevaluated basedonairandtheadiabaticwalltemperature.Duringtwo-phasejetimpingement,a thinliquidlmexistsontheheatersurface,andthinliquidlmdynamicsdominatethe heattransferphysics.Thusthewaterthermalconductivitybasedontheadiabaticwall temperatureisusedforNusseltnumbercalculationsofthetwo-phasejet.Typicalvalues forairviscosity,waterthermalconductivity,andairthermalconductivityrespectivelyare 6.45 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(6 Pa-swithlessthan1%variation,0.588W/m-Kwitha3%variation,and 0.0243W/m-Kwitha5%variation. 3.2.2Single-PhaseExperiments Toprovideacomparisonforreferencetothetwo-phasejetresultssingle-phase experimentsarecarriedoutwiththejetimpingementfacilityusingonlyairexpanding throughthenozzle,waterowiscutoff.Theexperimentalprocedureisessentially identicaltothatforthetwo-phasejetexpecttheappliedheatuxisreduced.Theheat uxesusedduringexperimentsarelowerthanthoseinthetwo-phaseexperimentto ensurethattheheaterdoesnotover-heatandde-laminatefromtheGarolitebase. Table3-1displaysthecorrespondingheatuxesforagivenReynoldsnumber. 3.3ExperimentalResults 3.3.1UncertaintyAnalysis UncertaintyanalysisfortheexperimentsisdoneusingthemethodofKlineand McClintock[54].TheuncertaintyofthecalculatedNusseltnumberrangesfrom2.0to 4.0%forthesingle-phasejetand2.5to18%forthetwo-phasejetnearthecenterline 62


Table3-1.Reynoldsnumberandcorrespondingheatuxes. Nominallowestmidhighest Re D HeatFluxHeatFluxHeatFlux kW/m 2 kW/m 2 kW/m 2 4.5 10 5 355065 7.3 10 5 6080100 1.0 10 6 80100120 and0.3to1.0%attheouterextentsofthedomain.TheReynoldsnumberuncertainty rangesfrom2to3%anddoesnotvaryappreciablebetweenthesingleandtwo-phase experiments.Theuncertaintyinthetemperaturemeasurementsare0.2 C. Tohelpascertaintheerrorintheexperimentsbyassumingthatthethreedimensional effectswereneglected,anumericalanalysiswasconducted.Thisanalysisusesa secondorderaccuratenitedifferenceschemetosolveEquation3onathree dimensionalgridof2 n 2 n 2 n wheren=4,5,6,and7.Thetopboundaryismodeled ashavingaNusseltnumberdistributionfoundintheexperimentsandtheremaining sidesaremodeledasbeinginsulated.AftercompletionofeachsimulationtheNusselt numberiscalculatedusingEquation3.Tocharacterizetheerrorinusingaone dimensionalassumptiontheroot-mean-squareerroriscalculatedviathefollowing equation error rms = v u u u u u u t L R 0 W R 0 Nu exp )]TJ/F39 11.9552 Tf 11.956 0 Td [(Nu sim 2 dxdy L R 0 W R 0 Nu exp 2 dxdy theuseofthedifferentgridsizesallowstheextrapolationoftheerrorusingRichardson's extrapolationmethod[55,56].UsingthehighestNusseltnumberdistributionfound duringtheexperiments,resultsinanrmserrorof1.21%withapeakerrorof3%located neartheorigin,whileusingthelowestNusseltnumberdistributionresultsinanrmserror of0.83%withapeakerrorof3.25%locatedneartheedgeofthedomain.Theseerrors 63


Figure3-5.Measuredsingle-phaseNu D spatialvariationatdifferentheatuxes. arelessthantheuncertaintycontainedintheexperimentalmeasurementsandthusthe one-dimensionaltreatmentforevaluatingtheNusseltnumberisdeemedsatisfactory. 3.3.2Single-PhaseResults Duringthecourseoftheexperimentsitwasobservedthattheheattransfer coefcientisindependentofheatux,asexpected.Resultsforatypicalexperiment areshowninFigure3-5.Toillustratetheamountofuncertaintyinthedataerrorbars havebeenincludedinthisgure.However,tofacilitateeaseofviewingtheyarenot shownintherestofthisSection. Figure3-6showsthemeasuredthermocoupletemperatureprolesatvariousheat uxesforanozzlespacingof4nozzlediametersforasingle-phaseexperiment.Note thattheadiabaticwalltemperaturecorrespondstothatmeasuredwithzeroapplied heatux.Figure3-7showsthatNusseltnumberscaleswith p Re D asreportedby 64


Figure3-6.Spatialheatertemperaturevariationatdifferentappliedheatuxes. Donaldsonetal.[46]andRahimetal.[51],amongothers.Asamatterofreference, asingle-phaseNusseltnumberontheorderof2,500correspondstoaheattransfer coefcientontheorderof11,000W/m 2 -Kinthepresentstudy.Asinglenozzlewasused intheexperimentsandthustheover-expansionpressureratioandReynoldsnumber arenotindependentofeachotherandtheeffectsofover-expansionratiocouldnotbe isolated. Figure3-8comparesthelocalNusseltnumberfornozzleheightsandReynolds numbersusedduringtheexperiments.Forr/D > 0.5andH/D > 2theNusseltnumber distributionisnotstronglydependentonthenozzleheight.However,forr/D < 0.5there isasmallvariationinNusseltnumber.Thisbehaviorresultsfromthecomplexshock structureatthenozzleexitanditsinteractionwiththeheatersurface.ForH/D 2 Nusseltnumberisslightlyelevated,butthiseffectappearstolessenathigherReynolds 65


a b Figure3-7.SpatialvariationofNu D atdifferentRe D ,aunscaledandbscaled. numbers.Apossibleexplanationforthisbehavioristhatduetothelowtemperatureof theowingair,thenozzlebecomescooled.Thiswillcausemoistureinthesurrounding airtocondenseonthenozzlewhichcanbecomeentrainedinthejet.Thisentrained moisturewillincreasetheamountofheatremovedfromthesurfaceoftheheaterand thuselevatethemeasuredNusseltnumber.Tocombatthisissuethenozzleisinsulated tothebestextentpossible.Withtheaddedinsulation,itisbelievedthatthemoisture condensationeffectsareminimalbut,non-zero. 66


a b c Figure3-8.Single-phaseNu D atvariousnozzleheighttodiameterratios,aRe D =4.57 10 5 ,bRe D =7.55 10 5 ,andcRe D =1.05 10 6 3.3.3Two-PhaseResults Thetwo-phasejetexperimentsareperformedinthesamemannerasthe single-phasejetwiththeexceptionofwaterbeingaddedtotheair-stream.Inorder toquantifytheeffectofthewaterontheheattransferproperties,themassfractionof waterinthejetiscalculatedas w = m l m l +_ m air 67


Figure3-9.Measuredtwo-phaseNu D spatialvariationatdifferentheatuxeswithoutice formation. Ingeneral,thetwo-phaseheattransfercoefcientisfoundtobeindependentof heatux.However,aspreviouslymentioned,whentheheatuxattheheatersurface istoolow,iceformationaffectstheheattransfermeasurements.Inordertocombat iceformationaminimumheatuxof315kW/m 2 isused.Nevertheless,therearea fewcaseswhereicingisobservedinheatuxesup350kW/m 2 .Toidentifyandhelp mitigatetheseeffects,themeanandstandarddeviationoftheheattransfercoefcient asafunctionofspaceistaken.Whenthestandarddeviationoftheexperimentalvalues exceeded20%,thenheatuxesof470,430,and390kW/m 2 areusedintheaveraging calculations.Theseheatuxesareselectedbecausethehigherheatuxeswillresult inhighersurfacetemperaturesandinhibiticeformation.Alsothehighertemperatures willresultinalarger Tandlessuncertaintyinthecomputedheattransfercoefcient. Approximately20%ofthemeasurementstakenrequirethesecorrectivemeasures, andinallcasestheresultingstandarddeviationislessthan20%ofthemean.See Figure3-9foranexampleofanexperimentwheretheheattransfercoefcientisclearly independentofheatuxandFigure3-10whereareductionintheheatuxusedwas necessary. 68


a b Figure3-10.Measuredtwo-phaseNu D spatialvariationatdifferentheatuxes,aice effectspresentandbafterremovaloflowestheatuxes. Figure3-11showstheradialvariationofmeasuredthermocoupletemperaturefor variousheatuxesforatwo-phaseexperiment.Notethatthezeroheatuxcondition representstheadiabaticwalltemperature.Figure3-12showstheradialvariationof Nusseltnumberforthetwo-phasejetatdifferentwatermassfractionsandconstant Reynoldsnumberandnozzleheight,Figure3-13showsthevariationwithnozzleheight withaconstantReynoldsnumberwithanominallyconstantmassfractionofliquid.Note thatitisnotpossibleinthecurrentstudytovaryReynoldsnumberandtheliquidmass 69


Figure3-11.Spatialheatertemperaturevariationatdifferentappliedheatuxes, two-phasejetresults. fractionindependentlyofeachother;thusitisnotpossibletoshowhowtheNusselt numberscaleswithReynoldsnumber. NusseltnumbergenerallyincreaseswithincreasingReynoldsnumberand increasingwatermassfractionneartheinteriorofthejet.Forr/D 1.5theredoes notappeartobeanoticeabledependenceofNusseltnumberonthenozzleheight. ThereissomevariationofNusseltnumberwithnozzleheightinthejetinterior,buta denitetrendisnotapparent.Forreferencepurposes,atwo-phaseNusseltnumber ontheorderof2,000correspondstoaheattransfercoefcientontheorderof200,000 W/m 2 -K.MoreexperimentalresultsthanthosepresentedinthisChapterarepresented inAppendixA. 70


a b c Figure3-12.Two-phaseNu D atvariousliquidmassfractions.aZ/D=2.0,Re D =4.42 10 5 ,bZ/D=6.0,Re D =4.45 10 5 ,andcZ/D=6.0,Re D =7.24 10 5 Heattransfercoefcientsexceeding400,000W/m 2 -KareobservedinFigure3-13c, whichareonthesameorderasthehighestliquidjetheattransfercoefcients,see[2], totheauthor'sknowledge.Whilethereismoreexperimentaluncertaintyatthesehigh heattransferratesto18%,theefcacyofthetwo-phasejetforhighheattransfer applicationsisclearlydemonstrated. Itisbrieynotedthattheoriceforthe0.37mmoricehadadefectandhenceis notperfectlycircular;theliquidowratedeliveredwaslessthanthatforthe0.33mm orice.TheNusseltnumberresultsforthe0.37mmoricearenoticeablysmallerthat 71


a b c Figure3-13.Two-phaseNu D numberatvariousnozzleheighttodiameterratios.aw= 0.0375,Re D =4.42 10 5 ,bw=0.0273,Re D =7.23 10 5 ,andcw= 0.0248,Re D =1.01 10 6 thatofthe0.33mmoriceanddonotfollowtheexpectedtrend.Thisisbelievedtobe duetotheeccentricityoftheoricecausingdifferentbehaviorinthemixingchamber andeffectingtheresultingdropletsize/distributionatthenozzleexit.Whilethe0.51 mmoricedoeshavesomeeccentricity,itisnotassevereasthatfoundinthe0.37 mmorice,anditdoesnotseemtohaveanoticeableeffectontheNusseltnumber measurements.PicturesofeachoriceareshowninAppendixC. 72


3.3.4EvaporationEffects Tohelpquantifytheeffectofevaporationontheheattransfercoefcientthe saturatedhumidityratioattheimpingementsiter=0andattheedgeofthemeasurement locationr=50.8mmiscarriedout.Thesaturatedhumidityratioiscalculatedfrom sat =0.622 P v sat P )]TJ/F39 11.9552 Tf 11.955 0 Td [(P v sat Thevaporsaturationpressure,P v,sat iscalculatedfrom[57] P v sat =exp 647.096 T )]TJ/F22 11.9552 Tf 9.298 0 Td [(7.85951783 v +1.84408259 v 1.5 )]TJ/F22 11.9552 Tf 11.955 0 Td [(11.78664977 v 3 +22.6807411 v 3.5 )]TJ/F22 11.9552 Tf 11.955 0 Td [(15.9618719 v 4 +1.80122502 v 7.5 where v =1 )]TJ/F39 11.9552 Tf 28.595 8.088 Td [(T 647.096 ThasunitsofKelvin,PhasunitsofPascals,andvisnon-dimensional.Atthejet impingementzonethetemperatureisontheorderof10 Candthepressureis approximatelythestagnationpressure.Resultsforthesaturatedhumidityratiofor thethreeseparatestagnationpressuresusedareallontheorderof10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 ;thusany effectsduetoevaporationnearthecenterlineareconsiderednegligible. Thepressureattheedgeoftheheateraswellasthetemperatureatthesurface oftheliquidlmareunknownandasimilaranalysiscannotbeperformed.However, novisualobservationofphasechangeatthehighesttemperaturesseenduring experimentsisseen.ItisobservedinFigures3-12and3-13,thatNusseltnumber remainsessentiallyconstantneartheedgeoftheheater.Evaporationwouldfurther enhanceheattransferresultinginanincreaseinNusseltnumberinthisregionthusitis believedthatevaporationislikelynegligibleinthisareaaswell. 73


3.4ComparisonbetweenSingleandTwo-PhaseJets Togainanappreciationofthetwo-phasejetheattransferenhancement,the measuredheattransfercoefcientiscomparedtothethatforthesingle-phasecase. Theheattransferenhancementfactorisdenedhereas = h mix h air a b c Figure3-14.Heattransferenhancementratioatvariousliquidmassfractions.aZ/D= 2.0,Re D =4.42 10 5 ,bZ/D=4.0,Re D =7.35 10 5 ,andcZ/D=2.0, Re D =1.01 10 6 74


Figure3-14showtheheattransferenhancementwiththevariationinwatermass fractionandReynoldsnumber.Itisobservedthattheenhancementincreaseswith increasingmassfraction;theincreaseisdiminishedwithincreasingReynoldsnumber. ItisobservedthatinFigure3-14thatforr/D < 0.5thereisamarkedincreaseinthe measuredheattransferenhancement.AthigherReynoldsnumbersthemaximum enhancementoccursneartheedgeofthejetr/D=0.5. ItisobservedinFigure3-15thatthevariationoftheheattransferenhancementwith nozzleheightissimilartothatforwatermassfraction,itshowsanincreasingtrendat lowerReynoldsnumbersbuttheeffectisdampedathigherReynoldsnumbers. 3.5Discussion Oneofthefeaturesapparentinallexperimentsisthatatradialdistancesof1.5to2 nozzlediameterstheNusseltnumberandheattransferenhancementratiosapproacha nearconstantvaluewhichisindicativeoflmowheattransfer.Variousstudies[34,40] havenotedthatthereisalargeadversepressuregradientinthisregionwhichlikely causesboundarylayerseparation[35].Duringthecourseofthepresentexperiments, tworingsindicativeofaseparationregionareobservedontheheatersurface.Oneof whichcorrespondstotheedgeofthenozzlewherethereisashockwavepresent,and theotherislocatedapproximately1.5nozzlediametersfromthejetcenterline.Insideof theseregiontheheattransfercoefcientisaffectedbynozzleheightandthewatermass fractionindicatingthatjetimpingementisthedominatingheattransfermechanismfor r/D < 1.5. Alloftheexperimentsreportedarecarriedoutatrelativelylowsurfacetemperatures; thehighesttemperatureisontheorderof70 C.Phasechangeduetomasstransferof theliquidintotheimpingingair-streamisconsiderednegligibleforreasonsdiscussed inSection3.3.4.AdditionallytheworkofBenardinandMudawar[58]explorethe Leidenfrostmodelforimpingingdropsandsprays.Theirmodelpredictsthepressurein dropletsusingone-dimensionalelasticimpacttheory[59],andacorrectionfactordueto 75


a b c Figure3-15.Heattransferenhancementratioatvariousnozzleheighttodiameterratios. aw=0.0205,Re D =4.54 10 5 ,bw=0.0290,Re D =7.24 10 5 ,andc w=0.0233,Re D =1.02 10 6 Engel[60,61]givesgoodresults.Thepressureriseattheimpingementsurfacecanbe modeledas P =0.20 l u o u snd whereu o isthedropletvelocityandu snd isthespeedofsoundintheliquid.Using thisequationitisseenthatforanydropletvelocityaboveapproximately7%ofthe speedoftheairintheimpingingjetapproximatelyMach3willyieldsurfacepressures 76


abovethatofthecriticalpressureforwater.Thusevenifphasechangeoccursatthe impingementpoint,thelatentheatofvaporizationiszeroandnoenhancementofheat transferwilloccur.Becauseofthecomplexshockstructuresoccurringwhenthejet impingesontothesurface,makingsimilarargumentsforregionsfarremovedfromthe impingementzonearenotreliableandthusarenotattempted.However,itisnotedthat mostoftheliquiddropletsimpingingontothesurfacewilloccurnearthecenterline;thus thepressurefarremovedfromthejetcenterlinewillbelowerandevaporationmaystill bepossibleatelevatedsurfacetemperaturesandheatuxes. Oneofthecurrentlimitationsofthecurrentresultsistheylackinformationonthe liquiddropsizedistribution.Suchmeasurementsarenotavailableatthecurrenttime andfutureworkisplannedtoaddressthisdeciency. Thecurrentheattransfermeasurementsarecomparedtothesingle-phaseliquidjet heattransfermeasurementsofOhetal.[2],andthemeasuredheattransfercoefcients areonthesameorderofmagnitude.Theliquidowrateinthecurrentexperimentsis verysmallwhencomparedtothoseexperiments,upto0.7g/s.5g/m 2 s,referenced toheatedareacomparedwith4.3kg/s.55 10 6 g/m 2 s,referencedtoheated area,afeaturewhichhassignicantindustrialadvantages.Inthestudyperformedby Ohetal.,liquidtovaporphasechangeisnotobserved,andthoseexperimentswere performedatmuchhigherheatuxesupto30timestheheatuxesreportedinthe currentstudy.Futureinvestigationswillexplorehigherheatuxregimes. 3.6Summary InthisChapterheattransferenhancementmeasurementsusingtwo-phase overexpandedsupersonicimpingingjetswerepresentedforawiderangeofReynolds numbers.Thesejetstwo-phasejetsaregeneratedbytheadditionofwaterdroplets upstreamofaconverging-divergingnozzle.Heattransfermeasurementsusinga single-phasejetisusedforcomparison.Itisobservedthattheadditionofwaterdroplets intotheairowsignicantlyenhancestheheattransferrate.Enhancementissignicant 77


nearthejetcenterlinetheenhancementfactorexceeds10inmostcases.Themass fractionofwateraddedtothejetisobservedtobyanimportantparameterforheat transfer,generallyincreasingNusseltfollowsincreasingwatermassfraction.However, itsinuencediminishesathigherReynoldsnumbers.Nozzleheightappearstohavea smallimpactontheobservedheattransferrates. 78


CHAPTER4 DETERMINATIONOFHEATTRANSFERCOEFFICIENTUSINGANINVERSEHEAT TRANSFERANALYSIS AswasseeninChapter3theuseofsteadystatemeasurementtechniquesyields lowsurfacetemperatureswhicharenotsuitableforevaporatingtheliquidlmandthe low Tbetweenthewallandliquidlmwhichcreatesuncertaintyinevaluatingtheheat transfercoefcient.Inordertoalleviatesomeoftheseeffectsatransientapproach involvinganinverseheattransferquenchingproblemisdeveloped. 4.1InverseProblems Therearetwobasicparadigmsinheattransfer.Themostwellknownparadigm isthesolutionofthetemperatureeldwithinamediumsubjecttoconstraintssuch asagiventhermalconductivity,thermaldiffusivity,andknownboundaryconditions. Ifalloftheconstraintsareknown,thentheresultingtemperatureeldwithinthe mediumofinterestcanbesolved;inmanycasesananalyticalsolutioncanbe determined.However,iftheseconditionsarenotknownthentheproblemisnotunique, isunder-specied,andnosolutioncanbedetermined. Thesecondparadigminheattransfer,namedinverseheattransfer,iswhen thetemperatureatspecicpointsinsideofamediumareknownandaconstraint needstobedeterminede.g.contactresistancebetweentwosurfacesoraboundary condition.Becauseofunavoidablemeasurementerrorsinthetemperatureeldthis problemisill-posedandcanbedifculttosolve.Thedifcultiesofthisproblemcanbe circumventedinveryspecialcircumstances,forexampleifonedesirestodetermine theheatuxappliedataboundaryofaone-dimensionalbaratsteadystateonecan ensembleaveragetemperaturemeasurementsatafewlocationsalongthelengthofthe baranddeterminethetemperaturegradientvialinearregression.Withaknownthermal conductivity,Fourier'slawcanbeusedtodeterminetheappliedheatuxandtheresults canbequiteaccurate.Unfortunatelythesesimpleproblemsdonotcomeaboutin 79


practiceoften.Forexampleiftheheatuxvariesintime,thentheabovemethodwould notbeapplicableandadifferentmethodwouldbeneeded. InverseProblems,ingeneral,fallintooneoftwocategories:parameterestimation, inwhichoneormoredesiredparametersaredeterminedusingexperimentaldata e.g.thermalconductivityofasolidandaappliedheatuxandfunctionestimation,in whichadesiredfunctionistobeestimatedusingexperimentaldatae.g.aboundary conditionwhichvariesinspaceandtime.Itshouldbenotedthatmanyfunction estimationproblemscanbeformulatedintermsofaparameterestimationproblemifthe functionalformoftheofdesiredfunctionisknown,forinstanceifthermalconductivity isaquadraticfunctionoftemperaturetheproblemcanbereducedtodeterminingthe coefcientsofthegoverningequation.Thisapproachcanyieldgoodresults,seeFlach and Ozisik[62]forexample,however,iftheformoftheequationisnotknown apriori thenthisapproachmaynotbeuseful. InverseProblemsandInverseHeatTransferIHTproblemshavebeenstudied extensivelyintheliteratureandhavebeeninusesinceatleastthe1950's.Tikhonov, [6365]amongothers,wasoneofthersttotacklethechallengeofinverseproblems andtakeintoaccountmeasurementerrors.HistechniquetitledTikhonov'sregularization minimizedtheleastsquareerrorbyaddingaregularizationtermthatpenalizes unwantedoscillationsintheestimatedfunction.Tikhonov'smethodcanberelated todampedleastsquaresmethods,mostnotablythemethodduetoLevenberg[66] andMarquardt[67],knownastheLevenberg-Marquardtmethod.Thesemethodsare onlysuitableforparameterestimation.Stoltz[68]usedafunctionestimationtechnique basedonDuhamel'sprincipleandtwosimultaneousthermocouplemeasurementsto determinethesurfaceheatuxinaone-dimensionalproblem.Thisprocessisknown asexactmatchinganddoesnottakeintoaccountanymeasurementerrors.Beck [6971]usedamethodsimilartothatbyStotlz;however,temperaturesatfuturetimes areusedtoprovideregularizationandreduceinstabilitiesinthemethod.Thismethod 80


canbeusedforparameterorfunctionestimationbut,canbecomeunstableforsmall timestepsandthushighlytransientphenomenacannotbeaccuratelyreproduced. TheMonteCarlomethodcanbeusedtoestimateaparameterorfunctionaswas demonstratedbyHaji-SheikhandBuckingham[72];agoodreviewofthetechniquecan befoundin[73].Amethodthatissuitableforsmalltimestepsandperformsparameter orfunctionestimationisAlifanvo'sIterativeRegularizationMethod[74].Thismethodis alsoknownasparameter/functionestimationwiththeadjointproblemandconjugate gradientmethod,andisthemethodusedforthepresentstudy.Thismethodwillbe abletodetermineatimeandspacevaryingheattransfercoefcientproducedbya multiphasesupersonicimpingingjet,aswellasanytemperaturedependenceduetoany evaporationoftheliquidlm. 4.2IntroductiontoInverseProblemSolutionUsingtheConjugateGradient MethodwithAdjointProblem Dealingwithinverseproblems,whichbytheirnatureareill-posed,usuallyinvolves sometypeofregularizationtechniqueoranoptimizationtechniquewhichinherently regularizesthesolution.Thetechniqueusedinthecurrentstudyisanoptimization techniqueknownasfunctionestimationusingtheconjugategradientmethodwith adjointproblem.Asthenameimpliesthismethodusestheconjugategradientmethod tominimizetheerrorintheleastsquaressensebetweentheestimatedoutputofan equation/systemofequationsandthemeasuredoutputwhichhasbeencorruptedwith noise.Themethodwillbedescribedbelowinitsgeneralformtofamiliarizethereader. Manyreferencesexistforfunctionalestimationwiththeadjointproblemandconjugate gradientmethodincluding Ozisik[75], OzisikandOrlande[76],Alifanov[74],andthe ChapterbyJarny[77].MuchofthefollowinganalysisfollowsthatofJarnyastheauthor foundthatparticularreferencetobemathematicallyrigorous,thorough,generalin nature,andeasytofollow.Specicimplementationsofthismethodwillbediscussed whereneeded. 81


4.2.1TheDirectProblem Thedirectproblemisthemodelequationsforthesystemofinterest.Itcanbe analgebraic,integral,ordinarydifferential,orpartialdifferentialequationorsystemof equationsorsomecombinationtherein. y x t = f x t whereyistheoutputofthesystem, istakentobeaparametersorfunctiontobe estimatedandxandtaretheindependentvariables.Notethatalthoughbothspace andtimeareindependentvariablesinthisexampleitisnotnecessaryfortheoutputto dependonbothofthem. 4.2.2TheMeasurementEquation Themeasurementequationexistsduetothediscretenatureofasamplingprocess andduetochangesbroughtaboutindataduetosensordynamics.Althoughinmodern dataacquisitionsystemsitispossibletomeasurequantitiesatanearcontinuousrate takingmeasurementsstillisaninherentlydiscreteprocess.BendatandPiersol[78] havewrittenagoodreferenceondatameasurementandanalysiswhichincludes sensor/systemdynamics. Sensordynamicscangreatlyeffectthemeasurementsofasystemandtheireffects canbequitesignicant.Thisprocesscanbesimpliedifthesensordynamicscanbe approximatedbyalineartimeinvariantLTIsystem,whichmostsensorsfallunder.In anLTIsystemtheoutputofasensoristheresultofaconvolutionofitsinputwiththe sensor'simpulseresponsefunction.Theimpulseresponsefunctionistheresponseof thesensor,initiallyatrestorzero,toanimpulseinput.Themeasurementequationcan bemathematicallyexpressedas Y m = t Z 0 h t )]TJ/F25 11.9552 Tf 11.955 0 Td [( y d 82


whereY m isthemeasuredoutput,histheimpulseresponsefunction,andyisthetrue outputofthesystem.Ifthesensorisperfectandthegoalistosimplydenoteitsdiscrete naturetheimpulseresponsefunctionwouldsimplybeadeltafunction.Foreaseof viewingthemeasurementequationcanalsobediscussedinanoperatorformsuchthat Equation4isequalto Y m = Cy 4.2.3TheIndirectProblem Theindirectproblemisactuallythestatementoftheleastsquarescriteria.When solvinganinverseproblemwiththecurrentmethodtheparameterorfunctionsoughtis theonewhichminimizestheleastsquarescriteria.Simplystatedtheindirectproblemis S = M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t ] 2 dt whereSistheintegratedsquaresnotethatinthecaseofdiscretedatathiswould bethesumofsquares,iisthesensornumber,andMisthetotalnumberofsensors. ThespatialdependenceofyisleftoutofEquation4becauseitisassumedthat thesensorsareplacedatvaryingdistancesinspace,thusthemeasurementoperator, C i wouldonlyoperateonmeasurementsatalocation, x i .Itcanbeusefultothinkof theleastsquarescriteriaintheformofanormoperator, k u k orsometimes h u v i ,for instanceEquation4isequalto S = k Y m )]TJ/F39 11.9552 Tf 11.956 0 Td [(Cy t k 4.2.4TheAdjointProblem Formulatingtheadjointproblemcorrectlyisacrucialstepinthesolutionprocess. Essentiallythisiswheretheoptimizationportionoftheproblemcomesintoplay.Todo 83


thistheindirectproblemisconsideredthemodelingequationandthedirectproblemis consideredasaconstraintsuchthatthefollowingequationholds, R y = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(f x t ThesearethenjoinedtogetherthroughtheuseofaLagrangemultiplier. L y = k Y m )]TJ/F39 11.9552 Tf 11.955 0 Td [(Cy t k)-222(h R y i whereListheLagrangianvariableand istheadjointvariablealsoknownasthe Lagrangemultiplierwhich,ingeneral,canbeafunctionofspaceandtime. Whenthecorrectparameter/function isinsertedintoEquation4theresulting Lagrangianiszeroforperfectmeasurements.Realworldmeasurementswillbe corruptedwithnoiseandtheresultingLagrangianwillbetheminimumleastsquares criteria. Todeterminetheproper theLagrangianmustbeminimized.Iftheadjointvariable istreatedasxedthethedifferentialoftheLagrangianis dL = hr y S y i)-222(h r y R y y i)-222(h r R y i orexpressedinamoreconvenientform dL = hr y S )]TJ/F25 11.9552 Tf 11.955 0 Td [( [ r y R y ] y i)-222(h [ r R y ] i Becausethechoiceoftheadjointvariableisnotconstraineditischosentobethe solutionof r y S )]TJ/F25 11.9552 Tf 11.955 0 Td [( [ r y R y ] =0 84


Equation4isknownastheadjointequation.Notethatthisisimplicitin throughmathematicaloperationusuallyinvolvingintegrationbypartsitcanbe expressedasanexplicitfunctionoftheadjointvariable. 4.2.5GradientEquation NotethattheadjointequationrenderstherstterminEquation4tobezero. Atthesolutionpointwhere isequaltothetruevalue,theLagrangianisequaltothe minimumoftheleastsquarescriteria dL = dS = hr S i andcomparingtheremainderofEquation4toEquation4thegradient equationresults r S = )]TJ/F25 11.9552 Tf 9.299 0 Td [( [ r R y ] Thegradientequationisusedintheconjugategradientminimizationalgorithmto determineastepsizeanddescentdirectioninordertominimizetheleastsquares criteria. 4.2.6SensitivityEquation Asmentionedoneoftheparametersneededtondtheminimumoftheindirect problemisthestepsize.Thisparametercantakeafewdifferentformsdepending onwhethertheinverseproblemislinearornon-linear.Intheinterestofpresentinga generalmethod,theformfornon-linearproblemsarepresented. Thestepsizetobedeterminedisaperturbationintheparameter/function ,which istobeestimated.Toderivethisquantitywesimplyperturbthedirectproblem y + y = f x t + ; generallytherighthandsideofEquation4islinearizedsuchthat 85


y + y = f x t + f x t WhenEquation4issubtractedfromtheaboveequationthesensitivityequationis theresult y = f x t NotethatthesecondterminEquation4andtherighthandsideofEquation4 containboth and .Inverseproblemsinwhichthesensitivityequationcontainsboth parameters/functions and arenon-linear.Notallinverseproblemsarenon-linearin nature,andthisformisusedhereforthesakeofgenerality.Thesensitivityequation simplystatesthataperturbationintheparametertobeestimatedwillresultina perturbationofthecomputedoutput. 4.2.7TheConjugateGradientMethod Theconjugategradientmethodisanoptimizationproblemforsolvinglinearor non-linearequations.Thereareseveralreferenceswhichdetailthemathematicsbehind thistool,forinstancethebooksbyRao[79]andFletcher[80],amongothers.Assuch readersinterestedinarigorousderivationofthemethodareencouragedtoconsult thesereferences. Theessentialstepsofthemethodarethataguessfor ischosen,theabove equationsaresolvedandasearchdirection, d whichisC-conjugatetotheprevious directioniscalculatedusingaconjugationcoefcient, .Thesearchdirectionisthen multipliedbythestepsize, andisaddedtothepreviousguessfor .Thisiterative processcontinuesuntiltheerrorbetweenthemeasuredoutputandcalculatedoutput reachesapredeterminedtolerance. Thereareseveraldifferentformsoftheconjugationcoefcient, intheliterature suchastheHestenes-Stiefel[81],Polak-Ribi ere[82],andFletcher-Reeves[83],among others.Allofthementionedformsareequivalentforlinearequations;howeveritis 86


discussedintheliterature[84,85]thatthePolak-Ribi ereformoftheequationhasbetter convergencepropertiesfornon-linearequationsandassuchwillbeusedinthepresent studyunlessotherwisenoted.ThePolak-Ribi ereformoftheconjugationcoefcientis k = M P i =1 hr S k r S k )-222(r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 i kr S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k for k =1,2,3... and k =0 for k =0 Thesearchdirectionisthencalculatedbythefollowing k = r S k + k k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 Thestepsizeisdenedbythefollowing =argmin [ S )]TJ/F25 11.9552 Tf 11.955 0 Td [( ] =argmin M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ] 2 dt TheoutputofthedirectproblemisexpandedinaTaylorseriesas y t )]TJ/F25 11.9552 Tf 11.956 0 Td [( y t )]TJ/F25 11.9552 Tf 11.955 0 Td [( @ y t @ y t )]TJ/F25 11.9552 Tf 11.955 0 Td [( y t andsubstitutingtheaboveintoEquation4yields =argmin M X i =1 t f Z 0 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i y i t + C i y t ] 2 dt Performingthedifferentiationwithrespectto ,settingtheresultequaltozeroand solvingfor yieldsthenalformoftheequationforthestepsize 87


k = M P i =1 t f R 0 C i y i x t k )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t C i y i t k dt M P i =1 t f R 0 [ C i y i t k ] 2 dt Itisnotedthattheaboveequationcanbesimpliedforlinearproblemswhenthe leastsquarescriteriatheindirectproblemiscastinquadraticform.However,forthe sakeofgenerality,theaboveequationwillbeusedthroughoutthecurrentChapter. 4.3FactorsInuencingInverseHeatTransferProblems Thereareseveralfactorswhichcaninuencethesolutionofaninverseproblem. Someofthesefactorsarediscussedbelow. 4.3.1BoundaryConditionFormulationEffects Toperformafunctionestimationinverseproblemtodetermineaspatiallyand temporallyvaryingheattransfercoefcient,thechoiceoftheboundarycondition formulationisveryimportant.TheboundaryconditioncanbeformulatedasaDirichlet speciedtemperatureboundaryconditionwherethesurfacetemperatureisdetermined andtheresultingheatuxiscalculatedinordertodeterminetheheattransfer coefcient,asaNeumannspeciedheatuxboundaryconditionwheretheheat uxisdeterminedandtheresultingsurfacetemperatureiscalculated,orasaRobin convectiontypeboundaryconditionwheretheheattransfercoefcientisdirectly determined.Atrstglance,theRobintypeboundaryconditionseemstobethebest choiceastheunderlyingphysicstakingplaceareconvectiveinnature.Uponfurther analysisthisisactuallytheworstchoice.Inordertodemonstratethisanexample usingaone-dimensionalheattransferproblemwithatimevaryingheattransfer coefcientwillbeusedbecauseoftheeaseofcalculation.Thesameconceptsapply toatwo-dimensionalproblemwithaspatiallyandtemporallyvaryingheattransfer coefcient. Thefollowinganalysiscanbefoundin[86]butisreproducedhereforclarityandto correctsomeerrorscontainedtherein.Supposeatimevaryingboundaryconditionis 88


Figure4-1.1-dimensionalsolidforthesensitivityproblem. appliedtothex=0surfaceofaonedimensionalsolidwithaninsulatedboundaryatx= L,seeFigure4-1.ThesolutionofthisproblemcanbedeterminedbyusingDuhamel's principleforeachtypeofboundaryconditionpreviouslydiscussed.Essentiallythetime varyingboundaryisconvolvedwiththeimpulseresponsefunctionoftheslab.The impulseresponsefunctionisdeterminedbysolvingtheheatequationforthesolidwitha boundaryconditionofunityastepresponsefunctionandthentakingthederivativeof thatfunctionwithrespecttotime.Forinstancethesolutionforatimevaryingheatuxin dimensionlessformis x t = o + Z t 0 q @ q x t )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ t d where q indimensionlessformis: q x t = t + 1 3 + x x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 13.547 8.088 Td [(2 1 X n =1 1 n 2 exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F39 11.9552 Tf 9.299 0 Td [(n 2 2 t cos n x 89


Inordertoanalyzewhichtypeofboundaryconditionshouldbeusedintheinverse problemformulation,asensitivityanalysisshouldbeperformed.Thesensitivity analysisisaccomplishedthroughtheuseofrelativestepsensitivitycoefcients. Firstthegoverningequationsandboundaryconditionsarenon-dimensionalized andtheirsolutionobtained.Thederivativeofthissolutionistakenwithrespectto thenon-dimensionalinputparameterfortheboundaryconditionnon-dimensional temperature,heatux,orBiotnumber.Theresultoftheseoperationsisthestep sensitivitycoefcient,althoughthemagnitudeofthecoefcientfortheconvectioncase variesdependingonthemagnitudeoftheinputBiotnumber.Toallowdirectcomparison ofthesesensitivitycoefcientstheyaremultipliedbytheirboundaryconditioninputs transformingthemtorelativestepsensitivitycoefcients,denotedasX input .With non-dimensionalizationoftheproblem,theneteffectisonlyseenintheconvectioncase. X q x t = q x t a X x t =1 )]TJ/F22 11.9552 Tf 13.547 8.088 Td [(2 1 X n =1 1 n )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(1 2 sin n )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(1 2 x exp )]TJ/F30 11.9552 Tf 11.291 16.856 Td [( n )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(1 2 2 2 t # b 90


X Bi x t = Bi @ @ Bi = Bi 1 X n =1 exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 n t n @ C n @ Bi cos n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x )]TJ/F39 11.9552 Tf -229.465 -35.714 Td [(C n @ n @ Bi [2 n t cos n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x + )]TJ/F39 11.9552 Tf 11.955 0 Td [(x sin n )]TJ/F39 11.9552 Tf 11.955 0 Td [(x ] o where @ n @ Bi = 1 tan n + n sec 2 n and @ C n @ Bi = @ n @ Bi 4 cos n 2 n + sin n )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(8 sin n [ 1+ cos n ] [ 2 n + sin n ] 2 n tan n = Bi c NotethatEquation4cdependsontheinputparameterBiandthustheinverse problemisnon-linearinnatureandcanbedifculttosolve.Alsonotethatthisequation isdifferentthanthatfoundin[86];theequationinthatreferencecontains x asopposed to 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x Aplotofthesensitivitycoefcientsat x =0.1 ispresentedinFigure4-2;itshould bementionedthatthemagnitudesofthesensitivitycoefcientsareplottedandthe coefcientsforBiareactuallynegativeandthisisnotclariedfortheplotinReference [86].Afewpointsofinterestsshouldbepointedout.First,notethatthecoefcients forBiarelowerthanalloftheothersandthatasBinumberincreasesitssensitivity coefcientdecreases.Thecoefcientsforatemperatureandheatuxinputaremuch largerthanthoseforBinumberwiththecoefcientsfortemperaturebeinglargerthan thoseforheatuxuntilavalueof t 0.75 .Clearlyaninverseproblemformulatedin termsofanunknownconvectioncoefcientisnotagoodchoice. 4.3.2SensorLocationEffects Thesensitivitycoefcientsalsodependonposition.Thesensitivitycoefcients foraheatuxinputareplottedinFigures4-3and4-4.Itiseasilyseenthatthecloser 91


Figure4-2.Relativestepsensivitycoefcientsatx =0.1asafunctionoftime. atemperaturesensorisplacedtotheboundaryofinterestthemoresensitiveitis tochangesofthatboundarycondition,asonewouldexpectfromsimplephysical reasoning. ComparingFigures4-3and4-4onecanseethatthemagnitudeoftherelative stepsensitivitycoefcientislargeratthebackwallforanunknownsurfacetemperature formulationthanforanunknownheatuxformulation.Thischaracteristicwillbe exploitedinthisstudyinordertominimizetheeffectsofthermocoupleinsertiononthe inverseproblem. 4.3.3ThermocoupleInsertionEffects Inordertoperformtemperaturemeasurements,solidthermocouplesarecommonly usedastheyarearobust,inexpensivemethodtothemeasurement.Aswasdemonstrated inSection4.3.2,theclosertothesurfaceofinterestasensorisplaced,better sensitivitiestochangesintheboundaryconditionareachieved.Thiscanbeaccomplished 92


Figure4-3.Relativestepsensivitycoefcientforaheatuxinput. Figure4-4.Relativestepsensivitycoefcientforatemperatureinput. bydrillingholesinthesolidandinsertingthermocouplesinsidethesolid.Placingholes inthesolidcanhaveadverseeffectsontheheattransferdynamics. 93


Severalresearchershavestudiedtheproblemofthermocouplesinsertedintoa solidandhowtheydistortthethermaleldaswellashowthisaffectsinverseproblems. ChenandLi[87]studiedtheproblemnumericallyandfoundthetheerrorproducedby thethermocoupleinsertionisproportionaltotheholesizeandthatthemagnitudeofthe errordecreasesintime.ChenandDanh[88]expoundedupontheresearchin[87]by performingexperimentswhichconrmedsomeofthepredictedresultsfromnumerical simulations,thesestudiedfocusedonthermocouplesinsertedparalleltothedirection ofheatow.Beck[89]usedDuhamel'stheoremtodetermineacorrectionkernelfor thermocouplesinsertednormaltoalowthermalconductivitysurfacetocompensate fortheinsertioneffects.WoodburyandGupta[90]usednumericalmethodstostudy thermocoupleinsertionandtheeffectsoninverseheattransferproblems.Woodbury andGupta[91]alsodevelopedasimpleone-dimensionalsensormodeltonumerically correcttheeffectsforthethermocoupleholes;thisstudyalsoincludedtheneffectfrom thethermocouplewiresandisapplicabletoathermocoupleofanyorientationtothe surface.Attiaetal.[92]performedaverycomprehensivenumericalandexperimental studywhichhelpedquantifytheerrorthatthethermocoupleinsertionproduceson measurementsincludingwireeffects,llermaterialeffects,andnon-idealcontact situationsinwhichthethermocoupleisinsertedatanangleinthehole.LiandWells [93]performednumericalandexperimentalworkstudyingthedifferentfactorsaffecting theerrorduetothermocoupleinsertion.Interestinglytheyfoundthatforathermocouple insertedperpendiculartothedirectionofheatowi.e.paralleltothesurfaceofinterest therewouldbenoeffectonthetemperaturemeasurementsbut,thermocouplesoriented paralleltothedirectionofheatowwouldhavenoticeableeffectsonmeasurements oronaninverseheattransferanalysis.Caron,Wells,andLi[94]continuedthisstudy andfoundacorrectionmodelcalledtheequivalentdepth.Thismodelimpliesthatthe temperaturesmeasuredfromaninsertedthermocouplecanbeputintoaninverse analysisasmeasurementstakenfromadifferentposition;thisnewpositionistheone 94


whichwouldwouldexperiencethetemperaturetransientsrecordediftherewereno thermocouplesinserted.Thecorrectionmodelwasonlyabletoaccuratelyreproduce surfaceheatuxhistories.Franco,Caron,andWells[95]continuedtheworkand developedcorrectionmodelswhichaccuratelyreproducesurfacetemperatures. AtrstinspectiontheworkofLiandWells[93]appearstogiveanidealresult, thatorientingthethermocoupleinaspecieddirectionwillcauseittohaveno noticeableimpact.Theauthorattemptedtousethisinformationanddesignedan inverseexperimentwithsheathedthermocouplesinserted2mmbelowthetopsurface ofacoppercylinderataangularspacingevery45 .Aftermanytrialstodetermine theimpulseresponsefunctionsoftheseembeddedthermocouplesitwasconcluded thatthermocouplessignicantlyimpactedtheheatowandtemperatureeld.This experimentalndingiscontrarytothestudybyLiandWells,butitcouldbedueto differingfactorssuchasdifferenttypesofthermocouplesused,differentsolidmaterial copperfortheauthor'sexperiment,aluminumforLiandWells,andthefacttherewere manythermocouplesinsertedversusoneforLiandWells. Onecommonalityfortheaboveworkcitedisthatthecorrectionmodelscanbequite complicatedandtheyonlyassesstheeffectsofasinglethermocouplebeinginserted intothesolid.Becauseofthesedifcultiesitwasdecidedtousesimplewelded-bead typethermocouplesandsilversolderthemtothebackofthesolidnoinsertion.This congurationeliminatedallinsertioneffectsbecausetherearenoholesdrilled.Thiswill affectthenatureoftheinverseproblembecausemeasurementsperformedattheback surfacewillcausethesensitivityoftheinversemethodtodecrease.Thislimitationcan beovercomebyformulatingtheproblemasanunknownsurfacetemperatureinsteadof anunknownsurfaceheatuxorconvectioncoefcient. 4.4InverseHeatTransferProblemFormulation AswasdemonstratedinSection4.3thebestchoiceforformulatinganinverse problemfordeterminingtheheattransfercoefcientisaspeciedtemperature 95


Figure4-5.Illustrationoftheheattransferphysicsoftheinverseproblemformulation. formulationwiththermocouplesmeasuringtemperatureatthebackwall.Oncethe temperatureattheimpingementsurfaceisknowntheheatuxatthesurfaceandheat transfercoefcientcanbedetermined.Aschematicdiagramillustratingtheproblem formulationisshowninFigure4-5.TheinverseproblemequationsfromSection4.2 willnowbecastintotheproperfromforanIHTproblemforacylinderataninitial temperaturethatisexposedtoatimeandspacevaryingsurfacetemperature. 4.4.1DirectProblem TheDirectprobleminnon-dimensionalformisformulatedas, 96


@ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 a r z =0, t = s r t b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r z t =0 =0 f wherethefollowingnon-dimensionalizationisused r = r R z = z L t = t R 2 = T o )]TJ/F39 11.9552 Tf 11.955 0 Td [(T T o s r t = T o )]TJ/F39 11.9552 Tf 11.956 0 Td [(T s r t T o 4.4.2MeasurementEquation Themeasurementequationis,initsrigorousform m i = t Z =0 1 Z r =0 L R Z z =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( r z r )]TJ/F39 11.9552 Tf 11.956 0 Td [(r i z )]TJ/F39 11.9552 Tf 11.955 0 Td [(z i r dr dz d NotethatthedeltafunctionsinEquation4merelytakeintoaccountthediscrete natureofthemeasurements.Notingthispoint,themeasurementequationcannowbe castas m i = t Z 0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( i d 97


Thesubscriptiintheequationdenotesthemeasurementlocation,ofwhichthere are7totalmeasurementpoints.Alsonotethateachthermocouplecanhaveitsown impulseresponsefunctionandhencethesubscript.Thisequationcantakeonan operatorformsimilartoEquation4. 4.4.3IndirectProblem Thecorrespondingindirectproblemis S s = t f Z 0 [ Z m t )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 dt Notethattheoperatorfromofthemeasurementequationisused. 4.4.4AdjointProblem Thedevelopmentoftheadjointproblemisquiteinvolvedmathematically.Because ofthesensordynamicsinvolvedinthemeasurementequationtheformoftheadjoint equationwilllookdifferentthanmanyofthosefoundintheliteraturesuchas[96100] forexample.Totheauthor'sknowledgetherearenoreferencesintheliteraturethat explicitlytakeintoaccountthesensordynamics,except[86],whichmerelydiscussesthe convolutionofthedeltafunctiontoaccountforthediscretenatureofthemeasurements. AlsoMarquardt'sanalysis[101]whichaccountsforsensordynamics,butusesastate anddisturbanceobserversmodel,whichisdifferentthanusingtheadjointproblemsuch asusedforthecurrentanalysis. Tobeginthederivationoftheadjointproblem,thenecessarysubstitutionsare carriedoutforEquation4 L s = t f Z 0 M X i =1 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.088 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 15.358 8.088 Td [(@ 2 @ z 2 v dt HerethenorminthesecondtermofEquation4isequalto 98


h u v i v = 1 Z r =0 L R Z z =0 uvr dr dz NextthesecondtermofEquation4isintegratedbyparts.Thisallowsforan explicitfunctionoftheadjointvariabletoappear.Afterusingtheboundaryandinitial conditionsofthedirectproblem,Equation4,theresultisthefollowing L s = t f Z 0 M X i =1 [ Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s ] 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.087 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 14.833 8.087 Td [(@ 2 @ z 2 v )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ @ r r =1 + @ @ r r =0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( @ @ z z = L R + s @ @ z z =0 )]TJ/F25 11.9552 Tf 13.151 0 Td [( @ @ z z =0 + t = t f dt NextthederivativeofEquation4istakenwithrespectto and s dL s = t f Z 0 M X i =1 h)]TJ/F22 11.9552 Tf 13.948 0 Td [(2 Y m i )]TJ/F39 11.9552 Tf 11.955 0 Td [(C i r z t s C i i )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( @ @ t )]TJ/F22 11.9552 Tf 15.133 8.088 Td [(1 r @ @ r r @ @ r )]TJ/F25 11.9552 Tf 14.833 8.088 Td [(@ 2 @ z 2 v )]TJ/F22 11.9552 Tf 11.955 0 Td [( @ @ r r =1 + @ @ r r =0 )]TJ/F22 11.9552 Tf 11.955 0 Td [( @ @ z z = L R + s @ @ z z =0 )]TJ/F25 11.9552 Tf 13.15 0 Td [( @ @ z z =0 + t = t f dt ThegoalistospecifytheadjointequationasthesolutiontothetermsinEquation4 involving .However,therstterminvolvesthemeasurementoperatorand .To rectifythistheadjointofthemeasurementequationissoughtsuchthat 99

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h e i t C i i = h C i e i t i where e i t = )]TJ/F22 11.9552 Tf 9.299 0 Td [(2 [ Y m i t )]TJ/F25 11.9552 Tf 11.956 0 Td [( m t ] Theoperator C i isknownastheadjointoperatorof C i .Tosolveforthisoperator examinationofthelefthandsideofEquation4gives h e i t C i i = t f Z t =0 e i t t Z =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d dt = t f Z =0 t Z t =0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dtd ComparingEquations4and4itisobservedthat C i e i t = t Z 0 h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dt Takingintoaccountcausality,itisknownthat for > t h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( =0. Thereforetheequationfortheoperator C i is C i e i t = t Z h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( e i t dt Nowknowingtheadjointoperatorofthemeasurementequation,theadjointproblem isselectedtobethesolutionof )]TJ/F25 11.9552 Tf 14.829 8.088 Td [(@ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 + C i [ Y m t )]TJ/F25 11.9552 Tf 11.955 0 Td [( m t ] a 100

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z =0 =0 b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r z t = t f =0, f whereEquation4isanalboundaryvalueproblem.Totransformittoaninitial boundaryvalueproblemthefollowingsubstitutioncanbeperformed, = t )]TJ/F39 11.9552 Tf 11.955 0 Td [(t f 4.4.5GradientEquation AftertheadjointproblemhasbeenspeciedthedifferentialoftheLagrangian becomes dL = t f Z 0 s @ @ z z =0 dt UsingEquation4theequationforthegradientis r S = @ @ z z =0 4.4.6SensitivityProblem UsingtheoperationssetoutinSection4.2.6thesensitivityproblemis @ @ t = 1 r @ @ r r @ @ r + @ 2 @ z 2 a z =0 = s r t b @ @ z z = L R =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e 101

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r z t =0 =0 f Notethattheformofsensitivityproblemisthesameasthatoftheadjointproblemand thedirectproblem,thusthesamenumericalsolvercanbeusedforallthreeproblems. 4.4.7ConjugateGradientMethod ThetheorybehindtheconjugategradientfromSection4.2.7remainsunchanged. Thefollowingaretheequationsspecictotheproblemathand.Therstequation iteratesforthesurfacetemperature k +1 s r t = k s r t )]TJ/F25 11.9552 Tf 11.955 0 Td [( k s r t andthenextequationprovidesforthesearchdirection k s r t = r S k r t + k k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 s r t Thenextequationgivestheconjugationcoefcient, k = t f R t =0 1 R r =0 r S k r t r S k r t )-222(r S k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 r t r dr dt t f R t =0 1 R r =0 [ r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 r t ] 2 r dr dt and k =0 for k =0. Thenalequationgivesthestepsize, k = M P i =1 t f R 0 C i r z t k s )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t C i i t k s dt M P i =1 t f R 0 [ C i i t k s ] 2 dt 102

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4.4.8StoppingCriteria Theconjugategradientmethodisaniterativeprocedureandtheterminationpoint mustbepredened,thediscrepancyprincipleisusedforthispurpose.Thediscrepancy principlestatesthatthestoppingpointforthecalculationiswhenthevalueoftheleast squaresfunctionseeEquation4isequaltothenormoftheuncertaintyofthe inputtemperatures.Usingthisprinciplethefollowingcriterionisdened: 2 = M X i =0 t f Z 0 2 i t dt where i isthestandarddeviationofthedataatmeasurementpointi.Forconstant uncertaintythefollowingholds 2 = M 2 t f andthusthealgorithmisterminatedwhen S k s 2 Onethingofnoteisthattheadjointproblemisanalvalueproblemwithanal conditionequaltozero;thustheLagrangemultiplierfortheoptimizationproblemiszero andchangestotheinitialguessatthenaltimearenotpossible.Thiscancausesome errorinthedeterminationoftheinverseproblemasshowninFigure4-6,noticethat nearthenaltimetheerrorbetweentheactualtemperatureversusthatreturnedbythe inversealgorithmislarge.Thislargeerrorcancausethealgorithmtohaveconvergence problems.Toovercomethisissue,theerrorcalculationofEquation4usesa truncatedsampleofthedata.Forinstanceiftheinputdatais60,000timestepslong only50,000timestepswouldbeusedintheerrorcalculation,andthusonlytheresults inthetruncatedsampleareconsideredreliable. 103

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Figure4-6.Comparisonofthetruetemperatureversusthetemperaturereturnedbythe inversealgorithm.Notethedisagreementnearthenaltime. 4.4.9Algorithm Alloftheequationsneededtosolvetheinverseheattransferproblemhavebeen developed.Thefollowingcomputationalalgorithmhasbeendevelopedtoobtaina solution: 1.Set s r t totheinitialguessusually1andset k =0 2.Solvethedirectproblem,Equation4usingthecurrentvalueof s r t and recordthetemperaturesatthemeasurementpoints. 3.SolvethemeasurementEquation4 m t 4.DetermineifthestoppingcriterionismetusingEquations4and4; terminatethealgorithmifthecriterionismet,otherwisecontinue. 5.Usingthemeasuredandpredictedtemperaturessolvetheadjointproblem, Equation4. 6.DeterminethevalueofthegradientusingEquation438. 7.Determine k fromEquation4and s r t fromEquation4. 104

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8.Solvethesensitivityproblem,Equation4andobtain atthemeasurement points. 9.Determine k fromEquation4 10.Determine k +1 s r t viaEquation4,set k = k +1 andreturntostep2. 4.5NumericalMethodandLimitations ThesolutionoftheDirect,Sensitivity,andAdjointproblemsneedtobefound, whileanalyticalsolutionsfortheseproblemsmayexistinsomespecialcircumstances generallysuchsolutionsarenotavailableandhencetheyaresolvednumerically. 4.5.1AlternatingDirectionImplicitMethod TheAlternatingDirectionImplicitADImethodofPeacemanandRachford[102] isacommonmethodusedforsolvingtheheatequation.TheADImethodissecond orderaccurateinspaceandtime,unconditionallystable,andiswellsuitedforsolving theseproblems.However,becauseofitsimplicitnatureitcanconsumeagreatdealof computingpoweriftheselectedtimestepisverysmall.Thebasicalgorithmforsolving thedirectproblembeginswiththerststep. n +1 = 2 i j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n i j t 2 = 1 i )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 r n +1 = 2 i +1, j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r + n +1 = 2 i +1, j )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 n +1 = 2 i j + n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r 2 + n i j +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n i j + n i j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 z 2 andthesecondstepis n +1 i j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i j t 2 = 1 i )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 r n +1 = 2 i +1, j )]TJ/F25 11.9552 Tf 11.955 0 Td [( n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r + n +1 = 2 i +1, j )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 n +1 = 2 i j + n +1 = 2 i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1, j r 2 + n +1 i j +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 n +1 i j + n +1 i j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 z 2 whereiandjaretheradialandaxialgridpointsindexes,respectivelyandnmarksthe timestep.Notethattheseindexesstartat1fortheaboveequations. 105

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4.5.2GridStretchingintheZ-Direction Toaccuratelygetanestimateoftheheattransfercoefcienthighmeasurement samplingratesandcorrespondinglysmallnumericaltimestepsareused.Duringthe initialtransientthesolidbehaveslikeasemi-innitemediumastheinitialeffectsofthe quenchingarenotgreatlyfeltbeyondthethermaldiffusionlengthwhichisproportional to p t .Toaccuratelycapturethethermalgradientsnearthesurfaceanemeshnear thesurfaceisdesired.Toaccomplishthistask,thefollowinggridtransformationisused, z = L R 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( tan )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 [ )]TJ/F25 11.9552 Tf 11.955 0 Td [( tan = ] where isthetransformedcoordinateand isastretchingparameter.Whileitis theoreticallypossibletostretchthegridinthezdirectionagreatdeal,itisgenerally unwisetodosomorethanisnecessary.Thiscausesthegridspacing r and z todifferfromeachothersignicantlywhichcancreateerror.InthecaseoftheADE methodthiscancauseunusablesolutionstobegenerated[103].Forthecurrent problemastretchingparameterof =1.1 ischosen.Figure4-7illustratestheeffectsof gridstretching. UsingthegridstretchingtransformationgivenbyEquation4,thedirect problemwillbetransformedto, @ @ t = 1 r @ @ r r @ @ r + h 2 2 @ 2 @ 2 + g 2 @ @ a r =0, t = s r t b @ @ =1 =0 c @ @ r r =0 =0 d @ @ r r =1 =0 e r t =0 =0 f 106

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a b Figure4-7.Effectsofgridstretching.arealdomainandbcomputationaldomain. h 2 = @ @ z g 2 = @ 2 @ z 2 Similartransformationsoftheadjointandsensitivityproblemsresultaswell.Alsonote thatanyuxquantitywillbetransformedas @ u @ z z = z p = h 2 z = z p @ u @ z = z p where z p isthezcoordinatewheretheuxcalculationisbeingcarriedout. 107

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4.5.3Timestepsizecomplications Inordertochoosethesizeofthetimesteptouseintheinverseheattransfer algorithm,somesamplecalculationsarecarriedout.Asatestproblemaonedimensional slabataninitialtemperatureof0issubjectedtoanon-dimensionaltemperatureofunity. TheADImethodisusedona32x32gridwithtimestepsof5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 and5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(5 Courantnumberof0.512and0.0512respectivelyforatotalof800timestepsineach case.TheresultsofthecalculationsareshowninFigure4-8.Itisclearlyseenthat theADImethodreproducestheexactsolutionfortemperatureforbothtimesteps remarkablywell.Itisalsoevidentthatattimestepsof5 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4 oscillationsappearin theresultingheatux.TheoscillationswerenotedbyKropf[104]fortheADImethod duetolargetimestepsandthediscontinuityofthesurfacetemperatureattheinitialtime step.Theseoscillationsarenotpresentinthesmallertimestepcaseandamaximum timestepof5 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(5 isselectedforimplementingtheinverseheattransferalgorithm. 4.6DeconvolutionforThermocoupleImpulseResponseFunctions Aninherentproblemforsolvingtheinverseheattransferproblemjustsetforthis knowingtheimpulseresponsefunctionsfortheinstalledthermocouples.Classically thermocouplesarethoughtofasrstordersystemshowever,theycansometimesbe thoughtofassecondorhigherordersystems[105]. However,giventhetooloftheinverseproblemtheimpulseresponsefunctionof eachthermocouplecanbedeterminedwithoutan apriori knowledgeofthefunctional formofthesefunctions.ThefollowingSectionswillformulatethenecessaryinverse problem.Theproblemisformulatedsuchthattheinputintothethermocoupleis knownviaanexactsolutionandthethermocoupleoutputisrecorded.Morethan oneexperimentalrecordcanbeusedinthefollowinganalysissimultaneouslyifdesired, eachexperimentalrecordisdenotedbythesubscripti.Thefollowinganalysisisaslight modicationtothatfoundinreference[77] 108

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a b c d Figure4-8.ComputedtemperatureandheatuxfordifferenttimestepsusingtheADI method. 4.6.1DirectProblem Thedirectproblemissimplytheconvolutionoftheinput,x,withtheimpulse responsefunctionh,todeterminetheoutputsignal,y, y i t = t Z 0 x i h i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d NotethesubscriptiinEquation4denotesameasurementchannel. 109

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4.6.2IndirectProblem TheindirectproblemisessentiallyunchangedfromtheonederivedinSection4.2.3. However,forcompletenessisitincludedbelow. S = M X i =0 1 2 t f Z 0 [ Y m i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y i t ] 2 dt 4.6.3AdjointProblem OncetheLagrangianhasbeenformed,whichintheinterestofspaceisnot demonstrated,theadjointequationischosenas t i = Y m i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y i t NotethatEquation4isasimpleexpressionandtheaboveformwhaschoseto facilitateeaseofsolution. 4.6.4GradientEquation Similartothetheadjointofthemeasurementoperatorthegradientequationis r S i t = t f Z i t h t )]TJ/F25 11.9552 Tf 11.955 0 Td [( dt NotethatEquation4isnotaconvolutionoperation. 4.6.5SensitivityProblem Usingthemethodoutlinedearlierthesensitivityproblemis y i t = t f Z 0 x i t )]TJ/F25 11.9552 Tf 11.955 0 Td [( h d Note,onceagain,thatthesubscript,i,denotesthemeasurementchannel. 4.6.6ConjugateGradientMethod Theconjugategradientmethodforthedeconvolutionproblemissimilartothe previousderivation.However,thisparticularimplementationusestheFletcher-Reeves 110

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equationforthestepsize[83].Recallthatthedifferentstepsizesareequivalentfor linearproblems,ofwhichthecurrentdeconvolutionproblemisone.Theequationforthe iterationsis h k +1 t = h k t + k h k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ; theequationforthesearchdirectionis h k +1 t = )]TJ/F40 7.9701 Tf 15.895 14.944 Td [(M X i =1 r S k i t + k h k t ; theequationfortheconjugationcoefcientis k = M P i =1 t f R 0 r S k t 2 dt M P i =1 t f R 0 [ r S k )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ] 2 dt ; and k =0 for k =0 andtheequationforthesearchdirectionis k = M P i =1 t f R 0 [ x i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(Y m i t ] h k t dt t f R 0 [ h k t ] 2 dt 4.6.7StoppingCriteria Thestoppingcriteriaremainsunchanged.Equations4and4stillapply andarenotrepeatedhere. 4.6.8Algorithm Alloftheequationsneededtoperformthedeconvolutionproblemhavebeen developed.Thefollowingisthecomputationalalgorithm: 111

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1.Set h t toaninitialguessandset k =0 2.Solvethedirectproblem,Equation4usingthecurrentvalueof h t 3.DetermineifthestoppingcriteriaismetusingEquations4and4, terminatethealgorithmifthecriteriaismet,otherwisecontinue. 4.Usingthemeasuredandpredictedvaluesofysolvetheadjointproblem, Equation4. 5.DeterminethevalueofthegradientusingEquation4. 6.Determine k fromEquation4and h t fromEquation4. 7.Solvethesensitivityproblem,Equation4andobtain y i t 8.Determine k fromEquation4 9.Determine h k +1 t viaEquation4,set k = k +1 andreturntostep2. 4.6.9TestCase Inordertodemonstratethecapabilitiesoftheabovedeconvolutionmethodasimple exampleispresented.Inthisexamplearstorderimpulseresponsefunctionischosen torepresentthesensor h = 1 exp )]TJ/F39 11.9552 Tf 10.977 8.088 Td [(t whereatimeconstantof =0.2sisused.Threedifferentinputsarechosen x =sin t x =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(t x = p t theseinputsareconvolvedwiththeimpulseresponsefunction;Equation4and noisehavingastandarddeviationof =0.1isadded.Theresultsofthedeconvolution aredisplayedinFigures4-9,4-10,and4-11. 112

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Figure4-9.Trueandestimatedimpulseresponsefunction. Figure4-10.Convergencehistory. ClearlyasevidencedbyFigure4-9thetrueimpulseresponsefunctionisreproduced quiteaccuratelyforthegivennoiselevel.Figure4-10showstheconvergencehistory,S vsiterationnumberk.LastlyFigure4-11showstheresultswhentherealinputfunction, x,isconvolvedwiththeestimatedimpulseresponsefunction. 113

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Figure4-11.Simulatedoutputlinesandoutputofxconvolvedwithestimatedimpulse responsefunctionsymbols. Onenoteworthypointisthatwhileattemptingtodeterminetheimpulseresponse functionofthermocouples,multipleexperimentalrecordsaregenerallynotusedinthis analysis.Thereasonisthatitisnotfeasibletousemanydistinctinputsignalsintothe systemgenerallyastepchangeintemperatureatasurfaceisused.Thusmultiple experimentalsamplerecordswillbenearlyidenticalandtheadvantagesofusing multipleinputswillnotberealized. 4.7Summary InthisChapterinverseproblemsandinverseheattransferareintroduced.General factorsaffectinganinverseanalysissuchassensordynamics,sensorlocation,and problemtypeformulationwereexploredaswellaswaystobestusetheanalysis. Problemsspecictothepresentcasewerealsoexplored,suchashowathermocouple insertedintoadrilledholeinthesolidcandistortthethermaleldarounditandimpede heattransfer.Asacorrectiveactionaninverseheattransferproblemwasdesignedthat usesnoinsertionholes,andtheproblemhasbeenformulatedtobeofanunknowntime 114

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andspacevaryingsurfacetemperatureinordertoovercomelimitationsassociatedwith suchaformulation. Numericalmethodstosolvethedirect,adjoint,andsensitivityproblemshavebeen explored.Oscillationscausedmainlybyusinglargetimestepsforadiscontinuityin thesurfacetemperatureattheinitialtimewerenotedandasuitabletimesteplimit wasfound.TheAlternatingDirectionImplicitmethodwaschosentobethesolution techniqueduetothehighspatialandtemporalaccuracy. Adeconvolutiontechniqueusingthefunctionestimationwithadjointproblemand conjugategradienttechniquewasdeveloped.Atestcaseillustratingthepowerofthe techniquewasperformedusingseveraldifferentsimulatedinputstoarstordersensor withnoisydata.Thistechniquewillbeusefulindeterminingtheimpulseresponse functionsofthermocouplesusedinexperiments. 115

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CHAPTER5 SENSORDYNAMICSANDTHEEFFECTIVENESSOFTHEINVERSEHEAT TRANSFERALGORITHM ThepurposeofthisChapteristoinvestigateamodelforthesensordynamicsof thermocouplesandverifyitviaanexperiment.Theeffectivenessoftheinverseheat transferalgorithmwillalsobepresented.Theeffectsofdischeight,numberofsensors, magnitudeofthenoisepresentinthemeasurementsystem,magnitudeoftheBiot numberdistribution,andthemagnitudeandaccuracyoftheimpulseresponsefunction modelwillbeexplored.Lastlytheeffectsofnon-idealinsulationwhichcausesthediscto gainheatduringanexperimentwillalsobedetailed. 5.1ThermocoupleMeasurementDynamics AswasstatedinChapter4,measurementdynamicsareanimportantfactorin inverseheattransferproblems.Assuchmucheffortwastakentoinvestigatedifferent modelsforthermocoupledynamics. 5.1.1LowBiotNumberThermocoupleModels Theresponsemodelofweldedbeadtypethermocouplesusedinthepresentstudy isaeldthathasbeenextensivelystudied[106110].Forthecaseofthermocouples inauidenvironment,amodelfortheresponsetimecanbedevelopedusingrst principles.Firstitisassumedthatthethermocouplecanbeapproximatedasasphere withconstantthermalproperties.WhentheBiotnumberBi=hD/kismuchless thanunity,thetemperatureproleinsideoftheweldedbeadcanbeapproximatedas constantthroughoutitsradius[111];duetothelowBiotnumberandwithoutlossof generality,thespherecanbemodeledasaslab.Figure5-1showsanillustrationofthe rstordersystemforaslab.Performinganenergybalanceontheslabresultsinthe following Vc p dT dt = )]TJ/F39 11.9552 Tf 9.298 0 Td [(hA T )]TJ/F39 11.9552 Tf 11.955 0 Td [(T 1 116

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Thetemperatureisnon-dimensionalized,andtheinitialdimensionlesstemperature istakentobezeroattheinitialtime.Thesolutiontotheordinarydifferentialequation ODEaboveis =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(t where =T-T o /T 1 -T o isthedimensionlesstemperature,T o istheinitialtemperature, and = Vc p /hAisthetimeconstantofthethermocouple.Theaboveequationisknown asthestepresponseequation;todeterminetheimpulseresponseofarstorder system,thederivativewithrespecttotimeofthestepresponseequationistaken,and thustheimpulseresponseforarstorderthermocoupleis: h t = 1 exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(t Thetimeconstantcanbethoughtofasthetimeneededforthethermocoupletoreach 63.2%ofthevalueofT 1 .After5timeconstantsthetemperatureofthethermocoupleis essentiallyT 1 Foratime-varyingfreestreamtemperaturethethermocouplewillindicatea differenttemperatureaspertheconvolutionintegraldetailedinChapter4.Theequation governingthisbehaviorisrepeatedhereforcompleteness. t = t Z 0 1 exp )]TJ/F39 11.9552 Tf 10.494 8.088 Td [(t )]TJ/F25 11.9552 Tf 11.955 0 Td [( d ForasolidembeddedthermocouplewithaBiotnumbermuchlessthanunity,thesame logicisapplied,excepthisnolongertheconvectionheattransfercoefcient,butnow theproducthArepresentsthecontactconductance. Similartotheaboverstordersystem,asecondordersystemfortheresponseofa thermocoupleisderived.Manyofthedetailsforasecondorderthermalsystemcanbe foundin[112];onlythesalientdetailsareincludedhere.Whenthethermocouplebead 117

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Figure5-1.Illustrationoftherstorderslab. iscoatedwithsomesortofintermediatematerialepoxyforexamplethiscanformtwo rstordersystemsinseries,anillustrationofthissystemisshowninFigure5-2.The followingsecondorderODEwillresult: d 2 T 1 dt 2 +2 n dT 1 dt + 2 n T 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(T 1 =0 where 2 n = 1 1 + 1 2 + h 1 h 2 2 2 n = 1 1 2 i = Vc p i hA i Usuallythesystemdenedabovehastwotimeconstants,whichisotherwiseknownas over-damped[105].SolvingtheaboveODEusingtheprocedureoutlinedforarstorder systemwillresultinthefollowingimpulseresponsefunction: 118

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Figure5-2.Illustrationofthesecondorderslab. h t = 1 2 n p 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 [ exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 t )]TJ/F22 11.9552 Tf 11.955 0 Td [(exp )]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 t ] 1,2 = n p 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1. Thedifferenceindynamicsbetweenarstandsecondordersystemisquitesubstantial asillustratedinFigure5-3.Noticethatforordershigherthanonetheimpulseresponse functionisequaltozeroattheinitialtime.Thisobservationisusefulindetermining whetherthesystemisofhigherorderwhenameasurementoftheimpulseresponse functionofthesystemisavailable. Thereisathirdmodelofnoteandthatisadiffusiveelementwhichisfollowedbya 2 nd ordersystem,hereafterreferredtoasthe2exponentialmodel.Thedetailsofthis modelcanbefoundinreference[113].Thediffusiveelementwillcausetheresulting impulseresponsefunctiontoconsistofthefollowingform h t = aexp )]TJ/F39 11.9552 Tf 9.298 0 Td [(b + cexp )]TJ/F39 11.9552 Tf 9.299 0 Td [(d 119

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Figure5-3.Exampleofarstandsecondorderimpulseresponsefunction. wherea,b,c,anddareconstants.Thisimpulseresponsefunctionbehavesverysimilar tothatofarstordersystem. WhiletheabovemodelsarequitegoodwhenthethermocoupleBiotnumberis lessthanunity,itcanproduceresultsthatdeviatefromidealwhentheBiotnumber approachesorexceedsunity.ForthecurrentexperimentsusingE-typethermocouples, whichhaveathermalconductivityof19.5W/m-Kwithacharacteristiclengthof approximately1mm,theBiotnumberwillbe0.1orlesswhenheattransfercoefcient islessthanapproximately20,000W/m 2 -K.Theuseofsmallerthermocouplesenables theuseofthelowBiotnumberassumptionwithhigherlimitsofheattransfercoefcients. Inthepresentstudysilversolderisusedtoattachweldedbeadthermocouplestoa heatedsample.Theuseofsilversolderwillresultincontactconductancesoneormore ordersofmagnitudehigherthanthelimitingcasepreviouslymentioned;thustheuse ofamodelwhichreliesonasmallBiotnumberassumptionforthepresentstudy,may producesomeerror. 120

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5.1.2HighBiotNumberThermocoupleModels Developingameasurementmodelforthermocouplesthatdoesnotrelyonalow Biotnumberassumptionisdifcult.Thereareseveralparameterswhichcanaffectthe impulseresponsefunction,suchasthethermaldiffusivityofthethermocoupleaswell asthethatofthesolidthethermocoupleisembeddedin,amongothers.Rabinand Rittel[114]havecompletedanumericalstudywherethelowBiotnumberassumptionis notnecessary.Toaccomplishthistheymodeledathermocoupleasasphereforbead typethermocouplesorcylinderforthermocoupleprobesembeddedinasolidthat undergoesastepchangeintemperature.Thenumericalexperimentsarecarriedout overalargenumberofthermaldiffusivityratiosandacurvetisusedtoexpressthe resultingimpulseresponsefunction.Theimpulseresponsefunctionhasthefollowing form h t =exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(B D t R 2 n whereBandnarecurvettingparameters,Ristheradiusofthesphere/cylinder,and D isthethermaldiffusivityofthesolid.Atableofthevariousconstantsisfoundin Table5-1. Table5-1.CurveFittingConstantsforRabinandRittel'sthermocoupleimpulseresponse model,from[114] CylindricalCaseSphericalCase TC = D BnBn 11.5820.562.7990.61 101.7240.453.1930.52 1001.8210.453.2090.50 3001.8300.453.2290.50 10001.8330.453.2360.50 Afewnoteworthypointsarethattheindicatedthermocoupletemperatureused inthenumericalexperimentsisthevolumeaveragedtemperatureinsideofthe thermocouple.RabinandRittelnotedthatthetemperatureinsideofthethermocouple 121

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Figure5-4.ImpulseresponsefunctionsusingthemodelofRabinandRittel,adapted from[114]. wouldbeverynon-uniformforthermaldiffusivityratioslowerthanapproximately300. Alsoitisbrieynotedthattheheattransferphysicsbeingmodeleddonotincludeany effectsofthermalcontactresistancebetweenthesolidandthermocouple.Agraphof theimpulseresponsefunctionsforsphericalthermocouplesisshowninFigure5-4. 5.1.3DesignofExperiment Theexperimentusedtodeterminetheimpulseresponsefunctionsisperformed insitu .Thethermocouplesarelocatedonthebottomofadiscusedinaquenching experiment.Thethermocouplesconsistof7weldedbeadEtypethermocouplesmade from26AWGwire.405mmdiameter.Thesethermocouplesarespacedevery45 withtheinitialthermocouplelocated1.59mmfromthecenterofthecopperdiscwiththe remainderofthethermocouplesplacedevery3.18mmradiallyfromitsneighbor.There areatotalof8thermocouplessilversolderedtothebackofthecopperdisc.Oneofthe thermocouplesisusedasagroundforthesystem.Adiagramofthecopperdiscusedin theexperimentsisshowninFigure5-5.Thediscwasinsulatedonallsidesexceptthe topbyaceramicinsulationmaterialfromCotronicsInc.knownasRescor750,aSiO 2 122

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Figure5-5.Diagramofthecopperdiscassembly. basedceramicwithathermaldiffusivityof8 10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(7 m 2 /sandathermalconductivityof 0.58W/m 2 -K. Oneofthechallengesfacedindeterminingtheimpulseresponsefunctionof thethermocouplesisaccuratelydeterminingtheinputtothethermocouples.Several differenttechniqueswereused.Oneoftheearlytechniquesistohaveasingle-phase liquid,turbulentjetimpingeonthesurfaceofthediscafterithadbeenheatedto approximately90 C.Theheattransfercoefcientforsuchajetisdeterminedusingthe correlationofLiuetal.[11].However,oneoftheconditionsassumedforthecorrelation isthatthesurfaceheatuxremainsconstantthroughouttheexperiment;thisconditionis notmet,eveninaquasi-steadysense.Theseexperimentsdidnotproducereliabledata. Anothermethodutilizedistousedryicetosimulateaconstanttemperatureboundary conditionatthesurfaceafterheatingthedisctoapproximately100 C.Whilethisis goodintheory,inpracticethesublimationofCO 2 wouldcauseabuildupofgasnear thesurfacewhichseemedtocausesomethermalresistance.Thenalexperiment 123

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Figure5-6.Illustrationoftheexperimentalsetup. attemptedistouseiceH 2 Otoproduceaconstantsurfacetemperature.Initiallya cylinderoficeformedwithinaStyrofoamcupmoldthatislargerindiameterthanthe copperdiscwasused;however,itwasrealizedafterexperimentsthattheicewould meltandleaveanimpressionofthecopperdiscintheice.Thismeltingoftheicewould causenon-uniformcontactonthecopperdiscwhichwouldrendertheassumptionof aconstantboundarytemperatureinvalid.Toovercomethischallenge,acylinderof iceofthesamediameterasthecopperdisc,madeusingaPVCpipe,isused.The moldofthecylinderhousedtheiceandaplungerisusedtohelpensureevenand constantpressureontheicecylinder.Adiagramofthisexperimentalsetupisshownin Figure5-6. Theheattransferphysicsgoverningthisexperimentareone-dimensional,transient conduction,andassuchthetemperatureexperiencedinsideofthecopperismodeled 124

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Figure5-7.Comparisonofbackwalltemperaturesofthecopperdisc,accountingforthe effectsofnon-idealinsulation. asinEquation4b.Toassesstheeffectsoftheinsulationonthesystemdynamics, anumericalstudyiscarriedoutmodelingthediscasatwo-dimensionalslabandis comparedtotheresultsofanideal,one-dimensionalslabwithperfectinsulation.The resultsofthisstudyareshowninFigure5-7.Theeffectsofnon-idealinsulationare essentiallynegligibleforshorttimeshowever,thereissomeerrorpresentattimes approaching0.5secondsanditvarieswiththeradialpositionofthethermocouples.This deviationfromidealcouldcausesomeerrorinthedeterminationofimpulseresponse functions. Thenalissueindeterminingtheimpulseresponsefunctionsistodetermine whentheiceisactuallyrstpressedagainstthesurface,thereissometimedelayin theresponseofthethermocoupleswhichprecludesusingtherstsignoftemperature changingastheinitialpointintheexperiment.Toovercomethisissueamicrophoneis placedontheceramicinsulationusedanditsoutputsignalisrecorded.Whentheice makestheinitialcontactwiththecopperdiscadistinctsoundisheardasmicro-fractures 125

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Figure5-8.Resultsoftwoseparateimpulseresponseexperiments,notethe repeatabilityoftheresultsforshorttimes. appearintheice.Usingtheoutputofthemicrophonethetimedelayforthethermal wavetopropagatetothethermocouplesisdeterminedtobeapproximately80ms. 5.1.4ExperimentalResults Oncethedataarerecorded,theinversedeconvolutiontechniquediscussedin Section4.6isusedtodeterminethethermocoupleimpulseresponsefunctions.The samplingrateusedintheexperimentsissetat8kHz.AswasmentionedinSection4.6 theuseofmultipleexperimentalrunsisnotnecessaryastheyallcloselyresembled eachother;thisisexpectedsincethesameinputsignalisessentiallyusedineach experiment.TherepeatabilityoftheexperimentsisclearlydemonstratedinFigure5-8 forshorttimeswheretheconstanttemperatureboundaryconditionismostlikelymet. Figure5-9showstheimpulseresponsefunctionasobtainedusingtheinverse deconvolutiontechniquefortherstchannelintheDAQ.Similarresultsareseenfor eachchannel,andintheinterestofavoidingredundancytheyarenotshown.Oneof theimmediatefeaturesnoticeableinthisgraphisthattheimpulseresponsefunctionis 126

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Figure5-9.Inversemethoddeconvolutionresults. non-zeroattheinitialtime;thussecondorderandhigherimpulseresponsefunctions canberuledout. 5.1.5ComparisontoEstablishedModels Todeterminethebestimpulseresponsemodelforthethermocouples,acomparison ismadetoarstorderresponse,a2exponentialresponse,aswellasthemodelof RabinandRittel.Thisisaccomplishedviatheuseofanonlinearleastsquarescurve tusingthefunction lsqcurvet availableinMatLab,whichusesatrustregionreective algorithm.Aftertheparametersofthemodelaredetermined,acomparisonofthe outputoftheimpulseresponsemodelwiththeactualtemperaturemeasuredduring experimentsismade;acomparisonofthemodelwiththeimpulseresponsefunction determinedviatheinversedeconvolutionalgorithmisalsomade. Toremaingeneral,therstordersystemmodelismodiedsuchthattwoparameters areusedinsteadofthesingletimeconstantoftherigorousmodelofEquation5,this newequationis h t = a exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(bt 127

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Figure5-10.Comparisonoftherstorderresponsefunctiontothede-convolved impulseresponsefunction. TheresultingparametersdeterminedfortherstchanneloftheDAQarea=0.532 andb=0.623;thisisequivalenttimeconstantofapproximately1.75seconds.A comparisonoftheresultingfunctionwiththatproducedviatheinversealgorithm isshowninFigure5-10.Itisevidentthatthebehaviorofthefunctioniscaptured marginallybyarstorderresponse.Thedifferencesbetweentheresponsemodel outputandtheexperimentaldataareshowninFigure5-11.Whiletheoveralltrend iscapturedquitewell,thereisnoticeableerrorbetweenthepredictedandmeasured temperatureresults. Toremaingeneral,themodelofRabinandRittelismodiedaswellsuchthatthree parametersareusedinsteadofthetwoinEquation5.Thisnewequationis h t = c exp )]TJ/F39 11.9552 Tf 9.299 0 Td [(dt n Theresultingparametersfortherstthermocouplechannelafterdeconvolutionarec= 1.775,d=0.216,andn=0.369assumingthattheradiusofthethermocoupleis0.5mm andusingthethermaldiffusivityofcopper.Acomparisonoftheresultingfunctionwith 128

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Figure5-11.Besttresultsusingarstorderimpulseresponsefunction. Figure5-12.Comparisonofthemodelof[114]tothede-convolvedimpulseresponse function. thatproducedviatheinversealgorithmfortherstchannelisshowninFigure5-12.This modelcompareswellwiththeinversemethoddeconvolution.However,thereissome disagreementneart=0.Thedifferencesbetweentheresponsemodeloutputandthe experimentaldataareshowninFigure5-13.Thetrendiscapturedquitewell. 129

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Figure5-13.Besttresultsusingthemodelof[114]responsefunction. ComparisonoftheparameterdtoBinTable5-1showsthatthereissome departurefromthemodelofRabinandRittelastheyareofdifferentordersofmagnitude. Thevalueoftheexponent,nisofthesameorderofmagnitude.Itisnotedthatthe thermalconductivityratioforthethermocoupletothecopperdiscisontheorderof10 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 welloutsidetherecommendedrange,andsothecomparisonisgeneralinnature. AsobservedinFigure5-13themodelofRabinandRittelproducesgoodresults despitethedifferencesinthecoefcientsofthecurvet.Therearesomeconditions intheexperimentsthatarenotaccountedforinthemodel.Namely,therewillbesome nitethermalresistancebetweenthethermocoupleandthecopperdisc,althoughit isbelievedtobesmallduetotheuseofsilversolderinattachingthethermocouples. Additionally,thethermocouplesintheexperimentsarenotfullyembeddedinthesolid, buttheyaresolderedtothebackofthecopperdisc.Thisassumptionmaynotbebad duetothefactthatthesilversolderwilltendtoencasethethermocouples. Thenalcomparisonisthatofthe2exponentialmodel.Theresultingmodel parametersfortherstchanneloftheDAQarea=0.658,b=0.402,c=0.228,and 130

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Figure5-14.Comparisonofthe2exponentialmodeltothedeconvolutionalgorithm results. d=2.961.Acomparisonoftheresultingfunctionwiththeresultsofthedeconvolution algorithmisshowninFigure5-14;theagreementoftheresultsisexcellent.These resultstranslateintogoodagreementbetweentheoutputmodelandtheexperimental dataasshowninFigure5-15. TheresultsofthecomparisonofthethreemodelsshowthatthemodelofRabinand Rittelandthe2exponentialmodelproducethebestresults.However,the2exponential modelisselectedasthebestduetothefactthatexcellentagreementisobserved betweenthecurvettedanddeconvolutionresults. 5.2InverseHeatTransferAlgorithmVerication Thepurposeofthisportionofthecurrentstudyistodevelopaninverseheat transferalgorithmtodeterminetheheattransfercoefcientofanimpingingjet.However, duetosensitivityissuesassociatedwithhavingsensorsplacedonthebacksurfaceof thetestsample,directestimationofthesurfaceheattransfercoefcientisdifcult,if notimpossible,forhighBiotnumbers.Assuchtheinverseheattransferalgorithmwas formulatedsuchthatthesurfacetemperatureisdeterminedandthesurfaceheatuxis 131

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Figure5-15.Besttresultsusingthe2exponentialmodel. thenestimated,afterwhichthesurfaceheattransfercoefcientisestimated.Inorderto determinetheaccuracyoftheproposedalgorithmaparametricstudyisconsidered. 5.2.1InverseQuenchingParametricStudySetup Thereareseveraldifferentfactorswhichcanaffecttheinverseheattransferstudy whichare:thematerialpropertiesofthedisci.e.thethermaldiffusivity,theimpulse responsefunctions,theheight,L,ofthedisc,theaspectratioofthediscL/R,theBiot numberdistribution,thenumberofmeasurementpointsonthebacksideofthedisc,and themagnitudeofthenoiseoftheDAQ.Differentvaluesforthesevariablesareselected fortheparametricstudyasdetailedbelow. Thesetupfortheinverseheattransferparametricstudywillsimulatethecopper discassemblyoutlinedinSection5.1.3.Thecopperdiscisataninitialconditionofzero indimensionlesstemperatureandthenthejetisinitiatedwithaspeciedBiotnumber distributionandanadiabaticwalltemperatureofunity.Theheattransfercoefcientofan impingingjetcanbemodeledasaGaussianfunctionasseenbelow 132

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Bi = Bi max 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(ratio exp )]TJ/F39 11.9552 Tf 9.298 0 Td [(r 2 2 2 + ratio whereBi max isthemaximumBiotnumberseen,ristheradialcoordinate, isthe standarddeviationwhichaffectsthewidthoftheGaussianpeak,andratioistheratioof themaximumtominimumBiotnumber.Figure5-16showstheBiotnumberdistribution fordifferentvaluesofBi max ,whileFigure5-17showstheBiotnumberdistributionfor differentvaluesof ,inbothguresthevalueofratiois0.1.ThreedifferentBi max values of15,5,and0.5wereusedintheparametricstudy,theserepresentpeakheattransfer coefcientsof234,78,and7.8kW/m 2 K,respectively.Onepointofnoteisthatfor highervaluesof theBiotnumberdistributionapproachesthatofaconstantBiot number,makingtheproblembehaveone-dimensional,whichisasimplerestimation problem.Thiseffectisalsoseenforhighervaluesoftheparameter,ratio.Toadequately determinetheeffectivenessoftheinverseheattransferalgorithmtheratioquantity wassetto0.1andformostnumericalexperiments wassetto0.1aswell,although studiesathighervaluesof werecarriedoutinordertoverifythatgoodresultscouldbe obtained. Theeffectsoftheimpulseresponsefunction,andthemagnitudeofthenoise arealsoexplored.WhileitwasfoundinSection5.1.5thatthe2exponentialmodel producedthebestresultsexperimentally,itisdifculttoassesstheeffectofvarying eachparameterofthemodel.Assuchtherstorderimpulseresponsefunctionin Equation5waschosentosimulatethethermocouplemeasurementdynamicsas itdoesafairjobofapproximatingthedynamics,anditonlycontainsoneconstantto vary.Thedifferenttimeconstantschosenfortheparametricstudyare0.1,1,and5 secondswhichallowforcomparisontobemadeforshortandlongtimedelaysofthe thermocouples. Todeterminetheeffectofthedischeightandaspectratioontheinverseheat transferproblem,themaximumradiusofthedisc,R,wassetto25.4mmandthree 133

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Figure5-16.BiotnumberdistributionshowingtheeffectsofBi max differentvaluesoftheheightwereselected:15,10,and5mmgivingaspectratiosof 0.591,0.394,and0.197,respectively.Whileperformingtheparametricstudyitwas foundthatforheightsof15mm,theinverseheattransferalgorithmwouldnotconverge; thusnoresultsfromthisheightarereportedinthestudy.Theeffectofthediscthermal diffusivity,theimpulseresponsefunction,andtheheightofthedisccanbecollapsed intoadimensionlesstimeconstantdenedbelow = L 2 Toassesstheeffectsofmeasurementnoise,whiteGaussiannoisewasadded tothesimulatedmeasurements.Themagnitudeofthenoiseindimensionlessspace dependsontwoseparatevariables,theinitialtemperatureandthemagnitudeofthe noiseintheDAQ.Thisrelationshipisshownbelow = DAQ T o 134

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Figure5-17.Biotnumberdistributionshowingtheeffectsof where isthedimensionlessnoise, DAQ isthestandarddeviationofthemeasurement noiseoftheDAQ,and T o istheinitialtemperature.Theinitialtemperaturewasset to150 CandtwodifferentstandarddeviationsoftheDAQwerechosen:0.2 C,the measurementnoiseseenintheDAQusedfordeterminingtheimpulseresponse function,and1 C. Inadditiontotheaboveconditions,otherconditionsweresetfortheparametric study.Thesamplingfrequencywassetto8kHzandthegridresolutionwassetto32 by32forL=10mm.OscillationsofheatuxsimilartotheonesseeninSection4.5.3 werepresentintheL=5mmcaseandwereeliminatedbyusingagridresolutionof 64by64.Thetotaltimedurationofthesimulatedexperimentswassetto30,000time steps,25,000ofwhichwereusedinthestoppingcriteria,exceptfortheBi max =0.5,L =5mmsimulationswhere60,000timestepswereused,50,000ofwhichwereusedin thestoppingcriteria.ThereasonforthisdisparityisthatthelowerBiotnumbercaused aslowerresponseinthemeasuredtemperature.Additionally,thenumberofsimulated 135

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measurementstakenwasselectedtobe8and16,themeasurementlocationswere denedbythefollowingformula r i meas = R i pts )]TJ/F22 11.9552 Tf 21.407 8.088 Td [(1 2 pts i =1,2,3,... M where r i meas isthelocationofmeasurementi,Ristheradiusofthedisc,ptsisthe numberofmeasurementpoints,andMisthetotalnumberofmeasurements. 5.2.2ErrorAssessmentMethods Inordertoquantifytheerrorbetweentheinverseheattransferalgorithmresults andthetrueBiotnumberanerrordenitionmustbemade.Thealgorithmistechnically designedtoestimatethesurfacetemperatureandthusasurfacetemperatureerror shouldbeusedtoestimatetheeffectiveness.Additionally,becausetheBiotnumberis thequantityofinterestitmustbeestimatedaswell.Theerrorassessmentchosenis thatofaroot-mean-squareerror,thermserrorequationforthesurfacetemperatureand Biotnumberdistributionisdenedbelow e rms = r R t R r act )]TJ/F25 11.9552 Tf 11.956 0 Td [( inv 2 r dr dt r R t R r act 2 r dr dt where iseitherthedimensionlesssurfacetemperatureorBiotnumber,thesubscript actdenotestheinputvariable,andthesubscriptinvdenotestheoutputvariablereturned bytheinversemethod. WhileperformingtheparametricstudyitwasfoundthattheestimationoftheBiot numbernear t =0 wasgrosslyinerror,beingseveralordersofmagnitudegreaterthan theactualBiotnumber.ThelargeBiotnumbersneartheinitialtimearenotbelievable. However,afterthisinitialspike,theBiotnumberestimatedwillgenerallymaintainanear constantvalueforaportionoftheresultsanditwasdecidedthattheBiotnumberrms errorwouldbeassessedsuchthatonlythisportionoftheBiotnumberdistributionwould beusedintheerrorcalculations,thisisshowninFigure5-18. 136

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Figure5-18.ComparisonoftheinverseresultsofBiotnumberatthecenterlineofthe discforBi max =15, =0.1.Theasterisksdenotewheretheresultsare truncated. Additionally,becauseofthenearconstantBiotnumberaftertheinitialtimeitwas decidedthattheBiotnumberwouldbeaveragedtemporallybetweenthetruncation points.ThisaverageBiotnumber,denotedasBi ave ,wasalsousedtodetermineanrms errorasdenedbelow e Bi ave rms = r R r Bi ave real )]TJ/F39 11.9552 Tf 11.955 0 Td [(Bi ave inv 2 r dr r R r Bi ave real 2 r dr 5.2.3ParametricStudyResults TheresultsfortheparametricstudyforL=10mmand DAQ =0.2 Carelistedin Table5-2.Itisclearlyevidentthattheinversemethodaccuratelydeterminesthesurface temperaturedistribution,whichitisdesignedtodo.Theaccuracyrangesfrom1to3 %anddoesnotseemtobeaffectedbyincreasingthenumberofmeasurementpoints. ThetruncatedBiotnumbererrorisnoticeablyhigher,rangingfrom15to48%forthe8 137

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Table5-2.RMSerrorsforL=10mm,and DAQ =0.2 C. e s rms e Bi rms truncatede Bi ave rms 0.1121.1235.6170.1121.1235.6170.1121.1235.617 M=8 150.0120.0180.0180.1530.1670.1760.1390.1530.162 Bi max 50.0130.0170.0190.2610.3040.3800.2510.2930.341 0.50.0150.0070.0140.4290.3960.4750.3300.3650.424 M=16 150.0120.0160.0200.1370.1510.3030.1290.0880.258 Bi max 50.0160.0170.0250.3120.2950.3740.3090.2770.344 0.50.0180.0100.0170.4970.6080.7160.4420.5940.654 measurementpointscase;theerrortendstoincreaseforalowerBi max .Contourplots oftheerrorsfortheL=10mm, DAQ =0.2 C,and8measurementpointsareshownin Figure5-19;intheinterestofneatnesscontourplotsoftheothercasesarenotshownin thisChapterandcanbefoundinAppendixB.For16measurementpoints,theerroris inthesamerangeforthetwohighestBi max cases,butisnoticeablyhigherforthelowest case.TheerrorforthetemporallyaveragedBiotnumbererrorisslightlylowerthanthat ofthetruncatedBiotnumbererrorforboth8and16measurementpoints.Fromthese dataitappearsthatforL=10mm,increasingthemeasurementpointsbeyond8has onlyanmarginalincreaseintheaccuracyoftheinversemethodfortheseconditions. DatafortheL=10mm, DAQ =1 C,and8measurementpointsisfoundin Table5-3.Themagnitudeforallthreeerrorsisnearthesamerangeasthatfoundforthe DAQ =0.2 Ccase;thustheinversemethodisinsensitivetonoiseofapproximately0.7 %.Simulationsfor16measurementpointswerenotperformedforthiscaseduetothe resultsseenfor DAQ =0.2 C. TheerrorresultsforL=5mmand DAQ =0.2 CisfoundinTable5-4.Itisseen thatthesurfacetemperatureerrorforboth8and16measurementpointsisverylow,1 %orless.However,theerrorforBi max =0.5isanorderofmagnitudegreaterthanthe errorfoundfortheothertwocases.TheerrorforthetruncatedBiotnumberdistribution 138

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a b c Figure5-19.ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints. a e Bi ave rms ,b e Bi rms truncated,andc e s rms Table5-3.RMSerrorsforL=10mm, DAQ =1 C,andM=8. e s rms e Bi rms truncatede Bi ave rms 0.1121.1235.6170.1121.1235.6170.1121.1235.617 150.0170.0220.0150.2040.2530.2230.1660.1650.173 Bi max 50.0170.0170.0190.3410.3020.3710.3280.2870.327 0.50.0120.0120.0150.5020.6970.5290.4080.3280.475 with8measurementpointsrangesfrom10to30%formaximumBiotnumbersof 5and15,buttheerrorisverymuchincreasedfortheBi max =0.5case,essentially demonstratingthattheestimatedBiotnumberdistributionisuseless.Thetruncated 139

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Table5-4.RMSerrorsforL=5mmand DAQ =0.2 C. e s rms e Bi rms truncatede Bi ave rms 0.4494.49322.470.4494.49322.470.4494.49322.47 M=8 150.0030.0070.0110.1430.0940.3160.0220.0520.108 Bi max 50.0030.0050.0060.1660.1560.1210.0300.0600.051 0.50.0120.0110.0141.3811.0061.0711.1140.9600.984 M=16 150.0020.0070.0100.1700.1330.2340.0300.0360.136 Bi max 50.0020.0040.0060.1030.1000.1580.0210.0390.094 0.50.0050.0060.0090.3340.2080.2910.1740.1440.190 Table5-5.RMSerrorforL=10mm, DAQ =0.2 C,M=8,andvariousvaluesof forthe Biotnumberdistribution. Bi max Bi e s rms e Bi rms e Bi ave rms truncated 150.255.620.0180.1300.039 50.255.620.0150.0700.023 50.55.620.0190.0560.021 50.755.620.0210.0900.033 Biotnumberresultsfor16measurementpointsshowanerrorrangeof10to33% fortheentirerangeofmaximumBiotnumbers,indicatingthatforlowBiotnumber distributionsanincreaseinthenumberofmeasurementpointsiswarranted.The temporallyaveragedBiotnumbererrorshowssimilartrends,asexpected. Table5-5showserrorresultsfortheBiotnumberdistributionwith 6 = 0.1.The magnitudeoftheerrorforsurfacetemperatureissimilartothatseeninthe =0.1case. However,thetruncatedBiotnumberandtemporallyaveragedBiotnumbershowalarge increaseinaccuracy.Thereasonforthischangewasmentionedpreviously.Witha largervalueof theproblembehavesmorelikeaone-dimensionalestimationproblem, whichlesscomplex. Whendeterminingtheimpulseresponsefunctionsforthermocouplesthereis apotentialforerrorbetweentherealimpulseresponsefunctionofthesensorand 140

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Table5-6.RMSerrorforL=10mmandM=8withdifferentactualandsimulatedtime constants. Bi max Bi act sim e s rms e Bi rms e Bi ave rms truncated 150.10.1120.1070.0120.1480.133 150.10.1120.1180.0120.1470.129 150.11.1231.0110.0470.2900.262 150.11.1231.0670.0370.2410.207 150.15.6175.0550.0790.5990.568 150.15.6175.5050.0260.3930.346 150.255.6175.5050.0520.5040.426 150.255.6175.7290.0450.2020.135 50.11.1231.1800.0410.5040.473 50.11.1231.0670.0400.4750.459 theresultsofexperiments.Theseerrorscanbeproducedduetonotbeingableto accuratelydeterminewhentheexperimentwasinitiated,orerrorsduetotheboundary conditiononthetopsurfacenotbeingpreciselyknown.Todeterminetheaffectthese errorshaveondeterminingtheBiotnumberdistribution,simulationswerecarried outsuchthatthetimeconstantfortheinversealgorithm sim wouldbedifferentthan theactualtimeconstantforthesensors act .ThisstudywasrestrictedtotheL=10 mm, DAQ =0.2 Ccase,andtheerrorssimulatedrangedfrom1to10%.Theresults fromthisstudycanbefoundinTable5-6.Itisnoticedthatforsmalltimeconstants, thedifferenceisnotverynoticeable.However,asthemagnitudeofthetimeconstant increases,sodoesthethemagnitudeoftheerror.Foratimeconstantof5secondsthe resultsareessentiallyunusable.Clearlycaremustbetakentoensurethattheimpulse responsefunctionsareestimatedaccurately,especiallywhenthetimeconstantsare large. 5.2.4HeatLoss/GainEffects Whenperformingatransientjetimpingementheattransferexperimenttheeffectsof non-idealinsulationwillbepresent,asbrieydiscussedinSection5.1.3.Theseeffects willcausethedisctogainheatfromtheinsulationafterthejetisinitiated,withmore 141

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heatenteringonthebottomnearthemaximumradialdistanceduetotherebeingmore insulationtouchingthediscinthatarea.Whilethisheatgainisverylowinmagnitude andcausesonaslighterrorapproximately3%att=1seconditwilltendtohave alargerimpactonthedeterminationofthesurfacetemperatureandhenceheatux andBiotnumber.Theseeffectshavebeendemonstratedbysimulatingthediscbeing ataninitialdimensionlesstemperatureofzeroandhavingasurfacetemperatureof unitybeingappliedtoitandrecordingthetemperatureat8locationsonthebackside ofthedisc;thissimulationincludestheceramicinsulationandallowsheatexchange betweenitandthecopperdisc.Thesimulatedmeasurementsaretheninputintorst orderimpulseresponsemodel,andwhiteGaussiannoiseof0.2 Cthedimensional initialtemperatureis150 Cisadded.Thesedataaretheninputintoaninverseheat transferalgorithmwhichassumesidealinsulation. Theerrorresultsforthesurfacetemperatureare0.050,0.044,and0.047fortime constantsof0.1,1,and5seconds,respectively.Whilethiserrorissmalltherealeffect liesintheestimationoftheheatux,whichhasanon-uniformdistribution.Figure5-20 showsthedifferencebetweentheinverseandexactsolutions.Clearlytheerrorwill producenoticeableerrorinthecalculationofBiotnumber. 5.2.5EffectivenessoftheInverseHeatTransferAlgorithm Allofthemajorfactorsaffectingtheinverseheattransferhavebeenexploredand quantied.Fromtheresultsitisseenthatfortimesneart=0thereissignicanterrorin thereturnedsurfacetemperaturewhichiscarriedoverinthedeterminationofsurface heatuxandBiotnumber.Thisshowsthatthedevelopedheattransferalgorithmwillnot beeffectiveatdetectingphasechangeheattransferneartheinitialinstantintime,where itismostlikelytooccur,withoutmodication.Apossiblemethodtoovercomesomeof theseproblemswouldbetoinsertthethermocouplesintothethediscandhavethem locatedclosertothetopsurface.However,caremustbetakentoensurethatanyeffects ofthethermocoupleinsertionareeitherminimizedorareexplicitlytakenintoaccountin 142

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Figure5-20.Comparisonoftheresultsbetweenexactsolutionandinverseresultsfor thesurfaceheatuxduetoastepchangeinsurfacetemperature.Time= 0.375seconds, =1second. theinverseheattransferalgorithm.Whenthethermocouplesarelocatedclosertothe topsurfacethentheproblemcanalsobeformulatedintermsofanunknownsurface heatux,whichwillprovideamoreaccurateestimateofthesurfaceheatux,andboth inverseproblemscanbesolved.TheBiotnumbercanthenbedeterminedusingthis moreaccurateinformation. Theeffectsduetoerrorsintheestimatedimpulseresponsefunctionwereexplored anditisimperativethattheybedeterminedaccurately,especiallyifthetimeconstant ofthesystemisabove1secondorelsegrosserrorwillengulftheresults.Theeffects duetothemagnitudeofthenoisepresentwerefoundtobenegligibleformagnitudes approximatingthosefoundinalaboratorysetting.Thenumberofsensorsdoesnot seemtobeacontributingfactorunlessthediscisverythinhasasmallL/Rratio.The timeconstantofthemeasurementsystemdoesnothaveamajoreffectontheaccuracy ofthemethodunlessithassomeerrorinitsestimate. 143

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Figure5-21.CenterlineNu D forZ/D=6.0andRe D =7.25 10 5 asreturnedfromthe inverseheattransferalgorithm. Measurementsforthesupersonicimpingingjetfacilityweretakenandanattempt wasmadetodeterminetheBiotnumberdistribution.Figure5-21showsresultsforthe centerlineNu D withZ/D=6.0andRe D =7.25 10 5 ,Figure5-22showsthespatial distributionofNu D forthesameexperiment,noticethatduetotheheatgainedfrom insulationalongthesideofthedisccausesalargeovershootnearr =1.Similarresults werefoundinallexperimentsperformedandtheestimatesofNusseltnumberare grosslyinerror. 5.3Summary InthisChapterthethermocoupledynamicswereinvestigated.Fourseparate modelsforthemeasurementdynamicswereexplored:rstorder,secondorder,Rabin andRittel,andthetwoexponentialmodelwereintroduced.Anexperimentalsetup todeterminetheproperimpulseresponsemodelwasconductedandthemodelof RabinandRittelandthetwoexponentialordermodelproducedexcellentresultsby comparingwellbetweenthetheoreticaloutputandtheexperimentalmeasurement.The 144

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Figure5-22.SpatialdistributionofNu D forZ/D=6.0,Re D =7.25 10 5 ,andt =2.11as returnedfromtheinverseheattransferalgorithm. twoexponentialordermodelwaschosenasthebestmodeldoitsagreementwiththe impulseresponsefunctionasdeterminedbytheinversedeconvolutionmethod. Aparametricstudyoftheinversemethodwasconductedtoexploretheeffects of:discheight,theimpulseresponsefunction,themagnitudeandshapeoftheBiot numberdistribution,thenumberoftemperaturesensorsused,andthemagnitude ofthenoisefromthedataacquisitionsystem.Theerrorwasassessedusingthe root-mean-squareerrorofthesurfacetemperaturedistributionaswellasatruncated BiotnumberdistributionandatemporallyaveragedBiotnumber.Itwasfoundthat increasingthenumberofthermocouplesforadischeightof10mmhadlittleinuence ontheaccuracyofthealgorithm.Foradischeightof5mmitwasseenthatincreasing thenumberofthermocoupleswoulddecreasetheerroroftheinversemethod.However, estimatingBiotnumberdistributionwithasmallmagnitudeprovedtohavesignicant error. Therelativeslownessofthesensordynamics,characterizedbyatimeconstant, didnotseemtohaveaneffectontheaccuracyoftheinversealgorithm.However,itwas 145

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seenthatforslowresponsetimes,asmallamountoferrorinthetimeconstantestimate willproducelargeerrorsintheBiotnumberdistribution.Thiseffectwasnotpresentin thecaseofsmalltimeconstants,itwasseentohaveanegligibleeffect.Theeffectofthe dataacquisitionsystemnoisehadnoeffectatmagnitudesexperiencedinalaboratory setting. Theeffectsofheatgainontheinverseproblemwereexplored.Itwasfoundthat althoughtheerrorinthesurfacetemperatureestimatewaslowapproxiamtely5%the errorinthesubsequentsurfaceheatuxcalculationwasverynoticeable.Measurements performedonthesupersonicjetimpingementfacilitywereunabletoproducereliable estimatesoftheBiotnumberdistribution. 146

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CHAPTER6 CONCLUSIONS Amulti-phasesupersonicimpingingjetfacilityforthermalmanagementhasbeen constructed.Thefacilityoperatesbytakinghighpressureairandreducingittoa pressurebetween1.0and2.4MPawhereitactsontopofawaterstoragetankand passesthroughamassowmeterandamixingchamber.Differentialpressurebetween thetopofthewaterstoragetankandthemixingchamberforcesthewaterthrough aregulatingoricewhereitbecomesmixedwiththeairinthemixingchamber;the resultingmixtureisthenexpandedthroughaconverging-divergingnozzle,withadesign Machnumberof3.24.Itisthendirectedtoasurfacewhereitimpingesandremoves heat.Thefacilitysub-systemsinclude:airstorage,waterstorageandowcontrol, airpressurecontrol,airmassowmeasuring,converging-divergingnozzle,anddata acquisition.Temperature,pressure,andMachnumberwerecalculatedthroughout thefacilityusingone-dimensionalgasdynamicrelations.Itwasfoundthatatnormal operatingconditionsnoshockwavewouldbepresentintheconverging-divergingnozzle andthustheexitingair/liquidmixtureisatsupersonicspeeds.However,becausethe operationalsupplypressureislowerthanideal,thenozzleoperatesinanoverexpanded manner. Steadystateheattransfermeasurementswereperformedfortheimpinging jetfacility.Theseexperimentswereperformedbyallowingthejettoimpingeon anichromestripwithcurrentpassingthroughitandthermocouplesrecordingthe backwalltemperatures.Thedatawerethentimeaveragedtoeliminatenoiseand turbulentuctuations.Bothsingle-phaseandmulti-phaseexperimentswereperformed inordertoassesstheheatremovalcapabilitiesofthemulti-phasejetandtoquantify heattransferenhancementwiththeadditionofthesmallamountofdispersedliquid droplets.Oncethedropletsimpingeonthesurface,athinliquidlmformsontopofthe surface.Theheattransfercharacteristicsofthejetaredifferentneartheimpingement 147

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zone,wheredropletimpactdominates.Awayfromtheimpingementzone,thinlm dynamicsdominatetheheattransferprocess.Thepeakheattransfercoefcientsfor themulti-phasejetexceed200,000W/m 2 -Knearthecenterline,matchingsomeof thehighestheattransferratesreportedintheliterature,whilesimultaneouslyhaving asignicantlyreducedliquidowrate.Thisreducedowratecanbeadvantageousin someindustrialsettings. ItwasfoundthatincreasingtheReynoldsnumberofthejetandtheliquidmass fractionofwaterincreasestheNusseltnumberofthejetnearthecenterline.These effectsarediminishedfarremovedfromtheimpingementzone.Nozzleheightisseen tohaveaslighteffectoftheNusseltnumberneartheinteriorofthejet,butnodenite trendisapparent,andisseentohavenosignicantimpactawayformthecenterline. ThesinglephaseimpingingjetNusseltnumberscomparewellwithdataintheliterature. Comparisonbetweenthesingleandmulti-phasejetsshowsthattheadditionoftheliquid dropletstothejetenhancesheattransferbyanorderofmagnitudeintheinteriorofthe jetandbyafactoroftwotoveawayformthecenterline. Thereisnoevidenceofsignicantphasechangeheattransferoccurringduringthe jetimpingementstudies.Analysisofthesaturatedhumidityrationearthecenterlineof thejetshowsthatlittleevaporationispossibleanditseffectsareessentiallynegligible. Determinationofthehumidityratiofarremovedformthecenterlineofthejetisnot possibleduetothelackofinformationregardingthesurfacetemperatureofthelm. However,theNusseltnumberinthisregionremainsessentiallyconstant;signicant evaporationwouldresultinincreasedNusseltnumberinthisregion.Thelackofthistype oftrendsupportsthecontentionthatthereisnosignicantphasechangeheattransfer takingplace.Theuseofdropletimpacttheorysuggeststhatheattransferenhancement inthejetinteriorduetophasechangemaynotbepossible.Thesaturationpressure issignicantlyelevatedduetothehighimpactvelocityofthedropletsandwillexceed thecriticalpressureforthenominalexitvelocityofthenozzle.Thusanyphasechange 148

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occurringwouldnotresultinadditionalheatremovalasthelatentheatofvaporization abovethecriticalpressureiszero. Aninverseheattransferalgorithmusingtheconjugategradientmethodwithadjoint problemwasdevelopedwhichcandeterminetheNusseltnumberforanimpinging jet.Thismethodexplicitlytakesintoaccountsensordynamics,afeaturethatisnot presentininversemethodsfoundintheliterature.Aproceduretodeterminetheimpulse responsefunctionofthermocouplesattachedtothesurfaceofasolidwasdeveloped usingicetocauseastepchangeinsurfacetemperature.Theresultingdataisthen inputintoadeconvolutionalgorithmwhichreturnstheimpulseresponsefunctionviaan inversemethod.Thisallowsinsightintotheunderlyingfunctionalformandallowsthe properimpulseresponsefunctionmodeltobefound.Fourdifferentimpulseresponse functionmodelswereexplored:arstordermodel,asecondordermodel,themodel ofRabinandRittel,andthetwoexponentialmodel.Itwasfoundthatthemodelof RabinandRittelandthetwoexponentialmodelproducedthebestagreementwhen comparedtoexperimentalresults.However,duetothefactthatthetwoexponential modelproducedbetteragreementbetweentheimpulseresponsefunctionsdetermined viacurve-tandinversealgorithm,itwaschosenasthebestmodelforthesurface mountedthermocouplesusedinthoseexperiments. Aparametricstudywasperformedusingsimulateddatatoexploretheeffectsofthe discheightandaspectratio,themagnitudeandshapeoftheBiotnumberdistribution, thenumberofsensorsused,theimpulseresponsefunction,andthemagnitudeof thenoiseofthedataacquisitionsystemontheerrorassociatedwiththeIHTmethod. Toassesstheerrorintheinverseheattransferalgorithmthreedifferenterrorswere considered,thermserrorbetweeninputsurfacetemperatureandthatreturnedbythe inversemethod,thermserrorbetweentheinputBiotnumberandthatreturnedbythe inversemethodduringatruncatedtimedomainwhenthebehaviorofthereturnedBiot numberwasnearlyconstant,andthermserrorbetweentheinputBiotnumberandthe 149

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temporallyaverageBiotnumberoverthesametruncatedtimedomain.Theinverseheat transferalgorithmisspecicallydesignedtoestimatethesurfacetemperatureandas suchtheerrorinitsestimatewasverysmall,ontheorderofafewpercent.Theerror forestimatingBiotnumberwasnoticeablyhigher,rangingfrom10%toover80%in somecases.Mostofthefactorsinvestigatedappeartohavelittleeffectontheinverse heattransfermethod.However,themagnitudeoftheBiotnumberdistributionhada largeeffect.AlowerBiotnumberproducessmallchangesinmeasuredtemperatureand thusisadifcultestimationproblemandanyerrorinthesurfacetemperatureestimation iscarriedoverandampliedinthesurfaceheatuxcalculation.Errorsintheimpulse responsefunctionwereseentohavealargeeffectincertaincases.Iftheoveralldelay intheimpulseresponsefunctionisshort,thenthemethodcantoleratesomeerror. However,iftheoveralldelayisashighas5seconds,thenevenanerrorassmallasone percentcanrendertheresultsoftheinverseheattransferalgorithmuseless. Finallytheeffectsofheatloss/gainfrominsulationontheimpingementtarget wereinvestigated.Thedifferenceintemperatureatthesimulatedmeasurementpoints betweenanidealizeddiscandonewithheattransferto/fromitsinsulationwassmall, approximately3%.However,thissmallerrorismagniedbytheinversealgorithm, causingittooverestimatetheheatuxneartheedgeofthedisc.Thesendingwere veriedbyperformingexperimentsusingthemulti-phasejetimpingementfacility. Inorderfortheinversemethodtobeeffective,theheatexchangewiththe insulationmustbetakenintoaccount.Theeffectsoftheheatexchangecouldalso belessenedbylocatingthetemperaturesensorsclosertothetopsurface.However, insertingthermocouplesinsideofthediscislikelytodistorttheheatowandsosome compromisemustbemade. 150

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APPENDIXA COMPLETESTEADYSTATETWO-PHASEHEATTRANSFERJETRESULTS Therearemanyheattransferdatasetsforthesteadystateexperimentsdiscussed inChapter3whicharenotpresentedduetospaceconsiderationsandbecauseallof theresultsfollowthetrendsdiscussedtherein.Forthesakeofcompleteness,theheat transferresultsforthesteadystatetwo-phaseimpingingjetresultsarepresentedinthis Appendix. ForFiguresA-1throughA-3thesameoriceisusedforeachexperiment,thus keepingthewatermassfractionnominallyconstant.JetReynoldsnumberisheld constantaswell,andthenozzlespacingisvaried. ForFiguresA-4throughA-7thenozzlespacingandReynoldsnumberareconstant. Theoriceischangedtovarywatermassfraction. ForFiguresA-8throughA-10thesameoriceisusedforeachexperiment,thus keepingthewatermassfractionandjetReynoldsnumbernominallyconstant,andthe nozzlespacingisvaried. ForFiguresA-11throughA-14thenozzlespacingandReynoldsnumberare constant,andtheoriceischangedtovarywatermassfraction. 151

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a b c d FigureA-1.Two-phaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0205,Re D =4.54 10 5 ,bw=0.0240,Re D =4.47 10 5 ,cw=0.0357, Re D =4.42 10 5 ,anddw=0.0375,Re D =4.42 10 5 152

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a b c d FigureA-2.Two-phaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0156,Re D =7.34 10 5 ,bw=0.0189,Re D =7.29 10 5 ,cw=0.0273, Re D =7.23 10 5 ,anddw=0.0290,Re D =7.24 10 5 153

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a b c d FigureA-3.Two-phaseNu D resultsforvariousnozzleheighttodiameterratios.aw= 0.0131,Re D =1.02 10 6 ,bw=0.0163,Re D =1.01 10 6 ,cw=0.0234, Re D =1.02 10 6 ,anddw=0.0248,Re D =1.01 10 6 154

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a b c FigureA-4.Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=2.0.a Re D =4.42 10 5 ,bRe D =7.20 10 5 ,andcRe D =1.01 10 6 155

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a b c FigureA-5.Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0.a Re D =4.43 10 5 ,bRe D =7.35 10 5 ,andcRe D =1.03 10 6 156

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a b c FigureA-6.Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=4.0.a Re D =4.45 10 5 ,bRe D =7.24 10 5 ,andcRe D =1.02 10 6 157

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a b c FigureA-7.Two-phaseNu D resultsforvariousliquidmassfractionsandZ/D=6.0.a Re D =4.56 10 5 ,bRe D =7.31 10 5 ,andcRe D =1.01 10 6 158

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a b c d FigureA-8.Two-phaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0205,Re D =4.54 10 5 ,bw=0.0240,Re D =4.47 10 5 cw=0.0357,Re D =4.42 10 5 ,anddw=0.0375,Re D =4.42 10 5 159

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a b c d FigureA-9.Two-phaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0156,Re D =7.34 10 5 ,bw=0.0189,Re D =7.29 10 5 cw=0.0273,Re D =7.23 10 5 ,anddw=0.0290,Re D =7.24 10 5 160

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a b c d FigureA-10.Two-phaseenhancementratioresultsforvariousnozzleheighttodiameter ratios.aw=0.0131,Re D =1.02 10 6 ,bw=0.0163,Re D =1.01 10 6 cw=0.0234,Re D =1.02 10 6 ,anddw=0.0248,Re D =1.01 10 6 161

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a b c FigureA-11.Two-phaseenhancementresultsforvariousliquidmassfractionsandZ/D =2.0.aRe D =4.42 10 5 ,bRe D =7.20 10 5 ,andcRe D =1.01 10 6 162

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a b c FigureA-12.Two-phaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=4.0.aRe D =4.43 10 5 ,bRe D =7.35 10 5 ,andcRe D =1.03 10 6 163

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a b c FigureA-13.Two-phaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=4.0.aRe D =4.45 10 5 ,bRe D =7.24 10 5 ,andcRe D =1.02 10 6 164

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a b c FigureA-14.Two-phaseenhancementratioresultsforvariousliquidmassfractionsand Z/D=6.0.aRe D =4.56 10 5 ,bRe D =7.31 10 5 ,andcRe D =1.01 10 6 165

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APPENDIXB COMPLETEINVERSEHEATTRANSFERALGORITHMERRORASSESSMENT CONTOURPLOTS Therearemanydatasetsforthethreedifferenterrorsusedtoassessthe effectivenessoftheinverseheattransferalgorithmandassuchallofthemwere notincludedinChapter5.Intheinterestofcompletenesstheyareincludedinthis Appendix.FigureB-1showstheerrorcontoursfortheL=5mm, DAQ =0.2 C,and8 measurementpointscase.FigureB-2showstheerrorcontoursfortheL=5mm, DAQ = 0.2 C,and16measurementpointscase.FigureB-3showstheerrorcontoursfortheL =10mm, DAQ =0.2 C,and8measurementpointscase.Lastly,FigureB-5showsthe errorcontoursfortheL=10mm, DAQ =1 C,and8measurementpointscase. 166

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a b c FigureB-1.ErrorContoursforL=5mm, DAQ =0.2 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 167

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a b c FigureB-2.ErrorContoursforL=5mm, DAQ =0.2 C,and16measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 168

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a b c FigureB-3.ErrorContoursforL=10mm, DAQ =0.2 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 169

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a b c FigureB-4.ErrorContoursforL=10mm, DAQ =0.2 C,and16measurementpoints. a e Bi ave rms ,b e Bi rms truncated,andc e s rms 170

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a b c FigureB-5.ErrorContoursforL=10mm, DAQ =1 C,and8measurementpoints.a e Bi ave rms ,b e Bi rms truncated,andc e s rms 171

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APPENDIXC IMAGESOFORIFICESUSEDDURINGEXPERIMENTS Duringoperationofthejetfacilityitwasnoticedthatthe0.38mmoriceprovided alowerliquidowratethanthe0.33mmorice.Afterexaminingthedifferentorices underanopticalmicroscopeitwasdiscoveredthatthe0.38and0.51mmoriceswere notperfectlycircular.Thiseccentricityisverynoticeableinthe0.38mmoriceand mayresultindifferentspraycharacteristics,whichcouldhaveaffectedtheheattransfer results.Theeccentricityisnotassevereinthe0.51mmoriceanddidnotseemto affecttheheattransferresults.FiguresC-1throughC-4showimagesoftheorices under4Xmagnication. FigureC-1.The0.33mmorice. 172

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FigureC-2.The0.37mmorice,notethehighdegreeofeccentricityintheorice FigureC-3.The0.41mmorice. 173

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FigureC-4.The0.51mmorice,notetheeccentricityintheorice 174

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BIOGRAPHICALSKETCH RichardParkerwasborninOrlando,FLandpromptlymovedtotheLouisville,KY area.AfterhisfamilyreceivedatransfertoColumbia,SCwhenhewas9yearsoldhe spenttheremainderofhischildhoodthere.Duringhissenioryearinhighschoolhe decidedtojointheUnitedStatesNavyasitwouldhelphimattainsomemuchneeded discipline. WhileintheUSNavyRichardwaspartoftheNavalNuclearPowerProgramand graduatedfromthenuclearmechanicprogramandwasthenselectedfornuclear chemistryschool.Aftercompletinghisnearlytwoyearlongschoolprogramhewas stationedinPearlHarbor,HIonboardtheUSSKeyWestSSN-722.Whileservingin ReactorLaboratoriesDivisionheexcelledathisdutiesandwasawardedthreeNavy andMarineCorpsAchievementMedalsalongwithotheraccolades,includingbeing promotedtoPettyOfcerFirstClassE6inunder6yearsofservice.Whileonboard theKeyWestRichardtookpartinOperationEnduringFreedomaftertheSeptember11, 2001terrorattacksonUSsoil. RealizingthathelikedtoexplorescienceandmathledRichardtoleavetheNavy afterhisinitialcommitmentwasconcluded.Havingfoundheenjoyedtheareaof uidmechanicsandheattransferhedecidedtostudymechanicalengineeringatthe UniversityofSouthCarolinawherehegraduated summacumlaude in2006.Realizing therewasmuchmoretolearnRicharddecidedtopursuegraduatestudiesatthe UniversityofFloridainmechanicalengineeringwhereheearnedhisPh.D.inthermal anduidsciences. 184

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