Analysis of Mass Transfer by Jet Impingement and Heat Transfer by Convection

Analysis of Mass Transfer by Jet Impingement and Study of Heat Transfer in a Trapezoidal Microchannel

Ejiro Stephen Ojada

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering University of South Florida

Major Professor: Muhammad M. Rahman, Ph.D. Frank Pyrtle, III, Ph.D. Rasim Guldiken, Ph.D.

Date of Approval: November 5, 2009

Keywords: Fully-Confined Fluid, Sherwood number, Rotating Disk, Gadolinium, Heat Sink

TABLE OF CONTENTS

LIST OF FIGURES ii

LIST OF SYMBOLS v

ABSTRACT viii

CHAPTER 1: INTRODUCTION AND LITERATURE REVIEW 1 1.1 Introduction (Mass Transfer by Jet Impingement ) 1 1.2 Literature Review (Mass Transfer by Jet Impingement) 2 1.3 Introduction (Heat Transfer in a Microchannel) 7 1.4 Literature Review (Heat Transfer in a Microchannel) 8

CHAPTER 2: ANALYSIS OF MASS TRANSFER BY JET IMPINGEMENT 15 2.1 Mathematical Model 15 2.2 Numerical Simulation 19 2.3 Results and Discussion 21

CHAPTER 3: ANALYSIS OF HEAT TRANSFER IN A TRAPEZOIDAL MICROCHANNEL 42 3.1 Modeling and Simulation 42 3.2 Results and Discussion 48

CHAPTER 4: CONCLUSION 65

REFERENCES 67

APPENDICES 79 Appendix A: FIDAP Code for Analysis of Mass Transfer by Jet Impingement 80 Appendix B: FIDAP Code for Fluid Flow and Heat Transfer in a Composite Trapezoidal Microchannel 86

LIST OF FIGURES

Figure 2.1a Confined liquid jet impingement between a rotating disk and an impingement plate, two-dimensional schematic. 18

Figure 2.1b Confined liquid jet impingement between a rotating disk and an impingement plate, three-dimensional schematic. 18

Figure 2.2 Mesh plot for a grid spacing of 20 x 500 in the axial and radial directions. 21

Figure 2.3 Dimensionless interface temperature distributions for different number of elements in r and z directions (Rej=1000, Rer=2310, =1.0, and Sc=2315). 22

Figure 2.4 Local Sherwood number and dimensionless concentration for different jet Reynolds numbers (=1.0, Rer=2310, nickel disk, and Sc=2315). 26

Figure 2.5 Local Sherwood number and dimensionless concentration for different rotational Reynolds numbers (=1.0, nickel disk, Rej=800, and Sc=2315). 27

Figure 2.6 Local Sherwood number and dimensionless concentration for different dimensionless heights (Rej=1500, Rer=2310, and Sc=2315). 28

Figure 2.7 Average Sherwood numbers with jet Reynolds numbers at various rotational Reynolds numbers (=1.0, nickel disk, and ferricyanide electrolyte and Sc=2315). 29

Figure 2.8 Average Sherwood number with jet Reynolds number at different dimensionless nozzle to target spacing (Rer=2310, and Sc=2315). 32

Figure 2.9 Average Sherwood number with jet Reynolds number at different Schmidt number (=1.15, nickel disk, ferricyanide electrolyte, and Rer=5082). 33

Figure 2.10 Dimensionless concentration for different electrolytes - 3 NaOH, HCl, Fe(CN)6 (Rej=650, Rer=0, and =1.0). 34

Figure 2.11 Local Sherwood number and dimensionless concentration over dimensionless radial distance for different impingement plate geometries, nickel disk with ferricyanide electrolyte (Rej=1000, Rer=5082, =1.15, and Sc=2315). 35

Figure 2.12 Average Sherwood number comparison with the experimental results obtained by Arzutuğ et al. [13] under various jet Reynolds and Schmidt numbers (left: =1.0, Sc=2315, Rer=2310; right: (=1.0, Rej=1000, and Rer=5082). 37

Figure 2.13 Average Sherwood number comparison with other studies within the core region for various Schmidt numbers ( =1.0, Rej=800, Rer=2310). 38

Figure 2.14 Comparison of predicted average Sherwood number using Equation 17 with present numerical data. 39

Figure 3.1 Schematic of the trapezoidal microchannel. 46

Figure 3.2 Average dimensionless interface temperature over the dimensionless axial coordinate for different grid sizes (magnetic field = 5T, Re =1600, D=150 μm, L=2.3 cm). 49

Figure 3.3 Variation of peripheral average Nusselt number along the channel dimensionless axial coordinate for different Reynolds number (Magnetic field = 5T, D=150 μm, L=2.3 cm). 50

Figure 3.4 Variation of peripheral average dimensionless interface temperature along the channel dimensionless axial coordinate for different Reynolds number and different magnetic fields (D=150 μm, L=2.3 cm). 51

Figure 3.5 Variation of Nusselt number along the channel dimensionless axial coordinate for different heat generation rates (Re = 1600, 2400, 3000, D=150 μm, L=2.3 cm). 52

Figure 3.6 Variation of peripheral average dimensionless interface temperature along the channel dimensionless axial coordinate for different thickness of the magnetic slab (Re= 1600, Magnetic field = 5T, D=150 μm, L=2.3 cm). 54

iii

Figure 3.7 Variation of Nusselt number along the channel dimensionless axial coordinate for different depths of the channel (Magnetic field =5T, Constant velocity). 55

Figure 3.8 Average dimensionless interface temperature numbers with Reynolds numbers at various channel depths (Magnetic field =5T, 1000 < Re < 3000). 56

Figure 3.9 Variation of Nusselt number over the dimensionless axial coordinate for different spacing between the channels (Re = 2400, magnetic field = 5T, D=150 μm, L=2.3 cm). 57

Figure 3.10 Comparison between simulation and standard convection relation (Dh=154μm, Magnetic field = 5T, D=150 μm, L=2.3 cm). 59

Figure 3.11 Variation of dimensionless pressure difference between inlet and outlet of the channel with Reynolds number (Magnetic field = 5T, L=2.3 cm). 60

Figure 3.12 Variation of the Nusselt numbers for different working fluids (Magnetic field = 5T, D=150 μm, L=2.3 cm). 61

Figure 3.13 Variation of dimensionless pumping power for different Reynolds numbers (Magnetic field = 5T, L=2.3 cm). 63

Figure 3.14 Average Nusselt number comparison with the experimental results obtained by Wu and Chen [38] under various jet (Magnetic field = 5T, 40 < Re < 200, D=150 μm, L=4.0 cm). 64

LIST OF SYMBOLS b Height of gadolinium slab, m B Substrate width, m

Bc Channel width, m

Bs Half of channel width (at bottom), m r C Average interface concentration, kg/m3, 2 d avg  C  r  dr 2  rd 0

Cp Constant pressure specific heat, J/kg-K C Solute concentration, kg/m3 dn Diameter of nozzle, m dd Diameter of rotating disk, m D Diffusion coefficient, m2/s Dp Dimensionless pump power

Dh Hydraulic diameter, m g Acceleration due to gravity, m/s2 go Heat generation rate within the solid, W/m3 G Mass flux, kg/m2.s 2 h Heat transfer coefficient, qint / (Tint−Tb), W/m K H Distance of the nozzle from the rotating disk, m H Height of substrate, m k Thermal conductivity, W/m K; or Turbulent kinetic energy, m2/s2

K Mass transfer coefficient, G/ (Cint–Cj), m/s L Channel length, m Nu Nusselt number P Pressure, N/m2 v

Pd Dimensionless pressure

Pp Pumping power

Prt Turbulent Prandtl number q Heat flux, W/m2

Q Dimensionless Concentration, 2(Cint – Cj) / (Gdn) r Radial coordinate, m

R Radius of rotating disk, m

Re Reynolds number, (Vjdn)/ or ρf .u in .Dh /μ

Sh Sherwood number, (kdn)/D S Spacing between channels, m T Temperature, ⁰C ΔT Temperature difference between inlet and outlet, °C

Vj Jet velocity, m/s

Vr, z,  Velocity component in the r, z and –direction, m/s, in cylindrical coordinate u Velocity component in x-direction, m/s v Velocity component in y-direction, m/s w Velocity component in z-direction, m/s z Axial coordinate, m Greek Symbols α Thermal diffusivity, m2/s

 Dimensionless nozzle to target spacing, H/dn ϵ Dissipation rate of turbulent kinetic energy, m2/s3  Dynamic viscosity, kg/m–s  Kinematic viscosity, m2/s 2 νt Eddy diffusivity, m /s  Angular coordinate, rad

 Dimensionless interface temperature  Density, kg/m3  Angular velocity, rad/s vi

Subscripts b Bulk avg. Average f Fluid g Gadolinium i Species i in Inlet int Interface j jet inlet max Maximums si Silicon s Solid

vii

Analysis of Mass Transfer by Jet Impingement and Study of Heat Transfer in a Trapezoidal Microchannel

Ejiro S. Ojada

ABSTRACT

This thesis numerically studied mass transfer during fully confined liquid jet impingement on a rotating target disk of finite thickness and radius. The study involved laminar flow with jet Reynolds numbers from 650 to 1500. The nozzle to plate distance ratio was in the range of 0.5 to 2.0, the Schmidt number ranged from 1720 to 2513, and rotational speed was up to 325 rpm. In addition, the jet impingement to a stationary disk was also simulated for the purpose of comparison. The electrochemical fluid used was an electrolyte containing 0.005moles per liter potassium ferricyanide (K3(Fe(CN6)),

-4 0.02moles per liter ferrocyanide (FeCN6 ), and 0.5moles per liter potassium carbonate

(K2CO3). The rate of mass transfer of this electrolyte was compared to Sodium

Hydroxide (NaOH) and Hydrochloric acid (HCl) electrochemical solutions. The material of the rotating disk was made of 99.98% nickel and 0.02% of chromium, cobalt and aluminum. The rate of mass transfer was also examined for different geometrical shapes of conical, convex, and concave confinement plates over a spinning disk. The results obtained are found to be in agreement with previous experimental and numerical studies.

viii

The study of heat transfer involved a microchannel for a composite channel of trapezoidal cross-section fabricated by etching a silicon <100> wafer and bonding it with a slab of gadolinium. Gadolinium is a magnetic material that exhibits high temperature rise during adiabatic magnetization around its transition temperature of 295K. Heat was generated in the substrate by the application of magnetic field. Water, ammonia, and FC-

77 were studied as the possible working fluids. Thorough investigation for velocity and temperature distribution was performed by varying channel aspect ratio, Reynolds number, and the magnetic field. The thickness of gadolinium slab, spacing between channels in the heat exchanger, and fluid flow rate were varied. To check the validity of simulation, the results were compared with existing results for single material channels.

Results showed that Nusselt number is larger near the inlet and decreases downstream.

Also, an increase in Reynolds number increases the total Nusselt number of the system.

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

1.1 Introduction ( Mass Transfer by Jet Impingement)

Jet impingement is a technique used in the industry to enhance heat and/or mass transfer processes. It provides the opportunity to control temperature and/or concentration to the desired needs. A few examples are paper drying process, material removal in steel mills, tempering of glass, cooling of high temperature gas turbines and electronic fabrication of printed wiring board components. The rotation of a disk also plays a role in enhancing the heat and mass transfer by inducing a secondary flow. The rotating disk enhances the wall jet effect at the interface which adds more complexity to the flow field and more mixing with the impinging jet flow.

1.2 Literature Review (Mass Transfer by Jet Impingement)

An early research work on mass transfer from a rotating disk was performed by

Kreith et al. [1] who studied the effect of a shroud on the mass transfer rate from a rotating disk in the laminar regime, having rotational Reynolds numbers ranging from

70000–140000. The correlation obtained did not account for the distance between the disk and the shroud. Nakoryakov et al. [2] studied theoretically and experimentally the hydrodynamics and mass transfer of a submerged liquid jet impinging onto a horizontal plane. They measured the wall shear stress and local mass transfer coefficients by an electro-diffusion method for a wide range of liquid flow rates. Chin and Tsang [3] studied the mass transfer from an impinging jet to the stagnation region on a circular disk electrode using the method of perturbation. They found out that within the radius, r/dd, from 0.1 to 1.0 turbulent nozzle flow and from 0.1 to 0.5 for laminar nozzle flow, the electrode has a “uniform accessibility” to the diffusion ions. The mass transfer rate begins to decrease beyond the uniform accessibility region. The impingement of two dimensional slot jet flows for high speed selective electroplating was studied by Alkire and Ju [4]. They measure local mass transfer coefficient for the system when it is submerged and when it is not. They also developed correlations for three regions: impingement, transition and wall jet flow regions. Chin and Agarwal [5] studied the local mass transfer rate of a submerged oblique impinging slot jet by electrochemical limiting current technique for the reduction of ferricyanide ion at isolated microelectrodes on the impinged surface. An electrochemical probe was used to measure the mass transfer coefficient. Moreno et al. [6] studied the mass transfer of an impinging liquid jet

2 confined between two parallel plates theoretically and by experiments. The mass transfer rate was characterized by an etching method in a cupric chloride etching solution and jet instability on the etching rate within the central impingement zone was discussed. Chen et al. [7] investigated experimentally the mass transfer between an impinging jet and a rotating disk. The naphthalene sublimation technique was used in the experiment. The experimental results showed that heat/mass transfer are divided into three regions which are the impingement dominated region, the mixed region and the rotation dominated region. It was concluded that the Sherwood number of a rotating disk with jet impingement was the sum of two components governed by the impinging jet and the rotating disk. Pekdemir and Davies [8] studied the mass transfer behavior of an isothermal system when a rotating circular cylinder is exposed to a two dimensional slot jet of air with a laminar flow. In the impingement dominated regime, they observed that the rotation of disk did not influence heat transfer characteristics of the system, while the jet impingement had a strong effect on the local heat transfer of the rotating disk. Chen and Modi [9] investigated the mass transfer characteristics of a turbulent slot jet impinging normally on a target wall with a confinement plate placed parallel to the target plate examined using numerical simulations. The flow was modeled using a k-w turbulence model. The Reynolds number simulated ranged from 450 to 20000, Prandtl or

Schmidt numbers from 0 to 2400 and the slot jets varied between 2 and 8 times the width of the slot jet. Chen et al. [10] conducted experiment on mass and heat transfer for high

Schmidt numbers with a laminar jet impingement flow onto rotating and stationary disks.

The experiment used naphthalene sublimation technique where three regimes where

3 observed, namely the impingement dominated regime, the mixed regime and the rotation dominated regime. Using a conical shaped impingement plate, Miranda and Campos [11] investigated mass transfer in a laminar by impinging jet. The distance between the nozzle and the plate was less than one nozzle diameter, the laminar flow was less than 1600, and the Schmidt was up to 50000. Oduoza [12] worked on mass transfer on a heated electrode by simulating high speed wire plating with simultaneous heat transfer in the laminar region. The working fluid used for the study was ferricyanide. In the simulation, it showed a distinct effect of thermally driven natural convection at a lower Reynolds number and but as the Reynolds number increased, it merged with the Leveque solution.

Arzutuğ et al. [13] compared the mass transfer distribution from a jet to a plate between a submerged conventional impinging jet (CIJ) and multichannel conventional impinging jet

(MCIJ). Electrochemical limiting diffusion current technique (ELDCT) was used to measure the local mass transfer coefficients. The values that were obtained for the mean mass transfer coefficients over the surface for CIJ and MCIJ were found to be relatively close to each other with MCIJ having slightly higher values. Quiroz et al. [14] also used the electrochemical limiting diffusion current technique (ELDCT) to measure the mass transfer between parallel disk cells with the help of the Levique relation. Sedahmed et al.

[15] studied the rate of mass transfer between two immiscible liquids, an aqueous layer and a mercury pool upon which an axial jet was impinging under turbulent flow conditions and measured by electrochemical limiting diffusion limiting current technique

(ELDCT).

Sara et al. [16] measured the mass transfer coefficient using the ELDCT method of an electrochemical system from an impinging liquid jet to a rotating disk in a fully confined environment. The study used a rotational Reynolds number of up to 120000, and a jet Reynolds number of up to 53000 with a non-dimensional jet-to-disk spacing of

2-8. They found out that the jet impingement had a considerable effect on the enhancement of the mass transfer compared to the case of the rotating disk without jet.

The effects on mass/heat transfer on rotation by impingement jet were also studied by

Hong et al. [17]. Their research covered a wide range of rotational Reynolds numbers

(400 to 10,000) including laminar, turbulent and transitional regimes. Hong et al. [18] investigated the mass transfer characteristics on a concave surface for rotating impinging jets. A jet with Reynolds number of 5,000 was applied to the concave surface and a flat surface. They found out that compared to flat surface, the heat/mass transfer on the concave surface is enhanced with increasing the span-wise direction due to the curvature effect, providing a higher averaged Sherwood value.

Research has also been done involving heat transfer in jet impingement processes.

Lallave et al [19] studied the characterization of conjugate heat transfer for a confined liquid jet impinging on a rotating and uniformly heated solid disk of finite thickness and radius. The study showed that the plate materials with higher thermal conductivity had a more uniform temperature distribution at the solid–fluid interface, and the local heat transfer coefficient increased with an increasing in Reynolds number which reduced the wall to fluid temperature difference over the entire interface. Lallave and Rahman [20]

5 worked on conjugate heat transfer characterization of a partially–confined liquid jet impinging on a rotating and uniformly heated solid disk of finite thickness and radius.

Even though a number of publications have considered the heat/mass transfer rate effect of numerous parameters, not enough research has been done on mass transfer during laminar jet impingement on a rotating disk in a fully confined environment using an electrolyte. The intent of this research is to investigate the mass transfer effect in a uniform laminar flow from the jet nozzle onto a rotating disk in a fully confined space.

The study parameter includes five jets Reynolds numbers, five rotational Reynolds numbers and stationary disk, five heights measured from the nozzle to the target disk, five Schmidt numbers, and different confinement plate shapes such as conical, convex, and concave.

Present results offer a better understanding of the fluid mechanics and mass transfer behavior of liquid jet impingement under confinement on top of a spinning target because of the incorporation of the varying parameters. Even though no new numerical technique has been developed, results obtained in this investigation are entirely new. The numerical results showing the quantitative effects of different parameters as well as the correlation for average Sherwood numbers will be practical guides for enhancement of mass transfer during the electrolyte synthesis and etching processes.

1.3 Introduction (Heat Transfer in a Microchannel)

Microchannels of trapezoidal cross section are widely used in silicon-based microsystems. The study of fluid flow and heat transfer is critical to the development of these microsystems. This thesis presents a systematic analysis of fluid flow and heat transfer processes during the magnetic heating of a magnetocaloric material which is bonded to the substrate. The substrate has an array of trapezoidal channels through which heat is transferred to the working fluid. When a magnetic field is imposed on a magnetocaloric material, heat is generated. This results in increase in temperature of the material. Similarly, the temperature drops during demagnetization when the field is removed. The purpose of this thesis is to study the effects of change in different geometrical and thermal parameters on fluid flow and heat transfer when a magnetic field is applied to the substrate material.

1.4 Literature Review (Heat Transfer in a Microchannel)

Wu and Little [31] measured the friction factors of laminar gas flow in the trapezoidal silicon/glass microchannels, and found that the surface roughness affected the values of the friction factors even in the laminar flow, which is different from the conventional macrochannel flow. Harley et al. [32] presented experimental and theoretical results of low Reynolds number, high subsonic Mach number, compressible gas flow in channels. Nitrogen, helium, and argon gases were used. Detailed data on velocity, density and temperature distributions were obtained. The effect of the Mach number on profiles of axial and transversal velocities and temperature were revealed.

Chen and Wu [33] investigated the microchannel flow in miniature TCDs (thermal conductivity detectors). Effects of channel size and boundary conditions were examined in details. It was found that the change in heat transfer rate in the entrance region depends primarily on the thermal conductivity change in the conduction-dominant region. Qu et al. [34] investigated heat transfer characteristics of water flowing through trapezoidal silicon microchannels. A numerical analysis was carried out by solving a conjugate heat transfer problem.

Rahman [35] presented new experimental measurements for pressure drop and heat transfer coefficient in microchannel heat sinks. Tests were performed with devices fabricated using standard Silicon <100> wafers. Channels of different depths (or aspect ratios) were studied. Tests were carried out using water as the working fluid. The fluid flow rate as well as the pressure and temperature of the fluid at the inlet and outlet of the device, and temperature at several locations in the wafer were measured. These

8 measurements were used to calculate local and average Nusselt number and coefficient of friction in the device. Toh et al. [36] studied the fluid flow and heat transfer in a microchannel by number computation. The results of the numerical computations where compared to experimental data for validation. Their research revealed that heat input lowers frictional losses at mostly lower Reynolds numbers since an increase in temperature leads to a decrease in viscosity thereby leading to smaller frictional losses.

Qu and Mudawar [37] also investigated the heat transfer behavior in a rectangular microchannel. They observed that when the thermal conductivity of a substrate is increased in which the fluid flows through while keeping all parameters constant, the temperature at the base surface of the heat sink reduces. They concluded that a higher laminar Reynolds number at 1400 will not be a fully developed flow in a microchannel and as a result will lead to enhanced heat transfer. Wu and Cheng [38] observed the same behavior of an approximate linear correlation between the Nusselt number and Reynolds number at Re < 100. They studied what effect the surface roughness of the microchannel and surfaces’ affinity for water (hydrophilic property) has on the Nusselt number. The investigation showed that there is an increase in the laminar Nusselt number when the surface roughness or hydrophilic property is increased. The apparent friction constant also increased with an increase in the surface roughness.

Wu and Cheng [39] measured the friction factor of laminar flow of deionized water in smooth silicon micro-channels of trapezoidal cross-section. The experimental data were found to be in agreement within ±11% with an existing analytical solution for an incompressible, fully developed, laminar flow under the no-slip boundary condition. It

9 was confirmed that Navier–Stokes equations are still valid for the laminar flow of deionized water in smooth micro-channels having hydraulic diameter as small as 25.9

μm. For smooth channels with larger hydraulic diameters of 103.4–291.0 μm, transition from laminar to turbulent flow occurred at Re = 1500–2000. Li et al. [40] conducted a numerical simulation on a silicon-based microchannel heat sink. The finite difference numerical code developed to solve the governing equations was the Tri-Diagonal Matrix

Algorithm. The behavior flow and the heat transfer were investigated to observe how the geometric parameters of the channel and thermo-physical properties affect them. The outcome of this study revealed that the thermo-physical properties of the liquid used in the analysis can considerably affect both flow and heat transfer in the microchannel heat sink. Mo et al. [41] studied the flow of nitrogen gas in a rectangular channel by forced convection. The different parameter varied during the study showed considerable effect on the heat transfer characteristic in the channel. The main parameters were temperature, hydraulic diameter, and aspect ratio. The research revealed that heat addition had the most influence on the system, followed by the channel aspect ratio, Reynolds number which is a function of the hydraulic diameter, and Prandtl number.

Owhaib and Palm [42] experimentally investigated the heat transfer characteristics of single-phase forced convection flow through circular microchannels.

The results were compared to correlations for heat transfer in macroscale channels. The results showed good agreement between classical correlations and experimentally measured data. Wu and Cheng [43] carried out a series of experiments to study different boiling instability modes of water flowing in microchannels at various heat flux and mass

10 flux with the outlet of the channels at atmospheric pressure. Eight parallel silicon microchannels with an identical trapezoidal cross-section were used in this experiment.

Morini et al. [44] investigated the rarefaction effects on the pressure drop for an incompressible flow through silicon microchannels having a rectangular and trapezoidal cross section. The roles of Knudsen number and the cross-section aspect ratio on the friction factor reduction due to the rarefaction were pointed out. Chen and Cheng [45] performed a visualization study on condensation of steam in microchannels etched in a silicon <100> wafer that was bonded by a thin Pyrex glass plate from the top. Saturated steam flowed through these parallel microchannels, whose walls were cooled by natural convection of air at room temperature. Stable droplet condensation was observed near the inlet of the microchannel. It was predicted that the droplet condensation heat flux increases as the diameter of the microchannel is decreased. The experimental investigation of heat transfer in a rectangular microchannel was also performed by Lee et al. [46]. They explored the validity of classical correlations based on conventionalized channels for predicting the thermal behavior in a single-flow. This study also showed that at a given flow rate within the laminar region, the heat transfer coefficient will increase with a decreasing channel size. In applying a uniform heat flux to a trapezoidal microchannel, Cao et al. [47] showed the effect of velocity slip on the Nusselt number and friction coefficient of the system. It was discovered that values of Nusselt number for a slip flow was larger than that of a no-slip flow and an increase in aspect ratio will result in an increase in fully Nusselt number. Hetsroni et al. [48] compared experimental result based on theoretical and numerical results for heat transfer in a microchannel at

11 small Knudsen numbers. The effect of the geometric and axial heat flux parameters on the system was analyzed. The thermal conduction through the working fluid, channel walls and energy dissipation was observed with regards to the parameters.

Zhuo et al. [49] also studied the heat transfer behavior in both triangular and trapezoidal microchannel by numerical and experimental processes. The intersection angle between the temperature and velocity gradient was observed and the synergy for

Reynolds numbers below 100 was much better. The field synergy principle was confirmed with an almost linear relationship between the Reynolds number and their corresponding Nusselt number for Re < 100. Li et al. [50] showed through studies that

Nusselt number and heat transfer coefficient will reduce along the flow of a microchannel with the least values at the outlet. Husain and Kim [51] used numerical methods in order to optimized microchannel heat sink using a surrogate analysis and evolutionary algorithm. In the optimization, the objective functions of thermal resistance and pumping power in the microchannel where formulated to evaluate the performance of the heat sink. Rahman et al. [52] investigated the convective heat transfer related to a magnetic field in a circular microchannel with rectangular substrate. The heat source was from gadolinium, a magnetocaloric material that generates heat within a magnetic field and different parameters where varied to see the influence on the heat transfer coefficient. Li and Kleinstreuer [53] compared the thermal conductivity model for nanofluids; one involved the application of a model based on the Brownian motion induced micro-mixing and the other was based on Navier-Stokes. The study was done on the flow of nanofluids pure water and CuO-water through a trapezoidal microchannel. Their research revealed

12 that nanofluids improved thermal performance of microchannel mixture flow but with a pressure drop.

Hooman [54] presented investigation on the convective characteristics of a rectangular microchannel with a porous medium while factoring parameters such as temperature jump, velocity profile, duct geometry, friction factor and slip coefficient.

Their influence on the Nusselt number was analyzed. Hasan et al. [55] studied the effect of channel geometry on a microchannel heat exchanger. Numerical simulations were carried out to solve developing flow and conjugate heat transfer. The shapes investigated include square, rectangular, trapezoidal and iso-triangle. Their investigation showed that with the parameters used, when the volume of a channel is decreased or the number of channels are increase, heat transfer increases, pressure drops and pumping power increases. This study within the parameters used showed the circular channel having the most effective thermal efficiency. Hsieh and Lin [56] performed experiment to determine the thermal characteristics of a fluid in rectangular microchannel. The fluids used in the experiments were deionized water, methanol and ethanol solutions. The parameters were aspect ratio, hydraulic diameter, Reynolds numbers, surface conditions, thermal properties and the different fluids. From within the extent of these parameters, it was observed that the hydrophilic surfaces had higher local heat transfer coefficients than that of hydrophobic for all test fluids. Chen et al. [57] studied the thermal behavior of heat transfer in different shapes of microchannels. These shapes include trapezoidal, rectangular and triangular shapes. In the study, the Nusselt number was seen to be highest at the inlet of the heat sink and least at the outlet. In comparison with the

13 different shapes of microchannel that were studied, it was observed that the triangular shaped model had the most thermal efficiency.

DeGregoria et al. [58] tested an experimental magnetocaloric refrigerator designed to operate within a temperature range of about 4 to 80 K. Helium gas was used as the heat transfer fluid. A single magnet was used to charge and discharge two in-line beds of magnetocaloric material. Zimm et al. [59] investigated magnetic refrigeration for near room temperature cooling. Water was used as the heat transfer fluid. A porous bed of magnetocaloric material was used in the experiment. It was found that using a 5T magnetic field, a refrigerator reliably produces cooling powers exceeding 500W at coefficient of performance 6 or more. Pecharsky and Gschneidner [60] discussed new magnetocaloric materials with respect to their magnetocaloric properties. Recent progress in magnetocaloric refrigerator design was reviewed.

The objective of the present investigation is to take a step ahead in study of micro-channels of trapezoidal cross section by investigating composite trapezoidal channels. A composite trapezoidal microchannel structure is formed by bonding a slab of gadolinium with silicon wafer where microchannels of trapezoidal cross section have been etched out of the silicon substrate. The study presents different parametric variations and its effect on the fluid flow and heat transfer characteristics of the channel.

CHAPTER 2

ANALYSIS OF MASS TRANSFER BY JET IMPINGEMENT

2.1 Mathematical Model

The diagram of figure 2.1 is a schematic of the problem being analyzed. It involves an axi-symmetric feature with the ejection of liquid jet from the nozzle which impinges on a rotating disk. Figure 2.1a and 2.1b are the 2D and 3D schematics respectively. The nozzle diameter, dn is 0.15cm which is kept as a constant function of

Rej used in simulation and calculations of local and average Sherwood number in equations (13 and 14). The rotating disk has a diameter, dd of 1.5cm where a 10 to 1 ratio with dn was intended. Dimensionless height,  is calculated H/dn. In analysis,  is made to vary to observe the effect it has on the mass transfer rate. The numerical model parameters include a Newtonian fluid with constant properties, an incompressible flow under laminar and steady state conditions. As part of this computational analysis, the system under study was under isothermal conditions neglecting the heat transfer effects of the energy equation. The equations describing the conservation of mass, momentum

(r,, and z directions respectively) can be written as [28].

V V V r  r  Z  0 (1) r r z

2 2 2 Vr Vθ Vr 1 p  Vr 1 Vr  Vr Vr  Vr   VZ    ν 2   2  2  (2) r r z ρf r  r r r z r 

V V V V  2V 1 V  2V V  θ r θ θ θ θ θ θ (3) Vr   VZ  ν 2   2  2  r r z  r r r z r 

V V 1 p  2V 1 V  2V  Z Z Z Z Z (4) Vr  VZ  g   ν 2   2  r z ρf z  r r r z  The mass transport equation of momentum accommodates the chemical species in the following form:

C C C    C   2C  2C  V i  V i  V i  D r i  i  i  (5) r θ z    2 2 2  r rθ z  rr  r  r θ z  In addition, the ion mass flux (moles/area-time) which involves the transfer of ions between an electrolyte and the electrode is defined by:

N  K(c  cint ) (6)

Where N is ion mass flux, which is related to the mass transfer coefficient, K, c∞ is the concentration of ions in the bulk fluid and cint is the concentration of ions on the interface. The following boundary conditions were used.

V C At r  0, 0  z  H : V  V  0, z  0, f  0 (7) θ r r r

At r  rd , 0  z  δ : p  patm (8)

At z  0, 0  r  r : V  V  0, V  Ω  r (9) d r z θ d At z  H, 0  r  : V  V , V  V  0, T  T (10) 2 z j r θ f j d C At z  H,  r  r : V  V  V  0, f  0 (11) 2 p θ r z z The boundary conditions applied to the flow is a no–slip condition where the velocity parallel to the walls and on the wall is zero. The formula used to determine the

16 average mass transfer coefficient has the same form that the average heat transfer coefficient equation used by Rahman and Lallave [20].

r 2 d   (12) K   K  rC  C dr av 2    int j  r  C  C  0 d  int j 

Where Cint is the average concentration at the solid–liquid interface, Cj and Cint are the jet and interface concentrations, respectively. The local and average Sherwood numbers are calculated according to the following expressions:

K  d Sh  n (13) D K  d Sh  av n (14) av D

Figure 2.1a Confined liquid jet impingement between a rotating disk and an impingement plate, two-dimensional schematic.

Figure 2.1b Confined liquid jet impingement between a rotating disk and an impingement plate, three-dimensional schematic. 18

2.2 Numerical Simulation In calculating the numerical computation, a few conditions have to be met which include the continuity equation (1), momentum, mass transport, and ion flux equations

(2-6), and boundary conditions (7–11). The above equations were solved using the

Newton–Raphson method through a finite element program, FIDAPTM [26]. This numerical analysis method helped to accommodate the non–linearity of the velocity and concentration computations. The finite element analysis was done using four node quadrilateral elements. Even though temperature did not play a big role in this simulation, an approximate value close to the room temperature and inlet velocity was assigned at the jet nozzle which corresponds to several Reynolds numbers. In addition, the velocity, pressure, and concentration were factored into the computations of each element; taking into account the boundary conditions for the electrolyte concentration at the jet nozzle.

To enhance the accuracy of the numerical model the mesh elements of the electrolyte region close to the solid interface were smaller than those above in the bulk region. Since, it is an electrolyte used in the simulation; the electrochemical system is best used in a controlled system of etching. The one-electron model as seen in equation (15) was used in the simulation as part of the cation’s concentration distribution at the spinning disk

(cathode) as presented by Moreno et al. [6].

3  4 Fe(CN)6  e  Fe(CN)6 (15)

An additional assumption made as part of this numerical study includes the absence of chemical reaction in the bulk fluid. The species of electrolyte in the system are assumed to be independent of one another in the fluid and therefore the system is uncoupled for the simulation. This uncoupling implies that the cathodic reaction of the 19 rotating disk is independent of the flow or diffusion of the fluid and because of this assumption, the etch rate or chemical reaction can be ignored in the analysis. The properties of the following electrolytes (NaOH), K3[Fe(CN)6], and HCl, were obtained from Moreno et al.[6], Sara et al. [16], Quiroz et al.[14], Guggenheim [21] and Fary

[22]. For this simulation, the Soret effect is negligible therefore, the flow was assumed to have only a mass transfer by convection and a mass transfer by diffusion which is as a result of a concentration gradient which can best be described by Fick’s law.

G  DiCi (16)

During the iteration, the values begin to converge relative to their previous values and the residuals are summed up for each variable which is less that 10-6. To verify that the conservation of mass was met, the flow rate at the outlet was compared with the flowrate at the nozzle of the jet to make sure their sum is zero. The suitable number of element to be used in the simulation was determined by an independent systematic pick.

A graph of the best of meshes can be seen in figure 2.3. The most accurate mesh of model shows a grid size of 20 x 500 divisions of elements in the axial (z) and radial (r) directions, respectively. Numerical results for this grid compared to the others gave almost identical results with an average margin error of 1.2%. The result of the electrolyte interface concentration distribution obtained from the finite element analysis is used to calculate the mass transfer coefficient, local and average Sherwood numbers.

2.3 Results and Discussion

The mesh used for simulation can be seen in figure 2.2 and figure 2.3 which shows the best of all meshes plotted together. Figure 2.3 facilitates the process of choosing an optimum mesh with the lowest number of percentage difference. The mesh used is the 20x500 mesh grid spacing having the smallest number of divergence when compared to the others at an average of 1.21%. The amount of grid spacing in the vertical direction was made denser under the jet nozzle to accommodate the mass transport equation. The Schmidt number focused on in this study results in a thin boundary layer and for this reason, the closer the grid spacing was to the interface the smaller grids got in the horizontal direction to facilitate the transport equation of the ions.

Figure 2.2 Mesh plot for a grid spacing of 20 x 500 in the axial and radial directions.

0.40

NZ x NR = 10 x 280

) 3 0.35 NZ x NR = 20 x 500

NZ x NR = 30 x 290 (g/cm

int int 0.30 NZ x NR = 35 x 320 NZ x NR = 40 x 340 0.25

0.20

0.15

0.10 Interface Interface Concentration, C

0.05 . 0.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Dimensionless Radial Distance, r/R

Figure 2.3 Dimensionless interface temperature distributions for different number of elements in r and z directions (Rej=1000, Rer=2310, =1.0, and Sc=2315).

Figure 2.4 plots the dimensionless radial distance, r/R versus local Sherwood number and dimensionless concentration. This figure involved analysis at jet Reynolds numbers limited to the laminar region ranging from 600 to 1500 with a spin rate of 125 rpm and  = 1.0. Figure 2.4 shows that Sh is highest at the stagnation point and then it quickly decreases as r/R increases. The local Sherwood becomes almost a constant at r/R

= 0.2 after the rapid drop but continues to decrease. This corresponds approximately to the boundary between the impingement dominated and mixed region. A similar behavior was reported by Arzutuğ et al. [13] for impinging jet on a rotating disk and by Metzger et al. [27]. The dynamics that affect the behavior of the local Sherwood number include the geometry of the system, flow factor such as the shape of the nozzle, Reynolds number, turbulence or laminar level at the jet nozzle, jet-to-impinging surface gap and rotation of the disk. The plot in figure 2.4 shows the highest Reynolds number 1500 with the highest local Sherwood number at the stagnation point and continues like that to r/R = 1.0. The lower the Reynolds number, the lower the local Sherwood numbers from the stagnation point to the outer limit of r/R. This shows that an increase in velocity at the nozzle increases the mass transfer rate of the species. For the behavior of the interface concentration, one can see three regions which are the impingement dominated, mixed region where the jet impingement and rotational effect of the disk have equal influence on the flow and then rotational dominated region where the expansion of the fluid takes place as r/R increases away from the center of the disk. The impingement dominated region has the slightly average positive slope until approximately r/R = 0.075, the mixed region starts with a sudden steep average positive slope up to approximately r/R = 0.18,

23 and the rotational region also has a steep slope but not as much as the mixed region. The lowest Reynolds number 600 has the highest concentration at the center of the stagnation point and continues like that to r/R = 1.0 because the lower the velocity of the flow over the disk or lower the Reynolds number at the nozzle, the more species component interact with the rotating disk interface. Also, the velocity of the fluid as it flows out reduces which leads to an increase in concentration on the outer edge of the disk.

Figure 2.5 is a plot of the local Sherwood number and interface dimensionless concentration which reflect the behavior of the rotational Reynolds number, Rer as the disk rotates. The range of the rotational Reynolds is from 0 to 6007, zero being a stationary disk. It was plotted with Rej = 800,  = 1.0 and Sc = 2315. This figure shows that the changes to the rotational Reynolds number do not have a significant effect on the mass transfer rate. It can be seen from the plot of the dimensionless interface concentration, the different rotational condition depicts the existence of three regions as pointed out by Sara et al. [16]. It can further be deduced that the dimensionless concentration in figure 2.5 has an expansion region that begins when Θ ≈ 0.2 because up to that point, they all lie on the same curve and begin to expand thereafter; this is where the different plots of rotational Reynolds diverge from each other. As stated by Miranda and Campos [11], the velocity profile inside the mass boundary layer explains this behavior where they are linear along the impingement and mixed regions but deviate from linearity at the expansion region. From the data collected, the lower the jet Reynolds number the more pronounced the deviation of the rotational Reynolds for the concentration would be. When the disk is stationary at Rer = 0, it has the highest

24 concentration at the outer radius of the disk compared to when the disk is rotating. The velocity can once again be used to explain this action. An increase in velocity as a result of the rotational speed leads to less concentration of the species on the disk. Therefore, the increment of spinning rate or rotational Reynolds number (Rer) will decrease the dimensionless concentration, (Θ) as seen in figure 2.5.

The distance between the nozzle and the rotating disk is simulated at different heights. Figure 2.6 shows the result of this action on the local Sherwood number and dimensionless concentration over the dimensionless radial distance for  = 0.5 to 20. The conditions applied to the simulation include Rej = 1500 and Rer = 2310. From the data, the highest dimensionless height  = 2.0 has the higher mass transfer rate than when  =

0.5 at the center of the disk, but towards the end of the disk,  = 0.5 has a higher mass transfer rate than when it is 2.0. This occurs because of the difference in the length of their potential core and the characteristics of the jet. The graph shows that an increment of the dimensionless height () will cause the species concentration on the disk to increase along the dimensionless radial distance (r/R). The rotation dominated region on the concentration data begins at a point that corresponds with proportionality to the dimensionless height () at the dimensionless radial distance of r/R  0.2. The local

Sherwood number shows the same behavior at figure 2.4 and 2.5 but as the dimensionless height increases, the local Sherwood number increases for each radius in the expansion region. The axis of the disk has the highest local Sherwood value which decreases rapidly with a very steep slope and when c/C ≈ 2.2, there is a rapid change to a gentle slope which is as a result of the increase in thickness of the mass boundary layer in the mixed

25 region. As the flow begins to expand, a minimum value of local Sherwood number is reached.

3500 400 Sh,Sh, Re =Re j= 650

650 ) Sh,Sh, Re =Re j=800j j 3000 800Sh, Re =1000 350 /C Sh, Re = j j int 1000 Sh,Sh, Re =Re j=1250j 1250 Sh,Sh, Re =Re j=1500 300 2500 1500 Q, Re Re= =650 750 j Q, Re Re= =800 800 j j 250 QQ, Re, Re= =1000 2000 1000 j j QQ, Re, Re = j=1250j 1250 200 QQ, Re, Re= j=1500 1500 1500 150 1000

100

Dimensionless Concentration, Q (C Q Concentration, Dimensionless Local Sherwood Number, Sh SherwoodNumber, Local

500 50

0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless Radial Distance (r/R)

Figure 2.4 Local Sherwood number and dimensionless concentration for different jet

Reynolds numbers (=1.0, Rer=2310, nickel disk, and Sc=2315).

1300 300

0 Sh, Rer= 0

231 )

1170 j 0Sh, Rer = 2310 415 Sh, Re = 4159 /C 8 r 250 508Sh, Re = 5083 int 1040 2 r 600Sh, Re = 6007 6 r c0 910 Q, Rer= 0 c231Q, Re = 2310 0 r j 200 c415Q, Re = 4159 8 r j 780 c508Q, Re = 5083 2 r c600Q, Rer = 6007 6 650 150

520 100

390 Local Sherwood Number, Sh SherwoodLocalNumber,

260 Concentration, Dimensionless (C Q 50 130

0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless Radial Distance, (r/R)

Figure 2.5 Local Sherwood number and dimensionless concentration for different rotational Reynolds numbers (=1.0, nickel disk, Rej=800, and Sc=2315).

4000 350 Sh,Sh, ß == 00.5.5 Sh,Sh, ß == 00.75.75

3500 ) Sh,Sh, ß == 11.0.0 300 j

Sh,Sh, ß == 11.5.5 /C int 3000 Sh,Sh, ß == 22.0.0 H/dQ,  =0=. 50c.5 250 H/dQ,  = 0=. 750.75c 2500 H/dQ,  = 1=. 01c.0 H/dQ,  = 1=. 51c.5 200 H/dQ,  = 2=. 02c.0 2000 ` 150 1500

100

Local Sherwood Number, Sh SherwoodLocal Number, 1000 Dimensionless Dimensionless Concentration, Q (C 500 50

0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless Radial Distance (r/R)

Figure 2.6 Local Sherwood number and dimensionless concentration for different dimensionless heights (Rej=1500, Rer=2310, and Sc=2315).

50 650 48 Rej = 650 J800Rej = 800

46 J1000Rej = 1000 avg J1250Rej = 1250 44 1500Rej = 1500

34 Average Sherwood Number, Sh Sherwood Number, Average

30 0 2000 4000 6000

Rotational Reynolds, Rer

Figure 2.7 Average Sherwood numbers with jet Reynolds numbers at various rotational Reynolds numbers (=1.0, nickel disk, and ferricyanide electrolyte and Sc=2315).

In figure 2.7, the average Sherwood number was plotted for different values of jet

Reynolds at  = 1.0 against rotational Reynolds’ values. The average Sherwood number increases with a relatively small difference when the rotational speed is increased which points out the fact that it has little effect on the outcome of the average Sherwood number. The result of the average Sherwood number shows that the effect of the mass transfer influenced by the rotational Reynolds is relatively trivial compared to the influence by the jet Reynolds. This shows that the impingement on the disk dominates the flow of the fluid with the spin rates that were used for the simulations and that the critical velocity of the rotational speed was not attained. Figure 2.7 also shows that the jet

Reynolds greatly affects the results of the Shavg. The results obtained in figure 2.7 are similar to those of previous works. It is quite obvious that the three regions of flow previously mentioned can no longer be observed because the curves are smoother since average Sherwood number is now cumulative, and the rapid changes are attenuated by the sum of the previous values.

Figure 2.8 shows the investigation of how  affects Shavg for 600  Re j 1500 where  13.09 rad/s . The length of the potential core has an effect on the local

Sherwood and as a result it has an influence on the average Sherwood number because as

 increases, Shavg decreases. Another factor causing the decrease in Shavg is the increased decay in average velocity as height of the nozzle from the rotating disk increases. As the nozzle to target spacing increases, there are more instances where there is mixing of induced turbulence occurs and the fluid does not expand as much as when the spacing is smaller. With the increasing distance with the nozzle of the jet, a large portion of the

30 disk will stay inside the low velocity region of the jet. Therefore, the mass transfer rate will decrease with an increasing distance from the nozzle of the jet.

Figure 2.9 shows the investigation of how β 1.15 for Schmidt numbers

1720  Sc  2472 affects Shavg over 600  Re j 1500 at  13.09 rad/s . It depicts Shavg increasing with Sc. The Schmidt number can be used to optimize the kinematic viscosity or diffusion coefficient needed for a system. It can also be interpreted as the average

Sherwood number increasing with an increase in kinematic viscosity or decrease in diffusion coefficient which is suitable in simulating fluids at specific properties. The lower the Schmidt number the less the local Sherwood number over the radial direction will be and therefore lead to a decline in the average mass transfer rate over the disk.

Figure 2.10 is a comparison between electrolytes. These electrolytes are potassium ferricyanide (K3[Fe(CN)6] ) being replaced with Sodium Hydroxide (NaOH) and the other is Hydrochloric acid (HCl). Their properties from Fary [22] and

Guggenheim [21] give us Sc = 791 and 655 at 25deg ⁰C for HCl and NaOH respectively because of their much larger diffusion coefficient. The difference in Sc between

K3[Fe(CN)6 at 2315 and the other two electrolytes being compared to it is very different from the ferricyanide having the highest concentration distribution followed by

Hydrochloric acid and then Sodium Hydroxide. The trend seen from this figure shows that the lower the Schmidt number, the lower the distribution of concentration.

70 J650Re = 650 65 J800Re = 800 J1000Re = 1000 avg 60 J1250Re = 1250 55 J1500Re = 1500

Average Sherwood Number, Sh Sherwood Number, Average 25

20 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Dimensionless Nozzle to Target Spacing, 

Figure 2.8 Average Sherwood number with jet Reynolds number at different dimensionless nozzle to target spacing (Rer=2310, and Sc=2315).

45 JRe650j = 650 43 JRe800= 800 J1000j JRe1250j = 1000

avg 41 JRe1500j = 1250 39

29 Average Sherwood Number, Sh SherwoodNumber, Average 27

25 1700 1900 2100 2300 2500 Schimdt Number, Sc

Figure 2.9 Average Sherwood number with jet Reynolds number at different Schmidt number (=1.15, nickel disk, ferricyanide electrolyte, and Rer=5082).

300 Conc,Q, Ferricyanide Ferricyanide Conc,Q, Hydrochloric Hydrochloric Acid Acid Conc,Q, Sodium Sodium HydroxideHydroxide 250

200

150

100 Local Sherwood Number, Sh SherwoodLocal Number,

0 0 0.2 0.4 0.6 0.8 1 Dimensionless Radial Distance (r/R)

Figure 2.10 Dimensionless concentration for different electrolytes - NaOH, HCl,

3 Fe(CN)6 (Rej=650, Rer=0, and =1.0).

3000 250

Sh, Parallel )

Sh, Conical j

2500 Sh, Concave /C 200 int

Sh, Convex C ( Q, Parallel 2000 Q, Conical Q, Concave 150 Q, Convex 1500

100

1000 Local Sherwood Number, Sh SherwoodLocal Number, 50

500 Dimensionless Dimensionless Concenctration, Q

0 0 0 0.2 0.4 0.6 0.8 1 Dimensionless Radial Distance, r/R

The graph in fig. 2.11 shows the distribution of mass transfer of local Sherwood numbers and dimensionless concentration (Θ) over r/R for different confinements plates at Rej = 1000 and Sc = 2315. The change in geometries is only with the confinement plate which are conical, convex and concave shaped relative to the rotating disk. Figure

2.11 plot shows the three geometries being compared to the parallel confinement plate used in this study. For the range of parameters used, the values of the Sh coincide for most part of the disk except for the range 0  r / R  0.025 where the conical shaped impingement plate starts with highest local Sherwood number and the parallel begins with the lowest. The dimensionless radial distance of r/R = 0.025 is equivalent to a radius

0.035cm. When comparing a parallel confinement plate to a conical plate, Miranda and

Campos [11] study showed that lower Reynolds and Schmidt numbers create a clear distinction of local Sherwood number results at the expansion region.

At the outer edge of the disk, the mass transfer rates are still almost the same with the parallel disk still having the lowest Sherwood numbers and the convex confinement plate having the highest. The conical shape had the highest Shavg followed by the concave with the parallel impingement plate having the lowest. For the dimensionless concentration part of the graph, the behavior is very much different. Starting with the impingement region, they all start out alike and this can be explained by the fact that they approximately have the same impingement plate shape within this region. At the region that is dominated by rotation, the interface concentration begin to behave differently and at this region the shape on the confinement plate begins to play a big role on the concentration distribution on the disk. From the various confinement plate geometries

36 that were examined, the conical shaped showed the best mass transfer rate which agrees with a previous study done by Miranda and Campos [11].

Schmidt Number, Sc

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 50 40 Present results rrrbrbrbrArzutug et al. [8] avg 39 Sc Present results 45 Sc ArzutuğArzutug etet al.al. [13] [8] 38 [13] 40 37

35 36

Present results 35 30 - Arzutuğ et al. [13] 34 25

Average Sherwood Number, Sh Sherwood Number, Average 33

20 32 700 900 1100 1300 1500 rrrbrbrbrbrg1700 1900 2100 Jet Reynolds Number, Re

Figure 2.12 Average Sherwood number comparison with the experimental results obtained by Arzutuğ et al. [13] under various jet Reynolds and Schmidt numbers (left:

=1.0, Sc=2315, Rer=2310; right: (=1.0, Rej=1000, and Rer=5082).

360

320 avg 280

240

200

160

120

Average Sherwood Number, Sh Sherwood Number, Average Present Study 80 Chin & Tsang[1]

40 Sara[12] Wang[27] 0 1700 1900 2100 2300 2500

Schmidt Numbers, Sc

Figure 2.13 Average Sherwood number comparison with other studies within the core region for various Schmidt numbers ( =1.0, Rej=800, Rer=2310).

80 avg

20 Actual Average Sherwood Number, Sh Sherwood Number, Average Actual

0 0 20 40 60 80

Predicted Average Sherwood Number, Shavg

Figure 2.14 Comparison of predicted average Sherwood number using Equation 17 with present numerical data.

Figure 2.12 shows two different average Sherwood number comparison for a range of jet Reynolds and Schmidt numbers. The right plot shows a varying jet Reynolds number and the left shows a range of Schmidt number. This is an attempt to show how close the present result obtained comes close to the correlation gotten by Arzutuğ et al.

[13]. The results of the plot on the left side were obtained for 800  Re j 1500 with Sc =

2315 at  13.09 rad/s . The average difference with the results from this plot was an absolute 4.9% with a maximum of 10.3%. On the right graph, the comparison was done at Rej = 1000 and Rer = 5082. The average difference here was 0.6% with the highest deviation of 1% from Arzutuğ et al. [13].

The illustrations shown in fig. 2.13 are used to discuss the behavior of the average

Sherwood number within the impingement dominated zone. Figure 2.13 shows the average Sherwood data in the impingement zone of present study, correlations obtained by Chin and Tsang [3], Sara et al. [16], and Wang et al. [25] on a flat surface. The

Sherwood number is used to show the mass transfer in a dimensionless parameter versus the Schmidt number of the electrolyte. The thicker line shows the finite element analysis results of present study. The computations involved had a jet Reynolds number (Rej=800) with a range of1720  Sc  2472. In this impingement zone with the parameters used, the

Sherwood number were close to being proportional the square root of Rej while being proportional to the cubic root of the Sc. The average Sherwood number in the core region compared to Chin and Tsang [3], Sara et al. [16], and Wang et al. [25] shows an absolute difference that range from 1 to 17% and had an average absolute difference of 7.73%.

A correlation for the average Sherwood number developed used the following parameters Rej, Rer, , and Sc as variables. In this correlation, various parameters were used and the data point employed corresponded with the electrolyte having a Sc =

2315(where =0.01275 cm2/s). The Prandtl number exponent of 0.4 from Martin’s equation [23] for a single confined liquid jet impingement was kept constant as part of present numerical correlation. The least square curve-fitting technique was adopted to develop the correlation and the least square fit of the corresponding logarithmic equation.

The behavior of the average Sherwood number with the various parameters determined the signs for the exponents. The correlation obtained for the present study can be seen in equation (17) and illustrated in figure 2.14.

. 0.286 0.0074 -0.714 0.33 Shavg = 0.404 Rej Rer  Sc (17)

The highest absolute percentage difference between the actual Shavg and the predicted results was 10.20% with an average absolute percentage difference of 3.89%.

The data of dimensionless parameters used for the correlation of this study take into

account jet Reynolds number 650  Re j 1500 , rotational Reynolds number, 0  Rer  6007

, dimensionless height, 0.5    2.0 , Schmidt number, 1720  Sc  2472 , and geometrical confinement plates: conical, convex, and concave layout. The present study correlation of a liquid jet impingement on a rotating disk in a fully confined space can help in predicting the mass transfer rates during the electrolyte synthesis as a process, especially for etching processes.

CHAPTER 3

ANALYSIS OF HEAT TRANSFER IN A TRAPEZOIDAL

MICROCHANNEL

3.1 Modeling and Simulation

The physical configuration of the system used in the present investigation is schematically shown in figure 3.1. A slab of gadolinium is placed on the top of the channel and bonded with the silicon wafer in such a way that a part of heat generated in gadolinium is directly dissipated to the working fluid whereas part is conducted through the silicon structure. Neglecting the effects of inlet and outlet plenums, it was assumed that the fluid enters the channel with a uniform velocity and temperature. The applicable differential equations for the conservation of mass and momentum in the Cartesian coordinate system are [61],

u v w    0 (18) x y z

u u u 1 p  u  u  u u  v  w    [(v  vt ) ]  [(v  vt ) ] [(v  vt ) ] (19) x y z ρ x x x y y z z

v v v 1 p  v  v  v u  v  w    [(v  vt ) ]  [(v  vt ) ] [(v  vt ) ] (20) x y z ρ y x x y y z z

w w w 1 p  w  w  w u  v  w    [(v  vt ) ] [(v  vt ) ] [(v  vt ) ] (21) x y z ρ z x x y y z z

The k-ε model was used for the simulation of turbulence. In this model, equations governing the conservation of turbulent kinetic energy and its rate of dissipation were solved. These equations can be expressed as,

k k k  v k  v k  v k u  v  w  [(v  t ) ]  [(v  t ) ]  [(v  t ) ] x y z x ζk x y ζk y z ζk z (22) u 2 u 2 v 2 v 2 w 2 w 2  vt[( )  ( )  ( )  ( )  ( )  ( ) ]  ε y z x z x y

ε ε ε  v ε  v ε  v ε u  v  w  [(v  t ) ]  [(v  t ) ]  [(v  t ) ] x y z x ζ x y ζ y z ζ z ε ε ε (23) 2 ε u 2 u 2 v 2 v 2 w 2 w 2 ε  C vt[( )  ( )  ( )  ( )  ( )  ( ) ]  C 1 k y z x z x y 2 k

2 vt  Cμk /ε (24)

The empirical constants appearing in equations (22)-(24) are given by the following values, Cμ=0.09, C1=1.44, C2=1.92, σk=1.0, σε=1.3. These values hold good for high

Reynolds numbers. For low Reynolds number the values of constants are [62],

C  1 exp(0.01y  0.008y3 ) (24a) μ λ λ

Where, 1/ 2 (24b) y  y /(k /)

 1.4 1.1exp(y /10) (24c) k 

 1.31.0 exp(y /10) (24d)

The energy equation in the fluid region is

2 2 2 T T T v  T  T  T u f  v f  w f  α  t f  f  f ( )[ 2 2 2 ] (25) x y z Prt x y z

The equation for steady state heat conduction for gadolinium is [63],

2 2 2  T  T  T g g g g0     0 2 2 2 (26a) x y z k g

The equation for steady state heat conduction for silicon is [63],

2 2 2  T  T  T s s s    0 2 2 2 (26b) x y z

To complete the physical model equations (18) – (26) are subjected to following boundary conditions,

At z=0, at fluid inlet, u=0, v=0, w=win, T= Tin (27)

At z=0 on solid surfaces of silicon and gadolinium

Ts Tg  0 ,  0 (28) z z

At z=L, at fluid outlet, p=0 (29)

At z=L, on solid surfaces of silicon and gadolinium,

T Tg s  0 ,  0 (30) z z

v w Tf At x=0, (H-D)

T At x=0, 0

Tg At x=0, H

T At x=B, 0

T At x=B, H

T At y=0, 0

T T At y=H, 0

T T At y=H, Bc

T At y=(H+Hg), 0

Tf Ts At y = (H-D), 0

At the inclined channel surface between fluid and silicon, 0

Tf Ts u=0, v=0, w=0, Tf = Ts, k  k (41) f n s n

Figure 3.1 Schematic of the trapezoidal microchannel.

The governing equations along with the boundary conditions were solved by using the Galerkin finite element method. Four-node quadrilateral elements were used. In each element, the velocity, pressure, and temperature fields were approximated which led to a set of equations that defined the continuum. The successive substitution algorithm was used to solve the nonlinear system of discretized equations. An iterative procedure was used to arrive at the solution for the velocity and temperature fields. The solution was considered converged when the field values did not change from one iteration to the next by 0.05%.

Figure 3.2 shows a grid independence study carried out to determine the optimum grid size. In order to ensure that an accurate solution was obtained, the number of elements that were used to mesh the geometry had to be deemed adequate. This was done by performing computations for several combinations of elements in all directions. The interface temperature plot was obtained. It was noted that the numerical simulation became grid independent at 18*18*90 elements. Computation with 18*18*90 elements produced results that were very close to the results produced by 24*24*110 and 8*8*42 elements. The difference between the values obtained with 24*24*110 and 18*18*90 elements was 0.36%. Therefore, 18*18*90 elements were chosen for the simulation.

3.2 Results and Discussion

For all the configurations studied the length of the microchannel (L) was kept constant at 23 mm. Magnetic field was varied from 2.5T to 10T. Reynolds number used was between 1000-3000. Height of the gadolinium slab was varied between 1.5 mm to 5 mm and the depth of the channel was varied between 100 μm to 300 μm. The roughness of microchannel surfaces was neglected in the turbulent analysis.

Figure 3.3 shows the variation of peripheral Nusselt number with dimensionless axial coordinate for different Reynolds number for magnetic field of 5T. Nusselt number is seen to be increasing with increase in Reynolds number. The temperature difference between fluid and solid is more at higher Reynolds number. Thus higher heat transfer coefficient is obtained at higher Reynolds number. Fluid gets heated as it passes through the channel. The temperature difference between fluid and solid decreases as one moves along the length of the channel. Thermal boundary layer grows until fully developed flow is established. Therefore, the Nusselt number is higher at the entrance and decreases downstream. The variation is larger at the entrance because of the rapid development of thermal boundary layer near the leading edge.

Figure 3.4 shows variation of peripheral dimensionless interface temperature for different Reynolds number and different magnetic fields. The solid-fluid interface temperature increases as the fluid moves downstream due to the development of thermal boundary layer starting with the entrance section as the leading edge. Interface temperature values increase with increase in magnetic field. It can be seen that, as

Reynolds number increases, the interface temperature decreases. For low Reynolds

48 number fluid remains in contact with the solid for longer time. Thus, high dimensionless interface temperature values are obtained at lower Reynolds number.

20 x * y * z = 8 * 8 * 42 18 x * y * z = 12 * 12 * 72 16 x * y * z = 18 * 18 * 90 14 x * y * z = 24 * 24 * 110 12

Nusselt Number, Nu Number, Nusselt 6

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Dimensionless Axial Coordinate, z/L

Figure 3.2 Average dimensionless interface temperature over the dimensionless axial coordinate for different grid sizes (magnetic field = 5T, Re =1600, D=150 μm, L=2.3 cm).

18 Re = 1000 16 Re = 1600 14 Re = 2400 12 Re = 3000 10

6 Nusselt Number, Nu Number, Nusselt

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Dimensionless Axial Coordinate, z/L

Figure 3.3 Variation of peripheral average Nusselt number along the channel dimensionless axial coordinate for different Reynolds number (Magnetic field = 5T, D=150 μm, L=2.3 cm).

6 2.5T, Re = 1600 2.5T, Re = 2400 2.5T, Re = 3000

Θ 5.0T, Re = 1600 5.0T, Re = 2400 5.0T, Re = 3000 5 10.T, Re = 1600 10.T, Re = 2400 10.T, Re = 3000

2 Dimensionless Interface Temperature, Temperature, Interface Dimensionless 1

0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless Axial Coordiate, z/L

18 5T, Re = 1600

16 5T, Re = 2400

14 5T, Re = 3000

12 10T, Re = 1600 10T, Re = 2400 10 10T, Re = 3000 8

Nusselt Number, Nu Number, Nusselt 6

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Dimensionless Axial Coordinate, z/L

Figure 3.5 Variation of Nusselt number along the channel dimensionless axial coordinate for different heat generation rates (Re = 1600, 2400, 3000, D=150 μm, L=2.3 cm).

Figure 3.5 shows variation of Nusselt number for different magnetic fields for

Reynolds number of 1600, 2400 and 3000. Nusselt number at a particular cross section in the channel remains almost the same for different magnetic fields.

Figure 3.6 shows variation of dimensionless interface temperature along the length of the channel for different thickness of magnetic slab. Reynolds number of fluid is 1600 and magnetic field is 5T. As shown in figure 1, the slab of magnetic material is placed on top of the microchannel. As thickness of the slab increases, more heat is generated in the magnetic material for the same magnetic field. The part of generated heat is directly dissipated to the working fluid from gadolinium whereas; part is conducted through the silicon structure and reaches the working fluid. It can be seen that interface temperature values increase with increase in slab thickness. It was found that

Nusselt number at a particular cross section in the channel remains almost the same for different thickness of magnetic material slab.

Figure 3.7 shows the dimensionless axial coordinate distribution of the Nusselt number for different channel depth and constant inlet velocity whose combined effect changes the Reynolds number. In all these plots, the local Nusselt number is large near the entrance and decreases downstream due to the development of thermal boundary layer. For the channel depth > 250 μm, the effect of channel depth on Nusselt number is less significant. At channel depth lower than 250 μm, the Nusselt number increases with channel depth over the entire length of the channel. For the channel depth of 250 μm and

300 μm the height of silicon wafer was used were 300μm and 350μm respectively.

4.0 Re = 1600, b = 0.05mm 3.5

Re = 1600, b = 1.5mm Θ 3.0 Re = 1600, b = 2.5mm Re = 1600, b = 5mm 2.5

2.0

1.5

1.0

0.5 Dimensionless Interface Temperature, Temperature, Interface Dimensionless

0.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless Axial Coordinate, z/L

20 D = 100 μm 18 D = 150 μm 16 D = 200 μm 14 D = 250 μm

12 D = 300 μm

Nusselt Number, Nu Number, Nusselt 6

0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless Axial Coordinate, z/L

Figure 3.7 Variation of Nusselt number along the channel dimensionless axial coordinate for different depths of the channel (Magnetic field =5T, Constant velocity).

Figure 3.8 shows the variation of the total average dimensionless interface temperature of the system for different Reynolds number and different channel depths for a magnetic field of 5T. It can be seen that as the Reynolds number increases, the average temperature decreases. A faster moving fluid carries heat at the faster rate, presenting lower values of average dimensionless interface temperature. For the same Reynolds number, as the depth of the channel increases, average temperature decreases.

3.0

avg D = 161 μm Θ 2.5 D = 400 μm D = 1820 μm

2.0

1.5

1.0

0.5

Average Dimensionless Interface Temperature, Temperature, Interface Dimensionless Average 0.0 750 1250 1750 2250 2750 3250 Reynolds Number, Re

Figure 3.8 Average dimensionless interface temperature numbers with Reynolds numbers at various channel depths (Magnetic field =5T, 1000 < Re < 3000). 56

14 S = 2.7mm 12 S = 2.8mm S = 2.9mm 10 S = 3.0mm

Nusselt Number, Nu Number, Nusselt 4

0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensioless Axial Coordinate, z/L

Figure 3.9 Variation of Nusselt number over the dimensionless axial coordinate for different spacing between the channels (Re = 2400, magnetic field = 5T, D=150 μm, L=2.3 cm).

Figure 3.9 shows variation of Nusselt number along the length of the channel for different spacing between the channels. The results show that increasing the solid path decreases the local Nusselt number along the channel degrading the thermal performance of the heat sink. The temperature gradient in the solid increases and the average solid temperature is slightly increased.

Figure 3.10 shows the comparison between the standard convection correlation and the present simulation model. Nusselt number for laminar flow in a smooth pipe is calculated by [64],

0.0668(Re.Pr.D / L) h (42) Nu  3.66  2 / 3 1 0.4(Re.Pr.Dh / L)

Nusselt number for turbulent flow in a pipe is calculated by [64],

Nu  0.023Re0.8.Pr0.4 (43)

The values of average Nusselt number for a circular channel with Dh=154μm were compared to that of the standard convection correlation for a trapezoidal channel with Dh=154μm. As the average Nusselt number values for circular channel were compared to those for the trapezoidal channel, there was a difference of 2.23% to 6.52% between the simulation results and the correlation results.

Figure 3.11 shows variation of dimensionless pressure difference between inlet and outlet of the channel for different Reynolds numbers. It can be seen that the value of dimensionless pressure difference between the inlet and outlet of the channel increases as

Reynolds number increases. This is expected because the velocity of the flow increases with Reynolds number. High velocity fluid creates higher pressure difference between inlet and outlet. For the same Reynolds number, as the depth of the channel increases, 58 velocity of the flow decreases. This results in lower values of dimensionless pressure difference between inlet and outlet of the channel.

Simulation

avg 5 Standard Convection Correlation [66]

2 Average Nusselt Number, Nu Number, Nusselt Average 1

0 500 1000 1500 2000 2500 3000 3500

Reynolds Number, Re

Figure 3.10 Comparison between simulation and standard convection relation (Dh=154μm, Magnetic field = 5T, D=150 μm, L=2.3 cm).

14 D = 100 μm

d 12 D = 150 μm D = 200 μm 10 D = 250 μm

4 Dimensionless Pressure Difference, ∆P Difference, Pressure Dimensionless 2

0 800 1300 1800 2300 2800 3300 Reynolds Number, Re

Figure 3.11 Variation of dimensionless pressure difference between inlet and outlet of the channel with Reynolds number (Magnetic field = 5T, L=2.3 cm).

14 Water

12 Helium

FC-72 10

6 Nusselt Number, Nu Number, Nusselt 4

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Dimensionless Axial Coordinate, z/L

Figure 3.12 Variation of the Nusselt numbers for different working fluids (Magnetic field = 5T, D=150 μm, L=2.3 cm).

Figure 3.12 shows variation of average Nusselt number along the dimensionless axial coordinate of the channel for different working fluids. Thermal conductivity of water (~ 0.6 W/m-K) is higher than that of helium (~ 0.15 W/m-K) and FC-72 (~ 0.057

W/m-K). This results in better Nusselt number values for water as compared to helium and FC-72.

Figure 3.13 shows variation of dimensionless pumping power required to pass the fluid through the channel for different Reynolds numbers. Pumping power is calculated from volumetric flow rate and pressure difference between inlet and outlet of the channel. As Reynolds number of fluid increases, the volumetric flow rate and dimensionless pressure difference increases. For the same Reynolds number, as the depth of the channel increases, velocity of the flow decreases. Thus, fluid flowing at a higher velocity requires more pumping power than fluid flowing at a lower velocity.

Figure 3.14 shows two different data of average Nusselt number comparison for a range of jet Reynolds. This is an attempt to show how close results of simulation from the present study come close to the experimental data gotten by Wu and Cheng [38]. The results of the plot were obtained for 40  Re  200, L=4cm, Bc=79μm and D=200μm.

The average difference between the results from this plot was 7.17% with a maximum of

13.32%.

3.0E+06

D = 100 μm p 2.5E+06 D = 150 μm

D = 200 μm

2.0E+06 D = 250 μm

1.5E+06

1.0E+06 Dimensionless Pumping Power, D Power, Pumping Dimensionless

5.0E+05

0.0E+00 750 1250 1750 2250 2750 3250 Reynolds Number, Re

Figure 3.13 Variation of dimensionless pumping power for different Reynolds numbers (Magnetic field = 5T, L=2.3 cm).

2.1

Simulation 1.8

Wu & Chen [8] avg

1.5

1.2

0.9

Average Nusselt Number, Nu Number, Nusselt Average 0.6

0.3

0 0 50 100 150 200 250 Reynolds Number, Re

Figure 3.14 Average Nusselt number comparison with the experimental results obtained by Wu and Chen [38] under various jet (Magnetic field = 5T, 40 < Re < 200, D=150 μm, L=4.0 cm).

CHAPTER 4

CONCLUSION

Analysis of impinging jet on a rotating disk was performed. The parameters utilized include Reynolds number, nozzle height to diameter ratio, rotational speed, geometry, and fluid property by various Schmidt numbers. The CFD simulation presented here could be a great tool in understanding the character of a flow in engineering applications involving jet impingement especially for in etching processes.

The behavior of local and average mass transfer through the dimensionless parameter

Sherwood was investigated using finite element analysis.

The conclusion of this investigation is that the local Sherwood number decreases with an increasing radius of a rotating or stationary disk which is also the rate of mass transfer decreasing towards the edge of the disk. An increase in rotational speed of the disk or Reynolds number at the jet nozzle will lead to an increase in mass transfer rate but the jet Reynolds has more of an impact. Therefore the results from the investigation for a low rotational speed of a rotating disk can be approximately equivalent to that of a stationary disk. Also, with the Reynolds numbers been dealt with, an increase in the nozzle to target spacing will reduce the mass transfer rate of the species. One of the most important engineering applications that this study can lend itself is to wet process equipments and the analysis from this study can be used in design when trying to determine etching rates.

For the microchannel, the simulation was performed by varying the channel aspect ratio, Reynolds number, heat generation rate and spacing between channels. At a smaller flow rate outlet temperature increased as the low velocity fluid remained in contact with the solid for longer time. Heat transfer coefficient and Nusselt number at a particular cross section in the channel remains almost the same for different heat generation rates. The peripheral average heat transfer coefficient and Nusselt number decreases along the length of the channel due to the development of thermal boundary layer. Large variation in the Nusselt number near the entrance can be attributed to large growth rate of thermal boundary layer near the leading edge. It is seen that the temperature in the channel drops down as the hydraulic diameter decreases. For the same channel, the maximum temperature decreases as the Reynolds number increases. The pressure drop in the channel increases as the Reynolds number increases. It is also seen that, as the Reynolds number increases, the power required for pumping the fluid through the channel increases. Nusselt number increases as the depth of the channel is increased.

For the same channel, Nusselt number increases with Reynolds number.

REFERENCES

[1] F. Kreith, J. H. Taylor, and J. P. Chong, “Heat and Mass Transfer from a Rotating

Disk,” Journal of Heat Transfer 81 (1959) 95–104.

[2] V.E. Nakoryakov, B.G. Pokusaev, and E.N. Troyan, “Impingement of an

axisymmetric liquid jet on a barrier,” International Journal of Heat and Mass

Transfer 21 (1978) 1175–1184.

[3] D.T. Chin, and C.H. Tsang, “Mass Transfer to an Impinging Jet Electrode,”

Journal of the Electrochemical Soc. Electrochemical Science and Technology 125

(9) (1978) 1461–1470.

[4] R. Alkire, and J. Ju, “High Speed Selective Electroplating with Impinging Two–

Dimensional Slot Jet Flow,” Journal of the Electrochemical Society 134 (2)

(1987) 294–299.

[5] D.T Chin and M. Agarwal, “Mass Transfer from an Oblique Impinging Slot Jet,”

Journal of the Electrochemical Society 138 (1991) 2643–2650.

[6] O.A. Moreno, R.H. Katyl, J.D. Jones, and P.A. Moschak, “Mass Transfer of an

Impinging Jet Confined between Parallel Plates,” IBM J. Res. Develop. 37 (2)

(1993) 143–154.

[7] Y.M. Chen, W.T. Lee, and S.J. Wu, “Heat/Mass Transfer between an Impinging

Jet and Rotating Disk,” Heat and Mass Transfer 34 (1998) 195–201.

[8] T. Pekdemir, and T.W. Davies, “Mass Transfer from Rotating Circular Cylinders

in a Submerged Slot Jet of Air,” International Journal of Heat and Mass Transfer

41 (1998) 3441–3450.

[9] Q. Chen and V. Modi, “Mass transfer in turbulent impinging slot jets,”

International Journal of Heat and Mass Transfer 42 (1999) 762–776.

[10] M. Chen, R. Chalupa, A. C. West, V. Modi, “High Schmidt Mass Transfer in a

Laminar Impinging Slot Jet Flow,” International Journal of Heat and Mass

Transfer 43 (2000) 3907–3915.

[11] J. M. Miranda, and B.L.M. Campos, “Impinging Jets Confined by a Conical Wall

- High Schmidt Mass Transfer Predictions in Laminar Flow,” 44 (2001) 1269–

1284.

[12] C.F. Oduoza, “Electrochemical Mass Transfer at a Heated Electrode in a Vertical

Annular Flow Cell,” Chemical Engineering and Processing 43 (2004) 921–928.

[13] M. E. Arzutuğ, S. Yapici, and M.M. Kocakerim, “A comparison of Mass Transfer

between a Plate and Submerged Conventional and Multichannel Impinging Jets,”

International Communications in Heat and Mass Transfer 32 (2005) 842–854.

[14] M.A. Quiroz, U.A. Martinez-Huitle, and C.A. Martinez-Huitle, “Mass Transfer

Measurement in a Parallel Disk Cell Using the Limiting Current Technique,”

Journal Mex. Chem. Soc 49 (3) (2005) 279–283.

[15] G.H. Sedahmed, M.S. Abdo, M.S. Hassan, A.H. Konsowa, “Intensification of

mass transfer controlled processes at the interface between two immiscible liquids

by an axial turbulent submerged jet,” Chemical Engineering and Processing 46

(2007) 10–16.

[16] O.N. Şara, J. Erkmen, S. Yapici, and M. Çopur, “Electrochemical Mass Transfer

between an Impinging Jet and a Rotating Disk in a Confined System,”

International Communications in Heat and Mass Transfer 35 (2008) 289–298.

[17] S. K. Hong, D. H. Lee and H. H. Cho, “Heat/Mass transfer measurement on

concave surface in rotating jet impingement,” Journal of Mechanical Science and

Technology 22 (2008) 1952–1958.

[18] S. K. Hong, D.H. Lee, and H.H. Cho, “Effect of Jet Direction on Heat/Mass

Transfer of Rotating Impingement Jet,” Applied Thermal Engineering 29 (2009)

2914–2920.

[19] J. C. Lallave, M. M. Rahman, A. Kumar, “Numerical analysis of heat transfer on

a rotating disk surface under confined liquid jet impingement,” International

Journal of Heat and Fluid Flow 28 (2007) 720–734.

[20] J. C. Lallave, and M. M. Rahman, “Heat Transfer from a Spinning Disk during

Semi–Confined Axial Impingement from a Rotating Nozzle,” International

Journal of Heat and Mass Transfer 51 (2008) 4400–4414.

[21] E.A. Guggenheim, “The Diffusion Coefficient of Sodium Chloride,” Dept. of

Chemistry, The University, Reading 1 (1954) 1048 – 1049.

[22] A.D. Fary, Jr., “The Diffusional Properties of Sodium Hydroxide,” The Institute

of Paper Chemistry, Appleton, Wisconsin, Doctor's Dissertation, January (1966)

73 -74.

[23] H. Martin, “Heat and Mass Transfer between Impinging Jets and Solid Surfaces,”

Academic Press, New York, Advances in Heat Transfer 13 (1977) 1–60.

[24] C.O. Popiel, and L. Boguslawski, “Local Heat Transfer from a Rotating Disk in

an Impinging Round Jet,” Journal of Heat Transfer 108 (1986) 357–364.

[25] X.S. Wang, Z. Dagan, and L.M. Jiji, “Heat Transfer between a Circular Free

Impinging Jet and a Solid Surface with Non-uniform Wall Temperature or Wall

Heat Flux–2. Solution for the Boundary Layer Region,” International Journal

Heat Mass Transfer 32 (1989) 1361.

[26] FIDAP Theoretical Manual, Fluid Dynamics International, Inc., Evanston, IL,

(1991).

[27] D.M. Metzger, R.S. Bunker, and G. Bosch, “Transient Liquid Crystal

Measurement of Local Heat Transfer on a Rotating Disk with Jet impingement,”

Turbomachinery 113 (1991) 52–59.

[28] A. Bejan, “Convection Heat Transfer, second edition,” Wiley, New York, 1995

pp. 1–17, pp. 595–600.

[29] V.L. Okulovl, S. Martemianov, and T.O. Murakhtina1, “Generalization of the

Leveque Problem for Mass Transfer in Swirling Flows in the Entrance Region of

a Cylindrical Section,” Doklady Physics 47(9) (2002) 685–689. Translated from

Doklady Akademii Nauk 386 (1) (2002) 51–55.

[30] S.Z. Shuja, B.S. Yilbas, M. Rashid, “Confined Swirling Jet Impingement onto an

Adiabatic Wall,” International Journal of Heat and Mass Transfer 46 (2003)

2947–2955.

[31] P.Y. Wu and W.A. Little, “Measurement of friction factors for the flow of gases

in very fine channels used for microminiature Joule-Thomson refrigerators,”

Cryogenics 24 (1983) 273-277.

[32] J.C. Harley, Y. Huang, H.H. Bau and J.N. Zewel, “Gas flow in micro-channels,”

Journal of Fluid Mechanics 284 (1995) 257-274.

[33] K. Chen and Y. Wu, “Thermal analysis and simulation of the microchannel flow

in miniature thermal conductivity detectors,” Sensors and Actuators 79 (2000)

211-218.

[34] W. Qu, G. M. Mala and D. Lee, “Heat transfer for flow in trapezoidal silicon

microchannels,” Int. Journal of Heat and Mass Transfer 21 (2000) 3925-36.

[35] M.M. Rahman, “Measurements of heat transfer in microchannel heat sinks,”

International Communications in Heat and Mass Transfer 27 (1) (2000) 495-506.

[36] K.C. Toh, X.Y. Chen and J.C. Chai, “Numerical computation of fluid flow and

heat transfer in microchannels,” International Journal of Heat and Mass Transfer

45 (2002) 5133–5141.

[37] W. Qu and I. Mudawar, “Analysis of three-dimensional heat transfer in micro-

channel heat sinks,” International Journal of Heat and Mass Transfer 45 (2002)

3973–3985.

[38] H.Y. Wu and P. Cheng, “An experimental study of convective heat transfer in

silicon microchannels with different surface conditions,” International Journal of

Heat and Mass Transfer 46 (2003) 2547–2556.

[39] H.Y. Wu and P. Cheng, “Friction factors in smooth trapezoidal silicon micro-

channels with different aspect ratios,” International Journal of Heat and Mass

Transfer 46 (2003) 2519-2525.

[40] J. Li, G.P. Peterson, and P. Cheng, “Three-dimensional analysis of heat transfer in

a micro-heat sink with single phase flow,” International Journal of Heat and Mass

Transfer 47 (2004) 4215–4231.

[41] H.L. Mo, Y.X. Zhou, T.Y. Zhu and T.W. Guo, “Forced convection of low

temperature nitrogen gas in rectangular channels with small aspect ratio,”

Cryogenics 44 (2004) 301–307.

[42] W. Owhaib and B. Palm, “Experimental investigation of single-phase convective

heat transfer in circular microchannels,” Experimental Thermal and Fluid

Sciences 28 (2004) 105-110.

[43] H.Y. Wu and P. Cheng, “Boiling instability in parallel silicon microchannels at

different heat flux,” International Journal of Heat and Mass Transfer 47 (2004)

3631-3641.

[44] G.L. Morini, M. Spiga and P. Tartarinic, “The rarefaction effect on the friction

factor of gas flow in microchannels,” Superlattices and Microstructures 35 (2004)

587-599.

[45] Y. Chen and P. Cheng, “Condensation of steam in silicon microchannels,”

International Communications in Heat and Mass Transfer 32 (2005) 175-183.

[46] P. Lee, S. V. Garimella and D. Liu, “Investigation of heat transfer in rectangular

microchannels,” International Journal of Heat and Mass Transfer 48 (2005) 1688–

1704.

[47] B. Cao, G. W. Chen and Q. Yuan, “Fully developed laminar flow and heat

transfer in smooth trapezoidal microchannel,” International Communications in

Heat and Mass Transfer 32 (2005) 1211–1220.

[48] G. Hetsroni, A. Mosyak, E. Pogrebnyak and L.P. Yarin, “Heat transfer in micro-

channels: Comparison of experiments with theory and numerical results,”

International Journal of Heat and Mass Transfer 48 (2005) 5580–5601.

[49] L. Zhuo, T. Wen-Quan and H. Ya-Ling, “A numerical study of laminar

convective heat transfer in microchannel with non-circular cross-section,”

International Journal of Thermal Sciences 45 (2006) 1140–1148.

[50] Z. Li, X. Huai, Y. Tao and H. Chen, “Effects of thermal property variations on the

liquid flow and heat transfer in microchannel heat sinks,” Applied Thermal

Engineering 27 (2007) 2803–2814.

[51] A. Husain and K. Kim, “Optimization of a microchannel heat sink with

temperature dependent fluid properties,” Applied Thermal Engineering 28 (2008)

1101–1107.

[52] M. M. Rahman, A. A. Gari and S. Shevade, “Heat transfer in circular

microchannels during volumetric heating with magnetic field,” Heat Mass

Transfer 44 (2008) 463–472.

[53] J. Li and C. Kleinstreuer, “Thermal performance of nanofluids flow in

microchannels,” International Journal of Heat and Fluid Flow 29 (2008) 1221–

1232.

[54] K. Hooman, “Heat and fluid flow in a rectangular microchannel filled with a

porous medium,” International Journal of Heat and Mass Transfer 51 (2008)

5804–5810.

[55] M.I. Hasan, A.A. Rageb, M. Yaghoubi and H. Homayoni, “Influence of channel

geometry on the performance of a counter flow microchannel heat exchanger,”

International Journal of Thermal Sciences 48 (2009) 1607–1618.

[56] S. Hsieh and C. Lin, “Convective heat transfer in liquid microchannels with

hydrophobic and hydrophilic surfaces,” International Journal of Heat and Mass

Transfer 52 (2009) 260–270.

[57] Y. Chen, C. Zhang, M. Shi and J. Wu, “Three-dimensional numerical simulation

of heat and fluid flow in noncircular microchannel heat sinks,” International

Communications in Heat and Mass Transfer 36 (2009) 917–920.

[58] A.J. DeGregoria, L.J. Feuling, J.F. Laatsch, J.R. Rowe, J.R. Trueblood and A.A.

Wang, “Test results of an active magnetic regenerative refrigerator,” Advances in

Cryogenic Engineering 37 (1992) 875-882.

[59] C. Zimm, A. Jastrab, A. Sternberg, V. Pecharsky, K. Gschneidner, M. Osborne

and I. Anderson, “Description and performance of a near-room temperature

magnetic refrigerator,” Advances in Cryogenic Engineering 43 (1998) 1759-1766.

[60] V.K. Pecharsky and K.A. Gschneidner, “Magnetocaloric effect and magnetic

refrigeration,” Journal of Magnetism and Magnetic Materials 200 (1999) 44-56.

[61] K.K. Ambatipudi and M.M. Rahman, “Analysis of conjugate heat transfer in

microchannel sinks,” Numerical Heat Transfer 37 (A) (2000) 711-731.

[62] Hwang C.B. and Lin C.A., “A low Reynolds number two equation model to

predict thermal fields,” International Journal of Heat and Mass Transfer 42 (2000)

3217-3230.

[63] M.N. Ozisik, “Heat Conduction”, 2nd edition, Wiley (1993) 403-420.

[64] A. Bejan, “Convection Heat Transfer”, 2nd edition, Interscience (1994) 797-857.

APPENDICES

Appendix A: FIDAP Code for Analysis of Mass Transfer by Jet Impingement

TITLE( ) JET IMPINGMENT ON ROTATING DISK R1550/W13.09/FLUX FI-GEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) WINDOW(CHANGE= 1, MATRIX ) 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 -10.00000 10.00000 -7.50000 7.50000 -7.50000 7.50000 //ADD POINTS //POINTS 1,2,3,4,5,6 POINT( ADD, COOR ) 0, 0 -0.01, 0 -0.16, 0 -0.16, 0.075 -0.16, 1.5 -0.16, 1.85 -0.01, 1.85 0, 1.85 0, 1.5 0, 0.075 -0.01, 0.075 -0.01, 1.5 -0.01, 1.5 //CONNECT POINTS WITH LINES //LINES 1,2,3,4,5,6 POINT( SELE, ID ) 1 2 3 4 5 6 7 8 9 10 1 CURVE( ADD, LINE ) POINT( SELE, ID ) 2 11 12 7 CURVE( ADD, LINE ) POINT( SELE, ID ) 12 80

Appendix A: (Continued)

9 CURVE( ADD, LINE ) //USE CORNER POINTS TO MAKE SURFACE // POINT( SELE, ID = 1 ) POINT( SELE, ID = 3 ) POINT( SELE, ID = 8 ) POINT( SELE, ID = 6 ) SURFACE( ADD, POIN, ROWW = 2, NOAD ) //CREATE MESH EDGES //CREATE MESH EDGES //MEDGE 1,2,3,4,5,6,7,8 CURVE( SELE, ID = 1 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 2 ) MEDGE( ADD, LSTF, INTE = 20, RATI = 4, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 3 ) MEDGE( ADD, SUCC, INTE = 30, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 4 ) MEDGE( ADD, SUCC, INTE = 460, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 5 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 6 ) MEDGE( ADD, FRST, INTE = 20, RATI = 4, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 7 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 8 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 9 ) MEDGE( ADD, SUCC, INTE = 460, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 10 ) MEDGE( ADD, SUCC, INTE = 30, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 11 ) MEDGE( ADD, SUCC, INTE = 30, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 12 ) MEDGE( ADD, SUCC, INTE = 460, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 13 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 14 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) //MAKE A LOOP OUT THE LINES //LOOP 1 CURVE( SELE, ID ) 1 11 12 14 9 10 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 2, EDG3 = 1, EDG4 = 2 ) //LOOP 2 CURVE( SELE, ID ) 81

Appendix A: (Continued)

14 13 7 8 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) //LOOP 3 CURVE( SELE, ID ) 2 3 4 5 6 13 12 11 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) //ADD A FACE //FACE 1 SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //FACE 2 SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //FACE 3 SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) MFACE( ADD ) //ADD MESH //MESH 1 MFACE( SELE, ID = 1 ) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "nickel" ) //MESH 2 MFACE( SELE, ID = 2 ) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "water" ) //MESH 3 MFACE( SELE, ID = 3 ) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "water" ) //MESH MAP ELEMENT ID // ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( SELE, ID = 1 ) MEDGE( MESH, MAP, ENTI = "outside" ) MEDGE( SELE, ID = 2 ) MEDGE( MESH, MAP, ENTI = "axis" ) MEDGE( SELE, ID = 3 ) MEDGE( MESH, MAP, ENTI = "inflow" ) MEDGE( SELE, ID = 4 ) 82

Appendix A: (Continued)

MEDGE( SELE, ID = 5 ) MEDGE( MESH, MAP, ENTI = "stationary disk" ) MEDGE( SELE, ID = 6 ) MEDGE( SELE, ID = 7 ) MEDGE( MESH, MAP, ENTI = "wall" ) MEDGE( SELE, ID = 8 ) MEDGE( MESH, MAP, ENTI = "outflow" ) MEDGE( SELE, ID = 9 ) MEDGE( SELE, ID = 10 ) MEDGE( MESH, MAP, ENTI = "bottom" ) MEDGE( SELE, ID = 11 ) MEDGE( SELE, ID = 12 ) MEDGE( MESH, MAP, ENTI = "interface" ) END( ) FIPREP( ) //FLUID AND SOLID PROPERTIES //PROPERTIES OF FLUID @ 25 deg Celcius DENSITY( ADD, SET = "water", CONS = 1.085 ) CONDUCTIVITY( ADD, SET = "water", CONS = 0.001434034 ) VISCOSITY( ADD, SET = "water", CONS = 0.013832 ) SPECIFICHEAT( ADD, SET = "water", CONS = 0.999521 ) SURFACETENSION( ADD, SET = "water", CONS = 72 ) //PROPERTIES OF SOLID DENSITY( ADD, SET = "nickel", CONS = 8.88 ) CONDUCTIVITY( ADD, SET = "nickel", CONS = 0.144979459 ) SPECIFICHEAT( ADD, SET = "nickel", CONS = 0.109869112 ) //DEFININING ENTITIES ENTITY( ADD, NAME = "outside", PLOT ) ENTITY( ADD, NAME = "axis", PLOT ) ENTITY( ADD, NAME = "inflow", PLOT ) ENTITY( ADD, NAME = "stationary disk", PLOT ) ENTITY( ADD, NAME = "wall", PLOT ) ENTITY( ADD, NAME = "outflow", PLOT ) ENTITY( ADD, NAME = "bottom", PLOT ) ENTITY( ADD, NAME = "interface", ESPE = 1, ATTA = "water", NATT = "nickel" ) ENTITY( ADD, NAME = "water", FLUI ) ENTITY( ADD, NAME = "nickel", SOLI ) //VELOCITY BOUNDARY CONDITIONS BCNODE( ADD, URC, ENTI = "axis", ZERO ) BCNODE( ADD, URC, ENTI = "inflow", ZERO ) BCNODE( ADD, UZC, ENTI = "inflow", CONS = 127.48 ) BCNODE( ADD, VELO, ZERO, ENTI = "stationary disk" ) BCNODE( ADD, VELO, ZERO, ENTI = "wall" ) BCNODE( ADD, VELO, ZERO, ENTI = "bottom" ) BCNODE( ADD, VELO, ZERO, ENTI = "interface" ) BCNODE( ADD, VELO, ZERO, ENTI = "outside" ) /DETERMINES THE CONCENTRATION OF THE ELECTROLYTE AT THE INFLOW BCNODE( SPEC = 1, CONS = 0.07734, ENTI = "inflow" ) //THERMAL BOUNDARY CONDITIONS BCNODE( ADD, TEMP, CONS = 20, ENTI = "inflow" ) /BCFLUX( ADD, HEAT, CONS = 2.9855, ENTI = "bottom" ) 83

Appendix A: (Continued)

//MASS FLUX BCFLUX( ADD, SPEC = 1, ENTI = "interface", CONS = 0.0001624 ) //THIS BOUNDARY CONDITION IS EQUIVALENT TO 1436 RPM BCNODE( UTHE, POLY = 1, ENTI = "nickel" ) 0, 13.09, 0, 1, 0 BODYFORCE( ADD, CONS, FX = 981, FY = 0, FZ = 0 ) PRESSURE( ADD, MIXE = 1e-11, DISC ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) OPTIONS( ADD, UPWI ) UPWINDING( ADD, STRE ) PROBLEM( ADD, CYLI, INCO, TRAN, LAMI, NONL, NEWT, MOME, ISOT, FIXE, SING, SPEC = 1 ) /PROBLEM( ADD, CYLI, INCO, TRAN, LAMI, NONL, NEWT, MOME, ENERGY, FIXE, SING, SPECIES = 1 ) SOLUTION( ADD, N.R. = 50, KINE = 20, VELC = 0.0001, RESC = 0.0001 ) TIMEINTEGRATION( ADD, BACK, NSTE = 301, TSTA = 0, DT = 1e-05, VARI, WIND = 1, NOFI = 10 ) POSTPROCESS( NBLO = 2 ) 1, 95, 47 95, 301, 1 CLIPPING( ADD, MINI ) 0, 0, 0, 0, 20, 0, 0, 0, 0.07734 ICNODE( ADD, URC, ENTI = "water", CONS = 10 ) ICNODE( ADD, UTHE, ENTI = "water", CONS = 5 ) //ADDED FOR MASS TRANSFER AND DIFFUSION COEFFICIENTS RESPECTIVELY //SPTRANSFER( ADD, SET = 1, CONSTANT = 0.0000002029, POWER=1.0, TEMPERATURE ) DIFFUSIVITY( SET = 1, CONS = 5.508e-06, ISOT, TEMP ) CAPACITY( CONS = 1 ) END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP ) / File closed at Fri Aug 7 18:01:35 2009. / File opened for append Fri Aug 7 18:48:23 2009. FIPOST( ) TIMESTEP( STEP = -1 ) TIMESTEP( STEP = 95 ) LINE( SPEC = 1, ENTI = "interface" ) END( ) END( ) FIPOST( ) DEVICE( POST, FILE = "MESHPLOT" ) MESH( ) END( ) END( ) FIPOST( ) DEVICE( POST, FILE = "VECTPLOT" ) VECTOR( VELO, FACT = 50 ) 84

Appendix A: (Continued)

END( ) END( ) / File closed at Sun Aug 23 19:15:27 2009. / File opened for append Mon Aug 24 01:39:20 2009. FIPOST( ) DEVICE( POST, FILE = "VECT3PLOT" ) VECTOR( VELO, FACT = 50 ) END( ) FIPOST( ) DEVICE( POST, FILE = "STRM3PLOT" ) CONTOUR( STRE, AUTO = 40 ) END( )

Appendix B: FIDAP Code for Fluid Flow and Heat Transfer in a Composite Trapezoidal Microchannel

TITLE( ) MICROCHANNEL FI-GEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) WINDOW(CHANGE= 1, MATRIX ) 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 -10.00000 10.00000 -7.50000 7.50000 -7.50000 7.50000 /^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ / COORDINATES FOR POINTS /^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ POINT( ADD, COOR ) 0, 0, 0 0.305, 0, 0 0.305, 0.01, 0 0.305, 0.025, 0 0.305, 0.525, 0 0, 0.525, 0 0, 0.025, 0 0, 0.01, 0 0.3006, 0.01, 0 0.29, 0.025, 0 0, 0.01, 2.3 0.305, 0.025, 2.3 0, 0, 0.5 0.305, 0, 0.5 0.305, 0.01, 0.5 0.305, 0.025, 0.5 0.305, 0.525, 0.5 0, 0.525, 0.5 0, 0.025, 0.5 0, 0.01, 0.5 0.3006, 0.01, 0.5 0.29, 0.025, 0.5 0, 0, 1 0.305, 0, 1 0.305, 0.01, 1 0.305, 0.025, 1 0.305, 0.525, 1 0, 0.525, 1 0, 0.025, 1 0, 0.01, 1 0.3006, 0.01, 1 0.29, 0.025, 1 0, 0, 1.5 0.305, 0, 1.5 86

Appendix B: (Continued)

0.305, 0.01, 1.5 0.305, 0.025, 1.5 0.305, 0.525, 1.5 0, 0.525, 1.5 0, 0.025, 1.5 0, 0.01, 1.5 0.3006, 0.01, 1.5 0.29, 0.025, 1.5 0, 0, 2 0.305, 0, 2 0.305, 0.01, 2 0.305, 0.025, 2 0.305, 0.525, 2 0, 0.525, 2 0, 0.025, 2 0, 0.01, 2 0.3006, 0.01, 2 0.29, 0.025, 2 0, 0, 2.3 0.305, 0, 2.3 0.305, 0.01, 2.3 0.305, 0.025, 2.3 0.305, 0.525, 2.3 0, 0.525, 2.3 0, 0.025, 2.3 0, 0.01, 2.3 0.3006, 0.01, 2.3 0.29, 0.025, 2.3 /^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ / CONNECTING POINTS WITH LINES /^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ //PART ONE /LINES 1 - 8 POINT( SELE, ID ) 1 2 3 4 5 6 7 8 1 CURVE( ADD, LINE ) /LINES 9 POINT( SELE, ID ) 9 3 CURVE( ADD, LINE ) /LINES 10 - 11 POINT( SELE, ID ) 7 87

Appendix B: (Continued)

10 4 CURVE( ADD, LINE ) /LINES 12 POINT( SELE, ID ) 9 10 CURVE( ADD, LINE ) /LINES 13 POINT( SELE, ID ) 8 9 CURVE( ADD, LINE ) /LINES 14 POINT( SELE, ID ) 8 11 CURVE( ADD, LINE ) /LINES 15 POINT( SELE, ID ) 4 12 CURVE( ADD, LINE ) //PART TWO /LINES 16 - 23 POINT( SELE, ID ) 13 14 15 16 17 18 19 20 13 CURVE( ADD, LINE ) /LINES 24 POINT( SELE, ID ) 21 15 CURVE( ADD, LINE ) /LINES 25 - 26 POINT( SELE, ID ) 19 22 16 CURVE( ADD, LINE ) /LINES 27 POINT( SELE, ID ) 21 22 CURVE( ADD, LINE ) 88

Appendix B: (Continued)

/LINES 28 POINT( SELE, ID ) 20 21 CURVE( ADD, LINE ) //PART THREE /LINES 29 - 36 POINT( SELE, ID ) 23 24 25 26 27 28 29 30 23 CURVE( ADD, LINE ) /LINES 37 POINT( SELE, ID ) 31 25 CURVE( ADD, LINE ) /LINES 38 - 39 POINT( SELE, ID ) 29 32 26 CURVE( ADD, LINE ) /LINES 40 POINT( SELE, ID ) 31 32 CURVE( ADD, LINE ) /LINES 41 POINT( SELE, ID ) 30 31 CURVE( ADD, LINE ) //PART FOUR /LINES 42 - 49 POINT( SELE, ID ) 33 34 35 36 37 38 39 40 33 CURVE( ADD, LINE ) 89

Appendix B: (Continued)

/LINES 50 POINT( SELE, ID ) 41 35 CURVE( ADD, LINE ) /LINES 51 POINT( SELE, ID ) 39 42 36 CURVE( ADD, LINE ) /LINES 53 POINT( SELE, ID ) 41 42 CURVE( ADD, LINE ) /LINES 54 POINT( SELE, ID ) 40 41 CURVE( ADD, LINE ) //PART FIVE /LINES 55 - 62 POINT( SELE, ID ) 43 44 45 46 47 48 49 50 43 CURVE( ADD, LINE ) /LINES 63 POINT( SELE, ID ) 51 45 CURVE( ADD, LINE ) /LINES 64 POINT( SELE, ID ) 49 52 46 CURVE( ADD, LINE ) /LINES 66 POINT( SELE, ID ) 51 52 CURVE( ADD, LINE ) /LINES 67 POINT( SELE, ID ) 90

Appendix B: (Continued)

50 51 CURVE( ADD, LINE ) //PART SIX /LINES 68 - 75 POINT( SELE, ID ) 53 54 55 56 57 58 59 60 53 CURVE( ADD, LINE ) /LINES 76 POINT( SELE, ID ) 61 55 CURVE( ADD, LINE ) /LINES 77 POINT( SELE, ID ) 59 62 56 CURVE( ADD, LINE ) /LINES 79 POINT( SELE, ID ) 61 62 CURVE( ADD, LINE ) /LINES 80 POINT( SELE, ID ) 60 61 CURVE( ADD, LINE ) /////////////////////////////////////////////////////////////////////// / //CREATING SURFACES CURVE( SELE, ID ) 1 2 3 4 5 6 7 8 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) CURVE( SELE, ID ) 16 91

Appendix B: (Continued)

17 18 19 20 21 22 23 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) CURVE( SELE, ID ) 29 30 31 32 33 34 35 36 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) CURVE( SELE, ID ) 42 43 44 45 46 47 48 49 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) CURVE( SELE, ID ) 55 56 57 58 59 60 61 62 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) CURVE( SELE, ID ) 68 69 70 71 72 73 74 75 SURFACE( ADD, WIRE, EDG1 = 1, EDG2 = 3, EDG3 = 1, EDG4 = 3 ) /////////////////////////////////////////////////////////////////////// /// //CREATING MESH EDGES CURVE( SELE, ID = 1 ) 92

Appendix B: (Continued)

MEDGE( ADD, SUCC, INTE = 18, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 2 ) MEDGE( ADD, SUCC, INTE = 6, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 3 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 4 ) MEDGE( ADD, SUCC, INTE = 4, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 5 ) MEDGE( ADD, SUCC, INTE = 18, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 6 ) MEDGE( ADD, SUCC, INTE = 4, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 7 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 8 ) MEDGE( ADD, SUCC, INTE = 6, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 9 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 10 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 11 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 12 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 13 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 14 ) MEDGE( ADD, SUCC, INTE = 90, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, ID = 15 ) MEDGE( ADD, SUCC, INTE = 90, RATI = 0, 2RAT = 0, PCEN = 0 ) /////////////////////////////////////////////////////////////////////// //// //CREATING LOOPS /LOOP 1 CURVE( SELE, ID ) 1 2 9 13 8 MLOOP( ADD, MAP, EDG1 = 1, EDG2 = 1, EDG3 = 2, EDG4 = 1 ) /LOOP 2 CURVE( SELE, ID ) 12 13 7 10 MLOOP( ADD, MAP ) /LOOP 3 CURVE( SELE, ID ) 12 11 3 93

Appendix B: (Continued)

9 MLOOP( ADD, MAP ) /LOOP 4 CURVE( SELE, ID ) 4 5 6 10 11 MLOOP( ADD, MAP, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 2 ) /////////////////////////////////////////////////////////////////////// ////// //CREATING FACE FOR MESH SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //CREATING FACE FOR MESH SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //CREATING FACE FOR MESH SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) MFACE( ADD ) //CREATING FACE FOR MESH SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 4 ) MFACE( ADD ) /////////////////////////////////////////////////////////////////////// ///// //CREATING SOLID FOR THE MESH MFACE( SELE, ID = 1 ) CURVE( SELE, ID = 14 ) MSOLID( PROJ ) MFACE( SELE, ID = 2 ) CURVE( SELE, ID = 14 ) MSOLID( PROJ ) MFACE( SELE, ID = 3 ) CURVE( SELE, ID = 15 ) MSOLID( PROJ ) MFACE( SELE, ID = 4 ) CURVE( SELE, ID = 15 ) MSOLID( PROJ ) /////////////////////////////////////////////////////////////////////// ///// //CREATION OF THE MESH AND ASSIGNING CONTINUOUS ENTITIES MSOLID( SELE, ID = 1 ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, ENTI = "SILICON" ) MSOLID( SELE, ID = 2 ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, ENTI = "SILICON" ) 94

Appendix B: (Continued)

MSOLID( SELE, ID = 3 ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, ENTI = "FLUID" ) MSOLID( SELE, ID = 4 ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, ENTI = "GADOLINIUM" ) /////////////////////////////////////////////////////////////////////// //// //ASSIGNING ENTITIES TO VARIOUS BOUNDARIES MFACE( SELE, ID = 3 ) MFACE( MESH, MAP, ENTI = "Fin" ) MFACE( SELE, ID = 15 ) MFACE( MESH, MAP, ENTI = "Fout" ) /MFACE( SELE, ID = 12 ) /MFACE( SELE, ID = 17 ) /MFACE( MESH, MAP, ENTI = "HF" ) MFACE( SELE, ID = 8 ) MFACE( MESH, MAP, ENTI = "Fbot" ) MFACE( SELE, ID = 17 ) MFACE( MESH, MAP, ENTI = "Ftop" ) MFACE( SELE, ID = 16 ) MFACE( MESH, MAP, ENTI = "Faxis" ) MFACE( SELE, ID = 14 ) MFACE( MESH, MAP, ENTI = "Fleft" ) MFACE( SELE, ID = 6 ) MFACE( MESH, MAP, ENTI = "Sbot" ) MFACE( SELE, ID = 7 ) MFACE( MESH, MAP, ENTI = "Saxis" ) MFACE( SELE, ID = 19 ) MFACE( MESH, MAP, ENTI = "Gaxis" ) MFACE( SELE, ID = 20 ) MFACE( MESH, MAP, ENTI = "Gtop" ) MFACE( SELE, ID = 21 ) MFACE( MESH, MAP, ENTI = "Gleft" ) MFACE( SELE, ID = 13 ) MFACE( MESH, MAP, ENTI = "Sleft1" ) MFACE( SELE, ID = 10 ) MFACE( MESH, MAP, ENTI = "Sleft2" ) MEDGE( SELE, ID = 23 ) MEDGE( MESH, MAP, ENTI = "RTedge" ) MEDGE( SELE, ID = 22 ) MEDGE( MESH, MAP, ENTI = "RBedge" ) MEDGE( SELE, ID = 24 ) MEDGE( MESH, MAP, ENTI = "LTedge" ) MEDGE( SELE, ID = 21 ) MEDGE( MESH, MAP, ENTI = "LBedge" ) END( ) FIPREP( ) /////////////////////////////////////////////////////////////////////// //// //PROPERTY OF SILICON CONDUCTIVITY( ADD, SET = "SILICON", CONS = 0.29637, ISOT ) 95

Appendix B: (Continued)

DENSITY( ADD, SET = "SILICON", CONS = 2.329 ) SPECIFICHEAT( ADD, SET = "SILICON", CONS = 0.16778 ) //PROPERTY OF FLUID CONDUCTIVITY( ADD, SET = "FLUID", CONS = 0.0014435, ISOT ) DENSITY( ADD, SET = "FLUID", CONS = 0.9974 ) SPECIFICHEAT( ADD, SET = "FLUID", CONS = 0.9988 ) VISCOSITY( ADD, SET = "FLUID", CONS = 0.0098 ) //PROPERTY OF GADOLINIUM CONDUCTIVITY( ADD, SET = "GADOLINIUM", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "GADOLINIUM", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "GADOLINIUM", CONS = 0.054971 ) /////////////////////////////////////////////////////////////////////// ///// //DEFINING ENTITIES ENTITY( ADD, NAME = "SILICON", SOLI ) ENTITY( ADD, NAME = "GADOLINIUM", SOLI ) ENTITY( ADD, NAME = "FLUID", FLUI ) ENTITY( ADD, NAME = "Fin", PLOT ) ENTITY( ADD, NAME = "Fout", PLOT ) /ENTITY( ADD, NAME = "HF", PLOT, ATTA = "FLUID", NATTA = "GADOLINIUM" ) ENTITY( ADD, NAME = "Fbot", PLOT, ATTA = "FLUID", NATT = "SILICON" ) ENTITY( ADD, NAME = "Ftop", PLOT, ATTA = "FLUID", NATT = "GADOLINIUM" ) ENTITY( ADD, NAME = "Faxis", PLOT ) ENTITY( ADD, NAME = "Fleft", PLOT, ATTA = "FLUID", NATT = "SILICON" ) ENTITY( ADD, NAME = "Sbot", PLOT ) ENTITY( ADD, NAME = "Saxis", PLOT ) ENTITY( ADD, NAME = "Gaxis", PLOT ) ENTITY( ADD, NAME = "Gtop", PLOT ) ENTITY( ADD, NAME = "Gleft", PLOT ) ENTITY( ADD, NAME = "Sleft1", PLOT ) ENTITY( ADD, NAME = "Sleft2", PLOT ) ENTITY( ADD, NAME = "RTedge", PLOT ) ENTITY( ADD, NAME = "RBedge", PLOT ) ENTITY( ADD, NAME = "LTedge", PLOT ) ENTITY( ADD, NAME = "LBedge", PLOT ) /////////////////////////////////////////////////////////////////////// ///// //INITIAL AND BOUNDARY CONDITION BCNODE( ADD, UX, ENTI = "Fin", ZERO ) BCNODE( ADD, UY, ENTI = "Fin", ZERO ) BCNODE( ADD, UZ, ENTI = "Fin", CONS = 1425.34 ) BCNODE( ADD, TEMP, ENTI = "Fin", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "Fbot", ZERO ) BCNODE( ADD, VELO, ENTI = "Ftop", ZERO ) BCNODE( ADD, VELO, ENTI = "Fleft", ZERO ) BCNODE( ADD, UX, ENTI = "Faxis", ZERO ) /BCFLUX( ADD, HEAT, ENTI = "HF", CONS = 14.49 ) BCFLUX( ADD, HEAT, ENTI = "Sbot", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Saxis", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gaxis", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gtop", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gleft", CONS = 0 ) 96

Appendix B: (Continued)

BCFLUX( ADD, HEAT, ENTI = "Sleft1", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Sleft2", CONS = 0 ) SOURCE( ADD, HEAT, CONS = 6.080715996, ENTI = "GADOLINIUM" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "SILICON" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "GADOLINIUM" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "FLUID" ) /////////////////////////////////////////////////////////////////////// // //IF THE FLOW IT TURBULENT AND K-E MODEL IS USED, ADD THE FOLLOWING //LINES OF CODES /VISCOSITY( ADD, SET = "FLUID", TWO-, CONS = 0.0098 ) /ICNODE( KINE, CONS = 0.003, ALL ) /ICNODE( DISS, CONS = 0.00045, ALL ) /BCNODE( KINE, CONS = 0.001, ENTI = "Fin" ) /BCNODE( DISS, CONS = 0.00045, ENTI = "Fin" ) /////////////////////////////////////////////////////////////////////// //// //EXECUTION COMMANDS DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) PROBLEM( ADD, 3-D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, SEGR = 10, PREC = 21, ACCF = 0, PPRO ) END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP )

END( ) FIPOST( ) PRINT( PRES, POIN, FILE = "comb.txt", SCRE ) 324 0.3006, 0.01, 0 0.301088889, 0.01, 0 0.301577778, 0.01, 0 0.302066667, 0.01, 0 0.302555556, 0.01, 0 0.303044444, 0.01, 0 0.303533333, 0.01, 0 0.304022222, 0.01, 0 0.304511111, 0.01, 0 0.305, 0.01, 0 0.305, 0.025, 0 0.303333333, 0.025, 0 0.301666667, 0.025, 0 0.3, 0.025, 0 0.298333333, 0.025, 0 0.296666667, 0.025, 0 0.295, 0.025, 0 0.293333333, 0.025, 0 0.291666667, 0.025, 0 0.29, 0.025, 0 97

Appendix B: (Continued)

0.291326715, 0.023126213, 0 0.29265343, 0.021252426, 0 0.293980145, 0.019378639, 0 0.29530686, 0.017504853, 0 0.296633575, 0.015631066, 0 0.29796029, 0.013757279, 0 0.299287005, 0.011883492, 0 0.3006, 0.01, 0.5 0.301088889, 0.01, 0.5 0.301577778, 0.01, 0.5 0.302066667, 0.01, 0.5 0.302555556, 0.01, 0.5 0.303044444, 0.01, 0.5 0.303533333, 0.01, 0.5 0.304022222, 0.01, 0.5 0.304511111, 0.01, 0.5 0.305, 0.01, 0.5 0.305, 0.025, 0.5 0.303333333, 0.025, 0.5 0.301666667, 0.025, 0.5 0.3, 0.025, 0.5 0.298333333, 0.025, 0.5 0.296666667, 0.025, 0.5 0.295, 0.025, 0.5 0.293333333, 0.025, 0.5 0.291666667, 0.025, 0.5 0.29, 0.025, 0.5 0.291326715, 0.023126213, 0.5 0.29265343, 0.021252426, 0.5 0.293980145, 0.019378639, 0.5 0.29530686, 0.017504853, 0.5 0.296633575, 0.015631066, 0.5 0.29796029, 0.013757279, 0.5 0.299287005, 0.011883492, 0.5 0.3006, 0.01, 1 0.301088889, 0.01, 1 0.301577778, 0.01, 1 0.302066667, 0.01, 1 0.302555556, 0.01, 1 0.303044444, 0.01, 1 0.303533333, 0.01, 1 0.304022222, 0.01, 1 0.304511111, 0.01, 1 0.305, 0.01, 1 0.305, 0.025, 1 0.303333333, 0.025, 1 0.301666667, 0.025, 1 0.3, 0.025, 1 0.298333333, 0.025, 1 0.296666667, 0.025, 1 0.295, 0.025, 1 0.293333333, 0.025, 1 98

Appendix B: (Continued)

0.291666667, 0.025, 1 0.29, 0.025, 1 0.291326715, 0.023126213, 1 0.29265343, 0.021252426, 1 0.293980145, 0.019378639, 1 0.29530686, 0.017504853, 1 0.296633575, 0.015631066, 1 0.29796029, 0.013757279, 1 0.299287005, 0.011883492, 1 0.3006, 0.01, 1.5 0.301088889, 0.01, 1.5 0.301577778, 0.01, 1.5 0.302066667, 0.01, 1.5 0.302555556, 0.01, 1.5 0.303044444, 0.01, 1.5 0.303533333, 0.01, 1.5 0.304022222, 0.01, 1.5 0.304511111, 0.01, 1.5 0.305, 0.01, 1.5 0.305, 0.025, 1.5 0.303333333, 0.025, 1.5 0.301666667, 0.025, 1.5 0.3, 0.025, 1.5 0.298333333, 0.025, 1.5 0.296666667, 0.025, 1.5 0.295, 0.025, 1.5 0.293333333, 0.025, 1.5 0.291666667, 0.025, 1.5 0.29, 0.025, 1.5 0.291326715, 0.023126213, 1.5 0.29265343, 0.021252426, 1.5 0.293980145, 0.019378639, 1.5 0.29530686, 0.017504853, 1.5 0.296633575, 0.015631066, 1.5 0.29796029, 0.013757279, 1.5 0.299287005, 0.011883492, 1.5 0.3006, 0.01, 2 0.301088889, 0.01, 2 0.301577778, 0.01, 2 0.302066667, 0.01, 2 0.302555556, 0.01, 2 0.303044444, 0.01, 2 0.303533333, 0.01, 2 0.304022222, 0.01, 2 0.304511111, 0.01, 2 0.305, 0.01, 2 0.305, 0.025, 2 0.303333333, 0.025, 2 0.301666667, 0.025, 2 0.3, 0.025, 2 0.298333333, 0.025, 2 0.296666667, 0.025, 2 99

Appendix B: (Continued)

0.295, 0.025, 2 0.293333333, 0.025, 2 0.291666667, 0.025, 2 0.29, 0.025, 2 0.291326715, 0.023126213, 2 0.29265343, 0.021252426, 2 0.293980145, 0.019378639, 2 0.29530686, 0.017504853, 2 0.296633575, 0.015631066, 2 0.29796029, 0.013757279, 2 0.299287005, 0.011883492, 2 0.3006, 0.01, 2.3 0.301088889, 0.01, 2.3 0.301577778, 0.01, 2.3 0.302066667, 0.01, 2.3 0.302555556, 0.01, 2.3 0.303044444, 0.01, 2.3 0.303533333, 0.01, 2.3 0.304022222, 0.01, 2.3 0.304511111, 0.01, 2.3 0.305, 0.01, 2.3 0.305, 0.025, 2.3 0.303333333, 0.025, 2.3 0.301666667, 0.025, 2.3 0.3, 0.025, 2.3 0.298333333, 0.025, 2.3 0.296666667, 0.025, 2.3 0.295, 0.025, 2.3 0.293333333, 0.025, 2.3 0.291666667, 0.025, 2.3 0.29, 0.025, 2.3 0.291326715, 0.023126213, 2.3 0.29265343, 0.021252426, 2.3 0.293980145, 0.019378639, 2.3 0.29530686, 0.017504853, 2.3 0.296633575, 0.015631066, 2.3 0.29796029, 0.013757279, 2.3 0.299287005, 0.011883492, 2.3 0.3006, 0.008333333, 0 0.301088889, 0.008333333, 0 0.301577778, 0.008333333, 0 0.302066667, 0.008333333, 0 0.302555556, 0.008333333, 0 0.303044444, 0.008333333, 0 0.303533333, 0.008333333, 0 0.304022222, 0.008333333, 0 0.304511111, 0.008333333, 0 0.305, 0.008333333, 0 0.305, 0.15, 0 0.303333333, 0.15, 0 0.301666667, 0.15, 0 0.3, 0.15, 0 100

Appendix B: (Continued)

0.298333333, 0.15, 0 0.296666667, 0.15, 0 0.295, 0.15, 0 0.293333333, 0.15, 0 0.291666667, 0.15, 0 0.257777778, 0.025, 0 0.259104493, 0.023126213, 0 0.260431208, 0.021252426, 0 0.261757923, 0.019378639, 0 0.263084638, 0.017504853, 0 0.264411353, 0.015631066, 0 0.265738068, 0.013757279, 0 0.267064783, 0.011883492, 0 0.3006, 0.008333333, 0.5 0.301088889, 0.008333333, 0.5 0.301577778, 0.008333333, 0.5 0.302066667, 0.008333333, 0.5 0.302555556, 0.008333333, 0.5 0.303044444, 0.008333333, 0.5 0.303533333, 0.008333333, 0.5 0.304022222, 0.008333333, 0.5 0.304511111, 0.008333333, 0.5 0.305, 0.008333333, 0.5 0.305, 0.15, 0.5 0.303333333, 0.15, 0.5 0.301666667, 0.15, 0.5 0.3, 0.15, 0.5 0.298333333, 0.15, 0.5 0.296666667, 0.15, 0.5 0.295, 0.15, 0.5 0.293333333, 0.15, 0.5 0.291666667, 0.15, 0.5 0.257777778, 0.025, 0.5 0.259104493, 0.023126213, 0.5 0.260431208, 0.021252426, 0.5 0.261757923, 0.019378639, 0.5 0.263084638, 0.017504853, 0.5 0.264411353, 0.015631066, 0.5 0.265738068, 0.013757279, 0.5 0.267064783, 0.011883492, 0.5 0.3006, 0.008333333, 1 0.301088889, 0.008333333, 1 0.301577778, 0.008333333, 1 0.302066667, 0.008333333, 1 0.302555556, 0.008333333, 1 0.303044444, 0.008333333, 1 0.303533333, 0.008333333, 1 0.304022222, 0.008333333, 1 0.304511111, 0.008333333, 1 0.305, 0.008333333, 1 0.305, 0.15, 1 0.303333333, 0.15, 1 101

Appendix B: (Continued)

0.301666667, 0.15, 1 0.3, 0.15, 1 0.298333333, 0.15, 1 0.296666667, 0.15, 1 0.295, 0.15, 1 0.293333333, 0.15, 1 0.291666667, 0.15, 1 0.257777778, 0.025, 1 0.259104493, 0.023126213, 1 0.260431208, 0.021252426, 1 0.261757923, 0.019378639, 1 0.263084638, 0.017504853, 1 0.264411353, 0.015631066, 1 0.265738068, 0.013757279, 1 0.267064783, 0.011883492, 1 0.3006, 0.008333333, 1.5 0.301088889, 0.008333333, 1.5 0.301577778, 0.008333333, 1.5 0.302066667, 0.008333333, 1.5 0.302555556, 0.008333333, 1.5 0.303044444, 0.008333333, 1.5 0.303533333, 0.008333333, 1.5 0.304022222, 0.008333333, 1.5 0.304511111, 0.008333333, 1.5 0.305, 0.008333333, 1.5 0.305, 0.15, 1.5 0.303333333, 0.15, 1.5 0.301666667, 0.15, 1.5 0.3, 0.15, 1.5 0.298333333, 0.15, 1.5 0.296666667, 0.15, 1.5 0.295, 0.15, 1.5 0.293333333, 0.15, 1.5 0.291666667, 0.15, 1.5 0.257777778, 0.025, 1.5 0.259104493, 0.023126213, 1.5 0.260431208, 0.021252426, 1.5 0.261757923, 0.019378639, 1.5 0.263084638, 0.017504853, 1.5 0.264411353, 0.015631066, 1.5 0.265738068, 0.013757279, 1.5 0.267064783, 0.011883492, 1.5 0.3006, 0.008333333, 2 0.301088889, 0.008333333, 2 0.301577778, 0.008333333, 2 0.302066667, 0.008333333, 2 0.302555556, 0.008333333, 2 0.303044444, 0.008333333, 2 0.303533333, 0.008333333, 2 0.304022222, 0.008333333, 2 0.304511111, 0.008333333, 2 0.305, 0.008333333, 2 102

Appendix B: (Continued)

0.305, 0.15, 2 0.303333333, 0.15, 2 0.301666667, 0.15, 2 0.3, 0.15, 2 0.298333333, 0.15, 2 0.296666667, 0.15, 2 0.295, 0.15, 2 0.293333333, 0.15, 2 0.291666667, 0.15, 2 0.257777778, 0.025, 2 0.259104493, 0.023126213, 2 0.260431208, 0.021252426, 2 0.261757923, 0.019378639, 2 0.263084638, 0.017504853, 2 0.264411353, 0.015631066, 2 0.265738068, 0.013757279, 2 0.267064783, 0.011883492, 2 0.3006, 0.008333333, 2.3 0.301088889, 0.008333333, 2.3 0.301577778, 0.008333333, 2.3 0.302066667, 0.008333333, 2.3 0.302555556, 0.008333333, 2.3 0.303044444, 0.008333333, 2.3 0.303533333, 0.008333333, 2.3 0.304022222, 0.008333333, 2.3 0.304511111, 0.008333333, 2.3 0.305, 0.008333333, 2.3 0.305, 0.15, 2.3 0.303333333, 0.15, 2.3 0.301666667, 0.15, 2.3 0.3, 0.15, 2.3 0.298333333, 0.15, 2.3 0.296666667, 0.15, 2.3 0.295, 0.15, 2.3 0.293333333, 0.15, 2.3 0.291666667, 0.15, 2.3 0.257777778, 0.025, 2.3 0.259104493, 0.023126213, 2.3 0.260431208, 0.021252426, 2.3 0.261757923, 0.019378639, 2.3 0.263084638, 0.017504853, 2.3 0.264411353, 0.015631066, 2.3 0.265738068, 0.013757279, 2.3 0.267064783, 0.011883492, 2.3 END( ) FIPOST( ) WINDOW( CHAN = 0, FRON ) CONTOUR( UX, AUTO = 50 ) CONTOUR( TEMP, AUTO = 50 ) CONTOUR( TEMP, AUTO = 25 ) CONTOUR( TEMP, AUTO = 10 ) CONTOUR( TEMP, AUTO = 15 ) 103

Appendix B: (Continued)

CONTOUR( TEMP, AUTO = 20 ) DEVICE( POST, FILE = "TEMPPLOT" ) CONTOUR( TEMP, AUTO = 20 ) END( )

104