Simulation and Optimization of cryogenic heat sink for superconducting power cable applications

THE FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING

SIMULATION AND OPTIMIZATION OF CRYOGENIC HEAT SINK FOR SUPERCONDUCTING POWER CABLE APPLICATIONS

By DARSHIT SHAH

A Thesis submitted to the College of Engineering in partial fulfillment of the requirements for the degree of Master of Science

Degree Awarded: Summer Semester, 2013

Darshit Rajiv Shah defended this thesis on June 12, 2013. The members of the supervisory committee were:

Juan C Ordonez Professor Directing Thesis Patrick J Hollis Committee Member Jose Vargas Committee Member Wei Guo Committee Member

The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements.

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To my parents, whom I acknowledge, now, more than ever

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ACKNOWLEDGMENTS I would like to acknowledge the constant support and guidance of Dr. Juan Ordonez throughout my degree and for introducing me to this project. He has been patient with me and pushed me to achieve success for which I am grateful to him. I would also like to acknowledge him for his guidance and support in making this project a wonderful and enjoyable experience. I would like to acknowledge the constant support of Dr. Alejandro Rivera – Alvarez and Dr. Lukas Graber, who got the ball rolling initially and without whose help this thesis work would have been impossible. I would also like to acknowledge the support of Dr. Sastry Pamidi and Dr. Chul Kim for their inputs and insights. I would like to acknowledge my family and friends for giving me the support and making me feel at home always. My research group has been engaging: giving thoughtful suggestions, throwing dinner parties and making life much better. For this, I would like to acknowledge Michael Coleman, Sam Yang, Piero Caballero, Julian Ramirez, David Delgado, Matt Vedrin and Obie Abakparo. Finally I would like to acknowledge Electric Ship Research and Development Consortium (ESRDC), Office of Naval Research (ONR) and the Department of Mechanical Engineering at Florida State University for their steady financial support.

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TABLE OF CONTENTS List of Tables ......................................................................................................................vii List of Figures ................................................................................................................... viii Abstract ................................................................................................................................. x 1. INTRODUCTION................................................................................................................. 1 1.1

Background on the Research Program........................................................................ 1

1.2

Superconductivity and Superconducting Cable System .............................................. 2

1.3

Bibliographical Review.............................................................................................. 4 1.3.1 Albany HTS Cable Project ..................................................................................... 5 1.3.2 Long Island Transmission Level HTS Cable Project .............................................. 6 1.3.3 ORNL-Southwire Company Demonstration Project ............................................... 7 1.3.4 Fault Current Limiting HTS cable with Con Edison ............................................... 8

1.4

Challenges and Objectives ......................................................................................... 9 1.4.1 Benefits and Challenges....................................................................................... 10 1.4.2 Objectives ........................................................................................................... 11

1.5

Overview ................................................................................................................. 12

2. SIMULATION OF SUPERCONDUCTING CABLE TERMINATION FOR LAMINAR FORCED CONVECTION................................................................................................... 15 2.1

Laminar Forced Convective Flow ............................................................................ 15

2.2

Numerical Verification and Simulation .................................................................... 16 2.2.1 Verification Cases ............................................................................................... 17 2.2.2 Two-dimensional FEM model ............................................................................. 24

2.3

Experimental Setup and Results ............................................................................... 32

2.4

Laminar 3-D FEM model ......................................................................................... 40

3. SIMULATION, MODEL VALIDATION AND OPTIMIZATION OF CABLE TERMINATION GEOMETRY ........................................................................................... 46 3.1

Introduction ............................................................................................................. 46 v

3.2

Turbulent 3-D Simulation using κ-ε Model .............................................................. 47

3.3

Model Validation with Experimental Results ........................................................... 52

3.4

Geometric Optimization of the Heat Sink ................................................................. 56

4. CONCLUSION ................................................................................................................... 61 4.1

Summary of Research Efforts .................................................................................. 61

4.2

Suggestions for Future Work ................................................................................... 63

REFERENCES ......................................................................................................................... 67 BIOGRAPHICAL SKETCH ..................................................................................................... 70

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LIST OF TABLES Table 2.1 Dimension of plate fin arrangement ........................................................................... 18 Table 2.2 Prototype heat sink dimensions .................................................................................. 33 Table 2.3 Heater wire configuration to obtain various heat load values ...................................... 36 Table 2.4 Experimental results for heat sink setup ..................................................................... 37 Table 2.5 Flow conditions for various experimental cases ......................................................... 39 Table 3.1 Pressure drop validation results with experimental data ............................................. 53 Table 3.2 Optimum values for objective function under given constraints ................................. 58 Table 4.1 Relative error percentages between numerical and experimental results ..................... 62

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LIST OF FIGURES Figure 1.1 Comparison of overhead power lines to HTS cables (http://www.doe.gov) .................3 Figure 1.2 Schematic view of Albany cable system .....................................................................4 Figure 1.3 3-in-one cable termination for Albany project cooled by LN2 .....................................5 Figure 1.4 LIPA cable thermal budget .........................................................................................6 Figure 1.5 Schematic of pressurized termination concept ............................................................8 Figure 1.6 A typical HTS cable termination in operation .............................................................9 Figure 1.7 A schematic diagram of the superconducting cable termination with heat sink (red), cable (green) and copper conductor (yellow). ........................................................... 14 Figure 2.1 Plain fin arrangements in a compact plate heat exchanger ......................................... 18 Figure 2.2 (a) Interferograms for plain fin arrangements for r =1 mm, Re=500 (b) Results obtained using COMSOL with isotherm temperatures in K .................... 20 Figure 2.3 Axial temperature variations for heat transfer in a tube ............................................. 21 Figure 2.4 Axial temperature variations obtained using FEM technique .................................... 23 Figure 2.5 Design of the prototype cable termination (total view) and cut views to show the internal fin structure (vertical and horizontal cut) ..................................................... 25 Figure 2.6 Variation of density of gaseous Helium with temperature at constant pressure .......... 27 Figure 2.7 2-Dimensional FEM model of the heat sink with important boundary conditions ...... 30 Figure 2.8 Surface temperature distribution (in Kelvin) with h = 90 W/m2 K and ṁ = 1.5 g/s ..... 31 Figure 2.9 Tpeak and Δp curves for various mass flow rates ........................................................ 32 Figure 2.10 Copper heat sink prototype used for experimentation.............................................. 34 Figure 2.11 Experimental setup with flow lines and heat sink with heater attached .................... 34 Figure 2.12 Heat sink wrapped in MLI before insertion into the cryostat ................................... 35 Figure 2.13 Mesh structure for the laminar FEM model............................................................. 42 Figure 2.14 GMRES solution curve for each iteration ............................................................... 43 viii

Figure 2.15 Error curve for non-linear solver using default settings ........................................... 43 Figure 2.16 Heat sink surface temperature (in Kelvin) with the fluid velocity field shown by black arrow heads .................................................................................................... 44 Figure 3.1 The average velocity component and the fluctuating velocity component (32) .......... 47 Figure 3.2 Mesh structure obtained by separately meshing the domains and the interface boundaries ............................................................................................................... 50 Figure 3.3 Convergence curve for stationary turbulent flow solver ............................................ 50 Figure 3.4 Streamline velocity field in the heat sink for case no. 6 ............................................. 51 Figure 3.5 Temperature profile in the heat sink for case no.6 ..................................................... 51 Figure 3.6 Schematic diagram for gHe flow system ................................................................... 52 Figure 3.7 Comparison of model results with experimental data for fluid outlet temperature ..... 54 Figure 3.8 Model validation for peak temperature of solid copper block .................................... 54 Figure 3.9 Model validation results for pressure drop across the experimental setup .................. 55 Figure 3.10 Vertical cut section showing geometrical optimization parameters considered for the study .................................................................................................................. 57 Figure 3.11(a) Variation of heat sink peak temperature with fin spacing (b) Contour plot showing optimized geometry for minimum peak temperature ........ 59 Figure 3.12(a) Variation of pressure loss across the heat sink with fin spacing (b) Contour plot showing optimized geometry for minimum pressure ...................... 60 Figure 4.1 Optimal topology and temperature distribution slices of 3D design domain .............. 64 Figure 4.2 Schematic representation of the superconducting cable volume elements in radial (r) and axial (z) direction .............................................................................................. 65

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ABSTRACT Superconducting power devices require cable terminations to intercept the heat inleaks from the ambient temperature, thus, maintaining the superconducting cable within operating cryogenic temperature limits. Owing to the possible safety hazards such as asphyxiation and cold burns resulting from the use of liquid cryogen, use of gaseous Helium as a possible cooling medium for superconducting power devices is being considered. Also, the use of helium gas facilitates operation of the superconducting device at temperatures much lower than even subcooled liquid Nitrogen, thereby, increasing their critical current density. A model is being developed using the finite element method (FEM) to study the feasibility of a helium gas cooled heat sink to be used as a cable termination. The results obtained from the simulation model are validated with an experimental setup. The numerical coolant temperature and pressure drop as well as the heat sink temperature correspond well with the experimental results. Furthermore, the heat sink is geometrically optimized for given mass flow rate and input conditions to produce a better thermal and fluid performance in terms of temperature gain and pressure loss. It is found that an unequally spaced heat sink, which distributes the flow uniformly across all the channels, is more effective and gives better performance.

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CHAPTER ONE INTRODUCTION This chapter reviews the basic applications of cryogenics and superconductivity in power systems. It also gives a bibliographical review of the various types of cryogenic superconducting cable terminations presently being used in demonstration and research projects worldwide. It lists the objectives and challenges of the work done in this thesis. An overview and scope of this thesis is discussed at the end of this chapter. 1.1.

Background on the Research Program

A collaborative research program between Office of Naval Research (ONR) and Center for Advanced Power Systems (CAPS) is ongoing at Florida State University. The primary objective of this program is to investigate the feasibility of high temperature superconductor (HTS) based degaussing system. This novel degaussing system promises a 75% reduction in system weight and a 80% reduction in installed cable length (1). Following up, it has been envisioned that HTS power devices cooled by gaseous helium be used for naval and airborne applications (2), (3). Continuing with this, a research program has been setup for understanding the feasibility of HTS applications in power transmission devices. The author is involved in this program as a graduate research assistant. Most HTS power devices and systems are cooled by liquid nitrogen. As a small part of the entire HTS cable system, the author was involved in setting up a thermal and fluid flow model to perform basic convection phenomenon studies on gaseous helium cooled superconducting cable termination. With the help of the collaborative research group, the present thesis develops a Finite Element 1

Method (FEM) model and the simulation results, thus obtained, have been validated with experimental data. 1.2.

Superconductivity and Superconducting Cable System

The science and technology of producing low temperature environment is generally referred to as cryogenics. A special property, of certain materials, that only appears at cryogenic temperatures is called “Superconductivity”. It is defined by the simultaneous disappearance of all electric resistance and appearance of perfect diamagnetism. Superconducting power cables are the most common application of superconductivity in the electric power system. Electric power is becoming the standard to how the society is developed well and its demand is increasing rapidly over the world. However, in most power systems, there are several difficulties from generation to distribution. Long transmission and distribution lines drive the use of cheap, reliable and efficient conductors like aluminum and copper. However, such conductors have ohmic losses and restrict the capability of thermal rating of electric facilities. On the other hand, superconducting cables in distribution class can deliver about 5 times more power than conventional XLPE (Cross Link Poly Ethylene) cable of same dimension. Comparing 66kV, 3 kA, 350 MVA class cables, the loss in a superconducting cable is approximately half that of a conventional cable (4). DC superconducting cable can even eliminate AC loss in superconductor, thereby, further increasing the system efficiency. Their study is in research stage and will be applied to HVDC transmission system. After McFee developed the idea of a superconducting cable in 1961, much work in Low temperature Superconductor (LTS) power cable cooled by liquid Helium have been done in 1970’s and 1980’s (5). In 1986, Bednorz and Muller developed High Temperature Superconductor (HTS) cable cooled by liquid Nitrogen. Research on this topic has progressed much and it is currently in industrial application stage. A schematic 2

comparison of overhead conventional power lines to underground HTS cables is shown in Figure 1.1. The figure demonstrates the relative less space occupied and less construction costs incurred by a HTS superconducting cable system. These advantages are further amplified in highly populated cities and mega power plants.

Figure 1.1 Comparison of overhead power lines to HTS cables (http://www.doe.gov) The main components of a HTS cable system (4) are HTS cable, cooling facility, cable termination and monitoring system as shown as an example in Figure 1.2. Different types of HTS cables are used for different purposes viz. single core for transmission, tri-axial type for sub transmission and co-axial type for distribution. The superconductive nature of the material

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vanishes when it reaches a particular critical temperature characterized by various parameters. The cooling station cools the HTS cable-termination system and is required to maintain the superconductive property with operating temperatures lower than corresponding critical temperatures. The termination locates both ends of HTS cables and connects the HTS cable to ambient temperature power line. Because of large difference of temperature between HTS cable and outer weather, termination has to sustain temperature difference and pump out heat inleaks into the system.

Figure 1.2 Schematic view of Albany cable system

1.3.

Bibliographical Review

The U.S. Department of Energy (DOE) is partnering with industry to fund various projects to demonstrate the use of HTS cable technology in power system devices. Various research and demonstration projects on such HTS technology are going on worldwide. Prominent among them are the ones in Albany, NY, Columbus, Ohio and Long Island, New York (6). Apart from this, research and demonstration projects of various types and scales in HTS superconductive power devices have been going on at Oak Ridge National Laboratory

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(ORNL) (7), Con Edison (8), JAPAN (9) and others. Case studies reported here focus on cable termination design and testing results reported by respective research agencies.

1.3.1. Albany HTS Cable Project In June 2005, the Albany project (10), (11) planned to place a HTS cable with 34.5 kV and 800 A between two substations in the existing Niagara Mohawk power grid. In this project, liquid Nitrogen is circulated in a loop, through the HTS cable, joints, terminations and a return pipe by a cooling system.

Figure 1.3 3-in-one cable termination for Albany project cooled by LN2 The entire 3-in-one type cable termination with current leads is housed inside a cryostat cooled with liquid Nitrogen. Figure 1.3 shows the basic structure of the HTS termination wherein the porcelain connection houses the normal copper conductor and it is connected to the HTS 5

cable at its outlet. The coolant fluid is circulated using a cryogenic pump with a second pump in parallel for redundancy. Normal operation consists of sub-cooled LN2 loop being continuously refrigerated using a cryocooler with the operating temperature being in the range of 67-77 K. The coolant flows at a maximum rate of 50 L/min at a gauge pressure of 0.5 MPa. During the test phase, the total heat loss measured, throughout the 350 m HTS cable system including the terminations and piping system, was 3.1 kW. 1.3.2. Long Island Transmission Level HTS Cable Project The U. S. DOE funds the design, development and demonstration of first long length transmission HTS cable to be installed in the Long Island Power Authority (LIPA) grid. It is designed to carry 2400 A at 138 kV (12), (13).

Figure 1.4 LIPA cable thermal budget

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In this project, unlike Albany project, each cable phase is connected at both ends with terminations that act as an interface between the HTS cable and the grid. Figure 1.4 shows the thermal budget of various elements of the system being operated under different conditions. The effect of various conditions and defects in the system on the performance of the HTS cable can be seen in the figure. The cable cooling system uses a Brayton cycle refrigerator with helium gas as working fluid. This cold helium gas at the refrigerator outlet cools the liquid nitrogen to minimum temperature of 65 K. At full current load, the nominal heat load on the HTS cable including the terminations is about 12 kW. This heat load is removed with the help of LN 2 flowing in at 0.375 kg/s at 18 bar maximum pressure and a maximum operating temperature of 72 K. 1.3.3. ORNL-Southwire Company Demonstration Project A joint program between ORNL and Southwire Company is set up to develop a HTS cable test facility and evaluate the performance of prototype HTS power transmission cables at different lengths between 1-5 m with support components like terminations and cryogenic cooling (7). The facility provides cooling to the cable and terminations of up to 1 kW with the help of boiling LN2 with an operating temperature range of 70-77 K. The pressurized termination uses two warm bushings. Between these two bushings and the ends of the HTS cable is a concentric arrangements of copper pipes designed to minimize heat load on to the system as shown in Figure 1.5. Each termination has two main feedthroughs, one making transition from ambient (295 K) to vacuum (295 K) and the second making transition from vacuum to subcooled LN2 (67-77 K) at 1-10 bar. The LN2 skid provides a nominal cooling of 1 kW at 67-77 K for the HTS cable and the two cable terminations (14).

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Figure 1.5 Schematic of pressurized termination concept 1.3.4. Fault Current Limiting HTS cable with Con Edison This project is part of a Secure Super Grid program which addresses the need to inter-tie distribution links at a low cost to enhance the system power capacity and limit fault currents. The cable system to be installed at the Consolidated Edison grid has a length of 300 m operating at 4000 A and 13.8 kV and consists of three phase Triax cable, vacuum jacketed piping and terminations (8). The terminations provide a transition to cryogenic temperatures from ambient conditions; serve as a connection point for the liquid nitrogen cooling system and also serve as an entry point for all the connectors and wiring system as shown in Figure 1.6. The cable and termination is cooled by circulating sub-cooled LN2 facility operating within 72 K to 75 K.

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Figure 1.6 A typical HTS cable termination in operation 1.4.

Challenges and Objectives

The primary cryogen in most studies to cool superconducting cable and their support terminations is liquid Nitrogen (6)-(10). Low cost, high specific heat capacity, ease of pumping and ease of availability are the major factors that drive the use of LN 2 in superconducting power devices. However, as visible in the case studies described earlier, the operating temperature of all cooling media (i.e. sub-cooled liquid Nitrogen) is 65-77 K because under ambient conditions it solidifies at 63 K. However, in the case of HTS materials, the critical current increases with decrease in their operating temperatures: for commercial HTS materials, critical current increases by 10% for every degree temperature lowered (15). Hence, lower operating temperatures facilitate higher operating currents resulting in smaller and light weight superconducting power 9

devices. Moreover, a wider operating temperature range enables larger current density variations. This facilitates operation of devices at the temperature most appropriate for a given current density suitable for a particular application. In some military applications, compact and lightweight power devices that offer a wide operating current density window are beneficial. At Florida State University, the research consortium is assigned the task to investigate the feasibility of the use of HTS cables cooled by gaseous Helium for naval applications. Generally, a liquid cryogen is circulated through a distribution system, in a typical degaussing system on board naval ships, passing through the living spaces. One litre of liquid nitrogen, for example, during boil-off can occupy 682 litres of space. Thus, in the event of a system breach, a leaking liquid cryogen would present asphyxiation and/or explosion hazards. Due to its high heat capacity, it could result in cold burns to nearby people and equipment. During the research for the possible use of HTS devices for naval and commercial applications at Florida State University, the optimum system weight was determined based on several factors. One of the key driving factors is the system operating temperature. The ideal temperature is found to be 55 K by Fitzpatrick et al. (1). For these safety and operational reasons, the use of gaseous cryogen becomes necessary. 1.4.1. Benefits and Challenges The only cryogens in gaseous state at around 55 K are Helium and Neon. However, Helium has some benefits over Neon. 1. Neon is more expensive and has lower thermal heat capacity than Helium. Haugan et al. (16) presented the benefits and analysis of compact and lightweight power transmission

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devices cooled by gaseous Helium for specialized high power airborne applications operating at temperatures of 50–80 K. 2. Another benefit of the use of gaseous Helium is that there exists no phase change during its operation allowing for much larger temperature gradient during the operation of the device. Hence, cold Helium gas is considered as the cooling fluid to maintain the operating temperature of the HTS cable, terminations and other support systems. There are, however, certain challenges with the use of gaseous Helium as cooling medium. 1. Gaseous Helium has much lower heat capacity and inferior dielectric strength. 2. The efficiency of commercial cryocoolers, used for cooling gaseous Helium to the operating temperature range, is low as compared to those available for liquid cryogens. 3. Another additional challenge is the difficulty in achieving the required mass flow rates due to low density of helium gas. 1.4.2. Objectives The above mentioned challenges drive the need to investigate and determine the feasibility of such a concept in commercial and industrial applications. The following are the objectives and goals set for this study: 

General Objective: To simulate and optimize a cryogenic heat sink cooled by gaseous Helium for superconducting power cable applications in order to study the feasibility of such a concept.



Specific Objectives: 

Develop a 2D FEM model to determine the number of fins in the heat sink.

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

Develop a fully coupled 3D model to simulate the heat sink behavior under various operating conditions.



Validate the model with experimental data from tests conducted on a prototype heat sink.



Optimize the geometry for fixed flow conditions in order to improve its performance. 1.5.

Overview

Power cables have terminations on either end to guarantee dielectric integrity. In case of a superconducting power cable, the terminations additionally need to link the operating cryogenic environment in the cable with the room temperature environment of the nonsuperconducting elements of the power system, such as copper cables, power transformers, circuit breakers, instrumentation transformers, or disconnect switches. The higher temperatures surrounding such terminations cause substantial heat influx into the superconducting cable. It is of utmost importance to minimize the heat influx to maintain the operating temperature of the superconducting cable as well as to minimize the capacity of the cryogenic system and operating costs of the superconducting cable system. Hence, a copper heat sink, also acting as the cable termination, is required to intercept the heat leak from the room temperature components to the superconducting cable. The schematic diagram of the proposed superconducting cable termination is as shown in Figure 1.7. Copper is the material proposed for this heat sink. This is so because copper has the highest thermal conductivity amongst metals and can be easily incorporated into the copper current leads to be placed into the system. Many numerical optimization studies for heat sinks of various types have been carried out in order to obtain high system performance or least flow resistance (17), (18) and (19). However, numerically optimized results for heat sinks using helium gas as coolant for cryogenic applications are not readily available. Also, there exist various numerical techniques for finding 12

approximate solutions to boundary value problems. Finite Element method (FEM), Finite Volume Method (FVM) and Finite Difference Method (FDM) are some of them. However, for better quality of approximations and discretization of the problem into larger number of cells/grid points, the Finite Element Method (FEM) is the choice for solving the problem presented in this thesis. Here, commercially available finite element analysis package, COMSOL Multiphysics is used to study the feasibility and, after validation with experimental data, optimize the geometry of the heat sink under manufacturing and overall dimensional constraints. The ease of availability of a licensed version of the software and the computational power to simulate both heat transfer and fluid flow mechanism are the driving factors for the choice of this software. The dissertation is composed of two major parts. The first part describes the problem statement and the thermodynamic and fluid flow model developed to solve the problem. A couple of verification cases have been used and modeled in order to demonstrate the ability of the user to model a system correctly. The simulation results for these cases were validated with known analytical and experimental values. Then, a two dimensional FEM model is developed to determine the optimum number of fins inside the heat sink for varying mass flow rates. This model simulates the heat transfer mechanism whereas the pressure drop across the heat sink is calculated using standard correlations for fluid flow inside a channel. Based on that, a heat sink is manufactured and tested experimentally. The test results are reported here. A laminar three dimensional FEM model is developed in order to simulate the laminar flows reported in the experimental results. The second part models the more unpredictable turbulent flow results reported during the experimental tests on the heat sink. The laminar and turbulent models are validated with the

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experimental results. There seem to be a good agreement between the numerical and experimental values.

Figure 1.7 A schematic diagram of the superconducting cable termination with heat sink (red), cable (green) and copper conductor (yellow). After model validation, optimization studies are carried out for the heat sink. The heat sink is optimized for various geometrical parameters keeping the flow conditions and surrounding temperature and pressure boundary conditions fixed. It is found that an unequally spaced heat sink that uniformly distributes the coolant flow leads to better performance in terms of decreased pressure drop and peak temperature of heat sink. Finally results of this research are summarized and the future works are suggested. 14

CHAPTER TWO SIMULATION OF SUPERCONDUCTING CABLE TERMINATION FOR LAMINAR FORCED CONVECTION 2.1

Laminar Forced Convective Flow

This thesis focuses on studying cable termination for a superconducting power cable cooled by gaseous Helium at 40-60 K range. The idea is to develop a model for the heat sink using Finite Element Method (FEM) and then validate the model with experimental results with a prototype heat sink. Initial studies concentrate on two and three dimensional modeling of the convective cooling phenomenon in forced flow. In laminar flow regime, fluid motion is highly ordered and it is possible to identify streamlines along which particles move. Surface friction and convection heat transfer rates highly depend on whether the flow is laminar or turbulent. When dealing with an internal flow problem, it is necessary to be concerned with the existence of entrance and fully developed regions. However, for simplicity, all flows have been assumed to be in the fully developed region. This assumption is valid in this case, because the geomtery of the heat sink is so small that the entrance region effects can be neglected. The Reynolds number (Re) for flow in a circular tube is defined as

(2.1)

where

is the mean fluid velocity over the tube cross section,

is the fluid viscosity and

is the fluid density,

is the tube diameter. In a fully developed flow, the critical reynolds 15

number for the onset of turbulence is

(20). For thermal simulation of heat sink in

laminar regime forced convection flow, we make sure that 2.2

.

Numerical Verification and Simulation

COMSOL Multiphysics 4.3 is used to perform finite element analysis for the heat sink. All numerical computer simulations are carried out using the in-built Heat Transfer module. The simulations are run on a computer with eight CPU dual - cores (2 × Intel Xeon X5570) with 24 GB RAM and 2.93 GHz processor clock speed. Due to the system symmetry, only half of the heat sink is simulated to reduce the total of number of finite elements and thus reduce the overall computational time to arrive at steady state results. In the course of laminar forced convection flow model development, two points are of great interest in order to achieve the final objective: 1. To determine the ideal number of fins for a given overall width of the base plate of the heat sink to have the best balance between heat transfer enhancement and pressure losses. This is done with the help of a two-dimensional (2D) steady state model of the heat sink. 2. To validate the simulation results with experimental test runs on the prototype of the heat sink. For this, a three dimensional (3D) model is computed to give a fully coupled analysis of the fluid flow and heat transfer mechanism using the design parameters obtained in 2D analysis and the space constraints associated with the prototype. The 3D model gives a more accurate representation of the actual flow field in order to validate it with experimental results.

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2.2.1 Verification Cases Flow and heat transport phenomena in heat sinks of various types have been thoroughly studied; theoretically, experimentally and numerically, mostly because of the common occurrence of such devices in various thermodynamics systems. However, before any FEM model, is deemed suitable for further simulations and analysis; the process of FEM modeling used by the author,

should be thoroughly verified and if possible validated with known

experimental and analytical results. Here two verification cases have been presented. They are compared with results obtained from the FEM model developed for the system represented by these cases. These specific verification cases closely relate to work done in this thesis. The results for both the experimental case and the analytical case match with those obtained with the FEM model. Since COMSOL is the FEM modeling tool used here, the match indicates the ability of the author of this thesis to correctly discretize the system into various elements and apply the relevant boundary conditions to arrive at steady state solutions. 2.2.1.1 Verification with experimental data. Fehle et al. (21) conducted a study aiming at enhancing the heat transfer in a compact heat exchanger. In order to have the exact knowledge of the temperature distribution in the heat exchanger, they applied holographic interferometry to visualize the temperature field. Figure 2.1 shows the heat exchanger prototype along with the necessary dimensions, as given by the author, shown in Table 2.1. The flow conditions for air and the thermal and fluid boundary conditions are applied using the in-built module for the geometry as described above. The interferograms produced are processed using a digital image processing system. The interference lines in the interferograms approximately resemble isotherms of the investigated duct flow.

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Table 2.1 Dimension of plate fin arrangement Parameter

Dimension (mm)

Height of fins, e

10

Width of the duct, b

10

Length, lr

300

Fin thickness, tf

2

Radius, r

1

Figure 2.1 Plain fin arrangements in a compact plate heat exchanger

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Governing Equations: The governing energy equation for heat exchanger solid domain is as follows: (

)

(2.14)

The governing equations for incompressible fluid flow domain are:

(

)

(

(

(

) )

(

)

)

(2.15)

(2.16) (

)

(2.17)

where the symbols stand for their usual meanings. The dependent variables in this type of analysis are T, temperature, P, pressure and u, velocity. Boundary Conditions: 1.

Initial Temperature Guess for the Non-Linear Solver: T= 298.13 K

2.

Inlet Temperature: 298.13 K

3.

Inlet Flow Reynolds Number: 500 with Laminar inflow

4.

Air Inlet Pressure: 1 atm

5.

Temperature Boundary Conditions: Applied at top and bottom surfaces maintained at constant temperature by heating plates

6.

Thermal Insulation: Applied to all solid and liquid interfaces not covered by other boundary conditions.

7.

No slip: This condition prescribes that the fluid at the wall is not moving.

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The test section is itself supplied with six water-supplied heating plates. The temperature of each plate is measured by thermocouples in order to maintain a uniform test surface temperature. Figure 2.2 show the close resemblance for the results obtained by holographic interferometry by the author and COMSOL. Fehle et al. report that the temperature difference between two neighboring isotherms is approximately 2.3 K. This particular observation can also be seen in the results obtained with the FEM model developed for this particular system.

Figure 2.2 (a) Interferograms for plain fin arrangements for r =1 mm, Re=500 (b) Results obtained using COMSOL with isotherm temperatures in K

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2.2.1.2 Verification with known analytical results. A known problem in the field of heat transfer is temperature distribution in the flow of a fluid stream inside a solid object. Because the internal flow is completely enclosed, an energy balance is applied to determine how the mean fluid temperature, Tm, and the solid surface temperature, Ts, vary with position along the enclosed space, a tube in this case. The solution to this problem with constant surface heat flux is given by Equation 2.2 and shown in Figure 2.3 (20).

( ) ̇

(

)

(2.2)

Figure 2.3 Axial temperature variations for heat transfer in a tube Governing Equations: The governing equations for the incompressible internal fluid flow domain are:

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(

)

(

(

(

) )

(

)

)

(2.15)

(2.16) (

)

(2.17)

Boundary Conditions: 1.

Initial Temperature Guess for the Non-Linear Solver: T= 298.15 K

2.

Inlet Temperature: 298.15 K

3.

Mass flow rate: 0.1 kg/s

4.

Total Heat Flux: Enters the total heat flux across the boundaries where the node is active. In this case, applied to pipe surface, = 2000 W

5.

No slip: This condition prescribes that the fluid at the wall is not moving.

The mean temperature thus varies linearly along the tube and the temperature difference (Ts-Tm) also varies along the length. This difference is initially small but increases due to decrease in h (convection heat transfer co-efficient) in the entrance region. However, in the fully developed region, h is constant and hence, the difference remains the same. In order to simulate this problem using FEM modeling technique, a system of heating water from an inlet temperature of 298.15 K is considered. The water passes through a thick walled tube of inner and outer diameters of 20 mm and 40 mm respectively. It is assumed that the outer surface of the tube is well insulated and electrical heating provides a constant heat flux distributed uniformly over the entire tube periphery. For a water mass flow rate of 0.1 kg/s, Figure 2.4 shows the results obtained during this analysis. The same trend in temperature variation along the axis is seen as explained by the analytical solution. 22

360

350

Ts

Temperature (K)

340

330

320

310

Tm

300

290 0

0.1

0.2

0.3

0.4

0.5

Axial Length , x (m)

Figure 2.4 Axial temperature variations obtained using FEM technique The analytical and experimental cases are considered for simply verifying the proper use of the software in general and the heat transfer module in particular to develop an FEM model for various systems with known, reported results. Chapters 2 and 3, further talk about validating the FEM model for superconducting cable termination with results obtained from the experimental setup consisting of the prototype heat sink manufactured specifically for this purpose.

23

2.2.2 Two-dimensional FEM model A superconducting cable termination essential consists of, for the sake of simplicity and the purview of this thesis, a heat sink that is required to intercept the heat leak from the room temperature components to the cryogenic temperature components of the superconducting cable. In order to validate the computational FEM models, a prototype heat sink is manufactured as shown in Figure 2.5. This is a scaled down version of the actual heat sink required to be installed in the superconducting cable system. The heat sink designed and modeled in this study is made of copper and features 18 fins of 10 cm length (22). It is integrated in a cylindrical copper tube, flattened on the bottom side. The flat surface allows for known and variable thermal load to be applied to the heat sink. Cryogenic helium gas is injected at high pressure by an external helium circulation system (15). The temperature and pressure at the inlet to the heat sink can be adjusted depending on cooling requirements. The assembled heat sink was wrapped in Aluminized Mylar foil and enclosed in a vacuum chamber to reduce the conduction and radiation heat inleaks into the system. The simplest Finite Element Method model is designed to determine the optimum number of fins required for the heat sink under the constraint that the overall base width and fin geometrical parameters remain constant. The model takes a vertical cross section of the heat sink as shown in Figure 2.6, and hence the focus of 2D numerical simulation is solely on the heat transfer mechanism. The pressure loss across the length is calculated analytically taking into account the flow between the parallel fins (plates) of the heat sink along with the entrance-exit and acceleration-deceleration effects.

24

Figure 2.5 Design of the prototype cable termination (total view) and cut views to show the internal fin structure (vertical and horizontal cut)

The density of helium gas varies largely with temperature and pressure. Hence, average density is calculated and used based on the inlet and outlet temperatures of the fluid. Heat Transfer in Solids (ht) module available in COMSOL is used over a parameterized geometry so as to easily allow sweeping over a number of fins. The software calculates the properties of copper, namely thermal conductivity, κ; specific heat at constant pressure, ϲp, and density, ρ, which are temperature dependent as given by (23). The helium properties are calculated using Engineering Equation Solver (EES) software package at the required temperature and pressure. The idea of using EES is to find the value of convective heat transfer co-efficient, һ, which is used for applying the convective cooling boundary condition in the 2 dimensional models. This co-efficient is found using inbuilt EES 25

functions and thermophysical property tables for gaseous Helium. EES uses an implementation of (24), (25) for calculating thermophysical properties except for thermal conductivity which is computed using (26). Reynolds number (

) is calculated using the hydraulic diameter

concept for parallel plate fins consistent with the geometry and mass flow rate assumed. The calculations are carried out for various mass flow rates so that flow stays mostly in the laminar regime. Correlations for both laminar and turbulent flow (if any) between smooth parallel plates, as given in (20), are used to find the heat transfer co-efficient and pressure losses inside the heat sink.

(

(2.3)

)

(

(2.4)

)

For a smooth surface the friction factor for laminar and turbulent regime respectively is given by �=96���

(Laminar Flow) (

(2.5) )

(2.6)

The pressure loss due to drag experienced by fluid flow is estimated as

(2.7)

where usselt

fin s acing

mean velocity of gHe

umbe

26

ength of heat sin

10

P = 8 bar

9.5 9

ρ (kg/m3)

8.5 8 7.5 7

6.5 6 5.5 5 40

45

50

55

60

65

Temperature (K)

Figure 2.6 Variation of density of gaseous Helium with temperature at constant pressure Figure 2.6 shows an instance of density variations across various temperature domains for gaseous Helium under 8 bar pressure obtained from RefProp which a standard software to calculate thermophysical properties of various fluids at different temperatures and pressures. From this figure, with a 20 K increase in the temperature of gaseous Helium, its density reduces by approximately 33%. Hence in order to account for the pressure drop due to acceleration and deceleration of the fluid stream due to density variations, an additional term is evaluated as given by (27).

(

)

27

(2.8)

where G is the mass velocity of the stream.

̇

; A = channel cross sectional area

and ̇ = gaseous Helium mass flow rate. In order for the two dimensional model to be more precise in calculating the pressure drop across the channel, entrance and exit effects due to sudden expansion and contraction are included as given by (28).

[

]

[

where

(

]

and

(

)

)

(2.9)

(2.10)

are sudden expansion and sudden contraction coefficients

respectively; d, D are the smaller and larger diameters of the connecting pipes Total pressure drop across the heat sink for a given flow parameter is given by (2.11) For the various cases of different mass flow rates and different geometries, the helium flow is found to be in laminar regime mostly and is modeled in COMSOL by providing the convective cooling boundary on the fin walls. In the case of the model with 9 fins, for example, using the above relations, an effective heat transfer coefficient value of h = 90.25 W/ (m2 K) for the convective boundary condition is calculated. Separate values of h are calculated for different flow conditions and geometries. All the calculations for h and the simulations are carried out in COMSOL. 28

are performed using EES and

Governing Equation: The steady state heat transfer equation for heat flow in the solid block of copper is governed by (

)

(

)

(2.12)

where f and , convective heat transfer coefficient. Boundary Conditions: 1.

Initial Temperature Guess for the Non-Linear Solver: 50 K

2.

Heat Flux: A heat influx boundary condition of 50 W is applied at the base of the 2D model appearing as a line at the bottom in the front view as shown in Figure 2.7

3.

Convective Cooling: It adds the convective term of Equation (2.12) to the boundaries wherein h is defined using the correlations discussed above.

4.

Thermal Insulation: The thermal insulation condition is applied across all other boundaries to mimic the experimental setup. It follows the following equation indicating no heat flux crosses the boundary. (

)

(2.13)

The mesh size is chosen as “no mal” with the default values as available in the gene al physics category. The normal mesh with 2986 elements is sufficient to satisfy mesh independence. Stationary Linear Solver produced the results as shown below in Figure 2.8 which indicates the surface temperature distribution across a vertical cross-section of the heat sink for a particular case with 9 fins in one half of the heat sink.

29

Figure 2.7 2-Dimensional FEM model of the heat sink with important boundary conditions The objective of the 2D computation was to find the number of fins required for optimum performance of the heat sink, i.e., a best tradeoff between temperature gradient and pressure drop across the heat sink. The entire heat sink is modeled for varying mass flow rates and varying number of fins (incremented in steps of 3) for a fixed fin thickness. It can be seen from Figure 2.9 that the 9 fin heat sink model (corresponding to half of the actual design) provides a good system balance for the heat sink performance. This primary two dimensional study forms the basis for further detailed three dimensional analyses and experimental validation.

30

Figure 2.8 Surface temperature distribution (in Kelvin) with h = 90 W/m2 K and ṁ = 1.5 g/s

31

140

80

ṁ = 0.5 g/s ṁ = 1 g/s ṁ = 1.5 g/s

120

70

60 100

80 40

60

Δ P (Pa)

Tpeak (K)

50

30 40 20

20

10

0

0 0

3

6

9

12

15

Number of Fins

Figure 2.9 Tpeak and Δp curves for various mass flow rates 2.3

Experimental Setup and Results

An experiment, conducted at the Center for Advanced Power Systems (CAPS), as described in detail in (22) is used to validate the simulation model. The prototype heat sink consists of four parts: The base block with fins, two end plates, and the surrounding enclosure (partially shown in Figure 2.10). All parts except for the cuts between the fins are machined using mechanical manufacturing processes. The cuts for the fins are machined by electrical

32

discharge machining (EDM). The four parts are joined through silver brazing for maximum conductivity and excellent structural strength. The supply tubes are soldered to the end plates using tin-lead solder. A heater with a nown esistance of 10.09615 Ω is attached to the bottom plate. Two temperature sensors are attached to the side walls of the heat sink to determine the temperature of the solid. Table 2.2 Prototype heat sink dimensions Heat Sink Geometrical Parameters

Dimensions (inch)

Total Length

6

Fin Spacing

0.03-0.06

Outer Radius of Curvature

1.49

Fin Thickness

0.03

Inlet/Outlet Pipe Diameters

0.5

Entry/Exit Chamber Length

1

Two additional temperature sensors are attached to the supply and exit tubes to measure the inlet and outlet fluid temperatures. An adjustable DC voltage source is used to control the heat influx to the heat sink. The helium circulation system allows adjusting the pressure and temperature of the helium flow at the inlet of the experimental setup. A differential pressure gauge is used to measure the pressure drop across the heat sink. The experimental setup with the gaseous Helium flow tubing, sensors attachments and heater wire within the cryostat is as shown in Figure 2.11. The heat sink, wrapped in aluminized Mylar (multi-layer insulation, MLI), is shown in Figure 2.12. 33

Figure 2.10 Copper heat sink prototype used for experimentation

Figure 2.11 Experimental setup with flow lines and heat sink with heater attached

34

Figure 2.12 Heat sink wrapped in MLI before insertion into the cryostat The experiment is conducted for different gas mass flow rates entering the system at different temperatures and pressures. Three different total heat flux values of 30 W, 50 W and 100 W are applied at the bottom of the heat sink in order to obtain a good range of experimental data in both laminar and turbulent regime. The pressure drop across the heat sink is located at the cryocooler. Table 2.3 shows the configuration settings for the heater wire in order to achieve different thermal load values whereas Table 2.4 shows the experimental data obtained by operating the heat sink at various temperatures and pressures and under different flow conditions. The entire experimental setup is initially cooled to the inlet temperature specified in the table before starting the experiment.

35

Table 2.3 Heater wire configuration to obtain various heat load values

Thermal heat load (W)

Current (A)

Voltage (V)

30

1.72

17.4

50

2.23

22.5

100

3.15

31.77

Resistance Ω

10.09615

An essential first step in any convection problem is to determine whether the flow is laminar or turbulent. Surface friction (hence pressure drop) and the convection heat transfer rates depend strongly on which of these conditions exists. Table 2.5 relates the experimentally measured gaseous Helium flow rates with the corresponding Reynolds number calculated using the standard definition as given by Equation 2.1. The wide range of Reynolds number indicates the range of experimental data available for further validation with simulation results. This chapter only focuses on developing a FEM model for gaseous Helium flows through the heat sink in the laminar regime. The turbulent flows result in fluctuations that enhance the heat transfer rates and lead to increase in pressure drop. These fluctuations have to be dealt with differently and thus the turbulent flow FEM model for cryogenic heat sink is presented in the next chapter.

36

37 50

100

50

100

5

6

7

8

50

3

100

100

2

4

50

1

12.01

10.49

7.18

5.98

9.73

8.57

10.7

9.3

Applied Inlet Case No. Heat Load Pressure (W) (bar)

48.0

42.8

50.3

42.3

65.5

58.6

61.3

56.8

Tin (K)

52.3

44.8

57.6

45.6

71.7

62.0

75.4

64.5

Tout (K)

62.8

55.0

70.8

58.0

81.8

74.8

88.0

82.0

0.702

0.680

0.712

0.666

0.890

0.844

0.144

0.115

Volume T peak (K) Flow rate (m3/h)

2.39

2.50

1.17

1.26

1.7

1.5

0.24

0.18

Approxim ate mass flow rate (kg/s)

3.34

3.34

1.85

1.83

2.97

2.94

0.51

0.39

Measure pressure drop (mbar)

Table 2.4 Experimental results for heat sink setup

38 30

50

30

50

12

13

14

50

10

11

30

9

12.39

12.14

9.28

8.85

12.5

12.06

Applied Inlet Case No. Heat Load Pressure (W) (bar)

40.3

39.7

42.6

41.1

50.4

48.4

Tin (K)

41.5

40.8

45.7

43.4

56.4

53.5

Tout (K)

46.6

45.2

50.7

47.6

62.5

60.4

1.664

1.688

0.712

0.699

0.307

0.307

Volume T peak (K) Flow rate (m3/h)

8.35

8.56

2.38

2.29

0.98

0.98

25.59

25.56

3.17

3.12

1.02

1.02

Approxima Measure te mass pressure flow rate drop (kg/s) (mbar)

Table 2.4 - continued

Table 2.5 Flow conditions for various experimental cases Case No.

Mas flow rate (kg/s)

Re

1

0.18

540.4

2

0.24

684.1

9

0.98

1978

10

0.98

1913

3

1.50

4522

4

1.70

4226

5

1.26

4592

6

1.17

3857

7

2.50

8162

Flow Regime

Laminar Flow

Turbulent 8

2.39

7263

11

2.29

7800

12

2.38

7650

13

8.56

27306

14

8.35

26491

39

Flow

2.4

Laminar 3 Dimensional FEM Model

Cryogenic circulation systems have been typically limited by the pressure loss handling capabilities. Two-dimensional analysis gives us a rough estimate of the pressure losses inside the heat sink. But, the fluid flow behavior could not be predicted accurately and can only be estimated in two dimensional studies since the entrance and exit chambers substantially impact the flow pattern. Therefore a three dimensional FEM model has been developed using the Conjugate Heat Transfer physics in COMSOL Multiphysics 4.3 to simulate a steady state, three dimensional fluid flow and determine its effects on the thermal performance of the heat sink. The fluid velocity field in certain cases is low enough to be assumed as laminar flow. Steady state results, assuming laminar flow consistent with experimental mass flow rates, are obtained. The entire geometry is divided into solid (copper) domain and fluid (helium gas) domain. The copper and helium properties are temperature and/or pressure dependent. The property functions are implemented in COMSOL using (23), (29), (30), (31). For temperatures below 140K, COMSOL does not provide any temperature dependence density function in the module present with us. For this, EES was used to calculate a piecewise function for temperature dependent Helium properties. Helium density is evaluated at average fluid operating temperature and provided as input to COMSOL. Governing Equations: The governing energy equation for copper domain is as follows: (

)

The governing momentum and energy equations for fluid flow domain are:

40

(2.14)

(

)

(

(

(

) )

(

)

)

(2.15)

(2.16) (

)

(2.17)

where the symbols stand for their original meanings as explained earlier. The dependent variables in this type of analysis are T, temperature, P, pressure and u, velocity. Boundary Conditions: 8.

Initial Temperature Guess for the Non-Linear Solver: T= 58.6 K

9.

Inlet Temperature: 58.6 K

10. Inlet Flow Rate: 0.78 g/s with Laminar inflow 11. Helium Outlet Pressure: 857 kPa 12. Total Heat Flux: Enters the total heat flux across the boundaries where the node is active. In this case, = 50 W 13. Thermal Insulation: As described earlier, the thermal insulation boundary condition is applied to all solid and liquid interfaces not covered by other boundary conditions. 14. No slip: This condition prescribes that the fluid at the wall is not moving. Due to the laminar inflow assumption, a parabolic velocity distribution at the entrance is assumed, and a surface heat flux at the bottom is used to simulate the heat influx into the heat sink from the ambient. The meshing is carried out with an aim to keep the computational time as sho t as ossible and yet not affect the esults thus obtained. The solid is meshed by a “no mal”

41

sized mesh whereas the fluid and its interface with the solid a e meshed with a “fine ” mesh wherein the meshing technique chosen is default as provided in the software.. The heat sink geometry consists of 9 fins, which are very closely spaced, forming 18 boundary layers on either side. In order to capture all effects, the meshing density is higher than usual.

Figure 2.13 Mesh structure for the laminar FEM model This amounts to a total of 1.54 million elements taking 156 minutes for convergence. A non-uniform mesh with higher mesh density towards the fluid/solid interfaces is chosen to ensure greater computational accuracy. The results are computed using the stationary solvers, which incorporate a GMRES solver and a non-linear solver at default settings. GMRES required 240 iterations and the non-linear solver 45 iterations to arrive at steady state results as shown in Figure 2.14 and Figure 2.15 respectively.

42

Figure 2.14 GMRES solution curve for each iteration

Figure 2.15 Error curve for non-linear solver using default settings

43

In both the curves, steady state is achieved when the relative error is less than 10 -3. The temperature and velocity fields (shown by arrow heads) obtained from the computation are shown in Figure 2.16. The thicknesses of the arrow indicate the magnitude of the velocity and the arrow heads indicate the direction.

Figure 2.16 Heat sink surface temperature (in Kelvin) with the fluid velocity field shown by black arrow heads Figure 2.16 clearly show the effect that entrance and exit chambers have on the nature of fluid flow as predicted and calculated by 2 dimensional analyses. The heat sink heats up as you go downstream. The results obtained by the laminar model, as explained above, are presented in 44

the next chapter. After formulating a 3 dimensional turbulent model, all simulations results from both the models are validated with the experimental results for the prototype heat sink. In order to increase the effectiveness of the heat sink, geometrical optimization studies are carried out by keeping the inlet fluid flow conditions and overall dimensions fixed.

45

CHAPTER THREE SIMULATION, MODEL VALIDATION AND OPTIMIZATION OF CABLE TERMINATION GEOMETRY 3.1

Introduction

The results, obtained from the experimental setup, consist of flows in laminar or turbulent regimes. In laminar flow regime fluid motion in highly ordered whereas for turbulent regimes the fluid motion is highly irregular and is characterized by velocity fluctuations. These fluctuations have two effects: 1. They enhance the transfer of momentum and energy thereby increasing the convection heat transfer rates. 2. They also increase the surface friction resulting in higher resistance to flow and hence higher pressure drop across the device. Generally, the Navier-Stokes equations can be used for turbulent flow simulations, although this would require a large number of elements to capture the wide range of scales in the flow. An alternative approach, widely used, is to divide the flow in large resolved scales and small unresolved scales. The small scales are them modeled using a turbulence model with the goal that the model is computationally less time consuming and hence less expensive. Different turbulence models invoke different assumptions. COMSOL has a turbulence interface using k-ε model in the heat transfer module. This model includes Reynolds-averaged Navier-Stokes

46

(RANS) most commonly used in industrial flow application (32). The RANS model divides the flow quantities into ̅

(3.1)

where ̅ is a mean value of a scalar quantity of flow obtained by time averaging over a long time and

is the fluctuating component that averages to zero over time as shown in Figure

3.1. The Turbulent Flow, κ-ε interface uses a RANS turbulence model type as explained in the next section.

Figure 3.1 The average velocity component and the fluctuating velocity component (32) 3.2

Turbulent 3-D Simulation using κ-ε Model

Various experimental cases, reported here, lie in the turbulent flow regime. The corresponding FEM model developed

ovides fo a tu bulent κ-ε model as desc ibed in (33).

This model assumes that the flow is incompressible and Newtonian and the Navier stokes equation is as given below. Also, for κ-ε model two additional transport equations and two de endent va iables a e added: the tu bulent

inetic ene gy κ and the dissi ation ate of 47

tu bulence ene gy ε . The equations fo tu bulent viscosity μ T, transport equation fo κ and t ans o t equation fo ε ead as given below. Dependent Variables: T, P, u, κ and ε Governing Equations: (3.2)

(

)

(

(

) )

(3.3)

where the symbols have their usual meanings as explained earlier.

(3.4)

((

((

(

(

where

,

,

(

)

)

) )

)

)

(

(3.5)

(3.6)

) )

(3.7)

are constants obtained from experimental data (33).

Boundary Conditions: The same boundary conditions as described in the laminar case are applicable for the turbulent case with specific changes pertaining to turbulence.

48

The three dimensional FEM model, developed using the above equations, is integrated with the help of the turbulent non-isothermal flow model. The values of various constants, turbulent length scale and intensity are default as provided by the interface. Helium density averaged over the entire operating temperature range is again given as input and the boundary conditions remain the same as explained in the laminar case. A different meshing technique has been carried out in order to obtain the same accuracy with a lesser number of elements and hence lesser computational time. Firstly, the interior interface boundaries between the solid and fluid domain are meshed. Then the fluid domain is meshed and finally the solid domain is discretized. The co

e domain is meshed with a “no mal” sized mesh whe eas the fluid is meshed with a

“fine ” mesh size. The inte face between the two domains is meshed with a t iangula mesh of default “no mal” size. This esults in limiting the total number of elements in the mesh structure to 371842, as shown in Figure 3.2. The total computational time for any turbulent case is approximately 185 min in order to arrive at steady state results. Figure 3.3 shows the convergence curve for the simulation case. Segregated group 1 solves for velocity, temperature and pressure at all the modes/grid points whereas segregated group 2 solves for turbulent kinetic ene gy κ and dissi ation ate of tu bulence ene gy ε. The results obtained, for case no. 6, for temperature and velocity field as as shown in Figure 3. 4 and 3.5. The streamline velocity profile clearly indicates a little bit of separation at the entry chamber of the heat sink and highly turbulent mixing at the exit chamber of the heat sink. The flow remains almost laminar between the fins due to small gap between each fin. The temperature profile for case no. 6 is shown here.

49

Figure 3.2 Mesh structure obtained by separately meshing the domains and the interface boundaries

Figure 3.3 Convergence curve for stationary turbulent flow solver

50

Figure 3.4 Streamline velocity field in the heat sink for case no. 6

Figure 3.5 Temperature profile in the heat sink for case no.6

51

3.3

Model Validation with Experimental Results

The experimental results, as reported earlier, are used to validate both the simulation models. In the experimental setup, the Helium gas pressure drop, reported, is measured at the cryocooler end as shown in Figure 3.6. This cryocooler supplies helium gas at 40-70 K to the experimental setup with the help of special cryogenic pipes under vacuum. The total length of these 1 inch cryopipes is about 10 ft. Also the pressure drop across the 25cm long, 0.4 inch diameter copper tubing, for inflow and outflow of gas through the heat sink, needs to be taken into consideration. Hence, an additional pressure drop term is calculated and added to the simulation results in order to compensate for the same. The additional term is calculated using Equations 2.4-2.6 and using the temperature and pressure dependent properties of fluid flow as given by ΔPflow_system in Table 3.1.

Figure 3.6 Schematic diagram for gHe flow system

52

The resulting temperature of gaseous Helium at the outlet of heat sink, T out, temperature of the solid copper block, T peak and

essu e d o ac oss the heat sin

ΔP a e lotted as shown

in Figure 3.7-3.9.

Table 3.1 Pressure drop validation results with experimental data Vol. Flow

ΔP (mbar)

ΔPmodel

ΔPflow_system

(mbar)

(mbar)

Exp.

Numerical

0.307

0.46

0.42

1.02

0.88

0.307

0.46

0.43

1.02

0.89

0.666

1.09

0.77

1.83

1.86

0.680

1.91

1.22

3.34

3.13

0.699

1.91

1.17

3.12

3.06

0.702

1.97

1.30

3.34

3.27

0.712

1.08

0.88

1.85

1.96

0.713

1.98

1.19

3.17

3.17

0.844

1.66

1.25

2.94

2.91

0.890

1.56

1.40

2.97

2.96

1.664

17.26

7.08

25.59

24.34

1.688

17.09

7.26

25.56

24.35

0.144

0.15

0.30

0.51

0.45

0.115

0.03

0.29

0.39

0.32

Rate (m3/h)

Due to lack of available data of the measuring device such as temperature sensors, pressure gauges, flow meters, etc. uncertainty analysis cannot be carried out. However, as a rule of thumb, the uncertainty of a measuring device is 20 % of the least count (34). Hence error bars, accordingly, have been plotted on the graphs using this rule. 53

80

Inlet Pressure: 5.98-12.50 bar Heat Influx: 30, 50, 100 W Inlet Temperature: 39.7-65.5 K

75

Temperature (K)

70 65

T_out Exp

60

T_out Numerical

55 50 45

40 35 0

0.2

0.4

0.6

0.8 Volume Flow

1

1.2

1.4

1.6

Rate (m3/h)

Figure 3.7 Comparison of model results with experimental data for fluid outlet temperature

85

Inlet Pressure: 5.98-12.50 bar Heat Influx: 30, 50, 100 W Inlet Temperature: 39.7-65.5 K

80 75 Temperature (K)

70 65

T_peak Exp

60

T_peak Numerical

55 50 45 40 0

0.2

0.4

0.6

0.8 1 1.2 Volume Flow Rate (m3/h)

1.4

1.6

Figure 3.8 Model validation for peak temperature of solid copper block

54

30

Inlet Pressure: 5.98-12.50 bar Heat Influx: 30, 50, 100 W Inlet Temperature: 39.7-65.5 K

25

ΔP (mbar)

20

ΔP Exp

15

ΔP Numerical 10

5

0 0

0.2

0.4

0.6

0.8

Volume Flow

1

1.2

1.4

1.6

Rate (m3/h)

Figure 3.9 Model validation results for pressure drop across the experimental setup The data plotted in these Figures do not show any particular trend or transition from laminar to turbulent regimes. This is so because each and every case has a unique set of pressure and flow conditions and also a unique matching value of heat flux is applied to each case as reported earlier in Table 2.4. The maximum relative error for Tout, Tpeak and ΔP are 1.97%, 6% and 17.94% respectively. These error percentages show good agreement of the numerical results with the experimental data. The numerical results mostly lay within the experimental measurement error bars as shown in the Figures.

55

3.4

Geometric Optimization of the Heat Sink

After validating the model, optimization studies are carried out for various geometrical parameters of the 9 fin (half section) model. Many optimization studies can be carried out since this heat sink incorporates a vast variety of variables such as geometrical parameters, gaseous helium mass flow rate, thermal mass of copper, fluid operating pressure, etc. Here, the focus is put on geometrical parameters, particularly on spacing between fins, keeping all the other input parameters constant. Objective Function: To minimize the pressure drop across the system and the peak temperature acquired by the heat sink. Constraints: Fixed overall base width and fin thickness of the heat sink; fixed flow parameters such as gHe mass flow rate, inlet temperature and pressure; fixed heat influx; fixed overall heat sink geometry. The corresponding values are as given below: Heat influx = 100 W gHe ṁ = 0.54 g/s, w = 17.8 mm, t = 0.79 mm, Helium density = 6.3 kg/m3. Problem Formulation: The various important geometrical parameters are shown in Figure 3. 9. Equation 3.8 binds all of them together. (3.8)

where

56

d2 d3 d1

w

Figure 3.10 Vertical cut section showing geometrical optimization parameters considered for the study Non-dimensionalizing the above equation with respect to d3 and

ee ing ‘w’ as a

constant known value we get, (3.9)

Keeping the thickness constant, the value of d3 can be determined for various values of x = d1/d3 and y = d2/d3 thereby satisfying the overall constraint on the heat sink geometry. Keeping all the other input parameters constant, optimization studies are performed plugging in 57

the values of d1, d2 and d3 in each case into the validated 3D COMSOL model. Initial studies pointed out the worst case scenario when x ≥ y ≥ 1. Hence, the further studies concentrated efforts on x ≤ y ≤ 1. Figure 3.11 and Figure 3.12 show the optimization curves for a given set of input conditions as described earlier. The contour plots help in determining the minimization values for the objective function. The corresponding optimum values are shown in Table 3.2. The results show that the performance of the heat sink can be increased by incorporating an unequally spaced fin structure that well distributes the coolant flow evenly across the fin structure in the design. This performance increase will be considerable for operating conditions and system parameters much higher than those considered here and in operation of the test apparatus. Table 3.2 Optimum values for objective function under given constraints

Optimum Values for

Optimum Values for

Peak Temperature

Pressure Loss

Parameter

x = d1/d3

0.75

0.75

y = d2/d3

1

0.85

d3

t

1.25 mm

0.79 mm

58

1.32 mm

0.79 mm

(a)

(b)

Figure 3.11 (a) Variation of heat sink peak temperature with fin spacing (b) Contour plot showing optimized geometry for minimum peak temperature

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(a)

(b)

Figure 3.12 (a) Variation of pressure loss across the heat sink with fin spacing (b) Contour plot showing optimized geometry for minimum pressure

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CHAPTER FOUR CONCLUSION The results presented in this thesis have some important applications to the design of cryogenic heat sink cooled by gaseous Helium for superconducting power device applications and to the basic understanding of heat transfer and fluid flow phenomena in forced convection type heat exchange devices. After summarizing the results here, suggestions for future work including integrating the present work with on-going research efforts at CAPS, Florida State University are given.

4.1

Summary of Research Efforts

A FEM model approach is used to simulate and optimize the problem presented in this thesis. COMSOL software package is used as the simulations tool. Before using COMSOL for model development, standard verification problems are presented with known and reported analytical and experimental results. The numerical results match with those obtained by the respective reporting agencies. A two dimensional model is developed in order to determine the ideal number of fins required to be made inside the prototype heat sink. It follows that a prototype heat sink with 18 fins has a good balance between the fluid pressure loss and the thermal performance of the heat sink.

A three dimensional FEM, laminar and turbulent, models are developed and appropriate boundary conditions are applied in order to achieve steady state results. These simulations results are compared with the experimental measured values. The simulations results for helium 61

temperature difference, the peak copper temperature and pressure losses across the heat sink reasonably match with the experimental values under the same input conditions. The relative error percentages between the numerical and experimental results are shown in Table 4.1. With the model validated, further optimization studies were carried out under the constraints of fixed overall geometry and input conditions. The results show that unequally spaced fin structure with distance ratios amount to an increase in the heat sink performance.

Table 4.1 Relative error percentages between numerical and experimental results

Case No.

% Error in Tout

% Error in Tpeak

% E o in ΔP

1

1.1025

3.8412

5.8946

2

0.9874

2.5162

6.4156

3

0.07097

6.219251

0.931973

4

1.12971

2.07824

0.20202

5

1.44079

4.960345

1.551913

6

1.44792

1.233051

5.794595

7

0.71339

3.882364

6.389222

8

0.9826

1.121019

2.06018

9

1.97009

6.006623

13.53725

10

0.25532

0.64

12.74902

11

1.6129

1.470588

2.039744

12

1.93654

0.861933

0.118612

13

1.81127

4.524336

4.730829

14

1.4759

4.838627

4.895662

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The maximum heat sink temperature difference between the equally spaced heat sink (worst case scenario) and the optimized geometry leading to unequal spacing (best case scenario) is about 4.5 K for the fixed constraints. This results in about 5.7 % decrease in the peak temperature of the copper heat sink. However, the effect of uniform flow distribution by unequally spacing the fins can be seen in the relative pressure drop difference. The difference in the pressure drop across the heat sink between the best and the worst case scenarios, as described earlier, is about 0.0899 mbar which amounts to 15.2 % decrease. These relative percentages show the importance of regulating the flow field in order to improve the performance of the heat sink. 4.2

Suggestions for Future Work

While conducting optimization studies and collaborating with the research consortium at CAPS, numerous ideas for future work presented themselves. Dede (35), (36) has conducted studies on numerical simulation based topology optimization. Topological optimization consists of an iterative loop in which finite element analysis, sensitivity analysis and optimization steps, in order to update design variables, are performed. Using FEM modeling technique, they have reported to have obtained an optimal cooling topology with fluid streamlines in branching channels as shown in Figure 4.1. Taking inspiration from this, this thesis work could be extended to include optimal topology studies for varied flow characteristics and varied thermal constraints as future work. Studies can also be done to know the effects of changing the design from a simple continuous flat plate fin to a more complex discontinuous geometry and optimizing its topology. This would then prove to be an ideal model to design and optimize a cryogenic heat sink, using helium gas as coolant, which could be used for varied applications.

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Ordonez et al. (37) developed a numerical model of a superconducting DC cable contained in a flexible cryostat. This model uses the Volume Element Method (VEM) numerical technique in order to simulate the steady state behavior of the superconducting cable to various situations such as quenching, constant heat load, point source heat load, etc. VEM is a conservative method for representing and solving partial differential equations in the form of algebraic equations. Volume Element (VE) refers to a finite region/space surrounding each node wherein the respective values are evaluated. Figure 4.2 shows the schematic representation of the volume elements considered in the numerical model.

Figure 4.1 Optimal topology and temperature distribution slices of 3D design domain

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The advantage of a VEM approach is that it reduces the entire geometry into fewer elements instead of millions or more elements discretized by FEM. This reduces the computational time drastically.

Figure 4.2 Schematic representation of the superconducting cable volume elements in radial (r) and axial (z) direction

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The heat sink cable termination design could also be modeled using VEM numerical technique. For a FEM model which takes about 150 min for a system of equations to arrive at steady state results, the VEM takes about 50 s to do the same job but with relatively less accuracy. This model could then be integrated with the superconducting DC cable in order to have a computationally inexpensive thermal model for the entire superconducting power system. Such a system could then be useful to carry out optimization studies with less computational effort.

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28. White, Frank M. Fluid MEchanics. 4th: McGraw-Hill. pp. 367-376. 29. American Society of Heating, Refrigeration and Air Conditioning Engineers. ASHRAE Handbook of Fundamentals. New York : ASHRAE, 1993. 30. Eckhert, E R G and Darke, M. Analysis of Heat and Mass Transfer. Bristol : Hemisphere, 1987. 31. Vargnaftik, N B. Tables of Thermophysical Properties of Liquids and Gases. 2nd. Bristol : Hemisphere, 1975. 32. COMSOL Inc., COMSOL 4.3 manual. 2012. 33. Wilcox, D C. Turbulence Modeling for CFD. 2nd. s.l. : DCW Industries, 1998. 34. Univeristy, Clemson. [Online] http://www.clemson.edu/ces/phoenix/tutorials/uncertain/. 35. Dede, E M., Experimental Investigation of the Thermal Performance of a Manifold Hierarchical Microchannel Cold Plate., ASME, 2011. ASME Conference Proceedings. pp. 59-67. 36. Dede, Ercan M., Multiphysics Topology Optimization of Heat Transfer and Fluid Flow Systems., Boston : COMSOL, 2009. COMSOL Conference. 37. Ordonez, J C, et al., Temperature and Pressure drop model for gaseous Helium cooled superconducting DC cables., IEEE Transactions on Applied Superconductivity, 23(3), 2013

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BIOGRAPHICAL SKETCH Darshit Rajiv Bhavana Shah Darshit R. Shah was born on August 30, 1988 as the third child in Mumbai, India. He obtained his Bachelors degree In Mechanical Engineering from Victoria Jubilee Technological Institute (VJTI), India in 2010. He was employed as research fellow in the Refrigeration and Cryogenics Laboratory, Indian Institute of Technology (IIT), Bombay between July, 2010 and June, 2011. During this term, he specialized in innovative refrigeration technologies and led the design and development of room temperature magnetic refrigerator at IIT Bombay. In August 2011, he joined the Florida State University, Tallahassee, Florida in M.Sc. with Thesis program and immediately joined the Thermal Management group at Center for Advanced Power Systems (CAPS) under the advisement of Dr. Juan Ordonez. At CAPS, he worked on optimal heat transfer area allocation for Vapor Compression Refrigeration for cooling periodic heat loads in naval applications. As a part of his thesis, he worked on simulation and optimization of cryogenic heat sink for superconducting power cable applications. Darshit is an active mentor-volunteer at Leon County Schools and a member of the Congress of Graduate Students at Florida State University. He is also an active member of the Cryogenic Society of America (CSA).

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