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

ii

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.

iii

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

iv

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

vi

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

vii

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

viii

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/m2K 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

ix

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

x

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 sub-

cooled 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.

1

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

2

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

3

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.

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

Figure 1.1 Comparison of overhead power lines to HTS cables (http://www.doe.gov)

4

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.

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

Figure 1.2 Schematic view of Albany cable system

5

(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.

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

Figure 1.3 3-in-one cable termination for Albany project cooled by LN2

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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

7

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 LN2

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).

8

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.

Figure 1.5 Schematic of pressurized termination concept

9

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 LN2 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

Figure 1.6 A typical HTS cable termination in operation

10

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

11

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.

12

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 non-

superconducting 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

13

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

14

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

experimental values.

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.

Figure 1.7 A schematic diagram of the superconducting cable termination with heat sink (red),

cable (green) and copper conductor (yellow).

15

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 density,

is the fluid viscosity and is the tube diameter. In a fully developed flow, the critical reynolds

16

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.

17

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.

18

Parameter Dimension (mm)

Height of fins, e 10

Width of the duct, b 10

Length, lr 300

Fin thickness, tf 2

Radius, r 1

Table 2.1 Dimension of plate fin arrangement

Figure 2.1 Plain fin arrangements in a compact plate heat exchanger

19

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.

20

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

21

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)

Governing Equations: The governing equations for the incompressible internal fluid

flow domain are:

Figure 2.3 Axial temperature variations for heat transfer in a tube

22

( ) ( ( ) )

( ) (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.

23

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.

290

300

310

320

330

340

350

360

0 0.1 0.2 0.3 0.4 0.5

Tem

per

atu

re (K

)

Axial Length , x (m)

Ts

Tm

Figure 2.4 Axial temperature variations obtained using FEM technique

24

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.

25

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

Figure 2.5 Design of the prototype cable termination (total view) and cut views to show the

internal fin structure (vertical and horizontal cut)

26

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 fin s acing mean velocity of gHe ength of heat sin

usselt umbe

27

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).

(

) (2.8)

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

40 45 50 55 60 65

ρ (

kg/m

3)

Temperature (K)

P = 8 bar

Figure 2.6 Variation of density of gaseous Helium with temperature at constant pressure

28

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).

[

]

( ) (2.9)

[

]

( ) (2.10)

where and 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/ (m2K) 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 are performed using EES and

the simulations are carried out in COMSOL.

29

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.

30

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.

Figure 2.7 2-Dimensional FEM model of the heat sink with important boundary conditions

31

Figure 2.8 Surface temperature distribution (in Kelvin) with h = 90 W/m2K and ṁ = 1.5 g/s

32

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

0

10

20

30

40

50

60

70

80

0

20

40

60

80

100

120

140

0 3 6 9 12 15

Δ P

(P

a)

Tp

eak

(K

)

Number of Fins

ṁ = 0.5 g/s ṁ = 1 g/s ṁ = 1.5 g/s

Figure 2.9 Tpeak and Δp curves for various mass flow rates

33

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.

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.

Table 2.2 Prototype heat sink dimensions

34

Figure 2.10 Copper heat sink prototype used for experimentation

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

35

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.

Figure 2.12 Heat sink wrapped in MLI before insertion into the cryostat

36

Thermal heat load (W) Current (A) Voltage (V) Resistance Ω

30 1.72 17.4

10.09615 50 2.23 22.5

100 3.15 31.77

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.

Table 2.3 Heater wire configuration to obtain various heat load values

37

Mea

sure

pre

ssure

dro

p

(mbar

)

0.3

9

0.5

1

2.9

4

2.9

7

1.8

3

1.8

5

3.3

4

3.3

4

Appro

xim

ate

mas

s

flow

rat

e

(kg/s

)

0.1

8

0.2

4

1.5

1.7

1.2

6

1.1

7

2.5

0

2.3

9

Volu

me

Flo

w r

ate

(m3/h

)

0.1

15

0.1

44

0.8

44

0.8

90

0.6

66

0.7

12

0.6

80

0.7

02

T p

eak (

K)

82.0

88.0

74.8

81.8

58.0

70.8

55.0

62.8

To

ut (

K)

64.5

75.4

62.0

71.7

45.6

57.6

44.8

52.3

Tin

(K

)

56.8

61.3

58.6

65.5

42.3

50.3

42.8

48.0

Inle

t

Pre

ssure

(bar

)

9.3

10.7

8.5

7

9.7

3

5.9

8

7.1

8

10.4

9

12.0

1

Appli

ed

Hea

t L

oad

(W)

50

100

50

100

50

100

50

100

Cas

e N

o.

1 2

3

4 5

6

7

8

Table 2.4 Experimental results for heat sink setup

38

Mea

sure

pre

ssure

dro

p

(mbar

)

1.0

2

1.0

2

3.1

2

3.1

7

25.5

6

25.5

9

Appro

xim

a

te m

ass

flow

rat

e

(kg/s

)

0.9

8

0.9

8

2.2

9

2.3

8

8.5

6

8.3

5

Volu

me

Flo

w r

ate

(m3/h

)

0.3

07

0.3

07

0.6

99

0.7

12

1.6

88

1.6

64

T p

eak (

K)

60.4

62.5

47.6

50.7

45.2

46.6

To

ut (

K)

53.5

56.4

43.4

45.7

40.8

41.5

Tin

(K

)

48.4

50.4

41.1

42.6

39.7

40.3

Inle

t

Pre

ssure

(bar

)

12.0

6

12.5

8.8

5

9.2

8

12.1

4

12.3

9

Appli

ed

Hea

t L

oad

(W)

30

50

30

50

30

50

Cas

e N

o.

9

10

11

12

13

14

Table 2.4 - continued

39

Case No. Mas flow rate (kg/s) Re Flow Regime

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

8 2.39 7263

11 2.29 7800

12 2.38 7650

13 8.56 27306

14 8.35 26491

Table 2.5 Flow conditions for various experimental cases

Laminar Flow

Turbulent

Flow

40

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:

( ) (2.14)

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

41

( ) ( ( ) )

( ) (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”

42

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.

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.

Figure 2.13 Mesh structure for the laminar FEM model

43

Figure 2.14 GMRES solution curve for each iteration

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

44

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 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

Figure 2.16 Heat sink surface temperature (in Kelvin) with the fluid velocity field shown by

black arrow heads

45

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.

46

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

47

(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.

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

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

48

tu bulence ene gy ε . The equations fo tu bulent viscosity μT, 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)

((

) ) (3.5)

((

) )

(3.6)

( ( ( ) )

( ) )

(3.7)

where , , 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.

49

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.

50

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

51

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

52

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

53

The resulting temperature of gaseous Helium at the outlet of heat sink, Tout, temperature

of the solid copper block, Tpeak and essu e d o ac oss the heat sin ΔP a e lotted as shown

in Figure 3.7-3.9.

Vol. Flow

Rate

(m3/h)

ΔPmodel

(mbar)

ΔPflow_system

(mbar)

ΔP (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

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.

Table 3.1 Pressure drop validation results with experimental data

54

35

40

45

50

55

60

65

70

75

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tem

pe

ratu

re (K

)

Volume Flow Rate (m3/h)

T_out Exp

T_out Numerical

Inlet Pressure: 5.98-12.50 bar

Heat Influx: 30, 50, 100 W

Inlet Temperature: 39.7-65.5 K

40

45

50

55

60

65

70

75

80

85

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tem

pe

ratu

re (K

)

Volume Flow Rate (m3/h)

T_peak Exp

T_peak Numerical

Inlet Pressure: 5.98-12.50 bar

Heat Influx: 30, 50, 100 W

Inlet Temperature: 39.7-65.5 K

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

Figure 3.8 Model validation for peak temperature of solid copper block

55

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.

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

ΔP

(m

bar

)

Volume Flow Rate (m3/h)

ΔP Exp

ΔP Numerical

Inlet Pressure: 5.98-12.50 bar

Heat Influx: 30, 50, 100 W

Inlet Temperature: 39.7-65.5 K

Figure 3.9 Model validation results for pressure drop across the experimental setup

56

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

57

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

Figure 3.10 Vertical cut section showing geometrical optimization parameters considered for

the study

d2

d3

d1

w

58

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.

Parameter

Optimum Values for

Peak Temperature

Optimum Values for

Pressure Loss

x = d1/d3 0.75 0.75

y = d2/d3 1 0.85

d3 1.25 mm 1.32 mm

t 0.79 mm 0.79 mm

Table 3.2 Optimum values for objective function under given constraints

59

Figure 3.11 (a) Variation of heat sink peak temperature with fin spacing

(b) Contour plot showing optimized geometry for minimum peak temperature

(a)

(b)

60

Figure 3.12 (a) Variation of pressure loss across the heat sink with fin spacing

(b) Contour plot showing optimized geometry for minimum pressure

(a)

(b)

61

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

62

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.

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

Table 4.1 Relative error percentages between numerical and experimental results

63

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.

64

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

65

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

66

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.

67

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70

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).