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Numerical Evaluation of Heat Transfer and Pressure Drop in Open Cell Foams

Permanent Link: http://ufdc.ufl.edu/UFE0021793/00001

Material Information

Title: Numerical Evaluation of Heat Transfer and Pressure Drop in Open Cell Foams
Physical Description: 1 online resource (79 p.)
Language: english
Creator: Bai, Mo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: analytical, cell, computational, drop, dynamics, fluid, foam, heat, media, metal, method, model, numerical, open, porous, pressure, simulation, transfer
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: As society pursues the space travel, advanced propulsion for the next generation of spacecraft will be needed. These new propulsion systems will require higher performance and increased durability, despite current limitations on materials. A break-through technology is needed in the thrust chamber. One of the most challenging problems is to cool the hot chamber wall more without creating additional pressure drops in the coolant passage. A promising method is using open cell foam to enhance the heat transfer rate from chamber wall to coolant. However, the pressure drop induced by foams is relatively large and thus becomes a critical issue. The focus of this thesis is the evaluation of heat transfer and pressure drop of open cell foams. A simplified analytical model has been developed to evaluate the heat transfer capability of the foamed channel, which is based on a diamond-shaped unit cell model. The predicted heat transfer results by the analytical model have been compared with experimental data of different Reynolds numbers from other researchers and favorable agreements have been obtained. For the evaluation of pressure drop in open-cell metal foams, direct numerical simulation models of the foam heat exchanger have been built using GAMBIT/FLUENT. The model is based on a structure of sphere-centered open cell tetrakaidecahedron. This model is very similar to the actual metal foams' microstructure of thin ligaments that form a network of interconnected open-cells. Grid independence of solution is investigated and simulation results are further compared with experiments. Finally, the feasibility of applying foam filled cooling channel on rocket thrust chamber is investigated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mo Bai.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Chung, Jacob N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021793:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021793/00001

Material Information

Title: Numerical Evaluation of Heat Transfer and Pressure Drop in Open Cell Foams
Physical Description: 1 online resource (79 p.)
Language: english
Creator: Bai, Mo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: analytical, cell, computational, drop, dynamics, fluid, foam, heat, media, metal, method, model, numerical, open, porous, pressure, simulation, transfer
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: As society pursues the space travel, advanced propulsion for the next generation of spacecraft will be needed. These new propulsion systems will require higher performance and increased durability, despite current limitations on materials. A break-through technology is needed in the thrust chamber. One of the most challenging problems is to cool the hot chamber wall more without creating additional pressure drops in the coolant passage. A promising method is using open cell foam to enhance the heat transfer rate from chamber wall to coolant. However, the pressure drop induced by foams is relatively large and thus becomes a critical issue. The focus of this thesis is the evaluation of heat transfer and pressure drop of open cell foams. A simplified analytical model has been developed to evaluate the heat transfer capability of the foamed channel, which is based on a diamond-shaped unit cell model. The predicted heat transfer results by the analytical model have been compared with experimental data of different Reynolds numbers from other researchers and favorable agreements have been obtained. For the evaluation of pressure drop in open-cell metal foams, direct numerical simulation models of the foam heat exchanger have been built using GAMBIT/FLUENT. The model is based on a structure of sphere-centered open cell tetrakaidecahedron. This model is very similar to the actual metal foams' microstructure of thin ligaments that form a network of interconnected open-cells. Grid independence of solution is investigated and simulation results are further compared with experiments. Finally, the feasibility of applying foam filled cooling channel on rocket thrust chamber is investigated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mo Bai.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Chung, Jacob N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021793:00001


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NUMERICAL EVALUATION OF HEAT TRANSFER AND PRESSURE DROP IN OPEN
CELL FOAMS
























By

MO BAI














A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE


UNIVERSITY OF FLORIDA


2007



































(D2007 Mo Bai



































To my parents









ACKNOWLEDGMENTS

I express my sincere appreciation to my advisor, Dr. Jacob N. Chung, for his believing me

and providing me the opportunity to work on many interesting and challenging researches. His

invaluable patience, wisdom, and encouragement helped me throughout my two years' study at

the University of Florida. Without his unfailing support, this work would not have been

possible.

Drs. William E. Lear, Jr and Bhavani V. Sankar offered valuable suggestions on my

research while serving on my supervisory committee. Doctoral candidate Junqiang Wang

graciously gave up his time to help me when I had questions. Their suggestions and help have

shaped this work considerably.

My fellow graduate students, Renqiang Xiong and Kun Yuan, have offered invaluable help

on my study and research. My friends have given me a memorable time at University of

Florida and made my life here enjoyable. Also, I would like to thank my parents and extended

family, they were always there when I need help and encouragement. Finally, I'm grateful to

my fiancee Wenwen Zhang, for her years of support.









TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ............................................................................................................. iv

L IST O F T A B L E S ........ .... .......... .................... ............ ............................................. vii

L IST O F FIG U R E S ............. ......... ......... ............................. .................................viii

NOMENCLATURE ..................... ....... ... .. ... .... ..................x

A B S T R A C T ................................ .................. ...................................... x iii

CHAPTER


1 INTRODUCTION ROCKET THRUST CHAMBER COOLING.................. ..........1

1.1 H history of the R ocket..................................... ........................................... ...... .............1
1.2 R ocket Structure ....................................................... ..................... 1
1.3 R ocket Thrust C ham ber C ooling .......................................................................... .... ...2
1.3.1 R degenerative Cooling ....................................... ........... .... .. ........ ..
1.3.2 Challenges on R degenerative C cooling ............................................ .....................4

2 PREVIOUS W ORK ON OPEN-CELL FOAM S ........................................ .....................9

2.1 H eat Transfer E nhancem ent ............................................................... ....................... 9
2 .2 E x p erim ents .................................... ............................................................10
2.3 CFD Simulation and Numerical M odel................................ ......................... ........ 11
2 .4 O their O pen-C ell F oam s ................................................................................. .......... 13
2 .4 .1 P polyurethane F oam s ....................................................................... .................. 13
2.4.2 Carbon Foam s .................. ...................................... ................. 13

3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS ................... 15

3.1 Geometry Simplification for Open-Cell Foam Filled Channels ..................................... 15
3.2 Mathematical Transport Model and Heat Transfer Equations ......................................16
3.2.1 V -type Struts.................................... ......................... ..... ... ........ 16
3.2.2 H -type Struts ...................... .............................. ......... ........ .... 17
3.2.3 Fluid Temperature Prediction (Coolant Temperature)............... .... ....... .....20
3.2.4 Total H eat Transfer ........ .................................... ....... ......... ............. 23
3.2.5 Evaluation of Heat Transfer Coefficient ........................................... 23
3.2.6 Equivalent Heat Transfer Coefficient................. ........ ...............25
3.3 Investigation of Cylinder Diameter and Surface Area Density .......................................27
3.4 Verification of the Analytical Model with Experimental Data .............. .. ................28
3.4.1 Validity of Analytical Prediction (Re=5* 103 2*104) ......................................... 29
3.4.2 Validity of Analytical Prediction (Re=l* 104 6*104) ............................... .....30









4 CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS .............................39

4.1 Introduction to Single C ell M odel ......................................................... .....................39
4.2 Mesh Generation and Grid Independent Study ..................................... .................41
4.3 Sim ulation R results and V erification........................................... .......................... 43

5 FEASIBILITY STUDY OF FOAMED COOLING CHANNELS FOR ROCKET
T H R U ST C H A M B E R ........................................................................ ... ........................... 53

5.1 Feasibility Study and Comparison with Open Cooling Channel.............................. 53
5.2 U uncertainty A analysis .................................. .. .......... .. ............55
5.2.1 H eat Transfer M odel........ ......... ......... .......... ....................... ............... 55
5.2.2 Pressure D rop Sim ulation......... ................. .................................. ............... 56
5.2.3 Rocket Condition Prediction ............................................................................ 56

6 C O N C L U SIO N S ................. ......................................... ........ ........ ..... .... .. ..6 1

L IST O F R E F E R E N C E S .......... .................... .......................................... .....................................63

B IO G R A PH IC A L SK E T C H .............................................................................. .....................65









LIST OF TABLES


Table page

3-1 Constants of Equation (3-22) ........................................... ........ .................................. 31

3-2 Parameters of experiments from Calmidi and analytical model.............. ... .............32

3-3 Foam parameters comparison between experiments from Calmidi and analytical
m o d e l ................... ...................3...................2..........

4-1 Com prison of different m eshes' results ........................................ ....................... 44

5-1 Micro open channel and foam filled channel model requirements............................... 57

5-2 Head-to-head comparison of open channel and foamed channel ......................................58

5-3 Heat transfer enhancement of foamed channel over open channel............................... 58

5-4 V elocity ratio at equal pressure drop ........................................ ........................... 58

5-5 Head-to-head comparison under rocket condition.................................. ............... 59

5-6 Comparison of open and foamed channels' performance.................... ...............59









LIST OF FIGURES


Figure page

1-1 Construction of a regenerative cooling tubular thrust chamber........................................5

1-2 Cutaway of a tubular cooling jacket ................................................... .... ...........6

1-3 Typical heat transfer rate intensity distribution for liquid propellant rocket....................

1-4 Simplified schematic of regenerative cooling system of liquid propellant rocket...............7

1-5 Section A-A of Fig. 1-4 and details of cooling channel .............. ................... ......... ...... 7

1-6 Different configurations of the cooling channel in thrust chamber ..............................8

2-1 P hotos of alum inum foam ................................................................................ ...... ... 14

3-1 Schem atic of a single cell in the simplified m odel ................................. ................32

3-2 M odel details................................. ........................................................... ............... 33

3-3 3-D scheme atic of the m odel ........................................................ ...... ...........................33

3-4 Heat transfer network of analytical model....................... ..........................34

3-5 Schematic of vertical strut fin model ............................ ................................ 34

3 -6 H -stru t m o d e l ............................................................................................................... 3 5

3-7 M odel for coolant temperature evaluation................................... .......................... 35

3-8 Cylinder diameter as function of relative foam density predicted by analytical model,
comparing with ERG's data of aluminum foams ..........................................................36

3-9 Surface area density as function of relative foam density predicted by analytical
model, comparing with ERG's data of aluminum foams ...............................................36

3-10 Nusselt number prediction made by analytical model compared with Calmidi and
Mahaj an's experimental data for 5 ppi aluminum foam...................................................37

3-11 Nusselt number prediction made by analytical model compared with Calmidi and
Mahaj an's experimental data for 10 ppi and 20 ppi aluminum foam ...............................37

3-12 Nusselt number prediction made by analytical model compared with Calmidi and
Mahajan's experimental data for 5 ppi and 40 ppi low porosity aluminum foam......3....8

4-1 Schematic of boundary cell and interior cell in open-cell foam.....................................44









4-2 Comparison of single cell model and real foam structure ...........................................45

4-3 Geom etry creation of a single cell ............................................................................... 45

4-4 2-D periodic model ........................ ........ .. ... ... .. .................. 46

4-5 1-D periodic model ........................ ........ .. ... ... .. ................. 46

4-6 M esh of a single cell m odel (coarse grids) ............................................. ............... 47

4-7 Details of the meshes on filaments (medium grids)................ .................47

4 -8 G rid s d istrib u tio n ........................................ .................. ........................ ................. .. 4 8

4-9 Velocity profile along flow direction through the cell..................... ............. ............... 48

4-10 Pressure distribution along flow direction through the cell......................... ............49

4-11 Velocity contours in three planes around the cell............... ........................................... 50

4-12 Static pressure contours in three planes around the cell ....................................... 51

4-13 Pressure drop versus inlet velocity and comparison with experimental data ....................52

5-1 Notional design strategy for foam-filled channels............................................... 59

5-2 Comparison of heat transfer coef vs. pressure drop of open and foamed channels..........60









NOMENCLATURE

a Cell size

A Surface area of foam

A, Area of cylinder's cross section

Aw Area of heated wall

C1 Constant related to the geometry of channel, can be looked up from tables

Cp Specific heat of coolant

d: Diameter of the cylinder.

d/ Inner diameter of test section

do Outer diameter of test section

f Friction factor

H Height of cooling channel

hi: Heat transfer coefficient between vertical cylinder and the cooling fluid.

h2: Heat transfer coefficient between horizontal cylinder and the cooling fluid.

heuqal Equivalent heat transfer coefficient of foam filled cooling channel

hw: Heat transfer coefficient between bare wall and the cooling fluid.

kf: Thermal conductivity of the coolant.

ks: Thermal conductivity of the cylinder.

1 Some constant defined in Eq.(3-18)

L Length of cooling channel

m Constant related to the geometry of channel, can be looked up from tables

mi Constant calculated from hi, ks, and d

M, Constant calculated from hi, ks, and d

m2 Constant calculated from h2, ks, and d

M2 Constant calculated from h2, ks, and d









mi Mass flow rate of coolant

Nh Number of horizontal cylinders per unit width

Nu Nusselt number

Nv Number of vertical cylinders per unit width

p Pressure

Pr Prandtl number

Q Total heat transfer rate to coolant

qh Heat transfer rate from a single horizontal (H-cylinder) to the coolant

qv Heat transfer rate from a single vertical (V-cylinder) to the coolant

qw Heat transfer rate from bare heated wall to the coolant

Re Reynolds number

To Inlet temperature of coolant

Ti Temperature of vertical (V-type) cylinders at x

T2 Temperature of vertical (V-type) cylinders at x+a

,znlet Inlet air temperature

T,outlet Outlet air temperature

Tb Bulk fluid temperature

To Coolant temperature

Th Temperature of horizontal (H-type) cylinder

Ts Temperature of vertical (V-type) cylinder

Tw Constant temperature of heated wall

V Average inlet velocity of coolant or volume of the foam

Vmax Maximum velocity of the coolant

x X coordinate or direction

y Y coordinate or direction










Z coordinate or direction


Greek Symbols

aA

0






Ps



1


Vf



Subscripts

1

2

c

h

v

w


Surface area density

Non-dimensional variable defined by Ts, To, and Tw

Non-dimensional variable defined by Ts, Tc, Tw, and To

Relative foam density

Density of foam

Density of solid

Ratio of bare wall surface area to the total wall surface area

Kinematic viscosity of coolant





Vertical (V-type) cylinder

Horizontal (H-type) cylinder

Coolant

Horizontal (H-type) cylinder

Vertical (V-type) cylinder

Wall









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

NUMERICAL EVALUATION OF HEAT TRANSFER AND PRESSURE DROP IN OPEN
CELL FOAMS

By

Mo Bai

December 2007

Chair: Jacob N. Chung
Major: Mechanical Engineering

As society pursues the space travel, advanced propulsion for the next generation of

spacecraft will be needed. These new propulsion systems will require higher performance and

increased durability, despite current limitations on materials. A break-through technology is

needed in the thrust chamber. One of the most challenging problems is to cool the hot chamber

wall more without creating additional pressure drops in the coolant passage. A promising

method is using open cell foam to enhance the heat transfer rate from chamber wall to coolant.

However, the pressure drop induced by foams is relatively large and thus becomes a critical issue.

The focus of this thesis is the evaluation of heat transfer and pressure drop of open cell foams.

A simplified analytical model has been developed to evaluate the heat transfer capability

of the foamed channel, which is based on a diamond-shaped unit cell model. The predicted

heat transfer results by the analytical model have been compared with experimental data of

different Reynolds numbers from other researchers and favorable agreements have been obtained.

For the evaluation of pressure drop in open-cell metal foams, direct numerical simulation models

of the foam heat exchanger have been built using GAMBIT/FLUENT. The model is based on a

structure of sphere-centered open cell tetrakaidecahedron. This model is very similar to the

actual metal foams' microstructure of thin ligaments that form a network of interconnected









open-cells. Grid independence of solution is investigated and simulation results are further

compared with experiments. Finally, the feasibility of applying foam filled cooling channel on

rocket thrust chamber is investigated.









CHAPTER 1
INTRODUCTION ROCKET THRUST CHAMBER COOLING

1.1 History of the Rocket

The history of rocketry is at least more than 700 years. The first rocket is said to be

invented by a Chinese scientist named Feng Jishen in 970 A.D., who used bamboo tubes and

black powder to generate great thrust power by expanding hot exhaust gas. That is the

prototype of today's firecracker and fireworks. The use of black powder to propel projectiles

was a precursor to the development of the first solid rocket. The principal idea of obtaining

thrust by reaction is thought to be founded by Hero of Alexandria in 67 A.D. He invented

many mechanisms which utilize the reaction principle that is thought to be the theory basis for

rockets. Rocket technologies first become known to Europeans by Genghis Khan when the

Mongols conquered Russia, Eastern and Central Europe. The Mongols got the technologies

from Chinese and they also employed Chinese rocketry experts. The first serious scientific

book on space travel is published by Konstantin Tsiolkovsky, a Russian high school mathematics

teacher, in 1903.[1] In 1920, Robert Goddard published A Method of Reaching Extreme

Altitudes, the first serious work on using rockets in space travel after Tsiolkovsky. Goddard

was a professor at Clarkson University in Massachusetts. He attached a supersonic nozzle to a

liquid rocket's combustion chamber, which became the first modern rocket. Hot gas in the

combustion chamber is expanded through the nozzle, and turns into cooler, hypersonic, highly

directed jet of gas, which greatly improves the thrust and efficiency. Goddard had more than

214 patents on rockets that were later bought by United States.

1.2 Rocket Structure

Most current rockets are chemically powered rockets, an internal combustion engines that

obtain thrust from expanding hot exhaust gas. From propellant's point of view, there are gas









propellant, solid propellant, liquid propellant, and even a mixture of both solid and liquid

propellant. Typically, a rocket engine' structure consists of injectors, combustion chamber and

the converging diverging nozzle, which can be seen in Figure 1-1. The injectors are used to

introduce fuel and oxidizer to combustion chamber. The combustion chamber is where the fuel

and oxidizer are mixed and burned. The nozzle is usually designed as an integral part together

with combustion chamber, its purpose is to regulate and direct exhaust gas to reach a supersonic

speed and get maximized thrust. In this study, the word thrust chamber is used to present the

integral structure of rocket combustion chamber and nozzle.

The thrust chamber is the key component of a rocket engine, here the propellant is injected,

vaporized, mixed, and burned to transform into hot exhaust gas. The combustion reaction can

fairly reach the temperature up to 3500K, which is much higher than the melting point of the

material used in thrust chamber. Thus, it's critical to make sure the thrust chamber won't melt,

vaporize, or combust. Some rockets chamber use ablative material or high temperature material,

such as carbon based materials graphite, diamond, and carbon nanotubes. Other rocket

chambers use conventional materials like aluminum, steel, or copper alloys. These kinds of

rocket then need a cooling system to prevent the chamber wall become to hot.

1.3 Rocket Thrust Chamber Cooling

Generally speaking, there are two major methods of cooling rocket thrust chamber today.

The first one is steady state method, which is the heat transfer rate through thrust wall and

temperature on the wall are constant, in other words, there's a thermal equilibrium. The steady

state method includes regenerative cooling and radiation cooling. The regenerative cooling is

done by attaching a cooling jacket onto the thrust wall and circulating one of the propellants

through the cooling channel before it is injected into chamber for combustion. Usually,

regenerative cooling is used for bipropellant rockets having medium to large thrust, and it is









effective for thrust chamber having high pressure and high heat transfer rate. The radiation

cooling is using an extension attached to the thrust nozzle exit to get extra radiation heat transfer

to the ambient space. Radiation cooling is primarily used in monopropellant rocket, which have

relatively low pressure and requires moderate heat transfer rate.

The second method to cool rocket thrust chamber is unsteady state method or transient heat

transfer method. For this method, there is no thermal equilibrium and the temperature on thrust

wall continues to increase. The total heat transfer absorbing capacity is determined by the

hardware. The rocket engine has to be stopped before the temperature reaches the hardware's

critical point. Ablative materials are commonly used in unsteady state cooling method and

solid propellant rocket, for which chamber pressures is lower and heat transfer rate is also low

[2].

1.3.1 Regenerative Cooling

This study is mainly about the steady state method using regenerative cooling. For

regenerative cooling, a cooling jacket is constructed in the thrust wall to allow the coolant to

circulate in the cooling channels. Usually, one of the propellants (commonly the fuel) is used

as the coolant. A typical tubular cooling jacket is shown in Figure 1-2. The fuel enters

through the inlets of every other tube, flow to the nozzle exit, and then enters the alternate tubes,

flow back to the injectors for combustion. There are also other rockets' coolant inlets are at the

nozzle throat area, coolant flows up and down in the nozzle exit region and flows up in the

chamber region. This design is considering heat transfer intensity of the rocket thrust chamber.

Because the heat transfer rate peak is often at the nozzle throat area, which is shown in Figure

1-3, letting the coolest coolant entering at throat area can greatly enhance the heat transfer

efficiency. Another method to enhance the heat transfer rate at throat area is increase coolant

flow velocities at that area. From Figure 1-2, cross-section area at section B is the smallest









which can generate the largest coolant velocities there. Figure 1-4 and 1-5 show schematics of

a liquid propellant rocket's thrust chamber and details of cooling channels [3].

Regenerative cooling method has many merits in the sense of heat transfer efficiency and

structure optimization. First, using fuel as the coolant greatly enhances heat transfer efficiency.

Because for liquid propellant rocket, the fuel is cryogenic, large temperature difference between

coolant and combustion gas can make great heat transfer rate. In addition, after flow through

the cooling channel, the fuel has higher temperature and become ready for combustion. Second,

the tubular cooling jacket reduces weight of the rocket thrust chamber and also the total weight

of rocket, thus greatly increases efficiency. Third, the cooling jacket structure transform thick

thrust wall into thin walls of cooling channels, which can reduce thermal stresses.

1.3.2 Challenges on Regenerative Cooling

Today, the needs for longer and faster space travel require rockets with more powerful

thrust and also bring challenging requirements to the cooling system. Much higher heat transfer

rate is needed for next generation rockets. Even with new advances in high-temperature and

high conductivity materials, thrust increases for large liquid propellant rocket engines are limited

by the cooling capacity of the cooling jacket. Cooling limits have been extended with the use of

film cooling, injector biasing, and transpiration cooling. However, these methods are costly to

engine performance since they require that some of the fuel pass through the thrust chamber

throat without contributing to thrust.

Currently, the vast majority of regenerative cooling rocket engines use either tube bundles

or milled rectangular passages as heat exchangers. Several improvements based on the tubular

cooling system of rocket thrust chamber are shown in Figure 1-6. The conventional

micro-channel heat exchanger is shown in Figure 1-6A. The partition walls serve as fins to

increase surface area thus enhance heat transfer rate and also support the hot wall. The high











aspect ratio heat exchanger is shown in Figure 1-6B, which has larger surface area so can


increase the cooling effectiveness. For Figure 1-6C, metal foam inserts are used in the channel


to get even larger heat transfer rate. This study is focused on evaluation of foam filled


channel's heat transfer and pressure drop, which has potential application in rocket thrust


chamber's cooling system.


Gaw feww
OX"" efts






moufilt"'





pyrotediPk ihl




Thrnletu m luI
Manda IOPm n

sea" time l ro~


rc Exit

in T Ooult


ChIfinbk stala bindl -~


Fu rld am



Figure 1-1. Construction of a regenerative cooling tubular thrust chamber, its nozzle internal
diameter is about 15 inch and thrust is about 165,000 lbf It was originally used in
the Thor missile. Recreated from reference Sutton [2].


I*Wect pla(<
Iri WO i











Reinforcing
tension


Injector i
I _





Top view without
manifold


cc- Exit (Section C)

___ Chamber
(Section B)
M Throat
(SectUf A)


A C


Figure 1-2. Cutaway of a tubular cooling jacket. The cooling tubes have variable
cross-section area to allow the same number of tubes at nozzle throat and nozzle exit.
Recreated from reference Sutton [2].


C4
E

5)l


Thrust chamber contour


Figure 1-3. Typical heat transfer rate intensity distribution for liquid propellant rocket. Peak
is at the thrust nozzle throat and nadir is usually at the nozzle exit. Recreated from
reference Sutton [2].




































Combustion Nozzle
Chamber


Figure 1-4. Simplified schematic of regenerative cooling system of liquid propellant rocket.
Recreated from reference Carlos [3].



Cooling
Channel -----"--, ~ external environment










noZle interl wall
center








Section A-A Channel details

Figure 1-5. Section A-A of Figure 1-4 and details of cooling channel. Recreated from
reference Carlos [3].













A B C
Figure 1-6. Different configurations of the cooling channel in thrust chamber. A)
Conventional micro cooling channel. B) High aspect ratio cooling channel. C)
Metal foamed cooling channel.


IIII









CHAPTER 2
PREVIOUS WORK ON OPEN-CELL FOAMS

2.1 Heat Transfer Enhancement

Open-cell foam is a kind of porous medium that is emerging as an effective method of heat

transfer enhancement, due to its large surface area to volume ratio, high thermal conductivity,

and intensified fluid (coolant) fixing. Figure 2-1 shows several pictures of typical aluminum

foam.

The use of open-cell foam to enhance heat transfer has been investigated widely. Koh

and Colony [4] and Koh and Stevens [5] investigated the heat transfer enhancement of

forced-convection in a channel filled with high thermal conductivity open-cell foam. In their

theoretical study, Koh and Colony [4] found that for a fixed wall temperature case, the heat

transfer rate increased by a factor of three. For a constant heat flux case, the wall temperature

and the temperature difference between the wall and the coolant can be drastically reduced.

Koh and Stevens (1975) performed experimental work to verify the numerical results of Koh and

Colony [4]. Koh and Stevens [5] used a stainless steel cylindrical annulus (1.5" ID and 2.1"

OD) with a length of 8 inches to experiment with heat transfer enhancement by porous filler.

The annulus was filled with peen shot (steel particles) whose diameters ranging from 0.08 inch to

0.11 inch. Nitrogen gas was used as the coolant. They found the heat flux increased from 17

to 37 Btu/ft2s for a constant wall temperature case and the wall temperature dropped from 1450

F to 350 F for the constant heat flux case. Hunt and Tien [6] utilized foam-like material and

fibrous media to enhance forced-convection for potential application to electronics cooling.

Their results showed that a factor of two to four times enhancement is achievable as compared to

laminar slug flow in a duct. Maiorov et al. [7] found empirically that the heat transfer rates in









channels with a high-thermal conductivity filler, compared to empty channels, reached a factor

of 25-40 enhancement for water and 200-400 for nitrogen gas.

Bartlett and Viskanta [8] developed a mathematical model to predict the enhancement by

high thermal conductivity porous media in forced-convection duct flows. They concluded that

a 5-30 times increase in heat transfer is feasible for most engineering conditions. It is believed

that the enhancement is mainly due to the micro turbulent mixing in the pores and super heat

transfer through high thermal conductivity porous structure.

Kuzay et al. [9] have reported liquid nitrogen convective heat transfer enhancement with

copper matrix inserts in tubes. They proved that the insertion of porous copper mesh into plain

tubes enhances the heat transfer by large amounts with a single-phase coolant. However, in

boiling, with tubes in which the porous insert is brazed to the tube wall for the best thermal

contact, the heat enhancement is to be on the order of four-fold relative to a plain tube. They

conclude that porous matrix inserts offer a significant advantage in cooling, providing a

jitter-free operation and a much higher effective heat transfer, at grossly reduced flow rates

relative to plain tubes.

More recently, Boomsma et al. [10] used a open-cell aluminum alloy metal foam

measuring 40 mm x 40 mm x 2mm as a compact heat exchanger. With liquid water as the

working fluid, they found that the heat exchanger generated resistances that are two to three

times lower than those of the open channel heat exchanger while requiring the same pumping

power.

2.2 Experiments

Many researchers investigated important characteristics of open-cell metal foams through

experiments. Leong and Jin [11] performed experiments to investigate characteristics of

oscillating flow through metal foams. They got detailed experimental data of flow pressure









drop versus flow velocities. They found the oscillating flow characteristics in metal foam are

governed by a hydraulic ligament diameter based Reynolds number and the dimensionless flow

displacement amplitude. And the Reynolds number has more significant effect on pressure

drop and velocities' relationship.

Kim et al. [12] experimentally investigated the impact of presence of aluminum foam on

the flow and convective heat transfer in an asymmetrically heated channel. The aluminum

foam they use has a porosity of 0.92, but with different permeability. They placed foam inside

a channel and keep the upper wall at constant temperature while the lower wall is thermally

insulated. They got correlations of the friction factor and Nusselt number with Reynolds

number.

Yuan et al. [13] investigated heat transfer enhancement and pressure drop in an annular

channel with nickel foams. They used air as the coolant and constant heat flux heaters inside

the inner tube of the annulus. They found the heat transfer enhancement was on the order of

twenty times over open channel. Correlations of pressure drop, Nusselt number and heat

transfer coefficient with Reynolds number were obtained.

2.3 CFD Simulation and Numerical Model

Many scholars investigated open-cell foams by numerical methods, both analytical and

computational. Lu et al. [14] developed an analytical model to mimic metal foams. It based

on cubic unit cells consisting of heated slender cylinders, and took advantage of existing heat

transfer data on convective crossflow over bank of cylinders. They solved out the overall heat

transfer coefficient of a heat exchanger analytically and also the pressure drop. A process to

optimize foam structure so as to maximize heat transfer rate was proposed. However, their

model maybe oversimplified the metal foam and leaded to overestimates.









Krishnan et al. [15] carried out a direct simulation of the transport phenomenon in

open-cell metal foam using a single unit cell structure. The unit cell is created by assuming the

void pore is spherical, and the pores are located at the vertices and center of a unit cell. The

final geometry is obtained by subtracting the spheres from the unit cell cube. They further used

that model to perform CFD simulation using Fluent/Gambit. Periodic conditions are used thus

only one cell is needed in simulation which greatly saved computational time. Total thermal

conductivity, pressure drop and heat transfer coefficient are obtained and compared with

experimental data. Yet, this model is only suitable for foams that has porosity larger than 0.94.

Krishnan et al. [16] then created other models to extend the model's capability to simulate lower

porosities (down to 0.80). Besides the body-centered cubic model [15], they developed other

models based on face-centered cubic (FCC), and A15 lattice, which is similar to Weaire-Phelan

structure. Good agreement to other researchers' experimental data is obtained on Nusselt

number and friction factor.

Chung et al. [17] predicted and evaluated heat transfer enhancement for liquid rocket

engine using metal foams. They developed a unit cell structure based on Kelvin's

tetrakaidecahedron. The ligaments of unit cell structure are simplified as cylinders.

Comparison of pressure drop predicted by that model with experimental data shows favorable

agreements. They further performed CFD simulation using that structure and also open channel

to predict pressure drop under rocket conditions, in which Reynolds number is up to 1 million

and coolant is hydrogen. They also provided some experiment data on copper and nickel foams

under lab conditions. The heat transfer enhancement of foams inserted channel over

conventional channel is 130%-170%. They believed that the enhancement is independent of

pressure drop and increases with decreasing pore size.









Boomsma et al. [18] developed a new approach to modeling flow through open-cell foams

and defined a new cell structure. Their new model was based on Weaire-Phelan structure.

This structure reduced the surface energy by 0.3% compared to tetrakaidecahedron [18]. The

Weaire-Phelan structure was further "wetted" by Surface Evolver. Boomsma et al. [18] used

that model to investigate pressure drop and velocity field in open-cell foams. They also

compared their CFD prediction with experimental data and found their results were 25% lower.

It's believed that the underestimates were due to the lack of pressure drop increasing wall effects

in the simulations.

2.4 Other Open-Cell Foams

2.4.1 Polyurethane Foams

Some researchers investigated other open-cell foams other than metal foams. Mills [19]

used CFD simulation to investigate the permeability of polyurethane foams. The unit cell

structure he used is Kelvin's tetrakaidecahedron, which is widely used in the simulation of metal

foams. He also used the Surface Evolver to get wetted structure of the Kelvin's model. He

concluded that the foam permeability is a function of the area of largest hole in the cells [19].

2.4.2 Carbon Foams

Carbon foams generally have better heat transfer performance than metal foams but induce

larger pressure drop, which is due to their smaller pore size and lower porosity. Yu et al. [20]

developed a unit cube-based model for carbon foam modeling. This structure allows lower

porosity which is a major property of carbon foam, compared to conventional metal foams.

Assumed that the entire foam has uniform pore diameter and pores are considered to be spherical

and centered, their model was obtained by subtracting a sphere from a unit cube. They used

that model to evaluate carbon foam's heat transfer and pressure drop analytically and compared

their results with experimental data.



















A B













C
Figure 2-1. Photos of aluminum foam. A) photo of aluminum foam brazed to a metal. B)
view from a different angle. C) SEM photo of typical aluminum foam









CHAPTER 3
ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS

3.1 Geometry Simplification for Open-Cell Foam Filled Channels

This transport model is based on the microscopic structure of the metal foam whose cells

can be approximated as in diamond shapes as illustrated by the model presented in Figure 3-1.

The ligament structure is composed of two types of struts. The vertical struts called V-type struts,

are perpendicular to the flow direction (x) while the horizontal struts, called the H-type struts are

on the plans (x-y) that is parallel to the flow direction.

Figure 3-2 shows the detailed infrastructure of the model. The picture on the left illustrates

the arrangement of the ligaments and their connection with the walls. The top wall is the heated

surface which represents the heat source from the combustion chamber. The bottom wall is

insulated as it stands for the outer wall for the cooling channel. The plot on the right is a top view,

which gives the horizontal cross section and the flow direction.

A 3-D schematic of the foam model is given in Figure 3-3 where two rows in the

downstream direction and four columns for each row in the cross-stream direction are shown to

illustrate the foam structure.

The heat transfer mechanisms are explained in terms of a network as shown in Figure 3-4.

The heated top wall is the heat source that interacts with the V-type foam ligaments (fins)

through conduction and also supplies heat to the coolant by convection through un-finned

surfaces. The V-type struts, which act as fins receive heat from the wall and then pass the heat to

the coolant by convection and to the H-type struts by conduction. The convection heat transfer

between the flow and the V-type struts will be modeled as heat transfer for flow over tube banks.

The H-type struts will lose heat to the flow by convection and also transfer some heat to

downstream V-struts by conduction.









3.2 Mathematical Transport Model and Heat Transfer Equations


3.2.1 V-type Struts

A standard "fin" analysis is applied for a vertical cylinder as shown in Figure 3-5. The

governing equation for a fin is given as below:

d 2 Ts 4hl
d2T T4h T)= 0 (3-1)
dy2 kd

* hi : heat transfer coefficient between vertical cylinder and the cooling fluid
* ks : thermal conductivity of the cylinder
* d : diameter of the cylinder
* Ts(y) : local temperature as a function ofy along the V-strut
* To : coolant temperature


Boundary Conditions: dT
dy y-H


Solve for

cosh m, (H y)
cosh m,H


where mi =


From Eq. (3-2), the rate of heat dissipated to the coolant from the strut q, in watts, is


(3-2)


q i-kAOT
Y-O


kd k (T, T) tanh(mH) = M, tanh(mH)(T, T7)
4 k d


whereM, = hkjr2d3 /4, hi is evaluated based on flow over a cylinder or tube bank. It's further

assumed that the heat transfer coefficient is the same for all the vertical struts.

In order to get an analytical solution, the average strut temperature over y is needed.


e -T
Let0 =
T, -To


cosh m, (H y)
cosh(mH) integrate Ts from y=0 to H, the average can be shown to be
cosh(mH)









S i cosh m, (H y) dy
H 0 cosh mH
1 1 1 H
S 1 (- )sinhm, (H y)IH
H cosh mH m"
tanhmH


mtanhmH
Thus, T = (T -T)+T (3-3)
mH

The average over y direction eliminates the y variable in Ts. Since the cross section of

rocket chamber is often annular, the coolant and foam can be treated as uniform in z direction.

Thus, coolant temperature is only a function ofx in the flow direction [14].

cosh m, (H y)
T,(x, y) = T,(x)+o (T T (x)) (3-4)
cosh mH

tanh m,H
T, (x) = (T, T,(x)) + T (x) (3-5)
mH

Heat transferred from a single V-cylinder to the coolant is

q, = idH h [T(x)- T(x)] (3-6)

3.2.2 H-type Struts

A similar "fin" analysis as the vertical cylinder is applied for the horizontal cylinders that

connect the vertical cylinders (Figure 3-6).

The governing equation should have a similar form as a V-cylinder though extra terms

need to be introduced to account for the angle (less than 90 degree) between H-cylinder and

coolant. Such dependence can be assumed to be very weak when x>10a, as stated in the paper

by Lu et al. [14]. There are several methods to solve the flow over H-type cylinders. Here

proposes two methods.









First method


The following equation will be used:

d2TT 4h
d-- 4h -Tc) = 0 (3-7)
dx2 kd

* h2 : heat transfer coefficient between horizontal cylinder and the cooling fluid
S ks : thermal conductivity of the cylinder
* d : diameter of the cylinder
* Th : Temperature of a H-strut
* To: coolant temperature

o h lx=0o
Boundary condition: 0
Th T2

T1 and T2 are the temperature of V-type cylinders at x and x+a. They can be evaluated

from Ts solved in part a. For a specific H-type cylinder, T1 and T2 are constant. Solution of

Eq.(3-7) is:

(T T) sinh[m2 ( x)] + (T T) sinh(m2x)
Th(x) Tc+ (3-8)
sinh(m2a)

The heat flux entering the cylinder at x=0

( TI -) cosh(m2a)- (- Tc)
sinh(m2a)

The heat flux leaving the cylinder at x=a

q2 2 (T -T)-(T2 T)cosh(m2a)
sinh(m2a)

r4h--
where m ,M2 hkfd3 /4
k,.d

The heat transfer to the coolant, is

< q1 q M T + Tcosh(m2a)-l
qh = ql-q2 = M2(7 + T2 2Tc) (3-9)
sinh(m2a)









Let 7, = T(x), T = T(x + a). For the coolant, since its temperature is function of x, we


simply choose its value at the middle point of H-type cylinder, thus T = (T, (x) + T (x + a)).
2

The heat transferred to the coolant from a single H-cylinder is:

cosh(m2a)-l 1 _
qh =M2 sinh2a) (T(x)+T(x + a)-T(x)-T(x+a)) (3-10)
sinh(na)

The same correlation of finding hi is used to evaluate h2, because it is still a cross flow

over a bank of cylinders. A correction on the free stream velocity is needed as the flow is not at

900 to the cylinder.

Second method

Another method is to assume the H-type cylinders have identical temperature distribution

with V-type cylinders along x direction. Thus, for a single H-type cylinder, the heat transfer

rate from it to coolant can be represented as,


qh = rd -a-h,(Th(x)- T(x)) (3-11)
2


where rd--a is a H-type cylinder's surface area, h2 is heat transfer coefficient for H-type
2

cylinders, and Ts(x)-Tc(x) is the temperature difference between cylinders and coolant. From

Eq.(3-5), the heat transfer rate can be further written in the form,

tanh mH
qh = )id- ah2 (7 T(x)) (3-12)
2 mH

The reason two methods are proposed is because the first method is found to have

unfavorable agreement with experimental data at low Reynolds number region, which will be

discussed later.









3.2.3 Fluid Temperature Prediction (Coolant Temperature)

The coolant temperature profile as a function of the downstream coordinate, x, is estimated

based on the following energy balance equation. Figure 3-7 shows the schematic.

mC [T (x + Ax)- T =(x)]= Nq, + Nq, + q (3-13)

Ax-1 Ax-1 H
where N, -2 is the number of vertical struts per unit width. And Nh = 2- is the
a a a

number of horizontal struts per unit width for a channel of height H.

Heat transfer from V-type struts q, can be evaluated from Eq. (3-6).

Heat transfer from H-type struts q, can be evaluated from Eq. (3-10).

Heat transfer from bare wall surface can be calculated from:

q, = 7Axh[T, T(x)] (3-14)

where rI is the ratio of bare wall surface area to the total wall surface area, and hw can be

evaluated from open channel heat transfer coefficient correlation.

First method (high Re)

Eq. (3-10) will be used for relatively high Reynolds number (>2*104). From Eq. (3-5),

- tanhm1H
Tx) -= H (T,- T(x))+ T(x). Let's further assume T,(x)= (T, -T)e-i + T where 1 is
mH

to be determined. Plug into Eq. (3-5),

I(x)- x ( tanhmH tanh mH (- )e
T,(x3- T1)(xc = (T, Th(x)) =o(r,-
mH mH

So Eq. (3-10) can be rewritten in the form:









cosh(ma)- -
qh=M2 sinh(a) (T(x)- T(x) + T(x+a)- Te(x+a))
sinh(na)


Scosh(m2a) -1 tanh(m,H) ((
sinh(2a) mH ((
sinh(na) mH


,O)e x +(T


Scosh(m2a) -1 tanh(mH) (1+ e l)(
2 1 (1 + e( )(T
sinh(m2a) mH
Scosh(m2a) -1 tanh(mH) (1+ e-la)
sinh(m2a) mH


l7 )e 'x


Tc)


q cosh(m2a)- 1 tanh(mH) 1+ e')(T T)
= ( )(T-T)
sinh(m2a) mH

Plugging Eq. (3-6), (3-14), and (3-15) into Eq. (3-13) yields:


mC,[[T(x + x)- (x)]


T (x)) tanh mH + rAxh, [T


x-1
a


T(x)]


+2Axl H Mcosh(M2a)- tanhmH (+)( -T())
a a sinh(m2a) mH


Add up similar terms,

(x + Ax)- T(x)
T T (x)


Ax 1 H cosh(m2a)-1 tanh mH (l+e- ")+h ]
[ tanhmH+2 3 sinh(m2a) mH
mC a2 a' sinh(nza) mH


1 1 H .cosh(m2a) -1 tanh mH (le )+hjw]
Let 1'= [ MI tanh mH M cosh( 1 tanh (1+e)+ ]
ihC a2 a3 sinh(m2a) mH

we have:

T (x + Ax)- T7 (x)
(T T(x))

Integrate (3-16) as Ax 0


Tc(x) dT
ST-T


J' dx


T- f x)d(T ) T

To


/l'dx
0


T,)e- (xa))


(3-15)


(3-16)












T T-TO
_>T


-'x


-> T(x)= (,- T)e-x +

Since we assume T7(x) = (T 7)e i + T in the beginning of this derivation, thus '= 1.

That also proves the previous assumption is correct. So,

TO(x) = (T, )eix + T (3-17)


where,


1 1 tah H cosh(ma) -1 tanh mH + e ,
= -[-M, tanh mH -+ M (1+esi)+n )
mC a a sinh(mza) mlH


(3-18)


1 can be determined by iterative method.

Second method (low Re)


Eq.(3-12) will be used for relatively low Reynolds number (<2* 104).


The energy balance


Eq.(3-13) still holds for this case. Plug in Eq.(3-6), (3-12), and (3-14) to Eq.(3-13) yields,


mC, [Te(x + Ax)- Te(x)]


So, T(x + Ax)- T(x)
T -T (x)


T (x)) tanh mH + 7Axh [T T- (x)]


Ax-1
a


Ax-1 H tanhmH
+2 x2 -H rd-- a-h2 t (T T(x))
a a 2 mlH


1[M, tanh + 7, + -5dH htanh mH]Ax
[v-C, tanhH +h a h2
1C a a mlH


Let constant 1 to be in the form,

1 1 ,JJ5rdH tanh mlH
/ = [- M tanh mH+h + h h2 ]
mC a a maH

Integrate Eq.(3-19) over x, we have:


(3-19)









T (x) = (T )e-x +T (3-20)

Eq.(3-20) has the same form with Eq.(3-17) in the first method, the difference is that, for

the second method, 1 can be calculated directly, and there's no need to iterate.

3.2.4 Total Heat Transfer

The total heat transferred to the coolant through the cylinders (V and H types) and the

inner wall is

Q(x) = (T (x)- TO)mCP (3-21)

Since T(x) = (T, -T)e + T+ from Eq. (3-17) and (3-20), so

Q(x)= mCp(T TO)(1-e X) (3-22)

where 1 can be determined from Eq. (3-18) or Eq.(3-19).

For a channel of length L, the total heat transfer is

Q(L) = mC,(Tw TO)(- e-L) (3-23)

3.2.5 Evaluation of Heat Transfer Coefficient

The heat transfer coefficient evaluation in the foregoing analytical method is critical and

will be discussed in this section. It's mentioned in previous sections that the heat transfer

coefficients of V-type cylinders, H-type cylinders, and bare wall are assumed to be identical,

respectively. Based on Reynolds number, all the heat transfer coefficients of cylinders can be

evaluated by the empirical correlations for flow over a tube bank correlation. And heat transfer

coefficient for bare wall can be calculated from correlation for open channels.

V-type cylinders hi

Flow over a bank of tubes has been widely investigated by researchers for many years, and

several correlations are available for heat transfer. For a staggered mesh, the average heat









transfer coefficient h for the entire tubes in the bank as defined in the Nusselt number,


Ad
Nud = can be obtained from the correlation below [21]:
kf

Nd = 1.13C1 Rem Pr1/3 (3-24)

where Ci and m are constants, they can be looked up from Incropera and DeWitt [21].

Reynolds number is defined as,

dV
Red max vmax (3-25)
vf

1.13C, Re max Prl/3 k
So, =dx f
d

Vmax is the maximum velocity of the coolant, kf is the thermal conductivity of coolant, vf is

kinematic viscosity of coolant, and d is the diameter of the cylinder, m and C1 are constants and

related to the geometry of channel, which can be obtained by tables.

According to Incropera and DeWitt [21], the maximum velocity occurs at the transverse

a
plane. It can be calculated as Vv max V, V is the incoming velocity.
a-d

Eq. (3-24) is valid for Re from 2000 to 40000. For smaller Re number, correlation for

flow over a single cylinder is used. Because the diameter of cylinder d is much smaller than

cell size a, that is their ratio d/a is about 0.2, the influence of inter-cylinders is neglected for low

Re number cases in this study. Equation for flow over a single cylinder from [21] is

Nu = CRemax Pr13 (3-26)

where C and m can be found from Table 3-1.









H-type cylinders h2

For H-type cylinders, it can also be treated as flow over a single cylinder. The difference

is that the flow direction is not perpendicular to the cylinder. So, the component of velocity

that is perpendicular to cylinder is considered. From geometry, the equation is shown below


Vhmax -2(a /) V (3-27)
2*(a* 5/2- d)

The other parameters are calculated as the same as V-type cylinders.

Bare wall hw

To evaluate the heat transfer coefficient for the bare wall, the following correlation of open

channel can be applied:

Nu = 0.021Pr 5 Reo' (3-28)

Reynolds number and Nusselt number are defined as


Re = (3-29)
vf

hD
Nu =h (3-30)
kf

For this case, only the bottom wall is heated, so for Eq. (3-29) and (3-30) D=2H, where H

is the height of the metal foamed cooling channel. And hw can be calculated from

Nukf
h= Nk (3-31)
2H

3.2.6 Equivalent Heat Transfer Coefficient

To calculate Nusselt number and equivalent heat transfer coefficient of open-cell foams is

the ultimate aim of the analytical model. The equivalent heat transfer coefficient is defined as,


hequai (3-32)
A (T1 -T)









such that the heat transfer from the foamed channel is equivalent to that carried away by a

coolant having average temperature of T, which flows through a open but otherwise identical

channel. As is surface area of heated wall, in this model, the width is 1, so A, = 1 *L. For the

mean value of coolant temperature, or bulk temperature, the arithmetic mean value of coolant

over x direction is used. From Eq.(3-20), T1(x) = (T, -T)e + T, let 0 =T T( =7 e ix
TT

SO

T 1 L 1 L 1- exp(-lL)
0= I = O1 0 (x)Jx =d e x 1fe = x)
T -TO L L IL

1- exp(-lL)
Thus, T, -T e = (T To), plugging into Eq.(3-32) yields,
IL

QlL
eq"al L(1- exp(-L))(T To)

Plug in Eq. (3-23) for Q,


equal C(T- T)(1- e)lL)
S L(1-e L)(T- TO)
= Mi2CP

If the second method (low Reynolds number) is used for H-type cylinders heat transfer,

Eq.(3-19) can be used for 1. Thus, hequal has the final form

1 -5r1idH tanh mH
hei= -M tanh mH +77hw +- h2 (3-33)
a2 a2 mH

From Eq.(3-33), the equivalent heat transfer coefficient is a function of foam geometry a, d,

channel height H, and heat transfer coefficient hi, h2, hw. It's not a function of inlet temperature,

wall temperature, or channel length.









3.3 Investigation of Cylinder Diameter and Surface Area Density

The relative foam density and surface area density are two most important properties of

foam. Relative foam density is closely related to permeability and pressure drop induced by

foams, defined by the following equation,

p*
P,

where p is relative foam density, p is density of foam, and p, is density of solid. Another

important property, porosity, is equal to 1- p.

The surface area density is defined by this equation,

A
a'A -


where A is surface area of foam, and V is the volume of the foam. The surface area density is

an important property of foam which is related to heat transfer capacity of a foam.

In order to verify the foregoing diamond shaped cell structure, it need to be made sure that

the structure represents the real metal foams well by retaining the relative foam density and

surface area density.

For the foregoing diamond shaped model, from geometry calculations, the structure's

relative foam density can be represented as:


p = (d)2_ d ( ; )3 (3-34)
4 a 2a

where a is cell size and d is the diameter of cylinders. To simplify calculation and derivation,

and because d/a is about 0.2, Eq. (3-34) can be rewritten as,

(5S +3)r d
p = ( )2 (3-35)
20 a












d 20 /2 (3-36)
a (5 + 3),-

The filament diameter d is calculated from Eq.(3-36) based on a=2mm(10ppi),

lmm(20ppi), and 0.5mm(40ppi) with different relative density, and further compared with

experimental data from ERG Duocel aluminum foams (Figure 3-8). Reasonable agreement is

obtained.

For surface area density,

ilda + ([5a 2d),Td
A a3

[(5 + 1 )-2dla]7rd
a2
(,5 + 0.6);r d
a a

From Eq. (3-36), surface area density can be represented as,

5.97 /2
A p71/2 (3-37)
a

Eq (3-37) is plotted in Figure 3-9 and compared with data from ERG. Duocel aluminum

foams. Good agreement is obtained except for 40ppi case.

3.4 Verification of the Analytical Model with Experimental Data

To verify the heat transfer analytical model, heat transfer predictions on certain metal

foams by the model are compared with experimental data from other researchers. Because two

methods are developed for different Reynolds number, the author made two comparisons with

other experiments with Reynolds number ranging from 5*103 to 2*104, and 104 to 6*104,

respectively, using both methods stated in Section 3.2.3.









3.4.1 Validity of Analytical Prediction (Re=5*103 2*104)

Calmidi and Mahajan [22] tested several aluminum metal foams using air as the coolant.

Nusselt number data is obtained as function of pore Reynolds number. The pore Reynolds

numbers are transformed into Reynolds number based on channel height in this study. The

foam samples Calmidi used have dimensions of 114mm*63mm*45mm, and they placed two

heaters onto both the top and the bottom of foams. The Reynolds number is relatively low, and

the second method in Section 3.2.3 is used for this comparison. For the analytical model in this

study, the top wall of foamed channel is assumed to be adiabatic. So to predict Calmidi's data,

the height of channel in analytical model can be treated as half of the height of Calmidi's sample,

which is 45/2=22.5mm. Table 2-2 shows details of the experiment from Calmidi and Mahajan

[22] and parameters used for analytical model in this study.

To mimic the real foams, the filament diameter and pore diameter are two important

parameters for specific foams. In the analytical model, cylinder diameter "d" represents the

foam filament diameter and cell size "a" represents pore diameter. Because the diamond

shaped cell in analytical model is a simplified structure for real foams, the parameters a and d

used in analytical model can be slightly different from the real filament diameter and pore

diameter. Table 2-3 shows the parameters used in models and also their comparison with the

experiment samples' data. Different values of d and a are tested and the values shown in Table

2-3 are the ones providing best agreements with experimental data.

The predictions for the 5 types of foams ranging from 5PPI to 40PPI are plotted in Figure

3-10, Figure 3-11, and Figure 3-12, and compared with experimental data from Calmidi [22].

Favorable agreements are obtained. The Nusselt number and Reynolds number are defined in

the following equations,









Re HV (3-38)
vf


Nu= -ual (3-39)
kf

where H is height of foam, V is inlet velocity of coolant, hequa, is equivalent heat transfer

coefficient of foam defined in Eq. (3-32), kf is coolant's thermal conductivity, and vf is

kinematic viscosity of coolant.

3.4.2 Validity of Analytical Prediction (Re=1*104 ~ 6*104)

To verify the validity of the analytical model with relatively high Reynolds number, the

second method stated in Section 3.2.3 is utilized. A set of experimental data is used to compare

with the prediction by the model (an insulated heat flux case is used for this calculation). The

data are from an experiment made by our lab, testing heat transfer and pressure drop of copper

foam. The details of this experiment can be obtained from Chung et al. [17].

For experiment, the total heat transfer rate to the air flow is defined by the energy balance:

Q= iCCp (Ta,outlet- Tinlet) (3-40)

Here, T,,outlet and T,,nzt are the outlet and inlet air temperature, respectively, and Cp is the

specific heat under constant pressure.

The bulk fluid temperature is defined as:

Tb =(Ttet +,,nlet )/2 (3-41)

Effective heat transfer coefficient is defined as:

hqual =Q/ A(T T) (3-42)

where A is the total heated surface area and T7 is the mean surface temperature

Reynolds number is defined as:










Re =(d -) (3-43)
/Vf

where, do and da are the outer and inner diameter of the test section, respectively. v is the

kinematic viscosity of the fluid and V is the mean velocity.

For analytical model, the same geometry is used, and the height of channel is defined as


H =d (3-44)
2

Reynolds number is defined as:


Re 2VH (3-45)
Vf

The cell size a is set to be 2mm, and the filament diameter d is set to be 0.5mm, which is

approximately a 10 PPI (pores per inch), relative density 8%'s foam.

Figure 3-13 shows the analytical model's prediction of heat transfer coefficient of the

copper channel used in the experiment and compares them with experimental data. It should be

pointed out that the data from analytical model is scaled by a factor of 0.7 as a correction, which

maybe due to a different dimensional scale between the model and experiment. The analytical

model predicts the heat transfer coefficient nicely from the plot. But for high Re number, the

analytical model underestimate the heat transfer coefficient. We found that for Reynolds

number less than 105, the insulated boundary condition (at y=H) model gives good prediction.

For extremely large Re numbers (>105), constant temperature model at both walls should be used.

More details can be found in Chapter 5.

Table 3-1. Constants of Equation (3-22), recreated from [21]
ReD C m
0.4-4 0.989 0.330
4-40 0.911 0.385
40-4,000 0.683 0.466
4000-40,000 0.193 0.618









Table 3-1. Continued
40,000-400,000 0.027 0.805


Table 3-2. Parameters of experiments from Calmidi [22] and analytical model
Experiment [22] Analytical Model
Geometry L/W/H (mm) 114/63/45 114/unit length/22.5
Coolant Air Air
Foam Aluminum Aluminum
Coolant Inlet Temperature (K) z300 300
Heated Wall Temperature (K) z350 350


Table 3-3. Foam parameters comparison between experiments from Calmidi [22] and analytical
model
Ligament Pore
i t e Results Comparison
Diameter Diameter
5PPI Experiment 0.50mm 4.02mme 3-
5PPI A Figure 3-10
Model 0.70mm 4.02mm
Experiment 0.40mm 3.13mm
Model 0.55mm 3.13mm
20I Experiment 0.30mm 2.70mm
Model 0.45mm 2.70mm
5I Experiment 0.55mm 3.80mm
Model 0.70mm 3.10mm
40PI Experiment 0.25mm 1.80mm
Model 0.20mm 1.50mm


Figure 3-1. Schematic of a single cell in the simplified model












To














InsulatimO



Figure 3-2. Model details





















Coolant













Heated Wall -


I lOm, 5000 rows of V-type cylinders





Figure 3-3. 3-D schematic of the model










Conduction
Heat transfer


-- I G 'Conveclion I
Heat transfer
Figure 3-4. Heat transfer network of analytical model




T=Tw


I y=o


y=H

Figure 3-5. Schematic of vertical strut fin model


Conduction
Heat transfer


Convection
Heat transfer













x












Flow Dirci On'


-a -------------

x+a







a


Figure 3-6. H-strut model




0


Coolant


Entry -


V-type Cyli der



- a

-TO


I_ _, )Side Wall
Figure 3-7. Model for coolant temperature evaluation












O l0ppi(Experiment)
A 20ppi(Experiment)


40ppi(Experment)
- lppi(Model)
- 20ppi(Model) O
- 40ppi(Model) O






O -- -------A

--~--.".-o------ -''-


0 'I
0 0.02


0.06 0.08
Relative foam density


0.1 0.12


Figure 3-8. Cylinder diameter as function of relative foam density predicted by analytical model,
comparing with ERG's data of aluminum foams, a=2mm, 1mm, and 0.5mm,
respectively, for lOppi, 20ppi, and 40ppi foams.


0 10ppi(Experiment)
O 20ppi(Experiment)
A 40ppi(Experiment)
- l0ppi(Model)
20ppi(Model)
- 40ppi(Model)


^^ A
A
A


0 0.02 0.04


0.06 0.08


0.1 0.12 0.14 0.16


Relative foam density

Figure 3-9. Surface area density as function of relative foam density predicted by analytical
model, comparing with ERG's data of aluminum foams, a=2mm, 1mm, and 0.5mm,
respectively, for lOppi, 20ppi, and 40ppi foams.


0
E


5 0

4


3
c2
C)


0.14


5000

4500

4000

3500

3000

2500

2000

1500

1000


oc '' ~6
.---Cr
.---O
,~0~












4000


3500

3000

2500


2000


1500

1000

500


5000


10000


15000


20000


25000


Re

Figure 3-10. Nusselt number prediction made by analytical model compared with Calmidi and
Mahajan's [22] experimental data for 5 ppi aluminum foam


4000


3500

3000

2500


2000


1500

1000

500


5000


10000


15000


20000


25000


Re

Figure 3-11. Nusselt number prediction made by analytical model compared with Calmidi and
Mahajan's [22] experimental data for 10 ppi and 20 ppi aluminum foams


- 5ppi(Model)
O 5ppi(Calmidi and Mahajan [23])





0 0 Q]

5..


10ppi(Model)
----- 20ppi(Model)

O 10ppi(Calmidi and Mahajan [23])
A 20ppi(Calmidi and Mahajan [23])


n.A.D...^.












4000


3500

3000

2500

S2000

1500

1000

500

0


5000


10000


15000


20000


25000


Re

Figure 3-12. Nusselt number prediction made by analytical model compared with Calmidi and
Mahajan's [22] experimental data for 5 ppi and 40 ppi low porosity aluminum foam


3000


2500


2000


S1500


S1000
'S


0 10000 20000 30000 40000 50000
Re


60000


Figure 3-13. Heat transfer coefficient predicted by analytical model
[17] experimental data for 10ppi copper foam


compared with Chung et al.


- 5ppi(Model)
----- 40ppi(Model)
O 5ppi(Calmidi and Mahajan [23])
A 40ppi(Calmidi and Mahajan [23])




S -- - - - -

LO*









CHAPTER 4
CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS

4.1 Introduction to Single Cell Model

Open-cell foams have been investigated by many researchers, both experimentally and

numerically. In chapter 3, the analytical heat transfer model deals with the whole foamed

cooling channel, and uses volume-averaged, semi-empirical equations. That is a macroscopic

approach, which neglects small-scale details of open-cell foams. With rapid developing

computing power, using a model with more foam's cell details becomes feasible in

computational fluid dynamics. Although the computer stations are still not powerful enough to

simulate the whole foam inserted channel at this stage, efforts can be made to investigate a single

cell in open-cell foams due to their property of repeated cell structure. That is the microscopic

approach.

Using microscopic approach to simulate pressure drop in open-cell foams takes advantage

of the repeated cell structure of foams and also the properties of flow through porous media.

For a specific type of foam, in which the porosity, pore per inch, and other material properties

are fixed, the pressure drop induced by the foam is only function of velocity of flow. And the

velocity profile in open-cell foam is almost unified, because the multi-filament in foam greatly

increases the intensity of turbulence in flow which flattens out the velocity gradient and makes

the boundary layer very thin (Figure 4-1). Thus, because of the unit cell structure and nearly

unified velocity in open-cell foam, the pressure drop evaluation process can be simplified

without modeling the whole foam inserted channel.

A strategy has been developed to focus on two typical cells as illustrated in Figure 4-1.

The first type is named interior cell, which is located relatively far away from the wall and in the

uniform velocity region. Since the velocities in all interior cells are identical and all cells have









the same structure, only one cell is needed to be modeled to evaluate pressure drop contribution

by interior cells. The second type cell is named boundary cell, which is distinguished from

interior cells and used to capture the pressure drop occurring at the wall. The pressure drop

induced by boundary cell is expected to be larger than that of a interior cell because the no-slip

condition at wall and velocity at boundaries has much larger velocity gradient.

To simulate the micro-structure of open-cell foam (typically metal foams), a

sphere-centered tetrakaidecahedron structure is constructed (Figure 4-2A). That structure is

very similar to the real micro-structure of metal foam (Figure 4-2B aluminum foam). A

tetrakaidecahedron is a polyhedron consisting of six quadrilateral faces and eight hexagons. It's

found by Lord Kelvin that the tetrakaidecahedron (Kelvin structure) is optimal structure for

packing cell, which has minimum surface-area to volume ration. Tetrakaidecahedron is seen in

reality when soap foam is observed [18]. The sphere-centered Kelvin cell can mimic the real

metal foam's micro-structure because of the foaming process of metal foam. A common

method used to foam metal such as aluminum is blowing a kind of foaming gas through molten

metal. The gas bubbles generated are free to move around. The liquid metal and gas bubbles

tend to attain an equilibrium state, i.e., a minimum surface energy state [15]. Thus, after the

solidifying process, the optimal tetrakaidecahedron structure is formed by metal and gas bubbles

generate pores which are similar to spheres. So, the sphere-centered tetrakaidecahedron can

represent the real micro-structure of metal foams very well.

In order to generate the sphere-centered Kelvin structure, a tetrakaidecahedron is generated

first by cutting off the six corners of a regular octahedron. Then build a sphere at the center of

the tetrakaidecahedron and subtracting the sphere from it yields the sphere-center Kelvin

structure. Figure 4-3 shows the process schematic.









As stated before, two types of cells are needed for the pressure drop simulation, interior

cell and boundary cell. Two computational models have been created for the two cells,

respectively. The first represents a typical interior cell and is termed the "2D-periodic" model

because periodic or symmetric boundary conditions are applied in 2 directions (Y, Z directions)

except in the stream-wise direction (X direction). A diagram of this is shown in Figure 4-4.

The second model treats the cell that is attached to the wall and is termed the "iD-periodic"

model. Here, periodicity is applied in only one direction (Y direction). In the other direction

one boundary was set as a wall and the remaining boundary as a symmetry plane. This is shown

in Figure 4-5.

The coolant used for the pressure drop simulation is air, which is assumed to be ideal gas

with constant density and viscosity. Energy equation is not considered at this stage which

means the temperature is constant. The air comes into the inlet of the channel and goes out

through the outlet (Figure 4-4, 4-5). The inlet was set as velocity inlet boundary, and different

inlet velocities were tested. The outlet was set as pressure outlet boundary having the

atmosphere pressure. No-slip conditions were imposed at the wall and cell surfaces.

4.2 Mesh Generation and Grid Independent Study

The Kelvin structure and channel models were created and meshed by GAMBIT, the

preprocessing meshing generation software. The whole channel was divided into three parts,

the inlet region, the outlet region, and the cell (central) region due to their different geometry

properties. The cell region in the middle was meshed using TGrid in GAMBIT, which

generated tetrahedral elements that can fit into the complex structure of Kelvin's cell. The inlet

and outlet regions were meshed by Cooper method in GAMBIT. Because flow at those regions

is less complicated than in the cell region, much less elements were generated at inlet and outlet

regions to save computing time. Figure 4-6 shows the meshed Kelvin cell. Figure 4-7









provides the mesh details at cell's filaments. Figure 4-8 presents that fine mesh is used at the

cell region and relatively coarse mesh is used at the inlet and outlet regions. The cell size is

about 2.54mm*2.54mm*2.54mm, which is about the cell size of a 10ppi foam made by ERG.

And the sphere centered in the cell has a diameter of 2.61mm. The porosity of the cell is thus

about 97.4%.

To examine the dependence of solution on meshes, three different meshes were generated

with different fineness. The coarse mesh consists of 451383 tetrahedral cells and 127010 nodes.

That model was then refined by the medium mesh, which consists of 708955 cells and 183056

nodes. The most delicate model was further refined to 1187729 cells and 335766 nodes, which

is named the fine mesh in this study. All the three different fineness models have the same cell

size, porosity, and channel geometries. The mesh independent study was done for a 2-D

periodic model in which inlet velocity is 4m/s and cell size is 10ppi. Figure 4-9 shows the

average x velocity profiles along the flow direction (x direction) of the three meshes. From the

figure there are no apparent differences among the three meshes with different number of

elements. Figure 4-10 provides comparison of simulation results made by coarse, medium, and

fine meshes. The differences among them are visible although slight. Pressure drop is

calculated from the following equation,

Ap p2 p
Ap p p (4-1)
a x2 X,

where p represents pressure, a is cell size, and x represent the x coordinate in flow direction.

The pressure drops simulated from those three models are shown in Table 4-1. The

relative error between coarse mesh and fine mesh is 3.5%, and relative error between medium

mesh and fine mesh is only 0.6%. Thus, the author thinks the coarse mesh is fine enough to









capture the pressure drop in foams and the coarse mesh was chosen to perform all the following

simulations.

Some more statements can be made on Figure 4-9 and 4-10. There are three regions

where the pressure drop is very significant, from Figure 4-10. The three regions are inlet of cell,

center of cell and outlet of cell. That agrees with the velocity profile in Figure 4-9, in the sense

that the regions having larger velocities induce more pressure drop. The reason is that potential

energy from pressure is transferred into kinetic energy.

4.3 Simulation Results and Verification

Simulations were performed using coarse mesh (Section 4.2). The cell size is set to be

about 2.54mm which is 10ppi and its porosity is about 97%.

Figure 4-11 shows the velocity magnitudes contours of several chosen planes in a case

with inlet velocity of 4m/s. There are three planes, the first one is at about y= -0.8mm, the

horizontal one is at the center of cell and the last one is at the left side of the channel. Figure

4-1 1(A) is a 3-D view of the three planes' contour, and (B)-(D) represents the three planes

respectively. The velocities between ligaments are relatively high and wakes can be found at

ligaments, which is evident especially in Figure 4-11(B). Figure 4-12 provides static pressure

contours of the same three planes. High pressure can be found where the flow encounters with

the ligaments (Figure 4-12(B), (C)).

More data were obtained for 2-D periodic and 1-D periodic models for several inlet

velocities to get pressure drop profiles for interior cells and wall cells. Experimental data from

Leong and Jin [11] were chosen to compare with the simulation data. The comparison was

shown in Figure 4-13 and the pressure drop was plotted as function of inlet velocity. Both

pressure drop profiles for interior cell and wall cell were compared with experiments and very









nice agreement was obtained. It can be found that the wall cell induces a little more pressure

drop because the no-slip condition of wall also contributes to the pressure drop.

It can be concluded that the Kelvin structure unit cell can capture the important

phenomenon of pressure drop occurring in metal cells and can be used to predict foam's pressure

drop.

Table 4-1. Comparison of different meshes' results
Mesh Pressure Drop Relative Error
Coarse 5.26 Pa/mm 3.5%
Medium 5.11 Pa/mm 0.6%
Fine 5.08 Pa/mm



Boundary layer







Wall =-Middle of channel





A Typical / Fnw direction
Boundary Cell
S A Typical
Interior Cell




Figure 4-1. Schematic of boundary cell and interior cell in open-cell foam


























Figure 4-2. Comparison of single cell model and real foam structure. A) single cell model used in
this study. B) SEM photo of aluminum foam.













Figure 4-3. Geometry creation of a single cell











Outlet


Inlet



S.x Periodic Directions Y, Z

Figure 4-4. 2-D periodic model



Symmetric Plane


Outlet


Inlet -


Periodic Dire
Figure 4-5. -D per ion
Figure 4-5. 1-D periodic model


Wall


Jv"

































Figure 4-6. Mesh of a single cell model (coarse grids)






.. -.:- .i:i-.:..j-.




-l /r
. .'



I .. .'
--- '. ----- M'












Figure 4-7. Details of the meshes on filaments (medium grids)




47
~ ~ ~ ~ ~ ~~~~~ ,: .,.. -, -,' ., TTTr
:t~~~~~~f::,-.,,~~~~~~~~~~ ",,r,,. .,.- _', "; ,: l
:', it I:, : : .. ,' ---., _. ,.., :, : :. '. -- '. : J :; .




Fiur 4-7.3' Deal fte ehso ilaet (mdu gis

.""47













4.4- _____ i---------4------
-..
-, .- - --- -- -C -- .-s -
































4 35
--- : ..- M m M h-















S3 -A-Fine Mesh-
.............




















4. -



4.1
r .15 4.
-', '.',,, -----


.. .. _: -- -_. .. -----_-- .













4.35








4.25







3.95 ''
3.95 ---------------------------------------------
-1.27 -0.77 -0.27 0.23 0.73 1.23
x position (mm)

Figure 4-9. Velocity profile along flow direction through the cell











12
1- Coarse Mesh
10
a-- Medium Mesh
8 A------ Fine Mesh




4

2

0

-2

-4
i4 ---------------------------

-1.27 -0.77 -0.27 0.23 0.73 1.23
x position (mm)

Figure 4-10. Pressure distribution along flow direction through the cell




















Y-X


5.9e00 (A)
5.58e+00
5.27e+00
4.96e+00
4.65e+00
4.34e+00
4.03e+00
3.72e+00
3.41e+00
3.10e+00 A
2.79e+00
2.48e+00
2.17e+00
1.86e*00
1.55e+00
1.24e+00
9.30e-01
.2) -01,.
3.10e-01 (
O.OOe+00












(C)













(D)


Figure 4-11. Velocity contours in three planes around the cell. A) 3-D view. B) Plane at y=
-0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the channel,
y=1.27mm



















Y-X


2,47+01 (A)
2. 24e+01
2.01e+01
1.77e+01
1.548+01
1.318+01
1.08e+01
6.12e+00
3.S0e+00
1.,47a+00 '
.8,49-01 Z "
317e+00 ,, -'
-S.50B+O0 ,,'-, .,,
-7,828+00
-1.01e+01
1 .25e+01
S 1.25f0+01
-I48eo+ 0 (B)
-1.71e+01
-1.94e+01

"- .... -.




)-X "













'I



(D)

Figure 4-12. Static pressure contours in three planes around the cell. A) 3-D view. B) Plane
at y= -0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the
channel, y=1.27mm















25
-- Interior Cell

20 -A-Wall Cell
2 20


C 15


8 10


5


0
0 2 4 6 8 10 12
Velocity (m/s)

Figure 4-13. Pressure drop versus inlet velocity and comparison with experimental data









CHAPTER 5
FEASIBILITY STUDY OF FOAMED COOLING CHANNELS FOR ROCKET THRUST
CHAMBER

To investigate the feasibility of the foamed cooling channel for rocket chamber, high Re

number cases were studied for both the open channel and foamed channel. Empirical equations

were used for the open channel. The analytical model derived in Chapter 3 was applied to

foamed channel to predict its heat transfer rate, and the CFD simulation method in Chapter 4 and

also some data for hydrogen from Chung et al. [17] was used to get the pressure drop prediction

correlation of rocket condition pressure drop. The author used the parameters of 10PPI foam to

perform all the calculation in analytical heat transfer model.

5.1 Feasibility Study and Comparison with Open Cooling Channel

The average velocity in open channel is set to be up to 250m/s (Re=106), which is under

rocket condition. Due to the high pressure drop gradient, the velocity in foamed channel can

not reach that high, but has about 1/5 of that. In order to keep the same mass flow rate, larger

cross section area is used. The idea is summarized in Table 5-1.

The coolant mass flow rates and pressure drops are set to be equal, which make sure that

the amount of coolant needed and the work needed to push the coolant are the same. Under that

requirement, if a higher heat transfer is obtained, the application of foamed channel will be

meaningful. Figure 5-1 shows the scheme. Table 5-2 lists parameters of open channel of

foam channel used in this comparison

For open channel, the following correlations suggested by Incropera and DeWitt [21] are

used.

Pressure drop:

-(dp / dx)H
f= (5-1)
pu /2









f is found to be constant 0.05 at high Re numbers for commercial metals.

Heat transfer:

NuD = 0.021Re8 Pro 5 (5-2)

hH
where Re= Hu/, Nu=
k

Comparisons of heat transfer between an open channel and foam-filled channels are shown

in Figure 5-2, Table 5-3 and Table 5-4. Figure 5-2 shows foamed channel has significant heat

transfer enhancement over open channel, when they have the same mass flow rate and pressure

drop.

Table 5-3 compares foamed channel's heat transfer coefficient with that of an open

channel at equal pressure drops. For instance, when the pressure drop is 841 kPa/m for both of

channels, the heat transfer coefficient increases from 18567 W/m2K to 36951 W/m2K, that's an

increase of 99%. Similar increases are found for other pressure drops. The enhancement gets

smaller with increasing pressure drop. That is due to the rapidly increasing heat transfer

coefficient of open channel. But the enhancement is still significant at Re=106.

Table 5-4 shows the velocities in the two types of channels with the same pressure drop.

To keep the mass flow rate be equal in two channels, the foamed channel area has to be

increased to compensate the low velocity. The results indicate that the foam channel should be

5.3 times of the open channel. If we keep the same base width, the height of the foam filled

channel therefore should be extended according to that ratio. From the data shown in the table,

the velocities ratio of open and foamed channels is approximately 5.3, and getting slightly

smaller with larger pressure drop.

A CFD simulation of open channel under rocket conditions has been accomplished by

Chung et al. [17]. A head to head comparison of open channel and 10PPI foamed channel









under that rocket condition is performed to show the feasibility of applying foam channel to the

rocket chamber. The details are shown in Table 5-5.

In order to keep the same pressure drop and mass flow rate, the velocity ratio in open and

foamed channels is kept 4:1, and the height of foamed channel is thus 4 times of open channel.

The results of CFD simulation of open channel and analytical prediction of foamed channel are

shown in Table 5-6. It's shown that the foamed channel's heat transfer coefficient will be 49581

W/m2K, which is more than 110% enhancement, compared to open channel's 23464 W/m2K.

That means, under the same pressure drop and mass flow rate, the foamed channel has a

significant capability to enhance the heat transfer efficient of the rocket's cooling chamber.

Actually, if higher PPI foams (like 20 or 40 PPI) are used, more enhancement of heat transfer is

expected, although it's not shown in this study due to the lack of data of higher PPI foams.

5.2 Uncertainty Analysis

To analyze the certainty of 110% enhancement predicted by analytical model and

simulation, an error analysis is performed in this section. The prediction error comes from both

the heat transfer model and the CFD pressure drop simulation. So the error of the prediction is

some combination of error from the analytical model and error made by the CFD simulation.

5.2.1 Heat Transfer Model

From the comparison of model and experimental data in Section 3.4.2, the uncertainty of

the heat transfer coefficient h prediction made by analytical model is calculated from Figure 3-13.

Predictions of h made by analytical model were compared with experimental data. The relative

error is about 30%, with a confidence of 90%.

h 30% (10tol) (5-3)
h









5.2.2 Pressure Drop Simulation

From the comparison of simulation and experimental data in Section 4.3, the uncertainty of

the pressure drop p prediction made by CFD simulation is calculated from Figure 4-13. The

relative error is about 30%, with a confidence of 90%.


p
=10% (10 to 1) (5-4)
P

5.2.3 Rocket Condition Prediction

Because the pressure drop is kept the same to find the coolant velocity in foamed channel,

under rocket conditions, the uncertainty of velocity can be evaluated.

Because p ~ v2, so the uncertainty of velocity can be calculated from


v = lo= 3.3% (10to 1) (5-5)
v

Since Re ~ v, so

A Re
-- =3.3% (10tol) (5-6)
Re

From the heat transfer model uncertainty analysis and Figure 3-13, hequa A Re03001

From regression analysis, A=1088, and Re=304000. The uncertainty of A can be treated as the

same as 30% from Eq. (5-3). So the uncertainty of A is AA=0.3*1088=326.4. And from Eq.

(5-6), the relative uncertainty of Re is 3.3%, with confidence of 10%, so

ARe=0.033*304000=10032. So, the uncertainty of equivalent heat transfer coefficient of

rocket can be calculated, after considering the uncertainty of pressure drop simulation, as


h h( equal AA)2 + equall R)2
equal (A A Re)
SA aRe


(Re 03001 A4)2 +(0.3001 A 9 Re)2
ReV 6999









14434W/m2K


So, the uncertainty of the equivalent heat transfer coefficient of rocket's foamed cooling is,

Ah 14434
equal = 29.1% (5-7)
ha 49581

The heat transfer coefficient of metal foamed channel can be represented as,

hqua = 49581W/m2K 29.1% (5-8)

If we take a close look of uncertainty equation of equal,


h qual A)2 + (q A Re)2
"-a'ARe
Equal OA 8Re
h^q A Re0 3001

(Re03001)2 + (0.3001 0 6999 ARe)2
Re
A Re0 3001

S(AA)2 +(0.3001 Re)2
A Re

AA
From the above equation, the uncertainty of equal comes mainly from which is 30%,
A

ARe
compared to -= 3.3%. Thus, the need for improve the precision of heat transfer analytical
Re

model is critical for this process.

Table 5-1. Micro open channel and foam filled channel model requirements
Open Channel Foamed Channel

Channel width = 2 mm Channel width = 2 mm

Channel height = 4 mm Channel height = x mm

Pressure drop = A Pressure drop = A

Coolant flow rate = B Coolant flow rate = B

Heat transfer = Q1 Heat transfer = Q2>Q1









Table 5-2. Head-to-head comparison of open channel and foamed channel
Working Inlet Temperature Inlet Channel Geometry
fluid Temperature of heated base Velocity Length Width Height

Open 10-250
nnel H2 100K(180R) 800K(1440R) 10250 m 2mm 4mm
Channel m/s
Foamed 21-22m
ame H2 100K(180R) 800K(1440R) 2-48m/s Im 2mm
Channel m


Table 5-3. Heat transfer enhancement of foamed channel over open channel
Open Channel
Foamed Channel Heat Heat Transfer
Pressure Drop (kPa/m) Heat Trans. Coef. 2
(W/m2K) Trans. Coef. (W/mK) Enhancement Percentage
841 18567 36951 99%
987 19795 37962 92%
1145 21004 38929 85%
1314 22196 39858 80%
1495 23372 40753 74%
1688 24534 41617 70%
1892 25682 42453 65%
2108 26817 43265 61%
2336 27940 44053 58%
2576 29053 44820 54%
2827 30154 45568 51%
3090 31246 46298 48%


Table 5-4. Velocity ratio at equal pressure drop
Pressure Drop Velocity in Velocity in
Pressure Drop
(kpa) Open Channel Foamed Ratio
pa/m) (m/s) Channel (m/s)


374
584
841
1145
1495
1892
2336
2576
2827
3090
3364
3650


80 14.9
100 18.7
120 22.6
140 26.4
160 30.2
180 34.1
200 37.9
210 39.8
220 41.8
230 43.7
240 45.6
250 47.6


5.36
5.34
5.32
5.31
5.30
5.29
5.28
5.27
5.27
5.26
5.26
5.26









Table 5-5. Head-to-head comparison under rocket condition
Working Inlet Temperature Inlet Channel Geometry
fluid Temperature of heated base Velocity Length Width Height
Open
annel H2 100K(180R) 800K(1440R) 207m/s 508mm 2mm 4mm
Channel
Foamed
hanel H2 100K(180R) 800K(1440R) 52m/s 508mm 2mm 16mm
Channel


Table 5-6. Comparison of open and foamed channels' performance
Pressure drop Mass flow rate Heat Trans. Coef.
Open Channel 0.0155kg/s 3 W/m2
4303 kPa/m 6 23464 W/m2K
(CFD results) (Re= 1*106)
Foamed Channel
(Analytical 4303 kPa/m 0.0155kg/s 49581 W/m2K
predictions)


< Current micro channel design >






2mm


4mm


Figure 5-1. Notional design strategy for foam-filled channels


< metal foam >


2mm


Height: x











60000


S50000

40000

u 30000

20000

10000 --- Open Channel
S-- Foamed Channel
0 1 11
0 500 1000 1500 2000 2500 3000 3500 4000

Pressure Drop (Kpa/m)

Figure 5-2. Comparison of heat transfer coef. vs. pressure drop of open and foamed channels









CHAPTER 6
CONCLUSIONS

An analytical heat transfer model and a CFD based pressure drop simulation method for

open-cell foams have been investigated and the feasibility of using foamed cooling channel for

rocket is studied.

The analytical heat transfer model has provided favorable agreement with some

experimental data and it can provide valuable prediction on heat transfer of foam filled cooling

channels. The remaining defect of that model is that it doesn't have a universal form. That is,

there have to be different equations for different Reynolds number ranges, as stated in Chapter 3.

This author believes that the reason is due to the heat transfer coefficient correlations the model

uses. The analytical model uses correlations of flow over bank of tubes and flow over single

cylinders, which don't have inter-cylinder or inter-tube effects. However, the real open-cell

foam's ligaments are connected to each other which may induce significant variation of

temperature distribution on ligaments and heat transfer enhancement over that of flow over tubes.

That's the reason why the model tends to underestimates the heat transfer coefficient when the

Reynolds number increases. The author believes that more experiments on different kinds of

foams and correlations are needed before a universal heat transfer model can be obtained and

currently the heat transfer model in this study can be useful on evaluation of some kind of

open-cell foams application. Also, an optimum design of foam's porosity, pore per inch and

ligament diameter to get maximum heat transfer rate can be investigated by the analytical heat

transfer model.

The CFD simulation of a single cell in metal foam is a feasible method to evaluate pressure

drop in foams. The Kelvin structure is very similar to the real micro-structure of metal foams

which can capture the most important flow phenomenon in metal foams. A remaining problem









with that model is the single cell model tends to overestimate the pressure drop a little bit.

That's because the pressure drop when the flow enters the cell is significant for a single cell but

is negligible for whole foams which contain thousands of cells in a line. That is a problem

caused by under-developed flow. A solution for it is to use periodic boundary also in the flow

direction. In future, this author would like to do some simulations on a single cell with 3

dimensional periodic boundaries and also couples the model with energy equation, in the hope of

solving the heat transfer and pressure drop in one model.









LIST OF REFERENCES


[1] Turner, M.J.L., 2000, Rocket and Spacecraft Propulsion, Praxis Publishing, Chichester, UK.
[2] Sutton, G.P., and Biblarz, O., 2001, Rocket Propulsion Elements, 7th ed., Wiley, New York.
[3] Carlos, H. M., Fernando, L., Antonio, F. C. da Silva and Jose, N. H., 2004, "Numerical
Solutions of Flows in Rocket Engines with Regenerative Cooling," Numer. Heat Transfer A,
45, pp. 699-717.
[4] Koh, J.C.Y. and Colony, R., 1974, "Analysis of Cooling Effectiveness for Porous Material in
a Coolant Passage," J. Heat Transfer, 96, pp. 324-330.
[5] Koh, J.C.Y. and Stevens, R.L., 1975, "Enhancement of Cooling Effectiveness by Porous
Materials in Coolant Passage," J. Heat Transfer, 97, pp. 309-311.
[6] Hunt, M.L. and Tien, C.L., 1988, "Effects of Thermal Dispersion on Forced Convection
Fibrous Media," Int. J. Heat Mass Transfer, 31, pp. 301-309.
[7] Maiorov, V.A, Polyaev, V.M., Vasilev, L.L. and Kiselev, A.I., 1984, "Intensification of
Convective Heat Exchange in Channels with a Porous High-Thermal-Conductivity Filler.
Heat Exchange with Local Thermal Equilibrium Inside the Permeable Matrix," J.
Engineering Physics Thermophysics, 47, pp. 748-757.
[8] Bartlett, R.F. and Viskanta, R., 1996, "Enhancement of Forced Convection in an
Asymmetrically Heated Duct Filled with High Thermal Conductivity Porous Media," J.
Enhanced Heat Transfer, 6, pp. 1-9.
[9] Kuzay, T.M., Collins and Koons, J., 1999, "Boiling Liquid Nitrogen Heat Transfer in
Channels with Porous Copper Inserts," Int. J. Heat Mass Transfer, 42, pp. 1189-1204.
[10] Boomsma, K., Poulikakos, D. and Zwick, F., 2003, "Metal Foams as Compact High
Performance Heat Exchangers," Mechanics of Materials, 35, pp. 1161-1176.
[11] Leong, K.C. and Jin, L.W., 2006, "Effect of Oscillatory Frequency on Heat Transfer in
Metal Foam Heat Sink of Various Pore Densities," Int. J. Heat Mass Transfer, 49, pp.
671-681.
[12] Kim, S.Y., Kang, B.H. and Kim, J., 2001, "Forced Convection from Aluminum Foam
Materials in an Asymmetrically Heated Channel," Int. J. Heat Mass Transfer, 44, pp.
1451-1454.
[13] Yuan, K., Avenall, J.N. Chung, J.N., Carroll, B.F., and Jones, G.W., 2005, "Enhancement of
Thrust Chamber Cooling with Porous Metal Inserts," 41nd AIAA/ASME/SAE/ASEE Joint
Propulsion Conference and Exhibit, Tucson, Arizona.
[14] Lu, T.J., Stone, H.A. and Ashby, M.F., 1998, "Heat Transfer in Open-Cell Metal Foams",
Acat. Mater., 46, pp. 3619-3635.
[15] Krishnan, S., Murthy, J.Y. and Garimella, S.V., 2006, "Direct simulation of Transport in
Open-Cell Metal foam," J. Heat Transfer, 128(8), pp. 793-799
[16] Krishnan, S., Garimella, S.V. and Murthy, J.Y., 2006, "Simulation of Thermal Transport in
Open-Cell Metal Foams: Effect of Periodic Unit Cell Structure," ASME International
Mechanical Engineering Congress and Exposition, Chicago, Illinois.
[17] Chung, J.N., Tully, L. and Kim, J.H., 2006, "Evaluation of Open Cell Foam Heat Transfer
Enhancement for Liquid Rocket Engines," 42nd AIAA/ASME/SAE/ASEE Joint Propulsion
Conference and Exhibit, Sacramento, California.
[18] Boomsma, K., Poulikakos, D., and Ventikos, Y., 2003, "Simulation of Flow through Open
Cell Metal Foams Using an Idealized Periodic Cell Structure," Int. J. Heat Fluid Flow, 24, pp.
825-834.
[19] Mills, N.J., 2005, "The Wet Kelvin Model for Air Flow through Polyurethane Open-Cell









Foams", J. Mater. Sci., 40, pp. 5845-5851
[20] Yu, Q., Thompson, B. E., and Straatman, A. G., 2006, "A Unit-Cube Based Model for Heat
Transfer and Pressure Drop in Porous Carbon Foam," J. Heat Transfer, 128(4), pp. 352-360
[21] Incropera, F. and DeWitt, D., 2003, Fundamentals of Heat and Mass Transfer, Wiley, New
York.
[22] Calmidi, V.V. and Mahajan, R.L., 2000, "Forced Convection in High Porosity Metal
Foams," J. Heat Transfer, 122, pp. 557-565.









BIOGRAPHICAL SKETCH

Mo Bai was born on January 21, 1983, in Liaoning, China. He graduated from Tsinghua

High School, Beijing, China, in 2001. He attended Tsinghua University and received his

Bachelor of Engineering, majoring in hydraulic engineering in the summer of 2005. Since then,

he has been pursuing a Master of Science degree in mechanical engineering while working as a

graduate research/teaching assistant.





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NUMERICAL EVALUATION OF HEAT TRAN SFER AND PRESSURE DROP IN OPEN CELL FOAMS By MO BAI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007

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2007 Mo Bai

PAGE 3

To my parents

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iv ACKNOWLEDGMENTS I express my sincere appreciation to my advi sor, Dr. Jacob N. Chung, for his believing me and providing me the opportunity to work on ma ny interesting and challe nging researches. His invaluable patience, wisdom, and encouragemen t helped me throughout my two years study at the University of Florida. Without his unfai ling support, this work would not have been possible. Drs. William E. Lear, Jr and Bhavani V. Sankar offered valuable suggestions on my research while serving on my supervisory committee. Doctoral candidate Junqiang Wang graciously gave up his time to help me when I had questions. Their suggestions and help have shaped this work considerably. My fellow graduate students, Renqiang Xiong a nd Kun Yuan, have offered invaluable help on my study and research. My friends have gi ven me a memorable time at University of Florida and made my life here enjoyable. Also I would like to thank my parents and extended family, they were always there when I need help and encouragement. Finally, Im grateful to my fiance Wenwen Zhang, for her years of support.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.............................................................................................................iv LIST OF TABLES................................................................................................................. .......vii LIST OF FIGURES................................................................................................................ .....viii NOMENCLATURE................................................................................................................... .....x ABSTRACT....................................................................................................................... ..........xiii CHAPTER 1 INTRODUCTION ROCKET TH RUST CHAMBER COOLING........................................1 1.1 History of the Rocket...................................................................................................... ....1 1.2 Rocket Structure........................................................................................................... ......1 1.3 Rocket Thrust Chamber Cooling........................................................................................2 1.3.1 Regenerative Cooling...............................................................................................3 1.3.2 Challenges on Regenerative Cooling.......................................................................4 2 PREVIOUS WORK ON OPEN-CELL FOAMS.....................................................................9 2.1 Heat Transfer Enhancement...............................................................................................9 2.2 Experiments................................................................................................................ ......10 2.3 CFD Simulation and Numerical Model............................................................................11 2.4 Other Open-Cell Foams....................................................................................................13 2.4.1 Polyurethane Foams...............................................................................................13 2.4.2 Carbon Foams.........................................................................................................13 3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS...................15 3.1 Geometry Simplification for Open-Cell Foam Filled Channels.......................................15 3.2 Mathematical Transport Model and Heat Transfer Equations.........................................16 3.2.1 V-type Struts...........................................................................................................16 3.2.2 H-type Struts...........................................................................................................17 3.2.3 Fluid Temperature Predic tion (Coolant Temperature)...........................................20 3.2.4 Total Heat Transfer.................................................................................................23 3.2.5 Evaluation of Heat Tr ansfer Coefficient................................................................23 3.2.6 Equivalent Heat Transfer Coefficient.....................................................................25 3.3 Investigation of Cylinder Diam eter and Surface Area Density........................................27 3.4 Verification of the Analytical Model with Experimental Data........................................28 3.4.1 Validity of Analytical Prediction (Re=5*103 ~ 2*104)..........................................29 3.4.2 Validity of Analytical Prediction (Re=1*104 ~ 6*104)..........................................30

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vi 4 CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS.............................39 4.1 Introduction to Single Cell Model....................................................................................39 4.2 Mesh Generation and Grid Independent Study................................................................41 4.3 Simulation Results and Verification.................................................................................43 5 FEASIBILITY STUDY OF FOAMED COOLING CHANNELS FOR ROCKET THRUST CHAMBER............................................................................................................53 5.1 Feasibility Study and Comparison with Open Cooling Channel......................................53 5.2 Uncertainty Analysis....................................................................................................... .55 5.2.1 Heat Transfer Model...............................................................................................55 5.2.2 Pressure Drop Simulation.......................................................................................56 5.2.3 Rocket Condition Prediction..................................................................................56 6 CONCLUSIONS....................................................................................................................61 LIST OF REFERENCES............................................................................................................. ..63 BIOGRAPHICAL SKETCH.........................................................................................................65

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vii LIST OF TABLES Table page 3-1 Constants of Equation (3-22).............................................................................................31 3-2 Parameters of experiments from Calmidi and analytical model........................................32 3-3 Foam parameters comparison between e xperiments from Calmidi and analytical model.......................................................................................................................... ........32 4-1 Comparison of different meshes results...........................................................................44 5-1 Micro open channel and foam f illed channel mode l requirements....................................57 5-2 Head-to-head comparison of open channel and foamed channel......................................58 5-3 Heat transfer enhancement of foamed channel over open channel....................................58 5-4 Velocity ratio at equal pressure drop.................................................................................58 5-5 Head-to-head comparison under rocket condition.............................................................59 5-6 Comparison of open and foamed channels performance..................................................59

PAGE 8

viii LIST OF FIGURES Figure page 1-1 Construction of a regenerative cooling tubular thrust chamber...........................................5 1-2 Cutaway of a tubular cooling jacket....................................................................................6 1-3 Typical heat transfer rate intensity distribution for liquid propellant rocket.......................6 1-4 Simplified schematic of regenerative c ooling system of liquid propellant rocket...............7 1-5 Section A-A of Fig. 1-4 and details of cooling channel......................................................7 1-6 Different configurations of the cooling channel in thrust chamber.....................................8 2-1 Photos of aluminum foam..................................................................................................14 -1 Schematic of a single cell in the simplified model............................................................32 -2 Model details............................................................................................................... .......33 3-3 3-D schematic of the model...............................................................................................33 -4 Heat transfer networ k of analytical model.........................................................................34 3-5 Schematic of vertical strut fin model.................................................................................34 3-6 H-strut model.............................................................................................................. .......35 3-7 Model for coolant temperature evaluation.........................................................................35 3-8 Cylinder diameter as function of relative foam density predicted by analytical model, comparing with ERGs data of aluminum foams..............................................................36 3-9 Surface area density as f unction of relative foam density predicted by analytical model, comparing with ERGs data of aluminum foams..................................................36 3-10 Nusselt number prediction made by analyt ical model compared with Calmidi and Mahajans experimental data for 5 ppi aluminum foam....................................................37 3-11 Nusselt number prediction made by analyt ical model compared with Calmidi and Mahajans experimental data for 10 ppi and 20 ppi aluminum foam................................37 3-12 Nusselt number prediction made by analyt ical model compared with Calmidi and Mahajans experimental data for 5 ppi and 40 ppi low porosity aluminum foam.............38 4-1 Schematic of boundary cell and in terior cell in open-cell foam........................................44

PAGE 9

ix 4-2 Comparison of single cell mode l and real foam structure.................................................45 4-3 Geometry creation of a single cell.....................................................................................45 4-4 2-D periodic model......................................................................................................... ...46 4-5 1-D periodic model......................................................................................................... ...46 4-6 Mesh of a single cell model (coarse grids)........................................................................47 4-7 Details of the meshes on filaments (medium grids)...........................................................47 4-8 Grids distribution......................................................................................................... ......48 4-9 Velocity profile along flow direction through the cell.......................................................48 4-10 Pressure distribution along fl ow direction through the cell...............................................49 4-11 Velocity contours in th ree planes around the cell..............................................................50 4-12 Static pressure contours in three planes around the cell....................................................51 4-13 Pressure drop versus inlet velocity and comparison with experimental data....................52 5-1 Notional design strategy for foam-filled channels.............................................................59 5-2 Comparison of heat transfer coef. vs. pressure drop of open and foamed channels..........60

PAGE 10

x NOMENCLATURE a Cell size A Surface area of foam Ac Area of cylinders cross section Aw Area of heated wall C1 Constant related to the geometry of ch annel, can be looked up from tables Cp Specific heat of coolant d: Diameter of the cylinder. id Inner diameter of test section od Outer diameter of test section f Friction factor H Height of cooling channel h1: Heat transfer coefficient between ve rtical cylinder and the cooling fluid. h2: Heat transfer coefficient between hor izontal cylinder and the cooling fluid. heuqal Equivalent heat transfer coefficient of foam filled cooling channel hw: Heat transfer coefficient between bare wall and the cooling fluid. kf: Thermal conductivity of the coolant. ks: Thermal conductivity of the cylinder. l Some constant defined in Eq.(3-18) L Length of cooling channel m Constant related to the geometry of ch annel, can be looked up from tables m1 Constant calculated from h1, ks, and d M1 Constant calculated from h1, ks, and d m2 Constant calculated from h2, ks, and d M2 Constant calculated from h2, ks, and d

PAGE 11

xi m Mass flow rate of coolant Nh Number of horizontal cylinders per unit width Nu Nusselt number Nv Number of vertical cy linders per unit width p Pressure Pr Prandtl number Q Total heat transfer rate to coolant qh Heat transfer rate from a single ho rizontal (H-cylinde r) to the coolant qv Heat transfer rate from a single vertical (V-cylinder) to the coolant qw Heat transfer rate from bare heated wall to the coolant Re Reynolds number T0 Inlet temperature of coolant T1 Temperature of vertical (V-type) cylinders at x T2 Temperature of vertical (V-type) cylinders at x+a ainletT Inlet air temperature aoutletT Outlet air temperature bT Bulk fluid temperature Tc Coolant temperature Th Temperature of horizontal (H-type) cylinder Ts Temperature of vertical (V-type) cylinder Tw Constant temperature of heated wall V Average inlet velocity of c oolant or volume of the foam Vmax Maximum velocity of the coolant x X coordinate or direction y Y coordinate or direction

PAGE 12

xii z Z coordinate or direction Greek Symbols A Surface area density Non-dimensional variable defined by Ts, Tc, and Tw c Non-dimensional variable defined by Ts, Tc, Tw, and T0 Relative foam density Density of foam s Density of solid Ratio of bare wall surface area to the total wall surface area vf Kinematic viscosity of coolant Subscripts 1 Vertical (V-type) cylinder 2 Horizontal (H-type) cylinder c Coolant h Horizontal (H-type) cylinder v Vertical (V-type) cylinder w Wall

PAGE 13

xiii Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NUMERICAL EVALUATION OF HEAT TRAN SFER AND PRESSURE DROP IN OPEN CELL FOAMS By Mo Bai December 2007 Chair: Jacob N. Chung Major: Mechanical Engineering As society pursues the space travel, advan ced propulsion for the next generation of spacecraft will be needed. These new propulsion systems will require higher performance and increased durability, despite current limitations on materials. A break-through technology is needed in the thrust chamber. One of the most challenging problems is to cool the hot chamber wall more without creating addi tional pressure drops in the coolant passage. A promising method is using open cell foam to enhance the heat transfer rate from chamber wall to coolant. However, the pressure drop induced by foams is relatively large and t hus becomes a critical issue. The focus of this thesis is the evaluation of heat transfer and pressure drop of open cell foams. A simplified analytical model has been develope d to evaluate the heat transfer capability of the foamed channel, which is based on a diamond-shaped unit cell model. The predicted heat transfer results by the analytical model ha ve been compared with experimental data of different Reynolds numbers from other researchers and favorable agreements have been obtained. For the evaluation of pressure dr op in open-cell metal foams, dire ct numerical simulation models of the foam heat exchanger have been built using GAMBIT/FLUENT. The model is based on a structure of sphere-centered open cell tetrakaidecahedron. This model is very similar to the actual metal foams microstructure of thin li gaments that form a network of interconnected

PAGE 14

xiv open-cells. Grid independence of solution is in vestigated and simulatio n results are further compared with experiments. Finally, the feasib ility of applying foam filled cooling channel on rocket thrust chamber is investigated.

PAGE 15

1 CHAPTER 1 INTRODUCTION ROCKET THRUST CHAMBER COOLING 1.1 History of the Rocket The history of rocketry is at least more than 700 years. The first rocket is said to be invented by a Chinese scientist named Feng Ji shen in 970 A.D., who used bamboo tubes and black powder to generate great thrust power by expanding hot exhaust gas. That is the prototype of todays firecracker and fireworks. The use of black powder to propel projectiles was a precursor to the development of the first solid rocket. The principal idea of obtaining thrust by reaction is thought to be founded by Hero of Alexandr ia in 67 A.D. He invented many mechanisms which utilize the reaction principle that is thought to be the theory basis for rockets. Rocket technologies first become known to Europeans by Genghis Khan when the Mongols conquered Russia, Eastern and Central Europe. The Mongols got the technologies from Chinese and they also employed Chinese rock etry experts. The first serious scientific book on space travel is published by Konstantin Tsiolkovsky, a Russian high school mathematics teacher, in 1903.[1] In 1920, Robert Goddard published A Method of Reaching Extreme Altitudes, the first serious work on using rock ets in space travel after Tsiolkovsky. Goddard was a professor at Clarkson University in Massac husetts. He attached a supersonic nozzle to a liquid rockets combustion chamber, which became the first modern rocket. Hot gas in the combustion chamber is expanded through the nozzl e, and turns into cooler, hypersonic, highly directed jet of gas, which greatly improves the thrust and efficiency. Goddard had more than 214 patents on rockets that were later bought by United States. 1.2 Rocket Structure Most current rockets are chemically powered ro ckets, an internal combustion engines that obtain thrust from expanding hot exhaust gas. From propellants point of view, there are gas

PAGE 16

2 propellant, solid propellant, liquid propellant, and even a mixture of both solid and liquid propellant. Typically, a rocket engine structure consists of in jectors, combustion chamber and the converging diverging nozzle, which can be se en in Figure 1-1. The injectors are used to introduce fuel and oxidizer to co mbustion chamber. The combustion chamber is where the fuel and oxidizer are mixed and burned. The nozzle is us ually designed as an in tegral part together with combustion chamber, its purpo se is to regulate and direct exhaust gas to reach a supersonic speed and get maximized thrust. In this study, th e word thrust chamber is used to present the integral structure of rocket combustion chamber and nozzle. The thrust chamber is the key component of a ro cket engine, here the pr opellant is injected, vaporized, mixed, and burned to transform into hot exhaust gas. The combustion reaction can fairly reach the temperature up to 3500K, which is much higher than the melting point of the material used in thrust chamber. Thus, its cri tical to make sure the th rust chamber wont melt, vaporize, or combust. Some rockets chamber us e ablative material or high temperature material, such as carbon based materials graphite, di amond, and carbon nanotubes. Other rocket chambers use conventional materials like alumin um, steel, or copper alloys. These kinds of rocket then need a cooling system to prevent the chamber wall become to hot. 1.3 Rocket Thrust Chamber Cooling Generally speaking, there are tw o major methods of cooling rocket thrust chamber today. The first one is steady state method, which is th e heat transfer rate through thrust wall and temperature on the wall are constant, in other wo rds, theres a thermal equilibrium. The steady state method includes regenerative cooling and radiation cooling. The regenerative cooling is done by attaching a cooling jacket onto the thru st wall and circulating one of the propellants through the cooling channel befo re it is injected into chamber for combustion. Usually, regenerative cooling is used fo r bipropellant rockets having medium to large thrust, and it is

PAGE 17

3 effective for thrust chamber having high pressure and high heat transfer rate. The radiation cooling is using an extension attach ed to the thrust nozzle exit to get extra radiation heat transfer to the ambient space. Radiation cooling is prim arily used in monopropellant rocket, which have relatively low pressure and require s moderate heat transfer rate. The second method to cool rocket thrust chambe r is unsteady state method or transient heat transfer method. For this method, there is no th ermal equilibrium and the temperature on thrust wall continues to increase. The total heat tr ansfer absorbing capacity is determined by the hardware. The rocket engine has to be stopped before the temperature reaches the hardwares critical point. Ablative materials are commonl y used in unsteady state cooling method and solid propellant rocket, for which chamber pressure s is lower and heat transfer rate is also low [2]. 1.3.1 Regenerative Cooling This study is mainly about the steady stat e method using regenerative cooling. For regenerative cooling, a cooling j acket is constructed in the thru st wall to allow the coolant to circulate in the cooling channels. Usually, one of the propellants (commonly the fuel) is used as the coolant. A typical tubular cooling jack et is shown in Figure 1-2. The fuel enters through the inlets of every other tube, flow to the nozzle exit, and then ente rs the alternate tubes, flow back to the injectors for combustion. There ar e also other rockets c oolant inlets are at the nozzle throat area, coolant flows up and down in the nozzle exit region and flows up in the chamber region. This design is considering heat tran sfer intensity of the ro cket thrust chamber. Because the heat transfer rate p eak is often at the nozzle throat area, which is shown in Figure 1-3, letting the coolest coolant en tering at throat area can greatly enhance the heat transfer efficiency. Another method to enhance the heat tr ansfer rate at throat area is increase coolant flow velocities at that area. From Figure 1-2, cross-section area at section B is the smallest

PAGE 18

4 which can generate the largest coolant velocities there. Figure 1-4 and 1-5 show schematics of a liquid propellant rockets thrust chamber and details of cooling channels [3]. Regenerative cooling method has many merits in the sense of heat transfer efficiency and structure optimization. First, using fuel as the co olant greatly enhances heat transfer efficiency. Because for liquid propellant rocket, the fuel is cryogenic, large temperature difference between coolant and combustion gas can make great heat transfer rate. In addition, after flow through the cooling channel, the fuel has higher temp erature and become ready for combustion. Second, the tubular cooling jacket reduces weight of the rocket thrust chamber and also the total weight of rocket, thus greatly increases efficiency. Th ird, the cooling jacket structure transform thick thrust wall into thin walls of cooling ch annels, which can reduce thermal stresses. 1.3.2 Challenges on Regenerative Cooling Today, the needs for longer and faster space travel require rockets with more powerful thrust and also bring challenging requirements to the cooling system. Much higher heat transfer rate is needed for next generation rockets. Even with new advances in high-temperature and high conductivity materials, thrust increases for large liquid propell ant rocket engines are limited by the cooling capacity of the coo ling jacket. Cooling limits have been extended with the use of film cooling, injector biasing, and transpiration cooling. However, these methods are costly to engine performance since they require that so me of the fuel pass through the thrust chamber throat without contributing to thrust. Currently, the vast majority of regenerative cooling rocket engines use either tube bundles or milled rectangular passages as heat excha ngers. Several improvements based on the tubular cooling system of rocket thrust chamber are shown in Figure 1-6. The conventional micro-channel heat exchanger is shown in Figur e 1-6A. The partition walls serve as fins to increase surface area thus enhance heat transfer rate and also support the hot wall. The high

PAGE 19

5 aspect ratio heat exchanger is shown in Fi gure 1-6B, which has larger surface area so can increase the cooling effectiveness. For Figure 16C, metal foam inserts are used in the channel to get even larger heat transfer rate. This study is focused on evaluation of foam filled channels heat transfer and pressure drop, whic h has potential applicati on in rocket thrust chambers cooling system. Figure 1-1. Construction of a regenerative cooli ng tubular thrust chambe r, its nozzle internal diameter is about 15 inch and thrust is a bout 165,000 lbf. It was originally used in the Thor missile. Recreated from reference Sutton [2].

PAGE 20

6 Figure 1-2. Cutaway of a tubular cooling j acket. The cooling tubes have variable cross-section area to allow the same number of tubes at nozzle throat and nozzle exit. Recreated from reference Sutton [2]. Figure 1-3. Typical heat transfer rate intensit y distribution for liquid pr opellant rocket. Peak is at the thrust nozzle throat and nadir is usually at the nozzle exit. Recreated from reference Sutton [2].

PAGE 21

7 Figure 1-4. Simplified schematic of regenerative cooling system of liquid propellant rocket. Recreated from reference Carlos [3]. Figure 1-5. Section A-A of Figure 1-4 and deta ils of cooling channel. Recreated from reference Carlos [3].

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8 Figure 1-6. Different configurations of the cooling channel in thrust chamber. A) Conventional micro cooling channel. B) High aspect ratio cooling channel. C) Metal foamed cooling channel. A B C

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9 CHAPTER 2 PREVIOUS WORK ON OPEN-CELL FOAMS 2.1 Heat Transfer Enhancement Open-cell foam is a kind of porous medium that is emerging as an effective method of heat transfer enhancement, due to its large surf ace area to volume ratio, high thermal conductivity, and intensified fluid (coolant) fixing. Figure 21 shows several pictures of typical aluminum foam. The use of open-cell foam to enhance heat tr ansfer has been investigated widely. Koh and Colony [4] and Koh and Stevens [5] inves tigated the heat transfer enhancement of forced-convection in a channel filled with hi gh thermal conductivity open-cell foam. In their theoretical study, Koh and Colony [4] found that for a fixed wall temperature case, the heat transfer rate increased by a factor of three. For a constant heat flux case, the wall temperature and the temperature difference between the wall and the coolant can be drastically reduced. Koh and Stevens (1975) performed e xperimental work to verify th e numerical results of Koh and Colony [4]. Koh and Stevens [5] used a stainl ess steel cylindrical a nnulus (1.5 ID and 2.1 OD) with a length of 8 inches to experiment with heat transfer enhancement by porous filler. The annulus was filled with peen shot (steel pa rticles) whose diameters ranging from 0.08 inch to 0.11 inch. Nitrogen gas was used as the coolant. They found the heat flux increased from 17 to 37 Btu/ft2s for a constant wall temperature case and the wall temperature dropped from 1450 oF to 350 oF for the constant heat flux case. Hunt and Tien [6] utilized foam-like material and fibrous media to enhance forced-convection for po tential application to electronics cooling. Their results showed that a factor of two to four times enhancement is achievable as compared to laminar slug flow in a duct. Maiorov et al. [7] found empirically that the heat transfer rates in

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10 channels with a high-thermal conductivity filler, compared to empty channels, reached a factor of 25-40 enhancement for water and 200-400 for nitrogen gas. Bartlett and Viskanta [8] developed a mathem atical model to predict the enhancement by high thermal conductivity porous media in forced-c onvection duct flows. They concluded that a 5-30 times increase in heat transfer is feasible for most engineering conditions. It is believed that the enhancement is mainly due to the micr o turbulent mixing in the pores and super heat transfer through high thermal conductivity porous structure. Kuzay et al. [9] have reported liquid nitrogen convective heat transfer enhancement with copper matrix inserts in tubes. They proved that the insertion of porous copper mesh into plain tubes enhances the heat transf er by large amounts with a single phase coolant. However, in boiling, with tubes in which the porous insert is brazed to the tube wall for the best thermal contact, the heat enhancement is to be on the orde r of four-fold relative to a plain tube. They conclude that porous matrix inserts offer a significant advantage in cooling, providing a jitter-free operation and a much higher effective heat transfer, at grossly reduced flow rates relative to plain tubes. More recently, Boomsma et al. [10] used a open-cell aluminum alloy metal foam measuring 40 mm x 40 mm x 2mm as a compact heat exchanger. With liquid water as the working fluid, they found that the heat exchange r generated resistances that are two to three times lower than those of the open channel heat exchanger while requiring the same pumping power. 2.2 Experiments Many researchers investigated important char acteristics of open-cell metal foams through experiments. Leong and Jin [11] performed ex periments to investigat e characteristics of oscillating flow through metal foams. They got detailed experimental data of flow pressure

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11 drop versus flow velocities. They found the osc illating flow characteristics in metal foam are governed by a hydraulic ligament diameter base d Reynolds number and the dimensionless flow displacement amplitude. And the Reynolds number has more significant effect on pressure drop and velocities relationship. Kim et al. [12] experimentally investigated the impact of presence of aluminum foam on the flow and convective heat transfer in an as ymmetrically heated channel. The aluminum foam they use has a porosity of 0.92, but with diffe rent permeability. They placed foam inside a channel and keep the upper wall at constant temperature while the lower wall is thermally insulated. They got correlations of the fr iction factor and Nusselt number with Reynolds number. Yuan et al. [13] investigated heat transfer enhancement and pressure drop in an annular channel with nickel foams. They used air as the coolant and constant heat flux heaters inside the inner tube of the annulus. They found the heat transfer enhancement was on the order of twenty times over open channel. Correlations of pressure drop, Nusselt number and heat transfer coefficient with Re ynolds number were obtained. 2.3 CFD Simulation and Numerical Model Many scholars investigated open-cell foams by numerical methods, both analytical and computational. Lu et al. [14] developed an an alytical model to mimic metal foams. It based on cubic unit cells consisting of heated slender cylinders, and took advantage of existing heat transfer data on convective crossflow over bank of cylinders. They solved out the overall heat transfer coefficient of a heat exchanger analytically and also the pressure drop. A process to optimize foam structure so as to maximize heat transfer rate was proposed. However, their model maybe oversimplified the metal fo am and leaded to overestimates.

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12 Krishnan et al. [15] carried out a direct simulation of the transport phenomenon in open-cell metal foam using a single unit cell stru cture. The unit cell is created by assuming the void pore is spherical, and the por es are located at the vertices and center of a unit cell. The final geometry is obtained by subtracting the sphe res from the unit cell cube. They further used that model to perform CFD simulation using Flue nt/Gambit. Periodic conditions are used thus only one cell is needed in simulation which gr eatly saved computational time. Total thermal conductivity, pressure drop and h eat transfer coefficient are obtained and compared with experimental data. Yet, this model is only suita ble for foams that has po rosity larger than 0.94. Krishnan et al. [16] then create d other models to extend the models capability to simulate lower porosities (down to 0.80). Besi des the body-centered cubic model [15], they developed other models based on face-centered c ubic (FCC), and A15 lattice, which is similar to Weaire-Phelan structure. Good agreement to other researcher s experimental data is obtained on Nusselt number and friction factor. Chung et al. [17] predicted and evaluated he at transfer enhancement for liquid rocket engine using metal foams. They develope d a unit cell structure based on Kelvins tetrakaidecahedron. The ligaments of unit cel l structure are simplified as cylinders. Comparison of pressure drop pred icted by that model with experimental data shows favorable agreements. They further performed CFD simula tion using that structure and also open channel to predict pressure drop under rocket conditions in which Reynolds num ber is up to 1 million and coolant is hydrogen. They also provided some experiment data on copper and nickel foams under lab conditions. The heat transfer e nhancement of foams inserted channel over conventional channel is 130%-170%. They believ ed that the enhancement is independent of pressure drop and increases with decreasing pore size.

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13 Boomsma et al. [18] developed a new appro ach to modeling flow through open-cell foams and defined a new cell structure. Their new model was based on Weaire-Phelan structure. This structure reduced the surface energy by 0.3% compared to tetrakaidecahedron [18]. The Weaire-Phelan structure was furt her wetted by Surface Evolver. Boomsma et al. [18] used that model to investigate pressure drop and velo city field in open-cell foams. They also compared their CFD prediction with experimental data and found their results were 25% lower. Its believed that the underestimates were due to the lack of pressure drop increasing wall effects in the simulations. 2.4 Other Open-Cell Foams 2.4.1 Polyurethane Foams Some researchers investigated other open-cell foams other than metal foams. Mills [19] used CFD simulation to investigate the permeab ility of polyurethane foams. The unit cell structure he used is Kelvins tetrakaidecahedron, which is widely used in the simulation of metal foams. He also used the Surface Evolver to ge t wetted structure of the Kelvins model. He concluded that the foam permeability is a function of the area of largest hole in the cells [19]. 2.4.2 Carbon Foams Carbon foams generally have better heat tran sfer performance than metal foams but induce larger pressure drop, which is due to their smalle r pore size and lower porosity. Yu et al. [20] developed a unit cube-based model for carbon fo am modeling. This structure allows lower porosity which is a major property of carbon foam, compared to conventional metal foams. Assumed that the entire foam ha s uniform pore diameter and pores are considered to be spherical and centered, their model was obtained by subtrac ting a sphere from a unit cube. They used that model to evaluate carbon foams heat transf er and pressure drop analytically and compared their results with experimental data.

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14 A B C Figure 2-1. Photos of aluminum foam. A) photo of aluminum foam brazed to a metal. B) view from a different angle. C) SEM photo of typical aluminum foam

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15 CHAPTER 3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS 3.1 Geometry Simplification for Op en-Cell Foam Filled Channels This transport model is based on the microsc opic structure of the metal foam whose cells can be approximated as in diamond shapes as i llustrated by the model presented in Figure 3-1. The ligament structure is composed of two types of struts. The vertical struts called V-type struts, are perpendicular to the flow dire ction (x) while the horizontal struts, called the H-type struts are on the plans (x-y) that is para llel to the flow direction. Figure 3-2 shows the detailed infrastructure of the model. The picture on the left illustrates the arrangement of the ligaments and their connection with the wa lls. The top wall is the heated surface which represents the heat source from the combustion chamber. The bottom wall is insulated as it stands for the outer wall for the coo ling channel. The plot on the right is a top view, which gives the horizontal cross se ction and the flow direction. A 3-D schematic of the foam model is gi ven in Figure 3-3 where two rows in the downstream direction and four columns for each ro w in the cross-stream direction are shown to illustrate the foam structure. The heat transfer mechanisms are explained in terms of a network as shown in Figure 3-4. The heated top wall is the heat source that in teracts with the V-type foam ligaments (fins) through conduction and also s upplies heat to the coolant by convection through un-finned surfaces. The V-type struts, which act as fins receiv e heat from the wall and then pass the heat to the coolant by convection and to the H-type stru ts by conduction. The convection heat transfer between the flow and the V-type struts will be m odeled as heat transfer for flow over tube banks. The H-type struts will lose heat to the flow by convection and also transfer some heat to downstream V-struts by conduction.

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16 3.2 Mathematical Transport Model and Heat Transfer Equations 3.2.1 V-type Struts A standard fin analysis is applied for a ve rtical cylinder as show n in Figure 3-5. The governing equation for a fin is given as below: 2 1 24 ()0s sc sdTh TT dykd (3-1) h1 : heat transfer coefficient between vertical cylinder and the cooling fluid ks : thermal conductivity of the cylinder d : diameter of the cylinder Ts(y) : local temperature as a function of y along the V-strut Tc : coolant temperature Boundary Conditions: 00 s w y s yHTT dT dy Solve for 1 1cosh() () cosh s cwcmHy TTTT mH (3-2) where 1 14sh m kd From Eq. (3-2), the rate of h eat dissipated to the coolant fr om the strut q, in watts, is 2 1 111 04 ()tanh()tanh()() 4s s cswcwc s yT h d qkAkTTmHMmHTT ykd where 23 11/4sMhkd, h1 is evaluated based on flow over a cylinder or tube bank. Its further assumed that the heat transfer coefficient is the same for all the vertical struts. In order to get an analytical solution, the average strut temperature over y is needed. Let 1 1cosh() cosh()sc wcTT mHy TTmH integrate Ts from y=0 to H, the average can be shown to be

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17 1 1 0 1 0 11 1 1cosh() 1 cosh 111 ()sinh() cosh tanhH H mHy dy HmH mHy HmHm mH mH Thus, 1 1tanh () s wccmH TTTT mH (3-3) The average over y direction elim inates the y variable in Ts. Since the cross section of rocket chamber is often annular, the coolant and fo am can be treated as uniform in z direction. Thus, coolant temperature is only a function of x in the flow direction [14]. 1 1cosh() (,)()(()) coshscwcmHy TxyTxTTx mH (3-4) 1 1tanh ()(())()swccmH TxTTxTx mH (3-5) Heat transferred from a single V-cylinder to the coolant is 1[()()]VscqdHhTxTx (3-6) 3.2.2 H-type Struts A similar fin analysis as the vertical cylinde r is applied for the horizontal cylinders that connect the vertical cylinders (Figure 3-6). The governing equation should have a similar form as a V-cylinder though extra terms need to be introduced to account for the angle (less than 90 degree) between H-cylinder and coolant. Such dependence can be assumed to be very weak when x>10a, as stated in the paper by Lu et al. [14]. There are several methods to solve the flow over H-type cylinders. Here proposes two methods.

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18 First method The following equation will be used: 2 2 24 ()0h hc sdTh TT dxkd (3-7) h2 : heat transfer coefficient between horizontal cylinder and the cooling fluid ks : thermal conductivity of the cylinder d : diameter of the cylinder Th : Temperature of a H-strut Tc : coolant temperature Boundary condition: 1 0 2 h x h xaTT TT T1 and T2 are the temperature of V-type cylinders at x and x+a. They can be evaluated from Ts solved in part a. For a specific H-type cylinder, T1 and T2 are constant. Solution of Eq.(3-7) is: 1222 2()sinh[()]()sinh() () sinh()cc hcTTmaxTTmx TxT ma (3-8) The heat flux entering the cylinder at x=0 122 12 2()cosh()() sinh()ccTTmaTT qM ma The heat flux leaving the cylinder at x=a 122 22 2()()cosh() sinh()ccTTTTma qM ma where 2 24sh m kd 23 22/4sMhkd The heat transfer to the coolant, is 2 12212 2cosh()1 (2) sinh()hcma qqqMTTT ma (3-9)

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19 Let 12(),()ssTTxTTxa For the coolant, since its temperature is function of x, we simply choose its value at the middl e point of H-type cylinder, thus 1 (()()) 2cccTTxTxa The heat transferred to the cool ant from a single H-cylinder is: 2 2 2cosh()1 (()()()()) sinh()hssccma qMTxTxaTxTxa ma (3-10) The same correlation of finding h1 is used to evaluate h2, because it is still a cross flow over a bank of cylinders. A correction on the free stre am velocity is needed as the flow is not at 90o to the cylinder. Second method Another method is to assume the H-type cylin ders have identical te mperature distribution with V-type cylinders along x direction. Thus, fo r a single H-type cylind er, the heat transfer rate from it to coolant can be represented as, 25 (()()) 2hscqdahTxTx (3-11) where 5 2 da is a H-type cylinders surface area, h2 is heat transfer coefficient for H-type cylinders, and Ts(x)-Tc(x) is the temperature difference between cylinders and coolant. From Eq.(3-5), the heat transfer rate can be further written in the form, 1 2 1tanh 5 (()) 2hwcmH qdahTTx mH (3-12) The reason two methods are proposed is b ecause the first method is found to have unfavorable agreement with experimental data at low Reynolds number region, which will be discussed later.

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20 3.2.3 Fluid Temperature Prediction (Coolant Temperature) The coolant temperature profile as a function of the downstream coordinate, x, is estimated based on the following energy balance equa tion. Figure 3-7 shows the schematic. [()()]pccvvhhwmCTxxTxNqNqq (3-13) where 21v x N a is the number of vertical struts per unit width. And 21 2h x H N aa is the number of horizontal struts per un it width for a channel of height H. Heat transfer from V-type struts vq can be evaluated from Eq. (3-6). Heat transfer from H-type struts hq can be evaluated from Eq. (3-10). Heat transfer from bare wall surface can be calculated from: [()]wwwcqxhTTx (3-14) where is the ratio of bare wall surface ar ea to the total wall surface area, and hw can be evaluated from open channel heat transfer coefficient correlation. First method (high Re) Eq. (3-10) will be used for relatively high Reynolds number (>2*104). From Eq. (3-5), 1 1tanh ()(())()swccmH TxTTxTx mH Lets further assume 0()()lx cwwTxTTeT where l is to be determined. Plug into Eq. (3-5), 11 0 11tanhtanh ()()(())()lx scwcwmHmH TxTxTTxTTe mHmH So Eq. (3-10) can be rewritten in the form:

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21 2 2 2 () 21 200 21 21 20 21 2 2 2cosh()1 (()()()()) sinh() cosh()1tanh() (()()) sinh() cosh()1tanh() (1)() sinh() cosh()1tanh( sinh()hscsc lxlxa ww lalx wma qMTxTxTxaTxa ma mamH MTTeTTe mamH mamH MeTTe mamH ma M ma 1 1) (1)()la wcmH eTT mH So, 21 2 21cosh()1tanh() (1)() sinh()la hwcmamH qMeTT mamH (3-15) Plugging Eq. (3-6), (3-14), and (3 -15) into Eq. (3-13) yields: 11 2 21 2 2 211 [()()](())tanh[()] cosh()1tanh 1 2(1)(()) sinh()pccwcwwc la wcx mCTxxTxMTTxmHxhTTx a mamH xH M eTTx aamamH Add up similar terms, 21 112 23 21()() cosh()1tanh 1 [tanh2(1)] ()sinh()la cc w wcpTxxTx mamH xH M mHMeh TTxmCaamamH Let 21 112 23 21cosh()1tanh 11 '[tanh2(1)] sinh()la w pmamH H lMmHMeh mCaamamH we have: ()() (())cc wcTxxTx lx TTx (3-16) Integrate (3-16) as 0 x 0() 0'Tcx x c wc TdT ldx TT 0() 0() 'cTx x wc wc TdTT ldx TT

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22 0ln()'Tc wc TTTlx 0()lx wc wTTx e TT 0()()lx cwwTxTTeT Since we assume 0()()lx cwwTxTTeT in the beginning of this derivation, thus ll That also proves the previous assumption is correct. So, 0()()lx cwwTxTTeT (3-17) where, 21 112 23 21cosh()1tanh 11 [tanh2(1)] sinh()la w pmamH H lMmHMeh mCaamamH (3-18) l can be determined by iterative method. Second method (low Re) Eq.(3-12) will be used for rela tively low Reynolds number (<2*104). The energy balance Eq.(3-13) still holds for this case. Plug in Eq .(3-6), (3-12), and (3-14) to Eq.(3-13) yields, 11 2 1 2 2 11 [()()](())tanh[()] tanh 15 2(()) 2pccwcwwc wcx mCTxxTxMTTxmHxhTTx a mH xH dahTTx aamH So, 1 112 22 1()() tanh 115 [tanh] ()cc w wcpTxxTx mH dH M mHhhx TTxmCaamH Let constant l to be in the form, 1 112 22 1tanh 115 [tanh]w pmH dH lMmHhh mCaamH (3-19) Integrate Eq.(3-19) over x, we have:

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23 0()()lx cwwTxTTeT (3-20) Eq.(3-20) has the same form with Eq.(3-17) in the first method, the difference is that, for the second method, l can be calculated dire ctly, and theres no need to iterate. 3.2.4 Total Heat Transfer The total heat transferred to the coolant th rough the cylinders (V and H types) and the inner wall is 0()(())cPQxTxTmC (3-21) Since 0()()lx cwwTxTTeT from Eq. (3-17) and (3-20), so 0()()(1)lx PwQxmCTTe (3-22) where l can be determined fr om Eq. (3-18) or Eq.(3-19). For a channel of length L, the total heat transfer is 0()()(1)lL PwQLmCTTe (3-23) 3.2.5 Evaluation of Heat Transfer Coefficient The heat transfer coefficient evaluation in th e foregoing analytical method is critical and will be discussed in this section. Its mentioned in previous sections that the heat transfer coefficients of V-type cylinders H-type cylinders, and bare wall are assumed to be identical, respectively. Based on Reynolds number, all the heat transfer coefficients cylinders can be evaluated by the empirical correl ations for flow over a tube bank correlation. And heat transfer coefficient for bare wall can be calcul ated from correlation for open channels. V-type cylinders h1 Flow over a bank of tubes has been widely i nvestigated by researchers for many years, and several correlations are availabl e for heat transfer. For a staggered mesh, the average heat

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24 transfer coefficient 1h for the entire tubes in the bank as defined in the Nusselt number, 1 d f hd Nu k can be obtained from the correlation below [21]: 1/3 1,max1.13RePrdm dNuC (3-24) where C1 and m are constants, they can be l ooked up from Incropera and DeWitt [21]. Reynolds number is defined as, ,max ,maxRev d fdV (3-25) So, 1/3 1,max 11.13RePrm dfCk h d Vmax is the maximum velocity of the coolant, kf is the thermal conduc tivity of coolant, vf is kinematic viscosity of coolant, and d is the diameter of the cylinder, m and C1 are constants and related to the geometry of channel, which can be obtained by tables. According to Incropera and DeWitt [21], the ma ximum velocity occurs at the transverse plane. It can be calculated as ,max va VV ad V is the incoming velocity. Eq. (3-24) is valid for Re from 2000 to 40000. For smaller Re number, correlation for flow over a single cylinder is used. Because the diameter of cylinder d is much smaller than cell size a, that is their ratio d/ a is about 0.2, the influence of inte r-cylinders is neglected for low Re number cases in this study. Equation fo r flow over a single cylinder from [21] is 1/3 ,maxRePrdm dNuC (3-26) where C and m can be found from Table 3-1.

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25 H-type cylinders h2 For H-type cylinders, it can also be treated as flow over a si ngle cylinder. The difference is that the flow direction is not perpendicular to the cylinder. So, th e component of velocity that is perpendicular to cylinder is considere d. From geometry, the equation is shown below ,max* 2*(*5/2)ha VV ad (3-27) The other parameters are calculated as the same as V-type cylinders. Bare wall hw To evaluate the heat transfer coefficient fo r the bare wall, the following correlation of open channel can be applied: 0.50.80.021PrReNu (3-28) Reynolds number and Nusselt number are defined as Re f DV (3-29) w f hD Nu k (3-30) For this case, only the bottom wall is heated, so for Eq. (3-29) and (3-30) D=2H, where H is the height of the metal foamed cooling channel. And hw can be calculated from 2 f wNuk h H (3-31) 3.2.6 Equivalent Heat Transfer Coefficient To calculate Nusselt number and equivalent heat transfer coefficient of open-cell foams is the ultimate aim of the analytical model. The equi valent heat transfer co efficient is defined as, ()equal wwcQ h ATT (3-32)

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26 such that the heat transfer from the foamed ch annel is equivalent to that carried away by a coolant having average temperature of cT, which flows through a ope n but otherwise identical channel. As is surface area of heated wall, in this model, the width is 1, so 1*sAL For the mean value of coolant temperature, or bulk temp erature, the arithmetic mean value of coolant over x direction is used. From Eq.(3-20), 0()()lx cwwTxTTeT let 0()lx wc c wTTx e TT so 00 0111exp() ()LL lx wc cc wTT lL xdxedx TTLLlL Thus, 01exp() ()wcwlL TTTT lL plugging into Eq.(3-32) yields, 0(1exp())()equal wQlL h LlLTT Plug in Eq. (3-23) for Q, 0 0()(1) (1)()lL Pw equal lL w PmCTTelL h LeTT mCl If the second method (low Reynolds number) is used for H-type cylinders heat transfer, Eq.(3-19) can be used for l. Thus, hequal has the final form 1 112 22 1tanh 15 tanhequalwmH dH hMmHhh aamH (3-33) From Eq.(3-33), the equivalent heat transfer coe fficient is a function of foam geometry a, d, channel height H, and heat transfer coefficient h1, h2, hw. Its not a function of inlet temperature, wall temperature, or channel length.

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27 3.3 Investigation of Cylinder Di ameter and Surface Area Density The relative foam density and surface area density are two mo st important properties of foam. Relative foam density is closely relate d to permeability and pressure drop induced by foams, defined by the following equation, s where is relative foam density, is density of foam, and s is density of solid. Another important property, porosity, is equal to 1 The surface area density is defined by this equation, AA V where A is surface area of foam, and V is the volume of the foam. Th e surface area density is an important property of foam which is rela ted to heat transfer capacity of a foam. In order to verify the foregoing diamond shaped ce ll structure, it need to be made sure that the structure represents the real metal foams well by retaining the relative foam density and surface area density. For the foregoing diamond shaped model, from geometry calculations, the structures relative foam density can be represented as: 23(51) ()() 42 dd aa (3-34) where a is cell size and d is the diameter of cy linders. To simplify calculation and derivation, and because d/a is about 0.2, Eq (3-34) can be rewritten as, 2(553) () 20 d a (3-35)

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28 So 1/220 (553) d a (3-36) The filament diameter d is calculated from Eq.(3-36) based on a=2mm(10ppi), 1mm(20ppi), and 0.5mm(40ppi) with different rela tive density, and further compared with experimental data from ERG Duocel aluminum foams (Figure 3-8). Reasonable agreement is obtained. For surface area density, 3 2(52) [(51)2/] (50.6)Adaadd a dad a d aa From Eq. (3-36), surface area density can be represented as, 1/25.97Aa (3-37) Eq (3-37) is plotted in Figure 3-9 and compared w ith data from ERG. Duocel aluminum foams. Good agreement is obtained except for 40ppi case. 3.4 Verification of the Analytical Model with Experimental Data To verify the heat transfer analytical mode l, heat transfer predictions on certain metal foams by the model are compared with experimental data from other researchers. Because two methods are developed for different Reynolds num ber, the author made two comparisons with other experiments with Reynolds number ranging from 5*103 to 2*104, and 104 to 6*104, respectively, using both methods stated in Section 3.2.3.

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29 3.4.1 Validity of Analytical Prediction (Re=5*103 ~ 2*104) Calmidi and Mahajan [22] tested several aluminum metal foams using air as the coolant. Nusselt number data is obtained as function of pore Reynolds number The pore Reynolds numbers are transformed into Reynolds number ba sed on channel height in this study. The foam samples Calmidi used have dimensions of 114mm*63mm*45mm, a nd they placed two heaters onto both the top and the bottom of foams. The Reynolds number is relatively low, and the second method in Section 3.2.3 is used for this comparison. For the analytical model in this study, the top wall of foamed channel is assumed to be adiabatic. So to predict Calmidis data, the height of channel in analytical model can be tr eated as half of the height of Calmidis sample, which is 45/2=22.5mm. Table 2-2 shows details of the experiment from Calmidi and Mahajan [22] and parameters used for analytical model in this study. To mimic the real foams, the filament di ameter and pore diameter are two important parameters for specific foams. In the analytic al model, cylinder diameter d represents the foam filament diameter and cell size a repr esents pore diameter. Because the diamond shaped cell in analytical model is a simplified st ructure for real foams, the parameters a and d used in analytical model can be slightly diffe rent from the real filament diameter and pore diameter. Table 2-3 shows the parameters used in models and also their comparison with the experiment samples data. Different values of d and a are tested and the values shown in Table 2-3 are the ones providing best ag reements with experimental data. The predictions for the 5 types of foams rangi ng from 5PPI to 40PPI are plotted in Figure 3-10, Figure 3-11, and Figure 3-12, and compared with experimental data from Calmidi [22]. Favorable agreements are obtained. The Nusse lt number and Reynolds number are defined in the following equations,

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30 Re f HV (3-38) equal fhH Nu k (3-39) where H is height of foam, V is inlet velocity of coolant, equalh is equivalent heat transfer coefficient of foam defined in Eq. (3-32), f k is coolants thermal conductivity, and f is kinematic viscosity of coolant. 3.4.2 Validity of Analytical Prediction (Re=1*104 ~ 6*104) To verify the validity of the analytical m odel with relatively high Reynolds number, the second method stated in Section 3.2.3 is utilized. A set of experimental data is used to compare with the prediction by the model (an insulated heat flux case is used for this calculation). The data are from an experiment made by our lab, testing heat transf er and pressure drop of copper foam. The details of this experiment can be obtained from Chung et al. [17]. For experiment, the total heat transfer rate to the air flow is define d by the energy balance: Pa,outleta,inlet QC(TT) m (3-40) Here, aoutletT and ainletT are the outlet and inlet air temperature, respectively, and Cp is the specific heat under constant pressure. The bulk fluid temperature is defined as: ,,2baoutletainletTTT (3-41) Effective heat transfer coefficient is defined as: /()equalssbhQATT (3-42) where s A is the total heated surface area and s T is the mean surface temperature Reynolds number is defined as:

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31 Reoi fVdd (3-43) where, od and id are the outer and inner diameter of the test section, respectively. f is the kinematic viscosity of the fluid and V is the mean velocity. For analytical model, the same geometry is us ed, and the height of channel is defined as 2oidd H (3-44) Reynolds number is defined as: 2 Re f VH (3-45) The cell size a is set to be 2mm, and the filament diameter d is set to be 0.5mm, which is approximately a 10 PPI (pores per in ch), relative density 8%s foam. Figure 3-13 shows the analytical models pred iction of heat transfer coefficient of the copper channel used in the experiment and compares them with experimental data. It should be pointed out that the data from analytical model is scaled by a factor of 0. 7 as a correction, which maybe due to a different dimensional scale betw een the model and experiment. The analytical model predicts the heat transfer coefficient nice ly from the plot. But for high Re number, the analytical model underestimate the heat transf er coefficient. We found that for Reynolds number less than 105, the insulated boundary condition (at y=H) model gives good prediction. For extremely large Re numbers (>105), constant temperature model at both walls should be used. More details can be found in Chapter 5. Table 3-1. Constants of Equatio n (3-22), recreated from [21] ReD C m 0.4-4 0.989 0.330 4-40 0.911 0.385 40-4,000 0.683 0.466 4000-40,000 0.193 0.618

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32 Table 3-1. Continued 40,000-400,000 0.027 0.805 Table 3-2. Parameters of experiments from Calmidi [22] and analytical model Experiment [22] Analytical Model Geometry L/W/H (mm) 114/63/45 114/unit length/22.5 Coolant Air Air Foam Aluminum Aluminum Coolant Inlet Temperature (K) 300 300 Heated Wall Temperature (K) 350 350 Table 3-3. Foam parameters comparison between experiments from Calmidi [22] and analytical model Ligament Diameter Pore Diameter Results Comparison Experiment 0.50mm 4.02mm 5PPI Model 0.70mm 4.02mm Figure 3-10 Experiment 0.40mm 3.13mm 10PPI Model 0.55mm 3.13mm Experiment 0.30mm 2.70mm 20PPI Model 0.45mm 2.70mm Figure 3-11 Experiment 0.55mm 3.80mm 5PPI Model 0.70mm 3.10mm Experiment 0.25mm 1.80mm 40PPI Model 0.20mm 1.50mm Figure 3-12 Figure 3-1. Schematic of a single cell in the simplified model

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33 Figure 3-2. Model details Figure 3-3. 3-D schematic of the model

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34 Figure 3-4. Heat transfer ne twork of analytical model Figure 3-5. Schematic of vertical strut fin model

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35 Figure 3-6. H-strut model Figure 3-7. Model for coolant temperature evaluation

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36 0 1 2 3 4 5 6 00.020.040.060.080.10.120.14 Relative foam densityCylinder diameter(10-4m) 10ppi(Experiment) 20ppi(Experiment) 40ppi(Experiment) 10ppi(Model) 20ppi(Model) 40ppi(Model) Figure 3-8. Cylinder diamet er as function of relative foam density predicted by analytical model, comparing with ERGs data of alumi num foams, a=2mm, 1mm, and 0.5mm, respectively, for 10ppi, 20ppi, and 40ppi foams. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 00.020.040.060.080.10.120.140.16 Relative foam densitySurface area density(m2/m3) 10ppi(Experiment) 20ppi(Experiment) 40ppi(Experiment) 10ppi(Model) 20ppi(Model) 40ppi(Model) Figure 3-9. Surface area density as function of relative foam density predicted by analytical model, comparing with ERGs data of al uminum foams, a=2mm, 1mm, and 0.5mm, respectively, for 10ppi, 20ppi, and 40ppi foams.

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37 0 500 1000 1500 2000 2500 3000 3500 4000 0500010000150002000025000 ReNu 5ppi(Model) 5ppi(Calmidi and Mahajan [23]) Figure 3-10. Nusselt number prediction made by analytical model compared with Calmidi and Mahajans [22] experimental da ta for 5 ppi aluminum foam 0 500 1000 1500 2000 2500 3000 3500 4000 0500010000150002000025000 ReNu 10ppi(Model) 20ppi(Model) 10ppi(Calmidi and Mahajan [23]) 20ppi(Calmidi and Mahajan [23]) Figure 3-11. Nusselt number prediction made by analytical model compared with Calmidi and Mahajans [22] experimental data for 10 ppi and 20 ppi aluminum foams

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38 0 500 1000 1500 2000 2500 3000 3500 4000 0500010000150002000025000 ReNu 5ppi(Model) 40ppi(Model) 5ppi(Calmidi and Mahajan [23]) 40ppi(Calmidi and Mahajan [23]) Figure 3-12. Nusselt number prediction made by analytical model compared with Calmidi and Mahajans [22] experimental data for 5 pp i and 40 ppi low porosity aluminum foam 0 500 1000 1500 2000 2500 3000 0100002000030000400005000060000 ReHeat Transfer Coef (W/m2K) Experimantal Data Analytical Model Figure 3-13. Heat transfer coeffi cient predicted by analytical m odel compared with Chung et al. [17] experimental data for 10ppi copper foam

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39 CHAPTER 4 CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS 4.1 Introduction to Single Cell Model Open-cell foams have been investigated by many researchers, both experimentally and numerically. In chapter 3, the analytical heat transfer model deals with the whole foamed cooling channel, and uses volume-averaged, semi-e mpirical equations. That is a macroscopic approach, which neglects small-scale details of open-cell foams. With rapid developing computing power, using a model with more foams cell details becomes feasible in computational fluid dynamics. Although the com puter stations are still not powerful enough to simulate the whole foam inserted ch annel at this stage, efforts can be made to investigate a single cell in open-cell foams due to their property of re peated cell structure. That is the microscopic approach. Using microscopic approach to simulate pressure drop in open-cell foams takes advantage of the repeated cell structure of foams and also the properties of flow through porous media. For a specific type of foam, in which the porosity pore per inch, and othe r material properties are fixed, the pressure drop induced by the foam is only function of velocity of flow. And the velocity profile in open-cell foam is almost uni fied, because the multi-filament in foam greatly increases the intensity of turbulence in flow whic h flattens out the velocity gradient and makes the boundary layer very thin (Figure 4-1). Th us, because of the unit cel l structure and nearly unified velocity in open-cell foam, the pressure drop evaluation process can be simplified without modeling the whole foam inserted channel. A strategy has been developed to focus on two typical cells as illustrated in Figure 4-1. The first type is named interior cell, which is located relatively far away from the wall and in the uniform velocity region. Since the velocities in a ll interior cells are identical and all cells have

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40 the same structure, only one cell is needed to be modeled to evaluate pressure drop contribution by interior cells. The second type cell is name d boundary cell, which is distinguished from interior cells and used to captu re the pressure drop occurring at the wall. The pressure drop induced by boundary cell is expected to be larger than that of a interior cell because the no-slip condition at wall and velocity at boundar ies has much larger velocity gradient. To simulate the micro-structure of open-cell foam (typically metal foams), a sphere-centered tetrakaidecahedron structure is constructed (Figur e 4-2A). That structure is very similar to the real micro-structure of metal foam (Figure 4-2B aluminum foam). A tetrakaidecahedron is a polyhedron consisting of six quadrilateral faces and eight hexagons. Its found by Lord Kelvin that the tetrakaidecahedro n (Kelvin structure) is optimal structure for packing cell, which has minimum surface-area to volume ration. Tetrakaidecahedron is seen in reality when soap foam is observed [18]. The sphere-centered Kelvin cell can mimic the real metal foams micro-structure because of th e foaming process of metal foam. A common method used to foam metal such as aluminum is blowing a kind of foaming gas through molten metal. The gas bubbles generated are free to move around. The liquid metal and gas bubbles tend to attain an equilibrium state, i.e., a mi nimum surface energy state [15]. Thus, after the solidifying process, the optimal tetrakaidecahed ron structure is formed by metal and gas bubbles generate pores which are similar to spheres. So, the sphere-centered tetrakaidecahedron can represent the real micro-structure of metal foams very well. In order to generate the sphere -centered Kelvin structure, a tetrakaidecahedron is generated first by cutting off the six corners of a regular oc tahedron. Then build a sphere at the center of the tetrakaidecahedron and subtracting the sphe re from it yields the sphere-center Kelvin structure. Figure 4-3 shows the process schematic.

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41 As stated before, two types of cells are need ed for the pressure drop simulation, interior cell and boundary cell. Two computational mo dels have been created for the two cells, respectively. The first represents a typical inte rior cell and is termed the D-periodic model because periodic or symmetric boundary conditions are applied in 2 directions (Y, Z directions) except in the stream-wise direction (X direction). A diagram of this is shown in Figure 4-4. The second model treats the cell that is attached to the wall and is termed the D-periodic model. Here, periodicity is applied in only one direction (Y direction). In the other direction one boundary was set as a wall and the remaining boundary as a symmetry plane. This is shown in Figure 4-5. The coolant used for the pressure drop simulatio n is air, which is assumed to be ideal gas with constant density and viscosity. Energy eq uation is not considered at this stage which means the temperature is constant. The air come s into the inlet of the channel and goes out through the outlet (Figure 4-4, 45). The inlet was set as veloc ity inlet boundary, and different inlet velocities were tested. The outlet wa s set as pressure out let boundary having the atmosphere pressure. No-slip conditions were imposed at the wall and cell surfaces. 4.2 Mesh Generation and Grid Independent Study The Kelvin structure and channel models were created and meshed by GAMBIT, the preprocessing meshing generation software. The whole channel was divided into three parts, the inlet region, the outlet region, and the cell (c entral) region due to their different geometry properties. The cell region in the middle was meshed using TGrid in GAMBIT, which generated tetrahedral elements that can fit into th e complex structure of Kelvins cell. The inlet and outlet regions were meshed by Cooper method in GAMBIT. Because flow at those regions is less complicated than in the cell region, much less elements were generated at inlet and outlet regions to save computing time. Figure 4-6 shows the meshed Kelvin cell. Figure 4-7

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42 provides the mesh details at cells filaments. Fi gure 4-8 presents that fine mesh is used at the cell region and relatively coarse mesh is used at the inlet and outlet regions. The cell size is about 2.54mm*2.54mm*2.54 mm, which is about the cell size of a 10ppi foam made by ERG. And the sphere centered in the cell has a diameter of 2.61mm. The porosity of the cell is thus about 97.4%. To examine the dependence of solution on mesh es, three different meshes were generated with different fineness. The coarse mesh cons ists of 451383 tetrahedral cells and 127010 nodes.That model was then refined by the medium mesh, which consists of 708955 cells and 183056 nodes. The most delicate model was further re fined to 1187729 cells and 335766 nodes, which is named the fine mesh in this study. All the three different fineness mo dels have the same cell size, porosity, and channel geometries. The mesh independent study was done for a 2-D periodic model in which inlet velocity is 4m/s and cell size is 10ppi. Figure 4-9 shows the average x velocity profiles along the flow direction (x direction) of the three meshes. From the figure there are no apparent differences among the three meshes with different number of elements. Figure 4-10 provides comparison of simulation results made by coarse, medium, and fine meshes. The differences among them are visible although slight. Pressure drop is calculated from the following equation, 21 21 p p p axx (4-1) where p represents pressure, a is cell size, and x represent the x coordinate in flow direction. The pressure drops simulated from those three models are shown in Table 4-1. The relative error between coarse mesh and fine me sh is 3.5%, and relativ e error between medium mesh and fine mesh is only 0.6%. Thus, the au thor thinks the coarse mesh is fine enough to

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43 capture the pressure drop in foams and the coarse mesh was chosen to perform all the following simulations. Some more statements can be made on Fi gure 4-9 and 4-10. There are three regions where the pressure drop is very si gnificant, from Figure 4-10. Th e three regions are inlet of cell, center of cell and outlet of cell. That agrees with the velocity profile in Figure 4-9, in the sense that the regions having larger velocities induce mo re pressure drop. The reason is that potential energy from pressure is tran sferred into kinetic energy. 4.3 Simulation Results and Verification Simulations were performed using coarse mesh (Section 4.2). The cell size is set to be about 2.54mm which is 10ppi and its porosity is about 97%. Figure 4-11 shows the velocity magnitudes co ntours of several chosen planes in a case with inlet velocity of 4m/s. There are three pl anes, the first one is at about y= -0.8mm, the horizontal one is at the center of cell and the last one is at the left side of the channel. Figure 4-11(A) is a 3-D view of the three planes co ntour, and (B)-(D) represents the three planes respectively. The velocities between ligaments are relatively high and wakes can be found at ligaments, which is evident especially in Figure 411(B). Figure 4-12 provides static pressure contours of the same three planes. High pressu re can be found where the flow encounters with the ligaments (Figure 4-12(B), (C)). More data were obtained for 2-D periodic and 1-D periodic models for several inlet velocities to get pressure drop profiles for interior cells and wall cells. Experimental data from Leong and Jin [11] were chosen to compare with the simulation data. The comparison was shown in Figure 4-13 and the pressure drop wa s plotted as function of inlet velocity. Both pressure drop profiles for interior cell and wa ll cell were compared with experiments and very

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44 nice agreement was obtained. It can be found that the wall cell induces a little more pressure drop because the no-slip condition of wall also contributes to the pressure drop. It can be concluded that the Kelvin stru cture unit cell can capture the important phenomenon of pressure drop occurring in metal cells and can be used to predict foams pressure drop. Table 4-1. Comparison of different meshes results Mesh Pressure Drop Relative Error Coarse 5.26 Pa/mm 3.5% Medium 5.11 Pa/mm 0.6% Fine 5.08 Pa/mm Figure 4-1. Schematic of boundary cell and interior cell in open-cell foam

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45 A B Figure 4-2. Comparison of single cell model and r eal foam structure. A) single cell model used in this study. B) SEM photo of aluminum foam. Figure 4-3. Geometry creation of a single cell

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46 Figure 4-4. 2-D periodic model Figure 4-5. 1-D periodic model

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47 Figure 4-6. Mesh of a single cell model (coarse grids) Figure 4-7. Details of the meshes on filaments (medium grids)

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48 Figure 4-8. Grids distribution. Cell region has more delicate grid s and other regions use coarse ones to save computation time 3.95 4 4.05 4.1 4.15 4.2 4.25 4.3 4.35 4.4 -1.27-0.77-0.270.230.731.23 x position (mm)Velocity (m/s) Coarse Mesh Medium Mesh Fine Mesh Figure 4-9. Velocity profile along flow direction through the cell

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49 -4 -2 0 2 4 6 8 10 12 -1.27-0.77-0.270.230.731.23 x position (mm)Pressure (pascal) Coarse Mesh Medium Mesh Fine Mesh Figure 4-10. Pressure di stribution along flow di rection through the cell

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50 Figure 4-11. Velocity contours in three planes around the cell. A) 3-D view. B) Plane at y= -0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the channel, y=1.27mm

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51 Figure 4-12. Static pressure contours in three pl anes around the cell. A) 3-D view. B) Plane at y= -0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the channel, y=1.27mm

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52 0 5 10 15 20 25 30 024681012 Velocity (m/s)Pressure Drop (kPa/m) Experimental Data from Leong [12] Interior Cell Wall Cell Figure 4-13. Pressure drop versus inlet veloc ity and comparison with experimental data

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53 CHAPTER 5 FEASIBILITY STUDY OF FOAMED COOLING CHANNELS FOR ROCKET THRUST CHAMBER To investigate the feasibility of the foamed cooling channel for rocket chamber, high Re number cases were studied for both the open cha nnel and foamed channel. Empirical equations were used for the open channel. The analytic al model derived in Chapter 3 was applied to foamed channel to predict its heat transfer ra te, and the CFD simulation method in Chapter 4 and also some data for hydrogen from Chung et al. [ 17] was used to get the pressure drop prediction correlation of rocket condition pressure drop. The author used the parameters of 10PPI foam to perform all the calculation in an alytical heat transfer model. 5.1 Feasibility Study and Comparison with Open Cooling Channel The average velocity in open channel is set to be up to 250m/s (Re=106), which is under rocket condition. Due to the high pressure drop gradient, the velocity in foamed channel can not reach that high, but has about 1/5 of that. In order to keep the same mass flow rate, larger cross section area is used. The idea is summarized in Table 5-1. The coolant mass flow rates and pressure drops are set to be equal, which make sure that the amount of coolant needed and the work needed to push the coolant are the same. Under that requirement, if a higher heat transfer is obtai ned, the application of foamed channel will be meaningful. Figure 5-1 shows the scheme. Ta ble 5-2 lists parameters of open channel of foam channel used in this comparison For open channel, the following correlations suggested by Incropera and DeWitt [21] are used. Pressure drop: 2(/) /2 dpdxH f u (5-1)

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54 f is found to be constant 0.05 at high Re numbers for commercial metals. Heat transfer: 0.80.50.021RePrDNu (5-2) where Re/ Hu hH Nu k Comparisons of heat transfer between an open channel and foam-filled channels are shown in Figure 5-2, Table 5-3 and Table 5-4. Figure 5-2 shows foamed channel has significant heat transfer enhancement over open channel, when th ey have the same mass flow rate and pressure drop. Table 5-3 compares foamed channels heat tr ansfer coefficient with that of an open channel at equal pressure drops. For instance, when the pressure drop is 841 kPa/m for both of channels, the heat transfer coe fficient increases from 18567 W/m2K to 36951 W/m2K, thats an increase of 99%. Similar increases are found for other pressure drops. The enhancement gets smaller with increasing pressure drop. That is due to the rapidly increasing heat transfer coefficient of open channel. But the enhancement is still significant at Re=106. Table 5-4 shows the velocities in the two types of channels with the same pressure drop. To keep the mass flow rate be equal in two channels, the foamed channel area has to be increased to compensate the low velocity. The re sults indicate that the foam channel should be 5.3 times of the open channel. If we keep the same base width, the height of the foam filled channel therefore should be extended according to th at ratio. From the data shown in the table, the velocities ratio of open and foamed channe ls is approximately 5.3, and getting slightly smaller with larger pressure drop. A CFD simulation of open channel under ro cket conditions has b een accomplished by Chung et al. [17]. A head to head comparison of open channel and 10PPI foamed channel

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55 under that rocket condition is perf ormed to show the feasibility of applying foam channel to the rocket chamber. The details are shown in Table 5-5. In order to keep the same pressure drop and mass flow rate, the velocity ratio in open and foamed channels is kept 4:1, and the height of foamed channel is thus 4 times of open channel. The results of CFD simulation of open channel and analytical pr ediction of foamed channel are shown in Table 5-6. Its shown that the foamed channels heat transfer coefficient will be 49581 W/m2K, which is more than 110% enhancement, compared to open channels 23464 W/m2K. That means, under the same pressure drop a nd mass flow rate, the foamed channel has a significant capability to enhance the heat transfer efficient of the rockets cooling chamber. Actually, if higher PPI foams (like 20 or 40 PPI) ar e used, more enhancement of heat transfer is expected, although its not shown in this study du e to the lack of data of higher PPI foams. 5.2 Uncertainty Analysis To analyze the certainty of 110% enhan cement predicted by analytical model and simulation, an error analysis is performed in th is section. The prediction error comes from both the heat transfer model and the CFD pressure drop simulation. So the error of the prediction is some combination of error from the analytical model and error made by the CFD simulation. 5.2.1 Heat Transfer Model From the comparison of model and experimental data in Section 3.4.2, the uncertainty of the heat transfer coefficient h prediction made by analytical model is calcul ated from Figure 3-13. Predictions of h made by analyt ical model were compared with experimental data. The relative error is about 30%, with a confidence of 90%. 30% h h (10 to 1) (5-3)

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56 5.2.2 Pressure Drop Simulation From the comparison of simulation and experiment al data in Section 4.3, the uncertainty of the pressure drop p prediction made by CFD simu lation is calculated from Figure 4-13. The relative error is about 30%, with a confidence of 90%. 10% p p (10 to 1) (5-4) 5.2.3 Rocket Condition Prediction Because the pressure drop is kept the same to find the coolant velocity in foamed channel, under rocket conditions, th e uncertainty of velocity can be evaluated. Because 2 p v, so the uncertainty of velocity can be calculated from 10%3.3% v v (10 to 1) (5-5) Since Re v so Re 3.3% Re (10 to 1) (5-6) From the heat transfer model un certainty analysis and Figure 3-13,0.3001ReequalhA. From regression analysis, A=1088, and Re=304000. The uncertainty of A can be treated as the same as 30% from Eq. (5-3). So the uncertainty of A is A =0.3*1088=326.4. And from Eq. (5-6), the relative uncertainty of Re is 3.3%, with confidence of 10%, so Re =0.033*304000=10032. So, the uncertainty of equivalent heat transfer coefficient of rocket can be calculated, after considering the uncertainty of pressure drop simulation, as 22()(Re) Reequalequal equalhh hA A 0.300122 0.6999(Re)(0.3001Re) Re A A

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57 214434W/mK So, the uncertainty of the equivalent heat transf er coefficient of rockets foamed cooling is, 14434 29.1% 49581equal equalh h (5-7) The heat transfer coefficient of metal foamed channel can be represented as, 249581W/mK29.1%equalh (5-8) If we take a close look of uncertainty equation of hequal, 22 0.3001 0.300122 0.6999 0.3001 22()(Re) Re Re (Re)(0.3001Re) Re Re Re ()(0.3001) Reequalequal equal equalhh A h A hA A A A A A From the above equation, the uncertainty of hequal comes mainly from A A which is 30%, compared to Re 3.3% Re Thus, the need for improve the prec ision of heat transfer analytical model is critical for this process. Table 5-1. Micro open channel and fo am filled channel model requirements Open Channel Foamed Channel Channel width = 2 mm Channel width = 2 mm Channel height = 4 mm Channel height = x mm Pressure drop = A Pressure drop = A Coolant flow rate = B Coolant flow rate = B Heat transfer = Q1 H eat transfer = Q2>Q1

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58 Table 5-2. Head-to-head comparison of open channel and foamed channel Channel Geometry Working fluid Inlet Temperature Temperature of heated base Inlet Velocity Length Width Height Open Channel H2 100K(180R) 800K(1440R) 10-250 m/s 1m 2mm 4mm Foamed Channel H2 100K(180R) 800K(1440R)2-48m/s 1m 2mm 21-22m m Table 5-3. Heat transfer enhancement of foamed channel over open channel Pressure Drop (kPa/m) Open Channel Heat Trans. Coef. (W/m2K) Foamed Channel Heat Trans. Coef. (W/m2K) Heat Transfer Enhancement Percentage 841 18567 36951 99% 987 19795 37962 92% 1145 21004 38929 85% 1314 22196 39858 80% 1495 23372 40753 74% 1688 24534 41617 70% 1892 25682 42453 65% 2108 26817 43265 61% 2336 27940 44053 58% 2576 29053 44820 54% 2827 30154 45568 51% 3090 31246 46298 48% Table 5-4. Velocity ratio at equal pressure drop Pressure Drop (kpa/m) Velocity in Open Channel (m/s) Velocity in Foamed Channel (m/s) Ratio 374 80 14.9 5.36 584 100 18.7 5.34 841 120 22.6 5.32 1145 140 26.4 5.31 1495 160 30.2 5.30 1892 180 34.1 5.29 2336 200 37.9 5.28 2576 210 39.8 5.27 2827 220 41.8 5.27 3090 230 43.7 5.26 3364 240 45.6 5.26 3650 250 47.6 5.26

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59 Table 5-5. Head-to-head comparison under rocket condition Channel Geometry Working fluid Inlet Temperature Temperature of heated base Inlet Velocity Length Width Height Open Channel H2 100K(180R) 800K(1440R)207m/s 508mm 2mm 4mm Foamed Channel H2 100K(180R) 800K(1440R)52m/s 508mm 2mm 16mm Table 5-6. Comparison of open and foamed channels performance Pressure drop Mass flow rateHeat Trans. Coef. Open Channel (CFD results) 4303 kPa/m 0.0155kg/s (Re=1*106) 23464 W/m2K Foamed Channel (Analytical predictions) 4303 kPa/m 0.0155kg/s 49581 W/m2K Figure 5-1. Notional design strategy for foam-filled channels

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60 0 10000 20000 30000 40000 50000 60000 05001000150020002500300035004000 Pressure Drop (Kpa/m)Heat Transfer Coef. (W/m2K ) Open Channel Foamed Channel Figure 5-2. Comparison of heat transfer coef vs. pressure drop of open and foamed channels

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61 CHAPTER 6 CONCLUSIONS An analytical heat transfer model and a CF D based pressure drop simulation method for open-cell foams have been investigated and the f easibility of using foamed cooling channel for rocket is studied. The analytical heat transfer model has provided favorable agreement with some experimental data and it can provide valuable pr ediction on heat transfer of foam filled cooling channels. The remaining defect of that model is that it doesnt have a universal form. That is, there have to be different equations for different Reynolds number ra nges, as stated in Chapter 3. This author believes that the reason is due to th e heat transfer coeffici ent correlations the model uses. The analytical model uses correlations of flow over bank of tubes and flow over single cylinders, which dont have inter-cylinder or in ter-tube effects. Howeve r, the real open-cell foams ligaments are connected to each othe r which may induce significant variation of temperature distribution on ligamen ts and heat transfer enhancement over that of flow over tubes. Thats the reason why the model tends to underes timates the heat transfer coefficient when the Reynolds number increases. The author believes that more experiments on different kinds of foams and correlations are needed before a universal heat transfer model can be obtained and currently the heat transfer model in this stu dy can be useful on evaluation of some kind of open-cell foams application. Also, an optimum design of foams porosity, pore per inch and ligament diameter to get maximum h eat transfer rate can be investigated by the analytical heat transfer model. The CFD simulation of a single cell in metal foam is a feasible method to evaluate pressure drop in foams. The Kelvin structure is very similar to the real micro-structure of metal foams which can capture the most important flow phe nomenon in metal foams. A remaining problem

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62 with that model is the single cell model tends to overestimate the pressure drop a little bit. Thats because the pressure drop when the flow en ters the cell is significant for a single cell but is negligible for whole foams which contain thousands of cells in a line. That is a problem caused by under-developed flow. A solution for it is to use periodic boundary also in the flow direction. In future, this author would like to do some simulations on a single cell with 3 dimensional periodic boundaries and also couples the model with energy equation, in the hope of solving the heat transfer and pressure drop in one model.

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63 LIST OF REFERENCES [1] Turner, M.J.L., 2000, Rocket and Spacecraft Propulsion, Praxis Publishing, Chichester, UK. [2] Sutton, G.P., and Biblarz, O., 2001, Rocket Propulsion Elements, 7th ed., Wiley, New York. [3] Carlos, H. M., Fernando, L., Antonio, F. C. da Silva and Jose, N. H., 2004, Numerical Solutions of Flows in Rocket Engines with Rege nerative Cooling, Numer. Heat Transfer A, 45, pp. 699-717. [4] Koh, J.C.Y. and Colony, R., 1974, Analysis of Cooling Effectiveness for Porous Material in a Coolant Passage, J. Heat Transfer, 96, pp. 324-330. [5] Koh, J.C.Y. and Stevens, R.L., 1975, Enha ncement of Cooling Effectiveness by Porous Materials in Coolant Passage, J. Heat Transfer, 97, pp. 309-311. [6] Hunt, M.L. and Tien, C.L., 1988, Effects of Thermal Dispersion on Forced Convection Fibrous Media, Int. J. Heat Mass Transfer, 31, pp. 301-309. [7] Maiorov, V.A, Polyaev, V.M., Vasilev, L.L. and Kiselev, A.I., 1984, Intensification of Convective Heat Exchange in Channels with a Porous High-Thermal-Conductivity Filler. Heat Exchange with Local Thermal Equilib rium Inside the Permeable Matrix, J. Engineering Physics Thermophysics, 47, pp. 748-757. [8] Bartlett, R.F. and Viskanta, R., 1996, Enhancement of Forced Convection in an Asymmetrically Heated Duct Filled with High Thermal Conductivity Porous Media, J. Enhanced Heat Transfer, 6, pp. 1-9. [9] Kuzay, T.M., Collins and Koons, J., 1999, Boiling Liquid Nitrogen Heat Transfer in Channels with Porous Copper Inserts, Int. J. Heat Mass Transfer, 42, pp. 1189-1204. [10] Boomsma, K., Poulikakos, D. and Zwic k, F., 2003, Metal Foams as Compact High Performance Heat Exchangers, Mechanics of Materials, 35, pp. 1161-1176. [11] Leong, K.C. and Jin, L.W., 2006, Effect of Oscillatory Frequency on Heat Transfer in Metal Foam Heat Sink of Various Pore Densities, Int. J. Heat Mass Transfer, 49, pp. 671-681. [12] Kim, S.Y., Kang, B.H. and Kim, J., 200 1, Forced Convection from Aluminum Foam Materials in an Asymmetrically Heated Channel, Int. J. Heat Mass Transfer, 44, pp. 1451-1454. [13] Yuan, K., Avenall, J.N. Chung, J.N., Carro ll, B.F., and Jones, G.W., 2005, Enhancement of Thrust Chamber Cooling with Porous Meta l Inserts, 41nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Tucson, Arizona. [14] Lu, T.J., Stone, H.A. and Ashby, M.F., 199 8, Heat Transfer in Open-Cell Metal Foams, Acat. Mater., 46, pp. 3619-3635. [15] Krishnan, S., Murthy, J.Y. and Garimella, S.V., 2006, Direct simulation of Transport in Open-Cell Metal foam, J. Heat Transfer, 128(8), pp. 793-799 [16] Krishnan, S., Garimella, S.V. and Murthy, J. Y., 2006, Simulation of Thermal Transport in Open-Cell Metal Foams: Effect of Periodic Unit Cell Stru cture, ASME International Mechanical Engineering Congress a nd Exposition, Chicago, Illinois. [17] Chung, J.N., Tully, L. and Kim, J.H., 2006, Evaluation of Open Cell Foam Heat Transfer Enhancement for Liquid Rocket Engines, 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Sacramento, California. [18] Boomsma, K., Poulikakos, D., and Ventikos, Y., 2003, Simulation of Flow through Open Cell Metal Foams Using an Idealized Periodic Cell Structure, Int. J. Heat Fluid Flow, 24, pp. 825. [19] Mills, N.J., 2005, The Wet Kelvin Model for Air Flow through Polyurethane Open-Cell

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64 Foams, J. Mater. Sci., 40, pp. 5845 [20] Yu, Q., Thompson, B. E., and Straatman, A. G., 2006, A Unit-Cube Based Model for Heat Transfer and Pressure Drop in Porous Carbon Foam, J. Heat Transfer, 128(4), pp. 352 [21] Incropera, F. and DeWitt, D., 2003, Fundamentals of Heat and Mass Transfer, Wiley, New York. [22] Calmidi, V.V. and Mahajan, R.L., 2000, Forced Convection in High Porosity Metal Foams, J. Heat Transfer, 122, pp. 557.

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65 BIOGRAPHICAL SKETCH Mo Bai was born on January 21, 1983, in Liaoning, China. He graduated from Tsinghua High School, Beijing, China, in 2001. He atte nded Tsinghua University and received his Bachelor of Engineering, majoring in hydraulic engineering in the summer of 2005. Since then, he has been pursuing a Master of Science degree in mechanical engineering while working as a graduate research/teaching assistant.


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