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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 OPENCELL 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 penC 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 OPENCELL FOAMS ................... 15 3.1 Geometry Simplification for OpenCell 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 OPENCELL 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 31 Constants of Equation (322) ........................................... ........ .................................. 31 32 Parameters of experiments from Calmidi and analytical model.............. ... .............32 33 Foam parameters comparison between experiments from Calmidi and analytical m o d e l ................... ...................3...................2.......... 41 Com prison of different m eshes' results ........................................ ....................... 44 51 Micro open channel and foam filled channel model requirements............................... 57 52 Headtohead comparison of open channel and foamed channel ......................................58 53 Heat transfer enhancement of foamed channel over open channel............................... 58 54 V elocity ratio at equal pressure drop ........................................ ........................... 58 55 Headtohead comparison under rocket condition.................................. ............... 59 56 Comparison of open and foamed channels' performance.................... ...............59 LIST OF FIGURES Figure page 11 Construction of a regenerative cooling tubular thrust chamber........................................5 12 Cutaway of a tubular cooling jacket ................................................... .... ...........6 13 Typical heat transfer rate intensity distribution for liquid propellant rocket.................... 14 Simplified schematic of regenerative cooling system of liquid propellant rocket...............7 15 Section AA of Fig. 14 and details of cooling channel .............. ................... ......... ...... 7 16 Different configurations of the cooling channel in thrust chamber ..............................8 21 P hotos of alum inum foam ................................................................................ ...... ... 14 31 Schem atic of a single cell in the simplified m odel ................................. ................32 32 M odel details................................. ........................................................... ............... 33 33 3D scheme atic of the m odel ........................................................ ...... ...........................33 34 Heat transfer network of analytical model....................... ..........................34 35 Schematic of vertical strut fin model ............................ ................................ 34 3 6 H stru t m o d e l ............................................................................................................... 3 5 37 M odel for coolant temperature evaluation................................... .......................... 35 38 Cylinder diameter as function of relative foam density predicted by analytical model, comparing with ERG's data of aluminum foams ..........................................................36 39 Surface area density as function of relative foam density predicted by analytical model, comparing with ERG's data of aluminum foams ...............................................36 310 Nusselt number prediction made by analytical model compared with Calmidi and Mahaj an's experimental data for 5 ppi aluminum foam...................................................37 311 Nusselt number prediction made by analytical model compared with Calmidi and Mahaj an's experimental data for 10 ppi and 20 ppi aluminum foam ...............................37 312 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 41 Schematic of boundary cell and interior cell in opencell foam.....................................44 42 Comparison of single cell model and real foam structure ...........................................45 43 Geom etry creation of a single cell ............................................................................... 45 44 2D periodic model ........................ ........ .. ... ... .. .................. 46 45 1D periodic model ........................ ........ .. ... ... .. ................. 46 46 M esh of a single cell m odel (coarse grids) ............................................. ............... 47 47 Details of the meshes on filaments (medium grids)................ .................47 4 8 G rid s d istrib u tio n ........................................ .................. ........................ ................. .. 4 8 49 Velocity profile along flow direction through the cell..................... ............. ............... 48 410 Pressure distribution along flow direction through the cell......................... ............49 411 Velocity contours in three planes around the cell............... ........................................... 50 412 Static pressure contours in three planes around the cell ....................................... 51 413 Pressure drop versus inlet velocity and comparison with experimental data ....................52 51 Notional design strategy for foamfilled channels............................................... 59 52 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.(318) 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 (Hcylinder) to the coolant qv Heat transfer rate from a single vertical (Vcylinder) 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 (Vtype) cylinders at x T2 Temperature of vertical (Vtype) cylinders at x+a ,znlet Inlet air temperature T,outlet Outlet air temperature Tb Bulk fluid temperature To Coolant temperature Th Temperature of horizontal (Htype) cylinder Ts Temperature of vertical (Vtype) 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 Nondimensional variable defined by Ts, To, and Tw Nondimensional 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 (Vtype) cylinder Horizontal (Htype) cylinder Coolant Horizontal (Htype) cylinder Vertical (Vtype) 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 breakthrough 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 diamondshaped 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 opencell 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 spherecentered open cell tetrakaidecahedron. This model is very similar to the actual metal foams' microstructure of thin ligaments that form a network of interconnected opencells. 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 11. 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 12. 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 13, 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 12, crosssection area at section B is the smallest which can generate the largest coolant velocities there. Figure 14 and 15 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 hightemperature 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 16. The conventional microchannel heat exchanger is shown in Figure 16A. 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 16B, 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 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 11. 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 12. Cutaway of a tubular cooling jacket. The cooling tubes have variable crosssection 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 13. 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 14. Simplified schematic of regenerative cooling system of liquid propellant rocket. Recreated from reference Carlos [3]. Cooling Channel ", ~ external environment noZle interl wall center Section AA Channel details Figure 15. Section AA of Figure 14 and details of cooling channel. Recreated from reference Carlos [3]. A B C Figure 16. 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 OPENCELL FOAMS 2.1 Heat Transfer Enhancement Opencell 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 21 shows several pictures of typical aluminum foam. The use of opencell foam to enhance heat transfer has been investigated widely. Koh and Colony [4] and Koh and Stevens [5] investigated the heat transfer enhancement of forcedconvection in a channel filled with high thermal conductivity opencell 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 foamlike material and fibrous media to enhance forcedconvection 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 highthermal conductivity filler, compared to empty channels, reached a factor of 2540 enhancement for water and 200400 for nitrogen gas. Bartlett and Viskanta [8] developed a mathematical model to predict the enhancement by high thermal conductivity porous media in forcedconvection duct flows. They concluded that a 530 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 singlephase 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 fourfold relative to a plain tube. They conclude that porous matrix inserts offer a significant advantage in cooling, providing a jitterfree 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 opencell 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 opencell 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 opencell 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 opencell 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 bodycentered cubic model [15], they developed other models based on facecentered cubic (FCC), and A15 lattice, which is similar to WeairePhelan 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 opencell foams and defined a new cell structure. Their new model was based on WeairePhelan structure. This structure reduced the surface energy by 0.3% compared to tetrakaidecahedron [18]. The WeairePhelan structure was further "wetted" by Surface Evolver. Boomsma et al. [18] used that model to investigate pressure drop and velocity field in opencell 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 OpenCell Foams 2.4.1 Polyurethane Foams Some researchers investigated other opencell 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 cubebased 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 21. 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 OPENCELL FOAMS 3.1 Geometry Simplification for OpenCell 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 31. The ligament structure is composed of two types of struts. The vertical struts called Vtype struts, are perpendicular to the flow direction (x) while the horizontal struts, called the Htype struts are on the plans (xy) that is parallel to the flow direction. Figure 32 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 3D schematic of the foam model is given in Figure 33 where two rows in the downstream direction and four columns for each row in the crossstream direction are shown to illustrate the foam structure. The heat transfer mechanisms are explained in terms of a network as shown in Figure 34. The heated top wall is the heat source that interacts with the Vtype foam ligaments (fins) through conduction and also supplies heat to the coolant by convection through unfinned surfaces. The Vtype struts, which act as fins receive heat from the wall and then pass the heat to the coolant by convection and to the Htype struts by conduction. The convection heat transfer between the flow and the Vtype struts will be modeled as heat transfer for flow over tube banks. The Htype struts will lose heat to the flow by convection and also transfer some heat to downstream Vstruts by conduction. 3.2 Mathematical Transport Model and Heat Transfer Equations 3.2.1 Vtype Struts A standard "fin" analysis is applied for a vertical cylinder as shown in Figure 35. The governing equation for a fin is given as below: d 2 Ts 4hl d2T T4h T)= 0 (31) 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 Vstrut * To : coolant temperature Boundary Conditions: dT dy yH Solve for cosh m, (H y) cosh m,H where mi = From Eq. (32), the rate of heat dissipated to the coolant from the strut q, in watts, is (32) q ikAOT YO 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 (33) 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)) (34) cosh mH tanh m,H T, (x) = (T, T,(x)) + T (x) (35) mH Heat transferred from a single Vcylinder to the coolant is q, = idH h [T(x) T(x)] (36) 3.2.2 Htype Struts A similar "fin" analysis as the vertical cylinder is applied for the horizontal cylinders that connect the vertical cylinders (Figure 36). The governing equation should have a similar form as a Vcylinder though extra terms need to be introduced to account for the angle (less than 90 degree) between Hcylinder 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 Htype cylinders. Here proposes two methods. First method The following equation will be used: d2TT 4h d 4h Tc) = 0 (37) 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 Hstrut * To: coolant temperature o h lx=0o Boundary condition: 0 Th T2 T1 and T2 are the temperature of Vtype cylinders at x and x+a. They can be evaluated from Ts solved in part a. For a specific Htype cylinder, T1 and T2 are constant. Solution of Eq.(37) is: (T T) sinh[m2 ( x)] + (T T) sinh(m2x) Th(x) Tc+ (38) 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 = qlq2 = M2(7 + T2 2Tc) (39) 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 Htype cylinder, thus T = (T, (x) + T (x + a)). 2 The heat transferred to the coolant from a single Hcylinder is: cosh(m2a)l 1 _ qh =M2 sinh2a) (T(x)+T(x + a)T(x)T(x+a)) (310) 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 Htype cylinders have identical temperature distribution with Vtype cylinders along x direction. Thus, for a single Htype cylinder, the heat transfer rate from it to coolant can be represented as, qh = rd ah,(Th(x) T(x)) (311) 2 where rda is a Htype cylinder's surface area, h2 is heat transfer coefficient for Htype 2 cylinders, and Ts(x)Tc(x) is the temperature difference between cylinders and coolant. From Eq.(35), the heat transfer rate can be further written in the form, tanh mH qh = )id ah2 (7 T(x)) (312) 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 37 shows the schematic. mC [T (x + Ax) T =(x)]= Nq, + Nq, + q (313) Ax1 Ax1 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 Vtype struts q, can be evaluated from Eq. (36). Heat transfer from Htype struts q, can be evaluated from Eq. (310). Heat transfer from bare wall surface can be calculated from: q, = 7Axh[T, T(x)] (314) 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. (310) will be used for relatively high Reynolds number (>2*104). From Eq. (35),  tanhm1H Tx) = H (T, T(x))+ T(x). Let's further assume T,(x)= (T, T)ei + T where 1 is mH to be determined. Plug into Eq. (35), I(x) x ( tanhmH tanh mH ( )e T,(x3 T1)(xc = (T, Th(x)) =o(r, mH mH So Eq. (310) 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+ ela) sinh(m2a) mH l7 )e 'x Tc) q cosh(m2a) 1 tanh(mH) 1+ e')(T T) = ( )(TT) sinh(m2a) mH Plugging Eq. (36), (314), and (315) into Eq. (313) yields: mC,[[T(x + x) (x)] T (x)) tanh mH + rAxh, [T x1 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 (316) as Ax 0 Tc(x) dT STT J' dx T f x)d(T ) T To /l'dx 0 T,)e (xa)) (315) (316) T TTO _>T 'x > T(x)= (, T)ex + 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 (317) where, 1 1 tah H cosh(ma) 1 tanh mH + e , = [M, tanh mH + M (1+esi)+n ) mC a a sinh(mza) mlH (318) 1 can be determined by iterative method. Second method (low Re) Eq.(312) will be used for relatively low Reynolds number (<2* 104). The energy balance Eq.(313) still holds for this case. Plug in Eq.(36), (312), and (314) to Eq.(313) yields, mC, [Te(x + Ax) Te(x)] So, T(x + Ax) T(x) T T (x) T (x)) tanh mH + 7Axh [T T (x)] Ax1 a Ax1 H tanhmH +2 x2 H rd ah2 t (T T(x)) a a 2 mlH 1[M, tanh + 7, + 5dH htanh mH]Ax [vC, 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.(319) over x, we have: (319) T (x) = (T )ex +T (320) Eq.(320) has the same form with Eq.(317) 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 (321) Since T(x) = (T, T)e + T+ from Eq. (317) and (320), so Q(x)= mCp(T TO)(1e X) (322) where 1 can be determined from Eq. (318) or Eq.(319). For a channel of length L, the total heat transfer is Q(L) = mC,(Tw TO)( eL) (323) 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 Vtype cylinders, Htype 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. Vtype 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 (324) 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 (325) 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. ad Eq. (324) 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 intercylinders is neglected for low Re number cases in this study. Equation for flow over a single cylinder from [21] is Nu = CRemax Pr13 (326) where C and m can be found from Table 31. Htype cylinders h2 For Htype 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 (327) 2*(a* 5/2 d) The other parameters are calculated as the same as Vtype 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' (328) Reynolds number and Nusselt number are defined as Re = (329) vf hD Nu =h (330) kf For this case, only the bottom wall is heated, so for Eq. (329) and (330) D=2H, where H is the height of the metal foamed cooling channel. And hw can be calculated from Nukf h= Nk (331) 2H 3.2.6 Equivalent Heat Transfer Coefficient To calculate Nusselt number and equivalent heat transfer coefficient of opencell foams is the ultimate aim of the analytical model. The equivalent heat transfer coefficient is defined as, hequai (332) 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.(320), 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.(332) yields, IL QlL eq"al L(1 exp(L))(T To) Plug in Eq. (323) for Q, equal C(T T)(1 e)lL) S L(1e L)(T TO) = Mi2CP If the second method (low Reynolds number) is used for Htype cylinders heat transfer, Eq.(319) can be used for 1. Thus, hequal has the final form 1 5r1idH tanh mH hei= M tanh mH +77hw + h2 (333) a2 a2 mH From Eq.(333), 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 (334) 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. (334) can be rewritten as, (5S +3)r d p = ( )2 (335) 20 a d 20 /2 (336) a (5 + 3), The filament diameter d is calculated from Eq.(336) 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 38). 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. (336), surface area density can be represented as, 5.97 /2 A p71/2 (337) a Eq (337) is plotted in Figure 39 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 22 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 23 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 23 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 310, Figure 311, and Figure 312, 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 (338) vf Nu= ual (339) kf where H is height of foam, V is inlet velocity of coolant, hequa, is equivalent heat transfer coefficient of foam defined in Eq. (332), 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) (340) 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 (341) Effective heat transfer coefficient is defined as: hqual =Q/ A(T T) (342) where A is the total heated surface area and T7 is the mean surface temperature Reynolds number is defined as: Re =(d ) (343) /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 (344) 2 Reynolds number is defined as: Re 2VH (345) 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 313 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 31. Constants of Equation (322), recreated from [21] ReD C m 0.44 0.989 0.330 440 0.911 0.385 404,000 0.683 0.466 400040,000 0.193 0.618 Table 31. Continued 40,000400,000 0.027 0.805 Table 32. 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 33. 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 310 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 31. Schematic of a single cell in the simplified model InsulatimO Figure 32. Model details Coolant Heated Wall  I lOm, 5000 rows of Vtype cylinders Figure 33. 3D schematic of the model Conduction Heat transfer  I G 'Conveclion I Heat transfer Figure 34. Heat transfer network of analytical model T=Tw I y=o y=H Figure 35. Schematic of vertical strut fin model Conduction Heat transfer Convection Heat transfer x Flow Dirci On' a  x+a a Figure 36. Hstrut model 0 Coolant Entry  Vtype Cyli der  a TO I_ _, )Side Wall Figure 37. 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 38. 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 39. 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 310. 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 311. 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 312. 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 313. 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 OPENCELL FOAMS 4.1 Introduction to Single Cell Model Opencell 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 volumeaveraged, semiempirical equations. That is a macroscopic approach, which neglects smallscale details of opencell 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 opencell foams due to their property of repeated cell structure. That is the microscopic approach. Using microscopic approach to simulate pressure drop in opencell 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 opencell foam is almost unified, because the multifilament in foam greatly increases the intensity of turbulence in flow which flattens out the velocity gradient and makes the boundary layer very thin (Figure 41). Thus, because of the unit cell structure and nearly unified velocity in opencell 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 41. 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 noslip condition at wall and velocity at boundaries has much larger velocity gradient. To simulate the microstructure of opencell foam (typically metal foams), a spherecentered tetrakaidecahedron structure is constructed (Figure 42A). That structure is very similar to the real microstructure of metal foam (Figure 42B 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 surfacearea to volume ration. Tetrakaidecahedron is seen in reality when soap foam is observed [18]. The spherecentered Kelvin cell can mimic the real metal foam's microstructure 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 spherecentered tetrakaidecahedron can represent the real microstructure of metal foams very well. In order to generate the spherecentered 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 spherecenter Kelvin structure. Figure 43 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 "2Dperiodic" model because periodic or symmetric boundary conditions are applied in 2 directions (Y, Z directions) except in the streamwise direction (X direction). A diagram of this is shown in Figure 44. The second model treats the cell that is attached to the wall and is termed the "iDperiodic" 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 45. 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 44, 45). 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. Noslip 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 46 shows the meshed Kelvin cell. Figure 47 provides the mesh details at cell's filaments. Figure 48 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 2D periodic model in which inlet velocity is 4m/s and cell size is 10ppi. Figure 49 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 410 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 (41) 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 41. 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 49 and 410. There are three regions where the pressure drop is very significant, from Figure 410. The three regions are inlet of cell, center of cell and outlet of cell. That agrees with the velocity profile in Figure 49, 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 411 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 41 1(A) is a 3D 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 411(B). Figure 412 provides static pressure contours of the same three planes. High pressure can be found where the flow encounters with the ligaments (Figure 412(B), (C)). More data were obtained for 2D periodic and 1D 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 413 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 noslip 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 41. 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 41. Schematic of boundary cell and interior cell in opencell foam Figure 42. Comparison of single cell model and real foam structure. A) single cell model used in this study. B) SEM photo of aluminum foam. Figure 43. Geometry creation of a single cell Outlet Inlet S.x Periodic Directions Y, Z Figure 44. 2D periodic model Symmetric Plane Outlet Inlet  Periodic Dire Figure 45. D per ion Figure 45. 1D periodic model Wall Jv" Figure 46. Mesh of a single cell model (coarse grids) .. .: .i:i.:..j. l /r . .' I .. .'  '.  M' Figure 47. Details of the meshes on filaments (medium grids) 47 ~ ~ ~ ~ ~ ~~~~~ ,: .,.. , ,' ., TTTr :t~~~~~~f::,.,,~~~~~~~~~~ ",,r,,. .,. _', "; ,: l :', it I:, : : .. ,' ., _. ,.., :, : :. '.  '. : J :; . Fiur 47.3' Deal fte ehso ilaet (mdu gis .""47 4.4 _____ i4 .. , .    C  .s  4 35  : .. M m M h S3 AFine 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 49. 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 410. Pressure distribution along flow direction through the cell YX 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 9.30e01 .2) 01,. 3.10e01 ( O.OOe+00 (C) (D) Figure 411. Velocity contours in three planes around the cell. A) 3D 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 YX 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 ' .8,4901 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 412. Static pressure contours in three planes around the cell. A) 3D 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 AWall Cell 2 20 C 15 8 10 5 0 0 2 4 6 8 10 12 Velocity (m/s) Figure 413. 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 51. 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 51 shows the scheme. Table 52 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= (51) pu /2 f is found to be constant 0.05 at high Re numbers for commercial metals. Heat transfer: NuD = 0.021Re8 Pro 5 (52) hH where Re= Hu/, Nu= k Comparisons of heat transfer between an open channel and foamfilled channels are shown in Figure 52, Table 53 and Table 54. Figure 52 shows foamed channel has significant heat transfer enhancement over open channel, when they have the same mass flow rate and pressure drop. Table 53 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 54 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 55. 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 56. 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 313. 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) (53) 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 413. The relative error is about 30%, with a confidence of 90%. p =10% (10 to 1) (54) 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) (55) v Since Re ~ v, so A Re  =3.3% (10tol) (56) Re From the heat transfer model uncertainty analysis and Figure 313, 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. (53). So the uncertainty of A is AA=0.3*1088=326.4. And from Eq. (56), 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% (57) ha 49581 The heat transfer coefficient of metal foamed channel can be represented as, hqua = 49581W/m2K 29.1% (58) 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 51. 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 52. Headtohead comparison of open channel and foamed channel Working Inlet Temperature Inlet Channel Geometry fluid Temperature of heated base Velocity Length Width Height Open 10250 nnel H2 100K(180R) 800K(1440R) 10250 m 2mm 4mm Channel m/s Foamed 2122m ame H2 100K(180R) 800K(1440R) 248m/s Im 2mm Channel m Table 53. 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 54. 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 55. Headtohead 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 56. 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 51. Notional design strategy for foamfilled 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 52. 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 opencell 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 intercylinder or intertube effects. However, the real opencell 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 opencell 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 microstructure 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 underdeveloped 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. 699717. [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. 324330. [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. 309311. [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. 301309. [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 HighThermalConductivity Filler. Heat Exchange with Local Thermal Equilibrium Inside the Permeable Matrix," J. Engineering Physics Thermophysics, 47, pp. 748757. [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. 19. [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. 11891204. [10] Boomsma, K., Poulikakos, D. and Zwick, F., 2003, "Metal Foams as Compact High Performance Heat Exchangers," Mechanics of Materials, 35, pp. 11611176. [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. 671681. [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. 14511454. [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 OpenCell Metal Foams", Acat. Mater., 46, pp. 36193635. [15] Krishnan, S., Murthy, J.Y. and Garimella, S.V., 2006, "Direct simulation of Transport in OpenCell Metal foam," J. Heat Transfer, 128(8), pp. 793799 [16] Krishnan, S., Garimella, S.V. and Murthy, J.Y., 2006, "Simulation of Thermal Transport in OpenCell 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. 825834. [19] Mills, N.J., 2005, "The Wet Kelvin Model for Air Flow through Polyurethane OpenCell Foams", J. Mater. Sci., 40, pp. 58455851 [20] Yu, Q., Thompson, B. E., and Straatman, A. G., 2006, "A UnitCube Based Model for Heat Transfer and Pressure Drop in Porous Carbon Foam," J. Heat Transfer, 128(4), pp. 352360 [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. 557565. 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. PAGE 1 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 PAGE 2 2007 Mo Bai PAGE 3 To my parents PAGE 4 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. PAGE 5 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 OPENCELL FOAMS.....................................................................9 2.1 Heat Transfer Enhancement...............................................................................................9 2.2 Experiments................................................................................................................ ......10 2.3 CFD Simulation and Numerical Model............................................................................11 2.4 Other OpenCell Foams....................................................................................................13 2.4.1 Polyurethane Foams...............................................................................................13 2.4.2 Carbon Foams.........................................................................................................13 3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPENCELL FOAMS...................15 3.1 Geometry Simplification for OpenCell Foam Filled Channels.......................................15 3.2 Mathematical Transport Model and Heat Transfer Equations.........................................16 3.2.1 Vtype Struts...........................................................................................................16 3.2.2 Htype 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 PAGE 6 vi 4 CFD SIMULATION OF PRESSURE DROP IN OPENCELL 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 PAGE 7 vii LIST OF TABLES Table page 31 Constants of Equation (322).............................................................................................31 32 Parameters of experiments from Calmidi and analytical model........................................32 33 Foam parameters comparison between e xperiments from Calmidi and analytical model.......................................................................................................................... ........32 41 Comparison of different meshes results...........................................................................44 51 Micro open channel and foam f illed channel mode l requirements....................................57 52 Headtohead comparison of open channel and foamed channel......................................58 53 Heat transfer enhancement of foamed channel over open channel....................................58 54 Velocity ratio at equal pressure drop.................................................................................58 55 Headtohead comparison under rocket condition.............................................................59 56 Comparison of open and foamed channels performance..................................................59 PAGE 8 viii LIST OF FIGURES Figure page 11 Construction of a regenerative cooling tubular thrust chamber...........................................5 12 Cutaway of a tubular cooling jacket....................................................................................6 13 Typical heat transfer rate intensity distribution for liquid propellant rocket.......................6 14 Simplified schematic of regenerative c ooling system of liquid propellant rocket...............7 15 Section AA of Fig. 14 and details of cooling channel......................................................7 16 Different configurations of the cooling channel in thrust chamber.....................................8 21 Photos of aluminum foam..................................................................................................14 1 Schematic of a single cell in the simplified model............................................................32 2 Model details............................................................................................................... .......33 33 3D schematic of the model...............................................................................................33 4 Heat transfer networ k of analytical model.........................................................................34 35 Schematic of vertical strut fin model.................................................................................34 36 Hstrut model.............................................................................................................. .......35 37 Model for coolant temperature evaluation.........................................................................35 38 Cylinder diameter as function of relative foam density predicted by analytical model, comparing with ERGs data of aluminum foams..............................................................36 39 Surface area density as f unction of relative foam density predicted by analytical model, comparing with ERGs data of aluminum foams..................................................36 310 Nusselt number prediction made by analyt ical model compared with Calmidi and Mahajans experimental data for 5 ppi aluminum foam....................................................37 311 Nusselt number prediction made by analyt ical model compared with Calmidi and Mahajans experimental data for 10 ppi and 20 ppi aluminum foam................................37 312 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 41 Schematic of boundary cell and in terior cell in opencell foam........................................44 PAGE 9 ix 42 Comparison of single cell mode l and real foam structure.................................................45 43 Geometry creation of a single cell.....................................................................................45 44 2D periodic model......................................................................................................... ...46 45 1D periodic model......................................................................................................... ...46 46 Mesh of a single cell model (coarse grids)........................................................................47 47 Details of the meshes on filaments (medium grids)...........................................................47 48 Grids distribution......................................................................................................... ......48 49 Velocity profile along flow direction through the cell.......................................................48 410 Pressure distribution along fl ow direction through the cell...............................................49 411 Velocity contours in th ree planes around the cell..............................................................50 412 Static pressure contours in three planes around the cell....................................................51 413 Pressure drop versus inlet velocity and comparison with experimental data....................52 51 Notional design strategy for foamfilled channels.............................................................59 52 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.(318) 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 (Hcylinde r) to the coolant qv Heat transfer rate from a single vertical (Vcylinder) 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 (Vtype) cylinders at x T2 Temperature of vertical (Vtype) cylinders at x+a ainletT Inlet air temperature aoutletT Outlet air temperature bT Bulk fluid temperature Tc Coolant temperature Th Temperature of horizontal (Htype) cylinder Ts Temperature of vertical (Vtype) 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 Nondimensional variable defined by Ts, Tc, and Tw c Nondimensional 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 (Vtype) cylinder 2 Horizontal (Htype) cylinder c Coolant h Horizontal (Htype) cylinder v Vertical (Vtype) 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 breakthrough 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 diamondshaped 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 opencell 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 spherecentered 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 opencells. 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 11. 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 12. 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 13, 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 12, crosssection area at section B is the smallest PAGE 18 4 which can generate the largest coolant velocities there. Figure 14 and 15 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 hightemperature 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 16. The conventional microchannel heat exchanger is shown in Figur e 16A. 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 16B, 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 11. 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 12. Cutaway of a tubular cooling j acket. The cooling tubes have variable crosssection area to allow the same number of tubes at nozzle throat and nozzle exit. Recreated from reference Sutton [2]. Figure 13. 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 14. Simplified schematic of regenerative cooling system of liquid propellant rocket. Recreated from reference Carlos [3]. Figure 15. Section AA of Figure 14 and deta ils of cooling channel. Recreated from reference Carlos [3]. PAGE 22 8 Figure 16. 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 PAGE 23 9 CHAPTER 2 PREVIOUS WORK ON OPENCELL FOAMS 2.1 Heat Transfer Enhancement Opencell 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 opencell 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 forcedconvection in a channel filled with hi gh thermal conductivity opencell 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 foamlike material and fibrous media to enhance forcedconvection 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 PAGE 24 10 channels with a highthermal conductivity filler, compared to empty channels, reached a factor of 2540 enhancement for water and 200400 for nitrogen gas. Bartlett and Viskanta [8] developed a mathem atical model to predict the enhancement by high thermal conductivity porous media in forcedc onvection duct flows. They concluded that a 530 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 fourfold relative to a plain tube. They conclude that porous matrix inserts offer a significant advantage in cooling, providing a jitterfree 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 opencell 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 opencell 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 PAGE 25 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 opencell 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. PAGE 26 12 Krishnan et al. [15] carried out a direct simulation of the transport phenomenon in opencell 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 bodycentered cubic model [15], they developed other models based on facecentered c ubic (FCC), and A15 lattice, which is similar to WeairePhelan 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. PAGE 27 13 Boomsma et al. [18] developed a new appro ach to modeling flow through opencell foams and defined a new cell structure. Their new model was based on WeairePhelan structure. This structure reduced the surface energy by 0.3% compared to tetrakaidecahedron [18]. The WeairePhelan structure was furt her wetted by Surface Evolver. Boomsma et al. [18] used that model to investigate pressure drop and velo city field in opencell 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 OpenCell Foams 2.4.1 Polyurethane Foams Some researchers investigated other opencell 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 cubebased 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. PAGE 28 14 A B C Figure 21. 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 PAGE 29 15 CHAPTER 3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPENCELL FOAMS 3.1 Geometry Simplification for Op enCell 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 31. The ligament structure is composed of two types of struts. The vertical struts called Vtype struts, are perpendicular to the flow dire ction (x) while the horizontal struts, called the Htype struts are on the plans (xy) that is para llel to the flow direction. Figure 32 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 3D schematic of the foam model is gi ven in Figure 33 where two rows in the downstream direction and four columns for each ro w in the crossstream direction are shown to illustrate the foam structure. The heat transfer mechanisms are explained in terms of a network as shown in Figure 34. The heated top wall is the heat source that in teracts with the Vtype foam ligaments (fins) through conduction and also s upplies heat to the coolant by convection through unfinned surfaces. The Vtype struts, which act as fins receiv e heat from the wall and then pass the heat to the coolant by convection and to the Htype stru ts by conduction. The convection heat transfer between the flow and the Vtype struts will be m odeled as heat transfer for flow over tube banks. The Htype struts will lose heat to the flow by convection and also transfer some heat to downstream Vstruts by conduction. PAGE 30 16 3.2 Mathematical Transport Model and Heat Transfer Equations 3.2.1 Vtype Struts A standard fin analysis is applied for a ve rtical cylinder as show n in Figure 35. The governing equation for a fin is given as below: 2 1 24 ()0s sc sdTh TT dykd (31) 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 Vstrut Tc : coolant temperature Boundary Conditions: 00 s w y s yHTT dT dy Solve for 1 1cosh() () cosh s cwcmHy TTTT mH (32) where 1 14sh m kd From Eq. (32), 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 PAGE 31 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 (33) 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 (34) 1 1tanh ()(())()swccmH TxTTxTx mH (35) Heat transferred from a single Vcylinder to the coolant is 1[()()]VscqdHhTxTx (36) 3.2.2 Htype Struts A similar fin analysis as the vertical cylinde r is applied for the horizontal cylinders that connect the vertical cylinders (Figure 36). The governing equation should have a similar form as a Vcylinder though extra terms need to be introduced to account for the angle (less than 90 degree) between Hcylinder 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 Htype cylinders. Here proposes two methods. PAGE 32 18 First method The following equation will be used: 2 2 24 ()0h hc sdTh TT dxkd (37) 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 Hstrut Tc : coolant temperature Boundary condition: 1 0 2 h x h xaTT TT T1 and T2 are the temperature of Vtype cylinders at x and x+a. They can be evaluated from Ts solved in part a. For a specific Htype cylinder, T1 and T2 are constant. Solution of Eq.(37) is: 1222 2()sinh[()]()sinh() () sinh()cc hcTTmaxTTmx TxT ma (38) 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 (39) PAGE 33 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 Htype cylinder, thus 1 (()()) 2cccTTxTxa The heat transferred to the cool ant from a single Hcylinder is: 2 2 2cosh()1 (()()()()) sinh()hssccma qMTxTxaTxTxa ma (310) 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 Htype cylin ders have identical te mperature distribution with Vtype cylinders along x direction. Thus, fo r a single Htype cylind er, the heat transfer rate from it to coolant can be represented as, 25 (()()) 2hscqdahTxTx (311) where 5 2 da is a Htype cylinders surface area, h2 is heat transfer coefficient for Htype cylinders, and Ts(x)Tc(x) is the temperature difference between cylinders and coolant. From Eq.(35), the heat transfer rate can be further written in the form, 1 2 1tanh 5 (()) 2hwcmH qdahTTx mH (312) 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. PAGE 34 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 37 shows the schematic. [()()]pccvvhhwmCTxxTxNqNqq (313) 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 Vtype struts vq can be evaluated from Eq. (36). Heat transfer from Htype struts hq can be evaluated from Eq. (310). Heat transfer from bare wall surface can be calculated from: [()]wwwcqxhTTx (314) 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. (310) will be used for relatively high Reynolds number (>2*104). From Eq. (35), 1 1tanh ()(())()swccmH TxTTxTx mH Lets further assume 0()()lx cwwTxTTeT where l is to be determined. Plug into Eq. (35), 11 0 11tanhtanh ()()(())()lx scwcwmHmH TxTxTTxTTe mHmH So Eq. (310) can be rewritten in the form: PAGE 35 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 (315) Plugging Eq. (36), (314), and (3 15) into Eq. (313) 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 (316) Integrate (316) as 0 x 0() 0'Tcx x c wc TdT ldx TT 0() 0() 'cTx x wc wc TdTT ldx TT PAGE 36 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 (317) where, 21 112 23 21cosh()1tanh 11 [tanh2(1)] sinh()la w pmamH H lMmHMeh mCaamamH (318) l can be determined by iterative method. Second method (low Re) Eq.(312) will be used for rela tively low Reynolds number (<2*104). The energy balance Eq.(313) still holds for this case. Plug in Eq .(36), (312), and (314) to Eq.(313) 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 (319) Integrate Eq.(319) over x, we have: PAGE 37 23 0()()lx cwwTxTTeT (320) Eq.(320) has the same form with Eq.(317) 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 (321) Since 0()()lx cwwTxTTeT from Eq. (317) and (320), so 0()()(1)lx PwQxmCTTe (322) where l can be determined fr om Eq. (318) or Eq.(319). For a channel of length L, the total heat transfer is 0()()(1)lL PwQLmCTTe (323) 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 Vtype cylinders Htype 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. Vtype 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 PAGE 38 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 (324) 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 (325) 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. (324) 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 rcylinders 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 (326) where C and m can be found from Table 31. PAGE 39 25 Htype cylinders h2 For Htype 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 (327) The other parameters are calculated as the same as Vtype 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 (328) Reynolds number and Nusselt number are defined as Re f DV (329) w f hD Nu k (330) For this case, only the bottom wall is heated, so for Eq. (329) and (330) D=2H, where H is the height of the metal foamed cooling channel. And hw can be calculated from 2 f wNuk h H (331) 3.2.6 Equivalent Heat Transfer Coefficient To calculate Nusselt number and equivalent heat transfer coefficient of opencell foams is the ultimate aim of the analytical model. The equi valent heat transfer co efficient is defined as, ()equal wwcQ h ATT (332) PAGE 40 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.(320), 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.(332) yields, 0(1exp())()equal wQlL h LlLTT Plug in Eq. (323) 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 Htype cylinders heat transfer, Eq.(319) can be used for l. Thus, hequal has the final form 1 112 22 1tanh 15 tanhequalwmH dH hMmHhh aamH (333) From Eq.(333), 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. PAGE 41 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 (334) 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 (334) can be rewritten as, 2(553) () 20 d a (335) PAGE 42 28 So 1/220 (553) d a (336) The filament diameter d is calculated from Eq.(336) 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 38). Reasonable agreement is obtained. For surface area density, 3 2(52) [(51)2/] (50.6)Adaadd a dad a d aa From Eq. (336), surface area density can be represented as, 1/25.97Aa (337) Eq (337) is plotted in Figure 39 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. PAGE 43 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 22 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 23 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 23 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 310, Figure 311, and Figure 312, 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, PAGE 44 30 Re f HV (338) equal fhH Nu k (339) where H is height of foam, V is inlet velocity of coolant, equalh is equivalent heat transfer coefficient of foam defined in Eq. (332), 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 (340) 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 (341) Effective heat transfer coefficient is defined as: /()equalssbhQATT (342) where s A is the total heated surface area and s T is the mean surface temperature Reynolds number is defined as: PAGE 45 31 Reoi fVdd (343) 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 (344) Reynolds number is defined as: 2 Re f VH (345) 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 313 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 31. Constants of Equatio n (322), recreated from [21] ReD C m 0.44 0.989 0.330 440 0.911 0.385 404,000 0.683 0.466 400040,000 0.193 0.618 PAGE 46 32 Table 31. Continued 40,000400,000 0.027 0.805 Table 32. 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 33. 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 310 Experiment 0.40mm 3.13mm 10PPI Model 0.55mm 3.13mm Experiment 0.30mm 2.70mm 20PPI Model 0.45mm 2.70mm Figure 311 Experiment 0.55mm 3.80mm 5PPI Model 0.70mm 3.10mm Experiment 0.25mm 1.80mm 40PPI Model 0.20mm 1.50mm Figure 312 Figure 31. Schematic of a single cell in the simplified model PAGE 47 33 Figure 32. Model details Figure 33. 3D schematic of the model PAGE 48 34 Figure 34. Heat transfer ne twork of analytical model Figure 35. Schematic of vertical strut fin model PAGE 49 35 Figure 36. Hstrut model Figure 37. Model for coolant temperature evaluation PAGE 50 36 0 1 2 3 4 5 6 00.020.040.060.080.10.120.14 Relative foam densityCylinder diameter(104m) 10ppi(Experiment) 20ppi(Experiment) 40ppi(Experiment) 10ppi(Model) 20ppi(Model) 40ppi(Model) Figure 38. 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 39. 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. PAGE 51 37 0 500 1000 1500 2000 2500 3000 3500 4000 0500010000150002000025000 ReNu 5ppi(Model) 5ppi(Calmidi and Mahajan [23]) Figure 310. 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 311. Nusselt number prediction made by analytical model compared with Calmidi and Mahajans [22] experimental data for 10 ppi and 20 ppi aluminum foams PAGE 52 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 312. 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 313. Heat transfer coeffi cient predicted by analytical m odel compared with Chung et al. [17] experimental data for 10ppi copper foam PAGE 53 39 CHAPTER 4 CFD SIMULATION OF PRESSURE DROP IN OPENCELL FOAMS 4.1 Introduction to Single Cell Model Opencell 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 volumeaveraged, semie mpirical equations. That is a macroscopic approach, which neglects smallscale details of opencell 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 opencell foams due to their property of re peated cell structure. That is the microscopic approach. Using microscopic approach to simulate pressure drop in opencell 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 opencell foam is almost uni fied, because the multifilament 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 41). Th us, because of the unit cel l structure and nearly unified velocity in opencell 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 41. 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 PAGE 54 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 noslip condition at wall and velocity at boundar ies has much larger velocity gradient. To simulate the microstructure of opencell foam (typically metal foams), a spherecentered tetrakaidecahedron structure is constructed (Figur e 42A). That structure is very similar to the real microstructure of metal foam (Figure 42B 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 surfacearea to volume ration. Tetrakaidecahedron is seen in reality when soap foam is observed [18]. The spherecentered Kelvin cell can mimic the real metal foams microstructure 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 spherecentered tetrakaidecahedron can represent the real microstructure 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 spherecenter Kelvin structure. Figure 43 shows the process schematic. PAGE 55 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 Dperiodic model because periodic or symmetric boundary conditions are applied in 2 directions (Y, Z directions) except in the streamwise direction (X direction). A diagram of this is shown in Figure 44. The second model treats the cell that is attached to the wall and is termed the Dperiodic 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 45. 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 44, 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. Noslip 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 46 shows the meshed Kelvin cell. Figure 47 PAGE 56 42 provides the mesh details at cells filaments. Fi gure 48 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 2D periodic model in which inlet velocity is 4m/s and cell size is 10ppi. Figure 49 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 410 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 (41) 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 41. 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 PAGE 57 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 49 and 410. There are three regions where the pressure drop is very si gnificant, from Figure 410. Th e three regions are inlet of cell, center of cell and outlet of cell. That agrees with the velocity profile in Figure 49, 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 411 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 411(A) is a 3D 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 412 provides static pressure contours of the same three planes. High pressu re can be found where the flow encounters with the ligaments (Figure 412(B), (C)). More data were obtained for 2D periodic and 1D 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 413 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 PAGE 58 44 nice agreement was obtained. It can be found that the wall cell induces a little more pressure drop because the noslip 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 41. 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 41. Schematic of boundary cell and interior cell in opencell foam PAGE 59 45 A B Figure 42. 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 43. Geometry creation of a single cell PAGE 60 46 Figure 44. 2D periodic model Figure 45. 1D periodic model PAGE 61 47 Figure 46. Mesh of a single cell model (coarse grids) Figure 47. Details of the meshes on filaments (medium grids) PAGE 62 48 Figure 48. 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 1.270.770.270.230.731.23 x position (mm)Velocity (m/s) Coarse Mesh Medium Mesh Fine Mesh Figure 49. Velocity profile along flow direction through the cell PAGE 63 49 4 2 0 2 4 6 8 10 12 1.270.770.270.230.731.23 x position (mm)Pressure (pascal) Coarse Mesh Medium Mesh Fine Mesh Figure 410. Pressure di stribution along flow di rection through the cell PAGE 64 50 Figure 411. Velocity contours in three planes around the cell. A) 3D 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 PAGE 65 51 Figure 412. Static pressure contours in three pl anes around the cell. A) 3D 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 PAGE 66 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 413. Pressure drop versus inlet veloc ity and comparison with experimental data PAGE 67 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 51. 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 51 shows the scheme. Ta ble 52 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 (51) PAGE 68 54 f is found to be constant 0.05 at high Re numbers for commercial metals. Heat transfer: 0.80.50.021RePrDNu (52) where Re/ Hu hH Nu k Comparisons of heat transfer between an open channel and foamfilled channels are shown in Figure 52, Table 53 and Table 54. Figure 52 shows foamed channel has significant heat transfer enhancement over open channel, when th ey have the same mass flow rate and pressure drop. Table 53 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 54 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 PAGE 69 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 55. 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 56. 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 313. 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) (53) PAGE 70 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 413. The relative error is about 30%, with a confidence of 90%. 10% p p (10 to 1) (54) 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) (55) Since Re v so Re 3.3% Re (10 to 1) (56) From the heat transfer model un certainty analysis and Figure 313,0.3001ReequalhA. From regression analysis, A=1088, and Re=304000. The uncertainty of A can be treated as the same as 30% from Eq. (53). So the uncertainty of A is A =0.3*1088=326.4. And from Eq. (56), 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 PAGE 71 57 214434W/mK So, the uncertainty of the equivalent heat transf er coefficient of rockets foamed cooling is, 14434 29.1% 49581equal equalh h (57) The heat transfer coefficient of metal foamed channel can be represented as, 249581W/mK29.1%equalh (58) 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 51. 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 PAGE 72 58 Table 52. Headtohead 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) 10250 m/s 1m 2mm 4mm Foamed Channel H2 100K(180R) 800K(1440R)248m/s 1m 2mm 2122m m Table 53. 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 54. 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 PAGE 73 59 Table 55. Headtohead 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 56. 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 51. Notional design strategy for foamfilled channels PAGE 74 60 0 10000 20000 30000 40000 50000 60000 05001000150020002500300035004000 Pressure Drop (Kpa/m)Heat Transfer Coef. (W/m2K ) Open Channel Foamed Channel Figure 52. Comparison of heat transfer coef vs. pressure drop of open and foamed channels PAGE 75 61 CHAPTER 6 CONCLUSIONS An analytical heat transfer model and a CF D based pressure drop simulation method for opencell 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 intercylinder or in tertube effects. Howeve r, the real opencell 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 opencell 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 microstructure of metal foams which can capture the most important flow phe nomenon in metal foams. A remaining problem PAGE 76 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 underdeveloped 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. PAGE 77 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. 699717. [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. 324330. [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. 309311. [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. 301309. [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 HighThermalConductivity Filler. Heat Exchange with Local Thermal Equilib rium Inside the Permeable Matrix, J. Engineering Physics Thermophysics, 47, pp. 748757. [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. 19. [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. 11891204. [10] Boomsma, K., Poulikakos, D. and Zwic k, F., 2003, Metal Foams as Compact High Performance Heat Exchangers, Mechanics of Materials, 35, pp. 11611176. [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. 671681. [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. 14511454. [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 OpenCell Metal Foams, Acat. Mater., 46, pp. 36193635. [15] Krishnan, S., Murthy, J.Y. and Garimella, S.V., 2006, Direct simulation of Transport in OpenCell Metal foam, J. Heat Transfer, 128(8), pp. 793799 [16] Krishnan, S., Garimella, S.V. and Murthy, J. Y., 2006, Simulation of Thermal Transport in OpenCell 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 OpenCell PAGE 78 64 Foams, J. Mater. Sci., 40, pp. 5845 [20] Yu, Q., Thompson, B. E., and Straatman, A. G., 2006, A UnitCube 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. PAGE 79 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. 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