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

Transcription

1 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 007

2 007 Mo Bai

3 To my parents

4 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 fiancée Wenwen Zhang, for her years of support. iv

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS... iv LIST OF TABLES... vii LIST OF FIGURES... viii NOMENCLATURE...x ABSTRACT... xiii CHAPTER INTRODUCTION ROCKET THRUST CHAMBER COOLING.... History of the Rocket.... Rocket Structure....3 Rocket Thrust Chamber Cooling Regenerative Cooling Challenges on Regenerative Cooling...4 PREVIOUS WORK ON OPEN-CELL FOAMS...9. Heat Transfer Enhancement...9. Experiments CFD Simulation and Numerical Model....4 Other Open-Cell Foams Polyurethane Foams Carbon Foams ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS Geometry Simplification for Open-Cell Foam Filled Channels Mathematical Transport Model and Heat Transfer Equations V-type Struts H-type Struts Fluid Temperature Prediction (Coolant Temperature) Total Heat Transfer Evaluation of Heat Transfer Coefficient Equivalent Heat Transfer Coefficient Investigation of Cylinder Diameter and Surface Area Density Verification of the Analytical Model with Experimental Data Validity of Analytical Prediction (Re=5*0 3 ~ *0 4 ) Validity of Analytical Prediction (Re=*0 4 ~ 6*0 4 )...30 v

6 4 CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS Introduction to Single Cell Model Mesh Generation and Grid Independent Study Simulation Results and Verification FEASIBILITY STUDY OF FOAMED COOLING CHANNELS FOR ROCKET THRUST CHAMBER Feasibility Study and Comparison with Open Cooling Channel Uncertainty Analysis Heat Transfer Model Pressure Drop Simulation Rocket Condition Prediction CONCLUSIONS...6 LIST OF REFERENCES...63 BIOGRAPHICAL SKETCH...65 vi

7 LIST OF TABLES Table page 3- Constants of Equation (3-) Parameters of experiments from Calmidi and analytical model Foam parameters comparison between experiments from Calmidi and analytical model Comparison of different meshes results Micro open channel and foam filled channel model requirements Head-to-head comparison of open channel and foamed channel Heat transfer enhancement of foamed channel over open channel Velocity ratio at equal pressure drop Head-to-head comparison under rocket condition Comparison of open and foamed channels performance...59 vii

8 LIST OF FIGURES Figure page - Construction of a regenerative cooling tubular thrust chamber Cutaway of a tubular cooling jacket Typical heat transfer rate intensity distribution for liquid propellant rocket Simplified schematic of regenerative cooling system of liquid propellant rocket Section A-A of Fig. -4 and details of cooling channel Different configurations of the cooling channel in thrust chamber Photos of aluminum foam Schematic of a single cell in the simplified model Model details D schematic of the model Heat transfer network of analytical model Schematic of vertical strut fin model H-strut model Model for coolant temperature evaluation Cylinder diameter as function of relative foam density predicted by analytical model, comparing with ERG s data of aluminum foams Surface area density as function of relative foam density predicted by analytical model, comparing with ERG s data of aluminum foams Nusselt number prediction made by analytical model compared with Calmidi and Mahajan s experimental data for 5 ppi aluminum foam Nusselt number prediction made by analytical model compared with Calmidi and Mahajan s experimental data for 0 ppi and 0 ppi aluminum foam 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 Schematic of boundary cell and interior cell in open-cell foam...44 viii

9 4- Comparison of single cell model and real foam structure Geometry creation of a single cell D periodic model D periodic model Mesh of a single cell model (coarse grids) Details of the meshes on filaments (medium grids) Grids distribution Velocity profile along flow direction through the cell Pressure distribution along flow direction through the cell Velocity contours in three planes around the cell Static pressure contours in three planes around the cell Pressure drop versus inlet velocity and comparison with experimental data Notional design strategy for foam-filled channels Comparison of heat transfer coef. vs. pressure drop of open and foamed channels...60 ix

10 NOMENCLATURE a A A c A w C C p Cell size Surface area of foam Area of cylinder s cross section Area of heated wall Constant related to the geometry of channel, can be looked up from tables Specific heat of coolant d: Diameter of the cylinder. d i d o f H h : h : h euqal h w : k f : k s : l L m m M m M Inner diameter of test section Outer diameter of test section Friction factor Height of cooling channel Heat transfer coefficient between vertical cylinder and the cooling fluid. Heat transfer coefficient between horizontal cylinder and the cooling fluid. Equivalent heat transfer coefficient of foam filled cooling channel Heat transfer coefficient between bare wall and the cooling fluid. Thermal conductivity of the coolant. Thermal conductivity of the cylinder. Some constant defined in Eq.(3-8) Length of cooling channel Constant related to the geometry of channel, can be looked up from tables Constant calculated from h, k s, and d Constant calculated from h, k s, and d Constant calculated from h, k s, and d Constant calculated from h, k s, and d x

11 m N h Nu N v p Pr Q q h q v q w Re T 0 T T Mass flow rate of coolant Number of horizontal cylinders per unit width Nusselt number Number of vertical cylinders per unit width Pressure Prandtl number Total heat transfer rate to coolant Heat transfer rate from a single horizontal (H-cylinder) to the coolant Heat transfer rate from a single vertical (V-cylinder) to the coolant Heat transfer rate from bare heated wall to the coolant Reynolds number Inlet temperature of coolant Temperature of vertical (V-type) cylinders at x Temperature of vertical (V-type) cylinders at x+a T ainlet, Inlet air temperature T a, outlet Outlet air temperature T b T c T h T s T w V V max x y Bulk fluid temperature Coolant temperature Temperature of horizontal (H-type) cylinder Temperature of vertical (V-type) cylinder Constant temperature of heated wall Average inlet velocity of coolant or volume of the foam Maximum velocity of the coolant X coordinate or direction Y coordinate or direction xi

12 z Z coordinate or direction Greek Symbols α A Surface area density θ Non-dimensional variable defined by T s, T c, and T w θ c Non-dimensional variable defined by T s, T c, T w, and T 0 ρ Relative foam density ρ * Density of foam ρ s η v f Density of solid Ratio of bare wall surface area to the total wall surface area Kinematic viscosity of coolant Subscripts Vertical (V-type) cylinder Horizontal (H-type) cylinder c h v w Coolant Horizontal (H-type) cylinder Vertical (V-type) cylinder Wall xii

13 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 Chair: Jacob N. Chung Major: Mechanical Engineering By Mo Bai December 007 As society pursues the space travel, advanced propulsion for the next generation of spacecraft will be needed. These new propulsion systems will require higher performance and increased durability, despite current limitations on materials. A break-through technology is needed in the thrust chamber. One of the most challenging problems is to cool the hot chamber wall more without creating additional pressure drops in the coolant passage. A promising method is using open cell foam to enhance the heat transfer rate from chamber wall to coolant. However, the pressure drop induced by foams is relatively large and thus becomes a critical issue. The focus of this thesis is the evaluation of heat transfer and pressure drop of open cell foams. A simplified analytical model has been developed to evaluate the heat transfer capability of the foamed channel, which is based on a diamond-shaped unit cell model. The predicted heat transfer results by the analytical model have been compared with experimental data of different Reynolds numbers from other researchers and favorable agreements have been obtained. For the evaluation of pressure drop in open-cell metal foams, direct numerical simulation models of the foam heat exchanger have been built using GAMBIT/FLUENT. The model is based on a structure of sphere-centered open cell tetrakaidecahedron. This model is very similar to the actual metal foams microstructure of thin ligaments that form a network of interconnected xiii

14 open-cells. Grid independence of solution is investigated and simulation results are further compared with experiments. Finally, the feasibility of applying foam filled cooling channel on rocket thrust chamber is investigated. xiv

15 CHAPTER INTRODUCTION ROCKET THRUST CHAMBER COOLING. 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 903.[] In 90, 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 4 patents on rockets that were later bought by United States.. 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

16 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 -. 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..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

17 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 []..3. 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 -. 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 -3, letting the coolest coolant entering at throat area can greatly enhance the heat transfer efficiency. Another method to enhance the heat transfer rate at throat area is increase coolant flow velocities at that area. From Figure -, cross-section area at section B is the smallest 3

18 which can generate the largest coolant velocities there. Figure -4 and -5 show schematics of a liquid propellant rocket s thrust chamber and details of cooling channels [3]. Regenerative cooling method has many merits in the sense of heat transfer efficiency and structure optimization. First, using fuel as the coolant greatly enhances heat transfer efficiency. Because for liquid propellant rocket, the fuel is cryogenic, large temperature difference between coolant and combustion gas can make great heat transfer rate. In addition, after flow through the cooling channel, the fuel has higher temperature and become ready for combustion. Second, the tubular cooling jacket reduces weight of the rocket thrust chamber and also the total weight of rocket, thus greatly increases efficiency. Third, the cooling jacket structure transform thick thrust wall into thin walls of cooling channels, which can reduce thermal stresses..3. Challenges on Regenerative Cooling Today, the needs for longer and faster space travel require rockets with more powerful thrust and also bring challenging requirements to the cooling system. Much higher heat transfer rate is needed for next generation rockets. Even with new advances in high-temperature and high conductivity materials, thrust increases for large liquid propellant rocket engines are limited by the cooling capacity of the cooling jacket. Cooling limits have been extended with the use of film cooling, injector biasing, and transpiration cooling. However, these methods are costly to engine performance since they require that some of the fuel pass through the thrust chamber throat without contributing to thrust. Currently, the vast majority of regenerative cooling rocket engines use either tube bundles or milled rectangular passages as heat exchangers. Several improvements based on the tubular cooling system of rocket thrust chamber are shown in Figure -6. The conventional micro-channel heat exchanger is shown in Figure -6A. The partition walls serve as fins to increase surface area thus enhance heat transfer rate and also support the hot wall. The high 4

19 aspect ratio heat exchanger is shown in Figure -6B, which has larger surface area so can increase the cooling effectiveness. For Figure -6C, metal foam inserts are used in the channel to get even larger heat transfer rate. This study is focused on evaluation of foam filled channel s heat transfer and pressure drop, which has potential application in rocket thrust chamber s cooling system. Figure -. Construction of a regenerative cooling tubular thrust chamber, its nozzle internal diameter is about 5 inch and thrust is about 65,000 lbf. It was originally used in the Thor missile. Recreated from reference Sutton []. 5

20 Figure -. Cutaway of a tubular cooling jacket. The cooling tubes have variable cross-section area to allow the same number of tubes at nozzle throat and nozzle exit. Recreated from reference Sutton []. Figure -3. Typical heat transfer rate intensity distribution for liquid propellant rocket. Peak is at the thrust nozzle throat and nadir is usually at the nozzle exit. Recreated from reference Sutton []. 6

21 Figure -4. Simplified schematic of regenerative cooling system of liquid propellant rocket. Recreated from reference Carlos [3]. Figure -5. Section A-A of Figure -4 and details of cooling channel. Recreated from reference Carlos [3]. 7

22 A B C Figure -6. Different configurations of the cooling channel in thrust chamber. A) Conventional micro cooling channel. B) High aspect ratio cooling channel. C) Metal foamed cooling channel. 8

23 CHAPTER PREVIOUS WORK ON OPEN-CELL FOAMS. Heat Transfer Enhancement Open-cell foam is a kind of porous medium that is emerging as an effective method of heat transfer enhancement, due to its large surface area to volume ratio, high thermal conductivity, and intensified fluid (coolant) fixing. Figure - shows several pictures of typical aluminum foam. The use of open-cell foam to enhance heat transfer has been investigated widely. Koh and Colony [4] and Koh and Stevens [5] investigated the heat transfer enhancement of forced-convection in a channel filled with high thermal conductivity open-cell foam. In their theoretical study, Koh and Colony [4] found that for a fixed wall temperature case, the heat transfer rate increased by a factor of three. For a constant heat flux case, the wall temperature and the temperature difference between the wall and the coolant can be drastically reduced. Koh and Stevens (975) performed experimental work to verify the numerical results of Koh and Colony [4]. Koh and Stevens [5] used a stainless steel cylindrical annulus (.5 ID and. 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. inch. Nitrogen gas was used as the coolant. They found the heat flux increased from 7 to 37 Btu/ft s for a constant wall temperature case and the wall temperature dropped from 450 o F to 350 o F for the constant heat flux case. Hunt and Tien [6] utilized foam-like material and fibrous media to enhance forced-convection for potential application to electronics cooling. Their results showed that a factor of two to four times enhancement is achievable as compared to laminar slug flow in a duct. Maiorov et al. [7] found empirically that the heat transfer rates in 9

24 channels with a high-thermal conductivity filler, compared to empty channels, reached a factor of 5-40 enhancement for water and for nitrogen gas. Bartlett and Viskanta [8] developed a mathematical model to predict the enhancement by high thermal conductivity porous media in forced-convection duct flows. They concluded that a 5-30 times increase in heat transfer is feasible for most engineering conditions. It is believed that the enhancement is mainly due to the micro turbulent mixing in the pores and super heat transfer through high thermal conductivity porous structure. Kuzay et al. [9] have reported liquid nitrogen convective heat transfer enhancement with copper matrix inserts in tubes. They proved that the insertion of porous copper mesh into plain tubes enhances the heat transfer by large amounts with a single phase coolant. However, in boiling, with tubes in which the porous insert is brazed to the tube wall for the best thermal contact, the heat enhancement is to be on the order of four-fold relative to a plain tube. They conclude that porous matrix inserts offer a significant advantage in cooling, providing a jitter-free operation and a much higher effective heat transfer, at grossly reduced flow rates relative to plain tubes. More recently, Boomsma et al. [0] used a open-cell aluminum alloy metal foam measuring 40 mm x 40 mm x mm 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.. Experiments Many researchers investigated important characteristics of open-cell metal foams through experiments. Leong and Jin [] performed experiments to investigate characteristics of oscillating flow through metal foams. They got detailed experimental data of flow pressure 0

25 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. [] 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 0.9, 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. [3] investigated heat transfer enhancement and pressure drop in an annular channel with nickel foams. the inner tube of the annulus. They used air as the coolant and constant heat flux heaters inside 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..3 CFD Simulation and Numerical Model Many scholars investigated open-cell foams by numerical methods, both analytical and computational. Lu et al. [4] 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.

26 Krishnan et al. [5] carried out a direct simulation of the transport phenomenon in open-cell metal foam using a single unit cell structure. The unit cell is created by assuming the void pore is spherical, and the pores are located at the vertices and center of a unit cell. The final geometry is obtained by subtracting the spheres from the unit cell cube. They further used that model to perform CFD simulation using Fluent/Gambit. Periodic conditions are used thus only one cell is needed in simulation which greatly saved computational time. Total thermal conductivity, pressure drop and heat transfer coefficient are obtained and compared with experimental data. Yet, this model is only suitable for foams that has porosity larger than Krishnan et al. [6] then created other models to extend the model s capability to simulate lower porosities (down to 0.80). Besides the body-centered cubic model [5], they developed other models based on face-centered cubic (FCC), and A5 lattice, which is similar to Weaire-Phelan structure. Good agreement to other researchers experimental data is obtained on Nusselt number and friction factor. Chung et al. [7] 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 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 30%-70%. They believed that the enhancement is independent of pressure drop and increases with decreasing pore size.

27 Boomsma et al. [8] developed a new approach to modeling flow through open-cell foams and defined a new cell structure. Their new model was based on Weaire-Phelan structure. This structure reduced the surface energy by 0.3% compared to tetrakaidecahedron [8]. The Weaire-Phelan structure was further wetted by Surface Evolver. Boomsma et al. [8] used that model to investigate pressure drop and velocity field in open-cell foams. They also compared their CFD prediction with experimental data and found their results were 5% lower. It s believed that the underestimates were due to the lack of pressure drop increasing wall effects in the simulations..4 Other Open-Cell Foams.4. Polyurethane Foams Some researchers investigated other open-cell foams other than metal foams. Mills [9] 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 [9]..4. 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. [0] developed a unit cube-based model for carbon foam modeling. This structure allows lower porosity which is a major property of carbon foam, compared to conventional metal foams. Assumed that the entire foam has uniform pore diameter and pores are considered to be spherical and centered, their model was obtained by subtracting a sphere from a unit cube. They used that model to evaluate carbon foam s heat transfer and pressure drop analytically and compared their results with experimental data. 3

28 A B C Figure -. 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 4

29 CHAPTER 3 ANALYTICAL MODEL FOR HEAT TRANSFER IN OPEN-CELL FOAMS 3. Geometry Simplification for Open-Cell Foam Filled Channels This transport model is based on the microscopic structure of the metal foam whose cells can be approximated as in diamond shapes as illustrated by the model presented in Figure 3-. The ligament structure is composed of two types of struts. The vertical struts called V-type struts, are perpendicular to the flow direction (x) while the horizontal struts, called the H-type struts are on the plans (x-y) that is parallel to the flow direction. Figure 3- shows the detailed infrastructure of the model. The picture on the left illustrates the arrangement of the ligaments and their connection with the walls. The top wall is the heated surface which represents the heat source from the combustion chamber. The bottom wall is insulated as it stands for the outer wall for the cooling channel. The plot on the right is a top view, which gives the horizontal cross section and the flow direction. A 3-D schematic of the foam model is given in Figure 3-3 where two rows in the downstream direction and four columns for each row in the cross-stream direction are shown to illustrate the foam structure. The heat transfer mechanisms are explained in terms of a network as shown in Figure 3-4. The heated top wall is the heat source that interacts with the V-type foam ligaments (fins) through conduction and also supplies heat to the coolant by convection through un-finned surfaces. The V-type struts, which act as fins receive heat from the wall and then pass the heat to the coolant by convection and to the H-type struts by conduction. The convection heat transfer between the flow and the V-type struts will be modeled as heat transfer for flow over tube banks. The H-type struts will lose heat to the flow by convection and also transfer some heat to downstream V-struts by conduction. 5

30 3.. V-type Struts 3. Mathematical Transport Model and Heat Transfer Equations A standard fin analysis is applied for a vertical cylinder as shown in Figure 3-5. The governing equation for a fin is given as below: dts 4h T s T c dy ksd ( ) = 0 (3-) h : heat transfer coefficient between vertical cylinder and the cooling fluid k s : thermal conductivity of the cylinder d : diameter of the cylinder T s (y) : local temperature as a function of y along the V-strut T c : coolant temperature Ts = T y= 0 w Boundary Conditions: Solve for dts dy y= H = 0 cosh m ( H y) s c ( w c ) cosh mh T = T + T T (3-) 4h where m =. kd s From Eq. (3-), the rate of heat dissipated to the coolant from the strut q, in watts, is T q= k A = k T T mh = M mh T T s π d 4 h s c s ( w c ) tanh( ) tanh( )( w c ) y 4 k y 0 sd = 3 where M = hk π d, h is evaluated based on flow over a cylinder or tube bank. It s further s /4 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. Ts Tc cosh m ( H y) Letθ = =, integrate T s from y=0 to H, the average can be shown to be T T cosh( mh) w c 6

31 θ = H H 0 cosh m ( H y) dy cosh mh = ( )sinh m ( H y) H cosh mh m tanh mh = mh H 0 tanh mh Thus, Ts = ( Tw Tc) + Tc (3-3) mh The average over y direction eliminates the y variable in T s. 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 of x in the flow direction [4]. cosh m ( H y) T x y = T x + T T x (3-4) s(, ) c( ) ( w c( )) cosh mh tanh mh T x = T T x + T x (3-5) s( ) ( w c( )) c( ) mh Heat transferred from a single V-cylinder to the coolant is q [ ( ) ( )] V = π dh h Ts x Tc x (3-6) 3.. H-type Struts A similar fin analysis as the vertical cylinder is applied for the horizontal cylinders that connect the vertical cylinders (Figure 3-6). The governing equation should have a similar form as a V-cylinder though extra terms need to be introduced to account for the angle (less than 90 degree) between H-cylinder and coolant. Such dependence can be assumed to be very weak when x>0a, as stated in the paper by Lu et al. [4]. There are several methods to solve the flow over H-type cylinders. Here proposes two methods. 7

32 First method The following equation will be used: dth 4h T h T c dx ksd ( ) = 0 (3-7) h : heat transfer coefficient between horizontal cylinder and the cooling fluid k s : thermal conductivity of the cylinder d : diameter of the cylinder T h : Temperature of a H-strut T c : coolant temperature Th = T x= 0 Boundary condition: Th = T x= a T and T are the temperature of V-type cylinders at x and x+a. They can be evaluated from T s solved in part a. For a specific H-type cylinder, T and T are constant. Solution of Eq.(3-7) is: T ( x) T h ( T T )sinh[ m ( a x)] + ( T T )sinh( m x) c c = c + (3-8) sinh( ma ) The heat flux entering the cylinder at x=0 q = M ( T Tc)cosh( ma) ( T Tc) sinh( ma) The heat flux leaving the cylinder at x=a q = M ( T Tc) ( T Tc)cosh( ma) sinh( ma) 4h where m =, M = h ksπ d kd s 3 /4 The heat transfer to the coolant, is cosh( ma) h = = ( + c) (3-9) sinh( ma ) q q q M T T T 8

33 Let T = Ts( x), T = Ts( x+ a). For the coolant, since its temperature is function of x, we simply choose its value at the middle point of H-type cylinder, thus Tc = ( Tc ( x ) + Tc ( x+ a )). The heat transferred to the coolant from a single H-cylinder is: cosh( ma ) h ( s ( ) s ( ) c ( ) c ( )) sinh( ma ) q = M T x + T x+ a T x T x+ a (3-0) The same correlation of finding h is used to evaluate h, 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 90 o to the cylinder. Second method Another method is to assume the H-type cylinders have identical temperature distribution with V-type cylinders along x direction. Thus, for a single H-type cylinder, the heat transfer rate from it to coolant can be represented as, 5 qh = π d a h ( Ts( x) Tc( x)) (3-) where 5 π d a is a H-type cylinder s surface area, h is heat transfer coefficient for H-type cylinders, and T s (x)-t c (x) is the temperature difference between cylinders and coolant. From Eq.(3-5), the heat transfer rate can be further written in the form, 5 tanh mh q = π d a h ( T T ( x)) (3-) h w c 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. 9

34 3..3 Fluid Temperature Prediction (Coolant Temperature) The coolant temperature profile as a function of the downstream coordinate, x, is estimated based on the following energy balance equation. Figure 3-7 shows the schematic. mc [ T ( x +Δx) T ( x)] = N q + N q + q (3-3) p c c v v h h w Δx Δx H where Nv = is the number of vertical struts per unit width. And N h = is the a a a number of horizontal struts per unit width for a channel of height H. Heat transfer from V-type struts Heat transfer from H-type struts q v can be evaluated from Eq. (3-6). q h can be evaluated from Eq. (3-0). Heat transfer from bare wall surface can be calculated from: q = ηδxh [ T T ( x)] (3-4) w w w c where η is the ratio of bare wall surface area to the total wall surface area, and h w can be evaluated from open channel heat transfer coefficient correlation. First method (high Re) Eq. (3-0) will be used for relatively high Reynolds number (>*0 4 ). From Eq. (3-5), tanh mh lx T x = T T x + T x. Let s further assume Tc( x) = ( T0 Tw) e + Tw where l is s( ) ( w c( )) c( ) mh to be determined. Plug into Eq. (3-5), tanh mh tanh mh T x T x T T x T T e s( ) c( ) = ( w c( )) = ( w 0) mh mh lx So Eq. (3-0) can be rewritten in the form: 0

35 cosh( ma ) h = ( s ( ) c ( ) + s ( + ) c ( + )) sinh( ma ) cosh( ma ) tanh( mh ) ( ) (( 0 ) lx ( 0 ) l x+ a = M Tw T e + Tw T e ) sinh( ma ) mh q M T x T x T x a T x a cosh( ma) tanh( mh) = M ( + e )( T T ) e la w 0 sinh( ma ) mh cosh( ma) tanh( mh) ( la + e )( T T ) = M sinh( ma ) mh w c lx So, cosh( ma ) tanh( mh ) ( la h )( w c ) sinh( ma ) mh q = M + e T T (3-5) Plugging Eq. (3-6), (3-4), and (3-5) into Eq. (3-3) yields: Δx mc p[ Tc( x+δx) Tc( x)] = M ( Tw Tc( x))tanh mh + ηδxhw[ Tw Tc( x)] a Δx H cosh( ma) tanh mh + M + e T T x la ( )( w c( )) a a sinh( ma) mh Add up similar terms, Tc( x+δx) T ( x) Δx H cosh( ma) tanh mh M mh M e ηh T T x mc a a m a mh c la = [ tanh + 3 ( + ) + w] w c( ) p sinh( ) H cosh( ma ) tanh mh la Let l' = [ M tanhmh + M 3 ( + e ) + ηhw ], mc a a sinh( m a) m H p we have: Tc( x+δx) Tc( x) = l' Δx ( T T ( x)) w c (3-6) Integrate (3-6) as Δx 0 Tc( x) x dtc T T0 w T = c 0 ldx ' Tc ( x) x dt ( w Tc) = T T T0 w c 0 ldx '

36 Tc ln( T T ) = l' x w c T0 T T x = e T T ( ) w c lx ' w 0 T ( x) = ( T T ) e + T lx ' c 0 w w lx Since we assume Tc( x) = ( T0 Tw) e + Tw in the beginning of this derivation, thus l' = l. That also proves the previous assumption is correct. So, T ( x) = ( T T ) e + T (3-7) lx c 0 w w where, H cosh( ma) tanh mh l = M mh + M + e + ηh mc a a m a m H la [ tanh 3 ( ) w] p sinh( ) (3-8) l can be determined by iterative method. Second method (low Re) Eq.(3-) will be used for relatively low Reynolds number (<*0 4 ). The energy balance Eq.(3-3) still holds for this case. Plug in Eq.(3-6), (3-), and (3-4) to Eq.(3-3) yields, Δx mc p[ Tc( x+δx) Tc( x)] = M ( Tw Tc( x))tanh mh + ηδxhw[ Tw Tc( x)] a Δ x H 5 tanh mh + π d a h T T x ( w c( )) a a mh Tc( x+δx) Tc( x) 5π dh tanh mh So, = [ M tanh mh + ηhw + h ] Δx T T ( x) mc a a mh w c p Let constant l to be in the form, 5π dh tanh mh l = M mh + ηh + h mc a a m H [ tanh w ] p (3-9) Integrate Eq.(3-9) over x, we have:

37 T ( x) = ( T T ) e + T (3-0) lx c 0 w w Eq.(3-0) has the same form with Eq.(3-7) in the first method, the difference is that, for the second method, l can be calculated directly, and there s no need to iterate Total Heat Transfer The total heat transferred to the coolant through the cylinders (V and H types) and the inner wall is Qx ( ) = ( T( x) T) mc (3-) c 0 P lx Since T ( x) = ( T0 T ) e + T from Eq. (3-7) and (3-0), so c w w lx Qx ( ) = mc ( T T)( e ) (3-) P w 0 where l can be determined from Eq. (3-8) or Eq.(3-9). For a channel of length L, the total heat transfer is ll QL ( ) = mc ( T T)( e ) (3-3) P w Evaluation of Heat Transfer Coefficient The heat transfer coefficient evaluation in the foregoing analytical method is critical and will be discussed in this section. It s mentioned in previous sections that the heat transfer coefficients of V-type cylinders, H-type cylinders, and bare wall are assumed to be identical, respectively. Based on Reynolds number, all the heat transfer coefficients of cylinders can be evaluated by the empirical correlations for flow over a tube bank correlation. And heat transfer coefficient for bare wall can be calculated from correlation for open channels. V-type cylinders h 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 3

38 transfer coefficient h for the entire tubes in the bank as defined in the Nusselt number, Nu d hd k =, can be obtained from the correlation below []: f Nu d m /3 =.3C Red,max Pr (3-4) where C and m are constants, they can be looked up from Incropera and DeWitt []. Reynolds number is defined as, Re d,max dv v,max = (3-5) ν f So, m.3c Red,max Pr h = d /3 k f V max is the maximum velocity of the coolant, k f is the thermal conductivity of coolant, v f is kinematic viscosity of coolant, and d is the diameter of the cylinder, m and C are constants and related to the geometry of channel, which can be obtained by tables. According to Incropera and DeWitt [], the maximum velocity occurs at the transverse plane. It can be calculated as Vv,max = a V a d, V is the incoming velocity. Eq. (3-4) is valid for Re from 000 to 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., the influence of inter-cylinders is neglected for low Re number cases in this study. Equation for flow over a single cylinder from [] is Nu d m /3 = C Red,max Pr (3-6) where C and m can be found from Table 3-. 4

39 H-type cylinders h For H-type cylinders, it can also be treated as flow over a single cylinder. The difference is that the flow direction is not perpendicular to the cylinder. So, the component of velocity that is perpendicular to cylinder is considered. From geometry, the equation is shown below V a = *( a* 5/ d) h,max * V (3-7) The other parameters are calculated as the same as V-type cylinders. Bare wall h w To evaluate the heat transfer coefficient for the bare wall, the following correlation of open channel can be applied: Nu = 0.0Pr Re (3-8) Reynolds number and Nusselt number are defined as DV Re = (3-9) ν f Nu hd k w = (3-30) f For this case, only the bottom wall is heated, so for Eq. (3-9) and (3-30) D=H, where H is the height of the metal foamed cooling channel. And h w can be calculated from h w Nuk f = (3-3) H 3..6 Equivalent Heat Transfer Coefficient To calculate Nusselt number and equivalent heat transfer coefficient of open-cell foams is the ultimate aim of the analytical model. The equivalent heat transfer coefficient is defined as, h equal Q A ( T T ) w w c (3-3) 5

40 such that the heat transfer from the foamed channel is equivalent to that carried away by a coolant having average temperature of T c, which flows through a open but otherwise identical channel. A s is surface area of heated wall, in this model, the width is, so A = * L. For the mean value of coolant temperature, or bulk temperature, the arithmetic mean value of coolant s lx over x direction is used. From Eq.(3-0), Tc( x) = ( T0 Tw) e + Tw, let T ( ) w Tc x lx θc = = e, T T w 0 so T L L w Tc lx exp( ll) = θc = θc( xdx ) = e dx= T T L 0 L 0 ll w 0 exp( ll) Thus, Tw Tc = ( Tw T0 ), plugging into Eq.(3-3) yields, ll h equal = QlL L( exp( ll))( T T ) w 0 Plug in Eq. (3-3) for Q, h equal ll mc P( Tw T0 )( e ) ll = ll L( e )( T T ) = mc l P w 0 If the second method (low Reynolds number) is used for H-type cylinders heat transfer, Eq.(3-9) can be used for l. Thus, h equal has the final form 5π dh h M mh h h tanh mh equal = tanh + η w + (3-33) a a mh From Eq.(3-33), the equivalent heat transfer coefficient is a function of foam geometry a, d, channel height H, and heat transfer coefficient h, h, h w. It s not a function of inlet temperature, wall temperature, or channel length. 6

41 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, ρ * ρ ρ s where ρ is relative foam density, ρ * is density of foam, and ρ s is density of solid. Another important property, porosity, is equal to ρ. The surface area density is defined by this equation, α A A V 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: ρ ( 5+ ) π d π d 4 a a 3 = ( ) ( ) (3-34) where a is cell size and d is the diameter of cylinders. To simplify calculation and derivation, and because d/a is about 0., Eq. (3-34) can be rewritten as, ρ ( ) π d 0 a ( ) (3-35) 7

42 So d a 0 ρ ( ) π / (3-36) The filament diameter d is calculated from Eq.(3-36) based on a=mm(0ppi), mm(0ppi), and 0.5mm(40ppi) with different relative density, and further compared with experimental data from ERG Duocel aluminum foams (Figure 3-8). Reasonable agreement is obtained. For surface area density, π da + ( 5a d) π d α A 3 a [( 5 + ) d/ a] π d = a ( ) π d a a From Eq. (3-36), surface area density can be represented as, α 5.97 a / A ρ (3-37) Eq (3-37) is plotted in Figure 3-9 and compared with data from ERG. Duocel aluminum foams. Good agreement is obtained except for 40ppi case. 3.4 Verification of the Analytical Model with Experimental Data To verify the heat transfer analytical model, heat transfer predictions on certain metal foams by the model are compared with experimental data from other researchers. Because two methods are developed for different Reynolds number, the author made two comparisons with other experiments with Reynolds number ranging from 5*0 3 to *0 4, and 0 4 to 6*0 4, respectively, using both methods stated in Section

43 3.4. Validity of Analytical Prediction (Re=5*0 3 ~ *0 4 ) Calmidi and Mahajan [] 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 4mm*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..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/=.5mm. Table - shows details of the experiment from Calmidi and Mahajan [] 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 -3 shows the parameters used in models and also their comparison with the experiment samples data. Different values of d and a are tested and the values shown in Table -3 are the ones providing best agreements with experimental data. The predictions for the 5 types of foams ranging from 5PPI to 40PPI are plotted in Figure 3-0, Figure 3-, and Figure 3-, and compared with experimental data from Calmidi []. Favorable agreements are obtained. The Nusselt number and Reynolds number are defined in the following equations, 9

44 HV Re = (3-38) ν f Nu h equal = (3-39) k f H where H is height of foam, V is inlet velocity of coolant, h equal is equivalent heat transfer coefficient of foam defined in Eq. (3-3), k f is coolant s thermal conductivity, and ν f is kinematic viscosity of coolant Validity of Analytical Prediction (Re=*0 4 ~ 6*0 4 ) To verify the validity of the analytical model with relatively high Reynolds number, the second method stated in Section 3..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. [7]. For experiment, the total heat transfer rate to the air flow is defined by the energy balance: Q= m C (T - T ) (3-40) P a,outlet a,inlet Here, aoutlet, T and T ainlet, are the outlet and inlet air temperature, respectively, and C p is the specific heat under constant pressure. where The bulk fluid temperature is defined as: ( ) T = T + T (3-4) b a, outlet a, inlet Effective heat transfer coefficient is defined as: h = Q/ A ( T T ) (3-4) equal s s b A s is the total heated surface area and Reynolds number is defined as: T s is the mean surface temperature 30

45 ( d ) V do i Re = (3-43) ν f where, d o and d i are the outer and inner diameter of the test section, respectively. ν f 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 do di = (3-44) Reynolds number is defined as: VH Re = (3-45) ν f The cell size a is set to be mm, and the filament diameter d is set to be 0.5mm, which is approximately a 0 PPI (pores per inch), relative density 8% s foam. Figure 3-3 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 0 5, the insulated boundary condition (at y=h) model gives good prediction. For extremely large Re numbers (>0 5 ), constant temperature model at both walls should be used. More details can be found in Chapter 5. Table 3-. Constants of Equation (3-), recreated from [] Re D C m , ,

46 Table 3-. Continued 40, , Table 3-. Parameters of experiments from Calmidi [] and analytical model Experiment [] Analytical Model Geometry L/W/H (mm) 4/63/45 4/unit length/.5 Coolant Air Air Foam Aluminum Aluminum Coolant Inlet Temperature (K) Heated Wall Temperature (K) Table 3-3. Foam parameters comparison between experiments from Calmidi [] and analytical model Ligament Pore Diameter Diameter Results Comparison 5PPI Experiment 0.50mm 4.0mm Model 0.70mm 4.0mm Figure 3-0 0PPI Experiment 0.40mm 3.3mm Model 0.55mm 3.3mm 0PPI Experiment 0.30mm.70mm Model 0.45mm.70mm Figure 3-5PPI Experiment 0.55mm 3.80mm Model 0.70mm 3.0mm 40PPI Experiment 0.5mm.80mm Model 0.0mm.50mm Figure 3- Figure 3-. Schematic of a single cell in the simplified model 3

47 Figure 3-. Model details Figure D schematic of the model 33

48 Figure 3-4. Heat transfer network of analytical model Figure 3-5. Schematic of vertical strut fin model 34

49 Figure 3-6. H-strut model Figure 3-7. Model for coolant temperature evaluation 35

50 Cylinder diameter(0-4m) ppi(Experiment) 0ppi(Experiment) 40ppi(Experiment) 0ppi(Model) 0ppi(Model) 40ppi(Model) Relative foam density Figure 3-8. Cylinder diameter as function of relative foam density predicted by analytical model, comparing with ERG s data of aluminum foams, a=mm, mm, and 0.5mm, respectively, for 0ppi, 0ppi, and 40ppi foams. Surface area density(m/m3) ppi(Experiment) 0ppi(Experiment) 40ppi(Experiment) 0ppi(Model) 0ppi(Model) 40ppi(Model) Relative foam density Figure 3-9. Surface area density as function of relative foam density predicted by analytical model, comparing with ERG s data of aluminum foams, a=mm, mm, and 0.5mm, respectively, for 0ppi, 0ppi, and 40ppi foams. 36

51 ppi(Model) 5ppi(Calmidi and Mahajan [3]) Nu Figure 3-0. Nusselt number prediction made by analytical model compared with Calmidi and Mahajan s [] experimental data for 5 ppi aluminum foam Re 4000 Nu ppi(Model) 0ppi(Model) 0ppi(Calmidi and Mahajan [3]) 0ppi(Calmidi and Mahajan [3]) Figure 3-. Nusselt number prediction made by analytical model compared with Calmidi and Mahajan s [] experimental data for 0 ppi and 0 ppi aluminum foams Re 37

52 ppi(Model) 40ppi(Model) 5ppi(Calmidi and Mahajan [3]) 40ppi(Calmidi and Mahajan [3]) Nu Re Figure 3-. 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 Experimantal Data Analytical Model Heat Transfer Coef (W/mK) Re Figure 3-3. Heat transfer coefficient predicted by analytical model compared with Chung et al. [7] experimental data for 0ppi copper foam 38

53 CHAPTER 4 CFD SIMULATION OF PRESSURE DROP IN OPEN-CELL FOAMS 4. Introduction to Single Cell Model Open-cell foams have been investigated by many researchers, both experimentally and numerically. In chapter 3, the analytical heat transfer model deals with the whole foamed cooling channel, and uses volume-averaged, semi-empirical equations. That is a macroscopic approach, which neglects small-scale details of open-cell foams. With rapid developing computing power, using a model with more foam s cell details becomes feasible in computational fluid dynamics. Although the computer stations are still not powerful enough to simulate the whole foam inserted channel at this stage, efforts can be made to investigate a single cell in open-cell foams due to their property of repeated cell structure. That is the microscopic approach. Using microscopic approach to simulate pressure drop in open-cell foams takes advantage of the repeated cell structure of foams and also the properties of flow through porous media. For a specific type of foam, in which the porosity, pore per inch, and other material properties are fixed, the pressure drop induced by the foam is only function of velocity of flow. And the velocity profile in open-cell foam is almost unified, because the multi-filament in foam greatly increases the intensity of turbulence in flow which flattens out the velocity gradient and makes the boundary layer very thin (Figure 4-). Thus, because of the unit cell structure and nearly unified velocity in open-cell foam, the pressure drop evaluation process can be simplified without modeling the whole foam inserted channel. A strategy has been developed to focus on two typical cells as illustrated in Figure 4-. 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 39

54 the same structure, only one cell is needed to be modeled to evaluate pressure drop contribution by interior cells. The second type cell is named boundary cell, which is distinguished from interior cells and used to capture the pressure drop occurring at the wall. The pressure drop induced by boundary cell is expected to be larger than that of a interior cell because the no-slip condition at wall and velocity at boundaries has much larger velocity gradient. To simulate the micro-structure of open-cell foam (typically metal foams), a sphere-centered tetrakaidecahedron structure is constructed (Figure 4-A). That structure is very similar to the real micro-structure of metal foam (Figure 4-B aluminum foam). A tetrakaidecahedron is a polyhedron consisting of six quadrilateral faces and eight hexagons. It s found by Lord Kelvin that the tetrakaidecahedron (Kelvin structure) is optimal structure for packing cell, which has minimum surface-area to volume ration. Tetrakaidecahedron is seen in reality when soap foam is observed [8]. The sphere-centered Kelvin cell can mimic the real metal foam s micro-structure because of the foaming process of metal foam. A common method used to foam metal such as aluminum is blowing a kind of foaming gas through molten metal. The gas bubbles generated are free to move around. The liquid metal and gas bubbles tend to attain an equilibrium state, i.e., a minimum surface energy state [5]. Thus, after the solidifying process, the optimal tetrakaidecahedron structure is formed by metal and gas bubbles generate pores which are similar to spheres. So, the sphere-centered tetrakaidecahedron can represent the real micro-structure of metal foams very well. In order to generate the sphere-centered Kelvin structure, a tetrakaidecahedron is generated first by cutting off the six corners of a regular octahedron. Then build a sphere at the center of the tetrakaidecahedron and subtracting the sphere from it yields the sphere-center Kelvin structure. Figure 4-3 shows the process schematic. 40

55 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 D-periodic model because periodic or symmetric boundary conditions are applied in directions (Y, Z directions) except in the stream-wise direction (X direction). A diagram of this is shown in Figure 4-4. The second model treats the cell that is attached to the wall and is termed the D-periodic model. Here, periodicity is applied in only one direction (Y direction). In the other direction one boundary was set as a wall and the remaining boundary as a symmetry plane. This is shown in Figure 4-5. The coolant used for the pressure drop simulation is air, which is assumed to be ideal gas with constant density and viscosity. Energy equation is not considered at this stage which means the temperature is constant. The air comes into the inlet of the channel and goes out through the outlet (Figure 4-4, 4-5). The inlet was set as velocity inlet boundary, and different inlet velocities were tested. The outlet was set as pressure outlet boundary having the atmosphere pressure. No-slip conditions were imposed at the wall and cell surfaces. 4. Mesh Generation and Grid Independent Study The Kelvin structure and channel models were created and meshed by GAMBIT, the preprocessing meshing generation software. The whole channel was divided into three parts, the inlet region, the outlet region, and the cell (central) region due to their different geometry properties. The cell region in the middle was meshed using TGrid in GAMBIT, which generated tetrahedral elements that can fit into the complex structure of Kelvin s cell. The inlet and outlet regions were meshed by Cooper method in GAMBIT. Because flow at those regions is less complicated than in the cell region, much less elements were generated at inlet and outlet regions to save computing time. Figure 4-6 shows the meshed Kelvin cell. Figure 4-7 4

56 provides the mesh details at cell s filaments. Figure 4-8 presents that fine mesh is used at the cell region and relatively coarse mesh is used at the inlet and outlet regions. The cell size is about.54mm*.54mm*.54mm, which is about the cell size of a 0ppi foam made by ERG. And the sphere centered in the cell has a diameter of.6mm. 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 tetrahedral cells and 700 nodes. That model was then refined by the medium mesh, which consists of cells and nodes. The most delicate model was further refined to 8779 cells and 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 -D periodic model in which inlet velocity is 4m/s and cell size is 0ppi. Figure 4-9 shows the average x velocity profiles along the flow direction (x direction) of the three meshes. From the figure there are no apparent differences among the three meshes with different number of elements. Figure 4-0 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, Δ p p p = a x x (4-) where p represents pressure, a is cell size, and x represent the x coordinate in flow direction. The pressure drops simulated from those three models are shown in Table 4-. 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 4

57 capture the pressure drop in foams and the coarse mesh was chosen to perform all the following simulations. Some more statements can be made on Figure 4-9 and 4-0. There are three regions where the pressure drop is very significant, from Figure 4-0. The three regions are inlet of cell, center of cell and outlet of cell. That agrees with the velocity profile in Figure 4-9, in the sense that the regions having larger velocities induce more pressure drop. The reason is that potential energy from pressure is transferred into kinetic energy. 4.3 Simulation Results and Verification Simulations were performed using coarse mesh (Section 4.). The cell size is set to be about.54mm which is 0ppi and its porosity is about 97%. Figure 4- shows the velocity magnitudes contours of several chosen planes in a case with inlet velocity of 4m/s. There are three planes, the first one is at about y= -0.8mm, the horizontal one is at the center of cell and the last one is at the left side of the channel. Figure 4-(A) is a 3-D view of the three planes contour, and (B)-(D) represents the three planes respectively. The velocities between ligaments are relatively high and wakes can be found at ligaments, which is evident especially in Figure 4-(B). Figure 4- provides static pressure contours of the same three planes. High pressure can be found where the flow encounters with the ligaments (Figure 4-(B), (C)). More data were obtained for -D periodic and -D periodic models for several inlet velocities to get pressure drop profiles for interior cells and wall cells. Leong and Jin [] were chosen to compare with the simulation data. Experimental data from The comparison was shown in Figure 4-3 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 43

58 nice agreement was obtained. It can be found that the wall cell induces a little more pressure drop because the no-slip condition of wall also contributes to the pressure drop. It can be concluded that the Kelvin structure unit cell can capture the important phenomenon of pressure drop occurring in metal cells and can be used to predict foam s pressure drop. Table 4-. Comparison of different meshes results Mesh Pressure Drop Relative Error Coarse 5.6 Pa/mm 3.5% Medium 5. Pa/mm 0.6% Fine 5.08 Pa/mm Figure 4-. Schematic of boundary cell and interior cell in open-cell foam 44

59 A B Figure 4-. Comparison of single cell model and real foam structure. A) single cell model used in this study. B) SEM photo of aluminum foam. Figure 4-3. Geometry creation of a single cell 45

60 Figure D periodic model Figure D periodic model 46

61 Figure 4-6. Mesh of a single cell model (coarse grids) Figure 4-7. Details of the meshes on filaments (medium grids) 47

62 Figure 4-8. Grids distribution. Cell region has more delicate grids and other regions use coarse ones to save computation time Velocity (m/s) Coarse Mesh Medium Mesh Fine Mesh x position (mm) Figure 4-9. Velocity profile along flow direction through the cell 48

63 0 8 Coarse Mesh Medium Mesh Fine Mesh Pressure (pascal) x position (mm) Figure 4-0. Pressure distribution along flow direction through the cell 49

64 Figure 4-. Velocity contours in three planes around the cell. A) 3-D view. B) Plane at y= -0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the channel, y=.7mm 50

65 Figure 4-. Static pressure contours in three planes around the cell. A) 3-D view. B) Plane at y= -0.8mm. C) Horizontal plane at the center of cell z=0. D) Left side of the channel, y=.7mm 5