Thermal Transport Phenomena in Porvair Metal Foams and Sintered Beds

Transcription

1 Final Report Thermal Transport Phenomena in Porvair Metal Foams and Sintered Beds C. Y. Zhao, T. Kim, T.J. Lu and H. P. Hodson Micromechanics Centre & Whittle Lab Department o Engineering University o Cambridge August 2001

2 Abstract This report presents experimental and numerical results on pressure drop and heat transer or Porvair metal oams and sintered bronze particle beds. The metal oams considered in this study are made o two dierent materials - FeCrAlY and copper. The microstructures o the oams have been characterized with SEM and image analysis, and experiment measurements on heat transer and pressure drop have been perormed or eight FeCrAlY samples and six copper samples with dierent pore sizes (ppi) and dierent relative densities. The experimental results show that the heat transer o FeCrAlY samples is more sensitive to relative densities than cell size, whereas or copper samples, the heat transer is more sensitive to the cell size than the relative density. This dierence in transport phenomena is attributed to the thermal resistance on the solid side and is heavily inluenced by the solid conductivity. A numerical model is developed to consider the transport o heat based on the measured microstructural parameters or Porvair metal oams; the eects o oam cellular microstructure on overall heat transer are predicted. Generally, the heat transer will increase with increasing relative density, and decrease with increasing pore size (ppi). Finally, orced convection through a sintered bronze bed is studied numerically. The sintered bronze bed has a much smaller porosity compared with the metal oams, and water was used as the coolant instead o air or metal oams. The results indicate that the heat transer o the sintered bed is about 30% higher than that o a packed bed. 1

3 Contents Nomenclature (4) 1 Introduction (6) 1.1 Background (6) 1.2 Objectives o the study (9) 2 Porvair metal oams (10) 2.1 Introduction (10) 2.2 Metal oam processing (13) Metal oam abrication and capabilities (13) Capability o manuacturing complex assemblies (14) 2.3 Microstructure o the Porvair metal oams (15) Speciication o the microscopic parameters (15) Measurement o Microstructures (19) Procedures o the measurements Results o the measured microstructures Veriication o the relationship among the microscopic parameters (25) 2.4 Experimental study or heat transer in Porvair metal oams (26) Experimental acility and procedures (26) Measurement uncertainties (28) Experimental results and analysis (28) Pressure drop Heat transer Product uncertainties (42) 2.5 Numerical modelling on orced convection in Porvair metal oams (44) Mathematical ormulations (44) Boundary conditions (46) Modelling on Porvair metal oams (47) Numerical procedure (49) Code validation (49) Numerical results or Porvair metal oams (51) 2

4 2.5.7 Eect o boundary conditions (58) Eects o microstructural parameters: Optimisation (60) Solid conductivity (k s ) Relative density (ρ r ) Porosity (ε) Pore size (d p ) 3 Sintered particle bed (65) 3.1 Introduction (65) 3.2 Mathematical ormulations (66) 3.3 Modelling on sintered bed (68) 3.4 Results and discussions (70) 4 Conclusions (73) Appendix A SEM pictures o oam samples (79) Appendix B Experimental data o Porvair metal oams (82) 3

5 Nomenclature α ~ speciic surace area per unit volume, m -1 C C I d d p heat capacity o luid inertial coeicient diameter o the ibre o the metal oam diameter o the pore size or metal oam; particle diameter or sintered bed D h hydraulic diameter o the channel, 2WH /( H + W ) F I h s h H k d k k e k s k se K L Nu,b Nu P Pr q w r R T T T,b T in T s T w riction resistance inertial variable interacial heat transer coeicient overall heat transer coeicient channel height thermal dispersion conductivity thermal conductivity o luid eective thermal conductivity o luid thermal conductivity o solid eective thermal conductivity o solid permeability length o the heat sink local Nusselt number overall Nusselt number pressure Prandtl number heat lux over the bottom surace inner-to-outer diameter ratio simpliication quantity temperature luid temperature bulk mean luid temperature inlet luid temperature solid temperature bottom wall temperature 4

6 u W X Y velocity width o heat sink dimensionless streamwise coordinate, y/h dimensionless vertical coordinate, y/h Greek symbols ε porosity ε opt optimised porosity ρ density o luid µ viscosity o luid s volume-averaged value over the luid region volume-averaged value over the solid region 5

7 1 Introduction 1.1 Background Recent advances in low-costing processes have enabled porous metals to be manuactured at high volume. In many engineering applications, it is attractive to design a primary mechanical structure to perorm other unctions such as heat dissipation. For such multiunctional structures, a robust methodology allowing the mechanical perormance (stiness, strength, weight) and thermal properties (heat dissipation, pumping power) to be optimized simultaneously is desirable. On the other hand, it is recognized that there is a diversity o thermal management issues. For example, the increasing demand or execution speed in modern computers has led to high heat luxes (> 100 W/cm 2 ) at the chip level. The high level o heat luxes provides a challenge o removing heat rom the junctions o power electronics. Thereore the use o porous metals as eicient compact heat exchangers or heat removal has become one o the most promising cooling techniques. This is attributed to the high surace-area-to-volume ratio intrinsic to these materials, resulting in enhanced heat transport and miniaturization o thermal systems. Forced convective heat transport in porous materials represents a rapidly growing branch o thermal science. Several thermal engineering applications can beneit rom a better understanding o convection through porous materials exempliied by geothermal systems, thermal insulations, microelectronic cooling system, iltering devices and products manuactured in the chemical industry. The initial investigation o luid low through a porous medium can be traced back to the nineteenth century. Darcy [1] was the irst to perorm recorded experiments and to produce ormulations pertaining to a porous medium. He discovered that the area-averaged luid velocity through a column o porous material is proportional to the pressure gradient and inversely proportional to the viscosity (µ) o the luid seeping through the porous material, represented by the Darcy low law as ollows, K dp u = µ dx where K is a material constant called permeability. (1.1) 6

8 The Darcy low model has subsequently been employed in numerous engineering applications related to luid low and heat transer in porous materials. However, although the Darcy model is popular in porous medium convective heat transer investigations, it neglects several key physical eects o importance in channel lows. For example, the Darcy low model does not satisy the no-slip condition on a solid boundary by neglecting riction due to macroscopic shear, and the inertial orces which are signiicant or relatively ast lows are disregarded. To thoroughly understand the luid low and heat transer characteristics in porous medium is a challenging task. In this respect, the complex microscopic transport phenomena at the pore level is important as these determine such macroscopic phenomena as heat transer augmentation and pressure loss increase. However, the complexity o the cellular morphology typically ound in commercial porous metals usually precludes a detailed microscopic investigation o the transport phenomena at the pore level. Thereore, the general transport equations are commonly integrated over a representative elementary volume, which accommodates the luid and the solid states within a porous structure. Though the loss o inormation with respect to the microscopic transport phenomena is inevitable with this approach, the integrated quantities, coupled with a set o proper constitutive equations representing the eects o microscopic interactions on the integrated quantities, do provide a rigorous and eective basis or analysing the transport phenomena in porous materials. There are two approaches available in applying the volume-averaging technique or heat transer investigations: one is averaging over a representative elementary volume containing both the luid and solid phases, and the other is averaging separately over each o the phases, thus resulting in a separate energy equation or each individual phase. These two models are reerred to as the one-equation model and the twoequation model, respectively. The one-equation model is valid when the thermal communication is suiciently eective so that the local temperature dierence is negligibly small between the luid and the solid phases. In some applications, however, the temperature dierences between phases cannot be neglected. In these situations the eects o the interacial surace and interstitial heat transer coeicients, which are related to the internal heat exchange between the solid and luid phases are 7

9 major actors causing heat transer augmentation in porous materials. In such cases, the two-equation model needs to be utilized. Although extensive studies have been conducted on heat transer in a porous medium channel [2-18], most o the research mainly ocused on the packed bed as the porous medium, oten assuming a linear relationship between its eective conductivity and porosity. In recent years, with advances in the processing technology, metal oams and sintered metal particle beds can be mass-produced with high quality. The microstructure o a metal oam produced via the metal sintering route by Porvair is shown in Fig. 1.1, whereas Fig. 1.2 depicts a typical sintered particle bed. Because metal oams and sintered beds are relatively new class o materials, investigations on thermal transport in these materials are scarce compared to packed beds, and hence the knowledge is not suicient or engineering applications. For metal oams, due to the complex cellular microstructures (Fig. 1.1), how to build a mathematical model which can properly account or the heat transport remains a problem to be addressed. For a sintered bed, how good is its heat transer perormance compared to that o a packed bed i water is used as the coolant luid? In addition, reliable test data is urgently needed to provide the base or industrial design as well as validation o the numerical models. Fig. 1.1 Microstructure o a typical Porvair metal oam 8

10 Fig. 1.2 Schematic o a sintered particle bed 1.2 Objectives o the study The aim o this research is to experimentally and numerically investigate the transport phenomena in Porvair metal oams and sintered bronze beds, and hence to provide guidance on materials processing and compact heat exchanger designs. The ocus o this report will be placed on Porvair metal oams, or which both experimental and numerical investigations have been perormed. As or the sintered bed, only numerical predictions have been carried out, with the aim to build a good heat transer model to assess the easibility o using such porous material as eicient heat dissipation medium. 9

11 2 Porvair Metal Foams 2.1 Introduction With recent developments in the processing technology, a range o novel materials have been manuactured or advanced, compact, and lightweight thermal systems. Metal oams with open cells, as one o the most promising emerging materials, have received much attention in recent years. The microstructure o a typical Porvair oam, shown in Fig. 2.1, consists o ligaments orming a network o inter-connected dodecahedral-like cells. The cells are randomly oriented, and mostly homogeneous in size and shape. Pore size may be varied rom approximately 0.1 mm to 7 mm. The relative density can be varied rom 3% to 15%. Alloys and single-element materials are available. Common materials include copper, aluminum, stainless steel, and high temperature iron-based alloys (FeCrAlY). The distinctive eature o the Porvair metal oams processed by the metal sintering technique is that the struts (ligaments) are hollow (see Fig. 2. 1b) compared to the solid struts o ERG oams manuactured via the expensive investment casting route. (a) (b) Figure 2.1 A typical Porvair metal oam: (a) cellular morphology; (b) cross-section o an individual strut In an attempt to enhance convective thermal transport, the metal oam materials can be used as an advanced compact heat exchanger. The motivation is attributed to the high surace area to volume ratio as well as enhanced low mixing due to the 10

12 tortuosity o the metal oam. Thereore, it is believed that the overall perormance o a thermal system can be substantially enhanced by using metal oams. Furthermore, metallic oams have attractive stiness/strength properties and can be processed in large quantity at low cost. From the heat transer point o view, metal oams can be considered as one type o porous medium, so that the study on metal oams can be classiied as thermal transport in porous media. The transport phenomena in porous media have been o continuing interest or the past ive decades [2]. This interest stems rom the complicated and interesting phenomena associated with transport processes in porous media. However, most studies on porous media have been restricted to the packed beds and granular materials with porosities in the range [2-18]. Relatively ew investigations o the transport phenomena have been conducted or very high porosity media (porosity ε > 0.9) such as metal oams. Even though metal oams can be broadly classiied as porous media, they have very distinctive eatures such as high porosities (ε > 0.9) and a unique open-celled structure. Consequently, most o previous studies on packed beds and granular porous media are not applicable to metal oams. Only during the last iteen years, transport phenomena in metal oams have started to receive attention [19-29]. Under the assumption o local thermal equilibrium, Hunt and Tien [19] studied the eects o thermal dispersion on orced convection in metal oams with water as the luid phase, and concluded that conduction o the metal oam may not be signiicant due to its thin cell ligaments, and dispersion may dominate the heat transport or metal oams with high porosity. Sathe et al. [20] studied combustion in metal oams as applied to porous radiant burners. Younis and Viscanta [21] have measured the volumetric heat transer o ceramic oam materials, and developed a Nusselt number correlation it to the experimental data. The volumetric heat transer rates measured were higher than those or packed beds. Lee et al. [22] investigated the application o metal oams as high-perormance air-cooled heat sinks in electronics packaging. In their experimental study, they demonstrated that aluminum oams could dissipate heat luxes up to 100 W/cm 2. Using the in approach, Lu et al. [23] have developed an analytical model to predict the metal oam-assisted heat transer, where oam is modelled as inter- 11

13 connected cylinders. Bastarows et al. [24] studied the single-sided heating o a oamilled channel or electronics cooling applications. The experimental method utilized both conductive thermal epoxy bonding and brazing o the metal oam to a heated plate. The test results revealed that brazed oam materials are much more eective at heat removal than epoxy-bonded samples, and that the heat exchange perormance is three times more eicient compared to a conventional in-pin array. Recently, Calmidi and Mahajan [25,26] proposed an eective thermal conductivity model or high porosity metal oams, and conducted an experimental and numerical investigation on the orced conduction in ERG aluminum oams with air as the luid phase. In their numerical study, they employed two-equation heat transer model and ound that the thermal dispersion eect was extremely small i the luid phase was air, which was quite dierent rom the conclusion derived by Hunt and Tien [19] who used water as the coolant. Kim et al. [27] experimentally studied laminar heat transport in aluminum oams, and their results showed that the oam material oers a better heat transer perormance compared to that o a louvered array, but at a greater pressure drop. Paek et al. [28] studied the eective conductivity and permeability o aluminum oams, and indicated that the eective conductivity (k e ) increases as the porosity decreases; however, no noticeable changes o k e were detected by varying the cell size o the metal oam at a ixed porosity (ε). The permeability K is substantially aected by both porosity and cell size. Boomsma and Poulikakos [29] put orward an eective thermal conductivity model based on a three-dimensional structure o the oam geometry. The results showed that despite the high porosity o the oam, the heat conductivity o the solid phase controls the overall eective thermal conductivity to a large extent, a act that must be dealt with in the oam manuacturing process i speciic ranges o the oam eective conductivity are desired. Though the above investigations have been conducted or the heat transport in metal oams, the study is still incomplete. Firstly, because these studies mainly ocused on ceramic or aluminum oams with solid cell struts, or orced convection in metal oams made o dierent materials with dierent cellular morphologies, there is little inormation in the open literature and there is a lack o reliable experimental data. Secondly, various models on the eective thermal conductivity o metal oams have been proposed [25, 28, 29], but their eects on predicting the overall heat transer 12

14 perormance o the oam in orced convection remain to be quantitatively studied. In other words, question like i the eective thermal conductivity can dominate the orced convection, whether heat is taken away directly by coolant or irstly conducted by solid and then transerred to the coolant low, need to be answered. Finally, no investigation has been conducted or Porvair metal oams with hollow struts (ligaments). Compared to oams with solid struts, how the hollowness aects the overall heat transer perormance needs to be addressed. In this section, the microstructures o Porvair metal oams will be characterized. Experimental measurements on orced convection will subsequently be carried out or eight FeCrAlY and six copper Porvair oams. A numerical model will then be developed incorporating the measured microstructural parameters. The eect o boundary conditions as well as the eects o microstructural oam parameters on overall heat transer perormance will be studied. In other words, the parameters which characterize thermal transport in metal oams need to be identiied, and how these parameters aect overall heat transer will be examined. 2.2 Metal oam processing Metal oam abrication and capabilities Metal oams have been manuactured or many years using a variety o novel techniques. Metallic sintering, metal deposition through evaporation, electrodeposition or chemical vapor decomposition (CVD), and investment casting (among numerous other methods) have created open cell oams. In oam creation through metal sintering, metallic particles are suspended in slurry and coated over a polymeric oam substrate. The oam skeleton vaporizes during heat treatment and the metallic particles sinter together to create the product. This method is thought to be the most cost-eective method, and the most amenable to mass production. The CVD method utilizes chemical decomposition o a reactive gas species in a vacuum chamber onto a heated substrate (polymer or carbon/graphite, depending upon the temperature o the deposition process). Production rates are limited in this method due to the rate at which material may be deposited on the substrate. However, highly reractory metals and ceramics may be created with this method (including rubidium and silicon carbide), with high quality. Molten metal iniltration is utilized to make 13

15 aluminum and copper oam materials [30]. In this method, the oam precursor is coated with a mold casting and packed into casting sand. The casting assembly is heated to decompose the precursor and harden the casting matrix. Molted metal is then pressure iniltrated into the casting, illing the voids o the original matrix. Ater solidiication, the material is broken ree rom the mold. The method has the advantage o being capable o producing a product o highly useul materials and alloys (such as aluminum), and generate a product with solid struts. However, the process is complex and expensive, requiring several processing steps and highly specialized equipment. O the methods suitable to produce metal oam materials, only the metal sintering method oers promise as a method that is capable o economically producing millions o components annually. The process is similar to the production o ceramic oam materials that are used in molten metal iltration, except in the heat treatment process. Heat treatment needs, however, are identical to those required in power metal industry or sintering pressed and injection molded materials. Production-designed equipment may be used eectively with automated lines, eliminating the need or handling in the process Capability o manuacturing complex assemblies To eectively use metal oam materials in heat exchange devices it is necessary to combine the material with tubes and sheets or low separation and heat transer. Development eorts have taken place at Porvair Fuel Cell Technology to successully combine a variety o metal oam materials with solid structures. Several proprietary articles have been constructed combining tubes and sheets to construct advanced, multiunctional heat exchange devices or a variety o customers. An important consideration in the ormation o the advanced heat exchangers is the quality o the bond joint between oam and tube. Figure 2.2 is a photograph o a developmental component consisting o tubes imbedded in a metal oam matrix. Metallurgical bonding between the tube wall and the oam matrix was achieved in this example by direct sintering. Figure 2.3 is an SEM micrograph o the joint region. Assemblies have also been manuactured through a proprietary co-sintering technique. Figure 2.4 shows an example o a oam-illed tube manuactured with this method. Complex assemblies combining metal oam with metal packaging are in the design stage to create an advanced two-phase heat exchange component or use in uel cell uel processing systems at Porvair Fuel Cell Technology. 14

16 Figure 2.2 Metal oam compact heat exchanger or high temperature service (oam material is PFCT's FeCrAlY) Fig. 2.3 SEM micrograph o a oam strut Sintered to a solid tube. Fig. 2.4 Example assemblies manuactured in a proprietary co-sintering technique 2.3 Microstructure o the Porvair metal oams Speciication o the microscopic parameters A typical Porvair metal oam structure shown in Fig. 2.1 is characterized by several key geometrical and physical parameters, namely, porosity (ε, void volume raction), pore size (d p ), ibre diameter (d ), inner-outer diameter ratio (r), and relative density (ρ r ). It should be noted that these parameters are not all independent o each other. 15

17 Fig. 2.5 Close-up o a single open cell (rom ONR workshop, Cambridge, UK) d p Fig. 2.6 Model o the tetrakaidecahedron (rom ONR workshop, Cambridge, UK) 16

18 With a closer look at a single cell o a typical open-celled metal oam (Fig. 2.5), it can be ound that the cell has the approximate shape o a tetrakaidecahedron with roughly pentagonal or hexagonal aces (Fig. 2.6). The ibres (ligaments) orm the edges o the tetrakaidecahedron, and there is a lumping o material at joints where the ligaments meet. The cross-section o the ligaments is circular only or ε 0.85 or less. As the porosity increases rom this value, the cross-section o the ligament changes rom circular to triangular due to the dierent rate o metal sintering at high porosity levels [31]. Since the structure is considerably complex, it may be approximated by an cross-cylinder representation [23,31] as shown in Fig ' d p Fig. 2.7 Open-cell representation o metal oam structure [23] First, consider the idealised representation o an open-celled metal oam as shown in Fig The ratio o the ibre diameter d d ' ' p = 2 ( 1 ε ) 3π ' d to the pore size ' d p can be derived as [23], (2.1) Now consider the more reasonable metal oam representation o Fig Calmidi [31] obtained the relationship between ' p d p ' d p and the pore size d p, as d = (2.2) 17

19 In other words, the pore size d p associated with the model o Fig. 2.6 can be converted into an equivalent cell size ' d p or the cell model o Fig. 2.7 by using Eq. (2.2). In the derivation o Eq. (2.1), it has been assumed that the ibres are circular. However, as previously discussed, in practice the ibres o a metal oam are not circular at porosities higher than It is thereore necessary to introduce a shape-actor S to account or this discrepancy, with [31]: (( 1 ε ) / 0.04 ) S = 1 e (2.3) Thus, rom Eqs. (2.1) to (2.3), an appropriate equation or metal oam structure is written as [27], d d 1 ε 3π 1 e = 1.18 ε 04 p 1 (( ) ) 1 / 0. (2.4) where d p and d in Eq. (2.4) now reer to the measured pore size and ibre diameter. It should be pointed out that Eq. (2.4) is suicient or ERG oams whose ibres (ligaments) are solid and or which the simple relation between porosity ( ε) and relative density (ρ r ) exits, ρ r = 1 ε. Thus, or ERG oams, there are only two independent microscopic parameters, i.e., pore size (d p ) or ibre diameter (d ), and porosity (ε) or relative density (ρ r ). For Porvair metal oams, however, because the ligaments are hollow, another parameter - the inner-to-outer diameter ratio (r) - is needed. The relationship between the porosity and relative density then becomes r 2 ( 1 )( 1 r ) ρ = ε (2.5) Consequently, the ollowing cross-relationships exist between a ERG oam and a Porvair oam: 2 ( ) ρ at same porosity (ε) (2.6) r, Porvair = ρ r, ERG 1 r 2 ERG r ε porvair = ε at same relative density (ρ 2 r ) (2.7) 1 r Both Eqs. (2.6) and (2.7) will be used in later calculations. From Eqs (2.4) to (2.7), it can be seen that there are three independent parameters characterizing Porvair metal oams, namely, porosity (ε) or relative density (ρ r ), pore size (d p ) or ibre diameter (d ), and inner-to-outer diameter ratio o the ibre (r). 18

20 2.3.2 Measurement o the microstructures Accurate values o cell parameters are crucial to the numerical calculations to be presented later. In this section, the relative density (ρ r ), pore size (d p ), ibre diameter (d ), and inner-to-outer diameter ratio o the ibre (r) will be careully measured. From the previous section, it is known that only three independent parameters are needed, so the pore size (d p ) and ibre diameter (d ) are dependent on each other. The aim o measuring both o them is to use the test data to check the applicability o Eq. (2.4) or Porvair metal oams. In this study, eight FeCrAlY samples and six copper samples provided by Porvair are measured. The samples have been speciied with industrial terminologies: pore size in terms o ppi ( pores per inch), and relative density (deined as the ratio o oam density to the density o the solid o which the oam is made). Table 2.1 lists all samples and their industrial speciications. Material pore size (ppi) relative density Sample #1 10 ppi 5% Sample #2 10 ppi 10% Sample #3 30 ppi 5% Sample #4 FeCrAlY 30 ppi 10% Sample #5 60 ppi 5% Sample #6 60 ppi 10% Sample #7 30 ppi 7.5% Sample #9 10 ppi 5% Sample #10 10 ppi 10% Sample #11 Copper 30 ppi 5% Sample #12 30 ppi 10% Sample #13 60 ppi 5% Sample #14 60 ppi 10% Table 2.1. Industrial speciications o Porvair oam samples Measurement procedures The measurement o cell and oam parameters was conducted by using the SEM (scanning electronic microscope) and an image analysis sotware KS 400 v3.0, rom Karl Zeiss Vision GmbH. In order to get good images with the SEM, the Electrical 19

21 Discharge Machine (EDM) was used to cut the samples, which can provide a good quality cutting section with smooth suraces. Twenty or more images o each sample were taken with the SEM. Measurements o the pore size (d p ), and the diameter o the ibres (ligaments) as well as the inner holes in the ligaments were then made by using the image analysis sotware. Ater a geometric calibration, the sotware can give the area and the perimeter o the selected open cell, as illustrated in Fig Fig. 2.8 The image analysis sotware used in this study From the measured area and perimeter, the equivalent pore size can be calculated. Similarly, the average diameters o the ibres and the inner holes can be obtained. Comparing dierent hand-drawn perimeters or the same cell, the uncertainty is estimated to be less than 10%. A summary o the SEM images or each sample is given in Appendix A. As or the relative density, it is obtained straightorwardly by weighing the oam sample and measuring its overall dimensions. Results o the measured microstructures a) Cell size (d p ) The nominal cell (pore) size o a metal oam can be calculated directly rom the product speciications based on ppi (Table 2.1). In this project, metal oams having three dierent industrial speciications are studied: 10 ppi, 30 ppi and 60 ppi. The corresponding nominal cell sizes are 2.54 mm, mm and mm, respectively. 20

22 For each sample, ten open cells were measured, thus the averaged cell size with its standard deviation can be obtained. The measured results or seven FeCrAlY samples and six copper samples are shown in Table 2.2 and Table 2.3, respectively. Sample eight or the FeCrAlY oam has identical industrial speciications as those o sample seven but manuactured rom a dierent batch job, and will be examined later or uncertainties in product quality due to processing. Sample #1 Sample #2 Sample #3 Sample #4 Sample #5 Sample #6 Sample #7 Nominal Average Std deviation Table 2.2 Cell size (mm) o FeCrAlY samples Sample #9 Sample #10 Sample #11 Sample #12 Sample #13 Sample #14 Nominal Average Std deviation Table 2.3 Cell size (mm) o copper samples. It is noted rom Tables 2.2 and 2.3 that there is signiicant dierence between the nominal and measured cell sizes, and the cell size o the FeCrAlY sample is quite dierent rom that o a copper sample with the same ppi. The dierence can be attributed to the manuacturing process and properties o dierent materials. b) Diameters o the ibre and its inner hole The other parameter to be measured rom the SEM images is the diameter ratio or the hollow struts. This is more diicult due to the small dimensions o the ibres. For FeCrAlY samples, reasonably good measurements can be made because o the images taken or various struts have relatively good quality. However, or copper samples, SEM images show that nearly all the struts are solid, with no presence o inner holes, as shown in Fig This may be attributed to a better sintering process o copper ater the evaporation o the polymeric skeleton. As a result, only the averaged ibre diameters were measured or copper samples. The measured results are summarized in Table 2.4 and Table 2.5 or FeCrAlY and copper samples, respectively. 21

23 Figure 2.9 Solid struts o a copper sample Sample #1 Sample #2 Sample #3 Sample #4 Sample #5 Sample #6 Sample #7 hole diameter strut diameter ratio 56% 44% 51% 40% 35% 27% 46% Table 2.4 Strut diameters (µm) and hole ratio o FeCrAlY samples. Sample #9 Sample #10 Sample #11 Sample #12 Sample #13 Sample #14 strut diameter Table 2.5 Strut diameters (µm) o copper samples. c) Relative density The oam samples used in the heat transer experiments are o the sandwich type, consisting o a oam core brazed onto two thin copper plates. Depending on the quality o processing and the material being used, the weight o the brazing material could take up as much as 25% o the weight o the oam itsel: two relative densities were thereore measured or each sample. The irst measurement represents the 22

24 relative density o the whole oam core with the presence o the brazing material. The second measurement was made by cutting a small cubic sample rom the central portion o the oam core in order to eliminate the eect o brazing. The measured results are given in Table 2.6. Materials Sample # pore size Relative density Nominal Measurement Measurement (core only) Sample #1 10 ppi 5% 5.7% 4.6% Sample #2 10 ppi 10% 14.3% 12.5% FeCrAlY Sample #3 30 ppi 5% 6.1% 4.1% Sample #4 30 ppi 10% 10.2% 9.3% Sample #5 60 ppi 5% 9.0% 5.5% Sample #6 60 ppi 10% 13.7% 9.2% Sample #7 30 ppi 7.5% 8.1% 5.4% Sample #9 10 ppi 5% 6.7% 7.44% Sample #10 10 ppi 10% 8.2% 11.5% Copper Sample #11 30 ppi 5% 4.4% 6.0% Sample #12 30 ppi 10% 10.0% 11.9% Sample #13 60 ppi 5% 5.7% 7.3% Sample #14 60 ppi 10% 9.5% 8.5% Table 2.6 Measured relative density or all samples Note that substantial discrepancy exists between the two relative densities or a given sample. Interestingly, the irst measurement is consistently larger than the second measurement or FeCrAlY samples, whereas the reverse holds or copper samples. This indicates that whilst there is a high concentration o brazing material near the oam-skin interace or FeCrAlY samples, the residues o the brazing material have migrated away rom the interace to the central portion o the oam core in copper 23

25 samples. For subsequent calculations, only the irst measurement will be used to represent the relative density o each sample. The above study on microstructures has led to a better physical picture o the Porvair metal oams. Table 2.7 summarizes all the results on microstructural parameters or both FeCrAly and copper samples. Properties FeCrAlY S-1 S-2 S-3 S-4 S-5 S-6 S-7 Pore size (ppi) Nominal relative density (%) measured relative density (%) measured relative density (%)-core only Nominal cell size (mm) Measured cell size (mm) Hole diameter (µm) Fibre diameter (µm) Ratio (%) Properties Copper S-9 S-10 S-11 S-12 S-13 S-14 Pore size (ppi) Nominal relative density (%) measured relative density (%) measured relative density (%)-core only Nominal cell size (mm) Measured cell size (mm) Hole diameter (µm) n/a n/a n/a n/a n/a n/a strut diameter(µm) ratio (%) n/a n/a n/a n/a n/a n/a Table 2.7 Summary o measured microstructures o all oam samples 24

26 2.3.3 Veriication o the relationship among the microscopic parameters From the measured relative densities (ρ r ), diameter ratio (r) and ibre diameters (d ) o the seven FeCrAlY and six copper samples, the pore size (d p ) can be calculated by using Eqs. (2.4) and (2.5). The comparison between the calculation and test data is shown in Fig measured value Eq. (3.4) d p d Fig Comparison o measured pore size and that calculated rom Eq. (2.4) From Fig. 2.10, it can be seen that there is certaindeviation between the measured and calculated pore sizes, but the deviation is considered reasonable given the complexity o the oam microstructures and the uncertainty o the measurements. 25

27 2.4 Experimental study or heat transer in Porvair metal oams The pressure drop and heat transer o the seven FeCrAlY oams and six copper samples listed in Table 2.7 are measured in this section. The oam shown in Fig was sandwiched between two 1 mm thick copper plates using nickel based brazing, and was subsequently trimmed to it into the test section o a heat sink channel o size m (W) m (L) m (H). Fig Sandwiched oam sample ready or heat transer experiment Experimental procedures and data acquisition The experimental apparatus consists o our sections: coolant supplier, test section, low channel, and data acquisition system. Air is used as a coolant and is orced through the channel inlet by a suction type air blower. One wire screen and one honeycomb are inserted in the channel beore its cross-section starts to contract. The coolant then lows through a 9:1 contraction section and a low developing channel region to ensure that the low is hydraulically ully-developed when it reaches the test section. As shown in Fig. 2.12, the oam sample is encapsulated by two acrylic sidewalls. A low rate regulator is located between the exit o the test section and the suction device. 26

28 Figure 2.12 Photograph o test rig or pressure drop and heat transer experiment For pressure drop measurements, 4 static pressure taps o inner diameter 0.68 mm were placed on each sample along the low direction on the upper cover plate with a uniorm spacing o 25 mm. A Scanivalve is calibrated with a digital micromanometer beore perorming measurements. For heat transer experiments, an asymmetrical isolux (constant wall heat lux) boundary condition was imposed on the lower copper skin by a heating element (silicone-rubber etched oil rom Watlow Inc.). The amount o heat released rom the heating element was adjusted by changing the voltage o V6HMTF Zenith AC variac, which was monitored by a Fluke 73 series II digital multimeter. The variac controls the heating element with a voltage range o V, corresponding to a heat input o W. For this investigation, input voltages o 120 and 168 V were used, generating 75 W o heat (or 4000 W/m 2 heat lux intensity) and 105 W (or 8000 W/m 2 ). The external surace o the heating element was covered by a 45mm thick Tancast 8 thermal insulation material to minimize heat loss rom direct contact. Because the electrical resistance oil was etched in a zigzag pattern on the silicone rubber pads, there is a possible gap (unspeciied by the manuacturer) between two columns o the oil. Pure copper heat spreader plate (0.9 mm) was inserted between the heating element and the metal oam skin to ensure uniormity o heat lux entering the heat sink. Four thin oil (0.05 mm thickness) T-type copper-constantan thermocouples (rom Rhopoint Inc.) were installed on the lower copper plate along the longitudinal direction (i.e. the low direction). There were two additional T-type thermocouples, positioned separately at the inlet and the outlet o the test section to measure the coolant temperature at each location. A temperature scanner with reading 27

29 resolution o ± 0.1 K was used to measure the thermocouple output, with the capability to record temperature rom all thermocouples simultaneously. All measurements were perormed under steady state conditions, and it usually takes minutes to reach steady state ater each change o the Reynolds number. A Pitot tube was positioned beore the test section to measure stagnation pressure and static pressure at the inlet o the test section. Because the blockage ratio, i.e., the ratio o channel height (12 mm) to the tube outer diameter (0.51 mm), is 23.5, wall intererence to the Pitot tube is expected to be negligible Measurement uncertainties An uncertainty analysis is perormed ollowing the method o Kline and McClintock [32]. The maximum heat loss through the insulation materials is estimated to be less than 2% (2.8W) o the 150W heat input. The heat loss through the side-walls is assumed to be negligible due to the small conduction area. Here, only random errors are considered or heat input, temperature and pressure measurements; analysis o systematic errors has not been conducted due to the shortage o relevant inormation and its complexity. The thermal conductivity k o aire varies slightly in the operating temperature range o to K. An arithmetic mean value is used or k, with uncertainty estimated to be within 6.6%. From these, the uncertainty in the measured heat transer coeicient and Nusslet number is estimated to be less than 7.0% and 9.6%, whilst the uncertainety in the pressure drop and iction actor measurements is estimated to be less than 5.0% and 7.8%, respectively Experimental results and analysis The pressure drop and heat transer results will be presented in this section. Pressure drop The measured results o pressure drop are plotted in Fig and Fig or FeCrAlY and copper samples, respectively, as a unction o the mean low velocity U m o air. 28

30 140, ,000 S - 6 FeCrAlY samples 100,000 dp Pa ( ) dl m 80,000 60,000 S - 5 S - 4 S ,000 S - 7 S ,000 S U m (m/s) Fig Pressure drop results or all FeCrAlY samples 140, ,000 Copper samples S ,000 S - 12 dp dl Pa m 80,000 60,000 S - 14 S - 13 S ,000 S , U m (m/s) Fig Pressure drop results or all copper samples 29

31 The pressure drop increases with relative density, while decreases with pore size (d p ). It can also be noted that the pressure drop o a copper sample is in general much higher than that o a FeCrAlY sample with identical nominal relative density and ppi. This may be attributed to the much irregular and smaller cell size o the copper sample compared to the FeCrAlY sample FeCrAlY S - 1 S - 2 S - 3 S - 4 S - 5 S - 6 S , Re Dh Fig Friction actors o all FeCrAlY samples 500 Copper 200 S S S - 12 S - 11 S S , , Re Dh Fig Friction actors o all copper samples 30

32 From the pressure drop measurements, the riction actor or each sample, deined as P Dh = L ρu 2, can be calculated. The results are shown in Fig and Fig / 2 m or FeCrAlY and copper samples, respectively. It can be seen that the riction actor approach to a constant or each sample, as expected in turbulence lows at relatively high Reynolds numbers. Heat Transer Skin temperature T W (x) measured rom thermocouples placed on the oam skin plates was recorded. The temperature distribution or one case (Sample #4, FeCrAlY with 30 ppi and 10% relative density) at an input heat lux q o 8000 W/m 2 is plotted in Fig or selected Reynolds numbers. The typical linear variation o wall surace temperature along the low direction is observed or ully developed convection with isolux boundary condition T W (x) [K] Re Dh = 3360 Re Dh = 5450 Re Dh = 7590 Re Dh = Re Dh = Re Dh = x [m] Fig Substrate surace temperature distribution along the low direction or FeCrAlY S-4 (q = 8000 W/m 2 ). 31

33 The thermal perormance o Porvair metal oams as a heat sink medium can be assessed by calculating the overall Nusselt number deined as ollows: 1 Nu = L where 0 L Nu Nu( x) dx (2.8) q T ( x) T h ( x) = (2.9) w in D k Here, q is the applied heat lux, T W is the local substrate temperature, T in is the coolant inlet temperature, k is the thermal conductivity o the coolant, and D h is the hydraulic diameter o the heat sink channel. The overall Nusselt numbers or seven FeCrAlY samples are summarized in Fig Nu FeCrAlY S - 1 S - 2 S - 3 S - 4 S - 5 S - 6 S ,000 5,000 10,000 2,0000 Fig Overall Nusselt numbers o all FeCrAlY samples Sample #1 (10 ppi, 5% nominal relative density) has the lowest value o the overall Nusselt number, while sample #2 (10 ppi, 10% nominal relative density) achieves the highest value. At a ixed nominal cell size or FeCrAlY samples, the eect o relative density on the overall Nusselt numbers is shown in Figs to 2.21 or 10ppi, 30ppi and 60 ppi, respectively. Similarly, at a given nominal relative density, the eect o cell size on the overall heat transer is presented in Figs and 2.23 or 5% and 10 % nominal relative densities, respectively. 32

34 FeCrAlY ppi S-2, 10% relative density (14.3% measured) Nu 200 S-1, 5% relative density (5.7% measured) 100 3,000 5,000 10,000 2,0000 Re Dh Fig Eect o relative density on overall Nusselt number or a ixed cell size (10 ppi) 300 FeCrAlY ppi S-4, 10% (10.2% measured) Nu S-3, 5% relative density (6.1% measured) 100 S-7, 7.5% (8.1% measured) 3,000 5,000 10,000 2,0000 Re Dh Dh Fig Eect o relative density on overall Nusselt numbers or a ixed cell size (30 ppi) 33

35 FeCrAlY 60 ppi S-6, 10% relative density (13.7% measured) Nu S-5, 5% relative density (9.0% measured) 100 3,000 5,000 10,000 2,0000 Re Dh Fig Eect o relative density on overall Nusselt numbers or a ixed cell size (60 ppi) FeCrAlY Nominal relative density = 5% 60 ppi (9.0% measured) Nu 10 ppi (5.7% measured) 30 ppi (6.1% measured) ,000 5,000 10,000 2,0000 Re Dh Fig Eect o cell size on overall Nusselt numbers or a ixed nominal relative density (5%) 34

36 FeCrAlY Nominal relative density = 10% 10 ppi (14.3% measured) Nu ppi (13.7% measured) 30 ppi (10.2% measured) 100 3,000 5,000 10,000 2,0000 Re Dh Fig Eect o cell size on overall Nusselt numbers or a ixed nominal relative density (10%) From the above results, it can be seen that the overall heat transer in general increases with increasing relative density and decreasing cell size. but the Fig doesn t show the same situation. However, Fig suggests that increasing the relative density is more important than reducing the cell size or heat transer enhancement. In other words, variations in the cell size inluence the pressure drop more than heat transer, whilst heat transer is more sensitive to the variations in the relative density. The overall Nusselt numbers or six copper samples are presented in Fig As or the FeCrAlY samples, the eects o relative density on the overall heat transer o copper samples are shown in Figs to 2.27, and eects o cell size are given in Fig and Fig

37 Copper 300 Nu S-9 S-10 S-11 S-12 S-13 S ,000 2,000 4,000 6,000 10,000 2,0000 Re Dh Fig Overall Nusselt numbers o all copper samples Copper 10 ppi Nu S-9, 5% relative density (6.7% measured) S-10, 10% relative density (8.2% measured) 100 1,000 2,000 4,000 6,000 10,000 2,0000 Re Dh Fig Eect o relative density on overall Nusselt numbers or a ixed cell size (10 ppi) 36

38 Nu Copper 30 ppi S-12, 10% relative density (10% measured) S-11, 5% relative density (4.4% measured) ,000 2,000 4,000 6,000 10,000 2,0000 Re Dh Fig Eect o relative density on overall Nusselt numbers or a ixed cell size (30 ppi) Nu Copper 60 ppi S-13, 5% relative density (5.7% measured) 100 S-14, 10% relative density (9.5% measured) 50 1,000 2,000 6,000 4,000 8,000 Re Dh Fig Eect o relative density on overall Nusselt numbers or a ixed cell size (60 ppi) 37

39 Copper 400 Nominal Relative density = 5% Nu ppi (5.7% measured) 30 ppi (4.4% measured) ppi (6.7% measured) 50 1,000 2,000 4,000 6,000 10,000 2,0000 Re Dh Fig Eect o cell size on overall Nusselt numbers or a ixed nominal relative density (5%) Copper Nominal relative density = 10% ppi (9.5% measured) Nu ppi (8.2% measured) ppi (10% measured) 50 1,000 2,000 4,000 6,000 10,000 2,0000 Re Dh Fig Eect o cell size on overall Nusselt numbers or a ixed nominal relative density (10%) 38

40 From these results, it can be seen that the overall heat transer o a copper sample is not as sensitive to the relative density as that or a FeCrAlY samples, whereas the cell size eect on heat transer o the copper sample is a bit more signiicant than that or the FeCrAlY sample. Again, smaller cell sizes (at a ixed relative density) lead to higher overall heat transer. Some explanations will be given below. For FeCrAlY samples, the solid thermal conductivity (k s ) is around 20 W/mK, so the thermal resistance at the solid side is large. Consequently, increasing the relative density o these oams can cause signiicant reduction o the thermal resistance in the solid part, resulting in a strong eect on the overall heat transer. However, or copper samples, the thermal conductivity (k s ) is approximately 300 W/mK, implying that the thermal resistance in the solid part is relatively small and the resistance in luid part is relatively large. Thereore, increasing the relative density o copper sample would not lead to a dramatic eect on heat transer, whereas reducing the cell size could to a certain extent enhance overall heat transer duo to the reduction o thermal resistance in the luid part. Fig compares the heat transer between FeCrAlY and copper samples having the same measured relative density (10%) and ppi (30). It is seen that the slope o the Nu versus Re Dh curve or the copper sample is larger than that or the corresponding FeCrAlY sample, conirming the above explanation ppi, 10% relative density 300 Nu 200 S-12, Copper (10% measured) S-4, FeCrAlY (10.2% measured) 100 1,000 2,000 5,000 10,000 2,0000 Re Dh Fig Comparison o overall Nusselt number or FeCrAlY and copper samples 39

41 At a given Reynolds number, the Nusselt numbers o copper samples are approximately 2 to 4 times those o FeCrAlY samples. For both types o oam, increasing the relative density and decreasing the cell size will lead to heat transer enhancement. However, it should be pointed out that the increase in pressure drop will become even larger as a consequence o the heat transer enhancement. To give an overall assessment, it is oten necessary to introduce a nondimensional eiciency index, % j Nu, which represents the ratio between heat transer and low resistance. Fig and Fig plot ~ j as a unction o the Reynolds number or FeCrAlY and copper samples, respectively. Both igures indicate that the ~ j number decreases while increasing relative density and decreasing cell size, conirming that pressure drop increases at a aster pace than heat transer does. The comparison between FeCrAlY and copper samples are given in Fig Note that sample 1 (FeCrAlY) and sample 9 (copper) achieve the highest eiciency while sample 6 (FeCrAlY) exhibits the lowest eiciency. On the whole, there is no major dierence between FeCrAlY and copper samples with similar relative density and pore size. 30 ~ j S - 1 S - 2 S - 3 S - 4 S - 5 S ,000 8,000 12,000 16,000 Re Dh Fig The eiciency numbers or FeCrAlY samples 40

42 ~ j S - 9 S - 10 S - 11 S - 12 S - 13 S ,000 8,000 12,000 16,000 Re Dh Fig The eiciency numbers or copper samples 30 ~ j S - 1 S - 2 S - 3 S - 4 S - 5 S - 6 S - 9 S - 10 S - 11 S - 12 S - 13 S ,000 8,000 12,000 16,000 Re Dh Fig The comparison o the eiciency numbers between FeCrAlY and copper samples 41

43 2.4.4 Product uncertainties Up until now the experiments on orced convection have been conducted or seven FeCrAlY samples and six copper samples. It is noted that the measured microstructures are quite dierent rom the industrial speciications. It would also be interesting to compare the perormance o two FeCrAlY samples (S-7 and S-8) with identical industrial speciications, i.e., 7.5% relative density and 30 ppi, but manuactured rom two dierent batches. With the eect o brazing, the measured relative densities are 8.1% and 8.9% or sample 7 and sample 8, respectively. Because o the dierence in relative densities, their pore sizes will also be dierent (although this is yet to be measured). As a result, the measured pressure drop and heat transer are somewhat dierent or the two samples, as shown in Fig and Fig Thereore, how to improve the product quality and reduce the variability rom dierent batches is a problem to be addressed. 60,000 50,000 FeCrAlY samples dp Pa ( ) dl m 40,000 30,000 S - 8 S ,000 10, U (m/s) Fig Comparison o pressure drop results between samples 7 and 8. 42

44 FeCrAlY 30 ppi S - 8, 7.5% (8.9% measured) Nu S-7, 7.5% (8.1% measured) 0 3,000 6,000 9,000 12,000 15,000 Re Dh Dh Fig Comparison o overall Nusselt numbers between samples 7 and 8 The experimental data o pressure drop, Nusselt number and eiciency index or all 14 Porvair metal oam samples are listed in Appendix B. 43

45 2.5 Numerical modelling on orced convection in Porvair metal oams In this section the modelling and numerical simulation will be conducted or Porvair metal oams. With the measured relative density, pore size and strut hollowness as input, the predicted heat transer perormance will be compared with test data to check the validity o the numerical model Mathematical ormulations The problem under investigation is orced convection o incompressible luid low through open-celled metal oams. As previously discussed, the complexity o the cellular structure usually precludes a detailed microscopic investigation o the transport phenomena at the pore level in porous media. Thereore, the general transport equations are commonly integrated over a representative elementary volume, which accommodates the luid and the solid phases within a porous structure. There are two approaches available in applying the volume-averaging technique or heat transer investigations: one is averaging over a representative elementary volume containing both the luid and the solid phases, and the other is averaging separately over each o the phases, thus resulting in a separate energy equation or each individual phase. These two models are reerred to as the one-equation model and the two-equation model, respectively. The one-equation model is valid when the local temperature dierence between the luid and solid phases is negligibly small. Because the temperature dierence between the solid and luid phases cannot be neglected in metal oams with air as the coolant luid, the two-equation model will be used below. The geometry o the compact heat sink with metal oams is depicted in Fig and Fig For simplicity, the width o the channel is assumed to be suiciently long such that the problem can be considered as two dimensional, and the other assumptions upon which the numerical model is based are summarized as ollows: (1) The medium is homogeneous and isotropic. (2) Forced convection dominates in the metal oams, i.e., natural convection eects are negligible. (3) Variation o the thermophysical properties with temperature is ignored. (4) Due to the relatively low operating temperature (< 100C) considered in the present study, radiation heat transer is neglected. 44

46 (5) Fluid low and heat transer reach steady state in the channel. Under these assumptions, the governing equations or the velocity and temperature ields in the metal oam material are established by applying the volume-averaging technique. The extended Darcy equation proposed by Vaai and Tien [3,4] is used in place o the Darcy equation in order to account or the boundary and inertial eects. The governing equations and boundary conditions are Continuity equation V = 0 (2.10) Momentum equation ρ µ 2 ε ( V ) V = P + V V ρ F [ V V ]J ε µ K I (2.11) The last term in Eq. (2.11) was irst introduced by Forchheimer to account or the inertial eects (non-darcy low). Similarly, the second term accounts or the boundary eects on the velocity distribution, and was irst introduced by Brinkman. Solid phase energy equation {( k + k ) T } h a( T T ) 0 = ~ (2.12) se Fluid phase energy equation d s s s {( k + k ) T } + h a( T T ) ρ C V T = ~ (2.13) e d where means a volume-averaged value and k se, T, hs, a ~, ρ, µ, C and k e are eective thermal conductivity o the solid, temperature, interacial heat transer coeicient, wetted area per volume, density, viscosity, heat capacity and eective thermal conductivity o luid, respectively; k d is the thermal dispersion conductivity, K is the permeability o the porous medium, ε is the porosity, V is the velocity vector, s s and J = V / V is the unit vector aligned along the pore velocity vector, V p ; F I is the P P inertial variable with unit m -1, which depends on the microstructure o the porous medium. I the Reynolds number is small such that laminar low prevails, F = C K, where C I is the inertial coeicient. I I / An order o magnitude analysis on the momentum equation shows that the momentum boundary layer thickness is o the order o ( ) 1/ 2 K /ε and that the convective term ( V )V responsible or boundary layer growth is signiicant only over a length o 45

47 the order o ( /ν ) Ku [3,9]. Thereore, a ully developed momentum boundary layer c is in orce beyond a very short developing length. In the present study, the ully developed velocity ield is assumed, and hence the momentum equation (2.11) and energy equations (2.12) and (2.13) can be simpliied as µ µ 0 u ε K 2 2 = P + u u ρ FI (2.14) ( T T ) Ts Ts 0 = k se + k se hs a~ s (2.15) x x y y T T T ρ C u = ( k e kd ) ( k e k d ) + hs a( Ts T ) x x + x + y + y ~ (2.16) For brevity, the bracket has been dropped in these equations Boundary conditions When a heat lux is directly applied to the outer surace o the metal oam, the applied heat is transerred to the solid and luid phases by conduction and convection. As discussed in reerences [33, 34], the wall heat lux boundary condition may be viewed in two dierent ways. The irst is to assume that each representative elementary volume at the wall surace receives a prescribed heat lux that is equal to the wall heat lux q w. As a result, the heat will be divided between the two phases on the basis o the physical values o their eective conductivities and their corresponding temperature gradients. The second approach is to assume that each o the individual phases at the wall surace will receive an equal amount o heat lux q w [33]. In the present study, a 1mm copper substrate with high thermal conductivity is attached to the metal oam as shown in Fig. 2.11, and the heat lux is applied to the external wall o the substrate instead o being applied directly to the outer surace o the metal oam. In this case, the temperature at the interace between the metal oam and the copper substrate can be considered to be uniorm regardless o whether it is in contact with the solid or luid phase due to the high thermal conductivity o the copper cover plate. Consequently, the boundary condition at the heating side o the channel can be written as T y= H Ts y= H T W (2.17) 46

48 where T w implies the temperature at the interace. This temperature is not known a priori and must be obtained as part o the solution. Finally, the boundary conditions are speciied as ollows, T T q = qw = k e + k and T s = T at y = 0; y s se y y= 0 y=0 q = 0 at y = H (2.18) T T = T in and s = 0 at x = 0 x T s T = 0 and = 0 x x at x = L where H and L are the height and length o the channel, respectively Modelling on Porvair metal oams Beore proceeding urther, the permeability K and inertial variable F I appearing in the momentum equation (2.14) must be known in order to calculate the velocity ield. The permeability K and inertial variable F I o a metal oam have been investigated by several researchers [19,28,31,35], although only Calmidi [31] gave speciic ormulations o K and F I based on experimental data. In this study, the ormulations proposed by Calmidi [31] will be used, with K d F = p ( ) d ε (2.19) d p 1.63 d = ( 1 ε ) K (2.20) d p I / Similarly, in energy equations (2.15) and (2.16), the eective conductivity k se, k e and the dispersion conductivity k d need to be determined in order to close the equations. An analytical model or the eective conductivity o open-celled metal oams based on the three-dimensional cellular morphology has recently been proposed by Boomsma and Poulikakos [29], yielding k e where = 2 2 ( R + R + R + R ) A B C D (2.21) 47

49 R A = 2 2 ( 2e + πλ( 1 e) ) k + 4 2e πλ( 1 e) s 4λ ( ) k 2 ( e 2λ) 2 ( e 2λ) e k + 2e 4λ ( e 2λ) RB 2 s R C ( e ) k (2.22a) = (2.22b) = 2πλ RD 2 2 e k s + with 2 2 ( 2 2e) 2 ( 1 2e 2) k s + 2( 2 2e πλ ( 1 2e 2 ) k ( 4 e ) k (2.22c) 2e = (2.22d) 3 ( 2 ( 5/8) e 2 2ε ) 2 λ =, e = (2.23) π ( 3 4e 2 e) Thus, the eective solid conductivity, k se, is obtained by setting k = 0 in (2.21) and (2.22). Similarly, the luid conductivity, k e, is obtained by setting k s = 0 in both equations. It should be noted that the above eective conductivity model is developed or metal oams having solid struts (e.g., ERG oams). For Porvair oams, the cell ligaments are typically hollow, and hence the porosity and eective solid conductivity need to be changed accordingly. I the Porvair oams has the same relative density as the ERG material, then its porosity can be calculated according to: 2 r ε porvair = ε (2.24) 2 1 r where r is the inner and out diameter ratio o the hollow struts in Porvair oams. Similarly, the eective solid conductivity o a Porvair oam should be modiied as k se ( ) 2, porvair = k se 1 r (2.25) As or the dispersion conductivity, k d, a widely adopted expression is [26], k d where u = CD ( Re k Pre ) ke (2.26) u Re k = u m ν m K, the Reynolds number based on permeability, µc Pr e = is the k Prandtl number based on eective conductivity, u m is the average low velocity entering the oam sample, and C D ( 0.1) is the thermal dispersion coeicient. e Prior to numerical calculations, the surace area density, a ~, and interstitial heat transer coeicient, h s, need to be known or the porous medium. The solid-luid 48

50 interacial surace area or arrays o parallel cylinders intersecting in three mutually perpendicular directions (Fig. 2.7) is ~ 3πd a = (2.27) d 2 p For metal oams, this expression is modiied by taking the 3D microstructure into account (i. e., open cells shaped like dodecahedra, and noncircular iber crosssections). By doing so, the pore size d p and solid strut diameter, d, are multiplied by a actor 0.59 and 1 e (( 1 ε )/ 0.04), respectively [31]. As or the interstitial heat transer coeicient, h s, Wakao et al. [10] proposed one o the most comprehensive models or packed beds. For oamed materials, however, no such general model exists. Here, based on a correlation developed by Zukauskas [36] or staggered cylinders in crosslow, the ollowing correlation [26] is employed in the present study, h = 0.52Re Pr k / d (2.28) s where, ( εν ) Re = ud Numerical procedure Because a ully developed velocity condition is assumed at the cross-section, the momentum equation (2.14) becomes a second-order nonlinear ODE, and its solution can be numerically calculated. The iteration will be terminated when the condition that the integral o the non-dimensional velocity over the height o the channel is unity within a speciied error o The energy equations were solved by using the ADI inite dierence scheme, with uniorm grid spacings used in the x and y directions, respectively. The convergence criterion was that the change in the solid-phase and luid-phase temperatures was less than 10-5 between successive iterations Code validation In order to validate the numerical code, the heat transer perormance o a ERG metal oam is calculated and compared with the experimental data obtained by Calmidi and Mahajan [22]. In their experiment, the boundary condition was same as in the present study, and the permeability and eective solid conductivity k se were given. The given values o K and k se are directly input the numerical programme. The predicted 49

51 velocity distribution and Nusselt number or the ERG oam tested in [22] are shown in Fig and Fig. 2.37, respectively. The agreement between prediction and experimental data [26] is good (Fig. 2.37). 5 4 Rek = 140 u Fig Fully developed velocity distribution along the vertical direction o a ERG oam channel Y Nu 10 5 Experimental data [26] Numerical result Re k Fig Comparison between numerical calculation and experimental data [26] or a ERG oam channel 50

52 2.5.6 Numerical results or Porvair metal oams For Porvair metal oam sample 1, Fig shows the solid and luid temperature distributions at the cross-section o X = 0.5. It can be seen that the temperature dierence between the solid and luid cannot be neglected, especially or smaller Reynolds numbers and small values o Y (i.e., near the thermal boundary zone). 160 T Sample 1 X = 0.5 Re Dh = T T s Fig Solid and luid temperature distributions along the vertical direction at X = 0.5 Y Sample 1 Ts,b T Re = 2000 Dh T,b Re = Dh T s,b T,b X Fig Averaged solid and luid temperature distributions along the streamwise direction 51

53 The streamwise distributions o solid and luid temperatures averaged over the crosssection o the channel, T sb, and T, b, are presented in Fig or sample 1. As expected rom a constant heat lux boundary condition, the solid and luid temperature lines are parallel to each other. The variation o local Nusselt number, Nu q D w h, b =, along the streamwise direction is shown in Fig or three Tw T, b k Reynolds numbers Sample 1 Nu, b Re = 2000 Dh x Fig Variation o local Nusselt number along streamwise direction From Fig. 2.40, it is seen that the eect o thermal entrance is signiicant i X is small (< 0.2), and the thermal entrance length becomes longer with increasing Reynolds number. The calculated results or the overall Nusselt number ( Nu ) are shown in Figs to 2.51 or all FeCrAlY and copper oam samples and compared with the corresponding experimental data. In these igures, the solid lines represent predictions based on the solid conductivity k s = 16 W/mK or FeCrAlY samples and k s = 372 W/mK or copper samples. Given the complexity o heat transer in metal oams, the predictions appear to be in reasonable agreement with measured values, although the predictions somewhat underestimate the Nusselt numbers or FeCrAlY samples and overestimate the overall heat transer rate or copper samples. The reason may be attributed to the 52

54 eects o nickel based brazing on the solid conductivity k s. Brazing tend to increase the solid conductivity or FeCrAlY samples and decrease the solid thermal conductivity or copper samples. In the aorementioned igures, the dashed lines represent the predicted results by simply changing the solid conductivity k s or both FeCrAlY and copper. It was ound that the solid conductivity k s ranging rom 16 ~ 26 W/mK or dierent FeCrAlY samples and 200 ~ 310 W/mK or dierent copper samples led to improved predictions in comparison with test data FeCrAlY 10 ppi S-2, 10% relative density (14.3% measured) Nu 200 k = 20 W/mK s k = 26 s 100 S-1, 5% relative density (5.7% measured) 0 3,000 5,000 7,000 9,000 11,000 13,000 15,000 Re Dh Fig Nusselt numbers or sample 1 and sample 2: predictions versus experiments FeCrAlY 30 ppi S-3, 5% relative density (6.1% measured) Nu k s = 26 W/mK ,000 6,000 9,000 12,000 15,000 Re Dh Fig Nusselt numbers or sample 3: predictions versus experiments 53

55 300 FeCrAlY ppi S-4, 10% (10.2% measured) Nu k = 20 W/mK s ,000 5,000 7,000 9,000 11,000 13,000 Re Dh Dh Fig Nusselt number or sample 4: prediction versus experiment FeCrAlY 60 ppi S-6, 10% relative density (13.7% measured) k s = 18 W/mK Nu 150 S-5, 5% relative density (9.0% measured) ,000 5,000 7,000 9,000 11,000 13,000 Re Dh Fig Nusselt numbers or sample 5 and sample 6: predictions versus experiments 54

56 FeCrAlY 30 ppi S-7, 7.5% (8.1% measured) Nu k = 23 W/mK s ,000 6,000 9,000 12,000 15,000 Re Dh Dh Fig Nusselt number or sample 7: prediction versus experiment 800 Copper ppi Nu 400 S-9, 5% relative density (6.7% measured) k = 310 W/mK s ,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 Re Dh Fig Nusselt numbers or sample 9: prediction versus experiment 55

57 Copper 10 ppi k s = 272 W/mK Nu S-10, 10% relative density (8.2% measured) 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 Re Dh Fig Nusselt numbers or sample 10: prediction versus experiment 800 Copper ppi S-11, 5% relative density (4.4% measured) Nu k = 300 W/mK s 0 1,000 3,000 5,000 7,000 9,000 11,000 Re Dh Fig Nusselt number or sample 11: prediction versus experiment 56

58 800 Copper ppi Nu 400 k s = 200 W/mK 200 S-12, 10% relative density (10% measured) 0 1,000 3,000 5,000 7,000 9,000 Re Dh Fig Nusselt number or sample 12: prediction versus experiment Copper 60 ppi S-13, 5% relative density (5.7% measured) Nu k = 320 w/mk s ,000 2,000 3,000 4,000 5,000 6,000 Re Dh Fig Nusselt number or sample 13: prediction versus experiment 57

59 Copper 60 ppi S-14, 10% relative density (9.5% measured) Nu k = 230 w/mk s ,000 2,000 3,000 4,000 5,000 6,000 Re Dh Fig Nusselt number or sample 14: prediction versus experiment Eect o boundary conditions The numerical calculations thus ar have been conducted or the constant heat lux boundary condition. In this section, the eect o changing the boundary condition to the constant wall temperature condition upon heat transer will be examined briely. For constant wall temperature, the boundary will become more direct and easy to implement in the numerical programme. The predicted local and overall Nusselt numbers or sample 1 are shown in Fig and Fig. 2.53, respectively, together with the results based on the constant heat lux condition. Note that the boundary eect is small, as expected. In general, the heat removal capability o the oam with constant heat lux is 5 ~ 6% larger than that associated with the constant wall temperature boundary condition. 58

60 Sample 1 Re = 6000 Dh Nu, b constant heat lux 100 constant wall temperature x Fig Eect o boundary conditions on local heat transer in sample Sample constant heat lux Nu constant wall temperature ,000 4,000 6,000 8,000 10,000 12,000 Re Dh Fig Eect o boundary conditions on overall heat transer in sample 1 59

61 2.5.8 Eects o microstructural parameters: Optimisation Solid thermal conductivity (k s ) From Eqs. (2.21) to (2.25), it is known that the solid conductivity (k s ) will directly aect the eective solid conductivity (k se ) o a metal oam, and subsequently aect its heat transer behavior. Fig presents the variations o the eective solid conductivity with the solid material conductivity or a ixed oam porosity ε = ε = k se k s Fig Linear relationship between k s and k se The eect o solid conductivity k s on the overall Nusselt number is shown in Fig or a ixed microstructure (sample 4) and three values o the Reynolds number. The results all reveal that the overall Nusselt number increases sharply when k s is small, and then gradually approaches a plateau with increasing k s. However, it should be noted that, or small Reynolds numbers ( Re= 1000), heat transer saturation occurs at small thermal conductivity levels (k s 50 W/mK), while or larger Reynolds numbers Re = 4000, the overall heat transer reaches the saturation stage when k s 200 w/mk. This implies that the main thermal resistance or small Reynolds numbers lies on the luid side when the solid thermal conductivity k s exceeds a critical value ( 50). Here, we deine this value as k s, max, beyond which the overall heat transer is independent o the solid thermal conductivity. For higher Reynolds number, k s, max 60

62 becomes larger because the thermal resistance in the luid side decreases. Consequently, in practical applications, there is no need to use oam materials with high thermal conductivities, i the Reynolds number is small. The variation o k s,max with the Reynolds number or sample 4 is presented in Fig , Re = Dh Nu k s Fig Eect o solid conductivity on overall Nusselt number or sample Sample 4 microstructure k s,max ,000 4,000 7,000 10,000 13,000 16,000 Fig Variation o Re Dh k s, max with Reynolds number or sample 4 61

63 Pore size (d p ) For ixed values o porosity ( ε = 0.95) and Reynolds number ( Re 10000), the Dh = eect o varying pore size (d p ) on the overall Nusselt number is presented in Fig or one FeCrAlY sample, but with the assumption that the inner-to-outer diameter ratio r = 0. The igure shows that the heat transer increases with decreasing pore size. 160 FeCrAlY (r = 0) 140 Re = Dh ε = 0.95 Nu d p (mm) Fig Eect o pore size (d p ) on overall heat transer 40 ~ j FeCrAlY (r = 0) Re = Dh ε = d p (mm) Fig Eect o pore size (d p ) on heat transer eiciency index 62

64 However, because the pressure drop is reduced more signiicantly by increasing the pore size (d p ), the eiciency index ( ~ j = Nu / ) still increases with increasing pore size (d p ), as shown in Fig Porosity (ε) For simplicity, assume the struts are solid (i.e., the inner-to-outer diameter ratio r = 0). The relationship between porosity (ε) and relative density (ρ r ) can then be simpliied as ρ r = 1-ε. For a ixed pore size o d p = 4 mm and Re 10000, the Dh = eect o porosity on the overall Nusselt number is shown in Fig The igure shows that overall heat transer decreases as the porosity is increased. In other words, heat removal will be enhanced by increasing the relative density (ρ r ) Nu Re = d p Dh = 4 mm FeCrAlY (r = 0) Fig Eect o porosity (ε) on overall heat transer The eects o porosity on the eiciency index ( j ~ ) are shown in Fig or three Reynolds numbers, with r = 0 and d = 4 mm. For a given Reynolds number, there p exists an optimised porosity, ε opt, which maximizes the eiciency index (Fig. 2.60). This optimised porosity decreases with increasing Reynolds number. The variation o ε opt with the Reynolds number is presented in Fig ε 63

65 Re Dh = d p = 4 mm FeCrAlY (r = 0) ~ j ε Fig Eect o the porosity (ε) on eiciency number ( j ~ ) d p = 4 mm FeCrAlY (r = 0) ε opt ,000 4,000 6,000 8,000 10,000 12,000 Re Dh Fig Variations o the ε opt with Reynolds numbers 64

66 3 Sintered Particle Bed 3.1 Introduction Packed particle beds have been widely used in the chemical industry and or energy storage purposes. Among the common applications are catalytic reactors, absorption and adsorption operations, as well as packed bed heat storage units. Due to their intrinsic high surace area ratio, the heat transer rate will be increased by several orders o magnitude. Thereore, they are now also being used as ixed-bed nuclear propulsion systems, spacecrat thermal management systems and heat sinks or high perormance cooling in microelectronics. The orced convection in packed bed has been extensively investigated [2-18] due to its many important engineering applications. Sintered beds with bronze spheres or other materials can now be produced in a large scale with high quality. A sintered bed can provide much better contact between the beads than a packed bed, and hence its eective thermal conductivity (k se ) could be dramatically improved. As a result, heat transer could be signiicantly enhanced by using sintering techniques. However, there is little study on orced convection in sintered beds in comparison with the extensive investigations on packed beds. Hwang and his co-workers [37,38] conducted experimental and numerical investigations on the transport o heat in sintered bronze beads. However, in their numerical calculations, the measured eective conductivity k e was used as the input: no model was put orward to calculate the eective thermal conductivity or sintered particle beds. In addition, in their experimental and numerical studies, air was used as the coolant luid. Since the porosity (ε) o a sintered bed is relatively small (ε ), its surace area density should be large and hence the thermal resistance in the solid part is relatively small compared to that on the luid side. Consequently, i water is used as the coolant, the thermal resistance in the luid part will be reduced dramatically, resulting in signiicant heat transer enhancement. As very ew studies have been conducted or orced water convection in sintered beds, a thorough investigation combining both modelling and testing is warranted. 65

67 The objective o this chapter is to build a numerical model, and study the eect o eective solid conductivity on the overall heat transer o a sintered bed. Predictions on orced water convection will be obtained or sintered beds made with bronze beads having our dierent diameters. 3.2 Mathematical ormulations The problem under investigation is orced convection o incompressible luid low through the sintered porous channel, as shown in Fig H q w Fig. 3.1 Forced convection in sintered bronze bed The width o the channel is assumed to be long enough that the problem will essentially be two dimensional and a ully developed velocity ield is assumed, because the momentum boundary layer becomes ully developed over a very short developing length in the porous medium. The governing equations or the velocity and temperature ields in the sintered bed are established by applying the volumeaveraging technique. The extended Darcy equation proposed by Vaai and Tien [3,4], is used in place o the Darcy equation in order to account or the boundary eect and inertial eect. The two-equation heat transer model [4,6] is employed in the present analysis. The governing equations and boundary conditions are thereore identical to those described in Chapter 2, but will be presented below or easy reerence: Momentum equation µ µ ρ C 0 u ε K K 2 I 2 = P + u u (3.1) Solid phase energy equation ( T T ) Ts Ts 0 = k se + k se hs a~ s (3.2) x x y y Fluid phase energy equation 66

68 T T T ρ C u = ( k e kd ) ( k e k d ) + hs a( Ts T ) x x + x + y + y ~ (3.3) Boundary conditions T T q = qw = k e + k and T s = T at y = 0; y s se y y= 0 y=0 q = 0 at y = H (3.4) T T = T in and s = 0 at x = 0 x T s T = 0 and = 0 x x at x = L It should be pointed out that although equations (3.1) to (3.4) are the same as those o chapter 2 or metal oams, the permeability K, inertial coeicient C I, eective solid and luid conductivity (k se, k e ), dispersion conductivity k d, interacial heat transer coeicient h s, and surace area density a ~ are dierent rom those o metal oams. These parameters need to be separately modelled in order to close the equations. 3.3 Modelling on sintered bed The permeability and inertial coeicient or a packed spherical particle bed were obtained rom the experimental results [39,40] as unctions o porosity ε and particle diameter d p : 3 2 ε d p K = 150 ε ( 1 ) C I = (3.5) 3 / 2 150ε These semi-empirical relations have been extensively validated, and can give reliable predictions on the permeability and inertial coeicient o a packed bed. For a sintered particle bed, Hwang and Chao [37] obtained its permeability and inertial coeicient, and the results obtained appear to be less than those predicted rom the packed bed correlations o (3.5). So ar, the test data are not suicient to correlate a ormulation similar to (3.5) or sintered beds. However, it is known that the permeability and inertial coeicient have small eects on the heat transer o a packed or sintered bed, although they have large eects on pressure drop. In the 67

69 present study only heat transer will be studied, and hence the permeability and inertial coeicient correlations o Eq. (3.5) will be adopted or numerical calculations. The eective thermal conductivity (k e ) o a luid saturated particle bed has signiicant eect on its heat transer, and hence must be studied careully. So ar, most researchers employed the linear relations k se ( ε ) k s = 1 and k e = εk in their investigations; however, the linear relationship or eective solid conductivity ( k se ) could lead to overestimations o k e, as demonstrated below. An analytical model or the eective conductivity o a packed bed was proposed by Zehner and Schluender [41], yielding k = k ( 1 k ) ( ) ks Bp + 1 B p 1 B p 1 ln 1 B B p k ks B pk / k 2 1 s p k k s ε + (3.6) 1 B p k ks e 1 ε * 2 10 / 9 where B = 1.25[ ( 1 ε )/ ε] p. Because the eective conductivity k e o a sintered particle bed is expected to be higher than that o a packed bed * k e, a linear relationship * k e = Ck e will be assumed in the present investigation, with the coeicient C determined rom experiments. Based on the measured data on equivalent conductivity or a sintered bed [37], it is ound that the selection o C = 1.8 leads to the best agreement with the experimental data. Finally, the eective conductivity o the solid part and luid part in a sintered particle bed can be obtained as ollows, k se = ke k e and k e εk =. (3.7) Next consider the thermal dispersion eects in a porous channel. Thermal dispersion results rom the existence o the solid matrix, which orces the low to undergo a tortuous path around the solid particles. Several researchers [5-8] have investigated the thermal dispersion eects on the heat transer o packed bed. It is ound that the thermal dispersion conductivity is in general proportional to a product o the local velocity and mixing length, namely, k d where = C ρc d ul (3.8) t p C t is an empirical constant, and l is the wall unction or thermal dispersion introduced by Cheng and co-workers [5,14]. To account or the wall eect on the 68

70 reduction o lateral mixing o the luid, the Van Driest type o wall unction l can be written as l = 1 exp = 1- exp - [ y /( wd p )] 0 y H / 2 [ ( H y) /( wd )] H / 2 y H p (3.9) where w is an empirical constant. In the present investigation, the empirical constants C = and w = 1. 5 as suggested in [14] are used. t Finally, the heat transer coeicient between the solid and luid phases, h s, modelled by the ollowing correlations [42], will be used in the present study: h s = = ( k / d v ) Pr Re Re d < ( k / d ) Pr Re Re 100 v d (3.10) where d = 4 ε / a% is the average void diameter, Re and Re d are the Reynold numbers v based on average void diameter d v and partical diameter, d p, respectively. The surace area density o a sintered bed with spherical particles is given by 2 a~ ε = (3.11) ( ε ) d p 69

71 3.4 Results and discussions For water-cooled convection in a packed bronze particle bed [13], Fig. 3.2 shows the predictions on local heat transer coeicient by using dierent eective solid conductivity k se models. It can be seen that whilst the packed bed model (3.6) gives quite good agreement with the experimental data, the commonly used linear model k se = (1 ε) k results in signiicant overestimations. This overestimation by the linear s conductivity model was also observed by Jiang and Ren [13], but they attributed it to the boundary condition used, and attempted to remedy the discrepancy between model and experiment by using dierent boundary conditions while keeping the linear eective solid conductivity model. This approach is nonetheless diicult to justiy, as it will lead to the inconsistency o the eective solid conductivity k se used in the boundary condition and energy equations. 40,000 30,000 packed bronze bed partical diameter = mm porosity = G = kg/s Experimental data [5] h x (W/m 2 K) 20,000 k se = (1-ε) ε)ks sintered model 10,000 packed model (Eq. 4.6) Fig. 3.2 Eect o eective solid conductivity on local heat transer coeicient in a packed particle bed For the water-cooled heat transer in a sintered bronze bed, the eect o k se on the overall Nusselt number is shown in Fig It can be seen that the linear model dramatically overestimates the Nusselt number. The results also show that sintering can increase the heat transer o a packed bed by approximately 30%, and increase its eective solid conductivity k se by about 80%. x / H 70

72 sintered bronze bed partical diameter = 1.5 mm porosity = 0.38 k se = (1- ε) k s Nu sintered model 200 packed model ( Eq. 4.6) 100 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Re Dh Fig. 3.3 Eect o eective solid conductivity k se on overall Nusselt number o sintered bed 0.3 partical diameter = 1.5 mm porosity = 0.38 Red = Tw T qh / k in se x / H Fig. 3.4 Variation o dimensionless wall temperature o a sintered porous channel along streamwise direction or dierent Reynolds numbers 71

73 Sintered copper Porosity = 0.38 d p = 0.5 mm 1.0 mm 1.5 mm Nu mm ,000 4,000 6,000 8,000 10,000 12,000 Fig. 3.5 Eect o particle diameter on overall Nusselt number o a sintered bronze particle bed The predicted dimensionless wall temperature distributions are shown in Fig. 3.4 or a sintered bronze bed. Fig. 3.5 gives the predictions on the eect o particle diameter upon overall heat transer o the sintered bed. It is seen that overall heat transer increases as the particles become smaller. The numerical predictions will be compared with experiment data in a uture study. Re Dh 72

74 4 Conclusions In this report, orced convection in heated porous channels made o metal oam and sintered bed have been investigated. The microstructures o FeCrAlY and copper oams have been measured, and the results show that the microstructures are quite dierent rom the industrial speciications, and the microstructures o FeCrAlY samples are dierent rom those o copper samples with same industrial speciications. The experimental study on orced convection was conducted or eight FeCrAlY and six copper oam samples. The results show that the heat transer o FeCrAlY samples is more sensitive to the relative density than the cell size, whereas or copper samples, the eect o relative density is less signiicant in comparison with the cell size eect. This can be attributed to the dierent thermal resistances on the solid side or FeCrAlY and copper samples. A numerical model was developed to incorporate the measured microscopic parameters, and the selection o suitable boundary conditions was discussed. The eects o microscopic parameters on overall heat transer perormance were numerically investigated and optimizations or designing metal oam heat exchangers were carried out. The optimal oam relative density increases as the Reynolds number is increased. The predictions imply that there is no need to use high thermal conductivity materials i the Reynolds number is small. Finally, numerical modelling on orced convection across sintered particle beds was carried out, and the eect o eective solid conductivity on heat transer was examined. The model was used to study the orced water convection in sintered bronze particle beds The results show that overall heat transer in a sintered bed is about 30 percent higher than that o a packed bed, and that reducing the particle size leads to enhanced heat removal capability o the porous medium. 73

75 Reerences [1] Darcy, H. 1856, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris. [2] Kaviany, M., 1991, Principles o Heat Transer in Porous Media, Springer- Verlag, New York. [3] Vaai, K. and Tien, C. L., 1981, Boundary and inertial eects on low and heat transer in porous media, Int. J. Heat Mass Transer, Vol.24, pp [4] Vaai, K. and Sozen, M., 1990, Analysis o energy and momentum transport or luid low through a porous bed, J. o Heat Transer, Vol. 112, pp [5] Hsu, C. T. and Cheng, P., 1990, Thermal dispersion in a porous medium, Int. J. Heat Mass Transer, Vol. 33, No. 8, pp [6] Amiri, A. and Vaai, K. 1994, Analysis o dispersion eects and non-thermal equilibrium, non-darcian, variable porosity incompressible low through porous media, Int. J. Heat Mass Transer, Vol. 37, No.6, pp [7] Kuznetsov, A. V., 2000, Investigation o the eect o the transverse thermal dispersion on orced convection in porous media, Acta Mechanica, Vol. 145, pp. [8] Hunt, M. L. and Tien, C. L., 1988, Non-Darcian convection in cylindrical packed beds, ASME Journal o Heat Transer, Vol. 110, pp [9] Vaai, K. and Thiyagaraja, R. 1987, Analysis o low and heat transer at the interace region o a porous medium, Int. J. Heat Mass Transer, Vol. 30, No.7, pp [10] Wakao, N., Kaguei, S., and Funazkri, T., 1979, Eect o luid dispersion coeicient on particle-to-luid heat transer coeicients in packed beds, Chem. Eng. Sci., Vol.34, pp [11] Renken, K. J. and Poulikakos, D., 1988, Experiment and analysis o orced convective heat transport in a packed bed o spheres, Int. J. Heat Mass Transer, Vol. 31, No.7, pp [12] Poulikakos, D. and Renken, K., 1987, Forced convection in a channel illed with porous medium, including the eects o low inertia, variable porosity, and Brinkman riction, ASME Journal o Heat Transer, Vol. 109, pp [13] Jiang, P. X. and Ren, Z. P., 2001, Numerical investigation o orced convection heat transer in porous media using a thermal non-equilibrium model, Int. J. o Heat and Fluid Flow, Vol. 22, pp

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77 [26] Calmidi, V. V. and Mahajan, R. L., 2000, Forced convection in high porosity metal oams, J. o Heat Transer, Vol. 122, pp [27] Kim, S. Y., Paek, J. W., and Kang, B. H., 2000, Flow and heat transer correlations or porous in in a plate-in heat exchanger, J. o Heat Transer, Vol. 122, pp [28] Paek, J. W., Kang, B. H., Kim, S. Y. and Hyun, J. M., 2000, Eective thermal conductivity and permeability o aluminium oam materials, Vol. 21, No.2, pp [29] Boomsma, K. and Poulikakos, D., 2001, On the eective thermal conductivity o a three-dimensionally structured luid-saturated metal oam, Int. J. Heat Mass Transer, Vol. 44, pp [30] Ashby, M. F., Evans, A. G., Fleck, N. A., Gibson, L. J., Hutchinson, J. W., Wadley, H. N. G., 2000, Metal Foams, A Design Guide, Butterworth-Heinemann, Boston, ISBN [31] Calmidi, V. V., Transport phenomena in high porosity ibrous metal oams, 1998, Ph. D thesis, University o Colorado. [32] Kline, S.J. and McClintock, F.A., 1953, Describing Uncertainties in Single- Sample Experiments, Mechanical Engineering, pp [33] Lee, D. Y. and Vaai, K., 1999, Analytical characterization and conceptual assessment o solid and luid temperature dierentials in porous media, Int. J. o Heat Mass Transer, Vol. 42, pp [34] Amiri, A., Vaai, K. and Kuzay, T. M., 1995, Eects o boundary conditions on non-darcian heat transer through porous media and experimental comparisons, Numerical Heat Transer, Part A, pp [35] Ngo, N. D. and Tamma, K. K., 2001, Microscale permeability predictions o porous ibrous media, Int. J. Heat Mass Transer, Vol. 44, pp [36] Zukauskas, A. A., 1987, Convective heat transer in cross-low, Handbook o Single-Phase Heat Transer, Wiley, New York. [37] Hwang, G. J. and Chao, C. H., 1994, Heat transer measurement and analysis or sintered porous channels, Journal o Heat Transer, Vol. 116, pp [38] Hwang, G. J., Wu, C. C. and Chao, C. H., 1995, Investigation o non-darcian orced convection in an asymmetrically heated sintered porous channel, Journal o Heat Transer, Vol. 117, pp

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79 Acknowledgments This work is supported partially by Porvair PLC, partly by the U.K. Engineering and Physical Sciences Research Council (EPSRC grant number EJA/U83), and partly by the U.S. Oice o Naval Research (ONR/ONRIFO grant number N ). Alberic Du Chene carried out most o the microstructural studies with SEM and image analysis, and participated in heat transer measurements. 78

80 Appendix A SEM pictures o the oam samples FeCrAlY samples #1-2 : 10 ppi, 5% (let) and 10% (right) FeCrAlY samples #3-4 : 30 ppi, 5% (let) and 10% (right) FeCrAlY samples #5-6 : 60 ppi, 5% (let) and 10% (right) 79

81 FeCrAlY sample #7 : 30 ppi, 7.5% Copper samples #9-10 : 10 ppi, 5% (let) and 10% (right) Copper samples #11-12 : 30 ppi, 5% (let) and 10% (right) 80

82 Copper samples #13-14 : 60 ppi, 5% (let) and 10% (right) 81