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PAGE 1 1 PORE LEVEL ENGINEERING OF MICRO S TRUCTURED FUNCTIONAL MEDIA WITH APPLICATIONS TO THERMAL PROPERTIES By ANUPAM AKOLKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQ UIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 Anupam Akolkar PAGE 3 3 To A ll good men (and women) who came to the aid of the party PAGE 4 4 ACKNOWLEDGMENTS Acknowled gments are due to Dr. Petrasch, my committee chairman, wh ose unwavering faith in me, coupled with his impeccable knowledge subject matter, made this level of research possible within a short time. I would like to thank Drs. Hahn and Klausner for contributing not only to the technical aspects of this study but al so to my general well being and focus. I would like to thank my lab mates Abhishek Singh, Midori Takagi and Benjamin Erickson for aiding me and encouraging me throughout the time spent at the Energy Research and Education Park. Lastly, I would like to than k my parents, my brother and my extended family for the support they have given me over the duration of my education, my friends for constantly reminding me that I am capable of exceeding my earlier achievements and my fellow students who chipped in with s uggestions and solutions whenever approached. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURES ................................ ................................ ................................ ......................... 8 LIST OF ABBREVIATIONS ................................ ................................ ................................ ........ 10 LIST OF SYMBOLS ................................ ................................ ................................ ..................... 11 ABSTRACT ................................ ................................ ................................ ................................ ... 15 CHAPTER 1 POROUS MEDIA FUNDAMENTALS AND APPLICATIONS ................................ ...... 17 1.1 Porous Media ................................ ................................ ................................ .................... 17 1.2 Tomography Data and Pore Level Ge ometry ................................ ................................ ... 18 1.2.1 Tomographic Reconstruction of Porous Medium Datasets ................................ .... 18 1.2.2 Manipulating Tomography Datasets ................................ ................................ ...... 21 1.2.2.1 Removing noise and meso porosity ................................ ............................. 21 1.2.2.2 Erosi ons and dilations to simulate porosity variation ................................ .. 21 1.3 Summary ................................ ................................ ................................ ........................... 25 2 RADIATIVE HEAT TRANSFER ................................ ................................ ........................ 27 2.1 Radiative Transfer in Porous Media ................................ ................................ ................. 27 2.2 Methodology ................................ ................................ ................................ ..................... 28 2.2.1 The Monte Carlo Method ................................ ................................ ....................... 29 2.2.2 Analytical Model ................................ ................................ ................................ .... 32 2.2.3 Convergence Study ................................ ................................ ................................ 34 2.3 Results ................................ ................................ ................................ ............................... 35 2.4 Summary ................................ ................................ ................................ ........................... 41 3 FLUID FLOW ................................ ................................ ................................ ....................... 43 3.1 F luid Flow in Porous Media ................................ ................................ ............................. 43 3.2 Theory ................................ ................................ ................................ ............................... 44 Forchheimer Coefficient ......................... 44 3.2.2 Analytical Flow Models ................................ ................................ ......................... 46 3.2.2.1 Models for permeability ................................ ................................ ............... 46 3.2.2.2 Models for Dupuit Forchheimer coefficient (non Darcy coefficient): ........ 50 3.3 Methodology ................................ ................................ ................................ ..................... 51 3.3.1 Mesh Generation and Flow Simulation ................................ ................................ .. 52 PAGE 6 6 3.3.2 Representative Mesh and Sample Length Scales ................................ ................... 55 3.4 Results ................................ ................................ ................................ ............................... 58 3.5 Summary ................................ ................................ ................................ ........................... 64 4 SCOPE AND OUTLOOK ................................ ................................ ................................ ...... 66 4.1 Accomplishments ................................ ................................ ................................ ............. 66 4.2 Future Scope ................................ ................................ ................................ ..................... 66 4.2.1 Solid Phase Conduction and Combined Conduction, Convection and Radiation Modeling ................................ ................................ ................................ ..... 66 4.2.2 Multi objective Optimization and Transient Process Simulation .......................... 67 4.3 Tailored Media ................................ ................................ ................................ .................. 67 LIST OF REFERENCES ................................ ................................ ................................ ............... 68 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ......... 74 PAGE 7 7 LIST OF TABLES Table page 1 1 Tomography data for model media. ................................ ................................ ................... 20 2 1 Extinction correlation coefficient c, forward scattering fraction, k, and asymmetry factor, g, for the media studied ................................ ................................ .......................... 37 3 1 Porosities and specific surface areas of the samples employed ................................ ......... 52 3 2 Permeability and Dupuit Forchheimer coefficient values for the RPC and CaCO 3 packed bed samples as identified from direct pore level simulation. ................................ 58 3 3 RMS err ors of the results for permeability for different models, across all porosities, for the sample media. ................................ ................................ ................................ ......... 60 3 4 RMS errors of the results for predicted Dupuit Forchheimer coefficient, across al l porosities, for the sample media ................................ ................................ ........................ 62 PAGE 8 8 LIST OF FIGURES Figure page 1 1 Representative Elementary Volume calculations of original samples for a) RPC, 0.913 and b) CaCO 3 ................................ ................................ ......................... 20 1 2 2 CaCO 3 ................................ ................................ ................................ .............. 21 1 3 Morphological operations performed on RPC: (a) segmented sample image obtained from original tomography data, (b) image after iterative openings and closings to remove meso after ................................ ................................ ....................... 23 1 4 Morphological operations performed on tomography data for the packed CaCO3 particle bed: (a) original data, (b) after segmentation and openings to remove noise ...... 24 1 5 Specific surface area vs. porosity for a) RPC and b) CaCO 3 samples. .............................. 25 2 1 Simulated system and boundary conditions ................................ ................................ ....... 31 2 2 Surface reflections in RPC sample of = 0.913 at = 0.65, with specular surfaces. Notice the penetration of radiation in the irradiated direction (+X). Even at high porosity, the transmission through the sample is low. ................................ ....................... 31 2 3 MC convergence study for an RPC sample of poros ity = 0.8351 and a CaCO 3 packed particle bed sample of porosity = 0.4577 for diffuse surfaces, irradiated with diffuse radiation. ................................ ................................ ................................ ........ 35 2 4 Transmittance and reflectance for diffusely irradiate d RPC sample, in case of (a) diffuse solid phase (b) specular solid phase. Here, 1 = 0.727, 2 = 0.863, 3 = 0.964. ..... 36 2 5 Transmittance and reflectance for RPC sample irradiated with collim ated radiation, ................................ ................................ ................................ ............... 38 2 6 Transmittance and reflectance for CaCO3 particle packed bed sampl e irradiated with ................................ ................................ ........................ 38 2 7 Transmittance and reflectance of CaCO 3 parti cle packed bed for sample irradiated with collimated radiation, in case of (a) diffuse solid phase (b) specular solid phase. Here, 1 = 0.159, 2 = 0.604, 3 = 0.904. ................................ ................................ ............ 40 PAGE 9 9 2 8 Absorptance RPC ( a) and CaCO 3 particle packed bed (b), for diffuse radiation and diffuse solid phases, with solid phase = 0.65. The dotted line indicates the local optimal porosity as a function of L / d nom ................................ ................................ ........... 40 3 1 Computational domain and boundary conditions. ................................ ............................. 53 3 2 Sectional views of tetrahedral grids generated in pore space of (a) RPC sample R1 and (b) CaCO 3 particle bed sample C3. It can be s een that the grid is very fine and dense at the solid void interface and element sizes rapidly converge to the preset maximum. ................................ ................................ ................................ .......................... 54 3 3 Dimensionless pressure along the length of a sample for i ncreasing Reynolds numbers: (a) RPC sample R3 and (b) CaCO 3 particle packed bed sample C3. For Re 0.1 1.0, the profiles overlap as the pressure drop is governed by permeability alone, at higher Re, the Dupuit Forchheimer term starts to become distinct, and dimensionless pressure drop increases. ................................ ................................ .............. 54 3 4 Pressure drop convergence vs. grid refinement for (a) RPC sample R1 and (b) CaCO 3 particle packed bed sample C3. ................................ ................................ ............. 56 3 5 Convergence of pressure drop per unit length vs. sample size for (a) RPC sample R1 and (b) CaCO 3 particle packed bed sample C3. ................................ ................................ 57 3 6 DPLS results an d fitting results for normalized pressure drops for (a) RPC samples and (b) CaCO 3 packed bed samples. ................................ ................................ .................. 59 3 7 Permeability vs. porosity for (a) RPC samples and (b) CaCO 3 packed bed samples. ....... 59 3 8 Permeability vs. porosity for (a) RPC and (b) CaCO 3 packed bed for corrected Kozeny constants. Also plotted are the DPLS results. ................................ ....................... 61 3 9 Dupuit Forchheimer coefficient vs. porosity for (a) RPC samples and (b) CaCO 3 samples. ................................ ................................ ................................ .............................. 62 PAGE 10 10 LIST OF ABBREVIATION S CFD Computational Fluid Dynamics DPLS Direct, pore level numerical simulation MCRT Monte Ca rlo Ray Tracing NRMSE Normalized RMS Error REV Representative Elementary Volume RMSE Root Mean Square Error RPC Reticulated porous ceramic RTE Radiative Transfer Equation PAGE 11 11 LIST OF SYMBOLS Alphabetical symbols a Particle size distribution constant ( ) A ove rall absorptance of medium sample ( ) A 0 Specific surface area (m 1) b Forchheimer coefficient correlation ( ) c correlation constant for extinction coefficient ( ) c 0 Inverse dimensionless permeability ( ) c 1 Dimensionless Dupu it Forchheimer coefficient ( ) C Relative convergence of pressure drop with mesh refinement ( ) C V Relative convergence of pressure drop with sample size ( ) d Diameter (hydraulic) (m) d f Fiber diameter (m) d nom nominal diameter (m) d p Particle diameter ( m) d solid Solid phase hydraulic diameter (m) d void Void phase hydraulic diameter (m) f d Friction factor due to fiber deflection ( ) F Dupuit Forchheimer coefficient (m 1) g asymmetry factor ( ) intensity in the forward direction along normal to irradiated plane ( ) intensity in the backward direction along normal to irradiated plane ( ) k forward scattering fraction ( ) k 4 k 5 ) k K Kozeny constant ( ) PAGE 12 12 K permeability (m2) L Sample length (m) L REV edge length of Representative Elementary Volume (REV) (m) m Forchheimer coefficient correlation ( ) n generic constant in relations for Dupuit Forchheimer coefficient correlation ( ) surface normal N e Effective pore number ( ) N rays number of stochastic rays irradiating the volume N x N y N z number of voxels in x y and z directions P Pressure (Pa) P 0 starting point of a ray in fluid phase P i point of incidence of a ray on the s olid void interface q constants of integration r Radial distance in sample volume (m) R overall reflectance of medium sample ( ) Re Reynolds number ( ) s s 2 2 point correlation function ( ) initial direction o f travel of a ray T overall transmittance of medium sample ( ) u D Darcean velocity vector (ms 1) u D Darcean velocity (ms 1) V Sample volume (m3) V eigenvectors PAGE 13 13 z Distance along sample axis (m) Greek Symbols absorptivity ( ) extinction coefficient of t he medium (m 1) eigenvalues Dirac Delta ( ) voxel edge length (m) medium porosity ( ) i angle of incidence (rad) s scattering angle (rad) wavelength of incoming radiation (m) cosine of scattering angle ( ) Dynamic viscosity (Pa s) size pa rameter ( ) hemispherical reflectivity ( ) directional hemispherical reflectivity ( ) standard deviation for 3 D Gaussian filtering (m) s isotropic scattering coefficient ( ) 0 interface gray value (bin)ijk segmented binary dataset for porous mat erial space ijk gray value dataset obtained from tomography data (x) continuous, 3 D gray value function azimuthal angle (rad) scattering phase function ( ) solid angle (sr) PAGE 14 14 Subscripts Br B rinkman drag model Ch Chen fibrous bed model CK Carman Ko zeny model Cond Conduit flow model Dav Davies fibrous bed model DPLS Direct Pore Level Simulation E Ergun correlation for Dupuit Forchheimer coefficient G Geertsma correlation HB Happel Brenner parallel flow model k K ernel Ky Kyan fibrous bed model M Modif ied Ergun (Macdonald et al) correlation s2a 2 point correlation approximation W Ward correlation porosity variation band voxel edge rel relative value 10 8 value calculated for 10 8 rays PAGE 15 15 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Sc ience PORE LEVEL ENGINEERING OF MICRO S TRUCTURED FUNCTIONAL MEDIA WITH APPLICATIONS TO THERMAL PROPERTIES By Anupam Akolkar May 2012 Chair: Jrg Petrasch Cochair: David Hahn Major: Aerospace Engineering Tomography data of porous media is used to determi ne thermal properties of these media through direct, pore level numerical simulation (DPLS) of radiative heat transfer and fluid flow. Two kinds of porous media have been studied: a) a 10 pores per inch (ppi) reticulated porous ceramic (RPC) and b) a packe d bed of CaCO 3 particles. Digital image processing routines are applied to the tomography data to simulate porosity variation in the media and extend the results over a range of porosities in each case. Radiative transfer in the media is studied for an op aque, diffuse or specular solid phase within a non participating void phase and the results were applied to an analytical two flux model to determine the extinction coefficient and forward scattering fraction. Porosity for optimum absorption of incident ra diation at a given sample thickness is determined. For the RPC, the forward scattering fraction varies between 0.38 and 0.57, and the extinction correlation coefficient varies between 9.56 and 7.03. For the packed CaCO 3 particle bed, the forward scattering fraction varies between 0.6 and 0.72, and the extinction coefficient varies between and 2.84 and 2.14. PAGE 16 16 Medium permeability and Dupuit Forchheimer coefficients have been determined for selected samples of the media through fluid flow simulations at Reynold s numbers varying from 0.1 200. The results have then been co mpared to existing flow models to determine their applicability to the media. An adjusted Kozeny constant has been determined for each medium For the RPC, the Kozeny constant is evaluated at 7.7 3 and for the CaCO 3 packed bed, it is found to be 6.10, leading to predictions of the permeability with an NRMSE of 4.16% and 3.37% respectively. PAGE 17 17 CHAPTER 1 POROUS MEDIA FUNDAMENTALS AND APPLICATIONS 1. 1 P orous Media Heat and mass transfer in porous media is of considerable interest in a variety of engineering applications [1,2] These applica tions tend to utilize unique characteristics of transport through the pore space in these media [3] Modern applications of porous media include radiant burners [4,5] reactive media in hydrogen generation from nat urally occurring hydrocarbons [6,7] and solar thermochemical production of hydrogen [8] catalytic converters for flue gases [9,10] and so on. Desi gn and optimization of applications using porous media often relies on continuum models of transport within these media [1] These models use volume averaging [11] and effective transport properties of the media as determined through experimentation and analytical modeling. Experimental determination of properties is restricted to a casewise basis a nd endures errors and limits imposed by experimental conditions overall a loss of generality occurs. It is also very difficult to analyze the effect of a particular topological parameter, say porosity, on a given transport property of the chosen porous m edium, since close control on the properties of a manufactured specimen is very difficult [12] Analytical models require simplification of the pore level geometry through assumptions on the porous medium morphology and although they deliver wider applicability, they lack in accuracy [2] Material from this chapter has been published in: A. Akolkar, J. Petrasch, Tomography based pore level optimiz ation of radiative transfer in porous media, International Journal of Heat and Mass Transfer. 54 (2011) 4775 4783. Material from this chapter has been submitted for publication in: A. Akolkar, J. Petrasch, Tomography Based Characterization and Optimizati on Of Fluid Flow Through Porous Media, Transport in Porous Media submitted March 2012. PAGE 18 18 The application of high resolution tomography to acquire 3D geometry of porous media with a view to using this data in pore level nu merical simulation of transport phenomena in these media has been discussed in detail in [13] This method allows for accurate representation of morphologies of media and also offers the possibility of studying t he effect of variation of morphology, accomplished in this case through image processing routines, on the transport characteristics of a given medium. The present study employs this technique in parametric modeling of radiative heat transfer and fluid flo w through two types of porous media: a) a 10 pores per inch (ppi) reticulated porous ceramic (RPC) a high porosity artificial ceramic structure [3] and b) a packed bed of CaCO 3 particles. The media have been treated as macroporous [3] media with a rigid solid phase distributed in a connected void phase. Solid interconnectivity exists in the RPC whereas the CaCO 3 particle packed bed consists of randoml y shaped, disconnected particles. The pore level geometry of the media is incrementally modified using digital image processing routines of erosion and dilation [14] to simulate porosity variation in the samples. 1 2 Tom ography D ata and P ore L evel G eometry 1. 2.1 Tomographic R econstruction of P orous M edium D atasets Tomography data used in [15,16] has been used in this thesis for the RPC and the CaCO 3 particle packed bed respectively. In each case the data has been obtained as a N z 8 bit graysc ale images of N x N y pixels each, so that a 3D set of gray values ijk of N x N y N z voxels, each of edge length is available. Collection of micron and submicron scale tomography data has been discussed in [13] Defining the porous medium as an intersection of two compact sets F (fluid or void phase) and S (solid phase) in 3 the solid void interface is defined as I = F S An interface gray value 0 is obtained using the method of modes [14] Also defined is the binary gray value data set, (bin)ijk by segmenting the dataset: PAGE 19 19 ( 1 1 ) From this, the porosity, defined as the fraction of the void phase volume to the total medium volume, is calculated as: ( 1 2 ) The gray value dataset is smoothed using a Gaussian kernel with standard deviation k and length k whereupon a smoothed gray level function ( r ) is obtained. The two point correlation function is defined as the probability that two arbitrary points in the porous medium separated by a distance r are both in the void (fluid) phase. This i s calculated using a large number of realizations n as: ( 1 3 ) Here, r i are randomly and uniformly distributed points in the medium space, i are uniformly distributed direction vectors and the distance between the two points where is checked, r is also uniformly distributed [13] The specific surface of the porous medium, defined as the interfac e area A ( I ) per unit medium volume, is found using [17] : ( 1 4 ) A subset of a porous medium that is large enough, so that the medium can be considered a continuum is called a Representative Elementary Volume (REV). The edge length of the smallest cuboidal subset of the medium sample beyond which the porosity of the medium stays PAGE 20 20 within a small band is denoted as L REV This value is calculated by statistical sampling over a large number of randomly originating sample cubes within the medium [17] Relevant tomography data for the original datasets of the model media are given in table 1 1 below. Figure s 1 1 1 2 show sample plots for L REV and the s 2 correlation for the original samples of the model media. (a) (b) Figure 1 1 Representative Elementary Volume calculations of original samples for a) RPC, 0.913 and b) CaCO 3 Table 1 1. Tomography data for model media. Material Sample size in voxels Sample size (mm 3 ) (mm) k (mm) k (mm) L REV,05 (mm) d nom (mm) RPC 768768768 23.0423.0423.04 0.030 1.5 5 8.9 2.54 CaCO 3 packed bed 5125 12512 23.0423.0423.04 0.045 1.5 5 5.4 1.41 PAGE 21 21 (a) (b) Figure 1 2 2 3 1. 2.2 Manipulating Tomography Datasets 1. 2.2.1 Removing noise and meso porosity Origin al sample tomography data contains grayscale noise which can cause significant error in calculation of the interface gray value 0 as well as distort the morphology representation. In case of the RPC, hollow struts present in the solid matrix structure [17] are a significant source of meso scale porosity which affects the outcome of morphological addition/subtraction operations. To remove these sources of error, iterative openings and closings [14] a re performed on the images prior to segmentation. 1. 2.2.2 Erosions and dilations to simulate porosity variation The addition of material to the solid phase and subtraction of material from it is simulated using erosions and dilations on the binary datasets p rior to Gaussian filtering. These are PAGE 22 22 implemented using a a 2 D, isotropic, disc structuring element of incrementally varying size on each of the N z images in the datasets for the media. Since the medium is initially isotropic and erosions and dilations ar e Minkowski set operations, in which the structuring element behaves as a vector positioned at the interfacial pixels, the isotropy of the medium in 3 D is retained after image manipulation. Erosions cause reduction in the white ( (bin),ijk = 1) region of the image, i.e. result in removal of a small layer from the solid phase, thus increasing the porosity, whereas dilations augment it, reducing the porosity. For the RPC sample, each dilation/erosion step corresponds to addition/removal of a layer of materi CaCO 3 3 particle packed bed, since there is no significant connectivity between the particles, subsequent erosions will lead to disc onnected particles typical for a fluidized bed. Figures 1 3 and 1 4 show samples of the image manipulation applied to each of the media. With the image processing routines applied, 20 sample volumes of the RPC, with porosity varying from 0.727 to 0.969 w ere generated and 25 sample volumes of the CaCO 3 particle packed bed with porosities between 0.134 and 0.918. Figure 1 5 shows the variation of specific surface area in the resulting samples. It is clear from the description of the process that the to simu late small thickness steps, either a very small disc size may be used (not recommended). A better way to simulate this change is by first expanding the image (using nearest neighbor interpolation) and then using a sufficiently large disc size so that circu lar symmetry is retained. Then the image is again reduced to the normal size again using the nearest neighbor interpolation. The disc element use is justified as long as the operations are performed on a larger image set of which the chosen dataset is a su bset. PAGE 23 23 (a) (b) (c) (d) Figure 1 3 Morphological operations performed on RPC: (a) segmented sample image obtained from original tomography data, (b) image after iterative openings and closings to remove meso scale porosity, (c) after addition o PAGE 24 24 (a) (b) (c) (d) Figure 1 4 Morphological operations performed on tomography data for the packed CaCO3 particle bed: (a) original data, (b) after segmentation and openings to remove noise (c) PAGE 25 25 (a) (b) Figure 1 5 Specific surface area vs. porosity for a) RPC and b) CaCO 3 samples. 1 3 Summary Porous media have a wide range of engineering ap plications, many of which require accurately determined effective transport properties of the media. Experimental determination of these properties is limited by manufacturability of graded samples of the media and inaccuracies in experimental data collect ion, while analytical modeling delivers wide ranging but inaccurate results due to underlying simplifications. High resolution computer tomography data applied towards accurately capturing morphological information of porous media for use in numerical simu lation of heat transfer phenomena has emerged as a feasible solution This study extends the techniques for tomography based property determination towards parameter based characterization of transport properties. This is made possible by the use of image processing routines of opening closing erosion and dilation through an isotropic 2 D structuring element, PAGE 26 26 to simulate porosity variation. A wide variation in sample porosity and specific surface can be achieved while still retaining the morphological ch aracteristics unique to the medium. PAGE 27 27 CHAPTER 2 RADIATIVE HEAT TRANSFER 2 1 R adiative Transfer in Porous Media Radiative transfer in micro structured porous media occurs in a range of technical applications such as porous radiant burners [4,18] packed beds or suspensions of reactive particles (e.g., in coal gasification and cracking of natural gas [6,7] ), high temperature processes involving porous catalyst carriers [19] volumetr ic radiation receivers for solar thermal and solar thermochemical applications [8,20] etc. Design and optimization of microstructured materials require the accurate modeling of th e effect of pore level geometry on radiative transfer. Two approaches are conceivable: 1) using analytical models linking pore level geometry to radiative continuum properties and subsequently solving the radiative transfer equation (RTE) on the continuum level; and 2) direct pore level simulation of radiative transfer. The former approach involves using theoretically determined radiative properties of porous media [21] or incorp orating reference measurements [22 24] The latter approach, has been implemented via analytical approximations of porous media geometries [7,25 28] or the use of actual pore level geometrical data from computer tomography [15,16,29] paired with MCRT [30 32] Baillis et al. [21] describe techniques for the determination of thermal radiative properties of porous or disperse media. Ni sipeanu et al. [33] provide a comparative study of continuum and direct simulations of radiative transfer. Petrasch et al. [34] have numerically demonstrated the equivalence of continuum and direct modeling f or certain materials. Efforts to identify and improve application specific radiative properties of porous media have been previously conducted both experimentally [35] and through theoretical models [36] In Material from this chapter has been published in: A. Akolkar, J. Petrasch, Tomography based pore level optimization of radiative transfer in porous media, Internation al Journal of Heat and Mass Transfer. 54 (2011) 4775 4783. PAGE 28 28 solar thermochemical processing [8] and volumetric solar thermal receivers [20] porous absorbers need to have high absorptance for maximum energy collection. The present study applies tomography based methods to analyze and optimize the overall absorptance of slabs of the model media described in ch apter 1 It is assumed that the media are non homogeneous, two phase materials, with one phase (void) completely transparent and the other phase (solid) completely opaque and gray. Both phases are assumed to be isothermal and the medium temperature is low, i.e., the contribution of emission from within the medium to radiative intensity is negligible compared to that of the irradiating source [15,27,29] Geometric optics is valid. Only surface exchanges are considered in the MCRT mode l. In this chapter, t he effect of varying pore level geometry, via addition and removal of material, on the absorptance of slabs subjected to direct radiation, is studied. Further, the effect of surface reflectivity and directional distribution of incident radiation (collimated/diffuse) is studied. Diffuse and specular reflections are considered as the limiting cases for the directional surface characteristics. Then, a parametric two flux model is fitted onto the direct simulation data. Values of the extinc tion coefficient, forward scattering fraction and asymmetry factor [32] are calculated for each of the cases above. The model is used to optimize porous slabs for absorption. Optimum porosities increase with sample thickn ess. For given relative sample thicknesses the optimum porosity for RPCs is seen to be much higher than for CaCO3 bed. 2. 2 Methodology The process of generating samples of varying porosity from the original tomography data has been discussed in chapter 1. Samples thus generated are used in MCRT simulations. The surface reflectivity and sample thickness along irradiated direction are treated as parametric values in the simulations. PAGE 29 29 For this study, we consider L REV, 0.05 sufficiently accurate. The pore space is also quantified by the nominal pore diameter, d nom For the RPC, the pores per inch (ppi) define d nom whereas for the CaCO 3 particle packed bed sample d nom is the hydraulic diameter at The relevant values have been given in chapter 1 table 1 1 Thickness variation in samples is simulated in steps of d nom by simply selecting the appropriate numbers of voxels in the irradiated direction when computing MCRT results. Among the sim plifying assumptions used in the MCRT setup is independent scattering [32] The smallest geometrical feature resolved is of the size of one voxel. The size parameter based on the size of one voxel, = / is 62.5 for th e RPC and 93.75 for the CaCO 3 particle packed bed, for a wavelength = 500 nm. In this case, Figure 11 2 in [32] justifies neglecting dependent scattering even for very small porosities The following sections describe the MCRT model used, boundary conditions and the analytical model development. 2. 2. 1 The Monte Carlo M ethod Non energy partitioning [37] Monte Carlo ray tracing is employed to study the interaction of radiation with the model media. A large number (10 6 ) of stochastic ray paths are generated on a parallel plane just outside the sample volume. Incident radiation is either collimated or diffuse Surfaces may either be diffuse or specularly r eflecting. For a specularly reflecting surface, the equation [27] ( 2 1 ) where i is the angle of incidence of the ray on the surface and is the hemispherical absorptivity. PAGE 30 30 The point of incidence of a ray with a starting point P 0 and direction on the solid phase is calculated by ( 2 2 ) where s is the distance travelled from the starting point to the interface and 0 is the interface gray val ue. At the solid surface, whether the ray is reflected or absorbed is determined by comparing a uniform random number with the reflectivity [32] The surface normal is calculated as the normalized gradient of the Gaussian smoothed continuous gray level function [13,15] ( 2 3 ) where P i is the point of incidence of the ray. The history of each ray is recorded until it is either absorbed or exit s the sample. Absorptance, reflectance, and transmittance of the slab are then calculated accordingly. Separate simulations are carried out for all six entry planes to average out the statistical fluctuations. A perfectly reflecting, specular boundary con dition is assumed at the four faces parallel to the direction of incidence [13,30] thus simulating an infinite slab. Figure 2 1 depicts the simulated situation. Figure 2 2 shows the points of reflection on a sample of the RPC of porosity 0.913 with a specular solid phase surface reflectivity 0.65, under the simulation conditions with diffuse irradiation of the sample It can be seen in the figure 2 2 that the incoming radiation fails to penetrate significantly through to the end of the sample. Also, the density of reflected rays decreases along the sample thickne ss. For diffuse radiation, the probability of grazing rays from the specular solid phase being reflected back is higher than for collimated radiation. This causes the peculiar pattern. PAGE 31 31 Figure 2 1. Simulated system and boundary conditions Figure 2 2 Surface reflections in RPC sample of = 0.913 at = 0.65, with specular surfaces. Notice the penetration of radiation in the irradiated direction (+ X ). Even at high porosity, the transmission through the sample is low. PAGE 32 32 2. 2 2 Analytical M odel A two flux model analogous to the modified Schuster Schwarzschild approximation used by Sagan and Pollack [38] is used. For purely absorbing and scattering media with unbalanced forward backward scattering, the following set of linear diffe rential equations is obtained: ( 2 4 ) where I + and I are norma lized intensities in the forward and backward hemispheres of the sample, respectively; is the extinction coefficient and k is a forward scattering fraction For geometric optics, according to [32] the scattering coefficient is: ( 2 5 ) A geometric optics model for the extinction coefficient developed by Hsu and Howell [39] by modeling a porous ceramic as a suspension of monodispersed, independently scattering spherical par ticles, and as used by Hendricks and Howell in [40] is used: ( 2 6 ) where is the hemispherical reflectivity and c is a correlation constant. The general solution of Eq. 2 4 is: ( 2 7 ) The eigenvalues are given by: ( 2 8 ) PAGE 33 33 which yield the rate of change in intensity The eigenvectors V 1,2 are given by: ( 2 9 ) Finally, the constants of integration, q 1 and q 2 are obtained by applying the following boundary conditions: ( 2 10 ) where L is the sample length selected. Transmittance, T and reflectance, R are then obtained as: ( 2 11 ) with being the hemispherical reflectivity at the sample face plane. Finally, the absorptance of the material sample is simply given as: ( 2 12 ) The Schuster Schwarzschild approximation does not assume a specific phase functi on. However, a binary, asymmetric scattering phase function ( )) defined below using the ( 2 13 ) where is the cosine of the scattering angle, s the azimuthal angle, Using this, the asymmetry factor [32] or sc attering anisotropy factor is calculated as, PAGE 34 34 ( 2 14 ) with Then, the unknowns, c and k are obtained via least square fitting of the model output ( Eq. 2 11 ) onto direct numerical simulation results for transmittance and reflectance using the L / d nom ratio, sample porosity, and the hemispherical reflectivity, as parameters. For the media given, we have used 16 values of thicknesses (RPC: 0.57 < L / d nom < 9.1, CaCO 3 particle packed bed: 1.21< L / d nom <19.4) and have varied the hemispherical reflectivities from 0.45 to 0.9. The data contains 20 values of porosity (0.73 < < 0.97) for the RPC and 25 values of porosity (0.13 < < 0.92) for the CaCO 3 particle packed bed sample. 2. 2.3 Convergence Study A convergence study is carried out for the RPC and the CaCO 3 particle packed bed sample, with a diffuse solid phase and of reference porosity = 0.8351 for the RPC and = 0.4577 for the CaCO 3 particle packed bed, with = 0.65 for 10 3 10 4 10 5 10 6 10 7 and 10 8 stochastic rays. The relative change in values of absorptance and reflectance with increased number of irradiating rays is depicted in the figure 2 3 Here, ( 2 15 ) PAGE 35 35 (a) (b) Figure 2 3. MC convergence study for an RPC sample of porosity = 0.835 1 and a CaCO 3 packed particle bed sample of porosity = 0.457 7 for diffuse surfaces, irradiated with diffuse radiation. Since MCRT is a statistical method, the convergence for the MC shows significant fluctuations for some runs. As indicated b y the trend line for N rays 1/2 the convergence follows the Central Limit Theorem. The convergence error at 10 6 rays is well below 0.001 for both cases, and this is below the tolerance on the L REV Therefore, a maximum likelihood estimation of the extincti on correlation coefficient and the forward scattering fraction was not considered and the results were accepted as accurate within the limits of the stochastic geometry fluctuations. 2 3 Results Transmittance and reflectance are plotted for diffuse and col limated incident radiation, for diffuse and specular solid phases; for the RPC and the CaCO 3 particle packed bed as a function of sample thickness. In each case, the maximum sample dimensions are 23.04 mm PAGE 36 36 23.04 mm 23.04 mm and the sample thickness is v aried along the normal to the irradiated plane. Figure 2 4 shows the variation of transmission and reflectance with the L / d nom ratio, in the case of a diffusely irradiated RPC sample volume, with the hemispherical reflectivity of the solid phase, = 0.65, for the case of both diffuse (fig. 2 4 ( a ) ) and specular (fig. 2 4 ( b ) ) solid phases. For samples of low and moderate porosity, the transmittance becomes negligible at lengths approaching 6 d nom However, it remains comparable to th e reflectance for highly porous ( =0.96) samples. The above value of is chosen to represent the average spectral behavior of a variety of sample media. (a) (b) Figure 2 4. Transmittance and reflectance for diffusely irradiated RPC sample, in case of (a) diffuse solid phase (b) specular solid phase. Here, 1 = 0.72 7 2 = 0.863, 3 = 0.96 4 Table 2 1 summarizes the correlation coefficient and the forward scattering fraction for the cases considered in this paper. From figure 2 4 and table 2 1 for the specular solid phase, the PAGE 37 37 value of the correlation coefficient, c is lower At the same time, the forward scattering fraction k takes a larger value, indicating greater forward propagation of radiation. Hendricks and Howell [40] have predicted c for RPCs as 4.4, Petrasch [13] has exp erimentally and tomographically determined c as 5.5 for a specific RPC sample. The difference presented here is attributed to the wider range of porosities considered and the limits of the 1D 2 flux method. Figure 2 5 and table 2 1 give RPC results for col limated radiation. The extinction correlation coefficient is smaller than in case of diffuse irradiation. A diffuse solid phase again tends to cause more radiation to be back than a specular phase. However, overall, the RPC behaves as a better absorber of collimated radiation than of diffuse radiation at typical porosities. Tabl e 2 1. Extinction correlation coefficient, c, forward scattering fraction, k, and asymmetry factor, g, for the media studied Medium Irradiation Solid phase c ( ) k ( ) g ( ) RMS E ( ) RPC Diffuse Diffuse 9.561 0.37 9 0.24 3 5.11 10 4 Specular 9.48 2 0.44 4 0.11 3 5.14 10 4 Collimated Diffuse 7.12 6 0.48 7 0.026 5.77 10 4 Specular 7.032 6 0.574 0.148 6.51 10 4 CaCO 3 particle packed bed Diffuse Diffuse 2.78 3 0.608 0.21 7 1.40 10 4 Specular 2.742 0.643 0.286 1.76 10 4 Collimated Diffuse 2.19 9 0.69 4 0.384 2.28 10 4 Specular 2.143 0.72 1 0.441 2.72 10 4 PAGE 38 38 (a) (b) Fig ure 2 5. Transmittance and reflectance for RPC sample irradiated with collimated radiatio n, in case of (a) diffuse solid phase (b) specular solid phase 7 4 (a) (b) Figure 2 6. Transmittance and reflectance for CaCO3 particle packed bed sample irradiated with 0.15 9 4 PAGE 39 39 The packed bed of CaCO 3 particles exhibits a higher overall transmittance for a given porosity than the RPC. The extinction correlation coefficient is smaller than that of the RPC. Figure 2 6 gives results for the packed CaCO 3 particle b ed subjected to diffuse radiation. The difference in the behaviors of the CaCO 3 particle packed bed and the RPC can be explained by the geometry of the solid phases. The RPC consists of a highly connected, finely structured geometry. This yields a large su rface to volume ratio in the solid phase. This is not the case with the CaCO 3 particle packed bed, which features a disconnected solid phase with a much within t he RPC structure, with more reflections per ray within a given volume as compared to the CaCO 3 particle packed bed. This, in turn, enhances the probability of absorption of radiation within the RPC volume. Table 2 1 and fig ure 2 7 give results for the CaC O 3 particle packed bed irradiated with collimated radiation. As in the case of the RPC, the sample performs better if exposed to collimated radiation. The CaCO 3 particle packed bed has a higher value of k than the RPC, indicating a stronger tendency to sca tter radiation forward. Therefore, larger sample L / d nom ratios are required for optimal performance of the CaCO 3 particle packed bed as an absorber. Maximization of the absorptance of finite thickness media samples is carried out based on the two flux mode l introduced in section 3. Figure 2 8 shows the results of the optimization process for diffuse radiation on a diffuse solid phase, in the case of the RPC and the packed bed of CaCO 3 particles respectively. Absorptance is shown as a function of sample L / d n om ratio and porosities. PAGE 40 40 (a) (b) Figure 2 7. Transmittance and reflectance of CaCO 3 particle packed bed for sample irradiated with collimated radiation, in case of (a) diffuse solid phase (b) specular solid phase. Here, 1 = 0.15 9 2 = 0.604, 3 = 0 .90 4 (a) (b) Fig ure 2 8. Absorptance RPC (a) and CaCO 3 particle packed bed (b), for diffuse radiation and diffuse solid phases, with solid phase = 0.65. The dotted line indicates the local optimal porosity as a function of L / d nom PAGE 41 41 Figure 2 8 indic ates that optimum porosities increase with sample thickness. Also, the CaCO 3 particle packed bed sample is less efficient as an absorber for comparable porosities and sample lengths than the RPC. 2 4 Summary This study extends the direct po re level simulation approach used in [15,16] to optimizing the micro geometry of porous structures for radiative performance. Representative samples of a disconnected solid phase medium (a packed bed of CaCO 3 particles) and connected solid phase medium (an RPC) are studied. Image processing concepts of dilation and erosion are utilized to simulate a stepwise change in medium porosity. A two flux model, based on a modified version of the Schuster Schwarzschild approximation has been applied to characterize the medium radiative behavior. Resu lts obtained from the model were in good agreement with the simulation data, with a maximum RMS error of 0.0006 across all cases examined. The MC model exhibited steady convergence. The extinction coefficients for the RPC are seen to be higher than for th e packed CaCO 3 particle bed. Also, the RPC has a lesser tendency for forward scattering of radiation as compared to the CaCO 3 particle packed bed sample. Thus, for the packed particle bed to perform as a good absorbing medium, a larger sample thickness wil l be required than that for an RPC sample of comparable porosity and solid phase reflectivity. While the results provided in this paper do not cover the entire range of pore level geometries in the model media, they make a strong case for tailoring their p orosities to develop optimum absorber structures. They are also applicable to the continuum modeling and design of these media. Medium porosities can be tailored to optimize for desired radiative behavior, by PAGE 42 42 carefully controlling the viscosity of ceramic slurry [12] in case of porous ceramics, or by managing the bed packing density and fluidization parameters in case of particle beds [36] PAGE 43 43 CHAPTER 3 FLUID FLOW 3 1 F luid Flow in Porous Media The diverse engineering applications of porous media, such as particulate filters [10] structured catalytic media [9] volumetric radiation receivers for solar thermal applications [8] etc., all exploit the heat and mass transfer characteristics peculiar to these media [3] Fluid flow through porous media is of considerable interest in these applications [1] The modeling of such microstructured media usually relies on volume averaged models [11] These models typically require one or several effective transport properties, which in turn depend on the porous microgeometry. Various phenomenological, geometrical and statistical models to predict these transport properties exist [1,2,41] Either conduit flow or drag flow, two opposite approaches of flow analysis, are typically employed in these models, with conduit flow models performing better at lower porosities and drag flow models performing better at higher porosities [2] High resolution computer tomograph y in conjunction with direct pore level numerical simulations (DPLS) [13] provides a rigorous and accurate alternative to these traditional empirical or statistical methods. The accuracy of permeabilities and Du puit Forchheimer coefficients determined by this method is only limited by the size of the tomographic dataset, its resolution and the discretization error of the numerical fluid solver employed. Tomography data has been successfully used in determining me dium permeabilities via random walk simulations [42] in Lattice Boltzmann flow simulations [43] for creating pore scale network models useful in transport simulations [44] for obtaining lineal path functions [45] Material from this chapter has been submitted for publication in: A. Akolkar, J. Petrasch, Tomography Based Characterization and Optimization Of Fluid Flow Through Porous Media, Transport in Poro us Media submitted March 2012. PAGE 44 44 etc. Characterization of overall convection heat transfer properties of RPCs using tomog raphy based methods has been carried out in [13] ; similar characterization of a packed particle bed has been done in [46] With the use of image processing operations on tomography data, topology variation can be affected, thereby extending the results of numerical simulations over a wider range of cho sen topological parameters such as porosity. This was demonstrated for thermal radiative transfer in [47] and is extended to fluid flow in the current work. Samples of incrementally varying porosity are created from tomography datasets of a 10 pores per inch (ppi) RPC and a packed bed of CaCO 3 particles respectively. Stationary 3 D, single phase fluid flow simulations with quasi 1D boundary conditions and Reynolds numbers ranging from 0.1 200 are carried out for each sample. The measured pressure gradient is used to calculate sample permeability and [1,2] Results are compared with traditional flow models. Models best suited to each medium are identified. An adjusted Kozeny constant is then calculated by fitting the Carman Kozeny equation to the results across porosities, for either medium. 3 2 Theory 3. Forchheimer Coefficient [2] for macroscopic flow through an isotropic porous medium, is expressed as, ( 3 1 ) Here, the pressure gradient ( p ) within the fluid phase is related to the mean fluid velocity ( u D ) (Darcean velocity) through the fluid viscosity, and the permeability, K the PAGE 45 45 fundamental property governing low Re transport through porous materials. At higher average velocities, a n additional quadratic term was proposed independently by Dupuit [48] and Forchheimer [49] to account for higher Re effects [2] The one dimensional modified Darcy equation is: ( 3 2 ) where F is the Dupuit Forchheimer coefficient. The non dimensionalized viscous pressure gradient, ( 3 3 ) can hence be expressed as a linear function of the flow Reynolds number by rewriting E q. 3 2, as, ( 3 4 ) Here, d is the fluid flow characte ristic length scale of the porous medium, (e.g. pore diameter), c 0 is the inverse dimensionless permeability, and c 1 is the dimensionless Dupuit Forchheimer coefficient: ( 3 5 ) Hence the permeabilities and the Dupuit Forchheimer coefficients can be determined based on numerical simulations of quasi 1D flow at multiple Reynolds numbers from the zero offset and the slope of t he best fit line to Eq. 3 4 The method has been described in [50] it will be used to examine the applicability of a range of property models to porous structures with a wide range of porosities. PAGE 46 46 3. 2.2 Analytical Flow Models A number of geometrical and phenomenological flow models exist for predicting the permeability and Dupuit Forchheimer coefficient (non Darcy coefficient). Detailed reviews of these models have been provided in [1,2] 3. 2.2.1 Models for permeability Models of permeability are typically based on simplifying assumptions applied to the flow path geomet ry The two main classes of permeability models are conduit flow models (flow inside a channel) and drag flow models (flow around a submerged body) [2] Both types of models rely on a simplified characterization of the medium topology via porosity, and the specific surface area, A 0 Empirical correlations generalizing results for a specific geometry and porosity range are limited in scope [1] Geometric and semi heuristic relationships typically have wider range of applicability but lower overall accuracy. As such, no universally applicable relationship in terms of the morphological parameters, and A 0 for predicting permeability exists [1] In general, drag models a re known to become less effective in permeability prediction with decreasing porosity, whereas conduit models become less effective at high porosities [2] Conduit Flow Hagen Poiseuille relation : Considering an arrangement of parallel straight tubes of diameter d and integrating the one dimensional Navier Stokes equation for the pressure drop across the length o f a tube bundle, a simple relationship for permeability is obtained as [1] [2] : ( 3 6 ) Noting that d = d void we can rewrite this as : PAGE 47 47 ( 3 7 ) Hydraulic Radius The Carman Kozeny Model : The Carma n Kozeny [51] theory models a porous medium as consisting of conduits of varying shape but, on an average, a constant cross sectional area. The permeabil ity, according to this theory, is given by [1] : ( 3 8 ) where k K is known as the Kozeny constant, and is derived from the average path lengths [2] Noting again, that d = d h,void we can rewrite this equation as: ( 3 9 ) The Kozeny constant is approximated as 5 for packed beds of spheres; various models describing its dependence on porosity and specific surface have been developed. Some are discussed below. Happel and Brenner Flow over cylinders : Happel and Brenner [52 ] derived solutions to Navier Stokes equations for flow along the length of assemblies of cylinders. They arrived at the following value of Kozeny constant: ( 3 10 ) PAGE 48 48 Kyan Fibrous Bed Model : Kyan et al. [53] derived an expression for the Kozeny constant based on a correlation between friction factor and Reynolds number obtained considering viscous losses, form drag and the elasticity of the fibers. The Kozeny constant is given as: ( 3 11 ) where N e ( 3 12 ) and f d is the friction factor due to fiber deflection, which is equal to zero in the case of rigid material, which is the case here. Packed bed of particles Rumpf Gupte : Rumpf and Gupte proposed an empirical correlation for a packed bed of uniformly rand om particles [54] over a wide range of porosities. The permeability in their case is given by: ( 3 13 ) where a is a constant, approximated as 1.05 for a wider particle size distribution and 1 for a narrower one. The value of particle diameter, d p is a volume averaged value of particle diameters across th e medium and for nonspherical particles, it is calculated as [1] : ( 3 14 ) PAGE 49 49 Empirical correlations for flow through fibrous beds : Two commonly cited correlations are the following Davies equation : Davies proposed an empirical correlation for permeability in flow through fibrou s beds [2] from a friction factor formulation: ( 3 15 ) Chen correlation : Another expression for flow through fibrous beds was derived by Chen [55] : ( 3 16 ) with k 4 = 6.1, k 5 = 0.64. Brinkman drag model : Brinkman [56] developed an expression for permeability by modeling viscous forces on sp herical particles embedded in a porous mass. The permeability is given by: ( 3 17 ) With d p given by Eq. 3 14 we can rewrite this as: ( 3 18 ) It has been pointed out [2] that this equation yields K = 0 for = 1/3. However, this model provides good permeability estimates for higher porosities. PAGE 50 50 2 point correlation bound (Berryman Milton) : From a rigorous formulation of the variational principle for viscous flow through porous media, Berryman and Milton [57] proposed a permeability bound given by: ( 3 19 ) whe re s 2 ( r) is the 2 point correlation function of the medium [17] The two point correlation, if approximated by two straight lines, given below: ( 3 20 ) yields, upon integration, ( 3 21 ) 3. 2.2.2 Models for Dupuit Forchheimer coefficient (non Darcy coefficient): Relations predicting the Dupuit Forchheimer coefficient (non Darcy coefficient) are often proportio nal to the inverse of the square root of the medium permeability. Some models consider an explicit dependence of the coefficient on porosity while others have suggested an implicit dependence through the inclusion of higher indices of inverse permeability. The commonly used models are given below. Ergun equation and the Modified equation : Ergun [58] calculated the pressure drop and friction factor for arrangements of granu lar columns From his PAGE 51 51 friction factor equations, the following correlation for the Forchheimer coefficient can be arrived at: ( 3 22 ) Macdonald et al. [59] tested the Ergun equation for friction factor of a bed of particles f p [60] to arrive at the following relations for Forchheimer coefficient, for a packed bed of smooth particles: ( 3 23 ) correlation : Ward [61] suggested an implicit dependence on porosity through dimensional analysis and proposed the following correlation : ( 3 24 ) relation : Geertsma [62] studied various consolidated and unconsolidated media and arrived at a relation combining the effe ct of porosity and permeability on the Dupuit Forchheimer coefficient: ( 3 25 ) 3. 3 Methodology Five samples are selected for the RPC and CaCO 3 packed bed respectively. The porosity and specific surface areas for the samples are given in Table 3 1 The samples in both cases cover a wide range of porosities, allowing for both conduit flow and drag flow effects to come into consideration. PAGE 52 52 Table 3 1 Porosities and specific surface areas of the samples employed Material Sample number Porosity ( ) Specific surface area A 0 (10 3 m 1 ) RPC R1 0.727 0.929 R2 0.792 0.858 R3 0.863 0.738 R4 0.913 0.636 R5 0.954 0.459 CaCO 3 particle packed bed C1 0.329 1.38 C2 0.457 1.45 C3 0.604 1.25 C4 0.712 1.03 C5 0.824 0.763 3. 3. 1 Mesh Generation and Flow Simulation Unstructured tetrahedral, body fitted grids are generated for each of the samples using an in house grid generator making use of the pore space indicator function [13] The mesh generator initially tessellates the entire sample volume using identical tetr ahedral elements. Then, by placing additional nodes at the solid void interfacial surfaces, it separates the subvolume to be discretized in this case, the pore space from the remaining volume. The mesh is then refined in the regions close to the solid void interface, using algorithms described in [63] Finally, by projecting nodes from interface crossing edges onto the interface i.e. vertex rounding and by cutting interface crossing elements into smaller elements, the desired degree of refinement is achieved. This mesh generator has been previously successfully used in [50] and [64] A paper describing the algorithms used in the mesh generator in detail is currently under review [65] In the present work, surface adaptive grid refinement is used, i.e., the size of a tetrahedral element, defined as the height of the tetrahedron, is directly proportional to its distance fr om the PAGE 53 53 solid void interface. This kind of meshing allows high resolutions at the interface, where large pressure and velocity gradients occur. Away from the interface, the cell size rapidly increases to a preset maximum size. Stationary, incompressible co ntinuity and momentum equations are solved in the pore space for Reynolds numbers ranging from 0.1 to 200, using ANSYS CFX. Since the flow is laminar, no turbulence model is necessary. The domain for the solution consists of the porous medium sample embedd ed in a square channel. A uniform velocity boundary condition is imposed at the inlet and a zero relative pressure condition is imposed at the outlet. The channel walls are assigned a free slip boundary condition, to simulate isotropy of the porous medium. The solid surfaces within the porous medium feature a no slip boundary condition. Figure 3 1 shows a schematic of the computational domain. Figure 3 2 shows sectional views of domain mesh samples for the RPC and the CaCO 3 particle packed bed. Figure 3 1 Computational domain and boundary conditions. PAGE 54 54 (a) (b) Figure 3 2. Sectional views of tetrahedral grids generated in pore space of (a) RPC sample R1 and (b) CaCO 3 particle bed sample C3. It can be seen that the grid is very fine and dense at the so lid void interface and element sizes rapidly converge to the preset maximum. (a) (b) Figure 3 3. Dimensionless pressure along the length of a sample for increasing Reynolds numbers: (a) RPC sample R3 and (b) CaCO 3 particle packed bed sample C3. PAGE 55 55 I n each case, the cross sectional mean pressure profile in the fluid phase is obtained and subsequently used to determine the mean pressure gradient across the sample. From the dimensionless pressure gradients, the inverse dimensionless permeability and the dimensionless Dupuit Forchheimer coefficient are calculated. Sample pressure profiles through the media are shown in Figure 3 3. For Re 0.1 1.0, the profiles overlap as the pressure drop is governed by permeability alone, at higher Re, the Dupuit Forchhei mer term starts to become distinct, and dimensionless pressure drop increases. 3. 3. 2 Representative Mesh and Sample Length Scales Grid refinement and sample size are determined through convergence studies. For both the RPC and the CaCO 3 packed bed samples, grid refinement was studied on a sample of size 128 128 128 voxels. For the RPC, grids with representative mesh length scales ranging from 3 packed bed, the representative lengt pressure drop across the sample for each level is calculated for flow at Re=200 and the relative convergence of pressure drop, defined as: ( 3 26 ) p ref is the pressure drop across the sample with the finest grid. The relative convergence is plotted against the ratio of sample nominal diameter to mesh representative length scale, in Figure 3 4. PAGE 56 56 (a) (b) Figure 3 4. Pressure drop convergence vs. grid refinement for (a) RPC sample R1 and (b) CaCO 3 particle packed bed sample C3. Bas corresponding to 0.018 d nom was selected for the RPC. The mesh length scale for CaCO 3 also was d nom In order to determine a representative but not ov erly large sample cube sample convergence studies were carried out. RPC sample cubes ranging from 128 128 128 voxels to 640 640 640 voxels from sample R1, were studied in Re=200 flow simulations. For the CaCO 3 bed, samples ranging from 64 64 64 voxels to 384 384 384 voxels were studied. Convergence of the pressure drop per unit length of the sample is examined: ( 3 27 ) PAGE 57 57 p ref / L ref ) is the pressure drop per unit length across the largest sample in each case. The convergence plots are shown in figure 3 5. (a) (b) Figure 3 5. Convergence of pres sure drop per unit length vs. sample size for (a) RPC sample R1 and (b) CaCO 3 particle packed bed sample C3. In the determination of permeability and Dupuit Forchheimer coefficient, for the RPC samples, a cube of edge length 640 voxels, corresponding to L / L REV = 2.2 was selected. For the CaCO 3 particle bed samples, the cube used was of edge length 384 voxels, i.e. L / L REV = 3.2. This resulted in meshes with 4.10 10 7 tetrahedrals for the sample with the lowest specific surface, A 0 and 7.07 10 7 tetrahedr als for the sample with the highest specific surface, in case of the RPC, and meshes with number of tetrahedrals equal to 4.47 10 7 and 6.80 10 7 respectively for the CaCO 3 samples. PAGE 58 58 3 4 Results The DPLS results for permeability and Dupuit Forchheimer c oefficient for the RPC and CaCO 3 are given in table 3 2 Normalized RMS errors (NRMSE) for the straight line fit of pressure drop with Eq. 3 4 have also been tabulated in table 3 2 Figure 3 6 shows the curves for the least squares fit values of c 0 and c 1 pg for the RPC and CaCO 3 samples. The results for the permeability, K calculated from the models listed in section 2.2, have been plotted in figure 3 7 below. In general, Dupuit Forchheimer type equat ions fit the CFD results well. Table 3 2 Permeability and Dupuit Forchheimer coefficient values for the RPC and CaCO 3 packed bed samples as identified from direct pore level simulation. Material Sample K DPLS (10 7 m 2 ) c 0 ( ) F DPLS (m 1 ) c 1 ( ) NRMSE (%) RPC R1 0.59 110 852 2.16 0.73 R2 0.89 72.7 548 1.39 0.70 R3 1.39 46.1 342 0.87 0.69 R4 2.05 31.5 227 0.58 0.64 R5 3.74 17.3 102 0.26 1.15 CaCO 3 particle packed bed C1 0.03 665 1.28 10 4 18.1 2.47 C2 0.09 224 3.29 10 3 4.65 2.53 C3 0.27 73.5 867 1.22 1.91 C4 0.60 33 422 0.59 0.85 C5 1.45 13.7 203 0.29 0.54 The values for CaCO 3 are similar for comparable porosity to those for RPC. As such, at high porosity, both the RPC and CaCO 3 structures interact with flow in the form of submerged bo dies rather than conduit structures. Then, the pressure drop relates to the drag force exerted on the flow by these structures. PAGE 59 59 (a) (b) Figure 3 6. DPLS results and fitting results for normalized pressure drops for (a) RPC samples and (b) CaCO 3 packe d bed samples. (a) (b) Figure 3 7. Permeability vs. porosity for (a) RPC samples and (b) CaCO 3 packed bed samples. PAGE 60 60 The Happel Brenner [52] relation best predicts the trend of the permeability with porosity for the RPC, while the Brinkman [56] model applies well to the packed bed of CaCO 3 particles, especially at high porosities. With a porosity depe ndent relationship for the Kozeny constant, the Kyan [53] fibrous bed model also effectively predicts the permeability for the RPC. The empirical Chen [55] and Davies [2] fibrous bed models are reasonably accurate for the RPC whereas the Rumpf Gupte [54] model performs well within the porosity range considered for the CaCO 3 packed bed. This agrees well with the data given in [1,2] The s 2 permeability bound approximation [57] results describe very high permea bility bounding values. NRMSE of K from model predictions with respect to K DPLS are shown in table 3 3 for the sample media. Table 3 3 RMS errors of the results for permeability for different models, across all porosities, for the sample media. Model Normalized RMS error (%) RPC CaCO 3 Conduit Flow 273 121 Carman Kozeny 70.3 15.1 Happel Brenner, Parallel Flow 11.8 10.7 Kyan, Fibrous Beds 39.4 15.9 Rumpf Gupte, Packed Beds 50.9 11.7 Davies, Fibrous Beds 17.9 Chen, Fibrous Beds 27.5 Brinkman Drag Model 19.3 8.26 2 point Correlation Approximation 40.4 65.9 PAGE 61 61 The Kozeny constant for each of the media is corrected using a least squares fit to the permeability data. The corrected Kozeny constant for the RPC is found to be equal to 7.73, with a N RMS error of 4.16 % and that of the CaCO 3 sample is evaluated at 6.10, with a NRMS error of 3.37 %. The plots for the Carman Kozeny relationship with the corrected constants are given in figure 3 8. (c) (d) Figure 3 8. Permeability vs. porosity for (a ) RPC and (b) CaCO 3 packed bed for corrected Kozeny constants. Also plotted are the DPLS results. The Dupuit Forchheimer coefficient is plotted for both media using the correlations described in equations 3 22 to 3 25 in figure 3 9 below. Satisfactory fi ts for this correlation are difficult to find in case of many of the analytical models, especially for the RPC. This is in part PAGE 62 62 media, such as sheet rock, as ea rly developments in this direction were entirely due to water drainage and petroleum extraction research. (c) (d) Figure 3 9 Dupuit Forchheimer coefficient vs. porosity for (a) RPC samples and (b) CaCO 3 samples. The NRMS errors of the results fo r the Dupuit Forchheimer coefficient are given in table 3 4 for both the media. Table 3 4 RMS errors of the results for predicted Dupuit Forchheimer coefficient, across all porosities, for the sample media Model Normalized RMS error (%) RPC CaCO 3 Ergu n 17.4 7.93 Modified Ergun (Macdonald) 13.5 5.94 Ward 153 17.7 Geertsma 57.4 101 PAGE 63 63 It is seen that in case of, both, the permeability and the Dupuit Forchheimer coefficient, hydraulic radius models predict the trend of variation reasonably well, for both sample media. For the RPC, the permeability is well predicted by the implicit inclusion of a drag model through [53] The behavior of the CaCO 3 packed bed is also predicted with high accuracy by the Rumpf Gupte [54] model, as described in [2] The correlations for the Dupuit Forchheimer coefficient suffer from lack of accuracy in both cases, since the constants used in the ge neric equation of the form, ( 3 28 ) are empirical [66] and need adjustment per case. Cooke [67] suggested using a correlation independent of the porosity, of the form: ( 3 29 ) From least squares fitting of the results for the Dupuit Forchheimer coefficient to equation (29), we get values of b = 1.41 10 5 a nd m = 1.08 for the RPC sample, with NR MSE of fitted coefficients 0.96 % and b = 4.06 10 7 and m = 1.23 for the CaCO 3 packed bed, with NRMSE of fi t equal to 0.53 %. Cooke [67] suggested using m = 1.24 for particle packs with particle sizes ranging fr om 1.68 mm 2.38 mm (mesh sizes 8 12). Conversely, Thauvin and Mohanty [68] used a network model to suggest that in a porosity independent relation for the Dupuit Forchheimer coefficient, m = 1. Using m = 1 in equation (29), we get b = 4.92 10 5 m for the RPC with an NRMSE of 2.22% and b = 3.72 10 5 m for the CaCO 3 packed bed with an NRMSE of 3.92%, while Thauvin and Mohanty predict a value of 2.47 10 5 Nevertheless, the PAGE 64 64 inverse of the permeability is seen to reasonably predict the behavior of the Dupuit Forchheimer coefficient with porosity variatio n. 3 5 Summary Direct numerical simulation of incompressible, stationary flow in reticulate porous ceramics and packed beds of CaCO 3 has been carried out based on tomographic data that has been modified using image processing techniques to obtain a wide range of porosities. Unstructured finite volume grids have been generated for porous media samples based on the modified tomography data. Flow in the Reynolds number range from 0.1 to 200 has been simulated. From the Dupuit Forchheimer equat ion (i.e., the modified Darcy equation) and using the numerical results for pressure gradient, the medium permeability and Dupuit Forchheimer coefficient have been determined. Various empirical and geometrical models for permeability and Dupuit Forchheimer coefficient have been compared to the numerical results. It has been shown that the Happel Brenner (parallel flow over cylinders) model predicts the permeability of the RPC samples within an NRMSE of 11.8 % and the Brinkman model predicts the permeabilit y of the packed bed within 8.26%. The Brinkman model, however, is not applicable at low porosities, approaching a permeability of zero at = 1/3. The Kyan model with a porosity dependent Kozeny constant and the Rumpf Gupte model for packed beds predict th e permeability of both materials satisfyingly over a wide range of porosities. Adjustment of the Kozeny constant extends the applicable range of porosity. The adjusted Kozeny constant of the RPCs has been determined as 7.73 and the Kozeny constant of packe d beds of CaCO 3 has been determined as 6.10. In case of both, the RPC and the CaCO 3 packed bed, the Modified Ergun (Macdonald et al.) model shows the leas t normalized RMS error, of 13.5% and 5.94 %, respectively, in predicting the Dupuit Forchheimer coeffic ient. The Geertsma correlation PAGE 65 65 produces valid results at high porosities for the CaCO 3 particle packed bed sample, but not for the RPC sample. The Ward correlation fails to predict the Dupuit Forchheimer coefficient with adequate accuracy in case of both m aterials. In conclusion, all permeability and Dupuit Forchheimer models investigated show at least some degree of deviation from direct pore level numerical simulation. The techniques described in this paper can be used to accurately characterize the flow parameters of a porous medium without the need for expensive experimentation. The reliance on models, with their inherent underlying assumptions, can thus be reduced. At the same time, for materials with similar morphological characteristics, this techniq ue can be used to develop models for fluid flow specific to that class of materials. PAGE 66 66 CHAPTER 4 SCOPE AND OUTLOOK 4 1 Accomplishments The unique transport properties of macroporous media, arising from their complex geometrical characteristics, offer scope for novel engineering applications of these media [3] A model based appr oach typically used for continuum analysis a nd design of such applications relies heavily on the accurate determination of effective transport properties. Tomography based numerical simulation of transport phenomena [13] has been shown to be a plausible mea ns of approaching this problem. Manipulation of tomography data through image processing techniques adds the capability of parameter based analysis of thermal properties [47] This has been shown in case of radiative heat transfer (chapter 2) and Darc ean fluid flow (chapter 3) through such media. 4 2 Future Scope 4. 2.1 Solid Phase Conduction and C ombined C onduction, C onvection and R adiation M odeling The analysis of multi scale conduction heat transfer in fluid saturated RPCs using tomography based methods has been demonstrated in [69] Using similar methods and by applying the manipulations to tomography data as in earlier chapters porosity dependent relationships for the effective thermal conductivity of the porous media can be obtained. The solution can fu rther be extended to cover combined conductive, convective and radiative heat transfer within the media. This will be useful in a ddressing, for example, reactive flows occurring in processes such as catalytic conversion, hydrogen generation etc. more rigorously. PAGE 67 67 4. 2.2 Multi objective O ptimization and T ransient P rocess S imulation Multi objective optimization to determine the ideal pore structure and porosity of a medium sample to be used in a process involving all three modes of heat transfer, e.g. solar thermochemical hydrogen production [8] can be a further extension of the capabilities of to mography data based modeling. While an engineered porous structure predicted through such optimization may not be readily manufacturable, naturally occurring or synthetic porous media with morphology closely approaching the optimized medium may be selected to achieve higher process efficiency. Transient modeling of heat transfer within the porous medium space, e.g. [7] is another key area of interest in thermochemical processes. In most cases, analytical models for porous media, su ch as packed beds of spheres, are employed to cover their geometrical character. While this may be computationally expensive, erosions and dilations dependent on reaction kinetics can be included in a combined simulation of heat transfer processes to close ly mimic the actual system. The results can be used to guide process parameter settings 4 3 Tailored M edia Within a limit, porous ceramics can be manufactured at a preset porosity by adjusting ceramic slurry viscosity [12] Similar manipulation can be effected, by controlling particle sizes and/or fluidizatio n parameters, for packed beds [36] Graded porosity media can also be obtained by combined adjustments to the manufacturing process. In order to achieve the optimum medium configuration as suggested by tomography based numerical simulations, the manufacturing process can be adapted t o include some of such tailoring. 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Schrader, P. Wyss, A. Steinfeld, Tomography Based Determination of the Effective Thermal Conductivity of Fluid Saturated Reticulate Porous Ceramics, Journal of Heat Transfer. 130 (2008) 03260 2. PAGE 74 74 BIOGRAPHICAL SKETCH exploration. Although an engineer by choice, he still actively enjoys a number of artistic pursuits while furthering his engineering research. Anu pam began his professional career in 2006 as a Maintenance Engineer in the Heavy Engineering Workshops of Larsen and Toubro Limited, an infrastructure multinational based in India. His work ethic and detail orientation resulted in a quick promotion to the Design and Engineering Centre of the Weapon Systems Division of the same company. Till mid 2008, he performed the dual du ties of a design engineer and a manufacturing liason on a project for a mid size Destroyer Class vessel of the Indian Navy. Returni ng to academia, Anupam enrolled in the g raduate program in Mechanical Engineering at the University of Florida. Specializing in s olid m echanics and d esign, he conducted research in finite element based structural design and fibre reinforced compos ites over a two degree, he was accepted as a Ph.D. student at the University, doing research in the Renewable Energy Laboratory, in the Thermal Sciences division of the Mechanic al and Ae rospace Engineering Department. His research in radiative heat transfer in porous media has since been published in the International Journal of Heat and Mass Transfer. At present, he is engaged in extending his research in porous media in the area of mass tran sport. Anupam completed his Bachelor of Mechanical Engineering, First Class, from the University of Pune, India, in 2006. His f inal y ear s pecialization topic was Robotics and Synthesis of Mechanisms. In keeping with his theme of exploration, he complet ed a f inal y ear p roject in diesel injection valve design at the Automotive Research Association of India. This project was program Anupam dabbled in a numbe r of co curricular and extra curricular activities. He is a PAGE 75 75 lifetime member of the Boat Club Quiz Club of Pune, an organization active in organizing and promoting trivia quizzing events in Pune. As a member of the Social Activities Committee of his college Anupam organized remedial tutoring for middle school and high school children from underprivileged backgrounds in Pune. He was also the Organizing Secretary of the Literary Society of his college. Anupam is an avid reader, enjoying both classic and p opular fiction. He has earned certification in oil and water colour painting. He is also a self professed wine and draft beverage connoisieur. 