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COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES By REZA MOHAMMADI ZIAZI Bachelor of Science in Aerospace Engineering Tehran Polytechnic University Tehran, Iran 2009 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2012ii COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES Thesis Approved: Dr. Frank W. Chambers Thesis Advisor Dr. Andrew S. Arena Dr. Bruce J. Ackerson iii ACKNOWLEDGEMENTS It is like a dream come true for me pursuing my Master’s degree in Mechanical engineering and doing critical research here in a fast developing school like Oklahoma State University. However, this dream was made true only because of continuous support, motivation, and kindness I received from my helpful advisor Dr. Frank W. Chambers. I have had the opportunity and pleasure of working with and learning from an advisor who not only cares to share his precious knowledge, but also motivates the students with his endless new ideas. I express my gratitude toward Dr. Chambers who I have been most comfortable consulting and asking questions. His helpful suggestions and comments will remain with me for my life. So thank you, Dr. Chambers, for everything. I would like to thank Dr. Andrew S. Arena Jr. and Dr. Bruce J. Ackerson for being in my thesis committee, guiding me through preparation of this thesis, and during my master’s. Also, I appreciate their time and patience for reviewing the thesis document and attending my examination. I also thank my lab partner Shyam, and thank all my friends specially Mehran, Amir, Nabil, and Hossein who supported me during the preparation of this thesis. I Studying away from your family sometimes can get really hard, but one thing made it easier for me; my parents and my sister. Thank you for your love, support, and kindness. Acknowledgements reflect the views of the author and are not endorsed by committee members or Oklahoma State University.iv Name: REZA MOHAMMADI ZIAZI Date of Degree: DECEMBER, 2012 Title of Study: COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES Major Field: MECHANICAL ENGINEERING Abstract: Near infrared sensors project light into particleladen flows and use light returned via diffuse reflectance and scattering phenomena to measure solids concentration and chemical composition. An important measurement parameter is the depth of penetration returning the light used in the measurements. In this work several optical modeling approaches were evaluated for predicting the depth of penetration of nearinfrared (NIR) light in slurry flow in a pipe. The goal was to find a model suitable for predicting the depth of penetration for slurry flows with arbitrary solid concentration profiles. The models which were considered fall in two categories; continuum and discontinuum theories. Continuum models consider the sample as one integral layer while discontinuum models divide the sample into layers one particle thick. Hence, discontinuum models are well suited for slurries with varying concentrations which can be adjusted layer by layer. The continuum theories used were Shuster, Kubelka–Munk, radiation transfer, three flux, and diffusion. The discontinuum theories used were plane parallel layer, assembly of sheets, and representative layer theory. Discontinuum models were applied to cases which included uniform, hypothetical, and realistic concentration profiles predicted using Computational Fluid Dynamics (CFD). For the CFD predictions, the slurry was xylene liquid and ADP solid flowing in a 50.8 mm diameter horizontal pipe at mean velocities between approximately 1.8 and 4.0 m/s, corresponding to Reynolds numbers between 1.2x105 and 2.8x105. Monodisperse spherical slurry particles with diameters ranging from 38 to 150 microns were considered with solid volume fractions ranging from 5% to 35%. Numerical versions of the optical models were created for the theories using MATLAB R2012a. Each model simulated the passage of NIR radiation through the sample based on light fractions. Light fractions were computed for each layer of the discontinuous models, and the depth of penetration was evaluated. Validation was performed with results in the literature for optical experiments for similar cases. Comparisons of the results suggest that the discontinuum theories are more accurate than continuum theories for these applications, with representative layer theory used with Stokes formula providing the best results. It is concluded that the models are suitable for predicting the depth of penetration for NIR measurements. v TABLE OF CONTENTS Chapter Page I. INTRODUCTION AND BACKGROUND ............................................................1 1.1 Introduction ........................................................................................................1 1.2 Motivation ..........................................................................................................2 1.3 Background ........................................................................................................4 1.3.1 NearInfrared Technology ........................................................................5 1.3.2 Slurries ......................................................................................................7 1.4 Statement of the Problem ...................................................................................9 1.5 Goals ................................................................................................................10 II. PAST CONTRIBUTIONS ...................................................................................13 2.1 Introduction ......................................................................................................13 2.2 History of past contributions ............................................................................14 2.3 Fundamental Definitions ..................................................................................18 2.3.1 Geometrical Optics .................................................................................19 2.3.2 Wavefront and Ray .................................................................................19 2.3.3 Radiant Energy ........................................................................................21 2.3.4 Solid Angle .............................................................................................21 2.3.5 Radiant flux .............................................................................................22 2.3.6 Radiance ..................................................................................................23 2.3.7 Irradiance ................................................................................................23 2.3.8 Radiant Intensity .....................................................................................24 2.3.9 Refractive Index ......................................................................................24 2.3.10 Diffraction .............................................................................................26 2.3.11 Refraction ..............................................................................................26 2.3.12 Reflectance (reflection) and reflectivity ...............................................29 2.3.13 Scattering ..............................................................................................33 2.3.14 Reflection (remission) Coefficient ........................................................35 2.3.15 Absorptance (absorption) ......................................................................36 2.3.16 Absorption Coefficient ..........................................................................37 2.3.17 Absorption Coefficient ..........................................................................40 2.4 Continiuum Theories of Diffuse Reflection ....................................................42vi Chapter Page 2.4.1 Introduction .............................................................................................42 2.4.2 Lambert Cosine Law ...............................................................................43 2.4.3 Mie Scattering .........................................................................................45 2.4.4 Radiation Transfer Treatments ...............................................................47 2.4.5 Schuster’s Theory ...................................................................................49 2.4.6 KubelkaMunk Theory ............................................................................49 2.4.7 Diffusion Theory .....................................................................................52 2.5 Discontinuum Theories of Diffuse Reflection .................................................52 2.5.1 Representative Layer Theory ..................................................................52 III. OPTICAL MODELING .....................................................................................57 3.1 Overview and Organization .............................................................................57 3.2 Development of Continuum Optical Models ...................................................60 3.2.1 Schuster’s Model ....................................................................................61 3.2.2 KubelkaMunk Model .............................................................................64 3.2.3 Radiation Transfer Approximations .......................................................72 3.2.4 Three Flux Model. ..................................................................................74 3.2.4 Diffusion Model. .....................................................................................75 3.3 Development of Discontinuum Optical Models ..............................................76 3.3.1 Model of Plane Parallel Layers ...............................................................76 3.3.2 Model for Assembly of Sheets ................................................................78 3.3.3 Representative Layer Model ...................................................................82 3.4 Code Implementation .......................................................................................93 IV. RESULTS AND DISCUSSION ..........................................................................98 4.1 Brief Recapitulation .........................................................................................98 4.2 Results of Continuum Optical Models ...........................................................101 4.2.1 Schuster’s Model ..................................................................................102 4.2.2 KubelkaMunk Model ...........................................................................107 4.2.3 Radiation Transfer Approximation .......................................................114 4.2.4 Three Flux Model .................................................................................118 4.2.5 Diffusion Model ....................................................................................122 4.3 Results of Discontinuum Optical Models ......................................................123 4.3.1 Model of Plane Parallel Layers .............................................................124 4.3.2 Model for Assembly of Sheets ..............................................................131 4.3.3 Representative Layer Model .................................................................134 4.4 Comparison of Continuum and Discontinuum Models .................................151 4.5 Depth of Penetration ......................................................................................153 vii Chapter Page 4.6 Summary ........................................................................................................154 V. CONCLUSIONS AND RECOMMENDATIONS ...........................................156 5.1 Conclusion of continuum theories .................................................................157 5.2 Conclusions of discontinuum theories ...........................................................159 5.3 Recommendations ..........................................................................................160 R. REFERENCES ...................................................................................................161 A. APPENDICES ....................................................................................................165 A.1 MATLAB Script for continuum models and theory for assembly of sheets 165 A.2 MATLAB scripts for the Representative Layer Theory ...............................172 A.2.1 Main code of representative layer model .............................................172 A.2.2 MATLAB input script for 38 micron ...................................................180 A.2.3 MATLAB input script for 75 micron ...................................................182 A.2.4 MATLAB input script for 150 micron .................................................186 viii LIST OF TABLES Table Page 1.1 Description of different fields and applications of nearinfrared radiation has already been used in the previous works ...................................................... 3 1.2 Examples of single and multicomponent, multiphase fluids ............................. 8 2.1 Overview of the development of modeling techniques for diffuse reflection. .......................................................................................................... 15 2.2 Variables used in the development of representative layer theory ................... 54 3.1 Variables used in the development of Kubelka and Munk’s simplified solution to the radiation transfer equation. (Neither or are exactly the same as the corresponding parameters defined by Schuster) ............................ 66 4.1 The total reflectance and ratio of absorption and scattering coefficients of radiative transfer equation ( ) which is calculated based on a three flux approximation by Burger et. al (1997) .................................................... 118 4.2 The values of two coefficients of Legendre polynomials for spherical disks suggested by Mudgett and Richards (1971). ......................................... 120 4.3 Description of computing absorption and remission coefficients by extrapolating from a known sample of thickness d to a sample of infinitesimal thickness. ................................................................................... 131 4.4 Refractive index of xylene and ADP .............................................................. 132 4.5 Different cases of hypothetical concentrations ............................................... 136 4.6 Different cases of CFD results ........................................................................ 142 ix LIST OF FIGURES Figure Page 1.1 Schematic representation of divisions of the infrared electromagnetic spectrum. ............................................................................................................. 5 1.2 The geometry of the light generator, detectors, and the sample ....................... 10 2.1 (a) Schematic representation of solid angle. (McCluney, 1994); (b) Solid angle relationships ............................................................................................ 22 2.2 Geometric description for the refracted waves approached the interface at equal time intervals ........................................................................................... 28 2.3 The geometry of the light generator, detectors, and the sample ....................... 31 2.4 (a) Diffuse reflection from a sample composed of particles (SPECAC Limited, n.d.); (b) Specular versus diffuse reflectance on the surface of a slurry fluid. (c) Diffuse and specular reflectance. Diffuse reflectance (left) sends beams in all directions regardless of the uniform parallel incident rays. Specular reflectance (right) incident cone is preserved (Dilaura, 2011) .................................................................................................. 32 2.5 Illustration of BouguerLambert law ................................................................ 39 2.6 (A) Absorption fraction; (B) transmission fraction ( ; (C) straight line representing Kd which lies on the transmission curve in smaller thicknesses of the sample or layer of the sample which is illuminated (Dahm & Dahm, 1999) ..................................................................................... 41 2.7 Diagram showing the variables used in the Lambert cosine law (Griffiths & Olinger, 2002) ............................................................................................... 44 2.8 (a) Experimental apparatus for the measurement of R( ), the local diffuse reflectance (b) Apparatus for the measurement of R, the total diffuse reflectance (Patterson et al., 1989) .................................................................... 49 2.9 Schematic representation of a layer of absorbing and light scattering particles ............................................................................................................. 50 3.1 Diagram of a sample of slurry representing the KubelkaMunk model. There is a random distribution of particles. (Chalmers & Griffiths, 2002) ..... 65x Figure Page 3.2 Reflectance and transmittance of a thin layer of thickness d within a sample ............................................................................................................... 70 3.3 Absorption, reflection and transmission possibilities for a sample composed of two layers .................................................................................... 77 3.4 The flow chart of the model of assembly of sheets .......................................... 81 3.5 Absorption (A), remission (R), and transmission (T) fractions as a function of thickness for the case A1 = 0.1, R1 = 0.1, T1 = 0.8 ......................... 84 3.6 The most linear region for absorption fraction based on the least square error between linear absorption with equation A = K.d and absorption curve derived from plane parallel mathematics ................................................ 85 3.7 Schematic of the algorithm used to computerize the representative layer theory and the Stokes formula for the integral light fractions .......................... 92 3.8 Concentration profiles derived from Fluent simulation for the initial concentration of 5% and Reynolds number of 140000 for three particle sizes of 38, 75, and 150 μm .............................................................................. 94 4.1 The geometry of the light generator, detectors, and the sample ....................... 99 4.2 Illustration of diffuse reflectance of thick layers at different depths through the sample. X values represent the distance from the edge of the sample through its depth. ................................................................................ 102 4.3 Scattering and absorption coefficients for the models of Schuster and KubelkaMunk. X values represent imaginary thickness inside the sample. ............................................................................................................ 104 4.4 Total diffuse reflectance for different initial fraction of light as a boundary condition for the sample. ................................................................ 105 4.5 Diffuse reflections for four different cases of initial fraction of light. ........... 106 4.6 Reflectance and transmittance of a thin layer of thickness d within a sample ............................................................................................................. 108 4.7 Intensity of incident beam through the sample at different initial boundary conditions ........................................................................................ 109 4.8 Intensity of reflected beam from the sample at different initial boundary conditions ........................................................................................................ 109 4.9 Absorption, remission, and transmission fractions calculated from KubelkaMunk function .................................................................................. 110 4.10 The log of reciprocal reflectance plotted as a function of the A/R fraction. .. 112 4.11 The log of reciprocal reflectance plotted versus A/R fraction derived by Bull (1990) ...................................................................................................... 112 xi Figure Page 4.12 Polynomial curve fitted to the log of reciprocal reflectance versus equivalent particle size in units of ........................................................... 113 4.13 Exponential curve fitted to the log of reciprocal reflectance .......................... 113 4.14 Diffuse reflection of radiative transfer approximation for different cosine of scattering angle ........................................................................................... 114 4.15 Diffuse reflection of radiation transfer model of Equation 4.1 for seven different cases of initial fraction of light. Model of Patterson et al. (1989). .. 116 4.16 Diffuse reflection of radiation transfer model for seven different cases of initial fraction of light based on the radiative transfer approximation proposed by Griffiths and Dahm (2008). ........................................................ 117 4.17 Diffuse reflectance for the direct and diffuse illuminations plotted versus the ratio of absorption to scattering coefficients of RTE approximate model developed by Burger (1997). ............................................................... 119 4.18 Diffuse reflectance of direct and diffuse illuminations plotted versus the ratio of absorption to scattering coefficients of KM theory for particles with the shape of spherical disks. .................................................................. 120 4.19 Diffuse reflectance of three different models of diffuse reflectance derived by Griffiths and Dahm (2008)............................................................ 121 4.20 Diffuse reflectance of three different theories of diffuse reflectance derived by Griffiths and Dahm (2008)............................................................ 122 4.21 Diffuse reflection of an isotropic, optically thick sample according to the three flux approximation, the diffusion approximation, and KubelkaMunk different models continuum theories. ................................................... 123 4.22 Absorption, reflection and transmission probabilities in two layers of a sample consisting of n+1 layers with different thicknesses changing sequentially based on a geometrical progression. ........................................... 125 4.23 Local absorption, reflection and transmission probabilities in the sample for initial condition of ................................ 126 4.24 The illustration of linear region in the model of plane parallel layers determined from sum of squared regression. .................................................. 128 4.25 Local absorption fraction for different initial conditions ................................ 129 4.26 Local transmission fractions for different initial conditions ........................... 129 4.27 Refractive index of ADP as a function of wavelength. The data is collected from Refractive Index Database (N.d.) ........................................... 132 4.28 Total absorption fraction through the layers of the sample for different initial conditions. ............................................................................................. 133 xii Figure Page 4.29 Total absorption fractions through the layers of the sample for different initial conditions .............................................................................................. 134 4.30 Hypothetical concentration profiles in a sample with particles of 75 micron ............................................................................................................. 136 4.31 Local and integral absorption fraction for the case of uniform concentrations ................................................................................................. 137 4.32 Local and integral remission fraction for the case of uniform concentrations ................................................................................................. 138 4.33 Local and integral transmission fraction for the case of uniform concentrations ................................................................................................. 139 4.34 Local and integral Absorption fraction for the case of arbitrary linear and parabolic concentration profiles ...................................................................... 140 4.35 Local and integral remission fraction for the case of arbitrary linear and parabolic concentration profiles ...................................................................... 141 4.36 Local and integral transmission fraction for the case of arbitrary linear and parabolic concentration profiles ............................................................... 141 4.37 CFD concentration profiles ............................................................................. 143 4.38 Local absorption fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ........................................ 144 4.39 Local remission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ........................................ 144 4.40 Local transmission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ................................... 145 4.41 Integral absorption fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 .................................... 146 4.42 Integral remission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 .................................... 146 4.43 Integral transmission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ............................. 147 4.44 Diffuse reflectance obtained based on of Dahm equation for different absorption to remission fractions .................................................................... 148 4.45 Fitted to the reflectance derived from Dahm equation in all initial fractions showed in Figure 4.44. ..................................................................... 149 4.46 Illustration of the void fraction effects on the absorption/remission function. .......................................................................................................... 149 xiii NOMENCLATURE A Absorption fraction Symbolic representation of the fraction of incident light absorbed by a layer A(R,T) Absorption/remission function B Apparent remission coefficient b Symbolic representation of fraction of incident light remitted from the surface of a particle C Concentration of particles inside the slurry sample computed from CFD Symbolic representation of remission power of the material comprising particles in layer i D Diffusion coefficient in radiative transfer model d The thickness of a sample or a layer of sample through which light is passing Symbolic representation of thickness of a particle inside a layer known as i E Irradiance Absorption coefficient obtained from radiative transfer equation Scattering coefficient obtained from radiative transfer equation g Cosine of scattering angles Extinction coefficient based on terminology of radiative transfer equation Intensity of incident light inside the sample xiv Intensity of incident radiation Intensity of reflected beam inside the sample Intensity of reflected beam out of sample K Absorption coefficient k Absorption power of material The effective absorption coefficient for L Radiance m Mass of the particle or layer of sample N Refractive index Q The radiant energy in Joules r Reflectance of a layer inside the material Reflectance of a particle inside a layer of material Reflectance from a surface of material based on Fresnel law R Remission fraction The reflectance of a layer so thick that further increase in thickness fails to change the reflectance s KubelkaMunk scattering coefficient Symbolic representation of cross sectional surface fraction that belongs to particles xv Cross sectional surface in a sample comprised of particle type i t Transmission fraction of a layer of material Volume fraction of particles Void fraction Vol Volume of a particle or a layer of the sample Weight fraction of a particle X Distance from the edge of the sample Greek symbols Absorption coefficient used in Mie, and three flux approximation and the Absorption probability for representative layer theory Fraction of incident light absorbed by a layer Sum of fractions of light inside a sample used in Stokes formula Fraction of radiation absorbed per unit pathlength in the sample The cosine of the angle between the direction of solid angle and the surface normal of the sample Solid angle Ratio of fractions of light inside a sample used in Stokes formula xvi Photon fluence rate or radiant flux with the unit of Joules per second A parameter for calculating integral fractions of light inside a sample used in Stokes formula Symbolic representation of density of particle type i The angle of incidence with the surface normal Fraction of radiation scattered per unit pathlength in the sample The average path length of light when passes through Subscripts i Type of particles d Properties of a layer of thickness d f Of the fluid p Of the particles j Number of particles in a layer s Measured quantity 1 Initial conditions; initial fraction of light for discontinuum theories and lower boundary of integration for continuum theories 1 CHAPTER I INTRODUCTION AND BACKGROUND 1.1 Introduction One of the main goals of the thesis is to answer the fundamental question “How far does light penetrate into the slurry fluid flow in a pipe during NearInfrared (NIR) irradiation by a NIR sensor?” For a particular case, a quick answer to this question would be to run an experiment. Moreover, an experiment requires an equipped laboratory with imaging instruments, probes inside the pipe, and other facilities to visualize the flow field. This might not be affordable. Since numerical approaches allow solutions for many cases, it is preferred to use a computational approach to simulate the travel of light inside the pipe using the newest and the most efficient theoretical approaches. By choosing the numerical method, a model is built based on the characteristics of the fluid at rest. Then the results from the model can be compared to the NIR diffuse reflectance theories. Calculating the penetration depth requires answers to related questions: “What is the concentration of particles inside the field and how is it going to be predicted through optical theories?”, and “What is the light distribution in terms of fractions of the incident radiation inside the particulate slurry?” There are related works in the literature, however, most of them are experimental techniques to measure concentrations of slurries either online or offline using optical NIR sensors. However, almost none of the theoretical approaches have been applied to predict light interactions with particles inside a slurry sample. It means that the theories found in the literature have not yet been applied to the case of a slurry sample. However,2 experimental methods to model light interaction with slurries are available in the literature. Study and research in the field of optical size and concentration monitoring of slurries have been performed using analytical methods, which required some parameters from experiments. But, there are approaches that do not need any experiment. Those methods use experimentdependent models which require some input parameters that must be provided through an experiment. In this thesis, the details of both of these two types of methods along with their experimental requirements, assumptions, and mathematical models will be reviewed. 1.2 Motivation In recent years, nearinfrared (NIR) optical modeling has gained extensive acceptance in different scientific and industrial areas by means of its advantages over other analytical techniques. This type of spectroscopy is prominent because of its ability to record spectra for solid and liquid samples. Modern applications of nearinfrared use it to measure the composition of unknown samples using techniques invented by scientists at the United States Department of Agriculture (Workman and Weyer, 2008). It has become a very popular technique in a wide variety of industries due to its speed, precision, broad applicability and avoidance of extraneous chemicals. It is widely used in agricultural, chemical, pharmaceutical, combustion products, textile and many other industries. Moreover, advances in instrumentation have caused the manufacture of spectrophotometers capable of rapidly providing spectra that are flexible enough for use in different applications. NIR spectroscopy requires some input parameters such as the intensity of radiation, the fractions of diffuse light in the medium, and the characteristics of the slurry fluid. A wide range of theories has been developed to support experimental capabilities of methods that use NIR diffuse reflectance and transmittance. There are a number of distinct mathematical models capable of characterizing nearinfrared light diffusion in onephase mediums or one phase with a mixture of two or more types of different components. These models are different, however, most of them implement the fundamental technique 3 of applying nearinfrared radiation to monitor different materials. The applications of NIR spectroscopy are summarized in Table 1.1. Table 1.1 Description of different fields and applications of nearinfrared radiation has already been used in the previous works Field Applications Model/Technique Clinical sector Human blood cells, In vivo imaging Human body tissues, Optical topography of brain Monte Carlo Simulation, Diffusion approximations, Steady state diffusion theory, Transport theory, Addding doubling method, DeltaEddington approximation, Astronomical sector Astrophysics, Detection of infrared galaxies, Interstellar molecules Dual field interferometry Agriculture Agricultural products monitoring, Analysis of food crops, Analysis of processed food like baking products, and Analysis of nonfood agricultural products like paper, wool, etc. Representative layer theory, KubelkaMunk theory, Diffuse reflectance theory, Schuster’s theory Remote monitoring NIR spectroscopic imaging, Hyperspectral imaging in the remote investigation of plants and soils Experimental techniques to collect data from airplanes or satellites to investigate the soil chemistry on the ground Material Science Film thickness measurements Nearinfrared measurement in the diffuse reflectance mode Particle measurement Slurry flow, granular flow, fluidized beds, powders and ground materials Continuum theories of diffuse reflectance such as classical KubelkaMunk theory, BeerLambert’s law, etc., MonteCarlo Simulation Industrial uses Study of industrial chemicals, Process control applications, Rapid laboratory measurements, Realtime online analyses ParticleLaden slurry transport in mines, Process control for food, agricultural, combustion, textile, and pharmaceutical products, Before examining the techniques shown in Table 1.1 in detail consider a brief introduction to NIR models. Based on the foundations of nearinfrared models there are two different paths indicated by these theoretical treatments: continuum theories and discontinuum theories. These two types of theories are separated by the two distinct types of mathematical models on which they are based. For example, the theory of representative layers is applicable in many industrial processes. Among the 4 theories of diffuse reflectance, the representative layer theory is the latest efficient method to model nearinfrared phenomena inside particulate, turbid, in vivo, powdered, and generally uniform and nonuniform mediums using discontinuous mathematics. As it was depicted in the Table 1.1, none of the nearinfrared theoretical treatments have analyzed slurries with discontinuum mathematical techniques. Reviewing the literature, discontinuous mathematical methods have not yet been used for slurries. Continuum theories are imperfect due to their weakness in exhibiting the light interactions with particles in real nonuniform mixtures that scatter light and are not continuous. On the other hand, the only method that addresses light propagation inside a slurry has been found in the work of Rabhi, Masion, Roze, Dussouillez, and Bottero (2010) that uses an experimentaltheoretical algorithm and some parameters such as refractive index to find particle size distribution (PSD), and the slurry concentration to be fed into a simulation. Hence, their approach is not pure theoretical. Meeten and Wood (1993) introduced another approximation for measuring the diffuse reflectance of fluids using optical fiber methods, but the classic KubelkaMunk theory is applied to interpret the experimental data. KubelkaMunk theory is a powerful tool to describe light diffusion but is only applicable to continuous mediums: it underestimates the optical properties of a real discontinuous sample. This introduction presents the approaches that are going to be discussed in the present thesis. In the following chapters the latest approach has been used to characterize the optical properties of slurries that naturally have discontinuity in their composition. 1.3 Background Nearinfrared technology and slurry mixtures are two separate academic, technological, and industrial areas for which at first it may seem very difficult to find a connection, but after a brief introduction one may see the relationship between them. Hence, studying an overview about the NearInfrared and the slurry is essential to reading and understanding this thesis. 5 1.3.1 Near Infrared Technology Nearinfrared is the electromagnetic radiation with a spectrum of wavelengths just beyond visible light. Nearinfrared is a part of infrared radiation which usually is divided into three spectral regions: near, mid and farinfrared. However, another categorization is done by Center for Measurement, Vegetation, and Health [MVH] (2006) at the University of New Hampshire: the infrared region is divided into five parts: near, shortwave, midwave, longwave, and farinfrared . This spectrum is shown in Figure 1.1. Figure 1.1 Schematic representation of divisions of the infrared electromagnetic spectrum. The near infrared region corresponds to a wavelength range of between 700 and 2500nm. The midinfrared region is further divided into short, mid and long wave infrared and corresponds to a wavelength range of 2500 – 5x104 nm. Finally the far infrared region corresponds to a wavelength range of 5x 104 – 1x106 nm. (MVH, 2006) The boundaries between near, mid and farinfrared regions are not agreed upon and can vary. The main factor that decides which wavelengths are included in each of these three infrared regions is the type of detector technology used for gathering infrared beams of light. The infrared region of light has longer wavelengths than visible light, making it invisible to the human eye, however, the nearinfrared spectrum has shorter wavelengths with proximity to visible light. Spectroscopy is defined as the study 6 of the wavelength spectrum of radiated energy interacts with matter – it is transmitted, reflected or absorbed. Although a theoretical analysis underlies NIR spectroscopy, it is mostly known as an experimental branch of NIR technology. Theoretical techniques have been developed to interpret the data gathered from the spectrum. Modeling is absent in the spectroscopic techniques. On the other hand, theories of diffuse reflectance provide an opportunity to model the nearinfrared penetration of light inside a sample. Recently, more sophisticated theorems have been applied to model the nearinfrared phenomena. The earliest application of NearInfrared spectroscopy goes back to the 1800’s, when organic compounds were investigated by NIR photography. Thus, the first application of NIR technology was studying chemicals in compounds. In the 1940s, the availability of new commercial instruments led to one of the first industrial applications of NIR in chemical and polymer industries. Until that time applications were confined to academic or research projects (Workman and Weyer, 2008). Norris in USDA in 1949 was the first person who began to apply NIR in the food industries. During the 1950s and 1960s, progress in this field of chemical monitoring became stagnant, however, diffuse reflection theories propelled NIR spectroscopy to a wide range of applications in agriculture, pharmaceuticals, and many other industries. With the advent of new instrumentation and methods, the interest in NIR industrial applications has become greater. Infrared technology is not limited to physical applications monitoring materials in samples; it can be applied in astrophysics and astronomical observations. NIR observations have been made from ground based observatories since the 1960's. They are done in much the same way as visible light observations for wavelengths less than 1 micron, but require special infrared detectors beyond 1 micron. Mid and farinfrared observations can only be made by observatories which can get above our atmosphere because spaceborne telescopes are very sensitive to heat. The reason is that the telescope itself has a certain temperature and continually radiates heat that would interfere with the measurements taken by a NIR sensor. Far infrared waves are thermal, therefore, we experience this type of infrared radiation every day in the form of heat. The heat that we 7 feel from sunlight, a fire, an oven or a warm coffee is infrared. These observations require the use of special cooled detectors containing crystals like germanium whose electrical resistance is very sensitive to heat. Despite all of the mid and far infrared applications, NIR is more popular among industries. Industries strive to produce quality commercial products at the lowest cost and in the shortest time. Introducing analytical methods that offer technical and costsaving advantages over conventional approaches can lead to businessrelated improvements in the manufacture of chemical products. In recent years, nearinfrared (NIR) spectroscopy has been incorporated into several analytical protocols for obtaining qualitative and quantitative information on incoming materials, mixtures, and products. One of the most important applications in industries is slurries. NIR spectroscopy is used in these industries to monitor composition and control the process. Therefore, NIR light and slurries are connected through this widespread application. 1.3.2 Slurries Despite much progress in techniques of conveying materials discovered in mines, the semiindustrial transport of minerals out of mines had been an issue until the early sixteenth century, when a solution was found. Hydraulic conveying over short distances for alluvial mineral separation was documented in the sixteenth century. The hydraulic term refers to the technique of transporting those minerals; the process that is performed with the stream of fluid inside pipelines. The particulate medium by which the hydraulic transport of minerals took place has an academic name; a slurry. Particleladen slurry transport through pipelines is one of the important industrial applications of slurries. The scientific interpretation of the word introduces the slurry as a solidliquid mixture. The slurry has been categorized in the group of multiphase mixtures. Another meaning states a thick suspension of solids in a liquid. Therefore, this type of flow can be categorized in the group of dispersed phase and separated flows. When there is more than one phase participating in a fluid 8 mixture, the combination is known as a multiphase mixture. As shown in Table 1.2, there are four types of multiphase flows (Crowe, 2012). Due to the presence of two phases, this flow is more complex than single phase flows. Twophase flows consist of discrete elements such as droplets in a gas or solids in a liquid. Thus, they are not attached to each other, therefore, in a separated flow, two phases are dispersed by a line of contact. Table 1.2 Examples of single and multicomponent, multiphase fluids Single Component Multicomponent Singlephase Water, Oxygen Air, Emulsions Multiphase SteamWater, FreonFreon Vapor AirXylene, Slurry Slurries are applicable in many engineering, industrial, chemical, and fossil energy areas like transport of bulk quantities of materials such as coals and ores or flow of mud. The significance of particle deposition results from its key role in industrial processes such as filtration, separation, particle transport, combustion, air and water pollution, metallurgical processes and many others (Hossain, Naser, and Imteaz, 2011). The fluid that carries the particles through the pipe may be Newtonian or nonNewtonian, and usually influenced by turbulence, drag, hindered velocity, virtual mass effects, phase interaction, crossingtrajectories, and particle inertia (Chen, 1994). Moreover, the process of transporting solidliquid mixtures through pipelines is common in the mining and petrochemical industries. Furthermore, turbulent slurry flows with high solid volumetric concentrations are an important part of many chemical or mineral applications. In addition, the need to measure the pressure drop and concentration profiles in pipelines made researchers study this kind of fluid and build more accurate models (Lahiry and Ghanta, 2010). Slurry transportation in the horizontal direction may be classified into three major flow patterns: (1) pseudohomogeneous slurry (or homogeneous); (2) heterogeneous and sliding bed slurry (or moving bed), and (3) saltation and stationary bed slurries, as described by Ling et al. (2003). A pseudohomogeneous slurry is a pattern 9 in which particles are uniformly distributed along the crosssection of the pipe since the velocity is so high to prevent the particles from accumulating in the bottom of pipe crosssection due to gravity effects. This process is practical since the high flow rates prevent the particles from falling due to the influence of gravity. However, heterogeneity happens in a slurry flow when the flow rate cannot overcome the transportation of the particles to the lower part of the crosssection. By further decreasing the velocity, the particles will accumulate in the bottom of the pipe crosssection. This region is called a moving bed layer (Ling et al., 2003). This phenomenon occurs in the saltation and stationary bed flows. In conclusion, when one thinks about the combination of a liquid with insoluble solids inside, there are many different examples. 1.4 Statement of the Problem The main goal can be expressed in this form: the nearinfrared region of a light beam is chosen to penetrate into a slurry for detecting the slurry particles by sensing the fractions of light in each section through the sample depth, with reflectance or transmittance to a NIR sensor. The problem is to estimate the fractions of light at each distance inside the sample to answer the question of the depth inside the slurry at which the particles will be detected by the sensor. Therefore, the fractions of light at each possible point inside the sample must be known to estimate the layer within the media where the light is able to be transmitted to the detector or is blocked by the heavy concentration of particles. As it is depicted in Figure 1.2, when the generated light reaches the surface of the medium, a portion is transmitted into the sample, and the rest is reflected. Then the transmitted light goes through the sample, which may be represented as many layers of particulate slurry. The reflectance, transmittance, and abrorptance occurs in each layer, until the remaining transmitted portion of light is detected by the detector. This geometry can be shown with another configuration where the detector is located in the same place as the generator, detecting the reflectance of the light instead of transmission. 10 Figure 1.2 The geometry of the light generator, detectors, and the sample Now, the problem is to predict the amount of the light through the sample and the penetration depth of light inside the slurry. Transmission, reflection, and absorption inside the sample gives us information about the concentration of particles inside the sample. Moreover, it can predict the shape of particle accumulation in each layer inside the slurry. 1.5 Goals The primary goal of this thesis is to permit accurate estimates of how much light is detectable by the sensor, and what portion of the sample contributes to the signal. Answering these two fundamental 11 questions will describe the sample from a spectroscopic perspective. This information will help us choose the best location for installing the NIR sensor outside of an slurry sample. Tools for modeling the NIR light interactions with particles inside a slurry are theories of diffuse reflectance. These theories were applied to the problem with their limitations and assumptions considered. The purpose for using different theories to model the optical interaction is to compare them and find the most computationally efficient approach for modeling the NIR diffuse reflectance phenomena in slurries. Moreover, the problem is divided into two groups of NIR diffuse reflectance theorems based on their mathematical treatments: continuous and discontinuous functions. Each of these two groups requires different arrangements for the configuration of the sample. Therefore, they have different results. Continuum theories aim to find the light fractions integrally inside the sample, so the accumulative fractions are estimable without using any other theory in any distance from the boundaries of the sample where the light first was encountered. This means that continuum theories directly calculate the cumulative fractions of light in each distance from the edge of the sample, and the methodology of these theories is different from discontinuum ones in which the light fractions are estimated locally, so one must combine them with another method to find light fractions at each depth inside the sample. The details of these methods will be discussed in the following chapters. The purpose of implementing discontinuous theories is to find the same fractions but locally. This gives us an opportunity to have an estimation about optical properties of each possible point and layer inside the sample when the effects of other layers are not present. It means that the reaction of each particular layer to the incident beam is considered in isolation and only based on the local configuration of the particles inside each particular layer of the sample. The main goal is achieved by considering these objectives and steps: (i) for continuum mathematical approaches such as plane parallel mathematics, KubelkaMunk theory, Schuster’s theory, three flux approximation, and diffusion theory, and (ii) for discontinuum mathematical 12 approaches such as the theory for plane parallel mathematics, assembly of sheets, representative layer theory, and combined representative layer theory and the Stokes formulas. In order to meet the goals discussed above, the following tasks must be done: (i) reviewing all of the theoretical approaches appropriate for the case of this problem in terms of assumptions, limitations, advantages, etc, (ii) implementing the theories by developing a mathematical model based on them, and (iii) comparing the results of the models. The first task will be done in Chapter 2, where continuum and discontinuum diffuse reflectance theories are introduced. In Chapter 3 the models created based on these two groups of theories are discussed respectively according to the mathematics of the methods. In Chapter 4, the results are compared with each other. Hence, the most compatible model will be introduced based on the results. Finally, in Chapter 5, the conclusions and recommendations are presented. 13 CHAPTER II PAST CONTRIBUTIONS 2.1 Introduction Optical methods for modeling NIR radiation phenomena inside a medium, whether it is composed of onephase continuous components, onephase powdered sample, or multiphase elements are available in the literature which will be addressed in this chapter. Although, this work connects the fluid dynamics of a slurry flow to an optical phenomena that is the interaction of light with particles in a slurry sample, most of the work has been concentrated on optical theories to model the NIR radiation inside a slurry. Hence, this chapter is dedicated to surveying the theoretical methods pertaining to optical modeling inside slurries. Before describing the approaches, one needs to know what has been done in past contributions to NIR technology in analyzing samples. Then, acquiring a preliminary knowledge about the definitions of fundamental expressions and terms that are related to optical models and are mostly used in this thesis is essential. After introducing the optical nomenclature, NIR optical methods are reviewed. These approaches are gathered by reviewing these major concepts in the available literature: nearinfrared spectroscopy, online and offline measurement of slurries, NIR diffuse reflectance and transmittance, particle size analysis, slurry concentration measurements, and theories used in optics. All of the optical methods examined in this study have certain fields of application based on their assumptions. The assumptions make constraints, however, by simplifying the analytical model14 one may produce a new model consistent with the restrictions. Therefore, measuring the capabilities of each theory in the literature through a thorough study may facilitate the choice of the approach which best matches the requirements of our problem. In this part optical methods are briefly introduced and then their assumptions and basic concepts, as well as the mathematics beyond those concepts will be discussed. 2.2 History of past contributions History of past contribution to the theories of diffuse reflectance helps to a better understanding of the theorems. A summary is gathered in Table 2.1. Bouguer in 1760, and scientists start the works on diffuse reflectance and engineers have been involved in the measurement of light since the early experiments and instruments described by him. Pierre Bouguer (1729) and Johann Lambert (1760) developed the principle underlying the model for the absorption of light that relates the absorption to the properties of the material through which the light is traveling (Dahm & Dahm, 2007). This law will be introduced in detail in this chapter. Moreover, Bouguer’s elementary mirror hypothesis (1729) represents an application of the absorption by glass and water for various thicknesses (Darrigol, 2012). Darrigol also reviewed the absorption law pointed out by Bouguer stating that the successive equal layers of a homogenous transparent medium absorb a constant ratio of light that they receive. Kortum (1969) delineated Bouguer’s elementary mirror hypothesis which has been employed to explain diffuse reflection from surfaces. Lambert’s inverse square law, additivity law, and extinction laws are all based on Bouguer’s work. Lambert’s hypotheses were more accepted than Bouguer’s findings since there were some contradictions in Bouguer’s observations in the laws of diffuse reflection. In fact, Bouguer’s model of diffuse reflection was found to fail later (Darrigol, 2012). 15 Table 2.1 Overview of the development of modeling techniques for diffuse reflection. 1760 P. Bouguer described diffuse reflection by elementary mirror hypothesis for surfaces with random orientations 1860 G.G. Stokes produced a twoflux theory of diffuse reflectance. The theory considers individual particles as plane parallel layers, and provides an analytical tool to the mathematics of plane parallel layers. 1887 E. Lommel and R.V. Seeliger developed a complicated theory of diffuse reflection concerning interactive irradiation of distinct elements inside a sample. 1905 A. Schuster proposed a special part of diffuse reflection based on continuous mathematics by publishing “Radiation through a foggy atmosphere”, which involves particles in sample. 1908 G. Mie devised a novel method involving the scattering of light by isolated spherical particles, which uses the mathematics of complex variables and numbers. 1926 T.H. Gronwall proposed an article that emphasized on a simplified version of plane parallel mathematics. 1931 Kubelka and Munk published their classic paper that demonstrates the Kubelka–Munk equation, derived using continuous mathematics. They displayed that the reflection which is calculated by the absorption/remission formula from an infinitely thick sample, is equivalent to the ratio of absorption to remission coefficients for a medium. Their work produced many controversies over the failure of their model at certain conditions, and confusion over the effects of: direct and diffuse illumination; front surface reflection; and the relationship of the absorption and scattering coefficients of a sample to the coefficients of the material making up the sample. (Chalmers & Griffiths, 2002) 1946 F. Benford produced a very complete theory describing the mathematics of plane parallel layers. His work was very creative, novel, and independent of previous contributions. 1963 N.T. Melamed published a paper proving his method for modeling diffuse reflectance inside particulate samples, which is independent from plane parallel mathematics. 1966 Wendlandt and Hecht published the first “handbook of diffuse reflection” in 1966, which was the first handbook in the field of spectroscopy. 1969 G. Kortum published ‘Reflectance Spectroscopy’, a book which is heavily mathematical and covers continuum and discontinuum theories rather thoroughly. (Chalmers & Griffiths, 2002) 19701975 E.L. Simmons proposed several articles including simplified version of plane parallel mathematics to model particles inside a sample, and a modified particle theory (Chalmers & Griffiths, 2002). He proved the proportionality of the discontinuous and continuous theories in the case of twoflux approximations. 1997 T. Burger et al. presented a method in their publication pointing out application of the radiation transfer model to characterize diffuse reflectance through more advanced methods such as using three (or more) fluxes to separate absorption and scattering coefficients. 19952000 Dahm and Dahm conducted a carefully planned research that results in: a method to use plane parallel mathematics to separate absorption and remission coefficients; presenting an absorption/remission function which is constant for all sample thickness; and an equation relating the absorption and remission of one thickness of sample to that of any other thickness. They clarified the confusions over Kubelka–Munk theory, developed the representative layer theory that models a complex sample by defining layers, each one representative of the sample. They also developed a simple particle model. 16 The calculation of the amounts of light absorbed, transmitted and remitted by several layers is a very complicated process. Some fraction of light will be absorbed by a layer, which reduces the transmitted fraction to the next consecutive layer. In addition, some may return to the first layer as the result of reflection from the next layer, which again a portion may again be absorbed, transmitted, or remitted back to the next consecutive layer. The mathematics that is able to model this complex physical situation is called mathematics of plane parallel layers, which was developed by Sir George Stokes in 1860. Dahm and Dahm (2007) assert that Stokes theory is an exact estimation for plane parallel particles. The Lommel – Seeliger reflectance law is a time honoredlaw, which has been used until today. It is based on a model, which is founded based on diffuse reflection by considering the optical characteristic of every single element inside the sample. Therefore, this model was very complex comparing to similar ones in that ages. However, it has some limitations such as predicting only the scattering of only collimated incident light (Fairbairn, 2004). Schuster proposed a method of scattering and absorption within a sample by visualizing a dilute suspension of particles in which the particles were luminescent, absorbing, and scattering. He published a paper in 1905 describing a particle theory that developed to determine a solution for particular problems (Griffiths & Dahm, 2008). He developed his particle theory using continuous mathematics. Gustav Mie designed a solution to Maxwell’s equations to describe the scattering of electromagnetic radiation by a sphere. The solution is in the form of an analytical infinite series. The Mie scattering solution finds exact solutions for Maxwell’s equations by separating equations for radial and angular dependence of the solution in the problems dealing with scattering (Bohren & Huffman, 1998). Thomas Hakon Grönwall published a paper in 1926 about new method in geometrical optics and presents a simpler method than the plane parallel mathematics. 17 Kubelka and Munk (1931) published their classic paper in the field of optics, which revealed a formulation for the scattering, and absorption of infinitely thick samples. Kubelka and Munk published their paper which is a twoflux version of a multiflux method of solving radiation transfer problems. This method is popular because of its simple analytical equation, which is known as the KubelkaMunk equation. However, this theory has some assumptions that limit the cases to which it can be applied. Benford (1946) described mathematics of plane parallel layers, which is very complete and novel. In 1946, he published a detailed analysis of the absorption and scattering of light in the article of ‘Radiation in a diffusing medium. He used discontinuous mathematics and assumed that the sample was divided into a series of plane parallel layers (Dahm & Dahm, 2008). Benford’s calculations were developing a system of relations that can be used to compute the reflection, absorption and transmission fractions for a sample if the number of layers and those fractions for each are known. Melamed proposed a complete statistical theory for the absolute diffuse reflectance of powders about a half century ago (Mandelis, Boroumand, & Bergh, 1990). In the past half century many spectroscopists performing quantitative diffuse reflectance spectroscopy of powdered materials have used the Melamed model. Melmad’s theorem was successful in the field of modeling diffuse reflectance since unlike other discontinuum mathematics, Melamed applied statistical summations on the sample consisting of discrete particles reflecting light diffusely. The laws of geometrical optics (Mandelis et al., 1990) limited his model. Mandelis et al. deduced that Melamed’s statistical approach had prominence over other theories that are based on plane parallel mathematics but with some limiting assumptions. Proceeding Melamed’s unique theory, Wendlandt and Hecht (1966) published the first handbook in the field of spectroscopy. In 1969, Kortum presented his book in the field of reflectance spectroscopy that is a breakthrough in using 18 cumbersome mathematics to embellish continuum and discontinuum theories of diffuse reflectance (Griffiths & Dahm, 2008). Simmons has used a simplified particle model (plane parallel mathematics) to relate diffuse reflectance to some optical parameters of the model without the use of heavy equations, which result from the more modified Melamed’s theory (Mielenz et al., 1977). A more recent study in the field of diffuse reflectance was done by Burger et al. (1995) who developed a diffuse reflectance model using three or more fluxes (Griffiths & Dahm, 2008). Dahm and Dahm (1999) presented a more complete method to study the diffuse reflection phenomena inside samples containing nonhomogeneous particles. They used the plane parallel layer model of Benford (1946) to develop a more accurate approximation to calculate the fractions of light for each layer. This accuracy is a result of relating the physical properties of the elements of the medium to the optical properties of the sample. They also developed an equation for relating absorption/remission of one layer to that of the sample. This equation is comparable to the KubelkaMunk function but is a descriptive tool for more complex samples (Griffiths & Dahm, 2008). Moreover, throughout their systematic work, they presented a series of explanations about the KubelkaMunk theory and its controversies. 2.3 Fundamental Definitions In this section fundamental definitions are presented that are necessary for the appreciation of the concepts proposed not only in this chapter but also in the following chapters. There is a chain of prerequisites for conveying the information about topics from basic elementary components in optics to more rigorous concepts. Some of the definitions are not found in the theories but introducing them is essential since these fundamental terms have used in some more advanced definitions. Hence, they 19 are explained before separately. Each part is described sufficiently for applying the theorems into models. 2.3.1 Geometrical Optics Geometrical optics is the study of light without diffraction or interference (Greivenkamp, 2004). It is essential to explain the fundamental definition of geometrical optics to briefly introduce this significant part of classical optics. Geometric optics refers to the ray optics and describe the light propagation in terms of rays. In this part of classical optics, the rays are explained to spread out in rectilinear path in a homogenous medium. Rays are able to be bent, transmitted by splitting in two or more parts at the interface between two distinct media, may be refracted in a medium where the refractive index changes, or are absorbed and reflected. Geometrical optics provides rules, which depend on the wavelength of the electromagnetic wave, for propagating these rays through an optical system. Therefore, geometric optics is a crucial simplification of optical sciences that fails to account for some optical phenomena such as diffraction and interference. It provides an outstanding estimation about the travel of light into materials, however, when the wavelength is very small compared with the size of objects or materials with which the light interacts. Geometric optics can be used to describe the geometrical aspects of imaging, including optical aberrations. 2.3.2 Wavefront and Ray In physics, a wavefront is the locus of points having the same phase: a two dimensional line, curve, or surface for a wave propagating in 3D space (Schuster, 1904). Greivenkamp (2004) reported a very concise description about the geometrical optics based on the optical path length. Optical path length is proportional to the time required for light to travel between two points (Greivenkamp, 2004). The relation is: (2.1) 20 where is the refractive index which varies along the path. If it is constant the integration is not required. Mouroulis and Macdonald (1997) suggested that the propagation of waves is commonly described by means of wavefronts. In geometrical optics the concept of wavefront is fundamental. Because of the higher frequencies of the waves such as infrared, optical, xray and gammaray, the temporal component of electromagnetic waves is usually neglected for the corresponding wavelengths, therefore, the phase of the spatial oscillation is the only parameter that is described. Moreover, most optical system and detectors are indifferent to polarization, so this property of the wave is also usually neglected. Maxwell's equations are a powerful descriptor for the optical system electromagnetic waves, and linear propagating waves such as sound waves that have similar wave equations. There are also geometrical wavefronts which are described similar to the physical wavefronts but are not propagating and are frozen in space (Mouroulis & Macdonald, 1997). Introducing the wavefront is essential to define another fundamental element in optical physics which is the ray. Rays are defined as the lines normal to a family of wavefronts. The direction of electromagnetic propagation as suggested by Mouroulis and Macdonald (1997) is given by the Poynting vector, which is the cross product of the electric and magnetic field vectors. This direction of energy flow is conveniently described by rays. One might think that a very narrow beam of light is a ray but it is not. The divergance in the beams of light contributes the idea that the crosssection of beams are not constant in their lengths. In this thesis, it has been assumed that the media is isotropic and our definition of rays is consistent with this assumption. However, there are some materials in which the direction of electromagnetic waves such as visible or nearinfrared light is not always normal to the wavefronts such as crystals or nonlinear media where it is possible for the wave to exit the material at a different wavelength from which it entered. 21 2.3.3 Radiant Energy McCluney (1994) stated that radiation is the scattering of energy in several forms through space. Following this definition, the radiant energy can be described as the quantity of energy propagating onto, through, or emerging from, a specified surface of given area in a given period of time. For estimating the energy, all the wavelength spectrum contained in the radiation are must be included. The radiant energy is known by Q and its unit when a limited wavelength range is to be considered is shown in joule. Nearinfrared light has a radiant energy calculated by integrating the radiant flux over time. 2.3.4 Solid Angle Solid angle is a two dimensional angle in three dimensional space and is described by a closed curve and a point in space. According to Figure 2.1 (a), the solid angle is the projection of curve C in space, and curve C subtends solid angle at point P at the center of a unit sphere. It extends the two dimensional concept of a plane angle to three dimensions. The magnitude of this angle is calculated by projecting the area of a closed curve in space on a sphere of unit radius, therefore, the solid angle of an object is equivalent to the area of the segment of a unit sphere, centered at the angle's vertex or center of sphere, that the object covers. Another relation that has been set forth by Palmer and Grant (2010) is the ratio of projected area of the closed curve on a sphere with arbitrary radius, , over the square of the sphere radius. This defining the solid angle along with the corresponding relationships have been presented schematically by Figure 2.1 (b). The unit of this angle is measured by steradian. The relationship associated with solid angle is: (2.2) where is the solid angle, is the segment of the projected area on the sphere, and r is the radius of sphere. 22 (a) (b) Figure 2.1 (a) Schematic representation of solid angle. (McCluney, 1994); (b) Solid angle relationships Hemispherical solid angle is another important definition. The three solid angles are directional, conical, and hemispherical. McCluney (1994) distinguished between these three types and claimed that the directional one refers to an infinitesimally small solid angle in a particular direction from arbitrary point p in space like Figure 2.1 (a) and (b). Directional solid angles are only used in mathematical models of reflectance and transmittance but are not applied in experiments and measurements since the associated flux with this infinitesimal angle is unmeasureable in quantity. Conical solid angles are measures of intermediatesized solid angles in the shape of circular cones. The mostly used solid angle is the hemispherical one that has the size of steradians. The shape of this angle is the same as a hemisphere. 2.3.5 Radiant flux The radiant flux is the time rate of flow of radiant energy with the unit of watt. This flow of radiation is characterized by the quantity of energy transferring through a region in space or usually a surface per unit of time. The defining equation is: (2.3) 23 where is the radiant energy in Joules and is the radiant flux with the unit of Joules per second or watt (McCluney, 1994). According to this definition, the radiant flux is radiated from a source over optical wavelengths, which are defined to be from to Hz. This region is equivalent to the electromagnetic spectrum commonly referred to as ultraviolet, visible, and infrared. 2.3.6 Radiance We started defining the fundamental quantities with radiant energy which has as its unit the joule (J). Radiant power introduced as radiant flux is energy per unit time ( ) with the symbol and is measured in Watts. Palmer and Grant (2010) deduced that these two definitions are not informational about the spatial distribution of power in terms of area or direction. Hence, another parameter is required that is an elemental quantity in radiometry, and is defined as the power per unit area, and per unit projected solid angle. It is known as radiance (Palmer & Grant, 2010). Since the radiance can be integrated along a beam, it is possible to calculate it in any location of an optical system. The radiance depends upon the direction and it can come from many points on a surface. Moreover, it can exist everywhere since it applies to a field quantity. The defining equation is (Palmer & Grant, 2010): (2.4) where, L is the radiance, is the solid angle, is the radiant flux and s is the surface. 2.3.7 Irradiance Irradiance (radiant incidence) is defined as the power of electromagnetic radiation per unit area. Described by McCluney, irradiance is the area density of radiant flux, the radiant flux per unit area in a specified surface that is incident on, passing through, or emerging from a point in the specified surface. The defining equation is: (2.5) 24 where is an element of radiant flux and is an element of area in the surface. It is essential to mention that radiance and irradiance are quite different. Radiance is the angular distribution of radiation emitted per unit area into a cone having unit solid angle, while irradiance sums up all this angular distribution over a specified solid angle . Hence, the relation associated with the summation over the solid angle comes from the following integration: (2.6) where, L is the radiance and is the solid angle. 2.3.8 Radiant Intensity McCluney (1994) characterized radiant intensity, I, as the density of solid angle or flux of energy radiated per unit solid angle (steradian) incident on, passing through, or emerging from a point in space. The defining equation is: (2.7) where is an element of radiant flux and is the element of a solid angle in the specified direction. The intensity is a strong function of direction from or toward the point for which it is defined. Also, it changes with distance from its source except for a collimated beam (Dahm & Dahm, 2007). Intensity can be derived from radiance by integrating over area: (2.8) 2.3.9 Refractive Index Index of refraction is a very fundamental parameter in optics, and it can be measured with a very high accuracy of (Chartier, 2005). Refractive index is a property of a substance which is a number indicating the level of electromagnetic wave propagation in a medium. Before introducing the relation 25 for refractive index, it is essential to know about the source of this concept. Fundamental equations of wave propagation into a medium are known as Maxwell’s equations. Using Maxwell’s equations for a dielectric medium and then simplifying the integral forms and implementing some mathematical operations and manipulating, one can obtain the general wave equations for the propagation of a periodic disturbance, which is shown below (Hecht, 2002): (2.9) (2.10) where E is the electric field vector, B is the magnetic field vector, is the electric permittivity of free space, and is the permeability of the free space. The phase velocity of Maxwell’s equations is: (2.11) Therefore, based on the Wendlandt and Hecht (1966) analysis, the refractive index, defined as the phase velocity in a vacuum which is equal to the speed of light compared to that in a given medium, is: (2.12) where c is the speed of light or phase velocity in vacuum and is the phase velocity in the material. and are permittivity and permeability of the material, while and are permittivity and permeability of the vacuum. When light passes through a medium, some part of it will always be absorbed. This can be conveniently considered by introducing a complex index of refraction: (2.13) where the real part of the refractive index indicates the phase speed, whereas the imaginary part is the amount of absorption loss when the electromagnetic wave propagates through the medium. 26 2.3.10 Diffraction Diffraction appears as a limitation to the rectilinear propagation of light (Chartier, 2005). Diffraction is a complicated process, and occurs when a light beam collides with an obstacle. After collision, some parts of the light deviate in different directions. In classical physics the diffraction is described as the apparent bending of waves around small obstacles and the spreading out of waves past small orifices. Diffraction is observed for any kind of propagating phenomenon. All types of waves can be diffracted. Pedrotti et al. (2007) have presented a compelete study on this important phenomenon in classical optics. They suggested a simple defintion for this term that is any deviation from geometrical optics which results from the obstruction of wavefront of the light. The diffraction is observed even if the obstacle is not opaque and the transmitted wavefronts are affected by change of amplitude and phase. 2.3.11 Refraction Refraction is the change in direction of a wave due to a change in its medium. This phenomenon occurs because of an inhomogeniety and discontinuity of the interface atoms which reflects light backward or transmit it forward through the medium. Hence, the fact that the incident rays are bent is called refraction (Hecht, 2002). Wendlandt and Hecht (1966) pointed out a simpler definition for refraction. They supposed that the electromagnetic radiation starts its path through a medium with a specific index of refraction while it reaches to another medium with a different refractive index. Whenever it encounters the separating region between two different media, a part of the beam is reflected back into the first medium, however part continues into the second medium, but with an altered direction of propagation. Hence, the refraction is essentially a surface phenomenon. Electromagnetic radiation is an element in the refraction process, hence the governing equations of electromagnetic radiation are valid when the light refracts from one medium to another. When the 27 light passes through the medium, the phase velocity of the wave is changed but its frequency remains constant. The refraction occurs for any type of electromagnetic wave; from light to any other electromagnetic waves such as sound. Refraction is described by Snell's law, which constructs an expression based on the law of refraction for the relationship between the two incident and refraction angles and the properties of the wave which passes through the boundary of two different isotropic materials. The formula is derived for a given pair of media and a wave with a specified frequency stating that the ratio of the sines of the angle of incidence and angle of transmitted wave after refraction is equivalent to the ratio of phase velocities ( ) in the two media. This ratio is also equivalent to the opposite ratio of the indices of refraction ( ): (2.14) To prove the expression above, one can start from sketching the refraction phenomenon at an interface like Figure 2.2 by drawing the wavefronts at an instant of time. Each of them is a surface that has a constant phase. Because of the change in the phase velocity at the interface, the waves bend (Hecht, 2002). In fact, Figure 2.2 is a representation of several exposures of a single wavefront that shows it in equal time intervals of . When the wavefront OB reaches the interface, it refracts at point A. Assuming that it takes for the wavefront OAB to get to the surface RED with phase velocity of , different points on the surface of the wave traverse with different phase velocities after this time interval because one portion of the wave is in the incident medium with phase velocity and another part is in the transmitting medium with phase velocity of . In this thesis, the analysis is done on the twophase flow of Xylene as the liquid and ADP particles as the solids. Nearinfrared radiation is employed as an electromagnetic wave that penetrates the solidliquid mixture by a sensor which is located in air outside of the mixture. The refractive index of 28 xylene and ADP are greater than the air or the phase velocity in air is greater than that in a liquidsolid mixture . Figure 2.2 Geometric description for the refracted waves approached the interface at equal time intervals Hence, according to Figure 2.2, for the case of nearinfrared waves in air as the incident medium striking on the xyleneADP sample as the transmitting medium, the vector BD is longer than AE because and time intervals are equal. In Figure 2.2, the two triangles ABD and AED share a common hypotenuse which is AD. Therefore, (2.15) where, and , so (2.16) By multiplying both sides by c and since and , one has: (2.17) Incident medium B R E O Transmitting medium A D 29 The equation above is the general law of refraction or Snell’s law. 2.3.12 Reflectance (reflection) and reflectivity Introducing this important concept in optics, demands an understanding of the difference between three distinct terminologies which are common for all light fractions. As pointed out by Siegel and Howell (1981) there are differences between the fractions ending –itivity and –ance. It is common to refer the ending –ivity to intensive, coherent, or bulk properties of matter. The ending –ance is used to assign for the extensive properties of a material. Reflection is a phenomenon by which a part of the light rays that are incident on the medium or particle leave it from the same side or scatter backward. This was a preliminary definition of reflectance that was addressed by Rea (2000). Hecht (2002) explained that reflection occurs when a beam of electromagnetic radiation impinges on a surface of material some light will scatter backward. Reflection, usually refers to a surface reflection not internal or external reflections within the medium. Hecht also clarified this issue for the gradual change in the mediums. In this situation, there is a continuous change of interface and very small amounts of reflection will be observed since the interface is effectively vanished. In contrast, any abrupt change in the interface exerts a discontinuity to the light intensity since a portion is reflected. Internal and external reflections will be discussed later in this section. There is no difference between the concept of reflection and reflectance in the literature, and it is possible to use reflectance in place of reflection or vice versa. However, in some of the references the meaning of reflectance and reflection are separated. For instance, Rea (2000) clearly delineated their boundaries by defining the reflectance as the ratio of reflected flux to the incident flux, while the reflection is a general term for the process by which the incident flux leaves a medium from the incident side or with a reflection angle less than 90 degrees, and without any change in frequency. According to CIE (1987), reflectance is the ratio of the reflected radiant to the incident flux that is radiated on the sample. Reflectivity is generally known as reflectance since it refers to the 30 incident electromagnetic power that is reflected at an interface. However, there is a slightly difference. The bureau of CIE (1987) specified a definition for reflectivity which is distinguished from reflectance. They suggested that reflectivity is the reflectance of a layer of the material with a thickness that does not change the reflectance if it is increased. This means that the sample or object is thick enough that the reflectance is not influenced by adding more thickness to it. This definition for reflectivity includes both internal scattering and surface reflectance. In other words, reflectivity is the restrictive value of reflectance as the surface becomes thick; it can be attributed to the inherent reflectance of the surface. McCluney (1994) claimed that sometimes it would be desirable to limit the definition of reflectivity to include only interface effects. By interpreting this definition, one can say that reflectivity is property of material, while reflection is the property of a particular sample of that material or surface. Dilaura (2011) deduced that reflection is affected by the geometry, wavelength, and polarization of the incident flux. There are three types of surface reflections depend on the radiation flux; specular, mixed, and diffuse reflection. Specular reflection happens on a polished material when the beams are reflected specularly. Specular reflection, also called regular reflection, occurs when all the parallel incident flux is reflected with reflection angles equal to the incidence angle (McCluney, 1994). Hecht (2002) pointed out that the specular reflection process occurs when the light remitted by millions of atoms combines as a single beam. As is obvious from Figure 2.3, what is important is that the scattering effects are absent in the specular reflections. In this Figure, specular angle and incident angles are shown. Specular angle is the angle between the reflected ray and the normal to the surface. For regular (specular) reflections these two angles are equal. Perfectly specular reflections have no diffuse element. 31 Figure 2.3 The geometry of the light generator, detectors, and the sample Another type of reflectance is diffuse reflection. Diffuse reflectance or irregular reflection occurs on samples and materials that consist of particles and crystals or have rough surfaces. Each incident ray encounters an infinitesimal particle which obeys the basic definition and rule of reflection. However, since the shapes of particles in a solidliquid mixture may not be uniform, there are some particles that do not have flat surfaces. Therefore, the reflected rays leave the particle surface in other planes with many different reflection angles and distinct directions. This process is depicted in Figure 2.4(a) where the incident beam strikes a particulate sample but it reflects diffusely in many different angles and in different planes. The intensities of reflected rays are not constant as is shown in Figure 2.4(a) and (b). The illuminated spot has the most energy and the reflected ray from this point has the greatest intensity, then it diminishes as the viewing angle to any side increases (Juds, 1988). Moreover, some of the energy of the incident illumination is attenuated by being transmitted through the sample or being absorbed by the liquid or solid particles inside the slurry which changes into heat and rises the temperature. A beam of parallel rays incident on a surface or medium that has a diffusely reflecting characteristic will be converted to a collection of rays distributed over a hemispherical solid angle. The diffusivity of materials are different. Some produce strong forward scattering whereas others diffuse the light almost in all directions. McCluney (1994) emphasized that if the reflection radiance is constant for all directions, then the surface type is called a Lambertian surface. The term diffuse can also be used for the nonLambertian surfaces. Perfectly diffuse reflectance is special case 32 for the diffuse reflections. This phenomenon occurs when the density and distribution of reflected beams are higher near the surface normal and decreases as the cosine of the angle of the reflected direction. An important point is that perfect diffusion does not mean perfect reflection; it does not mean that the reflection is 1.0. Most surfaces have a mixture of diffuse and specular characteristics. Rea (2000) described this feature on some special surfaces which diffuse most of the light in all directions except for a ray that reflects specularly. Therefore, this type of reflection is partly regular (specular) and partly diffuse. In real situations, a specularly reflecting material has less quantified capability to diffuse the light than reflect it specularly. On the other hand, a diffusely reflecting material reflects the light specularly less than the amount that it diffuses, quantitatively (McCluney, 1994). (a) (b) (c) Figure 2.4 (a) Diffuse reflection from a sample composed of particles (SPECAC Limited, n.d.); (b) Specular versus diffuse reflectance on the surface of a slurry fluid. (c) Diffuse and specular reflectance. Diffuse reflectance (left) sends beams in all directions regardless of the uniform parallel incident rays. Specular reflectance (right) incident cone is preserved (Dilaura, 2011) 33 All types of reflections which occur at interfaces including specular, diffuse, and mixed reflections, are external reflections. Total reflection has two categories; external and internal reflections. Considering the passage of an incident beam from medium 1 (incident medium) to medium 2 (transmitting medium), one can construe the occurrence of an external reflection when the refractive index of the first medium is less than the one for the second or transmitting medium. The reason is that the light is initially traveling from a less to a more optically dense medium. This phenomena happens since the compression of atoms in a dense material does not allow the photons to find a way through the medium, hence the reflection at the interface is strong. In fact, specular and diffuse reflections which are surface reflections are categorized as external reflections. On the other hand, internal reflection occurs when the light going from a region with high refractive index to a region with lower index. Dahm and Dahm (1999) also defined internal reflections as light that has been transmitted through one or more particles, returned from an internal particle, and transmitted through any particle encountered on the way back to the detector. In some occasions, neither reflection nor transmission takes place when incident beam encounters the interface. From a two dimensional perspective, the beam reflects exactly at the line of the interface or in a 3D view it will be reflected in the plane of the interface (Pedrotti et al., 2007). In this thesis the remission is used as the reflection in agreement with the notation used in the literature. Remission is the summation of external reflection, internal reflection, and backward scattering (Dahm & Dahm, 1999). The remission fraction is the ratio of the remitted intensity to incident light; the light that strikes a particle (Dahm & Dahm, 2007). 2.3.13 Scattering Scattering is a form of propagation of electromagnetic waves in a way that energy is attenuated through the scattering medium and remission to many directions (Pedrotti, 2007). Therefore, the scattering will be a result of deviation of light from the medium. Scattering is also called diffuse 34 scattering by Dahm and Dahm (1999). They referred to a process in which light interacts with a particle and changes its path in many directions regardless of the orientation of particle surface. This light is capable of being absorbed or remitted immediately. Mie theory is a powerful tool to exactly predict the scattering. The intensity of scattered light relies on the particle geometry (Dahm & Dahm, 1999). For inferring the difference between scattering and diffuse reflection, Kortum (1969) clarified this very important issue. Before going through the difference, a concise explanation about the dependency of particle size and wavelength is required. Reflection, refraction, and diffraction are three different elements by which the interaction of the wave with a particle is characterized. These three phenomena occurs differently if the particle size is greater than the wavelength of the electromagnetic wave. However, if the dimension of the particle is less than or comparable to the wavelength of light, then it would be impossible to separate the contribution of the concepts of reflection, refraction, and diffraction during the collision of waves with particles. This phenomenon is known as scattering (Kortum, 1969). In contrast, diffuse reflection is the isotropic angular distribution of light in densely packed materials. Single and Multiple Scattering It is very important to introduce two different phenomena about scattering; single and multiple scattering. Single scattering happens when radiation is scattered locally by a local center. If the numbers of scattering centers are high, the scattering will occur many times which is called multiple scattering. The difference between the single and multiple scattering is that single scattering is similar to a random phenomenon; however, multiple scattering is less random. The location of single scattering center is not determined, therefore, it tends to depend on the incident trajectory, which looks random to an observer. Single scattering is described by probability distributions (Bohren, 1998). 35 The reason that multiple scattering is less random than single scattering is because of the averaging of a large number of scattering events. Hence the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by a light beam passing through thick fog. Multiple scattering is analogous to diffusion. Optical scattering centers designed to produce multiple scattering are thus known as diffusers. Forward and Backward Scattering Forward and backward scattering are two types of scattering events. Forward scattering refers to the scattered beams that leave a sample travelling in a direction that produces a component with the same direction as the incident beam. Backward scattering is defined as the scattered light that leaves a sample in the opposite direction of the radiation. 2.3.14 Reflection (remission) Coefficient Reflection coefficient is defined as the difference between the propagation velocities of the two media forming the interface (Guenther, 1990). He offered the following equation for the reflection coefficient between two media: (2.18) where b is the reflection coefficient and and are the index of refractions for the incident and transmitted media, respectively. Or, Pedrotti et al. (2007) suggested the following formula which is the same as Equation 2.18: (2.19) where n is the relative refractive index. Dahm and Dahm (2007) suggested the following definition for the apparent linear remission coefficient of a sample which is the fraction of light remitted from a small thickness of the material divided by the magnitude of that small thickness. 36 (2.20) where is the remission fraction for light that is reflected from a small thickness of material. Chartier (2005) defined the reflection coefficient as a complex number where its modulus is equal to unity. He deduced this definition based on doing some calculation on Fresnel formulas. For modeling purposes in this thesis the formulation of Dahm has been used since it is consistent with the number of different layers. 2.3.15 Absorptance (absorption) In physics, absorption of an electromagnetic wave is a phenomenon in which the energy of a photon in the ray is taken up by the material. This energy is consumed in a nano scale by the electrons of an atom. The activities of the electrons due to radiation heat up the matter. In spectroscopy, absorptance, absorbance or absorption, also called optical density of a material is the logarithmic ratio of the radiation strike on a material to the radiation transmitted through a material. In the literature, the absorption of light through a material has been also called attenuation. When speaking of the linear absorption, the dependency of the attenuation is independent of the intensity of the beam, but in certain conditions, the medium changes its transparency due to the intensity of waves going through, and nonlinear absorption happens (Venkatram, Rao, & Akundi, 2005). McCluney (1994) defined the absorptance as the ratio of the absorbed radiant to the incident flux which has the symbol . Absorbance measurements are often carried out in analytical chemistry. Considering a light beam having an intensity impinging on a sample and an intensity incident on the detector, Dahm and Dahm (2007) suggested that absorbance function is the negative logarithm of the fraction of the incident intensity that strikes a detector. According to this definition the following formula is derived: (2.21) 37 where A is the absorption fraction is the incident intensity, and is the intensity exits the sample. Sometimes it is said that the absorbance is defined for the case of transmission through nonscattering sample as , where T is the transmission fraction (Dahm & Dahm, 2007). In contrast, the absorption function has certainly been useful in the field of nearinfrared reflectance spectroscopy in which it is defined as , where R is the remission fraction of light. But what is suggested by Dahm and Dahm (2007) and is used in this thesis is the meaning of absorbance function for the case of transmission. The reason for using this interpretation of absorption is because of instruments which use logbased absorbance. Dahm and Dahm (2007) stated different usages of logbased transmission and reflection cases. In a transmission measurement, since the fraction of light transmitted is calculated from , absorbance function can be written in the form of . In the experiments that are based on remission, the absorbance is calculated in terms of remission in the form of: (2.22) where R is the remission fraction measured from incident light. However, the exponential form of the absorption fraction is used for quantitative purposes. This form will be introduced completely in Chapter Three. 2.3.16 Absorption Coefficient The traditional definition for the absorption coefficient is the ratio of absorbed light by a thin layer in the material to the length of that small thickness (Dahm & Dahm, 2007). The following equation represents the absorption coefficient for a linear type of material: (2.23) K is the absorption coefficient, d is the thickness of the small layer of sample, and A is the absorption fraction of light inside the medium. An important point about the absorption is related to nonlinear 38 situations. When a specific thickness of the sample for which the absorption fraction is calculated is thick – the thickness criterion is explained in detail in Chapter Three – then the absorption fraction is not a linear function of the depth of the layer. Dahm and Dahm (2007) asserted that as the thickness of the layer under measurement gets smaller, the absorption fraction falls in the linear regions. If the sample is a small layer of material, the absorption is called linear and the value will be linearly proportional to the sample thickness. Dahm and Dahm (1999) also described this linearity for the material through their proposed hypothesis. According to their descriptions, the absorption is linear if all internal parts of the material are equally illuminated. Dahm and Dahm (2007) asserted that absorption coefficient is another definition for the absorption power of the material while there is no scattering in the sample. In a homogeneous sample any particular thickness which absorbs light, has a limit of capacity for absorption. The amount of absorption is in constant or linear proportion to the intensity of radiation. This fraction is independent of: the absolute value of intensity; and the location of that special thickness or layer of the sample. This rule is known as BouguerLambert Law in optical physics. The formulation is: (2.24) where k is the absorption coefficient, T is the transmitted intensity, is the intensity of the light a which passes the thickness of the material in the location of line 0 which is shown in Figure 2.5, and is the intensity of light at the point d, d is the distance between two lines, and d is the absorbing power of the material. Penetrating through the sample, some portion is absorbed and some will be transmitted. Pierre Bouguer (1760) discovered the law of absorption of light through the material, and later it was developed by Johann Lambert (Dahm & Dahm, 2007). This law describes the absorbed light by a homogenous material as a linear fraction of intensity on such a layer. The limitation of this technique is its homogeneity. 39 The absorbing power of material is known as the negative of the natural log of the intensity transmitted through a particular distance d of the sample (Dahm & Dahm, 2007). The absorbing power of a material is the same as the absorbance of a homogeneous sample of the material of unit thickness. For example, the absorbing power of a material is the same as the absorbance of a homogeneous sample of the material that is 1 cm thick: (2.25) Figure 2.5 Illustration of BouguerLambert law Therefore, the absorbing power of the material or k is calculated by the absorbance of the homogeneous sample of the material. It should be noticed that kd does not have any units. If one divides the relation above by the distance that light travels, d, the result will be the absorption power which is derived from the following equation: (2.26) d k is independent of thickness 40 For a portion of sample that is under direct illumination, the absorbing power of the material is the ratio of the amount of light that would be absorbed if every absorption portion were under incident intensity equal to the incident ray, to the intensity of the incident beam (Dahm & Dahm, 2007). The thickness of the layers is calculated in the units of . In this thesis will refer to the absorption coefficient of a sample and is the absorption coefficient of a material which is also known as absorption power of material. 2.3.17 Transmittance (transmission) Transmittance is the fraction of electromagnetic wave at a specified wavelength that passes through a sample. A more concise definition is the movement of light through a sample of material without being absorbed (Dahm & Dahm, 1999). The important point about transmission is that it is incorporated with the fact that suggests the amount of light that is not absorbed, is reflected or transmitted. The only difference between the reflection and transmission is the angle of incidence and the angle that light makes with the normal to the surface of interface. If the light is transmitted through the medium with an angle less than the critical angle – at which the remitted beam falls exactly on the interface line – the light is transmitted, whether internally within the layers of sample or externally outside of the medium. In this situation the light is refracted and it is transmitted to the other layers inside the sample or it is coming out of the sample. However, when the incident light strikes the sample at an angle more than the critical angle, the beam that is leaving the medium is said to be reflected. Therefore, the boundary between the reflectance and transmittance is very close. Dahm and Dahm (2007) pointed out that the transmission fraction is calculated from the following equation which is derived based on the Bouguer – Lambert law for transmission. The law says that the fraction of light beam transmitted through the sample while moving through the distance d inside a continuum medium is calculated from: 41 (2.27) where k is the linear absorption coefficient in the absence of scattering inside the sample. Considering this limiting assumption, one can conclude that this expression is designed for the samples with high concentration of particles. The reason is that by increasing the number of particles the chance of light scattering will increase, which is not consistent with the assumption of this law. But fortunately this equation is valid for the small particles according, as shown in Figure 2.6. The straight line in Figure 2.6 is a tangent line to the relation that represents the deviation of transmission curve from the line C with the relation kd. This line can be used instead of the exponential function in the linear region. The linear region is essential to be found since the validity of some theories such as BouguerLambert law and plane parallel layers are dependent upon this assumption. In Chapter Three, this linearity assumption is introduced. Furthermore, least squares method to find threshold for the minimum deviation from the straight line is presented not only for transmission but also for absorption. Figure 2.6 (A) Absorption fraction; (B) transmission fraction ( ; (C) straight line representing Kd which lies on the transmission curve in smaller thicknesses of the sample or layer of the sample which is illuminated (Dahm & Dahm, 1999) 42 2.4 Continiuum Theories of Diffuse Reflection 2.4.1 Introduction Nearinfrared (NIR) diffuse reflection (DR) has become extensively applicable for the quantitative analysis of many products and commodities since it 40 years ago when it was first used in spectroscopy. As was mentioned in Chapter 1, midinfrared diffuse reflectance is used more for investigating and characterizing than for the determining the quantity of particulate samples. Considering the mentioned cases, investigating the diffuse reflectance process is useful if the very large amount of information is to be collected from the spectrum. Those theories of diffuse reflectance that use simple mathematics has connections with the reflectance of a sample which is known by . In this situation, the assumption which says all light being reflected from both a sample and a a standard reflecting reference with 100% reflectance can be measured (Chalmers & Griffiths, 2002). Chalmers and Griffiths (2002) suggested that the relative diffuse reflectance can be measured by estimating the ratio of the spectrum of the radiation received by the detector from a particulate sample and the corresponding spectrum which can be estimated experimentally after reflection from a nonabsorbing reference matter. In the cases which the material thickness is large, any change in the thickness does not influence on the power of the radiation which is going to be received by the detector. Therefore, the power will not change. The measured spectrum is called the reflectance spectrum at infinite depth, (Chalmers & Griffiths, 2002). Chalmers and Griffiths (2002) asserted that all algorithms for multivariate analysis obtain the most accurate calculations about the concentration when the intensity of each spectral parameter is proportional to the concentration of the anlayte in a linear fashion. Hence, diffuse reflectance will be changed into the linear form if an ideal standard for the analysis of diffuse reflectance is required by applying theories. In the linear condition, the dependence of intensity of the spectrum on the concentration of each analyte is linear which is analogous to the conversion of transmittance spectra to 43 absorbance for considering Beer’s law (Chalmers & Griffiths, 2002). For many cases in nearinfrared diffuse reflectance spectrometry, a conversion which is equivalent to what is mentioned is performed. For instance, before applying the theorems to the sample and model the light interactions, one may require to change the spectrum in to the form of –log or log . It is significant to mention that most NIR/DR spectra is usually estimated using log1/R which, for thick samples, is the same as log (Chalmers & Griffiths, 2002). Fortunately, for many materials the concentration of the sample under anlysis will not change most of the time by even a factor of two. Therefore, the linear assumption is valid and the deviation from linearity is not considerable. Hence, Chalmers and Griffths (2002) concluded this approach usually have excellent results. Applying the parameter of log 1/R as the preferred ordinate for nearinfrared diffuse reflection is in opposite side to what most physical scientists would consider appropriate for experimental part of a diffuse reflectance measurement (Griffiths & Dahm, 2008). The theory of Kubelka and Munk is one of the mostly used theories of diffuse reflectance spectroscopy. In this Chapter, this theory is introduced with a summary of other continuum theories of diffuse reflection. The assumptions of the theories are very important since they produce limitations. These constraints only affect the accuracy of the results. However, all of these methods have been validated through experiments. The last part of this thesis contains a discussion of some studies and measurements about the depth of penetration of light. The information and models are investigated based on the available theories in the literature. Modeling the penetration depth in a twophase flow of xylene and ADP is achieved using the theorems, which are to be discussed in this Chapter. 2.4.2 Lambert Cosine Law The diffuse reflectance phenomenon is a widely applicable usages of nearinfrared radiation, which is easily noticed in everyday life. Griffiths and Dahm (2002) made an example for the intensity of radiation reflected from a completely matte surface. The reflected or remitted radiation has equal 44 intensity in all directions. Also, it is not important to know what is the angle of incidence or angle of observation of light from the surface. This experiment made Lambert to be the first person who created a mathematical formula and theory based on the diffuse reflectance of light (Chalmers & Griffiths, 2002). He assumed that the reflected flux which is known by , in an area , and solid angle steradians (sr), is proportionate to the cosine of the angle, which the incident beam builds with the normal to the surface, , and the angle of observation that are shown in Figure 2.7. This is obtained from the Equation 2.28: (2.28) Figure 2.7 Diagram showing the variables used in the Lambert cosine law (Griffiths & Olinger, 2002) where B is the radiation density or surface brightness in W , is the irradiation intensity in W for normal incidence, and the constant C is the fraction of the reflected intensity of radiation. C is less than unity because some portions of radiation flux are always absorbed with the surface. Equation 2.28 is famous since it is known as the Lambert cosine law (Chalmers & Griffiths, 2002). Kortum (1969) suggested that the Lambert cosine law can be derived from the second law of thermodynamics, although this opinion was not held by Wendlandt and Hecht (1966). Kortum (1966) pointed out that it is rigorously valid only for a material which has the characteristics of a blackbody. 45 These black body radiators acting as an ideal diffuse reflector. Griffiths and Dahm (2008) pointed out that the angular distribution of the reflected or remitted light is not dependent of the angle of incident radiation. This fact is in contradiction with tehe features of blackbody radiator since ideal diffuse reflectors has some different characteristics. This contradiction is made since all incident radiation is absorbed by a black body, however, an ideal diffuse reflector does not absorpb all the radiation. An ideal diffuse reflector is not realistic and has never been found in practice. Hence, the Lambert cosine law is an ideal descriptor of diffuse reflection which always there are deviations from that. According to what Chalmers and Griffiths (2002) stated, there are many authors that have reported the results of experimental investigations that were prepared to prove or disprove this law. They found that in the law is only valid when both the angle of incidence and the angle of observation are small. 2.4.3 Mie Scattering One of the most applicable and more accepted theories of the scattering of light was developed around by Mie. Mie scattering theory models the travel of light into the isolated particles, and predicts the absorption by and scattering from that particle. Only a very brief introduction is given here. Kortum (1969) has pointed out a summarized description about Mie scattering theory. Mie developed and derived a model to predict the following factors: (i) the angular distribution of the intensity, and (ii) the angular distribution of polarization of scattered radiation for a plane wave which is scattered once through a phenomena called single scattering by a particle (Griffiths & Dahm, 2008). These two phenomena can be both dielectric and absorbing. Mie assumes that the particle is spherical, and there is no limit on its size. He presented that the angular distribution of scattered radiation for single scattering will not be isotropic. The basic equation of Mie scattering theory is: (2.29) 46 where , is the scattered intensity at a distance R from the center of the sphere; is the intensity of the incident radiation, and is the wavelength of the incident radiation (Chalmers & Griffiths, 2002). The variables and are functions of the angle of the scattered radiation , the refractive index difference between particle and medium, and the ratio of the particle circumference to wavelength. The ratio of the refractive index of the sphere to its environment is known by m, and the ratio of the particle circumference to wavelength is known by p (Chalmers & Griffiths, 2002). Equation 2.29 applies only to the case of a dielectric nonabsorbing particle. If the particle absorption is high, the complex refractive index must be used in the determination of and . Mie theory is general for spherical particles of any size. However, it is valid only for single scattering, therefore, is applicable only to systems in which particles are well separated (Chalmers & Griffiths, 2002). For instance, when the particles in the atmosphere scatter light, the special case of Mie theory is obtained where the particle is much smaller than the wavelength of incident radiation. Most analyses in the range of midinfrared and nearinfrared spectroscopy the multiple scattering is a definite assumption since it happens in real samples (Chalmers & Griffiths, 2002). Theissing (1950) improved Mie theory by assuming that the multiple scattering from particles which nonetheless are still supposed to be sufficiently well dispersed and separated in the medium. Therefore, the interference and phase differences between the scattered fluxes from the particles are not important and can be neglected. The number of times when a photon is scattered and distributed in the medium is defined as the scattering order (Chalmers & Griffiths, 2002). By increasing the order of scatter the value of forward scattering in the sample is doing an opposite behavior and will decrease, therefore, the angular distribution of scattered light does not change with direction and in all directions is isotropic. He also discovered that the if the ratio of particle circumference to wavelength is larger the order of scattering must increase to generate an isotropic scattering. For example, if p is 0.6 and m is 1.25, twofold scattering is required for an isotropic distribution of the reflected radiation. However, 47 while p = 5 and m = 1.25, a scattering order of 8 is needed for isotropic reflection of radiation (Chalmers & Griffiths, 2002). In the NIR region, band absorptions are so low that large particle sizes, of the order of 100 microns, can be calculated using Mie theory (Griffiths & Dahm, 2008). Hence, p, the ratio of the particle circumference to wavelength, is large. In this case, the order of scattering must also be large in order to have an isotropic variation of the scattered radiation. In the midinfrared range, this feature is changed where it is quite common for p. Hence, it will be approximately around 1 (Griffiths & Dahm, 2008). If the number of particles in a sample is large, multiple scattering does take place for most samples of the type used for infrared reflection analysis, therefore, the diffusely reflected radiation has an isotropic distribution that should at least be reached. Hence, to predict multiple scattering in a densely packed medium is required to model the variation of reflectance with the change in concentration (Griffiths & Dahm, 2008). In most of the samples for which the nearinfrared diffuse reflection can be used, the scattering density is large, the ratio of particle circumference to wavelength is higher than 1, and the concentration of particles are high enough that phase relations and interferences between scattered beams do exist (Chalmers & Griffiths, 2002). Hence, in this case, there is no solution to the problem of multiple scattering in the sample. Therefore the scientist must resort to the phenomenological analysis may be required. 2.4.4 Radiation Transfer Treatments Theories that have been developed to predict the diffuse reflection of radiation are built based on the radiation transfer equation. A radiation transfer equation can be written as if one simplify it: (2.30) 48 Equation 2.30 is a description of the change in intensity dI of radiant flux in a specific wavelength in a sample with the density and with the pathlength of ds. describes the attenuation coefficient for the total radiation loss. This dissipation can be due to scattering or absorption. The radiation transfer equation that is used in the derivation of most phenomenological analyses considers only plane parallel layers of particles within the sample and can be written as: (2.31) where is the cosine of the angle with respect to the outward surface normal; m is the cosine of the angle with respect to the inward surface normal; is the optical thickness and is equal to , where dx is the distance between the boundaries of one planeparallel layer; I is the intensity of the beam of radiation striking the layer; is the albedo for single scattering, with the scattering and absorption coefficients and (Chalmers & Griffiths, 2002). The scattering phase function is the probability for scattering from direction into . If each beam scatters isotropically, and is not dependent to the angle between the incident radiation and the scattered radiation. The phenomenological analyses that have been developed can be deemed as continuum theories or discontinuum theories (Chalmers & Griffiths, 2002). In continuum theories, the absorption and scattering coefficients are the properties of the irradiated material. However, in discontinuum theories a layer of the material is made up of several partial layers whose thickness is limited by the size of the scattering and absorbing particles. It means that based on the size of particles the thicknesss of the layers must change (Chalmers & Griffiths, 2002). Optical constants can then be found from the scattering and absorption of these particles. This section only considers a sample as a continuum. In our modeling, we have used the method of approximation for the radiation transfer equation proposed by Patterson, Schwartz, and Wilson (1989). Figure 2.8 is an experimental apparatus that they used in their method. When the light reflects diffusely from the tissue containing a dye, the 49 changes in the reflectance is identified by the concentration of dye. In Figure 2.8, the measurement setup is shown. In this figure, an experiment based on radiative transfer approximation is illustrated. Figure 2.8 (a) Experimental apparatus for the measurement of R( ), the local diffuse reflectance (b) Apparatus for the measurement of R, the total diffuse reflectance (Patterson et al., 1989) 2.4.5 Schuster’s Theory Schuster’ theory was one of the earliest theories of diffuse reflectance. He was interested in astrophysical problems that were related to radiation transfer. From the general laws of radiation, Schuster suggested that an atmosphere is required for a star to show emission lines of light instead of absorption lines. The characteristics of this atmosphere is that the scattering overcomes the absorption. As mentioned by Nobbs (1985), the approach adopted by Kubelka and Munk was first introduced by Schuster in 1905 and was developed to calculate the radiation from the selfluminous opaque dust cloud surrounding a star. This theorem will be modeled with details in Chapter Three. 2.4.6 KubelkaMunk Theory KubelkaMunk is a widely used theory in the field of diffuse reflectance. The theory was basically developed in the field of spectroscopy. However, recently there were many debates on the revisions 50 over it to increase its accuracy and the number of different applications. The theory developed by Kubelka and Munk is obtained based on solutions to the radiative transfer problem. This theorem was first used to investigate the optics of paint layers. Kubelka and Munk developed their models based on the remission, not scattering. Kublka – Munk equations are produced from radiative transfer and are in the form of differential equations. The KubelkaMunk equation is a two flux version of a multiflux method used to solve the radiation transfer problem (Nobbs, 1985). Kubelka and Munk made several assumptions in their derivation of a simplified solution to the radiation transfer equation. Note that none of them are exactly the same as the corresponding parameters defined by Schuster. Figure 2.9 shows the type of system for which Kubelka and Munk derived their solution. Figure 2.9 Schematic representation of a layer of absorbing and light scattering particles The assumptions are listed below: The radiation fluxes ( and ) travel in two opposite directions. 51 The sample is illuminated with monochromatic radiation of intensity I0. The distribution of scattered radiation is isotropic so that all regular (specular) reflection is ignored. The particles in the sample layer (defined as the region between x = 0 and x = d) are randomly distributed. The particles are very much smaller than the thickness of the sample layer d. The sample layer is subject only to diffuse irradiation. The breadth of the macroscopic sample surface (in the plane) is great compared to the depth of the sample and the diameter of the beam of incident radiation (to discriminate against edge effects). The scattering particles are distributed homogeneously throughout the entire sample. Either an exponential or hyperbolic solution may be developed, although only the exponential derivation is shown here. A detailed description of the hyperbolic solution is given elsewhere. Kubelka and Munk arrived at the two fundamental differential equations which, once solved, give the simplified solution to Schuster’s fundamental function. As it is feasible that the angle through , , which the radiation might follow could be between and , the average pathlength for radiation passing through dx in the +x direction can be found by the integral: (2.32) After some mathematical operations and manipulations, which are discussed in detail in Chapter Three, one can find the two fundamental differential equations for KubelkaMunk theory: (2.33) (2.34) 52 2.4.7 Diffusion Theory Kienle et al. (1998) presented a different approach to investigate the radiation transfer equation. This approach is applicable in biomechanics and biomedical engineering. In the paper that is published by Kienle et al. (1998), the diffusion approximation is used to determine the optical properties of biological tissue. Griffiths and Dahm (2008) pointed out that the propagation of photons in a thick sample with scattering coefficients which are larger than the absorption coefficients, are described by diffusion approximation. The details of this approach are described in Chapter Three and the results are discussed in Chapter Four. 2.5 Discontinuum Theories of Diffuse Reflection 2.5.1 Representative Layer Theory A layer is representative of a sample if each one of the particles in the sample provide the same volume and surface area fractions in each layer as in the sample (Dahm & Dahm, 2007). The reason for using the volume fraction of the particles is because the absorption fraction of the particle is completely related to volume. On the other hand, the remission is strongly dependent on the surface area fraction of the particle. The important question is what makes a layer be representative of a sample. Dahm and Dahm (2007) demonstrated three reasons for a layer to be representative. First, the thickness of the layer should have no more than the thickness of one particle since one of the assumptions of representative layer theory was related to the scattering and absorption properties of the layer by combining the mentioned properties for each particle in the layer. Therefore, it would be more suitable to use the thickness for the layer to be nowhere more than one particle thick. Second, another feature of the representative layer introduced by Dahm and Dahm (2007) is that the volume fraction of each particle in the layer is the same as the sample. One of the most 53 important assumptions of representative layer theory is related to the voids. Thus, the media that represents the voids does not absorb or scatter light. This means that if the media is a fluid its absorption and scattering must be neglected. Moreover, the void fraction of the representative layer must be the same as the sample. Finally, the third reason is that the surface area fraction of each particle in the layer will be the same as the sample. Surface area fraction is a property that is more difficult to work with than volume fraction because the shape and size of the particles are required to estimate the total surface area fraction. In the model of the representative layer that we present here, a particle is of a different type from another
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Title  Computational Approaches to Optical Modeling Nir Radiation Phenomena in Slurries with Arbitrary Solids Concentration Profiles 
Date  20121201 
Author  Mohammadi Ziazi, Reza 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  Near infrared sensors project light into particleladen flows and use light returned via diffuse reflectance and scattering phenomena to measure solids concentration and chemical composition. An important measurement parameter is the depth of penetration returning the light used in the measurements. In this work several optical modeling approaches were evaluated for predicting the depth of penetration of nearinfrared (NIR) light in slurry flow in a pipe. The goal was to find a model suitable for predicting the depth of penetration for slurry flows with arbitrary solid concentration profiles. The models which were considered fall in two categories; continuum and discontinuum theories. Continuum models consider the sample as one integral layer while discontinuum models divide the sample into layers one particle thick. Hence, discontinuum models are well suited for slurries with varying concentrations which can be adjusted layer by layer. The continuum theories used were Shuster, KubelkaMunk, radiation transfer, three flux, and diffusion. The discontinuum theories used were plane parallel layer, assembly of sheets, and representative layer theory. Discontinuum models were applied to cases which included uniform, hypothetical, and realistic concentration profiles predicted using Computational Fluid Dynamics (CFD). For the CFD predictions, the slurry was xylene liquid and ADP solid flowing in a 50.8 mm diameter horizontal pipe at mean velocities between approximately 1.8 and 4.0 m/s, corresponding to Reynolds numbers between 1.2x105 and 2.8x105. Monodisperse spherical slurry particles with diameters ranging from 38 to 150 microns were considered with solid volume fractions ranging from 5% to 35%. Numerical versions of the optical models were created for the theories using MATLAB R2012a. Each model simulated the passage of NIR radiation through the sample based on light fractions. Light fractions were computed for each layer of the discontinuous models, and the depth of penetration was evaluated. Validation was performed with results in the literature for optical experiments for similar cases. Comparisons of the results suggest that the discontinuum theories are more accurate than continuum theories for these applications, with representative layer theory used with Stokes formula providing the best results. It is concluded that the models are suitable for predicting the depth of penetration for NIR measurements. 
Note  Thesis 
Rights  � Oklahoma Agricultural and Mechanical Board of Regents 
College  Engineering, Architecture, and Technology 
Transcript  COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES By REZA MOHAMMADI ZIAZI Bachelor of Science in Aerospace Engineering Tehran Polytechnic University Tehran, Iran 2009 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2012ii COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES Thesis Approved: Dr. Frank W. Chambers Thesis Advisor Dr. Andrew S. Arena Dr. Bruce J. Ackerson iii ACKNOWLEDGEMENTS It is like a dream come true for me pursuing my Master’s degree in Mechanical engineering and doing critical research here in a fast developing school like Oklahoma State University. However, this dream was made true only because of continuous support, motivation, and kindness I received from my helpful advisor Dr. Frank W. Chambers. I have had the opportunity and pleasure of working with and learning from an advisor who not only cares to share his precious knowledge, but also motivates the students with his endless new ideas. I express my gratitude toward Dr. Chambers who I have been most comfortable consulting and asking questions. His helpful suggestions and comments will remain with me for my life. So thank you, Dr. Chambers, for everything. I would like to thank Dr. Andrew S. Arena Jr. and Dr. Bruce J. Ackerson for being in my thesis committee, guiding me through preparation of this thesis, and during my master’s. Also, I appreciate their time and patience for reviewing the thesis document and attending my examination. I also thank my lab partner Shyam, and thank all my friends specially Mehran, Amir, Nabil, and Hossein who supported me during the preparation of this thesis. I Studying away from your family sometimes can get really hard, but one thing made it easier for me; my parents and my sister. Thank you for your love, support, and kindness. Acknowledgements reflect the views of the author and are not endorsed by committee members or Oklahoma State University.iv Name: REZA MOHAMMADI ZIAZI Date of Degree: DECEMBER, 2012 Title of Study: COMPUTATIONAL APPROACHES TO OPTICAL MODELING NIR RADIATION PHENOMENA IN SLURRIES WITH ARBITRARY SOLIDS CONCENTRATION PROFILES Major Field: MECHANICAL ENGINEERING Abstract: Near infrared sensors project light into particleladen flows and use light returned via diffuse reflectance and scattering phenomena to measure solids concentration and chemical composition. An important measurement parameter is the depth of penetration returning the light used in the measurements. In this work several optical modeling approaches were evaluated for predicting the depth of penetration of nearinfrared (NIR) light in slurry flow in a pipe. The goal was to find a model suitable for predicting the depth of penetration for slurry flows with arbitrary solid concentration profiles. The models which were considered fall in two categories; continuum and discontinuum theories. Continuum models consider the sample as one integral layer while discontinuum models divide the sample into layers one particle thick. Hence, discontinuum models are well suited for slurries with varying concentrations which can be adjusted layer by layer. The continuum theories used were Shuster, Kubelka–Munk, radiation transfer, three flux, and diffusion. The discontinuum theories used were plane parallel layer, assembly of sheets, and representative layer theory. Discontinuum models were applied to cases which included uniform, hypothetical, and realistic concentration profiles predicted using Computational Fluid Dynamics (CFD). For the CFD predictions, the slurry was xylene liquid and ADP solid flowing in a 50.8 mm diameter horizontal pipe at mean velocities between approximately 1.8 and 4.0 m/s, corresponding to Reynolds numbers between 1.2x105 and 2.8x105. Monodisperse spherical slurry particles with diameters ranging from 38 to 150 microns were considered with solid volume fractions ranging from 5% to 35%. Numerical versions of the optical models were created for the theories using MATLAB R2012a. Each model simulated the passage of NIR radiation through the sample based on light fractions. Light fractions were computed for each layer of the discontinuous models, and the depth of penetration was evaluated. Validation was performed with results in the literature for optical experiments for similar cases. Comparisons of the results suggest that the discontinuum theories are more accurate than continuum theories for these applications, with representative layer theory used with Stokes formula providing the best results. It is concluded that the models are suitable for predicting the depth of penetration for NIR measurements. v TABLE OF CONTENTS Chapter Page I. INTRODUCTION AND BACKGROUND ............................................................1 1.1 Introduction ........................................................................................................1 1.2 Motivation ..........................................................................................................2 1.3 Background ........................................................................................................4 1.3.1 NearInfrared Technology ........................................................................5 1.3.2 Slurries ......................................................................................................7 1.4 Statement of the Problem ...................................................................................9 1.5 Goals ................................................................................................................10 II. PAST CONTRIBUTIONS ...................................................................................13 2.1 Introduction ......................................................................................................13 2.2 History of past contributions ............................................................................14 2.3 Fundamental Definitions ..................................................................................18 2.3.1 Geometrical Optics .................................................................................19 2.3.2 Wavefront and Ray .................................................................................19 2.3.3 Radiant Energy ........................................................................................21 2.3.4 Solid Angle .............................................................................................21 2.3.5 Radiant flux .............................................................................................22 2.3.6 Radiance ..................................................................................................23 2.3.7 Irradiance ................................................................................................23 2.3.8 Radiant Intensity .....................................................................................24 2.3.9 Refractive Index ......................................................................................24 2.3.10 Diffraction .............................................................................................26 2.3.11 Refraction ..............................................................................................26 2.3.12 Reflectance (reflection) and reflectivity ...............................................29 2.3.13 Scattering ..............................................................................................33 2.3.14 Reflection (remission) Coefficient ........................................................35 2.3.15 Absorptance (absorption) ......................................................................36 2.3.16 Absorption Coefficient ..........................................................................37 2.3.17 Absorption Coefficient ..........................................................................40 2.4 Continiuum Theories of Diffuse Reflection ....................................................42vi Chapter Page 2.4.1 Introduction .............................................................................................42 2.4.2 Lambert Cosine Law ...............................................................................43 2.4.3 Mie Scattering .........................................................................................45 2.4.4 Radiation Transfer Treatments ...............................................................47 2.4.5 Schuster’s Theory ...................................................................................49 2.4.6 KubelkaMunk Theory ............................................................................49 2.4.7 Diffusion Theory .....................................................................................52 2.5 Discontinuum Theories of Diffuse Reflection .................................................52 2.5.1 Representative Layer Theory ..................................................................52 III. OPTICAL MODELING .....................................................................................57 3.1 Overview and Organization .............................................................................57 3.2 Development of Continuum Optical Models ...................................................60 3.2.1 Schuster’s Model ....................................................................................61 3.2.2 KubelkaMunk Model .............................................................................64 3.2.3 Radiation Transfer Approximations .......................................................72 3.2.4 Three Flux Model. ..................................................................................74 3.2.4 Diffusion Model. .....................................................................................75 3.3 Development of Discontinuum Optical Models ..............................................76 3.3.1 Model of Plane Parallel Layers ...............................................................76 3.3.2 Model for Assembly of Sheets ................................................................78 3.3.3 Representative Layer Model ...................................................................82 3.4 Code Implementation .......................................................................................93 IV. RESULTS AND DISCUSSION ..........................................................................98 4.1 Brief Recapitulation .........................................................................................98 4.2 Results of Continuum Optical Models ...........................................................101 4.2.1 Schuster’s Model ..................................................................................102 4.2.2 KubelkaMunk Model ...........................................................................107 4.2.3 Radiation Transfer Approximation .......................................................114 4.2.4 Three Flux Model .................................................................................118 4.2.5 Diffusion Model ....................................................................................122 4.3 Results of Discontinuum Optical Models ......................................................123 4.3.1 Model of Plane Parallel Layers .............................................................124 4.3.2 Model for Assembly of Sheets ..............................................................131 4.3.3 Representative Layer Model .................................................................134 4.4 Comparison of Continuum and Discontinuum Models .................................151 4.5 Depth of Penetration ......................................................................................153 vii Chapter Page 4.6 Summary ........................................................................................................154 V. CONCLUSIONS AND RECOMMENDATIONS ...........................................156 5.1 Conclusion of continuum theories .................................................................157 5.2 Conclusions of discontinuum theories ...........................................................159 5.3 Recommendations ..........................................................................................160 R. REFERENCES ...................................................................................................161 A. APPENDICES ....................................................................................................165 A.1 MATLAB Script for continuum models and theory for assembly of sheets 165 A.2 MATLAB scripts for the Representative Layer Theory ...............................172 A.2.1 Main code of representative layer model .............................................172 A.2.2 MATLAB input script for 38 micron ...................................................180 A.2.3 MATLAB input script for 75 micron ...................................................182 A.2.4 MATLAB input script for 150 micron .................................................186 viii LIST OF TABLES Table Page 1.1 Description of different fields and applications of nearinfrared radiation has already been used in the previous works ...................................................... 3 1.2 Examples of single and multicomponent, multiphase fluids ............................. 8 2.1 Overview of the development of modeling techniques for diffuse reflection. .......................................................................................................... 15 2.2 Variables used in the development of representative layer theory ................... 54 3.1 Variables used in the development of Kubelka and Munk’s simplified solution to the radiation transfer equation. (Neither or are exactly the same as the corresponding parameters defined by Schuster) ............................ 66 4.1 The total reflectance and ratio of absorption and scattering coefficients of radiative transfer equation ( ) which is calculated based on a three flux approximation by Burger et. al (1997) .................................................... 118 4.2 The values of two coefficients of Legendre polynomials for spherical disks suggested by Mudgett and Richards (1971). ......................................... 120 4.3 Description of computing absorption and remission coefficients by extrapolating from a known sample of thickness d to a sample of infinitesimal thickness. ................................................................................... 131 4.4 Refractive index of xylene and ADP .............................................................. 132 4.5 Different cases of hypothetical concentrations ............................................... 136 4.6 Different cases of CFD results ........................................................................ 142 ix LIST OF FIGURES Figure Page 1.1 Schematic representation of divisions of the infrared electromagnetic spectrum. ............................................................................................................. 5 1.2 The geometry of the light generator, detectors, and the sample ....................... 10 2.1 (a) Schematic representation of solid angle. (McCluney, 1994); (b) Solid angle relationships ............................................................................................ 22 2.2 Geometric description for the refracted waves approached the interface at equal time intervals ........................................................................................... 28 2.3 The geometry of the light generator, detectors, and the sample ....................... 31 2.4 (a) Diffuse reflection from a sample composed of particles (SPECAC Limited, n.d.); (b) Specular versus diffuse reflectance on the surface of a slurry fluid. (c) Diffuse and specular reflectance. Diffuse reflectance (left) sends beams in all directions regardless of the uniform parallel incident rays. Specular reflectance (right) incident cone is preserved (Dilaura, 2011) .................................................................................................. 32 2.5 Illustration of BouguerLambert law ................................................................ 39 2.6 (A) Absorption fraction; (B) transmission fraction ( ; (C) straight line representing Kd which lies on the transmission curve in smaller thicknesses of the sample or layer of the sample which is illuminated (Dahm & Dahm, 1999) ..................................................................................... 41 2.7 Diagram showing the variables used in the Lambert cosine law (Griffiths & Olinger, 2002) ............................................................................................... 44 2.8 (a) Experimental apparatus for the measurement of R( ), the local diffuse reflectance (b) Apparatus for the measurement of R, the total diffuse reflectance (Patterson et al., 1989) .................................................................... 49 2.9 Schematic representation of a layer of absorbing and light scattering particles ............................................................................................................. 50 3.1 Diagram of a sample of slurry representing the KubelkaMunk model. There is a random distribution of particles. (Chalmers & Griffiths, 2002) ..... 65x Figure Page 3.2 Reflectance and transmittance of a thin layer of thickness d within a sample ............................................................................................................... 70 3.3 Absorption, reflection and transmission possibilities for a sample composed of two layers .................................................................................... 77 3.4 The flow chart of the model of assembly of sheets .......................................... 81 3.5 Absorption (A), remission (R), and transmission (T) fractions as a function of thickness for the case A1 = 0.1, R1 = 0.1, T1 = 0.8 ......................... 84 3.6 The most linear region for absorption fraction based on the least square error between linear absorption with equation A = K.d and absorption curve derived from plane parallel mathematics ................................................ 85 3.7 Schematic of the algorithm used to computerize the representative layer theory and the Stokes formula for the integral light fractions .......................... 92 3.8 Concentration profiles derived from Fluent simulation for the initial concentration of 5% and Reynolds number of 140000 for three particle sizes of 38, 75, and 150 μm .............................................................................. 94 4.1 The geometry of the light generator, detectors, and the sample ....................... 99 4.2 Illustration of diffuse reflectance of thick layers at different depths through the sample. X values represent the distance from the edge of the sample through its depth. ................................................................................ 102 4.3 Scattering and absorption coefficients for the models of Schuster and KubelkaMunk. X values represent imaginary thickness inside the sample. ............................................................................................................ 104 4.4 Total diffuse reflectance for different initial fraction of light as a boundary condition for the sample. ................................................................ 105 4.5 Diffuse reflections for four different cases of initial fraction of light. ........... 106 4.6 Reflectance and transmittance of a thin layer of thickness d within a sample ............................................................................................................. 108 4.7 Intensity of incident beam through the sample at different initial boundary conditions ........................................................................................ 109 4.8 Intensity of reflected beam from the sample at different initial boundary conditions ........................................................................................................ 109 4.9 Absorption, remission, and transmission fractions calculated from KubelkaMunk function .................................................................................. 110 4.10 The log of reciprocal reflectance plotted as a function of the A/R fraction. .. 112 4.11 The log of reciprocal reflectance plotted versus A/R fraction derived by Bull (1990) ...................................................................................................... 112 xi Figure Page 4.12 Polynomial curve fitted to the log of reciprocal reflectance versus equivalent particle size in units of ........................................................... 113 4.13 Exponential curve fitted to the log of reciprocal reflectance .......................... 113 4.14 Diffuse reflection of radiative transfer approximation for different cosine of scattering angle ........................................................................................... 114 4.15 Diffuse reflection of radiation transfer model of Equation 4.1 for seven different cases of initial fraction of light. Model of Patterson et al. (1989). .. 116 4.16 Diffuse reflection of radiation transfer model for seven different cases of initial fraction of light based on the radiative transfer approximation proposed by Griffiths and Dahm (2008). ........................................................ 117 4.17 Diffuse reflectance for the direct and diffuse illuminations plotted versus the ratio of absorption to scattering coefficients of RTE approximate model developed by Burger (1997). ............................................................... 119 4.18 Diffuse reflectance of direct and diffuse illuminations plotted versus the ratio of absorption to scattering coefficients of KM theory for particles with the shape of spherical disks. .................................................................. 120 4.19 Diffuse reflectance of three different models of diffuse reflectance derived by Griffiths and Dahm (2008)............................................................ 121 4.20 Diffuse reflectance of three different theories of diffuse reflectance derived by Griffiths and Dahm (2008)............................................................ 122 4.21 Diffuse reflection of an isotropic, optically thick sample according to the three flux approximation, the diffusion approximation, and KubelkaMunk different models continuum theories. ................................................... 123 4.22 Absorption, reflection and transmission probabilities in two layers of a sample consisting of n+1 layers with different thicknesses changing sequentially based on a geometrical progression. ........................................... 125 4.23 Local absorption, reflection and transmission probabilities in the sample for initial condition of ................................ 126 4.24 The illustration of linear region in the model of plane parallel layers determined from sum of squared regression. .................................................. 128 4.25 Local absorption fraction for different initial conditions ................................ 129 4.26 Local transmission fractions for different initial conditions ........................... 129 4.27 Refractive index of ADP as a function of wavelength. The data is collected from Refractive Index Database (N.d.) ........................................... 132 4.28 Total absorption fraction through the layers of the sample for different initial conditions. ............................................................................................. 133 xii Figure Page 4.29 Total absorption fractions through the layers of the sample for different initial conditions .............................................................................................. 134 4.30 Hypothetical concentration profiles in a sample with particles of 75 micron ............................................................................................................. 136 4.31 Local and integral absorption fraction for the case of uniform concentrations ................................................................................................. 137 4.32 Local and integral remission fraction for the case of uniform concentrations ................................................................................................. 138 4.33 Local and integral transmission fraction for the case of uniform concentrations ................................................................................................. 139 4.34 Local and integral Absorption fraction for the case of arbitrary linear and parabolic concentration profiles ...................................................................... 140 4.35 Local and integral remission fraction for the case of arbitrary linear and parabolic concentration profiles ...................................................................... 141 4.36 Local and integral transmission fraction for the case of arbitrary linear and parabolic concentration profiles ............................................................... 141 4.37 CFD concentration profiles ............................................................................. 143 4.38 Local absorption fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ........................................ 144 4.39 Local remission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ........................................ 144 4.40 Local transmission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ................................... 145 4.41 Integral absorption fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 .................................... 146 4.42 Integral remission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 .................................... 146 4.43 Integral transmission fraction for particles with initial concentration of 5% in three different sizes of 38 , 75 , and 150 ............................. 147 4.44 Diffuse reflectance obtained based on of Dahm equation for different absorption to remission fractions .................................................................... 148 4.45 Fitted to the reflectance derived from Dahm equation in all initial fractions showed in Figure 4.44. ..................................................................... 149 4.46 Illustration of the void fraction effects on the absorption/remission function. .......................................................................................................... 149 xiii NOMENCLATURE A Absorption fraction Symbolic representation of the fraction of incident light absorbed by a layer A(R,T) Absorption/remission function B Apparent remission coefficient b Symbolic representation of fraction of incident light remitted from the surface of a particle C Concentration of particles inside the slurry sample computed from CFD Symbolic representation of remission power of the material comprising particles in layer i D Diffusion coefficient in radiative transfer model d The thickness of a sample or a layer of sample through which light is passing Symbolic representation of thickness of a particle inside a layer known as i E Irradiance Absorption coefficient obtained from radiative transfer equation Scattering coefficient obtained from radiative transfer equation g Cosine of scattering angles Extinction coefficient based on terminology of radiative transfer equation Intensity of incident light inside the sample xiv Intensity of incident radiation Intensity of reflected beam inside the sample Intensity of reflected beam out of sample K Absorption coefficient k Absorption power of material The effective absorption coefficient for L Radiance m Mass of the particle or layer of sample N Refractive index Q The radiant energy in Joules r Reflectance of a layer inside the material Reflectance of a particle inside a layer of material Reflectance from a surface of material based on Fresnel law R Remission fraction The reflectance of a layer so thick that further increase in thickness fails to change the reflectance s KubelkaMunk scattering coefficient Symbolic representation of cross sectional surface fraction that belongs to particles xv Cross sectional surface in a sample comprised of particle type i t Transmission fraction of a layer of material Volume fraction of particles Void fraction Vol Volume of a particle or a layer of the sample Weight fraction of a particle X Distance from the edge of the sample Greek symbols Absorption coefficient used in Mie, and three flux approximation and the Absorption probability for representative layer theory Fraction of incident light absorbed by a layer Sum of fractions of light inside a sample used in Stokes formula Fraction of radiation absorbed per unit pathlength in the sample The cosine of the angle between the direction of solid angle and the surface normal of the sample Solid angle Ratio of fractions of light inside a sample used in Stokes formula xvi Photon fluence rate or radiant flux with the unit of Joules per second A parameter for calculating integral fractions of light inside a sample used in Stokes formula Symbolic representation of density of particle type i The angle of incidence with the surface normal Fraction of radiation scattered per unit pathlength in the sample The average path length of light when passes through Subscripts i Type of particles d Properties of a layer of thickness d f Of the fluid p Of the particles j Number of particles in a layer s Measured quantity 1 Initial conditions; initial fraction of light for discontinuum theories and lower boundary of integration for continuum theories 1 CHAPTER I INTRODUCTION AND BACKGROUND 1.1 Introduction One of the main goals of the thesis is to answer the fundamental question “How far does light penetrate into the slurry fluid flow in a pipe during NearInfrared (NIR) irradiation by a NIR sensor?” For a particular case, a quick answer to this question would be to run an experiment. Moreover, an experiment requires an equipped laboratory with imaging instruments, probes inside the pipe, and other facilities to visualize the flow field. This might not be affordable. Since numerical approaches allow solutions for many cases, it is preferred to use a computational approach to simulate the travel of light inside the pipe using the newest and the most efficient theoretical approaches. By choosing the numerical method, a model is built based on the characteristics of the fluid at rest. Then the results from the model can be compared to the NIR diffuse reflectance theories. Calculating the penetration depth requires answers to related questions: “What is the concentration of particles inside the field and how is it going to be predicted through optical theories?”, and “What is the light distribution in terms of fractions of the incident radiation inside the particulate slurry?” There are related works in the literature, however, most of them are experimental techniques to measure concentrations of slurries either online or offline using optical NIR sensors. However, almost none of the theoretical approaches have been applied to predict light interactions with particles inside a slurry sample. It means that the theories found in the literature have not yet been applied to the case of a slurry sample. However,2 experimental methods to model light interaction with slurries are available in the literature. Study and research in the field of optical size and concentration monitoring of slurries have been performed using analytical methods, which required some parameters from experiments. But, there are approaches that do not need any experiment. Those methods use experimentdependent models which require some input parameters that must be provided through an experiment. In this thesis, the details of both of these two types of methods along with their experimental requirements, assumptions, and mathematical models will be reviewed. 1.2 Motivation In recent years, nearinfrared (NIR) optical modeling has gained extensive acceptance in different scientific and industrial areas by means of its advantages over other analytical techniques. This type of spectroscopy is prominent because of its ability to record spectra for solid and liquid samples. Modern applications of nearinfrared use it to measure the composition of unknown samples using techniques invented by scientists at the United States Department of Agriculture (Workman and Weyer, 2008). It has become a very popular technique in a wide variety of industries due to its speed, precision, broad applicability and avoidance of extraneous chemicals. It is widely used in agricultural, chemical, pharmaceutical, combustion products, textile and many other industries. Moreover, advances in instrumentation have caused the manufacture of spectrophotometers capable of rapidly providing spectra that are flexible enough for use in different applications. NIR spectroscopy requires some input parameters such as the intensity of radiation, the fractions of diffuse light in the medium, and the characteristics of the slurry fluid. A wide range of theories has been developed to support experimental capabilities of methods that use NIR diffuse reflectance and transmittance. There are a number of distinct mathematical models capable of characterizing nearinfrared light diffusion in onephase mediums or one phase with a mixture of two or more types of different components. These models are different, however, most of them implement the fundamental technique 3 of applying nearinfrared radiation to monitor different materials. The applications of NIR spectroscopy are summarized in Table 1.1. Table 1.1 Description of different fields and applications of nearinfrared radiation has already been used in the previous works Field Applications Model/Technique Clinical sector Human blood cells, In vivo imaging Human body tissues, Optical topography of brain Monte Carlo Simulation, Diffusion approximations, Steady state diffusion theory, Transport theory, Addding doubling method, DeltaEddington approximation, Astronomical sector Astrophysics, Detection of infrared galaxies, Interstellar molecules Dual field interferometry Agriculture Agricultural products monitoring, Analysis of food crops, Analysis of processed food like baking products, and Analysis of nonfood agricultural products like paper, wool, etc. Representative layer theory, KubelkaMunk theory, Diffuse reflectance theory, Schuster’s theory Remote monitoring NIR spectroscopic imaging, Hyperspectral imaging in the remote investigation of plants and soils Experimental techniques to collect data from airplanes or satellites to investigate the soil chemistry on the ground Material Science Film thickness measurements Nearinfrared measurement in the diffuse reflectance mode Particle measurement Slurry flow, granular flow, fluidized beds, powders and ground materials Continuum theories of diffuse reflectance such as classical KubelkaMunk theory, BeerLambert’s law, etc., MonteCarlo Simulation Industrial uses Study of industrial chemicals, Process control applications, Rapid laboratory measurements, Realtime online analyses ParticleLaden slurry transport in mines, Process control for food, agricultural, combustion, textile, and pharmaceutical products, Before examining the techniques shown in Table 1.1 in detail consider a brief introduction to NIR models. Based on the foundations of nearinfrared models there are two different paths indicated by these theoretical treatments: continuum theories and discontinuum theories. These two types of theories are separated by the two distinct types of mathematical models on which they are based. For example, the theory of representative layers is applicable in many industrial processes. Among the 4 theories of diffuse reflectance, the representative layer theory is the latest efficient method to model nearinfrared phenomena inside particulate, turbid, in vivo, powdered, and generally uniform and nonuniform mediums using discontinuous mathematics. As it was depicted in the Table 1.1, none of the nearinfrared theoretical treatments have analyzed slurries with discontinuum mathematical techniques. Reviewing the literature, discontinuous mathematical methods have not yet been used for slurries. Continuum theories are imperfect due to their weakness in exhibiting the light interactions with particles in real nonuniform mixtures that scatter light and are not continuous. On the other hand, the only method that addresses light propagation inside a slurry has been found in the work of Rabhi, Masion, Roze, Dussouillez, and Bottero (2010) that uses an experimentaltheoretical algorithm and some parameters such as refractive index to find particle size distribution (PSD), and the slurry concentration to be fed into a simulation. Hence, their approach is not pure theoretical. Meeten and Wood (1993) introduced another approximation for measuring the diffuse reflectance of fluids using optical fiber methods, but the classic KubelkaMunk theory is applied to interpret the experimental data. KubelkaMunk theory is a powerful tool to describe light diffusion but is only applicable to continuous mediums: it underestimates the optical properties of a real discontinuous sample. This introduction presents the approaches that are going to be discussed in the present thesis. In the following chapters the latest approach has been used to characterize the optical properties of slurries that naturally have discontinuity in their composition. 1.3 Background Nearinfrared technology and slurry mixtures are two separate academic, technological, and industrial areas for which at first it may seem very difficult to find a connection, but after a brief introduction one may see the relationship between them. Hence, studying an overview about the NearInfrared and the slurry is essential to reading and understanding this thesis. 5 1.3.1 Near Infrared Technology Nearinfrared is the electromagnetic radiation with a spectrum of wavelengths just beyond visible light. Nearinfrared is a part of infrared radiation which usually is divided into three spectral regions: near, mid and farinfrared. However, another categorization is done by Center for Measurement, Vegetation, and Health [MVH] (2006) at the University of New Hampshire: the infrared region is divided into five parts: near, shortwave, midwave, longwave, and farinfrared . This spectrum is shown in Figure 1.1. Figure 1.1 Schematic representation of divisions of the infrared electromagnetic spectrum. The near infrared region corresponds to a wavelength range of between 700 and 2500nm. The midinfrared region is further divided into short, mid and long wave infrared and corresponds to a wavelength range of 2500 – 5x104 nm. Finally the far infrared region corresponds to a wavelength range of 5x 104 – 1x106 nm. (MVH, 2006) The boundaries between near, mid and farinfrared regions are not agreed upon and can vary. The main factor that decides which wavelengths are included in each of these three infrared regions is the type of detector technology used for gathering infrared beams of light. The infrared region of light has longer wavelengths than visible light, making it invisible to the human eye, however, the nearinfrared spectrum has shorter wavelengths with proximity to visible light. Spectroscopy is defined as the study 6 of the wavelength spectrum of radiated energy interacts with matter – it is transmitted, reflected or absorbed. Although a theoretical analysis underlies NIR spectroscopy, it is mostly known as an experimental branch of NIR technology. Theoretical techniques have been developed to interpret the data gathered from the spectrum. Modeling is absent in the spectroscopic techniques. On the other hand, theories of diffuse reflectance provide an opportunity to model the nearinfrared penetration of light inside a sample. Recently, more sophisticated theorems have been applied to model the nearinfrared phenomena. The earliest application of NearInfrared spectroscopy goes back to the 1800’s, when organic compounds were investigated by NIR photography. Thus, the first application of NIR technology was studying chemicals in compounds. In the 1940s, the availability of new commercial instruments led to one of the first industrial applications of NIR in chemical and polymer industries. Until that time applications were confined to academic or research projects (Workman and Weyer, 2008). Norris in USDA in 1949 was the first person who began to apply NIR in the food industries. During the 1950s and 1960s, progress in this field of chemical monitoring became stagnant, however, diffuse reflection theories propelled NIR spectroscopy to a wide range of applications in agriculture, pharmaceuticals, and many other industries. With the advent of new instrumentation and methods, the interest in NIR industrial applications has become greater. Infrared technology is not limited to physical applications monitoring materials in samples; it can be applied in astrophysics and astronomical observations. NIR observations have been made from ground based observatories since the 1960's. They are done in much the same way as visible light observations for wavelengths less than 1 micron, but require special infrared detectors beyond 1 micron. Mid and farinfrared observations can only be made by observatories which can get above our atmosphere because spaceborne telescopes are very sensitive to heat. The reason is that the telescope itself has a certain temperature and continually radiates heat that would interfere with the measurements taken by a NIR sensor. Far infrared waves are thermal, therefore, we experience this type of infrared radiation every day in the form of heat. The heat that we 7 feel from sunlight, a fire, an oven or a warm coffee is infrared. These observations require the use of special cooled detectors containing crystals like germanium whose electrical resistance is very sensitive to heat. Despite all of the mid and far infrared applications, NIR is more popular among industries. Industries strive to produce quality commercial products at the lowest cost and in the shortest time. Introducing analytical methods that offer technical and costsaving advantages over conventional approaches can lead to businessrelated improvements in the manufacture of chemical products. In recent years, nearinfrared (NIR) spectroscopy has been incorporated into several analytical protocols for obtaining qualitative and quantitative information on incoming materials, mixtures, and products. One of the most important applications in industries is slurries. NIR spectroscopy is used in these industries to monitor composition and control the process. Therefore, NIR light and slurries are connected through this widespread application. 1.3.2 Slurries Despite much progress in techniques of conveying materials discovered in mines, the semiindustrial transport of minerals out of mines had been an issue until the early sixteenth century, when a solution was found. Hydraulic conveying over short distances for alluvial mineral separation was documented in the sixteenth century. The hydraulic term refers to the technique of transporting those minerals; the process that is performed with the stream of fluid inside pipelines. The particulate medium by which the hydraulic transport of minerals took place has an academic name; a slurry. Particleladen slurry transport through pipelines is one of the important industrial applications of slurries. The scientific interpretation of the word introduces the slurry as a solidliquid mixture. The slurry has been categorized in the group of multiphase mixtures. Another meaning states a thick suspension of solids in a liquid. Therefore, this type of flow can be categorized in the group of dispersed phase and separated flows. When there is more than one phase participating in a fluid 8 mixture, the combination is known as a multiphase mixture. As shown in Table 1.2, there are four types of multiphase flows (Crowe, 2012). Due to the presence of two phases, this flow is more complex than single phase flows. Twophase flows consist of discrete elements such as droplets in a gas or solids in a liquid. Thus, they are not attached to each other, therefore, in a separated flow, two phases are dispersed by a line of contact. Table 1.2 Examples of single and multicomponent, multiphase fluids Single Component Multicomponent Singlephase Water, Oxygen Air, Emulsions Multiphase SteamWater, FreonFreon Vapor AirXylene, Slurry Slurries are applicable in many engineering, industrial, chemical, and fossil energy areas like transport of bulk quantities of materials such as coals and ores or flow of mud. The significance of particle deposition results from its key role in industrial processes such as filtration, separation, particle transport, combustion, air and water pollution, metallurgical processes and many others (Hossain, Naser, and Imteaz, 2011). The fluid that carries the particles through the pipe may be Newtonian or nonNewtonian, and usually influenced by turbulence, drag, hindered velocity, virtual mass effects, phase interaction, crossingtrajectories, and particle inertia (Chen, 1994). Moreover, the process of transporting solidliquid mixtures through pipelines is common in the mining and petrochemical industries. Furthermore, turbulent slurry flows with high solid volumetric concentrations are an important part of many chemical or mineral applications. In addition, the need to measure the pressure drop and concentration profiles in pipelines made researchers study this kind of fluid and build more accurate models (Lahiry and Ghanta, 2010). Slurry transportation in the horizontal direction may be classified into three major flow patterns: (1) pseudohomogeneous slurry (or homogeneous); (2) heterogeneous and sliding bed slurry (or moving bed), and (3) saltation and stationary bed slurries, as described by Ling et al. (2003). A pseudohomogeneous slurry is a pattern 9 in which particles are uniformly distributed along the crosssection of the pipe since the velocity is so high to prevent the particles from accumulating in the bottom of pipe crosssection due to gravity effects. This process is practical since the high flow rates prevent the particles from falling due to the influence of gravity. However, heterogeneity happens in a slurry flow when the flow rate cannot overcome the transportation of the particles to the lower part of the crosssection. By further decreasing the velocity, the particles will accumulate in the bottom of the pipe crosssection. This region is called a moving bed layer (Ling et al., 2003). This phenomenon occurs in the saltation and stationary bed flows. In conclusion, when one thinks about the combination of a liquid with insoluble solids inside, there are many different examples. 1.4 Statement of the Problem The main goal can be expressed in this form: the nearinfrared region of a light beam is chosen to penetrate into a slurry for detecting the slurry particles by sensing the fractions of light in each section through the sample depth, with reflectance or transmittance to a NIR sensor. The problem is to estimate the fractions of light at each distance inside the sample to answer the question of the depth inside the slurry at which the particles will be detected by the sensor. Therefore, the fractions of light at each possible point inside the sample must be known to estimate the layer within the media where the light is able to be transmitted to the detector or is blocked by the heavy concentration of particles. As it is depicted in Figure 1.2, when the generated light reaches the surface of the medium, a portion is transmitted into the sample, and the rest is reflected. Then the transmitted light goes through the sample, which may be represented as many layers of particulate slurry. The reflectance, transmittance, and abrorptance occurs in each layer, until the remaining transmitted portion of light is detected by the detector. This geometry can be shown with another configuration where the detector is located in the same place as the generator, detecting the reflectance of the light instead of transmission. 10 Figure 1.2 The geometry of the light generator, detectors, and the sample Now, the problem is to predict the amount of the light through the sample and the penetration depth of light inside the slurry. Transmission, reflection, and absorption inside the sample gives us information about the concentration of particles inside the sample. Moreover, it can predict the shape of particle accumulation in each layer inside the slurry. 1.5 Goals The primary goal of this thesis is to permit accurate estimates of how much light is detectable by the sensor, and what portion of the sample contributes to the signal. Answering these two fundamental 11 questions will describe the sample from a spectroscopic perspective. This information will help us choose the best location for installing the NIR sensor outside of an slurry sample. Tools for modeling the NIR light interactions with particles inside a slurry are theories of diffuse reflectance. These theories were applied to the problem with their limitations and assumptions considered. The purpose for using different theories to model the optical interaction is to compare them and find the most computationally efficient approach for modeling the NIR diffuse reflectance phenomena in slurries. Moreover, the problem is divided into two groups of NIR diffuse reflectance theorems based on their mathematical treatments: continuous and discontinuous functions. Each of these two groups requires different arrangements for the configuration of the sample. Therefore, they have different results. Continuum theories aim to find the light fractions integrally inside the sample, so the accumulative fractions are estimable without using any other theory in any distance from the boundaries of the sample where the light first was encountered. This means that continuum theories directly calculate the cumulative fractions of light in each distance from the edge of the sample, and the methodology of these theories is different from discontinuum ones in which the light fractions are estimated locally, so one must combine them with another method to find light fractions at each depth inside the sample. The details of these methods will be discussed in the following chapters. The purpose of implementing discontinuous theories is to find the same fractions but locally. This gives us an opportunity to have an estimation about optical properties of each possible point and layer inside the sample when the effects of other layers are not present. It means that the reaction of each particular layer to the incident beam is considered in isolation and only based on the local configuration of the particles inside each particular layer of the sample. The main goal is achieved by considering these objectives and steps: (i) for continuum mathematical approaches such as plane parallel mathematics, KubelkaMunk theory, Schuster’s theory, three flux approximation, and diffusion theory, and (ii) for discontinuum mathematical 12 approaches such as the theory for plane parallel mathematics, assembly of sheets, representative layer theory, and combined representative layer theory and the Stokes formulas. In order to meet the goals discussed above, the following tasks must be done: (i) reviewing all of the theoretical approaches appropriate for the case of this problem in terms of assumptions, limitations, advantages, etc, (ii) implementing the theories by developing a mathematical model based on them, and (iii) comparing the results of the models. The first task will be done in Chapter 2, where continuum and discontinuum diffuse reflectance theories are introduced. In Chapter 3 the models created based on these two groups of theories are discussed respectively according to the mathematics of the methods. In Chapter 4, the results are compared with each other. Hence, the most compatible model will be introduced based on the results. Finally, in Chapter 5, the conclusions and recommendations are presented. 13 CHAPTER II PAST CONTRIBUTIONS 2.1 Introduction Optical methods for modeling NIR radiation phenomena inside a medium, whether it is composed of onephase continuous components, onephase powdered sample, or multiphase elements are available in the literature which will be addressed in this chapter. Although, this work connects the fluid dynamics of a slurry flow to an optical phenomena that is the interaction of light with particles in a slurry sample, most of the work has been concentrated on optical theories to model the NIR radiation inside a slurry. Hence, this chapter is dedicated to surveying the theoretical methods pertaining to optical modeling inside slurries. Before describing the approaches, one needs to know what has been done in past contributions to NIR technology in analyzing samples. Then, acquiring a preliminary knowledge about the definitions of fundamental expressions and terms that are related to optical models and are mostly used in this thesis is essential. After introducing the optical nomenclature, NIR optical methods are reviewed. These approaches are gathered by reviewing these major concepts in the available literature: nearinfrared spectroscopy, online and offline measurement of slurries, NIR diffuse reflectance and transmittance, particle size analysis, slurry concentration measurements, and theories used in optics. All of the optical methods examined in this study have certain fields of application based on their assumptions. The assumptions make constraints, however, by simplifying the analytical model14 one may produce a new model consistent with the restrictions. Therefore, measuring the capabilities of each theory in the literature through a thorough study may facilitate the choice of the approach which best matches the requirements of our problem. In this part optical methods are briefly introduced and then their assumptions and basic concepts, as well as the mathematics beyond those concepts will be discussed. 2.2 History of past contributions History of past contribution to the theories of diffuse reflectance helps to a better understanding of the theorems. A summary is gathered in Table 2.1. Bouguer in 1760, and scientists start the works on diffuse reflectance and engineers have been involved in the measurement of light since the early experiments and instruments described by him. Pierre Bouguer (1729) and Johann Lambert (1760) developed the principle underlying the model for the absorption of light that relates the absorption to the properties of the material through which the light is traveling (Dahm & Dahm, 2007). This law will be introduced in detail in this chapter. Moreover, Bouguer’s elementary mirror hypothesis (1729) represents an application of the absorption by glass and water for various thicknesses (Darrigol, 2012). Darrigol also reviewed the absorption law pointed out by Bouguer stating that the successive equal layers of a homogenous transparent medium absorb a constant ratio of light that they receive. Kortum (1969) delineated Bouguer’s elementary mirror hypothesis which has been employed to explain diffuse reflection from surfaces. Lambert’s inverse square law, additivity law, and extinction laws are all based on Bouguer’s work. Lambert’s hypotheses were more accepted than Bouguer’s findings since there were some contradictions in Bouguer’s observations in the laws of diffuse reflection. In fact, Bouguer’s model of diffuse reflection was found to fail later (Darrigol, 2012). 15 Table 2.1 Overview of the development of modeling techniques for diffuse reflection. 1760 P. Bouguer described diffuse reflection by elementary mirror hypothesis for surfaces with random orientations 1860 G.G. Stokes produced a twoflux theory of diffuse reflectance. The theory considers individual particles as plane parallel layers, and provides an analytical tool to the mathematics of plane parallel layers. 1887 E. Lommel and R.V. Seeliger developed a complicated theory of diffuse reflection concerning interactive irradiation of distinct elements inside a sample. 1905 A. Schuster proposed a special part of diffuse reflection based on continuous mathematics by publishing “Radiation through a foggy atmosphere”, which involves particles in sample. 1908 G. Mie devised a novel method involving the scattering of light by isolated spherical particles, which uses the mathematics of complex variables and numbers. 1926 T.H. Gronwall proposed an article that emphasized on a simplified version of plane parallel mathematics. 1931 Kubelka and Munk published their classic paper that demonstrates the Kubelka–Munk equation, derived using continuous mathematics. They displayed that the reflection which is calculated by the absorption/remission formula from an infinitely thick sample, is equivalent to the ratio of absorption to remission coefficients for a medium. Their work produced many controversies over the failure of their model at certain conditions, and confusion over the effects of: direct and diffuse illumination; front surface reflection; and the relationship of the absorption and scattering coefficients of a sample to the coefficients of the material making up the sample. (Chalmers & Griffiths, 2002) 1946 F. Benford produced a very complete theory describing the mathematics of plane parallel layers. His work was very creative, novel, and independent of previous contributions. 1963 N.T. Melamed published a paper proving his method for modeling diffuse reflectance inside particulate samples, which is independent from plane parallel mathematics. 1966 Wendlandt and Hecht published the first “handbook of diffuse reflection” in 1966, which was the first handbook in the field of spectroscopy. 1969 G. Kortum published ‘Reflectance Spectroscopy’, a book which is heavily mathematical and covers continuum and discontinuum theories rather thoroughly. (Chalmers & Griffiths, 2002) 19701975 E.L. Simmons proposed several articles including simplified version of plane parallel mathematics to model particles inside a sample, and a modified particle theory (Chalmers & Griffiths, 2002). He proved the proportionality of the discontinuous and continuous theories in the case of twoflux approximations. 1997 T. Burger et al. presented a method in their publication pointing out application of the radiation transfer model to characterize diffuse reflectance through more advanced methods such as using three (or more) fluxes to separate absorption and scattering coefficients. 19952000 Dahm and Dahm conducted a carefully planned research that results in: a method to use plane parallel mathematics to separate absorption and remission coefficients; presenting an absorption/remission function which is constant for all sample thickness; and an equation relating the absorption and remission of one thickness of sample to that of any other thickness. They clarified the confusions over Kubelka–Munk theory, developed the representative layer theory that models a complex sample by defining layers, each one representative of the sample. They also developed a simple particle model. 16 The calculation of the amounts of light absorbed, transmitted and remitted by several layers is a very complicated process. Some fraction of light will be absorbed by a layer, which reduces the transmitted fraction to the next consecutive layer. In addition, some may return to the first layer as the result of reflection from the next layer, which again a portion may again be absorbed, transmitted, or remitted back to the next consecutive layer. The mathematics that is able to model this complex physical situation is called mathematics of plane parallel layers, which was developed by Sir George Stokes in 1860. Dahm and Dahm (2007) assert that Stokes theory is an exact estimation for plane parallel particles. The Lommel – Seeliger reflectance law is a time honoredlaw, which has been used until today. It is based on a model, which is founded based on diffuse reflection by considering the optical characteristic of every single element inside the sample. Therefore, this model was very complex comparing to similar ones in that ages. However, it has some limitations such as predicting only the scattering of only collimated incident light (Fairbairn, 2004). Schuster proposed a method of scattering and absorption within a sample by visualizing a dilute suspension of particles in which the particles were luminescent, absorbing, and scattering. He published a paper in 1905 describing a particle theory that developed to determine a solution for particular problems (Griffiths & Dahm, 2008). He developed his particle theory using continuous mathematics. Gustav Mie designed a solution to Maxwell’s equations to describe the scattering of electromagnetic radiation by a sphere. The solution is in the form of an analytical infinite series. The Mie scattering solution finds exact solutions for Maxwell’s equations by separating equations for radial and angular dependence of the solution in the problems dealing with scattering (Bohren & Huffman, 1998). Thomas Hakon Grönwall published a paper in 1926 about new method in geometrical optics and presents a simpler method than the plane parallel mathematics. 17 Kubelka and Munk (1931) published their classic paper in the field of optics, which revealed a formulation for the scattering, and absorption of infinitely thick samples. Kubelka and Munk published their paper which is a twoflux version of a multiflux method of solving radiation transfer problems. This method is popular because of its simple analytical equation, which is known as the KubelkaMunk equation. However, this theory has some assumptions that limit the cases to which it can be applied. Benford (1946) described mathematics of plane parallel layers, which is very complete and novel. In 1946, he published a detailed analysis of the absorption and scattering of light in the article of ‘Radiation in a diffusing medium. He used discontinuous mathematics and assumed that the sample was divided into a series of plane parallel layers (Dahm & Dahm, 2008). Benford’s calculations were developing a system of relations that can be used to compute the reflection, absorption and transmission fractions for a sample if the number of layers and those fractions for each are known. Melamed proposed a complete statistical theory for the absolute diffuse reflectance of powders about a half century ago (Mandelis, Boroumand, & Bergh, 1990). In the past half century many spectroscopists performing quantitative diffuse reflectance spectroscopy of powdered materials have used the Melamed model. Melmad’s theorem was successful in the field of modeling diffuse reflectance since unlike other discontinuum mathematics, Melamed applied statistical summations on the sample consisting of discrete particles reflecting light diffusely. The laws of geometrical optics (Mandelis et al., 1990) limited his model. Mandelis et al. deduced that Melamed’s statistical approach had prominence over other theories that are based on plane parallel mathematics but with some limiting assumptions. Proceeding Melamed’s unique theory, Wendlandt and Hecht (1966) published the first handbook in the field of spectroscopy. In 1969, Kortum presented his book in the field of reflectance spectroscopy that is a breakthrough in using 18 cumbersome mathematics to embellish continuum and discontinuum theories of diffuse reflectance (Griffiths & Dahm, 2008). Simmons has used a simplified particle model (plane parallel mathematics) to relate diffuse reflectance to some optical parameters of the model without the use of heavy equations, which result from the more modified Melamed’s theory (Mielenz et al., 1977). A more recent study in the field of diffuse reflectance was done by Burger et al. (1995) who developed a diffuse reflectance model using three or more fluxes (Griffiths & Dahm, 2008). Dahm and Dahm (1999) presented a more complete method to study the diffuse reflection phenomena inside samples containing nonhomogeneous particles. They used the plane parallel layer model of Benford (1946) to develop a more accurate approximation to calculate the fractions of light for each layer. This accuracy is a result of relating the physical properties of the elements of the medium to the optical properties of the sample. They also developed an equation for relating absorption/remission of one layer to that of the sample. This equation is comparable to the KubelkaMunk function but is a descriptive tool for more complex samples (Griffiths & Dahm, 2008). Moreover, throughout their systematic work, they presented a series of explanations about the KubelkaMunk theory and its controversies. 2.3 Fundamental Definitions In this section fundamental definitions are presented that are necessary for the appreciation of the concepts proposed not only in this chapter but also in the following chapters. There is a chain of prerequisites for conveying the information about topics from basic elementary components in optics to more rigorous concepts. Some of the definitions are not found in the theories but introducing them is essential since these fundamental terms have used in some more advanced definitions. Hence, they 19 are explained before separately. Each part is described sufficiently for applying the theorems into models. 2.3.1 Geometrical Optics Geometrical optics is the study of light without diffraction or interference (Greivenkamp, 2004). It is essential to explain the fundamental definition of geometrical optics to briefly introduce this significant part of classical optics. Geometric optics refers to the ray optics and describe the light propagation in terms of rays. In this part of classical optics, the rays are explained to spread out in rectilinear path in a homogenous medium. Rays are able to be bent, transmitted by splitting in two or more parts at the interface between two distinct media, may be refracted in a medium where the refractive index changes, or are absorbed and reflected. Geometrical optics provides rules, which depend on the wavelength of the electromagnetic wave, for propagating these rays through an optical system. Therefore, geometric optics is a crucial simplification of optical sciences that fails to account for some optical phenomena such as diffraction and interference. It provides an outstanding estimation about the travel of light into materials, however, when the wavelength is very small compared with the size of objects or materials with which the light interacts. Geometric optics can be used to describe the geometrical aspects of imaging, including optical aberrations. 2.3.2 Wavefront and Ray In physics, a wavefront is the locus of points having the same phase: a two dimensional line, curve, or surface for a wave propagating in 3D space (Schuster, 1904). Greivenkamp (2004) reported a very concise description about the geometrical optics based on the optical path length. Optical path length is proportional to the time required for light to travel between two points (Greivenkamp, 2004). The relation is: (2.1) 20 where is the refractive index which varies along the path. If it is constant the integration is not required. Mouroulis and Macdonald (1997) suggested that the propagation of waves is commonly described by means of wavefronts. In geometrical optics the concept of wavefront is fundamental. Because of the higher frequencies of the waves such as infrared, optical, xray and gammaray, the temporal component of electromagnetic waves is usually neglected for the corresponding wavelengths, therefore, the phase of the spatial oscillation is the only parameter that is described. Moreover, most optical system and detectors are indifferent to polarization, so this property of the wave is also usually neglected. Maxwell's equations are a powerful descriptor for the optical system electromagnetic waves, and linear propagating waves such as sound waves that have similar wave equations. There are also geometrical wavefronts which are described similar to the physical wavefronts but are not propagating and are frozen in space (Mouroulis & Macdonald, 1997). Introducing the wavefront is essential to define another fundamental element in optical physics which is the ray. Rays are defined as the lines normal to a family of wavefronts. The direction of electromagnetic propagation as suggested by Mouroulis and Macdonald (1997) is given by the Poynting vector, which is the cross product of the electric and magnetic field vectors. This direction of energy flow is conveniently described by rays. One might think that a very narrow beam of light is a ray but it is not. The divergance in the beams of light contributes the idea that the crosssection of beams are not constant in their lengths. In this thesis, it has been assumed that the media is isotropic and our definition of rays is consistent with this assumption. However, there are some materials in which the direction of electromagnetic waves such as visible or nearinfrared light is not always normal to the wavefronts such as crystals or nonlinear media where it is possible for the wave to exit the material at a different wavelength from which it entered. 21 2.3.3 Radiant Energy McCluney (1994) stated that radiation is the scattering of energy in several forms through space. Following this definition, the radiant energy can be described as the quantity of energy propagating onto, through, or emerging from, a specified surface of given area in a given period of time. For estimating the energy, all the wavelength spectrum contained in the radiation are must be included. The radiant energy is known by Q and its unit when a limited wavelength range is to be considered is shown in joule. Nearinfrared light has a radiant energy calculated by integrating the radiant flux over time. 2.3.4 Solid Angle Solid angle is a two dimensional angle in three dimensional space and is described by a closed curve and a point in space. According to Figure 2.1 (a), the solid angle is the projection of curve C in space, and curve C subtends solid angle at point P at the center of a unit sphere. It extends the two dimensional concept of a plane angle to three dimensions. The magnitude of this angle is calculated by projecting the area of a closed curve in space on a sphere of unit radius, therefore, the solid angle of an object is equivalent to the area of the segment of a unit sphere, centered at the angle's vertex or center of sphere, that the object covers. Another relation that has been set forth by Palmer and Grant (2010) is the ratio of projected area of the closed curve on a sphere with arbitrary radius, , over the square of the sphere radius. This defining the solid angle along with the corresponding relationships have been presented schematically by Figure 2.1 (b). The unit of this angle is measured by steradian. The relationship associated with solid angle is: (2.2) where is the solid angle, is the segment of the projected area on the sphere, and r is the radius of sphere. 22 (a) (b) Figure 2.1 (a) Schematic representation of solid angle. (McCluney, 1994); (b) Solid angle relationships Hemispherical solid angle is another important definition. The three solid angles are directional, conical, and hemispherical. McCluney (1994) distinguished between these three types and claimed that the directional one refers to an infinitesimally small solid angle in a particular direction from arbitrary point p in space like Figure 2.1 (a) and (b). Directional solid angles are only used in mathematical models of reflectance and transmittance but are not applied in experiments and measurements since the associated flux with this infinitesimal angle is unmeasureable in quantity. Conical solid angles are measures of intermediatesized solid angles in the shape of circular cones. The mostly used solid angle is the hemispherical one that has the size of steradians. The shape of this angle is the same as a hemisphere. 2.3.5 Radiant flux The radiant flux is the time rate of flow of radiant energy with the unit of watt. This flow of radiation is characterized by the quantity of energy transferring through a region in space or usually a surface per unit of time. The defining equation is: (2.3) 23 where is the radiant energy in Joules and is the radiant flux with the unit of Joules per second or watt (McCluney, 1994). According to this definition, the radiant flux is radiated from a source over optical wavelengths, which are defined to be from to Hz. This region is equivalent to the electromagnetic spectrum commonly referred to as ultraviolet, visible, and infrared. 2.3.6 Radiance We started defining the fundamental quantities with radiant energy which has as its unit the joule (J). Radiant power introduced as radiant flux is energy per unit time ( ) with the symbol and is measured in Watts. Palmer and Grant (2010) deduced that these two definitions are not informational about the spatial distribution of power in terms of area or direction. Hence, another parameter is required that is an elemental quantity in radiometry, and is defined as the power per unit area, and per unit projected solid angle. It is known as radiance (Palmer & Grant, 2010). Since the radiance can be integrated along a beam, it is possible to calculate it in any location of an optical system. The radiance depends upon the direction and it can come from many points on a surface. Moreover, it can exist everywhere since it applies to a field quantity. The defining equation is (Palmer & Grant, 2010): (2.4) where, L is the radiance, is the solid angle, is the radiant flux and s is the surface. 2.3.7 Irradiance Irradiance (radiant incidence) is defined as the power of electromagnetic radiation per unit area. Described by McCluney, irradiance is the area density of radiant flux, the radiant flux per unit area in a specified surface that is incident on, passing through, or emerging from a point in the specified surface. The defining equation is: (2.5) 24 where is an element of radiant flux and is an element of area in the surface. It is essential to mention that radiance and irradiance are quite different. Radiance is the angular distribution of radiation emitted per unit area into a cone having unit solid angle, while irradiance sums up all this angular distribution over a specified solid angle . Hence, the relation associated with the summation over the solid angle comes from the following integration: (2.6) where, L is the radiance and is the solid angle. 2.3.8 Radiant Intensity McCluney (1994) characterized radiant intensity, I, as the density of solid angle or flux of energy radiated per unit solid angle (steradian) incident on, passing through, or emerging from a point in space. The defining equation is: (2.7) where is an element of radiant flux and is the element of a solid angle in the specified direction. The intensity is a strong function of direction from or toward the point for which it is defined. Also, it changes with distance from its source except for a collimated beam (Dahm & Dahm, 2007). Intensity can be derived from radiance by integrating over area: (2.8) 2.3.9 Refractive Index Index of refraction is a very fundamental parameter in optics, and it can be measured with a very high accuracy of (Chartier, 2005). Refractive index is a property of a substance which is a number indicating the level of electromagnetic wave propagation in a medium. Before introducing the relation 25 for refractive index, it is essential to know about the source of this concept. Fundamental equations of wave propagation into a medium are known as Maxwell’s equations. Using Maxwell’s equations for a dielectric medium and then simplifying the integral forms and implementing some mathematical operations and manipulating, one can obtain the general wave equations for the propagation of a periodic disturbance, which is shown below (Hecht, 2002): (2.9) (2.10) where E is the electric field vector, B is the magnetic field vector, is the electric permittivity of free space, and is the permeability of the free space. The phase velocity of Maxwell’s equations is: (2.11) Therefore, based on the Wendlandt and Hecht (1966) analysis, the refractive index, defined as the phase velocity in a vacuum which is equal to the speed of light compared to that in a given medium, is: (2.12) where c is the speed of light or phase velocity in vacuum and is the phase velocity in the material. and are permittivity and permeability of the material, while and are permittivity and permeability of the vacuum. When light passes through a medium, some part of it will always be absorbed. This can be conveniently considered by introducing a complex index of refraction: (2.13) where the real part of the refractive index indicates the phase speed, whereas the imaginary part is the amount of absorption loss when the electromagnetic wave propagates through the medium. 26 2.3.10 Diffraction Diffraction appears as a limitation to the rectilinear propagation of light (Chartier, 2005). Diffraction is a complicated process, and occurs when a light beam collides with an obstacle. After collision, some parts of the light deviate in different directions. In classical physics the diffraction is described as the apparent bending of waves around small obstacles and the spreading out of waves past small orifices. Diffraction is observed for any kind of propagating phenomenon. All types of waves can be diffracted. Pedrotti et al. (2007) have presented a compelete study on this important phenomenon in classical optics. They suggested a simple defintion for this term that is any deviation from geometrical optics which results from the obstruction of wavefront of the light. The diffraction is observed even if the obstacle is not opaque and the transmitted wavefronts are affected by change of amplitude and phase. 2.3.11 Refraction Refraction is the change in direction of a wave due to a change in its medium. This phenomenon occurs because of an inhomogeniety and discontinuity of the interface atoms which reflects light backward or transmit it forward through the medium. Hence, the fact that the incident rays are bent is called refraction (Hecht, 2002). Wendlandt and Hecht (1966) pointed out a simpler definition for refraction. They supposed that the electromagnetic radiation starts its path through a medium with a specific index of refraction while it reaches to another medium with a different refractive index. Whenever it encounters the separating region between two different media, a part of the beam is reflected back into the first medium, however part continues into the second medium, but with an altered direction of propagation. Hence, the refraction is essentially a surface phenomenon. Electromagnetic radiation is an element in the refraction process, hence the governing equations of electromagnetic radiation are valid when the light refracts from one medium to another. When the 27 light passes through the medium, the phase velocity of the wave is changed but its frequency remains constant. The refraction occurs for any type of electromagnetic wave; from light to any other electromagnetic waves such as sound. Refraction is described by Snell's law, which constructs an expression based on the law of refraction for the relationship between the two incident and refraction angles and the properties of the wave which passes through the boundary of two different isotropic materials. The formula is derived for a given pair of media and a wave with a specified frequency stating that the ratio of the sines of the angle of incidence and angle of transmitted wave after refraction is equivalent to the ratio of phase velocities ( ) in the two media. This ratio is also equivalent to the opposite ratio of the indices of refraction ( ): (2.14) To prove the expression above, one can start from sketching the refraction phenomenon at an interface like Figure 2.2 by drawing the wavefronts at an instant of time. Each of them is a surface that has a constant phase. Because of the change in the phase velocity at the interface, the waves bend (Hecht, 2002). In fact, Figure 2.2 is a representation of several exposures of a single wavefront that shows it in equal time intervals of . When the wavefront OB reaches the interface, it refracts at point A. Assuming that it takes for the wavefront OAB to get to the surface RED with phase velocity of , different points on the surface of the wave traverse with different phase velocities after this time interval because one portion of the wave is in the incident medium with phase velocity and another part is in the transmitting medium with phase velocity of . In this thesis, the analysis is done on the twophase flow of Xylene as the liquid and ADP particles as the solids. Nearinfrared radiation is employed as an electromagnetic wave that penetrates the solidliquid mixture by a sensor which is located in air outside of the mixture. The refractive index of 28 xylene and ADP are greater than the air or the phase velocity in air is greater than that in a liquidsolid mixture . Figure 2.2 Geometric description for the refracted waves approached the interface at equal time intervals Hence, according to Figure 2.2, for the case of nearinfrared waves in air as the incident medium striking on the xyleneADP sample as the transmitting medium, the vector BD is longer than AE because and time intervals are equal. In Figure 2.2, the two triangles ABD and AED share a common hypotenuse which is AD. Therefore, (2.15) where, and , so (2.16) By multiplying both sides by c and since and , one has: (2.17) Incident medium B R E O Transmitting medium A D 29 The equation above is the general law of refraction or Snell’s law. 2.3.12 Reflectance (reflection) and reflectivity Introducing this important concept in optics, demands an understanding of the difference between three distinct terminologies which are common for all light fractions. As pointed out by Siegel and Howell (1981) there are differences between the fractions ending –itivity and –ance. It is common to refer the ending –ivity to intensive, coherent, or bulk properties of matter. The ending –ance is used to assign for the extensive properties of a material. Reflection is a phenomenon by which a part of the light rays that are incident on the medium or particle leave it from the same side or scatter backward. This was a preliminary definition of reflectance that was addressed by Rea (2000). Hecht (2002) explained that reflection occurs when a beam of electromagnetic radiation impinges on a surface of material some light will scatter backward. Reflection, usually refers to a surface reflection not internal or external reflections within the medium. Hecht also clarified this issue for the gradual change in the mediums. In this situation, there is a continuous change of interface and very small amounts of reflection will be observed since the interface is effectively vanished. In contrast, any abrupt change in the interface exerts a discontinuity to the light intensity since a portion is reflected. Internal and external reflections will be discussed later in this section. There is no difference between the concept of reflection and reflectance in the literature, and it is possible to use reflectance in place of reflection or vice versa. However, in some of the references the meaning of reflectance and reflection are separated. For instance, Rea (2000) clearly delineated their boundaries by defining the reflectance as the ratio of reflected flux to the incident flux, while the reflection is a general term for the process by which the incident flux leaves a medium from the incident side or with a reflection angle less than 90 degrees, and without any change in frequency. According to CIE (1987), reflectance is the ratio of the reflected radiant to the incident flux that is radiated on the sample. Reflectivity is generally known as reflectance since it refers to the 30 incident electromagnetic power that is reflected at an interface. However, there is a slightly difference. The bureau of CIE (1987) specified a definition for reflectivity which is distinguished from reflectance. They suggested that reflectivity is the reflectance of a layer of the material with a thickness that does not change the reflectance if it is increased. This means that the sample or object is thick enough that the reflectance is not influenced by adding more thickness to it. This definition for reflectivity includes both internal scattering and surface reflectance. In other words, reflectivity is the restrictive value of reflectance as the surface becomes thick; it can be attributed to the inherent reflectance of the surface. McCluney (1994) claimed that sometimes it would be desirable to limit the definition of reflectivity to include only interface effects. By interpreting this definition, one can say that reflectivity is property of material, while reflection is the property of a particular sample of that material or surface. Dilaura (2011) deduced that reflection is affected by the geometry, wavelength, and polarization of the incident flux. There are three types of surface reflections depend on the radiation flux; specular, mixed, and diffuse reflection. Specular reflection happens on a polished material when the beams are reflected specularly. Specular reflection, also called regular reflection, occurs when all the parallel incident flux is reflected with reflection angles equal to the incidence angle (McCluney, 1994). Hecht (2002) pointed out that the specular reflection process occurs when the light remitted by millions of atoms combines as a single beam. As is obvious from Figure 2.3, what is important is that the scattering effects are absent in the specular reflections. In this Figure, specular angle and incident angles are shown. Specular angle is the angle between the reflected ray and the normal to the surface. For regular (specular) reflections these two angles are equal. Perfectly specular reflections have no diffuse element. 31 Figure 2.3 The geometry of the light generator, detectors, and the sample Another type of reflectance is diffuse reflection. Diffuse reflectance or irregular reflection occurs on samples and materials that consist of particles and crystals or have rough surfaces. Each incident ray encounters an infinitesimal particle which obeys the basic definition and rule of reflection. However, since the shapes of particles in a solidliquid mixture may not be uniform, there are some particles that do not have flat surfaces. Therefore, the reflected rays leave the particle surface in other planes with many different reflection angles and distinct directions. This process is depicted in Figure 2.4(a) where the incident beam strikes a particulate sample but it reflects diffusely in many different angles and in different planes. The intensities of reflected rays are not constant as is shown in Figure 2.4(a) and (b). The illuminated spot has the most energy and the reflected ray from this point has the greatest intensity, then it diminishes as the viewing angle to any side increases (Juds, 1988). Moreover, some of the energy of the incident illumination is attenuated by being transmitted through the sample or being absorbed by the liquid or solid particles inside the slurry which changes into heat and rises the temperature. A beam of parallel rays incident on a surface or medium that has a diffusely reflecting characteristic will be converted to a collection of rays distributed over a hemispherical solid angle. The diffusivity of materials are different. Some produce strong forward scattering whereas others diffuse the light almost in all directions. McCluney (1994) emphasized that if the reflection radiance is constant for all directions, then the surface type is called a Lambertian surface. The term diffuse can also be used for the nonLambertian surfaces. Perfectly diffuse reflectance is special case 32 for the diffuse reflections. This phenomenon occurs when the density and distribution of reflected beams are higher near the surface normal and decreases as the cosine of the angle of the reflected direction. An important point is that perfect diffusion does not mean perfect reflection; it does not mean that the reflection is 1.0. Most surfaces have a mixture of diffuse and specular characteristics. Rea (2000) described this feature on some special surfaces which diffuse most of the light in all directions except for a ray that reflects specularly. Therefore, this type of reflection is partly regular (specular) and partly diffuse. In real situations, a specularly reflecting material has less quantified capability to diffuse the light than reflect it specularly. On the other hand, a diffusely reflecting material reflects the light specularly less than the amount that it diffuses, quantitatively (McCluney, 1994). (a) (b) (c) Figure 2.4 (a) Diffuse reflection from a sample composed of particles (SPECAC Limited, n.d.); (b) Specular versus diffuse reflectance on the surface of a slurry fluid. (c) Diffuse and specular reflectance. Diffuse reflectance (left) sends beams in all directions regardless of the uniform parallel incident rays. Specular reflectance (right) incident cone is preserved (Dilaura, 2011) 33 All types of reflections which occur at interfaces including specular, diffuse, and mixed reflections, are external reflections. Total reflection has two categories; external and internal reflections. Considering the passage of an incident beam from medium 1 (incident medium) to medium 2 (transmitting medium), one can construe the occurrence of an external reflection when the refractive index of the first medium is less than the one for the second or transmitting medium. The reason is that the light is initially traveling from a less to a more optically dense medium. This phenomena happens since the compression of atoms in a dense material does not allow the photons to find a way through the medium, hence the reflection at the interface is strong. In fact, specular and diffuse reflections which are surface reflections are categorized as external reflections. On the other hand, internal reflection occurs when the light going from a region with high refractive index to a region with lower index. Dahm and Dahm (1999) also defined internal reflections as light that has been transmitted through one or more particles, returned from an internal particle, and transmitted through any particle encountered on the way back to the detector. In some occasions, neither reflection nor transmission takes place when incident beam encounters the interface. From a two dimensional perspective, the beam reflects exactly at the line of the interface or in a 3D view it will be reflected in the plane of the interface (Pedrotti et al., 2007). In this thesis the remission is used as the reflection in agreement with the notation used in the literature. Remission is the summation of external reflection, internal reflection, and backward scattering (Dahm & Dahm, 1999). The remission fraction is the ratio of the remitted intensity to incident light; the light that strikes a particle (Dahm & Dahm, 2007). 2.3.13 Scattering Scattering is a form of propagation of electromagnetic waves in a way that energy is attenuated through the scattering medium and remission to many directions (Pedrotti, 2007). Therefore, the scattering will be a result of deviation of light from the medium. Scattering is also called diffuse 34 scattering by Dahm and Dahm (1999). They referred to a process in which light interacts with a particle and changes its path in many directions regardless of the orientation of particle surface. This light is capable of being absorbed or remitted immediately. Mie theory is a powerful tool to exactly predict the scattering. The intensity of scattered light relies on the particle geometry (Dahm & Dahm, 1999). For inferring the difference between scattering and diffuse reflection, Kortum (1969) clarified this very important issue. Before going through the difference, a concise explanation about the dependency of particle size and wavelength is required. Reflection, refraction, and diffraction are three different elements by which the interaction of the wave with a particle is characterized. These three phenomena occurs differently if the particle size is greater than the wavelength of the electromagnetic wave. However, if the dimension of the particle is less than or comparable to the wavelength of light, then it would be impossible to separate the contribution of the concepts of reflection, refraction, and diffraction during the collision of waves with particles. This phenomenon is known as scattering (Kortum, 1969). In contrast, diffuse reflection is the isotropic angular distribution of light in densely packed materials. Single and Multiple Scattering It is very important to introduce two different phenomena about scattering; single and multiple scattering. Single scattering happens when radiation is scattered locally by a local center. If the numbers of scattering centers are high, the scattering will occur many times which is called multiple scattering. The difference between the single and multiple scattering is that single scattering is similar to a random phenomenon; however, multiple scattering is less random. The location of single scattering center is not determined, therefore, it tends to depend on the incident trajectory, which looks random to an observer. Single scattering is described by probability distributions (Bohren, 1998). 35 The reason that multiple scattering is less random than single scattering is because of the averaging of a large number of scattering events. Hence the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by a light beam passing through thick fog. Multiple scattering is analogous to diffusion. Optical scattering centers designed to produce multiple scattering are thus known as diffusers. Forward and Backward Scattering Forward and backward scattering are two types of scattering events. Forward scattering refers to the scattered beams that leave a sample travelling in a direction that produces a component with the same direction as the incident beam. Backward scattering is defined as the scattered light that leaves a sample in the opposite direction of the radiation. 2.3.14 Reflection (remission) Coefficient Reflection coefficient is defined as the difference between the propagation velocities of the two media forming the interface (Guenther, 1990). He offered the following equation for the reflection coefficient between two media: (2.18) where b is the reflection coefficient and and are the index of refractions for the incident and transmitted media, respectively. Or, Pedrotti et al. (2007) suggested the following formula which is the same as Equation 2.18: (2.19) where n is the relative refractive index. Dahm and Dahm (2007) suggested the following definition for the apparent linear remission coefficient of a sample which is the fraction of light remitted from a small thickness of the material divided by the magnitude of that small thickness. 36 (2.20) where is the remission fraction for light that is reflected from a small thickness of material. Chartier (2005) defined the reflection coefficient as a complex number where its modulus is equal to unity. He deduced this definition based on doing some calculation on Fresnel formulas. For modeling purposes in this thesis the formulation of Dahm has been used since it is consistent with the number of different layers. 2.3.15 Absorptance (absorption) In physics, absorption of an electromagnetic wave is a phenomenon in which the energy of a photon in the ray is taken up by the material. This energy is consumed in a nano scale by the electrons of an atom. The activities of the electrons due to radiation heat up the matter. In spectroscopy, absorptance, absorbance or absorption, also called optical density of a material is the logarithmic ratio of the radiation strike on a material to the radiation transmitted through a material. In the literature, the absorption of light through a material has been also called attenuation. When speaking of the linear absorption, the dependency of the attenuation is independent of the intensity of the beam, but in certain conditions, the medium changes its transparency due to the intensity of waves going through, and nonlinear absorption happens (Venkatram, Rao, & Akundi, 2005). McCluney (1994) defined the absorptance as the ratio of the absorbed radiant to the incident flux which has the symbol . Absorbance measurements are often carried out in analytical chemistry. Considering a light beam having an intensity impinging on a sample and an intensity incident on the detector, Dahm and Dahm (2007) suggested that absorbance function is the negative logarithm of the fraction of the incident intensity that strikes a detector. According to this definition the following formula is derived: (2.21) 37 where A is the absorption fraction is the incident intensity, and is the intensity exits the sample. Sometimes it is said that the absorbance is defined for the case of transmission through nonscattering sample as , where T is the transmission fraction (Dahm & Dahm, 2007). In contrast, the absorption function has certainly been useful in the field of nearinfrared reflectance spectroscopy in which it is defined as , where R is the remission fraction of light. But what is suggested by Dahm and Dahm (2007) and is used in this thesis is the meaning of absorbance function for the case of transmission. The reason for using this interpretation of absorption is because of instruments which use logbased absorbance. Dahm and Dahm (2007) stated different usages of logbased transmission and reflection cases. In a transmission measurement, since the fraction of light transmitted is calculated from , absorbance function can be written in the form of . In the experiments that are based on remission, the absorbance is calculated in terms of remission in the form of: (2.22) where R is the remission fraction measured from incident light. However, the exponential form of the absorption fraction is used for quantitative purposes. This form will be introduced completely in Chapter Three. 2.3.16 Absorption Coefficient The traditional definition for the absorption coefficient is the ratio of absorbed light by a thin layer in the material to the length of that small thickness (Dahm & Dahm, 2007). The following equation represents the absorption coefficient for a linear type of material: (2.23) K is the absorption coefficient, d is the thickness of the small layer of sample, and A is the absorption fraction of light inside the medium. An important point about the absorption is related to nonlinear 38 situations. When a specific thickness of the sample for which the absorption fraction is calculated is thick – the thickness criterion is explained in detail in Chapter Three – then the absorption fraction is not a linear function of the depth of the layer. Dahm and Dahm (2007) asserted that as the thickness of the layer under measurement gets smaller, the absorption fraction falls in the linear regions. If the sample is a small layer of material, the absorption is called linear and the value will be linearly proportional to the sample thickness. Dahm and Dahm (1999) also described this linearity for the material through their proposed hypothesis. According to their descriptions, the absorption is linear if all internal parts of the material are equally illuminated. Dahm and Dahm (2007) asserted that absorption coefficient is another definition for the absorption power of the material while there is no scattering in the sample. In a homogeneous sample any particular thickness which absorbs light, has a limit of capacity for absorption. The amount of absorption is in constant or linear proportion to the intensity of radiation. This fraction is independent of: the absolute value of intensity; and the location of that special thickness or layer of the sample. This rule is known as BouguerLambert Law in optical physics. The formulation is: (2.24) where k is the absorption coefficient, T is the transmitted intensity, is the intensity of the light a which passes the thickness of the material in the location of line 0 which is shown in Figure 2.5, and is the intensity of light at the point d, d is the distance between two lines, and d is the absorbing power of the material. Penetrating through the sample, some portion is absorbed and some will be transmitted. Pierre Bouguer (1760) discovered the law of absorption of light through the material, and later it was developed by Johann Lambert (Dahm & Dahm, 2007). This law describes the absorbed light by a homogenous material as a linear fraction of intensity on such a layer. The limitation of this technique is its homogeneity. 39 The absorbing power of material is known as the negative of the natural log of the intensity transmitted through a particular distance d of the sample (Dahm & Dahm, 2007). The absorbing power of a material is the same as the absorbance of a homogeneous sample of the material of unit thickness. For example, the absorbing power of a material is the same as the absorbance of a homogeneous sample of the material that is 1 cm thick: (2.25) Figure 2.5 Illustration of BouguerLambert law Therefore, the absorbing power of the material or k is calculated by the absorbance of the homogeneous sample of the material. It should be noticed that kd does not have any units. If one divides the relation above by the distance that light travels, d, the result will be the absorption power which is derived from the following equation: (2.26) d k is independent of thickness 40 For a portion of sample that is under direct illumination, the absorbing power of the material is the ratio of the amount of light that would be absorbed if every absorption portion were under incident intensity equal to the incident ray, to the intensity of the incident beam (Dahm & Dahm, 2007). The thickness of the layers is calculated in the units of . In this thesis will refer to the absorption coefficient of a sample and is the absorption coefficient of a material which is also known as absorption power of material. 2.3.17 Transmittance (transmission) Transmittance is the fraction of electromagnetic wave at a specified wavelength that passes through a sample. A more concise definition is the movement of light through a sample of material without being absorbed (Dahm & Dahm, 1999). The important point about transmission is that it is incorporated with the fact that suggests the amount of light that is not absorbed, is reflected or transmitted. The only difference between the reflection and transmission is the angle of incidence and the angle that light makes with the normal to the surface of interface. If the light is transmitted through the medium with an angle less than the critical angle – at which the remitted beam falls exactly on the interface line – the light is transmitted, whether internally within the layers of sample or externally outside of the medium. In this situation the light is refracted and it is transmitted to the other layers inside the sample or it is coming out of the sample. However, when the incident light strikes the sample at an angle more than the critical angle, the beam that is leaving the medium is said to be reflected. Therefore, the boundary between the reflectance and transmittance is very close. Dahm and Dahm (2007) pointed out that the transmission fraction is calculated from the following equation which is derived based on the Bouguer – Lambert law for transmission. The law says that the fraction of light beam transmitted through the sample while moving through the distance d inside a continuum medium is calculated from: 41 (2.27) where k is the linear absorption coefficient in the absence of scattering inside the sample. Considering this limiting assumption, one can conclude that this expression is designed for the samples with high concentration of particles. The reason is that by increasing the number of particles the chance of light scattering will increase, which is not consistent with the assumption of this law. But fortunately this equation is valid for the small particles according, as shown in Figure 2.6. The straight line in Figure 2.6 is a tangent line to the relation that represents the deviation of transmission curve from the line C with the relation kd. This line can be used instead of the exponential function in the linear region. The linear region is essential to be found since the validity of some theories such as BouguerLambert law and plane parallel layers are dependent upon this assumption. In Chapter Three, this linearity assumption is introduced. Furthermore, least squares method to find threshold for the minimum deviation from the straight line is presented not only for transmission but also for absorption. Figure 2.6 (A) Absorption fraction; (B) transmission fraction ( ; (C) straight line representing Kd which lies on the transmission curve in smaller thicknesses of the sample or layer of the sample which is illuminated (Dahm & Dahm, 1999) 42 2.4 Continiuum Theories of Diffuse Reflection 2.4.1 Introduction Nearinfrared (NIR) diffuse reflection (DR) has become extensively applicable for the quantitative analysis of many products and commodities since it 40 years ago when it was first used in spectroscopy. As was mentioned in Chapter 1, midinfrared diffuse reflectance is used more for investigating and characterizing than for the determining the quantity of particulate samples. Considering the mentioned cases, investigating the diffuse reflectance process is useful if the very large amount of information is to be collected from the spectrum. Those theories of diffuse reflectance that use simple mathematics has connections with the reflectance of a sample which is known by . In this situation, the assumption which says all light being reflected from both a sample and a a standard reflecting reference with 100% reflectance can be measured (Chalmers & Griffiths, 2002). Chalmers and Griffiths (2002) suggested that the relative diffuse reflectance can be measured by estimating the ratio of the spectrum of the radiation received by the detector from a particulate sample and the corresponding spectrum which can be estimated experimentally after reflection from a nonabsorbing reference matter. In the cases which the material thickness is large, any change in the thickness does not influence on the power of the radiation which is going to be received by the detector. Therefore, the power will not change. The measured spectrum is called the reflectance spectrum at infinite depth, (Chalmers & Griffiths, 2002). Chalmers and Griffiths (2002) asserted that all algorithms for multivariate analysis obtain the most accurate calculations about the concentration when the intensity of each spectral parameter is proportional to the concentration of the anlayte in a linear fashion. Hence, diffuse reflectance will be changed into the linear form if an ideal standard for the analysis of diffuse reflectance is required by applying theories. In the linear condition, the dependence of intensity of the spectrum on the concentration of each analyte is linear which is analogous to the conversion of transmittance spectra to 43 absorbance for considering Beer’s law (Chalmers & Griffiths, 2002). For many cases in nearinfrared diffuse reflectance spectrometry, a conversion which is equivalent to what is mentioned is performed. For instance, before applying the theorems to the sample and model the light interactions, one may require to change the spectrum in to the form of –log or log . It is significant to mention that most NIR/DR spectra is usually estimated using log1/R which, for thick samples, is the same as log (Chalmers & Griffiths, 2002). Fortunately, for many materials the concentration of the sample under anlysis will not change most of the time by even a factor of two. Therefore, the linear assumption is valid and the deviation from linearity is not considerable. Hence, Chalmers and Griffths (2002) concluded this approach usually have excellent results. Applying the parameter of log 1/R as the preferred ordinate for nearinfrared diffuse reflection is in opposite side to what most physical scientists would consider appropriate for experimental part of a diffuse reflectance measurement (Griffiths & Dahm, 2008). The theory of Kubelka and Munk is one of the mostly used theories of diffuse reflectance spectroscopy. In this Chapter, this theory is introduced with a summary of other continuum theories of diffuse reflection. The assumptions of the theories are very important since they produce limitations. These constraints only affect the accuracy of the results. However, all of these methods have been validated through experiments. The last part of this thesis contains a discussion of some studies and measurements about the depth of penetration of light. The information and models are investigated based on the available theories in the literature. Modeling the penetration depth in a twophase flow of xylene and ADP is achieved using the theorems, which are to be discussed in this Chapter. 2.4.2 Lambert Cosine Law The diffuse reflectance phenomenon is a widely applicable usages of nearinfrared radiation, which is easily noticed in everyday life. Griffiths and Dahm (2002) made an example for the intensity of radiation reflected from a completely matte surface. The reflected or remitted radiation has equal 44 intensity in all directions. Also, it is not important to know what is the angle of incidence or angle of observation of light from the surface. This experiment made Lambert to be the first person who created a mathematical formula and theory based on the diffuse reflectance of light (Chalmers & Griffiths, 2002). He assumed that the reflected flux which is known by , in an area , and solid angle steradians (sr), is proportionate to the cosine of the angle, which the incident beam builds with the normal to the surface, , and the angle of observation that are shown in Figure 2.7. This is obtained from the Equation 2.28: (2.28) Figure 2.7 Diagram showing the variables used in the Lambert cosine law (Griffiths & Olinger, 2002) where B is the radiation density or surface brightness in W , is the irradiation intensity in W for normal incidence, and the constant C is the fraction of the reflected intensity of radiation. C is less than unity because some portions of radiation flux are always absorbed with the surface. Equation 2.28 is famous since it is known as the Lambert cosine law (Chalmers & Griffiths, 2002). Kortum (1969) suggested that the Lambert cosine law can be derived from the second law of thermodynamics, although this opinion was not held by Wendlandt and Hecht (1966). Kortum (1966) pointed out that it is rigorously valid only for a material which has the characteristics of a blackbody. 45 These black body radiators acting as an ideal diffuse reflector. Griffiths and Dahm (2008) pointed out that the angular distribution of the reflected or remitted light is not dependent of the angle of incident radiation. This fact is in contradiction with tehe features of blackbody radiator since ideal diffuse reflectors has some different characteristics. This contradiction is made since all incident radiation is absorbed by a black body, however, an ideal diffuse reflector does not absorpb all the radiation. An ideal diffuse reflector is not realistic and has never been found in practice. Hence, the Lambert cosine law is an ideal descriptor of diffuse reflection which always there are deviations from that. According to what Chalmers and Griffiths (2002) stated, there are many authors that have reported the results of experimental investigations that were prepared to prove or disprove this law. They found that in the law is only valid when both the angle of incidence and the angle of observation are small. 2.4.3 Mie Scattering One of the most applicable and more accepted theories of the scattering of light was developed around by Mie. Mie scattering theory models the travel of light into the isolated particles, and predicts the absorption by and scattering from that particle. Only a very brief introduction is given here. Kortum (1969) has pointed out a summarized description about Mie scattering theory. Mie developed and derived a model to predict the following factors: (i) the angular distribution of the intensity, and (ii) the angular distribution of polarization of scattered radiation for a plane wave which is scattered once through a phenomena called single scattering by a particle (Griffiths & Dahm, 2008). These two phenomena can be both dielectric and absorbing. Mie assumes that the particle is spherical, and there is no limit on its size. He presented that the angular distribution of scattered radiation for single scattering will not be isotropic. The basic equation of Mie scattering theory is: (2.29) 46 where , is the scattered intensity at a distance R from the center of the sphere; is the intensity of the incident radiation, and is the wavelength of the incident radiation (Chalmers & Griffiths, 2002). The variables and are functions of the angle of the scattered radiation , the refractive index difference between particle and medium, and the ratio of the particle circumference to wavelength. The ratio of the refractive index of the sphere to its environment is known by m, and the ratio of the particle circumference to wavelength is known by p (Chalmers & Griffiths, 2002). Equation 2.29 applies only to the case of a dielectric nonabsorbing particle. If the particle absorption is high, the complex refractive index must be used in the determination of and . Mie theory is general for spherical particles of any size. However, it is valid only for single scattering, therefore, is applicable only to systems in which particles are well separated (Chalmers & Griffiths, 2002). For instance, when the particles in the atmosphere scatter light, the special case of Mie theory is obtained where the particle is much smaller than the wavelength of incident radiation. Most analyses in the range of midinfrared and nearinfrared spectroscopy the multiple scattering is a definite assumption since it happens in real samples (Chalmers & Griffiths, 2002). Theissing (1950) improved Mie theory by assuming that the multiple scattering from particles which nonetheless are still supposed to be sufficiently well dispersed and separated in the medium. Therefore, the interference and phase differences between the scattered fluxes from the particles are not important and can be neglected. The number of times when a photon is scattered and distributed in the medium is defined as the scattering order (Chalmers & Griffiths, 2002). By increasing the order of scatter the value of forward scattering in the sample is doing an opposite behavior and will decrease, therefore, the angular distribution of scattered light does not change with direction and in all directions is isotropic. He also discovered that the if the ratio of particle circumference to wavelength is larger the order of scattering must increase to generate an isotropic scattering. For example, if p is 0.6 and m is 1.25, twofold scattering is required for an isotropic distribution of the reflected radiation. However, 47 while p = 5 and m = 1.25, a scattering order of 8 is needed for isotropic reflection of radiation (Chalmers & Griffiths, 2002). In the NIR region, band absorptions are so low that large particle sizes, of the order of 100 microns, can be calculated using Mie theory (Griffiths & Dahm, 2008). Hence, p, the ratio of the particle circumference to wavelength, is large. In this case, the order of scattering must also be large in order to have an isotropic variation of the scattered radiation. In the midinfrared range, this feature is changed where it is quite common for p. Hence, it will be approximately around 1 (Griffiths & Dahm, 2008). If the number of particles in a sample is large, multiple scattering does take place for most samples of the type used for infrared reflection analysis, therefore, the diffusely reflected radiation has an isotropic distribution that should at least be reached. Hence, to predict multiple scattering in a densely packed medium is required to model the variation of reflectance with the change in concentration (Griffiths & Dahm, 2008). In most of the samples for which the nearinfrared diffuse reflection can be used, the scattering density is large, the ratio of particle circumference to wavelength is higher than 1, and the concentration of particles are high enough that phase relations and interferences between scattered beams do exist (Chalmers & Griffiths, 2002). Hence, in this case, there is no solution to the problem of multiple scattering in the sample. Therefore the scientist must resort to the phenomenological analysis may be required. 2.4.4 Radiation Transfer Treatments Theories that have been developed to predict the diffuse reflection of radiation are built based on the radiation transfer equation. A radiation transfer equation can be written as if one simplify it: (2.30) 48 Equation 2.30 is a description of the change in intensity dI of radiant flux in a specific wavelength in a sample with the density and with the pathlength of ds. describes the attenuation coefficient for the total radiation loss. This dissipation can be due to scattering or absorption. The radiation transfer equation that is used in the derivation of most phenomenological analyses considers only plane parallel layers of particles within the sample and can be written as: (2.31) where is the cosine of the angle with respect to the outward surface normal; m is the cosine of the angle with respect to the inward surface normal; is the optical thickness and is equal to , where dx is the distance between the boundaries of one planeparallel layer; I is the intensity of the beam of radiation striking the layer; is the albedo for single scattering, with the scattering and absorption coefficients and (Chalmers & Griffiths, 2002). The scattering phase function is the probability for scattering from direction into . If each beam scatters isotropically, and is not dependent to the angle between the incident radiation and the scattered radiation. The phenomenological analyses that have been developed can be deemed as continuum theories or discontinuum theories (Chalmers & Griffiths, 2002). In continuum theories, the absorption and scattering coefficients are the properties of the irradiated material. However, in discontinuum theories a layer of the material is made up of several partial layers whose thickness is limited by the size of the scattering and absorbing particles. It means that based on the size of particles the thicknesss of the layers must change (Chalmers & Griffiths, 2002). Optical constants can then be found from the scattering and absorption of these particles. This section only considers a sample as a continuum. In our modeling, we have used the method of approximation for the radiation transfer equation proposed by Patterson, Schwartz, and Wilson (1989). Figure 2.8 is an experimental apparatus that they used in their method. When the light reflects diffusely from the tissue containing a dye, the 49 changes in the reflectance is identified by the concentration of dye. In Figure 2.8, the measurement setup is shown. In this figure, an experiment based on radiative transfer approximation is illustrated. Figure 2.8 (a) Experimental apparatus for the measurement of R( ), the local diffuse reflectance (b) Apparatus for the measurement of R, the total diffuse reflectance (Patterson et al., 1989) 2.4.5 Schuster’s Theory Schuster’ theory was one of the earliest theories of diffuse reflectance. He was interested in astrophysical problems that were related to radiation transfer. From the general laws of radiation, Schuster suggested that an atmosphere is required for a star to show emission lines of light instead of absorption lines. The characteristics of this atmosphere is that the scattering overcomes the absorption. As mentioned by Nobbs (1985), the approach adopted by Kubelka and Munk was first introduced by Schuster in 1905 and was developed to calculate the radiation from the selfluminous opaque dust cloud surrounding a star. This theorem will be modeled with details in Chapter Three. 2.4.6 KubelkaMunk Theory KubelkaMunk is a widely used theory in the field of diffuse reflectance. The theory was basically developed in the field of spectroscopy. However, recently there were many debates on the revisions 50 over it to increase its accuracy and the number of different applications. The theory developed by Kubelka and Munk is obtained based on solutions to the radiative transfer problem. This theorem was first used to investigate the optics of paint layers. Kubelka and Munk developed their models based on the remission, not scattering. Kublka – Munk equations are produced from radiative transfer and are in the form of differential equations. The KubelkaMunk equation is a two flux version of a multiflux method used to solve the radiation transfer problem (Nobbs, 1985). Kubelka and Munk made several assumptions in their derivation of a simplified solution to the radiation transfer equation. Note that none of them are exactly the same as the corresponding parameters defined by Schuster. Figure 2.9 shows the type of system for which Kubelka and Munk derived their solution. Figure 2.9 Schematic representation of a layer of absorbing and light scattering particles The assumptions are listed below: The radiation fluxes ( and ) travel in two opposite directions. 51 The sample is illuminated with monochromatic radiation of intensity I0. The distribution of scattered radiation is isotropic so that all regular (specular) reflection is ignored. The particles in the sample layer (defined as the region between x = 0 and x = d) are randomly distributed. The particles are very much smaller than the thickness of the sample layer d. The sample layer is subject only to diffuse irradiation. The breadth of the macroscopic sample surface (in the plane) is great compared to the depth of the sample and the diameter of the beam of incident radiation (to discriminate against edge effects). The scattering particles are distributed homogeneously throughout the entire sample. Either an exponential or hyperbolic solution may be developed, although only the exponential derivation is shown here. A detailed description of the hyperbolic solution is given elsewhere. Kubelka and Munk arrived at the two fundamental differential equations which, once solved, give the simplified solution to Schuster’s fundamental function. As it is feasible that the angle through , , which the radiation might follow could be between and , the average pathlength for radiation passing through dx in the +x direction can be found by the integral: (2.32) After some mathematical operations and manipulations, which are discussed in detail in Chapter Three, one can find the two fundamental differential equations for KubelkaMunk theory: (2.33) (2.34) 52 2.4.7 Diffusion Theory Kienle et al. (1998) presented a different approach to investigate the radiation transfer equation. This approach is applicable in biomechanics and biomedical engineering. In the paper that is published by Kienle et al. (1998), the diffusion approximation is used to determine the optical properties of biological tissue. Griffiths and Dahm (2008) pointed out that the propagation of photons in a thick sample with scattering coefficients which are larger than the absorption coefficients, are described by diffusion approximation. The details of this approach are described in Chapter Three and the results are discussed in Chapter Four. 2.5 Discontinuum Theories of Diffuse Reflection 2.5.1 Representative Layer Theory A layer is representative of a sample if each one of the particles in the sample provide the same volume and surface area fractions in each layer as in the sample (Dahm & Dahm, 2007). The reason for using the volume fraction of the particles is because the absorption fraction of the particle is completely related to volume. On the other hand, the remission is strongly dependent on the surface area fraction of the particle. The important question is what makes a layer be representative of a sample. Dahm and Dahm (2007) demonstrated three reasons for a layer to be representative. First, the thickness of the layer should have no more than the thickness of one particle since one of the assumptions of representative layer theory was related to the scattering and absorption properties of the layer by combining the mentioned properties for each particle in the layer. Therefore, it would be more suitable to use the thickness for the layer to be nowhere more than one particle thick. Second, another feature of the representative layer introduced by Dahm and Dahm (2007) is that the volume fraction of each particle in the layer is the same as the sample. One of the most 53 important assumptions of representative layer theory is related to the voids. Thus, the media that represents the voids does not absorb or scatter light. This means that if the media is a fluid its absorption and scattering must be neglected. Moreover, the void fraction of the representative layer must be the same as the sample. Finally, the third reason is that the surface area fraction of each particle in the layer will be the same as the sample. Surface area fraction is a property that is more difficult to work with than volume fraction because the shape and size of the particles are required to estimate the total surface area fraction. In the model of the representative layer that we present here, a particle is of a different type from another 



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