This dissertation presents a methodology to simulate the dispersion of water droplets in the air flow typical of an Icing Tunnel. It is based on the understanding the physical parameters that influence the uniformity and the distribution of cloud of droplets in the airflow and to connect them with analytical parameters which may be used to describe the dispersion process. Specifically it investigates the main geometrical and physical parameters contributing to the droplets dispersion at different tunnel operative conditions, finding a consistent numerical approach to reproduce the local droplets dynamic, quantifying the possible limits of commercial CFD methods, pulling out the empirical parameters/constant needing to simulate properly the local conditions and validating the results with calibrated experiment. An overview of the turbulence and multiphase flow theories, considered relevant to the Icing Tunnel environment, is presented as well as basic concepts and terminology of particle dispersion. Taylor’s theory of particle dispersion has been taken as starting point to explain further historical development of discrete phase dispersion. Common methods incorporated in commercial CFD software are explained and relative shortcomings underlined. The local aerodynamic condition within tunnel, which are required to perform the calculation with the Lagrangian particle equation of motions, are generated numerically using different turbulent models and are compared to the historical K-ε model. Verification of the calculation is performed with grid independency studies. Stochastic Separated Flow methods are applied to compute the particle trajectories. The Discrete Random Walk, as described in the literature, has been used to perform particle dispersion analysis. Numerical settings in the code are related to the characteristics of the local turbulent condition such as turbulence intensity and length scales. Cont/d.

This dissertation presents a methodology to simulate the dispersion of water droplets in the air flow typical of an Icing Tunnel. It is based on the understanding the physical parameters that influence the uniformity and the distribution of cloud of droplets in the airflow and to connect them with analytical parameters which may be used to describe the dispersion process. Specifically it investigates the main geometrical and physical parameters contributing to the droplets dispersion at different tunnel operative conditions, finding a consistent numerical approach to reproduce the local droplets dynamic, quantifying the possible limits of commercial CFD methods, pulling out the empirical parameters/constant needing to simulate properly the local conditions and validating the results with calibrated experiment. An overview of the turbulence and multiphase flow theories, considered relevant to the Icing Tunnel environment, is presented as well as basic concepts and terminology of particle dispersion. Taylor’s theory of particle dispersion has been taken as starting point to explain further historical development of discrete phase dispersion. Common methods incorporated in commercial CFD software are explained and relative shortcomings underlined. The local aerodynamic condition within tunnel, which are required to perform the calculation with the Lagrangian particle equation of motions, are generated numerically using different turbulent models and are compared to the historical K-ε model. Verification of the calculation is performed with grid independency studies. Stochastic Separated Flow methods are applied to compute the particle trajectories. The Discrete Random Walk, as described in the literature, has been used to perform particle dispersion analysis. Numerical settings in the code are related to the characteristics of the local turbulent condition such as turbulence intensity and length scales. Cont/d.

Abdominal aortic aneurysm (AAA) disease involves a dilation of the aorta below the renal arteries. If the aneurysm becomes sufficiently dilated and tissue strength is less than vascular pressure, rupture of the aorta occurs entailing a high mortality rate. Despite improvements in surgical technique, the mortality rate for emergency repair remains high and so an accurate predictor of rupture risk is required. Inflammation and the associated recruitment of monocytes into the aortic wall are critical in the pathology of AAA disease, stimulating the degradation and remodeling of the vessel wall. Areas with high concentrations of macrophages may experience an increase in tissue degradation and therefore an increased risk of rupture. Determining the magnitude and distribution of monocyte recruitment can help us understand the pathology of AAA disease and add spatial accuracy to the existing rupture risk prediction models. In this study finite element computational fluid dynamics simulations of AAA haemodynamics are seeded with monocytes to elucidate patterns of cell deposition and probability of recruitment. Haemodynamics are first simulated in simplified AAA geometries of varying diameters with a patient averaged flow waveform inlet boundary condition. This allows a comparison with previous experimental investigations as well as determining trends in monocyte adhesion with aneurysm progression. Previous experimental investigations show a transition to turbulent flow occurring during the deceleration phase of the cardiac cycle. There has thus far been no investigation into the accuracy of turbulence models in simulating AAA haemodynamics and so simulations are compared using RNG κ − ε, κ − ω and LES turbulence models. The RNG κ − ε model is insufficient to model secondary flows in AAA and LES models are sensitive to inlet turbulence intensity. The probability of monocyte adhesion and recruitment depends on cell residence time and local wall shear stress. A near wall particle residence time (NWPRT)model is created incorporating a wall shear stress-limiter based on in vitro experimental data. Simulated haemodynamics show qualitative agreement with experimental results. Peaks of maximum NWPRT move downstream in successively larger geometries, correlating with vortex behaviour. Average NWPRT rises sharply in models above a critical maximum diameter. These techniques are then applied to patient-specific AAAs. Geometries are created from CT slices and velocity boundary conditions taken from Phase Contrast-MRI (PC-MRI) data for 3 patients. There is no gold standard for inlet boundary conditions and so simulations using 3 velocity components, 1 velocity component and parabolic flow profiles at the inlet are compared with each other and with PC-MRI data at the AAA midsection. The general trends in flow and wall shear stress are similar between simulations with 3 and 1 components of inlet velocity despite differences in the nature and complexity of secondary flow. Applying parabolic velocity profiles, however, can cause significant deviations in haemodynamics. Axial velocities show average to good correlation with PC-MRI data though the lower magnitude radial velocities produce high levels of noise in the raw data making comparisons difficult. Patient specific NWPRT models show monocyte infiltration is most likely at or around the iliac bifurcation.

In this thesis, trajectory computations of chaff particles ejected from a medium weight utility helicopter are performed using computational fluid dynamics. Since these chaff particles are ejected from a helicopter and carried by its flow field, it is necessary to compute and include the effects of the helicopter flow field in general and engine hot gases, main and tail rotor wakes in particular. The commercial code FLUENT is used for flow field and trajectory computations. Both main rotor and tail rotor are simulated by the so-called Virtual Blade Model in a transient fashion. Flows through the engine inlets and exhausts are treated via appropriate boundary conditions in the analysis. The generic ROBIN geometry is studied first in order to assess the accuracy of the Virtual Blade Model and various turbulence models. The computational solutions related to the ROBIN geometry are validated against the available experimental data. Flowfield and trajectory computations of chaff particles are done at a forward flight condition at which certain flight data and chaff trajectory data were acquired by ASELSAN, Inc. In the flight test, three successive chaff decoy ejections were conducted, and the chaff cloud distributions were recorded by two high-speed cameras positioned on two different locations on the helicopter. Numerical calculations employ the post-processed camera recordings for setting the initial distributions of the chaff particles. Then, the computational results related to the chaff particle trajectories are validated by comparing to the recorded transient chaff cloud distributions from the ASELSAN flight test. For post-processing of the recorded chaff distributions, an experimental analysis commercial code called TrackEye is used. It is found that the numerical simulations capture the trends of chaff particle distributions reasonably well.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / M.H. Hosni / Z.C. Zheng / In order to study the capability of computational methods in investigating the mechanisms associated with disease and contaminants transmission in aircraft cabins, the Computational Fluid Dynamics (CFD) models are used for the simulation of turbulent airflow, tracer gas diffusion, and particle dispersion in a generic aircraft cabin mockup. The CFD models are validated through comparisons of the CFD predictions with the corresponding experimental measurements. It is found that using Large Eddy Simulation (LES) with the Werner-Wengle wall function, one can predict unsteady airflow velocity field with relatively high accuracy. However in the middle region of the cabin mockup, where the recirculation of airflow takes place, the accuracy is not as good as that in other locations. By examining different k-ε models, the current study recommends the use of the RNG k-ε model with the non-equilibrium wall function as a Reynolds Averaged Navier Stokes (RANS) model for predicting the steady-state airflow velocity data. It is also found that changing the cabin air-inlet nozzle height has a significant effect on the flow behavior in the middle and upper part of the cabin, while the flow pattern in the lower part is not affected as much. Through the use of LES and species transport model in simulating tracer gas diffusion, very good agreement between predicted and measured tracer gas concentration data is observed for some monitoring locations, but the agreement level is not uniform for all the sampling point locations. The reasons for the deviations between predictions and measurements for those locations are discussed. The Lagrange-Euler approach is invoked in the particle dispersion simulations. In this approach, the equation of motion for the discrete phase is coupled with the continuous phase governing equations through the calculation of drag and buoyancy forces acting on particles. The continuous phase flow is turbulent and RANS is employed in order to calculate the continuous phase velocity field. A complete study on grid dependence for RANS simulation is performed through a controllable regional mesh refinement scheme. The grid dependence study shows that using unstructured grid with tetrahedral and hybrid elements in the refinement region are more efficient than using structured grid with hexahedral elements. The effect of turbulence on the particle dispersion is taken into account by using a stochastic tracking method (Discrete Random Walk model). One of the significant features of this study is the investigation of the effect of the number of tries on the accuracy of particle concentration predictions when Discrete Random Walk is used to model turbulent distribution of particles. Subsequently, the optimum number of tries to obtain the most accurate predictions is determined. In accordance with the corresponding experimental data, the effect of particle size on particle distribution is also studied and discussed through the simulation of two different sizes of mono-disperse particles in the cabin with straight injection tube, i.e., 3µm and 10µm. Due to the low particle loading, neglecting the effect of particles motion on the continuous phase flow-field seems to be a reasonable, simplifying assumption in running the simulations. However, this assumption is verified through the comparison of the results from 1-way and 2-way coupling simulations. Eventually through the simulations for the particle injection using the cone diffuser, the effects of cabin pressure gradient as well as the particle density on particles dispersion behavior are studied and discussed. In the last part of this dissertation, the turbulent airflow in a full-scale Boeing 767 aircraft cabin mockup with eleven rows of seats and manikins is simulated using steady RANS method. The results of this simulation cannot only be used to study the airflow pattern, but also can be used as the initial condition for running the tracer gas diffusion and particle dispersion simulations in this cabin mockup.

This master’s thesis deals with analytical and numerical description of surface gravity waves. Wave theories and their influence on water particle movement is described in the theoretical part of the thesis. Water particle moves in the same direction as wave propagation and this phenomenon is called Stokes drift. It has a significant influence on sediment transport and floating particle movement at water free surface. The experimental part consists of wave profile monitoring and water particle tracking in a wave flume with wave generator and beach model. The experimental results are compared with numerical simulation performed in the ANSYS Fluent software. Finally, the wave profiles obtained from simulation are compared with experimental wave profiles extracted by digital image processing.

Made available in DSpace on 2015-05-14T12:14:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 705049 bytes, checksum: 4794f7004746261efe9996d47989dd1f (MD5) Previous issue date: 2014-10-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we presented models of two state stochastic systems which interact through a global coupling, in a way that each population unit contributes to the state transition rates of the other units. We presented two models of global coupling in which is possible to observe a phase transition of a regime with units equally distributed on the two states to a phase where there is an agglomeration of units in one of the states. In the first coupling model this transition occurs in a continuous way as we increase the coupling parameter. Through a mean field approximation we shown that this phase transition occurs due to a subcritical pitchfork bifurcation where one of the phases is associated to a monostable regime (units equally distributed in the two states) and the other phase to a symmetric bistable regime (majority of the units agglomerated in one of the states). On the other hand the other model presents a discontinuous phase transition as we increase the coupling parameter, the mean field approach shows that this phase transition occurs due a supercritical pitchfork bifurcation where we have a monostable regime and a tristable regime presenting symmetry in relation to the central potential well, as the coupling parameter is increased the central stability reduces while the two other states becomes more stable. It was shown that for both coupling models, when we have a finite number of oscillators the system presents a multiplicative noise structure. This noise structure turns the stable states obtained with the mean field approximation on metastable states, also the fluctuations due to a finite number of units breaks the symmetry in the multistable regimes, this symmetry break occurs due to the asymmetric intensity of the fluctuations. We also obtained a Fokker-Planck equation for this system and the probability distribution of the number of units in each state, from this distribution it was possible to build a phase diagram for the phase transition from themonostable regime to the regime that presents multistability. This transition is characterized in terms of the coupling parameter and the number of units in the system. / Nesta tese, apresentamos modelos de sistemas estocásticos de dois níveis que interagem através de um acoplamento global, de forma que o estado ocupado por cada unidade da população influi na taxa de transição de estado das demais. Apresentamos dois modelos de acoplamento global onde é possível observar uma transição de fase de um regime onde as unidades estão distribuídas igualmente entre os dois estados para uma fase onde há a aglomeração de unidades em um dos estados. Em um dos modelos de acoplamento essa transição ocorre de forma continua com o parâmetro de acoplamento. Através de uma aproximação de campo médio mostramos que essa transição de fase ocorre devido a uma bifurcação de forquilha sub-crítica onde uma das fases ´e associada a um regime monopolista (unidades igualmente divididas entre os dois estados) e a outra fase a um regime bioestavel simétrico (maior parte das unidades aglomeradas em um dos estados). J´a o outro modelo apresenta uma transic¸ ao de fase descont´ınua com o par ametro de acoplamento. A abordagem de campo m´edio revela que essa transic¸ ao de fase ocorre atrav´es de uma bifurcac¸ ao de forquilha supercr´ıtica onde temos um regimemonoest´avel e um regime triest´avel apresentando simetria com relac¸ ao ao poc¸o de potencial central e a medida que o par ametro de acoplamento ´e aumentado a estabilidade central diminui enquanto os outros dois estados se tornam mais est´aveis. Foi mostrado que para ambos os modelos de acoplamento, quando temos uma quantidade finita de osciladores o sistema apresenta uma estrutura de ru´ıdo multiplicativo. Essa estrutura de ru´ıdo torna os estados est´aveis obtidos com a aproximac¸ ao de campo m´edio em estados metaest´aveis. Tamb´em foi mostrado que as flutuac¸ oes devido a quantidade finita de unidades quebra a simetria nos regimes com multiestabilidade, essa quebra de simetria ocorre devido a assimetrias da intensidade das flutuac¸ oes. Obtemos tamb´em uma equac¸ ao de Fokker-Planck para esse sistema. A soluc¸ ao da equac¸ ao de Fokker-Planck nos d´a a distribuic¸ ao de probabilidade da quantidade de unidades em cada estado. Essa distribuic¸ ao torna poss´ıvel a construc¸ ao de um diagrama de fases para a transic¸ ao de fase dos regimes monoest ´aveis para os regimes que apresentam multiestabilidade. Essa transic¸ ao ´e caracterizada em termos do par ametro de acoplamento e da quantidade de unidades do sistema.

Gas atomization is regarded as a reliable method for creating high-quality metal powders from molten metal. The liquid metal is fed into the chamber as a free-falling stream through a nozzle, where it is impinged by high-velocity gas jets, causing degeneration and production of metal droplets, which solidify to create metal powders. As the metal droplets fall lower towards the collection hoopers, the solidification process begins. As a result, having a strong handle on the process parameters helps to produce metal powders that are fine, spheroidized, and have good characteristics. A free fall atomizer with twelve discrete nozzles, having a cylindrical internal profile, arranged in two different levels has been employed to introduce high-velocity gas jets into the chamber, for the current study. A cross-sectional sketch created by Uddeholm AB provided the geometric dimensions, and CFD was used to generate a simulation experiment for the system. Fluent setup input values were derived from literature data. The primary objective of the study is to analyze the influence of varying inlet pressure and the number of discrete gas jet nozzles, on the flow behavior of the atomizing gas. Additionally, the Discrete Phase Model approach was adopted to study the interaction of particles with the gas flow. The simulation model was validated by carrying out the visualization experiment, Schlieren imaging. From the study, it was realized that the results of the numerical model showed a mismatch relative to the experimental value. This can be attributed to the discretization technique, input parameters and the numerical model employed in this study. However, the parametric study provided a qualitative analysis regarding the influence of input parameters on flow behavior. It was studied that with increasing the inlet pressure and number of discrete nozzles there is a subsequent increase in the maximum velocity attained by the atomizing gas, resulting in a decrease in velocity of melt introduced into the system. Additionally, a radial pressure gradient was observed to be present that increased in accordance with the parameters, resulting in reduction of the melt film thickness produced during pre-filming mechanism. However, the Discrete Phase Model provided evidence that with increment in the gas to melt ratio, the number of particles that get dispersed to make collision with the domain wall increased. Thus, a subsequent increase in downstream velocity was required to maintain the particles within the domain walls as the study parameters were increased. / Gasatomisering betraktas vara en tillförlitlig metod för att skapa högkvalitativt metallpulver från smält metall. Den flytande metallen matas in i en kammare som en fri fallande ström genom ett munstycke, där den trycks in av höghastighetsstrålar. Vilket skapar en degeneration och en bildandet av metalldroppar som stelnar till att metallpulver kan skapas. Stelningsprocessen börjar när metalldropparna faller emot uppsamlingsbågarna. Ett fint och sfärisk metall metallpulver med goda egenskaper kan produceras genom att ha en god kontroll på processparametrarna. I denna studie har en ”Free Fall Atomizer” med tolv diskreta munstycken med en cylindrisk profil arrangerade i två olika nivåer använts för att introducera höghastighetsstrålar i kammaren. De geometriska dimensionerna var försedda från en ritning i tvärsnitt skapad av Uddeholms AB och samt användes en CFD för att generera ett simulationsexperiment av systemet. Ingångsvärden för inställningarna av flödena härleddes från litteraturdata. Huvudsyftet med studien var att studera flödesbeteendet av den atomiserande gasen genom att analysera inflytandet av att variera ingångstrycket och antalet diskreta gasstrålmunstycken. Dessutom togs det till en diskret fasmodell för att studera partiklarnas interaktion med gasflödet. Simulationsmodellen validerades genom att utföra ett visualiseringsexperiment genom Schlierenfotografering. Det framgick i studien att den numeriska modellens resultat inte stämde överens med det experimentella värdet. Detta kan attribueras till diskretiseringstekniken, inmatningsparametrarna och den numeriska modellen som användes i studien. Hur som helst försedde den parametriska studien en kvalitativ analys angående inflytandet av inmatningsparametrarna på flödesbeteendet. Det framgick att en ökning av ingångstrycket och av antalet av diskreta munstycken gav en påföljande ökning i den maximala hastigheten som den atomiserande gasen kan erhålla. Vilket resulterar till en sänkning av hastigheten av smältan som introduceras till systemet. Dessutom observerades en radiell tryckgradient vara närvarande som ökade i enlighet med parametrarna. Det resulterade i en reduktion av smältfilmtjockleken som producerades under förfilmingsmekaniskmen. Trots det visade den diskreta fasmodellen att en ökning av gas till smältförhållandet också ökade antalet partiklar som sprids vidare för att kollidera med domänväggen. Således krävdes en påföljande ökning av nedströmningshastigheten för att bibehålla partiklarna inom domänväggar när studiens parametrar ökade.

[pt] O presente trabalho apresenta um estudo numérico da transferência de energia por radiação e sua interação com as propriedades radiantes cinzas e espectrais dos gases produtos da combustão e da fuligem que são formados em um processo de combustão turbulenta. Assim, utilizam-se sistemas de forno/- queimador que operam em regime de chama não pré-misturada de maneira a avaliar, através da dinâmica dos fluidos computacional (CFD), a influência que exercem os diversos modelos de propriedades radiantes sobre a representação da termoquímica do escoamento reativo. Com o objetivo de identificar as principais características e deficiências que apresentam cada um destes modelos, foram considerados dois cenários. O primeiro, correspondente a um problema de radiação unidimensional de um sistema homogêneo e não isotérmico onde são estudados, modelos de propriedades radiantes (i) disponíveis em um software comercial de CFD e (ii) aqueles que foram implementados neste trabalho. Além disso, foi empregado um código numérico que determina as propriedades radiantes espectrais de gases produtos da combustão e da fuligem através de uma abordagem de banda estreita. Para este fim, este código foi acoplado com o software de CFD. Em seguida, dois queimadores de porte laboratorial são empregados de forma a avaliar a capacidade preditiva dos modelos de propriedades radiantes: o primeiro queima propano gasoso e ar enriquecido com oxigênio e o segundo utiliza querosene líquido e oxigênio como reagentes. Dados experimentais de fluxo de calor radiante e de fração volumétrica da fuligem são utilizados para comparação com os resultados obtidos da simulação. Para ambas as configurações de queimador foi também estudado o modelo de Moss-Brookes para previsão da formação/consumo da fuligem. Os resultados obtidos demonstraram o bom desempenho da maioria dos modelos de propriedades radiantes estudados. Em particular, a abordagem de banda estreita foi o que melhor previu a radiação térmica. Além disso, a sua utilização com o modelo de Moss-Brookes levou à melhor previsão da fração volumétrica da fuligem. / [en] This work presents a numerical study of radiation heat transfer and its interaction with gray and spectral radiation of combustion products and soot that are formed in a turbulent combustion process. Different burner/furnace systems operating in a non-premixed combustion regime were used in order to evaluate, through computational fluid dynamics (CFD), the influence of several radiant properties models. Aiming to identify the key features and shortcomings that exhibit each of these models, two scenarios were considered. The first corresponds to a 1-D radiation problem where radiative properties models of a homogeneous non isothermal system are studied as (i) available CFD commercial software and (ii) those implemented in this work. Moreover, a numerical code was used in order to determine, through a narrow band approach, the spectral radiative properties of soot and combustion products. For this purpose, this code was coupled with the CFD software. Then, two laboratory-scale burners are used to assess the predictive capacity of radiative properties models: the first, burning propane and enriched air oxygen, and the second uses kerosene and oxygen as reactants. Measurements of radiant heat flux and soot volumetric fraction are used for comparison with simulation results. For both configurations, the performance of the Moss-Brookes model for predicting the soot production was also studied. The results of this study demonstrated the good performance of the majority of the radiant properties models studied. Particularly, the narrow band approach was the model that provided the best thermal radiation prediction. Moreover, the combination of the narrow band approach with the Moss-Brookes model lead to the best prediction of soot volume fraction.

Pour d non puissance d’un nombre premier, le nombre maximal de bases deux à deux décorrélées d’un espace de Hilbert de dimension d n’est pas encore connu. Dans ce mémoire, nous commençons par donner une construction de bases décorrélées en lien avec une famille de représentations irréductibles de l'algèbre de Lie su(2) et faisant appel aux sommes de Gauss.Puis nous étudions de façon systématique la possibilité de construire de telle bases au moyen des opérateurs de Pauli. 1) L’étude de la droite projective sur Zdm montre que, pour obtenir des ensembles maximaux de bases décorrélées à l’aide d'opérateurs de Pauli, il est nécessaire de considérer des produits tensoriels de ces opérateurs. 2) Les sous-modules lagrangiens de Zd2n, dont nous donnons une classification complète, rendent compte des ensembles maximalement commutant d'opérateurs de Pauli. Cette classification permet de savoir lesquels de ces ensembles sont susceptibles de donner des bases décorrélées : ils correspondent aux demi-modules lagrangiens, qui s'interprètent encore comme les points isotropes de la droite projective (P(Mat(n, Zd)²),ω). Nous explicitons alors un isomorphisme entre les bases décorrélées ainsi obtenues et les demi-modules lagrangiens distants, ce qui précise aussi la correspondance entre sommes de Gauss et bases décorrélées. 3) Des corollaires sur le groupe de Clifford et l’espace des phases discret sont alors développés.Enfin, nous présentons quelques outils inspirés de l’étude précédente. Nous traitons ainsi du rapport anharmonique sur la sphère de Bloch, de géométrie projective en dimension supérieure, des opérateurs de Pauli continus et nous comparons l'entropie de von Neumann à une mesure de l'intrication par calcul d'un déterminant. / For d not a power of a prime, the maximal number of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space is still unknown. In this thesis, we begin by an original building of MUBs by means of Gauss sums, in relation with a family of irreducible representations of the Lie algebra su(2).Then, we sytematically study the possibility of building such bases by means of Pauli operators. 1) The study of the projective line on Zdm shows that, in order to obtain maximal sets of MUBs, tensorial products of these operators are in order. 2) Lagrangian submodules of Zd2n, of which we give a complete classification, account for maximally commuting sets of Pauli operators. This classification enables to know which of these sets are likely to yield unbiased bases. They correspond to Lagrangian half-modules that can be interpreted as the isotropic points of the projective line (P(Mat(n, Zd)²),ω). Hence, we establish an isomorphism between the unbiased bases thus obtained and distant Lagrangian half-modules, which precises by the way the correspondance between Gauss sums and MUBs. 3) Corollaries on the Clifford group and the finite phase space are then developed.Finally, we present some tools inspired by the previous study. We deal with the cross-ratio on the Bloch sphere and projective geometry in higher dimension, Pauli operators with continuous exponents and we compare von Neumann entropy with a determinantal measure of entanglement