Dissertation submitted in fulfilment of the requirements for the degreerequirements Master of Technology: Civi Engineering in the Faculty of Engineering at the Cape Peninsula University of Technology Supervisor: APN. Sutherland Co-supervisor: Prof. R. Haldenwang Cape Town October 2013 / The reliable prediction of pressure drop versus flow rate for non-Newtonian pipe flow is important in many industrial processes. In laminar flow scale up is straightforward, but transitional velocity and turbulent flow predictions remain a practical problem. Various theoretical models exist, but nothing in literature shows conclusively which of these is the most reliable and consistent, nor is it evident what effect the choice of rheological model has on the predictions. The aim of this work was to i) evaluate the influence of different rheological models when used in existing prediction techniques for non-Newtonian flow ii) characterise each material type using selected (commonly used) rheological models and iii) predict laminar, transitional and turbulent pipe flow characteristics for each material type using existing prediction techniques, for comparison with experimental results. Only time-independent, homogeneous, non-Newtonian fluids in pipe sizes from 13mm to 200mm were investigated. Rheological models and laminar flow predictions used only the power law, Bingham plastic, Herschel-Bulkley, Casson and Hallbom yield plastic models. The techniques used to predict transitional velocity were Ryan & Johnson, Metzner-Reed, Hedström intersection method, Slatter and Hallbom. For turbulent flow the Newtonian approximation, Dodge & Metzner, Wilson & Thomas, Slatter, Hallbom modified Wilson & Thomas and the Bowen correlation methods were used. The study documents the relevant theory and presents an assessment of the influence of rheology on pipe flow predictions, summarised in terms of the practical performance of the various rheological model/prediction method combinations for the different materials. In laminar flow at practical pseudo shear rates (8V/D; taken as 40s-1) the choice of rheological model does not significantly influence pressure drop predictions. For yield-pseudoplastic materials (eg. kaolin) the Hedström intersection and the Slatter Reynolds number method with Bingham plastic or Casson rheology predicted transitional velocity most accurately. For Bingham plastic materials (eg. bentonite) the best predictions were obtained using the Metzner & Reed Reynolds number with Bingham plastic rheology, although similar results were observed for this technique with all rheologies. The transitional velocity for pseudoplastic materials (eg. CMC) was best predicted by the Slatter and Metzner & Reed Reynolds number methods, using power law or Casson rheology. For turbulent flow of yield pseudoplastic materials the Slatter method using the Casson rheology gave the most accurate predictions overall. Turbulent flow of Bingham plastic materials was best predicted by the Slatter, Hallbom pseudo fluid Nikuradse and Dodge & Metzner methods, using Bingham plastic, Casson or yield plastic rheology. For pseudoplastic materials the Slatter and Wilson & Thomas methods were the most accurate, when used with yield plastic or power law rheology. Transitionalal velocity and turbulent flow predictions for materials with a yield stress vary significantly with rheological model. Laminar data should therefore be examined thoroughly and rheological models fitted with care. For pseudoplastic fluids there is little difference in predictions between the various techniques as long as power law rheology is used.

Thesis (MTech (Chemical Engineering))--Cape Peninsula University of Technology, 2013 / Centrifugal pumps are the most commonly used pumps in slurry transport systems. The design of pumping systems dealing with liquids more viscous than water requires a reliable method of pump performance prediction for the pump selection. For Newtonian fluids, the Hydraulic Institute method is well established, but there is no generally accepted method for non-Newtonian fluids. Many authors have fallen back on using the Hydraulic Institute method for non-Newtonian fluids. This requires a constant viscosity while non-Newtonian fluid viscosity varies with the shear rate. The question arises: What viscosity should be used in this method for non-Newtonian fluids? Two approaches have been developed: the use of a Bingham plastic viscosity made by Walker and Goulas (1984) and the use of the apparent viscosity calculated using an “equivalent hydraulic pipe” diameter, designed by Pullum et al. (2007). Previous results obtained from these two approaches are not in agreement. Therefore, the aim of this study is to explore a suitable procedure to determine a representative non-Newtonian viscosity to be used in the Hydraulic Institute method to predict the pump performance. To achieve this goal, a set of data was experimentally obtained and the existing data were reused. Test work was conducted using the pump test rig in the Flow Process Research Centre at the Cape Peninsula University of Technology. A Warman 4/3 pump was tested, using four concentrations of kaolin suspension and three concentrations of CMC solution. Five pump speeds were chosen to run these tests: 1200, 1400, 1600, 1800, and 2000 rpm. An additional data set obtained by testing two submersible centrifugal pumps with eight concentrations of sludge, in Stockholm, Sweden, was also analysed. These sets of data were analysed firstly according to the Walker and Goulas (1984) approach and secondly according to the Pullum et al. (2007) approach. The use of the apparent viscosity led to the better pump head prediction. The results of this prediction were close to those obtained in the Pullum et al. (2007) work, and even better in some cases. On the other hand, the use of the Bingham plastic viscosity showed better pump efficiency prediction, although the Walker and Goulas (1984) efficiency prediction range was achieved only for one pump out of five. The apparent viscosity reflected the non-Newtonian behaviour but it could not represent alone the non-Newtonian viscosity because of the poor efficiency predictions and the sensitivity of the Pullum et al. (2007) approach to a change in viscosity. From the results of this work, it is advisable that the pump performance prediction be done using both apparent and Bingham plastic viscosity, the apparent viscosity for the head prediction and the Bingham plastic viscosity for the efficiency prediction.

A dissertation submitted to the Faculty of Engineering, Cape Peninsula University of Technology, Cape Town, in partial fulfilment of the requirements for the MTech Degree in Mechanical Engineering 2012 / Flow features provide considerable guidance for the rational selection of techniques to predict hydraulic behaviour and for suitable operating conditions for pipelines. Traditionally, water was used to transport coarse particles, and it was necessary to operate at velocities at which the flow was turbulent in order to avoid blockage. Consequently the friction losses were too high for economic operation. In addition, wear on pipes, fittings and pumps presented serious problems. Nowadays, it is well established that it is possible to operate at very high solids concentration in a heavy vehicle (carrier fluid). Similar solids throughputs may be achieved at very much lower velocities by operating in the laminar flow regime. This results not only in lower power requirement, but it also reduces wear and water consumption. In spite of these potential benefits, only a few studies dealing with the transport of coarse particles in heavy media have been reported. Since the distinction between different flow patterns is of paramount importance for modelling purposes, as equations are flow pattern dependent, and given the importance of avoiding excessive wear of pipes at low and high velocities, the present work was carried out in the context of dense or non-Newtonian carrier fluid. This project comprised analysis of existing data acquired at the Flow Process and Rheology Centre of the Cape Peninsula University of Technology. Kaolin in the range of 6% to 15% volumetric concentration was used as a carrier fluid and coarse material in the range of 10% to 30% volumetric concentration was simulated by silica sand ranging in size from 1 mm to 3 mm. For the purpose of this study flow patterns derived from resistance curves for various mixtures, particle concentrations, particle grading and flow conditions were compared with “concentration profiles” and images obtained from electrical resistance tomography (ERT). It appeared from this work that the sand concentration does not change the flow pattern but increases or reduces the pressure gradients depending on the case. The concentration of kaolin carrier can change the flow patterns from layered to homogeneous flow, inducing an increase in total pressure gradients as it increases. Flow patterns obtained from ERT compared reasonably well with those derived from pressure gradients profiles. The transition velocities from layered to heterogeneous flow obtained from both methods were similar, especially for low and moderate carrier concentrations. As the kaolin carrier concentration or as the sand concentration increased it became more difficult to distinguish the transition velocity between heterogeneous and layered flow. More work is still needed to improve the ERT instrument and its image reconstruction software.

The increasing activity of oil and gas extraction in deep water has stimulated several studies to solve problems encountered in drilling wells. During the drilling operation, a fluid (drilling mud) is pumped through the column to the bottom, carrying to the surface the drill cuttings generated, through the annular space formed between the column and the borehole wall. In this type of flow may appear a kind of hydrodynamic instability characterized by the appearance of toroidal vortices. This type of instability (Taylor-Couette) may profoundly alter the pressure drop of the flow, the shear stress at the borehole wall and the ability of carrying the solids. Moreover, during the process of drilling a well it is necessary promotes the well cementing and coating to provide a mechanical support, as well as, to isolate it from different rock formations traversed. For this step to be successful, the drilling mud must be completely removed from the annular space, and this removal may be impaired in wells which have varying eccentricity along the tube. Due to high costs of correction transactions and loss of drilling time, it is crucial to predict the mud flow around the annular. The effects of this variation of the eccentricity have not been much discussed in the literature, and they may have great influence on the displacement of the mud in the annular space. As an initial phase of work, numerical simulations were performed to study the flow and the emergence of Taylor- Couette instabilities in a concentric annular section, in order to compare them with literature data. Numerical simulations were developed in annular periodic sections, concentric and eccentric (E = 0.5) in order to obtain average profiles of axial and tangential velocities using different turbulence models, aiming at a comparison of simulated results with experimental data from literature. Later, it was made an experimental study and simulation to assess the effect of internal axis rotation on the pressure drop in the flow of non-Newtonian fluids (aqueous solutions of xanthan gum and carboxymethyl cellulose with 0.2% by weight) in a section annular concentric and the other with fixed eccentricity (E = 0.75). Finally, it was elaborated an experimental design (3k) with four variables, such as, xanthan gum concentration (0.05%, 0.10% and 0.15% by weight), eccentricity (0.0, 0.23 and 0.46), fluid flow rate (5, 7 and 9 m3/h) and internal rotation axis (0, 100 and 200 rpm). Following this planning, experimental data of pressure drop were collected, as well as, numerical simulations (CFD) in periodic annular sections to get results of axial velocity, in order to evaluate the effect of variable eccentric rotation on the fluid dynamics of non-Newtonian flows in annular spaces. / A crescente atividade de extração de petróleo e gás em águas cada vez mais profundas tem impulsionando diversos estudos para solucionar problemas encontrados na perfuração de poços. Durante a operação de perfuração, um fluido (lama de perfuração) é bombeado através da coluna até o fundo do poço, retornando à superfície carreando os cascalhos gerados pela broca, passando pelo espaço anular formado entre a coluna e a parede do poço. Neste tipo de escoamento pode ocorrer um tipo de instabilidade hidrodinâmica caracterizada pelo aparecimento de vórtices toroidais. Este tipo de instabilidade (Taylor-Couette), pode alterar profundamente a perda de carga do escoamento, a tensão cisalhante na parede do poço e a capacidade de carreamento de cascalho. Além disto, durante o processo de perfuração de um poço é necessário promover o revestimento e a cimentação do poço para fornecer a sua sustentação mecânica, bem como, para isolá-lo das diferentes formações rochosas atravessadas. Para esta etapa ser bem sucedida, a lama de perfuração deve ser completamente removida do anular, sendo que, esta remoção pode ser prejudicada em poços que apresentem excentricidade variando ao longo do tubo. Devido aos altos custos das operações de correção e a perda de tempo de perfuração, é fundamental prever este deslocamento da lama ao redor do anular. Os efeitos desta variação da excentricidade ainda não foram muito abordados na literatura e podem apresentar grande influência no deslocamento da lama no espaço anular. Como etapa inicial do trabalho, foram realizadas simulações numéricas para estudar o escoamento com surgimento de instabilidades do tipo Taylor-Couette em uma seção anular concêntrica, com intuito de compará-las com trabalhos da literatura. Depois foram desenvolvidas simulações numéricas em seções anulares periódicas, concêntrica e excêntrica (E = 0,5), a fim de obter perfis médios de velocidades axial e tangencial usando diferentes modelos de turbulência, visando uma comparação dos resultados simulados com os dados experimentais da literatura. Posteriormente, foi feito um estudo experimental e de simulação para avaliar o efeito da rotação do eixo interno sobre a queda de pressão no escoamento de fluidos não-Newtonianos (soluções aquosas de Goma Xantana e de Carboximetilcelulose a 0,2% em peso) numa seção anular concêntrica e outra com excentricidade fixa (E = 0,75). Finalmente, elaborou-se um planejamento fatorial de experimentos do tipo 3k com quatro variáveis, tais como, concentração de Goma Xantana (0,05%, 0,10% e 0,15%), excentricidade (0,0; 0,23 e 0,46), vazão volumétrica (5, 7 e 9 m3/h) e rotação do eixo interno (0, 100 e 200 rpm). Seguindo este planejamento foram levantados dados experimentais de queda de pressão, bem como, simulações numéricas (CFD) em seções periódicas para obtenção de resultados de velocidade axial, com intuito de avaliar efeito do movimento de rotação excêntrica variável sobre a dinâmica do escoamento de fluidos não-Newtonianos em espaços anulares. / Doutor em Engenharia Química

Fluidos Não-newtonianos estão presentes em grande quantidade na natureza e em muitas aplicações industriais. Tais fluidos apresentam uma variedade de comportamentos específicos. Dentre eles podemos citar: elasticidade, plasticidade, viscosidade variando com a taxa de deformação, tixotropia. Esta tese de doutorado apresenta um recente modelo para Fluidos Estruturados Tixotrópicos (de Souza Mendes, 2009), no qual baseasse nas tradicionais equações de balanço de momentum e massa, em uma equação constitutiva de Maxwell-b modificada e uma equação de evolução para o nível de estrutura do material. Na equação constitutiva de Maxwell-b modificada, tempo de relaxação e viscosidade estrutural são funções da estrutura do material. A equação evolutiva para o parâmetro de estruturação prevê efeitos de tixotropia, dependentes do tempo de equilíbrio do material. As equações que constituem o modelo mecânico para Fluidos Estruturados Tixotrópicos são aproximadas numericamente através do Método de Elementos Finitos de Galerkin Mínimos-quadrados (GLS). Tal método adicionou termos dependentes de malha ao método clássico de Galerkin, de forma a aumentar a estabilidade da formulação original, proporcionando compatibilizar os sub-espaços de velocidade e pressão. São investigados escoamentos creeping flow de Fluidos Estruturados Tixotrópicos em dois problemas clássicos de engenharia: em torno de um cilindro confinado entre duas placas e em uma expansão planar abrupta. Parâmetros relevantes ao modelo são variados em faixas pertinentes ao problema físico simulado e os resultados demonstraram uma boa capacidade de previsão. / Non-Newtonian fluids are present in large quantities in nature and in many industrial applications. Such fluids exhibit a variety of specific behaviors. Among them we can mention: elasticity, plasticity, viscosity varying with the strain rate, thixotropy, etc. This thesis presents a new model for thixotropic Structured Fluids (de Souza Mendes, 2009), in which were a basis in traditional balance equations of mass and momentum, the constitutive equation of Maxwell-type and an evolution equation for the level structure of the material. In the modified constitutive equation of Maxwell-b, relaxation time and structural viscosity are functions of material structure. The structure parameter evolutive equation provides effects of thixotropy, dependent on the equilibration time of the material. The equations which constitute the mechanical model for Structured Thixotropic fluids are approximated numerically using the Galerkin Least-squares Finite Element Method (GLS). Such a method added mesh dependent terms to the classical method of Galerkin in order to increase the stability of the original formulation, providing compatibility between sub-spaces of velocity and pressure. Are investigated Structured Thixotropic Fluids creeping flow in two classical problems in engineering: around a cylinder confined between two plates and in a planar abrupt expansion. Relevant parameters to the model are varied in pertinent ranges to physical problem simulated and the results demonstrated a good predictive capability.

The study of non-Newtonian flow is presents itself as relevant in bioengineering field, specially for design of devices that conduct blood, as arterial bypass grafts. Improvements in reducing energy dissipation and blood cell damage caused by artificial flows can be achieved by using numerical simulation and optimization methods. Thus, the present work proposes the study of design channels for steady, incompressible non-Newtonian flow, by using Topology Optimization Method based on the density method. The fluid flow is modeled with the Navier-Stokes equations coupled with Carreau-Yasuda constitutive equation for the dynamic viscosity to take into account the effects of the non-Newtonian blood properties. The Topology Optimization Method distributes regions of solid and fluid, given a volume constraint, within a specified domain in order to obtain a geometry and layout that minimizes energy dissipation, shear stress and vorticity by using the material pseudo-density as design variable. To apply this method to fluidic systems design, a fictional porous media based on Darcy equation is introduced. The flow model is implemented in its discrete form by using the Finite Element Method through the OpenSource platform FEniCS, applied to automate the solution of mathematical models based on differential equations. The optimization problem is solved by using the library DOLFIN-adjoint and IPOpt optimizer. Optimized topologies of channels for blood flow, focusing in arterial bypass grafts, are presented to illustrate the proposed method. / O estudo de escoamento de fluidos não-Newtonianos apresenta-se relevante no campo de bioengenharia, em especial no projeto de dispositivos para condução de sangue, como bypass arterial. Melhorias na redução de dissipação de energia e no dano às células sanguíneas causados por fluxos artificiais podem ser obtidas através do uso de técnicas de simulação e otimização numéricas. Deste modo, este trabalho propõe o estudo do projeto de canais para escoamentos incompressíveis em regime permanente de fluidos não-Newtonianos através do Método de Otimização Topológica baseado no método de densidade. O escoamento é modelado com as equações de Navier-Stokes acopladas com a equação constitutiva de Carreau-Yasuda para a viscosidade dinâmica, para que sejam considerados os efeitos das propriedades não-Newtonianas do sangue. O Método de Otimização Topológica distribui regiões de sólido e fluido, dada uma restrição de volume, dentro de um domínio especificado de modo a obter uma geometria e configuração que minimize a dissipação de energia, tensão de cisalhamento e vorticidade, utilizando a pseudo-densidade do material como variável de projeto. Para aplicar este método a sistemas fluidos, um meio poroso fictício, baseado na equação de Darcy, é introduzido. O modelo de escoamento é implementado em sua forma discreta utilizando o Método de Elementos Finitos através da plataforma OpenSource FEniCS, aplicada para automatizar a solução dos modelos matemáticos baseados em equações diferenciais, e o problema de otimização é resolvido utilizando a biblioteca DOLFIN-adjoint e otimizador IPOpt. Topologias otimizadas de canais para fluxo de sangue, com foco em bypass arterial, são apresentadas para ilustrar o método proposto.

Fluidos Não-newtonianos estão presentes em grande quantidade na natureza e em muitas aplicações industriais. Tais fluidos apresentam uma variedade de comportamentos específicos. Dentre eles podemos citar: elasticidade, plasticidade, viscosidade variando com a taxa de deformação, tixotropia. Esta tese de doutorado apresenta um recente modelo para Fluidos Estruturados Tixotrópicos (de Souza Mendes, 2009), no qual baseasse nas tradicionais equações de balanço de momentum e massa, em uma equação constitutiva de Maxwell-b modificada e uma equação de evolução para o nível de estrutura do material. Na equação constitutiva de Maxwell-b modificada, tempo de relaxação e viscosidade estrutural são funções da estrutura do material. A equação evolutiva para o parâmetro de estruturação prevê efeitos de tixotropia, dependentes do tempo de equilíbrio do material. As equações que constituem o modelo mecânico para Fluidos Estruturados Tixotrópicos são aproximadas numericamente através do Método de Elementos Finitos de Galerkin Mínimos-quadrados (GLS). Tal método adicionou termos dependentes de malha ao método clássico de Galerkin, de forma a aumentar a estabilidade da formulação original, proporcionando compatibilizar os sub-espaços de velocidade e pressão. São investigados escoamentos creeping flow de Fluidos Estruturados Tixotrópicos em dois problemas clássicos de engenharia: em torno de um cilindro confinado entre duas placas e em uma expansão planar abrupta. Parâmetros relevantes ao modelo são variados em faixas pertinentes ao problema físico simulado e os resultados demonstraram uma boa capacidade de previsão. / Non-Newtonian fluids are present in large quantities in nature and in many industrial applications. Such fluids exhibit a variety of specific behaviors. Among them we can mention: elasticity, plasticity, viscosity varying with the strain rate, thixotropy, etc. This thesis presents a new model for thixotropic Structured Fluids (de Souza Mendes, 2009), in which were a basis in traditional balance equations of mass and momentum, the constitutive equation of Maxwell-type and an evolution equation for the level structure of the material. In the modified constitutive equation of Maxwell-b, relaxation time and structural viscosity are functions of material structure. The structure parameter evolutive equation provides effects of thixotropy, dependent on the equilibration time of the material. The equations which constitute the mechanical model for Structured Thixotropic fluids are approximated numerically using the Galerkin Least-squares Finite Element Method (GLS). Such a method added mesh dependent terms to the classical method of Galerkin in order to increase the stability of the original formulation, providing compatibility between sub-spaces of velocity and pressure. Are investigated Structured Thixotropic Fluids creeping flow in two classical problems in engineering: around a cylinder confined between two plates and in a planar abrupt expansion. Relevant parameters to the model are varied in pertinent ranges to physical problem simulated and the results demonstrated a good predictive capability.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, foram estudados vÃrios tipos de escoamentos laminares de fluidos incompressÃveis Newtonianos e nÃo-Newtonianos. Isto foi feito atravÃs do estudo da interaÃÃo destes escoamentos com geometrias complexas atravÃs de modelagem computacional e da soluÃÃo numÃrica das equaÃÃes de conservaÃÃo do momento e continuidade de massa. Numa primeira etapa, a modelagem computacional de uma rede de poros foi usada para a gerar padrÃes de agregados granulares resultando de mecanismos de erosÃo-deposiÃÃo de grÃos leves. A geometria da rede de poros foi alterada dinamicamente de acordo com a transferÃncia de momento do escoamento para as partÃculas localizadas em cada vÃrtice da rede de poros. Os resultados mostraram que, para esse processo irreversÃvel, o modelo foi capaz de reproduzir padrÃes tÃpicos de processos de erosÃo bem-conhecidos. Numa segunda etapa, um separador de partÃculas semelhante à estrutura pulmonar foi proposto com base nas propriedades de escoamento em uma estrutura ramificada e nas propriedades de transporte inercial das partÃculas, quantificadas atravÃs do nÃmero de Stokes. Os resultados indicaram que a variaÃÃo dos parÃmetros de construÃÃo da estrutura ramificada leva a um regime eficiente do processo de separaÃÃo em um amplo espectro de valores do nÃmero de Stokes. Por Ãltimo, o escoamento de vÃrios fluidos nÃo-Newtonianos atravÃs de meios porosos desordenados em trÃs-dimensÃes foi estudado. Os resultados mostraram, para fluidos do tipo lei-de-potÃncia, que o escoamento pode ser descrito como uma curva universal se o nÃmero de Reynolds e a permeabilidade hidrÃulica forem redefinidos de maneira apropriada. Fluidos de Bingham tambÃm foram estudados atravÃs do modelo de Herschel-Bulkley. Neste caso, as simulaÃÃes revelaram que as interaÃÃes entre a geometria complexa do espaÃo poroso, as propriedades reolÃgicas do fluido e os efeitos inerciais do escoamento sÃo responsÃveis por uma melhora substancial da permeabilidade hidrÃulica do sistema em valores intermediÃrios do nÃmero de Reynolds. / In this work many types of incompressible laminar Newtonian and Non-Newtonian flows are studied. The interplay of these flows with complex geometries was investigated using computational modeling and numerical solution of the conservation of momentum and mass continuity equations. As a first step, the computational modeling of a network of pores was adopted to reveal the formation patterns caused by the mechanism of erosion-deposition of light grains. The geometry of the pore network was changed dynamically according to the flow momentum transfer for particles located on each vertex of the pore network. The results showed that, for this irreversible processes, the model is capable of reproducing patterns of formation of well-known erosion processes. In a second step, a particle separator inspired on the lung structure was proposed based on the flow properties in a branched structure and transport of inertial particles, quantified in terms of the Stokes number. The results indicated that the variation of construction parameters of the branched structure leads to an efficient design of the separation process in a wide range of values of the Stokes number. Finally, the flow of non-Newtonian fluids through three-dimensional disordered porous media has been studied. The results showed, for power-law fluids that the flow can de described as a universal curve if the Reynolds number and the hydraulic permeability are redefined properly. The flow of Bingham fluids was also studied using the model of Herschel-Bulkley. In this case, the simulations showed that the interaction between the complex geometry of the pore space, the rheological properties of the fluid and the inertial effects of the flow is responsible for a substantial improvement of the hydraulic permeability of the system at intermediate values of the Reynolds number.

[pt] Neste trabalho foi analisado o escoamento de fluidos não Newtonianos através de canais axisimétricos convergentes divergentes. A solução da conservação de massa e de conservação de momento foi obtida numericamente via volumes finitos utilizando o programa de computador Fluent. A equação constitutiva de fluidos Newtonianos generalizados foi utilizada para modelar o comportamento não Newtoniano, utilizando a equação constitucional de Shunk-Scriven para cálculo da viscosidade, que assume como sendo a média geométrica ponderada pelo classificador de escoamento R entre a viscosidade de cisalhamento e a viscosidade de extensão. Os resultados de perda de pressão e vazão são comparados com os resultados calculados pela relação simplificada proposta por Souza Mendes e Naccache, 2002 entre a perda de carga e vazão de fluidos viscoelásticos fluindo através do meio poroso, para analisar a sua performance. / [en] In this work, the flow of non-Newtonian fluids through axisimmetric convergingdiverging channels is analyzed. The solution of mass and momentum conservation equations is obtained numerically via finite volume technique using the Fluent software. The Generalized Newtonian Fluid constitutive equation was used to model the non- Newtonian fluid behavior, using the Shunk-Scriven model for the viscosity, where a weighted geometric mean by the flow classifier R between shear and extensional viscosities is assumed. The results of pressure drop and flow rate are compared to the ones predicted by a previously proposed simplified relation (Souza Mendes and Naccache, 2002) between pressure drop and flow rate, for viscoelastic fluids flow through porous media, in order to analyze its performance.

This thesis is concerned with a theoretical and experimental study of the hydrodynamics and heat transfer characteristics of viscous fluids flowing in tubes under laminar conditions. Particular attention has been given to the effects of the rheological properties and their variation with temperature. A review of problems of this type showed that in spite of the many potential applications of the results in a wide range of industries the subject had not been well developed and further work is justified in order to fill some of the gaps in our knowledge. The early part of the thesis considers the justification of the work in this way and sets down the scope and objectives. A computer progracune was then developed to allow the governing equations of the problem to be solved numerically to give the velocity and temperature profiles and pressure drop for both heating and cooling conditions. The results were also presented in the form of Nusselt numbers as a function of the Graetz numberp since this form is useful for engineering design purposes. The validity of the predictions were then checked by a programme of experimental work. Temperature and velocity profiles have been measured in order to provide a more severe test of the theory than could be imposed by the measurement of overall heat transfer rates. A combined thermocouple probe/Pitot tube was developed to allow simultaneous measurements of velocity and temperature to be made. A Newtonian oil and two non-Newtonian Carbopol solutions were studied. This is the first time that velocity and temperature profiles have been measured for non-Newtonian fluids in this type of situation. The results of the work heve shown that (a) the velocity and temperature profiles and pressure drops are greatly affected by the temperature dependence of the rheological properties and since real viscous fluids are normally very temperature-sensitive it is important that this effect is properly taken into account. (b) the engineering design correlations commonly used for the prediction of heat transfer coefficients can be seriously in error, especially for cooling conditions and when non-Nevitonian fluids are being considered. (c) a mathematical model can be developed which accurately describes all the phenomena and gives predictions which are very close to those observed experimentally. An important objective was to develop more accurate engineering design correlations for non-isothermal pressure drop and heat transfer rates. / University of Bradford