Publisher Summary This chapter provides an overview of the analytical and experimental hydrodynamics and heat transfer studies of Newtonian and non-Newtonian fluids in laminar and turbulent flow through rectangular tubes. The chapter in particular focuses on the rectangular duct geometry, with emphasis on the friction factor and heat transfer behavior of non-Newtonian fluids. It is recognized that non-Newtonian behavior is generally more complicated than Newtonian flow. In the case of non-Newtonian fluids, the theoretical predictions yield low estimates of the heat transfer under laminar flow conditions. The fact that the available experimental heat transfer measurements lie above the predictions could reflect an inadequacy in the analytical model. For non-Newtonians in turbulent flow through rectangular channels, the situation is even more complicated. Some non-Newtonian fluids act as pseudoplastics, showing some reduction in friction and heat transfer as compared with a Newtonian fluid. Other non-Newtonian fluids experience large reductions in the friction factor and in heat transfer under turbulent flow conditions.

Heat transfer to newtonian and non-newtonian fluids in rectangular ducts

Fully developed laminar flow of purely viscous non-Newtonian liquids through annuli, including the effects of eccentricity and inner-cylinder rotation

Pore blocking is one of the critical factors governing the overall performance of membrane filtration systems. Pore blockage leads to a large resistance to filtration, giving rise to a sharp decline in the filtrate flux rate under constant pressure conditions and a severe increase in the pressure for operation under constant flux conditions during membrane filtration. In membrane filtration work, blocking filtration laws are widely used to interpret the membrane fouling in membrane filtration. This article presents a systematic review on the theoretical developments of mechanistic models related to the blocking filtration laws for describing the membrane fouling in membrane filtration. Equations for constant pressure and constant flux separations in the interstices of a membrane and on the surface are reported for the filtrate flow of Newtonian and power-law non-Newtonian fluids. The blocking filtration laws are useful also in the evaluation of the reduced pore size, maximum filtrate volume, and fouling p...

A Review on Modeling of Pore-Blocking Behaviors of Membranes During Pressurized Membrane Filtration

Blocking filtration laws consist of four different filtration mechanisms: complete blocking, standard blocking, intermediate blocking, and cake filtration. Blocking filtration laws for describing both the pore blocking and cake formation have been extensively employed over the past several decades to evaluate the increase in filtration resistance with the progress of filtration in the field of classical particulate filtration. In recent years, blocking filtration laws become widely used also in membrane filtration such as microfiltration and ultrafiltration of colloids. This paper gives an overview of the developments of blocking filtration laws and equations under constant pressure and constant rate conditions reported for the filtrate flow of Newtonian and non-Newtonian fluids. The fouling index evaluating the degree of membrane fouling was examined on the basis of the blocking filtration equations. The blocking filtration laws were reexamined to extend the range of their application. Moreover, various combined models developed based on the blocking filtration laws were introduced for describing more rigorously the complicated filtration behaviors controlled by more than one mechanism which occurs successively or simultaneously.

Developments of Blocking Filtration Model in Membrane Filtration

During a flood event in a steep torrent, sediment may be transported by relatively clear water, by a hyperconcentrated slurry or in the form of a (pulsing) debris flow. The objective of this study was to examine the effect of an increasing fluid density and viscosity on the bed load transport capacity of the flow. A clay suspension was used to simulate the slurry consisting of water and fine particles. For the majority of the bed load transport tests, density effects were dominant and viscous effects were negligible. At steeper slopes, the transported grains occupy a considerable part of the (mixture) flow depth, thus influencing the flow resistance analysis. Depending on the known or assumed parameters, two different calculation procedures are proposed to determine the unknown parameters of the flow of the grain-fluid mixture

Bedload transport capacity of slurry flows at steep slopes

A general method is proposed for prediction of the flow rate and maximum velocity in the isothermal, steady, uniform, laminar flow of any incompressible, time-independent non-Newtonian fluid in straight open channels of arbitrary cross section. The method requires only a knowledge of two geometric coefficients and a function of shear stress, used to characterize the behavior of the fluid model. The slip effect at the solid boundary has been taken into consideration. Numerical values of the geometric parameters have been determined for flow through an inclined plane of infinite width, semi-circular, semi-elliptical, rectangular, and 90° and 60° symmetric triangular open channels. Applications have been made to various non-Newtonian fluid models such as the power-law, Bingham, Ellis, Meter and the Reiner-Rivlin general model. Numerical examples are presented.



A generalization of the Fanning friction factor — Reynolds number is also presented. The problem of determining the point of transition from laminar to turbulent flow in the general case is examined, as is the problem of prediction of the friction factor in turbulent flow.



On propose une methode generale pour predire la vitesse d'ecoulement et la velocite maximum l'ecoulement isothermique, constant, uniforme et laminaire de tout fluide incompressible, independant du temps et non-newtonien dans des canaux droits et ouverts dont la coupe transversale est arbitraire. La methode n'exige que la connaissance de deux coefficients geometriques d'une fonction de l'effort de cisaillement qui servent a caracteriser le comportement du modele fluide. On a pris en consideration l'effet de glissement a la limite solide. On a determine les valeurs numeriques des parametres geometriques dans le cas de l'ecoulement au moyen d'un plan incline de largeur infinie, semi-circulaire, semi-elliptique et rectangulaire et de canaux ouverts symetriques et triangulares de 90° et 60°. On a applique les donnees a divers modeles fluidcs et non-newtoniens tels que ceux de la loi de l'energie, de Bingham, d'Ellis, de Meter et du modele general Reiner-Rivlin. On a fourni des exemples numeriques.



On presente egalement une generalisation de la relation entre le facteur de friction de Fanning et le nombre de Reynolds. On etudie le probleme de la determination du point de transition entre un ecoulement laminaire et un turbulent dans le cas general, ainsi que celui de la prediction du facteur de friction dans un flot turbulent.

Non-newtonian flow through open channels

An improved parametric method is presented for characterization of the flow geometry in the rectilinear flow of non-Newtonian fluids in open and closed conduits of arbitrary crosssection and in purely viscous, inelastic flow of non-Newtonian fluids through packed beds and porous media. In the new formulation, an infinite number of geometric parameters characterize the flow geometry. The actual number required in a particular application is shown to be determined by the fluid model equation representing the rheological behavior of the fluid. For Ostwald-de-Waele and Ellis fluids with flow behavior indices s = 1/n and α integers, the number of geometric parameters required to represent the relationship between flow rate and pressure drop is given by s + 1 and α + 1, respectively. The efficacy of the present method is demonstrated by comparisons with available results for various fluid models and flow geometries.

Improved parametric characterization of flow geometries

Equations of motion characterizing the flow of incompressible, time-independent non-Newtonian fluids, exhibiting the anomalous surface effect, through compressible porous media are developed. Approximations are made to arrive at workable filtration equations for slurries of non-Newtonian (power-law) fluids. The constant-pressure and constant-rate filtration relationships developed are verified experimentally using slurries of calcium carbonate in water and dilute CMC solutions. The anomalous surface effect is found to exist in the filtration of the non-Newtonian fluids. The specific cake resistance in the case of the non-Newtonian sludge and the ratio of the effective slip velocity to the pore velocity are found to be functions of both the CMC concentration and the pressure drop across the filter bed.



On a mis au point des equations de mouvement caracterisant l'ecoulement des fluides non-newtoniens incompressibles et independents du temps, lesquels font voir l'effet d'une surface anormale dans des milieux poreux et compressibles. On a eu recours a des approximations pour mettre au point des equations assez satisfaisantes pour la filtration de suspensions epaisses de fluides non-newtoniens (loi d'energie). On a evalue experimentalement la correlation entre la filtration a pression constante et celle a vitesse constante, au moyen de coulis de carbonate de calcium dans l'eau et de solutions diluees de carboxymethyl-cellulose. On a trouve que l'effet de surface anormale existait dans la filtration de fluides non-newtoniens. On a aussi constate que la resistance specifique du cake de filtration, dans le cas d'un coulis non-newtonien et le rapport entre la velocite effective de glissement et la velocite des pores etaient fonctions a la fois de la concentration de la carboxymethylcellulose et de la chute de pression dans le lit filtrant.

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Filtration of non‐Newtonian fluids

A method was proposed earlier(1) for prediction of the average velocity or pressure drop in the flow of non- Newtonian fluids in conduits of arbitrary cross section. The method requires only a knowledge of two geometric parameters which characterize the flow geometry, and the fluid model equation. In this paper, evaluations of geometric parameters are presented for infinite and finite (enclosed)square and triangular arrays of cylinders, and for regular polygonal and star-shaped conduits. With these parameters, predictions of the average velocity or related variables for the Zlow of any arbitrary non-Newtonian fluid can now be made. Comparisons are made with recent analytical solutions and experimental data pertaining to non-Newtonian flow in concentric annuli, and equilateral and right isosceles triangular ducts, which give additional support to the reliability of the proposed method.



Dane un travail prkbdent(l) on a decrit une mbthode pour predire la vilociti moyenne ou l'abaissement de prerrsion dans l'icoulement des flnides non-newtoniens dans des eonduites de coupe arbitraire. La mbthode n'exige qn'une connaissance de deux paramktres giomitriques qui caract& risent la ghmbtrie de I'bcoulement et l'bquation modele fluide. On prbsente, dans le travail actuel, des bvaluations de parametres ghmbtriques dans le cas de rangbes infinies et finies (closes ), carrbes et triangulaires de cylindres, ainsi que pour des conduites en forme de polygones rbguliers et d'btoiles. GrPce i ces paramktres, on peut maintenant prddire la velocite moyenne ou les variables apparentees pour 1′ecoulement de n'importe quel fluide arbitraire et non-newtonien. On fait des comparaisons avec des solutions analytiques rkentes et des rdsultats expbri- mentaux qui ont trait P un bcoulement non-newtonien dans des anneaux coneentriques et des conduites en forme dtriangles bquilatdraux et isockles droits, ce qni appuie davantage la confiance qu'on accorde i la mbthode proposbe.

Carlos Tiu

Papers

Non-newtonian flow through open channels

Improved parametric characterization of flow geometries

Filtration of non‐Newtonian fluids

Geometric parameters for some flow channels

Flow of complex fluids in open or closed conduits, packed beds and porous media