Particle-fluid heat/mass transfer: Newtonian and non-Newtonian fluids

A new model based on the boundary layer analysis is developed for particle-fluid heat/mass transfer. The proposed model enables a unified consideration of spheres in granular beds as well as isolated spheres. Comparison of results with data for Newtonian and non-Newtonian fluids indicates that the proposed model provides a good prediction of particle-fluid heat/mass transfer.

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Partikel-Fluid Wärme- und Stoffübertragung newtonscher und nichtnewtonscher Fluids

Zusammenfassing Ein neues, auf einer Grenzschicht-Analyse beruhendes Modell für die Partikel-Fluid Wärme- und Stoffübertragung wird entwickelt. Das vorgeschlagene Modell ermöglicht eine einheitliche Betrachtung von Sphären in körnigen Schichten sowie von isolierten Sphären. Ein Vergleich der Ergebnisse mit Daten für newtonsche und nichtnewtonsche Fluids deutet darauf hin, daß das vorgeschlagene Modell eine gute Vorhersage der Partikel-Fluid Wärme- und Stoffübertragung ermöglicht.

     

Studies on heat transfer to Newtonian and non-Newtonian fluids in agitated vessel

Heat transfer studies to Newtonian and non-Newtonian fluids are carried out in a stirred vessel fitted with anchor/turbine impeller and a coil for heating/cooling with an objective of determining experimentally the heat transfer coefficient of few industrially important systems namely castor oil and its methyl esters, soap solution, CMC and chalk slurries. The effect of impeller geometry, speed and aeration is investigated. Generalized Reynolds and Prandtl numbers are calculated using an apparent viscosity for non-Newtonian fluids. The data is correlated using a Sieder–Tate type equation. A trend of increase in heat transfer coefficient with RPM in presence and absence of solids has been observed. Relatively high values of Nusselt numbers are obtained for non-Newtonian fluids when aeration is coupled with agitation. The contribution of natural convection to heat transfer has been accounted for by incorporating the Grashof number. The correlations developed based on these studies are applied for design of commercial scale soponification reactor. Power per unit volume resulted in reliable design of a reactor.     

The free surface flow of Newtonian and non-Newtonian fluids trapped by surface tension

In this paper the position of the free surface of a swirling fluid held in by surface tension is calculated by the finite element method. A new locally mass-conserving quadratic velocity, linear pressure triangular element is used to overcome non-physical solutions produced by the well known Taylor-Hood element.     

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.     

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.     

Similarity solutions of unsteady laminar incompressible boundary layer equations for flow,heat and mass transfer in non-Newtonian fluids around axisymmetric bodies

Summary The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter. In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis. The main analysis shows that the similarity solutions are possible only for the bodies for which where is a cross-sectional radius; and is the longitudinal distance from the nose point to the cross section. In case of rotating bodies, similarity solutions exist only for cones and disks. The analysis, as an example, has successfully been applied to the Powell-Eyring model. It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE. µ, B, andE are the material functions of the Powell-Eyring fluid.

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Zusammenfassung Es wird die mögliche Existenz von Ähnlichkeitslösungen für instationäre drei-dimensionale Grenzschichtströmungen rein-viskoser nicht-newtonscher Flüssigkeiten mit Wärme- und Stoffübertragung um voll eingetauchte stationäre achsensymmetrische Körper in allgemeiner Weise untersucht. Hierbei werden Transformationen verwendet, die einen einzigen linearen Parameter enthalten. Als Spezialfälle der allgemeinen Analyse werden Rotationsströmungen um ruhende und rotierende Körper diskutiert. Die Hauptanalyse ergibt, daß Ähnlichkeitslösungen nur existieren für Körper mit , wo den Abstand von der Achse und den longitudinalen Abstand auf der Oberfläche von der Nase des Körpers aus bedeuten. Im Falle rotierender Körper existieren solche Lösungen nur für Kegel und Kreisscheiben. Die Analyse läßt sich erfolgreich auf das Beispiel einer Powell-Eyring-Flüssigkeit anwenden. Man findet, daß bei gleicher Schergeschwindigkeit der Energieverbrauch zur Aufrechterhaltung der Körperrotation mit wachsender Eyring-Zahl Ey= 1/µBE ansteigt, wobeiµ, B undE Materialfunktionen der Powell-Eyring-Flüssigkeit bedeuten.

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With 1 figure and 1 table     

The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions.

This is the second part of a study examining the mechanical properties and capillary flow of fiber suspensions in Newtonian fluids and in polymer solutions. In part I results for the viscous and elastic properties of the fiber suspensions were presented and it was shown that the fiber suspensions exhibited normal stresses in Newtonian as well as in viscoelastic suspending media. It was thus expected that circulating secondary flows would occur near the entrance to a capillary. Four types of fillers (glass, carbon, nylon and vinylon fibers) suspended in glycerin, HEC solutions and Separan solutions were investigated. The entrance flow patterns were visualized and the pressure fluctuations measured. The visualization enabled the eddies occurring in the fiber suspensions in Newtonian fluids to be analysed and classified into two tpyes. The results from the flow visualization were correlated with the pressure fluctuations. Empirical equations for the tube length correction factor due to the elasticity were obtained.     

The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions.

The main purpose of this study was to examine the viscous and elastic properties and capillary flow of fiber suspensions in Newtonian fluids as well as in polymer solutions. The fillers used were glass, carbon, nylon and vinylon fibers. Glycerin was used as a Newtonian suspending medium and HEC and Separan solutions as viscoelastic suspending media. The viscosity and the first normal-stress difference were measured using a coaxial cylindrical rotating viscometer and a parallel-plate rheogoniometer respectively. The influence of the concentration, aspect ratio, diameter and flexibility of the fibers on the viscous and elastic properties of the fiber suspensions was investigated. Empirical equations were obtained for the relative viscosity and first normal-stress difference for the fiber suspensions in glycerin. The capillary flow of these suspensions is discussed in part II.     

The effect of bubble-induced liquid flow on mass transfer in bubble plumes

The use of microbubbles to enhance mass transfer in a compact bubble column has become a valuable topic recently. When the liquid flow induced by the presence of microbubbles is taken into account, the behavior of the microbubbles may differ widely from simple estimations. One example is the change of the residence time, which is determined not only by slip velocity but also the velocity of the surrounding liquid. In the present study the effect of the bubble-induced liquid flow on mass transfer in microbubble plumes is analyzed numerically. A two-way coupling Eulerian–Lagrangian approach is used to simulate oxygen bubble plumes with initial bubble diameters from 100 μm to 1 mm and a maximum local void fraction of less than 2% in compact rectangular tanks. The simulations illustrate that the effect of bubble-induced liquid velocity on the residence time of microbubbles increases with the decrease of initial bubble diameters, and also increases with the reduction of initial water depth. The differences between the concentrated and uniform bubble injections are compared. The results show that the uniform injection of microbubbles provides much better mass transfer efficiency than the concentrated injection, because the bubble-induced liquid flow is suppressed when bubbles are injected uniformly over the entire bottom of the tank.     

Cessation of the lid-driven cavity flow of Newtonian and Bingham fluids

We provide benchmark results for a transient variant of the lid-driven cavity problem, where the lid motion is suddenly stopped and the flow is left to decay under the action of viscosity. Results include Newtonian as well as Bingham flows, the latter having finite cessation times, for Reynolds numbers Re ∈ [1, 1000] and Bingham numbers Bn ∈ [0, 10]. The finite-volume method and Papanastasiou regularisation were employed. A combination of Re and Bn, the effective Reynolds number, is shown to convey more information about the flow than either Re or Bn alone. A time scale which characterises the flow independently of the geometry and flow parameters is proposed.     

Viscous dissipation effect on heat transfer characteristics of rectangular microchannels under slip flow regime and H1 boundary conditions

Viscous dissipation effect on heat transfer characteristics of a rectangular microchannel is studied. Flow is governed by the Navier–Stokes equations with the slip flow and temperature jump boundary conditions. Integral transform technique is applied to derive the temperature distribution and Nusselt number. The velocity distribution is taken from literature. The solution method is verified for the case where viscous dissipation is neglected. It is found that, the viscous dissipation is negligible for gas flows in microchannels, since the contribution of this effect on Nu number is about 1%. However, this effect should be taken into account for much more viscous flows, such as liquid flows. Neglecting this effect for a flat microchannel with an aspect ratio of 0.1 for Br=0.04 underestimates the Nu number about 5%.     

Axisymmetric stagnation point flow and mass transfer for power-law fluids II. Moderate schmidt number correction

The theory for axisymmetric stagnation point flow of power-law fluids has been extended to include the correction terms for convective diffusion at moderate Schmidt numbers. The dimensionless mass transfer rate is expressed as an asymptotic series that is valid for Re^{(1 ? n)/3(1 + n)}Sc^{?$\text{1}\text{3}$} < 1. The result can be used to predict accurate diffusion coefficients for dilute species in fluids with specified power-law characteristics.     

Heat and mass transfer of an individual vapor bubble in translational flow of an unbounded liquid volume

The condensational collapse of a spherical vapor bubble moving translationally through an unbounded incompressible liquid is investigated. The bubble moves at a varying velocity. The problem is solved within the framework of the axisymmetric formulation. The numerical investigation shows that the variability of the bubble rise velocity significantly affects the condensational collapse process. This effect is particularly prominent in the final stage, i.e., in the stage of thermal collapse. The results obtained agree well with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–115, May–June, 1994.     

Modeling the turbulent heat and momentum transfer in flows under different thermal conditions

Two-equation turbulence models for velocity and temperature (scalar) fields are developed to calculate wall shear flows under various flow conditions and related turbulent heat transfer under various wall thermal conditions. In the present models, we make the modified dissipation rates of both turbulent energy and temperature variance zero at a wall, though the wall limiting behavior of velocity and temperature fluctuations is reproduced exactly. Thus, the models assure computational expediency and convergence. Also, the present k- model is construted using a new type of expression for the Reynolds stress proposed by Abe et al. [Trans. JSME B 61 (1995) 1714–1721], whose essential feature lies in introducing the explicit algebraic stress model concept into the nonlinear k- formulation, and the present two-equation heat transfer model is constructed to properly take into account the effects of wall thermal conditions on the eddy diffusivity for heat. The models are tested with five typical velocity fields and four typical thermal fields. Agreement with experiment and direct simulation data is quite satisfactory.     

The flow field of Newtonian fluids in cone and plate viscometers with small gap angles

Summary The flow of a Newtonian fluid in cone-plate viscometers is determined by solving the Navier-Stokes equations. It is shown in an order-of-magnitude analysis that the governing equations can be simplified and that the flow field can be split into two parts, if the angle between cone and plate is small. A closedform solution for the inner flow field is established through an iteration procedure. The zeroth-order solution in which the convective terms are neglected is inserted into the governing equations which then can be integrated again. The outer flow field is also described analytically. The solution satisfies the boundary condition of the vanishing radial velocity component at the free surface. The flow is described throughout the whole gap. Velocity components, pressure distributions, torque and normal force due to inertia are calculated. Pressure distributions and the maximum radial velocity are measured. The free surface boundary condition is experimentally examined.

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Zusammenfassung Es wird die Strömung einer newtonschen Flüssigkeit in Kegel-Platte-Rheometern beschrieben. In einer Größenordnungsabschätzung wird gezeigt, daß die beschreibenden Gleichungen vereinfacht werden können und daß das Strömungsfeld in zwei Teile aufgeteilt werden kann, wenn der Winkel zwischen Kegel und Platte klein ist. Eine geschlossene Lösung für den inneren Bereich wird durch eine Iteration erhalten. Die Lösung nullter Ordnung, in der die Konvektion vernachlässigt ist, wird in die beschreibenden Gleichungen eingesetzt, die dann nochmals integriert werden. Das äußere Strömungsfeld wird ebenfalls analytisch bestimmt. Die Lösung erfüllt die Randbedingung der verschwindenden Radialkomponente an der freien Flüssigkeitsoberfläche. Die Strömung wird im gesamten Scherspalt beschrieben. Geschwindigkeitskomponenten, Druckverteilung, Drehmoment und Normalkraft aufgrund von Trägheitskräften werden berechnet. Druckverteilungen und maximale Radialgeschwindigkeiten werden gemessen. Die Randbedingung an der freien Flüssigkeitsoberfläche wird experimentell untersucht.

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With 14 figures and 4 tables     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Origin of regime transition to turbulent flow in bubble column: Orifice- and column-induced transitions

The possible events during bubble formation on an orifice were investigated using a rectangular bubble column (30 cm × 30 cm × 100 cm). The gas flow rate through a single orifice was adjusted from 0.1 dm³/min to 5.0 dm³/min covering a high flow rate regime. At the high gas flow rate, the bubble formation process was complicated by diverse events, such as wake effect, channeling, and orifice-induced turbulent flow. The detachment period could be used to discern the bubble formation steps because it was strongly affected by the above events. The bubble size distribution around the orifice was also analyzed to gain a clearer understanding of the bubble formation process. Above the rate of 3.0 dm³/min through a single orifice, the detachment period converged to a value of 25 ms irrespective of the orifice diameter. The bubble size distribution also showed little difference in this range of gas flow rate. This could be explained by the development of turbulent flow around the orifice. A 0.15 m in-diameter bubble column was tested to investigate the effect of orifice-induced turbulent flow on the regime transition in which the homogeneous flow regime is converted into the heterogeneous flow regime in the column. Obvious distinction between the orifice- and column-induced transitions was observed.     

Heat and mass transfer to a reacting particle in a gas flow for arbitrary temperature dependence of the transport coefficients

A study is made of simultaneous heat and mass transfer to a reacting particle of any shape in a translational (and shear) flow of a viscous heat conducting compressible gas for which the thermal conductivity and diffusion coefficient, and also the specific heat depend on the temperature. The first two terms of the asymptotic expansion with respect to the small Reynolds number are obtained for the mean Sherwood and Nusselt numbers. The case of a power-law temperature dependence of the gas viscosity is considered in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–119, January–February, 1984.     

Anomalous heat transfer and drag in laminar flow of viscoelastic fluids

A boundary-layer approach is used to derive an expression for the heat transfer coefficient when a viscoelastic fluid flows past a cylinder. The heat-transfer coefficient becomes independent of velocity at large values of the latter, thus explaining the experimental results of James and Acosta. The agreement with the observations of these authors is, however, only qualitative, since their experiments were carried out at relatively low Reynolds numbers. By changing the exponent of the Reynolds number from the boundary-layer approximation value to one which is valid in the range used by James and Acosta, a correlation is suggested which is in satisfactory agreement with their data. Similar considerations are used to explain the additional experimental observation of James and Acosta, i.e. that the drag coefficient for the flow past a cylinder also becomes independent of velocity at large values of the latter quantity.