Developing laminar forced convection in eccentric annuli

The paper presents a boundary-layer model describing the laminar forced convection heat transfer in the entry region of eccentric annuli. A finite-difference numerical algorithm is developed for solving this model. Numerical results are presented for the developing velocity profiles and the pressure drop in annuli of radius ratio 0.5 and 0.9 over a dimensionless eccentricity ranged from 0.1 to 0.8. Heat transfer parameters are presented for a fluid of Pr=0.7 under the conditions of an isothermally heated inner wall while the outer wall is kept at the inlet fluid temperature. Received on 18 March 1997     

Rarefied gas flow in concentric annular tube: Estimation of the Poiseuille number and the exact hydraulic diameter

The fully developed flow of rarefied gases through circular ducts of concentric annular cross sections is solved via kinetic theory. The flow is due to an externally imposed pressure gradient in the longitudinal direction and it is simulated by the BGK kinetic equation, subject to Maxwell diffuse-specular boundary conditions. The approximate principal of the hydraulic diameter is investigated for first time in the field of rarefied gas dynamics. For the specific flow pattern, in addition to the flow rates, results are reported for the Poiseuille number and the exact hydraulic diameter. The corresponding parameters include the whole range of the Knudsen number and various values of the accommodation coefficient and the ratio of the inner over the outer radius. The accuracy of the results is validated in several ways, including the recovery of the analytical solutions at the hydrodynamic and free molecular limits.     

Developing mixed convection in vertical eccentric annuli

This paper deals with the problem of combined (forced–free) convection in vertical eccentric annuli with simultaneously developing hydrodynamic and thermal boundary layers. A bipolar model has been developed and a numerical algorithm for solving this model is outlined. Results, not available in the literature, are presented for the developing velocity profiles, axial variation of pressure, full development length, and heat transfer parameters under thermal boundary conditions of having one of the annulus boundaries at a constant temperature while the other boundary is insulated. Both aiding and opposing free convection have been considered and possibilities of flow reversal occurrence have also been checked. After a distance from the channel entrance and provided that the value of Gr/Re is sufficiently large, aiding free convection can develop to overcome the fluid friction and the eccentric annular channel eventually works as a diffuser. The value of Gr/Re for which a vertical eccentric annular channel can work as a diffuser decreases as the eccentricity increases. The axial distance from the entrance at which the channel starts to work as a diffuser decreases as Gr/Re increases.     

Flow of Robertson-Stiff fluids through an eccentric annulus

I.IntroductionInthepetroleumindustry,itisusuallythecasethatthedrillingstring(orcasingpipe)isnotlocatedinsidethecenteroftheflowgeometryduringdrillingandcompletillgpl'ocess,this,inturll,willaltertile'11owingbehaviorofdrillingmudandcementslurl.ieswhichtlowinginaneccentricallllulus.OilaccoUlltofviscousofnon-Newtoniantluid,flowillaneccentricannulusdifTeresmuchwiththatinacollcentricannulus.Mailyinvestigatorshaveconductedresearchworksonthisstlbject.Earlyin1935,TaoandDollovallrealizedthattheimpel.t…     

Measurements of pulsating and oscillating flows of non-Newtonian fluids through concentric and eccentric cylinders

The unsteady flow of non-Newtonian fluids through concentric and eccentric cylinders was investigated experimentally. Two experiments were carried out; one was pulsating flow and the other was flow under a constant pressure gradient with the inner cylinder oscillating longitudinally. The flow enhancement was examined and its dependence on the frequency of the oscillations and the eccentricity of the apparatus was determined.     

A numerical investigation of developing flow in an eccentric curved annulus in the presence of gravity

In this article developing incompressible viscous flow in an eccentric curved annulus in the presence of gravity is numerically studied using a second order finite difference method based on the projection algorithm to solve the governing equations including the continuity and full Navier–Stokes equations. The equations written in a bipolar–toroidal coordinate system are discretized in a three dimensional staggered grid. The effects of governing non-dimensional parameters including the eccentricity, non-dimensional curvature ratio, Dean number, Froude number, aspect ratio, and the Reynolds number on the flow field in the entrance and fully developed region are investigated. The numerical results indicate that at the small Froude numbers, the flow field distorts from the symmetrical condition due to the larger body force effect and the axial velocity formation mostly takes place at the lower half of the annulus. In addition, at the constant Froude number, by decreasing the curvature radius, the peak axial velocity and its sharp gradient appear on the outer curvature region due to the larger centrifugal forces and by increasing the eccentricity the flow rate intensifies at the wider region and weakens at the narrower region due to the larger flow resistance. Furthermore, the friction factor increases by decreasing the Froude number and increasing the Dean number.     

Two-phase flow simulations at 0−4o inclination in an eccentric annulus

Co-current two-phase simulations of gas-liquid flow with mixture velocities from 1.2 to 4.2 m/s were run in a partially eccentric annulus and compared with entirely eccentric and concentric experimental data collected at the Institute for Energy Technology in Norway. The gas-phase was sulphur hexafluoride (SF₆) for all cases, while the liquid-phase was Exxsol D60 for the horizontal cases and a mixture of Exxsol D60 and Marcol 82 for the inclined case. The outer diameter of the annulus was 0.1 m for all cases, while the inner diameter was 0.05 m in the horizontal configuration and 0.04 m for the inclined configuration. The purpose of this paper is to explore the effect of the holdup fraction, mixture velocity, and interior pipe’s position on the pressure gradient and flow regime, in effect a study of the pressure gradient and holdup fraction transients. The comparisons between simulations and experiments indicate that moving the pipe from an entirely eccentric to the partially eccentric configuration has a drastic impact on the pressure gradient. In all 4 cases where the inner pipe was changed from a completely eccentric geometry in the experiments to a less eccentric configuration in the simulations, we notice an increase of 48–303% of the mean pressure gradient. Comparatively, the 4 cases where the pipe was moved from a concentric experimental configuration to a more eccentric configuration in the simulations result in less drastic pressure gradient changes. Two cases were within 22% of the experimental results for mean, maximum, and minimum pressure gradient, while the last two cases exceeded the minimum and mean pressure gradients by 25–250%, respectively. The flow regime is rarely significantly affected by a change in eccentricity; 2 out of the 8 horizontal cases indicate either a transition from wavy flow to slug flow or significantly larger waves. The most prominent and frequent discrepancies identified were altered slug and wave frequencies. The last case, a 4^o inclined, partially eccentric simulation was compared to an entirely eccentric experiment and results in a 0.2 Hz increase in wave frequency, up from the experimental 0.56 Hz and a 49% increase in the mean pressure gradient.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

DYNAMICS OF ROTORS IMMERSED IN ECCENTRIC ANNULAR FLOW. PART 2: EXPERIMENTS

In order to validate the theoretical approach described in Part 1, an extensive experimental study has been performed at Instituto Tecnológico e Nuclear (I.T.N.), using a test rig presenting some improvements over previous experiments by the authors. In agreement with the theoretical model, concentric configurations always display flutter instability of the forward mode only, at velocities much higher than the critical unbalance-excited rotor velocity. Also predicted by theory, an instability of the backward mode may occur at much lower spinning velocities, for some eccentric configurations—which is a conclusion of practical significance. Therefore, rotor-annulus eccentricity is a very important parameter, when stability of the system is addressed. The quantitative agreement of both modal frequencies and damping values, with respect to the rotor spinning velocity, is quite satisfactory at lower velocities. However, it deteriorates somewhat at higher velocities, even for concentric configurations. Discrepancies are discussed, in connection with experimental difficulties (some unavoidable three-dimensional flow effects in the test rig) or other possibly pertinent phenomena (dynamic flow nonlinearities)—which were not accounted for in the theoretical model. However, these problems should not be overstressed, as theoretical instability boundaries are usually fairly close to the experimental results.     

Performance of conical entrance orifice plates at low Reynolds numbers

Results are presented of an experimental investigation into the performance of conical entrance orifice plates manufactured according to BS 1042. Three plates, with diameter ratios of 0.247, 0.360 and 0.448, were tested in the region 100 < Re_D ? 1000 and in both the concentric and the fully eccentric position. The discharge coefficient, C_z, of the orifice was found to agree with that specified in BS 1042 for a diameter ratio of 0.247. For other diameter ratios, the discharge coefficient increased with the diameter ratio, as observed by other workers for the Kent plates. The eccentricity has no appreciable effect on the discharge coefficient, probably due to the effect of viscous action on the flow being more or less the same for the concentric and eccentric position of the orifice at low Reynolds numbers     

Conditions for pressure build-up due to buoyancy effects on forced convection in vertical eccentric annuli under thermal boundary condition of first kind

This paper presents a numerical investigation for the conditions at which the buoyancy effects (represented by the buoyancy parameter (Gr/Re)) result in pressure build-up due to mixed convection in vertical eccentric annuli under thermal boundary conditions of first kind. In this regard, the critical values of buoyancy parameter (Gr/Re)_crt at which the pressure gradient vanishes and starts to become positive leading to the pressure build-up are obtained numerically for radius ratio N=0.5 and eccentricity E=0.1–0.7. Results of practical applications such as the locations at which the negative pressure gradient becomes zero changing its sign to be positive, the locations of zero pressure defect and the fully developed length under different operating conditions are drawn and presented. For sufficiently large values of Gr/Re≫(Gr/Re)_crt, possibilities and locations of flow reversal incipient are determined. Information of technical relevance is presented.     

PERTURBATION SOLUTION OF NON-NEWTONIAN FLUID AXIAL LAMINAR FLOW THROUGH ECCENTRIC ANNULI

Using the perturbation method, the axial laminar flow of Non-Newtonian fluid through an eccentric annulus is studied in the present paper. The relative eccentricity ε is taken as a perturbation parameter, and the first order perturbation solutions of the problem, such as velocity field, limit velocity and pressure gradient, are all obtained.     

Pertubation solution of non-newtonian fluid axial laminar flow through eccentric annuli

Using the perturbation method, the axial laminar flow of Non-Newtonian fluid through an eccentric annulus is studied in the present paper. The relative eccentricity ε is taken as a perturbation parameter, and the first order perturbation solutions of the problem, such as velocity field, limit velocity and pressure gradient, are all obtained.     

Water-lubricated transport of high-viscosity oil in horizontal pipes: The water holdup and pressure gradient

This paper has investigated the water holdup and the pressure gradient of water-lubricated transport of high-viscosity oil flow in horizontal pipes. Experimental results on the water holdup and the pressure gradient of water-lubricated high-viscosity oil two-phase flow in a horizontal 1 in. pipe were discussed. Models for the prediction of the water holdup and/or the pressure gradient of core flow or water-lubricated flow were reviewed and evaluated. It was found that the water holdup of the water-lubricated flow is not only closely related to the input water volume fraction but also the degree of the oil phase eccentricity which is attributed to the oil phase Froude number. This can explain the inconsistency of the experimental results with regard to the relationship between the water holdup and the input water volume fraction in the literature. The applicability of the existing empirical or mechanistic models of water-lubricated high-viscosity oil flow were discussed and demonstrated. A modified correlation to the water holdup correlation of Arney et al. (1993) which was shown to be exclusively applicable for concentric core flow was introduced for stable water-lubricated flow, including both concentric and eccentric core flows. This correlation was evaluated and a fair applicability was shown. The accuracy of different models for the prediction of the pressure gradient of water-lubricated transport of high-viscosity oil was demonstrated to be not high in general. This is closely associated with the difficulty in accurately accounting for the influence of oil fouling on the pressure gradient.     

Analytical solutions of the helical flow of non newtonian fluid in eccentric annular space

ＡＮＡＬＹＴＩＣＡＬＳＯＬＵＴＩＯＮＳＯＦＴＨＥＨＥＬＩＣＡＬＦＬＯＷＯＦＮＯＮＮＥＷＴＯＮＩＡＮＦＬＵＩＤＩＮＥＣＣＥＮＴＲＩＣＡＮＮＵＬＡＲＳＰＡＣＥＺｈａｎｇＨａｉ－ｑｉａｏ（张海桥）ＷｕＪｉ－ｚｈｏｕ（吴继周）（ＴｈｅＤｅｐａｒｔｍｅｎｔｏｆ...     

圆柱管道内运动的小球受到的黏滞阻力的数值分析

为了更好地运用落球法测量研究流体的黏滞系数,研究小球在黏性流体中下落的受力情况,本文对小球在充满黏性流体的圆柱管道的下落过程进行分析。利用COMSOL4.4仿真模拟,建立了合理的仿真模型,并分析了小球受到的黏滞阻力与小球的大小、下落位置的关系。结果表明:选择速度项二阶近似、压强项一阶近似的离散化方法,可以得到和理论值非常相符的仿真结果;当下落过程中小球球心始终在圆柱轴线上时,小球受到的黏滞阻力相对于Stokes力的修正系数,是小球半径与圆柱管道半径的比例函数,本文得到了更大范围的符合理论解的修正系数;当下落过程中小球的球心偏离圆柱轴线时,对于同样大小的小球,黏滞阻力、压强力、黏性力均随着球心到轴线的距离先减小后增大,且具有不同的极小值点。     

Large-eddy simulation of counter-rotating Taylor–Couette flow: The effects of angular velocity and eccentricity

In the present work, turbulent flow in the annulus of a counter-rotating Taylor-Couette (CRTC) system is studied using large-eddy simulation. The numerical methodology employed is validated, for both the mean and second-order statistics, with the direct numerical simulation (DNS) data available in the literature, for a range of Reynolds numbers from 500 to 4000. Thereafter, turbulent flow occurring in this system at Reynolds numbers of 8000 and 16000 are studied, and the results obtained are analyzed using mean and second-order statistics, vortical structures, velocity vector plots and power energy spectra. Further, the spatio-temporal variation of azimuthal velocity, extracted near the inner cylinder, shows the existence of herringbone like patterns similar to that observed in the previous studies. The effect of eccentricity of the inner cylinder with respect to the outer cylinder is studied, on the turbulent flow in the CRTC system, for two different eccentricity ratios of 0.2 and 0.5 and for two different Reynolds numbers of 1500 and 4000. The results of the eccentric CRTC are analyzed using contours of pressure, mean and second-order statistics, velocity vectors, vortical structures, and turbulence anisotropy maps. It is observed from the eccentric CRTC simulations that the smaller-gap region seems to contain higher amplitude fluctuations and more vortical structures when compared with the larger-gap region. The mean turbulent kinetic energy contours do not change qualitatively with the Reynolds number, however, quantitatively a higher turbulent kinetic energy is observed in the higher Reynolds number case of 4000.     

One dimensional numerical simulation for steady annular condensation flow in rectangular microchannels

A one dimensional model for steady annular condensation flow in rectangular microchannels is developed and numerically solved under constant heat flux condition. The results indicate that the annular condensation length is determined by the contact angle, heat flux, vapor pressure, hydraulic diameter and aspect ratio of rectangular microchannels. A larger inlet vapor pressure and hydraulic diameter or a smaller heat flux and contact angle can all result in a longer annular condensation length. In addition, the simulation results of steady annular condensation flow in rectangular microchannels are compared with that in triangular microchannels. The differences in curvature radius, condensate pressure and velocity, vapor velocity distributions in rectangular and triangular microchannels under the same conditions verify the considerable influence of cross-section shape on micro flow condensation.     

Pressure losses in transitions between square and rectangular ducts of the same cross-sectional area

Experimental results are presented for the pressure loss in transitions between square and rectangular ducts where the two ends have the same cross-sectional area. The aspect ratios at the rectangular end ranged from 0.3 to 0.625, and the transition length from 1 to 2 times the hydraulic diameter. Reynolds numbers ranged from 50 000 to 125 000. The pressure drop may be divided into components arising from friction and velocity profile distortion. The friction component, which may be evaluated by normal pipe flow methods, accounts for the observed variation with Reynolds number. The velocity profile component increases as the aspect ratio of the rectangular end falls, and is significantly higher for rectangular to square than for square to rectangular transitions. There is an optimum length to hydraulic diameter ratio, for which the pressure loss is a minimum; it has not been found exactly, but is less than 2 and probably below 1.     

Laminar flow heat transfer for simultaneously developing velocity and temperature profiles in ducts of rectangular cross section

Summary A numerical method is used to solve the heat transfer equations for laminar flow in ducts of rectangular cross section with simultaneously developing temperature and velocity profiles, both for constant wall temperature and for constant heat input per unit length of the duct. Like the solutions for a fully developed velocity profile, the Nusselt number for each aspect ratio is found to increase from a limiting value at large distances from the entry plane to a maximum at the entry plane. The results also show a strong effect of the Prandtl number on the heat transfer coefficients with uniform and fully developed velocity profiles representing the upper and lower limits respectively. Comparisons are made with analytical solutions for circular ducts and parallel plates and with experimental data.