A unified fractional flow framework for predicting the liquid holdup in two-phase pipe flows

Two-phase pipe flow occurs frequently in oil&gas industry,nuclear power plants,and CCUS.Reliable calculations of gas void fraction(or liquid holdup)play a central role in two-phase pipe flow models.In this paper we apply the fractional flow theory to multiphase flow in pipes and present a unified modeling framework for predicting the fluid phase volume fractions over a broad range of pipe flow conditions.Compared to existing methods and correlations,this new framework provides a simple,approximate,and efficient way to estimate the phase volume fraction in two-phase pipe flow without invoking flow patterns.Notably,existing correlations for estimating phase volume fraction can be transformed and expressed under this modeling framework.Different fractional flow models are applicable to different flow conditions,and they demonstrate good agreement against experimental data within 5%errors when compared with an experimental database comprising of 2754 data groups from 14literature sources,covering various pipe geometries,flow patterns,fluid properties and flow inclinations.The gas void fraction predicted by the framework developed in this work can be used as inputs to reliably model the hydraulic and thermal behaviors of two-phase pipe flows.

Movement Properties of Pipe Flow Along Granite Slope of Three Gorges Area of Yangtze River in China

It is well known that, in most cases, soil water doesn't move in the form of laminar flow as described by Darcy law. Only when Reynolds number ( Re ) is no more than 10, does water movement follow Darcy law. A soil profile with 2 9 m long and 2 13 2 60 m deep was excavated on a lower slope located at Zigui County, Hubei Province, China. Field observation found that soil pipes were mainly distributed in the transient layer between horizon B with higher degree of granite weathering and horizon C with lower degree of granite weathering. At the foot of the slope, about 5 7 soil pipes per meter were observed along the vertical direction of the slope. The observed results, obtained by continuous observation of soil pipes and pipe flow processes at granite slope for many rainfall events, indicate that the relationship between velocity of pipe flow and hydraulic gradient along the pipe is parabolic rather than linear. Based on the investigated data of soil, landform, and land use etc., combined with observed data of pipe flow derived from many rainfall events, a pipe flow model was developed. For velocity V p, discharge Q p of pipe flow and radius r of soil pipe, great similarity was found between simulated and observed values. Particularly, the simulated length of soil pipes reflects the great difference among soil pipes as a result of its different position in the soil profile. The length values of 4 soil pipes were estimated to be 98 1%, 27 6%, 11 0% and 3 0% of the longest distance of the catchment, respectively. As a special case of water movement, soil pipe flow follows Darcy Weisbach law. Discharge of pipe flow is much greater than infiltration discharge in common. Only when the depth of groundwater is more than the diameter of soil pipe and water layer submerges soil pipes during rainfall, may pipe flow occur. Under these circumstances, discharge of pipe flow is directly proportional to the depth of groundwater.

Research on the orientation distribution of fibers immersed in a pipe flow

The computed orientation distribution of fibers immersed in laminar pipe flows showed that the longitudinal distributions are wide for small Reynolds numbers and become narrower with increasing Re . For low Re number, the axial orientation distributions are broad with almost no preferred orientations. For high Re number, the axial distribution becomes narrow, with sharp maxima. The mean values of the longitudinal orientation depend strongly on the Re number. The computed results are in qualitative agreement with relevant experimental results.

Numerical Simulation of Turbulent Swirling Pipe Flow with an Internal Conical Bluff Body

Turbulent swirling flow inside a short pipe interacting with a conical bluff body was simulated using the commercial CFD code Fluent.The geometry used is a simplified version of a novel liquid/gas separator used in multiphase flow metering.Three turbulence models,belonging to the Reynolds averaged Navier-Stokes(RANS)equations framework,are used.These are,RNG k-ε,SST k-ωand the full Reynolds stress model(RSM)in their steady and unsteady versions.Steady and unsteady RSM simulations show similar behavior.Compared to other turbulence models,they yield the best predictions of the mean velocity profiles though they exhibit some discrepancies in the core region.The influence of the Reynolds number on velocity profiles,swirl decay,and wall pressure on the bluff body are also presented.For Reynolds numbers generating a Rankine-like velocity profile,the width and magnitude of flow reversal zone decreases along the pipe axis disappearing downstream for lower Reynolds numbers.The tangential velocity peaks increase with increasing Reynolds number.The swirl decay rate follows an exponential form in accordance with the existing literature.These flow features would affect the performance of the real separator and,thus,the multiphase flow meter,noticeably.

Nanoparticle Migration in a Fully Developed Turbulent Pipe Flow Considering the Particle Coagulation

Numerical simulations of nanoparticle migration in a fully developed turbulent pipe flow are performed.The evolution of particle number concentration,total particle mass,polydispersity,particle diameter and geometric standard deviation is obtained by using a moment method to approximate the particle general dynamic equation.The effects of Schmidt number and Damkhler number on the evolution of the particle parameters are analyzed.The results show that nanoparticles move to the pipe center.The particle number concentration and total particle mass are distributed non-uniformly along the radial direction.In an initially monodisperse particle field,the particle clusters with various sizes will be produced because of coagulation.As time progresses,the particle cluster diameter grows from an initial value at different rates depending on the radial position.The largest particle clusters are found in the pipe center.The particle cluster number concentration and total particle mass decrease with the increase of Schmidt number in the region near the pipe center,and the particles with lower Schmidt number are of many dif-ferent sizes,i.e.more polydispersity.The particle cluster diameter and geometric standard deviation increase with the increase of Damkhler number at the same radial position.The migration properties for nano-sized particles are different from that for micro-sized particles.

RESEARCH ON THE MOTION OF PARTICLES IN THE TURBULENT PIPE FLOW OF FIBER SUSPENSIONS

The motion of fibers in turbulent pipe flow was simulated by 3-D integral method based on the slender body theory and simplified model of turbulence.The orientation distribution of fibers in the computational area for different Re numbers was computed.The results which were consistent with the experimental ones show that the fluctuation velocity of turbulence cause fibers to orient randomly.The orientation distributions become broader as the Re number increases.Then the fluctuation velocity and angular velocity of fibers were obtained.Both are affected by the fluctuation velocity of turbulence.The fluctuation velocity intensity of fiber is stronger at longitudinal than at lateral,while it was opposite for the fluctuation angular velocity intensity of fibers.Finally,the spatial distribution of fiber was given.It is obvious that the fiber dispersion is strenghened with the increase of Re numbers.

K-ε-T MODEL OF DENSE LIQUID-SOLID TWO-PHASE TURBULENT FLOW AND ITS APPLICATION TO THE PIPE FLOW

To predict the characteristics of dense liquid-solid two-phase flow, K-Ε-T model is established, in which the turbulent flow of fluid phase was described with fluid turbulent kinetic energy Kf and its dissipation rate Εf, and the particles random motion was described with particle turbulent energy Kp and its dissipation rate Εp and pseudothermal temperature Tp. The governing equations were also derived. With K-Ε-T model, numerical study of dense liquid-solid two-phase turbulent up-flow in a pipe is performed. The calculated results are in good agreement with experimental data of Alajbegovic et al. (1994), and some flow features are captured.

MAGNETOHYDRODYNAMIC PIPE FLOW IN DUCTS WITH PARTIAL CIRCULAR RING CROSS SECTION AND ANNULAR CROSS SECTION

In this paper we use the Green function method to solve the problem of steady one dimensional flow of an incompressible viscous, electrically conducting fluid through a pipe with partial circular ring cross sec- tion and one with annular cross section, in the presence of an applied transverse uniform magnetic field, We ob- tain analytic solutions and carry out some numerical calculations of the velocity distribution and induced magnet- ic field.

Optimal transient growth in turbulent pipe flow

The optimal transient growth process of perturbations driven by the pressure gradient is studied in a turbulent pipe flow. A new computational method is proposed, based on the projection operators which project the governing equations onto the sub- space spanned by the radial vorticity and radial velocity. The method is validated by comparing with the previous studies. Two peaks of the maximum transient growth am- plification curve are found at different Reynolds numbers ranging from 20 000 to 250 000. The optimal flow structures are obtained and compared with the experiments and DNS results. The location of the outer peak is at the azimuthal wave number n = 1, while the location of the inner peak is varying with the Reynolds number. It is observed that the velocity streaks in the buffer layer with a spacing of 100δv are the most amplified flow structures. Finally, we consider the optimal transient growth time and its dependence on the azimuthal wave length. It shows a self-similar behavior for perturbations of different scales in the optimal transient growth process.

On the helical pipe flow with a pressure-dependent viscosity

We address the flow of incompressible fluid with a pressure-dependent viscosity through a pipe with helical shape. The viscosity-pressure relation is defined by the Barus law. The thickness of the pipe and the helix step are assumed to be of the same order and considered as the small parameter. After transforming the starting problem, we compute the asymptotic solution using curvilinear coordinates and standard perturbation technique. The solution is provided in the explicit form clearly showing the influence of viscosity-pressure dependence and pipe＇s geometry on the effective flow.

Improved Dissipation Rate Equation for Swirling and Rotating Pipe Flows

The nature of turbulent swirling and rotating flow in a straight pipe is investigated using a family of near-wall two-equation models. Specifically, the viability of three different near-wall two-equation models is assessed. These models are asymptotically consistent near the wall. The first two models, one with isotropic and another with anisotropic eddy viscosity invoked, solved a dissipation rate equation with no explicit correction made to account for swirl and flow rotation. The third model assumes an isotropic eddy viscosity but solves an improved dissipation rate equation that has explicit corrections made to account for swirl and flow rotation. Calculations of turbulent flows in the swirl number range 0.25 - 1.3 with and without a central recirculation region reveal that, with the exception of the third model, neither one of the other two models can replicate the mean field at the swirl numbers tested. Furthermore, taking stress anisotropy into account also fails to model swirl effect correctly. Significant improvements can be realized from the third model, which is based on an improved dissipation rate equation and the assumption of isotropic eddy viscosity. The predicted mean flow and turbulence statistics correlate well with measurements at low swirl. At high swirl, the two-equation model with an improved dissipation rate equation can still be used to model swirling and rotating pipe flows with a central recirculation region. However, its simulation of flows without a central recirculation region is not as satisfactory.

Formation of Necklace-Type Vortex System Upstream of a Gate Valve in Pipe Flow

A gate valve is one of the main elements of a circular pipeline, but the flow characteristics around the gate valve are hardly known. In this study, clarification of the flow field in front of the gate valve model in a pipe flow via flow visualization and PIV analysis was attempted. As a result, four kinds of steady necklace-type vortex systems, 2-vortex, 4-vortex, 6-vortex and 8-vortex systems, were clearly observed in a Reynolds number between 290 and 2130. In addition, the main vortex was observed in the Reynolds number range between 2130 and 4870 with difficulty. On this account, both the center position and vorticity in the main vortex are presented against Reynolds number.

Simulation of Average Turbulent Pipe Flow: A Three-Equation Model

The aim of this study is to evaluate a three-equation turbulence model applied to pipe flow. Uncertainty is approximated by comparing with published direct numerical simulation results for fully-developed average pipe flow. The model is based on the Reynolds averaged Navier-Stokes equations. Boussinesq hypothesis is invoked for determining the Reynolds stresses. Three local length scales are solved, based on which the eddy viscosity is calculated. There are two parameters in the model;one accounts for surface roughness and the other is possibly attributed to the fluid. Error in the mean axial velocity and Reynolds stress is found to be negligible.

Process-Based Friction Factor for Pipe Flow

The Moody Diagram is widely used to determine the friction factor for fluid flow in pipes. The diagram combines the effects of Reynolds number and relative roughness to determine the friction factor. The relationship is highly non-linear and appears to have a complex interaction between viscous and boundary roughness effects. The Moody Diagram is based on predictions from an equation developed by Colebrook in 1939. The relationship requires an iteration process to make predictions. While empirical relationships have been developed that provide good predictions without an iteration process, no one has fully explained the cause for the observed results. The objective of this paper is to present a logical development for prediction of the friction factor. An equation has been developed that models the summed effect of both the laminar sublayer and the boundary roughness on the fluid profile and the resulting friction factor for pipes. The new equation does not require an iteration procedure to obtain values for the friction factor. Predicted results match well with values generated from Colebrook’s work that is expressed in the Moody Diagram. Predictions are within one percent of Colebrook values and generally less than 0.3 percent error from his values. The development provides insight to how processes operating at the boundary cause the friction factor to change.

FEM-BEM Coupling for the MHD Pipe Flow in an Exterior Region

In this study, the magnetohydrodynamic (MHD) flow through a circular pipe under the influence of a transverse mag- netic field when the outside medium is also electrically conducting is solved numerically by using FEM-BEM coupling approach. The coupled partial differential equations defined for the interior medium are transformed into homogenous modified Helmholtz equations. For the exterior medium on an infinite region, the Laplace equation is considered for the exterior magnetic field. Unknowns in the equations are also related with the corresponding Dirichlet and Neumann type coupled boundary conditions. Unknown values of the magnetic field on the boundary and for the exterior region are obtained by using BEM, and the unknown velocity and magnetic field inside the pipe are obtained by using SUPG type stabilized FEM. Computations are carried for very high values of magnetic Reynolds numbers Rm1, Reynolds number Re and magnetic pressure Rh of the fluid. The results show that using stabilized method enables us to get stable and accurate numerical approximations consistent with the physical configuration of the problem over rough mesh which also results a cheap computational cost.

Pipe Flow of Suspensions Containing Bubbles

The steady laminar pipe flow of a suspension with a gas volume fraction ∅≤0.5 and small or intermediate bubble deformations in long, and straight sections of a circular pipe is calculated. The calculations are based on the constitutive equation that was originally derived for dilute emulsions and further developed for concentrated suspensions containing bubbles. In contrast to the literature, an analytical procedure is used to determine the solution of a pipe flow more accurately. The results are presented and discussed with respect to the Reynolds number Re and capillary number Ca. If Ca 1, a bubble suspension has a parabolic velocity profile indicating a Newtonian rheology. If Ca ≈1, two regimes of flow are observed in agreement with the literature;that is, an inner plug flow where deformation rates are low and an outer flow where deformation rates are high. These results imply that, if Ca ∅?and that, if Ca ≥1, the opposite effect occurs;that is, the Reynolds number Re increases with increasing gas volume fraction.

The calculation of mechanical energy loss for incompressible steady pipe flow of homogeneous fluid

The calculation of the mechanical energy loss is one of the fundamental problems in the field of Hydraulics and Engineering Fluid Mechanics. However, for a non-uniform flow the relation between the mechanical energy loss in a volume of fluid and the kinematical and dynamical characteristics of the flow field is not clearly established. In this paper a new mechanical energy equation for the incompressible steady non-uniform pipe flow of homogeneous fluid is derived, which includes the variation of the mean turbulent kinetic energy, and the formula for the calculation of the mechanical energy transformation loss for the non-uniform flow between two cross sections is obtained based on this equation. This formula can be simplified to the Darcy-Weisbach formula for the uniform flow as widely used in Hydraulics. Furthermore, the contributions of the mechanical energy loss relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy in the turbulent uniform pipe flow are discussed, and the contributions of the mechanical energy loss in the viscous sublayer, the buffer layer and the region above the buffer layer for the turbulent uniform flow are also analyzed.

STABILITY IN THE CIRCULAR PIPE FLOW OF FIBER SUSPENSIONS

A linear stability analysis on the circular pipe flow of fiber suspensions ispresented. The fiber orientation tensors are used to describe the characteristics of theorientation dislribution of fibers. The constitutive equation for the fiber suspensions is set upand the modified Orr-Sommerfeld stability equation is derived. An improved finite difference methodwith high order accuracy is employed to solve the e-quation. The Newtonian pipe Poiseuille flowcorresponding to H = 0 is also analyzed for comparison. The results reveal that the fiber additiveswill enhance the flow stability, the degree of enhancement becomes high with the increases of theparameter H which accounts for the fiber resistance to the stretching along its axis. Fibersuspensions with large H can suppress the influence of high Reynolds number on the flow stability.Particularly, fibers give a higher attenuation of the short waves of disturbance.

DEPOSITION OF PARTICLES IN TURBULENT PIPE FLOW

The deposition of particles in turbulent pipe flow was investigated in terms of two mechanisms, turbulent and thermophoretic. A general equation incorporating these two mechanisms was formulated to calculate the deposition efficiency of aerosol particles in turbulent pipe flow together with thermophoretic deposition. The validity of the equation was confirmed by good agreement between calculated and measured results.

EFFECT OF NON-SPHERICAL PARTICLES ON THE FLUID TURBULENCE IN A PARTICULATE PIPE FLOW

In the non-spherical particulate turbulent flows, a set of new fluidfluctuating velocity equations with the non-spherical particle source term were derived, then a newmethod, which treats the slowly varying functions and rapidly varying functions separately, wasproposed to solve the equations, and finally the turbulent intensity and the Reynolds stress of theflu-id were obtained by calculating the fluctuating velocity statlsti-cally. The equations andmethod were used to a paniculate tur-bulent pipe flow. The results show that the turbulent intensityand the Reynolds stress are decreased almost inverse proportion-ally to the fluctuating velocityratio of particle to fluid. Non-spherical particles have a greater suppressing effect on thetur-bulence than the spherical particles. The particles with short re-laxation time reduce theturbulence intensity of fluid, while the particles with long relaxation time increase the turbulenceinten-sity of fluid. For fixed particle and fluid, the small particles sup-press the turbulence andthe large particles increase the turbu-ience.