Steady natural convection flow of a particulate suspension through a circular pipe

A continuum model for two-phase (fluid/particle) flow induced by natural convection is developed and applied to the problem of steady natural convention flow of a particulate suspension through an infinitely long pipe. The wall of the pipe is maintained at a constant temperature. The particle phase is endowed by an artificial viscosity which may be used to model particle-particle interaction in dension suspensions. Boundary conditions borrowed from rarefied gas dynamics are employed for the particle-phase wall conditions. Closed-form solutions for the velocity and temperature profiles are obtained. For the assumptions employed in the problem, the temperatures of both phases in the pipe are predicted to be uniform. A parametric study of some physical parameters involved in the problem is performed to illustrate the influence of these parameters on the velocity profiles of both the fluid and particle phases.     

Transient natural convection flow of a particulate suspension through a vertical channel

Continuum equations governing transient, laminar, fully-developed natural convection flow of a particulate suspension through an infinitely long vertical channel are developed. The equations account for particulate viscous effects which are absent from the original dusty-gas model. The walls of the channel are maintained at constant but different temperatures. No-slip boundary conditions are employed for the particle phase at the channel walls. The general transient problem is solved analytically using trigonometric Fourier series and the Laplace transform method. A parametric study of some physical parameters involved in the problem is performed to illustrate the influence of these parameters on the flow and thermal aspects of the problem.     

Transient non-Newtonian flow of a suspension with a compressible particle phase

Equations governing transient two-phase fluid-particle laminar flow over an infinite porous flat plate are developed. Both phases are assumed to behave as non-Newtonian power-law fluids. The mathematical model accounts for particle-phase viscous and diffusive effects. The particles are assumed spherical in shape and having a non-uniform density distribution. The resulting governing equations are nondimensionalized and solved numerically subject to appropriate initial and boundary conditions using an iterative, implicit, tri-diagonal finite-difference method. Graphical results for the displacement thicknesses and the skin-friction coefficients for both the fluid and particle phases are presented and discussed to illustrate special trends of the solutions.     

Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipes

The problem of unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of a uniform transverse magnetic field is formulated using a two-phase continuum model. Two different applied pressure gradient (oscillating and ramp) cases are considered. The general governing equations of motions (which include such effects as particulate phase stresses, magnetic force, and finite particle-phase volume fraction) are non-dimensionalized and solved in closed form in terms of Fourier cosine and Bessel functions and the energy equations for both phases are solved numerically since they are non-linear and are difficult to solve analytically. Numerical solutions based on the finite-difference methodology are obtained and graphical results for the fluid-phase volumetric flow rate, the particle-phase volumetric flow rate, the fluid-phase skin-friction coefficient and the particle-phase skin-friction coefficient as well as the wall heat transfer for plane and axisymmetric flows are presented and discussed. In addition, these numerical results are validated by favorable comparisons with the closed-form solutions. A comprehensive parametric study is performed to show the effects of the Hartmann magnetic number, the particle loading, the viscosity ratio, and the temperature inverse Stokes number on the solutions.     

Dusty-gas flow in a channel with horizontalwalls

An unsteady motion of a dilute gas-particle mixture in a plane channel under the action of a constant longitudinal pressure gradient and the transverse gravity force is studied theoretically. Within the Euler approach, a combined problem of finding the velocity components of the gas and the dust phase is formulated. The flow similarity parameters are found. The solution of the problem formulated is calculated using a finite-difference method. Asymptotic formulas are obtained, which describe the two-phase flow parameters for limiting values of the similarity parameters. The cases of both monodisperse and polydisperse particles are considered.     

Comparative analysis of CFD models for jetting fluidized beds: Effect of particle-phase viscosity

Under the Eulerian–Eulerian framework of simulating gas–solid two-phase flow, the accuracy of the hydrodynamic prediction is strongly affected by the selection of rheology of the particulate phase, for which a detailed assessment is still absent. Using a jetting fluidized bed as an example, this work investigates the influence of solid rheology on the hydrodynamic behavior by employing different particle-phase viscosity models. Both constant particle-phase viscosity model (CVM) with different viscosity values and a simple two-fluid model without particle-phase viscosity (NVM) are incorporated into the classical two-fluid model and compared with the experimental measurements. Qualitative and quantitative results show that the jet penetration depth, jet frequency and averaged bed pressure drop are not a strong function of the particle-phase viscosity. Compared to CVM, the NVM exhibits better predictions on the jet behaviors, which is more suitable for investigating the hydrodynamics of gas–solid fluidized bed with a central jet.     

Parametric analysis of two-phase flow instability in a channel with inlet and outlet hydraulic resistances

The mechanism of low-frequency self-oscillating instability of a one-dimensional two-phase flow in a channel with inlet and outlet hydraulic resistances is considered. The mechanism is based on the sensitivity of the inlet flow rate of the liquid to the pressure variation inside the channel and the sensitivity of the pressure to the variation of the outlet gas flow rate (with a constant mass rate of the liquid-gas phase transition per unit volume). A spectral analysis of the stability of the steady solution of the boundary-value problem for a hyperbolic-type nonlinear system of equations is performed within the framework of a two-velocity model of a gas-liquid flow. Parametric boundaries of the region of instability are obtained. The existence of self-oscillations in this range of parameters is supported by a numerical solution of the unsteady boundary-value problem. Institute of Catalysis, Siberian Division Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 47–53, January–February, 1998.     

Bifurcation phenomena in two-phase natural circulation

Multiple steady-state solutions have been found theoretically for a two-phase natural circulation loop. This bifurcation phenomenon is attributed to the nonmonotonic behavior of the buoyancy and friction forces, and the discontinuities due to transitions between two-phase flow patterns. The paper outlines a general and consistent modeling method to describe two-phase natural circulation at steady state. The solution of the one-dimensional governing conservation equations is performed by a combination of analytical and numerical stages. The method is implicit, to treat the highly nonlinear problem by multinested iteration loops. Results are presented and discussed for the multiple solutions of the flow rate, temperature, quality and void distributions and two-phase elevations. Pressure effects are also studied. A comparison with available data shows that the theoretical model simulates the important characteristics of two-phase thermosyphons.     

A THREE-FLUID MODEL OF THE SAND-DRIVEN FLOW

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Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

One-group interfacial area transport equation and its sink and source terms in narrow rectangular channel

The characteristics of two-phase flow in a narrow rectangular channel are expected to be different from those in other channel geometries, because of the significant restriction of the bubble shape which, consequently, may affect the heat removal by boiling under various operating conditions. The objective of this study is to develop an interfacial area transport equation with the sink and source terms being properly modeled for the gas–liquid two-phase flow in a narrow rectangular channel. By taking into account the crushed characteristics of the bubbles a new one-group interfacial area transport equation was derived for the two-phase flow in a narrow rectangular channel. The random collisions between bubbles and the impacts of turbulent eddies with bubbles were modeled for the bubble coalescence and breakup respectively in the two-phase flow in a narrow rectangular channel. The newly-developed one-group interfacial area transport equation with the derived sink and source terms was evaluated by using the area-averaged flow parameters of vertical upwardly-moving adiabatic air–water two-phase flows measured in a narrow rectangular channel with the gap of 0.993 mm and the width of 40.0 mm. The flow conditions of the data set covered spherical bubbly, crushed pancake bubbly, crushed cap-bubbly and crushed slug flow regimes and their superficial liquid velocity and the void fraction ranged from 0.214 m/s to 2.08 m/s and from 3.92% to 42.6%, respectively. Good agreement with the average relative deviation of 9.98% was obtained between the predicted and measured interfacial area concentrations in this study.     

Optimal systems of Lie subalgebras for a two-phase mass flow

We apply the Lie symmetry method to a two-phase mass flow model (Pudasaini, 2012 [18]) and construct one-, two- and three-dimensional optimal systems of Lie subalgebras corresponding to the non-linear PDEs. As an optimal system contains structurally important information about different types of invariant solutions, it provides precise insights into all possible invariant solutions emerging from infinitesimal symmetries. We use the optimal system of one-dimensional Lie subalgebras to reduce the two-phase mass flow model to other systems of PDEs. Using the fact that the Lie bracket contains information about further reduction, we further reduce to systems of ODEs and PDEs. We solve a system numerically and present results for different physical and Lie parameters. Simulations reveal fluid and solid dynamics are distinctly sensitive to different Lie parameters, whereas both phases are influenced by the solid and the fluid pressure parameters. Higher pressure gradients result in higher flow velocities and lower flow heights. Fluid velocities dominate solid velocities, but the solid heights are higher than the fluid heights. Results provide an overall picture of the physical process, and the coupled dynamics of the solid and fluid phase velocities and the flow heights. These are physically meaningful results in sheared inclined channel flow of coupled two-phase mixture. This confirms the consistency of the obtained similarity solutions and potential applicability of the models and the constructed optimal systems.     

Numerical investigation of static flow instability in a low-pressure subcooled boiling channel

A three-dimensional two-fluid model to predict subcooled boiling flow at low pressure is presented. The model is adopted to investigate the two-phase flow and heat transfer characteristics in a heated channel. The presence of bubbles as a consequence of heating flow through a vertical rectangular channel has a significant effect on the overall pressure drop along the channel. Numerical results were compared against a series experimental data performed at various conditions – mass flux, heat flux, inlet temperature and exit pressure. Good agreement on the overall pressure drop was achieved. The onset of flow instability velocity was also accurately determined when compared against measurements. Predicted results of void fraction provided useful information towards a more fundamental understanding of the occurrence of onset of nucleate boiling, onset of significant voiding and onset of flow instability. The phenomenon of boiling onset oscillations was also predicted through the use of the two-fluid model.     

Solution of the direct problem of a laval nozzle working with a two-phase medium

A study is made of a method of numerical solution of the system of ordinary differential equations describing the flow of a two-phase medium in a Laval nozzle in the one-dimensional approximation. The paper is concerned with the direct problem, in which the law of variation of the area of the transverse section of the nozzle and functional relationships between the entrance parameters are given and the unknown parameters of the two-phase flow are found along the length of the nozzle.     

Thermal hydraulic modeling of a natural circulation loop

 The experiment was carried out on the test loop HRTL-5, which simulates the geometry and system design of a 5 MW nuclear heating reactor. The analysis was based on a one-dimensional two-phase flow drift model with conservation equations for mass, steam, energy and momentum. Clausius–Clapeyron equation was used for the calculation of flashing front in the riser. A set of ordinary equations, which describes the behavior of two-phase flow in the natural circulation system, was derived through integration of the above conservation equations for the subcooled boiling region, bulk boiling region in the heated section and for the riser. The method of time-domain was used for the calculation. Both static and dynamic results are presented. System pressure, inlet subcooling and heat flux are varied as input parameters. The results show that subcooled boiling in the heated section and void flashing in the riser have significant influence on the distribution of the void fraction, mass flow rate and flow instability of the system, especially at low pressure. The response of mass flow rate, after a small disturbance in the heat flux is shown, and based on it the instability map of the system is given through experiment and calculation. There exists three regions in the instability map of the investigated natural circulation system, namely, the stable two-phase flow region, the unstable bulk and subcooled boiling flow region and the stable subcooled boiling and single phase flow region. The mechanism of two-phase flow oscillation is interpreted. Received on 24 January 2000     

Visualized study on specific points on demand curves and flow patterns in a single-side heated narrow rectangular channel

A simultaneous visualization and measurement study on some specific points on demand curves, such as onset of nucleate boiling (ONB), onset of significant void (OSV), onset of flow instability (OFI), and two-phase flow patterns in a single-side heated narrow rectangular channel, having a width of 40 mm and a gap of 3 mm, was carried out. New experimental approaches were adopted to identify OSV and OFI in a narrow rectangular channel. Under experimental conditions, the ONB could be predicted well by the Sato and Matsumura model. The OSV model of Bowring can reasonably predict the OSV if the single-side heated condition is considered. The OFI was close to the saturated boiling point and could be described accurately by Kennedy’s correlation. The two-phase flow patterns observed in this experiment could be classified into bubbly, churn, and annular flow. Slug flow was never observed. The OFI always occurred when the bubbles at the channel exit began to coalesce, which corresponded to the beginning of the bubbly–churn transition in flow patterns. Finally, the evolution of specific points and flow pattern transitions were examined in a single-side heated narrow rectangular channel.     

Analysis of forced-convection boiling flow instabilities in a single-channel upflow system

A numerical model is developed to predict the steady-state and transient behaviour of forced-convection boiling two-phase flow in a single channel. The model is based on the assumption of homogeneous two-phase flow and thermodynamic equilibrium of the phases. Compressibility effects in the two-phase region, motion of the bulk boiling interface and the thermal capacity of the heater wall have been included in the analysis. The model is used to study the effects of heat input, inlet subcooling and flow rate on the system behaviour. For comparison purposes, an experimental investigation was conducted using a single-channel, electrically heated, forced-convection upflow system. Steady-state operating characteristics, and stable and unstable regions, are determined as a function of heat flux, inlet subcooling and mass flow rate. Different modes of oscillation and their characteristics have been investigated. The model's predictions are in good agreement with the experimental results.     

The prediction of two-phase mixture level and hydrodynamically-controlled dryout under low flow conditions

A theory is given for the thermal-hydraulic phenomena during uncovery of a flow channel. This is relevant to a reactor core under typical small break or operational transient conditions. A distinct equivalent collapsed liquid level and a two-phase-mixture level are defined in the model. The former represents the liquid inventory in the channel, while the latter characterizes the heat transfer regimes. The definition of these levels are coupled through the mass and energy conservation equations, and the constitutive relations for void fraction and net vapor generation location.Analytical solutions are obtained for the transient variation of both the collapsed liquid and the two-phase mixture levels.The analyses have been compared with existing single-tube data with uniform heat flux, and rod bundle experiments with an axial power profile and inlet feedwater flow. The results demonstrate the potential of the present model for application to reactor conditions.     

Stratification of boiling two-phase flow in a single horizontal channel

The objective of this study is to investigate experimentally the stratification phenomena of boiling two-phase flow in a uniformly heated horizontal channel. Two-phase flow stratification due to gravity effects, and consequently its thermal and hydrodynamic behavior, under steady state conditions, have been determined by measuring 16 top and 16 bottom wall temperatures. Six distinct wall temperature profiles are found, and the corresponding flow patterns are discussed. A dimensionless number has been formulated for the prediction of the occurrence of different flow patterns.     

定床弯道内水沙两相运动的数值模拟

在适体同位网格中采用非正交曲线坐标系下的三维k-ε-kp固液两相双流体湍流模型研究弯道内水流和悬浮泥沙运动,主要计算了试验室S型水槽内清水流动的三维流场、120°弯道内水沙两相流动中底沙与底流的运动轨迹以及S型水槽内水沙两相流动的两相流场和泥沙浓度场. 对于S型水槽内清水流动,数值结果与试验结果吻合良好. 120°弯道内水沙两相流动中固液两相的运动轨迹在弯道直线段基本重合,在弯道内泥沙轨迹逐步偏离水体轨迹,其偏离程度随泥沙粒径增大而增大. 从S型水槽内水沙两相流动计算结果中发现泥沙纵向流速在壁面附近比水流纵向速度大,在远离壁面区域比水流纵向速度小;弯道内泥沙横向流速比水流横向流速小;垂向流速在直线段和泥沙沉速相当,在弯道内受螺旋水流影响而变化;两相流速差别随泥沙粒径增大而变大;泥沙浓度呈现下浓上稀的分布,在弯道内横向断面上呈现凸岸大凹岸小的分布,泥沙浓度随泥沙粒径增大而减小.