Mass transfer between particles and a flow in the presence of a volume chemical reaction

The exterior problem of the mass transfer between a spherical drop and a linear shear flow in the presence of a first-order volume reaction is solved in the diffusion boundary layer approximation. A simple approximate expression for calculating the average Sherwood number for a drop or solid particle of arbitrary shape is proposed. At large Péclet numbers this expression is applicable to any type of flow over the entire range of variation of the reaction rate constant. The problem of diffusion to a spherical drop in a translational Stokesian flow in the presence of a first-order volume reaction was investigated in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 109–113, November–December, 1987.     

页岩气藏压裂水平井试井分析

页岩气藏资源丰富,开发潜力巨大,已成为目前研究的热点.与常规气藏相比,页岩气藏运移机制复杂,流动模式呈非线性,有必要考虑页岩气的吸附解吸,天然微裂缝的应力敏感性,人工裂缝内的非达西流等非线性因素对压裂水平井压力响应的影响. 基于双重介质和离散裂缝混合模型,分别采用Langmuir等温吸附方程描述吸附解吸,渗透率指数模型描述应力敏感,Forchheimer方程描述非达西效应,建立页岩气藏压裂水平井数值试井模型. 运用伽辽金有限元法对模型进行求解.根据试井特征曲线,划分流动阶段,着重分析非线性因素对压力响应的影响.结果表明:页岩气藏压裂水平井存在压裂裂缝线性流、压裂裂缝径向流、地层线性流、系统径向流及封闭边界影响5 种流动阶段.吸附解吸的影响发生窜流之后,Langmuir吸附体积增大,拟压力导数曲线凹槽更加明显,系统径向流出现时间与压力波传播到边界时间均延迟;天然裂缝系统的应力敏感性主要影响试井曲线的晚期段,拟压力和拟压力导数曲线均表现为上翘,应力敏感效应越强,上翘幅度越大;高速非达西效应对早期段影响较大,非达西效应越强,拟压力降幅度越大,试井曲线上翘.与解析解的对比以及矿场实例验证了模型的正确性与适用性.      

环形喷管喷口气泡演化的实验研究

水下气泡的发展演化及气泡动力学行为是气液两相动力学的基础理论与水下射流应用的重要基础. 环形喷管/喷口形成的气泡及气体射流具有其不同于圆孔实心射流的特殊表现与规律机制,随着同心筒破水发射等特殊应用的出现,环形喷口气体射流/泡流的基础现象观测和机制分析成为迫切的需求. 基于环形喷管的设计和水下射流条件的分析,设计建立了一套环形喷管水箱实验系统,对水下环形喷管喷口气泡发展演化过程进行了初步的实验研究. 为观测研究气体通过环形喷管气泡生长发展过程,在较低压力、较低流速下,采用高速摄影仪记录气泡生长及发展演化过程. 结合对气泡发展演化过程的图像处理与分析,研究分析了环形喷口气泡形成区制、气泡生长过程形态发展特点、以及气泡形成时间及气泡体积变化特点. 研究表明:在本实验气体流量范围内(50.8~237.3 dm3/min),环形喷口气泡发展演化过程呈现较为明显的三周期区制,前泡尾流影响是环形气泡呈三周期区制的主要原因;不同周期内的气泡形成时间具有较稳定规律,并受到流量影响;气泡生长过程中有较为明显的下沉、回升特征;气泡表面张力、液体惯性与流动的共同作用,造成了典型的气泡顶部坍塌现象.      

Bubble formation in co-fed gas–liquid flows in a rotor-stator spinning disc reactor

The gas–liquid flow in a rotor-stator spinning disc reactor, with co-feeding of gas and liquid, is studied for high gas volumetric throughflow rates and high gas/liquid volumetric flow ratios. High speed imaging and spectral analysis of pressure drop signals are employed to analyse the flow. Two mechanisms of bubble formation are observed, one due to gas overpressure leading to large irregular bubbles, and one due to liquid turbulent vortices leading to small, well-defined bubbles. The two mechanisms lead to three distinct gas dispersion regimes, distinguished by their characteristic oscillations in pressure drop. At low rotational Reynolds numbers (Re_ω < 0.4 · 10⁶), in the gas spillover regime, the gas is dispersed as large bubbles only. Above this critical Re_ω, small bubbles are sheared off as well, thus forming a heterogeneous dispersion. At sufficiently high Re_ω, depending on the gas flow rate, the gas is homogeneously dispersed as small bubbles. The maximum gas flow that can be dispersed as small bubbles is linearly proportional to the local energy dissipation rate. The understanding of the bubble formation mechanisms and pressure signature allows prediction and detection of the prevailing hydrodynamic regime in scaled up spinning disc reactors and for different reaction fluids.     

A numerical study on drop formation of viscoelastic liquids using a nonlinear constitutive equation

In this paper, a numerical solution for viscoelastic drop formation from a nozzle into an ambient gas is presented. A volume of fluid (VOF) method is used to predict the formation and break-up process of viscoelastic drop. Here, Giesekus model is used as the constitutive equation. The major features of the phenomenon, such as instantaneous drop length, limiting length of a drop at breakup, minimum drop radius and the volume of the primary drop is determined for a range of the parameter space spanned by the appropriate dimensionless groups. The results reveal that enhancing the mobility factor, Wiessenberg number, and viscosity ratio causes a noticeable decrease in limiting drop length and a small decrease on the primary drop volume. Also, the increasing of gravitational bond number and capillary number causes the limiting drop length increases while the primary drop volume is reduced.     

颗粒物质在竖直管中下行蠕动流动的实验测量

颗粒流蠕动行为是颗粒物质在竖直管中流动时常见的一种流动现象,其产生机理较复杂。为此,本文在在内径为150mm、高为5000mm的竖直管实验装置上,以FCC催化剂为固体颗粒物料,采用PV6型颗粒速度测量仪,测量不同颗粒流率下竖直管中的颗粒下行蠕动流动速度以及颗粒固含率,系统地考察了颗粒物质在竖直管中下行流动时的蠕动流动特性及产生机理。实验结果表明,颗粒物质在竖直管中下行流动时的流动行为可划分为两种形式。在颗粒流率较小时,颗粒物质下行速度呈现脉冲式变化,有速度停滞,可称之为蠕动I型流动。随着颗粒流率的增加,颗粒下行速度停顿消失,但仍是起伏变化,为蠕动II型流动。当颗粒流率增大到一定值后,颗粒物质下行蠕动行为消失,转变为流化流动。颗粒物质下行的蠕动行为是出口区颗粒成拱与崩塌、颗粒与器壁滑动摩擦和颗粒力链作用的综合反映。     

Flow patterns and pressure drop of ionic liquid–water two-phase flows in microchannels

The two-phase flow of a hydrophobic ionic liquid and water was studied in capillaries made of three different materials (two types of Teflon, FEP and Tefzel, and glass) with sizes between 200 μm and 270 μm. The ionic liquid was 1-butyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl}amide, with density and viscosity of 1420 kg m⁻³ and 0.041 kg m⁻¹ s⁻¹, respectively. Flow patterns and pressure drop were measured for two inlet configurations (T- and Y-junction), for total flow rates of 0.065–214.9 cm³ h⁻¹ and ionic liquid volume fractions from 0.05 to 0.8. The continuous phase in the glass capillary depended on the fluid that initially filled the channel. When water was introduced first, it became the continuous phase with the ionic liquid forming plugs or a mixture of plugs and drops within it. In the Teflon microchannels, the order that fluids were introduced did not affect the results and the ionic liquid was always the continuous phase. The main patterns observed were annular, plug, and drop flow. Pressure drop in the Teflon microchannels at a constant ionic liquid flow rate, was found to increase as the ionic liquid volume fraction decreased, and was always higher than the single phase ionic liquid value at the same flow rate as in the two-phase mixture. However, in the glass microchannel during plug flow with water as the continuous phase, pressure drop for a constant ionic liquid flow rate was always lower than the single phase ionic liquid value. A modified plug flow pressure drop model using a correlation for film thickness derived for the current fluids pair showed very good agreement with the experimental data.     

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.     

A mathematical model for pressure drop of two-phase dry-plug flow in circular mini/micro channels

Dry-plug flow is a variation of two-phase plug flow that occurs in small scale channels and refers to the dry wall conditions at gas portions of the flow regime. Previous experimental studies found a significant increase in pressure drop in this flow regime from the two-phase wet-plug flow regime. In this work an analytical model for the pressure drop of this flow regime is developed and phenomena that influence pressure drop are examined. Unlike previous models, the proposed model seems to be applicable to a wide range of capillary numbers and give good estimations for all static contact angles. Contact angle hysteresis turned out to play a major role in inducing pressure drop in this flow regime. Model’s predictions were in good agreement with previous experimental data. Finally the model is applied to very low capillary number region and pressure drop predictions for this region are presented.     

Experiments on elongational flow of dilute polymer solutions: Part II: velocity field for the flow through small apertures

Following Part I which reported on the jet reaction and the excess pressure drop, this paper describes the results of flow velocities measured upstream of small apertures for dilute polymer solutions. The results suggest the existence of the two flow regimes reorted by Boger and Cable: the vortex growth regime and the divergent flow regime. The former corresponds to the regime of zero jet reaction and the latter to the regime of positive jet reaction. The axial velocity development for PEO solutions along the center line upstream of the aperture is found to yield an extremely high velocity rise at the onset of the vortex region and to give a nearly constant elongational rate within the vortex region. The constant elongational rate takes values such that the product with the relaxation time is about 0.5. Separan solutions do not provide so high a velocity rise at the onset of the vortex region but show an exponential rate of elongation in the vortex region.It is shown, by using the upper convected Maxwell model, that the steep velocity rise at the onset of the vortex region is given in the simplified flow model and the constant elongational rate within the vortex region holds the elongational stress nearly constant.     

微通道中应用"流动汇聚"形成分散体系

在微流控器件内集成具有流动汇聚功能的几何结构以研究液--液系统中液滴的形成. 相图显示液滴的大小为流量以及两种液体流量比的函数, 它包括两个区域, 在一个区域中液滴大小和孔隙宽度接近, 而在另一区域液滴大小则取决于``汇聚'后的细流直径, 从而能形成远小于孔隙的液滴. 单分散和多分散乳状液均可以被生成.     

Void fraction and pressure drop during external upward two-phase cross flow in tube bundles – part II: Predictive methods

The present paper is the Part II of a broad study concerning void fraction and pressure drop for air-water upward external flow across tube bundles. In the Part I, the experimental facility and the data regression procedures were described and the experimental results are presented and discussed. Initially, Part II presents a literature review concerning void fraction and pressure drop predictive methods available in the open literature for two-phase upward flow across tube bundles. Next, the methods from literature are compared among them and with the database presented in paper Part I. Significant discrepancies are observed among the predictive methods, and deviations as high as two orders of magnitude are verified among the predicted values of pressure drop. Then, a new void fraction predictive method is proposed based on the experimental results and on the minimum kinetic energy principle. This method provides satisfactory predictions of the results described in paper Part I and also of independent data from the literature. A new predictive method for frictional pressure drop during two-phase flow based on two-phase multiplier is also proposed. This method predicted 94% of the experimental data obtained in the present study within an error margin of ± 30%, and also provides accurate predictions of independent results for triangular tube bundles gathered in the open literature.     

Drop formation dynamics of constant low-viscosity,elastic fluids

The dynamics of drop formation under gravity has been investigated as a function of elasticity using a set of low-viscosity, ideal elastic fluids and an equivalent Newtonian glycerol–water solution. All solutions had the same shear viscosity, equilibrium surface tension, and density, but differed greatly in elasticity. The minimum drop radius in the early stages of drop formation (necking) was found to scale as expected from potential flow theory, independent of the elasticity of the solutions. Thus, during this stage of drop formation when viscous force is still weak, the dynamics are controlled by a balance between inertial and capillary forces, and there is no contribution of elastic stresses of the polymer. However, upon formation of the pinch regions, there is a large variation in the drop development to break-off observed between the various solutions. The elastic solutions formed secondary fluid threads either side of a secondary drop from the necked region of fluid between the upper and lower pinches, which were sustained for increasing amounts of time. The break-off lengths and times increase with increasing elasticity of the solutions. Evolution of the filament length is, however, identical in shape and form for all of the polymer solutions tested, regardless of differing elasticity. This de-coupling between filament growth rate and break-up time (or equivalently, final filament length at break-up) is rationalised. A modified force balance to that of Jones and Rees [48] is capable of correctly predicting the filament growth of these low-viscosity, elastic fluids in the absence of any elastic contributions due to polymer extension within the elongating filament. The elongation of the necked region of fluid (which becomes the filament) is dominated by the inertia of the drop, and is independent of the elasticity of the solution. However, elasticity does strongly influence the resistance of the pinch regions to break-off, with rapid necking resulting in extremely high rates of surface contraction on approach to the pinch point, initiating extension of the polymer chains within the pinch regions. This de-coupling phenomenon is peculiar to low-viscosity, elastic fluids as extension does not occur prior to the formation of the pinch points (i.e. just prior to break-up), as opposed to the high viscosity counterparts in which extension of polymers in solution may occur even during necking. Once steady-state extension of the polymers is achieved within the pinch at high extension rates, the thread undergoes elasto-capillary break-up as the capillarity again overcomes the viscoelastic forces. The final length at detachment and time-to-break-off (relative to the equivalent Newtonian fluid) is shown to be linearly proportional to the longest relaxation time of the fluid.     

An improved correlation of the pressure drop in stenotic vessels using Lorentz’s reciprocal theorem

A mathematical model of the human cardiovascular system in conjunction with an accurate lumped model for a stenosis can provide better insights into the pressure wave propagation at pathological conditions. In this study, a theoretical relation between pressure drop and flow rate based on Lorentz’s reciprocal theorem is derived, which offers an identity to describe the relevance of the geometry and the convective momentum transport to the drag force. A voxelbased simulator V-FLOW VOF3 D, where the vessel geometry is expressed by using volume of fluid(VOF) functions, is employed to find the flow distribution in an idealized stenosis vessel and the identity was validated numerically. It is revealed from the correlation that the pressure drop of NS flow in a stenosis vessel can be decomposed into a linear term caused by Stokes flow with the same boundary conditions, and two nonlinear terms. Furthermore, the linear term for the pressure drop of Stokes flow can be summarized as a correlation by using a modified equation of lubrication theory, which gives favorable results compared to the numerical ones. The contribution of the nonlinear terms to the pressure drop was analyzed numerically, and it is found that geometric shape and momentum transport are the primary factors for the enhancement of drag force. This work paves a way to simulate the blood flow and pressure propagation under different stenosis conditions by using 1D mathematical model.     

Effects of coalescence and breakup on the steady-state morphology of an immiscible polymer blend in shear flow

The steady-state morphology of an immiscible polymer blend in shear flow has been investigated by optical microscopy techniques. The blend is composed by poly-isobutylene (PIB) and poly-dimethylsiloxane (PDMS) of comparable viscosity. Experiments were performed by means of a home-made transparent parallel plate device. The two plates can be independently counterrotated, so that sheared droplets of the dispersed phase can be kept fixed with respect to the microscope point of view, and observed for long times. The distribution of drops and their average size were measured directly during flow at different shear rates and for different blend compositions. It was found that the average drop size in steady-state conditions is a decreasing function of the applied shear rate, and does not depend on blend composition for volume fractions up to 10%. Experiments have proved that, in the shear rate range which could be investigated, the stationary morphology is controlled only by coalescence phenomena, droplet breakup playing no role in determining the size of the dispersed phase. More generally, it has been shown that the steady-state morphology is a function not only of the physical parameters of the blend and of the shear rate, but also of the initial conditions applied to the blend. The steady-state results reported in this paper constitute the first direct experimental confirmation of theoretical models which describe the mechanisms of shear-induced drop coalescence.     

Drop-on-demand drop formation of colloidal suspensions

The drop formation dynamics in the drop-on-demand (DOD) inkjet process is studied for model inks including a Newtonian liquid and colloidal dispersions. The ink shear viscosity is a parameter often adjusted in tuning the DOD drop formation process. Apparent shear viscosity measured at low shear rates is currently used to characterize inkjet inks throughout both the inkjet industry and academia. However, during the ejection process in inkjet printing, very high shear rates (above 1 × 10⁵ s⁻¹) are involved. In this paper, the drop formation characteristics at 10 kHz drop formation rate in a DOD mode of a simple Newtonian liquid are compared with those of a colloidal suspension system which has the same low-shear-rate viscosity as the simple Newtonian liquid, but significantly different high-shear-rate viscosity. Under conditions of good jetting, the drop formation dynamics of the colloidal suspension is similar to that of the simple Newtonian liquid of similar low-shear viscosity, with only slight systematic differences observed. Good jetting is, however, difficult to obtain in the colloidal particle inks, with non-straight trajectories and non-axisymmetric ligaments commonly observed. These observations suggest that evaporation, nonuniform wetting, and particle-related changes in properties play a role when poor jetting behavior is observed for colloidal inks.     

Pressure drop prediction with a modified frictional-kinetic model for alumina in bypass pneumatic conveying system

A new frictional-kinetic model is proposed and modified for pressure drop prediction of alumina in a bypass pneumatic conveying system. This new model is based on the conventional Johnson–Jackson frictional-kinetic model. The critical value of solids volume fraction and maximum packing limit are modified based on the fluidized bulk density and tapped bulk density, respectively. In addition, an offset solid volume fraction is introduced into the frictional pressure model as well as into the radial distribution functions which represents the correction factors to modify the probability of collisions between particles when solid phase becomes excessively dense. For the application of the model, computational fluid dynamics (CFD) simulations were conducted by using kinetic theory, conventional frictional-kinetic model and modified frictional-kinetic model. The simulation results were then compared with the experimental results. It was found that the modified frictional-kinetic model showed the largest improvement on pressure drop prediction results compared with results obtained from applying the kinetic theory and the conventional frictional-kinetic model, especially for denser flows with low air mass flow rates and high solid loading ratios (SLR). In addition, the solids volume investigation of CFD simulations shows a strong comparison to the actual flow conditions in the pipe, as transient slug type flow of alumina is observed.     

Dynamics of formation and dripping of drops of deformation-rate-thinning and -thickening liquids from capillary tubes

Dynamics of formation of drops of non-Newtonian liquids from capillary tubes is studied computationally. The rheology of the drop liquids is described by a constitutive relation that accounts for both deformation-rate-thinning and -thickening. The analysis is expedited by reducing the original system of three-dimensional but axisymmetric equations to a system of one-dimensional slender-jet equations. The slender-jet equations are solved by a method of lines using a finite element method for spatial discretization and an adaptive finite difference method for time integration. The simulations follow the formation in time of thousands of drops in sequence, including any satellites that may be produced upon the breakup of a thin thread connecting an about-to-form primary drop to the rest of the liquid attached to the tube. Rate-thickening is shown to produce bead-on-string patterns, which are typically attributed to viscoelastic effects, along the thin threads as they near pinch-off. Rate-thinning, on the other hand, is demonstrated to reduce the length of such thin threads. Simulations are used to identify conditions that may lead to minimization and/or elimination of unwanted satellites. Analysis of dripping or leaky faucets of non-Newtonian liquids reveals rich nonlinear dynamical behavior. As with Newtonian liquids, simple periodic or P-1, where P stands for period, dripping at low flow rates gives way to more complex responses as flow rate is increased. In addition to P-1, P-2, and P-4 responses seen in recent computational analyses of dripping faucets of Newtonian liquids, the new non-Newtonian simulations have also uncovered difficult-to-find P-3 responses as well as chaotic states. Rate-thinning and low viscosities are shown to enhance the complexity of observed responses. Rate-thickening, on the other hand, lowers the critical value of the flow rate for the onset of complexity but narrows the range of flow rates over which the dynamics is complex. The possibility of hysteresis is demonstrated and the effect of fluid rheology on the value of the flow rate for transition from dripping to jetting is determined.     

Hydrodynamics of viscous fluid through porous slit with linear absorption

The exact solutions for the viscous fluid through a porous slit with linear ab-sorption are obtained. The Stokes equation with non-homogeneous boundary conditions is solved to get the expressions for the velocity components, pressure distribution, wall shear stress, fractional absorption, and leakage flux. The volume flow rate and mean flow rate are found to be useful in obtaining a convenient form of the longitudinal velocity component and pressure difference. The points of the maximum velocity components for a fixed axial distance are identified. The value of the linear absorption parameter is ran-domly chosen, and the rest available data of the rat kidney to the tabulate pressure drop and fractional absorption are incorporated. The effects of the linear absorption, uniform absorption, and flow rate parameters on the flow properties are discussed by graphs. It is found that forward flow occurs only if the volume flux per unit width is greater than the absorption velocity throughout the length of the slit, otherwise back flow may occur. The leakage flux increases with the increase in the linear absorption parameter. Streamlines are drawn to help the analysis of the flow behaviors during the absorption of the fluid flow through the renal tubule and purification of blood through an artificial kidney.     

Investigation of pressure drop in horizontal pipes with different diameters

The pressure drop has a significant importance in multiphase flow systems. In this paper, the effect of the volumetric quality and mixture velocity on pressure drop of gas-liquid flow in horizontal pipes of different diameters are investigated experimentally and numerically. The experimental facility was designed and built to measure the pressure drop in three pipes of 12.70, 19.05 and 25.40 mm. The water and air flow rates can be adjusted to control the mixture velocity and void fraction. The measurements are performed under constant water flow rate (CWF) by adding air to the water and constant total flow rate (CTF) in which the flow rates for both phases are changed to give same CTF. The drift-flux model is also used to predict the pressure drop for same cases. The present data is also compared with a number of empirical models from the literature. The results show that: i) the pressure drop increases with higher volumetric qualities for the cases of constant water flow rate but decreases for higher volumetric qualities of constant total flow rate due to the change in flow pattern. ii) The drift-flux model and homogenous model are the most suitable models for pressure drop prediction.