THE EFFECT OF FLOW PULSATIONS ON CORIOLIS MASS FLOW METERS

It has been reported that the accuracy of Coriolis mass flow meters can be adversely affected by the presence of pulsations (at particular frequencies) in the flow. A full analysis of the transient performance of a commercial Coriolis meter is only possible using finite element techniques. However, this is a transient, nonlinear problem in which the space and time variables are not (strictly) separable and the finite element techniques for tackling such problems make it desirable to have an analytical solution for a simplified meter, against which the finite element solution can be compared. This paper reports such a solution. The solution will also provide guidance for experiments. Existing analytical solutions for the performance of Coriolis meters in steady flow (a complex eigenvalue problem) are not easily extended to the transient flow case. The paper thus begins with the presentation of an alternative solution for steady flow through a simple, straight tube, Coriolis meter and it is notable that this solution gives a simple analytical expression for the experimentally observed small change in the resonant frequency of the meter, with flow rate, as well as an analytical expression for the meter sensitivity. The analysis is extended to the transient case, using classical, forced vibration, modal decomposition techniques. The solution shows that, unlike the steady flow case where the detector signals contain components at the drive frequency and the second mode frequency (Coriolis frequency), for pulsatile flow the detector signals will in general contain components involving at least four frequencies. It is demonstrated that the meter error depends on the algorithm used to estimate the phase difference from the detector signals. The particular flow pulsation frequencies which could possibly lead to large meter errors are identified.     

Analytical solution of velocity distribution for flow through submerged large deflection flexible vegetation

An analytical solution for predicting the vertical distribution of streamwise mean velocity in an open channel flow with submerged flexible vegetation is proposed when large bending occurs. The flow regime is separated into two horizontal layers: a vegetation layer and a free water layer. In the vegetation layer, a mechanical analysis for the flexible vegetation is conducted, and an approximately linear relationship between the drag force of bending vegetation and the streamwise mean flow velocity is observed in the case of large deflection, which differes significantly from the case of rigid upright vegetation. Based on the theoretical analysis, a linear streamwise drag force-mean flow velocity expression in the momentum equation is derived, and an analytical solution is obtained. For the free water layer, a new expression is presented, replacing the traditional logarithmic velocity distribution, to obtain a zero velocity gradient at the water surface. Finally, the analytical predictions are compared with published experimental data, and the good agreement demonstrates that this model is effective for the open channel flow through the large deflection flexible vegetation.     

Approximate solutions for power-law fluid flow past a particle at low Reynolds numbers

A simple perturbation approximation is proposed for describing flow behaviour of particles immersed in a uniform flow and an extensional flow of power-law fluids. The present solution for particles in a uniform flow field is in good agreement with the numerical solution in the literature. Theoretical predictions indicate that the effect of pseudoplasticity on flow around particles in an extensional flow field is small compared with that for particles in a uniform flow field.From the viewpoint of perturbation techniques, existing analytical solutions based on linearization of the equations of motion for particle in a power-law fluid are re-examined. Mass transfer to a power-law fluid from a particle is also discussed.     

Two dimensional analytical solution for a partially vegetated compound channel flow

The theory of an eddy viscosity model is applied to the study of the flow in a compound channel which is partially vegetated. The governing equation is constituted by analyzing the longitudinal forces acting on the unit volume where the effect of the vegetation on the flow is considered as a drag force item, The compound channel is divided into 3 sub-regions in the transverse direction, and the coefficients in every region＇s differential equations were solved simultaneously. Thus, the analytical solution of the transverse distribution of the depth-averaged velocity for uniform flow in a partially vegetated compound channel was obtained. The results can be used to predict the transverse distribution of bed shear stress, which has an important effect on the transportation of sediment. By comparing the analytical results with the measured data, the analytical solution in this paper is shown to be sufficiently accurate to predict most hydraulic features for engineering design purposes.     

低渗透多孔介质渗流动边界模型的解析与数值解

考虑启动压力梯度的低渗透多孔介质非达西渗流模型属于强非线性动边界问题, 分别利用相似变量变换方法和基于空间坐标变换的有限差分方法, 对内边界变压力情况下、考虑启动压力梯度的一维低渗透多孔介质非达西渗流动边界模型进行了精确解析与数值求解研究. 研究结果表明:该动边界模型存在唯一的精确解析解, 且所求得的精确解析解可严格验证数值解的正确性;且当启动压力梯度值趋于零时, 非达西渗流动边界模型的精确解析解将退化为达西渗流情况下的精确解析解. 由求解结果作出的非零无因次启动压力梯度下的地层压力分布曲线表现出紧支性特点, 其与达西渗流模型的有显著不同. 因此, 研究低渗透多孔介质中非稳态渗流问题时, 应该考虑动边界的影响. 研究内容完善了低渗透多孔介质的非达西渗流力学理论, 为低渗透油气藏开发的试井解释与油藏数值模拟技术提供了理论基础.      

An analytical solution for steady flow of a Quemada fluid in a circular tube

An analytical solution is obtained for steady flow of Quemada-type fluids in a circular tube driven by a constant pressure gradient. Expressions are derived for velocity distribution and for volumetric flow rate as a function of pressure gradient or wall shear stress.     

An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching

In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.     

On a Rigorous Resolvent Estimate for Plane Couette Flow

We derive a rigorous bound of the solution of the resolvent equation for plane Couette flow in three space dimensions. We combine analytical techniques with numerical computations. Compared to earlier results, our analytical techniques cover a larger part of the parameter domain consisting of wave numbers in two space directions and the Reynolds number. Numerical computations are needed only in a compact subset of the parameter domain. This research was supported by the Swedish Research Council grant 2003-5443.     

An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media

An analytical solution to the one-dimensional solute advection-dispersion equation in multi-layer porous media is derived using a generalized integral transform method. The solution was derived under conditions of steady-state flow and arbitrary initial and inlet boundary conditions. The results obtained by this solution agree well with the results obtained by numerically inverting Laplace transform-generated solutions previously published in the literature. The analytical solution presented in this paper provides more flexibility with regard to the inlet conditions. The numerical evaluation of eigenvalues and matrix exponentials required in this solution technique can be accurately and efficiently computed using the sign-count method and eigenvalue evaluation methods commonly available. The illustrative calculations presented herein have shown how an analytical solution can provide insight into contaminant distribution and breakthrough in transport through well defined layered column systems. We also note that the method described here is readily adaptable to two and three-dimensional transport problems.     

An analytical solution and analysis of characters for viscoelastic fluid flow in annular pipe

I.Introducti0nLuiCiqunandHuangJunqiI'I(l989),ZhuWeihuiandLuiCiquri1'l(l992)sequentiallystudiedtheaxialflowofsecondorderandMaxwellfluidsandanalyzedtheflowcharactersofthesefluids.Thispaperwillstudyunsteadyrotat0ryflowofsecondorderandMaxwellfluidsinannularpi…     

Effect of heat transfer on a third grade fluid in a flat channel

The heat transfer analysis on the laminar flow of an incompressible third grade fluid through a porous flat channel is examined. The lower plate is assumed to be at a higher temperature than the upper plate. Analytical solution for temperature distribution is obtained for various values of the controlling parameters and discussed. The obtained analytical solution is also compared with the numerical solution. The comparison shows the fact that the accuracy is remarkable. Copyright © 2009 John Wiley & Sons, Ltd.     

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

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Thermal stability for a reactive viscous flow in a slab

The paper deals with the effect of dimensionless non-Newtonian coefficient on the thermal stability of a reactive viscous liquid in steady flow between parallel heated plates. It is assumed that the liquid is symmetrically heated and the flow fully developed. Approximate analytical solution is obtained for the velocity of the flow and the criterion for which this solution is valid is determined. After the velocity distribution is known, the temperature distribution may be calculated. Disappearance of criticality (transition values) are obtained in the following cases: (i) bimolecular (ii) Arrhenius and (iii) sensitized temperature dependence. We have observed that non-linear effect from velocity and temperature fields introduced decaying for the transitional values of the dimensionless central temperature. Other effects of this non-linearity are reported. The results help to enhance understanding of the interplay between Newtonian and non-Newtonian thermal explosions.     

Temperature distribution in a storage tank

In this work the temperature distribution in a vertical storage tank fed at its top with warmer fluid at constant flow rate is investigated. The flow rate is considered to be so small that laminar flow conditions are established. The temperature of the admitted fluid is allowed to vary with time where step and ramp time functions are used in this work. A rigorous analytical solution is obtained which describes the temperature as function of vertical depth and time. Graphical representation of the temperature distribution is given for selected values of the flow parameter and the time constant. Special reference is made in selecting these numerical values to solar water heating systems. The results are discussed and also limitations are recalled.     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of −1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation.     

BASIC SOLUTION FOR THE THREE-PHASE COMPOSITE CONSTITUTIVE MODEL IN ANTIPLANE PIEZOELECTRICITY

A basic solution in series form for the three-phase composite cyclindrical model in antiplane piezoelectricity subjected to the action of a singularity in the intermediate matrix region is presented. The solution is obtained through the complex potential approach in conjuction with the techniques of analytical continuation, singularity analysis, Laurent series expansion in an annular region and Cauchy integral formulae, etc. Based on the complex potentials obtained, explicit expressions for the distribution of stress and electric displacement in the three regions are also derived.     

An approximate solution considering flow inertia between conical surfaces

In this paper, analytical calculation expressions of the pressure distribution, velocity distribution and the rate of the flow between conical surfaces are found by using the method of iterative approximate solution when the inertia terms of the Navier-Stokes equations in conical coordinates are taken into account. Furthermore, we compare the centrifugal flow with the centripetal flow of axisymmetrical passing flow.     

Gas Flow in Porous Media With Klinkenberg Effects

Gas flow in porous media differs from liquid flow because of the large gas compressibility and pressure-dependent effective permeability. The latter effect, named after Klinkenberg, may have significant impact on gas flow behavior, especially in low permeability media, but it has been ignored in most of the previous studies because of the mathematical difficulty in handling the additional nonlinear term in the gas flow governing equation. This paper presents a set of new analytical solutions developed for analyzing steady-state and transient gas flow through porous media including Klinkenberg effects. The analytical solutions are obtained using a new form of gas flow governing equation that incorporates the Klinkenberg effect. Additional analytical solutions for one-, two- and three-dimensional gas flow in porous media could be readily derived by the following solution procedures in this paper. Furthermore, the validity of the conventional assumption used for linearizing the gas flow equation has been examined. A generally applicable procedure has been developed for accurate evaluation of the analytical solutions which use a linearized diffusivity for transient gas flow. As application examples, the new analytical solutions have been used to verify numerical solutions, and to design new laboratory and field testing techniques to determine the Klinkenberg parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs from The Geysers geothermal field. We show that this new approach and the traditional method of Klinkenberg yield similar results of Klinkenberg constants for the laboratory tests; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Determination of Temperature due to Slip between the Wheel and the Rail Taking into Account Convective Cooling of Free Surfaces

A computational algorithm for determining temperature due to slip between the train wheel and the rail is proposed. The algorithm uses Ling's solution of the mixed twodimensional quasisteady'state thermalconductivity problem for a half'space heated locally by a fast moving distributed heat flow. This solution is calculated using the method of piecewise linear approximation by finite functions. An analytical solution of the problem is obtained for the particular case of uniform distribution of the frictional heat flow rate. The effect of different forms of heat flow rate distribution and the Biot number on the rail temperature field is studied.     

An approximate solution considering flow inertia between spherical surfaces

In this paper, the analytical expressions of the pressure distribution, velocity distribution and discharge of the flow between spherical surfaces are found by using the method of iterative approximate solution when the inertia terms of Navier-Stokes equations in spherical coordinates are taken into consideration. Furthermore, using these expressions, we can directly obtain the corresponding analytical expressions of the laminar radial flow between parallel disks, which are fully identical with corresponding results presented by refs. [3,4].