Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

Investigation of droplet deposition from a turbulent gas stream

Turbulent deposition of particles from two-phase flow onto the smooth wall of a tube has been studied theoretically and experimentally. A model is proposed for the deposition motion of large particles based on turbulent diffusion in the core followed by a free flight towards the wall. The theory shows that within the Stokes regime, the dimensionless deposition velocity $\text{k-}{}_{\mathrm{<mtext>d</mtext>}}\text{/}\text{u}$* depends on Re and $\text{τ}{}^{\mathrm{+}}$ only, where u* is the friction velocity, Re is the tube Reynolds number and $\text{τ}{}^{\mathrm{+}}$ is the dimensionless particle relaxation time. Deposition data are obtained for air-water droplet flow through a 12.7-mm i.d. acrylic tubing at $\text{Re}\text{= 52,500}$ and 94,600. The proposed theory satisfactorily describes the existing deposition data as well as present measurements, covering a wide range of Re and $\text{τ}{}^{\mathrm{+}}$.     

Homogenized model of reaction–diffusion in a porous mediumUn modèle homogénéisé de réaction–diffusion dans un milieu poreux

We study the initial boundary value problem for the reaction–diffusion equation,

$\text{?}{}_{\mathrm{t}}\text{u}{}^{\mathrm{\epsilon }}\text{??·(a}{}^{\mathrm{\epsilon }}\text{?u}{}^{\mathrm{\epsilon }}\text{)+g(u}{}^{\mathrm{\epsilon }}\text{)=h}{}^{\mathrm{\epsilon }}$

in a bounded domain $\text{Ω}$ with periodic microstructure $\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$, where a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and κ(ε) in $\text{M}{}^{\mathrm{(\epsilon )}}$ with κ(ε)→0 as ε→0. Combining the method of two-scale convergence and the variational homogenization we obtain effective models which depend on the parameter θ=lim_ε→0κ(ε)/ε². In the case of strictly positive finite θ the effective problem is nonlocal in time that corresponds to the memory effect. To cite this article: L. Pankratov et al., C. R. Mecanique 331 (2003).     

Non-linear boundary and eigenvalue problems for the emden-fowler equations by group methods

Two pragmatic boundary value and eigenvalue problems of the Emden-Fowler equation $\text{(t}{}^{\mathrm{\alpha }}\text{u′)′ + λt}{}^{\mathrm{\beta }}\text{?(u) = 0,?(u) = u}{}^{\mathrm{\gamma }}$ and e^u are studied using the simple one parameter group properties. In all cases boundary value problems are converted into initial value problems using the property of the invariance group. With $\text{?(u) = u}{}^{\mathrm{\gamma }}$ an eigenvalue problem is detailed and calculations presented.     

Bifurcations et solutions multiples en cavité 3D différentiellement chaufféeBifurcations and multiple solutions in a differentially heated cubic cavity

In this paper, a differentially heated square/cubic cavity is studied by performing three-dimensional direct numerical simulations. The first bifurcation observed at $\text{R}{}_{\mathrm{a}}\text{≈3.2×10}{}^7$ is due to the 3D vortex structures generated at the end regions of vertical boundary layers near the median plane. The main results of this Note are that the flow returns to a steady state for higher values of the Rayleigh number $\text{R}{}_{\mathrm{a}}$ (7×10⁷ and 10⁸ for example) still exhibiting these 3D vortex structures, and that multiple steady flows which differ by their symmetry properties, are obtained for $\text{R}{}_{\mathrm{a}}\text{=10}{}^8$. However, the flow reverts to unsteadiness for $\text{R}{}_{\mathrm{a}}\text{=3×10}{}^8$. In this latter case, the instability is due to the vertical boundary layers. To cite this article: G. de Gassowski et al., C. R. Mecanique 331 (2003).     

Boundary layer in compressible flow of wet water vapour

In this paper the influence of small droplets, with radius $\text{10}{}^{\mathrm{?8}}\text{m}\text{< r < 10}{}^{\mathrm{?6}}\text{m}$, on laminar and turbulent boundary layer behavior is considered. It is found that the laminar boundary layer in a two-phase flow with strongly dispersed liquid retains dissipation energy and that the recovery factor of enthalpy is greater than unity. In turbulent boundary layers small droplets are transported by turbulent diffusion and this leads to the recovery factor being less than unity. Its value in both cases depends mainly on the nondimensional number $\text{Ds}\text{= C}{}_{\mathrm{Le}}\text{L/(U}{}_{\mathrm{e}}{}^2\text{/2)}$. The laminar boundary layer solution for non-equilibrium two-phase flow is obtained. Profiles of the droplet mass fraction, vapour and droplets temperatures and droplet radius are computed for the case of a steady two-dimensional flow. The turbulent boundary layer is treated using a semi-empirical theory assuming thermodynamic equilibrium.     

Sur la fermeture des termes rapides pour des écoulements moyens rotationnelsOn the closure of rapid terms for rotational mean flows

The modelling of homogeneous turbulent flows with mean rotatio, considered in a previous Note, is handled under the form $\text{M}{}^1\text{=}\text{M}\text{[}\text{b}\text{,}\text{y}\text{,}\text{Q}{}^1\text{]}$ in terms of componental and dimensional anisotropies, and of the symmetrized stropholysis. A systematic technique of expansion is proposed. The necessary realisability conditions are then applied. It is shown that there exists no realisable functional $\text{M}$ which would be isotropic with respect to its arguments. To cite this article: J. Piquet, C. R. Mecanique 331 (2003).     

Homogenization limit for electrical conduction in biological tissues in the radio-frequency rangeLimite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation $\text{?}\text{div}\text{(σ}{}_0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+A}{}^0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+∫}{}_0{}^{\mathrm{t}}\text{A}{}^1\text{(t?τ)?}{}_{\mathrm{x}}\text{u}{}_0\text{(x,τ)}\text{d}\text{τ?}\text{F}\text{(x,t))=0}$ where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function $\text{F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).     

Fractional vapour content of a liquid pool through which vapour is bubbled

Data from a large number of Russian, American and German sources are examined and found to be correlated in general by

$\text{α}\text{1?α)}\text{1}\text{2}\text{= K[F}{}_{\mathrm{D}}\text{P}{}^{\mathrm{m}}\text{]}{}^{\mathrm{n}}$

where α is voidage or fractional vapour content, K is a constant, F_D is a Froude number and P is a physical properties group. However, the exponent m is found to vary from 0 to 0.3 and the exponent n from $\text{2}\text{3}$ to 0.79, depending upon the sources of the data. The most probable value for n is $\text{2}\text{3}$ but a firm choice cannot be made for m, which is either 0.16 or 0.3. The different values of m depend chiefly upon the method of measurement of the voidage.     

An experiment to measure the pressure dependence of the zero-shear-rate viscosity

A simple experimental method, based on Stokes' law for falling spheres, has been devised and used to measure the pressure-dependence of the zero-shear-rate viscosity of a polypropylene melt. The experiment was performed by maintaining three thick-walled test cylinders containing the polymer melt and the falling sphere at the same elevated temperature but different pressures for periods of time ranging from 20 to 48 hours.When compared with experiments using high-pressure capillary or rotational viscometers, this experimental method has the advantages that viscous heating is non-existent and the apparatus and data analysis are relatively simple. The principal disadvantage encountered here, thermal degradation at high temperatures, could probably be reduced by molding specimens under vacuum and by shortening the exposure time. Since the falling-sphere experiment provides data at very low shear rates and the capillary and rotational viscometers generate data at high shear rates, the two experimental methods are complementary.The pressure coefficient b [=d(In η₀/dp] was determined for Hercules Pro-fax 6523 polypropylene in two series of experiments at different temperatures. For seven experiments at 218.3°C and pressures up to 97.9 $\text{MN}\text{m}{}^2$ (14,200 psi), the average value of b ± 95% confidence limits was found to be 14.8 ($\text{GN}\text{m}{}^2$)^?1 ± 2.9.The average b was 12.6 ($\text{GN}\text{m}{}^2$)^?1 ± 1.4 in a series of eight experiments at 232.2°C and pressures up to 123 $\text{MN}\text{m}{}^2$ (17,800 psi).     

A simple stochastic model of two phase flow pressure drop accompanied by boiling inside circular tubes with in-line static mixers

A simple stochastic model has been developed for boiling pressure drop inside a circular tube with in-line static mixers. This model gives rise to the dimensionless correlating equation of the form:

$\text{[f] = a[Pr}{}_{\mathrm{m}}\text{]}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{[Re}{}_{\mathrm{L}}\text{]}{}^{\mathrm{?1}}\text{μm}\text{μ}{}_{\mathrm{L}}\text{ρ}\text{ρ}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{H}{}_{\mathrm{se}}\text{+ xH}{}_{\mathrm{LG}}\text{C}{}_{\mathrm{pm}}\text{ΔT}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

This correlation shows good agreement with the experimental data.     

Slow flow through a periodic array of spheres

Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction $\text{c}$ of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0($\text{c}{}^{10}$) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0($\text{c}{}^3$) on infinitely long cylinders in square and hexagonal arrays.     

Anharmonic oscillators revisited

A large class of anharmonic oscillators represented by the Hamiltonian $\text{H(p,q) = (}\text{l}\text{2}\text{)p}{}^2\text{+ (}\text{l}\text{2}\text{)q}{}^2\text{+ λq}{}^{\mathrm{\alpha }}\text{(α integer}\text{\$}\text{?}\text{2)}$ is considered. Owing to an integration tech using the Lagrange-Bürmann theorem the solution of the motion equation is given in terms of series of Gauss hypergeometric functions. The period and the action integral of bounded motions are finally expressed in terms of energy in the form of generalized hypergeometric functions.     

A line plastic-zone model for steady mode III crack growth in an elastic-plastic material

Steady-state quasi-static growth of a crack in the anti-plane shear mode through an elastic-plastic material is analyzed. The material is non-hardening and small-scale yielding conditions are assumed. The essential feature of the model is that the active plastic-zone is assumed to be a pair of discrete lines emanating from the crack tip out of the crack plane on which a suitable yield condition is satisfied. An exact solution is obtained for the plastic strain left in the wake of this active line plastic-zone. The extent of the plastic zone from the tip is determined to be 0.071 $\text{(}\text{k}\text{τ}{}_0\text{)}{}^2$ where k and τ₀ are the remote elastic stress intensity factor and the shear flow stress, respectively, and it is found that 36% of the elastic energy flowing into the crack-tip region during growth is dissipated through plastic work and 64% is trapped as residual elastic energy in the plastic-zone wake.     

Time-average local thickness measurement in falling liquid film flow

A method for monitoring time-varying local film thickness variation through the detection of laser scattering from suspended latex particles is briefly described. This method was used in conjunction with the Jeffreys theory of drainage from a flat plate to determine time-average local film thickness.Measurements were made at Reynolds numbers (equal to ($\text{4Q/ν}$)) from 145 to 4030 at varying distances along the direction of flow. At the bottom of the flow, 134 cm from the top, average film thickness is given by the expression: where $\text{a}{}_{\mathrm{i}}$ and $\text{n}{}_{\mathrm{i}}$ are constants unique to each of the three Reynolds number regions, wavy laminar, transitional and turbulent.     

A regime map for direct contact condensation

Direct-contact condensation is studied by injecting steam downward through a pipe and out of the submerged end into a pool of subcooled water. The motion of the steam/water interface is recorded by high-speed movies and systematically classified, based on the injection rate and the pool subcooling. The resulting regime map shows the existence of three main condensation modes as the injection rate is reduced: At a high rate of steam injection $\text{(>125}\text{kg/m}{}^2\text{s}\text{)}$ an oscillatory jet is observed. At a low rate of injection $\text{(<50}\text{kg/m}{}^2\text{s}\text{)}$ a phenomenon called “steam chugging,” in which the pool water periodically enters the injection pipe, is observed. At intermediate injection rates an oscillatory bubble exists continuously at the pipe exit.     

Viscous dissipation effects on thermophoretically augmented aerosol particle transport across laminar boundary layers

The sensitivity of aerosol particle motion to local temperature gradients has motivated this investigation of viscous dissipation effects on mass transport rates across nonisthermal, low mass-loading ‘dusty gas’ laminar boundary layers (lbl). From numerical lbl transfer calculations, including ‘ash’ particle thermophoresis and variable thermophysical properties, it has been found that for a specified wall temperature, $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$, and mainstream static temperature, $\text{T}{}_{\mathrm{<mtext>e</mtext>}}$ cous dissipation within the boundary layer increases total particle deposition rates, its relative importance being dependent on $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}$. For combustion turbine blades which operate at near-unity Mach number, neglect of viscous dissipation is found to cause about a 25% underestimate of the fouling rate at $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$ for particle diameters between $\text{0.6 × 10}{}^{\mathrm{?2}}\text{μ}\text{m}$ and 0.3 μm. Alternatively, for conditions of fixed adiabatic wall temperature, $\text{T}{}_{\mathrm{<mtext>aw</mtext>}}$, or fixed stagnation (reservoir) temperature, $\text{T}{}_0$, dusty gas acceleration to appreciable Mach numbers is associated with reduced particle arrival rates due, in part, to the associated reduction in mainstream gas temperature. Recently developed mass transfer rate correlations are extended and found to be successful when tested against the present numerical calculations.