Hydro-mechanical aspects of the sand production problem

This paper examines the hydro-mechanical aspect of the sand production problem and sets the basic frame of the corresponding mathematical modelling. Accordingly, piping and surface erosion effects are studied on the basis of mass balance and particle transport considerations as well as Darcy's law. The results show that surface erosion is accompanied by high changes of porosity and permeability close to the free surface. Quantities which can be measured in experiment, like the amount of produced solids or fluid discharge, can be used in an inverse way to determine the constitutive parameters of the problem.Notation dV Volume element - dV _s Volume of solids pt - dV _v Volume of voids - dV _ff Volume of fluid phase - dV _fs Volume of fluidized-particles - Volume of mixture - dM _s Mass of solids - dM _ff Mass of fluid phase - d _{M fs} Mass of fluidized-particles - Mass of mixture - s Density of solids - f Density of fluid - ff Density of fluid phase - fs Density of fluidized-particles - Density of mixture - _i ^ff Velocity of fluid - _i ^fs Velocity of fluidized-particles - _i ^s Velocity of solids - Velocity of mixture - q ^ff Volume-discharge of fluid - q ^fs Volume-discharge of fluidized-particles - Volume-discharge of mixture - m ^ff Mass-discharge of fluid - m ^fs Mass-discharge of fluidized-particles - Mass-discharge of mixture - _er Rate of mass-eroded - _dep Rate of mass-deposited - Mass generation term - dS _i Surface element - Pore-surface element - D _IJ Tensor of mechanical dispersion - x _i Location - t Time - Porosity - c Transport concentration - c _cr Critical value ofc - p Fluid-pressure - k Permeability coefficient - _k Kinematic viscosity - Spatial frequency of erosion starter points     

The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

Messung des ersten Normalspannungskoeffizienten von Kunststoffschmelzen im Bogenspalt

Zusammenfassung Bei einer stationären Schichtenströmung in einem Bogenspalt (azimutale Druckströmung im Ringspalt) bildet sich zwischen Innen- und Außenwand eine Druckdifferenz aus, deren Größe ein Maß für den 1. Normalspannungskoeffizienten der elastischen Flüssigkeit im Spalt ist. Die Strömung läßt sich zur Messung des 1. Normalspannungskoeffizienten verwenden. Der Schergeschwindigkeitsbereich der Messung liegt, wie bei der Kapillarrheometrie zur Bestimmung der Viskosität, zwischen 1 und 1000 s^–1. Die Auswertung der Messungen ist wegen des inhomogenen Scherfeldes relativ kompliziert. In der Arbeit wird ein besonders wirkungsvolles numerisches Auswerteverfahren hergeleitet und auf bestehende Messungen angewendet. Eine Besonderheit des Auswerteverfahrens ist die Freiheit der Wahl des Approximationsansatzes für die Viskositätskurve, während analytische Verfahren meist an einen bestimmten Ansatz gebunden sind. Außerdem braucht, im Gegensatz zu anderen derartigen Verfahren, die Position des schubspannungsfreien Stromfadensr ₀ nicht bestimmt zu werden.

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Summary The stress in steady viscometric flow of molten polymers is determined by the viscosity and by the two normal stress coefficients ₁ and ₂. The paper describes a method of measuring ₁ by means of steady circumferential shear flow in an annulus. The cylinders are stationary and the fluid flows due to a circumferential pressure gradient. The radial normal stresses at the outer and at the inner wall are different from each other. The pressure-differencep is a measure for the 1. normal stress coefficient of the viscoelastic fluid. Due to the inhomogeneous shear field, the evaluation of ₁ fromp measurements is quite complicated. A powerful numerical method of evaluation has been developed and applied to existing data. The method is not restricted to a special empirical formula for the flow curve (as an analytical method would be) and does not require the knowledge of the positionr ₀ of the stress-free stream line.

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a Pa s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten, s. Gl. [8] - AR _i — Koeffizient des Druckgefälles in-Richtung (Programm PFEIL) - AU _i — Koeffizient für Integration nach Simpson-Regel (Programm PFEIL) - b s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten - B _i — Koeffizient auf der rechten Seite des linearen Gleichungssystems (Programm PFEIL) - c — Exponent des Ansatzes des 1. Normalspannungskoeffizienten - CL _i CM _i CR _i — Koeffizienten der dimensionslosen Geschwindigkeit in dem linearen Gleichungssystem (Programm PFEIL) - F ₁,F ₂,F ₃ — Ableitungen der Summe der Fehlerquadrate nacha, b undc - G _k — Gewichtsfaktor - h m Spaltweite,r _a –r _i - H — dimensionslose Spaltweite, (r _a –r _i )/r _a - l m Länge des Bogenspaltes, 0,75(r _a +r _i ) - m — Exponent des Potenzansatzes, s. Gl. [13] - n — Dämpfungskonstante - N ₁ Pa 1. Normalspannungsdifferenz, _rr – - N ₂ Pa 2. Normalspannungsdifferenz - p Pa Druck - p Pa Druckgradient in-Richtung - P — dimensionsloser Druckgradient in-Richtung, s. Gl. [14] - p, p _k Pa Normalspannungsdifferenz zwischen Innen- und Außenwand im Bogenspalt, (– p + _rr ) _a – (–p + _rr ) _i - Q — Summe der Fehlerquadrate - r, R= r/r _a m, — Radiusvektor (Koordinate in Gradientenrichtung) - r ₀,R ₀=r ₀/r _a m, — Radius des neutralen Fadens - R — dimensionslose radiale Schrittweite - T, °C Temperatur bzw. Bezugstemperatur - v ms^–1 Geschwindigkeitskomponente in-Richtung - V ,V _,i — dimensionslose Geschwindigkeitskomponente in-Richtung - V _a ,V _k — dimensionslose Geschwindigkeit an der Außen- bzw. Innenwand - v _r ,v _z ms^–1 Geschwindigkeitskomponenten inr-undz-Richtung - ms ^–1 mittlere Geschwindigkeit in-Richtung - z m Koordinate in der indifferenten Richtung - K^–1 Temperaturkoeffizient der Viskosität - s^–1 Schergeschwindigkeit - s^–1 kritische Schergeschwindigkeit der Viskositätskurve, s. Gl. [13] - s^–1 Bezugsschergeschwindigkeit, - — dimensionslose Schergeschwindigkeit - — dimensionslose kritische Schergeschwindigkeit, - Pa s Viskosität - ₀ Pa s Nullviskosität - Pa s Bezugsviskosität, - — Radienverhältnis,r _i /r _a - ₁ Pa s ² 1. Normalspannungskoeffizient - Pa s² mittlerer 1. Normalspannungskoeffizient - ₂ Pa s² 2. Normalspannungskoeffizient - — Koordinate in Strömungsrichtung - Pa Spannung - a an der Außenwand - i, an der Innenwand - i laufender Index inr-Richtung - k Nummer des Meßpunktes - n Anzahl der Meßpunkte - n _i nord für Programm PFEIL - s _i süd für Programm PFEIL Mit 9 Abbildungen und 2 Tabellen     

On the forced motion of elastic solids

Summary This note develops a method for the solution of the elastokinetic boundary value problem for time dependent surface tractions and/or displacements, as well as body forces which are functions of time and space. The method of Williams is extended to resolve three-dimensional problems of elastodynamics by classical mathematical techniques.Nomenclature x _i position vector - t time - u _i displacement vector - _ij stress tensor - F _i vector characterizing body force per unit volume - stress vector acting on surface S with unit outer normal v _i - density - , Lamé's constants - _ij Kronecker delta     

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

Nonlinear rheology of a concentrated spherical silica suspension:

Time-dependent nonlinear flow behavior was investigated for a model hard-sphere suspension, a 50 wt% suspension of spherical silica particles (radius = 40 nm; effective volume fraction = 0.53) in a 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The suspension had two stress components, the Brownian stress ^B and the hydrodynamic stress ^H After start-up of flow at various shear rates , the viscosity growth function ₊ (t, ) was measured with time t until it reached the steady state. The viscosity decay function _– (t, ) was measured after cessation of flow from the steady as well as transient states. At low where the steady state viscosity ( ) exhibited the shear-thinning, the _– (t, ) and ⁺ (t, ) data were quantitatively described with a BKZ constitutive equation utilizing data for nonlinear relaxation moduli G (t, ). This result enabled us to attribute the thinning behavior to the decrease of the Brownian contribution ^B = ^B / (considered in the BKZ equation through damping of G (t, )). On the other hand, at high where ( ) exhibited the thickening, the BKZ prediction largely deviated from the ₊ (t, ) and ₊ (t, ) data, the latter obtained after cessation of steady flow. This result suggested that the thickening was due to an enhancement of the hydrodynamic contribution ^H = ^H / (not considered in the BKZ equation). However, when the flow was stopped at the transient state and only a small strain (<0.2) was applied, ^H was hardly enhanced and the _– (t, ) data agreed with the BKZ prediction. Correspondingly, the onset of thickening of ₊ (t, ) was characterized with a -insensitive strain ( 0.2). On the basis of these results, the enhancement of ^H (thickening mechanism) was related to dynamic clustering of the particles that took place only when the strain applied through the fast flow was larger than a characteristic strain necessary for close approach/collision of the particles.     

Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media

A mathematical model for transient contaminant transport resulting from the dissolution of a single component nonaqueous phase liquid (NAPL) pool in two-dimensional, saturated, homogeneous porous media was developed. An analytical solution was derived for a semi-infinite medium under local equilibrium conditions accounting for solvent decay. The solution was obtained by taking Laplace transforms to the equations with respect to time and Fourier transforms with respect to the longitudinal spatial coordinate. The analytical solution is given in terms of a single integral which is easily determined by numerical integration techniques. The model is applicable to both denser and lighter than water NAPL pools. The model successfully simulated responses of a 1,1,2-trichloroethane (TCA) pool at the bottom of a two-dimensional porous medium under controlled laboratory conditions.Notation a,a ₁ defined in (45a) and (45b), respectively - b defined in (45c) - b vector of true model parameters (n×1) - vector of estimated model parameters (n×1) - c liquid phase solute concentration (solute mass/liquid volume), M/L³ - c _s aqueous saturation concentration (solubility), M/L³ - C dimensionless liquid phase solute concentration, equal toc/c _s - molecular diffusion coefficient, L²/t - ^e effective molecular diffusion coefficient, equal to / ^*, L²/t - D _x longitudinal hydrodynamic dispersion coefficient, L²/t - D _z hydrodynamic dispersion coefficient in the vertical direction, L²/t - e random vector with zero mean (m×1) - erf[x] error function, equal to (2/ ^1/2) - f vector of fitting errors or residuals (m×1) - Fourier operator - _-1 Fourier inverse operator - g vector of model simulated data (m×1) - k mass transfer coefficient, L/t - average mass transfer coefficient, L/t - K _d partition or distribution coefficient (liquid volume/solids mass), L³/M - pool length, L - _o distance between the pool and the origin of the specified Cartesian coordinate system, L - Laplace operator - _-1 Laplace inverse operator - m number of observations - M Laplace/Fourier function defined in (38) - n number of model parameters - N Laplace/Fourier function defined in (39) - p defined in (46) - Pe _x Péclet number, equal toU _x /D _x - Pe _z Péclet number, equal toU _x /D _z - q defined in (47) - R retardation factor - s Laplace transform variable - S objective function - Sh local Sherwood number, equal tok/ _e - Sh _o overall Sherwood number, equal to l/ _e - t time,t - T dimensionless time, equal toU _x t/ - u dummy integration variable - u vector of independent variables - U _x average interstitial velocity, L/t - x spatial coordinate in the longitudinal direction, L - X dimensionless longitudinal length, equal to (x–)/ - y vector of observed data (m×1) - z spatial coordinate in the vertical direction, L - Z dimensionless vertical length, equal toz/ - Fourier transform variable - defined in (37) - defined in (50) - porosity (liquid volume/aquifer volume), L³/L³ - defined in (52a) and (52b), respectively - decay coefficient, t^–1 - dimensionless decay coefficient, equal to /U _x - bulk density of the solid matrix (solids mass/aquifer volume), M/L³ - dummy integration variable - ^* tortuosity     

Eine Methode der thermischen Berechnung von Gegen- und Gleichstrom-Spiralwärmeaustauschern

Zusammenfassung Eine neue Methode zur thermischen Berechnung wird entwickelt, die die Berücksichtigung der tatsächlichen Stromführung erlaubt. Es wird eine spezielle Spirale als Erzeugende der zylindrischen Oberfläche im geometrischen Modell eines Spiralwärmeaustauschers (SWA) betrachtet. Anstelle einer archimedischen Spirale wird die Erzeugende durch eine Spirale dargestellt, die auf der Basis der Evolvente eines gleichseitigen regelmäßigen Vielecks entsteht. Das Ergebnis ist, daß der SWA hypothetisch in viele Sektionen aufgeteilt ist. Die Folge dieser Änderung sind konstante Koeffizienten in einem linearen gewöhnlichen Differentialgleichungssystem. Die Austrittstemperaturen werden aus diesem System mit Hilfe der Laplace-Transformierten für jede Sektion getrennt bestimmt und durch Randbedingungen miteinander verknüpft.

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Method of thermal calculation for rating countercurrent and cocurrent spiral heat exchangers

A new method of thermal calculation is developed which takes the actual flow arrangement into account. A special spiral is considered as a trace of the cylindrical surface in the geometrical model of a Spiral Heat Exchanger (SHE). Instead of an Archimedic spiral, the trace is formed on the other spiral, which is drawn by involution of a regular manysided figure. As a result SHE is hypothetically divided into many sections. The consequences of this change are constant coefficients in a linear ordinary differential equation system. The sought temperatures are determined from this system by using Laplace's transform for each section separately and coupled by boundary conditions.

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Bezeichnungen b, b Kanalbreite (Kanalhöhe),b gemessen entlang dem Radius, der vom Anfang der Spirale ausgeht, undb gemessen senkrecht zu einer der beiden den Kanal begrenzenden Kurven - C _ijl Konstante Gl. (13) - E Einheitsmatrix - F(), F _i () verallgemeinerte Temperaturfunktion (Vektor und seine Komponenten) - Laplace-Transformierte vonF - h ₀ Höhe des SWA - k Wärmedurchgangskoeffizient - L Operator der Laplace-Transformation - m ₀ Anzahl der Seiten des regelmäßigen Vielecks - M Matrix im Gleichungssystem (8), Gl. (9) - n Anzahl der Kanäle - r, =r/b Radius, reduzierter Radius, reduzierter Krümmungs-radius, Koeffizienten im Gleichungssystem (6), (7) - r _m Radiusvektor der Kanäle im Sektorm mit den Komponenten ₁= ₁/b, ₂=r ₂/b ₃=r ₃/b, ..., _m =r _m /b - R = _I / _II Quotient der WärmekapazitätenW - s Bildvariable der Laplace-Transformation - t Fluidtemperatur - dimensionslose Temperatur des Fluids I - , Parameter im MatrixM (), Gl. (9) - dimensionslose Temperatur des Fluids II - Radius der archimedischen Spirale - Winkel - , geometrische Koordinaten - dimensionslose Temperaturänderungen (Betriebscharakteristik des Wärmeaustauschers) - Anzahl der Ubertragungseinheiten - = verallgemeinerter Winkel - )I betr. Fluid I - )II betr. Fluid II - ein betr. Eintritt des Fluids Verzeichnis der Abkürzungen SWA Spiralwärmeaustauscher - S-A archimedische Spirale - S-ERV Spirale aus der Evolvente eines regelmäßigen Vielecks     

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

The microscopic analysis of high forchheimer number flow in porous media

High Forchheimer number flow through a rigid porous medium is numerically analysed by means of the volumetric averaging concept. The microscopic flow mechanisms, which must be known in order to understand the macroscopic flow phenomena, are studied by utilising a periodic diverging-converging representative unit cell (RUC). The detailed information for the microscopic flow field, in association with the locally averaged momentum balance, makes it possible to quantitatively demonstrate that the microscopic inertial phenomenon, which leads to distorted velocity and pressure fields, is the fundamental reason for the onset of nonlinear (non-Darcy) effects as velocity increases. The hydrodynamic definitions for Darcy's law permeabilityk, the inertial coefficient and Forchheimer number Fo are obtained by applying the averaging theorem to the pore level Navier-Stokes equations. Finally, these macroscopic parameters are numerically calculated at various combinations of micro-geometry and flow rate, and graphically correlated with the relevant microscopic parameters.Nomenclature a _i body force acceleration (m/s²) - A viscous integral term defined in (4.6) - A _f area of entrance and exist of RUC (m²) - A _fs interfacial area between the fluid and solid phases (m²) - B pressure integral term defined in (4.4) - d throat diameter of RUC (m) - D pore diameter of RUC (m) - Fo Forchheimer number defined in (4.1) and (4.10) - g gravitational acceleration (m/s²) - i, j microscopic unit vector for RUC - k Darcy's law permeability (m²) - k _v velocity dependent permeability defined in (4.1) (m²) - L length of a unit cell (m) - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p microscopic fluid pressure (N/m²) - P macroscopic fluid pressure (N/m²) - _en mean pressure at entrance of RUC (N/m²) - _ex mean pressure at exit of RUC (N/m²) - r _i,r coordinate on the macroscopic scale (m) - Re _d Reynolds number defined in (4.5) - u _i,u microscopic velocity (m/s) - specific discharge (m/s) - _d mean velocity at the throat of RUC (m/s) - v microscopic velocity (m/s) - V _b representative elementary volume (REV) (m³) - V _f volume occupied by the fluid within REV (m³) - V _s volume occupied by the solid within REV (m³) - x _i,x coordinate on the microscopic scale (m) - X _i,X coordinate on the macroscopic scale (m) Greek the inertia coefficient (1/m) - viscosity coefficient (Ns/m²) - _i microscopic unit vector - areosity at the entrance and the exit cross-section of RUC - fluid density (kg/m³) - porosity - _f a general property of the fluid phase Symbols ^f intrinsic phase average - the fluctuating part of _f - the mean value of _f - _f ^* the dimensionless value of _f     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

A nonequilibrium theory of a slurry

A nonequilibrium theory of a slurry is developed and its practical use is illustrated by a simple stability analysis. Here a slurry is defined as a deformable continuum consisting of a liquid phase containing in suspension a large number of small solid particles which have formed by solidification from the liquid. The liquid is assumed to consist of two components and the solid to contain only one of the two. Consequently, the process of change of phase requires redistribution of material on the scale of the solid particles. This process is assumed to take a finite amount of time, requiring a nonequilibrium macroscopic theory. This theory contains four thermodynamic variables, three to represent the equilibrium state of the binary system and a fourth measuring the departure from thermodynamic equilibrium. The process of microscale diffusion of material is parameterized in the macroscale theory, leading to a Landau-type relaxation term in the equation of evolution of the fourth variable. The theory is simplified to yield a Boussinesq-like set of governing equations. Their practical use is illustrated by analyzing the stability of a simple steady solution of the equations and the effects of a non-zero relaxation time are discussed. A novel instability mechanism involving sedimentation of particles, previously found to occur in the equilibrium case, is found to persist in nonequilibrium, but disappears in the limit of no change of phase.Key to symbols a, b, c thermodynamic coefficients; see (3.36)–(3.38) - sedimentation coefficient; see (5.18) - C _p specific heat; see (3.24) - C _p ^de specific heat of the slurry; see (3.28) and (3.30) - c radius of solid particle (in §4) - D, D diffusive coefficients; see (3.40) and (3.41) - material diffusivity in liquid phase - D ^* modified diffusion coefficient; see (5.15) - d thermodynamic coefficient; see (3.39) - E specific internal energy - f, g, h thermodynamic coefficients; see (3.36)–(3.38) - g acceleration of gravity - reduced gravity; see (5.10) - i total diffusive flux vector of constituent 1 - i diffusive flux vector of constituent 1 in the liquid phase - j diffusive flux vector of solid phase - k thermal conductivity - k entropy flux vector - k _T, k_T thermodiffusion coefficients; see (3.40) and (3.41) - L latent heat of solidification per unit mass; see (3.7) and (3.24) - m wave number - m ^s rate of creation of mass of solid per unit volume through solidification - m ₁ ^s rate of creation of mass of solid constituent 1 per unit volume through solidification - mass rate of freezing per unit area per unit time - N number of solid particles per unit volume - p pressure - p _H hydrostatic component of pressure - p _m mechanical pressure - p ₁ dynamic component of pressure - q heat flux vector - Q _D rate of regeneration of heat through diffusive fluxes - Q _M rate of regeneration of heat through phase-change processes - Q _v rate of regeneration of heat through viscosity - Q vector defined by (3.16) - r heat externally supplied per unit mass (in §3); spherical radial coordinate (in §4) - S specific entropy of slurry - change of specific entropy with mass fraction of constituent 1; also change of chemical potential of liquid phase with temperature barring change of phase - change of chemical potential of liquid phase with temperature in phase equilibrium; see (3.28) and (3.30) - T temperature - t time - t ₀ relaxation time; see (5.30) - u barycentric velocity - u _H horizontal perturbation velocity - V sedimentation speed - w _a upward speed of simple state; see (6.5) and (6.12) - z upward vertical coordinate - upward unit vector - thermal expansion coefficient barring change of phase; see (3.23) - > ^* thermal expansion coefficient in phase equilibrium; see (3.27) and (3.30) - modified thermal expansion coefficient; see (5.1) and (5.4) - isothermal compressibility of slurry barring change of phase; see (3.23) - ^* isothermal compressibility of slurry in phase equilibrium; see (3.27) and (3.30) - dimensionless measure of departure from liquidus equilibrium; see (5.2) - _a deviation from phase equilibrium in simple state; see (6.6) and (6.13) - vertical wave number - volume expansion per unit mass upon melting; see (3.6) - change of chemical potential of liquid phase with pressure; see (3.25) - change of chemical potential of liquid phase with pressure for slurry; see (3.29) and (3.30) - compositional gradient in the static state; see (6.15) - vector defined by (3.35) - constant of integration; see (6.7) and (6.8) - coefficient defined by (6.23) - nonequilibrium expansion coefficient; see (5.1) and (5.4) - thermal diffusivity; =k/C _p - modified thermal diffusivity; see (5.33) - relaxation rate to phase equilibrium; see (2.2) - ₁ relaxation rate to solid-composition equilibrium; see (2.3) - sedimentation coefficient; see (4.29) - horizontal wave number vector - sedimentation coefficient; see (4.30) - ^L , ^s chemical potential of constituent 1 relative to constituent 2 in liquid and solid phase per unit mass; see (2.6) - change of chemical potential of liquid with liquid composition; see (3.8) - coefficient defined by (3.10) - kinematic shear viscosity - total mass fraction of constituent 1 (i.e., solute) - ^L, ^s mass fraction of constituent 1 in liquid and solid phases - density of slurry - _s density of solid phase - - - , growth rate of disturbance - stress tensor - deviatoric stress tensor - dimensionless temperature; see (5,3) - _a constant of integration; see (6.7) - mass fraction of solid phase in slurry - _b vertical gradient of mass fraction of solid; see (6.1) - dimensionless measure of _b; see (6.22) - _c temporal gradient of mass fraction of solid; see (6.1) - specific Gibbs free energy; see (3.13) - _L,_s specific Gibbs free energy of liquid and solid phases; see (2.12) - measure of departure from liquidus equilibrium; see (2.14) - measure of departure from solidus equilibrium; see (2.5) - spherical polar coordinate (in §4); see (4.20); wave angle (in §6); see (6.38)     

Dehnungsverhalten von Polyäthylen-Schmelzen

Zusammenfassung In einem Dehnungsrheometer werden Spannungs-Dehnungs-Diagramme von Polyäthylen-Schmelzen bei 150 °C und bei konstanter Dehnungsgeschwindigkeit gemessen ( zwischen 0,001 und 1 sec^–1). Weiterhin wird der reversible (elastische) Dehnungsanteil bestimmt. Messungen mit einem Dehnungstester für Kunststoff-Schmelzen ergänzen die Ausführungen.Die Ergebnisse zeigen deutlich, daß bei Dehnung mit zunehmender Verformungsgeschwindigkeit die Dehnungsviskosität nicht abnimmt, im Gegensatz zu dem bekannten strukturviskosen Verhalten bei Scherung.Bei Dehnungen bis zu=1 kann das Verhalten unabhängig von beschrieben werden, wenn als viskoelastische Materialfunktion die Dehnungs-Spannviskosität betrachtet wird. In diesem Bereich von gilt dabei die BeziehungT(t)=3 _s (t) mit _s (t) als zeitabhängige Scherviskosität im linear-viskoelastischen Bereich.Bei größeren Dehnungen und nicht zu kleinen Dehnungsgeschwindigkeiten zeigt verzweigtes Polyäthylen eine zusätzliche starke Spannungszunahme. In dem Bereich dieser zusätzlichen Verfestigung ist das Verhalten im wesentlichen eine Funktion der Dehnung und fast unabhängig von . Die zusätzliche Verfestigung scheint eine Folge der Verzweigungsstruktur des verzweigten Polyäthylens zu sein, da bei Linear-PE ein derartiger Verlauf des Spannungs-Dehnungs-Diagramms nicht beobachtet wird.Die Betrachtung des reversiblen Dehnungsanteils _R zeigt bei der ausführlich untersuchten Schmelze I (verzweigtes PE) drei verschiedene Bereiche: Unterhalb einer Grenzdehnungsgeschwindigkeit ist _R =0, unterhalb einer Versuchszeitt ^** ist _R =. Im dazwischenliegenden Bereich treten elastische und viskose Dehnungsanteile auf,= _R + _V , wobei für niedrige gilt, daß _R lg . Die Grenze wird der Frequenz der thermisch aktivierten Platzwechsel zugeordnet,t ^** erscheint als Zeit, innerhalb der die Verhakungen wie fixierte Vernetzungen wirken.In dem Anhang wird der Einfluß der Grenzflächenspannung zwischen PE-Schmelze und Silikonöl auf die Ergebnisse der Dehnungsversuche diskutiert.

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Summary Stress-strain relations for different PE melts are measured at 150 °C in an extensional rheometer under the condition of a constant extensional strain rate ( between 0,001 and 1 sec^–1). Further, the recoverable (elastic) portion _R of the total strain ( in Hencky's measure) is determined and additional measurements with a tensile tester for polymer melts are described.The results show clearly that in extension there is no decrease of the tensile viscosity with increasing deformation rate, in contrast to the well-known pseudoplastic behaviour in shear. Within total strains<1 the tensile behaviour can be described independently from by means of a viscoelastic material function called stressing viscosity . In this range of the relation _T (t)=3 _s (t) holds, where _s (t) is the stressing viscosity in shear in the linear viscoelastic range. For larger tensile strains and not too small branched PB shows a remarkable increase in stress. This work-hardening behaviour is mainly a function of and almost independent from . This additional hardening seems to be due to the branches in branched PE, because linear PE does not show this effect.The discussion of the recoverable tensile strain _R gives three regions of tensile rate: Below a critical there is _R =0. At times shorter thant ^** the equation _R = is valid. Within these limits both elastic and viscous portions of the total strain= _R + _V exist. may correlate with the frequency of the thermally activated position changes of the molecular segments.t ^** is assumed to be the time for the entanglements to act as fixed cross-links.In the appendix the influence of the interface tension between PE melt and silicone oil on the results of the tensile experiments is discussed.

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Vorgetragen auf der Deutschen Rheologen-Tagung, Berlin, 11.-13. Mai 1970.

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An der Weiterentwicklung des Dehnungsrheometers, an der Durchführung und Auswertung der Messungen waren die HerrenB. Kienle, F. Landmesser, M, Reuther undF. Scherr beteiligt. Herr Dr.F.Ramsteiner und HerrH. Schroeck haben sich um die Herstellung der Stränge aus Linear-PE bemüht. Herr Dr.W. Ball besorgte die GPC-Messungen und Herr Dr.P. Simak die Ultrarot-Untersuchung. Den vorgenannten Herren sei für ihre Hilfe beim Zustandekommen dieser Arbeit gedankt. Herrn Dr.H. Baur danke ich für wertvolle Diskussionen.     

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

Constitutive equations applied to dispersed systems

Summary It has been investigated whether constitutive equations, which have been proposed originally to describe the rheological behaviour of polymerlike materials, can be used to represent the rheology of dispersions. Such equations generally predict stresses that depend on both the shear () and a quantity ( ) which is the product of the shear rate ( ) and the time constant of the material ().The behaviour of dispersions depends in general on the concentration of the dispersed particles. The dissipative aspect of the rheological behaviour is almostNewtonian for very dilute dispersions while it becomes plastic for more densely packet dispersions. In the latter case the shear stress is practically independent of the shear rate at low shear rates. Such behaviour may be accounted for in the constitutive equations by assuming to be almost constant. This motivated us to choose the equation ofBogue where the relaxation time () depends on the shear rate ( ), according to 1/ = (1/ ₀) + a , where 1/ ₀ accounts for the viscous behaviour and a for the plastic behaviour.Comparing the actual rheological behaviour of dispersions of fat crystals in paraffin oil with the behaviour predicted by theBogue equation, it turns out that theBogue equation has some success in representing the stress overshoot in steady shear experiments. However, the predicted value of the normal stress for the concentrated dispersions is too low in comparison with the measured value. It is suggested that this discrepancy is due to the dilatant behaviour of these dispersions.Moreover, the values of the dynamic moduli measured in oscillatory shear are predicted incorrectly, due to considerable changes in particle network which already occur at very small deformations.With 10 figures     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - V large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for the -phase - mass density of the -phase, kg/m³ - mass density of the -phase, kg/m³ - viscosity of the -phase, N s/m² - viscosity of the -phase, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Zur Lösung des Differentialgleichungssystems aus Mengen- und Energiebilanzen bei der Kondensation eines reinen Stoffes aus einem Inertgasstrom

Zusammenfassung Für die Lösung des Differentialgleichungssystems aus Mengen- und Energiebilanzen bei der Kondensation eines reinen Stoffes aus einem Inertgasstrom unter Berücksichtigung des gekoppelten Wärme- und Stofftransports werden drei numerische Verfahren diskutiert. Eine vereinfachte und eine physikalisch exakte Methode ermöglichen die schrittweise Berechnung eines Kondensators im stationären Zustand durch örtliche Integration über den Apparat. Bei der dritten Methode wird eine mathematisch und physikalisch exakte Integration über die Zeit durchgeführt. Die Ergebnisse der Verfahren ohne Vereinfachungen sind identisch, die vereinfachte Methode ist nur eingeschränkt anwendbar.

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Numerical solution of a system of differential equations describing the condensation of a pure fluid from a stream of inert gas

This paper presents three numerical methods for solving a system of partial and coupled differential equations derived from energy and mass balances describing the condensation of a pure fluid from a stream of inert gas with consideration of coupled heat and mass transfer. A simplified and a physical exact method allow the stepwise calculation of a condenser in steady state by local integration over the apparatus. A mathematical and physical exact integration with respect to time is carried out in the third method. The methods without simplifications lead to identical results; the application of the simplified method is restricted.

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Formelzeichen Wärmeübergangskoeffizient Gas/Film - A Fläche - Stoffübergangskoeffizient - c _p spezifische isobare Wärmekapazität - E Ackermann-Korrektur - F Film - G Gasphase - h Enthalpie - h _v Verdampfungsenthalpie - i, (i) Stützstellennummer - I Phasengrenze (interface) - k Wärmedurchgangskoeffizient Film/Kältemittel - 1 Inertgas - 2,K kondensierende Komponente - KM Kältemittel - Le =a/D Lewiszahl = Temperaturleitfähigkeit / Diffusionskoeffizient - , Molmengenstrom, -dichte - p,p _s Druck, Dampfdruck - , Wärmestrom, -dichte - Dichte - Zeit - t,T Temperatur in °C, K - freie Gasströmung - V Volumen     

Laminar forced convection of power-law non-Newtonian fluids inside ducts

Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.

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Laminare Zwangskonvektion von nicht-Newtonschen Fluiden, die in Kanälen strömen und dem Potenzansatz folgen

Zusammenfassung Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.

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Nomenclature b radius of circular duct or half the spacing between parallel plates - C - D _h hydraulic diameter=4b for parallel plate, 2b for circular tube - f(r), F(r) temperature distribution at the inlet, dimensional and dimensionless, respectively - g (r), G (R) energy generation, dimensional and dimensionless, respectively - h (z) heat transfer coefficient - H _i(Z) defined by Eq. (11b) - k thermal conductivity - l₀,l _N reference lengths to nondimensionalizer andz coordinates respectively (l ₀=b andl _N= D_h) - n power-law index - Nu _av average Nusselt number=h _av D_h/k - Nu (Z) local Nusselt number=h (z)D _h/k - p 0 for parallel-plate duct, 1 for circular duct - r radial or normal coordinate - R = dimensionless radial coordinate - T(r, z) fluid temperature - T _av (z) average fluid temperature - T ^* reference temperature - T reference temperature difference - U(R) = normalized velocity profile - U ^*(R) - w (r) fully developed velocity profile - w _av average velocity - z axial coordinate - Z = dimensionless axial coordinate Greek letters thermal diffusivity - (R,Z) = dimensionless temperature - _w = dimensionless wall temperature - _i eigenvalues of the eigenvalue problem (10) - (_i,R) eigenfunctions of the eigenvalue problem (10)     

Pore structures and transport properties of sandstone

To quantitatively analyze the macroscopic properties of the flow in porous media by means of the continuum approach, detailed information (velocity and pressure fields) on the microscopic scale is necessary. In this paper, the numerical solution for incompressible, Newtonian flow in a diverging-converging representative unit cell (RUC) is presented. A new solution procedure for the problem is introduced. A review of the accuracy of the computational method is given.Nomenclature A _ff ^* area of entrance and exit of RUC - A _fs ^* interfacial area between the fluid and solid phases - d throat diameter of RUC (m) - D pore diameter of RUC (m) - i, j unit vector for RUC - L ^* wave length of a unit cell - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p ^* fluid pressure - ^* cross-sectional mean pressure - _en ^* entrance cross-sectional mean pressure - Re _d Reynolds number - x ^*, r^* cylindrical coordinates - u ^*, v^* velocity - u _cl ^* centerline velocity - _d mean velocity at the throat of RUC (m/s) - _D mean velocity at the large segment of RUC (m/s) Greek viscosity coefficient (Ns/m²) - _p excess momentum loss factor defined in (4.1) - fluid density (kg/m³) - ^* stream function - ^* vorticity - dimensionless circulation defined in (2.7) Symbols - the mean value - * dimensionless quantities     

Absorption von binären Gasblasen in einer Flüssigkeit mit unterschiedlicher Löslichkeit für die Gaskomponenten

Zusammenfassung Es wird die Absorption einer Einzelblase betrachtet, die aus zwei Gasen A und B besteht, wobei die Flüssigkeit in der Umgebung der Blase A und B in konstanten Konzentrationen enthält.Unter vereinfachten Bedingungen wird das gekoppelte System der Komponentenkontinuitätsgleichungen der Stoffe A und B in der näheren Umgebung der Blase numerisch gelöst. Es werden ausschließlich Fälle betrachtet, in welchen A sehr viel löslicher als B ist.Es stellt sich heraus, daß der Molenbruch an A in der Blase nach einem Anlauf einem stationären Wert zustrebt, der in der Regel nahe am Gleichgewichtswert für A liegt. Wenn dieser stationäre Wert erreicht ist, nimmt die Blasenoberfläche mit konstanter Geschwindigkeit ab oder zu, wobei diese Geschwindigkeit wesentlich von der Löslichkeit von B abhängt.Anhand einiger Beispiele wird dargestellt, in welcher Weise die Verläufe von Blasendurchmesser und-inhalt von den Stoffwerten und den Umgebungs- und Anfangsbedingungen abhängen.

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Absorption of a single two-component-bubble into liquid

The absorption of a single two-component-gas bubble is considered, the liquid in the vicinity of the bubble containing both gases A and B with constant concentrations. The system of the component continuity equations for A and B in the vicinity of the bubble is solved numerically under simplified conditions. The solubility of A is supposed to be much more important than that of B.It appears, that the mole fraction of A in the bubble tends towards a steady state value which is near its equilibrium value. When this steady state composition is established, the rate of variation of the bubble area becomes constant and depends mainly on the solubility of the less soluble gas B.Some examples show the influence of physical properties, boundary and initial conditions on the variation of the bubble diameter and composition with time.

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Bezeichnungen d Durchmesser - p_i Partialdruck der Komponente i - p Gesamtdruck - r Radialkoordinate - r*=r/R dimensionslose Radialkoordinate - t Zeit - x_i Molenbruch von i in der Flüssigkeit - x _i ^* Sättigungsmolenbruch von i - y_i Molenbruch von i in der Blase - y _i ^* Molenbruch von i in der Blase, der mit im Gleichgewicht stünde - Fourier-Zahl - H_i Henry-Koeffizient für i - R Blasen-Radius - R₀ Anfangsblasenradius - R*=R/R₀ -- - R_außen äußerer Radius der Grenzschicht - Sh Sherwood-Zahl - -- - -- - _Li Diffusionskoeffizient von i in der Flüssigkeit - -- - Molardichte Indices außerhalb der Grenzschicht - A leichter lösliches Gas - B schwerer lösliches Gas - G Gas - L Flüssigkeit