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.     

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 .     

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

Effects of couple-stresses on the dispersion of a soluble matter in a pipe flow of blood

Summary An analysis of the effects of couple-stresses on the effective Taylor diffusion coefficient has been carried out with the help of two non-dimensional parameters based on the concentration of suspensions and , a constant associated with the couple-stresses. It is observed that the concentration distribution increases with increasing or The effective Taylor diffusion coefficient falls rapidly with increasing when is negative.

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Zusammenfassung Der Einfluß der Momentenspannungen auf den effektiven Taylorschen Diffusionskoeffizienten wird untersucht. Dabei treten zwei dimensionslose Parameter and auf: Der erste bezieht sich auf die Suspensionskonzentration, der zweite kennzeichnet die Momentenspannungen. Man findet, daß die Verteilungsgeschwindigkeit mit wachsendem oder zunimmt. Dagegen fällt der Taylorsche Diffusionskoeffizient bei wachsendem stark ab, wenn negativ ist.

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a Tube radius - C Concentration - C _i Body moment vector - C ₀ Concentration at the axis of the tube - C _m Mean concentration - D Molecular diffusion coefficient - d _ij Symmetric part of velocity gradient - F Function of and characterising effective Taylor diffusion coefficient - f _i Body force vector - H A function of and - K ₂ Integration constant - K ^* Effective Taylor diffusion coefficient - k Radius of gyration of a unit cuboid with its sides normal to the spatial axes - I _n Modified Bessel's function ofnth order - L Length of the tube over which the concentration is spread - M Function ofH and - M _ij Couple stress tensor - P Function of - p Fluid pressure - Q Volume rate of the transport of the solute across a section of the tube - r Radial distance from the axis of the tube - T _ij Stress tensor - t Time coordinate - T _ij ^A Antisymmetric part of the stress tensor - u Relative fluid velocity - Average velocity - v _i Velocity vector - Fluid velocity at any point of the tube - v ₀ ⁿ Velocity of Newtonian fluid at the axis of the tube - _i Vorticity vector - x Axial coordinate - x ₁ Relative axial coordinate - z Non-Dimensional radial coordinate - Density - _ij Symmetric part of the stress tensor - µ Viscosity of the fluid - µ _ij Deviatoric part ofM _ij - , Constants associated with couple-stress With 3 figures     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Shear flow properties of concentrated solutions of linear and star branched polystyrenes

Summary The linear viscoelastic and viscometric functions have been determined for solutions of wellcharacterized monodisperse linear and star-branched polystyrenes and for commercial, polydisperse polystyrene. The value of the product c for these solutions was large and was obtained by using both high and low The effect of structure on the rheological properties was determined by examining how parameters in a modified Carreau viscosity equation (used to fit the data) varied with c, , and branching. No enhancement effects on the rheological properties were observed because of branching.The Cox-Merz rule was observed to describe the similarities between the viscosity and complex viscosity for most of the monodisperse samples studied. The broad molecular weight distribution polystyrene solutions did not follow this empiricism.With 17 figures and 4 tables     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

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)     

Single droplet combustion in a gravitational environment

A theoretical analysis is developed to study the combustion characteristics of a fuel droplet in a gravitational field. The normalized governing system consists of the complete conservation equations inr-z coordinates and includes finite-rate global kinetics. The Clausius-Clapeyron law is applied at the liquid-vapor interface to describe the evaporation process. A modified body-fitted grid generation technique is used to handle irregular boundaries. The effects of changing the droplet diameter and the gravity level are investigated. Under the variation of droplet diameter, flame structures, including isotherms, flame shape, velocity vector field, and mass burning rate are studied in detail. The predicted results exhibit good agreement with experimental data. When the gravity level increases, the computed results show that the flame shape is sensitive to variation in gravity. A simple correlation, , is found. Within the elevated gravity domain of experiment, the computed data agree well with measurements obtained by Okajima and Kumagai [10].In einer theoretischen Studie wird die Verbrennungscharakteristik eines Ethanoltröpfchens in einem Gravitationsfeld untersucht. Die normierten Grundbeziehungen umfassen die vollständigen Erhaltungsgleichungen inr-z-Koordinaten unter Einschluß einer Globalkinetik endlicher Umsatzrate. Das Clausius-Clapeyronsche Gesetz wird an der Flüssigkeits-Dampf-Phasengrenze angesetzt, um den Verdampfungsprozeß zu beschreiben. Eine spezielle Gittergenerierungstechnik ermöglicht die Behandlung irregulärer Berandungen. Die Einflüsse veränderlichen Tropfendurchmessers bzw. variablen Gravitationsniveaus werden untersucht. Bei variablem Tröpfchendurchmesser erfolgt eine Detailuntersuchung der Flammenstrukturen nach Isothermenfeld, Flammenform, Feld der Geschwindigkeitsvektoren und Verbrennungsrate. Die vorausberechneten Ergebnisse stimmen gut mit experimentellen Daten überein. Bei Anstieg des Gravitationsniveaus zeigen die Rechnungen eine empfindliche Veränderlichkeit der Flammenform. Es läßt sich eine einfache Beziehung aufstellen. Im Bereich höheren Gravitationspegels stimmen die berechneten Werte gut mit den Messungen von Okajima und Kumagai [10] überein.Financial support of this research by the National Science Council of the R.O.C., under project NSC 81-0401-E-009-531 is greatly appreciated. The authors also wish to express their gratitude to National Chiao Tung University for providing computer facility.     

Simulation of viscoelasticity for one case of material testing

Conclusions Similariry conditions have been established on the basis of which the viscosity can be simulated in testing viscoelastic materials for tension (compression) under hydrostatic pressure. It has been shown that criteria and account for the effect of viscosity, while the II number accounts for the effect of pressure. The criterion is, in form, identical to the analogous parameter in the theory of non-Newtonian fluid flow. It has been shown, furthermore, that criterion is the monodromic version of criterion (the similarity number). When P=0 or P is very small and the II number degenerates, then only criterion or criterion should be used.Institute of Problems in Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Prikladnaya Mekhanika, Vol. 11, No. 6, pp. 109–114, June, 1975.     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa     

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     

Pseudo-molecular dynamics study of grain boundary segregation

The segregation of bismuth atoms on the [101] tilt copper grain boundaries Σ3 ( ) 70.53°, Σ33 ( ) 58.99°, Σ11 ( ) 50.48° and Σ9 ( ) 38.94° has been studied by pseudo-molecular dynamics using the empiricalN-body potentials. The relationship between bismuth segregation and grain boundary structure has been discussed in detail. The subject supported by the Chinese Academy of Sciences and National Natural Science Foundation of China     

Mathematical model of the entry length in the flow of suspensions of solid particles in a gas

A mathematical model consisting of equations of mass and momentum and for the velocity field has been used for computing the entry length of the flow of non-Newtonian fluids in laminar, transition and turbulent regions. Experimental data measured in a vertical flow of a suspension of solid particles in air have been used for verifying the predictions. n flow index for laminar flow - Re Reynolds number defined for the flow of the carrier medium - q exponent for turbulent flow - ratio of core radius with a flat velocity profile to pipe radius - _c ratio of the axial component of local velocity in the core to mean velocity - w mean flow velocity - ratio of axial distance from the pipe entrance to the pipe radius - ratio of the entrance length to the pipe radius - relative mass fraction of particles - ratio of the distance from the pipe wall to the pipe radius - coefficient of pressure loss due to friction     

Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems

The concept of an interval stochastic matrix is introduced. We prove a combinatorial theorem which describes the network flow associated with an interval matrix. The semi-invariant vectors of are characterized in terms of eigenvectors with unit eigenvalue of stochastic matrices . These results are then applied to the approximation and machine computation of invariant measures of dynamical systems.Funded under Australian Research Council Grant A 8913 2609.     

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     

To construct a vector field with given curl function and divergence function

Two conclusions have been achieved in this paper. Firstly, a formal solution of the equations , has been derived with different point of view from commonly known classical method developed by Helmholtz(1), (2), (3).Secondly, a method to construct a vector field with given curl function and divergence function has been given in terms of the above solution.     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

The mathematically linear analysis of strain in continua

Summary One-one correspondence is postulated for two coordinate continua. One continuum is regarded as the initially undeformed state of a currently deformed continuum. The two continua are orthogonal trasformations each to the other. The square root of the quadratic metric, when the appropriate self-conjugate stretch dyadic is expressed in its principal form, gives the mathematically linear form to the analysis. The self-conjugate dyadic is expressed as =grad grad P in terms of a scalar potential function P. The physical and mathematical continuity of the strain dyadic is ensured by curl =0.A 4-vector analysis is evolved from a 4-vector quinternion analysis. The 4-vector analysis is of the same form as the usual 3-vector analysis that evolved from Hamilton's quaternion analysis.

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Sommario Si postula una corrispondenza biunivoca per due continui coordinati. Un continuo viene considerato come lo stato inizialmente non deformato di un continuo attualmente deformato.I due continui sono trasformazioni ortogonali uno dell'altro. La radice quadrata della metrica quadratica, quando la relativa diade di dilatazione autoconiugata venga espressa nella sua forma principale dà la forma matematicamente lineare alla analisi.La diade coniugata viene espresse come =grad grad P in termini di una funzione potenziale scalare P. La continuità fisica e matematica della diade di deformazione è assicurata da rot =0.Una analisi tetravettoriale viene sviluppata da analisi di quinternioni tetravettoriali. L'analisi tetravettoriale è della stessa forma dell'analisi trivettoriale che viene vsiluppata dalla analisi quaternionica trivettoriale di Hamilton.

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A generalisation that, when particularised, applies to the finite or infinitesimal straining of elastic bodies, incremental straining of plastic or fluidic bodies in 3-dimensional continua and to the space-time continuum as a 4-dimensional continuum.