Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.     

Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit

A hot-film probe has been used to measure slip of high-density polyethylene flowing through a conduit with a rectangular cross section. A transition from no slip to a stick-slip condition has been observed and associated with irregular extrudate shape. Appreciable extrudate roughness was initiated at the same flow rate as that at which the relationship between Nusselt number and Péclet number for heat transfer from the probe departed from the behavior expected for a no-slip condition at the conduit wall. A ₁ constant defined by eq. (A3) - C dimensionless group used in eq. (7) - C _p heat capacity - D constant in eq. (13) - f u _s/u - f _lin defined by eq. (A6) - G storage modulus - G loss modulus - k thermal conductivity - L length of hot film in thex-direction - L _eff effective length of large probe found from eq. (A3) - Nu _L Nusselt number, defined for a lengthL by eq. (2) - (Nu _L)₀ value ofNu _L atPe = 0 (eq. (A 1)) - Pe Péclet number,uL/ - Pe ₀ Péclet number in slip flow, eq. (6) - Pe ₁ Péclet number in shear flow, eq. (4) - q _L average heat flux over hot film of lengthL - R _i resistances defined by figure 8 - R _AB correlation coefficient defined by eq. (14) for signalsA andB - T temperature - T _s temperature of probe surface - T ambient temperature - T T _s –T - u average velocity - u _s slip velocity - V _b voltage indicated in figure 8 - W probe dimension (figure 6) - x distance in flow direction (figure 1) - y distance perpendicular to flow direction (figure 1) - thermal diffusivity,k/C _p - wall shear rate - _5% thickness of lubricating layer during probe calibration for introduction of an error no greater than 5%; see Appendix I - ^* complex viscosity - density - time - _c critical shear stress, eq. (13) - _w wall shear stress - frequency characterizing extrudate distortion (figures 12 and 13), or frequency of oscillation during rheometric characterization (figures 18–20) - * quantity obtained from normalized Nusselt number, eq. (A1), or complex viscosity ^* - A actual (small) probe (see Appendix I) - M model (large) probe (see Appendix I)     

Flow of a liquid metal coolant past a heat generating cylinder

The coupled problem whereby a solid heat generating cylinder is being cooled in steady state by a coolant in potential flow is investigated. An analytical technique for determining the temperature distributions in the solid and the fluid is presented. Numerical studies for six Péclet numbers (0.9<Pe<11.3) and three thermal conductivity ratios (0.31<K<3.1) were carried out.The surface hot-spot temperature and center temperature are presented graphically as functions of the Péclet number with the thermal conductivity ratio as a parameter. The average Nusselt number is found to be proportional to the Péclet number to approximately the one-half power. For the special case of constant surface temperature (uncoupled problem), the variation of local Nusselt number with angle measured from the forward stagnation point is in excellent agreement with the result presented by Grosh and Cess [6].Nomenclature ce _m(, –q) Mathieu function, periodic - D _n Fourier coefficient for solid temperature distribution - E _n Fourier coefficient for fluid temperature distribution - E() a term defined by equation (12), degree - F() a term defined by equation (13), degree - Fek _m(z, –q) modified Mathieu function, non-periodic - Fek _m(z, –q) derivative of Fek _m(z, –q) - h local heat transfer coefficient, energy/time area degree - average heat transfer coefficient, energy/time area degree - h _m mean heat-transfer coefficient, energy/time area degree - k _f thermal conductivity of fluid, energy/time length degree - K thermal conductivity ratio, k _f/k _s - k _s thermal conductivity of solid, energy/time length degree - Nu local Nusselt number, 2Rh/k _f - average Nusselt number defined by equation (55) - (Nu)_m mean Nusselt number defined by equation (57) - Pe Péclet number, 2RU/ _f - Q rate of heat generation per unit volume, energy/time volume - q parameter of Mathieu function, (Pe/4)² - q normal heat flux, energy/time area - R cylinder radius, length - Re Reynolds number, 2R/ - r radial position variable, length - T temperature, degree - T ₀ constant surface temperature, degree - T temperature of fluid at infinity, degree - T _e temperature at center of cylinder, degree - T _f temperature of fluid, degree - T _s temperature of solid, degree - T _w surface temperature, degree - surface hot-spot temperature, degree - reduced temperature, (T–)/E(1) - U approach velocity of flowing fluid, length/time - v velocity component in the direction, length/time - v _r velocity component in the r direction, length/time - z logarithm of Greek symbols _f thermal diffusivity of the fluid, (length)²/time - reduced radius, r/R - angular position variable measured from the trailing stagnation point, radians - kinematic viscosity, (length)²/time - angular position variable measured from the forward stagnation point, degree     

Über den Einfluß der veränderlichen Viskosität auf die Stabilität der ebenen Kanalströmung

Zusammenfassung Der Einfluß einer veränderlichen Viskosität auf die Stabilität der ebenen Kanalströmung wird untersucht. Um den Effekt der Viskositätsänderung besonders hervorzuheben, wird ein Materialgesetz ohne Relaxationseigenschaften zugrundegelegt. Außerdem wird nur das Verhalten von ebenen Störungen untersucht. Unter der Ausnutzung der Verwandtschaft der Problemstellung mit dem newtonschen Fall können die Näherungsgleichungen vonC. C. Lin in modifizierter Form übernommen werden. Die Stabilität wird durch die Änderung des Grundprofils infolge der veränderlichen Viskosität und die differentielle Viskosität in der kritischen Schicht bestimmt.

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Summary The influence of shear rate dependent viscosity on the stability of plane channel flow is investigated. In order to demonstrate the effect of the viscosity variation a constitutive model without relaxation properties is choosen. Furthermore only perturbations in the plane of flow are investigated. Since the problem is similar to the newtonian case, the approximate equations ofC. C. Lin can be appropriately modified. The stability depends on the change of the basic profile due to shear rate dependent viscosity and on differential viscosity in the critical layer.

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Liste der wichtigsten Symbole A Dimensionslose Kennzahl: - b Stoffkonstante - h Halbe Kanalhöhe - Druckgradient - Re Reynoldszahl - Re _k Kritische Reynoldszahl - Re _k Kritische Reynoldszahl für ein newtonsches Fluid mit der Viskosität - u _g(y) Grundgeschwindigkeitsprofil - U _M Maximale Geschwindigkeit - Viskosität - Viskosität im zweiten newtonschen Bereich - _D Differentielle Viskosität - Stoffkonstante - _k Kritischer Druckgradient _k = –(dp/dx)_k - _k Kritischer Druckgradient für ein newtonsches Fluid mit der Viskosität - Dichte des Fluids Mit 8 Abbildungen     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

A solitary wave in a porous medium

The one-phase Darcy continuity equation, including the quadratic gradient term, is considered. The exact linearization of the equation is found by a functional transformation for an arbitrary spatial dimension in the limit case where the constant fluid compressibility is much more dominant than the constant compressibilities of the reservoir parameters.The equation permits a solution representing a localized wave travelling through a one-dimensional reservoir without changing its form. This is the actual long-time limit of the transient solution for a constant sandface-rate injection of a compressible fluid with a constant compressibility if the fluid is much more compressible than the matrix. A solitary wave solution is not possible for production.A fully developed solitary wave would appear only for very high pressure increases, but the first signs of the emerging solitary wave are detectable at the sandface for moderate pressure increases which can appear under physical reservoir conditions.Latin symbols a Dimensionless wave propagation velocity - A _N Sandface area (N = 0, 1, 2) - c ₁, c ₂ Sums of compressibilities - c _x Generic (generalized) compressibility - c Fluid compressibility - c _h Reservoir height (i.e. bulk volume) compressibility (N = 0, 1) - c _k , c , c Generalized compressibilities - D Spatial reservoir dimensionality (D = 1, 2, 3) - f Fractional change of p _n1 due to nonlinear effects - h Reservoir height (proportional to bulk volume for N = 0, 1) - Horizontal reservoir width (N = 0) - k Reservoir permeability - K _N Constant with dimension of pressure (N = 0, 1, 2) - n Sum index - N Integer variable (N = D – 1) - p Reservoir pressure - p^* Overburden pressure - p _D Dimensionless (scaled) version of p - p ₀ Initial pressure - q Volumetric flow rate referred to sandface - r Radial (or linear) spatial distance from center of well - r _w Well radius - r _e External reservoir radius (or length) from center of well - t Time variable - t _f Injection/production time corresponding to fraction f - T Cole-Hopf-transformed version of dimensionless pressure y - u Rescaled (dimensionless) version of v _D - v Darcy velocity - v _d Dimensionless (scaled) version of v - x Generic symbol in compressibility expression (also used for auxiliary function and for auxiliary variable) - y Rescaled (dimensionless) version of p _D - z Dimensionless (scaled) version of r Greek symbols Coefficient of inertial resistance - Variable in wave solution for y - p _n1 Absolute change in physical sandface pressure due to production or injection - _p Pressure change over (dimensionless) distance behind and far away from front - _r Physical distance at constant time corresponding to - Characteristic (dimensionless) width of solitary wave - Formation porosity - ₁, ₂ Integration constants - Dimensionless (scaled) length of finite reservoir - Fluid viscosity - Fluid density - Dimensionless (scaled) version of t - Wave solution for dimensionless pressure y - Integer variable (±1) distinguishing between production and injection     

Spectral analysis of hydrodynamic tracer dispersion with an electrochemical measurement technique

We report tracer dispersion measurements in a capillary tube performed in the frequency domain using an electrochemical technique. Tracer Fe⁺⁺⁺ ions are produced by oxidizing Fe⁺⁺ ions at an emission anode; the inverse reaction allows to detect the tracer on a measurement electrode at the outlet of the sample. The amplitude and phase of the steady state signal detected at the outlet of the sample are measured as a function of the frequency of a sinusoidal concentration modulation induced at the inlet of the tube. Measurement results at two flow velocities are in agreement with predictions of the Taylor-Aris model.List of symbols A(f) output signal modulation amplitude at a modulation frequency f - a capillary tube radius - C _ox concentration of the solution ferricyanide ions - C _red concentration of the solution ferrocyanide ions - D _ox diffusion coefficient of tracer ions - D _m molecular diffusion coefficient - D longitudinal dispersion coefficient - E _e potential of emission electrode - E _d potential of detection electrode - F Faraday constant - J _m number of ions-g of tracer - I electrical current intensity on emitter electrode - I limiting current on detection electrode - k _x , k _y real and imaginary part of tracer concentration modulation wavevector - L total capillary tube length - Pe _L (= UL/D _m ) global Peclet number - S flow section - T _f characteristic exchange time with low velocity regions and dead zones - T ₀ mean transit time through the capillary - U mean fluid velocity - boundary layer thickness on detection electrode - phase shift between tracer concentration modulations at the inlet and the outlet of the sample - tracer concentration modulations spatial wavelength along the capillary tube - _a (= a ²/D _m ) characteristic diffusion time across the capillary section - tracer concentration modulation pulsation - _c cut-off frequency for concentration modulations at the capillary outlet     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank

A finite element method is used to solve the full Navier-Stokes and energy equations for the problems of laminar combined convection from three isothermal heat horizontal cylinders in staggered tube-bank and four isothermal heat horizontal cylinders in in-line tube-bank. The variations of surface shear stress, pressure and Nusselt number are obtained over the entire cylinder surface including the zone beyond the separation point. The predicted values of total, pressure and friction drag coefficients, average Nusselt number and the plots of velocity flow fields and isotherms are also presented.

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Die Finite-Elemente-Lösung von laminarer Strömung und kombinierter Konvektion von Luft in einer versetzten oder fluchtenden Rohranordnung

Zusammenfassung Eine Methode der finiten Elemente wird zur Lösung der vollständigen Navier-Stokes- und der Energiegleichung für die Probleme der laminaren kombinierten Konvektion an drei isothermen geheizten horizontalen Zylindern in versetzter Rohranordnung sowie für vier isotherme geheizte horizontale Zylinder in fluchtender Anordnung verwendet.Die Veränderung der Wandschubspannung, des Druckes und der Nusselt-Zahl werden für die gesamte Zylinderoberfläche, einschließlich des Bereiches nach dem Ablösepunkt, bestimmt. Die Werte des gesamten Widerstandsbeiwertes aufgrund von Druck und Reibung, die durchschnittliche Nusselt-Zahl und die Diagramme des Geschwindigkeitsfeldes und der Isothermen werden ebenfalls aufgezeigt.

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Nomenclature C specifie heat - C _D total drag coefficient - C _f friction drag coefficient - C _p pressure drag coefficient - D diameter of cylinder,L=2R ₀ - G, g gravitational acceleration - Gr Grashof number, g(T_w–T )D ³/v ² - h local heat transfer coefficient - K thermal conductivity - L spacing between the centers of cylinder - M _l shape function - N _i shape function - Nu, local and average Nusselt numbers - P dimensionless pressure, p^*/u ² - p ^*,p pressure, free stream pressure - Pe Peclet number,RePr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,Du /v - R ₀ radius of cylinder - T temperature - T _w temperature on cylinder surface with fixed value - T free stream temperature - v dimensionless x-direction component of velocity,v ^*/u - u ^* x-direction component of velocity - u free stream velocity - v dimensionless Y-direction component of velocity,v ^*/u - v ^* Y-direction component of velocity - X x-direction axis - x dimensionless x-direction coordinate,x ^*/D - x^* x-direction coordinate - Y Y-direction axis - y dimensionless Y-direction coordinate,y ^*/D - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - plane angle - dynamic viscosity - kinematic viscosity, / - density of fluid - _w dimensionless surface shear stress, _* ^w /u ² - skw/* surface shear stress - dimensionless temperature,      

Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

Forced convection heat transfer at the boundary layer of a packed bed

Numerical investigations of the nature of the fluid flow pattern and heat transfer at the boundary layer of a packed bed are reported. A volume averaged Navier-Stokes equation is used to predict the fluid flow and a volume averaged heat balance equation the heat transfer. A variable porosity in the packing is assumed in the region near the wall. Simulations are performed using a modified penalty Galerkin finite element method. The case of fully developed hydrodynamic flow and developing thermal flow is studied. The Nusselt number is found to depend on the Reynolds number, Graetz number and ratio of thermal conductivity of the solid and fluid phases. Comparison is made to some experimental literature values.Nomenclature A constant - [A] Navier-Stokes type matrix - B constant - [B] divergence matrix - C _p constant pressure heat capacity - d characteristic length - D _p particle diameter - D _t tube diameter - {F} solicitation vector - Gz Graetz number, z D _t ^–1 Pr _f Re _p - k permeability term - k _f Thermal conductivity of the fluid phase - k _s Thermal conductivity of the solid phase - [K] coefficient matrix for temperature equation - n normal vector - P pressure - Pr _f Prandtl number for the fluid _f C _p k _f ^-1 - r radial coordinate - R _t tube radius - R residual - R _m residual - Re _p Reynolds number for particle, - t tortuosity factor - T temperature - interstitial velocity - z axial coordinate - effective thermal conductivity - penalty parameter - boundary of solution domain - porosity - viscosity - density - test function - solution domain - test function     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

Viscosity of suspensions modeled with a shear-dependent maximum packing fraction

A large amount of data from the literature on viscosity of concentrated suspensions of rigid spherical particles are analyzed to support the new concept that the maximum packing fraction ( _M ) is shear-dependent. Incorporation of this behavior in a rheological model for viscosity () as a function of particle volume fraction () succeeds in describing virtually all non-Newtonian effects over the entire concentration range and also accounts for a yield stress. The most successful model is one proposed by Krieger and Dougherty for Newtonian viscosities, (, _M ), but with _M varying from a low-shear limit _M0 to a high-shear limit _M. Microstructural interpretations of this behavior are advanced, with arguments suggesting that similar rheological models should apply to suspensions of nonspherical and irregular particles.Symbols a particle size scale (for spheres, the diameter) - A lumped kinetic parameter in eqs. (23) and (24) - BS butadiene-styrene - C coefficient in Arrhenius model, eq. (2) - D coefficient in Mooney model, eq. (3) - e _i parameter representing one of the three electroviscous effects (i = 1, 2, or 3) - f fraction of total particulates that exist in the dispersed phase, eq. (22) - h solution factor, in Arrhenius model, eq. (2) - k crowding factor, in Mooney model, eq. (3) - k _D ,k _F kinetic rate coefficient for producing particles of dispersed or flocculated type, respectively - K Einstein coefficient for particles of any shape, eq. (1); equal to [] - KD Krieger-Dougherty model, eq. (6) - m exponent to characterize shear-dependence in viscosity models of Cross, eq. (10), and eq. (23), and also in yield stress prediction eq. (24) - N number of monodisperse components in a blend of spheres with different diameters - PD polydispersity (in size) parameter - S generalized shape parameter - T temperature - V _c volume of chamber in figure 6, representing the entire volume of the sample - V _P total volume of particles in the sample - V _D ,V _F sample volumes in which dispersed particles or flocculated particles, respectively, prevail; volumes of the dispersed phase or flocculated phase, containing both particles and carrier fluid - V _PD ,V _PF particle volume within the phase volumeV _D orV _F , respectively Greek coefficient in definition of _c in eq. (8); of order unity - coefficient regulating -sensitivity in eq. (10) - shear rate,dv ₁/dx ₂ in simple shear - shear viscosity of the suspension - ₀, low-shear and high-shear limiting values of - _s viscosity of the suspending fluid - [] intrinsic viscosity, - _r reduced viscosity,/ _s - Boltzmann's constant; in _c - shear stress - _c parameter characterizing sensitivity of viscosity to stress, in eq. (8) - _B dynamic yield stress in the floc model - _y yield stress - volume fraction occupied by solids in a suspension - _M maximum value of attainable by a given collection of particles under given conditions of flow - _M0, _M limiting values of _M at the low- and high- conditions, respectively     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

A light transmittance probe for interfacial area measurements of dispersed two phase flows with small particles

An optical probe measuring interfacial area () by light attenuation has been designed with a special emphasis on flows with sub-millimetric particles. It permits measurements in liquid-liquid or gas-liquid dispersions without need of introducing empirical correcting factors for the standard exponential decay law of light intensity while keeping an extended application range. This probe was successfully tested with an air-glass particle flow, the parameters of which were carefully determined basically by hold-up methods. The volume fraction of the dispersed phase was varied between 0.05% and 5%, and the particle size between 10 m and 300 m.List of symbols D diameter of spherical particle - D _S Sauter diameter - E ₀ irradiance on a surface perpendicular to light propagation 226E;=(1/l) averaged density function along y axis - f density function of a dispersion - f ₁, f ₂ focal length of the lenses L ₁, L ₂ - g granulometry function of a powder (probability density) - h granulometry function of a powder (unnormalized) - I ₀, I light beam intensity respectively before and after passing through the dispersion - j volumetric powder flow - K ₁, K ₂, K ₃ dimensionless constants - l optical path length of the beam in the dispersion - L experimental pipe width along x axis - m mass of a sample - n optical index of the continuous phase - p _a, p ₀ respectively slope of _a and ₀ straight line - r distance between particles - S _d scattering cross-section - V volume of dispersion - averaged particles velocity - x, y, z spatial coordinates - interfacial area - _a absolute interfacial area (by unit volume of dispersion) - ₀ interfacial area measured by light attenuation method - _d angle (around the initial direction of light propagation) within which a particle diffracts - _dr detector aperture angle - light wavelength - _d scattering cross section by unit volume of dispersion - light beam diameter - ₁, ₂ L₁, and L₂ lenses diameters - local volumetric fraction of dispersed phase - averaged fraction of dispersed phase along x axis - ₂ averaged fraction of dispersed phase along x and y axis - volumetric mass of particles     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²